**Computational Mechanics of Structures and Materials**

Editors

**Michele Bacciocchi Angelo Marcello Tarantino Raimondo Luciano Carmelo Majorana**

Basel • Beijing • Wuhan • Barcelona • Belgrade • Novi Sad • Cluj • Manchester

*Editors* Michele Bacciocchi University of San Marino Dogana Republic of San Marino

Carmelo Majorana University of Padova Padua Italy

Angelo Marcello Tarantino University of Modena and Reggio Emilia Modena Italy

Raimondo Luciano University of Napoli Parthenope Naples Italy

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Materials* (ISSN 1996-1944) (available at: https://www.mdpi.com/journal/materials/special issues/Comput Mech Struct Mater).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

Lastname, A.A.; Lastname, B.B. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-9712-6 (Hbk) ISBN 978-3-0365-9713-3 (PDF) doi.org/10.3390/books978-3-0365-9713-3**

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) license.

## **Contents**



## **About the Editors**

## **Michele Bacciocchi**

Michele Bacciocchi is an associate professor at the University of the Republic of San Marino (San Marino). He obtained his PhD in Civil, Chemical, Environmental, and Materials Engineering at the University of Bologna (Italy). He was a research-visiting scholar at Texas A&M University (United States). His research interests are the mechanics of solids, elasticity, and numerical modeling of structures. He co-authored more than 65 international peer-reviewed journal papers, 6 books, and more than 15 abstracts used at international and national conferences. He organized, as Conference Co-Chair, several international events on composite materials and structures.

#### **Angelo Marcello Tarantino**

Angelo Marcello Tarantino is a full professor of Structural Engineering at the University of Modena and Reggio Emilia (Italy). He is an expert in the theory of elasticity and structural engineering and the author of textbooks and 120 scientific publications, of which 90 were published in international journals, mainly with him billed as the sole author. He has carried out studies and research activities in France, England, the United States, Switzerland, and China. He is the scientist responsible for several research projects.

#### **Raimondo Luciano**

Raimondo Luciano is a full professor in Structural Mechanics at the University of Naples Parthenope and he is a member of the Comitato Nazionale dei Garanti per la Ricerca (CNGR). He was the Coordinator of the Faculty Activities of Civil and Environmental Engineering at the University of Cassino and Southern Lazio. He has been involved in many research activities since 1991 as an assistant professor at the University of Cassino. He has been the local coordinator of several research projects. He has more than 170 publications in national and international journals and conferences. He has collaborated with distinguished international scientists. He was the President of the National Committee of the MIUR for the PRIN Grants 2015. He was one of the four professors in the National Committee for Italian national qualification (ASN of MIUR) for Full and Associate professors in 08/B2 competitive sector. He collaborates for the realization of the Italian Design Guidelines CNR DT/200 and many others for the use of composite materials in Civil Engineering. He has been involved in over 20 mainly Italian consultancy projects pertaining to structures, namely in the diagnosis and strengthening of the design of bridges and buildings.

### **Carmelo Majorana**

Carmelo Majorana is a full professor of Solid and Structural Mechanics and Engineering at the University of Padova (Italy). Expert in Computational Mechanics and Structural Engineering, he is the author of textbooks and more than 200 scientific papers, of which 100 are in international journals. He has carried out studies and research activities in Europe and the United States. He has been the scientist responsible for several research projects.

## **Preface**

The idea of a Special Issue focused on "Computational Mechanics of Structures and Materials" arose from the enormous potential that characterizes different computational methods in many engineering fields. In particular, these methodologies, due to their ability to facilitate various tasks, have surely affected the approach to dealing with structures and materials, as can be observed from the body of pertinent literature, proving not only the attractiveness of this broad topic but also its potential in developing further advancements.

Therefore, the aim of the Special Issue is to gather innovative investigations dealing with accurate, reliable, and effective numerical approaches in the field of both structural mechanics and mechanics of materials. The collected papers present different computational techniques involving the achievement of solutions characterized by higher accuracy and reliability.

This Special Issue attracted many interesting submissions from many different countries of the world. Authors who decided to contribute and experts who reviewed the papers are all gladly acknowledged. The success of the collection has been made possible thanks to the constant support provided by Ms. Cecilia Zhang, the Section Managing Editor. The Guest Editors would like to thank her for her commitment to the Special Issue. Finally, the whole editorial team of *Materials*, including the Editors-in-Chief, must be mentioned.

**Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano, and Carmelo Majorana** *Editors*

## *Editorial* **Special Issue: "Computational Mechanics of Structures and Materials"**

**Michele Bacciocchi 1,\*, Angelo Marcello Tarantino <sup>2</sup> , Raimondo Luciano <sup>3</sup> and Carmelo Majorana <sup>4</sup>**


Computational methods have always affected many engineering fields due to their enormous potential and ability to facilitate various tasks. This statement is surely true as far as the research in structures and materials is concerned. In fact, it can be observed that the body of literature focused on computational and numerical methods for solving various structural problems and characterizing different constituents and materials is huge. This proves not only the attractiveness of this broad topic, but also its potential in developing further advancements in these contexts.

For these reasons, the Guest Editors decided to organize a Special Issue focused on "Computational Mechanics of Structures and Materials", to collect innovative investigations dealing with accurate, reliable, and effective numerical approaches in the field of both structural mechanics and mechanics of materials. To this aim, a broad scope has been defined, accepting not only finite element or finite-element-based methods, but also different computational techniques involving the achievement of solutions characterized by higher accuracy and reliability. The innovation demonstrated in dealing with advanced materials and constituents is a positive feature of all submitted papers that contributed to their acceptance.

Over several months, this Special Issues constantly proved to be a success, attracting and collecting many interesting submissions. The first heartfelt thanks is directed to all the authors from many different countries of the world who decided to contribute to the collection. Specifically, nineteen papers (out of the twenty-six submitted) have been published, passing through a meticulous review process. Sincere gratitude has to be expressed to the experts who reviewed the papers in spite of their many personal and academic responsibilities.

This great achievement has been made possible thanks to the constant support provided by Ms. Cecilia Zhang, the Section Managing Editor. The Guest Editors would like to thank her for the commitment given to the Special Issue. Finally, the whole editorial team of *Materials*, including the Editors-in-Chief, must be mentioned. They made the management of this Special Issue possible.

To celebrate the success of this commitment, a brief review of the works included in the collection is presented, highlighting the main advancements obtained in the field of computational mechanics.

Gawryluk presented a discussion on the appropriate choice of boundary conditions in structures subjected to a failure analysis. In particular, the research is focused on a thin-walled laminated angle column under compression. The results of both experimental and numerical tests are presented and compared, taking into account different damage criteria [1].

**Citation:** Bacciocchi, M.; Tarantino, A.M.; Luciano, R.; Majorana, C. Special Issue: "Computational Mechanics of Structures and Materials". *Materials* **2023**, *16*, 5617. https://doi.org/10.3390/ ma16165617

Received: 31 July 2023 Accepted: 9 August 2023 Published: 14 August 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1

Klimczak and Cecot developed an innovative multiscale finite element method. Their numerical approach proved to be a fast and flexible technique suitable for dealing with heterogeneous materials. The results were obtained in the context of the steady-state heat transfer problem [2].

Bogdan and Radosław employed the finite element method (FEM), including the Johnson–Cook model and the failure parameters of a peculiar class of steel, to simulate the resistance of structures to collisions, shelling, and the impact of pressure waves caused by explosions in water and air in relation to submarines [3].

By means of the finite element method, Wang et al. proposed an investigation on the bearing capacity of high-strength steel-reinforced concrete composite columns. In particular, their analysis emphasized the effect of the confinement of stirrups and steel, highlighting the influence of several parameters and discussing the role of different regulations [4].

Lee and Han studied an infinite isotropic solid embedding different kinds of isotropic and anisotropic spheroidal inclusions. To this aim, they introduced the volume integral equation method (VIEM), which has been demonstrated to be a versatile numerical approach for the three-dimensional elastostatic inclusion problem [5].

Mucha et al. proposed a numerical method to describe a propagating instability phenomenon that effects metals, known as the Portevin–Le Chatelier (PLC) effect. They performed several studies varying the model parameters, which was then efficaciously compared with experimental results [6].

Shi et al. presented a numerical investigation to discuss the effects of the direction and scale of microstructures on the tension problem of a composite plate with a circular hole, proving that these features in solids also have a significant influence on the development of advanced materials. In this context, a micropolar continuum (Cosserat) model was considered [7].

Badea et al. highlighted the limitations of usual finite element models in dealing with tubular structures. By means of a previously developed beam T-junction model, their results provided some strategies to improve the accuracy of beam-element-type approaches, taking into account the real junction stiffness [8].

The paper by Sokołowski and Kami´nski is focused on the problem of the topological optimization of corroding structures with uncertainties. They proposed a framework for the reliability-based design optimization (RBDO) of structural elements, considering a corrugated web I-girder as an example. Several numerical approaches have been compared in this context [9].

The work by Bochenek and Tajs-Zieli´nska also deals with topology optimization. They proposed the concept of the original heuristic topology generator, combining an algorithm with a commercial finite element software, characterized by a significant level of versatility [10].

Chan et al. investigated the mechanics of frozen particle fluid systems by means of a microscale simulation approach based on the discrete element method (DEM) and bonded-particle model (BPM) approach. The results provided by their methodology have been proven to be in good agreement with experimental results [11].

Alrayes et al. developed a numerical approach for simulating crack growth. To this aim, they used a scaled boundary finite element model (SBFEM), emphasizing the importance of simulating the fracture process zone when attempting to describe the cracking behavior of heterogeneous and quasi-brittle materials, taking concrete under monotonic and cyclic actions as an example [12].

Li et al. analyzed the behavior of a hydraulically expanded joint between a tubesheet and titanium tube by means of the finite element method. Their results highlighted the influence of several phenomena with many practical consequences [13].

Alrayes et al. studied mixed-mode crack propagation in concrete through some numerical simulations. They used the scaled boundary finite element method (SBFEM) to assess this phenomenon [14].

Kim et al. developed a nonlinear framework based on the modified couple stress theory to study the axisymmetric bending of circular and annular microplates subjected to thermal and mechanical loads. Their results, obtained via the finite element method, highlighted the effects of several parameters on the bending response [15].

Zhang et al. investigated creep at room temperature by means of a mechanical doublespring steering-gear load table. Their numerical results were successfully compared to experimental tests [16].

Andreotti et al. proposed a methodology to compute the resultant force of ballistic impacts resulting in a full fragmentation of the impactor with no penetration of the target. They discussed the accuracy of different discretization strategies for the corresponding finite element analysis. The results provide many useful suggestions to deal with this kind of problem [17].

The paper by Tahani et al. presents an investigation on the overall mechanical properties of ceramic–metal composites. This was achieved using a computational approach based on the molecular dynamics method. Their results highlighted the influence of several parameters [18].

Finally, the work by Giunta et al. deals with the free vibration analysis of variablestiffness composite plates. They expanded Carrera's unified formulation (CUF) plate finite elements to composite laminates reinforced by curvilinear fibers. The effectiveness of the solution was proven through a comparison with results available in the literature or obtained through other commercial codes [19].

The Guest Editors would like to congratulate all the authors for the remarkable achievements illustrated in these papers.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Article* **Impact of Boundary Conditions on the Behavior of Thin-Walled Laminated Angle Column under Uniform Shortening**

**Jarosław Gawryluk**

Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland; j.gawryluk@pollub.pl; Tel.: +48-815384144

**Abstract:** Determining the appropriate boundary conditions of a structure is a very important aspect in the failure analysis. In experimental tests, the method of compressing composite samples significantly influences the obtained results. In numerical studies, there is a problem of correctly defining the boundary conditions applied in real object. Therefore, many numerical tests on samples should be undertaken to observe their behavior and to determine ultimate load. The present work includes study to determine the impact of boundary conditions on the thin-walled laminated angle column under compression. The phenomenon of buckling and the post-buckling bahavior of columns were investigated experimentally and numerically. First, the real simply supported angle columns subjected to uniform shortening are tested. Due to the stress concentration between the real sample and the grips, a flexible pads were used. Experimental tests are carried out on the universal testing machine. The deformations of columns were measured using the non-contact Aramis System. The composite material condition was monitored by acoustic emission using the Vallen Systeme with piezoelectric sensors. Next, the numerical calculations in Abaqus software based on the finite element method are performed to validate the empirical results. To determine the influence of the boundary conditions, two numerical models of the system with and without flexible pads are developed. To estimate damage initiation load in numerical models a different damage criteria ( Tsai-Hill, Tsai-Wu, Azzi-Tsai-Hill, Hashin) are used. Based on the results specified that the model with elastic pads more accurately reflects the actual behavior of the L-profile element under compression. It was supported, i.e., by good agreement of flanges deflection (the equilibrium paths) with experimental results. Furthermore, a qualitative and quantitative agreement of damage initiation load were obtained using Hashin criteria (error 4.61%).

**Keywords:** flexible pad; laminated angle column; compresion; FEM; experiment

## **1. Introduction**

Loss of stability is an important process during operation of thin-walled structures, which can lead to the complete damage of the structure. Therefore, a detailed analysis of the structure failure under the load is extremely necessary from an engineering point of view. In particular, the problem is encountered in the aerospace industry [1,2], where isotropic materials are replaced with modern composite materials, which are characterized by excellent strength parameters with a significant weight reduction of these elements. An important issue is description the damage process of thin-walled laminated structures [3,4]. In thin-walled structures with flat walls made of composite material, the system works after the loss of stability (the system has stable equilibrium paths) [5,6]. This means that apart from knowledge the critical load value, the behaviour of the structure after loss of stability also has an important role. In most cases, the analysis of post-critical states requires taking into account non-linear relationships between displacements and deformations. Nonlinear calculations are usually performed for structures with initialized geometrical imperfections corresponding to the selected buckling mode of the structure, which was determined in the linear stability analysis. This makes it possible to determine the relationship between the load on the structure and the geometrical parameters

**Citation:** Gawryluk, J. Impact of Boundary Conditions on the Behavior of Thin-Walled Laminated Angle Column under Uniform Shortening. *Materials* **2021**, *14*, 2732. https:// doi.org/10.3390/ma14112732

Academic Editors: Carmelo Majorana, Angelo Marcello Tarantino, Raimondo Luciano and Michele Bacciocchi

Received: 24 April 2021 Accepted: 17 May 2021 Published: 21 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

determining the displacement of its nodes—i.e., we will obtain post-critical equilibrium paths [7]. By carrying out nonlinear studies in a post-critical state, the moment of failure initiation in a composite structure can be determined. This is particularly important as this phenomenon exhausts the load capacity of the structure. It is quite a complicated and ambiguous problem, as the failure model depends on many factors, i.e., the type of load, geometry and size of the analyzed element, mechanical properties of the materials used, preloads and damages. Moreover, the method of producing the actual structure significantly influences the course of the destruction process. As a result, it leads to many failure mechanisms, i.e., matrix/fiber cracking, local buckling of the fibers or the entire laminate layer, and delamination. In order to describe the respective damage mechanism, many failure models have been developed and various failure criteria have been introduced into the literature. The most popular criteria in practical analyses describe the failure of the first layer: the maximum stress criterion, the maximum deformation criterion, the Tsai-Hill criterion, the Tsai-Wu criterion, and the Azzi-Tsai-Hill criterion. The destruction process occurs when at least one criteria is fulfilled [8]. All the above-mentioned criteria provide information that the moment of damage initiation has caused, but they are not able to clearly define which element of the structure was destroyed. For this purpose, the Hashin's criterion can be used, which determines the destruction of fibers or matrix [9]. Interesting research describing the influence of boundary conditions on the buckling of beams is described in [10]. The Cold-Formed Steel beams with staggered slotted perforations were analyzed. These beams are being used in light gauge steel construction aiming to enhance both the fire and energy performance. However, these web perforations affect the bending capacity. In order to determine this effect, numerical and experimental studies were carried out. It was found that the influence of staggered slotted perforations on local buckling strength of the CFS beams is relatively small (11%).

In the literature, there are a lot of papers describing destruction process of thin-walled structures [11].The analytical methods are used for the post-buckling analysis, in which the thin-walled structure is described with orthotropic material [12]. In order to verify the theoretical models, it was necessary to conduct laboratory tests [13]. One of the experimental methods allowing a description of the damage process of the laminated structure is the acoustic emission method, which enables monitoring of the damage initiation and development of damage up to the complete destruction of the element. Experimental test of thin-walled chanell section profile subjected to compression are described in the paper [14]. Six different configurations of the glass fibers arrangement in the epoxy matrix, considering only symmetrical laminates were investigated. The tests were carried out on a testing machine with the use of an environment for recording acoustic emissions. This applied method made it possible to observe the structure's behaviour, thanks to which it was possible to estimate the damage initiation load. Similar studies for carbon-epoxy laminate were presented in the work [6]. The authors experimentally tested four variants of laminate fiber arrangement. The AE method was used experimentally. From the conducted experimental studies, it was possible to estimate the loads initiating the damage of structures. Teter et al. [15] presented compression tests of a thin-walled omega profile made of carbon-epoxy laminate. Experimental tests with the use of acoustic emission were considered. Additionally strain gauges were used to record deformations of selected surfaces. This allowed estimating the damage initiation load and post-buckling equilibrium path.The research carried out in the above works has shown that the acoustic emission method is effective in monitoring structural damage. In the latest works devoted to the damage of thin-walled structures, experimental and numerical studies using the finite element method can be found [16]. Nonlinear buckling analyses have been carried out to predict the initial buckling loads of the pultruded glass reinforced plastic wide flange columns under compression in paper [17]. The authors used new criteria for the web-flange junction, because the failure behaviour of this junction is quite different from web or flanges. Numerical results were compared with experimental, and good correlation was obtained. Rozylo et al. [18] described the numerical and experimental tests of thin-walled profiles

with an omega cross-section under axial compression. The non-linear range of structures with geometrical imperfection was calculated in the Abaqus environment by the Newton-Raphson method. The Hashin criterion was used to identify the initiation of structure failure. A high agreement of numerical and experimental results was obtained. Detailed research on the post-buckling analysis of thin-walled laminated profiles with a C-section is presented in [19]. Numerical studies in the Ansys environment and experimental studies were carried out. A high agreement of the results was obtained. Samples subjected to axial compression were analyzed. Similar studies on chanell section profiles were presented in [20], where the Abaqus environment was used, and samples were tested under eccentric loading. In the experimental tests, acoustic emission was used to determine the damage initiation. Teter et al. [21] conducted detailed studies of thin-walled angular columns in various configurations of fiber arrangement under uniform compression. The nonlinear research was carried out with the analytical-numerical and finite element method. The influence of geometrical imperfection on the value of the smallest buckling load was determined. The authors proposed the *P* − ∆ intersection method and the *P* − ∆ *<sup>n</sup>* method, thanks to which it was possible to estimate the value of the buckling load with high and small amplitude of the initial deflection with high accuracy. Similar numerical studies were compared with the semi-analytical method (SAM) based on the Byskov-Hutchinson method in [22]. The load-shortening diagrams of the column in the nonlinear range were developed as well as the influence of imperfection on the buckling load were determined.

Taking the above-mentioned into account, it can be said that there are still too few papers with results presenting the behaviour of thin-walled composite structures with L-profile subjected to compressive load. Therefore, it was decided to conduct experimental and numerical tests. This articule is a continuation of previous studies described in paper [23], where the numerical model of the elastic pad was validated and the influence of its parameters on the obtained results was determined. In the present study, the impact of boundary conditions on the thin-walled laminated angle column in post-buckling behavior is analyzed. Therefore, two numerical models with different boundary conditions were compared with the results obtained experimentally. Linear buckling and nonlinear postbuckling analysis were considered to investigate the post-buckling behavior of the beam under uniform compression. In order to estimate damage initiation load in numerical models a different damage criteria were used. Finally, the equilibrium paths for all analysed cases were determined. Based on the results, it was found that the model without elastic pads characterized a much more stiffness then the second one with pads, as shown by: a greater bifurcation load, damage initiation load in all analyzed criteria, lower shortening and less deflection of the column in its center. However, model with flexible pads shows a similar character to the experimental results, i.e., high convergence of the equilibrium paths and a small error in the damage initiation load (Hashin criteria).

#### **2. Problem Statement**

Thin-walled composite channel-section column with L-profile subject to axial compression were considered. Buckling and post-buckling of such beams were analyzed. Those investigations were mainly aimed at:


The thin-walled profiles under consideration were made of a carbon-epoxy unidirectional prepreg tape using autoclaving technique. The thickness of each ply was approx. 0.045 mm. This column consist of 18 layers with following configuration: [60, 0(2), −60(2), 60(3), −60(2), 0(3), −60(2), 0, 60(2)]T. The considered beam was 300 mm long and the width of flanges was 40 mm. The material properties of the beams were determined in tensile and compression tests described in paper [6]. The following material properties (Table 1) Young's modulus in two orthogonal directions *E*<sup>1</sup> and *E*<sup>2</sup> , Poisson's ratio *υ*12, and the

shear modulus *G*<sup>12</sup> were determined. Additionally, the limit properties of the composite, i.e., tensile strength *FTU* in two orthogonal directions, compression strength *FCU* in two orthogonal direction and shear strength *FSU* were determined. These material properties are presented in Table 2.

**Table 1.** Properties of used material.


**Table 2.** Limit properties of the composite as determined in compliance with relevant ISO standards.


To eliminate the effect of stress concentration in experimental test, flexible pads between the thin-walled profile and the testing machine handle were used. To check an impact of the flexible pad on post-buckling behaviour of the thin-walled beam under consideration, numerical model with and without pads were investigated.

#### **3. Laboratory Setup**

Experimental tests consisted in static compression of the fabricated L-profiles on the universal testing machine (Figure 1). In order to eliminate the stress concentration effect between the sample and handle of testing machine, the flexible pads with a thickness of 5.2 mm were used. Laboratory test were performed at a constant velocity of the cross-bar equal to 1 mm/min. The tested columns were loaded with the force value from zero to load, in which damage initiation load was observed. The composite material condition was monitored by acoustic emission. Signals were recorded using the Vallen Systeme, provided with a two piezoelectric sensors. In AE method usually uses the following parameters: the number of hits, the number of counts, the signal amplitude or energy [6,14,15]. In addition, to determine deflection of the sample in the whole range of load, a Aramis system was employed. The ARAMIS is a non-contact measuring system based on digital image correlation doing high-precision measurements with a 3D measurement resolution in the sub-micrometer range, regardless of the specimen's geometry and temperature. For this purpose, the system used a series of digital photos done at regular intervals during measurements time by two cameras positioned at the appropriate distance from the tested object. I used this system to recording the behavior of the real object, in particular to calculate the shortening of the sample and its flange deflections. The values of the load applied to the system were obtained directly from the testing machine. This allowed to determine a post-buckling equilibrium curves for all tested samples and the value of the load at which the first layer of the laminate was damaged. The experiments were conducted under laboratory conditions at room temperature of 23 ◦C.

**Figure 1.** Experimental setup.

Additionally, verification of the material constants for the flexible pad was carried out in experimental tests. Detailed research was described in paper [23]. However, the obtained substitute material constants are presented in Table 1.

#### **4. Numerical Nonlinear Analyses**

The numerical nonlinear stability problem was solved with the finite element method (FEM)—ABAQUS software [24]. Two models with different boundary conditions were considered. The first geometrical model (denoted as M-1) consists of only a part of the column situated between the rigid grips (see Figure 2a). The second one (denoted as M-2) has flexible pads between sample and rigid grips are additionally included (Figure 2b). The second assumed model was closer to the structure tested experimentally.

The L-profile laminate column was modelled using S8R shell finite elements. They are 8-node elements with a second order shape function with reduced integration (with six degrees of freedom in each node). The layup-ply technique is used to define the sequence of the laminate layers. Each layer is made of a carbon-epoxy unidirectional prepreg with material parameters presented in the previous section. However, the model of grips are shown as a rigid body (without predefined material properties), made of 4-node, bilinear quadrilateral discrete rigid elements (R3D4) with six degrees of freedom in each node. In the second model (M-2), the flexible pad consists of C3D20R solid element with reduced integration. These are 20-node 2nd order elements (with square shape function) with three translational degrees of freedom at each node. The number of elements was assumed on the basis of the previous experience (e.g., [23]). However, the mesh density was assumed in such a way as not to limit column deformation. The convergence of the model by selecting the size of the elements used (especially the column and pad) was investigated numerically. In the case of the pad, it was determined based on simulations that the number of 3 elements in thickness is sufficient. In both FE models, the thickness of elements discretizing a part of the model corresponding to the considered laminated beam is assumed as 0.81 mm. The system of the L-profile beam with grips was simply supported at two ends. Contact relations were defined between the rigid plate and the profile edge (M-1) or between the flexible pads and the profile edge (M-2) in the normal and tangential directions (friction coefficient set equal to 0.6).

**Figure 2.** Numerical models of the column: (**a**) M-1; (**b**) M-2.

The top rigid plate was locked at the first reference point (RP1), while the bottom rigid plate was allowed to move in the Z direction (at the second reference point (RP2)). The compressive load was realized by displacement of bottom plate along the longitudinal axis of the column. The displacement in the directions perpendicular to the plane of the angle column is set to zero in all nodes located on the profile edges (at the contact points). The specific boundary conditions used in the models are presented in Figure 2.

The above-described numerical models were employed in the numerical calculations to perform an eigenbuckling analysis and a nonlinear buckling analysis to investigate the post-buckling behavior of the beam under consideration. To estimate the moment of the first laminate layer damage, the following initiation criteria were used: maximum stress criterion, Tsai-Hill, Tsai-Wu, Azzi-Tsai-Hill, and Hashin criterions. The value of the damage initiation load and the place where it occurred, according to the above criteria were determined.

### **5. Comparison of Results and Discussion**

The results of the laboratory tests were used to validate the FE models employed to determine the post-buckling behavior of thin-walled structures. The results of the numerical calculations obtained with two models were compared to the results of the experimental investigations. In the linear numerical analysis, the lowest bifurcation load and the corresponding mode of buckling were determined. The same buckling mode was obtained for two models, while the bifurcation load differed by approx 14%. The first model (M-1) obtained a greater bifurcation load, which proves that this model had greater stiffness. However, when the stiffness of flexible pads significantly increased (e.g., E <sup>×</sup>10<sup>6</sup> ), the same bifurcation load was obtained in both models. Furthermore, the same mode (along the length of the column one buckling half-wave) in an experimental case was observed (Figure 3).

**Figure 3.** Modes of buckling under axial compression: (**a**) M-1; (**b**) EXP; (**c**) M-2.

The maps of the laminate failure parameter for initiation criteria obtained in the numerical calculations shows a different qualitative character in M-1 and M-2 models. Namely, in the first model, due to lack of flexible pads, the damage initiation in all criteria concerns only areas at the profile edges. However, in the second model the damage initiation concerns areas at the profile edges and additionally at the corner of the angle section in 1/2 of its height and in the middle part of one of the flanges. Using the Tsai-Hill, Tsai-Wu, Azzi-Tsai-Hill, and maximum stress criterions, it is possible to catch the moment of damage initiation, but it is not possible to determine the failure mechanism

(i.e., what elements of laminate have been damaged). Therefore, the damage initiation criteria based on Hashin theory was used,that allow independent assessment of damage initiation in individual components of the material, i.e., the fibre tensile/compressive initiation criterion (HSNFTCRT/HSNFCCRT) or the matrix tensile/compressive initiation criterion (HSNMTCRT/HSNMCCRT). The initiation of damage occurs when any of the above criteria reaches the critical value of 1. It is used to determine the composite material damage initiation load. In the analyzed sample, the obtained results demonstrate the fulfillment of the Hashin criterion when tensile the matrix (HSNMTCRT). Maps of damage parameter for M-1 and M-2 models are presented in (Figure 4). However, the condition was not achieved in the other failure parameters.

**Figure 4.** Maps of damage parameter for an angle column: (**a**) M-1; (**b**) M-2.

The damage initiation in M-2 model occurred mainly in the middle part of one of the flanges, while the place of damage initiation with the Hashin criterion did not change in the first model. Additionally, it was checked in the M-1 model whether the damage mechanism occurring in the M-2 model would appear in further analysis. After completing the necessary research, it was found that the damage mechanism from second model was not observed in the first model. From a practical point of view, the angle section under compression should damage closer to the center than at the edge of the profile. Furthermore, the maximall deflection in the middle of the column was observed in the experimental tests (Figure 5a). It was compared with the second numerical model Figure 5b.

Additionally, the deflection curves of two flanges (*u*<sup>1</sup> deflection in the X direction and *u*<sup>2</sup> deflection in the Y direction) from two numerical models were compared as a function of the compressive load. Moreover, the above-described curves have been compiled with the deformations recorded by the Aramis system. The deflection of the angle column near the middle, recorded from the experimental and two numerical cases are presented in Figure 6. The deflection of the flange is shown in absolute value, but it should be remembered that not every variant (real sample or numerical model) has a deflection in the same direction. The positioning of the test sample did not allow simultaneous observation of two L-profile column flanges. Therefore, the deflection only one of the sample flanges was recorded.

**Figure 5.** The deflection of an angle column under axial shortening: (**a**) EXP; (**b**) M-2.

**Figure 6.** The equilibrium paths, i.e., load-flange deflection relations.

The equilibrium paths clearly show that the first numerical model cis haracterized by a different nature of damage. The deflection in the same node at the damage initiation load

differed by about 120%. It relate to both X and Y directions. However, comparing the M-2 model with the experiment, a much smaller error in the range of 11–31% was obtained. Furthermore, the nature of the experimental curves (EXP-1 and EXP-2) complies with that obtained in the second numerical case. In experimental tests, using the AE method, the value of the damage initiation load of the first layer was determined. According to the literature [6,14,15], this type of damage occurs at the first local increase of the acoustic signal energy, which was significantly greater than the previous ones. In Figure 7, such a situation was observed for approx. 560 s, where the energy increased almost eightfold. Thus, the damage initiation load from the load signal at 560s. was assumed. The damage initiation load for other samples in a similar way were determined. Earlier registered lower energy readings could occur as a result of matching the sample to the flexible pads.

**Figure 7.** The load and acoustic energy signal versus time for the second real column (EXP-2): Aramis signal—red line; energy signal of AEM—blue mark.

All the values obtained from the experimental tests were presented in the collective diagram together with the results obtained from the numerical tests (Figure 8),where the following symbols have been adopted: TSAIW—Tsai-Wu criteria, TSAIH—Tsai-Hill'a criteria, AZZIT—Azzi-TsaiHill'a criteria, MSTRS—maximum stress criteria and HASHIN-Hashin's criteria.

**Figure 8.** The damage initiation load from numerical and experimental cases.

The results obtained for the first model were about 28 to 34% higher than for the second one. This confirms the significant influence of the boundary conditions on the behavior of laminated thin-walled columns. Such a large discrepancy in the results may indicate that both numerical models do not analyze the same damage mechanism. Hashin's criterion on the second model (M-2) almost perfectly defined the damage initiation load compared to the experimental.Based on the three experimental results, the median was determined to be 1361N (EXP-2). Next, relative errors for the numerical results with respect to the median of the experimental results were calculated (Figure 9). The smallest value overestimation was for model M-1 in the case of the Tsai-Wu criterion, while in the other criteria the error increased until it obtained the highest overestimation of the value for the Hashin criteria. However, the smallest underestimation of the value was for the M-2 model in the case of Hashin criterion (4.61%), while in the other criteria the error increased until it obtained the largest underestimation of the value for the Tsai-Wu criterion. In the first case (M-1), the Tsai-Wu criterion allows identifying the damage initiation load, while in the second model, the Hashin criterion definitely helps to identify the damage initiation load. It is worth remembering that in two variants this applies to different damage mechanisms. Considering the above, Hashin's criterion most accurately describes the actual damage initiation load.

**Figure 9.** The relative error between numerical and experimental results: *δP* = ((*PMES* − *PEXP*)/*PMES*) × 100 *percent*.

Dimensionless equilibrium paths determined from experimental and numerical tests are presented in Figure 10. A very good agreement of the results was obtained for the M-2 model, while the results from the M-1 model demonstrate a different damage mechanism. It is also clearly visible in Figure 11, where the equilibrium paths obtained from numerical models are presented. The shortening of the M-1 model was around 550% lower than for the second one. Moreover, the damage initiation load obtained in the first model is higher by about 50% compared to the results of the second model. In addition, it is worth noting how much influence the flexible pad (i.e., the appropriate stiffness of the support) has on the behavior of the thin-walled laminate L-shaped column. It can be suspected that, by increasing the stiffness of the pads, the equilibrium curve should be placed between the presented results.

**Figure 10.** Dimensionless equilibrium paths of L-profile column under shortening.

**Figure 11.** Equilibrium paths of L-profile column under shortening-numerical results.

### **6. Conclusions**

The main aim of the study was to validate the numerical models (M-1, M-2) with the results of the experimental tests. The first model M-1 consists of laminated column situated between the rigid grips, while in the second model the flexible pads between sample and rigid grips are additionally included.Based on the obtained results, the following conclusions have been drawn:


Therefore, the model with elastic pads more accurately reflects the actual behavior of the L-profile element under compression. However, further development in numerical models (especially checking more detailed boundary conditions) together with their experimental validation is necessary.

**Funding:** The project/research was financed in the framework of the project Lublin University of Technology—Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract No. 030/RID/2018/19).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


## *Article* **Higher Order Multiscale Finite Element Method for Heat Transfer Modeling**

**Marek Klimczak \* and Witold Cecot**

Faculty of Civil Engineering, Cracow University of Technology, Warszawska 24 Street, 31-155 Cracow, Poland; plcecot@cyf-kr.edu.pl

**\*** Correspondence: mklimczak@L5.pk.edu.pl

**Abstract:** In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than *p* = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the *p*-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

**Keywords:** heat transfer; multiscale finite-element method; homogenization

## **1. Introduction**

Numerical modeling of the heterogeneous materials is a very active research field [1–7]. This is due to the fact that such materials are widely used in many important industry branches, e.g., civil engineering [1,2,6,7], automotive engineering [3,4], aerospace engineering [8] and many others. The composite's superior performance is due to its manifold mechanisms, among others:


The major aspects, listed above, are analyzed at their respective scale, i.e., atomistic, micro-, meso- or macroscale, in order to optimize the overall composite performance at the higher scale. The design process of the new composites is enhanced with numerical analyses that are complementary to laboratory experimentsm and provide an evident speed-up. A number of numerical methods are used to model composites. The most frequent applications are based on the finite-element method [1,11], the finite-difference method [2], the discrete element method [12] and other methods [5,13]. In our research, we make use of the higher-order, finite-element method. In this paper, only the *p*-convergence is studied, i.e., we keep the same mesh and sequentially increase the approximation order

**Citation:** Klimczak, M.; Cecot, W. Higher Order Multiscale Finite Element Method for Heat Transfer Modeling. *Materials* **2021**, *14*, 3827. https://doi.org/10.3390/ma14143827

Academic Editors: Michele Bacciocchi, Raimondo Luciano, Angelo Marcello Tarantino and Carmelo Majorana

Received: 9 June 2021 Accepted: 2 July 2021 Published: 8 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

*p* within all elements. However, in our previous studies [14], we presented the application of the automatic *hp*-adaptivity [15,16] coupled with the multiscale finite-element method. The automatic *hp*-adaptivity enhances the whole framework due to its expected exponential convergence. In this paper, the scope is narrower concerning this aspect. The benefits of the higher-order multiscale-finite element method with the modified hierarchical shape functions in the multiscale analysis are demonstrated in Section 4.

A direct numerical analysis of the composite at the heterogeneity scale is frequently infeasible or unnecessary [17,18]. The analysis itself would be challenging in the context of the computational resources, as well as time-consuming. When completed, the results would provide an amount of data that could be impractical to present and demanding to store. Thus, a variety of upscaling methods was used to reduce the computation cost with the simultaneous incorporation of the lower-scale information in analyses carried out at the higher scale. Extensive revisions of the most popular upscaling methods can be found, for instance, in [19–22]. In our research, we made use of the multiscale finite-element method (MsFEM) in a form developed in [14,18,23,24]. A concise description of this is provided in Section 3.1. The outline and the specific features of the method are provided therein. Selected other multiscale finite-element methods [19] and similar approaches used for heat transfer analysis (or Poisson equations in general) are also presented in Section 3.1.

We mainly use MsFEM due to its flexibility. In the case of the composites' analysis, the method exhibits a very important advantage that should be stressed here. Namely, it is free of the assumption regarding the distinct separation of the bridged scales. In some applications (e.g., asphalt pavement), this feature is useful in effective multiscale analysis. This is due to the relationship between aggregate particle diameters and layer thicknesses.

The natural parallelization, without any special amendments, introduced to the independent problems solved within neighboring subdomains, guarantees a substantial speed-up to the computations. A number of observations, which are thoroughly discussed in Section 3.1, give rise to additional computational time savings, which facilitate MsFEM implementation.

The main novelty of this paper consists of the application of the higher-order, multiscale, finite-element method to the steady-state heat transfer problem. The approach is based on a modified shape function construction. The auxiliary boundary value problem was introduced, and used for the heat transfer problem. In our previous papers [14,18,23,24], we applied a higher-order MsFEM to other partial differential equations (PDEs), namely, to the elasticity and viscoelasticity.

In this paper, we demonstrate an effective MsFEM performance and also test its *p*convergence in order to illustrate the benefits of the higher-order approximation in a steadystate heat transfer problem. It is additionally discussed in terms of the computational time.

Finally, the proposed method was tested on very challenging numerical problems with a large contrast in terms of its material parameters (several orders of magnitude).

Ensuring the effective and reliable heat transfer analysis of the composites is one of the challenging numerical problems. Multiscale methods are the common approach in this case, due to the reasons given in the above paragraph. State-of-the-art papers devoted to the review and classification of multiscale methods developed for the heat transfer problem are present in the literature [25–27]. The main distinction refers to the analysis scale [27]; therefore, one can distinguish:


In order to avoid the problem of the excessively large amount of the information obtained in the direct composite analyses at any scale, a number of hybrid and multiscale methods are used for the heat transfer problem [27,28,30,31]. They can be classified in a manifold manner. In [27], they are mainly grouped according to the scales they bridge, e.g., molecular dynamics—continuum description, molecular dynamics—particle-based description. Below, following the distinction used in [27], we briefly present the selected methods using a description of the respective resolution, in order to subsequently upscale this. However, it should be noted that the coupling of scales is not exclusively performed for two neighboring scales. For instance, a hierarchical bridging between several scales can be performed, as in [33]. Thus, the coupling possibilities are not limited to those presented in the next section.

In this paper, the focus is only on the macroscale modeling. Namely, at both bridged scales, the assumption of the continuum of the domain is used. The purpose of describing the methods referring to lower analysis scales is to place the multiscale, finite-element method among other approaches. In the case of micro- and mesoscale analysis, we arbitrarily selected the most representative methods. For an extensive review of these, we refer the reader to [27,28,30,31]. In the case of the macroscale modeling, the description is slightly different. We present several representative methods, as well as a group of approaches that share similar concepts, as the multiscale finite-element method used in this study.

#### *1.1. Microscale Modeling*

In molecular dynamics, we analyze the motion of every single atom. An atom's motion is described by Newton's second law and interactions between atoms are described by the potential functions [27,30]. In the case of large domains, this approach is very time-consuming and provides an excessive amount of information. Thus, the research interest was to simplify the modeling.

The first coupled MD-continuum analysis was described in [34], and referred to the flow in a channel. The whole domain was divided into an atomistic and continuum region with a hybrid solution interface (HSI), where both of the descriptions were valid. The development of the methodology presented in [34] was mainly associated with the numerical treatment of the atomistic region and its coupling with the continuum region [35] (*coupling by states*). A coupling strength parameter *ξ* was introduced, due to the constrained dynamics used to transfer the data at the HSI. The study on this parameter led to further improvements [36,37] in the methodology presented in [34]. In [38], these two regions (continuum and MD) were coupled, introducing the fluxes in the HSI (*coupling by fluxes*). The USHER algorithm was also proposed [39] for both types of coupling to conserve the overall energy, momentum and mass by manipulating the number of atoms at the HSI.

The methods presented above constitute a group of hybrid methods, which use two regions with a different description resolution. Although these approaches are not purely heat-transfer-oriented, their methodologies can also be applied to the leading problem of this paper and to other elliptic problems. Due to the necessity of partial MD analysis in the domain, they remain relatively computationally demanding.

In order to actually bridge the MD scale with the continuum scale, a number of methods that originated from the heterogeneous multiscale method (HMM) [40] were developed. The main analysis was carried out at the continuum level with the local transfer of information from the atomistic scale.

#### *1.2. Mesoscale (Particle-Based) Modeling*

Unlike in the molecular dynamics, where we analyze the motion of every single atom/molecule, mesoscale modeling is based on a coarse-grained analysis [26,27]. This approach can be regarded as a kind of microscale analysis upscaling. A group of molecules is represented by the computational particle, and the mechanisms of such particles' evolution were assumed. The differences between the main representatives of this group consist of a description of the particle evolution mechanism. The macroscale description was based on the continuum assumption at the higher scale, and the microscopic description at the lower scale is discrete. The mesoscale modeling is situated between these, i.e., the description is discrete but the scale of analysis (spatial and temporal) is substantially reduced.

When the particle motion is characterized by a probabilistic description, the direct simulation Monte Carlo method (DSMC) was established [27,30]. When the hydrodynamic behavior of the particles was mainly modeled by the addition of the dissipative force, dissipative particle dynamics (DPD) are used. Two other main representatives of this approach are the lattice gas automata (LGA) and the lattice Boltzmann method (LBM), which are based on the simplifications introduced to the collision term in the Boltzmann equation [27].

For the multiscale techniques bridging the microscale and mesoscale modeling, one can look at, e.g., the methods transferring the velocities between the scales [41] (MD, DPD and the continuum) or their distributions [42].

## *1.3. Macroscale Modeling*

As far as the heat transfer modeling is concerned, the continuum assumption is typically used. It is also the main scale of interest within this paper. The methods which are most frequently [27] used to numerically solve the heat transfer problem at this scale are the finite-element method (FEM), the finite-difference method (FDM), the finite-volume method (FVM) and a number of further modifications. In our research, we make use of the higher-order finite-element method due to its expected exponential convergence, especially when combined with the automatic *hp*-adaptivity, as mentioned in Section 1.

The methods used to bridge between lower observation scales and the continuum description were mentioned in Sections 1.1 and 1.2. It is natural that the processes occurring at these scales affect the macroscale response. The scope of this paper is narrower. Namely, we are only interested in multiscale analysis at the continuum level. The constituent properties are assumed to be known (e.g., from the laboratory test), but we analyze the effective response of a composite.

The very basic engineering approach is made to assess the effective properties of the material (the conductivity, in this case) on the basis of its underlying microstructure. In further analysis, the material is numerically modeled as the homogeneous one. For instance, direct or inverse mixture rule is used for this purpose.

In the case of the periodic domains, a theoretical approach to the asymptotic homogenization [43,44] was primarily used. This is based on the auxiliary BVP solution for a unit cell. Practically, a number of such BVPs are solved for the increasing unit cell sizes. This version of the homogenization for the Poisson equation was used, e.g., in [45,46]. Unlike this method, the improved MsFEM used in this study does not require the periodicity of the domain. It needs an auxiliary BVP to be solved, but the subdomain (a coarse mesh element) used for this purpose is typically selected only once.

The most popular approach used in the numerical analyses is the computational homogenization [5,22], which is based on the representative volume element (RVE) approach. Using the auxiliary BVP solution for every RVE (typically associated with the Gauss integration point), the quantities at the macroresolution are assessed using the Hill–Mandel condition. It is also free of the assumption of domain periodicity. If the finite element method is used at both analysis levels, the approach is known as FE<sup>2</sup> homogenization [47,48]. The advantage of computational homogenization is that we do not need to assume a constitutive equation at the macroresolution, but can transfer the tangent stiffness tensor from the lower scale. The main limitation to this is the separation of the scales' condition in the RVE analysis. Namely, the ratio of the characteristic dimensions of the micro- and macroscale should not exceed 0.1. In some structures, this condition does not hold. The improved MsFEM, used in this study, does not require this condition to be fulfilled. The computational homogenization for the heat transfer problem is discussed, e.g., in [49]. In a recursive manner, macro- to microscale transitions are performed. The pointwise macroscale response is transferred to the lower scale in a form of the boundary conditions for the BVP solved in the RVE (and associated with this point). From this microscale level, the averaged quantities are transferred to the macroscale.

In [13,31], the multiscale seamless-domain method (SDM) is presented. It has some similarities to both the computational homogenization and the multiscale finite-element method used in this paper. The macroscale solution is sought at the so-called coarsegrained point. Its surrounding subdomain is discretized arbitrarily (regardless of the mesh generated for a subdomain associated with neighboring coarse-grained point). The solutions sought are interpolated within these subdomains. Subsequently, the final solution is obtained using the SDM scheme to "average" the solutions resulting from the overlapping subdomains. In MsFEM, discretizations for the neighboring subdomains need to be compatible. Moreover, MsFEM is the FE<sup>2</sup> scheme, unlike the SDM or the computational homogenization, which can be employed for any numerical method at the microscale level.

For instance, using the proper orthogonal decomposition (POD) at the RVE level constitutes the FEPOD method [50]. Therein, POD is used to reduce the basis and to speed-up the computations.

MsFEM shares a similar substructuring concept to the superelements [51]. This is a smart way of obtaining the fine mesh solution, using static condensation and solving the reduced problem only for the nodes associated with the skeleton. Consequently, the discretizations within the neighboring superelements need to be compatible. Unlike this concept, MsFEM first delivers the upscaled coarse-mesh solution, which can be transferred elementwise to the corresponding fine meshes.

As was noted in [27], not all the multiscale frameworks can be easily classified using the distinction on the bridged scales. The multiscale finite-element method improved in this paper does not fall within the above-mentioned classification. Like, e.g., the computational homogenization [22], it uses the same material description level (continuum) at both bridged scales.

The similarities between the MsFEM presented in this paper and the heterogeneous multiscale method (HMH) [40] should be underlined, since, in both cases, the microscale coupling *u* and macroscale *U* variables can be performed mutually by the respective operators. The means of their assessment constitutes the main part of the aforementioned methods. In HMH, the underlying microstructure is taken into account during the integration of the respective entries of the macroscale stiffness matrix and load vector. MsFEM enables the effective computation of these quantities through the multiplication operations of the assembled (within a coarse element) fine-mesh quantities.

A short comment on the nomenclature is necessary. There is a variety of so-called multiscale methods. The MsFEM version we use is based on a special shape function concept. There are also other approaches with a similar name. In [52], for instance, a multiscale finite-element method, based on the asymptotic expansion, is presented in applications to a periodic microstructure.

For a comprehensive description of the other existing homogenization methods, we refer to [20–22].

The remaining part of this paper is organized as follows. Section 2 constitutes a brief description of the analyzed heat transfer problem. For the sake of clarity, we limit this paper to the steady-state heat-transfer problem. In Section 3 an outline of the developed MsFEM version is presented, together with the formulation of the boundary value problem used to assess the special shape functions. Additionally, some comments on the numerical implementation are provided. In Section 4, the numerical results are shown to illustrate the performance of the introduced method. Finally, in Section 5, the findings of this paper are recapitulated and discussed.

#### **2. Problem Formulation**

We selected the steady-state heat-transfer equation as the test problem for the developed version of MsFEM. Thus, we limit our problem to the resulting Poisson equation of the following form:

$$\frac{\partial q\_x}{\partial x} + \frac{\partial q\_y}{\partial y} + \frac{\partial q\_x}{\partial z} = f \tag{1}$$

Herein, *f* denotes the internal heat source; *qx*, *q<sup>y</sup>* and *q<sup>z</sup>* denote the respective heat flux components. Using Fourier's law, the latter can be expressed in terms of the temperature *T*(*x*, *y*, *z*), as:

$$\begin{aligned} q\_x &= -k \frac{\partial T}{\partial x} \\ q\_y &= -k \frac{\partial T}{\partial y} \\ q\_z &= -k \frac{\partial T}{\partial z} \end{aligned} \tag{2}$$

where *k* is the thermal conductivity coefficient. We limit this study to the isotropic material, so *k* is independent of direction. This assumption is not necessary; therefore, we used it to simplify the description. Moreover, the materials analyzed in this study exhibit such behavior. Inserting (2) to Equation (1), we obtain:

$$-\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) - \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right) - \frac{\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) = f \tag{3}$$

for every subdomain with C<sup>1</sup> -continuous *k*, with both temperature and normal flux component continuity on the subdomain interfaces. In this paper, we use only two types of boundary conditions, i.e.,


## **3. Upscaling**

In this chapter, the MsFEM outline is described and illustrated. Some comments on the implementation of the presented approach are also provided, to emphasize the potential benefits of its application in the numerical modeling of composites.

#### *3.1. Idea*

As was mentioned in the Introduction, MsFEM shares a similar concept to HMM [40]. Namely, there is a distinct hierarchy of scales. Let us name them the microscale and the macroscale. Unlike in the Introduction, they both refer to the continuum level, but the microscale resolves the heterogeneous material structure and the macroscale is the scale of the analyzed effective material response. The mappings between the micro- (*u*) and macroscale (*U*) degrees of freedom must be determined in the first step. As a result of the analysis, we obtain the macroscale solution, and the microscale one can be derived in the post-processing phase.

The construction of the prolongation operator *P*, which allows us to express the microscale solution in terms of the macroscale one (*u* = *PU*), as well as in the other way (*U* = *P <sup>T</sup>u*), is the core of the argued methodology.

We proceed with two sets of compatible meshes. The whole domain is discretized at the macroresolution level with a coarse mesh. Then, each of its elements is substantially refined in order to capture all the heterogeneities at the microresolution level. In this manner, the coarse mesh is naturally compatible with this set of the corresponding fine meshes. In [14], we demonstrated the potential of *hp*-adaptivity application at the macroresolution level. Thus, this is also used here.

In order to generate the prolongation operator *P*, we modify the standard coarseelement shape functions to account for the microstructure, by the solution of the auxiliary problem in coarse elements [14,23]. In Figure 1, we present a standard bilinear shape function and its modified counterpart, which was obtained as an exemplary composite. Knowledge of the operator *P* is used to compute the effective stiffness matrices and load vectors. Instead of integrating the modified shape functions over the coarse elements, we can immediately compute the effective stiffness matrix *K<sup>H</sup>* and load vector *f<sup>H</sup>* using the

corresponding fine mesh quantities *K<sup>h</sup>* and *fh*, which are assembled locally within a single coarse mesh element, since

**Figure 1.** Exemplary bilinear standard (**a**) and modified (**b**) shape functions for the unit square coarse element.

For a number of tests on the special shape functions, we refer to [18]. Therein, MsFEM was used to solve the linear elasticity and viscoelasticity problems.

The general algorithm of the MsFEM application to the steady-state heat-transfer problem is analogous to the one presented in [18] for the linear elasticity. The difference, beside the different governing partial differential equation (PDE) to be solved (Equation (3)), consists of the other auxiliary boundary value problem (Equation (5)), which was consequently used for the special shape function assessment. This is marked in blue in Algorithm 1.

**Algorithm 1** Solve a heat transfer problem within a heterogeneous domain.

**Require:** define the problem (heterogeneous domain and boundary conditions) **Ensure:** a coarse mesh and an appropriate refinement of each coarse element **for** *n*=1 to *Nel* **do** {loop over coarse mesh elements} **for** *m*=1 to *M* **do** {loop over *n*-th element shape functions} solve local problem (5) in the *n*-th element for the *m*-th shape function **end for** compute *K<sup>H</sup>* and *f <sup>H</sup>* for the *n*-th element **end for** solve the coarse mesh problem using the effective matrices *K<sup>H</sup>* and vectors *f <sup>H</sup>*

#### *3.2. Formulation*

The problem formulation for the assessment of the modified shape function is presented below. This is solved within a subdomain Ω*<sup>s</sup>* (occupied by a single coarse mesh element) of the whole analyzed domain Ω. We need to solve problem (5) for every coarseelement shape function. The degrees of freedom obtained for the *m*-th standard shape function Ψ are the *m*-th column of the prolongation operator *P*. The problem presented below expresses the equality of the residuum of the solution (temperature) interpolant and residuum of the interpolated function Φ*m*.

*Given* Ψ*m, which is a coarse mesh scalar-valued shape function (m = 1, . . . , M), we look for its scalar-valued counterpart* Φ*m, which is a discrete solution of the following Dirichlet boundary value problem*

$$\begin{cases} \frac{\partial}{\partial \mathbf{x}\_i} \Big( k \frac{\partial \Phi\_{\mathbf{m}}}{\partial \mathbf{x}\_i} \Big) = -\frac{\partial^2}{\partial \mathbf{x}\_i^2} \Psi\_{\mathbf{m}} \quad \forall i = 1, 2, 3, \ \mathbf{x} \in \Omega\_s^j \subset \Omega\_s \subset \Omega\\ \Phi\_{\mathbf{m}} = \hat{\Phi}\_{\mathbf{m}} \quad \text{on} \quad \partial \Omega\_s\\ + \text{interface continuity conditions} \end{cases} \tag{5}$$

*where k is the thermal conductivity of the material at a given location x,* Ω *j <sup>s</sup> denotes the j-th constituent of the composite*.

For the finite-element-method computations, we need a variational formulation, which is presented below.

*Find* <sup>Φ</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>V</sup>*<sup>0</sup> <sup>+</sup> <sup>Φ</sup><sup>ˆ</sup> *such that*

$$\int\_{\Omega} k \nabla \boldsymbol{\upsilon} \cdot \nabla \Phi\_{\mathfrak{m}} d\Omega = \int\_{\Omega} \boldsymbol{\upsilon} \Delta \mathbb{V}\_{\mathfrak{m}} d\Omega \quad \forall \boldsymbol{\upsilon} \in V\_0 \quad \text{where} \quad V\_0 = \{ \boldsymbol{\upsilon} \in H^1(\Omega\_{\boldsymbol{\delta}}) : \boldsymbol{\upsilon} = 0 \text{ on } \partial \Omega\_{\boldsymbol{\delta}} \} \tag{6}$$

This was obtained via the multiplication of (5) by the test function *v*, and the further integration of both equation sides over Ω*<sup>s</sup>* (integrating by parts was also used at the lefthand side to transfer the derivative to the test function *v*). Practically, we additionally scaled the bubble functions in such a way that the extreme values were equal to 1.

Definition of the Dirichlet boundary conditions Φˆ *<sup>m</sup>* needs special treatment regarding the dimensionality of the problem:


**Figure 2.** Exemplary "bubble" (**a**) standard and (**b**) modified shape functions for the unit 1D coarse element.

#### *3.3. Implementation*

We implemented [14,18] MsFEM for the linear elasticity and viscoelasticity problems. In this paper (Section 4), we present the results of the 2D analysis of the steady-state heat transfer using modified MsFEM.

To facilitate the multiscale computations, one can take advantage of two approaches, which are almost equivalent in terms of time saved:


The second option is general, and the only time losses are due to the transfer of the local fine mesh computation (*K<sup>H</sup>* and *fH*) to the global solver.

Another advantage of MsFEM, in the context of computational efficiency, is the possible low cost of Equation (5) solution. In fact, this is the main computational cost of the whole approach, since it has to be solved for every standard coarse-element shape function Ψ*m*. However, it can be observed that Equation (5) leads to a system of algebraic equations with multiple right-hand sides. Let us denote the number of the functions Ψ*<sup>m</sup>* as *M*, and the time required for the solution of Equation (5) for a single Ψ*<sup>m</sup>* as *t m* Ψ . The overall computational time *T<sup>P</sup>* leading to the full operator *P* assessment is not, therefore, equal to *M* × *t m* Ψ but *T<sup>P</sup> M* × *t m* Ψ , and it is of the order of *t m* Ψ . This is due to the fact that the computational time necessary to solve the system of linear algebraic equations with multiple right-hand sides does not significantly exceed the computational time used for a single system of linear algebraic equations.

In addition to the above discussion, one can observe that the matrix *K<sup>h</sup>* , present in (4), is the same as the left-hand side of (5) for the arbitrary coarse element. Thus, the assembly can be performed once and is primarily used for Equation (5), and subsequently for Equation (4), which can also provide a further speed-up to the computations.

Preserving the global continuity of the solution is necessary in the MsFEM workflow. In the case of the irregularly shaped constituents (see Figure 3a), the solution to Equation (5) along the edges of the coarse elements (denoted with numbers from 1 to 4) is straightforward. For instance, one of the modified linear shape functions for the common edge of elements 2 and 4 (marked with a red dotted line) is of the form shown in Figure 3b.

**Figure 3.** Exemplary shape function assessment: (**a**) analyzed domain—colors denote various thermal conductivities of the constituent, (**b**) modified linear shape function along the dotted edge.

A situation where the coarse element edge coincides with different material interfaces needs special attention. When solving Equation (5) along an edge, thermal conductivity *k* needs to be used for a given *x*. Using different values for the common edge in a loop over coarse elements would not preserve the continuity of the sought function Φ within a 2D domain. To overcome this problem, we always take the average thermal conductivity at a given Gauss point ( *<sup>k</sup>*(*x*+*ε*)+*k*(*x*−*ε*) 2 , where *ε* is a small vector, orthogonal to the edge). This operation was performed to provide the continuity of the solution. If this was skipped, one could observe "jumps" in the solution of the interface (in the

discussed situation). The accuracy would decrease with the increasing contrast between the constituents' parameters.

#### **4. Numerical Results**

#### *4.1. Asphalt Concrete*

In this test, we analyzed a sample with the dimensions 10 cm × 15 cm, made of asphalt concrete (AC), as shown in Figure 4b. AC is a standard asphalt mixture that can be applied for all asphalt layers constituting the flexible pavement structure. It comprises two main phases: mastic (the asphalt binder mixed with the filler) and the aggregate particles. The detailed procedure of the AC microstructure generation used for this test is presented in [24]. Given the prescribed gradation curve in asphalt concrete, we used the approach based on the shrunk Voronoi cells to generate a microstructure realizing this requirement. Using the method presented in [24], we generated a non-periodic microstructure in a coarse element with "periodic boundary conditions", which enabled us to easily multiply this geometry. The temperature along the bottom edge was fixed and equal to 15 °C. The upper edge was subjected to heating with *q* = 30 W/m. The remaining edges were insulated (*q* = 0). Thermal conductivity was equal to 4 W/(mK) for the aggregate particles and 0.8 W/(mK) for the asphalt binder. In this test, we neglected the presence of the air voids.

In Figure 5, we present the temperature distributions obtained using direct FEM analysis (*p* = 1) and MsFEM (*p* = 5). The corresponding cross-section plots, performed at half the specimen height, are shown in Figure 6. The modified shape functions for the coarse elements are similar to those presented in Figure 1b. The temperature distributions for the lower approximation orders, used at the macroscale, were skipped for the sake of brevity. They are visually indistinguishable from the solution presented in Figure 5b. Instead, the convergence plots are presented in Figure 7. They illustrate the applicability of MsFEM with respect to both the number of degrees of freedom and computational time.

We measure the error using the following formula

$$\frac{|T^{\hbar} - T^{H}|}{|T^{\hbar}|} \tag{7}$$

where *T h* is the value of temperature at a given position obtained using the fine mesh and *T <sup>H</sup>* is the corresponding temperature obtained using the coarse mesh. By the fine mesh solution, we mean the reference solution plotted in Figure 5a. This was obtained using about 100,000 degrees of freedom.

**Figure 4.** Synthetic asphalt concrete microstructure: (**a**) within a single coarse element, (**b**) within a whole domain.

(**a**) (**b**)

**Figure 5.** Temperature distribution [°C]: (**a**) direct FEM solution, (**b**) upscaled solution (plotted for *p* = 5).

**Figure 6.** Temperature [°C] along the selected segment (at the half of the specimen height).

**Figure 7.** Convergence test for the increasing approximation orders at the macroscale (logarithmic scale) w.r.t. (**a**) number of degrees of freedom and (**b**) relative time (the ratio of the computational time used for the upscaled and reference solutions).

The additional modeling error introduced by the upscaling is very small; even for the bilinear shape functions used at the macroresolution its norm, it is smaller than 0.4%. It should be noted that, in this case, we used only 12 degrees of freedom. For the approximation order equal to 5 at this scale, the number of degrees of freedom (NDOF) is equal to 176 and the error norm drops to about 0.1%. The number of coarse elements is kept constant and equal to 6. The reduction in degrees of freedom between the reference and upscaled solutions range from about 570 to about 8300. The results of this test confirm the *p*-convergence of MsFEM. The *h*-convergence was not studied in this paper. The focus was on the verification of the higher-order approximation applicability at the macroscale. In our previous papers [14,23], we also tested this type of convergence for the elasticity problem.

In our academic code, we only implemented the approximation of the order *p*, equal up to 5, for the heat transfer problem. The code would have to be modified to numerically verify the further shape of the convergence plot for this example. It should be noted that the macroscale *p*-convergence is affected by the microscale, and the corresponding discretization at this scale. A theoretical example of *p* → ∞ would be cumbersome for this method. A prohibitively expensive fine mesh would have to be generated in order to approximate, with linear shape functions (typically used at the microscale level), a macroscale shape function of a very high order. This would be the case even for a very simple geometry. Practically, the order *p* = 5 is very rarely exceeded in numerical applications.

#### *4.2. Metal Foam*

In the second test, we present the application of MsFEM in steady-state heat-transfer analysis of the metal foam. This lightweight material is used in many industry branches. In the context of the numerical modeling of heat transfer, the metal foam, analyzed with the application of any upscaling methods, can be regarded as a very challenging problem. This is due to the large difference between the constituents' thermal conductivity. In this test, we analyzed the idealized aluminum foam sample with a matrix thermal conductivity equal to 236 W/(mK) and the thermal conductivity of the air was set as 0.0262 W/(mK). The rectangular sample of dimensions 2 cm × 8 cm was heated along the upper edge with *q* = 300 W/m. Along the bottom edge, the temperature is equal to 15 °C and the remaining edges are insulated. The material distribution, as well as the reference and upscaled solutions, are presented in Figure 8. The corresponding cross-section plots, performed at half of the specimen height, are shown in Figure 9. The air voids are modeled as circles with a random radii distribution.

In this test, the reference solution was obtained using more than 69,000 degrees of freedom. The fine mesh within a single coarse element is shown in Figure 10.

The upscaled solutions were obtained using 10 ÷ 126 degrees of freedom for the approximation order *p* = 1 ÷ 5, consecutively. The number of coarse elements is equal to 4. In Figure 11, we present the error convergence plots for the increasing approximation orders used at the macroscale. The convergence is demonstrated with respect to both the number of degrees of freedom and the computational time.

In this test, the *L*<sup>2</sup> norm of error (see Figure 11), measured according to 7, is equal to about 1.6% for *p* = 1 at the macroscale, and drops below 1.2% for the approximation order *p* = 5. The reduction in the number of degrees of freedom spans from about 550 (*p* = 5) to about 6900 (*p* = 1).

In Figure 12, we present the upscaling error distribution for this test (for *p* = 1, 3, 5). We compute the error according to Formula (7). The differences span from about 0.76% (*p* = 5) to about 2% (*p* = 1). These discrepancies are acceptable from the engineering perspective. The character of error distribution around the air voids closest to the boundary is the result of a small number of finite elements, generated between this boundary and the void (see Figure 10).

**Figure 8.** Material distribution (**a**) and temperature maps [°C]: (**b**) direct FEM solution, (**c**) upscaled solution (plotted for *p* = 5).

**Figure 10.** Fine mesh within a coarse element.

**Figure 11.** Convergence test for the increasing approximation orders at the macroscale (logarithmic scale) w.r.t. (**a**) number of degrees of freedom and (**b**) relative time (the ratio of the computational time used for the upscaled and reference solutions).

Both convergence plots were created using the logarithmic scale to demonstrate the impact of the higher-order approximation at the macroscale on the result correctness. Technically, the comparisons between the reference and upscaled solutions are justified, since the mesh was generated as follows. First, the mesh was generated for a single coarse element. Then, it was copied several times to model the whole domain. In this manner, we compare the solutions obtained using meshes of the same densities.

A short discussion regarding the error convergence plots shown in Figures 7b and 11b is necessary. Compared to the NDOF reduction in the upscaled solution, the observed speed-up is not equally impressive. This is due to the additional time necessary for the prolongation operator *P* assessment. It should be noted that this is highly affected by the implementation itself, as well as the discretizations at both scales. Moreover, MsFEM and other upscaling methods are addressed for problems, where the direct solutions are prohibitively computationally expensive or unfeasible to obtain. In this latter case, the computational time is less important than the ability to obtain the solution.

**Figure 12.** Error distribution for the increasing approximation order: (**a**) *p* = 1, (**b**) *p* = 3, (**c**) *p* = 5, consecutively.

#### **5. Concluding Remarks**

In this paper, we proposed a higher-order MsFEM for the steady-state heat-transfer problem. The main novelty consists of the modified shape function assessment for this problem, presented in Section 3.2, and numerical confirmation of the upscaled solution *p*-convergence for a very large material contrast. We demonstrated the convergence of this method with respect to both the number of degrees of freedom and the computational time.

The proposed approach was verified on two tests, which were performed for the realistic materials. The first presents the application of MsFEM to the steady-state heat-transfer analysis of the asphalt concrete sample. In this example, the focus was on the method performance for a non-trivial microstructure, and there was not a large difference between the constituent thermal conductivity. The second test presented the application of MsFEM to a very challenging numerical problem. The steady-state heat transfer analysis of the aluminum foam was presented. In order to illustrate the superiority of the method, the air voids were included in the numerical modeling. Its presence implies the necessity of special composite modeling. Namely, we deal with a material with constituent parameters varying by several orders of magnitude. This is a very difficult problem for any upscaling method.

In both of the presented tests, a large reduction in the degrees of freedom was observed for the upscaled solutions. Compared to the reference fine mesh solutions, this was equal to 570 ÷ 8300 (asphalt concrete sample) and 550 ÷ 6900 (metal foam sample). It should be noted that, even for the harder test (metal foam) and the bilinear approximation, the maximum error is equal to about 2%. In both tests, the *p*-convergence was observed. Considering the large disproportion of the material parameters and the obtained NDOF reduction, this result is acceptable.

In terms of the computational time, the applicability of MsFEM was also demonstrated. The relative time, i.e., the ratio of the computational time necessary for the upscaled and reference solutions, is much smaller than the unity for both tests. The possible speed-up is highly affected by the implementation and discretizations at both scales. Thus, this can be increased. However, this was not the focus of this study.

Our further research effort is to extend the proposed framework to thermoelastic analysis. Such an approach is necessary, e.g., in the context of asphalt concrete modeling, where the thermal effects are very significant.

**Author Contributions:** Conceptualization, M.K. and W.C.; methodology, M.K.; investigation, M.K. and W.C.; writing—original draft preparation, M.K.; writing—review and editing, M.K. and W.C. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Narodowe Centrum Nauki (Poland) grant number UMO-2017/25/B/ST8/02752. The APC was funded by the Dean of the Civil Engineering Faculty (Cracow University of Technology).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

### **References**


## *Article* **Material Properties of HY 80 Steel after 55 Years of Operation for FEM Applications**

**Szturomski Bogdan and Kici ´nski Radosław \***

Mechanical and Electrical Engineering Department, Polish Naval Academy, 81-103 Gdynia, Poland; bsztur@gmail.com

**\*** Correspondence: r.kicinski@amw.gdynia.pl; Tel.: +48-505-710-486

**Abstract:** The paper presents the results of testing the properties of HY 80 steel from the hull of a Kobben class 207 submarine after 60 years of operation in extreme sea conditions. Steels from the HY family in the post-war period were used to build American and German submarines. For the obtained fragment of steel from the hull of the Polish submarine ORP Jastrz ˛ab (ORP-Boat of the Republic of Poland), static tensile tests were performed on an MTS testing machine. Dynamic tensile tests were carried out on a rotary hammer for the strain rate in the range of 500~2000 s−<sup>1</sup> . Results: Based on the obtained results, the Johnson–Cook model and the failure parameters of HY 80 steel in terms of the finite element method (FEM) were developed. Conclusion: This model can be used to simulate fast-changing processes such as resistance of structures to collisions, shelling, and the impact of pressure waves caused by explosions in water and air related to submarines.

**Keywords:** HY 80 steel; static tensile test; dynamic tensile test; rotary hammer; plastic characteristics; Johnson–Cook material model

**Citation:** Bogdan, S.; Radosław, K. Material Properties of HY 80 Steel after 55 Years of Operation for FEM Applications. *Materials* **2021**, *14*, 4213. https://doi.org/10.3390/ma14154213

Academic Editor: Michele Bacciocchi

Received: 5 July 2021 Accepted: 26 July 2021 Published: 28 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Steels of increased strength have been used to construct the hulls of submarines, the structure of which is exposed to high loads caused by water pressure when submerged, impacts when lying on the bottom, and the effects of explosions of underwater mines and depth bombs. Commonly used by the United States for shipbuilding in the post-war years were HY 80, HY 100, HY 130, and HY 200 [1,2]. In 1960, HY 80 steel was used to make the hull of the first nuclear-powered submarine USS Thresher (SSN-593), which unfortunately ended its service tragically in April 1963.

The hulls of Los Angeles type units-USS Providence (SSN-719) [3] were made of HY 80 steel [4]. German shipyards also used this steel for the construction of project 205 submarines and their modifications. Until 2019, there were four Kobben-class (project 207) submarines in operational use by the Polish Navy. These were (S-306 Skolpen) ORP "Sep", (S-308 Stord) ORP "Sokol", (S-309 Swenner) ORP "Bielik", and (S-319 Kunna) ORP "Kondor" (Figure 1). The fifth submarine, (S-318 Kobben) ORP "Jastrzab", is used as a crew training simulator at the Polish Naval Academy. They are the last ships of this series worldwide. In 1964–1967, 15 such units were built at the German shipyard Rheinstahl Nordseewerke to modify the 205 project submarines for Norway's Navy. Five of them were transferred to the Polish Navy in 2002–2003 [5]. These ships end their service in the Polish Navy. Due to the decommissioning, their tactical data can be declassified, and therefore, their documentation can be used for scientific purposes.

HY 80 (High Yield 80) steel can withstand a load of 80 pounds per square inch which is equivalent to approximately 551.5 MPa. Their catalogue yield point is 552 MPa [4]. It is an iron alloy with a content of 0.12–0.2% carbon, 2–3.5% nickel with the addition of chromium, molybdenum, and copper. Owing to these additives, they are characterised by increased strength, good plasticity, impact strength, and corrosion resistance. Moreover,

steels of this type show good weldability, making it possible to build ships in sections and then join them [2]. *Materials* **2021**, *14*, x FOR PEER REVIEW 2 of 14

**Figure 1.** Submarines of project 207 (Kobben) in the home port in Gdynia [6]. **Figure 1.** Submarines of project 207 (Kobben) in the home port in Gdynia [6].

HY 80 (High Yield 80) steel can withstand a load of 80 pounds per square inch which is equivalent to approximately 551.5 MPa. Their catalogue yield point is 552 MPa [4]. It is an iron alloy with a content of 0.12–0.2% carbon, 2–3.5% nickel with the addition of chromium, molybdenum, and copper. Owing to these additives, they are characterised by increased strength, good plasticity, impact strength, and corrosion resistance. Moreover, In recent years, the number of accidents to submarines made of HY80 steel has increased. As a result of this study, submarine engineers have the opportunity to analyse the strength of submarine hulls. They can decide to allow them for further use. The conducted tests will allow the properties of the steel to be assessed after 55 years of operation to discern whether they have changed, or whether the ship's hull can be further used.

#### steels of this type show good weldability, making it possible to build ships in sections and then join them [2]. **2. Materials and Methods**

In recent years, the number of accidents to submarines made of HY80 steel has increased. As a result of this study, submarine engineers have the opportunity to analyse the strength of submarine hulls. They can decide to allow them for further use. The conducted tests will allow the properties of the steel to be assessed after 55 years of operation The adaptation of ORP "Jastrz ˛ab" (S-318 Kobben, built: 1966) (Figure 2) as a submarine simulator at the Naval Academy in Gdynia required numerous modifications to the hull, including vents and air conditioning connections. The material that had been removed to make holes for conditioning purposes was obtained as strength test specimens.

to discern whether they have changed, or whether the ship's hull can be further used. **2. Materials and Methods**  The adaptation of ORP "Jastrząb" (S-318 Kobben, built: 1966) (Figure 2) as a submarine simulator at the Naval Academy in Gdynia required numerous modifications to the The obtained material was used to make standardised samples for the quasi-static tensile test on the MTS testing machine with a diameter of 8 mm in accordance with EN ISO 6892-1: 2016 [7] (Figure 3). For the dynamic tensile test on a rotary hammer, round specimens with a thread with a working part diameter of 5 mm and a length of 40 mm were made. The samples were made along the ship's axis–axial.

hull, including vents and air conditioning connections. The material that had been removed to make holes for conditioning purposes was obtained as strength test specimens.

**Figure 2.** ORP "Jastrząb" (S-318 Kobben). Places from which material was taken for test samples. **Figure 2.** ORP "Jastrz ˛ab" (S-318 Kobben). Places from which material was taken for test samples.

**Figure 3.** HY 80 steel specimens for static and dynamic tests: (**a**) samples for static tests on the MTS machine; (**b**) samples for static tests after fracture; (**c**) samples for dynamic tests on a rotary hammer; (**d**) samples for dynamic tests after fracture. **Figure 3.** HY 80 steel specimens for static and dynamic tests: (**a**) samples for static tests on the MTS machine; (**b**) samples for static tests after fracture; (**c**) samples for dynamic tests on a rotary hammer; (**d**) samples for dynamic tests after fracture.

**3. Results** 

ure 4.

*3.1. Uniaxial Static Tensile Test* 

#### **3. Results**

#### *3.1. Uniaxial Static Tensile Test*

Samples were prepared from the obtained material, and a static tensile test was performed. The test was carried out on four samples. The test results are summarised in Figure 4. *Materials* **2021**, *14*, x FOR PEER REVIEW 5 of 14

**Figure 4.** Tensile test results; nominal σ–*ε* of HY 80 steel, strain rate ሶ = 0.0001 ିଵ. **Figure 4.** Tensile test results; nominal <sup>σ</sup>–*<sup>ε</sup>* of HY 80 steel, strain rate . *ε* = 0.0001 *s* −1 .

#### *3.2. The Study of Dynamic Mechanical Properties Using a Rotary Hammer*

*3.2. The Study of Dynamic Mechanical Properties Using a Rotary Hammer*  The Fundamentals of Technology Laboratory of the Naval Academy in Gdynia has a unique stand-a rotary hammer (Figure 5) that enables performing the dynamic tensile test at speeds in the range of 10–50 m/s. With a sample length of 20 mm, this allows the strain The Fundamentals of Technology Laboratory of the Naval Academy in Gdynia has a unique stand-a rotary hammer (Figure 5) that enables performing the dynamic tensile test at speeds in the range of 10–50 m/s. With a sample length of 20 mm, this allows the strain rate to be equal to 500–2000 s−<sup>1</sup> . The measurement results are presented in Table 1.

rate to be equal to 500–2000 s−1. The measurement results are presented in Table 1. At the rotary hammer laboratory stand, the sample breaking force is recorded at a given strain rate in the range of 0–2000 s−<sup>1</sup> . The maximum breaking force is then converted into the stress corresponding to the ultimate strength in dynamic tensile strength. The strain rate is defined as the ratio of the tearing speed of the sample to its measured length [8] as follows:

$$
\dot{\varepsilon} = \frac{\mathbf{d}\varepsilon}{\mathbf{d}t} = \frac{\mathbf{d}}{\mathbf{d}t} \left(\frac{v \cdot t}{l}\right) = \frac{v}{l} \tag{1}
$$

#### *3.3. HY 80 True Characteristics*

The relationship between the true stresses *σ*true and nominal stresses *σ*nom obtained from the tensile test is obtained assuming that the volume of the stretched sample during stretching is constant; thus,

$$l\_0 \cdot A\_0 = l \cdot A(F) \tag{2}$$

Hence,

$$
\sigma\_{\text{true}} = \frac{F}{A(F)} = \frac{F}{A\_0} \frac{l}{l\_0} = \sigma\_{\text{nom}} \left(\frac{l}{l\_0}\right) \tag{3}
$$

Since

$$\frac{l}{l\_0} = 1 + \varepsilon\_{\text{nom}} \tag{4}$$

we have, therefore,

$$
\varepsilon\_{\text{true}} = \ln(1 + \varepsilon\_{\text{nom}}).\tag{5}
$$

$$
\sigma\_{\text{true}} = \sigma\_{\text{nom}} (1 + \varepsilon\_{\text{nom}}) \tag{6}
$$

Plastic deformation is the difference between the true deformation *ε*true, and the elastic deformation *ε*el.

$$
\varepsilon\_{\rm pl} = \varepsilon\_{\rm true} - \varepsilon\_{\rm el} = \varepsilon\_{\rm true} - \frac{\sigma\_{\rm true}}{E} \tag{7}
$$

According to the above formulas, the true and plastic characteristics for the tested HY 80 steel samples were developed (Figure 6). The basic material constants describing the tested steel are summarised in Table 2. *Materials* **2021**, *14*, x FOR PEER REVIEW 6 of 14

**Figure 5.** (**a**) Rotary hammer station (Fundamentals of Technology Laboratory, Polish Naval Academy), (**b**) scheme of dynamic tensile test on a rotary hammer, and (**c**) dimensions of a sample. **Figure 5.** (**a**) Rotary hammer station (Fundamentals of Technology Laboratory, Polish Naval Academy), (**b**) scheme of dynamic tensile test on a rotary hammer, and (**c**) dimensions of a sample.


At the rotary hammer laboratory stand, the sample breaking force is recorded at a **Table 1.** Summary of test results on a rotary hammer.

**mm mm mm2 kN m/s s−1 MPa** 

**Hammer Rotational Speed** 

**Strain Rate** 

**Dynamic Ultimate Strength Rm** 

**Breaking** 

*Fm*

HY 80\_d1\_v10 5.03 18.69 19.86 25.13 10.00 535 1265.28

**Table 1.** Summary of test results on a rotary hammer.

**Area**  *A***<sup>0</sup>**

**Sample Name** *<sup>φ</sup>* **Measuring** 

**Length** 

HY 80\_d2\_v20 5.03 19.36 19.86 30.35 20.00 1033 1528.10 HY 80\_d3\_v30 5.02 19.33 19.78 30.76 30.00 1552 1554.92 HY 80\_d4\_v40 5.07 18.53 20.18 31.41 40.00 2159 1556.62

୲୰୳ୣ <sup>=</sup>

() <sup>=</sup>

 

The relationship between the true stresses *σ*true and nominal stresses *σ*nom obtained from the tensile test is obtained assuming that the volume of the stretched sample during

Plastic deformation is the difference between the true deformation *ε*true, and the elas-

୮୪ = ୲୰୳ୣ − ୣ୪ = ୲୰୳ୣ <sup>−</sup> ୲୰୳ୣ

According to the above formulas, the true and plastic characteristics for the tested HY 80 steel samples were developed (Figure 6). The basic material constants describing

 

= ୬୭୫ ൬

∙ =∙ () (2)

 

=1+୬୭୫ (4)

୲୰୳ୣ = ln(1+୬୭୫) (5)

୲୰୳ୣ = ୬୭୫(1+୬୭୫) (6)

൰ (3)

(7)

*3.3. HY 80 True Characteristics* 

stretching is constant; thus,

Hence,

Since

we have, therefore,

tic deformation *ε*el.

**Figure 6.** HY 80 σtrue-*ε*pl true characteristics, strain rate ሶ = 0.0001 ିଵ. **Figure 6.** HY 80 <sup>σ</sup>true-*ε*pl true characteristics, strain rate . *ε* = 0.0001 *s* −1 .


**Table 2.** Material constants describing HY 80 steel based on Formulas (3)–(7).

the tested steel are summarised in Table 2.

The chemical composition of the breakthrough structure and the material model proposal was also considered in the paper [2]. However, in CAE programs, functions are used to describe the plastic characteristic depending on the strain rate and temperature *<sup>σ</sup>*true <sup>=</sup> *<sup>σ</sup>*true *ε*pl, . *ε*, *θ* . In the case of metals, the Johnson–Cook constitutive model has become the most frequently used standard [9]. In this model, the plastic Huber–Mises– Hencky (HMH) reduced stresses σpl are described by the following equation:

$$\sigma\_{\rm pl} = \left(\mathbf{A} + \mathbf{B}\mathbf{\dot{e}}\_{\rm pl}^{\rm n}\right) \left[1 + \mathbf{C}\ln\left(\frac{\dot{\boldsymbol{\varepsilon}}}{\dot{\boldsymbol{\varepsilon}}\_{0}}\right)\right] \left[1 - \left(\frac{\theta - \theta\_{0}}{\theta\_{\rm melt} - \theta\_{0}}\right)^{\rm m}\right] \tag{8}$$

where

A–elastic range of the material *σ*pl=<sup>0</sup> (it is often simplified in form A = *R*e);

B–hardening parameter;

n–hardening exponent;

C–strain rate coefficient;

*ε*pl–true plastic strain; .

*ε* –strain rate; .

*ε*0–quasi-static strain rate (0.0001 s−<sup>1</sup> );

*θ*–current material temperature;

*θ*0–ambient temperature;

*θ*tmelt–melting temperature; m–thermal softening exponent.

The above values for this model are determined based on the static tensile test and Hopkinson or Taylor tests [10]. However, with strain rates ranging up to 2000 s−<sup>1</sup> , these data can be obtained from a rotary hammer tensile test.

The parameters A, B, C, n, and m can be determined in many other ways [11]. One of the ways is the so-called engineering formula, according to which the parameters of the first term A, B, and n are determined based on the results of the static tensile test according to the following algorithm:

*Rm, εm, E* should be determined from true characteristics, along with the values of the A point corresponding to *σ*pl=<sup>0</sup> , εpl=<sup>0</sup> , which constitute the elastic range of the material behaviour; then, according to the Formulas (9)–(12), calculate the *R*e,true*, R*m,true*, ε*m,true*, ε*m,pl values as follows:

$$\mathbf{A} = \sigma\_{\text{pl}=0} \tag{9}$$

$$R\_{\rm m,true} = R\_{\rm m}(1 + \varepsilon\_{\rm m}).\tag{10}$$

$$
\varepsilon\_{\rm m,true} = \ln(1 + \varepsilon\_{\rm m}) \tag{11}
$$

$$
\varepsilon\_{\rm m,pl} = \varepsilon\_{\rm m,true} - \frac{R\_{\rm m,true}}{E} \tag{12}
$$

determine the parameters B, n [11] according to the following Formulas (13) and (14): determine the parameters B, n [11] according to the following Formulas (13) and (14):

$$\mathbf{m} = \frac{R\_{\text{m,true}} \cdot \varepsilon\_{\text{m,pl}}}{R\_{\text{m,true}} - \mathbf{A}} \tag{13}$$

$$\mathbf{B} = \frac{\mathbf{R}\_{\rm m,true} - \mathbf{A}}{\varepsilon\_{\rm m,pl}^n} \tag{14}$$

) or a

൰ (15)

Taking the average values from Table 2 and using the Formulas (9), (13) and (14), the coefficients for the first component of the Johnson–Cook constitutive model were determined, which are the following: Taking the average values from Table 2 and using the Formulas (9), (13) and (14), the coefficients for the first component of the Johnson–Cook constitutive model were determined, which are the following:

$$\begin{array}{l} \text{A = 559 MPa;} \\ \text{B = 518 MPa;} \\ \text{n = 0.379.} \end{array}$$

Figure 7 shows the compilation of the nominal characteristic from the MTS machine (red), the true characteristic determined from Equations (5) and (6) (blue), and the JC model (the first part of the Equation (8) (green). Figure 7 shows the compilation of the nominal characteristic from the MTS machine (red), the true characteristic determined from Equations (5) and (6) (blue), and the JC model (the first part of the Equation (8) (green).

To determine the C parameter, it is necessary to know the value ୫,୲୰୳ୣ(ሶ

=ቆ ୫,୲୰୳ୣ(ሶ

୫,୲୰୳ୣ(

,(ሶ)**, MPa** 782.00 1047 1115 1130 1140

**C** - 0.021873 0.026366 0.026877 0.027108

Based on the calculations, the mean value of the C parameter was determined, C = 0.0268. In Figure 8, the influence of the C parameter on material behaviour is shown. Figure 8 shows the behaviour of the material as a function of strain rate in the Johnson–Cook

given strain rate determined during the dynamic tensile test on a rotary hammer (Table

)

− 1ቇ / ln ൬ ሶ

= .  **s−<sup>1</sup>** ሶ =  **s−<sup>1</sup>** ሶ =  **s−<sup>1</sup>** ሶ =  **s−<sup>1</sup>** ሶ =  **s−<sup>1</sup>**

ሶ 

ሶ )

**Figure 7.** Comparison of the nominal, true characteristics and the proposed Johnson Cook HY 80 steel model, strain rate ሶ = 0.0001 ିଵ. **Figure 7.** Comparison of the nominal, true characteristics and the proposed Johnson Cook HY 80 steel model, strain rate . *ε* = 0.0001 *s* −1 .

**Table 3.** Ultimate strength for various strain rates.

**Strain Rate,**  ሶ ሶ

To determine the C parameter, it is necessary to know the value *<sup>R</sup>*m, true . *ε* or a given strain rate determined during the dynamic tensile test on a rotary hammer (Table 3). From transforming Equation (8), we obtain

$$\mathbf{C} = \left(\frac{R\_{\mathrm{m,true}}(\dot{\varepsilon})}{R\_{\mathrm{m,true}}(\dot{\varepsilon}\_0)} - 1\right) / \ln\left(\frac{\dot{\varepsilon}}{\dot{\varepsilon}\_0}\right) \tag{15}$$

**Table 3.** Ultimate strength for various strain rates.


Based on the calculations, the mean value of the C parameter was determined, C = 0.0268. In Figure 8, the influence of the C parameter on material behaviour is shown. Figure 8 shows the behaviour of the material as a function of strain rate in the Johnson– Cook model (first and second term of Equation (8) against the background of the real characterisation determined from Equation (6). *Materials* **2021**, *14*, x FOR PEER REVIEW 10 of 14 model (first and second term of Equation (8) against the background of the real characterisation determined from Equation (6).

The values for the temperature component can be taken based on the literature [1,4], The values for the temperature component can be taken based on the literature [1,4], and they are similar for most steels; thus,

**Figure 8.** Johnson–Cook model of HY 80 steel as a function of strain rate, *C* = 0.0268.

principal stresses to the Huber–Mises–Hencky reduced stress *σ*HMH [12,13]

 = 1

and they are similar for most steels; thus, Ambient temperature θ<sup>0</sup> = 293.15 K;

Ambient temperature θ0 = 293.15 K; Melting temperature θtop = 1733÷1793; Melting temperature θtop = 1733~1793; Thermal coefficient m = 0.75 ÷ 1.15.

(Table 4).

## Thermal coefficient m = 0.75÷1.15. *3.4. HY 80 Steel Failure at Uniaxial Tension*

*3.4. HY 80 Steel Failure at Uniaxial Tension*  The material failure model used in CAE programs is detailed in several studies The material failure model used in CAE programs is detailed in several studies [6,8,12–14]. The value of the destructive deformation is a function of the so-called stress state indicator

[6,8,12–14]. The value of the destructive deformation is a function of the so-called stress state indicator *η*TRIAX (stress triaxiality). It is the ratio of the pressure being the mean of the

ୖ୍ଡ଼ <sup>=</sup>

For the uniaxial stretching state, the value of the triaxiality coefficient is equal to 0.33

ୌୌ

<sup>3</sup> (ଵ + ଶ + ଷ) (17)

(16)

In a three-dimensional state of stress, the pressure is

*η*TRIAX (stress triaxiality). It is the ratio of the pressure being the mean of the principal stresses to the Huber–Mises–Hencky reduced stress *σ*HMH [12,13]

$$
\eta\_{\text{TRIAX}} = \frac{p}{\sigma\_{\text{HMH}}} \tag{16}
$$

In a three-dimensional state of stress, the pressure is

$$p = \frac{1}{3}(\sigma\_1 + \sigma\_2 + \sigma\_3). \tag{17}$$

For the uniaxial stretching state, the value of the triaxiality coefficient is equal to 0.33 (Table 4). *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14 *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14 *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14 *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14 *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14 *Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14

> **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3]. **Table 4.** *η*TRIAX values for selected 3D cases [3].

#### **4. Discussion 4. Discussion 4. Discussion 4. Discussion 4. Discussion 4. Discussion 4. Discussion**

mation ௨

mation ௨

mation ௨

mation ௨

mation ௨

mation ௨

strain, ε

Nom True JC 3-4 2-7 3-6

Nom True JC 3-4 2-7 3-6

Nom True JC 3-4 2-7 3-6

Nom True JC 3-4 2-7 3-6

Nom True JC 3-4 2-7 3-6

Nom True JC 3-4 2-7 3-6

displacement of finite elemnt, *u*, mm

strain, ε

strain, ε

displacement of finite elemnt, *u*, mm

displacement of finite elemnt, *u*, mm

strain, ε

strain, ε

strain, ε

displacement of finite elemnt, *u*, mm

displacement of finite elemnt, *u*, mm

displacement of finite elemnt, *u*, mm

0

100

100

0

0

100

100

100

200

300

400

500

200

300

400

500

600

700

800

900

0

0

0

100

200

200

200

200

300

300

300

300

400

stress,

400

stress,

400

stress,

400

stress,

stress,

stress,

σ, MPa

σ, MPa

σ, MPa

500

σ, MPa

500

σ, MPa

500

σ, MPa

500

600

600

600

600

700

700

700

700

*1*

700

800

900

600

*1*

*R*e *A*

*1*

*R*e *A*

*1*

*R*e *A*

*R*e *A*

*R*e *A*

*R*e *A*

*1*

*1*

800

800

800

800

900

900

900

900

The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate. The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces disappear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degradation or failure curve defined by the parameter d, which is the damage evolution coefficient taking values from 0 to 1. The stress on the degradation curve is appropriate.

$$
\sigma = (1 - d)\overline{\sigma}.\tag{18}
$$

*2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m *2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m *2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m *2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m *2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m *2 d*=0, *u*pl=0 *5* − = 1− ത ത ത *R*m The material fracture occurs in point 4 after reaching the value of the fracture deformation *ε pl f ailure* However, if during the degradation of the material on the curve 2–4 the element breaks or the forces loading the element disappear, e.g., in point 3, then the remaining elastic forces will reduce its deformation to point 6 along the 3–6 path, which is not parallel to the 0–1 path. The evolution of failure determines the degree of degradation at which failure of the material will occur. The value of *d* = 0 means that the plastic stress has reached the value of *R*m, but the material has not yet been degraded, while the value of *d* = 1 means the complete degradation of the material. The failure evolution is described as a function of the plastic displacement of the *u*pl, defined as follows [13]:

The material fracture occurs in point 4 after reaching the value of the fracture defor-

The material fracture occurs in point 4 after reaching the value of the fracture defor-

. However, if during the degradation of the material on the curve 2–4 the

The material fracture occurs in point 4 after reaching the value of the fracture defor-

The material fracture occurs in point 4 after reaching the value of the fracture defor-

. However, if during the degradation of the material on the curve 2–4 the

. However, if during the degradation of the material on the curve 2–4 the

The material fracture occurs in point 4 after reaching the value of the fracture defor-

. However, if during the degradation of the material on the curve 2–4 the

The material fracture occurs in point 4 after reaching the value of the fracture defor-

. However, if during the degradation of the material on the curve 2–4 the

. However, if during the degradation of the material on the curve 2–4 the

*6*

*6*

*6*

*6*

*6*

*6*

*3*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

*u*failure=(ୟ୧୪୳୰ୣ−୫)⋅*L*

$$
\mu\_{\rm pl} = L \cdot \varepsilon\_{\rm pl} \tag{19}
$$

*4* ୟ୧୪୳୰ୣ

*4* ୟ୧୪୳୰ୣ

*4* ୟ୧୪୳୰ୣ

*4* ୟ୧୪୳୰ୣ

*4* ୟ୧୪୳୰ୣ

*4* ୟ୧୪୳୰ୣ

*d*=1, *u*pl= *u*fail

*d*=1, *u*pl= *u*fail

*d*=1, *u*pl= *u*fail

*d*=1, *u*pl= *u*fail

*d*=1, *u*pl= *u*fail

*d*=1, *u*pl= *u*fail

*E E E E E* where *L* is the characteristic length of the FEM element.

*E*

*7*

*7*

*7*

*7*

0.000 0.060 0.120 0.180 0.240

0.000 0.060 0.120 0.180 0.240

0.000 0.060 0.120 0.180 0.240

0.000 0.060 0.120 0.180 0.240

0.000 0.060 0.120 0.180 0.240

0.000 0.060 0.120 0.180 0.240

*u*pl=୮୪⋅*L*

*u*pl=୮୪⋅*L*

*u*pl=୮୪⋅*L*

*u*pl=୮୪⋅*L*

*u*pl=୮୪⋅*L*

*u*pl=୮୪⋅*L*

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).

**3D Cases** 

**Principal stresses** 

**4. Discussion** 

*σ*<sup>x</sup> = *σ*<sup>y</sup> *σ*z = 0

σy

> σx

*σ*<sup>x</sup> = *σ*<sup>y</sup> = *σ*<sup>z</sup> >0

σz σy

> σx

> > The rate of evolution of failure describes the path along which material degradation develops. In CAE programs, linear, exponential, and tabular descriptions are adopted. The linear relationship is expressed as the ratio of plastic displacement to failure displacement [13]. appear in point 2, then the elastic forces will reduce the deformation to point 7 along path 2~7 and parallel to path 0~1. In the model with failure, point 5 corresponds to point 3 on the curve 2~4, where strength loss (softening) occurs. The 2~4 curve is called the degrada-

> > The failure mechanism for HY 80 steel is shown in the true characteristic diagram *σtrue-ε true* (Figure 9). The elastic range is between points 0 and 1. Between points 1 and 2, there is a plastic range (hardening). In point 2, the destruction process is initiated. After crossing point 2 in the material model without failure criteria, the stresses would continue to increase with the strain increase towards point 5 and further. If the loading forces dis-

*σ*y < 0 *σ*x =*σ*z=0

σy

$$d = \frac{u\_{\rm p1}}{u\_{\rm failure}}\tag{20}$$

−*σ*x = −*σ*<sup>y</sup> *σ*<sup>z</sup> = 0

σy

> σx

> > *σ*x = *σ*<sup>y</sup> = *σ*<sup>z</sup> <0

σz σy

> σx

Table 5 lists the points from the diagram in Figure 9, based on which the failure parameters for tensile strength of HY 80 steel were determined.

*Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 14

**Table 4.** *η*TRIAX values for selected 3D cases [3].

σy

*σ*y > 0 *σ*<sup>x</sup> = *σ*z = 0

*η***TRIAX** ∞ 0.66 0.33 −0.33 −0.66 −∞

**Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true). **Figure 9.** Failure diagram on the true characteristics of HY 80 steel (*σ*true-*ε*true).



Following these parameters, calculations were carried out for uniaxial stretching as follows:

$$
\varepsilon\_{failure} = \varepsilon\_4 - \varepsilon\_7 = 0.2280 - 0.0991 = 0.1289.
$$

$$
d\overline{\sigma} = \sigma\_5 - \sigma\_3 = 836.0 - 489.9 = 346.1. \text{ MPa}
$$

$$
\text{since } \sigma = (1 - d)\overline{\sigma}. \text{ so } d = 1 - \frac{\sigma}{\overline{\sigma}} = 1 - \frac{489.9}{836} = 0.414.
$$

$$E' = (1 - d)E = (1 - 0.414) \cdot 211 = 124 \text{ GPa}$$

$$u\_{failure} = 0.1289 \cdot L$$

Summarising the tested HY-80 steel can be described by the following equations: Young modulus: *E* = 211 GPa;

Johnson–Cook model:

$$\sigma = \left(559 + 518 \cdot \varepsilon^{0.379}\right) \left[1 + 0.0268 \cdot \ln\left(\frac{\dot{\varepsilon}}{0.0001}\right)\right] \left[1 - \left(\frac{\theta - 293.15}{1.470}\right)^{1.14}\right]$$

Failure parameters:

$$d = 0.414; \; \varepsilon\_{failure} = 0.1289; \; \eta\_{Triax} = 0.33$$

#### **5. Conclusions**

Johnson–Cook HY 80 steel characteristics and material model were developed based on the static and dynamic tensile tests on the rotary hammer. Tensile tests performed on a rotary hammer allowed us to determine the mechanical properties of steel in the range of deformation speed 0–2000 s−<sup>1</sup> . The knowledge of the behaviour of steel for increased deformation rates enables the simulation of fast-changing processes such as a collision, projectile fire, impact of a shock wave (pressure from the explosion) on the tested object, or modelling of submarine implosion. The obtained data should be verified by an appropriate simulation and experiment, which will be the subject of the subsequent study.

The results of the tests of HY 80 steel after 55 years of operation show that the several decades of exploitation of this material in challenging sea conditions did not adversely affect its mechanical properties. They are close to catalogue values. The yield point of this steel is catalogued at 80 KSI (552 MPa). From the tests performed, the yield point of *R*<sup>e</sup> = 605.6 MPa (*R*<sup>02</sup> = 444.5 MPa) was obtained, and the strength limit was *R<sup>m</sup>* = 782 MPa with a deformation of 0.1, which proves that good plastic and strength properties were maintained.

By analysing the mechanical properties, it can be concluded that the ship's hull made of this steel without significant corrosion and operational losses could be used for the next years. One should be aware that it is still subject to erosive wear, which changes the overall strength of the hull. That may have an impact on limiting the maximum operational depth of the submarine.

Tests with a rotary hammer showed an increase in the strength of the steel with a reduced deformation. Unfortunately, due to the dynamic nature of the test and the possibility of potential damage to the extensometers, it was not possible to measure the deformation during the trial. This problem will be solved in the future with the use of high-speed cameras.

Increasing the strain rate in the range of up to 2000 s−<sup>1</sup> increases the strength of the tested steel to 1140 MPa. That is a typical phenomenon in high-quality steel.

The study determined the failure parameters for the uniaxial tensile case (*η* = 0.33). The compression/tensile diagram for steel is symmetrical, which allows for the assumed failure criterion also for *η* = −0.33. The obtained amount of material did not qualify for a greater number of tests in which the failure parameters could be determined for the remaining characteristic values of the triaxiality coefficient.

**Author Contributions:** Conceptualisation, S.B. and K.R.; methodology, S.B. and K.R.; software, S.B. and K.R.; validation, S.B. and K.R.; formal analysis, K.R.; investigation, S.B. and K.R.; resources, S.B. and K.R.; data curation, S.B. and K.R.; writing—original draft preparation, S.B.; writing—review and editing, K.R.; visualisation S.B. and K.R.; supervision, S.B.; project administration, K.R.; funding acquisition, S.B. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

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## *Article* **Analysis and Modification of Methods for Calculating Axial Load Capacity of High-Strength Steel-Reinforced Concrete Composite Columns**

**Jun Wang \*, Yuxin Duan \*, Yifan Wang, Xinran Wang and Qi Liu**

School of Civil Engineering, Northeast Forestry University, Harbin 150000, China; sunwangyifan@163.com (Y.W.); xinran19972021@163.com (X.W.); liu\_q\_wangyi@163.com (Q.L.)

**\*** Correspondence: jun.w.619@nefu.edu.cn (J.W.); yuxind666@163.com (Y.D.); Tel.: +86-188-4589-2847 (Y.D.)

**Abstract:** To investigate the applicability of the methods for calculating the bearing capacity of high-strength steel-reinforced concrete (SRC) composite columns according to specifications and the effect of confinement of stirrups and steel on the bearing capacity of SRC columns. The axial compression tests were conducted on 10 high-strength SRC columns and 4 ordinary SRC columns. The influences of the steel strength grade, the steel ratio, the types of stirrups and slenderness ratio on the bearing capacity of such members were examined. The analysis results indicate that using high-strength steel and improving the steel ratio can significantly enhance the bearing capacity of the SRC columns. When the slenderness ratio increases dramatically, the bearing capacity of the SRC columns plummets. As the confinement effect of the stirrups on the concrete improves, the utilization ratio of the high-strength steel in the SRC columns increases. Furthermore, the results calculated by AISC360-19(U.S.), EN1994-1-1-2004 (Europe), and JGJ138-2016(China) are too conservative compared with test results. Finally, a modified formula for calculating the bearing capacity of the SRC columns is proposed based on the confinement effect of the stirrups and steel on concrete. The results calculated by the modified formula and the finite element modeling results based on the confinement effect agree well with the test results.

**Keywords:** high-strength steel; steel-reinforced concrete column; axial bearing capacity; confinement effect; simulation analysis

## **1. Introduction**

Due to enjoying the advantages of strength, plasticity, toughness and weldability, high-strength steel has become one of the most important building materials. In recent years, high-strength steel with a yield strength ranging from 460 to 960 MPa has been used in building structures [1–3], such as the Bird's Nest and Water Cube in China, and the Sony Center in Germany [4]. However, the applicability of the current design and calculation methods to high-strength steel-reinforced concrete (SRC) composite structures has become challenging since they have gradually been applied to high-rise buildings and long-span structures.

In 2014, Q460 and Q690 high-strength steel were used to replace Q345 ordinary steel in SRC composite columns in the upper part of the structure in the high-rise project of Zhengzhou Greenland Central Plaza in Henan Province, China, to enhance the bearing capacity and reduce the self-weight of the structures. Thus, it is necessary to develop the design of SRC structures. However, in Specifications for Structural Steel Buildings (AISC360-16, the USA) [5], Design of Composite Steel and Concrete Structures (EN1994-1- 1:2004, Eurocode) [6] and Code for Design of Composite Structures (JGJ138-2016, China) [7], the design methods of SRC composite members are primarily based on the research on ordinary steel. Yang et al. performed model tests on Q460 high-strength SRC columns with a circular section to solve this issue and found that Q460 high-strength steel enhanced the

**Citation:** Wang, J.; Duan, Y.; Wang, Y.; Wang, X.; Liu, Q. Analysis and Modification of Methods for Calculating Axial Load Capacity of High-Strength Steel-Reinforced Concrete Composite Columns. *Materials* **2021**, *14*, 6860. https:// doi.org/10.3390/ma14226860

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 8 October 2021 Accepted: 11 November 2021 Published: 14 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Specimen Designation**

C1Q2S4T1R1 (C1) C2Q2S4T2R1 (C2) C3Q2S5T1R1 (C3) C4Q2S7T1R1 (C4) C5Q4S4T1R1 (C5) C6Q4S4T2R1 (C6) C7Q4S5T1R1(C7) C8Q4S7T1R1 (C8) C9Q4S7T1R2 (C9) C10Q4S7T1R3 (C10) C11Q6S4T1R1 (C11) C12Q6S4T2R1 (C12) C13Q6S5T1R1 (C13)

**Steel Grade** **Steel Ratio to Concrete**

ends of the columns.

bearing capacity of the SRC columns remarkably. Nevertheless, the calculation results by the relevant specifications were only about 60% of the test results, which led to a waste of materials [8]. Zhao et al. conducted tests on the Q460 SRC-core columns to find the reason why calculation results are too conservative [9]. It was found that the confinement effect of steel and stirrups on bearing capacity was neglected in the methods for calculating the bearing capacity of SRC composite columns in current specifications, and only the load-bearing capacity of the steel, the concrete and the longitudinal reinforcements was taken into account. Moreover, the theoretical stress–strain model of confined concrete on concrete was developed, but neither the full verification of the model nor specific calculation suggestions was proposed for the bearing capacity of SRC structures.

In recent years, there has been some research on the confinement theory of SRC members. In 1992, Mirza and Skrabek conducted tests on slender composite beam-columns with ordinary strength materials [10]. It was found that concrete in a SRC cross-section can be divided into three parts according to the confining pressure level, that is: unconfined concrete (UCC), partially confined concrete (PCC) and highly confined concrete (HCC). For PCC, the confinement effect can be considered as normal reinforced concrete, which has been extensively studied by Sheikh and Uzumeri, Mander et al. [11,12]. For HCC, Chen and Wu proposed an analytical method for predicting the axial compressive behavior of the SRC column using a cross-shaped steel section with flanges [13]. However, Wang and Su carried out tests on slender SRC columns; 270 MPa~600 MPa steel was used in the specimens. It was found that when the confinement of steel and stirrup was calculated, the bearing capacity of slender SRC columns increased by only 2% compared with only considering the confinement of stirrups [14]. Despite all this research, there are still some codes for SRC columns which do not take into account the strength of confined concrete, leading to the calculations being too conservative, such as AISC, Eurocode 4 and JGJ138- 2016. Therefore, more experiments are needed to analyze the confinement of stirrups and steel on concrete, especially when high-strength steel is used in SRC columns. The codes also need more suggestions to improve the accuracy of the calculations.

In this paper, structural tests were carried out on SRC columns under an axial load to verify the applicability of the current specifications to high-strength SRC composite columns and provide a calculation method considering the confinement effect for such composite columns. Then, the finite element models were utilized to prove the accuracy of the prediction method using the theoretical stress–strain model of confined concrete for engineering applications.

#### **2. Experimental Investigation**

#### *2.1. Test Specimens*

On the basis of the theory proposed by Zhao et al. [15] and the formulas described in EN1994-1-1:2004, Eurocode, the strength of steel, the steel ratio, the stirrups and the slenderness ratio are the primary factors influencing the bearing capacity of SRC columns. Thus, 10 high-strength SRC columns and 4 ordinary SRC columns were analyzed to verify the confinement effect and determine the magnitude of the known factors. Table 1 lists the main parameters of the specimens, and Figure 1 shows the labelling rule of specimens. *Materials* **2021**, *14*, 6860 3 of 24

**Stirrup Type**

Four kinds of sections were selected for this study, as shown in Figure 2. Moreover, stirrup encryption and reinforcement with carbon fiber cloth were performed within 1/6 height of both ends of the column to prevent local pressure failure at the upper and lower

**Slenderness** 

**Ratio Height/mm Stirrups**

**Spacing**

Q235 4.12% 130 × 70 × 5 × 5 1 17 1200 80 Q235 4.12% 130 × 70 × 5 × 5 2 17 1200 80 Q235 5.63% 140 × 75 × 5 × 8 1 17 1200 80 Q235 7.17% 140 × 80 × 8 × 8 2 17 1200 80 Q460 4.12% 130 × 70 × 5 × 5 1 17 1200 80 Q460 4.12% 130 × 70 × 5 × 5 2 17 1200 80 Q460 5.63% 140 × 75 × 5 × 8 1 17 1200 80 Q460 7.17% 140 × 80 × 8 × 8 2 17 1200 80 Q460 7.17% 140 × 80 × 8 × 8 1 29 2000 80 Q460 7.17% 140 × 80 × 8 × 8 2 40 2800 80 Q690 4.12% 130 × 70 × 5 × 5 1 17 1200 80 Q690 4.12% 130 × 70 × 5 × 5 2 17 1200 80 Q690 5.63% 140 × 75 × 5 × 8 1 17 1200 80

**Table 1.** Main parameters of specimens.

C14Q6S7T1R1 (C14) Q690 7.17% 140 × 80 × 8 × 8 2 17 1200 80

**Figure 2.** Sectional dimensions of specimens (millimeters).

C3Q2S5T1R1 (C3) C4Q2S7T1R1 (C4)

**Steel Grade** **Steel Ratio to Concrete**

**Specimen Designation**


**Table 1.** Main parameters of specimens. Q235 7.17% 140 × 80 × 8 × 8 2 17 1200 80 Q460 4.12% 130 × 70 × 5 × 5 1 17 1200 80

**Table 1.** Main parameters of specimens.

**Steel Geometric Size (h × bf ×t1 ×t2)/mm**

crete; **T**: Type of stirrups; **R**: Slenderness ratio).

*Materials* **2021**, *14*, 6860 3 of 24

Four kinds of sections were selected for this study, as shown in Figure 2. Moreover, stirrup encryption and reinforcement with carbon fiber cloth were performed within 1/6 height of both ends of the column to prevent local pressure failure at the upper and lower ends of the columns. stirrup encryption and reinforcement with carbon fiber cloth were performed within 1/6 height of both ends of the column to prevent local pressure failure at the upper and lower ends of the columns.

Four kinds of sections were selected for this study, as shown in Figure 2. Moreover,

**Figure 1.** Labeling rule of specimens. (**C**: Serial number; **Q**: Strength of steel; **S**: Steel ratio to con-

Q235 4.12% 130 × 70 × 5 × 5 1 17 1200 80 Q235 4.12% 130 × 70 × 5 × 5 2 17 1200 80 Q235 5.63% 140 × 75 × 5 × 8 1 17 1200 80

**Stirrup Type**

**Slenderness** 

**Ratio Height/mm Stirrups**

**Spacing**

**Figure 2. Figure 2.** Sectional dimensions of specimens (millimeters). Sectional dimensions of specimens (millimeters).

#### *2.2. Material Properties*

Steel sheets of grade Q235, Q460, Q690, with a reinforcement of HRB400, were selected to manufacture the specimens. Tensile coupon tests were carried out to obtain the material properties of steel. According to the Chinese Standard GB/T 228-2010 [16], the measured material properties of the steel presented in Table 2 are the mean values of results.


C50 concrete is used in this test. After finishing the preparation work, all the specimens were poured at one time and maintained for 28 days under the condition of a temperature no less than 5 ◦C. According to the standard for test methods of concrete structures (GB/T 50152-2012) [17], 9 concrete cubes (150 × 150 × 150 mm) were maintained under the same conditions as the specimens. The compressive strength test was carried out before the formal loading began, as shown in Figure 3. The results are listed in Table 3.

**Figure 3.** Concrete cubes and the failure mode of concrete tubes.


**Table 3.** Results of compressive strength test.

*f* 0 *cu* is the cubic compressive strength of concrete by the test, respectively; *f* 0 *cu*,*<sup>m</sup>* is the average cubic compressive strength; *f* 0 *c* is the calculation of the prism compressive strength, which is used in the finite element models; *E* 0 *c* is the elastic modulus measured by the test, respectively.

#### *2.3. Test Setup and Procedure*

A 30,000 kN servo-hydraulic testing machine was employed for loading, and the loading device is shown in Figure 4a. The accuracy of the instrument is 1/1000, which is allowed according to GB50152-2012 [17]. The specimens were subjected to multi-stage loading, and before formal loading, a 50 kN load was applied in advance to confirm whether the test apparatus functioned normally. During the formal testing, force-controlled loading was first conducted at a rate of 200 kN/min. When the load reached 60% of the estimated ultimate bearing capacity, the loading rate was reduced to 150 kN/min. When the load reached 80% of the estimated ultimate bearing capacity, the loading mode was changed from the force-controlled loading to the displacement-controlled loading, and the loading rate was set as 0.4 mm/min. After the ultimate load (ultimate bearing capacity), the testing was terminated when the load decreased to 70% of the ultimate bearing capacity. The arrangement of the measuring points is depicted in Figure 4b. The strain gauges were arranged on two adjacent sides of the steel, on the surfaces of stirrups and the longitudinal reinforcement, and four sides of concrete in the middle height of the specimens. The vertical displacement of the specimens was recorded by a displacement meter in the loading device.

**Figure 4.** Test setup and arrangement of strain gauges. (**a**) test setup and instrumentations; (**b**) strain gauges position. **Figure 4.** Test setup and arrangement of strain gauges. (**a**) test setup and instrumentations; (**b**) strain gauges position.

#### *2.4. Test Results and Analysis 2.4. Test Results and Analysis*

*2.3. Test Setup and Procedure*

in the loading device.

2.4.1. Failure Mode 2.4.1. Failure Mode

The specimen was in the elastic deformation stage during the initial loading, and no cracks formed on its surface. With an increase in the load, apparent longitudinal cracks appeared in the middle of the specimen and continuously extended to both ends of the column. When the ultimate bearing capacity was reached, both the steel and longitudinal reinforcement of the specimen reached the yield strength, the length and width of the cracks increased rapidly. In addition, the cover concrete at the middle height of the spec-The specimen was in the elastic deformation stage during the initial loading, and no cracks formed on its surface. With an increase in the load, apparent longitudinal cracks appeared in the middle of the specimen and continuously extended to both ends of the column. When the ultimate bearing capacity was reached, both the steel and longitudinal reinforcement of the specimen reached the yield strength, the length and width of the cracks increased rapidly. In addition, the cover concrete at the middle height of the specimen crushed and peeled off, and the longitudinal reinforcement bulged outward.

A 30,000 kN servo-hydraulic testing machine was employed for loading, and the loading device is shown in Figure 4a. The accuracy of the instrument is 1/1000, which is allowed according to GB50152-2012 [17]. The specimens were subjected to multi-stage loading, and before formal loading, a 50 kN load was applied in advance to confirm whether the test apparatus functioned normally. During the formal testing, force-controlled loading was first conducted at a rate of 200 kN/min. When the load reached 60% of the estimated ultimate bearing capacity, the loading rate was reduced to 150 kN/min. When the load reached 80% of the estimated ultimate bearing capacity, the loading mode was changed from the force-controlled loading to the displacement-controlled loading, and the loading rate was set as 0.4 mm/min. After the ultimate load (ultimate bearing capacity), the testing was terminated when the load decreased to 70% of the ultimate bearing capacity. The arrangement of the measuring points is depicted in Figure 4b. The strain gauges were arranged on two adjacent sides of the steel, on the surfaces of stirrups and the longitudinal reinforcement, and four sides of concrete in the middle height of the specimens. The vertical displacement of the specimens was recorded by a displacement meter

imen crushed and peeled off, and the longitudinal reinforcement bulged outward. Figure 5 illustrates the failure modes of the specimens. As shown in Figure 5a, when the specimen with built-in Q235 steel failed, it had a relatively low degree of surface cracking. At a steel ratio of 7.17%, when the strength grade of the steel improved from Q235 to Q460 and Q690, the vertical displacement of the specimen during failure increased from 7.53 mm to 10.5 mm and 11.7 mm, respectively. Furthermore, the cracking and crushing degree of the cover concrete intensified accordingly, as illustrated in Figure 5b,c, respectively.

As presented in Figure 5d, specimen C9 with a slenderness ratio of 29 showed strength failure characteristics. However, when the slenderness ratio of the specimen increased to 40, specimen C10 displayed evident instability failure characteristics with a significant lateral displacement, as shown in Figure 5e.

Figure 6 demonstrates the crushing patterns of the concrete in high-strength SRC composite columns with different stirrup configurations. Compared with the specimens with complex stirrups, the concrete crushing depth of specimens C1, C5 and C11 with standard rectangular stirrups reached the stirrup confinement concrete during failure. On the contrary, the stirrup confinement concrete of specimens C2, C6, and C12 with complex stirrups remained almost intact during failure.

tively.

Figure 5 illustrates the failure modes of the specimens. As shown in Figure 5a, when the specimen with built-in Q235 steel failed, it had a relatively low degree of surface cracking. At a steel ratio of 7.17%, when the strength grade of the steel improved from Q235 to Q460 and Q690, the vertical displacement of the specimen during failure increased from 7.53 mm to 10.5 mm and 11.7 mm, respectively. Furthermore, the cracking and crushing degree of the cover concrete intensified accordingly, as illustrated in Figure 5b,c, respec-

**Figure 5.** Failure modes of the specimens: (**a**) failure mode of the specimen with Q235 steel, steel ratio is 7.17%; (**b**) failure mode of the specimen with Q460 steel, steel ratio is 7.17%; (**c**) failure mode of the specimen with Q690 steel, steel ratio is 7.17%; (**d**) failure mode of the specimen C9; (**e**) failure mode of the specimen C10. **Figure 5.** Failure modes of the specimens: (**a**) failure mode of the specimen with Q235 steel, steel ratio is 7.17%; (**b**) failure mode of the specimen with Q460 steel, steel ratio is 7.17%; (**c**) failure mode of the specimen with Q690 steel, steel ratio is 7.17%; (**d**) failure mode of the specimen C9; (**e**) failure mode of the specimen C10. *Materials* **2021**, *14*, 6860 7 of 24

**Figure 6.** Concrete crushing modes of specimens with different stirrup configurations. (C1, C5, C<sup>11</sup> are the specimens with rectangular stirrup; C2, C6, C<sup>12</sup> are the specimens with complex stirrup). **Figure 6.** Concrete crushing modes of specimens with different stirrup configurations. (C<sup>1</sup> , C<sup>5</sup> , C<sup>11</sup> are the specimens with rectangular stirrup; C<sup>2</sup> , C<sup>6</sup> , C<sup>12</sup> are the specimens with complex stirrup).

#### 2.4.2. Ultimate Bearing Capacity

ing capacity of the members.

2.4.2. Ultimate Bearing Capacity Table 4 lists the ultimate bearing capacity of the specimens. Compared with the specimen with built-in Q235 steel, the ultimate bearing capacity of the specimens with builtin Q460 and Q690 steel increased by 17.1% and 35.3%, respectively, indicating a marked increase in the bearing capacity of the columns. Figure 7a plots the load–displacement curves of the specimens at different strength grades of the steel and steel ratios. For the specimens with the same strength grade of steel, the maximum improvement in the ultimate bearing capacity of the specimens with a steel ratio of 5.63% and 7.17% was 13.2% and 28.3%, respectively, compared with that of the specimen with a steel ratio of 4.12%, Table 4 lists the ultimate bearing capacity of the specimens. Compared with the specimen with built-in Q235 steel, the ultimate bearing capacity of the specimens with builtin Q460 and Q690 steel increased by 17.1% and 35.3%, respectively, indicating a marked increase in the bearing capacity of the columns. Figure 7a plots the load–displacement curves of the specimens at different strength grades of the steel and steel ratios. For the specimens with the same strength grade of steel, the maximum improvement in the ultimate bearing capacity of the specimens with a steel ratio of 5.63% and 7.17% was 13.2% and 28.3%, respectively, compared with that of the specimen with a steel ratio of 4.12%,

which implied that increasing the steel ratio could noticeably enhance the ultimate bear-

(**a**)


**Table 4.** Comparison of test results with calculation results calculated codes.

*Materials* **2021**, *14*, 6860 7 of 24

capacity of the members.

which implied that increasing the steel ratio could noticeably enhance the ultimate bearing

Figure 7b delineates the load–displacement curves of the specimens at different slenderness ratios. Raising the slenderness ratio from 17 to 29 while keeping the other parameters unchanged reduced the ultimate bearing capacity of the column by 4.9%, implying a negligible reduction in the bearing capacity of the members. In contrast, when the slenderness ratio increased from 17 to 40, the ultimate bearing capacity of the column declined to 90.1%, indicating a remarkable reduction. Specimen C10, with a slenderness ratio of 40 experienced a small vertical displacement when the load reached the ultimate bearing capacity. Moreover, the ultimate bearing capacity of the specimen plummeted, and the specimen presented evident brittle failure characteristics. **Figure 6.** Concrete crushing modes of specimens with different stirrup configurations. (C1, C5, C<sup>11</sup> are the specimens with rectangular stirrup; C2, C6, C<sup>12</sup> are the specimens with complex stirrup). 2.4.2. Ultimate Bearing Capacity Table 4 lists the ultimate bearing capacity of the specimens. Compared with the specimen with built-in Q235 steel, the ultimate bearing capacity of the specimens with builtin Q460 and Q690 steel increased by 17.1% and 35.3%, respectively, indicating a marked

Figure 7c draws the load–displacement curves of the specimens with different types of stirrups. When the type of stirrup was changed from standard rectangular stirrups to complex stirrups, the ultimate bearing capacity of the column with built-in Q235, Q460 and Q690 steel increased by 7.3%, 13.2% and 11.6%, respectively. The strength grade of the steel also raised the bearing capacity of the high-strength SRC composite columns with complex stirrups, which was due to the profound confinement effect of the complex stirrups, bringing the steel into full play. increase in the bearing capacity of the columns. Figure 7a plots the load–displacement curves of the specimens at different strength grades of the steel and steel ratios. For the specimens with the same strength grade of steel, the maximum improvement in the ultimate bearing capacity of the specimens with a steel ratio of 5.63% and 7.17% was 13.2% and 28.3%, respectively, compared with that of the specimen with a steel ratio of 4.12%, which implied that increasing the steel ratio could noticeably enhance the ultimate bearing capacity of the members.

**Figure 7.** *Cont.*

**Specimens**

**Figure 7.** Load–displacement curves of specimens: (**a**) the load–displacement curves of the specimens at different strength grades of the steel and steel ratios; (**b**) the load–displacement curves of the specimens at different slenderness ratios; (**c**) the load–displacement curves of the specimens with different types of stirrups. **Figure 7.** Load–displacement curves of specimens: (**a**) the load–displacement curves of the specimens at different strength grades of the steel and steel ratios; (**b**) the load–displacement curves of the specimens at different slenderness ratios; (**c**) the load–displacement curves of the specimens with different types of stirrups.

#### **Table 4.** Comparison of test results with calculation results calculated codes. **3. Modification of Bearing Capacity Calculation**

#### **Test Results AISC360-16 (** *3.1. Current Calculation Methods*

 **<sup>−</sup> 1)% Eurocode 4 ( <sup>−</sup> 1)% JGJ138-2016 ( − 1)% /kN /kN /kN /kN** C1Q2S4T1R1 4200 2633 37.3% 2713 35.4% 2943 29.9% C2Q2S4T2R1 4506 2633 41.6% 2713 39.8% 2943 34.7% C3Q2S5T1R1 4340 2887 33.5% 2968 31.6% 3166 27.1% In the American National Standard Specifications for Structural Steel Buildings (AISCI360-16), the wrapped reinforced concrete part is considered to be equivalent to the steel. Thus, the formula for calculating the axial compression by utilizing the steel structure design method is defined as:

$$P\_n = \begin{cases} \begin{array}{c} P\_{n0} \left( 0.658 \frac{P\_{00}}{P\_\ell} \right) & \frac{P\_{n0}}{P\_\ell} \le 2.25\\ 0.877 P\_\ell & \frac{P\_{n0}}{P\_\ell} > 2.25 \end{array} \end{cases} \tag{1}$$

$$P\_{n0} = F\_y A\_s + F\_{ysr} A\_{sr} + 0.85 f'\_c A\_c \tag{2}$$

$$P\_{\varepsilon} = \pi^2 \left( E I\_{\text{eff}} \right) / L\_{\varepsilon}^2 \tag{3}$$

C12Q6S4T2R1 5407 3666 32.2% 3810 29.5% 3954 26.9% C13Q6S5T1R1 5487 4299 21.7% 4461 18.7% 4546 17.1% C14Q6S7T1R1 6220 4953 20.4% 5126 17.6% 5152 17.2% where *A<sup>s</sup>* , *Asr* and *A<sup>c</sup>* are the cross-sectional area of the section steel, longitudinal reinforcement and concrete, respectively. *Fy*, *Fysr* and *f* 0 *c* represent the compressive strength of the steel, longitudinal reinforcement and concrete, respectively; *EIe f f* stands for the effective stiffness of the section; *L<sup>c</sup>* is the effective length of the member.

The Code Design of Composite Steel and Concrete Structures (EN1994-1-1:2004), defines the formula for calculating the bearing capacity of biaxially symmetric SRC columns under axial compression as:

$$N\_{Ed} \le \chi N\_{pl,Rd} \tag{4}$$

$$N\_{pl,Rd} = A\_d f\_{yd} + 0.85 A\_c f\_{cd} + A\_s f\_{sd} \tag{5}$$

where *Aa*, *A<sup>c</sup>* and *A<sup>s</sup>* denote the cross-sectional area of the section steel, concrete and longitudinal reinforcement; *fyd*, *fcd* and *fsd* are the compressive strength of the section steel, concrete and reinforcement, respectively; *χ* is the buckling reduction factor considering the relative slenderness ratio, and is expressed by Equation (7), as described in the Section 6.3 of Eurocode 3: Design of Steel Structures (EN 1993-1-1:2005),

$$\chi = \frac{1}{\Phi + \sqrt{\Phi^2 - \overline{\lambda}^2}}, \text{ when } \chi \le 1 \tag{6}$$

$$\Phi = 0.5\left[1 + a(\overline{\lambda} - 0.2) + \overline{\lambda}^2\right] \tag{7}$$

where *α* is the section type; *λ* represents the relative slenderness ratio.

The Code for Design of Composite Structures (JGJ138-2016), defines the formula for calculating the bearing capacity of axially compressed SRC columns under axial compression as:

$$N \le 0.9 \varphi(f\_c A\_c + f\_y' A\_s' + f\_a' A\_a') \tag{8}$$

where *Ac*, *A* 0 *<sup>s</sup>* and *A* 0 *a* indicate the cross-sectional area of the concrete, reinforcement and section steel, respectively; *fc*, *f* 0 *<sup>y</sup>* and *f* 0 *<sup>a</sup>* are the design value of the compressive strength of concrete, reinforcement and section steel, respectively; *ϕ* is the coefficient of axial compression stability and can be determined according to the slenderness ratio presented in a specific table in the code JGJ138-2016.

#### *3.2. Comparison between Test Results and Calculations*

Table 4 lists the test results and the calculations of the different specifications. Figure 8 compares them at various parameters. Figure 8a,b demonstrate that as the strength grade of steel improves from Q235 to Q460 and Q690 at a constant steel ratio, the results calculated by the different codes are far smaller than the test results, indicating that the calculations are too conservative. The results calculated according to code JGJ138-2016 are the closest to the test results, whereas those calculated according to code AISC360-16 are the most conservative.

**Figure 8.** *Cont.*

**Figure 8.** Effect of parameters on bearing capacity of specimens: (**a**) Comparison of test results and calculations for the specimens with steel ratio 5.63%; (**b**) Comparison of test results and calculations for the specimens with steel ratio 7.17%; (**c**) Comparison of test results and calculations for the specimens with Q690 steel; (**d**) Comparison of test results and calculations for the specimens with different slenderness ratios.

Figure 8c shows that, at a constant strength grade of the steel, the slope of the growth of the test results is similar to that of the results when the steel ratio enlarges.

As shown in Figure 8d, when the slenderness ratio increases from 17 to 40, the changing trend of the bearing capacity of the column determined by the test is similar to the one calculated by code AISC360-16: the higher the slenderness ratio is, the more profound its impact on the bearing capacity of the SRC composite columns becomes. However, the calculation results of codes JGJ138-2016 and Eurocode 4 show a small decreasing trend in the bearing capacity of the SRC columns with an increase in the slenderness ratio.

According to Figure 9, since the influence of the type of stirrups on the bearing capacity of composite columns is not considered in the methods proposed by different codes, the calculations according to different codes are generally similar. Changing the types of stirrups and strengthening the stirrup confinement effect on SRC composite columns can enhance the ultimate bearing capacity of the high-strength SRC columns more than that of the ordinary SRC columns. The ultimate bearing capacity of the Q690 SRC composite columns is relatively low when rectangular stirrups are configurated.

**Figure 9.** Effect of types of stirrups on bearing capacity of specimens. (C<sup>1</sup> , C<sup>5</sup> , C<sup>11</sup> are the specimens with rectangular stirrup; C<sup>2</sup> , C<sup>6</sup> , C<sup>12</sup> are the specimens with complex stirrup).

## *3.3. Modification of Formula for Calculating Bearing Capacity Considering Confinement Effect of Stirrup*

Specifications ignore the confinement effect of the stirrups and steel on concrete when calculating the bearing capacity of reinforced concrete columns under axial compression. The bearing capacity calculated by the different specifications is smaller than the measured bearing capacity, thus they produce a conservative result. At present, there are two types of methods for analyzing the confinement effect: one only considers the confinement effect of stirrups [12], and the other takes account of the confinement effect of the stirrups and steel [15,18].

Figure 10a illustrates the section considering the confined effect of the stirrups on concrete, and Figure 10b shows the section considering the confinement effect of the stirrups and the steel on concrete. It is found that the confinement effect of the steel (open section) on concrete increases the bearing capacity of the composite columns marginally; that is, by less than about 2% [14]. Hence, this theoretical analysis only takes the confinement effect of the stirrups on the concrete strength into account. There are two methods to calculate the stress–strain relationship of stirrup confined concrete.

**Figure 10.** Effectively confined region and ineffectively confined region of concrete: (**a**) Ordinary reinforced concrete column; (**b**) SRC column with cross-section steel.

One is according to the research of Uzumeri and Mander [11,12], the maximum restraint stress on the stirrups only plays a role in the core confined area. Thus, Mander proposed the following calculation formula for the effective restraint stress on the stirrups:

$$f\_l' = \frac{1}{2} k\_\epsilon \rho\_s f\_{yh} \tag{9}$$

where *ρ<sup>s</sup>* is the stirrup ratio, *fyh* represents the yield strength of the stirrup and *k<sup>e</sup>* indicates the effective restraint coefficient of the stirrup and is given by Equation (10). The effect of the stirrup on the core concrete confinement area is regarded as the one on all concrete areas within the stirrup area.

$$k\_{\varepsilon} = \frac{\left(1 - \sum \frac{(w')^2}{6b\_{\varepsilon}d\_{\varepsilon}}\right)\left(1 - \frac{s'}{2b\_{\varepsilon}}\right)\left(1 - \frac{s'}{2d\_{\varepsilon}}\right)}{1 - \rho\_{cc}}\tag{10}$$

where *ω*0 is the net distance between the adjacent longitudinal reinforcement; *b<sup>c</sup>* and *d<sup>c</sup>* represent the length and width of the rectangular stirrup, respectively; *s* 0 denotes the net distance between the stirrups; *ρcc* stands for the ratio of the area of the longitudinal reinforcement to that of the confinement area.

Then, Mander developed the calculation method for the peak stress improvement coefficient *k* of concrete in the confinement area:

$$k = -1.254 + 2.254\sqrt{1 + 7.94\frac{f\_l'}{f\_{c0}}} - 2\frac{f\_l'}{f\_{c0}}\tag{11}$$

where *fc*<sup>0</sup> is the axial compressive strength of concrete.

The other is the calculation method in the Section 7.2.3.1.6 of fib-CEB Model Code 2010 [19].

$$\sigma\_2 = w\_c f\_{cd} (1 - \frac{s\_c}{a\_c}) (1 - \frac{s\_c}{b\_c}) (1 - \frac{\sum b\_i^2 / 6}{a\_c b\_c}) \tag{12}$$

$$k = \frac{f\_{ck,c}}{f\_{ck}} = 1 + 3.5 \left(\frac{\sigma\_2}{f\_{ck}}\right)^{\frac{3}{4}} \tag{13}$$

$$
\varepsilon\_{c2,c} = \varepsilon\_{c2} \left[ 1 + 5 \left( \frac{f\_{ck,c}}{f\_{ck}} - 1 \right) \right] \tag{14}
$$

where *σ*2(= *σ*3) is the effective lateral compressive stress at the ULS due to confinement; *fck* stands for the characteristic compressive strength of concrete; *fck*,*<sup>c</sup>* is the value of confined concrete.

In terms of application, the above equations are very complicated, especially for square section members. To simplify, Yu Xiaolai (a scholar) proposed simplified formulas through a number of tests based on Mander's theory [20]. The equations are as follows,

$$f\_{\rm fc} = f\_{\rm c0} (2.254\sqrt{1 + 3.85\lambda\_{\rm v}} - 0.97\lambda\_{\rm v} - 1.254) \,\, k = \frac{f\_{\rm c0}}{f\_{\rm c0}} \tag{15}$$

$$
\varepsilon\_{\rm c\varepsilon} = \varepsilon\_{\rm c0} (1 + 3.5 \lambda\_{\rm v}) \tag{16}
$$

$$
\lambda\_v = \rho\_s f\_{y\text{h}} / f\_{c0} \tag{17}
$$

where *εc*<sup>0</sup> is the strain corresponding to the peak stress on the unconfined concrete, *ρ<sup>s</sup>* indicates the stirrup ratio; *fc*<sup>0</sup> denotes the axial compressive strength of concrete; *fyh* stands for the yield strength of the stirrup; *λ<sup>v</sup>* stands for the stirrup eigenvalue.

Based on the research by Yu, Equations (15)–(17) also apply to circular section members. The calculations of the simplified formulas are in good agreement with those of the above two methods, so the method for calculating the bearing capacity of the SRC column of rectangular section is modified by the simplified formulas.

These test results show that the confinement effect of the stirrup on concrete is noticeable, especially when the high-strength steel is configurated. The influence coefficient of steel strength is proposed based on Mander's model. The modified calculation formula for the bearing capacity of SRC columns under axial compression according to code JGJ138-2016 is as follows:

$$N \le 0.9\varphi(kA\_{\mathfrak{C}}f\_{\mathfrak{c}} + A\_{\mathfrak{a}}f\_{\mathfrak{a}} + A\_{\mathfrak{z}}f\_{\mathfrak{z}}) \tag{18}$$

For the SRC columns of circular or square section, Equations (15)–(17) can be selected to calculate. When the section is rectangular, Equations (9)–(14) should be used.

Figure 11 compares the bearing capacity of the specimens measured by the test results, with that calculated by the modified formulas, Equations (15)–(18), and calculations according to code JGJ 138-2016. The results confirm that the bearing capacity of the column calculated by Equations (15)–(18) deviates from the test results by only around 10%, and the bearing capacity calculated by code JGJ138-2016 deviates from the test results by 17–35%, respectively. Thus, the modified formula considering stirrup confinement can more accurately predict the ultimate bearing capacity of such members.

According to the work of Kim [21,22], it is necessary to ensure that the steel strain corresponding to the peak stress on concrete is not smaller than the yield strain of steel so as to achieve the full mechanical performance of high-strength steel in structures, that is:

$$
\varepsilon\_{\infty} \ge f\_a / \mathcal{E}\_a \tag{19}
$$

where *f<sup>a</sup>* and *E<sup>a</sup>* represent the yield stress and elastic modulus of section steel, respectively.

**Figure 11.** Comparison among test results, modified calculation results of formulas and the calculation results by JGJ138-2016.

According to Equations (16) and (17), when the high-strength steel is applied to SRC columns with a square section, this study determines the stirrup configuration conditions to ensure the full utilization of the strength of the steel as follows:

$$
\lambda\_v \ge \frac{f\_a - E\_a \varepsilon\_{c0}}{3.5 E\_a \varepsilon\_{c0}} \tag{20}
$$

According to Equation (16), the minimum stirrup eigenvalues are, respectively, 0.13 and 0.35, when Q460 and Q690 steel give full play to their strength in this test. When the rectangular stirrups are configured, the stirrup eigenvalue of specimen C5 is 0.15, which is higher than the minimum stirrup eigenvalue of the SRC columns with Q460 steel, 0.13; thus, the yield strength of Q460 steel can be brought into full play. However, for specimen C11, the stirrup eigenvalue of the rectangular stirrups is far lower than 0.35, thus the effective strength of Q690 steel is only 55.1% of its yield strength.

When the complex stirrups are configured in the SRC column, the stirrup eigenvalue of the specimen C6 is 0.34, much higher than 0.13, and the stress on the Q460 steel can reach the yield strength of the Q460 steel. Nevertheless, the stirrup eigenvalue of the specimen C12 is still slightly lower than the minimum stirrup eigenvalue, 0.35. The maximum stress on the Q690 steel can only reach as high as 89% of its yield strength, as shown in Figure 12. To summarize, the test data collected in this test, the requirements in code JGJ138-2016 and the calculations of Equation (16) can provide a reference for improving the stirrup design of high-strength SRC composite columns.

**Figure 12.** Axial load-strain relationships of C14.

## **4. Verification of Finite Element Model and Parametric Study**

*4.1. Establishment of Finite Element Model*

In Section 3.2, the reason for Q690 steel (in the specimen C11) unyielding under ultimate bearing capacity is well explained, and the design suggestion of the high-strength SRC column is put forward. However, it is found that not only the type of stirrups but also the spacing of stirrups affect the value of *ρ<sup>s</sup>* and *λv*. In this test, the value of *λ<sup>v</sup>* was increased by changing the stirrup type, the contribution of Q690 steel was increased, but the stirrup spacing was not considered. Therefore, the influence range of stirrup spacing on the bearing capacity of the specimens was studied by the analysis of expanding parameters with finite element models.

On the basis of the material property test, the finite element model of the test specimens was established using ABAQUS software to verify the applicability of the confinement theory to high-strength SRC columns in this test and study the influence of stirrup spacing on bearing capacity for high-strength SRC columns. Figure 13 displays the typical specimen models according to the test parameters. Moreover, eight-node hexahedral linear reduction integral stress element (C3D8R) were used for the concrete and steel in the finite element model, and two-node three-dimensional truss elements (T3D2) were employed for the reinforcement. Friction was defined at the interface between the concrete and the steel to account for their bonding. The grid was divided according to the length/Width/depth = 1.0:1.0:2.5, and the confinement effect was imposed according to the actual loading device.

**Figure 13.** Typical specimen models.

Table 2 presents the material properties of the steel obtained from the coupon tests. As for the concrete, three types of confinement effects were considered. Figure 14 also presents the three types of concrete sections. Figure 14a represents the section with no confinement; Figure 14b stands the section with the confinement effect of the stirrups and Figure 14c represents the section with the confinement effect of stirrups and steel. The uniaxial compression stress–strain curve of the concrete without any confinement effect was determined using the design of Concrete Structures (GB50010-2010) [23]. The

stress–strain relationship of the concrete with confinement effect of the stirrups or of both stirrups and steel was calculated by Equation (21) [12]:

$$
\sigma = \frac{f\_{cc}xr}{r - 1 + x^r} \tag{21}
$$

$$\mathbf{x} = \varepsilon/\varepsilon\_{\rm c\mathcal{C}} \cdot \mathbf{r} = \mathbf{E}\_{\rm c}/(\mathbf{E}\_{\rm c} - \mathbf{E}\_{\rm sec}) \cdot \mathbf{E}\_{\rm sc\mathcal{C}} = f\_{\rm c\mathcal{C}}/\varepsilon\_{\rm c\mathcal{C}} \cdot f\_{\rm c\mathcal{C}} = kf\_{\rm c\mathcal{O}} \cdot \varepsilon\_{\rm c\mathcal{C}} = [1 + \mathbf{5}(k - 1)]\varepsilon\_{\rm c\mathcal{O}} \tag{22}$$

where *σ* is the stress of confined concrete; *fcc* is the compressive strength of confined concrete; *Ec*, *Esec* are the elasticity modulus and secant modulus of concrete, respectively; *fc*<sup>0</sup> is the axial compressive strength of the unconfined concrete; *k* denotes the coefficient of improvement in the strength and strain.

**Figure 14.** Three kinds of confinement effect on concrete: (**a**) the section with no confinement; (**b**) the section with the confinement effect of the stirrups (**c**) the section with the confinement effect of stirrups and steel.

### *4.2. Verification of Finite Element Model*

Table 5 tabulates the ultimate bearing capacity of the specimens simulated by three finite element models. When the confinement effect is not taken into account, the difference between simulation data and the test results of the specimens with built-in Q235 steel is less than 5%. Figure 15a compares the simulation results with test result of C4. Figure 15b shows the load–displacement of simulation results and test result of C6, which also demonstrates the difference of specimens with built-in Q460 is between 4% and 8%, according to Table 5. Based on Table 5 and Figure 15c, the difference between the simulation data and the test results of Q690 high-strength SRC composite columns is in the range of 9% to 13%.



*N<sup>u</sup>* is the simulation result based on no confinement, *Nup* is the simulation result only considering the confinement effect of the stirrups, *Nuh* is the simulation result considering the confinement effect of the stirrups and steel.

**Figure 15.** Comparison between test results and the simulation results: (**a**) the load-displacement curves of simulation results and test result for C<sup>4</sup> ; (**b**) the load-displacement curves of simulation results and test result for C<sup>6</sup> . (**c**) the load-displacement curves of simulation results and test result for C14.

In general, the difference between the simulated data and the test results are both less than 11% for the models considering only the confinement effect of the stirrups and the models considering the confinement effect of the stirrups and the steel. The simulated value of the model considering the confinement effect of both the stirrups and the steel increases by about 2% compared with that of the model only considering the confinement effect of the stirrups. Therefore, the confinement effect of the stirrups and the steel on the bearing capacity of high-strength SRC composite columns cannot be ignored.

#### *4.3. Finite Element Parametric Study*

In order to expand the parametric analysis, it is necessary to verify the accuracy and applicability of the finite element model to high-strength SRC columns; C14 and C6 were taken as examples to compare the failure mode determined by the numerical simulation with the tested one, as shown in Figures 16 and 17. It can be seen that when the ultimate bearing capacity of the SRC column is reached, the maximum stress on the reinforcement is 460 MPa. However, the maximum stress on the Q690 steel is far from its yield strength, with slight buckling. The concrete on the buckling side crushes first. For specimen C6 with complex stirrups and Q460 steel, when the bearing capacity is reached, almost the entire length of the reinforcement yields with lateral expansion. The strain of the confinement concrete increases, and it functions well together with steel, which improves the utilization of the steel, concrete and reinforcement. As a result, the maximum stress on Q460 steel exceeds its yield strength, but no apparent buckling is noticed. According to the above comparison, both the bearing capacity and the failure mode obtained from the simulation show good agreement with the test results.

**Figure 16.** Stress–field nephogram of specimen C14 (considering only confinement effect of the stirrups). (**a**) Stress–field nephogram of C14, (**b**) Stress–field nephogram of Q690 steel in specimen C14 and (**c**) Stress–field nephogram of reinforcement.

**Figure 17.** Stress–field nephogram of specimen C6 (considering only confinement effect of the stirrups). (**a**) Stress–field nephogram of C6, (**b**) Stress–field nephogram of Q690 steel in specimen C6 and (**c**) Stress–field nephogram of reinforcement.

Changing the stirrup type enlarges the contribution of Q690 steel effectively. Another way to improve the stirrup confinement is to reduce the spacing of stirrups. The parameters of expanded specimens and the simulation results of bearing capacity are listed in Table 6.



For the four specimens with rectangular stirrups, the increasing magnitude of ultimate bearing capacity is similar to the increasing trend of *λ<sup>v</sup>* with the reduction in stirrups spacing, as shown in Figure 18a. However, although the ultimate bearing capacity of the three specimens with complex stirrups increases as spacing reduces, the increasing magnitude of ultimate bearing capacity decreases when the stirrup spacing decreases from 70 to 60 mm, as shown in Figure 18b. In general, if the value of is *λv*, too much larger than the minimum stirrup eigenvalues, the increasing magnitude of the bearing capacity will also decrease.

**Figure 18.** Influence of stirrup spacing on ultimate bearing capacity of the specimens: (**a**) specimens with rectangular stirrup (**b**) specimens with complex stirrup.

According to the stress Nephogram in Figure 19, the value of *λ<sup>v</sup>* of the four specimens with rectangular stirrups does not reach 0.35, and the steel still does not yield under the ultimate load. However, the increasing value of ultimate bearing capacity includes both that of confined concrete and steel. For the specimens with complex stirrups, reducing the spacing from 70 to 60 mm, the increasing value of ultimate bearing capacity is mainly from the bearing capacity of confined concrete. The contribution of Q690 steel is minimal. Therefore, when the materials have been selected, the design of high-strength SRC columns should be carried out in terms of stirrup type and stirrup spacing to ensure material utilization and save materials.

**Figure 19.** Q690 steel contribution of specimens with different stirrup spacing, type. (**a**) Rectangular stirrup, spacing = 50 mm (**b**) Complex stirrup, spacing = 70 mm.

#### **5. Conclusions**

An experimental study of SRC columns with high-strength steel was carried out to investigate the applicability of the formula for the bearing capacity described in specifications to high-strength SRC columns. It was concluded that the results calculated by the specifications were too conservative, and a modified formula considering the confinement effect of the stirrups was proposed. Furthermore, the comparison of the test results with results calculated according to the modified formula proved that considering the confinement effect of the stirrups on concrete for calculating the bearing capacity of high-strength SRC columns was accurate and effective. Finally, the finite element models considering different confinement levels were established, and their simulation results agree well with the test results. In general, an accurate calculation and design method for practical application was provided. The main conclusions that follow from the findings of the current work are that:


bearing capacity is obvious. When the stirrups eigenvalues *λ<sup>v</sup>* exceed the minimum stirrup eigenvalues, the improvement of bearing capacity is not apparent.

**Author Contributions:** Conceptualization, J.W.; methodology, J.W. and Y.D.; software, Y.D. and X.W.; validation, Y.W., Y.D. and Q.L.; formal analysis, J.W.; investigation, Y.D.; resources, J.W.; data curation, Y.D.; writing—original draft preparation, J.W. and Y.D.; writing—review and editing, Y.D.; visualization, X.W.; supervision, Q.L.; project administration, Y.W.; funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is financially supported by the Natural Science Foundation of Heilongjiang Province in China (ZD2019E001) 'Study on bearing capacity and seismic performance of Q460 high strength steel-reinforced concrete columns', and the National College Students' innovation and entrepreneurship training program 'Experimental study on mechanical properties of steel reinforced concrete columns under axial compression' (202110225185).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Jungki Lee \* and Mingu Han**

Department of Mechanical and Design Engineering, Hongik University, Sejong City 30016, Korea; pooh9265@naver.com

**\*** Correspondence: inq3jkl@hongik.ac.kr

**Abstract:** In this paper, the volume integral equation method (VIEM) is introduced for the numerical analysis of an infinite isotropic solid containing a variety of single isotropic/anisotropic spheroidal inclusions. In order to introduce the VIEM as a versatile numerical method for the threedimensional elastostatic inclusion problem, VIEM results are first presented for a range of single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under uniform remote tensile loading. We next considered single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix under remote shear loading. The authors hope that the results using the VIEM cited in this paper will be established as reference values for verifying the results of similar research using other analytical and numerical methods.

**Keywords:** volume integral equation method (VIEM); isotropic/anisotropic inclusion problems; boundary element method (BEM); standard finite element method (FEM)

### **1. Introduction**

The matrix and fibers in composites are usually made of isotropic material. However, in order to have higher strength and stiffness for commercial use, especially in the aerospace and automobile sectors, some constituents of metal matrix composites can be anisotropic. Since anisotropic materials are able to enhance mechanical properties toward orientation, certain mechanical properties (e.g., tensile strength) of anisotropic materials thus depend on orientation. As an example, in titanium-silicon carbide (Ti-SiC) composites, the matrix is nearly isotropic, but the SiC fiber has strong anisotropy and a multilayered structure including an interphase and a core.

A number of analytical techniques for solving inclusion problems are available when the inclusions are simple two-dimensional shapes (cylindrical and elliptical) or simple threedimensional shapes (spherical and ellipsoidal) and when they are well-separated [1–5]. In particular, Eshelby developed a simple and elegant method for solving the inclusion problem in isotropic solids in 1957 [1]. Eshelby first pointed out that the resulting elastic field can be found with the help of a sequence of imaginary cutting, straining and welding operations [1]. Eshelby also found that the strain and stress field inside the ellipsoidal inclusion is uniform and has a closed-form solution, regardless of the material properties and initial eigenstrain [1]. Eshelby's findings significantly influenced the mechanics of composites.

In the micromechanical analysis of composite materials, it is often assumed that the inclusions are periodically distributed in the matrix. Then, the unit-cell model with periodic boundary conditions is used to evaluate the overall, microstructure-insensitive, material properties of the composite. However, in real composites, the distribution of the inclusions is not periodic. Thus, the unit-cell model may not provide accurate estimates of the failure and damage mechanisms in composites [6–8].

Therefore, stress analysis of heterogeneous solids often requires the use of numerical approaches based on the standard finite element or boundary element formulations. How-

**Citation:** Lee, J.; Han, M. Volume Integral Equation Method Solution for Spheroidal Inclusion Problem. *Materials* **2021**, *14*, 6996. https:// doi.org/10.3390/ma14226996

Academic Editor: Michele Bacciocchi

Received: 27 September 2021 Accepted: 15 November 2021 Published: 18 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

ever, both methods present difficulties in dealing with problems involving infinite media or multiple anisotropic inclusions. In response to this concern, it has been demonstrated that the volume integral formulation can overcome both of these limitations in heterogeneous problems involving infinite media [9–11].

In comparison to the boundary element method (BEM), the volume integral equation method (VIEM) does not require the use of the Green's function for anisotropic inclusions and is not sensitive to the geometry of the inclusions. Moreover, as opposed to the standard finite element method (FEM), where it is necessary to discretize the full domain, the multiple inclusions only need to be discretized in the VIEM.

In this paper, three-dimensional elastostatic inclusion problems using the volume integral equation method (VIEM) will be investigated.

In order to introduce the VIEM as a versatile numerical method for the three-dimensional elastostatic inclusion problem, we first examine single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix subject to uniform remote tensile loading. Two different prolate and oblate spheroidal inclusions with an aspect ratio of 0.5 and 0.75 are considered, respectively. The matrix is assumed to be isotropic. Eight isotropic and five orthotropic inclusions with different characteristics are considered in the numerical calculation. The normalized tensile stress inside the inclusions is investigated in two different directions. Next, we examine single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix subject to remote shear loading. Two different prolate and oblate spheroidal inclusions with an aspect ratio of 0.5 and 0.75 are considered, respectively. The matrix is assumed to be isotropic. Three isotropic and two orthotropic inclusions with different characteristics are considered in the numerical calculation. The normalized shear stress inside the inclusions is investigated in two different directions.

The authors hope that the present solutions using the parallel volume integral equation method for the single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions with different material properties under uniform remote tensile loading or remote shear loading will be established as reference values for verifying the results of other analytical and numerical methods.

Since the VIEM is a combination of two powerful general-purpose numerical methods, the standard finite element method (FEM) and the boundary element method (BEM), it is also a highly beneficial tool in the field of numerical analysis and can play a very important role in solving inclusion problems. Subsequently, the purpose of this paper is to introduce the parallel volume integral equation method (PVIEM) as an accessible, versatile and powerful numerical method for solving inclusion problems in the areas of computational mechanics and mechanics of composite materials.

#### **2. Governing Equations of Volume Integral Equation Formulation**

The geometry of the general elastodynamic problem is shown in Figure 1a, where an infinite homogeneous, isotropic and linearly elastic solid containing a number of isotropic or anisotropic inclusions of arbitrary number and shape are subjected to prescribed dynamic loading at infinity.

In Figure 1a, V and S represent the volume and surface of the inclusion respectively, and n is the outward unit normal to S while V<sup>o</sup> and S<sup>o</sup> represent the infinite volume and surface, respectively.

The symbols ρ (1) and cijkl (1) denote the density and the elastic stiffness tensor of the inclusion, while ρ (2) and cijkl (2) denote the density and the elastic stiffness tensor of the infinite homogeneous, isotropic and linearly elastic matrix material, respectively. Therefore, cijkl (2) is a constant isotropic tensor, while cijkl (1) can be arbitrary, i.e., the inclusions may, in general, be inhomogeneous and anisotropic. The isotropic or anisotropic inclusions are assumed to be perfectly bonded to the matrix.

**Figure 1.** Geometry of the general (**a**) elastodynamic and (**b**) elastostatic problem. (**c**) A remote shear loading, σoxy. (**d**) A remote shear loading, σoxz. (**e**) A remote shear loading, σoyz. **Figure 1.** Geometry of the general (**a**) elastodynamic and (**b**) elastostatic problem. (**c**) A remote shear loading, σ o xy. (**d**) A remote shear loading, σ o xz. (**e**) A remote shear loading, σ o yz.

In Figure 1a, V and S represent the volume and surface of the inclusion respectively, and n is the outward unit normal to S while Vo and So represent the infinite volume and Mal and Knopoff [12] showed that the elastodynamic displacement, um(**x**), in the composite satisfies the volume integral equation:

$$\mathfrak{u}\_{\mathfrak{m}}(\mathbf{x}) = \mathfrak{u}\_{\mathfrak{m}}^{\rho}(\mathbf{x}) + \int\_{V} [\delta\rho\omega^{2}g\_{i}^{\mathfrak{m}}(\boldsymbol{\xi},\mathbf{x})\boldsymbol{u}\_{i}(\boldsymbol{\xi}) - \delta\boldsymbol{c}\_{i\bar{j}k\bar{l}}g\_{i,j}^{\mathfrak{m}}(\boldsymbol{\xi},\mathbf{x})\boldsymbol{u}\_{k\bar{l}}(\boldsymbol{\xi})]d\boldsymbol{\xi} \tag{1}$$

infinite homogeneous, isotropic and linearly elastic matrix material, respectively. Therefore, cijkl(2) is a constant isotropic tensor, while cijkl(1) can be arbitrary, i.e., the inclusions may, in general, be inhomogeneous and anisotropic. The isotropic or anisotropic incluwhere the integral is over the domain V occupied by the isotropic or anisotropic inclusions, δρ = ρ (1) <sup>−</sup> <sup>ρ</sup> (2) and δcijkl = cijkl (1) <sup>−</sup> <sup>c</sup>ijkl (2), and g<sup>i</sup> <sup>m</sup>(ξ,**x**) is the elastodynamic Green's function for the infinite homogeneous, isotropic and linearly elastic matrix material.

sions are assumed to be perfectly bonded to the matrix. Mal and Knopoff [12] showed that the elastodynamic displacement, um(**x**), in the composite satisfies the volume integral equation: <sup>2</sup> ( ) ( ) [ ( , ) ( ) ( , ) ( )] , , **x x ξx ξ ξx ξ ξ** *om m u u g u cg u d m m i i ijkl i j k l <sup>V</sup>* =+ − δρω δ (1) In Equation (1), u<sup>m</sup> o (**x**, ω)e−iω<sup>t</sup> represents the *m*th component of the displacement vector due to the incident field at **x** in the absence of the inclusions, while um(**x**, ω)e−iω<sup>t</sup> denotes the same quantity in the presence of the isotropic or anisotropic inclusions, where ω is the circular frequency of the waves. In what follows, the explicit dependence on the circular frequency, and the common time factor, e−iω<sup>t</sup> , for all field quantities will be suppressed.

surface, respectively.

The geometry of the general elastostatic problem is shown in Figure 1b–e. It has been shown by Lee and Mal [9] that the corresponding elastostatic displacement, um(**x**), within the composite, fulfills the volume integral equation as:

$$u\_m(\mathbf{x}) = u\_m^o(\mathbf{x}) - \int\_V \delta c\_{i\bar{j}k\bar{l}} g\_{i,j}^m(\boldsymbol{\xi}, \mathbf{x}) u\_{k,l}(\boldsymbol{\xi}) d\boldsymbol{\xi} \tag{2}$$

where the integral is over the space V occupied by the isotropic or anisotropic inclusions and δcijkl = cijkl (1) <sup>−</sup> <sup>c</sup>ijkl (2). The value g<sup>i</sup> <sup>m</sup>(ξ,**x**) represents the elastostatic Kelvin's solution (or Green's function) for the infinite homogeneous, isotropic and linearly elastic matrix material.

In Equations (1) and (2), the differentiations are with respect to the integration variable, ξi , and the summation convention and comma notation have been utilized. The integrand is non-zero within the isotropic or anisotropic inclusions only, since δcijkl = 0 outside the inclusions.

If **x** lies inside the inclusions, then Equations (1) and (2) are integro-differential equations for the unknown displacement vector **u**(**x**) within the inclusions. It should be noted that an algorithm was developed by Lee and Mal [9,10] to numerically calculate the unknown displacement vector **u**(**x**) by discretizing the inclusions only using standard finite elements. Once **u**(**x**) within the inclusions is determined, the displacement field outside the inclusions can be obtained from Equations (1) and (2) by evaluating the corresponding integrals respectively, and the stress field within and outside the inclusions can also be readily determined.

The volume integral equation method (VIEM) was originated from Lee and Mal [10] in 1995. Since 1995, Lee and his co-workers (e.g., [9–11,13–17]) have been developing a more engineering-oriented VIEM, while Buryachenko (e.g., [18–20]) has been examining a more mathematically oriented VIEM since 2000. Additionally, Dong has conducted research on the volume integral equation method since 2003 [21]. Therefore, the VIEM is broadening its influence on computational fields of study.

Furthermore, Section 4.3 entitled 'Volume Integral Equation Method' of the book "Micromechanics of Heterogeneous Materials" by Buryachenko [18] also explains further mathematical formulation of the elastostatic volume integral equation method. In particular, a general description of the volume integral equation method is presented in Chapter 4 entitled 'Volume Integral Equation Method (VIEM)' of the book "Advances in Computers and Information in Engineering Research, Vol. 2" by Michopoulos et al. (eds.) [22]. In addition, complete descriptions of the fundamental numerical technique of Equation (2) can be found in [17] for three-dimensional elastostatic problems.

Although each numerical method has certain advantages, specific disadvantages have led to further discussion and research. For example, in Section 3.1 of Reference [20], Buryachenko points out that the VIEM is quite time-consuming. Moreover, no optimized commercial software exists for its application.

Firstly, in order to resolve this 'time-consuming' problem, we propose the parallel volume integral equation method and implement MPI-based code. Such method allows us not only to solve the large domain but also to speed up computation in the volume integral equation method. The FORTRAN 90 (Version 1.1, IBM, Armonk, NY, USA) source code containing about 9000 lines for the three-dimensional VIEM of the previous paper [17] was parallelized and optimized for this paper, with support from the Korea Institute of Science and Technology Information (KISTI, Daejeon, Korea). Figure 2 shows the procedures of a representative MPI parallelization approach ("pvi3ds01\_sm7560xx.f90") for the sequential three-dimensional VIEM code ("svi3ds01\_sm4320xx.f"). As a result, the program execution time has been greatly reduced. Furthermore, we could use more finite elements (31,857 nodes and 7560 elements) in the VIEM model of this paper than those (18,109 nodes and 4320 elements) in the VIEM model of the previous paper [17]. The parallel FORTRAN source code for the three-dimensional VIEM is presently being processed in the KISTI-5. It is referred to as "Nurion", which is a system consisting of compute nodes, CPU-only nodes, Omni-Path interconnect networks, Burst Buffer high-speed storage, a Luster-based parallel

file system and a water-cooling device based on a Rear Door Heat Exchanger (RDHx). The CPU-only nodes consist of 132 Intel Xeon 6148 2.4 GHz processors (named "Skylake"). The total theoretical performance is 25.7 petaflops, which ranked 11th in the world in June 2018 (http://www.top500.org, accessed on 3 May 2021). It should be noted that, in order to investigate three-dimensional stress problems with multiple inclusions, in addition to parallelization and optimization of the sequential three-dimensional VIEM code, a domain decomposition method (DDM) was applied to the parallel three-dimensional VIEM code, with support from the Korea Institute of Science and Technology Information (KISTI). The domain decomposition method allows decomposition of large-sized problem solutions to solutions of several smaller-sized problems [23]. Therefore, the parallel volume integral equation method (PVIEM) using the domain decomposition method enables us to investigate more complicated multiple inclusion problems elastostatically or elastodynamically. Exchanger (RDHx). The CPU-only nodes consist of 132 Intel Xeon 6148 2.4 GHz processors (named "Skylake"). The total theoretical performance is 25.7 petaflops, which ranked 11th in the world in June 2018 (http://www.top500.org, accessed on 3 May, 2021). It should be noted that, in order to investigate three-dimensional stress problems with multiple inclusions, in addition to parallelization and optimization of the sequential three-dimensional VIEM code, a domain decomposition method (DDM) was applied to the parallel threedimensional VIEM code, with support from the Korea Institute of Science and Technology Information (KISTI). The domain decomposition method allows decomposition of largesized problem solutions to solutions of several smaller-sized problems [23]. Therefore, the parallel volume integral equation method (PVIEM) using the domain decomposition method enables us to investigate more complicated multiple inclusion problems elastostatically or elastodynamically.

and Technology Information (KISTI, Daejeon, Korea). Figure 2 shows the procedures of a representative MPI parallelization approach ("pvi3ds01\_sm7560xx.f90") for the sequential three-dimensional VIEM code ("svi3ds01\_sm4320xx.f"). As a result, the program execution time has been greatly reduced. Furthermore, we could use more finite elements (31,857 nodes and 7560 elements) in the VIEM model of this paper than those (18,109 nodes and 4320 elements) in the VIEM model of the previous paper [17]. The parallel FORTRAN source code for the three-dimensional VIEM is presently being processed in the KISTI-5. It is referred to as "Nurion", which is a system consisting of compute nodes, CPU-only nodes, Omni-Path interconnect networks, Burst Buffer high-speed storage, a Luster-based parallel file system and a water-cooling device based on a Rear Door Heat

*Materials* **2021**, *14*, x FOR PEER REVIEW 5 of 44

**Figure 2. Figure 2.** Procedures of 'pvi3ds01\_sm7560xx. Procedures of 'pvi3ds01\_sm7560xx.f90' using MPI parallelization. f90' using MPI parallelization.

Secondly, in order to resolve the 'no optimized commercial software' problem, we plan to develop a semi-commercial VIEM software called the "Volume Integral Equation Method Application Program" (VIEMAP). Table 1 shows the analysis capabilities of VIEMAP including a pre-processor (ViemMesh), a solver (VIEM) and a post-processor (ViemPlot) adapted to solve multiple isotropic/anisotropic inclusion problems in a computationally tractable manner. Figure 3 shows the registered trademark for the VIEMAP. The authors aim to help both university students and researchers create VIEM models using the VIEMAP more easily than using the standard finite element method (FEM), as well as solve multiple isotropic/anisotropic inclusion problems in an unbounded isotropic medium more accurately and conveniently than the boundary element method (BEM).

ViemPlot (Post-Processor)


#### **Table 1.** Capabilities of VIEMAP. medium more accurately and conveniently than the boundary element method (BEM). Secondly, in order to resolve the 'no optimized commercial software' problem, we

well as solve multiple isotropic/anisotropic inclusion problems in an unbounded isotropic

Secondly, in order to resolve the 'no optimized commercial software' problem, we plan to develop a semi-commercial VIEM software called the "Volume Integral Equation Method Application Program" (VIEMAP). Table 1 shows the analysis capabilities of VIE-MAP including a pre-processor (ViemMesh), a solver (VIEM) and a post-processor (ViemPlot) adapted to solve multiple isotropic/anisotropic inclusion problems in a computationally tractable manner. Figure 3 shows the registered trademark for the VIEMAP. The authors aim to help both university students and researchers create VIEM models

*Materials* **2021**, *14*, x FOR PEER REVIEW 6 of 44

**Figure 3.** Registered trademark for VIEMAP. **Figure 3.** Registered trademark for VIEMAP. (1) Displacement contour plot

(2) Stress contour plot

#### **3. Three-Dimensional Elastostatic Problems Using the VIEM 3. Three-Dimensional Elastostatic Problems Using the VIEM**

In this section, we first examine single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix subject to uniform remote tensile loading, σoxx, as shown in Figure 4 (also see Figures 1b and 5). The remote applied load can be arbitrarily chosen and was assumed to be σoxx = 143.10 GPa for convenience purposes only. Two different prolate spheroidal inclusions were considered: (a) a/b = c/b = 0.5 and (b) a/b = c/b = 0.75 (see Figure 5). Additionally, two different oblate spheroidal inclusions were considered: (a) b/a = c/a = 0.5 and (b) b/a = c/a = 0.75 (see Figure 5). In this section, we first examine single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix subject to uniform remote tensile loading, σ o xx, as shown in Figure 4 (also see Figures 1b and 5). The remote applied load can be arbitrarily chosen and was assumed to be σ o xx = 143.10 GPa for convenience purposes only. Two different prolate spheroidal inclusions were considered: (a) a/b = c/b = 0.5 and (b) a/b = c/b = 0.75 (see Figure 5). Additionally, two different oblate spheroidal inclusions were considered: (a) b/a = c/a = 0.5 and (b) b/a = c/a = 0.75 (see Figure 5). **Figure 3.** Registered trademark for VIEMAP.

**Table 1.** Capabilities of VIEMAP.

The elastic constants for the isotropic matrix and the isotropic inclusions are listed in Table 2. The elastic constants for the isotropic matrix and the orthotropic inclusions are listed in Table 3. **3. Three-Dimensional Elastostatic Problems Using the VIEM**  In this section, we first examine single isotropic/orthotropic spherical, prolate and

We next examine single isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions in an infinite isotropic matrix subject to remote shear loading, σ o xy, σ o xz or σ o yz, as shown in Figure 6 (also see Figures 1c–e and 5) [24]. The remote applied load can be arbitrarily chosen and was assumed to be σ o xy = σ o xz = σ o yz = 75.76 GPa for convenience purposes only. We considered the same geometry of the single spherical, prolate (with an aspect ratio of 0.5 and 0.75) and oblate (with an aspect ratio of 0.5 and 0.75) spheroidal inclusions in an infinite isotropic matrix under remote shear loading (σ o xy, σ o xz and σ o yz). oblate spheroidal inclusions in an infinite isotropic matrix subject to uniform remote tensile loading, σoxx, as shown in Figure 4 (also see Figures 1b and 5). The remote applied load can be arbitrarily chosen and was assumed to be σoxx = 143.10 GPa for convenience purposes only. Two different prolate spheroidal inclusions were considered: (a) a/b = c/b = 0.5 and (b) a/b = c/b = 0.75 (see Figure 5). Additionally, two different oblate spheroidal inclusions were considered: (a) b/a = c/a = 0.5 and (b) b/a = c/a = 0.75 (see Figure 5).

(1) Displacement contour plot (2) Stress contour plot

**Figure 4.** *Cont*.

(σoxx).

(**c**) An oblate spheroidal inhomogeneity

**Figure 4.** (**a**) Spherical, (**b**) prolate spheroidal and (**c**) oblate spheroidal inclusions under uniform remote tensile loading **Figure 4.** (**a**) Spherical, (**b**) prolate spheroidal and (**c**) oblate spheroidal inclusions under uniform remote tensile loading (σ o xx).

**Table 2.** Material Properties of the Isotropic Matrix and the Isotropic Inclusions.

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**Figure 5.** The orientation of spherical, prolate spheroidal and oblate spheroidal inclusions. (**a**) Spheroidal coordinate system. (**b**) Cartesian coordinate system. **Figure 5.** The orientation of spherical, prolate spheroidal and oblate spheroidal inclusions. (**a**) Spheroidal coordinate system. (**b**) Cartesian coordinate system.

The elastic constants for the isotropic matrix and the isotropic inclusions are listed in Table 2. The elastic constants for the isotropic matrix and the orthotropic inclusions are

**Material λ (GPa) μ (GPa) E (GPa) ν** Matrix (Iso\_01) 67.3401 37.8788 100.0 0.32 Inclusion (Iso\_01) 176.060 176.060 440.15 0.25 Matrix (Iso\_02) 121.154 80.7692 210.0 0.30 Inclusion (Iso\_02) 83.1643 176.724 410.0 0.16 Matrix (Iso\_03) 75.0 37.5 100.0 0.3333 Inclusion (Iso\_03) 150.0 75.0 200.0 0.3333 Matrix (Iso\_04) 75.0 37.5 100.0 0.3333 Inclusion (Iso\_04) 375.0 187.5 500.0 0.3333 Matrix (Iso\_05) 75.0 37.5 100.0 0.3333 Inclusion (Iso\_05) 750.0 375.0 1000.0 0.3333 Matrix (Iso\_06) 121.154 80.7692 210.0 0.30

**Table 2.** Material Properties of the Isotropic Matrix and the Isotropic Inclusions.

listed in Table 3.

Three different material properties (Iso\_01, Iso\_05 and Iso\_06) in Table 2 were used in the numerical calculation. The elastic constants for the isotropic matrix and the orthotropic inclusions are listed in Table 4. Table 5 shows various characteristics of the material properties used in the numerical calculation. In order to demonstrate the capability of the volume integral equation method for the three-dimensional anisotropic inclusion problem, three independent elastic constants, c<sup>44</sup> (shear modulus in the yz plane), c<sup>55</sup> (shear modulus in the xz plane) and c<sup>66</sup> (shear modulus in the xy plane), were assumed to be different from each other [25]. Inclusion (Iso\_06) 87.2202 41.0448 110.0 0.34 Matrix (Iso\_07) 75.0 37.5 100.0 0.3333 Inclusion (Iso\_07) 15.0 7.5 20.0 0.3333 Matrix (Iso\_08) 75.0 37.5 100.0 0.3333 Inclusion (Iso\_08) 52.5 26.25 70.0 0.3333 **Table 3.** Material Properties of the Isotropic Matrix and the Orthotropic Inclusions.


**Table 3.** Material Properties of the Isotropic Matrix and the Orthotropic Inclusions. **Unit: GPa Orthotropic Inclusions Isotropic Matrix Ort\_01 Ort\_02 Ort\_03 Ort\_04 Ort\_05** 

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**Table 4.** Material properties of the isotropic matrix and the orthotropic inclusions. as shown in Figure 6 (also see Figures 1c–e and 5) [24]. The remote applied load can be arbitrarily chosen and was assumed to be σoxy = σoxz = σoyz = 75.76 GPa for convenience


**Figure 6.** *Cont*.

shear loading.

Inclusion (Ort\_01) Orthotropic c11 > c22 = c33

Matrix (Ort\_02) Isotropic No restriction in Poisson's ratio

(**c**) An oblate spheroidal inclusion (σoyz)

**Figure 6.** (**a**) Spherical, (**b**) prolate spheroidal and (**c**) oblate spheroidal inclusions under remote **Figure 6.** (**a**) Spherical, (**b**) prolate spheroidal and (**c**) oblate spheroidal inclusions under remote shear loading.

**Table 4.** Material properties of the isotropic matrix and the orthotropic inclusions. **Table 5.** Material Property Characteristics.


Matrix (Iso\_07) Isotropic ν = 1/3 E(Inclusion) < E(Matrix) Inclusion (Iso\_07) Isotropic ν = 1/3 *3.1. Single Spherical Inclusion Problems under Uniform Remote Tensile Loading* 3.1.1. VIEM Formulation Applied to Isotropic Inclusion Problems

Matrix (Iso\_08) Isotropic ν = 1/3 E(Inclusion) < E(Matrix) Inclusion (Iso\_08) Isotropic ν = 1/3; E(Iso\_08) > E(Iso\_07) Matrix (Ort\_01) Isotropic No restriction in Poisson's ratio The displacements in the volume integral Equation (2) for isotropic spherical, prolate and oblate spheroidal inclusions can be expressed in the form:

u1(x) = u o 1 (x) − R V {δ(λ + 2µ)g 1 1,1u1,1 + δλ(g 1 1,1u2,2 + g 1 2,2u1,1) + δλ(g 1 1,1u3,3 + g 1 3,3u1,1) +δ(λ + 2µ)g 1 2,2u2,2 + δλ(g 1 2,2u3,3 + g 1 3,3u2,2) + δ(λ + 2µ)g 1 3,3u3,3 +δµ[g 1 2,3(u2,3 + u3,2) + g 1 3,2(u2,3 + u3,2)] +δµ[g 1 1,3(u1,3 + u3,1) + g 1 3,1(u1,3 + u3,1)] +δµ[g 1 1,2(u1,2 + u2,1) + g 1 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> (3) u2(x) = u o 2 (x) − R V {δ(λ + 2µ)g 2 1,1u1,1 + δλ(g 2 1,1u2,2 + g 2 2,2u1,1) + δλ(g 2 1,1u3,3 + g 2 3,3u1,1) +δ(λ + 2µ)g 2 2,2u2,2 + δλ(g 2 2,2u3,3 + g 2 3,3u2,2) + δ(λ + 2µ)g 2 3,3u3,3 +δµ[g 2 2,3(u2,3 + u3,2) + g 2 3,2(u2,3 + u3,2)] (4)

+δµ[g 1,2(u1,2 + u2,1) + g 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> u3(x) = u o 3 (x) − R V {δ(λ + 2µ)g 3 1,1u1,1 + δλ(g 3 1,1u2,2 + g 3 2,2u1,1) + δλ(g 3 1,1u3,3 + g 3 3,3u1,1) +δ(λ + 2µ)g 3 2,2u2,2 + δλ(g 3 2,2u3,3 + g 3 3,3u2,2) + δ(λ + 2µ)g 3 3,3u3,3 +δµ[g 3 2,3(u2,3 + u3,2) + g 3 3,2(u2,3 + u3,2)] +δµ[g 3 1,3(u1,3 + u3,1) + g 3 3,1(u1,3 + u3,1)] +δµ[g 3 1,2(u1,2 + u2,1) + g 3 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> (5)

2

3,1(u1,3 + u3,1)]

1,3(u1,3 + u3,1) + g

2

+δµ[g 2

2

where u1(x), u2(x) and u3(x) are the three-dimensional displacements, δcαβ = cαβ(1) − cαβ(2) (α, β = 1, 6), where cαβ(1) represents the elastic stiffness constants of the isotropic inclusions, while cαβ(2) denotes those for the isotropic matrix material: δc<sup>11</sup> = δc<sup>22</sup> = δc<sup>33</sup> = (λ<sup>1</sup> + 2µ1) − (λ<sup>2</sup> + 2µ2), δc<sup>12</sup> = δc<sup>13</sup> = δc<sup>23</sup> = λ<sup>1</sup> − λ<sup>2</sup> and δc<sup>44</sup> = δc<sup>55</sup> = δc<sup>66</sup> = µ<sup>1</sup> − µ2.

In Equations (3)–(5), g<sup>i</sup> <sup>m</sup>(ξ,**x**) is the Green's function for the infinite isotropic matrix material and is stated by Banerjee [26] and Pao and Varatharajulu [27] as:

$$\begin{aligned} \mathbf{g}\_1^1 &= \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_1-\boldsymbol{\xi}\_1)^2}{\mathbf{r}^2} + (3-4\mathbf{v})\right]} \\ \mathbf{g}\_2^1 &= \mathbf{g}\_1^2 = \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_1-\boldsymbol{\xi}\_1)(\mathbf{x}\_2-\boldsymbol{\xi}\_2)}{\mathbf{r}^2}\right]} \\ \mathbf{g}\_3^1 &= \mathbf{g}\_1^3 = \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_1-\boldsymbol{\xi}\_1)(\mathbf{x}\_3-\boldsymbol{\xi}\_3)}{\mathbf{r}^2}\right]} \\ \mathbf{g}\_2^2 &= \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_2-\boldsymbol{\xi}\_2)^2}{\mathbf{r}^2} + (3-4\mathbf{v})\right]} \\ \mathbf{g}\_3^2 &= \mathbf{g}\_2^3 = \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_2-\boldsymbol{\xi}\_2)(\mathbf{x}\_3-\boldsymbol{\xi}\_3)}{\mathbf{r}^2}\right]} \\ \mathbf{g}\_3^3 &= \frac{1}{16\pi(1-\mathbf{v})\text{tr}\left[\frac{(\mathbf{x}\_3-\boldsymbol{\xi}\_3)^2}{\mathbf{r}^2} + (3-4\mathbf{v})\right]} \end{aligned} \tag{6}$$

where r = |x <sup>−</sup> <sup>ξ</sup>| = <sup>q</sup> (*x*<sup>1</sup> − *ξ* 1 ) <sup>2</sup> + (*x*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>* 2 ) <sup>2</sup> + (*x*<sup>3</sup> <sup>−</sup> *<sup>ξ</sup>* 3 ) 2 , ν is Poisson's ratio and µ is the shear modulus for the infinite isotropic matrix material.

## 3.1.2. VIEM Formulation Applied to Orthotropic Inclusion Problems

Let the coordinate axes x1(x), x2(y) and x3(z) be taken parallel to the symmetry axes of the orthotropic material, and c11, c12, c13, c22, c23, c33, c44, c<sup>55</sup> and c<sup>66</sup> denote the elastic constants. The displacements in Equation (2) for orthotropic spherical, prolate and oblate spheroidal inclusions can be expressed in the form:

u1(x) = u o 1 (x) − R V {δc11g 1 1,1u1,1 + δc12(g 1 1,1u2,2 + g 1 2,2u1,1) + δc13(g 1 1,1u3,3 + g 1 3,3u1,1) +δc22g 1 2,2u2,2 + δc23(g 1 2,2u3,3 + g 1 3,3u2,2) + δc33g 1 3,3u3,3 +δc44[g 1 2,3(u2,3 + u3,2) + g 1 3,2(u2,3 + u3,2)] +δc55[g 1 1,3(u1,3 + u3,1) + g 1 3,1(u1,3 + u3,1)] +δc66[g 1 1,2(u1,2 + u2,1) + g 1 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> (7)

u2(x) = u o 2 (x) − R V {δc11g 2 1,1u1,1 + δc12(g 2 1,1u2,2 + g 2 2,2u1,1) + δc13(g 2 1,1u3,3 + g 2 3,3u1,1) +δc22g 2 2,2u2,2 + δc23(g 2 2,2u3,3 + g 2 3,3u2,2) + δc33g 2 3,3u3,3 +δc44[g 2 2,3(u2,3 + u3,2) + g 2 3,2(u2,3 + u3,2)] +δc55[g 2 1,3(u1,3 + u3,1) + g 2 3,1(u1,3 + u3,1)] +δc66[g 2 1,2(u1,2 + u2,1) + g 2 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> (8) u3(x) = u o 3 (x) − R V {δc11g 3 1,1u1,1 + δc12(g 3 1,1u2,2 + g 3 2,2u1,1) + δc13(g 3 1,1u3,3 + g 3 3,3u1,1) +δc22g 3 2,2u2,2 + δc23(g 3 2,2u3,3 + g 3 3,3u2,2) + δc33g 3 3,3u3,3 +δc44[g 3 2,3(u2,3 + u3,2) + g 3 3,2(u2,3 + u3,2)] +δc55[g 3 1,3(u1,3 + u3,1) + g 3 3,1(u1,3 + u3,1)] +δc66[g 3 1,2(u1,2 + u2,1) + g 3 2,1(u1,2 + u2,1)]}dξ1dξ2dξ<sup>3</sup> (9)

where u1(x), u2(x) and u3(x) are the three-dimensional displacements, δcαβ = cαβ(1) − cαβ(2) (α, β = 1, 6), where = cαβ(1) represents the elastic stiffness constants of the orthotropic inclusions, while cαβ(2) denotes those for the isotropic matrix material: δc<sup>11</sup> = c<sup>11</sup> − (λ<sup>2</sup> + 2µ2), δc<sup>22</sup> = c<sup>22</sup> − (λ<sup>2</sup> + 2µ2), δc<sup>33</sup> = c<sup>33</sup> − (λ<sup>2</sup> + 2µ2), δc<sup>12</sup> = c<sup>12</sup> − λ2, δc<sup>13</sup> = c<sup>13</sup> − λ2, δc<sup>23</sup> = c<sup>23</sup> − λ<sup>2</sup> and δc<sup>44</sup> = c<sup>44</sup> − µ2, δc<sup>55</sup> = c<sup>55</sup> − µ2, δc<sup>66</sup> = c<sup>66</sup> − µ2.

In Equations (7)–(9), g<sup>i</sup> <sup>m</sup>(ξ,**x**) is the Green's function for the infinite isotropic matrix material and is stated in Equation (6). Thus, the VIEM does not require the use of the Green's function for the orthotropic material of the inclusion. In general, Green's function for an anisotropic material is much more complex than that of an isotropic material [28]. Furthermore, a closed form solution of the generalized Green's function for an anisotropic material is not available in the literature.

In contrast, in the BEM, Green's functions for both the isotropic matrix and the anisotropic inclusions must be specified in the formulation. In particular, special emphasis is placed on the fact that Green's function for the anisotropic material of the inclusions is not required in the VIEM.

#### 3.1.3. Numerical Formulations in the VIEM

The integrands in Equations (3)–(8) contain singularities with different orders due to the singular characteristics of the Green's function at x = ξ (i.e., r = 0). Thus, evaluation of the singular integrals requires special attention. In general, g<sup>i</sup> <sup>m</sup>(ξ,**x**) behaves as 1/r, while its derivatives behave as 1/r<sup>2</sup> as r <sup>→</sup> 0. It should be noted that only g<sup>i</sup> <sup>m</sup>(ξ,**x**) for the isotropic matrix and its derivatives are required in the VIEM. Furthermore, in the BEM, the Green's function for anisotropic inclusions and their derivatives must also be specified. As a result, this may be a critical drawback to the BEM when solving multiple anisotropic inclusion problems.

In contrast to the BEM, the singularities in the VIEM are integrable (weak). Thus, we have decided to utilize the direct integration scheme stated by Li et al. [29]. Finally, after suitable adjustments, we have succeeded in addressing these weak singular integrands in the volume integral equation formulations.

A comprehensive elaboration for the accurate evaluation of singular integrals using the tetrahedron polar co-ordinates shown in [29] was presented in [17].

#### 3.1.4. A Single Isotropic Spherical Inclusion

In order to examine the accuracy of the numerical results using the VIEM, the numerical results using the VIEM for a single isotropic spherical inclusion were first compared to the analytical solutions [21,30]. We considered a single isotropic spherical inclusion with a radius of 6 mm in an infinite isotropic matrix subject to uniform remote tensile loading, σxx o , as shown in Figure 4a. It should be noted that the length of the radius can be arbitrarily chosen. In Figure 7, standard 20-node quadratic hexahedral elements were used in the discretization [31]. The number of hexahedral elements, 7560, was determined based on a convergence test. For the seven different material properties (Iso\_2, Iso\_03, Iso\_04, Iso\_05, Iso\_06, Iso\_07 and Iso\_08) in Table 2, a comparison was made between the

numerical results using the volume integral equation method (VIEM) and the analytical solutions. As shown in Table 5, there was no restriction to Poisson's ratio in the inclusions and matrices of Iso\_02 and Iso\_06. However, Poison's ratio was 1/3 in both the inclusion and matrix of Iso\_03, Iso\_04, Iso\_05, Iso\_07 and Iso\_08. Furthermore, for Iso\_02, Iso\_03, Iso\_04 and Iso\_05, Young's modulus (E) in the isotropic inclusion was greater than that in the isotropic matrix. For Iso\_06, Iso\_07 and Iso\_08, Young's modulus (E) in the isotropic matrix was greater than that in the isotropic inclusion. Thus, seven material properties representing a diversity of materials were chosen. Excellent agreement was found between the analytical and numerical solutions using the VIEM for the seven different materials considered. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. It should also be noted that the normalized tensile stress (σxx/σ o xx) inside the isotropic spherical inclusions was found to be constant [1,30]. Tables 6–8 show that the percentage differences for the two sets of results are less than 0.1% in seven cases. Figure 8 shows numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the isotropic spherical inclusions with a radius of 6 mm under uniform remote tensile loading. **Material VIEM (Average) Analytical Solution Error (%)**  Iso\_01 1.5800 - - Iso\_02 1.2823 1.2822 0.0078 **Table 7.** Normalized tensile stress component (σxx/σoxx) within the isotropic spherical inclusion due to uniform remote tensile loading (σoxx). **Material VIEM (Average) Analytical Solution Error (%)**  Iso\_03 1.3090 1.3091 0.0076 Iso\_04 1.6171 1.6173 0.0124 Iso\_05 1.7582 1.7582 0.0 **Table 8.** Normalized tensile stress component (σxx/σoxx) within the isotropic spherical inclusion due to uniform remote tensile loading (σoxx). **Material VIEM (Average) Analytical Solution Error (%)**  Iso\_06 0.7200 0.7200 0.0 Iso\_07 0.3557 0.3556 0.0281 Iso\_08 0.8343 0.8343 0.0

**Table 6.** Normalized tensile stress component (σxx/σoxx) within the isotropic spherical inclusion due

and matrix of Iso\_03, Iso\_04, Iso\_05, Iso\_07 and Iso\_08. Furthermore, for Iso\_02, Iso\_03, Iso\_04 and Iso\_05, Young's modulus (E) in the isotropic inclusion was greater than that in the isotropic matrix. For Iso\_06, Iso\_07 and Iso\_08, Young's modulus (E) in the isotropic matrix was greater than that in the isotropic inclusion. Thus, seven material properties representing a diversity of materials were chosen. Excellent agreement was found between the analytical and numerical solutions using the VIEM for the seven different materials considered. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. It should also be noted that the normalized tensile stress (σxx/σoxx) inside the isotropic spherical inclusions was found to be constant [1,30]. Tables 6–8 show that the percentage differences for the two sets of results are less than 0.1% in seven cases. Figure 8 shows numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 7) ≤ 360°) of the isotropic spherical inclusions with a radius of 6 mm under uniform

*Materials* **2021**, *14*, x FOR PEER REVIEW 13 of 44

remote tensile loading.

to uniform remote tensile loading (σoxx).

**Figure 7.** A typical discretized spherical model in the volume integral equation method (VIEM). (**a**) An inside view of a spherical model. (**b**) A spherical model. **Figure 7.** A typical discretized spherical model in the volume integral equation method (VIEM). (**a**) An inside view of a spherical model. (**b**) A spherical model.

**Table 6.** Normalized tensile stress component (σxx/σ o xx) within the isotropic spherical inclusion due to uniform remote tensile loading (σ o xx).


**Table 7.** Normalized tensile stress component (σxx/σ o xx) within the isotropic spherical inclusion due to uniform remote tensile loading (σ o xx).


**Figure 8.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic spherical inclusions with a radius of 6 mm under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08. **Figure 8.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic spherical inclusions with a radius of 6 mm under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

In most references, the numerical results for this problem were obtained in one direction. Thus, in order to show the VIEM results more thoroughly, the normalized tensile stress (σxx/σoxx) using the VIEM was presented along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 7) ≤ 360°) of the isotropic In most references, the numerical results for this problem were obtained in one direction. Thus, in order to show the VIEM results more thoroughly, the normalized tensile stress (σxx/σ o xx) using the VIEM was presented along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the isotropic spherical

spherical inclusions. It was determined in Figure 8 that the normalized tensile stress (σxx/σoxx) inside the isotropic spherical inclusions is constant in all directions considered.

consider a single orthotropic spherical inclusion with a radius of 6 mm in an infinite isotropic matrix subject to uniform remote tensile loading, σoxx, as shown in Figure 4a. It should be noted that the length of the radius can be arbitrarily chosen. In Figure 7, standinclusions. It was determined in Figure 8 that the normalized tensile stress (σxx/σ o xx) inside the isotropic spherical inclusions is constant in all directions considered.

#### 3.1.5. A Single Orthotropic Spherical Inclusion

In order to show the advantages of the volume integral equation method (VIEM), we consider a single orthotropic spherical inclusion with a radius of 6 mm in an infinite isotropic matrix subject to uniform remote tensile loading, σ o xx, as shown in Figure 4a. It should be noted that the length of the radius can be arbitrarily chosen. In Figure 7, standard 20-node quadratic hexahedral elements were used in the discretization [31]. The number of hexahedral elements was 7560, determined based on a convergence test. For this problem, in comparison to the boundary element method (BEM), since the VIEM is not sensitive to the anisotropy of the inclusions, it does not require use of the Green's function for the anisotropic inclusions. Moreover, as opposed to the standard FEM, where it is necessary to discretize the full domain, the orthotropic inclusion only needs to be discretized in the VIEM.

Five different material properties (Ort\_1, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) in Table 5 were used in the numerical calculation. As shown in Table 5, it was assumed that c<sup>11</sup> > c<sup>22</sup> = c<sup>33</sup> for five orthotropic inclusions. Additionally, c<sup>11</sup> of the inclusion in Ort\_03 > c<sup>11</sup> of the inclusion in Ort\_02 > c<sup>11</sup> of the inclusion in Ort\_01. Furthermore, c<sup>11</sup> of the inclusion in Ort\_04 < c<sup>11</sup> of the inclusion in Ort\_05 < c<sup>11</sup> of the inclusion in Ort\_01. Thus, five material properties representing a diversity of materials were chosen. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. Moreover, the normalized tensile stress (σxx/σ o xx) inside the orthotropic spherical inclusions was found to be constant [1,30]. Table 9 shows the numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) inside the orthotropic spherical inclusions. For the inclusions in Ort\_01, Ort\_02 and Ort\_03, the normalized tensile stress (σxx/σ o xx) inside the inclusion was greater than 1.0. However, for the inclusions in Ort\_04 and Ort\_05, the normalized tensile stress (σxx/σ o xx) inside the inclusion was less than 1.0. Figure 9 shows the numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the orthotropic spherical inclusions with a radius of 6 mm under uniform remote tensile loading. It was determined in Figure 9 that the normalized tensile stress (σxx/σ o xx) inside the orthotropic spherical inclusions is constant in all directions considered.


**Table 9.** Normalized tensile stress component (σxx/σ o xx) within the orthotropic spherical inclusion due to uniform remote tensile loading (σ o xx).

in the VIEM.

sidered.

ard 20-node quadratic hexahedral elements were used in the discretization [31]. The number of hexahedral elements was 7560, determined based on a convergence test. For this problem, in comparison to the boundary element method (BEM), since the VIEM is not sensitive to the anisotropy of the inclusions, it does not require use of the Green's function for the anisotropic inclusions. Moreover, as opposed to the standard FEM, where it is necessary to discretize the full domain, the orthotropic inclusion only needs to be discretized

Five different material properties (Ort\_1, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) in Table 5 were used in the numerical calculation. As shown in Table 5, it was assumed that c11 > c22 = c33 for five orthotropic inclusions. Additionally, c11 of the inclusion in Ort\_03 > c11 of the inclusion in Ort\_02 > c11 of the inclusion in Ort\_01. Furthermore, c11 of the inclusion in Ort\_04 < c11 of the inclusion in Ort\_05 < c11 of the inclusion in Ort\_01. Thus, five material properties representing a diversity of materials were chosen. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. Moreover, the normalized tensile stress (σxx/σoxx) inside the orthotropic spherical inclusions was found to be constant [1,30]. Table 9 shows the numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σoxx) inside the orthotropic spherical inclusions. For the inclusions in Ort\_01, Ort\_02 and Ort\_03, the normalized tensile stress (σxx/σoxx) inside the inclusion was greater than 1.0. However, for the inclusions in Ort\_04 and Ort\_05, the normalized tensile stress (σxx/σoxx) inside the inclusion was less than 1.0. Figure 9 shows the numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0° ≤ Ө (see Figure 7) ≤ 360°) of the orthotropic spherical inclusions with a radius of 6 mm under uniform remote tensile loading. It was determined in Figure 9 that the normalized tensile stress (σxx/σoxx) inside the orthotropic spherical inclusions is constant in all directions con-

**Figure 9.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic spherical inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a radius of 6 mm under uniform remote tensile loading. **Figure 9.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic spherical inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a radius of 6 mm under uniform remote tensile loading. In order to introduce the VIEM as a versatile numerical method, we considered a single isotropic/orthotropic spheroidal inclusion in an infinite isotropic matrix subject to uniform remote tensile loading, σoxx, as shown in Figure 4b,c. Figure 5 shows an orienta-

#### **Table 9.** Normalized tensile stress component (σxx/σoxx) within the orthotropic spherical inclusion *3.2. A Single Spheroidal Inclusion Problem under Uniform Remote Tensile Loading*

due to uniform remote tensile loading (σoxx). **Material VIEM (Average)**  Ort\_01 1.1520 Ort\_02 1.4536 In order to introduce the VIEM as a versatile numerical method, we considered a single isotropic/orthotropic spheroidal inclusion in an infinite isotropic matrix subject to uniform remote tensile loading, σ o xx, as shown in Figure 4b,c. Figure 5 shows an orientation of the spheroidal inclusion. 3.2.1. A Single Isotropic Prolate Spheroidal Inclusion Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (= 6 mm) can be arbitrarily chosen.

Ort\_03 1.5910

#### Ort\_04 0.5836 3.2.1. A Single Isotropic Prolate Spheroidal Inclusion Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b =

tion of the spheroidal inclusion.

Ort\_05 0.8129 Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (=6 mm) can be arbitrarily chosen. c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a con-

Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a convergence test. vergence test. Eight different isotropic inclusions (from Iso\_01 to Iso\_08) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized tensile stress (σxx/σoxx) inside the isotropic prolate spheroidal inclusions was found to be constant [1,30].

**Figure 10.** A typical discretized prolate spheroidal model (a/b = c/b = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of a prolate spheroidal model. (**b**) A prolate spheroidal model. **Figure 10.** A typical discretized prolate spheroidal model (a/b = c/b = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of a prolate spheroidal model. (**b**) A prolate spheroidal model.

**Figure 11.** A typical discretized prolate spheroidal model (a/b = c/b = 0.75) in the volume integral equation method (VIEM). (**a**) An inside view of a prolate spheroidal model. (**b**) A prolate spheroidal model. **Figure 11.** A typical discretized prolate spheroidal model (a/b = c/b = 0.75) in the volume integral equation method (VIEM). (**a**) An inside view of a prolate spheroidal model. (**b**) A prolate spheroidal model.

Tables 10–12 show numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) inside the isotropic prolate spheroidal inclusions. For the inclusions in Iso\_01, Iso\_02, Iso\_03, Iso\_04 and Iso\_05, the normalized tensile stress (σxx/σoxx) inside the inclusion was greater than 1.0. However, for the inclusions in Iso\_06, Iso\_07 and Iso\_08, the normalized tensile stress (σxx/σoxx) inside the inclusion was less than Eight different isotropic inclusions (from Iso\_01 to Iso\_08) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized tensile stress (σxx/σ o xx) inside the isotropic prolate spheroidal inclusions was found to be constant [1,30].

1.0. Figure 12 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 10) ≤ 360°) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under uniform remote tensile loading. Figure 13 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 11) ≤ 360°) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under uniform remote tensile loading. It was determined in Figures 12 and 13 that the normalized tensile stress (σxx/σoxx) inside the isotropic prolate spheroidal inclusions is constant in all directions considered. Tables 10–12 show numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) inside the isotropic prolate spheroidal inclusions. For the inclusions in Iso\_01, Iso\_02, Iso\_03, Iso\_04 and Iso\_05, the normalized tensile stress (σxx/σ o xx) inside the inclusion was greater than 1.0. However, for the inclusions in Iso\_06, Iso\_07 and Iso\_08, the normalized tensile stress (σxx/σ o xx) inside the inclusion was less than 1.0. Figure <sup>12</sup> shows numerical solutions by the volume integral equation method forthe normalized tensile stress (<sup>σ</sup>xx/<sup>σ</sup> o xx) along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 10) ≤ 360◦ ) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under uniform remote tensile loading. Figure 13 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 11) ≤ 360◦ ) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under uniform remote tensile loading. It was determined in Figures 12 and 13 that the normalized tensile stress (σxx/σ o xx) inside the isotropic prolate spheroidal inclusions is constant in all directions considered.

**Table 10.** Normalized tensile stress component (σxx/σ o xx) within the isotropic prolate spheroidal inclusion due to uniform remote tensile loading (σ o xx).



**Table 11.** Normalized tensile stress component (σxx/σ o xx) within the isotropic prolate spheroidal inclusion due to uniform remote tensile loading (σ o xx).

**Table 12.** Normalized tensile stress component (σxx/σ o xx) within the isotropic prolate spheroidal inclusion due to uniform remote tensile loading (σ o xx).


#### 3.2.2. A Single Orthotropic Prolate Spheroidal Inclusion

Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (=6 mm) can be arbitrarily chosen.

Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a convergence test.

Five different orthotropic inclusions (from Ort\_01 to Ort\_05) in Table 3 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized tensile stress (σxx/σ o xx) inside the orthotropic prolate spheroidal inclusions was found to be constant [1,30]. Table 13 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) inside the orthotropic prolate spheroidal inclusions. For the inclusions in Ort\_01, Ort\_02 and Ort\_03, the normalized tensile stress (σxx/σ o xx) inside the inclusion was greater than 1.0. However, for the inclusions in Ort\_04 and Iso\_05, the normalized tensile stress (σxx/σ o xx) inside the inclusion was less than 1.0. Figure 14 shows numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (left) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (right) the circumferential direction (0◦ ≤ θ (see Figure 10) ≤ 360◦ ) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under uniform remote tensile loading.

**Table 13.** Normalized Tensile Stress Component (σxx/σ o xx) within the Orthotropic Prolate Spheroidal Inclusion due to Uniform Remote Tensile Loading (σ o xx).


model.

the isotropic prolate spheroidal inclusions is constant in all directions considered.

 (**a**) (**b**)

**Figure 11.** A typical discretized prolate spheroidal model (a/b = c/b = 0.75) in the volume integral equation method (VIEM). (**a**) An inside view of a prolate spheroidal model. (**b**) A prolate spheroidal

Tables 10–12 show numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) inside the isotropic prolate spheroidal inclusions. For the inclusions in Iso\_01, Iso\_02, Iso\_03, Iso\_04 and Iso\_05, the normalized tensile stress (σxx/σoxx) inside the inclusion was greater than 1.0. However, for the inclusions in Iso\_06, Iso\_07 and Iso\_08, the normalized tensile stress (σxx/σoxx) inside the inclusion was less than 1.0. Figure 12 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 10) ≤ 360°) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under uniform remote tensile loading. Figure 13 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 11) ≤ 360°) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under uniform remote tensile loading. It was determined in Figures 12 and 13 that the normalized tensile stress (σxx/σoxx) inside

**Figure 12.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions with a/b = c/b = 0.5 (b = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08. **Figure 12.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**i**) the x– axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions with a/b = c/b = 0.5 (b = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

Figure 15 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (left) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (right) the circumferential direction (0◦ ≤ θ (see Figure 11) ≤ 360◦ ) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under uniform remote tensile loading. It was determined in Figures 14 and 15 that the normalized tensile stress (σxx/σ o xx) inside the orthotropic prolate spheroidal inclusions is constant in all directions considered.

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

(**b**)

(**a**)

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

**Figure 12.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions with a/b = c/b = 0.5 (b = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04

(**c**)

and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

(**b**)

**Figure 13.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions with a/b = c/b = 0.75 (b = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08. **Figure 13.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**i**) the x– axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions with a/b = c/b = 0.75 (b = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

#### **Table 10.** Normalized tensile stress component (σxx/σoxx) within the isotropic prolate spheroidal inclusion due to uniform remote tensile loading (σoxx). 3.2.3. A Single Isotropic Oblate Spheroidal Inclusion

**Material VIEM (Average) a/b = c/b = 0.5 (see Figure 5) a/b = c/b = 0.75 (see Figure 5)**  In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be noted that the length of a (=6 mm) can be arbitrarily chosen.

Iso\_01 1.4268 1.5028 Iso\_02 1.2177 1.2500 **Table 11.** Normalized tensile stress component (σxx/σoxx) within the isotropic prolate spheroidal inclusion due to uniform remote tensile loading (σoxx). **Material VIEM (Average)**  Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence test.

> Iso\_03 1.2374 1.2736 Iso\_04 1.4502 1.5330

**Table 12.** Normalized tensile stress component (σxx/σoxx) within the isotropic prolate spheroidal in-

Iso\_06 0.7613 0.7397 Iso\_07 0.4042 0.3780 Iso\_08 0.8610 0.8471

Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that

Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a con-

**a/b = c/b = 0.5 (See Figure 5) a/b = c/b = 0.75 (See Figure 5)** 

**VIEM (Average) a/b = c/b = 0.5 (See Figure 5) a/b = c/b = 0.75 (See Figure 5)** 

3.2.2. A Single Orthotropic Prolate Spheroidal Inclusion

the length of b (= 6 mm) can be arbitrarily chosen.

vergence test.

clusion due to uniform remote tensile loading (σoxx).

**Table 13.** Normalized Tensile Stress Component (σxx/σoxx) within the Orthotropic Prolate Spheroidal

Ort\_01 1.1244 1.1385 Ort\_02 1.3546 1.4038 Ort\_03 1.4519 1.5202 Ort\_04 0.6246 0.6027 Ort\_05 0.8375 0.8246

**a/b = c/b = 0.5 (See Figure 5) a/b = c/b = 0.75 (See Figure 5)** 

**Material VIEM (Average)** 

Five different orthotropic inclusions (from Ort\_01 to Ort\_05) in Table 3 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized tensile stress (σxx/σoxx) inside the orthotropic prolate spheroidal inclusions was found to be constant [1,30]. Table 13 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) inside the orthotropic prolate spheroidal inclusions. For the inclusions in Ort\_01, Ort\_02 and Ort\_03, the normalized tensile stress (σxx/σoxx) inside the inclusion was greater than 1.0. However, for the inclusions in Ort\_04 and Iso\_05, the normalized tensile stress (σxx/σoxx) inside the inclusion was less than 1.0. Figure 14 shows numerical solution by the volume integral equation method for the normalized tensile stress (σxx/σoxx) along (left) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (right) the circumferential direction (0° ≤ Ө (see Figure 10) ≤ 360°) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b

= 6 mm) under uniform remote tensile loading.

Inclusion due to Uniform Remote Tensile Loading (σoxx).

**Figure 14.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a/b = c/b = 0.5 (b = 6 mm) under uniform remote tensile loading. **Figure 14.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a/b = c/b = 0.5 (b = 6 mm) under uniform remote tensile loading. *Materials* **2021**, *14*, x FOR PEER REVIEW 21 of 44

**Figure 15.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a/b = c/b = 0.75 (b = 6 mm) under uniform remote tensile loading. **Figure 15.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a/b = c/b = 0.75 (b = 6 mm) under uniform remote tensile loading. inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence

**Figure 16.** A typical discretized oblate spheroidal model (b/a = c/a = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal model. **Figure 16.** A typical discretized oblate spheroidal model (b/a = c/a = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal model.

(**a**) (**b**) Eight different isotropic inclusions (from Iso\_01 to Iso\_08) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized tensile stress (σxx/σ o xx) inside the isotropic oblate spheroidal inclusions was found to be constant [1,30]. Tables 14–16 show numerical solutions by the volume integral equation method for the normalized tensile stress

**Figure 16.** A typical discretized oblate spheroidal model (b/a = c/a = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal

(**a**) (**b**)

(**a**) (**b**)

model.

tensile loading.

(σxx/σ o xx) inside the isotropic oblate spheroidal inclusions. For the inclusions in Iso\_01, Iso\_02, Iso\_03, Iso\_04 and Iso\_05, the normalized tensile stress (σxx/σ o xx) inside the inclusion was greater than 1.0. However, for the inclusions in Iso\_06, Iso\_07 and Iso\_08, the normalized tensile stress (σxx/σ o xx) inside the inclusion was less than 1.0. Figure 18 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 16) ≤ 360◦ ) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under uniform remote tensile loading. Figure 19 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 17) ≤ 360◦ ) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under uniform remote tensile loading. It was determined in Figures 18 and 19 that the normalized tensile stress (σxx/σ o xx) inside the isotropic oblate spheroidal inclusions is constant in all directions considered. ures 16 and 17. The number of elements, 7560, was determined based on a convergence test. (**a**) (**b**) **Figure 16.** A typical discretized oblate spheroidal model (b/a = c/a = 0.5) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal model.

 **Figure 15.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with a/b = c/b = 0.75 (b = 6 mm) under uniform remote

In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be

Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Fig-

*Materials* **2021**, *14*, x FOR PEER REVIEW 21 of 44

3.2.3. A Single Isotropic Oblate Spheroidal Inclusion

noted that the length of a (= 6 mm) can be arbitrarily chosen.

**Figure 17.** A typical discretized oblate spheroidal model (b/a = c/a = 0.75) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal model.

**Table 14.** Normalized tensile stress component (σxx/σ o xx) within the isotropic oblate spheroidal inclusion due to uniform remote tensile loading (σ o xx).


**Table 15.** Normalized tensile stress component (σxx/σ o xx) within the isotropic oblate spheroidal inclusion due to uniform remote tensile loading (σ o xx).


3.2.4. A Single Orthotropic Oblate Spheroidal Inclusion

In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be noted that the length of a (=6 mm) can be arbitrarily chosen.

model.

**Table 16.** Normalized tensile stress component (σxx/σ o xx) within the isotropic oblate spheroidal inclusion due to uniform remote tensile loading (σ o xx). (σxx/σoxx) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 16) ≤ 360°) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under uniform remote tensile loading. Figure 19 shows numerical

**Figure 17.** A typical discretized oblate spheroidal model (b/a = c/a = 0.75) in the volume integral equation method (VIEM). (**a**) An inside view of an oblate spheroidal model. (**b**) An oblate spheroidal

Eight different isotropic inclusions (from Iso\_01 to Iso\_08) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized tensile stress (σxx/σoxx) inside the isotropic oblate spheroidal inclusions was found to be constant [1,30]. Tables 14–16 show numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σoxx) inside the isotropic oblate spheroidal inclusions. For the inclusions in Iso\_01, Iso\_02, Iso\_03, Iso\_04 and Iso\_05, the normalized tensile stress (σxx/σoxx) inside the inclusion was greater than 1.0. However, for the inclusions in Iso\_06, Iso\_07 and Iso\_08, the normalized tensile stress (σxx/σoxx) inside the inclusion was less than 1.0. Figure 18 shows numerical solutions by the volume integral equation method for the normalized tensile stress

*Materials* **2021**, *14*, x FOR PEER REVIEW 22 of 44

**Figure 18.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08. **Figure 18.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**i**) the x– axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

(**c**)

(**a**)

(**b**)

quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence test. **Figure 18.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

(**i**) x–axis (**ii**) circumferential direction

(**c**)

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**Figure 19.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**i**) the x– axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions with b/a = c/a = 0.75 (a = 6 mm) under uniform remote tensile loading. (**a**) Iso\_01 and Iso\_02. (**b**) Iso\_03, Iso\_04 and Iso\_05. (**c**) Iso\_06, Iso\_07 and Iso\_08.

Five different orthotropic inclusions (from Ort\_01 to Ort\_05) in Table 3 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized tensile stress (σxx/σ o xx) inside the orthotropic oblate spheroidal inclusions was found to be constant [1,30]. Table 17 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) inside the orthotropic oblate pheroidal inclusions. For the inclusions in Ort\_01, Ort\_02 and Ort\_03, the normalized tensile stress (σxx/σ o xx) inside the inclusion was greater than 1.0. However, for the inclusions in Ort\_04 and Iso\_05, the normalized tensile stress (σxx/σ o xx) inside the

inclusion was less than 1.0. Figure 20 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0◦ ≤ θ (see Figure 16) ≤ 360◦ ) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under uniform remote tensile loading. Figure 21 shows numerical solutions by the volume integral equation method for the normalized tensile stress (σxx/σ o xx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0◦ ≤ θ (see Figure 17) ≤ 360◦ ) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under uniform remote tensile loading. It was determined in Figures 20 and 21 that the normalized tensile stress (σxx/σ o xx) inside the orthotropic oblate spheroidal inclusions is constant in all directions considered. *Materials* **2021**, *14*, x FOR PEER REVIEW 25 of 44 integral equation method for the normalized tensile stress (σxx/σoxx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0° ≤ θ (see Figure 17) ≤ 360°) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under uniform remote tensile loading. It was determined in Figures 20 and 21 that the normalized tensile stress (σxx/σoxx) inside the orthotropic oblate spheroidal inclusions is constant in all directions considered.

**Table 17.** Normalized tensile stress component (σxx/σ o xx) within the orthotropic oblate spheroidal inclusion due to uniform remote tensile loading (σ o xx). **Table 17.** Normalized tensile stress component (σxx/σoxx) within the orthotropic oblate spheroidal inclusion due to uniform remote tensile loading (σoxx).


**Figure 20.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading. **Figure 20.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading.

**Figure 21.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.75 (a = 6 mm) under uniform remote tensile loading. From Figures 8, 9, 12–15 and 18–21 and Tables 6–17, it was determined that if the inclusion is harder than the matrix, the normalized tensile stress (σxx/σ o xx) inside the inclusion is greater than 1.0. Additionally, the normalized tensile stress (σxx/σ o xx) inside the prolate spheroidal inclusion (a/b = c/b = 0.75) is greater than that inside the prolate spheroidal inclusion (a/b = c/b = 0.5). However, the normalized tensile stress (σxx/σ o xx) inside the oblate spheroidal inclusion (b/a = c/a = 0.5) is greater than that inside the oblate spheroidal inclusion (b/a = c/a = 0.75). Thus, the normalized tensile stress (σxx/σ o xx) inside the inclusion can be arranged in ascending order of magnitude: (1) prolate spheroidal inclusion (a/b = c/b = 0.5), (2) prolate spheroidal inclusion (a/b = c/b = 0.75), (3) sphere, (4) oblate spheroidal inclusion (b/a = c/a = 0.75) and (5) oblate spheroidal inclusion (b/a = c/a = 0.5). From Figures 8, 9, 12–15 and 18–21 and Tables 6–17, it was also determined that if the inclusion is softer than the matrix, the normalized tensile stress (σxx/σ o xx) inside the inclusion is less than 1.0. Additionally, the normalized tensile stress (σxx/σ o xx) inside the prolate spheroidal inclusion (a/b = c/b = 0.5) is greater than that inside the prolate spheroidal inclusion (a/b = c/b = 0.75). However, the normalized tensile stress (σxx/σ o xx) inside the oblate spheroidal inclusion (b/a = c/a = 0.75) is greater than that inside the

From Figures 8, 9, 12–15 and 18–21 and Tables 6–17, it was determined that if the inclusion is harder than the matrix, the normalized tensile stress (σxx/σoxx) inside the inclu-

oblate spheroidal inclusion (b/a = c/a = 0.5). Thus, the normalized tensile stress (σxx/σ o xx) inside the inclusion can be arranged in ascending order of magnitude: (1) oblate spheroidal inclusion (b/a = c/a = 0.5), (2) oblate spheroidal inclusion (b/a = c/a = 0.75), (3) sphere, (4) prolate spheroidal inclusion (a/b = c/b = 0.75) and (5) prolate spheroidal inclusion (a/b = c/b = 0.5). **Figure 20.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.5 (a = 6 mm) under uniform remote tensile loading.

*Materials* **2021**, *14*, x FOR PEER REVIEW 25 of 44

constant in all directions considered.

inclusion due to uniform remote tensile loading (σoxx).

integral equation method for the normalized tensile stress (σxx/σoxx) along (left) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (right) the circumferential direction (0° ≤ θ (see Figure 17) ≤ 360°) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under uniform remote tensile loading. It was determined in Figures 20 and 21 that the normalized tensile stress (σxx/σoxx) inside the orthotropic oblate spheroidal inclusions is

**Table 17.** Normalized tensile stress component (σxx/σoxx) within the orthotropic oblate spheroidal

Ort\_01 1.2292 1.1833 Ort\_02 1.7864 1.5780 Ort\_03 2.1040 1.7745 Ort\_04 0.5006 0.5453 Ort\_05 0.7570 0.7882

**b/a = c/a = 0.5 (See Figure 5) b/a = c/a = 0.75 (See Figure 5)** 

**Material VIEM (Average)** 

**Figure 21.** VIEM results for the normalized tensile stress component (σxx/σoxx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.75 (a = 6 mm) under uniform remote tensile loading. **Figure 21.** VIEM results for the normalized tensile stress component (σxx/σ o xx) along (**left**) the x–axis inside and (**right**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_01, Ort\_02, Ort\_03, Ort\_04 and Ort\_05) with b/a = c/a = 0.75 (a = 6 mm) under uniform remote tensile loading.

From Figures 8, 9, 12–15 and 18–21 and Tables 6–17, it was determined that if the inclusion is harder than the matrix, the normalized tensile stress (σxx/σoxx) inside the inclu-Both the standard finite element method (FEM) and the boundary element method (BEM) are powerful general-purpose tools in the field of numerical analysis. Since the VIEM is a combination of these two methods, it is also highly beneficial to the field of numerical analysis and can play a very important role in solving "inclusion problems". The authors hope that the results using the VIEM cited in this paper will be used as benchmarked data for verifying the results of similar research using other analytical and numerical methods.

#### *3.3. Single Spherical Inclusion Problems under Remote Shear Loading*

#### 3.3.1. VIEM Formulation Applied to Isotropic/Orthotropic Inclusion Problems

The displacements for isotropic spherical, prolate and oblate spheroidal inclusions can be determined from volume integral Equations (3)–(5), while the displacements for orthotropic spherical, prolate and oblate spheroidal inclusions can be determined from volume integral Equations (6)–(8).

#### 3.3.2. A Single Isotropic Spherical Inclusion

We considered a single isotropic spherical inclusion with a radius of 6 mm in an infinite isotropic matrix subject to remote shear loading, σ o xy, σ o xz and σ o yz, as shown in Figure 6a [24]. It should be noted that the length of the radius can be arbitrarily chosen. In Figure 7, standard 20-node quadratic hexahedral elements were used in the discretization [31]. The number of hexahedral elements, 7560, was determined based on a convergence test. Three different material properties (Iso\_01, Iso\_05 and Iso\_06) in Table 2 were used in the numerical calculation. It should be noted that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic spherical inclusions were found to be constant, respectively [1]. It should also be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. Table 18 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic spherical inclusions. For the inclusions in Iso\_01 and Iso\_05, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Iso\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively. Figure 22 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential

direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the isotropic spherical inclusions with a radius of 6 mm under remote shear loading.

**Table 18.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the isotropic spherical inclusion due to remote shear loading (σ o xy, σ o xz and σ o yz).


In most references, spherical inclusion problems under uniform remote tensile loading were considered. Thus, in order to show the VIEM results more thoroughly, the normalized shear stresses, (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz, using the VIEM were presented along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the isotropic spherical inclusions.

It was determined in Figure 22 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the single isotropic spherical inclusions are constant in all directions considered and are identical to each other. Since isotropic materials have an infinite number of planes of symmetry, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the single isotropic spherical inclusions turned out to be identical to each other.

#### 3.3.3. A Single Orthotropic Spherical Inclusion

In order to show the advantages of the volume integral equation method (VIEM), we considered a single orthotropic spherical inclusion with a radius of 6 mm in an infinite isotropic matrix subject to remote shear loading, σ o xy, σ o xz and σ o yz, as shown in Figure 6a. It should be noted that the length of the radius can be arbitrarily chosen. In Figure 7, standard 20-node quadratic hexahedral elements were used in the discretization [31]. The number of hexahedral elements was 7560, determined based on a convergence test. For this problem, in comparison to the boundary element method (BEM), since the VIEM is not sensitive to the anisotropy of the inclusions, it does not require the use of the Green's function for the anisotropic inclusions. Moreover, as opposed to the standard FEM, where it is necessary to discretize the full domain, the orthotropic inclusion only needs to be discretized in the VIEM.

Two different material properties (Ort\_06 and Ort\_07) in Table 4 were used in the numerical calculation [25]. As shown in Table 5, it was assumed that c<sup>55</sup> > c<sup>66</sup> > c<sup>44</sup> for two orthotropic inclusions. Additionally, c44, c<sup>55</sup> and c<sup>66</sup> of the inclusion were assumed be greater than µ of the matrix in the Ort\_06 material, while µ of the matrix was assumed to be greater than c44, c<sup>55</sup> and c<sup>66</sup> of the inclusion in the Ort\_07 material. Thus, two material properties representing different characteristics were chosen. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figure 7. Moreover, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic spherical inclusions were found to be constant, respectively [1]. Table 19 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic spherical inclusions. For the inclusion in Ort\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Ort\_07, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively. Figure 23 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 7) ≤ 360◦ ) of the orthotropic spherical inclusions with a radius of 6 mm under remote shear loading. It was determined in

Figure 23 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic spherical inclusions are constant in all directions considered and are different from each other. Since orthotropic materials have three planes/axes of symmetry and the independent shear moduli in three planes of symmetry are different from each other (c<sup>55</sup> > c<sup>66</sup> > c44), the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic spherical inclusions turned out to be different from each other. Furthermore, since c<sup>55</sup> (shear modulus in the xz plane) is greater than c<sup>66</sup> (shear modulus in the xy plane) and c<sup>66</sup> is greater than c<sup>44</sup> (shear modulus in the yz plane) in the orthotropic inclusions of the Ort\_06 and Ort\_07 materials, it was determined that the normalized shear stress, σxz/σ o xz, was greater than the normalized shear stress, σxy/σ o xy. Furthermore, σxy/σ o xy was found to be greater than the normalized shear stress, σyz/σ o yz, inside the orthotropic spherical inclusions. *Materials* **2021**, *14*, x FOR PEER REVIEW 27 of 44 Figure 7) ≤ 360°) of the isotropic spherical inclusions with a radius of 6 mm under remote shear loading.

**Figure 22.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic spherical inclusions (Iso\_01, Iso\_05 and Iso\_06) with a radius of 6 mm under remote shear loading (σoxy, σoxz and σoyz). **Figure 22.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic spherical inclusions (Iso\_01, Iso\_05 and Iso\_06) with a radius of 6 mm under remote shear loading (σ o xy, σ o xz and σ o yz).

**Table 18.** Normalized shear stress components (σxy/σoxy, σxz/σoxz and σyz/σoyz) within the isotropic

Iso\_01 1.7109 1.7109 1.7109 Iso\_05 1.9231 1.9231 1.9231 Iso\_06 0.6636 0.6636 0.6636

spherical inclusion due to remote shear loading (σoxy, σoxz and σoyz).

**Table 19.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the orthotropic spherical inclusion due to remote shear loading (σ o xy, σ o xz and σ o yz).

**Figure 23.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic spherical inclusions (Ort\_06 and Ort\_07) with a radius of 6 mm under remote shear loading (σoxy, σoxz and σoyz). **Figure 23.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic spherical inclusions (Ort\_06 and Ort\_07) with a radius of 6 mm under remote shear loading (σ o xy, σ o xz and σ o yz).

#### **Table 19.** Normalized shear stress components (σxy/σoxy, σxz/σoxz and σyz/σoyz) within the orthotropic spherical inclusion due to remote shear loading (σoxy, σoxz and σoyz). *3.4. A Single Spheroidal Inclusion Problem under Remote Shear Loading*

**Material VIEM (Average)**  In order to introduce the VIEM as a versatile numerical method, we considered a single isotropic/orthotropic spheroidal inclusion in an infinite isotropic matrix subject to

In order to introduce the VIEM as a versatile numerical method, we considered a single isotropic/orthotropic spheroidal inclusion in an infinite isotropic matrix subject to remote shear loading, σoxy, σoxz and σoyz, as shown in Figure 6b,c. Figure 5 shows the ori-

Ort\_06 1.4006 1.5456 1.3537 Ort\_07 0.4356 0.5576 0.4030

**σxy/σoxy σxz/σoxz σyz/σoyz**

entation of the spheroidal inclusion.

*3.4. A Single Spheroidal Inclusion Problem under Remote Shear Loading* 

remote shear loading, σ o xy, σ o xz and σ o yz, as shown in Figure 6b,c. Figure 5 shows the orientation of the spheroidal inclusion.

#### 3.4.1. A Single Isotropic Prolate Spheroidal Inclusion

Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (=6 mm) can be arbitrarily chosen.

Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a convergence test.

Three different isotropic inclusions (Iso\_01, Iso\_05 and Iso\_06) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic prolate spheroidal inclusions were found to be constant, respectively [1]. Table 20 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic prolate spheroidal inclusions. For the inclusions in Iso\_01 and Iso\_05, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Iso\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively. Figure 24 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 10) ≤ 360◦ ) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. Figure 25 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 11) ≤ 360◦ ) of the isotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. It was determined in Figures 24 and 25 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic prolate spheroidal inclusions are constant in all directions considered. Furthermore, since, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the yz plane in the prolate spheroidal inclusion, the normalized shear stress, σxy/σ o xy, was identical to the normalized shear stress, σyz/σ o yz, inside the isotropic prolate spheroidal inclusion under remote shear loading.

**Table 20.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the isotropic prolate spheroidal inclusion due remote shear loading (σ o xy, σ o xz and σ o yz).


**Figure 24.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with a/b = c/b = 0.5 (b = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 24.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with a/b = c/b = 0.5 (b = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

#### 3.4.2. A Single Orthotropic Prolate Spheroidal Inclusion

Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (=6 mm) can be arbitrarily chosen.

Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a convergence test.

**Figure 25.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with a/b = c/b = 0.75 (b = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 25.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with a/b = c/b = 0.75 (b = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

(**i**) A prolate spheroidal inclusion (**ii**) An oblate spheroidal inclusion (**a**) Cross-section in the xy plane Two different orthotropic inclusions (Ort\_06 and Ort\_07) in Table 4 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic prolate spheroidal inclusions were found to be constant, respectively [1]. Table 21 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic prolate spheroidal inclusions. For the inclusion in Ort\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Ort\_07, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively. Figure 27 shows numerical solutions by the volume integral equation method for the normalized shear stresses

(a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 10) ≤ 360◦ ) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. Figure 28 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 11) ≤ 360◦ ) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. It was determined in Figures 27 and 28 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic prolate spheroidal inclusions are constant in all directions considered. Furthermore, even though, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the yz plane in the prolate spheroidal inclusion, since c<sup>55</sup> (shear modulus in the xz plane) is greater than c<sup>66</sup> (shear modulus in the xy plane) and c<sup>66</sup> is greater than c<sup>44</sup> (shear modulus in the yz plane) in the orthotropic inclusions of the Ort\_06 and Ort\_07 materials, the normalized shear stress, σxy/σ o xy, was different from the normalized shear stress, σyz/σ o yz, inside the isotropic prolate spheroidal inclusion under remote shear loading.

**Table 21.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the orthotropic prolate spheroidal inclusion due remote shear loading (σ o xy, σ o xz and σ o yz).


#### 3.4.3. A Single Isotropic Oblate Spheroidal Inclusion

In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be noted that the length of a (=6 mm) can be arbitrarily chosen.

Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence test.

Three different isotropic inclusions (Iso\_01, Iso\_05 and Iso\_06) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic oblate spheroidal inclusions were found to be constant, respectively [1]. Table 22 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic oblate spheroidal inclusions. For the inclusions in Iso\_01 and Iso\_05, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Iso\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively.

loading (σoxy, σoxz and σoyz).

(**i**) A prolate spheroidal inclusion (**ii**) An oblate spheroidal inclusion (**a**) Cross-section in the xy plane

(**a**) Shear stress component (σxy/σoxy)

(**b**) Shear stress component (σxz/σoxz)

(**c**) Shear stress component (σyz/σoyz)

(**i**) x–axis (**ii**) circumferential direction

**Figure 25.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic prolate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with a/b = c/b = 0.75 (b = 6 mm) under remote shear

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

(**i**) A prolate spheroidal inclusion (**ii**) An oblate spheroidal inclusion

(**b**) Cross-section in the xz plane

 (**i**) A prolate spheroidal inclusion (**ii**) An oblate spheroidal inclusion (**c**) Cross-section in the yz plane

**Figure 26.** Cross-section in the (**a**) xy plane, (**b**) xz plane and (**c**) yz plane of (**i**) prolate spheroidal (with an aspect ratio of 0.5) and (**ii**) oblate spheroidal (with an aspect ratio of 0.5) inclusions under remote shear loading. **Figure 26.** Cross-section in the (**a**) xy plane, (**b**) xz plane and (**c**) yz plane of (**i**) prolate spheroidal (with an aspect ratio of 0.5) and (**ii**) oblate spheroidal (with an aspect ratio of 0.5) inclusions under remote shear loading.

3.4.2. A Single Orthotropic Prolate Spheroidal Inclusion Two different prolate spheroidal inclusions are considered: (a) a/b = c/b = 0.5, where b = 6 mm, and (b) a/b = c/b = 0.75, where b = 6 mm (see Figure 5). It should be noted that the length of b (= 6 mm) can be arbitrarily chosen. Figures 10 and 11 show a typical discretized model for the single (a) prolate spheroidal inclusion (a/b = c/b = 0.5 where b = 6 mm) and (b) prolate spheroidal inclusion (a/b = c/b = 0.75 where b = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single prolate spheroidal inclusion in Figures 10 and 11. The number of elements, 7560, was determined based on a convergence test. Two different orthotropic inclusions (Ort\_06 and Ort\_07) in Table 4 were used in the numerical calculation. It should be noted that the VIEM results represent average values Figure 29 shows numerical results using the volume integral equation method (VIEM) for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x– axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 16) ≤ 360◦ ) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. Figure 30 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 17) ≤ 360◦ ) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. It was determined in Figures 29 and 30 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic oblate spheroidal inclusions are constant in all directions considered. Furthermore, since, as shown in Figure 26, the cross-section in the xy

of the normalized stresses in all the nodes of the VIEM model in Figures 10 and 11. It should also be noted that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the orthotropic prolate spheroidal inclusions were found to be constant, respectively

normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the orthotropic prolate spheroidal inclusions. For the inclusion in Ort\_06, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Ort\_07, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion were less than 1.0, respectively. Figure 27 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−3 mm ≤ x ≤ 3 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 10) ≤ 360°) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.5 where b = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. Figure 28

plane is identical to the cross-section in the xz plane in the oblate spheroidal inclusion, the normalized shear stress, σxy/σ o xy, was identical to the normalized shear stress, σxz/σ o xz, inside the isotropic oblate spheroidal inclusion under remote shear loading. greater than c66 (shear modulus in the xy plane) and c66 is greater than c44 (shear modulus in the yz plane) in the orthotropic inclusions of the Ort\_06 and Ort\_07 materials, the normalized shear stress, σxy/σoxy, was different from the normalized shear stress, σyz/σoyz, inside the isotropic prolate spheroidal inclusion under remote shear loading.

shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−4.5 mm ≤ x ≤ 4.5 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 11) ≤ 360°) of the orthotropic prolate spheroidal inclusions (a/b = c/b = 0.75 where b = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. It was determined in Figures 27 and 28 that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the orthotropic prolate spheroidal inclusions are constant in all directions considered. Furthermore, even though, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the yz plane in the prolate spheroidal inclusion, since c55 (shear modulus in the xz plane) is

*Materials* **2021**, *14*, x FOR PEER REVIEW 34 of 44

**Figure 27.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic prolate **Figure 27.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_06 and Ort\_07) with a/b = c/b = 0.5 (b = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

(σoxy, σoxz and σoyz).

spheroidal inclusions (Ort\_06 and Ort\_07) with a/b = c/b = 0.5 (b = 6 mm) under remote shear loading

**Figure 28.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_06 and Ort\_07) with a/b = c/b = 0.75 (b = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 28.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic prolate spheroidal inclusions (Ort\_06 and Ort\_07) with a/b = c/b = 0.75 (b = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

3.4.3. A Single Isotropic Oblate Spheroidal Inclusion In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a **Table 22.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the isotropic oblate spheroidal inclusion due remote shear loading (σ o xy, σ o xz and σ o yz).


**Table 21.** Normalized shear stress components (σxy/σoxy, σxz/σoxz and σyz/σoyz) within the orthotropic

Ort\_06 1.4239 1.4192 1.3735 1.4180 1.4828 1.3685 Ort\_07 0.4258 0.6010 0.3934 0.4282 0.5774 0.3957

Three different isotropic inclusions (Iso\_01, Iso\_05 and Iso\_06) in Table 2 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the isotropic oblate spheroidal inclusions were found to be constant, respectively [1]. Table 22 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the isotropic oblate spheroidal inclusions. For the inclusions in Iso\_01 and Iso\_05, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Iso\_06, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside

Figure 29 shows numerical results using the volume integral equation method (VIEM) for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 16) ≤ 360°) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. Figure 30 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 17) ≤ 360°) of the isotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. It was determined in Figures 29 and 30 that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the isotropic oblate spheroidal inclusions are constant in all directions considered. Furthermore, since, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the xz plane in the oblate spheroidal inclusion, the normalized shear stress, σxy/σoxy, was identical to the normalized shear stress, σxz/σoxz, inside the

**VIEM (Average) a/b = c/b = 0.5 (See Figure 5) a/b = c/b = 0.75 (See Figure 5) σxy/σoxy σxz/σoxz σyz/σoyz σxy/σoxy σxz/σoxz σyz/σoyz**

prolate spheroidal inclusion due remote shear loading (σoxy, σoxz and σoyz).

the inclusion were less than 1.0, respectively.

isotropic oblate spheroidal inclusion under remote shear loading.

**Material** 

**Figure 29.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with b/a = c/a = 0.5 (a = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 29.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with b/a = c/a = 0.5 (a = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

#### 3.4.4. A Single Orthotropic Oblate Spheroidal Inclusion

In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be noted that the length of a (=6 mm) can be arbitrarily chosen. Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence test.

(**i**) x–axis (**ii**) circumferential direction

(**a**) Shear stress component (σxy/σoxy)

(**b**) Shear stress component (σxz/σoxz)

(**i**) x–axis (**ii**) circumferential direction

(**b**) Shear stress component (σxz/σoxz)

(**i**) x–axis (**ii**) circumferential direction

(**i**) x–axis (**ii**) circumferential direction

**Figure 29.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with b/a = c/a = 0.5 (a = 6 mm) under remote shear

(**c**) Shear stress component (σyz/σoyz)

loading (σoxy, σoxz and σoyz).

**Figure 30.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with b/a = c/a = 0.75 (a = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 30.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the isotropic oblate spheroidal inclusions (Iso\_01, Iso\_05 and Iso\_06) with b/a = c/a = 0.75 (a = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

3.4.4. A Single Orthotropic Oblate Spheroidal Inclusion In this section, two different oblate spheroidal inclusions are considered: (a) b/a = c/a = 0.5, where a = 6 mm, and (b) b/a = c/a = 0.75, where a = 6 mm (see Figure 5). It should be noted that the length of a (=6 mm) can be arbitrarily chosen. Figures 16 and 17 show a typical discretized model for the single (a) oblate spheroidal inclusion (b/a = c/a = 0.5 where a = 6 mm) and (b) oblate spheroidal inclusion (b/a = c/a = 0.75 where a = 6 mm) used in the VIEM [31], respectively. A total of 7560 standard 20-node quadratic hexahedral elements were used for the single oblate spheroidal inclusion in Figures 16 and 17. The number of elements, 7560, was determined based on a convergence test. Two different orthotropic inclusions (Ort\_06 and Ort\_07) in Table 4 were used in the numerical calculation. It should be noted that the VIEM results represent average values Two different orthotropic inclusions (Ort\_06 and Ort\_07) in Table 4 were used in the numerical calculation. It should be noted that the VIEM results represent average values of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic oblate spheroidal inclusions were found to be constant, respectively [1]. Table 23 shows numerical solutions by the volume integral equation method for the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic oblate spheroidal inclusions. For the inclusion in Ort\_06, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Ort\_07, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were less than 1.0, respectively. Figure 31 shows numer-

of the normalized stresses in all the nodes of the VIEM model in Figures 16 and 17. It should also be noted that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the orthotropic oblate spheroidal inclusions were found to be constant, respectively [1]. Ta-

inclusions. For the inclusion in Ort\_06, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion were greater than 1.0, respectively. However, for the inclusion in Ort\_07, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion were less than 1.0, respectively. Figure 31 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ Ө (see Figure 16) ≤ 360°) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. Figure 32 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σoxy, (b) σxz/σoxz and (c) σyz/σoyz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0° ≤ θ (see Figure 17) ≤ 360°) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under remote shear loading, σoxy, σoxz and σoyz. It was determined in Figures 31 and 32 that the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the orthotropic oblate spheroidal inclusions are constant in all directions considered. Furthermore, even though, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the xz plane in the oblate spheroidal inclusion, since c55 (shear modulus in the xz plane) is greater than c66 (shear modulus in the xy plane) and c66 is greater than c44 (shear modulus in the yz plane) in the orthotropic inclusions of the Ort\_06 and Ort\_07 materials, the normalized shear stress,

ical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 16) ≤ 360◦ ) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.5 where a = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. Figure 32 shows numerical solutions by the volume integral equation method for the normalized shear stresses (a) σxy/σ o xy, (b) σxz/σ o xz and (c) σyz/σ o yz along (i) the x–axis inside (−6 mm ≤ x ≤ 6 mm) and (ii) the circumferential direction (0◦ ≤ θ (see Figure 17) ≤ 360◦ ) of the orthotropic oblate spheroidal inclusions (b/a = c/a = 0.75 where a = 6 mm) under remote shear loading, σ o xy, σ o xz and σ o yz. It was determined in Figures 31 and 32 that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic oblate spheroidal inclusions are constant in all directions considered. Furthermore, even though, as shown in Figure 26, the cross-section in the xy plane is identical to the cross-section in the xz plane in the oblate spheroidal inclusion, since c<sup>55</sup> (shear modulus in the xz plane) is greater than c<sup>66</sup> (shear modulus in the xy plane) and c<sup>66</sup> is greater than c<sup>44</sup> (shear modulus in the yz plane) in the orthotropic inclusions of the Ort\_06 and Ort\_07 materials, the normalized shear stress, σxy/σ o xy, was different from the normalized shear stress, σxz/σ o xz, inside the orthotropic oblate spheroidal inclusion under remote shear loading.

**Table 23.** Normalized shear stress components (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) within the orthotropic oblate spheroidal inclusion due remote shear loading (σ o xy, σ o xz and σ o yz).


From Figures 22–25 and 27–32 and Tables 17–22, it was determined that if the inclusion is harder than the matrix, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion are greater than 1.0, respectively. It was also determined that if the inclusion is softer than the matrix, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion are less than 1.0, respectively.

From Figure 26, notable similarities are observed for isotropic inclusions. First, the cross-section in the xy plane of the isotropic prolate spheroidal inclusion is identical to the cross-section in the yz plane and is symmetrical to the cross-sections in the xy and xz planes of the isotropic oblate spheroidal inclusion. Second, the normalized shear stress, σxy/σ o xy, inside the isotropic prolate spheroidal inclusion is identical to both the normalized shear stress, σyz/σ o yz, inside the isotropic prolate spheroidal inclusion and the normalized shear stresses, σxy/σ o xy and σxz/σ o xz, inside the isotropic oblate spheroidal inclusion under remote shear loading. Third, the cross-section in the xz plane of the isotropic prolate spheroidal inclusion is symmetrical to the cross-section in the yz plane of the isotropic oblate spheroidal inclusion. Fourth, the normalized shear stress, σxz/σ o xz, inside the isotropic prolate spheroidal inclusion is identical to the normalized shear stress, σyz/σ o yz, inside the isotropic oblate spheroidal inclusion under remote shear loading.

In contrast, certain differences can be seen for orthotropic inclusions. First, although the cross-section in the xy plane of the orthotropic prolate spheroidal inclusion is still symmetrical to the cross-section in the xy plane of the orthotropic oblate spheroidal inclusion, it is no longer identical to the cross-section in the yz plane of the orthotropic prolate spheroidal inclusion. Second, since the cross-section in the xy plane of the orthotropic prolate spheroidal inclusion is no longer symmetrical to the cross-section in the xz plane of the orthotropic oblate spheroidal inclusion, the normalized shear stress, σxy/σ o xy, inside the orthotropic prolate spheroidal inclusion is only identical to the normalized shear stress, σxy/σ o xy, inside the orthotropic oblate spheroidal inclusion under remote shear loading.

Third, since the cross-section in the xz plane of the orthotropic prolate spheroidal inclusion is no longer symmetrical to the cross-section in the yz plane of the orthotropic oblate spheroidal inclusion, the normalized shear stress, σxz/σ o xz, inside the orthotropic prolate spheroidal inclusion is not identical to the normalized shear stress, σyz/σ o yz, inside the orthotropic oblate spheroidal inclusion under remote shear loading. *Materials* **2021**, *14*, x FOR PEER REVIEW 39 of 44 σxy/σoxy, was different from the normalized shear stress, σxz/σoxz, inside the orthotropic oblate spheroidal inclusion under remote shear loading.

**Figure 31.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_06 and Ort\_07) with b/a = c/a = 0.5 (a = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 31.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_06 and Ort\_07) with b/a = c/a = 0.5 (a = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

**Figure 32.** VIEM results for the normalized shear stress components (**a**) σxy/σoxy, (**b**) σxz/σoxz and (**c**) σyz/σoyz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_06 and Ort\_07) with b/a = c/a = 0.75 (a = 6 mm) under remote shear loading (σoxy, σoxz and σoyz). **Figure 32.** VIEM results for the normalized shear stress components (**a**) σxy/σ o xy, (**b**) σxz/σ o xz and (**c**) σyz/σ o yz along (**i**) the x–axis inside and (**ii**) the circumferential direction of the orthotropic oblate spheroidal inclusions (Ort\_06 and Ort\_07) with b/a = c/a = 0.75 (a = 6 mm) under remote shear loading (σ o xy, σ o xz and σ o yz).

From Figures 22–25 and 27–32 and Tables 17–22, it was determined that if the inclusion is harder than the matrix, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion are greater than 1.0, respectively. It was also determined that if the inclusion is softer than the matrix, the normalized shear stresses (σxy/σoxy, σxz/σoxz and σyz/σoyz) inside the inclusion are less than 1.0, respectively. It should be noted that, through numerical analysis using the volume integral equation method, we could quantitatively verify two qualitative predictions: (1) the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the orthotropic spherical inclusions are different from each other, and (2) for orthotropic spheroidal inclusions, there exists only one symmetrical cross-section when the remote loadings are shear (σ o xy, σ o xz and σ o yz).

From Figure 26, notable similarities are observed for isotropic inclusions. First, the cross-section in the xy plane of the isotropic prolate spheroidal inclusion is identical to the cross-section in the yz plane and is symmetrical to the cross-sections in the xy and xz planes of the isotropic oblate spheroidal inclusion. Second, the normalized shear stress, σxy/σoxy, inside the isotropic prolate spheroidal inclusion is identical to both the normal-It was determined that values of the normalized tensile stress (σxx/σ o xx) or the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic spheroidal inclusions differed significantly from those inside the orthotropic spheroidal inclusions. Therefore, thorough investigation of spheroidal inclusion problems requires stress analysis for both anisotropic spheroidal inclusion problems and isotropic spheroidal inclusion problems.

ized shear stress, σyz/σoyz, inside the isotropic prolate spheroidal inclusion and the normalized shear stresses, σxy/σoxy and σxz/σoxz, inside the isotropic oblate spheroidal inclusion under remote shear loading. Third, the cross-section in the xz plane of the isotropic prolate We also considered multiple isotropic/anisotropic spheroidal inclusions in an infinite isotropic matrix subject to uniform remote tensile loading, σ o xx. In a future paper, the

authors will introduce the VIEM solutions of multiple isotropic/orthotropic spheroidal inclusions in an infinite isotropic matrix under arbitrary loading conditions. It is obvious that general characteristics of multiple isotropic/anisotropic inclusion problems cannot be fully analyzed from the basic characteristics of the corresponding single or two isotropic/anisotropic inclusion problems. Therefore, applying multiple inclusion problems to a wide class of real composite materials and structures requires extending the analysis to multiple isotropic/anisotropic inclusions of different shapes.

Both the standard finite element method (FEM) and the boundary element method (BEM) are powerful general-purpose tools in the field of numerical analysis. Since the VIEM is a combination of these two methods, it is also highly beneficial to the field of numerical analysis and can play a very important role in solving "multiple inclusion problems". The authors hope that the results using the VIEM cited in this paper will be used as benchmarked data for verifying the results of similar research using other analytical and numerical methods.

#### **4. Conclusions**

In order to introduce the VIEM as a versatile numerical method for the three-dimensional elastostatic inclusion problem, it was applied to a class of three-dimensional elastostatic inclusion problems. We first considered single isotropic/orthotropic spherical, prolate (with an aspect ratio of 0.5 and 0.75) and oblate (with an aspect ratio of 0.5 and 0.75) spheroidal inclusions in an infinite isotropic matrix under uniform remote tensile loading. Thirteen inclusions with different characteristics were considered in the numerical calculation. Excellent agreement was found between the analytical and numerical solutions using the VIEM for single isotropic spherical inclusion problems. It was determined that the normalized tensile stress (σxx/σ o xx) inside the isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions was constant in two different directions (x–axis and circumferential direction). When the inclusion is harder than the matrix, the normalized tensile stress (σxx/σ o xx) inside the inclusion can be arranged in ascending order of magnitude: (1) prolate spheroidal inclusion (a/b = c/b = 0.5), (2) prolate spheroidal inclusion (a/b = c/b = 0.75), (3) sphere, (4) oblate spheroidal inclusion (b/a = c/a = 0.75) and (5) oblate spheroidal inclusion (b/a = c/a = 0.5).

We next considered single isotropic/orthotropic spherical, prolate (with an aspect ratio of 0.5 and 0.75) and oblate (with an aspect ratio of 0.5 and 0.75) spheroidal inclusions in an infinite isotropic matrix under remote shear loading. Five inclusions with different characteristics were considered in the numerical calculation. It was determined that the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the isotropic/orthotropic spherical, prolate and oblate spheroidal inclusions were constant in two different directions (x–axis and circumferential direction), respectively. When the inclusion was harder than the matrix, the normalized shear stresses (σxy/σ o xy, σxz/σ o xz and σyz/σ o yz) inside the inclusion were greater than 1.0, respectively. Furthermore, for isotropic spheroidal inclusions, there existed two identical or symmetrical cross-sections, while for orthotropic spheroidal inclusions, there existed only one symmetric cross-section when the remote loadings were shear (σ o xy, σ o xz and σ o yz).

It is the authors' hope that the present solutions for various types of inclusions with different material properties under different loading conditions using the parallel volume integral equation method will be established as reference values for verifying the results of other analytical and numerical methods.

It was also determined that applying multiple inclusion problems to a wide class of real composite materials and structures requires extending the analysis to multiple isotropic/anisotropic inclusions of different numbers and shapes. The parallel volume integral equation method (PVIEM) is now generally more applicable and executable than the standard finite element or boundary element methods. Subsequently, the PVIEM can be used to calculate other quantities of practical interest in realistic models of composites containing isotropic or anisotropic inclusions of arbitrary shapes under arbitrary loading conditions.

It should also be pointed out that, since the VIEM is a combination of the FEM and the BEM, it may have an unknown advantage that neither the FEM model nor the BEM model alone possess. For example, although certain VIEM models are incorrect from the point of view of the standard FEM only, they can be correctly implemented in the VIEM. In a future paper, the authors will attempt to provide more distinct examples to support this new finding. Finally, as a new machine learning-based predictive framework has been proposed for the accurate and efficient evaluation of singular integrals in the boundary element method (BEM) [32], of particular interest to researchers going forward will be the development of a general-purpose machine learning framework for predicting singular integrals [29] in the volume integral equation method.

**Author Contributions:** Conceptualization, J.L.; Methodology, J.L.; VIEM analysis, J.L.; Investigation, J.L. and M.H.; Validation, J.L. and M.H.; Software, J.L. and M.H.; Writing—original draft preparation, J.L.; FEM modeling, M.H.; Supervision, J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT, Republic of Korea (Grant No. 2015-R1A2A2A01004531), and the 2020 Hongik University Research Fund. The authors would also like to acknowledge the support of the National Supercomputing Center with supercomputing resources, including technical support (No. KSC-2020-CRE-0107).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available upon request from the corresponding author.

**Acknowledgments:** The authors would like to express their sincere appreciation to Kyunghun Lim for parallelizing and optimizing the three-dimensional VIEM code and Oh-Kyoung Kwon for applying the domain decomposition method to the parallel three-dimensional VIEM code.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


## *Article* **Simulation of PLC Effect Using Regularized Large-Strain Elasto-Plasticity**

**Marzena Mucha \*, Balbina Wcisło and Jerzy Pamin**

Chair for Computational Engineering, Faculty of Civil Engineering, Cracow University of Technology, Warszawska 24, 31-155 Cracow, Poland; balbina.wcislo@pk.edu.pl (B.W.); jerzy.pamin@pk.edu.pl (J.P.) **\*** Correspondence: marzena.mucha@pk.edu.pl

**Abstract:** The purpose of this paper is to develop a constitutive description and to numerically simulate a propagating instability phenomenon called the Portevin–Le Chatelier (PLC) effect, which is observed for metallic materials. It manifests itself by moving plastic shear bands in the sample and serrations in the stress–strain diagram. In this paper, the PLC is modeled by geometrically non-linear thermo-visco-plasticity with the hardening function of the Estrin–McCormick type to reproduce a serrated response. To regularize softening, which in this model comes from thermal, geometrical and strain-rate effects, the viscosity and heat conductivity are incorporated. Plasticity description can additionally include degradation of the yield strength, and then the model is enhanced by higherorder gradients. Simulations are performed using AceGen/FEM. Two tensioned specimens are tested: a rod and a dog-bone sample. The first specimen is used for general verification. The results obtained for the second specimen are compared with the experimental results. Studies for different values of model parameters are performed. The results of the simulations are in good agreement with the experimental outcome and the sensitivity to model parameters is in line with the expectations for the pre-peak regime. In the presented tests, the gradient enhancement does not significantly influence the results.

**Keywords:** PLC effect; visco-plasticity; thermo-mechanical coupling; gradient enhancement; FEM

### **1. Introduction**

The Portevin–Le Chatelier (PLC) effect is an instability phenomenon that manifests itself in bands of localized plastic strain rate, propagating along a stressed specimen. It is related to stress jumps (serrations) in the load-displacement diagram, which represents a specimen response under tension or shear. The source of this behaviour lies in the microstructure evolution, in particular at the level of dislocation motion. It is specifically exhibited by steel and aluminium alloys and occurs for a certain range of strain rates and temperatures. The PLC can reduce ductility and formability of alloys; hence, its analysis is of both theoretical and practical importance.

The plastic flow in metals and alloys can be explained by nucleation and motion of dislocations. The motion can be blocked by other dislocations, causing dislocation pile-up which can be unlocked by a sufficiently large strain. In solid solutions, dislocations can also be stopped by diffused solute atoms. The dislocation pinning by the solutes, repeatedly followed by unpinning, produces instabilities in the plastic flow. They occur as serrations in the stress–strain diagram, related to the motion (or repeated occurrence and vanishing) of localized strain-rate bands along a stressed specimen.

The micro-structural phenomenon responsible for the PLC effect is so-called Dynamic Strain Aging (DSA) [1,2]. As explained above, DSA is related to dynamic interactions between the motion of mobile dislocations and the diffusion of solute atoms. The recurring decrease in the concentration of solute atoms at temporarily arrested dislocations is represented by a reduction in solute contribution to the flow stress.

**Citation:** Mucha, M.; Wcisło, B.; Pamin, J. Simulation of PLC Effect Using Regularized Large-Strain Elasto-Plasticity. *Materials* **2022**, *15*, 4327. https://doi.org/10.3390/ ma15124327

Academic Editor: Raimondo Luciano, Carmelo Majorana, Angelo Marcello Tarantino and Michele Bacciocchi

Received: 12 May 2022 Accepted: 10 June 2022 Published: 18 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The PLC effect was first reported by [3]. Experimental studies and analytical models of propagative instabilities, in particular of Lueders bands and the Portevin–Le Chatelier effect, were discussed in [4]. An extensive numerical study of the phenomena using smallstrain isothermal elasto-plasticity models is provided in [5,6]. Finite element models of the PLC effect are analyzed there in the context of regularized dynamics. An overview of the experiments showing the PLC phenomenon, including a classification of its types, is presented in [7]. Experimental analysis and modeling of the three types is performed in [8]. A review of modeling options for the phenomenon is provided in [9].

The PLC effect can be described, among others, by the Estrin–McCormick model [10–13]. In [10], the model is derived; in [11,12], it is applied in small-strain FE simulations of the phenomenon. In [13], the model is implemented within a large strain model of elastoplasticity, including parameter identification for an aluminium alloy based on experiments on tensile specimens under loading with different rates.

In recent years, several scientists have considered the phenomenon in their theoretical, experimental and numerical studies. In particular, steel specimens were examined in [14–18] (the last paper covers an experimental study of the PLC phenomenon in highstrength steel) and aluminium alloys in a larger number of works; for instance, [13,19–22].

The majority of specimens used in the studies were rectangular or dog-bone shape tensile plates (see [8,13,15,19,20,23]), sometimes notched (e.g., [17]), and tensile rods with circular cross-section (see for instance [23]). Some studies concerned shear specimens [21,24,25]. Several papers compare the experimental response (in some cases monitored using the DIC technique) with simulation results; see, for instance [17,19,22,26–28]. The final failure, involving necking and fracture, was examined in [15]. Moreover, in some papers, the influence of loading/strain rate has been examined, see [8,13,20,24,29].

A few studies considered temperature dependence of the PLC effect. These were [20,25,30,31]. The simulation of the temperature-dependent process zone at the crack tip was in the focus of [14]. Finally, it is mentioned that a constitutive model of discontinuous plastic flow for materials deformed at cryogenic temperatures was developed in [32] and further considered in [33]. In fact, these papers and the present one belong to the broad field of research on thermo-mechanics of heterogeneous/composite materials and structures. The thermo-mechanical couplings are constantly a subject of intensive scientific activity; see, for instance [34–41].

In the present paper, the PLC effect is simulated using a formulation of geometrically non-linear thermo-plasticity developed in [42,43]. The model includes full thermomechanical coupling involving thermal expansion, plastic heating, thermal softening in the plasticity function, and Fourier's law in the deformed configuration. Following [44,45], the thermo-elastic coupling is neglected in the energy balance because it is relatively insignificant for the metallic materials under consideration.

The plasticity description is based on [46] and includes the Huber–Mises–Hencky yield criterion. This model was recently extended to visco-plasticity and was employed by the authors to simulate Lueders bands in [47]. The rate-dependent hardening function of the Estrin–McCormick type, described in [13], is used to simulate DSA and the serrations induced by strain-rate softening, but it is enhanced with temperature dependence of the characteristic time of solute diffusion.

The models are implemented in an AceGen code generator developed in [48] within Wolfram Mathematica. One of the aims of the research is to examine the influence of temperature and strain rates on the PLC phenomenon, and the importance of regularization in the models involving recurring strain-rate-softening phases. It is stressed here that the stabilizing effect is provided by viscosity and heat conduction, but can also be provided by a gradient enhancement.

The paper is organized as follows. In Section 2, the theory of large strain thermo-viscoplasticity is summarized and then extended to include the Estrin–McCormick component of the yield strength, which makes it possible to simulate the PLC effect. Interest is limited to tension and moderate temperatures (20–200 degrees Celsius). A gradient enhancement

of the model is optionally incorporated to regularize the softening involved in the model. In Section 3 the coupled balance equations are presented in local and weak forms, and then the implementation of the model in AceGen/FEM for Mathematica is briefly discussed. In Section 4, the simulation results are presented. First, a one-dimensional rod model is considered and detailed response is shown for one serration. Then, a series of simulations for a dog-bone specimen under tension is presented. The computed model is based on the experimental research on aluminium dog-bone-type specimens, presented in [49]. Comparisons with laboratory test results are made, and some parametric studies are performed. Finally, in Section 5 some conclusions are drawn and directions of future work are proposed.

#### **2. Brief Description of Constitutive Models**

### *2.1. Thermo-Visco-Plasticity*

The material models used in this paper for the simulation of the PLC-type instabilities are described below. They are based on the large-strain description of elasto-plasticity [13,45,50,51].

The starting point of the formulation is standard. We consider a continuous deformable body and its material is assumed to be initially isotropic. Vector **X** identifies the reference location of a body particle at time *t* = 0 and in initial temperature *T*<sup>0</sup> (*T*<sup>0</sup> is assumed to be the reference temperature for a strain-free state), vector **x** points to the current position of the particle at time *t* and in temperature *T*. The motion of the body is described by function **x** = *ϕ*(**X**, *t*, *T*). The classical definition of the deformation gradient **F** is recalled:

$$\mathbf{F} = \frac{\partial \boldsymbol{\varrho}(\mathbf{X}, t, T)}{\partial \mathbf{X}}.\tag{1}$$

A multiplicative decomposition of **F** into mechanical and thermal parts denoted by **F** *<sup>m</sup>* and **F** *θ* , respectively, is performed [45,52]. In turn, **F** *<sup>m</sup>* is decomposed into elastic and plastic factors **F** *<sup>e</sup>* and **F** *p* (see [53–55])

$$\mathbf{F} = \mathbf{F}^{m}\mathbf{F}^{\theta} = \mathbf{F}^{e}\mathbf{F}^{p}\mathbf{F}^{\theta}.\tag{2}$$

The thermal factor **F** *θ* is assumed to be purely volumetric and defined as

$$\mathbf{F}^{\theta} = (f^{\theta})^{1/3} \mathbf{I}, \qquad f^{\theta} = \det(\mathbf{F}^{\theta}), \tag{3}$$

where **I** is the second-order identity tensor. The volumetric deformation caused by the temperature change *T* − *T*<sup>0</sup> is represented by [55]

$$J^{\theta} = \exp[\Im \alpha\_T (T - T\_0)],\tag{4}$$

where *α<sup>T</sup>* is the coefficient of linear thermal expansion. Based on Equations (2) and (4) the mechanical part of the deformation gradient is derived as

$$\mathbf{F}^{m} = \exp[-\boldsymbol{\alpha}\_{T}(T - T\_{0})]\mathbf{F}.\tag{5}$$

For the classical thermo-plasticity theory, the Helmholtz potential calculated per unit volume in the reference configuration is decomposed into elastic, plastic, and purely thermal components (see [45,50])

$$
\psi(\mathbf{b}^{\varepsilon}, \boldsymbol{\alpha}, T) = \psi^{\varepsilon}(\mathbf{b}^{\varepsilon}) + \psi^{p}(\boldsymbol{\alpha}) + \psi^{\theta}(T). \tag{6}
$$

The following definitions of the potential parts are employed.

$$\psi^{\varepsilon}(\mathbf{b}^{\varepsilon}) = \frac{1}{2}G\left[\text{tr}(\det(\mathbf{b}^{\varepsilon})^{-1/3}\mathbf{b}^{\varepsilon}) - 3\right] + \frac{1}{2}\text{Kln}(f^{\varepsilon})^2,\tag{7}$$

$$\psi^p(\alpha) = (\sigma\_{yf} - \sigma\_{y0}) \left[ \alpha + \frac{\exp(-\delta\alpha)}{\delta} \right],\tag{8}$$

$$
\psi^{\theta}(T) = c \left[ (T - T\_0) - T \ln \left( \frac{T}{T\_0} \right) \right]. \tag{9}
$$

In Equation (7), *G* and *K* are shear and bulk elastic moduli, **b** *<sup>e</sup>* = **F** *e* (**F** *e* ) T is the elastic left Cauchy–Green tensor, and *J <sup>e</sup>* = det(**F** *e* ).

The second component of the free energy represents plastic hardening and is assumed in the form relevant for saturation-type hardening with a scalar equivalent plastic strain *α*, *σy*<sup>0</sup> is an initial yield strength, *σy f* is a final yield strength, and *δ* is a saturation constant. In general, *ψ <sup>p</sup>* depends on adopted hardening specification. Moreover, in Equation (9), *c* is the heat capacity per unit of volume. According to [46] it can be defined as *c* = −*T ∂* 2*ψ <sup>∂</sup>T*<sup>2</sup> and therefore for the adopted form of free energy, *c* is constant.

The Kirchhoff stress tensor *τ* and hardening function *h*(*α*) are derived from the free energy potential

$$
\pi = 2 \frac{\partial \psi}{\partial \mathbf{b}^{\varepsilon}} \mathbf{b}^{\varepsilon}, \qquad h = \frac{\partial \psi}{\partial \alpha}. \tag{10}
$$

The constitutive relation for heat conduction is the classical Fourier law for isotropic materials. It is formulated according to [46] using the Kirchhoff heat flux vector **q**

$$\mathbf{q} = -k\nabla T\_{\prime} \tag{11}$$

where *k* is a heat conduction coefficient specified in the reference configuration and ∇*T* is a spatial gradient of temperature.

Further, the plasticity formulation is specified. The yield function is defined as

$$F\_p(\boldsymbol{\pi}, \boldsymbol{\alpha}, \boldsymbol{\aleph}) = f(\boldsymbol{\pi}) - \sigma\_y(\boldsymbol{\alpha}, \boldsymbol{\aleph}) \le 0,\tag{12}$$

where *f*(*τ*) is the Huber–Mises–Hencky (HMH) stress measure and *σ<sup>y</sup>* represents the evolving yield strength (flow stress) for the rate-dependent (viscoplastic) model, which is the starting point of the derivation. The viscoplasticy formulation follows the consistency concept, cf. [5]. The particular forms of *σ<sup>y</sup>* will be discussed in the next section for the Estrin–McCormick visco-plasticity model and a gradient-enhanced version of the model. The following definitions are used

*f*(*τ*) = p 2*J*2, (13)

$$\mathbf{J}\_2 = \frac{1}{2} \mathbf{\tau}\_{dev}^2 \cdot \mathbf{I},\tag{14}$$

where *τdev* is deviatoric part of the Kirchhoff stress tensor and **I** is the second order unit tensor.

The yield function presented in Equation (12) has a general form which can easily be modified to apply another yield criterion. The Huber–Mises–Hencky function is chosen in the work because it describes the behaviour of metals satisfactorily. This form of stress measure is independent of the hydrostatic pressure, which implies the isochoric plastic flow. In this approach, the volumetric-deviatoric split of large-strain measures does not need to be incorporated in the description. It is worth mentioning that the volume of the material can change due to thermal expansion and elastic deformation.

Following [50], the associated flow rule is adopted for the Lie derivative of **b** *e*

$$-\frac{1}{2}\mathcal{L}\_{\upsilon}\mathbf{b}^{\varepsilon} = \dot{\gamma}\frac{\partial F\_p}{\partial \mathbf{r}}\mathbf{b}^{\varepsilon},\tag{15}$$

where *γ*˙ denotes the plastic multiplier satisfying the standard Kuhn–Tucker conditions:

$$
\gamma \ge 0, \qquad F\_p \le 0, \qquad \gamma F\_p = 0. \tag{16}
$$

According to [56], the plastic multiplier plays the role of the plastic strain measure according to the relation

$$
\hbar = \sqrt{2/3}\hat{\gamma}.\tag{17}
$$

#### *2.2. Estrin–McCormick Model with Optional Gradient Enhancement*

The Estrin–McCormick model (further called the McCormick model or the EMC model in brief) is a phenomenological description of DSA based on an internal variable called effective strain aging time *ta*, cf. [10,12,13]. The evolution of *t<sup>a</sup>* introduces repeated negative strain-rate dependence, causing the serrations related to nucleating and propagating localization bands.

In this paper, the model is extended to include temperature dependence, cf. [9,25]. The yield function is defined as

$$F\_p(\tau, \mathfrak{a}, \mathfrak{k}, t\_{\mathfrak{a}}, T) = f(\tau) - \sqrt{2/3} \sigma\_y(\mathfrak{a}, \mathfrak{k}, t\_{\mathfrak{a}}, T) \le 0. \tag{18}$$

It is assumed for simplicity that the standard hardening has a saturation character, viscosity does not depend on temperature, and thermal softening is linear. The yield strength *σ<sup>y</sup>* depends on equivalent plastic strain *α*, its rate *α*˙ , strain aging time *ta*, and temperature *T*. It has three components related to strain hardening *σH*, strain rate sensitivity *σV*, and dynamic strain aging *σB*:

$$
\sigma\_y(\mathfrak{a}, \mathfrak{a}, t\_{a\prime}T) = \sigma\_H(\mathfrak{a}, T) + \sigma\_V(\mathfrak{a}) + \sigma\_B(\mathfrak{a}, t\_{a\prime}T). \tag{19}
$$

The first component represents the saturation hardening scaled by a thermal softening factor

$$\sigma\_H(\mathfrak{a}) = \left[\sigma\_{y0} + (\sigma\_{yf} - \sigma\_{y0})(1 - \exp\left(-\delta\mathfrak{a}\right))\right][1 - H\_T(T - T\_0)].\tag{20}$$

The part (1 − *HT*(*T* − *T*0)) corresponds to linear thermal softening, and *H<sup>T</sup>* is a thermal softening modulus. Alternative formulae for thermal softening and their discussion in the context of strain localization simulations can be found in [57].

The second component of the yield strength introduces positive strain rate influence (*ξ* is viscosity parameter)

$$
\sigma\_V(\dot{\mathfrak{a}}) = \mathfrak{F}\dot{\mathfrak{a}}.\tag{21}
$$

It is noted that the time derivative of *α* in the viscous term *ξα*˙ in Equation (21) is computed using the backward Euler scheme as follows

$$
\hbar = \frac{\mathfrak{a}\_{\mathfrak{n}} - \mathfrak{a}\_{\mathfrak{n}-1}}{\Delta t},
\tag{22}
$$

where *α<sup>n</sup>* and *αn*−<sup>1</sup> denote the values of the equivalent plastic strain at the current and previous time moments, respectively, and ∆*t* is a time step.

The third component represents the influence of the DSA according to [13]

$$
\sigma\_B(t\_{a\prime}\alpha) = \sigma\_{B0}(a) \left[ 1 - \exp\left(-\frac{t\_a}{t\_0}\right)^n \right]. \tag{23}
$$

The formula particularly expresses the solute concentration at temporarily stopped mobile dislocations, which involves negative strain-rate sensitivity. It is driven by the strain aging time *ta*. Moreover, *t*<sup>0</sup> is the characteristic time for the solute diffusion, which determines how fast the saturation of hardening component *σ<sup>B</sup>* is reached. The saturation factor *σB*<sup>0</sup> (the maximum value of contribution *σB*) is assumed to depend linearly on the accumulated plastic strain measure *α*:

$$
\sigma\_{B0} = \sigma\_{B00} + \sigma\_{B00}' a\_{\prime} \tag{24}
$$

where *σB*<sup>00</sup> and *σ* 0 *<sup>B</sup>*<sup>00</sup> are model parameters as well as exponent *n* in Equation (23).

The strain aging time *t<sup>a</sup>* is related to a waiting time *t<sup>w</sup>* (which a dislocation spends at an obstacle) by the differential equation:

$$t\_a = 1 - \frac{t\_a}{t\_w} \,\prime \tag{25}$$

where the waiting time is related to the plastic strain rate *α*˙

$$t\_w = \frac{\Omega(\alpha)}{\hbar},\tag{26}$$

and the plastic strain increment Ω associated with the motion of dislocations between two obstacles (pinned configurations) is also assumed to depend linearly on *α*:

$$
\Omega(\alpha) = \Omega\_0 + \Omega\_0'\alpha. \tag{27}
$$

In the above equation Ω<sup>0</sup> and Ω0 0 are model parameters.

Algorithmically, the evolution of *t<sup>a</sup>* depends on the plastic strain increment ∆*α* and can be computed for time increments as follows [13]:

$$t\_a = \frac{t\_a^n + \Delta t}{1 + \frac{\Delta a}{\Omega (a^n + \Delta a)}} \text{ } \tag{28}$$

where the plastic strain rate *α*˙ has been approximated according to Equation (22).

Now, the model depends on temperature in a couple of ways: due to thermal expansion, plastic heating and thermal softening. It is assumed that only the basic mechanical parameters (Young modulus, initial and final yield strength) depend on temperature. Additionally, to examine the sensitivity of the McCormick model to temperature, the following dependence of parameter *t*<sup>0</sup> on temperature is assumed:

$$t\_0(T) = t\_{02} \exp(t\_{01}T). \tag{29}$$

The parameters of this function *t*<sup>01</sup> and *t*<sup>02</sup> have been determined on the basis of experimental results presented in [25].

It is emphasized that, next to thermal softening assumed in Equation (20) and strainrate softening present in Equation (23), geometrical softening due to large deformations is also present in the description; see, for instance, [58]. This version of the McCormick model incorporates two regularizing effects, i.e., rate dependence and heat conduction.

In more detailed material modeling, the first component of the yield strength can additionally include a damage-type reduction to represent an increasing porosity of the material related to large strains and leading to fracture. This extension of the model is here based on [51] where a reducing factor exp(−*βz*) decays from one to zero with increasing material degradation (*β* is a ductility parameter) and scales the hardening part of the yield strength. In the local version of the model, *z* would be taken equal to *α*; however, to control the influence of the degradation coefficient on the plastic strain localization process *z* is rather an averaged plastic strain measure, obtained from the following averaging equation [59]

$$z - l^2 \nabla\_0^2 \mathbf{z} = \mathbf{a}\_\prime \tag{30}$$

in which *l* is an internal length scale and, since so-called Lagrange averaging is employed according to [60], ∇<sup>0</sup> is the gradient operator in the material description. Homogeneous natural boundary conditions are assumed for Equation (30).

The yield function then reads

$$F\_p(\tau, \mathfrak{a}, \mathfrak{h}, z, \mathfrak{t}\_a, T) = f(\tau) - \sqrt{2/3} \sigma\_y(\mathfrak{a}, \mathfrak{h}, z, \mathfrak{t}\_a, T) \le 0,\tag{31}$$

and the flow stress depends on equivalent plastic strain *α*, its rate *α*˙ , (non-local) degradation parameter *z*, strain aging time *ta*, and temperature *T*

$$
\sigma\_y(\mathfrak{a}, \mathfrak{a}, z, \mathfrak{t}\_\mathfrak{a}, \mathbf{T}) = \sigma\_H(\mathfrak{a}, z, \mathbf{T}) + \sigma\_V(\mathfrak{a}) + \sigma\_B(\mathfrak{a}, \mathfrak{t}\_\mathfrak{a}, \mathbf{T}), \tag{32}
$$

where

$$\sigma\_H(\mathfrak{a}) = \left[ (\sigma\_{y0} + (\sigma\_{yf} - \sigma\_{y0})(1 - \exp\left(-\delta\mathfrak{a}\right)) \right] \exp(-\beta z) [1 - H\_T(T - T\_0)],\tag{33}$$

represents saturation hardening scaled by the degradation coefficient exp(−*βz*) and by the linear thermal softening factor 1 − *HT*(*T* − *T*0). This last version of the model thus incorporates the rate and gradient dependence simultaneously; cf., for instance, [61,62].

#### **3. Balance Equations**

Due to the distinction between the reference and the current configurations in the large strain analysis, the governing equations can be formulated in the material or spatial description; see, for instance, [46,63], respectively. In the described model, spatial quantities are used, but they refer to the volume or surface in the reference configuration; see [64].

The first governing equation for the analyzed coupled problem imposes static equilibrium in the local form

$$\text{Jdv}(\mathfrak{r}/I) = \mathbf{0}.\tag{34}$$

In Equation (34) div(·) is the divergence computed with respect to spatial coordinates and body forces have been neglected. The equilibrium Equation (34) is completed with the boundary conditions for displacement vector **u** and for traction vector **t**:

$$\begin{aligned} \mathbf{u} &= \hat{\mathbf{u}} \qquad \text{on} \qquad \partial \mathcal{B}\_{\mu \nu} \\ \mathbf{t} &= \boldsymbol{\tau} \cdot \mathbf{n} = \hat{\mathbf{t}} \qquad \text{on} \qquad \mathcal{g}(\partial \mathcal{B}\_{\tau}) \, \end{aligned} \tag{35}$$

where **n** is the normal to the body surface.

The second governing equation represents the energy balance written in the temperature form for a non-stationary heat transport, as follows.

$$c\frac{\partial T}{\partial t} - f \text{div}(-\mathbf{q}/\mathbf{J}) - \mathcal{R} = 0. \tag{36}$$

In Equation (36), R is a heat source density per unit of volume. It includes so-called plastic heating, i.e., the source density due to plastic dissipation written in the simple form [45]

$$\mathcal{R} = \chi \sigma\_y \hbar,\tag{37}$$

where *χ* denotes a heat dissipation factor [65], assumed to be constant. The energy balance Equation (36) is completed with appropriate boundary conditions:

$$\begin{aligned} T &= \hat{T} \qquad \text{on} \qquad \partial \mathcal{B}\_{T\prime} \\ \mathbf{q} \cdot \mathbf{n} &= \hat{q} \qquad \text{on} \qquad \mathfrak{g}(\partial \mathcal{B}\_{\mathfrak{q}})\_{\prime} \end{aligned} \tag{38}$$

and an initial condition stating that for *t* = 0 we have *T* = *T*<sup>0</sup> in B.

For the gradient-enhanced version of the model, the averaging Equation (30) is an additional balance equation. The averaged plastic strain *z* is an additional fundamental unknown, discretized in addition to displacements and temperature, leading to a three-field formulation of the coupled BVP.

The weak forms of the governing equations are the basis for the finite element implementation. Multiplication of Equation (34) by test function *δ***u**, integration over the

volume of body B and application of the divergence theorem as well as Neumann boundary conditions lead to the weak form of the linear momentum balance

$$\int\_{\mathcal{B}} (\nabla \delta \mathbf{u} : \mathbf{r}) \mathrm{d}V + \int\_{\mathcal{q}(\partial \mathcal{B}\_{\mathrm{r}})} \delta \mathbf{u} \cdot \mathbf{f} \mathrm{d}a = 0. \tag{39}$$

The weak form of Equation (36) is obtained using the standard procedure and the backward Euler scheme for time integration. As a result, the following integral equation is required to be valid for the current time

$$\int\_{\mathcal{B}} \left[ \delta T \frac{c}{\Delta t} (T - T\_n) + k \nabla \delta T \cdot \nabla T - \delta T \mathcal{R} \right] dV + \int\_{\mathfrak{P}(\partial \mathcal{B}\_q)} \delta T \mathfrak{d}q \mathbf{d}a = 0,\tag{40}$$

where *T<sup>n</sup>* is the value of temperature at the previous time moment and ∆*t* is the time increment. Finally, the weak form of Equation (30) is written as follows

$$\int\_{\mathcal{B}} \left[ \delta z (z - a) + l^2 \nabla\_0 \delta z \cdot \nabla\_0 z \right] dV = 0. \tag{41}$$

Equations (39), (40), and optionally (41) are required to be valid for any admissible weighting functions *δ***u**, *δT* and *δz*, respectively. After the introduction of finite element approximations of the two or three fundamental unknowns according to the Galerkin approach, a set of algebraic equations can be obtained for a monolithic solution algorithm.

#### **4. Implementation and Computational Tool**

The numerical implementation and testing of the coupled model is performed in Wolfram Mathematica packages AceGen and AceFEM, developed by Korelc [66]. The first package is used to program user-supplied procedures for the finite element method, in particular the tangent and residual subroutine for the Newton-Raphson algorithm and the postprocessing subroutine. The code is prepared in a special meta-language and automatically translated by AceGen. The routine can then be transferred to a chosen finite element environment (e.g., ABAQUS, FEAP), but an integrated FE engine AceFEM can also be used, and this is the case here. AceFEM is equipped with convenient pre- and post-processing tools and perfectly cooperates with AceGen in the computational process.

The main advantage of AceGen, from the researcher's point of view, is its ability to perform automatic differentiation of symbolic expressions. It is worth emphasising that the material model which is developed to simulate the PLC phenomenon results in a highly non-linear two- (or three-) field problem which is solved using the iterative Newton– Raphson algorithm which requires linearization of the governing equations. This part of the model preparation is very often the most challenging part of the implementation process. The application of automatic differentiation in the AceGen package significantly improves this step. If the residual (with all explicit and nested dependencies between variables) is properly defined, then the tangent matrix components are computed automatically as derivatives of the residual with respect to the unknowns. What is more, the finite element subroutine produced by AceGen is efficient and robust, since the code generator simplifies the symbolic expressions and has built-in optimization tools. A detailed description of AceGen features can be found e.g., in [66].

In fact, following the recommendation of Korelc [66], instead of introducing discretization into the residual Equations (39), (40) and optionally (41), pseudo-potentials are formulated for the equilibrium, energy balance and plastic strain averaging, minimization of which is equivalent to the residual equations. Specific forms of the potentials related to the governing equations considered in this model can be found in [42]. It should be mentioned that the model under consideration involves large strain plasticity and requires a solution of the non-linear set of equations at the level of Gauss points to calculate the values of internal variables. Thus, the relations between the internal variables and the global unknown fields are not given as explicit functions and the process of the automatic

differentiation has to be affected by the definition of appropriate exceptions. For more details, the reader is referred to [42,66,67].

Discretization is introduced into the pseudo-potentials following the classical Ritz approach, which yields a more robust solution procedure. An extensive description of the AceGen implementation of a large strain (gradient-enhanced) thermo-plasticity models can be found in [42,43]. The codes for thermo-visco-plasticity with the McCormick extension given in Equation (23) are developed based on the same approach.

The user subroutines in AceGen are prepared for three-dimensional finite elements, in particular hexahedral elements H8 with linear interpolation of all fields (displacement, temperature and, if relevant, averaged strain) and eight Gauss points. The linear interpolation is favorable in terms of computational effort, but it is known that plasticity simulations are affected by volumetric locking if full integration is used. Therefore, the so-called F-bar enhancement, see e.g., [68], is employed for the mechanical part of the formulation.

#### **5. Simulation of PLC Effect**

#### *5.1. Test Description*

In the numerical simulations of the PLC effect, two different samples are considered. First, an example computation for a simple tensile rod sample is made to show how the propagative instability is reproduced; see Figure 1, left. The rod dimensions are 10 × 10 × 500 mm. One end of the rod is fixed and a longitudinal displacement increasing to 75 mm (15% of the sample length) is uniformly applied at the other end within 100 s (the strain rate is 1.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> s −1 ). Fifty identical hexahedral elements with linear interpolation of all fields are used and one element is used in the cross section.

**Figure 1.** Geometry of samples and meshes, rod (**left**) and bone-shape sample (**right**).

Then, simulations are performed for a configuration based on the experimental boneshape plate sample analyzed in [49]; see Figure 1, right. The dimensions of the computed configuration are as follows: total length 102 mm, length of the middle part 57 mm, thickness 4 mm, width of middle part 12.5 mm, radius of fillets 12.5 mm, width of broader parts 20 mm. The bone-shape sample is uniformly elongated by 14.25 mm in 285 s. The element size in the central part of the mesh is approximately 2 mm.

For both specimens, insulation thermal boundary conditions are applied. The basic set of material model parameters for our study are taken from [13,25,49] for room temperature 25 ◦C and listed in Table 1. The results of the simulations are compared with the experimental results from [49].

The two variants of the material model described in Section 2 are used: the thermovisco-plastic model and the thermo-visco-plastic model with the gradient enhancement. For comparison with experiments, two options for the characteristic time of solute diffusion *t*<sup>0</sup> (called solute diffusion time in brief) are considered: either constant or depending on temperature. After the comparison with experiments the parametric study is performed. For the thermo-visco-plastic model, four sets of computations are made for different values of viscosity, heat conductivity, solute diffusion time, and maximum tension time. For the gradient-enhanced model, two sets of computations are carried out for different values of internal length and ductility parameter.

In Figure 2, the relations between the strain aging time *t<sup>a</sup>* and the relative extension ∆*L*/*L* are plotted for the two analyzed tests and selected points in the configurations, showing abrupt jumps of *t<sup>a</sup>* according to the McCormick model, representing the DSA

phenomenon. The left plot is obtained for the tensile rod, and the right one for the boneshape sample.

**Figure 2.** Strain ageing time vs. longitudinal displacement for rod benchmark at the right end of the sample (**left**) and for the bone-shape plate in tension at the centre of the sample (**right**).

To avoid convergence problems, the thermo-plastic model with the gradient enhancement is not used without the viscosity part. Since the McCormick part of the yield strength introduces recurring strain rate softening stages, the non-linear simulation algorithm fails without viscosity and it seems the gradient term is insufficient to prevent this.


**Table 1.** Basic set of mechanical, thermal and McCormick model parameters

#### *5.2. Tensile Rod Benchmark Test*

To simulate the PLC effect in a one-dimensional tension benchmark a simple 3D rod test is performed, see Figure 1 left.

In Figure 3 stress vs. the rod extension is plotted for the whole process (left plot) and the magnification of a one serration (right plot). The place of the arbitrarily chosen serration is marked by the red box on the left plot. Small serrations are visible at the beginning of the process and they gradually grow. After the diagram peaks, the serrations grow extensively, when they actually should vanish. Obviously, the model needs some modification to prevent this kind of behavior at the final (failure) stage, but proper modeling of this stage is outside the focus of this work.

**Figure 3.** Stress vs. relative longitudinal displacement for rod benchmark: whole process (**left**) and one selected serration (**right**).

In the right diagram in Figure 3 selected states are numbered in red. In Figure 4 two columns of plots are presented for the serration and those states. Each plot shows the distribution of equivalent strain rate *γ*˙ along the rod. For steps 2 and 3 before the peak, the band has a distributed form, while in the previous step 1, the band is localized. In the steps after the peak, *γ*˙ localizes again at a different position.

1 **Figure 4.** Distributions of *γ*˙ at numbered states of selected serration for tensile rod test.

#### *5.3. Comparison with Experiments for Bone-Shape Sample*

The comparisons with experiments from [49] are made for the thermo-visco-plastic model and for its gradient-enhanced version, for three initial temperatures: 25 ◦C, 100 ◦C, and 200 ◦C. Table 2 contains the values of the Young modulus, initial and final yield strength for the three temperatures.


**Table 2.** Parameters for different temperatures

Figure 5 presents the results obtained for the former model and Figure 6 for the latter one. The EMC material model parameters are fitted for the case without the dependence of the solute diffusion time *t*<sup>0</sup> on temperature. The small differences in the elastic part of the response can be caused by neglecting the elongation of the broader sample parts in the computational experiment. Notice that the level of the yielding initiation is reduced with temperature, which is related to thermal softening, and the numerical model correctly reproduces the behaviour.

**Figure 5.** Stress vs. relative extension for thermo-visco plastic model for 25 ◦C (**left**, **top**), 100 ◦C (**right**, **top**), 200 ◦C (**left**, **bottom**).

When the constant value of *t*<sup>0</sup> = 0.125 s is assumed according to [13], we can observe in Figure 5 a good agreement in terms of global load–deformation response for the temperature equal to 25 ◦C and a partial agreement for higher temperatures. For 100 ◦C and 200 ◦C the blue lines are close to the experimental black lines at the beginning of the process, but they do not mimic the failure at the end for a similar extension as in the experiments. The red line for the model with the solute diffusion time depending on temperature enters softening a bit earlier for room temperature, but it shows a much softer response, far from the

experimental diagrams, for higher temperatures. For the temperature equal to 100 ◦C and 200 ◦C the softening stage is entered much earlier than in the experiment. In the latter case, all serrations have been smoothed, which is similar to the findings presented in [25]. The values of model parameters *t*<sup>01</sup> and *t*02, which control the dependence of *t*<sup>0</sup> on temperature, are based on [25], but obviously the exponential character of function *t*0(*T*) is not a suitable choice.

**Figure 6.** Stress vs. relative extension for thermo-visco plastic model with gradient enhacement for 25 ◦C (**left**, **top**), 100 ◦C (**right**, **top**), 200 ◦C (**left**, **bottom**).

A similar behaviour as for the thermo-visco-plastic model can be observed for the variant with the gradient enhancement for the internal length *l* = 5 mm and ductility parameter *β* = 0.1, see Figure 6. It seems that the gradient enhancement of the model with the assumed material parameters, related to an additional yield strength reduction, has a minor influence on the simulated stress-relative elongation diagrams. This aspect is further analyzed in the parametric study, as follows.

### *5.4. Parametric Study*

Parametric studies for the two model variants, without and with the gradient enhancement, are performed. Six parameters are taken into account. For the thermo-visco-plastic model, the following parameters are varied: viscosity *ξ* (5, 40, 80 MPa·s), conductivity *k* (0, 50, 121, 200 J/(s·K·m)), solute diffusion time *t*<sup>0</sup> (0.01, 0.025, 0.125, 0.5, 5, 1000 s), and the duration of the elongation process *t*MAX (28.5, 285, 2850 s).

For the gradient-enhanced model, the ductility *β* (0.1, 0.5, 1, 2 [-]) and the internal length *l* (0, 5, 10, 20 mm) are changed. The parametric studies are carried out for the reference temperature equal to 25 ◦C; thus, the mechanical material parameters (Young modulus, the initial and ultimate yield strengths) are appropriate for this assumption and are taken from [49]. As given in Table 1, the default values of varied parameters are: viscosity *ξ* = 40, conductivity *k* = 121, and solute diffusion time *t*<sup>0</sup> = 0.125. Moreover, the maximum tension time *t*MAX = 285.

#### 5.4.1. Thermo-Visco-Plastic Model

In Figure 7, stress–strain diagrams for different values of viscosity (left, top), conductivity (right, top), solute diffusion time (left, bottom) and maximum tension time (right, bottom) are shown. The experimental diagram (black) for 25 ◦C is added for reference. There are no significant differences between the plots for different values of viscosity in the examined range of values; however, in the case when the viscosity *ξ* is equal to zero (not displayed here), the computation stops at the beginning of the process.

**Figure 7.** Stress vs. relative extension for different values of viscosity (**top**, **left**), conductivity (**top**, **right**), solute diffusion time (**bottom**, **left**) and maximum tension time (**bottom**, **right**).

In the second plot, it can be observed that the response is more brittle for smaller values of heat conductivity and no significant differences can be observed for higher values of *k* than 50 J/(s·K·m). The conductivity equal to 121 J/(s·K·m) is the value characteristic for the aluminium alloy used in the experiments described in [49].

The characteristic time of the solute diffusion affects the diagram smoothness and the load-carrying capacity. For high and low values of *t*0, the diagrams are smoother and there are no excessive serrations at the end of the process. The diagrams are ordered from the largest value of *t*<sup>0</sup> (the most brittle response) to the smallest value (most ductile), which means the yield strength is lower for higher values of the solute diffusion time.

The last diagram in Figure 7 shows that the response is rate dependent and the higher the load rate (the smaller *tmax* is), the smaller the predicted load-carrying capacity. The diagram for the largest value of *tmax* is smooth and does not exhibit serrations at the end of the process contrary to the other diagrams.

The next Figures 8 and 9 are plotted for *ξ* = 40 MPa·s, *k* = 121 J/(s·K·m), *t*<sup>0</sup> = 0.125 s and *tmax* = 285 s. In Figure 9, two columns with distributions of *γ*˙ along the central longitudinal axis of the dog-bone sample are shown for the numbered states in the load– displacement plot part for the selected serration shown in Figure 8. In Figure 10 the distributions of *γ*˙ in the sample are shown corresponding to the results presented in Figure 9. Before the serration peak, a band is visible on the right-hand side of the sample; see Figure 9, first row left. Then, when the peak is approached, the band on the right starts to disappear and a band on the left-hand side appears. After the peak, see Figure 9 third row left, the traces of the right band vanish completely and only the left band is visible. The same sequence of states can be observed in Figure 10.

**Figure 8.** Stress vs. relative extension for one serration with step numeration (*ξ* = 40 MPa·s, *k* = 121 J/(s·K·m) and *t*<sup>0</sup> = 0.125 s).

1 **Figure 9.** Distributions of *γ*˙ along the specimen axis at numbered states for *ξ* = 40 MPa·s, *k* = 121 J/(s·K·m) and *t*<sup>0</sup> = 0.125 s.

**Figure 10.** Distributions of *γ*˙ at numbered states (*ξ* = 40 MPa·s, *k* = 121 J/(s·K·m) and *t*<sup>0</sup> = 0.125 s).

5.4.2. Thermo-Visco-Plastic Model with Gradient Enhancement

In Figure 11, diagrams for different values of the ductility parameter (left) and the internal length (right) are shown. For comparison, two additional diagrams are added; the black line is the experimental diagram for 25 ◦C, and the gray line is for the thermo-viscoplastic model with the following parameters: *ξ* = 40 MPa·s, *k* = 121 J/(s·K·m), *t*<sup>0</sup> = 0.125 s and *tmax* = 285 s.

**Figure 11.** Stress vs. relative extension for different values of ductility and *l* = 5 mm (**left**) and for different internal lengths and *β* = 0.1 (**right**).

When the value of the ductility grows, the influence of the exponential reduction factor increases. It can be observed in Figure 11 (left) that for larger values of ductility, the load-carrying capacity is smaller and softening starts to dominate faster. The diagrams for different values of the internal length are presented in Figure 11 (right). There are no significant differences for the values of internal length larger than zero. The diagrams are close to the diagram obtained for the thermo-visco-plastic model. The diagram for *l* = 0 (blue line ) is slightly more brittle and ends for ∆*L*/*L* ≈ 22 due to divergence of the simulation.

In Figure 12, the distributions of the plastic strain rate are compared for a series of states in the deformation history. The plots on the left are for *l* = 0 and the right ones for *l* = 20 mm. For small deformation, a uniform distribution of *γ*˙ is observed, then a localized band is formed, which resembles a cross pattern of shear bands diffused by regularization and/or re-hardening. The reason can also be a too-coarse finite element mesh used for the simulation. The band travels through the process zone of the sample in a similar way irrespective of the assumed internal length. The plots do not show the expected influence of the length scale on the widths of the propagating localization zones. This is probably caused by the fact that the viscosity and heat conductivity provide sufficient regularization and the gradients active on the softening parts of the serrations merely counteract the additional softening source related to the yield stress degradation involved in the gradient-enhanced model.

Figure 13 presents parts of the stress-strain diagrams of one selected serration for two values of internal length *l* = 0 (left) and *l* = 20 mm (right). Further, Figure 14 shows the evolution of the distribution of the plastic strain rate *γ*˙ for the two values of the internal length scale within one selected serration presented in Figure 13. The plots for *l* = 0 show disappearing and reappearing localization zones, while the plots for *l* = 20 mm present a moving band. However, also for one serration the maps show a negligible influence of the length scale on the widths of the propagating bands.

**Figure 12.** Distributions of *γ*˙ at selected states in the deformation history for *l* = 0 (**left column**) and *l* = 20 mm (**right column**). For the rows of figures from top ∆*L*/*L* is equal to 3, 5, 7.5, 10, 12.5, 15, 17.5, 20.

**Figure 13.** Stress vs. relative extension for selected serration and two values of internal length *l* = 0 (**left**) and *l* = 20 mm (**right**).

1 **Figure 14.** Distributions of *γ*˙ at selected states in the deformation history, marked by numbers in Figure 13, for *l* = 0 (**left column**) and *l* = 20 mm (**right column**).

#### **6. Conclusions**

The Portevin–Le Chatelier (PLC) effect has been simulated using two versions of the large strain thermo-plastic Estrin–McCormick model. The model is capable of reproducing the results of the Dynamic Strain Aging (DSA) phenomenon: serrations in the load–displacement diagrams (repetitive changes of softening and hardening) and propagating localization zones. The model takes into account visco-plasticity and the second version also includes a gradient enhancement via an averaging equation for the equivalent plastic strain.

Two different configurations are used in simulations. A simple rod is considered first to show how the adopted constitutive model represents the PLC phenomenon. Then, tension of the experimental bone-shape sample from [49] is simulated, instability formation and propagation are studied, and a comparison with experimental results is performed. For both tests, detailed analyses of the evolution of the plastic strain rate for a selected serration have been presented. It seems that the localization band moves, but it rather gradually disappears and then reappears at a different position.

Further, a parametric study is performed. Different values of viscosity, conductivity, the time of solute diffusion in the DSA model, and the maximum tension time (loading rate) have been considered for the thermo-visco-plastic model. For the model with the gradient enhancement, the ductility parameter and the internal length have varied.

On one hand, the results of simulations are quite satisfactory: the simulated load– extension diagrams are quite close to the experimental results and the sensitivity to model parameters is in agreement with expectations. On the other hand, excessive post-peak serrations are visible for most of the computation, so a method to reduce them is needed. The adopted dependence of the parameters of the McCormick model on temperature led to results far from the experimental ones for higher temperatures, so this aspect requires further research and model improvement. Moreover, experimental studies are necessary to compare the shear band evolution in PLC simulations (and not only load–displacement plots) and to identify material model parameters in a similar way to [13,25].

Finally, the distributions of the equivalent plastic strain rate in the specimen for the gradient-enhanced model are examined for two values of the internal length scale *l* = 0 and *l* = 20 mm. They are compared for a series of states showing no significant differences in the size of the localization bands. This can be caused by the relatively coarse discretization used, or by the fact that viscosity and heat conduction provide some regularization, which manifests itself in smoothing of the simulated bands and in a weak influence of gradients.

**Author Contributions:** All authors worked on the concepts of the research, formulation of the models and preparation of the manuscript. M.M. and B.W. programmed the models in AceGen, M.M. performed all computations and provided numerical results. The research outcomes were discussed by all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was supported by the National Science Centre of Poland within grant number 2018/31/N/ST8/03573.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available in the article.

**Acknowledgments:** The authors acknowledge valuable discussions on the research with A. Menzel from TU Dortmund, Germany/Lund University, Sweden.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


## *Article* **Stress Field Evaluation in Orthotropic Microstructured Composites with Holes as Cosserat Continuum**

**Farui Shi 1,2,3, Nicholas Fantuzzi 3,\*, Patrizia Trovalusci <sup>4</sup> , Yong Li 1,2,\* and Zuoan Wei 1,2**


**Abstract:** It is known that the presence of microstructures in solids such as joints and interfaces has an essential influence on the studies of the development of advanced materials, rock mechanics, civil engineering, and so on. However, microstructures are often neglected in the classical local (Cauchy) continuum model, resulting in inaccurate descriptions of the behavior of microstructured materials. In this work, in order to show the impact of microstructures, an implicit 'non-local' model, i.e., micropolar continuum (Cosserat), is used to numerically investigate the effects of direction and scale of microstructures on the tension problem of a composite plate with a circular hole. The results show that distributions of field variables (such as displacements and stresses) have an obvious directionality with respect to the microstructures' direction. As the scale of microstructures increases, such a direction effect becomes more evident. Unlike the isotropic material where stress concentration occurs at the vertex of the hole and the stress concentration factor is close to 3, for the microstructured composite, the stress concentration can be observed at any location depending on the microstructures' directions, and the concentration factor can exceed 3 to a maximum close to 9 as the increasing scale of microstructures. In addition, differences in the mechanical behavior between Cosserat and Cauchy models can be also observed; such differences are more evident for the material showing a pronounced orthotropic nature.

**Keywords:** composite materials; microstructure direction; Cosserat continuum; stress concentration; scale effect

## **1. Introduction**

Microstructure is one of the most critical factors that involves many kinds of materials such as rock, ceramic, alloy, human cortical bone, etc. [1–4]. As an internal structure, microstructure can play a crucial role in determining the gross behavior and mechanical response of materials [5]. However, the microstructures in materials distribute randomly with different lengths and directions, which complicates the understanding of the material's response. In general, materials with microstructure have weaker strength than in intact materials [6,7]. Guo et al. [8] experimentally investigated the effect of bedding angle in phyllite under unloading confining pressures, where the rock bedding joints can be regarded as microstructures. They found that the rock shows different strengths as the bedding angle changes. Numerical research on the tunnel surrounding rocks with different inclination angles also shows a directional effect on the distributions of displacements and stress around the tunnel [9]. By reviewing indentation tests at the micron scale, Bauer et al. [10] demonstrate that an obvious length scale effect (i.e., non-locality) can be

**Citation:** Shi, F.; Fantuzzi, N.; Trovalusci, P.; Li, Y.; Wei, Z. Stress Field Evaluation in Orthotropic Microstructured Composites with Holes as Cosserat Continuum. *Materials* **2022**, *15*, 6196. https:// doi.org/10.3390/ma15186196

Academic Editor: Alexander Yu Churyumov

Received: 2 August 2022 Accepted: 4 September 2022 Published: 6 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

found when the material's intrinsic length scales are comparable with the dimension of specimens. Therefore, it is of importance to describe the macroscopic response of these materials by considering the influences of the microstructures.

The existence of microstructure results in the heterogeneity characteristic of materials. There are various methods that can be used to model the behavior of microstructured materials. Discrete modeling with interactions of each constituent in materials is a good option because it can produce an accurate result; however, this approach is often computationally cumbersome [11–13]. Alternatively, homogenizing the heterogeneous material as an equivalent continuum could be an efficient approach because it is faster and takes less computational cost [14]. Nevertheless, the application of this approach depends on selections of the homogenization method and macroscopic continuum theory that need to reveal the presence of microstructures. As is known, the classical Cauchy continuum may have disadvantages in describing the gross behavior of microstructured materials since it lacks in internal length descriptions [15,16]. This calls for the application of the non-local continuum theory, as this approach can reveal the presence of internal lengths [17]. In the non-local theory, internal lengths can be represented by adding extra degrees of freedom or parameter as internal variables, corresponding to the so-called implicit and explicit non-local descriptions, respectively [18,19].

The Cosserat continuum theory is a widely used implicit non-local model to investigate the microstructured material's behavior. After the completed mathematical foundations of the micropolar continuum was achieved, this theory became very popular (since the 70s). The Cosserat model introduces to each material point an extra degree of freedom, termed microrotation, which is different from the local rigid rotation (i.e., macrorotation). As a result, the stress and strain fields become asymmetric in this model, which is different from the classical Cauchy model with symmetric measurements. Moraes et al. [1] found that the asymmetrical property in the Cosserat model can be helpful to improve the description of the mechanical behavior of the materials such as rocks. The asymmetric strains also correspond to the relative rotation between the microrotation and macrorotation. Pau and Trovalusci [20] found that the relative rotation is significant in anisotropic materials, whereas it can be negligible in orthotetragonal materials where the internal length trends to vanish. With the advantage of keeping the memory of the microstructure, the Cosserat continuum was used to study many kinds of materials such as layered materials [21,22], fiber-reinforced materials [23,24], granular materials [25], and composites [26–28]. Using a homogenization process for the Cosserat continuum, Trovalusci and Masiani [29] numerically and experimentally studied the mechanical behavior of an inclined masonry structure in which microstructures (interfaces) show a different direction from the ordinary masonry structure. However, the results of this literature only showed the micropolar effect resulted from the Cosserat continuum but the effect of microstructures direction was not further discussed.

In this work, the effects of the microstructure's direction and length scale in a composite are studied to extend the understanding of microstructured materials, such as advanced materials with various microstructured and layered rocks with inclined angles. The composite considered here is made of rectangular blocks interacting with each other through their elastic interfaces, and it is homogenized as a Cosserat continuum by an energeticequivalence-based homogenization technique [30]. Thus, the characteristic of non-locality is involved in this study. Six directions and four length scales of the microstructure are investigated by the finite element method (FEM) for a tension problem of a composite plate with a circular hole. Therefore, this work also focuses on the stress concentration problem of microstructured materials. This problem has been widely reported in previous studies [31–34]. Holes in materials can induce stress concentration around it and hence reduce the mechanical properties [35]. However, a number of solutions have been carried out for holes in isotropic plates [36]. With the increasing research interests on materials, especially with microstructures, it is essential to gain a better understanding in modeling the mechanical behavior of these materials.

This paper is structured as follows. After the introduction section, Section 2 introduces the Cosserat theory and its FEM implementation. Section 3 presents the model, parameters, methods, etc. used in the simulations of the tension problem for a plate with a circular hole. In Section 4, numerical simulations are conducted and results of displacements, stresses, and relative rotation are shown. The stress distribution around the hole is discussed and the simulation results are analyzed in Section 5. In the end, conclusions and remarks are drawn in Section 6.

#### **2. Cosserat Continuum and Its FEM Implementation**

The Cosserat continuum is considered to be a multi-scale tool [37] that can be used to investigate the mechanical behavior of materials where the microstructures and, in particular, internal lengths, play a crucial role. As an implicit' non-local continuum, it is equipped with additional degrees of freedom revealing the presence of microstructures. That is, for two-dimensional (2D) Cosserat media, each material point has two translation degrees of freedom *u*1, *u*<sup>2</sup> and an additional microrotation degree of freedom *ω*. The microrotation *ω* is an independent degree of freedom and it is different from the macrorotation *θ* which is defined as the skew-symmetric part of the gradient of displacement. Thus, a peculiar measurement, the relative rotation *θ* − *ω*, can be defined in this continuum. A general displacement vector for the Cosserat material point can be expressed as:

$$\mathbf{d}^{\top} = \begin{bmatrix} u\_1 & u\_2 & \omega \end{bmatrix} = \begin{bmatrix} \mathfrak{u}^{\top} & \omega \end{bmatrix} \tag{1}$$

where *u* > = - *u*<sup>1</sup> *u*<sup>2</sup> . Due to the introduction of *ω*, the tangential strains in the Cosserat model are not reciprocal, i.e., *ε*<sup>12</sup> 6= *ε*21, and the microcurvature component is introduced as an additional strain measure; therefore, the linear strain–displacement relation can be expressed as:

$$
\begin{bmatrix} \varepsilon \\ \mathcal{X} \end{bmatrix} = \begin{bmatrix} \mathcal{L} & \mathcal{M} \\ 0 & \nabla \end{bmatrix} \begin{bmatrix} \mu \\ \omega \end{bmatrix} \tag{2}
$$

where *ε* > = - *ε*<sup>11</sup> *ε*<sup>22</sup> *ε*<sup>12</sup> *ε*21 contains the normal and tangential strains and *χ* <sup>&</sup>gt; <sup>=</sup> - *χ*<sup>1</sup> *χ*<sup>2</sup> contains the microcurvatures. ∇ is the gradient operator, and

$$\begin{array}{rcl} L & = \begin{bmatrix} \frac{\partial}{\partial \mathbf{x}\_1} & 0 & \frac{\partial}{\partial \mathbf{x}\_2} & 0\\ 0 & \frac{\partial}{\partial \mathbf{x}\_2} & 0 & \frac{\partial}{\partial \mathbf{x}\_1} \end{bmatrix}^\top \\\ \mathbf{M} & = \begin{bmatrix} 0 & 0 & 1 & -1 \end{bmatrix}^\top \end{array} \tag{3}$$

With the strain measures, the stresses of the Cosserat continuum can be obtained by a linear elastic constitutive equation as:

$$
\begin{bmatrix} \sigma \\ \mu \end{bmatrix} = \begin{bmatrix} \mathbb{A} & \mathbb{B} \\ \mathbb{B}^{\top} & \mathbb{D} \end{bmatrix} \begin{bmatrix} \varepsilon \\ \mathcal{X} \end{bmatrix} \tag{4}
$$

where *σ* > = - *σ*<sup>11</sup> *σ*<sup>22</sup> *σ*<sup>12</sup> *σ*21 contains the normal and tangential stresses and *µ* <sup>&</sup>gt; <sup>=</sup> - *µ*<sup>1</sup> *µ*<sup>2</sup> contains the couple stresses. The tangential stress components are also not reciprocal (*σ*<sup>12</sup> 6= *σ*21). The constitutive sub-matrices A, B, and D collect the constitutive terms *Aijhk*, *Bijh*, and *Dij*, where *i*, *j*, *h* and *k* = 1, 2.

A detailed three-dimensional finite element formulation for the Cosserat continuum can be found in [38]. For the sake of simplicity, the displacement-based finite element implementation for 2D Cosserat theory is presented here to model the behavior of microstructured material. Firstly, in the finite element procedure, displacement and microrotation fields should be approximated by the nodal values of an element. In this study, for avoiding

the element locking problem, a bi-quadratic (*Nu*) and a bi-linear (*Nω*) shape function are, respectively, used for the displacement and microrotation approximation:

$$\begin{array}{rcl} \mu & = \mathcal{N}\_{\boldsymbol{\mu}} \tilde{\mu} \\ \omega & = \mathcal{N}\_{\boldsymbol{\omega}} \tilde{\omega} \end{array} \tag{5}$$

where *u*˜ and *ω*˜ are nodal displacement and microrotation values. In the present paper, nine-node quadrangular elements are considered for an element. All nine node values are used to approximate the displacements, whereas values at four corner nodes are used for the microrotation. *N<sup>u</sup>* and *N<sup>ω</sup>* can be expressed as:

$$\begin{array}{rcl} \mathbf{N\_{u}} &=& \begin{bmatrix} \mathbf{N\_{1}}^{\boldsymbol{u}} & \mathbf{0} & \dots & \mathbf{N\_{\theta}}^{\boldsymbol{u}} & \mathbf{0} \\ \mathbf{0} & \mathbf{N\_{1}}^{\boldsymbol{u}} & \mathbf{0} & \dots & \mathbf{N\_{\theta}}^{\boldsymbol{u}} \end{bmatrix} \\ \mathbf{N\_{\omega}} &=& \begin{bmatrix} \mathbf{N\_{1}}^{\boldsymbol{\omega}} & \dots & \mathbf{N\_{4}}^{\boldsymbol{\omega}} \end{bmatrix} \end{array} \tag{6}$$

Substituting Equation (5) into (2), the strain vectors become:

$$\begin{array}{rcl} \boldsymbol{\varepsilon} & = \begin{bmatrix} \mathbf{LN}\_{\boldsymbol{\mu}} & \mathbf{MN}\_{\boldsymbol{\omega}} \end{bmatrix} \begin{Bmatrix} \tilde{\boldsymbol{u}} & \tilde{\boldsymbol{\omega}} \end{Bmatrix}^{\top} = \mathbf{B}\_{\boldsymbol{\varepsilon}} \tilde{\boldsymbol{d}},\\ \boldsymbol{\chi} & = \begin{bmatrix} 0 & \nabla \mathbf{N}\_{\boldsymbol{\omega}} \end{bmatrix} \begin{Bmatrix} \tilde{\boldsymbol{u}} & \tilde{\boldsymbol{\omega}} \end{Bmatrix}^{\top} = \mathbf{B}\_{\boldsymbol{\chi}} \tilde{\boldsymbol{d}} \end{array} \tag{7}$$

where *B<sup>ε</sup>* and *B<sup>χ</sup>* are the derivatives of the shape functions. *d***˜** is the unknown nodal values collecting *u*˜ and *ω*˜ . Substituting Equation (7) into Equation (4), the constitutive relations become:

$$\begin{array}{rcl} \sigma & = \mathbb{A}\mathbb{B}\_{\varepsilon}\tilde{d} + \mathbb{B}\mathbb{B}\_{\chi}\tilde{d}, \\ \mu & = \mathbb{B}^{\top}\mathbb{B}\_{\varepsilon}\tilde{d} + \mathbb{D}\mathbb{B}\_{\chi}\tilde{d} \end{array} \tag{8}$$

Now, the stress and couple stress measures can be obtained from the nodal values. Considering a domain A and boundary Γ, the principle of virtual work can be expressed as:

$$\int\_{\mathcal{A}} \delta \boldsymbol{\varepsilon}^{\top} \boldsymbol{\sigma} + \delta \boldsymbol{\chi}^{\top} \boldsymbol{\mu} \, d\mathcal{A} = \int\_{\mathcal{A}} \delta \boldsymbol{u}^{\top} \mathbf{b} \, d\mathcal{A} + \int\_{\Gamma} \delta \boldsymbol{u}^{\top} \mathbf{\tilde{f}} + \delta \boldsymbol{\omega}^{\top} \overline{\boldsymbol{m}} \, d\Gamma \qquad \forall \delta \boldsymbol{u}, \delta \boldsymbol{\omega} \tag{9}$$

where *δ* is the variational operator, *b* is the body force vector. *t* and *m* are the traction and couple-traction vectors applied on the boundary Γ. The components (*t<sup>i</sup>* and *m<sup>i</sup>* ) of *t* and *m* should satisfy the equilibrium at external boundary as *t<sup>i</sup>* = *σijn<sup>j</sup>* and *m<sup>i</sup>* = *µjn<sup>j</sup>* , where *n<sup>j</sup>* is the components of the outward unit normal to the boundary. Substituting Equations (5), (7) and (8) into (9) and excluding body forces, we obtain:

$$\underbrace{\delta \vec{d}^{\top} \underbrace{\int\_{\mathcal{A}^{\epsilon}} \left( \mathbf{B}\_{\varepsilon} \, ^{\top} \boldsymbol{\Lambda} \mathbf{B}\_{\varepsilon} + \mathbf{B}\_{\varepsilon} \, ^{\top} \boldsymbol{\mathbb{B}} \mathbf{B}\_{\mathcal{X}} + \mathbf{B}\_{\mathcal{X}} \, ^{\top} \mathbf{B}^{\top} \mathbf{B}\_{\varepsilon} + \mathbf{B}\_{\mathcal{X}} \, ^{\top} \boldsymbol{\mathbb{D}} \mathbf{B}\_{\mathcal{X}} \right) d\mathcal{A}^{\epsilon}}\_{\mathcal{K}^{\epsilon}} \underbrace{\tilde{d} = \delta \vec{d}^{\top} \int\_{\Gamma^{\epsilon}} \underbrace{\int\_{\Gamma^{\epsilon}} \, \overset{\scriptstyle \mathcal{H}\_{\mathbf{u}} \, ^{\top} \overline{\tilde{t}}}\_{\mathcal{F}^{\epsilon}} \Big] d\Gamma^{\epsilon} }\_{\mathcal{F}^{\epsilon}} d\Gamma^{\epsilon} \qquad \forall \delta \vec{d} \tag{10}$$

where *K e* and *F e* are the element stiffness matrix and the element nodal force vector. They can be computed numerically by a Gauss–Legendre integration with 3 × 3 grid. If considering arbitrary *δd***˜** , we can obtain the standard finite element formulation as:

$$\mathbf{K}^{\varepsilon}\vec{d} = \mathbf{F}^{\varepsilon} \tag{11}$$

At the end, the unknown *d***˜** can be obtained by solving this equation. With this solution, in the post-processing stresses and strains are firstly computed at Gauss points for each element and then an extrapolation technique is used to get stresses and strains at element nodal points.

The above implementations are achieved by an updated MATLAB code based on codes of a classical 2D Cauchy continuum as presented in [39].

#### **3. Numerical Simulation**

In this section, we intend to numerically investigate the effects of direction and scale of microstructures for a composite material that can be considered as an assembly made of rigid rectangular blocks in contact with elastic interfaces (Figure 1), where each rectangular block has the width of *b* and height of *h*. The assembly is arranged as an interlocking structure and the interfaces of blocks form the microstructures of this composite material. A homogenization procedure presented in [30] can be used to describe the assembly as an equivalent Cosserat continuum. In this work, 7-block representative volume element (RVE) that is highlighted with orange color in Figure 1 is used for the homogenization procedure to produce the Cosserat constitutive matrix in Equation (4). In the highlighted RVE, the blocks' centroids are represented by green crosses, and red lines mean the outward unit normal vectors of the central block's interfaces. The direction of microstructures can be changed by transforming the assembly of an angle *β* from *x* − *y* coordinate system to *X* − *Y* coordinate system as shown in Figure 1. In this study, we select 6 values of *β* (0 ◦ , 30◦ , 60◦ , 90◦ , 120◦ , 150◦ ). Furthermore, to obtain various scales of microstructures, 4 different block sizes are used by fixing the height of block *h* = 0.1 m and changing the aspect ratio *ρ* = *b*/*h* = 1.5, 3, 7, and 15, where a greater *ρ* corresponds to a longer rectangular block, as a consequence, showing more orthotropic nature.

**Figure 1.** Schemes of the considered assembly and RVE.

The blocks interact among themselves through elastic common interfaces. The adopted spring stiffness at the interfaces is:

$$K = \begin{bmatrix} k\_{\text{fl}} & \mathbf{0} \\ \mathbf{0} & k\_{\text{f}} \end{bmatrix} \tag{12}$$

where *k<sup>n</sup>* and *k<sup>t</sup>* are the normal and tangential stiffness per unit length, respectively. Here we have *k<sup>n</sup>* = 576.58 MPa/m and *k<sup>t</sup>* = 288.29 MPa/m. The rotation stiffness of interface is computed *k<sup>r</sup>* = *kn*(*d*/2) 2 , where *d* is the length of interface. Therefore, the Cosserat constitutive matrices of the reference RVE when *β* = 0 ◦ can be obtained by the homogenization technique that is based on an equivalence energy criterion between the material's discrete system of and the continuum model [30]. The constitutive matrix of transformed assembly can be obtained as follows:

$$\mathcal{C} = \mathcal{Q}^\top \mathcal{C}\_\mathsf{0} \mathcal{Q} \tag{13}$$

where *C***<sup>0</sup>** is the constitutive matrix when *β* = 0 ◦ , *Q* is the usual transformation matrix [40].

For comparison, the Cauchy continuum is also considered here to carry out the same simulations as performed by the Cosserat continuum. Because of the lack in microstructures, the constitutive relation of the Cauchy continuum has the form of: *σ* = Aˆ *ε*, where the matrix Aˆ is obtained from A and their relationship can be found in the previous literature as [41]:

$$\begin{aligned} \hat{A}\_{1111} &= A\_{1111} \\ \hat{A}\_{1122} &= A\_{1122} \\ \hat{A}\_{2222} &= A\_{2222} \\ \hat{A}\_{1112} &= (A\_{1112} + A\_{1121})/2 \\ \hat{A}\_{2212} &= (A\_{2212} + A\_{2221})/2 \\ \hat{A}\_{1212} &= (A\_{1212} + A\_{2121} + 2A\_{1221})/4 \end{aligned} \tag{14}$$

Tables 1–4 list the constitutive components of all configurations, where components keeping zero for all configurations are not reported. It can be seen that there are more zero components when *β* = 0 ◦ and 90◦ . Actually, only diagonal components of the constitutive matrix exist and an orthotropic nature of the material is observed for these two transformation angles. For other angles, all components of matrices A and D appear. Consequently, materials with these RVEs show a centrosymmetric nature [30]. B = 0 for all configurations, meaning there is no coupling between stresses/microcurvatures and microcouples/strains. As the aspect ratio *ρ* increases, constitutive components change monotonously except for *A*2222, *A*<sup>1212</sup> of Cosserat continuum and *A*ˆ <sup>2222</sup> of Cauchy continuum when *β* = 0 ◦ . Because of the fixed height of blocks (*h*), these components stay the same with increasing *ρ*.

**Table 1.** Cosserat and Cauchy constitutive parameters for RVE with block parameter *ρ* = 1.5, *Aijkl*, *A*ˆ *ijkl* [MPa], *<sup>D</sup>ij* [MPa·m<sup>2</sup> ].



**Table 2.** Cosserat and Cauchy constitutive parameters for RVE with block parameter *ρ* = 3, *Aijkl*, *A*ˆ *ijkl* [MPa], *<sup>D</sup>ij* [MPa·m<sup>2</sup> ].

**Table 3.** Cosserat and Cauchy constitutive parameters for RVE with block parameter *ρ* = 7, *Aijkl*, *A*ˆ *ijkl* [MPa], *<sup>D</sup>ij* [MPa·m<sup>2</sup> ].



**Table 4.** Cosserat and Cauchy constitutive parameters for RVE with block parameter *ρ* = 15, *Aijkl*, *A*ˆ *ijkl* [MPa], *<sup>D</sup>ij* [MPa·m<sup>2</sup> ].

In the following, a classical tension problem of a square plate with a circular hole is studied for all above-mentioned configurations. Figure 2 shows the sketch of the problem and its finite element meshing. The plate has a side length of 10 m and the radius of the hole is 1.25 m. A total of 1440 elements is used for this model. Due to the singular nature resulting from the presence of the hole, stress concentration is more likely observed around the hole under tension force. To make sure the results are reliable, a finer mesh is applied near the hole. Uniform tensile stress *σ*<sup>0</sup> = 1 MPa is applied on the right side of the plate. The left side of the plate is fixed symmetrically in the *x*-direction and the bottom left point is additionally fixed in the *y*-direction. In the present paper, the direction and scale effect of microstructure is investigated by setting various *β* and *ρ*. The simulation results are shown below.

**Figure 2.** Sketch of the plate with hole problem and its finite element mesh.

### **4. Results**

The results of displacements and stresses for the Cosserat and Cauchy models and relative rotation for the Cosserat model are presented in this section to show the effects of microstructure's direction and scale on the behavior of microstructured composite materials. Figure 3 depicts the horizontal displacement results *u*<sup>1</sup> of Cosserat and Cauchy models. It can be seen that change in direction of microstructure has a significant effect on *u*<sup>1</sup> for both two models. The smallest *u*<sup>1</sup> can be observed when *β* = 0 ◦ . As *β* changes from 0 ◦ to 150◦ , *u*<sup>1</sup> increases to the greatest at *β* = 90◦ and then decreases but this is expected for the shortest block case *ρ* = 1.5. In the case of *ρ* = 1.5, the greatest *u*<sup>1</sup> occurs at *β* = 60◦ and 120◦ but that is also close to *u*<sup>1</sup> at *β* = 90◦ . For all *β* representing the directions of microstructure, *u*<sup>1</sup> has a reduction as *ρ* increases. For the plate with the shortest blocks, displacement localization can be observed at the middle of the right side of the plate. However, as *ρ* increases, displacement localization reduces and uniform displacement distribution can be observed at the right side of the plate.

It can be seen from Figure 3 that the difference in *u*<sup>1</sup> between Cosserat and Cauchy models is not obvious for the orthotropic materials (*β* = 0 ◦ and 90◦ ). Here we take the difference in the maximum *u*<sup>1</sup> between the two models as ∆*u*1. For various *ρ*, ∆*u*<sup>1</sup> is 3.02–8.63 mm when *β* = 0 ◦ and 2.13–5.85 mm when *β* = 90◦ . However, the difference is more evident for the centrosymmetric materials. ∆*u*<sup>1</sup> increases from 20 mm to 69 mm as *ρ* increases when *β* = 30◦ , 60◦ , 120◦ , and 150◦ .

**Figure 3.** Horizontal displacement *u*<sup>1</sup> , mm, (**a**) Cosserat, (**b**) Cauchy.

Figure 4 depicts the vertical displacement results *u*<sup>2</sup> for two models. The orthotropic materials (*β* = 0 ◦ and 90◦ ) show negligible *u*<sup>2</sup> under the horizontal tension stress for both models. However, the centrosymmetric materials with other transformation angles can produce clear *u*<sup>2</sup> with more or less directionality.

It should be noted that there is a big difference in *u*<sup>2</sup> between Cosserat and Cauchy models. *u*<sup>2</sup> of these two models has a similar distribution only when *ρ* = 1.5. As *ρ* increases, *u*<sup>2</sup> distributions of two models become different. When *β* = 30◦ and 150◦ , the Cosserat

continuum shows high-intensity *u*<sup>2</sup> distribution on the right plate as *ρ* increases, e.g., the maximum *u*<sup>2</sup> is up to 117 mm when *β* = 150◦ ; however, *u*<sup>2</sup> from the Cauchy continuum has no clear high-intensity distribution with *ρ* and the maximum *u*<sup>2</sup> is just 77 mm when *β* = 150◦ . An opposite difference between two models can be observed when *β* = 60◦ and 120◦ , that is high-intensity *u*<sup>2</sup> distribution can be observed by the Cauchy but not the Cosserat model.

**Figure 4.** Vertical displacement *u*2, mm, (**a**) Cosserat, (**b**) Cauchy.

The horizontal stresses *σ*<sup>11</sup> of the two models are shown in Figure 5. The results show directionality of distribution of *σ*<sup>11</sup> with respect to the direction *β*. For orthotropic materials, the high-stress area is parallel to the *x*-direction and the peak stress happens at the top and bottom points of the hole edge (i.e., *x* = 0). However, for centrosymmetric materials, the high-stress area is inclined with the *x*-direction by an angle that can be related to the direction of microstructure, and the location of peak stress changes. It can be seen the directionality of *σ*<sup>11</sup> is more evident for greater *ρ*. As the increase of *ρ*, the high-stress area becomes wider. The exception happens at *β* = 90◦ which shows the opposite behavior. The existence of the hole as a singularity can result in the concentration of stress around the hole edge. In the following, the stress concentration at the hole boundary will be discussed.

Vertical stress *σ*<sup>22</sup> (Figure 6) also shows directionality of stress distribution with respect to *β*. There can be seen a difference in *σ*<sup>22</sup> between two models for various *ρ*. For relative short blocks (*ρ* = 1.5 and 3), two models produce closed behavior of *σ*22. However, for longer blocks (*ρ* = 7 and 15) especially at *β* = 60◦ , 90◦ and 120◦ , *σ*<sup>22</sup> from Cauchy model is significantly greater than that from Cosserat model.

**Figure 5.** Horizontal stress *σ*11, MPa, (**a**) Cosserat, (**b**) Cauchy.

**Figure 6.** Vertical stress *σ*22, MPa, (**a**) Cosserat, (**b**) Cauchy.

The relative rotation, defined as the difference between macrorotation (*θ*) and microrotation (*ω*), is a peculiar measurement in the Cosserat model. Figure 7 shows the relative rotation *θ* − *ω* for the Cosserat model. The directionality of *θ* − *ω* distribution can be also observed and that is more evident as *ρ* increases. For orthotropic materials, *θ* − *ω* shows point symmetric behavior with respect to the hole center where positive and negative *θ* − *ω* can be both observed, but the value of *θ* − *ω* is close to 0. Thus, the Cosserat model is very close to the Cauchy one. For *β* = 30◦ and 60◦ , the plate domain mainly undergoes a positive relative rotation. Oppositely, for *β* = 120◦ and 150◦ , this domain mainly undergoes a negative *θ* − *ω*. The above-mentioned indicates that relative rotation can be affected by the direction of microstructure. There is less relative rotation when the microstructures are arranged along parallel and perpendicular to the direction of force, whereas the microstructures arranged along other directions would result in an obvious relative rotation acting at a certain orientation.

**Figure 7.** Relative rotation *θ* − *ω* of Cosserat model.

#### **5. Discussions**

The problem of stress concentration has always been focused on due to the presence of singularity [32,42,43]. Under horizontal tension in this study, it can be seen that the stresses *σ*<sup>11</sup> and *σ*<sup>22</sup> in the plate (Figures 5 and 6) is mainly concentrated around the boundary of the hole. To better show the stress distribution, in the following, by transforming the stress state from Cartesian coordinate to polar coordinate, the hoop stress *σ<sup>h</sup>* at the hole boundary is depicted for two models in the polar coordinate system as shown in Figure 8. In this way, *σh* represents *σ*<sup>22</sup> when the polar angle equals to 0 or *π*, whereas the hoop stress denotes *σ*<sup>11</sup> when the polar angle is *π*/2 or 3*π*/2. Therefore, the location and magnitude of *σ<sup>h</sup>* can be clearly observed. For both Cosserat and Cauchy models, it can be seen that the distribution of *σ<sup>h</sup>* at the hole boundary is point symmetric to the center of hole and also shows directionality that depends on the aspect ration *ρ* and angle *β*.

**Figure 8.** Distribution of the hoop stress *σ<sup>h</sup>* at the hole boundary, MPa, (**a**) Cosserat, (**b**) Cauchy.

For the orthotropic materials, the distribution of *σ<sup>h</sup>* is symmetrical along the vertical direction (*π*/2 − 3*π*/2) in the polar coordinate. The highest *σ<sup>h</sup>* can be observed at the top and bottom points of the hole boundary (i.e., polar angle equals to *π*/2 and 3*π*/2) for the Cosserat model and Cauchy model when *β* = 0 ◦ , indicating that the peak stress results from the horizontal stress *σ*11. As for the Cauchy model when *β* = 90◦ , it is consistent with the above results for small *ρ*. As *ρ* increases, the highest *σ<sup>h</sup>* trend to be occurred at the right and left points of the hole boundary (i.e., polar angle equals to 0 and *π*). Since we observed a significant *σ*<sup>22</sup> concentration in the Cauchy model in Figure 6b, the vertical stress *σ*<sup>22</sup> is able to result in the peak stress for these cases. For centrosymmetric materials, the distribution of *σ<sup>h</sup>* is no longer symmetrical along the vertical direction but deviates from it to more or less of an extent because of the transformation angle of rectangular blocks. Thus, the highest *σ<sup>h</sup>* does not occur at these special points, i.e., polar angle equals to 0, *π*/2, *π*, or 3*π*/2. This is consistent with the results by early study [9], which investigates the stress distribution of the layered surrounding rock tunnel by considering different angles of rock joints (microstructures). It is also shown that the stress distribution is symmetrical when angle equals to 0 ◦ and 90◦ . When the angle is 45◦ the stress presents an asymmetric distribution and the tunnel even comes into being eccentric-pressed.

It should be noted that the effect of *β* is less for smaller *ρ*. When *ρ* = 1.5, the distribution of *σ<sup>h</sup>* is close to each other, that is, the highest *σ<sup>h</sup>* is located near to the top and bottom point of hole boundary and its value close to 3, whereas the lowest *σ<sup>h</sup>* near to right and left points and its value is around 1. Such a result is close to the well-known analytical solution for an infinite isotropic plate with a circular hole [33,44]. However, as *ρ* increases, the directionality of *σ<sup>h</sup>* distribution becomes more evident and the extreme values of *σ<sup>h</sup>*

also vary. For example, when *β* = 0 ◦ , as increase in *ρ* the highest value of *σ<sup>h</sup>* can increase to 9. In addition, the difference between the Cosserat and Cauchy models gets bigger with increasing *ρ*. The smallest values of *σ<sup>h</sup>* are not lower than −3 for the Cosserat continuum, whereas lower *σ<sup>h</sup>* to −9 can be obtained from the Cauchy continuum, especially when *β* = 60◦ , 90◦ , and 120◦ .

By investigating measurements of displacements, stress, etc, and the stress distribution of the square plate with circular hole under horizontal tension stress, a direction effect of microstructures can be found for the present problem. With different transformation angle *β*, the plate can produce mechanical behavior showing obvious directionality. Such a direction effect of microstructures can be also found in previous studies on surrounding rock roadway and tunnels with different dip angles of surroundings [9,45], where the joints of surrounding rocks can represent the microstructure interfaces. In this present paper, similar behaviors are often observed when *β* = 0 ◦ and 90◦ . That is because the microstructures in these two cases are both parallel and perpendicular to the *x*-direction, showing an orthotropic nature. The difference between these two cases is actually due to the different values of *ρ* (*ρ* < 1 when *β* = 90◦ whereas *ρ* > 1 when *β* = 0 ◦ ); therefore, measurements of these two cases have similar behavior but different intensities under the horizontal tension stress. As for microstructures not parallel or perpendicular to the *x*-direction, i.e., *β* = 30◦ , 60◦ , 120◦ and 150◦ , it can be seen from the constitutive matrices (Tables 1–4) that more coupling in the constitutive components such as dilatancy components [46] appear for these cases, showing a centrosymmetric nature. The centrosymmetric material can show different behavior from the orthotropic material by coupling different stresses and strains as well as the couple stresses and curvatures. As the transformed constitutive matrix is related to *β* (Equation (13)), the behavior of the centrosymmetric material depends on *β*. Under horizontal tension stress, it is shown that *u*<sup>1</sup> first increases and then decreases with *β*. Oppositely, as *β* increases the maximum *σ<sup>h</sup>* first decreases and then increases. Such an effect of *β* can be compared with the previous study [8] where a similar effect was found, that is, the strength of phyllite decreases first and then increases with the increase of bedding angle.

The direction effect of microstructures is more evident for higher *ρ*. When *ρ* is small (e.g., *ρ* = 1.5), the width of the rectangular block *b* is close to its height *h*. Thus, the assembly made of such blocks can show a nearly orthotetragonal behavior (close to isotropic) that is less sensitive to change in the microstructure direction. This could also account for the small differences between Cosserat and Cauchy results when *ρ* is small since it was known that orthotetragonal materials are very close to Cauchy continua [30,47].

With the increase in *ρ*, the assembly becomes more orthotropic and the measurements can show more obvious directionality with respect to *β*. As the length of microstructure (i.e., internal length) is introduced to the Cosserat continuum, asymmetries are generated for the shear stress and shear strain fields. Therefore, in the Cosserat model each stress is coupled with asymmetric shear strains through two constitutive components (*Aij*<sup>12</sup> and *Aij*21), and vice-versa. In the Cauchy model, there is just one component (*A*ˆ *ij*12). When *ρ* is small, the difference between *Aij*<sup>12</sup> and *Aij*<sup>21</sup> is not evident. As *ρ* increases, such difference increases rapidly, showing a higher degree of asymmetry. However, *A*ˆ *ij*<sup>12</sup> of the Cauchy model is an arithmetic mean by *Aij*<sup>12</sup> and *Aij*<sup>21</sup> (Equation (14)), which cannot show the asymmetric behavior of continua. Thus, the difference between the results from Cosserat and Cauchy models may be induced, especially for higher *ρ*. Moreover, since the microstructures are considered in the Cosserat continuum, the additional sub-matrix D is involved in the Cosserat constitutive relation rather than the Cauchy one. The components of D are negligible for small *ρ* but become prominent as *ρ* increases. It has been found that the stress can be re-distributed within the Cosserat continuum [34]. For the smaller *ρ*, the Cosserat continuum behaves close to the Cauchy continuum as mentioned above. As a result, the re-distribution of stress can be neglected. However, as the scale of the microstructure increases, such a re-distribution can be more prominent for higher *ρ*. This can be used to

account for the difference in *σ*<sup>22</sup> between the Cosserat and Cauchy continuum, especially for large *ρ*.

### **6. Conclusions**

The present paper investigates the mechanical behavior of the microstructured composite treated as Cosserat continuum by considering various microstructure's directions (*β*) and scales (*ρ*). According to the constitutive parameters obtained from the Cosserat homogenization procedure, the composite studied here can be classified as orthotropic and centrosymmetric materials depending on the direction *β*. The simulations are conducted for a tension problem of a microstructured plate with a circular hole, so this paper also focuses on the stress distribution around the hole. The main conclusions are as follows:


From this present study, the effect of the microstructure's directions on mechanical behavior of microstructured composite can be found, especially for the composite with large scale of the microstructure. The area applying the development of this research can be for microstructured materials with various dimensions (from micromaterials to macromaterials), where the scale of the microstructure should be comparable to the material's dimension. For example, the layered rock with joints has different dip angle due to the geological formation. The stress distribution can be also used for such composite with singularity, not only the circular shape used here, but alternative shapes. The rectangular microstructures with standard interlocking give a basic and important research aspect of the effect of the microstructure's direction and scale. As more and more advanced materials are developed nowadays, various microstructures formations are interesting to be studied in future research.

**Author Contributions:** Conceptualization, N.F. and P.T.; methodology, F.S., N.F. and P.T.; software, F.S., N.F. and P.T.; validation, F.S.; formal analysis, F.S.; investigation, F.S.; resources, N.F. and P.T.; data curation, F.S.; writing—original draft preparation, F.S., N.F. and P.T.; writing—review and editing, F.S., N.F., Y.L., P.T. and Z.W.; visualization, Y.L.; supervision, N.F., P.T. and Z.W.; project administration, N.F. and P.T.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study is supported by: Italian Ministry of University and Research PRIN 2017, project No. 2017HFPKZY (B88D19001130001); Sapienza Research Grants "Progetti Grandi" 2021 (B85F21008380001); Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0570); Fundamental Research Funds for the Central Universities (2020CDJ-LHZZ-004, 2020CDJQY-A046); State Key

Laboratory of Coal Mine Disaster Dynamics and Control (2011DA105287-MS201903). The first author thanks the funds from China Scholarship Council (CSC) (No. 202006050138).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


## *Article* **Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures**

**Francisco Badea <sup>1</sup> , JoseLuis Olazagoitia 1,\* and JesusAngel Perez <sup>2</sup>**


**Abstract:** The finite element analysis of tubular structures is typically based on models constructed employing beam-type elements. This modeling technique provides a quick and computationally efficient option for calculation. Nevertheless, it shows a series of limitations related to the simplicity of this type of element, among which the inability of accounting for the stiffness behavior at the joint level is of notable importance when modeling complex tubular structures. Despite these limitations, the alternative of simulating complex tubular structures with shell- or volume-type elements is highly costly due to the complexity of the modeling process and the computational requirements. Previous research has proposed alternative beam models that improve the estimations when modeling these structures. These research validations were limited to simple models. This paper presents a validation process utilizing a previously developed beam T-junction model in a complex tubular structure, intended to be representative for buses' and coaches' upper structures. Results obtained reveal that the accuracy of beam element type models can be significantly improved with the adequate implementation of elastic elements to account for the real junction stiffness.

**Keywords:** finite element analysis; structural optimization; beam T-junctions; beam model validation

## **1. Introduction**

The study and analysis of structures using the finite element method (FEM) is key to the development of complex structural systems [1]. In fact, CAD design and FEM analysis allow for the quick evolution of a sketch from a design to a structure capable of withstanding the required stresses, and these methods are widely accepted in the industry.

In FEM programs, it is possible, depending on the model to be represented, to choose three main types of elements: beam, shell, and volume elements. Although there are certain rules, the selection of the best type of element for a particular application is a generally complicated process, in which several factors must be considered. For example, it is necessary to know the characteristics and complexity of the structures to be simulated, the limitations of existing computational resources, the type of simulation, and the expected accuracy of the computational results [2–4].

The analyst's experience is often key in determining the best path to take in each case. In the particular case of large tubular structures such as those that can be found in buses and coaches, it is very common, for example, to use beam-type elements, due to the flexibility, limited computational need, and speed that they allow in the analysis of complex structures [5–7]. Models based on beam elements are comparatively simpler than those based on shell or volume types.

Although beam elements are often the preferred choice for tubular profile structures, they have some fundamental limitations due to their simple formulation. One of the main limitations is the impossibility to faithfully reproduce the localized behavior at the

**Citation:** Badea, F.; Olazagoitia, J.; Perez, J. Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures. *Materials* **2022**, *15*, 6468. https:// doi.org/10.3390/ma15186468

Academic Editor: Roberto Citarella

Received: 5 August 2022 Accepted: 14 September 2022 Published: 17 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

joint level. In these elements, the attachment with the environment is reduced to a single infinitely rigid node [8,9], which leads to estimation errors in the calculation [10].

The use of shell and volume elements in tubular element joints makes it possible to obtain more realistic joint models [1,11]. This is possible because the joint topology at the joint level can be captured by these element types with higher accuracy. In this way, it is possible to obtain accurate models that overcome the limitations of beam-type elements in this type of tubular structures. Previous research has shown that the differences in the calculation of stiffness of tubular structures modeled with beam elements versus models made with shell or volume models can vary between 5 and 45% [10,12]. The final error depends on various parameters of the tubular elements used, such as the shape of the tube cross-section and its thickness, as well as the complexity of the overall structure. The greater the complexity of the structure, the greater the difference in stiffness calculation between models made with different types of elements, since the error induced at each joint will have a deviating effect in the global response.

Tubular structural elements can be used in many applications, such as buses and coaches. The stiffness of the structure in buses uses large tubular elements and their configuration is often considerably complex. It is usual to use beam elements in their calculation since modeling with shell or volume elements is usually very time-consuming due to the complexity of the models [9]. In addition, they require a higher computational power that on many occasions does not compensate the accuracy of the obtained results.

For this reason, it would be desirable for the modeling of this type of structure to have a calculation methodology that takes advantage of the simplicity of beam-type tubular structures but improves the results through the altered modeling of the joints. There can be found several modelling proposals in the bibliography to improve ordinary beam elements accuracy. For example, B. Horton et al. [13] proposed models with modified stiffness characteristics at the adjacent elements of the joints. The same methodology was applied in [14] to account for the local stiffness modification due to crack development in beam-type structures. Additionally, hybrid models, where shell or volume element types are used for the joints, and beam elements for the rest of the beam sections, can be found with significant improvements reported [15,16]. This shell-beam or volume-beam hybrid modelling technique is still time-consuming since connection elements are to be configured at zones where the element type is modified, and thus not attractive for larger models with an important number of joints to be configured. Finally, several approaches can be found that focus on modifying the stiffness characteristics of the joint by introducing elastic elements [8,17].

This article presents a complete methodology for comparing different modelling techniques of beam-type structures. In the study, volume, ordinary beam, and alternate beam modelling techniques, presented in [8], are analyzed. The results are validated against a real structure with respect to which the results are compared and the best methodology to obtain a model based on beam-type joints with improved T-junctions is proposed.

To understand the methodology used, it is important to present the way of modeling the junctions between tubular elements. For simplicity, two possibilities of T-junctions between tubular elements are presented, which will be called T1 and T2. Figure 1 presents graphically the effective connections between these tubular elements.

It is important to note that the type of actual T-joint (either T1 or T2) has a direct influence on the final behavior of the structure, determining its behavior as a function of the type, size, and direction of the load applied to it. Regardless of this reality, the pure equivalent model based on beam elements will in all cases be composed of four nodes and three beam elements. Therefore, these results in the T1 or T2 junctions are blurred in the model, resulting in an infinitely stiff joint in all cases and a loss of the starting information.

*Materials* **2022**, *15*, 6468 3 of 15

**Figure 1.** T-junctions equivalent beam-type element model. T1 junction with two elements (green and purple) and T2 with three elements (green, purple and dark blue). **Figure 1.** T-junctions equivalent beam-type element model. T1 junction with two elements (green and purple) and T2 with three elements (green, purple and dark blue). equivalent model based on beam elements will in all cases be composed of four nodes and three beam elements. Therefore, these results in the T1 or T2 junctions are blurred in the model, resulting in an infinitely stiff joint in all cases and a loss of the starting information.

It is important to note that the type of actual T-joint (either T1 or T2) has a direct influence on the final behavior of the structure, determining its behavior as a function of the type, size, and direction of the load applied to it. Regardless of this reality, the pure equivalent model based on beam elements will in all cases be composed of four nodes and three beam elements. Therefore, these results in the T1 or T2 junctions are blurred in the model, resulting in an infinitely stiff joint in all cases and a loss of the starting information. To illustrate how the T-joint model of Figure 1 is modified in [8], Figure 2 presents To illustrate how the T-joint model of Figure 1 is modified in [8], Figure 2 presents two joint schemas, where the junction of both tubular elements is rigid in its left image (a) and is flexible in the right image (b). This flexibility is represented by the introduction of stiffness k1 to k6 at the intersection point of both tubular elements. These stiffness values are experimentally pre-determined to faithfully represent real T-joints. These stiffness values do not change the dimensions of the structure in any way. The direct effect of theminto the joint causes the displacement <sup>δ</sup>1 > <sup>δ</sup> in all cases. To illustrate how the T-joint model of Figure 1 is modified in [8], Figure 2 presents two joint schemas, where the junction of both tubular elements is rigid in its left image (a) and is flexible in the right image (b). This flexibility is represented by the introduction of stiffness k1 to k6 at the intersection point of both tubular elements. These stiffness values are experimentally pre-determined to faithfully represent real T-joints. These stiffness values do not change the dimensions of the structure in any way. The direct effect of them into the joint causes the displacement δ1 > δ in all cases.

two joint schemas, where the junction of both tubular elements is rigid in its left image (a)

**Figure 2.** Diagrams comparing the behavior of a standard beam T-junction and the alternative beam T-junction model for the same load state. (**a**) Rigid junction (**b**) flexible junction **Figure 2.** Diagrams comparing the behavior of a standard beam T-junction and the alternative beam T-junction model for the same load state. (**a**) Rigid junction (**b**) flexible junction.

**Figure 2.** Diagrams comparing the behavior of a standard beam T-junction and the alternative beam T-junction model for the same load state. (**a**) Rigid junction (**b**) flexible junction The consequence of the implementation of the junction-specific stiffness is an improvement in the accuracy, as it was already shown and validated for simple structural components in [8]. Nevertheless, a more realistic validation against a complex structure was still pending. This paper covers this aspect, showing the capability of the proposed The consequence of the implementation of the junction-specific stiffness is an improvement in the accuracy, as it was already shown and validated for simple structural components in [8]. Nevertheless, a more realistic validation against a complex structure was still pending. This paper covers this aspect, showing the capability of the proposed methodology to increase beam-based FEM models' accuracy and describing some key aspects to take into account when scaling from single-joint to complete complex structures. The consequence of the implementation of the junction-specific stiffness is an improvement in the accuracy, as it was already shown and validated for simple structural components in [8]. Nevertheless, a more realistic validation against a complex structure was still pending. This paper covers this aspect, showing the capability of the proposed methodology to increase beam-based FEM models' accuracy and describing some key aspects to take into account when scaling from single-joint to complete complex structures. The application scope of this methodology is limited to the elastic deformation range of the structure, since the spring elements introduced are linear, i.e., k1 to k6 stiffnesses are constant. Equivalent methodology could be developed with nonlinear springs and damping elements to analyze structures responses beyond the elastic range.

methodology to increase beam-based FEM models' accuracy and describing some key aspects to take into account when scaling from single-joint to complete complex structures. In summary, the present work, applied together with the results supplied in [8], provides a complete methodology for improving beam-type element models accuracy

for complex structures typically used in buses and coaches, without incurring increased computational costs and model preparation times. complex structures typically used in buses and coaches, without incurring increased computational costs and model preparation times.

The application scope of this methodology is limited to the elastic deformation range of the structure, since the spring elements introduced are linear, i.e., k1 to k6 stiffnesses are constant. Equivalent methodology could be developed with nonlinear springs and damp-

In summary, the present work, applied together with the results supplied in [8], provides a complete methodology for improving beam-type element models accuracy for

*Materials* **2022**, *15*, 6468 4 of 15

ing elements to analyze structures responses beyond the elastic range.

#### **2. Methodology 2. Methodology**

The results of the alternative T-junction model were verified for the simple T-junctions analyzed [8]. In that work, the complete methodology to obtain the proper spring stiffness values for each type of junction is described. Nonetheless, extrapolating the results to more complex tubular structures require a specific experimental validation process to assess the improvement capability of the alternative beam model proposed when crossed influences take place among the joints of the structure. The results of the alternative T-junction model were verified for the simple T-junctions analyzed [8]. In that work, the complete methodology to obtain the proper spring stiffness values for each type of junction is described. Nonetheless, extrapolating the results to more complex tubular structures require a specific experimental validation process to assess the improvement capability of the alternative beam model proposed when crossed influences take place among the joints of the structure. As the research carried out was intended to be applied to buses' and coaches' upper

As the research carried out was intended to be applied to buses' and coaches' upper structures, it was sought to use structures of these characteristics for the validation experiments. Since these structures are hard to come upon, a validation structure was designed and built having representative characteristics of buses' and coaches' upper structures (Figure 3). structures, it was sought to use structures of these characteristics for the validation experiments. Since these structures are hard to come upon, a validation structure was designed and built having representative characteristics of buses' and coaches' upper structures (Figure 3).

**Figure 3.** Tubular structure for the experimental validation. **Figure 3.** Tubular structure for the experimental validation.

The response of the validation structure under the prescribed load is captured by means of analog dials and later compared to the following finite element models: The response of the validation structure under the prescribed load is captured by means of analog dials and later compared to the following finite element models:


A further description on the validation structure and the finite element models is given in the following section.

The measured displacements at the evaluation points are then compared to the predictions of the different models in order to assess the accuracy of the models with respect to reality and the improvements achieved with the alternative beam models. Results will be discussed in Section 4.

#### **3. Experimental Validation 3. Experimental Validation**  *3.1. Validation Structure Description*

**3. Experimental Validation** 

be discussed in Section 4.

be discussed in Section 4.

#### *3.1. Validation Structure Description 3.1. Validation Structure Description*  The tubular validation structure was built with two significantly different sides from

given in the following section.

given in the following section.

*Materials* **2022**, *15*, 6468 5 of 15

*Materials* **2022**, *15*, 6468 5 of 15

The tubular validation structure was built with two significantly different sides from a geometric point of view. One of the sides was configured in a very similar way to what is commonly found in buses and coaches (Figure 3, number 2), whereas the opposite side had a significantly asymmetric configuration (Figure 3, number 1), so that a wide range of joint configurations is analyzed during the tests. Figure 4 shows a detail of the dimensions of both sides of the structure. The tubular validation structure was built with two significantly different sides from a geometric point of view. One of the sides was configured in a very similar way to what is commonly found in buses and coaches (Figure 3, number 2), whereas the opposite side had a significantly asymmetric configuration (Figure 3, number 1), so that a wide range of joint configurations is analyzed during the tests. Figure 4 shows a detail of the dimensions of both sides of the structure. a geometric point of view. One of the sides was configured in a very similar way to what is commonly found in buses and coaches (Figure 3, number 2), whereas the opposite side had a significantly asymmetric configuration (Figure 3, number 1), so that a wide range of joint configurations is analyzed during the tests. Figure 4 shows a detail of the dimensions of both sides of the structure.

A further description on the validation structure and the finite element models is

A further description on the validation structure and the finite element models is

The measured displacements at the evaluation points are then compared to the predictions of the different models in order to assess the accuracy of the models with respect to reality and the improvements achieved with the alternative beam models. Results will

The measured displacements at the evaluation points are then compared to the predictions of the different models in order to assess the accuracy of the models with respect to reality and the improvements achieved with the alternative beam models. Results will

**Figure 4.** Geometric configuration of the sides of the tubular validation structure. Unit: mm. **Figure 4.** Geometric configuration of the sides of the tubular validation structure. Unit: mm. **Figure 4.** Geometric configuration of the sides of the tubular validation structure. Unit: mm.

The experiments with the tubular validation structure were done on a universal test bench (Figure 3, number 3). Additionally, a solution had to be found to properly constrain the structure during testing to avoid errors introduced by the displacement of the restriction points, since these displacements are not accounted for on virtual FEM models. To minimize these errors, 100 mm × 100 mm × 10 mm L-shaped profiles welded to the base profiles of the tubular structure (Figure 3, number 4) were used. In all, 8 restriction points were used. To ensure proper clamping at the restrictions' points, bolts were tightened to 70% of their yield strength limit. Figure 5 illustrates welds in one of the base profiles with the clamping devices and a rectangular profile. The experiments with the tubular validation structure were done on a universal test bench (Figure 3, number 3). Additionally, a solution had to be found to properly constrain the structure during testing to avoid errors introduced by the displacement of the restriction points, since these displacements are not accounted for on virtual FEM models. To minimize these errors, 100 mm × 100 mm × 10 mm L-shaped profiles welded to the base profiles of the tubular structure (Figure 3, number 4) were used. In all, 8 restriction points were used. To ensure proper clamping at the restrictions' points, bolts were tightened to 70% of their yield strength limit. Figure 5 illustrates welds in one of the base profiles with the clamping devices and a rectangular profile. The experiments with the tubular validation structure were done on a universal test bench (Figure 3, number 3). Additionally, a solution had to be found to properly constrain the structure during testing to avoid errors introduced by the displacement of the restriction points, since these displacements are not accounted for on virtual FEM models. To minimize these errors, 100 mm × 100 mm × 10 mm L-shaped profiles welded to the base profiles of the tubular structure (Figure 3, number 4) were used. In all, 8 restriction points were used. To ensure proper clamping at the restrictions' points, bolts were tightened to 70% of their yield strength limit. Figure 5 illustrates welds in one of the base profiles with the clamping devices and a rectangular profile.

**Figure 5.** Clamping weld details. **Figure 5.** Clamping weld details. **Figure 5.** Clamping weld details.

#### *3.2. Experimental Setup*

The following Figure 6 shows an overview of the experimental setup assembly of the validation structure over the test bench. The structure (Figure 6, number 8) was conceived to be tested on a universal test bench (Figure 6, number 5) for which clamping devices had to be used (Figure 6, number 9), which were first welded to the base profiles of the validation structure and then bolted to the universal test bench with M20 bolts and threaded metal blocks.

weight was used (Figure 6, number 6) as a load input.

tion of the H-shaped test bench supports in that direction.

ber 2) tied to the corresponding eyebolts (Figure 6, number 3).

*3.2. Experimental Setup* 

metal blocks.

**Figure 6.** Three-dimensional CAD model with the characteristics of the experimental assembly corresponding to the testing of the validation structure. **Figure 6.** Three-dimensional CAD model with the characteristics of the experimental assembly corresponding to the testing of the validation structure.

The following Figure 6 shows an overview of the experimental setup assembly of the validation structure over the test bench. The structure (Figure 6, number 8) was conceived to be tested on a universal test bench (Figure 6, number 5) for which clamping devices had to be used (Figure 6, number 9), which were first welded to the base profiles of the validation structure and then bolted to the universal test bench with M20 bolts and threaded

In order to minimize uncertainties with respect to the force applied, a calibrated

With the objective of achieving a wide range characterization of loading, composed load states were input to the structure. The direction of the force was accurately selected by changing the orientation of the pulley (Figure 6, number 1) that conveys the force of the nylon cord. Additionally, by using H-shaped test bench supports, the height of the pulley could be changed to modify the vertical angle of the force applied (Figure 6, number 7). Finally, a deviation in the transverse direction was obtained by changing the posi-

The load was transferred to the structure by means of a nylon thread (Figure 6, num-

To obtain a complete and representative characterization of the validation structure, a total of 14 measuring points were defined and distributed throughout its geometry so In order to minimize uncertainties with respect to the force applied, a calibrated weight was used (Figure 6, number 6) as a load input.

that representative information of the global response was obtained. It was also taken into consideration the necessity to define measuring sections at different distances from the joints themselves and/or the clamping devices, so that a clearer picture of the behavior of the beam-type elements could be studied. Figure 7 shows a 3D CAD model of the validation structure on which the defined measuring sections are indicated. Two dial gauges were used to capture the displacement at the measuring sections, With the objective of achieving a wide range characterization of loading, composed load states were input to the structure. The direction of the force was accurately selected by changing the orientation of the pulley (Figure 6, number 1) that conveys the force of the nylon cord. Additionally, by using H-shaped test bench supports, the height of the pulley could be changed to modify the vertical angle of the force applied (Figure 6, number 7). Finally, a deviation in the transverse direction was obtained by changing the position of the H-shaped test bench supports in that direction.

one to measure small displacements of 0–5 mm with a 0.001 mm resolution, and another for larger displacements of 0–25 mm with a 0.01 mm resolution. The load was transferred to the structure by means of a nylon thread (Figure 6, number 2) tied to the corresponding eyebolts (Figure 6, number 3).

Due to the number of sections and their different locations in the validation structure, specific supports were used to fix the dial gauges depending on the sections measured. To obtain a complete and representative characterization of the validation structure, a total of 14 measuring points were defined and distributed throughout its geometry so that representative information of the global response was obtained. It was also taken into consideration the necessity to define measuring sections at different distances from the joints themselves and/or the clamping devices, so that a clearer picture of the behavior of the beam-type elements could be studied. Figure 7 shows a 3D CAD model of the validation structure on which the defined measuring sections are indicated.

Two dial gauges were used to capture the displacement at the measuring sections, one to measure small displacements of 0–5 mm with a 0.001 mm resolution, and another for larger displacements of 0–25 mm with a 0.01 mm resolution.

Due to the number of sections and their different locations in the validation structure, specific supports were used to fix the dial gauges depending on the sections measured.

Of the 14 sections defined, in four of them (a3, a4, a8, and a9), only the displacements in the Y direction were characterized since these sections were in the base profiles of the structure in regions very close to the clamping devices, so the displacements in the other directions could be neglected. In the rest of the sections, displacements in the two most significant directions were characterized. For example, in section a1, the displacements were measured in directions X and Y, whereas in section a13, the displacements were measured in directions Z and Y.

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**Figure 7.** Sections used to characterize the validation structure, identified as a1 to a14. **Figure 7.** Sections used to characterize the validation structure, identified as a1 to a14. were measured in directions X and Y, whereas in section a13, the displacements were measured in directions Z and Y.

Of the 14 sections defined, in four of them (a3, a4, a8, and a9), only the displacements in the Y direction were characterized since these sections were in the base profiles of the As with the validation experiments for the simple junctions [8], the following set of criteria was adopted to minimize the sources of error and achieve better-quality results: As with the validation experiments for the simple junctions [8], the following set of criteria was adopted to minimize the sources of error and achieve better-quality results:


**Figure 8.** Mounting of the dial gauge for the displacement measurement of the a2 section. **Figure 8.** Mounting of the dial gauge for the displacement measurement of the a2 section.

**Figure 8.** Mounting of the dial gauge for the displacement measurement of the a2 section. The same preparation process was used for each of the sections defined in the validation structure. To accept the measurements, the same quality criteria were used as for the experimental analyses of the simple junctions [8]. In this way, the standard deviations for a set of measurements in the same section were less than 0.005 mm for the 0.001 mm resolution dial gauges, and less than 0.05 mm for the 0.01 mm resolution dial gauges. For each of the sections defined, load and unload cycles were carried out by means of a hydraulic actuator which gave and released support to the calibrated weights during each

cycle. A total of 15 measurements were taken for each section, which were found to be an optimal compromise for obtaining precise results with the least number of measurements per point. *Materials* **2022**, *15*, 6468 8 of 15 **Figure 9.** Detail of the region adjacent to the a13 measurement section.

**Figure 9.** Detail of the region adjacent to the a13 measurement section. **Figure 9.** Detail of the region adjacent to the a13 measurement section. *3.3. FEM Model of the Validation Structure Modeled with Beam-Type Elements* 

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#### *3.3. FEM Model of the Validation Structure Modeled with Beam-Type Elements* To model the validation structure with beam-type elements, the modeling principles

The same preparation process was used for each of the sections defined in the validation structure. To accept the measurements, the same quality criteria were used as for the experimental analyses of the simple junctions [8]. In this way, the standard deviations for a set of measurements in the same section were less than 0.005 mm for the 0.001 mm resolution dial gauges, and less than 0.05 mm for the 0.01 mm resolution dial gauges. For each of the sections defined, load and unload cycles were carried out by means of a hy-To model the validation structure with beam-type elements, the modeling principles set out in [8] were adopted. Following the methodology proposed in this sub-section, the principal axes for each of the tubular profiles of the structure were identified and extracted. The regions where there were separations between the beam elements were also identified and corrected. Figure 10 shows the validation structure model built with beam-type elements. set out in [8] were adopted. Following the methodology proposed in this sub-section, the principal axes for each of the tubular profiles of the structure were identified and extracted. The regions where there were separations between the beam elements were also identified and corrected. Figure 10 shows the validation structure model built with beamtype elements.

**Figure 10.** Validation structure modeled with beam-type elements. Beam colors refer to the following sections employed: Light blue: 80 mm × 80 mm × 3 mm, red: 80 mm × 60 mm × 3 mm, purple: 80 mm × 60 mm × 2 mm, green: 80 mm × 40 mm × 3 mm, lilac: 80 mm × 40 mm × 2 mm, dark blue: 40 mm × 40 mm × 3 mm. **Figure 10.** Validation structure modeled with beam-type elements. Beam colors refer to the following sections employed: Light blue: 80 mm × 80 mm × 3 mm, red: 80 mm × 60 mm × 3 mm, purple: 80 mm × 60 mm × 2 mm, green: 80 mm × 40 mm × 3 mm, lilac: 80 mm × 40 mm × 2 mm, dark blue: 40 mm × 40 mm × 3 mm.

This figure presents two superimposed representations of the validation structure. The original base model built with beam-type elements is represented with black dotted lines, to which the model generated by a graphic option of the program was superimposed to reveal the elements, based on the cross-section properties defined. Key points were This figure presents two superimposed representations of the validation structure. The original base model built with beam-type elements is represented with black dotted lines, to which the model generated by a graphic option of the program was superimposed to reveal the elements, based on the cross-section properties defined. Key points were defined at the locations of the measuring sections in order to ensure that a node would be present in these points and therefore nodal displacement results could be extracted.

The equivalent moments of inertia were calculated in the cross-section of the clamping devices and were assigned to the beam elements, as shown in Figure 11.

**Figure 10.** Validation structure modeled with beam-type elements. Beam colors refer to the following sections employed: Light blue: 80 mm × 80 mm × 3 mm, red: 80 mm × 60 mm × 3 mm, purple: 80 mm × 60 mm × 2 mm, green: 80 mm × 40 mm × 3 mm, lilac: 80 mm × 40 mm × 2 mm, dark blue:

This figure presents two superimposed representations of the validation structure. The original base model built with beam-type elements is represented with black dotted lines, to which the model generated by a graphic option of the program was superimposed to reveal the elements, based on the cross-section properties defined. Key points were

40 mm × 40 mm × 3 mm.

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**Figure 11.** Characteristics of the structure and calculation of the moment of inertia equivalent to the cross-section in the regions of the clamping devices. **Figure 11.** Characteristics of the structure and calculation of the moment of inertia equivalent to the cross-section in the regions of the clamping devices. **Figure 11.** Characteristics of the structure and calculation of the moment of inertia equivalent to the cross-section in the regions of the clamping devices.

ing devices and were assigned to the beam elements, as shown in Figure 11.

ing devices and were assigned to the beam elements, as shown in Figure 11.

defined at the locations of the measuring sections in order to ensure that a node would be present in these points and therefore nodal displacement results could be extracted.

The equivalent moments of inertia were calculated in the cross-section of the clamp-

defined at the locations of the measuring sections in order to ensure that a node would be present in these points and therefore nodal displacement results could be extracted.

The equivalent moments of inertia were calculated in the cross-section of the clamp-

#### *3.4. Characteristics of the Validation Structure Modeled with Alternative Beam T-Junctions 3.4. Characteristics of the Validation Structure Modeled with Alternative Beam T-Junctions 3.4. Characteristics of the Validation Structure Modeled with Alternative Beam T-Junctions*

Implementing the alternative beam T-junction elements in accordance with the developed methodology requires determining the dimensional characteristics and the configuration of each of the joints (T1 or T2). A total of 38 joints were identified in the validation structure, as can be seen in the diagram presented in Figure 12. In this figure, each one of the junctions was identified with a circle and a number; for clarity purposes, one of Implementing the alternative beam T-junction elements in accordance with the developed methodology requires determining the dimensional characteristics and the configuration of each of the joints (T1 or T2). A total of 38 joints were identified in the validation structure, as can be seen in the diagram presented in Figure 12. In this figure, each one of the junctions was identified with a circle and a number; for clarity purposes, one of the sides has red circles and the other side green circles. Implementing the alternative beam T-junction elements in accordance with the developed methodology requires determining the dimensional characteristics and the configuration of each of the joints (T1 or T2). A total of 38 joints were identified in the validation structure, as can be seen in the diagram presented in Figure 12. In this figure, each one of the junctions was identified with a circle and a number; for clarity purposes, one of the sides has red circles and the other side green circles.

**Figure 12.** Diagram identifying the total number of joints in the validation structure. **Figure 12.** Diagram identifying the total number of joints in the validation structure.

**Figure 12.** Diagram identifying the total number of joints in the validation structure. Implementing the alternative junctions into a complex tubular structure requires identifying each of the junctions of the structure in accordance with the analyzed joint configurations (T1 or T2). As presented in Figure 13, most of the junctions can be easily Implementing the alternative junctions into a complex tubular structure requires identifying each of the junctions of the structure in accordance with the analyzed joint configurations (T1 or T2). As presented in Figure 13, most of the junctions can be easily assimilated with T1 or T2, as is the case with points 1 to 3, but some others cannot be easily determined or just have no clear equivalence. Over the 38 joints, 33 of them were easily assimilated to T1 or T2, whereas for 5 joints (4, 8, 16, 20, and 38), this similitude was not obvious. Implementing the alternative junctions into a complex tubular structure requires identifying each of the junctions of the structure in accordance with the analyzed joint configurations (T1 or T2). As presented in Figure 13, most of the junctions can be easily assimilated with T1 or T2, as is the case with points 1 to 3, but some others cannot be easily determined or just have no clear equivalence. Over the 38 joints, 33 of them were easily assimilated to T1 or T2, whereas for 5 joints (4, 8, 16, 20, and 38), this similitude was not obvious.

assimilated with T1 or T2, as is the case with points 1 to 3, but some others cannot be easily determined or just have no clear equivalence. Over the 38 joints, 33 of them were easily assimilated to T1 or T2, whereas for 5 joints (4, 8, 16, 20, and 38), this similitude was not obvious. To further analyze the junctions with no clear equivalence to T1 or T2 configurations, different models with various combinations were developed and evaluated. Although a complete description of all the combinations analyzed is out of the scope of the article, the best three configurations will be described in the following paragraphs (Figure 14).

For the first variant (a), joint 8 was modeled as a single type T2 joint, whereas for joints 4, 16, 20, and 38, the modeling characteristics were adhered to (a T1 and a T2) but changed the direction of application of the type T2 junction. For the second variant (b), it was decided to model all the type T1 and T2 joints that could be clearly identified in the structure and to keep all the joints showing uncertainties (8, 4, 16, 20, 38) as normal junctions. Finally, for the third variant (c), all the conflictive joints were assimilated as type T1 joints, keeping all the continuous profiles unmodified.

**Figure 13.** Equivalence of the complex structure joints to the T1 and T2 junctions. **Figure 13.** Equivalence of the complex structure joints to the T1 and T2 junctions. best three configurations will be described in the following paragraphs (Figure 14).

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**Figure 14.** Joint configurations analyzed. Variant (**a**), Variant(**b**) and Variant (**c**). **Figure 14.** Joint configurations analyzed. Variant (**a**), Variant(**b**) and Variant (**c**).

For the first variant (a), joint 8 was modeled as a single type T2 joint, whereas for joints 4, 16, 20, and 38, the modeling characteristics were adhered to (a T1 and a T2) but changed the direction of application of the type T2 junction. For the second variant (b), it The stiffness values of the elastic elements used for each of the analyzed joints were found through comparative simulations using joints modeled with volume-type elements as reference models, following the recommendations of [8].

**Figure 14.** Joint configurations analyzed. Variant (**a**), Variant(**b**) and Variant (**c**). For the first variant (a), joint 8 was modeled as a single type T2 joint, whereas for joints 4, 16, 20, and 38, the modeling characteristics were adhered to (a T1 and a T2) but It should be highlighted that the stiffness values at the joint levels obtained with the methodology of [8] are significantly high, ranging between 1 <sup>×</sup> <sup>10</sup>5–1 <sup>×</sup> <sup>10</sup><sup>6</sup> (N/m). These stiffnesses in some manner quantify the contribution of the joint to the deformation of the T-junctions in the linear domain. Ideally, it would be desired to improve each junction

changed the direction of application of the type T2 junction. For the second variant (b), it

with sets of elastic elements so that the complex beam structure provides the most accurate possible results. Despite that the assimilation to a joint type in 5 out of the 38 joints of the validation structure is not obvious, it was demonstrated in [8] that although between the T1 and T2 junctions there are significant differences, these differences were comparatively lower than the ones in comparison to the regular beam junction. In other words, assigning a junction as T1 when it is a T2 would induce fewer deviations than just having it as a regular beam. with sets of elastic elements so that the complex beam structure provides the most accurate possible results. Despite that the assimilation to a joint type in 5 out of the 38 joints of the validation structure is not obvious, it was demonstrated in [8] that although between the T1 and T2 junctions there are significant differences, these differences were comparatively lower than the ones in comparison to the regular beam junction. In other words, assigning a junction as T1 when it is a T2 would induce fewer deviations than just having it as a regular beam.

was decided to model all the type T1 and T2 joints that could be clearly identified in the structure and to keep all the joints showing uncertainties (8, 4, 16, 20, 38) as normal junctions. Finally, for the third variant (c), all the conflictive joints were assimilated as type T1

The stiffness values of the elastic elements used for each of the analyzed joints were found through comparative simulations using joints modeled with volume-type elements

It should be highlighted that the stiffness values at the joint levels obtained with the methodology of [8] are significantly high, ranging between 1 × 105–1 × 106 (N/m). These stiffnesses in some manner quantify the contribution of the joint to the deformation of the T-junctions in the linear domain. Ideally, it would be desired to improve each junction

#### *3.5. Characteristics of the Validation Structure Modeled with Volume-Type Elements 3.5. Characteristics of the Validation Structure Modeled with Volume-Type Elements*

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joints, keeping all the continuous profiles unmodified.

as reference models, following the recommendations of [8].

An additional model of the validation structure was constructed using volume-type elements. Although modeling rectangular beam section structures with volume-type elements has been demonstrated to be an excessively costly approach, it was decided to configure this model to be used for comparison purposes, due to its intrinsic high accuracy. An additional model of the validation structure was constructed using volume-type elements. Although modeling rectangular beam section structures with volume-type elements has been demonstrated to be an excessively costly approach, it was decided to configure this model to be used for comparison purposes, due to its intrinsic high accuracy.

The following Figure 15 illustrates the structure modelled with volume-type elements. It was modelled using linear hexahedral elements of size 3 mm, which corresponds to the minimum thickness of the rectangular beams employed in the validation structure. Tube joints were modeled by means of bonded-type contacts, without considering the weld seam geometry. The following Figure 15 illustrates the structure modelled with volume-type elements. It was modelled using linear hexahedral elements of size 3 mm, which corresponds to the minimum thickness of the rectangular beams employed in the validation structure. Tube joints were modeled by means of bonded-type contacts, without considering the weld seam geometry.

**Figure 15.** Detailed model of the validation structure, modeled with volume-type elements. **Figure 15.** Detailed model of the validation structure, modeled with volume-type elements.

#### **4. Results and Discussion 4. Results and Discussion**

Comparative analyses were performed for the evaluation of the improvements between the validation structure and the different finite element models of the validation Comparative analyses were performed for the evaluation of the improvements between the validation structure and the different finite element models of the validation structure. Table 1 summarizes the displacement relative deviations for each of the finite element models evaluated with respect to the values measured in the experimentally analyzed validation structures.

From the observation of the results Table 1, the following aspects can be observed:


3. Focusing on alternative beam models, variant (c) shows the best approximations to the experimental validation structure and to the detailed volume element model. The calculations are even better displacement estimations than those of the latter at a5 and a10 sections.


**Table 1.** Deflection relative differences of the FEM models with respect to the experimental results.

Since the ultimate objective of the implementation of alternative beam models is to improve the accuracy of tubular structures modeled in this manner, it was considered necessary to perform an additional evaluation of the capability of the models to predict stress distributions.

It was observed in [8] that the stress distributions resulting from analyzing the simple junctions modeled with beam or alternative beam type elements showed no significant differences. Nevertheless, it seems interesting to extend the analysis to complex structures, where the effect of the local stiffness modification at the joint level might show crossed influences.

Due to the inherent limitation of beam-type elements to account for local stress raisins at the joint level, the analysis is focused on the comparison of the general stress distribution maps. The volume element type model was used as a reference for the evaluation, since this type of model, although discarded for general use due to the high computational costs that it entails, can predict stress maps with high precision.

The resulting Von Mises stress distributions obtained for volume, ordinary beam, and the alternative beam with the best results (c) are showed in Figure 16. It is observed that both ordinary and alternative beam models show rather similar stress distributions, which are also similar to the reference volume model.

Although a detailed stress analysis of the models is not in the scope of the article, it is noted that the significant improvement in terms of deflection observed on the alternative beam model has limited influence on the general stress distribution for complex structures.

which are also similar to the reference volume model.

**Figure 16.** Von Mises stress maps for the validation structure modeled with volume, beam, and alternative beam type elements [MPa]. **Figure 16.** Von Mises stress maps for the validation structure modeled with volume, beam, and alternative beam type elements [MPa].

distribution maps. The volume element type model was used as a reference for the evaluation, since this type of model, although discarded for general use due to the high com-

The resulting Von Mises stress distributions obtained for volume, ordinary beam, and the alternative beam with the best results (c) are showed in Figure 16. It is observed that both ordinary and alternative beam models show rather similar stress distributions,

Although a detailed stress analysis of the models is not in the scope of the article, it is noted that the significant improvement in terms of deflection observed on the alternative beam model has limited influence on the general stress distribution for complex struc-

putational costs that it entails, can predict stress maps with high precision.

#### **5. Conclusions 5. Conclusions**

tures.

In the present paper, a test of specific complex structures was designed and built in order to conduct the validation of the alternative beam element type modeling technique presented in [8]. In the present paper, a test of specific complex structures was designed and built in order to conduct the validation of the alternative beam element type modeling technique presented in [8].

The structure was subjected to a controlled load, and displacements were captured at a total of 14 measuring sections for comparison purposes. The structure was subjected to a controlled load, and displacements were captured at a total of 14 measuring sections for comparison purposes.

For such complex structures, the application of the stiffness accuracy improvement by means of elastic elements becomes less obvious, since the assignation of T1 or T2 joints' configuration is not evident anymore for some joints. In the case of the validation of the structure used in this work, a total of five junctions were identified in which the selection was not clear and so the optimization of those particular junctions implied a detailed specific analysis for which different configurations were modelled for these junctions and compared to the experimental data, together with an ordinary beam element type and a volume element type model. For such complex structures, the application of the stiffness accuracy improvement by means of elastic elements becomes less obvious, since the assignation of T1 or T2 joints' configuration is not evident anymore for some joints. In the case of the validation of the structure used in this work, a total of five junctions were identified in which the selection was not clear and so the optimization of those particular junctions implied a detailed specific analysis for which different configurations were modelled for these junctions and compared to the experimental data, together with an ordinary beam element type and a volume element type model.

For the ordinary-type elements, the deviations with respect to the validation structure were found to be notable, with the beam model being approximately 58% stiffer. For the ordinary-type elements, the deviations with respect to the validation structure were found to be notable, with the beam model being approximately 58% stiffer.

By using the developed alternative T-junctions, it was shown that the characteristics of beam models' behavior can be modified. Evaluating the differences between the experimental structure and the models built, and ignoring the measurements corresponding to the support beams of the structure (sections a3, a4, a8, and a9), a reduction in the average deviations from almost 49% to 14% was achieved. It was also noticed that these models are indeed affected by the proper choice of elastic element configuration, ranging from an average error of 27% to 14%. In any case, even the worst selection of the elastic elements By using the developed alternative T-junctions, it was shown that the characteristics of beam models' behavior can be modified. Evaluating the differences between the experimental structure and the models built, and ignoring the measurements corresponding to the support beams of the structure (sections a3, a4, a8, and a9), a reduction in the average deviations from almost 49% to 14% was achieved. It was also noticed that these models are indeed affected by the proper choice of elastic element configuration, ranging from an average error of 27% to 14%. In any case, even the worst selection of the elastic elements for the junctions shows a significant accuracy improvement with respect to the ordinary beam model. Additionally, by analyzing the stress distributions in the different finite element models, it was found that the elastic elements inserted into the joints did not show significant influence on the stress distribution.

From the results presented in this work, it can be concluded that the utilization of the alternative beam T-junction model for the behavioral optimization of tubular structures represents a feasible methodology throughout which significant improvements of the analyzed model estimations can be obtained.

The authors would like to remark that, when performing studies throughout finite element analysis, it is necessary to take into consideration the fact that the beam-type elements represent a simplified element derived from the shell- and volume-type elements, which implies a series of intrinsic limitations determined by their own formulation.

**Author Contributions:** Conceptualization and methodology, F.B. and J.P.; software, F.B. and J.O.; validation, F.B.; formal analysis, F.B., J.P. and J.O.; investigation, F.B.; resources, F.B. and J.O.; writing original draft preparation, F.B., J.P. and J.O.; writing—review and editing, F.B., J.P. and J.O.; visualization and supervision, J.O. and J.P.; project administration, F.B.; funding acquisition, J.O. and F.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by the Cátedra Global Nebrija-Santander en Recuperación de Energía en el Transporte de Superficie.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Stochastic Reliability-Based Design Optimization Framework for the Steel Plate Girder with Corrugated Web Subjected to Corrosion**

**Damian Sokołowski \* and Marcin Kami ´nski**

Chair of Structural Reliability, Faculty of Civil Engineering, Architecture and Environmental Engineering, Lodz University of Technology, Al. Politechniki 6, 90-924 Lodz, Poland

**\*** Correspondence: damian.sokolowski@p.lodz.pl

**Abstract:** This paper proposes the framework for reliability-based design optimization (RBDO) of structural elements with an example based on the corrugated web I-girder. It tackles the problem of topological optimization of corroding structures with uncertainties. Engineering restrictions follow a concept of the limit states (LS) and extend it for stability and eigenfrequency assessment. The reliability constraints include all the LS; they are computed according to first- and secondorder reliability methods. The RBDO example minimizes the bridge girder cross-section while satisfying the structural reliability level for the ultimate and the serviceability limit states, stability, and eigenfrequency. It takes into consideration two uncorrelated random effects, i.e., manufacturing imperfection and corrosion. They are both Gaussian; the first of them is applied at assembly time, while the second is applied according to the time series. The example confronts three independent FEM models with an increasing level of detailing, and compares RBDO results for three concurrent probabilistic methods, i.e., the iterative stochastic perturbation technique (ISPT), the semi-analytical method, and the Monte Carlo simulation. This study proves that the RBDO analysis is feasible even for computationally demanding structures, can support automation of structural design, and that the level of detailing in the FEM models influences its results. Finally, it exemplifies that reliability restrictions for LS are much more rigorous than for their deterministic counterparts, and that the fastest ISPT method is sufficiently accurate for probabilistic calculations in this RBDO.

**Keywords:** reliability-based design optimization; stochastic perturbation technique; Monte Carlo simulation; semi-analytical method; topological optimization; corrugated web; corrosion

## **1. Introduction**

Contemporary structural designs require powerful tools for optimization purposes, which must be effective, fast, and easy to use. Together with an exponential increase in computational power, the traditional analytical approach to the optimization of Civil Engineering structures has become significantly outdated. In the majority of designs, this approach has already been replaced by more accurate deterministic methods, among which the finite element method plays a crucial role, and they are largely implemented in commercial software. Such an optimization strategy is applied for example in [1]; its goal is commonly focused on structural topology [2,3]. The traditional deterministic design appears to be suboptimal when significant uncertainties must be taken into account, such as climatic loads, material uncertainties, assembly errors, and corrosion or soil conditions, just to name a few. They cannot be avoided in structural and especially civil engineering designs. This is why a new concept called reliability-based design optimization (RBDO) arose, where the uncertainties are directly included in the design. An acceptable contrast between the efficiency of deterministic and reliability optimization is given in [4] and an exhaustive review of the concepts of RBDO is available [5] or in [6]. RBDO is well researched, especially for steel truss structures e.g., [7,8], and frames e.g., [9,10], where the computation effort of

**Citation:** Sokołowski, D.; Kami ´nski, M. Stochastic Reliability-Based Design Optimization Framework for the Steel Plate Girder with Corrugated Web Subjected to Corrosion. *Materials* **2022**, *15*, 7170. https://doi.org/10.3390/ma15207170

Academic Editors: Luigi Calabrese, Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 8 September 2022 Accepted: 10 October 2022 Published: 14 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

each optimization loop is acceptable. They are not so common in more complex problems involving the plate, shell, or composite structures, where the computation is much more demanding and the amount of strength and serviceability checks much higher; such construction is considered in this study. This extreme computational intensiveness is well depicted in [11] and still constitutes a major issue. Computational intensiveness is caused not only by the number of limit states but also by the high number of deterministic and random design variables, whose impact on these states is not always known a priori to RBDO. This aspect is commonly tackled by an initial sensitivity analysis, which determines the susceptibility of structural response to a variation of design parameters. It has been used with success for many years e.g., in [12], and eliminates unnecessary variables from further computation. Application of the RBDO could also include corrosion effects [13], which constitute a major topic of this paper.

Although some new concepts are still being put forward [14,15], the RBDO methods are quite mature now and allow a very efficient (but computationally demanding) design. What they still lack is the ability to define a reliable life of structures, so that the optimization is not aimed purely at modifications of the pristine materials, but also takes into consideration the degradation of its work throughout the service life. This is what we propose in our concept of RBDO, which allows a service life optimization with a reliability-based design approach for the determination of the reliable service life of steel structures that are subject to corrosion. Corrosion affects the strength and serviceability of steel structures, including their capacity, stability, and durability. Its effects are depicted for example in [16,17]. It is also a major reason for the careful and costly maintenance of steel structures [18] and together with fire softening it constitutes the main weakness of this material. Corrosion affects steel already at erection time, which is quite different from reinforced concrete where the onset of corrosion is shifted from this initial time [19] and calculated for example according to Fick's second law. This is because steel structures are directly affected by chloride attack and are not covered by other materials. Of course, there exist a variety of covers such as special paints or chrome plating and various additives (see for instance weathering steel), but even such techniques do not prevent this phenomenon completely; they require repetitive application during the service life of steel constructions. Corrosion especially affects thin-walled structures, where small pitting corrosion placed in a susceptible place, or a small reduction in thickness may cause serious loss of capacity. A fine example of such structures are those with sinusoidally corrugated webs currently applied in girders (also arched ones [20]) and also composite structures, where the deck is made of concrete and the web of constructional steel [21].

The corrugated web considerably increases rigidity and shear capacity [22], and decreases sensitivity to a local stability loss in the web [23], thus reducing the occurrence of local buckling. It also allows for reduced self-weight [24], as compared to conventional flat web I-girders, and does not increase the complexity of execution, as trusses do. The usage of these assets has increased, especially in structural applications such as bridges, pedestrian walkways, hangars, and industrial buildings. Interestingly, corrugated webs also outperform flat webs in energy dissipation and could be used in anti-seismic structures [25]. The first studies of the SIN web girders were conducted in the late 1990s [26], yet they are still new in civil engineering applications and have several disadvantages for such applications. They offer a weaker contribution to bending [27] and cause an additional normal stress distribution in flanges coming from transverse bending [28] compared to the regular I-beams; they also have complex bending–shear interaction [29]. To make matters worse, there still is a lack of design standards or specifications dealing with the behavior of such webs and SIN web girders.

This is why in recent years there has been intensive research on their basic behavior connected with stress [30], elastic critical moment, buckling [31,32], and shear [33,34]; they are computed in this study within the concept of limit states. A very interesting problem is the capacity and serviceability of such constructions in fire conditions [35], which considerably reduce the bearing capacity of steel structures and also their reliability.

Reliability-based computations of structural elements with a corrugated web are also available [36,37] and even a weight optimization could be found [38], but RBDO has not yet been proposed for such structures; this is especially true when corrosion is the leading random effect. Consequently, there is a need to develop recommendations that properly address the reliability issues of such girders, which is also the reason for their choice as an example for the proposed framework.

The principal objective of this paper is to propose an RBDO framework with a special focus on structural elements and constructions that must satisfy complex restrictions of engineering standards. The secondary goal is to successfully apply this framework to a computationally challenging example of the sinusoidally corrugated web I-girder. The study assumes the possibility of optimizing the topology of civil engineering structures subjected to uncertain corrosion evolution and engineering restrictions for a specified reliable service life. Further, it assumes that FEM modeling accuracy and probabilistic solver play a crucial role in the accuracy and timing of RBDO results. Next, it assumes the importance of the steel and environment type as well as the choice of the WLSM weighing function for the solution.

Therefore, a design optimization framework and additional theoretical background for time-dependent reliability civil engineering analysis have been presented and applied for the case study of the steel plate girder with a sinusoidally corrugated I web. The results obtained in this paper would allow for more optimal designing of such structures and may be applied to other steel and concrete structures.

#### **2. Theoretical Background**

According to the current design codes, the structural elements' durability period adopted is 50 years, and this is usually ensured by repeated design trials with the goal of minimizing weight or cost, alternatively optimization of its capacity for different limit states. In such terms, a designer solves an optimization problem with a clearly defined set of restrictions, which are first of all (1) the minimum capacity of the element, secondly (2) geometrical constraints, (3) material restrictions, (4) maximum deformations, and (5) other physical constraints. The most tedious work is usually required in the first of these, i.e., in ensuring a minimum capacity of the element. This is because the element (and an entire structure) is subjected to multiple loads of different morphology and with different placement (wind, snow, traffic load, vibrations, self-weight, machinery, etc.), which may or may not occur at the same time. That is why an engineer must check the structural capacity for multiple combinations, which almost always require a different computational approach. This work proposes a framework, according to which such optimization could be utilized with a specified goal of optimization. It takes into consideration a classical approach of limit states proposed by the Eurocode 0 [39] and also a more refined, higher-order probabilistic design method directly using the reliability theory, where the judgment of service life is done based on the reliability indices *β<sup>g</sup>* of the designed elements. Such refinement is advised in Appendices B and C of this design norm [39]. Within this framework, multiple structural elements may be assessed, and the entire structure could be optimized. The objective function may be purely topological, as in the below example, but it may also be cost-oriented, when the total cost of an element is optimized, or may minimize the difference of all the indices *βg*. The method selected for optimization purposes is deterministic non-gradient regular search, in which the main loop encompasses the subsequent steps of (1) deterministic verification of restrictions, (2) verification of the given limit states, and finally, (3) checks of the reliability indices. This is shown in Figure 1, where the optimization problem depends on the design time *t* and input uncertainty *ω*; at each step, a new topology *W*(*ω*; *t*) is proposed. A non-gradient approach is selected because in most civil engineering designs gradient cannot be explicitly computed. This is because each of the combinations defined in multiple limit states would be different and movement in any direction of the design domain may give converse results for multiple combinations. The main objective of this algorithm is structural optimization and the main

loop depicted in Figure 1 starts with an arbitrary point in the optimization domain (starting geometry of the element) that should be selected according to the engineering practice. At first, the most straightforward conditions are checked, i.e., geometrical, material, and physical constraints. Then, a cross-sectional class is determined, because its change demands different checks of the limit states and the reliability indices. Upon this, the finite element method is utilized to solve the mechanical problem and its results are used in the determination of both, the limit states and reliability indices. Verification of the physical restrictions and cross-sectional class is straightforward and depends strictly on the selected material, geometry, and static scheme of the specific element. The FEM must be carefully formulated for each part so that the parameters existent in the objective function could be easily (or preferably automatically) ameliorated and the subsequent model solved; this will be explained in detail for the specific context of the example. loop depicted in Figure 1 starts with an arbitrary point in the optimization domain (starting geometry of the element) that should be selected according to the engineering practice. At first, the most straightforward conditions are checked, i.e., geometrical, material, and physical constraints. Then, a cross-sectional class is determined, because its change demands different checks of the limit states and the reliability indices. Upon this, the finite element method is utilized to solve the mechanical problem and its results are used in the determination of both, the limit states and reliability indices. Verification of the physical restrictions and cross-sectional class is straightforward and depends strictly on the selected material, geometry, and static scheme of the specific element. The FEM must be carefully formulated for each part so that the parameters existent in the objective function could be easily (or preferably automatically) ameliorated and the subsequent model solved; this will be explained in detail for the specific context of the example.

because each of the combinations defined in multiple limit states would be different and movement in any direction of the design domain may give converse results for multiple combinations. The main objective of this algorithm is structural optimization and the main

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**Figure 1.** Principal algorithm of the method and calculation process. **Figure 1.** Principal algorithm of the method and calculation process.

Designing processes requiring a detailed explanation include the 'Calculate Limit States', 'Calculate Reliability indices' *β g* , and 'Validate stop criteria'. According to a common definition, a limit state is a state in which the construction element or an entire Designing processes requiring a detailed explanation include the 'Calculate Limit States', 'Calculate Reliability indices' *βg*, and 'Validate stop criteria'. According to a common definition, a limit state is a state in which the construction element or an entire structure will fail due to a specified external action. The design codes in their basic form define the ultimate limit state (ULS) and serviceability limit state (SLS), which describe either the conditions in which the construction will fail (ULS) or will stop working in an

acceptable way (SLS); the hierarchy of LS is shown on Figure 2. If the design is made based on the FEM, not the analytical approach, each element must meet five fundamental conditions for the linear regime of the structural materials. First, (1) the general stresses *σij* and (2) the reduced (commonly Huber–Mises) stresses *σred* must be lower than the plastic limit. In the case of the linear civil engineering structures, these are limited to the longitudinal normal stresses *σ*<sup>11</sup> and the shear stresses *σ*<sup>12</sup> (or *τ*). Additionally, (3) the structure must be stable and buckle neither locally nor globally (but local buckling is sometimes permitted when additional structural elements are added to the designed part); this constraint is depicted in Figure 2 by *ξ* (*W*, *t*). The last condition in the ULS encompasses (4) the first eigenfrequencies, which must be high enough not to be triggered by the wind or traffic; usually, the minimum is set to 5 Hz. In the SLS, (5) deformations of this element must be limited, which include the deflections *u* (*W*, *t*) and also displacements at its ends (or borders) and connections to other parts *δ<sup>i</sup>* (*W*, *t*). acceptable way (SLS); the hierarchy of LS is shown on Figure 2. If the design is made based on the FEM, not the analytical approach, each element must meet five fundamental conditions for the linear regime of the structural materials. First, (1) the general stresses *σij* and (2) the reduced (commonly Huber–Mises) stresses *σred* must be lower than the plastic limit. In the case of the linear civil engineering structures, these are limited to the longitudinal normal stresses *σ*<sup>11</sup> and the shear stresses *σ*<sup>12</sup> (or *τ*). Additionally, (3) the structure must be stable and buckle neither locally nor globally (but local buckling is sometimes permitted when additional structural elements are added to the designed part); this constraint is depicted in Figure 2 by *ξ (W,t)*. The last condition in the ULS encompasses (4) the first eigenfrequencies, which must be high enough not to be triggered by the wind or traffic; usually, the minimum is set to 5 Hz. In the SLS, (5) deformations of this element must be limited, which include the deflections *u (W,t)* and also displacements at its ends (or borders) and connections to other parts *δ<sup>i</sup> (W,t)*.

structure will fail due to a specified external action. The design codes in their basic form define the ultimate limit state (ULS) and serviceability limit state (SLS), which describe either the conditions in which the construction will fail (ULS) or will stop working in an

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**Figure 2.** Procedure for the determination of the limit states. **Figure 2.** Procedure for the determination of the limit states.

One may also include requirements for exceptional states, such as collision, fire loads, or checks for fatigue, but they are out of the scope of this work; this is important because of their rare inclusion in design purposes in civil engineering practice. All the mentioned states could be checked implicitly using the results of the subsequent FEM analyses, which are static, eigenfrequency, and critical load tasks available in the most common programs such as ABAQUS, ANSYS, Catia, or DIANA. Please note, that some of the limit states may be checked globally for each part or an entire structure, such as *ξ (W,t)* or *ϖ (W,t)*, while the others must be met for all the points of the structure (in FEM, for all the elements). These include the stresses, strains, deflections, deformations, and displacements, all of them depending strictly on the objective function—the topology of the element. If all restrictions in both limit states are met, the optimization could proceed to the next process a check of the reliability indices *β g* . One may also include requirements for exceptional states, such as collision, fire loads, or checks for fatigue, but they are out of the scope of this work; this is important because of their rare inclusion in design purposes in civil engineering practice. All the mentioned states could be checked implicitly using the results of the subsequent FEM analyses, which are static, eigenfrequency, and critical load tasks available in the most common programs such as ABAQUS, ANSYS, Catia, or DIANA. Please note, that some of the limit states may be checked globally for each part or an entire structure, such as *ξ* (*W*, *t*) or *v* (*W*, *t*), while the others must be met for all the points of the structure (in FEM, for all the elements).These include the stresses, strains, deflections, deformations, and displacements, all of them depending strictly on the objective function—the topology of the element. If all restrictions in both limit states are met, the optimization could proceed to the next process—a check ofthe reliability indices *<sup>β</sup>g*.

A process of the determination of reliability indices *β g* serves for a final check of the restrictions and could be started only if all the previous ones are met. This is because the limit functions *g* are almost directly taken from the limit states and used in the numerator of the reliability indices *β g* . This process is depicted in Figure 3, which defines the flow of A process of the determination of reliability indices *β<sup>g</sup>* serves for a final check of the restrictions and could be started only if all the previous ones are met. This is because the limit functions *g* are almost directly taken from the limit states and used in the numerator of the reliability indices *βg*. This process is depicted in Figure 3, which defines the flow of computations at this stage of optimization. It starts with optimization of the response function, then it determines the probabilistic coefficients, and finally calculates the reliability indices serving as the final restrictions. These indices are computed first at time *t* = 0 with an initial uncertainty *ω*<sup>0</sup> and only then for *t e* (0; *td*) and *ω*, where *t<sup>d</sup>* is designed service

life. This is because the design may be already unacceptable when no aging (or corrosion) of the material is included. The checks of reliability are performed for all the restrictions from the limit states separately and each of them must be met so that the current topology can be accepted. In this process, neither the amount nor the probability density function (PDF) is strictly limited, and the total number of uncertainty sources depends upon the probabilistic methods applied, as does the PDF for each of these. with an initial uncertainty <sup>0</sup> and only then for *t* ϵ (*0;t<sup>d</sup>* ) and ω, where *t<sup>d</sup>* is designed service life. This is because the design may be already unacceptable when no aging (or corrosion) of the material is included. The checks of reliability are performed for all the restrictions from the limit states separately and each of them must be met so that the current topology can be accepted. In this process, neither the amount nor the probability density function (PDF) is strictly limited, and the total number of uncertainty sources depends upon the probabilistic methods applied, as does the PDF for each of these.

computations at this stage of optimization. It starts with optimization of the response function, then it determines the probabilistic coefficients, and finally calculates the reliability indices serving as the final restrictions. These indices are computed first at time *t =* 0

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**Figure 3.** Procedure for the determination of reliability indices. **Figure 3.** Procedure for the determination of reliability indices.

Checks for *t =* 0 are performed for both the first-order reliability method (FORM) and the second-order reliability method (SORM) in sequential order; this order of analysis is defined by O in Figure 3. The last computation of reliable service life is computed solely by the FORM (for *O =* 1) and plotted in the service time domain. Unlike the majority of papers that propose only one method of probabilistic calculus (see for example [40]), we propose three concurrent methods, the iterative stochastic perturbation technique, the semi-analytical method (AM), and the crude Monte Carlo simulation (MCS). Please note that the computation of reliability restrictions is not limited to the above methods. Checks for *t* = 0 are performed for both the first-order reliability method (FORM) and the second-order reliability method (SORM) in sequential order; this order of analysis is defined by *O* in Figure 3. The last computation of reliable service life is computed solely by the FORM (for *O* = 1) and plotted in the service time domain. Unlike the majority of papers that propose only one method of probabilistic calculus (see for example [40]), we propose three concurrent methods, the iterative stochastic perturbation technique, the semi-analytical method (AM), and the crude Monte Carlo simulation (MCS). Please note that the computation of reliability restrictions is not limited to the above methods.

Determination of the response function, the first task in reliability-focused computations, serves as an inner optimization problem. It is solved at each optimization step just before the determination of probabilistic characteristics. It is devoted to the determination of a continuous response function of the capacity of the girder *f c* (*W*) from its discrete representation solved via the FEM. This is required in computations of subsequent limit Determination of the response function, the first task in reliability-focused computations, serves as an inner optimization problem. It is solved at each optimization step just before the determination of probabilistic characteristics. It is devoted to the determination of a continuous response function of the capacity of the girder *fc*(*W*) from its discrete representation solved via the FEM. This is required in computations of subsequent limit functions *g* defined as the difference of this capacity and the reaction to external action or an engineering limit *fe*(*W*).

$$\lg(\mathcal{W}) = f\_{\mathcal{C}}(\mathcal{W}) - f\_{\mathcal{C}}(\mathcal{W}).\tag{1}$$

Please note that the analytical formula relating the objective function and the functions of girder capacity cannot be explicitly derived analytically, which is why the FEM is used for its retrieval. The inner optimization problem is solved by deterministic non-gradient search in the discrete domain of the order of the response polynomial and the number of terms in this polynomial. Restrictions include several terms *nAi* > 0 *e* N and the order of polynomial *P<sup>O</sup>* > 0 *e* N. The optimization aim is twofold—minimization of the weighted least squares method (WLSM) variance and maximization of correlation of the polynomial and the FEM results.

$$\min(Var(\overline{r})) \land \max\left(Corr\left(m\_{FEM}, A\_l h^i\right)\right) \tag{2}$$

where higher precedence is set to the correlation. The stop function is generally not required in this problem because of a finite number of allowable points in the discrete optimization domain. *P<sup>O</sup>* is limited to 10 ÷ 30, firstly because of no real correlation gain for higher order polynomials, and secondly because of difficulties with its behavior outside or neighboring to the probing range of the FEM. Such optimization is performed for each of the limit functions. In the exemplary problem, only *umax* was taken into consideration, whose an explicit mathematical formulation is proposed as

$$u(\omega; t) = N \cdot q(\omega; t) = N \cdot A\_i(t) \cdot h^i(\omega; t); \ r^2 = \left(m\_{FEM} - A\_i h^i\right)^2; \quad \min\left(\sum\_{j=1}^n r\_j^2\right) \to A\_j; \tag{3}$$

$$\min(Var(r)) \wedge \max\left(Corr\left(m\_{FEM}, A\_i h^i\right)\right)$$

where *u*(*ω;t*) is the maximum deflection, *N* stands for shape function, *A<sup>i</sup>* are defined as the coefficients of approximating polynomial, *h <sup>i</sup>* define the subsequent powers of the design variable, and *r* is a residuum coming from a difference of the FEM result *m*FEM and the result coming from the polynomial response function *Aih i* .

The weighted least squares method (WLSM) solved at each optimization step uses the following polynomial approximation:

$$
\mu(b) \cong D^{(P\_o)} b^{P\_o} = f(\mathbf{D}, b) \quad P\_o = 1, \dots, s; s < n. \tag{4}
$$

where the polynomial basis of the *s*th order *P<sup>O</sup>* is used and solved around the web thickness of the current optimization step. This web thickness also serves as a mean value of the main random parameter included in the probabilistic calculus, indexed here by *b*. As a result *n* different pairs *b* (*α*) , **u** (*α*) for *α* = 1, . . . , *n* are returned, whose arguments belong to the neighborhood of expectation of b itself. The residuals in each trial point are introduced to get an algebraic condition for these expansion coefficients. They are next minimized. After relevant modifications, the following regular matrix equations are obtained

$$\left(\left(\mathbf{J}\right)^{T}\mathbf{w}\mathbf{J}\right)\mathbf{D}=\left(\mathbf{J}\right)^{T}\mathbf{w}\mathbf{u}\tag{5}$$

Such a system of equations (with the dimensions *n x s*) is solved symbolically in MAPLE [41].

The last process in the design loop is a stop condition and it is depicted in Figure 4. It evaluates if the optimized solution is found or not. Success occurs when at least one step finds a more optimal solution than *W*0. Failure is identified when (1) one of the indices from restrictions is within the margin of *β<sup>T</sup>* at design service time *t<sup>d</sup>* and (2) *n<sup>s</sup>* subsequent steps do not decrease the objective function (fail to optimize the *W*). An additional stop is defined for *iT*,*max* = 1000 steps to ensure a cutoff of the optimization with weak correlation; its fulfillment may lead either to optimization success when at least one of the previous *W* fulfilled all restrictions or optimization failure. A more elaborate stop condition may also be applied, but its inclusion would increase the computation time, which proves critical for engineering purposes. Please note that the solution and optimization convergence

will depend on the starting structural configuration *W*<sup>0</sup> and it is recommended to repeat optimization with different *W*0, using a different kind of initial cross-section for example.

**Figure 4.** Definition of stop criteria.

The reliability calculus at the initial time is only according to initial imperfections. A formulation of such a problem could be found in [42]. The final reliable life check is performed according to two random variables, namely the corrosion penetration depth and fabrication imperfection. This is possible in the stochastic context with the introduction of the relevant resulting functions of corrosion penetration depth into both, the web thickness mean value and its initial variation. Expected value of random web thickness *b* can be computed as

$$E[b] = \int\_{-\infty}^{+\infty} b \, p\_b(\mathbf{x}) d\mathbf{x} \equiv \frac{1}{M} \sum\_{i=1}^{M} b^{(i)} \tag{6}$$

while its variance as

$$\begin{split} Var(b) &= \int\_{-\infty}^{+\infty} \left( b - E[b] \right)^2 p\_b(\mathbf{x}) d\mathbf{x} + Var(d) - 2Corr(b, d) \\ &\equiv \frac{1}{M - 1} \sum\_{i=1}^{M} \left( b^{(i)} - E[b] \right)^2 + Var(d) - 2Corr(b, d) \end{split} \tag{7}$$

where *E*[*d*] is the expected value of corrosion penetration depth directly affecting the mean value of web thickness *b*; the correlation between these two stochastic variables *Corr*(*b*, *d*) is set to 0.

The reliability indices *βFORM* and *βSORM* may serve for initial reliability as well as structural health monitoring. The first of these could be computed in the following way:

$$
\beta\_{\mathcal{S}\_{\mathcal{I}} \text{ FORM}} = \frac{E[\mathcal{g}]}{\sigma[\mathcal{g}]} \tag{8}
$$

where *β<sup>g</sup>* stands for the reliability index of a specific limit function, *E*[*g*] is the expectation of this function *g*, and *σ*[*g*] is its standard deviation. A reliability index *βg*,*FORM* assumes a normal probability distribution of a given random (response) function; *βg*,*SORM* is defined as [37]

$$\beta\_{\mathcal{S},\text{CORM}} = -\Phi^{-1}\left(P\_{f2}\right) \tag{9}$$

where *P<sup>f</sup>* <sup>2</sup> denotes the probability of failure for the chosen probability distribution of the function relative to *βg*, FORM in the following manner:

$$P\_{f2} = \frac{\Phi(\beta\_{\mathcal{S}, \text{FORM}})}{\sqrt{1 + \beta\_{\mathcal{S}, \text{FORM}} \mathcal{K}}} \text{} \tag{10}$$

where *κ* is the curvature approximating the primary surface defined by the following formula:

$$\kappa = \frac{\frac{\text{du}^{(m)}}{\text{d}\text{b}\_0^{k\_2}}}{\left(1 + \left(\frac{\text{du}^{(m)}}{\text{d}\text{b}\_0^k}\right)^2\right)^{\frac{3}{2}}}.\tag{11}$$

### **3. Numerical Illustration**

Let us consider a corrugated web I-beam girder that is suspected of corrosion, as in Figure 5.

**Figure 5.** The layout of the girder in [mm].

In such girders, the web is predominantly affected by this phenomenon, which leads to loss of bearing capacity during its service life. This girder is subjected to topological optimization, whose goal is to optimize the cross-section that ensures reliability for 50 years of service according to Eurocode 0, Appendix B [39]. Optimization is performed within the framework proposed in Figure 1, where the limit states are checked based on three FEM models; the indices *β<sup>g</sup>* are verified for stresses, deflections, eigenfrequencies, and stabilities, while the final constraints of reliability are based on displacement; the following objective function *W* (*v*; *t*) is proposed:

$$\begin{split} W(\omega;\omega\_0;t) &= \left(A\_w(\omega;\omega\_0;t) + 2A\_f\right) \cdot L = \left(t\_w(\omega;\omega\_0;t) \cdot h\_w + 2A\_f\right) \cdot L \\ &= \left(\left(t\_{w\_0}(\omega\_0) - 2 \cdot \left(A(\omega) + t^{B(\omega)}\right)\right) \cdot h\_w + 2A\_f\right) \cdot L \end{split} \tag{12}$$

As a function of *v*0—the coefficient of variation of the web thickness at time *t* = 0, *v*—coefficient of variation of corrosion as well as of time *t*; *t* = 0 stands for an assembly time, in which exposure to the external environment begins. In this function, *A<sup>w</sup>* and *A<sup>f</sup>* are the cross-sectional areas of the web and the flange, *h<sup>w</sup>* stands for the height of the web, and *t<sup>w</sup>* denotes the thickness of the web. Let us note that in common civil engineering designs the flanges are much thicker than the web and are placed horizontally so that they are not so susceptible to corrosion. This is why the time dependence of this topology is generally based on the web, whose thickness decreases with corrosion. This thickness is uncertain already after its fabrication process, which is imposed by an initial coefficient of

variation *v*<sup>0</sup> and is then subjected to stochastic corrosion of the form *A* (*v*)+*t B*(*v*) . In this term *tw*= *tw*0(*ω*0) − 2· *A*(*ω*)+*t B*(*v*) , where the quotient 2 depicts the susceptibility of *t<sup>w</sup>* to corrosion from both sides. The corrosion function strictly depends on the environment and steel type, to which it is subjected, and is taken from Melchers 2002. Its coefficients are shown in Table 1, which summarizes the corrosion parameters for rural, urban, and marine environments separately for the carbon and weathering steels. This table shows that the marine environment is specified by the highest CoV, and has expectations a little smaller than the urban environment, while the rural environment is the least invasive. The weathering steel is much less affected by corrosion, as both parameters, A and B, have smaller mean values and CVS than the carbon steel in a corresponding environment type. Nonetheless, the downside of this type of steel is its cost.


**Table 1.** Statistical parameters of corrosion for various steel types [43].

A corrosion model applied in this work comes from the additional experiments reported in the literature and is applied with the parameters adjacent to the carbon steel in an urban environment. Its expected value reads

$$E[D(t)] = 3.52 \cdot 10^{-3} e^{2.81 \cdot 10^{-24} (1.05 \cdot 10^{12} + 1.00 \cdot 10^{11} \ln(t))} \tag{13}$$

The variance is introduced as

$$\begin{split} Var(D(t)) &= 7.57 \cdot 10^{-3} t^{1.19} e^{1.12 \cdot 10^{-1} \ln(t)^2} - 5.65 \cdot 10^{-4} t^{1.19} e^{5.63 \cdot 10^{-2} \ln(t)^2} + 3.13\\ &\quad + 1.24 \cdot 10^{-5} t^{1.19} e^{6.24 + 5.63 \cdot 10^{-2} \ln(t)^2}. \end{split} \tag{14}$$

They are both truncated by the third vital number for the reader's convenience.

The principal restrictions for this objective function come solely from the requirements of the bearing capacity, stability, eigenfrequency, and allowable deformation of the girder, which are a cross-section within its design service life. This, in turn, is defined in the current civil engineering design code, Eurocode 0 [39], which in its annexes proposes the limits of reliability index *β<sup>g</sup>* for each of the limit states (LS). They are divided into the ultimate limit state (ULS), in which the girder must withstand the normal, reduced, and shear stresses, have high enough first eigenfrequency, as well as not be susceptible to buckling. The second LS is the serviceability limit state (SLS), in which the deflection of this girder must be limited by the value of *l*/250. In such terms, there exist six restrictions for its reliability:

$$\begin{array}{l} \beta\_{\tau}(t) - \beta\_{\hat{\tau}}(t) \ge 0; \beta\_{\sigma\_{\text{red}}}(t) - \beta\_{\hat{\nu}}(t) \ge 0; \beta\_{\Omega}(t) - \beta\_{\hat{\Omega}}(t) \ge 0;\\ \beta\_{\sigma\_{\text{cr}}}(t) - \beta\_{\hat{\sigma}\_{\text{cr}}}(t) \ge 0; \beta\_{u\_{\text{max}}}(t) - \beta\_{\hat{\nu}}(t) \ge 0; \beta\_{\hat{\xi}}(t) - \beta\_{\hat{\xi}}(t) \ge 0; \end{array} \tag{15}$$

Further restrictions are connected to the geometry of the girder, its volume *V*, and cross-sectional area *A*, which must be all positive.

$$\text{If } \varrho\_{\prime} \hbar\_{w\prime\prime} \text{ t}\_{w\prime\prime} \text{ t}\_{f\prime\prime} \hbar\_{f\prime\prime} \text{ A}\_{\prime} \text{ V } > 0 \tag{16}$$

The geometry of this girder includes the height of the web *h<sup>w</sup>* , its width *t<sup>w</sup>* as well as the height *h<sup>f</sup>* and width *t<sup>f</sup>* of the flange. Let us note that the resulting checks of bearing capacity required by the Eurocode changes together with an increase of cross-sectional class and therefore its slenderness must be limited to keep the checks unified in terms of both, the flange and the web 0 < *h<sup>w</sup>* /*t<sup>w</sup>* < 72, 0 < *hf* <sup>2</sup> − *t<sup>w</sup>* − *a* /*t<sup>f</sup>* < 9. Otherwise, the procedure of determination of the limit states and the reliability indices would have to be changed substantially each time the section class changes.

The material selected for the design purposes is constructional steel. This choice determines all the material restrictions, including the density *\$* <sup>∈</sup> 7.75–8.05 [g/cm<sup>3</sup> ] and the plastic limit of this steel *fu*, which is here narrowed to the most common steels available on the international market, i.e., *f<sup>u</sup>* ∈ {195, 235, 275, 355, 420, 460} MPa with corresponding Young modulus *E* = 210 GPa and Poisson ratio of *µ* = 0.3. The material model applied in all the computations for constructional steel is linear with plastic limit *fu*.

Additional restrictions proceed directly from the external requirements or the investor and include static schemes and loads. These are the designed length of the girder *L* = 40 m, external load in form of a uniform pressure applied on the upper flange *q* = 150 kN/m, and degrees of freedom restricted at the ends of the girder as simple supports—although they are not directly included in the objective function but in the FEM model.

The inner optimization problem is solved here with the order of response polynomial in the range of 16 > *P<sup>O</sup>* > 0 *e N*. A maximum polynomial order is set, because previous optimization problems show that the solution starts to degrade already at *P*<sup>0</sup> > 10.

A shortened version of the results for one of these optimizations devoted to maximum deflection is given in Table 2. It shows that the accuracy of the WLSM approximation measured by its total error *EWLSM*, variance *αWLSM*, and correlation coefficient of the response function and the discrete FEM results *CWLSM* does not necessarily increase with an increase of the polynomial order or number of terms included. Moreover, the optimum order is not possible to determine a priori to solving the optimization problem. Interestingly, the limitation of the terms with a constant *P<sup>O</sup>* has a minor influence on *CWLSM* but increases both *EWLSM* and *αWLSM*. It must be noted that together with an increase of the *P<sup>O</sup>* and *n<sup>A</sup>* the computation complexity and length are also increased. Generally, the optimum order is in the range of *P<sup>O</sup>* ∈ {5; 10} and full polynomials are preferred. An additional problem in this inner optimization is the type of weights in the WLSM *WS*. The considered weighting schemes are equal *WSE*, triangular *WST*, and Dirac *WSD*; the last weighting scheme places greater importance on the realizations around the mean (or middle) of the uncertain parameter, and the equal weighting scheme puts the same weight on all the discrete data points but has problems when they are not the smooth and triangular type of weights, which decreases the importance of the data points with an increase of their distance from the mean. It is quite efficient for the low *P<sup>O</sup>* but for this weighting scheme *EWLSM* and *αWLSM* increase very fast together with an increase in *PO*. Dirac-type of weight puts the same importance on the mean as for all the other data points. It ensures the best *CWLSM* at the highest order, keeps a very small error and low variance for a high span of *PO*, and returns the smoothest approximation. This is why it was selected for further optimizations and the other weighting schemes were removed from the checks of a reliable life prediction. From Table 2, it could be also concluded that the optimum approximation was reached for a full polynomial of the ninth order and WLSM with a Dirac-type of weighting.


**Table 2.** Selected WLSM polynomial approximations of the extreme displacement versus random input.

Such an optimized third-order maximum deflection function valid for the SLS is given below:

$$E[R\_{\rm SLS}] = 8.09 \ - \ 3.13 \cdot 10^{-2} (E[t\_w]) + 2.91 \cdot 10^{-4} (E[t\_w])^2 - 9.71 \cdot 10^{-7} (E[t\_w])^3. \tag{17}$$

In this expression, the expectation of web thickness includes an influence of fabrication error *E*[*tw*0] existent at *t* = 0 and corrosion *E*[*D*(*t*)] so that *E*[*tw*]= *E*[*tw*0] − 2·*E*[*D*(*t*)]. Let us note that together with the total allowed deflection this maximum deflection function serves as a numerator in an expression of *βg*, *FORM* as *E*[*g*SLS]= *L*/350 − *E*[*R*SLS].

A variance of each limit state is available simply as diff(*E*[*g*], *tw*) 2 ·Var(*tw*), where *E*[*g*] represents the expected value of each limit state; for SLS it is *E*[*g*SLS]. The variance of web thickness could be obtained as *Var*(*tw*) =*Var*(*tw*0)+2·*Var*(*D*(*t*)); fabrication error and corrosion phenomenon are considered here as uncorrelated. An initial coefficient of variation of fabrication error is assumed as *α*(*tw*0)= 0.05.

The indices *βσ*<sup>ˆ</sup> , *βσ*ˆ*cr* , *βu*<sup>ˆ</sup> , *βτ*<sup>ˆ</sup> and *β*Ω<sup>ˆ</sup> define the threshold for each of the reliability restrictions, while *βσred* , *βumax* , *β*Ω, *β<sup>τ</sup>* , and *βσcr* denote the indices of reliability computed according to the first-order reliability method (FORM, see Equation (8)) or second-order reliability method (SORM, see Equation (9)). They strictly follow the types of checks made in the limit states of the framework but include the uncertainty disregarded in LS; a detailed formulation for a FORM and SORM is given in the theoretical background. Restrictions of *β<sup>g</sup>* are always in the form of a difference between the threshold indices *βg*<sup>ˆ</sup> and the resulting indices defining the girder *βg*. The threshold indices are piecewise constant functions with required service time *t<sup>s</sup>* , which by default is 50 years [39]. The minimum values corresponding to *ts*= 50 for *βσ*<sup>ˆ</sup> , *βσ*ˆ*cr* , *βτ*<sup>ˆ</sup> = 3.8 and *βu*<sup>ˆ</sup> , *β*Ω<sup>ˆ</sup> , *β* <sup>ˆ</sup>*<sup>ξ</sup>* = 1.5. This is basically because the former defines the ULS and the latter the SLS. In its first-order formulation, *β<sup>g</sup>* is a simple quotient of the expected value of the limit function *g* − *E*[*g*] and the standard deviation of *g* − *σ*[*g*]. In turn, the limit function is a difference between the capacity of the girder and its response to an external action or an engineering limit; there may exist multiple limit functions for a single engineering structure (such as for this girder). Owing to this, there exist also multiple formulations of the limiting indices, whose definition is very close to the limit states existent in the approach of Eurocode; for the context of this example, the following indices must be defined:


In this example *βT*= 0.3, *is*= 1, *iu*= 5, and *iT*= 100. Additionally, the process of determination of *β<sup>g</sup>* was postponed until the limit states were optimized, which allowed substantial optimization time savings.

#### *3.1. Numerical Model Description*

The FEM simulations are provided by the use of an FEM system ABAQUS [44] with the use of three 3D full-scale models reported below:


Their discretization has been shown in Figure 6. This is done to show the importance of the FEM models for optimization purposes and to contrast the results coming from different levels of detailing in the numerical model. The details available in these models are summarized in Table 3. This table firstly shows the type of elements used in the three models, their total number, and the total number of nodes. The highest number of nodes and elements are given in the first model, which is because of the FEM formulation based on the 3D elements. The third model has more than five times fewer elements and the second is a little less than the third. This is very close for the number of nodes, which are also the highest for the first model and the lowest for the second model; the highest level of detail is provided in the first model. They include the web flanges, ribs, and welds, the second model does not include welds, and the third postpones welds and ribs. This is strictly related to the number of parts and instances (provided in brackets of Table 3) created in these models—only 5 were required in the simplest third model and 563 in the first, mostly because of the very sophisticated welding required for the SIN web I-beams. This is also why the quantity of tied connections in the different models differs dramatically—only 6 for the third model and almost 1400 for the first. The latter part of Table 3. summarizes the constitutive models, analysis types, and the type and number of interactions between the modeled parts. The constitutive model applied, and the types of analyses performed for these models are the same, because of the optimization requirements. Static and static general analyses return stresses *σij*, Huber–Mises stress *σred*, and ultimate deflection *umax*.

**Table 3.** Details of the full-scale FEM models.


**Figure 6.** ABAQUS discretization of the girder, (**a**) with ribs and welds (volumetric, model 1), (**b**) with ribs (shell, model 2), (**c**) without ribs (shell, model 3).

The buckling analysis outputs the critical load *ξ* and the frequency analysis returns the eigenfrequency Ω. All of them are used in the checks of LS and *βg*. The constitutive relation is set to linear with a plastic limit to conform to the standard approach of the Eurocode [39].

The topics that require further attention are the details of discretization and features included in different models. The details of discretization are shown in Figure 6, which brings us closer to the mesh used in all three studies. The first two shell models (models 2 and 3) have a quad-dominated mesh with a free meshing technique, allowing the best adaptivity of the elements to the geometry, while the volumetric model—a mixture of hexahedral and tetrahedral finite elements of both structured and unstructured meshes with different sizes. This variability is provided to optimize the time effort and computational accuracy. A structured hexahedral mesh is applied to the web and flanges, while the unstructured meshing technique is preserved in the welds; this is because of their complex geometry. The mesh of ribs and webs in the shell models is structured and composed of quad elements, while one of the flanges is a mixture of quadratic and triangular elements that adapt to the sinusoidal pattern of the web; this is visualized in Figure 6. Elements used in all computations are conventional stress-displacement-based FEs. The C3D8 is a linear brick, with eight nodes, reduced integration, and a single integration point. The C3D10 is a second-order 10-node tetrahedral element with four integration points at each tetrahedral vertex. S4R is a shell with four nodes, reduced integration, and a single central integration point. It has implemented hourglass control and finite membrane strains. S3 is a three-node triangular general-purpose shell with finite membrane strains. The simple support was modeled with linear constraints. They were placed directly below the middle of the support rib (models 1 and 2) or at the outer edge of the bottom flange (model 3) along the bottom flange width. On the left side only rotational DOF (UR1) was allowed; on the right, UR1 and displacement along the length of the girder were allowed (U2). The load was applied as the equal surface load on the entire upper flange of the girder. Its magnitude was equal to 107.14 kN/m<sup>2</sup> (equivalent to 150 kN/m) including the dead load. The geometry of this girder was adopted exactly as given in Figure 5. rectly below the middle of the support rib (models 1 and 2) or at the outer edge of the bottom flange (model 3) along the bottom flange width. On the left side only rotational DOF (UR1) was allowed; on the right, UR1 and displacement along the length of the girder were allowed (U2). The load was applied as the equal surface load on the entire upper flange of the girder. Its magnitude was equal to 107.14 kN/m<sup>2</sup> (equivalent to 150 kN/m) including the dead load. The geometry of this girder was adopted exactly as given in Figure 5.

relation is set to linear with a plastic limit to conform to the standard approach of the

The topics that require further attention are the details of discretization and features included in different models. The details of discretization are shown in Figure 6, which brings us closer to the mesh used in all three studies. The first two shell models (models 2 and 3) have a quad-dominated mesh with a free meshing technique, allowing the best adaptivity of the elements to the geometry, while the volumetric model—a mixture of hexahedral and tetrahedral finite elements of both structured and unstructured meshes with different sizes. This variability is provided to optimize the time effort and computational accuracy. A structured hexahedral mesh is applied to the web and flanges, while the unstructured meshing technique is preserved in the welds; this is because of their complex geometry. The mesh of ribs and webs in the shell models is structured and composed of quad elements, while one of the flanges is a mixture of quadratic and triangular elements that adapt to the sinusoidal pattern of the web; this is visualized in Figure 6. Elements used in all computations are conventional stress-displacement-based FEs. The C3D8 is a linear brick, with eight nodes, reduced integration, and a single integration point. The C3D10 is a second-order 10-node tetrahedral element with four integration points at each tetrahedral vertex. S4R is a shell with four nodes, reduced integration, and a single central integration point. It has implemented hourglass control and finite membrane strains. S3 is a three-node triangular general-purpose shell with finite membrane strains. The simple support was modeled with linear constraints. They were placed di-

The latter detail—features modeled—illustrates a development of numerical research. Each consecutive model made in ABAQUS brings new additional details, namely the ribs and welds depicted in Figure 7 for the first model. It is also quite important that this first model consists of three-dimensional elements. This enables not only a more accurate stress analysis through the thickness of the modeled parts but also makes possible a check of interaction between the different elements of the girder and its utilization in terms of internal stress. They both are not available in the simplified models. The latter detail—features modeled—illustrates a development of numerical research. Each consecutive model made in ABAQUS brings new additional details, namely the ribs and welds depicted in Figure 7 for the first model. It is also quite important that this first model consists of three-dimensional elements. This enables not only a more accurate stress analysis through the thickness of the modeled parts but also makes possible a check of interaction between the different elements of the girder and its utilization in terms of internal stress. They both are not available in the simplified models.

*Materials* **2022**, *15*, x FOR PEER REVIEW 15 of 31

Eurocode [39].

**Figure 7.** Additional views on the discretization for a volumetric model of the girder (model 1). (**a**) the details of the weld discretization; (**b**) welded connection of the ribs with the sinusoidal web. **Figure 7.** Additional views on the discretization for a volumetric model of the girder (model 1). (**a**) the details of the weld discretization; (**b**) welded connection of the ribs with the sinusoidal web.

#### *3.2. Deterministic Limit States Analysis*

Computation of the limit states is performed based on the results taken from the FEM simulations, which are summarized in Table 4, where *σ*cr*, τ, umax, v*, and *ξ* are shown for the vicinity of the optimal web thickness *tw*= 56 mm. This table firstly shows that the stresses and displacements increase together with an increase of *t<sup>w</sup>* (an increase of *W* (*tw*)), while the *v* and *ξ* increase. All these effects are desirable, because the smaller the stresses and strains, the lower the usage of the material, and, secondly, the higher the critical loads and eigenfrequencies, the higher the margin between stability loss and the current state. Moreover, an addition of the welds has rather a marginal stiffening effect, because the displacements from the second and the third model are very close, at least in the considered loading scheme. A slightly higher *umax* for the first model comes principally from the addition of the welds and the difference between volumetric and shell FEM formulation. This is not true for the ultimate stresses and stability, both of which are significantly affected by the ribs. They decrease the ultimate stresses in the FEM for *σcr* and *τ* that are returned for the models without ribs. Interestingly, the addition of ribs directly connected to the web does not essentially change the stress flow of shear in the web, yet causes its strong reduction. This happens especially in the first model, which additionally detects quite a high shear in the entire web-flange weld, even in the middle of the girder span.


**Table 4.** Comparison of the results from the three FEM models.

The stresses induced in the middle of the span on the outer surface of flanges are comparable in all the models, but their placement is at the flange in the shell models and in the weld of the volumetric model. This placement is also the cause for a little oscillatory effect of the maximum stresses in the first model, where small changes in the thickness must also result in a change of the mesh; this effect is not observed in the shell models. A stress state is determined in the FEM at the post-processing stage, which is the major cause of its susceptibility to all discretization changes. Such problems are not observed in displacements, being the direct results of the FEM, nor in the global characteristics of *ω* and *ξ*; the character of stresses and their pattern is considered in a separate study.

Further, it is seen that the critical loads (*CL*) returned from the three models differ significantly for the three models. This is first because of the stiffening effect of the ribs for different global modes of buckling, and secondly because of the volumetric FEM formulation of the first model. This is evidenced in Table 4, which defines the ratio of the load at stability loss *qmax* to the level of loading coming from external actions according to the Eurocode *qinitial* with a subtracted initial load:

$$\frac{q\_{\text{max}}}{q\_{\text{initial}}} = \text{CL} + 1.\tag{18}$$

*CL* is much lower for the first model than for the others. It is because it was the only one that returned a local buckling of the support rib. Other models returned only the global losses of stability, the first of which was always rotation-torsional. They occur for quite a high magnitude of the *CL* (critical load ratio of over 4.45 or 7.37) for a girder, which is initially loaded to 88.4% of its ultimate bending capacity according to the assumptions of the Eurocode. The critical load in the volumetric model is much lower, but all the critical modes of behavior until the 25th one can be easily avoided by an increase in the thickness of the support or, preferably, a change in its geometry from plate to corrugated (insensitive to local buckling). Quite interestingly, the addition of the ribs increased the *CL* by about 50% for the shell model. No buckling is detected in the web itself, either by stiffened or unstiffened models. This proves the high contribution of the wavy web to the overall stability of the girder and the reasonable significance of its thickness, whose increase enlarges the critical load. This increase is consistent with an engineering intuition, but it does not stop the underneath mechanism leading to the buckling of the girder. Contrary to the results of the critical load, these for eigenfrequency show almost perfect agreement for models 1–2 and give a slightly lower quantity for the third one. This difference is increasing for the higher eigenfrequency modes (see Table 4), which is caused by the change in the linear dead load of a girder due to the stiffening ribs and welds, yet does not significantly hinder the character of these modes.

It must be additionally mentioned that a choice of the proper ultimate stress or displacement for purposes of LS and reliability restrictions is not a trivial task, especially when a complex 3D structure is considered and modeled with the use of shell or volumetric FEM elements. The global ultimate stress does not necessarily give a proper condition for the ULS and may turn out to be inappropriate for restrictions. Even for such conceptually uncomplicated structural elements as a simply supported beam, the ultimate stress may lie in other places than those distinguished by the beam theory (as happens in this study). In such conditions, reliability shall be checked not only for the location of the ultimate stress but also for these other locations detected in the beam theory, because the material in these places may be susceptible to local instability (as in all plate structures). Because of this, the ultimate stress allowed in such locations may be highly reduced and conversely, the material can be locally stronger in a location of ultimate stress predicted by the FEM (e.g., when some confinement exists). Therefore, in all complex structures, one should always first determine its possible weak points and check the reliability for all of these, not only the one for the ultimate stress or displacement revealed by the FEM results.

## *3.3. Probabilistic Aspects*

The determination of the reliability indices *β<sup>g</sup>* and related optimization procedures have been both programmed and completed in the computer algebra system MAPLE. The WLSM is based on the discrete results of three FEM models (see Table 3) and for three types of weights, Dirac, equal, and triangular (see Table 2). The input random parameter is web thickness with a mean value taken from the objective function *W* and updated in each loop of the optimization. The probabilistic density function (PDF) of this thickness is Gaussian and we treat here two separate random problems. The first is at *t* = 0 and is connected with a fabrication error (manufacturing imperfection) of this thickness with coefficient of variation (COV) in the range of *α*(*ω*0) ∈ {*0*; *0.15*}, and a second, where in addition to *ω*0, a second random variable is corrosion penetration depth, where time *t* is a design variable in a range of *t*∈ h0; 50i years. This second check is also a final and most severe restriction in the optimization loop because it takes into consideration the degradation of the girder with time. The formulation of the probabilistic moments and coefficients is here threefold:


Such an approach is selected for purposes of comparison between these three methods and also for redundancy so that even if one method fails in a specific optimization step or for the specific *g*, the indices could still be calculated and compared. A crude MCS with <sup>5</sup>·10<sup>5</sup> trials is chosen to dissolve all the doubts about the accuracy of a more refined Monte Carlo method. The spectrum of the web thickness used in each step is *tw*, *<sup>i</sup>* ± 5 mm with a difference of 1 mm for each computation, so that the response function is optimized based on 11 FEM results around the mean value defined at each optimization loop. The algorithm developed in MAPLE fully encompasses the process of reliability indices *β* (see Figures 1 and 3) and *βσ*cr, and for *t*∈ h0; 50i only *βumax* is restricted, which is done for the computation efficiency and simplicity of this example.

## *3.4. Reliability Restrictions β<sup>g</sup>*

The final reliability calculations are shown for the optimized *W*, for which *tw*= 56 mm. The structure of the reliability assessment for this *W* is divided into three main sections:


This is done firstly to show the results for the optimized *W* and secondly to emphasize the most important factors in the optimization, i.e., the FEM model, order of reliability assessment, and type of the weighting scheme in the optimization of the WLSM. Three independent methods of computation, AM, MCS, and SPT are used for simultaneous verification of the results. The probabilistic moments and coefficients are shown only for one, most restrictive limit function and solely relative to the weighting scheme, which is done to show their general outlook and present relation with an input uncertainty. The other ones return analogous results.

## 3.4.1. Initial Restriction of *βg*—WLSM Weighting Scheme

Computations of reliability-induced restrictions for the girder at *t* = 0 include the determination of the first four probabilistic moments and coefficients of all the limit functions considered, i.e., *Eg*(*α*(*ω*0)), *αg*(*α*(*ω*0)), *βg*(*α*(*ω*0)), and *κg*(*α*(*ω*0)), and its reliability index *βg*(*α*(*ω*0)). They are depicted in Figures 8–17 and are computed relative to the coefficient of variation of web thickness inflicted by an uncertain fabrication error *α* (*ω*0). This ensures an easy way of check of these restrictions for the chosen level of uncertainty and avoids the need for repetition of the entire optimization process in the case when its level is slightly increased. The full results are shown only for the most restrictive condition, which is a critical load (CL). The results of all other ones are limited solely to the index of reliability according to SORM; this is done for the brevity of the results. *Materials* **2022**, *15*, x FOR PEER REVIEW 19 of 31 condition, which is a critical load (CL). The results of all other ones are limited solely to the index of reliability according to SORM; this is done for the brevity of the results.

**Figure 8.** Expectations of the critical load for different types of weights. **Figure 8.** Expectations of the critical load for different types of weights.

**Figure 9.** CoV of the critical load for different types of weights.

**Figure 8.** Expectations of the critical load for different types of weights.

condition, which is a critical load (CL). The results of all other ones are limited solely to the index of reliability according to SORM; this is done for the brevity of the results.

**Figure 9. Figure 9.** CoV of the critical load for different types of weights. CoV of the critical load for different types of weights.

**Figure 10.** The skewness of the critical load for different types of weights. **Figure 10.** The skewness of the critical load for different types of weights.

**Figure 11.** Kurtosis of the critical load for different types of weights.

**Figure 10.** The skewness of the critical load for different types of weights.

**Figure 11. Figure 11.** Kurtosis of the critical load for different types Kurtosis of the critical load for different types of weights. of weights.

**Figure 12.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—displacement. **Figure 12.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—displacement.

**Figure 13.** Reliability index of the corrugated I-beam for different types of weights according to the

second-order reliability method (SORM)—normal stress.

second-order reliability method (SORM)—displacement.

**Figure 13.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—normal stress. **Figure 13.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—normal stress. *Materials* **2022**, *15*, x FOR PEER REVIEW 22 of 31

**Figure 12.** Reliability index of the corrugated I-beam for different types of weights according to the

**Figure 14.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—shear. second-order reliability method (SORM)—eigenfrequency. **Figure 14.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—shear.

**Figure 15.** Reliability index of the corrugated I-beam for different types of weights according to the

second-order reliability method (SORM)—shear.

**Figure 15.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—eigenfrequency. **Figure 15.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—eigenfrequency. *Materials* **2022**, *15*, x FOR PEER REVIEW 23 of 31

**Figure 14.** Reliability index of the corrugated I-beam for different types of weights according to the

**Figure 16.** Reliability index of the corrugated I-beam for different types of weights according to the first-order reliability method (FORM)—critical load. **Figure 16.** Reliability index of the corrugated I-beam for different types of weights according to the first-order reliability method (FORM)—critical load.

**Figure 17.** Reliability index of the corrugated I-beam for different types of weights according to the

tainty caused by the manufacturing error. The expected values (Figure 8) decrease, and

The lower probabilistic coefficients for the limit state based on stability criterion

)) firstly show that they both are highly affected by the uncer-

second-order reliability method (SORM)—critical load.

)) and *αξ*(*α*(*ω*<sup>0</sup>

*Eξ*(*α*(*ω*<sup>0</sup>

first-order reliability method (FORM)—critical load.

**Figure 17.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—critical load. **Figure 17.** Reliability index of the corrugated I-beam for different types of weights according to the second-order reliability method (SORM)—critical load.

**Figure 16.** Reliability index of the corrugated I-beam for different types of weights according to the

The lower probabilistic coefficients for the limit state based on stability criterion *Eξ*(*α*(*ω*<sup>0</sup> )) and *αξ*(*α*(*ω*<sup>0</sup> )) firstly show that they both are highly affected by the uncertainty caused by the manufacturing error. The expected values (Figure 8) decrease, and The lower probabilistic coefficients for the limit state based on stability criterion *E<sup>ξ</sup>* (*α*(*ω*0)) and *α<sup>ξ</sup>* (*α*(*ω*0)) firstly show that they both are highly affected by the uncertainty caused by the manufacturing error. The expected values (Figure 8) decrease, and the coefficient of variation (Figure 9) increases together with an increase in this uncertainty and the rate of this change always increases. Interestingly, the changes in the *E<sup>ξ</sup>* are up to 50% and the coefficient of variation is multiple times higher than for the input. The three methods of computation show a perfect agreement.

The skewness *β<sup>ξ</sup>* (*α*(*ω*0)) and kurtosis *κ<sup>ξ</sup>* (*α*(*ω*0)) for the critical load depicted in Figures 10 and 11 represent a converse character. The skewness is predominantly negative, while the kurtosis is positive. They both have quite a strong relationship with the input uncertainty, and they reach very high magnitudes of up to 120 for *β<sup>ξ</sup>* and 13,000 for *κ<sup>ξ</sup>* . A scatter of the three stochastic methods is quite high, but still, the AM and MCS demonstrate a quite comparable trend. The SPT is effective here only up to *α* = 0.1, but it is the fastest.

The indices of reliability *βσ*, *βσred* , *βτ*, *βumax* , *β*Ω, and *β<sup>ξ</sup>* include all the relevant limit states and are all shown in Figures 12–17. They principally show a high converse dependence on the input uncertainty of manufacturing error and are also affected by the WLSM weighting scheme. The lowest initial reliability is reported for the critical load (see Figures 16 and 17). This is a direct effect of two main factors. The first of these is the fact that even the deterministic safety is quite small here (function close to one, which constitutes its lower limit, see Table 4). The second is connected with a magnitude of the *CoV*, which is very high for this state function (Figure 9) and has a strong exponential character relative to the input *CoV*. With an increasing input uncertainty, this smallest index rapidly decreases, but still from approx. *α* (*ω*0)= 0.07 the lower bound of reliability is governed by the normal stress shown in Figure 13. The indexes also depend upon the type of weight of WLSM. This difference is the strongest for the two indices constituting the lower bound of the reliability, namely the normal stress and the critical load, and additionally for the shear (see Figures 13, 14 and 16). This is because their first two probabilistic moments show the highest dependence on the type of weight applied.

The character of a relationship between the reliability index and an input uncertainty is not always smooth and without local inflection points. The indices for the ultimate displacement, as well as critical load, show small fluctuations through their course, especially around *α* (*ω*0)= 0.10. Such fluctuations are intriguing, but for now, their cause has not been determined.

The last comparison is the FORM vs. SORM index of reliability, which is given based on critical load, the most difficult reliability restriction at *t* = 0. It is based on Figures 16 and 17.

The graphs according to both orders have a very similar character. This is expected, because the applied SORM is based on the Gaussian input probability density and, therefore, these indices should be comparable. An important observation is that *β<sup>g</sup>* according to SORM is less dependent on the type of weight and has a little higher magnitude for an extensive input uncertainty. Finally, the SORM approach corrects the errors coming from the FORM, i.e., the diverging or scattered character of *β<sup>g</sup>* for some limit functions. An additional observation is concerned with the interchangeability of these three probabilistic methods applied in restrictions of the reliability index; all the methods show an almost perfect agreement for all the state functions. Due to this, when only one stochastic variable is taken into consideration, all the methods can be used alternatively. In such conditions, the most preferable one seems to be the stochastic perturbation method (SPT), which is not dependent on a direct derivation, does not require a considerable number of trials (as the MCS), and is also the swiftest.

The last, yet most important, observation is a total limit of input uncertainty coming from the fabrication imperfection that allows for fulfilling the restriction of reliability *βg*. This limit differs for all the state parameters and ranges from *α* (*ω*0)= 0.09 to around *α* (*ω*0)= 0.21. The lowest one constitutes a total limit, and therefore the objective function with optimum *tw*= 56 mm is *α* (*ω*0)= 0.09. This limit corresponds to the limit function of critical load (*ξ*). This is a reasonable result because for most of the constructional elements the fabrication error causes uncertainty in *t<sup>w</sup>* lower than (*ω*0)= 0.05. Nonetheless, reliability restrictions at *t* = 0 are not the most rigorous, which undoubtedly are the ones connected with a joint effect of corrosion and the considered fabrication error. Improving the resistance of the girder to stability loss would be enough to increase the overall reliability of this girder at *t* = 0, but it may not be sufficient to improve reliability in the corrosive environment.

## 3.4.2. Initial Restriction of *β<sup>g</sup>*

The differences in initial reliability restriction fulfillment for the three models are shown in Figures 18–21. They depict the FORM indices for the four most important limit states, i.e., *βσ*, *βumax* , *β*Ω, and *β<sup>ξ</sup>* in the function of the *α* (*ω*0) and for *t* = 0. These plots are principally presented to highlight the importance of the choice of the FEM model type and its accuracy in fulfillment of the reliability restrictions for the objective function. They are all computed with the use of the Dirac weighting scheme and with three alternative probabilistic methods, MCS, AM, and SPT. These indices perfectly justify the purpose, being vastly dependent on the model type and almost uniform for all the probabilistic methods. Therefore, it is highly recommended to put more effort during the modeling process of the FEM and into the collection of data coming from these simulations than into the choice and scrutiny of the probabilistic method. The probabilistic coefficients bring no more information for optimization purposes and therefore they are not included. One may refer to Figures 8–11 for the required information.

**Figure 18.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the deflection. **Figure 18.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the deflection. *Materials* **2022**, *15*, x FOR PEER REVIEW 26 of 31

critical load (*ξ*). This is a reasonable result because for most of the constructional elements

bility restrictions at *t =* 0 are not the most rigorous, which undoubtedly are the ones connected with a joint effect of corrosion and the considered fabrication error. Improving the resistance of the girder to stability loss would be enough to increase the overall reliability of this girder at *t =* 0, but it may not be sufficient to improve reliability in the corrosive

The differences in initial reliability restriction fulfillment for the three models are shown in Figures 18–21. They depict the FORM indices for the four most important limit

> α(*ω*<sup>0</sup>

in the function of the

are principally presented to highlight the importance of the choice of the FEM model type and its accuracy in fulfillment of the reliability restrictions for the objective function. They are all computed with the use of the Dirac weighting scheme and with three alternative probabilistic methods, MCS, AM, and SPT. These indices perfectly justify the purpose, being vastly dependent on the model type and almost uniform for all the probabilistic methods. Therefore, it is highly recommended to put more effort during the modeling process of the FEM and into the collection of data coming from these simulations than into the choice and scrutiny of the probabilistic method. The probabilistic coefficients bring no more information for optimization purposes and therefore they are not included. One may

) *=* 0.05. Nonetheless, relia-

) and for *t =* 0. These plots

the fabrication error causes uncertainty in *t<sup>w</sup>* lower than (*ω*<sup>0</sup>

, and *β ξ*

refer to Figures 8–11 for the required information.

environment.

states, i.e., *β*

3.4.2. Initial Restriction of

*σ* , *β umax* , *β<sup>Ω</sup>*

**Figure 19.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the ultimate normal stress. **Figure 19.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the ultimate normal stress.

**Figure 20.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM

and based on the eigenfrequencies.

and based on the ultimate normal stress.

**Figure 20.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the eigenfrequencies. **Figure 20.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the eigenfrequencies. *Materials* **2022**, *15*, x FOR PEER REVIEW 27 of 31

**Figure 19.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM

**Figure 21.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the critical load. **Figure 21.** Reliability index of the corrugated I-beam for three ABAQUS models according to FORM and based on the critical load.

From Figures 18–21 it is first seen that the reliability margin decreases together with an increase of α (*ω*<sup>0</sup> ). It is in the range of 50 ÷ 150 for a small α (*ω*<sup>0</sup> ) *<* 0.01, but it rapidly decreases and reaches zero for α (*ω*<sup>0</sup> ) ∈ (0.10;0.14); this is still much higher than typical uncertainty caused by the fabrication imperfection. The second observation concerns the inexistence of a limit state setting the reliability restriction for the entire α (*ω*<sup>0</sup> ). For each From Figures 18–21 it is first seen that the reliability margin decreases together with an increase of *α* (*ω*0). It is in the range of 50 ÷ 150 for a small *α* (*ω*0) < 0.01, but it rapidly decreases and reaches zero for *α* (*ω*0) ∈ (*0.10*; *0.14*); this is still much higher than typical uncertainty caused by the fabrication imperfection. The second observation concerns the inexistence of a limit state setting the reliability restriction for the entire

level of uncertainty, this limit is set in different LS, and therefore the topological optimization is not trivial—each limit state depends on multiple parameters of the model. The

is also the most detailed one, it should be the one taken for the final optimization. This is because it had the smallest margin in the LS. An alternative approach is the inclusion of results from all the models for optimization purposes, but this would cause almost threefold longer computations because the FEM solution is very computationally-intensive. The correspondence between the three methods is almost perfect for all the four considered reliability indices and for all the FEM models, with only one exception of Figure 20 for the third, most simplified model (shell without ribs), for which neither of the methods

One of the observations which should undoubtedly be highlighted is the fact that for all the limit states the reliability indices according to different models diverge not only in

is an index based on the deflection (Figure 18), which starts very high for the deterministic

the reliability index based on the ultimate normal stress (Figure 19) and computed according to the shell model (third model) which shows a negative value of the index. This is a direct outcome of an existence of the unacceptably high stress exceeding the plastic limit of the construction steel already in the FEM results. The existence of the negative index is incorrect and should by all means be excluded from further analysis. Nevertheless, unlike

), its negative value gives a piece of important information for optimization

α(*ω*<sup>0</sup> α(*ω*<sup>0</sup>

). One more interesting relation is unveiled by

) = 0.07 crosses all the other indices to become the lowest (to

). A very good example

their initial value but also in the strength of their relation to

α(*ω*<sup>0</sup>

constitute a lower limit) for all higher

converges.

in α(*ω*<sup>0</sup>

model and just over

*α* (*ω*0). For each level of uncertainty, this limit is set in different LS, and therefore the topological optimization is not trivial—each limit state depends on multiple parameters of the model. The highest restriction is decisive, which is not unique for the different models. On the other hand, the first model is also the most restrictive for a majority of the cases and because it is also the most detailed one, it should be the one taken for the final optimization. This is because it had the smallest margin in the LS. An alternative approach is the inclusion of results from all the models for optimization purposes, but this would cause almost threefold longer computations because the FEM solution is very computationally-intensive. The correspondence between the three methods is almost perfect for all the four considered reliability indices and for all the FEM models, with only one exception of Figure 20 for the third, most simplified model (shell without ribs), for which neither of the methods converges.

One of the observations which should undoubtedly be highlighted is the fact that for all the limit states the reliability indices according to different models diverge not only in their initial value but also in the strength of their relation to *α* (*ω*0). A very good example is an index based on the deflection (Figure 18), which starts very high for the deterministic model and just over *α* (*ω*0) = 0.07 crosses all the other indices to become the lowest (to constitute a lower limit) for all higher *α* (*ω*0). One more interesting relation is unveiled by the reliability index based on the ultimate normal stress (Figure 19) and computed according to the shell model (third model) which shows a negative value of the index. This is a direct outcome of an existence of the unacceptably high stress exceeding the plastic limit of the construction steel already in the FEM results. The existence of the negative index is incorrect and should by all means be excluded from further analysis. Nevertheless, unlike in *α* (*ω*0), its negative value gives a piece of important information for optimization purposes—that the element is not fulfilling restrictions already in the process of the limit states, and before reliability checks, a new optimization loop with a new value of *W* should have been already started.

## 3.4.3. Durability Analysis with *β<sup>g</sup>*

The last and final restriction of the optimization problem is the reliable service life of the constructional element, which is set as *t* = 50 years. This restriction is formulated with the use of the FROM index of *β<sup>g</sup>* and presented for the limit state of *umax*. It is computed for the joint impact of the corrosion penetration depth and the manufacturing imperfection, both being random and uncorrelated. Reliability is calculated with three probabilistic methods, i.e., AM, MCS, and SPT. Corrosion is modeled according to the third model from Table 1 with two random parameters *A* and *B*, while the uncertainty of initial imperfection is set within the following bounds of *α* (*ω*0) ∈ [0.05; 0.25] with a Gaussian PDF. This imperfection is introduced during the production of the beam and is considered timeindependent, while the corrosion process is described by a time series. The response function utilized for probabilistic calculations is calculated with an inner optimization problem and WLSM is based on the Dirac weighting scheme. The result is presented for the optimized objective function *W*(*tw*= 56 mm) and based on the most refined first FEM model. This final reliability restriction is depicted in Figure 22 as a function of service time *t* ∈ (0; 50) years. It firstly shows that the limit of 1.8 is reached at around 60 years, and the margin of the restriction is within the stop criterion defined for this example as *βT*= 0.3. Secondly, the uncertainty of an initial manufacturing imperfection only marginally affects the final result, and this is why its sole impact was checked in the preceding step of the optimization loop. This index starts at*β<sup>g</sup>* = 60, sharply decreases in a convex manner with a decreasing slope and has an apparent limit of *β<sup>g</sup>* ∼= 0. This index shows a very good coincidence of the two probabilistic methods, the SPT and MCS. The third, semi-analytical method (AM) diverges from the others, and this is why it is not reported in Figure 22. This exemplifies the usability of this triple redundant method, which allows a successful check of restriction even when one of the probabilistic methods turns out to be unavailable or divergent for a specific step.

**Figure 22.** Reliability index of the corrugated I-beam for different types of weights according to FORM—deflection (SLS), the second-order polynomial. **Figure 22.** Reliability index of the corrugated I-beam for different types of weights according to FORM—deflection (SLS), the second-order polynomial.

purposes—that the element is not fulfilling restrictions already in the process of the limit states, and before reliability checks, a new optimization loop with a new value of *W*

The last and final restriction of the optimization problem is the reliable service life of the constructional element, which is set as *t =* 50 years. This restriction is formulated with

for the joint impact of the corrosion penetration depth and the manufacturing imperfection, both being random and uncorrelated. Reliability is calculated with three probabilistic methods, i.e., AM, MCS, and SPT. Corrosion is modeled according to the third model from Table 1 with two random parameters *A* and *B*, while the uncertainty of initial imperfec-

> α (*ω*<sup>0</sup> )∈

imperfection is introduced during the production of the beam and is considered timeindependent, while the corrosion process is described by a time series. The response function utilized for probabilistic calculations is calculated with an inner optimization problem and WLSM is based on the Dirac weighting scheme. The result is presented for the optimized objective function *W*(*t<sup>w</sup> =* 56 mm) and based on the most refined first FEM model. This final reliability restriction is depicted in Figure 22 as a function of service time *t* ∈ (0;50) years. It firstly shows that the limit of 1.8 is reached at around 60 years, and the margin of the restriction is within the stop criterion defined for this example as *β*

Secondly, the uncertainty of an initial manufacturing imperfection only marginally affects the final result, and this is why its sole impact was checked in the preceding step of the

incidence of the two probabilistic methods, the SPT and MCS. The third, semi-analytical method (AM) diverges from the others, and this is why it is not reported in Figure 22. This exemplifies the usability of this triple redundant method, which allows a successful check of restriction even when one of the probabilistic methods turns out to be unavailable or

*g* ≅

*g*

and presented for the limit state of *umax.* It is computed

[0.05;0.25] with a Gaussian PDF. This

*=* 60, sharply decreases in a convex manner with

0. This index shows a very good co-

*T =* 0.3.

should have been already started.

3.4.3. Durability Analysis with

the use of the FROM index of *β*

tion is set within the following bounds of

optimization loop. This index starts at *β*

divergent for a specific step.

a decreasing slope and has an apparent limit of *β*

*g*

#### **4. Concluding Remarks**

This paper presents an optimization framework for topological problems in the domain of civil engineering. It is exemplified by a successfully optimized simply supported SIN web I-girder. The main novelty in this paper is the concept of reliable service life prediction and its application to a computationally demanding structure. It allows automation of the reliability-based design of custom structural elements. The principal objective of the proposed RBDO algorithm is a determination of the best topology that satisfies all the design restrictions applicable to civil engineering structures during their service life. Restrictions include the limit states, i.e., the ultimate limit state and serviceability limit state, stability, and vibrations in the deterministic and reliability context. The limit functions are applied directly after the FEM results. The optimization loop consists of subsequent verification of physical and geometrical restrictions, FEM problem solution, verification of all deterministic design restrictions, and finally verification of reliability-based restrictions. This is done for a specified service life of construction. It could be applied to a wide variety of structural elements and entire structures; the limit functions could be obtained analytically, by BEM, FEM, neural networks, or with the use of any other algorithm that outputs the required state parameters.

The most critical points of this algorithm include the calculation of the representative limit functions for local state parameters, such as stresses and strains, FEM detailing, and interpretation of FEM results. For this reason, three concurrent models were proposed and shortly contrasted in the optimized solution; in current maturity, the algorithm uses a deterministic non-gradient search. It is intended for replacement in future works by a more effective method, such as neural networks.

The proposed algorithm is applied to a practical example of the SIN web I-girder. This provides additional insight into the reliability of such elements, as well as their susceptibility to loss of stability, vibrations, and deformability. It also adds some valuable remarks to the FEM modeling in RBDO problems. The results obtained in this work confirm that numerical modeling precision significantly affects the optimization outcome. This is because its choice has a direct effect on the stress state of the girder and its buckling. Different FEM models applied in this study return important dissimilarities in stress distribution, its maximum values, and placement of peaks. This is also true for buckling loads and their patterns. It is also exemplified here that the choice of the FEM formulation (volumetric vs. shell), as well as the finite element type and order, significantly affects the optimization. Furthermore, it is difficult to set the required level of detail in the FEM model and the minimum amount of FEs before the RBDO. This is because the buckling load together with ultimate stresses and deflections have no clear correlation to the model accuracy. Therefore, a decrease in the level of geometric and computational precision may hide significant design problems. They include low buckling mode or high-stress peaks in the welds that are omitted by simplified models and lead to an overestimation of the overall reliability of this structure. Importantly, only the most detailed model revealed additional instabilities in the girder occurring at much smaller loads than the others and having a local, rather than global character.

The corrosion process affected the considered example in terms of both reliability and structural capacity. Its evolution increased the internal stresses and decreased critical loads. The influence of the initial fabrication error on the service life of the exemplary structure was marginal and may be omitted in future research. On the contrary, steel and environment type had a much more substantial effect on the service life of an exemplary structure. This is because of its direct effect on the evolution of corrosion depth and its uncertainty. Please note that maintenance, such as painting or plating, is not taken into consideration in the current algorithm. Performed regularly, it will increase the reliable life of the structure. Its inclusion is planned in future research.

The triple probabilistic calculations applied in this study provided a piece of important information on the convergence of probabilistic calculations that would be otherwise unavailable. On the other hand, their mutual application slowed down the optimization process. Computational time reduction could be achieved by limiting the RBDO to a single probabilistic method. The ISFEM is the most recommended in this case. It is the fastest and significantly accelerates the optimization process. Its accuracy is a little lower than that of the MCS, but the difference proved to be marginal for the considered example.

**Author Contributions:** Conceptualization, M.K.; methodology, D.S. and M.K.; data curation, D.S.; writing—original draft, D.S.; writing—review and editing, M.K.; software and methodology, D.S. and M.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This paper has been written in the framework of the research grant OPUS no. 2021/41/B/ST 8/02432 "Probabilistic entropy in engineering computations" sponsored by the National Science Center in Cracow, Poland.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Cellular Automaton Mimicking Colliding Bodies for Topology Optimization**

**Bogdan Bochenek and Katarzyna Tajs-Zieli ´nska \***

Faculty of Mechanical Engineering, Cracow University of Technology, 31-155 Krakow, Poland

**\*** Correspondence: katarzyna.tajs-zielinska@pk.edu.pl

**Abstract:** Needs and demands of contemporary engineering stimulate continuous and intensive development of design methods. Topology optimization is a modern approach which has been successfully implemented in a daily engineering design practice. Decades of progress resulted in numerous applications of topology optimization to many research and engineering fields. Since the design process starts already at the conceptual stage, innovative, efficient, and versatile topology algorithms play a crucial role. In the present study, the concept of the original heuristic topology generator is proposed. The main idea that stands behind this proposal is to take advantage of the colliding bodies phenomenon and to use the governing laws to derive original Cellular Automata rules which can efficiently perform the process of optimal topologies generation. The derived algorithm has been successfully combined with ANSYS, a commercial finite element software package, to illustrate its versatility and to make a step toward engineering applications. Based on the results of the tests performed, it can be concluded that the proposed concept of the automaton mimicking colliding bodies may be an alternative algorithm to other existing topology generators oriented toward engineering applications.

**Keywords:** topology optimization; cellular automaton; colliding bodies; heuristic update rules

## **1. Introduction**

As it has been observed over the years, topology optimization has been a dynamically developing research area with numerous applications to many research and engineering fields. The researchers community continuously works on innovative, efficient, and versatile topology optimization approaches, methods, and algorithms, whereas the spectrum of numerous solutions of topology optimization problems ranges from classic Michell structures to sophisticated contemporary engineering ones. The various approaches to the generation of optimal topologies have been presented along with emerging concepts which have been implemented in a broadly understood engineering area. The comprehensive discussion on various aspects of topology optimization has been provided by many survey papers: e.g., [1–4] recently complemented by Ribeiro et al. [5] and Logo and Ismail [6]. The long-lasting development of topology optimization confirms that it still remains one of the most important research fields within the area of structural and material design.

Along with the research issues of topology optimization, the practical aspects of engineering implementation of topology optimization techniques have become more and more important. As a result, the topology optimization tools are nowadays present in commercial engineering software. However, the black-box topology generators implemented into commercial software do not guarantee that the final results are the best available. Therefore, although remarkable achievements have been already made toward topology optimization application in engineering, there is still room for further investigations. Recently published papers [7–11] may serve here only as examples.

In the present study, the concept of the original heuristic topology generator is proposed. The main idea that stands behind this proposal is to take advantage of the colliding

**Citation:** Bochenek, B.; Tajs-Zieli ´nska, K. Cellular Automaton Mimicking Colliding Bodies for Topology Optimization. *Materials* **2022**, *15*, 8057. https://doi.org/10.3390/ ma15228057

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 12 October 2022 Accepted: 11 November 2022 Published: 15 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

bodies phenomenon, and use the governing laws to derive original Cellular Automata rules which can efficiently perform optimal topologies generation process. The inspiration for this proposal was the series of papers by Kaveh and co-workers [12–15] in which the concept of Colliding Bodies Optimization for a function minimization has been proposed. This paper proposes an original technique which is also inspired by the collision of bodies phenomenon but this time it is oriented toward optimization of structure topology. It is worth underlining that the rules are built so as to cope also with irregular finite element meshes. The derived algorithm has been combined with ANSYS 14.0, a commercial finite element software package, to illustrate its versatility.

As mentioned above, the applied approach is based on the concept of Cellular Automata (CA). Cellular Automata are built so as to mimic the behavior of complicated systems in a relatively easy way. From a computational point of view, the special local rules are implemented with a view to control the performance of a system under consideration. Hence, the local physical quantities are respectively updated, which allows us to describe the global behavior of the system. The concept of Cellular Automata has been known since the late 1940s when von Neumann and Ulam proposed this idea. Henceforth, this approach has been found interesting by researchers representing various fields but probably for the first time topology optimization has been discussed within the CA approach only in the paper by Inou et al. [16]. Many papers have been hereafter published on that subject, and the majority of them have appeared during the last two decades, see e.g., [17–20] or [21]. The efficient CA algorithm has been also proposed and then developed by Bochenek and Tajs-Zieli ´nska [22,23] and recently [24,25].

The outline of the paper is as follows. In Section 2, the topology optimization problem is formulated, then the concept of Cellular Automata mimicking colliding bodies is introduced, and finally the algorithm built based on this idea is described. Its implementation in the topology generation process is illustrated by an introductory example. Original examples of topology generation of selected 2D structures are discussed in the first part of Section 3 presenting performance of the topology generator. Next in this section, utilizing results of the preliminary computations, the Cellular Automaton is combined with ANSYS as the efficient structural analysis tool and its application to selected, both plane and spatial, engineering tasks is presented. With a view to cover a broad area of implementations, the discussed tasks include also irregular cell lattice. Based on the results of performed tests, the paper ends with concluding remarks in Section 4.

### **2. Methods and Concepts**

In this section, the topology optimization problem is formulated, the concept of Cellular Automata mimicking colliding bodies is introduced, and the algorithm built based on this idea is described. The introductory example illustrates the implementation of the proposed concept into the topology generation process.

#### *2.1. Structural Topology Optimization*

The most commonly formulated structural topology optimization problem is to generate material layout which within a design domain leads to a minimal value of the structure compliance *c*, Equation (1). Hence, one can follow the optimization problem formulated in a widely recognized paper by Sigmund [26]. The available material volume fraction *κ* is defined and treated in the optimization process as the constraint imposed on structure volume *V*, Equation (2). The finite element approach has been applied:

$$\text{minimize } c(d) = \mathbf{u}^T \mathbf{k} \mathbf{u} = \sum\_{i=1}^N d\_i^p \mathbf{u}\_i^T \mathbf{k}\_i \mathbf{u}\_i \tag{1}$$

$$\text{subject to } V(\mathbf{d}) = \kappa V\_0 \tag{2}$$

$$
\mathbf{k}\,\mathbf{u} = \mathbf{f} \tag{3}
$$

$$0 < d\_{\min} \le d\_i \le 1. \tag{4}$$

= (3)

subject to ሺ) = (2)

The quantity **u***<sup>i</sup>* denotes the displacement vector, **k***<sup>i</sup>* stands for the stiffness matrix, and both are defined for *N* elements. The design variable *d<sup>i</sup>* , which represents the relative material density, is assigned to each element. In Equation (3), **k** represents the global stiffness matrix, **u** stands for the global displacement vector, and **f** is the vector of forces. Due to the simple bounds imposed in Equation (4) on the design variables with *dmin* (e.g., 10−<sup>9</sup> ) as a non-zero minimum value of relative density, singularity is avoided. The quantity denotes the displacement vector, stands for the stiffness matrix, and both are defined for *N* elements. The design variable , which represents the relative material density, is assigned to each element. In Equation (3), represents the global stiffness matrix, stands for the global displacement vector, and is the vector of forces. Due to the simple bounds imposed in Equation (4) on the design variables with (e.g., 10−9) as a non-zero minimum value of relative density, singularity is avoided.

The SIMP (solid isotropic material with penalization) approach (e.g., [27]) in the form of power law is adapted as the material representation, see Equation (5). The modulus of elasticity *E<sup>i</sup>* for each finite element is a function of the design variable *d<sup>i</sup>* : The SIMP (solid isotropic material with penalization) approach (e.g., [27]) in the form of power law is adapted as the material representation, see Equation (5). The modulus of elasticity for each finite element is a function of the design variable :

$$E\_i = d\_i^p E\_0. \tag{5}$$

In Equation (5), the quantity *E*<sup>0</sup> stands for modulus of elasticity, defined for a solid material, whereas *p* (typically *p* = 3) is responsible for penalization of intermediate densities. This allows controlling the design process and leads to obtaining black-and-white resulting structures. During the topology generation process, a material is redistributed within the design domain, which results in removing parts unnecessary from design criteria viewpoint. In Equation (5), the quantity stands for modulus of elasticity, defined for a solid material, whereas *p* (typically *p* = 3) is responsible for penalization of intermediate densities. This allows controlling the design process and leads to obtaining black-and-white resulting structures. During the topology generation process, a material is redistributed within the design domain, which results in removing parts unnecessary from design criteria viewpoint.

#### *2.2. Concept of the Cellular Automaton Mimicking Colliding Bodies 2.2. Concept of the Cellular Automaton Mimicking Colliding Bodies*

*Materials* **2022**, *15*, x FOR PEER REVIEW 3 of 20

The selection of a proper method of topology generation determines the effectiveness of the topology optimization process. Heuristic optimization techniques become popular among researchers because they are easy to implement numerically, gradient information is not required, and one can easily combine this type of algorithm with any finite element structural analysis code. The selection of a proper method of topology generation determines the effectiveness of the topology optimization process. Heuristic optimization techniques become popular among researchers because they are easy to implement numerically, gradient information is not required, and one can easily combine this type of algorithm with any finite element structural analysis code.

In this paper, the original heuristic topology generator built as Cellular Automaton which mimics Colliding Bodies (CAmCB) is proposed. The idea is that the design domain of the structure is decomposed into a lattice of cells which are usually equivalent to finite elements. For each cell, the surrounding cells form a neighborhood. The bodies are distributed within this lattice (Figure 1). In this paper, the original heuristic topology generator built as Cellular Automaton which mimics Colliding Bodies (CAmCB) is proposed. The idea is that the design domain of the structure is decomposed into a lattice of cells which are usually equivalent to finite elements. For each cell, the surrounding cells form a neighborhood. The bodies are distributed within this lattice (Figure 1).

**Figure 1.** The cell lattice. A body is placed in each cell. The neighborhood, which is represented by the red circle, is identified around each body/cell. **Figure 1.** The cell lattice. A body is placed in each cell. The neighborhood, which is represented by the red circle, is identified around each body/cell.

Let us assume that the bodies have masses and velocities. Masses are proportional to cell areas whereas velocities are related to cell compliances. Furthermore, let us imagine that the neighboring bodies collide with the central one, which results in changing its status. In what follows, the central body can either be forced to remain in its position, or is pushed away (Figure 2).

*Materials* **2022**, *15*, x FOR PEER REVIEW 4 of 20

**Figure 2.** The cell neighborhood. The neighboring cells collide with the central one. **Figure 2.** The cell neighborhood. The neighboring cells collide with the central one.

**Figure 2.** The cell neighborhood. The neighboring cells collide with the central one. From a topology generation point of view, the interpretation is that the central cell From a topology generation point of view, the interpretation is that the central cell remains solid, or is driven to become a void one (Figure 3). From a topology generation point of view, the interpretation is that the central cell remains solid, or is driven to become a void one (Figure 3).

Let us assume that the bodies have masses and velocities. Masses are proportional to cell areas whereas velocities are related to cell compliances. Furthermore, let us imagine that the neighboring bodies collide with the central one, which results in changing its status. In what follows, the central body can either be forced to remain in its position, or

Let us assume that the bodies have masses and velocities. Masses are proportional to cell areas whereas velocities are related to cell compliances. Furthermore, let us imagine that the neighboring bodies collide with the central one, which results in changing its status. In what follows, the central body can either be forced to remain in its position, or

**Figure 3.** The topology generation. Some cells remain, some are eliminated. **Figure 3.** The topology generation. Some cells remain, some are eliminated. **Figure 3.** The topology generation. Some cells remain, some are eliminated.

#### *2.3. Local Update Rules 2.3. Local Update Rules 2.3. Local Update Rules*

is pushed away (Figure 2).

is pushed away (Figure 2).

While building Cellular Automaton, it is assumed that the interactions between bodies/cells take place only within the specified neighborhood, where they are governed by local rules which are identical for all cells, and are applied simultaneously to each of them. According to the concept of the paper, the local rules are derived so as to mimic collisions taking place between bodies/cells within each neighborhood. The governing equations based on the physics laws of momentum and energy are applied. Let us consider the central cell and a neighboring one colliding with it (Figures 4 and 5). While building Cellular Automaton, it is assumed that the interactions between bodies/cells take place only within the specified neighborhood, where they are governed by local rules which are identical for all cells, and are applied simultaneously to each of them. According to the concept of the paper, the local rules are derived so as to mimic collisions taking place between bodies/cells within each neighborhood. The governing equations based on the physics laws of momentum and energy are applied. Let us con-While building Cellular Automaton, it is assumed that the interactions between bodies/cells take place only within the specified neighborhood, where they are governed by local rules which are identical for all cells, and are applied simultaneously to each of them. According to the concept of the paper, the local rules are derived so as to mimic collisions taking place between bodies/cells within each neighborhood. The governing equations based on the physics laws of momentum and energy are applied. Let us consider the central cell and a neighboring one colliding with it (Figures 4 and 5). *Materials* **2022**, *15*, x FOR PEER REVIEW 5 of 20

**Figure 4.** A neighboring body (k) collides with the central one (0). Before collision. **Figure 4.** A neighboring body (k) collides with the central one (0). Before collision.

**Figure 5.** A neighboring body (k) collides with the central one (0). After collision.

<sup>ଶ</sup> <sup>=</sup> <sup>1</sup>

ଶ <sup>=</sup> ሺ െ )ଵ 2ଵ 

ଶ <sup>=</sup> ሺ െ )ଵ 2ଵ 

where represents the cell area and is a function associated with local compliances. It is proposed to select the quantity ଶ as the basis for building the update rule. Before

<sup>⎧</sup> െ if ൏ ଵ

 ଶ െ ଵ

if ଵ

if ଶ

A monotonically increasing function representing cell compliances is selected for the intermediate interval ଵ ଶ and then function values are assigned to the cells, respectively. Here, the linear function is selected to fulfill: ሺଵ) = െ and ሺଶ)=,

െ

ଶ ଵ ଶ െ ଵ

Based on the results obtained from a structural analysis, the values of local compliances are calculated for all cells/elements. The compliances are sorted then in the ascending order, and those having the lowest and the highest values are identified. In the next step, ଵ, ଶ are selected and values of are assigned to cells ሺ = 1,2 … ) accord-

As stated earlier, velocities are related to cell compliances and masses are propor-

Based on the above, the velocity of the central cell after collision equals:

tional to cell areas. Equation (8) can be rewritten in the form of Equation (9):

that, the details regarding how to calculate values are given.

=

⎩ ⎪ ⎨ ⎪

= 2

ing to Equation (10):

thus:

<sup>2</sup> ଶ <sup>ଶ</sup> 1 <sup>2</sup> ଶ

ଵ െ ଵ = െଶ ଶ (6)

<sup>ଶ</sup> . (7)

. (8)

, (9)

ଶ. (10)

. (11)

The governing equations are as follows:

1 <sup>2</sup> ଵ <sup>ଶ</sup> 1 <sup>2</sup> ଵ

**Figure 5.** A neighboring body (k) collides with the central one (0). After collision.

**Figure 4.** A neighboring body (k) collides with the central one (0). Before collision.

**Figure 5.** A neighboring body (k) collides with the central one (0). After collision. The governing equations are as follows:

ing to Equation (10):

$$m\_k v\_{k1} - m\_0 v\_{01} = -m\_k v\_{k2} + m\_0 v\_{02} \tag{6}$$

$$
\frac{1}{2}m\_kv\_{k1}^2 + \frac{1}{2}m\_0v\_{01}^2 = \frac{1}{2}m\_kv\_{k2}^2 + \frac{1}{2}m\_0v\_{02}^2.\tag{7}
$$

1 <sup>2</sup> ଵ <sup>ଶ</sup> 1 <sup>2</sup> ଵ <sup>ଶ</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> ଶ <sup>ଶ</sup> 1 <sup>2</sup> ଶ <sup>ଶ</sup> . (7) Based on the above, the velocity of the central cell after collision equals:

$$v\_{02} = \frac{(m\_k - m\_0)v\_{01} + 2m\_k v\_{k1}}{m\_0 + m\_k}.\tag{8}$$

ଶ <sup>=</sup> ሺ െ )ଵ 2ଵ . (8) As stated earlier, velocities are related to cell compliances and masses are proportional to cell areas. Equation (8) can be rewritten in the form of Equation (9):

$$F\_{02} = \frac{(A\_k - A\_0)F\_{01} + 2A\_k F\_{k1}}{A\_0 + A\_k},\tag{9}$$

ଶ <sup>=</sup> ሺ െ )ଵ 2ଵ , (9) where represents the cell area and is a function associated with local compliances. where *A* represents the cell area and *F* is a function associated with local compliances. It is proposed to select the quantity *F*<sup>02</sup> as the basis for building the update rule. Before that, the details regarding how to calculate *F* values are given.

It is proposed to select the quantity ଶ as the basis for building the update rule. Before that, the details regarding how to calculate values are given. Based on the results obtained from a structural analysis, the values of local compliances are calculated for all cells/elements. The compliances are sorted then in the as-Based on the results obtained from a structural analysis, the values of local compliances are calculated for all cells/elements. The compliances are sorted then in the ascending order, and those having the lowest and the highest values are identified. In the next step, *N*1, *N*<sup>2</sup> are selected and values of *F* are assigned to cells (*i* = 1, 2 . . . *n*) according to Equation (10):

$$F\_{\overline{i}} = \left\{ \begin{array}{c} -\mathbb{C} \text{ if } i < N\_{\text{1}} \\\ f\_{i} & \text{if } \quad N\_{\text{1}} \le i \le N\_{\text{2}}. \\\ \mathbb{C} \text{ if } i > N\_{\text{2}} \end{array} \right. \tag{10}$$

 = ⎪ <sup>⎧</sup> െ if ൏ ଵ if ଵ ଶ. (10) A monotonically increasing function representing cell compliances is selected for the intermediate interval *N*<sup>1</sup> ≤ *i* ≤ *N*<sup>2</sup> and then function values are assigned to the cells, respectively. Here, the linear function is selected to fulfill: *fi*(*N*1) = −*C* and *fi*(*N*2) = *C*, thus:

$$f\_i = 2C \frac{i}{N\_2 - N\_1} - C \frac{N\_2 + N\_1}{N\_2 - N\_1}.\tag{11}$$

A monotonically increasing function representing cell compliances is selected for the intermediate interval ଵ ଶ and then function values are assigned to the cells, respectively. Here, the linear function is selected to fulfill: ሺଵ) = െ and ሺଶ)=, The quantity *C* in Equation (11) is a user-specified parameter, usually equal to 1. The above described compliance mapping technique, represented by Equations (10) and (11), has been discussed also in [25].

thus: = 2 െ ଶ ଵ . (11) Having finished with data preparation, the update rule can be built. Hence, repeating collisions for all neighboring cells, the average quantity *F*<sup>02</sup> can be calculated based on Equation (12):

$$\overline{F}\_{02} = \frac{1}{M} \sum\_{k=1}^{M} F\_{02(k)}.\tag{12}$$

Finally, the design variables can be updated according to Equation (13):

$$d\_{new}^{(i)} = d^{(i)} + m\overline{F}\_{02\prime} \tag{13}$$

where *m* denotes the move limit (e.g., *m* = 0.2).

#### *2.4. Algorithm*

In order to implement the above-proposed design rule, a numerical algorithm was built. The sequential approach was adapted for the optimization procedure, meaning that for each iteration, the structural analysis performed for the optimized element is followed by a local updating process. Simultaneously, for a specified volume fraction, a global volume constraint is applied. As a result, during the optimization process, the generated topologies preserve a specified volume fraction of a solid material.

The issue to discuss regards the form of Equation (9). In the case of a regular lattice of cells/elements, the first component of the numerator vanishes. In order to preserve the influence of the central cell compliance on the final result during the iteration process, it is proposed to modify the cells area representation:

$$A\_{\bar{l}} = A[1 + b(2r - 1)],\tag{14}$$

where

$$b = b\_0 \left( 1 - \frac{t - 1}{t\_{\max} - 1} \right). \tag{15}$$

In Equations (14) and (15), *b*<sup>0</sup> is a small value, *r* is a random number taken from [0, 1] interval, *t* stands for the current iteration number, whereas *tmax* is a selected number of iterations. As a result, *A<sup>i</sup>* = *A* only for *t* = *tmax*.

In order to control the topology generation, the threshold values *N*<sup>1</sup> and *N*<sup>2</sup> can be modified so as to adjust the width of the interval [*N*1, *N*2] during the iteration process. It is proposed to start with a relatively wide interval, and then to reduce it successively. As a result, at the beginning of the topology generation process, the large design domain is searched by the Automaton, and the majority of void cells is eliminated. Then, during the iterative process while reducing the interval [*N*1, *N*2], the so-called gray cells of intermediate densities are eliminated, which finally results in obtaining distinct solid/void structures.

#### *2.5. Introductory Example*

The rectangular structure shown in Figure 6 has been chosen as the introductory example. The mesh of 3200 (80 × 40) square elements/cells has been generated to perform structural analysis and topology optimization for the data: *E*<sup>0</sup> = 10 GPa, *ν* = 0.3, *P* = 100 N, *a* = 40 mm, *κ* = 0.5, *b*<sup>0</sup> = 0.05. As for the topology generation, the Moore type neighborhood, i.e., cells having common vertices with the central one, has been applied. *Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 20

**Figure 6.** The rectangular structure with applied load and support. **Figure 6.** The rectangular structure with applied load and support.

The CAmCB algorithm found the final topology, which is shown in Figure 7, whereas the iteration history is given in Figure 8. The strategy of , see Equation (10), implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration The CAmCB algorithm found the final topology, which is shown in Figure 7, whereas the iteration history is given in Figure 8. The strategy of *F<sup>i</sup>* , see Equation (10), implementation was as follows: one started with *N*<sup>1</sup> = *N*·0.02, and then from iteration 25 *N*<sup>1</sup> = *N*·0.5,

25 ଵ = ∙ 0.5, where is the number of cells. Simultaneously, ଶ = ∙ 0.6 has re-

**Figure 8.** The compliance history. Minimal value: 13.62 Nmm. Black dots on the red line represent

The compliance value found for this structure is equal to 13.62 Nmm. This outperforms the solution reported in [25] where compliance of 14.02 Nmm has been obtained

mained fixed for the entire iteration process.

the compliance values for subsequent iterations.

for the final topology.

**Figure 7.** The final topology.

entire iteration process. mained fixed for the entire iteration process. mained fixed for the entire iteration process.

**Figure 6.** The rectangular structure with applied load and support.

**Figure 6.** The rectangular structure with applied load and support.

*Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 20

*Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 20

where *N* is the number of cells. Simultaneously, *N*<sup>2</sup> = *N*·0.6 has remained fixed for the

The CAmCB algorithm found the final topology, which is shown in Figure 7, whereas the iteration history is given in Figure 8. The strategy of , see Equation (10), implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.5, where is the number of cells. Simultaneously, ଶ = ∙ 0.6 has re-

The CAmCB algorithm found the final topology, which is shown in Figure 7, whereas the iteration history is given in Figure 8. The strategy of , see Equation (10), implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.5, where is the number of cells. Simultaneously, ଶ = ∙ 0.6 has re-

**Figure 7.** The final topology. **Figure 7.** The final topology.

**Figure 7.** The final topology.

**Figure 8.** The compliance history. Minimal value: 13.62 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 8.** The compliance history. Minimal value: 13.62 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 8.** The compliance history. Minimal value: 13.62 Nmm. Black dots on the red line represent the compliance values for subsequent iterations.

The compliance value found for this structure is equal to 13.62 Nmm. This outperforms the solution reported in [25] where compliance of 14.02 Nmm has been obtained for the final topology. The compliance value found for this structure is equal to 13.62 Nmm. This outperforms the solution reported in [25] where compliance of 14.02 Nmm has been obtained for the final topology. The compliance value found for this structure is equal to 13.62 Nmm. This outperforms the solution reported in [25] where compliance of 14.02 Nmm has been obtained for the final topology.

#### **3. Results and Discussion**

The original examples of topology generation are discussed in this section, presenting the performance of the algorithm. With a view to cover a broad area of implementations, the discussed tasks regard plane and spatial structures. The case of irregular cell lattice is also considered.

In what follows, to illustrate more thoroughly how the proposed CAmCB algorithm works, some numerical examples have been selected. The algorithm performance is presented first for plane test structures, and then for plane and spatial engineering structures. For the test structures, a Matlab-based algorithm has been applied, whereas for engineering structures, the topology generator has been combined with the ANSYS system, which was responsible for performing the structural analysis.

#### *3.1. Topology Generation for the Test Structures*

The results of topology generation performed for four plane test structures are presented below.

#### 3.1.1. Test Structure 1 Test Structure 1 Test Structure 1

sented below.

sented below.

**3. Results and Discussion** 

**3. Results and Discussion** 

cell lattice is also considered.

cell lattice is also considered.

*3.1. Topology Generation for the Test Structures* 

*3.1. Topology Generation for the Test Structures* 

To perform the first test, the structure shown in Figure 9 has been selected. The mesh of 60,000 (400 × 150) square elements/cells has been implemented, and structural analysis and topology optimization have been performed for the data: *E*<sup>0</sup> = 10 GPa, *ν* = 0.3, *P* = 100 N, *a* = 50 mm, *κ* = 0.25, *b*<sup>0</sup> = 0.05. The Moore type neighborhood has been applied. To perform the first test, the structure shown in Figure 9 has been selected. The mesh of 60,000 (400 × 150) square elements/cells has been implemented, and structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 50 mm, = 0.25, = 0.05. The Moore type neighborhood has been applied. To perform the first test, the structure shown in Figure 9 has been selected. The mesh of 60,000 (400 × 150) square elements/cells has been implemented, and structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 50 mm, = 0.25, = 0.05. The Moore type neighborhood has been applied.

The original examples of topology generation are discussed in this section, presenting the performance of the algorithm. With a view to cover a broad area of implementations, the discussed tasks regard plane and spatial structures. The case of irregular

The original examples of topology generation are discussed in this section, presenting the performance of the algorithm. With a view to cover a broad area of implementations, the discussed tasks regard plane and spatial structures. The case of irregular

In what follows, to illustrate more thoroughly how the proposed CAmCB algorithm works, some numerical examples have been selected. The algorithm performance is presented first for plane test structures, and then for plane and spatial engineering structures. For the test structures, a Matlab-based algorithm has been applied, whereas for engineering structures, the topology generator has been combined with the ANSYS

In what follows, to illustrate more thoroughly how the proposed CAmCB algorithm works, some numerical examples have been selected. The algorithm performance is presented first for plane test structures, and then for plane and spatial engineering structures. For the test structures, a Matlab-based algorithm has been applied, whereas for engineering structures, the topology generator has been combined with the ANSYS

The results of topology generation performed for four plane test structures are pre-

The results of topology generation performed for four plane test structures are pre-

*Materials* **2022**, *15*, x FOR PEER REVIEW 8 of 20

*Materials* **2022**, *15*, x FOR PEER REVIEW 8 of 20

system, which was responsible for performing the structural analysis.

system, which was responsible for performing the structural analysis.

**Figure 9.** The test structure 1 with applied loads and support. **Figure 9.** The test structure 1 with applied loads and support. **Figure 9.** The test structure 1 with applied loads and support.

The algorithm found the final topology, which is shown in Figure 10, whereas the iteration history is given in Figure 11. The strategy of implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.75, and from iteration 75 ଵ = ∙ 0.9. The quantity ଶ = ∙ 0.98 has remained fixed for the whole iteration process. The algorithm found the final topology, which is shown in Figure 10, whereas the iteration history is given in Figure 11. The strategy of *F<sup>i</sup>* implementation was as follows: one started with *N*<sup>1</sup> = *N*·0.02, and then from iteration 25 *N*<sup>1</sup> = *N*·0.75, and from iteration 75 *N*<sup>1</sup> = *N*·0.9. The quantity *N*<sup>2</sup> = *N*·0.98 has remained fixed for the whole iteration process. The algorithm found the final topology, which is shown in Figure 10, whereas the iteration history is given in Figure 11. The strategy of implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.75, and from iteration 75 ଵ = ∙ 0.9. The quantity ଶ = ∙ 0.98 has remained fixed for the whole iteration process.

**Figure 10.** The final topology found for the test structure 1. **Figure 10.** The final topology found for the test structure 1. **Figure 10.** The final topology found for the test structure 1.

**Figure 11.** The compliance history for the test structure 1. Minimal value: 151.84 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 11.** The compliance history for the test structure 1. Minimal value: 151.84 Nmm. Black dots on the red line represent the compliance values for subsequent iterations.

For the structure shown in Figure 12, the mesh of 80,000 (400 × 200) square ele-

The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 13 and 14, respectively. As for the strategy of implementation of , one started with ଵ = ∙ 0.35, and then from iteration 25 ଵ =∙ 0.5, from iteration 50 ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. ଶ =∙

been performed for the data: = 10 GPa, = 0.3, = 100 N, = 100 mm, = 0.3, =

0.05. The Moore type neighborhood has been applied.

**Figure 12.** The test structure 2 with applied load and support.

0.98 has remained fixed for all iterations.

3.1.2. Test Structure 2

#### 3.1.2. Test Structure 2 3.1.2. Test Structure 2

For the structure shown in Figure 12, the mesh of 80,000 (400 × 200) square elements/cells has been generated. The structural analysis and topology optimization have been performed for the data: *E*<sup>0</sup> = 10 GPa, *ν* = 0.3, *P* = 100 N, *a* = 100 mm, *κ* = 0.3, *b*<sup>0</sup> = 0.05. The Moore type neighborhood has been applied. For the structure shown in Figure 12, the mesh of 80,000 (400 × 200) square elements/cells has been generated. The structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 100 mm, = 0.3, = 0.05. The Moore type neighborhood has been applied.

**Figure 11.** The compliance history for the test structure 1. Minimal value: 151.84 Nmm. Black dots

on the red line represent the compliance values for subsequent iterations.

*Materials* **2022**, *15*, x FOR PEER REVIEW 9 of 20

**Figure 12.** The test structure 2 with applied load and support. **Figure 12.** The test structure 2 with applied load and support.

The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 13 and 14, respectively. As for the strategy of implementation of , one started with ଵ = ∙ 0.35, and then from iteration 25 ଵ =∙ 0.5, from iteration 50 ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. ଶ =∙ 0.98 has remained fixed for all iterations. The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 13 and 14, respectively. As for the strategy of implementation of *F<sup>i</sup>* , one started with *N*<sup>1</sup> = *N*·0.35, and then from iteration 25 *N*<sup>1</sup> = *N*·0.5, from iteration 50 *N*<sup>1</sup> = *N*·0.75, and finally from iteration 75 *N*<sup>1</sup> = *N*·0.9. *N*2= *N*·0.98 has remained fixed for all iterations. *Materials* **2022**, *15*, x FOR PEER REVIEW 10 of 20 *Materials* **2022**, *15*, x FOR PEER REVIEW 10 of 20

**Figure 13.** The final topology found for the test structure 2. **Figure 13.** The final topology found for the test structure 2. **Figure 13.** The final topology found for the test structure 2.

3.1.3. Test Structure 3

3.1.3. Test Structure 3

applied.

applied.

**Figure 14.** The compliance history for the test structure 2. Minimal value: 125.07 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 14.** The compliance history for the test structure 2. Minimal value: 125.07 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 14.** The compliance history for the test structure 2. Minimal value: 125.07 Nmm. Black dots on the red line represent the compliance values for subsequent iterations.

To perform the third test, the structure shown in Figure 15 has been proposed. The

The algorithm found the final topology, which is shown in Figure 16, whereas the iteration history is given in Figure 17. The strategy of implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.75, from iteration

The algorithm found the final topology, which is shown in Figure 16, whereas the iteration history is given in Figure 17. The strategy of implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.75, from iteration

analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 10 mm, = 0.3, = 0.05. The Moore type neighborhood has been

analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 10 mm, = 0.3, = 0.05. The Moore type neighborhood has been

To perform the third test, the structure shown in Figure 15 has been proposed. The

**Figure 15.** The test structure 3 with applied load and support.

**Figure 15.** The test structure 3 with applied load and support.

#### 3.1.3. Test Structure 3 3.1.3. Test Structure 3

To perform the third test, the structure shown in Figure 15 has been proposed. The mesh of 80,000 (400 × 200) square elements/cells has been implemented and structural analysis and topology optimization have been performed for the data: *E*<sup>0</sup> = 10 GPa, *ν* = 0.3, *P* = 100 N, *a* = 10 mm, *κ* = 0.3, *b*<sup>0</sup> = 0.05. The Moore type neighborhood has been applied. To perform the third test, the structure shown in Figure 15 has been proposed. The mesh of 80,000 (400 × 200) square elements/cells has been implemented and structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 10 mm, = 0.3, = 0.05. The Moore type neighborhood has been applied.

**Figure 14.** The compliance history for the test structure 2. Minimal value: 125.07 Nmm. Black dots

on the red line represent the compliance values for subsequent iterations.

*Materials* **2022**, *15*, x FOR PEER REVIEW 10 of 20

**Figure 13.** The final topology found for the test structure 2.

**Figure 15.** The test structure 3 with applied load and support. **Figure 15.** The test structure 3 with applied load and support.

The algorithm found the final topology, which is shown in Figure 16, whereas the iteration history is given in Figure 17. The strategy of implementation was as follows: one started with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.75, from iteration The algorithm found the final topology, which is shown in Figure 16, whereas the iteration history is given in Figure 17. The strategy of *F<sup>i</sup>* implementation was as follows: one started with *N*<sup>1</sup> = *N*·0.02, and then from iteration 25 *N*<sup>1</sup> = *N*·0.75, from iteration 50 *N*<sup>1</sup> = *N*·0.75, and finally from iteration 75 *N*<sup>1</sup> = *N*·0.9. The quantity *N*<sup>2</sup> = *N*·0.98 remained fixed for the whole iteration process. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 20 50ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. The quantityଶ = ∙ 0.98 remained fixed for the whole iteration process. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 20 50ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. The quantityଶ = ∙ 0.98 remained fixed for the whole iteration process.

**Figure 16.** The final topology found for the test structure 3. **Figure 16.** The final topology found for the test structure 3. **Figure 16.** The final topology found for the test structure 3.

**Figure 17.** The compliance history for the test structure 3. Minimal value: 342.96 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 17.** The compliance history for the test structure 3. Minimal value: 342.96 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 17.** The compliance history for the test structure 3. Minimal value: 342.96 Nmm. Black dots on the red line represent the compliance values for subsequent iterations.

ments/cells has been applied. The structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 50 mm, = 0.25, =

ments/cells has been applied. The structural analysis and topology optimization have been performed for the data: = 10 GPa, = 0.3, = 100 N, = 50 mm, = 0.25, =

0.05. The Moore type neighborhood has been applied.

0.05. The Moore type neighborhood has been applied.

3.1.4. Test Structure 4

3.1.4. Test Structure 4

## 3.1.4. Test Structure 4

For the structure shown in Figure 18, the mesh of 137,500 (250 × 550) square elements/cells has been applied. The structural analysis and topology optimization have been performed for the data: *E*<sup>0</sup> = 10 GPa, *ν* = 0.3, *P* = 100 N, *a* = 50 mm, *κ* = 0.25, *b*<sup>0</sup> = 0.05. The Moore type neighborhood has been applied. *Materials* **2022**, *15*, x FOR PEER REVIEW 12 of 20 *Materials* **2022**, *15*, x FOR PEER REVIEW 12 of 20

**Figure 18.** The test structure 4 with applied load and support. **Figure 18.** The test structure 4 with applied load and support. **Figure 18.** The test structure 4 with applied load and support.

The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 19 and 20, respectively. As for the strategy of implementation: one started for ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.5, from iteration 50 ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. ଶ = ∙ 0.98 has remained fixed for all iterations. The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 19 and 20, respectively. As for the strategy of *F<sup>i</sup>* implementation: one started for *N*<sup>1</sup> = *N*·0.02, and then from iteration 25 *N*<sup>1</sup> = *N*·0.5, from iteration 50 *N*<sup>1</sup> = *N*·0.75, and finally from iteration 75 *N*<sup>1</sup> = *N*·0.9. *N*<sup>2</sup> = *N*·0.98 has remained fixed for all iterations. The final structure topology found by the algorithm and the illustration of the compliance history are given in Figures 19 and 20, respectively. As for the strategy of implementation: one started for ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.5, from iteration 50 ଵ = ∙ 0.75, and finally from iteration 75 ଵ = ∙ 0.9. ଶ = ∙ 0.98 has remained fixed for all iterations.

**Figure 19.** The final topology found for the test structure 4. **Figure 19. Figure 19.** The final topology found for the test structure 4. The final topology found for the test structure 4.

**Figure 20.** The compliance history for the test structure 4. Minimal value: 22.71 Nmm. Black dots on the red line represent the compliance values for subsequent iterations. **Figure 20.** The compliance history for the test structure 4. Minimal value: 22.71 Nmm. Black dots on the red line represent the compliance values for subsequent iterations.

As can be seen from the above, the original CAmCB algorithm can effectively generate minimal compliance topologies. It is also worth comparing the obtained results with the ones which can be found for the considered structures when using other existing and popular approaches. The top88 algorithm [28] based on the optimality criterion and the PTOc one [29], utilizing the concept of proportional topology optimization have been selected for this purpose. The above papers provide Matlab codes of topology generators and these have been used to perform computations for the test structures defined earlier in this section. Table 1 gathers the results of these computations. **Table 1.** Comparison of minimum compliance values [Nmm] found for the three algorithms. As can be seen from the above, the original CAmCB algorithm can effectively generate minimal compliance topologies. It is also worth comparing the obtained results with the ones which can be found for the considered structures when using other existing and popular approaches. The top88 algorithm [28] based on the optimality criterion and the PTOc one [29], utilizing the concept of proportional topology optimization have been selected for this purpose. The above papers provide Matlab codes of topology generators and these have been used to perform computations for the test structures defined earlier in this section. Table 1 gathers the results of these computations.


**Algorithm Test Structure 1 Test Structure 2 Test Structure 3 Test Structure 4 Table 1.** Comparison of minimum compliance values [Nmm] found for the three algorithms.

results which can be better in terms of objective function values than the ones obtained with the use of other approaches selected for this comparison. *3.2. Engineering Applications*  One can observe that the CAmCB algorithm proposed in this paper allows us to find results which can be better in terms of objective function values than the ones obtained with the use of other approaches selected for this comparison.

#### A series of illustrative engineering examples has been selected to examine the effec-*3.2. Engineering Applications*

tiveness of the introduced concept of the CAmCB topology generator. Both regular and irregular cell lattices are considered to show the algorithm performance and the versatility of the approach. As mentioned earlier, the proposed topology generator can be easily combined with any solver built on finite element methods. Hence, the optimization module has been linked to the professional system ANSYS to perform structural analyses. It is worth noting that the proposed algorithm does not require additional density filtering, the so-called gray elements are eliminated, and the checkerboard effect has not been observed for generated topologies. A series of illustrative engineering examples has been selected to examine the effectiveness of the introduced concept of the CAmCB topology generator. Both regular and irregular cell lattices are considered to show the algorithm performance and the versatility of the approach. As mentioned earlier, the proposed topology generator can be easily combined with any solver built on finite element methods. Hence, the optimization module has been linked to the professional system ANSYS to perform structural analyses. It is worth noting that the proposed algorithm does not require additional density filtering, the so-called gray elements are eliminated, and the checkerboard effect has not been observed for generated topologies.

#### 3.2.1. Mechanical Part 3.2.1. Mechanical Part

The model of a control arm structure presented in Figure 21 has been chosen for this purpose. The mesh of 16,304 elements/cells has been generated to perform structural analysis and topology optimization for the data: = 210 GPa, = 0.28, = 0.4, = 0.01. The structure consists of a non-optimized region presented in Figure 22 as a gray area whereas the design domain is presented as a red area. The structure is loaded by two The model of a control arm structure presented in Figure 21 has been chosen for this purpose. The mesh of 16,304 elements/cells has been generated to perform structural analysis and topology optimization for the data: *E*<sup>0</sup> = 210 GPa, *ν* = 0.28, *κ* = 0.4, *b*<sup>0</sup> = 0.01. The structure consists of a non-optimized region presented in Figure 22 as a gray area whereas the design domain is presented as a red area. The structure is loaded

by two concentrated forces: a horizontal force equal to 7000 N and a vertical one equal to 2700 N. The horizontal displacement of nodes in the inner bound of the round hole A are equal to zero, while all nodes in area B are fixed. remains fixed for all iterations. This strategy has been applied for all presented engineering examples. It is worth pointing out that in order to complete the optimization process about 50 iterations are needed.

concentrated forces: a horizontal force equal to 7000 N and a vertical one equal to 2700 N. The horizontal displacement of nodes in the inner bound of the round hole A are equal to

concentrated forces: a horizontal force equal to 7000 N and a vertical one equal to 2700 N.

As for the strategy of implementation: one starts with ଵ = ∙ 0.02, and then from iteration 25 ଵ = ∙ 0.5, and from iteration 50 ଵ = ∙ 0.75, whereas ଶ = ∙ 0.98

ଵ

*Materials* **2022**, *15*, x FOR PEER REVIEW 14 of 20

zero, while all nodes in area B are fixed.

**Figure 21.** The control arm: dimensions in mm. **Figure 21.** The control arm: dimensions in mm.

**Figure 22.** The design domain, loads, and supports of the control arm.

The algorithm found the final topology, which is shown in Figure 23. The resulting compliance equals 11,949 Nmm. Referring to the prior comparison of the results, the value of 12,372 Nmm was obtained when the algorithm [28] was utilized. compliance equals 11,949 Nmm. Referring to the prior comparison of the results, the **Figure 22.** The design domain, loads, and supports of the control arm. As for the strategy of *F<sup>i</sup>* implementation: one starts with *N*<sup>1</sup> = *N*·0.02, and then from iteration 25 *N*<sup>1</sup> = *N*·0.5, and from iteration 50 *N*<sup>1</sup> = *N*·0.75, whereas *N*<sup>2</sup> = *N*·0.98 remains fixed for all iterations. This strategy has been applied for all presented engineering examples. It is worth pointing out that in order to complete the optimization process about 50 iterations are needed.

The algorithm found the final topology, which is shown in Figure 23. The resulting compliance equals 11,949 Nmm. Referring to the prior comparison of the results, the value of 12,372 Nmm was obtained when the algorithm [28] was utilized.

**Figure 23.** The final topology found for the control arm. **Figure 23.** The final topology found for the control arm.

The CAmCB algorithm codes for the example considered in this section are provided in the Supplementary Files. The CAmCB algorithm codes for the example considered in this section are provided in the Supplementary Files.

#### 3.2.2. The Frame Structure-Generation of Topology for Irregular Cell Lattices 3.2.2. The Frame Structure-Generation of Topology for Irregular Cell Lattices

The aim of this example is to extend the presentation of the proposed algorithm toward an irregular grid of cells related to a non-regular mesh of finite elements. Resizing a traditional uniform grid of cells allows us to obtain flexible solutions, for e.g., extremely irregular design domains where it is difficult or impossible to cover them with uniform cells. Additionally, regions with stress concentrations, such as around holes or sharp edges, should be covered with a fine mesh, which is not necessary for the structure as a whole. The procedure of refining a mesh in selected regions can be used in order to achieve an accurate solution without an excessive increase of the number of elements caused by using a fine mesh implemented for the whole structure. The aim of this example is to extend the presentation of the proposed algorithm toward an irregular grid of cells related to a non-regular mesh of finite elements. Resizing a traditional uniform grid of cells allows us to obtain flexible solutions, for e.g., extremely irregular design domains where it is difficult or impossible to cover them with uniform cells. Additionally, regions with stress concentrations, such as around holes or sharp edges, should be covered with a fine mesh, which is not necessary for the structure as a whole. The procedure of refining a mesh in selected regions can be used in order to achieve an accurate solution without an excessive increase of the number of elements caused by using a fine mesh implemented for the whole structure.

The example illustrating this case is the portal frame presented in Figure 24. The data is as follows: = 200 GPa, = 0.25, = 0.5, = 0.01. The irregular lattice of cells is distributed according to Figure 25. For the irregular lattice of 14,024 cells (two-dimensional 6-node triangular elements—Plane82) ANSYS software was utilized for static analysis in the optimization process. The optimization has been performed and the obtained final topology is presented in Figure 26. Loads of 100N each have been ap-The example illustrating this case is the portal frame presented in Figure 24. The data is as follows: *E*<sup>0</sup> = 200 GPa, *ν* = 0.25, *κ* = 0.5, *b*<sup>0</sup> = 0.01. The irregular lattice of cells is distributed according to Figure 25. For the irregular lattice of 14,024 cells (two-dimensional 6-node triangular elements—Plane82) ANSYS software was utilized for static analysis in the optimization process. The optimization has been performed and the obtained final topology is presented in Figure 26. Loads of 100 N each have been applied. The resulting compliance is equal to 5.03 <sup>×</sup> <sup>10</sup>−<sup>3</sup> Nmm.

plied. The resulting compliance is equal to 5.03 × 10−3 Nmm. The algorithm found the final topology, which is shown in Figure 26.

#### 3.2.3. The Box Tube-Generation of Topology for Spatial Structure

The box tube shown in Figure 27 has been selected as the final example. The box tube cross section with 3 mm wall thickness is a square (100 mm × 100 mm), the tube is 250 mm long. Loads of 1000 N each have been applied as shown in Figure 28. The data is as follows: *E*<sup>0</sup> = 200 GPa, *ν* = 0.3, *κ* = 0.4, *b*<sup>0</sup> = 0.01. A regular mesh of 11,088 three-dimensional 8-node elements (Solid45) has been applied for a static analysis made by ANSYS software (the length of the element edge is 3 mm). For the example of this section, the algorithm utilizes the von Neumann type of neighborhood. The resulting topology is presented in Figure 29, for which the final compliance equals 278.8 Nmm.

The algorithm found the final topology which is shown in Figure 29.

The algorithm performance was additionally tested based on the same example, repeating computations for low volume fraction *κ* = 0.25. The resulting topology for which the final compliance reaches the value equal to 586.3 Nmm is presented in Figure 30.

*Materials* **2022**, *15*, x FOR PEER REVIEW 16 of 20

*Materials* **2022**, *15*, x FOR PEER REVIEW 16 of 20

**Figure 24.** The portal frame: dimensions in mm. **Figure 24.** The portal frame: dimensions in mm. **Figure 24.** The portal frame: dimensions in mm. **Figure 24.** The portal frame: dimensions in mm.

**Figure 25.** The design domain, loads, and supports of the portal frame. **Figure 25.** The design domain, loads, and supports of the portal frame. **Figure 25.** The design domain, loads, and supports of the portal frame.

tube cross section with 3 mm wall thickness is a square (100 mm × 100 mm), the tube is

The algorithm found the final topology, which is shown in Figure 26.

The algorithm found the final topology, which is shown in Figure 26.

The box tube shown in Figure 27 has been selected as the final example. The box tube cross section with 3 mm wall thickness is a square (100 mm × 100 mm), the tube is

The box tube shown in Figure 27 has been selected as the final example. The box tube cross section with 3 mm wall thickness is a square (100 mm × 100 mm), the tube is

The box tube shown in Figure 27 has been selected as the final example. The box **Figure 26.** The final topology found for the portal frame. **Figure 26.** The final topology found for the portal frame. **Figure 26.** The final topology found for the portal frame.

*Materials* **2022**, *15*, x FOR PEER REVIEW 17 of 20

250 mm long. Loads of 1000 N each have been applied as shown in Figure 28. The data is as follows: = 200 GPa, = 0.3, = 0.4, = 0.01. A regular mesh of 11,088 three-dimensional 8-node elements (Solid45) has been applied for a static analysis made by ANSYS software (the length of the element edge is 3 mm). For the example of this section, the algorithm utilizes the von Neumann type of neighborhood. The resulting topology is presented in Figure 29, for which the final compliance equals 278.8 Nmm.

250 mm long. Loads of 1000 N each have been applied as shown in Figure 28. The data is as follows: = 200 GPa, = 0.3, = 0.4, = 0.01. A regular mesh of 11,088 three-dimensional 8-node elements (Solid45) has been applied for a static analysis made by ANSYS software (the length of the element edge is 3 mm). For the example of this section, the algorithm utilizes the von Neumann type of neighborhood. The resulting topology is presented in Figure 29, for which the final compliance equals 278.8 Nmm.

**Figure 27.** The box tube: dimensions in mm. **Figure 27.** The box tube: dimensions in mm. **Figure 27.** The box tube: dimensions in mm.

**Figure 28.** The design domain, loads, and supports of the box tube. **Figure 28.** The design domain, loads, and supports of the box tube.

**Figure 28.** The design domain, loads, and supports of the box tube.

*Materials* **2022**, *15*, x FOR PEER REVIEW 18 of 20

no need to calculate gradients. In order to illustrate the effectiveness of the proposed **Figure 29.** The final topology for the box tube for volume fraction *κ* = 0.4.

The algorithm found the final topology which is shown in Figure 29.

The algorithm performance was additionally tested based on the same example,

The discussion regarding the proposed algorithm and its performance is summed up in this section. In the presented study, the original concept of Cellular Automaton mimicking Colliding Bodies (CAmCB) has been applied for topology optimization using the minimum compliance as the objective function. The CAmCB algorithm combines Cellular Automata heuristic with Colliding Bodies phenomenon to create a fast convergent technique which provides black-and-white topologies, without gray regions and the checkerboard effect. Moreover, additional density filtering is not necessary and there is

which the final compliance reaches the value equal to 586.3 Nmm is presented in Figure

**Figure 29.** The final topology for the box tube for volume fraction = 0.4.

**Figure 30.** The final topology for the box tube for volume fraction = 0.25.

30.

**4. Concluding Remarks** 

30.

**Figure 29.** The final topology for the box tube for volume fraction = 0.4.

The algorithm found the final topology which is shown in Figure 29.

The algorithm performance was additionally tested based on the same example, repeating computations for low volume fraction = 0.25. The resulting topology for which the final compliance reaches the value equal to 586.3 Nmm is presented in Figure

**Figure 30.** The final topology for the box tube for volume fraction = 0.25. **Figure 30.** The final topology for the box tube for volume fraction *κ* = 0.25.

#### **4. Concluding Remarks 4. Concluding Remarks**

The discussion regarding the proposed algorithm and its performance is summed up in this section. In the presented study, the original concept of Cellular Automaton mimicking Colliding Bodies (CAmCB) has been applied for topology optimization using the minimum compliance as the objective function. The CAmCB algorithm combines Cellular Automata heuristic with Colliding Bodies phenomenon to create a fast convergent technique which provides black-and-white topologies, without gray regions and the checkerboard effect. Moreover, additional density filtering is not necessary and there is no need to calculate gradients. In order to illustrate the effectiveness of the proposed The discussion regarding the proposed algorithm and its performance is summed up in this section. In the presented study, the original concept of Cellular Automaton mimicking Colliding Bodies (CAmCB) has been applied for topology optimization using the minimum compliance as the objective function. The CAmCB algorithm combines Cellular Automata heuristic with Colliding Bodies phenomenon to create a fast convergent technique which provides black-and-white topologies, without gray regions and the checkerboard effect. Moreover, additional density filtering is not necessary and there is no need to calculate gradients. In order to illustrate the effectiveness of the proposed CAmCB algorithm, selected numerical examples have been investigated. The algorithm performance is presented for plane test structures and for plane and spatial engineering structures. In the latter case, the proposed optimizer was combined with professional FEM analysis codes. The advantage of the developed algorithm is that it is a versatile technique which allows implementation of rectangular or triangular lattices, adaptation to highly non-uniform finite element lattices, as well as consideration of the total volume constraint with large and small volume fraction which is important especially for lightweight topology optimization. Preliminary studies reveal the possibility of applying CAmCB algorithm into uncommon but interesting issues such as the consideration of design-dependent loading (self-weight) or topology optimization of multi-material structures. The results of the tests performed so far are encouraging, which allows us to consider the proposed concept of the automaton mimicking colliding bodies phenomenon as an alternative algorithm to other existing topology generators suited for engineering applications.

**Supplementary Materials:** The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/ma15228057/s1.

**Author Contributions:** As far as conceptualization, methodology, software, investigation, and writing the paper are concerned B.B. and K.T.-Z. contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by the Faculty of Mechanical Engineering Cracow University of Technology.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


## *Article* **Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method**

**Tsz Tung Chan 1,\*, Stefan Heinrich <sup>1</sup> , Jürgen Grabe <sup>2</sup> and Maksym Dosta <sup>1</sup>**


**Abstract:** An inventive microscale simulation approach is applied to investigate the mechanics of frozen particle fluid systems (PFS). The simulation is based on the discrete element method (DEM) and bonded-particle model (BPM) approach. Discrete particles connected by solid bonds represent frozen agglomerates. Uniaxial compression experiments were performed to gather data for material modeling and further simulation model validation. Different typical mechanical behavior (brittle, ductile, dilatant) were reviewed regarding strain rates, saturation levels, and particle mechanical or surface properties. Among all these factors, strain rate significantly affects the mechanical behavior and properties of the agglomerates. A new solid bond model considering strain-dependent and time-dependent behavior is developed for describing the rheology of the frozen particle fluid systems. Without alternating Young's modulus and Poisson's ratio of the bond material, the developed solid model provides a suitable agreement with the experimental results regarding different strain rates.

**Keywords:** discrete element method; bonded-particle model; frozen particle fluid systems; material modeling; material micromechanics; creep

## **1. Introduction**

Frozen particle fluid systems (PFS), such as frozen agglomerates or frozen soils (grounds), are classified as composite materials that have been vastly investigated in academic and industrial fields. Regarding natural science, frozen soils have been studied for centuries. Due to building projects being developed closer to the arctic region, artificial ground freezing (AGF) was introduced as a temporary stabilizing technique for mining and construction projects [1]. In the technical particle field, the exploration of the interplay of granular and liquid phases at freezing temperature, as well as the resulting micromechanical behavior of the final composite, plays an essential role in many processes, starting from the storing of humid materials at low temperatures in the silo, ending with cryogenic grinding of temperature-sensitive materials.

A considerable portion of frozen PFS is ice, which has been studied in the academic field for centuries. Mechanics of ice were well investigated throughout different scopes, such as the mechanical properties of polycrystalline ice and columnar-grained ice [2,3], viscoelastic properties [4], creep behavior [5], temperature effects on creep behavior [6], the influence of surface properties on bond interface performance [7], critical factor influence the formation of ice bond [8]. Recent research has attempted to develop an artificial neural network to establish a predictive database for ice mechanical behavior regarding ice type, temperature, and strain rate [9].

Nonetheless, simulation tools were applied to investigate the mechanical behavior of ice. For instance, the finite element model (FEM) was used for the high dynamic behavior of ice [10], the interaction between ship structure with broken ice by the elastic ice model [11],

**Citation:** Chan, T.T.; Heinrich, S.; Grabe, J.; Dosta, M. Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method. *Materials* **2022**, *15*, 8505. https://doi.org/10.3390/ ma15238505

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 13 October 2022 Accepted: 22 November 2022 Published: 29 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

or the discrete element method (DEM) used for interaction between a conical structure with sea ice in the arctic region [12].

Apart from the bond material alone, the composite of frozen PFS has been investigated for centuries due to different applications related to frozen soils. Available literature has investigated rheology on frozen soil and the application of AGF [13–15].

Similarly, numerous simulation works have been carried out on frozen soil and particle-reinforced composite failure progress with FEM and xFEM (extended finite element method) [15,16]. Resolving crack propagation with xFEM showed promising results. However, re-meshing was required after every consecutive step due to the changing geometry. FEM coupling with a thermo-hydro-mechanical model has been conducted to tackle the interaction between thermal, hydraulic, and mechanical loads [17]. Similar works using DEM to resolve the mechanical problems for frozen soils have been carried out [18,19], with creditable results revealed from the simulation.

The particle-based mesh-free discrete element method (DEM) is a numerical model for understanding particle dynamics introduced by Cundall and Strack [20]. The bondedparticle model (BPM) extends the soft-sphere formulation of DEM [21], in which solid bonds are created to connect primary particles and form the agglomerates. During the simulation, each bond is treated as a separate entity and can be removed or created to mimic fracture or even material sintering. Both BPM and DEM have been applied for tackling different mechanical problems, including damage progress of concrete or high-performance concrete [22,23], cemented sand [24], rock mechanics [25], or mechanics of biopolymer aerogel [26], and many other materials. The main advantages of BPM simulation are:


The application of BPM demands high computational power due to the massive number of objects considered in the simulation and the small simulation time step. However, different parallelization techniques, especially focused on applying graphic process units (GPU), have efficiently compensated for such deficiencies [27].

In this contribution, a new solid bond model that combines strain-dependent linear elastic behavior with time-dependent creep behavior has been developed and integrated into the open-source DEM framework (MUSEN) [27]. The uniaxial compression experiment has proceeded for material parameter calibration and simulation model validation. Detail of the solid bond model and comparison between experimental and simulation results are discussed.

#### *1.1. Ice Rheology*

Ice rheology has been investigated for decades, with different literature analyzing the mechanical properties, viscoelastic properties, and creep behavior. Young's modulus of ice ranges between 9.7 and 11.2 GPa and Poisson's ratio from 0.29 to 0.32, obtained by the biaxial bending of ice plates at approximately −10 ◦C [28,29]. Ice tensile strength and compression strength react differently concerning temperature and strain rate. Ice's tensile strength ranges from 0.7 to 3.1 MPa, and compressive strength ranges from 5 to 25 MPa. The temperature-weakening effect on tensile strength is less than compressive strength [30].

Compared to temperature, strain rate has almost no effect on tensile strength but vastly alternates the compressive strength of ice. From 10−<sup>8</sup> s −1 to 10−<sup>3</sup> s −1 , the compressive strength increases with an increase in strain rate, surpassing 10−<sup>3</sup> s −1 compressive strength decreases with an increase in strain rate [29].

Apart from mechanical properties, mechanical behavior is crucial to be identified. Ductile, dilatant, and brittle behavior can be identified in ice, mainly characterized by strain rate. In the case of high strain rates, brittle behavior prevails. Under tensile stress with a

high strain rate, the resistance to ice damage can be described by the nucleation and growth of cracks. The strength is limited by the grain size of the ice, which can vastly alternate the crack propagation. Under compressive strain, abrupt collapses occur at around 0.5% strain, in which the shear plane is located around 30◦ to the maximum principal stress plane [31].

Ductile behavior is mainly identified by strain-rate hardening and thermal softening, in which the activation energy almost doubled above −10 ◦C. The dislocation-based process dominates the primary behavior under low strain rate deformation. The deformation relationship is quantified by quasi-static creep [32], where the material experience permanent deformation under stress, which is well below the yield stress for a prolonged duration. Three different creep phases can be identified with decreasing, steady, and increasing strain rates [33]. The power law can be used to formulate a rheological model for the creep behavior of ice [5]. From this numerical relationship, the main dominance of ice under low strain rate loading depends on applied stress and temperature [31]. The power law for creep (Equation (1)) describes the creep strain rate . *εn*,*cr*, which depends on the applied stress *σ* and two model parameters *A* and *m*. These parameters should be adjusted to considered different material and temperatures:

$$
\dot{\varepsilon}\_{\text{\tiny n,cr}} = A \cdot \sigma^m \tag{1}
$$

### *1.2. Rheology of Frozen Soil*

Frozen soil shares remarkable similarities with ice in mechanical behavior, which is closely related to frozen PFS. Suitable insight into frozen PFS can be given by interpreting the characteristic of frozen soil. It has four typical mechanical behaviors: brittle failure, brittle behavior with failure just after the yield point, ductile behavior with strain hardening, and strain softening [34]. Literature regarding permafrost soil samples and artificially frozen soil has confirmed such behavior [35,36]. Apart from the strain rate, volumetric ice content plays a vital role in frozen soil's mechanical behavior. The ice content influences the frozen soil's ductility or brittleness [37]. Furthermore, the temperature, salinity, dynamic load, and refreezing slightly influence the mechanical strength and behavior of the frozen soil [38–42].

Regarding the unfrozen water in the frozen soil, various works of the literature showed a remarkable effect on the frozen soil's mechanical and creep behavior [43,44]. Due to the incompressible nature of water, unfrozen water in the frozen soil can transfer both negative and positive pressure. Still, water can be discharged with drainage, and the frozen soil's original form of stress state can be reestablished. However, due to its complexity, unfrozen water is not considered in both the experimental or simulation stages.

### **2. Materials and Methods**

#### *2.1. Uniaxial Compression Test*

A universal texture measurement system, T.A. X.T. plus Texture Analyzer (Stable Micro System Ltd., Surrey, United Kingdom), was used to perform uniaxial compression tests. This setup was coupled with a self-constructed climate chamber, which aimed to maintain the ambient temperature of the inner cavity under the freezing point, thus preventing any thermal failure of specimens. It consisted of 3D-printed parts, coupled with radiators and radial fans, connected with a cryogenic unit (IKA Temperature Control, RC2 basic, lowest temperature: −20 ◦C). The chamber could maintain the ambient temperature at −10 ◦C, with less than 0.3 ◦C deviations. This experimental setup required a metal punch and base to protect the load cell and Texture Analyzer base, which were cooled down passively by the cool air in the climate chamber. Additionally, a 3D-printed fragment container is created to capture any fragment generated during the experiment, preventing the ice-particle pieces from being trapped in the radial fan. Three-dimensional CAD drawings of the climate chamber and coupled configuration are shown in Figure 1.

passively by the cool air in the climate chamber. Additionally, a 3D-printed fragment container is created to capture any fragment generated during the experiment, preventing the ice-particle pieces from being trapped in the radial fan. Three-dimensional CAD drawings

of the climate chamber and coupled configuration are shown in Figure 1.

**Figure 1.** Texture Analyzer system equipped with a climate chamber: (**Left**) Climate chamber coupled with Texture Analyzer; (**Right**) CAD design. **Figure 1.** Texture Analyzer system equipped with a climate chamber: (**Left**) Climate chamber coupled with Texture Analyzer; (**Right**) CAD design.

#### *2.2. Specimen Preparation 2.2. Specimen Preparation*

The experiments have been carried out for pure ice samples and afterward for the frozen PFS. Here, accurate specimen preparation has played a crucial role in the credibility of the experiment data since undesired defects influence the data's accuracy. For this purpose, special molds constructed with silicon have been used for the ice and PFS freezing process, as silicon is excellent for specimen extraction. The experiments have been carried out for pure ice samples and afterward for the frozen PFS. Here, accurate specimen preparation has played a crucial role in the credibility of the experiment data since undesired defects influence the data's accuracy. For this purpose, special molds constructed with silicon have been used for the ice and PFS freezing process, as silicon is excellent for specimen extraction.

Nevertheless, different challenges have been faced during the preparation of samples. For example, freezing water in silicon mold was not acceptable. Defects, such as bulging, have been observed. Bulging was caused due to water expansion, as water decreases in density during the phase transition stage. The phase transition occurs from outside to inwards, forming a rigid shell of frozen water surrounding the inner unfrozen part. As the inner part expands during further freezing, such expansion breaks the outer shell in different directions. It creates different bulging or defects, leading to the sample's unpredictable geometry. Nevertheless, different challenges have been faced during the preparation of samples. For example, freezing water in silicon mold was not acceptable. Defects, such as bulging, have been observed. Bulging was caused due to water expansion, as water decreases in density during the phase transition stage. The phase transition occurs from outside to inwards, forming a rigid shell of frozen water surrounding the inner unfrozen part. As the inner part expands during further freezing, such expansion breaks the outer shell in different directions. It creates different bulging or defects, leading to the sample's unpredictable geometry.

Hence, polycrystalline ice [6] was produced for the experiment based on the method proposed by [45], formed by the compression of snow in the initial stages. Firstly, crushed ice (1.12 mm–1.7 mm) formed after the freezing of distilled water was packed into the mold, and 0 °C distilled water was then injected from the bottom. Then, the specimen froze in the household refrigerator under −18 °C. The ice grain inside is randomly oriented to create a homogeneous structure. In addition, ice specimens produced by such a method were mainly formed from pre-frozen ice. The expansion rate of the specimen was kept under control, which allowed more delicate monitoring of stress and strain calculation. Hence, polycrystalline ice [6] was produced for the experiment based on the method proposed by [45], formed by the compression of snow in the initial stages. Firstly, crushed ice (1.12 mm–1.7 mm) formed after the freezing of distilled water was packed into the mold, and 0 ◦C distilled water was then injected from the bottom. Then, the specimen froze in the household refrigerator under −18 ◦C. The ice grain inside is randomly oriented to create a homogeneous structure. In addition, ice specimens produced by such a method were mainly formed from pre-frozen ice. The expansion rate of the specimen was kept under control, which allowed more delicate monitoring of stress and strain calculation.

Different particle types were considered for the frozen PFS specimen to cover the broadest range of interest. Samples of 10 ± 0.05 mm and 8 ± 0.05 mm in diameter were created, with a height-to-diameter ratio ranging around 1.6 ± 0.1. The 8 mm mold was designated to achieve higher stress, as the Texture Analyzer load cell can only withstand 500 N. The maximum compressive pressure achieved with the setup is 6.37 MPa (10 mm Different particle types were considered for the frozen PFS specimen to cover the broadest range of interest. Samples of 10 ± 0.05 mm and 8 ± 0.05 mm in diameter were created, with a height-to-diameter ratio ranging around 1.6 ± 0.1. The 8 mm mold was designated to achieve higher stress, as the Texture Analyzer load cell can only withstand 500 N. The maximum compressive pressure achieved with the setup is 6.37 MPa (10 mm diameter) and 9.95 MPa (8 mm diameter), respectively.

diameter) and 9.95 MPa (8 mm diameter), respectively. An overview of primary particles utilized for sample preparations and their classification is provided in Table 1. Agglomerates are constructed by primary natural (obtained from nature without any shape or surface alternating processing) or technical (manufactured according to predefined geometry and surface properties) particles.


Quartz sand X X 13.416 49.623 0.5

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**Table 1.** Classification of primary particles used in experiments according to their properties. tured according to predefined geometry and surface properties) particles.

An overview of primary particles utilized for sample preparations and their classification is provided in Table 1. Agglomerates are constructed by primary natural (obtained from nature without any shape or surface alternating processing) or technical (manufac-

The saturation level of investigated PFS was either 100% or 75%. The definition of saturation level is calculated according to the remaining volume, which is not occupied by particles. The number of particles in the specimen is the same across different saturation levels. The saturation level governed by what portion of the remaining volume was occupied by bond material is calculated as follows: The saturation level of investigated PFS was either 100% or 75%. The definition of saturation level is calculated according to the remaining volume, which is not occupied by particles. The number of particles in the specimen is the same across different saturation levels. The saturation level governed by what portion of the remaining volume was

$$Saturation\,level = \frac{Volume\_{bond}}{Volume\_{specific} - Volume\_{particle}} \cdot 100\% \tag{2}$$

The samples with 100% saturation were prepared as follows. Firstly, primary particles were poured into the mold. Afterward, deionized water was injected from the mold's bottom and passed to the degassing chamber to avoid forming gas bubbles; hence no undesired defects existed in the agglomerates. For 75% saturation samples, a predefined amount of water was sprayed onto the particles, and then the particles were thoroughly mixed and filled into the mold. The samples with 100% saturation were prepared as follows. Firstly, primary particles were poured into the mold. Afterward, deionized water was injected from the mold's bottom and passed to the degassing chamber to avoid forming gas bubbles; hence no undesired defects existed in the agglomerates. For 75% saturation samples, a predefined amount of water was sprayed onto the particles, and then the particles were thoroughly mixed and filled into the mold.

Both types of PFS were then frozen overnight in a household refrigerator at −18 ◦C. PFS specimens were extracted from the silicon molds, and necessary adjustment with a utility knife blade was applied if the contact surface was not perpendicular to the sidewall. Two final samples with a diameter of 10 mm are shown in Figure 2. Both types of PFS were then frozen overnight in a household refrigerator at −18 °C. PFS specimens were extracted from the silicon molds, and necessary adjustment with a utility knife blade was applied if the contact surface was not perpendicular to the sidewall. Two final samples with a diameter of 10 mm are shown in Figure 2.

**Figure 2.** Two exemplary samples of frozen PFS: (**Left**) aggregate with alpha-alumina primary particles at 75% saturation level; (**Right**): glass bead PFS with 100% saturation. **Figure 2.** Two exemplary samples of frozen PFS: (**Left**) aggregate with alpha-alumina primary particles at 75% saturation level; (**Right**): glass bead PFS with 100% saturation.

#### *2.3. Investigated Parameter Space 2.3. Investigated Parameter Space*

Since the strain rate has a decisive influence on the rheological behavior of ice and frozen soil [1,31], two different strain rates have been tested to review different PFS mechanical behavior. A total of 10−<sup>3</sup> s <sup>−</sup><sup>1</sup> was applied for a low strain rate, and 10−<sup>2</sup> s <sup>−</sup><sup>1</sup> was

used for a high strain rate. The strain rate is controlled by the compression speed, specified according to specimen height. The compression speed of 0.02 mm/s, which is two times larger than the minimal compression speed of the Texture Analyzer, was chosen as the low strain rate for all specimens. For high strain rate, compression speed is calculated by *specimen height* × 0.01. All experiments have been performed at a temperature of around −10 ◦C. The possibility of unfrozen water inside the agglomerate can be minimized with such a temperature. for a high strain rate. The strain rate is controlled by the compression speed, specified according to specimen height. The compression speed of 0.02 mm/s, which is two times larger than the minimal compression speed of the Texture Analyzer, was chosen as the low strain rate for all specimens. For high strain rate, compression speed is calculated by ℎℎ × 0.01. All experiments have been performed at a temperature of around −10 °C. The possibility of unfrozen water inside the agglomerate can be minimized with such a temperature.

Since the strain rate has a decisive influence on the rheological behavior of ice and frozen soil [1,31], two different strain rates have been tested to review different PFS me-

−1 was applied for a low strain rate, and 10−2 s

<sup>−</sup><sup>1</sup> was used

#### *2.4. Ice Creep Behavior 2.4. Ice Creep Behavior*

chanical behavior. A total of 10

The polycrystalline ice specimen was loaded to a particular force, and the applied pressure was held for 240 s to review the creep behavior. Figure 3 shows the transition from the primary creep (decreasing in strain rate) to the secondary creep (steady in strain rate) in the experiment. However, an utterly constant strain rate was not yet achieved, which is crucial for calibrating the creep parameters for the simulation. In the literature, a steady strain rate was achieved after 100 min of force holding [32], which was impossible from the current experimental setup. The polycrystalline ice specimen was loaded to a particular force, and the applied pressure was held for 240 s to review the creep behavior. Figure 3 shows the transition from the primary creep (decreasing in strain rate) to the secondary creep (steady in strain rate) in the experiment. However, an utterly constant strain rate was not yet achieved, which is crucial for calibrating the creep parameters for the simulation. In the literature, a steady strain rate was achieved after 100 min of force holding [32], which was impossible from the current experimental setup.

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−3 s

**Figure 3.** Polycrystalline ice specimen subjected to constant force load for 240 s under three different forces. **Figure 3.** Polycrystalline ice specimen subjected to constant force load for 240 s under three different forces.

Compressive Young's modulus was estimated according to the primary loading part, in which a linear stress-strain relationship was observed. Several demonstration stressstrain curves are presented in Figure 4. Representative stress-strain curve of polycrystalline ice during creep experiment (primary loading phase) A linear relationship is fitted and the slope is obtained from the linear portion of the curve. The fitted curve is marked with a straight line on the figure. The compressive Young's modulus estimation does not consider the primary loading portion. As this portion usually has a lower stiffness. It is related to the surface of the ice column is not entirely in touch with the punch, and the punch was deforming the surface rather than compressing the complete column of ice. The compressive Young's modulus is 346.575 ± 48.2687 MPa. Compressive Young's modulus was estimated according to the primary loading part, in which a linear stress-strain relationship was observed. Several demonstration stressstrain curves are presented in Figure 4. Representative stress-strain curve of polycrystalline ice during creep experiment (primary loading phase) A linear relationship is fitted and the slope is obtained from the linear portion of the curve. The fitted curve is marked with a straight line on the figure. The compressive Young's modulus estimation does not consider the primary loading portion. As this portion usually has a lower stiffness. It is related to the surface of the ice column is not entirely in touch with the punch, and the punch was deforming the surface rather than compressing the complete column of ice. The compressive Young's modulus is 346.575 ± 48.2687 MPa.

**Figure 4.** Representative stress-strain curve of polycrystalline ice during creep experiment (primary loading phase). **Figure 4.** Representative stress-strain curve of polycrystalline ice during creep experiment (primary loading phase).

#### *2.5. Fracture Patterns of Frozen PFS 2.5. Fracture Patterns of Frozen PFS*

Figure 5 shows the breakage pattern of the frozen PFS of different saturation levels during uniaxial compression under a high strain rate (10−2 s −1). The fully saturated PFS samples can better transmit the pressure through the specimen. Thus, the cracks propagate from top to bottom, breaking the specimen into relatively large fragments containing numerous primary particles. In this case, the complete failure of the specimen occurred vigorously. In contrast, no large fragments were formed for PFS samples with 75% saturation. Only tiny pieces with several primary particles detached from the main structure during loading. Figure 5 shows the breakage pattern of the frozen PFS of different saturation levels during uniaxial compression under a high strain rate (10−<sup>2</sup> s −1 ). The fully saturated PFS samples can better transmit the pressure through the specimen. Thus, the cracks propagate from top to bottom, breaking the specimen into relatively large fragments containing numerous primary particles. In this case, the complete failure of the specimen occurred vigorously. In contrast, no large fragments were formed for PFS samples with 75% saturation. Only tiny pieces with several primary particles detached from the main structure during loading.

One additional phenomenon observed was the inhomogeneity of 75% saturation level PFS. The PFS is prepared by freezing particle water mixture. As the fridge temperature is kept at −18 °C, the water in the agglomerate cannot be frozen instantly. During the slow freezing process, part of the water around the particle performs a phase transition, and the remaining water is concentrated toward the lower portion of the agglomerate, forming an inhomogeneous specimen. Such a slow freezing process also imposes internal and bond structure differences concerning different primary particles. As the particle surface topography, particles' shape, separation distance, and liquid bridge size influence the successfulness of liquid bridge formation [46–48]. Thus, only the successfully formed liquid bridge forms a solid bond. This causes the differences in internal structure and bond structure to differ concerning different types of primary particles. However, it is impossible to gather the difference in the bond structure and internal structure in 75% saturation level PFS between different primary particles with the current experimental setup. One additional phenomenon observed was the inhomogeneity of 75% saturation level PFS. The PFS is prepared by freezing particle water mixture. As the fridge temperature is kept at −18 ◦C, the water in the agglomerate cannot be frozen instantly. During the slow freezing process, part of the water around the particle performs a phase transition, and the remaining water is concentrated toward the lower portion of the agglomerate, forming an inhomogeneous specimen. Such a slow freezing process also imposes internal and bond structure differences concerning different primary particles. As the particle surface topography, particles' shape, separation distance, and liquid bridge size influence the successfulness of liquid bridge formation [46–48]. Thus, only the successfully formed liquid bridge forms a solid bond. This causes the differences in internal structure and bond structure to differ concerning different types of primary particles. However, it is impossible to gather the difference in the bond structure and internal structure in 75% saturation level PFS between different primary particles with the current experimental setup.

**Figure 5.** The fracture pattern for agglomerates at the strain rate is 10−2 s −1: (**a**) 100% saturation, alphaalumina PFS; (**b**) 75% saturation, glass PFS. *2.6. Mechanical Behavior of Frozen PFS* **Figure 5.** The fracture pattern for agglomerates at the strain rate is 10−<sup>2</sup> s −1 : (**a**) 100% saturation, alpha-alumina PFS (Supplementary material: Video S1); (**b**) 75% saturation, glass PFS (Supplementary material: Video S2).

#### All possible mechanical behaviors can be observed in the experiments performed un-*2.6. Mechanical Behavior of Frozen PFS*

der different conditions, such as varied types of primary particles, strain rate, and saturation levels. Figure 6 shows the stress-strain curve of some representative experiments for PFS All possible mechanical behaviors can be observed in the experiments performed under different conditions, such as varied types of primary particles, strain rate, and saturation levels.

with sand and alpha-alumina primary particles under high strain (HS) and low strain (LS) rates. The results show that the sand PFS under high strain rate loading reveals brittle failure. In the case of alpha-alumina and high strain rates, brittle behavior with failure is observed just after the yield point. Finally, ductile behavior with strain hardening is typical for sand PFS under low strain rate loading and strain softening for alpha-alumina PFS under low strain rates. Figure 6 shows the stress-strain curve of some representative experiments for PFS with sand and alpha-alumina primary particles under high strain (HS) and low strain (LS) rates. The results show that the sand PFS under high strain rate loading reveals brittle failure. In the case of alpha-alumina and high strain rates, brittle behavior with failure is observed just after the yield point. Finally, ductile behavior with strain hardening is typical for sand PFS under low strain rate loading and strain softening for alpha-alumina PFS under low strain rates.

**Figure 6.** Representative stress-strain characteristics for different frozen PFS under a high strain (HS) rate of 10−2 s <sup>−</sup><sup>1</sup> and low strain (LS) rate of 10−3 s −1 . **Figure 6.** Representative stress-strain characteristics for different frozen PFS under a high strain (HS) rate of 10−<sup>2</sup> s <sup>−</sup><sup>1</sup> and low strain (LS) rate of 10−<sup>3</sup> s −1 .

#### *2.7. Bonded-Particle Model Approach 2.7. Bonded-Particle Model Approach*

Open-source GPU-accelerated DEM framework MUSEN was used to describe frozen PFS behavior. Among others, this system supports calculations with the bonded-particle model (BPM) [27]. In BPM, the spherical primary particles can be connected through solid bonds. Every single bond can have its unique dimensions and material properties. Bonds can be created or destroyed during simulation to mimic sintering or fracture behavior. The bond is modeled as a virtual cylindrical linkage between the particles. Bonds are treated as virtual objects with no volume or mass, and the internal force is calculated according to strain changes; in our case, time is also considered in the solid bond mathemat-Open-source GPU-accelerated DEM framework MUSEN was used to describe frozenPFS behavior. Among others, this system supports calculations with the bonded-particlemodel (BPM) [27]. In BPM, the spherical primary particles can be connected through solid bonds. Every single bond can have its unique dimensions and material properties. Bonds can be created or destroyed during simulation to mimic sintering or fracture behavior. The bond is modeled as a virtual cylindrical linkage between the particles. Bonds are treated as virtual objects with no volume or mass, and the internal force is calculated according to strain changes; in our case, time is also considered in the solid bond mathematical model.

ical model. Agglomerate is generated with two consecutive steps. The packing of primary particles is generated in a virtual volume according to the force-bias algorithm [26]. The virtual volume represents the geometry of the specimen. All particles are randomly generated in this pseudo space, which means no particles were generated outside the virtual volume. The number of particles is governed by the porosity preset in the particle generation progress. In each iteration, the overlaps between particles are detected, and force, which is proportional to the overlap, is calculated. Afterward, the primary particles are shifted according to this force. Generation completes if the maximum overlap is smaller than the target value. Finally, to build the agglomerate, particles are connected with solid bonds. The generation of bonds is governed by the minimum ( ) and maximum distance between the surface of spheres ( ). If the distance between the surfaces ranges between the preset value ( and ), bond is created. As mentioned, the bonds are treated as virtual objects; thus, overlapping between bonds is allowed. By alternating the limiting value , different amounts of bonds can be generated inside the same particle packing. In most cases, minimum distance ( ) is set to a negative value, as particle overlapping Agglomerate is generated with two consecutive steps. The packing of primary particles is generated in a virtual volume according to the force-bias algorithm [26]. The virtual volume represents the geometry of the specimen. All particles are randomly generated in this pseudo space, which means no particles were generated outside the virtual volume. The number of particles is governed by the porosity preset in the particle generation progress. In each iteration, the overlaps between particles are detected, and force, which is proportional to the overlap, is calculated. Afterward, the primary particles are shifted according to this force. Generation completes if the maximum overlap is smaller than the target value. Finally, to build the agglomerate, particles are connected with solid bonds. The generation of bonds is governed by the minimum (*L min gen* ) and maximum distance between the surface of spheres (*L max gen* ). If the distance between the surfaces ranges between the preset value (*L min gen* and *L max gen* ), bond is created. As mentioned, the bonds are treated as virtual objects; thus, overlapping between bonds is allowed. By alternating the limiting value *L max gen* , different amounts of bonds can be generated inside the same particle packing. In most cases, minimum distance (*L min gen* ) is set to a negative value, as particle overlapping is allowed during package generation.

#### is allowed during package generation. *2.8. Solid Bond Model Considering Creep Behavior*

*2.8. Solid Bond Model Considering Creep Behavior* The newly developed solid bond model aims to tackle the strain rate-dependent behavior of the bond material. Therefore, the component to consider the creep behavior has been included in the model. The particle-particle and particle-wall interactions are calcu-The newly developed solid bond model aims to tackle the strain rate-dependent behavior of the bond material. Therefore, the component to consider the creep behavior has been included in the model. The particle-particle and particle-wall interactions are calculated according to the Hertz–Mindlin model, whereby the normal force is calculated

lated according to the Hertz–Mindlin model, whereby the normal force is calculated

according to Hertzian theory [49], and the tangential force is calculated according to the model proposed by Mindlin et al. [50].

During simulation, the calculation of bond force is separated into the normal and tangential directions. The newly developed model coupled the strain-dependent elastic bond model with the time-dependent creep model. As schematically shown in Figure 6, the primary deformation stage of frozen PFS is in a linear elastic relationship [51]. The linear elastic strain-dependent relationship describes the primary loading phases of bonds. The total strain *εn*,*to* in normal direction is calculated based on the initial *L<sup>I</sup>* and current bond length *Lc*:

$$
\varepsilon\_{n,to} = \frac{L\_c - L\_I}{L\_I} \tag{3}
$$

Total strain in the normal direction can be decomposed into two parts:

$$
\varepsilon\_{\mathfrak{n}, \mathfrak{o}} = \varepsilon\_{\mathfrak{n}, \mathfrak{e} \mathfrak{l}} + \varepsilon\_{\mathfrak{n}, \mathfrak{c} \mathfrak{r}} \tag{4}
$$

where *εn*,*el* is the elastic strain and *εn*,*cr* is the irreversible deformation due to creep. The simulation automatically replaces plastic deformation with creep deformation [52]. Plastic deformation is regarded as an inelastic deformation, effectively the deformation due to creep under viscoelastic conditions. Normal bond stress *σ<sup>n</sup>* is calculated by:

$$
\sigma\_n = \begin{cases}
\text{E-}\varepsilon\_{n,\text{el}} & \text{if } \sigma\_n < \sigma\_{n,y} \\
\sigma\_{n,y} & \text{if } \sigma\_n \ge \sigma\_{n,y}
\end{cases}
\tag{5}
$$

where *E* is Young's modulus of the bond material and the *σn*,*<sup>y</sup>* is the yield strength.

According to Norton [53], the power law describes the creep behavior of solid bonds (Equation (1)). The power law can provide approximately the same behavior concerning different applied stresses, which means the equation can provide an approximate same "shape" regardless of different applied stress [54]. Change of temperature during simulation is not considered for the model simplicity. The irreversible creep strain in the solid bonds is calculated iteratively in each time step as:

$$
\varepsilon\_{\mathfrak{n},\mathcal{C}}(t+\Delta t) = \varepsilon\_{\mathfrak{n},\mathcal{C}}(t) + \Delta t \cdot A \cdot \left(\sigma\_{\mathfrak{n}}(t)\right)^{\mathfrak{m}} \tag{6}
$$

∆*t* denotes the simulation time step. The resulting bond force in the normal direction *Fn*,*<sup>b</sup>* calculated by:

$$F\_{n,b}\left(t+\Delta t\right) = E \cdot A\_b \cdot \left(\varepsilon\_{n,to} - \varepsilon\_{n,cr}(t+\Delta t)\right) \tag{7}$$

where *A<sup>b</sup>* is the bond's cross-cut area. With strain applied to the bond, normal stress response increased according to an approximately linear relationship. The calculation of bond normal stress during the loading part combines the linear strain-dependent relationship with the not linear time-dependent relationship. Bond normal stress increases until yield strength and holds. If strain remains unchanged after a specific time, force response decreases gradually, calculated according to the creep parameter and the previous time step's bond normal stress. A demonstration of the stress-time relationship for a single bond is shown in Figure 7.

**Figure 7.** Schematic stress-strain diagram for solid bond under load in the normal direction. **Figure 7.** Schematic stress-strain diagram for solid bond under load in the normal direction.

Creep behavior is considered in both normal and tangential directions. The same power law of creep describes the creep behavior in the tangential direction. Unlike the calculation of strain in the normal direction, the bond deformation in the tangential direction ⃗ in every simulation step is updated according to the previous time step relative Creep behavior is considered in both normal and tangential directions. The same power law of creep describes the creep behavior in the tangential direction. Unlike the calculation of strain in the normal direction, the bond deformation in the tangential direction <sup>→</sup> *δ t* in every simulation step is updated according to the previous time step relative motion:

$$
\stackrel{\rightarrow}{\delta}\_{t}(t+\Delta t) = T \cdot \stackrel{\rightarrow}{\delta}\_{t}(t) + \stackrel{\rightarrow}{v}\_{t,rel} \cdot \Delta t \tag{8}
$$

 () + ⃑, ∙ ∆ (7) where is the rotation matrix to consider the motion of connected particles between the current and previous time step [55], and ⃑, is the relative velocity at the contact point where *T* is the rotation matrix to consider the motion of connected particles between the current and previous time step [55], and \* *v <sup>t</sup>*,*rel* is the relative velocity at the contact point in the tangential direction.

in the tangential direction. The tangential stress, which is crucial for the calculation of the tangential creep strain rate, is calculated according to the following: The tangential stress, which is crucial for the calculation of the tangential creep strain rate, is calculated according to the following:

$$
\sigma\_t = \frac{\left| \stackrel{\rightarrow}{\delta}\_I \right|}{L\_I} \cdot \frac{E}{2 \cdot (1 + v)} \tag{9}
$$

where is the Poisson's ratio of the bond material. The same power law with the same creep parameters A and m are used for the creep behavior in the tangential direction. Strain change due to creep in the tangential direction after every consecutive time step is calculated according to: where *v* is the Poisson's ratio of the bond material. The same power law with the same creep parameters A and m are used for the creep behavior in the tangential direction. Strain change due to creep in the tangential direction after every consecutive time step is calculated according to:

$$
\stackrel{\rightarrow}{\varepsilon}\_{t}(t+\Delta t) = \stackrel{\rightarrow}{\varepsilon}\_{t}(t) + \varepsilon\_{t,\mathcal{cr}}\left(\Delta t\right) \cdot \stackrel{\rightarrow}{r}\_{t} \tag{10}
$$

⃗ () where <sup>→</sup> *r t* is the unit vector of the bond in the tangential direction, which is defined by:

is the unit vector of the bond in the tangential direction, which is defined by:

$$\stackrel{\rightarrow}{r}\_{t} = \frac{\stackrel{\rightarrow}{\delta}\_{t}(t)}{\left|\stackrel{\rightarrow}{\delta}\_{t}(t)\right|}\tag{11}$$

(11)

⃗ , = ⃗ ( + ∆) ∙ 2 ∙ (1 + ) The resulting tangential force can be calculated by:

motion:

where ⃗

$$
\stackrel{\rightarrow}{F}\_{t,b} = \frac{\stackrel{\rightarrow}{\delta}\_{I}(t+\Delta t)}{L\_{I}} \cdot \frac{E}{2 \cdot (1+v)} \cdot A\_{b} \tag{12}
$$

∙

 as well as comparing the total strain and critical breakage strain , as: Apart from the agglomerate mechanical behavior, the agglomerate fracture is considered in the model. This is accomplished by comparing the individual bond stresses under

loading with material properties such as normal *σmax* and tangential strength *τmax* as well as comparing the total strain and critical breakage strain *εn*,*max* as: ⃗ , + ⃗⃗ , ∙ ≥ (12)

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$$\frac{\stackrel{\rightarrow}{F}\_{n,b}}{A\_b} + \frac{\stackrel{\rightarrow}{M}\_{n,b} \cdot R\_b}{I} \ge \sigma\_{\text{max}}\tag{13}$$

$$\frac{\stackrel{\rightarrow}{F}\_{t,b}}{A\_b} + \frac{\stackrel{\rightarrow}{\dot{M}}\_{t,b} \cdot R\_b}{J} \ge \tau\_{\text{max}}\tag{14}$$

$$
\varepsilon\_{n,to} \ge \varepsilon\_{n,max} \tag{15}
$$

$$
\to \qquad \to \qquad
$$

where *R<sup>b</sup>* is the bond radius, → *Mn*,*<sup>b</sup>* and → *Mt*,*<sup>b</sup>* are the bending and torsional moments of the bond, respectively, *I* is the moment of inertia, and *J* is the polar (torsional) moment of inertia of the bond's cross-section. of inertia of the bond's cross-section. Pressure melting and recreation of bonds are not considered in this solid bond model, as it vastly increases the complexity of the mathematical model, which has to consider the

Pressure melting and recreation of bonds are not considered in this solid bond model, as it vastly increases the complexity of the mathematical model, which has to consider the temperature, pressure, and factors governing the recreation of the bond. temperature, pressure, and factors governing the recreation of the bond. The normal compressive strength alternated according to strain rate [29]. However, due to the nature of the model, implementing a relationship for calculating normal and

The normal compressive strength alternated according to strain rate [29]. However, due to the nature of the model, implementing a relationship for calculating normal and tangential strength according to strain rate causes massive fluctuation in both values. The newly developed solid bond model has not considered the strain rate-dependent compressive strength relationship. Variance in bond normal strength due to strain rate or particle surface properties is compensated by alternating the bond normal and tangential strength concerning PFS constructed by different primary particles under different strain rates. tangential strength according to strain rate causes massive fluctuation in both values. The newly developed solid bond model has not considered the strain rate-dependent compressive strength relationship. Variance in bond normal strength due to strain rate or particle surface properties is compensated by alternating the bond normal and tangential strength concerning PFS constructed by different primary particles under different strain rates.

#### **3. Result and Discussion 3. Result and Discussion**

where is the bond radius, ⃗⃗

#### *3.1. Experimental Result 3.1. Experimental Result*

The overview of Young's modulus and breakage stresses of frozen PFS under different saturation levels are presented in Figure 8. A reduction in the proportion of the bond material in the agglomerates vastly weakens the mechanical structure. This phenomenon can be observed from the value of the average Young's modulus and breakage stress, in which the reduction in saturation level decreases both values, incredibly massive in breakage stress. The reduction ratio in mechanical strength is significantly higher than the reduction ratio in the bond material volume fraction. The overview of Young's modulus and breakage stresses of frozen PFS under different saturation levels are presented in Figure 8. A reduction in the proportion of the bond material in the agglomerates vastly weakens the mechanical structure. This phenomenon can be observed from the value of the average Young's modulus and breakage stress, in which the reduction in saturation level decreases both values, incredibly massive in breakage stress. The reduction ratio in mechanical strength is significantly higher than the reduction ratio in the bond material volume fraction.

**Figure 8.** Average Young's modulus and breakage stress of different PFS: (**a**) Young's modulus; (**b**) Breakage stress. **Figure 8.** Average Young's modulus and breakage stress of different PFS: (**a**) Young's modulus; (**b**) Breakage stress.

For 100% saturation level PFS samples, Young's modulus increased with the increase in strain rate, except for the frozen PFS with glass particles. The strain rate of PFS with glass particles does not significantly influence Young's modulus. One of the possible reasons for that is the different creep behavior of ice in the contact zone between particle and bond. From Table 1, the glass bead has the lowest value in both Ra and Rz, which means the glass bead has the smoothest surface among all the primary particles. The smooth surface affects the contact zone behavior. This phenomenon can also be observed for the breakage stress of glass PFS with a 100% saturation level, where the aggregate reveals the lowest breakage strength under both strain rates. Overall, for all investigated materials, it can be observed that with an increased roughness, the strength increases, and the highest strength reveal aggregates containing non-spherical sand particles.

Young's modulus for sand PFS samples at different strain rates is also a vast difference. The surface properties of sand particles are similar to Polyethene particles. The much higher Young's modulus at a high strain rate is related to the primary particle's mechanical properties but is mainly associated with a lower porosity and, as a result, a more significant number of contacts for each particle. Consequently, a more compact and refined internal bond structure is formed inside such samples, which increases stiffness.

On the other hand, the lower value of Young's modulus at a low strain rate might be related to the pressure melting phenomenon, when the ice melts under pressure and then freezes again due to particle realignment. Due to the massive number of bonds in the sand PFS and the viscoelasticity of ice, this process can escalate, which causes a significant drop in Young's modulus at a low strain rate. However, the current experiment setup cannot prove such an assumption. The strain rate for saturation level 75% only slightly affects Young's modulus. However, for an unambiguous conclusion, it is necessary to conduct more experiments here, especially crucial data regarding the internal structure under pressure.

The average breakage stress of 100% and 75% saturation level PFS provided additional information about the mechanical behavior, presented in Figure 8. At 100% saturation level, an increase in strain rate leads to a higher fracture strength, except for the glass PFS, since the glass particle reveals a much smoother surface than other particles. The smooth surface leads to a weak interface between the bond material and the particle surface. At 75% saturation level, the high deviation within the experiment data does not allow for observing a clear trend except for the sand PFS, where it can be concluded that strain rate causes an increase in breakage stress. Apart from quantitative analysis of mechanical behavior, which is based on breakage stress, strain, and Young's modulus, the overall rheological behavior should be considered. Table 2 gives an insight into agglomerate behavior for different saturation levels, temperatures, strain rates, and particle properties.


**Table 2.** Mechanical behavior overview concerning saturation levels, temperatures, strain rates, and particle properties.

Particle surface properties and strain rate can influence mechanical behavior. Furthermore, the 75% saturation samples yielded more stepwise breakage under loading.

#### *3.2. Simulation Setup*

The agglomerates with 10 mm in diameter and 16 mm in height have been generated according to the section "Bonded-Particle Model Approach" procedure. The diameter of primary particles was tailored to the particles used in the experiment. Additionally, the driving velocity of the upper moving geometry (metal punch) was increased to 200 times (low strain rate: 4 mm/s, high strain rate: 32 mm/s) compared to the experiments to reduce the total computation time. The increase in moving geometry speed can lead to emerging of artificial elastic waves propagating through the material. The restitution coefficient for all particles and particle-wall interactions was reduced to 0.1 to prevent such a phenomenon. The restitution coefficient controls the amount of kinetic energy dissipated during simulation. In addition, the creep parameter has been adjusted accordingly.

For the spherical particles (polyethene, glass, alpha-alumina), spheres were used to represent the particles in the simulation. For non-spherical sand particles, the simulation has also used spheres to reproduce the particles due to contact detection occupying significant calculation power in DEM simulation. Computing the contact for spheres is less demanding, determined by the center distance being more or less than the distance of the combined radius. With non-spherical particles, not only is the distance considered, but also the relative rotation of particles needs to be taken into account. The simulation of non-spherical particles in DEM is tremendously more demanding than spherical particles [56].

The main model parameters and material properties of primary particles are listed in Table 3. The polymer particles, glass beads, and alpha-alumina Young's modulus were determined from self-performed experiments. Here the force-displacement characteristics obtained from uniaxial compression tests were used to adjust Young's modulus by fitting the Hertz model to experimental results [57]. Additionally, due to the non-spherical nature of sand particles, a uniaxial compression test for material modeling parameter calibration is impossible. In contrast, Young's modulus of sand particles was taken from the literature [58]. Both density and Poisson's ratio of different particles were taken from the literature [59–62].


**Table 3.** Main agglomerates properties used for simulation of different types of PFS with 100% saturation level.

The particle-particle, particle-wall sliding, and rolling friction were referenced from different literature, with further fine adjustment per trial-and-error procedure [62–64]. According to Gu et al. [65], the non-spherical PFS particle-particle and particle-wall sliding and rolling friction coefficients have been adjusted. In the DEM simulations, the shape of particles was neglected, and all investigated particles were modeled as spheres. Due to such a simplified representation, the value of rolling friction for non-spherical sand particles was much higher compared to spherical particles.

The material parameters for ice bonds have been adjusted according to experiments with 100% saturation. The simulation results are compared to averaged experiment stressstrain curve.

Young's modulus maintained the same for all different sets of experiments and was estimated at 350 MPa. The Poisson's ratio was taken from the literature as 0.31 [29]. Only the normal strength, tangential strength, and creep parameters *A* and *m* were tuned to individual experimental sets. The adjusted parameters for different sets of experiments are listed in Table 4. Particles investigated in this study have been selected due to their significant mechanical and surface properties differences. The creep parameters are adjusted by a trial-and-error procedure. It is related to two reasons. First, the creep parameter in the works of literature is calibrated concerning pure ice samples. The value cannot be applied to the simulation, as the contact zone influences the creep parameter concerning different primary particles. The second is the increased speed of moving geometry. Since the creep parameter is a time-dependent material parameter, any adjustment toward the simulation scene related to time influences the creep parameter setup.

**Table 4.** Material properties for ice bonds used for modeling different agglomerates at different loading rates.


Similar to the creep parameters, bond normal and tangential strength are tuned by the trial-and-error procedure. Consequently, the contact zone between ice bridges and particles was significantly varied. It is expected that the strength of the contact zone for smooth glass particles was much smaller than the similar strength for rough and highly porous alpha-alumina particles. Therefore, the bonds' normal and tangential strengths were varied to consider that effect. Primary sets of simulations were performed during the trial-and-error material parameter tuning procedure. Young's modulus and Poisson's ratio have kept unchanged throughout the process. Simulation results were transferred into the stress-strain curves, which were compared with the experimental result. The material parameters were then narrowed down to achieve accurate simulation results compared to the experiment result. Creep parameters concerning different primary particles are compared to validate the model capability. As shown in Table 4, only alpha-alumina PFS has a different creep parameter, which is correlated to its aggregated surface topography.

For modeling agglomerates with lower saturation degrees, the same material parameters for solid bonds were taken in Table 4. However, the bond generation parameters were alternated. New diameter and new maximal generation distance between particles *L max gen* were specified.

Compared to the 100% saturation degree, for the bond generation in the case of agglomerates with a low saturation degree of 75%, the maximal distance *L max gen* has been reduced to 0.01 mm. Such a value assumes that an ice bond is only formed when the particles are in contact or adjacent with a minimal distance. As mentioned, parameters constraining the successfulness of liquid bridge formation were related to the geometry, surface topography, separation distance, and size of the liquid bridge. As shown in Table 5, the total number of bonds for 75% saturation degree was almost halved compared to the 100% saturation level.


*Materials* **2022**, *15*, x FOR PEER REVIEW 16 of 22

the 100% saturation level.

**Table 5.** Agglomerates bond generation set up for different types of PFS in 75% saturation level. structure. A quarter facilitates the bond diameter for agglomerates with spherical and non-spherical primary particles.

surface topography, separation distance, and size of the liquid bridge. As shown in Table 5, the total number of bonds for 75% saturation degree was almost halved compared to

The cross-section area was also reduced to mimic the alternation in the internal bond

The cross-section area was also reduced to mimic the alternation in the internal bond structure. A quarter facilitates the bond diameter for agglomerates with spherical and non-spherical primary particles. different from the 75% saturation level. Due to the difference in initial geometry, the deviation of the ice bond after phase transformation was accumulated. The ice bond of the 75% saturation is entirely different from the 100% saturation level. According to such an

geometry of ice bonds before phase transformation at 100% saturation level is entirely

The assumption behind such a setup was the ice bridge formation deviation between particles in 100% and 75% saturation levels. At 100% saturation level, the cavity was occupied by water or particles, and both are incompressible substances. Solid bonds can completely enclose all primary particles inside the agglomerates. However, at a 75% saturation level, due to the volume reduction in bond substance, the individual geometry of the bonds between particles differs from the 100% saturation level agglomerate. The ice bridges are formed by the phase transition of the capillary liquid bridges. The starting geometry of ice bonds before phase transformation at 100% saturation level is entirely different from the 75% saturation level. Due to the difference in initial geometry, the deviation of the ice bond after phase transformation was accumulated. The ice bond of the 75% saturation is entirely different from the 100% saturation level. According to such an assumption, the reduction in bond diameter is applied to the simulation model. assumption, the reduction in bond diameter is applied to the simulation model. **Table 5.** Agglomerates bond generation set up for different types of PFS in 75% saturation level. **Parameter Polyethylene/Glass/Alpha-Alumina** (**Spherical**) **Sand** (**Non-Spherical**) Bond diameter (mm) 0.75 0.22 Maximal bond generation distance (mm) 0.01 0.01 Number of particles ≈230 ≈11,200 Number of bonds ≈550 ≈34,000 In Figure 9, the agglomerates generated for different initial conditions are shown.

In Figure 9, the agglomerates generated for different initial conditions are shown. The top part shows the agglomerates with spherical particles, and the lower part only shows the internal bond structure. The top part shows the agglomerates with spherical particles, and the lower part only shows the internal bond structure.

Frozen PFS with spherical particles at 100% saturation level

Frozen PFS with non-spherical particles at 100% saturation level Frozen PFS with non-spherical particles at 75% saturation level

**Figure 9.** Representative agglomerate with diameter 10 mm, height 16 mm (**upper** part: agglomerates in complete form; **lower** part: agglomerates' internal structure). **Figure 9.** Representative agglomerate with diameter 10 mm, height 16 mm (**upper** part: agglomerates in complete form; **lower** part: agglomerates' internal structure).

### *3.3. Comparison of Simulation and Experimental Results*

The simulation and experimental results for the uniaxial compression test for different agglomerates with varied strain rates are shown in Figure 10. It can be seen that simulation provides acceptable values in breakage stress, strain, and Young's modulus compared to experimental results. The proposed solid bond model can review different mechanical behavior regarding different strain rates.

the shape deviation of simulation and experiment particles has proven feasible.

The simulation and experimental results for the uniaxial compression test for different agglomerates with varied strain rates are shown in Figure 10. It can be seen that simulation provides acceptable values in breakage stress, strain, and Young's modulus compared to experimental results. The proposed solid bond model can review different me-

Without the variation of bonds' Young's modulus and Poisson's ratio, the proposed solid bond model can tackle the strain rate-dependent behavior with compensation of normal and tangential strength toward the particle surface properties. Last but not least, using alternated particle-particle and particle-wall friction coefficients to compensate for

*3.3. Comparison of Simulation and Experimental Results*

chanical behavior regarding different strain rates.

**Figure 10.** Experimental and simulation results for 100% saturation level for high strain (HS) rates (10<sup>−</sup><sup>2</sup> s −1) and low strain (LS) rates (10−3 s −1). **Figure 10.** Experimental and simulation results for 100% saturation level for high strain (HS) rates (10−<sup>2</sup> s −1 ) and low strain (LS) rates (10−<sup>3</sup> s −1 ).

Apart from comparing the stress-strain data, the comparison between the experiment's average Young's modulus and breakage stress and simulation result is presented in Figure 11. All the simulation result lies within or very close to the average experiment value in both Young's modulus and breakage stress. Without the variation of bonds' Young's modulus and Poisson's ratio, the proposed solid bond model can tackle the strain rate-dependent behavior with compensation of normal and tangential strength toward the particle surface properties. Last but not least, using alternated particle-particle and particle-wall friction coefficients to compensate for the shape deviation of simulation and experiment particles has proven feasible.

Apart from comparing the stress-strain data, the comparison between the experiment's average Young's modulus and breakage stress and simulation result is presented in Figure 11. All the simulation result lies within or very close to the average experiment value in both Young's modulus and breakage stress. *Materials* **2022**, *15*, x FOR PEER REVIEW 18 of 22

**Figure 11.** Comparison of experimental and simulation results for 100% saturation at −10 °C: (**a**) Young's modulus; (**b**) Breakage stress. **Figure 11.** Comparison of experimental and simulation results for 100% saturation at −10 ◦C: (**a**) Young's modulus; (**b**) Breakage stress.

The deviation between every experimental data was enormous in lower saturation level PFS. The experimental data and data obtained by simulation were compared by com-

The comparison of breakage stress and Young's modulus of agglomerates with lower

Young's modulus from simulation reveals more significant deviations to experimental data than breakage stress. The main reason is the lack of detailed information on the contact zone between the bond and the particle. Such information is particularly essential for the particle-bond interface creep behavior, thus alternating the elastic behavior

Contrary to the assumptions made in the model, the internal structures of glass bead PFS and alpha-alumina PFS might not be the same. The capillary properties of glass and alpha-alumina are different, leading to the formation of varying bond networks. Furthermore, another critical role may play in the particle size and shape variability for the sand PFS. Thus, using the same settings for the bond generation applied for spherical and nonspherical PFS imposes out-of-range discrepancies in simulation and experimental results

exact value, as the material parameter of particles and bonds for different conditions were not alternated. Only the internal bond structure is adjusted to compensate for the deviation of saturation level. Breakage stress obtained from the simulation yielded acceptable agreement with the experiment, as most of the simulation result lies within or close to the standard deviation range, except for the alpha-alumina PFS. The internal structure of the lower saturation level is generated according to arbitrary values, leading to a big difference compared to the realistic structure. The simulation and experiment comparison shows that the weakening effect of reducing saturation level differs remarkably toward

alpha-alumina particles.

of the agglomerate.

for non-spherical PFS.

paring the values of breakage stress and Young's modulus.

The deviation between every experimental data was enormous in lower saturation level PFS. The experimental data and data obtained by simulation were compared by comparing the values of breakage stress and Young's modulus.

The comparison of breakage stress and Young's modulus of agglomerates with lower saturation degrees is shown in Figure 12. Simulations were not aimed at achieving the exact value, as the material parameter of particles and bonds for different conditions were not alternated. Only the internal bond structure is adjusted to compensate for the deviation of saturation level. Breakage stress obtained from the simulation yielded acceptable agreement with the experiment, as most of the simulation result lies within or close to the standard deviation range, except for the alpha-alumina PFS. The internal structure of the lower saturation level is generated according to arbitrary values, leading to a big difference compared to the realistic structure. The simulation and experiment comparison shows that the weakening effect of reducing saturation level differs remarkably toward alpha-alumina particles. *Materials* **2022**, *15*, x FOR PEER REVIEW 19 of 22

**Figure 12.** Comparison of experimental and simulation results for 75% saturation at −10 °C: (**a**) Young's modulus; (**b**) Breakage stress. **Figure 12.** Comparison of experimental and simulation results for 75% saturation at −10 ◦C: (**a**) Young's modulus; (**b**) Breakage stress.

**4. Conclusions** Both experiment and simulation studies for frozen PFS were performed in this contribution. The experiments presented results that are in suitable agreement with previous literature. Primary particles with different surface and mechanical properties were constructed to validate the influence on the agglomerate mechanical properties and behavior. Young's modulus from simulation reveals more significant deviations to experimental data than breakage stress. The main reason is the lack of detailed information on the contact zone between the bond and the particle. Such information is particularly essential for the particle-bond interface creep behavior, thus alternating the elastic behavior of the agglomerate.

The saturation level was alternated during the experiment phase, which reviews a vast weakening effect on the agglomerate. The bonded-particle model, an extension of DEM, has been used to tackle the problem. A new solid bond model considering creep behavior has been developed and implemented into the MUSEN simulation framework. The developed model can simulate strain rate-dependent material, formulating the different mechanical responses under different strain rates. Nonetheless, particle composite material experiencing creep behavior or dam-Contrary to the assumptions made in the model, the internal structures of glass bead PFS and alpha-alumina PFS might not be the same. The capillary properties of glass and alpha-alumina are different, leading to the formation of varying bond networks. Furthermore, another critical role may play in the particle size and shape variability for the sand PFS. Thus, using the same settings for the bond generation applied for spherical and non-spherical PFS imposes out-of-range discrepancies in simulation and experimental results for non-spherical PFS.

#### age can also be simulated. **4. Conclusions**

of the manuscript.

Throughout all different simulations, the bond Young's modulus and Poisson's ratio have kept the same, with creep parameters, normal and tangential strength tuned according to primary particles and strain rate. The simulation model has proven to be capable of considering the strain rate-dependent behavior of the frozen PFS. In the lower saturation levels of frozen PFS, the bond's material parameters were kept the same, and only the internal structure of the agglomerate in the simulation was changed. That shows a more considerable deviation in Young's modulus but acceptable values in breakage stress. Lack Both experiment and simulation studies for frozen PFS were performed in this contribution. The experiments presented results that are in suitable agreement with previous literature. Primary particles with different surface and mechanical properties were constructed to validate the influence on the agglomerate mechanical properties and behavior. The saturation level was alternated during the experiment phase, which reviews a vast weakening effect on the agglomerate.

of data regarding the internal structure leads to a higher variation of simulation results

**Author Contributions:** Conceptualization, T.T.C. and M.D.; methodology, T.T.C. and M.D.; soft-

tion, T.T.C.; writing—original draft preparation, T.T.C. and M.D.; writing—review and editing, T.T.C., J.G., S.H., and M.D.; visualization, T.T.C.; supervision, S.H. and M.D.; project administration, S.H. and J.G.; funding acquisition, J.G. All authors have read and agreed to the published version

**Funding:** The authors acknowledge the funding of this research by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in the framework of Research Training Group GRK2462: Processes in natural and technical Particle Fluid Systems (PintPFS) [DFG PintPFS, 2019] at Hamburg University of Technology (TUHH). Publishing fees funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 491268466 and the Hamburg University of Technology (TUHH) in the funding programme \*Open Access Publishing\*.

under a lower saturation level in PFS experiments.

**Institutional Review Board Statement:** Not applicable

The bonded-particle model, an extension of DEM, has been used to tackle the problem. A new solid bond model considering creep behavior has been developed and implemented into the MUSEN simulation framework. The developed model can simulate strain ratedependent material, formulating the different mechanical responses under different strain rates. Nonetheless, particle composite material experiencing creep behavior or damage can also be simulated.

Throughout all different simulations, the bond Young's modulus and Poisson's ratio have kept the same, with creep parameters, normal and tangential strength tuned according to primary particles and strain rate. The simulation model has proven to be capable of considering the strain rate-dependent behavior of the frozen PFS. In the lower saturation levels of frozen PFS, the bond's material parameters were kept the same, and only the internal structure of the agglomerate in the simulation was changed. That shows a more considerable deviation in Young's modulus but acceptable values in breakage stress. Lack of data regarding the internal structure leads to a higher variation of simulation results under a lower saturation level in PFS experiments.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/ma15238505/s1, Video S1: 100% saturation, alpha-alumina PFS; Video S2: 75% saturation, glass PFS.

**Author Contributions:** Conceptualization, T.T.C. and M.D.; methodology, T.T.C. and M.D.; software, T.T.C. and M.D.; validation, T.T.C., S.H. and M.D.; investigation, T.T.C. and M.D.; data curation, T.T.C.; writing—original draft preparation, T.T.C. and M.D.; writing—review and editing, T.T.C., J.G., S.H. and M.D.; visualization, T.T.C.; supervision, S.H. and M.D.; project administration, S.H. and J.G.; funding acquisition, J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge the funding of this research by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in the framework of Research Training Group GRK2462: Processes in natural and technical Particle Fluid Systems (PintPFS) [DFG PintPFS, 2019] at Hamburg University of Technology (TUHH). Publishing fees funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 491268466 and the Hamburg University of Technology (TUHH) in the funding programme \*Open Access Publishing\*.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request.

**Conflicts of Interest:** Maksym Dosta has contributed to this work during his employment at TUHH. Other authors declare no conflict of interest.

#### **References**


## *Article* **Modeling Cyclic Crack Propagation in Concrete Using the Scaled Boundary Finite Element Method Coupled with the Cumulative Damage-Plasticity Constitutive Law**

**Omar Alrayes <sup>1</sup> , Carsten Könke <sup>1</sup> , Ean Tat Ooi <sup>2</sup> and Khader M. Hamdia 3,\***

	- Ballarat, VIC 3350, Australia

**Abstract:** Many concrete structures, such as bridges and wind turbine towers, fail mostly due to the fatigue rapture and bending, where the cracks are initiated and propagate under cyclic loading. Modeling the fracture process zone (FPZ) is essential to understanding the cracking behavior of heterogeneous, quasi-brittle materials such as concrete under monotonic and cyclic actions. The paper aims to present a numerical modeling approach for simulating crack growth using a scaled boundary finite element model (SBFEM). The cohesive traction law is explored to model the stress field under monotonic and cyclic loading conditions. In doing so, a new constitutive law is applied within the cohesive response. The cyclic damage accumulation during loading and unloading is formulated within the thermodynamic framework of the constitutive concrete model. We consider two common problems of three-point bending of a single-edge-notched concrete beam subjected to different loading conditions to validate the developed method. The simulation results show good agreement with experimental test measurements from the literature. The presented analysis can provide a further understanding of crack growth and damage accumulation within the cohesive response, and the SBFEM makes it possible to identify the fracture behavior of cyclic crack propagation in concrete members.

**Keywords:** crack propagation; cohesive zone method; constitutive modelling; cyclic loading; scaled boundary finite element

## **1. Introduction**

Concrete structural elements very often fail due to fatigue fractures, in which repeated loading can lead to the growth of existing cracks [1–4]. To better understand the fatigue fracturing under cyclic loading, a detailed analysis of the fatigue behavior and the associated crack propagation is required for economical and reliable design of concrete structures.

The advanced studies on cyclic crack propagation are mostly empirical, wherein large number of data samples from experiments are used for fitting the relationship. The most commonly used approach to predict fatigue life and crack growth rate is the well-known Paris law [5,6]. This phenomenological law relates the amplitude of the stress state (defined by stress intensity factor K) and the crack growth rate *da*/*dN*, which can be considered a valuable tool for engineering fatigue analysis. However, it has been shown that Paris law loses much of its prediction ability when conditions are not ideal, such as with non-constant amplitude loading and short cracks [7,8]. Nevertheless, advanced numerical models have been developed widely to capture the phenomena behind the cyclic crack propagation under subcritical loading levels. Numerical simulations are more flexible in the sense

**Citation:** Alrayes, O.; Könke, C.; Ooi, E.T.; Hamdia, K.M. Modeling Cyclic Crack Propagation in Concrete Using the Scaled Boundary Finite Element Method Coupled with the Cumulative Damage-Plasticity Constitutive Law. *Materials* **2023**, *16*, 863. https://doi.org/10.3390/ ma16020863

Academic Editors: Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 22 November 2022 Revised: 29 December 2022 Accepted: 4 January 2023 Published: 16 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

that they can predict fatigue life and crack growth under general loading conditions and geometries. They can be applied to study design variations in early design stages.

Several modeling approaches for crack propagation under cyclic and fatigue loading are well documented in the literature [9,10]. The cohesive zone model (CZM) has been implemented in classical fracture mechanics by [11,12] to reduce the mesh quality required for crack simulation. The CZM is based on elastic damage material for both monotonic and fatigue crack growth [13,14]. For concrete material, the softening damage, whose localization is governed numerically by finite element simulation, is aimed at simulating the propagation of the fatigue fracture in the cohesive process zone [15,16]. However, these types of models are used to accumulate damage only along the damaged locations of the loading/unloading paths.

The second type of crack simulation model is the phase field model (PFM). The concept of the PFM approach is to regularize free energy of degradation, which effectively reduces material fracture resistance under fatigue loading [17]. It was developed to predict quasistatic and dynamic fracturing in brittle and ductile regimes considering isotropic and anisotropic toughness [18]. This method introduces the degradation of the fracture energy as a function of a local energy-accumulation variable. As a result of repeated loading, the structural loading history is taken into consideration [19]. Similar approaches have been published recently in [20], which simulated fatigue crack growth. A nonlinear kinematic and isotropic hardening were considered. Differently, simulations of molecular dynamics can be used to evaluate the interfacial strength [21].

Additionally, discrete lattice models have many features of the discrete element method (DEM) to simulate the heterogeneous microstructure and crack propagation [22]. The formulation combines the damage mechanics and plasticity theory with a cyclic damage evolution law. The model characterizes the critical response of concrete material undergoing cyclic loading. The behavior obtained by the DEM simulations is a collective response constituted from all contacts and particles in the domain.

Many models in the literature [23–31] are dedicated to simulating the quasi-brittle behavior, including a set of constitutive equations for the monotonic, fatigue, and hysterical material responses. Furthermore, several calculation schemes also exist to predict tensile, flexural monotonic, and fatigue behavior [32,33]. The established damage law allows a damage accumulation process for random cycles. The damage model concludes the primary dissipative phenomenon, which is activated during unloading and reloading.

The scaled boundary finite element method (SBFEM) is a very attractive approach to modeling crack nucleation and propagation under general loading conditions [34–37]. The cohesive fracture and stress field can be determined using interface elements with zero thickness, which are inserted directly into the SBFEM [38–40]. The cohesive traction forces and the stress field close to the crack tip are accurately computed as they are defined analytically. This enables the onset of crack propagation to obtain the correct load-deflection response. Yang [41] developed the SBFEM to solve linear crack propagation in brittle materials under monotonic loading. He benefited from the salient feature of the high accuracy of the stress intensity factor (SIF) in SBFEM computed directly from singular stress solutions and flexible substructuring of each domain. The crack simulation of concrete slabs based on a cohesive zone model in an explicit SBFEM-FEM frame for seismic cyclic loading was reported in [42] to facilitate dynamic analysis. However, the calculation of coupled SBFEM-FEM analysis can be very computationally intensive. For cyclic loading, the crack evolution can also be simulated using quasi-static analysis. The accuracy of the method was validated by a cyclic damage test with a concrete beam. A fully automatic modeling methodology characterized by a simple remeshing algorithm was developed, and the mixed-mode crack propagation problem was efficiently solved. Yang and Deeks [43] further coupled the procedure of SBFEM with the FEM for quasi-brittle materials. An extended polygon scaled boundary finite element method [44] was developed to simulate nonlinear dynamic analysis. A direct remeshing algorithm for crack propagation was obtained for quasi-brittle materials. The study of dynamic fracture modeling by SBFEM

was developed in [45] to model the crack propagation of impact-test specimens. The stress intensity factor, displacement, and stresses were extracted from the dynamic solution.

In the present paper, we further extend the SBFEM for modeling cyclic-damageinduced cracks' behavior within the SBFEM framework. The model considers the cumulative crack opening/sliding measure to dominate the damage mechanism at the subcritical loading levels. Similar approaches have been proposed in [40] for the numerical simulation of concrete under monotonic loading. The novelty of our approach is to establish a link between the cyclic damage rate and the efficiency of the SBFEM in modeling crack propagation. By comparing the thermodynamic softening law of the constitutive model for fracture, several aspects have been provided, which include the loading–unloading path, the damage evolution during the load cycle, and the crack-opening traction behavior.

The paper is organized as follows. The theoretical formulation of the cohesive crack model inside SBFEM is represented in Section 2. The behavior of the constitutive material model is studied at the level of material point (Gauss point) in Section 3. The performance of the cohesive cyclic crack model within the thermodynamic framework is then reported, which was applied in [46]. In Section 4, we present the calibration and validation of the model based on the results of the cyclic flexural bending test of plain concrete published in the literature. We present numerical investigations focused on the effect of the loading sequence on the material behavior.

#### **2. Scaled Boundary Finite Element (SBFEM)**

#### *2.1. Fundamentals*

Figure 1 shows the basic concept of the cohesive crack model in the scaled boundary method for a typical bounded domain. The mesh is represented by a discretized collection of arbitrary-sided polygons, or (as in Figure 1a) quadtrees elements. Each element is maintained by a curve relative to a scaling center (*x*0,*y*0). This condition is satisfied by dividing the domain into many sub-domains, which can be made visible for each boundary. The boundary is discretized by one-dimensional finite elements with a local coordinate *η* in an interval of −1 ≤ *η* ≤ 1; see Figure 1b. Let (*x*0,*y*0) be the scaling center, and *ξ* is the radial coordinate with *ξ* = 0 at the center and *ξ* = 1 at the boundary. The coordinates on the boundary are interpolated by *x<sup>b</sup>* = [*N*(*η*)]{*xbn*}, and *y<sup>b</sup>* = [*N*(*η*)]{*ybn*}, where [*N*(*η*)] is the vector of nodal shape functions, and {*xbn*}, {*ybn*} are the nodal coordinates. The displacement field, *u*(*ξ*, *η*), can be defined semi-analytically as

$$\{\mu(\xi,\eta)\} = \left[N\_{\mu}(\eta)\right] \{\mu(\xi)\} \tag{1}$$

We calculate the nodal displacement functions *u*(*ξ*) at the radial lines, *ξ*. Meanwhile, they are interpolated by the linear shape functions [*Nu*(*η*)] in the direction of *η*, which are obtained by multiplying a suitable identity matrix with each element in [*N*]. Thus, the strain and the stress fields are formulated as:

$$\{\varepsilon(\xi,\eta)\} = [B\_1(\eta)]\{\mu(\xi)\}\_{\xi} + (1/\xi)[B\_2(\eta)]\{\mu(\xi)\} \tag{2}$$

$$\{\sigma(\xi,\eta)\} = [D]\{\varepsilon(\xi,\eta)\} = [D][B\_1(\eta)]\{u(\xi)\}\_{\xi} + (1/\xi)[B\_2(\eta)]\{u(\xi)\} \tag{3}$$

where *B*1(*η*) and *B*2(*η*) are the strain matrices, and *D* is the constitutive matrix [39]. The weak form of the elastic equilibrium of forces is obtained according to the principle of virtual work [47], or from the weighted residual technique; see ref. [34]. The governing equations can be written as follows:

$$\{E\_0\} \xi^2 \{\mu(\xi)\}\_{,\xi\overline{\xi}} + ([E\_0] + [E\_1] + [E\_1^T]) \xi \{\mu(\xi)\}\_{,\xi} - [E\_2] \{\mu(\xi)\} = 0 \tag{4}$$

$$\{P(\mathfrak{f})\} = [E\_0] \mathfrak{f} \{\mathfrak{u}(\mathfrak{f})\}\_{\mathfrak{f}} + [E\_1]^T \{\mathfrak{u}(\mathfrak{f})\} \tag{5}$$

with {*P*} being the load vector. Equation (4) includes second-order Cauchy–Euler equations, called the scaled boundary finite element equation in the displacement with the coefficient matrices [*E*0], [*E*1], [*E*2]. Furthermore, Equation (4) is a homogeneous secondorder differential Equation (in case there is no side face or body loads) with *n* unknowns. By introducing a new variable [*χ*(*ξ*)] with Hamiltonian matrix *Z*, the system becomes a first-order ordinary differential equation [48] as

$$
\langle \mathfrak{f} | \chi(\xi) | \_{,\xi} = -|Z| | \chi(\xi) | \tag{6}
$$

and

$$\left[\chi(\xi)\right] = \left[\{\mu(\xi)\}\{q(\xi)\}\right]^T \tag{7}$$

where *q*(*ξ*) are analytical functions that represent the internal nodal forces vector:

$$\{\boldsymbol{q}(\boldsymbol{\xi})\} = [E\_0] \mathfrak{F} \{\boldsymbol{\mu}(\boldsymbol{\xi})\}\_{\boldsymbol{\xi}} + [E\_1]^T \{\boldsymbol{\mu}(\boldsymbol{\xi})\} \tag{8}$$

and the Hamitonian matrix is calculated as a function of [*E*0], [*E*1], [*E*2]:

$$[Z] = \begin{bmatrix} [E\_0]^{-1}[E\_0]^T & -[E\_0]^{-1} \\ -[E\_2] + [E\_1][E\_0]^{-1}[E\_1]^T & -[E\_1][E\_0]^{-1} \end{bmatrix} \tag{9}$$

An eigenvalue decomposition of [Z] follows [49]:

$$
\begin{bmatrix} [Z] \begin{bmatrix} [\phi\_{\boldsymbol{u}}^{(n)}] & [\phi\_{\boldsymbol{u}}^{(p)}] \\ [\phi\_{\boldsymbol{q}}^{(n)}] & [\phi\_{\boldsymbol{q}}^{(p)}] \end{bmatrix} = \begin{bmatrix} [\phi\_{\boldsymbol{u}}^{(n)}] & [\phi\_{\boldsymbol{u}}^{(p)}] \\ [\phi\_{\boldsymbol{q}}^{(n)}] & [\phi\_{\boldsymbol{q}}^{(p)}] \end{bmatrix} \times \begin{bmatrix} [\lambda^{(n)}] & 0 \\ 0 & [\lambda^{(p)}] \end{bmatrix} \tag{10}
$$

where [*λ*] is the diagonal matrix of *λ* (*p*) and *λ* (*n*) . The superscripts p and n refer to positive and negative. [*φ* (*p*) *q* ], [*φ* (*p*) *<sup>u</sup>* ], and [*φ* (*n*) *<sup>u</sup>* ] are the eigenvectors corresponding to *λ* (*p*) , [*φ* (*n*) *q* ], and [*λ* (*n*) ], respectively. The solution of Equation (6) yields:

$$\{\boldsymbol{q}(\xi)\} = [\phi\_q^{(n)}] \mathfrak{F}^{-[\boldsymbol{\lambda}^{(n)}]} \{\boldsymbol{c}^{(n)}\} + [\phi\_q^{(p)}] \mathfrak{F}^{-[\boldsymbol{\lambda}^{(p)}]} \{\boldsymbol{c}^{(p)}\} \tag{11}$$

$$\{\mu(\xi)\} = [\phi\_{\mu}^{(n)}]\mathfrak{f}^{-[\lambda^{(n)}]}\{\mathfrak{c}^{(n)}\} + [\phi\_{\mu}^{(p)}]\mathfrak{f}^{-[\lambda^{(p)}]}\{\mathfrak{c}^{(p)}\} \tag{12}$$

{*c* (*p*)} and {*<sup>c</sup>* (*n*)} are the integration constants. For a bounded domain, the boundary condition at {*ξ* = 0} produces {*c* (*p*)} <sup>=</sup> 0. In this case, the modes of non-positive real components of eigenvalue [*λ*] contribute to the solution of finite displacement at the scaling center.

The equivalent nodal forces on the boundary and the stiffness matrix of the domain are formulated, respectively, as

$$\{P\} = [\phi\_q^{(n)}]\{\mathcal{c}^{(n)}\} = [\phi\_q^{(n)}][\phi\_u^{(n)}]^{-1}\{\mu\_b\} \tag{13}$$

$$[\mathbb{K}] = [\phi\_q^{(n)}][\phi\_u^{(n)}]^{-1} \tag{14}$$

At the boundaries, the nodal displacements {*ub*} can be calculated from the global stiffness matrix *K* and load vector *P*.

Meanwhile, substituting Equation (11) into Equation (1) yields the displacement field in the bulk domain as

$$\{\boldsymbol{\mu}(\mathfrak{f},\boldsymbol{\eta})\} = \left[\mathcal{N}\_{\boldsymbol{\mu}}(\boldsymbol{\eta})\right] \sum\_{i=1}^{n} \mathfrak{f}^{-\boldsymbol{\lambda}^{(n)}i} \boldsymbol{c}\_{i} \{\boldsymbol{\phi}\_{i}\} \tag{15}$$

Hence, the stress field is formulated by

$$\{\sigma(\mathfrak{F},\eta)\}\rangle = [D]\sum\_{i=1}^{n} \mathfrak{F}^{-\lambda^{(n)\_{i}}} ([-\lambda^{(n)}][B\_{1}(\eta)] + [B\_{2}(\eta)]) \{\phi\_{i}\} \tag{16}$$

**Figure 1.** The concept of a cohesive crack model using the scaled boundary finite element method.

#### *2.2. Stress Field at Crack Tip with Cohesive Tractions*

The fracture process zone in a quasi-brittle material can transfer the cohesive forces between the crack faces. This is attributed to the interlocking of the aggregate, in addition to the surface friction. The cohesive traction representing the crack faces is applied as side-face forces. The equilibrium condition (Equation (4)) in a polygon containing a crack tip is augmented to include the load vector containing the side-face tractions, as in [43].

$$\mathbb{E}\left[E\_0\right]\xi^2\{\boldsymbol{\mu}(\boldsymbol{\xi})\}\_{\boldsymbol{\xi}\boldsymbol{\tilde{\xi}}}+(\left[E\_0\right]+\left[E\_1\right]+\left[E\_1^T\right])\xi\{\boldsymbol{\mu}(\boldsymbol{\xi})\}\_{\boldsymbol{\tilde{\xi}}}-\left[E\_2\right]\{\boldsymbol{\mu}(\boldsymbol{\xi})\}-\left\{F\_l(\boldsymbol{\xi})\right\}=0\tag{17}$$

In this work, the cohesive force on the crack faces {*Ft*(*ξ*)} will be computed based on the shadow domain procedure, which has been introduced by [40].

The concept of the cohesive cyclic crack model, as depicted in Figure 1, is shown in the following steps:


three corresponding edges (two edges, *L*<sup>1</sup> and *L*2, for each crack face, and one edge, *L*3, to split the crack cell into two). Knowing the crack angle, *θ*, the orientation of *L*<sup>3</sup> is projected in a way that a straight line is extended from the crack tip with an angle *θ*. Then, the node closest to the intersection point at edge of the cracked cell is employed to split the polygon.


In the fracture process zone, cohesive tractions *tn*, *t<sup>s</sup>* are expressed as a function of relative opening and sliding displacements *d*. In the local coordinate system, the stiffness matrix reads:

$$[k\_{int}] = \frac{A}{2} \sum\_{i=1}^{n\_{\mathcal{S}}} w\_i \mathcal{M}\_i^T [k] \mathcal{M}\_i \tag{18}$$

where *k* is the stiffness of the softening laws, *A* is the crack surface area, *w<sup>i</sup>* is the onedimensional Gaussian weight, *n<sup>g</sup>* is the number of integration points, and *M<sup>i</sup>* is the linear shape function matrix [40]. The stiffness matrices of the interface element *kint* can be assembled attractively. In this case, the local coordinates (*ξ <sup>p</sup>*, *ηp*) in the shadow domain are defined first to obtain the coordinates (*x*, *y*) for a new node in the new crack cell. For this purpose, we use a search algorithm to determine the element in the shadow domain that includes the point (*x*, *y*). In doing so, the nodal displacements and the cohesive tractions are calculated along the crack. These are then mapped back to the crack cell to calculate the stress intensity factors required to determine if the crack propagates. The SIF considering the cohesive forces on the crack face is calculated by representing the cohesive forces as a power function in *ξ* following from the form of the side face traction vector *Ft*(*ξ*), as in [43].

Linearly varying or constant distributed loads are approaches to representing a force over a particular distance. According to [47], when the side-face loads are distributed by a power function, then the modal displacement loads are

$$\{\mathfrak{u}\_t(\mathfrak{f})\} = \mathfrak{f}^{t+1}\{\phi\_l\} \tag{19}$$

Substitution of Equation (19) into Equation (17) yields

$$\left[ (t+1)^2 [E\_0] + (t+1)([E\_1^T] - [E\_1]) - [E\_2] \right]^{-1} \{ \phi\_l \} + \{ F\_l \} = \{ 0 \} \tag{20}$$

Rearranging will give the nodal displacements for the side-face load mode {*φt*} as

$$\{\phi\_l\} = -\left[ (t+1)^2 [E\_0] + (t+1)([E\_1^\top] - [E\_1]) - [E\_2] \right]^{-1} \{F\_l\} = [B\_1(t)] \{F\_l\} \tag{21}$$

In order to express the cohesive tractions as a power of function, the normal traction distribution *σ*(*ξ*) is assumed to be the summation of *M* raised to the power of function *ξ*:

$$\sigma(\mathfrak{f}) = f\_l \sum\_{i=1}^{M} e\_i \mathfrak{f}^{ti} \tag{22}$$

where *e<sup>i</sup>* is coefficient to be calculated. Considering a parameter *µ*, the exponents *t<sup>i</sup>* are determined as *t<sup>i</sup>* = (*i* − 1 + *µ*) .

The tractions at the crack tip, the Gaussian points, and the crack mouth *σ<sup>j</sup>* (*j* = 1, *M*) are used to generate *M* number of equations as

$$\sigma\_{\dot{j}} = \sigma(\mathfrak{f}\_{\dot{j}}) = f\_l \sum\_{i=1}^{M} e\_i \mathfrak{f}\_{\dot{j}}^{ti} \tag{23}$$

where *ξ<sup>j</sup>* = *lj*/*L* is the distance from the jth point on the crack to the crack tip *l<sup>j</sup>* and the length of the crack *L*. The coefficients {*e*} = {*e*<sup>1</sup> *e*<sup>2</sup> ... *eM*} *<sup>T</sup>* are then calculated as

$$\{e\} = [\mathbb{S}]^T f\_t^{-1} \{\sigma\} \tag{24}$$

where {*σ*} = {*σ*<sup>1</sup> *σ*<sup>2</sup> ... *σM*} *T* , and the matrix [*S*] is

$$[S] = \begin{bmatrix} \mathfrak{z}\_1^{t\_1} & \mathfrak{z}\_1^{t\_2} & \cdots & \mathfrak{z}\_1^{t\_M} \\ \mathfrak{z}\_2^{t\_1} & \mathfrak{z}\_2^{t\_2} & \cdots & \mathfrak{z}\_2^{t\_M} \\ \vdots & \vdots & \ddots & \vdots \\ \mathfrak{z}\_M^{t\_1} & \mathfrak{z}\_M^{t\_2} & \cdots & \mathfrak{z}\_M^{t\_M} \end{bmatrix} \tag{25}$$

The nodal side-face load vector becomes

$$\{F\_l(\xi)\} = \sum\_{i=1}^{M} \xi^{t\_i} \{F\_{t\_i}\} \tag{26}$$

with

$$\{F\_l(\xi)\} = Af\_l e\_l \{R\_1\} \tag{27}$$

and

$$\{R\_1\} = \{-sin\delta\ \cos\delta\ 0\ \cdots \ 0\ \sin\delta\ \ -cos\delta\}^T\tag{28}$$

where *A* = is the area of crack surface.

The displacement solution is thus expressed by two components: the modes of normal displacement due to external loading and the modes of side-face displacement due to cohesive tractions as

$$\{\mu(\mathfrak{f},\eta)\} = [N(\eta)][\sum\_{i=1}^{N} c\_i \mathfrak{f}^{\lambda\_i} \{\phi\_i\} + \sum\_{i=1}^{M} e\_i \mathfrak{f}^{t\_i+1} A f\_t[\mathcal{B}\_1(t\_i)] \{\mathcal{R}\_1\}] \tag{29}$$

On the subdomain boundary, the nodal displacement *ubs* is calculated as

$$\{\mu\_{\rm bs}\} = [\phi]\{\mathcal{c}\} + [\phi\_l]\{\mathcal{e}\} \tag{30}$$

where [*φ*] and [*φ<sup>i</sup>* ] are given in Equation (10), and the matrix [*φ<sup>t</sup>* ] is transformed as:

$$\begin{Bmatrix} \{\phi\_l\} = Af\_l[B\_1(t\_1)\ \ B\_1(t\_2)\ \cdots \ \ B\_1(t\_M)]\{\mathcal{R}\_1\} \end{Bmatrix} \tag{31}$$

The nodal displacements *ubs* in Figure 2 are gained by mapping the mesh from the shadow domain, as shown in Figure 2b. The constants {*c*} are given by

$$\{\circledast\} = [\phi]^{-1}(\{u\_{\rm bs}\} - [\phi\_l]\{e\}) \tag{32}$$

Subsequently, Equation (29) is read as:

$$\{\boldsymbol{u}(\xi,\eta)\} = [N(\eta)] \sum\_{i=1}^{N+M} c\_i \xi^{(\overline{\lambda\_i}-1)} \{\overline{\phi\_i}\} \tag{33}$$

where

$$\begin{cases} \overline{\phi\_i} = \phi\_{i\prime} \ \overline{\lambda\_i} = \lambda\_i \qquad \text{for } i = 1, \cdots, N\\ \overline{\phi\_i} = \phi\_{t\prime} \ \overline{\lambda\_i} = t\_i + 1 \quad \text{for } i = N + 1 \cdots \prime, N + M \end{cases}$$

The stress field can be calculated similarly to Equation (16) as

$$\{\sigma(\mathfrak{f},\eta)\} = \sum\_{i=1}^{N+M} c\_i \mathfrak{f}^{(\overline{\lambda\_i}-1)} \{\psi\_i(\eta)\} \tag{34}$$

where each term in Equation (34) can be interpreted as a stress mode and

$$\{\psi\_i(\eta)\} = [D](\overline{\lambda\_i}[B\_1(\eta)] + [B\_2(\eta)])\{\overline{\phi\_i}\} \tag{35}$$

Comparing Equation (15) and Equation (33), and Equation (16) and Equation (34) shows that when the cohesive traction is evaluated, an extra number (*M*) of displacement nodes and the same of stress modes are added to the displacement field and stress field, respectively.

The direction of crack propagation is then determined based on [43]. In order to consider a perfect crack path prediction, the SIFs of the semi-analytical SBFEM stress solutions are calculated.

#### *2.3. Stress Intensity Factor (SIF) for Scaling Center at Crack Tip*

The SBFEM has the advantage of accurately representing the crack zone's stress field without needing a more discretized mesh [38,50]. This tool enables the SIFs to be directly calculated from the semi-analytical solutions of the stresses. In this work, two SIFs are determined. The first is obtained from the linear elastic fracture mechanics solution at a generic load step and is used to determine the crack propagation direction. The side-face tractions are not considered in this case. The second concerns the crack cell considering the effect of the cohesive tractions obtained from the shadow domain. In both cases, the procedure to calculate the SIFs is the same. The only difference is the equation used to represent the stress field, i.e., Equation (16) in case 1 and Equation (34) in case 2. The procedure is outlined as follows:

Figure 1c shows the cracked domain modeled by the SBFEM. The location of the scaling center should be at the crack tip. There is no need to discretize the side faces connected to the scaling center. The SIF could be accurately computed from the semi-analytical solutions of the stresses [51]. The stress intensity factors solutions can be extracted from their definitions as follows.

$$\begin{Bmatrix} \begin{Bmatrix} K\_I \\ K\_{II} \end{Bmatrix} = \lim\_{r \to 0} \begin{Bmatrix} \sqrt{2\pi r} \sigma\_{yy}|\_{\theta=0} \\ \sqrt{2\pi r} \sigma\_{xy}|\_{\theta=0} \end{Bmatrix} \tag{36}$$

where *r* and *θ* represent the polar coordinates. As illustrated in Figure 1, *r* and *θ* originate at the crack tip and are correlated by

$$
\sigma = \mathfrak{J}L(\theta) \tag{37}
$$

where *L*(*θ*) is the distance between any point A at the cracked domain and the crack tip (*L*(*θ*) = *L*<sup>3</sup> in Figure 1c). Substituting Equation (37) in Equation (36) leads to

$$\begin{Bmatrix} \begin{Bmatrix} K\_I \\ K\_{II} \end{Bmatrix} = \lim\_{r \to 0} \begin{Bmatrix} \sqrt{2\pi L(\theta)} \sum\_{i=0}^n c\_i \mathfrak{z}^{-\lambda i - 1} \sigma\_{yy}|\_{\theta = 0} \\ \sqrt{2\pi L(\theta)} \sum\_{i=0}^n c\_i \mathfrak{z}^{-\lambda i - 1} \sigma\_{xy}|\_{\theta = 0} \end{Bmatrix} \tag{38}$$

From Equation (38), when *ξ* → 0, all the corresponding stress modes that have *λ<sup>i</sup>* ≥ 1 will disappear. When *λ<sup>i</sup>* = 0.5, singular stresses are obtained in mode I and mode II. An analytical solution of the limits in Equation (38) yields

$$\begin{Bmatrix} \begin{Bmatrix} K\_I \\ K\_{II} \end{Bmatrix} \end{Bmatrix} = \sqrt{2\pi L\_0} \sum\_{i=I,II} \left( c\_i \begin{Bmatrix} \mathfrak{F}^{-\lambda i - 1} \sigma\_{yy} \vert\_{\theta=0} \\ \mathfrak{F}^{-\lambda i - 1} \sigma\_{xy} \vert\_{\theta=0} \end{Bmatrix} \right)\_i \tag{39}$$

#### *2.4. Crack Growth Criterion*

The zero-K condition based on [52] is used to determine crack propagation in the crack domain. Therefore, when the stress at the crack tip is finite, a cohesive crack propagates, and accordingly, no stress singularity exists. The crack will propagate in the condition

**Figure 2.** Calculation of *k<sup>I</sup>* using shadow domain method: (**a**) cohesive crack model (CCM) in SBFEM; (**b**) subdomain discretization.

The crack length ∆*a* and its angle *θ* are used to define the new location of the crack tip. Figure 2a displays the discretised SBFEM polygon and cracked subdomains of the cohesive crack model (CCM) after the first round of growth. In this shadow domain concept, the crack surfaces is discretized first, and then crack cell elements (CIEs) are inserted into the mesh. This will partition the crack subdomain S1 into two (S1 and S2 in Figure 2b). The CIEs are then used to calculate side-face traction along the crack, upon which the SIFs *KI*(*θ*) can be defined to calculate the crack growth criterion. We apply the mesh mapping technique to calculate the nodal displacements of the cracked subdomain S1. The remeshing procedure during crack propagation is performed based on [40].

#### **3. Cumulative Damage-Plasticity Based Constitutive Law**

The constitutive behavior describing cyclic damage in the process zone is embedded in the definition of the interface elements. It has been defined using the thermodynamicsbased uniaxial interface model proposed in [46,53,54]. The model assumes that the development of cyclic load is dominated by a cumulative level of the inelastic relative displacement within the interface. The uniaxial model can be applied for the normal behavior (*σ*<sup>N</sup> − *w*) and for the shear behavior (*τ* − *s*) of the interface as a unified constitutive model.

#### *3.1. Brief Summary of the Model's Formulation*

The regular formulation of the thermodynamically interface model is described briefly in this section. The Helmholtz free energy is defined as

$$
\rho \psi(u, u^P, \omega, u, z) = \frac{1}{2} (1 - \omega) E (u - u^P)^2 + \frac{1}{2} \gamma u^2 + \frac{1}{2} Kz^2 \tag{41}
$$

where *ρ* is the density; *E* is the elastic stiffness; *u* represents the relative displacement at the interface (i.e., opening displacement *u* = *w* in the normal direction and slip *u* = *s* in the tangential/shear direction); *K* and *γ* represent the isotropic and kinematic hardening moduli, respectively. The state variables of the interface model are the inelastic displacement *u P* , the damage variable *ω*, and the hardening variables *z*, *α*.

The thermodynamic forces, *X* and *Z*, and the related energy release rate, *Y*, can be calculated by differentiating Equation (41) with respect to each state variable as follows.

$$
\sigma^P = \sigma = -\frac{\partial \rho \psi}{\partial u^P} = (1 - \omega)E(u - u^P) \tag{42}
$$

$$X = \frac{\partial \rho \psi}{\partial a} = \gamma a, \qquad Z = \frac{\partial \rho \psi}{\partial z} = Kz \tag{43}$$

$$Y = -\frac{\partial \rho \psi}{\partial \omega} = \frac{1}{2}E(\mu - \mu^P)^2\tag{44}$$

where *σ* represents the stress components (i.e., normal stress *σ*<sup>N</sup> in the case of opening displacement *w*, and shear stress *τ* in the case of slip displacement *s*). A yield function similar to plasticity theory, which defines the boundary between elastic and inelastic domains, is introduced into the effective stress space as follows.

$$f(\mathfrak{v}, \mathbf{X}, \mathbf{Z}) = |\mathfrak{v} - \mathbf{X}| - \mathbf{Z} - \sigma\_0 \tag{45}$$

with *σ*˜ being the effective stress given as *σ*˜ = *σ*/(1 − *ω*) and *σ*<sup>0</sup> being the elastic stress limit. The flow potential determining the damage evolution augments the threshold function (Equation (45)) with an extra term as

$$\phi = f(\vartheta, X, Z) + \frac{S(1 - \omega)^c}{(r + 1)} \left(\frac{Y}{S}\right)^{r + 1} \tag{46}$$

where *S* is the damage strength parameter, and *c* and *r* are exponential rate parameters. The evolution equations can be obtained by differentiating (Equation (46))

$$
\dot{\mu}^P = \dot{\lambda} \frac{\partial \phi}{\partial \sigma^P} = \frac{\dot{\lambda}}{1 - \omega} \text{sign}(\mathfrak{v} - X) \tag{47}
$$

$$\dot{\mathfrak{A}} = -\dot{\lambda} \frac{\partial \phi}{\partial X} = \dot{\lambda} \text{sign}(\mathfrak{v} - X), \qquad \dot{\mathfrak{z}} = -\dot{\lambda} \frac{\partial \phi}{\partial \overline{Z}} = \dot{\lambda} \tag{48}$$

$$
\dot{\omega} = \dot{\lambda} \frac{\partial \phi}{\partial Y} = \dot{\lambda} \left(1 - \omega\right)^{c} \left(\frac{Y}{S}\right)^{r} \tag{49}
$$

This model can be implemented as a time-stepping algorithm, as described in [46]. The damage accumulation under both monotonic and cyclic loading is described through the modified flow potential by [46,54].

#### *3.2. Elementary Studies of the Cohesive Model*

To illustrate the phenomenological behavior of the used constitutive model and its applicability for modeling cyclic and fatigue behavior, a material model of crack behavior at the point level (Gauss point) under opening and shear displacement is presented in this section.

The described parameters of monotonic and cyclic response material behavior are plotted in Figure 3. The exponential parameter *c* was used to control the dropped-down part of the crack opening (*COD*) and sliding (*CSD*) curve. The parameters *c* and *r* were applied for tuning the accumulation of the damage due to cyclic loading. The damage strength parameter *S*, however, could control the brittleness in the response. The model parameters for a common combination of concrete matrix C30/37 were identified using the parametric study reported in [46]. The setup of the study is provided in Figure 3 for monotonic loading and for cyclic loading, as the cohesive model parameters are summarized. The cohesive model stiffness (*E*) was set equal to Young's modulus of concrete. The parameters *σ*, *K*, *γ*, *S*, *r*, and *c* were identified using a black line for the monotonic response and a blue line for the monotonic response.

**Figure 3.** Characterization of the crack behavior under cyclic loading (blue lines) and monotonic loading (black lines) at the material-point level: (**a**) crack opening, (**b**) crack sliding.

The cyclic loading curves of the crack opening versus cyclic loading can be compared with the corresponding curves obtained numerically for monotonic loading. The described model was implemented using zero-thickness interface elements inside the SBFEM framework in Equation (18)). For the monotonic and the cyclic loading, the damage evolution for the loaded and unloaded responses is depicted in Figure 3 for crack opening and crack sliding. The accumulation of the damage parameter is nonlinear. The traction opening/sliding cohesive models for two loading scenarios are studied.

#### **4. Numerical Validation**

#### *4.1. Test Setup*

Three-point bending (TPB) tests were studied to validate the numerical method in this study. The contributions of both traction modes, *k<sup>n</sup>* and *k<sup>s</sup>* , in the cohesive zone model, were investigated. The investigations performed by [8] have shown that the inclusion of the normal energy dissipation dominated the response of post-peak crack mouth sliding displacement (CMOD). The nonlinear equilibrium equations were solved using Newton– Raphson iteration [55], which is characterized by strain softening in the process zone. The benchmark examples are TPB tests with a single-edge notch (Figure 4). Two sizes of the beam were considered in the tests: small beams with a cross-section height of *h* = 200 mm, and large beams with *h* = 400 mm. The beam width was *b* = 100 mm. The lengths of small and large beams were 600 and 1200 mm, respectively. For the notch depth, *h*<sup>0</sup> = *h*/6, whereas the maximum grain size (*d*0) was 8 mm. The experimental measurements for the concrete beams were provided by Baktheer and Becks [8], and the material properties were adopted from [8], as listed in Table 1.

**Table 1.** Parameters of concrete [8].


**Figure 4.** A single-notched concrete beam subjected to a three-point load. (**a**) Geometry, (**b**) initial mesh.

#### *4.2. Loading Scenarios*

Experimentally, the crack opening displacements and the mid span deflection of the tested beam were recorded, along with the applied force *F*. The TPB supported beam was tested symmetrically by displacement-controlled loading at the top edge. The typical two different loading scenarios are shown in Figure 5. In the SBFEM simulation, an incrementally increased monotonic load (Figure 5a) was applied with an increment size of 0.0005; there were 200 load steps. The load was controlled by the crack tip opening displacement (CMOD) until failure. In the second loading scenario, Figure 5b, a sequence of loading and unloading cycles was applied to define the CMOD. In this way, detailed characteristics of the post-peak loading and unloading of the load–CMOD curve were obtained. This can help to analyze the damage mechanism involved in the cyclic flexural behavior of concrete.

**Figure 5.** Typical loading scenarios of the studied tested beams: (**a**) monotonic behavior, (**b**) cyclic behavior.

### *4.3. Monotonic Loading*

The softening curve parameters to model the fracture process zone are presented, and a range of values were applied based on the parametric study in Section 3. The material parameters were calibrated for two examples under monotonic loading. Then, the material model was validated using the size-effect calculations. The obtained numerical results for cyclic loading were obtained in additional to validating the method. For this investigation, the properties of the concrete and cohesive interface element are listed in Table 2.


**Table 2.** Model parameters for concrete cohesive interface element.

The tracked points for the notched pattern and the initial mesh were defined as shown in Figure 4b. For the small beam of cross-section height of *h* = 200 mm, the mesh consisted of 481 polygons and 584 nodes. Meanwhile, for the large beam (*h* = 400 mm), the initial mesh comprised 1483 polygons and 1628 nodes. Plane stress conditions were assumed.

Figure 6 compares the predicted load-crack mouth opening displacement (CMOD) of the TBP small beam with the experimental results reported by [8] under monotonic loading. The corresponding curve of the numerical predictions by SBFEM is depicted in Figure 6, plotted as a blue dashed line. The numerical results of the load–CMOD curve are in a good agreement with the experimental measurements. A maximum load of 18.75 kN was obtained at a CMOD of 0.017 mm. Interestingly, the load–CMOD curve of the numerical was not influenced by the length of crack propagation.

**Figure 6.** Numerical predictions of load–CMOD curves and the corresponding experimental curves for the single-notched three-point bending test under monotonic loading.

The crack propagation due to increasing load with initial ∆*<sup>a</sup>* = 3.0 mm is shown in Figure 7. Our results show a straight crack path in the direction of the point of external load (*F*). Th fracture process zone extends up in the middle of the beam Figure 7b at peak load before cracking. At a load of 5.763 kN, the crack propagates in the post-peak region (Figure 7c). For this load level, the cohesive force vanishes. Finally, as the actual crack's length is increased, the fracture zone is shortened, as expected, by increasing the load level; see Figure 7d. The influence of the size of stiffness degradation is depicted for both small and large tests in Figure 8, which shows the numerical predictions, along with experimental measurements of monotonic tests based on [8]. The nominal strength (*σN*) of SBFEM numerical results were determined in the same way in [8] under monotonic behavior. It is calculated by [1,52]:

$$
\sigma\_N = \frac{c\_n F\_u}{bh} \,\prime\tag{50}
$$

where *F<sup>u</sup>* is the ultimate force and *c<sup>n</sup>* = 3*L*0/(2*h* − *h*0) is determined by the bending theory for notched beams. Figure 8 depicts a log–log plot of the the relative size of the beam (horizontal axis) and the nominal strength (vertical axis). The numerical results and the experiments of [8] indicate that the nominal strength is increased by decreasing beam size. The numerical results of nominal strength and the experimental data have a ratio of 1.01–1.04 for small beams, and a ratio of 0.98–1.02 for large beams. In addition, less scatter in the predictions of the large beams was obtained.

**Figure 8.** The effect of the size of the beam on the nominal strength under monotonic behavior.

#### *4.4. Cyclic Loading*

Figure 5 shows the numerical predictions and the experimental measurements for cyclically increasing loading. The loading was controlled by the CMOD for three unloading cycles and applied until failure. Good agreement of the numerical predictions (Figure 9b) with respect to the experiment tests (Figure 9a) is obvious.

**Figure 9.** Comparison of numerical predictions (**a**) and experimental measurements (**b**) of Cyclic-CMOD curves for the single-notched three-point bending test.

Furthermore, in our analysis we explore the main dissipative mechanisms. For this purpose, the TPB beams were subjected to a few loading cycles with an incremental increase in the CMOD values. The obtained cyclic responses for both small and large beams are plotted in Figure 10a,b, respectively. One of the principal noticeable effects during the cyclic loading in the post-peak regime is the degradation of the unloading stiffness, which defines the value of the damage. From the damage evolution, it was observed that the damage parameter had a value larger than 0.5 at the first post-peak cyclic load for a small beam; the damage started to progress in the pre-peak subcritical load levels. Furthermore, the damage parameter *ω* has a value larger than 0.65 for a large beam.

**Figure 10.** Post-peak cyclic behavior of SBFEM analysis and corresponding damage evolution for (**a**) a small beam and (**b**) a large beam.

This is explained by knowing that the developed crack showed a rough surface that is not fully closed during the unloading of the specimen. This was confirmed for the cyclic behavior in the simulation of SBFEM and experiment tests. Additionally, the stiffness degradation and the growth of unclosed crack openings were characterized for both sizes.

Plots of *K<sup>I</sup>* -*CMOD* are shown in Figure 11 for monotonic and cyclic applied loads. In Figure 11a, the points that represent the initial mesh of Figure 4 were calculated once *K<sup>I</sup>* ≥ 0. Then, the crack opened gradually based on a crack-propagation criterion. The

numerical calculation of *K<sup>I</sup>* by SBFEM with a fewer degrees of freedom (DOFs) manifested good crack trajectory predictions.

**Figure 11.** *K<sup>I</sup>* − *CMOD* and loading-point curves for mode-I bending beam for: monotonic loading (**a**) and cyclic loading (**b**).

Since the goal of the present study was to apply the constitutive law with a cumulative damage feature within SBFEM, we considered only mode-I cyclic crack propagation in our analysis. Further studies with applications to mixed modes loading are planned for future publications, where more advanced constitutive cohesive zone models could be used, e.g., [14,56].

#### **5. Conclusions**

Cracks in concrete can occur when the tensile stresses imposed by actions exceed the tensile strength of the material. Furthermore, the cracks can also be initiated under repeated loads with stress levels below the tensile strength. In this work, the cyclic cohesive crack procedure-based SBFEM was implemented to study the crack propagation in concrete. The proposed model showed the ability to simulate the monotonic and cyclic behavior of a cohesive crack interface element, e.g., a concrete interface. It provided a realistic prediction of cyclic damage behavior for up to several load cycles. The output for the numerical simulation of monotonic loading analysis showed full agreement with experimental data from the literature. The results differed 5% for the maximum peak force. Regarding the nominal strength, the ratio of the numerical results to the experimental data under monotonic loading varied between 1.01 and 1.04 for small beams. The ratio was 0.98–1.02 for large beams.

Additionally, the proposed procedure has been proved to be an efficient tool for estimating the damage level. The level of damage accumulation (*ω*) and material plasticity variables were calculated based on thermodynamics. The described damage model has been successfully implemented to describe the cyclic behavior of cohesive interface elements using SBFEM. The damage parameter *ω* has a value larger than 0.5 at the first post-peak cyclic load for a small beam, and has a value larger than 0.65 for a large beam. The cyclic responses obtained by SBFEM for both small and large beams presented good agreement with the experimental data.

The predicted load–CMOD responses in the validation examples were within the range measured in the cyclic and monotonic loading experiments. Testing results demonstrated that the most important factors for the overall simulation were the thermodynamic hardening modulus *γ* and the damage strength parameter *S*. The simulations executed to study the effect of the loading sequence offered successful results and demonstrated the effect of damage accumulation for realistic predictions for concrete structures.

**Author Contributions:** Methodology, O.A., C.K., E.T.O. and K.M.H.; Validation, O.A., C.K., E.T.O. and K.M.H.; Writing—original draft, Omar Alrayes, Carsten Könke, E.T.O. and K.M.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)- Projektnummer 492535144.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Khader M. Hamdia thanks the support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 492535144.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


## **Thermodynamic Parameters**


### **References**


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## *Article* **Investigation on Performance of Hydraulically Expanded Joint of Titanium–Steel Clad Tubesheet**

**Jia Li <sup>1</sup> , Juan Li <sup>1</sup> , Yuyan Zhang <sup>1</sup> and Changyu Zhou 2,\***

<sup>2</sup> School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China

**\*** Correspondence: changyu\_zhou@163.com

**Abstract:** The performance of a hydraulically expanded joint between tubesheet and titanium tube was analyzed using a finite element numerical calculation. The connection strength of Q345R tubesheet and TA2-Q345R clad tubesheet was studied using a tight expansion method. The results proved that the residual contact pressure and pullout force of the tight expansion joint of TA2-Q345R clad tubesheet were greater than those of the Q345R tubesheet. However, the residual contact pressure of the expanded joint without a groove for the TA2-Q345R tubesheet and the pullout force failed to meet the requirement of connection strength. Hence, the groove was employed on the contact surface. The influences of groove position and groove width on the connection strength of the expanded joint with grooves in tubesheet hole were studied. The results show that the residual contact pressure of the clad tubesheet of grooving in the cladding layer was higher than that of grooving in the base layer. The effect of the position of groove in the cladding layer and base layer on the residual contact pressure could be neglected. A wider groove led to a higher residual contact pressure, which increased significantly when the groove width was 4 mm.

**Keywords:** TA2-Q345R clad tubesheet; expanded joint; residual contact pressure; groove width

## **1. Introduction**

Titanium is an alloy with low density, high strength, and strong corrosion resistance, which is widely used in the chemical industry, marine ships, aviation, medical instruments, automotive industry, livelihood supplies, and other fields [1,2]. Because of its good economic and applicability, titanium-steel clad tubeplate structures are applied in titanium shells and tube heat exchangers.

The joint between the tube and tubesheet is the key part of the shell and tube heat exchanger, which is also prone to failure. Expansion is an important type of connection between the tube and tubesheet, which can be divided into hydraulic expansion, mechanical expansion, and explosion expansion. Among them, hydraulic expansion is a uniform flexible expansion method. The expansion pressure is easy to control and does not damage the inner wall surface of the expansion; thus, the joint is of good quality.

How to improve the sealing performance and tensile resistance of hydraulic expanded joints has always been the focus of research. Therefore, many research achievements have been made on the influence of material properties [3,4], machining accuracy [5,6], geometric parameters [7,8], manufacturing process [9,10], and other factors on the performance of joints. In order to improve the connection strength and sealing performance of the tube-to-tubesheet joints, tubesheet holes are grooved [11,12]. Material properties, especially plasticity, play a crucial role in obtaining accurate results close to true values [7]. Before finite element analysis, it is important to determine material properties such as yield stress [13]. Previous studies have shown that strain hardening is an important factor influencing the accuracy of joint numerical analysis results [14,15]. Previous studies mainly focused on the combination of a carbon steel tube and low-alloy steel tube sheet, or the

**Citation:** Li, J.; Li, J.; Zhang, Y.; Zhou, C. Investigation on Performance of Hydraulically Expanded Joint of Titanium–Steel Clad Tubesheet. *Materials* **2023**, *16*, 1106. https:// doi.org/10.3390/ma16031106

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 13 December 2022 Revised: 20 January 2023 Accepted: 22 January 2023 Published: 27 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

combination of a titanium tube and single-material tube sheet. For the combination of titanium tube and titanium–steel clad tubesheet, we can find reports on the stress analysis [16,17], creep of titanium [18], etc. However, there are few reports on the impact of the titanium–steel clad tubesheet cladding layer and grooving parameters on the performance of expanded joints. Jawad et al. [7] used experimental methods to study the expanded joint of titanium–steel clad tubesheet; the best result was obtained when the groove width was equal to 1.56√ *r*o*t*, where r<sup>o</sup> and t are the outside radius and thickness of the tube, respectively. The residual contact stress of titanium–steel clad tubesheet with a groove in the base layer was calculated by numerical simulation [19]. The above studies did not discuss the influence of groove form on titanium–steel clad tubesheet joint performance. By means of an experiment, Ma Qiulin [20] found that TA2 had the characteristic of elastic hysteresis after loading, and its residual contact pressure decreased with the increase in time after the completion of titanium tube expansion. Subsequently, the experimental results of the same research team [21] showed that, under 260 MPa hydraulic expansion pressure, after 96 h of unloading the expansion pressure, the pullout force of the joint decreased by 60% and 23%, corresponding to the expanded joint without grooves in the tubesheet hole and with two grooves in the tubesheet hole, respectively. Therefore, the residual contact pressure at the moment of expansion completion should not be used as the only basis for analyzing the performance of expanded joints, and the negative effect caused by elastic hysteresis should also be considered. Therefore, in this paper, the hydraulic expanded tube joint of a titanium–steel clad tubesheet is taken as the research object. Through numerical simulation, the sealing performance and connection strength of the tight expansion (joint without groove) are studied; furthermore, the joint with grooves is analyzed. The influence of the expansion method and structural parameters, including groove form, groove width, and groove location in the base, is investigated, which provides a basis for the optimization of the expansion structure of the clad tubesheet. sheet, or the combination of a titanium tube and single-material tube sheet. For the combination of titanium tube and titanium–steel clad tubesheet, we can find reports on the stress analysis [16,17], creep of titanium [18], etc. However, there are few reports on the impact of the titanium–steel clad tubesheet cladding layer and grooving parameters on the performance of expanded joints. Jawad et al. [7] used experimental methods to study the expanded joint of titanium–steel clad tubesheet; the best result was obtained when the groove width was equal to <sup>o</sup> 1.56 *r t* , where ro and t are the outside radius and thickness of the tube, respectively. The residual contact stress of titanium–steel clad tubesheet with a groove in the base layer was calculated by numerical simulation [19]. The above studies did not discuss the influence of groove form on titanium–steel clad tubesheet joint performance. By means of an experiment, Ma Qiulin [20] found that TA2 had the characteristic of elastic hysteresis after loading, and its residual contact pressure decreased with the increase in time after the completion of titanium tube expansion. Subsequently, the experimental results of the same research team [21] showed that, under 260 MPa hydraulic expansion pressure, after 96 h of unloading the expansion pressure, the pullout force of the joint decreased by 60% and 23%, corresponding to the expanded joint without grooves in the tubesheet hole and with two grooves in the tubesheet hole, respectively. Therefore, the residual contact pressure at the moment of expansion completion should not be used as the only basis for analyzing the performance of expanded joints, and the negative effect caused by elastic hysteresis should also be considered. Therefore, in this paper, the hydraulic expanded tube joint of a titanium–steel clad tubesheet is taken as the research object. Through numerical simulation, the sealing performance and connection strength of the tight expansion (joint without groove) are studied; furthermore, the joint with grooves is analyzed. The influence of the expansion method and structural parameters, including groove form, groove width, and groove location in the base, is investigated, which provides a basis for the optimization of the expansion structure of the clad tubesheet.

studies mainly focused on the combination of a carbon steel tube and low-alloy steel tube

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#### **2. Finite Element Model Analysis**

#### *2.1. Material Performance* **2. Finite Element Model Analysis**

The tube was made of commercial pure titanium TA2, and the titanium–steel clad tubesheet material was TA2-Q345R, in which the base material was Q345R and the cladding material was TA2. The physical properties of the two materials are shown in Table 1. The true stress and strain of material were simulated using multilinear isotropic reinforcement material, and their mechanical property curves are shown in Figure 1. *2.1. Material Performance* The tube was made of commercial pure titanium TA2, and the titanium–steel clad tubesheet material was TA2-Q345R, in which the base material was Q345R and the cladding material was TA2. The physical properties of the two materials are shown in Table 1. The true stress and strain of material were simulated using multilinear isotropic reinforce-

**Table 1.** Material properties. ment material, and their mechanical property curves are shown in Figure 1.


**Figure 1.** Tensile property curves of materials. **Figure 1.** Tensile property curves of materials.

#### *2.2. Finite Element Model 2.2. Finite Element Model* The tubes were arranged in regular triangles. Due to the periodicity of pipe drainage,

The tubes were arranged in regular triangles. Due to the periodicity of pipe drainage, a seven-hole model was adopted, and 1/12 of the circle, i.e., a 30◦ area, was taken as the research object, as shown in Figure 2a. The size of the tube was ϕ25 mm × 1.5 mm, and the length of the tube was 180 mm. The diameter of the tubesheet hole was 25.3 mm, the distance between the tube centers was 32 mm, the outer diameter of the tubesheet was 480 mm, and the total thickness of the tubesheet was 50 mm with a 38 mm base layer and a 12 mm cladding layer, conforming to the recommended scope of the standard [22]. a seven-hole model was adopted, and 1/12 of the circle, i.e., a 30° area, was taken as the research object, as shown in Figure 2a. The size of the tube was φ25 mm × 1.5 mm, and the length of the tube was 180 mm. The diameter of the tubesheet hole was 25.3 mm, the distance between the tube centers was 32 mm, the outer diameter of the tubesheet was 480 mm, and the total thickness of the tubesheet was 50 mm with a 38 mm base layer and a 12 mm cladding layer, conforming to the recommended scope of the standard [22].

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**Figure 2.** Finite element analysis: (**a**) analysis scope; (**b**) finite element mesh model. **Figure 2.** Finite element analysis: (**a**) analysis scope; (**b**) finite element mesh model.

Because the overall structure of all joints was similar, only the finite element model of the joint with grooved clad tubesheet holes is shown in this manuscript. The SOLID185 element was used to establish the finite element model of the expanded joint, as shown in Figure 2b. The mesh was finely divided near the expansion surface. The combination of the outer wall of the titanium tube and the inner wall of the tubesheet hole was a nonlinear contact problem. The outer wall of the titanium tube was the contact surface, and the CONTA173 contact surface element was selected. The inner wall of the tubesheet hole was the target surface, and the TARGE170 target surface element was selected. The contact algorithm adopted the Augmented Lagrange method, and the appropriate solution value was guaranteed by controlling the contact stiffness FKN and the maximum allowable penetration value FTOLN [23,24]. Because the overall structure of all joints was similar, only the finite element model of the joint with grooved clad tubesheet holes is shown in this manuscript. The SOLID185 element was used to establish the finite element model of the expanded joint, as shown in Figure 2b. The mesh was finely divided near the expansion surface. The combination of the outer wall of the titanium tube and the inner wall of the tubesheet hole was a nonlinear contact problem. The outer wall of the titanium tube was the contact surface, and the CONTA173 contact surface element was selected. The inner wall of the tubesheet hole was the target surface, and the TARGE170 target surface element was selected. The contact algorithm adopted the Augmented Lagrange method, and the appropriate solution value was guaranteed by controlling the contact stiffness FKN and the maximum allowable penetration value FTOLN [23,24].

During expansion, expansion pressure was applied on the inner surface of tube; the tube end was subjected to axial and circumferential symmetric constraints, whereas the outer cylindrical surface of the tubesheet was axially constrained, and the remaining surfaces were free. During expansion, expansion pressure was applied on the inner surface of tube; the tube end was subjected to axial and circumferential symmetric constraints, whereas the outer cylindrical surface of the tubesheet was axially constrained, and the remaining surfaces were free.

After expansion, the expansion pressure on the tube was removed. Under the pulling condition, the axial displacement constraint of the tube was removed, axial (Z-direction) displacement was applied, and the other boundary conditions remained unchanged. During calculation, an automatic step and large deformation were chosen, and a linear search to stabilize the calculation and the complete Newton–Rapson method were used. After expansion, the expansion pressure on the tube was removed. Under the pulling condition, the axial displacement constraint of the tube was removed, axial (Z-direction) displacement was applied, and the other boundary conditions remained unchanged. During calculation, an automatic step and large deformation were chosen, and a linear search to stabilize the calculation and the complete Newton–Rapson method were used.

#### **3. Tight Expansion Analysis 3. Tight Expansion Analysis**

#### *3.1. Theoretical Calculation Method 3.1. Theoretical Calculation Method*

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Yan Huigeng's [25] theoretical calculation method of residual contact pressure based on the double-cylinder model is widely used for the calculation of residual contact pressure of the tight expansion, as shown in Equation (1). Yan Huigeng's [25] theoretical calculation method of residual contact pressure based on the double-cylinder model is widely used for the calculation of residual contact pressure of the tight expansion, as shown in Equation (1).

$$p\_\mathbf{c}^\* = (1 - 2c)p\_\mathbf{i} - \frac{2}{\sqrt{3}}\sigma\_\mathbf{s} \ln K\_\mathbf{i} \tag{1}$$

Here, Here,

$$c = 1/\left\{\mathcal{K}\_{\rm t}^2 (1 - \mu\_{\rm t}) + 1 + \mu\_{\rm t} + \frac{E\_{\rm t} (\mathcal{K}\_{\rm t}^2 - 1)}{E\_{\rm s} (\mathcal{K}\_{\rm s}^2 - 1)} [1 - \mu\_{\rm s} + \mathcal{K}\_{\rm s}^2 (1 + \mu\_{\rm s})] \right\}.$$

where *σ*st is the yield strength of the tube material (MPa), *K*<sup>t</sup> = *r*o/*r*<sup>i</sup> is the diameter ratio of the heat exchanger tube, *ro, r<sup>i</sup>* are respectively the inner and outer diameters of the heat exchanger tube, *K*<sup>s</sup> = *R*o/*R*<sup>i</sup> is the diameter ratio of the equivalent cylinder, *Ro, R<sup>i</sup>* are respectively the inner and outer diameters of the equivalent cylinder, *µ*<sup>t</sup> , *µ*<sup>s</sup> are the Poisson's ratios of the tube and tubesheet material, respectively, and *E*<sup>t</sup> , *E*<sup>s</sup> are respectively the elastic moduli of the tube and tubesheet material (MPa). where *σ*st is the yield strength of the tube material (MPa), *K*t = *r*o/*r*<sup>i</sup> is the diameter ratio of the heat exchanger tube, *ro, ri* are respectively the inner and outer diameters of the heat exchanger tube, *K*s =*R*o/*R*<sup>i</sup> is the diameter ratio of the equivalent cylinder, *Ro, Ri* are respectively the inner and outer diameters of the equivalent cylinder, *µ*t, *µ*<sup>s</sup> are the Poisson's ratios of the tube and tubesheet material, respectively, and *E*t, *E*<sup>s</sup> are respectively the elastic moduli of the tube and tubesheet material (MPa).

However, Yan Huigeng's theoretical calculation method assumes that the tube material is an ideal elastic–plastic material, and *σ*st in Equation (1) is the yield strength of the tube material without considering the strain strengthening of the material; thus, the theoretically calculated value of *p*<sup>c</sup> \* is greater than the real value. Hao Junwen [13] proposed the concept of pipe equivalent yield strength *σ*seq according to the research methods of predecessors [25], i.e., replacing *σ*st in Equation (1) with *σ*seq, as shown in Figure 3. However, Yan Huigeng's theoretical calculation method assumes that the tube material is an ideal elastic–plastic material, and *σ*st in Equation (1) is the yield strength of the tube material without considering the strain strengthening of the material; thus, the theoretically calculated value of *p*c\* is greater than the real value. Hao Junwen [13] proposed the concept of pipe equivalent yield strength *σ*seq according to the research methods of predecessors [25], i.e., replacing *σ*st in Equation (1) with *σ*seq, as shown in Figure 3.

**Figure 3.** Schematic diagram for determining the equivalent yield strength. **Figure 3.** Schematic diagram for determining the equivalent yield strength.

As shown in Figure 3, the curve *OABCD* represents the true stress–strain curve of the titanium tube. The section *OA* is the elastic deformation stage, the section *AB* is the partial plastic deformation stage, and the section *BD* is the full strain strengthening stage. Assuming that the tube contacts the tubesheet when the stress and strain develop to point *C*, and that the abscissa *C*x = *g*/*r*<sup>o</sup> corresponding to point *C* is the strain value of the outer wall of the tube, *g* is the clearance between the outer wall of the tube and the inner wall of the hole. By making a straight line with a slope of *Et* through the *C*<sup>x</sup> point, the straight line intersects the curve at point *Q*. Then, the ordinate *Q*<sup>y</sup> corresponding to point *Q* is the equivalent yield strength of the tube material *σ*seq. According to the tensile curve of TA2 in Figure 1, *σ*seq = 415 MPa, which is 35 MPa different from the original value of 380 MPa. As shown in Figure 3, the curve *OABCD* represents the true stress–strain curve of the titanium tube. The section *OA* is the elastic deformation stage, the section \_ *AB* is the partial plastic deformation stage, and the section *BD* is the full strain strengthening stage. Assuming that the tube contacts the tubesheet when the stress and strain develop to point *C*, and that the abscissa *C*<sup>x</sup> = *g*/*r*<sup>o</sup> corresponding to point *C* is the strain value of the outer wall of the tube, *g* is the clearance between the outer wall of the tube and the inner wall of the hole. By making a straight line with a slope of *E*<sup>t</sup> through the *C*<sup>x</sup> point, the straight line intersects the curve at point *Q*. Then, the ordinate *Q*<sup>y</sup> corresponding to point *Q* is the equivalent yield strength of the tube material *σ*seq. According to the tensile curve of TA2 in Figure 1, *σ*seq = 415 MPa, which is 35 MPa different from the original value of 380 MPa.

#### *3.2. Comparison of Simulation Results and Theoretical Calculation Results*

The experiment and numerical simulation results show that the friction coefficient *f* is related to many factors such as the accuracy of the machined surface and the properties of materials. According to the experimental data of the expansion connection between TA2

tube and Q345R tubesheet in the literature [20], the friction coefficient *f* was set to 0.28. The *p*c \* with tube material TA2 and tubesheet material Q345R was simulated, and the results were respectively compared with the theoretical results of Yan [25] and Hao [13], as shown in Figure 4. It can be seen that the simulation value was much smaller than the value of Yan's theoretical method, and it was mainly consistent with the result of Hao's theoretical method, which is closer to the real situation. Therefore, the parameters of the simulation and the results were considered reasonable. tube and Q345R tubesheet in the literature [20], the friction coefficient *f* was set to 0.28. The *p*c\* with tube material TA2 and tubesheet material Q345R was simulated, and the results were respectively compared with the theoretical results of Yan [25] and Hao [13], as shown in Figure 4. It can be seen that the simulation value was much smaller than the value of Yan's theoretical method, and it was mainly consistent with the result of Hao's theoretical method, which is closer to the real situation. Therefore, the parameters of the simulation and the results were considered reasonable. The *p*c\* with tube material TA2 and tubesheet material Q345R was simulated, and the results were respectively compared with the theoretical results of Yan [25] and Hao [13], as shown in Figure 4. It can be seen that the simulation value was much smaller than the value of Yan's theoretical method, and it was mainly consistent with the result of Hao's theoretical method, which is closer to the real situation. Therefore, the parameters of the simulation and the results were considered reasonable.

The experiment and numerical simulation results show that the friction coefficient *f* is related to many factors such as the accuracy of the machined surface and the properties of materials. According to the experimental data of the expansion connection between TA2

The experiment and numerical simulation results show that the friction coefficient *f* is related to many factors such as the accuracy of the machined surface and the properties of materials. According to the experimental data of the expansion connection between TA2 tube and Q345R tubesheet in the literature [20], the friction coefficient *f* was set to 0.28.

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*3.2. Comparison of Simulation Results and Theoretical Calculation Results*

*3.2. Comparison of Simulation Results and Theoretical Calculation Results*

**Figure 4.** Comparison of calculation results [13,25]. **Figure 4.** Comparison of calculation results [13,25]. **Figure 4.** Comparison of calculation results [13,25].

#### *3.3. Performance Analysis of Tight Expansion Joint 3.3. Performance Analysis of Tight Expansion Joint 3.3. Performance Analysis of Tight Expansion Joint*

In order to investigate the influence of the cladding surface on the sealing performance of the expansion joint, the expansion and pulling processes of the joints between the TA2 tube and Q345R tubesheet, and between the TA2 tube and TA2-Q345R clad tubesheet under different expansion pressures were simulated. The average *p*c\* on the contact surface and pullout force *F* were obtained, as shown in Figure 5. In order to investigate the influence of the cladding surface on the sealing performance of the expansion joint, the expansion and pulling processes of the joints between the TA2 tube and Q345R tubesheet, and between the TA2 tube and TA2-Q345R clad tubesheet under different expansion pressures were simulated. The average *p*<sup>c</sup> \* on the contact surface and pullout force *F* were obtained, as shown in Figure 5. In order to investigate the influence of the cladding surface on the sealing performance of the expansion joint, the expansion and pulling processes of the joints between the TA2 tube and Q345R tubesheet, and between the TA2 tube and TA2-Q345R clad tubesheet under different expansion pressures were simulated. The average *p*c\* on the contact surface and pullout force *F* were obtained, as shown in Figure 5.

**Figure 5.** Tight expansion joints performance of two kinds of tubesheets. **Figure 5.** Tight expansion joints performance of two kinds of tubesheets. **Figure 5.** Tight expansion joints performance of two kinds of tubesheets.

It can be seen from Figure 5 that *p*<sup>c</sup> \* and *F* of expanded joints of different materials increased with the increase in expansion pressure, and *p*<sup>c</sup> \* and *F* of the TA2-Q345R clad tubesheet were both higher than those of the Q345R tubesheet under the same expansion pressure. It can be seen from Figure 5 that *p*<sup>c</sup> \* and *F* of expanded joints of different materials increased with the increase in expansion pressure, and *p*<sup>c</sup> \* and *F* of the TA2-Q345R clad tubesheet were both higher than those of the Q345R tubesheet under the same expansion pressure. The pullout force provided by the expanded joint is expressed in Equation (2). It can be seen from Figure 5 that *p*<sup>c</sup> \* and *F* of expanded joints of different materials increased with the increase in expansion pressure, and *p*<sup>c</sup> \* and *F* of the TA2-Q345R clad tubesheet were both higher than those of the Q345R tubesheet under the same expansion pressure.

The pullout force provided by the expanded joint is expressed in Equation (2). The pullout force provided by the expanded joint is expressed in Equation (2).

$$F > \pi dl[q]. \tag{2}$$

According to GB/T151-2014 "Heat Exchanger" [26], the allowable pullout force [*q*] of the steel heat exchange tube of tight expansion is 2 MPa; thus, the pullout force should satisfy *F* > 7.23 kN according to Equation (2). For titanium expansion joints, the pullout force decreases with the time after the expansion due to the elastic hysteresis effect. A previous study [21] gave the relationship curve between the pullout force and time after the

expansion of the titanium tube and steel tubesheet. Considering that the pullout force of the joint decreases by about 75% during the period from the completion of manufacturing until service, the corresponding pullout force of the titanium tube should be greater than 28.92 kN at the moment of completion of the expansion. As can be seen from Figure 5, the pullout force of the joint still failed to meet this requirement even when the expansion pressure exceeded 320 MPa. ing until service, the corresponding pullout force of the titanium tube should be greater than 28.92 kN at the moment of completion of the expansion. As can be seen from Figure 5, the pullout force of the joint still failed to meet this requirement even when the expansion pressure exceeded 320 MPa. Figure 6 shows the contact pressure distribution on the contact surface of the TA2-

According to GB/T151-2014 "Heat Exchanger" [26], the allowable pullout force [*q*] of the steel heat exchange tube of tight expansion is 2 MPa; thus, the pullout force should satisfy *F* > 7.23 kN according to Equation (2). For titanium expansion joints, the pullout force decreases with the time after the expansion due to the elastic hysteresis effect. A previous study [21] gave the relationship curve between the pullout force and time after the expansion of the titanium tube and steel tubesheet. Considering that the pullout force of the joint decreases by about 75% during the period from the completion of manufactur-

*Materials* **2023**, *16*, x FOR PEER REVIEW 6 of 14

Figure 6 shows the contact pressure distribution on the contact surface of the TA2- Q345R clad tubesheet or Q345R tubesheet after loading and after unloading. According to Figure 6a,c, when the expansion pressure reached 320 MPa, there were two upper and lower sealing rings on the contact surface of the two tubesheet conditions, consistent with the simulation results in [4,27]. When the expansion pressure was unloaded, it can be seen from Figure 6b,d that there was a residual contact pressure ring on the contact surface of the tube side, but this was not obvious on the shell side. This is because the stiffness of the tube on the shell side was greater than that on the tube side; hence, it was not easy to expand and deform. It can also be found from Figure 6b,d that the residual contact pressure of the sealing ring of the TA2-Q345R clad tubesheet after unloading was greater than that of the Q345R tubesheet (57.9 MPa > 24.7 MPa) because the cladding layer of the clad tubesheet and the tube were the same material TA2. When the expansion pressure was removed, the tubesheet rebounded with the tube; thus, the contact pressure of the cladding part could maintain high values. Therefore, although the contact pressure of the base part decreased substantially because the elastic modulus of TA2 was less than that of Q345R, the average residual contact pressure in the whole contact range of the clad tubesheet was higher because the residual contact pressure of the cladding part was high, and the corresponding pullout force was also higher. Q345R clad tubesheet or Q345R tubesheet after loading and after unloading. According to Figure 6a,c, when the expansion pressure reached 320 MPa, there were two upper and lower sealing rings on the contact surface of the two tubesheet conditions, consistent with the simulation results in [4,27]. When the expansion pressure was unloaded, it can be seen from Figure 6b,d that there was a residual contact pressure ring on the contact surface of the tube side, but this was not obvious on the shell side. This is because the stiffness of the tube on the shell side was greater than that on the tube side; hence, it was not easy to expand and deform. It can also be found from Figure 6b,d that the residual contact pressure of the sealing ring of the TA2-Q345R clad tubesheet after unloading was greater than that of the Q345R tubesheet (57.9 MPa > 24.7 MPa) because the cladding layer of the clad tubesheet and the tube were the same material TA2. When the expansion pressure was removed, the tubesheet rebounded with the tube; thus, the contact pressure of the cladding part could maintain high values. Therefore, although the contact pressure of the base part decreased substantially because the elastic modulus of TA2 was less than that of Q345R, the average residual contact pressure in the whole contact range of the clad tubesheet was higher because the residual contact pressure of the cladding part was high, and the corresponding pullout force was also higher.

**Figure 6.** Contact pressure distribution on the contact surface of two kinds of tubesheets: (**a**) TA2- Q345R clad tubesheet after loading; (**b**) TA2-Q345R clad tubesheet after unloading; (**c**) Q345R tubesheet after loading; (**d**) Q345R tubesheet after unloading. **Figure 6.** Contact pressure distribution on the contact surface of two kinds of tubesheets: (**a**) TA2- Q345R clad tubesheet after loading; (**b**) TA2-Q345R clad tubesheet after unloading; (**c**) Q345R tubesheet after loading; (**d**) Q345R tubesheet after unloading.

Three axial paths A–A', B–B', and C–C' (in Figure 6) were taken at 0°, 15°, and 30° of the circular direction of the contact surface, and the residual contact pressure along the three paths was extracted as shown in Figure 7. It can be seen from Figure 7 that the highest contact pressure of the two tubeplates was at the same position in the axial direction (Z-direction). Compared with the Q345R tubesheet, the effective sealing range of the tube side of the clad tubesheet was closer to the expansion edge, the *p*c\* value at the sealing ring was much higher than that of the Q345R tubesheet, and the *p*c\* value of the middle and shell side of the two kinds of tubeplates was similar. Three axial paths A–A0 , B–B0 , and C–C0 (in Figure 6) were taken at 0◦ , 15◦ , and 30◦ of the circular direction of the contact surface, and the residual contact pressure along the three paths was extracted as shown in Figure 7. It can be seen from Figure 7 that the highest contact pressure of the two tubeplates was at the same position in the axial direction (Z-direction). Compared with the Q345R tubesheet, the effective sealing range of the tube side of the clad tubesheet was closer to the expansion edge, the *p*<sup>c</sup> \* value at the sealing ring was much higher than that of the Q345R tubesheet, and the *p*<sup>c</sup> \* value of the middle and shell side of the two kinds of tubeplates was similar.

Interface

60

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**Figure 7.** Axial distribution of residual contact pressure of the two kinds of tubesheets. **Figure 7.** Axial distribution of residual contact pressure of the two kinds of tubesheets. Figure 8 indicates the *p*c\* on the circumferential paths D–D′ (tube side) and E–E′ (shell

Figure 8 indicates the *p*c\* on the circumferential paths D–D′ (tube side) and E–E′ (shell side) on two sealing rings The results show that the *p*c\* of the shell side of the two tubesheets was low, and the *p*c\* at some point on the TA2-Q345R clad tubesheet was zero, indicating that, with the rebound of the TA2 tube after unloading, the sealing ring at some positions on the contact surface of the shell side gradually disappeared. When the heat exchanger is in service, the shell side medium would leak along the contact surface and extend to the pipe side, which would damage the contact surface and cause hidden dangers to the safe operation of the equipment. Therefore, the method of ungrooved hole tight expansion was not suitable for the titanium–steel clad tubesheet, and it was necessary to Figure 8 indicates the *p*<sup>c</sup> \* on the circumferential paths D–D<sup>0</sup> (tube side) and E–E0 (shell side) on two sealing rings. The results show that the *p*<sup>c</sup> \* of the shell side of the two tubesheets was low, and the *p*<sup>c</sup> \* at some point on the TA2-Q345R clad tubesheet was zero, indicating that, with the rebound of the TA2 tube after unloading, the sealing ring at some positions on the contact surface of the shell side gradually disappeared. When the heat exchanger is in service, the shell side medium would leak along the contact surface and extend to the pipe side, which would damage the contact surface and cause hidden dangers to the safe operation of the equipment. Therefore, the method of ungrooved hole tight expansion was not suitable for the titanium–steel clad tubesheet, and it was necessary to groove the tubesheet hole for strength expansion to improve the joint connection strength. side) on two sealing rings The results show that the *p*c\* of the shell side of the two tubesheets was low, and the *p*c\* at some point on the TA2-Q345R clad tubesheet was zero, indicating that, with the rebound of the TA2 tube after unloading, the sealing ring at some positions on the contact surface of the shell side gradually disappeared. When the heat exchanger is in service, the shell side medium would leak along the contact surface and extend to the pipe side, which would damage the contact surface and cause hidden dangers to the safe operation of the equipment. Therefore, the method of ungrooved hole tight expansion was not suitable for the titanium–steel clad tubesheet, and it was necessary to groove the tubesheet hole for strength expansion to improve the joint connection strength.

**Figure 8.** Circumferential distribution of residual contact pressure of Q345R tubesheet and TA2- Circumferential position/(°) **Figure 8.** Circumferential distribution of residual contact pressure of Q345R tubesheet and TA2- Q345R clad tubesheet. **Figure 8.** Circumferential distribution of residual contact pressure of Q345R tubesheet and TA2- Q345R clad tubesheet.

#### Q345R clad tubesheet. **4. Expanded Joint with Grooved Clad Tubesheet Holes**

**4. Expanded Joint with Grooved Clad Tubesheet Holes** In the standard GB/T 151-2014, the structure and dimension of the circumferential grooves in clad tubesheet holes are as shown in Figure 9. In order to study the effect of groove form on residual contact pressure and pullout force, four groove forms were designed, as depicted in Figure 10. Figure 10a–d feature one groove in the cladding layer only, one groove in the base layer only, one groove in the cladding layer and one groove in the base layer, and one groove in the cladding layer and two grooves in the base layer, respectively. The effects of groove width *w*, groove distance *s*, and groove spacing *b* on the **4. Expanded Joint with Grooved Clad Tubesheet Holes** In the standard GB/T 151-2014, the structure and dimension of the circumferential grooves in clad tubesheet holes are as shown in Figure 9. In order to study the effect of groove form on residual contact pressure and pullout force, four groove forms were designed, as depicted in Figure 10. Figure 10a–d feature one groove in the cladding layer only, one groove in the base layer only, one groove in the cladding layer and one groove in the base layer, and one groove in the cladding layer and two grooves in the base layer, respectively. The effects of groove width *w*, groove distance *s*, and groove spacing *b* on the connection strength and sealing performance are also investigated. The groove dimen-In the standard GB/T 151-2014, the structure and dimension of the circumferential grooves in clad tubesheet holes are as shown in Figure 9. In order to study the effect of groove form on residual contact pressure and pullout force, four groove forms were designed, as depicted in Figure 10. Figure 10a–d feature one groove in the cladding layer only, one groove in the base layer only, one groove in the cladding layer and one groove in the base layer, and one groove in the cladding layer and two grooves in the base layer, respectively. The effects of groove width *w*, groove distance *s*, and groove spacing *b* on the connection strength and sealing performance are also investigated. The groove dimensions are listed in Table 2.

connection strength and sealing performance are also investigated. The groove dimen-

sions are listed in Table 2.

sions are listed in Table 2.

**Figure 9.** The structure and dimension of grooves. **Figure 9.** The structure and dimension of grooves. **Figure 9.** The structure and dimension of grooves.

**Figure 10.** Four schemes for grooving: (**a**) one groove in the cladding layer only; (**b**) one groove in the base layer only; (**c**) one groove in the cladding layer and one groove in the base layer; (**d**) one groove in the cladding layer and two grooves in the base layer. **Figure 10.** Four schemes for grooving: (**a**) one groove in the cladding layer only; (**b**) one groove in the base layer only; (**c**) one groove in the cladding layer and one groove in the base layer; (**d**) one groove in the cladding layer and two grooves in the base layer. **Figure 10.** Four schemes for grooving: (**a**) one groove in the cladding layer only; (**b**) one groove in the base layer only; (**c**) one groove in the cladding layer and one groove in the base layer; (**d**) one groove in the cladding layer and two grooves in the base layer.


#### 6 12 6 12 *4.1. Effect of Groove Form*

7 14 *4.1. Effect of Groove Form* When a single groove was in the base layer or the cladding layer, the groove width *w*<sup>1</sup> or *w*<sup>2</sup> was 8 mm; when both the cladding layer and the base layer were grooved, the 7 14 *4.1. Effect of Groove Form* When a single groove was in the base layer or the cladding layer, the groove width *w*<sup>1</sup> or *w*<sup>2</sup> was 8 mm; when both the cladding layer and the base layer were grooved, the When a single groove was in the base layer or the cladding layer, the groove width *w*<sup>1</sup> or *w*<sup>2</sup> was 8 mm; when both the cladding layer and the base layer were grooved, the base layer groove width *w*<sup>2</sup> was 8 mm, and the cladding layer groove width *w*<sup>1</sup> was 6 mm. Figures 11 and 12 illustrate the residual contact pressure and pullout force at the moment of the expansion pressure being unloaded.

base layer groove width *w*<sup>2</sup> was 8 mm, and the cladding layer groove width *w*<sup>1</sup> was 6 mm. Figures 11 and 12 illustrate the residual contact pressure and pullout force at the moment

base layer groove width *w*<sup>2</sup> was 8 mm, and the cladding layer groove width *w*<sup>1</sup> was 6 mm. Figures 11 and 12 illustrate the residual contact pressure and pullout force at the moment

of the expansion pressure being unloaded.

of the expansion pressure being unloaded.

**Figure 11.** Effect of groove location on residual contact pressure. **Figure 11.** Effect of groove location on residual contact pressure. **Figure 11.** Effect of groove location on residual contact pressure.

**Figure 12.** Effect of groove location on pullout force. **Figure 12.** Effect of groove location on pullout force. **Figure 12.** Effect of groove location on pullout force.

Figure 11 shows that joints with grooved holes had a higher residual contact pressure *p*c\* than those without grooved holes, and *p*c\* was positively correlated with expansion pressure. Lines A and B coincided, indicating that grooving in the base layer or the cladding layer had little effect on the average residual contact pressure. When there was a groove in the cladding layer, the grooving in the base layer significantly improved *p*c\* , and the double grooving in the base layer was best. Figure 11 shows that joints with grooved holes had a higher residual contact pressure *p*c\* than those without grooved holes, and *p*c\* was positively correlated with expansion pressure. LinesA and B coincided, indicatingthat grooving in the base layer or the cladding layer had little effect on average residual pressure. When there groove in the cladding layer, the grooving in the base layer significantly improved *p*\*, and the double grooving in the base layer was best. Figure 11 shows that joints with grooved holes had a higher residual contact pressure *p*c \* than those without grooved holes, and *p*<sup>c</sup> \* was positively correlated with expansion pressure. Lines A and B coincided, indicating that grooving in the base layer or the cladding layer had little effect on the average residual contact pressure. When there was a groove in the cladding layer, the grooving in the base layer significantly improved *p*<sup>c</sup> \* , and the double grooving in the base layer was best.

As shown in Figure 12, the pullout force *F* almost increased with *p*i, and *p*<sup>i</sup> had a weak influence on *F* when the number of grooves was two or three. Under the same expansion pressure, the relationship of pullout force at different grooving locations was as follows: *F* for one groove in the cladding layer and two grooves in the base layer > *F* for one groove in the cladding layer and one groove in the base layer > *F* for one groove in the cladding layer only > *F* for one groove in the base layer only > *F* for no grooving. The pullout force of grooved joints was greater than 28.92 kN; therefore, when the strength expansion with a groove width of 8 mm was applied to titanium tubes, even when considering the decline in the pullout force caused by elastic hysteresis, it could still meet the requirements of the standard for the pullout force when the equipment is in service. In addition, during grooved expansion, the pullout force drop caused by elastic hysteresis is smaller than it is in the joint without groove [20]; thus, the pullout force value is more conservative. Although lines A and B coincided in Figure 11, the values of the pulling force in these two cases were quite different, indicating that the pullout force is related not only to the residual contact pressure but also to the deformation of the heat exchange tube near the groove. As shown in Figure 12, the pullout force *F* almost increased with *p*i, and *p*<sup>i</sup> had a weak influence on *<sup>F</sup>* when the number of grooves was two or three. Under the same expansion the relationship pullout at different grooving was as follows: *F* for one groove in the cladding layer and two grooves in the base layer > *F* for one groove in the cladding layer and one groove in the base layer > *F* for one groove in the cladding layer only > *F* for one groove in the base layer only > *F* for no grooving. The pullout force of grooved joints was greater than 28.92 kN; therefore, when the strength expansion with a width of mm was applied to titanium tubes, in the force caused by it still the requirements of standard for the pullout force when the equipment is in service. In addition, during grooved expansion, the pullout force drop caused by elastic hysteresis is smaller than it is in the joint without groove [20]; thus, the pullout force value is more conservative. Although lines A and B coincided in Figure 11, the values of the pulling force in these two were indicating that the pullout force related not only to the residual contact pressure but also to the deformation of the heat exchange tube near the groove. As shown in Figure 12, the pullout force *F* almost increased with *p*<sup>i</sup> , and *p*<sup>i</sup> had a weak influence on *F* when the number of grooves was two or three. Under the same expansion pressure, the relationship of pullout force at different grooving locations was as follows: *F* for one groove in the cladding layer and two grooves in the base layer > *F* for one groove in the cladding layer and one groove in the base layer > *F* for one groove in the cladding layer only > *F* for one groove in the base layer only > *F* for no grooving. The pullout force of grooved joints was greater than 28.92 kN; therefore, when the strength expansion with a groove width of 8 mm was applied to titanium tubes, even when considering the decline in the pullout force caused by elastic hysteresis, it could still meet the requirements of the standard for the pullout force when the equipment is in service. In addition, during grooved expansion, the pullout force drop caused by elastic hysteresis is smaller than it is in the joint without groove [20]; thus, the pullout force value is more conservative. Although lines A and B coincided in Figure 11, the values of the pulling force in these two cases were quite different, indicating that the pullout force is related not only to the residual contact pressure but also to the deformation of the heat exchange tube near the groove.

Figure 13 depicts the distribution of residual contact pressure along the axial path at the position of circumfluence 0° when the expansion pressure was 280 MPa. Only two cases are considered in Figure 13: a single groove in the cladding and a single groove in the base. It can be seen from Figure 13 that, at the corner of each side of the groove, the residual contact pressure was high because deformed tubes sank into the groove during expansion. Both the tube and the cladding layer were titanium materials, and the rebound Figure 13 depicts the distribution of residual contact pressure along the axial path at the position of circumfluence 0° when the expansion pressure was 280 MPa. Only two cases are consideredin Figure 13: a single groove in the cladding and a single groove in can be Figure that, the corner side groove, residual contact pressure was high because deformed tubes sank into the groove during expansion. Both the tube and the cladding layer were titanium materials, and the rebound Figure 13 depicts the distribution of residual contact pressure along the axial path at the position of circumfluence 0◦ when the expansion pressure was 280 MPa. Only two cases are considered in Figure 13: a single groove in the cladding and a single groove in the base. It can be seen from Figure 13 that, at the corner of each side of the groove, the residual contact pressure was high because deformed tubes sank into the groove during expansion. Both the tube and the cladding layer were titanium materials, and the rebound step was

consistent. Therefore, the maximum residual contact pressure between the titanium tube and the titanium cladding layer was greater than that between the titanium tube and the Q345R base layer. The titanium pipe sank more deeply in the cladding groove than in the base groove, and the pullout force was correspondingly greater. step was consistent. Therefore, the maximum residual contact pressure between the titanium tube and the titanium cladding layer was greater than that between the titanium tube and the Q345R base layer. The titanium pipe sank more deeply in the cladding groove than in the base groove, and the pullout force was correspondingly greater. nium tube and the titanium cladding layer was greater than that between the titanium tube and the Q345R base layer. The titanium pipe sank more deeply in the cladding groove than in the base groove, and the pullout force was correspondingly greater.

step was consistent. Therefore, the maximum residual contact pressure between the tita-

Interface

**Figure 13.** Axial distribution of residual contact pressure. **Figure 13.** Axial distribution of residual contact pressure. **Figure 13.** Axial distribution of residual contact pressure.

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#### *4.2. Effect of Groove Width 4.2. Effect of Groove Width 4.2. Effect of Groove Width*

Figure 14 shows the influence of groove width on joint performance when the expansion pressure was 280 MPa, the groove depth was 0.5 mm, and four grooving locations were considered. If the expansion length allowed, the groove width was as large as possible. It can be seen from the results in Figure 14 that the groove width had a great influence on *p*c\* . With the increase in groove width, *p*c\* showed an overall upward trend. When the groove width was 10 mm, *p*c\* approached the maximum, and the optimal groove width for uniform expansion was generally 8–10 mm [13]. It is worth noting that, when the groove width was 4 mm, *p*c\* increased significantly and peaked earlier. In this paper, the expansion pressure of 290–320 MPa was also studied. The results show that a larger *p*c\* was obtained when the groove width was 4 mm. Figure 14 shows the influence of groove width on joint performance when the expansion pressure was 280 MPa, the groove depth was 0.5 mm, and four grooving locations were considered. If the expansion length allowed, the groove width was as large as possible. It can be seen from the results in Figure 14 that the groove width had a great influence on *p*c \* . With the increase in groove width, *p*<sup>c</sup> \* showed an overall upward trend. When the groove width was 10 mm, *p*<sup>c</sup> \* approached the maximum, and the optimal groove width for uniform expansion was generally 8–10 mm [13]. It is worth noting that, when the groove width was 4 mm, *p*<sup>c</sup> \* increased significantly and peaked earlier. In this paper, the expansion pressure of 290–320 MPa was also studied. The results show that a larger *p*<sup>c</sup> \* was obtained when the groove width was 4 mm. Figure 14 shows the influence of groove width on joint performance when the expansion pressure was 280 MPa, the groove depth was 0.5 mm, and four grooving locations were considered. If the expansion length allowed, the groove width was as large as possible. It can be seen from the results in Figure 14 that the groove width had a great influence on *p*c\* . With the increase in groove width, *p*c\* showed an overall upward trend. When the groove width was 10 mm, *p*c\* approached the maximum, and the optimal groove width for uniform expansion was generally 8–10 mm [13]. It is worth noting that, when the groove width was 4 mm, *p*c\* increased significantly and peaked earlier. In this paper, the expansion pressure of 290–320 MPa was also studied. The results show that a larger *p*c\* was obtained when the groove width was 4 mm.

**Figure 14.** Effects of groove width on residual contact pressure. Groove width, *w*1 and *w*2 /mm **Figure 14.** Effects of groove width on residual contact pressure. **Figure 14.** Effects of groove width on residual contact pressure.

The reasons were as follows: in the expansion process, the external surface of the tube and the grooved structure always contacted from the edge of the groove, followed by the bottom of the groove, and finally filled the side of the groove. When the groove width was 2 mm, a small part of the titanium tube "sank" into the groove, the concentrated line contact pressure was generated between the tube wall and the groove edge, and the residual contact pressure after unloading increased compared with that without grooving. When the width of the groove increased to 4 mm, the "sinking" effect was greater. However, because the slot width was smaller, the middle of the depression was not in contact with the bottom of the groove, and the titanium tube was close to the edge of the groove. At The reasons were as follows: in the expansion process, the external surface of the tube and the grooved structure always contacted from the edge of the groove, followed by the bottom of the groove, and finally filled the side of the groove. When the groove width was 2 mm, a small part of the titanium tube "sank" into the groove, the concentrated line contact pressure was generated between the tube wall and the groove edge, and the residual contact pressure after unloading increased compared with that without grooving. When the width of the groove increased to 4 mm, the "sinking" effect was greater. However, because the slot width was smaller, the middle of the depression was not in contact with the bottom of the groove, and the titanium tube was close to the edge of the groove. At The reasons were as follows: in the expansion process, the external surface of the tube and the grooved structure always contacted from the edge of the groove, followed by the bottom of the groove, and finally filled the side of the groove. When the groove width was 2 mm, a small part of the titanium tube "sank" into the groove, the concentrated line contact pressure was generated between the tube wall and the groove edge, and the residual contact pressure after unloading increased compared with that without grooving. When the width of the groove increased to 4 mm, the "sinking" effect was greater. However, because the slot width was smaller, the middle of the depression was not in contact with the bottom of the groove, and the titanium tube was close to the edge of the groove. At this time, the *p*<sup>c</sup> \* of the outer wall of the tube at the corresponding groove corner increased

significantly. When the groove width was 6 mm and 8 mm, the titanium tube and the bottom of the groove contacted, and the corresponding position of the tube outer wall generated residual contact pressure. At the same time, "bottoming" led to a decrease in the titanium tube and groove edge line contact pressure, and the overall effect was a slight decrease in *p*<sup>c</sup> \* . When the groove width was 10 mm, with the increase in contact area between the titanium tube and the groove bottom, the contact was more sufficient, and the *p*<sup>c</sup> \* increased. When the groove width was greater than 10 mm, *p*<sup>c</sup> \* showed a slight downward trend. this time, the *p*c\* of the outer wall of the tube at the corresponding groove corner increased significantly. When the groove width was 6 mm and 8 mm, the titanium tube and the bottom of the groove contacted, and the corresponding position of the tube outer wall generated residual contact pressure. At the same time, "bottoming" led to a decrease in the titanium tube and groove edge line contact pressure, and the overall effect was a slight decrease in *p*c\* . When the groove width was 10 mm, with the increase in contact area between the titanium tube and the groove bottom, the contact was more sufficient, and the

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The elastic modulus of titanium was small, and the titanium tube could easily "sink" into the groove. Before contact with the bottom of the groove, the titanium tube contacted the edge line of the groove. The groove edge produced a high radial force on the outer surface of the titanium tube, which corresponded to a large *p*<sup>c</sup> \* . Figure 15 shows the radial force of the outer surface of the titanium tube at the joint with a single groove in the base layer only, and groove widths of 2 mm, 4 mm, 6 mm, and 8 mm. It can be observed that, when the groove width was 4 mm, the radial force was the maximum. *p*c\* increased. When the groove width was greater than 10 mm, *p*c\* showed a slight downward trend. The elastic modulus of titanium was small, and the titanium tube could easily "sink" into the groove. Before contact with the bottom of the groove, the titanium tube contacted the edge line of the groove. The groove edge produced a high radial force on the outer surface of the titanium tube, which corresponded to a large *p*c\* . Figure 15 shows the radial force of the outer surface of the titanium tube at the joint with a single groove in the base layer only, and groove widths of 2 mm, 4 mm, 6 mm, and 8 mm. It can be observed that, when the groove width was 4 mm, the radial force was the maximum.

**Figure 15.** Effects of groove width on radial force at groove edge. (**a**) 2 mm; (**b**) 4 mm; (**c**) 6 mm; (**d**) 8 mm. **Figure 15.** Effects of groove width on radial force at groove edge. (**a**) 2 mm; (**b**) 4 mm; (**c**) 6 mm; (**d**) 8 mm.

As shown in Figure 16, in general, the pullout force increased with the groove width.

This is because, with the increase in groove width, the heat exchange tube "sank" into the groove more fully, and the shear force against the tube wall increased; hence, the pullout force increased. However, when the groove width continued to increase, equivalent to the increase in the initial expansion gap, the pullout force declined. As shown in Figure 16, in general, the pullout force increased with the groove width. This is because, with the increase in groove width, the heat exchange tube "sank" into the groove more fully, and the shear force against the tube wall increased; hence, the pullout force increased. However, when the groove width continued to increase, equivalent to the increase in the initial expansion gap, the pullout force declined. *Materials* **2023**, *16*, x FOR PEER REVIEW 12 of 14

Figure 17 represents the influence of the distance from the single groove in the base

in the base layer *w*2 was 10 mm. As depicted in Figure 17, with the increase in distance

engineering practice, the distance from the single groove in the base layer to the interface

**Figure 17.** Effects of groove distance from the single groove in base to interface on residual contact

clad groove *w*1 was 6 mm, the width of the double groove in the base layer *w*2 was 10 mm, and the distance from the first groove in the base layer to the interface was 4 mm. As can

Figure 18 shows the influence of double groove space b on *p*c\*

be seen from the figure, the double groove space *b* had little influence on *p*c\*

exhibited the highest value.

**Figure 18.** Effects of groove space on residual contact pressure of joints.

2 4 6 8 10

*p*i = 280 MPa

*p*i = 290 MPa

*p*i = 300 MPa

*p*i = 310 MPa

*p*i = 320 MPa

groove space, *b*/mm

can be appropriately increased to improve the performance of the expanded joint.

, where *w*1 was 6 mm and the width of the single groove

showed an upward trend. In

. Here, the width of the

. When the

**Figure 16.** Effects of groove width on pullout force. **Figure 16.** Effects of groove width on pullout force.

*4.3. Effects of Groove Location in the Base* 

4 8 12 16 20

Groove distance, *s*/mm

pressure of joints.

20

Residual contact pressure,

*p*\*

c/MPa

22

24

Residual contact pressure,

*p*\*

c/MPa

26

28

*p*i = 280 MPa

*p*i = 290 MPa

*p*i = 300 MPa

*p*i = 310 MPa

*p*i = 320 MPa

spacing was 6 mm, *p*c\*

from the single groove in the base layer to the interface, *p*c\*

#### *4.3. Effects of Groove Location in the Base 4.3. Effects of Groove Location in the Base 4.3. Effects of Groove Location in the Base*

Pull-out force, *F*/kN

Pull-out force, *F*/kN

**Figure 16.** Effects of groove width on pullout force.

**Figure 16.** Effects of groove width on pullout force.

0 2 4 6 8 10 12 14

Groove width, *w*1 and *w*2 /mm

0 2 4 6 8 10 12 14

 A'-1 groove in the cladding layer only B'-1 groove in the base layer only C'-1 groove in the cladding layer & 1 groove in the base layer D'-1 groove in the cladding layer & 2 grooves in the base layer

 A'-1 groove in the cladding layer only B'-1 groove in the base layer only C'-1 groove in the cladding layer & 1 groove in the base layer D'-1 groove in the cladding layer & 2 grooves in the base layer

Groove width, *w*1 and *w*2 /mm

*Materials* **2023**, *16*, x FOR PEER REVIEW 12 of 14

*Materials* **2023**, *16*, x FOR PEER REVIEW 12 of 14

Figure 17 represents the influence of the distance from the single groove in the base layer to the clad interface on *p*<sup>c</sup> \* , where *w*<sup>1</sup> was 6 mm and the width of the single groove in the base layer *w*<sup>2</sup> was 10 mm. As depicted in Figure 17, with the increase in distance from the single groove in the base layer to the interface, *p*<sup>c</sup> \* showed an upward trend. In engineering practice, the distance from the single groove in the base layer to the interface can be appropriately increased to improve the performance of the expanded joint. Figure 17 represents the influence of the distance from the single groove in the base layer to the clad interface on *p*c\* , where *w*<sup>1</sup> was 6 mm and the width of the single groove in the base layer *w*<sup>2</sup> was 10 mm. As depicted in Figure 17, with the increase in distance from the single groove in the base layer to the interface, *p*c\* showed an upward trend. In engineering practice, the distance from the single groove in the base layer to the interface can be appropriately increased to improve the performance of the expanded joint. Figure 17 represents the influence of the distance from the single groove in the base layer to the clad interface on *p*c\* , where *w*<sup>1</sup> was 6 mm and the width of the single groove in the base layer *w*<sup>2</sup> was 10 mm. As depicted in Figure 17, with the increase in distance from the single groove in the base layer to the interface, *p*c\* showed an upward trend. In engineering practice, the distance from the single groove in the base layer to the interface can be appropriately increased to improve the performance of the expanded joint.

Figure 18 shows the influence of double groove space b on *p*c\* . Here, the width of the clad groove *w*<sup>1</sup> was 6 mm, the width of the double groove in the base layer *w*<sup>2</sup> was 10 mm, and the distance from the first groove in the base layer to the interface was 4 mm. As can be seen from the figure, the double groove space *b* had little influence on *p*c\* . When the spacing was 6 mm, *p*c\* exhibited the highest value. Figure 18 shows the influence of double groove space b on *p*<sup>c</sup> \* . Here, the width of the clad groove *w*<sup>1</sup> was 6 mm, the width of the double groove in the base layer *w*<sup>2</sup> was 10 mm, and the distance from the first groove in the base layer to the interface was 4 mm. As can be seen from the figure, the double groove space *b* had little influence on *p*<sup>c</sup> \* . When the spacing was 6 mm, *p*<sup>c</sup> \* exhibited the highest value. Figure 18 shows the influence of double groove space b on *p*c\* . Here, the width of the clad groove *w*<sup>1</sup> was 6 mm, the width of the double groove in the base layer *w*<sup>2</sup> was 10 mm, and the distance from the first groove in the base layer to the interface was 4 mm. As can be seen from the figure, the double groove space *b* had little influence on *p*c\*. When the spacing was 6 mm, *p*c\* exhibited the highest value.

**Figure 18.** Effects of groove space on residual contact pressure of joints. **Figure 18.** Effects of groove space on residual contact pressure of joints. **Figure 18.** Effects of groove space on residual contact pressure of joints.

#### **5. Conclusions**

In this study, the finite element method was used to explore the influence of different expansion parameters on the performance of the expanded joint between a titanium–steel tubesheet and a titanium tube. The difference between a Q345R tubesheet and TA2-Q345R tubesheet in tight expansion was analyzed, and the effects of groove position and width on residual contact pressure *p*<sup>c</sup> \* and pullout force *F* in the TA2-Q345R tubesheet expanded joint were analyzed. The following conclusions could be drawn:


the cladding layer was higher than that in the base layer. When the number of grooves in the base layer was two, the optimal *p*<sup>c</sup> \* and *F* could be obtained.

(3) The expansion performance of the joint was gradually enhanced with the increase in groove width within the range of 2–14 mm. When the groove width was 4 mm, the residual contact pressure increased significantly due to the higher radial force of the groove edge. The position of the groove had no obvious effect on the joint performance in the base layer or in the cladding layer.

**Author Contributions:** Conceptualization, J.L. (Jia Li) and C.Z.; data curation, J.L. (Jia Li) and J.L. (Juan Li); formal analysis, J.L. (Jia Li); funding acquisition, C.Z.; investigation, J.L. (Jia Li); methodology, J.L. (Jia Li) and C.Z.; project administration, C.Z.; resources, C.Z.; software, J.L. (Jia Li); supervision, C.Z.; validation, J.L. (Jia Li); visualization, J.L. (Jia Li), J.L. (Juan Li), and Y.Z.; writing—original draft, J.L. (Jia Li); writing—review and editing, J.L. (Jia Li), J.L. (Juan Li) and Y.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant numbers 51975271 and 51475223).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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**Omar Alrayes <sup>1</sup> , Carsten Könke <sup>1</sup> and Khader M. Hamdia 2,\***


**\*** Correspondence: hamdia@ikm.uni-hannover.de

**Abstract:** In quasi-brittle materials such as concrete, numerical methods are frequently used to simulate the crack propagation for monotonic loading. However, further research and action are required to better understand the fracture properties under cyclic loading. For this purpose, in this study, we present numerical simulations of mixed-mode crack propagation in concrete using the scaled boundary finite element method (SBFEM). The crack propagation is developed based on a cohesive crack approach combined with the thermodynamic framework of a constitutive concrete model. For validation, two benchmark crack-mode examples are modelled under monotonic and cyclic loading conditions. The numerical results are compared against the results from available publications. Our approach revealed good consistency compared to the test measurements from the literature. The damage accumulation parameter was the most influential variable on the loaddisplacement results. The proposed method can provide a further investigation of crack growth propagation and damage accumulation for cyclic loading within the SBFEM framework.

**Keywords:** mixed mode crack propagation; cohesive zone method; cyclic loading; SBFEM

**Citation:** Alrayes, O.; Könke, C.; Hamdia, K.M. A Numerical Study of Crack Mixed Mode Model in Concrete Material Subjected to Cyclic Loading. *Materials* **2023**, *16*, 1916. https://doi.org/10.3390/ ma16051916

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 5 February 2023 Revised: 19 February 2023 Accepted: 22 February 2023 Published: 25 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

## **1. Introduction**

The application of fatigue fractures is essential in analysing the performance of concrete structures. In fracture mechanics, concrete discontinuities also have the most significant investigation in the field of engineering [1,2]. To better understand the rapid failure of concrete structures under cyclic loading, a detailed procedure of fatigue crack propagation is required. The prediction of the direction of crack propagation and orientation of quasi-brittle material as concrete is essential for the robust and reliable design of concrete structures.

In concrete material, modelling of crack propagation and the numerical simulation of crack growth remains an outstanding issue and a critical topic of ongoing research. Primarily, the finite element technique is mainly used to simulate the crack behaviour numerically. Still, discontinuities in material simulation cannot be fully demonstrated, since the finite element method (FEM) is based on a continuum approach.

The cracks are typically mapped by areas of high strain rates when using the smeared crack approach, as in Ref. [3]. The division of the crack opening into an equivalent element length of a finite element causes the effect of smeared crack formation. This method has a drawback in that it cannot accurately reflect the actual fracture pattern because the distortion and discontinuity in the displacement field are not mapped. Alternately, discontinuities are added at the element edges in the discrete crack approach [4]. This method is affiliated with a high numerical effort since each iteration step has a continuous re-meshing process.

Based on the extensions of the conventional FEM, cohesive numerical approaches in modelling crack propagation have been developed to avoid this disadvantage [5–8]. Particular crack tip components were created to reduce the mesh quality essential for crack

simulation by the FEM [9]. However, many difficulties have been reported by Ref. [10] for material modelling using the FEM framework. The nodal displacements for finite elements at the crack tip should be omitted when calculating the stress intensity factor (SIFs). To determine the crack propagation path, several theories have been put forward [11]. Since it has an approximate explicit solution for the crack growth direction (*θ*) as a function of the stress intensity factor under pure tension (*K<sup>I</sup>* ) and mixed-mode condition (*KI I*), the maximum tangential stress criterion is frequently used in FEM simulations of cyclic crack propagation. In this case, it should be noted that the procedure of mesh refinement is typically needed in the vicinity of the crack tip. The global or local re-meshing technique is the most standard method to describe monotonic and cyclic crack propagation under linear elastic fracture mechanics (LEFM) assumptions [11].

The cohesive zone model of Ref. [12] is most commonly used to model the process zone. The process zone is often modelled in FEM utilizing the zero-thickness interface elements. Interface elements are used in a variety of modelling techniques in the literature, such as placing them along the crack paths [13,14], inserting them along all element interfaces in the mesh [15–17], and placing them along the crack surfaces as the crack propagates [18–20]. While some methods are derived based on a priori information of the crack paths obtained from experiments [13,14,21], sophisticated re-meshing algorithms to propagate the crack with high mesh densities or particular finite elements were implemented to model the singular stress fields around crack tips [22,23]. High mesh densities are needed to achieve smooth and precise predicted crack paths, even though the methods developed in Refs. [15–17,24] do not. An additional nodal enrichment with special stress functions was included by using the extended finite element method (XFEM) to simulate singular stress zones around crack tip [25]. The same appealing property that does not require re-mesh to describe crack propagation is shared by XFEM and embedded crack models. The cohesive tractions at the crack edges are included in the governing equations to account for their work. Many research investigations have discussed cohesive crack propagation for statics and dynamics issues using XFEM [26,27].

Meanwhile, the scaled boundary finite element method (SBFEM) has been proposed recently to facilitate the dilemma of the computation burdens [28–31]. The SBFEM is a very efficient method in solving problems with unbounded media and singularities. The method's effectiveness in handling singularities and unbounded domain problems has prompted researchers to extend its applications to solve diverse problems in various engineering fields, such as fracture mechanics [32,33], dam reservoir interaction [34], electromagnetic [35] and image-based analysis [36]. In order to minimize computation costs, the SBFEM assigns no discretization of side-face boundaries [33]. Using polygon elements created by the SBFEM, Ref. [37] has developed an automatic LEFM-based crack propagation modelling technique. By utilizing SBFEM's appealing feature, the particular stress fields near crack tips were analytically represented [28]. The SBFEM has shown a considerable efficiency compared to the classical FEM in calculating the stress singularities [38]. Cracktip mesh refinement, as in FEM, is avoided since the SBFEM calculates SIFs from the stress solutions at the edges of subdomains and the nodes on the domain boundary. Furthermore, a domain can be divided into subdomains in any required way, and the accuracy of the stress and SIFs solutions is specified based on the re-meshing procedure. Accordingly, this feature is more flexible in simulation crack propagation than in the FEM. In addition, the re-meshing procedure can be as simple as used in the boundary element method (BEM). Egger et al. [39] examined the computational efficiency of the SBFEM for solving linear elastic fracture mechanics problems. A comparison between the SBFEM, Extended finite element method (XFEM), and FEM was constructed by introducing different examples for calculating the SIFs. The output showed that the SBFEM reached the exact solution faster than XFEM. An extended finite element method by Ref. [40] was developed to simulate nonlinear dynamic analysis. A direct remeshing algorithm for crack propagation has been obtained in quasi-brittle materials. However, more investigations are required in modelling crack propagation for concrete under cyclic and fatigue loading. As the singularities of

cracks in the material interface are analytically calculated, the initial crack and the stress state can be easily defined.

In this study, the SBFEM framework developed in Ref. [32] is implemented to model cohesive crack propagation in quasi-brittle materials under cyclic loading. A new constitutive model based on Ref. [41] is implemented to model the propagating cracks that depend on the direction of applied loads. This model introduces an efficient simulation of concrete material under cyclic behaviour. At sub-critical loading levels, the model relies on the cumulative measure of propagation as a key damage-driving mechanism. The results of the proposed approach are validated against the methods in Refs. [42,43]. Concerning the concrete elements' fracture and material response, analyses are performed using the thermodynamic constitutive material law for concrete. The material model assumes a combination of plasticity and damage theory in Refs. [44–46]. Similarly, as in Ref. [47] the proposed method aims to simulate the crack propagation of concrete under cyclic loading. However, we further extend the SBFEM framework to simulate the mixed mode crack damage behaviour under various loading scenarios. As results, two mixed-mode crack propagation problems are modelled for monotonic and cyclic loading. The results are discussed and compared with the data available in publications.

The paper is structured as follows. In Section 2, the principle of the constitutive material model at the cohesive interface element is explained. In addition, the constitutive material model at the cohesive interface element for mixed-mode material response is investigated. Section 3 introduces the proposed mixed-mode crack procedure for concrete material using SBFEM. Additionally, the re-meshing procedure of crack propagation is introduced. Section 4 introduces a nonlinear crack model for cohesive interactions, and a flowchart for solving the SIFs is given. In Section 5, two numerical simulations are modelled to validate the nonlinear model. Based on the findings of the cyclic bending test of plain concrete that were published in the literature, we provide the calibration and validation of the proposed model. The effect of the loading sequence on the material's stiffness was the main focus of the numerical investigations.

#### **2. Constitutive Relation under Cyclic Loading**

The advanced material models define a direct relationship between the invariant of the strain and stress by linking the damage evolution with the strain, as in Refs. [10,48,49]. In order to reflect the opening/closure and growth of the micro-cracks and/or the frictional sliding along their length, the formulation of the dissipative mechanisms has been refined by introducing the internal sliding strain as a damage-driving variable within the framework of isotropic damage and internal sliding strain. This way, a unified model for monotonic and low cycle fatigue loads was proposed by Refs. [50,51].

Comparing the thermodynamic softening law of the constitutive model for the fracture, the proposed model has the ability to simulate the plastic deformation of the experimental results based on the plasticity and damage variables. Therefore, a numerical approach is introduced in this work to consider monotonic and cyclic behaviour. The proposed approach considers the cumulative measure of slip as an essential damage-driving mechanism at the subcritical loading levels, as illustrated in Figure 1. The constitutive behaviour of the embedded interface elements that represent the fracture process zone has been identified utilizing the thermodynamic-based interface model [1,47].

A cumulative measure of the inelastic displacement inside the interface governs the damaged evolution of concrete material. The evolution law in Equation (1) introduces the cumulative opening and/or sliding as the fundamental source of damage. The adopted failure criterion is identified based on the evolution law of the threshold function of the damage plasticty law, as described in Ref. [41]. In a manner similar to how Lemaitre's damage potential was presented in Ref. [52], this feature has been introduced through the modified flow potential in Refs. [50,53].

**w**

**Figure 1.** Modelling approach of the cohesive concrete zone for both crack opening and sliding.

$$
\dot{\omega} = (1 - \omega)^{c+1} \left(\frac{\chi}{S}\right)^r \left(\frac{\sigma\_n}{\sigma\_n - m\sigma\_0}\right) |\dot{u}^P| \tag{1}
$$

with

$$\dot{\mu}^P = \frac{\dot{\lambda}}{1 - \omega} \text{sign}(\mathcal{o}\_n - X) \tag{2}$$

where *u*˙ *P* is a representation of the relative displacement at the interface (i.e., opening displacement *w* = *COD* in the normal direction and slip *s* = *CSD* in the tangential/ shear direction, see Figure 1), *Y* is the energy release rate related to the damage mechanism, and *X* represent the thermodynamic force. The state variables of the damage variable *ω*, *S* are the damage strength parameter, and *c*,*r* are the exponential parameters controlling the accumulation rate of the damage. The *σ*˜*n*, *σ*<sup>0</sup> and *σ<sup>n</sup>* are the effective stress limits.

The introduced material model has been integrated as an implicit time-stepping method into the Scaled Boundary Finite Element Framework. The return mapping procedure is used to correct the internal variables after an elastic trial step, and the incremental multiplier at each time step is computed numerically from the consistency condition, ˙ *f* . The exact process has been applied to the finite element framework by Ref. [41].

The incremental value ∆*λ* can be obtained by substituting the evolution equations into the consistency condition as in Refs. [47,51].

$$
\Delta\lambda = \frac{f\_{n+1}^{\text{trial}}}{E/(1-\omega\_n) + \gamma + K} \tag{3}
$$

where *E* is the elastic stiffness, *K* and *γ* represent the isotropic and kinematic hardening moduli, respectively. Due to the implicit form of the damage evolution Equation (1), the iterative Newton scheme is applied to identify an admissible state. For fatigue and cyclic simulations, this might be too expensive. As a result, we adopt the assumption of a damage quasi-constant over a time step in Refs. [41,54], which significantly speeds up the simulation without sacrificing accuracy. The proposed model needs a consistent algorithmic stiffness to ensure a reliable and effective numerical implementation. The algorithmic stiffness establishes a relationship between the rates of stress and displacement by

$$
\dot{\sigma} = (E^{alg})\dot{\mathfrak{u}}\tag{4}
$$

The stress rate then can be expressed as

$$
\sigma = (1 - \omega)E(\boldsymbol{\upmu} - \boldsymbol{\upmu}^P) - \boldsymbol{\upomega}E(\boldsymbol{\upmu} - \boldsymbol{\upmu}^P) \tag{5}
$$

The algorithmic stiffness is obtained by substituting the evolution equations for damage and displacement with the incremental multiplier in Equation (3) as

$$\begin{split} E^{alg} &= (1 - \omega)E - \frac{(1 - \omega)E^2}{E + (\gamma + K)(1 - \omega)} \\ &- \frac{(1 - \omega)^c E^2 (u - u^p) (\frac{\overline{\sigma}}{\overline{\sigma} - m\overline{\sigma}\_0}) (\frac{Y}{S})^r \text{sign}(\hat{\sigma}\_{n+1}^{p, \text{trial}} - \gamma u\_n)}{(E/(1 - \omega) + \gamma + K)} \end{split} \tag{6}$$

Following the formulation of the equilibrium condition on a zero-thickness element of the interface, the described material model is embedded into the initial boundary value problem of the SBFEM in a usual manner.

#### **3. Modelling Crack Propagation**

#### *3.1. Crack Tip Stress Field in the Presence of Cohesive Traction*

Fracture in quasi-brittle materials such as concrete involves a process zone [55]. The numerical models and simulations consider the cracking phenomenon that can be detected physically. Due to surface friction and aggregate interlocking, normal and shear tractions can be transferred across crack surfaces. In this study, the interface elements are utilized to model the cohesive cracks that result from mixed-mode loading scenarios. Figure 2 illustrates a typical bounded domain of an interface element at the crack tip. The crack displacement in Figure 2b along the interface elements consists of crack opening displacement (*COD*) and crack sliding displacement (*CSD*). The nonlinear cohesive tractions for normal traction and tangential cohesion are *σ* and *τ*, respectively.

**Figure 2.** Crack propagation of interface SBFEM element: (**a**) interface cohesive model in SBFEM, (**b**) distribution of the cohesive forces.

The governing equations of SBFEM for an element containing a crack tip with side face tractions is motivated by the works in Ref. [38]. Along the radial lines, *ξ* nodal displacement functions *u*(*ξ*) are used, while the displacement functions in the *η* direction are interpolated by the shape functions [*N*(*η*)]. The displacement field *u*(*ξ*, *η*) is scaled as boundary coordinates and expressed including the normal displacement modes and the sideface displacement modes as

$$\{u(\mathfrak{f},\eta)\} = [N(\eta)] \sum\_{i=1}^{N+M} c\_i \mathfrak{f}^{(\overline{\lambda\_i}-1)} \{\overline{\phi\_i}\} \tag{7}$$

where *φ<sup>i</sup>* is the side-face load mode, *c<sup>i</sup>* is the integration constants, and *λ<sup>i</sup>* is the eigenvalue matrix.

The stress field can be calculated in the presence of cohesive traction as

$$\{\sigma(\xi,\eta)\} = \sum\_{i=1}^{N+M} c\_i \xi^{(\overline{\lambda\_i}-1)} \{\psi\_i(\eta)\} \tag{8}$$

where each term in Equation (8) can be interpreted as a stress mode and

$$\{\psi\_i(\eta)\} = [D](\overline{\lambda\_i}[B\_1(\eta)] + [B\_2(\eta)])\{\overline{\Phi\_i}\} \tag{9}$$

where [*D*] is the material constitutive matrix [43], [*B*1(*η*)] and [*B*2(*η*)] are the SBFEM straindisplacement matrices. In addition, it indicates that *N* is the number of displacement modes and stress modes, where an extra *M* are added to both fields when the cohesive traction is considered.

The stress intensity factors for the homogeneous material square root singular problem are defined as

$$\begin{Bmatrix} \begin{Bmatrix} K\_I \\ K\_{II} \end{Bmatrix} = \sqrt{2\pi L\_0} \sum\_{i=I,II} \left( c\_i \begin{Bmatrix} \xi^{-\lambda\_i - 1} \sigma\_{yy} |\_{\theta=0} \\ \xi^{-\lambda\_i - 1} \sigma\_{xy} |\_{\theta=0} \end{Bmatrix} \right)\_i \tag{10}$$

where *L*<sup>0</sup> = *L*<sup>3</sup> is the distance between the crack tip and the point A at the crack surface direction on the boundary, see Figure 2a. *c<sup>i</sup>* are integration constants. As *ξ* → 0, two modes can yield singular stresses with *λ<sup>i</sup>* = 0.5 . These two stress modes will be considered as mode I and mode II.

#### *3.2. Crack Propagation*

The crack initiation in the SBFEM domain is determined according to the zero-K condition [56]. Once the stress at the crack tip is finite, a cohesive crack will propagate. Any crack that satisfies the zero-K condition at the end of each load step will be identified using the SBFEM-based algorithm created by Ref. [57]. This hypothesis assumes that a cohesive crack will propagate if there is no stress singularity and finite stress at the crack tip. The crack propagates under the following condition

$$\mathbf{K}\_{\mathbf{I}}(\boldsymbol{\theta}) \ge \mathbf{0} \tag{11}$$

The procedure of crack propagation is illustrated in Figure 3 and is described as follows. In order to locate the new crack tip in each crack propagation step, it is necessary to first identify the crack propagation direction (Θ*c*) and the specified crack propagation length (∆*a*). Once the stress intensity factors (SIFs) have been calculated from Equation (10), the Θ*<sup>c</sup>* can be computed as in Ref. [58].

**Figure 3.** Crack propagation re-meshing procedure.

The re-meshing procedure is outlined in Figure 3a,b for one crack propagation step. Given ∆*a* and Θ*c*, the location of the new crack tip, shown in Figure 3a, is calculated and located in the cracked subdomain (point A is used to compute the SIFs). Two new vertices (*V*<sup>1</sup> and *V*<sup>2</sup> in Figure 3b) have been created from the former crack tip. Four new edges (*E*1–*E*4) are constructed along with the creation of two new subdomains (1 and 2). All edges of the newly cracked subdomain (3) must be visible from the new crack tip, and the new edges and subdomains are utilized to track the crack path.

The crack propagation criteria are examined when the external load increases. Once it is satisfied at a particular load, the crack length ∆*a* and the crack angle *θ* are utilized to pinpoint the location of the new crack tip in the mesh as in Ref. [33].

The cohesive tractions along the crack are obtained based on the condition *K<sup>I</sup>* ≥ 0. The crack subdomain is split up into standard subdomains called cell interface elements

(CIEs). The new CIEs are then coupled with SBFEM normal cells. The system stiffness matrix can be created by assembling the stiffness matrices and equivalent nodal forces of the subdomains and CIEs. The SIFs can then be calculated once the nodal displacements and cohesive tractions along the crack are identified. The material softening is represented by the constitutive material model, explained in Section 2.

#### **4. Implementation Procedure by SBFEM**

The flow chart for the numerical process is shown in Figure 4. A further explanation of the numerical procedure is presented as follows:

**Figure 4.** Key steps of the stress field domain and stress intensity factor SIFs.


4. Repeat step 3 until the structure fails and the numerical simulation is terminated. Output the necessary parameters, such as the crack propagation path, the number of cyclic loads *N*, and *CMOD* and *CMSD* displacements.

The above modelling methodology has been implemented in a computer program using MATLAB software. Figure 4 shows a proposed flowchart of the program. The preprocessing step is to define the input set of the tested problem. The constitutive law is inserted into the SBFEM framework as an interface element at the crack tip. The nonlinear consistent interface model is solved using the displacement control algorithm to obtain the post-processing findings for monotonic and cyclic loading. The cyclic damage accumulation during loading and unloading is formulated within the constitutive model.

Based on the constitutive model at the material point level, the relative displacement of the crack surface *u<sup>i</sup>* is calculated, including the opening displacement (*COD*) and the sliding displacement (*CSD*) of the crack surface. The key concept behind this method is the linear superposition of an iterative methodology applied to the relative displacement of the crack surface in order to solve and estimate the cohesive tractions on the crack surface.

The standard SBFEM solution of the stress intensity factor formula can calculate all three stress intensity factors.

$$K\_I = K\_I^P + K\_I^C \tag{12}$$

where *K<sup>I</sup>* is the total stress intensity factor and *K P I* and *K C I* are the components related to the external and cohesive forces, respectively. Thus, *K P <sup>I</sup>* > 0 when the crack expands due to the external force of the model, while *K C <sup>I</sup>* < 0 when the crack tends to close due to the cohesive force. When force balance is achieved as a result of the aspects covered by the external and cohesive forces, *K<sup>I</sup>* = 0, equivalently. Therefore, *K<sup>I</sup>* ≥ 0 can be utilized as the criterion for considering whether the crack will continue to propagate or not as in Ref. [59]. The solution of the cohesive tractions is summarized with the following steps:


$$\{t\} = [k] \{\mu\} \tag{13}$$

where *k* is the stiffness of the softening laws.

The stiffness matrix of an interface element in the local coordinate system is:

$$\left[k\_{int}\right] = \frac{A}{2} \sum\_{i=1}^{n\_{\mathcal{S}}} w\_i M\_i^{\;T} [k] M\_i \tag{14}$$

where *A* is the crack surface area, *w<sup>i</sup>* is the one-dimensional Gaussian weight, *n<sup>g</sup>* is the number of integration points, and *M<sup>i</sup>* is the linear shape function matrix [33]. Based on Equation (14) the solution of the displacement and stress equations is calculated in Equations (7) and (8), respectively;

(c) Proceed until the variation depicted in Figure 4 is consistent with the relationship between *t<sup>i</sup>* and *ui*+<sup>1</sup> .

#### **5. Numerical Simulation and Model Verification**

#### *5.1. Three-Point Bending Beam*

The mixed mode I-II in the TPB beam under monotonic and cyclic loading was predicted based on the numerical procedure presented in Figure 4. The experimental test

results were done by Ref. [60] for concrete beams under mixed mode fracture. Table 1 summarizes the material parameters of the tested concrete specimens. The geometric dimensions of the specimen; TPB specimen with cross-section height of *h* = 160 mm. The beam height was scaled to the beam length and span, as shown in Figure 5. The beam width is kept constant with *b* = 80 mm. The notch depth was set to *h*<sup>0</sup> = *h*/2.

**Figure 5.** (**a**) Loading scenarios used in the simulation. Monotonic loading (**left**) and cyclic loading (**right**); (**b**) three-point bending beam for mixed mode crack propagation.

In the analyses, two systematic sets of loading scenarios are used. The first loading scenario introduces a typical monotonically increasing loading Figure 5a. In the second loading scenario, the sequence of unloading cycles are applied in Figure 5b.


**Table 1.** The parameters of the material of the experimental test in Ref. [60].

The properties of the concrete and cohesive interface element for COD and CSD responses are listed in Table 2. The proposed constitutive model has a set of plastic parameters *γ*, *K* and the damage strength *S*, as reported in Ref. [47]. In this study, the nonidentical material response of the proposed model and the experimental data is caused by the unified parametric prediction of the calibrated material behaviour for both monotonic and cyclic loading scenarios. In this calibration, the unified parameters of monotonic and cyclic material response at the Gauss point are plotted in Figure 6. A comparison of the experimental [60] and numerical simulations is depicted in Figure 7 for the traction stress curve of the mixed-mode crack displacement under monotonic loading. It should be noted that more advanced cohesive constitutive laws with coupled normal and tangential damage evolution can be used, e.g., see Ref. [61].

**Table 2.** Model parameters for the concrete cohesive interface element.

**Figure 6.** Cohesive traction response under cyclic loading (blue lines) and monotonic loading (gray lines) at the material point level: (**a**) traction-crack opening, (**b**) traction-crack sliding.

**Figure 7.** A comparison between the experimental measurements in Ref. [60] and the modelling results of the cohesive traction for mixed mode I-II fracture.

The two-dimensional SBFEM modelling was utilized to establish the mesh of the TPB beam. A total of 205 elements were used. The mesh refinement near the crack tip was refined, as depicted in Figure 8a. Based on COD and CSD derived from the SBFEM calculation, the SIFs were computed. Eventually, the complete mixed mode I-II cyclic and monotonic crack propagation in TPB beam was simulated, as illustrated in Figure 8b. The crack propagation due to the increasing load is shown in Figure 8b. The results showed a curved crack path in the direction of the point of the external load (F). The distribution of the traction forces is shown in Figure 8c.

(**a**)

(**b**)


(**c**)

**Figure 8.** Predicted final crack paths of a TPB beam; (**a**) SBFEM mesh with 313 elements and boundary conditions, (**b**) predicted final crack path, (**c**) cohesive traction distribution.

To demonstrate the effectiveness of the numerical method, the SBFEM results of test simulation were compared with the experimental results in Ref. [60]. Figure 9 shows the comparison of the crack propagation paths for monotonic loading, where the shaded region contains the experimentally measured crack paths. As can be shown, there is a reasonable agreement between the numerically predicted paths in SBFEM and both numerical FEM, as well as the experimental results in Ref. [60].

**Figure 9.** Predicted final crack paths of a TPB beam; (**a**) experimental shadow results in Ref. [60], (**b**) SBFEM simulation.

Figure 10 compares the predicted load-crack mixed-displacement of the TBP beam with the experimental results reported by Ref. [60] under monotonic loading. The SBFEM numerical predictions' related curve is shown in Figure 10, plotted in a black dashed line. The numerical results of the load-displacement curve are in good agreement with the experimental measurements. A maximum load of 53.1 kN is obtained at CMOD of 0.027 mm. Figure 10 compares the predicted load-crack mixed-displacement of the TBP beam with the experimental results reported by Ref. [60] under monotonic loading. The SBFEM numerical predictions' related curve is shown in Figure 10, plotted in a black dashed line. The numerical results of the load-displacement curve are in good agreement with the experimental measurements. A maximum load of 53.1 kN is obtained at CMOD of 0.027 mm.

**Figure 10.** Numerical predictions of load-CMSD curves and the corresponding experimental data in Ref. [60] for the three-point bending test under monotonic loading. **Figure 10.** Numerical predictions of load-CMSD curves and the corresponding experimental data in Ref. [60] for the three-point bending test under monotonic loading.

Figure 11 shows the results of the monotonic SIFs for both mode I and mode II, where the numerically measured SIFs are plotted. In Figure 11a, the points representing the initial mesh of Figure 10 are calculated once *K<sup>I</sup>* ≥ 0. Then, the crack opens gradually based on a crack propagation criterion. The numerical calculation of *K<sup>I</sup>* by SBFEM with a fewer number of degrees of freedom (DOFs) manifests good crack trajectory predictions. Figure 11 shows the results of the monotonic SIFs for both mode I and mode II, where the numerically measured SIFs are plotted. In Figure 11a, the points representing the initial mesh of Figure 10 are calculated once *K<sup>I</sup>* ≥ 0. Then, the crack opens gradually based on a crack propagation criterion. The numerical calculation of *K<sup>I</sup>* by SBFEM with a fewer number of degrees of freedom (DOFs) manifests good crack trajectory predictions.

**Figure 11.** Mixedmode I–II crack displacement for monotonic loading: *KI*-CMOD (**a**) and *KII*-CMSD (**b**). The experimental observations and numerical calculations for a cyclically mixed-**Figure 11.** Mixedmode I–II crack displacement for monotonic loading: *K I*-CMOD (**a**) and *K I I*-CMSD (**b**).

mode loading are shown in Figure 12. The loading is controlled by the CMOD/CMSD, including eight unloading cycles. The results show that the numerical comparison of crack propagation CMSD is relatively stable while the CMOD increases as the N of the cycle increases. As a result, there is a good agreement between the experiment results and the numerical predictions of CMSD in Figure 12. The experimental observations and numerical calculations for a cyclically mixedmode loading are shown in Figure 12. The loading is controlled by the CMOD/CMSD, including eight unloading cycles. The results show that the numerical comparison of crack propagation CMSD is relatively stable while the CMOD increases as the N of the cycle

increases. As a result, there is a good agreement between the experiment results and the numerical predictions of CMSD in Figure 12.

#### *5.2. Four-Point Bending Beam*

The proposed method is next verified using the numerical results of the four-point bending concrete beam under mixed mode fracture [62]. The geometry, loads, and support conditions are illustrated in Figure 13. The width and height of the specimen are denoted by 100 × 100 [mm]. The material properties of the concrete beam are summarized in Table 3.

**Figure 13.** A single-notched concrete beam under monotonic mixed-mode loading. (**a**) Geometry, (**b**) initial mesh with boundary conditions.

(**b**)


**Table 3.** Parameters of the material in the experimental test by Ref. [63].

A particular loading condition is implemented to generate mode II crack initiation for both monotonic and cyclic loading. The calculation is performed under displacementcontrolled loading. For the analysis of the concrete beam, the following material properties are used, see Table 4 below.


**Table 4.** Model parameters for the concrete cohesive interface element.

The initial mesh of the tested beam is illustrated in Figure 13. The mesh consists of 1069 elements. The crack propagation length of ∆*a* = 22 mm is adopted in the crack propagation simulation. Figure 14 compares the predicted traction point displacement response of the developed method with the numerical results of Ref. [62] in the literature. Overall, there is good agreement between the results of the developed method and the experimental measurements. The pre-peak response compares very well with the previously reported numerical results. The numerical SBFEM predicted a maximum load of 36.2 kN and is closest to both the experimental and FEM predictions of Ref. [62].

**Figure 14.** Comparison of the tested beam in Ref. [62] and the proposed cohesive traction for mixed modes I–II fracture.

Figure 15 represents the predicted crack propagation process. During the simulation, a crack propagates from the tip notch towards the loading point on the bottom surface of the four point bending beam. The paths of the crack in the experimental results and SBFEM are curved. In Figure 15, the predicted crack path of the FEM simulation in Ref. [62] seems to be more efficient, however, the mesh adaptive procedure of the SBFEM will produce more efficient results and reduce the computational time costs.

**Figure 15.** Predicted final crack paths of a tested beam; (**a**) the experimental shadow results in Ref. [62] and the numerical results and (**b**) SBFEM simulation.

Figure 16 compares the predicted load-crack displacement response of the developed method with the numerical FEM results of Ref. [62]. Overall, the results obtained from the developed method agree well with the numerical for the monotonic loading. The pre-peak response compares well with the previously published numerical results. All the numerical predictions below estimate the maximum load. The SBFEM predicted a maximum load of 34.8 kN and is closest to the FEM predictions.

**Figure 16.** Numerical predictions of load-CMSD curves and the corresponding experimental data in Ref. [62] for the tested beam under monotonic loading.

Figure 17 shows SBFEM numerical predictions and FEM results for a cyclically mixedmode loading. The loading is controlled by the CMSD, also including eight unloading cycles applied until failure. The results in Figure 17a show that the traction forces of SBFEM have a very good agreement with the numerical results in FEM. In Figure 17b, the crack displacement of the SBFEM simulation is more underestimated in comparison to the FEM measurements. The calculation of the mixed crack displacement at the material point, the number of load steps, and the democratization of the applied mesh have a significant effect on the SBFEM numerical results. However, more experimental data of cyclic mixed mode tests are required to validate the proposed numerical method.

### **6. Conclusions**

In this paper, a newly developed SBFEM numerical method for mixed mode crack propagation in concrete under cyclic loading was proposed. The proposed procedure allowed accurate SIFs to be calculated directly from the SBFEM analytical framework without more discretization at crack-tip meshes or by using singular elements, as in FEM. Comparing the thermodynamic softening law of the constitutive model for fracture, several aspects have been provided, which incorporate the loading-unloading path, the damage evolution during the load cycle, and the crack traction displacement behaviour.

The cyclic behaviour of interfaces using SBFEM has been successfully described using the damage accumulation hypothesis. The proposed method showed the ability to simulate both monotonic and cyclic behaviour of a cohesive crack interface element, e.g., concrete interface, utilizing a consistent set of material parameters. The cyclic loading simulations' output agreed well with experimental data from the literature. The proposed method performed to study the effect of fatigue loading provides promising results and establishes a damage accumulation hypothesis for the simulation of multiple cohesive cracks under 1000 load cycles.

**Author Contributions:** Methodology, O.A., C.K. and K.M.H.; Validation, O.A., C.K. and K.M.H.; Writing—original draft, O.A., C.K. and K.M.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)- Projektnummer 492535144.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Khader M. Hamdia thanks the support by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projectnumber 492535144.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Nomenclature**


#### **References**


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## *Article* **Nonlinear Finite Element Model for Bending Analysis of Functionally-Graded Porous Circular/Annular Micro-Plates under Thermomechanical Loads Using Quasi-3D Reddy Third-Order Plate Theory**

**Jinseok Kim 1,\*, Enrique Nava <sup>1</sup> and Semsi Rakici <sup>2</sup>**


**Abstract:** A nonlinear finite element model for axisymmetric bending of micro circular/annular plates under thermal and mechanical loading was developed using quasi-3D Reddy third-order shear deformation theory. The developed finite element model accounts for a variation of material constituents utilizing a power-law distribution of a two-constituent material, three different porosity distributions through plate thickness, and geometrical nonlinearity. The modified couple stress theory was utilized to account for the strain gradient effects using a single material length scale parameter. Three different types of porosity distributions that have the same overall volume fraction but different enhanced areas were considered as a form of cosine functions. The effects of the material and porosity distribution, microstructure-dependency, the geometric nonlinearity, and various boundary conditions on the bending response of functionally-graded porous axisymmetric microplates under thermomechanical loads were studied using the developed nonlinear finite element model.

**Keywords:** nonlinear finite element analysis; axisymmetric plates; quasi-3D Reddy third-order theory; functionally-graded porous materials; modified couple stress theory

## **1. Introduction**

Functionally-graded materials (FGMs) are advanced engineering materials composed of two or more constituents with a continuous variation in their compositions. Unlike FGMs, laminated composites exhibit immediate changes in thermal and mechanical properties of the constituents, resulting in stress concentrations at the interfaces where two discrete materials bond together. This leads to delamination problems and the presence of residual stresses in conventional composites working under severe conditions. FGMs were developed by researchers in Japan in 1984 to overcome these issues encountered in a thermal coating material requirement of a hypersonic space plane project [1]. Since then, FGMs have been used in various fields such as aerospace, automobile, electronic, and medical industries due to their advantages over laminated composites and their flexibility to be designed according to the needs of the application field and working environment. The reader is referred to the following review articles [2–5] for details of the historical development of these materials, manufacturing techniques, and optimization of their functionality.

The FGMs have great potential for improving the performance of various components in engineering structures, especially circular and annular plates. Over the last few decades, researchers have extensively studied the behavior of functionally-graded (FG) circular and annular plates under thermal, mechanical, and combined thermomechanical loadings. Since FGMs were initially designed to withstand extreme thermal environments, most of the literature focuses on their thermal analysis. Typical FGMs used in these studies are

**Citation:** Kim, J.; Nava, E.; Rakici, S. Nonlinear Finite Element Model for Bending Analysis of Functionally-Graded Porous Circular/Annular Micro-Plates under Thermomechanical Loads Using Quasi-3D Reddy Third-Order Plate Theory. *Materials* **2023**, *16*, 3505. https://doi.org/10.3390/ ma16093505

Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana

Received: 3 April 2023 Revised: 24 April 2023 Accepted: 28 April 2023 Published: 2 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

made from a mixture of ceramics for their low thermal conductivity and metals for their ductility and resistance to fracture caused by stresses likely to occur in high-temperature gradients. Additionally, the majority of studies on FG plates employ a power law or exponential distribution of materials through the thickness direction of the plates

In 1998, Reddy and Chin [6] conducted a numerical study to investigate thermomechanical responses of FG cylinders and plates under extreme thermal loading conditions using the first-order shear deformation plate theory (FSDT). In their study, the effects of thermomechanical coupling on the response of FGMs subjected to thermal shock were investigated. For the functionally-graded axisymmetric cylinder subjected to high thermal loading, the temperature distribution obtained from both coupled and uncoupled formulations did not show significant differences. However, it was observed that the radial stresses were more affected than the hoop stresses in the FG cylinder.

Using the FSDT, exact solutions of the static bending analysis of FG circular and annular plates having various boundary conditions were presented by Reddy et al. [7]. They derived the solutions of deflections, forces, and the moments of the FG plates based on FSDT in terms of the associated quantities for the isotropic plates based on the classical plate theory (CPT). Hence, the bending solutions of the FG circular plate became readily available whenever the CPT solution was known. Ma and Wang [8] studied the axisymmetric nonlinear bending and post buckling response of functionally-graded circular plates under thermal, mechanical and combined thermomechanical loading conditions. In this study, governing equations were derived using the von Kármán plate theory and the numerical solutions were obtained with the help of the shooting method. The results of this study showed that temperature distribution, deflection values, critical buckling temperature, and post buckling behavior of the functionally-graded circular plates were significantly affected by the volume fraction index.

Praveen and Reddy [9] introduced the finite element formulations that account for the transverse shear strains, rotary inertia, and von Kármán nonlinear strains to perform static and dynamic thermoelastic analysis of the functionally-graded ceramic-metal plates based on FSDT. In 2000, Reddy [10] presented the formulation and analytical solution of simply-supported rectangular FG plates using third order shear deformation plate theory (TSDT) including thermomechanical coupling, time dependency, and von Kármán geometric nonlinearity. From these two studies, it was concluded that the distribution of material constituents in the functionally-graded plates had a significant influence on the resulting thermoelastic response of FG plates. Najafizadeh and Heydari [11] investigated the thermal buckling analysis of functionally-graded circular plates under both uniform and non-uniform temperature changes by employing the TSDT.

Prakash and Ganapathi [12] employed the finite element method to carry out asymmetricfree vibration and thermoelastic stability analysis of functionally-graded circular plates. Nie and Zhong [13] studied the three-dimensional free and forced vibration analysis of functionally-graded circular plates and found that the lowest nondimensional frequency and circumferential wave number of the plate increased as the thickness-to-width ratio increased. They also observed that the magnitudes of the displacements and stresses became larger as the forcing frequency approached the natural frequency of the FG circular plate.

Efraim and Eisenberger [14] presented the free vibration analysis of variable thickness thick annular plates using the exact element method and the dynamic stiffness method. They used FSDT in their formulations and varied Poisson ratio according to the power law distribution in addition to elastic modulus and mass density. Golmakani and Kadkhodayan [15] presented another study that accounted for the gradation of Poisson ratio. They investigated the nonlinear bending analysis of annular FG plates based on both FSDT and TSDT. The same authors [16] later performed a large deflection analysis of circular and annular FG plates subjected to thermomechanical loading within the framework of FSDT, including von Kármán nonlinearity.

Saidi et al. [17] employed unconstrained third order shear deformation theory to analyze the axisymmetric bending and buckling behavior of thick FG circular plates.

Nosier and Fallah [18] reformulated governing equations of the FSDT into interior and edge-zone equations for functionally-graded circular plates. By introducing two sets of equations to define the edge-zone problem, they uncoupled the bending and extension equations, which made it possible to obtain analytical solutions for the asymmetric behavior of functionally-graded circular plates with various boundary conditions under mechanical and thermal loading. Later, they included the von Kármán nonlinear strains into their formulations and investigated the axisymmetric and asymmetric nonlinear bending of functionally-graded circular plates subject to linearly-varying transverse loading [19]. The axisymmetric bending analysis of FG circular plates under arbitrary transverse loads was studied by Yun et al. [20]. They obtained the analytical solutions for the FG circular plates with elastic simple and rigid slipping supports cases when the material property of the FG plate was varying with an exponential distribution. Another analytical study was conducted to solve for in-plane and out-of-plane free vibrations of thick FG circular and annular plates embedded in piezoelectric layers by Talabi and Saidi [21], employing TSDT. The effects of both electrical and mechanical boundary conditions, geometrical parameters of the plate, and in-plane displacements on the middle plane on the natural frequencies of FG circular and annular plates were discussed. Zur [ ˙ 22] applied the Neumann series method to investigate the free vibration behavior of discrete-continuous FG circular plates that may have several ring attachments such as masses, springs and damping elements.

The FG circular and annular plates can be further improved by adding porosity into their composition to decrease the weight of the structure and/or increase the insulation properties. Hence, it is important to examine the mechanical and thermal responses of FG porous plates under different loading and boundary conditions. A general solution of a porous FG circular plate that is supported by a non-uniform Kerr elastic foundation and subjected to non-axisymmetric, non-uniform shear and normal tractions, and a magnetic actuation was developed by Rad and Shariyat [23]. Their results showed that the radial displacement component was more prone to being affected by the induced magnetic actuation. Additionally, because of the presence of incompressible fluid in the pores in this study, as the porosity increased, the plates became stiffer. The buckling behavior of porous circular plate between piezoelectric layers under thermal loading was investigated by Jabbari et al. [24]. They showed that, as the porosity increased, the critical temperature decreased and the plate whose pores were saturated with fluid became unstable. On the other hand, the critical temperature of the plates can be decreased by increasing the thermal expansion coefficient of the fluid filling the pores and the piezoelectric layers. Zhao et al. [25] studied the free and forced vibration analysis of FG porous circular, annular, and sector plates with general elastic restraints using FSDT.

These extensive studies conducted on FG circular and annular plates show that these structures have an intrinsic advantage resulting from the non-homogeneity and smooth variations of the material properties. It is shown that the deflections and tensile stresses of FG circular and annular plates can be lower and critical buckling loads can be higher as compared to the homogeneous ones, depending on the predetermined variation of material properties of FG circular and annular plates. It is also possible to adjust the natural frequencies of these structures by changing the variation of the material distribution. Hence, all these conclusions make it attractive to examine the performance of the FGMs for the micro-scale structures. However, conventional continuum mechanics cannot capture the size dependency that is experimentally observed at the micro-scale [26–29]. Therefore, a higher order continuum theory is required for the accurate modeling and analysis of these structures. Couple stress theories [30–32], Erigen nonlocal elasticity theory [33] and the strain gradient elasticity theories [26,34,35] are some of the higher order continuum theories that take the size dependency into account. The modified couple stress theory is the most commonly employed theory because only a single length scale parameter is needed to include size effect.

Ke et al. [36] investigated the bending, buckling, and free vibration analyses of FG annular microplates with hinged–hinged and clamped–clamped boundary conditions. Their size-dependent annular microplate model was based on the Mindlin plate theory and the modified couple stress theory. This study showed that elastic buckling analysis was more sensitive to size effect than the free vibration analysis. Similar analyses were presented by Ansari et al. [37] for FG circular and annular microplates. They also employed Mindlin plate theory, but different to the previous study, size dependency was included using modified strain gradient elasticity theory. Both studies agreed that the smaller the dimensionless length scale parameter they had, the smaller the deflection but the higher the critical buckling load and natural frequencies that they obtained.

Reddy and Berry [38] presented the classical and the first order plate theories for axisymmetric bending of circular micro-plates including von Kármán nonlinear strains. Size dependency was captured with the modified couple stress theory. Later, Reddy et al. [39] used this theory to develop nonlinear finite element models for FG circular plates.

An analytical solution for the free vibration of FG circular and annular nanoplates was obtained by Hosseini-Hashemi et al. [40] based on Mindlin plate theory and Eringen nonlocal elasticity theory. Beni et al. [41] studied the same problem for FG cylindrical nanoshells using FSDT in conjunction with the modified couple stress theory. They presented the effects of the material length scale, distribution of the FGMs, nanotube thickness, and length on the fundamental frequencies. Eshraghi et al. [42] studied the bending and the free vibration analysis of FG annular and circular microplates subjected to thermal loading using the modified couple stress theory. They unified the displacement fields such that results for Kirchoff plate theory, Mindlin plate theory, and third order shear deformation plate theory can be generated. Additionally, not only the mechanical and thermal properties of the FG plates but also the material length scale parameter were not kept constant but were changed through the thickness direction, obeying a power law distribution. The transverse deflections, normalized circumferential and radial stresses, and the natural frequencies were presented for different thermal loading, material, and geometrical parameters. Ji et al. [43] developed a plate model capturing the size dependency for FG circular micro-plates based on the strain gradient theory of Zhou. They analyzed the bending and free vibration of a simply-supported circular micro-plate and the results were compared with those obtained by employing the strain gradient theory of Lam, the modified couple stress theory, and the CPT.

A free vibration and thermal buckling analysis of an FG porous circular micro-plate was conducted by Shojaeefard et al. [44] based on CPT and FSDT with modified couple stress theory. The effects of the temperature change, distribution of the material properties, size-dependency, and porosity on the fundamental frequencies and critical temperature were investigated. Kim et al. [45] presented the analytical solutions of bending, free vibration, and the buckling problem for FG porous micro-plates using CPT and FSDT in conjunction with the modified couple stress theory. Recently, Wang and Zhang [46] studied the thermal buckling and postbuckling responses of GPL-reinforced nanocomposite beams using the higher order shear deformation theory with temperature-dependent properties. Zhang et al. [47] carried out analytical studies on thermo-mechanical responses of porous functionally-graded, graphene-reinforced cylindrical panels based on a third order shear deformation theory. The acoustic characteristics of functionally-graded porous graphene reinforced nano composite plates (FG-PGRC) were studied by Xu et al. [48]. In their study, a higher order shear deformation theory was utilized to study the vibration and noise reduction of an FG-PGRC plate.

This study aimed to investigate the behavior of FG porous circular microplates under thermal and mechanical loadings, which has not been studied in the literature. To this end, a nonlinear finite element model was developed based on quasi-3D Reddy third-order shear deformation theory and the modified couple stress theory, taking into account von Kármán nonlinear strains to consider geometrical nonlinearity. The FGM was composed of two constituents based on a power law distribution through the thickness direction, and three different porosity profiles were considered. Parametric analyses were conducted to investigate the effects of the distribution of material properties and porosity, size-dependency, geometric nonlinearity, and different boundary conditions on the static bending analysis of FG porous circular microplates.

#### **2. Constitutive Models**

#### *2.1. Functionally-Graded Porous Materials*

The model considers isotropic axisymmetric plates composed of two constituents with varying material properties and internal porosity through the thickness, modeled using a power-law distribution and cosine variation, respectively. The typical material properties of functionally-graded porous materials (FGPM) are thus captured in the model, as shown in Equation (1).

$$P(z) = \left[ (P\_t - P\_b) f(z) + P\_b \right] (1 - \psi(z)), \qquad f(z) = \left( \frac{z}{h} + \frac{1}{2} \right)^n,\tag{1}$$

where *P<sup>t</sup>* and *P<sup>b</sup>* are material properties on the top and bottom surfaces of plates, *n* is powerindex, *f*(*z*) is a volume fraction function, and *ψ*(*z*) is a porosity distribution function. Three different types of porosity distributions were considered in this study.

$$\begin{aligned} \text{Type 1}: \psi(z) &= \phi \cos \left[ \pi \left( \frac{z}{h} \right) \right] \\ \text{Type 2}: \psi(z) &= \phi \cos \left[ \frac{\pi}{2} \left( \frac{z}{h} + 0.5 \right) \right] \\ \text{Type 3}: \psi(z) &= \phi \cos \left[ \frac{\pi}{2} \left( \frac{z}{h} - 0.5 \right) \right] \end{aligned} \tag{2}$$

where *φ* is the maximum porosity value along thickness direction. The distribution of porosity through the thickness of the plates was normalized to have the same porous volume, and it is important to investigate the effect of different porosity distributions [45]. Figure 1a displays the normalized porosity distribution throughout the thickness of the plate. Figure 1b–d show the effects of porosity distributions on the variation of typical materials properties. As an example, a porosity value was set to *φ* = 0.5, three different power-law index values *n* = 0, 0.5, and 5.0 were set. The ratio of material properties on the top and bottom surfaces was assumed to be *<sup>E</sup><sup>t</sup> Eb* = 10. The Type 1, Type 2, and Type 3 porosity distributions are a symmetric and center-enhanced, a bottom area enhanced, and a top area enhanced porosity distributions, respectively.

#### *2.2. Modified Couple Stress Theory*

The motion of the material particles in classical couple stress theory [30,49] is described to rotate the material particles in addition to forces in the classical continuum mechanics. The size-dependent effect was captured using two additional material constants in the classical couple stress theory. These two material constants are difficult to determine because of their indeterminacy. Eringen [33] proposed a micropolar theory and defined the motion of a particle using the location vector and inner product of a rigid vector. A modified couple stress theory using the concept of the representative volume elements and a higher order equilibrium condition was proposed by Yang et al. [32]. According to the modified couple stress theory, the deviatoric part of a couple stress tensor is only associated with the symmetric part of rotation gradient and it contributes to the total strain energy along with the classical strain energy. The strain energy potential of an axisymmetric plate based on the modified couple stress theory can be expressed as

$$\begin{split} \mathcal{U} &= \frac{1}{2} \int\_{V} (\boldsymbol{\sigma} : \boldsymbol{\varepsilon} + \boldsymbol{m} : \boldsymbol{\chi}) dV \\ &= \frac{1}{2} \int\_{r\_{i}}^{r\_{o}} \left[ \int\_{-\frac{h}{2}}^{\frac{h}{2}} (\boldsymbol{\sigma} : \boldsymbol{\varepsilon} + \boldsymbol{m} : \boldsymbol{\chi}) dz \right] r dr \end{split} \tag{3}$$

where *r<sup>i</sup>* and *r<sup>o</sup>* are the inner and outer radii of the plate, *σ* and *ε* are the Cauchy stress tensor and Von Kámán nonlinear strain tensor, *m* and *χ* are the deviatoric part of the symmetric couple stress tensor and the symmetric curvature tensor. Note that the differential volume element *dV* can be written as *dV* = *rdrdθdz* and 2*π* from the integration with respect to *θ* being omitted in Equation (3). The curvature tensor and the deviatoric part of the symmetric couple stress tensor are defined as [32]

$$\mathbf{x} = \frac{1}{2} \left[ \nabla \boldsymbol{\omega} + \left( \nabla \boldsymbol{\omega} \right)^{\mathsf{T}} \right] \tag{4}$$

$$
\mathfrak{m} = 2\mu \ell^2 \mathfrak{x} \tag{5}
$$

where *ω* is the rotation vector, *ω* = <sup>1</sup> <sup>2</sup>∇ × *u*, *µ* is the shear modulus, and ` is a length scale parameter.

**Figure 1.** Distribution of porosity and a typical material property [45]. (**a**) Distribution of porosity through thickness. (**b**) Distribution of typical material property (*n* = 0.5). (**c**) Distribution of typical material property (*n* = 1.0). (**d**) Distribution of typical material property (*n* = 5.0).

1

1

1

1

In this study, an isotropic linear elastic material was assumed and the stress and strain relation [50] for an axisymmetric plate is

$$
\begin{Bmatrix}
\sigma\_{rr} \\
\sigma\_{\theta\theta} \\
\sigma\_{zz} \\
\sigma\_{rz}
\end{Bmatrix} = \Lambda \begin{bmatrix}
1 - \nu & \nu & \nu & 0 \\
\nu & 1 - \nu & \nu & 0 \\
\nu & \nu & 1 - \nu & 0 \\
0 & 0 & 0 & \frac{1}{2}(1 - 2\nu)
\end{bmatrix} \begin{Bmatrix}
\varepsilon\_{rr} - a(T - T\_0) \\
\varepsilon\_{\theta\theta} - a(T - T\_0) \\
\varepsilon\_{zz} - a(T - T\_0) \\
\varepsilon\_{rz}
\end{Bmatrix},\tag{6}
$$

where Λ = *E*/[(1 + *ν*)(1 − 2*ν*)], *E* is Young's modulus , which varies along the plate's thickness, *ν* is a constant Poisson's ratio in the elastic stiffness matrix. *α* is the thermal expansion coefficient, and *T* and *T*<sup>0</sup> are the temperature at a material point and the reference temperature of the undeformed body.

The nonzero curvatures and modified couple stresses are

$$\begin{Bmatrix} \begin{Bmatrix} m\_{r\theta} \\ m\_{\theta z} \end{Bmatrix} = 2\ell^2 \mu \begin{Bmatrix} \chi\_{r\theta} \\ \chi\_{\theta z} \end{Bmatrix} . \tag{7}$$

#### **3. Quasi-3D Reddy Third-Order Plate Theory**

#### *3.1. Displacement and Strains*

The displacement field of quasi-3D Reddy third-order plate theory can be derived from an assumption of a cubic variation of in-plane displacements and a quadratic variation of deflection (i.e., out-of-plane displacement) with zero tangential traction on top and bottom surfaces. The displacement field of cubic variation of in-plane displacement and a quadratic variation of deflection through thickness direction for axisymmetric plates takes the form of

$$\begin{aligned} u\_r(r, z, t) &= u\_0(r, t) + z \theta\_r(r, t) + z^2 \phi\_r(r, t) + z^3 \psi\_r(r, t) \\ u\_z(r, z, t) &= w\_0(r, t) + z \theta\_z(r, t) + z^2 \phi\_z(r, t) .\end{aligned} \tag{8}$$

With the assumption of zero tangential traction on top and bottom surfaces, the displacement (8) can be written in the form of

$$
\varepsilon\_{rz}\left(r,\frac{h}{2},t\right) = \varepsilon\_{rz}\left(r,-\frac{h}{2},t\right) = 0.\tag{9}
$$

The form of quasi-3D Reddy third-order plate theory for axisymmetric plates takes

$$\begin{aligned} u\_{\mathbf{r}}(\mathbf{r},z,t) &= u\_0(\mathbf{r},t) + z\theta\_{\mathbf{r}}(\mathbf{r},t) - \frac{z^2}{2}\frac{\partial \theta\_z}{\partial r} - z^3 c\_1 \left[ \theta\_{\mathbf{r}}(\mathbf{r},t) + \frac{\partial \lambda(\mathbf{r},t)}{\partial r} \right] \\ u\_z(\mathbf{r},z,t) &= w\_0(\mathbf{r},t) + z\theta\_z(\mathbf{r},t) + z^2 \phi\_z(\mathbf{r},t), \end{aligned} \tag{10}$$

where *u*<sup>0</sup> is the membrane displacement, *θ<sup>r</sup>* is the rotation of a transverse normal about *θ* direction, *w*<sup>0</sup> is the deflection, *θ<sup>z</sup>* and *φ<sup>z</sup>* are the thickness stretch, *λ* = *w*<sup>0</sup> + *<sup>h</sup>* 2 4 *φ<sup>z</sup>* and *c*<sup>1</sup> = <sup>4</sup> 3*h* 2 .

Based on the assumption of small strains and moderate rotations, nonzero von Kámán nonlinear strain for the axisymmetric plate is given by [39].

$$\begin{aligned} \varepsilon\_{rr} &= \frac{\partial u\_r}{\partial r} + \frac{1}{2} \left( \frac{\partial u\_z}{\partial r} \right)^2 \\\\ \varepsilon\_{\theta\theta} &= \frac{u\_r}{r} \\ \varepsilon\_{zz} &= \frac{\partial u\_z}{\partial z} \\ \varepsilon\_{rz} &= \frac{1}{2} \left( \frac{\partial u\_r}{\partial z} + \frac{\partial u\_z}{\partial r} \right) . \end{aligned} \tag{11}$$

The non-zero strains with the displacement field (10) of quasi-3D Reddy third-order plate theory are defined as

$$
\begin{aligned}
\begin{Bmatrix}
\varepsilon\_{rr} \\
\varepsilon\_{\theta\theta} \\
\varepsilon\_{zz} \\
2\varepsilon\_{rz}
\end{Bmatrix} &= \begin{Bmatrix}
\frac{\partial u\_{0}}{\partial r} + \frac{1}{2} \left(\frac{\partial w\_{0}}{\partial r}\right)^{2} \\
& \frac{u\_{0}}{\theta} \\
& \frac{\partial}{\partial z} \\
& \theta\_{r} + \frac{\partial w\_{0}}{\partial r}
\end{Bmatrix} + z \begin{Bmatrix}
\frac{\partial \theta\_{r}}{\partial r} \\
\frac{\theta\_{r}}{\theta} \\
\frac{\theta\_{r}}{2\theta\_{\theta}} \\
0
\end{Bmatrix} - z^{2} \begin{Bmatrix}
& \frac{1}{2}\frac{\partial^{2}\theta\_{z}}{\partial r^{2}} \\
& \frac{1}{2r}\frac{\partial\theta\_{z}}{\partial r} \\
& 0 \\
\end{Bmatrix} \\ & \qquad - z^{3} \begin{Bmatrix}
& \left(\frac{1}{\partial r} + \frac{\partial\lambda}{\partial r}\right) \\
& \frac{c\_{1}}{r} \left(\theta\_{r} + \frac{\partial\lambda}{\partial r}\right) \\
& 0 \\
& 0 \\
& 0
\end{Bmatrix},
\end{aligned} \tag{12}$$

where *c*<sup>2</sup> = <sup>4</sup> *h* 2 . The symmetric part of the curvature tensor for axisymmetric plates is defined as

$$\begin{aligned} \chi\_{r\theta} &= \frac{1}{2} \left( \frac{\partial \omega\_{\theta}}{\partial r} - \frac{\omega\_{\theta}}{r} \right) \\ \chi\_{\theta z} &= \frac{1}{2} \frac{\partial \omega\_{\theta}}{\partial z} \end{aligned} \tag{13}$$

where *ω<sup>θ</sup>* = 1 2 *∂u<sup>r</sup> ∂z* − *∂u<sup>z</sup> ∂r* ! . Thus, the *χr<sup>θ</sup>* and *χθ<sup>z</sup>* in terms of the displacements in Equation (10) take the form of

$$
\begin{Bmatrix} \chi\_{\theta\theta} \\ \chi\_{\theta z} \end{Bmatrix} = \frac{1}{4} \begin{Bmatrix} \frac{\partial \theta\_r}{\partial r} - \frac{\partial^2 w\_0}{\partial r^2} - \frac{1}{r} \left( \theta\_r - \frac{\partial w\_0}{\partial r} \right) \\ -2 \frac{\partial \theta\_z}{\partial r} \end{Bmatrix} - \frac{z}{2} \begin{Bmatrix} \frac{\partial^2 \theta\_z}{\partial r^2} - \frac{1}{r} \frac{\partial \theta\_z}{\partial r} \\ c\_2 \left( \theta\_r + \frac{\partial \lambda}{\partial r} \right) + \frac{\partial \phi\_z}{\partial r} \end{Bmatrix} \\ - \frac{z^2}{4} \left\{ c\_2 \left[ \frac{\partial \theta\_r}{\partial r} + \frac{\partial^2 \lambda}{\partial r^2} - \frac{1}{r} \left( \theta\_r + \frac{\partial \lambda}{\partial r} \right) \right] + \frac{\partial^2 \phi\_z}{\partial r^2} - \frac{1}{r} \frac{\partial \phi\_z}{\partial r} \right\} . \tag{14}$$

#### *3.2. Governing Equations*

In this study, the soft-coupled thermoelastical behavior of functionally-graded porous materials was analyzed using the finite element method. The equations of equilibrium and the weak form finite element model for static bending problems of axisymmetric plate were obtained using the principle of virtual displacement.

$$\begin{split} 0 = & -\int\_{V} \left( \sigma\_{rr} \delta \varepsilon\_{rr} + \sigma\_{\theta\theta} \delta \varepsilon\_{\theta\theta} + \sigma\_{zz} \delta \varepsilon\_{zz} + 2 \sigma\_{rz} \delta \varepsilon\_{rz} + 2 m\_{r\theta} \delta \chi\_{r\theta} + 2 m\_{\theta z} \delta \chi\_{\theta z} \right) dV \\ & + \int\_{V} \left( \bar{f}\_{l} \delta u\_{i} + \mathfrak{c}\_{i} \delta \omega\_{i} \right) dV + \int\_{S} \left( \bar{t}\_{l} \delta u\_{i} + \mathfrak{s}\_{l} \delta \omega\_{i} \right) dS \\ & + \int\_{\Omega^{\sharp}} \left( q\_{l}^{t} \delta u\_{i} + p\_{i}^{t} \delta \omega\_{i} \right) d\Omega^{t} + \int\_{\Omega^{b}} \left( q\_{l}^{b} \delta u\_{i} + p\_{i}^{b} \delta \omega\_{i} \right) d\Omega^{b} . \end{split} \tag{15}$$

where *σij* and *mij* are the symmetric part of the stress tensor and the deviatoric part of the couple stress tensor. ¯ *f<sup>i</sup>* and *c*¯*<sup>i</sup>* are the body forces and couples. ¯*t<sup>i</sup>* , *s*¯*i* , and ¯*d* are the surface forces and couples on the side surfaces. *q α i* and *p α i* are the surface forces and couples on top (*α* = *t*) and bottom (*α* = *b*) surfaces.

The governing equations of quasi-3D Reddy third order theory are

0 = 1 *r* " *∂ ∂r rN*(0) *rr* − *N* (0) *θθ* # + *F* (0) *r* (16) 0 = 1 *r* ( *∂ ∂r rN*(1) *rr* − *N* (1) *θθ* − *c*<sup>1</sup> *∂ ∂r rN*(3) *rr* − *N* (3) *θθ* ! − *r N* (0) *rz* − *c*2*N* (2) *rz* + 1 2 " *∂ ∂r rM*(0) *rθ* + *M* (0) *<sup>r</sup><sup>θ</sup>* − *c*<sup>2</sup> *∂ ∂r rM*(2) *rθ* + *M* (2) *<sup>r</sup><sup>θ</sup>* <sup>−</sup> <sup>2</sup>*rM*(1) *θz* !#) + *F* (1) *<sup>r</sup>* − *c*1*F* (3) *r* (17) 0 = 1 *r* ( *∂ ∂r* " *rN*(0) *rr ∂w ∂r* !# + *c*<sup>1</sup> *∂ ∂r* " *∂ ∂r rN*(3) *rr* <sup>−</sup> *rN*(3) *θθ* # + *∂ ∂r rN*(0) *rz* − *c*<sup>2</sup> *∂ ∂r rN*(2) *rz* + 1 2 " *∂ ∂r ∂ ∂r rM*(0) *rθ* + *M* (0) *rθ* ! + *c*<sup>2</sup> *∂ ∂r ∂ ∂r rM*(2) *rθ* + *M* (2) *rθ* ! − 2*c*<sup>2</sup> *∂ ∂r rM*(1) *θz* #) + *F* (0) *<sup>z</sup>* + *c*1 *r ∂ ∂r rF*(3) *r* (18) 0 = 1 *r* ( 1 2 *∂ ∂r* " *∂ ∂r rN*(2) *rr* − *N* (2) *θθ* # − *N* (0) *zz* + " *∂ ∂r ∂ ∂r rM*(1) *rθ* + *M* (1) *<sup>r</sup><sup>θ</sup>* <sup>−</sup> *rM*(0) *θz* !#) + *F* (1) *<sup>z</sup>* + 1 2 *∂ ∂r rF*(2) *r* (19)

$$\begin{split} 0 &= \frac{1}{r} \Bigg[ \frac{1}{3} \frac{\partial}{\partial r} \Bigg[ \frac{\partial}{\partial r} \Big( r N\_{rr}^{(3)} \Big) - N\_{\theta\theta}^{(3)} \Bigg] - 2r N\_{zz}^{(1)} \\ &+ \frac{\partial}{\partial r} \Bigg[ \frac{\partial}{\partial r} \Big( r M\_{r\theta}^{(2)} \Big) + M\_{r\theta}^{(2)} - 2r M\_{\theta z}^{(1)} \Bigg] \Bigg) + F\_z^{(2)} + \frac{1}{3} \frac{\partial}{\partial r} \Big( r F\_r^{(3)} \Big), \end{split} \tag{20}$$

where <sup>n</sup> *N* (*k*) *ij* , *M* (*k*) *ij* o = R *h* 2 − *h* 2 *z k σij*, *mij dz* and *F* (*k*) *<sup>i</sup>* = R *h* 2 − *h* 2 *z k* n *f<sup>i</sup>* + h *q t <sup>i</sup>* − 1 *k q b i* io*dz*. Note that the body couple *c*¯*<sup>θ</sup>* is omitted in the governing equation.

The temperature distribution through thickness direction can be determined by solving the steady state energy equation,

$$-\frac{d}{dz}\left(k(z)\frac{dT}{dz}\right) = 0,\tag{21}$$

where *k*(*z*) is heat conductivity and *T* is the temperature. The effective thermal conductivity is defined using the Maxwell–Eucken model described by Deng et al. [51]:

$$k(z) = k\_s(z) \left[ \frac{k\_f + 2k\_s(z) + 2\Phi\left(k\_f - k\_s(z)\right)}{k\_f + 2k\_s(z) - \Phi\left(k\_f - k\_s(z)\right)} \right],\tag{22}$$

where *ks*(*z*) and *k <sup>f</sup>* are the thermal conductivity of the solid and fluid phases, respectively, and Φ is the porosity. In this study, the thermal conductivity of the solid is obtained using a power-law distribution described in previous section.

### **4. Finite Element Model**

A weak from Galerkin finite element model for the circular plate bending is developed using the principle of virtual displacement (15) and a weak form is directly developed from the energy Equation (21) for steady state heat conduction problem. The details of weak form Galerkin finite element model can be found in Reddy [52].

The temperature *T* and generalized displacements (*u*0, *θ<sup>r</sup>* , *w*0, *θz*, *φz*) are approximated in following form:

$$T(z) = \sum\_{j=1}^{n} T\_j \hat{\Psi}\_j(z) \tag{23}$$

$$
\mu\_0(r) = \sum\_{j=1}^n \mu\_j \psi\_j(r) \tag{24}
$$

$$\theta\_r(r) = \sum\_{j=1}^n \theta\_j \psi\_j(r) \tag{25}$$

$$w\_0(r) = \sum\_{I=1}^{2n} \Delta\_I^{(1)} \phi\_I(r) \tag{26}$$

$$\theta\_z(r) = \sum\_{I=1}^{2n} \Delta\_I^{(2)} \phi\_I(r) \tag{27}$$

$$\phi\_z(r) = \sum\_{J=1}^{2n} \Delta\_f^{(3)} \phi\_I(r),\tag{28}$$

where *T<sup>j</sup>* are nodal temperatures through thickness direction; *u<sup>j</sup>* , *θj* , and *w<sup>j</sup>* are nodal displacements in the radial direction; *ψ*ˆ *<sup>j</sup>* and *ψ<sup>j</sup>* are the Lagrange interpolation functions; *φ<sup>J</sup>* are the Hermite interpolation functions; ∆ (*i*) *J* are generalized deflections and *i* = 1, 2, 3 correspond to *w*0, *θz*, *φz*, respectively; *n* is the number of nodes in an element.

The finite element model of the steady state heat conduction problem is given by

$$\left[\left[\hat{\mathbb{K}}\right]^{\varepsilon}\left\{T^{\varepsilon}\right\}=\left\{\hat{\mathbb{F}}\right\}^{\varepsilon},\tag{29}$$

where the stiffness matrix and external heat flux are

$$\hat{\mathcal{K}}\_{ij} = \int\_{-\frac{\hbar}{2}}^{\frac{\hbar}{2}} k(z) \left( \frac{d\hat{\psi}\_i}{dz} \frac{d\hat{\psi}\_i}{dz} \right) dz \tag{30}$$

$$
\mathbf{f}\_{\mathbf{i}} = q\_{\mathbf{i}}.\tag{31}
$$

The finite element model of an axisymmetric plate static bending is given by

$$
\begin{bmatrix}
K^{11} & K^{12} & K^{13} & K^{14} & K^{15} \\
K^{21} & K^{22} & K^{23} & K^{24} & K^{25} \\
K^{31} & K^{32} & K^{33} & K^{34} & K^{35} \\
K^{41} & K^{42} & K^{43} & K^{44} & K^{45} \\
K^{51} & K^{52} & K^{53} & K^{54} & K^{55}
\end{bmatrix}
\begin{Bmatrix}
\{u\_0\} \\ \{\theta\_r\} \\ \{w\_0\} \\ \{\theta\_z\} \\ \{\theta\_\ell\} \\ \emptyset \\
\emptyset^T
\end{Bmatrix}^{\varepsilon} = \begin{Bmatrix}
\{F^1\} \\ \{F^2\} \\ \{F^3\} \\ \{F^4\} \\ \emptyset^T
\end{Bmatrix}^{\varepsilon}.\tag{32}
$$

The elements of the stiffness matrix, *K lm*, and the elements of force vector, *F l* , are defined in Appendix A.

The solution of the nonlinear finite element model (32) is obtained using Newton's iteration procedure. The linearized element equations take the form of

$$\mathbf{T}^{\varepsilon} \left( \Delta^{(i-1)} \right) \delta \Delta^{(i)} = -\mathbf{R}^{\varepsilon} \left( \Delta^{(i-1)} \right),\tag{33}$$

where **T** *e* is the tangent stiffness matrix, *δ***∆** (*i*) is incremental displacements at the *i*th iteration, and **R** *e* is the residual vector. The tangent matrix and residual are defined as [52]

$$\mathbf{T}^{\varepsilon} = \frac{\partial \mathbf{R}^{\varepsilon}}{\partial \mathbf{A}^{\varepsilon}} \qquad \mathbf{R}^{\varepsilon} = \left(\mathbf{K}^{\varepsilon}\mathbf{A}^{\varepsilon} - \mathbf{F}^{\varepsilon}\right)^{(i-1)}.\tag{34}$$

By solving the assembled global system equation, the global incremental displacement vector at *i*th iteration, *δ***U**(*i*) is obtained.

$$\delta \mathbf{U}^{(i)} = -\left[\mathbf{T}\left(\mathbf{U}^{(i-1)}\right)\right]^{-1} \mathbf{R}^{(i-1)}.\tag{35}$$

The total displacement at the *i*th iteration is obtained by adding the incremental solution at the *i*th iteration to the previous solution at the (*i* − 1)th iteration [39].

$$\mathbf{U}^{(i)} = \mathbf{U}^{(i-1)} + \delta \mathbf{U}^{(i)}.\tag{36}$$

In this study, we considered geometrical nonlinearity with elastic material behavior. For this purpose, the Newton's iteration is sufficient to obtain the converged solutions. However, when limit load, softening branches, or snap-through behavior are considered, another solution procedure, such as the arc length method, should be considered. These solution procedures can be used in conjunction with various numerical methods such as isogeometric techniques [53,54] or the Rayleigh Ritz method [55] in addition to the finite element method.

#### **5. Numerical Results**

In the numerical examples, we considered several examples of annular circular plates with various boundary conditions such as simply-supported and clamped boundary conditions. To validate the developed finite element model, we compared our results with available studies in the literature. We also conducted convergence studies to obtain optimal mesh size and different quadrature rules to make sure we avoided any locking phenomena. In this study, we used 16 elements and full quadrature rules for linear parts of the stiffness matrix and reduced quadrature rules for shear, nonlinear, and couple stress parts of the stiffness matrix.

Figure 2 shows the annular plate we studied. The numerical parameters for the validation study were adapted from the study of Reddy et al. [39]: *h* = 0.1, *r<sup>o</sup>* = 10*h*, *r<sup>i</sup>* = 0.25*r*0, *E*<sup>1</sup> = 10<sup>6</sup> , and *E*<sup>2</sup> = 10<sup>5</sup> .

**Figure 2.** An axisymmetric FGM annular plate [39].

Figures 3–5 show that maximum deflection versus the load parameter *P* = *q*0*h* 4 *Ecr* 4 *o* at the free edges, where *q*<sup>0</sup> is a distributed load on the top surface, *h* is the plate thickness, *E<sup>c</sup>* is the Young's modulus of ceramic materials on the top surface, and *r<sup>o</sup>* is the outer radius of the annular plate. The developed finite element model shows a good agreement with the study of Reddy et al. [39]. In this figure, the effects of the material variations based on the change of the power-law index, and the effect of length scale parameter are presented with various boundary conditions.

**Figure 3.** Maximum deflection at outer edge with clamped inner edge.

**Figure 4.** Maximum deflection at inner edge with clamped outer edge.

**Figure 5.** Maximum deflection at inner edge with simply-supported outer edge.

With the validated finite element model, we evaluated the effects of various parameters such as the length scale parameters, the shape of porosity distribution, power law index, and boundary conditions. In this study, we considered a porous functionally-graded material with Monel and zirconia and the material properties of them follow [56]:


where *K<sup>i</sup>* is the bulk modulus, *µ<sup>i</sup>* is the shear modulus, *α<sup>i</sup>* is the thermal expansion coefficient, *k<sup>i</sup>* is the thermal conductivity, and the subscription *m* and *c* indicate metal and ceramic, respectively. We assumed that the porous is filled with the air and the thermal conductivity of the air is assumed to be *k<sup>a</sup>* = 0.025572 W/mK.

To induce thermal load, two different temperatures were applied on the top and bottom surfaces; 500 K was applied on the top surface and 300 K was applied on the bottom surface. Figure 6 shows the temperature distribution through plate thickness depending on the variation of material constituents and porosity distribution. The temperature distribution was obtained by solving the energy Equation (21). Three different types of porosity distributions and the variation of material constituents were considered. In the area where the volume fraction for porosity is larger, the thermal resistance becomes larger and the temperature change through the thickness is less than the area where the volume fraction of porosity is lesser. With a larger power-law index, the effective thermal conductivity is increased and thermal resistance becomes smaller because the volume fraction of metal is increased.

For illustration purposes, the same plate geometry as Reddy et al. [39] was used, and the effects of various parameters with clamped and simply-supported boundary conditions were considered.

Figures 7 and 8 show effects of the length scale parameter on the maximum deflection of FGM plates. When the power-law index is larger, the volume fraction of Monel is larger and the FGM plate becomes stiffer. The length scale parameter can capture the stiffening

behavior in micro scale structures. The deflections of FGM plates with various length scale parameters are shown in Figures 9 and 10.

**Figure 6.** Temperature distribution of porous FGM.

**Figure 7.** Maximum deflection of FGM with clamped outer edge with nonzero length scale parameters.

**Figure 8.** Maximum deflection of FGM with simply-supported outer edge with nonzero length scale parameters.

**Figure 9.** Deflection of FGM with clamped outer edge with nonzero length scale parameters.

**Figure 10.** Deflection of FGM with simply-supported outer edge with nonzero length scale parameters.

The overall volume fraction of porosity in three porosity distributions is the same, but the enhanced porous areas are mid, bottom, and top surfaces with Type 1, Type 2, and Type 2 porosity distributions. The porous FGM is softer than non-porous FGM, and Type 1 results in the stiffest plates because the materials on the top and bottom surfaces remain. There are no differences in the plate bending stiffness between Type 1 and Type 2 distribution when a homogeneous material is assumed. When the power-law index is larger than zero, the volume fraction of stiffer material becomes larger in the FGMs. In the case of Type 2 distribution, the volume fraction of the stiffer material is decreased, and in the case of Type 3, the volume fraction of the softer material is decreased. Therefore, Type 2 will be softer than Type 3 in the case of FGMs. Figures 11 and 12 show the effects of three different porosity distributions and material variations on the maximum deflections with clamped and simply-supported outer edges, respectively. The deflections along the radial direction are shown in Figures 13 and 14.

Figures 15–18 show a normal stress distribution through the plate thickness. In the case of the clamped outer edge, the normal stress in the area where the volume fraction of porosity is larger is smaller than the area where the volume fraction of porosity is smaller because the area with larger porosity is softer than the other areas. It is clearly shown that the normal stress at the bottom surface (*z* = −*h*/2) with porosity distribution Type 3 is larger than porosity distribution Type 2, which enhances the porosity distribution in the lower area of the plates. In the case of the simply-supported outer edge, the normal stress distribution is a parabolic shape unlike the case of the clamped outer edge. This is because the thickness stretch is not constrained in the case of simply-supported boundary conditions. Only the mid plane deflection, *w*0, is constrained. The nonzero length scale parameters make the FGM plate stiffer, but there are no material property changes, which results in smaller stresses.

**Figure 11.** Maximum deflection of FGM with clamped outer edge with nonzero porosity.

**Figure 12.** Maximum deflection of FGM with simply-supported outer edge with nonzero porosity.

**Figure 13.** Deflection of FGM with clamped outer edge with nonzero porosity.

**Figure 14.** Deflection of FGM with simply-supported outer edge with nonzero porosity.

**Figure 15.** Normal stress variation through the thickness of plate with clamped outer edge with nonzero length scale parameters.

**Figure 16.** Normal stress variation through the thickness of plate with simply-supported outer edge with nonzero length scale parameters.

**Figure 17.** Normal stress variation through the thickness of plate with clamped outer edge with nonzero porosity.

**Figure 18.** Normal stress variation through the thickness of plate with simply-supported outer edge porosity.

Figures 19–22 show transverse shear stress distributions through the plate thickness. Similar effects of porosity distribution and length scale parameter on the transverse shear stresses are observed. The length scale parameter makes the FGM plate stiffer and smaller stress values are obtained. However, the porosity distribution affects the material properties and larger stress values are present in stiffer areas. The proposed quasi-3D Reddy third-order plate theory can capture a parabolic variation of the transverse shear stresses, and it does not require a shear correction factor that is present in low order shear deformation theories.

**Figure 19.** Transverse stress variation through the thickness of plate with clamped outer edge with nonzero length scale parameters.

**Figure 20.** Transverse stress variation through the thickness of plate with simply-supported outer edge with nonzero length scale parameters.

**Figure 21.** Transverse stress variation through the thickness of plate with clamped outer edge with nonzero porosity.

**Figure 22.** Transverse stress variation through the thickness of plate with simply-supported outer edge porosity.

Figures 23–26 show the effect of thermal load. The thermal load is induced by temperature boundary conditions; 500 K is applied on top surface and 300 K is applied on bottom surfaces. In the case of the clamped outer edge, the deflection is due to thermal load in the negative direction because the plate bends down due to the thermal load. This is clearly

shown in Figure 25. In the case of the simply-supported outer edge, the plate bends down due to the thermal load at the same place; the plate rotates about the outer edge which results in the positive deflection due to the thermal load.

**Figure 23.** Maximum deflection of FGM under thermal load with clamped outer edge.

**Figure 24.** Maximum deflection of FGM under thermal load with simply-supported outer edge.

**Figure 25.** Maximum deflection of porous FGM under thermal load with clamped outer edge.

**Figure 26.** Maximum deflection of porous FGM under thermal load with simply-supported outer edge.

#### **6. Conclusions**

In this study, a nonlinear finite element (FE) model for axisymmetric circular/annular plates was presented. The developed finite element model accounts for geometric nonlinearity, variation of material constituents, microstructure size effects, and effects of porosity distributions. Using the developed FE model, the bending behavior of functionally-graded axisymmetric annular plates under thermomechanical loads was analyzed.

Numerical analysis results for an axisymmetric annular plate with various boundary conditions were presented. A parametric study was conducted to understand the effects of porosity distributions, the variation of material properties, and microstructure size on the bending behavior of axisymmetric annular plates. In summary, the following results were observed:


The presented finite element model can be extended to an asymmetric circular/annular plate.

**Author Contributions:** Conceptualization, J.K.; Data curation, E.N.; Formal analysis, J.K.; Methodology, J.K.; Project administration, J.K.; Software, J.K. and E.N.; Validation, J.K. and E.N.; Writing—original draft, J.K. and S.R.; Writing—review & editing, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Western Michigan University Research Start-Up.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

$$K\_{ij}^{11} = \int\_{r\_1}^{r\_2} \left[ A\_{11}^{(0)} \left( \frac{\partial \psi\_i}{\partial r} \frac{\partial \psi\_j}{\partial r} + \frac{1}{r^2} \psi\_i \psi\_j \right) + \frac{A\_{12}^{(0)}}{r} \left( \frac{\partial \psi\_i}{\partial r} \psi\_j + \psi\_i \frac{\partial \psi\_j}{\partial r} \right) \right] r dr \tag{A1}$$
 
$$\dots \quad \epsilon\_1 \gamma \left[ \psi\_1 \gamma\_1 \dots \psi\_{(2)} \gamma\_1 \gamma\_2 \psi\_i \gamma\_3 \dots \gamma\_n \right] \tag{A1}$$

$$\begin{split} K\_{ij}^{12} = \int\_{r\_1}^{r\_2} \left[ \left( A\_{11}^{(1)} - c\_1 A\_{11}^{(3)} \right) \left( \frac{\partial \psi\_i}{\partial r} \frac{\partial \psi\_j}{\partial r} + \frac{1}{r^2} \psi\_i \psi\_j \right) \right. \\ \left. + \frac{1}{r} \left( A\_{12}^{(1)} - c\_1 A\_{12}^{(3)} \right) \left( \frac{\partial \psi\_i}{\partial r} \psi\_j + \psi\_i \frac{\partial \psi\_j}{\partial r} \right) \right] r dr \\ \leq \int\_{r\_1}^{r\_2} \left( A\_{11}^{(0)} / \gamma\_{rr} \right) \gamma\_{rr} \gamma\_{rr} \end{split} \tag{A2}$$

$$K\_{l\bar{l}}^{13} = \int\_{r\_1}^{r\_2} \left[ \frac{A\_{11}^{(0)}}{2} \left( \frac{\partial w\_0}{\partial r} \right) \frac{\partial \psi\_i}{\partial r} \frac{\partial \phi\_f}{\partial r} - c\_1 A\_{11}^{(3)} \left( \frac{\partial \psi\_i}{\partial r} \frac{\partial^2 \phi\_f}{\partial r^2} + \frac{1}{r^2} \psi\_i \frac{\partial \phi\_f}{\partial r} \right) \right. \\
$$

$$+ \frac{A\_{12}^{(0)}}{2r} \left( \frac{\partial w\_0}{\partial r} \right) \psi\_i \frac{\partial \phi\_f}{\partial r} - c\_1 \frac{A\_{12}^{(3)}}{r} \left( \frac{\partial \psi\_i}{\partial r} \frac{\partial \phi\_f}{\partial r} + \psi\_i \frac{\partial^2 \phi\_f}{\partial r^2} \right) \right] r dr \tag{A3}$$

$$K\_{l}^{14} = \int\_{r\_1}^{r\_2} \left[ -\frac{A\_{11}^{(2)}}{2} \left( \frac{\partial \psi\_l}{\partial r} \frac{\partial^2 \phi\_f}{\partial r^2} + \frac{1}{r^2} \psi\_l \frac{\partial \phi\_f}{\partial r} \right) + A\_{12}^{(0)} \left( \frac{\partial \psi\_l}{\partial r} \phi\_l + \frac{1}{r} \psi\_l \phi\_f \right) \right. \tag{A4}$$
 
$$-\frac{A\_{12}^{(2)}}{2r} \left( \frac{\partial \psi\_l}{\partial r} \frac{\partial \phi\_f}{\partial r} + \psi\_l \frac{\partial^2 \phi\_f}{\partial r^2} \right) \Big] r dr \tag{A4}$$

*K* 15 *iJ* = Z *r*<sup>2</sup> *r*1 − *A* (3) 11 3 *∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *ψi ∂φJ ∂r* ! + 2*A* (1) <sup>12</sup> *c*<sup>2</sup> *∂ψ<sup>i</sup> ∂r φ<sup>J</sup>* + 1 *r ψiφ<sup>J</sup>* ! − *A* (3) 12 3*r ∂ψ<sup>i</sup> ∂r ∂φJ ∂r* + *ψ<sup>i</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! *rdr* (A5) *K* 21 *ij* = " *A* (1) <sup>11</sup> − *c*1*A* (3) 11 *∂ψ<sup>i</sup> ∂r ∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *ψiψ<sup>j</sup>* ! + 1 *r A* (1) <sup>12</sup> − *c*1*A* (3) 12 *∂ψ<sup>i</sup> ∂r ψ<sup>j</sup>* + *ψ<sup>i</sup> ∂ψ<sup>j</sup> ∂r* !#*rdr* (A6) *K* 22 *ij* = Z *r*<sup>2</sup> *r*1 ( *A* (2) <sup>11</sup> − 2*c*1*A* (4) <sup>11</sup> + *c* 2 1*A* (6) 11 *∂ψ<sup>i</sup> ∂r ∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *ψiψ<sup>j</sup>* ! + 1 *r A* (2) <sup>12</sup> − 2*c*1*A* (4) <sup>12</sup> + *c* 2 1*A* (6) 12 *∂ψ<sup>i</sup> ∂r ψ<sup>j</sup>* + *ψ<sup>i</sup> ∂ψ<sup>j</sup> ∂r* ! + *A* (0) <sup>55</sup> − 2*c*2*A* (2) <sup>55</sup> + *c* 2 <sup>2</sup>*A* (4) 55 *ψiψ<sup>j</sup>* + 1 4 " 1 *r* 2 *B* (0) <sup>−</sup> <sup>2</sup>*c*2*<sup>B</sup>* (2) + *c* 2 2*B* (4) *r* 2 *∂ψ<sup>i</sup> ∂r ∂ψ<sup>j</sup> ∂r* + *ψiψ<sup>j</sup>* ! + 4*c* 2 2*B* (2)*ψiψ<sup>j</sup>* − 1 *r B* (0) <sup>−</sup> <sup>2</sup>*c*2*<sup>B</sup>* (2) + *c* 2 2*B* (4) *ψi ∂ψ<sup>j</sup> ∂r* + *∂ψ<sup>i</sup> ∂r ψj* !#)*rdr* (A7) *K* 23 *iJ* = Z *r*<sup>2</sup> *r*1 ( 1 2 *∂w*<sup>0</sup> *∂r A* (1) <sup>11</sup> − *c*1*A* (3) <sup>11</sup> *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* − *c*<sup>1</sup> *A* (4) <sup>11</sup> − *c*1*A* (6) 11 *∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *ψi ∂φJ ∂r* ! + 1 2*r ∂w*<sup>0</sup> *∂r A* (1) <sup>12</sup> − *c*1*A* (3) 12 *ψi ∂φJ ∂r* − *c*1 *r A* (4) <sup>12</sup> − *c*1*A* (6) <sup>12</sup> *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* − *c*1 *r A* (4) <sup>12</sup> − *c*1*A* (6) 12 *ψi ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *A* (0) <sup>55</sup> − 3*c*2*A* (2) <sup>55</sup> + *c* 2 1*A* (4) 55 *ψi ∂φJ ∂r* − 1 4 " *B* (0) <sup>−</sup> *<sup>c</sup>* 2 2*B* (4) *∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂ψ<sup>i</sup> ∂r ∂φJ ∂r* ! − 4*c* 2 2*B* (2)*ψ<sup>i</sup> ∂φJ ∂r* − 1 *r B* (0) <sup>−</sup> *<sup>c</sup>* 2 2*B* (4) *ψi ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ψi ∂φJ ∂r* !#)*rdr* (A8) *K* 24 *iJ* = Z *r*<sup>2</sup> *r*1 ( − 1 2 *A* (3) <sup>11</sup> − *c*1*A* (5) 11 *∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *ψi ∂φJ ∂r* ! + 1 *r A* (1) <sup>12</sup> − *c*1*A* (3) 12 *ψiφ<sup>J</sup>* + *A* (1) <sup>12</sup> − *c*1*A* (3) <sup>12</sup> *∂ψ<sup>i</sup> ∂r φ<sup>J</sup>* − 1 2*r A* (3) <sup>12</sup> − *c*1*A* (5) 12 *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* + *ψ<sup>i</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 2 " 1 *r B* (1) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (3) *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* + *ψ<sup>i</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − *r ∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − 1 *r* 2 *B* (1) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (3) *ψi ∂φJ ∂r* + 2*c*2*B* (1)*ψ<sup>i</sup> ∂φJ ∂r* #)*rdr* (A9) *K* 25 *iJ* = Z *r*<sup>2</sup> *r*1 ( − 1 3 *A* (4) <sup>11</sup> − *c*1*A* (6) 11 *∂ψ<sup>i</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *ψi ∂φJ ∂r* ! + 2 *r A* (2) <sup>12</sup> − *c*1*A* (4) 12 *ψiφ<sup>J</sup>* + 2 *A* (2) <sup>12</sup> − *c*1*A* (4) <sup>12</sup> *∂ψ<sup>i</sup> ∂r φ<sup>J</sup>* − 1 3*r A* (4) <sup>12</sup> − *c*1*A* (6) 12 *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* + *ψ<sup>i</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 !

$$\begin{aligned} &+\frac{1}{2}\Big[\frac{1}{r}\Big(\mathcal{B}^{(2)} - c\_{2}\mathcal{B}^{(4)}\Big)\left(\frac{\partial\psi\_{i}}{\partial r}\frac{\partial\phi\_{f}}{\partial r} + \psi\_{i}\frac{\partial^{2}\phi\_{f}}{\partial r^{2}} - r\frac{\partial\psi\_{i}}{\partial r}\frac{\partial^{2}\phi\_{f}}{\partial r^{2}}\right) \\ &-\frac{1}{r^{2}}\Big(\mathcal{B}^{(2)} - c\_{2}\mathcal{B}^{(4)}\Big)\psi\_{i}\frac{\partial\phi\_{f}}{\partial r} + 4c\_{2}\mathcal{B}^{(2)}\psi\_{i}\frac{\partial\phi\_{f}}{\partial r}\Big]\Big)rdr\tag{A10} \end{aligned} \tag{A10}$$

*K* 31 *Ij* = " *A* (0) 11 *∂w*<sup>0</sup> *∂r* ! *∂φI ∂r ∂ψ<sup>j</sup> ∂r* − *c*1*A* (3) 11 *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψj* ! + *A* (0) 12 *r ∂w*<sup>0</sup> *∂r* ! *∂φI ∂r ψ<sup>j</sup>* − *c*<sup>1</sup> *A* (3) 12 *r ∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψj* ! *rdr* (A11) *K* 32 *Ij* = Z *r*<sup>2</sup> *r*1 ( *∂w*<sup>0</sup> *∂r A* (1) <sup>11</sup> − *c*1*A* (3) <sup>11</sup> *∂φ<sup>I</sup> ∂r ∂ψ<sup>j</sup> ∂r* − *c*<sup>1</sup> *A* (4) <sup>11</sup> − *c*1*A* (6) 11 *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψj* ! + 1 *r ∂w*<sup>0</sup> *∂r A* (1) <sup>12</sup> − *c*1*A* (3) <sup>12</sup> *∂φ<sup>I</sup> ∂r ψ<sup>i</sup>* − *c*1 *r A* (4) <sup>12</sup> − *c*1*A* (6) <sup>12</sup> *∂φ<sup>I</sup> ∂r ∂ψ<sup>j</sup> ∂r* − *c*1 *r A* (4) <sup>12</sup> − *c*1*A* (6) <sup>12</sup> *<sup>∂</sup>* 2*φI ∂r* 2 *ψ<sup>j</sup>* + *A* (0) <sup>55</sup> − 3*c*2*A* (2) <sup>55</sup> + *c* 2 1*A* (4) <sup>55</sup> *∂φ<sup>I</sup> ∂r ψi* − 1 4 " *B* (0) <sup>−</sup> *<sup>c</sup>* 2 2*B* (4) *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* − 1 *r ∂φI ∂r ∂ψ<sup>j</sup> ∂r* ! − 4*c* 2 2*B* (2) *∂φI ∂r ψj* − 1 *r B* (0) <sup>−</sup> *<sup>c</sup>* 2 2*B* (4) *∂* 2*φI ∂r* 2 *ψ<sup>j</sup>* − 1 *r ∂φI ∂r ψj* !#)*rdr* (A12) *K* 33 *I J* = Z *r* 2 *r*1 ( 1 2 *∂w*<sup>0</sup> *∂r* ! *∂w*<sup>0</sup> *∂r A* (0) 11 − *c*2 *r A* (3) 12 ! *∂φI ∂r ∂φJ ∂r* + *c* 2 1 *r A* (6) 12 *∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − *c*1*A* (3) 11 *∂w*<sup>0</sup> *∂r* ! *∂φ<sup>I</sup> ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 2 *∂* 2*φI ∂r* 2 *∂φJ ∂r* ! + *c* 2 1*A* (6) 11 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! + *A* (0) <sup>55</sup> − 3*c*2*A* (2) <sup>55</sup> + *c* 2 <sup>2</sup>*A* (4) <sup>55</sup> *∂φ<sup>I</sup> ∂r ∂φJ ∂r* + 1 4 " *c*2 2*B* (4) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! + *B* (0) + 2*c*2*B* (2) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r ∂φI ∂r ∂φJ ∂r* ! − 1 *r B* (0) + 2*c*2*B* (2) + *c* 2 2 *∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 !#)*rdr* (A13) *K* 34 *I J* = Z *r* 2 *r*1 − 1 2 *∂w*<sup>0</sup> *∂r* ! *A* (2) 11 *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *A* (2) 12 *r ∂φI ∂r ∂φJ ∂r* − 2*A* (0) 12 *∂φI ∂r φJ* + *c*1 2 *A* (5) 11 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − *c*1*A* (3) 12 1 *r ∂φI ∂r φ<sup>J</sup>* + *∂* 2*φI ∂r* 2 *φJ* ! + *c*1 2*r A* (5) 12 *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *∂* 2*φI ∂r* 2 *∂φJ ∂r* ! + 1 2 " 2*c*2*B* (1) *∂φI ∂r ∂φJ ∂r* + *B* (1) + *c*2*B* (3) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* − 1 *r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* !#)*rdr* (A14) *K* 35 *I J* = Z *r* 2 *r*1 − 1 3 *∂w*<sup>0</sup> *∂r* ! *A* (3) 11 *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *A* (3) 12 *r ∂φI ∂r ∂φJ ∂r* − 6*A* (1) 12 *∂φI ∂r φJ* + *c*1 3 *A* (6) 11 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − 2*c*1*A* (4) 12 1 *r ∂φI ∂r φ<sup>J</sup>* + *∂* 2*φI ∂r* 2 *φJ* ! − *c*1 3*r A* (6) 12 *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *∂* 2*φI ∂r* 2 *∂φJ ∂r* ! + 1 2 " 4*c*2*B* (2) *∂φI ∂r ∂φJ ∂r* + *B* (2) + *c*2*B* (4) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* − 1 *r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* !#)*rdr* (A15) *K* 41 *Ij* = Z *r*<sup>2</sup> *r*1 − *A* (2) 11 2 *∂* 2*φi ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψj* ! + *A* (0) 12 *φI ∂ψ<sup>j</sup> ∂r* + 1 *r φIψ<sup>j</sup>* !

− *A* (2) 12 2*r ∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψi* ! *rdr* (A16) *K* 42 *I J* = Z *r*<sup>2</sup> *r*1 ( − 1 2 *A* (3) <sup>11</sup> − *c*1*A* (5) 11 *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψj* ! + 1 *r A* (1) <sup>12</sup> − *c*1*A* (3) 12 *φIψ<sup>i</sup>* + *A* (1) <sup>12</sup> − *c*1*A* (3) <sup>12</sup> *<sup>φ</sup>I∂ψ<sup>j</sup> ∂r* − 1 2*r A* (3) <sup>12</sup> − *c*1*A* (5) 12 *∂ψ<sup>i</sup> ∂r ∂φJ ∂r* + *∂* 2*φI ∂r* 2 *ψj* ! + 1 2 " 1 *r B* (1) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (3) *∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψ<sup>j</sup>* − *r ∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* ! − 1 *r* 2 *B* (1) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (3) *∂φ<sup>I</sup> ∂r ψ<sup>j</sup>* + 2*c*2*B* (1) *∂φI ∂r ψj* #)*rdr* (A17) *K* 43 *I J* = Z *r* 2 *r*1 − 1 4 *∂w*<sup>0</sup> *∂r* ! *A* (2) 11 *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *A* (2) 12 *r ∂φI ∂r ∂φJ ∂r* − 2*A* (0) 12 *∂φI ∂r φJ* + *c*1 2 *A* (5) 11 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − *c*1*A* (3) 12 1 *r φI ∂φJ ∂r* + *φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + *c*1 2*r A* (5) 12 *∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 2 " 2*c*2*B* (1) *∂φI ∂r ∂φJ ∂r* + *B* (1) + *c*2*B* (3) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* !#)*rdr* (A18) *K* 44 *I J* = Z *r* 2 *r*1 *A* (0) <sup>11</sup> *φIφ<sup>J</sup>* + *A* (4) 11 4 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − *A* (2) 12 2 *∂* 2*φI ∂r* 2 *φ<sup>J</sup>* + *φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − *A* (2) 12 2*r ∂φI ∂r φ<sup>J</sup>* + *φ<sup>I</sup> ∂φJ ∂r* ! + *A* (4) 12 4*r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + *B* (0) *∂φI ∂r ∂φJ ∂r* +*B* (2) " *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* #)*rdr* (A19) *K* 45 *I J* = Z *r* 2 *r*1 *A* (1) <sup>11</sup> *φIφ<sup>J</sup>* + *A* (5) 11 6 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − *A* (3) 12 3 3 *∂* 2*φI ∂r* 2 *φ<sup>J</sup>* + *φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − *A* (3) 12 3*r* 3 *∂φI ∂r φ<sup>J</sup>* + *φ<sup>I</sup> ∂φJ ∂r* ! + *A* (5) 12 6*r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 2*B* (1) *∂φI ∂r ∂φJ ∂r* +*B* (3) " *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* #)*rdr* (A20) *K* 51 *Ij* = Z *r*<sup>2</sup> *r*1 − *A* (3) 11 3 *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψj* ! + 2*A* (1) <sup>12</sup> *c*<sup>2</sup> *φI ∂ψ<sup>j</sup> ∂r* + 1 *r φIψ<sup>j</sup>* ! − *A* (3) 12 3*r ∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψj* ! *rdr* (A21) *K* 52 *Ij* = Z *r*<sup>2</sup> *r*1 ( − 1 3 *A* (4) <sup>11</sup> − *c*1*A* (6) 11 *∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* + 1 *r* 2 *∂φI ∂r ψi* ! + 2 *r A* (2) <sup>12</sup> − *c*1*A* (4) 12 *φIψ<sup>j</sup>* + 2 *A* (2) <sup>12</sup> − *c*1*A* (4) 12 *φI ∂ψ<sup>j</sup> ∂r* − 1 3*r A* (4) <sup>12</sup> − *c*1*A* (6) 12 *∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψj* ! + 1 2 " 1 *r B* (2) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (4) *∂φI ∂r ∂ψ<sup>j</sup> ∂r* + *∂* 2*φI ∂r* 2 *ψ<sup>j</sup>* − *r ∂* 2*φI ∂r* 2 *∂ψ<sup>j</sup> ∂r* ! − 1 *r* 2 *B* (2) <sup>−</sup> *<sup>c</sup>*2*<sup>B</sup>* (4) *∂φ<sup>I</sup> ∂r ψ<sup>j</sup>* + 4*c*2*B* (2) *∂φI ∂r ψj* #)*rdr* (A22)

*K* 53 *I J* = Z *r* 2 *r*1 − 1 6 *∂w*<sup>0</sup> *∂r* ! *A* (3) 11 *∂* 2*φI ∂r* 2 *∂φJ ∂r* + *A* (3) 12 *r ∂φI ∂r ∂φJ ∂r* − 6*A* (1) <sup>12</sup> *φ<sup>I</sup> ∂φJ ∂r* + *c*1 3 *A* (6) 11 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − 2*c*1*A* (4) 12 1 *r φI ∂φJ ∂r* + *φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − *c*1 3*r A* (6) 12 *∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 2 " 4*c*2*B* (2) *∂φI ∂r ∂φJ ∂r* + *B* (2) + *c*2*B* (4) *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* !#)*rdr* (A23) *K* 54 *I J* = Z *r* 2 *r*1 *A* (1) <sup>11</sup> *φIφ<sup>J</sup>* + *A* (5) 11 6 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − *A* (3) 12 3 3*φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *∂* 2*φI ∂r* 2 *φJ* ! − *A* (3) 12 3*r* 3*φ<sup>I</sup> ∂φJ ∂r* + *∂φI ∂r φJ* ! + *A* (5) 12 6*r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *∂* 2*φI ∂r* 2 *∂φJ ∂r* ! + 2*B* (1) *∂φI ∂r ∂φJ ∂r* +*B* (3) " *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + *∂* 2*φI ∂r* 2 *∂φJ ∂r* ! + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* #)*rdr* (A24) *K* 55 *I J* = Z *r* 2 *r*1 4*A* (2) <sup>11</sup> *φIφ<sup>J</sup>* + *A* (6) 11 9 *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* ! − 2*A* (4) 12 3 *∂* 2*φI ∂r* 2 *φ<sup>J</sup>* + *φ<sup>I</sup> ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! − 2*A* (4) 12 3*r ∂φI ∂r φ<sup>J</sup>* + *φ<sup>I</sup> ∂φJ ∂r* ! + *A* (6) 12 9*r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 4*B* (2) *∂φI ∂r ∂φJ ∂r* +*B* (4) " *∂* 2*φI ∂r* 2 *∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 − 1 *r ∂* 2*φI ∂r* 2 *∂φJ ∂r* + *∂φI ∂r ∂* <sup>2</sup>*φ<sup>J</sup> ∂r* 2 ! + 1 *r* 2 *∂φI ∂r ∂φJ ∂r* #)*rdr* (A25) *F* 1 *<sup>i</sup>* = Z *r* 2 *r*1 ( *f* (0) *<sup>r</sup>* + *q* (0) *r ψ<sup>i</sup>* + *T* (0) *r ∂ψ<sup>i</sup> ∂r* + 1 *r T* (0) *θ ψi* ) *rdr* (A26)

$$F\_{l}^{2} = \int\_{r\_{1}}^{r^{2}} \left\{ \left[ \left( f\_{r}^{(1)} + q\_{r}^{(1)} \right) - c\_{1} \left( f\_{r}^{(3)} + q\_{r}^{(3)} \right) + \left( c\_{\theta}^{(0)} + p\_{\theta}^{(0)} \right) - c\_{2} \left( c\_{\theta}^{(2)} + p\_{\theta}^{(2)} \right) \right] \psi\_{l} \right. \tag{A27}$$
 
$$+ \left( T\_{r}^{(1)} - c\_{1} T\_{r}^{(3)} \right) \frac{\partial \psi\_{l}}{\partial r} + \frac{1}{r} \left( T\_{\theta}^{(1)} - c\_{1} T\_{\theta}^{(3)} \right) \psi\_{l} \right] r dr \tag{A27}$$

$$\begin{split} F\_{I}^{3} &= \int\_{r\_{1}}^{r^{2}} \left\{ \left( f\_{z}^{(0)} + q\_{z}^{(0)} \right) \Phi\_{I} - \left[ c\_{1} \left( f\_{r}^{(3)} + q\_{r}^{(3)} \right) + \left( c\_{\theta}^{(0)} + p\_{\theta}^{(0)} \right) + c\_{2} \left( c\_{\theta}^{(2)} + p\_{\theta}^{(2)} \right) \right] \frac{\partial \Phi\_{I}}{\partial r} \\ &\quad + \left( \frac{\partial w\_{0}}{\partial r} \right) T\_{r}^{(0)} \frac{\partial \Phi\_{I}}{\partial r} - c\_{1} T\_{r}^{(3)} \frac{\partial^{2} \Phi\_{I}}{\partial r^{2}} + \frac{1}{r} T\_{\theta}^{(3)} \frac{\partial \Phi\_{I}}{\partial r} \right] r dr \\ F\_{I}^{4} &= \int\_{r\_{1}}^{r^{2}} \left\{ \left( f\_{z}^{(1)} + q\_{z}^{(1)} \right) \Phi\_{I} - \left[ \frac{1}{2} \left( f\_{r}^{(2)} + q\_{r}^{(2)} \right) + 2 \left( c\_{\theta}^{(1)} + p\_{\theta}^{(1)} \right) \right] \frac{\partial \Phi\_{I}}{\partial r} \right. \end{split} \tag{A28}$$

$$\begin{split} & \quad \mathbf{J}\_{r\_{1}} \left\{ \left< \mathbf{J}^{\sharp z} + \mathbf{J}^{\sharp z} \right> \right. \\ & \left. \left. \left< \mathbf{J}^{(1)} \right|\_{\mathbf{r}} - \frac{1}{2} T\_{r}^{(2)} \frac{\partial^{2} \Phi\_{I}}{\partial r^{2}} - \frac{1}{2r} T\_{\theta}^{(2)} \frac{\partial \Phi\_{I}}{\partial r} \right> r dr \end{split} \tag{A29}$$

$$F\_I^5 = \int\_{r\_1}^{r^2} \left\{ \left( f\_z^{(2)} + q\_z^{(2)} \right) \phi\_I - \left[ \frac{1}{3} \left( f\_r^{(3)} + q\_r^{(3)} \right) + 3 \left( c\_\theta^{(2)} + p\_\theta^{(2)} \right) \right] \frac{\partial \phi\_I}{\partial r} \right. \tag{A30}$$

$$+ 2T\_z^{(1)} \phi\_I - \frac{1}{3} T\_r^{(3)} \frac{\partial^2 \phi\_I}{\partial r^2} - \frac{1}{3r} T\_\theta^{(3)} \frac{\partial \phi\_I}{\partial r} \right\} \, r dr \tag{A30}$$

where

$$A\_{11}^{(k)} = \int\_{-\frac{\hbar}{2}}^{\frac{\hbar}{2}} (z)^k (1 - \nu) \Lambda(z) \, dz, \qquad A\_{12}^{(k)} = \int\_{-\frac{\hbar}{2}}^{\frac{\hbar}{2}} (z)^k \nu \Lambda(z) \, dz.$$

$$\begin{split} &A\_{55}^{(k)} = \int\_{-\frac{h}{2}}^{\frac{h}{2}} (z)^k \frac{(1-2\nu)}{2} \Lambda(z) \, dz, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \begin{split} &A^{(k)} = \int\_{-\frac{h}{2}}^{\frac{h}{2}} (z)^k \, \ell^2(1-2\nu) \Lambda(z) \, dz, \\ &f\_{\xi}^{(k)} = \int\_{-\frac{h}{2}}^{\frac{h}{2}} (z)^k \, \bar{f}\_{\xi} \, dz, \qquad \qquad \qquad \qquad q\_{\xi}^{(k)} = \left(\frac{h}{2}\right)^{(k)} \left[q\_{\xi}^{t} + (-1)^{(k)} q\_{\xi}^{b}\right], \\ &\mathcal{L}\_{\theta}^{(k)} = \int\_{-\frac{h}{2}}^{\frac{h}{2}} (z)^k \, \bar{\sigma}\_{\theta} \, dz, \qquad \qquad p\_{\theta}^{(k)} = \left(\frac{h}{2}\right)^{(k)} \left[p\_{\theta}^{t} + (-1)^{(k)} p\_{\theta}^{b}\right], \\ &T\_{\xi}^{(k)} = \int\_{-\frac{h}{2}}^{\frac{h}{2}} (z)^k \sigma\_{lj}^{T} \, dz \end{split} \tag{A31}$$

for *k* = 0, 1, 2, 3, 4, 6, *ξ* = *r*, *θ*, *z*, and *σ T ij* is the thermal stress due to the temperature difference (*T* − *T*0).

#### **References**


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## *Article* **Research on the Spring Creep Based on the Load Simulator of the Double Torsion Spring Steering Gear**

**Bo Zhang, Peijie Ren \*, Zhuo Wang and Hongwen Ma**

College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China; zhangbo\_heu@hrbeu.edu.cn (B.Z.); wangzhuo\_heu@hrbeu.edu.cn (Z.W.); mahongwen@hrbeu.edu.cn (H.M.) **\*** Correspondence: rpj@hrbeu.edu.cn

**Abstract:** In this paper, creep at room temperature is studied using a mechanical double−spring steering−gear load table, and the results are used to determine the accuracy of theoretical and simulated data. A creep equation at room temperature, based on the parameters obtained by a new macroscopic tensile experiment method, is used to analyze the creep strain and creep angle of a spring under force. The correctness of the theoretical analysis is verified by a finite−element method. Finally, a creep strain experiment of a torsion spring is carried out. The experimental results are 4.3% lower than the theoretical calculation results, which demonstrates the accuracy of the measurement, with an error of <5% achieved. The results shows that the equation used for the theoretical calculation is highly accurate and can meet the requirements of engineering measurement.

**Keywords:** load simulator; double spring; pre-compression; creep effect; prototype experiment

## **1. Introduction**

During flight, a missile control system gives instructions to control the wing of the steering gear to rotate at a certain Angle, which changes the direction and magnitude of the gas force acting on the missile, thus changing the missile's flight trajectory [1]. Therefore, the performance of its steering gear greatly influences the performance of a missile [2]. As steering gear is an integral component of an aircraft system, it is impossible to carry out a steering gear test after the design and manufacture of the system [3]. Therefore, it is necessary to test the performance of steering gear during its manufacture. A simulation load table is mainly used to simulate the force of steering gear during actual use in the environment [4].

At present, large-scale servos are mainly tested using electric and electro-hydraulic servo load simulators [1], while mechanical load simulators are widely used in small servo testing because of their high loading accuracy, small residual torque [5,6], small size, low manufacturing cost, simple structure, and easy maintenance.

For a mechanical load table, a spring torsion bar is generally used as the core component to provide load force. A spring torsion bar has the advantages of small error and high reliability [7]. In order to eliminate the non-linearity of spring reverse loading and the change of stiffness, and to reduce the zero balance range of a spring due to residual stress, friction, zero hysteresis, and other factors, Zhang [8] has proposed a mechanical load table with a structure of a double-spring coaxial reverse arrangement and pre-compression, which can effectively reduce the zero balance range. Because of the moment produced by the pre-compression of a spring, the spring will creep.

Zhu [9] studied the creep process of a precision helical tensile spring, which provided a method for measuring the creep of a tensile spring at room temperature. T. H. Alden [10] studied the strain hardening of 304 stainless steel during low temperature creep and proposed a theory that can be used to predict the creep curve and the hardening effect caused by creep. A. Oehlert [11] studied the room-temperature creep of high strength steel and found that creep can occur at lower stresses and that creep strain increases with creep

**Citation:** Zhang, B.; Ren, P.; Wang, Z.; Ma, H. Research on the Spring Creep Based on the Load Simulator of the Double Torsion Spring Steering Gear. *Materials* **2023**, *16*, 3763. https:// doi.org/10.3390/ma16103763

Academic Editors: Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 12 April 2023 Revised: 12 May 2023 Accepted: 14 May 2023 Published: 16 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

time and stress, but decreases with an increase of number of cycles. B. Alfredsson [12] conducted low-temperature creep experiments on martensite and bainite microstructures of high-strength rolling bearing steel, and found that the two exhibited different primary creep behaviors. The research results have certain guiding significance for the design and application of high-strength steel. Paul R. Barrett [13] developed a modified Coble creep model to describe the experimental low-stress creep rates in alloys with thermally stable precipitate structures. Brian K. Milligan [14] has studied the creep behavior of Al-Cu alloys at certain temperatures, and found that increasing the thermal stability of the precipitates in Al-Cu alloys can significantly improve their creep properties. Hu [15] explained how the evolution of microstructure affects the creep properties of a material physically, and evaluated several secondary phenomena in the curve of creep rate versus time of 316H stainless steel, which is vital to the realistic life assessment of critical engineering components. Wu [16] established the creep constitutive equation of a stainless steel spring to study the creep of a spring, and found that the higher the ambient temperature, the greater the creep strain of a stainless steel spring. When the service temperature increases from 25 ◦C to 320 ◦C, the 24h creep strain increases by five times.

At present, the research on creep mainly focuses on the creep behavior of materials, or the establishment of creep models under high temperature, but the research on creep at room temperature is relatively scarce. Most of the research on the creep behavior of springs has been carried out at high temperature and mainly focus on the stress relaxation of a spring [17–22]. In this paper, the influence of creep effect of a spring due to pre-compression is studied. The specific research contents are as follows: Through deconstruction and reorganization, using the original room-temperature creep constitutive equation as a basis, a room-temperature creep constitutive equation of a torsion spring with relevant parameters is obtained based on a macro-tensile test, which is then compared and verified by a finite-element simulation. The stress and strain of a spring are analyzed, and an accurate stress expression of the spring is obtained. Experiments are designed to verify the creep performance of a spring, and the error between the theoretical calculation results of spring creep strain and experimental data is obtained.

### **2. Creep Equation and Experimental Method of a Spring at Room Temperature**

#### *2.1. Creep Equation of a Spring at Room Temperature*

In room-temperature creep, the creep deformation increases logarithmically with time, which is consistent with the first stage of typical creep. Strain hardening and fatigue models are usually used in creep theory at room temperature. The fatigue model is more accurate in fast loading, while the strain hardening model can be used in room-temperature creep under arbitrary loading [23].

According to the microscopic situation of room-temperature creep, Schoeck [24] proposed the constitutive equation of room-temperature creep:

$$
\dot{\varepsilon}\_{\rm c} = NAV\nu \exp\left(-\frac{U}{KT}\right),
\tag{1}
$$

where: . *εc*—room-temperature creep rate; *N*—dislocation density; *A*—the area of dislocations swept after passing an obstacle; *V*—activation volume; *ν*—vibration frequency of the dislocation line; *U*—thermal activation energy required to pass obstacles; *K*—Boltzmann constant; *T*—experimental temperature.

*U* can be expressed as the product of the thermal activation energy, *U*0, minus the effective stress, *σ*eff, acting on the dislocation line and the activation volume, *V*, namely:

$$
\mathcal{U} = \mathcal{U}\_0 - \sigma\_{\rm eff} V. \tag{2}
$$

In the strain hardening model, the external stress, *σ*, is constant, but due to the hardening effect, the effective stress, *σ*eff, decreases with the increase of creep value, *εc*, namely:

$$
\sigma\_{\rm eff} = \sigma - \theta \varepsilon\_{\rm cf} \tag{3}
$$

where: *θ*—hardening coefficient at room-temperature creep; *εc*—creep value.

The relationship between room-temperature creep and creep time can be obtained by introducing Equations (2) and (3) into Equation (1), namely:

$$
\varepsilon\_{\rm c} = \alpha \ln \left( \frac{t\_{\rm c}}{\tau} + 1 \right),
\tag{4}
$$

where:

$$\mathfrak{a} = \frac{KT}{\theta V}, \tau = \frac{KT}{\theta V} \frac{1}{NABV} \exp\frac{U\_0 - \sigma V}{KT},\tag{5}$$

where: *B*—Burgers vector; *θ*—strain hardening coefficient.

Derived from Equation (4), the relationship between creep rate, . *εc*, and creep time, *tc*, at room temperature can be obtained as follows:

.

$$
\dot{\varepsilon}\_{\text{c}} = \frac{\mathfrak{a}}{t\_{\text{c}} + \mathfrak{r}}.\tag{6}
$$

After the end of loading, creep just appears. At this time, *<sup>t</sup><sup>c</sup>* <sup>=</sup> <sup>0</sup> and . *ε<sup>c</sup>* = *α*/*τ* = . *εc*0, which can be substituted into Equation (6) to obtain the following relationship:

$$
\dot{\varepsilon}\_{\rm c} = \left(\frac{t\_{\rm c}}{a} + \frac{1}{\dot{\varepsilon}\_{\rm c0}}\right)^{-1}.\tag{7}
$$

It can be seen from Equation (7) that the main influencing factors of creep rate are *α* and . *εc*0, which can be directly obtained by experiment. By fitting the experimental data with Equation (7), the creep rate equation can be obtained.

According to Equation (6), the room-temperature creep rate, . *εc*, can be obtained only after obtaining the influencing factors *α* and *τ*. However, these two factors are a measure of micro performance, which are difficult to obtain and not suitable for the situation of large individual differences. Therefore, Xiao [24] adopted a method to calculate room-temperature creep only with macro parameters, and the relevant parameters can be obtained through routine experiments, which is a more simple and convenient method in engineering applications.

The initial creep rate, . *ε<sup>c</sup>* = *α*/*τ* = . *εc*0, at the beginning of creep can be combined with Equation (4) to obtain Equation (8):

$$
\varepsilon\_{\rm c} = \alpha \ln \left( \frac{t}{\alpha} \dot{\varepsilon}\_{\rm c0} + 1 \right). \tag{8}
$$

Therefore, the parameter *<sup>τ</sup>* is transformed into the initial creep rate, . *εc*0. There is no difference between creep loading at room temperature and tensile-test loading. Therefore, the strain rate at the moment when the room-temperature creep loading is completed is equal to the rate when the creep is just carried out, and the creep stress is equal to the stress at the end of the loading.

The Ramberg–Osgood model [25] is usually used to describe the stress–strain curve of steel. This model was put forward in 1943. The main idea is that the strain of a material is composed of elastic deformation and plastic deformation. The nominal flow limit, *σ*0.2, of a material is selected by the classical method, and the corresponding deformation *ε*0.2 = 0.002, then the equation for the Ramberg–Osgood model is:

$$
\varepsilon = \frac{\sigma}{E} + 0.002 \left( \frac{\sigma}{\sigma\_{0.2}} \right)^n \,\tag{9}
$$

where: *σ*0.2—nominal flow limit; *n*—strain hardening coefficient.

The strain hardening coefficient can be selected by the classical method. If *σ*= *σ*0.1 is used [26], the strain hardening coefficient is:

$$m = \frac{\ln(\varepsilon\_{0.2}/\varepsilon\_{0.1})}{\ln(\sigma\_{0.2}/\sigma\_{0.1})} \tag{10}$$

The above equation is accurate when the stress is less than the nominal flow limit, *σ*0.2, but when the stress exceeds the nominal flow limit the calculated result of this model is larger than the actual result.

On the basis of the Ramberg–Osgood model [25], Kim J. R. Rasmussen [27] put forward the method of subsection fitting through experimental research. The boundary point is the nominal flow limit, *σ*0.2. When the stress is less than the nominal flow limit, the Ramberg– Osgood model is used. After *σ*0.2 is exceeded, the Ramberg–Osgood model is calculated in the translation coordinate system. Through a large number of experimental calculations and statistical analysis, an improved Ramberg–Osgood model is obtained:

$$\varepsilon = \begin{cases} \left. \frac{\sigma}{E} + 0.002 \left( \frac{\sigma}{\sigma\_{0.2}} \right)^{n} \right| \sigma \le \sigma\_{0.2} \\\ \frac{\sigma - \sigma\_{0.2}}{E\_{0.2}} + \varepsilon\_{\mu} \left( \frac{\sigma - \sigma\_{0.2}}{\sigma\_{\mu} - \sigma\_{0.2}} \right)^{m} + \varepsilon\_{0.2} \sigma \ge \sigma\_{0.2} \end{cases} \tag{11}$$


$$E\_{0.2} = \frac{E}{1 + 0.002n/e'} \tag{12}$$

where: *e*—parameter, *e* = *σ*0.2/*E*; *εu*—total strain at final fracture; *σu*—stress at final fracture, i.e., tensile strength; *m*—index, m = 1 + 3.5*σ*0.2/*σu*; *ε*0.2—*σ*0.2 corresponding total engineering strain, *ε*0.2 = *σ*0.2/*E* + 0.002.

The strain rate at the loading stage can be obtained by deriving the time *t* from both sides of Equation (11) at the same time.

$$\frac{\mathbf{d}\varepsilon}{\mathbf{d}t} = \begin{cases} \left[\frac{1}{E} + 0.002n(\frac{\sigma}{\sigma\_{0.2}})^{n-1}\right] \frac{\mathbf{d}\sigma}{\mathbf{d}t}, \sigma \le \sigma\_{0.2} \\\ \left[\frac{1}{E\_{0.2}} + \varepsilon\_{\mathbf{u}}m(\frac{\sigma - \sigma\_{0.2}}{\sigma\_{\mathbf{u}} - \sigma\_{0.2}})^{m-1}\frac{1}{\sigma - \sigma\_{0.2}}\right] \frac{\mathbf{d}\sigma}{\mathbf{d}t}, \sigma \ge \sigma\_{0.2} \end{cases} \tag{13}$$

It is known that the state at the end of loading is the initial state at the beginning of creep, that is, . *εc*0<sup>−</sup> = . *εc*0<sup>+</sup> = . *<sup>ε</sup>*(*T*1), and . *εc*<sup>0</sup> = *α*/*τ*. By substituting Equation (8), Equation (14) can be obtained:

$$
\varepsilon\_{\rm c} = \mathfrak{a}(\sigma\_{\rm c}) \ln[1 + \frac{t}{\mathfrak{a}(\sigma\_{\rm c})} \dot{\varepsilon}\_{1}(\sigma\_{\rm c})],
\tag{14}
$$

where *σ<sup>c</sup>* is the constant stress in the creep stage of the material and its value is equal to the material stress at the completion of loading. Therefore, . *ε*(*σc*) = . *ε*(*T*1). By using Equations (12) and (13), the creep value increases with the increase of creep time.

#### *2.2. Spring Material and Size Parameters*

The spring material is 65 Mn, and its specific performance parameters are shown in Table 1.


**Table 1.** Mechanical properties of 65 Mn Steel.

The spring calculation process has been mentioned in another article [8], and the spring size parameters are shown in Table 2 below.

**Table 2.** Main parameters of a torsion spring bearing the torque of 300 N·mm.


#### *2.3. Calculation of Temperature Creep in Spring Chamber*

According to article [28], the maximum stress of a cylindrical helical torsional spring when it only withstands external torque T is:

$$\sigma\_{\rm bb} = -\frac{\cos^3 a}{z\_{\rm m} \mathcal{C}} T[0.154 + (0.246 \cos^2 a - 0.096 \sin^2 a) \frac{1}{\mathcal{C}}] \tag{15}$$

$$\sigma\_{\rm tt} = -\frac{\cos\alpha}{z\_{\rm mf}} T[1 + 0.871 \frac{\cos^2\alpha}{\mathcal{C}} + (0.032 \sin^2\alpha + 0.642 \cos^2\alpha) \frac{\cos^2\alpha}{\mathcal{C}}],\tag{16}$$

and

$$\tau\_{\rm bb} = \tau\_{\rm bt} = \frac{\sin a}{z\_l} T[1 + 0.635 \frac{\cos^2 a}{\mathcal{C}} + 0.163 \frac{\cos^4 a}{\mathcal{C}^2}] \,\tag{17}$$

where: *a*—torsion spring mounting Angle; *zm*—flexural section coefficient; *C*—spring index; *zt*—torsion section coefficient; *α*—helical angle.

According to Mohr's strength theory, the equivalent stress at the danger point of the spring is:

$$
\sigma^\* = \frac{1-m}{2}(\sigma\_{\rm tt} + \sigma\_{\rm bb}) + \frac{1+m}{2}\sqrt{(\sigma\_{\rm tt} - \sigma\_{\rm bb})^2 + 4\tau\_{\rm t}^2} \tag{18}
$$

and

$$m = \frac{\sigma\_{\rm st}}{\sigma\_{\rm sc}} \le 1 \tag{19}$$

where: *σst*—tensile yield point; *σsc*—compressive yield point.

When the spring is compressed by 30◦ , the moment of a single spring is 300 N·mm. By substituting the relevant values in Table 2 into Equations (15)–(17), *σbb* = 16.869 MPa, *σtt* = 539.33 MPa, and *τbt* = 20.99 MPa can be obtained. By substituting these three values into Equation (18) (where m = 0.9231), the equivalent stress, *σ* \* = 525.38 MPa, of the spring danger point can be obtained.

Since the equivalent stress *σ* <sup>∗</sup> = 525.38 MPa is ≤ *σ*0.2 at the spring danger point, the strain rate at the spring danger point under constant external load can be obtained by substituting the parameter into Equation (13).

$$\frac{\mathbf{d}\varepsilon}{\mathbf{d}t} = [\frac{1}{E\_0} + \frac{0.002n}{\sigma\_{0.2}}(\frac{\sigma}{\sigma\_{0.2}})^{n-1}]\frac{\mathbf{d}\sigma}{\mathbf{d}t} \tag{20}$$

When the equivalent stress *σ* ≤ *σ*0.2, in the loading stage, the relationship between stress and strain rate is given by Equation (20). Assuming that the loading rate is 1, the relationship between them is shown in Figure 1. When the equivalent stress ≤.ଶ, in the loading stage, the relationship between stress and strain rate is given by Equation (20). Assuming that the loading rate is 1, the relationship between them is shown in Figure 1.

σ

0 0.2 0.2 d 1 0.002 d [ ( )] d d

st sc *m* 1 σ

When the spring is compressed by 30°, the moment of a single spring is 300 N∙mm. By substituting the relevant values in Table 2 into Equations (15)-(17), = 16.869 MPa, ௧௧ = 539.33 MPa, and ௧ = 20.99 MPa can be obtained. By substituting these three values into Equation (18) (where m = 0.9231), the equivalent stress, ∗ = 525.38 MPa, of the

Since the equivalent stress ∗ = 525.38 MPa is ≤ .ଶ at the spring danger point, the strain rate at the spring danger point under constant external load can be obtained by


*tE t* (20)

σσ

*n <sup>n</sup>*

 σ

= ≤ (19)

σ

*Materials* **2023**, *16*, x FOR PEER REVIEW 6 of 15

where: ௦௧—tensile yield point; ௦—compressive yield point.

ε

= +

spring danger point can be obtained.

substituting the parameter into Equation (13).

and

**Figure 1.** Relationship between stress and strain rate. **Figure 1.** Relationship between stress and strain rate.

Upon substitution of the equivalent stress of the spring danger point into Equation (20), and applying the result to Equation (14), the calculation formula for creep strain under external load T = 300 N·mm is: Upon substitution of the equivalent stress of the spring danger point into Equation (20), and applying the result to Equation (14), the calculation formula for creep strain under external load T = 300 N·mm is:

$$\varepsilon\_{\rm c} = \mathfrak{a}(\sigma\_{\rm c}) \ln[1 + \frac{t}{\mathfrak{a}(\sigma\_{\rm c})} (\frac{1}{E\_0} + \frac{0.002n}{\sigma\_{0.2}} (\frac{\sigma}{\sigma\_{0.2}})^{n-1}) \frac{\mathbf{d}\sigma}{\mathbf{d}t}]. \tag{21}$$

c 0 0.2 0.2

In Equation (21), and α() can be obtained by fitting the data obtained from tensile and creep tests, where *n* = 1.113 and α() is: In Equation (21), *n* and α(*σc*) can be obtained by fitting the data obtained from tensile and creep tests, where *n* = 1.113 and α(*σc*) is:

c c

$$
\mathfrak{a}(\sigma\_{\mathfrak{c}}) = 8.1427 \times 10^{-9} \sigma\_{\mathfrak{c}}.\tag{22}
$$

According to the relevant experiment experience of loading rate, in the tensile test, the value of loading rate is generally set at 5~40 MPa/min. According to relevant literature [29], the magnitude of strain in the first stage of the material obtained from loading rates within the range of 5~40 MPa/min remains basically unchanged, with slight differences in subsequent stages, but the difference is not significant. Therefore, to simplify the calculation, the loading rate is selected as 20 MPa/min, that is, d/ = 0.333 MPa/s. According to the relevant experiment experience of loading rate, in the tensile test, the value of loading rate is generally set at 5~40 MPa/min. According to relevant literature [29], the magnitude of strain in the first stage of the material obtained from loading rates within the range of 5~40 MPa/min remains basically unchanged, with slight differences in subsequent stages, but the difference is not significant. Therefore, to simplify the calculation, the loading rate is selected as 20 MPa/min, that is, d*σ*/*dt* = 0.333 MPa/s.

By integrating Equations (14), (20) and (22), and taking the loading rate as 0.333 MPa/s, the expression formula of creep stress can be obtained as follows: By integrating Equations (14), (20) and (22), and taking the loading rate as 0.333 MPa/s, the expression formula of creep stress can be obtained as follows:

$$
\varepsilon\_{\rm c} = a(\sigma\_{\rm c}) \ln[1 + \frac{t}{a(\sigma\_{\rm c})} (\frac{1}{3E} + \frac{0.002n}{3\sigma\_{0.2}} (\frac{\sigma\_{\rm c}}{\sigma\_{0.2}})^{n-1})].\tag{23}
$$

It can be seen from the above formula that creep strain is mainly related to time and stress, and the relationship between the three is shown in Figure 2.

As can be seen in Figure 2, the room-temperature creep of a cylindrical helical torsion spring shows a typical creep curve trend when the stress is determined. In the case of low stress, the creep of the torsion spring enters the second stage of stable creep in a short time. In the stable creep stage, the creep strain rate is small, and there is little increase in creep strain with time. In the condition of high stress, the first stage of creep of a torsion spring ends after a longer time, and the creep of the torsion spring enters the second stage of stable creep after a longer time. It can be seen that, in the same case, the greater the stress produced by a torsion spring, the longer the time it experiences in the first creep stage. In the stable creep stage, compared with the lower stress condition, the creep strain rate is larger, and the increase of creep strain is larger for a long time.

After derivation of time t on both sides of Equation (23), the relationship between creep rate of a cylindrical helical torsional spring, stress, and time can be obtained, as shown in Figure 3. As can be seen from Figure 3, the creep rate decreases significantly with the increase of time. Compared with a state of low stress, the time of torsional spring creep in the first stage of creep increases obviously in a state of high stress, and the creep rate in the second stage of creep also increases obviously. c c c 0.2 0.2 1 0.002 ( )ln[1 ( ( ) )] ( )3 3 ε ασ ασ σ σ <sup>−</sup> =+ + *<sup>n</sup> t n <sup>E</sup>* . (23) It can be seen from the above formula that creep strain is mainly related to time and stress, and the relationship between the three is shown in Figure 2. **Figure 2.** Relation of creep strain to stress and time. As can be seen in Figure 2, the room-temperature creep of a cylindrical helical torsion spring shows a typical creep curve trend when the stress is determined. In the case of low

stress, the creep of the torsion spring enters the second stage of stable creep in a short time.

c 1

c 1

(23)

σ

 σ

*<sup>E</sup>* .

c 0.2 0.2

 σ

1 0.002 ( )ln[1 ( ( ) )] ( )3 3

<sup>−</sup> =+ + *<sup>n</sup> t n*

It can be seen from the above formula that creep strain is mainly related to time and

ασ

σ

*Materials* **2023**, *16*, x FOR PEER REVIEW 7 of 15

stress, and the relationship between the three is shown in Figure 2.

*Materials* **2023**, *16*, x FOR PEER REVIEW 7 of 15

c c

ε ασ

**Figure 2.** Relation of creep strain to stress and time. **Figure 2.** Relation of creep strain to stress and time. creep in the first stage of creep increases obviously in a state of high stress, and the creep rate in the second stage of creep also increases obviously.

shown in Figure 3. As can be seen from Figure 3, the creep rate decreases significantly **Figure 3.** Relation of creep rate to stress and time. **Figure 3.** Relation of creep rate to stress and time.

creep in the first stage of creep increases obviously in a state of high stress, and the creep rate in the second stage of creep also increases obviously. Since the stress values in each part of the spring are not equal and cannot be calculated in detail, it is not practical to calculate the creep strain. In order to simplify the calculation and improve the safety margin, the creep strain values at the spring danger points are chosen to replace the creep strain values at each part of the spring. It can be Since the stress values in each part of the spring are not equal and cannot be calculated in detail, it is not practical to calculate the creep strain. In order to simplify the calculation and improve the safety margin, the creep strain values at the spring danger points are chosen to replace the creep strain values at each part of the spring. It can be seen from the above that the equivalent stress of the spring danger point is *σ* ∗ = 525.38 MPa, and the relationship between creep strain value and time can be obtained by substituting it into Equation (23), as shown in Figure 4.

with the increase of time. Compared with a state of low stress, the time of torsional spring

It can be seen from Figure 4 that the creep strain curve after the torsional spring loading is in line with the first and second stages of the theoretical creep curve. With the increase of time, the creep strain continues to increase and the rate decreases to a fixed value. The creep strain of the torsional spring will not enter the third stage because it is at room temperature and the loading stress is not large.

In the elastic deformation stage, according to data [30], the stress–strain relationship in pure bending can be written as:

$$\varepsilon\_{\rm e} = \frac{(\rho + y)\mathrm{d}\theta - \rho \mathrm{d}\theta}{\rho \mathrm{d}\theta} = \frac{y}{\rho'} \tag{24}$$

Since the stress values in each part of the spring are not equal and cannot be calculated in detail, it is not practical to calculate the creep strain. In order to simplify the calculation and improve the safety margin, the creep strain values at the spring danger points are chosen to replace the creep strain values at each part of the spring. It can be where: *y* is the distance between the linear strain on the section and the neutral axis, and assuming that each fiber is only subject to axial tension and compression, it can be obtained according to Hooke's law:

$$
\sigma = E\varepsilon = E\frac{y}{\rho} \tag{25}
$$

1 *ρ* = *M EI<sup>z</sup>* . (26)

Therefore, the stress–strain relationship at the lower boundary of the section in the elastic stage of a torsion spring is: *Materials* **2023**, *16*, x FOR PEER REVIEW 8 of 15

$$
\varepsilon\_{\mathbf{e}} = \frac{yM}{EI\_z}.\tag{27}
$$

According to the relationship between creep strain and elastic strain of a torsion spring, the change of rotation angle during creep of a torsion spring can be obtained, as shown in Figure 5. seen from the above that the equivalent stress of the spring danger point is ∗ = 525.38 MPa, and the relationship between creep strain value and time can be obtained by substituting it into Equation (23), as shown in Figure 4.

**Figure 4.** Relationship between creep strain and time of torsion spring. **Figure 4.** Relationship between creep strain and time of torsion spring.

assuming that each fiber is only subject to axial tension and compression, it can be obtained according to Hooke's law: **Figure 5.** Relationship between creep angle and time of torsion spring. **Figure 5.** Relationship between creep angle and time of torsion spring.

σ ε ρ = = *<sup>y</sup> <sup>E</sup> <sup>E</sup>* (25) and In this paper, the creep process of a torsion spring is calculated theoretically, and the creep strain of a torsion spring at room temperature, and the relationship between creep angle and time, are obtained. The following uses the finite-element method to simulate the creep process of a torsion spring. In this paper, the creep process of a torsion spring is calculated theoretically, and thecreep strain of a torsion spring at room temperature, and the relationship between creep angle and time, are obtained. The following uses the finite-element method to simulate the creep process of a torsion spring.

ρ

(27)

1 ρ = *z M EI* . (26) Therefore, the stress-strain relationship at the lower boundary of the section in the The simulation was carried out by using Abaqus. One end of the torsion spring was fixed, and a torque of 300 N·mm was applied to the other end. In the first analysis step, a torque of 300 N·mm was applied to make the spring undergo elastic deformation, and the time was 1 s. The second analysis step was creep analysis, which lasted for 54,000 s. The displacement results obtained are shown in Figure 6. The simulation was carried out by using Abaqus. One end of the torsion spring was fixed, and a torque of 300 N·mm was applied to the other end. In the first analysis step, a torque of 300 N·mm was applied to make the spring undergo elastic deformation, and the time was 1 s. The second analysis step was creep analysis, which lasted for 54,000 s. The displacement results obtained are shown in Figure 6.

> e ε=

> > *z*

*EI* .

*yM*

According to the relationship between creep strain and elastic strain of a torsion spring, the change of rotation angle during creep of a torsion spring can be obtained, as

(**a**) (**b**) **Figure 6.** Torsion spring deformation diagram: (**a**) elastic deformation diagram (**b**) creep defor-

The shadow in Figure 6 is the image before the torsion spring is deformed. It can be seen from the figure that the displacement at the elastic deformation stage is 8.761 mm from the farthest point of the central axis of the spring. After 54,000 s, its deformation increases to 8.847 mm. Compared with the elastic stage, the creep deformation is 0.98% of the elastic deformation and the creep strain is 2.45 × 10-5. In order to display the creep curve more clearly, the curve at the elastic deformation stage is ignored and only the curve within a period of time at the beginning of creep is truncated. The point is selected as the lower endpoint of the torsional spring applying force, and the creep curve is shown in

and

elastic stage of a torsion spring is:

shown in Figure 5.

mation diagram.

Figure 7.

In this paper, the creep process of a torsion spring is calculated theoretically, and the creep strain of a torsion spring at room temperature, and the relationship between creep angle and time, are obtained. The following uses the finite-element method to simulate

The simulation was carried out by using Abaqus. One end of the torsion spring was fixed, and a torque of 300 N·mm was applied to the other end. In the first analysis step, a torque of 300 N·mm was applied to make the spring undergo elastic deformation, and the time was 1 s. The second analysis step was creep analysis, which lasted for 54,000 s. The

**Figure 5.** Relationship between creep angle and time of torsion spring.

displacement results obtained are shown in Figure 6.

the creep process of a torsion spring.

**Figure 6.** Torsion spring deformation diagram: (**a**) elastic deformation diagram (**b**) creep deformation diagram. **Figure 6.** Torsion spring deformation diagram: (**a**) elastic deformation diagram (**b**) creep deformation diagram.

The shadow in Figure 6 is the image before the torsion spring is deformed. It can be seen from the figure that the displacement at the elastic deformation stage is 8.761 mm from the farthest point of the central axis of the spring. After 54,000 s, its deformation increases to 8.847 mm. Compared with the elastic stage, the creep deformation is 0.98% of the elastic deformation and the creep strain is 2.45 × 10-5. In order to display the creep curve more clearly, the curve at the elastic deformation stage is ignored and only the curve within a period of time at the beginning of creep is truncated. The point is selected as the lower endpoint of the torsional spring applying force, and the creep curve is shown in The shadow in Figure 6 is the image before the torsion spring is deformed. It can be seen from the figure that the displacement at the elastic deformation stage is 8.761 mm from the farthest point of the central axis of the spring. After 54,000 s, its deformation increases to 8.847 mm. Compared with the elastic stage, the creep deformation is 0.98% of the elastic deformation and the creep strain is 2.45 <sup>×</sup> <sup>10</sup>−<sup>5</sup> . In order to display the creep curve more clearly, the curve at the elastic deformation stage is ignored and only the curve within a period of time at the beginning of creep is truncated. The point is selected as the lower endpoint of the torsional spring applying force, and the creep curve is shown in Figure 7. *Materials* **2023**, *16*, x FOR PEER REVIEW 10 of 15

**Figure 7.** Creep simulation displacement of torsion spring. **Figure 7.** Creep simulation displacement of torsion spring.

It can be seen from Figure 7 that the creep simulation curve of a torsion spring is similar to the theoretical calculation curve, and a comparison between the simulation curve and the theoretical curve is shown in Figure 8. It can be seen from Figure 7 that the creep simulation curve of a torsion spring is similar to the theoretical calculation curve, and a comparison between the simulation curve and the theoretical curve is shown in Figure 8.

It can be seen from Figure 8 that the theoretical calculation value of torsion spring creep is slightly less than the simulation value in the early stage, and the theoretical calculation value is slightly greater than the simulation result as time goes on. In the later period, the theoretical calculation is smaller than the simulation result. At 54,000 s, the theoretical calculation value of torsion spring creep is 0.2935◦ , while the simulation result is 0.3105◦ , which is 5.79% larger than the theoretical calculation, proving that the theoretical calculation formula is more accurate. Regarding the error between the theoretical calculation and simulation results: on the one hand, it may be because the software adopts traditional agehardening creep theory in the finite-element simulation process, without considering the influence of some material properties, such as interaction and microstructure. In addition, when using this theory, parameters such as creep strain rate, creep activation energy, and initial hardness of materials need to be determined. If the actual material parameters are different from those used in the theoretical calculation, the calculation results will be biased.

It can be seen from Figure 8 that the theoretical calculation value of torsion spring creep is slightly less than the simulation value in the early stage, and the theoretical calculation value is slightly greater than the simulation result as time goes on. In the later period, the theoretical calculation is smaller than the simulation result. At 54,000 s, the theoretical calculation value of torsion spring creep is 0.2935°, while the simulation result is 0.3105°, which is 5.79% larger than the theoretical calculation, proving that the theoretical calculation formula is more accurate. Regarding the error between the theoretical calculation and simulation results: on the one hand, it may be because the software adopts traditional age-hardening creep theory in the finite-element simulation process, without considering the influence of some material properties, such as interaction and microstructure. In addition, when using this theory, parameters such as creep strain rate, creep activation energy, and initial hardness of materials need to be determined. If the actual material parameters are different from those used in the theoretical calculation, the calculation results will be biased. On the other hand, it may be because some small quantities are omitted in the derivation of theoretical formulas, which leads to the change of calculation accuracy. However, on the whole, the error between the theoretical calculation results and the finite-element simulation results is within an acceptable error range, which shows that the theoretical calculation results are more accurate and can accurately predict and estimate the performance and life of a spring in use, which is of great significance for design-

ing high-performance and reliable spring components.

**Figure 8.** Comparison between theory and Simulation of torsion spring.

On the other hand, it may be because some small quantities are omitted in the derivation of theoretical formulas, which leads to the change of calculation accuracy. However, on the whole, the error between the theoretical calculation results and the finite-element simulation results is within an acceptable error range, which shows that the theoretical calculation results are more accurate and can accurately predict and estimate the performance and life of a spring in use, which is of great significance for designing high-performance and reliable spring components. **Figure 7.** Creep simulation displacement of torsion spring. It can be seen from Figure 7 that the creep simulation curve of a torsion spring is similar to the theoretical calculation curve, and a comparison between the simulation curve and the theoretical curve is shown in Figure 8.

*Materials* **2023**, *16*, x FOR PEER REVIEW 10 of 15

**Figure 8.** Comparison between theory and Simulation of torsion spring. **Figure 8.** Comparison between theory and Simulation of torsion spring.

#### It can be seen from Figure 8 that the theoretical calculation value of torsion spring *2.4. Experimental Method of Spring Chamber Temperature Creep 2.4. Experimental Method of Spring Chamber Temperature Creep*

creep is slightly less than the simulation value in the early stage, and the theoretical calculation value is slightly greater than the simulation result as time goes on. In the later period, the theoretical calculation is smaller than the simulation result. At 54,000 s, the theoretical calculation value of torsion spring creep is 0.2935°, while the simulation result is 0.3105°, which is 5.79% larger than the theoretical calculation, proving that the theoretical calculation formula is more accurate. Regarding the error between the theoretical calculation and simulation results: on the one hand, it may be because the software adopts traditional age-hardening creep theory in the finite-element simulation process, without The structure of the cylindrical helical torsional spring loading table is shown in Figure 9, and is mainly composed of two identical unilateral loading mechanisms, a bottom plate, and a steering gear fixed seat. The unilateral loading mechanisms are fixed onto the bottom plate through support legs, and the rudder wing is locked by a bolt onto the gripper. The steering gear is installed on the steering gear seat and is locked by bolts. The deflection of the rudder wing is driven by the clamping claw to rotate the rotating shaft of the two unilateral loading mechanisms, and the torque is provided by the torsion springs in the unilateral loading mechanisms. The structure of the cylindrical helical torsional spring loading table is shown in Figure 9, and is mainly composed of two identical unilateral loading mechanisms, a bottom plate, and a steering gear fixed seat. The unilateral loading mechanisms are fixed onto the bottom plate through support legs, and the rudder wing is locked by a bolt onto the gripper. The steering gear is installed on the steering gear seat and is locked by bolts. The deflection of the rudder wing is driven by the clamping claw to rotate the rotating shaft of the two unilateral loading mechanisms, and the torque is provided by the torsion springs in the unilateral loading mechanisms.

**Figure 9.** Torsion spring load table overall structure (1—unilateral loading mechanism; 2—gripper; 3—steering gear; 4—steering gear fixed seat; 5—bottom plate). **Figure 9.** Torsion spring load table overall structure (1—unilateral loading mechanism; 2—gripper; 3—steering gear; 4—steering gear fixed seat; 5—bottom plate).

The internal structure of a unilateral loading mechanism is shown in Figure 10. The torsional spring loading platform mainly realizes the change of loading torque by replacing the unilateral loading mechanism. Different torsional springs correspond to different torque. The internal structure of a unilateral loading mechanism is shown in Figure 10. The torsional spring loading platform mainly realizes the change of loading torque by replacing the unilateral loading mechanism. Different torsional springs correspond to different torque.

**Figure 10.** Unilateral loading structure internal structure (1—Angle sensor; 2—sensor seat; 3—fastening screws; 4—tail bearing seat; 5—stop cover; 6—torsional spring loading block; 7—torsional

Each unilateral loading mechanism contains two torsion springs that are arranged in a coaxial reverse direction, and the rotating ends of the two torsion springs wind in opposite directions. The installation diagram is shown in Figure 11. Both torsion springs are pre-compressed by 30°. When the axial rotation is on one side, the force of one torsion spring increases and the compression Angle increases, while the compression Angle of

spring; 8—front bearing seat).

the other torsion spring decreases.

torque.

3—steering gear; 4—steering gear fixed seat; 5—bottom plate).

*2.4. Experimental Method of Spring Chamber Temperature Creep* 

springs in the unilateral loading mechanisms.

The structure of the cylindrical helical torsional spring loading table is shown in Figure 9, and is mainly composed of two identical unilateral loading mechanisms, a bottom plate, and a steering gear fixed seat. The unilateral loading mechanisms are fixed onto the bottom plate through support legs, and the rudder wing is locked by a bolt onto the gripper. The steering gear is installed on the steering gear seat and is locked by bolts. The deflection of the rudder wing is driven by the clamping claw to rotate the rotating shaft of the two unilateral loading mechanisms, and the torque is provided by the torsion

**Figure 10.** Unilateral loading structure internal structure (1—Angle sensor; 2—sensor seat; 3—fastening screws; 4—tail bearing seat; 5—stop cover; 6—torsional spring loading block; 7—torsional spring; 8—front bearing seat). **Figure 10.** Unilateral loading structure internal structure (1—Angle sensor; 2—sensor seat; 3 fastening screws; 4—tail bearing seat; 5—stop cover; 6—torsional spring loading block; 7—torsional spring; 8—front bearing seat).

**Figure 9.** Torsion spring load table overall structure (1—unilateral loading mechanism; 2—gripper;

The internal structure of a unilateral loading mechanism is shown in Figure 10. The torsional spring loading platform mainly realizes the change of loading torque by replacing the unilateral loading mechanism. Different torsional springs correspond to different

Each unilateral loading mechanism contains two torsion springs that are arranged in a coaxial reverse direction, and the rotating ends of the two torsion springs wind in opposite directions. The installation diagram is shown in Figure 11. Both torsion springs are pre-compressed by 30°. When the axial rotation is on one side, the force of one torsion spring increases and the compression Angle increases, while the compression Angle of the other torsion spring decreases. Each unilateral loading mechanism contains two torsion springs that are arranged in a coaxial reverse direction, and the rotating ends of the two torsion springs wind in opposite directions. The installation diagram is shown in Figure 11. Both torsion springs are pre-compressed by 30◦ . When the axial rotation is on one side, the force of one torsion spring increases and the compression Angle increases, while the compression Angle of the other torsion spring decreases. *Materials* **2023**, *16*, x FOR PEER REVIEW 12 of 15 *Materials* **2023**, *16*, x FOR PEER REVIEW 12 of 15

**Figure 11.** Double torsion spring installation diagram. **Figure 11.** Double torsion spring installation diagram. **Figure 11.** Double torsion spring installation diagram.

The rotation of the steering gear is transmitted to the torsion springs through the clamping claw, which provides the corresponding torque. The change of the output voltage of the rotary potentiometer is measured by a multimeter, and the rotation angle can be obtained by a certain conversion formula. The rotation of the steering gear is transmitted to the torsion springs through the clamping claw, which provides the corresponding torque. The change of the output voltage of the rotary potentiometer is measured by a multimeter, and the rotation angle can be obtained by a certain conversion formula. The rotation of the steering gear is transmitted to the torsion springs through the clamping claw, which provides the torque. The change of the output voltage of the rotary potentiometer is by a multimeter, and the rotation angle can be obtained by a certain conversion formula.

After loading the weight, the output voltage value of the Angle sensor can be measured by keeping the weight unchanged. After converting the voltage value into Angle, the

After loading the weight, the output voltage value of the Angle sensor can be measured by keeping the weight unchanged. After converting the voltage value into Angle, the

After loading the weight, the angle change value can be obtained through the voltage

After loading the weight, the angle change value can be obtained through the voltage

change value of the voltmeter, and the experimental data are show in Table 3:

change value of the voltmeter, and the experimental data are show in Table 3:

**Time (s) Angle (**°**) Time (s) Angle (**°**)**  0 0 7500 0.174726 60 0.011457 8400 0.177591 210 0.054423 9600 0.180455 780 0.105982 12,420 0.211963 960 0.108846 12,600 0.214828 1080 0.11171 14,700 0.218987 1800 0.114575 18,700 0.226679 2100 0.126032 21,800 0.233096 2220 0.128897 24,900 0.245713 2280 0.131761 28,000 0.245677

**Time (s) Angle (**°**) Time (s) Angle (**°**)**  0 0 7500 0.174726

60 0.011457 8400 0.177591 210 0.054423 9600 0.180455

780 0.105982 12,420 0.211963 960 0.108846 12,600 0.214828

1080 0.11171 14,700 0.218987 1800 0.114575 18,700 0.226679

2100 0.126032 21,800 0.233096 2220 0.128897 24,900 0.245713

The specific experimental process is shown in Figure 12. The specific experimental process is shown in Figure 12. The specific experimental process is shown in Figure 12.

**Figure 12.** Creep test of spring on load table. **Figure 12.** Creep test of spring on load table. 2280 0.131761 28,000 0.245677 **Figure 12.** Creep test of spring on load table.

**3. Experimental Data and Analysis** 

**3. Experimental Data and Analysis** 

**Table 3.** Creep strain data sheet of torsion spring.

**Table 3.** Creep strain data sheet of torsion spring.

tained.

tained.

After loading the weight, the output voltage value of the Angle sensor can be measured by keeping the weight unchanged. After converting the voltage value into Angle, the relationship of the spring shape variable with time and the spring creep curve can be obtained.

#### **3. Experimental Data and Analysis**

After loading the weight, the angle change value can be obtained through the voltage change value of the voltmeter, and the experimental data are show in Table 3:


**Table 3.** Creep strain data sheet of torsion spring.

In order to facilitate direct observation, the data in the table were fitted and plotted, as shown in Figure 13. In order to facilitate direct observation, the data in the table were fitted and plotted, as shown in Figure 13.

**Figure 13.** Creep curve of torsion spring. **Figure 13.** Creep curve of torsion spring.

It can be seen from Figure 13 that the creep test data and its fitting to a curve of a torsion spring conform to a typical creep curve. In the early stage of creep, the data fluctuate greatly, but in the later stage, the data tend to be stable. It is possible that, in the process of spring creep, after the sliding dislocation moves to barriers such as the grain boundary or second phase particles, the movement of dislocation stops gradually under the obstruction, and the phenomenon of accumulation appears. At this time, the number of moving dislocations decreases, which is reflected in the decrease of creep rate, or even It can be seen from Figure 13 that the creep test data and its fitting to a curve of a torsion spring conform to a typical creep curve. In the early stage of creep, the data fluctuate greatly, but in the later stage, the data tend to be stable. It is possible that, in the process of spring creep, after the sliding dislocation moves to barriers such as the grain boundary or second phase particles, the movement of dislocation stops gradually under the obstruction, and the phenomenon of accumulation appears. At this time, the number of moving dislocations decreases, which is reflected in the decrease of creep rate, or even

stagnation. As more and more accretion occurs, the dislocation will climb and slide over the barrier, which may be the reason for the step phenomenon in the creep experiment. Figure 8, in the previous section, shows a comparison between a theoretical and sim-

**Figure 14.** Comparison of creep angle curve of torsion spring.

stagnation. As more and more accretion occurs, the dislocation will climb and slide over the barrier, which may be the reason for the step phenomenon in the creep experiment. of moving dislocations decreases, which is reflected in the decrease of creep rate, or even stagnation. As more and more accretion occurs, the dislocation will climb and slide over the barrier, which may be the reason for the step phenomenon in the creep experiment.

It can be seen from Figure 13 that the creep test data and its fitting to a curve of a torsion spring conform to a typical creep curve. In the early stage of creep, the data fluctuate greatly, but in the later stage, the data tend to be stable. It is possible that, in the process of spring creep, after the sliding dislocation moves to barriers such as the grain boundary or second phase particles, the movement of dislocation stops gradually under the obstruction, and the phenomenon of accumulation appears. At this time, the number

Figure 8, in the previous section, shows a comparison between a theoretical and simulated curve of torsion spring creep Angle. A comparison between theory, simulation, and experimental results is shown in Figure 14. Figure 8, in the previous section, shows a comparison between a theoretical and simulated curve of torsion spring creep Angle. A comparison between theory, simulation, and experimental results is shown in Figure 14.

*Materials* **2023**, *16*, x FOR PEER REVIEW 13 of 15

as shown in Figure 13.

**Figure 13.** Creep curve of torsion spring.

3300 0.134625 32,100 0.2519 3600 0.13749 36,120 0.264493 3900 0.140354 40,120 0.265124 5200 0.15754 44,120 0.271456 5300 0.163269 49,000 0.277328 5340 0.166133 50,800 0.279596 5520 0.171862 54,000 0.281319

In order to facilitate direct observation, the data in the table were fitted and plotted,

**Figure 14. Figure 14.** Comparison of creep angle curve of torsion spring. Comparison of creep angle curve of torsion spring.

As can be seen from Figure 14, the trend in creep strain for the theoretical calculation and finite-element simulation is the same as that of the experiment, and they all enter the stable creep stage at about 15,000 s. The theoretical calculation curve is highly consistent with the experimental fitting curve. At the initial stage of creep, the theoretical calculation curve basically coincides with the experimental data, and at the later stage of creep, the theoretical calculation creep rate is not much different from the experimental fitting curve. On the one hand, the source of error is the loss of precision caused by omitting some tiny quantities in the derivation of the theoretical calculation formula, on the other hand, it may be due to the slightly different properties of spring materials and materials in the data. At 54,000 s, the theoretical calculation is 4.3% larger than the experimental data creep value, and the error is acceptable in practical engineering application, which shows the accuracy of the theoretical calculation of torsion spring creep. As such it can be used to predict and estimate the performance and life of a spring in use, so as to avoid creep failure under high stress or long-term load conditions, thus ensuring the reliability and safety of mechanical components.

#### **4. Conclusions**

In this paper, the creep effect of a cylindrical helical torsion spring under pre-compression is studied. Firstly, through deconstruction and reorganization, using the original roomtemperature creep constitutive equation as a basis, a room-temperature creep constitutive equation of a torsion spring can be obtained based on macroscopic tensile tests. Through the stress–strain analysis of a spring, the relationship between creep strain and time and stress of a cylindrical helical torsion spring is obtained, and the correctness of the theoretical calculation is verified by finite-element simulation.

Finally, in order to determine the influence of spring creep effect, a creep strain experiment of a torsion spring is carried out. Through experiments, it can be concluded that the cylindrical helical torsion spring enters a stable creep stage after about 15,000 s, and the creep strain angle is 0.28◦ after 54,000 s. By comparing the theoretical calculation results with the experimental results, it can be concluded that the error between the theoretical formula calculation results and the experimental results of a torsion spring is 4.3%, which is less than the engineering error range of 5%, which meets the requirements of engineering measurement and is of great significance for more accurately predicting and estimating the performance and life of a spring in use, and designing high-performance and reliable spring elements.

**Author Contributions:** Methodology, P.R.; Validation, B.Z. and H.M.; Writing—review & editing, Z.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by NNSFC (National Natural Science Foundation of China), grant number 51879063.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Article* **Ballistic Impacts with Bullet Splash—Load History Estimation for .308 Bullets vs. Hard Steel Targets**

**Riccardo Andreotti 1,2,\*, Andrea Casaroli <sup>1</sup> , Ivan Colamartino <sup>1</sup> , Mauro Quercia <sup>2</sup> , Marco Virginio Boniardi <sup>1</sup> and Filippo Berto <sup>3</sup>**


**Abstract:** The study focuses on testing a simplified way of estimating the resultant force due to ballistic impacts resulting in a full fragmentation of the impactor with no penetration of the target. The method is intended to be useful for the parsimonious structural assessment of military aircrafts with integrated ballistic protection systems by means of large scale explicit finite element simulations. The research investigates the effectiveness of the method in allowing the prediction of the fields of plastic deformation collected by hard steel plates impacted by a wide range of semi-jacketed, monolithic, and full metal jacket .308 Winchester rifle bullets. The outcomes show the effectiveness of the method being strictly related to the full compliance of the considered cases with the bullet-splash hypotheses. The study therefore suggests the application of the load history approach only after careful experimental investigations on the specific impactor–target interactions.

**Keywords:** bullet splash; bullet-splash; ballistic impact; terminal ballistics; finite element method; ballistic protection; explicit solver; impact simulation; Creusabro®; Durostat®

## **1. Introduction**

For two decades, transient finite element simulation has represented a fundamental tool for the crashworthiness assessment and structural optimization of vehicles [1–3]. In the aerospace industry, airframe development passes through many steps of optimization against potential threats for the survivability of the vehicle such as bird impacts, hail impacts, emergency landing, ditching, and wire strike. All of these potential threats are faced by means of dedicated simulations involving extended parts or even the entire airframe, modelled in finite elements to verify the survivability of every layer of material and every single rivet connecting the structures. In this technological scenario, the development of a modern military aircraft follows the same methodological approach with the obvious complications due to the main aim of a military vehicle: to maximize the probability of surviving a real combat scenario. This implies that the critical components of the vehicle must be protected against the typical threat of warfare: ballistic impacts [4,5]. To do this, critical areas of the vehicle, such as the cockpit and the transmission, are surrounded by ballistic protection panels, which are experimentally verified to be able to withstand specific ballistic impacts without allowing any penetration of the bullets. In this field the aim of the simulations is therefore not to verify the ability of the protective panels to withstand the impact of the bullets but instead to verify the strength of the support structures to which the panels are attached against the impulsive loads due to several impact positions along the entire airframe, representing the random incidence of the shots during fire fights. In this scenario it is therefore fundamental to be able to model the stress propagation through the structure to verify the strength of every connection and part of the support system. This

**Citation:** Andreotti, R.; Casaroli, A.; Colamartino, I.; Quercia, M.; Boniardi, M.V.; Berto, F. Ballistic Impacts with Bullet Splash—Load History Estimation for .308 Bullets vs. Hard Steel Targets. *Materials* **2023**, *16*, 3990. https://doi.org/10.3390/ ma16113990

Academic Editors: Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 19 March 2023 Revised: 19 May 2023 Accepted: 23 May 2023 Published: 26 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

involves many different load cases involving significant parts of the airframe modelled in full detail.

Protective panels typically consist of an external layer of hard ceramic tiles intended to fragment the hitting bullets during the first phase of the impact, and an internal layer made of composite fabric to withstand the local impact forces and absorb the residual energy without allowing the bullet fragments to penetrate the protection panel [6]. Therefore, the typical scenario of a perfectly working protection panel consists of a total fragmentation of the bullet with no penetration of the target. This phenomenon is called bullet splash, due to the fact that the fragmentation of the bullet makes its kinematics similar to the flow of a mass of a fluid being deflected by the target surface [7,8]. In this scenario Andreotti et al. (2021) first proposed a simplified approach to reduce the computational cost needed to simulate the forces acting on the target due to bullet splash, by considering the phenomenon as a fluid-structure interaction where a mass of fluid represents the behavior of the bullet fragments. The approach proved to be effective in predicting the local and global effects of bullet splashes due to 9 × 21 mm full metal jacket bullets hitting 4 mm AISI 304 plates [9]. Based on that experience, Andreotti et al. (2022) [10] proposed and validated a further simplified method to make the finite elements simulations more efficient and equally effective by avoiding the hydrodynamic part of the calculation, by introducing an estimated load history approach that consists in decoupling the bullet fragmentation phenomenon and the target reaction to avoid the costs of modelling the bullet, and introducing an equivalent impact force as a load curve *F(t)* to be directly applied to the impact point on the structure; the intensity vs. time curve is calculated based on the initial impact velocity and the density distribution of the specific bullets. The approach proved to be effective and extremely efficient for 9 × 21 mm full metal jacket bullets hitting 4 mm AISI 304 plates as well as for monolithic copper .308 rifle bullets hitting high hardness steel plates. The validation confirmed the equivalence between the fluid structure interaction and the load history approach in terms of stress waves propagating from the epicenter of the target, resultant forces at the constraint, and residual deformation fields of the plates [11,12].

In this paper we propose the application and experimental validation of the estimated load history approach to cover a wide range of rifle bullet typologies, from monolithic to partially jacketed and full metal jacket bullets. The validation has been performed by comparing the residual deformation field measured on steel plates impacted by the bullets and the residual deformation field predicted by finite element simulations where the applied load history was estimated based on experimental measurements of the impact velocity and density distribution of the tested bullets. The work gives the reader a reference for the effective applicability of the load history method for finite element simulations intended to simulate the stress propagation from ballistic protection panels to the support structures to assess, without the need to model the bullets, their fragmentation, and the interactions with the targets, therefore allowing a more parsimonious approach to the problem. This method is based on the hypothesis that the protective panel is able to withstand the ballistic impacts of the considered bullets at a certain velocity, resulting in a complete fragmentation of the bullet with no penetration or fragmentation of the target. Within these hypotheses the method consists in estimating the force history due to the impacts based only on the bullet sections and their impact velocity, and applying the estimated load history directly to the elements of the structure to simulate the structural phenomenon. Section 2 provides the reader with all the information to reproduce the conducted tests. In Section 3 we provide all the main analytical, numerical, and experimental results involved in the validation process. In Section 4 the results are discussed. In Section 5 we summarize the conclusions and further developments of the research.

#### **2. Materials and Methods**

To validate the load-history approach on a wide range of bullet typologies, we took into consideration four different .308 rifle bullets hitting dual-layered steel plates made of two different steels. Both steels have high strength and significant ductility so that they can withstand the impacts without allowing the penetration of the bullets, and, at the same time, are able to collect residual deformations so that the field of residual displacements could be used as experimental evidence to compare with the numerical results for the validation of the approach. two different steels. Both steels have high strength and significant ductility so that they can withstand the impacts without allowing the penetration of the bullets, and, at the same time, are able to collect residual deformations so that the field of residual displacements could be used as experimental evidence to compare with the numerical results for the validation of the approach.

To validate the load-history approach on a wide range of bullet typologies, we took into consideration four different .308 rifle bullets hitting dual-layered steel plates made of

*Materials* **2023**, *16*, x FOR PEER REVIEW 3 of 33

## *2.1. Ballistic Test Procedures*

**2. Materials and Methods** 

The bullets were shot using a Remington 700 cal. 0.308 Winchester. The initial bullet velocity was measured using a Magnetospeed® V3 ballistic chronograph assembled at the rifle muzzle. Four different caliber .308 Winchester types of bullet were used: Soft Point (SP), Hollow Point (HP), Full Metal Jacket (FMJ) and Monolithic (MONO) bullets. Their main characteristics are summarised in Table 1. The external geometry and longitudinal section of the bullets was documented through a LEICA® M165 stereo microscope at 3.5× magnification. The bullets were sectioned using 120 grit abrasive paper, lubricated using water (Figure 1). *2.1. Ballistic Test Procedures*  The bullets were shot using a Remington 700 cal. 0.308 Winchester. The initial bullet velocity was measured using a Magnetospeed® V3 ballistic chronograph assembled at the rifle muzzle. Four different caliber .308 Winchester types of bullet were used: Soft Point (SP), Hollow Point (HP), Full Metal Jacket (FMJ) and Monolithic (MONO) bullets. Their main characteristics are summarised in Table 1. The external geometry and longitudinal section of the bullets was documented through a LEICA® M165 stereo microscope at 3.5× magnification. The bullets were sectioned using 120 grit abrasive paper, lubricated using water (Figure 1).

**Table 1.** Main data of the bullets used for experimentation as provided by the manufacturers. The nominal muzzle velocity is the expected velocity at the exit of the firearm. The nominal energy is the expected kinetic energy of the bullets at the exit of the firearm. **Table 1.** Main data of the bullets used for experimentation as provided by the manufacturers. The nominal muzzle velocity is the expected velocity at the exit of the firearm. The nominal energy is the expected kinetic energy of the bullets at the exit of the firearm.


**Figure 1.** External geometry and longitudinal section of the bullets used for experimentation; (**a**) Soft Point, (**b**) Hollow Point, (**c**) Full Metal Jacket and (**d**) Monolithic. **Figure 1.** External geometry and longitudinal section of the bullets used for experimentation; (**a**) Soft Point, (**b**) Hollow Point, (**c**) Full Metal Jacket and (**d**) Monolithic.

The targets were obtained by joining two 4 × 500 × 500 mm plates. Two different high-strength structural steels were tested; their mechanical and metallurgical properties are detailed in Section 2.2. Four tests were carried out on each metal plate using the bullets detailed in Figure 1 and in Table 1. The nominal impact positions are detailed in Section 2.3.

The targets were positioned at 100 m from the rifle muzzle, oriented normally to the bullet trajectory. The metal plates were bolted to a steel and wood support frame specially fabricated to carry out the tests. The support frame was designed to guarantee full visibility of the metal plate side from the cameras used to film the impacts. A panel with a chessboard pattern composed of black and white squares with a 20 mm side was installed on the opposite side. This solution was adopted to guarantee a dimensional reference for the high frequency camera shootings.

Upon completion of the tests, the bullet impact areas were accurately photographed both on the front and rear sides of the metal plates; the pictures are provided in Section 3.1. The dome-shaped bulges created by the bullets in the rear surface of the metal plates were also measured using a Borletti dial gauge with a sensitivity of 10 µm. The measurement procedure included tracing four reference lines slanted by 45◦ relative to each other. All the lines cross the apex of the bulges which was adopted as zero point. 14 measures of metal plate deformation were taken for each of the four lines, at a distance of ±5 mm, ±10 mm, ±15 mm, ±25 mm, ±35 mm, ±45 mm, and ±55 mm from the bulge apex. The values so obtained were plotted in diagrams (Section 3.4), each showing the minimum and maximum deformation of the metal plate measured at increasing distance from the bulge apex.

#### *2.2. Mechanical and Metallurgical Characterisation of the Metal Plates*

The targets were fabricated using high-yield stress structural steel plates. The two steel grades selected for the experimentation go under the trade names of Creusabro® 8000 [13–15] and Durostat® 400 [16]. They are characterised by a combination of very high tensile strength and optimum wear resistance. Their typical applications are ballistic shields and barriers, buckets for heavy equipment, dump truck beds, conveyor components, and rollers and rotors for grain mills and oil mills. The high level of hardness and tensile strength is obtained through controlled cooling after hot rolling. Their limited carbon content, together with a considerable manganese content (or manganese-nickel content in the case of Creusabro® 8000), allows them to have acceptable toughness and weldability characteristics, notwithstanding very high values of hardness and tensile strength. The significant amount of chromium and molybdenum guarantees the hardenability necessary to form a mixed martensite and bainite structure. Both steel grades are characterised by micrometric carbides finely dispersed in the metal and further improving abrasive wear resistance [14].

In comparison with Durostat® 400 steel, which is composed exclusively of martensite, bainite and carbides, Creusabro® 8000 steel also exhibits a small amount of residual austenite that turns into martensite when it is subjected to cold deformation. This property, known under the acronym TRIP (Transformation Induced by Plasticity), further improves the wear resistance and impact strength of Creusabro® 8000 steel compared to Durostat® 400 steel. The TRIP effect enhances steel hardness and its ability to absorb energy as the cold deformation increases [17,18].

The chemical composition of both steels was assessed by means of optical emission spectroscopy (OES). Tables 2 and 3 provide an overview of the results. Both steels meet the requirements of the applicable steel mill specifications. Comparing the two steels, it is clearly apparent that Creusabro® 8000 is much richer in alloying elements than Durostat® 400. Durostat® 400 does not contain molybdenum, and its carbon, chromium, and silicon content is less than half of the values measured in Creusabro® 8000. On the other hand, Durostat® 400 contains twice the amount of manganese, but it does not contain nickel which is present in a good quantity in Creusabro® 8000. The chemical composition clearly shows that Creusabro® 8000 is by far more hardenable but less weldable than Durostat® 400, compared to which it exhibits an equivalent carbon content that is higher by approximately 25% (CECreusabro®8000 = 0.63; CEDurostat®400 = 0.50). Having received the same heat treatment, Creusabro® 8000 is also harder and mechanically stronger thanks to its carbon content that is more than twice that of Durostat® 400.


shows that Creusabro® 8000 is by far more hardenable but less weldable than Durostat® 400, compared to which it exhibits an equivalent carbon content that is higher by approximately 25% (CECreusabro®8000 = 0.63; CEDurostat®400 = 0.50). Having received the same heat treatment, Creusabro® 8000 is also harder and mechanically stronger thanks

**Table 2.** Chemical analyses of Creusabro® 8000 steel (wt%). **Table 2.** Chemical analyses of Creusabro® 8000 steel (wt%).

*Materials* **2023**, *16*, x FOR PEER REVIEW 5 of 33

to its carbon content that is more than twice that of Durostat® 400.

**Table 3.** Chemical analyses Durostat® 400 steel (wt%). **Table 3.** Chemical analyses Durostat® 400 steel (wt%).


Both metal plates were subjected to metallographic analyses and Vickers HV1 hardness tests. The samples were mirror polished using progressively finer abrasive paper (120 grit, 180 grit, 320 grit, 400 grit, 600 grit, and 1200 grit) and polishing cloths with diamond-based synthetic abrasive (grain size 3 µm and 1 µm). Lubrication was guaranteed using water with abrasive paper and the suspension containing the abrasive matter with the cloths. Metallographic etching was obtained using the reagent "Nital 2", a 2% nitric acid/ethanol solution. The pictures were captured using a LEICA® DM4000M optical microscope at 500× magnification. The same samples were also used to evaluate core hardness in the metal plates. The tests were conducted using a Vickers LEITZ®-WETZLAR® (Leica Camera AG, Wetzlar, Germany) micro-durometer fitted with a digital camera and set at the HV1 hardness scale (1 kgf, 30 s). Both metal plates were subjected to metallographic analyses and Vickers HV1 hardness tests. The samples were mirror polished using progressively finer abrasive paper (120 grit, 180 grit, 320 grit, 400 grit, 600 grit, and 1200 grit) and polishing cloths with diamond-based synthetic abrasive (grain size 3 µm and 1 µm). Lubrication was guaranteed using water with abrasive paper and the suspension containing the abrasive matter with the cloths. Metallographic etching was obtained using the reagent "Nital 2", a 2% nitric acid/ethanol solution. The pictures were captured using a LEICA® DM4000M optical microscope at 500× magnification. The same samples were also used to evaluate core hardness in the metal plates. The tests were conducted using a Vickers LEITZ®- WETZLAR® (Leica Camera AG, Wetzlar, Germany) micro-durometer fitted with a digital camera and set at the HV1 hardness scale (1 kgf, 30 s).

The tests confirm what was previously suggested by the chemical composition, i.e., both steel grades form martensite and bainite with carbides finely dispersed in the metal (Figure 2) [19]. This type of investigation could not reveal the small amount of residual austenite prescribed for Creusabro® 8000 steel that, in any case, exhibits an average hardness higher than Durostat® 400 (540 HV1 compared to 420 HV1). The tests confirm what was previously suggested by the chemical composition, i.e., both steel grades form martensite and bainite with carbides finely dispersed in the metal (Figure 2) [19]. This type of investigation could not reveal the small amount of residual austenite prescribed for Creusabro® 8000 steel that, in any case, exhibits an average hardness higher than Durostat® 400 (540 HV1 compared to 420 HV1).

**Figure 2.** Metallographic images of the metal plates used for experimentation, made of (**a**) Creusabro® 8000 steel and (**b**) Durostat® 400 steel. Both steel grades form martensite and bainite with carbides finely dispersed in the metal (etching agent: Nital2—Magnification: 500×). This type of investigation could not reveal the small amount of residual austenite prescribed for Creusabro® 8000 steel. **Figure 2.** Metallographic images of the metal plates used for experimentation, made of (**a**) Creusabro® 8000 steel and (**b**) Durostat® 400 steel. Both steel grades form martensite and bainite with carbides finely dispersed in the metal (etching agent: Nital2—Magnification: 500×). This type of investigation could not reveal the small amount of residual austenite prescribed for Creusabro® 8000 steel.

The characterisation of the two steels was completed by conducting tensile tests at room temperature. Proportional specimens, obtained in compliance with the UNI EN ISO 6892-1:2020 standard, were used [20,21] (Figure 3). Results are summarised in Table 4, while the engineering stress/strain curves are shown in Figure 4.

while the engineering stress/strain curves are shown in Figure 4.

The characterisation of the two steels was completed by conducting tensile tests at room temperature. Proportional specimens, obtained in compliance with the UNI EN ISO

**Figure 3.** Geometry of the specimens used for the tensile tests. Proportional specimens obtained in compliance with the UNI EN ISO 6892-1:2020 standard were used [20]. **Figure 3.** Geometry of the specimens used for the tensile tests. Proportional specimens obtained in compliance with the UNI EN ISO 6892-1:2020 standard were used [20].

> **Table 4.** Summary of tensile tests results for the Creusabro® 8000 steel and Durostat® 400 plates used for experimentation.


 **E [Gpa] YS [Mpa] UTS [Mpa] YS/UTS Ag% A%**  Creusabro® 8000 210 1190 1735 0.68 5 11 **Figure 4.** Engineering stress/strain curves for Creusabro® 8000 steel (red lines) and Durostat® 400 steel (black lines) plates used for experimentation. **Figure 4.** Engineering stress/strain curves for Creusabro® 8000 steel (red lines) and Durostat® 400 steel (black lines) plates used for experimentation.

Durostat® 400 210 1100 1290 0.85 3 11 The tensile tests confirm what had been anticipated by the hardness tests. The ultimate tensile strength and yield strength of Creusabro® 8000 exceed those of Durostat® 400 steel (UTSCreusabro®8000 = 1735 Mpa; YSCreusabro®8000 = 1190 Mpa; UTSDurostat®400 = 1290 Mpa; YSDurostat®400 = 1100 Mpa). The YS/UTS ratio and percentage elongation under maximum load, Ag%, are, respectively, equal to 0.85 and 3% in Durostat® 400 and 0.68 and 5% in Creusabro® 8000. Both steels exhibit analogous values of percentage elongation after **Table 4.** Summary of tensile tests results for the Creusabro® 8000 steel and Durostat® 400 plates used for experimentation.  **E [Gpa] YS [Mpa] UTS [Mpa] YS/UTS Ag% A%**  Creusabro® 8000 210 1190 1735 0.68 5 11 Durostat® 400 210 1100 1290 0.85 3 11 The tensile tests confirm what had been anticipated by the hardness tests. The ultimate tensile strength and yield strength of Creusabro® 8000 exceed those of Durostat® 400 steel (UTSCreusabro®8000 = 1735 Mpa; YSCreusabro®8000 = 1190 Mpa; UTSDurostat®400 = 1290 Mpa; The tensile tests confirm what had been anticipated by the hardness tests. The ultimate tensile strength and yield strength of Creusabro® 8000 exceed those of Durostat® 400 steel (UTSCreusabro ® <sup>8000</sup> = 1735 Mpa; YSCreusabro ® <sup>8000</sup> = 1190 Mpa; UTSDurostat ® <sup>400</sup> = 1290 Mpa; YSDurostat ® <sup>400</sup> = 1100 Mpa). The YS/UTS ratio and percentage elongation under maximum load, Ag%, are, respectively, equal to 0.85 and 3% in Durostat® 400 and 0.68 and 5% in Creusabro® 8000. Both steels exhibit analogous values of percentage elongation after fracture (A%Creusabro ® <sup>8000</sup> = A%Durostat ® <sup>400</sup> = 11%). These values confirm another important aspect revealed by chemical analyses: Creusabro® 8000 steel's strain hardening ability is higher than that of Durostat® 400 steel. Creusabro® 8000 is also superior to Durostat® 400

> YSDurostat®400 = 1100 Mpa). The YS/UTS ratio and percentage elongation under maximum load, Ag%, are, respectively, equal to 0.85 and 3% in Durostat® 400 and 0.68 and 5% in

in terms of plastic deformation ability before reaching the necking point. Therefore, the Creusabro® 8000 steel exhibits superior performance compared to the Durostat® 400 steel in terms of tensile strength, hardness, and energy absorption capability. On the other hand, its weldability is poorer than in Durostat® 400.

#### *2.3. Testing Setup and Impact Speed Computation*

Two Phantom VEO 710 high speed cameras were used to capture the bullets' speed and trajectory. In particular, the first was positioned orthogonally with respect to the trajectory at a distance variable between 2250 and 2550 mm from the trajectory plane; the second one, instead, was positioned at the same orthogonal distance with an angle of 20◦ to obtain a perspective view of the shots. Data were acquired via two ethernet cables with a synchronized digital trigger. Further details for each shot are shown in Tables 5 and 6, while schemes of the data acquisition setup are shown in Figures 5 and 6.

**Table 5.** Data acquisition: camera 1.


**Table 6.** Data acquisition: camera 2.


While the perspective view was meant to capture the three-dimensional fragmentation dynamics of the projectile, the orthogonal videos were recorded to compute the projectile impact and exit speeds, via the well-known time-of-flight principle: knowing a priori the dimensions of a reference object, the video's resolution, and time history, the projectile trajectory is easily transformed into impact velocity. Such a method is one of the most used in ballistics to effectively evaluate the trajectory of bullets [22], with multiple applications in 2D and 3D analyses [23]. In the present work, even if the bullet itself could have been in principle used as reference for calibration, its small dimensions would have introduced a potentially high error; consequently, the reference object was a background reference plane, a chessboard panel attached perpendicularly to the frame. As described in Section 2.1, the chessboard was characterized by regular squares of 20 × 20 mm each, while the nominal trajectories were driven by four equally spaced targets in correspondence of the vertical midline of the plates. To minimize the error, the calibration procedure was performed on the highest possible number of squares, resulting in 0.8 mm/px. Once the calibration is performed, the bullet speed is computed by evaluating the distance travelled by the bullet and dividing it by the correspondent time interval. However, the fact that the reference plane used for calibration and the projectile trajectory plane were not coincident introduced a distortion factor between the measured and real speeds. The issue is easily solved if the position of the camera is fixed with respect to the reference plane: in that case the distortion factor is linear as a function of the distance of the bullet from that plane (Figure 7) and consequently the correction is

**Figure 5.** Scheme of the camera settings (top view). straightforward. **Figure 5.** Scheme of the camera settings (top view).

**Figure 6.** Scheme of the camera settings (front view).

**Figure 5.** Scheme of the camera settings (top view).

**Figure 6.** Scheme of the camera settings (front view). **Figure 6.** Scheme of the camera settings (front view).

While the perspective view was meant to capture the three-dimensional fragmentation dynamics of the projectile, the orthogonal videos were recorded to compute the projectile impact and exit speeds, via the well-known time-of-flight principle: knowing a priori the dimensions of a reference object, the video's resolution, and time history, the projectile trajectory is easily transformed into impact velocity. Such a method is one of the most used in ballistics to effectively evaluate the trajectory of bullets [22], with multiple applications in 2D and 3D analyses [23]. In the present work, even if the bullet itself could have been in principle used as reference for calibration, its small dimensions would have introduced a potentially high error; consequently, the reference object was a background reference plane, a chessboard panel attached perpendicularly to the frame. As described in Section 2.1, the chessboard was characterized by regular squares of 20 × 20 mm each, while the nominal trajectories were driven by four equally spaced targets in correspondence of the vertical midline of the plates. To minimize the error, the calibration procedure was performed on the highest possible number of squares, resulting in 0.8 mm/px. Once the calibration is performed, the bullet speed is computed by evaluating the distance travelled by the bullet and dividing it by the correspondent time interval. However, the fact that the reference plane used for calibration and the projectile trajectory plane were not coincident introduced a distortion factor between the While the perspective view was meant to capture the three-dimensional fragmentation dynamics of the projectile, the orthogonal videos were recorded to compute the projectile impact and exit speeds, via the well-known time-of-flight principle: knowing a priori the dimensions of a reference object, the video's resolution, and time history, the projectile trajectory is easily transformed into impact velocity. Such a method is one of the most used in ballistics to effectively evaluate the trajectory of bullets [22], with multiple applications in 2D and 3D analyses [23]. In the present work, even if the bullet itself could have been in principle used as reference for calibration, its small dimensions would have introduced a potentially high error; consequently, the reference object was a background reference plane, a chessboard panel attached perpendicularly to the frame. As described in Section 2.1, the chessboard was characterized by regular squares of 20 × 20 mm each, while the nominal trajectories were driven by four equally spaced targets in correspondence of the vertical midline of the plates. To minimize the error, the calibration procedure was performed on the highest possible number of squares, resulting in 0.8 mm/px. Once the calibration is performed, the bullet speed is computed by evaluating the distance travelled by the bullet and dividing it by the correspondent time interval. However, the fact that the reference plane used for calibration and the projectile trajectory plane were not coincident introduced a distortion factor between the measured and real speeds. The issue is easily solved if the position of the camera is fixed with respect to the reference plane: in that case the distortion factor is linear as a function of the distance of the bullet from that plane (Figure 7) and consequently the correction is straightforward.

measured and real speeds. The issue is easily solved if the position of the camera is fixed Naming this distance *d*<sup>1</sup> and the observer-reference plane distance *d*2, a correction factor *C* can be defined as follows:

$$\mathbf{C} = \mathbf{C} \tag{1}$$

$$\mathbf{C} = 1 - \frac{d\_1}{d\_2} \tag{1}$$

Consequently:

straightforward.

$$
v = \mathbb{C}v\_{meas} \tag{2}$$

Considering the relatively low resolution and number of available frames capturing the bullet during the pre-impact phases, multiple measurements were carried out mapping the position of three sections of the bullet (frustum, beginning of the ogive, and tip) for multiple combinations of frames. This redundant procedure allowed us to average out the human error introduced by manual recognition of the projectile shape. A minimum of six measures per shot were collected and averaged to obtain the resulting impact velocities.

**Figure 7.** Error on position vs. observer-reference distance. The worst-case scenario is identified for the Monolithic bullets, for which the distance from the trajectory and the reference plane is 400 mm: the apparent speed in such cases is 15.1% higher than the real speed. **Figure 7.** Error on position vs. observer-reference distance. The worst-case scenario is identified for the Monolithic bullets, for which the distance from the trajectory and the reference plane is 400 mm: the apparent speed in such cases is 15.1% higher than the real speed.

#### Naming this distance ଵ and the observer-reference plane distance ଶ, a correction *2.4. Bullet-Splash Load History Estimation*

experimentally:

factor *C* can be defined as follows: ଵ The load history estimation is based on two main hypotheses that must be verified experimentally:


 = ௦ (2) Considering the relatively low resolution and number of available frames capturing the bullet during the pre-impact phases, multiple measurements were carried out mapping the position of three sections of the bullet (frustum, beginning of the ogive, and tip) for multiple combinations of frames. This redundant procedure allowed us to average out the human error introduced by manual recognition of the projectile shape. A minimum of six measures per shot were collected and averaged to obtain the resulting impact velocities. *2.4. Bullet-Splash Load History Estimation*  The load histories were calculated according to the progressive fragmentation theory introduced by Andreotti et al. in 2022 [10–12] for 90 degree bullet splashes. The load history formula allows us to reconstruct the *F*(*t*) depending on the density distribution characterizing each section of the bullet and its impact speed. The interaction between the impactor and the target is treated as the interaction of a fluid flow, representing the flow of bullet debris, and a rigid, fixed plane normal to the axis of the bullet. The force at a generic time *t* is calculated as the time derivative of the elementary portion of the bullet ideally intersecting the target surface at that time. Considering the 90-degree deflection, the elementary variation in momentum can be expressed as:

$$dq = dm\upsilon\tag{3}$$

• The bullet encounters complete fragmentation during the interaction with the target; • The target can withstand the interaction without being penetrated, causing the deflection of the debris. The load histories were calculated according to the progressive fragmentation theory where *v* is the impact velocity of the bullet and *m* is its mass. Considering a homogeneous bullet with density *ρ*, assuming that the velocity variation only happens to the material ideally intersecting the impact surface, the elementary mass that is deflected in the elementary time *dt* is:

$$dm = \rho A(t)vdt\tag{4}$$

history formula allows us to reconstruct the *F*(*t*) depending on the density distribution characterizing each section of the bullet and its impact speed. The interaction between the impactor and the target is treated as the interaction of a fluid flow, representing the flow of bullet debris, and a rigid, fixed plane normal to the axis of the bullet. The force at a where *A(t)* represents the intersection between the bullet volume and the impact surface plane, and *vdt = ds* is the elementary translation of the bullet in *dt*. By substituting Equation (4) into Equation (3), and dividing both terms by *dt*, we obtain the expression of the impact force for a homogeneous bullet:

$$F(t) = \frac{d\eta(t)}{dt} = \rho A(t)v^2\tag{5}$$

*dq dmv* = (3)

which integration in time correctly equals the initial momentum of the impactor:

$$q = \int F(t)dt = v\rho \int A(t)vdt = v\rho \int A(s)ds = v\rho V = vm\tag{6}$$

where *V* is the volume of the homogeneous bullet.

For a generic impactor, composed by *M* materials, the load history due to the bullet splash can therefore be expressed as the sum of *M* terms:

$$F(t) = \sum\_{i=1}^{M} F\_i(t) \tag{7}$$

Therefore, the generic expression of the estimated load history due to bullet splash is:

$$F(t) = v^2 \sum\_{i=1}^{M} \rho\_i A\_i(t) \tag{8}$$

where *ρ<sup>i</sup>* is the density of the *i*-th material, and *A<sup>i</sup> (t)* is the section of that material ideally intersecting the surface of the target at the interaction time *t*.

To estimate the load-history due to bullet-splash of the tested impactors the formula (Equation (8)) was applied to the bullets section (Figure 1) by imposing an impact speed equal to those experimentally measured for the tested impacts.

To acquire the *A<sup>i</sup> (t)* of the bullets, the first step consisted in mapping its sections normally to the axis of the bullet *x*. At *N* discrete axial coordinates *x<sup>j</sup>* (30 to 50 points along the entire axis of the bullets, depending on their geometrical complexity) the radial coordinates of every material boundary were measured, so that, in general, the area of the i-th material at axial coordinate *x<sup>j</sup>* could be calculated as the area of a hollow circle:

$$A\_i(\mathbf{x}\_{\mathbf{j}}) = \pi (\mathrm{Re}\_{\mathbf{i}\mathbf{j}}^2 - \mathrm{Ri}\_{\mathbf{i}\mathbf{j}}^2) \tag{9}$$

where *Reij* and *Riij* are the external and internal radiuses of the *i*-th material at the *j*-th axial coordinate *x<sup>j</sup>* .

Once the materials sections *A<sup>i</sup> (x)* were mapped, the *A<sup>i</sup> (t)* of the specific shot was calculated by converting the spatial coordinate into time coordinates, by dividing the *x<sup>j</sup>* by the initial impact velocity *v*:

$$A\_i(t\_j) = A\_i(\frac{\mathcal{X}\_j}{v}) \tag{10}$$

The resulting load history is therefore a discretized curve to be automatically interpolated by the finite element solver at every time-integration step:

$$F(t\_j) = v^2 \sum\_{i=1}^{m} \rho\_i A\_i(t\_j) \tag{11}$$

#### *2.5. Finite Element Simulation Setup*

To verify the representativeness of estimated impulses, the considered impacts were simulated by applying the load histories to the plates by means of the explicit finite element solver LS-DYNA [24]. The plates were uniformly discretized with 2.5 mm fully integrated 4-node shell elements with 16 integration points in the thickness. The load was applied normally to the plates. Each shot was simulated by applying the corresponding load history as uniformly distributed on arbitrary areas, centered at the epicenter of the impacts, and with extensions comparable with the extension of the interaction marks visible on the impacted plates. The actual space distribution of the pressure fields acting during the impacts is in fact unknown. Therefore, to verify the sensitivity of the simulation results to the local pressure distribution, the same load history was applied on different arbitrarily defined distribution areas. The shots were simulated three times, by varying the extension of the loaded surface according to the experimental observations (Figures 12 and 13). The different tested areas had 100 mm<sup>2</sup> , 200 mm<sup>2</sup> , and 400 mm<sup>2</sup> extensions (see Figure 8), representing, respectively, the minimum and maximum extension of the compression area, and the extension of the sliding area as experimentally observed on the impacted plates (Figures 12 and 13). The plates were constrained in the load direction *x* by zero translation of the boundary nodes, and with double orthogonal symmetry planes intersecting at the epicenter of the impacts. The constraint between the two plates was assured by setting a penalty contact with a static and dynamic friction coefficient of 0.65, typical for clean dry steel-on-steel interactions [25]. arbitrarily defined distribution areas. The shots were simulated three times, by varying the extension of the loaded surface according to the experimental observations (Figures 12 and 13). The different tested areas had 100 mm2, 200 mm2, and 400 mm2 extensions (see Figure 8), representing, respectively, the minimum and maximum extension of the compression area, and the extension of the sliding area as experimentally observed on the impacted plates (Figures 12 and 13). The plates were constrained in the load direction *x* by zero translation of the boundary nodes, and with double orthogonal symmetry planes intersecting at the epicenter of the impacts. The constraint between the two plates was assured by setting a penalty contact with a static and dynamic friction coefficient of 0.65, typical for clean dry steel-on-steel interactions [25].

To verify the representativeness of estimated impulses, the considered impacts were simulated by applying the load histories to the plates by means of the explicit finite element solver LS-DYNA [24]. The plates were uniformly discretized with 2.5 mm fully integrated 4-node shell elements with 16 integration points in the thickness. The load was applied normally to the plates. Each shot was simulated by applying the corresponding load history as uniformly distributed on arbitrary areas, centered at the epicenter of the impacts, and with extensions comparable with the extension of the interaction marks visible on the impacted plates. The actual space distribution of the pressure fields acting during the impacts is in fact unknown. Therefore, to verify the sensitivity of the simulation results to the local pressure distribution, the same load history was applied on different

*Materials* **2023**, *16*, x FOR PEER REVIEW 11 of 33

*2.5. Finite Element Simulation Setup* 

**Figure 8.** Overview (**left**) and detail (**right**) of the finite element model of the plate highlighting the choice and progressive increase in the loaded elements characterizing the three arbitrary load distributions: 100 mm2 distribution (grey), 200 mm2 distribution (yellow), and 400 mm2 distribution (light blue); the three circles represent the area-equivalent circular extensions corresponding to the minimum and maximum limit of the compression area, and the external limit of the sliding area as experimentally observed on the impacted plates, respectively (Figures 12 and 13). Only a quarter of the plate was modeled, with orthogonal symmetry conditions. **Figure 8.** Overview (**left**) and detail (**right**) of the finite element model of the plate highlighting the choice and progressive increase in the loaded elements characterizing the three arbitrary load distributions: 100 mm<sup>2</sup> distribution (grey), 200 mm<sup>2</sup> distribution (yellow), and 400 mm<sup>2</sup> distribution (light blue); the three circles represent the area-equivalent circular extensions corresponding to the minimum and maximum limit of the compression area, and the external limit of the sliding area as experimentally observed on the impacted plates, respectively (Figures 12 and 13). Only a quarter of the plate was modeled, with orthogonal symmetry conditions.

#### 2.5.1. Development of the Finite Element Model The development of the finite element model here proposed was the result of three 2.5.1. Development of the Finite Element Model

main steps, aiming at finding the most efficient discretization strategy to eventually allow engineers to correctly simulate the propagation of the stress waves caused by bullet splashes on generic, extended, ballistic protection panels. The development started from a highly detailed simulation of the local effects of the impacts, and progressively evolved towards more parsimonious models guaranteeing the same effectiveness in terms of predicting the residual deformation fields and the stress waves generated by the impacts. The development of the finite element model here proposed was the result of three main steps, aiming at finding the most efficient discretization strategy to eventually allow engineers to correctly simulate the propagation of the stress waves caused by bullet splashes on generic, extended, ballistic protection panels. The development started from a highly detailed simulation of the local effects of the impacts, and progressively evolved towards more parsimonious models guaranteeing the same effectiveness in terms of predicting the residual deformation fields and the stress waves generated by the impacts.

The first step (Andreotti et al., 2021 [9]) consisted in simulating the bullet-splash phenomenon as a fluid structure interaction (FSI) to reproduce the progressive deflection of the bullet fragments interacting with the target. The model used for the FSI simulation was a 3D solid mesh composed of 0.3 mm hexahedral elements. The experimental validation of this model was based on the measurements of the micro-hardness and deformation fields across the impact surface of 4 mm AISI 304L steel plates impacted by 9 × 21 mm FMJ bullets. The validation demonstrated extreme accuracy in predicting the field of plastic strain as well as the overall residual deformation field.

The second step introduced the load history estimation method discussed in this paper. The method was first tested on the same experimental dataset analyzed by Andreotti et al. (2021) [9]. The simulation applied the estimated load history as a uniformly distributed load on arbitrary concentric circular areas, with the plate being discretized with the same 0.3 mm hexahedral elements mesh used for the validation of the FSI approach. The

comparison between the overall residual deformation field, the stress waves, and the resultant reaction forces validated the load history approach as an equally accurate way to simulate bullet-splash for macroscopic structural assessment purposes, allowing a reduction of the computational cost of 60% compared to FSI.

In the third step, to improve the efficiency of the method and its parsimony and suitability to real-world engineering applications, a progressive simplification of the structural model was introduced, leading to the full shell model here proposed. The validation of the 2.5 mm shell model was performed again on the 4 mm AISI 304L steel plates impacted by 9 × 21 mm FMJ bullets; the residual deformation field, the stress waves, and the resultant reaction forces predicted by the simulation were confirmed to be accurate in comparison with the experimental evidence and with the results of the most accurate simulation based on the 0.3 mm size solid finite element model (Andreotti et al., 2022 [10]).

Compared to the FSI simulation model, the 2.5 mm shell discretization allowed a reduction of the calculation cost of more than 99%, also guaranteeing accurate prediction of the structural effects of the impacts, thanks to a maximum stable integration time step of 0.46 µs, that guarantees an accurate reproduction of the considered impulses, whose duration is less than 50 µs, and is compatible with reasonable computational costs for real-world applications. The analyses discussed in the present paper were conducted with an initial time integration step equal to 90% of the maximum, i.e., 0.414 µs. It is important to notice that the 2.5 mm size was also identified as the maximum element size compatible with reasonable load application detail to distinguish the load distributions considered in the study. The optimization of the computational cost of the finite element model also considered the symmetries and the extension of the discretized plate. Preliminary tests were conducted on the entire model to verify the influence of the boundary effects on the different shot positions, concluding that no boundary effects were significant on the results; therefore, the overall dimensions of the finite element models of the plates were reduced to 125 × 125 mm, with two orthogonal symmetry planes, to represent a 250 × 250 mm plate impacted at the center. In fact, the dimensions of the plates are such that the local effects of the impacts cannot be influenced by the boundary constraints because the maximum duration of the considered load history is less than 50 µs while the time for the stress waves to travel back and forth from the epicenter to the constraints is around 100 µs (considering the sound speed in steel is equal to 5000 m/s).

#### 2.5.2. Constitutive Model Associated with the Plates

The constitutive model associated with the plates is a kinematic elastic-plastic model with damage (\*MAT\_81\\*MAT\_PLASTICITY\_WITH\_DAMAGE in Ls-Dyna kewords [26]). For Creusabro® 8000 the post-yield hardening is regulated by a constant modulus ETAN = 11,891.7 MPa; the initial necking strain is EPPF = 0.05827. For Durostat® 400 the hardening modulus is ETAN = 5759.59 MPa; the initial necking strain is 0.04402. The rate effect was implemented as a scale factor for the yield stress as a function of the strain rate according to Cowper–Symonds model (Equation (12)) [26] which parameters C = 396,500/s and *p* = 3.0745 were obtained from Boyce et al. (2007) [27]:

$$
\sigma = \sigma\_0 \left[ 1 + \left( \frac{\dot{\varepsilon}}{\overline{\mathcal{C}}} \right)^{1/p} \right] \tag{12}
$$

where *σ*<sup>0</sup> is the quasi-static value of the yield stress, and *σ* is its generic dynamic value at strain rate . *ε*. Table 7 summerizes the parameters of the constitutive models.

**Table 7.** Table of the constitutive parameters associated with the Creusabro® 8000 and Durostat® 400 models associated with the plates: RO = density; E = Young Modulus; PR = Poisson Ratio; SIGY = Yield Stress; ETAN = linear hardening modulus; EPPF = initial necking strain; C, p = rate effect parameters according to the Cowper–Symonds model; EPPFR = ultimate failure strain. **Table 7.** Table of the constitutive parameters associated with the Creusabro® 8000 and Durostat® 400 models associated with the plates: RO = density; E = Young Modulus; PR = Poisson Ratio; SIGY = Yield Stress; ETAN = linear hardening modulus; EPPF = initial necking strain; C, p = rate effect parameters according to the Cowper–Symonds model; EPPFR = ultimate failure strain.

according to Cowper–Symonds model (Equation (12)) [26] which parameters C = 396,500/s

<sup>0</sup> 1

where *σ*0 is the quasi-static value of the yield stress, and *σ* is its generic dynamic value at

. Table 7 summerizes the parameters of the constitutive models.

σ σ 1/

*C* ε

 = + 

*p*

(12)


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and *p* = 3.0745 were obtained from Boyce et al. (2007) [27]:

#### 2.5.3. Validation of the Finite Element Discretization 2.5.3. Validation of the Finite Element Discretization

strain rate

ε

To verify the consistency of the results in terms of stress waves and deformation fields, sensitivity tests were conducted on models with different mesh sizes. In this subsection we compare the results of a single Creusabro® 8000 steel plate impacted by a monolithic .308 bullet at 735 m/s, simulated by applying the corresponding estimated load history to the sharpest distribution (100 mm2, Figure 8). The results of the 2.5 mm shell model were compared with the results of an analogous model with 1.25 mm mesh size, therefore four times the nodal density. The comparisons show almost perfect adherence of the stress fields (Figure 9), stress waves (Figure 10), and residual displacement fields (Figure 11), validating the 2.5 mm shell discretization as an effective compromise between detailed load description, consistency of the results, and maximization of the time integration step for the containment of the computational cost in real-world industrial applications requiring the modelling of large structural systems. To verify the consistency of the results in terms of stress waves and deformation fields, sensitivity tests were conducted on models with different mesh sizes. In this subsection we compare the results of a single Creusabro® 8000 steel plate impacted by a monolithic .308 bullet at 735 m/s, simulated by applying the corresponding estimated load history to the sharpest distribution (100 mm2, Figure 8). The results of the 2.5 mm shell model were compared with the results of an analogous model with 1.25 mm mesh size, therefore four times the nodal density. The comparisons show almost perfect adherence of the stress fields (Figure 9), stress waves (Figure 10), and residual displacement fields (Figure 11), validating the 2.5 mm shell discretization as an effective compromise between detailed load description, consistency of the results, and maximization of the time integration step for the containment of the computational cost in real-world industrial applications requiring the modelling of large structural systems.

**Figure 9.** Stress field convergence. Comparison between the maximum principal stress field at 0.05 ms as predicted by the 2.5 mm model (**left**) and the 1.25 mm model (**right**). The difference in peak stress is less than 1%. No significant differences are visible. The stress wave is axially symmetric despite the squared shape of the arbitrary loaded surface. **Figure 9.** Stress field convergence. Comparison between the maximum principal stress field at 0.05 ms as predicted by the 2.5 mm model (**left**) and the 1.25 mm model (**right**). The difference in peak stress is less than 1%. No significant differences are visible. The stress wave is axially symmetric despite the squared shape of the arbitrary loaded surface.

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**Figure 10.** Maximum principal stress waves comparison at 50 mm from the origin. The predictions obtained by the higher nodal density model (red) and by the lower nodal density model (blue) are almost impossible to distinguish, even after 0.4 ms, which corresponds to around ten times the load application time. The maximum difference in peak stress is 10% after 0.23 ms, therefore 0.19 mm after the end of the loading time. This demonstrates that the stress prediction is consistent. **Figure 10.** Maximum principal stress waves comparison at 50 mm from the origin. The predictions obtained by the higher nodal density model (red) and by the lower nodal density model (blue) are almost impossible to distinguish, even after 0.4 ms, which corresponds to around ten times the load application time. The maximum difference in peak stress is 10% after 0.23 ms, therefore 0.19 mm after the end of the loading time. This demonstrates that the stress prediction is consistent. **Figure 10.** Maximum principal stress waves comparison at 50 mm from the origin. The predictions obtained by the higher nodal density model (red) and by the lower nodal density model (blue) are almost impossible to distinguish, even after 0.4 ms, which corresponds to around ten times the load application time. The maximum difference in peak stress is 10% after 0.23 ms, therefore 0.19 mm after the end of the loading time. This demonstrates that the stress prediction is consistent.

maximum difference is less than 0.5% at the peak: 0.03 mm over 7.3 mm of maximum deformation. **3. Results Figure 11.** Residual deformation fields at 20 ms after the impacts [mm]. Comparison between the 2.5 mm shell model (**left**) and the 1.25 mm shell model (**right**). The predictions are equivalent; the maximum difference is less than 0.5% at the peak: 0.03 mm over 7.3 mm of maximum deformation. **Figure 11.** Residual deformation fields at 20 ms after the impacts [mm]. Comparison between the 2.5 mm shell model (**left**) and the 1.25 mm shell model (**right**). The predictions are equivalent; the maximum difference is less than 0.5% at the peak: 0.03 mm over 7.3 mm of maximum deformation.

#### This section provides the experimental, analytical, and numerical results of the study. **3. Results 3. Results**

In Section 3.1 the pictures of the impacted plates and the frames capturing the evolution of the impacts are investigated to verify which of the considered impacts are compliant with the theoretical hypotheses of the bullet-splash. Section 3.2 summarizes the results of the measurements and estimations of the impact velocities. In Section 3.3 the load histories due to bullet-splash are plotted, having been calculated according to the theory from the measured impact velocities and the sections of the bullets. In Section 3.4, to verify the representativeness of the estimated load histories, the experimental fields of residual displacement are compared with the corresponding predictions obtained by means of explicit finite element simulations, where the discretized plates are loaded by direct application of the corresponding load curves. This section provides the experimental, analytical, and numerical results of the study. In Section 3.1 the pictures of the impacted plates and the frames capturing the evolution of the impacts are investigated to verify which of the considered impacts are compliant with the theoretical hypotheses of the bullet-splash. Section 3.2 summarizes the results of the measurements and estimations of the impact velocities. In Section 3.3 the load histories due to bullet-splash are plotted, having been calculated according to the theory from the measured impact velocities and the sections of the bullets. In Section 3.4, to verify the representativeness of the estimated load histories, the experimental fields of residual displacement are compared with the corresponding predictions obtained by means of explicit finite element simulations, where the discretized plates are loaded by direct application of the corresponding load curves. This section provides the experimental, analytical, and numerical results of the study. In Section 3.1 the pictures of the impacted plates and the frames capturing the evolution of the impacts are investigated to verify which of the considered impacts are compliant with the theoretical hypotheses of the bullet-splash. Section 3.2 summarizes the results of the measurements and estimations of the impact velocities. In Section 3.3 the load histories due to bullet-splash are plotted, having been calculated according to the theory from the measured impact velocities and the sections of the bullets. In Section 3.4, to verify the representativeness of the estimated load histories, the experimental fields of residual displacement are compared with the corresponding predictions obtained by means of explicit finite element simulations, where the discretized plates are loaded by direct application of the corresponding load curves.

#### *3.1. Verification of the Bullet Splash Hypotheses 3.1. Verification of the Bullet Splash Hypotheses*  The verification of the hypotheses was performed by means of analyses of the

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The verification of the hypotheses was performed by means of analyses of the experimental evidence. The inspection of the back plates allowed the verification of the non-penetration hypothesis. The inspection of the front plates allowed the verification of partial penetrations, here defined as the penetration of bullet material only through the front plate. The inspection of the full fragmentation of the bullets was performed by observation of the frames collected by means of the two high-frame-rate cameras in order to verify the gradual fragmentation of the bullet during the impact, the overall axial symmetry of the deflection kinematics, and the absence of major fragments rebounding from the plate and/or showing dimensions comparable with the bullets. A fragment was considered major if showing dimensions comparable with the calibre of the bullet. The presence of major fragments adhering to the impact surface was also verified by inspecting the plates. experimental evidence. The inspection of the back plates allowed the verification of the non-penetration hypothesis. The inspection of the front plates allowed the verification of partial penetrations, here defined as the penetration of bullet material only through the front plate. The inspection of the full fragmentation of the bullets was performed by observation of the frames collected by means of the two high-frame-rate cameras in order to verify the gradual fragmentation of the bullet during the impact, the overall axial symmetry of the deflection kinematics, and the absence of major fragments rebounding from the plate and/or showing dimensions comparable with the bullets. A fragment was considered major if showing dimensions comparable with the calibre of the bullet. The presence of major fragments adhering to the impact surface was also verified by inspecting the plates.

#### 3.1.1. Non-Penetration 3.1.1. Non-Penetration

The targets were able to withstand the ballistic impacts without allowing any part of the bullets to penetrate through the 8 mm thickness. Figures 12 and 13 show the effects of the impacts on the front and rear surfaces of the plates: the front views show the impact epicenter and the surrounding traces due to the radial debris deflection; the rear views show the bulges due to the residual plastic strain of the plates with no marks of penetration. It is worth noting that the Full Metal Jacket (FMJ) bullet caused the partial penetration of the first Durostat® 400 plate (Figure 14). The targets were able to withstand the ballistic impacts without allowing any part of the bullets to penetrate through the 8 mm thickness. Figures 12 and 13 show the effects of the impacts on the front and rear surfaces of the plates: the front views show the impact epicenter and the surrounding traces due to the radial debris deflection; the rear views show the bulges due to the residual plastic strain of the plates with no marks of penetration. It is worth noting that the Full Metal Jacket (FMJ) bullet caused the partial penetration of the first Durostat® 400 plate (Figure 14).

**Figure 12.** Front and rear surfaces of the Durostat® 400 plates impacted by the bullets. The impacted areas show the marks of the interaction with the bullets, which can be distinguished in a central area where the interaction appears to be more compressive, and a peripheral area where the radial marks show the effects of the sliding of the bullet debris. The sliding marks are evident from a diameter of approximately 11 to 16 mm (roughly 100 mm2 to 200 mm2), to a diameter of 20 to 25 mm around the epicenter (around 400 mm2). The detailed front view of the FMJ effects shows the melted filler mixed with parts of the jacket, still blocked inside the crater. **Figure 12.** Front and rear surfaces of the Durostat® 400 plates impacted by the bullets. The impacted areas show the marks of the interaction with the bullets, which can be distinguished in a central area where the interaction appears to be more compressive, and a peripheral area where the radial marks show the effects of the sliding of the bullet debris. The sliding marks are evident from a diameter of approximately 11 to 16 mm (roughly 100 mm<sup>2</sup> to 200 mm<sup>2</sup> ), to a diameter of 20 to 25 mm around the epicenter (around 400 mm<sup>2</sup> ). The detailed front view of the FMJ effects shows the melted filler mixed with parts of the jacket, still blocked inside the crater.

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**Figure 13.** Front and rear surfaces of the Creusabro® 8000 plates impacted by the bullets. The impacted areas show the marks of the interaction with the bullets, that can be distinguished in a central area where the interaction appears to be more compressive, and a peripheral area where the radial marks show the effects of the sliding of the bullet debris. The sliding marks are evident from a diameter of approximately 11 to 16 mm (roughly 100 mm2 to 200 mm2), to a diameter of 20 to 25 mm around the epicenter (around 400 mm2). **Figure 13.** Front and rear surfaces of the Creusabro® 8000 plates impacted by the bullets. The impacted areas show the marks of the interaction with the bullets, that can be distinguished in a central area where the interaction appears to be more compressive, and a peripheral area where the radial marks show the effects of the sliding of the bullet debris. The sliding marks are evident from a diameter of approximately 11 to 16 mm (roughly 100 mm<sup>2</sup> to 200 mm<sup>2</sup> ), to a diameter of 20 to 25 mm around the epicenter (around 400 mm<sup>2</sup> ). **Figure 13.** Front and rear surfaces of the Creusabro® 8000 plates impacted by the bullets. The impacted areas show the marks of the interaction with the bullets, that can be distinguished in a central area where the interaction appears to be more compressive, and a peripheral area where the radial marks show the effects of the sliding of the bullet debris. The sliding marks are evident from a diameter of approximately 11 to 16 mm (roughly 100 mm2 to 200 mm2), to a diameter of 20 to 25 mm around the epicenter (around 400 mm2).

3.1.2. Full Fragmentation **Figure 14.** Partial penetration of FMJ bullet material through the first plate of Durostat® 400. Back of the first plate showing the penetration of some melted filler material. **Figure 14.** Partial penetration of FMJ bullet material through the first plate of Durostat® 400. Back of the first plate showing the penetration of some melted filler material.

#### The analysis of the captured frames allowed to assess that all the bullets encountered full fragmentation and deflection, except for the FMJs. In fact, the FMJ impacting on 3.1.2. Full Fragmentation 3.1.2. Full Fragmentation

The analysis of the captured frames allowed to assess that all the bullets encountered full fragmentation and deflection, except for the FMJs. In fact, the FMJ impacting on The analysis of the captured frames allowed to assess that all the bullets encountered full fragmentation and deflection, except for the FMJs. In fact, the FMJ impacting on Durostat® 400 partially penetrated the first plate, while part of the FMJ impacting on Creusabro® 8000 (Figure 15) was observed to slowly rebound after the impact, as a

the first plate showing the penetration of some melted filler material.

solid volume (Figure 16). These cases are therefore not compliant with the proposed bullet-splash definition. Creusabro® 8000 (Figure 15) was observed to slowly rebound after the impact, as a solid volume (Figure 16). These cases are therefore not compliant with the proposed bulletsplash definition.

Durostat® 400 partially penetrated the first plate, while part of the FMJ impacting on

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**Figure 15.** The figure shows ten frames of the first 0.3 ms after the impact for the combination FMJ-Creusabro® 8000, taken from the footage recorded by the perspective camera (Camera 2). The first frames (**a**–**c**) show the crushing of the bullet at its first contact with the target, allowing the appreciation of a spark caused by the high hardness of the plate. Fragments are then projected radially from the impact point starting from small particles (**c**–**f**), with slightly bigger fragments **Figure 15.** The figure shows ten frames of the first 0.3 ms after the impact for the combination FMJ-Creusabro® 8000, taken from the footage recorded by the perspective camera (Camera 2). The first frames (**a**–**c**) show the crushing of the bullet at its first contact with the target, allowing the appreciation of a spark caused by the high hardness of the plate. Fragments are then projected radially from the impact point starting from small particles (**c**–**f**), with slightly bigger fragments later ricocheting in annular patterns (**f**–**j**). Frames shown here are processed to show the parts of interest, filtering out the noise otherwise preventing the appreciation of the bullet fragmentation.

interest, filtering out the noise otherwise preventing the appreciation of the bullet fragmentation.

**Figure 16.** The figure shows a large fragment of the Full Metal Jacket bouncing back from the impact area of the Creusabro® 8000 plate. The shot is therefore non-compliant with the bullet-splash definition. The fragment translates with perpendicular speed measured at 4.4 ± 2% m/s. Mild spinning can be appreciated. The three frames shown here, extracted from the second camera footage, were processed to isolate the fragment from the dust created after the impact: for each frame, only the fragment and the dent on the plate are visible; an approximate dimensional reference for the dent and the fragment is reported in frame (**a**–**c**). The fragment has a diameter of around 7.5 mm and thickness of about 4 mm, with the shape similar to the bulge created on the plate. **Figure 16.** The figure shows a large fragment of the Full Metal Jacket bouncing back from the impact area of the Creusabro® 8000 plate. The shot is therefore non-compliant with the bullet-splash definition. The fragment translates with perpendicular speed measured at 4.4 ± 2% m/s. Mild spinning can be appreciated. The three frames shown here (**a**–**c**), extracted from the second camera footage, were processed to isolate the fragment from the dust created after the impact: for each frame, only the fragment and the dent on the plate are visible; an approximate dimensional reference for the dent and the fragment is reported in frame (**a**). The fragment has a diameter of around 7.5 mm and thickness of about 4 mm, with the shape similar to the bulge created on the plate.

#### *3.2. Impact Velocity Estimation 3.2. Impact Velocity Estimation*

The frame-by-frame analysis of the bullet kinematics before the impacts allowed identification of the impact velocity with a maximum error of ±5.7% (Table 8), except for shot 2 where the lack of light made the approach too uncertain. In this case the impact velocity was therefore estimated based on the muzzle velocity and the average percentage of speed loss calculated on four other Hollow Point bullets shot the same day in the same conditions (Table 9). The maximum error for shot 2 was therefore calculated as the sum of the maximum error due to muzzle velocity measurement (±2.8%) and the maximum error due to frame-by-frame approach on hollow point bullets (±5.7%). The maximum error The frame-by-frame analysis of the bullet kinematics before the impacts allowed identification of the impact velocity with a maximum error of ±5.7% (Table 8), except for shot 2 where the lack of light made the approach too uncertain. In this case the impact velocity was therefore estimated based on the muzzle velocity and the average percentage of speed loss calculated on four other Hollow Point bullets shot the same day in the same conditions (Table 9). The maximum error for shot 2 was therefore calculated as the sum of the maximum error due to muzzle velocity measurement (±2.8%) and the maximum error due to frame-by-frame approach on hollow point bullets (±5.7%). The maximum error associated with impact velocity of Shot 2 is therefore ±8.5%.


**Table 8.** Estimated impact velocities.

associated with impact velocity of Shot 2 is therefore ±8.5%.

6 Hollow Point (HP) Creusabro® 8000 702 5.7 \* Estimated based on the average velocity loss of four Hollow Point bullets shot in the same conditions.

8 Monolithic (MONO) Creusabro® 8000 749 2.8 \* Estimated based on the average velocity loss of four Hollow Point bullets shot in the same **Table 9.** Average velocities at muzzle and at 100 m and relative losses for each bullet type. Averages are computed over a total of four shots per bullet type.


#### Soft Point (SP) 763 620 18.7 *3.3. Estimated Load Histories*

Hollow Point (HP) 814 727 10.7 Full Metal Jacket (FMJ) 843 731 13.3 Monolithic (MONO) 837 741 11.5 According to the proposed load history approach, eight load histories (Figures 17 and 18) were calculated, based on the bullets' section (Figure 1) and the mea-

7 Full Metal Jacket (FMJ) Creusabro® 8000 736 2.9

sured impact velocities (Table 8). The mass distribution was reconstructed based on the scaled sections associated with the density values summarized in Table 10, so that the volume integration of the mapped density fields exactly equals the nominal masses of the bullets, and therefore the time integration of the load histories correctly corresponds to the momentum of the impacting bullets. **Bullet [kg/m3] [kg/m3] Bullet Mass [g]**  Soft Point (SP) 8730 12,640 11.66 Hollow Point (HP) 8730 12,280 10.89 Full Metal Jacket (FMJ) 8730 11,325 9.53 Monolithic (MONO) 9425 - 9.59

**Table 10.** Density values associated with the bullet materials for the four bullet types so that the

volume integration of the density fields correspond to the nominal bullets' masses.

According to the proposed load history approach, eight load histories (Figures 17 and 18) were calculated, based on the bullets' section (Figure 1) and the measured impact velocities (Table 8). The mass distribution was reconstructed based on the scaled sections associated with the density values summarized in Table 10, so that the volume integration of the mapped density fields exactly equals the nominal masses of the bullets, and therefore the time integration of the load histories correctly corresponds to the momentum

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*3.3. Estimated Load Histories* 

of the impacting bullets.

**Figure 17.** Load histories estimated according to Equation (8), applied to the impacts on Durostat® 400. The SP, HP and MONO bullets have similar force peaks around 220–230 kN. The FMJ reaches instead 307 kN force. The duration of the impulses ranges from 36 to 47 µs. **Figure 17.** Load histories estimated according to Equation (8), applied to the impacts on Durostat® 400. The SP, HP and MONO bullets have similar force peaks around 220–230 kN. The FMJ reaches instead 307 kN force. The duration of the impulses ranges from 36 to 47 µs. *Materials* **2023**, *16*, x FOR PEER REVIEW 20 of 33

**Figure 18.** Load histories estimated according to Equation (8), applied to the impacts on Creusabro® 8000. The SP reaches 194 kN peak force, MONO reaches 243 kN, HP reaches 260 kN, and FMJ reaches 281 kN. The durations of the impulses range from 39 µs for FMJ to 49 µs for SP. **Figure 18.** Load histories estimated according to Equation (8), applied to the impacts on Creusabro® 8000. The SP reaches 194 kN peak force, MONO reaches 243 kN, HP reaches 260 kN, and FMJ reaches 281 kN. The durations of the impulses range from 39 µs for FMJ to 49 µs for SP.

In Figures 19–34 we provide the comparison between numerical and experimental

the bulges (Figures 19, 21, 23, 25, 27, 29, 31 and 33). The experimental measurements (represented with their variability due to a ±0.5 mm error in positioning the instrument at the bulge apex) are compared with the simulation results obtained with the three arbitrary load distributions, where the same load history was distributed on 100, 200, and 400 mm2 as described in Section 2.5.1. Based on the mapping of the residual displacements, three validation indexes have been considered: the apex displacement, the overall volume of the residual bulge, and the radial slope at 50 mm from the apex (the boundary of the measured range). The validation was conducted by dividing the indexes calculated on the simulation results by the indexes calculated on the experimental results. The validation results are presented as percentages, so that a validation higher than 100% means that the simulation overestimated the corresponding experimental index. To enhance the effects of the load distribution on the validation indexes, the three indexes were plotted as a

function of the loaded area (Figures 20, 22, 24, 26, 28, 30, 32 and 34).

*3.4. Field of Residual Displacements* 


**Table 10.** Density values associated with the bullet materials for the four bullet types so that the volume integration of the density fields correspond to the nominal bullets' masses.

### *3.4. Field of Residual Displacements*

In Figures 19–34 we provide the comparison between numerical and experimental results in terms of residual displacements of the plates. The fields of residual displacements have been represented as a function of the radial distance from the apex of the bulges (Figures 19, 21, 23, 25, 27, 29, 31 and 33). The experimental measurements (represented with their variability due to a ±0.5 mm error in positioning the instrument at the bulge apex) are compared with the simulation results obtained with the three arbitrary load distributions, where the same load history was distributed on 100, 200, and 400 mm<sup>2</sup> as described in Section 2.5.1. Based on the mapping of the residual displacements, three validation indexes have been considered: the apex displacement, the overall volume of the residual bulge, and the radial slope at 50 mm from the apex (the boundary of the measured range). The validation was conducted by dividing the indexes calculated on the simulation results by the indexes calculated on the experimental results. The validation results are presented as percentages, so that a validation higher than 100% means that the simulation overestimated the corresponding experimental index. To enhance the effects of the load distribution on the validation indexes, the three indexes were plotted as a function of the loaded area (Figures 20, 22, 24, 26, 28, 30, 32 and 34). *Materials* **2023**, *16*, x FOR PEER REVIEW 21 of 33

**Figure 19.** Residual displacements caused by the Soft Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) is near the maximum experimental value (blue dotted line), around 4 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) is near the minimum experimental value, around 3 mm. The predicted apex displacement for the 400 mm2 distribution (light blue line) is around 1 mm lower than the minimum experimental value. The radial extension of the bulge is underestimated. Over 35 mm radial distance from the apex the three load distributions are impossible to distinguish. **Figure 19.** Residual displacements caused by the Soft Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) is near the maximum experimental value (blue dotted line), around 4 mm. The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) is near the minimum experimental value, around 3 mm. The predicted apex displacement for the 400 mm<sup>2</sup> distribution (light blue line) is around 1 mm lower than the minimum experimental value. The radial extension of the bulge is underestimated. Over 35 mm radial distance from the apex the three load distributions are impossible to distinguish.

**Figure 20.** Soft Point bullet on Durostat® 400 plates: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (109%), while the 200 mm2 and 400 mm2 give 82% and 55%. The three distributions provide more consistent predictions in terms of

bulge volume and slope, all between 51 and 68%.

distinguish.

slope, all between 45% and 67%.

**Figure 19.** Residual displacements caused by the Soft Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) is near the maximum experimental value (blue dotted line), around 4 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) is near the minimum experimental value, around 3 mm. The

35 mm radial distance from the apex the three load distributions are impossible to distinguish.

**Figure 20.** Soft Point bullet on Durostat® 400 plates: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (109%), while the 200 mm2 and 400 mm2 give 82% and 55%. The three distributions provide more consistent predictions in terms of bulge volume and slope, all between 51 and 68%. **Figure 20.** Soft Point bullet on Durostat® 400 plates: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm<sup>2</sup> distribution gives the best estimation (109%), while the 200 mm<sup>2</sup> and 400 mm<sup>2</sup> give 82% and 55%. The three distributions provide more consistent predictions in terms of bulge volume and slope, all between 51 and 68%. *Materials* **2023**, *16*, x FOR PEER REVIEW 22 of 33

**Figure 21.** Residual displacements caused by the Hollow Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 1 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) coincides with the maximum experimental value, around 3.4 mm. The predicted apex displacement for the 400 mm2 distribution (light blue line) is almost coincident with the minimum experimental value. The radial extension of the bulge is slightly underestimated. Over 25 mm radial distance from the apex the three load distributions are almost impossible to **Figure 21.** Residual displacements caused by the Hollow Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 1 mm. The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) coincides with the maximum experimental value, around 3.4 mm. The predicted apex displacement for the 400 mm<sup>2</sup> distribution (light blue line) is almost coincident with the minimum experimental value. The radial extension of the bulge is slightly underestimated. Over 25 mm radial distance from the apex the three load distributions are almost impossible to distinguish.

**Figure 22***.* Validation for HP on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (119%), while the 100 mm2 and 400 mm2 give 157% and 80%. The three distributions provide more consistent predictions in terms of bulge volume and distinguish.

**Figure 21.** Residual displacements caused by the Hollow Point bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 1 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) coincides with the maximum experimental value, around 3.4 mm. The predicted apex displacement for the 400 mm2 distribution (light blue line) is almost coincident

**Figure 22***.* Validation for HP on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (119%), while the 100 mm2 and 400 mm2 give 157% and 80%. The three distributions provide more consistent predictions in terms of bulge volume and slope, all between 45% and 67%. **Figure 22.** Validation for HP on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm<sup>2</sup> distribution gives the best estimation (119%), while the 100 mm<sup>2</sup> and 400 mm<sup>2</sup> give 157% and 80%. The three distributions provide more consistent predictions in terms of bulge volume and slope, all between 45% and 67%. *Materials* **2023**, *16*, x FOR PEER REVIEW 23 of 33

**Figure 23.** Residual displacements caused by the Full Metal Jacket bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) underestimates the experimental range (blue and red dotted lines) of about 2.5 mm. The overall bulge extension is significantly underestimated. For radial distance over 25 mm from the apex, the three load distributions are almost impossible to distinguish. **Figure 23.** Residual displacements caused by the Full Metal Jacket bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) underestimates the experimental range (blue and red dotted lines) of about 2.5 mm. The overall bulge extension is significantly underestimated. For radial distance over 25 mm from the apex, the three load distributions are almost impossible to distinguish.

**Figure 24***.* Validation for FMJ on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (63%), while the 200 mm2 and 400 mm2 give 48% and 30%. The three distributions provide consistent predictions in terms of bulge volume around 30–

33%. In slope, the estimate increases as the distribution area, from 17.5% to 40%.

**Figure 23.** Residual displacements caused by the Full Metal Jacket bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) underestimates the

distributions are almost impossible to distinguish.

**Figure 24***.* Validation for FMJ on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (63%), while the 200 mm2 and 400 mm2 give 48% and 30%. The three distributions provide consistent predictions in terms of bulge volume around 30– 33%. In slope, the estimate increases as the distribution area, from 17.5% to 40%. **Figure 24.** Validation for FMJ on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm<sup>2</sup> distribution gives the best estimation (63%), while the 200 mm<sup>2</sup> and 400 mm<sup>2</sup> give 48% and 30%. The three distributions provide consistent predictions in terms of bulge volume around 30–33%. In slope, the estimate increases as the distribution area, from 17.5% to 40%. *Materials* **2023**, *16*, x FOR PEER REVIEW 24 of 33

**Figure 25.** Residual displacements caused by the Monolithic bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) is almost coincident with the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm2 distribution (yellow line) coincides with the minimum experimental value. The radial extension of the bulge is slightly underestimated. For radial distances over 35 mm from the apex, the three load distributions are impossible to distinguish. **Figure 25.** Residual displacements caused by the Monolithic bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) is almost coincident with the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) coincides with the minimum experimental value. The radial extension of the bulge is slightly underestimated. For radial distances over 35 mm from the apex, the three load distributions are impossible to distinguish.

**Figure 26***.* Validation for MONO on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (113%), while the 200 mm2 and 400 mm2 give 83% and 54%. The bulge volume predictions decrease from 76% to 63% as the area

increases. The slope predictions are more consistent, between 57% and 67%.

distributions are impossible to distinguish.

**Figure 25.** Residual displacements caused by the Monolithic bullet on Durostat® 400 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) is almost coincident with the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm2

**Figure 26***.* Validation for MONO on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (113%), while the 200 mm2 and 400 mm2 give 83% and 54%. The bulge volume predictions decrease from 76% to 63% as the area increases. The slope predictions are more consistent, between 57% and 67%. **Figure 26.** Validation for MONO on Durostat® 400: validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm<sup>2</sup> distribution gives the best estimation (113%), while the 200 mm<sup>2</sup> and 400 mm<sup>2</sup> give 83% and 54%. The bulge volume predictions decrease from 76% to 63% as the area increases. The slope predictions are more consistent, between 57% and 67%. *Materials* **2023**, *16*, x FOR PEER REVIEW 25 of 33

**Figure 27.** Residual displacements caused by the Soft Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) slightly exceeds the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm2 distribution (yellow line) is in the middle of the experimental range. The predicted apex displacement for the 400 mm2 distribution (light blue line) underestimates the minimum experimental value of about 0.5 mm. The radial extension of the bulge is slightly underestimated. Over 35 mm radial distance from the apex the three load distributions are impossible to distinguish. predictions are more consistent, between 80% and 88%. **Figure 27.** Residual displacements caused by the Soft Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) slightly exceeds the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) is in the middle of the experimental range. The predicted apex displacement for the 400 mm<sup>2</sup> distribution (light blue line) underestimates the minimum experimental value of about 0.5 mm. The radial extension of the bulge is slightly underestimated. Over 35 mm radial distance from the apex the three load distributions are impossible to distinguish.

**Figure 28.** Validation for SP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (98%), while the 100 mm2 and 400 mm2 give 128% and 66%. The bulge volume predictions decrease from 81% to 56% as the area increases. The slope

**Figure 27.** Residual displacements caused by the Soft Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) slightly exceeds the maximum experimental value (blue dotted line). The predicted apex displacement for the 200 mm2 distribution (yellow line) is in the middle of the experimental range. The predicted apex displacement for the

the apex the three load distributions are impossible to distinguish.

**Figure 28.** Validation for SP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (98%), while the 100 mm2 and 400 mm2 give 128% and 66%. The bulge volume predictions decrease from 81% to 56% as the area increases. The slope predictions are more consistent, between 80% and 88%. **Figure 28.** Validation for SP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm<sup>2</sup> distribution gives the best estimation (98%), while the 100 mm<sup>2</sup> and 400 mm<sup>2</sup> give 128% and 66%. The bulge volume predictions decrease from 81% to 56% as the area increases. The slope predictions are more consistent, between 80% and 88%. *Materials* **2023**, *16*, x FOR PEER REVIEW 26 of 33

**Figure 29.** Residual displacements caused by the Hollow Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 0.2 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) is in the middle of the experimental range. The predicted apex displacement for the 400 mm2 distribution (light blue line) is 0.5 lower than the minimum experimental value. The radial extension of the bulge is slightly underestimated. For radial distances over 25 mm from the apex, the three load distributions are impossible to distinguish. **Figure 29.** Residual displacements caused by the Hollow Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 0.2 mm. The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) is in the middle of the experimental range. The predicted apex displacement for the 400 mm<sup>2</sup> distribution (light blue line) is 0.5 lower than the minimum experimental value. The radial extension of the bulge is slightly underestimated. For radial distances over 25 mm from the apex, the three load distributions are impossible to distinguish.

**Figure 30.** Validation for HP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (98%), while the 100 mm2 and 400 mm2 give 125% and 63%. The bulge volume predictions decrease from 75% to 63% as the area increases. The slope

predictions increase from 60% (100 mm2) to 80% (200 and 400 mm2).

**Figure 29.** Residual displacements caused by the Hollow Point bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) of about 0.2 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) is in the middle of the experimental range. The predicted apex

over 25 mm from the apex, the three load distributions are impossible to distinguish.

**Figure 30.** Validation for HP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (98%), while the 100 mm2 and 400 mm2 give 125% and 63%. The bulge volume predictions decrease from 75% to 63% as the area increases. The slope predictions increase from 60% (100 mm2) to 80% (200 and 400 mm2). **Figure 30.** Validation for HP on Creusabro® 8000. Validation of the calculated residual displacement fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm<sup>2</sup> distribution gives the best estimation (98%), while the 100 mm<sup>2</sup> and 400 mm<sup>2</sup> give 125% and 63%. The bulge volume predictions decrease from 75% to 63% as the area increases. The slope predictions increase from 60% (100 mm<sup>2</sup> ) to 80% (200 and 400 mm<sup>2</sup> ). *Materials* **2023**, *16*, x FOR PEER REVIEW 27 of 33

**Figure 31.** Residual displacements caused by the Full Metal Jacket bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) underestimates the experimental (blue and red dotted lines) range of about 2 mm. The overall bulge extension is significantly underestimated. For radial distances over 15 mm from the apex, the three load distributions are difficult to distinguish. **Figure 31.** Residual displacements caused by the Full Metal Jacket bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) underestimates the experimental (blue and red dotted lines) range of about 2 mm. The overall bulge extension is significantly underestimated. For radial distances over 15 mm from the apex, the three load distributions are difficult to distinguish.

**Figure 32***.* Validation for FMJ on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (67%), while the 200 mm2 and 400 mm2 give 49% and 32%. The bulge volume predictions are consistent between 22% and 25%. The

slope predictions increase from 9% (100 mm2) to 19% (400 mm2).

distinguish.

**4. Discussion** 

distributions are difficult to distinguish.

**Figure 31.** Residual displacements caused by the Full Metal Jacket bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) underestimates the

**Figure 32***.* Validation for FMJ on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm2 distribution gives the best estimation (67%), while the 200 mm2 and 400 mm2 give 49% and 32%. The bulge volume predictions are consistent between 22% and 25%. The slope predictions increase from 9% (100 mm2) to 19% (400 mm2). **Figure 32.** Validation for FMJ on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 100 mm<sup>2</sup> distribution gives the best estimation (67%), while the 200 mm<sup>2</sup> and 400 mm<sup>2</sup> give 49% and 32%. The bulge volume predictions are consistent between 22% and 25%. The slope predictions increase from 9% (100 mm<sup>2</sup> ) to 19% (400 mm<sup>2</sup> ). *Materials* **2023**, *16*, x FOR PEER REVIEW 28 of 33

**Figure 33.** Residual displacements caused by the Monolithic bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) by 0.3 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) almost coincides with the middle range of the experimental values (blue and red dotted lines). The predicted apex displacement for the 400 mm2 distribution (light blue line) coincides with the minimum experimental value. The radial extension of the bulge is well estimated. For radial distances over 35 mm from the apex, the three load distributions are impossible to **Figure 33.** Residual displacements caused by the Monolithic bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm<sup>2</sup> distribution (grey line) exceeds the maximum experimental value (blue dotted line) by 0.3 mm. The predicted apex displacement for the 200 mm<sup>2</sup> distribution (yellow line) almost coincides with the middle range of the experimental values (blue and red dotted lines). The predicted apex displacement for the 400 mm<sup>2</sup> distribution (light blue line) coincides with the minimum experimental value. The radial extension of the bulge is well estimated. For radial distances over 35 mm from the apex, the three load distributions are impossible to distinguish.

**Figure 34***.* Validation for MONO on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (107%), while the 100 mm2 and 400 mm2 give 141% and 70%. The bulge volume predictions decrease with the area, from 113% to 76%.

All the observed shots showed massive fragmentation of the bullets, with no full penetration of the 8 mm thick targets. However, evidence of non-complete fragmentation was reported on both FMJ bullets: a large solid part of the bullet, with dimensions comparable with the caliber, was found bouncing back after the shot on Creusabro® 8000 (Figure 16), and a significant volume of filler material together with part of the jacket material was found having penetrated the first plate of Durostat® 400 (Figures 12 and 14). These two cases, therefore, do not satisfy the theoretical hypotheses supporting the

The slope predictions are more consistent, between 109% and 123%.

distinguish.

**4. Discussion** 

**Figure 33.** Residual displacements caused by the Monolithic bullet on Creusabro® 8000 plates. The predicted apex displacement for the 100 mm2 distribution (grey line) exceeds the maximum experimental value (blue dotted line) by 0.3 mm. The predicted apex displacement for the 200 mm2 distribution (yellow line) almost coincides with the middle range of the experimental values (blue and red dotted lines). The predicted apex displacement for the 400 mm2 distribution (light blue line)

**Figure 34***.* Validation for MONO on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm2 distribution gives the best estimation (107%), while the 100 mm2 and 400 mm2 give 141% and 70%. The bulge volume predictions decrease with the area, from 113% to 76%. The slope predictions are more consistent, between 109% and 123%. **Figure 34.** Validation for MONO on Creusabro® 8000. Validation of the calculated residual displacements fields in terms of apex displacement (blue), total bulge volume (red), and slope at 50 mm (yellow). In abscissa the loaded area of the three arbitrary distributions. In terms of apex displacement, the 200 mm<sup>2</sup> distribution gives the best estimation (107%), while the 100 mm<sup>2</sup> and 400 mm<sup>2</sup> give 141% and 70%. The bulge volume predictions decrease with the area, from 113% to 76%. The slope predictions are more consistent, between 109% and 123%.

#### **4. Discussion**

All the observed shots showed massive fragmentation of the bullets, with no full penetration of the 8 mm thick targets. However, evidence of non-complete fragmentation was reported on both FMJ bullets: a large solid part of the bullet, with dimensions comparable with the caliber, was found bouncing back after the shot on Creusabro® 8000 (Figure 16), and a significant volume of filler material together with part of the jacket material was found having penetrated the first plate of Durostat® 400 (Figures 12 and 14). These two cases, therefore, do not satisfy the theoretical hypotheses supporting the All the observed shots showed massive fragmentation of the bullets, with no full penetration of the 8 mm thick targets. However, evidence of non-complete fragmentation was reported on both FMJ bullets: a large solid part of the bullet, with dimensions comparable with the caliber, was found bouncing back after the shot on Creusabro® 8000 (Figure 16), and a significant volume of filler material together with part of the jacket material was found having penetrated the first plate of Durostat® 400 (Figures 12 and 14). These two cases, therefore, do not satisfy the theoretical hypotheses supporting the formula for the load history estimation demonstrated in Section 2.4, For this reason both FMJ shots should be considered control cases to evaluate the effectiveness of the method when its applicability conditions are not verified.

> Considering the fields of residual displacement, the simulation and the experimental results diverge significantly for both FMJ bullet cases. The predicted displacement fields are being consistently and significantly underestimated, particularly so in terms of bulge volume and slope, for which the best estimations for FMJ impacts are, respectively, 33% and 40% on Durostat® 400 and 25% and 19% for Creusabro® 8000 (Figures 23, 24, 31 and 32). The relevant underestimation in the fields of residual displacement in these two cases is due to the force peaks needed to decelerate the relevant solid parts of the bullets instead of gradually deflecting their mass as happens during the ideal bullet splash. In the case of the partial penetration of the Durostat® 400 plate, the local reduction in the stiffness of the target due to the loss of integrity of the front plate and the direct impact of parts of the bullet onto the back plate should also have an effect in increasing the real back plate deformation.

> The experimental verifications have instead confirmed that, on the contrary, the SP, HP, and MONO cases are fully compliant with the bullet-splash theoretical hypotheses; the comparisons between the predicted and experimental residual displacement fields for these cases are therefore considered significant for the validation of the load histories estimated with the proposed method.

> The validation was performed on the results obtained by simulating the impacts on the three different arbitrary load distributions considered representative of the extension of the marks left on the plates after the impacts. The 100 mm<sup>2</sup> and 200 mm<sup>2</sup> distributions were considered representative of the minimum and maximum observed areas where the interaction between bullet and target is mainly compressive, causing the formation of the

fragments and their 90-degree, almost instantaneous, deflection. The 400 mm<sup>2</sup> distribution area, instead, includes approximately the entire extension of the interaction areas observed on the plates, including the peripheral area where the radial marks show that the interaction between bullet and target only consisted in a sliding contact of the already formed and deflected fragments. This last distribution was therefore expected to be less representative of the local compressive phenomena, but still relevant as a sensitivity test case of the effects of the pressure distributions on the deformation fields.

The comparisons between the simulated and experimental fields of residual displacements (Figures 19, 21, 23, 25, 27, 29, 31 and 33) show that the effects of the load distributions are significant only within 25 to 35 mm from the apex. The differences in terms of residual displacement at the apex are due to the fact that the three distributions apply pressure intensities that are inversely proportional to their loaded areas to guarantee that the integration of the pressure fields is equivalent to the load history for all the arbitrary distributions. To apply the same force, the pressure applied on the 100 mm<sup>2</sup> area must be, respectively, two and four times higher than the intensity applied by the 200 mm<sup>2</sup> and 400 mm<sup>2</sup> uniform distributions; therefore, the loaded elements must withstand proportionally higher stress and consequently encounter higher fields of plastic strain compared to the same elements loaded with lower intensity distributions. This explains the local differences in terms of maximum residual displacement at the apex between the three force-equivalent distributions as well as the almost indistinguishable effects over 25 to 35 mm from the apex, since the peripheral elements encounter approximately the same stress waves for all the distributions. In fact, the differences in radial extension between the three loaded areas are very limited compared to the distance covered by a mechanical signal during the duration of the considered impulses, therefore the stress waves propagating from the 100 mm<sup>2</sup> distribution have a duration that is less than 3% shorter than the 400 mm<sup>2</sup> one, which has neglectable engineering effects.

The validation graphs (Figures 20, 22, 24, 26, 28, 30, 32 and 34) demonstrate that the estimated load histories applied on the 100 mm<sup>2</sup> and 200 mm<sup>2</sup> distributions allow correct predictions of the residual displacements at the apex for all the verified bullet splashes (SP, HP, and MONO), with the sharpest distribution giving percentage validations between 109% and 157% (average 126%) on Durostat® 400 and between 125% and 141% (average 131%) for Creusabro® 8000. Moreover, the 200 mm<sup>2</sup> distribution gives representative validations between 82% and 119% (average 95%) on Durostat® 400 and between 98% and 107% (average 101%) for Creusabro® 8000. It is crucial to notice that even though the two sharpest distributions have a 100% difference in pressure intensity and loaded surfaces, the average difference in the estimations of the maximum residual displacement between the two is limited to 31% (129% vs. 98%). Even the 400 mm<sup>2</sup> distribution gives an average validation in apex displacement of 65%, despite a pressure field intensity equal to 25% of the one imposed by the sharpest distribution.

In all of the cases, the validation in bulge volume is much more consistent between the distributions, with just a modest reduction from the sharpest to the flattest. The sharpest distribution gives average percentage validations of 65% on Durostat® 400 and 85% on Creusabro® 8000, while the 200 mm<sup>2</sup> distribution gives 63% on Durostat® 400 and 81% on Creusabro® 8000; the flattest gives 52% on Durostat® 400 and 65% on Creusabro® 8000. The slight reduction in bulge volume is due to the reduction in local stress that causes the reduction in the maximum residual displacement at the apex, which has a minor effect on the total bulge volume.

The validation in slope at 50 mm from the apex in most of the cases increases from the sharpest to the intermediate distribution and then slightly decreases or remains constant for the flattest: the 100 mm<sup>2</sup> gives an average validation of 54% on Durostat® 400 and 85% on Creusabro® 8000, while the 200 mm<sup>2</sup> distribution gives 66% on Durostat® 400 and 97% on Creusabro® 8000. The 400 mm<sup>2</sup> gives 66% on Durostat® 400 and 90% on Creusabro® 8000. The increase and slight decrease with the extension of the loaded areas is explained by the opposite effects of reducing the load intensity, which decreases the bulge volume, and reducing the average distance from the load application to the peripheral area where the slope is calculated, which tends to increase the peripheral deformation.

In terms of general bulge shape, the simulations tend to underestimate the half-depth diameter. The calculation of the bulge volumes shows that the simulations of the shots on Durostat® 400 estimate bulge volumes around 35% lower than the reality (Table 11), while on Creusabro® 8000 the estimation is better, with just 15% of underestimation on average between the cases (Table 12). A difference in shape between the plate materials is also observed in terms of slope of the bulge at 50 mm from the apex, with a 46% underestimation on Durostat® 400 plates (Table 11) and a 15% underestimation on Creusabro® 8000 (Table 12), again on average between the cases. These differences are due to the simplification hypotheses introduced with the method, in particular with the hypothesis of a 90-degree deflection of the bullet debris, which is an ideal reference case that is closer to reality when the impact surface keeps its perfect planarity. In reality, after the impact surface gradually bulges, the debris are subjected to a slightly higher deflection angle, interacting on a gradually wider surface on the target. This results in a radial component of the impact force that increases the width of the bulge and a slight increase in the normal force due to the higher angle of deflection of the debris. This is confirmed by the fact that the estimation of the bulge volume and slope is significantly better on Creusabro® 8000 plates, which encounter more contained residual displacements compared to the Durostat® 400 ones, thanks to its higher hardness and tensile strength. This confirms that the proposed method is well suited for targets much harder than the bullets, which represents the interactions between bullets and a perfectly working ballistic protection.

**Table 11.** Validation comparisons for Durostat® 400. Validation percentages are calculated as numerical simulation results obtained for the 100 mm<sup>2</sup> distribution divided by the experimental results.


\* Not compliant with bullet-splash hypotheses.

**Table 12.** Validation comparisons for Creusabro® 8000. Validation percentages are calculated as numerical simulation results obtained for the 100 mm<sup>2</sup> distribution divided by the experimental results.


\* Not compliant with bullet-splash hypotheses.

#### **5. Conclusions**

The study verified the effectiveness of the proposed simplified formula to estimate the load history due to bullet-splash in predicting the resultant interaction forces due to semi-jacketed, full metal jacket and monolithic .308 Winchester bullets impacting on double 4 mm-thick plates of Durostat® 400 and Creusabro® 8000 hard steels.

The experimental analyses reported that the SP, HP and MONO impacts were compliant with the bullet-splash theory, while both the FMJ cases failed to meet the theoretical hypotheses due to partial penetration of the target and partial fragmentation. None of the targets were fully penetrated.

The validation of the method was performed by comparing the experimental results, in terms of fields of residual displacement measured on the impacted plates, with the results of finite element explicit simulations to predict the residual deformations of the plates subjected to the load histories estimated according to the proposed theory. Sensitivity tests were conducted to verify the influence of the extension of the load distribution on the numerical results.

The validation in terms of residual displacements at the apex confirm the effectiveness of the method for all the shots fully compliant with the bullet-splash hypotheses (SP, HP, MONO). On the contrary, the simulations applying the load histories estimated for the FMJ bullets significantly underestimated the residual deformation fields, confirming that the effectiveness of method strictly depends on the validation of its theoretical hypotheses.

The validation in terms of bulge volume and slope at 50 mm from the apex of the bulges shows an underestimation of the overall residual deformation fields which has been related to the strength of the target: the more the target is deformed under the normal impact forces, the more the debris must be deflected, therefore probably resulting in a wider bulge than expected by considering only a 90-degree deflection as the method hypothesizes. This suggests a possible improvement and generalization of the formula to take into account deflection angles higher than 90 degrees, depending on the specific experimental evidence.

The sensitivity tests conducted to check the effects of the load distributions on the deformation fields demonstrate that even varying the area (or intensity) by two or four times only causes local effects, within 25 to 35 mm from the apex, while the bulge volume and the slope at 50 mm are just slightly influenced. It is worth noting that the results confirm that the two sharpest distributions (100 mm<sup>2</sup> and 200 mm<sup>2</sup> ) are the most representative, with validations in terms of apex displacements between 82% and 157%. These two distributions were identified as the range of areas subjected to mostly compressive interactions according to the marks left on the plates; this suggests that an equivalent application area could be effectively identified in this range for all the test cases. The consistency of the validation percentages for different bullets with significantly different mechanical and geometrical characteristics further confirms that the real space distribution of the pressure fields has neglectable effects, and therefore the resultant load history approach is appropriate for the purpose of the study.

All the evidence suggests that the method could be properly applied to the assessment of structural systems subjected to ballistic impacts, provided that previous experimental ballistic tests have been analyzed and demonstrated that the target surface is able to completely fragment and deflect the impactor without allowing any penetration. In case of significant deformation of the impact surface, the results suggest that the method should be applied more carefully, with conservative safety margins to be assessed considering the specific outcomes of the experimental tests.

Further developments of the research should therefore investigate the effects of high deflection angles and the influence of the debris dimensions on the intensity of the impulses, with the ultimate goal of identifying possible simplified procedures to generalize the load history approach and to allow the parsimonious simulation of a wider range of ballistic impacts causing partial penetrations and fragmentations or high deformation of the target.

**Author Contributions:** R.A. and M.Q. conceived the method and conducted all the numerical analyses. A.C. and I.C. conducted the experimental analyses. F.B. supervised the metallurgical characterization of the plates. M.V.B. leads the project. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Informed Consent Statement:** The study did not involve humans.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The ballistic tests were conducted at the shooting range of Fombio (Lodi, Italy) belonging to the Italian national shooting range network (Tiri a Segno Nazionali/TSN); the shots were taken by Giorgio Prandini. A special thanks from the authors is due to Prandini and the whole staff for the invaluable support, expertise, and patience.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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**Masoud Tahani 1,2,\*, Eligiusz Postek <sup>2</sup> and Tomasz Sadowski 3,\***

	- <sup>3</sup> Department of Solid Mechanics, Lublin University of Technology, 20-618 Lublin, Poland

**Abstract:** The equivalent characteristics of the materials' interfaces are known to impact the overall mechanical properties of ceramic–metal composites significantly. One technological method that has been suggested is raising the temperature of the liquid metal to improve the weak wettability of ceramic particles with liquid metals. Therefore, as the first step, it is necessary to produce the diffusion zone at the interface by heating the system and maintaining it at a preset temperature to develop the cohesive zone model of the interface using mode I and mode II fracture tests. This study uses the molecular dynamics method to study the interdiffusion at the interface of α-Al2O3/AlSi12. The hexagonal crystal structure of aluminum oxide with the Al- and O-terminated interfaces with AlSi12 are considered. A single diffusion couple is used for each system to determine the average main and cross ternary interdiffusion coefficients. In addition, the effect of temperature and the termination type on the interdiffusion coefficients is examined. The results demonstrate that the thickness of the interdiffusion zone is proportional to the annealing temperature and time, and Aland O-terminated interfaces exhibit similar interdiffusion properties.

**Keywords:** self-diffusion; interdiffusion; diffusion coefficient; Al2O3/AlSi12 interface; molecular dynamics

## **1. Introduction**

Metal matrix composites (MMCs) are increasingly employed in the automotive, aerospace, and biomedical industries owing to their exceptional specific strength, high stiffness, and remarkable wear resistance [1]. These composites commonly employ aluminum, titanium, or magnesium as matrix materials, while alumina, silicon carbide, or boron carbide are often utilized as reinforcing elements [2,3].

Aluminum oxide (Al2O3) is a versatile and widely used ceramic material with various applications due to its excellent properties and attractive price [4]. Some common uses of aluminum oxide include abrasive material used in grinding and polishing tools, hightemperature environment applications such as furnace linings and refractory materials, electrical insulators, dental and medical applications, and as a filter medium [3,5].

The eutectic aluminum–silicon (AlSi12) alloys, widely used in the transportation industry [6] and have high specific properties and good castability, can replace the pure Al metal matrix. AlSi12 alloy is an aluminum alloy that contains 12 wt.% silicon. It is commonly used in casting applications due to its good fluidity and ability to produce castings with fine details [6]. The high silicon content in the alloy also provides it with excellent thermal properties, making it suitable for use in engine parts and other high-temperature applications [2]. This alloy also has a low density and good corrosion resistance, which makes it useful in the aerospace and automotive industries.

Metal–ceramic composites may exhibit improved wear resistance and strength properties compared to the individual materials and can be used in various high-temperature

**Citation:** Tahani, M.; Postek, E.; Sadowski, T. Diffusion and Interdiffusion Study at Al- and O-Terminated Al2O3/AlSi12 Interface Using Molecular Dynamics Simulations. *Materials* **2023**, *16*, 4324. https://doi.org/10.3390/ ma16124324

Academic Editors: Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 28 April 2023 Revised: 2 June 2023 Accepted: 7 June 2023 Published: 12 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

applications. For example, a composite of Al2O<sup>3</sup> and AlSi12 alloy can be made by various techniques such as powder metallurgy [6,7], hot pressing, squeeze casting [7], or infiltration [1,7–10]. Interpenetrating phase composites (IPCs) are novel materials with possibly enhanced characteristics compared with traditional composites with discontinuous particles, whiskers, or short fibers [11–15]. The properties and performance of the composite can be tailored by the processing conditions, relative proportions of the two materials, microstructure, proportion of the components, and the interface's properties.

This study investigates the use of α-Al2O<sup>3</sup> reinforcement in the AlSi12 metal alloy matrix. The Al2O3/AlSi12 composite has demonstrated very good wear and abrasion resistance [16,17]. Therefore, this composite material has the potential to be used in brake disks in the automotive industry [7]. The mechanical characteristics of the interface constituents and the nature of the interface determine the general mechanical and failure behavior of MMCs [18–20]. To this end, the interface attributes in MMCs must be thoroughly investigated. Diffusion causes the interface between phases to exhibit a fuzzy region. Hence, the primary step toward deriving the cohesive zone model of the interface is to investigate the diffusion between the two phases.

Oishi and Kingery [21] first measured oxygen self-diffusion in single and polycrystalline Al2O<sup>3</sup> in 1960. They studied diffusion in temperatures above 1650 ◦C and observed enhanced diffusion for the polycrystalline specimens. Lagerlof et al. [22] also deduced oxygen self-diffusion coefficients using observations of the shrinking of tiny prismatic dislocation loops in sapphire crystals subjected to prior distortion at a temperature of 1400 ◦C. The diffusion coefficient was determined, and it was assumed that oxygen lattice diffusion was smaller than aluminum lattice diffusion. Paladino and Kingery [23] determined the self-diffusion coefficient of aluminum in coarse-grain polycrystalline aluminum oxide using aluminium-26 as a tracer in the temperature range of 1670–1905 ◦C. They found that the diffusivity of aluminum ions is greater than oxygen ions.

Furthermore, Gall et al. [24] measured aluminum self-diffusion in single-crystal α-Al2O<sup>3</sup> using aluminum-26 as a radioactive tracer in the temperature range of 1540–1697 ◦C. They obtained very different conclusions regarding the diffusion coefficients compared to Paladino and Kingery [23]. A review of the major diffusion processes in α-Al2O3, including aluminum and oxygen lattice diffusion, oxygen grain boundary diffusion, and pipe diffusion, was presented by Heuer [25]. Knowledge regarding the diffusion of aluminum and oxygen in aluminum oxide was found to be insufficient. Using the density functional theory, Milas et al. [26] investigated the diffusion of Al, O, Pt, Hf, and Y atoms on the α-Al2O3(0001) surface to study the diffusion mechanisms at the alumina grain boundaries in thermal barrier coatings. They discovered that the Al diffusion is significantly lower than the O diffusion barrier. The literature on the self-diffusion of single crystals and the impurity diffusion of some significant elements in alumina was reviewed by Pelleg [27]. Moreover, they discussed grain boundary diffusion and poly-crystalline alumina diffusion.

Unfortunately, the wettability of ceramic particles with liquid aluminum alloys is often weak. Many technological procedures have been suggested to improve the wetting of ceramic by liquid metal. These include raising the temperature of the metal liquid, pretreatment of ceramic particles or fibers, coating the ceramics, and incorporating some surface-active elements into the matrix. To the authors' knowledge, no previous investigations have been conducted on the diffusion behavior of the α-Al2O3/AlSi12 diffusion couple. Therefore, this study aims to explore the self-diffusion and interdiffusion phenomena at the interface by employing the molecular dynamics (MD) method by increasing the system's temperature to the specified level. The Al- and O-terminated interfaces of α-Al2O<sup>3</sup> with AlSi12 are considered. The influence of annealing temperature, annealing duration, and type of termination at the interface on the diffusion zone and interdiffusion coefficients are studied.

#### **2. Modeling Method and Simulation Technique**

Diffusion involves the migration of atoms or molecules from a region of higher concentration to a region of lower concentration. Atoms can diffuse across the interface, resulting in the movement of atoms between the two phases. The diffusion rate depends on several factors, including the temperature, chemical composition of the two materials, and the interface between the two phases. At high temperatures, the diffusion rate will be faster, and the atoms will have more energy to move through the material. It is intended to investigate the effect of raising the temperature on the diffusion region and interdiffusion coefficients at the α-Al2O3/AlSi12 interface.

The molecular dynamics method can study basic processes such as diffusion by using Newton's second law to calculate the acceleration of atoms by describing atomic interactions through interatomic potentials. In this study, MD simulations are performed utilizing the open-source MD program LAMMPS version 23Jun2022 (large-scale atomic/molecular massively parallel simulator) [28], and the OVITO version 3.8.4 (open visualization tool) software [29] is utilized to visualize the atomic structure's evolution. The interatomic potential must be precisely quantified because an interatomic potential energy model typically represents atomic interactions. Experimental data or ab initio calculations, such as cohesive energy and elastic modulus, can be used to determine the model parameters. The following section explores the interatomic potentials attributed to aluminum oxide, aluminum, silicon, and the interface.

#### *2.1. Potential Functions*

In the Al2O3/AlSi12 system, several atomic interactions are possible and should be taken into account during simulations. The atomic interactions between Al particles in an fcc crystal structure differ significantly from those between aluminum oxide ceramic particles. Metal atoms have electron clouds that determine the strength of their bonds, whereas ionic bonding is the primary factor in ceramics. The interface between metal and ceramic, where the atoms tend to create bonds between two dissimilar structures, introduces additional complexity.

The third-generation charge-optimized many-body potential (COMB3) [30] is a type of interatomic potential that can be used to describe interactions between atoms in aluminum– oxygen systems. The COMB3 potential uses a combination of pair potentials and electron density functions to describe the atomic interactions. The potential is fitted to experimental data and ab initio calculations. It has been shown to reproduce a wide range of properties of aluminum–oxygen systems, including the lattice constant, elastic constants, and the deformation of Al and Al2O<sup>3</sup> under tensile loading. The total energy per atom for the Al-O system, with a charge of *q* at position *r*, in the COMB3 potential can be expressed as [30]:

$$\mathcal{U}\_{\rm{ul}}(r,\,\boldsymbol{q}) = \mathcal{U}\_{\rm{es}}(\boldsymbol{q},\,\boldsymbol{r}) + \mathcal{U}\_{\rm{shrt}}(\boldsymbol{q},\,\boldsymbol{r}) + \mathcal{U}\_{\rm{vdW}}(\boldsymbol{r}) + \mathcal{U}\_{\rm{evr}}(\boldsymbol{r}) \tag{1}$$

where *Ues* denotes the energy required to create an atom's charge, as well as the energies involved in charge–charge interactions, charge–nuclear interactions, and polarizability. Furthermore, *Ushort* is the energy of pairwise attractive and repulsive functions, *UvdW* is long-range van der Waals interactions, and *Ucorr* is the correction terms employed to adjust energies associated with specific angles outside the bond order terms.

The Tersoff potential [31], an empirical function composed of two-body terms, is employed for silicon–oxygen interactions. The bonding between atoms *i* and *j* in the many-body Tersoff potential can be expressed as:

$$V\_{\rm ij} = f\_{\mathbb{C}}(r\_{\rm ij}) \left[ f\_{\mathbb{R}}(r\_{\rm ij}) + b\_{\rm ij} f\_{\mathbb{A}}(r\_{\rm ij}) \right] \tag{2}$$

where *fR*(*rij*), *fA*(*rij*), and *fC*(*rij*) are repulsive, attractive, and cut-off potential functions, *rij* is the atomic bond length between atom *i* and *j*, and *bij* is a function that adjusts the attractive interaction, respectively.

The ab initio data gathered by Zhao et al. [32] are consistent with the Morse potential, which best represents aluminum–silicon interactions. The Morse potential function is defined as:

$$V = D\_0 \left[ e^{-2a(r - r\_0)} - 2e^{-a(r - r\_0)} \right] \tag{3}$$

where *D*0, *α*, *r*, and *r*<sup>0</sup> represent the well depth of the potential, the width of the potential, the distance between atoms, and the equilibrium bond length, respectively. The Morse potential with parameters *D*<sup>0</sup> = 0.4824 eV, *α =* 1.322 1/Å, and *r*<sup>0</sup> = 2.92 Å [26] is employed in this study for aluminum–silicon interactions.

The elastic constants of α-Al2O<sup>3</sup> are determined with previously mentioned potential functions and then compared with the experimental [33], MD simulations [34], and innovative integration of metadynamics and kinetic Monte Carlo simulation techniques ref. [35] in Table 1. The same results for AlSi12 are also presented in this table. The lattice parameters of hexagonal α-Al2O<sup>3</sup> are *a = b =* 4.759 Å, *c =* 12.991 Å, *α = β =* 90<sup>o</sup> , and *γ =* 120<sup>o</sup> , and the lattice constant of fcc Al is 4.0495. AlSi12 single-crystal is formed by substituting 12 wt.% of Al atoms with Si atoms. The linear elastic constants *Cij* are obtained at zero temperature by analyzing the stress–strain relation *Cij* = *∂σij*/*∂εij*, where *σij* and *εij* are, respectively, the stress and strain components. General decent agreements between the present results and those of other investigators are observed in Table 1. Consequently, the potential functions utilized here demonstrate an accurate simulation of the interactions between atoms.

**Table 1.** The elastic constants determined through the current MD simulations and their comparison with values reported by other researchers.


### *2.2. Molecular Dynamics Model*

According to high-resolution transmission electron microscopy, it has been observed that the predominant orientation relationship at the Al2O3/Al interface is characterized by the parallel alignment of the Al(111) plane and the Al2O3(0001) basal plane [36]. Pilania et al. [37] also studied coherent and semi-coherent α-Al2O3(0001)/Al(111) interfaces with a mixed metallic–ionic atomistic model using MD simulations. Therefore, in this study, the lattice orientation alignment (0001)[2 1 1 0]α−Al2O<sup>3</sup> (111)[1 1 2]AlSi12 is taken into account according to the research of other investigators.

The current model comprises a bilayer nanocomposite composed of α-Al2O<sup>3</sup> and AlSi12. The initial α-Al2O3/AlSi12 interface shown in Figure 1 is considered a single crystal of AlSi12 at the bottom and a single crystal of α-Al2O<sup>3</sup> at the top with an initial gap of 2.0 Å which closely approximates the equilibrium atomic distance at the interface. To examine the impact of alumina terminations on diffusion, two configurations are modeled at the interface: one with Al-termination and another with O-termination. These cases allow a comprehensive exploration to occur of how different terminations affect diffusion behavior. The MD model has a typical size of about 119 × 58 × 184 Å, containing a total of 109,986 atoms.

are implemented for the sample in all three directions.

**Figure 1.** Model of the Al-terminated α-Al2O3/AlSi12 interface designed for the MD analyses. **Figure 1.** Model of the Al-terminated α-Al2O3/AlSi12 interface designed for the MD analyses.

The geometric arrangement of atoms is optimized through the utilization of the conjugate gradient (CG) energy minimization method. First, the *NVT* canonical ensemble (constant number of particles *N*, volume *V*, and temperature *T*) at a constant temperature of 1200 K is imposed on the sample for 10 ps. Second, the *NPT* ensemble (constant number of particles *N*, pressure *P*, and temperature *T*) at zero pressure and a constant temperature of 1200 K is used for 15 ps to regulate the volume and achieve relaxation in the assembled interface system. Subsequently, the sample is subjected to heating at a heating rate of 10 K/ps until it reaches a preset temperature. Finally, the temperature is held constant at the specified value for a duration of 2.0 ns to analyze interdiffusion while monitoring and recording the atomic movements throughout this period. All processes are conducted using the *NPT* ensemble at zero pressure, employing a time-step of 0.2 fs. The simulations are performed at 1500, 1600, 1800, and 2000 K temperatures. Periodic boundary conditions

**3. Results and Discussion**  To study the diffusion properties of the α-Al2O3/Al interface, the system is heated to a predetermined temperature and maintained there for 2.0 ns. The development of the interface diffusion for the Al-terminated Al2O3/AlSi12 interface after heating it to 2000 K is illustrated in Figure 2. The initial configuration illustrates the sharp interface between Al2O3 and AlSi12, considering an initial gap of 2 Å. Furthermore, after maintaining it for 2.0 ns at 2000 K, the system configuration represents the local movement of atoms and the creation of a diffusion zone. The diffusion front is shown in this figure with a dashed line. The geometric arrangement of atoms is optimized through the utilization of the conjugate gradient (CG) energy minimization method. First, the *NVT* canonical ensemble (constant number of particles *N*, volume *V*, and temperature *T*) at a constant temperature of 1200 K is imposed on the sample for 10 ps. Second, the *NPT* ensemble (constant number of particles *N*, pressure *P*, and temperature *T*) at zero pressure and a constant temperature of 1200 K is used for 15 ps to regulate the volume and achieve relaxation in the assembled interface system. Subsequently, the sample is subjected to heating at a heating rate of 10 K/ps until it reaches a preset temperature. Finally, the temperature is held constant at the specified value for a duration of 2.0 ns to analyze interdiffusion while monitoring and recording the atomic movements throughout this period. All processes are conducted using the *NPT* ensemble at zero pressure, employing a time-step of 0.2 fs. The simulations are performed at 1500, 1600, 1800, and 2000 K temperatures. Periodic boundary conditions are implemented for the sample in all three directions.

#### **3. Results and Discussion**

To study the diffusion properties of the α-Al2O3/Al interface, the system is heated to a predetermined temperature and maintained there for 2.0 ns. The development of the interface diffusion for the Al-terminated Al2O3/AlSi12 interface after heating it to 2000 K is illustrated in Figure 2. The initial configuration illustrates the sharp interface between Al2O<sup>3</sup> and AlSi12, considering an initial gap of 2 Å. Furthermore, after maintaining it for 2.0 ns at 2000 K, the system configuration represents the local movement of atoms and the creation of a diffusion zone. The diffusion front is shown in this figure with a dashed line.

#### *3.1. Self-Diffusion*

The mean square displacements (MSDs) of Al, O, and Si atoms after maintaining the system for a duration of 2.0 ns at different temperatures of 1500, 1600, 1800, and 2000 K for the Al- and O-terminated α-Al2O3/AlSi12 diffusion couples are depicted in Table 2. This table presents the MSD values for Al atoms in α-Al2O3, Al atoms in AlSi12, and all Al atoms in the system. As expected, due to the difference in ceramic and metal atomic bonding, the Al atoms in α-Al2O<sup>3</sup> have a significantly lower MSD than the Al atoms in AlSi12. It is also observed that the MSD of O atoms is smaller than the MSD of Al atoms in α-Al2O3. It is observed that the MSD of Al atoms is larger than the MSD of Si atoms, and the MSD of Si atoms is also larger than the MSD of O atoms.

Al-terminated α-Al2O3/AlSi12

**Figure 2.** Cross-sections of the atomic configurations of the Al-terminated Al2O3/AlSi12 interface. The figure shows the initial atomic structure before relaxation, as well as the configuration after the system is held at 2000 K for 2.0 ns. The dashed line indicates the front of the diffusion region. **Figure 2.** Cross-sections of the atomic configurations of the Al-terminated Al2O3/AlSi12 interface. The figure shows the initial atomic structure before relaxation, as well as the configuration after the system is held at 2000 K for 2.0 ns. The dashed line indicates the front of the diffusion region.

*3.1. Self-Diffusion*  The mean square displacements (MSDs) of Al, O, and Si atoms after maintaining the **Table 2.** Mean square displacement (nm<sup>2</sup> ) of Al, O, and Si atoms at different temperatures for the Aland O-terminated α-Al2O3/AlSi12 diffusion couples.


**Diffusion Couple Temperature (K) Atom Al (Al2O3) Al (AlSi12) Al O Si**  1500 0.65 24.95 13.46 0.35 2.22 The coefficients of self-diffusion for each atom type are obtained by analyzing the slope of the MSDs employing Einstein's relation [38]:

$$D\_A = \lim\_{t \to \infty} \frac{1}{N\_A} \sum\_{i=1}^{N\_A} \frac{\left< \left| \mathbf{r}\_i^A(t) - \mathbf{r}\_i^A(0) \right|^2 \right>}{6t} \tag{4}$$

O-terminated α-Al2O3/AlSi12 1500 0.50 25.11 13.57 0.34 2.34 1600 0.64 32.26 17.58 0.38 2.36 1800 1.04 46.98 25.42 0.45 2.40 2000 1.67 62.34 33.63 0.50 2.46 The coefficients of self-diffusion for each atom type are obtained by analyzing the slope of the MSDs employing Einstein's relation [38]: 2 1 ( ) (0) <sup>1</sup> lim 6 *A A A <sup>N</sup> i i <sup>A</sup> <sup>t</sup> <sup>A</sup> <sup>i</sup> t <sup>D</sup>* →∞ *N t* <sup>=</sup> − <sup>=</sup> **r r** (4) where *NA* represents the total number of atoms of type *A*, *<sup>A</sup> <sup>i</sup>***r** denotes the position vector where *N<sup>A</sup>* represents the total number of atoms of type *A*, **r** *A i* denotes the position vector of the *i*th atom belonging to type *A*, and h· · ·i signifies the average calculated across all atoms of the same type. The activation energy *Q* and pre-exponential factor *D*<sup>0</sup> of atoms can be obtained by fitting the self-diffusion coefficients to the Arrhenius equation *D* = *D*<sup>0</sup> exp(−*Q*/*RT*). The Arrhenius plots of Al, O, and Si atoms for the Al- and Oterminated interfaces are illustrated in Figure 3. Similar to the MSD, Al atoms in Al2O<sup>3</sup> have a significantly smaller self-diffusion coefficient than the Al atoms in AlSi12 because of the differences in atomic bonding between ceramic and metal. Additionally, the selfdiffusion coefficient of O atoms is less pronounced than Al atoms in Al2O3. As can be seen, Al atoms have a higher self-diffusion coefficient than Si atoms, and Si atoms also have a higher self-diffusion coefficient than O atoms. Table 3 also displays the outcomes of the atoms' activation energies and pre-exponential factors.

> of the *i*th atom belonging to type *A*, and signifies the average calculated across all atoms of the same type. The activation energy *Q* and pre-exponential factor *D*0 of atoms can be obtained by fitting the self-diffusion coefficients to the Arrhenius equation

atoms' activation energies and pre-exponential factors.

**Figure 3.** Plots of Arrhenius equation for Al, O, and Si atoms in the Al- and O-terminated α-Al2O3/AlSi12 interface. (**a**) Al in Al2O3, (**b**) Al in AlSi12, (**c**) Al, (**d**) O, and (**e**) Si atoms. **Figure 3.** Plots of Arrhenius equation for Al, O, and Si atoms in the Al- and O-terminated α-Al2O3/AlSi12 interface. (**a**) Al in Al2O<sup>3</sup> , (**b**) Al in AlSi12, (**c**) Al, (**d**) O, and (**e**) Si atoms.

*D D Q RT* = − <sup>0</sup> exp( ) . The Arrhenius plots of Al, O, and Si atoms for the Al- and Oterminated interfaces are illustrated in Figure 3. Similar to the MSD, Al atoms in Al2O3 have a significantly smaller self-diffusion coefficient than the Al atoms in AlSi12 because of the differences in atomic bonding between ceramic and metal. Additionally, the selfdiffusion coefficient of O atoms is less pronounced than Al atoms in Al2O3. As can be seen, Al atoms have a higher self-diffusion coefficient than Si atoms, and Si atoms also have a higher self-diffusion coefficient than O atoms. Table 3 also displays the outcomes of the

**Table 3.** Arrhenius parameters, *D*0 and *Q*, for self-diffusion of Al, O, and Si atoms for Al- and Cterminated α-Al2O3/AlSi12 diffusion couples. **Table 3.** Arrhenius parameters, *D*<sup>0</sup> and *Q*, for self-diffusion of Al, O, and Si atoms for Al- and C-terminated α-Al2O3/AlSi12 diffusion couples.


#### *3.2. Interdiffusion 3.2. Interdiffusion*

The interdiffusion flux of an *n-*component system is described by the following Onsager's formulation [39,40] of Fick's law: The interdiffusion flux of an *n-*component system is described by the following Onsager's formulation [39,40] of Fick's law:

1

−

$$
\widetilde{J}\_i = -\sum\_{j=1}^{n-1} \widetilde{D}\_{ij}^n \frac{\partial C\_j}{\partial z} \tag{5}
$$

where e*J<sup>i</sup>* , *C<sup>i</sup>* , and *∂Ci*/*∂z* are the interdiffusion flux, mole fraction, and concentration gradient of component *<sup>i</sup>*, respectively. Furthermore, *<sup>D</sup>*e*<sup>n</sup> ij* is the interdiffusion coefficient. According to Equation (5), the interdiffusion behavior in a ternary system can be described by four independent interdiffusion coefficients: *<sup>D</sup>*e<sup>3</sup> <sup>11</sup>, *<sup>D</sup>*e<sup>3</sup> <sup>12</sup>, *<sup>D</sup>*e<sup>3</sup> <sup>21</sup>, and *<sup>D</sup>*e<sup>3</sup> <sup>22</sup>. The Boltzmann– Matano [41,42] method can determine the interdiffusion coefficients.

In the present research, the average interdiffusion coefficients are determined using the approach proposed by Dayananda and Sohn [41]. The average main interdiffusion coefficients (i.e., *D*e 3 <sup>11</sup> and *D*e 3 <sup>22</sup>) and cross interdiffusion coefficients (i.e., *D*e 3 <sup>12</sup> and *D*e 3 <sup>21</sup>) are evaluated by computing the atomic interdiffusion flux using only the single diffusion couple under study. The concentration curve is fitted using the Gaussian error function for each component. The interested reader will find detailed explanations about the method in Refs. [42,43]. for each component. The interested reader will find detailed explanations about the method in Refs. [42,43]. The variations in Al, Si, and O atom concentrations with respect to the *z*-coordinate,

are evaluated by computing the atomic interdiffusion flux using only the single diffusion couple under study. The concentration curve is fitted using the Gaussian error function

, *Ci*, and *C z <sup>i</sup>* ∂ ∂ are the interdiffusion flux, mole fraction, and concentration

According to Equation (5), the interdiffusion behavior in a ternary system can be

In the present research, the average interdiffusion coefficients are determined using the approach proposed by Dayananda and Sohn [41]. The average main interdiffusion

) and cross interdiffusion coefficients (i.e., <sup>3</sup> *<sup>D</sup>*<sup>12</sup>

is the interdiffusion coefficient.

11 12 21 *DDD* , , , and <sup>3</sup> *<sup>D</sup>*<sup>22</sup>

. The

)

and <sup>3</sup> *<sup>D</sup>*<sup>21</sup>

The variations in Al, Si, and O atom concentrations with respect to the *z*-coordinate, which is normal to the interface plane, are shown in Figure 4 for a quantitative analysis of the diffusion process in the Al-terminated α-Al2O3/Al interface. To obtain the concentration profiles, the diffusion couple is divided into thin slices with a thickness of 2.0 Å along the interface plane. The count of atoms for each type is determined within each slice. Figure 4 shows the initial concentration profiles before diffusion, and the profiles observed after keeping the system at 2000 K for a duration of 2.0 ns. A grey region also depicts the diffusion zone. The variations in atom concentrations for the O-terminated interface, which are not shown here for conciseness, indicate that the diffusion zones in the Al- and O-terminated systems are not significantly different. which is normal to the interface plane, are shown in Figure 4 for a quantitative analysis of the diffusion process in the Al-terminated α-Al2O3/Al interface. To obtain the concentration profiles, the diffusion couple is divided into thin slices with a thickness of 2.0 Å along the interface plane. The count of atoms for each type is determined within each slice. Figure 4 shows the initial concentration profiles before diffusion, and the profiles observed after keeping the system at 2000 K for a duration of 2.0 ns. A grey region also depicts the diffusion zone. The variations in atom concentrations for the O-terminated interface, which are not shown here for conciseness, indicate that the diffusion zones in the Al- and O-terminated systems are not significantly different.

*Materials* **2023**, *16*, x FOR PEER REVIEW 8 of 12

gradient of component *i*, respectively. Furthermore, *<sup>n</sup> Dij*

and <sup>3</sup> *<sup>D</sup>*<sup>22</sup>

described by four independent interdiffusion coefficients: <sup>333</sup>

Boltzmann–Matano [41,42] method can determine the interdiffusion coefficients.

where *<sup>i</sup> J*

coefficients (i.e., <sup>3</sup> *D*<sup>11</sup>

**Figure 4.** The variations in Al, Si, and O atom concentrations along the *z*-axis during interdiffusion of the Al-terminated α-Al2O3/AlSi12 interface. The initial system before relaxation and after a 2.0 ns maintenance at 2000 K are illustrated. The diffusion zone is depicted by the gray region. **Figure 4.** The variations in Al, Si, and O atom concentrations along the *z*-axis during interdiffusion of the Al-terminated α-Al2O3/AlSi12 interface. The initial system before relaxation and after a 2.0 ns maintenance at 2000 K are illustrated. The diffusion zone is depicted by the gray region.

Figure 5 illustrates the variations in the interdiffusion flux *J* and 0 *Jz z* ( ) <sup>−</sup> for the Al- and O-terminated α-Al2O3/AlSi12 diffusion couples after keeping the systems at 2000 K for a duration of 2.0 ns. The position of the Matano plane, denoted by *z*0, is also shown in Figure 5 by a vertical dashed line. It is observed that the Matano plane corresponds to the point of highest interdiffusion flux. The independent variables are arbitrarily chosen as the Al and O atoms, while the Si atom is assigned as the dependent variable. The profile variations in the two diffusion couples appear to be very similar. However, it is worth noting that the maximum interdiffusion flux of the Al-terminated interface is slightly higher than that of the O-terminated interface. Figure 5 illustrates the variations in the interdiffusion flux e*J* and e*J*(*z* − *z*0) for the Aland O-terminated <sup>α</sup>-Al2O3/AlSi12 diffusion couples after keeping the systems at 2000 Kfor a duration of 2.0 ns. The position of the Matano plane, denoted by *<sup>z</sup>*0, is also shown in Figure 5 by a vertical dashed line. It is observed that the Matano plane corresponds to the point of highest interdiffusion flux. The independent variables are arbitrarily chosen as the Al and O atoms, while the Si atom is assigned as the dependent variable. The profile variations in the two diffusion couples appear to be very similar. However, it is worth noting that the maximum interdiffusion flux of the Al-terminated interface is slightly higher than that of the O-terminated interface. *Materials* **2023**, *16*, x FOR PEER REVIEW 9 of 12

**Figure 5.** Interdiffusion flux *J* and 0 *Jz z* ( ) <sup>−</sup> for diffusion systems of the Al-terminated and Oterminated α-Al2O3/AlSi12 heated to 2000 K and kept at that temperature for a duration of 2 ns. **Figure 5.** Interdiffusion flux e*J* and e*J*(*z* − *z*0) for diffusion systems of the Al-terminated and Oterminated α-Al2O3/AlSi12 heated to 2000 K and kept at that temperature for a duration of 2 ns.

**For Composition Range of the Lower Side of Matano Plane 3 11 2 10 (m /s)** *Dij* <sup>−</sup> ×

*Si DOAl*

*Si DAlO*

Table 4 presents the calculated average values of the main and cross ternary interdiffusion coefficients for the Al- and O-terminated α-Al2O3/AlSi12 diffusion couples. The diffusion couples are kept at annealing temperatures of 1500, 1600, 1800, and 2000 K for a duration of 2.0 ns. The coefficients are determined using the composition ranges on

interdiffusion coefficients increase as the annealing temperature increases, as expected. Furthermore, all cross ternary interdiffusion coefficients are significantly smaller, with at least four orders of magnitude lower than the main interdiffusion coefficients. Hence, the cross ternary interdiffusion coefficients do not significantly influence the current ternary systems. Moreover, based on the findings in this table, it is observed that the diffusivity of Si and O atoms shows a slight increase in the Al-terminated system compared to the Oterminated counterpart. However, generally speaking, there is no appreciable distinction between the average interdiffusion coefficients of the Al- and O-terminated systems.

**Table 4.** The average interdiffusion coefficients for the ternary systems on either side of the Matano plane. These values are determined after the system is maintained at the preset temperature for a

*Si DOO*

1500 0.489 −4.2 × 10<sup>−</sup>7 1.8 × 10−5 0.584 0.489 2.2 × 10−<sup>7</sup>−2.0 × 10<sup>−</sup>5 0.584 1600 0.518 −6.3 × 10<sup>−</sup>7 8.4 × 10−6 0.623 0.518 5.7 × 10−<sup>7</sup> −9.3 × 10<sup>−</sup>6 0.623 1800 0.608 −7.9 × 10<sup>−</sup>7 6.1 × 10−7 0.845 0.608 −1.2 × 10<sup>−</sup><sup>6</sup> −1.9 × 10<sup>−</sup>6 0.845 2000 0.753 1.3 × 10<sup>−</sup><sup>7</sup> −6.4 × 10<sup>−</sup>6 1.307 0.753 −2.5 × 10<sup>−</sup>7 2.8 × 10−6 1.307

1500 0.429 2.1 × 10<sup>−</sup>7 6.3 × 10−6 0.489 0.429 2.8 × 10−7 4.1 × 10−6 0.489 1600 0.452 −5.4 × 10<sup>−</sup><sup>7</sup> −1.3 × 10<sup>−</sup>6 0.531 0.452 7.6 × 10−7 3.8 × 10−6 0.531 1800 0.527 −4.9 × 10<sup>−</sup><sup>7</sup> −6.9 × 10<sup>−</sup>6 0.562 0.527 1.4 × 10−6 1.7 × 10−6 0.685 2000 0.696 4.9 × 10<sup>−</sup>5 2.9 × 10−4 1.012 0.696 −1.3 × 10<sup>−</sup>4 9.9 × 10−5 1.012

A molecular dynamics method was employed to investigate atomistic evolutions during the interdiffusion at the α-Al2O3/AlSi12 interface. The self-diffusion and interdiffusion coefficients were assessed at 1500, 1600, 1800, and 2000 K annealing

*Si DAlAl*

**For Composition Range of the Upper Side of Matano Plane 3 11 2 10 (m /s)** *Dij* <sup>−</sup> ×

*Si DOAl*

*Si DOO*

*Si DAlO*

duration of 2 ns.

*Si DAlAl*

**4. Conclusions** 

**Diffusion Couple Temperature** 

Al-terminated α-Al2O3/AlSi12

O-terminated α-Al2O3/AlSi12 **(K)** 

Table 4 presents the calculated average values of the main and cross ternary interdiffusion coefficients for the Al- and O-terminated α-Al2O3/AlSi12 diffusion couples. The diffusion couples are kept at annealing temperatures of 1500, 1600, 1800, and 2000 K for a duration of 2.0 ns. The coefficients are determined using the composition ranges on the lower and upper sides of the Matano plane. It is observed from Table 4 that the main interdiffusion coefficients increase as the annealing temperature increases, as expected. Furthermore, all cross ternary interdiffusion coefficients are significantly smaller, with at least four orders of magnitude lower than the main interdiffusion coefficients. Hence, the cross ternary interdiffusion coefficients do not significantly influence the current ternary systems. Moreover, based on the findings in this table, it is observed that the diffusivity of Si and O atoms shows a slight increase in the Al-terminated system compared to the O-terminated counterpart. However, generally speaking, there is no appreciable distinction between the average interdiffusion coefficients of the Al- and O-terminated systems.

**Table 4.** The average interdiffusion coefficients for the ternary systems on either side of the Matano plane. These values are determined after the system is maintained at the preset temperature for a duration of 2 ns.


## **4. Conclusions**

A molecular dynamics method was employed to investigate atomistic evolutions during the interdiffusion at the α-Al2O3/AlSi12 interface. The self-diffusion and interdiffusion coefficients were assessed at 1500, 1600, 1800, and 2000 K annealing temperatures for different diffusion couples. Based on the findings of this study, the following conclusions can be made:


Future studies may utilize the samples after diffusion and cooling to determine the effective mechanical properties of the Al2O3/AlSi12 interface through the cohesive zone model and, therefore, the mechanical properties of the MMC.

**Author Contributions:** All authors contributed to the study's conception and design. Data collection and analysis were performed by M.T. The first draft of the manuscript was written by M.T. and all authors commented on previous versions of the manuscript. The final manuscript was approved by all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Polish National Agency for Academic Exchange (NAWA) under grant number [BPN/ULM/2021/1/00115/U/DRAFT/00001]. National Science Center, Poland, grant No. UMO 2019/33/B/ST8/01263.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request due to restrictions, e.g., privacy.

**Acknowledgments:** E.P. and T.S. acknowledge the National Science Center, Poland, grant No. UMO 2019/33/B/ST8/01263. The calculations were performed at PLGrid national facilities—Academic Computer Centre Cyfronet in Krakow, Academic Computer Centre in Gda ´nsk, and Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Article* **A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modeling**

**Gaetano Giunta 1,\*, Domenico Andrea Iannotta 1,2 and Marco Montemurro <sup>3</sup>**


**Abstract:** Variable Angle Tow (VAT) laminates offer a promising alternative to classical straightfiber composites in terms of design and performance. However, analyzing these structures can be more complex due to the introduction of new design variables. Carrera's unified formulation (CUF) has been successful in previous works for buckling, vibrational, and stress analysis of VAT plates. Typically, one-dimensional (1D) and two-dimensional (2D) CUF models are used, with a linear law describing the fiber orientation variation in the main plane of the structure. The objective of this article is to expand the CUF 2D plate finite elements family to perform free vibration analysis of composite laminated plate structures with curvilinear fibers. The primary contribution is the application of Reissner's mixed variational theorem (RMVT) to a CUF finite element model. The principle of virtual displacements (PVD) and RMVT are both used as variational statements for the study of monolayer and multilayer VAT plate dynamic behavior. The proposed approach is compared to Abaqus three-dimensional (3D) reference solutions, classical theories and literature results to investigate the effectiveness of the developed models. The results demonstrate that mixed theories provide the best approximation of the reference solution in all cases.

**Keywords:** free vibration analysis; finite element method; variable angle tow plates; Carrera's unified formulation; Reissner's mixed variational theorem

## **1. Introduction**

Over the last decades, composite structures have gained significant attention across diverse application fields, including aerospace, automotive and construction, due to their unique properties. Due to their high stiffness-to-weight ratio, composites help to build light structures with interesting mechanical properties. Despite this, a common thought is that the potential of fiber-reinforced structures could be better exploited by improving the directional properties through the variation of the fiber angle along the in-plane directions. The choice to keep the fiber orientation constant in each layer is particularly restrictive for geometries that present geometrical discontinuities such as cut-outs. VAT plates are characterized by an in-plane variation of fiber angle, helping to expand the design space of a specific structure. This is particularly useful for optimization problems, where a wider design space can positively affect the search of an optimal solution. For example, in the context of vibrational analyses, the maximization of fundamental frequencies can be improved by using curvilinear fibers. VATs were originally obtained through automated tape placement (ATP) and automated fiber placement (AFP). ATP helps the automated placement of composite material tapes with a specific angle in order to reproduce a desired path. AFP is similar to ATP, since the main difference is related to the width of the material that is laid down: while ATP handles a tape with a width between 75 and

**Citation:** Giunta, G.; Iannotta, D.A.; Montemurro, M. A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modeling. *Materials* **2023**, *16*, 4643. https://doi.org/ 10.3390/ma16134643

Academic Editors: Angelo Marcello Tarantino, Carmelo Majorana, Raimondo Luciano and Michele Bacciocchi

Received: 17 May 2023 Revised: 14 June 2023 Accepted: 17 June 2023 Published: 27 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

300 mm, AFP involves the placement of the material with a typical width between 3.1 and 12.7 mm. By consequence, AFP allows for better control of fiber angles, achieving a wider design flexibility; see Dirk et al. [1]. However, automated processes show some limitations related to manufacturing defects, such as gaps and overlaps, or constraints such as the minimum steering radius. These aspects can be partially overcome due to new technologies such as additive manufacturing (AM), also known as 3D printing. AM involves the layer-by-layer deposition of materials to create a three-dimensional object. In the case of variable angle tow composites, AM techniques are used to deposit and cure layers of composite materials with varying fiber orientations; see Zhuo et al. [2]. VAT composites have diverse applications ranging from aerospace engineering and wind energy to automotive and construction contexts, offering enhanced structural performance, weight reduction, and tailored properties for improved efficiency and functionality in a wide range of industries. For example, these materials can be employed in the optimization of aircraft wings to enhance structural weight and fuel consumption, as presented in Brooks et al. [3]. In the space context, VATs can be used for the design of liquid oxygen in order to reduce the mass and increase the payload of space launchers, as discussed by Gren et al. [4]. Despite the significant advantages associated with curvilinear fiber composites, these materials have limitations from both manufacturing and design perspectives. The production of VAT composites can be more complex in comparison with traditional laminates, since the material behavior is strongly affected by process-induced defects. Moreover, not all fiber patterns can be realized, because of the technological limitations that characterize their production. The complexity of analysis is one of the main disadvantages of VATs, because a greater number of unknowns must be taken into account and unfeasible fiber patterns could be obtained during the optimization process.

Several methods for the study of VAT mechanical responses are available in the literature. In the following text, a brief review of these approaches is presented, with a particular focus on free vibration analyses. To the best of the authors' knowledge, the first works that have been presented on the topic are based on the assumption of a constant fiber angle within each element in a finite element method (FEM) solution. Therefore, the continuous variation of fiber direction was approximated in a step-wise discrete way. This approach can be used in commercial FEM software tools that, at the moment, cannot handle continuous fiber variation. Hyer and Charette [5] and Hyer and Lee [6] used this method to improve the VAT tensile strength and buckling response, respectively. One of the main disadvantages of this step-wise approach is that, as the true variation is continuous, the discrete representation of fiber angle variation imposes a further approximation. A p-version FEM based on the third-order shear deformation theory (TSDT) was applied by Akhavan and Ribeiro [7] to preform vibrational analyses. The results showed that fiber variation helps to increase (or decrease) natural frequencies and that thin plates are more affected by this phenomenon if compared with thick ones. Ribeiro and Akhavan [8] used the p-version FEM approach with elements based on the first-order shear deformation theory (FSDT) to perform non-linear vibration analyses. The advantage of the p-version of the FEM is that the accuracy of the approximation is improved by increasing the order of shape functions over the elements. Vibration analyses were performed on VAT plates with a central circular cut-out considering parabolic fibers by Hachemi et al. [9]. Zhao and Kapania [10] investigated the free vibration of prestressed VAT stiffened plates, where plates and stiffeners were modeled separately through Mindlin plate theory and Timoshenko beam theory, respectively. The compatibility conditions at the interface between the plate and stiffeners were satisfied by using a transformation matrix. Honda and Narita [11] used the classical plate theory within the Ritz method in order to evaluate the natural frequencies and vibrational modes. An experimental approach was used in Rodrigues et al. [12] for the free vibration analysis of a plate with free boundary conditions that was subjected to random excitation via an electromagnetic shaker. Subsequently, the results were compared to the ones obtained through FEM, where a four-node isoparametric element based on the Reissner–Mindlin theory was used. Stodieck et al. [13] showed that curvilinear fibers

can be useful for improving the aeroelastic response of composite wings. The Rayleigh– Ritz method and classical lamination theory were used to develop a 1D beam model, considering the assumption of null chamber deformation of the wing chord-wise section. The aeroelastic response was computed by introducing quasi-static aerodynamic forces in a model developed for the plate structural analysis. A parametric study showed that by using VATs, it is possible to influence wing response both positively and negatively.

Curvilinear fibers can improve the modal response, as shown in several works. Abdalla et al. [14] used the classical lamination theory in combination with a successive approximation method in order to solve an optimization problem. The results showed that curvilinear fibers increased the optimal fundamental frequency in comparison with straight ones. A similar approach was presented in Blom et al. [15], where the maximization of the first natural frequency considering manufacturing constraints was obtained for VAT conical shells. In Carvalho et al. [16], a genetic algorithm and shell elements based on FSDT were used for maximization of the fundamental frequency. The multi-scale two-level (MS2L) approach helps to split the optimization problem in two parts. The composite is modeled as an equivalent homogeneous anisotropic plate in the first step, which aims to find the ideal distribution of the polar parameters that represent the mechanical design variables. The main goal of a second step is to establish the best stacking sequence in relation to the mechanical property distribution that has been obtained in the first step. The MS2L method was applied by Montemurro and Catapano [17] to VAT plates in order to optimize the buckling response. In order to evaluate the polar parameters, B-spline surfaces were introduced, while manufacturing constraints were considered during the second step. More details about the MS2L approach can be found in Catapano et al. [18], Montemurro and Catapano [19] and Fiordilino et al. [20], where both stiffness and buckling optimization problems were solved.

VAT structures have also been studied by using Carrera's unified formulation. CUF is a mathematical framework that helps the derivation of different theories, such as classical lamination plate theory, higher-order shear deformation theories, or LW approaches, within a unique formulation; see Carrera [21,22]. The a priori approximation over the thickness (typical of plates' structural modeling) can also be freely assumed as a generic combination of functions whose number is a free parameter of the formulation. When polynomial functions are used, as in this article, the expansion order along the thickness of the plate is arbitrary in the formulation, and it can be set when performing a specific analysis. This flexibility is beneficial because it helps to tailor the accuracy and computational efficiency of the analysis to the specific requirements of the problem at hand. Carrera et al. [23] used CUF in order to develop a Navier closed-form solution for the static analysis of isotropic plates under several loading conditions. The same approach was used in Carrera and Giunta [24] in order to perform failure analyses on isotropic plates. A further extension of this method was shown in Giunta et al. [25], where a indentation failure analysis of composite sandwich plates was performed. Giunta et al. [26] performed free vibration analyses of composite beams. In Viglietti et al. [27] and Fallahi et al. [28], free vibration and buckling analyses of VATs were performed through the use of a 1D CUF model. Within this framework, shell models were developed as well for VAT cases in order to perform stress analyses; see Sánchez-Majano et al. [29]. In Pagani and Sánchez-Majano [30,31] and Sánchez-Majano et al. [32], manufacturing defects were taken into account by using stochastic techniques. Vescovini and Dozio [33] used the Ritz method within CUF for vibrational and buckling analyses. A generalization of CUF was developed in order to allow for the use of different expansions for every component of the displacement vector. Demasi et al. [34] applied this approach to the study of VAT plates with an ESL model. A further advantage of CUF is that it can be used in combination with different variational statements. An alternative to the classic PVD is represented by the RMVT, where both displacements and transverse out-of-plane stresses are considered as primary variables. RMVT has been widely used within CUF for the study of straight-fiber composite structures. For example, Carrera and Demasi [35,36] developed RMVT-based CUF models to perform the static analysis of straight fiber plates.

The free vibration analysis is an important problem in engineering (see Babaei et al. [37]), and within this context, CUF has been applied to the study of VATs considering PVD as the main variational statement. For this reason, this work aims to extend this framework with the RMVT formulation in order to develop a family of hierarchical plate finite elements. This will help to better predict the natural frequencies of composite plates characterized by curvilinear fibers. Section 2 shows the theoretical derivation for free vibration problems. Section 3 presents the numerical results where three cases are investigated. Analyses are performed that consider a varying side-to-thickness ratio in order to investigate thin and thick plates, and the differences between models are discussed regarding PVD or RMVT statements. The results are compared to reference solutions for validation. Concluding observations and remarks are presented in Section 4.

#### **2. Carrera's Unified Formulation**

A plate is a flat body whose material points lie in the Cartesian closed-point subset

$$\mathcal{P} = \Omega \times \mathcal{H} \tag{1}$$

of the three-dimensional space R<sup>3</sup> where:

$$\begin{aligned} \Omega &= \left\{ (x, y) \, : \, \frac{x}{a'} \, \frac{y}{b} \in [0, 1] \right\} \subset \mathbb{R}^2, \\ \mathcal{H} &= \left\{ z \, : \, \frac{2z}{h} \in [-1, 1] \right\} \, \end{aligned} \tag{2}$$

where *a* and *b* are the dimensions along the two in-plane axes, and *h* measures its thickness along the *z*-axis, where *z a* and *b*. The global reference system and plate geometry are presented in Figure 1.

**Figure 1.** Plate geometry and reference system.

The displacement field is expressed as:

$$\mathbf{u} = \left\{ \begin{array}{c} u\_x \\ u\_y \\ u\_z \end{array} \right\}. \tag{3}$$

The strain tensor components can be written in vector form. Two vectors are obtained, representing the in-plane and out-of-plane components:

$$\mathfrak{e}\_p = \left\{ \begin{array}{c} \mathfrak{e}\_{xx} \\ \mathfrak{e}\_{yy} \\ \mathfrak{e}\_{xy} \end{array} \right\} , \ \mathfrak{e}\_n = \left\{ \begin{array}{c} \gamma\_{xz} \\ \gamma\_{yz} \\ \gamma\_{zz} \end{array} \right\} . \tag{4}$$

The hypothesis of small displacements helps to use a linear strain–displacement relation:

$$\begin{array}{l} \boldsymbol{\varepsilon}\_{p} = \mathbf{D}\_{p} \mathbf{u} \\ \boldsymbol{\varepsilon}\_{n} = (\mathbf{D}\_{n\Omega} + \mathbf{D}\_{\boldsymbol{\varepsilon}\boldsymbol{z}}) \mathbf{u} \end{array} \tag{5}$$

where **D***p*, **D***n*<sup>Ω</sup> and **D***nz* are the following differential operators:

$$\mathbf{D}\_{p} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & 0\\ 0 & \frac{\partial}{\partial y} & 0\\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{bmatrix}, \mathbf{D}\_{n\Omega} = \begin{bmatrix} 0 & 0 & \frac{\partial}{\partial x} \\\ 0 & 0 & \frac{\partial}{\partial y} \\\ 0 & 0 & \frac{\partial}{\partial y} \end{bmatrix}, \mathbf{D}\_{nz} = \begin{bmatrix} \frac{\partial}{\partial z} & 0 & 0 \\ 0 & \frac{\partial}{\partial z} & 0 \\\ 0 & 0 & \frac{\partial}{\partial z} \end{bmatrix}. \tag{6}$$

The stress vector is expressed in a similar manner:

$$
\sigma\_p = \left\{ \begin{array}{c} \sigma\_{\rm xx} \\ \sigma\_{\rm yy} \\ \sigma\_{\rm xy} \end{array} \right\}, \; \sigma\_n = \left\{ \begin{array}{c} \sigma\_{\rm xz} \\ \sigma\_{\rm yz} \\ \sigma\_{\rm zz} \end{array} \right\}. \tag{7}
$$

Hooke's law reads:

$$\begin{array}{l} \sigma\_p = \mathbf{C}\_{pp}\mathbf{e}\_p + \mathbf{C}\_{pn}\mathbf{e}\_n \\ \sigma\_n = \mathbf{\tilde{C}}\_{np}\mathbf{e}\_p + \mathbf{\tilde{C}}\_{nn}\mathbf{e}\_n \end{array} \tag{8}$$

where the terms **C**e *pp*, **C**e *pn*, **C**e*np* and **C**e*nn* are subcomponents of a material stiffness matrix **C**e according to the stress and strain ordering in Equations (4) and (7), where the fibers lay in Ω and where they are not, in general, aligned with the *x*-axis. **C** stands for the stiffness matrix in the global reference system, and its components can be written in terms of the Young's moduli *E<sup>L</sup>* and *ET*, shear moduli *GLT* and *GTT* and Poisson's ratios *νLT* and *νTT*, where subscripts *L* and *T* stand for the directions parallel and perpendicular to the fibers, respectively. For further details, see Reddy [38].

#### *2.1. Variable Stiffness Composite Plates*

Laminated VAT structures are considered in this work. For this reason, the material stiffness coefficients can change layer-wise along the thickness and pointwise along the in-plane directions. The mapping of **C** into **C**e reads:

$$
\tilde{\mathbf{C}} = \mathbf{T} \mathbf{C} \mathbf{T}^T.\tag{9}
$$

Superscript *T* stands for the transpose operator. The matrix **T** represents a rotation matrix that depends on an in-plane rotation angle *θ*. For the sake of brevity, the components of **C**e and **T** are not reported here; they can be found in Reddy [38]. In a laminated VAT, the rotation angle *θ* is a bi-dimensional field in Ω. In this work, two different variation laws are considered for *θ*, a linear variation law and a parabolic one. The linear law can be expressed according to the following formula:

$$
\theta(\mathfrak{a}) = \Phi + T\_0 + \frac{T\_1 - T\_0}{d} |\mathfrak{a}|\,. \tag{10}
$$

The angle Φ describes the original direction along which *θ* varies, and *α* is a generic spatial variable defined as:

$$
\mathfrak{a} = \mathfrak{x}' \cos(\Phi) + \mathfrak{y}' \sin(\Phi) \,. \tag{11}
$$

*x* 0 and *y* 0 denote a generic in-plane reference system used for describing a fiber path, where *θ* is measured. The introduction of a new reference system is useful in order to represent the local fiber orientation independently from the global reference system identified by axes *x* and *y*. *T*<sup>0</sup> and *T*<sup>1</sup> are the angles between the *α*-axis and the tangent to a fiber for *α* equal to zero and *d*, respectively; see Figure 2.

**Figure 2.** Example of in-plane fiber orientation.

As shown in the Figure 2, the fiber angle is always measured with respect to the *x* 0 -axis, and it can change along a generic direction *α*, defined as a combination of *x* 0 and *y* 0 depending on the angle Φ. Further details about the fiber linear variation law can be found in Gürdal et al. [39]. The parabolic law can be expressed according to the following equation:

$$\theta(\mathfrak{a}) = \Phi + T\_0 + \tan^{-1}\left(\gamma \frac{\mathfrak{a}}{d}\right). \tag{12}$$

The parameter *γ* is used to control the shape of the parabola, and it is related to the final fiber angle *T*<sup>1</sup> as *T*<sup>1</sup> = tan−<sup>1</sup> (±*γ*). More details about the parabolic fiber path can be found in Hachemi et al. [9] and Honda et al. [40]. The following notation, based upon the above introduced parameters, is used in order to describe the in-plane linear and parabolic fiber behavior: Φ < *T*0, *T*<sup>1</sup> >.

#### *2.2. Variational Statements*

PVD and RMVT variational statements are considered to derive the governing equations for the free vibration problem for a laminated VAT plate. The fundamental distinction is that the RMVT considers the vector of the out-of-plane stresses *σ<sup>n</sup>* as a primary unknown, whereas the PVD considers only displacements as primary variables. For the PVD case, the following variational statement applies:

$$\int\_{\Omega} \int\_{\mathcal{H}} \left( \delta \boldsymbol{\varepsilon}\_{p\mathcal{G}}^T \boldsymbol{\sigma}\_{p\mathcal{H}} + \delta \boldsymbol{\varepsilon}\_{n\mathcal{G}}^T \boldsymbol{\sigma}\_{n\mathcal{H}} \right) dz \, d\Omega + \delta \mathcal{L}\_{\text{in}} = 0 \,\,\,\tag{13}$$

where the subscript *G* refers to the components obtained from the geometrical relations in Equation (5), and subscript *H* refers to the components obtained from Hooke's law in Equation (8). L*in* is the virtual work of the inertial forces, and *δ* stands for a virtual variation. For the RMVT case, the variational statement is:

$$\int\_{\Omega} \int\_{\mathcal{H}} \left[ \delta \boldsymbol{\varepsilon}\_{p\mathbf{G}}^{T} \boldsymbol{\sigma}\_{p\mathbf{H}} + \delta \boldsymbol{\varepsilon}\_{n\mathbf{G}}^{T} \boldsymbol{\sigma}\_{n\mathbf{M}} + \delta \boldsymbol{\sigma}\_{n\mathbf{M}}^{T} (\boldsymbol{\varepsilon}\_{n\mathbf{G}} - \boldsymbol{\varepsilon}\_{n\mathbf{H}}) \right] dz \, d\Omega + \delta \mathcal{L}\_{\text{in}} = 0 \,. \tag{14}$$

The *M* subscript refers to the transverse stress components considered as primary unknowns in the mixed formulation. For the RMVT formulation, Hooke's law is rewritten as follows:

$$\begin{array}{l} \sigma\_{pH} = \hat{\mathbf{C}}\_{pp} \boldsymbol{\varepsilon}\_{pG} + \hat{\mathbf{C}}\_{pn} \boldsymbol{\sigma}\_{nM} \\ \boldsymbol{\varepsilon}\_{nH} = \hat{\mathbf{C}}\_{np} \boldsymbol{\varepsilon}\_{pG} + \hat{\mathbf{C}}\_{nn} \boldsymbol{\sigma}\_{nM} \end{array} \tag{15}$$

where **Cˆ** *pp*, **Cˆ** *pn*, **Cˆ** *np* and **Cˆ** *nn* are (see Carrera and Demasi [35]):

$$\begin{array}{lcl}\mathbf{\tilde{C}}\_{pp} = \mathbf{\tilde{C}}\_{pp} - \mathbf{\tilde{C}}\_{pn}\mathbf{\tilde{C}}\_{nn}^{-1}\mathbf{\tilde{C}}\_{np} \\ \mathbf{\tilde{C}}\_{pn} = \mathbf{\tilde{C}}\_{pn}\mathbf{\tilde{C}}\_{nn}^{-1} \\ \mathbf{\tilde{C}}\_{np} = -\mathbf{\tilde{C}}\_{nn}^{-1}\mathbf{\tilde{C}}\_{np} \\ \mathbf{\tilde{C}}\_{nn} = \mathbf{\tilde{C}}\_{nn}^{-1} \end{array} \tag{16}$$

The superscript "−1" indicates the inverse of a matrix. The inertial work can be expressed as:

$$
\delta \mathcal{L}\_{\text{in}} = \int\_{\Omega} \int\_{\mathcal{H}} \delta \mathbf{u}^T \rho \,\ddot{\mathbf{u}} \, d\Omega \, dz \,\,\tag{17}
$$

where *ρ* is the plate material density, and **u**¨ represents the acceleration vector.

#### *2.3. Kinematic Assumptions*

CUF uses an axiomatic approach along the through-the-thickness direction to represent the primary unknowns; see Carrera [22]. The generic unknown component *f* = *f*(*x*, *y*, *z*) is approximated as:

$$f(\mathbf{x}, y, z) = F\_{\tau}(z) g\_{\tau}(\mathbf{x}, y) \; , \; \tau = 0, \; 1, \; \dots \; \; N \; , \tag{18}$$

where *f* is a displacement component in a formulation derived from the PVD, but it can also be an out-of-plane stress component when a RMVT formulation is considered. *F<sup>τ</sup>* is an approximation function in H, and *g<sup>τ</sup>* is an unknown two-dimensional function in Ω. According to Einstein's notation, a twice-repeated index implies a sum over the index range. Finally, *N* is the approximation order. Both *N* and *F<sup>τ</sup>* are a priori defined. This feature of CUF helps to obtain multiple theories in the same formulation. Within CUF, ESL or LW models can also be obtained depending on the support of *Fτ*. In an ESL model

$$F\_{\mathsf{T}} \colon \mathcal{H} \mapsto \mathbb{R} \text{, whereas for a LW model } F\_{\mathsf{T}} \colon \mathcal{H}^{k} \mapsto \mathbb{R} \text{ where } \mathcal{H}^{k} = \left\{ z^{k} \, : \, \frac{\mathfrak{Z}^{k}}{h^{k}} \in [-1, 1] \right\}$$

such that H = *N* S*l k*=1 <sup>H</sup>*<sup>k</sup>* and <sup>H</sup>*<sup>k</sup>* ∩ H*<sup>k</sup>* 0 = ∅ for *k* 6= *k* 0 with *k*, *k* <sup>0</sup> = 1, 2, . . . , *N<sup>l</sup>* , where *k*

*Nl* is the total number of laminae, and *h* is the thickness of a generic *k* lamina such that *k* = *Nl* ∑ *k*=1 *h k* . The number of unknowns in the ESL case is independent of the number of layers

in the lamination since the approximation is imposed globally over H. The total stiffness contributions can be seen as a weighted average of each layer stiffness along the thickness. Maclaurin's series approximation is considered for the ESL models as a linear combination of the power functions:

$$F\_{\tau}(z) = z^{\tau}, \quad \tau = 0, 1, \dots, N \text{ }, \tag{19}$$

where *N* is the expansion order. The computational cost of ESL models depends on *N* only, and for a given *N*, it is lower than a LW model since this latter model depends on the total number of layers in the lamination. ESL models are suitable for relatively thick laminates. However, they are unable to accurately predict the behavior of thick plates with a high degree of anisotropy. ESL models have *C* <sup>∞</sup>-continuity over <sup>H</sup> because of the used approximation functions, whereas laminated composites present a *C* 0 -continuity since the interface between the two consecutive layers of the different materials introduces a change in the slope of the displacements (also known as zig-zag displacement through-thethickness variation). This behavior can be accommodated within an ESL theory by means of Murakami's function. This approach is not considered here; for more details, refer to Carrera [41]. In an LW model, the kinematics of each layer are formulated independently:

$$f^k(\mathbf{x}, y, z) = \mathcal{F}\_b(z)\mathcal{g}\_b^k(\mathbf{x}, y) + \mathcal{F}\_l(z)\mathcal{g}\_r^k(\mathbf{x}, y) + \mathcal{F}\_l(z)\mathcal{g}\_l^k(\mathbf{x}, y) \ , \ \ r = 2, \dots, N \ \text{,} \tag{20}$$

where subscripts *b* and *t* stand for the bottom and top layers, respectively. Congruence at the interface is retrieved via a through-the-thickness assembly procedure similar to that used in the finite element method. For this reason, Lagrange polynomials (which ensure partition of unity), or the following linear combination of Legendre polynomials, which are represented as:

$$F\_l\left(z\left(\tilde{\boldsymbol{\xi}}^k\right)\right) = \frac{P\_0 + P\_1}{2}, \quad F\_b\left(z\left(\tilde{\boldsymbol{\xi}}^k\right)\right) = \frac{P\_0 - P\_1}{2}, \quad F\_r\left(z\left(\tilde{\boldsymbol{\xi}}^k\right)\right) = P\_r - P\_{r-2}, \quad r = 2, \dots, N \tag{21}$$

are typically used as approximation functions over <sup>H</sup>*<sup>k</sup>* . The use of Lagrange or Legendre polynomials along the thickness changes according to the used model, and this is specified at the end of the next subsection. In Equation (21), *ξ <sup>k</sup>* = 2*z k h k* ∈ [−1, 1] and *P<sup>i</sup>* = *P<sup>i</sup> ξ k* are an *i*-order Legendre polynomial. Equation (21) creates a base where *F<sup>t</sup>* and *F<sup>b</sup>* are the two linear Lagrange polynomials, and *F<sup>r</sup>* is a kind of *p*-version-enriching function since it does not contribute to a base linear combination for *ξ <sup>k</sup>* <sup>=</sup> <sup>±</sup>1, being, by definition, *<sup>F</sup>r*(±1) <sup>=</sup> 0. Since LW base functions have local support, inter-layer *C* 0 -continuity for layers made of different materials is ensured, but the computational costs are higher than for ESL models.

#### *2.4. Acronym System*

An acronym system is used in order to identify all the derived theories. Figure 3 shows this system.

**Figure 3.** Acronym system.

The first letter addresses the approximation level that is applied: 'E' denotes the ESL models, whereas 'L' denotes the LW models. The second letter denotes the variational statement: PVD or RMVT are denoted by 'D' or 'M', respectively. The last number is the order of expansion along the plate thickness. A number at the beginning of the acronym, when present, indicates how many virtual layers have been used to approximate each physical layer in an LW model to improve the results for a given approximation order. If this number is not present, only one virtual layer has been used to represent each physical layer.

As an example, in EDN models, the displacement field can be expressed as:

$$\begin{aligned} \mu\_{\mathbf{x}} &= \mu\_{\mathbf{x}0} + \mu\_{\mathbf{x}1}z + \mu\_{\mathbf{x}2}z^2 + \dots + \mu\_{\mathbf{x}N}z^N \\ \mu\_{\mathbf{y}} &= \mu\_{y0} + \mu\_{y1}z + \mu\_{y2}z^2 + \dots + \mu\_{yN}z^N \\ \mu\_{\mathbf{z}} &= \mu\_{z0} + \mu\_{z1}z + \mu\_{z2}z^2 + \dots + \mu\_{zN}z^N \end{aligned} \tag{22}$$

In vector form:

$$\mathbf{u} = F\_0 \mathbf{u}\_0 + F\_1 \mathbf{u}\_1 + \dots + F\_N \mathbf{u}\_N = F\_\mathsf{\tau} \mathbf{u}\_\mathsf{\tau} \quad \text{\(\tau = 0, 1, \dots, N\)}.\tag{23}$$

where *F<sup>τ</sup>* = *z <sup>τ</sup>* and **u***<sup>τ</sup>* = **u***τ*(*x*, *y*). Additionally, classical theories can be taken into account. Classical lamination theory (CLT) and first-order shear deformation theory are obtained as a particular case of a first-order ESL theory. FSDT is obtained through the penalization of the *uz*<sup>1</sup> term, while for CLT, transverse shear stresses are disregarded by using a fictitiously

high value of the material shear moduli. The material stiffness matrix needs to be reduced in a plane-stress sense to overcome thickness locking.

For LDN solutions, only displacements are considered as the primary variables:

$$\mathbf{u}^{k} = F\_0 \mathbf{u}\_0^k + F\_1 \mathbf{u}\_1^k + \dots + F\_N \mathbf{u}\_N^k = F\_\tau \mathbf{u}\_\tau^k, \qquad \tau = 0, \ 1, \dots, N, \ k = 1, 2, \dots, N\_l \ . \tag{24}$$

For LMN solutions, transverse stresses are treated as primary variables. The transverse stress field can be expressed as:

$$
\sigma\_n^k = F\_0 \sigma\_0^k + F\_1 \sigma\_1^k + \dots + F\_N \sigma\_N^k = F\_\mathsf{T} \sigma\_\mathsf{T}^k, \qquad \tau = 0, \ 1, \dots, N, \ k = 1, \ 2, \dots, N\_\mathsf{I}. \tag{25}
$$

It can be observed that ESL theories can be considered as a particular case for LW theories. While in the first case the integration along the thickness is performed in order to represent composite properties through a unitary layer, for the second case, the integration is computed layer by layer. This helps to represent the kinematics of each layer separately for LW models. LDN solutions are obtained with Lagrange polynomials with equally spaced nodes, whereas LMN ones are obtained with Legendre polynomials.

### *2.5. FE Stiffness Matrices*

As far as a FEM solution is concerned, the in-plane domain is discretized into *N<sup>e</sup>* subdomains such as Ω = *N* S*e e*=1 <sup>Ω</sup>*<sup>e</sup>* and <sup>Ω</sup>*<sup>e</sup>* <sup>∩</sup> <sup>Ω</sup>*<sup>e</sup>* 0 = ∅ for *e* 6= *e* 0 . Shape functions are then introduced for the approximation of the variation over Ω*<sup>e</sup>* . In the case of a bi-dimensional model, Equation (18) becomes:

$$f(\mathbf{x}, y, z) = F\_{\mathbf{7}}(z) N\_{\mathbf{i}}(\mathbf{x}, y) g\_{\mathbf{7}\mathbf{i}} \; \; \; \tau = 0, \; 1, \; \dots, \; N \; \; \; \; i = 1, \; \dots, N\_{\mathbf{n}} \; \; \prime \tag{26}$$

where *N<sup>i</sup>* stands for the shape functions, and *N<sup>n</sup>* is the number of nodes in the used finite element. Classical Lagrange shape functions are used. They are not presented here for the sake of brevity, but they can be found in Bathe [42]. FE stiffness matrices are obtained by the weak form of the variational principles. In the PVD case, considering Equation (26), the displacement field can be written as:

$$\mathbf{u} = F\_{\mathsf{T}} N\_{\mathrm{i}} \left\{ \begin{array}{c} q\_{x\mathsf{T}i} \\ q\_{y\mathsf{T}i} \\ q\_{z\mathsf{T}i} \end{array} \right\} = F\_{\mathsf{T}} N\_{\mathsf{i}} \mathbf{q}\_{\mathsf{T}i} \,. \tag{27}$$

Through the substitution of Equations (5), (8) and (27) into Equation (13), the weak PVD form can be obtained:

R Ω*e δq T τi* - **D***<sup>T</sup> p* (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ<sup>s</sup> pp***D***<sup>p</sup> Nj* **I** + **D***<sup>T</sup> p* (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ<sup>s</sup> pn***D***n*<sup>Ω</sup> *Nj* **I** + **D***<sup>T</sup> p* (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τs*,*<sup>z</sup> pn Nj* **I** +**D***<sup>T</sup> n*Ω (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ<sup>s</sup> np***D***<sup>p</sup> Nj* **I** + **D***<sup>T</sup> n*Ω (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ<sup>s</sup> nn***D***n*<sup>Ω</sup> *Nj* **I** + **D***<sup>T</sup> n*Ω (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τs*,*<sup>z</sup> nn Nj* **I** +(*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ*,*<sup>z</sup> <sup>s</sup> np* **D***<sup>p</sup> Nj* **I** + (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ*,*<sup>z</sup> <sup>s</sup> nn* **D***n*<sup>Ω</sup> *Nj* **I** + (*N<sup>i</sup>* **<sup>I</sup>**)**Z**e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn Nj* **I** *qsjd*Ω = − R Ω*e δq T τi* (*N<sup>i</sup>* **I**)*ρEτ<sup>s</sup> Nj* **I** *q***¨** *sjd*Ω , (28)

where:

$$\left(\widetilde{\bf Z}^{\rm res}\_{\rm wr}, \widetilde{\bf Z}^{\rm res}\_{\rm wr}, \widetilde{\bf Z}^{\rm res}\_{\rm var}, \widetilde{\bf Z}^{\rm res}\_{\rm var}, \widetilde{\bf Z}^{\rm res}\_{\rm var}\right) = \left(\widetilde{\bf C}\_{\rm av} E\_{\rm res}, \widetilde{\bf C}\_{\rm av} E\_{\rm r, \rm s}, \widetilde{\bf C}\_{\rm avr} E\_{\rm r, \rm s}, \widetilde{\bf C}\_{\rm avr} E\_{\rm r, \rm s}\right); \quad w, r = p, n, \tag{29}$$

$$
\left(\mathbb{E}\_{\mathsf{TS}}, \mathbb{E}\_{\mathsf{T}\_{\mathsf{T},\mathsf{S}}\prime}, \mathbb{E}\_{\mathsf{TS}\_{\mathsf{T},\mathsf{T}}\prime}, \mathbb{E}\_{\mathsf{T},\mathsf{S},\mathsf{z}}\right) = \int\_{\mathcal{H}} \left(\mathbb{F}\_{\mathsf{T}} \mathbb{F}\_{\mathsf{S}\prime} \mathbb{F}\_{\mathsf{T}\_{\mathsf{T},\mathsf{z}}} \mathbb{F}\_{\mathsf{T}} \mathbb{F}\_{\mathsf{T}} \mathbb{F}\_{\mathsf{S},\mathsf{z}\prime} \mathbb{F}\_{\mathsf{T},\mathsf{z}} \mathbb{F}\_{\mathsf{S},\mathsf{z}}\right) d\mathsf{z}\,\,.\tag{30}
$$

An axis coordinate as comma-preceded subscript stands for a derivative in that coordinate direction. In compact vector form, Equation (28) reads:

$$
\delta \boldsymbol{q}\_{\tau \mathbf{i}}^T \mathbf{K}^{\tau \text{sij}} \boldsymbol{q}\_{\mathbf{sj}} = -\delta \boldsymbol{q}\_{\tau \mathbf{i}}^T \mathbf{M}^{\tau \text{sij}} \boldsymbol{\vec{q}}\_{\mathbf{sj}} \,\tag{31}$$

where **<sup>K</sup>***τsij* and **<sup>M</sup>***τsij* <sup>∈</sup> <sup>R</sup>3×<sup>3</sup> are fundamental nuclei (FN) of the stiffness and mass matrices, respectively. Through the cycles on the indices *τ*, *s*, *i* and *j*, it is possible to build the stiffness and mass matrices of a finite element. The components of the stiffness FN for the PVD case can be written as:

*K τsij xx* = Z Ω*<sup>e</sup> Z*e*τs pp*11*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*16*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*16*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*66*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn*<sup>44</sup> *NjN<sup>i</sup> d*Ω , *K τsij xy* = Z Ω*<sup>e</sup> Z*e*τs pp*12*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*16*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*26*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*66*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn*<sup>45</sup> *NjN<sup>i</sup> d*Ω , *K τsij xz* = Z Ω*<sup>e</sup> <sup>Z</sup>*e*τs*,*<sup>z</sup> pn*13*NjNi*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τs*,*<sup>z</sup> pn*36*NjNi*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> nn*44*Nj*,*<sup>x</sup> <sup>N</sup><sup>i</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> nn*45*Nj*,*<sup>y</sup> N<sup>i</sup> d*Ω , *K τsij yx* = Z Ω*<sup>e</sup> Z*e*τs pp*12*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*26*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*16*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*66*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn*<sup>45</sup> *NjN<sup>i</sup> d*Ω , *K τsij yy* = Z Ω*<sup>e</sup> Z*e*τs pp*22*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*26*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*26*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> pp*66*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn*<sup>55</sup> *NjN<sup>i</sup> d*Ω , *K τsij yz* = Z Ω*<sup>e</sup> <sup>Z</sup>*e*τs*,*<sup>z</sup> pn*23*NjNi*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τs*,*<sup>z</sup> pn*36*NjNi*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> nn*45*Nj*,*<sup>x</sup> <sup>N</sup><sup>i</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> nn*55*Nj*,*<sup>y</sup> N<sup>i</sup> d*Ω , *K τsij zx* = Z Ω*<sup>e</sup> <sup>Z</sup>*e*τs*,*<sup>z</sup> nn*44*NjNi*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τs*,*<sup>z</sup> nn*45*NjNi*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> np*13*Nj*,*<sup>x</sup> <sup>N</sup><sup>i</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> np*36*Nj*,*<sup>y</sup> N<sup>i</sup> d*Ω , *K τsij zy* = Z Ω*<sup>e</sup> <sup>Z</sup>*e*τs*,*<sup>z</sup> nn*45*NjNi*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τs*,*<sup>z</sup> nn*55*NjNi*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> np*23*Nj*,*<sup>y</sup> <sup>N</sup><sup>i</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup> np*36*Nj*,*<sup>x</sup> N<sup>i</sup> d*Ω , *K τsij zz* = Z Ω*<sup>e</sup> Z*e*τs nn*44*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> nn*45*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> nn*45*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ<sup>s</sup> nn*55*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*e*τ*,*<sup>z</sup> <sup>s</sup>*,*<sup>z</sup> nn*<sup>33</sup> *NjN<sup>i</sup> d*Ω . (32)

The mass FN can be written as:

$$\mathbf{M}^{\mathsf{T}\mathrm{sij}} = \int\_{\Omega\_{\mathsf{f}}} (N\_{\mathsf{i}}\mathbf{I}) \rho E\_{\mathsf{T}\mathsf{s}} \left( N\_{\mathsf{j}}\mathbf{I} \right) d\Omega \,. \tag{33}$$

It is possible to observe that **M***τsij* is a diagonal matrix and that since the plate density is assumed to be constant, the term *ρEτ<sup>s</sup>* can be placed outside the integral.

In the RMVT case, transverse stresses are a priori approximated:

$$\sigma\_{\rm II} = F\_{\rm \tau} N\_{\rm i} \left\{ \begin{array}{c} \mathcal{g}\_{\rm xz \tau i} \\ \mathcal{g}\_{\rm yz \tau i} \\ \mathcal{g}\_{\rm z \tau i} \end{array} \right\} = F\_{\rm \tau} N\_{\rm i} \mathbf{g}\_{\rm \tau i} \,. \tag{34}$$

Through the substitution of Equations (5), (15), (27) and (34) into Equation (14), the RMVT weak form can be obtained:

Z Ω*<sup>e</sup> δq T τi* - **D***<sup>T</sup> p* (*N<sup>i</sup>* **I**)**Zˆ** *<sup>τ</sup><sup>s</sup> pp***D***<sup>p</sup> Nj* **I** *qsj* + *δq T τi* - **D***<sup>T</sup> p* (*N<sup>i</sup>* **I**)**Zˆ** *<sup>τ</sup><sup>s</sup> pn Nj* **I** + **D***<sup>T</sup> <sup>n</sup>*Ω(*N<sup>i</sup>* **I**)(*Eτs***I**) *Nj* **I** +(*N<sup>i</sup>* **I**)(*Eτ*,*<sup>z</sup> <sup>s</sup>***I**) *Nj* **I** *gsj* + *δg T τi* - (*N<sup>i</sup>* **I**)(*Eτs***I**)**D***n*<sup>Ω</sup> *Nj* **I** + (*N<sup>i</sup>* **I**)(*Eτs*,*<sup>z</sup>* **I**) *Nj* **I** −(*N<sup>i</sup>* **I**)**Zˆ** *<sup>τ</sup><sup>s</sup> np***D***<sup>p</sup> Nj* **I** *qsj* − *δg T τi* (*N<sup>i</sup>* **I**)**Zˆ** *<sup>τ</sup><sup>s</sup> nn Nj* **I** *gsjd*Ω = − Z Ω*<sup>e</sup> δq T τi* (*N<sup>i</sup>* **I**)*ρEτ<sup>s</sup> Nj* **I** *q***¨** *sjd*Ω , (35)

where:

$$\left(\hat{\mathbf{2}}\_{\text{uv}}^{\text{rs}}, \hat{\mathbf{2}}\_{\text{uv}}^{\text{r}\_{\text{c}}\text{s}}, \hat{\mathbf{2}}\_{\text{uv}}^{\text{rs}\_{\text{c}}}, \hat{\mathbf{2}}\_{\text{uv}}^{\text{r}\_{\text{c}}\text{s}}\right) = \left(\hat{\mathbf{C}}\_{\text{uv}} E\_{\text{rs}}, \hat{\mathbf{C}}\_{\text{uv}} E\_{\text{r}\_{\text{c}}\text{s}} E\_{\text{uv}} E\_{\text{rs}\_{\text{c}}\text{s}}, \hat{\mathbf{C}}\_{\text{uv}} E\_{\text{r}\_{\text{c}}\text{s}}\right); \quad w\_{\text{r}} r = p\_{\text{r}} n \; . \tag{36}$$

In a compact form:

$$\begin{split} \delta \boldsymbol{q}\_{\tau \boldsymbol{i}}^{\mathrm{T}} \mathbf{K}\_{\boldsymbol{u}\boldsymbol{u}}^{\mathrm{T}\mathrm{s}\boldsymbol{i}\boldsymbol{j}} \boldsymbol{q}\_{\mathrm{s}\boldsymbol{j}} + \delta \boldsymbol{q}\_{\tau \boldsymbol{i}}^{\mathrm{T}} \mathbf{K}\_{\boldsymbol{u}\boldsymbol{\sigma}}^{\mathrm{T}\mathrm{s}\boldsymbol{i}\boldsymbol{j}} \boldsymbol{q}\_{\mathrm{s}\boldsymbol{j}} &= -\delta \boldsymbol{q}\_{\tau \boldsymbol{i}}^{\mathrm{T}} \mathbf{M}^{\tau\mathrm{s}\boldsymbol{i}\boldsymbol{j}} \boldsymbol{q}\_{\mathrm{s}\boldsymbol{j}} \,. \\ \delta \mathbf{g}\_{\tau \boldsymbol{i}}^{\mathrm{T}} \mathbf{K}\_{\boldsymbol{\sigma}\boldsymbol{u}}^{\mathrm{T}\mathrm{s}\boldsymbol{i}} \boldsymbol{q}\_{\mathrm{s}\boldsymbol{j}} + \delta \mathbf{g}\_{\tau \boldsymbol{i}}^{\mathrm{T}} \mathbf{K}\_{\boldsymbol{\sigma}\boldsymbol{\sigma}}^{\mathrm{T}\mathrm{s}\boldsymbol{i}} \boldsymbol{q}\_{\mathrm{s}\boldsymbol{j}} &= \boldsymbol{0} \,. \end{split} \tag{37}$$

In this case, four fundamental nuclei are obtained. The components of the FN for the RMVT case can be written as:

*K τsij uuxx* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pp*11*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*31*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*13*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*33*Nj*,*<sup>y</sup> Ni*,*<sup>y</sup> d*Ω , *K τsij uuxy* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pp*12*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*32*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*13*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*33*Nj*,*<sup>x</sup> Ni*,*<sup>y</sup> d*Ω , *K τsij uuyx* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pp*21*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*31*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*23*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*33*Nj*,*<sup>y</sup> Ni*,*<sup>x</sup> d*Ω , *K τsij uuyy* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pp*22*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*32*Nj*,*<sup>y</sup> <sup>N</sup>i*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*23*Nj*,*<sup>x</sup> <sup>N</sup>i*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pp*33*Nj*,*<sup>x</sup> Ni*,*<sup>x</sup> d*Ω , *K τsij uuxz* = 0 , *K τsij uuyz* = 0 , *K τsij uuzx* = 0 , *K τsij uuzy* = 0 , *K τsij uuzz* = 0 , *K τsij <sup>u</sup>σxx* = Z Ω*<sup>e</sup> Eτ*,*<sup>z</sup> <sup>s</sup>NjN<sup>i</sup> d*Ω , *K τsij <sup>u</sup>σxz* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pn*13*NjNi*,*<sup>x</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pn*33*NjNi*,*<sup>y</sup> d*Ω , *K τsij <sup>u</sup>σyy* = Z Ω*<sup>e</sup> Eτ*,*<sup>z</sup> <sup>s</sup>NjN<sup>i</sup> d*Ω , *K τsij <sup>u</sup>σyz* = Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> pn*23*NjNi*,*<sup>y</sup>* + *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> pn*33*NjNi*,*<sup>x</sup> d*Ω , *K τsij <sup>u</sup>σzx* = Z Ω*<sup>e</sup> EτsNjNi*,*<sup>x</sup> d*Ω , *K τsij <sup>u</sup>σzy* = Z Ω*<sup>e</sup> EτsNjNi*,*<sup>y</sup> d*Ω , *K τsij <sup>u</sup>σzz* = Z Ω*<sup>e</sup> Eτ*,*<sup>z</sup> <sup>s</sup>NjN<sup>i</sup> d*Ω , *K τsij <sup>u</sup>σxy* = 0 , *K τsij <sup>u</sup>σyx* = 0 , *K τsij <sup>σ</sup>uxx* = Z Ω*<sup>e</sup> Eτs*,*<sup>z</sup> NjN<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uxz* = Z Ω*<sup>e</sup> EτsNj*,*<sup>x</sup> N<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uyy* = Z Ω*<sup>e</sup> Eτs*,*<sup>z</sup> NjN<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uyz* = Z Ω*<sup>e</sup> EτsNj*,*<sup>y</sup> N<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uzx* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> np*31*Nj*,*<sup>x</sup> <sup>N</sup><sup>i</sup>* <sup>−</sup> *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> np*33*Nj*,*<sup>y</sup> N<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uzy* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> np*32*Nj*,*<sup>y</sup> <sup>N</sup><sup>i</sup>* <sup>−</sup> *<sup>Z</sup>*<sup>ˆ</sup> *<sup>τ</sup><sup>s</sup> np*33*Nj*,*<sup>x</sup> N<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uzz* = Z Ω*<sup>e</sup> Eτs*,*<sup>z</sup> NjN<sup>i</sup> d*Ω , *K τsij <sup>σ</sup>uxy* = 0 , *K τsij <sup>σ</sup>uyx* = 0 , *K τsij σσxx* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> nn*11*NjN<sup>i</sup> d*Ω , *K τsij σσxy* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> nn*12*NjN<sup>i</sup> d*Ω , *K τsij σσyx* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> nn*21*NjN<sup>i</sup> d*Ω , *K τsij σσxx* = − Z Ω*<sup>e</sup> Z*ˆ *<sup>τ</sup><sup>s</sup> nn*22*NjN<sup>i</sup> d*Ω , *K τsij σσxz* = 0 , *K τsij σσyz* = 0 , *K τsij σσzx* = 0 , *K τsij σσzy* = 0 , *K τsij σσzz* = 0 . (38)

The mass FN is the same as the PVD case; see Equation (33). Since the in-plane integrals are calculated via Gauss quadrature, it is crucial to consider an appropriate number of Gauss points in accordance with the variational rule of the fiber angle.

#### **3. Results and Discussion**

Three cases are analyzed in this work: a cantilever monolayer plate, a clamped multilayer plate and a clamped multilayer plate with a central circular cut-out. For each case, a square geometry is considered (*a* = *b* = 1 m). Parametric studies are performed considering different side-to-thickness ratios (*a*/*h* = 100, 10, 5). Material properties are represented in Table 1 for all the considered analyzed cases.


Reference solutions are represented by an Abaqus 3D model where quadratic elements (C3D20R) were used. For CUF solutions, nine-node square elements were used. For each case study, a preliminary convergence analysis was carried out to identify the appropriate mesh for both CUF and Abaqus solutions.

#### *3.1. Monolayer Plate*

The first case corresponds to a cantilever monolayer plate with density *ρ* = 1540 kg/m<sup>3</sup> . For this problem, axes *x* 0 and *y* 0 of the angle reference system are coincident with axes *x* and *y* of the plate. This means that the origin of the angle reference system is the same as the global one and that *x* 0 and *y* 0 are parallel to *x* and *y*, respectively. It is assumed that the fiber angle is a linear function of *y* 0 ; see Equation (10). The length parameter corresponds to *d* = *b*, while the direction of fiber variation *α* corresponds to *y* 0 , which means that Φ = 90◦ . In this case, *T*<sup>0</sup> = 0 ◦ and *T*<sup>1</sup> = 90◦ . The fiber orientation changes only along *y* 0 from a value of *θ*(0) = Φ + *T*<sup>0</sup> = 90◦ to *θ*(*b*) = Φ + *T*<sup>1</sup> = 180◦ . The angle variational law in this case can be expressed as 90 < 0, 90 >, and it is presented in Figure 4.

**Figure 4.** Stacking sequence; case 1.

This law has been taken from Viglietti et al. [27]. The reference solution contains 80 elements along each in-plane side and 12 elements along the thickness. The only clamped side of the plate is the one that lies on the *xz* plane, corresponding to *y* 0 = 0. For the CUF results, a 10 × 10 mesh is considered. Table 2 shows the degrees Of freedom (DOF) for some considered solutions.


**Table 2.** Degrees of freedom; case 1.

It is possible to observe that higher-order CUF models allow for a DOF reduction of one order of magnitude in comparison with the Abaqus 3D reference solution. Table 3 shows the first five natural frequencies for *a*/*h* = 100.


**Table 3.** Natural frequencies (Hz), *a*/*h* = 100; case 1.

For this case, classic and higher-order theories show very good approximations of the reference solution, where the maximum difference from the reference solution is 0.4% for the second natural frequency computed via CLT. Table 4 shows the first five natural frequencies for *a*/*h* = 10.

**Table 4.** Natural frequencies (Hz), *a*/*h* = 10; case 1.


It is possible to observe that classical and lower-order ESL theories are now less accurate, especially for the prediction of higher frequencies. For example, CLT, FSDT and ED2 models, corresponding to the third natural frequency, present a percentage error equal to 8.1%, 1.0% and 1.1%, respectively. This can be explained by considering that the side-to-thickness ratio *a*/*h* = 10 corresponds to a thick plate. In this case, higher-order theories are needed to obtain an accurate approximation. Since a moderately thick plate is considered, transverse shear stresses affect the solution. This is the reason that CLT, which neglects those stresses, is less close to the reference solution. The best approximations of plate natural frequencies are given by 2LM2 and 3LM4 mixed theories, which show a maximum percentage error of 0.1% each for the fourth natural frequency. In particular, it is possible to observe that the 2LM2 solution is globally closer to Abaqus in comparison with the 3LD4 solution, even if the last one is characterized by a higher number of degrees of freedom. Table 5 shows the first five natural frequencies for *a*/*h* = 5.

Because of the low side-to-thickness ratio, a very thick plate is considered, and lowerorder theories do not provide a correct prediction of the natural frequencies. For CLT, the sixth mode is the same as the fifth mode of the reference solution, that is, the order of appearance is swapped. In this regard, mode tracking was performed by visually comparing the modes of each proposed solution with those of the reference solution obtained in Abaqus. The corresponding percentage error is as high as 27.1%. On the other hand, a 3LM4 model matches the Abaqus reference results.


**Table 5.** Natural frequencies (Hz), *a*/*h* = 5; case 1.

#### *3.2. Multilayer Plate*

The second case is taken from Viglietti et al. [27] and corresponds to a multilayer clamped plate with density *ρ* = 1540 kg/m<sup>3</sup> . The plate is composed of three layers with the same thicknesses. It is assumed that fiber angle is a function of *y* 0 only, which means that *α* is parallel to *y* 0 (Φ = 90◦ ). In this case, a linear law is considered for the fiber path, according to Equation (10). For this problem, axes *x* 0 and *y* 0 of the angle reference system are aligned with axes *x* and *y* of the plate, but their origin is placed on the center of the plate (*a*/2, *b*/2). In this case, *d* = *b*/2 is considered as the length parameter in Equation (10). *T*<sup>0</sup> and *T*<sup>1</sup> are set for each layer as follows: *T layer*1 <sup>0</sup> = *T layer*3 <sup>0</sup> = 0 ◦ , *T layer*2 <sup>0</sup> = −45◦ , *T layer*1 <sup>1</sup> = *T layer*3 <sup>1</sup> = 45◦ , *T layer*2 <sup>1</sup> = −60◦ . The lamination of the plate is 90 < 0, 45 > for layer 1, 90 < −45, −60 > for layer 2 and 90 < 0, 45 > for layer 3. The stacking sequence is presented in Figure 5.

**Figure 5.** Stacking sequence; case 2.

As for the previous case, the Abaqus reference solution contains 80 elements along each side and 12 elements along the thickness. For the CUF results, a 10 × 10 mesh is considered. Table 6 shows the first five natural frequencies for *a*/*h* = 100, together with the results presented in Viglietti et al. [27].

In this case, the best approximation is given by the LM2 and LM4 theories. The LM2 and LM4 models both have a maximum percentage error as high as 0.4% corresponding to the third frequency. In addition, classical and low-order theories provide good results since a thin plate is considered. For this reason, transverse stresses do not play an important role. For example, the maximum error given by CLT is 2.1% for the fifth frequency. The case for *a*/*h* = 10 is presented in Table 7.


**Table 6.** Natural frequencies (Hz), *a*/*h* = 100; case 2.

**Table 7.** Natural frequencies (Hz), *a*/*h* = 10; case 2.


Here, the CLT model shows that the inversion of the third and fourth modes can be observed by the corresponding values of the frequencies that are not in ascending order as the mode number increases. In comparison with the monolayer plate, in this case, the mode inversions of the CLT model can be seen for higher side-to-thickness ratios and lower frequencies. For the third mode, CLT shows a percentage error of 96.0%, while the best approximation is given by LM4, which has a percentage error of 0.17% for the same mode. Table 8 shows the first five frequencies for *a*/*h* = 5.

**Table 8.** Natural frequencies (Hz), *a*/*h* = 5; case 2.


In this case, lower-order theories have an evident loss of accuracy. The CLT model can predict only the first two modes. In addition, the FSDT and ED2 models show nonnegligible errors, which become bigger with the increase in frequency. On the other hand, mixed models are able to correctly predict the dynamic behavior of the plate for both low and high frequencies.

#### *3.3. Multilayer Plate with Central Hole*

Case 3 is taken from Hachemi et al. [9] and corresponds to a multilayer clamped plate that presents a circular cut-out. The center of the cut-out is placed at the plate center (*a*/2, *b*/2), and its radius is *r* = 0.2 m. It is assumed that the fiber angle is a parabolic function of *x* 0 , which means that *α* is parallel to *x* 0 (Φ = 0 ◦ ). As in the previous case, the *x* 0 and *y* 0 axes are parallel, respectively, to *x* and *y*, and their origin is placed at the center of the plate. The angle variational law is defined in Equation (12), considering *d* = *a*/2. The plate is composed of two layers that have the same thicknesses. The values of *T*<sup>0</sup> and *T*<sup>1</sup> are set for each layer as follows: *T layer*1 <sup>0</sup> = *T layer*2 <sup>0</sup> = 0 ◦ , *T layer*1 <sup>1</sup> = 30◦ , *T layer*2 <sup>1</sup> = −30◦ . The stacking sequence is 0 < 0, ±30 >; see Figure 6.

**Figure 6.** Stacking sequence; case 3.

In this case, the Abaqus reference solution is made of 73728 elements: 4608 elements are defined on the plate plane, and 16 elements are defined along the thickness. For the CUF results, 128 elements are used on the plate plane. The natural frequencies are expressed in the following dimensionless form:

*ω* = *ωa* 2 p *ρh*/*D*<sup>0</sup> , (39)

$$D\_0 = E\_2 h^3 / 12 (1 - \nu\_{\rm LT} \nu\_{\rm TL}) \,\,,\tag{40}$$

where *ω* is the natural frequency, while *D*<sup>0</sup> represents a reference bending stiffness. Table 9 presents the first five non-dimensional frequencies for *a*/*h* = 100.


**Table 9.** Non-dimensional frequencies *ω*, *a*/*h* = 100; case 3.

It is possible to observe that the theories show a good approximation of the reference results. In addition, the percentage errors of FSDT and CLT are less than 2%. Mixed theories match the Abaqus results. Table 10 shows the results for *a*/*h* = 10 in order to compare the Abaqus reference solution with the one presented in Hachemi et al. [9] and the solutions obtained with CUF.


**Table 10.** Non-dimensional frequencies *ω*, *a*/*h* = 10; case 3.

As already observed in previous cases, classical theories and, in general, low-order ones are not able to provide an accurate approximation of natural frequencies, because of the low side-to-thickness ratio value. It is also possible that this generates the inversion of modes four and five for the CLT case. On the other hand, the best approximation is given by mixed theories, which are closer to the Abaqus solution for high frequencies. The shapes of the modes are presented in Figures 7–11 for *a*/*h* = 10. The modal shapes obtained with the LM4 model are compared with those of Abaqus 3D.

**Figure 8.** Mode 2, *a*/*h* = 10; case 3.

(**a**) LM4 (**b**) Abaqus 3D

**Figure 11.** Mode 5, *a*/*h* = 10; case 3.

The first mode shows a simple bending of the plate on the *xy* plane with a single halfwave along each in-plane direction. The second and the third modes show two half-waves in the *y* and *x* directions, respectively. Mode number four shows three half-waves along the plate diagonally between the *x* and *y* axes. The fifth mode shows three half-waves along the *y* direction. Finally, Table 11 shows the frequencies for *a*/*h* = 5.


**Table 11.** Non-dimensional frequencies *ω*, *a*/*h* = 5; case 3.

Since a thick plate is considered, the effect of transverse stresses is not negligible, which causes the classical and lower-order theories to be inaccurate. This can be observed for CLT, which is not able to predict the fourth and fifth modes and has an error as high as 69.4% for the third mode. Considering the FSDT, ED4 and LD4 models, this error can be reduced to 4.4%, 0.5% and 0.1%, respectively.

#### **4. Conclusions**

In this paper, a new framework for the dynamic analysis of VAT structures is presented. RMVT is developed within CUF in order to obtain a new family of 2D models for the freevibration analysis of VAT plates. The results are obtained via either RMVT or PVD and are compared in order to show the effective capabilities of the proposed method in the prediction of VAT plates' natural frequencies. The Abaqus 3D reference solutions and results from Refs. [9,27] are also included to further validate the models proposed in this article. Linear and parabolic laws are both considered in order to describe the in-plane path of fiber variation. The possibility to use a polynomial order defined a priori through CUF and the introduction of the transverse stress field as a primary variable of the problem through RMVT both help to obtain a valid approach for the prediction of VAT dynamic behavior. After the results analysis, the following remarks can be made:


In conclusion, the application of RMVT within CUF has demonstrated significant potential for improving the accuracy and efficiency of modeling VAT plates for free-vibration

analyses. The promising results suggest, as future perspectives, the extension to buckling and failure analyses where an accurate and efficient modeling of VAT structures under various loading and operational conditions is required.

**Author Contributions:** Methodology, G.G., D.A.I. and M.M.; Software, D.A.I.; Validation, D.A.I.; Writing—original draft, D.A.I.; Writing—review & editing, G.G. and M.M.; Supervision, G.G. and M.M.; Funding acquisition, G.G. and M.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Luxembourg National Research Fund (FNR), grant reference INTER/ANR/21/16215936 GLAMOUR-VSC. For the purpose of open access, and in fulfillment of the obligations arising from the grant agreement, the author has applied a Creative Commons Attribution 4.0 International (CC BY 4.0) license to any Author Accepted Manuscript version arising from this submission. M. Montemurro is grateful to French National Research Agency for supporting this work through the research project GLAMOUR-VSC (Global–LocAl two-level Multi-scale optimisation strategy accOUnting for pRocess-induced singularities to design Variable Stiffness Composites) ANR-21-CE10-0014.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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