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Metals
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Constitutive Modelling for Metals
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Constitutive Modelling for Metals

A special issue of Metals (ISSN 2075-4701).

Deadline for manuscript submissions: closed (31 March 2019) | Viewed by 47350

Special Issue Editors

Prof. Dr. Robertt Valente

E-Mail Website1 Website2
Guest Editor
Department of Mechanical Engineering, Universidade de Aveiro, Aveiro, Portugal
Interests: Computational mechanics; Computational plasticity; Finite Element Method (FEM) / Isogeometric Analysis (IgA) / Finite Cell Method (FCM); Simulation of forming and joining processes
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Myoung-Gyu Lee

E-Mail Website
Guest Editor
Department of Materials Science and Engineering, Seoul National University, Seoul 08826, Korea
Interests: constitutive modeling for plasticity; multiscale modeling and simulation for structure materials; experimental mechanics; metal forming and shaping
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In a world facing constant technological evolution, and where a circular economy represents the dominant paradigm, the optimized use of raw materials with the lowest energetic impact is a strong (and increasingly important) requirement. Together with this rational use of resources, structural requirements for final products are a key factor for materials science and mechanical engineers. To this aim, physically-consistent, reliable and computationally-efficient constitutive modelling is the cornerstone of an efficient design.

Within this Special Issue on "Constitutive Modelling for Metals", we aim to provide a wide visibility for the most up-to-date and relevant works in this field, from both experimental and modelling/numerical simulation standpoints.

Following your research achievements in the field of constitutive modelling, we would like to invite you and your group to submit a contribution to this Special Issue of Metals (https://www.mdpi.com/journal/metals). Being an open access journal with a high impact factor (https://www.mdpi.com/journal/metals/stats) we are sure your work will have a strong impact for a wide range of readers.

The deadline for papers submission is 31 of December, 2018. After receiving a submission, it will be processed following the review process. After the revision stages, accepted papers in their final form will be immediately published with a specific label mentioning the Special Issue, with no delay (i.e., with no need to wait for other papers in the Special Issue).

We hope you accept this invitation, and help us to make a high-impact and high-quality Special Issue on "Constitutive Modelling for Metals". In the case of a positive answer, please let us know as soon as possible so that we can inform the Editorial Office about your inclusion.

Best,

Prof. Robertt Valente
Prof. Myoung-Gyu Lee
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Metals is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Anisotropic yield criteria
  • Anisotropic hardening
  • Computational constitutive modelling and experimental validation
  • Conventional and innovative numerical simulation techniques
  • Damage criterion and fracture propagation
  • Industrial applications (aeronautic, automotive, beverage cans, etc.)
  • Modelling of advanced joining technologies (FSW, SPR, mechanical joints, etc.)
  • Multi-scale computational models and their implementation
  • Springback, rupture and wrinkling predictions
  • Theoretical and numerical practice for novel forming technologies

Published Papers (11 papers)

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Research

30 pages, 5984 KiB  
Article
An Evolutionary Yield Function Model Based on Plastic Work and Non-Associated Flow Rule
by Taejoon Park, Fadi Abu-Farha and Farhang Pourboghrat
Metals 2019, 9(5), 611; https://doi.org/10.3390/met9050611 - 25 May 2019
Cited by 11 | Viewed by 4046
Abstract
A constitutive law was developed based on the evolutionary yield function to account for the evolution of anisotropy induced by the plastic deformation. For the effective description of anisotropy, the yield stress function and plastic potential were separately defined based on the non-associated [...] Read more.
A constitutive law was developed based on the evolutionary yield function to account for the evolution of anisotropy induced by the plastic deformation. For the effective description of anisotropy, the yield stress function and plastic potential were separately defined based on the non-associated flow rule. In particular, for the description of the equivalent status, the accumulated plastic work was employed as an alternative to the accumulated plastic strain. Numerical formulations based on the plastic work were also derived in case the hardening rule, as well as the evolution of the plastic potential and yield stress function, were defined in terms of the plastic work. The developed constitutive law was characterized using the mechanical properties of the multi-phase BAO QP980 steel and niobium sheets at room temperature. From the uniaxial tension tests and the balanced biaxial tension test, separate sets of anisotropic coefficients for each of the plastic potential and yield stress functions were obtained as a function of the plastic work. By comparing with non-evolving yield functions, the importance of the developed constitutive law to properly describe the evolution of the plastic potential and yield function were validated. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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17 pages, 18040 KiB  
Article
Determination of Johnson–Cook Constitutive Parameters for Cutting Simulations
by Michael Storchak, Philipp Rupp, Hans-Christian Möhring and Thomas Stehle
Metals 2019, 9(4), 473; https://doi.org/10.3390/met9040473 - 23 Apr 2019
Cited by 36 | Viewed by 6425
Abstract
The Johnson–Cook constitutive equation is very widely used for simulating cutting processes. Different methods are applied for establishing parameters of the constitutive equation. Based on the methods analysed in this study, two algorithms were worked out to determine the constitutive parameters for the [...] Read more.
The Johnson–Cook constitutive equation is very widely used for simulating cutting processes. Different methods are applied for establishing parameters of the constitutive equation. Based on the methods analysed in this study, two algorithms were worked out to determine the constitutive parameters for the prevailing conditions during cutting processes. In the first algorithm, all constitutive parameters were established simultaneously with standardized test methods. In the second algorithm, the constitutive parameters were established separately in accordance with the cutting conditions prevailing in machining processes. The developed methodology was verified with AISI 1045 heat-treatable steel and Ti10V2Fe3Al (Ti-1023) titanium alloy. The two materials were examined in standardized tensile and compression tests with varying strain rates and temperatures. In addition, the kinetic characteristics of the orthogonal cutting process were established. Based on the results obtained by experiment and the algorithms developed, the constitutive parameters for the cutting conditions were calculated. The parameters were used to determine the material model for simulating the orthogonal cutting process. The algorithms developed were verified by comparing the simulated and experimentally determined kinetic cutting characteristics, which confirmed their good quality. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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16 pages, 20784 KiB  
Article
On the Design of Innovative Heterogeneous Sheet Metal Tests Using a Shape Optimization Approach
by António Andrade-Campos, José Aquino, João M. P. Martins and Bernardete Coelho
Metals 2019, 9(3), 371; https://doi.org/10.3390/met9030371 - 22 Mar 2019
Cited by 7 | Viewed by 3381
Abstract
The development of full-field measurement methods has enabled a new trend of heterogeneous mechanical tests. The inhomogeneous strain fields retrieved from these tests are being widely used in the calibration of constitutive models for sheet metals. However, today, there is no mechanical test [...] Read more.
The development of full-field measurement methods has enabled a new trend of heterogeneous mechanical tests. The inhomogeneous strain fields retrieved from these tests are being widely used in the calibration of constitutive models for sheet metals. However, today, there is no mechanical test able to characterize the material in a large range of strain states. The aim of this work is to present a heterogeneous mechanical test with an innovative tool/specimen shape, capable of producing rich heterogeneous strain paths and thus providing extensive information on material behavior. The proposed specimen is found using a shape optimization process where an index that evaluates the richness of strain information is used. In this work, the methodology and results are extended to non-specimen geometry dependence and to the non-dependence of the geometry parametrization through the use of the Ritz method for boundary value problems. Different curve models, such as splines, B-splines, and NURBS, are used, and C1 continuity throughout the specimen is guaranteed. Moreover, several deterministic and stochastic optimization methods are used in order to find the method or the combination of methods able to minimize the cost function effectively. Results demonstrated that the solution is dependent on the geometry definition, as well as on the optimization methodology. Nevertheless, the obtained solutions provided a wider spectrum of strain states than standard tests. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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22 pages, 7035 KiB  
Article
Avrami Kinetic-Based Constitutive Relationship for Armco-Type Pure Iron in Hot Deformation
by Yan Zhang, Qichao Fan, Xiaofeng Zhang, Zhaohui Zhou, Zhihui Xia and Zhiqiang Qian
Metals 2019, 9(3), 365; https://doi.org/10.3390/met9030365 - 21 Mar 2019
Cited by 8 | Viewed by 3639
Abstract
The work presents a full mathematical description of the stress-strain compression curves in a wide range of strain rates and deformation temperatures for Armco-type pure iron. The constructed models are based on a dislocation structure evolution equation (in the case of dynamic recovery [...] Read more.
The work presents a full mathematical description of the stress-strain compression curves in a wide range of strain rates and deformation temperatures for Armco-type pure iron. The constructed models are based on a dislocation structure evolution equation (in the case of dynamic recovery (DRV)) and Avrami kinetic-based model (in the case of dynamic recrystallization (DRX)). The fractional softening model is modified as: X = ( σ 2 − σ r 2 ) / ( σ d s 2 − σ r 2 ) considering the strain hardening of un-recrystallized regions. The Avrami kinetic equation is modified and used to describe the DRX process considering the strain rate and temperature. The relations between the Avrami constant k ∗ , time exponent n ∗ , strain rate ε ˙ , temperature T and Z parameter are discussed. The yield stress σ y , saturation stress σ r s , steady stress σ d s and critical strain ε c are expressed as the functions of the Z parameter. A constitutive model is constructed based on the strain-hardening model, fractional softening model and modified Avrami kinetic equation. The DRV and DRX characters of Armco-type pure iron are clearly presented in these flow stress curves determined by the model. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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16 pages, 5537 KiB  
Article
Microstructure-Based Constitutive Modelling of Low-Alloy Multiphase TRIP Steels
by Álvaro Salinas, Diego Celentano, Linton Carvajal, Alfredo Artigas and Alberto Monsalve
Metals 2019, 9(2), 250; https://doi.org/10.3390/met9020250 - 20 Feb 2019
Cited by 7 | Viewed by 3680
Abstract
The microstructure of low-alloy multiphase transformation-induced plasticity (TRIP) steels consists of ferrite, bainite, and metastable retained austenite, which can be transformed into martensite by plastic deformation. In some cases, residual martensite can be present in the initial microstructure. The mechanical behavior of these [...] Read more.
The microstructure of low-alloy multiphase transformation-induced plasticity (TRIP) steels consists of ferrite, bainite, and metastable retained austenite, which can be transformed into martensite by plastic deformation. In some cases, residual martensite can be present in the initial microstructure. The mechanical behavior of these steels depends on the interaction between the intrinsic characteristics of the existing phases and the austenite stability. Due to these factors, the definition of their true stress-strain flow law is complex. This work presents the mechanical characterization of a phenomenological constitutive stress-strain flow law based on the Bouquerel et al. model, as evaluated for three TRIP steels of the same composition but undergoing different heat treatments. Morphological aspects of the existing phases, austenite stability, and suitable mixture laws between phases are considered. The model is found to accurately reproduce a true stress-strain flow law obtained under tensile uniaxial conditions and provide detailed information on the effective stress strain partition between the existing phases. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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13 pages, 3002 KiB  
Article
Hot Deformation Behavior of a 2024 Aluminum Alloy Sheet and its Modeling by Fields-Backofen Model Considering Strain Rate Evolution
by Zhubin He, Zhibiao Wang, Yanli Lin and Xiaobo Fan
Metals 2019, 9(2), 243; https://doi.org/10.3390/met9020243 - 18 Feb 2019
Cited by 14 | Viewed by 3957
Abstract
The deformation behavior of a 2024 aluminum alloy sheet at elevated temperatures was studied by uniaxial hot tensile tests over the nominal initial strain rate range of 0.001–0.1 s−1 and temperature range of 375–450 °C. In order to analyze the deformation behavior [...] Read more.
The deformation behavior of a 2024 aluminum alloy sheet at elevated temperatures was studied by uniaxial hot tensile tests over the nominal initial strain rate range of 0.001–0.1 s−1 and temperature range of 375–450 °C. In order to analyze the deformation behavior with higher accuracy, a digital image correlation (DIC) system was applied to determine the strain distribution during hot tensile tests. Local stress-strain curves for different local points on the specimens were calculated. The strain rate evolution of each point during the tensile tests was investigated under different deformation conditions. Then, an improved Fields–Backofen (FB) model, taking into account the local strain rate evolution instead of the fixed strain rate, was proposed to describe the constitutive behaviors. It has been found that obvious non-uniform strain distribution occurred when the true strain was larger than 0.3 during hot tensile tests. The strain rate distribution during deformation was also non-uniform. It showed increasing, steady, and decreasing variation tendencies for different points with the increasing of strain, which led to the local flow stress being different at different local points. The flow stresses predicted by the improved FB model showed good agreement with experimental results when the strain rate evolutions of local points during tensile tests were considered. The prediction accuracy was higher than that of traditional FB models. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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18 pages, 8601 KiB  
Article
Applying Machine Learning to the Phenomenological Flow Stress Modeling of TNM-B1
by Johan A. Stendal, Markus Bambach, Mark Eisentraut, Irina Sizova and Sabine Weiß
Metals 2019, 9(2), 220; https://doi.org/10.3390/met9020220 - 13 Feb 2019
Cited by 18 | Viewed by 4047
Abstract
Data-driven or machine learning approaches are increasingly being used in material science and research. Specifically, machine learning has been implemented in the fields of materials discovery, prediction of phase diagrams and material modelling. In this work, the application of machine learning to the [...] Read more.
Data-driven or machine learning approaches are increasingly being used in material science and research. Specifically, machine learning has been implemented in the fields of materials discovery, prediction of phase diagrams and material modelling. In this work, the application of machine learning to the traditional phenomenological flow stress modelling of the titanium aluminide (TiAl) alloy TNM-B1 (Ti-43.5Al-4Nb-1Mo-0.1B) is investigated. Three model types were developed, analyzed and compared; a physics-based phenomenological model (PM) originally developed for steel by Cingara and McQueen, a purely data-driven machine learning model (MLM), and a hybrid model (HM), which uses characteristic points predicted by a learning algorithm as input for the phenomenological model. The same amount of data was used to both fit the PM and train the MLM and HM. The models were analyzed and compared based on the accuracy of their predictions, development and computing time, and their ability to predict on interpolated and extrapolated inputs. The results revealed that for the same amount of experimental data, the MLM was more accurate than the PM. In addition, the MLM was better able to capture the characteristic peak stress in the TNM-B1 the flow curves, and could be developed and computed faster. Furthermore, the MLM was able to make realistic predictions for inputs outside the experimental data used for training. The HM showed comparable accuracy to the PM for the experimental conditions. However, the HM was able to produce a better fit for input conditions outside the training data. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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13 pages, 7872 KiB  
Article
A Comparative Study on Arrhenius and Johnson–Cook Constitutive Models for High-Temperature Deformation of Ti2AlNb-Based Alloys
by Zhubin He, Zhibiao Wang and Peng Lin
Metals 2019, 9(2), 123; https://doi.org/10.3390/met9020123 - 24 Jan 2019
Cited by 19 | Viewed by 3279
Abstract
In order to thoroughly understand the quantitative relationships between the flow stress and deformation conditions for Ti2AlNb-based alloys at elevated temperatures, the Arrhenius and Johnson–Cook constitutive models are analyzed and identified on the basis of the uniaxial tensile tests. The Johnson–Cook [...] Read more.
In order to thoroughly understand the quantitative relationships between the flow stress and deformation conditions for Ti2AlNb-based alloys at elevated temperatures, the Arrhenius and Johnson–Cook constitutive models are analyzed and identified on the basis of the uniaxial tensile tests. The Johnson–Cook model is modified so that the referenced temperature range can be randomly adjusted. By experimental verification, the Arrhenius model (including the Backofen model) is suitable for the deformation at relatively low strain-rate deformation, such as the superplastic forming, and the modified J–C model is applicable for the deformation within a wide range of strain rates. For deformation at high temperatures, the constitutive model enables a more precise description of the effect of strain on the flow stress through introducing as train-softening factor exp(sε). Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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16 pages, 6128 KiB  
Article
Grain Scale Representative Volume Element Simulation to Investigate the Effect of Crystal Orientation on Void Growth in Single and Multi-Crystals
by Woojin Jeong, Chang-Hoon Lee, Joonoh Moon, Dongchan Jang and Myoung-Gyu Lee
Metals 2018, 8(6), 436; https://doi.org/10.3390/met8060436 - 08 Jun 2018
Cited by 9 | Viewed by 3461
Abstract
Crystal plasticity finite element (CPFE) simulations were performed on the representative volume elements (RVE) modeling body centered cubic (bcc) single, bi- and tri-crystals. The RVE model was designed to include a void inside a grain, at a grain boundary and at a triple [...] Read more.
Crystal plasticity finite element (CPFE) simulations were performed on the representative volume elements (RVE) modeling body centered cubic (bcc) single, bi- and tri-crystals. The RVE model was designed to include a void inside a grain, at a grain boundary and at a triple junction. The effect of single crystal orientation on the flow strength and growth rate of the void was discussed under prescribed boundary conditions for constant stress triaxialities. CPFE analyses could explain the effect of inter-grain orientations on the anisotropic growth of the void located at the grain boundaries. The results showed that the rate of void growth had preferred orientation in a single crystal, but the rate could be significantly different when other orientations of neighboring crystals were considered. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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21 pages, 47075 KiB  
Article
An Incremental Physically-Based Model of P91 Steel Flow Behaviour for the Numerical Analysis of Hot-Working Processes
by Alberto Murillo-Marrodán, Eli Saúl Puchi-Cabrera, Eduardo García, Mirentxu Dubar, Fernando Cortés and Laurent Dubar
Metals 2018, 8(4), 269; https://doi.org/10.3390/met8040269 - 14 Apr 2018
Cited by 7 | Viewed by 5651
Abstract
This paper is aimed at modelling the flow behaviour of P91 steel at high temperature and a wide range of strain rates for constant and also variable strain-rate deformation conditions, such as those in real hot-working processes. For this purpose, an incremental physically-based [...] Read more.
This paper is aimed at modelling the flow behaviour of P91 steel at high temperature and a wide range of strain rates for constant and also variable strain-rate deformation conditions, such as those in real hot-working processes. For this purpose, an incremental physically-based model is proposed for the P91 steel flow behavior. This formulation considers the effects of dynamic recovery (DRV) and dynamic recrystallization (DRX) on the mechanical properties of the material, using only the flow stress, strain rate and temperature as state variables and not the accumulated strain. Therefore, it reproduces accurately the flow stress, work hardening and work softening not only under constant, but also under transient deformation conditions. To accomplish this study, the material is characterised experimentally by means of uniaxial compression tests, conducted at a temperature range of 900–1270 °C and at strain rates in the range of 0.005–10 s−1. Finally, the proposed model is implemented in commercial finite element (FE) software to provide evidence of the performance of the proposed formulation. The experimental compression tests are simulated using the novel model and the well-known Hansel–Spittel formulation. In conclusion, the incremental physically-based model shows accurate results when work softening is present, especially under variable strain-rate deformation conditions. Hence, the present formulation is appropriate for the simulation of the hot-working processes typically conducted at industrial scale. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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15 pages, 4963 KiB  
Article
Quadratic Midpoint Integration Method for J2 Metal Plasticity
by Inkyu Rhee and Woo Kim
Metals 2018, 8(1), 66; https://doi.org/10.3390/met8010066 - 18 Jan 2018
Cited by 1 | Viewed by 4681
Abstract
The quadratic variants of the generalized midpoint rule and return map algorithm for the J2 von Mises metal plasticity model are examined for the accuracy of deviatoric stress integration of the constitutive equation. The accuracy of stress integration using a strain rate [...] Read more.
The quadratic variants of the generalized midpoint rule and return map algorithm for the J2 von Mises metal plasticity model are examined for the accuracy of deviatoric stress integration of the constitutive equation. The accuracy of stress integration using a strain rate vector for arbitrary direction is presented in terms of an iso-error map for comparison with the exact solution. Accuracy and stability issues of the quadratic integration method are discussed using a two-dimensional metal panel problem with a single slit-like defect in the center. The scale factor and shape factor were introduced to a quadratic integration rule for assuming a returning directional tensor from a trial stress onto the final yield surface. Luckily enough, the perfectly plastic model is the only case where the analytical solution is possible. Thus, solution accuracies were compared with those of the exact solutions. Since the standard scale factor ranges from 0 to 1, which is similar to the linear α -method, the penalty scale factors that are greater than 1 were mainly explored to examine the solution accuracies and computational efficiency. A higher value of scale factor above five shows a better computational efficiency but a decreased solution accuracy, especially in the higher plastification zone. A well-balanced scale factor for both computational efficiency and solution accuracy was found to be between one and five. The trade-off scale factor was proposed to be five. The proper shape factor was also proposed to be {1,1,4}/6 among the different combinations of weight distribution over a time interval. This proposed scale factor and shape factor is also valid for relatively long time periods. Full article
(This article belongs to the Special Issue Constitutive Modelling for Metals)
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