**1. Introduction**

Woven compounds are being increasingly considered for lots of applications because they provide ease in making complex geometries, but the mechanical properties of the different weave material supports are less visible than non-woven (angle-ply) laminates [1]. Note that there is no mathematical modelling of structural fiber-reinforced composite structures, though one measurement was obtained here using estimates in which woven fibers were formed from two orthotropic unidirectional fiber results in a curved twist using a weave. Recently, there has been in increased focus on exploring integrated and complete mechanical properties and methods of textile composites tested for uniaxial or biaxial tension, pressure, flexibility, and short-beam cutting [2–5]. Various analytical techniques have been developed to predict the elastic properties of representative volume elements (RVEs) of textile composites that include 2D and 3D weave composite. Woven composites can provide a potential solution to the basic limitations of traditional laminated composites: delamination and production that requires more workers. The addition of binder yarns provides through-thickness reinforcement, leading to highly advanced interlaminar materials and fabric binding to allow near-net-shape preforms to be woven and handled [6].

However, apart from these benefits, the use of 2.5D woven compounds is very limited in niche applications. One of the main reasons for this is the lack of predictive numerical tools, which limit their use in the early stages of design. Another method of production is to cut a simple three-dimensional weave, consisting of two two-dimensional (2D) fabrics connected by twisted loops of yarn, to form a 'hairy' fabric. These 2.5D fabrics are then coated with epoxy resin in a standard way, laminated, and treated with autoclave [7]. Woven Composites (2.5D-WC) not only have a higher delamination resistance compared to 2D laminated composites, but also have a simpler structure than 3D textile composites.

**Citation:** Kaddaha, M.A.; Younes, R.; Lafon, P. New Geometrical Modelling for 2D Fabric and 2.5D Interlock Composites. *Textiles* **2022**, *2*, 142–161. https://doi.org/10.3390/ textiles2010008

Academic Editor: Laurent Dufossé

Received: 13 January 2022 Accepted: 24 February 2022 Published: 7 March 2022

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**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/).

Recently, most parts in the aero-engine sector, namely fan/compressor blades and casing, were made using 2.5D-WC [8].

#### *1.1. Objective*

The purpose of this paper is to introduce new content and a new, simple geometrical model for analyzing the mechanical properties of woven composites, i.e., 2D and 2.5D. In addition, the model will estimate the change in performance due to changing the weaving parameters that dictates the 2.5D weave architecture about the elastic properties; this is discussed in detail. Finally, important conclusions are reached.

An effective recursive algorithm, a matrix solidification method, is designed for multilayer media that is usually anisotropic. This algorithm has the efficiency and calculation of the standard transmission matrix method and is unconditional in terms of a high frequency calculation system and layer thickness. In this algorithm, the stiffness (compliance) matrix is calculated.

This algorithm allows researchers and end users to calculate the durability and compliance matrix of any woven compound material, of any type, shape, or thickness, while saving time and cost in finding the right solution.

This paper introduces a geometric modeling tool to predict the expandable stiffness features of 2.5D woven orthogonal interlock composites. The model is able to reproduce the behavior of woven composites formed by different fiber-reinforced types.

#### *1.2. State of the Art*

Various analytical techniques have been developed to predict the expansive properties of volume representation (RVEs) of textile composites that include 2D and 2.5D weave composites. Bogdanovich and Pastore [9], Whitcomb et al. [10], and Potluri and Thammandra [11] have conducted a complete review of the topic. Based on stiffness averaging and rigidity ratio, Ishikawa and Chou [12–14] managed a two-orthogonal set of fabric compositions such as laminate with two laminae placed at 0/90 position. They proposed different models for predictable expansion, such as (1) a mosaic model (where fiber continuity and retraction was left), (2) a crimp model (where fiber continuity and undulation geometry were defined using the functions of -trigonometric), and (3) bridge coupling model (rigid or compliant dimensions in two orthogonal directions and suitable for satin weave combinations). Naik et al. [15,16] analyzed fiber retreat to the two orthogonal directions and used detailed trigonometric functions to extend the crimson models of Ishikawa and Chou (i.e., parallel two-dimensional series models and series-parallel models) to test expandable structures different combinations of PWF. Saka and Harding [17] and Carey et al. [18] developed rectilinear and sinusoidal crimp models to predict expansion structures using the classic laminate plate theory applied in the context of an expandable base and using the same power system. Tong et al. [19] take fiber as a curved rod based on stretch elastic or shear base in a woven composite structure and present a curved bark model to investigate the impact of fiber flexibility on the strength of woven composite material based on the concept of classical laminate plate theory. Kollegal and Sridharan [20] employed hierarchic works to determine the original curved profile of the strings and plate displays and introduced the micro-plate model for PWF analysis.

A significant amount of research was performed using the principles of differential stress [11,21] or FE (FEA) analysis [22–24] to analyze the expandable tracts of woven compounds and a detailed stress field across the RVE from the limited equipment. Regarding basic properties, in order to produce FE models, precise vertical segments or continuous mathematical functions are used to demonstrate in detail the ideal PWF geometry and to represent the thread path and cross section shape.

Mesoscopic simulations of transverse compaction have attracted recent interest. Geometric modeling methods have been developed using fabric matching software such as TexGen [25,26] or WiseTex [27,28], which have been used in various articles [29–33]. Multi-chain digital element-based methods have also been used [34–38]. Green et al. [39,40]

proposed a numerical procedure for creating mesostructural geometric models in simulated fabrics by combining appropriate TexGen models with multi-chain digital materials. Nguyen et al. [41] followed the method developed by Hivet and Boisse [42] to produce mesostructural geometric models of 2D woven fabrics to mimic the opposite combination. Numerical simulations of hypo elastic material based on fiber rotation were well matched to experiments. Alternatively, based on the discrete homogenization method, Goda et al. [43] and Rahali et al. [44] can reproduce the functional properties of 2.5D machines and 3D interlock textiles. Zhang et al. [45] developed a model based on the optical aspect ratio to predict mechanical responses and failure location under uniaxial and biaxial loading.

However, these methods are based on appropriate geometrical models, therefore cannot explain mesostructural changes in fiber structure, i.e., a detailed study of how fibers degenerate under bonding. Consideration and simplicity are often needed in numerical matching, which reduces calculation costs, facilitates the modeling process, and provides computer estimations. However, this simplification may prevent the investigation of other events occurring in real systems, especially in the case of mesoscopic and microscopic disorders.

Specific studies of mesoscopic transformation of fibrous reinforcement can also be performed with minimally invasive imaging such as X-ray micro-computed tomography (Micro-CT). With this process, it is possible to measure, model, and analyze the geometric structure of fiber cables. Desplentere et al. [46] and Schell et al. [47,48] were among the first to demonstrate the power of Micro-CT to show the geometric features of 3D fabrics on a mesoscopic scale. In Pazmino et al.'s study [49], the geometric features are directly measured by the fiber pulse from small tomographic images to read a single layer of 3D non-crimp woven fabric. Naouar et al.'s [50,51] mesostructural geometry models successfully reconstructed 2D woven fabric and 3D orthogonal fabric from small tomographic images. Wang et al. [52] studied longitudinal compression and the Poisson number of fiber cables followed a similar pattern. Badel et al. [53] performed both numerical simulations and tomographic analyzes of 2D woven fabrics under biaxial tension and inplane shear deformation.

Combined with digital volume integration, Mendoza et al. [54] developed a mathematical model for the distortion of woven compounds, which was recently used to measure the flexibility caused by the production process [55]. However, no research studies to date have reported on the analysis of variables based on meso-structural models of fiberglass fabrics separated by these simplified methods at different levels of assembly.

Methodology:

*Textiles* **2022**, *2*, FOR PEER REVIEW 4

Model

2D Fabric / 2.5 D interlock

#### **2. Geometrical Modelling 2. Geometrical Modelling** *2.1. 2D Fabric*

#### *2.1. 2D Fabric 2.1. 2D Fabric*

The user—whether a student or a fellow researcher—can themselves design a fabric composite. To construct the whole structure of composite as he or she likes from the beginning and to make the whole process "user friendly", the process took the shape of building a puzzle by a simple "click and drop" game as shown in the figure below (Figure 1). The user—whether a student or a fellow researcher—can themselves design a fabric composite. To construct the whole structure of composite as he or she likes from the beginning and to make the whole process "user friendly", the process took the shape of building a puzzle by a simple "click and drop" game as shown in the figure below (Figure 1). The user—whether a student or a fellow researcher—can themselves design a fabric composite. To construct the whole structure of composite as he or she likes from the beginning and to make the whole process "user friendly", the process took the shape of building a puzzle by a simple "click and drop" game as shown in the figure below (Figure 1).

terized by orthogonal linking of two sets of threads, called warp and weft yarn. The warp

As 2D woven fabrics are made by interlacing yarns in a weaving loom, yarns are

terized by orthogonal linking of two sets of threads, called warp and weft yarn. The warp

divided into two components: one called the warp, running along the length of the loom, and the other is the weft, running in the cross direction. The woven structure is charac-**Figure 1.** Puzzle structure for 2D fabric. **Figure 1.** Puzzle structure for 2D fabric.

As 2D woven fabrics are made by interlacing yarns in a weaving loom, yarns are divided into two components: one called the warp, running along the length of the loom, and the other is the weft, running in the cross direction. The woven structure is characterized by orthogonal linking of two sets of threads, called warp and weft yarn. The warp threads are aligned with the direction of the fabric leaving the weaving equipment, which is also called the warp direction. As shown in Figure 2, 2D woven fabric has two yarn sets as warp (0◦ ) and filling (90◦ ) and interlaced to each other to form the surface. *Textiles* **2022**, *2*, FOR PEER REVIEW 5 threads are aligned with the direction of the fabric leaving the weaving equipment, which is also called the warp direction. As shown in Figure 2, 2D woven fabric has two yarn sets as warp (0°) and filling (90°) and interlaced to each other to form the surface. *Textiles* **2022**, *2*, FOR PEER REVIEW 5 threads are aligned with the direction of the fabric leaving the weaving equipment, which is also called the warp direction. As shown in Figure 2, 2D woven fabric has two yarn sets as warp (0°) and filling (90°) and interlaced to each other to form the surface. yarn sets as warp (0°) and filling (90°) and interlaced to each other to form the surface. **Figure 2.** 2D fabric components.

threads are aligned with the direction of the fabric leaving the weaving equipment, which is also called the warp direction. As shown in Figure 2, 2D woven fabric has two

*Textiles* **2022**, *2*, FOR PEER REVIEW 5

**Figure 2.** 2D fabric components. **Figure 2.** 2D fabric components.**Figure 2.** 2D fabric components.

To start from the beginning, the user is free to choose the settings of the puzzle in each dimension, i.e., number of columns and rows, and the number of fibers, as shown in Figure 3. To start from the beginning, the user is free to choose the settings of the puzzle in each dimension, i.e., number of columns and rows, and the number of fibers, as shown in Figure 3. To start from the beginning, the user is free to choose the settings of the puzzle in each dimension, i.e., number of columns and rows, and the number of fibers, as shown in Figure 3.


The next step is to construct the geometry as desired and shown in the following **Figure 3.** Puzzle settings for 2D fabric. **Figure 3.** Puzzle settings for 2D fabric. Plain weave is a very common and very strong basic weave where each warp fiber

examples in Figures 4–7. 2.1.1. Plain Weave Fabric [1/1] The next step is to construct the geometry as desired and shown in the following examples in Figures 4–7. The next step is to construct the geometry as desired and shown in the following examples in Figures 4–7. passes alternately below and above each weft. The fabric is symmetrical and has good stability and logical porosity. However, they are the most difficult to weave, and the high quality of fiber crimp offers lower mechanical features compared to other weaving styles.

**Figure 4.** Plain weave fabric. **Figure 4.** Plain weave fabric.

**Figure 4.** Plain weave fabric.

**Figure 4.** Plain weave fabric.

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strings will carry the load when the fabric is torn.

Twill is a type of textile weave with a pattern of corresponding diagonal ribs. It can be seen by observing the presence of diagonal lines that are pronounced along the width of the fabric. It has a higher resistance to cracking than plain weave because it has fewer wires that connect to each area, hence the greater the internal flow rate. In addition, two

Sateen is a fabric that usually has a shiny surface and a hidden back, one of the three basic types of fabric weaving, seamless weaving, and twill. Four or more full-length or single-stranded strands floating over a straight rope, and four straight strands floating on a single weft thread indicate satin weaving. Floating strings are missed interfaces, where the warp thread lies on the weft in a satin with a straight face and where the weft thread

2.1.2. Twill [2/1, 3/1]

**Figure 5.** Twill weave fabric. **Figure 5.** Twill weave fabric.

2.1.3. Sateen/Satin [5-end, 8-end]

lies on straight strands on satin with a weft face.

Basket weave is known also as Hopsack and Matt Weave. Hopsack weave, a variant of plain weave, uses two or more warp and/or two or more weft fibers joined as a single thread. This weave is obtained by doubling or repeating the combined points of a plain weave in both the direction of the wrap and weft. These fabrics are made of two or more strings placed in the same shed. The stitching pattern is similar to a plain weave, but two or more strands follow the same parallel pattern. Fabrics designed by Matt are flexible and cannot wrinkle as there are a few cutters that are a square inch. Fabrics look flatter than conventional weaving fabrics. However, longer floats are more flexible. Matt fabric has great anti-tear properties. Matt's design tends to offer more smooth fabrics. For a

**Figure 6.** Sateen/Satin weave fabric. **Figure 6.** Sateen/Satin weave fabric.

2.1.4. Basket Weave [2/2, 4/4]

repetitive Matt weave size, the warp numbers and weft threads are equal.

**Figure 7.** Basket weave fabric. **Figure 7.** Basket weave fabric.

#### 2.1.5. 2D Fabric Hybrid Composites 2.1.1. Plain Weave Fabric [1/1]

Hybrid yarns are where two or more fibers are brought together to combine the performance and aesthetic of both, as shown in Figure 8. Although environmental concerns are encouraging many manufacturers to focus on the use of a single fiber, there are still and will remain reasons for bringing fibers with different qualities together, partic-Plain weave is a very common and very strong basic weave where each warp fiber passes alternately below and above each weft. The fabric is symmetrical and has good stability and logical porosity. However, they are the most difficult to weave, and the high quality of fiber crimp offers lower mechanical features compared to other weaving styles.

#### ularly for performance and health and safety applications. 2.1.2. Twill [2/1, 3/1]

Twill is a type of textile weave with a pattern of corresponding diagonal ribs. It can be seen by observing the presence of diagonal lines that are pronounced along the width of the fabric. It has a higher resistance to cracking than plain weave because it has fewer wires that connect to each area, hence the greater the internal flow rate. In addition, two strings will carry the load when the fabric is torn.

### 2.1.3. Sateen/Satin [5-end, 8-end]

Sateen is a fabric that usually has a shiny surface and a hidden back, one of the three basic types of fabric weaving, seamless weaving, and twill. Four or more full-length or single-stranded strands floating over a straight rope, and four straight strands floating on a single weft thread indicate satin weaving. Floating strings are missed interfaces, where the warp thread lies on the weft in a satin with a straight face and where the weft thread lies on straight strands on satin with a weft face.
