*2.3. Construction Testing* - Symmetry test (not obligatory).

Tests should be performed as shown in Figures 10 and 11 to validate the shape suggested by the user. These tests are: These tests obtained by logical geometrical construction of the 2.5D interlock structure are as constrains that define whether the design (structure) from the user is valid–

Tests should be performed as shown in Figures 10 and 11 to validate the shape

*Textiles* **2022**, *2*, FOR PEER REVIEW 10

2 Through the Thickness Angle Interlock

**Figure 11.** Through the thickness angle interlock .


These tests obtained by logical geometrical construction of the 2.5D interlock structure are as constrains that define whether the design (structure) from the user is valid– applicable—as shown in Figure 10. and if there is a possibility to add a shape in the puzzle as shown in Figure 12. For example, Shape 3 cannot be similar to Shape 2's incident and reflected interlock at the same

As named, the function of the tests are to check if there is a continuity of structure and if there is a possibility to add a shape in the puzzle as shown in Figure 12. For example, Shape 3 cannot be similar to Shape 20 s incident and reflected interlock at the same time. The puzzle should be symmetrical if the user is constructing a symmetrical shape. If not, the test to check symmetry can be deactivated. Shapes 2, 3, and 4 cannot be repeated vertically. Shapes 2 and 4 cannot be repeated horizontally. Shapes 2 and 4 should be separated only by Shape 3. Shape 3 cannot follow Shape 1 or 2. At least one instance of Shape 1 should separate Shapes 2 and 4 horizontally. In addition, there are other structural and geometrical conditions and boundaries related to the shape. time. The puzzle should be symmetrical if the user is constructing a symmetrical shape. If not, the test to check symmetry can be deactivated. Shapes 2, 3, and 4 cannot be repeated vertically. Shapes 2 and 4 cannot be repeated horizontally. Shapes 2 and 4 should be separated only by Shape 3. Shape 3 cannot follow Shape 1 or 2. At least one instance of Shape 1 should separate Shapes 2 and 4 horizontally. In addition, there are other structural and geometrical conditions and boundaries related to the shape.

**Figure 12.** Validation tests.

**Figure 12.** Validation tests. Note that all these conditions/boundaries are tested automatically by the JavaScript code created by the author.

Note that all these conditions/boundaries are tested automatically by the JavaScript code created by the author. Moreover, if there is any structural mistake made by the designer (user) the code will identify the mistake with the relation of the mentioned tests. In addition, the code Moreover, if there is any structural mistake made by the designer (user) the code will identify the mistake with the relation of the mentioned tests. In addition, the code will specify exactly the place of the error with the messages: "Pattern problem at column 2—Follow up problem at (2,2)—Invalid structure" for the designer have to reshape its structure to continue and the sentences "No pattern problem—No follow up problem— Valid structure", as shown in Figure 13

will specify exactly the place of the error with the messages: "Pattern problem at column 2—Follow up problem at (2,2)—Invalid structure" for the designer have to reshape its structure to continue and the sentences "No pattern problem—No follow up prob-

*Textiles* **2022**, *2*, FOR PEER REVIEW 11

lem—Valid structure", as shown in Figure 13

**Figure 13.** Validation tests. **Figure 13.** Validation tests.

#### *2.4. Analytical Modelling 2.4. Analytical Modelling*

First, the user should fill fibers and matrix parameters such as Young's and shear modulus (Eᵢ,Gᵢ) and fiber volume fraction (Vf) as shown in Figure 14 as a user interface to the analytical modelling. First, the user should fill fibers and matrix parameters such as Young's and shear modulus (E<sup>i</sup> ,G<sup>i</sup> ) and fiber volume fraction (Vf) as shown in Figure 14 as a user interface to the analytical modelling. *Textiles* **2022**, *2*, FOR PEER REVIEW 12

> Then, to calculate the characteristics of the key elements and the contribution rate, they make a macroscopic layer and subsequently the whole unit cell. The current ap-

> To calculate the macroscopic stiffness of the whole unit cell, it must be broken down or discretized to the microscale. The microscale looks at the individual elements that make a cell and then the layers that make up the macroscopic unit cell. Therefore, the unit cell undergoes the first level discretization into layers, then the second level discretization into the individual elements that make up the layer. An element can be an individual, filler, binder, or matrix region within a layer. Having determined the stiffness of each element, the stiffness of whole row can be found. Once the stiffness of all rows is known, the model formulates and calculates the stiffness of the whole unit cell and finally outputs the elastic constants. The prediction of the unit cell or macroscopic stiffness begins

with the calculation at the micro scale (the constituent elements within a cell).

sentative volume fractions of a group of elements to the layer to improve the elastic stiffness speculation compared to existing analytical modeling methods that use the 2.5D woven composite as a composite component containing layers of unidirectional elements (which are fibrous tows encased in resin). The new modelling approach creates expres-


**Figure 14.** Fiber and matrix parameters. **Figure 14.** Fiber and matrix parameters.

sions that discretize the unit cell into elements

The algorithm applied is as follows:

Then, to calculate the characteristics of the key elements and the contribution rate, they make a macroscopic layer and subsequently the whole unit cell. The current approach creates expressions at the micro level with the aim of calculating more representative volume fractions of a group of elements to the layer to improve the elastic stiffness speculation compared to existing analytical modeling methods that use the 2.5D woven composite as a composite component containing layers of unidirectional elements (which are fibrous tows encased in resin). The new modelling approach creates expressions that discretize the unit cell into elements

To calculate the macroscopic stiffness of the whole unit cell, it must be broken down or discretized to the microscale. The microscale looks at the individual elements that make a cell and then the layers that make up the macroscopic unit cell. Therefore, the unit cell undergoes the first level discretization into layers, then the second level discretization into the individual elements that make up the layer. An element can be an individual, filler, binder, or matrix region within a layer. Having determined the stiffness of each element, the stiffness of whole row can be found. Once the stiffness of all rows is known, the model formulates and calculates the stiffness of the whole unit cell and finally outputs the elastic constants. The prediction of the unit cell or macroscopic stiffness begins with the calculation at the micro scale (the constituent elements within a cell). *Textiles* **2022**, *2*, FOR PEER REVIEW 13

The algorithm applied is as follows:

The fiber matrix (S) strength can be easily calculated. In the pocket of a pure resin matrix, it is generally regarded as isotropic substances. When the elastic modulus and Poisson's ratio of resin matrix are given, it is easy to determine its basic relationship. It is easy to obtain their compliance matrix in the local material coordinate systems (1- 2- 3) to

> 

ν ν <sup>1</sup> - - 000

νν 1 - - 000

<sup>1</sup> 0 0 0 00 G

<sup>1</sup> 0 0 00 0

<sup>1</sup> ν ν - - 000

f f 12 13

f ff 1 11 f f 21 23 ff f 22 2

E EE

EE E

E EE

f f 31 32 f ff 3 33

<sup>1</sup> 0 0 0 00

f 23

> f 31

G

f 12 (1)

G

ሾSᶠሿ =

The fiber matrix (S) strength can be easily calculated. In the pocket of a pure resin matrix, it is generally regarded as isotropic substances. When the elastic modulus and Poisson's ratio of resin matrix are given, it is easy to determine its basic relationship. It is easy to obtain their compliance matrix in the local material coordinate systems (1- 2- 3) to (x- y- z) by inverting their stiffness matrix (Sij) as shown in Equations (1) and (2).

h S f i = 1 E f 1 − νf 12 E f 1 − νf 13 E f 1 0 0 0 − νf 21 E f 2 1 E f 2 − νf 23 E f 2 0 0 0 − νf 31 E f 3 − νf 32 E f 3 1 E f 3 0 0 0 0 0 0 <sup>1</sup> G f 23 0 0 0 0 0 0 <sup>1</sup> G f 31 0 0 0 0 0 0 <sup>1</sup> G f 12 (1) [S <sup>m</sup>] = 1 E <sup>m</sup> <sup>−</sup>ν<sup>m</sup> E <sup>m</sup> <sup>−</sup>ν<sup>m</sup> E <sup>m</sup> 0 0 0 −ν<sup>m</sup> E m 1 E <sup>m</sup> <sup>−</sup>ν<sup>m</sup> E <sup>m</sup> 0 0 0 −ν<sup>m</sup> E <sup>m</sup> <sup>−</sup>ν*<sup>m</sup>* E m 1 E <sup>m</sup> 0 0 0 0 0 0 <sup>1</sup> G <sup>m</sup> 0 0 0 0 0 0 <sup>1</sup> G <sup>m</sup> 0 0 0 0 0 0 <sup>1</sup> G m (2)

Each type of cell unit consists of two types of elements, namely fiber and a pure resin matrix. First, the lamina compliance matrix was calculated. Like most micro-mechanical models, the fibers and the resin matrix assume transversely isotropic, and both of them assume to be linearly elastic in the model. In order to achieve the elastic properties of the composites, the Chamis-proposed fiber-matrix ROM was selected to compute the engineering elastic constants. The Chamis micro-mechanical is a widely used and reliable model, providing an equation of all five independent structures that stretch like this [56]:

$$\mathbf{E}\_{11} = \mathbf{V}^{\mathbf{f}} \mathbf{E}^{\mathbf{f}}{}\_{11} + \mathbf{V}^{\mathbf{m}} \mathbf{E}^{\mathbf{m}} \tag{3}$$

$$\mathbf{E\_{22} = E\_{33} = E^{\rm m}/(1 - \sqrt{\mathbf{V}^{\rm f}(1 - E^{\rm m}/(E^{\rm f}\_{22}))})} \tag{4}$$

$$\mathbf{G}\_{23} = \mathbf{G}^{\mathbf{m}} / (1 - \sqrt{\mathbf{V}^{\mathbf{f}} \left(1 - \mathbf{G}^{\mathbf{m}} / (\mathbf{G}^{\mathbf{f}} \mathbf{Z} \mathbf{3})\right)}) \tag{5}$$

$$\mathbf{G\_{12} = G\_{13} = G^m} / \left(1 - \sqrt{\mathbf{V}^\mathbf{f} \left(1 - \mathbf{G^m}/\left(\mathbf{G^f\_{22}}\right)\right)}\right) \tag{6}$$

$$\mathbf{v}\_{23} = \mathbf{V}^{\mathbf{f}} \mathbf{v}\_{23}^{\mathbf{f}} + \mathbf{V}^{\mathbf{m}} \left( 2\mathbf{v}^{\mathbf{m}} - \mathbf{v}\_{12} \left( \mathbf{E}\_{22} / \mathbf{E}\_{11} \right) \right) \tag{7}$$

$$\mathbf{v}\_{12} = \mathbf{v}\_{13} = \mathbf{v}^{\mathbf{m}} + \mathbf{v}^{\mathbf{f}} \left(\mathbf{v}\_{12}^{\mathbf{f}} - \mathbf{v}^{\mathbf{m}}\right) \tag{8}$$

where V<sup>f</sup> is the fiber volume fraction, E<sup>f</sup> <sup>11</sup> is the Young's elastic modulus of the fiber in principle axis 1, E<sup>f</sup> <sup>22</sup> is the Young's elastic modulus of the fiber in principle axis 2, G<sup>f</sup> <sup>12</sup> is the longitudinal shear modulus of the fiber, G<sup>f</sup> <sup>23</sup> is the transverse shear modulus of the fiber, ν f <sup>12</sup> is the primary Poisson's ratio of the fiber, and Em, ν <sup>m</sup>, and G<sup>m</sup> represent the Young's elastic modulus, Poisson's ratio, and shear modulus of the matrix, respectively.

The compliance matrix of the fiber can be easily calculated. For the pure resin matrix pocket, it is generally regarded as isotropic material. When the elastic modulus and Poisson's ratio of resin matrix are given, it is easy to determine its basic relationship.

After that, we can obtain the stiffness in the global coordinate system by transforming the stresses and strains with the generalized transformation matrix as the next form:

$$\mathbf{I}\left[\mathbf{C}^{\mathbf{b}}\right] = \left[\mathbf{T}\right]\_{\mathbf{k}}^{\mathbf{T}} \left[\mathbf{C}\right]\_{\mathbf{k}} \left[\mathbf{T}\right]\_{\mathbf{k}} \tag{9}$$

where k is the number of unit cells in the Puzzle structure and the angle defined as the cosines of the angle between the axes of the local and global coordinate systems before and after rotation:

$$\begin{aligned} \begin{bmatrix} \mathbf{T} \end{bmatrix} = \begin{bmatrix} \mathbf{c}^2 & \mathbf{s}^2 & 2\mathbf{s}\mathbf{c}\_1^2 & 0 & 0 & 0\\ \mathbf{s}^2 & \mathbf{c}^2 & -2\mathbf{s}\mathbf{c} & 0 & 0 & 0\\ -\mathbf{s}\mathbf{c} & \mathbf{s}\mathbf{c} & \mathbf{c}^2 - \mathbf{s}^2 & 0 & 0 & 0\\ 0 & 0 & 0 & \mathbf{c}^2 - \mathbf{s}^2 & 0 & 0\\ 0 & 0 & 0 & 0 & \mathbf{c}^2 - \mathbf{s}^2 & 0\\ 0 & 0 & 0 & 0 & 0 & \mathbf{c}^2 - \mathbf{s}^2 \end{bmatrix} \end{aligned} \tag{10}$$

$$\mathbf{c} = \text{Cos}\left(\boldsymbol{\theta}\right) \tag{11}$$

$$\mathbf{s} = \text{Sim}\left(\boldsymbol{\Theta}\right) \tag{12}$$

where θ is the angle of weaving for the yarns.

A unidirectional cell falls under the orthotropic material category. As the cell is thin and does not carry any out-of-plane loads, one can assume plane stress conditions for the cell.

Therefore, assuming σ3 = 0, τ23 = 0, and τ31 = 0, calculation of element contribution is required so that the stiffness of the whole cell can be calculated as well as the overall unit cell stiffness. In our case, the contribution is related to the percentage in volume of the bounder (interlock fiber) with reference to the matrix volume (for example 30% bounder, 70% resin).

$$\text{Mij} = \text{(p)} \, \text{C}^{\text{bK}} \text{ij} + (1 - \text{p}) \, \text{C}^{\text{K}} \text{mij} \tag{13}$$

where [C<sup>b</sup> ] and [Cm] are the bounder and resin stiffness matrix in the global coordinate system, respectively, p is the percentage in volume of the bounder and resin in the composite structure, and k is the number of element in the whole structure.

Calculating the total matrix/row by summation of matrices (series summation) as shown in Equation (14):

$$\mathbf{Mt} = \frac{\sum\_{i=1}^{m} \mathbf{M}\_i}{m} \tag{14}$$

where *n* is the number of rows in the puzzle and [M<sup>i</sup> ] are the inverse of matrices found in Equation (14).

The final step is the summation of the found matrices in column and in rows as shown in Equation (15):

$$\mathbf{[S]} = \frac{\sum\_{i=1}^{n} \mathbf{M} \mathbf{t}\_i^{-1}}{n} \tag{15}$$

where *m* is the number of columns in the puzzle and [Mt-<sup>1</sup> i ] are the matrices found in Equation (15).

#### **3. Materials**

*3.1. 2D Fabric*

To ensure the effectiveness of this model based on this work. In comparison between current results modeling, experimental data, and previously developed multi-scale modelling, four different examples are generated. As studies, different mechanical features are considered and results are released. Examples were chosen from the specimens studied by J.J. Xiong and his co-workers [57] to calculate the engineering elastic constants of 2D Fabric Woven composites.

The composites studied in this section are the following:

E-glass/epoxy—I; E-glass/epoxy—II; T300/epoxy; EW220/5284.


The fabric specifications and mechanical properties of the above four kinds of textile composites are listed in Table 1

**Table 1.** Fabric specifications and properties of 2D orthogonal fiber and resin.

#### *3.2. 2.5D Interlock*

The composites are made from carbon fibers T300J and resin matrix RTM6 (Table 2). Two examples were chosen from the specimens studied by Hallal and his co-workers on 3SHM (3 stages homogenization method) [58]. The architecture of these composites involves only warp weaver yarns and weft straight yarns, as shown in Figures 10 and 11. The REV of the composite-H2 is composed of 6 warp yarns and 12 weft yarns, while the REV of the composite 71 is composed of 3 warp yarns and 24 weft yarns. The REV of composite 69 contains 6 warp yarns and 6 weft yarns. Warp yarns have a mean inclination angle equal to 29◦ for the linear longitudinal part of weft yarns. The warp yarns have a linear plus undulated longitudinal parts and a flattened elliptical cross-section. They are interlocked with weft yarns in two steps. Undulated parts have a mean inclination angle equal to 24◦ for H2 and 12◦ for 71. The weft yarns have a linear longitudinal part while its cross-section is a flattened ellipse. In addition, the fiber volume fractions in both warp and weft yarns is taken equal to 0.6. Table 2 shows the mechanical properties of carbon fibers and matrix. Carbon fibers assumed transversely isotropic material, which gives the following assumption:

**Table 2.** Mechanical properties of carbon fibers and matrix [58].


#### **4. Results and Discussion**

With the purpose of validating the developed modeling technique, the results were compared with experimental data published in the open literature. A comparison between the results of the present modeling, experimental data, and previously developed multiscale modeling was conducted. As case studies, different mechanical properties were considered, and the results are shown in Tables 3–5, each of which shows a clear comparison between the experimental results and the presented model.

**Table 3.** Comparison between experiments and predictions for tension moduli (GPa).



**Table 4.** Analytical results of the iso-strain model compared to numerical results for the composites H2 [58].

**Table 5.** Analytical results of the iso-strain model compared to numerical results for the composites 71 [58].


For example, Table 3 shows the comparison of longitudinal and Young's moduli between four different materials and fibers studied by J.J. Xiong [57]. On the other hand, Tables 4–6 show a clear comparison for the section of 2.5D interlock composite with its different types.

**Table 6.** Analytical results of the iso-strain model compared to numerical results for the composites 69 [58].


#### *4.1. 2D Fabric*

As shown in Table 3, the comparison was focused on tension moduli from the specimens studied by J.J. Xiong and his co-workers [57]. The method present accurate and precise values with the percentages of error varied from 0.44% for longitudinal Young's modulus Ex, in E-glass/epoxy—II experiment up 1.91% for EW220/5284.

#### *4.2. 5D Interlock*

The following table presents the results found by a previous analytical model from the reference validation mentioned earlier

The comparison was made to the engineering properties found using the code. A few notes should be taken into consideration: The model does not have curvature shape between Shapes 2, 3, and 4. The structure is in the coordinate system (x, y) by the reference. The dimensions of fiber are considered to be a tube shaped form and not elliptical. By comparing the result, we see that there is a validation to the code and the new model with a marginal error due to the curvature shape missing and the real/theoretical shape relation between the numerical, analytical, and geometrical model. It was observed from Table 4 that the Young's modulus in direction 2 is the most accurate parameter, up to 1% in comparison 3SHM found in Hallal [58], and that the maximum average error is shown for the Young's modulus in direction 1 with 10% of error. From Table 4, the percentage of error of transverse Young's modulus is much higher than for the longitudinal Young's modulus, with a value of 4.13% as the percentage of error of longitudinal Young's modulus was 0.614%. In addition, the percentages of error for shear modulus in Tables 4 and 5 were 1.12% and 1.21%, respectively. Finally, from Table 6, the percentage of error of transverse Young's modulus was also much higher than for the longitudinal Young's modulus, with a value of 5.07% and the percentage of error of longitudinal Young's modulus was 1%. The percentage of error for shear modulus in Table 6 was 1.31%.

In addition to the high accuracy level of the current modelling technique, the very short runtime required of the modelling renders the developed multi-scale modelling as a cost-effective computational tool for estimating the Young's moduli of laminate composites. The results showed that the present model could be used to effectively evaluate the elastic properties for laminate composites.

#### **5. Conclusions**

A geometrical-modelling tool has been presented to predict the elastic stiffness characteristics of 2D fabric and 2.5D woven interlock composites with the ability to assess change in performance as a consequence of altering weaving parameters. The models presented in this paper are able to reproduce the behavior of woven composites formed by different fiberreinforced types. This approach has been validated against experimental data produced independently of this work for orthogonal interlock weaves and compared to existing modelling approaches. The present model performs better in all predictions compared to the existing modelling efforts. Good agreement was observed among numerical results, with small differences in the longitudinal strains, as shown in the results section that the shear modulus was the most accurate parameter up to 1% in the 2.5D interlock section, while the error in 2D fabric section was lower with values such as 0.44% in tension moduli. As for the longitudinal Young's modulus, the maximum average error is shown with 10% but with an average percentage of error of transversal strains of 4%. This may be due to a lack of precision in the mechanical properties of the components, i.e., the fibers, which are used as input data. The percentage is related to the lack of an accurate and precise database structure of the used material, but it shows a great ability to use the algorithm.

The suggested hypotheses are the simplest that can be applied for determining the engineering data with all required information, and specifications from the user are presented in a "user friendly" form that can make studying and searching a new level of excitement and interactivity. As shown in the methodology section, applying the algorithm is easy and simple. All this research can be accessed easily with all its codes and models, along with its regulations and explanations needed. Working on these codes will eliminate the hard work caused by the computational efforts using software such as Abaqus or Ansys, etc.

At any given time, the user can calculate required forces to have a known deformation or to predict the deformation for a known force; in addition, they would have easy access to all engineering properties such as stiffness or compliance matrices. A future scope is now set in each section to have more research and data analysis by using these codes and models without any super computers and losing time, money, and effort. Any researcher can now find the optimum set of materials for all levels of woven composites materials (2D Plain weave, Twill, Satin, or Basket), along with altering materials for 2.5D interlock composites such as layer to layer or through the thickness angle interlock. The model works best on other types of combinations, and further analysis and evaluation of the models is required. The introduced framework may apply to other fiber conditions as part of a fiber volume, but these conditions will require further development.

**Author Contributions:** R.Y.: Methodology and Supervision, P.L.: Data and spelling, M.A.K.: Writing, calculating, programming and analyzing. All 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.
