Multi-Objective Optimization of E-Glass/S-Glass and HS-Carbon Hybrid Laminates with Holes for Improved Torsional Performance

2.1 Theoretical

2.1.1 Mathematical formulas

The research works [2, 39, 40] demonstrate the mathematical relationship between Hooke's law, which describes the elastic stress-strain relationship, and the linear proportional behavior of stress and strain in elastic materials.

$\sigma_{i i}=C_{i j k l} \varepsilon_{k l}$    (1)

where, $C_{i j k l}$ is a fourth-order tensor with 81 elastic constants and is the inverted form of Hooke's law for orthotropic materials, as detailed in reference [40].

$\varepsilon_i=S_{i j} \sigma_j$    (2)

The coefficients $S_{i j}$ are called compliance coefficients, which are the inverse of the stiffness matrix [40].

$S_{i i}=C_{i j}^{-1}$    (3)

The constitutive Eq. (2) for an orthotropic material can be translated into matrix form [40].

$\left[\begin{array}{l}\varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12}\end{array}\right]=\left[\begin{array}{cccccc}S_{11} & S_{12} & S_{13} & 0 & 0 & 0 \\ S_{21} & S_{22} & S_{23} & 0 & 0 & 0 \\ S_{31} & S_{32} & S_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{66}\end{array}\right]\left[\begin{array}{l}\sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \tau_{23} \\ \tau_{13} \\ \tau_{12}\end{array}\right]$    (4)

where, the engineering constants' individual compliance coefficients are [40]:

$\begin{gathered}S_{11}=1 / E_1, \quad S_{12}=-v_{12} / E_2, \quad S_{13}=-v_{31} / E_3, \\ S_{21}=-v_{12} / E_1, S_{22}=1 / E_2, S_{32}=-v_{32} / E_3, \\ S_{31}=-v_{13} / E_1, \quad S_{32}=-v_{23} / E_2, \quad S_{33}=1 / E_3 \\ S_{44}=1 / G_{23}, S_{55}=1 / G_{13}, S_{66}=1 / G_{12} \\ S_{i i}=S_{i i}\end{gathered}$   (5)

$E_1, E_2$, and $E_3$ are the modulus of elasticity in the (1-2-3) coordinate system. G12, G13, and G23 are the shear modulus. The Poisson's ratios are given by $v_{i j}$ [40].

When the principal material coordinate system is rotated through an angle $\boldsymbol{\theta}$ about the common $z$-axis (3), but does not coincide with the global $x-y-z$ coordinate system, the transformed compliance matrix can be defined as [40].

$[\bar{S}]=\left[T_2\right]^{-1}[S]\left[T_1\right]$    (6)

$[\bar{S}]=\left[\begin{array}{cccccc}\bar{S}_{11} & \bar{S}_{12} & \bar{S}_{13} & 0 & 0 & \bar{S}_{16} \\ \bar{S}_{21} & \bar{S}_{22} & \bar{S}_{23} & 0 & 0 & \bar{S}_{26} \\ \bar{S}_{31} & \bar{S}_{32} & \bar{S}_{33} & 0 & 0 & \bar{S}_{36} \\ 0 & 0 & 0 & \bar{S}_{44} & \bar{S}_{45} & 0 \\ 0 & 0 & 0 & \bar{S}_{54} & \bar{S}_{55} & 0 \\ \bar{S}_{61} & \bar{S}_{62} & \bar{S}_{63} & 0 & 0 & \bar{S}_{66}\end{array}\right]$    (7)

where, the transformation matrices are $T_1$ and $T_2$.

$T_1=\left[\begin{array}{cccccc}\mathrm{c}^2 & \mathrm{~s}^2 & 0 & 0 & 0 & 2 c s \\ \mathrm{~s}^2 & \mathrm{c}^2 & 0 & 0 & 0 & -2 c s \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & c & -s & 0 \\ 0 & 0 & 0 & s & c & 0 \\ -c s & c s & 0 & 0 & 0 & \mathrm{c}^2-\mathrm{s}^2\end{array}\right]$    (8)

$T_2=\left[\begin{array}{cccccc}c^2 & s^2 & 0 & 0 & 0 & c s \\ s^2 & c^2 & 0 & 0 & 0 & -c s \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & c & -s & 0 \\ 0 & 0 & 0 & s & c & 0 \\ -2 c s & 2 c s & 0 & 0 & 0 & c^2-s^2\end{array}\right]$    (9)

where, $S$ is the compliance matrix of the lamina, $c=\cos (\theta)$ and $s=\sin (\theta)$. The three-dimensional version of the composite's compliance matrix is substituted into the Classical Laminated Theory (CLT) equations [3].

The fundamental equation for the load-deformation relationship of laminate analysis is presented in Eq. (10) as follows [41]:

$\left\{\begin{array}{l}N \\ M\end{array}\right\}=\left[\begin{array}{ll}A & B \\ B & D\end{array}\right]\left\{\begin{array}{c}\varepsilon^0 \\ k\end{array}\right\}$    (10)

$\{\mathrm{N}\},\{\mathrm{M}\},\left\{\varepsilon^0\right\}$, and $\{\mathrm{k}\}$ represent the vectors of resultant forces, moments, midplane strains, and curvatures, respectively. The matrices $[\mathrm{A}],[\mathrm{B}]$, and $[\mathrm{D}]$ correspond to extensional stiffness, coupling stiffness, and bending stiffness, respectively. For symmetrical laminates, the coupling matrix $[\mathrm{B}]$, which connects extension and bending, is zero. The loaddeformation relation in classical lamination theory (CLT) is presented in Eq. (10), according to references [42, 43]. Versions of the matrices [A], [B], and [D] can be written as follows:

$A_{i j}=\sum_{\mathrm{k}=1}^{\mathrm{n}}[\overline{\mathrm{Q}}]_{\mathrm{k}}\left(h_k-h_{k-1}\right)$   (11)

$B_{i j}=\frac{1}{2} \sum_{\mathrm{k}=1}^{\mathrm{n}}[\overline{\mathrm{Q}}]_{\mathrm{k}}\left(h_k^2-h_{k-1}^2\right)$    (12)

$D_{i j}=\frac{1}{3} \sum_{\mathrm{k}=1}^{\mathrm{n}}[\overline{\mathrm{Q}}]_{\mathrm{k}}\left(h_k^3-h_{k-1}^3\right)$    (13)

$h=\sum_{\mathrm{n}=1}^{\mathrm{k}} t_k$   (14)

$[\bar{Q}]=[\bar{s}]^{-1}$    (15)

where, $n$ is the total number of layers, $t_k$ is the thickness of each layer, and $[\overline{\mathrm{Q}}]_{\mathrm{k}}$ is the transformed reduced stiffness matrix [42]. The plane stress constitutive equation allows for calculating stresses at any z-position within the $\mathrm{k}^{\text {th }}$ layer [40].

$\{\sigma\}_x=[\overline{\mathrm{Q}}]_{\mathrm{k}}\{\varepsilon\}_x$    (16)

Converting stresses from principal coordinates ($x$ and $y$) to local coordinates is necessary for calculating von Mises stresses and the Tsai-Wu failure criterion [44]:

$\{\sigma\}_{12}=\left[T_2\right]\{\sigma\}_x$    (17)

Eq. (18) explains the conversion of strains from principal coordinates ($x$ and $y$) to local coordinates.

$\{\varepsilon\}_{12}=\left[T_1\right]\{\varepsilon\}_x$  (18)

2.1.2 Longitudinal Young's modulus ($E_x$)

To calculate the effective Young's modulus $E_x$ of a laminate, we must solve the laminate Eq. (10) by reversing it [41].

$\left\{\begin{array}{l}\varepsilon^0 \\ k\end{array}\right\}=\left[\begin{array}{ll}A & B \\ B & D\end{array}\right]^{-1}\left\{\begin{array}{l}N \\ M\end{array}\right\}$   (19)

Since this inversion can be complex, it is simpler to invert the three tensors $\mathrm{A}, \mathrm{B}$, and D from Eq. (19) one by one. This allows us to rewrite Eq. (19) as follows [41]:

$\left\{\begin{array}{c}\varepsilon^0 \\ k\end{array}\right\}=\left[\begin{array}{cc}a & b \\ b^T & d\end{array}\right]^{-1}\left\{\begin{array}{l}N \\ M\end{array}\right\}$     (20)

with

$\begin{aligned} a & =\left(A-B D^{-1} B\right)^{-1} \\ d & =\left(D-B A^{-1} B\right)^{-1} \\ b=-a B D^{-1} & =-\left(A-B D^{-1} B\right)^{-1} B D^{-1}\end{aligned}$      (21)

The following can be defined to determine the laminate's properties [41]:

${{a}^{\divideontimes }}=ha$   (22)

Thus, the effective Young's modulus of the laminate in the $x$ direction is [41]:

${{E}_{x}}=\text{ }\!\!~\!\!\text{ }{}^{1}/{}_{a_{11}^{\divideontimes }}$    (23)

2.1.3 Shear modulus $G_{x y}$

The in-plane shear modulus $G_{x y}$ of the laminate can be determined from the ${{A}_{66}}$ term of the extensional stiffness matrix [45]:

${{G}_{xy}}={}^{{{A}_{66}}}/{}_{h}$   (24)

where, $h$ is the total thickness of the laminate and ${{A}_{66}}$ is calculated as:

$A_{66}=\sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\bar{Q}_{66}\right]_{\mathrm{k}}\left(h_k-h_{k-1}\right)$     (25)

where, ${{\left[ {{{\bar{Q}}}_{66}} \right]}_{\text{k}}}$ is the transformed shear stiffness of the $k^{\text {th }}$ ply.

This study uses a genetic algorithm to optimize composite laminates based on CLT. The effective modulus of elasticity ($E_x$ and $G_{x y}$) obtained for each optimized configuration can be compared to the modulus calculated by FEM.

2.1.4 Shear stress and stress concentration factor

For this study, we focused on calculating two types of stress: shear stress and von Mises stress. The equation below defines the relationship for determining the shear stress near the holes in a composite laminate [46]:

$\tau_{x y, \max , \text { with holes }}=K_{x y} \times \tau_{x y, \max , \text { No hole }}$    (26)

The shear stress $\tau_{x y, \max , \text { No hole}}$ is calculated using Eq. (16), where $K_{x y}$ is the shear stress concentration factor for the laminate, and is defined as follows [47]:

$K_{x y}=K_t^{\infty}\left(1+\frac{d}{w}\right)$   (27)

where,$\text{ }\!\!~\!\!\text{ }K_{t}^{\infty }$ is the stress concentration factor for an infinite laminated plate containing a circular hole [48].

$K_t^{\infty}=1+\sqrt{2 / A_{11}\left\{\sqrt{A_{11} A_{22}}-A_{12}+\frac{A_{11} A_{22}-A_{12}^2}{2 A_{66}}\right\}}$    (28)

2.1.5 The von Mises stress

The von Mises stress is determined by the following formula [49]:

$\sigma_{\text {VonMises }}=\sqrt{\begin{array}{c}1 / 2\left(\left(\sigma_x-\sigma_y\right)^2+\left(\sigma_y-\sigma_z\right)^2+\left(\sigma_z-\sigma_x\right)^2\right) \\ +3\left(\tau_{x y}^2+\tau_{y z}^2+\tau_{x z}^2\right)\end{array}}$    (29)

where, $\sigma_x, \sigma_y$ and $\sigma_z$ show the normal stresses and $\tau_{x y}, \tau_{y z}$ and $\tau_{x z}$ are the shear stresses in the respective directions.

2.1.6 Tsai-Wu failure criterion

This study used the Tsai-Wu failure criterion to evaluate the progressive degradation of each layer of a hybrid composite laminate. The failure index (FI) for each composite layer was calculated using Eq. (30) [43, 50].

$\begin{gathered}F I=F_1 \sigma_1+F_2 \sigma_{21}+F_3 \sigma_3+F_{11} \sigma_1^2+F_{22} \sigma_2^2+ \\ F_{33} \sigma_3^2+F_{44} \sigma_4^2+F_{55} \sigma_5^2+F_{66} \sigma_6^2+2 F_{12} \sigma_1 \sigma_2+ \\ 2 F_{13} \sigma_1 \sigma_3+2 F_{23} \sigma_2 \sigma_3\end{gathered}$    (30)

$\begin{aligned} & F_1=1 / X t+1 / X c ; \quad F_2=1 / Y t+1 / Y c \\ & F_3=1 / Z t+1 / Z c ; \quad F_{11}=-1 / X t \times X c \\ & F_{22}=-1 / Y t \times Y c ; \quad F_{33}=-1 / Z t \times Z c \\ & F_{44}=1 / S_{23}^2 ; \quad F_{55}=1 / S_{13}^2 \\ & F_{66}=1 / S_{12}^2 ; \quad F_{12}=-\frac{1}{2} \sqrt{F_{11} \times F_{22}} ; \\ & F_{23}=-\frac{1}{2} \sqrt{F_{22} \times F_{33}} ; \quad F_{13}=-\frac{1}{2} \sqrt{F_{11} \times F_{33}}\end{aligned}$    (31)

In Eq. (31), $\mathrm{X}, \mathrm{Y}$, and Z represent the material strengths in the $\mathrm{X}, \mathrm{Y}$, and Z directions, respectively. The subscripts $t$ and $c$ denote tensile and compressive strengths, respectively. S denotes the shear strength of the composite laminate. Tensile, compressive, and shear strength values for SG-EP, EG-EP, and CF-EP composites are widely available in the literature, as indicated in references [51-53]. For this study, we used the values in the ANSYS Workbench 16.2 library, as shown in Table 1. The characteristics of the materials used also originate from this library. Negative values indicate compressive strength.

2.2 Problem statement

This study presents 3D numerical analyses of a hybrid composite laminate containing holes aligned along the $x$-axis. The laminate is composed of E-glass/epoxy (EG-EP), S-glass/epoxy (SG-EP), and HS-carbon/epoxy (CF-EP) composite layers. The analyses were performed using ANSYS Workbench with the Anisotropic Composite Preprocessor (ACP) module. The objective of this study is to optimize the laminate to maximize the effective elastic moduli, specifically the longitudinal Young's modulus $E_x$ and the shear modulus $G_{x y}$, while minimizing the von Mises stress and the shear stress near the holes and respecting the Tsai-Wu failure criterion ($\leq 0.6$). The optimization was carried out using an MOGA implemented in MATLAB. A torsional moment was applied to two opposite sides of the hybrid laminated composite plate around the $x$-axis.

The hybrid composite laminate consisted of six to twelve layers, with a total thickness ranging from two to four millimeters. The thickness of each layer was adjusted to meet this constraint. The percent perforated area ranged from $5 \%$ to $25 \%$, and the hole diameters ranged from 10 mm to 25 mm. Fiber orientations varied between $-90^{\circ}$ and $90^{\circ}$. A design constraint required that $25 \%$ of the laminate consist of HS-carbon/epoxy (CF-EP) and that all three materials (CF-EP, EG-EP, and SG-EP) be present. The Tsai-Wu failure index was $\mathrm{FI} \leq 0.6$, and the center distance between the holes was 45 mm. The materials used were EG-EP, SGEP, and CF-EP. The mechanical properties of the composites are detailed in Table 1.

Table 1. Material properties of reinforced epoxy laminates

2.3 Multi-objective genetic algorithm 

2.3.1 Genetic algorithm

MOGA is a computer method that solves problems with multiple goals [55]. Instead of finding one perfect solution, MOGA looks for a set of good solutions, known as the Pareto front (see Section 2.3.2). MOGA is useful in engineering and design for tasks such as optimizing structures, managing resources, and modeling complex systems. It helps decisionmakers select the optimal solution for their needs. A multi-criteria optimization problem can be described as follows [56]:

Maximize/minimize

$f_i(x)_i, i=1, \ldots, N$     (32)

Subject to constraints

$g_j(x)=0, \quad j=1, \ldots, M$    (33)

$h_k(x) \leq 0 k=1, \ldots, W$    (34)

where, $f_i, g_j$ and $h_k$ represent the objective functions, equality constraints, and inequality constraints, respectively. N denotes the number of objective functions, and $x$ is an n-dimensional vector.

A genetic algorithm (GA) is a computer method that mimics natural selection and genetics to find solutions. The main components are genes and chromosomes. A chromosome is a sequence of genes. When solving a real-world problem, the genes represent the aspects to be improved upon, while the chromosomes represent possible solutions. GAs search for the best solution by testing many chromosomes. GAs have two advantages: they can test multiple solutions at once, and they can temporarily accept suboptimal solutions, which prevents them from getting stuck on an average solution [57].

GAs are great for complex problems because they can easily convert variables into genetic sequences that form chromosomes. They explore different paths by maintaining a group of potential solutions. Through operations like crossover and mutation, GAs offer alternative solutions, even if the best solution isn't perfect for reasons not previously considered. This makes GAs useful for designing and optimizing composite structures with layers of materials that have complex properties [43, 58].

A genetic algorithm for multiple goals is called MOGA. MOGA is an improved version of NSGA-II, an algorithm that uses elitism to select optimal solutions. Here's how it works: First, a random group of solutions is created. Then, each solution is evaluated, the best solution is selected, and the solutions are mixed (crossover) and changed (mutation). Elitism helps ensure that the best solutions are used to create new ones. If the results aren't good enough, the process starts over. Ultimately, it provides a set of good solutions known as the Pareto front [59-61].

2.3.2 Pareto approach

Named after Vilfredo Pareto, the Pareto methodology identifies optimal solutions for multiobjective optimization by highlighting a Pareto front: a set of non-dominated solutions where improving one objective compromises another. This approach is useful in science and engineering for balancing conflicting objectives, such as cost/performance or efficiency/sustainability. Visualizations of the Pareto front, as illustrated in Deb's work [55], make it easier to understand the trade-offs involved in complex decision-making [62].

2.3.3 Mathematical formulation of the optimization problem

Objective function:

- Maximize the effective elastic modulus (stiffness).

- Minimize von Mises stress.

- Minimize shear stress.

Design's variables:

- Ply angles: $\left(-90^{\circ} \leq \theta \leq 90^{\circ}\right)$.

- Ply materials: HS-Carbon/Epoxy (CF-EP), E-Glass/Epoxy (EG-EP), S-Glass/Epoxy (SG-EP).

- Number of plies: Between 6 and 12.

- Thickness ranging from 2 to 4 mm.

- Number of holes.

- The percent perforated area (between $5 \%$ and $25 \%$).

- Hole diameters (between 10 mm and 25 mm).

Constraints:

- Tsai-Wu failure index: FI ≤ 0.6.

- 25% of the plies must be HS-Carbon/Epoxy (CF-EP).

- All three materials (CF-EP, EG-EP and SG-EP) must be present in the laminate.

- Center distance between the holes is 45 mm.

The geometric properties of the composite plate have been defined, including a length of 150 mm and a width of 50 mm. Three torsional tests with moments of 1000 N·mm, 2000 N·mm, and 4000 N·mm were performed on two opposite surfaces of the plate in the yz-plane around the $x$-axis.

The first step was configuring the parameters of the MOGA algorithm. An initial population of 100 individuals with a maximum of 100 generations was defined. A larger configuration (200 individuals and 200 generations) was subsequently evaluated, revealing a performance deviation ranging from 1.5% to 2%. Given this moderate discrepancy, the reduced parameters (100 individuals and 100 generations) were retained to limit computational time, which remains particularly high due to the large number of design variables being optimized simultaneously. Increasing the population size or the number of generations would have resulted in prohibitively long computation times without proportional improvement in the quality of the Pareto front. A Pareto fraction was set to retain optimal solutions that demonstrate trade-offs between objectives. To promote exploration of new solutions, a crossover rate of 95% was chosen, and an adaptive mutation function was used to maintain solution diversity. The program displays progress at each iteration, showing the best objective values.

The optimization produced a Pareto front that highlighted the trade-offs between the effective elastic moduli ($E_x$ and $G_{x y}$) and the maximum von Mises stress and shear stress. An optimal solution was extracted that describes the laminate configuration, including the number of plies, laminate thickness, fiber orientation angles for each ply, number of holes, percent perforated area, hole diameters, stress concentration in each hole, and materials. For this solution, stresses (von Mises and shear) under torsion moments, as well as the maximum TsaiWu failure index, were calculated and displayed. The results also include the effective elastic moduli ($E_x$ and $G_{\mathrm{xy}}$), the shear stress in each hole, and the maximum von Mises stress of the optimized hybrid laminated composite.

2.4 Finite element analysis

Finite element analysis (FEA) enables the modeling of composite materials with various simulation tools. ANSYS offers several options for modeling composite materials, including different simulation capabilities. ANSYS ACP is a module that simplifies the definition of required thicknesses and stacking sequences for composite laminates. It includes a layer thickness editor, a stacking sequence editor, a layer arrangement modeling tool, and failure criteria analysis for each layer. Part 2 in Figure 1 illustrates the process for finite element simulation of hybrid composite laminates.

Figure 1. Processing steps of the multi-objective genetic algorithm (MOGA) and finite element analysis (FEA)

This study involved modeling composite layers made of EG-EP, SG-EP, and CF-EP using ANSYS ACP. This tool allowed us to design composite layers by specifying the thickness of the plies and the orientation of the fibers according to the desired specifications. Next, the hybrid composite material was laminated.

In ANSYS ACP, composite material modeling is performed layer by layer. The process begins in the ACP (Pre) module with creating a sketch of the geometric shape. Then, the structure is refined using an appropriate mesh. Fiber orientations and the stacking sequence are defined using the "stack up" command. Each layer is specified with a distinct material, such as E-glass/epoxy (EG-EP), S-glass/epoxy (SG-EP), or HS-carbon/epoxy (CF-EP). Their mechanical properties, including elastic modulus, Poisson's ratio, and tensile strength, are listed in Table 1. After establishing the geometric model and layer properties, finite element simulations were conducted in ANSYS's "Static Structural" module. In this module, boundary conditions and loads were applied. After execution, the simulation results are displayed, including von Mises stresses, shear stresses, and other results. The Tsai-Wu failure index for the laminate and/or each ply, as well as the effective elastic modulus, is displayed in the ACP (Post) module.

Figure 2 shows the mesh structure and fiber orientations generated by the ACP module. This visualization shows the finite element discretization of the composite model and the precise definition of the ply orientations. These are essential for accurately analyzing the material's anisotropic properties.

Figure 2. (a) Mesh configuration, (b) fiber orientations in the Anisotropic Composite Preprocessor (ACP) module for the hybrid composite plate

2.5 Mesh independence

Under a torsional moment, a refined mesh with an element size of 1 mm (corresponding to 67470 elements) stabilizes the von Mises stresses at a value of 20.12 MPa, as illustrated in Figure 3. This element size was chosen because it marks the point at which convergence is achieved, thereby minimizing discretization errors and optimizing computational time for maximum efficiency.

To reproduce the torsion test conditions recommended in references [63, 64], two opposing torsional moments were applied to the end faces of the model around the longitudinal $x$-axis. The longitudinal edges were left free to allow warping of the section, as shown in Figure 4.

Figure 3. Convergence of von Mises stress as a function of the number of elements

Figure 4. Application of torsional moments on two opposite sides of the laminated composite plate around the x-axis

The presence of circular holes in composite laminates, as defined by their geometric parameters (number, diameter, and spacing), significantly alters the mechanical behavior and properties of these materials. This alteration primarily results from the stress concentration phenomenon that occurs near the holes, as demonstrated by Guo et al. [17] and Özaslan et al. [35]. The ultimate strength of such plates is influenced by numerous parameters that have been extensively studied, including loading type, stacking sequence, number of plies, and hole diameter. Configurations dominated by 0°-oriented plies are distinguished by the highest stress concentration factors near the holes [38]. Furthermore, increasing hole size leads to a significant reduction in failure load, as shown by experimental compression tests. Tensile test results also confirm that the structural strength of composite laminates decreases as the number of holes increases [65]. These observations are consistent with the challenges identified in the present study.

According to the results of the MOGA optimization, a clear trend emerges in the stacking sequences optimized for increasing torsional moments (from 1000 N·mm to 4000 N·mm). The number of plies in SG-EP (S-glass fiber impregnated with epoxy resin) decreases progressively (from six to four plies), while the number of plies in EG-EP (E-glass fiber) increases (from one to four plies). The number of plies in CF-EP (carbon fiber) remains fixed at three. This is in response to the constraint imposed to prevent the carbon fiber content from exceeding 25% of the total number of plies. This redistribution is due to the superior properties of S-glass fiber, particularly its significantly higher modulus of elasticity. This gives S-glass fiber a better capacity to withstand high local shear and tensile stresses around holes under moderate loads. However, as the torsional moment increases, E-glass fiber is favored in greater numbers to ensure more homogeneous load distribution and avoid excessive overall stiffness, which could lead to localized failure.

Figure 8. The Von Mises stress, Shear stress and Tsai-Wu failure index of conventional [±45]₁₁ E-glass/epoxy laminate subjected to torsional moment of 4000 N·mm. (a) Von Mises stress, (b) Shear stress, (c) Tsai-Wu failure index

The sequences optimized by MOGA significantly outperform the pure EG-EP [±45] laminate, which is considered the optimal baseline configuration for pure shear in torsion. Finite element simulations were performed on a pure EG-EP [±45] laminate with fair diameter holes under a torsional loading of 4000 N·mm. The laminate had the same dimensions and number of plies as the optimized sequence for this load level. The pure EG-EP [±45] laminate exhibits inferior performance, including a limited increase in longitudinal stiffness and higher von Mises and shear stress peaks, as well as a larger Tsai-Wu failure index. Figure 8 illustrates the finite element simulation results obtained for the conventional [±45]₁₁ laminate. The optimized sequences strategically place very high-modulus CF-EP plies at intermediate or high angles (for example, close to ±77° to ±81° in some cases, or ±43° to ±87° in others), while distributing the more compliant SG-EP and EG-EP plies (offering better shear compliance and higher strain-to-failure) in critical zones to dampen stress gradients. The optimized ply angles (±70° to ±81° at lower torsion moments, shifting closer to ±45° at higher moments) position the fibers in a configuration that minimizes direct loading in the regions of highest shear flow perturbation around the hole. This significantly reduces the local shear stress concentration factor in these critical zones. At the same time, these angles remain sufficiently distant from 0°/90° to maintain high in-plane shear stiffness, thereby enabling efficient uptake and more diffuse redistribution of the torsional shear flow across a broader area of the laminate. The alternation between high-modulus CF-EP plies (at intermediate/high angles) and more compliant SG-EP/EG-EP plies further smooths local shear strain gradients and mitigates sharp shear stress peaks. This non-classical angular dispersion accounts for the observed improvements over the pure [±45] EG-EP baseline: 64% increase in the longitudinal Young's modulus E_x, 17% reduction in the von Mises equivalent stress, 29% reduction in the maximum shear stress τ_xy, and 12% decrease in the Tsai-Wu failure index. This hybridization of materials, combined with an unconventional angular dispersion, generates several highly favorable synergistic effects under torsional loading. Indeed, fiber angles near ±45° to ±80° allow for a particularly efficient uptake of torsional shear stress due to the directional stiffness of the fibers, while avoiding direct alignment with the high-stress tensile zones generated around the holes. This configuration leads to a significant reduction in shear stress τ_xy peaks around the holes, promoting a much more uniform distribution of shear flow across the entire width of the laminate. Furthermore, the alternation between plies with very high longitudinal stiffness (CF-EP) and the more compliant plies (SG-EP and EG-EP) creates a progressive stiffness transition around the holes, which substantially reduces the effective stress concentration factor compared to single-material or non-hybrid laminates. The literature on notched hybrid composites often confirms that this type of hybridization delays the onset of local damage by smoothing out stress discontinuities between plies. Finally, the strategic placement of SG-EP plies with high shear strength in the outer layers or within critical thickness zones strongly minimizes matrix-dominated shear failure modes around the holes. The extremely low Tsai-Wu indices systematically achieved (always below 0.6 and validated by finite element analysis) thus demonstrate a particularly balanced and efficient utilization of the directional strengths of the different materials, without excessive stress localization.

The present work, therefore, aims to optimize hybrid composite laminates containing holes and subjected to torsional loading, using the Tsai-Wu failure criterion as the primary constraint to prevent premature laminate failure. Our analyses highlight the significant influence of fiber orientation, material stacking sequence (CF-EP, EG-EP, SG-EP), as well as hole diameter and number, on the mechanical behavior of perforated hybrid laminates under torsional moments. For example, the hybrid laminate with the stacking sequence [-77_SG-EP/79_EG-EP/77_CF-EP/-81_SG-EP/53_SG-EP/-51_SG-EP/24_SG-EP/0_CF-EP/68_CF-EP/47_SG-EP] exhibits distinct behavior under a torsional moment of 1000 N·mm (Figure 5), characterized by high strength, reduced stresses, and a very low Tsai-Wu failure index. This result, obtained via MOGA and compared to finite element results, shows excellent agreement, with percentage errors of 10.85% for the von Mises stress, 6.74% for the shear modulus G_xy, 4.45% for the longitudinal Young's modulus E_x, 21.80% for the Tsai-Wu index, and (11.8%, 16.27%, 9.21%) for the maximum shear stresses τ_xy around the holes at coordinates (-45 mm, 0 mm, 45 mm), respectively. Detailed comparisons for the entire study are presented in Figures 9 and 10. Similar trends are observed for higher torsional moments (2000 N·mm and 4000 N·mm) shows in Figures 6 and 7, respectively, with the corresponding optimized sequences generally yielding errors below 10 to 15% for key quantities, confirming the robustness of the approach.

Figure 9. Comparison between FEM and MOGA results for torsional moments. (a) Von Mises stress, (b) Shear moduli G_xy, (c) Longitudinal Young’s Moduli E_x

Figure 10. Comparison between FEM and MOGA results for torsional moments. (a) Tsai-Wu index, (b) Shear stress near the holes for torsion moment of 1000 N·mm, (c) Shear stress near the holes for torsion moment of 2000 N·mm

The FEM and MOGA methods are useful for studying hybrid composite laminates with holes. These methods provide similar results with minimal error for the Tsai-Wu index: 0.38% error in the torsional moment of 2000 N·mm and 1.60% and 2.04% error in the elastic modulus under the same loading. This proves that they accurately predict stress. Simulations performed using ANSYS Workbench and a genetic algorithm implemented in MATLAB enable robust analysis under various loadings and provide a solid foundation for optimization. These tools help us better understand laminate behavior, particularly as it relates to fiber orientations, stacking sequences, and the presence of holes. The diameter and number of holes significantly affect stress distribution and structural integrity; larger or more numerous holes increase stress concentrations and potentially elevate the risk of failure under torsional loads.