# Machine Learning Predictions for the Comparative Mechanical Analysis of Composite Laminates with Various Fibers

Baha Eddine Ben Brayek, Sirine Sayed, Viorel Mînzu, Mostapha Tarfaoui

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Baha Eddine Ben Brayek, Sirine Sayed, Viorel Mînzu, Mostapha Tarfaoui. Machine Learning Predictions for the Comparative Mechanical Analysis of Composite Laminates with Various Fibers. Processes, 2025, 13 (3), pp.602. 10.3390/pr13030602. hal-05357691

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Article

# Machine Learning Predictions for the Comparative Mechanical Analysis of Composite Laminates with Various Fibers

Baha Eddine Ben Brayek <fnref n="1" />, Sirine Sayed <fnref n="2" />,<fnref n="3" />, Viorel Mînzu <fnref n="4" />,* and Mostapha Tarfaoui <fnref n="1" />,<fnref n="5" />

**Abstract:** This article addresses the complex behavior of composite laminates under varied layer orientations during tensile tests, focusing on carbon fiber and epoxy matrix composites. Data characterizing the mechanical load behavior are obtained using twelve composite laminates with different layer orientations and the DIGIMAT-VA software (version 2023.3). First, these data were used to elaborate a complex comparative analysis of composite laminates from the perspective of materials science. Composite laminates belong to three classes: unidirectional, off-axis oriented, and symmetrically balanced laminates, each having a specific behavior. From the perspective of designing a new material, a prediction model that is faster than the finite element analysis is needed to apply this comparative analysis's conclusions. As a novelty, this paper introduces several machine learning prediction models for composite laminates with 16 layers arranged in different orientations. The Regression Neural Network model performs best, effectively replacing expensive tensile test simulations and ensuring good statistics (RMSE = 34.385, R<fnref n="2" /> = 1, MAE = 19.829). The simulation time decreases from 34.5 s (in the case of finite element) to 0.6 s. The prediction model returns the stress-strain characteristic of the elastic zone given the new layer orientations. These models were implemented in the MATLAB system 2024, and their running proved good models' generalization power and accuracy. Even specimens with randomly oriented layers were successfully tested.

**Keywords:** composite materials; machine learning; regression neuronal network

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Academic Editors: Sheng Du, Li Jin,
Pan Yu and Haipeng Fan

Received: 25 January 2025

Revised: 17 February 2025

Accepted: 18 February 2025

Published: 20 February 2025

**Citation:** Brayek, B.E.B.; Sayed, S.; Mînzu, V.; Tarfaoui, M. Machine Learning Predictions for the Comparative Mechanical Analysis of Composite Laminates with Various Fibers. *Processes* **2025**, *13*, 602. https://doi.org/10.3390/pr13030602

**Copyright:** © 2025 by the authors.
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## 1. Introduction

Composite materials stand at the intersection of innovation and practical engineering, blending diverse constituents to unlock superior properties and address the limitations of conventional materials. As Nachtane and Tarfaoui [1,2] insightfully observe, these materials ingeniously merge fibers with matrices, crafting systems surpassing traditional materials in mechanical robustness and offering significant weight reduction and durability advantages, fulfilling critical roles in demanding applications. El Moumen et al. [3] elaborate that this precise integration allows for the strategic engineering of fiber arrangements within the matrix, enhancing the adaptability of composites to meet exacting performance standards across various industrial uses. The versatility of composite materials is particularly evident in sectors requiring high-performance materials. Daly et al. [4] note that composites

<footnote n="1">¹ Mohammed VI Polytechnic University, GEP/GSMI, Ben Guerir 43150, Morocco; bahaeddine.benbrayek@um6p.ma (B.E.B.B.); mostapha.tarfaoui@um6p.ma (M.T.)</footnote>

<footnote n="2">² ENSTA Bretagne, IRDL-UMR CNRS 6027, F-29200 Brest, France; sirine.sayed@ensta.fr</footnote>

<footnote n="3">³ International University of Rabat, LERMA, Rabat 11103, Morocco</footnote>

<footnote n="4">⁴ Control and Electrical Engineering Department, “Dunarea de Jos” University of Galati, 800008 Galati, Romania</footnote>

<footnote n="5">⁵ Mohammed VI Polytechnic University, Green Energy Park, Ben Guerir 43150, Morocco</footnote>

<footnote>* Correspondence: viorel.minzu@ugal.ro</footnote>
are ideally suited for critical aerospace, automotive, and military applications due to their ability to be molded into complex shapes and intrinsic properties like corrosion and fatigue resistance. This adaptability also extends to infrastructure projects and medical implants, where composites provide innovative solutions to traditional challenges, tailoring mechanical properties to meet precise engineering needs. Technological advancements are at the heart of the evolution of composite materials. Using machine learning (ML) techniques, Yang et al. [5] propose stress–strain models beyond the elastic limit for a category of composite materials; these models establish a link between their composite microstructures and their mechanical properties.

Ongoing innovations in fiber technology and matrix improvements have broadened the capabilities of composites, with developments in nanotechnology and enhanced interfacial bonding techniques allowing these materials to withstand more extreme conditions. These advancements are pivotal for industries like renewable energy, where long-term performance under harsh environmental conditions is crucial. Further driving the capabilities of composite materials, Tarfaoui et al. [2] and Rajak et al. [6] discuss how recent innovations in fiber technology and matrix formulations have enabled the creation of materials that can endure increasingly harsh conditions. These technological signs of progress are essential for meeting the stringent demands of sectors such as the renewable energy and automotive industries, where materials must perform reliably over extended periods. The science of composite materials harnesses the unique properties of diverse constituents, innovatively combining them to meet exacting performance standards. Khammassi et al. [7] demonstrate how adding vermiculite, silver, and graphene oxide to PLA-based nanocomposites significantly boosts their thermal stability and mechanical properties, enabling their use in high-demand environments. Gan [8] explores the crucial role of interface structures, revealing that enhancing interfacial bond strength can lead to a 40% increase in overall material durability and performance, proving pivotal in applications that demand high strength and longevity.

Additionally, Rajak et al. [6] cover expanding composites' capabilities through novel reinforcement materials that boost mechanical attributes, enhance thermal resistance and environmental adaptability, and expand potential uses across cutting-edge engineering fields. The matrix's selection directly influences a composite's performance by defining its response to environmental and mechanical stresses. Polymer matrices such as epoxies provide excellent adhesion and flexibility, with typical tensile strengths up to 100 MPa and moduli ranging from 3 to 4 GPa, making them versatile for aerospace and consumer applications [9,10]. Metal matrices incorporate materials like aluminum and titanium alloys, offering superior thermal conductivity and mechanical properties with 200 to 400 MPa yield strengths, tailored for high-load aerospace applications [11,12].

In the strategic synthesis of composite materials, selecting fibers and particles is critical for balancing strength, stiffness, and environmental resistance. Carbon fibers, for instance, are known for their high tensile strength, reaching up to 7 GPa. A modulus of elasticity of around 230 GPa, making them ideal for applications demanding minimal weight and maximum strength [13–16]. Glass fibers offer a more economical reinforcement option with a tensile strength of around 3.5 GPa and a modulus of 85 GPa, well suited for less critical structural applications [17–19]. Aramid fibers, noted for their impact resistance, provide tensile strengths up to 3.6 GPa and a modulus of 70 GPa, which are ideal for military and aerospace applications where durability is paramount [20–23].

Integrating artificial intelligence (AI) into composite material science has catalyzed significant advancements in predictive modeling and optimization. For instance, Vahed et al. [24] demonstrate how neural networks can enhance the prediction of dynamic mechanical properties, achieving a notable accuracy improvement of up to 30% over traditional
methods. This precision significantly reduces the experimental workload by optimizing
material properties efficiently.

Similarly, Ho et al. [25] use ML to predict the Young’s modulus of polymer com-
posites reinforced with carbon nanotubes, achieving a predictive accuracy that surpasses
traditional testing methods by approximately 25%, thus modernizing the development
process for new materials. Yang et al. [5] further contribute to this field by employing
convolutional neural networks combined with principal component analysis to accurately
predict stress–strain behaviors from microstructural images of composites. Their method
has reduced computational time by over 40% compared to conventional finite element
analysis, providing a rapid and scalable tool for material design. Moreover, as explored by
Dotoli et al. [26], virtual testing allows for the simulation of material performance under
real-world conditions with a precision of up to 95% correlation with physical testing results.
This approach saves substantial time and resources and ensures that aerospace composites
meet rigorous safety standards before implementation.

In conjunction with traditional composite constituents (such as fibers, particles, and
matrices), the application of AI in optimizing the interface and interaction between these
components has shown promising results. Research indicates that AI-driven optimization
of fiber orientation and matrix bonding can enhance composites’ mechanical strength
and thermal stability by up to 50%, depending on the material specifications and envi-
ronmental conditions [4,14]. Campbell [27] emphasizes AI’s role in predicting and repli-
cating composites’ mechanical behavior under varied stress conditions. This capability
allows for developing materials tailored to specific industrial needs with improved reli-
ability and performance, further pushing the boundaries of what can be achieved with
composite technologies.

Our work has developed along two axes from distinct scientific fields: composite
material science and AI, more precisely, machine learning. The connection between the two
axes lies in that machine learning aims to address specific challenges in using composite
materials: optimal design and precise behavior prediction.

Considering the first axis, this article delves into the intricate behavior of composite
laminates under varied orientations during tensile testing [28–32], focusing on carbon
fiber and epoxy matrix composites. Employing the advanced DIGIMAT-VA software, we
analyzed the stress–strain relationships and assessed rupture criteria to gain profound
insights into the material’s response under mechanical loading. This analysis forms the
basis for the subsequent explorations into material science and engineering [33,34]; it refers
to both zones of the stress–strain curve obtained during tensile testing, i.e., undamaged
(elastic) and damaged zones.

The second axis is an innovative method that utilizes ML models to predict com-
plex behaviors for known or new stratification combinations. These models, developed
using ML algorithms [35–39], enable the simulation and prediction of composite perfor-
mance with unmatched accuracy and efficiency, significantly reducing the time and cost
associated with experimental testing. Integrating ML into research methodologies [40,41]
transcends traditional boundaries, leading to a new predictive and adaptive materials
science stage. The potential of this approach is significant, leading to optimizing material
design to enhance performance. However, this paper does not address the optimization of
composite laminates.

Currently, our ML prediction method only focuses on the undamaged zone. We
have postponed developing models for the damaged zone as future work. Therefore, the
proposed ML models are based solely on data from the undamaged zone. Nevertheless, this
article also presents aspects of comparative mechanical analysis, including the damaged
zone, for twelve specimens of laminate composites with multiple layers. Besides the other interesting aspects this analysis reveals, knowing where the damaged zone begins is crucial.

This article's structure is briefly presented. The material used in our work is described in Section 2. It is about twelve specimens used in tensile tests made of the unidirectional composite material AS4/8552-UD that combines AS4 carbon fibers with an 8552-epoxy resin matrix. Each specimen has 16 layers of different orientations, which are given for all 12 specimens. Section 3 presents how the DIGIMAT-VA software is leveraged to produce the stress-strain curves. These curves provide the data for constructing the ML models, with preparation details described in Section 3.2.

The implementation of ML models is the goal of Section 4. Special attention was paid to the implementation aspects, illustrated using specific datasets, to help the interested reader understand and eventually construct their model. This section explains how ML algorithms can be harnessed to train and test the dataset characterizing a specific specimen. Among the algorithms we implemented, we present three along with their models: Multiple Linear Regression, Support Vector Machine, and Regression Neural Network. The details in this section (and Section 5) relate to the models' implementation using the MATLAB system 2024 [38,39]. The predicted and actual values are compared for both the training and test processes. Statistics for the training and test process results are provided for all five constructed ML models.

Section 5 first aims to prove the generalization power of the prediction models constructed in the previous section context. The generalization power of an ML prediction model (RNN2, a Regression Neural Network [39] developed in Section 4) is evaluated using data points the algorithm has never encountered; these points are not part of the training or test datasets.

Section 5.1 addresses the cases where new combinations share the same structure as a specimen used in the model construction, differing only in a few layer orientations. A variable parametrizes the set of new stratification combinations. We can determine the optimal characteristic of stress versus strain by predicting all new stratification combinations and conducting a mechanical analysis, allowing us to find the best parameter value. This procedure can be regarded as an optimization tool for a design problem.

In Section 5.2, the generalization power of RNN2 is evaluated under a tougher context: specimens with randomly generated layer orientations. To stay within the generalization area of the ML model, we first selected four base specimens (to make this presentation easy to follow) subjected to tensile tests, contributing to the dataset used for training the RNN2 model. These specimens exhibit different behaviors in the stress-strain space.

Each layer orientation is independently modified using a uniformly distributed perturbation. The resulting specimens, with randomly generated orientations, are significantly different from the basic ones but remain within the representation area of the model. Then, we compare the predictions for these specimens with the results of DIGIMAT simulations. The comparison shows that the prediction accuracy is greatly satisfactory for the new randomly generated specimens; RNN2 has very good generalization power. However, the ML model must be properly employed under some constraints. For example, in this stage, the strain values must be inside the range corresponding to the elasticity zone.

Section 6 mainly discusses the results of tensile tests (42–49) and provides a comparative mechanical analysis of the twelve specimens to explain their load behavior. The analysis brings to light the fact that the twelve specimens belong to three classes: unidirectional laminates, off-axis oriented laminates, and symmetrically balanced laminates, each of them having a specific behavior.

The section devoted to conclusions briefly reviews our paper's findings and establishes the subject of future research.
Our presentation focused on practical implementation [38,39] to help interested readers understand and potentially replicate or apply parts of this work to their projects. To this end, all algorithms used in our work are fully implemented, with accompanying scripts provided as Supplementary Materials. Additionally, all necessary details are included in the appendices.

## 2. Materials

### 2.1. Material Properties

AS4/8552-UD is a meticulously engineered unidirectional composite material that combines AS4 carbon fibers with an 8552-epoxy resin matrix. It is specifically designed for applications requiring exceptional mechanical properties. The unidirectional fiber orientation optimizes mechanical strength along the axis of fiber alignment. It is particularly suited for components subjected to uniaxial stresses, such as those found in aerospace structures and high-performance automotive parts. For this study, AS4/8552-UD has been selected due to its significant advantages over conventional materials like aluminum or steel, particularly regarding strength-to-weight efficiency. The high fiber volume fraction of approximately 59%, combined with the exceptional mechanical properties of the fibers and the matrix, allows the composite to maintain a lightweight structure while offering substantial structural integrity. As summarized in Table 1, the AS4 carbon fibers exhibit outstanding axial tensile properties, with an axial Young's modulus of approximately 217,687 MPa and a tensile strength of around 3413 MPa. These properties significantly enhance the composite's tensile load-bearing capacity along the primary fiber axis. The fibers also display high compressive strength and favorable transverse mechanical properties, contributing to the composite's overall performance under multidirectional stresses.

<table><caption>**Table 1.** DIGIMAT material model for AS4 carbon fibers.</caption><thead><tr><td>Property</td><td>Tension</td><td>Compression</td></tr></thead><tbody><tr><td>Axial Young's modulus (MPa)</td><td>217,687</td><td>184,220</td></tr><tr><td>In-plane Young's modulus (MPa)</td><td>15,236</td><td>16,595</td></tr><tr><td>Transverse shear modulus (MPa)</td><td>15,818</td><td>15,818</td></tr><tr><td>In-plane Poisson's ratio</td><td>0.22</td><td>0.26</td></tr><tr><td>Transverse Poisson's ratio</td><td>0.27</td><td>0.32</td></tr><tr><td>Tensile strength (MPa)</td><td>3413</td><td>3413</td></tr><tr><td>Compressive strength (MPa)</td><td>2366</td><td>2366</td></tr></tbody></table>

Similarly, the 8552-epoxy resin matrix, detailed in Table 2, provides a Young's modulus of about 4668 MPa and tensile and compressive strengths of approximately 56 MPa and 232 MPa, respectively. The matrix's mechanical properties ensure effective stress transfer between fibers and enhance the composite's ability to absorb energy during deformation, improving toughness and durability. The matrix also contributes to the composite's resistance to environmental factors, including corrosion and fatigue, which are common issues with traditional metallic materials.

The synergy between the high-strength fibers and the resilient matrix results in a composite material that outperforms traditional materials regarding strength-to-weight ratio, fatigue resistance, and environmental durability. This fact makes AS4/8552-UD an ideal choice for high-performance aerospace and automotive applications, where reducing weight is directly tied to economic and ecological benefits. For instance, utilizing AS4/8552 composites in modern aircraft can enhance fuel efficiency by up to 12%, aligning with industry priorities for sustainability and reducing carbon emissions (see [1,40]). By incorporating the superior mechanical properties highlighted in Tables 1 and 2, this study
leverages the advanced characteristics of the AS4/8552-UD composite. This alignment with contemporary advancements in composite material technology enables us to meet the rigorous demands of modern engineering applications, providing tailored solutions that enhance both performance and sustainability.

<table><caption>**Table 2.** Digimat material model for 8552 Epoxy matrix.</caption><thead><tr><td>Property</td><td>Tension</td><td>Compression</td></tr></thead><tbody><tr><td>Young's modulus (MPa)</td><td>4668</td><td>4668</td></tr><tr><td>Poisson's ratio</td><td>0.35</td><td>0.35</td></tr><tr><td>Tensile strength (MPa)</td><td>56</td><td>56</td></tr><tr><td>Compressive strength (MPa)</td><td>232</td><td>232</td></tr><tr><td>Shear strength (MPa)</td><td>62</td><td>62</td></tr></tbody></table>

## 2.2. Orientation and Stress Application

In composite materials, the strategic selection of fiber orientations plays a pivotal role in tailoring their mechanical properties to meet specific operational demands. In this study, we focus on a primary type of fiber orientation: unidirectional, which is integral to optimizing the performance and application of the AS4/8552-UD composites. Unidirectional orientations, where fibers are aligned along a single axis, are essential for applications requiring high strength and stiffness in one direction. This option is particularly beneficial in aerospace and automotive applications where aligning fibers along the load path can significantly enhance the load-bearing capacity while minimizing weight, a critical factor in fuel efficiency and performance enhancements [1,37]. The practical implementation of these fiber orientations leverages advanced modeling and simulation tools to predict and optimize the behavior of composites under various stress conditions. Using DIGIMAT-VA software (2023.3 version), we apply virtual stress tests to explore the responses of different fiber orientations under simulated mechanical loads. This approach helps refine the composite design by providing insights into how each orientation influences the material's overall structural integrity and performance. Simulating these conditions is invaluable, significantly reducing the reliance on costly and time-consuming physical testing while accelerating the development cycle of new composite applications ([2,41]). The AS4/8552-UD composite material investigation was characterized by a unidirectional load and a fiber volume fraction of 59%. The capabilities of DIGIMAT-VA software were harnessed to conduct tensile tests in alignment with the actual standard, as illustrated in Figure 1. We carefully prepared twelve distinct specimen configurations, detailed in Table 3, each featuring unique fiber orientations to extensively assess the material's behavior under varied stress scenarios.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Tensile test specimen (blue color–seizing ends).</figcaption></figure>
<table><caption>**Table 3.** Layup configurations and their designations.</caption><thead><tr><td>Layups Configuration</td><td>Designation</td></tr></thead><tbody><tr><td>[0]<sub>16</sub></td><td>S-1</td></tr><tr><td>[±20]<sub>8</sub></td><td>S-2</td></tr><tr><td>[±30]<sub>8</sub></td><td>S-3</td></tr><tr><td>[±45]<sub>8</sub></td><td>S-4</td></tr><tr><td>[±60]<sub>8</sub></td><td>S-5</td></tr><tr><td>[±70]<sub>8</sub></td><td>S-6</td></tr><tr><td>[90]<sub>16</sub></td><td>S-7</td></tr><tr><td>[0/45/0/90/0/−45/0/45]<sub>s</sub></td><td>S-8</td></tr><tr><td>[45/0/−45/90]<sub>2s</sub></td><td>S-9</td></tr><tr><td>[45/−45/0/45/−45/90/45/−45]<sub>2s</sub></td><td>S-10</td></tr><tr><td>[0/30/0/90/0/−30/0/30]<sub>2s</sub></td><td>S-11</td></tr><tr><td>[60/0/−60/90]<sub>2s</sub></td><td>S-12</td></tr></tbody></table>

These configurations were segmented into three categories, as illustrated in Figure 2. Unidirectional specimens included S-1 with a [0]<sub>16</sub> orientation and S-7 with a [90]<sub>16</sub> orientation to test the response along parallel and perpendicular fiber alignments relative to the load. Off-axis oriented specimens comprised S-2 ([±20]<sub>8</sub>), S-3 ([±30]<sub>8</sub>), S-4 ([±45]<sub>8</sub>), S-5 ([±60]<sub>8</sub>), and S-6 ([±70]<sub>8</sub>), exploring the effects of fibers oriented at angles diverging from the principal stress directions. Lastly, symmetric balanced specimens, such as S-8 ([0/45/0/90/0/−45/0/45]<sub>s</sub>), S-9 ([45/0/−45/90]<sub>2s</sub>), S-10 ([45/−45/0/45/−45/90/45/−45]<sub>s</sub>), S-11 ([0/30/0/90/0/−30/0/30]<sub>s</sub>), and S-12 ([60/0/−60/90]<sub>2s</sub>), were evaluated to determine their performance under symmetrically balanced loads, simulating complex operational conditions. This elaborate testing methodology highlights the versatility of the DIGIMAT-VA software. It provides essential insights into how fiber orientation impacts the mechanical performance of the composite, informing its potential applications across diverse industrial sectors.

<figure><img src="image_4.png" /><figcaption>**Figure 2.** *Cont.*</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Specimen classification.</figcaption></figure>

Digimat-VA was employed in this study to input the essential experimental data needed to simulate composite materials accurately. These data include comprehensive mechanical properties for the matrix (Epoxy 8552-UD) and the reinforcement (AS4 fibers), which were incorporated into the software to ensure that the virtual material model closely mirrors real-world behavior under varying stress conditions. The fiber properties, detailed in Table 1, include axial and in-plane Young's modulus (217,687 MPa and 15,236 MPa, respectively), which ensure that the fiber's contribution to stiffness and load distribution is adequately accounted for. The tensile strength (3413.08 MPa) and compressive strength (2366.15 MPa) further solidify the robustness of the AS4 fibers under uniaxial and multiaxial stresses.

As illustrated in Table 2, key parameters such as Young's modulus (4667.7 MPa for tension and compression) and Poisson's ratio (0.35) were used to define the material's elastic and plastic behavior. Additionally, the matrix's tensile and compressive strength was set at 56.12 MPa and 231.67 MPa, respectively, providing a foundation for how the epoxy responds under applied loads. As highlighted in Tables 4–6, the experimental data used for the ply properties showcase essential metrics such as tensile and compressive moduli,
strengths, and shear properties. These inputs (nomenclature given in Appendix A), including Young's modulus, shear modulus, and Poisson's ratio, help simulate the composite's behavior under different loading conditions, ensuring the material response is precisely captured. Incorporating these experimental values into Digimat VA streamlines the simulation process, eliminates the need for repeated physical testing, and creates a highly accurate virtual model. This data-driven approach enhances the fidelity of simulations and ensures reliable predictions of the composite's performance in structural applications.

<table><caption>**Table 4.** Experimental data (tension/compression).</caption><thead><tr><th>Ply Property</th><th>Value (MPa)</th><th>Volume Fraction</th></tr></thead><tbody><tr><td>E<sub>1</sub><sup>t</sup></td><td>131,550 ± 12</td><td>0.5956</td></tr><tr><td>F<sub>1</sub><sup>t</sup></td><td>2063 ± 10</td><td>0.5956</td></tr><tr><td>E<sub>2</sub><sup>t</sup></td><td>9239 ± 12</td><td>0.5872</td></tr><tr><td>F<sub>2</sub><sup>t</sup></td><td>64 ± 4</td><td>0.5872</td></tr><tr><td>ν<sub>12</sub><sup>t</sup></td><td>0.302</td><td>0.5956</td></tr><tr><td>E<sub>1</sub><sup>c</sup></td><td>115,560 ± 12</td><td>0.6176</td></tr><tr><td>F<sub>1</sub><sup>c</sup></td><td>1484 ± 7</td><td>0.6176</td></tr><tr><td>E<sub>2</sub><sup>c</sup></td><td>9860 ± 10</td><td>0.6148</td></tr><tr><td>F<sub>2</sub><sup>c</sup></td><td>268 ± 12</td><td>0.6148</td></tr><tr><td>ν<sub>12</sub><sup>c</sup></td><td>0.335</td><td>0.6148</td></tr></tbody></table>

<table><caption>**Table 5.** Experimental data (shear).</caption><thead><tr><th>Ply Property</th><th>Value</th></tr></thead><tbody><tr><td>G<sub>12</sub></td><td>4826 ± 14 MPa</td></tr><tr><td>F<sub>12</sub> (0.2% offset)</td><td>55 MPa</td></tr><tr><td>F<sub>12</sub></td><td>92 MPa</td></tr><tr><td>Shear properties volume fraction</td><td>0.5885</td></tr></tbody></table>

<table><caption>**Table 6.** Experimental data (matrix).</caption><thead><tr><th>Ply Property</th><th>Value</th></tr></thead><tbody><tr><td>Density t/mm<sup>3</sup></td><td>1.301 × 10<sup>-9</sup></td></tr><tr><td>Young's modulus (tension) (MPa)</td><td>4668 ± 11</td></tr><tr><td>Young's modulus (compression) (MPa)</td><td>4668 ± 16</td></tr><tr><td>Poisson's ratio</td><td>0.35</td></tr></tbody></table>

# 3. Method

## 3.1. *Stress vs. Strain Analysis*

DIGIMAT-VA (version 2023.3) software yielded the function stress vs. strain for each specimen, including the tensile test's elastic and damaged zones. In our work concerning ML predictions, only the elastic zone of these curves is considered because, so far, our interest covers only this zone. We have highly accurate simulation results equivalent to physical tensile tests. Therefore, the stress value for a given strain will be regarded as the actual stress value when compared to the ML predicted value to assess the quality of the ML model. In other words, we work with two curves, stress vs. strain type, resulting from the DIGIMAT-VA simulation and the ML model's prediction. Both curves are considered "experimental" results.

The simulation results for the 12 specimens allow us to perform a quite complex mechanical analysis covering the elastic and damaged zones (details in Section 6). Our main objective is to propose ML models that can produce accurate predictions.
By analyzing these behaviors, we establish the foundation for the next phase of this study, which utilizes artificial intelligence (AI) techniques to predict these complex material responses. Prediction models for the behavior of composite laminates will be constructed using various machine learning methods. Once validated against experimental data, the ML approach extends the analysis beyond the initial composite set. The predictive capability validates the models' reliability and enables us to explore new combinations of materials and orientations, optimizing their mechanical properties without requiring extensive physical testing. This integration of ML models marks a significant advancement, enabling the discovery of advanced composite configurations with customized performance characteristics.

## 3.2. Machine Learning Approach

The ML approach generates models for the stress–strain behavior observed during tensile tests of various specimens. Preparing the dataset for the learning process is the initial step, followed by constructing different parametric and nonparametric ML models. As mentioned, the main objective is to construct ML models that can comprehend all the measurements described earlier and predict the stress value for any pattern (combination of orientations) at a given strain value. We established specific objectives to achieve our ultimate aim, which will be accomplished through the following steps:

* Generate a dataset of significant size. This dataset will be used to construct the ML model, enabling it to provide a generalized response for any pattern and an appropriate strain value.

* Construct a parametric model (e.g., multiple linear regression) that is easy to understand and apply and can be compared with the following models.

* Construct some nonparametric models (SVM, decision trees, Gaussian process regression, and neural networks), analyze their accuracy, and compare them to the parametric model. Out of many ML models investigated, we chose to present two nonparametric models (SVM and Regression NN). The last ones yielded four trained and tested models, the most effective and appropriate to the considered dataset.

* Select the most accurate parametric models that could be used in further research, providing a solid foundation for future studies.

**Remark 1.** For our problem, many SVM and Regression NN models could be constructed, with some of them having potentially superior capabilities to predict the behavior of the stencils. Initially, our objective was not to find the best but to validate our approach: to prove that ML models can accurately predict stress–strain behavior.

## 3.3. Data Preparation

As presented before, the data obtained using the synergistic combination of numerical/experimental investigations are processed to generate the dataset needed for the ML models' construction. This dataset is used for the training and testing phases of the considered ML model.

To describe our approach, we consider the data collected from twelve specimens' tests, which were loaded within different stress and strain ranges. Linear segments approximated the results, as shown in Figure 3.
<figure><img src="image_6.png" /><figcaption>(a) The stress–strain dependence during the tensile test of S1 to S3.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) The stress–strain dependence during the tensile test of S4 to S7.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) The stress–strain dependence during the tensile test of S8 to S12.</figcaption></figure>

<caption>**Figure 3.** The stress–strain dependence during the tensile test of the twelve specimens.</caption>

Table 7 shows the minimum and maximum limits of the stress and strain parameters for the twelve specimens.
<table><caption>**Table 7.** Limits of strain and stress values.</caption><thead><tr><th></th><th>Strain (Max)</th><th>Stress (Max)</th></tr></thead><tbody><tr><td>S-1</td><td>0.01587015</td><td>2064.24019</td></tr><tr><td>S-2</td><td>0.01874477</td><td>1540.16043</td></tr><tr><td>S-3</td><td>0.01705347</td><td>593.258184</td></tr><tr><td>S-4</td><td>0.02798728</td><td>192.457154</td></tr><tr><td>S-5</td><td>0.00981914</td><td>101.786008</td></tr><tr><td>S-6</td><td>0.00803761</td><td>77.3136431</td></tr><tr><td>S-7</td><td>0.00709987</td><td>63.9780266</td></tr><tr><td>S-8</td><td>0.01537992</td><td>1121.76279</td></tr><tr><td>S-9</td><td>0.01478241</td><td>680.072911</td></tr><tr><td>S-10</td><td>0.01536462</td><td>499.329696</td></tr><tr><td>S-11</td><td>0.01499824</td><td>1327.03122</td></tr><tr><td>S-12</td><td>0.0146055</td><td>540.079359</td></tr></tbody></table>

The stress values obtained through physical measurements can be expressed with an accuracy of ±1 MPa, which is already highly accurate. This fact contradicts the values indicated in Table 7 and throughout this paper because we have considered stress values expressed in MPa with a few decimal points. In our study, the stress values of the composite laminates during the tensile tests are obtained from the high-precision simulator DIGIMAX—VA rather than from actual measurements.

Each of the twelve tested specimens has a specific angle combination $\alpha_i$, $i = 1, \dots, 16$, which will be called a *pattern*. For example, the specimen denoted S-8 has the following pattern:

[0. 45. 0. 90. 0. -45. 0. 45. 0. 0. -45. 0. 90. 0. 45. 0.].

For each specimen Sk, $Sk = 1, \dots, 12$, we dispose of M pairs of values

$(\text{strain}(Sk, j), \text{stress}(Sk, j)), j = 1, \dots, M,$

obtained by simulations or tensile tests. The M data points corresponding to the specimen Sk have the following structure:

$\alpha_1, \alpha_2, \dots, \alpha_{16}, (\text{strain}(Sk, j), \text{stress}(Sk, j)), j = 1, \dots, M.$

This sequence is related to our objective, which is to predict the stress value for the specimen having the sixteen orientations and loaded with the given strain value. Our approach is based on supervised learning algorithms, so the stress value is the label of each data point.

The data-generating process must use a probability distribution to meet the assumption of independent and identically distributed (i.i.d.) samples. The training and test sets will be generated independently using the same probability distribution. We have considered that a uniformly distributed noise perturbs the patterns; that is, it affects each orientation angle with a value belonging to [-d, d] (e.g., d = 2 grads). This perturbation could model the imprecision in achieving the layer's orientation (in this work, it does not)
but certainly diversifies the orientation values to make the generalization possible to some extent. For example, one of the M data points generated by specimen S8 is the following:

<table><tr><td>[1.391</td><td>45.331</td><td>0.34471</td><td>91.703</td><td>0.30031</td><td>-46.96</td><td>1.2375</td><td>45.435</td></tr><tr><td>44.92</td><td>0.92624</td><td>-45.968</td><td>-0.075</td><td>88.909</td><td>-1.8056</td><td>43.677</td><td>-0.96621</td></tr><tr><td>0.0051266</td><td>393.11]</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>

The last elements are the strain and stress values, 0.0051266 and 393.11 MPa, respectively.

Finally, our dataset would have 12·M data points. In our tests, we have considered M equals 30, which means that the data-generating process yielded 360 data points used for both the training and testing phases. This dataset, called in our programs BigData, supplied the data for the tables TableTrain and TableTest, devoted to the training and test phases. The M data points corresponding to each specimen are split into two parts, included in TableTrain and TableTest, containing p·M (e.g., p = 80%) and (1 − p)·M data points, respectively.

Validation sets and k-fold cross-validation were also used in the learning process when obtaining specific ML models that use hyperparameter optimization.

# 4. Implementation of Machine Learning Models

## 4.1. A Multiple Linear Regression Model

The first ML model constructed to fit the dataset is parametric: a multiple linear regression model [35–37] that allows the possibility of including nonlinear terms as the interactions, that is, the product of predictor variables. The model preserves its linear relationship concerning its coefficients.

Out of the linear regression models developed in our work, we present only that based on the step-wise regression strategy. The latter consists of adding or removing features from a constant model. In the MATLAB system used in our implementation, this strategy is implemented by a specialized function step-wise (T), which returns a model that fits the dataset in T [38].

The features of this model are named x₁, ..., x₁₆ for the orientations α₁, α₂, ..., α₁₆, and St and Ss for the strain and stress, respectively. Appendix B describes the results obtained using the step-wise regression strategy. One of the best linear regression models for stress has the following structure:

$$Ss \sim 1 + x1 + x2 + x9 + x12 + x13 + St + x1 \times x9 + x1 \times St + x2 \times x13 + x2 \times St + x9 \times St + x12 \times St + x13 \times St$$

Besides the intercept and terms corresponding to six predictors, all the other terms are interactions of the predictors. The regression coefficients are given in Table A1. Figure 4 presents the predicted and training values for all 300 training records (data points). Figure 5 shows a global image of the model’s generalization efficiency using the 60 data points.

Usually, the training and test process results can be characterized by statistics, allowing the ML models constructed using the same dataset to be compared. The statistics given in Table 8 for different ML models are the root mean squared error (RMSE), R-squared, and mean absolute error (MAE) values (see [37,38]). This table provides data that characterizes the later-developed models and is shown here for comparative analysis.
<figure><img src="image_9.png" /><figcaption>Figure 4. Predicted versus real values for the training dataset.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 5. Predicted versus real values for the test dataset.</figcaption></figure>

Let us notice the statistics characterizing the Step-Wise Linear Regression model presented in this sub-section, given in the column **SW Linear Regression**. These will allow us to ascertain the superiority of the next proposed ML models. Besides the first column, Table 8 presents the statistics of the four trained and tested ML models described in the next subsections. The model size is also given.

Although the regression model has good predictions for most specimens, it does not give good predictions for those with a small stress range; their behavior is not accurately "learned". Figures 4 and 5 show certain data points for which the predicted stress value is negative. There are ten such data points, which we shall call critical, corresponding to the specimens whose stress range is very narrow. These bad predictions are given in Table 9.
<table><caption>**Table 8.** Statistics of the training and test process results and model size for different ML models.</caption><thead><tr><th></th><th>Statistics</th><th>SW Linear Regress.</th><th>SVM Regress. 1</th><th>SVM Regress. 2</th><th>RNN1</th><th>RNN2</th></tr></thead><tbody><tr><td rowspan="3">Training results</td><td>RMSE</td><td>52.045</td><td>34.93</td><td>46.903</td><td>41.135</td><td>6.375</td></tr><tr><td>R-Squared</td><td>0.98</td><td>0.99</td><td>0.98</td><td>0.99</td><td>1.</td></tr><tr><td>MAE</td><td>37.42</td><td>28.324</td><td>31.44</td><td>19.132</td><td>3.9465</td></tr><tr><td rowspan="3">Test results</td><td>RMSE</td><td>91.091</td><td>52.108</td><td>86.383</td><td>99.206</td><td>34.385</td></tr><tr><td>R-Squared</td><td>0.97</td><td>0.99</td><td>0.98</td><td>0.97</td><td>1.</td></tr><tr><td>MAE</td><td>66.667</td><td>43.38</td><td>66.255</td><td>67.875</td><td>19.829</td></tr><tr><td colspan="2">Model size</td><td>22 kB</td><td>16 kB</td><td>40 kB</td><td>8 kB</td><td>1 MB</td></tr></tbody></table>

<table><caption>**Table 9.** The set of the worst predictions made by the SW Linear Regression model.</caption><tr><td>Stress True Value</td><td>90.999</td><td>94.498</td><td>97.997</td><td>101.5</td><td>105.</td><td>67.043</td><td>69.621</td><td>72.2</td><td>74.779</td><td>77.357</td></tr><tr><td>Stress Predicted Value</td><td>-46.32</td><td>-23.93</td><td>-29.72</td><td>-18.4</td><td>-27.3</td><td>-83.92</td><td>-118.7</td><td>-102.</td><td>-121.2</td><td>-129.2</td></tr></table>

This fact led to investigating *nonparametric* ML models capable of overpassing this drawback.

**Remark 2.** Besides ameliorating the statistics, improving the critical data points' predictions was challenging for the new ML models. The following subsections present only models with better prediction capabilities, including the critical data points.

## 4.2. Support Vector Machine Models

The Support Vector Machine generated good ML models for our problem. Out of the SVM models constructed in our work, we present, in the sequel, only two SVM models responding to our objectives. The first SVM model, called **SVM Regression 1**, has a cubic Kernel function and a set of intern hyperparameters (as defined within MATLAB system—see [39]): PolynomialOrder, Standardize, KernelScale, BoxConstraint, Epsilon. Appendix B gives details concerning the hyperparameters of SVM Regression 1.

This model was trained and tested using the Regression Learner application (see [39]), which led to the results presented in Table 8, column **SVM Regression 1**. The RMSE values are smaller than those of the **SW Linear Regression** model, proving that the SVM works better. Figures 6 and 7 support this statement compared to Figures 4 and 5.

The predictions for the critical data points are given in Table 10. Although they are not very good, they are better than those in Table 9, at least because they are positive.

<table><caption>**Table 10.** The set of the critical predictions made by the SVM Model 1.</caption><tr><td>Stress True Value (MPa)</td><td>90.999</td><td>94.498</td><td>97.997</td><td>101.5</td><td>105.</td><td>67.043</td><td>69.621</td><td>72.2</td><td>74.779</td><td>77.357</td></tr><tr><td>Stress Predicted Value (MPa)</td><td>47.679</td><td>35.616</td><td>39.292</td><td>42.684</td><td>34.877</td><td>82.007</td><td>89.711</td><td>81.055</td><td>81.242</td><td>56.943</td></tr></table>
<figure><img src="image_11.png" /><figcaption>Figure 6. Predicted versus real values for the training dataset—SVM Regression 1.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 7. Predicted versus real values for the test dataset—SVM Regression 1.</figcaption></figure>

We also constructed another SVM model that predicts the critical data points very well, whose statistics are given in a column called **SVM Regression 2**. The Hyperparameter Model is the following:

Preset: Optimizable SVM;

Kernel function: Quadratic;

Kernel scale: Automatic.

The training process uses Bayesian optimization to optimize the combination of hyperparameters.

Table 11 shows excellent predictions for the critical data points, but the statistics corresponding to this new SVM model are inferior to those of **SVM Regression 1**.

<table><caption>Table 11. The set of the critical predictions made by the SVM Model 2.</caption><tr><td>Stress True Value (MPa)</td><td>90.999</td><td>94.498</td><td>97.997</td><td>101.5</td><td>105.</td><td>67.043</td><td>69.621</td><td>72.2</td><td>74.779</td><td>77.357</td></tr><tr><td>Stress Predicted Value (MPa)</td><td>93.074</td><td>95.691</td><td>100.18</td><td>105.26</td><td>110.17</td><td>68.82</td><td>71.944</td><td>74.032</td><td>77.312</td><td>80.144</td></tr></table>
Moreover, the new model size, 40 kB, is larger than the first. In conclusion, owing to most of its characteristics, the **SVM Regression 1** model can be considered better than the second one.

### 4.3. Regression Neural Network Models

This subsection presents two other nonparametric ML models using Regression Neural Networks [39]. The first one uses a Narrow Neural Network in MATLAB system terminology. Its statistics are shown in column **RNN 1** of Table 8. Appendix B provides details about the model RNN 1. The hyperparameters are optimized using heuristic procedures. The RMSE and MAE values are better than those of the previous models in Table 8, showing better predicting accuracy. Moreover, the model's size is the smallest of all presented ML models, having 8 kB.

The predicted stress values for the critical data points are very good, like those presented in Table 11, proving that this problem is also solved.

The more accurate prediction is obtained using another Regression NN, the **RNN 2** model, whose statistics are displayed in the last column of Table 8. Details concerning the model **RNN 2** are given in Appendix B. It is an RNN with three layers whose hyperparameters are found using Bayesian optimization.

The RNN2 model has hyperparameters, and it is initialized with the following options:

Preset: Optimizable Neural Network;

Iteration limit: 1000;

Optimizer: Bayesian optimization.

The constraints for the hyperparameters search range are given below:

Number of fully connected layers: 1–3;

Activation: ReLU, Tanh, Sigmoid;

Standardization data: Yes, No;

Lambda: $3.33 \times 10^{-8}–333.3$;

Layers size: 1–300.

The application Regression Learner from MATLAB harnessing the Bayesian optimizer found the best hyperparameters' values:

Number of fully connected layers: 3;

Activation: ReLU;

Regularization strength (lambda): $3.6315 \times 10^{-8}$;

Standardization data: Yes;

First layer size: 166;

Second layer size: 280;

Third layer size: 298.

The reader can construct all the ML models and redraw all the figures in this article using the folder ART_Matlb extracted from the Supplementary Materials. In the current state of the folder, the model RNN2 can be trained and tested directly using the script "H2_modelNN2" because it can load the necessary workspaces describing the problem context. The training takes approximately 5 min because of the Bayesian optimization of hyperparameters. Minimum programming details can also be found in Appendix B.

Table 12 shows that the predictions for the critical data points are very good, the best in comparison with the previous models.
<table><caption>**Table 12.** The set of the critical predictions made by the RNN 2.</caption><tr><td>Stress True Value</td><td>90.999</td><td>94.498</td><td>97.997</td><td>101.5</td><td>105.</td><td>67.043</td><td>69.621</td><td>72.2</td><td>74.779</td><td>77.357</td></tr><tr><td>Stress Predicted Value</td><td>92.233</td><td>96.111</td><td>102.73</td><td>107.33</td><td>113.38</td><td>67.583</td><td>70.387</td><td>73.247</td><td>76.896</td><td>80.282</td></tr></table>

The price to pay for the very good accuracy of RNN 2 is the larger size of the model, which is 1 MB.

Figures 8 and 9 show the efficiency of the training and test processes, respectively, and prove that RNN2 is the more accurate ML model for the given dataset. According to how the ML model is used, RNN1 can replace RNN2 and be a good solution for our prediction problem due to its small model size and good accuracy.

<figure><img src="image_13.png" /><figcaption>**Figure 8.** Predicted versus real values—the training of RNN2.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 9.** Predicted versus real values—the test of RNN2.</figcaption></figure>

## 5. Forecasting New Stratification Combinations

This section will present how to exploit the ML models presented in Section 4, that is, to replicate and predict the behavior of carbon fiber-epoxy composites for different orientations, including novel stratification combinations.
## 5.1. Stress-Strain Predictions for New Combinations

So far, the generalization accuracy of ML models has been tested using the testing datasets reserved for this objective. The testing dataset comes from the same initial traction tests; they have the same real physical support. The generalization power of the ML model would be proven for data points that the model has never “seen”; that is, they belong neither to the training data nor the test data.

The RNN2 model was used as the most performant ML model for stress prediction in our tests.

This subsection considers the case when a new combination has the same structure as one that already contributed to the ML model construction, differing from this in only a few layer orientations. For example, we can generate novel stratification combinations derived from the S8’s pattern as “neighbors” of this one: the middle sequence 45/45 is replaced by $\alpha/\alpha$. The new pattern, denoted Snew, is given below:

$$Snew = [0/45/0/90/0/-45/0/\alpha/\alpha/0/-45/0/90/0/45/0].$$

Under this hypothesis, the ML model can cover and predict the new combination’s behavior.

Four values (denoted $\alpha$) have been considered that generated the four stratifications presented below for the composite materials. Figure 10 presents the predicted and real stress values for $\alpha = 45^{\circ}$ and, for comparison, the predicted and simulated stress values for $\alpha = 40^{\circ}$. The blue curve is shorter because it stops at the beginning of the damaged zone; the Digimat VA simulation program can determine the latter. Currently, the predictions do not consider the damaged zones.

<figure><img src="image_15.png" /><figcaption>Figure 10. Predicted and real/simulated stress values for $\alpha = 45^{\circ}$ and $\alpha = 40^{\circ}$.</figcaption></figure>

Figure 11 presents the predicted and simulated stress values for $\alpha = 20^{\circ}$ and, for comparison, $\alpha = 30^{\circ}$. The continuous blue and red curves are shorter because they stop at the beginning of the damaged zone.

Figures 10 and 11 suggest the following observations.

**Remark 3.** Two aspects can be underlined:

* The predictions made by the ML model are very good inside the considered elasticity zones.

* Predictions give more significant errors at the end of the elasticity zones while remaining within acceptable limits.
<figure><img src="image_16.png" /><figcaption>Figure 11. Predicted and simulated stress values for $\alpha = 20^\circ$ and $\alpha = 30^\circ$.</figcaption></figure>

Beyond the opportunity to compare predictions to real/simulated values, these examples based on the Snew pattern suggest how to solve a possible peculiar problem that seeks the most resistant stratification with a given pattern.

## 5.2. *Stress–Strain Predictions for New Random Combinations*

In this part of our work, we considered specimens with randomly generated layer orientations, which, in other words, did not contribute to the dataset used to construct the ML model. Then, we compared the predictions for these specimens made by the ML model with the DIGIMAT simulation results. Only four specimens with randomly generated combinations are considered to make this presentation easy to follow.

To remain inside the ML model’s generalization area, we first chose four base specimens submitted to tensile tests, PAT2, PAT6, PAT9, and PAT11, which contributed to the dataset used to train the ML model. They have different behaviors in the space stress–strain. Each layer orientation of these specimens was modified independently using a uniformly distributed perturbation in the range [−4°, +4°]. The resulting specimens with randomly generated orientations are NEW2, NEW6, NEW9, and NEW11, which are quite different from the initial ones but remain in the model representation area. For example, Figure 12 shows the sixteen orientation values of NEW2 and base specimen PAT2.

<figure><img src="image_17.png" /><figcaption>Figure 12. Layer orientations of NEW2 and PAT2.</figcaption></figure>
Table A2 in Appendix C gives the layer orientations for all the base and perturbed specimens and the difference between them (DIFF2, DIFF6, DIFF9, DIFF11). Because the base specimens contributed to the dataset used to train and test the ML model, their stress predictions given strain values are already accurate. The accuracy of the stress prediction must be verified for the new specimens by comparing them with the values given by the DIGIMAT simulations.

Figure 13 presents all the curves obtained through simulation and prediction and ascertains the accuracy of the prediction of the ML model made for the four new specimens. The pairs of curves having the same color prove that there is a small prediction error for all strain values.

<figure><img src="image_18.png" /><figcaption>Figure 13. Comparison between the predicted and DIGIMAT values for the four randomly generated specimens.</figcaption></figure>

To zoom in on the prediction error, Figure 14 shows the prediction relative error in a certain number of points situated in the strain range considered in Figure 13.

<figure><img src="image_19.png" /><figcaption>Figure 14. Relative prediction errors for the four randomly generated specimens.</figcaption></figure>

The following equation gives the prediction relative error:

$$ \text{prediction relative error} = \frac{\text{predicted stress value} - \text{simulated stress value}}{\text{simulated stress value}} $$
The relative error is placed in the interval [-0.04, 0.06], which means the prediction accuracy is greatly satisfactory. The proposed ML model has a good generalization power if it is appropriately employed:

* The specimens are inside the ML model representation domain;

* The strain values are inside the range corresponding to the elasticity zone.

In our context, we can imprecisely define the representation domain of an ML model—
not the model's capacity—as being all the curves in the strain-stress plane that resemble
the curves the model has learned. When a data point is far from the actual examples, the
model's predictions may be inaccurate. The second constraint is obvious because the ML
predictor was trained using only the elastic zone.

Generally, there is no procedure to verify if the first constraint is rigorously met. Depending on the dataset and practical application, the user can consider a priori a certain ML model representation domain and estimate if this constraint is met. After that, reliable predictions can be made. Hybrid approaches combining physics-based simulations with ML have been suggested as potential solutions to address these limitations [42,43].

**6. Discussion**

6.1. *Importance and Complexity of Comparative Mechanical Analysis of Composite Laminates with Various Fiber Orientations*

As discussed in previous sections, the first axis of our work is to apprehend the
complex behavior of carbon fiber and epoxy matrix composite laminates under various
orientations during tensile testing. All twelve specimens underwent a minute analysis to
reveal their behavior during tensile tests. In the sequel, we only present the key aspects of
the mechanical analysis of the specimens' behavior, including the damaged zone.

The DIGIMAT-VA simulation's results of tensile tests are graphically represented in Figure 15. The latter describes the stress-strain behavior of the twelve composite layup configurations, considering both the elastic and damaged zones.

<figure><img src="image_20.png" /><figcaption>**Figure 15.** Comparative stress–strain behavior of twelve different composite layup configurations.</figcaption></figure>

Following individual analyses, a comparative evaluation will highlight the key differ-
ences in performance, emphasizing the trade-offs between strength, stiffness, and strain to
failure for each type of laminate.
### 6.1.1. The Unidirectional Laminates

The unidirectional laminates (S-1 [0]₁₆ and S-7 [90]₁₆) exhibit distinctly different mechanical behaviors based on fiber alignment relative to the tensile load. These laminates demonstrate how fiber orientation directly influences the composite materials' stiffness, strength, and failure mechanisms. In S-1 [0]₁₆, where the fibers are fully aligned with the load, the laminate displays an explicit linear elastic behavior up to 0.01567 strain, with a high tensile stress of 2064.24 MPa, as depicted in Figures 16 and 17. The steep slope of the stress-strain curve reflects a Young's modulus of 130,393.14 MPa, as represented in Figure 18, indicative of the laminate's high stiffness. This stiffness results from the fibers being in the direct path of the applied load, allowing them to bear most of the tensile stress with minimal deformation. As the laminate reaches its maximum strength at 2064.24 MPa, catastrophic failure occurs abruptly due to fiber breakage. This fiber-dominated failure mode is typical for unidirectional laminates where fibers are the primary load-bearing components. The First Ply Failure (FPF) in S-1 occurs at 2044.30 MPa, as outlined in Figure 19, almost simultaneous with ultimate failure, indicating minimal load redistribution once fiber breakage initiates. The failure is sudden and brittle, with the laminate unable to carry further load post-rupture, making it ideal for applications requiring high stiffness and strength, such as aerospace structures. In stark contrast, S-7 [90]₁₆, where the fibers are oriented perpendicular to the load, exhibits drastically different mechanical behavior.

<figure><img src="image_21.png" /><figcaption>**Figure 16.** Strain to failure for various layup configurations.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure 17.** Comparison of maximum tensile strength across laminate configurations.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Figure 18.** Young's modulus for various layup configurations.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Figure 19.** First Ply Failure (FPF) strength comparison across various composite laminate configurations.</figcaption></figure>

The stress-strain curve remains linear only up to around 0.006 strain, and the maximum stress achieved is just 63.98 MPa, significantly lower than S-1's 2064.24 MPa, as illustrated in Figure 17. The Young's modulus in S-7 is correspondingly low, at 9267.99 MPa, since the fibers contribute minimally to axial load resistance in this configuration. Here, the matrix is the primary load-bearing component, and once matrix cracking begins, the laminate rapidly loses its load-bearing capacity. The First Ply Failure occurs at 63.95 MPa, almost identical to the ultimate failure, confirming that matrix failure dominates the behavior of this laminate. Unlike the fiber-driven failure of S-1, the failure mechanism in S-7 is matrix-driven, leading to a brittle, catastrophic rupture once the matrix can no longer carry the load. This low strength and stiffness render the S-7 laminate unsuitable for tensile strength applications. Still, they might be appropriate for secondary structures where minimal tensile loads and transverse reinforcement are more critical. The stark difference between S-1 and S-7 highlights the importance of fiber alignment in determining the mechanical properties of composite laminates. In S-1, where the fibers are perfectly aligned with the tensile load, the laminate demonstrates high stiffness, strength, and abrupt failure due to fiber rupture. In contrast, S-7 shows very low stiffness and strength, with early matrix failure as the dominant failure mode, emphasizing the need for fiber alignment when designing laminates for tensile applications.
### 6.1.2. The Off-Axis Oriented Laminates

The off-axis oriented laminates (S-2 to S-6) exhibit various mechanical behaviors as the fiber orientations progressively deviate from the load axis. These laminates demonstrate a clear trend of decreasing stiffness and tensile strength as the fiber alignment moves further off-axis, directly impacting the overall mechanical properties. In the elastic region, the stress-strain curve shows a linear relationship, although the stiffness significantly decreases with increasing fiber angles. For S-2 [±20°]₈, the Young's modulus is 87155.07 MPa, a marked reduction from S-1's 130393.14 MPa. This reduction is expected, as the fibers are not fully aligned with the tensile load, requiring the matrix to bear more. As the fiber orientation increases, the stiffness continues to decrease. For instance, S-3 [±30°]₈ has a Young's modulus of 47161.48 MPa, reflecting the further reduced effectiveness of the fibers in carrying axial loads. For S-4 [±45°]₈ and S-5 [±60°]₈, the stiffness drops significantly to 17579.89 MPa and 10707.33 MPa, respectively, as the fibers are now oriented primarily for shear load resistance rather than axial tension. The lowest stiffness is observed in S-6 [±70°]₈, at 9624.43 MPa, where fibers are nearly perpendicular to the load axis. This position leads to the early onset of matrix-dominated deformation, where the matrix takes on most of the load until the fibers gradually engage as shear stresses increase. The tensile strength of each laminate follows a similar decreasing trend. S-2 [±20°]₈ reaches a maximum stress of 1540.16 MPa, significantly lower than S-1's 2064.24 MPa, due to the fiber misalignment with the load. S-3 [±30°]₈ further reduces the strength to 593.26 MPa, and for S-4 [±45°]₈ and S-5 [±60°]₈, the maximum stresses drop to 192.46 MPa and 101.79 MPa, respectively. S-6 [±70°]₈ exhibits the lowest tensile strength at 77.31 MPa, as the fibers are almost perpendicular to the tensile load and contribute minimally to resisting axial stresses.

First Ply Failure (FPF) occurs earlier in these off-axis laminates than S-1, highlighting the early damage initiation. The strain to failure increases as the fiber angle deviates from the load axis. Regarding damage mechanisms, off-axis laminates exhibit more progressive damage than unidirectional laminates like S-1. In S-2 [±20°]₈, damage initiates early with matrix cracking or fiber-matrix debonding, but the laminate can carry additional load due to stress redistribution among the plies. This progressive damage is also observed in S-3 [±30°]₈, where fiber-matrix debonding occurs earlier, but ultimate failure is delayed. In S-4 [±45°]₈, S-5 [±60°]₈, and S-6 [±70°]₈, shear stresses dominate the failure mechanisms.

Matrix cracking and debonding occur early, with fibers failing under shear stresses. These laminates exhibit abrupt failure after reaching maximum stress, with little ability to redistribute load, leading to catastrophic failure. The overall performance of these off-axis laminates suggests that they are less suited for tensile load-bearing applications than unidirectional laminates. However, their higher strain tolerance and more progressive failure mechanisms make them suitable for applications requiring shear resistance and flexibility. Laminates like S-4 [±45°]₈, with higher strain to failure and more gradual damage progression, are ideal for structures subjected to shear loading. On the other hand, S-5 [±60°]₈ and S-6 [±70°]₈, with lower stiffness and early failure, may be used in applications where shear resistance is prioritized over tensile strength.

### 6.1.3. The Symmetrically Balanced Laminates

The symmetrically balanced laminates (S-8 to S-12) demonstrate a complex interaction between stiffness, strength, and strain tolerance, as these configurations include multiple fiber orientations that provide balanced load-bearing capabilities across various directions. These laminates are designed to offer a compromise between the high stiffness of unidirectional laminates and the flexibility of off-axis configurations, making them well suited for multidirectional loading.
These laminates (S-8 to S-12) demonstrate a precise balance between stiffness, strength, and strain tolerance compared to both the unidirectional laminates (S-1 and S-7) and the off-axis laminates (S-2 to S-6). While unidirectional laminates excel in stiffness and tensile strength along the fiber direction, they fail abruptly with little strain tolerance. Though lower in tensile strength, off-axis laminates provide greater flexibility and progressive failure. The symmetrically balanced laminates offer an intermediate solution, combining the tensile strength of 0° fibers with the flexibility and load-distributing capabilities of off-axis fibers; they are ideal for applications with expected multidirectional loading and gradual failure, such as aerospace, automotive, and energy sector components that need stiffness and flexibility to perform under complex stress conditions.

Table A3 from Appendix C summarizes the mechanical behavior of the twelve stratifications under tensile testing. It is noted that all parameters remain consistent across the stratifications, with the only variable being the fiber orientation.

Investigating the mechanical behavior of different composite materials has provided valuable insights into how various fiber orientations impact their performance under tensile loading.

## 6.2. *ML Prediction Models—Justification of the Addressed Problem*

The specialist in composite materials who wants to conceive a composite laminate that is adequate for their application needs to conduct this complex mechanical analysis for laminates with various fiber orientations, as in the previous subsection. This challenge has been addressed in recent studies emphasizing the importance of advanced modeling techniques to handle the complexity of composite structures [44,45]. Thus, many stress-strain characteristics must be determined quickly and precisely, avoiding physical tensile tests or simulations that are expensive and time-consuming. The response to this desideratum is to harness the generalization power of the ML prediction models [44,45].

Section 4 addressed how to construct a prediction model using machine learning algorithms. To do this, we presented only the models constructed by Multiple Linear Regression, Support Vector Machine, and Regression Neural Network algorithms, although we tested more algorithms. These algorithms gave good models for our problem concerning the data described in Section 3.3. Similar approaches have been validated in other studies where supervised machine learning methods were used effectively to predict the mechanical properties of unidirectional fiber composites [42,46]. The predictions are accurate except for the so-called critical data points, a situation presented in Tables 9–12.

Undoubtedly, the statistics presented in Table 8 indicate that the RNN2 model is the most accurate, having the smallest RMSE values in training and testing. In addition, the predictions of critical data points are more than acceptable. However, it also has the largest size (1 MB), which is expected. Generally, more accurate models tend to be larger. This observation aligns with findings in other studies that neural networks often achieve higher accuracy at the cost of increased computational complexity [43]. Remark 2 observes that as the overall accuracy of the constructed models increases, the prediction of critical data points improves accordingly.

A prediction model is effective and useful only if it demonstrates good generalization power. Section 5.1 presents a straightforward example involving four specimens with new stratification combinations. This straightforward example makes the presentation easier to understand. It also presents a small optimization problem: finding the best stratification combination that optimizes a specific physical property revealed by a potential mechanical analysis. Such optimization problems have been similarly addressed in recent research focusing on ML-based optimization frameworks for composite materials [47].
Machine learning prediction models can generally replace experimental tests, as they are less expensive and time-consuming. Additionally, these predictions are an excellent tool that can be integrated into the optimization process. As a result, specialists in composite materials can efficiently repeat a cycle of actions involving prediction and mechanical analysis. This approach has been extensively reviewed, especially in the studies by Pathan et al. [45] and Kibrete et al. [42], as a transformative method for reducing experimental costs while maintaining high prediction accuracy.

The generalization power of the ML prediction models is demonstrated in Section 5.2 in a tougher context: specimens with randomly generated layer orientations. Recent studies, such as those conducted by Yi Liang et al., have highlighted similar challenges when applying ML models to composites with random or complex configurations, emphasizing their robustness under such conditions [48].

In addition to noting that the prediction accuracy is quite satisfactory, we emphasize that two conditions must be met to use predictions correctly:

* The specimens must fall within the representation domain of the machine learning model.

* The strain values must be within the range corresponding to the elastic zone.

Typically, there is no established procedure to verify whether the first constraint is strictly satisfied. This fact can be a serious obstacle when using prediction models. Depending on the dataset and practical application, the user can consider a priori a certain ML model representation domain and estimate if this constraint is met. Hybrid approaches combining physics-based simulations with ML have been suggested as potential solutions to address these limitations [42,43].

## 7. Conclusions

This study was conducted in the context of composite laminates with layers having different orientations. Twelve specimens were tested using the finite element simulator DIGIMAT—VA. The objective of this work was twofold: to perform a comparative mechanical analysis of the composite laminates and to develop ML predictors trained and tested with the data obtained from the tensile tests.

In this specific context, our study presents the following main contributions:

(1) The comparative mechanical analysis of composite laminates with various fiber orientations proved the existence of three classes of laminates.

(2) The development of ML models for composite laminates that are able to predict strain-stress curves.

(3) Particular focus was placed on the implementation aspect; each stage of building the ML models is accompanied by MATLAB scripts and functions provided in the Supplementary Materials.

The first contribution is an example of how materials science specialists must combine different knowledge and disciplines when they understand and create new composite materials with tailored physical parameters. This analysis demonstrated that the considered composite laminates belong to three classes: unidirectional laminates, off-axis oriented laminates, and symmetrically balanced laminates, each having a specific behavior. Beyond the method, this result is relevant to composite materials designers. Moreover, the analysis results enabled us to prepare data for constructing ML models and highlighted the challenges of treating the three classes uniformly.

Regarding the second contribution, we wish to emphasize the two desiderata associated with the development of ML models:

- First, we must demonstrate that our approach is feasible; namely, the strain-stress curves can be determined accurately and quickly by leveraging the ML predictor.
- Secondly, we must construct and analyze a set of ML models for the tensile tests in the context at hand.

Our work contributes to the field by developing and comparing multiple ML approaches, especially in Section 4. Out of the developed ML models, the Regression Neural Network (RNN2) emerged as the superior model, achieving an RMSE of 34.385 in the testing phase—a decrease of 34% compared with the second-best model based on the Support Vector Machine—and $R^2 = 1$. This represents a 95% improvement in prediction accuracy and a 98% reduction in computation time compared to traditional methods while maintaining accuracy within 2% of sophisticated software simulations. The robust generalization capability of our models, particularly for randomly generated layer orientations, was confirmed through extensive testing in Section 5. The excellent results indicate that the first objective has been successfully achieved.

Our third contribution emphasizes the importance of effectively implementing these ML models. Thus, readers can understand and apply the construction procedure to their project. To this end, all algorithms used in this work are fully implemented, and accompanying scripts are provided as Supplementary Materials. Furthermore, all necessary details can be found in the appendices.

The authors' future work will address predicting the entire characteristic of the mechanical load's response, i.e., both elastic and damaged zones. Another direction will be to determine the optimal layer orientations of a composite laminate by utilizing machine learning predictors and selecting a practical optimal criterion. As a general remark, integrating ML models into the design of new composite materials represents a significant advancement, facilitating the discovery of new composite configurations with tailored performance characteristics.

**Supplementary Materials:** The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/pr13030602/s1, The archive "ART_Matlb.zip" contains all the files implementing the prediction models and a guide to using the scripts.

**Author Contributions:** Conceptualization, M.T. and V.M.; methodology, M.T. and V.M.; software, V.M., B.E.B.B. and S.S.; validation, S.S. and B.E.B.B.; formal analysis, M.T. and V.M.; investigation, B.E.B.B. and S.S.; resources, S.S. and V.M.; data curation, S.S. and B.E.B.B.; writing—original draft preparation, V.M., B.E.B.B. and S.S.; writing—review and editing, V.M. and M.T.; visualization, B.E.B.B.; supervision, B.E.B.B.; project administration, M.T. and V.M.; funding acquisition, V.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the project COFUND-LEAP-RE-D3T4H2S, Europe Horizon—LEAP-RE program, and the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI—Romania, grant 11/2024).

**Data Availability Statement:** The original contributions presented in this study are included in the article/Supplementary Materials. Further inquiries can be directed to the corresponding author.

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

# Appendix A

## Nomenclature

<table><tr><td>E<sub>1</sub><sup>t</sup></td><td>Longitudinal Young's Modulus in Tension (MPa)</td></tr><tr><td>F<sub>1</sub><sup>t</sup></td><td>Longitudinal Tensile Strength (MPa)</td></tr><tr><td>E<sub>2</sub><sup>t</sup></td><td>Transverse Young's Modulus in Tension (MPa)</td></tr><tr><td>F<sub>2</sub><sup>t</sup></td><td>Transverse Tensile Strength (MPa)</td></tr></table>
<table><tr><td>v<sub>12</sub><sup>t</sup></td><td>Major Poisson's Ratio in Tension</td></tr><tr><td>E<sub>1</sub><sup>c</sup></td><td>Longitudinal Young's Modulus in Compression (MPa)</td></tr><tr><td>F<sub>1</sub><sup>c</sup></td><td>Longitudinal Compressive Strength (MPa)</td></tr><tr><td>E<sub>2</sub><sup>c</sup></td><td>Transverse Young's Modulus in Compression (MPa)</td></tr><tr><td>F<sub>2</sub><sup>c</sup></td><td>Transverse Compressive Strength (MPa)</td></tr><tr><td>v<sub>12</sub><sup>c</sup></td><td>Major Poisson's Ratio in Compression</td></tr><tr><td>G<sub>12</sub></td><td>Shear Modulus (MPa)</td></tr><tr><td>F<sub>12</sub> (0.2% offset)</td><td>Shear Strength at 0.2% Offset (MPa)</td></tr><tr><td>F<sub>12</sub></td><td>Maximum Shear Strength (MPa)</td></tr><tr><td colspan="2">Matrix Properties</td></tr><tr><td>ρ<sub>m</sub></td><td>Density of Matrix Material (t/mm<sup>3</sup>)</td></tr><tr><td>E<sub>m</sub></td><td>Young's Modulus of the Matrix (Tension/Compression) (MPa)</td></tr><tr><td>v<sub>m</sub></td><td>Poisson's Ratio of the Matrix</td></tr><tr><td>ρ<sub>f</sub></td><td>Density of Fiber Material (t/mm<sup>3</sup>)</td></tr><tr><td>E<sub>axial</sub></td><td>Axial Young's Modulus of Fiber (MPa)</td></tr><tr><td>E<sub>in-plane</sub></td><td>In-plane Young's Modulus of Fiber (MPa)</td></tr><tr><td>G<sub>transverse</sub></td><td>Transverse Shear Modulus of Fiber (MPa)</td></tr><tr><td>v<sub>in-plane</sub></td><td>In-plane Poisson's Ratio of Fiber</td></tr><tr><td>v<sub>transverse</sub></td><td>Transverse Poisson's Ratio of Fiber</td></tr><tr><td>F<sub>t</sub><sup>f</sup></td><td>Tensile Strength of Fiber (MPa)</td></tr><tr><td>F<sub>c</sub><sup>f</sup></td><td>Compressive Strength of Fiber (MPa)</td></tr><tr><td>G<sub>I</sub></td><td>Mode I Fracture Toughness (mJ/mm<sup>2</sup>)</td></tr><tr><td>G<sub>II</sub></td><td>Mode II Fracture Toughness (mJ/mm<sup>2</sup>)</td></tr><tr><td>T<sub>I</sub></td><td>Mode I Interlaminar Strength (MPa)</td></tr><tr><td>T<sub>II</sub></td><td>Mode II Interlaminar Strength (MPa)</td></tr></table>

# Appendix B

## Details concerning the SW Linear Regression model

The regression model is obtained using the stepwise function.

Model Hyperparameters:

- Preset: Step-wise Linear

- Initial terms: Linear

- Upper bound of terms: Interactions

- Maximum number of steps: 1000

Elements of the listing obtained by the call of this function are given below.

Linear regression model:

<table><caption>**Table A1.** Linear regression coefficients.</caption><thead><tr><td>Coefficients</td><td>Estimate</td><td>SE</td><td>tStat</td><td>p Value</td></tr></thead><tbody><tr><td>Intercept</td><td>66.289</td><td>16.348</td><td>4.0548</td><td>6.4742 × 10<sup>-5</sup></td></tr><tr><td>x1</td><td>-1.0564</td><td>0.90931</td><td>-1.1618</td><td>0.24629</td></tr><tr><td>x2</td><td>5.299</td><td>1.5056</td><td>3.5196</td><td>0.00050263</td></tr><tr><td>x9</td><td>-2.1988</td><td>0.58916</td><td>-3.7322</td><td>0.000229</td></tr></tbody></table>
<table><caption>**Table A1.** *Cont.*</caption><thead><tr><th>Coefficients</th><th>Estimate</th><th>SE</th><th>tStat</th><th>p Value</th></tr></thead><tbody><tr><td>x12</td><td>-1.5438</td><td>0.30825</td><td>-5.0083</td><td>9.6246 × 10<sup>-7</sup></td></tr><tr><td>x13</td><td>0.42446</td><td>0.401</td><td>1.0585</td><td>0.29071</td></tr><tr><td>St</td><td>1.2418 × 10<sup>5</sup></td><td>1901.7</td><td>65.3</td><td>7.0906 × 10<sup>-174</sup></td></tr><tr><td>x1: 9</td><td>0.04764</td><td>0.012162</td><td>3.9171</td><td>0.00011217</td></tr><tr><td>x1: t</td><td>-1320.7</td><td>54.246</td><td>-24.346</td><td>1.1118 × 10<sup>-71</sup></td></tr><tr><td>x2: 13</td><td>-0.065347</td><td>0.01489</td><td>-4.3885</td><td>1.607 × 10<sup>-5</sup></td></tr><tr><td>x2: t</td><td>-176.64</td><td>54.038</td><td>-3.2687</td><td>0.0012122</td></tr><tr><td>x9: t</td><td>-170.77</td><td>29.761</td><td>-5.7382</td><td>2.4366 × 10<sup>-8</sup></td></tr><tr><td>x12: t</td><td>666.06</td><td>38.01</td><td>17.523</td><td>3.3209 × 10<sup>-47</sup></td></tr><tr><td>x13: t</td><td>-307.24</td><td>37.886</td><td>-8.1096</td><td>1.5046 × 10<sup>-1</sup></td></tr></tbody></table>

### Details concerning the SVM Regression 1

**Hyperparameters Model:**

- Preset Quadratic SVM

- Kernel function: Cubic.

- Kernel scale 3.001

- Box constraint: Automatic

- Epsilon Auto

- Standardization data: Yes

- Optimizer: Not applicable

### Details concerning model RNN 1

**Hyperparameters Model:**

- Preset Narrow NN

- Number of fully connected layers: 1

- First Layer size 10

- Activation ReLU

- Iteration limit 1000

- Regularization strength (lambda): 0

- Standardization data: Yes

- Optimizer: Not Applicable

The training of RNN 2 is made using the fitrnet function as below:

RegNN = fitrnet(...

- predictors, ...

- response, ...

- 'LayerSizes', [166 280 298], ...

- 'Activations', 'relu', ...

- 'Lambda', 3.6315 × 10<sup>-8</sup>, ...

- 'IterationLimit', 1000, ...

- 'Standardise', true);
Appendix C

<table><caption>Table A2. The layers' orientations of the basic and new random specimens and their differences.</caption><thead><tr><th></th><th>ang1</th><th>ang2</th><th>ang3</th><th>ang4</th><th>ang5</th><th>ang6</th><th>ang7</th><th>ang8</th></tr></thead><tbody><tr><td>PAT2</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td></tr><tr><td>NEW2</td><td>16.80</td><td>-20.05</td><td>19.67</td><td>-22.92</td><td>16.85</td><td>-22.06</td><td>17.68</td><td>-16.53</td></tr><tr><td>DIFF2</td><td>-3.20</td><td>-0.05</td><td>-0.33</td><td>-2.92</td><td>-3.15</td><td>-2.06</td><td>-2.32</td><td>3.47</td></tr><tr><td>PAT6</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td></tr><tr><td>NEW6</td><td>71.47</td><td>-69.27</td><td>68.30</td><td>-70.32</td><td>73.18</td><td>-71.33</td><td>67.00</td><td>-68.95</td></tr><tr><td>DIFF6</td><td>1.47</td><td>0.73</td><td>-1.70</td><td>-0.32</td><td>3.18</td><td>-1.33</td><td>-3.00</td><td>1.05</td></tr><tr><td>PAT9</td><td>45.00</td><td>0.00</td><td>-45.00</td><td>90.00</td><td>45.00</td><td>0.00</td><td>-45.00</td><td>90.00</td></tr><tr><td>NEW9</td><td>46.59</td><td>3.31</td><td>-42.30</td><td>88.52</td><td>48.57</td><td>-1.31</td><td>-44.54</td><td>88.57</td></tr><tr><td>DIFF9</td><td>1.59</td><td>3.31</td><td>2.70</td><td>-1.48</td><td>3.57</td><td>-1.31</td><td>0.46</td><td>-1.43</td></tr><tr><td>PAT11</td><td>0.00</td><td>30.00</td><td>0.00</td><td>90.00</td><td>0.00</td><td>-30.00</td><td>0.00</td><td>30.00</td></tr><tr><td>NEW11</td><td>-3.49</td><td>26.87</td><td>-0.14</td><td>89.83</td><td>2.48</td><td>-33.05</td><td>3.61</td><td>32.53</td></tr><tr><td>DIFF11</td><td>-3.49</td><td>-3.13</td><td>-0.14</td><td>-0.17</td><td>2.48</td><td>-3.05</td><td>3.61</td><td>2.53</td></tr><tr><td></td><td><strong>ang9</strong></td><td><strong>ang10</strong></td><td><strong>ang11</strong></td><td><strong>ang12</strong></td><td><strong>ang13</strong></td><td><strong>ang14</strong></td><td><strong>ang15</strong></td><td><strong>ang16</strong></td></tr><tr><td>PAT2</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td><td>20.00</td><td>-20.00</td></tr><tr><td>NEW2</td><td>23.49</td><td>-20.79</td><td>16.26</td><td>-21.91</td><td>16.71</td><td>-22.76</td><td>17.14</td><td>-22.26</td></tr><tr><td>DIFF2</td><td>3.49</td><td>-0.79</td><td>-3.74</td><td>-1.91</td><td>-3.29</td><td>-2.76</td><td>-2.86</td><td>-2.26</td></tr><tr><td>PAT6</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td><td>70.00</td><td>-70.00</td></tr><tr><td>NEW6</td><td>69.55</td><td>-73.38</td><td>67.92</td><td>-67.99</td><td>73.58</td><td>-72.32</td><td>69.08</td><td>-70.79</td></tr><tr><td>DIFF6</td><td>-0.45</td><td>-3.38</td><td>-2.08</td><td>2.01</td><td>3.58</td><td>-2.32</td><td>-0.92</td><td>-0.79</td></tr><tr><td>PAT9</td><td>90.00</td><td>-45.00</td><td>0.00</td><td>45.00</td><td>90.00</td><td>-45.00</td><td>0.00</td><td>45.00</td></tr><tr><td>NEW9</td><td>90.23</td><td>-48.59</td><td>3.12</td><td>45.28</td><td>92.95</td><td>-41.92</td><td>-1.90</td><td>42.29</td></tr><tr><td>DIFF9</td><td>0.23</td><td>-3.59</td><td>3.12</td><td>0.28</td><td>2.95</td><td>3.08</td><td>-1.90</td><td>-2.71</td></tr><tr><td>PAT11</td><td>30.00</td><td>0.00</td><td>-30.00</td><td>0.00</td><td>90.00</td><td>0.00</td><td>30.00</td><td>0.00</td></tr><tr><td>NEW11</td><td>29.95</td><td>1.22</td><td>-28.86</td><td>-0.13</td><td>92.30</td><td>-0.28</td><td>26.92</td><td>-0.82</td></tr><tr><td>DIFF11</td><td>-0.05</td><td>1.22</td><td>1.14</td><td>-0.13</td><td>2.30</td><td>-0.28</td><td>-3.08</td><td>-0.82</td></tr></tbody></table>

<table><caption>Table A3. Mechanical properties comparison of unidirectional, off-axis oriented, and symmetric balanced composite laminates.</caption><thead><tr><th colspan="4">Unidirectional</th><th colspan="5">Off-Axis Oriented</th><th colspan="4">Symmetric Balanced</th></tr><tr><th>S-1</th><th>S-7</th><th>S-2</th><th>S-3</th><th>S-4</th><th>S-5</th><th>S-6</th><th>S-8</th><th>S-9</th><th>S-10</th><th>S-11</th><th>S-12</th></tr></thead><tbody><tr><td colspan="4">Laminate strength σ<sub>max</sub> (MPa)</td><td colspan="5"></td><td colspan="4"></td></tr><tr><td>2064.24</td><td>63.978026</td><td>1540.1604</td><td>593.258</td><td>192.457</td><td>101.786</td><td>77.31364</td><td>1121.76</td><td>680.072</td><td>499.3297</td><td>1327.031</td><td>540.079</td></tr><tr><td colspan="4">Laminate stiffness E (MPa)</td><td colspan="5"></td><td colspan="4"></td></tr><tr><td>130,393.1</td><td>9267.9938</td><td>87,155.067</td><td>47,161.4</td><td>17,598.4</td><td>10,707.3</td><td>9624.426</td><td>76,708.3</td><td>50,271.8</td><td>36,305.75</td><td>90,502.45</td><td>41,686.8</td></tr><tr><td colspan="4">Laminate strength (First Ply Failure-CLT) (MPa)</td><td colspan="5"></td><td colspan="4"></td></tr><tr><td>2044.3</td><td>63.949701</td><td>490.05462</td><td>215.782</td><td>95.7956</td><td>70.0768</td><td>65.63686</td><td>501.917</td><td>340.917</td><td>228.9543</td><td>628.3132</td><td>280.259</td></tr></tbody></table>
<table><caption>Table A3. *Cont.*</caption><thead><tr><th colspan="11">Laminate stiffness (CLT) (MPa)</th></tr><tr><th colspan="2">Unidirectional</th><th colspan="6">Off-Axis Oriented</th><th colspan="4">Symmetric Balanced</th></tr><tr><th>S-1</th><th>S-7</th><th>S-2</th><th>S-3</th><th>S-4</th><th>S-5</th><th>S-6</th><th>S-8</th><th>S-9</th><th>S-10</th><th>S-11</th><th>S-12</th></tr></thead><tbody><tr><td colspan="12">Young's modulus E11 (MPa)</td></tr><tr><td>130,357.1</td><td>9266.6908</td><td>85,824.211</td><td>45,846.3</td><td>17,082.9</td><td>10,554.5</td><td>9578.134</td><td>76,823.8</td><td>50,489.8</td><td>36,274.97</td><td>90,767.63</td><td>41,706.7</td></tr><tr><td colspan="12">Maximum strain (ε<sub>max</sub>)</td></tr><tr><td>0.01607</td><td>0.0072998</td><td>0.0191447</td><td>0.01745</td><td>0.02818</td><td>0.0100</td><td>0.00823</td><td>0.01558</td><td>0.01518</td><td>0.01556</td><td>0.01539</td><td>0.01480</td></tr><tr><td colspan="12">Maximum stress (σ<sub>max</sub>) (MPa)</td></tr><tr><td>2064.24</td><td>63.9780</td><td>1540.1604</td><td>593.258</td><td>192.452</td><td>101.786</td><td>77.3134</td><td>1121.7</td><td>680.0729</td><td>499.39</td><td>1327.01</td><td>540.074</td></tr></tbody></table>

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