Machine Learning Algorithms for Prediction and Characterization of Cohesive Zone Parameters for Mixed-Mode Fracture

Abstract

Fatigue and fracture prediction in composite materials using cohesive zone models depends on accurately characterizing the core and facesheet interface in advanced composite sandwich structures. This study investigates the use of machine learning algorithms to identify cohesive zone parameters used in the fracture analysis of advanced composite sandwich structures. Experimental results often yield non-unique solutions, complicating the determination of cohesive parameters. Numerical determination can be time-consuming due to fine mesh requirements near the crack tip. This research evaluates the performance of Support Vector Regression (SVR), Random Forest (RF), and Artificial Neural Network (ANN) machine learning methods. The study uses features extracted from load–displacement responses during the fracture of the Asymmetric Double-Cantilever Beam (ADCB) specimen. The inputs include the displacement at the maximum load (δ*), the maximum load ($P_{max}$), the total area under the load–displacement curve ($A_t$), and the initial slope of the linear region of the load–displacement curve (m). There are two objectives in this research: the first is to investigate which method performs best in identifying the interfacial cohesive parameters between the honeycomb core and carbon-epoxy facesheets, while the second objective is to reduce the dimensionality of the dataset by reducing the number of input features. Reducing the number of inputs can simplify the models and potentially improve the performance and interpretability. The results show that the ANN method produced the best results, with a mean absolute percentage error (MAPE) of 0.9578% and an R-squared (R²) value of 0.7932. These values indicate a high level of accuracy in predicting the four cohesive zone parameters: maximum normal contact stress (${\sigma }_I$), critical fracture energy for normal separation ($G_I$), maximum equivalent tangential contact stress (${\sigma }_{II}$), and critical fracture energy for tangential slip ($G_{II}$).

1. Introduction

Composite sandwich structures have become essential in structural engineering for applications where stiffness, strength, and weight optimization are crucial considerations. These adaptable structures, which offer compelling advantages in terms of efficiency and reduced environmental impact, find application in various industries, including wind energy, automotive, aerospace, and marine industries [1,2]. However, despite their outstanding performance, they face a significant obstacle: the complex interaction between the facesheet and core materials and debonding failure modes. Debonding at this interface can significantly affect structural integrity, potentially jeopardizing performance and safety [3]. Researchers have proposed advanced modeling techniques to comprehend and reduce facesheet-to-core debonding in composite sandwich structures.

Cohesive zone models (CZMs) have found wide application in the field for their predictive capabilities in simulating impact events and predicting damage in composite structures [4,5,6,7]. When simulating fracture processes numerically, CZMs are essential because they provide a framework for understanding the complex interactions between the different materials [8]. Fundamentally, the CZM identifies surfaces of discontinuity where abrupt changes in displacement occur and relates these changes to appropriate tractions via traction–separation laws. In addition to simplifying the examination of the bond–slip relationship comprehensively, this method also naturally considers debonding factors like crack initiation, propagation, and arrest. The CZM is adaptable in tackling a broad range of fracture-related problems and has practical applications in different fields, for instance, delamination in composite structures [9,10,11], adhesively bonded joints [12,13], and peeling [14]. The problem of accurately identifying CZM parameters in honeycomb/carbon-epoxy sandwich structures is addressed in this study. Using different ML algorithms, we seek to simplify the determination of these parameters, frequently obscured by non-unique solutions in experimental data and the challenging nature of numerical methods.

CZMs using different constitutive laws [15,16,17,18,19] have been proposed for various material systems. These CZMs offer a valuable toolbox for examining the fracture process in multiple materials because of their distinctive shapes and properties. In the context of composite structures, Akhmet et al. [20] utilized a mixed-mode bilinear CZM to study the stress distribution behavior of corrugated sandwich structures subjected to a three-point bending load. Airoldi et al. [21] applied a trilinear CZM to analyze composite structures’ delamination and crack advancement using a double-cantilever beam test. Han et al. [22] employed a mixed-mode exponential CZM to investigate debonding at the facesheet and core interfaces. Additionally, the authors examined the effects of various loading conditions. In CZMs, specific parameters are crucial for determining the cohesive law, and these parameters can be identified in multiple ways. Researchers have recently proposed various techniques to assess traction–separation characteristics and enable numerical simulations to predict cohesive failure. Several studies in the literature have analytically analyzed CZM parameters, contributing to a deeper understanding of these models.

By considering the elastic behavior in the adhesive, the double-cantilever beam test was proposed by Jumel et al. [23]. Using this modified model, they calculated the double-cantilever beam specimen’s stiffness and critical energy release rate. In addition, based on the beam on elastic foundation theory and assuming the hyperelastic behavior of the adhesive layer, Cabello et al. [24] proposed a model to investigate the mechanical responses of a double-cantilever beam specimen with a thick, flexible adhesive. Cabello et al. [25] proposed an analytical approach based on the elastic foundation approach to predict the fracture behavior of the double-cantilever beam specimen. Simplifying assumptions for some analytical techniques may lead to inaccuracies when dealing with complex geometry, such as sandwich structures with nonlinear materials or intricate failure modes [26]. Inverse identification is another approach for determining CZM parameters. This approach has attracted the attention of numerous researchers. Valoroso et al. [27] researched an inverse method for determining CZM parameters using the DCB test by analyzing the outcomes from various measurement approaches. Jaillon et al. [28] used an indirect method with uncertainty estimation of the CZM constants. Xu et al. [29] identified these mixed-mode fracture parameters using the inverse process. The authors used simulations and experiments to examine the load–displacement curves and the crack path in this investigation. Shen and Paulino [30] investigated using full-field displacement and an optimization strategy within an FE framework to identify the elastic characteristics and CZM parameters.

Recent research highlights the growing application of machine learning (ML) in composite mechanics, materials, and design. ML techniques have been employed to predict the mechanical properties of composites [31], optimize polymer composites [32], and enhance the process simulation [33]. Deep learning has shown promise in improving composite design efficiency [34,35], while micromechanics-based deep learning offers potential for multi-scale modeling [36]. ML has also been applied to experimental solid mechanics and structures [37,38] and alloy design [35,39]. These advanced techniques can enable researchers to address complex challenges such as delamination, damage detection, and vibration analysis [40,41,42,43,44,45,46] with increased accuracy and efficiency. By utilizing extensive datasets, ML algorithms can identify complex patterns and relationships, enhancing our understanding of the mechanical behavior of these materials. Furthermore, ML techniques can precisely characterize the behavior and properties of composite materials by analyzing the intricate relationships between various experimental parameters and the resulting mechanical performance. This approach has been demonstrated to be effective in various studies, showcasing the potential of ML solutions to surpass traditional analytical methods [47,48,49]. These studies demonstrate ML’s ability to handle complex nonlinear systems, process large datasets, and provide accurate predictions for material characterization and design processes. However, challenges remain, including data quality and quantity requirements, and the need for well-designed algorithms suitable for the problems at hand.

Among the various ML algorithms, Artificial Neural Networks (ANNs), Random Forest (RF), and Support Vector Regression (SVR) have shown remarkable efficacy in different applications. For instance, Balcıoğlu and Seçkin [50] compared ANN methods with traditional finite element analysis (FEA) for fracture behavior prediction, demonstrating superior performance in learning from real-world data. Zhang et al. [51] established a method coupling FEA with ML to predict the mechanical properties of composite laminates, including failure factors under random stress states and critical buckling eigenvalues. Their study demonstrated that both ANN and RF models could learn from large samples generated by parametric FEA models, providing predictions consistent with FEA values. Although ANN models achieved smaller root-mean-square errors, RF models offered faster learning processes, proving the effectiveness of both methods in predicting mechanical properties. Additionally, Liu et al. [52] employed SVR with multi-domain features to localize low-velocity impacts on composite plates, achieving precise predictions. Their method, which utilized empirical mode decomposition for signal preprocessing and the bat algorithm for optimization, demonstrated significant improvements in localization accuracy on a carbon fiber reinforced plastic (CFRP) plate using fiber Bragg grating sensors. Furthermore, Lu et al. [53] utilized SVR to predict damage degrees in CFRP structures, highlighting its robustness in handling complex datasets. These studies collectively underscore the versatility and effectiveness of ML algorithms in advancing composite material research, offering robust solutions that often outperform traditional methods. Moreover, Sikdar et al. [54] investigated the identification and categorization of the damage for carbon-fiber reinforced composite panels using neural networks and deep learning. The authors proposed a deep learning architecture based on neural networks to automatically extract damage characteristics from images and utilize them to categorize damage-source locations in the composite panel. Viotti and Gomes [55] studied delamination identification for composite sandwich structures using Artificial Neural Networks, Random Forests, decision trees, and Support Vector Machine approaches. The researchers used algorithms to detect, locate, and size damages in the structure’s core and facesheets. These studies show how machine learning can improve the analysis, design, and monitoring of composite structures.

A significant application of ML in composites is the optimization of cohesive zone model (CZM) parameters. Traditionally, determining these parameters required extensive computational resources and often led to non-unique solutions. However, ML approaches have revolutionized this process. For example, Poblete et al. [8] directly measured rate-dependent traction–separation laws for cohesive zone modeling, enhancing the accuracy of fracture simulations. Tao et al. [56] proposed a CZM driven by an ANN for simulating Mode-I damage propagation, where the traction–separation relationship is calculated by a multi-layer perceptron. Liu et al. [57] demonstrated that machine learning models, including regression trees and neural networks, can provide accurate results for fracture toughness measurements, with neural networks offering superior simplicity and reliability compared to regression trees. Furthermore, ML has been instrumental in enhancing the predictive maintenance and structural health monitoring of composite structures. Qing et al. [58] reviewed various ML applications in this area, illustrating how these techniques streamline inspection processes and improve the reliability of composite materials. Additionally, Freed et al. [12] implemented a probabilistic ML strategy to predict the failure of adhesively bonded joints using CZM, showcasing the potential of ML to handle complex adhesive behaviors under different loading conditions. By applying Artificial Neural Networks, Su et al. [59] researched CZM parameters for composites bonded to concrete using ANN to identify two parameters on the CZM model. Zhao et al. [60] introduced a novel method for determining CZM parameters using deep neural networks and die shear tests, demonstrating high accuracy in predicting and identifying key CZM parameters such as maximum shear traction strength, separation displacement of the interface, and interface stiffness.

In this study, we aim to improve the identification of CZM parameters for mixed-mode fractures in composite sandwich structures by employing and comparing three specific machine learning algorithms: SVR, RF, and ANN. We developed a finite element model of the Asymmetric Double-Cantilever Beam (ADCB) specimen to simulate mixed-mode fracture behavior and generate a database from characteristic features of the load–displacement responses when varying the cohesive zone parameters. This database is used to train and optimize the machine learning algorithms to predict four critical CZM parameters: maximum normal contact stress (${\sigma }_I$), critical fracture energy for normal separation ($G_I$), maximum equivalent tangential contact stress (${\sigma }_{II}$), and critical fracture energy for tangential slip ($G_{II}$). Our objective is to determine which machine learning method best predicts these parameters and identify the most influential input features, thereby enhancing the predictive capabilities of fracture analysis in composite sandwich structures.

2. Cohesive Zone Modeling

In this study, we identify the CZM parameters based on an analysis of the load–displacement curve from the ADCB specimen tests. The ADCB is a simple but effective specimen for measuring fracture properties where the position of the crack in the specimen results in a different percentage of model-I and mode-II load. Unlike linear relationships, this method uses FE analysis findings to construct a series of ML models. Our suggested process consists of four major stages: (1) Data collection: We collected data from the ADCB test using the FE model-generated load–displacement curves. These data serve as a basis for our further modeling attempts. (2) Data preprocessing: Before model construction, we preprocessed the data to ensure the quality and effectiveness of the data for ML algorithms. This process involves data cleaning, feature selection, and data scaling. Our models use the displacement at the maximum load (${\delta }^{*}$), maximum load ($P_{max}$), total area under load–displacement curve ($A_t$), and the initial slope in the linear region of the load-displacement curve ($m$) as the four input parameters for the ML algorithms. (3) Model training and optimization: In this stage, we investigate three different ML algorithms, namely, Support Vector Machine (SVM), Random Forest (RF), and Artificial Neural Network (ANN). These algorithms are trained and optimized to capture the most essential data correlations. (4) Prediction and identification: These ML methods are processed to identify the CZM parameters.

2.1. Cohesive Zone Model

In this study, we used the sandwich ADCB specimen to assess the fracture behavior occurring at the interface between the facesheet and core materials of the structure. Due to the geometrical characteristics of the sandwich structure and the differing materials in the facesheets and core, mixed-mode loading comprising both Mode I and Mode II is induced, rather than purely Mode I loading. Numerical simulations of the ADCB specimen with a linear softening law (Figure 1) for CZM are implemented in the ANSYS Workbench 2022R2 [61]. The CZM relied on a quadratic stress criterion to simulate the initiation of damage [62,63]:

$${\left({\displaystyle \frac{{\sigma }_I}{{\sigma }_{u, I}}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_{II}}{{\sigma }_{u, II}}}\right)}^2=1$$

(1)

(${\sigma }_I$, ${\sigma }_{II}$) represent the loading components for modes I and II, while (${\sigma }_{u, I}$, ${\sigma }_{u, II}$) are the local strengths. Also, the softening relation during propagation is as follows:

$${\sigma }_m=(1-d)k{\delta }_m$$

(2)

where

$${\delta }_m=\sqrt{{\delta }_I^2+{\delta }_{II}^2}$$

(3)

In the above equations, k is the interfacial stiffness, ${\delta }_m$ is the equivalent mixed-mode I + II relative displacement, and ${\sigma }_m$ is the resulting mixed-mode I + II traction. The calculation of the damage parameter d involves taking the ratio between the calculated actual accumulated dissipated energy ($G_{Td}$) and the critical total fracture energy ($G_{Tc}$) specific to the current mode ratio (Figure 1). The linear energetic loading criteria, including Modes I and II, simulated for damage propagation is:

$${\displaystyle \frac{G_I}{G_{Ic}}}+{\displaystyle \frac{G_{II}}{G_{IIc}}}=1$$

(4)

which enables the definition of the $G_{Tc}$ for each mode ratio $\beta =G_{II}/G_T$ as being:

$$G_{Tc}={\displaystyle \frac{G_{Ic}{G}_{IIc}}{\left(1-\beta \right)G_{IIc}+\beta G_{Ic}}}$$

(5)

Figure 1 provides a schematic representation of the Asymmetric Double-Cantilever Beam (ADCB) specimen used to study the fracture behavior of composite sandwich structures. The inset graph illustrates the linear softening cohesive law, which defines the relationship between applied traction and relative displacement. The maximum normal contact stress (${\sigma }_I$) is the peak value on the y-axis, indicating the highest stress before softening begins under normal separation (mode I). The critical fracture energy for normal separation ($G_I$) is represented by the area under the curve on the left side, up to the point where ${\delta }_{max}$ occurs on the x-axis, showing the energy needed to propagate a crack due to normal separation. Similarly, the maximum equivalent tangential contact stress (${\sigma }_{II}$) is the peak stress on the y-axis for tangential slip (mode II), while the critical fracture energy for tangential slip ($G_{II}$) is represented by the area under the curve up to ${\delta }_{um}$ for tangential displacement. The graph captures how stresses (${\sigma }_m$) vary with relative displacements (${\delta }_{om}$, ${\delta }_{max}$, and ${\delta }_{um}$), and how the areas under the curve correspond to the critical fracture energies ($G_I$ and $G_{II}$), providing a comprehensive understanding of the traction–separation relationship crucial for predicting fracture behaviors in composite sandwich structures.

2.2. Numerical Simulation of the ADCB Specimen

The finite element model with the cohesive zone model can effectively represent the fracture characteristics of asymmetric structures with different materials, including composite sandwich structures with honeycomb cores and carbon fiber facesheets. In this context, we have developed a nonlinear ADCB finite element model with CZM to capture the interfacial failure process of the ADCB specimen. The model is then compared to experimental results in the literature. The material properties for both core and facesheets are presented in

Table 1. Elastic properties of the facesheets and core of the CFRP/NOMEX sandwich structure; $L$ = 230 mm, $a_0$ = 50 mm, $h_c$ = 10 mm, $h_s$ = 50 mm [62].

Material	Properties
	${\mathit{E}}_{\mathbf{1}}$	${\mathit{E}}_{\mathbf{2}}$	${\mathit{E}}_{\mathbf{3}}$	${\mathit{v}}_{\mathbf{12}}$	${\mathit{v}}_{\mathbf{13}}$	${\mathit{v}}_{\mathbf{23}}$	${\mathit{G}}_{\mathbf{12}}$	${\mathit{G}}_{\mathbf{13}}$	${\mathit{G}}_{\mathbf{23}}$
CFRP	$130 \mathrm{G}\mathrm{P}\mathrm{a}$	$8200 \mathrm{M}\mathrm{P}\mathrm{a}$	$8200 \mathrm{M}\mathrm{P}\mathrm{a}$	$0.27$	$0.27$	$0.41$	$4100 \mathrm{M}\mathrm{P}\mathrm{a}$	$4100 \mathrm{M}\mathrm{P}\mathrm{a}$	$3200 \mathrm{M}\mathrm{P}\mathrm{a}$
NOMEX	$0.45 \mathrm{M}\mathrm{P}\mathrm{a}$	$0.45 \mathrm{M}\mathrm{P}\mathrm{a}$	$258 \mathrm{M}\mathrm{P}\mathrm{a}$	$0.9956$	$0.0005$	$0.0005$	$0.11 \mathrm{M}\mathrm{P}\mathrm{a}$	$38.62 \mathrm{M}\mathrm{P}\mathrm{a}$	$63.12 \mathrm{M}\mathrm{P}\mathrm{a}$

. Our approach involves a numerical simulation of the ADCB sandwich structure specimen, as shown in Figure 1, using the CZM with a linear softening law based on fracture energy. To effectively capture the mixed-mode fracture behavior of the sandwich structure, this tool requires the definition of four essential parameters: maximum normal contact stress (${\sigma }_I$), critical fracture energy for normal separation ($G_I$), maximum equivalent tangential contact stress (${\sigma }_{II}$), and critical fracture energy for tangential slip ($G_{II}$). In our analysis, the first phase involves using geometry tools to produce an accurate model of the structure to accurately depict the design, dimensions, and layer orientations. A uniform element size of 1.5 mm was selected, discretizing the model into 8405 elements and making it possible for the simulation to capture stress concentrations accurately and reliably. With the help of this module, composite materials can be made with the desired orientations, stacking sequences, and properties. The NOMEX^® Honeycomb core and facesheets made of carbon-fiber-reinforced polymer (CFRP) prepreg were precisely modeled using ACP, considering their material characteristics.

2.3. Geometry and Materials Properties

The sandwich structure consists of unidirectional facesheets comprising ten layers of prepreg carbon-fiber-reinforced polymer (CFRP) prepreg (SEAL^® Texipreg HS 160 RM). This type of material offers high strength-to-weight ratios, exceptional stiffness, and superior fatigue resistance. These properties are vital for applications where structural integrity, weight, and durability are paramount. Each skin has a thickness of 1.5 mm. The core material is NOMEX^® Honeycomb with a thickness of 10 mm. The honeycomb structure provides both lightweight and robust support to the facesheets. Table 1 shows the material parameters of the facesheets and core. Figure 1 illustrates a schematic of the ADCB test for composite sandwich structure. The height of the facesheets and core are $h_f$ and $h_c$, respectively, and the initial crack is depicted by $a_0$.

3. Machine Learning Algorithms

This study uses three ML algorithms (SVR, RF, and ANN) to train and predict the cohesive zone parameters. These algorithms have three main stages: data input, a learning algorithm, and output. The algorithm learns from historical data to generalize and predict new and unseen data, allowing continuous improvement. SVM algorithms are generally practical for binary classifications because they are based on mathematical optimization. On the other hand, RF is an ensemble method that uses multiple decision trees to improve accuracy and reduce overfitting. Finally, ANN is a deep learning method that uses multiple interconnected layers resembling the neuron connections in the human brain but typically requires large datasets. A description of the methods used is defined below.

3.1. Support Vector Regression

The Support Vector Regression (SVR) approach extends the Support Vector Machine (SVM) to build a machine learning model. SVM is known for its efficiency, and the most widely used SVR model is the ε-SVR introduced by Vapnik [64]. Notably, prior researchers have demonstrated the successful application of the SVR algorithm within the area of composite structures [52,53,65]. For our dataset, we have defined them as $\left\{\left(x_1,y_1\right),\left(x_2,y_2\right),\dots , \left(x_i,y_i\right),\dots , \left(x_N,y_N\right) \right\},$ where each $x_i$, is an input vector and its associated, $y_i$ vector represents the corresponding output value. It is used to find a mapping function $f\left(x\right)$ to describe the correlation of input and output values is the regression problem that needs to be solved. In our application, the input vector $x_i$ is a collection of four parameters: the displacement at the maximum load (${\delta }^{*}$), the maximum load ($P_{max}$), the total area under the load–displacement curve ($A_t$), and the initial slope of the linear region of the load–displacement curve ($m$). Correspondingly, the output $y_i$ is a set of four values: ${\sigma }_I, { G}_I, {\sigma }_{II},$ and $G_{II}$. The formulation for the SVR approach in the linear regression is defined as follows [65]:

$$y=f\left(x\right)=w\cdot {\phi }_x+b$$

(6)

where ${\phi }_x$, $w=\left\{w_1,\dots ,w_i,\dots ,w_N\right\}$, and $b$ are the mapping function, the adjustable weight vector, and the adjustable bias parameter, respectively. The SVR optimization problem can be written as follows:

$$\min {‖w‖}^2+C{\sum }_{i=1}^n\left({\xi }_i+{\xi }_i^{*}\right)$$

(7)

$$subject to \left\{\begin{array}{c}{y}_i-w\cdot \phi \left(x_i\right)-b\le \epsilon +{\xi }_i\\ {-y}_i+w\cdot \phi \left(x_i\right)+b\le \epsilon +{\xi }_i^{*}\\ {\xi }_i,{\xi }_i^{*}>0 \end{array}\right. i=1, 2, \dots , n$$

(8)

where $\epsilon$ corresponds to the width of the ε-tube, which defines a zone around the predicted values where errors are allowed without penalty. C is a penalty factor, and ${\xi }_i$ and ${\xi }_i^{*}$ are slack variables that measure the distances between the predicted value and the nearest boundary of the ε-tube. Figure 2 is a simplified schematic of the Support Vector Regression (SVR) model for multi-dimensional parameter analysis. The axes represent a combination of the multiple input parameters (${\delta }^{*}, P_{max}$, $A_t$, $m$) and the predicted output parameters (${\sigma }_I, G_I, {\sigma }_{II}, G_{II}$), respectively. Each data point represents a distinct combination of inputs that result in corresponding outputs. The regression line represents the prediction of the SVR model, with support vectors indicating the maximum margin. This figure is only a conceptual representation; the model handles each parameter in a multidimensional space.

3.2. Random Forest

Classification and regression problems can be addressed using the ensemble machine learning technique called Random Forest (RF) [66,67]. Bootstrapping is one of the fundamental concepts in RF. This procedure describes randomly selecting samples using replacement to create n subsamples from the original dataset. This process implies that some samples might be chosen more than once in a subset while others might not. All these bootstrapped datasets train a single decision tree for RF. The randomness that RF adds to the feature selection process is one of its primary characteristics. It merely considers a random subset of characteristics rather than all features at each node to identify the optimal split. Every bootstrapped subsample provides the foundation for an individual decision tree, like Tree 1, Tree 2, Tree 3, and so on. Though part of the same forest, these trees are distinct in their structure and decision pathways. When all the decision trees are built, the regression prediction for a new data point can be obtained by aggregating the predictions from individual trees.

In our problem, the algorithm draws a bootstrap sample from our dataset, which includes specific values for ${\delta }^{*}$, $P_{max}$, $A_t$, and $m$. Next, for each node in a tree, the model randomly selects a subset of our features (${\delta }^{*}$, $P_{max}$, $A_t$, and $m$). For instance, one node may prioritize splitting data depending on load values, while another may prioritize the area under the curve. Each tree is trained to understand the relationships between these features and the ${\sigma }_I, G_I, {\sigma }_{II}$ and $G_{II}$ values. The schematic representation of the RF model is indicated in Figure 3. The mathematical formulation for the regression method is as follows:

$$f^N\left(x\right)={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{y}_i$$

(9)

where $f^N\left(x\right)$ is the predicted output for a new data point, N is the number of trees in the RF, and $y_i$ is the prediction from the ith tree.

3.3. Artificial Neural Network

Artificial Neural Networks (ANNs) are popular machine learning algorithms widely used in engineering applications [68,69]. ANNs are networks of artificial or natural neurons connected by a computational or mathematical model used to process information. This algorithm consists of three essential parts: the input, hidden, and output layers. The input layer, the first part of an Artificial Neural Network, collects and encodes raw data or characteristics from external sources. Each neuron in this layer represents a different input characteristic and acts as the network’s entrance point for information. The hidden layers of an Artificial Neural Network are the intermediary levels placed between the input and output layers. Hidden layers use interconnected neurons to carry out extensive mathematical operations on the input data, enabling the network to identify complex patterns and correlations. The output layer is the last layer of an Artificial Neural Network. It is in the position of making predictions and classifications based on the data processed in the hidden layers. Each of our inputs (${\delta }^{*}$, $P_{max}$, $A_t$, $m$) is represented as activation units from $a_1^{in}$ to $a_4^{in}$ (Figure 4). Each unit will process the input through their weights $W_{m, d}^{in}$. Each neuron in the hidden layer, represented as $a_d^1$ to $a_f^3$, processes information from the previous layer through connection weights like $W_{d, e}^1$ and transmits it forward. These weights determine the significance or influence of one neuron’s output on another. By training this network on our dataset, these weights adjust themselves to optimize the prediction of outputs (${\sigma }_I, G_I, {\sigma }_{II}$ and $G_{II}$) based on the inputs (${\delta }^{*}$, $P_{max}$, $A_t$, and $m$).

This study uses a three-layer ANN machine-learning technique, as depicted in Figure 4. Within this framework, $x_i$ represents the ${\delta }^{*}$, $P_{max}$, $A_t$, and $m$ values, while $y_i$ denotes ${\sigma }_I, G_I, {\sigma }_{II}$, and $G_{II}$ values. The start of the $i$th neuron, in the $l$th level, is denoted as $a_i^l$, and the connections between the $k$th sector in level $l$ and the $j$th sector in the subsequent level ($l+1$) are represented by $w_{k,j}^l$. To illustrate, $a_i^{in}$ and $a_i^{out}$ correspond to the activation units. The output in level ($l+1$) is a function of the sum of their weighted inputs.

$$a^{l+1}=f(a^l{W}_k^l+b^l)$$

(10)

In Equation (6), the activation function denoted as $f$ determines whether to activate the neuron, while $b^l$ is a vector representing bias for the ($l+1$)th level. $a^l$ is a vector with the activation units for level l. Furthermore, Equation (8) defines $W^l$ as a weight matrix with dimensions $r\times s$, where $s$ is the number of units in the $(l+1)$th layer.

$$a^l=\left[a_1^l, a_2^l,\dots , a_r^l\right]$$

(11)

$$W^l=\left[\begin{array}{ccc}{w}_{1, 1}^l& \cdots & w_{1, s}^l\\ ⋮& \ddots & ⋮\\ w_{r, 1}^l& \cdots & w_{r, s}^l\end{array}\right]$$

(12)

4. Results and Discussion

4.1. Validation of Modeling Approach

The finite element (FE) results of the ADCB specimen with the cohesive zone model are compared with the experimental results to show the ability of the model to capture the fracture propagation behavior. Since the load–displacement curve is a required input for the ML methods utilized in the present study, we conducted validation to compare our FE outcomes for this curve with those in the existing literature. The load–displacement curve is a graphical representation that depicts the relationship between the applied load and the resulting displacement or deformation in the structure. It can be seen that in Figure 5, the present results are in reasonable agreement with the experimental test, which is presented in [62].

4.2. Dataset Collection for Cohesive Zone Model Parameters

The CZM requires the specifications of four parameters for mixed-mode fracture properties. The following parameter ranges were used based on previous research and practical experience: ${\sigma }_I$ runs from 0.15 MPa to 0.35 MPa, $G_I$ is between 280 $\mathrm{J}/{\mathrm{m}}^2$ and 600 $\mathrm{J}/{\mathrm{m}}^2$, ${\sigma }_{II}$ is between 1.6 MPa and 3.5 MPa, and $G_{II}$ is between 700 $\mathrm{J}/{\mathrm{m}}^2$ and 980 $\mathrm{J}/{\mathrm{m}}^2$. A series of finite element nonlinear analyses were performed to capture the debonding for this dataset. We extracted four key parameters from the load–displacement curves for use as input features in our machine learning algorithms: the maximum load, displacement at maximum load, the area under the curve, and the initial slope of the linear part of the load–displacement curve. Our database contains 51 load–displacement curves from this modeling, with some depicted in Figure 6.

4.3. Hyperparameters Tuning

The next step in our approach is hyperparameter tuning, which involves systematically adjusting the settings and configurations of a machine learning algorithm to optimize its performance on the given dataset. Hyperparameters are parameters not learned from the data but set before training. We discuss the techniques we used in this process for each ML algorithm. In SVR hyperparameter tuning, the $\mathit{\epsilon }$ value and the box constraint are essential for optimal performance. The error tolerance is controlled by $\epsilon$, which establishes the margin size so that errors do not penalize the model. This affects the model’s sensitivity to variations in the training set.

In contrast, the box constraint is pivotal in balancing the tradeoff between the model’s complexity and its ability to fit training data errors. Its correct calibration ensures that the model neither overfits by capturing unnecessary noise nor underfits by being too generalized. As shown in

Table 2. Hyperparameter tuning for the SVR model for predicting the cohesive zone parameters.

$\mathit{\epsilon }$	Mean R2	Mean MAPE	Box Constraint	Mean R2	Mean MAPE
0.01	0.5308	6.0558	0.1	0.4658	7.2371
0.02	0.5376	6.1988	0.7	0.5359	6.2742
0.03	0.5233	7.1526	1	0.5376	6.1987
0.04	0.5168	7.4750	2	0.5424	6.3128
0.05	0.5024	7.8319	3	0.5380	6.4652

, which thoroughly examines how these configurations affect the behavior of the SVR model, different combinations of epsilon and box constraint values are presented in detail. The MAPE and R² are shown and compared in this table.

The results of a K-fold cross-validation hyperparameter tuning procedure for the Random Forest method are presented in

Table 3. Cross-validated MAPE scores of the eight RFs with different tree levels for optimization of cohesive zone parameter determination.

No.	No. of Trees	Number of Folds (%)										Mean (%)
		1	2	3	4	5	6	7	8	9	10
1	5	8.4952	9.6051	11.9662	8.6994	7.4333	9.2395	6.1857	9.9078	11.5513	11.249	9.4332
2	10	8.7319	10.6674	9.7386	8.9025	9.3058	11.1510	9.1339	7.0797	9.5477	10.2851	9.4544
3	36	8.4919	10.4474	8.5306	10.7308	11.5945	8.2028	8.0728	10.4431	10.0279	10.8345	9.7376
4	82	8.4929	11.4597	9.1358	9.6219	9.7889	8.4278	11.4191	5.7201	10.1255	7.8462	9.2038
5	140	8.4924	9.8091	7.1622	10.7415	7.4258	10.4358	8.9860	10.9316	9.8295	10.3503	9.4167
6	260	8.4918	11.4592	11.1199	7.7427	9.6577	10.3049	7.2246	9.4013	9.8909	7.2202	9.2517
7	480	8.4918	10.2073	8.7420	8.9150	8.7455	8.3842	11.089	10.9269	13.2572	13.8415	10.2604
8	650	8.4918	10.7436	9.9257	8.5966	12.4392	6.6686	10.0478	6.5755	11.3680	8.3010	9.3161

. Finding the model configuration with the best Mean Absolute Percentage Error (MAPE) requires optimizing the hyperparameters. The MAPE measures predictive accuracy, whereby the lower the MAPE, the more effective the model. With MAPE as the primary assessment metric, we investigated how changing the number of trees in the Random Forest affected the model’s performance. The investigation used a grid search technique to examine the ensemble’s performance for tree counts ranging from 5 to 650. It is important to note that the quantity of trees directly impacts the stability of the predictions in the RF [70,71]. Although a single decision tree can be susceptible to noise in its training data, an ensemble of trees can mitigate individual errors, leading to a more robust and stable model. However, there is a tradeoff that a model with too few trees may have a high variation, and a model with too many trees may have computational inefficiencies without improving accuracy. One important finding from our analysis is that, based on the lowest MAPE value, 82 seems to be the ideal number of trees for this model. This suggests that the model’s average prediction accuracy is higher than for other configurations. The performance of each design is analyzed over ten distinct cross-validation folds.

Furthermore, as the number of trees in the model increases, we observe that the MAPE tends to stabilize when evaluated without employing the k-fold technique. However, the introduction of k-fold cross-validation introduces variability in the MAPE values. The reason for this variation is that the method used in k-fold cross-validation chooses different subsets of data for testing and training. This naturally introduces diversity into the results, emphasizing the impact of different data subsets on the model’s accuracy.

In the ANN algorithm, the configuration of network layers and the number of neurons in each layer are considered hyperparameters. Optimizing the model features is also called hyperparameter tuning. The dataset is initially divided into training and testing subsets to aid this process. In the current study, we use 80% of the dataset for the training. In our attempt at hyperparameter tuning, we use a tenfold cross-validation approach. This approach allows for avoiding the overfitting of the machine learning algorithm. With this validation method, the dataset for the training is randomly divided into ten unique, non-overlapping subsets, known as folds. During each iteration, nine folds are utilized to train the model, while one fold is an independent evaluation set. This process is iterated ten times, yielding ten performance assessments for the training dataset. Finding the model configuration with the best MAPE requires optimizing the hyperparameters for our three-hidden-layer Artificial Neural Network (ANN). We used k-fold cross-validation throughout the hyperparameter optimization process to ensure our findings were reliable and robust. We employed a random search strategy to manage the computational complexity of the extensive search space, randomly selecting hyperparameter combinations for evaluation. Using this method, we effectively explored the vast hyperparameter space and optimized the ANN’s architecture without considering every possible variety. It is a sensible option for hyperparameter tuning in complex machine-learning models because it balances computational resources and model performance. The best-performing hyperparameter configurations are listed in

Table 4. MAPE levels of the eight ANNs studied with three hidden layers.

Number	Design	Number of Folds (%)										Mean (%)
		1	2	3	4	5	6	7	8	9	10
1	(24, 136, 68)	0.9660	1.0791	1.0517	0.9845	1.0195	1.1307	1.0552	1.0303	1.0629	1.0409	1.0321
2	(128, 128, 64)	1.0464	1.0807	0.9970	1.0213	1.0448	1.0183	1.0091	1.0912	1.0286	1.0598	1.0417
3	(64, 128, 32)	0.9523	1.1049	1.0135	1.0064	0.9675	1.1326	1.1764	1.0116	1.0382	1.0859	1.0489
4	(64, 32, 16)	0.9965	0.9698	1.0894	1.1165	1.0789	0.9931	1.0096	1.0806	1.2289	1.0737	1.0518
5	(25, 14, 30)	1.0566	1.2809	1.0140	1.0373	0.9576	1.1982	1.0863	0.9965	1.0097	1.0804	1.0718
6	(6, 12, 18)	1.1260	1.1050	0.9924	1.4297	0.9799	1.0727	1.0918	1.0369	1.0854	1.0713	1.0991
7	(12, 12, 12)	1.1769	1.2491	3.1212	1.2077	1.1493	1.1373	1.1298	1.0434	0.9577	1.0871	1.3260
8	(4, 4, 16)	1.0675	1.0979	2.2337	1.4208	1.0566	2.6262	1.6814	1.1751	1.0834	1.0336	1.4476

, along with the corresponding MAPE values. The leading design (24, 136, and 68) comprises three layers, achieving an average MAPE of 1.0321%. Based on these results and in conjunction with the purelin transfer activation function, we selected the (24, 136, and 68) design as the machine learning model for our study.

4.4. Evaluation of Machine Learning Algorithms

In the evaluation of the performance of the ML algorithms, we compare the actual and predicted output for the training and test sets in the SVR, RF, and ANN algorithms, respectively. This comparison provides a direct perception of each algorithm’s accuracy in approximating the CZM parameters. Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 illustrate a comparative analysis between the CZM parameters predicted by the machine learning models and the true values during the training and testing phases. In the context of machine learning, regression lines are used to illustrate how well the predicted values from the models align with the actual observed values. By plotting the predicted values against the actual values and drawing a regression line, we can visually assess the accuracy and reliability of the models. A perfect model would have all points lying on the 45-degree line (y = x), indicating that the predicted values match the actual values perfectly. By comparing these plots, we can identify which algorithm provides the most accurate predictions.

The results display how closely the model matches the actual dataset and shed light on the model’s variability. Compared to the ground truth values for the training and testing sets, the results show that all the machine learning models produce remarkably accurate predictions. This substantial agreement highlights the effectiveness of the machine learning method for CZM parameter identification, which primarily relies on the load–displacement responses of the ADCB. Notably, it is important to note that the training sets are randomly chosen. It is also worth mentioning that the training set constitutes 80% of the data in this study.

These algorithms were chosen among many available options because they represent a diverse set of approaches—SVR for its margin maximization in high-dimensional spaces, RF for its ensemble learning and feature importance capabilities, and ANNs for their deep learning and non-linear modeling strengths. By combining these algorithms, the study aims to leverage their individual strengths to identify the most accurate and efficient method for predicting CZM parameters, ultimately enhancing the predictive capabilities for fracture analysis in composite sandwich structures. The computational resources required for each method vary significantly. SVR generally requires less computational power due to its efficiency in handling smaller datasets and fewer hyperparameters. RF, while being more resource-intensive due to the need to train multiple decision trees, provides a balance between computational load and model accuracy. The most computationally demanding are ANNs, which require substantial resources for training because of their complex architecture and numerous parameters.

Figure 7 and Figure 8 display comparative regression analyses of the SVR model for training and testing datasets, respectively. In these figures, there is a noticeable variance in the ${\sigma }_{II}$ parameter during both the training and testing. Certain areas in the training set where the data points diverge from the reference line suggest possible weaknesses in the model. Comparably, the testing set exhibits noticeable variations in particular regions, which may indicate that the model could not fully capture the intricacies of the training data. Although many of the training data points for $G_{II}$ closely match the reference line, there are some deviations, particularly as values rise. Similar trends may also be seen in the testing data, with predictions being less reliable in particular value ranges. These variations suggest the SVR model could face challenges in capturing specific ranges. Thus, the SVR model demonstrates good performance for ${\sigma }_I$ and $G_I$ while there are evident challenges with the ${\sigma }_{II}$ and $G_{II}$ parameters. These variations could be attributed to inherent complexities in the dataset. It is worth noting that this phenomenon can also be seen in Figure 13, which indicates the R² and MAPE values, and it reveals that SVR underperforms compared to ANN and RF for ${\sigma }_{II}$ and $G_{II}$.

The training set for the RF model is displayed in Figure 9, where the predicted and actual outputs for ${\sigma }_I$, $G_I$, ${\sigma }_{II}$, and $G_{II}$ are well aligned. The test set is shown in Figure 10, with data points largely following the reference line but showing minor variations in $G_{II}$. Both figures indicate the RF model’s predictive capability with slight variations in testing. While the Random Forest (RF) algorithm demonstrated satisfactory performance during the training phase, as indicated by the close alignment between the predicted and actual outputs for the training dataset, its performance declined noticeably during the testing phase, particularly for the parameters ${\sigma }_{II}$ and $G_{II}$. This discrepancy can be attributed to several factors. Firstly, the RF algorithm may have overfitted the training data, capturing noise and specific patterns that do not generalize well to new, unseen data. Overfitting is a common issue when the model is excessively complex or when there are not enough training data to robustly capture the underlying trends. Secondly, differences in the distribution of the training and testing datasets may have impacted the model’s accuracy. If the testing data include scenarios or parameter ranges not well represented in the training set, the model’s predictions may be less accurate. Additionally, the feature importance analysis revealed that certain features significantly influence the RF model’s predictions. The model’s reliance on these features might lead to variability in performance, especially if these features do not generalize well to the testing data. Despite hyperparameter tuning, the selected parameters for the RF model might not be optimal for capturing the complex relationships in the testing data. Further fine-tuning or alternative approaches such as cross-validation with a more diverse dataset may be necessary. These factors suggest that while the RF model can effectively learn from the training data, its generalization capability is limited. Future work can focus on addressing these issues by exploring techniques to mitigate overfitting, ensuring a more representative dataset, and further refining the model’s hyperparameters.

Figure 11 and Figure 12 show comparative regression analyses of the ANN model for training and testing datasets, respectively. Both figures indicate scatter plots of actual versus predicted outputs, with the red reference line showing an ideal one-to-one correspondence. Figure 11 demonstrates that throughout the training phase, the data points for outputs ${\sigma }_I, G_I, {\sigma }_{II},$ and $G_{II}$ are closely clustered around the reference line, indicating a well-fitting model. The closeness of data points to the reference line for each output in Figure 12 also shows the model’s resilience and capacity for generalization during testing.

A comparative analysis of the three ML algorithms uses the R-squared ($R^2$) and the Mean Absolute Percentage Error (MAPE) metrics. R² measures the proportion of variance in the dependent variable explained by the independent variables, indicating the model’s goodness of fit. In contrast, the MAPE (Mean Absolute Percentage Error) evaluates prediction accuracy by expressing the average absolute percentage error, providing a clear measure of forecasting performance. The normalized index $R^2$, shown in Equation (13), is frequently used to assess algorithms for various problems.

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{\sum _{i=1}^n{(y_i-\overline{y})}^2}}$$

(13)

where n is the number of data points, $y_i$ represents the actual CZM parameter values ${\widehat{y}}_i$ represents the predicted CZM parameter values, and $\overline{y}$ is the mean of the actual CZM parameter values. The method’s accuracy typically increases as the index $R^2$ gets closer to 1. Another important performance metric in evaluating the accuracy of our machine learning algorithms is the Mean Absolute Percentage Error (MAPE). The percentage deviation between predicted and actual CZM parameters is measured comprehensively by MAPE.

$$MAPE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|\times 100\%$$

(14)

It should be mentioned that the method’s accuracy tends to improve as the MAPE value approaches 0%. Notably, because the unit and magnitude of the CZM parameters differ, the comparison of Root Mean Square Error (RMSE) is not included in this study because RMSE is sensitive to the scale of the data [65]. Figure 13 compares the $R^2$ and MAPE values produced by three distinct ML methods. All three methods consistently have $R^2$ values greater than 0.83 for ${\sigma }_I$. Notably, ANN has an $R^2$ value of almost 0.98, which is the highest. $R^2$ values for all methods continue to exceed 0.7 in $G_I$. SVR has the greatest $R^2$ value of 0.85 in this output. However, for ${\sigma }_{II}$ and $G_{II}$, a noticeable trend appears. Only the ANN algorithm maintains an $R^2$ value near to one, whereas the $R^2$ values of the other two methods decline significantly. When the performance of these three methods is compared, ANN constantly surpasses the other two, especially for ${\sigma }_{II}$ and $G_{II}$, where it achieves near-perfect fit with a $R^2$ value close to 1. Overall, the average $R^2$ for all outputs is 0.7932 for ANN, 0.5224 for SVR, and 0.4908 for RF. Similar trends emerge when comparing the MAPE values generated by three machine learning approaches. The MAPE values for the ANN algorithm consistently outperform those of the other two methods across all outputs. Notably, the MAPE value for the RF algorithm increases by about 16% in ${\sigma }_{II}$, making it the highest among all MAPE values. When the average MAPE values for all outputs are considered, ANN has a score of 0.9578, whereas SVR and RF lag with values of 6.1093 and 8.4951, respectively.

Figure 14 visually represents the importance of the load–displacement curve parameters for the ML algorithms. Given the four inputs in this study, the figure shows how these inputs influence the overall responses. The impact of these features varies between algorithms. In the case of SVR and RF, for example, the area under the curve significantly influences the responses. In the ANN model, however, $P_{max}$, $A_t$, and $m$ of the curve all contribute roughly equally to shaping the responses. Furthermore, in SVR, the maximum load ($P_{max}$) and the initial slope of the curve ($m$) are roughly equal in importance in determining the responses.

Figure 15 compares the CZM parameters obtained via various ML algorithms and the ground truth values obtained through the simulation for the load–displacement curves of mixed-mode fractures in composite sandwich structures. We used four distinct points located among the gap of the dataset. The figure depicts excellent agreement between the CZM parameters predicted by the ML algorithms and those obtained through rigorous finite element simulations, confirming the reliability and precision of the ML-based approach for characterizing mixed-mode fractures in such complex structures.

We also chose four points outside the dataset to demonstrate the adaptability of our ML algorithms in predicting CZM parameters. Figure 16 shows the ML algorithms’ ability to effectively predict CZM parameters even when dealing with data points outside the dataset boundaries. It should be noted that all ML algorithms can produce accurate outcomes in predicting CZM parameters for this out-of-dataset point. This demonstrates the robustness and generalizability of our ML-based approaches in characterizing mixed-mode fractures beyond the boundaries of the training data.

These figures are intended to compare the performance of machine learning algorithms in predicting cohesive zone model (CZM) parameters for training and test conditions. Figure 15 shows the comparative analysis of load–displacement curves predicted by various machine learning (ML) models and finite element (FE) analysis for data points that are within the training dataset. Figure 16 shows the comparative assessment of load–displacement curves predicted by the same ML models and FE analysis for data points outside the training dataset (extrapolated data). It evaluates the generalization capability of each ML model in predicting load–displacement behavior for data not seen during training.

5. Conclusions

Support Vector Regression (SVR), Random Forest (RF), and Artificial Neural Networks (ANNs) are frequently evaluated alongside other techniques in machine learning applications. SVR and ANN often demonstrate comparable or superior performance to traditional methods in predicting material properties and process outcomes. The choice of optimal model depends on factors such as dataset size, interpretability requirements, and computational resources. While ANN models often excel with large datasets, SVR and RF may perform better with smaller samples. In this study, we evaluated the performance of three machine learning algorithms—SVR, RF, and ANN—in predicting cohesive zone model (CZM) parameters in composite sandwich structures. Using a finite element model of the Asymmetric Double-Cantilever Beam (ADCB) specimen, we generated a comprehensive dataset to train these algorithms. Our results demonstrate that the ANN model consistently provided the most accurate predictions, achieving an average Mean Absolute Percentage Error (MAPE) of 0.96% and an R-squared (R²) value of 0.79. This indicates a high level of precision in predicting the four critical CZM parameters: maximum normal contact stress (${\sigma }_I$), critical fracture energy for normal separation ($G_I$), maximum equivalent tangential contact stress (${\sigma }_{II}$), and critical fracture energy for tangential slip ($G_{II}$).

The SVR and RF models also performed well but with slightly lower accuracy compared to the ANN model. The SVR model showed significant robustness in handling small to medium-sized datasets, while the RF model excelled in handling large datasets with numerous input features and provided valuable insights into feature importance. In conclusion, the ANN model proved to be the most effective for this application, enhancing the predictive capabilities for fracture analysis in composite sandwich structures. These findings suggest that machine learning, particularly ANN, can significantly improve the efficiency and accuracy of identifying CZM parameters. The SVR can be applied to smaller datasets. The comprehensive comparison of SVR, RF, and ANN models in this context further establishes a foundation for selecting appropriate machine learning techniques based on dataset characteristics and prediction requirements.

Moreover, the feature importance analysis indicated that the displacement at maximum load (δ*), maximum load ($P_{max}$), the total area under the load–displacement curve ($A_t$), and the initial slope of the linear region (m) were all significant predictors, highlighting the comprehensive nature of the input data utilized. The ANN model also demonstrated excellent predictive capabilities for data points beyond the training range, affirming its robustness and generalizability. These findings suggest that the strategic selection of input features can significantly enhance the model’s performance.