Physics-Informed Machine Learning for Failure Mode Proportion Prediction in Composite Adhesive Joints Open Access

As the quality standards for joining composite materials continue to rise, adhesive bonding has gained prominence for its ability to uniformly distribute mechanical loads, minimize localized stress concentrations, and preserve the intrinsic properties of the bonded components. It has been recognized as a suitable solution for connecting both similar and dissimilar substrates across various applications such as aviation, aerospace, automotive, marine, infrastructure, civil structures, medical industries, and more [1,2]. In contrast to mechanical fastening methods such as riveting, bolting, or welding, which can introduce stress concentrations or thermal damage, adhesive bonding avoids these drawbacks while providing additional benefits. It preserves the structural integrity of the joined materials, significantly reduces overall assembly weight, and proves especially advantageous in large composite structures. By lowering manufacturing costs and enhancing damage tolerance, adhesive bonding meets the demanding requirements of lightweight and high-performance applications [3]. This versatility makes it indispensable in fields where lightweight and high-performance joints are critical.

Composite adhesive joints are produced through multistage manufacturing processes (MMPs), consisting of a series of precise, interconnected steps. The study follows a five-stage manufacturing process as illustrated in Fig. 1. A brief description of each stage is provided as follows: (1) material handling that begins with thawing and layup of the prepreg materials to ensure a suitable working temperature and designed property such as orthotropy; (2) part fabrication proceeding with the out-of-autoclave curing of the prepreg and resulting in standardized carbon fiber reinforced polymer (CFRP) laminates; (3) surface treatment and characterization including cutting, sanding, cleaning, the intentional introduction of contamination, and contact angle measuring to assess surface wettability, thereby ensuring optimal bonding conditions; (4) joining process that bonds the treated surfaces and cures the adhesive under controlled out-of-autoclave conditions to ensure adhesion and maintain mechanical integrity; and (5) mechanical testing, where lap shear tests determine joint strength, digital image correlation (DIC) analyzes strain distribution, and fracture surface observation offers a comprehensive assessment of bonding quality. The detailed manufacturing parameters from each stage are presented in Table 1. The importance of these manufacturing processes lies not only in their individual contributions to the material properties but also in their cumulative impact on the reliability. A flaw in the intermediate stage can propagate through the production chain to downstream tasks, leading to a significant reduction in performance or even massive failure. Understanding the interplay between manufacturing parameters and final quality measures is vital to advancing adhesive bonding technology and addressing the demands of modern engineering applications.

As the use of composite materials continues to grow in various applications, it becomes increasingly important to investigate and understand their behavior and failure mechanisms, particularly in scenarios involving structural damage to composites, such as in aircraft [4]. Also, mechanical characterization and failure analysis are important factors in the design of composite applications [5]. They highlight the importance of mechanical characterization and failure analysis as integral steps in understanding, enhancing MMPs, and improving durability.

Finite element analysis (FEA) has been widely applied in the composite material research. It has been used to model bending behaviors in laminated composites [6], evaluate stress distributions under dynamic loading, and optimize structural designs to improve performance and reliability [7]. The strength of FEA lies in its ability to provide precise, theory-driven simulations critical for analyzing and optimizing composite materials. In parallel, data-driven machine learning models have emerged as a complementary approach, excelling in capturing complex patterns and relationships within large datasets. These models have been used to predict key mechanical properties such as adhesive strength and modulus response, offering efficient alternatives to traditional experimental and computational methods [8]. In failure analysis, machine learning models have shown promise in predicting stress distributions and crack patterns, providing cost-effective and scalable solutions for analyzing composite materials [9,10], and showing their potential for advanced composite material analysis.

While FEA offers detailed, physics-based simulations, its high computational cost limits its use in complex systems. In contrast, machine learning efficiently handles large datasets but requires extensive data and lacks physical interpretability. Integrating both approaches combines their strengths: FEA generates high-fidelity data, which machine learning can use to predict properties and failure behaviors across wider design spaces. This integration has improved tasks such as stacking optimization in turbine blades [11], failure prediction in laminates [12], and fracture toughness analysis in hybrids [13], demonstrating its potential in composite research. Despite progress in bonded joint analysis, many studies focus narrowly on failure characterization or lack detail in distinguishing failure modes. Quantitative insights for real-world applications remain limited. To address this, we propose the physics-informed failure mode proportion prediction (PIFMP) model, combining manufacturing and simulation data to improve mixed-mode failure prediction and overcome the limitation of purely data-driven or FEA approaches. The framework solves the issue by (1) proposing a predictive model that captures complex interactions among failure modes, offering a detailed understanding of failure behaviors; (2) leveraging matched experimental and simulation data to enable early diagnosis of failure modes, predicting potential structural damage before it occurs; and (3) integrating data-driven models with FEA-derived features into a physics-informed machine learning (PIML) framework to address challenges related to data scarcity and physical interpretability.

The primary contribution of this article is summarized as follows: (1) It proposes the first framework that directly predicts mixed-mode failure proportion rather than focusing solely on a single dominant mode, offering deeper insights into adhesive joint behavior. (2) This article bridges data-driven and physics-based methods by integrating cohesive zone modeling (CZM), finite element simulations, and a time-series transformer, enhancing predictive accuracy while maintaining physics-informed consistency. (3) This article proposes a fully end-to-end integrated framework that leverages MMP parameters and physics-based simulations to enable rapid prediction of mixed-mode failure proportions.

The remainder of this article is organized as follows: Sec. 2 reviews existing methodologies in adhesive joint modeling. Section 3 introduces the proposed PIFMP framework. Section 4 presents its experimental validation, and Sec. 5 summarizes the key findings and discusses directions for future research.

The related work focuses on prior studies on the single-lap joint configuration, which aligns with the experimental approach adopted in this study and serves as a foundation for investigating joint failure mechanisms [14]. The performance of adhesive joints is critically influenced by the parameters and stages involved in the MMPs. Several studies have examined these factors through experimental and physics-based numerical approaches. Yang et al. [15] utilized FEA to enhance the ASTM D 5656 test, emphasizing that adhesive thickness and stress distribution across the bonded line significantly affect mechanical properties. Focusing on the CZM, Huang et al. [16] explored the interaction between lap length and adhesive thickness in single-lap joints by combining the cohesive zone model (CZM) with experimental validation and numerical simulations, which demonstrated that optimized parameters improve shear strength and reduce stress concentrations. Gerini-Romagnoli and Nassar [17] developed a two-dimensional shear stress-heat and moisture diffusion model to study heat and moisture diffusion in adhesive joints, concluding that these environmental factors significantly alter stress distribution and reduce joint reliability. These studies underscore the impact of manufacturing parameters on the mechanical performance and the reliability of adhesive joints.

Understanding the failure behavior of composite adhesive joints is critical to enhancing their mechanical performance and durability. FEA is a commonly used method to analyze failure behavior by simulating complex geometries and material behaviors under realistic conditions. For example, Huang et al. [18] used FEA to validate analytical models for stress and strain distributions in single-lap joints, incorporating laminated plate theory to analyze substrate failure. The simulations accurately correlated stress distributions with failure loads, providing insights into the influence of bond line thickness on joint strength. Similarly, Ferreira et al. [19] integrated FEA with CZM to study T-stiffeners under peel loads. Their parametric study evaluated geometric parameters such as overlap length and adherend thickness, enabling the optimization of joint design for maximum load capacity. Caltagirone et al. [20] utilized FEA to model the transition from cohesive to adhesive failure modes in single-lap joints with varying adhesive layer thicknesses. The study demonstrated how increasing bond line thickness altered failure mechanisms, providing a framework for empirical model development. Saleh et al. [21] combined DIC with FEA to study stress distributions and damage evolution in double-lap bi-material joints. The FEA results validated analytical shear-lag models and highlighted damage initiation sites, aiding in joint performance prediction. Furthermore, Cao et al. [22] employed explicit FEA models to simulate progressive damage and failure in bolted composite joints, incorporating CZM for delamination and matrix cracking. This approach provided detailed insights into damage initiation and propagation under high-bearing loads.

FEA, while accurate, often demands significant computational resources and time, especially for complex models. It also relies heavily on precise input parameters, making it sensitive to errors in material properties or boundary conditions. In contrast, data-driven methods like machine learning are faster, require less computational effort, and can uncover nonlinear relationships without explicit physical models. Kulisz and Biruk-Urban [23] implemented artificial neural networks (ANNs) to predict the shear strength of single-lap adhesive joints, modeling the relationship between adhesive composition and test temperatures. The study employed the Levenberg–Marquardt algorithm to optimize model accuracy, minimizing error and computational complexity while accurately predicting shear strength across varying conditions. Kang et al. [8] utilized ANNs to predict lap shear and impact peel strengths of epoxy adhesives based on formulation components, including catalyst, curing agent, and flexibilizer content. The study constructed a predictive ANN model optimized through linear regression and root-mean-square error (RMSE) analysis, identifying the formulation components with high linear correlations to adhesive strength. Tosun and Calık [24] developed a three-layer feedforward ANN to estimate failure loads in single-lap adhesive joints. The model used geometric inputs such as bond length and width, with weights optimized using the Levenberg–Marquardt algorithm, to perform failure load estimation under tensile loading. Kaiser et al. [25] integrated regression and classification machine learning models, including deep neural networks and random forests, to analyze damage modes and failure strengths in single-lap joints.

However, data-driven methods often require extensive datasets for reliable training and lack explicit incorporation of physical information, making their predictions less interpretable. PIML, on the other hand, offers a framework that integrates machine learning with physical information, addressing the limitations of purely data-driven models and traditional numerical methods [26]. By integrating physical information in the algorithm, this approach enhances the accuracy of predictions in adhesive joint analysis and provides a more robust understanding of failure mechanisms. For instance, Boumaiza et al. [27] developed a framework that combined FEA and machine learning to predict adhesive damage in composite-steel joints under hygrothermal effects. The study utilized the damage zone theory from FEA as input for support vector regression to predict damage ratios. Liang et al. [28] proposed an approach that combined FEA and machine learning to predict the strength of adhesive-bonded joints. The study utilized FEA to simulate stress distributions and trained machine learning models with the resulting data, embedding physical constraints from FEA into the predictive framework. Another important study, conducted by Wang et al. [29], proposed the Physics-Informed Neural Ordinary Differential Equation (ODE) with Heterogeneous control Inputs framework, which embeds physics-informed insights into a neural ODE [30] structure to predict the load–displacement curves of composite adhesive joints. The model integrates manufacturing parameters, directly linking process variables to joint performance metrics such as strength and stiffness.

The past studies about joint failure analysis have often focused on load–displacement and stress–strain responses, failure loads, or the prediction of specific failure modes. Among these approaches, analyzing failure modes offers the most direct understanding of failure mechanisms in the bonding area. Failure modes in bonded joints can be generally categorized into three types: adherend failure, where the substrate fails; adhesive failure, which occurs at the bond interface; and cohesive failure, occurring within the adhesive layer itself [31,32]. However, in many cases, failure does not occur as a single mode but rather as a combination of multiple modes, where adhesive, cohesive, and adherend failures interact. Due to the complexity of these interactions, most studies limit their scope to a single failure mode. This highlights the necessity for comprehensive models that can account for the complexity and interplay of these failure mechanisms to predict and enhance joint reliability more accurately. Existing methods have not provided a comprehensive approach to link process parameters with the detailed analysis of failure mode proportion, leaving this critical aspect unaddressed. This study is the first to integrate manufacturing parameters into the prediction of mixed failure modes in composite adhesive joints.

This article introduces a physics-informed, machine learning-enhanced framework, PIFMP, designed to predict failure mode proportions in composite adhesive joints. By addressing the shortcomings of traditional methods, this framework combines FEA and machine learning techniques to improve predictive performance through physics-guided design. The overall framework is shown in Fig. 2. Manufacturing parameters are first used to derive inputs for CZM and FEA simulations, as detailed in Sec. 3.1, which captures physical behaviors and generates high-dimensional time-series data. Section 3.2 focuses on the time-series transformer model, which processes these data to extract temporal features critical for identifying failure modes. Finally, Sec. 3.3 describes the ensemble regression model and its integration strategy, emphasizing its role in aggregating predictions to estimate failure mode proportions accurately.

The CZM [33] is a key approach for simulating the mechanical response of bonded joints, characterizing the initiation, evolution, and fracture energy distribution of the adhesive layer under normal and shear modes. To align with experimental observations, key CZM parameters and their permissible ranges must be defined. Drawing on previous experiments, this study selects eight core parameters with the exploration intervals indicated in Table 2. Conducting a uniform search within these specified ranges ensures that the optimization procedure can adequately explore each parameter space, thereby avoiding local optima that may arise from an overly narrow initial setting.

The workflow shown in Fig. 3 starts with defining parameter ranges and generating initial sets via Latin hypercube sampling (LHS). FEA results are then used to build a Gaussian process regression (GPR) surrogate model, where a loss function is employed to compare simulation outputs with experimental data. Bayesian optimization (BO) iteratively refines parameters until convergence, yielding optimal CZM values for accurate simulations.

Using LHS, a point is randomly sampled within each interval, i.e., $Intervalij$⁠. The interval indices are then permuted independently for each dimension j to enhance diversity in the sampling. The resulting sampled points are combined to form the initial candidate matrix $Xczm$⁠, where each row $xczmi$ represents a parameter vector, and each column corresponds to a dimension of the parameter space.

After the BO process identifies the optimal CZM parameters, the results are integrated with manufacturing parameters using a random forest regression model. This approach establishes the functional mapping from the manufacturing parameters $xmfg=[xmfg1,xmfg2,…,xmfgd]$ to the optimized CZM parameters $xczm*=[xczm1*,xczm2*,…,xczm8*]$⁠, described as $xczm*=1Nt∑i=1Ntgi(xmfg)$⁠, where $Nt$ denotes the number of decision trees in the random forest and $gi(⋅)$ denotes the prediction function of the $i$th tree. Averaging across multiple trees enables the model to capture the nonlinear relationship between manufacturing parameters and CZM parameters for simulation. Once the mapping is established, the model can predict CZM parameters directly from manufacturing inputs, eliminating the need for repetitive and time-consuming FEA simulations in practical applications.

In the FEA process, two key types of physical high-dimensional data are exported at each time-step. The first type is the scalar damage evolution gradient (SDEG), which represents progressive damage in the bonding area and adjacent adherends. It captures the degradation of stiffness and strength in the cohesive layer during loading. The second type is contact opening (COPEN), which measures the interface gap between bonded surfaces in the normal direction. SDEG quantifies the material's internal damage evolution, while COPEN provides spatial and mechanical details of interfacial separation. This dual perspective enables the comprehensive analysis of the failure mechanisms, which is critical for evaluating the structural integrity and performance of bonded joints.

The study focuses on a three-dimensional regression task, which distinguishes the proportion between different failure modes, and adopts a time-series transformer model that incorporates a self-attention mechanism. First, in the feature extraction stage, the input data $Xphy∈RM×F$ represent preprocessed time-series features, where M denotes the fixed sequence length and F is the feature dimension, comprising the total number of target elements and nodes from SDEG and COPEN, respectively. To enhance representational capacity, the model projects $Xphy$ into a high-dimensional latent space to obtain the latent representation $Z0∈RM×D$⁠, where D is the embedding dimension. This process is defined as $Z0=XphyW+b$⁠, where $W∈RF×D$ is the weight matrix and $b∈RD$ is the bias vector.

To optimize the model, the mean squared error loss function, which penalizes large errors and encourages the model to prioritize reducing significant deviations to improve accuracy, was applied in this study as $LMSE(θ)=1N∑i=1N(r^i−ri)2$⁠, where $r^i$ represents the predicted value for the $i$th sample and $ri$ denotes the corresponding ground truth value. To accelerate convergence and maintain stability, the Adam optimizer is adopted to update the model's parameters $θ.$

The output of the model is formally defined as $r^=$$[r^adr,r^adh,r^coh]∈R3$⁠, representing the predicted proportions of adherend, adhesive, and cohesive failures, respectively. By leveraging the temporal nature of SDEG and COPEN data, the model thus refines its ability to identify the proportion of failure types.

Beyond its regression capabilities, the model is also capable of handling a classification task by identifying the dominant failure mode from the predicted proportions. This involves converting the regression output into discrete classes by determining the failure mode corresponding to the highest predicted proportion.

The primary goal of this study is to estimate the proportions of failure modes that collectively characterize the overall fracture behavior. After obtaining the outputs $r^$ from the FEA-based transformer model, these predictive features are concatenated with the manufacturing parameter vector as $xens=xmfg⊕r^$⁠, where $⊕$ denotes feature concatenation. This approach preserves the relationship between probabilistic failure mode predictions and manufacturing parameters while minimizing information loss. Specifically, let $f(e):Rd′→R3$ represent a base regressor, where $d′$ denotes the dimension of the augmented features that include both manufacturing parameters and deep-model outputs. After each base regressor produces its inference result for the same input $xens$⁠, an unweighted ensemble approach is applied, where all base regressors contribute equally to the final prediction $r^ens=1|E|∑e∈Ef(e)(xens)$⁠, ensuring the collective contribution from all individual regressors. $E$ indexes all base regressors. The resulting ensemble output $r^ens∈R3$ represents the final predicted proportions of failure modes.

By aggregating diverse regression algorithms, the ensemble approach mitigates the variance and bias risks associated with individual models. Additionally, the incorporation of FEA-derived physical knowledge enhances the framework's interpretability through physics-based parameters (SDEG and COPEN), which inherently encode material degradation and interfacial separation mechanisms. Concurrently, deep time-series predictions further improve the accuracy of the estimated failure proportions. The integration of these components ensures that predictions align with both FEA-simulated failure progression patterns and data-driven insights.

This section describes the experimental setup and validation procedures for the proposed PIFMP framework. It begins by describing the data collection and labeling process, followed by the configuration of FEA and the machine learning models. This empirical evaluation demonstrates how the integration of physics-based simulations and manufacturing parameters improves predictive accuracy.

The dataset for this study was derived from the manufacturing and testing processes of composite adhesive joints composed of CFRP substrates and epoxy structural adhesive film. The detailed elastic properties of the CFRP and the mechanical properties of the adhesive are shown in Table 3 [34–36].

The study collected a total of 76 samples, each comprising 32 manufacturing parameters listed in Table 1, along with a 448 × 448 color image of the adhesive interface between the substrates, providing detailed visualization of the bonding region. Specifically, temperature was measured at four discrete points (top, bottom, left, and right) to capture any localized thermal variations during curing, and contact angles were evaluated at four distinct locations on each substrate to thoroughly assess surface wettability before bonding.

The label acquisition process is a critical step in ensuring objective and reliable failure mode predictions. A two-phase labeling methodology, combining manual assessment with numerical optimization techniques, was employed to ensure objectivity and reliability in failure mode predictions. This integration not only enhances the consistency of failure mode classification but also reduces the subjectivity inherent in human-based evaluations.

In the first phase of the labeling process, primary failure modes are defined based on expert judgment. CFRP is typically identified by its characteristic black fiber layers, while resin residue often displays a pinkish hue. Although these color distinctions assist in visual identification, the classification process is fundamentally grounded in precise material correspondence and mechanical observations. Therefore, the approach integrates visual assessments with an evaluation of the maximum reaction forces recorded during failure events. Each interface location is systematically analyzed to classify the three failure modes: (1) Adherend failure areas are identified where corresponding positions on both sides of the interface reveal exposed CFRP, indicating that there is no adhesive resin residue. (2) Adhesive failure is assigned to regions where one side displays adhesive resin residue and the corresponding position shows exposed CFRP, signifying the detachment of adhesive from the substrate. (3) Cohesive failure is defined in regions where both corresponding positions exhibit adhesive resin residue, reflecting failure occurring within the resin material.

The resulting label serves as the principal criterion for the subsequent optimization, ensuring a reliable foundation. Additionally, samples showing extreme dominance of failure proportions are recorded. These extreme cases act as boundary conditions and constraints for the following optimization process, ensuring the calibration respects these distinct scenarios while maximizing alignment with expert judgments.

To address the complexity of segmenting adhesive and cohesive regions, a dual color-space approach was implemented to identify resin residue based on its pinkish hue. HSV (Hue, Saturation, Value) thresholds were used for initial segmentation, while RGB thresholds acted as upper bounds to refine the results. This approach ensured precise differentiation between adhesive resin residue and exposed CFRP surfaces.

To address the cyclic nature of hue in HSV space, the color range was split into two regions. Pink tones, near the 0 deg and 180 deg boundaries, were captured using two HSV threshold sets: [0, 20, 17] to [20, 228, 215] and [133, 20, 22] to [180, 237, 225], ensuring full coverage of pink hues while excluding other colors. Complementary RGB thresholds [194, 255, 124] further improved differentiation between resin residue and black CFRP surfaces. An example of the result is shown in Fig. 4, where the optimized thresholds identify the distribution of resin residue and the black CFRP surfaces. The final label map is calculated based on the corresponding positions of the two bonding areas on the CFRP surfaces.

The finite element model of the single-lap joint was discretized with shell elements in the abaqus environment. The single-lap joint comprised two CFRP adherends and one adhesive layer, meshed with a total of 3989 elements. The minimum mesh sizes for the adherends and adhesive layer were approximately 0.2 mm and 0.1 mm, respectively, ensuring accurate stress distribution results. The model dimensions are shown in Fig. 5, where the CFRP adherends and adhesive layer were modeled with CPS4R and COH2D4 elements, respectively. The overlap length of the joint was 25 mm, the adherend length was 100 mm, the adherend thickness was 2 mm, and the adhesive thickness was 0.2 mm. The material properties of the CFRP adherends and adhesive layer in this simulation are consistent with the data presented in Table 3 of Sec. 4.1.

The two adherends were modeled as symmetric but opposite parts relative to the adhesive layer to ensure an accurate representation of the mechanical response under applied loads. The mechanical behavior of the CFRP adherends was represented using an orthotropic elastic material model in abaqus. The adhesive layer and its interfaces with the adherends were modeled using cohesive elements with a CZM to simulate the traction-separation behavior, which accurately captures the adhesive's mechanical and interfacial response under loading. Boundary conditions included fixing the left side of the joint while applying a velocity of 300 mm/s at the right reference point in the x-direction. The analysis was performed as a quasi-static simulation with an initial increment size of 0.05 and a maximum increment size of 0.1.

In FEA simulations, COPEN and SDEG are two key parameters used to characterize the behavior of adhesive interfaces in this study. A larger COPEN value indicates a higher degree of separation between the adhesive and adherend surfaces, signifying more severe interfacial opening or peeling. SDEG, ranging from 0 (intact) to 1 (fully damaged), represents the progressive damage state of the adhesive layer. Specifically, COPEN primarily reflects adhesive failure when interfacial opening dominates, while SDEG is critical for distinguishing cohesive failure, as it captures internal damage progression within the adhesive layer. The integration of COPEN and SDEG ensures that the three failure modes in bonded joints are effectively characterized.

The configuration of COPEN and SDEG in FEA is illustrated in Fig. 6. In the zoomed-in views of COPEN, the upper blue line represents the interface between the adhesive and the upper adherend, while the lower red line corresponds to the interface between the adhesive and the lower adherend. These lines capture the localized deformation and separation in the adhesive interface. The SDEG plot shows the damage evolution in cohesive elements, with the upper and lower layers indicating damage in the adherends, and the middle layer highlighting damage progression within the adhesive layer. Mesh refinement in the overlap region ensures the accurate representation of stress and damage gradients, capturing the critical interface behavior.

The spatial distribution of COPEN and SDEG at critical stress values is presented in Fig. 7. The results reveal that COPEN values significantly increase at the edge regions of the adhesive interface, indicating pronounced peeling effects in these areas. Simultaneously, SDEG values exhibit distinct distribution patterns in the adhesive layer and the adherend plates. As shown in Fig. 7(a), in the adhesive layer, the edge regions reach the critical damage level earlier than the central region, indicating that damage initiates more rapidly at the edges, while the central region accumulates damage more gradually under applied stress. The adhesive layer in Fig. 7(b) also shows higher damage evolution. On the other hand, in the adherend plates shown in Fig. 7(c), the SDEG values show less pronounced variation compared to those in the adhesive layer, because in the lap shear experiment, the adhesive serves as the primary load-bearing component, while the adherend plates experience lower stress levels and slower damage progression. The data further demonstrate that stress concentration at the adhesive edge leads to peak COPEN values, whereas material damage in the central region is primarily influenced by shear stress, as reflected in the high SDEG values. These distribution characteristics provide reliable support for the quantitative analysis of adhesive interface stress and damage behavior.

The error distribution is illustrated in Fig. 8 using a violin plot, where the relative error is predominantly concentrated in the range of −0.2 to 0.2, indicating good agreement between the simulation and experimental results in most cases. However, for smaller reaction forces, the error exhibits greater variability. This can be attributed to smaller reaction forces being associated with subtle material behaviors under low-stress conditions, which are more sensitive to parameter settings in the material model. As a result, even minor deviations in model parameters can amplify the discrepancies in simulating smaller reaction forces, thereby increasing the complexity of comparing simulation and experimental data. Nonetheless, the green line in the figure marks the median error value of 0.004, indicating minimal discrepancies between simulation and experimental results for most samples and confirming their agreement.

This section details the empirical validation setup, parameter configurations, and key results of the FEA-based prediction experiment. The models were trained on a system equipped with an Intel^® Core^™ i7-13700 CPU, 64 GB of RAM, and two NVIDIA RTX 4090 GPUs.

The time-series features were constructed by aggregating SDEG and COPEN values, representing material damage evolution and interfacial separation, respectively. For SDEG, elements within two specific ranges, the carbon fiber and epoxy, were included, contributing a total of 170 features. For COPEN, nodes from epoxy contributed an additional 86 features. This resulted in an input feature dimension $F=256$⁠. Each time-series was padded or truncated to a fixed length of $M=10$ to ensure uniformity across samples, with padding applied to sequences shorter than M. The Transformer encoder used in this study consisted of $ℓ=2$ layers, each comprising a multihead attention mechanism with $Hatt=4$ attention heads and a feedforward network. The embedding dimension for the input data was set to $D=128$⁠. Positional encodings were added to the input embeddings to preserve temporal order, following a sinusoidal-based encoding scheme. An Adam optimizer was employed, with a learning rate $η=10−4$⁠. Training was performed over 200 epochs with an early stopping mechanism, using a batch size of 8.

The outputs of this phase provide essential physics-informed insights into the material behavior. To emphasize the reliability of this information, a cross-validation process was conducted to confirm its consistency and utility in guiding subsequent predictive tasks. The FEA-based prediction results, summarized in Tables 4 and 5, demonstrate the dual capability of this approach to simultaneously predict failure mode proportions and identify the dominant failure mode classification. The proportion prediction task achieves an average RMSE of 0.1125, while the classification task achieves an accuracy exceeding 93% across fivefold cross-validation. This highlights the robustness of the model in accurately capturing the critical failure mode, which not only validates the reliability of the model but also ensures the credibility of the information incorporated into the final proportion prediction.

As this study is the first to address the prediction of mixed failure mode proportions in composite adhesive joints, no existing comparable studies are available for direct reference or benchmarking purposes. Therefore, four experimental setups were designed and compared to thoroughly evaluate the efficacy of PIFMP. The final step in the framework, the ensemble regression model, integrated random forest regressor, gradient boosting regressor, XGBoost regressor, and AdaBoost regressor. Each regressor was configured with 100 estimators to balance computational efficiency and predictive performance. The input variables for each experiment are defined as follows:

1. Manufacturing parameters only model (Mfg. model): This baseline model uses only manufacturing parameters and geometric features as inputs, denoted by $xmfg$⁠, serving as the reference point for performance comparison.

2. Data-driven enhanced model (DDE model): The model enhances the baseline by integrating failure mode predictions generated from a data-driven classification model. These predictions are encoded as one-hot vectors, $r^DDE=argmax([rDDadr,rDDadh,rDDcoh])$⁠, assigning a value of 1 to the highest predicted proportion, while setting the others to 0. The input to this model is the concatenation of manufacturing parameters and $r^DDE$⁠, represented as $xDDE=xmfg⊕r^DDE.$

3. Physics label augmented (PLA) model: Failure mode predictions generated from FEA simulations are used as discrete labels. These predictions are also encoded following the same rule as the DDE model, $r^FEA=argmax([r^adr,r^adh,r^coh])$⁠, where the FEA-derived failure mode with the highest proportion determines the label. The model input combines manufacturing parameters and $r^FEA$⁠, forming $xPLA=xmfg⊕r^FEA.$

4. PIFMP model: This model uses the full failure mode proportions derived from FEA simulations as continuous inputs, denoted as $r^=[r^adr,r^adh,r^coh].$ The input, $xPIFMP=xmfg⊕r^$⁠, incorporates these detailed proportions, offering a more detailed and probabilistic perspective on the failure mode interactions.

The study adopts the RMSE as the primary evaluation metric, defined as $RMSE=1N∑i=1N(yi−y^i)2$ because it penalizes large errors more heavily and ensures that the error remains on the same scale as the original data. The values presented in Table 6 are the mean RMSE obtained from fivefold cross-validation. Fig. 9 provides a visual comparison of the model predictions. The proposed PIFMP model achieves the lowest overall RMSE of 0.108, outperforming the other methods, particularly in predicting adhesive and cohesive failures, confirming the model's reliable performance. Since the adhesive layer bears most of the shear load in single-lap joint experiments, improvements in predicting adherend failure, which tends to be a smaller proportion, are limited. However, by reducing errors by more than 20% relative to the baseline in the key adhesive and cohesive modes, PIFMP demonstrates its superior capability in modeling and predicting interfacial failure behavior.

This advantage arises from embedding physically informed features into the transformer-based time-series model, yielding a more nuanced characterization of damage progression. Additionally, integrating multistage manufacturing parameters allows the model to reflect real-world process variations while retaining physics-informed consistency. These findings confirm that physics-informed integration of FEA simulation and machine learning is highly effective for precisely predicting failure mode proportions in composite adhesive joints, offering both accuracy and practical application value.

This study introduces the PIFMP model, a novel PIML framework tailored for predicting the proportions of mixed failure modes in composite adhesive joints. As the first approach specifically designed to predict mixed failure proportions, this research represents a pioneering approach in the field. Rather than solely identifying the predominant failure mode, assessing the proportions of mixed failure modes provides a more nuanced understanding of failure risks. For example, even if cohesive failure is predominant, a substantial presence of adhesive failure may indicate potential issues such as insufficient cleaning, inadequate surface roughness, or environmental contamination on the bonding surface. If these factors are not properly addressed, premature failure may occur under subsequent fatigue loading.

The proposed framework combines advanced simulation techniques with machine learning methods to deliver accurate and physics-consistent predictions of failure mode proportions. FEA and CZM form the foundation for generating physically informed, high-dimensional data that represent critical aspects of adhesive joint behavior. These features are analyzed using a time-series transformer to model temporal damage evolution, and ensemble regression integrates simulation insights with manufacturing parameters, enabling reliable and efficient failure predictions tied to process characteristics. Empirical validation confirms the framework's effectiveness and stability in predicting failure mode proportions.

Beyond its predictive capabilities, the framework also serves as a foundational tool for advancing failure analysis and quality assurance in composite systems. First, the ensemble regression maps the measured manufacturing parameters to the failure mode proportions, thereby identifying which stage or operation, such as surface preparation, adhesive chemistry, or curing temperature and pressure, is responsible for high adhesive or cohesive failure ratios. This provides a quantitative basis for the root cause analysis. Second, in the context of reverse engineering, parameter search algorithms within the mapping model can identify feasible process designs that yield the desired failure behavior. This approach, implemented at early prototyping stages, enables more efficient optimization of parameters such as surface treatment setting, curing temperature and pressure, or adhesive formulation.

In conclusion, this study provides a comprehensive and innovative solution for predicting mixed failure modes in composite adhesive joints. By unifying physics-based simulation with machine learning, it sets a new benchmark for failure mode analysis and paves the way for more efficient and reliable composite material manufacturing and design processes.

The authors acknowledge the generous support from NSF and member companies of the Composite and Hybrid Materials Interfacing (CHMI) IUCRC.

• The National Science Foundation (Grant EEC-2052714).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.