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Cited by 8Mechanical Behavior of CompositesFatigue and fracture mechanicsMachine Learning in Materials Science

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

Open access
Ramian, Arash, Elhajjar, Rani·2024·Journal of Composites Science
DOI 10.3390/jcs8080326
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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 (Pmax), the total area under the load–displacement curve (At), 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 (σI), critical fracture energy for normal separation (GI), maximum equivalent tangential contact stress (σII), and critical fracture energy for tangential slip (GII).

AI snapshot

Written from the paper’s full text.

TL;DR

This study compares SVR, RF, and ANN for identifying four mixed-mode cohesive zone parameters from ADCB load-displacement features in composite sandwich structures. ANN performed best, with average MAPE 0.9578% and R² 0.7932, while SVR and RF were less accurate, especially for σII and GII.

Key findings

  • ANN produced the best overall predictions: average MAPE 0.9578% and average R² 0.7932 across the four CZM parameters; SVR averaged MAPE 6.1093 and R² 0.5224, while RF averaged MAPE 8.4951 and R² 0.4908.
  • ANN maintained R² near 1 for σII and GII, whereas SVR and RF declined significantly for those outputs; all methods had R² greater than 0.83 for σI and greater than 0.7 for GI.
  • Hyperparameter tuning selected an ANN architecture of (24, 136, 68) with average MAPE 1.0321%; for RF, 82 trees was identified as optimal based on lowest MAPE.
  • Feature importance varied by algorithm: SVR and RF were strongly influenced by area under the load-displacement curve, while ANN used Pmax, At, and m roughly equally; all four inputs were reported as significant predictors.

Methods

  • A finite element model of the Asymmetric Double-Cantilever Beam (ADCB) sandwich specimen with a linear-softening cohesive zone model was built in ANSYS Workbench 2022R2, using 8405 elements at a uniform 1.5 mm size, CFRP facesheets, and a NOMEX honeycomb core; the load-displacement curve was validated against experimental data from the literature.
  • The dataset contained 51 load-displacement curves generated by varying CZM parameters over σI = 0.15–0.35 MPa, GI = 280–600 J/m², σII = 1.6–3.5 MPa, and GII = 700–980 J/m². Inputs were displacement at maximum load (δ*), maximum load (Pmax), total area under the curve (At), and initial linear slope (m); outputs were σI, GI, σII, and GII.
  • SVR, RF, and ANN were trained and optimized using 80% of the data for training, 10-fold cross-validation, and hyperparameter tuning; performance was evaluated with R² and MAPE.

Limitations

  • Stated: RF performance declined during testing for σII and GII, attributed to possible overfitting, differences between training and testing distributions, and hyperparameters that may not be optimal; the authors suggest more diverse datasets and further tuning.
  • Stated: RMSE was not compared because the CZM parameters differ in unit and magnitude.
  • Evident: the database consists of only 51 finite-element-generated curves, and training relies on simulations rather than direct experiments; validation uses one experimental load-displacement curve from the literature, so broader experimental generalizability is not established. Out-of-dataset tests use only four points.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.3390/jcs8080326

Cites

70 of 70 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

1 of 9 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    Simplifying assumptions in some analytical techniques may lead to inaccuracies when dealing with complex geometry, such as sandwich structures with nonlinear materials or intricate failure modes.

    “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].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [26] Ereiz, 2022, Review of finite element model updating methods for structural applications, Structures source

      The Introduction (page unavailable) says that structural idealization and simplification, as well as incorrect assumptions about geometry and nonlinear behavior, can create model uncertainty. It does not address sandwich structures or intricate failure modes, so those specific examples are not supported by our copy. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
  • Findingown data

    In SVR, the maximum load (Pmax) and the initial slope of the curve (m) are roughly equal in importance in determining 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.”
    Quote in the paperNot a substantial claim
  • Findingown data

    Based on the lowest MAPE value, 82 is the ideal number of trees for the Random Forest model.

    “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.”
    Quote in the paperNot a substantial claim
  • Findingown data

    The database generated by the finite element modeling contains 51 load-displacement curves.

    “Our database contains 51 load-displacement curves from this modeling, with some depicted in Figure 6.”
    Quote in the paperNot a substantial claim
  • Findingnone given

    The SVR model showed significant robustness in handling small to medium-sized datasets.

    “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.”
    Quote in the paperSubstantial claim
  • May combine two claims. 18
  • The quote only partly states this claim. 2
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning

    The Random Forest algorithm performed satisfactorily during the training phase but its performance declined noticeably during the testing phase, particularly for the σII and GII parameters.

    “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 σ I I and G I I .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    ANN constantly surpasses SVR and RF, especially for σII and GII, where it achieves a near-perfect fit with an R2 value close to 1.

    “When the performance of these three methods is compared, ANN constantly surpasses the other two, especially for σ I I and G I I , where it achieves near-perfect fit with a R 2 value close to 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The leading ANN hyperparameter configuration (24, 136, and 68) comprises three layers and achieves an average MAPE of 1.0321%.

    “The leading design (24, 136, and 68) comprises three layers, achieving an average MAPE of 1.0321%.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    R2 values for all methods exceed 0.7 for GI, with SVR having the greatest R2 value of 0.85 for this output.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The ANN model consistently provided the most accurate predictions among the three algorithms, achieving an average MAPE of 0.96% and an R2 value of 0.79.

    “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 (R 2 ) value of 0.79.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The 51 load-displacement curves generated by the finite element modeling constitute a dataset adequate for training and evaluating the machine learning algorithms.

  • May combine two claims.

    Findingown data

    The MAPE value for the RF algorithm increases by about 16% in σII, making it the highest among all MAPE values.

    “Notably, the MAPE value for the RF algorithm increases by about 16% in σ I I , making it the highest among all MAPE values.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The SVR model demonstrates good performance for the σI and GI parameters but evident challenges for the σII and GII parameters.

    “Thus, the SVR model demonstrates good performance for σ I and G I while there are evident challenges with the σ I I and G I I parameters.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The average R2 for all outputs is 0.7932 for ANN, 0.5224 for SVR, and 0.4908 for RF.

    “Overall, the average R 2 for all outputs is 0.7932 for ANN, 0.5224 for SVR, and 0.4908 for RF.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The 51 load-displacement curves generated by the finite element modeling constitute a dataset adequate for training and evaluating the machine learning algorithms.

  • May combine two claims.

    Findingnone given

    The RF model excelled in handling large datasets with numerous input features and provided valuable insights into feature importance.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For σII and GII, only the ANN algorithm maintains an R2 value near one, whereas the R2 values of SVR and RF decline significantly.

    “However, for σ I I and G I I , 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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    All three methods have R2 values greater than 0.83 for σI, with ANN having the highest R2 value of almost 0.98.

    “All three methods consistently have R 2 values greater than 0.83 for σ I .Notably, ANN has an R 2 value of almost 0.98, which is the highest.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The SVR and RF models performed well but with slightly lower accuracy than the ANN model for predicting cohesive zone parameters.

    “The SVR and RF models also performed well but with slightly lower accuracy compared to the ANN model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The average MAPE values for all outputs are 0.9578 for ANN, 6.1093 for SVR, and 8.4951 for RF.

    “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.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The 51 load-displacement curves generated by the finite element modeling constitute a dataset adequate for training and evaluating the machine learning algorithms.

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    Among the SVR, RF, and ANN methods evaluated, ANN produced the best results for predicting the four cohesive zone parameters, with a mean absolute percentage error (MAPE) of 0.9578% and an R-squared (R2) value of 0.7932.

    “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 (R 2 ) value of 0.7932.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    All ML algorithms can produce accurate outcomes in predicting CZM parameters for 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.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The finite element simulation results can serve as ground truth for the CZM parameters against which the ML predictions are judged.

  • May combine two claims.

    Findingnone given

    The ANN model demonstrated excellent predictive capabilities for data points beyond the training range, affirming its robustness and generalizability.

    “The ANN model also demonstrated excellent predictive capabilities for data points beyond the training range, affirming its robustness and generalizability.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The displacement at maximum load (δ*), maximum load (Pmax), the total area under the load-displacement curve (At), and the initial slope of the linear region (m) were all significant predictors in the feature importance analysis.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Artificial Neural Networks (ANNs), Random Forest (RF), and Support Vector Regression (SVR) have shown remarkable efficacy in different applications.

    “Among the various ML algorithms, Artificial Neural Networks (ANNs), Random Forest (RF), and Support Vector Regression (SVR) have shown remarkable efficacy in different applications.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [50] 2021, Comparison of machine learning methods and finite element analysis on the fracture behavior of polymer composites, Arch. Appl. Mech. source
    • cites [51] Zhang, 2022, Mechanical properties prediction of composite laminate with FEA and machine learning coupled method, Compos. Struct. source
    • cites [52] Liu, 2021, A hybrid support vector regression with multi-domain features for low-velocity impact localization on composite plate structure, Mech. Syst. Signal Process. source

  • May combine two claims.

    Interpretationreasoning

    While the RF model can effectively learn from the training data, its generalization capability is limited.

    “These factors suggest that while the RF model can effectively learn from the training data, its generalization capability is limited.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Prior researchers have demonstrated the successful application of the SVR algorithm within the area of composite structures.

    “Notably, prior researchers have demonstrated the successful application of the SVR algorithm within the area of composite structures [52,53,65].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [52] Liu, 2021, A hybrid support vector regression with multi-domain features for low-velocity impact localization on composite plate structure, Mech. Syst. Signal Process. source
    • cites [53] Lu, 2018, Damage degree prediction method of CFRP structure based on fiber Bragg grating and epsilon-support vector regression, Optik source
    • cites [65] Lyu, 2021, Prediction of the axial compressive strength of circular concrete-filled steel tube columns using sine cosine algorithm-support vector regression, Compos. Struct. source

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductreasoning

    K-fold (tenfold) cross-validation was used throughout the hyperparameter optimization process to ensure the findings were reliable and robust and to avoid overfitting.

    “We used k-fold cross-validation throughout the hyperparameter optimization process to ensure our findings were reliable and robust.”
    • Also stated in: § 4.3. Hyperparameters Tuning · plain
      “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    The comparison of SVR, RF, and ANN models establishes a foundation for selecting appropriate machine learning techniques based on dataset characteristics and prediction requirements.

    “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.”
    Quote in the paperNot a substantial claim
22 of 22

  • The paper relies on this without support.

    Experimental results often yield non-unique solutions, complicating the determination of cohesive parameters.

    “Experimental results often yield non-unique solutions, complicating the determination of cohesive parameters.”

  • The paper relies on this without support.

    Numerical determination of cohesive parameters can be time-consuming due to fine mesh requirements near the crack tip.

    “Numerical determination can be time-consuming due to fine mesh requirements near the crack tip.”

  • The paper relies on this without support.

    Reducing the number of input features can simplify the models and potentially improve their performance and interpretability.

    “Reducing the number of inputs can simplify the models and potentially improve the performance and interpretability.”

  • The paper relies on this without support.

    The ADCB is a simple but effective specimen for measuring fracture properties where the position of the crack results in a different percentage of Mode I and Mode II load.

    “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.”

  • The paper relies on this without support.

    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.

    “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.”

  • The paper relies on this without support.

    The finite element simulation results can serve as ground truth for the CZM parameters against which the ML predictions are judged.

    “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.”

  • The paper relies on this without support.

    The 51 load-displacement curves generated by the finite element modeling constitute a dataset adequate for training and evaluating the machine learning algorithms.

    “Our database contains 51 load-displacement curves from this modeling, with some depicted in Figure 6.”

  • The paper relies on this without support.

    The four load-displacement curve features (maximum load, displacement at maximum load, area under the curve, and initial slope) are sufficient to identify the four CZM parameters.

    “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.”
8 of 8

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • ANN's reported average R2 conflicts with its per-output description

    Major

    The paper describes ANN R2 as almost 0.98 for σ I, above 0.7 for G I, and near one for both remaining outputs. Those four values cannot plausibly average just 0.7932.

  • Claim that ANN always surpasses SVR conflicts with G I result

    Minor

    The text says ANN constantly surpasses the other methods, but it also says SVR has the greatest R2 for G I.

2 of 2

Cites

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2 of the 72 references couldn’t be matched to a known paper.

Cited by

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Related

Authors
Ramian, A., & Elhajjar, R.
Affiliations
University of Wisconsin System (US)
Corresponding author
Elhajjar, Rani
ORCID
Venue
Journal of Composites Science
Publisher
MDPI AG
Published
2024-08-17
Volume
8
Issue
8
Pages
326
ISSN
2504-477X (ISSN-L)
DOI
10.3390/jcs8080326
OpenAlex
W4401686590
Semantic Scholar
06d1586b89bd6506cc68f3212a8ede1d0c40fd63
Cited by
8
References
72
Citations by year
2026: 5 · 2025: 3
FWCI
1.24
Type
journal article
Language
en
Open access
Licence
Locations
Journal of Composites Science (publishedVersion, cc-by)
Topics
  • Physical Sciences › Engineering › Mechanics of Materials › Mechanical Behavior of Composites
  • Physical Sciences › Engineering › Mechanics of Materials › Fatigue and fracture mechanics
  • Physical Sciences › Materials Science › Materials Chemistry › Machine Learning in Materials Science
Keywords
cohesive zone model, mixed-mode fracture, composite sandwich structures, dimensionality reduction, honeycomb core, support vector regression, random forest, artificial neural networks, fracture energy, contact stress

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