Johannes Gerritzen, Andreas Hornig, and Maik Gude have established an end-to-end computational framework deriving a data-driven failure criterion for unidirectional fiber-reinforced composites under general multiaxial stress states. Predicting structural failure under arbitrary three-dimensional loading remains a persistent challenge in materials science. While deformation behavior is well understood, failure originates at the micro-scale through complex interactions between constituent fibers and matrix. Direct micro-scale modeling of full structures is currently infeasible due to severe computational limitations, necessitating meso-scale continua simplifications and scale-bridging criteria. Combining representative volume element simulations with deep neural networks, the team demonstrated that machine learning models trained entirely on synthetic micromechanical data reliably identify composite failure modes and critical load thresholds across complex multiaxial stress fields.

Classical failure models for fiber-reinforced composites were historically formulated and validated under simplified plane-stress conditions or restricted scenarios rarely involving more than two simultaneous non-zero stress components. Following established meso-scale frameworks such as Puck’s theory, analytical models typically distinguish between inter-ply delamination and intra-ply failure, subdividing the latter into fiber failure and inter-fiber failure. However, real-world structural components frequently induce complex multiaxial stress states near notches, mechanical joints, and localized impact zones. Under these general loading conditions, established failure criteria such as Tsai-Wu, LaRC03, or LaRC04 can exhibit unphysical predictions. Transitions between competing failure mechanisms often produce discontinuous failure envelopes with abrupt slope changes, yielding notable discrepancies when compared against experimental observations or multiscale analyses.

Developing empirical failure envelopes for fully three-dimensional multiaxial stress states through physical testing remains practically impossible due to experimental limits in controlling up to six simultaneous stress components. To circumvent this bottleneck, the researchers leveraged micromechanical simulations to generate a synthetic database mapping material response up to failure onset. The team selected a unidirectional glass fiber-reinforced epoxy resin composite as their reference material, drawing constituent properties from the World Wide Failure Exercise. The composite features E-glass fibers with 1200 tex yarns embedded in an MY750/HY917/DY063 epoxy matrix. In LS-DYNA, glass fibers were modeled as isotropic linear-elastic solids using MAT_1. An erosion criterion based on principal strains captured tension-compression asymmetry, defining tensile failure strain at 2.905 percent and compressive failure strain at 1.959 percent.

The epoxy resin matrix was represented through an elastic-plastic constitutive formulation coupling LS-DYNA MAT_24 with MAT_ADD_DAMAGE_GISSMO for pressure-dependent failure. To avoid the prohibitive computational cost of explicitly modeling fiber-matrix interface debonding across thousands of multiaxial simulations, micro-cracks and debonding were merged into an effective pseudo-plastic matrix response. This captured experimental resin strengths from the World Wide Failure Exercise while introducing higher deformability. To generate synthetic training data efficiently, the team constructed an idealized representative volume element with 60 percent fiber volume content in square packing. With a standard fiber diameter of 14 micrometers, the resulting volume element had edge lengths of 22.65 micrometers along all coordinate directions, including the longitudinal fiber axis.

To ensure the synthetic database was not compromised by spatial discretization artifacts, the authors conducted a mesh sensitivity study examining three element densities. The selected reference mesh maintained at least four elements across the shortest matrix bridges, yielding a target element edge length of 1.4 micrometers. This setup discretized the volume element into 36,864 hexahedral elements for fibers and 14,592 for matrix. Using 24 CPU cores and 48 GB RAM, running nine uniaxial strength simulations on the reference mesh took 11.1 minutes on average. A two-fold mesh refinement shifted predicted uniaxial strength by negative 5.4 percent while increasing computational cost by a factor of 26.5. Coarsening by a factor of two increased strength by 9.5 percent while reducing runtime by a factor of 17.1.

Periodic boundary conditions were applied to opposite faces of the volume element in LS-DYNA using INCLUDE_UNITCELL with master nodes and orthogonal spring tripods constrained against angular skew. To numerically sample the six-dimensional stress space under monotonic proportional loading without biasing sampling toward a specific failure geometry, loading directions were distributed evenly across unit k-spheres, with k from 1 to 5. Across all combinations of embedding k-dimensional subspaces into the six-dimensional stress space, the total surface area equaled approximately 830.58. The researchers selected 1,982 load cases to preserve uniform point density. Material effort was evaluated using the Cuntze criterion, scaling target stresses to convert them through the composite’s effective secant stiffness into displacement boundary conditions while eliminating unintended transverse stress components.

Finite element analyses were executed on the Barnard high-performance computing cluster at TU Dresden using implicit time integration in LS-DYNA. Twenty models ran in parallel, each utilizing 24 CPU cores and 48 GB RAM. Simulations terminated once the first element was deleted, signaling failure onset. To increase data resolution near decision boundaries, simulations restarted from their penultimate increment with the time step decreased by a factor of 100. Each simulation averaged 16.9 minutes, yielding 52 data points. To mitigate class imbalance, the dataset was augmented by extrapolating stress states past failure up to 1.3 times failure time using tangent stiffness, labeling each extrapolated state with the active failure mode, specifically fiber failure for fibers and inter-fiber failure for matrix.

The machine learning model was established as a supervised classification task in TensorFlow version 2.15.1, mapping six normalized stress components directly to one of three classes: no failure, fiber failure, or inter-fiber failure. Because loading was monotonic and proportional, strains were directly dependent on stresses and were omitted to minimize dimensionality. Input stresses were scaled to the unit interval using a MinMaxScaler fit exclusively on the training subset. The dataset was partitioned into training, validation, and test subsets using a 70:20:10 stratified split based on whole load cases. Fully connected neural networks with two, three, and four hidden layers were investigated, utilizing Swish activations, dropout, and softmax output. Hyperparameter optimization across 200 trials was conducted using the Bayesian tree-structured Parzen estimator in Hyperopt via OmniOpt.

Optimization targeted the macro area under the precision-recall curve on the validation set, tuning batch size, learning rate, dropout, and a class imbalance mitigation exponent alpha. Class weights were applied within a weighted categorical crossentropy loss, initialized via Glorot uniform and trained using the Adam optimizer with cosine decay following a five-epoch linear warm-up. Networks were retrained five times with random initializations using optimal parameters derived from kernel density estimates. The four-layer neural network achieved the highest overall performance on the test set, demonstrating robust generalization across all metrics. To benchmark physical validity, the baseline neural network and micromechanical model were compared against experimental data from the World Wide Failure Exercise across primary uniaxial loading axes.

The micromechanical model matched experimental benchmarks well in longitudinal tension (2.03 percent deviation), longitudinal compression (1.63 percent), transverse compression (1.64 to 2.12 percent), and longitudinal shear (7.32 to 9.12 percent). However, omitting explicit interface debonding to keep the required 13,398 CPU hours manageable led to larger deviations in transverse tension (91.73 to 92.65 percent) and transverse shear (33.04 percent). The baseline neural network initially underestimated transverse compressive strength by 14.9 percent, displaying a 9.7 percent directional discrepancy between transverse axes. To correct this without degrading multiaxial performance, the researchers executed an adaptive fine-tuning routine across nine uniaxial directions. Adding dynamically weighted uniaxial points whenever predictions drifted beyond plus or minus 5 percent of micromechanical values brought all uniaxial predictions within the 5 percent error bound after 12 epochs.

Evaluated across two-dimensional stress combinations along 250 evenly distributed planar directions, the fine-tuned data-driven failure criterion accurately reproduced the smooth failure envelopes determined by the micromechanical simulations. In the transverse normal stress plane combining sigma-two and sigma-three under equal sign loading, where micromechanical simulation data were unavailable, the neural network generalized effectively by predicting continuous closed failure surfaces. Assessing multiaxial performance across the withheld test set of 10,815 stress states required only 1.33 seconds total for neural network inference, equivalent to 1.23e-4 seconds per stress state. Compared to the sequential micromechanical simulation runtime of 19.5 seconds per stress state, the data-driven model achieved a computational speed-up factor of 1.59e5, evaluating failure states approximately 159,000 times faster.

Analysis of the test set confusion matrix showed that while distinguishing true no failure and fiber failure was highly reliable, separating no failure from inter-fiber failure presented the primary challenge due to the sharp decision boundary. To evaluate these misclassifications, the researchers introduced a scalar correction factor kappa measuring the proportional distance required to shift an erroneous prediction onto the true failure boundary. For the neural network, the vast majority of correction factors fell within 0.9 and 1.1, demonstrating that misclassifications occurred very close to the decision surface. Accepting an engineering load tolerance of plus or minus 2 percent increased the model’s effective binary accuracy from 88 percent to 95 percent. Traditional Tsai-Wu and Cuntze criteria evaluated on the identical benchmark required tolerance margins of 11 percent and 14 percent to match the network’s baseline agreement. The researchers concluded that synthetic micromechanical data provide a viable basis for rapid multiaxial failure modeling, with future efforts aimed at incorporating cohesive interface failure and expanding to woven composite architectures.

Subject of Research: Machine learning failure criterion for fiber-reinforced composites under multiaxial stress states

Article Title: A Micromechanics-Based Data-Driven Failure Criterion for Fiber-Reinforced Composites under Multiaxial Stress States

Article References: Gerritzen, J., Hornig, A., & Gude, M. (2026). A Micromechanics-Based Data-Driven Failure Criterion for Fiber-Reinforced Composites under Multiaxial Stress States. Advanced Composites and Hybrid Materials, 9(6), Article 490. https://doi.org/10.1007/s42114-026-02134-6

Image Credits: AI Generated

DOI: 10.1007/s42114-026-02134-6

Keywords: fiber-reinforced composites, data-driven failure criterion, micromechanics, representative volume element, multiaxial stress states, neural networks, inter-fiber failure, fiber failure, LS-DYNA, periodic boundary conditions, World Wide Failure Exercise, Cuntze criterion, Tsai-Wu criterion, hyperparameter optimization, synthetic failure data, Swish activation, finite element analysis, Barnard HPC cluster