A physics-informed data-driven framework for modeling hyperelastic materials with progressive damage and failure

TL;DR

This work introduces a two-stage physics-informed data-driven framework for hyperelastic materials undergoing progressive damage and failure. By combining energy-limiter theory with an irreducible tensor-basis decomposition, it learns intact elasticity and damage evolution separately via Gaussian Process Regression, enforcing thermodynamic and mechanical constraints. Stage I captures the undamaged response through invariant-based mappings, while Stage II learns a monotone, non-negative damage evolution function χ(W) that saturates to zero at large energies, ensuring energy-consistent failure behavior. The approach achieves high in-distribution accuracy and robust out-of-sample generalization (e.g., compression and shear) with limited data, and demonstrates practical applicability to brain tissue mechanics, offering interpretable internal variables and energy-based failure metrics for soft materials. This framework blends physical interpretability with data-driven flexibility, enabling reliable constitutive discovery under data scarcity and paving the way for physics-consistent material modeling in complex soft systems.

Abstract

This work presents a two-stage physics-informed, data-driven constitutive modeling framework for hyperelastic soft materials undergoing progressive damage and failure. The framework is grounded in the concept of hyperelasticity with energy limiters and employs Gaussian Process Regression (GPR) to separately learn the intact (undamaged) elastic response and damage evolution directly from data. In Stage I, GPR models learn the intact hyperelastic response through volumetric and isochoric response functions (or only the isochoric response under incompressibility), ensuring energetic consistency of the intact response and satisfaction of fundamental principles such as material frame indifference and balance of angular momentum. In Stage II, damage is modeled via a separate GPR model that learns the mapping between the intact strain energy density predicted by Stage I models and a stress-reduction factor governing damage and failure, with monotonicity, non-negativity, and complete-failure constraints enforced through penalty-based optimization to ensure thermodynamic admissibility. Validation on synthetic datasets, including benchmarking against analytical constitutive models and competing data-driven approaches, demonstrates high in-distribution accuracy under uniaxial tension and robust generalization from limited training data to compression and shear modes not used during training. Application to experimental brain tissue data demonstrates the practical applicability of the framework and enables inference of damage evolution and critical failure energy. Overall, the proposed framework combines the physical consistency, interpretability, and generalizability of analytical models with the flexibility, predictive accuracy, and automation of machine learning, offering a powerful approach for modeling failure in soft materials under limited experimental data.

Figures (9)

• Figure 1: Schematic of the proposed physics-informed data-driven constitutive modeling framework. The complete material response dataset (stress versus strain) comprises both undamaged (intact) and damaged regimes. In Stage I (Section \ref{['subsec:stage I model']}), the intact portion of the response is processed to compute strain invariants ($J,\bar{I}_1,\bar{I}_2$) and corresponding response functions ($\zeta, \Gamma_1, \Gamma_2$). These quantities serve as the input–output training data for two GPR models that separately characterize the volumetric and isochoric hyperelastic responses, denoted by $\mathcal{M}_{vol}$ and $\mathcal{M}_{iso}$, respectively. In Stage II (Section \ref{['subsec:stage II model']}), the trained Stage I GPR models are used to predict the intact (i.e., undamaged) stress response and the associated intact hyperelastic strain energy density $W$ over the full loading history. A stress-reduction factor $\raisebox{2pt}{$\chi$}(W)$, representing progressive damage and failure, is then inferred and used to train a third, constrained GPR model $\mathcal{M}_{dam}$. Together, the three GPR models $\mathcal{M}_{vol}$, $\mathcal{M}_{iso}$, and $\mathcal{M}_{dam}$ constitute the proposed two-stage physics-informed GPR-based model, enabling physically consistent stress prediction under arbitrary deformation paths.
• Figure 2: (a) Second Piola–Kirchhoff stress components $\mathbf{S}_{11}$ and $\mathbf{S}_{22}$ versus stretch $\lambda$ for the synthetic uniaxial tension dataset used in numerical validation. Inset shows a schematic illustration of the uniaxial tension deformation, highlighting the reference and deformed states and the 11-loading direction. (b) Axial second Piola–Kirchhoff stress $\mathbf{S}_{11}$ versus $\lambda$ for brain tissue specimens reported by Franceschini et al. Franceschini2006 and used to demonstrate model application to real experimental data. Solid curves denote gray matter (GM) of thalamus; dashed curves correspond to white matter from the occipital (WM-o) and parietal (WM-p) lobes.
• Figure 3: In-distribution model performance under uniaxial tension using synthetic data. (a) Axial and lateral second Piola--Kirchhoff stress components, $\mathbf{S}_{11}$ and $\mathbf{S}_{22}$, versus stretch $\lambda$ in the training regime, comparing the proposed two-stage physics-informed GPR-based model with the analytical constitutive model, a black-box GPR model, and a single-stage (direct) GPR-based model. (b) Corresponding percentage relative error versus stretch, computed using the Frobenius norm of the stress tensor.
• Figure 4: Process-level interpretation of the two-stage physics-informed GPR-based constitutive model under uniaxial tension (synthetic data). (a) Axial (11) and lateral (22) components of the predicted intact stress $\widetilde{\mathbf{S}}_{\mathrm{intact}}$ from the Stage I GPR models and the total stress $\widetilde{\mathbf{S}}$ from the full two-stage model, compared with training data. (b) Stress-reduction factor $\raisebox{2pt}{$\chi$}$ predicted by the proposed two-stage model as a function of stretch $\lambda$, showing monotonic, non-negative decay toward zero. (c) Predicted intact hyperelastic strain energy density $W$ (from Stage I GPR models) and total strain energy density $\psi$ (from the complete model) versus stretch, illustrating energy saturation in the damage regime while the intact energy continues to increase.
• Figure 5: Out-of-sample generalization under uniaxial compression using synthetic data. (a) Axial second Piola–Kirchhoff stress component $\mathbf{S}_{11}$ and (b) lateral stress component $\mathbf{S}_{22}$ versus stretch $\lambda \in [1,0.5]$, comparing predictions of the proposed two-stage physics-informed GPR-based model with the analytical constitutive model, black-box GPR model, and single-stage (direct) GPR-based model against the ground-truth response. (c) Corresponding pointwise percentage relative error as a function of stretch, computed using the Frobenius norm of the stress tensor.