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

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.

💡 Research Summary

The paper introduces a two‑stage physics‑informed, data‑driven constitutive modeling framework specifically designed for hyperelastic soft materials that undergo progressive damage and eventual failure. Traditional hyperelastic models, such as Neo‑Hookean, Mooney‑Rivlin, or Ogden, describe the strain‑energy density as a monotonically increasing function of deformation, which precludes any representation of material rupture. To overcome this limitation, the authors adopt the “energy limiter” concept originally proposed by Volokh, embedding a finite saturation energy (\psi_f) that caps the amount of strain energy a material element can store. As the intact strain‑energy density (W) approaches (\psi_f), the total energy (\psi(W)) asymptotically reaches the limiter, naturally modeling complete failure.

Stage I – Intact hyperelastic response
The first learning stage focuses exclusively on the undamaged response. The strain energy is split into a volumetric part (U(J)) (function of the Jacobian (J)) and an isochoric part (\bar W_{\text{iso}}(\bar I_1,\bar I_2)) (functions of the deviatoric invariants). Gaussian Process Regression (GPR) is employed separately for each part, using the deformation invariants as inputs and the scalar response functions (e.g., (\zeta(J)=J,dU/dJ), (\Gamma_1,\Gamma_2)) as outputs. GPR’s Bayesian nature guarantees an exact zero‑stress reference state, enforces material frame indifference, and automatically satisfies balance of angular momentum and thermodynamic consistency. Moreover, the probabilistic framework provides predictive uncertainty, which is valuable when experimental data are noisy or scarce.

Stage II – Damage evolution
In the second stage, damage is represented by a scalar stress‑reduction factor (\chi(W)=d\psi/dW) that multiplies the intact stress tensor. A second GPR model learns the mapping from the intact strain‑energy density (W) (computed from Stage I) to (\chi). Crucially, three physics‑based constraints are imposed during training via penalty terms: (i) monotonicity ((\chi) must be non‑increasing with increasing (W)), (ii) non‑negativity ((\chi\ge0)), and (iii) complete failure ((\chi\to0) as (W\to\infty)). These constraints ensure that the damage evolution is thermodynamically admissible and that the model reproduces the saturation behavior dictated by the energy limiter.

Mathematical formulation
The total second Piola–Kirchhoff stress is expressed as
\