A Differentiable Framework for Gradient Enhanced Damage with Physics-Augmented Neural Networks in JAX-FEM

TL;DR

This paper presents a differentiable, physics-augmented neural network framework within JAX-FEM for simulating damage in soft materials, addressing model flexibility and mesh dependence issues.

Contribution

It combines ICNN-based constitutive models with a gradient-enhanced damage formulation in a differentiable FEM framework, ensuring thermodynamic consistency and mesh independence.

Findings

Validated through stress-strain fits and parametric studies.

Demonstrated mesh-independent damage simulation in soft materials.

Open-source implementation available for broader adoption.

Abstract

Soft materials such as rubbers, hydrogels, and biological tissues undergo damage in the form of stiffness degradation without apparent changes in their stress-free geometry. Accurate simulation of this behavior is critical in applications ranging from soft robotics to the design of medical devices, yet two persistent challenges are the difficulty of constructing flexible, thermodynamically consistent constitutive models, and the mesh dependence of finite element solutions caused by strain softening. Here we address both challenges simultaneously by combining physics-augmented neural network constitutive models with a gradient-enhanced damage formulation implemented within the differentiable finite element framework JAX-FEM. The elastic strain energy and the damage yield function are each parameterized by input-convex neural networks (ICNNs), which enforce polyconvexity and satisfaction of the Clausius--Duhem inequality by design. The gradient-enhanced formulation introduces a non-local damage field governed by an additional partial differential equation, regularizing the spatial distribution of damage and eliminating mesh dependence. The implementation is validated through local stress-strain fits, single-element parametric studies, a mesh and solution strategy study for a uniform deformation case, and a notched plate simulation. The results demonstrate that the proposed framework enables flexible, data-driven, mesh-independent damage simulation for a broad class of soft materials. We anticipate that the open-source implementation will lower the barrier to adopting physics-augmented neural network constitutive models.