Advances in polyconvex anisotropic hyperelasticity

2026-05-26

AI Summary

This work addresses the challenge of simultaneously ensuring constitutive consistency and model flexibility in modeling anisotropic hyperelastic materials by proposing a physics-augmented neural network (PANN) framework grounded in group symmetrization and the construction of polyconvex invariants. For the first time, polyconvex integrity bases and function bases compatible with high-order crystallographic symmetry groups—such as tetragonal and cubic—are systematically constructed, guaranteeing that the model inherently satisfies fundamental mechanical admissibility conditions. By integrating this framework with a nonlinear homogenization data-driven approach, the authors successfully develop polyconvex PANN constitutive models spanning multiple symmetry classes, from triclinic to cubic. The accuracy and generalization capability of the proposed method are rigorously validated on highly nonlinear homogenization data from cubic metamaterials.

Abstract

A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new polyconvex PANN constitutive model for anisotropic hyperelasticity based on triclinic invariants and group symmetrization. For finite symmetry groups, this model fulfills all common mechanical constitutive conditions a priori. (2) We propose a group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We provide an extensive introduction to the construction of polyconvex integrity and functional bases, which form the basis of polyconvex invariant-based constitutive models. We discuss polyconvex bases for triclinic, isotropic, transversely isotropic, monoclinic, rhombic, tetragonal, and cubic symmetry groups. (4) We benchmark the polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials.

Problem

• polyconvexity
• anisotropic hyperelasticity
• constitutive modeling
• symmetry groups
• integrity basis

Innovation

• polyconvexity
• anisotropic hyperelasticity
• group symmetrization
• physics-augmented neural networks
• integrity basis