Construction of irreducible integrity basis for anisotropic hyperelasticity via structural tensors

TL;DR

This paper introduces a new analytical-numerical method to construct irreducible integrity bases for anisotropic hyperelastic materials, covering various symmetry types and facilitating advanced material modeling including machine learning approaches.

Contribution

It develops a comprehensive technique for deriving polynomially complete and irreducible invariant sets for all common anisotropies in hyperelasticity using structural tensors.

Findings

Derived integrity bases for 11 classical anisotropies.

Extended results to 4 non-crystal anisotropies.

Proven polynomial completeness and irreducibility of the bases.

Abstract

We present a straightforward analytical-numerical methodology for determining polynomially complete and irreducible scalar-valued invariant sets for anisotropic hyperelasticity. By applying the proposed technique, we obtain irreducible integrity bases for all common anisotropies in hyperelasticity via the structural tensor concept, i.e., invariants are formed from a measure of deformation (symmetric 2nd order tensor) and a set of structural tensors describing the material's symmetry. Our work covers results for the 11 types of anisotropy that arise from the classical 7 crystal systems, as well as findings for 4 additional non-crystal anisotropies derived from the cylindrical, spherical, and icosahedral symmetry systems. Polynomial completeness and irreducibility of the proposed integrity bases are proven using the Molien series and, in addition, with established results for scalar-valued invariant sets from the literature. Furthermore, we derive relationships between a set of multiple structural tensors that specify a symmetry group and a description using only a single structural tensor. Both can be used to construct irreducible integrity bases by applying the proposed analytical-numerical method. The provided invariant sets are of great importance for modeling anisotropic materials via the structural tensor concept using both classical models as well as modern approaches based on machine learning. Alongside the results presented, this article also aims to provide an introductory overview of the complex field of modeling anisotropic materials.