Polyconvex Model for Materials with Cubic Anisotropy

TL;DR

This paper develops a polyconvex strain energy model for hyperelastic materials with cubic anisotropy, ensuring mathematical well-posedness and capturing directional mechanical behavior.

Contribution

It introduces a novel framework using a structural tensor and polynomial basis to formulate polyconvex energy functions for cubic anisotropic materials.

Findings

Model exhibits physically relevant directional properties.

Strains vary with direction, reflecting anisotropy.

Lack of symmetry leads to shear stresses and displacements.

Abstract

Polyconvexity is one of the known conditions which guarantee existence of solutions of boundary value problems in finite elasticity. In this work we propose a framework for development of polyconvex strain energy functions for hyperelastic materials with cubic anisotropy. The anisotropy is captured by a single fourth order structural tensor for which the minimal polynomial basis is identified and used for the formulation of the strain energy functions. After proving the polyconvexity of some polynomial terms, we proceed to propose a model based on a simple strain energy function. We investigate the behavior of the model analytically in one dimension and numerically in two and three dimensions. These investigations allow us conclude that the model possesses the physically relevant directional properties, in particular, strains in different directions are different and the lack of any loading symmetries leads to development of ``shear'' stresses and displacements.