A polyconvex model for materials with cubic symmetry

Abstract

A framework for developing polyconvex strain energy functions for hyperelastic materials with cubic anisotropy is proposed. Polyconvexity of the strain energy density guarantees existence of solutions of boundary value problems in finite elasticity. The anisotropy is described 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, a model based on a simple strain energy function is proposed. The model response is investigated analytically under simple deformations and its versatility and robustness are demonstrated via numerical simulations of problems in two and three dimensions involving large anisotropic deformations. In particular, the model is verified in the infinitesimal limit against the exact solution for an anisotropic plate with a hole subjected to remote unidirectional loading.