A view on anisotropic finite hyper-elasticity

The paper presents a modular framework for the formulation of anisotropic hyper-elastic materials at large strains. Thereby, additional symmetric second order tensors are incorporated into the free Helmholtz energy density which allow, e.g., the interpretation as structural tensors. In order to prove the analogy between the material setting – usually based on the right Cauchy–Green tensor – and the spatial formulation – typically in terms of the Finger tensor – the general representation theorem of isotropic tensor functions is applied. As a result, the spatial formats of the stress tensors and the tangent operators within the anisotropic hyper-elastic case are given, whereby a specific additive structure of contributions due to the Finger tensor and the additional symmetric tensors is obtained.