Anisotropy in geometrically non-linear elasticity with generalized Seth-Hill strain tensors projected to invariant subspaces

The paper presents a formulation for geometrically non-linear anisotropic elastic material behaviour. To this end the space of symmetric second order tensors related to the undeformed reference configuration is decomposed into invariant subspaces characterizing the symmetry conditions of the specific material. The subspaces are equipped with tensor bases, which have been determined in the literature for several material classes, such as cubic symmetry, transverse isotropy and complete isotropy. Upon formulating a free energy function in terms of generalized Seth–Hill strain tensors projected onto the subspaces, the symmetry conditions characterizing the specific material are satisfied a priori. From standard results of continuum mechanics the corresponding Lagrangian stress tensor and elastic moduli are obtained by straightforward application of the chain rule. Representative examples in one-dimensional loading illustrate the material behaviour of cubic symmetry for elastic materials.