  In this paper we propose an anisotropic extension of the isotropic
exponentiated Hencky energy, based on logarithmic strain invariants. Unlike
other elastic formulations, the isotropic exponentiated Hencky elastic energy
has been derived solely on differential geometric grounds, involving the
geodesic distance of the deformation gradient F to the group of rotations. We
formally extend this approach towards anisotropy by defining additional
anisotropic logarithmic strain invariants with the help of suitable structural
tensors and consider our findings for selected case studies.
