Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, polyconvex functions which are always s.w.l.s. are usually considered. For isotropic as well as for transversely isotropic and orthotropic materials constitutive functions that are polyconvex already exist. The main goal of this contribution is to provide a new method for the construction of polyconvex hyperelastic models for more general anisotropy classes. The fundamental idea is the introduction of positive definite second-order structural tensors G=HHTG=HHT encoding the anisotropies of the underlying crystal. These tensors can be viewed as a push-forward of a cartesian metric of a fictitious reference configuration to the real reference configuration. Here the driving transformations HH in the push-forward operation are mappings of the cartesian base vectors of the fictitious configuration onto crystallographic motivated base vectors. Restrictions of this approach are based on the polyconvexity condition as well as on the usage of second-order structural tensors and pointed out in detail.