Polyconvex anisotropic hyperelastic energies

The mathematical treatment of boundary value problems is mainly based on the direct methods of variations, i.e. to find a minimizing deformation of the elastic free energy subject to the specific boundary conditions. Existence of minimizers of some variational principles in finite elasticity is based on the concept of quasiconvexity, introduced by Morrey (1952), which ensures that the functional to be minimized is weakly lower semi-continuous. This integral inequality condition is rather complicated to handle. Thus, amore important concept for practical use is the notion of polyconvexity in the sense of BALL [

1

]. Polyconvex functions are quasi- and rank-one convex. The latter condition ensures the ellipticity of the corresponding acoustic tensor for all deformations. For isotropic material response functions there exist some models, e.g. the Ogden-, Mooney-Rivlin- and Neo-Hooke-type models, which satisfy this concept. For transversely-isotropic materials a variety of polyconvex functions has been proposed in [

2

,

3

]. A detailed analysis in view of materially stability and the adjustment to experimental data for biological tissues can be found in [

4

,

5

], respectively. In this talk we recapitulate the notion of generalized convexity conditions and discuss the modeling of polyconvex anisotropic hyperelastic energies in the framework of the invariant theory. The main attention is turned on orthotropic functions, especially the invariant modeling in the concept of structural tensors and some essential steps of the mathematical analysis are presented. We conclude with some numerical examples.