Claudia Grabs
Besides physical assumptions that need to be made on the energy density function of an hyperelastic material, there are several requirements that arise from the mathematical perspective. In order to find minimizers of the total elastic energy with the direct method in the calculus of variations or to ensure strong ellipticity of the associated equations, certain notions of convexity play a crucial role, among them the notion of polyconvexity. I will introduce these, state their relations, and then examine certain hyperelastic models in this respect. We will see for example that the Saint Venant-Kirchhoff model is not polyconvex (and therefore not convex in the classical sense), but the Mooney-Rivlin models are polyconvex.
