Rank 1 Convexity for a Class of Incompressible Elastic Materials

Abstract

We consider the class of incompressible elastic solids for which the stored energy function W(・) depends on the deformation gradient F through |F|², W( F )=φ(κ), $\kappa=\sqrt {{{{\left|F \right|}^2}-3}}$. We show that W(・) is rank 1 convex at a given F ⇔ φ(・) is non-decreasing at $\sqrt {{{{\left|F \right|}^2}-3}}$ and has $\sqrt {{{{\left|F \right|}^2}-3}}$ as a point of convexity.