On isotropic rank 1 convex functions

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.

On a counterexample of a rank 1 convex function which is not polyconvex in the case N = 2

SynopsisJ. M. Ball has introduced the notion of polyconvexity to study nonlinear problems in elasticity and he has shown that polyconvexity implies rank 1 convexity. In this paper we prove by a counterexample that the converse of this implication is false in two dimensions.