Characterization of polyconvex isotropic functions

Abstract

Polyconvexity is an important concept in the analysis of energies related to elasticity. A function $f:{\mathbb{R}}^{d\times d}\to \mathbb{R}$ is called polyconvex if it can be written as a convex function in the minors of the argument. We show that for isotropic functions it suffices to consider diagonal matrices. For $d=3$, this leads to a dimension reduction for the convex representative of $f$ from ${\mathbb{R}}^{19}$ to ${\mathbb{R}}^7$. Moreover, we present a new result for the polyconvexity of functions formulated in the principal invariant of the left or right stretch tensor.