Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type

Abstract

{Let $\lambda_i = \nu_i(F)$ denote the three singular values of the deformation gradient $F \in {\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_\psi(F)=\sum_i \psi\left(|\!\log\lambda_i|\right)$ with $\psi:[0,\infty)\mapsto\mathbb{R}$. Set $g(s)=\psi\left({\rm arcosh}\frac{s}{2}\right)$ for all $s\geq2$. If $g$ has a convex and non-decreasing extension $\bar g$ to $[0,\infty)$, then $W_\psi$ is the restriction to ${\rm SL}(3)$ of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(\nu_i(F+{\rm Cof} F)\right). $} The proof uses the identity $\nu_i(F+{\rm Cof} F)=\lambda_i+\lambda_i^{-1}$, up to permutation, on ${\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!\lambda_i} $ is rank-one convex on ${\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\log \lambda_i$ alone is insufficient; the quadratic Hencky energy $\sum_i \log^2\lambda_i$ is not rank-one convex on ${\rm SL}(3)$.