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

{Let ${\lambda }_i={\nu }_i(F)$ denote the three singular values of the deformation gradient $F\in {\mathrm{G}\mathrm{L}}^{+}(3)$.} We consider the family of incompressible isotropic energies $W_{\psi }(F)={\sum }_i\psi (\mathrm{\mid }\text{\hspace{0.17em}⁣}\log {\lambda }_i\mathrm{\mid })$ with $\psi :[0,\mathrm{\infty })\mapsto \mathbb{R}$. Set $g(s)=\psi \left(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{s}{2}\right)$ for all $s\ge 2$. If $g$ has a convex and non-decreasing extension $\bar{g}$ to $[0,\mathrm{\infty })$, then $W_{\psi }$ is the restriction to $\mathrm{S}\mathrm{L}(3)$ of the explicit polyconvex function \ {$F\mapsto {\sum }_i\bar{g}({\nu }_i(F+\mathrm{C}\mathrm{o}\mathrm{f}F))\mathrm{.}$} The proof uses the identity ${\nu }_i(F+\mathrm{C}\mathrm{o}\mathrm{f}F)={\lambda }_i+{\lambda }_i^{-1}$, up to permutation, on $\mathrm{S}\mathrm{L}(3)$ and Ball’s convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in $\mathrm{S}\mathrm{L}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $F\mapsto {\sum }_i{e}^{{\log }^2\text{\hspace{0.17em}⁣}{\lambda }_i}$ is rank-one convex on $\mathrm{S}\mathrm{L}(3)$ 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 $\mathrm{S}\mathrm{L}(3)$.