A non-ellipticity result, or the impossible taming of the logarithmic strain measure

The logarithmic strain measures $\parallel \log U{\parallel }^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^{T}F}$ corresponding to the deformation gradient $F$ and $\parallel \mathrm{.}\parallel$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\mathrm{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W\text{\hspace{0.17em}⁣}:\mathrm{}(n)\to \mathbb{R}$ of the form \begin{equation} W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $\mathrm{\Psi }\text{\hspace{0.17em}⁣}:[0,\mathrm{\infty })\to \mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function $\mathrm{\Psi }$ such that $W$ of the form (1) is Legendre-Hadamard elliptic. Similarly, we consider the related isochoric strain measure $\parallel {\mathrm{dev}}_n\log U{\parallel }^2$, where ${\mathrm{dev}}_n\log U$ is the deviatoric part of $\log U$. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case $n=2$, we show that for $n\ge 3$, no strictly monotone function $\mathrm{\Psi }\text{\hspace{0.17em}⁣}:[0,\mathrm{\infty })\to \mathbb{R}$ exists such that $F\mapsto \Psi(\lVert\operatorname{dev}n\log U\rVert^2)$ is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form $F\mapsto \Psi(\lVert\operatorname{dev}n\log U\rVert^2) + W{$\mathrm{v}\mathrm{o}\mathrm{l}$}(\det F)$cannotberank-oneconvexforanyfunction$W{$\mathrm{v}\mathrm{o}\mathrm{l}$}\colon(0,\infty)\to$\mathbb{R}$$if$\Psi$ is strictly monotone.