Title: A non-ellipticity result, or the impossible taming of the logarithmic strain measure,
Abstract: The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the
principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding
to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius
matrix norm, arises naturally via the geodesic distance of $F$ to the special
orthogonal group $\operatorname{SO}(n)$. 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
$W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation}
W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable
function $\Psi\colon[0,\infty)\to\mathbb{R}$ 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 $\Psi$
such that $W$ of the form (1) is Legendre-Hadamard elliptic.
Similarly, we consider the related isochoric strain measure
$\lVert\operatorname{dev}_n\log U\rVert^2$, where $\operatorname{dev}_n \log U$
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 $n\geq3$, no strictly monotone function
$\Psi\colon[0,\infty)\to\mathbb{R}$ 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
$F\mapsto \Psi(\lVert\operatorname{dev}_n\log U\rVert^2) +
W_{\mathrm{vol}}(\det F)$ cannot be rank-one convex for any function
$W_{\mathrm{vol}}\colon(0,\infty)\to\mathbb{R}$ if $\Psi$ is strictly monotone.