A real-analytic polyconvex energy with monotone Cauchy stress and global stress inversion

Abstract

We construct a family $W_a$ of real-analytic, everywhere-defined, isotropic hyperelastic energies on $\GL(3)$. Each $W_a$ is polyconvex, vanishes at the identity, has a stress-free reference with positive infinitesimal shear and bulk moduli, and satisfies a global uniform true-stress--true-strain monotonicity condition (TSTS-$M^{++}$) with respect to logarithmic strain. In particular, every nonzero logarithmic-strain increment generates positive incremental Cauchy-stress power. We then prove that the Cauchy-stress map $V\mapsto\sigma_a(V)$ is a global smooth diffeomorphism from the positive-definite symmetric matrices onto all symmetric matrices. The construction is based on a real-analytic volumetric function satisfying two scalar curvature inequalities and a multiplicative coupling between $\|F\|^2$ and $\det F$. Full fourth-order tensor calculations, including changing-eigenframe and repeated-stretch modes, are supplied. The two-dimensional analogue and numerical verification results are also given. This provides an alternative to the Bai--Wu--Li--Zhang--Neff construction, not a restatement of it.