A globally defined compressible polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)

Abstract

We construct a compressible isotropic energy on ${\mathrm{GL}}^{+}(3)$ combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every $\mu >0$ and $k>0$, the energy $W_k(F)=\frac{\mu }{2k}[\exp (k(\Vert F{\Vert }^2+3J^{-1}+J-7))-1]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge 1/(8\sqrt{3})$. The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold $k_{\mathrm{B}}\approx 0.0083$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for $k_{\mathrm{B}}\le k<1/(8\sqrt{3})$, the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with $b\ge b_{\ast }$, including $W_k{|}_{J=1}$ for $k\ge k_{\ast }$ and the stress-free $I_1$-exponential energy, and for the exponentiated Hencky energy when $k_{\mathrm{eH}}>3/8$. Incompressible neo-Hooke instead has a limit point. A further compressible $I_1$ energy retains the global constitutive properties and reference moduli of $W_k$ but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.

arXiv comment39 pages, 8 figures. Revised following peer review; accepted for publication in European Journal of Mechanics - A/Solids