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

Abstract

Polyconvexity is a standard ingredient in the variational existence theory of finite elasticity, whereas true-stress-true-strain monotonicity (TSTS-M++) requires a positive incremental Cauchy-stress response. These two constitutive restrictions are independent, and Wollner, Holzapfel and Neff left open whether a compressible isotropic energy defined on the whole of $\mathrm{GL}^{+}(3)$ can satisfy both. We give an explicit affirmative answer. For every $\mu>0$ and $k>0$, the stored-energy function $W_k(F)=\frac{\mu}{2k}\bigl[\exp\bigl(k(\lVert F\rVert^2+3J^{-1}+J-7)\bigr)-1\bigr]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge 1/(8\sqrt{3})$. In this regime every symmetric Cauchy stress corresponds to a unique positive-definite stretch, while the reference stretch is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the strictly smaller sharp threshold $k_{\mathrm{B}}\approx 0.00827233304$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it the map is a global $C^{\infty}$ diffeomorphism. Thus, for $k_{\mathrm{B}}\le k<1/(8\sqrt{3})$, the map $V\mapsto\sigma(V)$ remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, every prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and its inner pressure increases smoothly and strictly from zero to infinity.