A compressible isotropic hyperelastic model combining polyconvexity and global true-stress-true-strain monotonicity

Abstract

Formulating of a compressible hyperelastic model that satisfies both polyconvexity and strict monotonicity of the true stress true strain relationship across the entire deformation domain has been a long-standing problem in isotropic nonlinear elasticity. This paper presents, an explicit two-parameter solution to this problem in both two- and three-dimensional cases, with $n\in\{2,3\}$ being the spatial dimension. The proposed strain energy is objective and isotropic, smooth and finite over the entire ${\rm GL}^{+}\,(n)$, and it has the rotation group ${\rm SO}\,(n)$ as its unique set of natural states. By constructing a jointly convex extension of the deformation gradient and its determinant, we rigorously prove that this energy is polyconvex and rank-one convex. The two independent parameters in the model correspond precisely to given (positive) shear modulus and (positive) bulk modulus at the natural state. By directly evaluating the complete fourth-order logarithmic stress tangent, we show that its self-adjoint symmetric part is strictly positive definite everywhere. We further demonstrate that the left stretch and the Cauchy stress form a globally smooth diffeomorphism. This paper provides a constructive solution to the unification of polyconvexity, true stress true strain monotonicity (TSTS-${\rm M^{++}}$), and global stress reversibility relevant to Truesdell's Hauptproblem.