Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

TL;DR

In two dimensions, the paper proves that rank-one convexity implies polyconvexity for a class of energy functions, providing a counterexample to Morrey's conjecture within this subclass.

Contribution

The paper establishes that for 2D objective, isotropic, and isochoric energies, rank-one convexity implies polyconvexity, thus negatively answering Morrey's conjecture in this context.

Findings

Rank-one convexity implies polyconvexity for 2D isochoric energies.

Provides criteria for convexity conditions via the deviatoric part of the logarithmic strain.

Offers representation formulas for objective and isotropic functions.

Abstract

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on $\mathrm{GL}^+(2)$ is already polyconvex on $\mathrm{GL}^+(2)$. Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.