Rank-one convexity implies polyconvexity in isotropic planar incompressible elasticity

TL;DR

This paper proves that in isotropic planar incompressible elasticity, rank-one convexity of the energy function guarantees its polyconvexity, simplifying the analysis of material stability.

Contribution

It establishes that rank-one convexity implies polyconvexity for objective, isotropic energy functions in 2D incompressible elasticity.

Findings

Rank-one convexity implies polyconvexity in the specified setting

Simplifies stability analysis of isotropic incompressible materials

Provides theoretical foundation for energy function design

Abstract

We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$ implies the polyconvexity of $W$.