Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations

TL;DR

This paper investigates the relationship between homogeneous Cauchy stress and strain in nonlinear elasticity, providing a counterexample where stress is homogeneous but strain is not, due to non rank-one convex energy.

Contribution

It demonstrates that in nonlinear elasticity, homogeneous stress does not necessarily imply homogeneous strain, challenging assumptions from linear elasticity.

Findings

Counterexample with inhomogeneous deformation and constant Cauchy stress

Homogeneous stress can occur with non-homogeneous strain in nonlinear elasticity

Non rank-one convex energy leads to non-uniqueness in stress-strain relations

Abstract

We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy.