A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity

TL;DR

This paper demonstrates that for certain convex hyperelastic energies, non-homogeneous deformations can produce homogeneous Cauchy stress, challenging previous assumptions that such stress states imply homogeneous deformations.

Contribution

It provides explicit examples showing that strictly rank-one convex energies can induce constant Cauchy stress under non-homogeneous deformations, extending prior results.

Findings

Explicit example of non-homogeneous deformation with homogeneous stress

Extension of convexity criteria to strictly rank-one convex energies

Counterexample to previous non-homogeneity restrictions

Abstract

It has recently been shown that for a Cauchy stress response induced by a strictly rank-one convex hyperelastic energy potential, a homogeneous Cauchy stress tensor field cannot correspond to a non-homogeneous deformation if the deformation gradient has discrete values, i.e. if the deformation is piecewise affine linear and satisfies the Hadamard jump condition. In this note, we expand upon these results and show that they do not hold for arbitrary deformations by explicitly giving an example of a strictly rank-one convex energy and a non-homogeneous deformation such that the induced Cauchy stress tensor is constant. In the planar case, our example is related to another previous result concerning criteria for generalized convexity properties of conformally invariant energy functions, which we extend to the case of strict rank-one convexity.