Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations

TL;DR

This paper demonstrates how non-homogeneous deformations can induce homogeneous Cauchy stress in isotropic hyperelastic bodies, providing explicit examples and exploring the implications beyond linear elasticity.

Contribution

It identifies compatible non-homogeneous deformations that produce homogeneous stress in hyperelastic bodies and provides explicit examples involving non-rank-one convex materials.

Findings

Homogeneous Cauchy stress can result from non-homogeneous strains in isotropic hyperelasticity.

Explicit deformation and stress solutions are provided for a cuboid geometry.

Non-rank-one convex materials can exhibit these stress-strain behaviors.

Abstract

In isotropic finite elasticity, unlike in the linear elastic theory, a homogeneous Cauchy stress may be induced by non-homogeneous strains. To illustrate this, we identify compatible non-homogeneous three-dimensional deformations producing a homogeneous Cauchy stress on a cuboid geometry, and provide an example of an isotropic hyperelastic material, which is not rank-one convex, and for which the homogeneous stress and the associated non-homogeneous strains on a domain similar to those analysed are given explicitly.