Universal Deformations in Compressible Isotropic Cauchy Elastic Solids with Residual Stress

TL;DR

This paper demonstrates that in compressible isotropic Cauchy elastic solids, the presence of residual stress prevents universal deformations unless the residual stress is zero, highlighting a fundamental restriction on deformation behavior.

Contribution

It proves that universal deformations in such solids require homogeneous residual stress, which implies that non-trivial residual stress cannot coexist with universal deformations.

Findings

Universal deformations must be homogeneous.

Residual stress must be homogeneous for universal deformations.

Non-trivial residual stress cannot admit universal deformations.

Abstract

We investigate universal deformations in compressible isotropic Cauchy elastic solids with residual stress, without assuming any specific source for the residual stress. We show that universal deformations must be homogeneous, and the associated residual stresses must also be homogeneous. Since a non-trivial residual stress cannot be homogeneous, it follows that residual stress must vanish. Thus, a compressible Cauchy elastic solid with a non-trivial distribution of residual stress cannot admit universal deformations. These findings are consistent with the results of \citet{YavariGoriely2016}, who showed that in the presence of eigenstrains, universal deformations are covariantly homogeneous and in the case of simply-connected bodies the universal eigenstrains are zero-stress (impotent).