Controllable Deformations in Compressible Isotropic Implicit Elasticity

TL;DR

This paper characterizes universal controllable deformations in compressible isotropic implicit elasticity, showing they are homogeneous but not all homogeneous deformations are universally controllable, highlighting material dependence.

Contribution

It provides a mathematical characterization of universal deformations in implicit elasticity, revealing their homogeneity and material dependence.

Findings

Universal deformations are homogeneous.

Not all homogeneous deformations are permissible.

Universal deformations depend on material properties.

Abstract

For a given material, \emph{controllable deformations} are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, \emph{universal deformations} are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress $\boldsymbol{\sigma}$ and the left Cauchy-Green strain $\mathbf{b}$, have the implicit form $\boldsymbol{\mathsf{f}}(\boldsymbol{\sigma},\mathbf{b})=\mathbf{0}$. We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is permissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.