Universal Deformations and Inhomogeneities in Isotropic Cauchy Elasticity

Abstract: For a given class of materials, \emph{universal deformations} are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities -- \emph{universal inhomogeneity constraints}. Those inhomogeneities compatible with the universal inhomogeneity constraints are called \emph{universal inhomogeneities}. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e., Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in the compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more strict than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well.