  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.
