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.
For a given class of materials, 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—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities.
Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this paper, we extend Ericksen's analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids.
