Universal stress fields for anti-plane strain problems for a new class of elastic bodies

In classical elasticity, universal deformations have traditionally been used to characterize the constitutive response of elastic materials. Within an implicit elastic framework, however, for the subclass of models for which the strain is expressed as a function of the stress, since a real stress field may not exist corresponding to a given deformation field, as well as to obtain explicit expressions for both the stress and the strain fields, universal stress fields may be more appropriate for the same purpose. In this paper, we consider a subclass of isotropic compressible implicit elastic constitutive models for which the linearized strain is a function of the Cauchy stress and investigate universal stress fields for anti-plane strain problems for this subclass. Explicit expressions are obtained for the universal stress fields, and it is found that both uniform and non-uniform stress fields are admissible. Boundary value problems are formulated based on the stress fields, and the predictions of a stress-softening model, a stress-stiffening model, and the classical linearized elastic model are studied and compared. It is found that the stress-softening and stress-stiffening models predict distinct material responses, both of which can differ considerably from the classical linearized elastic model.