The Anelastic Ericksen's Problem: Universal Eigenstrains and Deformations in Compressible Isotropic Elastic Solids

The elastic Ericksen's problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions.  In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modeled by a Riemannian material manifold whose metric depends on their distribution. In this framework, we show that the natural generalization of the concept of homogeneous deformations is the notion of  covariantly homogeneous deformations — deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possible universal eigenstrains. We show that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions two and three the universal eigenstrains are zero-stress.