Ericksen’s problem: Existence of additional inhomogeneous constant-principal invariant universal deformations

Determine whether there exist any additional inhomogeneous universal deformations with constant principal invariants (i.e., deformations for which the principal invariants of the left Cauchy–Green tensor are constant) in incompressible isotropic elastic solids beyond the currently known examples, notably beyond the fifth family of universal deformations.

Background

Universal deformations are those that can be sustained in the absence of body forces solely by boundary tractions for every material in a specified class. For incompressible isotropic elastic solids, Ericksen identified four families of such deformations, and subsequent work uncovered a fifth family characterized by constant principal invariants but with inhomogeneous deformation fields.

Despite these advances, it remains unresolved whether further inhomogeneous universal deformations with constant principal invariants exist for incompressible isotropic elastic solids. This question is classically referred to as Ericksen’s problem.