A local solution of Ericksen’s problem

Abstract

Ericksen’s problem asks for all universal deformations of homogeneous incompressible isotropic hyperelastic solids. Such deformations can be maintained in equilibrium, without body forces, for every isotropic strain energy function, while the pressure and surface tractions may depend on the constitutive law. Six classical families are known, but their completeness has remained open for three distinct constant principal stretches with unrestricted principal directions. We prove that every orientation preserving C³ incompressible local universal deformation in three dimensions belongs locally to one of these families, including their regular Cartesian continuations.
The main step is to prove that, for constant pairwise distinct principal stretches, the plane field orthogonal to one principal direction is locally integrable. Using the principal frame formulation, we reduce the universal equilibrium and compatibility equations to a finite differential system. Removing the length scale yields a normalized state map whose fibres of positive dimension induce common local Euclidean similarities of the reference and current configurations, with the same dilation factor. On a hypothetical region where all three principal abnormalities are nonzero, analysis of these similarities and the frame equations excludes ranks zero, one and two. For rank three, we bound the dimension of the Zariski closure of the state image by two, using estimates for the generic fibres of the projection onto the two abnormality ratios.
Real analyticity follows from neo-Hookean equilibrium. It selects a fixed principal label whose orthogonal plane field is integrable throughout a neighbourhood, and permits continuation of the classical deformations through critical points of the principal invariant map and removable coordinate singularities. Kafadar’s theorem and the earlier classifications then establish the completeness of the six classical families and resolve the local form of Ericksen’s problem.