A Note on Ericksen's Problem for Radially Symmetric Deformations

Abstract

The determination of exact deformations that apply to all homogeneous, isotropic, incompressible, hyperelastic materials is referred to in the literature as Ericksen's problem. For this class of materials, there remains only the case of constant strain invariants to be examined. The authors consider, in spherical polar coordinates, a radially symmetric deformation for which the constancy of the strain invariants arises as a natural part of the equilibrium conditions. In addition, we show that the remaining equilibrium equations admit two exact first integrals, which can be exploited to reduce the order of the problem. Finally, we show in certain complex coordinates (Z, Z) that the determination of further solutions of Ericksen's problem hinges on finding a real function P(Z, Z) that must satisfy two nonlinear partial differential equations. One of these partial differential equations admits linearization, and separable solutions may be determined involving associated Legendre functions. However, the remaining partial differential equation does not simplify under this transformation, and the authors have been unable to proceed further. Although the authors have been unable to determine any new solutions as such, the present analysis constitutes a major simplification of one of the remaining outstanding problems of nonlinear continuum mechanics.