On a non-symmetrical plane strain deformation for hyperelastic materials

Abstract

In finite elasticity, the number of exact solutions which apply to all materials of a general class, such as compressible or incompressible hyperelastic materials, is known to be limited. The question arises, however, as to whether or not we might determine the largest sub-class of such materials for which a particular deformation is a possible solution of the governing equations. In this note, for a certain family of non-symmetrical plane strain deformations for hyperelastic materials, we determine every strain-energy function which gives rise to a non-trivial deformation of this type. A previously obtained solution for the Mooney material and several recently obtained solutions applying to special strain-energy functions are all shown to be deduced from a general formulation. In addition, a new solution is derived here, also applying to a particular strain-energy function. The purpose of this note is to demonstrate that for physically meaningful response functions no further exact solutions of this type exist.