Symmetry classification of plane deformations of hyperelastic compressible materials

Abstract

The governing equations for plane deformations of isotropic compressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. For materials known as harmonic materials, only one solution of any generality is known. It is natural to ask whether the governing equations possess some special property for this particular material. In this paper, we show that the governing equations admit an extensive class of symmetries for these so-called harmonic materials and further show that there exists a second class of strain-energy functions, where the governing equations admit a general class of symmetries. We further show that for this strain-energy function, the governing equations are linearizable.