New families of exact solutions for finitely deformed incompressible elastic materials

Abstract

The governing partial differential equations for static deformations of homogeneous isotropic incompressible elastic materials are highly nonlinear, and consequently only a few exact solutions are known. For these materials, only one general solution involving a single arbitrary function is presently known, which is for plane deformations and is applicable to the so-called Varga strain-energy function. In this paper, new families of exact solutions are derived for plane and axially symmetric deformations of perfectly elastic materials. In the case of plane deformations, a different formulation of the known general solution for the Varga strain-energy function is presented, from which numerous new solutions may be obtained. In the case of axially symmetric deformations, the Varga material again arises as the privileged strain-energy function, and this constitutes the main result of the paper. For this material, a number of new simple exact solutions are derived for axially symmetric deformations. Finally, all the solutions obtained here are shown to also apply to a modified Varga strain-energy function involving the reciprocals of the principal stretches