Introduction

In this chapter problems of finite deformation elastostatics for isotropic hyperelastic solids are considered. Exact solutions for some problems of finite deformation of incompressible elastic solids have been obtained by inverse methods. In these methods a deformation field is assumed, and it is verified that the equilibrium equations and stress and displacement boundary conditions are satisfied. There are some elastostatic problems that have deformation fields that are possible in every homogeneous isotropic incompressible elastic body in the absence of body forces. These deformation fields are said to be controllable. In an important paper by Ericksen an attempt is made to obtain all static deformation fields that are possible in all homogeneous isotropic incompressible bodies acted on by surface tractions only. Ericksen further showed that the only deformation possible in all homogeneous compressible elastic bodies, acted on by surface tractions only, is a homogeneous deformation. In this chapter solutions are given for several problems of finite deformation of incompressible isotropic hyperelastic solids and two simple problems for an isotropic compressible solid. These solutions with one exception involve controllable deformation fields and are based on the physical components of stress, deformation gradient, and the left Cauchy-Green strain tensor. This is in contrast to the approach of Green and Zerna, who used convected coordinates and tensor components.

Rivlin, in the late 1940's, was the first to obtain most of the existing solutions, for deformation of an isotropic incompressible hyperelastic solid.