The purpose of this chapter is to focus on a variety of exact results applying to the perfectly elastic incompressible Varga materials. For these materials it is shown that the governing equations for plane strain, plane stress and axially symmetric deformations, admit certain first order integrals, which together with the constraint of incompressibility, give rise to various second order problems. These second order problems are much easier to solve than the full fourth order systems, and indeed some of these lower order problems admit elegant general solutions. Accordingly, the Varga elastic materials give rise to numerous exact deformations, which include the controllable deformations known to apply to all perfectly elastic incompressible materials. However, in addition to the standard deformations, there are many exact solutions for which the corresponding physical problem is not immediately apparent. Indeed, many of the simple exact solutions display unusual and unexpected behaviour, which possibly reflects non-physical behaviour of the Varga elastic materials for extremely large strains. Alternatively, these exact results may well mirror the full consequences of nonlinear theory. This chapter summarizes a number of recent developments.

Introduction

Natural and synthetic rubbers can accurately be modelled as homogeneous, isotropic, incompressible and hyperelastic materials, and which are sometimes referred to as perfectly elastic materials. The governing partial differential equations tend to be highly nonlinear and as a consequence the determination of exact analytical solutions is not a trivial matter.