In this chapter we describe how certain features of the nonlinear inelastic ehaviour of solids can be described using a theory of pseudo-elasticity. ecifically, the quasi-static stress softening response of a material can be described by allowing the strain-energy function to change, either continuously or discontinuously, as the deformation process proceeds. In particular, the strain energy may be different on loading and unloading, residual strains may be generated and the energy dissipated in a loading/unloading cycle may be calculated explicitly. The resulting overall material response is not elastic, but at each stage of the deformation the governing equilibrium equations are those appropriate for an elastic material. The theory is described in some detail for the continuous case and then examined for an isotropic material with reference to homogeneous biaxial deformation and its simple tension, equibiaxial tension and plane strain specializations. A specific model is then examined in order to illustrate the (Mullins) stress softening effect in rubberlike materials. Two representative problems involving non-homogeneous deformation are then discussed. The chapter finishes with a brief outline of the theory for the situation in which the stress (and possibly also the strain) is discontinuous.

Introduction

For the most part the chapters in this volume are concerned with elasticity per se. However, there are some circumstances where elasticity theory can be used to describe certain inelastic behaviour. An important example is deformation theory plasticity, in which nonlinear elasticity theory is used to describe loading up to the point where a material yields and plastic deformation is initiated.