Introduction

In this chapter, we assemble the basic field equations and jump conditions for a one-dimensional, purely mechanical theory of nonlinear elasticity; although thermal effects will be omitted, inertia will be taken into account. The theory presented here is general enough to describe nonlinearly elastic materials that, under suitable conditions of stress, are capable of existing in either of two phases. As we shall see, a key feature of this theory is that the potential energy of the material, as a function of strain at a fixed stress, has two local minima. The associated constitutive relation between stress and strain will then necessarily be nonmonotonic, possessing a maximum and a minimum connected by an unstable regime in which stress declines with increasing strain.

Experiments that provide the motivation for the theory about to be developed fall into two categories. The first of these involves slow tensile loading and unloading of slender bars or wires composed of materials such as shape-memory alloys. The model to be constructed to describe experiments of this kind is one of uniaxial stress in a one-dimensional nonlinearly elastic continuum, and the processes to be studied for this model are quasistatic. The stress-induced phase transitions in such experiments occur in tension, so the two minima in the potential energy density occur at positive – or extensional – values of strain, as do the extrema in the stress– strain relation.