Review Paper. Characterization and stability of two–phase piecewise–homogeneous deformations

Abstract

Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F±(p) of F(p) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.