Characterization and stability of two-phase piecewise-homogeneous deformations

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.