Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effects via global bifurcation

We consider a simplified model of a two-phase elastic solid with small interfacial energy in forced anti-plane shear. We propose a novel approach to finding meta-stable equilibria (local minima of the potential energy), based upon global bifurcation theory, a priori bounds and numerical path following. In particular, we treat the reciprocal of the capillarity coefficient as a continuation parameter, and we fully exploit the hidden symmetry of our elliptic boundary-value problem, both strategies of which are crucial for obtaining rigorous existence results and for performing reliable and efficient computation. We obtain branches of locally stable equilibria exhibiting phase nucleation (and anti-nucleation) and fine layering of mixtures.