Computational stability of phase-tip splitting in the presence of small interfacial energy in a simple two-phase solid

We report on further results for the problem studied in Healey and Miller (2007) [1], concerning stable equilibria for a two-phase model of an elastic solid in anti-plane shear in the presence of small interfacial energy. The existence and computation of global solution branches of equilibria for arbitrarily small interfacial energy is presented in Healey and Miller (2007) [1], and the computational stability (local energy minimization) is used as the selection criterion. However, the stability of important solutions associated with so-called phase-tip splitting (the formation of fine “needles” at the tips of phase stripes) was missed in Healey and Miller (2007) [1]. This is due to a delicate inaccuracy involved in the numerical diagnostics of the second variation of the energy, which is uncovered and resolved in this work.