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 [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 [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 [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. keywords: phase-tip splitting; solid-solid phase transitions; two-phase equilibri; interfacial energ; meta-stability; finite differences