Spherically symmetric two-phase deformations and phase transition zones

This paper is concerned with characterization and stability assessment of two-phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material. We study general properties of equilibrium two-phase spherically symmetric deformations. Then we specialize to phase transformations of a solid sphere that is subjected to an all-round tension/pressure. Two material models are used to demonstrate a variety of transformation behaviours and some common features. For both materials we construct phase transition zones (PTZs) formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface. Then we demonstrate how the PTZ can be used for the prediction of the number of two-phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the PTZ. We show that even in the simplest case of one interface the solution is not unique: two equilibrium two-phase solutions as well as one uniform one-phase solution are found under the same boundary conditions. For the three solutions we construct their load-deformation diagrams and compare the associated total energies. The stability of the two-phase states with respect to radial and small-wavelength perturbations is also examined. We observe how unstable solutions are related with the PTZ.