Normal modes and nonlinear stability behaviour of dynamic phase boundaries in elastic materials

We consider an ideal non-thermal elastic medium described by a stored-energy function. We study time-dependent configurations with subsonically moving phase boundaries across which, in addition to the classical jump relations of Rankine-Hugoniot type, some kinetic rule acts as a two-sided boundary condition. We establish a concise version of a normal-modes determinant that characterizes the local-in-time linear and nonlinear (in)stability of such patterns. Specific attention is given to the case where the enrgy function has two local minimizers which can coexist via a static planar phase boundary. Being dynamic perturbations of such configurations of particular interest, it is shown that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of the energy function and the kinetic rule at the minimizers.