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Normal modes and nonlinear stability behaviour of dynamic phase boundaries in elastic materials
Heinrich Freistühler and Ramón G. Plaza
This paper considers an ideal non-thermal elastic medium described by a stored-energy function $W$. It studies time-dependent configurations with subsonically moving phase boundaries across which, in addition to the jump relations (of Rankine-Hugoniot type) expressing conservation, some kinetic rule $g$ acts as a two-sided boundary condition. The paper establishes 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 $W$ has two local minimizers $U^A,U^B$ which can coexist via a static planar phase boundary. Dynamic perturbations of such configurations being of particular interest, the paper shows that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of $W$ and $g$ at $U^A$ and $U^B$.