Normal modes and nonlinear stability behaviour of dynamic phase boundaries in elastic materials
Heinrich Freistühler and Ramón G. Plaza
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Submission date: 29. Mar. 2006
published in: Archive for rational mechanics and analysis, 186 (2007) 1, p. 1-24
DOI number (of the published article): 10.1007/s00205-007-0051-y
Keywords and phrases: phase boundaries, Lopatinski determinant
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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 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 and .