Self-similarly expanding regions of phase change yield cavitational instabilities and model deep earthquakes

Abstract

The dynamical fields that emanate from self-similarly expanding ellipsoidal regions undergoing phase change (change in density, i.e., volume collapse, and change in moduli) under pre-stress, constitute the dynamic generalization of the seminal Eshelby inhomogeneity problem (as an equivalent inclusion problem), and they consist of pressure, shear, and M waves emitted by the surface of the expanding ellipsoid and yielding Rayleigh waves in the crack limit. They may constitute the model of Deep Focus Earthquakes (DFEs) occurring under very high pressures and due to phase change. Two fundamental theorems of physics govern the phenomenon: the Cauchy-Kowalewskaya theorem, which, based on dimensional analysis and analytic properties alone, dictates that there is zero particle velocity in the interior, and Noether's theorem, that extremizes (minimizes for stability) the energy spent to move the phase boundary so that it does not become a sink (or source) of energy, and which determines the self-similar shape (axes expansion speeds). The expression from Noether's theorem indicates that the expanding region can be planar, thus breaking the symmetry of the input and the phenomenon manifests itself as a newly discovered one of a “dynamic collapse/ cavitation instability”, where very large strain energy condensed in the very thin region can escape out. In the presence of shear, the flattened very thin ellipsoid (or band) will be oriented in space so that the energy due to phase change under pre-stress is able to escape out at minimum loss condensed in the core of dislocations gliding out on the planes where the maximum configurational force (Peach-Koehler) is applied on them. Phase change occurring planarly produces in a flattened expanding ellipdoid, a new defect present in the DFEs. The radiation patterns are obtained in terms of the equivalent to the phase change six eigenstrain components, which also contain effects due to planarity through the Dynamic Eshelby Tensor for the flattened ellipsoid. Some models in the literature of DFEs are evaluated and excluded on the basis of not having the energy to move the boundary of phase discontinuity. Noether's theorem is valid in anisotropy and nonlinear elasticity, and the phenomenon is independent of scales, valid from the nano to the very large ones, and applicable in general to other dynamic phenomena of stress induced martensitic transformations, shear banding, and amorphization.