The dynamic Eshelby problem: nucleation and growth of a phase change defect as the mechanism of deep earthquakes and failure waves

A defect of phase change in density and change in moduli modeled as a self-similarly dynamically expanding Eshelby ellipsoidal inhomogeneous inclusion can nucleate and grow under high pressure at a critical loading. The self-similarly expanding ellipsoid possesses the “lacuna” property, the particle velocity vanishing in the interior domain, which allows the constant stress Eshelby property to be valid in the interior and the inclusion to grow as a whole in the presence of inertia. The energetics for nucleation and growth are derived from the energy-momentum tensor and first principles and generalize in the presence of inertia the “force on an interface” obtained in statics by Eshelby based on a thought experiment. The solution obtains the flow of energies across the moving phase boundary of an inhomogeneous inclusion, at the balancing of which, corresponding to the vanishing of the M integral, the interface presents no obstacle, and an arbitrarily small inclusion of phase change nucleates and grows at constant potential energy. The solution explains the generation of a shear seismic source radiation in deep-focus earthquakes and the generation of failure waves producing a zone with micro-fractures under compression in lima glass, and has wider applications to amorphization defects, defects in alloys, laser additive manufacturing, etc.