Applied Math Seminar: The self-similarly expanding ellipsoid of phase change yields dynamic cavitational instabilities and models deep earthquakes

Seminar | September 19 | 11 a.m.-12 p.m. | 891 Evans Hall

 Xanthippi Markenscoff, UCSD

 Department of Mathematics

Dedicated to the memory of Prof. G.I. Barenblatt (1927-2018)

The solution of the elastodynamic problem with body-force equivalent to transformation strain suddenly applied and remaining constant inside an ellipsoidal inclusion expanding from zero dimension with constant axes speeds (i.e., self-similarly) constitutes the dynamic generalization of the Eshelby problem. The emitted dynamical fields consist of pressure, shear, and M waves emitted by the expanding surface of discontinuity and yielding Rayleigh waves in the crack limit. Dimensional analysis and analytic properties of the self-similar expansion of the ellipsoid from zero initial conditions dictate that the particle velocity (and, hence, the kinetic energy) vanishes in the interior domain where the stress is proven constant. Noether's theorem extremizes (minimizes for stability) the energy rate spent to move the boundary of phase discontinuity, with the expression indicating that the expanding region can be planar (as thin “pancake”) thus breaking the symmetry of the input, and the phenomenon manifests itself as a newly discovered “dynamic cavitation instability”. It is shown how three-dimensional change in density (“volume collapse”) can propagate planarly and produce a zero volume radiation. The solution provides a model for Deep Focus Earthquakes assumed to be due to the nucleation and growth of phase transformations under very high pressures. The far field radiation patterns are obtained in terms of the phase change and the shape of the ellipsoid, and are consistent with seismological observations.