Elastic Dislocation Theory for a Self-Gravitating Elastic Configuration with an Initial Static Stress Field

The linearized equations of motion and linearized boundary and continuity conditions governing small elastic-gravitational disturbances away from equilibrium of an arbitrary, uniformly rotating, self-gravitating, perfectly elastic Earth model with an arbitrary initial static stress field are derived. The appropriate form of Rayleigh's variational principle and of the Betti reciprocal theorem and the Volterra dislocation relation for such a configuration are given. The latter is then used to derive an explicit expression for the equivalent body forces to be applied in the absence of a seismic dislocation in order to produce a dynamical response of the Earth model equivalent to that produced by the dislocation. It is found that if the initial static stress in the vicinity of the dislocation is purely hydrostatic, then a point tangential displacement dislocation has as an exactly equivalent body force the familiar double couple of moment μ0, A0Δs0. If however the hypocentral static stress field has a deviatoric part, then additional equivalent body forces must be used properly to model a seismic dislocation. The necessary additional equivalent forces are explicitly exhibited; theoretically their existence provides a method of estimating hypocentral stresses, but the application of any such method is probably premature.