Seismic source time function of propagating longitudinal-shear cracks

An expression of stress components corresponding to a time-dependent dislocation (discontinuity of displacement) is obtained for the case of a two-dimensional longitudinal-shear crack, on the basis of the Green function representation theorem of the elastic-wave equation. This expression is used to examine various dislocations from the viewpoint of the boundary condition that should be satisfied on the fault plane. It is shown that the dislocation functions consistent with propagating cracks have quite different characteristics between the supersonic propagation with higher than shear-wave velocity and the subsonic propagation with lower than shear-wave velocity. For supersonic propagation, the source time function giving correct rupture propagation represents a step-function type of change in the particle velocity, whereas the propagating cracks have a singularity similar to the static cracks at the edge in the case of subsonic propagation. The source time function proposed by J. N. Brune is applicable to supersonic propagation if a suitable correction is made, but this time function does not represent the correct behavior of the cracks propagating at a subsonic velocity. A source time function that is applicable to subsonic rupture propagation is also presented in this paper.