A physical model for aftershocks triggered by dislocation on a   rectangular fault
  We find the static displacement, stress, strain and the modified Columb failure stress produced in an elastic medium by a finite size rectangular fault after its dislocation with uniform stress drop but a non uniform dislocation on the source. The time-dependent rate of triggered earthquakes is estimated by a rate-state model applied to a uniformly distributed population of faults whose equilibrium is perturbated by a stress change caused only by the first dislocation. The rate of triggered events in our simulations is exponentially proportional to the stress change, but the time at which the maximum rate begins to decrease is variable from fractions of hour for positive stress changes of the order of some MPa, up to more than a year for smaller stress changes. As a consequence, the final number of triggered events is proportional to the stress change. The model predicts that the total number of events triggered on a plane containing the fault is proportional to the 2/3 power of the seismic moment. Indeed, the total number of aftershocks produced on the fault plane scales in magnitude as 10^{M}. Including the negative contribution of the stress drop inside the source, we observe that the number of events inhibited on the fault is, at long term, nearly identical to the number of those induced outside, representing a sort of conservative natural rule. Considering its behaviour in time, our model doesn't completely match the popular Omori law; in fact it has been shown that the seismicity induced closely to the fault edges is intense but of short duration, while that expected at large distances (up to some tens times the fault dimensions) exhibits a much slower decay. 
R. Console and F. Catalli