Influence of the shape of the friction law and fault finiteness on the duration of initiation

We consider the two-dimensional (2-D) elastic problem of slip instability under slip dependent friction. This paper concentrates on the parameters that determine the duration of the initiation phase, that is, the delay between an initial small perturbation of the system at the metastable equilibrium and the onset of dynamic rupture propagation. We first consider the case of a homogeneous fault (i.e., with infinite length) with a slip dependent friction with varying weakening rate. We show that different laws associated with the same values of stress drop and critical slip lead to a broad range of initiation duration. The duration is mainly governed by the slope of the friction law at the origin. We interpret these results using the analytical solution proposed by Campillo and Ionescu [1997] for the case of a constant weakening rate. These late results suggest a definition of a characteristic length associated with the rate of weakening at the origin. When the region of slip is limited to a finite weak patch, we found that the duration of initiation varies rapidly when the fault length is of the order of the characteristic length. Under these conditions the initiation duration increases extremely rapidly with decreasing fault length up to about 100 s in the numerical experiments we carried out. These results suggest that very simple elastic models with slip dependent friction and realistic values of the parameters could explain a broad range of delay of the onset of rupture propagation.