Universal Nucleation Length for Slip-Weakening Rupture Instability under Non-Uniform Fault Loading

We consider the nucleation of instability on a slip-weakening fault subjected to a heterogeneous, locally peaked “loading” stress. That stress is assumed to gradually increase due to tectonic loading but to retain its peaked character. The case of a linear stress versus slip law is considered in the framework of two-dimensional quasi-static elasticity for a planar fault. Slip initiates when the peak of the loading stress first reaches the strength level of the fault to start slip weakening. Then the size of the slipping region grows under increased loading stress until finally a critical nucleation length is reached, at which no further quasi-static solution exists for additional increase of the loading. That marks the onset of a dynamically controlled instability. We prove that the nucleation length is independent of the shape of the loading stress distribution. Its universal value is proportional to an elastic modulus and inversely proportional to the slip-weakening rate, and is given by the solution to an eigenvalue problem. That is the same eigenvalue problem as introduced by Campillo, Ionescu and collaborators for dynamic slip nucleation under spatially uniform pre-stress on a fault segment of fixed length; the critical length we derive is the same as in their case. To illustrate the nucleation process, and its universal feature, in specific examples, we consider cases for which the loading stress is peaked symmetrically or non-symmetrically, and employ a numerical approach based on a Chebyshev polynomial representation. Laboratory-derived and earthquake-inferred data are used to evaluate the nucleation size.