Self-Similar Earthquake Nucleation on Rate-and-State Faults

We obtain self-similar solutions (two-dimensional and quasi-static) for the acceleration to instability of a fixed-length patch on a fault obeying rate-and-state friction. The solution is applicable in the limit Vθ /D_c≫1, so that the evolution of the state variable is well-approximated by ˙ {θ }=Vθ /D_c. For simulations on an infinite fault with a/b< ∼0.5, the nucleation zone spontaneously evolves to the size and velocity distribution of the self-similar solution for which the stress intensity factor K=0, for which the nucleation length L_ν =1.3774G^*D_c/bσ, independent of a, where G^* is the elastic stiffness. For a/b<0.3781, Vθ increases with time and the large Vθ /D_c solution remains applicable until elastodynamics comes into play. For larger a/b, Vθ at the crack center diminishes to a quasi-constant value modestly larger than 1, and the nucleation zone ultimately appears similar to an expanding slip-weakening crack with constant slip-weakening rate but time-varying peak and residual stresses. The nucleation length in these cases (defined as the minimum of the time-dependent size of the nucleation zone) generally increases with a/b but is very sensitive to the boundary and initial conditions. For sufficiently large values of Vθ /D_c upon localization, the nucleation zone can undergo velocity increases of many orders of magnitude before the self-similar solution becomes inapplicable; this is why this solution dominates the simulations of Dieterich [1992] even for a/b\sim0.9. For a/b$<0.3781, smaller nucleation zones are capable of reaching instability; these correspond to self-similar solutions with [˙ {Vθ }]\ge0 and K$>0, so they could be applicable to faults shorter than L_ν . The smallest viable nucleation zone L_min increases in size with increasing a/b and equals L_ν at a/b=0.3781. For a=0, which in the limit Vθ /D_c\gg1 corresponds to slip-weakening behavior, L_{min} equals the universal nucleation length of 0.579G^*D_c/b\sigma found for slip-weakening behavior by Uenishi and Rice [2003] (the slip-weakening rate is b\sigma/D_c). The family of self-similar solutions can thus be viewed as linking the observation of Dieterich [1992] that L_\nu scales as b^{-1} (the K=0 solution), with the expectation from stability analyses that L_{min} scales as (b-a)^{-1} (the K$>0 solutions for which [˙ {Vθ }]=0).