A Renormalization Group Approach to the Stick-Slip Behavior of Faults

We treat a fault as an array of asperities with a prescribed statistical distribution of strengths. When an asperity fails, the stress on the failed asperity is transferred to one or more adjacent asperities. For a linear array the stress is transferred to a single adjacent asperity and for a two-dimensional array to three adjacent asperities. Using a renormalization group (RG) method, we investigate the properties of a scale invariant hierarchical model for the stochastic growth of fault breaks through induced failure by stress transfer. An extrapolation to arbitrarily large scales shows the existence of a critical applied stress at which the solutions bifurcate. At stresses less than the critical stress, virtually no asperities fail on a large scale, and the fault is locked. Above the critical stress, asperity failure cascades away from the nucleus of failure; we interpret this catastrophic failure as an earthquake and it corresponds to the transition from stick to slip behavior on the fault. Thus the stick-slip behavior of most faults can be attributed to the distribution of asperities on the fault. We propose our stochastic mechanism as an alternative to the traditional hypotheses for the stick-slip behavior of faults. A major advantage of our approach is the inclusion of scale invariance. Thus the observed frequency-magnitude relation for seismicity is a natural consequence of our basic hypothesis.