Models for earthquake rupture propagation

The basic processes associated with earthquake rupture propagation are poorly understood. In particular the crack-tip problem is singular when considered in terms of a stress-intensity factor. We introduce the Barenblatt cohesive zone to remove this singularity and consider a uniformly propagating, mode III crack that bisects a strip. Downstream of the crack tip we consider both a stress-free condition and a viscous resistance on the crack surface. The technique of matched asymptotic expansions is used to obtain solutions. However, with a stress-free boundary condition a Griffith energy balance for the initiation of rupture in terms of cohesive forces is obtained but the solution does not determine a rupture speed. The available elastic energy must be greater than the energy required to break the cohesive bond. With a viscous resistance to slip on the crack surface, the tip singularity associated with the outer solution is reduced from 1/2 to a smaller value and a velocity of crack propagation is found. The rupture initiation criterion is unaffected by the viscosity while, as the viscous or cohesive forces are decreased, the rupture velocity increases towards the shear-wave velocity. Our results are similar to those obtained by Nakanishi [Nakanishi, H., 1994. Continuum model of mode-III crack propagation with surface friction. Phys. Rev. E49, 5412-5419.] applying a Wiener-Hopf technique to a related problem. We believe that our solution provides an explanation for the observation of Heaton (slip) pulses during earthquakes. We suggest that there are two slip-mode regimes during an earthquake rupture. In the immediate vicinity of the crack tip, slip velocities are Very small and cohesive forces dominate. This is the regime that has been studied experimentally in the laboratory; plastic deformation of the surfaces and gouge dominate and the drop in the frictional stress is small. At higher slip velocities, away from the crack tip, there is a second frictional mode with low frictional stresses. This may be due to the fluidization of the granular fault gauge and provides a rational basis for a transition from cohesion to viscous resistance on the crack surface. As the driving stress drops the slip velocity decreases, there is a return to the cohesive mode and the fault locks and heals.