Dynamic rupture modeling with laboratory‐derived constitutive relations

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Abstract
A laboratory‐derived state variable friction constitutive relation is used in the numerical simulation of the dynamic growth of an in‐plane or mode II shear crack. According to this formulation, originally presented by J. H. Dieterich, frictional resistance varies with the logarithm of the slip rate and with the logarithm of the frictional state variable as identified by A. L. Ruina. Under conditions of steady sliding, the state variable is proportional to (slip rate)−1. Following suddenly introduced increases in slip rate, the rate and state dependencies combine to produce behavior which resembles slip weakening. When rupture nucleation is artificially forced at fixed rupture velocity, rupture models calculated with the state variable friction in a uniformly distributed initial stress field closely resemble earlier rupture models calculated with a slip weakening fault constitutive relation. Additional rupture models are calculated in which rupture nucleation is achieved naturally, with numerical simulations of the quasi‐static response of the fault leading to the onset of unstable, dynamic rupture. When rupture nucleation with the state variable friction law takes place naturally, a large fraction of the fault accelerates before accelerating slip is concentrated in what ultimately becomes the rupture nucleation patch. The state evolution accompanying this accelerating slip leads to higher average rupture speeds or a more rapid rupture acceleration to nearPwave rupture speeds. Rupture models are also calculated for the seismological asperity problem, that is, the failure of a highly stressed fault patch surrounded by a region of zero stress drop. Dynamic overshoot of slip into the region of zero stress drop roughly agrees with a simple energy balance analysis; the final size of the rupture is proportional to the square of the size of the high stress patch. Earlier frictional stability analyses have led to the definition of a critical fault patch size for rupture nucleation. This critical patch size is generally different from critical crack lengths determined from crack tip energy balance considerations applied to a simpler slip weakening law. In the model calculations, dynamic rupture does not nucleate if the starting patch size is less than the critical patch size. This is consistent with the frictional stability analyses. Thus these model calculations suggest that dynamic rupture following a state variable friction relation is similar to that following a simpler fault slip weakening law. However, when modeling the full cycle of fault motions, rate‐dependent frictional responses included in the state variable formulation are important at low slip rates associated with rupture nucleation. The critical rupture nucleation dimension appropriate for a slip weakening fault does not predict the critical nucleation dimension for a state variable fault.

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