On rate-state and Coulomb failure models

We examine the predictions of Coulomb failure stress and rate-state frictio nal models. We study the change in failure time (clock advance) Delta t due to stress step perturbations (i.e., coseismic static stress increases) add ed to "background" stressing at a constant rate (i.e., tectonic loading) at time t(0). The predictability of Delta t implies a predictable change in s eismicity rate r(t)/r(0), testable using earthquake catalogs, where r(0) is the constant rate resulting from tectonic stressing. Models of r(t)/r(0), consistent with general properties of aftershock sequences, must predict an Omori law seismicity decay rate, a sequence duration that is less than a f ew percent of the mainshock cycle time and a return directly to the backgro und rate. A Coulomb model requires that a fault remains locked during loadi ng, that failure occur instantaneously, and that Delta t is independent of t(0). These characteristics imply an instantaneous infinite seismicity rate increase of zero duration. Numerical calculations of r(t)/r(0) for differe nt state evolution laws show that aftershocks occur on faults extremely clo se to failure at the mainshock origin time, that these faults must be "Coul omb-like," and that the slip evolution law can be precluded. Real aftershoc k population characteristics also may constrain rate-state constitutive par ameters; a may be lower than laboratory values, the stiffness may be high, and/or normal stress may be lower than lithostatic. We also compare Coulomb and rate-state models theoretically. Rate-state model fault behavior becom es more Coulomb-like as constitutive parameter a decreases relative to para meter b. This is because the slip initially decelerates, representing an in itial healing of fault contacts. The deceleration is more pronounced for sm aller a, more closely simulating a locked fault. Even when the rate-state D elta t has Coulomb characteristics, its magnitude may differ by some consta nt dependent on b. In this case, a rate-state model behaves like a modified Coulomb failure model in which the failure stress threshold is lowered due to weakening, increasing the clock advance. The deviation from a non-Coulo mb response also depends on the loading rate, elastic stiffness, initial co nditions, and assumptions about how state evolves.