Fault rupture between dissimilar materials: Ill‐posedness, regularization, and slip‐pulse response

Abstract

Abstract. Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of self-sustained slip pulses [Weertman, 1980] propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i.e., with a constant coefficient of friction) and a remote driving shear stress that is arbitrarily less than the corresponding frictional strength. Following Andrews and Ben-Zion [1980] analysis, the propagation occurs only in one direction, which is that of slip in the more compliant medium. When the generalized Rayleigh wave speed does not exist, similar self-sustained pulses propagate at about the slower S wave speed and in the same direction. RR also suggested that for sufficiently high coefficient of friction, another kind of (less unstable) self-sustained pulses, propagating at a velocity close to the slower P wave speed and in the opposite direction, could also exist. We numerically verify that prediction.

Rate and state dependent friction and the stability of sliding between elastically deformable solids

“…The linearized expressions, given above, of single-state variable laws were previously derived by Rice (1983) for the special case = constant = 0 . Also, if all velocity dependence of friction is neglected (a = a − b = 0) and if f = so that there is no instantaneous response to normal stress change, then the above expression corresponds to one which Ranjith and Rice (2001) and Cochard and Rice (2000) showed would regularize the ill-posedness of elastodynamic problems of sliding between dissimilar materials mentioned in the introduction.…”

“…The problem of stability of steady sliding between dissimilar materials has received much recent attention in theoretical modeling (Renardy, 1992;Martins et al, 1992Adams, 1995;Martins and Simões, 1995;Simões and Martins, 1998;Cochard and Rice, 2000;Ranjith and Rice, 2001). That work has, mostly, neglected rate and state e ects, instead assuming a constant coe cient of friction f, and has focused on the coupling between inhomogeneous slip and alteration of normal stress.…”

“…Andrews and Ben-Zion (1997) reported numerical simulations of highly unstable slip pulses along interfaces between dissimilar materials that were shear-loaded below the friction threshold. Cochard and Rice (2000) showed that the model which Andrews and Ben-Zion used, with constant friction coe cient, would not give numerical convergence with grid reÿnement, due to the ill-posedness discussed. Cochard and Rice (2000) also showed that a reformulation of the problem using the regularized friction law of Ranjith and Rice (2001) did lead to convergent solutions with properties somewhat like those suggested by Andrews and Ben-Zion (1997).…”

“…Cochard and Rice (2000) showed that the model which Andrews and Ben-Zion used, with constant friction coe cient, would not give numerical convergence with grid reÿnement, due to the ill-posedness discussed. Cochard and Rice (2000) also showed that a reformulation of the problem using the regularized friction law of Ranjith and Rice (2001) did lead to convergent solutions with properties somewhat like those suggested by Andrews and Ben-Zion (1997). Extensive further studies along those lines, using the same regularized friction law, are reported by Ben-Zion (2001).…”

“…The widely occurring unstable situations correspond toâ ¿ 0. For those, the problem of response to a generic perturbation is ill-posed (Renardy, 1992;Martins and Simões, 1995;Simões and Martins, 1998;Cochard and Rice, 2000;Ranjith and Rice, 2001); it has no solution for any positive time if the Fourier strength of the perturbation falls o less rapidly than a negative exponential in |k| at large |k|.…”