Slip dynamics at an interface between dissimilar materials

Abstract

It has been shown recently that steady frictional sliding along an interface between dissimilar elastic solids with Coulomb friction acting at the interface is ill-posed for a wide range of material parameters and friction coefficients. The ill-posedness is manifest in the unstable growth of interfacial disturbances of all wavelengths, with growth rate inversely proportional to the wavelength. We first establish the connection between the ill-posedness and the existence of a certain interfacial wave in frictionless contact, called the generalized Rayleigh wave. Precisely, it is shown that for material combinations where the generalized Rayleigh wave exists, steady sliding with Coulomb friction is ill-posed for arbitrarily small values of friction. In addition, intersonic unstable modes and supersonic steady-state modes exist for sufficiently large values of the friction coefficient. Secondly, regularization of the problem by an experimentally motivated friction law is studied. We show that a friction law with no instantaneous dependence on normal stress but a simple fading memory of prior history of normal stress makes the problem well-posed.

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

“…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 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.…”

“…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|.…”

“…That result has already put a focus on how the Coulomb friction law should be modiÿed to regularize such problems (Simões and Martins, 1998;Ranjith and Rice, 2001). 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.…”

Strategies for the Detection and Analysis of Space-Time Patterns of Earthquakes on Complex Fault Systems

“…Frictional resistance laws that range from the simplest Amontons-Coulomb stick-slip friction [17], to heuristic laws such as slip-or velocity-weakening laws [4,17,18], to laws based on recent laboratory friction experiments including rate-andstate [19][20][21][22][23][24][25][26][27][28] and leaky threshold laws [18,25,26,28], to other types of rupture and healing laws characterized by inner-and outer-physical scales [7].…”