We study the stability of steady sliding between elastically deformable continua using rate and state dependent friction laws. That is done for both elastically identical and elastically dissimilar solids. The focus is on linearized response to perturbations of steady-state sliding, and on studying how the positive direct e ect (instantaneous increase or decrease of shear strength in response to a respective instantaneous increase or decrease of slip rate) of those laws allows the existence of a quasi-static range of response to perturbations at su ciently low slip rate. We discuss the physical basis of rate and state laws, including the likely basis for the direct e ect in thermally activated processes allowing creep slippage at asperity contacts, and estimate activation parameters for quartzite and granite. Also, a class of rate and state laws suitable for variable normal stress is presented. As part of the work, we show that compromises from the rate and state framework for describing velocity-weakening friction lead to paradoxical results, like supersonic propagation of slip perturbations, or to ill-posedness, when applied to sliding between elastically deformable solids. The case of sliding between elastically dissimilar solids has the inherently destabilizing feature that spatially inhomogeneous slip leads to an alteration of normal stress, hence of frictional resistance. We show that the rate and state friction laws nevertheless lead to stability of response to su ciently short wavelength perturbations, at very slow slip rates. Further, for slow sliding between dissimilar solids, we show that there is a critical amplitude of velocity-strengthening above which there is stability to perturbations of all wavelengths. ?

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.

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