RSPA_CiardoViesca_SupplementaryMaterial.pdf from Nonlinear stability analysis of slip in a single degree of freedom elastic system with frictional evolution laws spanning ageing to slip

We present a nonlinear stability analysis of quasi-static slip in a spring-block model. The sliding interface is governed by rate-and-state friction, with an intermediate state evolution law spanning ageing and slip laws. We examine the robustness of prior results to changes in the evolution law, including the unconditional stability of the ageing law for spring stiffnesses above a critical value. We investigate two scenarios: a spring-block model with stationary and non-stationary point loading rate. In the former scenario, deviations from the ageing law lead to only conditional stability for spring stiffnesses above a critical value: finite perturbations can trigger instability, consistent with prior results for the slip law. For a given supercritical stiffness, the perturbation size required to induce instability grows as the state evolution law approaches the ageing law. By contrast, for a stationary point loading rate, there exists a maximum critical stiffness above which instability can never develop, for any perturbation size. This critical stiffness vanishes as the slip law is approached. For a limited extension of the results found for a spring-slider to continuum faults, we derive relations for an effective spring stiffness as a function of elastic moduli and a characteristic fault dimension or perturbation wavelength.