Non-linear stability analysis of slip in a single-degree-of-freedom elastic system with frictional evolution laws spanning aging to slip

We present a non-linear stability analysis of quasi-static slip in a spring-block model. The sliding interface is governed by rate- and state-dependent friction, with an intermediate state evolution law that spans between aging and slip laws using a dimensionless parameter \epsilon. Our results extend and generalize previous findings of Gu et al. (1984) and Ranjith and Rice (1999) that considered slip and aging laws, respectively. We examine the robustness of these prior results to changes in the evolution law, including the finding of unconditional stability of the aging law for spring stiffnesses above a critical value. Our analysis provides analytical trajectories of slip motion in a phase plane as function of dimensionless governing parameters. We investigate two scenarios: a spring-block model with stationary and non-stationary point loading rate. When the loading point is stationary, we find that deviations from the aging law lead to only conditional stability of the slider for spring stiffnesses above a critical value: finite perturbations can trigger instability, consistent with prior results for the slip law. We quantify these critical perturbations as a function of the governing parameters. We find that, for a given supercritical stiffness, the size of the perturbation required to induce instability grows as the state evolution law approaches the aging law. In contrast, when the point loading rate is stationary, our results suggest that there exists a maximum critical stiffness above which an instability can never develop, for any perturbation size. This critical stiffness is \epsilon-dependent and vanishes as the slip law is approached: conditional stability is then expected in the slip law limit. Finally, we derive relations for an effective spring stiffness as a function of the elastic moduli and a characteristic fault dimension or a characteristic perturbation wavelength.