The transition from steady frictional sliding to inertia-dominated instability with rate and state friction

Unstable frictional slip motions are investigated with a rate and state friction law across the transitions from stable, quasi-static slip to dynamic, stick-slip motion, and finally to, inertia dominated quasi-harmonic vibration. We use a novel numerical method to capture the full dynamics and investigate the roles of inertial and quasistatic factors of the critical stiffness defining the transition to instability, Kc. Our simulations confirm theoretical estimates of Kc, which is dependent on mass and velocity. Furthermore, we show that unstable slip motion has two distinct dynamic regimes with characteristic limit cycles: (i) stick-slip motions in the quasi-static (slowly loaded) regime and (ii) quasi-harmonic oscillations in the dynamic (rapidly loaded) regime. Simulation results show that the regimes are divided by the dynamic frictional instability coefficient, η = MV2/σaDc and stiffness of the system K. The quasi-static regime is governed by the ratio K/Kc and both the period and magnitude of stick-slip cycles decrease with increasing loading rate. In the dynamic regime, slip occurs in harmonic limit cycles, the frequency of which increases with loading velocity to a limit set by the natural frequency of the system. Our results illuminate the origin of the broad spectrum of slip behaviors observed for systems ranging from manufacturing equipment to automobiles and tectonic faults, with particular focus on the role of elasto-frictional coupling in dictating the transition from slow slip to dynamic instability. We highlight distinct characteristics of friction-induced slip motions (stick-slip and friction-induced vibration) and show that the dynamic frictional instability coefficient (η) is a key parameter that both defines the potential for instability and determines the dynamic characteristics of instability.