Precursory predictors of the onset of stick-slip frictional instability

Defining precursory phenomena that forewarn the onset of stick slip events remains an elusive but central goal in earthquake and other failure forecasting. We present a method to define the timing of instability by separately decoupling velocity and inertial effects in a spring-slider system. System state is defined in phase-space by velocity and inertia that in turn control critical stiffness and define the onset of instability while sliding velocity V = V_Lk/(k + dF/du) remains finite. A general energy analysis defines velocity and inertial effects (coefficients) that act against each other to promote/suppress instability by respectively increasing/decreasing critical stiffness. Repetitive stick-slip experiments define features of stick-slip cycles and identify a precursory trend in accelerating slip that precedes the onset of unstable sliding. We represent this precursory acceleration (dV/dt) immediately preceding instability by a general power-law relation dV/dt = AV^α that reduces to V = B(c_i + t_i-t)^-β. This represents the “true” timing of the onset of stick slip, t_i relative to that “projected” from the linearization of the precursory data, t_f, with t_f=c_i + t_i. c_i represents the small offset between the “predicted” and “true” timing of the onset of stick slip resulting from velocity and inertial effects and may be considered as the error in prediction. This error is one-order of magnitude smaller (∼3 ms) than the lead-time that forewarns of the event (∼40 ms) in our experiments and thus is useful as a predictor. Three independent methods using power-law, linear and criticality relationships confirm the fidelity of the timing of the slip instability transition evident in the stick slip data. This theoretical treatment suggests that the underlying physical meaning of the parameter c in the modified Omori law R∼(c + t_M-t)^-q is the time of the main shock in advance of the timing of the singularity predicted for an ideal response.