Dynamic motion of a single degree of freedom system following a rate and state dependent friction law

Sequences of dynamic instabilities are analyzed for a single degree of freedom elastic system which slides along a surface having frictional resistance depending on slip rate and slip rate history, in the manner of Dieterich, Ruina and others. The system is represented as a rigid block in contact with a fixed surface and having a spring attached to it whose opposite end is forced to move at a uniform slow speed. The resulting “stick‐slip” motions are well understood in the classical case for which there is an abrupt drop from “static” to “sliding” frictional resistance. We analyze them here on the basis of more accurate frictional constitutive models. The problem has two time scales, an inertial scale set by the natural oscillation period T of the analogous frictionless system as T /2π and a state relaxation scale L / V occurring in evolution, over a characteristic slip distance L , of frictional stress τ towards a “steady state” value τ ss ( V ) associated with slip speed V . We show that τ ≃ τ ss ( V ) during motions for which acceleration a satisfies aL / V 2 ≪ 1, and that this condition is met during an inertia controlled instability in typical circumstances for which the unstable slip is much greater than L . Since V / a is of order T /2π during inertia controlled motion, one has L / V ≪ T /2π, whereas L / V ≫ T /2π during much of the essentially quasi‐static “stick” part of the cycle when there is a sufficiently small imposed velocity at the load point. Thus the physically irrelevant time scale ( L / V during inertial controlled motion, T /2π during quasi‐static motion) is much shorter than the relevant scale, which is troublesome from a numerical point of view as it is the shorter time scale which constrains allowable step size. We propose efficient numerical procedures to deal with such response, in which the full equations with inertia and state relaxation are solved only in a transition regime when both time scales are significant. We show results for several friction laws, all having history dependence based on a single evolving state variable and all having properties that ∂τ/∂ V > 0 for instantaneous changes in V , that τ evolves towards τ ss ( V ) as exp (−δ/ L ) with ongoing slip δ when V = const, and that dτ ss ( V )/d V < 0 except possibly at high V . During the dynamic instabilities we find that motion continues at a nearly steady state condition, τ≃τ ss ( V ), until dynamic overshoot becomes so significant that “arrest” begins. In the arrest stage, V drops rapidly to very much lower values (never zero in our models) under nearly fixed state conditions, and then the long quasi‐static “stick” phase of the motion begins again.