On the microphysical foundations of rate-and-state friction

The rate-and-state formulation of friction is well established as a phenomenological yet quantitative description of friction dynamics, in particular the onset of stick-slip instabilities arising from an oscillatory bifurcation. We first discuss the physical origins of two theories for the derivation of friction coefficients used in rate-and-state models, both derived from thermally activated rate processes. Secondly, we propose a general expression for the state evolution law in the form of a first order kinetics which describes the relaxation to a velocity dependent equilibrium interfacial state ϕ ss ( v ) over a velocity dependent dynamic rejuvenation time-scale t ϕ ( v ) . We show that the unknown relation ϕ ss ( v ) , defined as the ratio of t ϕ to a constant interfacial stationary healing time-scale t ⁎ ⁎ , can be estimated directly from the experimental measurements of the steady-state friction coefficient and the critical stiffness for the onset of stick-slip behaviour of a spring-block system. Using a specific experimental dataset, we finally illustrate that this method provides the experimental measurements of the apparent memory length L a ( v ) = v t ⁎ ⁎ ϕ ss ( v ) and the constant characteristic relaxation time t ⁎ ⁎ from which a constant intrinsic memory length L = V ⁎ t ⁎ ⁎ can be defined once a slip rate of reference V ⁎ is chosen. As a result the complete state evolution law can be experimentally characterised.