Real contacts and evolution laws for rate and state friction



The well-known formalism of rate- and state-dependent friction represents the transition between static and dynamic friction. The instantaneous coefficient of friction is proportional to the logarithm of the sliding velocity V and the logarithm of a state variable ψ whose inverse is a measure of the current amount of damage within a shearing gouge layer or on a sliding surface. Evolution equations that represent the change of the state variable with time include damage from sliding and healing. The Dieterich (1979) evolution law contains separate terms for the two effects; the Ruina (1983) law groups the two effects into one term proportional to V ln(V). In addition, the state variable depends exponentially on porosity as exp[(ϕ − f)/Cɛ], where ϕ is the porosity at reference conditions and Cɛ is a dimensionless constant. I obtain these relationships as well as relationships between friction-law parameters and physical parameters by representing asperity contacts as rectangular deformable regions between two rigid sheets. The creep rate within an asperity depends exponentially on the second invariant of the deviatoric stress equation image. In terms of asperity-scale quantities the invariant includes normal traction and shear traction. The derivations assume that contact stresses scale with macroscopic stresses. This yields simple expressions analogous to those in rate and state friction. The Ruina (1983) evolution law in general and the decrease in compaction rate during a hold in a slider experiment in particular arise because a sudden decrease in sliding velocity implies that the shear traction and hence the invariant decrease. The porosity-state relationship arises when real stresses at contacts (or the inverse real contact area) depend linearly on the separation of the sheets and by analogy the porosity. Conversely, the Ruina (1983) evolution law is inapplicable when asperity-scale stresses within a deforming material do not scale with macroscopic stresses.