### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Rate and State Equations
- 3. Ruina Evolution Law
- 4. Discussion and Conclusions
- Acknowledgments
- References

The well-known formalism of rate and state dependent friction represents the transition between starting friction and sliding friction. It expresses the instantaneous coefficient of friction in macroscopic quantities as μ = μ_{0} + *a* ln (*V*/*V*_{0}) + *b* ln (ψ/ψ_{norm}), where μ_{0} is the first-order coefficient of friction, *a* and *b* are small (∼0.01) dimensionless constants, *V* is sliding velocity, *V*_{0} is a reference sliding velocity, and the inverse of the state variable 1/ψ represents damage. The normalizing state variable ψ_{norm} ≡ (*P*/*P*_{0})^{α/b} represents the effect of the normal traction *P* on stress concentrations, where *P*_{0} is a constant with dimensions of pressure and α ≫ *b* is another dimensionless constant. Evolution equations represent the combined effect of damage from sliding and healing on the state variable. The Ruina (1983) evolution equation implies that the state variable does not change (no healing) during holds when sliding is stopped. It arises from exponential creep within gouge when the concentrated stress at asperities scales with the macroscopic quantities. The parameter *b* − *a* being positive is a necessary condition for a spring-slider system becoming unstable. Microscopically, this parameter represents the tendency of asperities accommodating shear creep to persist longer than asperities of compaction creep at high sliding velocities. In the Dieterich (1979) evolution law, healing occurs when the sample is at rest. It is a special case where creep that produces shear and creep that produces compaction occur at different microscopic locations at a subgrain scale. It also applies qualitatively for compaction at a shear traction well below that needed for frictional sliding. Chemically, it may apply when shear sliding occurs within weak microscopic regions of hydrated silica while compaction creep occurs within comparatively anhydrous grains.