A New Microscopic Model of the Rate- and State- Friction Evolution

The Slip (Ruina) law and the Aging (Dieterich) law are the two most common descriptions of the evolution of "state" in rate- and state-dependent friction, behind which are the ideas of slip-dependent and time-dependent fault healing, respectively. Since the mid-1990's, friction experiments have been interpreted as demonstrating that fault healing in rock is primarily time-dependent, and that frictional strength is proportional to contact area (Dieterich and Kilgore, 1994; Beeler et al., 1994). However, a recent re-examination of the data of Beeler et al. (1994) suggests that the evidence for time-dependent healing is equivocal, while large step velocity decreases provide unequivocal evidence of slip-dependent healing (Bhattacharya et al., AGU 2016). Nonetheless, unlike the Aging law, for which see-through experiments showing growing contacts could serve as a physical model, there has been no corresponding physical picture for the Slip law. In this study, we develop a new microscopic model of friction in which each asperity has a heterogeneous strength, with individual portions "remembering" the velocity at which they came into existence. Such a scenario could arise via processes that are more efficient at the margin of a contact than within the interior (e.g., chemical diffusion). A numerical kernel for friction evolution is developed for arbitrary slip histories and an exponential distribution of asperity sizes. For velocity steps we derive an analytical expression that is essentially the Slip law. Numerical inversions show that this model performs as well as the Slip law when fitting velocity step data, but (unfortunately) without improving much the fit to slide-hold-slide data. Because "state" as defined by the Aging law has traditionally been interpreted as contact age, we also use our model to determine whether the "Aging law" actually tracks contact age for general velocity histories. As is traditional, we assume that strength increases logarithmically with age. For reasonable definitions of "age" we obtain results significantly different from the Aging law for velocity step increases. Interestingly, we can obtain an analytical solution for velocity steps that is very close to the Aging law if we adopt a definition of age that we consider to be non-physical.