On dynamic sliding with rate- and state-dependent friction laws

Abstract

We consider the dynamic motion of an elastic slab subject to non-linear friction on a rigid substratum. We consider two categories of friction laws. The first corresponds to rate-dependent models. This family of models naturally contains the steady-state models of Dieterich and Ruina. In the second category-regularized rate-dependent models-the friction law is mainly rate-dependent, but it is more complex and it is defined by a general differential relation. The regularized rate-dependent models include as a particular case the classical rate and single-state variable friction laws of Dieterich and Ruina. The two models of friction show very different mathematical behaviour. The rate-dependent models lead to a scalar equation, which has no unique solution in general. If the velocity weakening rate exceeds a certain value, many solutions exist. To overcome this difficulty, we have to define a formal rule of choice of the solution. To discriminate between solutions we propose using the perfect delay convention of the catastrophe theory. The second category of models, that is, the regularized rate-dependent models, leads to a differential equation, which has a unique solution. We give its condition of stability and we show that it corresponds to the condition of non-uniqueness of the first model. Considering the particular regularized rate-dependent model of Perrin et al. (1995), we show numerically that the limit solution when the characteristic slip L → 0 is the one corresponding to the rate-dependent model (the steady-state model) assuming the perfect delay convention. Hence, the perfect delay convention takes on a physical sense because it leads to a solution that is the limit of a regular problem. We suggest that the perfect delay convention may be used when pure rate (or mainly rate) dependence is involved. Finally, we analyse briefly the role of the other parameters, A and B, of the rate and state formulation in the context of the shearing slab.