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

We consider the dynamic motion of an elastic slab subject to non-linear fri ction on a rigid substratum. We consider two categories of friction laws. T he first corresponds to rate-dependent models. This family of models natura lly 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 differenti al relation. The regularized rate-dependent models include as a particular case the classical rate and single-state variable friction laws of Dieteric h and Ruina. The two models of friction show very different mathematical be haviour. The rate-dependent models lead to a scalar equation, which has no unique solution in general. If the velocity weakening rate exceeds a certai n value, many solutions exist. To overcome this difficulty, we have to defi ne a formal rule of choice of the solution. To discriminate between solutio ns we propose using the perfect delay convention of the catastrophe theory. The second category of models, that is, the regularized rate-dependent mod els, leads to a differential equation, which has a unique solution. We give its condition of stability and we show that it corresponds to the conditio n of nonuniqueness of the first model. Considering the particular regulariz ed rate-dependent model of Perrin et al. (1995), we show numerically that t he limit solution when the characteristic slip L-->0 is the one correspondi ng to the rate-dependent model (the steady-state model) assuming the perfec t 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 proble m. We suggest that the perfect delay convention may be used when pure rate (or mainly rate) dependence is involved. Finally, we analyse briefly the ro le of the other parameters, A and B, of the rate and state formulation in t he context of the shearing slab.