On the dynamic sliding with friction of a rigid block and of an infinite elastic slab

We consider dynamic motions of two elastic systems undergoing frictional slip. The first one is the classical model of the frictional slider loaded through an elastic spring. The second one is an infinite elastic slab bounded by two planes which is in contract with a rigid body and submitted to shearing. Slip weakening and slip rate weakening friction laws are both considered. The two simple systems show very different qualitative behaviors. In the case of the slip dependent friction a slider moves with a single slip event when a critical stress level is reached. Under the same conditions, a series of slip events occur for the infinite slab. This difference between the behavior of the two systems is due to the important part played by inertia in the mass concentrated block slider model. In the case of slip rate weakening, the analysis of the problem for the infinite slab indicates a major difficulty: this problem has no unique solution if the rate of weakening exceeds a limit that is explicitly given. Whatever is the selection rule chosen to discriminate the solution, shocks will occur. The slip history obtained for the slab is very different from the one obtained with a block slider. For the infinite elastic slab the slip velocity exhibits sharp variations (shocks). On the contrary, a block slider does not exhibit this behavior. It is a clear example of the limitation of the use of such a simple analogy to describe the actual properties of the relative motions of two media in contact with friction.