Slip dynamics at a bimaterial interface

The mathematical well-posedness of the problem of dynamic stability to perturbations from a state of steady frictional sliding along a planar interface between solids with dissimilar elastic properties is studied. It has been recently discovered that this problem is often ill-posed when a Coulomb friction law is taken to act on the interface, i.e. when the frictional strength $t$ is proportional to the normal stress at the interface σ through the (constant) friction coefficient f, $t$ = fσ.; Two experimentally motivated features of friction that make the stability problem well posed are identified. We show that a friction law of the form = −(V/L)($t$ − fσ), where V is the sliding velocity and L is a sliding length, regularizes the problem. The distinguishing feature of this law is that it has a memory effect, and no instantaneous effect, of normal stress on frictional strength. Such a response to normal stress changes is suggested by high speed sliding experiments (with speeds of the order of 1 m/sec) of hard steels against cutting tool materials. We also show that a friction law with an instantaneous dependence of friction on sliding velocity in the form $t$ ∼ aσ ln(V) with a > 0 makes the stability problem well-posed at low sliding speeds. Creep slippage experiments on a variety of materials—metals, rocks, paper, polymers—at slip speeds of up to a few mm/sec support such a positive, logarithmic instantaneous effect of sliding speed, as do models of thermally activated creep at the contacting asperities of the two solids.; Slip stability in a simple spring block model is also studied. Analytical non-linear stability results are derived for quasi-static slip of the block with an experimentally motivated friction law dependent on the slip velocity as well as on the state of the sliding surface. It is shown that the critical spring stiffness for stable steady sliding of the block obtained from a linear analysis fully determines the non-linear stability results. We argue that this implies a slip patch of a critical size has to develop in a continuum system before an instability can develop.