Hysteresis and Recovery Length in a Dry Solid Friction Experiment

We report an experiment on dry friction between two elastic surfaces made of millimetric steel spheres embedded in an elastic medium. We study the hysteresis cycle of the tangential force induced by a quasi-static cyclic displacement. At low velocities, hysteresis effects are fully characterized by a recovery length comparable to the size of the steel spheres. This length is independent of the velocity and of the normal load. We propose an interpretation for this recovery length with a simple model of discrete contacts. Dry solid friction occurs in a wide range of systems and is of enormous practical importance. The interest of the physicists in this subject has been renewed recently, due to the development of earth-quake models (1), as well as the analogies with pinning problems such as magnetic vortices, charge density waves, fracture fronts, wetting hysteresis, etc. From a macroscopic point of view solid friction is a phenomenon with a well defined threshold. It is very well known that two solids put in contact with each other remain at rest until the tangential force applied to them reaches a threshold value Fa = psFN, where ps is the static friction coefficient and FN the normal force applied to the bodies. Above this threshold the bodies enter into a relative motion. If the two bodies are moved periodically back and forth on a distance Ax, the existence of the threshold clearly produces a mechanical hysteresis because one needs to change the driving force from Fa to -Fa in order to reverse the motion. Thus the energy Ed, dissipated to do back and forth the relative displacement Ax, is just Ed ~ 2FaAz, independently of the amplitude of Ax. This value of Ed is a consequence of the fact that, in the standard model, there is no relative displacement of the two solids if the tangential force is below threshold. However it has been known for a long time that surfaces undergoing dry solid friction exhibit displacements at the microscopic scale, called micro-slip, before the macroscopic sliding occurs (2,3). Micro-slip is of interest, not only because of its engineering importance in computer modelling of solid friction, but also in relation with the theories of Inechanical friction based on elastic instabilities in a pinning potential (4-6). In these theories,