Phenomenological Theory of Friction in the Quasistatic Limit : Collective Pinning and Memory Effects

When a elastic body is moved quasistatically back and forth over a surface, the friction of the interface is experimentally observed to circulate through a hysteretic loop. The asymptotic behaviour of the hysteresis loop is approached exponentially. We describe how this behaviour is connected to the collective properties of the elastic instabilities suffered by the elastic body as it is displaced quasistatically. We express the length scale of the exponential in terms of the elasticity of the surface and the properties of the rough substrate. The predicted scaling are confirmed numerically.