  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 scalings are confirmed numerically.
