  We generalize the Persson contact mechanics and rubber friction theory to the
case where both surfaces have surface roughness. The solids can be rigid,
elastic or viscoelastic, and can be homogeneous or layered. We calculate the
contact area, the viscoelastic contribution to the friction force, and the
average interfacial separation as a function of the sliding speed and the
nominal contact pressure. We illustrate the theory with numerical results for a
rubber block sliding on a road surface. We find that with increasing sliding
speed, the influence of the roughness on the rubber block decreases, and for
typical sliding speeds involved in tire dynamics it can be neglected.
