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I have developed a theory of contact mechanics between randomly rough surfaces. The solids are assumed to deform elastically when the stress sigma is below the yield stress sigma(Y), and plastically when sigma reaches sigma(Y). I study the dependence of the (apparent) area of contact on the magnification. I show that in most cases the area of real contact A is proportional to the load. If the rough surface is self-affine fractal (Hurst exponent H) the whole way up to the lateral size L of the nominal contact area, then (assuming no plastic deformation) A approximately L(H).
B. N. J. Persson (Mon,) studied this question.