On the elastic energy and stress correlation in the contact between elastic solids with randomly rough surfaces

When two elastic solids with randomly rough surfaces are brought in contact, a very inhomogeneous stress distribution sigma(x) will occur at the interface. Here I study the elastic energy and the correlation function <sigma(q)sigma(-q)>, where sigma(q) is the Fourier transform of sigma(x) and where <...> stands for ensemble average. I relate <sigma(q})sigma(-q)> to the elastic energy stored at the interface, and I show that for self affine fractal surfaces, quite generally <sigma(q)sigma(-q)> \sim q^{-(1+H)}, where H is the Hurst exponent of the self-affine fractal surface.