A very simple estimate of adhesion of hard solids with rough surfaces based on a bearing area model

In the present note, we suggest a single-line equation estimate for adhesion between elastic (hard) rough solids with Gaussian multiple scales of roughness. It starts from the new observation that the entire DMT solution for “hard” spheres (Tabor parameter tending to zero) with the Maugis law of attraction can be obtained using the Hertzian relationship load-indentation and estimating the area of attraction as the increase of the bearing area geometrical intersection when the indentation is increased by the Maugis range of attraction. The bearing area model in fact results in a simpler and even more accurate solution than DMT for intermediate Tabor parameters, although it retains one of the assumptions of DMT, that elastic deformations are not affected by attractive forces. Therefore, a solution is obtained for random rough surfaces combining Persson’s adhesiveless asymptotic simple form solution with the bearing area model, which is trivially computed for a Gaussian. A comparison with recent data from extensive numerical computations involving roughness with wavelength from nano to micrometer scale shows that the approximation is quite good for the pull-off in the simulations, and it remarks the primary importance in this regime of a single parameter, the macroscopic well-defined quantity (rms) amplitude of roughness, and small sensitiveness to rms slopes and curvatures.