Fractal geometry of contacting patches in rough elastic contacts

Many naturally formed and processed surfaces are rough over a broad range of length scales. Surface roughness reduces the area of contact between solids, with ramifications for phenomena that depend on the geometry of the interface and the amount of direct contact, including friction and adhesion. In this work, we employ large-scale boundary-element simulations for nonadhesive, elastic solids to study the size dependence of contact patch mean pressure and geometry for patches formed between solids with self-affine fractal surface roughness across seven decades in patch area. Contact patches with diameters smaller than a crossover length scale of order the minimum wavelength of roughness are generally compact with simple geometries and bear pressures well described by Hertz theory. The patch pressure in contact patches larger than the crossover scale rises logarithmically before saturating at a finite value. Furthermore, the largest contact patches formed during our simulations are ramified and populated with regions out of contact, or bubbles, which reduce patch area and increase patch perimeter. As a result, we show that the mean contact diameter of the largest patches saturates, indicating that the patch contact area is proportional to the total patch perimeter. We quantify the effects of bubbles on patch area and perimeters as a function of Hurst exponent and contrast our findings with results of comparable bearing-area model calculations. The slow evolution of the mean patch pressure with patch size in our large-scale calculations explains the common observation that the global mean contact pressure depends on the structure of the roughness, the contact area, and even on system size.