Rough spheres in elastic contact - problem solving in the lazy way

When arbitrary rough or structured surfaces enter into contact, the first spots that touch are the highest peaks (asperities). They exert surface stresses to each other which cause the surfaces to deform. Depending on the bodies‘ curvatures and the roughness parameters, the shape of the contact spots and the resulting stresses can take many different forms. One problem of particular interest is the elastic contact of a sphere which appears smooth to the naked eye, but has some micro roughness which is relevant at low normal loads. I will show how the random roughness can be replaced by a simple regular shape that has the same statistical and contact mechanical properties. This new shape is superposed onto a perfectly smooth sphere to give an equivalent system. Solving the problem of what spots of the surface are actually in contact usually requires dealing with integral equations and nonlinear, floating boundary conditions. I will show how to overcome the hassle by applying Valentin Popov’s Method of Dimensionality Reduction (MDR). The combination of both shortcuts allows us to obtain force-displacement-curves for the whole range of normal loads using only basic algebra.