Elastic Multiscale Contact of Rough Surfaces: Archard’s Model Revisited and Comparisons With Modern Fractal Models

Abstract

Bowden and Tabor (1 BT, in the following) state that friction is dictated by “adhesion (cold weld)” and “ploughing (inelastic deformation term),” between asperities. Amonton’s law could easily be explained for the ploughing term, as the real area of contact would simply be A=P/H, where H is the hardness of the softer of the contacting bodies, and P is the applied load. However, for the elastic term, which in most cases would be the dominant one, Hertz’ theory would not predict linearity with load. During the 1950s, several articles appeared in prestigious journals (234) where multiscale models were introduced to explain Amonton’s and several connected well-known laws for friction, wear and electrical/thermal resistance, in terms of elastic deformations of multiscale, and rigorously ∞-scale model which we would now call a fractal (4), as depicted in Fig. 1—this is not the only possible choice, as the Archard model only takes into account of load redistribution and not of the actual geometry. These models found that the relation real contact area to total load for an ensemble of elastic asperities separated enough from each other to neglect interaction effects, is Display FormulaAn=KnWẼαnwhere Ẽ=E/(1−ν2), and Kn is a coefficient which depends on the number of scales introduced n, and was computed by Archard for the first few scales only of his model. Archard’s main finding was that αn tends rapidly to one as n is increased. No particular attention was, vice versa, paid by Archard to the coefficient Kn, which will be here recomputed in general and will be specialized for a fractal geometry.