A “re-vitalized” Greenwood and Williamson model of elastic contact between fractal surfaces

Greenwood and Williamson in 1966 (GW) proposed a theory of elastic contact mechanics of rough surfaces which is today the foundation of many theories in tribology (friction, adhesion, thermal and electrical conductance, wear, etc.). However, the theory has periodically received criticisms for the “resolution-dependence” of the asperity features. Greenwood himself has recently concluded that: “The introduction by Greenwood and Williamson in 1966 of the definition of a ‘peak’ as a point higher than its neighbours on a profile sampled at a finite sampling interval was, in retrospect, a mistake, although it is possible that it was a necessary mistake” [Greenwood and Wu, 2001. Surface roughness and contact: an apology. Meccanica 36 (6), 617–630]. We propose a “discrete” version of the GW model, keeping the approximation of a surface by quadratic functions near summits, where the summit arrangement is found from numerical realizations or real surfaces scans. The contact is then solved either summing the Hertzian relationships, or considering interaction effects to the first-order in a very efficient algorithm. We conduct experiments on Weierstrass–Mandelbrot fractal surfaces, concluding that:the real contact area–load relationship is well captured by the original GW theoretical model, once the correct mean radius is used. The relationship is robust and shows relatively little scatter;the conductance–load relationship is vice versa only approximately given by the original GW theoretical model. Significant deviations from linearity and significant scatter seem to be found, particularly at low fractal dimensions;the load, area and conductance dependences with separation show significant dependence on the actual phase arrangements, and hence significant scatter at large separations. Effect of interaction is seen strongly at low separations, where scatter is minimal. The discrete GW model permits to include these effects, except when the asperity description breaks down. Refinements of the original GW theory using the full random process theory (such as that by Bush Gibson and Thomas, BGT) result only in small improvements with a significant additional complication. However, the BGT relationship between contact area and load at low loads is more accurate than the more recent theory by Persson. The distribution derived from the original GW theory has been obtained, and shown to be closer to the numerical results than that predicted by the Persson model, even if the area error is removed. It is concluded that the original GW theory deserves the general success received so far, since the resolution-dependence of geometrical features is an intrinsic feature of “fractals” but not a problem for the GW theory, when interaction effects are included.