A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load

The author presents a very simple and slightly corrected version of the well-known Greenwood and Williamson (GW) model which closely follows the predictions of more complicated and computationally expensive theory such as the Bush, Gibson and Thomas (BGT) theory [Bush, A.W., Gibson, R.D., Thomas, T.R., 1975. Wear 35, 87]. This new model (which I call GW modified in the following) still treats the asperities of the rough surface as spherical cups, but, this time, the curvature of spheres is not constant and instead depends on the asperity height. The GW modified theory is, in particular, able to predict, in the limiting case of large separations, the same asymptotic linearity between contact area and load as in the BGT theory. This, in turn, proves that the BGT asymptotic linear area–load relation is not a consequence of having taken into account that the contact between the asperities is actually elliptic and, therefore, of having included the spread of asperity curvature at a given height (which incidentally makes the treatment very complicated), but a consequence of having included only the influence of asperity heights on the curvature distribution of the summits. I also give a simple explanation for why the GW modified model and the BGT theory follow exactly the same asymptotic behavior. Indeed, I show that the surface summits can be treated as perfectly spherical cups (all those at the same height having the same radius of curvature) as their height is increased to very large values. In fact, in the asymptotic limit of large separation, the mean curvature of summits is shown to increase proportionally to the summit height, whereas the difference of the two principal curvatures approaches a finite constant value. The consequence of this is that the Hertzian contact between the approaching elastic (initially flat) half-space and the summits exactly resemble that between an elastic half-space and a sphere.