An Assessment of the Greenwood-Williamson and Other Asperities Models, With Special Reference to Electrical Conductance

Although in principle simple and neat results are obtained with the classical Greenwood-Williamson (GW) model (linearity of real contact area and conductance with load), the definition of asperity as local maxima of the surface leads to uncertain results for multiscale surfaces, as suspected already by Greenwood in a recent self-assessment of his theory [Greenwood, J. A., and Wu, J.J., 2001, “Surface roughness and contact: an apology,” Meccanica 36(6), pp. 617–630]. Quoting the conclusions in the latter paper “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 suggest that an alternative definition of asperity captures the mechanics of the contact more correctly, that of Aramaki-Majumbdar-Bhushan (AMB). Here, numerical experiments confirm that with a Weierstrass series fractal profile (taken as a 2D slice of a true fractal surface but then used to define a set of circular asperities), load and conductance for numerically measured asperities defined “à la Greenwood-Williamson” (3PP, 3-point peaks) differ significantly from the results obtained with the Aramaki-Majumbdar-Bhushan definition of asperity. The AMB definition, which is based on the bearing area intersection best parabola fitting, gives finite limits for all quantities and varies very little with small scale terms, and tends to coincide with the 3PP method only at unrealistically large fractal dimensions $D$⁠, or at unrealistically large separations. However, it remains unclear how the AMB results compare with the proper treatment of the problem when interaction effects are fully taken into account.