Conductance of rough random profiles

Recently, the real contact area and the compliance and electrical resistance for a rough surface defined with a Weierstrass series have been studied under the assumption that superposed self-affine sine waves had well separated wavelengths, extending the celebrated procedures pioneered by Archard [Archard, J.F., 1957. Elastic deformation and the laws of friction. Proc. R. Soc. Lond. A 243, 190–205]. Here, more realistic fractal rough surface profiles are considered, by using the Weierstrass series with random phases, and with much lower separation of the various scales, using a full or a hybrid numerical/analytical technique. A non-linear layer algorithm is developed which is a very efficient approximate tool to study this problem, avoiding the need for averaging over various realizations of profiles with random phases. The multiscale problem is solved by a cascade of 2-scales problems, each of which is solved with a few elements for an imposed contact area, deriving load as a function of indentation and the conductance by differentiation using Barber’s analogy theorem.Dimensionless results for the conductance as a function of applied pressures show that the conductance seems to be close to a power law at low loads, and is nearly linear at intermediate loads (following the normalized single sinusoidal case except at the origin). At high loads, the conductance becomes strongly dependent on fractal dimension because of weak dependence on the contribution of small wavelength scales (higher order terms in the series). Since roughness tends to be squeezed out, the conductance tends to increase more than linearly (more so, the smaller is the fractal dimension). However, another limit could be found in terms of the finite size of the specimen, which may suggest reaching a finite limit. The resulting curves could then be sigmoidal, as confirmed by qualitative comparisons with experiments in the literature.