Elastic contact stiffness and contact resistance for the Weierstrass profile

The Weierstrass series comprises a system of superposed self-affine sine waves that can be used to define a simple idealization of a two-dimensional fractal rough surface profile. The load-compliance relation for the contact of this profile with a rigid plane is here estimated using Westergaard's solution for the contact of a single sine wave with a plane and various approximations concerning the interaction of the different terms in the series. These approximations are compared with a numerical solution for the contact of the profile defined by the first few terms of the series. Once the load-compliance relation is established, the electrical contact resistance can be determined, using an analogy between the conduction and incremental elastic contact problems. The results show that these simple estimates give quite good predictions of the relations between load, compliance and contact resistance. They also confirm that these relations are largely determined by the coarse scale features of the surface profile, in contrast to the predictions of classical asperity model theories.