Linear elastic contact of the Weierstrass profile

A contact problem is considered in which an elastic half-plane is pressed a gainst a rigid fractally rough surface, whose profile is defined by a Weier strass series. It is shown that no applied mean pressure is sufficiently la rge to ensure full contact and indeed there are not even any contact areas of finite dimension-the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to devel op a relation between the contact pressure distribution at scale n-1 and th at at scale n. Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same pro blem appropriate at large n is constructed following a technique due to Arc hard. This is found to give a very good approximation to the numerical resu lts even at small n, except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n, tending to a pow er-law behaviour at large n which corresponds to a limiting fractal dimensi on of (2 - D), where D is the fractal dimension of the surface profile. How ever, it is not a 'simple' fractal, in the sense that it deviates from the power-law form at low n, at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appro priately normalized size distribution tends to a limiting function at large n.