A contact problem is considered in which an elastic half-plane is pressed against a rigid fractally rough surface, whose pro le is de ned by a Weierstrass series. It is shown that no applied mean pressure is su¯ciently large to ensure full contact and indeed there are not even any contact areas of nite 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 develop a relation between the contact pressure distribution at scale n 1 and that at scale n. Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results 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 power-law behaviour at large n which corresponds to a limiting fractal dimension of (2 D), where D is the fractal dimension of the surface pro le. However, 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 appropriately normalized size distribution tends to a limiting function at large n.

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A method is developed for placing bounds on the electrical contact conductance between contacting elastic bodies with rough surfaces. An analogy is rst established between contact conductance and the incremental sti¬ness in the mechanical contact problem. Results from contact mechanics and the reciprocal theorem are then used to bracket the mechanical load{displacement curve between those for two related smooth contact problems. This enable bounds to be placed on the incremental sti¬ness and hence on the electrical conductance for the rough contact problem. The method is illustrated by two simple examples, but its greatest potential probably lies in establishing the maximum e¬ect of neglected microscales of roughness in a solution of the contact problem for bodies with multiscale or fractal roughness.

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