Linear elastic contact of the Weierstrass profile

Abstract

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.

Finite-element analysis of contact between elastic self-affine surfaces

“…Most surfaces are rough enough that atoms are only close enough to interact strongly in areas where peaks or asperities on opposing surfaces overlap. Experiments [1,2,3,4] and theory [5,6,7,8,9,10,11,12] show that this real area of contact A is often much smaller than the projected area A 0 of the surfaces. They have also correlated [1,2,3] the increase in friction with normal load W to a corresponding increase in A.…”

Some Issues of Rough Surface Contact Plasticity at Micro- and Nano-Scales

“…Motivated by the recent work in [3], we have carried out an elastic-plastic contact analysis of a fractal rough surface, which is idealized with a Weierstrass series. Scaling relations of the total contact area versus the applied load and roughness properties can be deduced [4], and compared with numerical simulations that have pre-specified spatial cutoff sizes [5,6].…”

The “Sport” of Rough Contacts and the Fractal Paradox in Wear Laws

“…In an attempt to solve the paradoxes of Ciavarella et al [4], Gao et al [19] added plasticity in the Weierstrass model. While the elastic model [4] showed that the -fractal limit‖ consists of an infinite number of contact spots, with zero size, subjected to infinite contact pressure, Gao et al [19]'s model found that while the total contact area and contact pressure are well defined, it remains impossible to predict the actual number of contacts or their size. They also suggest to add adhesion (but the problem with both plasticity and adhesion has not even been attempted yet) or truncating the fractal process where the fractal description breaks down.…”

“…In fact, he even anticipated -fractals‖ which were much later used in a more refined form in the -contact sport‖ of rough surfaces. Yet it was only much later that Ciavarella et al [4] remarked that linearity involves a linear coefficient which is scale-dependent, and in particular it goes to zero for realistic fractal geometry. This was done with the not very popular choice of a Weierstrass series as a fractal, but later Gaussian model theories [5] did not change the basic conclusion that -no applied mean pressure is sufficiently large 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‖.…”