Analytical Studies of Contact Problems for Fractal Surfaces

摘要

The analytical approaches to contact between fractal surfaces are discussed. It was found that the spectral density function of various surfaces has often the power law character. This behaviour is typical for fractal objects because the fractal box dimension of the surfaces can be calculated using the exponent of the power law of the spectral density. Two fractal models of contact presented in 1991 are discussed, namely the M-B (Majumdar-Bhushan) model based on Weierstrass-Mandelbrot (W-M) profile, and the B-M (Borodich-Mosolov) model based on the Cantor profile introduced by the author. Although the M-B model was practically withdrawn by the authors, it is still very popular in scientific literature. Often it is considered as a synonym for fractal contact models. The M-B model attracts the researchers by the simplicity of the approach. Indeed, it assumes that one can consider an appropriate W-M graph instead of a real rough surface, i.e. a scaled W-M graph having the same power law exponent of the spectral density function as the original surface. Then it is suggested to solve the contact problem for W-M profile by superposition of the Hertzian solutions for asperities of various amplitudes and spans. In fact, this approach is inconsistent. The Cantor profile model is also simple for analytical analysis. However, it has a minor drawback: all asperities of the profile have one-level character, while, as Archard showed, real roughness has hierarchical structure. Another class of fractal surfaces, namely parametric-homogeneous (PH) surfaces is considered. This class of functions was also introduced by the author. Problems of contact between PH-punches and elastic half-space obey a non-classical self-similarity and it is possible to give a strict mathematical treatment to the problems and obtain some strict results concerning general properties of the solutions. It is shown that the fractal dimension alone cannot characterize the contact features of rough surfaces. Moreover, if one considers two punches having the same fractal surface but situated either above or below the surface then he can obtain different asymptotics in both load-displacement and load-area relations. Using results of recent numerical simulations of contact for non-convex PH-punches, the validity of several recent approaches to fractal contact is checked. Finally, it is suggested to follow the Archard's idea by studying an alternative to the Cantor profile model, namely a multilevel prefractal model based on iterated function algorithms.