In this chapter, it was established that surface roughness plays a vital role in microtribology. Rough surfaces are ubiquitous in nature, and in fact it is rare that a surface is perfectly flat. Therefore, any realistic study of tribology must include the effects of surface roughness. Contact between two rough surfaces occurs in the form of discrete spots. It is at these contact spots that all the forces responsible for friction and wear are generated. These spots are not of the same size and are not uniformly distributed over the contact interface. Due to the occurrence of surface roughness over several decades of length scales, ranging from nanometer to millimeter scales, the contact spots have a wide distribution of sizes.
Since interfacial forces have been found to be size dependent, it is very important to determine the size distribution of contact spots. In addition, dynamic and force interactions between neighboring spots are also important thus requiring knowledge of the spatial distribution of contact spots.
In order to study the influence of surface roughness on microtribology, it is first necessary to charac- terize surface roughness in a way that can be easily used to develop models and theories of friction and wear. What is of vital importance is the size and spatial distribution of contact spots. Such distributions FIGURE 4.42 Comparison between the predictions of the Cantor set model (Warren and Krajcinovic, 1996b) and the experiments (Handzel-Powierza et al., 1992).
F∝δα; Ar ∝δα
α = −
+ −
1 1
D
D D
c c
are influenced both by roughness and the contact mechanics of surfaces. In this chapter it was shown that conventional methods of roughness characterization using rms values of surface height, slope, and curvature can depend on the resolution of the roughness-measuring instrument and are not unique to a surface. Therefore, theories based on such parameters can produce misleading results. Roughness measurements show that most surfaces are composed of small asperities that sit on larger asperities, which sit on even larger asperities in a hierarchical manner. To characterize this intrinsic multiscale structure of surface roughness, one must develop techniques that are independent of any length scale.
The hierarchical structure allows fractal geometry to be used to characterize a surface by two scale- independent parameters — fractal dimension D and amplitude scaling constant G. The latter has units of length. The relation between the fractal parameters (D and G) and the conventional rms quantities were also developed. Roughness measurements also showed that there exist some surfaces that do not follow the scaling hierarchical behavior that fractal characterization demands. A generalized technique was therefore developed which can characterize both fractal and nonfractal surfaces. This technique can be used directly in theories of contact mechanics that finally yield the size distribution of contact spots needed to develop theories of friction and wear.
A new relation for the size distribution of contact spots was developed in terms of surface height probability distribution and asperity curvature for different asperity sizes. This relation is general and can be applied to both fractal and nonfractal surfaces. It was encouraging to find that application of the fractal scaling law for asperity curvature in this general size distribution produced a well-known empirical power law behavior.
Since the size distribution of contact spots depends not only on the roughness but on the contact mechanics, the last section dealt with some contact mechanics theories. As representative of classical theories, the Greenwood–Williamson model was discussed and critiqued. It was shown that the GW model is applicable only when the surface contains a single dominant roughness scale. Machined surfaces with periodic grooves or polycrystalline surfaces with a narrow grain size distribution are typical exam- ples. However, roughness measurements show that many surfaces do not follow this behavior and are best described by fractal scaling laws. A fractal contact theory showed that for elastic contact between surfaces, the real area of contact Ar and the compressive load Fn follow a power law of the form Fn ≈Arα, where the exponent α varies between 0.75 and 1 depending on the fractal dimension D. Since several surfaces were found to be fractal in certain length regimes and nonfractal in others, there was a need to develop a generalized theory of contact mechanics. This was developed and was based on the generalized roughness characterization technique. To demonstrate the use of this theory, contact of a magnetic thin- film rigid disk was studied. The prediction for real area of contact as a function of compressive load was in close agreement with experimental results. The theory gave the load, real area of contact, and the size of the largest contact spot as a function of the surface separation. In addition, the size distribution of contact spots was also determined as a function of the real area of contact, but was found to be in poor agreement with experimental observations. The reasons for this disagreement were explored.
Despite recent progress in surface roughness characterization and contact mechanics, there still exist several unanswered questions and unaddressed issues. We have assumed that contact spots of isotropic surfaces are circular and those of anisotropic surfaces are elliptical. This is not necessarily true since even for isotropic surfaces, the spots can be elongated in one direction. However, the direction of elongation is random for isotropic surfaces and nonrandom in anisotropic surfaces. The noncircular contact spots for isotropic surfaces is clearly shown in the simulation of Figure 4.3. Therefore, the assumption of circular contact spots is not correct and must be changed. In this chapter we have considered asperity deformation due to a flat hard plane. The problem of asperity indentation into a flat but softer surface has not been addressed.
Although this chapter establishes the importance of size and spatial distributions of contact spots in the development of tribological theories, the issue of spatial distribution has not been addressed at all.
The spatial distribution is needed to study stress interactions and for dynamic interaction between contact spots. This is very important for high surface deformation cases and for the transition between static and kinetic friction. The problem lies not in developing a relation for the spatial distribution but
developing a method that can be used in a theory or model of neighboring asperity interaction. That itself has not been developed or properly understood. Therefore, the question of neighboring asperity interaction and the spatial distribution is an open problem that needs future attention.
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