The fractal characterization techniques, discussed in detail in Section 4.3.3, overcome some of the short- comings of conventional methods that use σ, σ′, and σ″. One of the requirements of the fractal technique is that the structure function or the power spectrum must follow power law scaling behavior, that is, S(τ) ≈ τ2(2–D) or P(ω) ≈ ω–(5–2D). If this is satisfied, then the asperity height, δ, and the base size, l, follow the scaling relation δ ≈ l(2–D). This is particularly useful in tribology since only two parameters, G and D, need to be known to study tribological phenomena at all length scales in the fractal regime. However, the experimental data in Figures 4.13 and 4.15 through 4.21 show that although the scaling behavior is followed in some cases, it is not universal. In addition, the power law can change at a transition length scale and is not universal over all length scales. Yet, the rms slope and curvature cannot be used to characterize them since the surface can have multiple scales, which, although they do not follow the scaling behavior δ ≈ l(2–D), can lead to scale-dependent rms values. A technique must therefore be developed that will work for both fractal and nonfractal surfaces and yet be scale independent. This section introduces a new method with these issues in mind.
It is necessary to identify first how the surface characteristics will be used. As discussed in Section 4.2, knowledge of the surface structure is important for predicting the size and spatial distributions of contact spots as well as for the mechanics of asperity sliding. Since these spots are formed by asperities, what is important is the asperity geometry at relevant length scales and its size and spatial distributions on the surface. The conventional techniques, which use rms height, slope and curvature, find an average asperity shape, whereas the fractal techniques determine the shape at all length scales by the scaling law. Both techniques can be combined to form a general method for roughness characterization as follows.
The roughness is characterized by two parameters — V(l) and K(l) — which are found in the x- and y-directions* by the following relations.
(4.3.31)
(4.3.32)
Here the 〈 〉 symbol implies averaging over the measured data points. It is evident that the function V(l) is the square root of the structure function S(l). For a fractal surface V(l) = G(D–1)l(2–D). The function K(l) is the rms curvature of asperities of lateral scale l which in the fractal model is assumed to vary as K(l) = G(D–1)/lD. In the generalized model, such power laws will not be assumed and instead the raw data
*The x- and y-directions are chosen to be the principal directions of an anisotropic surface. The principal directions can be found by first obtaining V in all directions to get a V vs. l surface. If one takes a horizontal cut of the surface for V = constant, then one can connect the loci of the intersecting l values into a curve which in general can be approximated by an ellipse. Then the x- and the y-directions correspond to the major and the minor axes of the ellipse. The assumption made in this model is that the major and minor axes remain the same at all length scales.
V z x y z x y
V z x y z x y
x
y
l l
l l
( )= [ ( + )− ( ) ]
( )= [ ( + )− ( ) ]
, ,
, ,
2
2
K z x y z x y
K z x y z x y z x y
x
y
l l l
l
l l l
l
( )= [ ( + )+ ( − ) ]
( )= [ ( + )+ ( − )− ( ) ]
, ,
, , ,
2
2
2
2
2
of V(l) and K(l) will be used. There is no need to prove whether the surface is fractal or nonfractal. If the surface is fractal, both these parameters will show scaling behavior. If the surface is nonfractal and yet contains multiple length scales, this technique will allow one to incorporate the scale-dependent information contained in V(l) and K(l). When the surface is perfectly periodic with wavelength, λ, then K(l) will show a peak when l = λ/2. In general, V(l) and K(l) neither follow scaling behavior nor do they show sudden jumps in the data but are a combination of both. It will be shown in Section 4.5 that these characteristics can be used to develop theories of contact mechanics and other tribological phe- nomena. Before that, it is necessary to understand how this technique can be used to characterize anisotropic surfaces.
Consider a typical plot of Vx, Vy, Kx, and Ky as a function of l for a general anisotropic surface in Figure 4.25. An asperity on this surface will have an elliptic base with major and minor axes lx and ly
and curvatures kx and ky as shown in Figure 4.26. The values of lx and ly can be found from Figure 4.25 by taking the intersection of a horizontal line with the Vx and Vy curves. Intersections of the vertical lines through lx and ly with the Kx and Ky curves give the respective curvature values of kx and ky. Note that it is possible to have more than one intersection of a horizontal line with the curves Vx and Vy producing FIGURE 4.25 Qualitative demonstration of a typical V vs. l and K vs. l plot for the generalized roughness characterization technique. Note that, in general, the surface is anisotropic such that the curves are different in the x- and y-directions.
FIGURE 4.26 (a) Schematic diagram of an ellipsoidal asperity of an anistropic surface. (b) Equivalent hemispherical asperity that can be used to study the mechanics of contact with a flat hard plane.
more than one value of lx and ly, as shown in Figure 4.25. The area of an asperity base is a = πlxly . For more than one value of lx and ly, the product lxly can be obtained for different combinations, and therefore for a particular value of V = constant (horizontal line) it is possible to have different asperity base areas. For each combination of lxly, the effective curvature of an ellipsoidal asperity of base area a is k(a) = 0.5(kx + ky). This will be used in the contact mechanics of ellipsoidal asperities (Timoshenko and Goodier, 1970) with a flat plate. The procedure followed above collapses the anisotropic roughness data in the x- and y-directions in Figure 4.25 into a plot of V and k vs. the asperity base area a.
Roughness data of a magnetic tape surface will be taken as an example to demonstrate the application of the generalized contact mechanics model. Figure 4.27a shows the plots of Vx, Vy, Kx, and Ky as a function of l for a magnetic tape, whereas Figure 4.27b shows the collapsed data of V(a) and K(a) as a function of asperity base area, a. The roughness was measured by an atomic force microscope and the details of the measurement as well as the surface images are described elsewhere(Oden et al., 1992).
It is evident that this generalized roughness characterization technique does not demand any require- ments for either a scaling fractal surface structure or a surface with single scale of asperities. In that respect, it is more powerful than the other techniques, as will be shown in Section 4.5. However, the simplicity of the rms characterization or the fractal techniques that use only G and D is more than that of the generalized technique. As will be shown later, numerical integration must be performed to get any meaningful results for contact mechanics and size distribution.
FIGURE 4.27 Variation of surface height V and curvature K for the magnetic tape A surface. (a) V and K vs. l for two orthogonal directions; (b) V and K vs. the asperity base area. This collapses the curves in (a) by the procedure discussed in the text.