4.5 Contact Mechanics of Rough Surfaces
4.5.3 Generalized Model for Fractal and Nonfractal Surfaces
The structure function data of several types of machined metals, thin-film disks, and magnetic tape in Section 4.3.3.3 show such surfaces have a multiscale structure and yet do not necessarily follow a fractal scaling behavior over all length scales. For such surfaces neither the GW model nor the fractal model can be used. A more generalized analysis is therefore necessary and is developed here for the first time.
In Section 4.4.2 we saw that the size distributions for elastically and plastically deformed asperities are not the same as was assumed in the fractal analysis. By using the size distributions for elastic
TABLE 4.2 Profile Structure Function, Critical Contact Spot, Size-Distribution of Spots, Real Area of Contact, Fraction of Elastic Contact Area and Total Load — Predictions of Fractal Contact Model
Surface profile structure function
Critical contact for spot area
Size-distribution of contact spots,n(a)
Real area of contact
Fraction of elastic contact area
Total elastic-plastic load (aı > ac)
Total plastic load (aı < a) Fn = HAr
Ff =∫0Lτ( ) ( )a n a
a G
H E
c= D
(π )
( )− 2
1 1
2
n a D a a
L D
( )= 2 D2 1+2
A D
Da
r= L
− 2
A A
Da D A
re r
c r
D
= −( )−
( )−
1 2
2 2
F DHa a
D
DEG a
D
a
a D
Ha a EG a a
a
n L D
c
D D
L D
c L
D
L c
L L
c
= − +
π − − ≠
= +
π
− − − −
( ) ( ) ( ) ( )
( ) ( )
2 2 2 1 3 2
3 2
3 2 2
3 4 1 4
1 2 3 4 3 2
2
4
3 3 2 1 1 5
3
.
ln
for
for DD=1 5.
(Equation 4.4.10) and plastic (Equation 4.4.20) contact spots, the total load can be found from Equation 4.5.23 to be
(4.5.25)
FIGURE 4.31 Effect of varying roughness parameter G on (a) real area of contact (b) fraction of real area of contact in elastic deformation.
F g d A H
k a k a dk da
a k a
dk da da
E k a a a
k a dk da da
n a
a
a a
c
c L
= ( ) π ( ) − π( ) + + ( )
+ π ( ) + ( )
∫
∫
0 3
5 2
2 1 2
2
1 2 1
4
3 1
φ
We saw earlier in Section 4.4.2 that the largest contact spot area, aL, is related to the surface mean separation d/σ by Equation 4.4.22. Therefore, Equation 4.5.25 establishes a relation between the total compressive load, Fn, and the separation d/σ. If a Gaussian height distribution is assumed for the magnetic tape, the real area of contact is related to the separation as
(4.5.26)
To solve the integrals in Equation 4.5.25 we need the relation between the curvature k(a) and the contact spot area, a. In the fractal model a scaling relation of the form k(a) = GD–1/aD/2 is assumed. In contrast, the generalized model does not make any assumption for the curvature k(a) but directly uses the experimental roughness data that is processed by the generalized characterization techniques FIGURE 4.32 Effect of varying roughness parameter D on (a) real area of contact (b) fraction of real area of contact in elastic deformation.
A A d
r = a
2 erfc 2 σ
described in Section 4.3.4. We demonstrate the application of the generalized analysis by studying the contact between magnetic thin-film rigid disk C and a hard Pyrex glass surface. The hardness of the magnetic film is H = 6.2 GPa and the elastic modulus of the equivalent surface is E = 52 GPa. Figure 4.16 shows the surface image of disk C, whereas Figure 4.17 shows its structure function. Figure 4.35 plots the surface height, V(a) and the curvature k(a) as a function of spot area, a, for this rough surface. This data can be used in Equation 4.5.25 along with numerical integration of the integrals to obtain the normal load. Figure 4.36 shows the elastic–plastic regime map for the this surface. Using the values of hardness and elastic modulus, the critical plasticity index is equal to φ = 0.035. The plasticity index, Ψ = V(a)k(a) is found to be always less than φ, thereby leading to the conclusion that all the asperities deform elastically.
Figure 4.37 compares the results of the generalized theory and the experiments (Bhushan and Dugger, 1990) in terms of the apparent pressure, Fn/Aa and the nondimensional real area of contact, Ar/Aa. Not only are the trends quite similar, the values for the predicted Ar/Aa are reasonably close to those measured.
FIGURE 4.33 Comparison between predictions of fractal model and experiments of Yamada et al. (1978).
FIGURE 4.34 Comparison among predictions of the fractal model, the GW model, and experiments of Bhushan and Dugger (1990) for contact of a magnetic rigid disk.
FIGURE 4.35 Plot of surface height, V(a) and curvature k(a) for the magnetic thin-film rigid disk C surface used in the computations of the generalized model.
FIGURE 4.36 Elastic–plastic regime of the magnetic thin-film rigid disk surface. The critical plasticity index φ = 0.035.
FIGURE 4.37 Comparison of the experimental data (Bhushan and Dugger, 1990) and the predictions of the generalized theory for the magnetic thin-film rigid disk C in terms of compressive apparent pressure, Fn/Aa, and the nondimensional real area of contact, Ar/Aa.
Keeping in mind the variability of surface mechanical properties, particularly for thin films, the agreement is quite good and any better agreement would only be by chance. Figure 4.38 shows the predicted compressive pressure and the diameter of the largest contact spot as a function of the nondimensional separation, d/σ. It is evident that the largest spot size decreases drastically as the separation is increased slightly.
Figure 4.39a shows histograms of the experimentally (Bhushan and Dugger, 1990) obtained size dis- tribution n(a)∆a/Aa per 1 mm2 of the apparent area, Aa, of rigid disk C. The observations were made by an optical microscope. The contact spots are placed into bins of different spot areas and then the numbers are counted for each bin. The two sets of data are for Ar/Aa = 1.64 and 2.51%. The general trend is that as the size of the contact spot decreases, the number of contact spots increases. To obtain the contribution of contact spots of a particular size to the real area of contact, Ar, Figure 4.39b plots the product of an(a)∆a/Aa as a percentage. Therefore, this plot shows that when Ar/Aa = 2.51%, contact spots of area 100 àm2 contribute 1.46%, those of area 67.83 àm2 contribute 0.35%, and those of 30.91 àm2 contribute 0.91%, and so on. If the percentages of each bin are added up, it should equal to the real area of contact Ar/Aa = 2.51%. However, an addition of the percentages of individual bins equals 3.33%. Similarly when the real area of contact is claimed to be Ar/Aa = 1.64% the addition shows the fraction to be 4.32%.
Therefore, there could be some margin of error in these measurements. Nevertheless, we compared the cumulative size distribution obtained from the experimental data with that of the predictions of the generalized theory in Figure 4.40. The agreement between the theory and experiments is quite poor. It is difficult to say which one is more accurate. The experiments may have flaws as discussed in Figure 4.39b.
In addition, contact spots smaller than 1 àm are near the limit of optical diffraction. Therefore, the accuracy in observing and counting spots may be poor in this range. This could perhaps explain the discrepancy for small length scales. With regard to the predictions, they depend on accurate statistics of the surface roughness. The statistics seem to be accurate for small length scales since the data set needed for making roughness averages described in Section 4.3.4 is quite large. In contrast, the roughness statistics may not be accurate for large length scales since the averages needed for accurate roughness character- ization are made over very few data points and therefore may not be good representative numbers.
However, the optical observations of the contact spots are quite accurate in this size range since this is well above the optical diffraction limit. In summary, the discrepancy for small and large spots could be attributed to measurements and observations. It is encouraging to see that at intermediate length scales, the agreement between theory and experiments is not that terrible.
FIGURE 4.38 Plot of the apparent pressure, Fn/Aa, and the diameter of the largest spots size as a function of the nondimensional surface separation, d/σ.