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TECHNICAL PAPERS

# A New Microcontact Model Developed for Variable Fractal Dimension, Topothesy, Density of Asperity, and Probability Density Function of Asperity Heights

[+] Author and Article Information
Jeng Luen Liou

Department of Military Meteorology Engineering, Air Force Institute of Technology, Kaohsiung 820, Taiwan, R.O.C.

Jen Fin Lin1

Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, R.O.C.jflin@mail.ncku.edu.tw

1

Author to whom correspondence should be addressed.

J. Appl. Mech 74(4), 603-613 (Apr 19, 2006) (11 pages) doi:10.1115/1.2338059 History: Received April 26, 2005; Revised April 19, 2006

## Abstract

In the present study, the fractal theory is applied to modify the conventional model (the Greenwood and Williamson model) established in the statistical form for the microcontacts of two contact surfaces. The mean radius of curvature $(R)$ and the density of asperities $(η)$ are no longer taken as constants, but taken as variables as functions of the related parameters including the fractal dimension $(D)$, the topothesy $(G)$, and the mean separation of two contact surfaces. The fractal dimension and the topothesy varied by differing the mean separation of two contact surfaces are completely obtained from the theoretical model. Then the mean radius of curvature and the density of asperities are also varied by differing the mean separation. A numerical scheme is thus developed to determine the convergent values of the fractal dimension and topothesy corresponding to a given mean separation. The topographies of a surface obtained from the theoretical prediction of different separations show the probability density function of asperity heights to be no longer the Gaussian distribution. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. The density of asperities is reduced by decreasing the mean separation. The contact load and the total contact area results predicted by variable $D$, $G*$, and $η$ as well as non-Gaussian distribution are always higher than those forecast with constant $D$, $G*$, $η$, and Gaussian distribution.

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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 5

The dimensionless topothesy varying with the dimensionless mean separation. The values of G* are obtained for the elastic, elastoplastic and fully plastic regimes. (a)(ψ)0=0.75 and (b)(ψ)0=2.0.

Figure 6

Density of asperities varying with the dimensionless mean separation

Figure 7

Probability density functions of asperity heights varying with the dimensionless asperity height: (a)(ψ)0=0.75 and (b)(ψ)0=2.0

Figure 8

Plasticity index of rough surfaces varying with the dimensionless mean separation

Figure 9

Variations of the dimensionless contact loads with the dimensionless mean separation. They are presented to compare the evaluations based on variable D, G*, η, and non-Gaussian g with the evaluations based on constant D, G*, η, and Gaussian g.

Figure 10

Variations of the dimensionless total contact area with the dimensionless mean separation. They are presented to compare the evaluations based on variable D, G*, η, and non-Gaussian g with the evaluations based on constant D, G*, η, and Gaussian g.

Figure 1

The schematic diagram of two contact surfaces with deformation

Figure 2

Flow chart for the numerical analyses of D(d*) and G*(d*)

Figure 3

The theoretical results of N∕An expressed as a function of the contact spot area a: (a)(ψ)0=0.75 and (b)(ψ)0=2.0

Figure 4

The fractal dimensions varying with the dimensionless mean separation. These data of D are obtained from the slope values of those five curves shown in Fig. 3. (a)(ψ)0=0.75 and (b)(ψ)0=2.0.

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