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RESEARCH PAPERS

# A New Method Developed for Fractal Dimension and Topothesy Varying With the Mean Separation of Two Contact Surfaces

[+] Author and Article Information
Jeng Luen Liou

Department of Mechanical Engineering, National Cheng Kung University, Tainan City, 70101, Taiwan, ROC

Jen Fin Lin1

Department of Mechanical Engineering, National Cheng Kung University, Tainan City, 70101, Taiwan, ROCjflin@mail.ncku.edu.tw

1

Corresponding author.

J. Tribol 128(3), 515-524 (Mar 03, 2006) (10 pages) doi:10.1115/1.2197839 History: Received August 31, 2005; Revised March 03, 2006

## Abstract

Instead of a general consideration of the fractal dimension $(D)$ and the topothesy $(G*)$ as two invariants in the fractal analysis of surface asperities, these two roughness parameters in the present study are varied by changing the mean separation $(d*)$ of two contact surfaces. The relationship between the fractal dimension and the mean separation is found first. By equating the structure functions developed in two different ways, the relationship among the scaling coefficient in the power spectrum function, the fractal dimension, and topothesy of asperity heights can be established. The variation of topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained from the experimental results of the number of contact spots and the power spectrum function at different mean separations. A numerical scheme is developed in this study to determine the convergent values of fractal dimension and topothesy corresponding to a given mean separation. The theoretical results of the contact spot number predicted by the present model show good agreement with the reported experimental results. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. Significant differences in the contact load or the total contact area are shown between the models of constant $D$ and $G*$ and variable $D$ and $G*$ as the mean separation is reduced to smaller values.

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## Figures

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

Figure 4

The fractal dimensions varying with the dimensionless mean separation. These data of D are obtained from the slope values of those four curves shown in Fig. 3.

Figure 5

Determination of the region satisfying Eqs. 30,31. The tribological in the present study is analyzed according curve 2(a) to 2(c).

Figure 6

The topothesy varying with the dimensionless mean separation

Figure 7

Probability density functions of asperity heights varying with the dimensionless asperity heights

Figure 8

Comparisons of the experimental results (14) and the theoretical results of N∕An as a function of the contact spot area a

Figure 9

Variations of the dimensionless contact load with the dimensionless mean separation. They are presented to compare the evaluations based on variable D and G* with the evaluations based on constant D and G*.

Figure 10

Variations of the dimensionless total contact area with the dimensionless contact load. They are presented to compare the evaluations based on variable D and G* with the evaluations based on constant D and G*.

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