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RESEARCH PAPERS

# $n$-Point Asperity Model for Contact Between Nominally Flat Surfaces

[+] Author and Article Information
A. Hariri, J. W. Zu, R. Ben Mrad

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8

J. Tribol 128(3), 505-514 (Mar 10, 2006) (10 pages) doi:10.1115/1.2194915 History: Received May 13, 2005; Revised March 10, 2006

## Abstract

For several decades, asperities of nominally flat rough surfaces were considered to be points higher than their immediate neighbors. Recently, it has been recognized that this model is incorrect. To address the issue, a new multiple-point asperity model, called the $n$-point asperity model, is introduced in this paper. In the new model, asperities are composed of $n$ neighboring sampled points with $n-2$ middle points being above a certain level. When the separation between two surfaces decreases, new asperities with higher number of sample points, $n$, will come into existence. Based on the above model, the height and curvature of $n$-point asperities are defined and their distributions are found. The model is developed for Gaussian surfaces and for the general case of an autocorrelation function (ACF). As a case study, the exponential ACF is applied to the new model, which is shown to produce remarkably good agreement with measurements from real and simulated surfaces.

###### FIGURES IN THIS ARTICLE
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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 1

A typical equivalent normalized profile of two nominally flat surfaces in contact

Figure 8

Theoretical curves of (a) fZ0∣S as a function of z0∕σ and (b) fC∣S as a function of cΔx2∕σ, drawn for various values of n, h, and for ρ=0.86

Figure 9

Comparison of dimensionless curvature from the new model (continuous lines) and the average curvature from the conventional three-point peak method (dashed lines), measured from profile a (ρ=0.1), nitride (ρ=0.55), and profile b (ρ=0.86)

Figure 2

A three-point asperity at level h (indicated with roman numerals), which will be transformed to a seven-point one at level h2 (indicated with numbers)

Figure 3

An equivalent parabolic asperity made from the first and last points and an equivalent middle point z0

Figure 4

(a)–(c) Normalized profiles a, b, and c with ρ=0.1, 0.86, and 0.954, respectively, generated by computer, (d) normalized Nitride profile sampled from a MUMPs chip, with ρ=0.55 and β*=690nm, and (e) the normalized WA profile with ρ=0.954(3)

Figure 5

(a) The probability density function, fZ, and (b) normalized ACF, ρ, of the profiles in Fig. 4

Figure 6

(a)–(c) Probability distribution of Z0, fZ0, as a function of z0∕σ. (d)–(e) Probability distribution of C, fC, as a function of cΔx2∕σ, drawn for n=3, 4 and 5, respectively, for the WA profile, profile c (Fig. 4) and theory (ρ=0.954).

Figure 7

(a)–(c) Probability density of Z0 given S, fZ0∣S, as a function of z0∕σ. (d)–(f) Probability density of C given S, fC∣S, as a function of cΔx2∕σ, for n=3, 4, and 5, respectively, and for profiles a and b, nitride (Fig. 4), and theory (for ρ=0.1, 0.55, and 0.86).

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