Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T14:56:44.593Z Has data issue: false hasContentIssue false
  • English
  • Français

A Microcontact Model Developed for Asperity Heights with a Variable Profile Fractal Dimension, A Surface Fractal Dimension, Topothesy, and Non-Gaussian Distribution

Published online by Cambridge University Press:  05 May 2011

J. L. Liou  and
J. F. Lin
J. L. Liou*
Affiliation:
Department of Aircraft Engineering, Air Force Institute of Technology, Kaohsiung, Taiwan 82047, R.O.C.
J. F. Lin*
Affiliation:
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Assistant Professor, corresponding author
**Professor
Get access

Abstract

The cross sections formed by the contact asperities of two rough surfaces at an interference are islandshaped, rather than having the commonly assumed circular contour. These island-shaped contact surface contours show fractal behavior with a profile fractal dimension Ds. The surface fractal dimension for the asperity heights is defined as D and the topothesy is defined as G. In the study of Mandelbrot, the relationship between D and Ds was given as D = Ds + 1 if these two fractal dimensions are obtained before contact deformation. In the present study, D, G, and Ds are considered to be varying with the mean separation (or the interference at the rough surface) between two contact surfaces. The D-Ds relationships for the contacts at the elastic, elastoplastic, and fully plastic deformations are derived and the inceptions of the elastoplastic deformation regime and the fully plastic deformation regime are redefined using the equality of two expressions established in two different ways for the number of contact spots (N). The contact parameters, including the total contact force and the real contact area, were evaluated when the size distribution functions (n) for the three deformation regimes were available. The results indicate that both the D and Ds parameters in these deformation regimes increased with increasing the mean separation (d*). The initial plasticity index before contact deformation (ψ)0 is also a factor of importance to the predictions of the contact load (F*t) and contact area (At*) between the model of variable D and G, non-Gaussian asperity heights and circular contact area and the present model of variable D and G, non-Gaussian asperity heights and fractal contact area.

Keywords

Profile fractal dimensionSurface fractal dimensionMicrocontact.
Type
Articles
Information
Journal of Mechanics , Volume 25 , Issue 1 , March 2009 , pp. 103 - 115
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Greenwood, J. A. and Williamson, J. B. P., “Contact of Nominally Flat Surfaces,” Proc. R. Soc. London, Ser. A, 295, pp. 300–319 (1966).Google Scholar
2.Mandelbrot, B. B., “How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science, 156, pp. 636–638 (1967).CrossRefGoogle Scholar
3.Majumdar, A. and Bhushan, B., “Fractal Model of Elastic-Plastic Contact Between Rough Surfaces,” ASMEJ. Tribol., 113, pp. 1–11 (1991).CrossRefGoogle Scholar
4.Bhushan, B. and Majumdar, A., “Elastic-Plastic Contact for Bifractal Surfaces,” Wear, 153, pp. 53–64 (1992).CrossRefGoogle Scholar
5.Blackmore, D. and Zhou, G., “A New Fractal Model for Anisotropic Surfaces,” Int. J. Mach. Tools Manuf., 38, pp. 551–557 (1998).CrossRefGoogle Scholar
6.Blackmore, D. and Zhou, J. G., “Fractal Analysis of Height Distributions of Anisotropic Rough Surfaces,” Fractals, 6, pp. 43–58 (1998).CrossRefGoogle Scholar
7.Zahouani, H., Vargiolu, R. and Loubet, J. L., “Fractal Models of Surface Topography and Contact Mechanics,” Math. Comput. Modell., 28, pp. 517–534 (1998).CrossRefGoogle Scholar
8.Yan, W. and Komvopoulos, K., “Contact Analysis of Elastic-Plastic Fractal Surfaces,” J. Appl. Phys., 84, pp. 3617–3624 (1998).CrossRefGoogle Scholar
9.Jackson, R. L. and Streator, J. L., “A Multi-Scale Model for Contact Between Rough Surfaces,” Wear, 261, pp. 1337–1347 (2006).CrossRefGoogle Scholar
10.Kogut, L. and Jackson, R. L., “A Comparison of Contact Modeling Utilizing Statistical and Fractal Approaches,” ASME J. Tribol., 128, pp. 213–217 (2005).CrossRefGoogle Scholar
11.Morag, Y. and Etsion, I., “Resolving the Contradiction of Asperities Plastic to Elastic Mode Transition in Current Contact Models of Fractal Rough Surfaces,” Wear, 262, pp. 624–629 (2007).CrossRefGoogle Scholar
12.Mandelbort, B. B., The Fractal Geometry of Nature, W. H. Freeman, New York (1982).Google Scholar
13.Othmani, A. and Kaminsky, C., “Three Dimensional Fractal Analysis of Sheet Metal Surfaces,” Wear, 214, pp. 147–150 (1998).CrossRefGoogle Scholar
14.Chung, J. C. and Lin, J. F., “Fractal Model Developed for Elliptic Elastic-Plastic Asperity Microcontacts of Rough Surfaces,” ASME J. Tribol., 126, pp. 646–654 (2004).CrossRefGoogle Scholar
15.Ausloos, M. and Berman, D., “A Multivariate Weierstrass-Mandelbrot Function,” Proc. R. Soc. London, Ser. A, 400, pp. 331–350 (1985).Google Scholar
16.Blackmore, D. and Zhau, G., “A General Fractal Distribution Function for Rough Surface Profiles,” SIAM J. Appl. Math., 56, pp. 1694–1719 (1996).CrossRefGoogle Scholar
17.Chung, J. C. and Lin, J. F., “Variation in Fractal Properties and Non-Gaussian Distributions of Microcontact Between Elastic-Plastic Rough Surfaces with Mean Surface Separation,” ASME J. Appl. Mech., 73, pp. 143–152 (2006).CrossRefGoogle Scholar
18.Liou, J. L. and Lin, J. F., “A New Method Developed for Fractal Dimension and Topothesy Varying with the Mean Separation of Two Contact Surfaces,” ASME J. Tribol., 128, pp. 515–524 (2006).CrossRefGoogle Scholar
19.Liou, J. L. and Lin, J. F., “A New Microcontact Model Developed for Variable Fractal Dimension, Topothesy, Density of Asperity and Probability Density Function of Asperity Heights,” ASME J. Appl. Mech., 74, pp. 603–613 (2007).CrossRefGoogle Scholar
20.Mandelbrot, B. B., “Self-Affine Fractals and Fractal Dimension,” Phys. Scr., 32, pp. 257–260 (1985).CrossRefGoogle Scholar
21.Johnson, K. L., Contact Mechanics, Cambridge University Press, Cambridge, UK (1987).Google Scholar
22.Chang, W. R., Etsion, I. and Bogy, D. B., “Static Friction Coefficient Model for Metallic Rough Surfaces,” ASMEJ. Tribol., 10, pp. 57–63 (1988).CrossRefGoogle Scholar
23.Kogut, L. and Etsion, I., “Elastic–Plastic Contact Analysis of a Sphere and a Rigid Flat,” ASME J. Appl. Mech., 69, pp. 657–662 (2002).CrossRefGoogle Scholar
24.Tabor, D., The Hardness of Metals, Clarendon Press, Oxford, UK (1951).Google Scholar
25.Komvopoulos, K. and Ye, N., “Three-Dimensional Contact Analysis of Elastic-Plastic Layered Media with Fractal Surface Topographies,” ASME Journal of Tribology, 123, pp. 632–640 (2001).CrossRefGoogle Scholar
26.Bhushan, B. and Dugger, M. T., “Real Contact Area Measurements on Magnetic Rigid Disks,” Wear, 137, pp. 41–50 (1990).CrossRefGoogle Scholar
27.Liou, J. L. and Lin, J. F., “Elastic-Plastic Microcontact Analysis of a Sphere and a Flat Plate,” Journal of Mechanics, 23, pp. 341–351 (2007).CrossRefGoogle Scholar
28.Mandelbrot, B. B., “Stochastic Models for the Earth's Relief, the Shape and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands,” Proc. Natl. Acad. Sci., USA, 72, pp. 3825–3828 (1975).CrossRefGoogle Scholar
29.Voss, R. F., Random Fractals: Characterization and Measurement. In Scaling Phenomena in Disordered Systems, Plenum Press, New York, pp. 1–11 (1985).Google Scholar
30.Nayak, P. R., “Random Process Model of Rough Surfaces,” J. Lubr. Technol., 93, pp. 398–407 (1971).CrossRefGoogle Scholar
31.Bhushan, B., Handbook of Micro/Nano Tribology, 2nd Ed., CRC Press LLC, Boca Raton, USA (1999).Google Scholar
32.Greenwood, J. A., “A Note on Nayak's Third Paper,” Wear, 262, pp. 225–227 (2007).CrossRefGoogle Scholar