Abstract

A fractal dimension and a fractal roughness parameter are usually used to characterize a fractal surface. For a fractal-regular surface, a fractal domain length is also included. Such a formulation is based on an approximation using a constant value of the fractal scaling parameter that represents the ratio of the spatial frequencies of adjacent harmonic components in the Weierstrass–Mandelbrot (W-M) function. Although there were some reasons for assuming a constant value of 1.5 for the fractal scaling parameter, it is still left more or less arbitrary to adopt this assumption in fractal modeling of solid contact. In the present study, the fractal scaling parameter was treated as a variable rather than a constant by using a form of the W-M function with randomized phases based on a random walk formulation. A simple numerical scheme with clear graphical interpretation was developed to determine the value of the fractal scaling parameter. The fractal dimension, fractal roughness parameter, and fractal scaling parameter were all recovered with reasonable accuracy from numerically generated surface profiles.

References

1.
Greenwood
,
J. A.
, and
Williamson
,
J. B. P.
, 1966, “
Contact of Nominally Flat Surfaces
,”
Proc. R. Soc. London, Ser. A
1364-5021,
A295
, pp.
300
319
.
2.
Greenwood
,
J. A.
, and
Wu
,
J. J.
, 2001, “
Surface Roughness and Contact: An Apology
,”
Meccanica
0025-6455,
36
, pp.
617
630
.
3.
Wang
,
S.
, 2004, “
Real Contact Area of Fractal-Regular Surfaces and Its Implications in the Law of Friction
,”
ASME J. Tribol.
0742-4787,
126
(
1
), pp.
1
8
.
4.
Majumdar
,
A.
, and
Bhushan
,
B.
, 1991, “
Fractal Model of Elastic-Plastic Contact Between Rough Surfaces
,”
ASME J. Tribol.
0742-4787,
113
(
1
), pp.
1
11
.
5.
Wang
,
S.
, and
Komvopoulos
,
K.
, 1994, “
A Fractal Theory of the Interfacial Temperature Distribution in the Slow Sliding Regime—Part I: Elastic Contact and Heat Transfer Analysis
,”
ASME J. Tribol.
0742-4787,
116
(
4
), pp.
812
823
;
Wang
,
S.
, and
Komvopoulos
,
K.
, 1994, “
A Fractal Theory of the Interfacial Temperature Distribution in the Slow Sliding Regime—Part II: Multiple Domains, Elastoplastic Contacts and Applications
,”
ASME J. Tribol.
0742-4787,
116
(
4
), pp.
824
-
832
.
6.
Wang
,
S.
, and
Komvopoulos
,
K.
, 1995, “
A Fractal Theory of the Temperature Distribution at Elastic Contacts of Fast Sliding Surfaces
,”
ASME J. Tribol.
0742-4787,
117
(
2
), pp.
203
215
.
7.
Komvopoulos
,
K.
, and
Yan
,
W.
, 1997, “
A Fractal Analysis of Stiction in Microelectromechanical Systems
,”
ASME J. Tribol.
0742-4787,
119
, pp.
391
400
.
8.
Majumdar
,
A.
, and
Tien
,
C. L.
, 1991, “
Fractal Network Model for Contact Conductance
,”
ASME J. Heat Transfer
0022-1481,
113
, pp.
516
525
.
9.
Wang
,
S.
, 2007, “
Development of Theoretical Contact Width Formulas and a Numerical Model for Curved Rough Surfaces
,”
ASME J. Tribol.
0742-4787,
129
(
4
), in press.
10.
Wang
,
S.
, and
Chan
,
W. K.
, 2005, “
Experimental Observation of Fractal-Regular Surfaces and a Transformation Scheme for Extracting Fractal Parameters
,”
Proc. World Tribol. Congress III
,
ASME
,
New York
, Paper No. WTC2005-63585.
11.
Majumdar
,
A.
, and
Bhushan
,
B.
, 1999, “
Characterization and Modeling of Surface Roughness and Contact Mechanics - Roughness Measurements
,”
Handbook of Micro/Nanotribology
,
B.
Bhushan
, ed.,
CRC Press
, Boca Raton, pp.
203
210
.
12.
Ganti
,
S.
, and
Bhushan
,
B.
, 1995, “
Generalized Fractal Analysis and Its Applications to Engineering Surfaces
,”
Wear
0043-1648,
180
, pp.
17
34
.
13.
Bhushan
,
B.
, 2001,
Modern Tribology Handbook
, Vol.
2
,
CRC Press
, Boca Raton, pp.
1428
1430
.
14.
Borodich
,
F. M.
, 2006, “
Fractals and Surface Roughness in EHL
,”
IUTAM Symposium on Elastohydrodynamics and Micro-Elastohydrodynamics
,
R. W.
Snidle
, and
H. P.
Evans
, eds.,
Springer
, Dordrecht, pp.
397
408
.
15.
Majumdar
,
A.
, and
Tien
,
C. L.
, 1990, “
Fractal Characterization and Simulation of Rough Surfaces
,”
Wear
0043-1648,
136
, pp.
313
327
.
16.
Persson
,
B. N. J.
,
Bucher
,
F.
, and
Chiaia
,
B.
, 2002, “
Elastic Contact Between Randomly Rough Surfaces: Comparison of Theory With Numerical Results
,”
Phys. Rev. B
0163-1829,
65
, p.
184106
.
17.
Ciavarella
,
M.
,
Demelio
,
G.
,
Barber
,
J. R.
, and
Jang
,
Y. H.
, 2000, “
Linear Elastic Contact of the Weierstrass Profile
,”
Proc. R. Soc. London, Ser. A
1364-5021,
A456
, pp.
387
405
.
18.
Ciavarella
,
M.
,
Murolo
,
C.
, and
Demelio
,
G.
, 2006, “
On Elastic Contact of Rough Surfaces: Numerical Experiments and Comparisons with Recent Theories
,”
Wear
0043-1648,
261
, pp.
1102
1113
.
19.
Persson
,
B. N. J.
, 2001, “
Elastoplastic Contact Between Randomly Rough Surfaces
,”
Phys. Rev. Lett.
0031-9007,
87
, p.
116101
.
20.
Whitehouse
,
D. J.
, 2001, “
Fractal or Fiction
,”
Wear
0043-1648,
249
, pp.
345
353
.
21.
Yao
,
J. J.
, and
Chang
,
M. F.
, 1995, “
A Surface Micromachined Miniature Switch for Telecommunications Applications With Signal Frequencies From DC up to 4GHz
,”
Proc. of 8th Int. Conf. on Solid-State Sensors and Actuators
,
IEEE
,
New York
, pp.
384
387
.
22.
Russ
,
J. C.
, 1994,
Fractal Surfaces
,
Plenum
, New York.
23.
Holm
,
R.
, 1967,
Electrical Contacts
, 4th ed.,
Springer
, New York.
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