Abstract

In this study, the normal stiffness of elastic contact between rough surfaces with asperities following Gaussian distribution is investigated using ubiquitiform theory, developed from fractal theory. In the generalized ubiquitiformal Sierpinski carpet model, the rough surface including contact asperity is controlled for, given the lower bound to scale invariance of rough surfaces. Considering the stiffness of a single asperity deduced from the Hertz contact model, we deduce the theoretical relation between the normal stiffness and the elastic contact of rough surfaces based on ubiquitiform theory. The results show that the normal contact stiffness of a rough surface increases as the normal load rises. If the ubiquitiformal complexity of a rough surface increases or the lower bound to scale invariance of a rough surface decreases, the normal contact stiffness of the rough surface should increase. The larger the ubiquitiformal complexity of a rough surface is, the more obvious the impact of the lower bound to scale invariance on the normal contact stiffness of the rough surface becomes. The results based on the ubiquitiformal model and the experimental results are in closer agreement. Therefore, the introduction of scale invariance is crucial to the surface contact problem.

References

1.
Sayles
,
R. S.
, and
Thomas
,
T. R.
,
1978
, “
Surface Topography as a Nonstationary Random Process
,”
Nature
,
271
(
5644
), pp.
431
434
. 10.1038/271431a0
2.
Persson
,
B. N. J.
,
2006
, “
Contact Mechanics for Randomly Rough Surface
,”
Surf. Sci. Rep.
,
61
(
4
), pp.
201
227
. 10.1016/j.surfrep.2006.04.001
3.
HAN
,
J. H.
, and
Ping
,
S.
,
2005
, “
Fractal Characterization and Simulation of Surface Profiles of Copper Electrodes and Aluminum Sheets
,”
Mater. Sci. Eng. A
,
403
(
1–2
), pp.
174
181
. 10.1016/j.msea.2005.05.026
4.
Jackson
,
R. L.
, and
Green
,
I.
,
2006
, “
A Statistical Model of Elasto-Plastic Asperity Contact Between Rough Surfaces
,”
Tribol. Int.
,
39
(
2
), pp.
906
914
. 10.1016/j.triboint.2005.09.001
5.
Stanley
,
H. M.
, and
Kato
,
T.
,
1997
, “
An FFT-Based Method for Rough Surface Contact
,”
ASME J. Tribol.
,
119
(
3
), pp.
481
485
. 10.1115/1.2833523
6.
Polycarpou
,
A.
, and
Etsion
,
I.
,
1999
, “
Analytical Approximations in Modeling Contacting Rough Surfaces
,”
ASME J. Tribol.
,
121
(
2
), pp.
234
239
. 10.1115/1.2833926
7.
Wu
,
J.
,
2000
, “
The Properties of Asperities of Real Surfaces
,”
ASME J. Tribol.
,
123
(
4
), pp.
872
883
. 10.1115/1.1353179
8.
Shi
,
J. P.
,
Ma
,
K.
, and
Liu
,
Z. Q.
,
2012
, “
Normal Contact Stiffness on Unit Area of a Mechanical Joint Surface Considering Perfectly Elastic Elliptical Asperities
,”
ASME J. Tribol.
,
134
(
3
), p.
031402
. 10.1115/1.4006924
9.
Greenwood
,
J. A.
, and
Williamson
,
J. B. P.
,
1966
, “
Contact of Nominally Flat Surfaces
,”
Proc. R. Soc. London
,
295
(
1442
), pp.
300
319
. 10.1098/rspa.1966.0242
10.
Whitehouse
,
D. J.
, and
Archard
,
J. F.
,
1970
, “
The Properties of Random Surfaces of Significance in Their Contact
,”
Proc. R. Soc. London
,
316
(
1524
), pp.
97
121
. 10.1098/rspa.1970.0068
11.
Kim
,
T. W.
,
Bhushan
,
B.
, and
Cho
,
Y. J.
,
2006
, “
The Contact Behavior of Elastic/Plastic Non-Gaussian Rough Surfaces
,”
Tribol. Lett.
,
22
(
1
), pp.
1
13
. 10.1007/s11249-006-9036-5
12.
Shi
,
J.
,
Cao
,
X.
, and
Hu
,
Y.
,
2015
, “
Statistical Analysis of Tangential Contact Stiffness of Joint Surfaces
,”
Arch. Appl. Mech.
,
85
(
12
), pp.
1997
2008
. 10.1007/s00419-015-1033-4
13.
Mandelbrot
,
B.
,
1967
, “
How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
,”
Science
,
156
(
3775
), pp.
636
638
. 10.1126/science.156.3775.636
14.
Borodich
,
F. M.
, and
Mosolov
,
A,B
,
1992
, “
Fractal Roughness in Contact Problems
,”
J. Appl. Math. Mech.
,
56
(
5
), pp.
681
690
. 10.1016/0021-8928(92)90054-C
15.
Warren
,
T. L.
, and
Krajcinovic
,
D.
,
1995
, “
Fractal Models of Elastic-Perfectly Plastic Contact of Rough Surfaces Based on the Cantor Set
,”
Int. J. Solids Struct.
,
32
(
19
), pp.
2907
2922
. 10.1016/0020-7683(94)00241-N
16.
Majumdar
,
A.
, and
Bhushan
,
B.
,
1991
, “
Fractal Model of Elastic-Plastic Contact Between Rough Surfaces
,”
ASME J. Tribol.
,
113
(
1
), pp.
1
11
. 10.1115/1.2920588
17.
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.
,
116
(
4
), pp.
812
823
. 10.1115/1.2927338
18.
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.
,
116
(
4
), pp.
824
832
. 10.1115/1.2927341
19.
Komvopoulos
,
K.
, and
Ye
,
N.
,
2000
, “
Three-Dimensional Contact Analysis of Elastic-Plastic Layered Media With Fractal Surface Topographies
,”
ASME J. Tribol.
,
123
(
3
), pp.
632
640
. 10.1115/1.1327583
20.
Kogut
,
L.
, and
Jackson
,
R. L.
,
2006
, “
A Comparison of Contact Modeling Utilizing Statistical and Fractal Approaches
,”
ASME J. Tribol.
,
128
(
1
), pp.
213
217
. 10.1115/1.2114949
21.
Morag
,
Y.
, and
Etsion
,
I.
,
2007
, “
Resolving the Contradiction of Asperities Plastic to Elastic Mode Transition in Current Contact Models of Fractal Rough Surfaces
,”
Wear
,
262
(
5
), pp.
624
629
. 10.1016/j.wear.2006.07.007
22.
Komvopoulos
,
K.
, and
Gong
,
Z.
,
2007
, “
Stress Analysis of a Layered Elastic Solid in Contact With a Rough Surface Exhibiting Fractal Behavior
,”
Int. J. Solids Struct.
,
44
(
7
), pp.
2109
2129
. 10.1016/j.ijsolstr.2006.06.043
23.
Jackson
,
R. L.
,
2010
, “
An Analytical Solution to an Archard-Type Fractal Rough Surface Contact Model
,”
Tribol. Trans.
,
53
(
4
), pp.
543
553
. 10.1080/10402000903502261
24.
You
,
J. M.
, and
Chen
,
T. N.
,
2010
, “
A Static Friction Model for the Contact of Fractal Surfaces
,”
P I Mech Eng E-J Pro
,
224
(
5
), pp.
513
518
. 10.1243/13506501JET760
25.
Jiang
,
S.
,
Zheng
,
Y.
, and
Zhu
,
H.
,
2009
, “
A Contact Stiffness Model of Machined Plane Joint Based on Fractal Theory
,”
ASME J. Tribol.
,
132
(
1
), p.
011401
. 10.1115/1.4000305
26.
Buczkowski
,
R.
,
Kleiber
,
M.
, and
Starzynski
,
G.
,
2014
, “
Normal Contact Stiffness of Fractal Rough Surfaces
,”
Archi. Mech.
,
66
(
6
), pp.
411
428
. 10.24423/aom.1286
27.
Falconer
,
K. J.
,
2003
,
Fractal Geometry: Mathematical Foundations and Applications
,
Wiley
,
New York
, p.
499
.
28.
Addison
,
P. S.
,
2000
, “
The Geometry of Prefractal Renormalisation: Application to Crack Surface Energies
,”
Fractals
,
8
(
2
), pp.
147
153
. 10.1142/S0218348X00000160
29.
Borodich
,
F. M.
,
1997
, “
Some Fractal Models of Fracture
,”
J. Mech. Phys. Solids
,
45
(
2
), pp.
239
259
. 10.1016/S0022-5096(96)00080-4
30.
Borodich
,
F. M.
,
1999
, “
Fractals and Fractal Scaling in Fracture Mechanics
,”
Int. J. Fract.
,
95
(
1–4
), pp.
239
259
. 10.1023/A:1018660604078
31.
Mandelbrot
,
B. B.
,
Passoja
,
D. E.
, and
Paullay
,
A. J.
,
1984
, “
Fractal Character of Fracture Surfaces of Metals
,”
Nature
,
308
(
5961
), pp.
721
722
. 10.1038/308721a0
32.
Ou
,
Z. C.
,
Li
,
G.-Y.
,
Duan
,
Z.-P.
, and
Huang
,
F.-L.
,
2014
, “
Ubiquitiform in Applied Mechanics
,”
J. Theor. Appl. Mech.
,
52
(
1
), pp.
37
46
.
33.
Mosolov
,
A. B.
, and
Dinariev
,
O. Y.
,
1988
, “
Fractals, Scales, and Geometry of Porous Materials
,”
Sov. Phys. Tech. Phys
,
33
(
2
), pp.
145
148
.
34.
Jackson
,
R. L.
,
2010
, “
An Analytical Solution to an Archard-Type Fractal Rough Surface Contact Model
,”
Tribol. Trans.
,
53
(
4
), pp.
543
553
. 10.1080/10402000903502261
35.
Wilson
,
W. E.
,
Angadi
,
S. V.
, and
Jackson
,
R. L.
,
2010
, “
Surface Separation and Contact Resistance Considering Sinusoidal Elastic–Plastic Multi-Scale Rough Surface Contact
,”
Wear
,
268
(
1
), pp.
190
201
. 10.1016/j.wear.2009.07.012
36.
Li
,
G. Y.
,
Ou
,
Z. C.
, and
Xie
,
R.
,
2016
, “
A Ubiquitiformal One-Dimensional Steady-State Conduction Model for a Cellular Material Rod
,”
Int. J. Thermophys.
,
37
(
4
), pp.
1
13
. 10.1007/s10765-015-2010-4
37.
Min
,
Y.
,
Ou
,
Z. C.
, and
Li
,
G. Y.
,
2016
, “
Ubiquitiformal Fracture Energy and Size Effect of Traditional Fracture Energy (in Chinese)
,”
Acta Armamentarii
,
37
(
1
), pp.
91
95
.
38.
Li
,
J. Y.
,
Ou
,
Z. C.
, and
Tong
,
Y.
,
2017
, “
A Statistical Model for Ubiquitiformal Crack Extension in Quasi-Brittle Materials
,”
Acta Mechanica
,
228
(
7
), pp.
1
8
. 10.1007/s00707-017-1859-7
39.
Khezrzadeh
,
H.
, and
Mofid
,
M.
,
2006
, “
Tensile Fracture Behavior of Heterogeneous Materials Based on Fractal Geometry
,”
Theor. Appl. Fract. Mech.
,
46
(
1
), pp.
46
56
. 10.1016/j.tafmec.2006.05.006
40.
Greenwood
,
J.
, and
Tripp
,
H. J.
,
1970
, “
The Contact of Two Nominally Flat Rough Surfaces
,”
Arch. Proc. Inst. Mech. Eng.
,
185
(
1
), pp.
625
634
. 10.1243/PIME_PROC_1970_185_069_02
41.
Chang
,
W. R.
,
Etsion
,
I.
, and
Bogy
,
D. B.
,
1987
, “
An Elastic-Plastic Model for the Contact of Rough Surfaces
,”
ASME J. Tribol.
,
109
(
2
), pp.
257
263
. 10.1115/1.3261348
42.
Kogut
,
L.
, and
Etsion
,
I.
,
2002
, “
Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat
,”
ASME J. Appl. Mech.
,
69
(
5
), pp.
657
662
. 10.1115/1.1490373
43.
Johnson
,
K. L.
,
1987
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge
.
44.
Shuting
,
W.
,
2011
, “
Interfacial Stiffness Characteristic Modeling of Mechanical Fixed Joints
,”
J. Huazhong Univ. Sci. Technol. (Natural Science Edition)
,
39
(
8
), pp.
1
5
. 10.1016/S1005-0302(11)60031-5
45.
Bhushan
,
B.
,
2008
,
“Nanotribology and Nanomechanics an Introduction
,” 2nd ed.,
Springer
,
Heidelberg
.
46.
Bhushan
,
B.
, and
Majumda
,
A.
,
1992
, “
Elastical-Plastic Contact Model for Bifractal Surfaces
,”
Wear
,
153
(
1
), pp.
53
64
. 10.1016/0043-1648(92)90260-f
47.
Mandelbrot
,
B. B.
,
1983
,
The Fractal Geometry of Nature
,
Freeman
,
New York
.
48.
Buczkowski
,
R.
, and
Kleiber
,
M.
,
1999
, “
A Stochastic Model of Rough Surfaces for Finite Element Contact Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
169
(
1–2
), pp.
43
59
. 10.1016/s0045-7825(98)00175-3
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