The deposition of layers is an effective way to improve the tribological performance of rough surfaces. The contact mechanics of layered rough surfaces needs to be studied to optimize layer parameters. Since 1995 a lot of progress has been made in the development of numerical contact models, which analyze the contact behavior of layered rough surfaces with no assumption concerning the roughness distribution as well as the effect of interfacial liquid film on the contact statistics. These models predict the contact pressure profile on the interface and contact statistics, namely fractional contact area, the maximum value of contact pressure, von Mises and principal tensile stresses, and relative meniscus force. The results allow the specification of layer properties to reduce friction, stiction, and wear of layered rough surfaces. A comprehensive review of these numerical contact models is presented here. Based on the formulation of contact problems, these models are classified into three categories: direct formulation, weighted residual formulation, and minimum total potential energy formulation. The numerical methods applied in these models include Finite Difference Method (FDM), Finite Element Method (FEM), and Boundary Element Method (BEM). Typical examples of layered rough surfaces of contact simulated by those models are presented. The examples contain data for various surface topographies, elastic and elastic-plastic material properties, normal and tangential loading conditions, and dry and wet interfaces. A 3D BEM model based on a variational principle is described in detail for its capability to analyze the layered rough surfaces of contact involving a large number of contact points. Applications of the model to magnetic storage devices and MicroElectroMechanical Systems (MEMS) are presented. This review article contains 92 references.

1.
Bowden FP and Tabor D (1950), The Friction and Lubrication of Solids, Part I, Clarendon Press, Oxford.
2.
Bowden FP and Tabor D (1964), The Friction and Lubrication of Solids, Part II, Clarendon Press, Oxford.
3.
Bhushan B (1996), Tribology and Mechanics of Magnetic Storage Devices, 2nd Edition, Springer-Verlag, New York.
4.
Bhushan B (1999), Principles and Applications of Tribology, John Wiley, New York.
5.
Bhushan B (1999), Handbook of Micro/Nanotribology, CRC Press, Boca Raton FL.
1.
Bhushan B (2001), Modern Tribology Handbook, Vol. 1: Principles of Tribology;
2.
Vol 2: Materials, Coatings, and Industrial Applications, CRC Press, Boca Raton FL.
1.
Bhushan B and Gupta BK (1991), Handbook of Tribology: Materials, Coating and Surface Treatments, McGraw-Hill, New York.
2.
Bhushan
B
(
1998
),
Contact mechanics of rough surfaces in tribology: Multiple asperity contact
,
Tribol. Lett.
4
,
1
35
.
3.
Greenwood
JA
and
Williamson
JBP
(
1966
),
Contact of nominally flat surfaces
,
Proc. R. Soc. London, Ser. A
295
,
300
319
.
4.
Majumdar
A
and
Bhushan
B
(
1990
),
Role of fractal geometry in roughness characterization and contact mechanics of rough surfaces
,
ASME J. Tribol.
112
,
205
216
.
5.
Majumdar
A
and
Bhushan
B
(
1991
),
Fractal model of elastic-plastic model for the contact of rough surfaces
,
ASME J. Tribol.
113
,
1
11
.
6.
Bhushan
B
and
Majumdar
A
(
1992
),
Elastic-plastic contact model of bifractal surfaces
,
Wear
153
,
53
64
.
7.
Bhushan
B
(
1996
),
Contact mechanics of rough surfaces in tribology: Single asperity contact
,
Appl. Mech. Rev.
49
,
275
298
.
8.
Chen
WT
and
Engel
PA
(
1972
),
Impact and contact stress analysis in multilayer media
,
Int. J. Solids Struct.
8
,
1257
1281
.
9.
Gupta
PK
and
Walowit
JA
(
1974
),
Contact stresses between an elastic cylinder and a layered elastic solid
,
ASME J. Lubr. Technol.
96
,
250
257
.
10.
O’Sullivan
TC
and
King
RB
(
1988
),
Sliding contact stress field due to a spherical indenter on a layered elastic half-space
,
ASME J. Tribol.
110
,
235
240
.
11.
Plumet
S
and
Dubourg
M-C
(
1998
),
A 3-D model for a multilayered body loaded normally and tangentially against a rigid body: Application to specific coatings
,
ASME J. Tribol.
120
,
668
676
.
12.
Chen
WT
(
1971
),
Computation of stresses and displacements in a layered elastic medium
,
Int. J. Eng. Sci.
9
,
775
800
.
13.
Burmister
DM
(
1945
),
The general theory of stresses and displacements in layered systems
,
J. Appl. Phys.
16
,
89
94
.
14.
King
RB
and
O’Sullivan
TC
(
1987
),
Sliding contact stresses in a two-dimensional layered elastic half-space
,
Int. J. Solids Struct.
23
,
581
597
.
15.
Gupta
PK
,
Walowit
JA
, and
Finkin
EF
(
1973
),
Stress distributions in plane strain layered elastic solids subjected to arbitrary boundary loading
,
ASME J. Lubr. Technol.
95
,
427
432
.
16.
Peng
W
and
Bhushan
B
(
2002
),
Sliding contact analysis of layered elastic/plastic solids
,
ASME J. Tribol.
124
,
46
61
.
17.
Cole
SJ
and
Sayles
RS
(
1992
),
A numerical model for the contact of layered elastic bodies with real rough surfaces
,
ASME J. Tribol.
114
,
334
340
.
18.
Komvopoulos
K
(
1988
),
Finite element analysis of a layered elastic solid in normal contact with a rigid surface
,
ASME J. Tribol.
110
,
477
485
.
19.
Komvopoulos
K
(
1989
),
Elastic-plastic finite element analysis of indented layered media
,
ASME J. Tribol.
111
,
430
439
.
20.
Diao
DF
,
Kato
K
, and
Hayashi
K
(
1994
),
The maximum tensile stress on a hard coating under sliding friction
,
Tribol. Int.
27
,
267
272
.
21.
Smith
JO
and
Liu
CK
(
1953
),
Stress due to tangential and normal loads on an elastic solid with application to some contact stress problems
,
ASME J. Appl. Mech.
201
,
157
166
.
22.
Kral
ER
,
Komvopoulos
K
, and
Bogy
DB
(
1993
),
Elastic-plastic finite element analysis of repeated indentation of a half-space by a rigid sphere
,
ASME J. Appl. Mech.
60
,
829
841
.
23.
Kral
ER
,
Komvopoulos
K
, and
Bogy
DB
(
1995
),
Finite element analysis of repeated indentation of an elastic-plastic layered medium by a rigid sphere, Part I: Surface results
,
ASME J. Appl. Mech.
62
,
20
28
.
24.
Kral
ER
,
Komvopoulos
K
, and
Bogy
DB
(
1995
),
Finite element analysis of repeated indentation of an elastic-plastic layered medium by a rigid sphere, Part II: Subsurface results
,
ASME J. Appl. Mech.
62
,
29
42
.
25.
Stephens
LS
,
Liu
Y
, and
Meletis
EI
(
2000
),
Finite element analysis of the initial yielding behavior of a hard coating/substrate system with functionally graded interface under indentation and friction
,
ASME J. Tribol.
122
,
381
387
.
26.
Kral
ER
and
Komvopoulos
K
(
1996
),
Three-dimensional finite element analysis of surface deformation and stresses in an elastic-plastic layered medium subjected to indentation and sliding contact loading
,
ASME J. Appl. Mech.
63
,
365
375
.
27.
Kral
ER
and
Komvopoulos
K
(
1996
),
Three-dimensional finite element analysis of subsurface stresses and shakedown due to repeated sliding on a layered
,
ASME J. Appl. Mech.
63
,
967
973
.
28.
Kral
ER
and
Komvopoulos
K
(
1997
),
Three-dimensional finite element analysis of subsurface stress and strain fields due to sliding contact on an elastic-plastic layered medium
,
ASME J. Tribol.
119
,
332
341
.
29.
Zienkiewicz OC (1977), The Finite Element Method, McGraw-Hill, New York.
30.
Beer G and Watson JO (1992), Introduction to Finite and Boundary Element Methods for Engineers, John Wiley, New York.
31.
Komvopoulos
K
and
Choi
DH
(
1992
),
Elastic finite element analysis of multi-asperity contact
,
ASME J. Tribol.
114
,
823
831
.
32.
Oden
JT
and
Martin
JAC
(
1985
),
Models and computational methods for dynamic friction phenomena
,
Comput. Methods Appl. Mech. Eng.
52
,
527
634
.
33.
Martin
JAC
,
Oden
JT
, and
Simoes
FMF
(
1990
),
A study of statix and kinetic friction
,
Int. J. Eng. Sci.
28
,
29
92
.
34.
Kikuchi N and Oden JT (1988), Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, PA.
35.
Man KW (1994), Contact Mechanics using Boundary Elements, Computational Mechanics Publications, Boston MA.
36.
Kannel JW and Dow TA (1985), Evaluation of contact stress between rough elastic and layered cylinders, Leed-Lyon Conf, Butterworths, 198–206.
37.
Merriman
T
and
Kannel
J
(
1989
),
Analyses of the role of surface roughness on contact stresses between elastic cylinders with and without soft surface coating
,
ASME J. Tribol.
111
,
87
94
.
38.
Johnson KL (1985), Contact Mechanics, Cambridge Univ. Press, Cambridge, UK.
39.
Mao
K
,
Sun
Y
, and
Bell
T
(
1996
),
A numerical model for the dry sliding contact of layered elastic bodies with rough surfaces
,
Tribol. Trans.
39
,
416
424
.
40.
Mao
K
,
Bell
T
, and
Sun
Y
(
1997
),
Effect of sliding friction on contact stresses for multilayered elastic bodies with rough surfaces
,
ASME J. Tribol.
119
,
476
480
.
41.
Tian
X
and
Bhushan
B
(
1996
),
A numerical three-dimensional model for the contact of rough surfaces by variational principle
,
ASME J. Tribol.
118
,
33
41
.
42.
Nogi
T
and
Kato
T
(
1997
),
Influence of a hard surface layer on the limit of elastic contact-Part I: Analysis using a real surface model
,
ASME J. Tribol.
119
,
493
500
.
43.
Polonsky
A
and
Keer
LM
(
2000
),
A fast and accurate method for numerical analysis of elastic layered contacts
,
ASME J. Tribol.
122
,
30
35
.
44.
Winther
R
(
1980
),
Some superlinear convergence results for the Conjugate Gradient Method
,
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
17
,
14
17
.
45.
Hestenes
MR
and
Stiefel
E
(
1952
),
Methods of conjugate gradients for solving linear systems
,
J. Res. Natl. Bur. Stand.
49
,
409
436
.
46.
Pshenichny BN and Danilin YM (1975), Numerical Methods in Optimization Problems, (in Russian) Nauka, Moscow.
47.
Polonsky
A
and
Keer
LM
(
1999
),
A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques
,
Wear
231
,
206
219
.
48.
Polonsky
A
and
Keer
LM
(
2000
),
Fast methods for solving rough contact problems: A comparative study
,
ASME J. Tribol.
122
,
36
41
.
49.
Richards TH (1977), Energy Methods in Stress Analysis: with an Introduction to Finite Element Techniques, Halsted Press, New York.
50.
Kalker
JJ
and
van Randen
Y
(
1972
),
A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems
,
J. Eng. Math.
6
,
193
206
.
51.
Kalker
JJ
(
1977
),
Variational principles of contact elastostatics
,
J. Inst. Math. Appl.
20
,
199
219
.
52.
Peng
W
and
Bhushan
B
(
2001
),
A numerical three-dimensional model for the contact of layered elastic/plastic solids with rough surfaces by variational principle
,
ASME J. Tribol.
123
,
330
342
.
53.
Peng
W
and
Bhushan
B
(
2001
),
Three-dimensional contact analysis of layered elastic/plastic solids with rough surfaces
,
Wear
249
,
741
760
.
54.
Stanley
HM
and
Kato
T
(
1997
),
An FFT-based method for rough surfaces contact
,
ASME J. Tribol.
119
,
481
485
.
55.
Ai
X
and
Sawamiphakdi
K
(
1999
),
Solving elastic contact between rough surfaces as an unconstrained strain energy minimization by using CGM and FFT techniques
,
ASME J. Tribol.
121
,
639
647
.
56.
Love
AEH
(
1929
),
The stress produced in a semi-infinite solid by pressure on part of the boundary
,
Proc. R. Soc. London, Ser. A
228
,
377
420
.
57.
Airy GB (1862), Brit. Assoc. Advan. Sci. Rept.
58.
Malvern LE (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall.
59.
Ju
Y
and
Farris
TN
(
1996
),
Special analysis of two-dimensional contact problems
,
ASME J. Tribol.
118
,
320
328
.
60.
Brandt
A
and
Lubrecht
AA
(
1990
),
Multilevel matrix multiplication and fast solution of integral equations
,
J. Comput. Phys.
90
,
348
370
.
61.
Yu
M
and
Bhushan
B
(
1996
),
Contact analysis of three-dimensional rough surfaces under frictionless and frictional contact
,
Wear
200
,
265
280
.
62.
Press WH, Teukolsky SA, Vetterling WT, and Flannery BP (1999), Numerical Recipes in FORTRAN, the Art of Scientific Computing, 2nd Edition, Cambridge Univ Press, Cambridge, UK.
63.
Powell
MJD
(
1977
),
Restart procedures for the conjugate gradient method
,
Math. Program.
12
,
241
254
.
64.
Derjaguin BV, Chuarev NV, and Muller VM (1987), Surface Forces, Consultants Bureau, New York.
65.
Adamson AW (1990), Physical Chemistry of Surfaces, 5th Edition, John Wiley, New York.
66.
Israelachvili JN (1992), Intermolecular and Surface Forces, 2nd Edition, Academic Press, San Diego, CA.
67.
Li Y and Talke FE (1990), A model for the effect of humidity on stiction of the head/disk interface, Tribology and Mechanics of Magnetic Storage Systems, B Bhushan (ed), SP-29, STLE, Park Ridge II, 79–84.
68.
Tian
H
and
Matsudaira
T
(
1993
),
The role of relative humidity, surface roughness and liquid build-up on static friction behavior of the head/disk interface
,
ASME J. Tribol.
115
,
28
35
.
69.
Gao
C
,
Tian
X
, and
Bhushan
B
(
1995
),
A meniscus model for optimization of texturing and liquid lubrication of magnetic thin film rigid disks
,
Tribol. Trans.
38
,
201
212
.
70.
Bhushan
B
,
Kotwal
CA
, and
Chilamakuri
SK
(
1998
),
Kinetic meniscus model for prediction of rest stiction
,
ASME J. Tribol.
120
,
42
53
.
71.
Tian
X
and
Bhushan
B
(
1996
),
The micro-meniscus effect of a thin liquid film on the static friction of rough surfaces contact
,
Bull. Obs. Astron. Belgrade
29
,
163
178
.
72.
Whitehouse
DJ
and
Archard
JF
(
1970
),
The properties of random surface of significance in their contact
,
Proc. R. Soc. London, Ser. A
316
,
97
121
.
73.
Hu
YZ
and
Tonder
K
(
1992
),
Simulation of 3-D random surface by 2-D digital filter and Fourier analysis
,
Int. J. Mach. Tools Manuf.
32
,
82
90
.
74.
Chilamakuri
SK
and
Bhushan
B
(
1998
),
Contact analysis of non-Gaussian random surfaces
,
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
212
,
19
32
.
75.
Onions
RA
and
Archard
JF
(
1973
),
The contact of surfaces having a random structure
,
Bull. Obs. Astron. Belgrade
6
,
289
304
.
76.
Peng W and Bhushan B (2002), Transient dynamic analysis of the sliding contact of layered elastic/plastic solids, ASME J. Tribol. (submitted).
77.
Poon
CY
and
Bhushan
B
(
1995
),
Comparison of surface roughness measurements by stylus profiler, AFM and non-contact optical profiler
,
Wear
190
,
76
88
.
78.
Peng
W
and
Bhushan
B
(
2000
),
Numerical contact analysis of layered rough surfaces for magnetic head slider-disk contact
,
J. Info. Storage Proc. Syst.
2
,
263
280
.
79.
Bhattacharya
AK
and
Nix
WD
(
1988
),
Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates
,
Int. J. Solids Struct.
24
,
1287
1298
.
80.
Oliver
WC
and
Pharr
GM
(
1992
),
An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments
,
J. Mater. Res.
7
,
1564
1583
.
81.
Stewart J (1995), Calculus, 3rd Edition, Brooks/Cole, Pacific Grove CA.
82.
Bhushan
B
(
1999
),
Chemical, mechanical and tribological characterization of ultra-thin and hard amorphous carbon coatings as thin as 3.5 nm: recent developments
,
Diamond Relat. Mater.
8
,
1985
2015
.
83.
Bhushan
B
,
Chyung
K
, and
Miller
RA
(
1996
),
Micromechanical property measurements of glass and glass-ceramic substrates for magnetic thin-film rigid disks for gigabit recording
,
Adv. Inf. Storage Syst.
7
,
3
17
.
84.
Poon
CY
and
Bhushan
B
(
1996
),
Numerical contact and stiction analyses of Gaussian isotropic surfaces for magnetic head slider/disk contact
,
Wear
202
,
68
82
.
85.
Bhushan
B
and
Venkatesan
S
(
1993
),
Mechanical and tribological properties of silicon for micromechanical applications: A review
,
Adv. Inf. Storage Syst.
5
,
211
239
.
1.
Bhushan
B
and
Kulkarni
A
(
1996
),
Effect of normal load on micro scale friction measurements
,
Thin Solid Films
278
,
49
56
;
2.
(
1997
) erratum,
293
,
333
333
.
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