A three-dimensional numerical model based on a semianalytical method in the framework of small strains and small displacements is presented for solving an elastic-plastic contact with surface traction. A Coulomb’s law is assumed for the friction, as commonly used for sliding contacts. The effects of the contact pressure distribution and residual strain on the geometry of the contacting surfaces are derived from Betti’s reciprocal theorem with initial strain. The main advantage of this approach over the classical finite element method (FEM) is the computing time, which is reduced by several orders of magnitude. The contact problem, which is one of the most time-consuming procedures in the elastic-plastic algorithm, is obtained using a method based on the variational principle and accelerated by means of the discrete convolution fast Fourier transform (FFT) and conjugate gradient methods. The FFT technique is also involved in the calculation of internal strains and stresses. A return-mapping algorithm with an elastic predictor∕plastic corrector scheme and a von Mises criterion is used in the plasticity loop. The model is first validated by comparison with results obtained by the FEM. The effect of the friction coefficient on the contact pressure distribution, subsurface stress field, and residual strains is also presented and discussed.

1.
Lamagnère
,
P.
,
Fougères
,
R.
,
Lormand
,
G.
,
Vincent
,
A.
,
Girodin
,
D.
,
Dudragne
,
G.
, and
Vergne
,
F.
, 1998, “
A Physically Based Model for Endurance Limit of Bearing Steels
,”
ASME J. Tribol.
0742-4787,
120
, pp.
421
426
.
2.
Nélias
,
D.
,
Jacq
,
C.
,
Lormand
,
G.
,
Dudragne
,
G.
, and
Vincent
,
A.
, 2005, “
A New Methodology to Evaluate the Rolling Contact Fatigue Performance of Bearing Steels With Surface Dents—Application to 32CrMoV13 (Nitrided) and M50 Steels
,”
ASME J. Tribol.
0742-4787,
127
, pp.
611
622
.
3.
Vincent
,
A.
,
Nélias
,
D.
,
Jacq
,
C.
,
Robin
,
Y.
, and
Dudragne
,
G.
, 2006, “
Comparison of Fatigue Performances of 32CrMoV13 and M50 Steels in Presence of Surface Indents
,”
J. ASTM Int.
1546-962X,
3
(
2
), pp.
1
9
.
4.
Jacq
,
C.
,
Nélias
,
D.
,
Lormand
,
G.
, and
Girodin
,
D.
, 2002, “
Development of a Three-Dimensional Semi-Analytical Elastic-Plastic Contact Code
,”
ASME J. Tribol.
0742-4787,
124
, pp.
653
667
.
5.
Boucly
,
V.
,
Nélias
,
D.
,
Liu
,
S.
,
Wang
,
Q. J.
, and
Keer
,
L. M.
, 2005, “
Contact Analyses for Bodies With Frictional Heating and Plastic Behavior
,”
ASME J. Tribol.
0742-4787,
127
, pp.
355
364
.
6.
Nélias
,
D.
,
Boucly
,
V.
, and
Brunet
,
M.
, 2006, “
Elastic-Plastic Contact Between Rough Surfaces: Proposal for a Wear or Running-in Model
,”
ASME J. Tribol.
0742-4787,
128
, pp.
236
244
.
7.
Wang
,
F.
, and
Keer
,
L. M.
, 2005, “
Numerical Simulation for Three Dimensional Elastic-Plastic Contact With Hardening Behavior
,”
ASME J. Tribol.
0742-4787,
127
, pp.
494
502
.
8.
Polonsky
,
I. A.
, and
Keer
,
L. M.
, 1999, “
A Numerical Method for Solving Rough Contact Problems Based on the Multi-Level Multi-Summation and Conjugate Gradient Techniques
,”
Wear
0043-1648,
231
, pp.
206
219
.
9.
Allwood
,
J.
, 2005, “
Survey and Performance Assessment of Solution Methods for Elastic Rough Contact Models
,”
ASME J. Tribol.
0742-4787,
127
, pp.
10
23
.
10.
Liu
,
S.
,
Wang
,
Q.
, and
Liu
,
G.
, 2000, “
A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses
,”
Wear
0043-1648,
243
, pp.
101
111
.
11.
Johnson
,
K. L.
, 1985,
Contact Mechanics
,
Cambridge University Press
,
London
.
12.
Kalker
,
J. J.
, 1990,
Three Dimensional Elastic Bodies in Rolling Contact
,
Kluwer Academic
,
Dordrecht
.
13.
Chiu
,
Y. P.
, 1977, “
On the Stress Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space
,”
ASME J. Appl. Mech.
0021-8936,
44
, pp.
587
590
.
14.
Chiu
,
Y. P.
, 1978, “
On the Stress Field and Surface Deformation in a Half-space With a Cuboidal Zone in Which Initial Strains Are Uniform
,”
ASME J. Appl. Mech.
0021-8936,
45
, pp.
302
306
.
15.
Brebbia
,
C. A.
,
Telles
,
J. C. F.
, and
Wrobel
,
L. C.
, 1984,
Boundary Element Techniques. Theory and Applications in Engineering
,
Springer-Verlag
,
Berlin
.
16.
Telles
,
J. C. F.
, and
Brebbia
,
C. A.
, 1979, “
On the Application of the Boundary Element Method to Plasticity
,”
Appl. Math. Model.
0307-904X,
3
, pp.
466
470
.
17.
Telles
,
J. C. F.
, and
Brebbia
,
C. A.
, 1979, “
The Boundary Element Method in Plasticity
,”
New Developments in Boundary Element Methods
,
C. A.
Brebbia
, ed.,
Computational Mechanics Centre
,
Southampton
, pp.
295
317
.
18.
Liu
,
S.
, and
Wang
,
Q.
, 2003, “
Transient Thermoelastic Stress Fields in a Half-Space
,”
ASME J. Tribol.
0742-4787,
125
, pp.
33
43
.
19.
Brebbia
,
C. A.
, 1980,
The Boundary Element Method for Engineers
,
Pentech
,
London
.
20.
Mayeur
,
C.
,
Sainsot
,
P.
, and
Flamand
,
L.
, 1995, “
A Numerical Elastoplastic Model for Rough Contact
,”
ASME J. Tribol.
0742-4787,
117
, pp.
422
429
.
21.
Fotiu
,
P. A.
, and
Nemat-Nasser
,
S.
, 1996, “
A Universal Integration Algorithm for Rate-Dependant Elastoplasticity
,”
Comput. Struct.
0045-7949,
59
, pp.
1173
1184
.
22.
Stephens
,
L. S.
,
Yan
,
L.
, and
Meletis
,
E. I.
, 2000, “
Finite Element Analysis of the Initial Yielding Behaviour of a Hard Coating∕Substrate System With Functionally Graded Interface Under Indentation and Friction
,”
ASME J. Tribol.
0742-4787,
122
, pp.
381
387
.
23.
Chang
,
W. R.
,
Etsion
,
I.
, and
Bogy
,
D. B.
, 1987, “
An Elastic-Plastic Model for the Contact of Rough Surfaces
,”
ASME J. Tribol.
0742-4787,
109
, pp.
257
263
.
24.
Jackson
,
R.
,
Chupoisin
,
I.
, and
Green
,
I.
, 2005, “
A Finite Element Study of the Residual Stress and Deformation in Hemispherical Contacts
,”
ASME J. Tribol.
0742-4787,
127
, pp.
484
493
.
25.
Kadin
,
Y.
,
Kligerman
,
Y.
, and
Etsion
,
I.
, 2006, “
Multiple Loading-Unloading of an Elastic-Plastic Spherical Contact
,”
Int. J. Solids Struct.
0020-7683,
43
, pp.
7119
7127
.
26.
Kogut
,
L.
, and
Etsion
,
I.
, 2002, “
Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat
,”
ASME J. Appl. Mech.
0021-8936,
69
, pp.
657
662
.
27.
Tallian
,
T. E.
,
Chiu
,
Y. P.
, and
Amerongen
,
E. V.
, 1978, “
Prediction of Traction and Microgeometry Effects on Rolling Contact Fatigue Life
,”
ASME J. Lubr. Technol.
0022-2305,
100
, pp.
156
166
.
28.
Nélias
,
D.
,
Dumont
,
M.-L.
,
Champiot
,
F.
,
Vincent
,
A.
,
Girodin
,
D.
,
Fougères
,
R.
, and
Flamand
,
L.
, 1999, “
Role of Inclusions, Surface Roughness and Operating Conditions on Rolling Contact Fatigue
,”
ASME J. Tribol.
0742-4787,
121
, pp.
240
251
.
29.
Howell
,
M. B.
,
Rubin
,
C. A.
, and
Hahn
,
G. T.
, 2004, “
The Effect of Dent Size on the Pressure Distribution and Failure Location in Dry Point Frictionless Rolling Contacts
,”
ASME J. Tribol.
0742-4787,
126
, pp.
413
421
.
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