A comprehensive numerical model is developed using Lagrangian finite element (FE) formulation for investigating the steady-state viscoelastic (VE) rolling contact response. Schapery's nonlinear viscoelastic (NVE) model is adopted to simulate the VE behavior. The model accounts for large displacements and rotations. A spatially dependent incremental form of the VE constitutive equations is derived. The dependence on the history of the strain rate is expressed in terms of the spatial variation of the strain. The Lagrange multiplier approach is employed. The classical Coulomb's friction law is used. The developed model is verified and its applicability is demonstrated.

References

References
1.
Lee
,
E. H.
,
1955
, “
Stress Analysis in Viscoelastic Bodies
,”
Q. Appl. Math.
,
13
, pp.
183
190
.
2.
Kalker
,
J. J.
,
1990
,
Three-Dimensional Elastic Bodies in Rolling Contact
,
Kluwer
,
Dordrecht, The Netherlands
.10.1007/978-94-015-7889-9
3.
Wang
,
G.
, and
Knothe
,
K.
,
1993
, “
Stress Analysis for Rolling Contact Between Two Viscoelastic Cylinders
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
310
317
.10.1115/1.2900794
4.
Kalker
,
J. J.
,
1991
, “
Viscoelastic Multilayered Cylinders Rolling With Dry Friction
,”
ASME J. Appl. Mech.
,
58
(
3
), pp.
666
679
.10.1115/1.2897247
5.
Golden
,
J. M.
,
1979
, “
The Problem of a Moving Rigid Punch on an Unlubricated Viscoelastic Half-Space
,”
Mech. Appl. Math.
,
32
, pp.
25
52
.10.1093/qjmam/32.1.25
6.
Hunter
,
S. C.
,
1961
, “
The Rolling Contact of a Rigid Cylinder With a Viscoelastic Half Space
,”
ASME J. Appl. Mech.
,
28
(
4
), pp.
611
617
.10.1115/1.3641792
7.
Morland
,
L. W.
,
1962
, “
A Plane Problem of Rolling Contact in Linear Viscoelasticity Theory
,”
ASME J. Appl. Mech.
,
29
(
2
), pp.
345
352
.10.1115/1.3640553
8.
Morland
,
L. W.
,
1967
, “
Exact Solutions for Rolling Contact Between Viscoelastic Cylinders
,”
Q. J. Mech. Appl. Math.
,
20
(
1
), pp.
73
106
.10.1093/qjmam/20.1.73
9.
Qiu
,
X.
,
2006
, “
Full Two-Dimensional Model for Rolling Resistance, I. Hard Cylinder on Viscoelastic Foundation of Finite Thickness
,”
J. Eng. Mech.
,
132
(
11
), pp.
1241
1251
.10.1061/(ASCE)0733-9399(2006)132:11(1241)
10.
Qiu
,
X.
,
2009
, “
Full Two-Dimensional Model for Rolling Resistance. II: Viscoelastic Cylinders on Rigid Ground
,”
J. Eng. Mech.
,
135
(
1
), pp.
20
30
.10.1061/(ASCE)0733-9399(2009)135:1(20)
11.
Usov
,
P. P.
, and
Danilov
,
V. D.
,
2009
, “
The Contact Problem for a Viscoelastic Layer and Rigid Cylinder During Regular Sliding
,”
J. Frict. Wear
,
30
(
4
), pp.
246
257
.10.3103/S1068366609040047
12.
Padyala
,
P.
,
2009
, “
Steady Rolling Contact and Resistance of a Cylinder on a Viscoelastic Foundation by an Integral Equation Algorithm
,” M.Sc. thesis, Aerospace Engineering, Iowa State University, Ames, IA.
13.
González
,
J. A.
, and
Abascal
,
R.
,
2006
, “
Efficient Stress Evaluation of Stationary Viscoelastic Rolling Contact Problems Using the Boundary Element Method: Application To Viscoelastic Coatings
,”
Eng. Anal. Boundary Elem.
,
30
(
6
), pp.
426
434
.10.1016/j.enganabound.2006.01.006
14.
Carbone
,
G.
, and
Putignano
,
C.
,
2013
, “
A Novel Methodology to Predict Sliding and Rolling Friction of Viscoelastic Materials: Theory and Experiments
,”
J. Mech. Phys. Solids
,
6
(
1
), pp.
1822
1834
.10.1016/j.jmps.2013.03.005
15.
Zéhil
,
G.-P.
, and
Gavin
,
H.
,
2013
, “
Three-Dimensional Boundary Element Formulation of a Viscoelastic Layer of Finite Thickness Applied to the Rolling Resistance of a Rigid Sphere
,”
Int. J. Solids Struct.
,
50
(
6
), pp.
833
842
.10.1016/j.ijsolstr.2012.11.020
16.
Lynch
,
F. S.
,
1969
, “
A Finite Element Method of Viscoelastic Stress Analysis With Application to Rolling Contact Problems
,”
Int. J. Numer. Methods Eng.
,
1
(
4
), pp.
379
394
.10.1002/nme.1620010405
17.
Batra
,
R. C.
,
1981
, “
Quasistatic Indentation of a Rubber-Covered Roll by a Rigid Roll
,”
Int. J. Numer. Methods Eng.
,
17
(
12
), pp.
1823
1833
.10.1002/nme.1620171207
18.
Bapat
,
C. N.
, and
Batra
,
R. C.
,
1984
, “
Finite Plane Strain Deformations of Nonlinear Viscoelastic Rubber Covered Rolls
,”
Int. J. Numer. Methods Eng.
,
20
(
10
), pp.
1911
1927
.10.1002/nme.1620201011
19.
Bapat
,
C. N.
, and
Batra
,
R. C.
,
1985
, “
Finite Deformations of a Viscoelastic Roll Cover Contacting a Rigid Plane Surface
,”
Commun. Appl. Numer. Methods
,
1
(
4
), pp.
169
176
.10.1002/cnm.1630010406
20.
Oden
,
J. T.
, and
Lin
,
T. L.
,
1986
, “
On the General Rolling Contact Problem for Finite Deformation of a Viscoelastic Cylinder
,”
Comput. Methods Appl. Mech. Eng.
,
57
(
3
), pp.
297
367
.10.1016/0045-7825(86)90143-X
21.
Bass
,
J. M.
,
1987
, “
Three-Dimensional Finite Deformation, Rolling Contact of a Hyper-Elastic Cylinder: Formulation of the Problem and Computational Results
,”
Comput. Struct.
,
26
(
6
), pp.
991
1004
.10.1016/0045-7949(87)90116-7
22.
Oden
,
J. T.
,
Lin
,
T. L.
, and
Bass
,
J. M.
,
1988
, “
A Finite Element Analysis of the General Rolling Contact Problem for a Viscoelastic Rubber Cylinder
,”
Tire Sci. Technol.
,
16
(
1
), pp.
18
43
.10.2346/1.2148795
23.
Kikuchi
,
N.
, and
Oden
,
J. T.
,
1988
,
Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods
, Vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
24.
Padovan
,
J.
, and
Paramadilok
,
O.
,
1985
, “
Transient and Steady State Viscoelastic Rolling Contact
,”
Comput. Struct.
,
20
(
1
), pp.
545
553
.10.1016/0045-7949(85)90102-6
25.
Padovan
,
J.
,
1987
, “
Finite Element Analysis of Steady and Transiently Moving/Rolling Nonlinear Viscoelastic Structure
,”
Comput. Struct.
,
27
(
2
), pp.
249
257
.10.1016/0045-7949(87)90093-9
26.
Hu
,
G. D.
, and
Wriggers
,
P.
,
2002
, “
On the Adaptive Finite Element Method of Steady-State Rolling Contact for Hyperelasticity in Finite Deformations
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
13–14
), pp.
1333
1348
.10.1016/S0045-7825(01)00326-7
27.
Gotoh
,
J.
,
Yu
,
Q.
, and
Shiratori
,
M.
,
1999
, “
Experimental/Numerical Analyses of a Viscoelastic Body Under Rolling Contact
,”
Mech. Time Depend. Mater.
,
3
(
3
), pp.
245
261
.10.1023/A:1009838504011
28.
Korunović
,
N.
,
Trajanović
,
M.
, and
Stojković
,
M.
,
2007
, “
Finite Element Model for Steady State Rolling Tire Analysis
,”
J. Serb. Soc. Comput. Mech.
,
1
(
1
), pp.
63
79
.
29.
Korunović
,
N.
,
Trajanović
,
M.
,
Stojković
,
M.
,
Mišić
,
D.
, and
Milovanović
,
J.
,
2011
, “
Finite Element Analysis of a Tire Steady Rolling on the Drum and Comparison With Experiment
,”
J. Mech. Eng.
,
57
(
12
), pp.
888
897
.10.5545/sv-jme.2011.124
30.
Qin
,
F.
,
Yu
,
Y.
, and
Rudolphi
,
T.
,
2010
, “
Finite Element Modeling of Viscoelastic Stress Analysis Under Moving Loads
,”
Int. J. Mech. Mater. Eng.
,
1
(
4
), pp.
226
233
.
31.
Zheng
,
Q.
,
Zhu
,
H.
, and
Yu
,
A.
,
2011
, “
Finite Element Analysis of the Rolling Friction of a Viscous Particle on a Rigid Plane
,”
Powder Technol.
,
207
(
1
), pp.
401
406
.10.1016/j.powtec.2010.11.026
32.
Mahmoud
,
F. F.
,
El-Shafei
,
A. G.
,
Attia
,
M. A.
, and
Abdelrahman
,
A. A.
,
2013
, “
Analysis of Quasistatic Frictional Contact Problems in Nonlinear Viscoelasticity With Large Deformations
,”
Int. J. Mech. Sci.
,
66
, pp.
109
119
.10.1016/j.ijmecsci.2012.11.001
33.
Chudzikiewicz
,
A.
, and
Myslinski
,
A.
,
2011
, “
On Wheel Rail Elastic Contact Problems for Multi-Layer Structure
,”
Computer Methods in Mechanics
,
CMM
,
Warsaw, Poland
.
34.
Yan
,
X.
,
2003
, “
Nonlinear Three-Dimensional Finite Element Analysis of Steady Rolling Radial Tires
,”
J. Reinf. Plast. Compos.
,
22
(
8
), pp.
733
750
.10.1177/0731684403022008004
35.
Schapery
,
R. A.
,
1969
, “
On the Characterization of Nonlinear Viscoelastic Materials
,”
Polym. Eng. Sci.
,
9
(
4
), pp.
295
310
.10.1002/pen.760090410
36.
Schapery
,
R. A.
,
1997
, “
Nonlinear Viscoelastic and Viscoplastic Constitutive Equations Based on Thermodynamics
,”
Mech. Time Depend. Mater.
,
1
(
2
), pp.
209
240
.10.1023/A:1009767812821
37.
Lai
,
J.
, and
Bakker
,
A.
,
1996
, “
3-D Schapery Representation for Nonlinear Viscoelasticity and Finite Element Implementation
,”
Comput. Mech.
,
18
(
3
), pp.
182
191
.10.1007/BF00369936
38.
Hildebrand
,
F. B.
,
1987
,
Introduction to Numerical Analysis
,
2nd ed.
,
McGraw-Hill
,
New York
.
39.
Zienkiewicz
,
O. C.
, and
Taylor
,
R. L.
,
2005
,
The Finite Element Method
,
6th ed.
,
McGraw-Hill
,
London
.
40.
Stupkiewicz
,
S.
,
2001
, “
Extension of the Node-to-Segment Contact Element for Surface-Expansion-Dependent Contact Laws
,”
Int. J. Numer. Methods Eng.
,
50
(
3
), pp.
739
759
.10.1002/1097-0207(20010130)50:3<739::AID-NME49>3.0.CO;2-G
41.
Nackenhorst
,
U.
,
Ziefle
,
M.
, and
Suwannachit
,
A.
,
2010
, “
Finite Element Techniques for Rolling Rubber Wheels
,”
Elastomere Friction (LNACM)
, Vol.
51
,
Springer
,
Berlin
, pp.
123
163
.10.1007/978-3-642-10657-6_5
You do not currently have access to this content.