The continuum theory of elasticity and/or homogeneously discretized finite element models have been commonly used to investigate and analyze subsurface stresses in Hertzian contacts. These approaches, however, do not effectively capture the influence of the random microstructure topology on subsurface stress distributions in Hertzian contacts. In this paper, a finite element model for analyzing subsurface stresses in an elastic half-space subjected to a general Hertzian contact load with explicit consideration of the material microstructure topology is presented. The random internal geometry of polycrystalline microstructures is modeled using a 3D Voronoi tessellation, where each Voronoi cell represents a distinct material grain. The grains are then meshed using finite elements, and an algorithm was developed to eliminate poorly shaped elements resulting from “near degeneracy” in the Voronoi tessellations. Hertzian point and line contacts loads are applied as distributed surface loads, and the model’s response is evaluated with commercial finite element software ABAQUS. Internal stress results obtained from the current model compare well with analytical solutions from theory of elasticity. The influence of the internal microstructure topology on the subsurface stresses is demonstrated by analyzing the model’s response to an over rolling element using a critical plane approach.

1.
Littmann
,
W. E.
, 1969, “
The Mechanism of Contact Fatigue
,”
NASA
Special Report No. SP-237.
2.
Littmann
,
W. E.
, and
Widner
,
R. L.
, 1966, “
Propagation of Contact Fatigue From Surface and Sub-Surface Origins
,”
ASME J. Basic Eng.
0021-9223,
88
, pp.
624
636
.
3.
Tallian
,
T. E.
, 1999,
Failure Atlas for Hertz Contact Machine Elements
,
ASME
,
New York
, pp.
179
231
.
4.
Hertz
,
H.
, 1882, “
On the Contact of Elastic Solids
,”
J. Reine Angew. Math.
0075-4102,
92
, pp.
156
171
.
5.
Sadeghi
,
F.
,
Jalalahmadi
,
B.
,
Slack
,
T.
,
Raje
,
N.
, and
Arakere
,
N. K.
, 2009, “
A Review of Rolling Contact Fatigue
,”
ASME J. Tribol.
0742-4787,
131
, p.
041403
.
6.
Österlund
,
R.
, and
Vingsbo
,
O.
, 1980, “
Phase Changes in Fatigued Ball Bearings
,”
Metall. Trans. A
0360-2133,
11A
, pp.
701
707
.
7.
Voskamp
,
A. P.
, 1985, “
Material Response to Rolling Contact Loading
,”
ASME J. Tribol.
0742-4787,
107
, pp.
359
364
.
8.
Zaretsky
,
E. V.
,
Parker
,
R. J.
, and
Anderson
,
W. J.
, 1969, “
A Study of Residual Stress Induced During Rolling
,”
ASME J. Lubr. Technol.
0022-2305,
91
, pp.
314
319
.
9.
Murakami
,
Y.
,
Kodama
,
S.
, and
Konuma
,
S.
, 1989, “
Quantitative Evaluation of Effects of Non-Metallic Inclusions on Fatigue Strength of High Strength Steels. I: Basic Fatigue Mechanism and Evaluation of Correlation Between the Fatigue Fracture Stress and the Size and Location of Non-Metallic Inclusions
,”
Int. J. Fatigue
0142-1123,
11
(
5
), pp.
291
298
.
10.
Nishioka
,
K.
, 1957, “
On the effect of Inclusion Upon the Fatigue Strength
,”
J. Soc. Mater. Sci. Jpn.
0514-5163,
6
, pp.
382
385
.
11.
Jalalahmadi
,
B.
, and
Sadeghi
,
F.
, 2009, “
A Voronoi Finite Element Study of Fatigue Life Scatter in Rolling Contacts
,”
ASME J. Tribol.
0742-4787,
131
, p.
022203
.
12.
Raje
,
N.
,
Sadeghi
,
F.
, and
Rateick
,
R. G.
, Jr.
, 2007, “
A Discrete Element Approach to Evaluate Stresses Due to Line Loading on an Elastic Half-Space
,”
Comput. Mech.
0178-7675,
40
, pp.
513
529
.
13.
Raje
,
N.
,
Sadeghi
,
F.
, and
Rateick
,
R. G.
, Jr.
, 2008, “
A Statistical Damage Mechanics Model for Subsurface Initiated Spalling in Rolling Contacts
,”
ASME J. Tribol.
0742-4787,
130
, p.
042201
.
14.
Raje
,
N.
,
Sadeghi
,
F.
,
Rateick
,
R. G.
, Jr.
, and
Hoeprich
,
M. R.
, 2008, “
A Numerical Model for Life Scatter in Rolling Element Bearings
,”
ASME J. Tribol.
0742-4787,
130
, p.
011011
.
15.
Okabe
,
A.
,
Boots
,
B.
,
Sugihara
,
K.
, and
Chiu
,
S. N.
, 2000,
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams
,
2nd ed.
,
Wiley
,
West Sussex, UK
.
16.
Callister
,
W. D.
, Jr.
, 2000,
Material Science and Engineering: An Introduction
,
5th ed.
,
Wiley
,
New York
, pp.
51
52
.
17.
Mücklich
,
F.
,
Osher
,
J.
, and
Schneider
,
G.
, 1997, “
Die Charakterisierung Homogener Polyedrischer Gefüge mit Hilfe des Räumlichen Poisson-Voronoi-Mosaiks und der Vergleich zur DIN 50 601 (The Characterization of Homogeneous Polyhedral Microstructures Applying the Spatial Poisson-Voronoi Tessellation Compared to the Standard DIN 50 601)
,”
Zeitschrift für Metalkunde
,
88
(
1
), pp.
27
32
.
18.
Kozaczek
,
K. J.
,
Petrovic
,
B. G.
,
Ruud
,
C. O.
,
Kurtz
,
S. K.
, and
McIlree
,
A. R.
, 1995, “
Microstructural Modelling of Grain-Boundary Stresses in Alloy 600
,”
J. Mater. Sci.
0022-2461,
30
, pp.
2390
2400
.
19.
Kumar
,
S.
, and
Kurtz
,
S. K.
, 1994, “
Simulation of Material Microstructure Using a 3D Voronoi Tessellation: Calculation of Effective Thermal Expansion Coefficient of Polycrystalline Materials
,”
Acta Metall. Mater.
0956-7151,
42
(
12
), pp.
3917
3927
.
20.
Kumar
,
S.
,
Kurtz
,
S. K.
, and
Agarwala
,
V. K.
, 1996, “
Micro-Stress Distribution Within Polycrystalline Aggregate
,”
Acta Mech.
0001-5970,
114
, pp.
203
216
.
21.
Nygårds
,
M.
, and
Gudmundson
,
P.
, 2002, “
Three-Dimensional Periodic Voronoi Grain Models and Micromechanical FE-Simulations of a Two-Phase Steel
,”
Comput. Mater. Sci.
0927-0256,
24
, pp.
513
519
.
22.
Song
,
X.
,
Zhang
,
S. Y.
,
Dini
,
D.
, and
Korsunsky
,
A. M.
, 2008, “
Finite Element Modelling and Diffraction Measurement of Elastic Strains During Tensile Deformation of HCP Polycrystals
,”
Comput. Mater. Sci.
0927-0256,
44
(
1
), pp.
131
137
.
23.
Dick
,
T.
,
Basseville
,
S.
, and
Cailletaud
,
G.
, 2008, “
Fatigue Modeling in Fretting Contact With a Crystal Plasticity Model
,”
Comput. Mater. Sci.
0927-0256,
43
(
1
), pp.
36
42
.
24.
Korsunsky
,
A. M.
,
Dini
,
D.
,
Dunne
,
F. P. E.
, and
Walsh
,
M. J.
, 2007, “
Comparative Assessment of Dissipated Energy and Other Fatigue Criteria
,”
Int. J. Fatigue
0142-1123,
29
, pp.
1990
1995
.
25.
Zhang
,
M.
,
McDowell
,
D. L.
, and
Neu
,
R. W.
, 2009, “
Microstructure-Sensitive Modeling: Application to Fretting Contacts
,”
Int. J. Fatigue
0142-1123,
31
, pp.
1397
1406
.
26.
Zhang
,
M.
,
McDowell
,
D. L.
, and
Neu
,
R. W.
, 2009, “
Microstructure Sensitivity of Fretting Fatigue Based on Computational Crystal Plasticity
,”
Tribol. Int.
0301-679X,
42
, pp.
1286
1296
.
27.
MathWorks, Inc.
, 2009, MATLAB 7.8.0.
28.
Barber
,
C. B.
,
Dobkin
,
D. P.
, and
Huhdanpaa
,
H. T.
, 1996, “
The Quickhull Algorithm for Convex Hulls
,”
ACM Trans. Math. Softw.
0098-3500,
22
(
4
), pp.
469
483
.
29.
Barbe
,
F.
,
Decker
,
L.
,
Jeulin
,
D.
, and
Cailletaud
,
G.
, 2001, “
Intergranular and Intragranular Behavior of Polycrystalline Aggregates. Part 1: FE Model
,”
Int. J. Plast.
0749-6419,
17
, pp.
513
536
.
30.
Diard
,
O.
,
Leclercq
,
S.
,
Rousselier
,
G.
, and
Cailletaud
,
G.
, 2005, “
Evaluation of Finite Element Based Analysis of 3D Multicrystalline Aggregates Plasticity Application to Crystal Plasticity Model Identification and the Study of Stress and Strain Fields Near Grain Boundaries
,”
Int. J. Plast.
0749-6419,
21
, pp.
691
722
.
31.
Dini
,
D.
,
Song
,
X.
,
Zhang
,
S. Y.
, and
Korsunsky
,
A. M.
, 2009, “
Residual Strain Analysis in Polycrystalline Aggregates Using Diffraction Measurement and Finite Element Modeling
,”
J. Strain Anal.
0022-4758,
44
, pp.
55
70
.
32.
Fritzen
,
F.
,
Böhlke
,
T.
, and
Schnack
,
E.
, 2009, “
Periodic Three-Dimensional Mesh Generation for Crystalline Aggregates Based on Voronoi Tessellations
,”
Comput. Mech.
0178-7675,
43
, pp.
701
713
.
33.
Dassault Systèmes
, 2009, ABAQUS 6.9 Documentation, Dassault Systèmes Simulia Corp., Providence, RI. http://abaqusdocs.ecn.purdue.edu:2080/v6.9/http://abaqusdocs.ecn.purdue.edu:2080/v6.9/
34.
Johnson
,
K. L.
, 1985,
Contact Mechanics
,
Cambridge University
,
Cambridge, UK
, p.
95
.
35.
Lundberg
,
G.
, and
Palmgren
,
A.
, 1947, “
“Dynamic Capacity of Rolling Element Bearings
,”
Acta Polytech. Scand., Mech. Eng. Ser.
0001-687X,
1
(
3
), pp.
7
53
.
36.
Ioannides
,
E.
, and
Harris
,
T. A.
, 1985, “
A New Fatigue Life Model for Rolling Bearings
,”
ASME J. Tribol.
0742-4787,
107
, pp.
367
378
.
37.
Zhou
,
R. S.
, 1993, “
Surface Topography and Fatigue Life of Rolling Contact Bearings
,”
Tribol. Trans.
1040-2004,
36
, pp.
329
340
.
38.
Lemaitre
,
J.
, and
Chaboche
,
J. L.
, 1990,
Mechanics of Solid Materials
,
Cambridge University
,
Cambridge, UK
.
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