A 3D graded coating/substrate model based on multigrid techniques within a finite difference frame work is presented. Localized refinement is implemented to optimize memory requirement and computing time. Validation of the solver is performed through a comparison with analytical results for (i) a homogeneous material and (ii) a graded material. The algorithm performance is analyzed through a parametric study describing the influence of layer thickness (0.01 < t/a < 10) and mechanical properties (0.005 < Ec/Es < 10) of the coating on the contact parameters (Ph, a). Three-dimensional examples are then presented to illustrate the efficiency and the large range of possibilities of the model. The influence of different gradations of Young’s modulus, constant, linear and sinusoidal, through the coating thickness on the maximum tensile stress is analyzed, showing that the sinusoidal gradation best accommodates the property mismatch of two successive layers. A final case is designed to show that full 3D spatial property variations can be accounted for. Two spherical inclusions of different size made from elastic solids with Young’s modulus and Poisson’s ratio are embedded within an elastically mismatched finite domain and the stress field is computed.

References

References
1.
Ju
,
Y.
, and
Liu
,
S.
, 1988, “
Parameters Affecting Thermomechanical Cracking in Coated Media Due to High Speed Friction Load
,”
Trans. ASME J. Tribol.
,
110
, pp.
222
229
.
2.
Leroy
,
J. M.
,
Floquet
,
A.
, and
Villechaise
,
B.
, 1989, “
Thermomecanical Behavior of Multilayered Media: Theory
,”
Trans. ASME J. Tribol.
,
111
, pp.
538
544
.
3.
Plumet
,
S.
, and
Dubourg
,
M. C.
, 1998, “
A 3D Model for a Multilayered Body Loaded Normally and Tangentially Against a Rigid Body: Application
,”
J. Tribol.
,
120
, pp.
668
676
.
4.
Polonsky
,
I. A.
, and
Keer
,
L. M.
, 2000, “
A Fast and Accurate Method for Numerical Analysis of Elastic Layered Contacts
Trans. ASME J. Tribol.
,
122
, pp.
30
35
.
5.
Polonsky
,
I. A.
, and
Keer
,
L. M.
, 2000, “
Fast Method for Solving Rough Contact Problems: A Comparative Study
,”
Trans. ASME J. Tribol.
,
122
, pp.
36
41
.
6.
Nogi
,
T.
, and
Kato
,
T.
, 1997, “
Influence of a Hard Surface Layer on the Limit of Elastic Contact: Part 1: Analysis Using a Real Surface Model
,”
Trans. ASME J. Tribol.
,
119
, pp.
493
500
.
7.
Kesler
,
O.
,
Finot
,
M.
,
Suresh
,
S.
, and
Sampath
,
S.
, 1996, “
Determination of Processing-Induced Stresses and Properties of Layered and Graded Coatings: Experimental Method and Results for Plasma-Sprayed Ni-Al2 03
,”
Acta Mater.
,
45
, pp.
3113
3134
.
8.
Suresh
,
S.
, and
Mortensen
,
A.
, 1997, “
Functionally Graded Metals and Metal-Ceramic Composites: Part 2 Thermomechanical Behaviour
,”
Int. Mater. Rev.
,
42
, pp.
85
116
.
9.
Giannakopoulos
,
A. E.
, and
Suresh
,
S.
, 1996, “
Indentation of Solids With Gradients in Elastic Properties: Part II. Axisymmetric Indentords
,”
Int. J. Solids Struct.
,
34
, pp.
2393
2428
.
10.
Prasad
,
A.
,
Dao
,
M.
, and
Suresh
,
S.
, 2008, “
Steady-State Frictional Sliding Contact on Surfaces of Plastically Graded Materials
,”
Acta Mater.
,
57
, pp.
511
524
.
11.
Holmberg
,
K.
,
Laukkanen
,
A.
,
Ronkainen
,
H.
,
Wallin
,
K.
,
Varjus
,
S.
, and
Koskinen
,
J.
, 2006, “
Tri-Bological Contact Analysis of a Rigid Ball Sliding on a Hard Coated Surface Part II: Material Deformations, Influence of Coating Thickness and Young’s Modulus
,”
Surf. Coat. Technol.
,
200
, pp.
3810
3823
.
12.
Ye
,
N.
, and
Komvopoulos
,
K.
, 2003, “
Three-Dimensional Finite Element Analysis of Elastic-Plastic Layered Me-Dia Under Thermomechanical Surface Loading
,”
J. Tribol.
,
125
, pp.
52
59
.
13.
Venner
,
C. H.
, and
Lubrecht
,
A. A.
, 2000,
Multi-Level Methods in Lubrication
,
Elsevier Tribology Series
,
Amsterdam
.
14.
Fretigny
,
C.
, and
Chateauminois
,
A.
, 2007, “
Solution for the Elastic Field in a Layered Medium Under Axisymmetric Contact Loading
,”
J. Phys. D
,
40
, pp.
5418
5426
.
15.
Perriot
,
A.
, and
Barthel
,
E.
, 2004, “
Elastic Contact to a Coated Half-Space: Effective Elastic Modulus and Real Penetration
,”
J. Mater. Res.
,
19
, pp.
600
608
.
16.
Watremetz
,
B.
,
Baietto Dubourg
,
M. C.
, and
Lubrecht
,
A. A.
, 2007, “
2D Thermo-Mechanical Contact Simulations in a Functionally Graded Material: A Multigrid-Based Approach
,”
Tribol. Int.
,
40
, pp.
754
762
.
17.
Boffy
,
H.
,
Baietto
,
M. C.
,
Sainsot
,
P.
, and
Lubrecht
,
A. A.
, 2012, “
Detailed Modelling of a Moving Heat Source Using Multigrid Methods
,”
Tribol. Int.
,
46
, pp.
279
287
.
18.
Johnson
,
K. L.
, 1985,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK.
19.
Bai
,
D.
, and
Brandt
,
A.
, 1987, “
Local Mesh Refinement Multilevel Techniques
,”
J. Sci. Comput.
,
8
, pp.
109
134
.
20.
Brandt
,
A.
, 1984,
Multigrid Techniques: Guide With Applications to Fluid Dynamics
,
Gesellschaft fur Mathematik und Datenverarbeitung
,
Munich, Germany
.
21.
Brandt
,
A.
, 1977, “
Multilevel Adaptative Solution to Boundary Value Problems
,”
Math. Comput.
,
31
, pp.
333
390
.
22.
Sainsot
,
P.
, and
Lubrecht
,
A. A.
, 2010, “
Efficient Solution of the Dry Contact of Rough Surfaces: A Comparison of FFT and MG Methods
,”
Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol.
,
225
, pp.
441
448
.
23.
Suresh
,
S.
, 2001, “
Graded Materials for Resistance to Contact Deformation and Damage
,”
Science
,
292
, pp.
2447
2451
.
24.
Eshelby
,
J. D.
, 1957, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London
,
241
, pp.
376
396
.
25.
Eshelby
,
J. D.
, 1959, “
The Elastic Field outside an Ellipsoidal Inclusion
,”
Proc. R. Soc. London Ser. A
,
252
, p.
561
.
26.
Leroux
,
J.
,
Fulleringer
,
B.
, and
Nélias
,
D.
, 2010, “
Contact Analysis in Presence of Spherical Inhomogeneities Within a Half-Space
,”
Int. J. Solids Struct.
,
47
, pp.
3034
3049
.
You do not currently have access to this content.