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Research Papers: Elastohydrodynamic Lubrication

Elastohydrodynamic Lubrication of Inhomogeneous Materials Using the Equivalent Inclusion Method

[+] Author and Article Information
Zhanjiang Wang

State Key Laboratory of
Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: wangzhanjiang001@gmail.com

Dong Zhu

School of Aeronautics and Astronautics,
Sichuan University,
Chengdu, Sichuan 610065, China

Qian Wang

State Key Laboratory of
Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received April 17, 2013; final manuscript received October 20, 2013; published online December 27, 2013. Assoc. Editor: Daniel Nélias.

J. Tribol 136(2), 021501 (Dec 27, 2013) (10 pages) Paper No: TRIB-13-1083; doi: 10.1115/1.4025939 History: Received April 17, 2013; Revised October 20, 2013

Solid materials forming the boundaries of a lubrication interface may be elastoplastic, heat treated, coated with multilayers, or functionally graded. They may also be composites reinforced by particles or have impurities and defects. Presented in this paper is a model for elastohydrodynamic lubrication interfaces formed with these realistic materials. This model considers the surface deformation and subsurface stresses influenced by material inhomogeneities, where the inhomogeneities are replaced by inclusions with properly determined eigenstrains by means of the equivalent inclusion method. The surface displacement or deformation caused by inhomogeneities is introduced to the film thickness equation. The stresses are the sum of those caused by the fluid pressure and the eigenstrains. The lubrication of a material with a single inhomogeneity, multiple inhomogeneities, and functionally graded coatings are analyzed to reveal the influence of inhomogeneities on film thickness, pressure distribution, and subsurface stresses.

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References

Ghosh, S., and Moorthey, S., 1995, “Elastic-Plastic Analysis of Arbitrary Heterogeneous Materials With the Voronoi Cell Finite Element Method,” Comput. Methods Appl. Mech. Eng., 121(1–4), pp. 373–409. [CrossRef]
Jalalahmadi, B., and Sadeghi, F., 2009, “A Voronoi Finite Element Study of Fatigue Life Scatter in Rolling Contacts,” ASME J. Tribol., 131(2), p. 022203. [CrossRef]
Raje, N. N., Sadeghi, F., Rateick, R. G., Jr., and Hoeprich, M. R., 2007, “Evaluation of Stresses Around Inclusions in Hertzian Contacts Using the Discrete Element Method,” ASME J. Tribol., 129(2), pp. 283–291. [CrossRef]
Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proc. R. Soc. London, Ser. A, 241, pp. 376–396. [CrossRef]
Mura, T., 1993, Micromechanics of Defects in Solids, 2nd ed., Kluwer, Dordrecht, Netherlands.
Mindlin, R. D., and Cheng, D. H., 1950, “Thermoelastic Stress in the Semi-Infinite Solid,” J. Appl. Phys.21, pp. 931–933. [CrossRef]
Seo, K., and Mura, T., 1979, “The Elastic Field in a Half Space Due to Ellipsoidal Inclusions With Uniform Dilatational Eigenstrains,” ASME J. Appl. Mech., 46(3), pp. 568–572. [CrossRef]
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., 45(2), pp. 302–306. [CrossRef]
Jacq, C., Nélias, D., Lormand, G., and Girodin, D., 2002, “Development of a Three-Dimensional Semi-Analytical Elastic–Plastic Contact Code,” ASME J. Tribol., 124(4), pp. 653–667. [CrossRef]
Zhou, K., Chen, W. W., Keer, L. M., and Wang, Q. J., 2009, “A Fast Method for Solving Three-Dimensional Arbitrarily Shaped Inclusions in a Half Space,” Comput. Methods Appl. Mech. Eng., 198(9–12), pp. 885–892. [CrossRef]
Leroux, J., Fulleringer, B., and Nélias, D., 2010, “Contact Analysis in Presence of Spherical Inhomogeneities Within a Half-Space,” Int. J. Solids Struct., 47(22–23), pp. 3034–3049. [CrossRef]
Liu, S. B., Jin, X. Q., Wang, Z. J., Keer, L. M., and Wang, Q., 2012, “Analytical Solution for Elastic Fields Caused by Eigenstrains in a Half-Space and Numerical Implementation Based on FFT,” Int. J. Plast., 35, pp. 135–154. [CrossRef]
Wang, Z. J., Jin, X. Q., Keer, L. M., and Wang, Q., 2012, “Numerical Methods for Contact Between Two Joined Quarter Spaces and a Rigid Sphere,” Int. J. Solids Struct., 49(18), pp. 2515–2527. [CrossRef]
Chen, W. W., Zhou, K., Keer, L. M., and Wang, Q., 2010, “Modeling Elasto-Plastic Indentation on Layered Materials Using the Equivalent Inclusion Method,” Int. J. Solids Struct., 47(20), pp. 2841–2854. [CrossRef]
Zhou, K., Keer, L. M., and Wang, Q., 2011, “Semi-Analytic Solution for Multiple Interacting Three-Dimensional Inhomogeneous Inclusions of Arbitrary Shape in an Infinite Space,” Int. J. Numer. Methods Eng., 87(7), pp. 617–638. [CrossRef]
Wang, Z. J., Jin, X. Q., Zhou, Q. H., Ai, X. L., Keer, L. M., and Wang, Q., 2013, “An Efficient Numerical Method With a Parallel Computational Strategy for Solving Arbitrarily Shaped Inclusions in Elasto-Plastic Contact Problems,” ASME J. Tribol., 135(3), p. 031401. [CrossRef]
Dowson, D., and Higginson, G. R., 1959, “A Numerical Solution to the Elastohydrodynamic Problem,” J. Eng. Sci., 1, pp. 6–15. [CrossRef]
Dowson, D., and Higginson, G. R., 1966, Elastohydrodynamic Lubrication: The Fundamentals of Roller and Gear Lubrication, Pergamon, New York.
Lubrecht, A. A., and Ioannides, E., 1991, “A Fast Solution of the Dry Contact Problem and the Associated Sub-Surface Stress Field, Using Multilevel Techniques,” ASME J. Tribol., 113(1), pp. 128–133. [CrossRef]
Venner, C. H., and Lubrecht, A. A., 2000, Multilevel Methods in Lubrication, Vol. 37, Elsevier, New York.
Liu, S. B., Wang, Q., and Liu, G., 2000, “A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses,” Wear, 243(1–2), pp. 101–111. [CrossRef]
Wang, W. Z., Wang, H., Liu, Y. C., Hu, Y. Z., and Zhu, D., 2003, “A Comparative Study of the Methods for Calculation of Surface Elastic Deformation,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 217(2), pp. 145–152. [CrossRef]
Zhu, D., and Wang, Q., 2011, “Elastohydrodynamic Lubrication (EHL): A Gateway to Interfacial Mechanics-Review and Prospect,” ASME J. Tribol., 133(4), p. 041001. [CrossRef]
Zhu, D., and Hu, Y., 1999, “The Study of Transition From Full Film Elastohydrodynamic to Mixed and Boundary Lubrication,” Proceedings of the 1999 STLE/ASME H. S. Cheng Tribology Surveillance, STLE, Park Ridge, IL, pp. 150–156.
Hu, Y. Z., and Zhu, D., 2000, “A Full Numerical Solution to the Mixed Lubrication in Point Contacts,” ASME J. Tribol., 122(1), pp. 1–9. [CrossRef]
Zhu, D., 2007, “On Some Aspects in Numerical Simulation of Thin Film and Mixed EHL,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 221, pp. 561–579. [CrossRef]
Slack, T. S., Raje, N., Sadeghi, F., Doll, G., and Hoeprich, M. R., 2007, “EHL Modeling for Non-Homogenous Materials: The Effect of Material Inclusions,” ASME J. Tribol., 129(2), pp. 256–273. [CrossRef]
Liu, Y. C., Chen, W. W., Liu, S. B., Zhu, D., and Wang, Q., 2007, “An Elastohydrodynamic Lubrication Model for Coated Surfaces in Point Contacts,” ASME J. Tribol., 129(3), pp. 509–516. [CrossRef]
Liu, Y. C., Zhu, D., and Wang, Q., 2008, “Effect of Stiff Coatings on EHL Film Thickness in Point Contacts,” ASME J. Tribol., 130(3), p. 031501. [CrossRef]
Wang, Z. J., Wang, W. Z., Wang, H., and Hu, Y. Z., 2009, “Stress Analysis on Layered Materials in Point Elastohydrodynamic–Lubricated Contacts,” Tribol. Lett., 35(3), pp. 229–244. [CrossRef]
Ren, N., Zhu, D., Chen, W. W., and Wang, Q. J., 2010, “Plasto-Elastohydrodynamic Lubrication (PEHL) in Point Contacts,” ASME J. Tribol., 132(3), p. 031501. [CrossRef]
Ren, N., Zhu, D., and Wang, Q. J., 2011, “Three-Dimensional Plasto-Elastohydrodynamic Lubrication (PEHL) for Surfaces With Irregularities,” ASME J. Tribol., 133(3), p. 031502. [CrossRef]
Bair, S., 2007, High Pressure Rheology for Quantitative Elastohydrodynamics, Elsevier, Amsterdam.
Liu, S. B., and Wang, Q., 2002, “Study Contact Stress Fields Caused by Surface Tractions With a Discrete Convolution and Fast Fourier Transform Algorithm,” ASME J. Tribol., 124(1), pp. 36–45. [CrossRef]
Ai, X., 1993, “Numerical Analyses of Elastohydrodynamically Lubricated Line and Point Contacts With Rough Surfaces by Using Semi-System and Multigrid Methods,” Ph.D. thesis, Northwestern University, Evanston, IL.
Liu, Y. C., Wang, W. Z., Zhu, D., Hu, Y. Z., and Wang, Q., 2013, “Effects of Differential Scheme and Mesh Density on EHL Film Thickness in Point Contacts,” ASME J. Tribol., 128(3), pp. 641–653. [CrossRef]
Wang, Z. J., Jin, X. Q., Keer, L. M., and Wang, Q., 2013, “Novel Model for Partial-Slip Contact Involving a Material With Inhomogeneity,” ASME J. Tribol., 135(4), p. 041401. [CrossRef]

Figures

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Fig. 1

EHL point contact between a moving ball and a smooth stationary flat with embedded elastic inhomogeneities

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Fig. 2

Flow chart for the numerical calculation

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Fig. 3

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of a cuboidal inhomogeneity entraining velocity U = 1 m/s. (a) E2 = E1, (b) E2 = 2E1, and (c) E2 = 0.5E1.

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Fig. 4

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of a cuboidal inhomogeneity at entraining velocity U = 10 m/s. (a) E2 = E1, (b) E2 = 2E1, and (c) E2 = 0.5E1.

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Fig. 5

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of a spherical inhomogeneity at entraining velocity U = 10 m/s. (a) E2 = 2E1 and (b) E2 = 0.5E1.

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Fig. 6

Dimensionless central film thickness and minimum film thickness affected by inhomogeneity size (fixed inhomogeneity depth 0.5a). (a) Central film thickness and (b) minimum film thickness.

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Fig. 7

Dimensionless central film thickness and minimum film thickness affected by inhomogeneity depth (fixed inhomogeneity length a). (a) Central film thickness and (b) minimum film thickness.

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Fig. 8

Contour plots of dimensionless film thickness for the cases of a cuboidal inhomogeneity with depth 0.5a and size 0.9a at entraining velocity U = 10 m/s. (a) E2 = E1, (b) E2 = 4E1, and (c) E2 = 0.25E1

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Fig. 9

Dimensionless film thickness and its components, i.e., surface geometry h0, displacement caused by fluid pressure uz, disturbed displacement caused by inhomogeneity u˜z, and normal approach between two bodies δz (inhomogeneity depth and length are both 0.5a). (a) Stiff inhomogeneity (E2 = 2E1) and (b) compliant inhomogeneity (E2 = 0.5E1).

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Fig. 10

Three-dimensional shapes of inhomogeneities, their locations, and sizes. (a) Cuboidal inhomogeneities and (b) spherical inhomogeneities.

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Fig. 11

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of multiple cuboidal inhomogeneities at entraining velocity U = 10 m/s. (a) E2 = 2E1 and (b) E2 = 0.5E1.

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Fig. 12

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of multiple spherical inhomogeneities at entraining velocity U = 10 m/s. (a) E2 = 2E1 and (b) E2 = 0.5E1.

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Fig. 13

Substrate with a functional graded coating; Young's modulus of the coating gradually decreases or increases along the depth direction

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Fig. 14

Dimensionless fluid pressure and film thickness along the x-axis and subsurface von Mises stress in the x-z plane for the cases of the flat surface modified by a functionally graded coating at entraining velocity U = 10 m/s. (a) E2 = 2E1 and (b) E2 = 0.5E1.

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Fig. 15

Computing time affected by the size of cuboidal inhomogeneities

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Fig. 16

Equivalent inclusion method, Cijkl and Cijkl* denote the elastic moduli of the matrix (D-Ω) and inhomogeneity (Ω); ɛkl0 is the homogeneous (i.e., in the absence of material inhomogeneity) solution caused by the fluid pressure; and ɛ˜kl is the perturbed strain caused by the inhomogeneity

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