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Research Papers: Contact Mechanics

An Efficient Numerical Method With a Parallel Computational Strategy for Solving Arbitrarily Shaped Inclusions in Elastoplastic Contact Problems

[+] Author and Article Information
Zhanjiang Wang

e-mail: wangzhanjiang001@gmail.com

Qinghua Zhou

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

Xiaolan Ai

The Timken Company,
Canton, OH 44706

Leon M. Keer

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

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 October 1, 2012; final manuscript received March 3, 2013; published online March 28, 2013. Assoc. Editor: James R. Barber.

J. Tribol 135(3), 031401 (Mar 28, 2013) (12 pages) Paper No: TRIB-12-1166; doi: 10.1115/1.4023948 History: Received October 01, 2012; Revised March 03, 2013

The plastic zone developed during elastoplastic contact may be effectively modeled as an inclusion in an isotropic half space. This paper proposes a simple but efficient computational method to analyze the stresses caused by near surface inclusions of arbitrary shape. The solution starts by solving a corresponding full space inclusion problem and proceeds to annul the stresses acting normal and tangential to the surface, where the numerical computations are processed by taking advantage of the fast Fourier transform techniques with a parallel computing strategy. The extreme case of a cuboidal inclusion with one facet on the surface of the half space is chosen to validate the method. When the surface truncation domain is extended sufficiently and the grids are dense enough, the results based on the new approach are in good agreement with the exact solutions. When solving a typical elastoplastic contact problem, the present analysis is roughly two times faster than the image inclusion approach and six times faster than the direct method. In addition, the present work demonstrates that a significant enhancement in the computational efficiency can be achieved through the introduction of parallel computation.

Copyright © 2013 by ASME
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References

Mura, T., 1993, Micromechanics of Defects in Solids, 2nd and revised edition, Kluwer Academic, Dordrecht, Netherlands.
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]
Mindlin, R. D., and Cheng, D. H., 1950, “Thermo Elastic Stress in the Semi-Infinite Solid,” J. Appl. Phys., 21(9), pp. 931–933. [CrossRef]
Seo, K., and MuraT., 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]
Owen, D. R. J., and Mura, T., 1967, “Periodic Dislocation Distributions in a Half-Space,” J. Appl. Phys., 38(5), pp. 1999–2009. [CrossRef]
Owen, D. R. J., 1971, “Solutions to Arbitrarily Oriented Periodic Dislocation and Eigenstrain Distributions in a Half-Space,” Int. J. Solids Struct., 7(10), pp. 1343–1361. [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]
Boucly, V., Nélias, D., Liu, S., Wang, Q., and KeerL. M., 2005, “Contact Analyses for Bodies With Frictional Heating and Plastic Behavior,” ASME J. Tribol., 127(2), pp. 355–364. [CrossRef]
Nélias, D., Boucly, V., and BrunetM., 2006, “Elastic-Plastic Contact Between Rough Surfaces: Proposal for a Wear or Running-In Model,” ASME J. Tribol., 128(2), pp. 236–244. [CrossRef]
Nélias, D., Antaluca, E., Boucly, V., and Cretu, S., 2007, “A Three-Dimensional Semi-Analytical Model for Elastic–Plastic Sliding Contacts,” ASME J. Tribol., 129(4), pp. 761–771. [CrossRef]
Chen, W. W., and Wang, Q., 2008, “Thermomechanical Analysis of Elasto-Plastic Bodies in a Sliding Spherical Contact and the Effects of Sliding Speed, Heat Partition, and Thermal Softening,” ASME J. Tribol., 130(4), p. 041402. [CrossRef]
Chen, W. W., Liu, S. B., and Wang, Q., 2008, “Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts With Nominally Flat Surface,” ASME J. Appl. Mech., 75(1), p. 011022. [CrossRef]
Wang, Z. J., Wang, W. Z., Hu, Y. Z., and WangH., 2010, “A Numerical Elastic-Plastic Contact Model for Rough Surfaces,” Tribol. Trans., 53(2), pp. 224–238. [CrossRef]
Liu, S. B., and Wang, Q., 2005, “Elastic Fields Due to Eigenstrains in a Half-Space,” ASME J. Appl. Mech., 72(6), pp. 871–878. [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]
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. Plasticity, 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]
Wang, Z. Q., Ghoniem, N. M., Swaminarayan, S., and LeSar, R., 2006, “A Parallel Algorithm for 3D Dislocation Dynamics,” J. Comput. Phys., 219(2), pp. 608–621. [CrossRef]
Chung, S. W., Lee, C. S., and Kim, S. K., 2006, “Large-Scale Simulation of Crack Propagation Based on Continuum Damage Mechanics and Two-Step Mesh Partitioning,” Mech. Mater., 38(1–2), pp. 76–87. [CrossRef]
Lee, H., Gillman, A. S., and Matouîs, K., 2011, “Computing Overall Elastic Constants of Polydisperse Particulate Composites From Microtomographic Data,” J. Mech. Phys. Solids., 59(9), pp. 1838–1857. [CrossRef]
Amdahl, G. M., 1967, “Validity of the Single Processor Approach to Achieving Large-Scale Computing Capabilities,” AFIPS Conf. Proc., 30, pp. 483–485. [CrossRef]
Chiu, Y. P., 1977, “Stress-Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space,” ASME J. Appl. Mech., 44(4), pp. 587–590. [CrossRef]
Chiu, Y. P., 1980, “On the Internal Stresses in a Half-Plane and a Layer Containing Localized Inelastic Strains or Inclusions,” ASME J. Appl. Mech., 47(2), pp. 313–318. [CrossRef]
Jin, X., Keer, L. M., and Wang, Q., 2009, “New Green's Function for Stress Field and a Note of Its Application in Quantum-Wire Structures,” Int. J. Solids Struct., 46(21), pp. 3788–3798. [CrossRef]
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]
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]
FFTW Package, available at http://www.fftw.org/
Rodin, G. J., 1996, “Eshelby's Inclusion Problem for Polygons and Polyhedral,” J. Mech. Phys. Solids, 44(12), pp. 1977–1995. [CrossRef]
Nozaki, H., and Taya, M., 2001, “Elastic Fields in a Polyhedral Inclusion With Uniform Eigenstrains and Related Problems,” ASME J. Appl. Mech., 68(3), pp. 441–452. [CrossRef]
Gao, X. L., and Liu, M. Q., 2012, “Strain Gradient Solution for the Eshelby-Type Polyhedral Inclusion Problem,” J. Mech. Phys. Solids, 60(2), pp. 261–276. [CrossRef]

Figures

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Fig. 1

New method: the elastic fields caused by an inclusion in a half space are the sum of results due to the inclusion and the surface tractions to create a surface free from their opposite tractions

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Fig. 2

Computational domain, mesh types, and grid points

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Fig. 3

Image inclusion approach: the elastic fields caused by an inclusion in a half space are the sum of results due to two inclusions with specified eigenstrains and results caused by surface normal stress which makes the surface free from surface tractions

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Fig. 4

Stresses caused by a cuboidal inclusion (2a × 2a × 2a) in the full space, along the x axis in the target plane at different distances h to the cuboidal center: (a) model, (b) h = a, (c) h = 1.5a, and (d) h = 2a

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Fig. 5

von Mises stress for case 1 to 3 (the computational domain varies but the grid size is held constant for different cases): (a) along the line passing through point (0, 0.1a, 0) and parallel to the x axis, and (b) along the z axis

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Fig. 6

von Mises stress for cases 4 to 6 (the computational domain is fixed but the grid size varies in different cases): (a) along the line passing through point (0, 0.1a, 0) and parallel to the x axis, and (b) along the z axis

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Fig. 7

Resultant stresses from different methods, i.e., the analytical solutions, the new method, and the approach of Zhou et al. [16]; (a) along the line passing through point (0, 0.1a, 0) and parallel to the x axis, and (b) along the z axis

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Fig. 8

Multi-inclusion model and the resultant dimensional von Mises stresses σvonr/ph in the xz plane using different methods: (a) multi-inclusion model, (b) from the new method, (c) based on Zhou et al. [16], and (d) based on Liu et al. [17]

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Fig. 9

Elastoplastic contact model

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Fig. 10

Results from different methods, i.e., the new method, the approach of Zhou et al. [16], and Liu et al. [17]: (a) dimensionless pressure along the x axis, and (b) equivalent plastic strain along the z axis

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Fig. 11

Dimensional residual von Mises stresses σvonr/ph from different methods: (a) the new method, (b) based on Zhou et al. [16], and (c) based on Liu et al. [17]

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Fig. 12

Computational time with different methods: (a) new method, (b) based on Zhou et al. [16], and (c) based on Liu et al. [17]

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Fig. 13

Parallel strategy for calculating stress caused by eigenstrains in the full space

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Fig. 14

Times for different 3D transform size with double-precision complex data type, where each execution time is obtained through repeating 3D-FFT transforms 10 times

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Fig. 15

Time for calculating the eigenstress by using different numbers of processors and different grid sizes

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Fig. 16

Time for computing an elastoplastic contact case by using different numbers of processors and different grid sizes

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