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TECHNICAL PAPERS

Studying Contact Stress Fields Caused by Surface Tractions With a Discrete Convolution and Fast Fourier Transform Algorithm

[+] Author and Article Information
Shuangbiao Liu, Qian Wang

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

J. Tribol 124(1), 36-45 (Jun 19, 2001) (10 pages) doi:10.1115/1.1401017 History: Received February 14, 2001; Revised June 19, 2001
Copyright © 2002 by ASME
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References

Liu,  G., Wang,  Q., and Lin,  C., 1999, “A Survey of Current Models for Simulating the Contact Between Rough Surfaces,” Tribol. Trans., 42, No. 3, pp. 581–591.
Liu,  S. B., and Wang,  Q., 2001, “A Three-Dimensional Thermomechanical Model of Contact Between Non-Conforming Rough Surfaces,” ASME J. Tribol., 123, pp. 17–26.
Morrison, N., 1994, Introduction to Fourier Analysis, John Wiley and Sons, Inc., New York.
O’Sullivan,  T. C., and King,  R. B., 1988, “Sliding Contact Stress Field Due to a Spherical Indenter on a Layered Elastic Medium,” ASME J. Tribol., 110, pp. 235–240.
Nogi,  T., and Kato,  T., 1997, “Influence of a Hard Surface Layer on the Limit of Elastic Contact; Part I—Analysis Using a Real Surface Model,” ASME J. Tribol., 119, pp. 493–500.
Hu,  Y. Z., Barber,  G. C., and Zhu,  D., 1999, “Numerical Analysis for the Elastic Contact of Real Rough,” Tribol. Trans., 42, No. 3, pp. 443–452.
Plumet,  S., and Dubourg,  M. C., 1998, “A 3-D Model for a Multilayered Body Loaded Normally and Tangentially Against a Rigid Body: Application to Specific Coatings,” ASME J. Tribol., 120, pp. 668–676.
Ai,  X. L., and Sawamiphakdi,  K., 1999, “Solving Elastic Contact Between Rough Surfaces as an Unconstrained Strain Energy Minimization by Using CGM and FFT Techniques,” ASME J. Tribol., 121, pp. 639–647.
Polonsky,  I. A., and Keer,  L. M., 2000, “A Fast and Accurate Method for Numerical Analysis of Elastic Layered Contacts,” ASME J. Tribol., 122, pp. 30–35.
Ju,  Y., and Farris,  T. N., 1996, “Spectral Analysis of Two-Dimensional Contact Problems,” ASME J. Tribol., 118, pp. 320–328.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992, Numerical Recipes in Fortran 77-The Art of Scientific Computing (second edition), Cambridge University Press, Cambridge, UK.
Liu,  S. B., Wang,  Q., and Liu,  G., 2000, “A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses,” Wear, 243, pp. 101–110.
Liu,  S. B., Rodgers,  M., Wang,  Q., and Keer,  L., 2000, “A Fast and Effective Method For Transient Thermoelastic Displacement Analyses,” ASME J. Tribol., 123, pp. 479–485.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Ahmadi, N., 1982, “Non-Hertzian Normal and Tangential Loading of Elastic Bodies in Contact,” Ph.D. dissertation, Northwestern University, Evanston, IL.
Kalker,  J. J., 1986, “Numerical Calculation of the Elastic Field in a Half-Space,” Commun. Appl. Numer. Methods, 2, pp. 401–410.
Hills, D. A., Nowell, D., and Sackfield, A., 1993, Mechanics of Elastic Contacts, Butterworth Heinemann Ltd., Oxford.
Sackfield,  A., and Hills,  D. A., 1983, “Research Note: A Note on the Hertz Contact Problem: A Correlation of Standard Formulas,” J. Strain. Anal., 18, No. 3, pp. 195–197.
Burmister,  D. M., 1945, “The General Theory of Stresses and Displacements in Layered Systems,” J. Appl. Phys., 16, pp. 89–94.
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Figures

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Elastic bodies and notations: (a) a halfspace; and (b) a halfspace with a single coating layer
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Discretization for computation: (a) in the z or zr direction; and (b) N1×N2 grid points on the x-y plane with z=zl or z=zrl
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The aliasing phenomenon in the frequency domain: (a) a spatial function with its discrete series; and (b) FT,FFT, the aliasing phenomenon, and the wrap-around order
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The aliasing phenomenon in the spatial domain (L=4,N=64; function f̃(m)=2/(1+m2) is illustrated, whose values outside the region [−201,201] are small enough to be neglected): (a) the function, its discrete series, and wrap-around order in the frequency domain; and (b) the spatial function f(x)=e−|x| and the aliasing phenomenon
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The Gibbs phenomenon shown by a rectangular pulse: (Dashed line: By conversion with AL=0,N=256; Solid line: By a Fourier series with 256 terms; and Bold solid line: By conversion with AL=8,N=256)
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The von Mises stress J2/P0 (x-z plane with y=0): (a) pressure loading; and (b) shear loading
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Relative errors of the Von Mises stress by comparing the solutions from the DC-FFT algorithm with those from Ref. 17: (a) x-z plane at y=0, pressure loading; (b) x-z plane at y=0, shear loading; (c) x-y plane at z=2/63 aH, pressure loading; and (d) x-y plane at z=2/63 aH, shear loading
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The von Mises stress J2/P0 (x-z plane at y=0): (a) μf=0.25; and (b) μf=0.5
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A halfspace with a rough surface in contact (RMS roughness=0.21 μm): (a) pressures distribution (H=1.8 GPa); and (b) the von Mises stress in the halfspace, μf=0.25
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A layered halfspace with a smooth surface in contact: (a) contact pressure profiles for different layers; (b) the von Mises stress J2/P0f=0.25,E1=2E2; and (c) the Von Mises stress J2/P0f=0.5,E1=E2/2
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The von Mises stress distribution in a layered halfspace (the loading is given in Fig. 9(a))

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