Stress Analysis of Layered Elastic Solids With Cracks Using the Fast Fourier Transform and Conjugate Gradient Techniques

[+] Author and Article Information
I. A. Polonsky, L. M. Keer

Department of Civil Engineering, Northwestern University, Evanston, IL 60208-3109

J. Appl. Mech 68(5), 708-714 (Mar 13, 2001) (7 pages) doi:10.1115/1.1381394 History: Received August 21, 2000; Revised March 13, 2001
Copyright © 2001 by ASME
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Schulz,  H., and Quinto,  D. T., 1999, “Technolgical Development of PVD Hard Coatings for Industry,” Z. Metallkd., 90, pp. 831–836.
Suchentrunk,  R., Fuesser,  H. J., Staudigl,  G., Jonke,  D., and Meyer,  M., 1999, “Plasma Surface Engineering—Innovative Processes and Coating Systems for High-Quality Products,” Surf. Coat. Technol., 112, pp. 351–357.
Ju,  Y., and Farris,  T. N., 1996, “Spectral Analysis of Two-Dimensional Contact Problems,” ASME J. Tribol., 118, pp. 320–328.
Polonsky,  I. A., Chang,  T. P., Keer,  L. M., and Sproul,  W. D., 1997, “An analysis of the Effect of Hard Coatings on Near-Surface Rolling Contact Fatigue Initiation Induced by Surface Roughness,” Wear, 208, pp. 204–219.
Nogi,  T., and Kato,  T., 1997, “Influence on 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.
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.
Chen,  W. T., and Engel,  P. A., 1972, “Impact and Contact Stress Analysis in Multilayer Media,” Int. J. Solids Struct., 8, pp. 1257–1281.
Chiu,  Y. P., and Hartnett,  M. J., 1983, “A Numerical Solution for Layered Solid Contact Problems With Applications to Bearings,” ASME J. Lubr. Technol., 105, pp. 585–590.
Cole,  S. J., and Sayles,  R. S., 1992, “A Numerical Model for the Contact of Layered Elastic Bodies With Real Rough Surfaces,” ASME J. Tribol., 114, pp. 334–340.
Kubo,  A., Okamoto,  T., and Kurokawa,  N., 1981, “Contact Stress Between Rollers With Surface Irregularity,” ASME J. Mech. Des., 103, pp. 492–498.
Francis,  H. A., 1983, “The Accuracy of Plane Strain Models for the Elastic Contact of Three-Dimensional Rough Surface,” Wear, 85, pp. 239–256.
Ren,  N., and Lee,  S. C., 1993, “Contact Simulation of Three-Dimensional Rough Surfaces Using Moving Grid Method,” ASME J. Tribol., 115, pp. 597–601.
Polonsky,  I. A., and Keer,  L. M., 1999, “A Numerical Method for Solving Rough Contact Problems Based on the Multi-Level Multi-Summation and Conjugate Gradient Techniques,” Wear, 231, pp. 206–219.
Ziegele,  H., Rebholz,  C., Voevodin,  A. A., Leyland,  A., Rohde,  S. L., and Matthews,  A., 1997 “Studies of the Tribological and Mechanical Properties of Laminated CrC-SiC Coatings Produced by rf and dc Sputtering,” Tribol. Int., 30, pp. 845–856.
Polonsky,  I. A., Chang,  T. P., Keer,  L. M., and Sproul,  W. D., 1998, “A Study of Rolling Contact Fatigue of Bearing Steel Coated With PVD TiN Films: Coating Response to Cyclic Contact Stress and Physical Mechanisms Underlying Coating Effect on the Fatigue Life,” Wear, 215, pp. 191–204.
Moulinec,  H., and Suquet,  P., 1994, “A Fast Numerical Method for Computing the Linear and Nonlinear Mechanical Properties of Composites,” C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers, 318, No. 2, pp. 1417–1423.
Moulinec,  H., and Suquet,  P., 1998, “A Numerical Method for Computing the Overall Response of Nonlinear Composites with Complex Microstructures” Comput. Methods Appl. Mech. Eng., 157, pp. 69–94.
Herrmann,  K. P., Muller,  W. H., and Neumann,  S., 1999, “Linear and Elastic-Plastic Fracture Mechanics Revisited by use of Fourier Transforms—Theory and Application,” Comput. Mater. Sci., 16, pp. 186–196.
Tian,  H., and Saka,  N. J., 1992, “Finite-Element Analysis of Interface Cracking in Sliding Contacts,” Wear, 155, pp. 163–182.
Eberhardt,  A. W., and Kim,  B. S., 1998, “Crack Face Friction Effects on Mode II Stress Intensities for a Surface-Cracked Coating in Two-Dimensional Rolling Contact,” Tribol. Trans., 41, pp. 35–42.
Souza,  R. M., Mustoe,  G. G. W., and Moore,  J. J., 1999, “Finite-Element Modeling of the Stresses and Fracture During the Indentation of Hard Elastic Films on Elastic-Plastic Aluminum Substrates,” Thin Solid Films, 356, pp. 303–310.
Lin,  W., and Keer,  L. M., 1989, “Analysis of a Vertical Crack in a Multilayered Medium,” ASME J. Appl. Mech., 56, pp. 63–69.
Lin,  W., and Keer,  L. M., 1989, “Three-Dimensional Analysis of Cracks in Layered Transversely Isotropic Media,” Proc. R. Soc. London, Ser. A, A424, pp. 307–322.
Kuo,  C. H., and Keer,  L. M., 1995, “Three-Dimensional Analysis of Cracking in a Multilayered Composite,” ASME J. Appl. Mech., 62, pp. 273–281.
Mura, T., 1982, Micromechanics of Defects in Solids, Kluwer, Dordrecht.
Murakami,  Y., and Nemat-Nasser,  S., 1983, “Growth and Stability of Interacting Surface Flaws of Arbitrary Shape,” Eng. Fract. Mech., 17, pp. 193–210.
Lee,  J. C., Farris,  T. M., and Keer,  L. M., 1987, “Stress Intensity Factors for Cracks of Arbitrary Shape Near an Interfacial Boundary,” Eng. Fract. Mech., 27, pp. 27–41.
Hanson,  M. T., Lin,  W., and Keer,  L. M., 1989, “Three-Dimensional Analysis of Cracking Through the Boundary of a Two-Phase Material,” ASME J. Appl. Mech., 56, pp. 850–857.
Mura,  T., 1964, “Periodic Distributions of Dislocations,” Proc. R. Soc. London, Ser. A, A280, pp. 528–544.
Chen,  W. T., 1971, “Computation of Stresses and Displacements in a Layered Elastic Medium,” Int. J. Eng. Sci., 9, pp. 775–800.
Gilbert,  F., and Backus,  G., 1966, “Propagator Matrices in Elastic Wave and Vibration Problems,” Geophysics, 31, pp. 326–332.
Pan,  E., 1991, “An Exact Solution for Transversely Isotropic, Simply Supported and Layered Rectangular Plates,” J. Elast., 25, pp. 101–116.
Moran,  B., and Shih,  C. F., 1987, “Crack Tip and Associated Domain Integrals From Momentum and Energy-Balance,” Eng. Fract. Mech., 27, pp. 615–642.
Murakami, Y., 1986, Stress Intensity Factors Handbook, 1st Ed., Pergamon, Oxford, UK.
Hutchinson,  J. W., 1987, “Crack Tip Shielding by Micro-Cracking in Brittle Solids,” Acta Metall. Mater., 35, pp. 1605–1619.


Grahic Jump Location
Layered solid containing contact-induced cracks (shown as thick black lines). Counterpart roughness is exaggerated. The y-direction is normal to the picture plane.
Grahic Jump Location
Discretization of the cracked layer. Grid nodes are shown as circles. Nodes carrying eigenstrain are filled.
Grahic Jump Location
Crack-opening displacement distribution along a crack radius for a penny-shaped crack: numerical solution (diamonds) and analytical solution (solid line)
Grahic Jump Location
Multiple cracks in a thin coating: (x,z) view (a) and (y,z) view (b)
Grahic Jump Location
Distribution of the normal stress σxx in the main crack plane for tensile loading; single crack case (a) and multiple crack cases (b)
Grahic Jump Location
Distribution of the shear stress σxz in the main crack plane for contact loading: single crack case (a) and multiple crack case (b)




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