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Numerical Treatment of Singularities in Elastic Contact Problems and Applications

  • Conference paper
Computational Mechanics ’86

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Abstract

New procedure is suggested for numerical and analytical treatment of elastic contact problems for non-classical domains. The numerical procedure is based on formulae derived for an accurate computer evaluation of singular integrals. These formulae define the value of the singular integral in the neighbourhood of the singularities while the regular part of the integral can be evaluated by any standard subroutine. The method allows practically exact solution of elastic contact problems for punches of various plan forms like, for example, rectangle, rhombus, triangle, oval, etc. The nature of the stress singularity at the boundary of the domain of contact is clarified. Numerical results presented for a rectangle indicate that the assumption of a square root singularity is wrong, and that the power of the singularity is not constant along a rectangle edge.

The new analytical approach is based on an integral representation for the kernel of the governing integral equation from which some simple yet accurate formulae are derived for the relationship between the applied forces and the genralized displacements of the punch. The method’s applicability is far beyond the field of elastic contact problems. It can be applied for the evaluation of the capacity of flat laminae, in the field of wave propagation it can be used for the evaluation the coefficients of electrical and magnetic polarizability, etc. Some specific examples of such use are given.

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References

  1. V.M. Aleksandrov and V.A. Babeshko, On the pressure on an elastic half-space by a wedge-shaped punch. Journal of Applied Mathematics and Mechanics (PMM), Vol. 36, 1972, pp. 78–83.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Z.P. Bažant, Three-dimensional harmonic functions near termination or intersection of gradient singularity lines: a general numerical method. International Journal of Solids and Structures, Vol. 12, 1974, pp. 221–243.

    MATH  Google Scholar 

  3. B.S Berger and P.S. Bernard, The numerical integration of Green’s functions. Trans. ASME, Journal of Applied Mechanics, Vol. 50, 1983, pp. 456–459.

    Article  ADS  MATH  Google Scholar 

  4. N.M. Borodachev, Contact problem for a stamp with a rectangular base. Journal of Applied Mathematics and Mechanics (PMM), Vol. 40, 1976 pp. 505–512.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. S.B. Cohn, Determination of aperture parameters by electrolitic-tank measurements. Proc. IRE, Vol. 39, 1951, pp. 1416–1421.

    Article  Google Scholar 

  6. S.B. Cohn, The electric polarizability of apertures of arbitrary shape. Proc. of the IRE, Vol. 40, 1952, pp. 1069–1071.

    Article  Google Scholar 

  7. E.T. Copson, On the problem of the electrified disc. Proc. Edinburgh Math. Soc., Vol. 8, 1947, PP. 14–19.

    Article  MathSciNet  Google Scholar 

  8. P.J. Davis and P. Rabinowitz, Methods of numerical integration. Academic Press, New York, 1975.

    MATH  Google Scholar 

  9. H. Engels, Numerical quadrature and cubature. Academic Press, New York, 1980.

    MATH  Google Scholar 

  10. V.I. Fabrikant, Effect of shearing force and tilting moment on a cylindrical punch attached to a transversely isotropic half-space. Journal of Applied Mathematics and Mechanics (PMM), Vol. 35, 1971, pp. 147–151.

    Article  ADS  MATH  Google Scholar 

  11. V.I. Fabrikant, Closed forms solution of a two-dimensional integral equation. Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, No. 2, 1971, pp. 102–104, (in Russian).

    MathSciNet  Google Scholar 

  12. R.G. Gabdulkhaev, Cubature formulas for multi-dimensional singular integrals. Izvestiya VUZ’ov, Matematika, Vol. 19, No. 4, 1975, pp. 3–13 (in Russian).

    Google Scholar 

  13. L.A Galin,Contact Problems in the Theory of Elasticity (in Russian), Gostekhteorizdat, 1953. Translated by H. Moss, Dept. of Math., North Carolina State College, Raleigh, 1961.

    Google Scholar 

  14. G.W.O. Howe, The capacity of rectangular plates and a suggested formula for the capacity of aerials. The Radio Review, Vol. 1, Oct. 1919 – June 1920, pp. 710–714.

    Google Scholar 

  15. L. Kraus, Diffraction by a Plane Angular Sector. Ph.D. Thesis, New York University, 1955.

    Google Scholar 

  16. E.A. Kraut, Diffraction of elastic waves by a rigid 90° wedge. Bull. Seism, Soc. Am., Vol. 58, 1968, pp. 1083–1115.

    Google Scholar 

  17. J.A. Liggett, Singular cubature over triangles. International Journal for Numerical Methods in Engineering, Vol. 18, 1982, pp. 1375–1384.

    Article  ADS  MATH  Google Scholar 

  18. F. De Meulenaere and J. Van Bladel, Polarizability of some small apertures. IEEE Trans, on Antennas and Propagation, Vol. AP-25, 1977, pp. 198–205.

    Article  ADS  Google Scholar 

  19. B. Noble, The numerical solution of the singular integral equation for the charge distribution on a flat rectangular lamina. Sympos. Numerical Treatment of Ordinary Differential Equations, Integral and Integrodifferential Equations. (Proc. Rome Sympos. 20–24 September 1960) Birkhauser, Berlin-Stuttgart, 1960.

    Google Scholar 

  20. B. Noble, The potential and charge distribution near the tip of a flat angular sector. NYU EM-135, July, 1959.

    Google Scholar 

  21. E.E. Okon and R.F. Harrington, The polarizabilities of electrically small apertures of arbitrary shape. IEEE Trans. on Electromagnetic Compatibility, Vol. EMC-23, 1981, pp.359–366.

    Article  ADS  Google Scholar 

  22. E.E. Okon and R.F. Harrington, The potential integral for a linear distribution over a triangular domain. International Journal for Numerical Methods in Engineering, Vol. 18, 1982, pp. 1821–1828.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. E.E. Okon and R.F. Harrington, The capacitance of discs of arbitrary shape. Tech. Rep. No.10, Contract No. N00014–76-C-0225. US Dep. Navy, Office of Naval Research, Rep. No. TR-79–3, April, 1970.

    Google Scholar 

  24. J. Radlow, Arch. Ration. Mechanics and Analysys, Vol. 19, 1965, pp.62-

    MathSciNet  ADS  MATH  Google Scholar 

  25. D.K. Reitan and T.J. Higgins, Accurate determination of the capacitance of a thin rectangular plate. Trans. AIEE, Vol. 75, pt. 1, 1957, pp. 761–766.

    Google Scholar 

  26. V.L. Rvachev, On the pressure on elastic half-space of a punch which has the planform of a wedge. Prikl. Mat. i Mekhanika (in Russian), Vol. 23, 1959, pp. 169–178.

    MathSciNet  Google Scholar 

  27. L. Solomon, Upon the geometrical punch-penetration rigidity. Lincei-Rend. Sc. fis. mat. e nat., Vol. 36, 1964, pp. 832–835.

    Google Scholar 

  28. L. Solomon, Une solution approchée du problème du poinçon rigide a base plane bornée convexe non elliptique. Compt. Rend. Acad. Sc. Paris, Vol. 258, 1964, pp. 64–66.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Concordia University, Montreal, H3G 1M8, Canada

    V. I. Fabrikant & T. S. Sankar

Authors
  1. V. I. Fabrikant
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  2. T. S. Sankar
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Editor information

Editors and Affiliations

  1. Department of Nuclear Engineering, The University of Tokyo, 3-1, Hongo 7-chome, Bunkyo-ku, Tokyo 113, Japan

    Genki Yagawa

  2. Center for the Advancement of Computational Mechanics, School of Civil Engineering, Georgia Institute of Technology, 225, North Avenue, Atlanta, GA, 30332, USA

    Satya N. Atluri

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© 1986 Springer Japan

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Fabrikant, V.I., Sankar, T.S. (1986). Numerical Treatment of Singularities in Elastic Contact Problems and Applications. In: Yagawa, G., Atluri, S.N. (eds) Computational Mechanics ’86. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68042-0_57

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  • DOI: https://doi.org/10.1007/978-4-431-68042-0_57

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-68044-4

  • Online ISBN: 978-4-431-68042-0

  • eBook Packages: Springer Book Archive

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