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A three-dimensional parabolic punch problem in linear elasticity

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Abstract

In this paper an analytic solution for a three-dimensional contact problem, in linear elasticity, is constructed through the separation of Laplace's equation in paraboloidal coordinates. A rigid punch under normal loading is applied to an isotropic elastic medium occupying an infinite half-space where the contact region is parabolic and the punch profile is prescribed. This treatment allows for a general punch profile provided it is physically reasonable so as to ensure the convergence of the solution.

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References

  1. F.M. Arscott, Paraboloidal co-ordinates and Laplace's equation, Proceedings of the Royal Society of Edinburgh 66 (1964) 129–139.

    Google Scholar 

  2. F.M. Arscott and A. Darai, Curvilinear co-ordinate systems in which the Helmholtz equation separates, I.M.A. Journal of Applied Mathematics 27 (1981) 33–70.

    Google Scholar 

  3. F.M. Arscott, R. Lacroix and W.T. Shymanski, A three-term recursion and the computation of Mathieu functions, Proc. 8th Manitoba Conference on Numerical Mathematics and Computing (1978) 107–115.

  4. A. Darai and F.M. Arscott, A potential problem arising from the strip-punch problem in elasticity, J. Engg. Maths. 22 (1988) 203–223.

    Google Scholar 

  5. A. Darai, Applications of Higher Special Functions to Some Three-Dimensional Contact Problems in the Classical Theory of Elasticity, Ph.D. Thesis, University of Manitoba (1985).

  6. A.H. England, Complex Variable Methods in Elasticity, Wiley-Interscience (1971).

  7. A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of Integral Transforms, Vol. 1, McGraw-Hill Book Company, Inc. (1954).

  8. G.M.L. Gladwell, Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff (1980).

  9. J.J. Kalker and Y. Van Randen, A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems, J. Engg. Maths. 6 (1972) 193–206.

    Google Scholar 

  10. M.K. Kassir, The distribution of stress around a flat parabolic crack in an elastic solid, Engg. Frac. Mech. 2 (1971) 373–385.

    Google Scholar 

  11. Macsyma Reference Manual Version 13, Symbolics Inc. (1988).

  12. N.W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press (1947).

  13. R.C. Shah and A.S. Kobayashi, On the parabolic crack in an elastic solid, Engg. Frac. Mech. 1 (1968) 309–325.

    Google Scholar 

  14. R. Shail, Lamé polynomial solutions to some elliptic crack and punch problems, Int. Journal of Engg. Sci. 16 (1978) 551–563.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Western Illinois University, 61455, Macomb, IL, USA

    A. Darai

  2. Department of Applied Mathematics, University of Manitoba, R3T 2N2, Winnipeg, Canada

    F. M. Arscott

Authors
  1. A. Darai
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  2. F. M. Arscott
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Darai, A., Arscott, F.M. A three-dimensional parabolic punch problem in linear elasticity. J Eng Math 24, 369–383 (1990). https://doi.org/10.1007/BF00130689

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  • DOI: https://doi.org/10.1007/BF00130689

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