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Abstract

When a smooth rigid punch of arbitrary shape is pressed onto an elastic body, an area of contact D is formed within which the contact pressure p is non-negative. Outside of the contact area D, p = 0. The problem is to find the contact pressure p and the shape of the contact region D for a given depth of indentation, or a given total force applied to the punch. In general, the governing equation can be reduced to the following integral equation:
$$w\left( {{x_1},{x_2}} \right) = \iint\limits_D {\left( {{x_1},{x_2},{\xi _1},{\xi _2}} \right)p\left( {{\xi _1},{\xi _2}} \right)d{\xi _1}d{\xi _2}}$$
(1a)
in which K(x 1, x 2, ξ1 ξ1) is known and w is given by
$$w\left( {{x_1},{x_2}} \right) = \alpha - f\left( {{x_1},x{}_2} \right)$$
(1b)
where a is the depth of indentation and f(x 1, x 2) is the function which describes the shape of the punch. Unless K and f are simple in form, the solutions of eq. [1] for p and D in general require a numerical approximation. Since the physical observation demands that p > 0 inside D and p = 0 outside D, the method of linear programming is probably the most effective one in solving this problem numerically. The purpose of this paper is to show that not only the contact problem of a rigid punch on an elastic body yields eq. [1], many contact problems of a rigid punch on a viscoelastic body and the contact problems of a moving punch on a viscoelastic body also reduce to eq. [1]. It should be noticed that depending on the shape of the punch, D may not be a simply-connected region. It should also be mentioned that for a viscoelastic body, D is in general not a constant but varies with time even in the case of a punch which is held stationary on the surface of a viscoelastic body.

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References

  1. 1).
    Sokolnikoff, I. S., Mathematical Theory of Elasticity (New York 1956).Google Scholar
  2. 2).
    Sneddon, I. N., Fourier Transforms (New York 1951).Google Scholar
  3. 3).
    Ting, T. C. T., J. Appl. Mech. 35, 248–254 (1968).ADSCrossRefzbMATHGoogle Scholar
  4. 4).
    Ting, T. C. T., J. Appl. Mech. 33, 845–854 (1966).ADSCrossRefzbMATHGoogle Scholar
  5. 5).
    Lee, E. H., Viscoelastic Stress Analysis, Structural Mechanics, Proceedings of the First Symposium on Naval Structural Mechanics (New York 1960).Google Scholar
  6. 6).
    Lee, E. H. and J. R. M. Radok, J. Appl. Mech. 27, 438–444(1960).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7).
    Hunter, S. C., J. Mech. Physics Solids 8, 219–234 (1960).ADSCrossRefzbMATHGoogle Scholar
  8. 8).
    Graham, G. A. C., Int. J. Eng. Sci., 3, 27–45 (1965).CrossRefzbMATHGoogle Scholar
  9. 9).
    Graham, G. A. C., Int. J. Eng. Sci. 5, 495–514 (1967).CrossRefzbMATHGoogle Scholar
  10. 10).
    Ting, T. C. T. and C. H. Wu, J. Appl. Mech. 39, 461–468 (1972).CrossRefzbMATHGoogle Scholar
  11. 11).
    Efimov, A.B., Vestnik Moskovskogo Universiteta, Seriya 1, Matematika-Mekhanika, No. 2, 120–127 (1966).MathSciNetGoogle Scholar
  12. 12).
    Tsai, Y. M., Q. Appl. Math. 27, 371–380 (1969).zbMATHGoogle Scholar
  13. 13).
    Hunier, S. C., J. Appl. Mech. 28, 611–617 (1961).ADSCrossRefGoogle Scholar
  14. 14).
    Morland, L. W., J. Appl. Mech. 29, 345–352 (1962).ADSCrossRefzbMATHGoogle Scholar
  15. 15).
    Yang, W. H., J. Appl. Mech. 33, 395–401 (1966).ADSCrossRefGoogle Scholar
  16. 16).
    Christensen, R. M., Theory of Viscoelasticity (New York 1971).Google Scholar
  17. 17).
    Gurtin, M. E. and E. Sternberg, Arch. Rat. Mech. Analysis 11, 291–356 (1962).ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • T. C. T. Ting
    • 1
  1. 1.Dept. of Materials EngineeringUniversity of IllinoisChicagoUSA

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