Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T14:00:36.990Z Has data issue: false hasContentIssue false
  • English
  • Français

An eigenvalue problem for elastic contact with finite friction

Published online by Cambridge University Press:  24 October 2008

D. A. Spence
D. A. Spence
Affiliation:
Mathematics Research Center, University of Wisconsin
Get access Rights & Permissions [Opens in a new window]

Abstract

The two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction μ between the surfaces. The contact area is then divided into an inner adhesive region −c < x < c in which the surface displacements are known, surrounded by regions c < |x| < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses σ and τ at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity π proportional to the difference τ – μσ in the adhesive region. The largest eigenvalue of this equation, for which π > 0, gives the adhesive boundary c in terms of μ and Poisson's ratio ν. A similarity transformation shows that c has the same value for both flat-faced and power law punches.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 249 - 268
Copyright
Copyright © Cambridge Philosophical Society 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Byrd, P. F. and Friedman, M. D.Handbook of elliptic integrals for engineers and physicieta (Berlin, Springer-Verlag, 1953).Google Scholar
(2)Carlemann, T.Sur la resolution de certaines equations integrales. Arkiv för Matematik, Astronomi och Fusik Band 16 (1922), No. 26.Google Scholar
(3)Erdelyi, A. (editor). Higher transcendental functions, vol. 1 (New York, McGraw-Hill, 1953).Google Scholar
(4)Galin, L. A.Akad. Nauk SSSR. Prikl. Mat. Mech. 9 (1945), 413–424; also in Contact problems in the theory of elasticity, ed. Sneddon (North Carolina State College Dep. Math. Report, 1961).Google Scholar
(5)Minlin, R. D.J. Appl. Mech. 16 (1949), 259–268.CrossRefGoogle Scholar
(6)Goodman, L. E.Trans. ASME. 84 (1962), (E), 515–522.Google Scholar
(7)Muskhelishpili, N. I.Some basic problems of the mathematical theory of elasticity (Translated from the third Russian edition, 1949, by Radok, J. R. M.) (Groningen: Noordhoff, 1953).Google Scholar
(8)Spence, D. A.Proc. R. Soc. Ser. A 305 (1968), 55–80.Google Scholar
(9)Tricomi, F. G.Integral equations (New York: Interscience, 1957).Google Scholar
(10)Abramowitz, M. and Segun, I. A.Handbook of mathematical functions (New York: Dover, 1965).Google Scholar
(11)Spence, D. A. (1972). The Hertz contact problem with finite friction. U. Wisconsin, Math. Research Center TR 1209: presented at 13th Intl. Congress of Theoretical and Applied Mechanics, Moscow.Google Scholar