Abstract
The dominant part of an integral equation arising in connection with boundary value problems for the circular disc is evaluated in terms of orthogonal polynomials. This relation leads to an efficient method for numerical solution of the complete integral equation even in the presence of a complicated bounded kernel. The static problem of a circular crack in an infinite elastic body under general loads is used to illustrate vector boundary conditions leading to two coupled integral equations, while the problem of a vibrating flexible circular plate in frictionless contact with an elastic half space is solved by use of the associated numerical method.
Similar content being viewed by others
References
I.N. Sneddon,Mixed boundary value problems in potential theory, North-Holland, Amsterdam, 1966.
C.J. Tranter,Integral transforms in mathematical physics, Chapman and Hall, London, 1971.
L.S. Fu and T.P. Tsai, "A numerical scheme for mixed boundary value problems in elasticity", Computers and Structures, Vol. 8, 41–49, 1978.
G.Ia. Popov, "Some properties of classical polynomials and their application to contact problems", Journal of Applied Mathematics and Mechanics, Vol. 27, 1255–1271, 1963.
M. Abramowitz and I. Stegun,Handbook of mathematical functions, Dover, New York, 1965.
G.N. Watson,Theory of Bessel functions, Cambridge University Press, 1966.
A. Erdelyi, Ed.,Higher transcendental functions, McGraw-Hill, New York, 1953.
G.M.L. Gladwell, "Polynomial solutions for an ellipse on an anisotropic elastic half-space", Quarterly Journal of Mechanics and Applied Mathematics, Vol. 31, 251–260, 1978.
M.K. Kassir and G.C. Sih,Three-dimensional crack problems, Noordhoff, Leyden, 1975.
G.B. Jeffery, "The relations between spherical, cylindrical and spheroidal harmonics", Proceedings of the London Mathematical Society, Vol. 16, 133–139, 1917.
S. Krenk, "A circular crack under asymmetric loads and some related integral equations", Journal of Applied Mechanics, Vol. 46, 821–826, 1979.
R.A. Westmann, "Simultaneous pairs of dual integral equations", SIAM Review, Vol. 7, 341–348, 1965.
F. Erdogan and L.Y. Bahar, "On the solution of simultaneous dual integral equations", Journal of SIAM, Vol. 12, 666–675, 1964.
S. Krenk,Polynomial solutions to singular integral equations, Dissertation, Risø National Laboratory, January 1981.
S. Krenk and H. Schmidt, "Vibration of an elastic circular plate on an elastic half space - A direct approach", Journal of Applied Mechanics, Vol. 48, 161–168, 1981.
H. Schmidt and S. Krenk, "Asymmetric vibrations of a circular elastic plate on an elastic half space", International Journal of Solids and Structures, Vol. 18, 91–105, 1982.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krenk, S. Some integral relations of Hankel transform type and applications to elasticity theory. Integr equ oper theory 5, 548–561 (1982). https://doi.org/10.1007/BF01694053
Issue Date:
DOI: https://doi.org/10.1007/BF01694053