Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



An integral equation approach to boundary value problems of classical elastostatics

Author: Frank J. Rizzo
Journal: Quart. Appl. Math. 25 (1967), 83-95
DOI: https://doi.org/10.1090/qam/99907
MathSciNet review: QAM99907
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Abstract | References | Additional Information

Abstract: The analogy between potential theory and classical elasticity suggests an extension of the powerful method of integral equations to the boundary value problems of elasticity. A vector boundary formula relating the boundary values of displacement and traction for the general equilibrated stress state is derived. The vector formula itself is shown to generate integral equations for the solution of the traction, displacement, and mixed boundary value problems of plane elasticity. However, an outstanding conceptual advantage of the formulation is that it is not restricted to two dimensions. This distinguishes it from the methods of Muskhelishvili and most other familiar integral equation methods. The presented approach is a real variable one and is applicable, without inherent restriction, to multiply connected domains. More precisely, no difficulty of the order of determining a mapping function is present and unwanted Volterra type dislocation solutions are eliminated a priori. An indication of techniques necessary to effect numerical solution of the resulting integral equations is presented with numerical data from a set of test problems.

References [Enhancements On Off] (What's this?)

  • [1] E. Betti, Il Nuovo Cimento 6-10, (1872)
  • [2] C. Somigliana, Il Nuovo Cimento 17-20, (1885)
  • [3] G. Lauricella, Sur l'intégration de l'équation relative à l'equilibre des plaques élastiques encastrées, Acta Math. 32, (1909)
  • [4] I. Fredholm, Solution d'un problème fondamental de la theorie de l'elasticité, Arch. Mat. Ast[ill]onom. Fysik 2, (1905)
  • [5] D. I. Sherman, Dokl. Akad. Nauk SSSR 27-28, (1940); 32, (1941)
  • [6] S. G. Mikhlin, Integral equations and their applications to certain problems in mechanics, mathematical physics and technology, Pergamon Press, New York-London-Paris-Los Angeles, 1957. Translated from the Russian by A. H. Armstrong. MR 0087877
  • [7] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, P. Noordhoff, Ltd., Groningen, 1953. Translated by J. R. M. Radok. MR 0058417
  • [8] M. A. Jaswon, Integral equation methods in potential theory. I, Proc. Roy. Soc. Ser. A 275 (1963), 23–32. MR 0154075
  • [9] M. A. Jaswon and A. R. Ponter, An integral equation solution of the torsion problem, Proc. Roy. Soc. Ser. A 273 (1963), 237–246. MR 0149729
  • [10] G. T. Symm, Integral equation methods in potential theory. II, Proc. Roy. Soc. Ser. A 275 (1963), 33–46. MR 0154076
  • [11] A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
  • [12] N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
  • [13] William Duncan MacMillan, The theory of the potential, MacMillan’s Theoretical Mechanics, Dover Publications, Inc., New York, 1958. MR 0100172

Additional Information

DOI: https://doi.org/10.1090/qam/99907
Article copyright: © Copyright 1967 American Mathematical Society

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