An integral equation approach to boundary value problems of classical elastostatics
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
G. Lauricella, Sur l’intégration de l’équation relative à l’equilibre des plaques élastiques encastrées, Acta Math. 32, (1909)
I. Fredholm, Solution d’un problème fondamental de la theorie de l’elasticité, Arch. Mat. Ast[ill]onom. Fysik 2, (1905)
D. I. Sherman, Dokl. Akad. Nauk SSSR 27–28, (1940); 32, (1941)
Mikhlin, S. G., 1957, Integral equations and their applications to certain problems in mechanics, mathematical physics and technology
Muskhelishvili, N. I., 1953, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending
Jaswon, M. A., 1963, Integral equation methods in potential theory. I, Proc. Roy. Soc. London Ser. A, 275, 23, 10.1098/rspa.1963.0152
Jaswon, M. A., 1963, An integral equation solution of the torsion problem, Proc. Roy. Soc. London Ser. A, 273, 237, 10.1098/rspa.1963.0085
Symm, G. T., 1963, Integral equation methods in potential theory. II, Proc. Roy. Soc. London Ser. A, 275, 33, 10.1098/rspa.1963.0153
Love, A. E. H., 1944, A treatise on the Mathematical Theory of Elasticity
Muskhelishvili, N. I., 1972, Singular integral equations
MacMillan, William Duncan, 1958, The theory of the potential