Abstract
This paper presents an exact analytical solution to the displacement boundary-value problem of elasticity for a torus. The introduced form of the general solution of elastostatics equations allows to solve exactly a broad class of boundary-value problems in coordinate systems with incomplete separation of variables in the harmonic equation. The original boundary-value problem for a torus is reduced to infinite systems of linear algebraic equations with tridiagonal matrices. An analytical technique for solving systems of diagonal form is developed. Uniqueness of the solutions of vector boundary-value problems involving the generalized Cauchy-Riemann equations is investigated, and it is shown that the obtained solution for the displacement boundary-value problem for a torus is unique due to the specific properties of the suggested general solution. The analogy between problems of elastostatics and steady Stokes flows is demonstrated, and the developed elastic solution is used to solve the Stokes problem for a torus.
Similar content being viewed by others
References
P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill (1953) 997 pp.
A. Wangerin, Ñber das Problem des Gleichgewichts elastischerRotationskörper. Arch. der Math. und Phys. 55 (1873) 113–146.
Yu. I. Soloviev, Theaxisymmetric problem of elasticity for a torus and a space with toroidal cave. Mekh. Tverd. Tela 3 (1969) 99–105 [in Russian].
Yu. N. Podil'chuk and V. S. Kirilyuk, Nonaxially symmetric deformation of atorus. Soviet Appl. Mech. 19 (1983) 743–748.
V. S. Kirilyuk, Stress concentration in anisotropic medium with an elastic toroidal inhomogeneity. Soviet Appl. Mech. 24 (1988) 11–14.
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. New York: Prentice Hall(1965) 556 pp.
W. H. Pell and L. E. Payne, On Stokes flow about a torus. Mathematika 7(1960) 78–92.
S. J. Wakiya, On the exact solution of the Stokes equations for a torus. J. Phys.Soc. Jap. 37 (1974) 780–783.
N. Lebedev, The functions associated with a ring of oval cross-section. Tech. Phys. 1 (1937) 3–24.
E. W. Hobson, Spherical and EllipsoidalHarmonics. Cambridge: University Press (1931) 500 pp.
H. Bateman and A. Erdelyi. HigherTranscendental Funcitons. Vol. 1. New York: McGraw-Hill (1953) 282 pp.
G. V. Kutsenko and A. F. Ulitko, An Exact Solution of the Axisymmetric problem of the theory of elasticity for a hollow ellipsoid of revolution. Soviet Appl. Mech. 11 (1975) 3–8.
F. D. Gakhov, Boundary Value Problems.New York: Pergamon Press (1966) 564 pp.
N. I. Muskhelishvili, Singular Integral Equations.Groningen: Noordhoff (1953) 447 pp.
G. G. Stokes, Mathematical and Physical Papers. Vol.1. Cambridge: University Press (1880) 352 pp.
Ghosh, S. On the steady motion of a viscousliquid due to translation of a tore parallel to its axis. Bull. of the Calcutta Math. Soc. 18 (1927) 185–194.
H. Takagi, Slow viscous flow due to the motion of a closed torus. J. Phys. Soc. Jap. 35(1973) 1225–1227.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krokhmal, P. Exact solution of the displacement boundary-value problem of elasticity for a torus. Journal of Engineering Mathematics 44, 345–368 (2002). https://doi.org/10.1023/A:1021253709644
Issue Date:
DOI: https://doi.org/10.1023/A:1021253709644