The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically
Similar content being viewed by others
References
T. B. Berezyuk, A. Ya. Grigorenko, and I. I. Dyyak, “Solving the stress problem for a finite cylinder by the Schwarz method with hybrid approximations,” Int. Appl. Mech., 39, No. 10, 1173–1178 (2003).
N. M. Borodachev, “Construction of exact solutions to three-dimensional elastic problems in stresses,” Int. Appl. Mech., 37, No. 6, 762–769 (2001).
A. N. Guz, “Design models in linearized solid mechanics,” Int. Appl. Mech., 40, No. 5, 506–516 (2004).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill Book Company (1968).
A. S. Kosmodamianskii and V. A. Shaldyrvan, Multiply Connected Thick Plates [in Russian], Naukova Dumka, Kyiv (1978).
Yu. A. Krutkov, Tensor of Stress Function and General Solutions in Static Elasticity [in Russian], Izd. AN SSSR, Moscow–Leningrad (1949).
A. I. Lurie, Theory of Elasticity, Springer, Berlin (2005).
O. V. Onishchuk and A. V. Tolkachev, “Complete system of polynominal solutions of the differential equations of elasticity,” Visn. Odesk. Derzh. Univ., Fiz.-Mat. Nauky, 4, No. 4, 148–153 (1999).
V. P. Revenko, “General solution of the three-dimensional Lamé equations of elasticity,” Dop. NAN Ukrainy, No. 7, 39–45 (2005).
V. P. Revenko, “Principal stress–strain state of an elastic cylinder,” Dop. NAN Ukrainy, No. 8, 49–55 (2005).
V. P. Revenko, “General solution of the three-dimensional Lamé equations in curvilinear coordinates,” Dop. NAN Ukrainy, No. 4, 49–55 (2006).
V. G. Rekach, Solving Problems of Elasticity: A Manual [in Russian], Vysshaya Shkola, Moscow (1977).
M. G. Slobodyanskii, “General solutions expressed in terms of harmonic functions to the equations of elasticity for simply connected and multiply connected domains,” Prikl. Mat. Mekh., 18, No. 1, 55–74 (1954).
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970).
A. F. Ulitko, Vector Series in the Three-Dimensional Theory of Elasticity [in Russian], Akadempereodika, Kyiv (2002).
L. I. Fridman, “General solutions of elastic problems and boundary-value problems,” Prikl. Mat. Mekh., 65, No. 2, 268–278 (2001).
F. S. Churikov, “A general solution to the equilibrium equations for displacements in the theory of elasticity,” Prikl. Mat. Mekh., 17, No. 6, 751–754 (1953).
K. T. Chau and X. X. Wei, “Finite solid circular cylinders subjected to arbitrary surface load. Part I: Analytic solution,” Int. J. Solids Struct., 37, 5707–5732 (2000).
Ya. M. Grigorenko and Ya. M. Avramenko, “Influence of geometrical and mechanical parameters on the stress–strain state of closed nonthin conical shells,” Int. Appl. Mech., 44, No. 10, 1119–1127 (2008).
A. Ya. Grigorenko, I. I. Dyyak, and I. I. Prokopyshin, “Domain decomposition method with hybrid approximations applied to solve problems of elasticity,” Int. Appl. Mech., 44, No. 11, 1213–1222 (2008).
Ya. M. Grigorenko, “Nonconventional approaches to static problems for noncircular cylindrical shells in different formulations,” Int. Appl. Mech., 43, No. 1, 35–53 (2007).
H. Neuber, “Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie,” ZAMM, 14, No. 4, 203–212 (1934).
P. F. Papkovich, “Solution gènèrale des èquations diffèrentielles fondamentales de èlasticitè, exprimèe par trois fonctiones harmoniques,” C. R. Acad. Sci. Paris, 195, No. 3, 513–515 (1932).
V. P. Revenko, “Numerical-analytical method to determine the stress state of an elastic rectangular plate,” Int. Appl. Mech., 44, No. 1, 73–80 (2008).
C. K. Youngdahl, “On the completeness of a set of stress functions appropriate to the solution of elasticity problems in general cylindrical coordinates,” Int. J. Eng. Sci., 7, 61–79 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 45, No. 7, pp. 52–65, July 2009.
Rights and permissions
About this article
Cite this article
Revenko, V.P. Solving the three-dimensional equations of the linear theory of elasticity. Int Appl Mech 45, 730–741 (2009). https://doi.org/10.1007/s10778-009-0225-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-009-0225-4