 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| E. V. Rozhkova, "On Solutions of the Problem in Stresses with the Use of Maxwell Stress Functions," Mech. Solids. 44 (4), 526-536 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
4 |
Pages |
526-536 |
| DOI |
10.3103/S0025654409040049 |
| Title |
On Solutions of the Problem in Stresses with the Use of Maxwell Stress Functions |
| Author(s) |
E. V. Rozhkova (Tashkent Institute of Railway Transport Engineers, Adylkhodzhieva 1, Mirabadskii r-n, Tashkent, 100167 Uzbekistan, rojkova-elena@mail.ru) |
| Abstract |
The spatial problems of elasticity are mainly solved in
displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method. |
| Keywords |
problem in stresses, compatibility equations, Saint-Venant identities |
| References |
| 1. | A. I. Lurie,
Spatial Problems of Theory of Elasticity
(Nauka, Moscow, 1955)
[in Russian]. |
| 2. | B. A. Bondarenko and A. N. Filatov,
Quasipolynomial Functions and Their Applications to Problems of Elasticity
(Fan, Tashkent, 1978)
[in Russian]. |
| 3. | B. E. Pobedrya,
"On the Problem in Stresses,"
Dokl. Akad. Nauk SSSR 240 (3), 564-567 (1978)
[Soviet Phys. Dokl. (Engl. Transl.)
23, 351-353 (1978)]. |
| 4. | B. E. Pobedrya, S. V. Sheshenin, and T. Kholmatov,
Problem in Stresses
(FAN, Tashkent, 1988)
[in Russian]. |
| 5. | V. N. Frolov,
Special Classes of Functions in Anisotropic Elasticity
(Fan, Tashkent, 1981)
[in Russian]. |
| 6. | Yu. L. Spivakov,
Special Classes of Solutions of Linear Differential Equations and Their Applications to Anisotropic and Inhomogeneous Elasticity
(Fan, Tashkent, 1986)
[in Russian]. |
| 7. | E. V. Rozhkova,
"On Completeness of Saint Venant Identities
and on Determining Displacements from Strains,"
in Proc. Intern. Conf. "Problems of Mechanics
and Seismic Resistance of Constructions"
(Tashkent, 2004),
pp. 614-617
[in Russian]. |
| 8. | B. F. Vlasov,
"On Equations of Continuity of Deformations,"
Prikl. Mekh.
6 (11), 85-90 (1970)
[Int. Appl. Mech. (Engl. Transl.)
6 (11), 1227-1231 (1970)]. |
|
| Received |
09 March 2007 |
| Link to Fulltext |
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