 | | 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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| V.I. Gorbachev and L.L. Firsov, "New Statement of the Elasticity Problem for a Layer," Mech. Solids. 46 (1), 89-95 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
89-95 |
| DOI |
10.3103/S0025654411010146 |
| Title |
New Statement of the Elasticity Problem for a Layer |
| Author(s) |
V.I. Gorbachev (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, vigorby@mail.ru)
L.L. Firsov (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia) |
| Abstract |
The problem on the equilibrium of an inhomogeneous anisotropic elastic layer is considered. The classical statement of the problem in displacements consists of three partial differential equations with variable coefficients for the three displacements and of three boundary conditions posed at each point of the boundary surface. Sometimes, instead of the statement in displacements, it is convenient to use the classical statement of the problem in stresses [1] or the new statement of the problem in stresses proposed by B. E. Pobedrya [2]. In the case of the problem in stresses, it is necessary to find six components of the stress tensor, which are functions of three coordinates. The choice of the statement of the problem depends on the researcher and, of course, on the specific problem. The fact that there are several statements of the problem makes for a wider choice of the method for solving the problem. In the present paper, for a layer with plane boundary surfaces, we propose a new statement of the problem, which, in contrast to the other two statements indicated above, can be called a mixed statement. The problem for a layer in the new statement consists of a system of three partial differential equations for the three components of the displacement vector of the midplane points. The system is coupled with three integro-differential equations for the three longitudinal components of the stress tensor. Thus, in the new statement, just as in the other statements in stresses, one should find six functions. In the new statement, three of these functions (the displacements of the midplane points) are functions of two coordinates, and the other three functions (the longitudinal components of the stress tensor) are functions of three coordinates. It is shown that all equations in the new statement are the Euler equations for the Reissner functional with additional constraints. After the problem is solved in the new statement, three components of the displacement vector and three transverse components of the stress tensor are determined at each point of the layer. The new statement of the problem can be used to construct various engineering theories of plates made of composite materials. |
| Keywords |
theory of elasticity, mechanics of composites, statement of problems, elastic layer, theory of plates |
| References |
| 1. | W. Nowacki,
Theory of Elasticity
(Mir, Moscow, 1975)
[in Russian]. |
| 2. | B. E. Pobedrya, S. V. Sheshenin, and T. Kholmatov,
Problem in Stresses
(FAN, Tashkent, 1988)
[in Russian]. |
| 3. | B. E. Pobedrya,
Lectures on Tensor Analysis
(Izd-vo MGU, Moscow, 1979)
[in Russian]. |
| 4. | V. G. Zubchaninov,
Foundations of the Theory of Elasticity and Plasticity
(Vyssh. Shkola, Moscow, 1979)
[in Russian]. |
| 5. | V. I. Gorbachev and V. A. Simakov,
"An Operator Method for Solving the Equilibrium Problem
for an Elastic Inhomogeneous Anisotropic Slab,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 2, 55-64 (2004)
[Mech. Solids (Engl. Transl.)
39 (2), 43-49 (2004)]. |
| 6. | K. Rektorys,
Variational Methods in Mathematics, Science, and Engineering
(Mir, Moscow, 1985)
[in Russian]. |
| 7. | Yu. N. Rabotnov,
Mechanics of Deformable Solids
(Nauka, Moscow, 1979)
[in Russian]. |
| 8. | B. E. Pobedrya,
Numerical Methods in the Theory of Elasticity and Plasticity
(Izd-vo MGU, Moscow, 1995)
[in Russian]. |
| 9. | B. E. Pobedrya and D. V. Georgievskii,
Lectures on Elasticity Theory
(Editorial URSS, Moscow, 1999)
[in Russian]. |
|
| Received |
20 August 2010 |
| Link to Fulltext |
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