 | | 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: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| K.A. Skoptsov and S.V. Sheshenin, "Asymptotic Analysis of Laminated Plates and Shallow Shells," Mech. Solids. 46 (1), 129-138 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
129-138 |
| DOI |
10.3103/S0025654411010195 |
| Title |
Asymptotic Analysis of Laminated Plates and Shallow Shells |
| Author(s) |
K.A. Skoptsov (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, arbrk1@gmail.com)
S.V. Sheshenin (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, shesheni@mech.math.msu.su) |
| Abstract |
It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.
In what follows, these methods are generalized to the case of combined bending and extension of a longitudinal laminated plate up to the third approximation, which permits finding all components of the stress tensor. The study of the plate behavior is based on the method of homogenization of the three-dimensional problem of linear elasticity and does not use any hypotheses. It turns out that the Kirchhoff-Love hypothesis for the entire packet of layers is simply a consequence of the method in the zeroth approximation, and the bending stresses corresponding to the classical theory of laminated plates [8] are obtained in the first approximation. The successive approximations describe the behavior of the normal and the stress more precisely.
In the present paper, the results obtained in [7] are refined, and the asymptotic solution is compared with the direct analysis of a laminated plate by the finite element method (FEM). The method is also generalized to the case of a shallow shell. |
| Keywords |
asymptotic analysis, homogenization method, laminated plate, shallow shell |
| References |
| 1. | A. A. Il'yushin and V. S. Lenskii,
Strength of Materials
(Fizmatgiz, 1959)
[in Russian]. |
| 2. | N. S. Bakhvalov and G. P. Panasenko,
Averaging of Processes in Periodic Media
(Nauka, Moscow, 1984)
[in Russian]. |
| 3. | B. E. Pobedrya,
Mechanics of Composite Materials
(Izd-vo MGU, Moscow, 1984)
[in Russian]. |
| 4. | R. V. Kohn and M. Vogelius,
"A New Model for Thin Plates with Rapidly Varying Thickness,"
Int. J. Solids Struct.
20 (4), 333-350 (1984). |
| 5. | T. Levinski and J. J. Telega,
Plates, Laminates, and Shells.
Asymptotic Analysis and Homogenization
(World Sci. Publ., Singapore, London, 2000). |
| 6. | S. V. Sheshenin,
"Asymptotic Analysis of Plates with Periodic Cross-Sections,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 6, 71-79 (2006)
[Mech. Solids (Engl. Transl.)
41 (6), 57-63 (2006)]. |
| 7. | S. V. Sheshenin,
"Application of Averaging Method to Plates Periodic in the Horizontal Projection,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.,
No. 1, 47-51 (2006)
[Moscow Univ. Math. Bull. (Engl. Transl.)]. |
| 8. | R. M. Jones,
Mechanics of Composite Materials
(Taylor and Francis, Philadelphia, 1998). |
| 9. | Yu. A. Amenzade,
Theory of Elasticity
(Vysshaya Shkola, Moscow, 1976)
[in Russian]. |
| 10. | N. G. de Bruijn,
Asymptotic Methods in Analysis
(North-Holland, Amsterdam, 1958; Inostr. Lit., Moscow, 1961). |
|
| Received |
09 September 2010 |
| Link to Fulltext |
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