 | | 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 |
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| S. V. Sheshenin, "Asymptotic analysis of plates with periodic cross-sections," Mech. Solids. 41 (6), 57-63 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
6 |
Pages |
57-63 |
| Title |
Asymptotic analysis of plates with periodic cross-sections |
| Author(s) |
S. V. Sheshenin (Moscow) |
| Abstract |
In the present paper, we use the homogenization method to analyze elastic plates with
periodic cross-sections in lateral bending. The asymptotic study of such plates can be
performed on the basis of the following two approaches. In the first approach, one applies
the homogenization method to the equations of shell theory (for example, the equations of
shallow shells or plates with initial deflection). The second approach starts from the
three-dimensional elasticity equations. An application of the homogenization method leads to
the two-dimensional equations of plate theory for the smooth components of the stress-strain
state and to local three-dimensional problems on the periodicity cell for the fluctuations.
We point out several papers in this direction. Bending of a homogeneous plate with
periodically rough surfaces was apparently studied for the first time in [3]. In [4],
longitudinal extension of a periodically inhomogeneous plate with plane boundaries was
considered. In [5, 6], an asymptotic representation of the solution was obtained both for
extension in the plate plane and for bending of an arbitrarily inhomogeneous plate with
irregular boundaries. The only condition is the periodicity of the plate shape and
properties. In [6], the planar stressed state and bending were studied separately. In [5],
it was shown that this separation is possible only if the plate has a plane of symmetry of
its shape and properties. Otherwise, the equations of joint plane bending state are
obtained. In all papers listed above, only the first terms of the asymptotic expansion were
actually studied. In [7], two subsequent approximations were obtained for bending state
without tension, which permits calculating the tangential and transverse stresses.
In the present paper, we completely analyze the bending-tension state of a periodic plate
subjected to a transverse load. This theory can be used to study various ribbed, pressed,
and honeycomb plates. The zero approximation (this terminology was introduced in [1]) gives
the homogenized equations of the plane-bending state. The local problems of the zero
approximation permit calculating the effective rigidities. After certain transformations,
these problems correspond to the experimental determination of effective rigidities. The
first- and second-order approximations permit calculating the tangential and transverse
stresses. To calculate these quantities, we obtain local problems on the periodicity cell
and show that these problems are solvable. We present an example of calculating the tension,
bending, and mutual rigidities for a model plate whose periodicity cell is a hexagon. The
resulting theory can be used to study laminated plates. In this case, the zero approximation
gives the classical theory of laminated plates described, for example, in [8, 9]. |
| References |
| 1. | B. E. Pobedrya, Mechanics of Composite Materials [in Russian], Izd-vo MGU, Moscow,
1984. |
| 2. | N. S. Bakhvalov and G. P. Panasenko, Homogenization of Processes in Periodic Media
[in Russian], Nauka, Moscow, 1984. |
| 3. | R. V. Kohn and M. Vogelius, "A new model of thin plates with rapidly varying
thickness," Int. J. Solids and Struct., Vol. 20, No. 4, pp. 333-350, 1984. |
| 4. | G. P. Panasenko and M. V. Reztsov, "Homogenization of the three-dimensional
elasticity problem for an inhomogeneous plate," Doklady AN SSSR, Vol. 294, No. 5,
pp. 1061-1065, 1986. |
| 5. | T. Levinski and J. J. Telega, Plates, Laminates, and Shells. Asymptotic Analysis and
Homogenization, World Scientific, Singapoore, London, 2000. |
| 6. | L. V. Muravleva and S. V. Sheshenin, "Averaging for thin-walled bodies," Izv. RAN.
MTT [Mechanics of Solids], No. 4, pp. 129-138, 2004. |
| 7. | S. V. Sheshenin, "Application of the homogenization method to plates with periodic
cross-sections," Vestnik MGU. Ser. I Mat. Mekh., No. 1, pp. 47-51, 2006. |
| 8. | R. M. Jones, Mechanics and Analysis of Composite Materials, Taylor and Francis,
Philadelphia, London, 1998. |
| 9. | V. V. Vasiliev and E. V. Morozov, Mechanics and Analysis of Composite Materials,
Elsevier, Oxford, 2001. |
|
| Received |
10 August 2006 |
| << Previous article | Volume 41, Issue 6 / 2006 | Next article >> |
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