Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
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IssuesArchive of Issues2004-4pp.102-109

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Total articles in the database: 10864
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L. V. Muravleva and S. V. Sheshenin, "Averaging for Thin-Walled Bodies," Mech. Solids. 39 (4), 102-109 (2004)
Year 2004 Volume 39 Number 4 Pages 102-109
Title Averaging for Thin-Walled Bodies
Author(s) L. V. Muravleva (Moscow)
S. V. Sheshenin (Moscow)
Abstract The well-known homogenization technique [1, 2] is developed here in the case of bodies whose structure is periodic in some directions and non-periodic in others. There are many thin-walled bodies with a periodic structure in one or two directions, but their properties in the third direction are non-periodic. Such structural elements have numerous technological applications. Among these are stampings or ribbed plates, reinforced concrete plates or beams, and automobile tire tread. For all such bodies it seems natural to use effective mechanical characteristics. In many cases, the averaged (or effective) characteristics can be found on the basis of engineering approach to effective properties, which in fact amounts to the experimental determination of elastic moduli in tests with material specimens. However, it is desirable not only to find averaged characteristics of periodic structures, such as mean strains and stresses, but also to determine the fluctuations of these fields in the components of a structure. This can be achieved by the averaging technique, which has a long history. In most publications, this technique is applied to structures that are periodic in all directions. However, in [3], this method was used for averaging the plane problem of elasticity for an inhomogeneous plate of constant thickness. In [4], this method is used in the case of homogeneous plates of variable thickness. Periodic ribbed plates were approached in [5, 6], but the techniques used there were not based on the method of averaging. In [7], attention is given to plates whose structure is inhomogeneous along their thickness.

In this paper, the averaging method is first applied to inhomogeneous plates of arbitrary thickness (for instance, stampings) for the determination of longitudinal and flexural rigidities. This approach follows the ideas of [3, 4]. An interesting property discovered in connection with the averaging of such plates is that the mean tensile and shear strains cannot be found by volume averaging. Next, the method of averaging is applied to calculate longitudinal and flexural rigidities of aluminum stampings.

The method developed here is also applied to reinforced concrete plates, with the reinforcement having a non-periodic structure in the direction across the thickness. For a specific plate (experimental data regarding lateral bending of such plates are known [8]), we find effective characteristics in the elastic and plastic regions.

Finally, the method is extended to the problem of finding effective properties of pneumatic tires. The tread pattern can be rather sophisticated, and the discretization of such a complex domain used for numerical analysis entails large computational errors and is unjustified. The utilization of the effective characteristics of the tread can reduce the computational effort.
1.  N. S. Bakhvalov and G. P. Panasenko, Averaging of Processes in Periodic Media [in Russian], Nauka, Moscow, 1984.
2.  B. E. Pobedrya, Mechanics of Composites [in Russian], Izd-vo MGU, Moscow, 1984.
3.  G. P. Panasenko and M. V. Reztsov, "Homogenization of a three dimensional elasticity problem for an inhomogeneous plate," Doklady AN SSSR, Vol. 294, No. 5, pp. 1061-1065, 1987.
4.  R. V. Kohn and M. Vogelius, "A new model of thin plates with rapidly varying thickness," Int. J. Solids. Struct., Vol. 20, No. 4, pp. 333-350, 1984.
5.  V. A. Lesnichnaya, L. I. Manevich, and I. V. Andrianov, Averaging Method in Static and Dynamic Problems for Ribbed Shells [in Russian], Nauka, Moscow, 1985.
6.  I. V. Amiro and V. A. Zarutskii, Theory of Ribbed Shells [in Russian], Naukova Dumka, Kiev, 1980.
7.  V. I. Gorbachev and V. A. Simakov, "An approach to the solution of elasticity problems for an anisotropic layer inhomogeneous along its thickness," in Elasticity and Inelasticity [in Russian], pp. 426-427, Izd-vo MGU, Moscow, 2001.
8.  H. Tsubota, J. Mizuno, K. Kusuma, K. Brandes, J. Hertter, E. Limberger, T. Yamashita, T. Fujita, and Y. Sagimoto, "Experimental studies of inelastic behavior of reinforced concrete plates under high-speed loading. Part 1. Effects of dynamic loading," in Structures under Shock and Impact, pp. 743-759, Thessaloniki, Greece, 1998.
9.  K. Brandes, "Strain rate effects appearing in mechanical behavior of reinforced concrete structures under impact load," Mech. Inelastic Med. and Struct.: Proc. Intern. Symp., pp. 73-85, Warszawa, Poznan, 1982.
10.  K. Brandes, "Blast-resistant structures," Intern. Workshop on Plast-Resistant Structures, Tsaighua Univ., Beijing, 1992.
11.  N. S. Bakhvalov and M. E. Eglit, "Effective moduli of composites reinforced by systems of plates and beams," Zh. Vychisl. Mat. i Mat. Fiz., Vol. 38, No. 5, pp. 813-834, 1998.
Received 07 February 2002
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