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 Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 218–228 (Mi tm3762)

Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

A. S. Shamaev, V. V. Shumilova

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis $Ox_1$ of periodically alternating $M$ elastic and $M$ viscoelastic layers parallel to the plane $Ox_2x_3$. It is shown that the spectrum of the boundary value problem is the union of roots of $M$ equations. The asymptotic behavior of the spectrum of the problem as $M\to\infty$ is analyzed; in particular, it is proved that not all sequences of eigenvalues of the original (prelimit) problem converge to eigenvalues of the corresponding homogenized (limit) problem.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968516040130

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 202–212

Bibliographic databases:

UDC: 517.958+517.984

Citation: A. S. Shamaev, V. V. Shumilova, “Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 218–228; Proc. Steklov Inst. Math., 295 (2016), 202–212

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3762
• https://doi.org/10.1134/S0371968516040130
• http://mi.mathnet.ru/eng/tm/v295/p218

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This publication is cited in the following articles:
1. T. N. Bobyleva, “Method of calculation of stresses in the layered elastic-creeping arrays”, 5th International Scientific Conference Integration, Partnership and Innovation in Construction Science and Education, MATEC Web Conf., 86, ed. V. Andreev, EDP Sciences, 2016, UNSP 01024
2. T. Bobyleva, A. Shamaev, “Stress distribution in layered elastic creeping array with a vertical cylindrical shaft”, RSP 2017 (XXVI R-S-P Seminar 2017 Theoretical Foundation of Civil Engineering), MATEC Web Conf., 117, ed. S. Jemiolo, A. Zbiciak, M. MitewCzajewska, M. Krzeminski, M. Gajewski, EDP Sciences, 2017, UNSP 00020
3. A. S. Shamaev, V. V. Shumilova, “Calculation of natural frequencies and damping coefficients of a multi-layered composite using homogenization theory”, IFAC-PapersOnLine, 51:2 (2018), 126–131
4. T. N. Bobyleva, A. S. Shamaev, “Method of approximate calculation of the stress tensor in layered elastic-creeping environments”, IFAC-PapersOnLine, 51:2 (2018), 138–143
5. T. Bobyleva, A. Shamaev, “The averaged model of layered elastic-creeping composite materials”, XXI International Scientific Conference on Advanced in Civil Engineering Construction - the Formation of Living Environment, IOP Conference Series-Materials Science and Engineering, 365, ed. y A. Askadskiy, A. Pustovgar, T. Matseevich, A. Adamtsevich, IOP Publishing Ltd, 2018, 042078
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