Abstract:
The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis Ox1 of periodically alternating M elastic and M viscoelastic layers parallel to the plane Ox2x3. 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→∞ 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.
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, Trudy Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 218–228; Proc. Steklov Inst. Math., 295 (2016), 202–212
\Bibitem{ShaShu16}
\by A.~S.~Shamaev, V.~V.~Shumilova
\paper Asymptotic behavior of the spectrum of one-dimensional vibrations in a~layered medium consisting of elastic and Kelvin--Voigt viscoelastic materials
\inbook Modern problems of mechanics
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2016
\vol 295
\pages 218--228
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968516040130}
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\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 295
\pages 202--212
\crossref{https://doi.org/10.1134/S0081543816080137}
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Linking options:
https://www.mathnet.ru/eng/tm3762
https://doi.org/10.1134/S0371968516040130
https://www.mathnet.ru/eng/tm/v295/p218
This publication is cited in the following 18 articles:
A. S. Shamaev, V. V. Shumilova, “Homogenization of motion equations for medium consisting of elastic material and incompessible Kelvin-Voigt fluid”, Ufa Math. J., 16:1 (2024), 100–111
Vladlena V. Shumilova, “Spectrum of one-dimensional eigenoscillations of two-phase layered composites”, Zhurn. SFU. Ser. Matem. i fiz., 16:1 (2023), 35–47
Tatiana Bobyleva, Alexey Shamaev, A. Ter-Martirosyan, D. Bazarov, “On the spectra of composite materials with different dissipation models”, E3S Web of Conf., 410 (2023), 01004
Tatyana Bobyleva, Alexey Shamaev, Lecture Notes in Civil Engineering, 282, Proceedings of FORM 2022, 2023, 211
V. V. Shumilova, “Spektr odnomernykh sobstvennykh kolebanii dvukhfaznykh sloistykh sred s periodicheskoi strukturoi”, Tr. IMM UrO RAN, 28, no. 4, 2022, 250–261
A. S. Shamaev, V. V. Shumilova, “Spectrum of One-Dimensional Eigenoscillations of a Medium Consisting of Viscoelastic Material with Memory and Incompressible Viscous Fluid”, J Math Sci, 257:5 (2021), 732
V. V. Shumilova, “Spektr sobstvennykh kolebanii sloistoi sredy, sostoyaschei iz materiala Kelvina–Foigta i vyazkoi neszhimaemoi zhidkosti”, Sib. elektron. matem. izv., 17 (2020), 21–31
A. S. Shamaev, V. V. Shumilova, “Spectrum of one-dimensional natural vibrations of layered medium consisting of elastic material and viscous incompressible fluid”, Moscow University Mathematics Bulletin, 75:4 (2020), 172–176
A. S. Shamaev, V. V. Shumilova, “Asymptotics of the Spectra of One-Dimensional Natural Vibrations in Media Consisting of Solid and Fluid Layers”, Dokl. Phys., 65:4 (2020), 153
T N Bobyleva, A S Shamaev, “Effective modules of a layered elastic creep medium with power creep kernels”, IOP Conf. Ser.: Mater. Sci. Eng., 913:3 (2020), 032059
A. S. Shamaev, V. V. Shumilova, “Asymptotics of the spectrum of one-dimensional natural vibrations in a layered medium consisting of viscoelastic material and viscous fluid”, Fluid Dyn., 54:6 (2019), 749–760
Tatyana Bobyleva, Tat'yana Bobyleva, “AVERAGED MODEL OF LAYERED ELASTIC-CREEPING COMPOSITE MATERI-ALS”, Bulletin of Belgorod State Technological University named after. V. G. Shukhov, 4:1 (2019), 45
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
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
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
Tatiana Bobyleva, Alexei Shamaev, A. Volkov, A. Pustovgar, A. Adamtsevich, “Effective characteristics of a layered tube consisting of elastic-creeping materials”, MATEC Web Conf., 251 (2018), 04039
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
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
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