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

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Abstract

The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis O x1 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.

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Authors and Affiliations

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

    A. S. Shamaev & V. V. Shumilova

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  1. A. S. Shamaev
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  2. V. V. Shumilova
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Correspondence to A. S. Shamaev.

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Original Russian Text © A.S. Shamaev, V.V. Shumilova, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 218–228.

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Shamaev, A.S., Shumilova, V.V. Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials. Proc. Steklov Inst. Math. 295, 202–212 (2016). https://doi.org/10.1134/S0081543816080137

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  • DOI: https://doi.org/10.1134/S0081543816080137

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