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Spectrum of One-Dimensional Eigenoscillations of a Medium Consisting of Viscoelastic Material with Memory and Incompressible Viscous Fluid

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We consider the initial-boundary value problem describing joint oscillations of periodically alternating layers of viscoelastic material with memory and layers of viscous incompressible fluid. Solving auxiliary problems on the periodicity cell, we write the corresponding homogenized problem and show that the eigenvalues of the boundary value problems involved in the spectra of one-dimensional eigenoscillations of the original and homogenized media are roots of transcendental and algebraic equations respectively. In the case where the eigenoscillations are perpendicular to the layers, we show that the spectra of the original problems Hausdorff converge to the spectrum of the homogenized problem.

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Authors and Affiliations

  1. Institute for Problems in Mechanics of RAS, 101-1, pr. Vernadskogo, Moscow, 119526, 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 V. V. Shumilova.

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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 161-172.

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Shamaev, A.S., Shumilova, V.V. Spectrum of One-Dimensional Eigenoscillations of a Medium Consisting of Viscoelastic Material with Memory and Incompressible Viscous Fluid. J Math Sci 257, 732–746 (2021). https://doi.org/10.1007/s10958-021-05513-0

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  • DOI: https://doi.org/10.1007/s10958-021-05513-0

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