Skip to main content
SpringerLink
Log in

Asymptotics of the Spectrum of One-Dimensional Natural Vibrations in a Layered Medium Consisting of Viscoelastic Material and Viscous Fluid

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

Abstract

The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It is found that the spectrum mentioned above consists of the roots of transcendental equations whose number is proportional to the number of the layers of original medium. As the initial approximations of the roots for solving these equations numerically, it is proposed to use the points of the spectrum of one-dimensional natural vibrations of the corresponding homogenized medium. These points represent the roots of linear fractional equations. It is shown that the points at which the denominators of fractions in the linear fractional equations vanish should be also taken as the initial approximations. The accuracy of the initial approximations is proved to increase when the number of layers of the original medium increases and the layer thickness decreases simultaneously.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akulenko, L.D. and Nesterov, S.V., Inertial and dissipative properties of a porous medium occupied by a viscous fluid, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 2005, vol. 40, no. 1, pp. 109–119.

    Google Scholar 

  2. Shamaev, A.S. and Shumilova, V.V., Spectrum of natural vibrations in a layered medium consisting of an elastic material and a viscous fluid, Dokl. Ross. Akad. Nauk, 2013, vol. 448, no. 1, pp. 43–46.

    Google Scholar 

  3. Zhikov, V.V., On an extension of the method of two-scale convergence and its applications, Mat. Sb., 2000, vol. 191, no. 7, pp. 31–72.

    Article  MathSciNet  Google Scholar 

  4. Oleinik, O.A., Shamaev, A.S., and Yosifian, G.A., Mathematical Problems in Elasticity and Homogenization, Stud. Math. Appl. 26, Amsterdam: North-Holland, 1992.

    MATH  Google Scholar 

  5. Kosmodem’yanskii, D.A. and Shamaev, A.S., Spectral properties of some problems in mechanics of strongly inhomogeneous media, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 2009, vol. 44, no. 6, pp. 75–114.

    Google Scholar 

  6. Shamaev, A.S. and Shumilova, V.V., Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin—Voigt viscoelastic materials, Tr. Mat. Inst. RAN im. V.A. Steklova, 2016, vol. 295, No. 1, pp. 202–212.

    Article  MathSciNet  Google Scholar 

  7. Shamaev, A.S. and Shumilova, V.V., Calculation of natural frequencies and damping coefficients of a multi-layered composite using homogenization theory, IFAC Papers On Line, Vol. 51, no. 2, pp. 126–131.

  8. Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Fundamentals of the Mathematical Theory of Thermal Viscoelasticity), Moscow: Nauka, 1970.

    Google Scholar 

  9. Shamaev, A.S. and Shumilova, V.V., Spectrum of one-dimensional vibrations of a composite consisting of layers of elastic and viscoelastic materials, 2012, vol. 15, no. 4, pp. 124–134.

    Google Scholar 

  10. Il’yushin, A.A., Mekhanika sploshnoi sredy (Continuum Mechanics), Moscow: Izd-vo MGU, 1990.

    MATH  Google Scholar 

  11. Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, New York: Springer-Verlag, 1980.

    MATH  Google Scholar 

  12. Shamaev, A.S. and Shumilova, V.V., Averaging of acoustic equations for a porous viscoelastic material with long-time memory filled with a viscous fluid, Dif. Uravneniya, 2012, vol. 48, no. 8, pp. 1174–1186.

    MATH  Google Scholar 

  13. Shumilova, V.V., Spectral analysis of a class of integro-differential equations in viscoelasticity theory, Probl. Mat. Anal., 2013, vol. 73, pp. 167–172.

    Google Scholar 

  14. Vlasov, V.V., Wu J., and Kabiriva, G.R., Well-posed solvability and the spectral properties of abstract hyperbolic equations with residual effect, Sovr. Matem. Fundam. Napravleniya, 2010, vol. 35, pp. 44–59.

    Google Scholar 

Download references

Funding

The work was carried out with financial support from the Russian Science Foundation under the grant no. 16-11-10343.

Author information

Authors and Affiliations

  1. Ishlinskii Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia

    A. S. Shamaev & V. V. Shumilova

Authors
  1. A. S. Shamaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Shumilova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. S. Shamaev or V. V. Shumilova.

Additional information

Russian Text © The Author(s), 2019, published in Izvestiya RAN. Mekhanika Zhidkosti i Gaza, 2019, No. 6, pp. 12–24.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shamaev, A.S., Shumilova, V.V. Asymptotics of the Spectrum of One-Dimensional Natural Vibrations in a Layered Medium Consisting of Viscoelastic Material and Viscous Fluid. Fluid Dyn 54, 749–760 (2019). https://doi.org/10.1134/S0015462819060107

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0015462819060107

Key words

Search

Navigation