Skip to main content
SpringerLink
Log in

Study of Kelvin–Voigt Models Arising in Viscoelasticity

  • Integral Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

An operator model of integro-differential equations arising in the theory of viscoelasticity is studied. The spectral analysis of operator functions which are the symbols of Gurtin–Pipkin-type integro-differential equations is carried out with the Kelvin–Voigt friction taken into account.

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. Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Foundations of Mathematical Theory of Thermoviscoelasticity), Moscow: Nauka, 1970.

    Google Scholar 

  2. Pipkin, A.C. and Gurtin, M.E., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 1968, vol.31, pp. 113–126.

    Article  MathSciNet  MATH  Google Scholar 

  3. Pandolfi, L., The controllability of the Gurtin–Pipkin equations: a cosine operator approach, Appl. Math. Optim., 2005, vol.52, pp. 143–165.

    Article  MathSciNet  MATH  Google Scholar 

  4. Muñoz Rivera, J.E. and Naso, M.G., On the decay of the energy for systems with memory and indefinite dissipation, Asymptot. Anal., 2006, vol.49, pp. 189–204.

    MathSciNet  MATH  Google Scholar 

  5. Pandolfi, L. and Ivanov, S., Heat equations with memory: lack of controllability to the rest, J. Math. Anal. Appl., 2009, vol.355, pp. 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  6. Amendola, G., Fabrizio., M., and Golden, J.M., Thermodynamics of Materials with Memory, Boston: Springer, 2012.

    Book  MATH  Google Scholar 

  7. Vlasov, V.V. and Rautian, N.A., Spectral analysis of integro-differential equations in a Hilbert space, Sovr. Mat. Fund. Napravleniya, 2012, vol.62, pp. 53–71.

    MATH  Google Scholar 

  8. Vlasov, V.V. and Rautian, N.A., Spektral’nyi analiz funktsional’no-differentsial’nykh uravnenii (Spectral Analysis of Functional-Differential Equations), Moscow: MAKS Press, 2016.

    Google Scholar 

  9. Vlasov, V.V., Rautian, N.A., and Shamaev, A.S., Investigation of operator models arizing in problems of hereditary mechanics, Sovr. Mat. Fund. Napravleniya, 2012, vol.45, pp. 43–61.

    Google Scholar 

  10. Eremenko, A. and Ivanov, S., Spectra of the Gurtin–Pipkin type equations, SIAM J. Math. Anal., 2011, vol.43, pp. 2296–2306.

    Article  MathSciNet  MATH  Google Scholar 

  11. Shkalikov, A.A. and Griniv, R.O., On an operator pencil arising in the problem of beam oscillation with internal damping, Math. Notes, 1994, vol.56, no. 2, pp. 840–851.

    Article  MathSciNet  MATH  Google Scholar 

  12. Miloslavskii, A.I., Instability spectrum of an operator pencil, Math. Notes, 1991, vol.49, no. 4, pp. 391–395.

    Article  MathSciNet  Google Scholar 

  13. Goluzin, G.M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo (Geometric Theory of Functions of Complex Variable), Moscow: Nauka, 1966.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. V. Davydov & Yu. A. Tikhonov

Authors
  1. A. V. Davydov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. A. Tikhonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Davydov.

Additional information

Original Russian Text © A.V. Davydov, Yu.A. Tikhonov, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 12, pp. 1663–1677.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davydov, A.V., Tikhonov, Y.A. Study of Kelvin–Voigt Models Arising in Viscoelasticity. Diff Equat 54, 1620–1635 (2018). https://doi.org/10.1134/S001226611812008X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S001226611812008X

Search

Navigation