Mechanics of Time-Dependent Materials

, Volume 19, Issue 2, pp 135–151 | Cite as

Low-frequency perturbations of rigid body motions of a viscoelastic inhomogeneous bar

  • J. Kaplunov
  • A. ShestakovaEmail author
  • I. Aleynikov
  • B. Hopkins
  • A. Talonov


This paper deals with a low-frequency analysis of a viscoelastic inhomogeneous bar subject to end loads. The spatial variation of the problem parameters is taken into consideration. Explicit asymptotic corrections to the conventional equations of rigid body motion are derived in the form of integro-differential operators acting on longitudinal force or bending moment. The refined equations incorporate the effect of an internal viscoelastic microstructure on the overall dynamic response. Comparison with the exact time-harmonic solutions for extension and bending of a bar demonstrates the advantages of the developed approach. This research is inspired by modeling of railcar dynamics.


Viscoelastic Microstructure Perturbation Rigid body Low-frequency 



J. Kaplunov and A. Shestakova gratefully acknowledge support from the industrial project with AMSTED Rail, USA. J. Kaplunov’s research in the area of mechanics of inhomogeneous solids was supported by National University of Science and Technology “MISiS”, Russia by grant K3-2014-052. The authors also grateful to Dr. D. Prikazchikov for a number of valuable comments.


  1. Addessamad, Z., Kostin, J., Panasenko, G., Smyshlyaev, V.P.: Memory effect in homogenisation of a viscoelastic Kelvin–Voigt model with time-dependent coefficients. Math. Models Methods Appl. Sci. 9, 1603–1630 (2009) CrossRefGoogle Scholar
  2. Andrianov, I.V., Awrejcewicz, J.: New trends in asymptotic approaches: summation and interpolation methods. Appl. Mech. Rev. 54(1), 69–92 (2001) CrossRefGoogle Scholar
  3. Ansari, M., Esmailzadeh, E., Younesian, P.: Longitudinal dynamics of freight trains. Int. J. Heavy Veh. Syst. 16(1/2), 102–131 (2009) CrossRefGoogle Scholar
  4. Babenkova, E., Kaplunov, J.: The two-term interior asymptotic expansion in the case of low-frequency longitudinal vibrations of an elongated elastic rectangle. In: Proc. of the IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics. Series Solid Mechanics and Its Applications, vol. 113, pp. 123–131. Kluwer Academic, Dordrecht (2003a) Google Scholar
  5. Babenkova, E., Kaplunov, J.: Low-frequency decay conditions for a semi-infinite elastic strip. Proc. R. Soc. Lond. Ser. A 460, 2153–2169 (2003b) MathSciNetCrossRefGoogle Scholar
  6. Chen, C., Han, M., Han, Y.: A numerical model for railroad freight car-to-car end impact. Discrete Dyn. Nat. Soc. 927592, 11 (2012) Google Scholar
  7. Craster, R.V., Joseph, L.M., Kaplunov, J.: Long-wave asymptotic theories: the connection between functionally graded waveguides and periodic media. Wave Motion 51, 581–588 (2013) MathSciNetCrossRefGoogle Scholar
  8. Cristensen, R.M.: Viscoelasticity: An Introduction, vol. 359, 2nd edn. Academic Press, San Diego (1982) Google Scholar
  9. Iwnicki, S.: Handbook of Railway Vehicle Dynamics, vol. 552. CRC Press, Boca Raton (2006) CrossRefGoogle Scholar
  10. Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. Lond. Ser. A 2079, 855–880 (2007) MathSciNetCrossRefGoogle Scholar
  11. Rabotnov, Yu.N.: Elements of Hereditary Solid Mechanics, vol. 387. Mir, Moscow (1980) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • J. Kaplunov
    • 1
  • A. Shestakova
    • 1
    Email author
  • I. Aleynikov
    • 2
  • B. Hopkins
    • 2
  • A. Talonov
    • 3
  1. 1.School of Computing and MathematicsKeele UniversityKeeleUK
  2. 2.Amsted RailGranite CityUSA
  3. 3.National University of Science and Technology “MISiS”MoscowRussia

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