International Applied Mechanics

, Volume 55, Issue 6, pp 601–619 | Cite as

Determining the Parameters of the Hereditary Kernels of Isotropic Nonlinear Viscoelastic Materials in Combined Stress State*

  • V. P. GolubEmail author
  • Yu. M. Kobzar’
  • P. V. Fernati

The relations between the heredity kernels of isotropic nonlinear viscoelastic materials in combined and one-dimensional stress states are derived. The constitutive equations are presented in a form corresponding to the proportional deviator hypothesis. The nonlinearity of viscoelastic properties is described by Rabotnov’s type models. The creep strains and stress relaxation in thin-walled tubular elements subject to a combination of tension and torsion are determined and tested experimentally.


nonlinear viscoelasticity isotropic material combined stress state proportional deviator hypothesis creep kernel stress relaxation kernel thin-walled tube element tension and torsion 


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  1. 1.
    I. I. Bugakov, Creep of Polymeric Materials. Theory and Application [in Russian], Nauka, Moscow (1973).Google Scholar
  2. 2.
    I. I. Gol’denblat., V. L. Bazhanov, and V. A. Kopnov, Long-Term Strength in Mechanical Engineering [in Russian], Mashinostroenie, Moscow (1977).Google Scholar
  3. 3.
    A. A. Koltunov, “Method of determining the volume and shear characteristics of elastico-viscous hereditary media from uniaxial-tension (compression) experiments,” Mech. Polym., 5, No. 4, 667–671 (1969).CrossRefGoogle Scholar
  4. 4.
    M. A. Koltunov, Creep and Relaxation [in Russian], Vysshaya Shkola, Moscow (1976).Google Scholar
  5. 5.
    A. F. Kregers and M. R. Kilevits, “Detailed examination of high-density polyethylene in the conditions of nonlinear creep and stress relaxation,” Mech. Comp. Mater., 21, No. 2, 117–123 (1985).CrossRefGoogle Scholar
  6. 6.
    M. N. Stepnov, Statistical Processing of Mechanical Test Data [in Russian], Mashinostroenie, Moscow (1972).Google Scholar
  7. 7.
    G. M. Fichtenholz, A Course in Differential and Integral Calculus [in Russian], Vol. 2, Nauka, Moscow (1960).Google Scholar
  8. 8.
    R. M. Christensen, Theory of Viscoelasticity. An Introduction, Academic Press Inc., New-York (1971).Google Scholar
  9. 9.
    J. D. Ferry, Viscoelastic Properties of Polymers. 2nd ed., John Willey and Sons, New-York (1971).Google Scholar
  10. 10.
    W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland Publishing Company, Amsterdam (1976).zbMATHGoogle Scholar
  11. 11.
    V. P. Golub, “Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity,” Int. Appl. Mech., 47, No. 6, 727–734 (2011).ADSCrossRefGoogle Scholar
  12. 12.
    V. P. Golub, P. V. Fernati, and Ya. G. Lyashenko, “Determining the parameters of the fractional exponential hereditary kernels of linear viscoelastic materials,” Int. Appl. Mech., 44, No. 9, 963–974 (2008).ADSCrossRefGoogle Scholar
  13. 13.
    V. P. Golub, Yu. M. Kobzar’, and V. S. Ragulina, “A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity,” Int. Appl. Mech., 47, No. 3, 290–301 (2011).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    V. P. Golub, Ya. V. Pavlyuk, and P. V. Fernati, “Determining parameters of fractional-exponential heredity kernels of nonlinear viscoelastic materials,” Int. Appl. Mech., 53, No. 4, 419–433 (2017).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    V. P. Golub, A. D. Pogrebnyak, and I. B. Romanenko, “Application of smoothing spline approximations in problems on identifications of creep parameters,” Int. Appl. Mech., 33, No. 6, 477–484 (1997).ADSCrossRefGoogle Scholar
  16. 16.
    Y. M. Kobzar’, “Models of long-term brittle fracture of rods in tension and compression under creep conditions,” Int. Appl. Mech., 53, No. 4, 444–453 (2017).ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    J. S. Y. Lai and W. N. Findley, “Behavior of nonlinear viscoelastic material under simultaneous stress relaxation in tension and creep in torsion,” ASME J. Appl. Mech., No. 36, 22–37 (1969).ADSCrossRefGoogle Scholar
  18. 18.
    H. Leaderman, Elastic and Creep Properties of Filaments Materials and Other High Polymers, Textile Foundation, Washington (1943).Google Scholar
  19. 19.
    B. P. Maslov, “Combined numerical and analytical determination of Poisson’s ratio for viscoelastic isotropic materials,” Int. Appl. Mech., 54, No. 2, 220–230 (2018).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    B. Persoz, “Le principle de superposition de Boltzman,” Cahier Groupe Frans, Ĕtudes Rhĕol, 2, No. 1, 237–245 (1957).Google Scholar
  21. 21.
    Y. N. Rabotnov, Creep Problems in Structural Members, North-Holland Publishing Company, Amsterdam (1969).zbMATHGoogle Scholar
  22. 22.
    R. O. Stafford, “On mathematical forms for the material functions in nonlinear viscoelasticity,” J. Mech. and Phys. Solids, 17, No. 5, 339–354 (1969).ADSCrossRefGoogle Scholar
  23. 23.
    I. M. Ward, Mechanical Properties of Solid Polymers, Willey and Sons, New York (1971).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. P. Golub
    • 1
    Email author
  • Yu. M. Kobzar’
    • 1
  • P. V. Fernati
    • 1
  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

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