Acta Mechanica

, Volume 229, Issue 6, pp 2539–2559 | Cite as

Modification of the iterative method for solving linear viscoelasticity boundary value problems and its implementation by the finite element method

  • Alexander Svetashkov
  • Nikolay Kupriyanov
  • Kayrat Manabaev
Original Paper


The problem of structural design of polymeric and composite viscoelastic materials is currently of great interest. The development of new methods of calculation of the stress–strain state of viscoelastic solids is also a current mathematical problem, because when solving boundary value problems one needs to consider the full history of exposure to loads and temperature on the structure. The article seeks to build an iterative algorithm for calculating the stress–strain state of viscoelastic structures, enabling a complete separation of time and space variables, thereby making it possible to determine the stresses and displacements at any time without regard to the loading history. It presents a modified theoretical basis of the iterative algorithm and provides analytical solutions of variational problems based on which the measure of the rate of convergence of the iterative process is determined. It also presents the conditions for the separation of space and time variables. The formulation of the iterative algorithm, convergence rate estimates, numerical computation results, and comparisons with exact solutions are provided in the tension plate problem example.


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  1. 1.
    Ataoglu, S.: A two dimensional mixed boundary-value problems in a viscoelastic medium. Struct. Eng. Mech. 32(3), 407–427 (2009)CrossRefGoogle Scholar
  2. 2.
    Brinsoh, L.C., Knauss, W.G.: Finite element analysis of multiphase viscoelastic solids. J. Appl. Mech. 59(4), 730–727 (1992)CrossRefGoogle Scholar
  3. 3.
    Carini, A., Gelfi, P., Marchina, E.: An energetic formulation for the linear viscoelastic problem. Part I: theoretical results and first calculations. Int. J. Numer. Methods Eng. 38(1), 37–62 (1995)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chazal, C., Pitti, R.M.: Integral approach for time dependent materials finite element method. J. Theor. Appl. Mech. 49(4), 1029–1048 (2011)Google Scholar
  5. 5.
    Chazal, C., Pitti, R.M.: Modeling of ageing viscoelastic materials in three dimensional finite element approach. Mecc. Int. J. Theor. Appl. Mech. 45(3), 439–441 (2010)zbMATHGoogle Scholar
  6. 6.
    Christensen, R.M.: Theory of Viscoelasticity, 2nd edn. Academic Press, New York (1982)Google Scholar
  7. 7.
    Cozzano, B.S., Rodriguez, B.S.: The Trefftz boundary method in viscoelasticity. Comput. Model. Eng. Sci. 20(1), 21–33 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Flugge, W.: Viscoelasticity. Blaisdell Press, New York (1967)zbMATHGoogle Scholar
  9. 9.
    Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Mech. Anal. 11(1), 291–356 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gurtin, M.E.: Variational principles in the linear theory of viscoelasticity. Ibid. 1(3), 179–191 (1963)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lahellec, N.: Effective behavior of linear viscoelastic composites: a time integration approach. Int. J. Solids Struct. 44(2), 507–529 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Matveenko, V., Trufanov, N.: Multi-operator boundary value problems of viscoelasticity of piecewise-homogeneous bodies. J. Eng. Math. 78(1), 119–129 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Janovsky, V., Shaw, S., Wardy, M.K., Whiteman, J.R.: Numerical methods for treating problems of viscoelastic isotropic solid deformation. J. Comput. Appl. Math. 63(1–3), 91–107 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pavlov, S.M., Svetashkov, A.A.: Iteration method for solving linear viscoelasticity problems. Russ. Phys. J. 36(4), 400–406 (1993)CrossRefGoogle Scholar
  15. 15.
    Pipkin, A.C.: Lectures on Viscoelasticity Theory. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  16. 16.
    Pobedrya, B.E.: Chislennye metody v teorii uprugosti i plastichnosti. MGU, Moscow (1995). [in Russian]zbMATHGoogle Scholar
  17. 17.
    Rabotnov, Y.N.: Elementy nasledstvennoy mekhaniki tverdykh tel. Nauka, Moscow (1977). [in Russian]Google Scholar
  18. 18.
    Reddy, J.N.: An Introduction to Continuum Mechanics. CUP, New York (2008)Google Scholar
  19. 19.
    Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1988)CrossRefzbMATHGoogle Scholar
  20. 20.
    Svetashkov, A.A.: Time-effective moduli of linear viscoelastic body. Mech. Compos. Mater. 36(1), 37–44 (2000a)CrossRefGoogle Scholar
  21. 21.
    Svetashkov, A.A.: Iteracionnye metody reshenija zadach linejnoj i nelinejnoj vjazkouprugosti, termovjazkouprugosti, termouprugosti. Thesis (DSci in Physics and Mathematics). Scientific Research Institute of Applied Mathematics and Mechanics, Tomsk State University, Tomsk (2000b). [in Russian]Google Scholar
  22. 22.
    Wang, H.N., Nie, G.H.: Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions. Acta Mech. 210(3), 315–330 (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Zienkiewicz, O.C., Watson, M., King, I.: A numerical method of viscoelastic stress. J. Mech. Sci. 10(10), 807–827 (1968)CrossRefGoogle Scholar
  24. 24.
    Zocher, M.A., Groves, S.E.: A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Methods Eng. 40, 2267–2288 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Alexander Svetashkov
    • 1
  • Nikolay Kupriyanov
    • 1
  • Kayrat Manabaev
    • 1
  1. 1.National Research Tomsk Polytechnic UniversityTomskRussia

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