Mechanics of Composite Materials

, Volume 54, Issue 6, pp 775–788 | Cite as

A Unified Approach to Determining the Effective Physicomechanical Characteristics of Filled Polymer Composites Based on Variational Principles

  • S. A. Bochkareva
  • N. Yu. Grishaeva
  • B. A. LyukshinEmail author
  • P. A. Lyukshin
  • N. Yu. Matolygina
  • S. V. Panin
  • Yu. A. Reutov

An approach to solving the known problem on determination of the effective physical and mechanical properties of dispersedly filled compositions is presented. The solution is based on the formulation and numerical implementation of boundary-value problems based on the fundamental equations of the theories of elasticity, thermal conductivity, electrostatics, and electrical conductivity. The general concept is to obtain detailed parameters of state of the composition under external actions with account of the real structure of the material. Averaging procedures are used to determine the effective properties of the compositions. The formulation and solution of boundary-value problems, in all cases, is reduced to the formulation of corresponding variational principles and their implementation by the methods of computational mechanics, in particular, by the finite element method. The effective characteristics of the composite under mechanical, thermal, and electromagnetic actions are calculated on the basis of energy considerations: the stored energy of a structurally inhomogeneous body is equal to the energy of a homogeneous comparison body under various actions. A comparison of calculated and experimental values of the effective characteristics attests to the applicability of the models proposed and the methods of their implementation.


deformation-strength characteristics effective properties thermophysical properties numerical model 



The was performed work within the framework of the PFI GAN project for 2013-2020 in the trend of basic research 23 [Registration number of research and development АААА-А16-116122010041-9].


  1. 1.
    B. E. Pobedrya, Mechanics of Composite Materials [in Russian], M., Izd. MGU (1984).Google Scholar
  2. 2.
    A. A. Tashkinov and V. E. Shavshukov, “Theoretical field approach to describing the deformation of multicomponent polycrystalline materials,” Vest. Samar Gos. Tekhn. Univ., Ser. Mat. Nauki, Iss.4, No. 33, 86-97 (2013).Google Scholar
  3. 3.
    A. S. Zharkov, I. I. Anisimov, A. V. Shchemilinin, S. A. Bochkarev, А. Lyukshin, and R. A. Zagorodnikov, “Investigation of the stress-strain state of a dispersedly filled polymer composite with use of volumetric models,” Mekh. Kompozit. Mater. Konstr., 18, No. 1, 16-34 (2012).Google Scholar
  4. 4.
    A. S. Zharkov, I. I. Anisimov, A. V. Litvinov, E. A. Chashchihin, V. I. Desyatykh, S. P. Ogorodnikov, S. A. Bochkarev, and B. A. Lyukshin, “Formation principles of the mechanical properties of high-energy filled polymer compositions,” Mekh. Kompozit. Mater. Konstr., 17, No. 2, 155-169 (2011).Google Scholar
  5. 5.
    S. V. Shil’ko and D. A. Chernous, “Modified Takayanaga deformation model of dispersegly filled composites, Part 4. Hyperelasticity and plastic flow,” Mekh. Kompozit. Mater. Konstr., 20, No. 3. 403-413 (2014).Google Scholar
  6. 6.
    S. V. Shil’ko and D. A. Chernous, “Modified Takayanaga deformation model of dispersegly filled composites, Part 2. Determinations of elastic moduli and yield point with account of an interphase layer,” Mekh. Kompozit. Mater. Konstr., 19, No. 2, 181-195 (2013).Google Scholar
  7. 7.
    I. I. Anisimov, E. A. Chashchihin, V. I. Desyatykh, S. P. Ogorodnikov, B. A. Lyukshin, S. A. Bochkareva, and N Yu. Grishaeva, “Development of calculation – experimental methods of optimization of the mechanical characteristics of high-energy filled polymer systems,” Mekh. Kompozit. Mater. Konstr., 17, No. 3, 293-305 (2011).Google Scholar
  8. 8.
    R. M. Christensen. Mechanics of Composite Materials, Jonh Wiley & Sons, New York-Chichester-Brisbane-Toronto (1982)??Google Scholar
  9. 9.
    V. V. Vasil’ev, Mehanics of Structures of Composite Materials [in Russian], M., Mashinostroenie (1988).Google Scholar
  10. 10.
    S. A. Ambartsumyan, Theory of Anisotropic Plates: Strength, Stability, and Vibrations [in Russian], M., Nauka (1987).Google Scholar
  11. 11.
    A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Strength of Polymer and Composite Materials [in Russian], Riga, Zinatne (1980).Google Scholar
  12. 12.
    B. A. Lyukshin, S. V. Shil’ko, S. V. Panin et al., “Dispersedly Filled Polymer Composites of Technical and Medical Purposes [in Russian], Novosibirsk, Izd. SO RAN (2017).Google Scholar
  13. 13.
    P. A. Lyukshin, B. A. Lyukshin, N. Yu. Matolygina, and S. V. Panin, “Determination of the effective thermophysical characteristics of a composite material,” Fiz. Mezomekh., 11, No. 5, 103-110 (2008).Google Scholar
  14. 14.
    M. Würkner, H. Berger, and U. Gabbert, “Numerical study of effective elastic properties of fiber reinforced composites with rhombic cell arrangements and imperfect interface,” Int. J. Eng. Sci., 63, 1-9 (2013).CrossRefGoogle Scholar
  15. 15.
    Yu.V. Sovetova, Yu. N. Sidorenko, and V. A. Skripnyak, “A multilevel approach to research of the effect of volume ratio of components of a fibrous unidirectional carbon-fiber plastic on its mechanical characteristics,” Vest. Tomsk Gos. Univ., Math. Mekh., No. 2, 77-89 (2014).Google Scholar
  16. 16.
    S. Konovalenko, A. Yu. Smolin, I. S. Konovalenko, V. V. Promakhov, and S. G. Psakhye, “Computer research of dependence of the mechanical properties of a brittle material on the partial concentration of different-size pores in its structure,” Vest. Tomsk Gos. Univ., Math. Mekh., No. 6, 79-87 (2013).Google Scholar
  17. 17.
    B. A., Lyukshin S. V., Panin S. A., Bochkareva N. Yu., Grishaeva P. A., Lyukshin and Yu. A. Reutov, “Modeling of filled polymeric composite materials in view of structural features,” Procedia Engineering, No. 113, 474-478 (2015).Google Scholar
  18. 18.
    J. Z. Liang and G. S. Liu. “A new heat transfer model of inorganic particulate-filled polymercomposites,” J. Mater. Sci., 44, No. 17, 4715-4720 (2009).CrossRefGoogle Scholar
  19. 19.
    J. Z. Liang and F. H. Li, “Simulation of heat transfer in hollow glass-bead-filled polypropylene composites by finite element method,” Polymer Testing, 26, No. 3, 419-424 (2007).CrossRefGoogle Scholar
  20. 20.
    I. V. Fleming and V. S. Kim, “Application of a calculation method of thermophysical properties of composite materials to cable rubbers,” Izv. Tomsk Polytekhn. Univ., 317, No. 4, 62-65 (2010).Google Scholar
  21. 21.
    V. A. Mikheyev, V. S. Sulaberidze, and V. D. Mushenko, “Research of the heat conductivity of composite materials on the basis of silicone with nanofillers,” Izv. Vuzov, Priborostroenie, 58, No. 7, 571-575 (2015).Google Scholar
  22. 22.
    P. A. Lyukshin, B. A. Lyukshin, N. Yu. Matolygina, S. V. Panin, “Determination of the effective thermophysical characteristics of a composite material,” Fiz. Mezomekh., 11, No. 5, 103-110 (2008).Google Scholar
  23. 23.
    S. A. Bochkareva, N. Yu. Grishaeva, B. A. Lyukchin, and P. A. Lyukchin, “Determination of the thermal conductivity coefficient of inhomogeneous media,” AIP Conf. Proc., No. 1623, 71-74 (2014).Google Scholar
  24. 24.
    N. Yu. Grishaeva, P. A. Lyukshin, B. A. Lyjukshin, S. V. Panin, S. A. Bochkarev, Yu. A. Reutov, and N. Yu. Matolygina, “Modification of thermophysical characteristics of polymers by introduction microfillers,” Mekh. Kompozit. Mater. Konstr., 22, No. 3, 342-361 (2016).Google Scholar
  25. 25.
    N. Yu. Grishaeva, P. A. Lyukshin, B. A. Lyukshin, Yu. A. Reutov, A. I. Reutov, and S. A. Bochkareva, “Calculation of heat conductivity of the wall of a multilayered pipe from inhomogeneous materials,” Mekh. Kompozit. Mater. Konstr., 23, No. 1,. 12-24 (2017).Google Scholar
  26. 26.
    A. K. Arzhnikov, M. P. Galanin, and A. V. Feoktistov,” Mathematical Model for Calculating the Physical Properties of a Nanocomposite with Tunnelling Electrical Conduction and Its Numerical Realization [in Russian], Preprints M. V. Keldysh IPM, ИПМ, No. 96 (2013). URL: (date of the reference{manipulation}: 03.07.2018).
  27. 27.
    G. I. Atabekov, et al., Theoretical Foundations of Electrical Engineering, Parts 2,3 [in Russuan], M., Energy (1979).Google Scholar
  28. 28.
    L. R. Neumann and K. S. Demirchyan, Theoretical Foundations of Electrical Engineering. [in Russuan], L., Energy (1975)Google Scholar
  29. 29.
    E. N. Yakovleva, V. B. Yakovlev, and I. V. SLavrov, “Comparative analysis of methods for calculating the dynamic characteristics composite dielectrics,” Int. Conf “ Intermatic-2012,” Moscow, December, 3-7, 2012, Moscow, MIREA, Part 3, 93-96 (2012).Google Scholar
  30. 30.
    A A. Snarskii and I. I. Zhenirovskii, “Percolation effects in thermoelectrical disordered biphase media,” Termoelektrichestvo, No. 3, 65-81 (2007).Google Scholar
  31. 31.
    N. Xie, W. Shao, L. Feng, L Lv, and L. Zhen, “Fractal analysis of disordered conductor-insulator composites with different conductor backbone structures near percolation threshold,” J. Phys. Chem. C. 116, Iss. 36, 19517-19525 (2012).Google Scholar
  32. 32.
    M.Ya.Sushko, and S. K. Кrys’kov, “Metod of compact groups in the theory of dielectric permeability of heterogeneous systems,” Zhurn. Tekhn. Fiz., 79, Iss. 3, 97-101 (2009).Google Scholar
  33. 33.
    Yu. I. Dimitrienko, A. P. Sokolov, and M. N. Markevich, “ Modeling the dielectric characteristics of composite materials on the basis of the method of asymptotic averaging,” Nauka Obrazov., Elektron. Zhurn., No. 1, 49-64 (2013).Google Scholar

Copyright information

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

Authors and Affiliations

  • S. A. Bochkareva
    • 1
    • 2
  • N. Yu. Grishaeva
    • 1
    • 2
  • B. A. Lyukshin
    • 1
    • 2
    • 3
    Email author
  • P. A. Lyukshin
    • 1
  • N. Yu. Matolygina
    • 1
  • S. V. Panin
    • 1
    • 4
  • Yu. A. Reutov
    • 2
    • 4
  1. 1.Institute of Physics and Strength and Materiology of the Siberian Branch of the Russian Academy of SciencesTomskRussia
  2. 2.Tomsk State University of Control Systems and RadioelectronicsTomskRussia
  3. 3.National Research Tomsk State UniversityTomskRussia
  4. 4.National Research Tomsk Polytechnical UniversityTomskRussia

Personalised recommendations