Variable-order fractional constitutive model for the time-dependent mechanical behavior of polymers across the glass transition

  • Ruifan Meng
  • Deshun YinEmail author
  • Siyu Lu
  • Guangjian Xiang
Regular Article
Part of the following topical collections:
  1. Focus Point on Fractional Differential Equations in Physics: Recent Advantages and Future Direction


In this paper, the variable-order fractional constitutive model is adopted to study the time-dependent mechanical behavior of polymers across the glass transition. The fractional order is assumed to vary with time to describe the evolution of the mechanical properties. In order to examine the proposed method, the stress responses of two representative polymers, PETG and PC, across the glass transition are experimentally obtained and compared with the model predictions. It is shown that, by adopting the parameter of critical strain, the variable-order fractional model is able to well fit the data at all temperatures, from below, through, to above the glass transition. Furthermore, the order functions are graphically plotted and analyzed to show the ability of the variable-order to capture the mechanical property evolution of polymers in all phases. It is observed that the features of the order curves at the glass transition temperature are a combination of those of the glassy phase and the rubbery phase. As a result, we demonstrate that the variable-order fractional model is efficient in describing the time-dependent behavior of polymers across the glass transition and the rule of order functions with temperature can intuitively predict the change of the mechanical properties.


  1. 1.
    R. Xiao, H. Sun, W. Chen, Int. J. Nonlinear Mech. 93, 7 (2017)ADSCrossRefGoogle Scholar
  2. 2.
    V. Srivastava, S.A. Chester, N.M. Ames, L. Anand, Int. J. Plasticity 26, 1138 (2010)CrossRefGoogle Scholar
  3. 3.
    R. Xiao, J. Choi, N. Lakhera, C.M. Yakacki, C.P. Frick, T.D. Nguyen, J. Mech. Phys. Solids 61, 1612 (2013)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Yu, G. Kang, K. Chen, Int. J. Plasticity 89, 29 (2017)CrossRefGoogle Scholar
  5. 5.
    M.C. Boyce, S. Socrate, P.G. Llana, Polymer 41, 2183 (2000)CrossRefGoogle Scholar
  6. 6.
    N. Billon, J. Appl. Polym. Sci. 125, 4390 (2012)CrossRefGoogle Scholar
  7. 7.
    G.Z. Voyiadjis, A. Samadi-Dooki, J. Appl. Phys. 119, 225104 (2016)ADSCrossRefGoogle Scholar
  8. 8.
    R.B. Dupaix, M.C. Boyce, Mech. Mater. 39, 39 (2007)CrossRefGoogle Scholar
  9. 9.
    D. Mathiesen, D. Vogtmann, R.B. Dupaix, Mech. Mater. 71, 74 (2014)CrossRefGoogle Scholar
  10. 10.
    H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, Commun. Nonlinear Sci. Numer. Simul. 64, 213 (2018)ADSCrossRefGoogle Scholar
  11. 11.
    D. Baleanu, A. Jajarmi, M. Hajipour, Nonlinear Dyn. 94, 397 (2018)CrossRefGoogle Scholar
  12. 12.
    J. Singh, D. Kumar, D. Baleanu, S. Rathore, Math. Methods Appl. Sci. 42, 1588 (2019)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Kumar, J. Singh, D. Baleanu, Physica A 492, 155 (2018)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Singh, D. Kumar, D. Baleanu, Math. Model. Nat. Phenom. 14, 303 (2019)CrossRefGoogle Scholar
  15. 15.
    D. Baleanu, A. Jajarmi, E. Bonyah, M. Hajipour, Adv. Differ. Equ. 2018, 230 (2018)CrossRefGoogle Scholar
  16. 16.
    R. Meng, D. Yin, C.S. Drapaca, Comput. Mech. 64, 163 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Hajipour, A. Jajarmi, D. Baleanu, H.G. Sun, Commun. Nonlinear Sci. 69, 119 (2019)CrossRefGoogle Scholar
  18. 18.
    A. Jajarmi, D. Baleanu, Chaos Solitons Fractals 113, 221 (2018)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Jajarmi, D. Baleanu, J. Vib. Control 24, 2430 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    K. Adolfsson, M. Enelund, P. Olsson, Mech. Time-Depend. Mater. 9, 15 (2005)ADSCrossRefGoogle Scholar
  21. 21.
    H. Khajehsaeid, Polym. Test. 68, 110 (2018)CrossRefGoogle Scholar
  22. 22.
    E. Kontou, S. Katsourinis, J. Appl. Polym. Sci. 133, 23 (2016)CrossRefGoogle Scholar
  23. 23.
    D. Ingman, J. Suzdalnitsky, M. Zeifman, J. Appl. Mech. 67, 383 (2000)ADSCrossRefGoogle Scholar
  24. 24.
    C.F. Lorenzo, T.T. Hartley, Nonlinear Dyn. 29, 57 (2002)CrossRefGoogle Scholar
  25. 25.
    C.F. Coimbra, Ann. Phys. 12, 692 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    D. Valério, J.S.D. Costa, Variable-order Fractional Derivatives and their Numerical Approximations (Elsevier North-Holland, Inc., 2011)Google Scholar
  27. 27.
    A.H. Bhrawy, M.A. Zaky, Nonlinear Dyn. 85, 1815 (2016)CrossRefGoogle Scholar
  28. 28.
    R. Almeida, D.F. Torres, Sci. World J. 2013, 915437 (2013)Google Scholar
  29. 29.
    J.P. Neto, M.C. Rui, D. Valerio, S. Vinga, D. Sierociuk, W. Malesza, M. Macias, A. Dzieliński, Comput. Math. Appl. 75, 3147 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Y. Bouras, D. Zorica, T.M. Atanacković, Z. Vrcelj, Appl. Math. Model. 55, 551 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    D. Ingman, J. Suzdalnitsky, J. Eng. Mech. 131, 763 (2005)CrossRefGoogle Scholar
  32. 32.
    L.E. Ramirez, C.F. Coimbra, Ann. Phys. 16, 543 (2007)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Z. Li, H. Wang, R. Xiao, S. Yang, Chaos, Solitons Fractals 102, 473 (2017)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    R. Meng, D. Yin, C. Zhou, H. Wu, Appl. Math. Model. 40, 398 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    W. Smit, H.D. Vries, Rheol. Acta 9, 525 (1970)CrossRefGoogle Scholar
  36. 36.
    L.E.S. Ramirez, C.F.M. Coimbra, Int. J. Differ. Eq. 2010, 846107 (2010)Google Scholar
  37. 37.
    H.G. Sun, C. Wen, Y.Q. Chen, Physica A 388, 4586 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of mechanics and materialsHohai UniversityNanjing JiangsuChina

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