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

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Abstract.

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.

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References

  1. R. Xiao, H. Sun, W. Chen, Int. J. Nonlinear Mech. 93, 7 (2017)

    Article  ADS  Google Scholar 

  2. V. Srivastava, S.A. Chester, N.M. Ames, L. Anand, Int. J. Plasticity 26, 1138 (2010)

    Article  Google Scholar 

  3. R. Xiao, J. Choi, N. Lakhera, C.M. Yakacki, C.P. Frick, T.D. Nguyen, J. Mech. Phys. Solids 61, 1612 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  4. C. Yu, G. Kang, K. Chen, Int. J. Plasticity 89, 29 (2017)

    Article  Google Scholar 

  5. M.C. Boyce, S. Socrate, P.G. Llana, Polymer 41, 2183 (2000)

    Article  Google Scholar 

  6. N. Billon, J. Appl. Polym. Sci. 125, 4390 (2012)

    Article  Google Scholar 

  7. G.Z. Voyiadjis, A. Samadi-Dooki, J. Appl. Phys. 119, 225104 (2016)

    Article  ADS  Google Scholar 

  8. R.B. Dupaix, M.C. Boyce, Mech. Mater. 39, 39 (2007)

    Article  Google Scholar 

  9. D. Mathiesen, D. Vogtmann, R.B. Dupaix, Mech. Mater. 71, 74 (2014)

    Article  Google Scholar 

  10. H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, Commun. Nonlinear Sci. Numer. Simul. 64, 213 (2018)

    Article  ADS  Google Scholar 

  11. D. Baleanu, A. Jajarmi, M. Hajipour, Nonlinear Dyn. 94, 397 (2018)

    Article  Google Scholar 

  12. J. Singh, D. Kumar, D. Baleanu, S. Rathore, Math. Methods Appl. Sci. 42, 1588 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  13. D. Kumar, J. Singh, D. Baleanu, Physica A 492, 155 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Singh, D. Kumar, D. Baleanu, Math. Model. Nat. Phenom. 14, 303 (2019)

    Article  Google Scholar 

  15. D. Baleanu, A. Jajarmi, E. Bonyah, M. Hajipour, Adv. Differ. Equ. 2018, 230 (2018)

    Article  Google Scholar 

  16. R. Meng, D. Yin, C.S. Drapaca, Comput. Mech. 64, 163 (2019)

    Article  MathSciNet  Google Scholar 

  17. M. Hajipour, A. Jajarmi, D. Baleanu, H.G. Sun, Commun. Nonlinear Sci. 69, 119 (2019)

    Article  Google Scholar 

  18. A. Jajarmi, D. Baleanu, Chaos Solitons Fractals 113, 221 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  19. A. Jajarmi, D. Baleanu, J. Vib. Control 24, 2430 (2018)

    Article  MathSciNet  Google Scholar 

  20. K. Adolfsson, M. Enelund, P. Olsson, Mech. Time-Depend. Mater. 9, 15 (2005)

    Article  ADS  Google Scholar 

  21. H. Khajehsaeid, Polym. Test. 68, 110 (2018)

    Article  Google Scholar 

  22. E. Kontou, S. Katsourinis, J. Appl. Polym. Sci. 133, 23 (2016)

    Article  Google Scholar 

  23. D. Ingman, J. Suzdalnitsky, M. Zeifman, J. Appl. Mech. 67, 383 (2000)

    Article  ADS  Google Scholar 

  24. C.F. Lorenzo, T.T. Hartley, Nonlinear Dyn. 29, 57 (2002)

    Article  Google Scholar 

  25. C.F. Coimbra, Ann. Phys. 12, 692 (2003)

    Article  MathSciNet  Google Scholar 

  26. D. Valério, J.S.D. Costa, Variable-order Fractional Derivatives and their Numerical Approximations (Elsevier North-Holland, Inc., 2011)

  27. A.H. Bhrawy, M.A. Zaky, Nonlinear Dyn. 85, 1815 (2016)

    Article  Google Scholar 

  28. R. Almeida, D.F. Torres, Sci. World J. 2013, 915437 (2013)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  30. Y. Bouras, D. Zorica, T.M. Atanacković, Z. Vrcelj, Appl. Math. Model. 55, 551 (2018)

    Article  MathSciNet  Google Scholar 

  31. D. Ingman, J. Suzdalnitsky, J. Eng. Mech. 131, 763 (2005)

    Article  Google Scholar 

  32. L.E. Ramirez, C.F. Coimbra, Ann. Phys. 16, 543 (2007)

    Article  MathSciNet  Google Scholar 

  33. Z. Li, H. Wang, R. Xiao, S. Yang, Chaos, Solitons Fractals 102, 473 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  34. R. Meng, D. Yin, C. Zhou, H. Wu, Appl. Math. Model. 40, 398 (2016)

    Article  MathSciNet  Google Scholar 

  35. W. Smit, H.D. Vries, Rheol. Acta 9, 525 (1970)

    Article  Google Scholar 

  36. L.E.S. Ramirez, C.F.M. Coimbra, Int. J. Differ. Eq. 2010, 846107 (2010)

    Google Scholar 

  37. H.G. Sun, C. Wen, Y.Q. Chen, Physica A 388, 4586 (2009)

    Article  ADS  Google Scholar 

Download references

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Authors and Affiliations

  1. College of mechanics and materials, Hohai University, No. 8 Fochengxi Road, Jiangning District, 211100, Nanjing Jiangsu, China

    Ruifan Meng, Deshun Yin, Siyu Lu & Guangjian Xiang

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  1. Ruifan Meng
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  2. Deshun Yin
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  3. Siyu Lu
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  4. Guangjian Xiang
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Correspondence to Deshun Yin.

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Meng, R., Yin, D., Lu, S. et al. Variable-order fractional constitutive model for the time-dependent mechanical behavior of polymers across the glass transition. Eur. Phys. J. Plus 134, 376 (2019). https://doi.org/10.1140/epjp/i2019-12767-x

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