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Journal of Mechanical Science and Technology

, Volume 33, Issue 1, pp 139–147 | Cite as

A fractional approach to the time-temperature dependence of dynamic viscoelastic behavior

  • Z. L. Li
  • Y. Qin
  • B. Sun
  • C. L. Jia
  • W. J. Zhang
  • B. J. Yan
  • Q. L. Shi
Article
  • 3 Downloads

Abstract

Fractional derivative and WLF equation are effective in describing the dynamic behavior and time-temperature effect of viscoelastic damping materials, respectively. These approaches have essentially evolved from the viscoelastic constitutive behavior. Based on such intrinsic relation, a fractional time-temperature superposition principle model (FTTSPM) that integrates the fractional constitutive relation and WLF equation was proposed. The parameters of this model were determined by performing tensile and DMA tests, and the master curves at 5 °C constructed by FTTSPM and WLF equation were compared. The theoretical prediction over the extended frequency span as the master curves was made by using the fractional standard linear solid model (FSLSM) to validate FTTSPM. The numerical results show that FTTSPM conforms to the time-temperature superposition principle. The parameters α and B′ in this model denote the impact of the material and environment on the shifted factor, respectively. For the storage and loss modulus, the extended frequency obtained by FTTSPM is broader than that obtained by the WLF equation. Moreover, the evaluation of the storage and loss modulus by FTTSPM is much closer to the theoretical prediction compared with that by the WLF equation. Therefore, FTTSPM is a concise and experiment-based approach with a higher precision and greater frequency-extended capacity compared with the WLF equation. However, FTTSPM inevitably faces a vertical shift when non-thermo-rheologically simple materials are considered. The physical mechanism and practical application of FTTSPM will be examined in further research.

Keywords

Dynamic behavior Fractional model Master curve Time-temperature dependence Viscoelastic modulus 

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References

  1. [1]
    W. Y. Jung and A. J. Aref, A combined honeycomb and solid viscoelastic material for structural damping applications, Mechanics of Materials, 35 (8) (2003) 831–844.CrossRefGoogle Scholar
  2. [2]
    C. H. Park, S. J. Ahn, H. C. Park and S. Na, Modeling of a hybrid passive damping system, Journal of Mechanical Science and Technology, 19 (1) (2005) 127–135.CrossRefGoogle Scholar
  3. [3]
    R. K. Singh, R. Kant, S. S. Pandey, M. Asfer, B. Bhattacharya, P. k. panigrahi and S. Bhattacharya, Passive vibration damping using polymer pads with microchannel arrays, Journal of Microelectromechanical Systems, 22 (3) (2013) 695–707.CrossRefGoogle Scholar
  4. [4]
    M. Mohammadimehr, A. A. Monajemi and M. Moradi, Vibration analysis of viscoelastic tapered micro-rod based onstrain gradient theory resting on visco-pasternak foundation using DQM, Journal of Mechanical Science and Technology, 29 (6) (2015) 2297–2305.CrossRefGoogle Scholar
  5. [5]
    Z. L. Zhang, S. Q. Li, and W. G. Zhu, Temperature spectrum model of dynamic mechanical properties for viscoelastic damping materials, Journal of Mechanical Engineering, 47 (20) (2011) 135.CrossRefGoogle Scholar
  6. [6]
    D. G. Fesko and N. W. Tschoegl, Time-temperature superposition in thermorheologically complex materials, Journal of Polymer Science Polymer Symposia, 35 (1) (2010) 51–69.CrossRefGoogle Scholar
  7. [7]
    P. Micha, On the use of the WLF model in polymer and foods, Critical Reviews in Food Science and Nutrition, 32 (1) (1992) 59–66.CrossRefGoogle Scholar
  8. [8]
    B. K. Ashokan and J. L. Kokini, Determination of the WLF constants of cooked soy flour and their dependence on the extent of cooking, Rheologica Acta, 45 (2) (2005) 192–201.CrossRefGoogle Scholar
  9. [9]
    A. Mathias, K. Ulrich and F. Petra, Evaluation of the relevance of the glassy state as stability criterion for freeze-dried bacteria by application of the Arrhenius and WLF model, Cryobiology, 65 (3) (2012) 308–318.CrossRefGoogle Scholar
  10. [10]
    P. Lima, S. Silva, J. Oliveira and V. Costa, Rheological properties of ground tyre rubber based thermoplasitc elastomeric blends, Polymer Testing, 45 (2015) 58–67.CrossRefGoogle Scholar
  11. [11]
    J. R. Lin and L. W. Chen, The mechanical-viscoelastic model and WLF relationship in shape memorized linear ether-type polyurethanes, Journal of Polymer Research, 6 (1) (1999) 35–44.CrossRefGoogle Scholar
  12. [12]
    J. Dudowicz, J. F. Douglas and K. F. Freed, The meaning of the “universal” WLF parameters of glass-forming polymer liquids, The Journal of Chemical Physics, 142 (1) (2015) 014905.CrossRefGoogle Scholar
  13. [13]
    D. W. Schaffner, The application of the WLF equation to predict lag time as a function of temperature for three psychrotrophic bacteria, International Journal of Food Microbiology, 27 (2–3) (1995) 107–115.CrossRefGoogle Scholar
  14. [14]
    J. L. Zheng, S T Lu and X. G. Tian, Viscoelastic damage characteristics of asphalt based on creep test, Engineering Mechanics, 25 (2008) 193–196.Google Scholar
  15. [15]
    F. Zhu, G. W. Xu and W. P. Ding, Tube theory based analysis on the rheological behavior of wheat gluten dough, Transactions of the Chinese Society of Agricultural Engineering, 23 (7) (2007) 24–29.Google Scholar
  16. [16]
    B. H. Liu, J. Zhou, Y. T. Sun, Y. Wang, J. Xu and Y. B. Li, An experimental study on the dynamic viscoelasticity of PVB film material for vehicle, Automotive Engineering, 34 (10) (2012) 898–904+927.Google Scholar
  17. [17]
    M. R. Permoon, J. Rashidinia, A. Parsa, H. Haddadpour and R. Salehi, Application of radial basis functions and sinc method for solving the forced vibration of fractional viscoelastic beam, Journal of Mechanical Science and Technology, 30 (7) (2016) 3001–3008.CrossRefGoogle Scholar
  18. [18]
    Z. H. Tang, G. H. Luo, W. Chen and X. G. Yang, Dynamic characteristics of vibration system including rubber isolator, Journal of Nanjing University of Aeronautics & Astronautics, 46 (2) (2014) 285–291.Google Scholar
  19. [19]
    Z. L. Li, D. G. Sun, B. Sun, B. J. Yan, B. H. Han and J. Meng, Fractional model of viscoelastic oscillator and application to a crawler tractor, Noise Control Engineering Journal, 64 (3) (2016) 388–402.CrossRefGoogle Scholar
  20. [20]
    K. X. Hu and K. Q. Zhu, A note on fractional Maxwell model for PMMA and PTFE, Polymer Testing, 30 (7) (2011) 797–799.CrossRefGoogle Scholar
  21. [21]
    A. Hernández-Jiménez, J. Hernández-Santiago, A. Macias-García and J. Sánchez-González, Relaxation modulus in PMMA and PTFE fitting by fractional Maxwell model, Polymer Testing, 21 (3) (2002) 325–331.CrossRefGoogle Scholar
  22. [22]
    A. W. Wharmby and R. L. Bagley, Modifying Maxwell’s equations for dielectric materials based on techniques from viscoelasticity and concepts from fractional calculus, International Journal of Engineering Science, 79 (2014) 59–80.CrossRefzbMATHGoogle Scholar
  23. [23]
    L. L. Cao, Y. Li, G. H. Tian, B. D. Liu and Y. Q. Chen, Time domain analysis of the fractional order weighted distributed parameter Maxwell model, Computers & Mathematics with Applications, 66 (5) (2013) 813–823.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J. S. Hwang and T. Y. Hsu, A fractional derivative model to include effect of ambient temperature on HDR bearings, Engineering Structures, 23 (5) (2001) 484–490.CrossRefGoogle Scholar
  25. [25]
    S. Y. Zhu, C. B. Cai and P. D. Spanos, A nonlinear and fractional derivative viscoelastic model for rail pads in the dynamic analysis of coupled vehicle-slab track systems, Journal of Sound and Vibration, 335 (2015) 304–320.CrossRefGoogle Scholar
  26. [26]
    F. Renaud, G. Chevallier, J. L. Dion and R. Lemaire, Viscoelasticity measurements and identification of viscoelastic parametric models, ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA (2011) 701–708.Google Scholar
  27. [27]
    M. L. Williams, R. F. Landel and J. D. Ferry, The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, Journal of the American Chemical Society, 77 (14) (1955) 3701–3707.CrossRefGoogle Scholar
  28. [28]
    P. Jindal, R. N. Yadav and N. Kumar, Dynamic mechanical characterization of PC/MWCNT composites under variable temperature conditions, Iranian Polymer Journal, 26 (6) (2017) 445–452.CrossRefGoogle Scholar
  29. [29]
    J. J. Espindola, M. S. Joao and M. O. Eduardo, A generalized fractional derivative approach to viscoelastic material properties measurement, Applied Mathematics and Computation, 164 (2) (2005) 493–506.CrossRefzbMATHGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Z. L. Li
    • 1
  • Y. Qin
    • 1
  • B. Sun
    • 2
  • C. L. Jia
    • 1
  • W. J. Zhang
    • 1
  • B. J. Yan
    • 1
  • Q. L. Shi
    • 1
  1. 1.School of Mechanical EngineeringTaiyuan University of Science and TechnologyTaiyuanChina
  2. 2.School of Applied ScienceTaiyuan University of Science and TechnologyTaiyuanChina

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