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Fractional viscoelastic models for interconverting linear viscoelastic functions of various polymeric structures

  • Stelios Katsourinis
  • Evagelia KontouEmail author
Original Contribution
  • 20 Downloads

Abstract

In this work, the capacity of the generalized fractional Maxwell (GFMM) and Kelvin-Voigt (GFKM) models for the interconversion of dynamic to static (creep and relaxation) functions, with regard to appropriate experimental data of various polymeric structures, is examined. The analysis is executed within the frame of linear viscoelasticity. Furthermore, a comparative study with the results produced by the implementation of the fractional Zener model has been performed. A good approximation of the generated viscoelastic functions by GFKM and GFMM model was postulated. Concerning the fractional Zener model, it can be extracted that its effectiveness to the interconversion of the viscoelastic functions is dependent on the specific material’s viscoelastic response, and the wideness of the time/frequency region examined. It was found that the incorporation of a correction factor in the calculation procedure of the creep compliance function can result in significantly better results, regardless of the model used.

Graphical abstract

Keywords

Interconversion Viscoelastic functions Fractional calculus 

Notes

References

  1. Alcoutlabi M, Martinez Vega JJ (2003) Modeling of the viscoelastic behavior of amorphous polymers by the differential and integration fractional method: the relaxation spectrum H(τ). Polymer 44:7199–7208CrossRefGoogle Scholar
  2. Alves NM, Gomez Ribelles JL, Gomez Tejedor JA, Mano JF (2004) Viscoelastic behavior of poly(methyl methacrylate) networks with different cross-linking degrees. Macromolecules 37:3735–3744CrossRefGoogle Scholar
  3. Berry GC, Plazek DJ (1997) On the use of stretched – exponential functions for both linear viscoelastic creep and stress relaxation. Rheol Acta 36:320–329CrossRefGoogle Scholar
  4. Emri I, Tschoegl NW (1994) Generating line spectra from experimental responses. Part 4. Application to experimental data. Rheol Acta 33:60–70CrossRefGoogle Scholar
  5. Emri I, von Bernstorff BS, Cvelbar R, NIkonov A (2005) Re-examination of the approximate methods for interconversion between frequency and time-dependent material functions. J Non-Newtonian Fluid Mech 129:75–84CrossRefGoogle Scholar
  6. Fernández P, Rodrίguez D, Lamela MJ, Fernández-Canteli A (2011) Study of the interconversion between viscoelastic behavior functions of PMMA. Mechanics of Time Dependent Materials 15:169–180CrossRefGoogle Scholar
  7. Ferry JD (1980) Viscoelastic behavior of polymers. Wiley, New YorkGoogle Scholar
  8. Georgiopoulos P, Kontou E, Niaounakis M (2014) Thermomechanical properties and rheological behavior of biodegradable composites. Polym Compos 35(6):140–1149.  https://doi.org/10.1002/pc22761 Google Scholar
  9. Heymans N, Bauwens JC (1994) Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol Acta 33:210–219CrossRefGoogle Scholar
  10. Katicha SW, Flintsch GW (2012) Fractional viscoelastic models: master curve construction, interconversion, and numerical approximation. Rheol Acta 51:675–689.  https://doi.org/10.1007/s00397-012-0625-y CrossRefGoogle Scholar
  11. Katicha SW, Apeagyei AK, Flintsch GW, Loulizi A (2014) Universal linear viscoelastic approximation property of fractional viscoelastic models with application to asphalt concrete. Mech Tim Dep Mater 18:555–571.  https://doi.org/10.1007/s11043-014-9241-9 CrossRefGoogle Scholar
  12. Katsourinis S, Kontou E (2018) Comparing interconversion methods between linear viscoelastic material functions. Mechanics of Time-Depend Materials 22(3):401–419.  https://doi.org/10.1007/s11043-017-9363-y CrossRefGoogle Scholar
  13. Kontou E, Katsourinis S (2016) Application of a fractional model for simulation of the viscoelastic functions of polymers. J Appl Polym Sci 133.  https://doi.org/10.1002/APP.43505
  14. Liu Y (2001) A direct method for obtaining discrete relaxation spectra from creep data. Rheol Acta 40:256–260CrossRefGoogle Scholar
  15. Mainardi F (2010a) An historical perspective on fractional calculus in linear viscoelasticity. Arxiv preprint arXiv:10072959Google Scholar
  16. Mainardi F (2010b) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. Imperial College, LondonCrossRefGoogle Scholar
  17. Mainardi F, Spada G (2011) Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur Phys J Spec Top 193(1):133–160CrossRefGoogle Scholar
  18. Ngai KL, Plazek DJ (1986) A quantitative explanation of the difference in the temperature dependences of the viscoelastic softening and terminal dispersions of linear amorphous polymers. J Polym Sci B Polym Phys 24(3):619–632.  https://doi.org/10.1002/polb.1986.090240310 CrossRefGoogle Scholar
  19. Ngai KL, Plazek DJ, Deo SS (1987) Physical origin of the anomalous temperature dependence of the steady-state compliance of low molecular weight polystyrene. Macromolecules 20(12):3047–3054.  https://doi.org/10.1021/ma00178a018 CrossRefGoogle Scholar
  20. Papoulia KD, Panoskaltsis VP, Kurup NV, Korovajchuk I (2010) Rheological representation of fractional order viscoelastic material models. Rheol Acta 49(4):381–400.  https://doi.org/10.1007/s00397-010-0436-y CrossRefGoogle Scholar
  21. Park SW, Schapery RA (1999a) Methods of interconversion between linear viscoelastic material functions. Part I. A numerical method based on Prony series. Int J Solids Struct 36:1653–1675CrossRefGoogle Scholar
  22. Park SW, Schapery RA (1999b) Methods of interconversion between linear viscoelastic material functions. Part II. An approximate analytical method. Int J Solids Struct 36(11):1677–1699CrossRefGoogle Scholar
  23. Pritz TJ (2003) Five-parameter fractional derivative model for polymeric damping materials. J Sound Vibr 265(5):935–952CrossRefGoogle Scholar
  24. Sane SB, Knauss WG (2001) The time-dependent bulk response of poly (methyl methacrylate). Mechanics of Time Dependent Materials 5:293–324CrossRefGoogle Scholar
  25. Saprunov I, Gergesova M, Emri I (2014) Prediction of viscoelastic material functions from constant stress- or stain-rate experiments. Mechanics of Time Dependent Materials 18:349–372CrossRefGoogle Scholar
  26. Schiessel H, Blumen A (1995) Mesoscopic pictures of the sol-gel transition: ladder models and fractal networks. Macromolecules 28(11):4013–4019.  https://doi.org/10.1021/ma00115a038 CrossRefGoogle Scholar
  27. Schiessel H, Metzler R, Blumen A, Nonnenmacher T (1995) Generalized viscoelastic models: their fractional equations with solutions. J Phys A Math Gen 28(23):6567–6584.  https://doi.org/10.1088/0305-4470/28/23/012 CrossRefGoogle Scholar
  28. Sorvari J, Malinen M (2007) Numerical interconversion between linear viscoelastic material functions with regularization. Int J Solids Struct 44:1291–1303CrossRefGoogle Scholar
  29. Tschoegl NW (1988) The phenomenological theory of linear viscoelastic behavior, an introduction. Springer-Verlag, New YorkGoogle Scholar
  30. Tschoegl NW, Knauss WG, Emri I (2002) Poisson’s ratio in linear viscoelasticity – a critical review. Mechanics of Time Dependent Materials 6:3–51CrossRefGoogle Scholar
  31. Williams G, Watts DC (1970) Non-symmetrical dielectric relaxation behavior arising from a simple empirical decay function. Trans Faraday Soc 66:80–85CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematical and Physical Sciences, Section of MechanicsNational Technical University of AthensAthensGreece

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