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Modelling “unusual” behaviour after strain reversal with hierarchical fractional models

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Abstract

Various experiments involving strain reversal have been described by Kitagawa et al. For example, after initially loading and unloading a polymer sample, stress “relaxation” showed an initial increase in stress. This behaviour was called “Unusual Behaviour after Strain Reversal” or UBASR by Kitagawa et al., and attributed to an effect of non-linear behaviour. This behaviour is however similar in kind to thermal expansion occurring before contraction after an indirect quench as observed some years ago by Kovacs. We show here that this behaviour can also occur in the range of linear viscoelasticity, and it is then simply a result of fading memory. Such fading memory is particularly well described by models containing β elements, or springpots. In this paper, Kitagawa’s data is analysed using a viscoelastic model containing springpots, described by a fractional differential constitutive equation. A novel method of integration based on Boltzmann superposition is developed. This method is shown to be applicable to non-linear viscoelasticity. The model affords a good phenomenological description of the changes in so-called “unusual” behaviour induced by non-linearity.

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References

  • Beris AN, Edwards BJ (1993) On the admissibility criteria for linear viscoelastic kernels. Rheol Acta 32:505–510

    CAS  Google Scholar 

  • Friedrich C, Schiessel H, Blumen A (1999) In: Siginer DA, De Kee D, Chhabra RP (eds) Advances in the flow and rheology of non-Newtonian fluids. Elsevier Amsterdam, p 429

  • Glöckle WG, Nonnenmacher TF (1991) Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24:6426–6434

    Google Scholar 

  • Heymans N (1996) Hierarchical models for viscoelasticity: dynamic behaviour in the linear range. Rheol Acta 35:508–519

    CAS  Google Scholar 

  • Heymans N (1998) Modelling non-linear and time-dependent behaviour of viscoelastic materials using hierarchical models. Fifth European Rheology Conference, Progress & Trends in Rheology, vol 5, pp 100–101

  • Heymans N (1999) Properties of hierarchical models for non-linear viscoelastic behaviour. CanCNSM 1st Canadian Conference on Nonlinear Solid Mechanics, Victoria, Canada, 16–20 June, pp 255–263

  • Heymans N (2000) Connection between threshold yield stress and double yield in semi-crystalline polymers. Eleventh International Conference on Deformation, Yield and Fracture of Polymers, Cambridge, UK, pp 109–112

  • Heymans N (2003) Implementation of fractional calculus using hierarchical models—application to the terminal transition of a complex polymer. DETC’03, Chicago, USA, Sept 2–6

  • Heymans N, Bauwens J-C (1994) Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol Acta 33:210–219

    CAS  Google Scholar 

  • Kitagawa M, Zhou D, Qiu J (1995) Stress-strain curves for solid polymers. Polym Eng Sci 35:1725–1732

    CAS  Google Scholar 

  • Kitagawa M, Zhou D, Murata M, Shimada K, Umeoka H (1999a) Unusual behavior of stress-strain relations during strain reversal. CanCNSM 1st Canadian Conference on Nonlinear Solid Mechanics, Victoria, Canada, 16–20 June, pp 321–330

  • Kitagawa M, Zhou D, Shimada K, Umeoka H (1999b) Anomalous behavior associated with unloading in polyethylene. J Soc Mater Sci Jpn 48:592–597

    CAS  Google Scholar 

  • Koeller RC (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51:299–307

    Google Scholar 

  • Kovacs AJ (1963) Transition vitreuse dans les polymères amorphes. Etude phénoménologique. Fortschr Hochpolym-Forsch 3:394–507

  • Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, San Diego

  • Schiessel H, Blumen A (1993) Hierarchical analogues to fractional relaxation equations. J Phys A Math Gen 26:5057–5069

    Article  Google Scholar 

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Authors and Affiliations

  1. Physique des Matériaux de Synthèse 259, Université Libre de Bruxelles, B-1050, Bruxelles, Belgium

    Nicole Heymans

  2. Kanazawa University, 920-8667, Kodatuno 2-40-20, Kanazawa, Japan

    Masayoshi Kitagawa

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  1. Nicole Heymans
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  2. Masayoshi Kitagawa
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Correspondence to Nicole Heymans.

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Heymans, N., Kitagawa, M. Modelling “unusual” behaviour after strain reversal with hierarchical fractional models. Rheol Acta 43, 383–389 (2004). https://doi.org/10.1007/s00397-003-0354-3

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  • DOI: https://doi.org/10.1007/s00397-003-0354-3

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