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A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics

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  • Lei, Dong
  • Liang, Yingjie
  • Xiao, Rui

Abstract

We develop a fractional model to describe the thermomechanical behavior of amorphous thermoplastics. The fractional model is composed of two parallel fractional Maxwell elements. The first fractional Maxwell model is used to describe the glass transition, while the second component is aimed at describing the viscous flow. We further derive the analytical solutions for the stress relaxation modulus and complex modulus through Laplace transform. We then demonstrate the model is able to describe the master curves of the stress relaxation modulus, storage modulus and loss modulus, which all show two distinct transition regions. The obtained parameters show that the modulus of the two fractional Maxwell elements differs in 2–3 orders of magnitude, while the relaxation time differs in 7–9 orders of magnitude. Finally, we apply the model to describe the stress response of constant strain rate tests. The model, together with the parameters obtained from fitting the master curve of stress relaxation modulus, can accurately predict the temperature and strain rate dependent stress response.

Suggested Citation

  • Lei, Dong & Liang, Yingjie & Xiao, Rui, 2018. "A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 465-475.
    • Handle: RePEc:eee:phsmap:v:490:y:2018:i:c:p:465-475
    DOI: 10.1016/j.physa.2017.08.037
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    References listed on IDEAS

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    1. Carrera, Y. & Avila-de la Rosa, G. & Vernon-Carter, E.J. & Alvarez-Ramirez, J., 2017. "A fractional-order Maxwell model for non-Newtonian fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 276-285.
    2. Yin, Deshun & Meng, Ruifan & Duan, Xiaomeng & Li, Yanqing, 2014. "Mechanism of complicated volume deformation in polymers and its fractional time-based description," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 404(C), pages 1-7.
    3. Rodríguez, R.F. & Fujioka, J. & Salinas-Rodríguez, E., 2015. "Fractional correlation functions in simple viscoelastic liquids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 326-340.
    4. Wu, Guo-Cheng & Baleanu, Dumitru & Xie, He-Ping & Chen, Fu-Lai, 2016. "Chaos synchronization of fractional chaotic maps based on the stability condition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 374-383.
    5. Fan, Wenping & Jiang, Xiaoyun & Qi, Haitao, 2015. "Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 40-49.
    6. Sun, HongGuang & Hao, Xiaoxiao & Zhang, Yong & Baleanu, Dumitru, 2017. "Relaxation and diffusion models with non-singular kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 590-596.
    7. Chakrabarti, Rajarshi, 2012. "Dynamics of end-to-end loop formation for an isolated chain in viscoelastic fluid," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5326-5331.
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    Cited by:

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