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A fractional derivative-based numerical approach to rate-dependent stress–strain relationship for viscoelastic materials

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Abstract

Strain/stress-controlled loading, loading–unloading, loading-relaxation (or creep), and corresponding cyclic tests are essential for characterizing the viscoelastic materials' rate-dependent stress–strain relationship. A three-parameter model is proposed based on the basic definition of fractional derivative viscoelasticity and time-varying viscosity. This model is applied to many complex loading conditions. The solutions for three monocyclic loading conditions are given and then further generalized to arbitrary linear loading conditions, which are assumed to be first-order functions of time. The generalized solution for the arbitrary linear loading path is validated by modelling the mechanical response of cyclic loading–unloading and loading–relaxation (or creep) tests. Four sets of experimental data for polymer materials are employed to demonstrate the proposed fractional derivative model's efficiency. The results show that it can accurately model strain/ stress-controlled response under various loading conditions using only three parameters. The model is then implemented in numerical software to explore its capacity further, and the simulation results show that it also succeeds in simulating cyclic loading–unloading tests.

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Acknowledgements

The present work is supported by the National Natural Science Foundation of China 51674266, the State Key Research Development Program of China 2016YFC0600704, and the Yueqi Outstanding Scholar Program of CUMTB 2017A03.

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Authors and Affiliations

  1. School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Teng Su, Jiawei Zhao & Zelin Liu

  2. School of Energy and Mining Engineering, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Hongwei Zhou

  3. Laboratory 3SR, Grenoble Alpes University, Grenoble, France

    Teng Su & Daniel Dias

  4. Antea Group, Antony, France

    Daniel Dias

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  1. Teng Su
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Correspondence to Hongwei Zhou.

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Su, T., Zhou, H., Zhao, J. et al. A fractional derivative-based numerical approach to rate-dependent stress–strain relationship for viscoelastic materials. Acta Mech 232, 2347–2359 (2021). https://doi.org/10.1007/s00707-021-02946-1

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