Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

Fractal and Fractional

Yang-Quan Chen¹  Jun-Sheng Duan²  Di-Chen Hu² 

[1] MESA Lab, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA;School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China;

关键词: constitutive model;    fractional calculus;    fractional derivative;    creep compliance;    hysteresis;    relaxation modulus;   

DOI  :  10.3390/fractalfract5020036

来源: DOAJ

【 摘 要 】

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale $log\left(\omega \right)$, and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale $log\left(\omega \right)$. The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders $\alpha$ and $\beta$ could be used to model real-world physical properties well.

【 授权许可】

Unknown