Circular arc rules of complex plane plot for model parameters determination of viscoelastic material
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Circular arc rules of complex plane plot for model parameters determination of viscoelastic material Yilin Yin1,2 · Zhenghong Yang1,2
· Meilun Shi1,2
Received: 2 October 2019 / Accepted: 28 August 2020 © Springer Nature B.V. 2020
Abstract We present a new approach to determine the rheological parameters of a mechanical model of viscoelastic materials. The fractional derivative solid model composed of a spring in series with a fractional derivative Kelvin–Voigt element has been employed to characterize the dynamic mechanical response of a real viscoelastic material. In dynamic mechanical analysis (DMA) measurements, the frequency-dependent loss modulus is plotted against the storage modulus in the form of complex plane plot. We find that the complex plane plot of the fractional derivative solid model is a depressed or distorted semicircle with its center below the real axis. The model parameters could be identified graphically via its complex plane curve, from which the spring constants can be obtained through the two intercepts of the extrapolated circular arc with the real axis, whereas the order of fractional dashpot can be estimated by the displacement of the semicircle center. The graphical method allows us to easily find rheological parameters, without using complicated calculation and special algorithms. Keywords Viscoelasticity · Fractional derivative model · Dynamic mechanical analysis (DMA) · Complex modulus · Complex plane · Parameter identification
1 Introduction Materials whose mechanical behavior exhibits the combination of elastic and viscous properties are generally addressed as viscoelastic. These materials show creep, stress relaxation,
B Z. Yang
[email protected] Y. Yin [email protected] M. Shi [email protected]
1
Key Laboratory of Advanced Civil Engineering Materials (Tongji University), Ministry of Education, 4800 Cao’an Road, Shanghai 201804, China
2
School of Materials Science and Engineering, Tongji University, 4800 Cao’an Road, Shanghai 201804, China
Mech Time-Depend Mater
and hysteresis. Polymers are the typical representatives of this special class of materials (Strobl 2007; van Krevelen and te Nijenhuis 2009). The mechanical behavior can be measured both in the time domain and in the frequency domain. In the time domain measurement, we can see creep and stress relaxation in the form of exponential decay with characteristic constant of time dimension. In the frequency domain measurement, the frequency of the input signal is varied over a certain range and the resulting steady-state harmonic response is measured. Viscoelastic behavior is usually characterized by mechanical model composed of linear spring and dashpot elements. The classical rheological models are characterized by two, three, and four elements, which we refer to as the Maxwell, Kelvin–Voigt, standard solid, Zener, standard liquid, and Burgers models (Ferry 1980; Tschoegl 1989; Makris and Kampas 2009; Mainardi and Spada 2011; Shaw and MacKnight 2018). These simple models, while providing qualitative guidan
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