Circular arc rules of complex plane plot for model parameters determination of viscoelastic material

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.