Fractional viscoelastic models: master curve construction, interconversion, and numerical approximation

We use fractional viscoelastic models that result from the application of fractional calculus to the linear viscoelastic theory to characterize thermorheologically simple linear viscoelastic materials. Model parameters are obtained through an optimization procedure that simultaneously determines the time–temperature shift factors. We present analytical interconversion based on the fractional viscoelastic model between the main viscoelastic functions (relaxation modulus, creep compliance, storage modulus, and loss modulus) and the analytical forms of the relaxation and retardation spectra. We show that the fractional viscoelastic model can be approximated by a Prony series to any desired level of accuracy. This property allows the efficient determination of the fractional viscoelastic model response to any loading history using the well-known recursive relationships of Prony series models.