On fractional modeling of viscoelastic foams

Abstract : Empiric models have been introduced to describe frequency dependence of dielectric permittivity. Simple exponential models are often not satisfactory, while advanced non-exponential models (usually referred as “anomalous relaxation”) are commonly required to better explain experimental observations of complex systems. For viscoelastic materials, the so-called fractional derivatives models are powerful for both dynamic and loss moduli prediction. In this paper, the analysis of the main models used in the characterization of dielectric and viscoelastic materials such as five-parameter fractional Zener model and empiric Havriliak–Negami model are analysed. The fractional shape parameters describing the symmetric and asymmetric broadening of the complex modulus don't have the same influence in low and high frequencies. In contrast to the five-parameter Zener model, the empiric model asymmetry parameter has an influence on complex modulus at low frequencies comparing to the loss modulus peak frequency. A no resonance technique based on a forced vibrations procedure is used to investigate the frequency dependent complex shear modulus of a polyurethane foam, not influenced by its fluid phase, in the range 0.1–500 Hz. It is shown that the Havriliak–Negami model can predict the frequency dependence for a wide frequency range.