Application of Fractional Calculus to Modeling the Non-Linear Behaviors of Ferroelectric Polymer Composites: Viscoelasticity and Dielectricity.

Ferroelectric polymer composites normally show non-linear mechanical and electrical behaviors due to the viscoelastic and dielectric relaxation of polymer matrixes. In this paper, a fractional calculus approach is used to describe the non-linear behavior of ferroelectric polymer composites from both viscoelastic and dielectric perspectives. The fractional elements for viscoelasticity and dielectricity are "spring-pot" and "cap-resistor", which can capture the intermediate properties between spring and dashpot or capacitor and resistor, respectively. For modeling the viscoelastic deformation, the "spring-pot" equation is directly used as the fractional mechanical model. By contrast, for the dielectricity of ferroelectric polymer composites, which is usually characterized by dielectric constants and dielectric losses, the "cap-resistor" equation is further formulated into the frequency domain by Fourier transform to obtain the fractional order dielectric model. The comparisons with experimental results suggest that the proposed models can well describe the viscoelastic deformation as well as the frequency dependence of the dielectric constant and dielectric loss of ferroelectric polymer composites. It is noted that the fractional order dielectric model needs to be separated into two regions at low and high frequencies due to the polarization effect. Additionally, when the dipole relaxations occur at higher frequencies, the proposed model cannot describe the rise of the dielectric loss curve.