Mathematical modeling of hysteresis loops in ferroelectric materials
  In published papers, the Gibbs free energy of ferroelectric materials has usually been quantified by the retention of 6th or 8th order polarization terms. In this paper, a newly analytical model of Gibbs free energy, thereout, a new model of polarization-electric field hysteresis loops in ferroelectric materials has been derived mathematically. As a model validation, four patterns of polarization-electric field hysteresis loops of ferroelectric materials have been depicted by using the model. The calculated results indicated that the self-similar model can characterize the various patterns of hysteresis loops in ferroelectric materials through adjusting the external excitation or the synthetically parameter (e.g., electric, temperature, and stress, etc.) employed in the model. 
Zhi Ma, Liying Xi, Xuming Wang, Yanan Ma, Fu Zheng, Hua Gao, Hongfei   Liu, Huanming Chen
cond-mat.mtrl-sci