A thermodynamic approach to hysteretic models in ferroelectrics

The purpose of the paper is to establish a constitutive model for the hysteretic properties in ferroelectrics. Both the polarization and the electric field are simultaneously independent variables so that the constitutive functions depend on both of them. This viewpoint is naturally related to the fact that an hysteresis loop is a closed curve surrounding the region of interest. For the sake of generality, the deformation of the material and the dependence on the temperature are allowed to occur. The constitutive functions are required to be consistent with the second law of thermodynamics. Among other results, the second law implies a general property on the relation between the polarization and the electric field via a differential equation. This equation shows a dependence fully characterized by the free energy and a dependence which is related to the dissipative character of the hysteresis. As a consequence, different hysteresis models may have the same free energy. Models compatible with thermodynamics are then determined by appropriate selections of the free energy and of the dissipative part. Correspondingly, major and minor hysteretic loops are plotted.