One model of polarization polycrystall ferroelectric materials in 3D case

One of the major problems in modeling polarizing processes in polycrystalline ferroelectrics materials is the problem of a formulation of constitutive equations between an electric field and polarization. There is a set of the models using power principles. For example, models and methods of the theory of plasticity (Landis C.M., Huber J.E., McMeeking R.M., Fleck N.A.), models of the theory of pinning sites (Jilys D.C., Atherton D.L., Smith, R.C., Ounaies Z., Hom C.L.), models of the theory of switching (Chen W., Hwang S.C., Huo Y., Jiang Q., Lu W., Fang D.N., Zhang Z.K.) and others. In the present work the method of the theory of pinning sites is generalized on a three-dimensional case. Constitutive equations are obtained in the form of system of the partial differential equations which manage to be reduced to system of the ordinary differential equations.