About Mathematical Models of Irreversible Polarization Processes of a Ferroelectric and Ferroelastic Polycrystals

Abstract

This chapter presents the prevalent mathematical models of irreversible processes in polycrystalline ferroelectric materials when they are subjected to intense electrical and mechanical influences. The main purpose of such models is to describe the dielectric hysteresis loops, with which the models of Rayleigh and Preisach coped well, though they were developed almost a 100 years ago. Nevertheless, in order to describe the whole gamut of material properties in irreversible polarization-depolarization processes, it was required in the last three decades to develop new approaches and methods that take into account the material structure and the physics of the process. In this chapter, we attempted to collect the most common one-dimensional models, with a view to give a brief description of the basics and approaches with the application of working formulas, algorithms and graphs of numerical calculations. On one-dimensional models, the basics of three-dimensional models are worked out, such as evolutionary laws, domains switching criteria, generalizations from “ hysteron ” to the polarization surface, and so on, so they are a necessary step in modeling. However, some of them proved to be so effective that they obtained the right to independent existence, as happened with the Preisach model, which found application in dynamic systems. This research is based on published articles, monographs, proceedings of conferences, and scientific reports of individual collectives published over the past 20 – 25 years. of polycrystalline ferroelectric media on the external effects of high-intensity electric and mechanical fields are considered. The bases of construction of each model are disassembled. The fundamentals of the construction of each of the well-known Rayleigh models, evolution models, models of plasticity theory, Preisach models, models of orientation switching, energy switching models, the Giles-Atherton model are analyzed and the results of their work in the form of hysteresis loops are presented. The main