1. Introduction

Polycrystalline ferroelectric materials or ceramics are active materials that, by virtue of their internal structure, have the ability to convert mechanical energy into electrical energy, and vice

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

versa. This means that by acting on the sample an electric field or mechanical stress, we observe a response in the form of an electric displacement and the strain [1]. When the intensity of external electric fields or a mechanical stresses are small, the deformations and electrical displacements caused by them are also small. Such processes are called reversible; they say that there is a linear response. The modeling of such response leads to the construction of constitutive relations in the form of linear algebraic equations, which, like the generalized Hooke law, connect external and internal parameters. We can say that the mathematical model of the behavior of such materials is described by the linear algebraic operators in which the elements of elastic, piezoelectric and dielectric constant tensors are found experimentally. The overwhelming majority of problems concerning the calculation of the physical characteristics of transducers with polarized before saturation piezoceramic elements are solved within the linear response of active materials. In this case, the complete formulation of the problem includes the equations of motion, the equations of electrostatics, geometric relationships, and the constitutive relations. The general solutions obey the corresponding initial and boundary conditions. Such problems in the mathematical plan are linear. In simple cases, they can be solved analytically, and in more complex cases, we have to use numerical methods, for example, the finite element method.

The situation changes dramatically as soon as external loads reach thresholds, and their intensity continues to increase. In this case, irreversible processes begin, expressed in the fact that the response of the material will already be nonlinear. The consequence of this is that under increasing loads we have some nonlinear equations, while for decreasing ones we have other nonlinear equations. The constitutive relations become nonlinear and ambiguous. Mathematically, they can no longer be described by simple algebraic relations, but it is necessary to use operator relations of hysteresis type.

Due to the small volume of the article, we confine ourselves of modeling irreversible polarization-depolarization processes by an electric field and mechanical stresses under isothermal processes. Irreversible processes associated with relaxation properties, with the influence of temperature, with the features of the influence of size the ferroelectric granules, the dynamics of processes, and some others will not be considered here. The main circle of questions will be connected with the analysis of existing mathematical models describing the response of the material to external influences of high intensity for the isothermal process: we will consider the principles of constructing the constitutive relations, analyze them, and formulate some conclusions.

2. Some experimental data

Figure 3. Dielectric hysteresis loop.

Figure 1. Deformation hysteresis loop.

Figure 2. Magnetic hysteresis loop.

The criterion for the correctness and adequacy of the work of any model is the acceptable coincidence of the predicted data with experimental data. It should be noted that a qualitative experiment is a very complex study, so most of them reflect only certain properties with simple effects. We note only those works that reflect the electric and elastic response due to the action of the electric field and mechanical stresses. Basically, these are the works where the properties of ferroelectric ceramics of the perovskite type are investigated: for example, BaTiO3, or a ceramics containing lead: PZT, PLZT 8/65/35. Interesting results [3] on the response of PLZT 8/65/35 on the effect of electric and mechanical fields and similar results [4–10] for complex loads show that the loops of electrical and deformation hysteresis depend significantly on the intensity of the operating fields. Uniaxial mechanical compressive stresses along the electric

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

http://dx.doi.org/10.5772/intechopen.78262

41

First, we note an interesting regularity: many irreversible processes have a similar response in the sense that the relationships between external and internal parameters are mathematically described by similar relationships. For example, in plasticity media, stresses cause elastic and residual deformations; in ferromagnets, the magnetic field leads to induced and remnant magnetization; in ferroelectrics, the electric field generates induced and residual polarization, etc. In the case of cyclic processes, the responses are described by hysteresis dependences, as shown in Figures 1–3. Therefore, it is not surprising that the mathematical methods being developed for the study of certain processes are often used to describe others. Mathematical modeling of nonlinear responses of polycrystalline ferroelectric materials plays a significant role [2], therefore, the creation and use of such models is based on experimental data.

About Mathematical Models of Irreversible Polarization Processes of a Ferroelectric and Ferroelastic… http://dx.doi.org/10.5772/intechopen.78262 41

Figure 1. Deformation hysteresis loop.

versa. This means that by acting on the sample an electric field or mechanical stress, we observe a response in the form of an electric displacement and the strain [1]. When the intensity of external electric fields or a mechanical stresses are small, the deformations and electrical displacements caused by them are also small. Such processes are called reversible; they say that there is a linear response. The modeling of such response leads to the construction of constitutive relations in the form of linear algebraic equations, which, like the generalized Hooke law, connect external and internal parameters. We can say that the mathematical model of the behavior of such materials is described by the linear algebraic operators in which the elements of elastic, piezoelectric and dielectric constant tensors are found experimentally. The overwhelming majority of problems concerning the calculation of the physical characteristics of transducers with polarized before saturation piezoceramic elements are solved within the linear response of active materials. In this case, the complete formulation of the problem includes the equations of motion, the equations of electrostatics, geometric relationships, and the constitutive relations. The general solutions obey the corresponding initial and boundary conditions. Such problems in the mathematical plan are linear. In simple cases, they can be solved analytically, and in more complex cases, we have to use

The situation changes dramatically as soon as external loads reach thresholds, and their intensity continues to increase. In this case, irreversible processes begin, expressed in the fact that the response of the material will already be nonlinear. The consequence of this is that under increasing loads we have some nonlinear equations, while for decreasing ones we have other nonlinear equations. The constitutive relations become nonlinear and ambiguous. Mathematically, they can no longer be described by simple algebraic relations, but it is necessary to

Due to the small volume of the article, we confine ourselves of modeling irreversible polarization-depolarization processes by an electric field and mechanical stresses under isothermal processes. Irreversible processes associated with relaxation properties, with the influence of temperature, with the features of the influence of size the ferroelectric granules, the dynamics of processes, and some others will not be considered here. The main circle of questions will be connected with the analysis of existing mathematical models describing the response of the material to external influences of high intensity for the isothermal process: we will consider the principles of constructing the constitutive relations, analyze them, and for-

First, we note an interesting regularity: many irreversible processes have a similar response in the sense that the relationships between external and internal parameters are mathematically described by similar relationships. For example, in plasticity media, stresses cause elastic and residual deformations; in ferromagnets, the magnetic field leads to induced and remnant magnetization; in ferroelectrics, the electric field generates induced and residual polarization, etc. In the case of cyclic processes, the responses are described by hysteresis dependences, as shown in Figures 1–3. Therefore, it is not surprising that the mathematical methods being developed for the study of certain processes are often used to describe others. Mathematical modeling of nonlinear responses of polycrystalline ferroelectric materials plays a significant

role [2], therefore, the creation and use of such models is based on experimental data.

numerical methods, for example, the finite element method.

use operator relations of hysteresis type.

mulate some conclusions.

40 Ferroelectrics and Their Applications

Figure 2. Magnetic hysteresis loop.

Figure 3. Dielectric hysteresis loop.
