Andrew J. Kurdila - Vibrations of Linear Piezostructures

However, with cooling below the Curie temperature , a thermodynamic process drives a structural phase transition so that the final crystalline phase has a lower symmetry. At the lower temperature it can be shown [18] that the lower symmetry crystal phase has at least two energetically equivalent configurations or variants. Furthermore, with the application of an external electric field, it must be the case that it is possible switch among these crystalline variants in a reversible process. The ferroelectric material forms domains that consist of these energetically equivalent crystalline variants. Figure 1.3depicts schematically the and domains [31] that can appear in single crystal barium titanate [31]. Note in the figure that the polarization vectors are opposite from one domain to the next, and their average polarization over a macroscale can have zero effective polarization. Because of the presence of these domains, below the Curie temperature the polarization versus applied electric field takes the form of a hysteresis loop as shown in Figure 1.4. Initially, the domains cancel their effects over the macroscopic specimen and at . The polarization increases as in Figure 1.2for a range of electric field . When a critical value , the coercive electric field strength, is reached, the domains abruptly switch so that they are approximately well‐aligned with the external electric field. With all domains having approximately aligned polarization vectors, the polarization again follows a nonlinear single valued curve until saturation is achieved. When the electric field is reversed, and reaches the opposite coercive electric field strength , the domains switch again so their polarization vectors are approximately aligned with the second variant. The result of this cyclic process is that after the transient response there is a nonzero polarization, the spontaneous polarization, for an electric field strength . At a macroscopic scale, then, the effective or average polarization can switch with the application of the external electric field.

Figure 1.2 Polarization versus applied electrical field for ferroelectric above the Curie temperature .

Figure 1.3 and domains in , [31].

Source: Walter J. Merz, Domain Formation and Domain Wall Motion in Ferro‐electric BaTiO 3Single Crystals, em Physical Review, Volume 95, Number 3, August 1, 1954, pp. 690–698.

Figure 1.4 Polarization versus electrical field hysteresis below the Curie temperature .

In view of these observations, at a fundamental level, the micromechanics of piezoelectricity is understood in terms of crystalline asymmetry. While the most general theory of linear piezoelectricity of material continua in three dimensions is discussed in Chapter 5, intuition can be built by considering a one dimensional example. Figure 1.5depicts the direct piezoelectric effect graphically, while the converse effect is shown in Figure 1.6. For the specimens shown, the mechanical variables are the stress and strain , and the electrical variables include the electric field , electric displacement , voltage , and the electrical potential . In Figure 1.5we suppose that the top and bottom of the specimen are free to displace. A thin film electrode, one that does not alter the mechanical properties of the specimen, is applied to the top and bottom surfaces by a deposition or sputtering process. An ideal current meter, over which the potential difference is approximately zero, is attached to the top and bottom electroded surfaces. A positive stress is applied as shown. As we discuss in Chapter 5the constitutive laws that couple the electrical and mechanical variables can take many forms. In this one dimensional example we choose to express the dependency among the electrical and mechanical variables as

( 1.1)