Piezoelectric, defined stress constants

The piezoelectric stress constant (e-constant) is defined as the polarization per unit volume induced by unit strain ... 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

The term piezoelectric nonlinearity is used here to describe relationship between mechanical and electrical fields (charge density D vs. stress a, strain x vs. electric field E) in which the proportionality constant d, is dependent on the driving field, Figure 13.1. Thus, for the direct piezoelectric effect one may write D = d(a)a and for the converse effect x = d(E)E. Similar relationships may be defined for other piezoelectric coefficients (g, h, and e) and combination of electro-mechanical variables. The piezoelectric nonlinearity is usually accompanied by the electro-mechanical (D vs. a or x vs. E) hysteresis, as shown in Figure 13.2. By hysteresis we shall simply mean, in the first approximation, that there is a phase lag between the driving field and the response. This phase lag may be accompanied by complex nonlinear processes leading to a more general definition of the hysteresis [2],... 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

Coefficients aij T) depend explicitly on temperature T. Coefficients afj, aijki, are supposed to be temperature independent, constants giju and Vijkimn determine the magnitude of the gradient energy. Tensors gijki, Oijki and positively defined. Tensor is the surface excess elastic moduli, p, p is the surface stress tensor [81,82], is the surface piezoelectric or piezomagnetic tensor [67], qijki are the bulk striction coefficients Ciju are components of elastic stiffness tensor [83]. 

The polarization P is a measure of the degree of piezoelectricity in a given material. In a piezoelectric material, a change in polarization AP results from an applied stress X or strain S under conditions of constant temperature and zero electric field. A linear relationship exists between AP and the piezoelectric constants. Because of material anisotropy, P is a vector that has three orthogonal components in the 1,2, and 3 directions. Alternatively, the piezoelectric constants can be defined as... 

The extent of piezoelectric activity depends upon the poling time, field strength, and temperature. The dipoles are aligned relative to the direction of the field during the poling process. The constants that characterize the activity of piezoelectric and pyroelectric films have been listed in Table 14.14. The axes are defined as length 1, width 2, and thickness 3. Two numeric subscripts are used to respectively denote the axis of polarization (applied electrical field) and the axis of mechanical strain or applied stress. 

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