Piezoelectric stress coefficients

The total pyroelectric response at constant stress, p , is the sum of the primary pyroelectic response, given by p, and the secondary pyroelectric response, which is the product of the direct piezoelectric stress coefficient gs, and the thermal expansion coefficients ... 

The parameter A is the effective area of the crystal, a is the stress, dq is the thickness of the quartz plate, and 25 is the piezoelectric stress coefficient [63]. Its value is 9.65 x 10 Cm for AT-cut quartz. Actually, putting down a number for the effective area of a quartz crystal. A, is not an easy task (end of Appendix A). The effective area is less than the total area of the plate because of energy trapping [22]. 

Piezoelectricity links the fields of electricity and acoustics. Piezoelectric materials are key components in acoustic transducers such as microphones, loudspeakers, transmitters, burglar alarms and submarine detectors. The Curie brothers [7] in 1880 first observed the phenomenon in quartz crystals. Langevin [8] in 1916 first reported the application of piezoelectrics to acoustics. He used piezoelectric quartz crystals in an ultrasonic sending and detection system - a forerunner to present day sonar systems. Subsequently, other materials with piezoelectric properties were discovered. These included the crystal Rochelle salt [9], the ceramics lead barium titanate/zirconate (pzt) and barium titanate [10] and the polymer poly(vinylidene fluoride) [11]. Other polymers such as nylon 11 [12], poly(vinyl chloride) [13] and poly (vinyl fluoride) [14] exhibit piezoelectric behavior, but to a much smaller extent. Strain constants characterize the piezoelectric response. These relate a vector quantity, the electrical field, to a tensor quantity, the mechanical stress (or strain). In this convention, the film orientation direction is denoted by 1, the width by 2 and the thickness by 3. Thus, the piezoelectric strain constant dl3 refers to a polymer film held in the orientation direction with the electrical field applied parallel to the thickness or 3 direction. The requirements for observing piezoelectricity in materials are a non-symmetric unit cell and a net dipole movement in the structure. There are 32-point groups, but only 30 of these have non-symmetric unit cells and are therefore capable of exhibiting piezoelectricity. Further, only 10 out of these twenty point groups exhibit both piezoelectricity and pyroelectricity. The piezoelectric strain constant, d, is related to the piezoelectric stress coefficient, g, by... 

The output forces Fi and F- are linearly proportional to the input current 7 and piezoelectric stress coefficient e and inversely proportional to the permittivity and circular frequency co. 

The Mason equivalent circuit may be derived directly from Eq. 19. It is sometimes called a transmission-line circuit model since the transcendental terms in the matrix appear in the same way when modeling power transmission lines. Most importantly, the circuit represents more than one resonance with these transcendental terms. Consider first an element that does not have piezoelectricity, implying the piezoelectric stress coefficient e = 0. The force-velocity relationships in the nonpiezoelectric element would then be... 

We estimated the elastic constants and piezoelectric coefficients numerically from changes in the free energy and polarization of the crystal at finite applied strains. The results are given in Table 11.1. Contributions to the direct piezoelectric stress coefficients, gs (/= 1, 2, 3), were estimated from numerical derivatives of Ap and of (cos cp). The direct piezoelectric strain coefficients were calculated from Eq. (12). 

Experiments of the type described have been reported for PHB and the results analysed in terms of a two-phase model.The overall piezoelectric stress coefficient was assumed to be composed of additive... 

The quantities dyjj and Cyk are known as the piezoelectric strain and piezoelectric stress coefficients, respectively. 

Table 2.1 Typical Piezoelectric Stress Coefficients for Selected Materials...

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