Piezoelectric constant

There are two variants of piezoelectric constant matrices, piezoelectric stress and piezoelectric strain. The second of these can be obtained from the former by multiplying by the inverse elastic constant matrix. For many materials the piezoelectric constants are zero by symmetry if there is a centre of inversion. The piezoelectric stress constants are derived from the second derivative matrices according to the relationship  

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The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

Table 4.1. Second- and third-order piezoelectric constants (after Davison and Graham [79D01]).

The ratio of third- to second-order piezoelectric constants has also been determined for x-cut quartz with the acceleration pulse loading method [77G05]. Two experiments yielded values for Cm/Cu of 15.0 and 16.6 compared to the ratio of 15.3 [72G03] determined from the fit to the 25 shock loading experiments. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

Several structural theories of piezoelectricity [72M01, 72M02, 72A05, 74H03] have been proposed but apparently none have been found entirely satisfactory, and nonlinear piezoelectricity is not explicitly treated. With such limited second-order theories, physical interpretations of higher-order piezoelectric constants are speculative, but such speculations may help to place some constraints on an acceptable piezoelectric theory. 

The piezoelectric constant studies are perhaps the most unique of the shock studies in the elastic range. The various investigations on quartz and lithium niobate represent perhaps the most detailed investigation ever conducted on shock-compressed matter. The direct measurement of the piezoelectric polarization at large strain has resulted in perhaps the most precise determinations of the linear constants for quartz and lithium niobate by any technique. The direct nature of the shock measurements is in sharp contrast to the ultrasonic studies in which the piezoelectric constants are determined indirectly as changes in wavespeed for various electrical boundary conditions. 

The most distinctive aspect of the shock work is the determination of higher-order piezoelectric constants. The values determined for the constants are, by far, the most accurate available for quartz and lithium niobate, again due to the direct nature of the measurements. Unfortunately it has not been possible to determine the full set of constants. Given the expense and destructive nature of the shock experiment, it is unlikely that a full set of higher-order piezoelectric constants can be determined. A less expensive investigation of higher-order constants could be conducted with the ramp wave or acceleration wave loading experiment described in the chapter. 

The measured pyroelectric coefficient can be represented as the sum of the first coefficient (real pyroelectric coefficient - pt ) and the second coefficient, which depends on the piezoelectric constant (dtJ), the thermal... 

Here, P is the piezoelectric constant of the piezoelectric tube, is the lateral stiffness of the cantilever. 

The geometry of the tube scanner also provides a purely electrical method to self-test and self-calibrate, especially for measuring the piezoelectric constants in a cryogenic environment. The piezoelectric constant varies with temperature in a complicated manner, and also with the particular batch of materials by the manufacturer and time (the aging effect). 

From Fig. 9.16, we obtain c/33 1.05 kN, a value consistent with the value listed in the catalog (1.27 kN). The value might be somewhat lower than the true value because the bonding of the tube ends is not perfectly rigid. If one end of the tube is free, or both ends are free, the deformation pattern varies significantly at the end(s). The net end effect is to reduce the value of the double piezoelectric response. Even if the end-bonding condition is unknown, an accurate measurement of the temperature or time variation of the piezoelectric constant can still be achieved. In other words, if the piezoelectric scanner is calibrated by a direct mechanical measurement or by the scale of images at one temperature, then its variation can be precisely determined by the electrical measurements based on double piezoelectric responses. 

Detach the power supply and check the piezoelectric constant. 

See Local density of states Lead zirconate titanate ceramics 217—220 chemical composition 218 coupling constants 220 Curie point 218 depoling field 219 piezoelectric constants 220 quality number 219 Leading-Bloch-waves approximation 123 Level motion-demagnifier 271 Liquid-crystal molecules 338 Living cell 341... 

See Lead zirconate titanate ceramics Piezoelectric constants 220 Piezoelectricity 213—221... 

A nanometer-sized silicon nitride, compacted into bulk sample, showed a strong piezoelectric constant 2613 X 10 12 [C/N]. It is interpreted by the charge accumulation in the interfaces and the surfaces of microvoids (67). 

Values of piezoelectric constants are, however, very scattered among polymers. In the case of oriented poly(y-methyl L-glutamate) film, the piezoelectric strain constant (d-constant) amounts to as much as 10 x 10 8 cgsesu when elongated in a direction at 45° to the draw-axis (Fukada, 1970), which is comparable with d = 6.5 x 10 8 cgsesu for X-cut... 

The piezoelectric constant of polymer films is usually a function of the frequency of the applied strain, and the constant is expressed by a complex quantity. In other words, the open-circuit voltage across the film surfaces is not in phase with the applied strain and the short-circuit current is not in phase with the strain rate. This effect, first pointed out by Fukada, Date and Emura (1968) and designated piezoelectric relaxation or dispersion, will be discussed in this review in terms of irreversible thermodynamics and composite-system theory. 

The piezoelectric constant eik is a third-rank tensor and vanishes when the material has a center of symmetry. When the strain is not uniform, however, a higher order piezoelectricity appears in proportion to the... 

When S varies with the frequency to under the condition E = 0, the variation of D arises only from the piezoelectric effect. When the material is biased with a d.c. field E0, the apparent piezoelectric constant is given by... 

In the usual experiment where E0 = 0, the term kSE in Eq. (9) does not make any contribution as far as the electrical response with the same frequency as the mechanical excitation is concerned. However, as will be described in 2.2 and 2.4, the piezoelectric constant of a polymer film is sometimes a function of the electrostriction constant which plays an important role in the anisotropy and relaxational behavior of the piezoelectric effect. 

Even for poly(y-methyl L-glutamate) with a high piezoelectric constant (d= 10 7 cgsesu), the electro-mechanical coupling constant k defined by... 

When an electric field E is applied, Eq changes by an amount proportional to E, as usually expressed by (a — 1) E/4n, where e is the dielectric constant. When a strain is applied to the film, changes in and hence changes in polarization are divided into two components 1. displacement equal to the macroscopic displacement and 2. residual displacement. The latter is the internal strain and causes the intrinsic piezoelectricity. The effect of internal strain on P is expressed by eu, where e is the intrinsic piezoelectric constant and u is the elongational strain along the x-axis. The electric displacement D can therefore be written... 

Dielectric constant and piezoelectric constant are assumed to depend on... 

III. Methods for Measuring the Piezoelectricity and Electrostriction Constant of Polymer Films 3.1. Measurement of the Piezoelectric Constant... 

B) When an alternating voltage (frequency = co) is applied to the film with a c. bias voltage Vo, the film is strained with a frequency co. The induced strain consists of two parts, one independent of V0 and the other proportional to V0. The former is the piezoelectric response and the latter the electrostrictive one, if the piezoelectric constant is assumed to be independent of V0 (Kawai (1), 1969). 

However, in contrast to the cases of complex elastic modulus G and dielectric constant e, the imaginary part of the piezoelectric constant, e", does not necessarily imply an energy loss (Holland, 1967). In the former two, G"/G and e"/e express the ratio of energy dissipation per cycle to the total stored energy, but e"/e does not have such a meaning because the piezoelectric effect is a cross-coupling effect between elastic and electric freedoms. As a consequence, e" is not a positive definite quantity in contrast to G" and e". In a similar way to e, however, the Kramers-Kronig relations (Landau and Lifshitz, 1958) hold for e ... 

In this section, we will derive a thermodynamic expression of the complex piezoelectric constant e for a single-phase system with a single relaxation process. 

Piezoelectric polymer film is usually partially crystalline and the crystallites are embedded in the amorphous phase, which exhibits mechanical relaxations. Therefore, the strain of each crystallite, S, may differ in both amplitude and phase from that of the film as a whole, S. In this case the complex piezoelectric constant of the film is written by putting S/S — K (complex quantity) in Eq. (62) as... 

Fig. 14. Estimation of relaxation strength from temperature dispersion of piezoelectric constant...

Fig. 27. Hysteresis of apparent piezoelectric constant of roll-drawn polyfvinylidene fluoride) film at room temperature plotted against dc bias field. Draw-ratio = 7 (Oshiki and Fukada, 1972)...

Fig. 27 indicates the apparent piezoelectric constant e of roll-drawn PVDF as a function of static bias field E0 (Oshiki and Fukada, 1972). The value of e at E0=0 represents the true piezoelectric constant e. The curve exhibits a hysteresis and the polarity of e changes according to the poling history. If the piezoelectricity in /)-form PVDF originates from the polarization charge due to spontaneous polarization, inversion of polarity of e would mean the inversion of the polarization by the external field and hence /S-form PVDF may be a ferroelectric material, as was first suggested by Nakamura and Wada (1971). 

Moreover, a large scatter of the piezoelectric constant among samples of different poling conditions, such as is indicated in Table 3, seems to suggest that the piezoelectricity of polarized PVDF does not have a single origin. Further work is required to settle this problem. 

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