Piezoelectric 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 material properties appearing in Eqs. (6)-(9) are defined by the partial derivatives of the dependent variables (P, c, e) with respect to the independent variables. At this point, to maintain consistency with the literature on the P-phase of PVDF, we label c as the 1 axis, a as the 2 axis, and, b as the 3 axis. In evaluating the piezoelectric and pyroelectric responses we consider changes in polarization along the 3 axis only polarization along the 1 and 2 axes remains zero, by symmetry, for all the cases considered here. The direct piezoelectric strain 03 , pC/N) and stress (gaj, C/iiE) coefficients are defined in Eqs. (10) and (11),... 

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... 

Fig. 9. Piezoelectric strain constant of uniaxially drawn poly(y-methyl L-glutamate) film (a-helical form) plotted against the angle 6 between draw-axis and stress direction. Draw-ratio = 2. Drawn after Fukada, Date, and Hirai [Nature 211, 1079 (1966)] by permission of Macmillan (Journals) Ltd.

Fig. 11. Complex piezoelectric strain constant (20 Hz), complex Young s modulus (30 Hz), and complex dielectric constant (1kHz) of uniaxially drawn poly(D-propylene oxide) film plotted against temperature. Draw-ratio = 1.5. Degree of crystallinity=40%. Drawn after Furukawa and Fukada [Nature 221,1235 (1969)] by permission of Macmillan (Journals) Ltd.

Fig. 12. Complex piezoelectric strain constant of uniaxially drawn cellulose triacetate film plotted against temperature. Draw-ratio = 2. Plasticizer content = 10%. Frequency = 20 Hz. Drawn after Fukada, Date, and Emura [J. Soc. Mat Sci. Japan 17,335 (1968)] by permission of the Society of Materials Science, Japan...

Fig. 26. Correlation between increment of spontaneous polarization from 80° C to 15° C and piezoelectric strain constant at room temperature for /9-form polarized poly(vinylidene fluoride) films. Poling temperature = 90° C. Poling field = 700 kV/cm (Murayama, 1972)...

Fig. 29. Temperature dependence of complex piezoelectric strain constant of composite film of polyester resin and powdered PZT (50% of the volume) polarized at room temperature under a d.c. field of 100 kV/cm. Reproduced from Fukada and Date [Polymer Journal, 1,410 (1970)] by permission of the Society of Polymer...

Table 3. Piezoelectric strain constant of polymer films at room temperature...

Date,M., Fukada,E. An apparatus for measuring piezoelectric strain and stress constants in polymers. Rep. Progr. Polymer Phys. Japan 13, 375 (1970). 

Fig. 4. The effect of temperature on the piezoelectric strain constant, d3v for A, nylon-11 B, nylon-7 and C, poly(vinylidene...

The origin of the nonlinearity and hysteresis in the films is most likely due to displacement of domain walls [4], If domain walls move in a medium with a random distribution of pinning center, the response of the material can be described, in the first approximation by Rayleigh relations. We next demostrate how optical interferometry can be sued to verify whether this particular model applies to the investigated pzt thin film. In the case of the converse piezoelectric effect, when the driving field E is varied between — Eo and Eo, the piezoelectric strain x is hysteretic and can be expressed by the following Rayleigh relations ... 

From tensor algebra, the tensor property relating two associated tensor quantities, of rank / and rank g, is of rank (/-b g). Hence, the physical property connecting /, and aj is the third-rank tensor known as the piezoelectric effect, and it contains 3 = 27 piezoelectric strain coefficients, dyk. The piezoelectric coefficients are products of electrostriction constants, the electric polarization, and components of the dielectric tensor. 

The piezoelectric strain coefficients now take the form d and the relation between the applied stress, o- , and induced polarization, P,-, can be expressed in final matrix-like form as ... 

A centrosymmetric stress cannot produce a noncentrosymmetric polarization in a centrosymmetric crystal. Electric dipoles cannot form in crystals with an inversion center. Hence, only the twenty noncentrosymmetric point groups are associated with piezoelectricity (the noncentrosymmetric cubic class 432 has a combination of other symmetry elements which preclude piezoelectricity). The piezoelectric strain coefficients, dj for these point groups are given in Table 8.7, where, as expected, crystal symmetry dictates the number of independent coefficients. For example, triclinic crystals require the full set of 18 coefficients to describe their piezoelectric properties, but mono-chnic crystals require only 8 or 10, depending on the point group. 

TABLE 8.7. Piezoelectric Strain Coefficients in the Noncentrosymmetric Point Groups... 

Class Point Group Piezoelectric Strain Coefficients... 

Here, die = e26/Gq = 3.1 x 10 mV is the piezoelectric strain coefficient. Note that the derivation assumes laterally infinite resonators it does not account for energy trapping. Equation 120 is therefore expected to miss a numerical factor of order unity. Inputting values ( 26 = 3.1 pm V ), we arrive at ... 

Tracy M, Chang FK. Identifying impacts in composite plates with piezoelectric strain sensors, part I theory. J Intell Mater Syst Struct 1998 9(11) 920—8. 

Zhao P, Pisani D, L3mch CS. Piezoelectric strain sensor/actuator rosettes. Smart Mater Struct 2011 20(10) 102002. http //dx.doi.Org/10.1088/0964-1726/20/10/102002. Martin T, Hudd J, Wells P, Tunnichffe D, Das-Gupta D. The use of low profile piezoelectric sensors for impact and acoustic emission (AE) detection in CFRC structures. J Intell Mater Syst Struct 2001 12(8) 537—44. http //dx.doi.org/10.1177/ 10453890122145339. 

Pyro- and Piezoelectric Properties The electric field application on a ferroelectric nanoceramic/polymer composite creates a macroscopic polarization in the sample, responsible for the piezo- and pyroelectricity of the composite. It is possible to induce ferroelectric behavior in an inert matrix [Huang et al., 2004] or to improve the piezo-and pyroelectricity of polymers. Lam and Chan [2005] studied the influence of lead magnesium niobate-lead titanate (PMN-PT) particles on the ferroelectric properties of a PVDF-TrFE matrix. The piezoelectric and pyroelectric coefficients were measured in the electrical field direction. The Curie point of PVDF-TrFE and PMN-PT is around 105 and 120°C, respectively. Different polarization procedures are possible. As the signs of piezoelectric coefficients of ceramic and copolymer are opposite, the poling conditions modify the piezoelectric properties of the sample. In all cases, the increase in the longitudinal piezoelectric strain coefficient, 33, with ceramic phase poled) at < / = 0.4, the piezoelectric coefficient increases up to 15 pC/N. The decrease in da for parallel polarization is due primarily to the increase in piezoelectric activity of the ceramic phase with the volume fraction of PMN-PT. The maximum piezoelectric coefficient was obtained for antiparallel polarization, and at < / = 0.4 of PMN-PT, it reached 30pC/N. 

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. 

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... 

Studies of ferroeleetrie substanees based on plastieized flnoropolymers show that seleetion of adeqnate formnlation is still an experimental ehallenge beeanse reasons of material behavior are not fully understood. For example, addition of 5% tricresyl phosphate incteased piezoelectric strain coefBdent but an addition of 10% tricresyl phosphate made the material to become nonferroelectric. 

Piezoelectric Field Piezoelectric Field Piezoelectric Stress Piezoelectric Strain... 

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