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Piezoelectric solids

Damping in piezoelectric solids: introduction
Piezoelectricity is a coupled field effect as is thermoelasticity. In piezoelectric materials stress and strain are coupled to electrical field and polarization. Not all materials are piezoelectric; only those materials lacking a center of symmetry on the atomic scale can be piezoelectric. Examples of piezoelectric materials include quartz, Rochelle salt, and lead titanate zirconate ceramics.
The piezoelectric contribution to the mechanical loss tangent of a piezoelectric solid is derived from its complex piezoelectric and dielectric coefficients [1]. This loss depends on specimen geometry as a result of differences in effects related to the electrical boundary conditions. Including a positive out-of-phase piezoelectric modulus results in reduced values of the predicted loss, which constitutes an improvement over earlier theories which predict losses exceeding measured losses by a factor greater than two.

Summary
Piezoelectric relaxation has been observed in many materials, including ceramics [2], composites [3], and bone [4]. Such relaxation can be represented with complex piezoelectric coefficients or by a piezoelectric loss tangent [2], [5]. Mechanical relaxation also occurs in piezoelectric materials and is important in applications: large damping is considered desirable in materials used to generate short acoustic pulses for flaw detection ; small damping (high mechanical Q) is desirable in stable resonators and high power transducers. Piezoelectric reactions influence the apparent stiffness of a solid, under conditions in which neither electric field E nor electric displacement D is constant [7]. One can consider a piezoelectric contribution to mechanical relaxation: experimental evidence for such a contribution in quartz under quasistatic loading has appeared at least as early as 1915 [8]. A connection between dielectric loss and mechanical loss in piezoelectric solids is to be expected on heuristic grounds in that dielectric relaxation entails dissipation of electrical energy; if this energy has come from the piezoelectric conversion of mechanical energy, then mechanical relaxation or anelasticity must also occur.
The piezoelectric contribution to the loss tangent of the thin plate and other shapes is calculated in [1]. The effect of the out-of phase piezoelectric relaxation term d" is much more pronounced in the contribution to the anelastic (mechanical) relaxation than in the contribution to the storage compliance. Including a positive out-of-phase piezoelectric modulus results in reduced values of the predicted loss, which constitutes an improvement over earlier theories which predict losses exceeding measured losses by a factor greater than two.

References
[1] Lakes, R. S., "Shape-dependent damping in piezoelectric solids," IEEE Trans. Sonics, Ultrasonics, SU27, 208-213, 1980.
[2] Martin, G. E. "Dielectric, piezoelectric, and elastic losses in longitudinally polarized segmented ceramic tubes," U.S. Navy J. Underwater Acoustics, 15, 329-332, Apr. 1965.
[3] Furukawa, T. and Fukada, E. "Piezoelectric relaxation in composite epoxy-PZT system due to ionic conduction, Japan. J. Appl. Phys., 16, 453-458, Mar. 1977.
[4] Bur, A. J., "Measurements of the dynamic piezoelectric properties of bone as a function of temperature and humidity," J. Biomechanics, 9, 495-507, 1976.
[5] Holland, R. "Representation of dielectric, elastic, and piezoelectric losses by complex coefficients," IEEE Trans. Sonics Ultrason., SU-14, 18-20, Jan. 1967.
[6] Jaffe, H. and Berlincourt, D. A., "Piezoelectric transducer materials," Proc. IEEE, 53, 1372-1386, 1965.
[7] Cady, W. G., Piezoelectricity. New York: Dover, 1964.
[8] Joffe, A. F., The Physics of Crystals. New York: McGraw-Hill, 1928.