Third-order elastic, piezoelectric, and dielectric constants

doi:10.1007/s10483-019-2550-7

Abstract

Abstract: The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail. These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are 0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry. These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.

Key words: third-order elastic constant, third-order piezoelectric constant, nonlinear, third-order dielectric constant, crystal, coordinate transformation, tensor

2010 MSC Number:

Yanming ZHANG, Jun JIN, Hongping HU. Third-order elastic, piezoelectric, and dielectric constants. Applied Mathematics and Mechanics (English Edition), 2019, 40(12): 1831-1846.

TrendMD

References 31

[1]	KOGAN, S. M. Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Soviet Physics-Solid State, 5(10), 2069-2070(1964)
[2]	ZHANG, Y. M., CHEN, H., DAI, L. X., HU, H. P., FAN, G. F., and LV, W. Z. Analysis on performance of flextensional piezoelectric hydrophone. 2017 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications, Sichuan, 235-239(2017)
[3]	WANG, H. R., XIE, J. M., XIE, X., HU, Y. T., and WANG, J. Nonlinear characteristics of circular-cylinder piezoelectric power harvester near resonance based on flow-induced flexural vibration mode. Applied Mathematics and Mechanics (English Edition), 35(2), 229-236(2014) https://doi.org/10.1007/s10483-014-1756-6
[4]	YANG, Z. T., YANG, J. S., and HU, Y. T. Optimal electrode shape and size of doubly rotated quartz plate thickness mode piezoelectric resonators. Applied Physics Letters, 92(10), 103516(2008)
[5]	HASHIMOTO, K. Y. Surface Acoustic Wave Devices in Telecommunications, Springer, Berlin (2000)
[6]	XIE, J. M. and HU, Y. T. Electric admittance analysis of quartz crystal resonator in thicknessshear mode induced by array of surface viscoelastic micro-beams. Applied Mathematics and Mechanics (English Edition), 38(1), 29-38(2017) https://doi.org/10.1007/s10483-017-2154-6
[7]	NAKAGAWA, R., KYOYA, H., SHIMIZU, H., KIHARA, T., and HASHIMOTO, K. Y. Study on generation mechanisms of second-order nonlinear signals in surface acoustic wave devices and their suppression. Japanese Journal of Applied Physics, 54(7S1), 07HD12(2015)
[8]	NAKAGAWA, R., SUZUKI, T., SHIMIZU, H., KYOYA, H., and HASHIMOTO, K. Y. Study on generation mechanisms of third-order nonlinearity in SAW devices. Ultrasonics Symposium, IEEE (2015)
[9]	YONG, Y. K. and PANG, X. Nonlinear frequency response of second harmonic generation in SAW IDT resonators. 2017 IEEE International Ultrasonics Symposium, IEEE, 1-4(2017)
[10]	YONG, Y., PATEL, M., and TANAKA, M. Effects of thermal stresses on the frequencytemperature behavior of piezoelectric resonators. Journal of Thermal Stresses, 30(6), 639-661(2007)
[11]	THURSTON, R., MCSKIMIN, H., and ANDREATCH, J. P. Third-order elastic coefficients of quartz. Journal of Applied Physics, 37(1), 267-275(1966)
[12]	THURSTON, R. and BRUGGER, K. Third-order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media. Physical Review, 133(6A), A1604(1964)
[13]	BOGARDUS, E. Third-order elastic constants of Ge, MgO, and fused SiO2. Journal of Applied Physics, 36(8), 2504-2513(1965)
[14]	CHO, Y. and YAMANOUCHI, K. Nonlinear, elastic, piezoelectric, electrostrictive, and dielectric constants of lithium niobate. Journal of Applied Physics, 61(3), 875-887(1987)
[15]	MAYER, A., MAYER, E., MAYER, M., JÄGER, P., RUILE, W., BLEYL, I., and WAGNER, K. Effective nonlinear constants for SAW devices from FEM calculations. IEEE International Ultrasonics Symposium, IEEE, 1-4(2015)
[16]	LIU, H., SHIN, K. C., LEE, J. J., and CAI, Z. M. Nonlinear acoustoelastic interactions of lamb waves with LiNbO3 films deposited on sapphire substrates. Key Engineering Materials, 261, 263-268(2004)
[17]	ARTIOLI, G., MONACO, H. L., VITERBO, D., FERRARIS, G., GILLI, G., ZANOTTI, G., and CATTI, M. Fundamentals of Crystallography, Oxford University Press, Oxford (2002)
[18]	HEARMON, R. "Third-order" elastic coefficients. Acta Crystallographica, 6(4), 331-340(1953)
[19]	MEITZLER, A., TIERSTEN, H., WARNER, A., BERLINCOURT, D., COQUIN, G., and WELSH, I. IEEE Standard on Piezoelectricity, American National Standards Institute, New York (1988)
[20]	TAGANTSEV, A. Piezoelectricity and flexoelectricity in crystalline dielectrics. Physical Review B, 34(8), 5883(1986)
[21]	RAY, M. Exact solutions for flexoelectric response in nanostructures. Journal of Applied Mechanics, 81(9), 091002(2014)
[22]	MURA, T. Micromechanics of Defects in Solids, Springer Science & Business Media, Dordrecht (1987)
[23]	HU, Y. T., WANG, J. N., YANG, F., XUE, H., HU, H. P., and WANG, J. The effects of first-order strain gradient in micro piezoelectric-bimorph power harvesters. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 58(4), 849-852(2011)
[24]	WANG, J. N., WANG, H. R., HU, H. P., LUO, B., and HU, Y. T. On the strain-gradient effects in micro piezoelectric-bimorph circular plate power harvesters. Smart Materials & Structures, 21, 015006(2012)
[25]	WANG, Y. and HERRON, N. Nanometer-sized semiconductor clusters:materials synthesis, quantum size effects, and photophysical properties. The Journal of Physical Chemistry, 95(2), 525-532(1991)
[26]	AULD, B. A. Acoustic Fields and Waves in Solids, Wiley, New York (1973)
[27]	NEWNHAM, R. E. Properties of Materials:Anisotropy, Symmetry, Structure, Oxford University Press on Demand, Oxford (2005)
[28]	LUAN, G. D., ZHANG, J. D., and WANG, R. Q. Piezoelectric Transducers and Array (in Chinese), Beijing University Press, Beijing (2005)
[29]	YANG, J. An Introduction to the Theory of Piezoelectricity, Springer, New York (2005)
[30]	TAGANTSEV, A. Theory of flexoelectric effect in crystals. Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 88(6), 2108-2122(1985)
[31]	HU, S. L. and SHEN, S. P. Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Science China Physics, Mechanics and Astronomy, 53(8), 1497-1504(2010)