Extraction of Nonlinear Thermo-Electroelastic Equations for High Frequency Vibrations of Piezoelectric Resonators with Initial Static Biases

Document Type: Research Paper


1 School of Mechanical Engineering, College of Engineering, University of Zanjan, Zanjan, Iran

2 College of Mechanical Engineering, University of Tehran, Tehran, Iran

3 Department of Electrical Engineering, Pennsylvania State University, USA


In this paper, the general case of an anisotropic thermo-electro elastic body subjected to static biasing fields is considered. The biasing fields may be introduced by heat flux, body forces, external surface tractions, and electric fields. By introducing proper thermodynamic functions and employing variational principle for a thermo-electro elastic body, the nonlinear constitutive relations and the nonlinear equation of motion are extracted. The equations have the advantage of employing the Lagrangian strain and second Piola-Kirchhoff stress tensor with symmetric characteristics. These equations are used to analyze the high frequency vibrations of piezoelectric resonators under finite biasing fields. A system of three dimensional equations is derived for initial and incremental fields on the body. Capability of the equations in numerical modelling of temperature-frequency and force-frequency effects in quartz crystal is demonstrated. The numerical results compare well with the data from experiments. These equations may be used in accurate modelling of piezoelectric devices subjected to thermo electro mechanical loads.


[1] Patel M. S., 2008, Nonlinear Behavior in Quartz Resonators and its Stability, PhD dissertation, New Brunswick, New Jersey University.
[2] Ebrahimi F., Rastgoo A.,2009, Temperature effects on nonlinear vibration of FGM plates coupled with piezoelectric actuators, Journal of Solid Mechanics 1(4): 271-288.
[3] Ultrasonics, Ferroelectrics, and Frequency Control, 2015, IEEE Transactions on 62 2015(6): 1104-1113.
[4] Baumhauer J. C., Tiersten H. F., 1973, Nonlinear electro elastic equations for small fields superposed on a bias, The Journal of the Acoustical Society of America 54(4): 1017-1034.
[5] Tichý J., Jiˇrí E., Erwin K., Jana P.,2010, Fundamentals of Piezoelectric Sensorics: Mechanical, Dielectric, and Thermodynamical Properties of Piezoelectric Materials, Springer Science & Business Media.
[6] Dulmet B., Roger B.,2001, Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. I. Theory, The Journal of the Acoustical Society of America 110(4): 1792-1799.
[7] Tiersten H. F., 1971, On the nonlinear equations of thermo-electro elasticity, International Journal of Engineering Science 9(7): 587-604.
[8] Tiersten H. F.,1975, Nonlinear electro elastic equations cubic in the small field variables, The Journal of the Acoustical Society of America 57(3): 660-666.
[9] Yang J. S., Batra R. C., 1995, Free vibrations of a linear thermo piezo electric body, Journal of Thermal Stresses 18(2): 247-262.
[10] Lee P. C. Y., Wang Y. S., Markenscoff X., 1975, High− frequency vibrations of crystal plates under initial stresses, The Journal of the Acoustical Society of America 57(1): 95-105.
[11] Lee P. C. Y., Yong Y. K., 1986, Frequency‐temperature behavior of thickness vibrations of doubly rotated quartz plates affected by plate dimensions and orientations, Journal of Applied Physics 60(7): 2327-2342.
[12] Yong Y. K., Wu W., 2000, Lagrangian temperature coefficients of the piezoelectric stress constants and dielectric permittivity of quartz, In Frequency Control Symposium and Exhibition, Proceedings of the 2000 IEEE/EIA International.
[13] Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 48 2001(5): 1471-1478.
[14] Yong Y. K., Mihir P., Masako T., 2007, Effects of thermal stresses on the frequency-temperature behavior of piezoelectric resonators, Journal of Thermal Stresses 30(6): 639-661.
[15] Mase G., Thomas R., Smelser E., George E. M., 2009, Continuum Mechanics for Engineers, CRC press.
[16] Lee P. C. Y., Yong Y. K., 1984, Temperature derivatives of elastic stiffness derived from the frequency‐temperature behavior of quartz plates, Journal of Applied Physics 56(5): 1514-1521.
[17] Yang J.,2013, Vibration of Piezoelectric Crystal Plates, World Scientific.
[18] Yang J.,2005, An Introduction to the Theory of Piezoelectricity, Springer Science & Business Media.
[19] Kuang Zh. B.,2011, Some Thermodynamic Problems in Continuum Mechanics, INTECH Open Access Publisher.
[20] Montanaro A., 2010, Some theorems of incremental thermo-electro elasticity, Archives of Mechanics 62(1): 49-72.
[21] Yang J., Yuantai H., 2004, Mechanics of electro elastic bodies under biasing fields, Applied Mechanics Reviews 57(3): 173-189.
[22] EerNisse E.P., 1980, Temperature dependence of the force frequency effect for the AT, FC, SC and rotated X-Cuts, 34th Proceedings of the Annual Symposium on Frequency Control.
[23] Beerwinkle A. D., 2011, Nonlinear Finite Element Modeling of Quartz Crystal Resonators, M.S. Thesis, Oklahoma State University, Stillwater, USA.
[24] Mohammadi M. M., Hamedi M.,2016, Experimental and numerical investigation of force-frequency effect in crystal resonators, Journal of Vibro-Engineering 18(6): 3709-3718.
[25] Thurston R. N., McSkimin H. J., Andreatch Jr P.,1966, Third‐order elastic coefficients of quartz, Journal of Applied Physics 37(1): 267-275.
[26] Kittinger E., Jan T., Wolfgang F.,1986, Nonlinear piezoelectricity and electrostriction of alpha quartz, Journal of Applied Physics 60(4): 1465-1471.
[27] Reider Georg A., Erwin K., Tichý J.,1982, Electro elastic effect in alpha quartz, Journal of Applied Physics 53(12): 8716-8721.
[28] Ratajski J. M., 1968, Force-frequency coefficient of singly rotated vibrating quartz crystals, IBM Journal of Research and Development 12(1): 92-99.