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Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

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  • 1. Key Laboratory of Impact and Safety Engineering, Ministry of Education, Ningbo University, Ningbo 315211, Zhejiang Province, China;
    2. Piezoelectric Device Laboratory, Faculty of Mechanical Engineering & Mechanics, Ningbo University, Ningbo 315211, Zhejiang Province, China

Received date: 2022-01-07

  Revised date: 2022-02-24

  Online published: 2022-05-05

Supported by

the National Natural Science Foundation of China (No.11702150),the Natural Science Foundation of Zhejiang Province of China (Nos.LY20A020002 and LY21A020003),the Natural Science Foundation of Ningbo (No.202003N4015),the Project of Key Laboratory of Impact and Safety Engineering (Ningbo University),the Ministry of Education (No.CJ202009),and the Technology Innovation 2025 Program of Municipality of Ningbo (No.2019B10122)

Abstract

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity. The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle. The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode. Only the shear strain gradient through the thickness is considered in the present model. With geometric nonlinearity, the governing equations are converted into differential equations as the function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation. Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent, and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates. The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly. The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.

Cite this article

Yang ZHENG, Bin HUANG, Lijun YI, Tingfeng MA, Longtao XIE, Ji WANG . Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales[J]. Applied Mathematics and Mechanics, 2022 , 43(5) : 653 -666 . DOI: 10.1007/s10483-022-2842-7

References

[1] WANG, W., LI, P., JIN, F., and WANG, J. Vibration analysis of piezoelectric ceramic circular nanoplates considering surface and nonlocal effects. Composite Structures, 140, 758-775(2016)
[2] ZHANG, Z. and JIANG, L. Size effects on electromechanical coupling fields of a bending piezoelectric nanoplate due to surface effects and flexoelectricity. Journal of Applied Physics, 116, 134308(2014)
[3] ZHENG, Y., HUANG, B., and WANG, J. Flexoelectric efiect on thickness-shear vibration of a rectangular piezoelectric crystal plate. Materials Research Express, 8, 115702(2021)
[4] AMIR, S., BABAAKBAR-ZAREI, H., and KHORASANI, M. Flexoelectric vibration analysis of nanocomposite sandwich plates. Mechanics Based Design of Structures and Machines, 48, 146-163(2019)
[5] ANSARI, R., FARAJI-OSKOUIE, M., NESARHOSSEINI, S., and ROUHI, H. Flexoelectricity effect on the size-dependent bending of piezoelectric nanobeams resting on elastic foundation. Applied Physics A, 127, 518(2021)
[6] CHEN, W., LIANG, X., and SHEN, S. Forced vibration of piezoelectric and flexoelectric Euler-Bernoulli beams by dynamic Green's functions. Acta Mechanica, 232, 449-460(2020)
[7] ZENG, S., WANG, K., WANG, B., and WU, J. Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 41(6), 859-880(2020) https://doi.org/10.1007/s10483-020-2620-8
[8] CHEN, W., WANG, L., and DAI, H. Stability and nonlinear vibration analysis of an axially loaded nanobeam based on nonlocal strain gradient theory. International Journal of Applied Mechanics, 11, 1950069(2019)
[9] HASHEMI-KACHAPI, S. H., DARDEL, M., DANIALI, H. M., and FATHI, A. Pull-in instability and nonlinear vibration analysis of electrostatically piezoelectric nanoresonator with surface/interface effects. Thin-Walled Structures, 143, 106210(2019)
[10] PRADHAN, N. and SARANGI, S. K. Nonlinear vibration analysis of smart functionally graded plates. Materials Today:Proceedings, 44, 1870-1876(2021)
[11] XIANG, S., LEE, K. Y., and LI, X. F. Elasticity solution of functionally graded beams with consideration of the flexoelectric effect. Journal of Physics D:Applied Physics, 53, 105301(2020)
[12] ROJAS, E. F., FAROUGHI, S., ABDELKEFI, A., and PARK, Y. H. Nonlinear size dependent modeling and performance analysis of flexoelectric energy harvesters. Microsystem Technologies, 25, 3899-3921(2019)
[13] FANG, K., LI, P., and QIAN, Z. Static and dynamic analysis of a piezoelectric semiconductor cantilever under consideration of flexoelectricity and strain gradient elasticity. Acta Mechanica Solida Sinica, 34, 673-686(2021)
[14] ZHAO, X., ZHENG, S., and LI, Z. Size-dependent nonlinear bending and vibration of flexoelectric nanobeam based on strain gradient theory. Smart Materials and Structures, 28, 075027(2019)
[15] YUAN, J., ZHANG, X., and CHEN, C. Nonlinear vibration analysis of damaged microplate considering size effect. Shock and Vibration, 2020, 1-13(2020)
[16] FAKHER, M. and HOSSEINI-HASHEMI, S. Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method. Journal of Vibration and Control, 27, 378-391(2020)
[17] ZAREPOUR, M., HOSSEINI, S. A. H., and AKBARZADEH, A. H. Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model. Applied Mathematical Modelling, 69, 563-582(2019)
[18] YANG, W., DENG, Q., LIANG, X., and SHEN, S. Lamb wave propagation with flexoelectricity and strain gradient elasticity considered. Smart Materials and Structures, 27, 085003(2018)
[19] YANG, W., LIANG, X., DENG, Q., and SHEN, S. Rayleigh wave propagation in a homogeneous centrosymmetric flexoelectric half-space. Ultrasonics, 103, 106105(2020)
[20] ZHU, J., CHEN, S., CHEN, Y., CHEN, J., HU, P., WU, H., and ZHOU, Y. Thickness-twist waves in the nanoplates with flexoelectricity. Mechanics of Advanced Materials and Structures, 28, 2343-2350(2020)
[21] CHEN, Y. and YAN, Z. Nonlinear analysis of axially loaded piezoelectric energy harvesters with flexoelectricity. International Journal of Mechanical Sciences, 173, 105473(2020)
[22] CHEN, Y., ZHANG, M., SU, Y., and ZHOU, Z. Coupling analysis of flexoelectric effect on functionally graded piezoelectric cantilever nanobeams. Micromachines, 12, 595(2021)
[23] DAI, H. L., YAN, Z., and WANG, L. Nonlinear analysis of flexoelectric energy harvesters under force excitations. International Journal of Mechanics and Materials in Design, 16, 19-33(2019)
[24] DENG, Q., LYU, S., LI, Z., TAN, K., LIANG, X., and SHEN, S. The impact of flexoelectricity on materials, devices, and physics. Journal of Applied Physics, 128, 080902(2020)
[25] SHI, J., FAN, C., ZHAO, M., and YANG, J. Variational analysis of thickness-shear vibrations of a quartz piezoelectric plate with two pairs of electrodes as an acoustic wave filter. International Journal of Applied Electromagnetics and Mechanics, 47, 951-961(2015)
[26] SHI, J., FAN, C., ZHAO, M., and YANG, J. Thickness-shear vibration characteristics of an AT-cut quartz resonator with rectangular ring electrodes. International Journal of Applied Electromagnetics and Mechanics, 51, 1-10(2016)
[27] ZHAO, Z., QIAN, Z., and WANG, B. Effects of unequal electrode pairs on an x-strip thicknessshear mode multi-channel quartz crystal microbalance. Ultrasonics, 72, 73-79(2016)
[28] YUAN, L., WU, R., DU, J., WANG, J., and YANG, J. Thickness-shear and thickness-twist vibrations of rectangular quartz crystal plates with nonuniform thickness. Mechanics of Advanced Materials and Structures, 24, 937-942(2016)
[29] WU, R., WANG, W., CHEN, G., DU, J., MA, T., and WANG, J. Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations. Ultrasonics, 65, 338-344(2016)
[30] WU, R., ZHANG, W., MA, T., DU, J., and WANG, J. Thickness-shear frequencies of an infinite quartz plate with graded material properties across the thickness. Acta Mechanica Solida Sinica, 33, 361-367(2020)
[31] LI, P., JIN, F., and YANG, J. Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour. IEEE Transactions on Ultrasonics Ferroelectrics&Frequency Control, 59, 1006-1012(2012)
[32] LIU, B., XING, Y. F., EISENBERGER, M., and FERREIRA, A. J. M. Thickness-shear vibration analysis of rectangular quartz plates by a numerical extended Kantorovich method. Composite Structures, 107, 429-435(2014)
[33] MA, T., WANG, J., DU, J., and YANG, J. Resonances and energy trapping in AT-cut quartz resonators operating with fast shear modes driven by lateral electric fields produced by surface electrodes. Ultrasonics, 59, 14-20(2015)
[34] WU, R., WANG, J., DU, J., HUANG, D., and HU, Y. The non-linear thickness-shear vibrations of quartz crystal plates under an electric field. International Journal of Non-Linear Mechanics, 61, 32-38(2014)
[35] SHEN, S. and HU, S. A theory of flexoelectricity with surface effect for elastic dielectrics. Journal of the Mechanics and Physics of Solids, 58, 665-677(2010)
[36] WU, R., WANG, J., DU, J., HUANG, D., WEI, Y., and HU, Y. An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method. IEEE Transactions on Ultrasonics Ferroelectrics&Frequency Control, 59, 30-39(2012)
[37] BASKARAN, S., HE, X., CHEN, Q., and FU, J. Y. Experimental studies on the direct flexoelectric efiect in α-phase polyvinylidene fluoride films. Applied Physics Letters, 98, 242901(2011)
[38] SHEN, Z. and CHEN, W. Converse flexoelectric effect in comb electrode piezoelectric microbeam. Physics Letters A, 376, 1661-1663(2012)
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