Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

doi:10.1007/s10483-020-2620-8

摘要/Abstract

摘要： A nonlocal strain gradient theory (NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients, which makes it benefit from both hardening and softening effects in small-scale structures. In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported (SS) boundary, the clamped-clamped (CC) boundary, the clamped-free (CF) boundary, and the clamped-simply supported (CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam. The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

关键词: piezoelectric nanobeam, sandwich structure, flexoelectric, nonlocal strain gradient theory (NSGT)

Abstract: A nonlocal strain gradient theory (NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients, which makes it benefit from both hardening and softening effects in small-scale structures. In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported (SS) boundary, the clamped-clamped (CC) boundary, the clamped-free (CF) boundary, and the clamped-simply supported (CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam. The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Key words: piezoelectric nanobeam, sandwich structure, flexoelectric, nonlocal strain gradient theory (NSGT)

中图分类号:

74F15

Shan ZENG, Kaifa WANG, Baolin WANG, Jinwu WU. Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(6): 859-880.

参考文献

[1] WANG, B. L., GUO, Y. B., and ZHANG, C. W. Cracking and thermal shock resistance of a Bi₂Te₃ based thermoelectric material. Engineering Fracture Mechanics, 152, 1-9(2016)
[2] XU, H. B., ZHANG, C. W., LI, H., and OU, J. P. Real-time hybrid simulation approach for performance validation of structural active control systems:a linear motor actuator based active mass driver case study. Structural Control and Health Monitoring, 21(4), 574-589(2014)
[3] SUN, L., LI, C., LI, J., ZHANG, C. W., and DING, X. S. Strain transfer analysis of a clamped fiber Bragg grating sensor. Applied Sciences, 7(2), 188(2017)
[4] ZHANG, C. W. and OU, J. P. Control strategies and experimental verifications of the electromagnetic mass damper system for structural vibration control. Earthquake Engineering and Engineering Vibration, 7(2), 181-192(2008)
[5] ZHANG, C. W. Control force characteristics of different control strategies for the wind-excited 76-story benchmark building structure. Advances in Structural Engineering, 17(4), 543-559(2014)
[6] XUE, Q. C., ZHANG, C. W., HE, J., ZOU, G. P., and ZHANG, J. C. An updated analytical structural pounding force model based on viscoelasticity of materials. Shock and Vibration, 2016, 2596923(2016)
[7] XU, H. B., ZHANG, C. W., LI, H., TAN, P., OU, J. P., and ZHOU, F. L. Active mass driver control system for suppressing wind-induced vibration of the Canton Tower. Smart Structures and Systems, 13(2), 281-303(2014)
[8] POURKIAEE, S. M., KHADEM, S. E., SHAHGHOLI, M., and BAB, S. Nonlinear modal interactions and bifurcations of a piezoelectric nanoresonator with three-to-one internal resonances incorporating surface effects and van der Waals dissipation forces. Nonlinear Dynamics, 88(3), 1785-1816(2017)
[9] HSU, Y. J., JIA, Z., and KYMISSIS, I. A locally amplified strain sensor based on a piezoelectric polymer and organic field-effect transistors. IEEE Transactions on Electron Devices, 58(3), 910-917(2011)
[10] CHOI, H. W., JEON, C. W., LIU, C., WATSON, I. M., DAWSON, M. D., EDWARDS, P. R., MARTIN, R. W., TRIPATHY, S., and CHUA, S. J. InGaN nano-ring structures for high-efficiency light emitting diodes. Applied Physics Letters, 86(2), 021101(2004)
[11] WANG, Z. L. Triboelectric nanogenerators as new energy technology for self-powered systems and as active mechanical and chemical sensors. ACS Nano, 7(11), 9533-9557(2013)
[12] SONG, M., ZHANG, Y., PENG, M. Z., and ZHAI, J. Y. Low frequency wideband nano generators for energy harvesting from natural environment. Nano Energy, 6, 66-72(2014)
[13] MOHAMMADIMEHR, M. and ROSTAMI, R. Bending and vibration analyses of a rotating sandwich cylindrical shell considering nanocomposite core and piezoelectric layers subjected to thermal and magnetic fields. Applied Mathematics and Mechanics (English Edition), 39(2), 219240(2018) https://doi.org/10.1007/s10483-018-2301-6
[14] ZHOU, Y. G., CHEN, Y. M., and DING, H. J. Analytical modeling of sandwich beam for piezoelectric bender elements. Applied Mathematics and Mechanics (English Edition), 28(12), 1581-1586(2007) https://doi.org/10.1007/s10483-007-1204-z
[15] ZHAO, X., QIAN, Z. H., LIU, J. X., and GAO, C. F. Effects of electric/magnetic impact on the transient fracture of interface crack in piezoelectric-piezomagnetic sandwich structure:anti-plane case. Applied Mathematics and Mechanics (English Edition), 41(1), 139-156(2020) https://doi.org/10.1007/s10483-020-2552-5
[16] AREFI, M., KARROUBI, R., and IRANI-RAHAGHI, M. Free vibration analysis of functionally graded laminated sandwich cylindrical shells integrated with piezoelectric layer. Applied Mathematics and Mechanics (English Edition), 37(7), 821-834(2016) https://doi.org/10.1007/s10483-016-2098-9
[17] EBRAHIMI, F. and KARIMIASL, M. Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams. Mechanics of Advanced Materials and Structures, 25(11), 943-952(2018)
[18] AMIR, S., KHORASANI, M., and BABAAKBAR-ZAREI, H. Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and firstorder shear deformation theory. Journal of Sandwich Structures and Materials (2018) https://doi.org/10.1177/1099636218795385
[19] KARAMI, B., SHAHSAVARI, D., LI, L., KARAMI, M., and JANGHORBAN, M. Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory. Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science, 233(1), 287-301(2019)
[20] ARANI, A. G., BABAAKBAR-ZAREI, H., and HAGHPARAST, E. Vibration response of viscoelastic sandwich plate with magnetorheological fluid core and functionally graded-piezoelectric nanocomposite face sheets. Journal of Vibration and Control, 24(21), 5169-5185(2018)
[21] ZENG, S., WANG, B. L., and WANG, K. F. Nonlinear vibration of piezoelectric sandwich nanoplates with functionally graded porous core with consideration of flexoelectric effect. Composite Structures, 207, 340-351(2019)
[22] YUE, Y. M., XU, K. Y., and CHEN, T. A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects. Composite Structures, 136, 278-286(2016)
[23] WANG, K. F. and WANG, B. L. An analytical model for nanoscale unimorph piezoelectric energy harvesters with flexoelectric effect. Composite Structures, 153, 253-261(2016)
[24] WANG, K. F., WANG, B. L., and ZENG, S. Analysis of an array of flexoelectric layered nanobeams for vibration energy harvesting. Composite Structures, 187, 48-57(2018)
[25] RUPA, N. S. and RAY, M. C. Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity. International Journal of Mechanics and Materials in Design, 13(3), 453-467(2017)
[26] WANG, K. F., WANG, B. L., and ZHANG, C. W. Surface energy and thermal stress effect on nonlinear vibration of electrostatically actuated circular micro-/nanoplates based on modified couple stress theory. Acta Mechanica, 228(1), 129-140(2017)
[27] GHADIRI, M., SHAFIEI, N., and SAFARPOUR, H. Influence of surface effects on vibration behavior of a rotary functionally graded nanobeam based on Eringen's nonlocal elasticity. Microsystem Technologies, 23(4), 1045-1065(2017)
[28] SHAFIEI, N. and KAZEMI, M. Nonlinear buckling of functionally graded nano-/micro-scaled porous beams. Composite Structures, 178, 483-492(2017)
[29] SHAFIEI, N., MOUSAVI, A., and GHADIRI, M. On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. International Journal of Engineering Science, 106, 42-56(2016)
[30] AZIMI, M., MIRJAVADI, S. S., SHAFIEI, N., HAMOUDA, A. M. S., and DAVARI, E. Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution. Mechanics of Advanced Materials and Structures, 25(6), 467-480(2018)
[31] AKBAŞ, Ş. D. Forced vibration analysis of functionally graded porous deep beams. Composite Structures, 186, 293-302(2018)
[32] SAHMANI, S., AGHDAM, M. M., and AKBARZADEH, A. H. Size-dependent buckling and postbuckling behavior of piezoelectric cylindrical nanoshells subjected to compression and electrical load. Materials & Design, 105, 341-351(2016)
[33] SARVESTANI, H. Y., AKBARZADEH, A. H., and MIRABOLGHASEMI, A. Structural analysis of size-dependent functionally graded doubly-curved panels with engineered microarchitectures. Acta Mechanica, 229, 2675-2701(2018)
[34] SAHMANI, S., MOHAMMADI-AGHDAM, M., and AKBARZADEH, A. Surface stress size dependency in nonlinear instability of imperfect piezoelectric nanoshells under combination of hydrostatic pressure and lateral electric field. AUT Journal of Mechanical Engineering, 2(2), 177-190(2018)
[35] SHAHVERDI, H. and BARATI, M. R. Vibration analysis of porous functionally graded nanoplates. International Journal of Engineering Science, 120, 82-99(2017)
[36] ROMANO, G., BARRETTA, R., DIACO, M., and MAROTTI DE SCIARRA, F. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences, 121, 151-156(2017)
[37] KOUTSOUMARIS, C. C., VOGIATZIS, G. G., THEODOROU, D. N., and TSAMASPHYROS, G. J. Application of bi-Helmholtz nonlocal elasticity and molecular simulations to the dynamical response of carbon nanotubes. AIP Conference Proceedings, 1702(1), 190011(2015)
[38] SHAAT, M. and ABDELKEFI, A. New insights on the applicability of Eringen's nonlocal theory. International Journal of Mechanical Sciences, 121, 67-75(2017)
[39] LIM, C. W., ZHANG, G., and REDDY, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313(2015)
[40] SAHMANI, S. and FATTAHI, A. M. Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Applied Mathematics and Mechanics (English Edition), 39(4), 561-580(2018) https://doi.org/10.1007/s10483-018-2321-8
[41] SHARIFI, Z., KHORDAD, R., GHARAATI, A., and FOROZANI, G. An analytical study of vibration in functionally graded piezoelectric nanoplates:nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 40(12), 1723-1740(2019) https://doi.org/10.1007/s10483-019-2545-8
[42] WANG, J., ZHU, Y. L., ZHANG, B., SHEN, H. M., and LIU, J. Nonlocal and strain gradient effects on nonlinear forced vibration of axially moving nanobeams under internal resonance conditions. Applied Mathematics and Mechanics (English Edition), 41(2), 261-278(2020) https://doi.org/10.1007/s10483-020-2565-5
[43] ZHANG, B., SHEN, H. M., LIU, J., WANG, Y. X., and ZHANG, Y. R. Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects. Applied Mathematics and Mechanics (English Edition), 40(4), 515-548(2019) https://doi.org/10.1007/s10483-019-2482-9
[44] LU, L., GUO, X. M., and ZHAO, J. Z. A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Applied Mathematical Modelling, 68, 583-602(2019)
[45] LU, L., ZHU, L., GUO, X. M., ZHAO, J. Z., and LIU, G. Z. A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells. Applied Mathematics and Mechanics (English Edition), 40(12), 1695-1722(2019) https://doi.org/10.1007/s10483-019-2549-7
[46] SAHMANI, S. and AGHDAM, M. M. A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells. Composite Structures, 178, 97-109(2017)
[47] EBRAHIMI, F. and BARATI, M. R. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Composite Structures, 159, 174-182(2017)
[48] EBRAHIMI, F., BARATI, M. R., and DABBAGH, A. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. International Journal of Engineering Science, 107, 169-182(2016)
[49] LU, L., GUO, X. M., and ZHAO, J. Z. A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms. International Journal of Engineering Science, 119, 265-277(2017)
[50] SAHMANI, S., AGHDAM, M. M., and RABCZUK, T. A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Materials Research Express, 5, 045048(2018)
[51] LIANG, X., HU, S. L., and SHEN, S. P. Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart Materials and Structures, 23(3), 035020(2014)
[52] MEHRALIAN, F. and BENI, Y. T. Vibration analysis of size-dependent bimorph functionally graded piezoelectric cylindrical shell based on nonlocal strain gradient theory. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(1), 27(2018)
[53] ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710(1983)
[54] LU, L., GUO, X. M., and ZHAO, J. Z. Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. International Journal of Engineering Science, 116, 12-24(2017)
[55] CHANTHANUMATAPORN, S. and WATANABE, N. Free vibration of a light sandwich beam accounting for ambient air. Journal of Vibration and Control, 24(16), 3658-3675(2018)
[56] KE, L. L., WANG, Y. S., and WANG, Z. D. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Composite Structures, 94, 2038-2047(2012)