Size-dependent bending and vibration analysis of piezoelectric nanobeam based on fractional-order kinematic relations

doi:10.1007/s10483-025-3274-9

Abstract

Abstract:

In this paper, a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams. The general nonlocal strains in the Euler-Bernoulli piezoelectric beam are defined by a frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives. The strain energy, the work done by external loads, and the kinetic energy based on the fractional-order kinematic model are derived and expressed in explicit forms. The boundary conditions for the nonlocal Euler-Bernoulli beam are derived through variational principles. Furthermore, a finite element model for the fractional-order system is developed in order to obtain the numerical solutions to the integro-differential equations. The effects of the fractional order and the vibration order on the static bending and vibration responses of the Euler-Bernoulli piezoelectric beams are investigated numerically. The results from the present model are validated against the existing results in the literature, and it is demonstrated that they are theoretically consistent. Although this fractional finite element method (FEM) is presented in the context of a one-dimensional (1D) beam, it can be extended to higher dimensional fractional-order boundary value problems.

Key words: scale effect, Riesz-Caputo fractional-order derivative, Euler-Bernoulli piezoelectric beam, fractional-order finite element method (FEM)

2010 MSC Number:

O342

Zhiwen FAN, Hai QING. Size-dependent bending and vibration analysis of piezoelectric nanobeam based on fractional-order kinematic relations. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1261-1272.

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Figures/Tables 9

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References 44

[1]	WAN, Q., LI, Q. H., CHEN, Y. J., WANG, T. H., HE, X. L., LI, J. P., and LIN, C. L. Fabrication and ethanol sensing characteristics of ZnO nanowire gas sensors. Applied Physics Letters, 84(18), 3654–3656 (2004)
[2]	LAZARUS, A., THOMAS, O., and DEÜ, J. F. Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS. Finite Elements in Analysis and Design, 49(1), 35–51 (2012)
[3]	SU, W. S., CHEN, Y. F., HSIAO, C. L., and TU, L. W. Generation of electricity in GaN nanorods induced by piezoelectric effect. Applied Physics Letters, 90(6), 063110 (2007)
[4]	WANG, X. D., ZHOU, J., SONG, J. H., LIU, J., XU, N. S., and WANG, Z. L. Piezoelectric field effect transistor and nanoforce sensor based on a single ZnO nanowire. Nano Letters, 6(12), 2768–2772 (2006)
[5]	LI, C., GUO, W., KONG, Y., and GAO, H. J. Size-dependent piezoelectricity in zinc oxide nanofilms from first-principles calculations. Applied Physics Letters, 90(3), 033108 (2007)
[6]	HADJESFANDIARI, A. R. Size-dependent piezoelectricity. International Journal of Solids and Structures, 50(18), 2781–2791 (2013)
[7]	SHEN, S. and HU, S. A theory of flexoelectricity with surface effect for elastic dielectrics. Journal of the Mechanics and Physics of Solids, 58(5), 665–677 (2010)
[8]	HU, S. D., LI, H., and TZOU, H. S. Distributed flexoelectric structural sensing: theory and experiment. Journal of Sound and Vibration, 348, 126–136 (2015)
[9]	MAJDOUB, M. S., SHARMA, P., and CAGIN, T. Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Physical Review B, 77(12), 125424 (2008)
[10]	YAN, Z. and JIANG, L. Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity. Journal of Physics D: Applied Physics, 46(35), 355502 (2013)
[11]	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)
[12]	LIU, C., KE, L. L., WANG, Y. S., YANG, J., and KITIPORNCHAI, S. Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Composite Structures, 106, 167–174 (2013)
[13]	REN, Y. and QING, H. On the consistency of two-phase local/nonlocal piezoelectric integral model. Applied Mathematics and Mechanics (English Edition), 42(11), 1581–1598 (2021) https://doi.org/10.1007/s10483-021-2785-7
[14]	REN, Y. and QING, H. Elastic buckling and free vibration of functionally graded piezoelectric nanobeams using nonlocal integral models. International Journal of Structural Stability and Dynamics, 22(5), 2250047 (2022)
[15]	REN, Y. and QING, H. Bending and buckling analysis of functionally graded Timoshenko nanobeam using two-phase local/nonlocal piezoelectric integral model. Composite Structures, 300, 116129 (2022)
[16]	REN, Y. M., SCHIAVONE, P., and QING, H. On well-posed integral nonlocal gradient piezoelectric models for static bending of functionally graded piezoelectric nanobeam. European Journal of Mechanics-A/Solids, 96, 104735 (2022)
[17]	GHORBANPOUR ARANI, A., ABDOLLAHIAN, M., and KOLAHCHI, R. Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory. International Journal of Mechanical Sciences, 100, 32–40 (2015)
[18]	CHU, L., DUI, G., and JU, C. Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory. Composite Structures, 186, 39–49 (2018)
[19]	LI, Y. S., FENG, W. J., and CAI, Z. Y. Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory. Composite Structures, 115, 41–50 (2014)
[20]	MOHAMMADIMEHR, M., ROUSTA NAVI, B., and GHORBANPOUR ARANI, A. Modified strain gradient Reddy rectangular plate model for biaxial buckling and bending analysis of double-coupled piezoelectric polymeric nanocomposite reinforced by FG-SWNT. Composites Part B: Engineering, 87, 132–148 (2016)
[21]	FATTAHIAN DEHKORDI, S. and TADI BENI, Y. Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory. International Journal of Mechanical Sciences, 128, 125–139 (2017)
[22]	LI, Y. S. and PAN, E. Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory. International Journal of Engineering Science, 97, 40–59 (2015)
[23]	RAZAVI, H., BABADI, A. F., and TADI BENI, Y. Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Composite Structures, 160, 1299–1309 (2017)
[24]	YU, X., MAALLA, A., and MORADI, Z. Electroelastic high-order computational continuum strategy for critical voltage and frequency of piezoelectric NEMS via modified multi-physical couple stress theory. Mechanical Systems and Signal Processing, 165, 108373 (2022)
[25]	YAN, Z. and JIANG, L. Y. The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects. Nanotechnology, 22(24), 245703 (2011)
[26]	YAN, Z. and JIANG, L. Y. Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 468(2147), 3458–3475 (2012)
[27]	ZHU, C. S., FANG, X. Q., LIU, J. X., and LI, H. Y. Surface energy effect on nonlinear free vibration behavior of orthotropic piezoelectric cylindrical nano-shells. European Journal of Mechanics-A/Solids, 66, 423–432 (2017)
[28]	ZHANG, C., CHEN, W., and ZHANG, C. Two-dimensional theory of piezoelectric plates considering surface effect. European Journal of Mechanics A/Solids, 41, 50–57 (2013)
[29]	ZHANG, C., ZHU, J., CHEN, W., and ZHANG, C. Two-dimensional theory of piezoelectric shells considering surface effect. European Journal of Mechanics-A/Solids, 43, 109–117 (2014)
[30]	SUMELKA, W., BLASZCZYK, T., and LIEBOLD, C. Fractional Euler Bernoulli beams: theory, numerical study and experimental validation. European Journal of Mechanics-A/Solids, 54, 243–251 (2015)
[31]	RAHIMI, Z., RASH AHMADI, S., and SUMELKA, W. Fractional Euler-Bernoulli beam theory based on the fractional strain-displacement relation and its application in free vibration, bending and buckling analyses of micro/nanobeams. Acta Physica Polonica A, 134(2), 574–582 (2018)
[32]	RAHIMI, Z., SUMELKA, W., and SHAFIEI, S. The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model. Bulletin of the Polish Academy of Sciences-Technical Sciences, 66(5), 737–745 (2018)
[33]	STEMPIN, P. and SUMELKA, W. Space-fractional Euler-Bernoulli beam model-theory and identification for silver nanobeam bending. International Journal of Mechanical Sciences, 186, 105902 (2020)
[34]	LAZOPOULOS, K. A. and LAZOPOULOS, A. K. On fractional bending of beams. Archive of Applied Mechanics, 86(6), 1133–1145 (2016)
[35]	LAZOPOULOS, K. A. and LAZOPOULOS, A. K. On fractional bending of beams with Λ-fractional derivative. Archive of Applied Mechanics, 90(3), 573–584 (2020)
[36]	LAZOPOULOS, K. A., LAZOPOULOS, A. K., and KARAOULANIS, D. On Λ-fractional buckling and post-buckling of beams. Archive of Applied Mechanics, 94(7), 1829–1840 (2024)
[37]	PATNAIK, S., SIDHARDH, S., and SEMPERLOTTI, F. A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity. International Journal of Solids and Structures, 202, 398–417 (2020)
[38]	SIDHARDH, S., PATNAIK, S., and SEMPERLOTTI, F. Geometrically nonlinear response of a fractional-order nonlocal model of elasticity. International Journal of Non-Linear Mechanics, 125, 103529 (2020)
[39]	SIDHARDH, S., PATNAIK, S., and SEMPERLOTTI, F. Fractional-order structural stability: formulation and application to the critical load of nonlocal slender structures. International Journal of Mechanical Sciences, 201, 106443 (2021)
[40]	DING, W., PATNAIK, S., and SEMPERLOTTI, F. Transversely heterogeneous nonlocal Timoshenko beam theory: a reduced-order modeling via distributed-order fractional operators. Thin-Walled Structures, 197, 111608 (2024)
[41]	YANG, J. An Introduction to the Theory of Piezoelectricity, 2nd ed., Springer, New York (2018)
[42]	SUMELKA, W. and BLASZCZYK, T. Fractional continua for linear elasticity. Archives of Mechanics, 66(3), 147–172 (2014)
[43]	WANG, Q. On buckling of column structures with a pair of piezoelectric layers. Engineering Structures, 24(2), 199–205 (2002)
[44]	REN, Y. M. and QING, H. Elastic buckling and free vibration of functionally graded piezoelectric nanobeams using nonlocal integral models. International Journal of Structural Stability and Dynamics, 22(5), 2250047 (2022)