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On well-posed local-nonlocal mixed integral model of piezoelectricity for dynamic stability and vibration analysis of piezoelectric Timoshenko nanobeams with general boundary constraints
Received date: 2025-10-24
Revised date: 2026-01-29
Online published: 2026-03-31
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12502187, 52378195, and 12172169), the National Key Research and Development Program of China (No. 2023YFF006001), the Xi’an Young and Middle-aged Science and Technology Innovation Leading Talent Project (No. 25ZORC00008), the Natural Science Basic Research Program of Shaanxi (Nos. 2025JC-YBQN-018 and 2025JC-YBQN-028), the Scientific Research Program Funded by Education Department of Shaanxi Provincial Government (No. 24JK0519), and the Natural Sciences and Engineering Research Council of Canada via a Discovery Grant (No. NSERC RGPIN-2023-03227)
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Existing research has shown that nonlocal piezoelectric differential models often yield inconsistent dynamic responses for nanostructures. To address this issue, the two-phase local-nonlocal integral formulation has been proposed and has garnered increasing scholarly attention as an effective alternative. This study presents the first implementation of this theoretically consistent and paradox-free framework to investigate the size-dependent dynamic stability and free vibration behavior in piezoelectric Timoshenko nanobeams. The generalized boundary conditions are simulated through elastic constraints incorporating both translational and rotational springs at both beam ends. Departing from conventional approaches, the present formulation simultaneously accounts for size effects in both bending deformation and axial deformation caused by external voltages via the derivation of an equivalent differential representation of the well-posed local-nonlocal integral piezoelectric model. This formulation is rigorously complemented by a complete set of constitutive constraint conditions, ensuring mathematical well-posedness throughout the analytical framework. The generalized differential quadrature method (GDQM) is used to discretize the governing differential equations, enabling numerical determination of dynamic instability regions (DIRs) for various boundary configurations. Following comprehensive validation through comparative analyses, we systematically examine the influence of nonlocal parameters, static force factors, and boundary stiffness characteristics on the DIRs of the beams. Furthermore, this investigation underscores the significance of incorporating nonlocal effects into voltage-induced axial loading, addressing a critical gap in the current understanding of electromechanical coupling at nanoscale dimensions.
Pei ZHANG , P. SCHIAVONE , Luke ZHAO , Dongbo LI , Yanming REN , Hai QING . On well-posed local-nonlocal mixed integral model of piezoelectricity for dynamic stability and vibration analysis of piezoelectric Timoshenko nanobeams with general boundary constraints[J]. Applied Mathematics and Mechanics, 2026 , 47(4) : 815 -838 . DOI: 10.1007/s10483-026-3370-8
| [1] | EOM, C. B. and TROLIER-MCKINSTRY, S. Thin-film piezoelectric MEMS. MRS Bulletin, 37, 1007–1017 (2012) |
| [2] | MOHITH, S., UPADHYA, A. R., NAVIN, K. P., KULKARNI, S. M., and RAO, M. Recent trends in piezoelectric actuators for precision motion and their applications: a review. Smart Materials and Structures, 30, 013002 (2021) |
| [3] | PILLAI, G. and LI, S. S. Piezoelectric MEMS resonators: a review. IEEE Sensors Journal, 21, 12589–12605 (2021) |
| [4] | MOTZ, C., WEYGAND, D., SENGER, J., and GUMBSCH, P. Micro-bending tests: a comparison between three-dimensional discrete dislocation dynamics simulations and experiments. Acta Materialia, 56, 1942–1955 (2008) |
| [5] | PENG, C., ZHAN, Y., and LOU, J. Size-dependent fracture mode transition in copper nanowires. Small, 8, 1889–1894 (2012) |
| [6] | ERINGEN, A. C. and EDELEN, D. G. B. On nonlocal elasticity. International Journal of Engineering Science, 10, 233–248 (1972) |
| [7] | MINDLIN, R. D. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1, 417–438 (1965) |
| [8] | 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) |
| [9] | 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) |
| [10] | 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 |
| [11] | KR?NER, E. Elasticity theory of materials with long range cohesive forces. International Journal of Solids and Structures, 3, 731–742 (1967) |
| [12] | ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21, 313–342 (1987) |
| [13] | ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710 (1983) |
| [14] | THAI, H. T., VO, T. P., NGUYEN, T. K., and KIM, S. E. A review of continuum mechanics models for size-dependent analysis of beams and plates. Composite Structures, 177, 196–219 (2017) |
| [15] | SHAAT, M., GHAVANLOO, E., and FAZELZADEH, S. A. Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mechanics of Materials, 150, 103587 (2020) |
| [16] | ZHOU, Z. G. and WANG, B. The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory. International Journal of Engineering Science, 40, 303–317 (2002) |
| [17] | 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) |
| [18] | WANG, W. J., 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) |
| [19] | LI, C., ZHU, C., ZHANG, N., SUI, S., and ZHAO, J. Free vibration of self-powered nanoribbons subjected to thermal-mechanical-electrical fields based on a nonlocal strain gradient theory. Applied Mathematical Modelling, 110, 583–602 (2022) |
| [20] | ZHOU, S. S., QI, L., ZHANG, R. M., LI, A. Q., QIAO, J. W., and ZHOU, S. J. Electro-mechanical responses of transversely isotropic piezoelectric nano-plate based on the nonlocal strain gradient theory with flexoelectric effect. Acta Mechanica, 234, 5647–5672 (2023) |
| [21] | MAO, J. J., LU, H. M., ZHANG, W., and LAI, S. K. Vibrations of graphene nanoplatelet reinforced functionally gradient piezoelectric composite microplate based on nonlocal theory. Composite Structures, 236, 111813 (2020) |
| [22] | BAKHTIARI-NEJAD, F. and NAZEMIZADEH, M. Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory. Acta Mechanica, 227, 1363–1379 (2016) |
| [23] | ZENKOUR, A. M. and SOBHY, M. Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin-Voigt viscoelastic nanoplate embedded in a viscoelastic medium. Acta Mechanica, 229, 3–19 (2017) |
| [24] | HE, Q. L., ZHU, C. S., HAN, B. H., FANG, X. Q., and LIU, J. X. Size-dependent free vibration of piezoelectric semiconductor plate. Acta Mechanica, 234, 4821–4836 (2023) |
| [25] | FANG, X. Q., ZOU, Y. H., and HE, Q. L. Nonlinear vibration of five-layered functionally graded piezoelectric semiconductor nano-plate on Pasternak foundation. Mechanics Based Design of Structures and Machines, 52, 10761–10782 (2024) |
| [26] | FANG, X. Q., DUAN, J. Q., ZHU, C. S., and LIU, J. X. Vibration analysis of piezoelectric semiconductor beams with size-dependent damping characteristic. Materials Today Communications, 36, 106929 (2023) |
| [27] | CHALLAMEL, N. and WANG, C. M. The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology, 19, 345703 (2008) |
| [28] | LI, C., YAO, L. Q., CHEN, W. Q., and LI, S. Comments on nonlocal effects in nano-cantilever beams. International Journal of Engineering Science, 87, 47–57 (2015) |
| [29] | ELTAHER, M. A., ALSHORBAGY, A. E., and MAHMOUD, F. F. Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37, 4787–4797 (2013) |
| [30] | ZHANG, P. and QING, H. On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams. Applied Mathematics and Mechanics (English Edition), 42(7), 931–950 (2021)https://doi.org/10.1007/s10483-021-2750-8 |
| [31] | ZHANG, P., SCHIAVONE, P., and QING, H. Two-phase local/nonlocal mixture models for buckling analysis of higher-order refined shear deformation beams under thermal effect. Mechanics of Advanced Materials and Structures, 29, 7605–7622 (2022) |
| [32] | ROMANO, G. and BARRETTA, R. Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, 14–27 (2017) |
| [33] | ROMANO, G. and BARRETTA, R. Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Composites Part B: Engineering, 114, 184–188 (2017) |
| [34] | ROMANO, G., BARRETTA, R., and DIACO, M. On nonlocal integral models for elastic nano-beams. International Journal of Mechanical Sciences, 131-132, 490–499 (2017) |
| [35] | KE, L. L. and WANG, Y. S. Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Materials and Structures, 21, 025018 (2012) |
| [36] | ZHANG, D. P., LEI, Y. J., and ADHIKARI, S. Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mechanica, 229, 2379–2392 (2018) |
| [37] | SHARIATI, M., SHISHESAZ, M., SAHBAFAR, H., POURABDY, M., and HOSSEINI, M. A review on stress-driven nonlocal elasticity theory. Journal of Computational Applied Mechanics, 52, 535–552 (2021) |
| [38] | TIAN, Y., XU, B., YU, D., MA, Y., WANG, Y., JIANG, Y., HU, W., TANG, C., GAO, Y., LUO, K., ZHAO, Z., WANG, L. M., WEN, B., HE, J., and LIU, Z. Ultrahard nanotwinned cubic boron nitride. nature, 493, 385–388 (2013) |
| [39] | LI, X., WEI, Y., LU, L., LU, K. and GAO, H. Dislocation nucleation governed softening and maximum strength in nano-twinned metals. nature, 464, 877–880 (2010) |
| [40] | SCHI?TZ, J. and JACOBSEN, K. W. A maximum in the strength of nanocrystalline copper. Science, 301, 1357–1359 (2003) |
| [41] | WANG, Y. B., HUANG, K., ZHU, X. W., and LOU, Z. M. Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model. Mathematics and Mechanics of Solids, 24, 559–572 (2019) |
| [42] | FAKHER, M. and HOSSEINI-HASHEMI, S. Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution. Engineering with Computers, 38, 231–245 (2020) |
| [43] | QING, H. Well-posedness of two-phase local/nonlocal integral polar models for consistent axisymmetric bending of circular microplates. Applied Mathematics and Mechanics (English Edition), 43(5), 637–652 (2022)https://doi.org/10.1007/s10483-022-2843-9 |
| [44] | ZHANG, P., SCHIAVONE, P., and QING, H. Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation. Applied Mathematics and Mechanics (English Edition), 44(1), 89–108 (2023)https://doi.org/10.1007/s10483-023-2948-9 |
| [45] | FERNáNDEZ-SáEZ, J. and ZAERA, R. Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. International Journal of Engineering Science, 119, 232–248 (2017) |
| [46] | ZHU, X. W., WANG, Y. B., and DAI, H. H. Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model. International Journal of Engineering Science, 116, 130–140 (2017) |
| [47] | FAKHER, M., BEHDAD, S., NADERI, A., and HOSSEINI-HASHEMI, S. Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium. International Journal of Mechanical Sciences, 171, 105381 (2020) |
| [48] | ZHANG, P., SCHIAVONE, P., and QING, H. Local-nonlocal integral theories of elasticity with discontinuity for longitudinal vibration analysis of cracked rods. Acta Mechanica, 235, 7419–7440 (2024) |
| [49] | NADERI, A., FAKHER, M., and HOSSEINI-HASHEMI, S. On the local/nonlocal piezoelectric nanobeams: vibration, buckling, and energy harvesting. Mechanical Systems and Signal Processing, 151, 107432 (2021) |
| [50] | 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, 2250047 (2022) |
| [51] | WANG, Q. On buckling of column structures with a pair of piezoelectric layers. Engineering Structures, 24, 199–205 (2002) |
| [52] | ALTAN, S. B. Uniqueness of initial-boundary value problems in nonlocal elasticity. International Journal of Solids and Structures, 25, 1271–1278 (1989) |
| [53] | 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) |
| [54] | WANG, K. F. and WANG, B. L. The electromechanical coupling behavior of piezoelectric nanowires: surface and small-scale effects. Europhysics Letters, 97, 66005 (2012) |
| [55] | WANG, X. W. Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Applied Mathematics Letters, 77, 94–100 (2018) |
| [56] | AL-SHUJAIRI, M. and MOLLAMAHMUTO?LU, ?. Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect. Composites Part B: Engineering, 154, 292–312 (2018) |
| [57] | TORNABENE, F., FANTUZZI, N., UBERTINI, F., and VIOLA, E. Strong formulation finite element method based on differential quadrature: a survey. Applied Mechanics Reviews, 67, 020801 (2015) |
| [58] | ZHANG, P., SCHIAVONE, P., and QING, H. Stress-driven local/nonlocal mixture model for buckling and free vibration of FG sandwich Timoshenko beams resting on a nonlocal elastic foundation. Composite Structures, 289, 115473 (2022) |
| [59] | ZHANG, P., SCHIAVONE, P., and QING, H. A unified local-nonlocal integral formulation for dynamic stability of FG porous viscoelastic Timoshenko beams resting on nonlocal Winkler-Pasternak foundation. Composite Structures, 322, 117416 (2023) |
| [60] | KOLAHCHI, R., HOSSEINI, H., and ESMAILPOUR, M. Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories. Composite Structures, 157, 174–186 (2016) |
| [61] | PHAM, Q. H. and NGUYEN, P. C. Dynamic stability analysis of porous functionally graded microplates using a refined isogeometric approach. Composite Structures, 284, 115086 (2022) |
| [62] | CHEN, H. Y., LI, W., and YANG, H. Dynamic stability in parametric resonance of vibrating beam micro-gyroscopes. Applied Mathematical Modelling, 103, 327–343 (2022) |
| [63] | LI, H. N., LI, C., SHEN, J. P., and YAO, L. Q. Vibration analysis of rotating functionally graded piezoelectric nanobeams based on the nonlocal elasticity theory. Journal of Vibration Engineering and Technologies, 9, 1155–1173 (2021) |
| [64] | ZHU, X. W. and LI, L. A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect. Applied Mathematics and Mechanics (English Edition), 40(11), 1561–1588 (2019)https://doi.org/10.1007/s10483-019-2541-5 |
| [65] | JIANG, J. N. and WANG, L. F. Analytical solutions for the thermal vibration of strain gradient beams with elastic boundary conditions. Acta Mechanica, 229, 2203–2219 (2018) |
| [66] | LIU, H. B., WEI, Z. G., TAN, G. J., HAN, Y. Y., and LIU, Z. Y. Vibratory characteristics of cracked non-uniform beams with different boundary conditions. Journal of Mechanical Science and Technology, 33, 377–392 (2019) |
| [67] | TANG, Y. and QING, H. Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models. Communications in Nonlinear Science and Numerical Simulation, 116, 106808 (2023) |
| [68] | ZHANG, P., SCHIAVONE, P., and QING, H. Hygro-thermal vibration study of nanobeams on size-dependent visco-Pasternak foundation via stress-driven nonlocal theory in conjunction with two-variable shear deformation assumption. Composite Structures, 312, 116870 (2023) |
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