Pull-in Instability Analysis of a Nanocantilever Based on the ‎Two-Phase Nonlocal Theory of Elasticity

Document Type : Research Paper


1 Department of Bio- and Nanomechanics, Belarusian State University, 4 Nezavisimosti Avenue, Minsk, 220140, Belarus

2 Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Via Amendola 2, Reggio Emilia, 42122, Italy‎


This paper deals with the pull-in instability of cantilever nano-switches subjected to electrostatic and intermolecular forces in the framework of the two-phase nonlocal theory of elasticity. The problem is governed by a nonlinear integro-differential equation accounting for the external forces and nonlocal effects. Assuming the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to a sixth-order differential one and derive a pair of additional boundary conditions. Aiming to obtain a closed-form solution of the boundary-value problem and to estimate the critical intermolecular forces and pull-in voltage, we approximate the resultant lateral force by a linear or quadratic function of the axial coordinate. The pull-in behavior of a freestanding nanocantilever as well as its instability under application of a critical voltage versus the local model fraction are examined within two models of the load distribution. It is shown that the critical voltages calculated in the framework of the two-phase nonlocal theory of elasticity are in very good agreement with the available data of atomistic simulation. 


Main Subjects

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[1] Bruschi, P., Nannini, A., Paci, D., Pieri, F., A method for cross-sensitivity and pull-in voltage measurement of MEMS two-axis accelerometers, Sensors and Actuators A: Physical, 123-124, 2005, 185-193.
[2] Matmat, M., Koukos, K., Coccetti, F., Idda, T., Marty, A., Escriba, C. et al., Life expectancy characterization of capacitive RF MEMS switches, Microelectronics Reliability, 50(9-10), 2010, 1692-1696.
[3] De Volf, Van Spengen, W.M., Techniques to study the reliability of metal RF MEMS of capacitive switches, Microelectronics Reliability, 42(9-11), 2002, 1789-1794.
[4] Lin, W.H., Zhao, Y.P., Nonlinear behavior for nanoscale electrostatic actuators with Casimir force, Chaos, Solitons and Fractals, 23, 2005, 1777-1785.
[5] Cai1, T., Fang, Yu., Fang, Yi., Li, R., Yu, Y., Huang, M., Electrostatic pull-in application in flexible devices: A review, Beilstein Journal of Nanotechnology, 13, 2022, 390–403.
[6] Ke, C.H., Pugno, N., Peng, B., Espinosa, H.D., Experiments and modeling of carbon nanotube-based NEMS devices, Journal of the Mechanics and Physics of Solids, 53(6), 2005, 1314-1333.
[7] Loh, O.Y., Espinosa, H.D., Nanoelectromechanical contact switches, Nature Nanotechnology, 7(5), 2012, 283-295.
[8] Basu, S., Prabhakar, A., Bhattacharya, E., Estimation of stiction force from electrical and optical measurements on cantilever beams, Journal of Microelectromechanical Systems, 16(5), 2007, 1254-1262.
[9] Ramezani, A., Alasty, A., Akbari, J., Closed-form solutions of the pull-in instability in nanocantilevers under electrostatic and intermolecular surface forces, International Journal of Solids and Structures, 44, 2007, 4925-4941.
[10] Gusso, A., Delben, G.J., Dispersion force for materials relevant for micro-and nanodevices fabrication, Journal of Physics D: Applied Physics, 41 (17) (2008) 175405.
[11] Lamoreaux, S., The Casimir force: background, experiments, and applications, Reports on Progress in Physics, 68, 2005, 201236.
[12] Lin, W.H., Zhao, Y.P. Casimir effect on the pull-in parameters of nanometer switches, Microsystem Technologies, 11, 2005, 80-85.
[13] Soroush, R., Koochi, A., Kazemi, A.S., Noghrehabadi, A., Haddadpour, H., Abadyan, M., Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators, Physica Scripta, 82, 2010, 045801.
[14] Koochi, A., Kazemi, A.S., Beni, Y.T., Yekrangi, A., Abadyan, M., Theoretical study of the effect of Casimir attraction on the pull-in behavior of beam-type NEMS using modified Adomian method, Physica E: Low-dimensional Systems and Nanostructures, 43(2), 2010, 625–632.
[15] Radi, E., Bianchi, G., di Ruvo, L., Upper and lower bounds for the pull-in parameters of a micro- or nanocantilever on a flexible support, International Journal of Non-Linear Mechanics, 92, 2017, 176-186.
[16] Radi, E., Bianchi, G., di Ruvo, L., Analytical bounds for the electromechanical buckling of a compressed nanocantilever, Applied Mathematical Modelling, 59, 2018, 571-582.
[17] Bianchi, G., Radi, E., Analytical estimates of the pull-in voltage for carbon nanotubes considering tip-charge concentration and intermolecular forces, Meccanica, 55(1), 2020, 193-209.
[18] Radi, E., Bianchi, G., Nobili, A., Bounds to the pull-in voltage of a MEMS/NEMS beam with surface elasticity, Applied Mathematical Modelling, 91, 2021, 1211-1226.
[19] Goharimanesh, M., Koochi, A., Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: The energy balance method, Reports in Mechanical Engineering, 2(1), 2021, 41-50.
[20] Malikan, M., Eremeyev, V.A., Sedighi, H.M., Buckling analysis of a non-concentric double-walled carbon nanotube, Acta Mechanica, 231, 2020, 5007-5020.
[21] Dastjerdi, S., Malikan, M., Mechanical analysis of eccentric defected bilayer graphene sheets considering the van der Waals force, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 235(1-2), 2021, 44-51.
[22] Sedighi, H.M., Malikan, M., Valipour, A., Żur, K.K., Nonlocal vibration of carbon/boron-nitride nano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method, Journal of Computational Design and Engineering, 7(5), 2020, 591-602.
[23] Jena, S.K., Chakraverty, S., Malikan, M., Sedighi, H., Implementation of Hermite–Ritz method and Navier’s technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity, Journal of Mechanics of Materials and Structures, 15(3), 2020, 405-434.
[24] Koochi, A., Goharimanesh, M., Gharib, M.R., Nonlocal electromagnetic instability of carbon nanotube-based nano-sensor, Mathematical Methods in the Applied Sciences, 2021, https://doi.org/10.1002/mma.7216.
[25] Abouelregal, A.E., Sedighi, H., Faghidian, S.A, Shirazi, A.H., Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load, Facta Universitatis, Series: Mechanical Engineering, 19(4), 2021, 633-656.
[26] Malikan, M., Eremeyev, V.A., On nonlinear bending study of a piezo-flexomagnetic nanobeam based on an analytical-numerical solution, Nanomaterials, 10(9), 2020, 1762.
[27] Malikan, M., Eremeyev, V.A., On a flexomagnetic behavior of composite structures, International Journal of Engineering Science, 175, 2022, 103671.
[28] Abdi, J., Koochi, A., Kazemi, A.S., Abadyan, M., Modeling the effects of size dependence and dispersion forces on the pull-in instability of electrostatic cantilever NEMS using modified couple stress theory, Smart Materials and Structures, 20, 2011, 055011.
[29] Rahaeifard, M., Kahrobaiyan, M., Asghari, M., Ahmadian, M., Static pull-in analysis of microcantilevers based on the modified couple stress theory, Sensors and Actuators A: Physical, 171, 2011, 370-374.
[30] Taati, E.,  Sina, N., Static Pull-in Analysis of Electrostatically Actuated Functionally Graded Micro- Beams Based on the Modified Strain Gradient Theory, International Journal of Applied Mechanics, 10(3), 2018, 1850031.
[31] Yang, J., Jia, X.L., Kitipornchai, S., Pull-in instability of nano-switches using nonlocal elasticity theory, Journal of Physics D: Applied Physics, 41, 2008, 035103-035111.
[32] Mousavi, T., Bornassi, S., Haddadpour, H., The effect of small scale on the pull-in instability of nano-switches using DQM, International Journal of Solids and Structures, 50(9), 2013, 11931202.
[33] Sedighi, H.M., Sheikhanzadeh, A., Static and dynamic pull-in instability of nano-beams resting on elastic foundation based on the nonlocal elasticity theory, Chinese Journal of Mechanical Engineering, 30, 2017, 385-397.
[34] Eringen, A.C., Theory of Nonlocal Elasticity and Some Applications, Technical report, Princeton Univ NJ Dept of Civil Engineering, 1984.
[35] Eringen, A.C., Nonlocal Continuum Field Theories, Springer Science & Business Media, 2002.
[36] Sadeghian, H., Yang, C.K., Goosen, J.F.L., van der Drift, E., Bossche, A., French, P.J., van Keulen, F., Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability, Applied Physics Letters, 94, 2009, 221903.  
[37] Mikhasev, G.I., Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area, Continuum Mechanics and Thermodynamics, 33, 2021, 1299-1312.
[38] Mikhasev, G., Nobili, A., On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory, International Journal of Solids and Structures, 190, 2020, 47-57.
[39] Mikhasev, G., Avdeichik, E., Prikazchikov, D., Free vibrations of nonlocally elastic rods, Mathematics and Mechanics of Solids, 24(5), 2019, 1279-1293.
[40] Romano, G., Barretta, R., Diaco, M., de Sciarra, F.M., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, International Journal of Mechanical Sciences, 121, 2017, 151-156.
[41] Dequesnes, M., Rotkin, S.V., Aluru, N.R., Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches, Nanotechnology, 13, 2002, 120-131.
[42] Fernández-Sáez, J., Zaera, R., Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory, International Journal of Engineering Sciences, 119, 2017, 232-248.
[43] Mikhasev, G.I., A study of free high-frequency vibrations of an inhomogeneous nanorod, based on the nonlocal theory of elasticity, Vestnik St. Petersburg University: Mathematics, 54(2), 2021, 125-134.
[44] Akita, S., Nakayama, Y., Nanotweezers consisting of carbon nanotubes operating in an atomic force microscope, Applied Physics Letters, 79, 2001, 1691-1693.
[45] Wong, E.W., Sheehan, P.E., Lieber, C.M., Nanobeam mechanics elasticity, strength, and toughness of nanorods and nanotubes, Science, 277, 1997, 1971-1975.
[46] Van Lier, G., Van Alsenoy, C., Van Doren, V., Geerlings, P., Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene, Chemical Physics Letters, 326(1-2), 2000, 181-185.
[47] Treacy, M.M.J., Ebbesen, T.W., Gibson, J.M., Exceptionally high Young’s modulus observed for individual carbon nanotubes, Nature, 381, 1996, 678-680.
[48] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 1983, 4703-4710.
[49] Mikhasev, G., Botogova, M., Free localized vibrations of a long double-walled carbon nanotube introduced into an inhomogeneous elastic medium, Vestnik St. Petersburg University: Mathematics, 49(1), 2016, 85–91.