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International Journal of Mechanics and Materials in Design

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17-07-2019

Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory

Authors: Van-Hieu Dang, Dong-Anh Nguyen, Minh-Quy Le, The-Hung Duong

Published in: International Journal of Mechanics and Materials in Design | Issue 2/2020

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Abstract

The nonlinear vibration of a nanobeam under electrostatic force is investigated through the nonlocal strain gradient theory. Using Galerkin method, the partial differential equation of motion is reduced to an ordinary nonlinear differential one. The equivalent linearization method with a weighted averaging and a variational approach are used independently to establish the frequency–amplitude relationship under closed-forms for comparison purpose. Effects of material and operational parameters on the frequency ratio (the ratio of nonlinear frequency to linear frequency), on the nonlinear frequency, and on the stable configuration of the nanobeam are studied and discussed.

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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 Mater. Struct. 20, 055011 (2011). https://doi.org/10.1088/0964-1726/20/5/055011 CrossRef

Akgöz, B., Civalek, Ö.: Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch. Appl. Mech. 82, 423–443 (2012). https://doi.org/10.1007/s00419-011-0565-5 CrossRefMATH

Akgöz, B., Civalek, Ö.: A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int. J. Eng. Sci. 70, 1–14 (2013). https://doi.org/10.1016/j.ijengsci.2013.04.004 MathSciNetCrossRefMATH

Anh, N.D.: Short Communication Dual approach to averaged values of functions: a form for weighting coefficient. Vietnam J. Mech. 37(2), 145–150 (2015). https://doi.org/10.15625/0866-7136/37/2/6206 CrossRef

Anh, N.D., Hai, N.Q., Hieu, D.V.: The equivalent linearization method with a weighted averaging for analyzing of nonlinear vibrating systems. Latin Am. J. Solids Struct. 14, 1723–1740 (2017). https://doi.org/10.1590/1679-78253488 CrossRef

Aranda-Ruiz, J., Loya, J., Fernández-Sáez, J.: Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos. Struct. 4(9), 2990–3001 (2012). https://doi.org/10.1016/j.compstruct.2012.03.033 CrossRef

Batra, R.C., Porfiri, M., Spinello, D.: Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct. 16(6), R23 (2007). https://doi.org/10.1088/0964-1726/16/6/R01 CrossRef

Chaterjee, S., Pohit, G.: A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams. J. Sound Vib. 322(4–5), 969–986 (2009). https://doi.org/10.1016/j.jsv.2008.11.046 CrossRef

Chong, A.C.M., Yang, F., Lam, D.C.C., Tong, P.: Torsion and bending of micron-scaled structures. J. Mater. Res. 16(04), 1052–1058 (2001). https://doi.org/10.1557/JMR.2001.0146 CrossRef

Chuang, W.C., Lee, H.L., Chang, P.Z., Hu, Y.C.: Review on the modeling of electrostatic MEMS. Sensors (Basel) 10(6), 6149–6171 (2010). https://doi.org/10.3390/s100606149. Epub 2010 CrossRef

Dean, R.N., Luque, A.: Applications of microelectromechanical systems in industrial processes and services. IEEE Trans. Ind. Electron. 56(4), 913–925 (2009). https://doi.org/10.1109/tie.2009.2013691 CrossRef

Duan, J.S., Rach, R.: A pull-in parameter analysis for the cantilever NEMS actuator model including surface energy, fringing field and Casimir effects. Int. J. Solids Struct. 50(22–23), 3511–3518 (2013). https://doi.org/10.1016/j.ijsolstr.2013.06.012 CrossRef

Eom, K., Park, H.S., Yoon, D.S., Kwon, T.: Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles. Phys. Rep. 503(4–5), 115–163 (2011). https://doi.org/10.1016/j.physrep.2011.03.002 CrossRef

Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972a). https://doi.org/10.1016/0020-7225(72)90050-X MathSciNetCrossRefMATH

Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972b). https://doi.org/10.1016/0020-7225(72)90070-5 MathSciNetCrossRefMATH

Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972). https://doi.org/10.1016/0020-7225(72)90039-0 MathSciNetCrossRefMATH

Fakhrabadi, M.M.S., Khorasani, P.K., Rastgoo, A., Ahmadian, M.T.: Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone–Wales defects. Solid State Commun. 157, 38–44 (2013). https://doi.org/10.1016/j.ssc.2012.12.016 CrossRef

Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994). https://doi.org/10.1016/0956-7151(94)90502-9 CrossRef

Fu, Y., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies. Appl. Math. Model. 35, 941–951 (2011). https://doi.org/10.1016/j.apm.2010.07.051 MathSciNetCrossRef

He, J.-H.: Variational approach for nonlinear oscillators. Chaos, Solitons Fractals 34, 1430–1439 (2007). https://doi.org/10.1016/j.chaos.2006.10.026 MathSciNetCrossRefMATH

Hieu, D.V., Hai, N.Q.: Analyzing of nonlinear generalized duffing oscillators using the equivalent linearization method with a weighted averaging. Asian Res. J. Math. 9(1), 1–14 (2018). https://doi.org/10.9734/ARJOM/2018/40684 CrossRef

Hieu, D.V. Hai, N.Q., Hung, D.T.: Analytical approximate solutions for oscillators with fractional order restoring force and relativistic oscillators. Int. J. Innov. Sci. Eng. Technol. 4(12), 28–35 (2017)

Hieu, D.V., Hai, N.Q., Hung, D.T.: The equivalent linearization method with a weighted averaging for solving undamped nonlinear oscillators. J. Appl. Math. 2018, Article ID 7487851 (2018). https://doi.org/10.1155/2018/7487851 MathSciNetCrossRef

Koiter, W.T.: Couple-stresses in the theory of elasticity: I and II. Philos. Trans. R. Soc. Lond. B 67, 17–44 (1964)MathSciNetMATH

Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli-Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008). https://doi.org/10.1016/j.ijengsci.2007.10.002 CrossRefMATH

Kroner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1967). https://doi.org/10.1016/0020-7683(67)90049-2 CrossRefMATH

Krylov, S.: Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures. Int. J. Non-Linear Mech. 42(4), 626–642 (2007). https://doi.org/10.1016/j.ijnonlinmec.2007.01.004 CrossRef

Krylov, N., Bogoliubov, N.: Introduction to Nonlinear Mechanics. Princenton University Press, New York (1943)MATH

Li, L., Hu, Y.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015). https://doi.org/10.1016/j.ijengsci.2015.08.013 MathSciNetCrossRefMATH

Li, L., Hu, Y., Ling, L.: Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos. Struct. 133, 1079–1092 (2015). https://doi.org/10.1016/j.compstruct.2015.08.014 CrossRef

Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015). https://doi.org/10.1016/j.jmps.2015.02.001 MathSciNetCrossRefMATH

Loh, O.Y., Espinosa, H.D.: Nanoelectromechanical contact switches. Nat. Nanotechnol. 7, 283–295 (2012). https://doi.org/10.1038/nnano.2012.40 CrossRef

Lu, L., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017). https://doi.org/10.1016/j.ijengsci.2017.03.006 MathSciNetCrossRefMATH

Luo, A.C.J., Wang, F.Y.: Chaotic motion in a micro-electro-mechanical system with non-linearity from capacitors. Commun. Nonlinear Sci. Numer. Simul. 7, 31–49 (2002). https://doi.org/10.1016/S1007-5704(02)00005-9 CrossRefMATH

Ma, H.M., Gao, X.L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008). https://doi.org/10.1016/j.jmps.2008.09.007 MathSciNetCrossRefMATH

Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of NEMS electrostatic switches. Nanotechnology 21(50), 505708 (2010a). https://doi.org/10.1088/0957-4484/21/50/505708 CrossRef

Ma, H.M., Gao, X.L., Reddy, J.N.: A nonclassical Reddy-Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010b). https://doi.org/10.1615/IntJMultCompEng.v8.i2.30 CrossRef

Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N.: Nonlocal and strain gradient based model for electrostatically actuated silicon nanobeams. Microsyst. Technol. 21, 457–464 (2015). https://doi.org/10.1007/s00542-014-2110-2 CrossRefMATH

Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3(1), 1–7 (1963). https://doi.org/10.1007/BF02327219 CrossRef

Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964). https://doi.org/10.1007/BF00248490 MathSciNetCrossRefMATH

Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965). https://doi.org/10.1016/0020-7683(65)90006-5 CrossRef

Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415 (1962). https://doi.org/10.1007/BF00253946 MathSciNetCrossRefMATH

Nejad, M.Z., Hadib, A., Rastgoo, A.: Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nanobeams based on nonlocal elasticity theory. Int. J. Eng. Sci. 103, 1–10 (2016). https://doi.org/10.1016/j.ijengsci.2016.03.001 CrossRefMATH

Oh, K.W., Ahn, C.H.: A review of microvalves. J. Micromech. Microeng. 16(5), R13 (2006). https://doi.org/10.1088/0960-1317/16/5/R01 CrossRef

Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355 (2006). https://doi.org/10.1088/0960-1317/16/11/015 CrossRef

Qian, Y.H., Ren, D.X., Lai, S.K., Chen, S.M.: Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam. Commun. Nonlinear Sci. Numer. Simul. 17, 1947–1955 (2012). https://doi.org/10.1016/j.cnsns.2011.09.018 MathSciNetCrossRefMATH

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. Appl. Phys. Lett. 94, 221903 (2009). https://doi.org/10.1063/1.3148774 CrossRef

Sadeghzadeh, S., Kabiri, A.: Application of higher order Hamiltonian approach to the nonlinear vibration of micro electro mechanical systems. Latin Am. J. Solids Struct. 13, 478–497 (2016). https://doi.org/10.1590/1679-78252557 CrossRef

Sedighi, H.M.: Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut. 95, 111–123 (2014). https://doi.org/10.1016/j.actaastro.2013.10.020 CrossRef

Şimşek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016). https://doi.org/10.1016/j.ijengsci.2016.04.013 MathSciNetCrossRefMATH

Stolken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998). https://doi.org/10.1016/S1359-6454(98)00153-0 CrossRef

Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)MathSciNetCrossRef

Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 29, 591–599 (2010). https://doi.org/10.1016/j.euromechsol.2009.12.005 CrossRef

Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002). https://doi.org/10.1016/S0020-7683(02)00152-X CrossRefMATH

Yiming, F., Zhang, J., Wan, L.: Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr. Appl. Phys. 11, 482–485 (2011). https://doi.org/10.1016/j.cap.2010.08.037 CrossRef

Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12(5), 672–680 (2003). https://doi.org/10.1109/JMEMS.2003.818069 CrossRef

Zhang, W.-M., Yan, H., Peng, Z.-K., Meng, G.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014). https://doi.org/10.1016/j.sna.2014.04.025 CrossRef

Title: Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory
Authors: Van-Hieu Dang
Dong-Anh Nguyen
Minh-Quy Le
The-Hung Duong
Publication date: 17-07-2019
Publisher: Springer Netherlands
Published in: International Journal of Mechanics and Materials in Design / Issue 2/2020
Print ISSN: 1569-1713
Electronic ISSN: 1573-8841
DOI: https://doi.org/10.1007/s10999-019-09468-8

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