In this work, a microbeam model based on the Euler–Bernoulli beam theory and the nonlocal strain gradient theory was developed to study nonlinear vibration and stability of a functionally graded porous microbeam under electrostatic actuation. The electrostatic force is the driving force per unit length, resulting from electrostatic excitation. The material properties of the microbeam change continuously along thickness direction according to simple power-law distribution in terms of the fractions of constituents. Two cases of even and uneven porosity distributions are considered. The Hamilton principle is used to obtain the governing equations of the motion for the microbeam. The equation of motion for the microbeams is reduced to a nonlinear ordinary differential equation with rational restoring force by applying Galerkin method. The frequency–amplitude relationship is given in the closed form by employing He’s Hamiltonian approach. The expression of the static pull-in voltage is also derived. Effects of the material length scale parameter, the nonlocal parameter, the power-law index, the porosity distribution factor, the applied voltage and the initial amplitude on nonlinear vibration and stability of the functionally graded porous microbeam are examined and discussed.
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Chuang, W.C., Lee, H.L., Chang, P.Z., Hu, Y.C.: Review on the modeling of electrostatic MEMS. Sensors 10, 6149–6171 (2010)
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)
Fu, Y., Du, H., Huang, W., Zhang, S., Hu, M.: TiNi-based thin films in MEMS applications: a review. Sens. Actuators A Phys. 112, 395–408 (2004)
Clarke, D.R., Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10(4), 853–863 (1995)
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)
Stölken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998)
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)
Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(1), 417–438 (1965)
Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)
Yang, F., Chong, A.C.M., Lam, D.C.C., et al.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)
Lam, D.C.C., Yang, F., Chong, A.C.M., et al.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)
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)
Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)
Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)
Ansari, R., Oskouie, M.F., Gholami, R., Sadeghi, F.: Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Compos. B 89, 316–327 (2016)
Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)
Farokhi, H., Ghayesh, M.H.: Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int. J. Eng. Sci. 91, 12–33 (2015)
Ghayes, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)
Dang, V.H., Nguyen, D.A., Le, M.Q., Ninh, Q.H.: Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging. Arch. Appl. Mech. 90, 87–106 (2020)
Li, Li., Yujin, Hu.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)
Lu, Lu., 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)
Arefi, M., Pourjamshidian, M., Arani, A.G.: Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment. Appl. Phys. A 123, 323 (2017)
Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511 (2008)
De Rosa, M.A., Lippiello, M.: Nonlocal frequency analysis of embedded single-walled carbon nanotube using the differential quadrature method. Compos. B 84, 41–51 (2016)
Zhen, Y.X., Wen, S.L., Tang, Y.: Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E 105, 116–124 (2019)
Atashafrooz, M., Bahaadini, R., Sheibani, H.R.: Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow. Mech. Adv. Mater. Struct. 27(7), 586–598 (2020)
Wang, B.L., Shen, S.J., Zhao, J.F., Chen, X.: A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 30, 517–524 (2011)
Thanh, C.L., Loc, V.T., Huu, T.V., Abdel-Wahab, M.: The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 350(15), 337–361 (2019)
Shahsavari, D., Karami, B., Mansouri, S.: Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. Eur. J. Mech. A. Solids 67, 200–214 (2018)
Thai, H.T., Vo, T.P., Nguyen, T.K., Kim, S.-E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017)
Farajpour, A., Ghayesh, M.H., Farokhi, H.: A review on the mechanics of nanostructures. Int. J. Eng. Sci. 133, 231–263 (2018)
Li, Li., Yujin, Hu.: Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 107, 77–97 (2016)
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)
Tounsi, A., Al-Basyouni, K.S., Mahmoud, S.R.: Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos. Struct. 125, 621–630 (2015)
Allam, M.N.M., Radwan, A.F.: Nonlocal strain gradient theory for bending, buckling, and vibration of viscoelastic functionally graded curved nanobeam embedded in an elastic medium. Adv. Mech. Eng. 11, 1–15 (2019)
Arefi M, Pourjamshidian M, Arani AG, Rabczuk T (2019) Influence of flexoelectric, small-scale, surface and residual stress on the nonlinear vibration of sigmoid exponential and power-law FG Timoshenko nano-beams. J. Low Freq. Noise Vib. Act. Control; 38(1):122–142.
Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R., Darabi, M.A.: Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress theory. Compos. Struct. 114, 124–134 (2014)
Ke, L.L., Yang, J., Kitipornchai, S., Bradford, M.A.: Bending, buckling and vibration of size-dependent functionally graded annular microplates. Compos. Struct. 94(1), 3250–3257 (2012)
Motezaker, M., Jamali, M., Kolahchi, R.: Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory. J. Comput. Appl. Math. 369, 112625 (2020)
Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331–1342 (2015)
Liu, Hu., Liu, H., Yang, J.: Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation. Compos. B 155, 244–256 (2018)
Akbaş, S.D.: Forced vibration analysis of functionally graded porous deep beams. Compos. Struct. 186, 293–302 (2018)
Karami, B., Janghorban, M., Rabczuk, T.: Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos. B Eng. 182, 107622 (2020)
Li, Q., Wu, D., Chen, X., Liu, L., Yu, Y., Gao, W.: Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler-Pasternak elastic foundation. Int. J. Mech. Sci. 148, 596–610 (2018)
Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–117 (2003)
Fu, Y.M., Zhang, J., Wan, L.J.: Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr. Appl. Phys. 11, 482–485 (2011)
Dang, V.H., Nguyen, D.A., Le, M.Q., Duong, T.H.: Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory. Int. J. Mech. Mater. Des. 16, 289–308 (2020)
Mojahedi, M., Zand, M.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1041 (2010)
Fu, Y., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies. Appl. Math. Model. 35, 941–951 (2011)
Rahaeifard, M., Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sens. Actuators A 171, 370–374 (2011)
Baghani, M.: Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int. J. Eng. Sci. 54, 99–105 (2012)
Ouakada, H.M., Hasan, M.H., Jaber, N.R., Hafiz, M.A.A., Alsaleem, F., Younis, M.: On the double resonance activation of electrostatically actuated microbeam based resonators. Int. J. Non-Linear Mech. 121, 103437 (2020)
Li, L., Tang, H., Hu, Y.: The effect of thickness on the mechanics of nanobeams. Int. J. Eng. Sci. 123, 81–91 (2018)
Tang, H., Li, L., Hu, Y.: Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams. Appl. Math. Model. 66, 527–547 (2019)
Chen, W., Wang, L., Dai, H.: Stability and nonlinear vibration analysis of an axially loaded nanobeam based on nonlocal strain gradient theory. Int. J. Appl. Mech. 11(07), 1950069 (2019)
Chen, W., Wang, L., Dai, H.: Nonlinear free vibration of nanobeams based on nonlocal strain gradient theory with the consideration of thickness-dependent size effect. J. Mech. Mater. Struct. 14(1), 119–137 (2019)
Hieu, D. V., Duong, T. H., Bui, G. P.: Nonlinear vibration of a functionally graded nanobeam based on the nonlocal strain gradient theory considering thickness effect. Adv. Civ. Eng. Vol. 2020, Article ID 9407673
Tang, H., Li, L., Hu, Y., Meng, W., Duan, K.: Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Struct. 137, 377–391 (2019)
He, J.H.: Hamiltonian approach to nonlinear approach. Phys. Lett. A 374, 2312–2314 (2010)
He, J.H.: Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun. 29, 107–111 (2002)
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2020.03. The authors are grateful for this support.
The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.
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Dang, VH., Do, QC. Nonlinear vibration and stability of functionally graded porous microbeam under electrostatic actuation. Arch Appl Mech (2021). https://doi.org/10.1007/s00419-021-01884-7
- Functionally graded porous
- Electrostatic actuation
- Nonlocal strain gradient