Journal of Applied Nonlinear Dynamics
Buckling and Nonlinear Vibration of SizeDependent Nanobeam based on the NonLocal Strain Gradient Theory
Journal of Applied Nonlinear Dynamics 9(3) (2020) 427446  DOI:10.5890/JAND.2020.09.007
Van  Hieu Dang
Department of Mechanics, Faculty of Automotive and Power Machinery Engineering, Thai Nguyen University of Technology, Thainguyen, Vietnam
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Abstract
Based on the nonlocal strain gradient theory, a EulerBernoulli nanobeam model subjected to the compressive axial force and resting on the WinklerPasternak layer is developed to study buckling and free nonlinear vibration problems. Critical buckling force and
nonlinear frequency of simply supported nanobeam are analytically derived. Comparison of obtained analytical solutions with published and numerical ones shows accuracy of the present solutions. Effects of the scale factor, the aspect ratio, the Winkler parameter and the Pasternak parameter on the critical buckling force ratio and the vibration response of the nanobeam are studied in this work.
Acknowledgments
This work is supported by Thai Nguyen University of Technology.
References

[1]  Chuang, W.C., Lee, H.L., Chang, P.Z., and Hu, Y.C. (2010), Review on the modeling of electrostatic MEMS, Sensors, 10, 61496171. 

[2]  Loh, O.Y., and Espinosa, H.D. (2012), Nanoelectromechanical contact switches, Nature Nanotechnology, 7, 283295. 

[3]  Zhang, W.M., Yan, H., Peng, Z.K., and Meng, G. (2014), Electrostatic pullin instability in MEMS/NEMS: A review, Sensors Actuators A: Physical, 214, 187218. 

[4]  Zhang, W.M., Yan, H., Peng, Z.K., and Meng, G. (2014), Electrostatic pullin instability in MEMS/NEMS: A review, Sensors and Actuators A: Physical, 214, 187218 

[5]  Clarke, D.R., Ma, Q., and Clarke, D.R. (1995), Size dependent hardness of silver single crystals, Journal of Materials Research, 10(4), 853863. 

[6]  Fleck, N.A., Muller, G.M., Ashby, M.F., and Hutchinson, J.W. (1994), Strain gradient plasticity: Theory and experiment, Acta Metallurgica et Materialia, 42( 2), 475487. 

[7]  Stölken, J.S. and Evans, A.G. (1998), A microbend test method for measuring the plasticity length scale, Acta Materialia, 46(14), 51095115 

[8]  Eringen, A.C. (1983), On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54(9), 47034710. 

[9]  Reddy, J.N. (2007), Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45, 288307. 

[10]  Aydogdu, M. (2009), A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E, 41, 16511655. 

[11]  Ke, L.L., Wang, Y.S., and Wang, Z.D. (2012), Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Composite Structures, 94, 20382047. 

[12]  Șimşek, M. (2014), Large Amplitude Free Vibration of Nanobeams with Various Boundary Conditions Based on the Nonlocal Elasticity Theory, Composites Part B: Engineering, 56, 621628. 

[13]  Togun, N. and Bagdatlı, S.M. (2016), Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on NonLocal EulerBernoulli Beam Theory, Mathematical and Computational Applications, 21(3), doi:10.3390/mca21010003 

[14]  Toupin, R. (1962), Elastic materials with couplestresses, Archive for Rational Mechanics and Analysis, 11, 385414. 

[15]  Mindlin, R.D. and Tiersten, H.F. (1962), Effects of couplestresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 41548. 

[16]  Mindlin, R. (1964), Microstructure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 5278. 

[17]  Mindlin, R. (1965), Second gradient of strain and surfacetension in linear elasticity, International Journal of Solids and Structures, 1, 414438. 

[18]  Koiter, W.T. (1964), Couplestresses in the theory of elasticity: I and II. Koninklijke Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Arts and Sciences), B67, 1744. 

[19]  Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., and Tong, P. (2003), Experiments and theory in strain gradient elasticity, Journal of Mech Phys Solids, 51, 477508. 

[20]  Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong P. (2002), Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 27312743. 

[21]  PapargyriBeskou, S., Tsepoura, K.G., Polyzos, D., and Beskos, D.E. (2003), Bending and stability analysis of gradient elastic beams, International Journal of Solids and Structures, 40(2), 385400. 

[22]  Kong, S.L., Zhou, S.J., Nie, Z.F., and Wang, K. (2009), Static and dynamic analysis of microbeams based on strain gradient elasticity theory, International Journal of Engineering Science, 47, 487498. 

[23]  Wang, B., Zhao, J., and Zhou, S. (2010), A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics  A/Solids, 29(4), 5919. 

[24]  Rajabi, F. and Ramezani, S. (2013), A nonlinear microbeam model based on strain gradient elasticity theory, Acta Mechanica Solida Sinica, 26(1), 2134 

[25]  Salamattalab, M., Shahabi, F., and Assadi, A. (2013), Size dependent analysis of functionally graded microbeams using strain gradient elasticity incorporated with surface energy, Applied Mathematical Modelling, 37(12), 507526. 

[26]  Xia, W., Wang, L., and Yin, L. (2010), Nonlinear nonclassical microscale beams: Static bending, postbuckling and free vibration, International Journal of Engineering Science, 48, 20442053. 

[27]  Simsek, M. (2014), Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Composite Structures, 112, 26472. 

[28]  Farokhi, H., Ghayesh, M.H., and Amabili, M. (2013), Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory, International Journal of Engineering Science, 68, 1123. 

[29]  Hieu, D.V. (2018), Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation, Mathematical Problems in Engineering, Volume 2018, Article ID 1031237, 17 pages. 

[30]  Ma, H.M., Gao, X.L., and Reddy, J.N. (2008), A microstructuredependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56(12), 33793391. 

[31]  Akgoz, B. and Civalek, O. (2013), Free vibration analysis of axially functionally graded tapered Bernoulli Euler microbeams based on the modified couple stress theory, Composite Structures, 98, 31422. 

[32]  Belardinelli, P., Lenci, S., and Brocchini, M. (2014), Modeling and analysis of an electrically actuated microbeambased on nonclassical beam theory, Journal of Computational and Nonlinear Dynamics, 9(3), 031016, 10 pages. 

[33]  Belardinelli, P., Lenci, S., and Brocchini, M. (2014), A comparison of different semianalytical techniques to determine the nonlinear oscillations of a slender microbeam, Meccanica, 49, 18211831. 

[34]  Belardinelli, P., Lenci, S., and Brocchini, M. (2015), Vibration frequency analysis of an electricallyactuated microbeam resonator accounting for thermoelastic coupling effects, International Journal of Dynamics and Control, 3(2), 157172. 

[35]  Thai, H.T., Vo, T.P., Nguyen, T.K., and Kim, S.E. (2017), A review of continuum mechanics models for sizedependent analysis of beams and plates, Composite Structures, 177, 196219. 

[36]  Lim, C.W., Zhang, G., and Reddy, J.N. (2015), A HigherOrder Nonlocal Elasticity and Strain Gradient Theory and Its Applications in Wave Propagation, Journal of the Mechanics and Physics of Solids, 78, 298313. 

[37]  Li, L. and Hu, Y.J. (2015), Buckling analysis of sizedependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, 97, 8494. 

[38]  Li, L., Li, X.B., and Hu, Y.J. (2016), Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, 102, 7792. 

[39]  Simsek, M. (2016), Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105, 1227. 

[40]  Ebrahimi, F. and Barati, M.R. (2017), Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232, issue 11, 20672078. 

[41]  Arefi, M., Pourjamshidian, M., and Ghorbanpour, A. (2017), Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nanobeam with FGCNTRCs facesheets in electrothermal environment, Applied Physics A, 123, 323. 

[42]  Li, L., Hu, Y.J., Li, X.B., and Ling, L. (2016), Sizedependent effects on critical flow velocity of fluidconveying microtubes via nonlocal strain gradient theory, Microfluid Nanofluid, 20, 76. 

[43]  Atashafrooz, M., Bahaadini, R., and Sheibani, H.R. (2018), Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow, Mechanics of Advanced Materials and Structures, 25, DOI: 10.1080/15376494.2018.1487611 . 

[44]  Ghayesh, M.H. and Farajpour, A. (2018), Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory, International Journal of Engineering Science, 129, 8495. 

[45]  Ghayesh, M.H. and Farokhi, H. (2018), On the viscoelastic dynamics of fluidconveying microtubes, Interna tional Journal of Engineering Science, 127, 186200. 

[46]  Mohammadi, K., Rajabpour, A., and Ghadiri, M. (2018), Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation, Computational Materials Science, 148, 104115. 

[47]  Anh, N.D., Hai, N.Q., and Hieu, D.V. (2017), The equivalent linearization method with a weighted averaging for analyzing of nonlinear vibrating systems, Latin American Journal of Solids Structures, 14(9), 17231740. 

[48]  Hieu, D.V., Hai, N.Q., and Hung, D.T. (2018), The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators, Journal of Applied Mathematics, Volume 2018, Article ID 7487851. 

[49]  Hieu, D.V. and Hai, N.Q. (2018), Analyzing of Nonlinear Generalized Duffing Oscillators Using the Equivalent Linearization Method with a Weighted Averaging, Asian Research Journal of Mathematics, 9(1), 114. 

[50]  Barari, A., Kaliji, H.D., Ghadami, M., and Domairry, G. (2011), Nonlinear vibration of EulerBernoulli beams, Latin American Journal of Solids Structures, 8, 139148 

[51]  Nayfeh, A.H. and Emam, S.A. (2008), Exact solution and stability of postbuckling configurations of beams, Nonlinear Dynamics, 54, 395408 

[52]  Nguyen, D.K., Trinh, T.H., and Sthenly, G.B. (2012), Postbuckling response of elasticplastic beam resting on an elastic foundation to eccentric axial load, The IES Journal Part A: Civil & Structural Engineering, 5(1), 4349. 

[53]  Gan, B.S. and Kien, N.D. (2018), Large Deflection Analysis of Functionally Graded Beams Resting on a TwoParameter Elastic Foundation, Journal of Asian Architecture and Building Engineering, 13(3), 649656, DOI: 10.3130/jaabe.13.649. 

[54]  Bayat, M., Pakar, I., and Domairry, G. (2012), Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review, Latin American Journal of Solids Structures, 9(2), 145234. 