Abstract
The nonlinear electroelastic vibration behavior of viscoelastic nanoplates is investigated based on nonlocal elasticity theory. Employing nonlinear strain–displacement relations, the geometrical nonlinearity is modeled while governing equations are derived through Hamilton’s principle and they are solved applying semi-analytical generalized differential quadrature (GDQ) method. Eringen’s nonlocal elasticity theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Based on Kelvin–Voigt model, the influence of the viscoelastic coefficient is also discussed. It is demonstrated that the GDQ method has high precision and computational efficiency in the vibration analysis of viscoelastic nanoplates. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as electric voltage, small-scale effects, van der Waals interaction, Winkler and Pasternak elastic coefficients, the viscidity and aspect ratio of the nanoplate on its nonlinear vibrational characteristics. It is explicitly shown that the electroelastic vibration behavior of viscoelastic nanoplates is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of viscoelastic nanoplates which are fundamental elements in nanoelectromechanical systems.
Similar content being viewed by others
References
C.L. Kane, E.J. Mele, Size, shape, and low energy electronic structure of carbon nanotubes. Phys. Rev. Lett. 78, 1932–1935 (1997)
A. Maiti, A. Svizhenko, M.P. Anantram, Electronic transport through carbon nanotubes: effects of structural deformation and tube, chirality. Phys. Rev. Lett. 88, 126805 (2002)
Z.L. Wang, Zinc oxide nanostructures: growth, properties and applications. Mater. Today 7(6), 26–33 (2004)
A.P. Chauhan, D.G. Ashwell, Small and large amplitude free vibrations square shallow shells. Int. J. Mech. Sci. 11, 337–350 (1969)
S.C. Pradhan, A. Kumar, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput. Mater. Sci. 50, 239–245 (2010)
S.C. Pradhan, J.K. Phadikar, Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206–223 (2009)
T. Murmu, S.C. Pradhan, Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory. J. Appl. Phys. 105, 064319 (2009)
R. Ansari, B. Arash, H. Rouhi, Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos. Struct. 93, 2419–2429 (2011)
F. Scarpa, S. Adhikari, R. Chowdhury, The transverse elasticity of bilayer graphene. Phys. Lett. A 374, 2053–2057 (2010)
R.K. Bhangale, N. Ganesan, Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates. Int. J. Solids Struct. 43, 3230–3253 (2006)
L.-L. Ke, Y.-S. Wang, J. Yang, S. Kitipornchai, Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mech. Sin. 30, 516–525 (2014)
Y.S. Li, Z.Y. Cai, S.Y. Shi, Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory. Compos. Struct. 111, 522–529 (2014)
S.R. Asemi, A. Farajpour, Vibration characteristics of double-piezoelectric-nanoplate-systems. Micro Nano Lett. 9, 280–285 (2014)
Belabed et al., An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Compos. B 60, 274–283 (2014)
A. Yahia et al., Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct. Eng. Mech. 53(6), 1143–1165 (2015)
A. Mahi, E.A. Adda Bedia, A. Tounsi, A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic functionally graded, sandwich and laminated composite plates. Appl. Math. Model. 39, 2489–2508 (2015)
M. Bennoun, M.S.A. Houari, A. Tounsi, A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech. Adv. Mater. Struct. 23(4), 423–431 (2016)
Bousahla et al., A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates. Int. J. Comput. Methods 11(6), 1350082 (2014)
A.A. Meziane et al., An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J. Sandw. Struct. Mater. 16(3), 293–318 (2014)
H. Bellifa, K.H. Benrahou, L. Hadji, M.S.A. Houari, Tounsi, A Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position. J. Braz. Soc. Mech. Sci. Eng. 38, 265–275 (2016)
Bouderba et al., Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations. Steel Compos. Struct. 14(1), 85–104 (2013)
A. Hamidi, M.S.A. Houari, S.R. Mahmoud, A. Tounsi, A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates. Steel Compos. Struct. 18(1), 235–253 (2015)
H. Hebali, A. Tounsi, M.S.A. Houari, A. Bessaim, E.A.A. Bedia, New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. J. Eng. Mech. 140, 374–383 (2014)
M. Zidi, A. Tounsi, M.S.A. Houari, E.A.A. Bedia, O. Anwar Beg, Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory. Aerosp. Sci Technol. 34, 24–34 (2014)
A. Tounsi, M.S.A. Houari, S. Benyoucef, E.A.A. Bedia, A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp. Sci Technol. 24, 209–220 (2013)
Belkorissat et al., On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable models. Steel Compos. Struct. 18(4), 1063–1081 (2015)
F. Bounouara, K.H. Benrahou, I. Belkorissat, A. Tounsi, A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel Compos. Struct. 20(2), 227–249 (2016)
M. Bourada, A. Kaci, M.S.A. Houari, A. Tounsi, A new simple shear and normal deformations theory for functionally graded beams. Steel Compos. Struct. 18, 409–423 (2015)
Tounsi et al., Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes. Adv. Nano Res. 1(1), 1–11 (2013)
Besseghier et al., Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix. Adv. Nano Res. 3(1), 29–37 (2015)
L. Chaht et al., Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos. Struct. 18(2), 425–442 (2015)
Ahouel et al., Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept. Steel Compos. Struct. 20(5), 963–981 (2016)
A. Zemri, M.S.A. Houari, A.A. Bousahla, A. Tounsi, A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct. Eng. Mech. 54, 693–710 (2015)
S. Benguediab, A. Tounsi, M. Zidour, A. Semmah, Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes. Compos. B 57, 21–24 (2014)
N. Yamaki, Influence of large amplitudes on flexural vibrations of elastic plates. ZAMM 41, 601–610 (1961)
L.E. Shen, H.S. Shen, C.L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput. Mat. Sci. 48, 680–685 (2010)
P. Malekzadeh, Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM. Thin-wall Struct. 46, 11–26 (2008)
E. Jomehzadeh, A.R. Saidi, The small scale effect on nonlinear vibration of single layer graphene sheets. World Acad. Sci. Eng. Technol. 5, 06–29 (2011)
A.R. Setoodeh, P. Malekzadeh, A.R. Vosoughi, Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory. J. Mech. Eng. Sci. 7, 1896–1906 (2011)
A. Ghorbanpour Arani, R. Kolahchi, A.M. Barzoki, M.R. Mozdianfard, S.M. Noudeh Farahani (2012) Elastic foundation effect on nonlinear thermo-vibration of embedded double-layered orthotropic grapheme sheets using differential quadrature method. J. Mech. Eng. Sci. 1–18
S. Razavi, A. Shooshtari, Nonlinear free vibration of magneto-electro-elastic rectangular plates. Compos. Struct. 119, 377–384 (2015)
A. Ghorbanpour Arani, M.J. Maboudi, R. Kolahchi, Nonlinear vibration analysis of visoelastically coupled DLAGS-systems. Eur. J. Mech. A/Solids 45, 185–197 (2014)
F. Ebrahimi, S.H.S. Hosseini, Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. J. Therm. Stress. 39(5), 606–625 (2016)
F. Ebrahimi, S.H.S. Hosseini, Double nanoplate-based NEMS under hydrostatic and electrostatic actuations. Eur. Phys. J. Plus 131(5), 1–19 (2016)
Y. Wang, F.M. Li, Y.Z. Wang, Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Phys. E 67, 65–76 (2015)
A.C. Eringen, Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
J.N. Reddy, An Introduction to Continuum Mechanics (Cambridge University Press, NewYork, 2008)
J.N. Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd edn. (Taylor & Francis, Philadelphia, 2007)
L.L. Ke, Y.S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater. Struct. 21, 025018 (2012)
R. Lakes, Viscoelastic Materials (Cambridge University Press, New York, 2009)
W.N. Findley, F.A. Davis, Creep and relaxation of nonlinear viscoelastic materials (Courier Corporation, 2013)
J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48, 1507–1518 (2010)
I.S. Raju, G.V. Rao, K. Raju, Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates. J. Sound Vib. 46, 415–425 (1976)
C.Y. Chia, Nonlinear analysis of plates (McGraw-Hill, New York, 1980)
S. Kitipornchai, X.Q. He, K.M. Liew, Continuum model for the vibration of multilayered grapheme sheets. Phys. Rev. B 72, 075443 (2005)
Z. Zong, Y. Zhang, Advanced Differential Quadrature Methods (Taylor & Francis Group, New York, 2009)
W. Chen, C. Shu, W. He, The applications of special matrix products to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates. Comput. Struct. 74, 65–76 (2000)
P. Lancaster, M. Timenetsky, The Theory of Matrices with Applications, 2nd edn. (Academic Press, Orlando, 1985)
A.G. Arani, R. Kolahchi, A.A.M. Barzoki, M.R. Mozdianfard, S.M.N. Farahani, Elastic foundation effect on nonlinear thermo-vibration of embedded double-layered orthotropic grapheme sheets using differential quadrature method. Proc. I Mech. E Part C: J. Mech. Eng. Sci. 227, 862–879 (2013)
H.N. Chu, G. Herrman, Influence of large amplitude free flexural vibrations of rectangular elastic plates. J. Appl. Mech. 23, 532–540 (1956)
T. Wah, Large amplitude flexural vibrations of rectangular plates. Intern. J. Mech. Sci. 5, 425–438 (1963)
C. Mei, K. Decha-umphai, A finite element method for nonlinear forced vibration of rectangular plates. AIAA J. 23, 1104–1110 (1985)
C.Y. Chia, M.K. Prabhakara, A general mode approach to nonlinear flexural vibrations of laminated rectangular plates. J. Appl. Mech. 45, 623–628 (1978)
T. Manoj, M. Ayyappan, K.S. Krishnan, B. Nageswara Rao, Nonlinear vibration analysis of thin laminated rectangular plates on elastic foundations. ZAMM Z. Angew. Math. Mech. 3, 183–192 (2000)
Q. Wang, Axi-symmetric wave propagation in a cylinder coated with a piezoelectric layer. Int. J. Solids Struct. 39, 3023–3037 (2002)
L. Chen, L.L. Ke, Y.S. Wang, Jie Yang, S. Kitipornchai, Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Compos. Struct. 106, 167–174 (2013)
H. Niknam, M.M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos. Struct. 119, 452–462 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ebrahimy, F., Hosseini, S.H.S. Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates. Appl. Phys. A 122, 922 (2016). https://doi.org/10.1007/s00339-016-0452-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00339-016-0452-6