Nano-Scaled Plate Free Vibration Analysis by Nonlocal Integral Elasticity Theory

Document Type : Research Article

Authors

Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran

Abstract

In the current study, a finite element method is developed using the principle of total potential energy based on nonlocal integral elasticity theory to investigate the free vibration behavior of nano-scaled plates. The classical plate theory is considered for deriving the formulations of the plate. The eigenvalue problem is extracted by using the variational principle and corresponding natural frequencies of free vibration are obtained. Different boundary conditions and various geometries can now be appropriately analyzed by using the nonlocal finite element method proposed in the current article. The results of the current study are compared with those available in the literature. Then the effects of nonlocal parameters, geometrical parameters, various boundary conditions and surface effects on the free vibration behavior of nano-scaled plates are investigated.

Keywords

Main Subjects

References

[1] S. Iijima, Helical microtubules of graphitic carbon, Nature. 354 (1991) 56–58.
[2] S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter, Nature. 363 (1993) 603–605. doi:10.1038/363603a0.
[3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A.A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science (80-. ). 306 (2004) 666–669. doi:10.1126/science.1102896.
[4] A.K. Geim, K.S. Novoselov, The rise of graphene, Nat. Mater. 6 (2007) 183–191. doi:10.1038/nmat1849.
[5] B. Babić, J. Furer, S. Sahoo, S. Farhangfar, C. Schönenberger, Intrinsic Thermal Vibrations of Suspended Doubly Clamped Single-Wall Carbon Nanotubes, Nano Lett. 3 (2003) 1577–1580. doi:10.1021/nl0344716.
[6] H.-Y. Chiu, P. Hung, H.W.C. Postma, M. Bockrath, Atomic-Scale Mass Sensing Using Carbon Nanotube Resonators, Nano Lett. 8 (2008) 4342–4346.
[7] M. Pimenta, a. Marucci, S. Empedocles, M. Bawendi, E. Hanlon, a. Rao, P. Eklund, R. Smalley, G. Dresselhaus, M. Dresselhaus, Raman modes of metallic carbon nanotubes, Phys. Rev. B. 58 (1998) R16016–R16019. doi:10.1103/PhysRevB.58.R16016.
[8] W.H. Duan, C.M. Wang, Y.Y. Zhang, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, J. Appl. Phys. 101 (2007) 1–7. doi:10.1063/1.2423140.
[9] Y.Y. Zhang, C.M. Wang, W.H. Duan, Y. Xiang, Z. Zong, Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes., Nanotechnology. 20 (2009) 395707–395714. doi:10.1088/0957-4484/20/39/395707.
[10] Y. Xiang, H.-S. Shen, Compressive Buckling of Rippled Graphene via Molecular Dynamics Simulations, Int. J. Struct. Stab. Dyn. 16 (2016) 1550071. doi:10.1142/S0219455415500716.
[11] F. Mehralian, Y. Tadi Beni, M. Karimi Zeverdejani, Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations, Phys. B Condens. Matter. 521 (2017) 102–111. doi:10.1016/j.physb.2017.06.058.
[12] F. Mehralian, Y. Tadi Beni, M. Karimi Zeverdejani, Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes, Phys. B Condens. Matter. 514 (2017) 61–69. doi:10.1016/j.physb.2017.03.030.
[13] Y. Chen, J.D. Lee, A. Eskandarian, Atomistic viewpoint of the applicability of microcontinuum theories, Int. J. Solids Struct. 41 (2004) 2085–2097. doi:10.1016/j.ijsolstr.2003.11.030.
[14] R.A. Toupin, Elastic materials with couple-stresses, Arch. Ration. Mech. Anal. 11 (1962) 385–414. doi:10.1007/BF00253945.
[15] A.C. Eringen, E.S. Suhubi, Nonlinear theory of simple micro-elastic solids—I, Int. J. Eng. Sci. 2 (1964) 189–203. doi:10.1016/0020-7225(64)90004-7.
[16] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703–4710. doi:10.1063/1.332803.
[17] S. Kitipornchai, X.Q. He, K.M. Liew, Continuum model for the vibration of multilayered graphene sheets, Phys. Rev. B - Condens. Matter Mater. Phys. 72 (2005) 1–6. doi:10.1103/PhysRevB.72.075443.
[18] P. Lu, P.Q. Zhang, H.P. Lee, C.M. Wang, J.N. Reddy, Non-local elastic plate theories, Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2007) 3225–3240. doi:10.1098/rspa.2007.1903.
[19] S.C. Pradhan, J.K. Phadikar, Nonlocal elasticity theory for vibration of nanoplates, J. Sound Vib. 325 (2009) 206–223. doi:10.1016/j.jsv.2009.03.007.
[20] 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 (2009). doi:10.1063/1.3091292.
[21] R. Aghababaei, J.N. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, J. Sound Vib. 326 (2009) 277–289. doi:10.1016/j.jsv.2009.04.044.
[22] R. Ansari, S. Sahmani, B. Arash, Nonlocal plate model for free vibrations of single-layered graphene sheets, Phys. Lett. Sect. A Gen. At. Solid State Phys. 375 (2010) 53–62. doi:10.1016/j.physleta.2010.10.028.
[23] M. Şimşek, Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Phys. E Low-Dimensional Syst. Nanostructures. 43 (2010) 182–191. doi:10.1016/j.physe.2010.07.003.
[24] M. Aydogdu, A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Phys. E Low-Dimensional Syst. Nanostructures. 41 (2009) 1651–1655. doi:10.1016/j.physe.2009.05.014.
[25] A. Alibeigloo, Free vibration analysis of nano-plate using three-dimensional theory of elasticity, Acta Mech. 222 (2011) 149–159. doi:10.1007/s00707-011-0518-7.
[26] A. Alibeigloo, Three-dimensional free vibration analysis of multi-layered graphene sheets embedded in elastic matrix, J. Vib. Control. 19 (2013) 2357–2371. doi:10.1177/1077546312456056.
[27] D. Karličić, S. Adhikari, T. Murmu, M. Cajić, Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system, Compos. Part B Eng. 66 (2014) 328–339. doi:10.1016/j.compositesb.2014.05.029.
[28] M. Şimşek, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Compos. Part B Eng. 56 (2014) 621–628.
[29] M. Aydogdu, A nonlocal rod model for axial vibration of double-walled carbon nanotubes including axial van der Waals force effects, J. Vib. Control. 21 (2015) 3132–3154. doi:10.1177/1077546313518954.
[30] Z. Zhang, C.M. Wang, N. Challamel, M. Asce, Eringen’s Length-Scale Coef fi cients for Vibration and Buckling of Nonlocal Rectangular Plates with Simply Supported Edges, J. Eng. Mech. 141 (2015) 1–10. doi:10.1061/(ASCE)EM.1943-7889.0000838.
[31] L. Behera, S. Chakraverty, Effect of scaling effect parameters on the vibration characteristics of nanoplates, J. Vib. Control. 22 (2016) 2389–2399. doi:10.1177/1077546314547376.
[32] F. Mehralian, Y.T. Beni, A Nonlocal Strain Gradient Shell Model for Free Vibration Analysis of Functionally Graded Shear Deformable Nanotubes, Int. J. Eng. Appl. Sci. 9 (2017) 88–102.
[33] H. Zeighampour, Y.T. Beni, I. Karimipour, Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory, Microfluid. Nanofluidics. 21 (2017). doi:10.1007/s10404-017-1918-3.
[34] H. Zeighampour, Y.T. Beni, I. Karimipour, Material length scale and nonlocal effects on the wave propagation of composite laminated cylindrical micro / nanoshells, Eur. Phys. J. Plus. 132 (2017) 1–14. doi:10.1140/epjp/i2017-11770-7.
[35] H. Zeighampour, Y. Tadi Beni, M. Botshekanan Dehkordi, Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory, Thin-Walled Struct. 122 (2018) 378–386. doi:10.1016/j.tws.2017.10.037.
[36] J.K. Phadikar, S.C. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Comput. Mater. Sci. 49 (2010) 492–499. doi:10.1016/j.commatsci.2010.05.040.
[37] C. Polizzotto, Nonlocal elasticity and related variational principles, Int. J. Solids Struct. 38 (2001) 7359–7380. doi:10.1016/S0020-7683(01)00039-7.
[38] A.A. Pisano, A. Sofi, P. Fuschi, Nonlocal integral elasticity: 2D finite element based solutions, Int. J. Solids Struct. 46 (2009) 3836–3849. doi:10.1016/j.ijsolstr.2009.07.009.
[39] M. Taghizadeh, H.R. Ovesy, S.A.M. Ghannadpour, Beam Buckling Analysis by Nonlocal Integral Elasticity, Int. J. Struct. Stab. Dyn. 16 (2016) 1–19. doi:10.1142/S0219455415500157.
[40] M. Taghizadeh, H.R. Ovesy, S.A.M. Ghannadpour, Nonlocal integral elasticity analysis of beam bending by using finite element method, Struct. Eng. Mech. 54 (2015) 755–769. doi:10.12989/sem.2015.54.4.755.
[41] M. Naghinejad, H.R. Ovesy, Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory, J. Vib. Control. 24 (2018) 3974–3988. doi:10.1177/1077546317717867.
[42] M. Naghinejad, H.R. Ovesy, Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach, J. Vib. Control. (2018) 1–15.
[43] D.G.B. Edelen, A.E. Green, N. Laws, Nonlocal continuum mechanics, Arch. Ration. Mech. Anal. 43 (1971) 36–44. doi:10.1007/BF00251544.
[44] A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972) 233–248. doi:10.1016/0020-7225(72)90039-0.
[45] Z.P. Bažant, M. Jirásek, Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress, J. Eng. Mech. 128 (2002) 1119–1149. doi:10.1061/(ASCE)0733-9399(2002)128:11(1119).
[46] M. Karimi, A.R. Shahidi, Nonlocal, refined plate, and surface effects theories used to analyze free vibration of magnetoelectroelastic nanoplates under thermo-mechanical and shear loadings, Appl. Phys. A Mater. Sci. Process. 123 (2017) 0. doi:10.1007/s00339-017-0828-2.
[47] M. Karimi, H.A. Haddad, A.R. Shahidi, Combining surface effects and non-local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates, Micro Nano Lett. 10 (2015) 276–281. doi:10.1049/mnl.2014.0651.
[48] M. Karimi, A.R. Shahidi, S. Ziaei-Rad, Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings, Microsyst. Technol. (2017). doi:10.1007/s00542-017-3395-8.
[49] M. Karimi, H.R. Mirdamadi, alireza Shahidi, Shear vibration and buckling of double ‑layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects, Springer Berlin Heidelberg, 2015. doi:10.1007/s00542-015-2744-8.
[50] M. Karimi, H.R. Mirdamadi, A. Shahidi, Positive and negative surface effects on the buckling and vibration of rectangular nanoplates under biaxial and shear in ‑plane loadings based on nonlocal elasticity theory, J. Brazilian Soc. Mech. Sci. Eng. (2016). doi:10.1007/s40430-016-0595-6.
[51] M.H. Shokrani, M. Karimi, M.S. Tehrani, H.R. Mirdamadi, Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method, J. Brazilian Soc. Mech. Sci. Eng. 38 (2016) 2589–2606. doi:10.1007/s40430-015-0370-0.
[52] K.F. Wang, B.L. Wang, A finite element model for the bending and vibration of nanoscale plates with surface effect, Finite Elem. Anal. Des. 74 (2013) 22–29. doi:10.1016/j.finel.2013.05.007.
AUT Journal of Mechanical Engineering
Volume 3, Issue 1
May 2019
Pages 77-88

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