Buckling and Vibration Analysis of a Double-layer Graphene ‎Sheet Coupled with a Piezoelectric Nanoplate

Document Type : Research Paper


1 Malek Ashtar University of Technology, Tehran, Iran‎

2 Department of Mechanical Engineering, Semnan University, Semnan, Semnan, Iran‎

3 Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Hamedan, Iran

4 Department of Mechanical Engineering, Khatmol Anbia Air Defense, Tehran, Iran


In this article, the vibration and buckling of a double-layer Graphene sheet (DLGS) coupled with a piezoelectric nanoplate through an elastic medium (Pasternak and Winkler models) are investigated. DLGS are subjected to biaxial in-plane forces and van der Waals force existing between each layer. Polyvinylidene fluoride (PVDF) piezoelectric nanoplate is subjected to an external electric potential. For the sake of this study, sinusoidal shear deformation theory of orthotropic plate expanded with Eringen’s nonlocal theory is selected. The results indicate that nondimensional frequency and nondimensional critical buckling load rise when the ratio of width to thickness increases. Furthermore, incrementing the effect of elastic medium parameter results in increasing the stiffness of the system and, consequently, rising nondimensional frequency and critical buckling load.


Main Subjects

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[1] Hashemi S.H., Samaei A.T., Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory. Physica E: Low-dimensional Systems and Nanostructures, 43, 2011, 1400-1404.
[2] Murmu T., Pradhan S.C., Buckling of biaxially compressed orthotropic plates at small scales. Mechanics Research Communications, 36, 2009, 933-938.
[3] Haghshenas A., Arani A.G., Nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via Mindlin plate theory. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 2014, 907-920.
[4] Jomehzadeh E., Saidi A.R., A study on large amplitude vibration of multilayered graphene sheets. Computational Materials Science, 50, 2011, 1043-1051.
[5] Ghorbanpour Arani A., Fereidoon A., Kolahchi R., Nonlocal DQM for a nonlinear buckling analysis of DLGSs integrated with Zno piezoelectric layers. Journal of Computational Applied Mechanics, 45, 2014, 9-22.
[6] Cao Q., Xiao G., Huaipeng W., Pengjie W., Aaron L., Yucheng L., Qing P., A review of current development of graphene mechanics. Crystals, 8, 2018, 357.
[7] Eringen A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 1983, 4703-4710.
[8] Eringen A.C., Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1972, 1-16.
[9] Eringe A.C., Nonlocal continuum field theories. New York, Springer, 2002.
[10] Arani A.G., Shiravand, A., Rahi, M., Kolahchi, R., Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation. Physica B: Condensed Matter, 407, 2012, 4123-4131.
[11] Pradhan S.C., Kumar A., Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Composite Structures, 93, 2011, 774-779.
[12] Arani A.G., Kolahchi R., Vossough H., Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory. Physica B: Condensed Matter, 407, 2012, 4458-4465.
[13] Ke L.L., Liu C., Wang Y.S., Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Physica E: Low-dimensional Systems and Nanostructures, 66, 2015, 93-106.
[14] Ebrahimi F., Barati M.R., Damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate based on nonlocal strain gradient theory. Smart Materials and Structures, 26, 2017, 065018.
[15] Shen Z.B., Tang H.L., Li D.K., Tang G.J., Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Computational Materials Science, 61, 2012, 200-205.
[16] Malekzadeh P., Setoodeh A.R., Beni A.A., Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates. Composite Structures, 93, 2011, 1631-1639.
[17] Pradhan S.C., Phadikar J.K., Nonlocal elasticity theory for vibration of nanoplates. Journal of Sound and Vibration, 325, 2009, 206-223.
[18] Malikan M., Buckling analysis of a micro composite plate with nano coating based on the modified couple stress theory. Journal of Applied and Computational Mechanics, 4, 2018, 1-15.
[19] Zhu X., Li, L., Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Composite Structures, 178, 2017, 87-96.
[20] Zhu, X., Li, L., Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. International Journal of Mechanical Sciences, 133, 2017, 639-650.
[21] Allahyari E., Asgari M., Thermo-mechanical vibration of double-layer graphene nanosheets in elastic medium considering surface effects; developing a nonlocal third order shear deformation theory. European Journal of Mechanics-A/Solids, 75, 2019, 307-321.
[22] Kadari B., Bessaim A., Tounsi A., Heireche H., Bousahla A.A., Houari M.S.A., Buckling analysis of orthotropic nanoscale plates resting on elastic foundations. International Journal of Nano Research, 55, 2018, 42-56.
[23] Shaat M., A general nonlocal theory and its approximations for slowly varying acoustic waves. International Journal of Mechanical Sciences, 130, 2017, 52-63.
[24] Ghorbani K., Rajabpour A., Ghadiri M., Determination of carbon nanotubes size-dependent parameters: molecular dynamics simulation and nonlocal strain gradient continuum shell model. Mechanics Based Design of Structures and Machines, 2019, 1-18.
[25] Thai H.T., Vo T.P., Nguyen T.K., Lee J., A nonlocal sinusoidal plate model for micro/nanoscale plates. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 2014, 2652-2660.
[26] Touratier M., An efficient standard plate theory. International Journal of Engineering Science, 29, 1991, 901-916.
[27] Khajehdehi Kavanroodi M., Fereidoon A., Mirafzal A.R., Buckling analysis of coupled DLGSs systems resting on elastic medium using sinusoidal shear deformation orthotropic plate theory. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, 2017, 2817-2829.
[28] Ghorbanpour Arani A., Jamali M., Ghorbanpour-Arani A.H., Kolahchi R., Mosayyebi M., Electro-magneto wave propagation analysis of viscoelastic sandwich nanoplates considering surface effects. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231, 2017, 387-403.
[29] Zenkour A.M., Sobhy M., Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium. Physica E: Low-dimensional Systems and Nanostructures, 53, 2013, 251-259.
[30] Chemi A., Zidour M., Heireche H., Rakrak K., Bousahla A.A., Critical buckling load of chiral double-walled carbon nanotubes embedded in an elastic medium. Mechanics of Composite Materials, 53, 2018, 827-836.
[31] Narendar S., Gopalakrishnan S., Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory. Acta Mechanica, 223, 2012, 395-413.
[32] Lindahl N., Midtvedt D., Svensson J., Oleg A. Nerushev, Lindvall N., Isacsson A., Eleanor EB Campbell, Determination of the bending rigidity of graphene via electrostatic actuation of buckled membranes. Nano Letters, 12, 2012, 3526-3531.
[33] Pradhan S.C., Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Physics Letters A, 373, 2009, 4182-4188.
[34] Mohammadimehr M., Navi B.R., Arani A.G., Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures, 131, 2015, 654-671.
[35] Ansari R., Hasrati E., Faghih Shojaei M., Gholami R., Mohammadi V., Shahabodini A., Size-dependent bending, buckling and free vibration analyses of microscale functionally graded mindlin plates based on the strain gradient elasticity theory. Latin American Journal of Solids and Structures, 13, 2016, 632-664.
[36] Farajpour A., Solghar A.A., Shahidi A., Postbuckling analysis of multi-layered graphene sheets under non-uniform biaxial compression. Physica E: Low-dimensional Systems and Nanostructures, 47, 2013, 197-206.
[37] Arani A.G., Kolahchi R., Barzoki A.A.M., Mozdianfard M.R., Farahani S.M.N., Elastic foundation effect on nonlinear thermo-vibration of embedded double-layered orthotropic graphene sheets using differential quadrature method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 227, 2013, 862-879.
[38] Arani A.G., Abdollahian M., Kolahchi R., Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory. International Journal of Mechanical Sciences, 100, 2015, 32-40.
[39] Mohammadimehr M., Rousta-Navi B., Ghorbanpour-Arani A., Biaxial buckling and bending of smart nanocomposite plate reinforced by CNTs using extended mixture rule approach. Mechanics of Advanced Composite Structures, 1, 2014, 17-26.
[40] Thai H.T., Vo T.P., A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Applied Mathematical Modeling, 37, 2013, 3269-3281.
[41] Mohammadimehr M., Mohandes M., Moradi M., Size dependent effect on the buckling and vibration analysis of double-bonded nanocomposite piezoelectric plate reinforced by boron nitride nanotube based on modified couple stress theory. Journal of Vibration and Control, 22, 2016, 1790-1807.
[42] Xiang S., Wang J., Ai Y.T., Li G.C., Buckling analysis of laminated composite plates by using various higher-order shear deformation theories. Mechanics of Composite Materials, 51, 2015, 645-654.
[43] Mohammadimehr M., Mohandes M., The effect of modified couple stress theory on buckling and vibration analysis of functionally graded double-layer boron nitride piezoelectric plate based on CPT. Journal of Solid Mechanics, 7, 2015, 281-298.
[44] Kim S.E., Thai H.T., Lee J., Buckling analysis of plates using the two variable refined plate theory. Thin-Walled Structures, 47, 2009, 455-462.
[45] Arani A.G., Jalaei M.H., Niknejad S., Arani, A.A.G., Size-Dependent Analysis of Orthotropic Mindlin Nanoplate on Orthotropic Visco-Pasternak Substrate with Consideration of Structural Damping. Journal of Solid Mechanics, 11, 2019, 236-253.