Investigation of Linear and Nonlinear Buckling of Orthotropic Graphene Sheets with Nonlocal Elasticity Theories

Document Type: Persian


1 MSc Student, Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

2 Assistant Prof., Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran


In this paper, analysis of linear and nonlinear buckling of relatively thick orthotropic graphene sheets is carried out under mechanical load based on elasticity theories. With the help of  nonlocal elasticity theory, the principle of virtual work, first order shear theory and Von-Karman nonlinear strain, the dominant relationship in terms of obtained displacements has been obtained, and the method of differential quadrature (DQ) with non-uniform distribution points (Chebyshev -Gauss-Lobato) has been used. To check the validity, the obtained results have been compared with other resources, and the effects of nonlocal coefficient, thickness, radius and elastic base on the dimensionless buckling loads were calculated and investigated. Moreover, the results of analysis using nonlocal and local theory were compared together. It can be noticed that the dimensionless buckling loads on graphene sheets increased more with a decrease in flexibility as far boundary condition is concerned. Additionally, with an increase in the sheet radius, the variation between nonlocal and local analysis results will be more. 


[1] Taniguchi N.,On the Basic Concept of Nanotechnolog, Proceedings of the International Conference of Production Engineering, London, 1974, pp.18-23.

[2] Ma M., Tu J.P., Yuan Y.F., Wang X.L., Li K.F., Mao F., Zeng Z.Y., Electrochemical Performance of ZnONanoplates as Anode Materials for Ni/Zn Secondary Batteries, Journal of Power Source, Vol. 179, 2008, pp. 395-400.

[3]  J. Yguerabide, E. E. Yguerabide , Resonance Light Scattering Particles as Ultrasensitive Labels for Detection of Analytes in a wide Range of Applications, Journal of Cellular Biochemistry-Supplement, 37, 2001, pp.71-81.

[4] Agesen M., Sorensen C.B., Nanoplates and Their Suitability for Use as Solar Cells, Proceeding of Clean Technology, Boston Secondary Batteries, Journal of Power Source, Vol. 179, 2008, pp. 395-400.

[5] Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubonos S.V., Grigorieva I.V., Firsov A.A., Electric Field Effect in Atomically Thin Carbon Films, Science, vol. 306, 2004, pp. 666–669.

[6] Xu Z.P., Buehler M.J., Geometry controls conformation of graphene sheets: membranes, ribbons, and scrolls, ACS-Nano, Vol. 4, 2010, pp. 3869–3876.

[7] Chiu H.Y., Hung P., Postma H.W.Ch., Bockrath M., Atomic-Scale Mass Sensing Using Carbon Nanotube Resonators, Nano Letters, Vol.8, 2008, pp. 4342–4346.

[8] Hernandez E., Goze C., Bernier P., Rubio A.,  Elastic Properties of C and BxCyNz Composite Nanotubes, Physics Review Letters, Vol. 80, 1998, pp. 4502–4505.

[9] Li C.Y., Chou T.W., Elastic wave velocities in single-walled carbon nanotubes, Physics Review B, Vol. 73, 2006, pp. 245-407.

[10] Li C., Chou T.W., Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators, Physics Review B, Vol. 68, 2003, pp. 073405.

[11]Eringen A.C., Nonlocal Continuum Field Theories, Newyork, Springer-Verlag.

[12] Fleck N.A., Hutchinson J.W., Strain Gradient Plasticity, Advance applied mechanics, Vol. 33, 2002, pp. 295-361.

[13]F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong , Couple Stress Based Strain Gradient Theory for Elasticity, International journal of solid structs, 39, 2002, pp. 2731-2743.

[14]Parnes R., Chiskis A., Buckling of nano-fibre reinforced composites: a re-examination of elastic buckling, Journal ofMechanics and Physics of Solids, Vol. 50, 2002, pp. 855–879.

[15]Pradhan S.C., Murmu T., Small Scale Effect onthe Buckling of Single-Layered GrapheneSheetsunder Biaxial Compression via Nonlocal Continuum Mechanics, Computational Materials Science, Vol. 47, 2009, pp. 268-274.

[16]Samaei A.T., Abbasion S., Mirsayar M.M., Buckling Analysis of a Single-Layer Graphene Sheet Embedded in an Elastic Medium Based on Nonlocal Mindlin Plate Theory, Mechanics Research Communications, Vol. 38, 2011, pp.481-485.

[17]Farajpour A., Danesh M., Mohammadi M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, PhysicaE., Vol. 44, 2011, pp.719–727.

[18]Narendar S., Gopalakrishnan S., Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics, Physica E., Vol. 43, 2011, pp. 1185–1191.

[19] Lim C.W., Yang Q., Zhang J.B., Thermal buckling of nanorod based on non-local elasticity theory, International Journal of  Non-Linear Mechanics, Vol. 47, 2012, pp. 496-505.

[20] Farajpour A., Shahidi A.R., Mohammadi M., Mohzoon M., Buckling of Orthotropic Micro/Nanoscale Plates under Linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures, Vol. 94, 2012, pp. 1605-1615.

[21] Prasanna Kumar T.J., Narendar S., Gopalakrishnan S., Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics. Composite Structures, Vol. 100, 2013, pp. 332–342.

[22] Emam S.A., A general nonlocal nonlinear model for buckling of nanobeams, Applied Mathematical Modelling, Vol. 37, 2013, pp. 6929–6939.

[23] Mohammadi M., Farajpour A.,  Moradi A., Ghayour M., Shear buckling of  orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites: Part B, Vol. 56 , 2014, pp. 629–637.

[24] Sarrami-Foroushani S., Azhari M., Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects, Physica E., Vol. 57, 2014, pp. 83–95.

[25] Golmakania M.E., Rezatalaba J., Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory, Composite Structures, Vol. 119, 2015, pp. 238–250.

[26] Farajpour A., Mohammadi M., Shahidi A.R., Mahzoon M., Axisymmetric Buckling of the Circular Graphene Sheets with the Nonlocal continuum plate model, Physica E., Vol. 43, 2011, pp. 1820–1825.



[27] KaramoozRavari M.R., Shahidi A.R., Axisymmetric buckling of the circularannularnanoplates using finite difference method, Mechanica, Vol. 48, 2013, pp. 135–144.

[28] Bedroud M., Hosseini-Hashemi S., Nazemnezhad R., Buckling of circular/annular Mindlinnanoplates via nonlocal lasticity,Acta Mechanics, Vol. 224, 2013, pp. 2663-2676.

[29] Nosier A., Fallah F., Non-linear Analysis of Functionally Graded Circular Plates under Asymmetric Transverse Loading, International journal of non-Linear mechanics, Vol. 44 , 2009, pp. 928-942.

[30]Naderi A., Saidi A.R., Exact solution for stability analysis of moderately thick functionally graded, Composite Structures, Vol. 93, 2011, pp. 629–638.

[31]Shu C.,Differential Quadrature and Its Application in Engineering, Berlin, Springer.