Analytical predictions for the buckling of a nanoplate subjected to non-uniform compression based on the four-variable plate theory

Document Type: Research Paper


Department of Mechanical Engineering, Faculty of Engineering, Islamic Azad University, Mashhad, Iran


In the present study, the buckling analysis of the rectangular nanoplate under biaxial non-uniform compression using the modified couple stress continuum theory with various boundary conditions has been considered. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using the Hamilton’s principle. An analytical approach has been applied to obtain exact results from various boundary conditions. Due to the fact that there is not any research about the buckling of nanoplates based on the S-FSDT including the couple stress effect, the obtained results have been compared with the molecular dynamic simulation and FSDT papers which use the Eringen nonlocal elasticity theory. At the end, the results have been presented by making changes in some parameters such as the aspect ratio, the effect of various non-uniform loads and the length scale parameter.


Main Subjects

[1] de La Fuente, J., “CEO Graphenea”

[2] Walker, L.S., Marotto, V.R., Rafiee, M.A., Koretkar, N., Corral, E.L. “Toughening in graphene ceramic composites”, ACS Nano. 5, pp. 3182-90, 2011.

[3] Kvetkova, L., Duszova, A., Hvizdos, P., Dusza, J., Kun, P., Balazsi, C. “Fracture toughness and toughening mechanisms in graphene platelet reinforced Si 3 composites”, Scripta Materialia. 66, pp. 793-796, 2012.

[4] Liang, J., Huang, Y., Zhang, L., Wang, Y., Ma, Y., Guo, T., Chen, Y. “Molecular‐level dispersion of graphene into poly (vinyl alcohol) and effective reinforcement of their nanocomposites”, Advanced Functional Materials. 19, pp. 2297-2302, 2009.

[5] Rafiee, M.A., Rafiee, J., Srivastana, I., Wang, Z., Song, H., Yu, Z-Z., Koratkar, “Fracture and fatigue in    graphene nanocomposites”, Small. 6, pp. 179-83, 2010.

[6] Civalek, O., Demir, Ç. Akgöz, B. “Free Vibration and Bending Analysis of Cantilever Microtubules Based on Nonlocal Continuum Model”, Mathematical and Computational Applications. 15, pp. 289-298, 2010.

[7] Akgöz, B., Civalek, o. “Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams”, International Journal of Engineering Science. 49, pp. 1268-1280, 2011.

[8] Malekzadeh, P., Setoodeh, A.R., Beni, A.A. “Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium”, Composite Structures. 93, pp. 2083-2089, 2011.

[9] Zenkour, A.M., Sobhy, M. “Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium”, Physica E. 53, pp. 251-259, 2013.

[10] Murmu, T., Sienz, J., Adhikari, S., Arnold, C. “Nonlocal buckling of double-nanoplate-systems under biaxial compression”, Composites: Part B. 44, pp. 84-94, 2013.

[11] Wang, Y-Z., Cui, H-T., Li, F-M., Kishimoto, K., “Thermal buckling of a nanoplate with small-scale effects”, Acta Mechanical. 224, pp. 1299-1307, 2013.

[12] Malekzadeh, P., Alibeygi, A. “Thermal Buckling Analysis of Orthotropic Nanoplates on Nonlinear Elastic Foundation”, Encyclopedia of Thermal Stresses, pp. 4862-4872, 2014.

[13] 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. 56, pp. 629-637, 2014.

[14] Radic, N., Jeremic, D., Trifkovic, S., Milutinovic, M. “Buckling analysis of double-orthotropic nanoplates embedded in Pasternak elastic medium using nonlocal elasticity theory”, Composites: Part B. 61, pp. 162-171, 2014.

[15] Karlicic, D., Adhikari, S., Murmu, T. “Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system”, Composites: Part B. 66, pp. 328-339, 2014.

[16] Anjomshoa, A., Shahidi, A.R., Hassani, B., Jomehzadeh, E. “Finite Element Buckling Analysis of Multi-Layered Graphene Sheets on Elastic Substrate Based on Nonlocal Elasticity Theory”, Applied Mathematical Modelling, 38. pp. 1-22, 2014.

[17] Radebe, I.S., Adali, S. “Buckling and sensitivity analysis of nonlocal orthotropic nanoplates with uncertain material properties”, Composites: Part B. 56, pp. 840-846, 2014.

[18] Nguyen, T.K., T. P., Nguyen, B.D., Lee, J., “An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory”, Composite Structures, compstruct.2015. pp. 074, 2015.

[19] Golmakani, M.E., Rezatalab, J. “Non uniform biaxial buckling of orthotropic Nano plates embedded in an elastic medium based on nonlocal Mindlin plate theory”, Composite Structures. 119, pp. 238-250, 2015.

[20] Jamali, M., Shojaee, T., Mohammadi, B. “Uniaxial buckling analysis comparison of nanoplate and nanocomposite plate with central square cut out using domain decomposition method”, Journal of Applied and Computational Mechanics. 2, pp. 230-242, 2016.

[21] Zarei, M. Sh., Hajmohammad, M. H., Nouri, A. “Dynamic buckling of embedded laminated nanocomposite plates based on sinusoidal shear deformation theory”, Journal of Applied and Computational Mechanics. 2, pp. 254-261, 2016.

[22] Malikan, M., Jabbarzadeh, M., Dastjerdi, Sh. “Non-linear Static stability of bi-layer carbon nanosheets resting on an elastic matrix under various types of in-plane shearing loads in thermo-elasticity using nonlocal continuum”, Microsystem Technologies, DOI: 10.1007/s00542, pp. 016-3079-9, 2016.

[23] Mindlin, R. D. “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates”, Transaction of the ASME. 73, pp. 31-38, 1951.

[24] Thai, H-T., Choi, D-H. “A simple first-order shear deformation theory for laminated composite plates”, Composite Structures. 106, pp. 754-763, 2013.

[25] Mindlin, R. D. “Tiersten HF. Effects of couple-stresses in linear elasticity”, Archive for Rational Mechanics and Analysis. 11, pp. 415-48, 1962.

[26] Toupin, R. A., “Elastic materials with couple stresses”, Archive for Rational Mechanics and Analysis. 11, pp. 385-414, 1962.

[27] Koiter, W. T., “Couple stresses in the theory of elasticity”, I and II. Proc K Ned Akad Wet (B. 67, pp. 17-44, 1964.

[28] Cosserat, E., Cosserat, F., “Theory of deformable bodies”, Scientific Library, 6. Paris: A. Herman and Sons, Sorbonne 6, 1909.

[29] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., “Couple stress based strain gradient theory for elasticity”, International Journal of Solids and Structures. 39, pp. 2731-43, 2002.

[30] Akgöz, B., Civalek, O., “Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory”, Materials and Design. 42, pp. 164-171, 2012.

[31] Thai, H-T., Thuc, P., Nguyen, T-K., Lee, J., “Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory”, Composite Structures. 123, pp. 337-349, 2015.

[32] Dey, T., Ramachandra, L.S., “Buckling and postbuckling response of sandwich panels under non-uniform mechanical edge loadings”, Composites: Part B. 60, pp. 537-545, 2014.

[33] Leissa, A.W., Kang, Jae-Hoon, “Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses”, International Journal of Mechanical Sciences. 44, pp. 1925-1945, 2002.

[34] Hwang, I., Seh Lee, J., “Buckling of Orthotropic Plates under Various Inplane Loads”, KSCE Journal of Civil Engineering. 10, pp. 349–356, 2006.

[35] Malikan, M. “Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory”, Applied Mathematical Modelling. 48, pp. 196-207, 2017.

[36] Golmakani, M.E., Sadraee Far, M.N. “Buckling analysis of biaxially compressed double layered graphene sheets with various boundary conditions based on nonlocal elasticity theory”, Microsystem Technologies, DOI 10.1007/s00 .pp,542-016-3053-6, 2016.

[37] Ansari, R., Sahmani, S. “Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations”, Applied Mathematical Modeling. 37, pp. 7338–7351, 2013.