Differential Quadrature Method for Dynamic Buckling of Graphene Sheet Coupled by a Viscoelastic Medium Using Neperian Frequency Based on Nonlocal Elasticity Theory

Document Type: Research Paper


1 Department of Mechanical engineering, faculty of engineering, Islamic Azad University, Mashhad branch, Iran

2 Department of mechanical engineering, Islamic Azad university, Mashhad


In the present study, the dynamic buckling of the graphene sheet coupled by a viscoelastic matrix was studied. In light of the simplicity of Eringen's non-local continuum theory to considering the nanoscale influences, this theory was employed. Equations of motion and boundary conditions were obtained using Mindlin plate theory by taking nonlinear strains of von Kármán and Hamilton's principle into account. On the other hand, a viscoelastic matrix was modeled as a three-parameter foundation. Furthermore, the differential quadrature method was applied by which the critical load was obtained. Finally, since there was no research available for the dynamic buckling of a nanoplate, the static buckling was taken into consideration to compare the results and explain some significant and novel findings. One of these results showed that for greater values of the nanoscale parameter, the small scale had further influences on the dynamic buckling.


Main Subjects

[1] Shijie, C., Hong, H., Hee Kiat, Ch., Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact, International Journal of Impact Engineering, 25(2), 2001, 147-167.

[2] Hosseini-Ara, R., Mirdamadi, H.R., Khademyzadeh, H., Salimi, H., Thermal effect on dynamic stability of single-walled Carbon Nanotubes in low and high temperatures based on Nonlocal shell theory, Advanced Materials Research, 622-623, 2013, 959-964.

[3] Haftchenari, H., Darvizeh, M., Darvizeh, A., Ansari, R., Sharma, C.B., Dynamic analysis of composite cylindrical shells using differential quadrature method (DQM), Composite Structures, 78(2), 2007, 292–298.

[4] Tamura, Y.S., Babcock, C.D., Dynamic stability of cylindrical shells under step loading, Journal of Applied Mechanics, 42(1), 1975, 190-194 

[5] Jabareen, M., Sheinman, I., Dynamic buckling of a beam on a nonlinear elastic foundation under step loading, Journal of Mechanics of Materials and Structures, 4, 2009, 7-8.

[6] Ramezannezhad Azarboni, H., Darvizeh, M., Darvizeh, A., Ansari, R., Nonlinear dynamic buckling of imperfect rectangular plates with different boundary conditions subjected to various pulse functions using the Galerkin method, Thin-Walled Structures, 94, 2015, 577–584.

[7] Wang, X., Yang, W.D., Yang, S., Dynamic stability of carbon nanotubes reinforced composites, Applied Mathematical Modelling, 38(11-12), 2014, 2934-2945.

[8] Petry, D., Fahlbusch, G., Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Structures, 38(3), 2000, 267–283.

[9] Kubiak, T., Criteria of dynamic buckling estimation of thin-walled structures, Thin-Walled Structures, 45(10-11), 2007, 888–892.

[10] Reddy, J.N., Srinivasa, A.R., Non-linear theories of beams and plates accounting for moderate rotations and material length scales, International Journal of Non-Linear Mechanics, 66, 2014, 43-53.

[11] Ghorbanpour Arani, A., Shiravand, A., Rahi, M., Kolahchi, R., Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation, Physica B, 407, 2012, 4123–4131.

[12] Eringen, A.C., Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.

[13] Eringen, A.C., Linear theory of non-local elasticity and dispersion of plane waves, International Journal of Engineering Science, 10(5), 1972, 425-435.

[14] Duan, W.H., Wang, C.M., Zhang, Y.Y., Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics, 101(2), 2007, 24305-24311.

[15] Duan, W.H., Wang, C.M., Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology, 18(38), 2007, 385704.

[16] https://www.slideshare.net/zead28/concept-ofcomplexfrequency.

[17] Franco, S., Electric Circuits Fundamentals, Oxford University Press, Inc., 1995.

[18] Bellman, R., Kashef, B.G., Casti, J., Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equation, Journal of Computational Physics, 10(1), 1972, 40–52.

[19] Shu, C., Differential Quadrature and Its Application in Engineering, Springer, Berlin, 2000.

[20] Bellman, R., Casti, J., Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications, 34(2), 1971, 235–238.

[21] Chen, W., Differential Quadrature Method and its Applications in Engineering, Shanghai Jiao Tong University, 1996.

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

[23] Murmu, T., Pradhan, S.C., Buckling analysis of a single-walled carbon nanotubes embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E, 41(7), 2009, 1232–9.

[24] 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, 23(6), 2017, 2145-2161.

[25] 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 Modelling, 37(12-13), 2013, 7338–7351.

[26] Malikan, M., Jabbarzadeh, M., Sh. Dastjerdi, 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, 23(7), 2017, 2973-2991.

[27] 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, 2017, 196–207.

[28] Malikan, M., Analytical predictions for the buckling of a nanoplate subjected to non-uniform compression based on the four-variable plate theory, Journal of Applied and Computational Mechanics, 3(3), 2017, 218–228.

[29] Malikan, M., Buckling analysis of micro sandwich plate with nano coating using modified couple stress theory, Journal of Applied and Computational Mechanics, 4(1), 2018, 1-15.

[30] Civalek, Ö., Korkmaz, A.,  Demir, Ç., Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges, Advances in Engineering Software, 41(4), 2010, 557-560.

[31] Dastjerdi, S., Jabbarzadeh, M., Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics, Composites Part B: Engineering, 87, 2016, 161-175.

[32] Karličić, D., Adhikari, S., Murmu, T., Cajić, M., Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi- nanoplate system, Composites Part B, 66, 2014, 328-339.