Vibration Behavior of Nanocomposite Plate Reinforced by Pristine and Defective Graphene Sheets; an Analytical Approach


Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran


Free vibration characteristics of polymer composite plates reinforced by graphene nanosheets employing the Eringen nonlocal elasticity theory were investigated. Theoretical formulations are derived based on Hamilton’s principle implementing linear orthotropic constitutive equations of lamina while the behavior of nanostructure points affected by all other nonlocal points is also taken into account. For obtaining the mechanical properties, a new modified Halpin–Tsai model is employed. Governing equations are solved by developing an efficient analytical solution. The accuracy of the presented method is examined, by comparing the results with literature in which a good agreement is observed. Effects of different boundary conditions, volume fraction, graphene sheets orientation angle and Eringen nonlocal parameter on frequency of nanocomposite are analyzed. Effects of the presence of vacancy defects in the nanosheet on the behavior of reinforced composites were also studied. The results illustrate that by increasing the nonlocal parameter the natural frequency showed a decreasing trend while by increasing the graphene sheet’s volume fraction, natural frequencies significantly increased. It could be concluded that the orientation angle variations in graphene sheets, did not play an important role on the natural frequency of nanocomposite as well as degradation of properties resulted in from vacancy defects.


1.     Eringen, A.C., "Nonlocal continuum field theories, Springer Science & Business Media,  (2002).

2.     Karličić, D., Kozić, P., Adhikari, S., Cajić, M., Murmu, T. and Lazarević, M., " Nonlocal mass-nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field ", International Journal of Mechanical Sciences,  Vol. 96, (2015), 132-142.

3.     Pradhan, S. and Phadikar, J., "Nonlocal elasticity theory for vibration of nanoplates", Journal of Sound and Vibration,  Vol. 325, No. 1-2, (2009), 206-223.

4.     Hashemi, S.H. and Khaniki, H.B., "Analytical solution for free vibration of a variable cross-section nonlocal nanobeam", International Journal of Engineering-Transactions B: Applications,  Vol. 29, No. 5, (2016), 688-696.

5.     Shariyat, M., Sarvi, Z. and Asgari, M., "A unit-cell-based three-dimensional molecular mechanics analysis for buckling load, effective elasticity and poisson's ratio determination of the nanosheets", Molecular Simulation,  Vol. 42, No. 5, (2016), 353-369.

6.     Montazeri, A. and Rafii-Tabar, H., "Multiscale modeling of graphene-and nanotube-based reinforced polymer nanocomposites", Physics Letters A,  Vol. 375, No. 45, (2011), 4034-4040.

7.     Kitipornchai, S., Chen, D. and Yang, J., "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Materials & Design,  Vol. 116, (2017), 656-665.

8.     Jalali, S., Jomehzadeh, E. and Pugno, N., "Influence of out-of-plane defects on vibration analysis of graphene: Molecular dynamics and non-local elasticity approaches", Superlattices and Microstructures,  Vol. 91, (2016), 331-344.

9.     Nazemnezhad, R., "Nonlocal timoshenko beam model for considering shear effect of van der waals interactions on free vibration of multilayer graphene nanoribbons", Composite Structures,  Vol. 133, (2015), 522-528.

10.   Arani, A.G., Haghparast, E. and Zarei, H.B., "Nonlocal vibration of axially moving graphene sheet resting on orthotropic visco-pasternak foundation under longitudinal magnetic field", Physica B: Condensed Matter,  Vol. 495, (2016), 35-49.

11.   Allahyari, E. and Fadaee, M., "Analytical investigation on free vibration of circular double-layer graphene sheets including geometrical defect and surface effects", Composites Part B: Engineering,  Vol. 85, (2016), 259-267.

12.   Yao, H., Hawkins, S.A. and Sue, H.-J., "Preparation of epoxy nanocomposites containing well-dispersed graphene nanosheets", Composites Science and Technology,  Vol. 146, (2017), 161-168.

13.   Ragavan, K. and Rastogi, N.K., "Β-cyclodextrin capped graphene-magnetite nanocomposite for selective adsorption of bisphenol-a", Carbohydrate polymers,  Vol. 168, (2017), 129-137.

14.   Gharib, A., Karimi, M.S. and Arani, A.G., "Vibration analysis of the embedded piezoelectric polymeric nano-composite panels in the elastic substrate", Composites Part B: Engineering,  Vol. 101, No., (2016), 64-76.

15.   Mohammadimehr, M., Navi, B.R. and 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,  Vol. 131, No., (2015), 654-671.

16.   Eringen, A.C., "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", Journal of applied physics,  Vol. 54, No. 9, (1983), 4703-4710.

17.   Affdl, J.H. and Kardos, J., "The halpin‐tsai equations: A review", Polymer Engineering & Science,  Vol. 16, No. 5, (1976), 344-352.

18.   Shen, H.-S., "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Composite Structures,  Vol. 91, No. 1, (2009), 9-19.

19.   Reddy, J.N., "Theory and analysis of elastic plates and shells, CRC press,  (2006).

20.   Hao, F., Fang, D. and Xu, Z., "Mechanical and thermal transport properties of graphene with defects", Applied physics letters,  Vol. 99, No. 4, (2011), 041901.

21.   Zhu, P., Lei, Z. and Liew, K.M., "Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory", Composite Structures,  Vol. 94, No. 4, (2012), 1450-1460.

22.   Alibeigloo, A. and Emtehani, A., "Static and free vibration analyses of carbon nanotube-reinforced composite plate using differential quadrature method", Meccanica,  Vol. 50, No. 1, (2015), 61-76.

23.   Wu, C.-P. and Li, H.-Y., "Three-dimensional free vibration analysis of functionally graded carbon nanotube-reinforced composite plates with various boundary conditions", Journal of Vibration and Control,  Vol. 22, No. 1, (2016), 89-107.