Large deformation analysis for functionally graded carbon nanotube-reinforced composite plates using an efficient and simple refined theory

  • Bakhti, K. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Kaci, A. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Bousahla, A.A. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Houari, M.S.A. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Tounsi, A. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Adda Bedia, E.A. (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes)
  • Received : 2012.10.08
  • Accepted : 2013.02.20
  • Published : 2013.04.25


In this paper, the nonlinear cylindrical bending behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) is studied using an efficient and simple refined theory. This theory is based on assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTCRs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The fundamental equations for functionally graded nanocomposite plates are obtained using the Von-Karman theory for large deflections and the solution is obtained by minimization of the total potential energy. The numerical illustrations concern the nonlinear bending response of FG-CNTRC plates under different sets of thermal environmental conditions, from which results for uniformly distributed CNTRC plates are obtained as comparators.


functionally graded materials;nanocomposites;nonlinear behavior;refined plate theory


  1. Thostenson, E.T., Ren, Z.F. and Chou, T.W. (2001), "Advances in the science and technology of carbon nanotubes and their composites: A review", Compos. Sci. Tech., 61(13), 1899-1912.
  2. Thostenson, ET. and Chou, TW. (2003), "On the elastic properties of carbon nanotube-based composites: modelling and characterization", J. Phys. B: Appl. Phys., 36(5), 573-582.
  3. Vodenitcharova, T. and Zhang, LC. (2003), "Effective wall thickness of a single-walled carbon nanotube", Phys. Rev. B, 68, 165401.
  4. Vodenitcharova, T and, Zhang, LC. (2006), "Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube", Int'l J. Solids Struct., 43(10), 3006-3024.
  5. Wu, T.L., Shukla, K.K. and Huang, J.H. (2007), "Post-buckling analysis of functionally graded rectangular plates", Compos. Struct., 81(1), 1-10.
  6. Wuite, J. and Adali, S. (2005), "Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis", Compos. Struct., 71(3-4), 388-396.
  7. Yang, J. and Chen, Y. (2008), "Free vibration and buckling analyses of functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60.
  8. Yang, J., Kitipornchai, S. and Liew, K.M. (2003), "Large amplitude vibration of thermo-electromechanically stressed FGM laminated plates", Comput. Methods Appl. Mech. Eng., 192(36), 3861-3885.
  9. Yang, J. and Shen, H.S. (2003), "Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions", Compos. Part B, 34(2), 103-115.
  10. Zhang, C.L. and Shen, H.S. (2006a), "Temperature-dependent elastic properties of single-walled carbon nanotubes: prediction from molecular dynamics simulation", Appl. Phys. Lett., 89(8), 081904.
  11. Zhang, C.L. and Shen, H.S. (2006b), "Buckling and postbuckling analysis of single-walled carbon nanotubes in thermal environments via molecular dynamics simulation", Carbon, 44(13), 2608-2616.
  12. Hu, N., Fukunaga, H., Lu, C., Kameyama, M., Yan, B. (2005), "Prediction of elastic properties of carbon nanotube reinforced composites", Proc. R. Soc. A, 461(2058), 1685-1710.
  13. Jin, Y., Yuan, FG. (2003), "Simulation of elastic properties of single-walled carbon nanotubes", Compos. Sci. Tech., 63(11), 1507-1515.
  14. Kaci, A., Tounsi, A., Bakhti, K. and Adda Bedia, E.A. (2012), "Nonlinear cylindrical bending of functionally graded carbon nanotube-reinforced composite plates", Steel and Composite Structures, 12(6), 491- 504.
  15. Ke, L.L., Yang, J. and Kitipornchai, S. (2009), "Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening", Compos. Struct., 90(2), 152-160.
  16. Matsunaga, H. (2009), "Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory", Compos. Struct., 88(4), 519-531.
  17. Mokashi, VV., Qian, D. and Liu, YJ. (2007), "A study on the tensile response and fracture in carbon nanotube-based composites using molecular mechanics", Compos. Sci. Tech., 67(3-4), 530-540.
  18. Na, K.S. and Kim, J.H. (2009), "Three-dimensional thermomechanical buckling analysis for functionally graded composite plates", Compos. Struct., 73(4), 413-422.
  19. Odegard, G.M., Gates, T.S., Wise, K.E., Park, C. and Siochi, E.J. (2003), "Constitutive modelling of nanotube-reinforced polymer composites", Compos. Sci. Tech., 63(11), 1671-1687.
  20. Ray, M.C. and Batra, RC. (2007), "A single-walled carbon nanotube reinforced 1-3 piezoelectric composite for active control of smart structures", Smart Mater. Struct., 16(5), 1936-1947.
  21. Salehi-Khojin, A. and Jalili, N. (2008), "Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electro-thermomechanical loadings", Compos. Sci. Tech., 68(6), 1489-1501.
  22. Sallai, B.O., Tounsi, A., Mechab, I., Bachir Bouiadjra, M., Meradjah, M. and Adda Bedia, E.A. (2009), "A theoretical analysis of flexional bending of $Al/Al_2O_3$ S-FGM thick beams", Comput. Mater. Sci., 44(4), 1344-1350.
  23. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19.
  24. Shen, H.S. and Zhang, C.L. (2010), "Thermal buckling and post buckling behavior of functionally graded carbon nanotube-reinforced composite plates", Mater. Des., 31(7), 3403-3411.
  25. Shen, H.S. (2011). "Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells", Compos. Struct., 93(10), 2496-2503.
  26. Suresh, S. and Mortensen, A. (1998), "Fundamentals of functionally graded materials: processing and thermomechanical behavior of graded metals and metalceramic composites", Int'l Mater. Review, 42(3),85-116.
  27. Ajayan, P.M., Stephan, O., Colliex, C. and Trauth, D. (1994), "Aligned carbon nanotube arrays formed by cutting a polymer resin - nanotube composite", Science, 256(5176), 1212-1214.
  28. Benachour, A., Tahar, H.D., Atmane, H.A., Tounsi, A. and Ahmed, M.S. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Composites: Part B, 42(6), 1386-1394.
  29. Benatta, M.A., Mechab, I., Tounsi, A. and Adda bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 675-776.
  30. Cadek, M., Coleman, J.N., Barron, V., Hedicke, K. and Blau, W.J. (2002), "Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites", Appl. Phys. Lett., 81(27), 5123-5125.
  31. Chang, T., Geng, J. and Guo, X. (2005), "Chirality- and size-dependent elastic properties of single-walled carbon nanotubes", Appl. Phys. Lett., 87(25), 251929.
  32. Chen, C.S. (2005), "Nonlinear vibration of a shear deformable functionally graded plate", Compos. Struct., 68(3), 295-302.
  33. Elliott, J.A., Sandler, J.K.W., Windle, A.H., Young, R.J. and Shaffer, M.S.P. (2004), "Collapse of single wall carbon nanotubes is diameter dependent", Phys. Rev. Lett., 92, 095501.
  34. Esawi, A.M.K. and Farag, M.M. (2007), "Carbon nanotube reinforced composites: potential and current challenges", Mater. Des., 28(9), 2394-2401.
  35. Fidelus, J.D., Wiesel, E., Gojny, F.H., Schulte, K. and Wagner, H.D. (2005), "Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites", Compos. Part A: Appl. Sci. Manufact., 36(11), 1555-1561.
  36. Fukuda, H. and Kawata, K. (1974), "On Young's modulus of short fibre composites", Fibre Sci Technol, 7(3), 207-222.
  37. Griebel, M., Hamaekers, J. (2004), "Molecular dynamics simulations of the elastic moduli of polymer - carbon nanotube composites", Comput. Methods Appl. Mech. Eng., 193(17-20), 1773-1788.
  38. Hadji, L., Ait Atmane, H., Tounsi, A., Mechab, I. and Adda Bedia, E.A. (2011), "Free vibration of functionally graded sandwich plates using four variable refined plate theory", Appl. Math. Mech., 32(7), 925-942.
  39. Han, Y. and Elliott, J. (2007), "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci., 39(2), 315-323.

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