On the vibration of postbuckled functionally graded-carbon nanotube reinforced composite annular plates

Document Type : Article


1 Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, Lahijan, P.O. Box 1616, Iran

2 Faculty of Mechanical Engineering, University of Guilan, Rasht, P.O. Box 3756, Iran


This paper studies the free vibration charachterstics of post-buckled functionally graded nanocomposite annular plates reinforced by single-walled carbon nanotubes (SWCNTs). The analysis is performed by employing a generalized differenitail quadrature (GDQ)-type numerical technique and psedue arc-length continuation scheme. The SWCNT reinforcement is considered to be either uniformly distributed (UD) or functionally graded (FG) in the thickness direction. The material properties of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are estimated using an equivalent continuum model based on the modified rule of mixture. The vibration problem is formulated on the basis of the first-order shear deformation theory for moderately thick laminated plates and von Kármán geometric nonlinearity. By employing Hamilton’s principle and a variational approach, the governing equations and the associated boundary conditions (BCs) are derived which are then discretized via the GDQ method. The postbuckling characteristics of FG-CNTRC annular plates are investigated by plotting the equilibrium postbuckling path as the load-deflection curves. Thereafter, the free vibration behavior of FG-CNTRC annular plates in pre- and post-buckled states is examined. Effects of different parameters including type of BCs, CNT volume fraction, outer radius-to-thickness ratio and inner-to-outer radius ratio are investigated in detail.


Main Subjects

1. Iijima, S. Helical microtubules of graphitic carbon", Nature, 354, pp. 56-58 (1991). 2. Bianco, A., Kostarelos, K., and Prato, M. Applications of carbon nanotubes in drug delivery", Current Opinion in Chemical Biology, 9, pp. 674-679 (2005). 3. Treacy, M.J., Ebbesen, T., and Gibson, J. Exceptionally high Young's modulus observed for individual carbon nanotubes", Nature, 381, p. 678 (1996). 4. Baughman, R.H., Zakhidov, A.A., and De Heer, W.A. Carbon nanotubes-the route toward applications", Science, 297, pp. 787-792 (2002). 5. Salvetat, J.-P., Bonard, J.-M., Thomson, N., et al. Mechanical properties of carbon nanotubes", Applied Physics A, 69, pp. 255-260 (1999). 6. Li, Y., Wang, Q., and Wang, S. A review on enhancement of mechanical and tribological properties of polymer composites reinforced by carbon nanotubes and graphene sheet: Molecular dynamics simulations", Composites Part B: Engineering, 160, pp. 348-361 (2018). 7. Deep, N. and Mishra, P. Evaluation of mechanical properties of functionalized carbon nanotube reinforced PMMA polymer nanocomposite", Karbala International Journal of Modern Science, 4, pp. 207-215 (2018). 8. Hassanzadeh-Aghdam, M.K., Ansari, R., and Mahmoodi, M.J. Thermo-mechanical properties of shape memory polymer nanocomposites reinforced by carbon nanotubes", Mechanics of Materials, 129, pp. 80-98 (2019). 9. Ajayan, P., Stephan, O., Colliex, C., et al. Aligned carbon nanotube arrays formed by cutting a polymer resin-nanotube composite", Science, 265, pp. 1212- 1214 (1994). 10. Hassanzadeh-Aghdam, M. and Mahmoodi, M. A comprehensive analysis of mechanical characteristics of carbon nanotube-metal matrix nanocomposites", Materials Science and Engineering: A, 701, pp. 34- 44 (2017). 11. Hassanzadeh-Aghdam, M.K., Ansari, R., and Mahmoodi, M.J. Thermal expanding behavior of carbon nanotube-reinforced metal matrix nanocomposites-A micromechanical modeling", Journal of Alloys and Compounds, 744, pp. 637-650 (2018). 12. Hassanzadeh-Aghdam, M., Ansari, R., and Mahmoodi, M. Micromechanical estimation of biaxial thermomechanical responses of hybrid _ber-reinforced metal matrix nanocomposites containing carbon nanotubes", Mechanics of Materials, 119, pp. 1-15 (2018). 13. Foroughi, M.R., Khoroushi, M., Nazem, R., et al. The e_ect of carbon nanotubes/bioglass nanocomposite on mechanical and bioactivity properties of glass ionomer cement", Scientia Iranica, 23, pp. 3123-3134 (2016). 14. AfzaliTabara, M., Alaeib, M., Khojasteha, R.R., et al. Preference of nanoporous graphene to Single- Walled Carbon Nanotube (SWCNT) for preparing silica nanohybrid Pickering emulsion for potential Chemical Enhanced Oil Recovery (C-EOR)", Scientia Iranica, 24, pp. 3491-3499 (2017). 15. Ra_ee, M., Nitzsche, F., and Labrosse, M. E_ect of functionalization of carbon nanotubes on vibration and damping characteristics of epoxy nanocomposites", Polymer Testing, 69, pp. 385-395 (2018). 16. Nasihatgozar, M., Daghigh, V., Eskandari, M., et al. Buckling analysis of piezoelectric cylindrical composite panels reinforced with carbon nanotubes", International Journal of Mechanical Sciences, 107, pp. 69-79 (2016). 17. Wu, H., Kitipornchai, S., and Yang, J. Imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite beams", Applied Mathematical Modelling, 42, pp. 735- 752 (2017). 18. Wu, H.L., Kitipornchai, S., and Yang, J. Thermal buckling and postbuckling analysis of functionally graded carbon nanotube-reinforced composite beams", Applied Mechanics and Materials, 846, p. 182 (2016). 19. Wang, M., Li, Z.-M., and Qiao, P. Semi-analytical solutions to buckling and free vibration analysis of carbon nanotube-reinforced composite thin plates", Composite Structures, 144, pp. 33-43 (2016). 20. Alibeigloo, A. and Liew, K. Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using di_erential quadrature method", International Journal of Applied Mechanics, 7, pp. 1550002 (2015). 21. Jalali, S. and Heshmati, M. Buckling analysis of circular sandwich plates with tapered cores and functionally graded carbon nanotubes-reinforced composite face sheets", Thin-Walled Structures, 100, pp. 14- 24 (2016). 22. Mehar, K., Panda, S.K., and Mahapatra, T.R. Thermoelastic deection responses of CNT reinforced sandwich shell structure using _nite element method", Scientia Iranica, 25, pp. 2722-2737 (2018). 23. Hassanzadeh-Aghdam, M., Mahmoodi, M., and Ansari, R. Micromechanical characterizing the e_ective elastic properties of general randomly distributed CNT-reinforced polymer nanocomposites", Probabilistic Engineering Mechanics, 53, pp. 39-51 (2018). 24. Hassanzadeh-Aghdam, M., Mahmoodi, M., and Ansari, R. Micromechanics-based characterization of mechanical properties of fuzzy _ber-reinforced composites containing carbon nanotubes", Mechanics of Materials, 118, pp. 31-43 (2018). 25. Griebel, M. and Hamaekers, J. Molecular dynamics simulations of the elastic moduli of polymercarbon nanotube composites", Computer Methods in Applied Mechanics and Engineering, 193, pp. 1773- 1788 (2004). R. Gholami and R. Ansari/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3857{3874 3873 26. Han, Y. and Elliott, J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Computational Materials Science, 39, pp. 315-323 (2007). 27. Bonnet, P., Sireude, D., Garnier, B., et al. Thermal properties and percolation in carbon nanotubepolymer composites", Applied Physics Letters, 91, p. 201910 (2007). 28. Meguid, S. and Sun, Y. On the tensile and shear strength of nano-reinforced composite interfaces", Mater. Design., 25, pp. 289-296 (2004). 29. Vodenitcharova, T. and Zhang, L. Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube", International Journal of Solids and Structures, 43, pp. 3006-3024 (2006). 30. Gholami, R., Ansari, R., and Gholami, Y. Nonlinear resonant dynamics of geometrically imperfect higherorder shear deformable functionally graded carbonnanotube reinforced composite beams", Composite Structures, 174, pp. 45-58 (2017). 31. Formica, G., Lacarbonara, W., and Alessi, R. Vibrations of carbon nanotube-reinforced composites", Journal of Sound and Vibration, 329, pp. 1875-1889 (2010). 32. Ansari, R., Shojaei, M.F., Mohammadi, V., et al. Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams", Composite Structures, 113, pp. 316- 327 (2014). 33. Shen, H.-S. and He, X. Large amplitude free vibration of nanotube-reinforced composite doubly curved panels resting on elastic foundations in thermal environments", Journal of Vibration and Control, 23, pp. 2672-2689 (2017). 34. Ansari, R., Hasrati, E., Shojaei, M.F., et al. Forced vibration analysis of functionally graded carbon nanotube-reinforced composite plates using a numerical strategy", Physica E: Low-dimensional Systems and Nanostructures, 69, pp. 294-305 (2015). 35. Ansari, R. and Gholami, R. Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes", Composite Structures, 154, pp. 707-723 (2016). 36. Ansari, R., Pourashraf, T., Gholami, R., and Shahabodini, A. Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers", Composites Part B: Engineering, 90, pp. 267-277 (2016). 37. Lin, F. and Xiang, Y. Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams", International Journal of Structural Stability and Dynamics, 14, p. 1350056 (2014). 38. Song, Z., Zhang, L., and Liew, K. Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments", International Journal of Mechanical Sciences, 115, pp. 339- 347 (2016). 39. Mehrabadi, S.J., Aragh, B.S., Khoshkhahesh, V., et al. Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes", Composites Part B: Engineering, 43, pp. 2031-2040 (2012). 40. Lei, Z., Liew, K., and Yu, J. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment", Composite Structures, 106, pp. 128-138 (2013). 41. Shen, H.-S., Wang, H., and Yang, D.-Q. Vibration of thermally postbuckled sandwich plates with nanotubereinforced composite face sheets resting on elastic foundations", International Journal of Mechanical Sciences, 124, pp. 253-262 (2017). 42. Ahmadi, M., Ansari, R., and Rouhi, H. Studying buckling of composite rods made of hybrid carbon _ber/carbon nanotube reinforced polyimide using multiscale FEM", Scientia Iranica, Transactions B (In Press). DOI:10.24200/sci.2018.5722.1444 43. Gholami, R. and Ansari, R. Geometrically nonlinear resonance of higher-order shear deformable functionally graded carbon-nanotube-reinforced composite annular sector plates excited by harmonic transverse loading", The European Physical Journal Plus, 133, p. 56 (2018). 44. Gholami, R., Ansari, R., and Gholami, Y. Numerical study on the nonlinear resonant dynamics of carbon nanotube/_ber/polymer multiscale laminated composite rectangular plates with various boundary conditions", Aerospace Science and Technology, 78, pp. 118-129 (2018). 45. Wang, Q., Pang, F., Qin, B., et al. A uni_ed formulation for free vibration of functionally graded carbon nanotube reinforced composite spherical panels and shells of revolution with general elastic restraints by means of the Rayleigh-Ritz method", Polymer Composites, 39, pp. E924-E944 (2018). 46. Esawi, A.M. and Farag, M.M. Carbon nanotube reinforced composites: potential and current challenges", Mater. Design., 28, pp. 2394-2401 (2007). 47. Fidelus, J., Wiesel, E., Gojny, F., et al. Thermomechanical properties of randomly oriented carbon/ epoxy nanocomposites", Composites Part A: Applied Science and Manufacturing, 36, pp. 1555-1561 (2005). 48. Shen, H.-S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Composite Structures, 91, pp. 9-19 (2009). 49. Librescu, L., Oh, S.-Y., and Song, O. Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability", Journal of Thermal Stresses, 28, pp. 649-712 (2005). 50. Shen, H.-S. and Wang, Z.-X. Assessment of Voigt and Mori-Tanaka models for vibration analysis of 3874 R. Gholami and R. Ansari/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3857{3874 functionally graded plates", Composite Structures, 94, pp. 2197-2208 (2012). 51. Shen, H.-S. Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium", Composite Structures, 94, pp. 1144-1154 (2012). 52. Wang, Z.-X. and Shen, H.-S. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments", Computational Materials Science, 50, pp. 2319-2330 (2011). 53. Liew, K.-M., Xiang, Y., Kitipornchai, S., et al., Vibration of Mindlin Plates: Programming the p-Version Ritz Method, Elsevier (1998). 54. Reddy, J.N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press (2004). 55. Gurarslan, G. and Sari, M. Numerical solutions of linear and nonlinear di_usion equations by a di_erential quadrature method (DQM)", International Journal for Numerical Methods in Biomedical Engineering, 27, pp. 69-77 (2011). 56. Bellman, R. and Casti, J. Di_erential quadrature and long-term integration", Journal of Mathematical Analysis and Applications, 34, pp. 235-238 (1971). 57. Shu, C., Generalized Di_erential-Integral Quadrature and Application to the Simulation of Incompressible Viscous Flows Including Parallel Computation, University of Glasgow (1991). 58. Shu, C. and Richards, B.E. Application of generalized di_erential quadrature to solve two-dimensional incompressible Navier-Stokes equations", International Journal for Numerical Methods in Fluids, 15, pp. 791- 798 (1992). 59. Ansari, R., Gholami, R., Shojaei, M.F., et al. Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded _rst-order shear deformable micro-and nano-beams based on the Mindlin's strain gradient theory", Applied Mathematical Modelling, 40, pp. 9872-9891 (2016). 60. Chen, W. and Zhong, T. The study on the nonlinear computations of the DQ and DC methods", Numerical Methods for Partial Di_erential Equations: An International Journal, 13, pp. 57-75 (1997). 61. Shen, H.-S. and Zhang, C.-L. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates", Mater. Design., 31, pp. 3403-3411 (2010). 62. Ke, L.-L., Yang, J., Kitipornchai, S., et al. Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane", International Journal of Engineering Science, 81, pp. 66-81 (2014).