Authors

Department of Mechanical and Aerospace Engineering, Shiraz University of Technology

Abstract

The present work derives the exact analytical solutions for buckling and post-buckling analysis of nano-composite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Euler-Bernoulli beam theory and principle of virtual work. The reinforcements are considered to be aligned in the polymeric matrix either uniformly distributed (UD) or functionally graded (FG) distributed through the thickness direction of the beam. In FG beams, material properties vary gradually along the thickness direction. The effective material properties of the nano-composite beam are predicted based on the extended rule of mixture. Also, by applying von Kármán assumptions, the geometric nonlinearities are taken into consideration. The developed governing equations are solved by utilizing analytical methods and exact closed form solutions for buckling and post-buckling loads of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam with different boundary conditions are obtained. By comparing the present post-buckling load results with the ones reported in the literature, the accuracy and reliability of the current method are demonstrated. Eventually, the numerical results are provided and the effects of CNTs distribution, CNTs volume fraction, slenderness ratio, maximum deflection of the beam and boundary conditions on the post-buckling characteristics of the CNTRC beam are discussed.

Keywords

[1] S. Iijima, “Helical microtubules of graphitic carbon”, Nature, vol. 354, no. 6348, pp. 56-58, November 1991.
[2] B.G. Demczyk, Y.M. Wang, J. Cumings, M. Hetman, W. Han, A. Zettl and R.O. Ritchie, “Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes,” Materials Science and Engineering, vol. 334, no. 1-2, pp. 173-178, September 2002.
[3] H. Dai, “Carbon nanotubes: opportunities and challenges,” Surface Science, vol. 500, no. 1-3, pp. 218-241, March 2002.
[4] R. Ansari, H. Rouhi and S. Sahmani, “Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models,” Journal of Vibration and Control, vol. 20, no. 5, pp. 670-678, April 2014.
[5] H. Wan, F. Delale and L. Shen, “Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites,” Mechanics Research Communications, vol. 32, no. 5, pp. 481-489, September 2005.
[6]     Y. Han and J. Elliot, “Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites: modeling and characterization,” Computational Materials Science, vol. 39, no. 2, pp. 315-323, April 2007.
[7]     E.T. Thostenson and T.W. Chou, “On the elastic properties of carbon nanotube-based composites: Modeling and characterization”, Journal of Physics D: Applied Physics, vol. 36, no. 5, pp. 573-582, February 2003.
[8]     V. Mittak, Polymer nanotube nanocomposite: synthesis, properties and applications, John Wiley and Sons, New Jersey, 2010.
[9]      A.M.K. Esawi and M.M. Farag, “Carbon nanotube reinforced composites: potential and current challenges,” Materials and Design, vol. 28, no. 9, pp. 2394-2401, November 2007.
[10] K.M. Liew, Z.X. Lei and L.W. Zhang, “Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review”, Composite Structures, vol. 120, pp. 90-97, February 2015.
[11] B.S. Aragh, A.H.N. Barati and H. Hedayati, “Eshelby-Mori-Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels”, Composites Part B: Engineering, vol. 43, no. 4, pp. 1943-1954, June 2012.
[12] R. Ansari, M.F. Shojaei, V. Mohammadi, R. Gholami and F. Sadeghi, “Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams”, Composite Structures, vol. 113, pp. 316-327, July 2014.
[13] H. Shafiei and A.R. Setoodeh, “Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation”, Steel and Composite Structures, vol. 24, no. 1, pp. 65-77, May 2017.
[14] M. Shojaee, A.R. Setoodeh and P. Malekzadeh, “Vibration of functionally graded CNTs-reinforced skewed cylindrical panels using a transformed differential quadrature method”, Acta Mechanica, vol. 228, no. 7, pp. 2691-2711, July 2017.
[15]  D.G. Ninh, “Nonlinear thermal torsional post-buckling of carbon nanotube-reinforced composite cylindrical shell with piezoelectric actuator layers surrounded by elastic medium”, Thin-Walled Structures, vol. 123, pp. 528-538, February 2018.
[16]  L.L. Ke, J. Yang and S. Kitipornchai, “Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams,” Composite Structures, vol. 92, no. 3, pp. 676-683, February 2010.
[17]  L.W. Zhang, Z.X. Lei and K.M. Liew, “Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach,” Composite Part B: Engineering, vol. 75, no. 15, pp. 36-46, June 2015.
[18]  A.R. Setoodeh and M. Shojaee, “Application of TW-DQ method to nonlinear free vibration analysis of FG carbon nanotube-reinforced composite quadrilateral plates,” Thin-Walled Structures, vol. 108, pp. 1-11, November 2016.
[19]  H.S. Shen, X.Q. He and D.Q. Yang, “Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations”, International Journal of Non-Linear Mechanics, vol. 91, pp. 69-75, May 2017.
[20]  M.H. Yas and N. Samadi, “Free vibration and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation,” International Journal of Pressure Vessels and Piping, vol. 98, pp. 119-128, October 2012.
[21]  H.L. Wu, J. Yang, and S. Kitipornchai, 2016. “Imperfection sensitivity of postbuckling behavior of functionally graded carbon nanotube-reinforced composite beams,” Thin-Walled Structures, vol. 108, pp. 225-233, November 2016.
[22]  H.L. Wu, S. Kitipornchai and J. Yang, “Thermal buckling and postbuckling analysis of functionally graded carbon nanotube-reinforced composite beams,” Applied Mechanics and Materials, vol. 846, pp. 182-187, July 2016.
[23]  S. Pouresmaeeli and S.A. Fazelzadeh, “Uncertain buckling and sensitivity analysis of functionally graded carbon nanotube-reinforced composite beam”, International Journal of Applied Mechanics, vol. 9, no. 5, pp. 1750071, July 2017.
[24]  S.A. Emam and A H. Nayfeh, “Postbuckling and free vibration of composite beams”, Composite Structures, vol. 88, no. 4, pp. 636-642, May 2009.
[25]  A.R. Setoodeh, M. khosrownejad and P. Malekzadeh, “Exact nonlocal solution for postbuckling of single-walled carbon nanotubes,” Physica E, vol. 43, no. 9, pp. 1730-1737, July 2011.
[26]  A. Fallah and M.M. Aghdam, “Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation,” European Journal of Mechanics A/Solids, vol. 30, no. 4, pp. 571-583, July-August 2011.
[27]  R.K. Gupta, J.B. Gunda, G.R. Janardhan and G.V. Rao, “Post-buckling analysis of composite beams: Simple and accurate closed-form expressions,” Composite Structures, vol. 92, no. 8, pp. 1947-1956, July 2010.