Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 194529, 13 pages
http://dx.doi.org/10.1155/2014/194529
Research Article

Free Vibration Analysis of DWCNTs Using CDM and Rayleigh-Schmidt Based on Nonlocal Euler-Bernoulli Beam Theory

1School of Engineering, Viale dell’Ateneo Lucano 10, 85100 Potenza, Italy
2Department of Structures for Engineering and Architecture, Via Forno Vecchio 36, 80134 Naples, Italy

Received 14 August 2013; Accepted 20 October 2013; Published 25 February 2014

Academic Editors: G. J. Gibbons, P. Lonetti, G. Nikas, and B. F. Yousif

Copyright © 2014 Maria Anna De Rosa and Maria Lippiello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Iijima, “Helical microtubules of graphitic carbon,” Nature, vol. 354, no. 6348, pp. 56–58, 1991. View at Google Scholar · View at Scopus
  2. A. Krishnan, E. Dujardin, T. W. Ebbesen, P. N. Yianilos, and M. M. J. Treacy, “Young's modulus of single-walled nanotubes,” Physical Review B, vol. 58, no. 20, pp. 14013–14019, 1998. View at Google Scholar · View at Scopus
  3. B. G. Demczyk, Y. M. Wang, J. Cumings et al., “Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes,” Materials Science and Engineering A, vol. 334, no. 1-2, pp. 173–178, 2002. View at Publisher · View at Google Scholar · View at Scopus
  4. R. S. Ruoff, D. Qian, and W. K. Liu, “Mechanical properties of carbon nanotubes: theoretical predictions and experimental measurements,” Comptes Rendus Physique, vol. 4, no. 9, pp. 993–1008, 2003. View at Publisher · View at Google Scholar · View at Scopus
  5. M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, “Exceptionally high Young's modulus observed for individual carbon nanotubes,” Nature, vol. 381, no. 6584, pp. 678–680, 1996. View at Google Scholar · View at Scopus
  6. J. Tersoff and R. S. Ruoff, “Structural properties of a carbon-nanotube crystal,” Physical Review Letters, vol. 73, no. 5, pp. 676–679, 1994. View at Publisher · View at Google Scholar · View at Scopus
  7. M. B. Nardelli, B. I. Yakobson, and J. Bernholc, “Brittle and ductile behavior in carbon nanotubes,” Physical Review Letters, vol. 81, no. 21, pp. 4656–4659, 1998. View at Google Scholar · View at Scopus
  8. C. Q. Ru, “Column buckling of multiwalled carbon nanotubes with interlayer radial displacements,” Physical Review B, vol. 62, no. 24, pp. 16962–16967, 2000. View at Publisher · View at Google Scholar · View at Scopus
  9. Q. Wang, T. Hu, and Q. J. Chen, “Bending instability characteristics of double walled nanotubes,” Physical Review B, vol. 71, no. 4, pp. 045403–045412, 2005. View at Google Scholar
  10. Y. Y. Zhang, C. M. Wang, and V. B. C. Tan, “Buckling of multiwalled carbon nanotubes using Timoshenko beam theory,” Journal of Engineering Mechanics, vol. 132, no. 9, pp. 952–958, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. R. F. Gibson, E. O. Ayorinde, and Y.-F. Wen, “Vibrations of carbon nanotubes and their composites: a review,” Composites Science and Technology, vol. 67, no. 1, pp. 1–28, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. R. Ansari, S. Ajori, and B. Arash, “Vibrations of single- and double-walled carbon nanotubes with layerwise boundary conditions: a molecular dynamics study,” Current Applied Physics, vol. 12, no. 3, pp. 707–711, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. J. Yoon, C. J. Ru, and A. Mioduchowski, “Non-coaxial resonance of an isolated multiwall carbon nanotubes,” Physical Review B, vol. 66, pp. 233402–233409, 2002. View at Google Scholar
  14. J. Yoon, C. Q. Ru, and A. Mioduchowski, “Vibration of an embedded multiwall carbon nanotube,” Composites Science and Technology, vol. 63, no. 11, pp. 1533–1542, 2003. View at Publisher · View at Google Scholar · View at Scopus
  15. K.-Y. Xu, E. C. Aifantis, and Y.-H. Yan, “Vibrations of double-walled carbon nanotubes with different boundary conditions between inner and outer tubes,” Journal of Applied Mechanics, vol. 75, no. 2, pp. 0210131–0210139, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Y. Xu, X. N. Guo, and C. Q. Ru, “Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces,” Journal of Applied Physics, vol. 99, no. 6, pp. 064303–064307, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. I. Elishakoff and D. Pentaras, “Fundamental natural frequencies of double-walled carbon nanotubes,” Journal of Sound and Vibration, vol. 322, no. 4-5, pp. 652–664, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. T. Natsuki, Q.-Q. Ni, and M. Endo, “Analysis of the vibration characteristics of double-walled carbon nanotubes,” Carbon, vol. 46, no. 12, pp. 1570–1573, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. C. T. Sun and H. Zhang, “Size-dependent elastic moduli of platelike nanomaterials,” Journal of Applied Physics, vol. 93, no. 2, pp. 1212–1218, 2003. View at Publisher · View at Google Scholar · View at Scopus
  20. A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” Journal of Applied Physics, vol. 54, no. 9, pp. 4703–4710, 1983. View at Publisher · View at Google Scholar · View at Scopus
  21. J. N. Reddy, “Nonlocal theories for bending, buckling and vibration of beams,” International Journal of Engineering Science, vol. 45, no. 2-8, pp. 288–307, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. J. Peddieson, G. R. Buchanan, and R. P. McNitt, “Application of nonlocal continuum models to nanotechnology,” International Journal of Engineering Science, vol. 41, no. 3-5, pp. 305–312, 2003. View at Publisher · View at Google Scholar · View at Scopus
  23. S. A. M. Ghannadpour, B. Mohammadi, and J. Fazilati, “Bending buckling and vibration problems of nonlocal Euler beams using Ritz method,” Composite Structures, vol. 96, pp. 584–589, 2013. View at Google Scholar
  24. S. C. Pradha and J. K. Phadikar, “Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory,” Structural Engineering and Mechanics, vol. 33, no. 2, pp. 193–213, 2009. View at Google Scholar · View at Scopus
  25. Q. Wang and V. K. Varadan, “Vibration of carbon nanotubes studied using nonlocal continuum mechanics,” Smart Materials and Structures, vol. 15, no. 2, pp. 659–666, 2006. View at Publisher · View at Google Scholar · View at Scopus
  26. R. Ansari and S. Sahmani, “Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1965–1979, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Shakouri, R. M. Lin, and T. Y. Ng, “Free flexural vibration studies of double-walled carbon nanotubes with different boundary conditions and modeled as nonlocal Euler beams via the Galerkin method,” Journal of Applied Physics, vol. 106, no. 9, Article ID 094307, pp. 094307–094316, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. H. Ehteshami and M. A. Hajabasi, “Analytical approaches for vibration analysis of multi-walled carbon nanotubes modeled as multiple nonlocal Euler beams,” Physica E, vol. 44, no. 1, pp. 270–285, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. R. Ansari and H. Rouhi, “Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theory,” Journal of Engineering Materials and Technology, vol. 134, no. 1, Article ID 011008, pp. 011008–011016, 2012. View at Publisher · View at Google Scholar · View at Scopus
  30. C. M. Wang, Y. Y. Zhang, and X. Q. He, “Vibration of nonlocal Timoshenko beams,” Nanotechnology, vol. 18, no. 10, Article ID 105401, pp. 105401–105409, 2007. View at Publisher · View at Google Scholar · View at Scopus
  31. M. Hemmatnezhad and R. Ansari, “Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory,” Journal of Theoretical and Applied Physics, vol. 44, no. 5, pp. 270–285, 2013. View at Google Scholar
  32. L. L. Ke, Y. Xiang, J. Yang, and S. Kitipornchai, “Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory,” Computational Materials Science, vol. 47, no. 2, pp. 409–417, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. J. N. Reddy and S. D. Pang, “Nonlocal continuum theories of beams for the analysis of carbon nanotubes,” Journal of Applied Physics, vol. 103, no. 2, Article ID 023511, pp. 023511–023526, 2008. View at Publisher · View at Google Scholar · View at Scopus
  34. S. Adali, “Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal euler-bernoulli beam model,” Nano Letters, vol. 9, no. 5, pp. 1737–1741, 2009. View at Publisher · View at Google Scholar · View at Scopus
  35. M. A. De Rosa and M. Lippiello, “Natural vibration frequencies of tapered beams,” Engineering Transactions, vol. 57, no. 1, pp. 44–66, 2009. View at Google Scholar
  36. A. Raithel and C. Franciosi, “Dynamic analysis of arches using Lagrangian approach,” Journal of Structural Engineering, vol. 110, no. 4, pp. 847–858, 1984. View at Google Scholar · View at Scopus
  37. S. Wolfram, The Mathematica 8, Wolfram Media/Cambridge University Press, 2010.
  38. C. W. Bert, “Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes, and plates,” Journal of Sound and Vibration, vol. 119, no. 2, pp. 317–326, 1987. View at Google Scholar · View at Scopus
  39. P. A. A. Laura, “Optimization of variational methods,” Ocean Engineering, vol. 22, no. 3, pp. 235–250, 1995. View at Google Scholar · View at Scopus
  40. M. A. De Rosa and C. Franciosi, “The optimized Rayleigh method and mathematica in vibrations and buckling problems,” Journal of Sound and Vibration, vol. 191, no. 5, pp. 795–808, 1996. View at Publisher · View at Google Scholar · View at Scopus
  41. M. A. De Rosa and M. Lippiello, “Non-classical boundary conditions and DQM for double-beams,” Mechanics Research Communications, vol. 34, no. 7-8, pp. 538–544, 2007. View at Publisher · View at Google Scholar · View at Scopus