Transverse Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in an Elastic Medium Using Bernoulli-Fourier Method

ABSTRACT

Based on the Timoshenko beam theory and Bernoulli-Fourier method, a single-elastic beam model is developed for transverse vibrations of single-walled carbon nanotubes under additional axial load, which includes the effects of the elastic medium around them. Explicit expressions are derived for the natural frequencies and transversal responses of simply supported single-walled carbon nanotubes. The influence of addition axial load and the properties of elastic medium on the vibrations are discussed. The results showed that the effects of addition axial load on the lower natural frequencies of single-walled carbon nanotubes are sensitive to the lower vibration modes and the stiff elastic medium. The lower natural frequencies depend on the axial load; they become smaller with increasing axial load and vary with the vibration modes. In addition, except for the first mode, the effects of the axial load on the stiff elastic medium are considerably greater than those on the flexible one. However, the constants of the elastic medium have little effect on the first mode. The critical axial buckling stress and strain for simply-supported single-walled carbon nanotubes are also obtained.

Based on the Timoshenko beam theory and Bernoulli-Fourier method, a single-elastic beam model is developed for transverse vibrations of single-walled carbon nanotubes under additional axial load, which includes the effects of the elastic medium around them. Explicit expressions are derived for the natural frequencies and transversal responses of simply supported single-walled carbon nanotubes. The influence of addition axial load and the properties of elastic medium on the vibrations are discussed. The results showed that the effects of addition axial load on the lower natural frequencies of single-walled carbon nanotubes are sensitive to the lower vibration modes and the stiff elastic medium. The lower natural frequencies depend on the axial load; they become smaller with increasing axial load and vary with the vibration modes. In addition, except for the first mode, the effects of the axial load on the stiff elastic medium are considerably greater than those on the flexible one. However, the constants of the elastic medium have little effect on the first mode. The critical axial buckling stress and strain for simply-supported single-walled carbon nanotubes are also obtained.

Cite this paper

T. Horng, "Transverse Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in an Elastic Medium Using Bernoulli-Fourier Method,"*Journal of Surface Engineered Materials and Advanced Technology*, Vol. 2 No. 3, 2012, pp. 203-209. doi: 10.4236/jsemat.2012.223031.

T. Horng, "Transverse Vibration Analysis of Single-Walled Carbon Nanotubes Embedded in an Elastic Medium Using Bernoulli-Fourier Method,"

References

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[3] D. Srivastava, M. Menon and K. Cho, “Computational Nanotechnology with Carbon Nanotubes and Fullerenes,” Computing in Science & Engineering, Vol. 3, No. 4, 2001, pp. 42-55. doi:10.1109/5992.931903

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[6] B. I. Yakobson, C. J. Brabec and J. Bernholc, “Nanome-chanics of Carbon Tubes: Instabilities beyond Linear Response,” Physical Review Letters, Vol. 76, No. 14, 1996, pp. 2511-2514. doi:10.1103/PhysRevLett.76.2511

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[10] Y. Q. Zhang, G. R. Liu and J. S. Wang, “Small-Scale Effects on Buckling of Multiwalled Carbon Nanotubes under Axial Compression,” Physical Review B, Vol. 70, No. 20. 2004, p. 205430. doi:10.1103/PhysRevB.70.205430

[11] Y. Zhang, G. Liu and X. Han, “Transverse Vibrations of Double-Walled Carbon Nanotubes under Compressive Axial Load,” Physics Letters A, Vol. 340, No. 1-4, 2005, pp. 258-266. doi:10.1016/j.physleta.2005.03.064

[12] K. M. Liew and Q. Wang, “Analysis of Wave Propaga- tion in Carbon Nanotubes via Elastic Shell Theories,” International Journal of Engineering Science, Vol. 45, No. 2-8, 2007, pp. 227-241. doi:10.1016/j.ijengsci.2007.04.001

[13] J. C. Hsu, R. P. Chang and W. J. Chang, “Resonance Frequency of Chiral Single-Walled Carbon Nanotubes Using Timoshenko Beam Theory,” Physics Letters A, Vol. 372, No. 16, 2008, pp. 2757-2759. doi:10.1016/j.physleta.2008.01.007

[14] J. Yoon, C. Q. Ru and A. Mioduchowski, “Vibration of an Embedded Multiwall Carbon Nanotube,” Composites Science and Technology, Vol. 63, No.11, 2003, pp. 1533- 1542. doi:10.1016/S0266-3538(03)00058-7

[15] C. M. Wang, V. B. C. Tan and Y. Y. Zhang, “Timoshenko Beam Model for Vibra-tion Analysis of Multi-Walled Carbon Nanotubes,” Journal of Sound and Vibration, Vol. 294, No. 4-5, 2006, pp. 1060-1072. doi:10.1016/j.jsv.2006.01.005

[16] S. C. Pradhan and T. Murmu, “Small-Scale Effect on Vibration Analysis of Sin-gle-Walled Carbon Nanotubes Embedded in an Elastic Me-dium Using Nonlocal Elasticity Theory,” Journal of Applied Physics, Vol. 105, No. 2, 2009, p. 114309.

[17] M. X. Shi and Q. M. Li, “Mode Bernoulli-Fourier in Single-Walled Carbon Nanotubes,” International Journal of Mechanical Sciences, Vol. 52, No. 5, 2010, pp. 663-671. doi:10.1016/j.ijmecsci.2009.09.018

[18] R. B. Chen, C. H. Lee, C. P. Chang and M. F. Lin, “Electronic and Optical Properties of Finite Carbon Nanotubes in an Electric Field,” Nanotechnology, Vol. 18, No. 7, 2007, p. 075704. doi:10.1088/0957-4484/18/7/075704

[19] M. A. Moreles, S. Botello and R. Salinas, “A Root-Finding Technique to Compute Eigenfrequencies for Elastic Beams,” Journal of Sound and Vibration, Vol. 284, No. 3-5, 2005, pp. 1119-1129. doi:10.1016/j.jsv.2004.07.028

[20] J. Avsec and M. Oblak, “Thermal Vibrational Analysis for Simply Supported Beam and Clamped Beam,” Journal of Sound and Vibration, Vol. 308, No. 3-5, 2007, pp. 514-525. doi:10.1016/j.jsv.2007.04.002

[21] F. P. Beer and E. R. Johnston, “Mechanics of Materials,” McGraw-Hill, New York, 1981.

[1] A. Bianco, K. Kostarelos and M. Prato, “Applications of Carbon Nanotubes in Drug Delivery,” Current Opinion in Chemical Biology, Vol. 9, No. 6, 2005, pp. 674-679. doi:10.1016/j.cbpa.2005.10.005

[2] R. S. Ruoff and D. C. Lorents, “Mechanical and Thermal Properties of Carbon Nanotubes,” Carbon, Vol. 33, No. 7, 1995, pp. 925-930. doi:10.1016/0008-6223(95)00021-5

[3] D. Srivastava, M. Menon and K. Cho, “Computational Nanotechnology with Carbon Nanotubes and Fullerenes,” Computing in Science & Engineering, Vol. 3, No. 4, 2001, pp. 42-55. doi:10.1109/5992.931903

[4] R. Saito, G. Dresselhaus and M. S. Dresselhaus, “Physical Properties of Carbon Nano-tubes,” Imperial College, London, 1998.

[5] P. J. F. Harris, “Carbon Nanotubes and Related Structures,” Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9780511605819

[6] B. I. Yakobson, C. J. Brabec and J. Bernholc, “Nanome-chanics of Carbon Tubes: Instabilities beyond Linear Response,” Physical Review Letters, Vol. 76, No. 14, 1996, pp. 2511-2514. doi:10.1103/PhysRevLett.76.2511

[7] P. Zhang, H. Jiang, Y. Huang, P. H. Geubelle and K. C. Hwang, “An atomistic-Based Continuum Theory for Carbon Nanotubes: Analysis of Fracture Nucleation,” Journal of the Mechanics and Physics of Solids, Vol. 52, No. 5, 2004, pp. 977-998. doi:10.1016/j.jmps.2003.09.032

[8] E. W. Wong, P. E. Sheehan and C. M. Lieber, “Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nano-rods and Nanotubes,” Science, Vol. 277, No. 5334, 1997, pp. 1971-1975. doi:10.1126/science.277.5334.1971

[9] O. Lourie, P. M. Co and H. D. Wagner, “Buckling and Collapse of Embedded Carbon Nanotubes,” Physical Review Letters, Vol. 81, No. 8, 1998, pp. 1638-1641. doi:10.1103/PhysRevLett.81.1638

[10] Y. Q. Zhang, G. R. Liu and J. S. Wang, “Small-Scale Effects on Buckling of Multiwalled Carbon Nanotubes under Axial Compression,” Physical Review B, Vol. 70, No. 20. 2004, p. 205430. doi:10.1103/PhysRevB.70.205430

[11] Y. Zhang, G. Liu and X. Han, “Transverse Vibrations of Double-Walled Carbon Nanotubes under Compressive Axial Load,” Physics Letters A, Vol. 340, No. 1-4, 2005, pp. 258-266. doi:10.1016/j.physleta.2005.03.064

[12] K. M. Liew and Q. Wang, “Analysis of Wave Propaga- tion in Carbon Nanotubes via Elastic Shell Theories,” International Journal of Engineering Science, Vol. 45, No. 2-8, 2007, pp. 227-241. doi:10.1016/j.ijengsci.2007.04.001

[13] J. C. Hsu, R. P. Chang and W. J. Chang, “Resonance Frequency of Chiral Single-Walled Carbon Nanotubes Using Timoshenko Beam Theory,” Physics Letters A, Vol. 372, No. 16, 2008, pp. 2757-2759. doi:10.1016/j.physleta.2008.01.007

[14] J. Yoon, C. Q. Ru and A. Mioduchowski, “Vibration of an Embedded Multiwall Carbon Nanotube,” Composites Science and Technology, Vol. 63, No.11, 2003, pp. 1533- 1542. doi:10.1016/S0266-3538(03)00058-7

[15] C. M. Wang, V. B. C. Tan and Y. Y. Zhang, “Timoshenko Beam Model for Vibra-tion Analysis of Multi-Walled Carbon Nanotubes,” Journal of Sound and Vibration, Vol. 294, No. 4-5, 2006, pp. 1060-1072. doi:10.1016/j.jsv.2006.01.005

[16] S. C. Pradhan and T. Murmu, “Small-Scale Effect on Vibration Analysis of Sin-gle-Walled Carbon Nanotubes Embedded in an Elastic Me-dium Using Nonlocal Elasticity Theory,” Journal of Applied Physics, Vol. 105, No. 2, 2009, p. 114309.

[17] M. X. Shi and Q. M. Li, “Mode Bernoulli-Fourier in Single-Walled Carbon Nanotubes,” International Journal of Mechanical Sciences, Vol. 52, No. 5, 2010, pp. 663-671. doi:10.1016/j.ijmecsci.2009.09.018

[18] R. B. Chen, C. H. Lee, C. P. Chang and M. F. Lin, “Electronic and Optical Properties of Finite Carbon Nanotubes in an Electric Field,” Nanotechnology, Vol. 18, No. 7, 2007, p. 075704. doi:10.1088/0957-4484/18/7/075704

[19] M. A. Moreles, S. Botello and R. Salinas, “A Root-Finding Technique to Compute Eigenfrequencies for Elastic Beams,” Journal of Sound and Vibration, Vol. 284, No. 3-5, 2005, pp. 1119-1129. doi:10.1016/j.jsv.2004.07.028

[20] J. Avsec and M. Oblak, “Thermal Vibrational Analysis for Simply Supported Beam and Clamped Beam,” Journal of Sound and Vibration, Vol. 308, No. 3-5, 2007, pp. 514-525. doi:10.1016/j.jsv.2007.04.002

[21] F. P. Beer and E. R. Johnston, “Mechanics of Materials,” McGraw-Hill, New York, 1981.