Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory
Publication: Journal of Engineering Mechanics
Volume 132, Issue 9
Abstract
A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.
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References
Barnholc, J., Brabec, C., Nardelli, M. B., Maiti, A., Roland, C., and Yakobson, B. I. (1998). “Theory of growth and mechanical properties of nanotubes.” Appl. Phys. A: Mater. Sci. Process., 67, 39–46.
Baughman, R. H., et al. (1999). “Carbon nanotubes actuators.” Science, 284, 1340–1344.
Chopra, N. G., and Zettl, A. (1998). “Measurement of the elastic modulus of multiwall boron nitride nanotubes.” Solid State Commun., 105, 297–300.
Cornwell, C. F., and Wille, L. T. (1997). “Elastic properties of single-walled carbon nanotubes in compression.” Solid State Commun., 101, 555–558.
Dai, H. J., Hafner, J. H., Rinzler, A. G., Colbert, D. H., and Smalley, R. E. (1996). “Nanotubes as nanoprobes in scanning probe microscopy.” Nature (London), 384, 147–150.
Falvo, M. R., et al. (1997). “Bending and buckling of carbon nanotubes under large strain.” Nature (London), 389, 582–584.
Feng, J. T., Xu, K. Y., and Ru, C. Q. (2004). “Curvature effect of interlayer van der Waals forces on axial buckling of double-walled carbon nanotubes.” Int. J. Struct. Stab. Dyn., 4, 515–526.
Govindjee, S., and Sackman, J. L. (1999). “On the use of continuum mechanics to estimate the properties of nanotubes.” Solid State Commun., 110, 227–230.
Hahn, H. T., and Williams, J. G. (1984). “Compression failure mechanisms in unidirectional composites.” Compos. Mater.--Test. Des. Conf., 7, 115–139.
Han, Q., and Lu, G. X. (2003). “Torsional buckling of a double-walled carbon nanotube embedded in an elastic medium.” Eur. J. Mech. A/Solids, 22, 875–883.
He, X. Q., Kitipornchai, S., and Liew, K. M. (2005). “Buckling analysis of multi-walled carbon nanotubes: A continuum model accounting for van der Waals interaction.” J. Mech. Phys. Solids, 53, 303–326.
Iijima, S. (1991). “Helical microtubules of graphitic carbon.” Nature (London), 354, 56–58.
Kim, P., and Lieber, C. M. (1999). “Nanotube nanotweezers.” Science, 286, 2148–2150.
Lanir, Y., and Fung, Y. C. B. (1972). “Fiber composite columns under compressions.” J. Compos. Mater., 6, 387–401.
Lourie, O., Cox, D. M., and Wagner, H. D. (1998). “Buckling and collapse of embedded carbon nanotubes.” Phys. Rev. Lett., 81, 1638–1641.
Popov, V. N., and Doren, V. E. N. (2000). “Elastic properties of single-walled carbon nanotubes.” Phys. Rev. B, 61, 3078–3084.
Ru, C. Q. (2000a). “Column buckling of multiwalled carbon nanotubes with interlayer radial displacements.” Phys. Rev. B, 62, 16962–16967.
Ru, C. Q. (2000b). “Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube.” J. Appl. Phys., 87, 1712–1715.
Ru, C. Q. (2001a). “Axially compressed buckling of double-walled carbon nanotubes embedded in an elastic medium.” J. Mech. Phys. Solids, 49, 1265–1279.
Ru, C. Q. (2001b). “Degraded axial buckling strain of multiwalled carbon nanotubes due to interlayer slips.” J. Appl. Phys., 89, 3426–3433.
Ru, C. Q. (2004). “Elastic models for carbon nanotubes.” Encyclopedia of nanoscience and nanotechnology, H. S. Nalwa, ed., American Scientific, Stevenson Ranch, Calif., Vol. 2, 731–744.
Sears, A., and Batra, R. C. (2004). “Macroscopic properties of carbon nanotubes from molecular-mechanics simulations.” Phys. Rev. B, 69, 235406.
Shames, I. H., and Dym, C. L. (1985). Energy and finite element methods in structural mechanics, Hemisphere, Washington, D.C.
Shen, H. S. (2003). “Postbuckling prediction of double-walled carbon nanotubes under hydrostatic pressure.” Int. J. Solids Struct., 41, 2643–2657.
Snow, E. S., Campbell, P. M., and Novak, J. P. (2002a). “Atomic force microscopy using single-wall C nanotubes probes.” J. Vac. Sci. Technol. B, 20, 822–827.
Snow, E. S., Campbell, P. M., and Novak, J. P. (2002b). “Single-wall carbon nanotubes atomic force microscope probes.” Appl. Phys. Lett., 80, 2002–2004.
Sudak, L. J. (2003). “Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics.” J. Appl. Phys., 94, 7281–7287.
Timoshenko, S. P. (1921). “On the correction for shear of the differential equation for transverse vibration of prismatic bars.” Philos. Mag., 41, 744–746.
Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability, McGraw–Hill, New York.
Treacy, M. M. J., Ebbesen, T. W., and Gibson, J. M. (1996). “Exceptionally high Young’s modulus observed for individual carbon nanotubes.” Nature (London), 381, 678–680.
Wang, C. M., Reddy, J. N., and Lee, K. H. (2000). Shear deformable beams and plates: Relationships with classical solutions, Elsevier, Oxford, U.K.
Wang, C. Y., Ru, C. Q., and Mioduchowski, A. (2003a). “Axially compressed buckling of pressured multi-walled carbon nanotubes.” Int. J. Solids Struct., 40, 3893–3911.
Wang, C. Y., Ru, C. Q., and Mioduchowski, A. (2003b). “Elastic buckling of multiwall carbon nanotubes under high pressure.” J. Nanosci. Nanotechnol., 3, 1–10.
Wong, E. W., Sheehan, P. E., and Lieber, C. M. (1997). “Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes.” Science, 277, 1971–1975.
Yakobson, B. I., Brabec, C. J., and Bernholc, J. (1996). “Nanomechanics of carbon tubes: Instabilities beyond linear response.” Phys. Rev. Lett., 76, 2511–2514.
Yakobson, B. I., and Smalley, R. E. (1997). “Fullerene nanotubes: and beyond.” Am. Sci., 85, 324–337.
Yoon, J., Ru, C. Q., and Mioduchowski, A. (2003). “Vibration of an embedded multiwall carbon nanotube.” Compos. Sci. Technol., 63, 1533–1542.
Yu, M. F., Lourie, O., Dyer, M. J., Moloni, K., Kelly, T. F., and Ruoff, R. S. (2000). “Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load.” Science, 287, 637–640.
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© 2006 ASCE.
History
Received: Jan 21, 2005
Accepted: Dec 29, 2005
Published online: Sep 1, 2006
Published in print: Sep 2006
Notes
Note. Associate Editor: Bojan B. Guzina
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