The Very‐Low Shear Modulus of Multi‐Walled Carbon Nanotubes Determined Simultaneously with the Axial Young's Modulus via in situ Experiments

The natural frequencies (f) as a function of the length (L) of single, multi‐walled carbon nanotubes (CNTs) are measured using the electric‐field‐induced resonance method together with the “nanoknife” technique for cutting nanotubes to the desired length. The experimental f‐L data for short CNTs are found to be adequately described by the Timoshenko beam model, but not by the widely‐used Euler‐Bernouilli beam model. The failure of the Euler‐Bernouilli beam model is due to its neglect of the significant effect of shear deformation caused by the extremely‐anisotropic mechanical properties of CNTs. The axial Young's modulus and radial shear modulus of CNTs are obtained simultaneously through fitting the experimental f‐L data with the Timoshenko beam model.

[1]  Keh Chih Hwang,et al.  An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes , 2008 .

[2]  John A Rogers,et al.  Molecular scale buckling mechanics in individual aligned single-wall carbon nanotubes on elastomeric substrates. , 2008, Nano letters.

[3]  Qing Chen,et al.  Cutting and sharpening carbon nanotubes using a carbon nanotube ‘nanoknife’ , 2007 .

[4]  W. Wan,et al.  Simultaneous measurement of Young’s and shear moduli of multiwalled carbon nanotubes using atomic force microscopy , 2007 .

[5]  Zhongfan Liu,et al.  Two distinct buckling modes in carbon nanotube bending. , 2007, Nano letters.

[6]  K. Hwang,et al.  Thickness of graphene and single-wall carbon nanotubes , 2006 .

[7]  T. D. Yuzvinsky,et al.  Ultrahigh frequency nanotube resonators. , 2006, Physical review letters.

[8]  Chien Ming Wang,et al.  Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes , 2006 .

[9]  Y. S. Zhang,et al.  Size dependence of Young's modulus in ZnO nanowires. , 2006, Physical review letters.

[10]  V. Varadan,et al.  Wave characteristics of carbon nanotubes , 2006 .

[11]  J. Gaillard,et al.  Determination of the bending modulus of an individual multiwall carbon nanotube using an electric harmonic detection of resonance technique. , 2005, Nano letters.

[12]  Quanshui Zheng,et al.  Size dependence of the thin-shell model for carbon nanotubes. , 2005, Physical review letters.

[13]  J. Gaillard,et al.  Mechanical properties of chemical vapor deposition-grown multiwalled carbon nanotubes , 2005 .

[14]  S. Arepalli,et al.  Transmission-electron-microscopic studies of mechanical properties of single-walled carbon nanotube bundles , 2004 .

[15]  T. Belytschko,et al.  Nonlinear mechanical response and rippling of thick multiwalled carbon nanotubes. , 2003, Physical review letters.

[16]  Enge Wang,et al.  Dual-mode mechanical resonance of individual ZnO nanobelts , 2003 .

[17]  Philippe H. Geubelle,et al.  The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials , 2002 .

[18]  W. D. de Heer,et al.  Nanomechanics of individual carbon nanotubes from pyrolytically grown arrays. , 2000, Physical review letters.

[19]  R. Ruoff,et al.  Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load , 2000, Science.

[20]  W. D. Heer,et al.  Electrostatic deflections and electromechanical resonances of carbon nanotubes , 1999, Science.

[21]  K. Méténier,et al.  Elastic Modulus of Ordered and Disordered Multiwalled Carbon Nanotubes , 1999 .

[22]  G. A. D. Briggs,et al.  Elastic and shear moduli of single-walled carbon nanotube ropes , 1999 .

[23]  Charles M. Lieber,et al.  Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes , 1997 .

[24]  J. Bernholc,et al.  Nanomechanics of carbon tubes: Instabilities beyond linear response. , 1996, Physical review letters.

[25]  T. Ebbesen,et al.  Exceptionally high Young's modulus observed for individual carbon nanotubes , 1996, Nature.

[26]  Leonard Meirovitch,et al.  Elements Of Vibration Analysis , 1986 .

[27]  R. E. D. Bishop,et al.  Vibration Problems in Engineering, 4th Edition. S. P. Timoshenko, D. H. Young and W. Weaver. Wiley, Chichester. 1974. 538 pp. Illustrated. £8.95. , 1975, The Aeronautical Journal (1968).

[28]  O. L. Blakslee,et al.  Elastic Constants of Compression-Annealed Pyrolytic Graphite , 1970 .

[29]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[30]  T. Huang,et al.  The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions , 1961 .