Effect of bending instabilities on the measurements of mechanical properties of multiwalled carbon nanotubes

Physical Review B - Condensed Matter and Materials Physics

Volumn 67, Issue 7, 2003, Pages

  Liu, Jefferson Z ^a   Zheng, Quanshui ^a   Jiang, Qing ^b  

^a TSINGHUA UNIVERSITY   (China)

^b University of California Riverside   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CARBON;

ANISOTROPY; ARTICLE; ELASTICITY; FORCE; MATHEMATICAL ANALYSIS; MOLECULAR DYNAMICS; NANOPARTICLE; PREDICTION; THEORY;

EID: 0037441302     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.67.075414     Document Type: Article

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32. Our further numerical simulations (J. Z. Liu, Studies on several mechanical problems of carbon nanotubes, Ph.D. thesis, Tsinghua University, 2002 based on a shell-cobweb model, for bending a 40-nm-length, 5-nm-diameter carbon nanotubes show that the rippling mode always emerges at severe bending for nanotubes of 6 or more walls, while for nanotubes of fewer walls, one, two or four kinks, instead of rippling, develop at severe bending. Our analysis and simulations suggest that rippling appears to be associated with the multiplicity of walls of MWNT’s and the ultralow interwall sliding resistance strength (ISRS) (Refs. 910111718). The bilinear relation (3) captures, approximately, the constitutive behavior of multiwalled carbon nanotubes near rippling although the dimensionless parameters (formula presented) and (formula presented) vary with configurations of individual tubes. For instance, we find that the predicted responses of the lateral force and the strain energy versus the deflection fit the measurements of Wong et al. (Ref. 13) remarkably well, if we set (formula presented) and (formula presented) for a six-walled nanotube of 40-nm in length and 5-nm in diameter, taking the interwall spacing to be 0.34 nm.
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34. The loading P is applied by controlling the displacement (formula presented) of the AFM probe, whose tip is located at the free end of the AFM’s cantilevered beam. The relative displacement (formula presented) of the probe tip to the controlled displacement (formula presented) is recorded and it is proportional to P in the form (formula presented) with the known AFM elastic stiffness parameter (formula presented) This leads to the deflection (formula presented) When (formula presented) or equivalently, (formula presented) the classical prediction (formula presented) with (formula presented) is valid. As (formula presented) is increasingly crossing the critical value (formula presented) with respect to (formula presented) and (formula presented) the bending mode is transiting from the classical mode to the rippling one. Prior to the emergence of the first rippling period (which is about one fourth of the nanobeam height), the nanobeam becomes kinked at the cantilevered end (formula presented) with a certain kinked angle denoted by (formula presented) Denoting by (formula presented) and (formula presented) respectively, the loading/displacement pairs immediately before and after the kinking and neglecting the dynamic effect, we have (formula presented)It thus yields (formula presented) (formula presented)In other words, as the AFM probe displacement (formula presented) increases to exceed (formula presented) a kink develops at the fixed end, causing the loading to drop (formula presented) and correspondingly, the displacement to rise (formula presented) The normalized slope is (formula presented)The above result is consistent with the force-displacement measurment of Wong et al. (Ref. 13) in their Fig. 4A. For a qualitative comparison, the dotted line in Fig. 22 is plotted having taken into account of the above kinking effect.