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

It has been reported that a multiwalled carbon nanotube in bending may exhibit an unusal elastic mode that corresponds to the wavelike distortion or ripple along the inner arc of the bent nanotube, called the rippling mode, which cannot be predicted by the linear theories. The present analysis shows that the rippling mode is permissible by the theory for highly anisotropic elastic materials of finite deformation and that the dependence of the bending moment upon the bending curvature can be well approximated by a bilinear relation, in which the transition from one linear branch to the other corresponds to the emergence of the rippling mode. With this bilinear relation, the authors show that the deflection response to a transverse force is consistent with the unusual behavior reported by Wong, Sheehan, and Liebert [Science 277, 1971 (1997)]. Furthermore, their analysis indicates that there exists a critical diameter, for given load and length, which corresponds to the emergence of rippling mode, and that the effective Young's modulus of multiwalled carbon nanotubes at the vibration resonance drops sharply as the diameter increases to surpass this critical value, confirming a phenomenon observed by Poncharal, Wang, Ugarte, and de Heer [Science 283, 1513 (1999)].