Effect of a rippling mode on resonances of carbon nanotubes

A recent study determined the Young's modulus of carbon nanotubes by measuring resonance frequency and using the modulus-frequency relation resulting from the linear vibration theory. It leads to the report that the Young's modulus decreases sharply, from about 1 to 0.1 TPa with the diameter D increasing from 8 to 40 nanometers, and the investigators attributed this decrease to the emergence of an unusual bending mode during the measurement that corresponds to rippling on the inner arc of the bent nanotubes. The nonlinear analysis presented in this paper that captures the rippling mode suggests that the effective Young's modulus can indeed decrease substantially with increasing diameter, and that the results from the classical linear theory may be invalid in such measurements.