Effects of a small length scale on vibrations of anembedded double-walled carbon nanotube

Extensive continuum analyses are carried out to estimate the influence of matrix stiffness, a small length scale, and intertubular radial displacements on free vibrations of an individual double-walled carbon nanotybe. The analyses are based on both local and classical Euler-Bernoulli and Timoshenko elasticity theories with concentricity and nonconcentricity assumptions. The effect of a small length scale is incorporated in the formulations. New intertubular resonant frequencies are calculated based on these theories. Detailed results are demonstrated for the resonant frequencies as functions of matrix stiffness and the small length scale. The results in dicate that the internal radial dis placement and the stiffness of the surrounding matrix cangreatly affect the resonant frequencies, especially at higher frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. Moreover, at high frequencies and small aspect ratios, the effect of the small length scale becomes more significant.