On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium



Free axisymmetric vibrations of a single-walled carbon nanotube (SWCNT) embedded in a nonhomogeneous elastic matrix are studied on the base of the nonlocal continuum shell theory. The effect of the surrounding elastic medium are considered using the Winkler-type spring constant which is assumed to be variable along the tube axis. The tube may be prestressed by external tensile forces. The Flügge type shell equations, including the initial membrane hoop and axial stresses, are used as the governing ones. The constitutive equations are formulated by considering the small-scale effects. Using the asymptotic approach, the SWCNT eigenmodes are constructed in the form of functions decreasing rapidly away from some “weakest” line which is assumed to be far from the tube edges. This study shows that introducing the small length scale parameter into the tube model allows to take into account inclusions in the surrounding elastic matrix which may results in the strong localization of some eigenmodes. The dependence of the natural frequencies and the power of the localization of the corresponding modes upon the constant of non-locality, tensile stresses and the nanotube radius is analyzed.