Low-frequency vibrations of single-walled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders–Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh–Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders–Koiter equations are strongly simplified neglecting in-plane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort

No abstract

New phenomenon of a weak energy localization of optical low-frequency oscillations in carbon nanotubes (CNT) is analytically predicted in the framework of continuum shell theory. This phenomenon takes place for CNTs of finite length with medium aspect ratio. The origin of localization is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the MD simulation of simply supported CNT.PACS numbers: 61.48. De, 63.22.Gh, From a modern point of view, carbon nanotubes are twice the exciting subject of scientific researches. On the one hand, they are associated with great hopes for creation of super-small and ultra-fast electronic and electromechanical devices [1][2][3][4]. On the other hand, they are quasi-one-dimensional objects that allow to check out some of the funamentals of modern solid-state physics. In particular, variuos computational and in-situ measurements of thermoconductivity of CNT [5][6][7][8][9] are directly related with the problem of finiteness of thermoconductivity of one-dimensional anharmonic lattices. This problem has been formulated more than fifty years ago in the famous work by Fermi, Pasta and Ulam [10]. The widearea study of nonlinear lattices dynamics led to discovery of new class of elementary excitations -solitons, the main feature of those is the self-localization in the homogeneous lattices [11,12]. From the point of view of energy trasfer, solitons take place dual role. Being very effective energy carriers they also provide the effective scattering of small-amplitude phonons [13][14][15][16][17][18][19]. From the mathematical point of view, the solution like a breather exists only in the infinite systems with a continuous spectrum, while the nanoscale objects can be rather considered as finite ones. In such a case the problem of definition of nonlinear localized excitations has significant pecularities.It was recently shown [20][21][22] that the finite systems of weakly coupled oscillators exhibit strongly modulated non-stationary oscillations characterized by the maximum possible energy exchange between the groups of the oscillators or the maximum energy transfer from the external source of energy to the system [23]. The solutions describing these processes are referred to as Limiting Phase Trajectories (LPTs). The development and the use of the analytical framework based on the LPT concept is motivated by the fact that resonant non-stationary processes occurring in a broad variety of finite dimensional physical models are beyond the well-known paradigm of nonlinear normal modes (NNMs), fully justified only for quasi-stationary processes and non-stationary processes in non-resonant case. While the NNMs approach has been proved to be an effective tool for the analysis of instability and bifurcations of stationary processes (see, e.g., [24]), the use of the LPTs concept provides the adequate procedures for studying strongly modulated regimes as well as the transitions to energy localization and chaotic behavior [20]. Such an approach...

No abstract

No abstract