The lowfrequency oscillations and energy localization of SingleWalled Carbon Nanotubes (SWNTs) are studied in the framework of the SandersKoiter shell theory. The circumferential flexure modes (CFMs) are analysed. Simply supported, clamped and free boundary conditions are considered. Two different approaches are proposed, based on numerical and analytical models. The numerical model uses in the linear analysis a double mixed series expansion for the displacement fields based on Chebyshev polynomials and harmonic functions. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations, which is solved by the RungeKutta numerical method. The analytical model considers a reduced version of the SandersKoiter shell theory obtained by assuming small circumferential and tangential shear deformations. These two assumptions allow to condense the longitudinal and circumferential displacement fields into the radial one. A nonlinear fourthorder partial differential equation for the radial displacement field is derived, which allows to calculate the natural frequencies and to estimate the nonlinearity effect. An analytical solution of this equation is obtained by the multiple scales method. The previous models are validated in linear field by means of comparisons with experiments, molecular dynamics simulations and finite element analyses retrieved from the literature. The concept of energy localization in SWNTs is introduced, which is a strongly nonlinear phenomenon. The lowfrequency nonlinear oscillations of the SWNTs become localized ones if the intensity of the initial excitation exceeds some threshold which depends on the SWNTs length. This localization results from the resonant interaction of the zoneboundary and nearest nonlinear normal modes leading to the confinement of the vibration energy in one part of the system. The value of the initial excitation corresponding to this energy confinement is referred to as energy localization threshold. The effect of the aspect ratio on the analytical and numerical values of the energy localization threshold is investigated; different boundary conditions are considered.
Dynamics and Stability of Carbon Nanotubes / Strozzi, Matteo; Barbieri, Marco; Zippo, Antonio; Pellicano, Francesco.  (2015). (Intervento presentato al convegno AIMETA 2015 XXII Congresso dell’Associazione Italiana di Meccanica Teorica e Applicata tenutosi a Genova, Italy nel September 1417, 2015).
Dynamics and Stability of Carbon Nanotubes
STROZZI, MATTEO^{};BARBIERI, MARCO;ZIPPO, Antonio;PELLICANO, Francesco
2015
Abstract
The lowfrequency oscillations and energy localization of SingleWalled Carbon Nanotubes (SWNTs) are studied in the framework of the SandersKoiter shell theory. The circumferential flexure modes (CFMs) are analysed. Simply supported, clamped and free boundary conditions are considered. Two different approaches are proposed, based on numerical and analytical models. The numerical model uses in the linear analysis a double mixed series expansion for the displacement fields based on Chebyshev polynomials and harmonic functions. The RayleighRitz method is applied to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are reexpanded by using approximate eigenfunctions. An energy approach based on Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations, which is solved by the RungeKutta numerical method. The analytical model considers a reduced version of the SandersKoiter shell theory obtained by assuming small circumferential and tangential shear deformations. These two assumptions allow to condense the longitudinal and circumferential displacement fields into the radial one. A nonlinear fourthorder partial differential equation for the radial displacement field is derived, which allows to calculate the natural frequencies and to estimate the nonlinearity effect. An analytical solution of this equation is obtained by the multiple scales method. The previous models are validated in linear field by means of comparisons with experiments, molecular dynamics simulations and finite element analyses retrieved from the literature. The concept of energy localization in SWNTs is introduced, which is a strongly nonlinear phenomenon. The lowfrequency nonlinear oscillations of the SWNTs become localized ones if the intensity of the initial excitation exceeds some threshold which depends on the SWNTs length. This localization results from the resonant interaction of the zoneboundary and nearest nonlinear normal modes leading to the confinement of the vibration energy in one part of the system. The value of the initial excitation corresponding to this energy confinement is referred to as energy localization threshold. The effect of the aspect ratio on the analytical and numerical values of the energy localization threshold is investigated; different boundary conditions are considered.File  Dimensione  Formato  

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