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A numerical approach is adopted for the multiscale analysis of vibrations of single-walled carbon nanotubes (SWCNTs). The SWCNT is modeled by a hyperelastic membrane whose kinematics is described using the higher-order Cauchy-Born rule. The constitutive model is formulated exclusively in terms of the interatomic potential, so, it inherits the atomistic information and involves no other phenomenological input. The variational differential quadrature (VDQ) method is employed in which the continuum model is discretized using DQ, and a weak form of equation of motion is obtained via a variational approach. VDQ is computationally advantageous since it has a fast rate of convergence and can reproduce the results of molecular dynamics simulations. Detailed investigations into frequencies and mode shapes of SWCNTs with different geometrical parameters, boundary conditions and chiralities are carried out. It is found that short nanotubes display a coupling between the axial/torsional and bending modes. Also, as the tube diameter or length increases, mode transitions are made at several critical points. If the edge supports are more flexible and tube length is longer, the critical diameters are larger. Eventually, the vibration characteristics of axially strained nanotubes are analyzed, and it is concluded that SWCNTs with smaller radii have higher strain sensitivity. [ABSTRACT FROM AUTHOR]

In this research, a continuum-based model is presented to explore potential energy, force distribution and oscillatory motion of ions, and in particular chloride ion, inside carbon nanotubes (CNTs) decorated by functional groups at two ends. To perform this, van der Waals (vdW) interactions between ion and nanotube are modeled by the 6-12 Lennard-Jones (LJ) potential, whereas the electrostatic interactions between ion and functional groups are modeled by the Coulomb potential and the total interactions are analytically derived by summing the vdW and electrostatic interactions. Making the assumption that carbon atoms and charge of functional groups are all uniformly distributed over the nanotube surface and the two ends of nanotube, respectively, a continuum approach is utilized to evaluate the related interactions. Based on the actual force distribution, the equation of motion is also solved numerically to arrive at the time history of displacement and velocity of inner core. With respect to the proposed formulations, comprehensive studies on the variations of potential energy and force distribution are carried out by varying functional group charge and nanotube length. Moreover, the effects of these parameters together with initial conditions on the oscillatory behavior of system are studied and discussed in detail. It is found out that chloride ion escapes more easily from negatively charged CNTs which is followed by uncharged and positively charged ones. It is further shown that the presence of functional groups leads to enhancing the operating frequency of such oscillatory systems especially when the electric charges of ion and functional groups have different signs. [ABSTRACT FROM AUTHOR]

This article aims to present a comprehensive study on the oscillatory behavior of concentric carbon nanocones (CNCs) inside carbon nanotubes (CNTs) using a continuum approach. To this end, the optimum radius of nanotube for which the nanocone lies on the tube axis is determined based on the distribution of suction energy. Using the Runge–Kutta numerical integration scheme, the equation of motion is solved numerically to attain the time history of displacement and velocity of nanocone. It is observed that the oscillation of nanocone occurs with respect to its axial equilibrium distance which moves further away from the middle axis of nanotube as the number of pentagons increases. A novel semi-analytical expression as a function of geometrical parameters, initial conditions and cone vertex direction is also proposed for the precise evaluation of oscillation frequency. With respect to the proposed frequency expression, a detailed parametric study is conducted to get an insight into the effects of number of pentagons, cone vertex direction and initial conditions on the oscillatory behavior of CNC–CNT oscillators. It is found that nanocones with more pentagons generate greater maximum frequencies inside nanotubes. Furthermore, it is shown that higher maximum frequencies can be achieved if the nanocone enters the nanotube from base. [ABSTRACT FROM AUTHOR]