Finite Element Modelling And Simulations Of Carbon Nanotubes Based On Atomic Potentials

Carbon nanotubes (CNTs) have very amazing mechanical, electronic, thermal and optical properties, which enable them many potential applications. Since the discovery of CNTs, great attention has been paid to the prediction of their mechanical behaviors. In this dissertation, a finite element model of CNTs is proposed basing on their atomic potentials, and several key mechanical problems of CNTs are studied by the model. The major contents are following:(1) Finite element modeling of CNTs. In order to capture the distinguishing in-planeσ-σand out-of-planeσ-πbond angle bending rigidities of C-C bonds in CNTs, we propose a novel equivalent beam element, which can perfectly replace the harmonic potentials to describe the bond stretching, bond angle variance, inversion angle variance and torsion of the C-C bonds. The parameters of the equivalent beam element are extracted from the constitutive relations of a graphite sheet under different load conditions. Using the equivalent beam elements, the frame structure finite element model of single-walled carbon nanotubes (SWNTs) is developed. Then nonlinear spring elements are used to replace the Lennard–Jones potentials to represent the interlayer van der Waals interactions of multi-walled carbon nanotubes (MWNTs) and the corresponding finite element model of MWNTs is constructed. The present model can be incorporated into any standard commercial finite element software. Since the commercial finite element softwares are comprehensively developed and have very powerful calculation abilities, many mechanical problems of CNTs can be solved by the models in prospect.(2) Calculations of the elastics moduli of SWNTs. The five independent elastic moduli of SWNTs with arbitrary chirality and diameter are evaluated systematically. It is found that the elastic properties of SWNTs are transversely isotropic when the tube diameter is small. The smaller the tube diameter, the stronger the dependence of the elastic properties on the tube size and chirality, while when the tube diameter is large enough, the SWNTs degenerate from transversely isotropic to isotropic and the elastic moduli tend to that of a graphite sheet. This work extends the previously reported results and provides a full understanding of the anisotropic elastic properties of SWNTs(3) Simulation of the elastic buckling of CNTs. The compression buckling behavior of CNTs is predicted by using the eigenvalue buckling analysis. It is found that the critical buckling modes and the critical buckling strains are varied with the aspect of the CNTs. The results also indicate that there exists an optimum diameter for SWNTs with the same lengths at which the critical compression buckling strain reaches its maximum value. The bending buckling of SWNTs is also been studied elementarily through the automatic addition of artificial damping to the model. The results show good agreement with previous molecular dynamics calculations.(4) Analysis of the vibration modes of CNTs. By using the eigenvalue extraction method, the vibration modes of SWNTs and MWNTs are studied, and the rigid coaxial vibration modes, which are mainly determined by the interlayer van der Waals interations, are especially concerned. It is shown that in our calculation range, all the vibration frequencies of CNTs are above the order of GHz, indicating that CNTs may be an ideal material for high frequency nano devices.(5) Calculation of the Raman-active modes of CNTs. The Raman modes of SWNTs are extracted by the eigenvalue extraction method. It is shown that the present calculated frequencies of the low-order Raman modes agree well with the related experimental and theoretical results, especially that of the radial breathing modes (RBMs) are greatly consistent with existing ab initio results. This indicates that the present method is applicable for the calculation of Raman-active modes of CNTs, especially for the calculation of RBMs. So we conduct detailed prediction of the RBMs of MWNTs. Results show that RBM number of a MWNT equals to its layer number, and the frequency of each RBM is higher than the RBM frequency of the corresponding SWNT of each MWNT layer.(6) Investigation on the GHz oscillation driven by Casimir or electrostatic force. A mechanical oscillator model of double-walled coaxial cylindrical tubes, which is analogous to double-walled carbon nanotubes, is proposed and the oscillation behavior of the model driven by Casimir or electrostatic force is studied by energy approach It is shown that when the model is at nano-scale, its oscillation frequency can reach or even exceed the order of GHz, indicating that GHz high frequency oscillation may be quite common phenomenon at nano-scale.