|
|
||
|
|
|
|
From:
Prediction of buckling characteristics of carbon nanotubes, International Journal of Solids and Structures, Vol. 4, No. 20, October 2007, pp. 6535 - 6550 by N. Hu (a, b), K. Nunoya (b), D. Pan (c), T. Okabe (b), H. Fukunaga (b)
(a) Department of Engineering Mechanics, Chongqing University, Chongqing 400011, PR China
(b) Department of Aerospace Engineering, Tohoku University, Aramaki-Aza-Aoba 6-6-01, Aoba-ku, Sendai 980-8579, Japan
(c) Institute for Materials Research, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8579, Japan
ABSTRACT: In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first constructed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiffness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der-Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agreement with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler’s beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results.
Partial Conclusions:
For the single-walled carbon nanotubes, we can conclude:
(1) The detailed molecular configuration of carbon nanotubes has very small influence on the buckling behavior of carbon nanotubes when the dimensions of nanotubes are large.
(2) When the aspect ratio of nanotubes is very large, the Euler buckling mode occurs, and the buckling load of carbon nanotubes can be predicted by using Euler’s beam buckling theory.
(3) When the aspect ratio of nanotubes is very low, the shellbuckling mode occurs, and the buckling behaviors of nanotubes are similar to those of thin-walled shells.
(4) The cap of carbon nanotubes has no significant influence on the buckling behaviors when the Euler beam buckling mode happens. And, when the boundary condition is pin-fixed, the effect of the cap is weak. However, when the free-fixed boundary condition is used, for the cases of shellbuckling, the addition of cap will lead to 50% increase of buckling load. The effectiveness of cap is equivalent to that the boundary condition is changed from the free-fixed to the pin-fixed. To model the buckling of nanotubes with cap under the free-fixed boundary condition, the nanotubes without cap, but with the pin-fixed boundary condition can be employed for simplicity.
For the multi-walled carbon nanotubes, we can conclude:
(1) When the Euler beam buckling mode happens, the effect of increase of walls is equivalent to that of the increase of the moment of inertia of the cross-section. Then, for multi-walled carbon nanotubes, if the total thickness of nanotubes is known, the buckling load of nanotubes can be simply evaluated from Euler’s beam buckling theory.
(2) When the shellbuckling happens, the buckling load of multi-walled carbon nanotubes can be obtained from the product of the buckling load of a representative single-walled carbon nanotubes and the number of walls.
Page 33 / 76