Transversely isotropic elastic properties of single-walled carbon nanotubes by a rectangular beam model for the C-C bonds

Continuum mechanics modeling of carbon nanotubes has long been an attractive issue, but how to reflect exactly the physics essential of the atomic bonds still remains to be a challenging problem. To capture the distinguishing in-plane - and out-of-plane -π bond angle bending rigidities of C-C bonds in carbon nanotubes, an equivalent beam element with rectangular section is proposed and a corresponding frame structure model for a single-walled carbon nanotube (SWNT) is developed. By using the model, the five independent elastic moduli of SWNTs with arbitrary chirality and diameter are evaluated systematically. It is found that the elastic properties of the SWNTs are transversely isotropic when the tube diameter is small. The smaller the tube diameter is, the stronger the dependence of the elastic properties on the tube size and chirality is, 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. The present model can be incorporated into any standard finite element software directly, providing an extremely versatile and powerful tool for the study of nanostructures that beyond the computational capability of current atomistic approaches.