ABSTRACT
At the beginning of this considerations, let introduce basic relations describing carbon-carbon (C-C) interactions. They characterize physical behavior of carbon nanotubes, being in fact atomistic structures, and are fundamental in the further transformation from molecular dynamics relations to continuum (shell) mechanics. The structure of nanotubes is obtained by conformational mapping of a graphene sheet onto a cylindrical surface. The nanotubes radius is estimated by using the relation
wherer0 = 0.141 nm is the carbon-carbon distance. The integer’s n and m denote the number of unit vectors a1 and a2 along two directions in the honeycomb crystal lattice of graphene. If m = 0, they are called “zigzag” nanotubes; if n =m, they are called “armchair” nanotubes. For any other values of n and m, the nanotubes are called “chiral” because the chains of atoms spiral around the tube axis instead of closing around the circumference. To capture the essential feature of chemical bonding in graphite, Brenner (1990) established an inter atomic potential (called as the REBO potential) for carbon in the following form
The parameter Bij in Eq. (2)represents the multi body coupling between the bond from the atoms i and j and the local environment of the atom i, and is given by
where θijk is the angle between bonds i−j and i−k, and the function G is given by
and the term Bij is expressed in the symmetric form
The set of material parameters is adopted here as follows
In contrast to the REBO potential function, in which the bond stretch and bond angle are coupled in the potential, Belytschko et al. [185] proposed the modified Morse potential function, which can be expressed as the sum of energies that are associated with the variance of the bond length V stretch, and the bond angle Vangle, that is,
The material constants are following
By differentiating Eq. (2) or (8), the stretching force of atomic bonds is obtained. The force variations with the bond length are almost the same while the bond angle is kept constant as 2π/3. However, for the REBO
potential, the force varies with the bond angle variations, whereas for the modified Morse potential it is always constant. Thus, the inflection point (force peak) is not constant for the REBO potential. As it is reported, both bond lengths and bond angles vary as CNTs are stretched. Therefore, in our numerical model, it is necessary to consider two possible formulations of inter atomic potentials to analyze and compare the influence of those effects on the non-linear behavior and fracture strain.
