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# Numerical Simulation of the Mechanical Behavior

DOI link for Numerical Simulation of the Mechanical Behavior

Numerical Simulation of the Mechanical Behavior book

# Numerical Simulation of the Mechanical Behavior

DOI link for Numerical Simulation of the Mechanical Behavior

Numerical Simulation of the Mechanical Behavior book

## ABSTRACT

The high symmetry of SWCNTs permits the parametric generation of the atomic coordinates for any (n1, n2) tubule. The helical and rotational symmetries [254, 255] are used to define the full Euclidean symmetry group of infinite SWCNTs and, hence, to obtain the coordinates of the atoms. Line groups [255] are the group of Euclidean symmetries of the systems that exhibit translational periodicity in one direction. Typical examples of these structures are the quasi-one-dimensional crystals like SWCNTs. The monomers are the elementary structural units of the lattice and their regular arrangement is obtained by pure translations combined with operations on the screw axis. Monomers are clustered into larger local units: the elementary cells. Let a SWCNT be axially aligned on the z-axis in the Euclidean space and let P be an axial point group. Each line group L is a weak-direct product L = ZP of a group Z of the generalized translations and the axial point group P. The former arranges the monomers, while the latter implies the symmetry of the monomers. The axial point group P leaves the z-axis invariant and the infinite cyclic group Z is either a screw axis or a glide plane group. In the Koster-Seitz notation, the generator of the glide plane group is denoted as

whereadenotes the translational period of the group L and σv is the vertical mirror plane. The generator of the screw axis group, the latter denoted by Trq(a), is

where q and r are non-negative integers such that q = αn, for α ∈N. The choice of r is not unique, given r any multiple of q/n may be added, with no effect on the resulting group L. In order to establish a fixed value of r, two different conventions can be used [256]:

r is coprime with q/n, r is the minimal allowed value that is coprime with q.