Tensile Deformation of Single-Wall Carbon Nanotube Based on a Quasi-Continuous Analysis

In this paper, a quasi-continuous analysis based on the VIB method [1] for the tensile deformation of single-wall carbon nanotubes is presented trough taking the hyper elasticity theory [2] as a theoretical foundation. As a result of the noncentrosymmetric atomic structure of some nanometer materials, the effects of local nonuniform deformation on the calculation of the stress and strain can be basically restrained by means of searching the relative balance positions of every atoms at different stages of deformation. The tension response of a single-wall carbon nanotube is analyzed, and their axial force-strain curves can be obtained. Furthermore, the strain and the stress at the onset of fracture of the single-wall carbon nanotubes under tension can be estimated, which are in agreement with the results of MD simulation. Introduction Nanotechnology is considered to be a crucial technical field with much influence on our lives in the 21 st century. Materials and components at nano-scales have some noticeable properties, such as extremely high rigidity and intensity. One of the present research aims is to develop an analytical method to analyze the mechanical behavior of materials at nano-scales. Nanometer is at the scale of atoms, and the interactions among atoms show decisive influence on the behavior of nanometer materials at this scale, so nanometer materials must be analyzed as materials with certain atomic configurations and atomic potentials, but not a simple continuum. Molecular dynamics (MD) is on the basis of the interactions among atoms, and is able to make relatively precise and matter-of-fact simulations for mechanical behaviors of nanometer materials; but it is limited when the atom number is so large that computers cannot support the computation capability. The hyper elasticity theory [2] proposes a good idea to solve this problem. Its main idea is: in the course of the deformation of the material, all of the interaction potentials among atoms are directly summated as the strain energy of the material, with the strain energy density differentiated by strain, the constitutive model of the material is deduced. Summating atom potentials as strain energy is on the basis of the Cauchy-Born rule [3] , but the Cauchy-Born rule cannot be applied to materials with noncentrosymmetric atomic structure such as carbon nanotubes. In order to apply the Cauchy-Born rule, a modification is required according to the atomic structure of the carbon nanotube [3] . The schematic diagram of the atomic structure of a carbon nanotube within the tube surface (a node represents an carbon atom) is shown in Fig. 1. Disassemble this hexagon lattice into two sub-lattices A and B (marked in Fig. 1), atoms-A and atoms-B are in two different environments, respectively. When a carbon nanotube is subjected to a macroscopically uniform deformation (such as axial tension), the same type of atoms (A or B) is in the same environments, and in the same state of loading and deformation, the Cauchy-Born rule is applicable. Key Engineering Materials Online: 2004-10-15 ISSN: 1662-9795, Vols. 274-276, pp 325-330 doi:10.4028/www.scientific. net/KEM.274-276.325 © 2004 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific. net. (Semanticscholar.org-13/03/20,17:14:07) However, in the total lattice (A+B), different types of atoms are in different environments, and in different states of loading and deformation, so the Cauchy-Born rule is not applicable, and continuum theory cannot be applied. Zhang and Huang [3] introduced a vector ξ relative to the rigid body displacement between these two sub-lattices, besides the Green strain tensor E for describing the deformation within each of the sub-lattices. The vector ξ is calculated according to the minimum energy principle at every step of deformation, and it is actually introduced to describe the local nonuniform deformation within the total lattice (A+B). In fact, the geometric parameters introduced when searching the initial balance positions of undeformed nanotube (in section 2) have the same function with the vector ξ . Therefore, the method of Zhang and Huang is equivalent to searching the relative balance positions of atoms once again at every step of deformation. In this paper, a quasi-continuous analysis based on the VIB method [1] for the tensile deformation of single-wall carbon nanotubes is presented trough taking the hyper elasticity theory [2] as a theoretical foundation. As a result of the noncentrosymmetric atomic structure of some nanometer materials, the effects of local nonuniform deformation on the calculation of the stress and strain can be basically restrained by means of searching the relative balance positions of every atom at different stages of deformation. 1.Basic theory model. The potential between two carbon atoms i and j is described with the Tersoff-Brenner potential [5] : ) ( ) , ( ) ( ij A ijk ij ij ij R ij r V r B r V U θ − = , (1) Here: )) ( 2 exp( )] 1 /( ) ( [ ) ( ) ( ) ( e ij ij ij ij e ij ij ij ij R R r S S D r f r V − − − = β , )) ( / 2 exp( )] 1 /( ) ( [ ) ( ) ( ) ( e ij ij ij ij ij e ij ij ij ij A R r S S S D r f r V − − − = β ) (r fij is the cutoff function with break points at ) 1 ( ij R and ) 2 ( ij R . If the summation of the T-B potential within every C-C bond is taken as the strain energy directly, it can be found that there is already some strain energy before the nanotube deforms, which is not reasonable in the sense of mechanical conception. In order to get rational prediction, it is neccessary to calculate the strain energy in the following way: Figure 2 A simple diatomic system Figure 3 The cell picked within the carbon nanotube surface Figure 1 A schematic diagram of the atomic structure of a carbon nanotube within the tube 326 Advances in Engineering Plasticity and Its Applications