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This paper aims at developing a mathematic model to characterize the mechanical properties of single-walled carbon nanotubes (SWCNTs). The carbon-carbon (C–C) bonds between two adjacent atoms are modeled as Euler beams. According to the relationship of Tersoff-Brenner force theory and potential energy acting on C–C bonds, material constants of beam element are determined at the atomic scale. Based on the elastic deformation energy and mechanical equilibrium of a unit in graphite sheet, simply form ED equations of calculating Young's modulus of armchair and zigzag graphite sheets are derived. Following with the geometrical relationship of SWCNTs in cylindrical coordinates and the structure mechanics approach, Young's modulus and Poisson's ratio of armchair and zigzag SWCNTs are also investigated. The results show that the approach to research mechanical properties of SWCNTs is a concise and valid method. We consider that it will be useful technique to progress on this type of investigation.

Since the discovery by Iijima in 1991 [

In the past, researchers used experimental method to measure mechanical properties of CNTs. Treacy et al. [

Meanwhile, for researching mechanical properties of CNTs, a number of researchers solved the difficulties in nanosized experiments in terms of computer simulation. For the analysis of nanostructural materials, atomic simulation methods such as first-principle quantum-mechanical methods [

Therefore, there is a demand of developing a modeling technique that could analyze the mechanical properties of CNTs at the atomic scale. Considering CNTs as a rolled cylindrical graphite sheet, we step from Young’s modulus of the C–C bonds counted as Euler beam at atomic scale and extend the theory of classical structural mechanics into the modeling of carbon graphite sheet. The effects of tube curvature on the mechanical properties of SWCNTs are considered in closed-form solutions. The mechanical properties of SWCNTs, including Young’s modulus, Poisson’s ratio, the length of C–C bonds and the angle between the adjacent C–C bonds are discussed as functions of nanosized structure.

CNTs can be considered as graphite sheets rolled into cylindrical shape. The one-atom-thick graphite sheet looks like chicken wire which is made of a single-carbon-atom thickness. The structure of CNTs as shown in Figure

Schematic diagram of hexagonal graphite sheet.

CNTs are classified into three categories named as zigzag (

From the viewpoint of molecular mechanics, CNTs are treated as a large array of molecules consisting of carbon atoms. According to the Tersoff-Brenner force field theory [

According to classical structural mechanics, the strain energy of a uniform beam in graphite sheet is expressed as

Based on energy conservation law, a linkage between the force constants in molecular mechanics and the sectional stiffness parameters in structural mechanics is established. Equations (

As long as the force constants

Knowledge of Young’s modulus (

Force analysis of armchair graphite sheet unit subjected to axial tension loading.

According to Castigliano’s Law, the rotation angel on point

Substituting (

For an armchair graphite sheet being subjected to the tension stress

Based on Hooke’s law, the relationship of the tension stress and the strain is

For a zigzag graphite sheet, the analytical approach is similar to that of the armchair graphite sheet. The unit of a zigzag graphite sheet is drawn in Figure

Force analysis of zigzag graphite sheet unit subjected to axial tension loading.

Then the strain of per unit length is obtained as follows:

For a zigzag graphite sheet being subjected to the tension stress

When

SWCNTs can be ideally constructed starting from a graphite sheet. According to the chiral vector, there are three kinds of structure, as shown in Figure

Classification of SWCNTs by chiral vector and chiral angle (a) armchair SWCNT, (b) zigzag SWCNT, and (c) chiral SWCNT.

Figure

Because of the effect of

In cylindrical coordinates, on account of the

Substituting (

Figure

The distances of the bonds in vector space are as follows:

For zigzag SWCNTs, the analysis method resembles armchair SWCNTs; we calculate the included angle for zigzag SWCNTs as follows:

Considering space curvature, Young’s modulus of SWCNTs in cylindrical coordinates depends on bond length and included angle between two bonds. The method for calculating Young’s modulus of armchair SWCNTs in three-dimensions resembles that in two dimensions. Affecting factors in three dimensional coordinates of Young’s modulus of armchair SWCNTs are mentioned in (

We obtain Young’s modulus of zigzag SWCNTs given in the following in the same way

For armchair SWCNTs, the stretch deformations of the bonds caused by concentrated force and bending moment are schematically signed in Figure

Analysis of axial deformation and angular displacement of armchair and zigzag SWCNTs.

According to geometric properties and elastic theory, taking the included angle between two bonds in three dimensions into consideration, we obtain the equilibrium equations about extension variation of the C–C bonds

Poisson’s ratio of armchair SWCNTs can be defined as the ratio between circumferential strain, and axial strain, substituting (

For zigzag SWCNTs, the analysis step is similar to that for armchair SWCNTs. To analyze Poisson’s ratio of zigzag SWCNTs, Figure

The equilibrium equations about extension variation of the bond are described as

The strains in axial direction and circumferential direction are defined as follows, respectively:

Poisson’s ratio of armchair SWCNTs can be defined as the ratio between circumferential strain and axial strain; then we obtain

The atomic-based continuum mechanic approach described in the previous section was implemented for studying the effective elastic properties of graphite sheets and SWCNTs. In this section, the mechanical characteristics of graphite sheet and SWCNTs are examined.

In the present simulation,

Geometrical and material properties of C–C bonds.

Beam element | Abbreviation | Value | Unit |
---|---|---|---|

Diameter | 0.146 | nm | |

Cross-section area |
1.678 ^{-2} | nm^{2} | |

Moment of inertia |
2.241 ^{-5} | nm^{4} | |

Young’s modulus | 5.530 | TPa | |

Shear elastic modulus | 0.871 | TPa |

Comparing (

Since SWCNTs are defined as rolled graphite sheets, the lengths of bonds in vector space are changed owing to the effect of curvature. For armchair SWCNTs rolled by armchair graphite sheets, all of the lengths of C–C bonds become shorter because of the connection with curvature in circumferential direction, while, for zigzag SWCNTs rolled by zigzag graphite sheet, the length

C–C bonds lengths of armchair SWCNTs and zigzag SWCNTs.

When graphite sheets are rolled into SWCNTs, the lengths and spatial relations of C–C bonds change obviously. Figure

Included angles of adjacent C–C bonds in armchair SWCNTs and zigzag SWCNTs.

Young’s modulus of armchair SWCNTs and zigzag SWCNTs.

Comparing (

Considering the variations of length and included angles of two adjacent C–C bonds, from graphite sheets into SWCNTs, Figure

Poisson’s ratio of armchair SWCNTs and zigzag SWCNTs.

The mechanical properties of both armchair and zigzag SWCNTs are characterized by using continuum mechanics in the atomic scale. In terms of the conjunction of Tersoff-Brenner force field method and energy conservation law, the graphite sheet is of isotropic property and Young’s modulus of graphite sheet is obtained to be 1.04 TPa. Furthermore, considering the variations of the length and the included angle of two adjacent C–C bonds rolled from graphite sheets into SWCNTs, Young’s modulus and Poisson’s ratio of SWNCTs with armchair and zigzag structures are investigated as a function of SWCNT radius. We predict that Young’s modulus and Poisson’s ratio of SWNCTs are influenced obviously by relatively smaller radius while being little affected by larger radius. We are confident that this model provides a useful method to analyze mechanical properties of CNTs and other nanosized structures at the atomic scale.

This work was supported by Grant-in-Aid for Global COE Program by the Ministry of Education, Culture, Sports, Science and Technology.