## Abstract

Molecular mechanics has been widely used to analytically study mechanical behaviour of carbon nanotubes. However, explicit expressions for elastic properties of carbon nanotubes are so far confined to some special cases due to the lack of fully constructed governing equations for the molecular mechanics model. In this paper, governing equations for an analytical molecular mechanics model are fully established. The explicit expressions for five in-plane elastic properties of a chiral single-walled carbon nanotube are derived, which make properties at different length-scales directly connected. The effects of tube chirality and tube diameter are investigated. In particular, the present results show that the classic relationship from the isotropic elastic theory of continuum mechanics between Young's modulus and shear modulus of a single-walled carbon nanotube is not retained. The present analytical results are helpful to the understanding of elastic properties of carbon nanotubes, and also useful to the topic of linking molecular mechanics with continuum mechanics.

## 1. Introduction

Carbon nanotubes have become ideal candidates for multifarious applications, including super-strong materials and nanomechanical devices, partially due to their amazing mechanical properties, such as exceptional high stiffness and tensile strength (Ajayan & Zhou 2001; Sinnott & Andrews 2001; Qian *et al*. 2002; Cheng 2002; Dai 2002; Fisher *et al*. 2003; Gao & Kong 2004). There are two categories of carbon nanotubes: multi-walled carbon nanotubes (MWNTs) and single-walled carbon nanotubes (SWNTs). A MWNT consists of two or more concentric cylindrical shells of graphene sheets arranged coaxially around a central hollow with interlayer separation as in graphite (0.34 nm), whereas a SWNT is made of a single layer graphene cylinder. Although a graphene layer is isotropic, the elastic properties of a SWNT showed significant chirality- and size-dependence (Hernandez *et al*. 1998; Sanchez-Portal *et al*. 1999). For example, quantum mechanics-based calculations indicated that, with the tube diameter decreasing from 2 to 0.4 nm, a maximum reduction of Young's modulus of a SWNT is more than 10% (Hernandez *et al*. 1998). This is difficult to understand within the framework of elastic theories. On the other hand, such a deviation of Young's modulus would result in a significant effect on the mechanical behaviour of some carbon nanotube-based materials or devices, such as the elastic properties of nanotube-reinforced composites (Thostenson *et al*. 2001; Fisher *et al*. 2003; Guz & Rushchitskii 2003; Lau *et al*. 2004; Hu *et al*. 2005), the vibration frequency of a nanotube oscillator (Yoon *et al*. 2002, 2005; Li & Chou 2003*b*; Jiang *et al*. 2004), and so on. Hence, a clear understanding of elastic properties of carbon nanotubes is very important for the engineering applications of carbon nanotubes.

The chirality- and size-dependent elastic properties of SWNTs have been extensively studied by numerical calculations (Hernandez *et al*. 1998; Sanchez-Portal *et al*. 1999; Arroyo & Belytschko 2002, 2003; Li & Chou 2003*a*) and analytical approaches (Popov *et al*. 2000; Chang & Gao 2003; Shen & Li 2004; Xiao *et al*. 2005). Compared to the numerical methods, analytical models usually give explicit solutions to the problems considered. This is particularly helpful to the topic of linking theories at different length-scales, which is in fact a very important subject in solid mechanics, because the properties at different length-scales of nanotubes can be directly connected by the obtained expressions. Popov *et al*. (2000) derived analytical formulae to predict Young's modulus and Poisson's ratio of SWNTs using Born's perturbation technique within a lattice dynamics framework. However, no mathematically explicit expressions for the elastic properties were obtained in their work. The first closed-form expressions for the elastic properties of SWNTs were obtained by Chang & Gao (2003) by establishing a ‘stick-spiral’ model based on a molecular mechanics approach. The expressions are concise but capable of relating Young's moduli and Poisson's ratios of armchair and zigzag nanotubes to their molecular structures. Xiao *et al*. (2005) extended the ‘stick-spiral’ model to the torsion loading condition and also studied the nonlinear behaviour of SWNTs by incorporating a modified Morse potential into the model. Shen & Li (2004), based on the molecular mechanics approach too, obtained closed-form expressions for the elastic properties of SWNTs under various loading conditions. The results from the above-mentioned studies agree well with the existing numerical results, but one needs much less computer time to yield them. Despite much effort being made in the above-mentioned studies, the closed-form expressions for the elastic properties of carbon nanotubes are confined to achiral nanotubes, i.e. armchair and zigzag nanotubes, till a recent paper by the authors (Chang *et al*. 2005*a*). In that paper, the axial elastic modulus and Poisson's ratio for chiral nanotubes were obtained using the ‘stick-spiral’ model. However, the closed-form expressions for the circumferential elastic properties and the shear modulus of chiral carbon nanotubes are still not available, because the governing equations of the ‘stick-spiral’ model are not fully established.

In this paper, complementary to the previous works, governing equations of the ‘stick-spiral’ model developed by Chang & Gao (2003) are fully established and closed-form expressions for Young's modulus, Poisson's ratio (along both axial and circumferential directions) and shear modulus of chiral nanotubes are obtained. The present governing equations may make it possible for many of the current molecular mechanics-based analytical works about achiral nanotubes, such as in Shen & Li (2004), Xiao *et al*. (2005), Chang *et al*. (2005*b*,*c*) and Shen & Li (2005*a*,*b*), to be extended to chiral nanotubes, and thus are very useful to the topic of linking molecular mechanics with continuum mechanics (Odegard *et al*. 2002; Leung *et al*. 2005).

## 2. Geometry of SWNTs

Carbon nanotubes can be classified into MWNTs and SWNTs. A MWNT consists of two or more concentric cylindrical shells of graphene sheets arranged coaxially around a central hollow with interlayer separation as in graphite (0.34 nm), whereas a SWNT is made of a single layer graphene cylinder.

A SWNT can be viewed as a graphene sheet rolled into a tube. In principle, infinite numbers of nanotube geometries can exist, because a graphene sheet can be rolled up with different angles. Different rolling angles result in different chiralities, or helicities, of SWNTs. A common approach is using a chiral vector or chiral angle to identify a SWNT. Figure 1 shows a schematic of a graphene sheet. A vector ** C** in the graphene plane can be described as a combination of base vectors

**and**

*a***of the hexagon by(2.1)with**

*b**n*and

*m*being two integers. If the head of the vector

**touches its tail when the graphene sheet is rolled into a tube, we call**

*C***the chiral vector, or roll-up vector of the nanotube. The magnitude of the chiral vector, , represents the circumference of the nanotube, where**

*C**r*

_{0}is the carbon–carbon bond length. A SWNT can thus be uniquely indexed by a pair of integers (

*n*,

*m*) (White

*et al*. 1993) to represent its chirality or helicity. The chirality of a SWNT can also be indicated by the chiral angle

*ϕ*(see figure 1), which is given by(2.2)The two limiting cases of nanotubes are (

*n*, 0) (whose chiral angle is 0) and (

*n*,

*n*) (whose chiral angle is

*π*/6), which are usually known as zigzag and armchair tubes based on the geometry of carbon bonds around the circumference of the nanotube. Zigzag and armchair tubes are achiral nanotubes because of the highly geometry symmetry, whereas SWNTs with a chiral angle of 0<

*ϕ*<

*π*/6 are chiral nanotubes.

Another important geometrical parameter of SWNTs is the translation vector ** T**, which is directed along the SWNT axis and perpendicular to the chiral vector

**(see figure 1). In the graphene plane,**

*C***is given by(2.3)The magnitude of the translation vector, , corresponds to the length of the SWNT unit cell (which is marked in grey in figure 1).**

*T*## 3. Governing equations of the stick-spiral model

In the framework of molecular mechanics, the total potential energy, *E*_{t}, of a SWNT at small strains can be expressed as a sum of energies associated with the variance of bond length, *U*_{ρ}, and bond angle, *U*_{θ}, in the form of Hooke's law (Chang & Gao 2003), i.e.(3.1)where d*r*_{i} is the bond elongation of bond *i* and d*θ*_{j} is the variance of bond angle *j* and *K*_{ρ} and *K*_{θ} are the related force constants.

Similar to the previous work by Chang & Gao (2003), we adopt the ‘stick-spiral’ model to analyse the equilibrium situation of the local structure of the SWNT. In the model, an elastic stick with an axial stiffness of *K*_{ρ} and an infinite bending stiffness is used to model the force–stretch relationship of the carbon–carbon bond, and a spiral spring with a stiffness of *K*_{θ} is used to model the twisting moment resulting from an angular distortion of the bond angle.

We consider a (*n*, *m*) SWNT subjected to an axial force *F*, an internal pressure *P* and an axial torque *M*_{T}, as shown in figure 2*a*. The relationships between the external forces and the internal forces yield(3.2)(3.3)(3.4)where *R* is the tube radius and *f*_{i} and *s*_{i} are forces contributed on carbon bonds along axial and circumferential directions, respectively (see figure 2*b*).

Equilibrium of the local structure of the SWNT needs(3.5)(3.6)

Force equilibrium to bond extension leads to(3.7)From the moment equilibrium of three bonds, we have(3.8)where *ω*_{ij}, the torsion angle between the plane though *r*_{i} parallel to the nanotube axis and the plane of *θ*_{j}, can be calculated by(3.9)The structural parameters *φ*_{i}, *ϕ*_{i} and *θ*_{i} are defined in figure 2.

The geometrical relationships of a SWNT satisfy(3.10)With the use of equations (3.9) and (3.10), the variation of bond angle can be obtained as(3.11)Substituting equations (3.11) into (3.8), the moment equilibrium equations can be rewritten as(3.12)with(3.13)

The cylindrical structure of a SWNT always needs its chiral vector to keep a closed ring, i.e. the dislocation between the head and the tail of the chiral vector should be zero no matter how the SWNT is deformed unless bond breaking occurs. This feature actually gives compatible equations of a deformed SWNT as follows:(3.14)

We now have 12 independent equations given by equations (3.2)–(3.6), (3.7), (3.12), (3.14), and 15 independent variables *F*, *P*, *M*_{T}, *f*_{1}, *f*_{2}, *f*_{3}, *s*_{1}, *s*_{2}, *s*_{3}, d*r*_{1}, d*r*_{2}, d*r*_{3}, d*ϕ*_{1}, d*ϕ*_{2}, d*ϕ*_{3}. We note here that *F*, *P* and *M*_{T} are applied external forces. Once these forces are given, the present problem is solvable. To obtain the closed-form expressions for elastic properties of carbon nanotubes, we can solve the problem in the following way. First, we chose d*ϕ*_{3} as self-variable, and then set two of the three applied external forces *F*, *P* or torque *M*_{T} to be zero. Thus, the other 12 parameters can be determined by the 12 governing equations as linear functions of d*ϕ*_{3}. The related elastic properties can thus be calculated with the use of these parameters. Details will be shown in the successive sections.

## 4. General solutions to the governing equations

We first introduce two series of internal forces *p*_{i} and *q*_{i}, via equations (3.7) and (3.12), such that(4.1)(4.2)

The internal forces *f*_{i} and *s*_{i} can thus be given by(4.3)

(4.4)

Equations (3.2)–(3.6) can be written as(4.5)(4.6)(4.7)(4.8)(4.9)Once two of the external forces (e.g. *P* and *M*_{T}) are known, *p*_{i} can be represented as the functions of *q*_{i} from the corresponding two equations of (4.5)–(4.7) (e.g. (4.6) and (4.7)) and (4.8) in the form of(4.10)We rewrite equation (3.14) and (4.9) as(4.11)(4.12)with(4.13)

(4.14)

(4.15)

If the assumption of *r*_{1}=*r*_{2}=*r*_{3}=*r*_{0} for the undeformed SWNT is used, from equations (4.2), (4.11) and (4.12), we can obtain(4.16)where(4.17)and(4.18)with *i*, *j*, *k* in clockwise order.

The parameters *p*_{i}, d*r*_{i}, *f*_{i} and *s*_{i} can then be calculated via equations (4.1), (4.3), (4.4) and (4.10), and another unknown external force (e.g. *F*) can be determined by the one unused equation of (4.5)–(4.7) (e.g. (4.5)).

## 5. Chirality- and size-dependent elastic properties of SWNTs

### (a) Longitudinal Young's modulus and Poisson's ratio

To obtain the longitudinal Young's modulus and Poisson's ratio of a SWNT, we assume that the SWNT is subjected only to an axial force *F*, thus we have internal pressure *P*=0 and axial torque *M*_{T}=0. Using equations (4.6)–(4.8), together with the definition of the matrix ** M** in (4.10), we have(5.1)

The axial and circumferential strains can be calculated by the geometrical deformation of the SWNT as follows:(5.2)

(5.3)

When a SWNT is treated as a cylindrical shell, the surface Young's modulus *Y*_{S} and Poisson's ratio can be defined by (Chang & Gao 2003)(5.4)(5.5)It should be noted that the self-variable d*ϕ*_{3} will be eliminated in the final expressions of Young's modulus and Poisson's ratio.

In the present analysis, for simplicity, we may take the angles *ϕ*_{i} for the undeformed (*n*, *m*) SWNT as follows:(5.6)This means that the SWNT is reasonably assumed to be formed by an ideal rolling from graphene sheet according to some existing results (Sanchez-Portal *et al*. 1999). The torsion angles *φ*_{i} can be calculated by(5.7)

With the use of equations (5.6) and (5.7), equations (5.4) and (5.5) will be simplified as(5.8)(5.9)with(5.10)

(5.11)

(5.12)

For a SWNT, the possible index numbers should be *n*≥3 and 0≤*m*≤*n*. We numerically find that(5.13)where the maximum value is reached for a (3, 1) tube. According to equation (5.13), equations (5.10)–(5.12) can be approximated by(5.14)(5.15)(5.16)with maximum error less than 0.5% for all parameters in the whole range of *n* and *m* pairs.

In particular, we can obtain the values of the above parameters for some limit cases as follows:(5.17)(5.18)(5.19)These values coincide with those given by Chang & Gao (2003). It should be noted that a limit analysis should be conducted when the present equations are used to obtain the elastic properties for a (*n*, 0) zigzag tube because of the occurring of cot *ϕ*_{1} in the element *M*_{11} of the matrix * M* (see equations (5.1)).

Figure 3*a*,*b* shows the present predictions of chirality- and size-dependent Young's modulus and Poisson's ratio of nanotubes. Here, *K*_{ρ}=742 nN N^{−1} m^{−1} and *K*_{θ}=1.42 nN nm (which were obtained by Chang & Gao (2003) from the graphite data of *Y*_{S}=0.36 TPa nm and *ν*=0.16) and *r*_{0}=0.142 nm are used. It is seen that Young's modulus increases while Poisson's ratio decreases with increasing tube diameter and increasing tube chiral angle. The smaller the tube diameter and the smaller the tube chiral angle, the stronger the dependence of Young's modulus and Poisson's ratio on the tube size and chirality. The chirality- and size-dependence may be ignored when the tube diameter is larger than 2.0 nm, and both the two elastic properties approach the limit values for graphite.

The present predictions for Young's modulus and Poisson's ratio are consistent with many of those given by various methods, such as tight bonding calculations (Hernandez *et al*. 1998; Sanchez-Portal *et al*. 1999), atomistic-based continuum analysis (Arroyo & Belytschko 2002) and the molecular mechanics approach (Li & Chou 2003*a*; Xiao *et al*. 2005). Because the present closed-form expressions give essentially identical results with those given by our previous work (Chang *et al*. 2005*a*), we do not repeat the discussions here.

### (b) Circumferential Young's modulus and Poisson's ratio

To obtain the circumferential elastic modulus of a SWNT, we assume that the SWNT is subjected only to an internal pressure *P*, thus we have *F*=0 and *M*_{T} =0. Using equations (4.5), (4.7) and (4.8), together with the definition of the matrix ** M** in (4.10), we obtain(5.20)

The circumferential surface Young's modulus and Poisson's ratio can be defined as(5.21)(5.22)where the axial strain *ϵ* and circumferential strain *ϵ*′ are calculated by (5.2) and (5.3). We note that the self-variable d*ϕ*_{3} will be eliminated in the final expressions of the circumferential Young's modulus and Poisson's ratio.

Again, we reasonably assume that the SWNT is formed by an ideal rolling from graphene sheet according to some existing results (Sanchez-Portal *et al*. 1999). With the use of equations (5.6) and (5.7), equations (5.21) and (5.22) will be simplified as(5.23)(5.24)Much to our surprise, *λ*, *η* and *ξ* can be expressed by (5.10)–(5.12) too. This means that Young's moduli and Poisson's ratios along axial and circumferential directions of a chiral SWNT are exactly the same. This implies that the in-plane elastic property of a SWNT is actually isotropic. In other words, the isotropy of a graphene sheet is maintained when it was rolled into a nanotube.

### (c) Longitudinal shear modulus

To obtain the longitudinal shear modulus of a SWNT, we assume that the SWNT is subjected only to a torque *M*_{T}, thus we have *F*=0 and *P*=0. Using equations (4.5), (4.6) and (4.8), together with the definition of the matrix ** M** in (4.10), we obtain(5.25)

The shear strain can be calculated by the geometrical deformation of the SWNT as(5.26)

When a SWNT is viewed as a continuum cylindrical shell, the surface shear modulus *G*_{S} can be defined by(5.27)We note that the self-variable d*ϕ*_{3} will be eliminated in the final expressions of shear modulus.

Again, we reasonably assume that the SWNT is formed by an ideal rolling from graphene sheet according to some existing results (Sanchez-Portal *et al*. 1999). With the use of equations (5.6) and (5.7), equations (5.27) will be simplified as(5.28)with(5.29)

(5.30)

In particular, we can obtain the values of the above parameters for some limit cases as follows:(5.31)

(5.32)

(5.33)

Figure 4*a* shows the predictions of the surface shear modulus *G*_{S}. It is seen that the surface shear modulus *G*_{S} decreases with decreasing tube diameter and decreasing tube chiral angle. The smaller the tube diameter and the larger the tube chiral angle, the stronger the dependence of the shear modulus on the tube size and chirality. When the tube diameter is larger than 2.0 nm, the values of the shear modulus approaches the limit value for graphite, and the chirality- and size-dependence may be ignored. A maximum reduction of *G*_{S} from its limit value is more than 20%, which may cause severe effects on the mechanical behaviour of carbon nanotubes. We note that the present predictions are in reasonable agreement with those given by Popov *et al*. (2000) using a lattice dynamics model.

A SWNT is often modelled as an continuum cylindrical shell, in which the discrete nature of atomic structure may be merged. In the theory of continuum mechanics, the shear modulus *G* can be related to Young's modulus *Y* and Poisson's ratio *ν* by(5.34)

Obviously, we can also obtain from equation (5.28) a surface shear modulus of a SWNT with the use of equations (5.8) and (5.9). In order to be distinguished from the one we obtained from molecular mechanics, *G*_{S}, we name the surface shear modulus based on the continuum mechanics theory as a *counterfeit* surface shear modulus, , which is given by(5.35)

The variation of the counterfeit surface shear modulus against the tube diameter is shown in figure 4*b*. It is seen that decreases with decreasing tube chiral angle, which is quite different from the surface shear modulus *G*_{S}. To evaluate the difference between and , we show the ratio of for SWNTs with different chiralities in figure 5. It is found that is larger than for SWNTs with larger chiral angles, while for SWNTs with smaller chiral angles, is smaller than . The maximum difference between and is about 15%. This finding indicates that the shear modulus cannot be directly calculated by equation (5.28). In other words, the classic relationship between the Young's modulus and the shear modulus is *not* retained for a SWNT due to the discrete nature of nanostructures, although the in-plane elastic property of a SWNT is isotropic. Therefore, there should be caution about the direct use of the relationships from the theory of continuum mechanics at nanoscales, in spite of many successful examples of modelling CNTs using continuum shell theory.

## 6. Conclusions

On the basis of the molecular mechanics approach, we have established governing equations for the so called ‘stick-spiral’ model to investigate chirality- and size-dependent elastic properties of SWNTs. Closed-from expressions were obtained for five elastic properties of a SWNT as functions of its atomic structure. The present model is useful to the topic of linking molecular mechanics with continuum mechanics, and helpful to the further understanding of elastic properties of carbon nanotubes. Some main conclusions are drawn as follows:

Young's moduli and Poisson's ratios along axial and circumferential directions of a chiral SWNT are exactly the same. This indicates that the in-plane elastic property of a SWNT is actually isotropic. In other words, the isotropy of a graphene sheet is maintained when it is rolled into a nanotube.

The smaller the tube diameter, the stronger the dependence of the elastic properties on the tube size and chirality. When the tube diameter is larger than 2.0 nm, the values of the elastic properties approach the limit values for graphite, and the chirality- and size-dependence may be ignored. The smaller the tube chiral angle, the stronger the size effects on Young's modulus and Poisson's ratio, but the weaker the size effect on the shear modulus. The deviations of Young's modulus and Poisson's ratio and the shear modulus from their limit values (i.e. for graphene sheets) are up to 8, 44 and 20%, respectively.

The classic relationship between Young's modulus and the shear modulus in the elastic theory of continuum mechanics is

*not*retained for a SWNT. If the classic formula is used, the shear modulus will be overestimated for SWNTs with larger chiral angles, but will be underestimated for those with smaller chiral angles. The maximum deviation of about 15% can be observed in the calculations. Hence, there should be caution about the direct use of some relationships from the continuum mechanics at nanoscales, in spite of many successful examples of modelling carbon nanotubes using continuum shell theory.

## Acknowledgments

We gratefully acknowledge the financial support from the National Natural Science Foundation of China (10402019, 10472061), Shanghai Rising-Star Program (05QMX1421), Shanghai Leading Academic Discipline Project (Y0103), Development Fund of Shanghai Committee of Education and Key Project of Shanghai Committee of Science and Technology (04JC14034).

## Footnotes

- Received October 2, 2005.
- Accepted January 19, 2006.

- © 2006 The Royal Society