## 1. Introduction

Since single-walled carbon nanotube (SWCNT) and multi-walled carbon nanotube (MWCNT) are found by Iijima (1991, 1993), these nanomaterials have stimulated extensive interest in the material research communities in the past decades. It has been found that carbon nanotubes possess many interesting and exceptional mechanical and electronic properties (Ruoff et al., 2003; Popov, 2004). Therefore, it is expected that they can be used as promising materials for applications in nanoengineering. In order to make good use of these nanomaterials, it is important to have a good knowledge of their mechanical properties.

Experimentally, Tracy et al. (1996) estimated that the Young’s modulus of 11 MWCNTs vary from 0.4TPa to 4.15TPa with an average of 1.8TPa by measuring the amplitude of their intrinsic thermal vibrations, and it is concluded that carbon nanotubes appear to be much stiffer than their graphite counterpart. Based on the similar experiment method, Krishnan et al. (1998) reported that the Young’s modulus is in the range of 0.9TPa to 1.70TPa with an average of 1.25TPa for 27 SWCNTs. Direct tensile loading tests of SWCNTs and MWCNTs have also been performed by Yu et al. (2000) and they reported that the Young’s modulus are 0.32-1.47TPa for SWCNTs and 0.27-0.95TPa for MWCNTS, respectively. In the experiment, however, it is very difficult to measure the mechanical properties of carbon nanotues directly due to their very small size.

Based on molecular dynamics simulation and Tersoff-Brenner atomic potential, Yakobson et al. (1996) predicted that the axial modulus of SWCNTs are ranging from 1.4 to 5.5 TPa (Note here that in their study, the wall thickness of SWNT was taken as 0.066nm); Liang & Upmanyu (2006) investigated the axial-strain-induced torsion (ASIT) response of SWCNTs, and Zhang et al. (2008) studied ASIT in multi-walled carbon nanotubes. By employing a non-orthogonal tight binding theory, Goze et al. (1999) investigated the Young’s modulus of armchair and zigzag SWNTs with diameters of 0.5-2.0 nm. It was found that the Young’s modulus is dependent on the diameter of the tube noticeably as the tube diameter is small. Popov et al. (2000) predicted the mechanical properties of SWCNTs using Born’s perturbation technique with a lattice-dynamical model. The results they obtained showed that the Young’s modulus and the Poisson’s ratio of both armchair and zigzag SWCNTs depend on the tube radius as the tube radius are small. Other atomic modeling studies include first-principles based calculations (Zhou et al., 2001; Van Lier et al., 2000; Sánchez-Portal et al., 1999) and molecular dynamics simulations (Iijima et al., 1996). Although these atomic modeling techniques seem well suited to study problems related to molecular or atomic motions, these calculations are time-consuming and limited to systems with a small number of molecules or atoms.

Comparing with atomic modeling, continuum modeling is known to be more efficient from computational point of view. Therefore, many continuum modeling based approaches have been developed for study of carbon nanotubes. Based on Euler beam theory, Govinjee and Sackman (1999) studied the elastic properties of nanotubes and their size-dependent properties at nanoscale dimensions, which will not occur at continuum scale. Ru (2000a,b) proposed that the effective bending stiffness of SWCNTs should be regarded as an independent material parameter. In his study of the stability of nanotubes under pressure, SWCNT was treated as a single-layer elastic shell with effective bending stiffness. By equating the molecular potential energy of a nano-structured material with the strain energy of the representative truss and continuum models, Odegard et al. (2002) studied the effective bending rigidity of a graphite sheet. Zhang et al. (2002a,b,c, 2004) proposed a nanoscale continuum theory for the study of SWCNTs by directly incorporating the interatomic potentials into the constitutive model of SWCNTs based on the modified Cauchy-Born rule. By employing this approach, the authors also studied the fracture nucleation phenomena in carbon nanotubes. Based on the work of Zhang (2002c), Jiang et al. (2003) proposed an approach to account for the effect of nanotube radius on its mechanical properties. Chang and Gao (2003) studied the elastic modulus and Poisson’s ratio of SWCNTs by using molecular mechanics approach. In their work, analytical expressions for the mechanical properties of SWCNT have been derived based on the atomic structure of SWCNT. Li and Chou (2003) presented a structural mechanics approach to model the deformation of carbon nanotubes and obtained parameters by establishing a linkage between structural mechanics and molecular mechanics. Arroyo and Belytschko (2002, 2004a,b) extended the standard Cauchy-Born rule and introduced the so-called exponential map to study the mechanical properties of SWCNT since the classical Cauchy-Born rule cannot describe the deformation of crystalline film accurately. They also established the numerical framework for the analysis of the finite deformation of carbon nanotubes. The results they obtained agree very well with those obtained by molecular mechanics simulations. He et al. (2005a,b) developed a multishell model which takes the van der Waals interaction between any two layers into account and reevaluated the effects of the tube radius and thickness on the critical buckling load of MWCNTs. Gartestein et al. (2003) employed 2D continuum model to describe a stretch-induced torsion (SIT) in CNTs, while this model was restricted to linear response. Using the 2D continuum anharmonic anisotropic elastic model, Mu et al. (2009) also studied the axial-induced torsion of SWCNTs.

In the present work, a nanoscale continuum theory is established based on the higher order Cauchy-Born rule to study mechanical properties of carbon nanotubes (Guo et al., 2006; Wang et al., 2006a,b, 2009a,b). The theory bridges the microscopic and macroscopic length scale by incorporating the second-order deformation gradient into the kinematic description. Our idea is to use a higher-order Cauchy-Born rule to have a better description of the deformation of crystalline films with one or a few atom thickness with less computational efforts. Moreover, the interatomic potential (Tersoff 1988, Brenner 1990) and the atomic structure of carbon nanotube are incorporated into the proposed constitutive model in a consistent way. Therefore SWCNT can be viewed as a macroscopic generalized continuum with microstructure. Based on the present theory, mechanical properties of SWCNT and graphite are predicted and compared with the existing experimental and theoretical data.

The work is organized as follows: Section 2 gives Tersoff-Brenner interatomic potential for carbon. Sections 3 and 4 present the higher order Cauchy-Born rule is constructed and the analytical expressions of the hyper-elastic constitutive model for SWCNT are derived, respectively. With the use of the proposed constitutive model, different mechanical properties of SWCNTs are predicted in Section 5. Finally, some concluding remarks are given in Section 6.

## 2. The interatomic potential for carbon

In this section, Tersoff-Brenner interatomic potential for carbon (Tersoff, 1988; Brenner, 1990), which is widely used in the study of carbon nanotubes, is introduced as follows.

Where

with the constants given in the following.

## 3. The higher order cauchy-born rule

Cauchy-Born rule is a fundamental kinematic assumption for linking the deformation of the lattice vectors of crystal to that of a continuum deformation field. Without consideration of diffusion, phase transitions, lattice defect, slips or other non-homogeneities, it is very suitable for the linkage of 3D multiscale deformations of bulk materials such as space-filling crystals (Tadmor et al., 1996; Arroyo and Belytschko, 2002, 2004a,b). In general, Cauchy-Born rule describes the deformation of the lattice vectors in the following way:

where

where

In order to alleviate the limitation of Cauchy-Born rule for the description of the deformation of curved atom films, we introduce the higher order deformation gradient into the kinematic relationship of SWCNT. The same idea has also been shown by Leamy et al. (2003).

From the classical nonlinear continuum mechanics point of view, the deformation gradient tensor
*infinitesimal* material line element

As in Leamy et al. (2003), by taking the finite length of the initial lattice vector

Assuming that the deformation gradient tensor

Retaining up to the second order term of

Comparing with the standard Cauchy-Born rule, it is obvious that with the use of this higher order term, we can pull the vector

## 4. The hyper-elastic constitutive model for SWCNT

With the use of the above kinematic relation established by the higher order Cauchy-Born rule, a constitutive model for SWCNTs can be established. The key idea for continuum modeling of carbon nanotube is to relate the phenomenological macroscopic strain energy density

Assuming that the energy associated with an atom

And

where

Based on the strain energy density

where

The corresponding strain energy density can also be rewritten as:

Where

denotes the total energy of the representative cell related to atom

We can also define the generalized stiffness

where the subscripts

From (14) and (15), the tangent modulus tensors can be derived as:

where

Just as emphasized by Cousins(1978a,b), Tadmor (1999), Zhang (2002c), Arroyo and Belytschko (2002a), since the atomic structure of carbon nanotube is not centrosymmetric, the standard Cauchy-Born rule can not be used directly since it cannot guarantee the inner equilibrium of the representative cell. An inner shift vector

Substituting (24) into

Then the modified tangent modulus tensors can be obtained as:

Where

(37) |

(40) |

(41) |

(42) |

where

## 5. Mechanical properties of SWCNTs

It is usually thought that SWCNTs can be formed by rolling a graphite sheet into a hollow cylinder. To predict mechanical properties of SWCNTs, a planar graphite sheet in equilibrium energy state is here defined as the undeformed configuration, and the current configuration of the nanotube can be seen as deformed from the initial configuration by the following mapping:

where

### 5.1. The energy per atom for graphene sheet and SWCNTs

First, based on the present model, the energy per atom of the graphite sheet is calculated and the value of -1.1801

The energy per atom as the function of diameters for armchair and zigzag SWNTs relative to that of the graphene sheet is shown in Figure 4. The trend is almost the same for both armchair and zigzag SWNTs. The energy per atom decreases with increase of the tube diameter with

For larger tube diameter, the energy per atom approaches that of graphite. On the whole, it can be shown that the energy per atom depends obviously on tube diameters, but does not depend on tube chirality. For comparison, the results obtained by Robertson et al. (1992) with the use of both empirical potential and first-principle method based on the same interatomic potential are also shown in Figure 4. It can be found the present results are not only in good agreement with Robertson’s results, but also with those obtained by Jiang et al. (2003) based on incorporating the interatomic potential (Tersoff-Brenner potential) into the continuum analysis.

Figure 5 shows the energy per atom for different chiral SWCNTs ((2n, n), (3n, n), (4n, n), (5n, n) and (8n, n)) as a function of tube radius relative to that of the graphene sheet. As is expected, the energy per atom of chiral SWCNTs decreases with increasing tube radius and the limit value of this quantity is -7.3756 eV when the radius of tube is large. From Figure 5, it can be clearly found again that the strain energy per atom depends only on the radius of the tube and is independent of the chirality of SWCNTs, which is similar to armchair and zigzag SWCNTs.

### 5.2. Young’s modulus and Poisson ratio for graphene sheet and SWCNTs

As shown by Zhang et al. (2002c), the Young’s modulus and the Poisson’s ratio of planar graphite can be defined from

For SWCNTs, we also use the above expressions to estimate their mechanical properties along the axial direction although the corresponding elasticity tensors are no longer isotropic as in planar graphite case. Note that all calculations performed here are based on the Cartesian coordinate system and the Young’s modulus

As for the graphite, the resulting Young’s modulus is 0.69TP (see the dashed line in Figure 6a), which agrees well with that suggested by Zhang et al (2002c) and Arroyo and Belytschko (2004b) based on the same interatomic potential (represents by the horizontal solid line in Figure 6a). The Poisson’s ratio predicted by the present approach is 0.4295 (see the dashed line shown in Figure 6c), which is also very close to the value of 0.4123 given by Arroyo and Belytschko (2004b) using the same interatomic potential.

As for armchair and zigzag SWCNTs, Figure 6a displays the variations of the Young’s modulus with different diameters and chiralities. It can be observed that the trend is similar for both armchair and zigzag SWNTs and the influence of nanotube chirality is not significant. For smaller tubes whose diameters are less than 1.3 nm, the Young’s modulus strongly depends on the tube diameter. However, for tubes diameters larger than 1.3 nm, the dependence becomes very weak. As a whole, it can be seen that for both armchair and zigzag SWNTs the Young’s modulus increases with increase of tube diameter and a plateau is reached when the diameter is large, which corresponds to the modulus of graphite predicted by the present method. The existing non-orthogonal tight binding results given by Hernández et al.(1998), lattice-dynamics results given by Popov et al. (2000) and the exponential Cauchy-Born rule based results given by Arroyo and Belytschko (2002b) are also shown in Figure 6a for comparison. Comparing with the results given by Hernández et al. (1998) and Popov et al. (2000), it can be seen that although their data are larger than the corresponding ones of the present model, the general tendencies predicted by different methods are in good agreement. From the trend to view, the present predicted trend is also in reasonable agreement with that given by Robertson et al. (1992), Arroyo and Belytschko (2002b), Chang and Gao (2003) and Jiang et al. (2003). As for the differences between the values of different methods, it may be due to the fact that different parameters and atomic potential are used in different theories or algorithms (Chang and Gao, 2003). For example, Yakobson’s (1996) result of surface Young’s modulus of carbon nanotube based on molecular dynamics simulation with Tersoff-Brenner potential is about 0.36TPa nm, while Overney’s (1993) result based on Keating potential is about 0.51 TP nm. Recent *ab initio* calculations by Sánchez-Portal et al.(1999) and Van Lier et al. (2000) showed that Young’s modulus of SWNTs may vary from 0.33 to 0.37TPa nm and from 0.24 to 0.40 TPa nm, respectively. Furthermore, it can be found that our computational results agree well with that given by Arroyo and Belytschko (2002b) with their exponential Cauchy-Born rule. They are also in reasonable agreement with the experimental results of

Figure 6b depicts the size-dependent Young’s moduli of different chiral SWCNTs ((2n, n), (3n, n), (4n, n), (5n, n) and (8n, n)). It can be seen that Young’s moduli for different chiral SWCNTs increase with increasing tube radius and approach the limit value of graphite when the tube radius is large. For a given tube radius, the effect of tube chirality can almost be ignored. The Young’s modulus of different chiral SWCNTs are consistent in trends with those for armchair and zigzag SWCNTs. For chiral SWCNTs, the trends of the present results are also in accordance with those given by other methods, including lattice dynamics (Popov et al., 2000) and the analytical molecular mechanics approach (Chang & Gao, 2003).

From Figure 6c, the effect of tube diameter on the Poisson’s ratio is also clearly observed. It can be seen that, for both armchair and zigzag SWNTs, the Poisson’s ratio is very sensitive to the tube diameters especially when the diameter is less than 1.3 nm. The Poisson’s ratio of armchair nanotube decreases with increasing tube diameter but the situation is opposite for that of the zigzag one. However, as the tube diameters are larger than 1.3 nm, the Poisson’s ratio of both armchair and zigzag SWNTs reach a limit value i.e. the Poisson’s ratio of the planar graphite. For comparison, the corresponding results suggested by Popov et al. (2000) are also shown in Figure 6c. It can be observed that the tendencies are very similar between the results given by Popov et al. (2000) and the present method although the values are different. Moreover, it is worth noting although many investigations on the Poisson’s ratio of SWNTs have been conducted, there is no unique opinion that is widely accepted. For instance, Goze et al. (1999) showed that the Poisson’s ratio of (10,0), (20,0), (10,0) and (20,0) tubes are 0.275, 0.270, 0.247 and 0.256, respectively. Based on a molecular mechanics approach, Chang and Gao (2003) suggested that the Poisson’s ratio for armchair and zigzag SWNTs will decrease with increase of tube diameters from 0.19 to 0.16, and 0.26 to 0.16, respectively. In recent *ab initio* studies of Van Lier et al. (2000), even negative Poisson’s ratio is reported.

It also can be seen from Figure 6c that the obtained Poisson’s ratio is a little bit high when tube diameter is less than 0.3nm. It may be ascribed to the fact that when tube diameter is less than 0.3nm, because of the higher value of curvature, higher order (

### 5.3. Shear modulus for SWCNTs

As for the shear moduli of SWCNTs, to the best of our knowledge, only few works studied this mechanical property systematically since it is difficult to measure them with experiment techniques. Most of these works focus only on the armchair and zigzag SWCNTs.(Popov et al., 2000; Li & Chou, 2003) Thus, the shear moduli of achiral (i.e., armchair and zigzag) SWCNTs are firstly investigated and compared with the existing results (Li & Chou, 2003) for validation of the present model. Then the shear modulus of SWCNTs with different chiralities including (2n, n), (3n, n), (4n, n), (5n, n) and (8n, n) are studied systematically. For determining the shear modulus of SWCNT, it is essential to simulate its pure torsion deformation which can be implemented by incrementally controlling

Figure 7a shows the variations of the shear modulus of achiral SWCNTs with respect to the tube radius. It can be found that shear modulus of armchair and zigzag SWCNTs increase with increasing tube radius and approach the limit value 0.24 TPa when the tube radius is large. It is also observed that, similar to the results given by Li & Chou (2003) and Xiao et al. (2005), the present predicted shear moduli of armchair and zigzag SWCNTs hold similar size-dependent trends and the chirality-dependence of shear moduli is not significant. Figure 7b shows the normalized shear moduli obtained with different methods. The normalization is achieved by using the values of 0.24 TPa and 0.48 TPa which are the limiting values of graphite sheet obtained by the present approach and molecular structural mechanics (Li & Chou, 2003), respectively. Although there is a discrepancy in limit values, it can be found that the size effect obtained by the present study is in good agreement with that of Li and Chou (2003). The difference among the limit values may be attributed to the different atomistic potential and/or force field parameters used in the computation model.

The size-dependent shear modulus of different chiralities SWCNTs are displayed in Figure 8. It is observed that, similar to achiral SWCNTs, the shear moduli of chiral SWCNTs increase with increasing tube radius and a limit value of 0.24 TPa is approaching when the tube radius (also n) is large. For (2n, n) SWCNT, the maximum difference of shear modulus is up to 42%. The dependence of tube chirality is not obvious for chiral SWCNTs. With reference to Figure 7a and Figure 8, it can be found that, at small radius (<1nm), the shear modulus of SWCNTs are sensitive to the tube radius, while at larger radius (>1nm), the size and chirality dependency can be ignored.

### 5.4. Bending stiffness for graphene sheet and SWCNTs

In present study, the so-called bending stiffness for graphene sheet refers to the resistance of a flat graphite sheet or the curved wall of CNT with respect to the infinitesimal local bending deformation. The bending stiffness for SWCNTs refers to the bending resistance of the cylindrical tube formed by rolling up graphite sheet with respect to the infinitesimal global bending deformation (see Figure 9 for reference). It should be pointed out that for the first definition, the bending stiffness is an intrinsic material property solely determined by the atomistic structure of the mono-layer crystalline membrane. The second definition, however, is a *structural property* which is determined not only by the bending stiffness of the single atom layer crystalline membrane*,* but also by the geometry dimensions, such as the diameter of the tube. Unfortunately, these two issues are not well addressed in the past literatures (Kudin et at., 2001; Enomoto et al., 2006).

Based on the higher order Cauchy-Born rule and Equation (51), the strain energy per atom (energy relative to a planar graphite sheet) as a function of the radius of bending curvature can

be obtained. By fitting the data of the strain energy and the bending curvature radii with respect to the equation
^{2}/atom, which is almost independent of its rolling direction. This indicates that the flat graphite sheet is nearly isotropic with regard to bending. The current result agrees well with the effective bending stiffness of graphite sheet 2.20 eVÅ^{2}/atom reported by Arroyo and Belytschko (2004a) with membrane theory and the same interatomic potential under the condition of infinitesimal bending. It is also in good agreement with the result of 2.32 eVÅ^{2}/atom obtained by Robertson et al. (1992) with atomic simulations.

To explore the effective bending stiffness of carbon nanotube based on the higher order Cauchy-Born rule, the following map is used to describe the pure bending deformation of the tube

where

Figure 10 show the bending strain energy of zigzag (10,0) SWCNT as a function of bending angle. Here the bending strain energy is defined as the difference between the energy of the deformed tube and that of its straight status. It can be found that the present results obtained with much less computational effort are in good agreement with those of MD simulations.

where

Once the bending strain energy

Just like the derivation of the bending stiffness of the flat graphite sheet, here no representative thickness of the tube is required to obtain the effective bending stiffness of CNTs.

## 6. Conclusion

In this charpter, a higher order Cauchy-Born rule has been constructed for studying mechanical properties of graphene sheet and carbon nanotubes. In the present model, by including the second order deformation gradient tensor in the kinematic description, we can alleviate the limitation of the standard Cauchy-Born rule for the modeling of nanoscale crystalline films with less computational efforts. Based on the established relationship between the atomic potential and the macroscopic continuum strain energy, analytical expressions for the tangent modulus tensors are derived. From these expressions, the hyper-elastic constitutive law for this generalized continuum can be obtained.

With the use of this constitutive model and the Tersoff-Brenner atomic potential for carbon, the size and chirality dependent mechanical properties (including strain energy, Young’s modulus, Poisson’s ratio, shear modulus, bending stiffness) of graphene sheet and carbon nanotube are predicted systematically. The present investigation shows that except for Poisson’s ratio other mechanical properties (such as Young’s modulus, shear modulus, bending stiffness and so on) for graphene sheet and SWCNTs are size-dependent and their chirality-dependence is not significant. With increasing of tube radius, Young’s modulus and shear modulus of SWCNTs increase and converge to the corresponding limit values of graphene sheet. As for Poisson’s ratio, it can be found that it is very sensitive to the radius and the chirality of SWCNTs when the tube diameter is less than 1.3 nm. The present results agree well with those obtained by other experimental, atomic modeling and continuum concept based studies.

Besides, the present work also discusses some basic problems on the study of the bending stiffness of CNTs. It is pointed out that the bending stiffness of a flat graphite sheet and that of CNTs are two different concepts. The former is an intrinsic material property while the later is a structural one. Since the smeared-out model of CNTs is a generalized continuum with microstructure, the effective bending stiffness of it should be regarded as an independent structural rigidity parameter which can not be determined simply by employing the classic formula in beam theory. It is hoped that the above findings may be helpful to clarify some obscure issues on the study of the mechanical properties of CNTs both theoretically and experimentally.

It should be pointed out that the present method is not limited to a specific interatomic potential and the study of SWCNTs. It can also be applied to calculate the mechanical response of MWCNTs. The proposed model can be further applied to other nano-film materials. The key point is to view them as generalized continuum with microstructures.