A molecular-mechanics based finite element model for strength prediction of single wall carbon nanotubes

Materials Science and Engineering A 454–455 (2007) 170–177

A molecular-mechanics based finite element model for strength prediction of single wall carbon nanotubes M. Meo ∗ , M. Rossi Material Research Center, Department of Mechanical Engineering, University of Bath, Bath, Ba2 7AY, UK Received 1 August 2005; accepted 1 November 2006

Abstract The aim of this work was to develop a finite element model based on molecular mechanics to predict the ultimate strength and strain of single wallet carbon nanotubes (SWCNT). The interactions between atoms was modelled by combining the use of non-linear elastic and torsional elastic spring. In particular, with this approach, it was tried to combine the molecular mechanics approach with finite element method without providing any not-physical data on the interactions between the carbon atoms, i.e. the CC-bond inertia moment or Young’s modulus definition. Mechanical properties as Young’s modulus, ultimate strength and strain for several CNTs were calculated. Further, a stress–strain curve for large deformation (up to 70%) is reported for a nanotube Zig-Zag (9,0). The results showed that good agreement with the experimental and numerical results of several authors was obtained. A comparison of the mechanical properties of nanotubes with same diameter and different chirality was carried out. Finally, the influence of the presence of defects on the strength and strain of a SWNT was also evaluated. In particular, the stress–strain curve a nanotube with one-vacancy defect was evaluated and compared with the curve of a pristine one, showing a reduction of the ultimate strength and strain for the defected nanotube. The FE model proposed demonstrate to be a reliable tool to simulate mechanical behaviour of carbon nanotubes both in the linear elastic field and the non-linear elastic field. © 2007 Published by Elsevier B.V. Keywords: Carbon nanotubes; Molecular mechanics; Young’s modulus; Mechanical properties; Finite element analysis

1. Introduction Since their discovery carbon nanotubes [1] have attracted considerable attention in scientific communities. This is partly due to their remarkable mechanical, electrical and thermal properties. In particular, material composites such as carbon nanotube, nanoparticle-reinforced polymers and metals have shown potentially wide application. Specifically to mechanical properties, single wall nanotubes (SWNTs) have the highest Young’s modulus about 1 TPa, if normalized to their diameter, and this is one of the main reason why carbon nanotubes (CNTs) have attracted much interest for low weight structural composites [2]. A detailed summary of CNTs mechanical properties can be found in [3]. A Young’s modulus for SWNTs and multi wall nanotubes (MWNTs) was reported to be 1.25 TPa, while a Poisson

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ratio around 0.14–0.28 was reported depending on the approach and the energy potential used. Further, experimental data of 15 SWNT bundles under tensile load showed that, the Young’s modulus ranged from 0.32 up to 1.47 TPa with an average of 1.02 TPa. The tensile strength ranged from 13 to 53 GPa [4]. In the case of MWNTs a Young’s modulus of 0.9 TPa was estimated by conducting pulling and bending tests [5]. Computational simulation for predicting mechanical properties of CNTs has been recognised to be a powerful tool to overcome the difficulties arising from the measurements of nanoscale dimensions. Several approaches can be used to evaluate the mechanical properties of SWNT and MWNT [6]. Xiao et al. [8] found a tensile strength for Armchair (126.2 GPa) and Zig-Zag (94.5 GPa) with a maximum strain of 23.1% and 15.6–17.5%, respectively. Natsuki and Endo [12], predicted the maximum stress to be around 70 GPa at 11% of strain for the ZigZag nanotube and 88 GPa at 15% for Armchair. Sun [13] found a tensile strength from 77 GPa (Zig-Zag) up to 101 GPa (Armchair). Further, an independence of the tensile strength from nanotube diameter was found.

M. Meo, M. Rossi / Materials Science and Engineering A 454–455 (2007) 170–177

In [15], an atomistic simulation of nanotube fracture was carried out. The relationship between CC-bond stretching curve shape and nanotube fracture was investigated. Further, the influence of one molecular structure vacancy defect on the stress–strain curve, was calculated. Specifically, a reduction of tensile strength and strain, referring to pristine nanotube, was observed. Finally, a moderate chirality effect on the tensile strength was found. The molecular structure vacancy defect can occur, for example, in a TEM environment [15], after a high-dose irradiation aiming a desired functionality or at the purification stage [16]. In [16], a study on SWNT mechanical properties with a single, a double and a triple vacancy, was proposed. Further, is stated here that, carbon nanotube show the ability to heal vacancies by saturating dandling bonds. From the calculation proposed, this effect seems to reduce the vacancy defect influence on nanotube mechanical properties. In [17], the effect of single and double vacancies on nanotube mechanical properties, was investigated. A reduced failure stress and strain was observed due to the vacancy defect. In the present work, a finite element model, based on molecular mechanics theory, was used to predict the strength of SWNT. The elements chosen were both non-linear elastic springs and linear elastic torsional spring. By the non-linear elastic spring elements proposed, it was tried to overcome the lack of information about the sectional properties of carbon–carbon bond. Moreover, by linear elastic torsional springs, it was tried to overcome the not rigorously definition in bond angles bending typical of the beam-like approach [18]. Using this model, the SWNT ultimate mechanical properties were calculated under uniaxial tension. This study was performed on several kinds of nanotubes with different chirality and size. The effect of nanotube diameter and chirality on the tensile stress and related strain was calculated. Finally, the influence of the vacancy defect on the mechanical properties was investigated.

Fig. 1. 2D graphene sheet with nanotube parameters.

“Chiral nanotubes”. An helicity angle is defined as: √ 3m tan θ = 2n + m

dt =

L π

(4)

√ where a is the unit vector length and a = 3acc with acc = 0.1421 nm. Single wall carbon nanotube’s unit cell is defined with an honeycomb lattice ideal cutting along the line OAB’B defined

A single wall nanotube (SWNT) can be viewed as a rolled graphene sheet. A complete description of the relations governing the geometry of SWNT and a graphical description (Fig. 1) of the variables involved can be found in [19].Starting from a graphene sheet two lattices vectors a1 and a2 are defined. The chiral vector Ch is defined in terms of the integers (n, m) and the basis vectors a1 and a2 of the honeycomb lattice: (1)

Generally, n > m is used and the remaining SWNT’s geometrical characteristics are defined starting from chiral index n and m. When (n, n) or (n, 0) pair is chosen to describe the SWNT, a particular configuration come out as showed in Fig. 2. The structure with (n, n) is usually labelled as Armchair, while the structure with (n, 0) is usually labelled as Zig-Zag. SWNTs with Armchair or Zig-Zag structure are generally named “achiral nanotubes”. Nanotubes risen from (n, m) pair are labelled as

(2)

It is simple to evaluate that θ = 0◦ for Zig-Zag configuration and θ = 30◦ for Armchair configuration. The nanotubes’ circumference and diameter are defined as:  L = a n2 + nm + m2 (3)

2. Geometry

Ch = n × a1 + m × a2

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Fig. 2. Carbon nanotube: (a) Armchair; (b) Zig-Zag; (c) Chiral.

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by Ch vector (defined above) and the T vector, T = t1 × n + t 2 × m

(5)

with t1 and t2 integer defined as: t1 =

2m + n dr

t2 = −

2n + m dr

(6) (7)

where dr is the highest common divisor of (2m + n, 2n + m). After the ideal cutting the honeycomb can be ideally rolled to create a nanotube unit cell. Once the unit cell is obtained, it can be translated along the T vector to create chosen length nanotube. A simple rule to relate honeycomb lattice’s Cartesian coordinate and SWNT from this generated is given in [20].  x x  (X, Y, Z) = R cos , R sin ,y (8) R R where X, Y, Z are the nanotube coordinates, x, y are the graphene coordinates and R is nanotube radius. 3. Molecular mechanics theory In order to predict the SWNT mechanical properties, the Molecular Mechanics theory can be used. Molecular Mechanics studies the mechanics of atomic nuclei moving around, in molecule or in an assembly of molecules and no attention is paid explicitly to the movements of electron. The total force on each atomic nuclei or nucleus is the sum of the force generated by the electrons and electrostatic force between the positively charged nuclei themselves. A nucleus can be considered as a material point. The potential energy depends solely on the relative positions of nuclei and the corresponding force field acting on the moving nuclei is called the molecular force field. For a random assembly of molecules, many physical quantities cannot be expressed analytically and to evaluate them, the motion of the whole system under an assumed potential function can be simulated according to the classical or the quantum mechanical equation of motion. This approach has been taken to form a large branch of computational molecular science, i.e. molecular dynamics, which is traditionally distinguished from molecular mechanics [21]. In its general formula, the potential energy (Fig. 3) is described as:       V= Vr + VΘ + Vϕ + Vω + VvdW + Vel

Fig. 3. Interatomic interaction in molecular mechanics theory. Ball and stick representation.

In some paper regarding CNTs, the effects of Vϕ , Vω , VvdW , are neglected under the uniaxial loading and small strain hypothesis [7,8,23,24]. Regarding the bond stretching, the accurate shapes of the experimental bond energy curve of diatomic molecules, have become available from the spectroscopy. An attempt to represent the experimentally determined bond energy curves of diatomic molecules by simple analytical functions is the Morse function [21]: Vr = Dije [e(−2aij rij ) − 2e(−aij rij ) ]

(10)

where Dije represents the energy required to stretch the bond rij from its equilibrium distance to infinity, rij is the bond length variation and aij is an empirical coefficient. From this, another expression for the Morse potential with parameters for hybridised sp2 bonds is given: Vr = De {[1 − e−β(r−r0 ) ]2 − 1}

(11)

where r0 is the bond equilibrium length, De is the dissociation energy and β is a fitting parameter. The following parameters were used both in [15] and in the present work: r0 = 0.139 nm, De = 6.03105 × 10−10 N nm, β = 2.625 × 1010 m−1 = 26.25 nm−1 . For the C–C–C bond angle energy this expression is given: Vθ = 21 kθ (θ)2 [1 + ksextic (θ)4 ]

(12)

with: kθ = 0.9 × 10−18 N m/rad2 , θ = θ − θ 0 θ 0 = 2.094 rad, ksextic = 0.754 rad−4 [15].

with

4. Carbon–carbon bond—molecular mechanics based finite element approach

(9) where Vr is the bond stretching, VΘ is the bond angle bending, Vϕ is the dihedral angle torsion, Vω is the inversion terms, VvdW is the Van der Walls interaction and Vel is the electrostatic interaction. Various functional forms may be used for these potential energy terms depending on the particular material and loading conditions considered [22].

Based on the theory mentioned before, a carbon–carbon bond was modeled. The model was built using a finite element commercial code. For the axial stretching the Morse potential with the parameter set described in [15] was used (11). On deriving this expression it is possible to have the relation force/bond length variation: F (r) = 2βDe (1 − e−βr )e−βr

(13)

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Fig. 4. Carbon–carbon bond: (a) physical model and (b) FE model.

Fig. 7. Momentum vs. rotation curve calculated using a torsional spring.

Fig. 5. Force vs. displacement curve calculated using the Morse potential.

An elastic non-linear spring element [25] was chosen to simulate the carbon–carbon stretching behaviour described from Eq. (11). This element, normally, has three degrees of freedom for each node (i, j): displacement Ux , displacement Uy , displacement Uz . In this study, the degrees of freedom (DOF) were extended from three to six, including the rotational DOF: θ x , θ y , θ z (Fig. 4). This was useful to link together non-linear spring element with the torsional spring, as will be later explained. Moreover, because the chemical bond always remains straight regardless to the applied loads, the spring bending had to be neglected. In Fig. 5, it is reported the force displacement curve used, obtained from Eq. (11), with the results of the simulated behaviour superimposed. On deriving Eq. (12) it was possible to find the relationship between momentum and C–C–C angle variation reported below: M(θ) = kθ θ[1 + 3ksextic (θ)4 ]

(14)

To simulate the bending behaviour a linear elastic torsional spring was used. The proper bending stiffness was set according to Belytschko [15]. Fig. 6 shows the finite element model for angle bending simulation. This element is defined by two nodes

Fig. 6. Torsional elastic spring between two non-linear elastic spring.

i, k and has 6 degree of freedom for each node (Ux , Uy , Uz , Rot X, Rot Y, Rot Z). A simulation of angle bending variation θ in x–y plane was performed. In Fig. 7, the momentum/rotation curve obtained by Eq. (14) is shown, with the results of the simulated behaviour superimposed. It is obvious from the diagram, that there is an underestimation of the analytical value, especially for large angle variation. With the proper set of constant, it is possible to simulate the other bond deformations, as torsion and out-of-plane angle variations, but as several author stated [6–8] this effect is negligible in small strain conditions. Moreover, in [8,12,20], these effects were neglected also for carbon nanotube ultimate strength evaluation. Also in this work, the torsional potential, the out-of-plane potential, the Van der walls forces and electrostatic forces were not considered. This assumption was confirmed by some numerical tests performed to evaluate the effective negligible influence of these parameters on the simulation results. 5. Single wall carbon nanotube model The next step of this work, was the construction of a carbon nanotube model based on the proposed modelling technique. A MATLAB® software was developed to build Armchair, Zig-Zag and Chiral nanotubes geometry. The coordinates generated by the program were used to build nanotubes finite element model. In Fig. 8, two representative nanotube models are represented. 6. SWCNT model Young’s modulus evaluation In order to be validate the model proposed, various simulations were performed to calculate the Young’s modulus of

Fig. 8. (a) Nanotube (15,0) Zig-Zag and (b) nanotube (15,15) Armchair.

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Fig. 9. Nanotube top view. Cross-sectional area definition.

several carbon nanotubes and then the results were compared with experimental data. To calculate the Young’s modulus, one nanotube extremity was totally restrained and on the opposite extremity a displacement was imposed. The nanotube Young’s modulus was evaluated with the equation:  σ Ftot /A0 EYoung = = (15) ε L/L0 where Ftot is the sum of the reaction force generated after the displacement imposed, A0 = πDn t (with Dn = nanotube diameter, thickness t = 0.34 nm interlayer graphite distance), L0 is the nanotube length and L is the displacement imposed. The cross sectional area is considered the same of a hollow cylinder (Fig. 9). With the approach mentioned above, several carbon nanotubes under uniaxial load were simulated. In agreement with WenXing et al. [11], a length to radius ratio smaller than 10 could affect the results, thus, most of the tests were performed with a ratio greater, to limit the edge effects. The results (average ±standard deviation) were: 0.920 ± 0.005 TPa for Armchair nanotube, 0.912 ± 0.009 TPa for Zig-Zag nanotube and 0.915 ± 0.009 TPa for Chiral nanotube. These results were in good agreement with the result of Natsuki and Endo [12] where, an elastic modulus equal to 0.94 TPa for the nanotubes (10,10) and (17,0), was calculated. Further, a good agreement was found with the results of WenXing et al. [11] where, an average equal to 929.8 ± 11.5 GPa is reported. This average was calculated on every nanotubes under investigation. 7. SWCNT model tensile strength evaluation By the model proposed, the ultimate strength of several carbon nanotubes was calculated. Looking at the CC-bond stretching curve (Fig. 5), the strain at which the tensile force is at its maximum will be labelled as inflection point [15]. The

Fig. 10. Nanotube Armchair (12,12) under uniaxial load.

strain at which the tensile force vanishes will be called as bondbreaking strain. Moreover, as stated in [15], the failure process can be considered completed when all bonds around the nanotube circumference are broken. But, on the other hand, once a single bond failed, no equilibrium solution can be easily found. Further, it was found, that the shape of the CC-bond stretching curve after the inflection point does not affect the fracture behaviour. Based on these observations, in the present work, it was assumed that the nanotube fracture starts when one or more bonds overcome the inflection point. In fact, looking at the stretching curve, if a force greater than the maximum is applied, no equilibrium force reaction can be generated from the bond. To calculate the ultimate tensile strength, one nanotube extremity was totally restrained and on the opposite extremity, a displacement was imposed. The reaction forces were stored. The stress value, for each displacement, was calculated by the cross sectional area described in the previous paragraph. In order to be confident with the hypothesis described above, an Armchair carbon nanotubes (12,12) was tested, and the results compared with the work of Belytschko [15] and Liew [26]. The stress–strain diagram is reported in Fig. 10, both the ultimate strength and curve shape are in good agreement with [15] and [26]. After the reliable results obtained for the nanotube (12,12) other nanotubes were tested. The results are summarized in Tables 1 and 2. The results showed no tensile strength dependence on SWNT radius. On the other hand, the Zig-Zag configuration lead to a less tensile strength value than the Armchair configuration. These results are in agreement with [13]. Further, in Fig. 11,

Table 1 Armchair nanotube ultimate properties Chiral index (n, m)

Diameter (nm)

Lenght (nm)

L/d

Ultimate strength (GPa)

Strain (%)

Armchair (5,5) (6,6) (7,7) (10,10) (12,12)

0.678 0.814 0.949 1.360 1.628

14.644 17.106 19.567 13.475 7.261

∼21.6 ∼21.0 ∼20.6 ∼9.9 ∼4.5

117.3 117.1 121.3 117.2 117.9

19.95 19.60 20.68 20.00 20.00

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Table 2 Zig-Zag nanotube ultimate properties Chiral index (n, m)

Diameter (nm)

Length (nm)

L/d

Ultimate strength (GPa)

Strain (%)

Zig-Zag (5,0) (7,0) (8,0) (9,0) (10,0) (15,0) (20,0)

0.391 0.548 0.626 0.705 0.783 1.178 1.566

8.313 11.723 13.286 14.707 16.697 12.576 12.434

∼21.0 ∼21.4 ∼21.2 ∼20.8 ∼21.3 ∼10.8 ∼8.0

93.8 94.4 94.7 94.3 94.7 94.5 94.5

17.20 18.00 18.40 18.40 18.40 18.00 18.00

Table 3 Nanotube ultimate properties results Reference number

Method

SWNT chiral index

Ultimate strength (GPa)

Ultimate strain

Present work Present work Present work Present work [8] [8] [12] [12] [17] [17] [26] [26]

MM MM MM MM MM MM MM MM DFT DFT MD MD

5,5 10,10 10,0 20,0 12,12 20,0 10,10 17,0 5,5 10,0 10,10 12,12

117.3 117.2 94.7 94.5 126,2 93,5 88 70 110 105 140.4 ∼110

0.199 0.200 0.184 0.180 0.231 0.156 0.15 0.11 0.3 0.20 0.280 ∼0.20

MD, molecular dynamics; MM, molecular mechanics; DFT, density functional theory.

the stress–strain diagram of two nanotubes with different chirality and similar radius are reported and compared. Due to a more accurate (and time consuming) simulation, it is possible to observe for the nanotube (5,5) a non-linear behaviour at the end of the curve. The same behaviour was observed in [26] with a molecular dynamics simulation. Yet, it is important to point out that in [26] the Stone-Wales defect was taken in consideration. In the present work, this kind of defect was not considered according to the work of [16]. The results carried out by the simulation performed, are in good agreement with the results of other author. Specifically, a comparison between the results of other authors and the present work is reported in Table 3.

Fig. 11. Stress/strain curve of nanotubes (5,5) and (9,0).

8. Stress/strain curve after maximum strength point Further simulations were conducted to the study the stress–strain curve behavior after the maximum strength in order to validate the assumption that the fracture starts when the first bond overcome the inflection point. A global test was performed on a Zig-Zag nanotube (9,0). It was performed a purely static computation as other authors did with a different Finite Element approach [18]. The results are shown in Fig. 12. The results are in good agreement with [18] where a nanotube (17,0) was deformed up to 100%. A similar maximum stress was found after a results normalization to the nanotube section.

Fig. 12. Stress/strain curve of nanotube 9,0 up to 70% of deformation.

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ied. Further, estimation of MWNTs mechanical properties are already under investigation. 10. Conclusion

Fig. 13. Stress/Strain curve of nanotube 5,5 Pristine and 5,5 with a one-vacancy defect.

9. Vacancy defect effect on mechanical properties Vacancy defects in a nanotube structure can occur due to various reasons such as a TEM environment [15], an electron irradiaton [16] or oxidative purification [17]. Various studies by different approach were carried out by several authors to understand the influence of this kind of defect on the SWNT mechanical properties. Specifically, in [15] the same Young’s modulus and lower failure stress and strain was found for a one-vacancy-defect nanotube compared with a pristine one. In [16], a study on the effects of a single, double and triple vacancy on the mechanical properties, was developed. Further, it was stated that CNT show the ability to heal vacancies by saturating dandling bonds. So, the study was developed on reconstructed and non-reconstructed vacancies and it was found that vacancy reconstruction, partially reduce the tensile strength loss compared to non-reconstructed vacancy defect. In the present work, the non-reconstructed one-vacancy defect effect was studied. One carbon atom and related bonds, were removed from the nanotube structure. The same uniaxial loading test was performed both on the pristine nanotube and on a one-vacancy defect nanotube. The results were stored and compared. In Fig. 13, it is possible to observe the stress/strain curve for a 5,5 pristine nanotube and one-vacancy defect 5,5 nanotube. As it is possible to observe, the curve of the pristine and defected nanotube are superimposed. No significant Young’s modulus change was found. A tensile strength reduction from 117.3 GPa (at strain 0.199) to 89.8 GPa (at strain 0.123) was found. Although for a (20,0) nanotube, the same curve superimposition was found in [15]. Focusing the attention to the properties reduction, in [16] a reduction of 15% and 30% of the maximum tensile strength and strain of a one-vacancy defect (5,5) nanotube was found. In the present work a reduction of 23% was found for the tensile strength and 38% for the critical strain. Further, it is possible to observe in [16] that reconstructed and no-reconstructed mono-vacancy has a similar effect on nanotube mechanical properties. With the model proposed, the influence of different kind of defects on the nanotubes’ mechanical properties will be stud-

A new finite element approach, based on the molecular mechanics theory and on the use of non-linear and torsional springs, was proposed. With this approach, it was tried to implement the molecular mechanics theory in a finite element code without any not-physical hypothesis, i.e. the CC-bond inertia moment or Young’s modulus definition. The model proposed was able to simulate carbon nanotube behaviour under uniaxial stress both in elastic field and up to fracture. Further, it was possible to study the effects of atomic structure defect on the mechanical properties. Based on the assumption that, the fracture phenomena starts when one or more bonds overcome the inflection point, the ultimate strength and strain was calculated for several both Armchair and ZigZag nanotubes. The results found, were in good agreement with other works on the same subject [8–12,17,26] even if different approaches were used. Moreover, with the model proposed, it was possible to deform one nanotube up to 70%. The results found were in agreement with Nasdala and Ernst [18]. The only difference we have found, comparing the results of the present work to literature, was that the difference between the ultimate strain for Armchair and Zig-Zag nanotube was only 2%. Finally, a one-vacancy defect was introduced in a pristine nanotube and its effect on the mechanical properties studied. A ultimate strength and strain lowering was found for defected nanotube, according to the work of [15,17]. The difference between theoretical and experimental data could be caused by atomic clamp slippage during the test or by defect induced in the structure [15]. Acknowledgements We are very grateful to Mr. G. Zumpano for his useful technical advices. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [15]

S. Iijima, Nature 354 (1991) 56–58. J. Robertson, Mater. Today 7 (10) (2004) 46–52. J.D. Salvetat, A. Rubio, Carbon 40 (2002) 1729–1734. M.F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Phys. Rev. Lett. 84 (24) (2000) 5552–5555. B.G. Demczyk, Y.M. Wang, J. Cumings, M. Hetman, W. Han, A. Zettl, R.O. Ritchie, Mater. Sci. Eng. A A334 (2002) 173–178. T. Natsuki, K. Tantrakan, M. Endo, Appl. Phys. A 79 (2004) 117–124. T. Chang, H. Gao, J. Mech. Phys. solids 51 (2003) 1059–1074. J.R. Xiao, B.A. Gama, J.W. Gillespie jr, Int. J. Solids Struct. 42 (2005) 3075–3092. C. Li, T.W. Chou, Int. J. Solids Struct. 40 (2003) 2487–2499. C. Li, T.W. Chou, Compos. Sci. Technol. 63 (2003) 1517–1524. B. WenXing, Z. ChangChun, C. WanZhao, Physica B 352 (2004) 156– 163. T. Natsuki, M. Endo, Carbon 42 (2004) 2147–2151. X. Sun, W. Zhao, Mater. Sci. Eng. A 390 (2005) 366–371. T. Belytschko, S.P. Xiao, G.C. Schatz, R.S. Ruoff, Phys. Rev. B 65 (235430) (2002) 1–8.

M. Meo, M. Rossi / Materials Science and Engineering A 454–455 (2007) 170–177 [16] M. Sammalkorpi, A. Krasheninnikov, A. Kuronen, K. Nordlund, K. Kaski, Phys. Rev. B 70 (2004) 245416. [17] S.L. Mielke, D. Troya, S. Zhang, Li Je-Luen, S. Xiao, R. Car, R.S. Ruoff, G.C. Schatz, T. Beliytschko, Chem. Phys. Lett. 390 (2004) 413–420. [18] L. Nasdala, G. Ernst, Comput. Mater. Sci. 33 (2005) 443–458. [19] M.S. Dresselhaus, G. Dresselhaus, R. Saito, Carbon 33 (7) (1995) 883–891. [20] J. Koloczek, K. Young-Kyun, A. Burian, J. Alloys Compd. 328 (2001) 222–225. [21] K. Machida, Principles of Molecular Mechanics, Wiley ed., Wiley and Kodansha, 1999.

177

[22] A.K. Rappe’, C.J. Casewit, Molecular Mecahnics Across Chemistry, University Science Books ed., University Science Books, Suasalito, CA, 1997. [23] K.L. Lau, M. Chipara, H.Y. Ling, D. Hui, Composites B: Eng. 35 (2004) 95–101. [24] G.M. Odegard, T.S. Gates, L.M. Nicholson, K.E. Wise, Compos. Sci. Technol. 62 (2002) 1869–1880. [25] ANSYS Inc. Theory Manual, SAS IP Inc., 2003. [26] K.M. Liew, X.Q. He, C.H. Wong, Acta Mater. 52 (2004) 2521–2527.