nanomechanics modeling and simulation of carbon nanotubes
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Nanomechanics Modeling and Simulationof Carbon Nanotubes
Xi Chen1 and Yonggang Huang2
Abstract: Carbon nanotubes �CNTs� have been perceived as having a great potential in nanoelectronic and nanomechanical devices.Recent advances of modeling and simulation at the nanoscale have led to a better understanding of the mechanical behaviors of carbonnanotubes. The modeling efforts incorporate atomic features into the continuum or structural mechanics theories, and the numericalsimulations feature quantum mechanical approach and classical molecular dynamics. Multiscale and multiphysics modeling and simula-tion tools have also been developed to effectively bridge the different lengths and time scales, and to link basic scientific research withengineering application. The general approaches of the theoretical and numerical nanomechanics of CNTs are briefly reviewed. This paperis not intended to be a comprehensive review, but to introduce readers �especially those with traditional civil engineering or engineeringmechanics backgrounds� to the new, interdisciplinary, or emerging fields in engineering mechanics, in this case the rapidly growingfrontier of nanomechanics through the example of carbon nanotubes.
DOI: 10.1061/�ASCE�0733-9399�2008�134:3�211�
CE Database subject headings: Simulation; Models; Mechanical properties.
Introduction
Since their discovery in 1991 �Iijima 1991�, carbon nanotubes�CNTs� have been the subject of intensive research, thanks to awide range of their potential applications: �1� as light-weightstructural materials with extraordinary mechanical properties suchas stiffness and strength �Treacy et al. 1996�, which may be in theform of either nanoropes �Thess et al. 1996� or the CNT rein-forced composites �Thostenson et al. 2001; Thostenson and Chou2002�; �2� in nanoelectronic components as the next-generation ofnanotransistors �Postma et al. 2001�, semiconductors, and nano-wires �Friedman et al. 2005; Tans et al. 1999�; �3� as probes inscanning probe microscopy and atomic force microscopy �AFM��Dai et al. 1996� with the added advantage of a chemically func-tionalized tip; �4� as high-sensitivity microbalances �Poncharal etal. 1999�; �5� as gas and molecule sensors �Kong et al. 2000� ornanostrain sensors �Cao et al. 2005�; �6� in hydrogen storage de-vices by using its high surface–volume ratio �Dillon et al. 1997�;�7� as field-emission type displays �Wang et al. 1998�; �8� aselectrodes in organic light-emitting diodes �Lee et al. 1999�; �9�as tiny tweezers for nanoscale manipulation �Kim and Lieber1999�; �10� as nanofluidic components �Hummer et al. 2001� andnanovalves �Grujicic et al. 2005; Solares et al. 2004�; �11� asactuators such as artificial muscles �Baughman et al. 1999�; and
1Associate Professor, Dept. of Civil Engineering and Engineering Me-chanics, Columbia Univ., New York, NY 10027.
2Professor, Dept. of Civil/Environmental Engineering, and Dept. ofMechanical Engineering, Northwestern Univ., Evanston, IL 60208.
Note. Associate Editor: Ross Barry Corotis. Discussion open untilAugust 1, 2008. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on July 18, 2007;approved on July 27, 2007. This paper is part of the Journal of Engi-neering Mechanics, Vol. 134, No. 3, March 1, 2008. ©ASCE, ISSN
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�12� as energy absorption and storage devices �Qiao et al. 2007�,to name a few. In general, the single-walled carbon nanotube�SWCNT� can be regarded as a roll of graphene sheet, and dif-ferent chiralities arise by varying the rolling vector. The atomicstructures of the zigzag, armchair, and chiral SWCNTs are shownin Fig. 1; the multiwalled CNTs �MWCNTs� are assemblies ofcoaxial SWCNTs where the neighboring layers are separated bythe van der Waals equilibrium distance ��0.34 nm�.
The mechanical properties of the CNTs must be fully under-stood in order to fulfill their promises �Chen and Cao 2007;Huang and Wang 2003; Qian et al. 2002�. Perhaps the most fun-damental phenomenological mechanical property of the CNT isits effective Young’s modulus. A variety of experimental attemptshave been put together to measure the Young’s moduli of carbonnanotubes. From thermal vibration tests, the moduli of theSWCNTs and MWCNTs are found to be in the ranges of0.40–4.15 TPa �Treacy et al. 1996� and 0.9–1.9 TPa �Krishnan etal. 1998�, respectively. By using an AFM tip to impose lateralforces to bend a MWCNT cantilever deposited on a low-frictionsubstrate, the Young’s moduli of MWCNTs are found to be1.28±0.59 TPa �Wong et al. 1997�. The elasticity of the CNTs isalso found to be size dependent: by measuring the electrome-chanical resonances of the CNTs, Poncharal et al. �1999� havediscovered that the stiffness of the MWCNTs decreases quicklywhen the diameter exceeds 10 nm. Such a phenomenon was notobserved in the static bending experiments with an AFM �Wonget al. 1997�. These diverse experimental measurements suggestthat the CNT mechanical properties vary with different chiralitiesand lengths, and sometimes defects.
Besides experimental efforts, theoretical and numerical studiesof the mechanical properties of the CNTs have attracted wideattentions. These studies provide critical insights on the deforma-tion and strength of materials and structures made by the CNTs.For example, the effects of waviness, weak bonding, and agglom-eration of the CNTs have limited the strength of the CNT rein-forced composite �Fisher et al. 2002, 2003; Shi et al. 2004a,b�, the
presence of geometrical imperfections and defects may signifi-NAL OF ENGINEERING MECHANICS © ASCE / MARCH 2008 / 211
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cantly reduce their buckling strength �Cao and Chen 2006b,c,d�,and thermal vibration causes apparent thermal contraction ofthese compliant quasi-one-dimensional structures �Cao et al.2006b�. Understanding of these critical factors is very importantto explain the experiments and to guide the practical applicationsof the CNTs.
The modeling and simulation approaches can be divided intothree main categories: �1� atomistic simulation at the nanoscale;�2� continuum and structural mechanics modeling which incorpo-rates atomistic characteristics; and �3� multiscale modeling andsimulation that bridges the length and time scales.
The atomic simulations of the CNTs are mainly based on �1�the classical molecular dynamics �MD� with empirical inter-atomic potentials �Iijima et al. 1996; Yakobson et al. 1996�; �2�the tight-binding and other semiempirical methods �Hernandezet al. 1998�; and �3� the first-principle quantum mechanics �QM�calculations �Sanchez-Portal et al. 1999�. At the most detailedlevel, the electric charge density distribution and electrostatic en-ergy can be deduced precisely by solving the Schrödinger equa-tion with high-level density functional theories �DFT� or ab initio�AB� methods. Thus, the QM approach provides accurate solu-tions for atomic and molecular structural deformation and frac-ture, as well as any chemical reaction that may occur �Cao et al.2006a; Kundin et al. 2001; Sanchez-Portal et al. 1999�. However,the QM calculation is computationally expensive and only effec-tive to small systems containing several hundreds of carbonatoms, with the CNT length up to several nanometers. In order toreduce the computational cost and still keep the QM characteris-tics, a semiempirical �SE� method such as tight binding �TB� wasdeveloped to solve the molecular orbital functions by replacingcomplex integrals with simpler empirical parameters and func-tions �Porezag et al. 1995�. The SE/TB approach can be applied tolarger systems �with the CNT length less than 100 nm� �Nardelliand Bernholc 1999; Zhang et al. 1998�. Although computation-ally, it is still about 100–1,000 times more expensive than theMD. The more popular classical MD simulations may be readilyemployed to explore the CNTs containing several million atoms,by ignoring the electron motions and expressing the system po-tentialenergy as a function of the nuclear positions of atoms, thus re-ceiving wide applications.
With the development of a better force field �i.e., less empiri-
Fig. 1. Atomic structures of zigzag �12,0�, armchair �7,7�, and chiral�8,4� SWCNTs, all have similar radii. Chirality is a key factorgoverning the mechanical properties of CNTs.
cal �Sun 1998�� and numerical algorithms, the classical MD simu-
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lations have been shown to play an important role in revealing theconstitutive mechanisms of the CNTs, including tension, bending,torsion �Chen and Cao 2006; Gao et al. 1998; Lu 1997; Robertsonet al. 1992; Yakobson et al. 1996�, radial deformation �Cao et al.2006c�, thermomechanical and vibration behaviors �Fig. 2� �Caoet al. 2005; Cao et al. 2006b�, as well as self-folding �Buehleret al. 2006� and buckling instabilities caused by compression�Buehler et al. 2004; Iijima et al. 1996; Liew et al. 2004; Ozakiet al. 2000; Srivastava and Barnard 1997; Yakobson et al. 1996�,indentation �Cao and Chen 2006a�, and torsion and bending �Caoand Chen 2006c; Pantano et al. 2004; Shibutani and Ogata 2004�.In all these studies, it has been verified that the mechanical prop-erties of the CNTs vary with their chirality and length. Specifi-cally, the radius of the CNT is governed by its chirality and if thethickness of a nanotube is a constant, then the CNT exhibitsbeamlike behaviors with increasing length and decreasing radius;on the other hand, the nanotube behaves more like shells withincreasing radius and decreasing length. Moreover, the size, dis-tribution, and accumulation of the geometrical imperfections�which arise during the equilibrium optimization of the atomicstructure� and defects �e.g., atomic vacancies or the Stone–Walesdefect where the bonds rearrange into pentagon or heptagon pairs�play a critical role in the mechanical integrity �Cao and Chen2006b,d, 2007�. Note that classical MD simulations alone cannotbe used to study the fracture of the CNTs �with the exception ofusing the more advanced reactive potentials that are also morecomputationally expensive�, and the TB-based MD or multiscalecoupling between the TB and MD are needed �in order to simu-late larger systems�, discussed below.
Although the MD approach yields results that are in manycases explicit in nature, due to the limitation in time scales�typically, several ns or less� and length scales of the models,they become less practical in problems involving a large numberof CNTs �e.g., the long MWCNTs or CNT bundles�, or to com-pare with experiments that are conducted at a much longer timescale. Moreover, for practical applications, phenomenologicalcontinuum-based material parameters such as the Young’s modu-lus need to be defined and measured for the CNTs. Therefore, it isof fundamental value to develop reliable and efficient continuummodels for the CNTs, which should closely duplicate the consti-tutive behaviors obtained from atomistic simulations �Arroyo andBelytschko 2005�.
The continuum studies are based on structural mechanics in-volving classic truss, beam, or shell theories incorporating atomicpotentials. Two main types of continuum approaches are avail-able, where the SWCNT is taken to be either a continuum tube�pipe�, or a geometrical space frame. In the first type of approach,the SWCNTs may be conveniently modeled as continuous shells�Pantano et al. 2004; Yakobson et al. 1996� with a fixed wallthickness, t, and a fixed Young’s modulus, E, for all CNTs—theonly variable used to distinguish different chiralities is the radiusof the shell, R. The uniaxial stretching energy of a nanotube andthe energy required to roll a graphite sheet into a SWCNT can bereadily derived from thin plate theory; by comparing with the MDsimulation and ab initio methods �Kundin et al. 2001; Yakobsonet al. 1996; Zhou et al. 2000�, E�3.9–7 TPa and t�0.066–0.089 nm can be fitted. Recently, Chen and Cao �2006� havejustified these parameters as E=6.85 TPa and t=0.08 nm, bymatching continuum shell deformation and vibration with MDsimulations for a variety of CNTs. To assemble SWCNTs into aMWCNT, the nonbonded interaction between neighboring tubelayers can be modeled as a Lennard-Jones type of interface po-
tential, which introduces repulsive or attractive pressure along the34:211-216.
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shell surface as the system is deformed �Cao et al. 2006c; Pantanoet al. 2004�. A similar approach can also be readily extended toCNT bundles. The shell model can be easily implemented intocommercial finite-element packages and it has been widely usedto explore the static, dynamic, and bifurcation behaviors of theSWCNTs, MWCNTs, and CNT reinforced composites.
For an individual SWCNT using the thin shell model, the mo-ment of inertia is I=�R3t, and the cross-sectional area is A=2�Rt. The bending stiffness and tension stiffness then become�EtR3 and 2�ERt, respectively, i.e., the SWCNT Young’s modu-lus E and thickness t always appear together via their product Et.Huang et al. �2006� showed that it is unnecessary to define theSWCNT Young’s modulus and thickness separately since all ex-perimentally measurable or theoretically calculable properties in-volve Et, not E nor t separately. In fact, the prior modeling andsimulations of SWCNT can be grouped to two types: one takesthe interlayer spacing of graphite 0.34 nm as the SWCNT thick-ness, and the resulting effective Young’s modulus is around 1 TPa�Hernandez et al. 1998; Li and Chou 2003; Lu 1997�; and theother is based on the continuum shell modeling, which gives theeffective thickness around 0.066 nm and Young’s modulus around5.5 TPa as discussed above �Chen and Cao 2006; Kudin et al.2001; Pantano et al. 2003; Yakobson et al. 1996; Zhou et al.2000�. These two types, however, give approximately the sameproduct Et, about 3.4 TPa·nm.
The main disadvantage of the shell model is that it has ne-glected the atomic characteristics. The effect of chirality is onlymodeled through different radii, which is insufficient in many
Fig. 2. Example of the MD simulation of the SWCNT thermomechantemperature will lead to apparent thermal contraction in the axial direcmodes that contribute significantly to the thermal contraction; �b� timseveral lowest vibration modes shown in �a�; and �c� coefficient ofdirections from 100 to 800 K, which is primarily caused by the therm
cases since nanotubes with similar radii but different chiralities
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�such as those in Fig. 1� may exhibit distinct behaviors �Jianget al. 2003�. Moreover, it cannot be used to study the effect ofdefects on the mechanical properties, and is unsuitable to accountfor the forces acting on individual atoms �Jiang et al. 2004a�. Toincorporate the effect of chirality, Zhang et al. �2002a,b� havedeveloped a continuum theory by directly incorporating the inter-atomic potentials into the constitutive model for CNTs. This con-tinuum theory agrees well with the atomistic simulations, such asthe fracture strain of CNTs �Jiang et al. 2006; Zhang et al. 2002c,2004� and CNT reinforced composites �Shi et al. 2005�, defectnucleation in CNTs �Song et al. 2006a� and boron-nitride nano-tubes �Song et al. 2006b�. Such an atomistic-based continuumtheory can account for the finite temperature �Jiang et al. 2004b,2005� as well as the coupled electromechanical behavior �Johnsonet al. 2004; Liu et al. 2004a�. In a complementary continuumstudy, Arroyo and Belytschko developed continuum theories ofCNTs based on the Cauchy–Born rule �Arroyo and Belytschko2004, 2005�.
Another direct approach incorporating the atomic properties isto approximate C–C bonds by structural beams or trusses. Forexample, Li and Chou �2003, 2004� have replaced all C–C bondsin the SWCNT by beams with circular cross sections. Chang andGao �2003� have used an elastic rod �truss� to model the bondstretching and a spiral �rotational� spring to model the twistingmoment resulted from the angular distortion of the bond angle.Chen and Cao �2006� developed a space-frame beam structurewhere beam elements are used to bridge the nearest and next-nearest carbon atom pairs—the first type accounts for bond
operty, adapted from �Cao et al. 2006b�: �a� thermal vibration at finitehere the lateral vibration may be decoupled into several fundamentalry of lateral vibration, which can be expressed as the summation of
al expansion of SWCNT remains negative in both radial and axialration effect
ical prtion, we histothermal vib
stretching and the second type accounts for bond angle variation.
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In all these structural mechanics models, the cross-section andelastic properties of the structural elements are obtained eitherdirectly from parameters used in certain force fields of molecularmechanics, or by fitting the MD strain energies of the CNTs undervarious deformation modes. Similar models are developed byother researchers, for example, Leung et al. �2005�, Nasdala andErnst �2005�, and Odegarda et al. �2002�.
Since the fundamental assumption of the continuum approxi-mation breaks down in many cases of nanomechanics and all-atom simulations are computationally expensive, it is highly de-sirable to carry out multiscale simulations to take advantage ofboth approaches and overcome the length and time scale limits inan efficient manner. Depending on the ways of coupling multiplescales, the multiscale simulation can be either concurrent or hier-archical. In the concurrent simulation, all scales �QM �or TB/SE�,MD, continuum� are coupled within one unified numerical codewhere the bridging technique provides the link among them. Inthe hierarchical approach, simulations are carried out at separatescales, which provides critical insights for improved modeling inthe next larger scale.
In the concurrent simulation, a portion of the electronicallyimportant regions �e.g., the crack tip� needs to be treated quantummechanically, which is embedded in a classical MD particle sys-tem �the particles can be either atoms or coarse grains�, and thefar field can be sufficiently described by continuum regions basedon the finite-element method �FEM� or meshless particle method�Liu et al. 2004c; Park and Liu 2004�. Such a multiscale schemeis able to capture the quantities that vary quickly in the criticalatomic region, while significantly reducing the computational costby treating the surrounding material in an averaged sense. A greatchallenge in the multiscale simulations concerns the treatment ofthe area linking methods of different scales, and various strategieshave been proposed, such as the quasicontinuum method �Knapand Ortiz 2001; Tadmor et al. 1996�, handshaking method �Abra-ham et al. 1998; Broughton et al. 1999�, and bridge scale method�Kadowaki and Liu 2004; Wagner and Liu 2003�. The quasicon-tinuum method treats the atomic lattice in a continuum manner,where a triangulation of lattice subset deduces the finite-element-like shape functions at lattice points, and the quantities at inter-mediate points can be obtained from interpolation. In thehandshaking method, at the QM/MD or MD/continuum interface,the Hamiltonian can be represented by the average of the twodescriptions, and the total system Hamiltonian is the summationof the three scales and that of the interface. In the bridge scalemethod, both continuum and atomic representations are allowedto coexist in the same region of interest, and the total scale isgiven by the summation of the FEM interpolations and MD dis-placements, subtract off the projection of the MD part onto theFEM basis. The concurrent simulations have been used to studythe deformation, fracture, and defects of the CNTs and CNT re-inforced polymers �Dumitrica et al. 2003; Gates et al. 2005;Mielke et al. 2004; Zhang et al. 2005�.
In the hierarchical study of the CNTs, Liu et al. �2004b, 2005,2006� developed an order-N, atomic-scale finite-element method�AFEM� that is as accurate as molecular mechanics. AFEM canhandle discrete atoms characterized by pair potentials or multi-body potentials. It is an order-N method �i.e., the computationscales linearly with the system size�, and is therefore much fasterthan the �order-N2� conjugate gradient method widely used inatomistic studies. The vast increase in speed of AFEM makes itpossible to study large-scale problems �Fig. 3� that would take anunbearable amount of time with the existing methods. Further-
more, AFEM can be linked seamlessly with the continuum FEM214 / JOURNAL OF ENGINEERING MECHANICS © ASCE / MARCH 2008
J. Eng. Mech. 2008.1
since they are within the same theoretical framework �of FEM�.This linkage between discrete atoms and continuum solids pro-vides a powerful concurrent computation alternative that signifi-cantly reduces the degree of freedom and enables the computationfor a much larger scale �possibly macroscale� systems.
To summarize, modeling and simulation at multiple scalesplay an indispensable role towards the understanding of the me-chanical properties of the CNTs as well as to fulfill their promises.The general atomistic, continuum, and multiscale strategies of thetheoretical and numerical nanomechanics of the CNTs are brieflyreviewed in this paper, which provide a critical link between thescience underpinning the CNTs and their engineering applica-tions. The topic is rapidly evolving and there are still numerousmethodologies and other likely applications that we have notmentioned due to space limitations. Moreover, the nanomechanicsprinciples discussed above can be readily extended to othernanoscale materials and structures. It is conceivable that theunique structures and properties of the CNTs and other nanoma-terials will bring a profound impact to the technology, where thenanomechanics plays a critical role.
Acknowledgments
The first writer acknowledges support from NSF-CMS-0407743and NSF-CAREER-CMMI-0643726.
References
Abraham, F. F., Broughton, J. Q., Bernstein, J., and Kaxiras, E. �1998�.“Spanning the length scales in dynamic simulation.” Comput. Phys.,12, 538–546.
Fig. 3. Examples from the multiscale simulation by using the AFEM,adapted from Liu et al. �2005�: �a� schematic of the AFEM for azigzag SWCNT, where the position change of central atom 1influences the energy stored in nine atomic bonds within this element,from which the element stiffness matrix can be derived based onatomic potentials. Application of the AFEM to woven nanostructuremade by ten �5,5� SWCNTs; �b� woven nanostructure subject to apoint force of 50 eV /nm as indicated; �c� structure with a brokenCNT away from the same point force; �d� structure with a brokenCNT underneath the same point force. The contours are used todistinguish displacements of atoms.
Arroyo, M., and Belytschko, T. �2004�. “Finite crystal elasticity of carbon
34:211-216.
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nloa
ded
from
asc
elib
rary
.org
by
UN
IV O
F C
ON
NE
CT
ICU
T L
IBR
AR
IES
on 0
3/18
/13.
Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
nanotubes based on the exponential Cauchy–Born rule.” Phys. Rev. B,69, 115415.
Arroyo, M., and Belytschko, T. �2005�. “Continuum mechanics modelingand simulation of carbon nanotubes.” Mechanica, 40, 455–469.
Baughman, R. H., et al. �1999�. “Carbon nanotube actuators.” Science,284, 1340–1344.
Broughton, J. Q., Abraham, F. F., Bernstein, N., and Kaxiras, E. �1999�.“Concurrent coupling of length scales: Methodology and applica-tions.” Phys. Rev. B, 60, 2391–2403.
Buehler, M., Kong, Y., and Gao, H. �2004�. “Deformation mechanisms ofvery long single-wall carbon nanotubes subject to compressive load-ing.” J. Eng. Mater. Technol., 126, 245–249.
Buehler, M., Kong, Y., Gao, H., and Huang, Y. �2006�. “Self-folding andunfolding of carbon nanotubes.” J. Eng. Mater. Technol., 128, 3–10.
Cao, G., and Chen, X. �2006a�. “Mechanisms of nanoindentation onsingle-walled carbon nanotubes: The effect of nanotube length.” J.Mater. Res., 21, 1048–1070.
Cao, G., and Chen, X. �2006b�. “Buckling behaviors of the single-walledcarbon nanotubes and a targeted-molecular dynamics simulation.”Phys. Rev. B, 74, 165422.
Cao, G., and Chen, X. �2006c�. “Buckling of single-walled carbon nano-tubes under bending: Molecular dynamics and finite element simula-tions.” Phys. Rev. B, 73, 155435.
Cao, G., and Chen, X. �2006d�. “The effect of displacement increment onthe axial compressive buckling behavior of single-walled carbonnanotubes.” Nanotechnology, 17, 3844–3855.
Cao, G., and Chen, X. �2007�. “The effects of chirality and boundaryconditions on the mechanical properties of single-walled carbon nano-tubes.” Int. J. Solids Struct., 44, 5447–5465.
Cao, G., Chen, X., and Kysar, J. W. �2005�. “Strain sensing with carbonnanotubes: Numerical analysis of the vibration frequency of deformedsingle-walled carbon nanotubes.” Phys. Rev. B, 72, 195412.
Cao, G., Chen, X., and Kysar, J. W. �2006a�. “Numerical analysis of theradial breathing mode of individual armchair and zigzag single-walledcarbon nanotubes under deformation.” J. Appl. Phys., 100, 124305–124314.
Cao, G., Chen, X., and Kysar, J. W. �2006b�. “Thermal vibration andapparent thermal contraction of single-walled carbon nanotubes.” J.Mech. Phys. Solids, 54, 1206–1236.
Cao, G., Tang, Y., and Chen, X. �2006c�. “Elastic properties of carbonnanotubes in radial direction.” J. Nanoeng. Nanosyst., 219, 73–88.
Chang, T., and Gao, H. �2003�. “Size-dependent elastic properties of asingle-walled carbon nanotube via a molecular mechanics model.” J.Mech. Phys. Solids, 51, 1059–1074.
Chen, X., and Cao, G. �2006�. “A structural mechanics approach ofsingle-walled carbon nanotubes generalized from atomistic simula-tion.” Nanotechnology, 17, 1004–1015.
Chen, X., and Cao, G. �2007�. “Review: Atomistic studies of mechanicalproperties of carbon nanotubes.” J. Theor. Comput. Nanosci., 4, 823–839.
Dai, H., Hafner, J. H., Rinzler, A. G., Colbert, D. T., and Smalley, R. E.�1996�. “Nanotubes as nanoprobes in scanning probe microscopy.”Nature (London), 384, 147–150.
Dillon, A. C., Jones, K. M., Bekkedahl, T. A., Kiang, C. H., Bethune, D.S., and Heben, M. J. �1997�. “Storage of hydrogen in single-walledcarbon nanotubes.” Nature (London), 386, 377–379.
Dumitrica, T., Belytschko, T., and Yakobson, B. I. �2003�. “Bond-breaking bifurcation states in carbon nanotube fracture.” J. Chem.Phys., 118, 9485–9488.
Fisher, F. T., Bradshaw, R. D., and Brinson, L. C. �2002�. “Effects ofnanotube waviness on the mechanical properties of nanoreinforcedpolymers.” Appl. Phys. Lett., 80, 4647–4649.
Fisher, F. T., Bradshaw, R. D., and Brinson, L. C. �2003�. “Fiber wavi-ness in nanotube-reinforced polymer composites: I. Modulus predic-tions using effective nanotube properties.” Compos. Sci. Technol., 63,1689–1703.
Friedman, R. S., McAlpine, M. C., Ricketts, D. S., Ham, D., and Lieber,C. M. �2005�. “High-speed integrated nanowire circuits.” Nature
(London), 434, 1085.JOUR
J. Eng. Mech. 2008.1
Gao, G. H., Cagin, T., and Goddard, W. A. �1998�. “Energetics, structure,mechanical, and vibrational properties of single-walled carbon nano-tubes.” Nanotechnology, 9, 184–191.
Gates, T. S., Odegard, G. M., Frankland, S. J. V., and Clancy, T. C.�2005�. “Computational materials: Multiscale modeling and simula-tion of nanostructured materials.” Compos. Sci. Technol., 65, 2416–2434.
Grujicic, M., Cao, G., Pandurangana, B., and Royb, W. N. �2005�.“Finite-element analysis-based design of a fluid-flow control nanov-alve.” Mater. Sci. Eng., B, 117, 53–61.
Hernandez, E., Goze, C., Bernier, P., and Rubio, A. �1998�. “Elastic prop-erties of C and BxCyNz composite nanotubes.” Phys. Rev. Lett., 80�,4502–4505.
Huang, Y., and Wang, Z. L. �2003�. Mechanics of carbon nanotubes,Elsevier Science, Amsterdam, The Netherlands.
Huang, Y., Wu, J., and Hwang, K. �2006�. “Thickness of graphene andsingle-wall carbon nanotubes.” Phys. Rev. B, 74�, 245413
Hummer, G., Rasalah, J. G., and Noworyta, J. P. �2001�. “Water conduc-tion through the hydrophobic channel of a carbon nanotube.” Nature(London), 414, 188–190.
Iijima, S. �1991�. “Helical microtubules of graphitic carbon.” Nature(London), 354, 56–58.
Iijima, S., Brabec, C., Maiti, A., and Bernholc, J. �1996�. “Structuralflexibility of carbon nanotubes.” J. Chem. Phys., 104, 2089–2092.
Jiang, H., et al. �2003�. “The effect of nanotube radius on the constitutivemodel for carbon nanotubes.” Comput. Mater. Sci., 28, 429–442.
Jiang, H., Feng, X. Q., Huang, Y., Hwang, K. C., and Wu, P. �2004a�.“Defect nucleation in carbon nanotubes under tension and torsion:Stone–Wales transformation.” Comput. Methods Appl. Mech. Eng.,193, 3419–3429.
Jiang, H., Huang, Y., and Hwang, K. C. �2005�. “A finite-temperaturecontinuum theory based on the interatomic potential.” J. Eng. Mater.Technol., 127, 408–416.
Jiang, H., Huang, Y., Zhang, P., and Hwang, K. C. �2006�. Fracturenucleation in single-wall carbon nanotubes: The effect of nanotubecharality, Springer, Dordrecht, The Netherlands.
Jiang, H., Liu, B., Huang, Y., and Hwang, K. C. �2004b�. “Thermal ex-pansion of single-wall carbon nanotubes.” J. Eng. Mater. Technol.,126, 265–270.
Johnson, H. T., Liu, B., and Huang, Y. �2004�. “Electron transport indeformed carbon nanotubes.” J. Eng. Mater. Technol., 126, 222–229.
Kadowaki, H., and Liu, W. K. �2004�. “Bridging multiscale method forlocalization problems.” Comput. Methods Appl. Mech. Eng., 193,3267–3302.
Kim, P., and Lieber, C. M. �1999�. “Nanotube nanotweezers.” Science,286, 2148–2150.
Knap, J., and Ortiz, M. �2001�. “An analysis of the quasicontinuummethod.” J. Mech. Phys. Solids, 49, 1899–1923.
Kong, J., et al. �2000�. “Nanotube molecular wires as chemical sensors.”Science, 287, 622.
Krishnan, A., Dujardin, E., Ebbesen, T. W., Yianilos, P. N., and Treacy,M. M. J. �1998�. “Young’s modulus of single-walled nanotubes.”Phys. Rev. Lett., 58, 14013–14019.
Kundin, K. N., Scuseria, G. E., and Yakobson, B. I. �2001�. “C2F, BN,and C nanoshell elasticity from ab initio computations.” Phys. Rev. B,64, 235406.
Lee, C. J., et al. �1999�. “Synthesis of uniformly distributed carbon nano-tubes on a large area of Si substrates by thermal chemical vapordeposition.” Appl. Phys. Lett., 75, 1721–1723.
Leung, A. Y. T., Guo, X., He, X. Q., and Kitipornchai, S. �2005�. “Acontinuum model for zigzag single-walled carbon nanotubes.” Appl.Phys. Lett., 86, 083110–083112.
Li, C. Y., and Chou, T. W. �2003�. “A structural mechanics approach forthe analysis of carbon nanotubes.” Int. J. Solids Struct., 40, 2487–2499.
Li, C. Y., and Chou, T. W. �2004�. “Modeling of elastic buckling ofcarbon nanotubes by molecular structural mechanics approach.”Mech. Mater., 36, 1047–1055.
Liew, K. M., Wong, C. H., He, X. Q., Tan, M. J., and Meguid, S. A.
NAL OF ENGINEERING MECHANICS © ASCE / MARCH 2008 / 215
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�2004�. “Nanomechanics of single- and multiwalled carbon nano-tubes.” Phys. Rev. B, 69, 115429.
Liu, B., Huang, Y., Jiang, H., Qu, S., and Hwang, K. C. �2004a�. “Theatomic-scale finite-element method.” Comput. Methods Appl. Mech.Eng., 193, 1849–1864.
Liu, B., Jiang, H., Huang, Y., Qu, S., Yu, M.-F., and Hwang, K. C. �2005�.“Atomic-scale finite-element method in multiscale computation withapplications to carbon nanotubes.” Phys. Rev. B, 72, 035435.
Liu, B., Jiang, H., Huang, Y., Qu, S., Yu, M.-F., and Hwang, K. C. �2006�.“Finite-element method: From discrete atoms to continuum solids.”Handbook of theoretical and computational nanotechnology, Ameri-can Scientific, Stevenson Ranch, Calif.
Liu, B., Jiang, H., Johnson, H. T., and Huang, Y. �2004b�. “The influenceof mechanical deformation on the electrical properties of single-wallcarbon nanotubes.” J. Mech. Phys. Solids, 52, 1–26.
Liu, W. K., Karpov, E. G., Zhang, S., and Park, H. S. �2004c�. “Anintroduction to computational nanomechanics and materials.” Com-put. Methods Appl. Mech. Eng., 193, 1529–1578.
Lu, J. P. �1997�. “Elastic properties of carbon nanotubes and nanoropes.”Phys. Rev. Lett., 79, 1297–1300.
Mielke, S. L., et al. �2004�. “The role of vacancy defects and holes in thefracture of carbon nanotubes.” Chem. Phys. Lett., 390, 413–420.
Nardelli, M., and Bernholc, J. �1999�. “Mechanical deformations andcoherent transport in carbon nanotubes.” Phys. Rev. B, 60, 16338–16341.
Nasdala, L., and Ernst, G. �2005�. “Development of a four-node finiteelement for the computation of nanostructured materials.” Comput.Mater. Sci., 33, 443–458.
Odegarda, G. M., Gatesb, T. S., Nicholsonc, L. M., and Wised, K. E.�2002�. “Equivalent-continuum modeling of nanostructured materi-als.” Compos. Sci. Technol., 62, 1869–1880.
Ozaki, T., Iwasa, Y., and Mitani, T. �2000�. “Stiffness of single-walledcarbon nanotubes under large strain.” Phys. Rev. Lett., 84�8�, 1712–1715.
Pantano, A., Boyce, M. C., and Parks, D. M. �2003�. “Nonlinear struc-tural mechanics based modeling of carbon nanotube deformation.”Phys. Rev. Lett., 91, 145501–145504.
Pantano, A., Parks, D. M., and Boyce, M. C. �2004�. “Mechanics ofdeformation of single- and multiwall carbon nanotubes.” J. Mech.Phys. Solids, 52, 789–821.
Park, H. S., and Liu, W. K. �2004�. “An introduction and tutorial onmultiple-scale analysis in solids.” Comput. Methods Appl. Mech.Eng., 193, 1733–1772.
Poncharal, P., Wang, Z. L., Ugarte, D., and de Heer, W. A. �1999�. “Elec-trostatic deJections and electromechanical resonances of carbon nano-tubes.” Science, 283, 1513–1516.
Porezag, D., Frauenheim, T., Kohler, T., Seifert, G., and Kaschner, R.�1995�. “Construction of tight-binding-like potentials on the basis ofdensity-functional theory—Application to carbon.” Phys. Rev. B, 51,12947–12957.
Postma, H. W. C., Teepen, T., Yao, Z., Grigoni, M., and Dekker, C.�2001�. “Carbon nanotube single-electron transistors at room tempera-ture.” Science, 293, 76.
Qian, D., Wagner, G. J., Liu, W. K., Yu, M.-F., and Ruoff, R. S. �2002�.“Mechanics of carbon nanotubes.” Appl. Mech. Rev., 55, 495–533.
Qiao, Y., Cao, G., and Chen, X. �2007�. “Effect of gas molecules onnanofluidic behaviors.” J. Am. Chem. Soc., 129, 2355–2359.
Robertson, D. H., Brenner, D. W., and Mintmire, J. W. �1992�. “Energet-ics of nanoscale graphitic tubules.” Phys. Rev. B, 45, 12592–12595.
Sanchez-Portal, D., Artacho, E., Solar, J. M., Rubio, A., and Ordejon, P.�1999�. “Ab initio structural, elastic, and vibrational properties of car-bon nanotubes.” Phys. Rev. B, 59, 12678–12688.
Shi, D. L., Feng, X. Q., Huang, Y., and Hwang, K. C. �2004a�. “Criticalevaluation of the stiffening effect of carbon nanotubes in composites.”Key Eng. Mater., 261, 1487–1492.
Shi, D. L., Feng, X. Q., Huang, Y., Hwang, K. C., and Gao, H. �2004b�.“The effect of nanotube waviness and agglomeration on the elasticproperty of carbon nanotube-reinforced composites.” J. Eng. Mater.Technol., 126, 250–257.
216 / JOURNAL OF ENGINEERING MECHANICS © ASCE / MARCH 2008
J. Eng. Mech. 2008.1
Shi, D. L., Feng, X. Q., Jiang, H., Huang, Y., and Hwang, K. C. �2005�.“Multiscale analysis of fracture of carbon nanotubes embedded incomposites.” Int. J. Fract., 134, 369–386.
Shibutani, Y., and Ogata, S. �2004�. “Mechanical integrity of carbonnanotubes for bending and torsion.” Modell. Simul. Mater. Sci. Eng.,12, 599–610.
Solares, S., Blanco, M., and Goddard, W. A. �2004�. “Design of a nano-mechanical fluid control valve based on functionalized silicon canti-levers: Coupling molecular mechanics and classical engineering de-sign.” Nanotechnology, 15, 1405–1415.
Song, J., et al. �2006b�. “Stone–Wales transformation: Precurson of frac-ture in carbon nanotubes.” Int. J. Mech. Sci., 48, 1464–1470.
Song, J., Huang, Y., Jiang, H., Hwang, K. C., and Yu, M. F. �2006a�.“Deformation and bifurcation analysis of boron-nitride nanotubes.”Int. J. Mech. Sci., 48, 1197–1207.
Srivastava, D., and Barnard, S. �1997�. Molecular dynamics simulation oflarge-scale nanotubes on a shared-memory architecture, ACM, NewYork.
Sun, H. �1998�. “COMPASS: An ab Initio force field optimized forcondensed-phase applications. Overview with details on alkane andbenzene compounds.” J. Phys. Chem. B, 102, 7338–7364.
Tadmor, E. B., Ortiz, M., and Phillips, R. �1996�. “Quasicontinuumanalysis of defects in solids.” Philos. Mag. A, 73, 1529–1541.
Tans, S. J., Verschueren, R. M., and Dekker, C. �1999�. “Room-temperature transistor based on a single carbon nanotube.” Nature(London), 393, 40.
Thess, A., et al. �1996�. “Crystalline ropes of metallic carbon nanotubes.”Science, 273, 483–487.
Thostenson, E. T., and Chou, T.-W. �2002�. “Aligned multiwalled carbonnanotube-reinforced composites: Processing and mechanical charac-terization.” J. Phys. D, 35, L77–L80.
Thostenson, E. T., Ren, Z. F., and Chou, T. W. �2001�. “Advances in thescience and technology of carbon nanotubes and their composites: Areview.” Compos. Sci. Technol., 61, 1899-1912.
Treacy, M. M. J., Ebbesen, T. W., and Gibson, J. M. �1996�. “Exception-ally high Young’s modulus observed for individual carbon nano-tubes.” Nature (London), 381, 678.
Wagner, G. J., and Liu, W. K. �2003�. “Coupling of atomistic and con-tinuum simulations using a bridging scale decomposition.” J. Comput.Phys., 190, 249–274.
Wang, Q. H., Yan, M., and Chang, R. P. H. �1998�. “A nanotube-basedfield-emission flat-panel display.” Appl. Phys. Lett., 72, 2912–2914.
Wong, E. W., Sheehan, P. E., and Lieber, C. M. �1997�. “Nanobeammechanics: Elasticity, strength, and toughness of nanorods and nano-tubes and carbon nanotubes.” Science, 277, 1971–1973.
Yakobson, B., Brabec, C., and Bernholc, J. �1996�. “Nanomechanics ofcarbon tubes: Instabilities beyond the linear response.” Phys. Rev.Lett., 76, 2511–2514.
Zhang, P., Huang, Y., Gao, H., and Hwang, K. C. �2002a�. “Fracturenucleation in single-wall carbon nanotubes under tension: A con-tinuum analysis incorporating interatomic potentials.” J. Appl. Mech.Rev., 69, 454–458.
Zhang, P., Huang, Y., Geubelle, P. H., and Hwang, K. C. �2002b�. “On thecontinuum modeling of carbon nanotubes.” Acta Mech. Sin., 18, 528–536.
Zhang, P., Huang, Y., Geubelle, P. H., Klein, P. A., and Hwang, K. C.�2002c�. “The elastic modulus of single-wall carbon nanotubes: Acontinuum analysis incorporating interatomic potentials.” Int. J. Sol-ids Struct., 39, 3893–3906.
Zhang, P., Jiang, H., Huang, Y., Geubelle, P. H., and Hwang, K. C.�2004�. “An atomistic-based continuum theory for carbon nanotubes:Analysis of fracture nucleation.” J. Mech. Phys. Solids, 52, 977–998.
Zhang, P., Lammert, P. E., and Crespi, V. H. �1998�. “Plastic deformationsof carbon nanotubes.” Phys. Rev. Lett., 81, 5346–5349.
Zhang, S., et al. �2005�. “Mechanics of defects in carbon nanotubes:Atomistic and multiscale simulations.” Phys. Rev. B, 71, 115403.
Zhou, X., Zhou, J. J., and Ou-Yang, Z. C. �2000�. “Strain energy andYoung’s modulus of single-wall carbon nanotubes calculated fromelectronic energy-band theory.” Phys. Rev. B, 62, 13692–13696.
34:211-216.