research article study of the nanomechanics of cnts under

Research ArticleStudy of the Nanomechanics of CNTs under Tension byMolecular Dynamics Simulation Using Different Potentials

S K Deb Nath12 and Sung-Gaun Kim1

1 Division of Mechanical and Automotive Engineering Kongju National University Republic of Korea2Department of Mechanical Science and Bioengineering Graduate School of Engineering Science Osaka UniversityToyonaka Osaka 560-8531 Japan

Correspondence should be addressed to S K Deb Nath sankar 20005yahoocom

Received 17 September 2013 Accepted 23 October 2013 Published 13 March 2014

Academic Editors E Liarokapis and M Razeghi

Copyright copy 2014 S K Deb Nath and S-G Kim This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

At four different strain rates the tensile stress strain relationship of single-walled 12-12 CNT with aspect ratio 91 obtained byRebo potential (Brenner 1990) Airebo potential (Stuart et al 2000) and Tersoff potential (Tersoff 1988) is compared with that ofBelytschko et al (2002) to validate the present model Five different empirical potentials such as Rebo potential (Brenner 1990)Rebo potential (Brenner et al 2002) Inclusion LJ with Rebo potential (Brenner 1990) Airebo potential (Stuart et al 2000) andTersoff potential (Tersoff 1988) are used to simulate CNT subjected to axial tension differing its geometry at high strain rate InRebo potential (Mashreghi andMoshksar 2010) only bond-order term is used and in Rebo potential (Brenner et al 2002) torsionalterm is included with the bond-order term At high strain rate the obtained stress strain relationships of CNTs subjected to axialtension differing its geometries using five different potentials are compared with the published results and from the comparison ofthe results the drawback of the published results and limitations of different potentials are evaluated and the appropriate potentialis selected which is the best among all other potentials to study the elastic elastic-plastic properties of different types of CNTsThepresent study will help a new direction to get reliable elastic elastic-plastic properties of CNTs at different strain rates Effects oflong range Van der Waals interaction and torsion affect the elastic elastic-plastic properties of CNTs and why these two effects arereally needed to consider in bond-order Rebo potential (Brenner 1990) to get reliable elastic elastic-plastic properties of CNTs isalso discussed Effects of length-to-diameter ratio layering of CNTs and different empirical potentials on the elastic elastic-plasticproperties of CNTs are discussed in graphical and tabular forms with published results as a comparative manner to understand thenanomechanics of CNTs under tension using molecular dynamics simulation

1 Introduction

A variety of new intriguing materials have been discoveredand synthesized in the last two decades which have causedphenomenal change in the area of materials science andamong them one is the class of carbon compounds referredto fullerene nanotubes Carbon nanotubes can be made ofas rolling up sheets of graphite that are sometimes crappedon each end with structures that vary depending on theconditions under which they are synthesizedThey are single-walled [1 2] with diameters as small as about 1 nm or multi-walled [3 4] with outer diameters ranging from 5 to 350 nm

Since their discovery by Lijima and Lchihashi [1] carbonnanotubes (CNTs) have been the subject of extensive researchdue to their unique structural and electronic properties thatlead to a wide range of potential applications from electronics[5] to mechanical reinforcements in composites [6] Earlystudies on the mechanical properties of CNT were carriedout considering compression test [7 8] Yakobson et al [7]first investigated the instability of single wall CNT underaxial compression bending and torsional deformation usingcontinuum shellmodel Yakobson et al [9] studied the behav-ior of CNTs under a high strain rate using Tersoff-Brennerrsquosreactive empirical bond-order (REBO) potential [10 11]

Hindawi Publishing CorporationISRN Condensed Matter PhysicsVolume 2014 Article ID 606017 18 pageshttpdxdoiorg1011552014606017

2 ISRN Condensed Matter Physics

Sinnott et al [12] estimated the theoretical Youngrsquosmodulifor carbon fibers composed of single-walled fullerene nan-otubes aligned in the direction of the tubule axis They alsoinvestigated a new carbon composite composed of layerednanotubule fibers and diamond and they observed that thiscomposite had a high-modulus low-densitymaterial that wasquite stable to shear and other distortions Liew et al [13]examined the elastic and elastic-plastic properties of carbonnanotubes under axial tension using second generation ofreactive empirical bond-order potential coupled with theLennard-Jones potential at high strain rate In their analysisplastic regions are not clear because in most of the analysisof CNT it was observed that before breaking the stiffness isthe highest which contradicts the reliability of their solutiondue to the mismanagement of the application of the Van derWaals interactions They did not minimize CNT structureto reach its minimum energy before applying tensile loadingat its both ends Besides Liew et al [13] did not includetemperature effect in the theoretical Rebo potential [11]and Van der Waals interaction for analyzing the elasticelastic-plastic properties of CNTs by molecular dynamicsimulation So the elastic-plastic properties obtained byLiew et al [13] by molecular dynamics simulation are notbeyond doubtful quality of solutions Molecular dynamicssimulations of tensile pulling of carbon nanotubes (SWCNTs)(both armchair and Zigzag configurations) were conductedusing the Brenner potential to investigate the variation ofsix-carbon bond lengths and bond angles of the hexagons insingle-walled carbon nanotubes (SWCNT) as a function oftensile strain [14] For armchair CNTs the brittle fracture isobserved and the ultimate strength of CNTs is many timesto that of CNTS obtained by Liew et al [13] The stressstrain relationship of armchair CNT using Rebo potentialby the tension test in molecular dynamics environment byAgrawal et al [14] contradicts the elastic-plastic region ofCNT by Liew et al [13] although in his analysis Van derWaals interactions are absent Wang and Vijayaraghavan [15]studied the buckling characteristics of several curved formsof single-walled carbon nanotubes by molecular dynamicssimulations using Rebo potential [11] and Van der Waalsinteraction Tserpes and Papanikos [16] proposed a three-dimensional finite element model for armchair Zigzagand chiral single-walled carbon nanotubes which is basedon the assumption that carbon nanotubes when subjectedto loading behave like space-frame structures The bondsbetween carbon atoms are considered as connecting loadcarrying members while the carbon atoms as joints of themembers [16] To create the FE models nodes are placedat the locations of carbon atoms and the bonds betweenthem are modeled using three-dimensional elastic beamelements and the elastic properties of armchair Zigzag andchiral single-walled carbon nanotubes are obtainedwhich arecompared with those of experimental results to verify theFE model [16] Salvetat-Delmotte and Rubio [17] concludedthat CNTs have indeed a great potential as reinforcingelements for composites Coto et al [18] studied the effect ofcarboxylation on axial Youngrsquos modulus of carbon nanotubesusing molecular dynamics simulation COMPASS force fieldis used to model the interatomic interactions in single wall

(SWCNT) and multiwall carbon (MWCNT) with differentamount of ndashCOOH group attached to their surfaces [18]Mashreghi and Moshksar [19] determined the structuralparameters of several armchair single-walled carbon nan-otubes such as potential energy as a function of bond lengthand bond angles respectively using molecular dynamicssimulations Tang et al [20] synthesized carbon nanotubepolymer composites mixing carbon nanotubes and highdensity polyethylene by using the melt processing methodand they observed that the stiffness peak load and work tofailure for the composite films increase with the increase ofMWCNT content Sammalkorpi et al [21] studied how theYoungrsquos modulus and tensile strength of nanotubes dependson defects employing molecular dynamics simulation andcontinuum theory The Youngrsquos modulus of nanotubes withdefects will essentially be the same unless the number ofvacancy concentration is extremely high [21] On the otherhand the tensile strength will substantially drop due to thequasi-one-dimensional atomic structure of SWNTs already ifa single vacancy is present the tensile strength of a SWNT isgoverned by the weakest segment of the tube [21] Wang etal [22] observed that composites showed higher mechanicalproperties such as theYoungrsquosmodulus and tensile strength ofthe SWNTepoxy composite rise with the increase of SWNTconcentration The Youngrsquos modulus and yield strength havebeen doubled and quadrupled for composites with respec-tively 1 and 4wt of nanotubes compared to the pure resinmatrix samples [23] A recent research illustrates qualitativerelationship between Youngrsquos modulus of a CNT and theamount of disorder in the atomic structure of the walls[24] Lin and Chen [25] evaluated the effective mechanicalproperties of CNT-based composites using a 3D nanoscalerepresentative volume element based continuum mechanicsand the finite element method Hernandez-Perez and Aviles[26] investigated the influence of the interphase on the effec-tive properties of carbon nanotube composites using finiteelement analysis and elasticity solutions for representativevolume elements Micro Pt wire exhibits higher strength ascompared to its bulk due to decreasing size which is observedfrom the experimental study of bending test of Pt microwirewith the help of finite element analysis [27] In the very shortrange of diameters in Au nanowires Youngrsquos modulus andyield strength increase with the decreasing of their diameterswhich are clearly observed in the bending and tension test ofAu nanowires by molecular dynamics simulations [28] Dueto having small sizes of diameters in CNTs and as a resultof increasing their surface energy their strength and stiffnessare many times to those of their conventional bulk materialswhich are made of C elements

Due to having high stiffness and light weight CNT isa promising candidate in nanocomposites to increase theirstrength and stiffness Nowadays people are concentratingto evaluate the actual elastic elastic-plastic properties ofCNTs experimentally and theoretically But handling CNTin experimental way is a difficult task and the obtainedmechanical properties by the experimental way are beyondactual quantitative measures To obtain the actual quantita-tive mechanical properties theoretical approach is only analternative approach Researchers are trying to study CNT by

ISRN Condensed Matter Physics 3

quantum ab initio calculations at a very small scale and on theother hand classical molecular dynamics based on empiricalpotentials are being used to study large size CNT Besidesvery recently people are trying to study CNTs using finiteelement and finite volume approach considering continuumapproach Classical molecular dynamics are the only reliablemethod to get reliable mechanical properties of CNTs butit depends on the potentials There are different potentials[10 11 29 31] which are being widely used to obtain elasticelastic-plastic properties of CNT To predict actual elastic-plastic and fracture region selections of good potentials arenecessary In most of the studies temperature effect is notincluded which is beyond practical condition Firstly theelastic elastic-plastic response of single-walled 12-12 CNTswith aspect ratio 91 subjected to a uniform axial loading atits both ends at four different strain rates obtained by Rebopotential [11] Airebo potential [29] and Tersoff potential [10]is compared with that of Belytschko et al [30] to validate thepresent model In the present study we study the mechanicalbehaviors of CNTs subjected to a uniform tensile load at itsopposite ends at high strain rate using five different potentialsby molecular dynamics simulation as a comparative mannerwith the existing published results and the limitations of thepublished results as well as our results obtained by differentpotentials are highlighted and from our analysis we try toselect appropriate potential which can give more accurateelastic elastic-plastic response of CNTs at different strainrates Besides effects of usual parameters length-to-diameterratio layering ofCNTs on the elastic elastic-plastic propertiesof CNTs are shown by figures and tables We try to establishthe superiority of the present technique and results of CNTs

2 Simulation Procedure

At four different strain rates such as very low strain rate(stretching velocity at both ends of CNTs is 009ms) lowstrain rate (stretching velocity at both ends of CNTs is05ms) moderate strain rate (stretching velocity at bothends of CNTs is 5ms) and high strain rate (stretchingvelocity at both ends of CNTs is 20ms) the stress strainrelationship of single-walled 12-12 CNT with aspect ratio 91subjected to uniform tensile load obtained by Rebo potential[11] Airebo potential [29] and Tersoff potential [10] is com-pared with that of Belytschko et al [30] to validate the modelTo study the elastic elastic-plastic properties of CNTwe needits stress strain relationship up to its fracture In this studywe also simulate single-walled 10-10 12-12 CNTs double-walled (55) 10-10 CNTs three-walled (55) 10-10 and 15-15 CNTs four-walled (55) 10-10 15-15 and 20-20CNTs usingmolecular dynamics simulation considering five differentpotentials such as Tersoff [10] Rebo potential [11] Rebopotential [31] Inclusion LJ with Rebo potential [11] andAirebo potential [29] at high strain rate Firstly the CNTstructure is minimized to get its minimum energy usingdifferent potentials before applying velocity to two groupsof atoms at both ends of CNTs for all strain rates At highstrain rate tension test is carried out applying a constantvelocity 20ms at some atoms of both ends of the CNT

in opposite directions by molecular dynamics simulationBefore applying velocity the total force on some groupedatoms at both ends of the CNT is kept null so that extraforce is not counted during stress strain relationship Force iscounted after applying velocity to a group of atoms at bothend of the CNT And the number of atoms in each groupis kept equal In the microcanonical ensemble moleculardynamics simulations of CNTs the strain is computed by120576 = (119871 minus 119871

0)1198710in which 119871

0and 119871 are the initial and current

length of CNTs respectively and the stress is obtained by120590 = 119865119878 The axial force 119865 is obtained by summing theinteratomic force for the atoms at the end of CNTs and thecross-sectional area is 119878 = 120587119889ℎ where 119889 is the diameter ofthe CNT and ℎ is the thickness of the CNT that is taken asℎ = 0335 nm for a 10-10 single-walled CNT ℎ = 067 nmfor a two-walled (55) and 10-10 CNT ℎ = 1005 nm for athree-walled (55) 10-10 and 15-15 CNT and ℎ = 134 nmfor a four-walled (55) 10-10 15-15 and 20-20CNT 119871

0119889

ratio of CNTs is considered as 91 and 45 respectively inthe present study Used time step is 0001 ps and the wholesimulation is considered at a temperature 300K Using NVTcanonical ensemble temperature is controlled Details of theused empirical potentials in molecular dynamics simulationof CNTs are discussed in the following sections for clearunderstanding of the readers Due to absence of long rangeVan der Waals interactions and torsional effects in Tersoffpotential [10] and Rebo potential [11] Rebo potential [31]and Inclusion LJ in Rebo potential [10] there is possibility ofhaving some inaccuracies of the elastic-plastic properties ofCNT when the simulation is carried out considering Tersoffpotential [10] Rebo potential [11] Rebo potential [31] andInclusion LJ with Rebo potential [11]

3 Theoretical Formulations

31 Tersoff Potential [10] The energy 119864 as a function of theatomic coordinates by Tersoff [10] is taken to be

119864 = sum

119894

119864119894=

1

2

sum

119894 = 119895

119881119894119895 (1)

Here119864 is the total energy of the system which is decomposedfor convenience into a site energy 119864

119894and a bond energy 119881

119894119895

119881119894119895= 119891119862(119903119894119895) [119891119877(119903119894119895) + 119887119894119895119891119860(119903119894119895)] (2)

The indices 119894 and 119895 run over the atoms of the system and 119903119894119895

is the distance from atom 119894 to 119895 The function 119891119877represents a

repulsive pair potential which includes the orthogonalizationenergy when atomic wave functions overlap and 119891

119860repre-

sents an attractive pair potential associated with bonding

119891119877(119903119894119895) = 119860 exp (minus120582

1119903119894119895)

119891119860(119903119894119895) = minus119861 exp (minus120582

2119903119894119895)

(3)

The extra term 119891119862is merely a smooth cut-off function to

limit the range of the potential since for many applicationsshort ranged functions permit a tremendous reduction incomputational effort


Here the cut-off function is simply taken as

119891119862(119903119894119895)

=

1 119903119894119895lt 119877 minus 119863

1

2

minus

1

2

sin[(1205872) (119903

119894119895minus 119877)

119863

] 119877 minus 119863 lt 119903119894119895lt 119877 + 119863

0 119903119894119895gt 119877 + 119863

(4)

where 119887119894119895represents a measure of the bond order and is for

now assumed to be a monotonically decreasing function ofthe coordination of atoms 119894 and 119895 Details of 119887

119894119895refer to Tersoff

[10]

32 Rebo Potential [11] The binding energy for the hydrocar-bon potential is given as a sum over bounds as

119864REBO

= sum

119894

sum

119895(gt119894)

[119881119877(119903119894119895) minus 119887119894119895119881119860(119903119894119895)] (5)

where the repulsive and attractive pair terms are given by

119881119877(119903119894119895) = 119891119862(119903119894119895)119881119877

119900(119903119894119895)

119881119860(119903119894119895) = 119891119862(119903119894119895)119881119860

119900(119903119894119895)

119891119862(119903119894119895)

=

1 119903119894119895lt 119863min

[

1

2

+

1

2

cos(120587 (119903119894119895minus 119863min)

119863max minus 119863min)] 119863min lt 119903119894119895 lt 119863max

0 119903119894119895gt 119863max

(6)

Here 119903119894119895is the distance between 119894th and 119895th atoms 119887

119894119895is a

bond-order term and 119881119877119900

and 119881119860119900

are functions of 119903119894119895 For

more details of these terms refer to Brenner et al [10 11]

33 Airebo Potential [29] Stuart et al [29] developed areactive potential for hydrocarbons with intermolecularinteractionsWith the adaptive treatment of dispersion inter-molecular repulsion and torsional interactions the entireenergy is given by the expression

119864AIREBO

=

1

2

sum

119894

sum

119895 = 119894

[

[

119864REBO119894119895

+ 119864119871119869

119894119895+ sum

119896 = 119894119895

sum

119897 = 119894119895119896

119864tors119896119894119895119897]

]

(7)

The REBO interaction is based on the form proposed byTersoff [10]

119864REBO119894119895

= 119881119877

119894119895+ 119887119894119895119881119860

119894119895(8)

in which repulsive and attractive contributions are combinedin a ratio determined by the bonding term 119887

119894119895

The repulsive term has the form used by Brenner et al[11 31]

119881119877

119894119895= 119908119894119895(119903119894119895) [1 +

119876119894119895

119903119894119895

]119860119894119895119890minus120572119894119895119903119894119895

(9)

where the parameters 119876119894119895 119860119894119895 and 120572

119894119895depend on the atom

types 119894 and 119895 Values for these and all other potentialparameters are given in Table II in [29] The 119908

119894119895term is a

bond-weighing factor

119908119894119895(119903119894119895) = 119878119905(119905119888(119903119894119895)) (10)

which switches off the REBO interactions when the atompairs exceed typical bonding distances The switching func-tion takes the form

119878119905(119905) = Θ (minus119905) + Θ (119905)Θ (1 minus 119905)

1

2

[1 + cos (120587119905)] (11)

where the switching region for each type of bond is given bya scaling function

119905119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(12)

The attractive pair interaction in (8) is given by a tripleexponential

119881119860

119894119895= minus119908119894119895(119903119894119895)

3

sum

119899=1

119861(119899)

119894119895119890minus120573(119899)

119894119895119903119894119895 (13)

which is switched off smoothly for non-short-ranged inter-actions through the use of bond weight The 119887

119894119895term in (8)

specifies the bond order for the interaction between 119894 and 119895

119887119894119895=

1

2

[119901120590120587

119894119895+ 119901120590120587

119895119894] + 120587119903119888

119894119895+ 120587119889ℎ

119894119895 (14)

This term is only roughly equivalent to the usual chemicalconcept of a bond order and is simply a means of modifyingthe strength of a bond due to changes in the local environ-ment

The principal contribution to 119887119894119895is the covalent bond

interaction given by the terms 119875120590120587119894119895

and 119875120590120587119895119894

119875120590120587

119894119895=[

[

1 + sum

119896 = 119894119895

119908119894119896(119903119894119896) 119892119894(cos 120579

119895119894119896) 119890120582119895119894119896

+ 119875119894119895]

]

minus12

119875120590120587

119895119894=[

[

1 + sum

119896 = 119895119894

119908119895119896(119903119895119896) 119892119895(cos 120579

119894119895119896) 119890120582119894119895119896

+ 119875119895119894]

]

minus12

(15)

The penalty function 119892119894imposes a cost on bonds that are too

close to one another Its functional form is a fifth-order splineWhen the central atom is a carbon the spline also dependson the local coordination number defined as the sum of the


carbon-only and hydrogen-only coordination numbers 119875119894119895is

a function of119873C119894119895and119873H

119894119895

119873119894119895= 119873

C119894119895+ 119873

H119894119895 (16)

where 119873C119894119895= (sum

119896 = 119894

120575119896119862119908119894119896(119903119894119896)) minus 120575

119895119862119908119894119895(119903119894119895) (17)

counts a carbon-only coordination number with 120575119894119895repre-

senting a Kronecker delta The hydrogen-only coordinationnumber 119873H

119894119895is defined similarly Using this coordination

number the angle-bending penalty function 119892119894switches

smoothly between a form 119892(1)

C appropriate for covalentcompounds with low coordination and another form 119892

(2)

Csuitable for highly coordinated bulk materials

119892C (cos 120579119895119894119896) = 119892(1)

C (cos 120579119895119894119896) + 119878119905(119905119873(119873119894119895))

times [119892(2)

C (cos 120579119895119894119896) minus 119892(1)

C (cos 120579119895119894119896)

(18)

At intermediate value of 119873 the switching function 119878119905(119905119873)

provides for a smooth transition with 119878119905 given by (11) andthe scaling function 119905

119873given by

119905119873(119873119894119895) =

119873119894119895minus 119873

min119894119895

119873max119894119895

minus 119873min119894119895

(19)

The 119890120582119895119894119896 term is added to improve the potential energy surfacefor abstraction of hydrogen atoms from hydrocarbons with

120582119895119894119896= 4120575119894H [(120575119896H120588HH + 120575119896C120588CH minus 119903119894119896)

minus (120575119895H120588HH + 120575119895C120588CH minus 119903119894119895)]

(20)

where 120575119894119895represents the Kronecker delta for atom types 119894

and 119895 The 119875119894119895term is a two-dimensional cubic spline [32]

in 119873C119894119895and 119873H

119894119895 whose coefficients are chosen to reproduce

the values of 119875119894119895 These terms are included in the REBO

model to give accurate bond energies for small hydrocarbonsAlthough most REBO parameters were not modified indeveloping the AIREBO model the 119875

119894119895values were modified

at two points to counteract the additional torsion energiesin the AIREBO potential for unsaturated systems such asethylene and graphite In addition to the bonding interactionsgiven by (15) the REBO potential also includes contributionsto the bond order from radial and conjugation effects Theseenter the potential through the 120587119903119888

119894119895term which is a three-

dimensional cubic spline in the variables119873119894119895119873119895119894 and119873conj

119894119895

The indices119873119894119895and119873

119895119894are the coordination numbers defined

in (16) and119873conj119894119895

is a local measure of conjugation in the 119894-119895bond

119873conj119894119895

= 1 +[

[

sum

119896 = 119894119895

120575119896C119908119894119896 (119903119894119896) 119878

119905(119905conj (119873119896119894))

]

]

2

+[

[

sum

119897 = 119894119895

120575119897C119908119895119897 (119903119895119897) 119878

119905(119905conj (119873119897119895))

]

]

2

(21)

with 119905conj specifying the range of coordination numbers underwhich a bond is assumed to be part of a radical or conjugatednetwork

119905conj (119873) =119873 minus 119873

min

119873max

minus 119873min

120587119903119888

119894119895= 119865119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

(22)

The remaining contribution to the bond-order 119887119894119895is 120587119889ℎ119894119895

120587119889ℎ

119894119895= 119879119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

times sum

119896 = 119894119895

sum

119897 = 119894119895

(1 minus cos2120596119896119894119895119897) times 119908119905

119894119896(119903119894119896) 119908119905

119895119897(119903119895119897)Θ

times (sin (120579119895119894119896) minus 119904

min) times Θ (sin (120579

119894119895119897) minus 119904

min)

cos120596119896119894119895119897

=

119903119895119894times 119903119894119896

10038161003816100381610038161003816119903119895119894times 119903119894119896

10038161003816100381610038161003816

sdot

119903119894119895times 119903119895119897

10038161003816100381610038161003816119903119894119895times 119903119895119897

10038161003816100381610038161003816

(23)

The bond-weighting function is

1199081015840

119905(119903119894119895) = 1198781015840(1199051015840

119888(119903119894119895)) (24)

The scaling function 1199051015840119888is

1199051015840

119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(25)

The LJ contribution to the 119894-119895 pair energy is

119864119871119869

119894119895= 119878 (119905

119903(119903119894119895)) 119878 (119905

119887(119887lowast

119894119895)) 119862119894119895119881119871119869

119894119895(119903119894119895)

+ [1 minus 119878 (119905119903(119903119894119895))] 119862119894119895119881119871119869

119894119895(119903119894119895)

(26)

includes the traditional LJ term

119881119871119869

119894119895(119903119894119895) = 4120576

119894119895[(

120590119894119895

119903119894119895

)

12

minus (

120590119894119895

119903119894119895

)

6

] (27)

modified by several sets of switching functionsThe switchingfunction 119878(119905) is

119878 (119905) = Θ (minus119905) + Θ (119905)Θ (1 minus 119905) [1 minus 1199052(3 minus 2119905)] (28)

Both have continuous first derivatives at the switching regionboundaries

Below a certain separation distance 119903119871119869max119894119895

themagnitudeof the LJ term depends on the bonding environment Thegradual exclusion of LJ interactions as 119903

119894119895changes is con-

trolled by the 119905119903 scaling function

119905119903(119903119894119895) =

119903119894119895minus 119903119871119869min119894119895

119903119871119869max119894119895

minus 119903119871119869min119894119895

(29)


At intermolecular distances the LJ interaction is includedonly if there is no significant bonding interaction betweentwo atoms as specified by 119905

119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)

And if the atoms 119894 and 119895 are not connected by two or fewerintermediate atoms this latter switch is controlled by bondweights

119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)

The torsional potential for the dihedral angle determinedby atoms 119894 119895 119896 and 119897 is controlled by the term

119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)

4 Results and Discussion

At different strain rates we simulate CNTs subjected to auniform axial loading at its both ends using Tersoff poten-tial [10] Rebo potential [11] and Airebo potential [29] tounderstand the effect of strain rates on their potential energyand elastic elastic-plastic properties To understand the effectof Van der Waals interaction and torsion with bond-orderpotential per atom potential energy with strain of single-walled 12-12 CNTs subjected to a uniform tension at theirboth ends obtained by different potentials are compared toeach other At high strain rate we simulate CNTs subjectedto an axial tension at their both ends using five differentpotentials such as Tersoff potential [10] Rebo potential [11]Rebo potential [31] Inclusion LJ with Rebo potential [11]and Airebo potential [29] to study the effect of potentials ontheir elastic elastic-plastic properties Tersoff potential [10]and Rebo potential [11] are bond-order potential by shortrange cut-off functions Out of the cut-off distance carbonatoms in CNT in different planes and layers do not interactalthough they experience weak Van der Waals interactionFor nonbonded atoms although the Van derWaals attractionand repulsion exhibit at a short distance but for the bondedatoms these Van der Waals attraction and repulsion occurat a long distance When bond-order potential is usedtorsional term must be included to get higher accuracy ofthe solution because during simulation position of atomschanges due to interacting forces among bonded atoms andbond experiences some angular displacement Stuart et al[29] introduced long range Van der Waals and torsionalterms in Rebo potential [11] and obtained elastic constantsbond energy vacancy formation energy enthalpy of carbonin diamond and graphite structures using Rebo potential [11]and modified Rebo potential that is Airebo potential [29]

and compared their results obtained by Airebo potential [29]and Rebo potential [11] with experimental resultsThe resultsobtained by Airebo potential [29] show better agreementthan that of Rebo potential [11] with experimental resultsFrom the study of diamond and graphite structure by Stuartet al [29] it is clear that Airebo potential [29] is moresuitable for getting highly accurate solution of CH structuresthan that of Rebo potential [11] Firstly we obtain per atompotential energy of a single-walled 12-12 CNT as a functionof axial strain at a very low strain rate using three differentpotentials such as Airebo potential [29] Rebo potential[11] and Tersoff potential [10] to understand the effect ofVan der Waals interaction and torsional effects on bond-order potential whose effects are reflected in the elasticelastic-plastic properties of CNTs We obtain the stress strainrelationship of a single-walled 12-12 CNT subjected to auniaxial loading by molecular dynamics simulation usingthree different potentials at four different strain rates that isfrom very low strain rate to high strain rate and comparethese with that obtained by Belytschko et al [30] to getreliable potential for studying the mechanics of CNTs Effectsof strain rates on the elastic elastic-plastic properties of asingle-walled 12-12 CNT subjected to an axial loading usingdifferent potentials are investigated For very low to highstrain rates a suitable potential is selected for the study ofnanomechanics of CNTs from the comparative study of peratom potential energy and tensile stress as a function of axialstrain using three different potentials with those of publishedresults To understand the nanomechanics of single-walleddouble-walled three-walled and four-walled CNTs at highstrain rate five different potentials are also used which areRebo potential [11] Rebo potential [31] Airebo potential [29]Tersoff potential [10] and Inclusion LJ in Rebo potential [11]In Rebo potential [11] only bond-order term is included InRebo potential [31] torsion term is added with bond-orderterm In Airebo potential [29] Van der Waals interaction andtorsional terms are added with bond-order term In Tersoffpotential [10] only bond order is added In Inclusion LJ inRebo potential [11] Van der Waals interaction is added withbond-order term Effects of layering and aspect ratio on theelastic elastic-plastic properties of CNTs are studied at highstrain rate considering five different potentials as comparativemanners with those of published results We compare theelastic elastic-plastic properties of different types of CNTssubjected to a tensile load at its opposite end at a high strainrate using five different potentials with that of Liew et al[13] Youngrsquos modulus of CNTs is obtained from the slopeof the stress strain relationship of CNTs till 5 strain Yieldstrength and yield strain of CNTs are measured from a pointof the stress strain curvewhere its slope is the lowest Ultimatestrength and ultimate strain of CNTs are measured from apoint of stress strain curve where the stress is the highestPoissonrsquos ratio of CNTs is calculated considering the change ofits diameter and length just before fracture using the ratio oflateral to longitudinal strain Besides we present the elasticelastic-plastic properties of CNTs at high strain rate usingdifferent potentials with the published results as a tabularform We also obtain Youngrsquos modulus yield strength yieldstrain ultimate strength and ultimate strain of single-walled


FF

(a)

(b)

Figure 1 Physical configuration of a single-walled (12 12) CNTat length-to-diameter ratio 451 (a) original configuration (b)snapshot just before fracture obtained by molecular dynamicssimulation using Airebo potential [29]

Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)

Rebo potential [11]Airebo potential [12]Tersoff potential [10]

Tension at a very low strain rate

minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64

Figure 2 Per atom potential energy of a single-walled (12 12) CNTwith length-to-diameter ratio 91 as a function of axial strain whenboth ends of CNT are stretched at a velocity 009 ms (with very lowstrain rate)

double-walled three-walled and four-walled CNTs from theaxial stress strain relationships of different types of CNTsusing different potentials by molecular dynamics simulationand present in tabular form

Figure 1 presents the original and final physical config-uration just before breaking of a single-walled 12-12 CNThaving length-to-diameter ratio 451 using Airebo potential[29] Figure 2 illustrates per atom potential energy of a single-walled 12-12 CNT with aspect ratio 91 as a function of strainat a very low strain rate (stretching velocity at both ends ofa CNT is 009ms) obtained by Rebo potential [11] Airebopotential [29] and Tersoff potential [10] as a comparativemanner From Figure 2 it is clearly observed that per atom

potential energy of CNTs obtained by Tersoff potential is thelowest per atom potential energy of CNTs obtained by Rebopotential [11] is the highest and per atom potential energyobtained by Airebo potential [29] remains in between themdue to addition of Van der Waals and torsional interactionswith bond-order term in Airebo potential [29] Although theslope of per atom potential energy as a function of strain forRebo potential [11] and Airebo potential [29] is almost thesame the slope of per atom potential energy as a function ofaxial strain of single-walled 12-12 CNTs obtained by Tersoffpotential [10] is higher than that of Rebo potential [11] andAirebo potential [29] From Figure 2 it is observed thatfracture strain of a single-walled 12-12 CNT obtained by Rebopotential [11] and Airebo potential [29] is almost the samebut fracture strain of single-walled 12-12 CNTs obtained byTersoff potential [10] is the lowest

Figure 3 illustrates the comparative study of the ten-sile stress strain relationship of single-walled 12-12 CNTssubjected to an axial tension using Rebo potential [11]Airebo potential [29] and Tersoff potential [10] at fourdifferent strain rates with that of Belytschko et al [30]From Figure 3(a) it is observed that the tensile stress strainrelationships of single-walled 12-12 CNTs obtained by Rebopotential [11] and Airebo potential [29] at a very small strainrate (stretching velocity of both ends of CNTs is 009ms) arevery close to that of single-walled 12-12 CNTs by Belytschkoet al [30] but the tensile stress strain relationship obtainedby Tersoff potential [10] deviates a large extent from thatof Belytschko et al [30] From Figure 3(b) it is observedthat tensile stress strain relationship of a single-walled 12-12 CNT obtained by Rebo potential [11] and Airebo potential[29] at a small strain rate (stretching velocity at both ends ofCNTs is 05ms) is almost the same and up to yield strengthand the tensile stress strain relationship obtained by Rebopotential [11] and Airebo potential [29] matches with thatof Belytschko et al [30] but after yielding the tensile stressstrain relationship of a single-walled 12-12 CNT obtainedby Rebo potential [11] and Airebo potential [29] disagreeswith that of Belytschko et al [30] At a very low (stretchingvelocity at both ends of CNTs is 009ms) and low strain(stretching velocity at both ends of CNTs is 05ms) rate theslope of stress strain relationship obtained byTersoff potentialis higher than that of Rebo potential [11] Airebo potential[29] and Belytschko et al [30] because decreasing rate ofper atom potential energy of a single-walled 12-12 CNT withaxial strain obtained by Tersoff potential is higher than thatof Rebo potential [11] and Airebo potential [29] as shownin Figure 2 Ultimate stress and strain of single-walled 12-12 CNT obtained by Rebo potential [11] and Airebo potential[29] are higher than that of Belyschko et al [30] when thestretching velocity at both ends of CNTs is 05ms FromFigure 3(c) it is observed that ultimate strength and strain ofa single-walled 12-12 CNT at moderate strain rate (stretchingvelocity at both ends of CNTs is 5ms) obtained by Rebopotential [11] are higher than those of Airebo potential [29]because at moderate strain rate Van der Waals and torsionaleffect decrease per atom potential energy of CNTs with strainfor which it takes less strain and time to reach the lowestper atom potential energy for fracture but for fracture of


CNTs opposite phenomenon is observed when tension testof a CNT is carried out using Rebo potential [11] at moderatestrain rate At high strain rate the deviation between ultimatestrength and strain of single-walled 12-12 CNTs obtained byRebo potential [11] and Airebo potential [29] is higher thanthat of moderate strain rate At moderate and high strainrate ultimate strength of single-walled 12-12 CNTs obtainedby Airebo potential [29] is closer to that of Belytschkoet al [30] than that of Rebo potential [11] In all strainrates the slope of the stress strain relationship of a single-walled 12-12 CNT obtained by Tersoff potential [10] is higherthan that of Rebo potential [11] Airebo potential [29] andBelytschko et al [30] As at very low strain rate the stressstrain relationship of a single-walled 12-12 CNT subjected toaxial loading obtained by Rebo potential [11] and Airebopotential [29] is very close to that of Belytschko et al [30]Airebo potential [29] and Rebo potential [11] are suitable forthe study of nanomechanics of CNTs at a very low strainrate At a very low strain rate per atom potential energywith strain of a single-walled 12-12 CNT obtained Airebopotential [29] ismore accurate than that of Rebo potential [11]because Stuart et al [29] observed that the vacancy formationenergy bond energy and enthalpy of graphite and diamondby Airebo potential [29] is closer to those of experimentalresult than those obtained byRebo potential [11] Atmoderateand high strain rates Airebo potential [29] will give moreaccurate per atom potential energy and elastic elastic-plasticresponse of CNTs during loading till fracture than those ofother potentials because each bonded atom experiences VanderWaals interaction and torsional effect during loading andthese two terms are not included in the rest of other potentialstogether

Figure 4 describes the stress strain relationship of asingle-walled 12-12 CNT having length-to-diameter ratio 91subjected to uniform tension at its opposite ends usingAirebo potential [29] by molecular dynamics simulation athigh strain rate (stretching velocity at both ends of CNTs is20ms) The stress strain relationship curve gives three onsetcut-off diameters where the elastic elastic-plastic propertieschange suddenly In Figure 4 the slope of the curve at acertain range is constant that is the stiffness of the CNTremains constant and then the slope of the stress strainrelationships of the CNT gradually decreases to a zero valuethat is the stiffness of the CNT gradually decreases and whenthe diameter of the CNT suddenly drops the slope of thestress strain curve becomes zero and this is the phase changeof CNT due to strain hardening and decreasing its diametersuddenly enhances its stiffness to a greater extent till fracture

Figures 5(a) and 5(b) illustrate the effect of differentpotentials on per atom potential energy and per atom totalenergy of CNTs as a function of strain with the tensile loadat their both ends by molecular dynamic simulation at highstrain rate Per atom potential energy of loaded CNTs at aspecific strain before fracture obtained by Rebo potential [11]is higher than that obtained byAirebo potential [29] as shownin Figure 5(a) due to the absence of the torsional and Vander Waals interaction in Rebo potential [11] and the fracturestrain of CNTs obtained by Rebo potential [11] is higher thanthat of Airebo potential [29] although the per atom potential

energy of CNTs during fracture obtained by Rebo potential[11] and Airebo potential [29] is almost the same As theslope of per atom potential energy as a function of axialstrain of loaded CNTs obtained by Rebo potential [11] andAirebo potential [29] is nearly the same and fracture strain ofCNTs obtained by Rebo potential [11] considering per atompotential energy is higher than that of Airebo potential [29]the slope of the stress strain relationship of CNTs obtainedby Rebo potential [11] and Airebo potential [29] should bethe same and ultimate fracture strength of CNTs obtainedby Rebo potential [11] should be higher than that of Airebopotential [29] Although Tersoff potential [10] is also a bond-order potential like Rebo potential [11] the slope of per atompotential energy of CNTs as a function of strain obtainedby Tersoff potential [10] is the highest among those of otherpotentials whichmeans the slope of the stress strain of loadedCNTs obtained by Tersoff potential [10] should be higherthan that of other potentials From Figure 5(a) it is clearthat ultimate strength of CNTs obtained by Tersoff potential[10] should be the lowest of all other potentials Figure 5(b)illustrates per atom total energy as a function of strain ofCNTs obtained by five different potentials and after fractureper atom total energy of CNTs obtained by these potentialdecreases a little amount due to its increased kinetic energyEffect of kinetic energy on per atom total energy of CNTsobtained by all of the mentioned potentials is small

Liew et al [13] obtained tensile stress strain relationship ofsingle-walled double-walled three-walled and four-walledCNTs with aspect ratios 91 and 451 respectively usingRebo potential [11] with Van der Waals interaction withoutminimizing the initial configuration of CNTs before applyingtensile loading at high strain rate (stretching velocity atboth ends of the CNTs is 20ms) Before loading CNTs itremains in minimized energy level So to get accurate elasticelastic-plastic response of CNTs we should minimize energybefore applying loading because afterminimization of energyapplying potentials in molecular dynamics simulation theposition of C atoms changes and little changes are observedin shapes of CNTs Besides Agrawal et al [14] studied elasticelastic-plastic response of single wall CNTs subjected to auniform axial loading varying CndashC bond length and bondangle at high strain rate using only bond-order Rebo potential[11] In the present simulation of CNTs weminimize differenttypes of CNTs with aspect ratios 91 and 451 respectivelybefore loading using five different potentials and then applyload at its both ends (stretching velocity at both ends of CNTsis 20ms) like Liew et al [13] and Agrawal et al [14] Wecompare the tensile stress strain relationships of single-walledCNTs obtained by us usingRebo potential [11] Rebo potential[31] Airebo potential [29] Inclusion LJ with Rebo potential[11] and Tersoff potential [10] with those of Agrawal et al[14] and Liew et al [13] Besides we compare the tensilestress strain relationships of double-walled three-walledand four-walled CNTs obtained by us using Rebo potential[11] Rebo potential [31] Airebo potential [29] InclusionLJ with Rebo potential [11] and Tersoff potential [10] withthose of Liew et al [13] From the comparative studies wetry to obtain accurate tensile stress strain relationships ofdifferent types of CNTs and suitable potential for finding


Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

Belytschko et al [16] Airebo potential [12]

Rebo potential [11]Tersoff potential [10]


(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120


Tersoff potential [10]Rebo potential [11]

Tension at a low strain rate

(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180

Tension at moderatestrain rate

Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240

Tension at high strain rate

Axial strain

Tens

ile st

ress

(GPa

)



(d)

Figure 3 Comparison of the stress strain relationship of a single-walled 12-12 CNT subjected to a uniform tensile load at its both ends for thecase of different strain rates using different potentials with those obtained by DFT calculation by Belytschko et al [30] (a) at very low strainrate (stretching velocity of both ends of a CNT = 009ms) (b) at low strain rate (stretching velocity of both ends of a CNT = 05ms) (c) atmoderate strain rate (stretching velocity of both ends of CNTs = 5ms) (d) at high strain rate (stretching velocity of both ends of a CNT =20ms)


Table 1 Comparison of some parameters in Airebo potential by Stuart et al [29] with those of Sinnott et al [12] which were used by Liew etal [13] in their simulation of CNTs during coupling Van der Waals interaction with Rebo [11 31] potentials

Parameters used for the carbon carbon Van der Waalsinteractions (excluding the spline coefficients) Sinnott et al [12] Airebo potential Stuart et al [29] Units

120576CC 00042038 000284 ev120576CH mdash 000206 ev120576HH mdash 000150 ev120590CC 337 34 A120590CH mdash 3025 A120590HH mdash 265 A119903119871119869minCC 228 34 A

119903119871119869mediumCC 340 mdash A119903119871119869maxCC 10 10 A119903119871119869minCH mdash 3025 A119903119871119869maxCH mdash 3396228 A119903119871119869minHH mdash 265 A119903119871119869maxHH mdash 2975208 A

Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset

Figure 4 Stress strain curves for an ideal single-walled 12-12 CNThaving length-to-diameter ratio 91 at 300K temperature obtainedby Airebo potential [29] The onset of the cut-off function can beperceived as an artificial peakThe flat regime depicts a sudden bondelongation to a length corresponding to the cut-off After this flatregion the diameter of CNT drops

more accurate elastic elastic-plastic response of CNTs duringloading at high strain rate Figures 6(a) and 6(b) illustratethe tensile stress strain relationship of single-walled 12-12 CNThaving length diameter ratios 91 and 45 respectivelyby five different potentials with two published results as acomparative manner at high strain rate (stretching velocity atboth ends of CNTs is 20ms)The nature of the elastic-plasticregion obtained by Agrawal et al [14] is totally different fromthat of Liew et al [13] The elastic-plastic region obtained bythe present five different potentials agrees well with that ofAgrawal et al [14] although Agrawal et al [14] solved the

CNT problem using Rebo potential [11] On the other handLiew et al [13] solved the CNT problem using Rebo potentialincluding Van der Waals interactions and we also solved theCNT problem using Airebo potential [29] Inclusion LJ withRebo potential [11] in which Airebo potential [11] consist oflong range Van der Waals interaction and torsional effect ofbond aswell as Inclusion LJ with Rebo potential [11] consist ofRebo potential [11] andVanderWaals interactionThe elastic-plastic region of CNT obtained by Airebo potential agreeswell with that of Agrawal et al [14] and the ultimate strengthof CNT obtained by Agrawalet al [14] is higher from that ofAirebo potential [29] because they did not consider Van derWaals interactions and torsional effect From the comparativestudy it is clear that the elastic elastic-plastic properties ofCNT obtained by Airebo potential [29] are more reliable andaccurate The elastic elastic-plastic response of CNT by Liewet al [13] does not agree well with other results due to lackof minimization of CNT structure before loading Besidesthe magnitude of the potential depth for Van der Waalsinteraction as shown in Table 1 is nearly two times that of oursin Airebo potential [29] and it is another reason to deviate thestress strain relationship of single-walled 12-12 CNTs by Liewet al [13] from that of ours and Agrawal et al [14] Effects ofVan derWaals interaction and torsional effect on the per atompotential energywith strain stress strain relationship ofCNTssubjected to uniform tension at high strain rate are clearlyobserved in Figures 5 and 6 Figures 7(a) and 7(b) illustratethe tensile stress as a function of axial strain of a single-walled 10-10 CNT with length-to-diameter ratios 91 and 451respectively using five different potentials with the publishedresults by Liew et al [13] Up to elastic region the stressstrain relationship obtained by five different potentials agreeswell with Liew et al [13] but in the elastic-plastic region theshapes of the curves of the stress strain relationship of CNTby us for five different potentials are different from that ofLiew et al [13] which contradicts the reliability of the stress


Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom

Airebo potential [12]Rebo potential [11]

Rebo potential [15] Tersoff potential [10]

Single-walled 12-12 CNT

minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91

Inclusion LJ with Rebo potential [11]

(a)

Tota

l ene

rgy

per a

tom

Single-walled 12-12 CNTLD = 91

Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)

Figure 5 Effect of empirical potentials (a) on the average potential energy as a function of axial strain (b) per atom total energy as a functionof axial strain of a single-walled 12-12 CNT under tension with steps at 300K


Airebo potential [12]

Rebo potential [11] Rebo potential [15]

Tersoff potential [10]

Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)

Agrawal et al [17] Dmin = 160 AAgrawal et al [17] Dmin = 165 AAgrawal et al [18] Dmin = 170 A


(a)

0

50

100

150

200

250

300

Rebo potential [11]Rebo potential [15]Tersoff potential [10]


Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)

Liew et al [14]Agrawal et al [17] Dmin = 160 AAgrawal et al [17] Dmin = 165 AAgrawal et al [18] Dmin = 170 A


(b)

Figure 6 Comparison of the elastic elastic-plastic response of a single-walled 12-12 CNT using different potentials with the published results(a) LD = 91 (b) LD = 451


Airebo potential [12]Rebo potential [11]Rebo potential [15]

Tersoff potential [10]Liew et al [14]

0

50

100

150

200

250


Axial strain00 01 02 03 04

LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)Inclusion LJ with Rebo potential [11]

(b)


strain relationship obtained by Liew et al [13] because thestress strain relationship of CNTs for five different potentialsshows three onset cut-off which ensure sudden change indiameter of the CNT which is also reflected in the recentpublished results byAgrawal et al [14] and Sammalkorpi et al[21] Variations of stress strain relationships of single-walledCNTs are observed if length-to-diameter ratio and diameterof CNTs are varied as shown in Figures 6 and 7

From Figures 8(a) and 8(b) it is observed that the stressstrain relationship of a double-walled CNT differs for fivedifferent potentials although the trends of the shape of thestress strain relationship are similar and the elastic range thestress strain relationship obtained by five different potentialsare nearly equal to that of Liew et al [13] In the elastic-plasticrange of double-walled CNTs our results obtained by fivedifferent potentials disagree with that of CNTs obtained byLiew et al [13] Figures 9 and 10 illustrate the same stressstrain relationship of three-walled and four-walled CNTs intwo aspect ratios 91 and 45 using five different potentialswith the published results by Liew et al [13] as a comparativemanner at high strain rate the stress strain relationship in theelastic range of three-walled and four-walled CNTs obtainedby five different potentials agrees with that of Liew et al [13]but the stress strain relationship in the elastic-plastic range ofthree-walled and four-walled CNTs obtained by five differentpotentials does not agree with that of Liew et al [13] Amongthe stress strain relationships obtained by Tersoff potential[10] Rebopotential [11] Rebopotential [31] InclusionLJwithRebo potential [11] and Airebo potential [29] there is the

highest reliability in the stress strain relationship obtained byusing Airebo potential [29] because this potential includeslong range Van der Waals interaction and torsional effect ofbonds with Rebo potential [11] Naturally every bonded atomexperiences weak Van der Waals interaction after a certaindistance when bonded atom displaces its position there issome angular displacement for which CndashC bonds of CNTsexperience torsion At very low strain range the stress strainrelationship of a single-walled 12-12 CNT obtained by Rebopotential [11] and Airebo potential [29] agrees well with thatof Belytschko et al [30] Although at very low strain rangethe effects of Van derWaals interaction and torsional effect onthe elastic elastic-plastic response are negligible at moderateand high strain rate the effects of Van der Waals interactionand torsion effect play important roles which are observedin the graphical results of stress strain relationships of CNTsduring loading (see Figures 3 and 6ndash10) Effects of Van derWaals interaction and torsional effects on per atom potentialenergy of CNTs during tension at different strain rates areobserved (see Figures 2 and 5) Effects of layering on thestress strain relationship of CNTs subjected to tensile loadat a high strain rate are also observed as shown in Figures8 to 10 Effects of Van der Waals interaction and torsion onthe stress strain relationships of double-walled three-walledand four-walled CNT are observed from their comparativestudy of those of Rebo potential [11] Rebo potential [31]Inclusion LJ with Rebo potential [11] and Airebo potential[29] as observed in Figures 8 to 10 Figure 11 illustrates thephysical configuration of four-walled CNTs at different steps


Double-walled 5-5 10-10



0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)

Double-walled 5-510-10 CNT



0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)

Figure 8 Comparison of the elastic elastic-plastic response of a double-walled 5-5 10-10 CNT using different potentials with the publishedresults (a) LD = 91 (b) LD = 451

Three-walled 5-5 10-10 15-15 CNT



0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140

160Three-walled 5-5 10-10 15-15 CNT



0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)

Figure 9 Comparison of the elastic elastic-plastic response of a three-walled 5-5 10-10 and 15-15 CNT using different potentials with thepublished results (a) LD = 91 (b) LD = 451


Table 2 Elastic elastic-plastic properties of a single-walled CNT by the tension test (tension test is carried out by applying velocity of someatoms at both ends of the CNT 20ms)

119871119863 Potentials 119864 GPa 120590119910 GPa 120576

119910120590119906 GPa 120576

119906V


91

Airebo [29] 91220 1128916 0223 1697363 03167 035680Tersoff [10] 10975 1150750 01754 15532130 026110 026110Rebo [11] 88342 1108912 02136 2145082 0357 035457Rebo [31] 89668 1126569 0225 19618111 035500 mdashInclusion 119871119869 with Rebo [11] 89792 1145128 02342 18186681 031920 mdash

451

Airebo [29] 93507 11445180 023530 16125031 031590 029046Tersoff 11077 1203995 02005 1477876 02697 01744Rebo [11] 89977 11147040 021830 19681650 034170 026955Rebo [31] 92077 11398380 023500 19424210 035590Inclusion 119871119869 with Rebo [11] 91315 11463750 023670 17287730 031780


91

Airebo [29] 10384 11329630 021190 17272411 029830 022345Tersoff [10] 12369 1202628 01903 1733022 02713 01535Rebo [11] 10365 11081970 019820 23228951 034530 03350Rebo [31] 10401 11308620 021250 21239920 034550Inclusion 119871119869 with Rebo [11] 10405 11366160 021330 18336310 029990

451

Airebo 10839 11669020 022680 19944099 032050 018104Tersoff 11204 1180722 01862 1918131 03041 02267Rebo [11] 10844 11119750 019570 24791631 035530Rebo [31] 10841 11629670 022620 22250110 035370Inclusion 119871119869 with Rebo [11] 10889 11167250 019620 21141220 032130 03637

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT


Inclusion LJ with Rebo potential [11]Tersoff potential [10]Liew et al [14]

0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)

Figure 10 Comparison of the elastic elastic-plastic response of a four-walled 5-5 10-10 15-15 and 20-20CNT using different potentials withthe published results (a) LD = 91 (b) LD = 451


Table 3 Elastic elastic-plastic properties of multiwall CNT by the tension test (tension test is carried out by applying velocity of some atomsat both ends of the CNT 20ms)


119910120590119906 GPa 120576

119906

Double-walled 5-5 10-10 CNT

91

Airebo [29] 10145 11282160 021750 17550349 032870Tersoff [10] 11529 1123452 016460 16250680 027450Rebo [11] 97041 11263770 022090 19880260 033950Rebo [31] 10009 11244860 021870 19285950 034840Inclusion 119871119869 with Rebo [11] 98305 11317720 022080 16855659 030890

451



91

Airebo [29] 10712 11403640 021340 17642340 030660Tersoff 11121 1177347 01846 1809215 02946Rebo [11] 96032 11311430 021650 22430141 034430Rebo [31] 10113 11175010 020460 21210780 034500Inclusion 119871119869 with Rebo [11] 11185 11095180 019090 17431090 029360

451

Airebo [29] 75797 8074040 020330 12315620 031920Tersoff [10] 91749 8403300 017470 10858530 027420Rebo [11] 7557 8283280 022410 14203690 032090Rebo [31] 75659 8030010 020370 11780770 030060Include 119871119869 with Rebo [11] 7594 8085670 020450 12786900 030110

Four-walled 5-5 10-0 15-15 20-20 CNT

91


451


till fracture In the outer layer of a four-walled CNT how 5-7-7-5 defects generate during fracture which are reflected inFigure 11(c)

Table 2 lists the computed Youngrsquos modulus poisonrsquosration yield strength ultimate strength yield strain andultimate strain of single-walled 12-12 and 10-10 CNTs withlength-to-diameter ratios equal to 91 and 451 respec-tively using five different potentials Youngrsquos modulus yieldstrength and yield strain of single-walled 12-12 CNTs havingaspect ratio 91 obtained by five different potentials separatelyare lower than those of aspect ratio 451 but the oppositephenomenon is observed for its ultimate strength becauseper atom potential energy change with strain is higher forCNTs with lower aspect ratio than from its higher aspectratio Youngrsquos modulus and yield strength of a single-walled10-10 CNT having aspect ratio 91 obtained by Rebo potential[11] Rebo potential [31] and Airebo potential [29] are lowerthan those of aspect ratio 451 on the other hand ultimatestrength and strain having aspect ratio 91 obtained by Rebopotential [11] Rebo potential [31] Airebo potential [29]

Inclusion LJ with Rebo potential [11] and Tersoff potential[10] are lower than those of aspect ratio 451 separately Peratompotential energywith strain is dramatically changed dueto change of aspect ratio of CNTs which change the slope ofstress strain curve of CNTs subjected to a tensile load usingdifferent potentials and for which effects of aspect ratio areobserved on the elastic elastic-plastic properties of CNTsFrom Table 2 it is observed that Youngrsquos modulus yieldstrength yield strain ultimate strength and ultimate strainof 12-12 single-walled CNTs with the same aspect ratio differfrom those of single-walled 10-10 CNTs due to the size effectin diameters because diameter of single-walled 12-12 CNTis higher than that of single-walled 10-10 CNTs Effects ofdifferent potentials on Youngrsquos modulus yield strength yieldstrain ultimate strength ultimate strain and poisonrsquos rationof single-walled 12-12 and 10-10 CNTs are also observed in thetabular results

Table 3 lists the computed Youngrsquos modulus yieldstrength ultimate strength yield strain and ultimate strainof double-walled three-walled and four-walled CNTs with


(a)

(b)

(c)

Figure 11 Physical configuration of CNT at different state (a)original configuration (b) before fracture (c) 5-7-7-5 defects duringfracture of a four-walled 5-5 10-10 15-15 and 20-20 CNT (LD =451)

length-to-diameter ratios 91 and 451 respectively using fivedifferent potentialswith the published results Youngrsquosmoduliof double-walled CNT having aspect ratio 91 obtained byAirebo potential [29] Rebo potential [11] Rebo potential[31] and Inclusion LJ with Rebo potential [11] are lower thanthose of aspect ratio 451 separately but yield strengths ofa double-walled CNT having aspect ratio 451 using Rebopotential [11] Rebo potential [31] Airebo potential [29]Inclusion LJ with Rebo potential [11] and Tersoff potential[10] are higher than those of aspect ratio 951 separately Yieldstrains ultimate strengths and ultimate strains of double-walled CNTs having aspect ratio 451 using Rebo potential[11] Rebo potential [31] and Airebo potential [29] are lowerthan those of aspect ratio 91 Although Youngrsquos modulus ofCNTs with aspect ratio 91 obtained by Tersoff potential ishigher than that of aspect ratio 451 for yield strength yieldstrain ultimate strength and ultimate strain the oppositephenomenon is observed Youngrsquos moduli yield strengthsand ultimate strengths of three-walled CNT with aspect ratio91 obtained by all potentials mentioned in the Table 3 arehigher than those of aspect ratio 451 separately Yield strainand ultimate strain of three-walled CNTs obtained by Rebopotential [11] Rebo potential [31] Inclusion LJ with Rebopotential [11] and Airebo potential [29] mentioned in Table 3also show the effect of aspect ratio of CNTs Youngrsquos moduliyield strengths ultimate strengths of four-walled CNT with

aspect ratio 451 obtained by all potentials mentioned in theTable 3 are higher than those of aspect ratio 91 separatelyThere are also observed effects of aspect ratio on the yieldstrains and yield strengths of four-walled CNTs obtainedby all potentials mentioned in Table 3 From Table 3 it isclearly observed that Youngrsquos moduli yield strengths yieldstrains ultimate strengths and ultimate strains of double-walled three-walled and four-walledCNTswith aspect ratios91 and 451 obtained by Airebo potential [29] and InclusionLJ with Rebo potential [11] differ from those of all of the otherpotentials due to layering effects of CNTs because in Airebopotential [29] and Inclusion LJ with Rebo potential [11] Vander Waals interaction is added which interact with interlayercarbon atoms during loading Besides FromTables 2 and 3 itis clearly observed that Youngrsquos moduli yield strengths yieldstrains ultimate strengths and ultimate strains of single-walled double-walled three-walled and four-walled CNTswith aspect ratios 91 and 451 obtained by Rebo potential[31] and Airebo potential [29] differ from those of all of theother potentials due to torsional effects of CndashC bonds ofCNTs because in Rebo potential [31] and Airebo potential[29] torsional interaction is added which include angularmovement of CndashC bond during loading

From the tabular results it is clearly observed that Youngrsquosmodulus of single-walled double-walled three-walled andfour-walled CNTs with LD ratios 91 and 451 respectivelyobtained by Tersoff potential [10] is greater than that obtainedby Rebo potential [11] Rebo potential [31] Inclusion LJ withRebo potential [11] and Airebo potential [29] because theslope of per atom potential energy with strain of CNTsobtained by Tersoff potential is higher than those of otherpotentials (see Figures 2 and 5) On the other hand theultimate strength of single-walled double-walled three-walled and four-walled CNTs with LD ratios 91 and 451respectively obtained by Rebo potential [11] is greater thanthat obtained by Tersoff potential [10] Rebo potential [31]Inclusion LJ with Rebo potential [11] and Airebo potential[29] because per atom potential energy of CNTs with strainobtained by Rebo potential is higher than those of otherpotentials and it takes the highest stress and strain to reachminimumper atompotential energy for the breaking ofCNTsduring tensile loading using Rebo potential [11] (see Figures 2and 5) For single- double- three- and four-walledCNTs theultimate strength obtained by Airebo [29] potential is lowerthan that of Rebo potential [11] because in Airebo potential[29] torsional and Van der Waals interactions are presentwhich reduce per atom potential energy for which ultimatestrength of CNTs obtained byAirebo potential [29] is lower ascompared to Rebo potential [11] For single- double- three-and four-walled CNTs the effects of torsion and Van derWaals interactions are observed clearly when their ultimatestrength obtained by Rebo potential [11] is compared withthat of Rebo potential [31] Inclusion LJ with Rebo potential[11] and Airebo potential [29] From the tabular results itis clear that torsion and Van der Waals interactions play animportant role in their mechanical properties in the plasticrange during loading

When tensile load is applied at its both ends of CNTsusing different potentials by molecular dynamics simulation


per atom potential energy decreases and during fracture peratom potential energy reaches aminimum value As per atompotential energy of CndashC structure using Tersoff potential islower (Table 1 in [33]) than that of Rebo potential [11] andRebo potential [31] CNT will break at lower strain than thatof other potentials and this phenomenon is already reflectedin the above graphical and Tabular results As bond radiusand bond length of CndashC in Tersoff potential are higher(see Table 1 in [33]) than those of Rebo [11] and Rebo [31]potential the stiffness of CNT and the slope of the stressstrain curves using Tersoff potential should be higher thanthose of Rebo potential [11] and Rebo potential [31] andthis phenomenon is clearly observed in the graphical andTabular results As per atom potential energy of CndashC in Rebopotential [11] is higher (see Table 1 in [33]) than that ofRebo [31] and Tersoff potential [10] the ultimate strengthof CNT using Rebo potential [11] should be higher thanthat of Rebo [31] and Tersoff potential [10] because to reachminimum per atom potential energy for its breaking it needsthe highest tensile stress among all of the other potentialsand from the graphical results this phenomenon is alsoreflected in the graphical and tabular results When Van derWaals interaction is present in Airebo potential [29] duringtension some neighboring carbon atoms come in repulsiveand attractive zone As a result C atoms experience extrastress during loading which enhance decreasing the ultimatestrength of CNT The other reason is that per atom potentialenergy of CNT structure obtained by Airebo potential [29] islower than that of Rebo potential [11] and greater than that ofTersoff potential [10] So for this reason the ultimate strengthof different types CNT using Airebo potential [29] is higherthan that of Tersoff potential [10] and less than that of Rebopotential [11] which is reflected in the graphical and Tabularresults

From the tabular results as shown in Tables 2 and 3 theeffects of layering length-to-diameter ratio of CNTs on theirelastic elastic-plastic properties are clearly observed whenthe tension tests are carried out using different potentialsBesides there is an observed effect of empirical potentialson Youngrsquos modulus yield strength yield strain ultimatestrength ultimate strain and Poissonrsquos ratio of CNT undertension test using molecular dynamics simulation Effects ofpotentials on the reliability and correctness of the solution arealso reflected in the graphical and tabular results Effects oflayers and length-to-diameter ratio of CNT are also reflectedin the tabular results

5 Conclusions

We study per atom potential energy of single-walled CNTsubjected to axial tensile load at its opposite ends at a verylow strain rate using Rebo potential [11] Rebo potential [31]Airebo potential [29] and Tersoff potential [10] We comparethe stress strain relationship of single-walled 12-12 CNTsobtained by Rebo potential [11] Rebo potential [31] Airebopotential [29] and Tersoff potential [10] at a very low strainrate with that of Belytschko et al [30] obtained by densityfunctional theory and the stress strain relationship obtainedby Rebo potential [11] and Airebo potential [29] agree with

that of Belytschko et al [30] At high strain rate we also studythe elastic elastic-plastic properties of single andmultiwalledCNT subjected to uniform tension at its opposite ends usingfive different potentials such as Rebo potential [11] Rebopotential [31] Inclusion LJ with Rebo potential [11] Tersoffpotential [10] and Airebo potential [29] with the publishedresults as a comparative manner to understand the effect ofpotentials on the solution From the comparative study it isclearly understood that the reliability and accuracy of Airebopotential [29] are higher than other potentials for the elasticelastic-plastic analysis of CNT because in Airebo potential[29] long range Van derWaals interaction and torsional effectof bond are included To test the reliability of elastic elastic-plastic properties of CNTs obtained by us using differentpotentials we compared our results with some publishedresults and found out major limitations of such results whichwill be helpful for the further study of the elastic elastic-plastic properties of CNTs under tension Effects of numberof layers length-to-diameter ratio and empirical potentialsare also observed on the elastic elastic-plastic properties ofCNTs

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

References

[1] S Lijima and T Lchihashi ldquoSingle-shell carbon nanotubes of1-nm diameterrdquo Nature vol 363 no 6430 pp 603ndash605 1993

[2] D S Bethune C H Kiang M S De Vries et al ldquoCobalt-catalysed growth of carbon nanotubes with single-atomic-layerwallsrdquo Nature vol 363 no 6430 pp 605ndash607 1993

[3] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

[4] T W Ebbesen and P M Ajayan ldquoLarge-scale synthesis ofcarbon nanotubesrdquoNature vol 358 no 6383 pp 220ndash222 1992

[5] L Merhari Hybrid Nanocomposites for Nanotechnology Elec-tronic Optical Magnetic and Biomedical Applications SpringerNew York NY USA 2009

[6] O Breuer and U Sundararaj ldquoBig returns from small fibersa review of polymercarbon nanotube compositesrdquo PolymerComposites vol 25 no 6 pp 630ndash645 2004

[7] B I Yakobson C J Brabec and J Bernholc ldquoNanomechanicsof carbon tubes instabilities beyond linear responserdquo PhysicalReview Letters vol 76 pp 2511ndash2524 1996

[8] A Garg J Han and S B Sinnott ldquoInteractions of carbon-nanotubule proximal probe tips with diamond and graphenerdquoPhysical Review Letters vol 81 no 11 pp 2260ndash2263 1998

[9] B I Yakobson M P Campbell C J Brabec and J BernholcldquoHigh strain rate fracture and C-chain unraveling in carbonnanotubesrdquo Computational Materials Science vol 8 no 4 pp341ndash348 1997

[10] J Tersoff ldquoNew empirical approach for the structure and energyof covalent systemsrdquo Physical Review B vol 37 no 12 pp 6991ndash7000 1988

[11] D W Brenner ldquoEmpirical potential for hydrocarbons for usein simulating the chemical vapor deposition of diamond filmsrdquoPhysical Review B vol 42 no 15 pp 9458ndash9471 1990


[12] S B Sinnott O A Shenderova C TWhite and DW BrennerldquoMechanical properties of nanotubule fibers and compositesdetermined from theoretical calculations and simulationsrdquoCarbon vol 36 no 1-2 pp 1ndash9 1998

[13] K M Liew X Q He and C H Wong ldquoOn the study ofelastic and plastic properties of multi-walled carbon nanotubesunder axial tension usingmolecular dynamics simulationrdquoActaMaterialia vol 52 no 9 pp 2521ndash2527 2004

[14] P M Agrawal B S Sudalayandi L M Raff and R KomandurildquoMolecular dynamics (MD) simulations of the dependenceof C-C bond lengths and bond angles on the tensile strainin single-wall carbon nanotubes (SWCNT)rdquo ComputationalMaterials Science vol 41 no 4 pp 450ndash456 2008

[15] C H Wong and V Vijayaraghavan ldquoNanomechanics of imper-fectly straight single walled carbon nanotubes under axialcompression by using molecular dynamics simulationrdquo Com-putational Materials Science vol 53 no 1 pp 268ndash277 2012

[16] K I Tserpes and P Papanikos ldquoFinite element modeling ofsingle-walled carbon nanotubesrdquoComposites Part B vol 36 no5 pp 468ndash477 2005

[17] J-P Salvetat-Delmotte and A Rubio ldquoMechanical properties ofcarbon nanotubes a fiber digest for beginnersrdquo Carbon vol 40no 10 pp 1729ndash1734 2002

[18] B Coto I Antia M Blanco et al ldquoMolecular dynamics studyof the influence of functionalization on the elastic propertiesof single and multiwall carbon nanotubesrdquo ComputationalMaterials Science vol 50 no 12 pp 3417ndash3424 2011

[19] A Mashreghi and M M Moshksar ldquoBond lengths and bondangles of armchair single-walled carbon nanotubes throughmolecular dynamics and potential energy curve approachesrdquoComputational Materials Science vol 49 no 4 pp 871ndash8752010

[20] W Tang M H Santare and S G Advani ldquoMelt processing andmechanical property characterization of multi-walled carbonnanotubehigh density polyethylene (MWNTHDPE) compos-ite filmsrdquo Carbon vol 41 no 14 pp 2779ndash2785 2003

[21] M Sammalkorpi A Krasheninnikov A Kuronen K Nord-lund andKKaski ldquoMechanical properties of carbon nanotubeswith vacancies and related defectsrdquo Physical Review B vol 70no 24 Article ID 245416 8 pages 2005

[22] Q Wang J Dai W Li Z Wei and J Jiang ldquoThe effects of CNTalignment on electrical conductivity andmechanical propertiesof SWNTepoxy nanocompositesrdquo Composites Science andTechnology vol 68 no 7-8 pp 1644ndash1648 2008

[23] A Allaoui S Bai H M Cheng and J B Bai ldquoMechanical andelectrical properties of aMWNTepoxy compositerdquoCompositesScience and Technology vol 62 no 15 pp 1993ndash1998 2002

[24] J-P Salvetat J-M Bonard N B Thomson et al ldquoMechanicalproperties of carbon nanotubesrdquo Applied Physics A vol 69 no3 pp 255ndash260 1999

[25] Y J Liu and X L Chen ldquoEvaluations of the effective mate-rial properties of carbon nanotube-based composites using ananoscale representative volume elementrdquo Mechanics of Mate-rials vol 35 no 1-2 pp 69ndash81 2003

[26] A Hernandez-Perez and F Aviles ldquoModeling the influenceof interphase on the elastic properties of carbon nanotubecompositesrdquo Computational Materials Science vol 47 no 4 pp926ndash933 2010

[27] S K Deb Nath H Tohmyoh and M A Salam AkandaldquoEvaluation of elastic elastic-plastic properties of thin Pt wireby mechanical bending testrdquo Applied Physics A vol 103 no 2pp 493ndash496 2011

[28] S K Deb Nath and S -G Kim ldquoOn the elastic elastic-plasticproperties of Au nanowires in the range of diameters 1ndash200nmrdquoJournal of Applied Physics vol 112 no 12 Article ID 123522 10pages 2012

[29] S J Stuart A B Tutein and J A Harrison ldquoA reactive potentialfor hydrocarbons with intermolecular interactionsrdquo Journal ofChemical Physics vol 112 no 14 pp 6472ndash6486 2000

[30] T Belytschko S P Xiao G C Schatz andR S Ruoff ldquoAtomisticsimulations of nanotube fracturerdquo Physical Review B vol 65 no23 Article ID 235430 8 pages 2002

[31] D W Brenner O A Shenderova J A Harrison S J Stuart BNi and S B Sinnott ldquoA second-generation reactive empiricalbond order (REBO) potential energy expression for hydrocar-bonsrdquo Journal of Physics Condensed Matter vol 14 no 4 pp783ndash802 2002

[32] W H Pres B P Flannery S A Teukoisky and W T VatteringNumerical Recipes Cambridge University Press CambridgeUK 1986

[33] R Smith and K Beardmore ldquoMolecular dynamics studies ofparticle impacts with carbon-basedmaterialsrdquoThin Solid Filmsvol 272 no 2 pp 255ndash270 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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OpticsInternational Journal of



AstronomyAdvances in

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Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


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Journal of

Biophysics


ThermodynamicsJournal of


Sinnott et al [12] estimated the theoretical Youngrsquosmodulifor carbon fibers composed of single-walled fullerene nan-otubes aligned in the direction of the tubule axis They alsoinvestigated a new carbon composite composed of layerednanotubule fibers and diamond and they observed that thiscomposite had a high-modulus low-densitymaterial that wasquite stable to shear and other distortions Liew et al [13]examined the elastic and elastic-plastic properties of carbonnanotubes under axial tension using second generation ofreactive empirical bond-order potential coupled with theLennard-Jones potential at high strain rate In their analysisplastic regions are not clear because in most of the analysisof CNT it was observed that before breaking the stiffness isthe highest which contradicts the reliability of their solutiondue to the mismanagement of the application of the Van derWaals interactions They did not minimize CNT structureto reach its minimum energy before applying tensile loadingat its both ends Besides Liew et al [13] did not includetemperature effect in the theoretical Rebo potential [11]and Van der Waals interaction for analyzing the elasticelastic-plastic properties of CNTs by molecular dynamicsimulation So the elastic-plastic properties obtained byLiew et al [13] by molecular dynamics simulation are notbeyond doubtful quality of solutions Molecular dynamicssimulations of tensile pulling of carbon nanotubes (SWCNTs)(both armchair and Zigzag configurations) were conductedusing the Brenner potential to investigate the variation ofsix-carbon bond lengths and bond angles of the hexagons insingle-walled carbon nanotubes (SWCNT) as a function oftensile strain [14] For armchair CNTs the brittle fracture isobserved and the ultimate strength of CNTs is many timesto that of CNTS obtained by Liew et al [13] The stressstrain relationship of armchair CNT using Rebo potentialby the tension test in molecular dynamics environment byAgrawal et al [14] contradicts the elastic-plastic region ofCNT by Liew et al [13] although in his analysis Van derWaals interactions are absent Wang and Vijayaraghavan [15]studied the buckling characteristics of several curved formsof single-walled carbon nanotubes by molecular dynamicssimulations using Rebo potential [11] and Van der Waalsinteraction Tserpes and Papanikos [16] proposed a three-dimensional finite element model for armchair Zigzagand chiral single-walled carbon nanotubes which is basedon the assumption that carbon nanotubes when subjectedto loading behave like space-frame structures The bondsbetween carbon atoms are considered as connecting loadcarrying members while the carbon atoms as joints of themembers [16] To create the FE models nodes are placedat the locations of carbon atoms and the bonds betweenthem are modeled using three-dimensional elastic beamelements and the elastic properties of armchair Zigzag andchiral single-walled carbon nanotubes are obtainedwhich arecompared with those of experimental results to verify theFE model [16] Salvetat-Delmotte and Rubio [17] concludedthat CNTs have indeed a great potential as reinforcingelements for composites Coto et al [18] studied the effect ofcarboxylation on axial Youngrsquos modulus of carbon nanotubesusing molecular dynamics simulation COMPASS force fieldis used to model the interatomic interactions in single wall

(SWCNT) and multiwall carbon (MWCNT) with differentamount of ndashCOOH group attached to their surfaces [18]Mashreghi and Moshksar [19] determined the structuralparameters of several armchair single-walled carbon nan-otubes such as potential energy as a function of bond lengthand bond angles respectively using molecular dynamicssimulations Tang et al [20] synthesized carbon nanotubepolymer composites mixing carbon nanotubes and highdensity polyethylene by using the melt processing methodand they observed that the stiffness peak load and work tofailure for the composite films increase with the increase ofMWCNT content Sammalkorpi et al [21] studied how theYoungrsquos modulus and tensile strength of nanotubes dependson defects employing molecular dynamics simulation andcontinuum theory The Youngrsquos modulus of nanotubes withdefects will essentially be the same unless the number ofvacancy concentration is extremely high [21] On the otherhand the tensile strength will substantially drop due to thequasi-one-dimensional atomic structure of SWNTs already ifa single vacancy is present the tensile strength of a SWNT isgoverned by the weakest segment of the tube [21] Wang etal [22] observed that composites showed higher mechanicalproperties such as theYoungrsquosmodulus and tensile strength ofthe SWNTepoxy composite rise with the increase of SWNTconcentration The Youngrsquos modulus and yield strength havebeen doubled and quadrupled for composites with respec-tively 1 and 4wt of nanotubes compared to the pure resinmatrix samples [23] A recent research illustrates qualitativerelationship between Youngrsquos modulus of a CNT and theamount of disorder in the atomic structure of the walls[24] Lin and Chen [25] evaluated the effective mechanicalproperties of CNT-based composites using a 3D nanoscalerepresentative volume element based continuum mechanicsand the finite element method Hernandez-Perez and Aviles[26] investigated the influence of the interphase on the effec-tive properties of carbon nanotube composites using finiteelement analysis and elasticity solutions for representativevolume elements Micro Pt wire exhibits higher strength ascompared to its bulk due to decreasing size which is observedfrom the experimental study of bending test of Pt microwirewith the help of finite element analysis [27] In the very shortrange of diameters in Au nanowires Youngrsquos modulus andyield strength increase with the decreasing of their diameterswhich are clearly observed in the bending and tension test ofAu nanowires by molecular dynamics simulations [28] Dueto having small sizes of diameters in CNTs and as a resultof increasing their surface energy their strength and stiffnessare many times to those of their conventional bulk materialswhich are made of C elements

Due to having high stiffness and light weight CNT isa promising candidate in nanocomposites to increase theirstrength and stiffness Nowadays people are concentratingto evaluate the actual elastic elastic-plastic properties ofCNTs experimentally and theoretically But handling CNTin experimental way is a difficult task and the obtainedmechanical properties by the experimental way are beyondactual quantitative measures To obtain the actual quantita-tive mechanical properties theoretical approach is only analternative approach Researchers are trying to study CNT by






0)1198710in which 119871



0119889




119864 = sum

119894

119864119894=

1

2

sum

119894 = 119895

119881119894119895 (1)



119894119895

119881119894119895= 119891119862(119903119894119895) [119891119877(119903119894119895) + 119887119894119895119891119860(119903119894119895)] (2)




119860repre-


119891119877(119903119894119895) = 119860 exp (minus120582

1119903119894119895)

119891119860(119903119894119895) = minus119861 exp (minus120582

2119903119894119895)

(3)





119891119862(119903119894119895)

=

1 119903119894119895lt 119877 minus 119863

1

2

minus

1

2

sin[(1205872) (119903

119894119895minus 119877)

119863

] 119877 minus 119863 lt 119903119894119895lt 119877 + 119863

0 119903119894119895gt 119877 + 119863

(4)




[10]


119864REBO

= sum

119894

sum

119895(gt119894)

[119881119877(119903119894119895) minus 119887119894119895119881119860(119903119894119895)] (5)


119881119877(119903119894119895) = 119891119862(119903119894119895)119881119877

119900(119903119894119895)

119881119860(119903119894119895) = 119891119862(119903119894119895)119881119860

119900(119903119894119895)

119891119862(119903119894119895)

=

1 119903119894119895lt 119863min

[

1

2

+

1

2

cos(120587 (119903119894119895minus 119863min)


0 119903119894119895gt 119863max

(6)


119894119895is a


and 119881119860119900




119864AIREBO

=

1

2

sum

119894

sum

119895 = 119894

[

[

119864REBO119894119895

+ 119864119871119869

119894119895+ sum

119896 = 119894119895

sum

119897 = 119894119895119896

119864tors119896119894119895119897]

]

(7)


119864REBO119894119895

= 119881119877

119894119895+ 119887119894119895119881119860

119894119895(8)


119894119895


119881119877

119894119895= 119908119894119895(119903119894119895) [1 +

119876119894119895

119903119894119895

]119860119894119895119890minus120572119894119895119903119894119895

(9)




119894119895term is a


119908119894119895(119903119894119895) = 119878119905(119905119888(119903119894119895)) (10)


119878119905(119905) = Θ (minus119905) + Θ (119905)Θ (1 minus 119905)

1

2

[1 + cos (120587119905)] (11)


119905119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(12)


119881119860

119894119895= minus119908119894119895(119903119894119895)

3

sum

119899=1

119861(119899)

119894119895119890minus120573(119899)

119894119895119903119894119895 (13)


119894119895term in (8)


119887119894119895=

1

2

[119901120590120587

119894119895+ 119901120590120587

119895119894] + 120587119903119888

119894119895+ 120587119889ℎ

119894119895 (14)




and 119875120590120587119895119894

119875120590120587

119894119895=[

[

1 + sum

119896 = 119894119895

119908119894119896(119903119894119896) 119892119894(cos 120579

119895119894119896) 119890120582119895119894119896

+ 119875119894119895]

]

minus12

119875120590120587

119895119894=[

[

1 + sum

119896 = 119895119894

119908119895119896(119903119895119896) 119892119895(cos 120579

119894119895119896) 119890120582119894119895119896

+ 119875119895119894]

]

minus12

(15)






119894119895

119873119894119895= 119873

C119894119895+ 119873

H119894119895 (16)

where 119873C119894119895= (sum

119896 = 119894

120575119896119862119908119894119896(119903119894119896)) minus 120575

119895119862119908119894119895(119903119894119895) (17)







(2)


119892C (cos 120579119895119894119896) = 119892(1)

C (cos 120579119895119894119896) + 119878119905(119905119873(119873119894119895))

times [119892(2)

C (cos 120579119895119894119896) minus 119892(1)

C (cos 120579119895119894119896)

(18)



119873given by

119905119873(119873119894119895) =

119873119894119895minus 119873

min119894119895

119873max119894119895

minus 119873min119894119895

(19)


120582119895119894119896= 4120575119894H [(120575119896H120588HH + 120575119896C120588CH minus 119903119894119896)


(20)



in 119873C119894119895and 119873H








119894119895

The indices119873119894119895and119873


in (16) and119873conj119894119895


119873conj119894119895

= 1 +[

[

sum

119896 = 119894119895

120575119896C119908119894119896 (119903119894119896) 119878

119905(119905conj (119873119896119894))

]

]

2

+[

[

sum

119897 = 119894119895

120575119897C119908119895119897 (119903119895119897) 119878

119905(119905conj (119873119897119895))

]

]

2

(21)


119905conj (119873) =119873 minus 119873

min

119873max

minus 119873min

120587119903119888

119894119895= 119865119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

(22)


120587119889ℎ

119894119895= 119879119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

times sum

119896 = 119894119895

sum

119897 = 119894119895

(1 minus cos2120596119896119894119895119897) times 119908119905

119894119896(119903119894119896) 119908119905

119895119897(119903119895119897)Θ

times (sin (120579119895119894119896) minus 119904


119894119895119897) minus 119904

min)

cos120596119896119894119895119897

=

119903119895119894times 119903119894119896

10038161003816100381610038161003816119903119895119894times 119903119894119896

10038161003816100381610038161003816

sdot

119903119894119895times 119903119895119897

10038161003816100381610038161003816119903119894119895times 119903119895119897

10038161003816100381610038161003816

(23)


1199081015840

119905(119903119894119895) = 1198781015840(1199051015840

119888(119903119894119895)) (24)


1199051015840

119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(25)


119864119871119869

119894119895= 119878 (119905

119903(119903119894119895)) 119878 (119905

119887(119887lowast

119894119895)) 119862119894119895119881119871119869

119894119895(119903119894119895)

+ [1 minus 119878 (119905119903(119903119894119895))] 119862119894119895119881119871119869

119894119895(119903119894119895)

(26)


119881119871119869

119894119895(119903119894119895) = 4120576

119894119895[(

120590119894119895

119903119894119895

)

12

minus (

120590119894119895

119903119894119895

)

6

] (27)








119905119903(119903119894119895) =

119903119894119895minus 119903119871119869min119894119895

119903119871119869max119894119895

minus 119903119871119869min119894119895

(29)



119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)


119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)


119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)





FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








0)1198710in which 119871



0119889




119864 = sum

119894

119864119894=

1

2

sum

119894 = 119895

119881119894119895 (1)



119894119895

119881119894119895= 119891119862(119903119894119895) [119891119877(119903119894119895) + 119887119894119895119891119860(119903119894119895)] (2)




119860repre-


119891119877(119903119894119895) = 119860 exp (minus120582

1119903119894119895)

119891119860(119903119894119895) = minus119861 exp (minus120582

2119903119894119895)

(3)





119891119862(119903119894119895)

=

1 119903119894119895lt 119877 minus 119863

1

2

minus

1

2

sin[(1205872) (119903

119894119895minus 119877)

119863

] 119877 minus 119863 lt 119903119894119895lt 119877 + 119863

0 119903119894119895gt 119877 + 119863

(4)




[10]


119864REBO

= sum

119894

sum

119895(gt119894)

[119881119877(119903119894119895) minus 119887119894119895119881119860(119903119894119895)] (5)


119881119877(119903119894119895) = 119891119862(119903119894119895)119881119877

119900(119903119894119895)

119881119860(119903119894119895) = 119891119862(119903119894119895)119881119860

119900(119903119894119895)

119891119862(119903119894119895)

=

1 119903119894119895lt 119863min

[

1

2

+

1

2

cos(120587 (119903119894119895minus 119863min)


0 119903119894119895gt 119863max

(6)


119894119895is a


and 119881119860119900




119864AIREBO

=

1

2

sum

119894

sum

119895 = 119894

[

[

119864REBO119894119895

+ 119864119871119869

119894119895+ sum

119896 = 119894119895

sum

119897 = 119894119895119896

119864tors119896119894119895119897]

]

(7)


119864REBO119894119895

= 119881119877

119894119895+ 119887119894119895119881119860

119894119895(8)


119894119895


119881119877

119894119895= 119908119894119895(119903119894119895) [1 +

119876119894119895

119903119894119895

]119860119894119895119890minus120572119894119895119903119894119895

(9)




119894119895term is a


119908119894119895(119903119894119895) = 119878119905(119905119888(119903119894119895)) (10)


119878119905(119905) = Θ (minus119905) + Θ (119905)Θ (1 minus 119905)

1

2

[1 + cos (120587119905)] (11)


119905119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(12)


119881119860

119894119895= minus119908119894119895(119903119894119895)

3

sum

119899=1

119861(119899)

119894119895119890minus120573(119899)

119894119895119903119894119895 (13)


119894119895term in (8)


119887119894119895=

1

2

[119901120590120587

119894119895+ 119901120590120587

119895119894] + 120587119903119888

119894119895+ 120587119889ℎ

119894119895 (14)




and 119875120590120587119895119894

119875120590120587

119894119895=[

[

1 + sum

119896 = 119894119895

119908119894119896(119903119894119896) 119892119894(cos 120579

119895119894119896) 119890120582119895119894119896

+ 119875119894119895]

]

minus12

119875120590120587

119895119894=[

[

1 + sum

119896 = 119895119894

119908119895119896(119903119895119896) 119892119895(cos 120579

119894119895119896) 119890120582119894119895119896

+ 119875119895119894]

]

minus12

(15)






119894119895

119873119894119895= 119873

C119894119895+ 119873

H119894119895 (16)

where 119873C119894119895= (sum

119896 = 119894

120575119896119862119908119894119896(119903119894119896)) minus 120575

119895119862119908119894119895(119903119894119895) (17)







(2)


119892C (cos 120579119895119894119896) = 119892(1)

C (cos 120579119895119894119896) + 119878119905(119905119873(119873119894119895))

times [119892(2)

C (cos 120579119895119894119896) minus 119892(1)

C (cos 120579119895119894119896)

(18)



119873given by

119905119873(119873119894119895) =

119873119894119895minus 119873

min119894119895

119873max119894119895

minus 119873min119894119895

(19)


120582119895119894119896= 4120575119894H [(120575119896H120588HH + 120575119896C120588CH minus 119903119894119896)


(20)



in 119873C119894119895and 119873H








119894119895

The indices119873119894119895and119873


in (16) and119873conj119894119895


119873conj119894119895

= 1 +[

[

sum

119896 = 119894119895

120575119896C119908119894119896 (119903119894119896) 119878

119905(119905conj (119873119896119894))

]

]

2

+[

[

sum

119897 = 119894119895

120575119897C119908119895119897 (119903119895119897) 119878

119905(119905conj (119873119897119895))

]

]

2

(21)


119905conj (119873) =119873 minus 119873

min

119873max

minus 119873min

120587119903119888

119894119895= 119865119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

(22)


120587119889ℎ

119894119895= 119879119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

times sum

119896 = 119894119895

sum

119897 = 119894119895

(1 minus cos2120596119896119894119895119897) times 119908119905

119894119896(119903119894119896) 119908119905

119895119897(119903119895119897)Θ

times (sin (120579119895119894119896) minus 119904


119894119895119897) minus 119904

min)

cos120596119896119894119895119897

=

119903119895119894times 119903119894119896

10038161003816100381610038161003816119903119895119894times 119903119894119896

10038161003816100381610038161003816

sdot

119903119894119895times 119903119895119897

10038161003816100381610038161003816119903119894119895times 119903119895119897

10038161003816100381610038161003816

(23)


1199081015840

119905(119903119894119895) = 1198781015840(1199051015840

119888(119903119894119895)) (24)


1199051015840

119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(25)


119864119871119869

119894119895= 119878 (119905

119903(119903119894119895)) 119878 (119905

119887(119887lowast

119894119895)) 119862119894119895119881119871119869

119894119895(119903119894119895)

+ [1 minus 119878 (119905119903(119903119894119895))] 119862119894119895119881119871119869

119894119895(119903119894119895)

(26)


119881119871119869

119894119895(119903119894119895) = 4120576

119894119895[(

120590119894119895

119903119894119895

)

12

minus (

120590119894119895

119903119894119895

)

6

] (27)








119905119903(119903119894119895) =

119903119894119895minus 119903119871119869min119894119895

119903119871119869max119894119895

minus 119903119871119869min119894119895

(29)



119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)


119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)


119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)





FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119891119862(119903119894119895)

=

1 119903119894119895lt 119877 minus 119863

1

2

minus

1

2

sin[(1205872) (119903

119894119895minus 119877)

119863

] 119877 minus 119863 lt 119903119894119895lt 119877 + 119863

0 119903119894119895gt 119877 + 119863

(4)




[10]


119864REBO

= sum

119894

sum

119895(gt119894)

[119881119877(119903119894119895) minus 119887119894119895119881119860(119903119894119895)] (5)


119881119877(119903119894119895) = 119891119862(119903119894119895)119881119877

119900(119903119894119895)

119881119860(119903119894119895) = 119891119862(119903119894119895)119881119860

119900(119903119894119895)

119891119862(119903119894119895)

=

1 119903119894119895lt 119863min

[

1

2

+

1

2

cos(120587 (119903119894119895minus 119863min)


0 119903119894119895gt 119863max

(6)


119894119895is a


and 119881119860119900




119864AIREBO

=

1

2

sum

119894

sum

119895 = 119894

[

[

119864REBO119894119895

+ 119864119871119869

119894119895+ sum

119896 = 119894119895

sum

119897 = 119894119895119896

119864tors119896119894119895119897]

]

(7)


119864REBO119894119895

= 119881119877

119894119895+ 119887119894119895119881119860

119894119895(8)


119894119895


119881119877

119894119895= 119908119894119895(119903119894119895) [1 +

119876119894119895

119903119894119895

]119860119894119895119890minus120572119894119895119903119894119895

(9)




119894119895term is a


119908119894119895(119903119894119895) = 119878119905(119905119888(119903119894119895)) (10)


119878119905(119905) = Θ (minus119905) + Θ (119905)Θ (1 minus 119905)

1

2

[1 + cos (120587119905)] (11)


119905119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(12)


119881119860

119894119895= minus119908119894119895(119903119894119895)

3

sum

119899=1

119861(119899)

119894119895119890minus120573(119899)

119894119895119903119894119895 (13)


119894119895term in (8)


119887119894119895=

1

2

[119901120590120587

119894119895+ 119901120590120587

119895119894] + 120587119903119888

119894119895+ 120587119889ℎ

119894119895 (14)




and 119875120590120587119895119894

119875120590120587

119894119895=[

[

1 + sum

119896 = 119894119895

119908119894119896(119903119894119896) 119892119894(cos 120579

119895119894119896) 119890120582119895119894119896

+ 119875119894119895]

]

minus12

119875120590120587

119895119894=[

[

1 + sum

119896 = 119895119894

119908119895119896(119903119895119896) 119892119895(cos 120579

119894119895119896) 119890120582119894119895119896

+ 119875119895119894]

]

minus12

(15)






119894119895

119873119894119895= 119873

C119894119895+ 119873

H119894119895 (16)

where 119873C119894119895= (sum

119896 = 119894

120575119896119862119908119894119896(119903119894119896)) minus 120575

119895119862119908119894119895(119903119894119895) (17)







(2)


119892C (cos 120579119895119894119896) = 119892(1)

C (cos 120579119895119894119896) + 119878119905(119905119873(119873119894119895))

times [119892(2)

C (cos 120579119895119894119896) minus 119892(1)

C (cos 120579119895119894119896)

(18)



119873given by

119905119873(119873119894119895) =

119873119894119895minus 119873

min119894119895

119873max119894119895

minus 119873min119894119895

(19)


120582119895119894119896= 4120575119894H [(120575119896H120588HH + 120575119896C120588CH minus 119903119894119896)


(20)



in 119873C119894119895and 119873H








119894119895

The indices119873119894119895and119873


in (16) and119873conj119894119895


119873conj119894119895

= 1 +[

[

sum

119896 = 119894119895

120575119896C119908119894119896 (119903119894119896) 119878

119905(119905conj (119873119896119894))

]

]

2

+[

[

sum

119897 = 119894119895

120575119897C119908119895119897 (119903119895119897) 119878

119905(119905conj (119873119897119895))

]

]

2

(21)


119905conj (119873) =119873 minus 119873

min

119873max

minus 119873min

120587119903119888

119894119895= 119865119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

(22)


120587119889ℎ

119894119895= 119879119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

times sum

119896 = 119894119895

sum

119897 = 119894119895

(1 minus cos2120596119896119894119895119897) times 119908119905

119894119896(119903119894119896) 119908119905

119895119897(119903119895119897)Θ

times (sin (120579119895119894119896) minus 119904


119894119895119897) minus 119904

min)

cos120596119896119894119895119897

=

119903119895119894times 119903119894119896

10038161003816100381610038161003816119903119895119894times 119903119894119896

10038161003816100381610038161003816

sdot

119903119894119895times 119903119895119897

10038161003816100381610038161003816119903119894119895times 119903119895119897

10038161003816100381610038161003816

(23)


1199081015840

119905(119903119894119895) = 1198781015840(1199051015840

119888(119903119894119895)) (24)


1199051015840

119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(25)


119864119871119869

119894119895= 119878 (119905

119903(119903119894119895)) 119878 (119905

119887(119887lowast

119894119895)) 119862119894119895119881119871119869

119894119895(119903119894119895)

+ [1 minus 119878 (119905119903(119903119894119895))] 119862119894119895119881119871119869

119894119895(119903119894119895)

(26)


119881119871119869

119894119895(119903119894119895) = 4120576

119894119895[(

120590119894119895

119903119894119895

)

12

minus (

120590119894119895

119903119894119895

)

6

] (27)








119905119903(119903119894119895) =

119903119894119895minus 119903119871119869min119894119895

119903119871119869max119894119895

minus 119903119871119869min119894119895

(29)



119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)


119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)


119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)





FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119894119895

119873119894119895= 119873

C119894119895+ 119873

H119894119895 (16)

where 119873C119894119895= (sum

119896 = 119894

120575119896119862119908119894119896(119903119894119896)) minus 120575

119895119862119908119894119895(119903119894119895) (17)







(2)


119892C (cos 120579119895119894119896) = 119892(1)

C (cos 120579119895119894119896) + 119878119905(119905119873(119873119894119895))

times [119892(2)

C (cos 120579119895119894119896) minus 119892(1)

C (cos 120579119895119894119896)

(18)



119873given by

119905119873(119873119894119895) =

119873119894119895minus 119873

min119894119895

119873max119894119895

minus 119873min119894119895

(19)


120582119895119894119896= 4120575119894H [(120575119896H120588HH + 120575119896C120588CH minus 119903119894119896)


(20)



in 119873C119894119895and 119873H








119894119895

The indices119873119894119895and119873


in (16) and119873conj119894119895


119873conj119894119895

= 1 +[

[

sum

119896 = 119894119895

120575119896C119908119894119896 (119903119894119896) 119878

119905(119905conj (119873119896119894))

]

]

2

+[

[

sum

119897 = 119894119895

120575119897C119908119895119897 (119903119895119897) 119878

119905(119905conj (119873119897119895))

]

]

2

(21)


119905conj (119873) =119873 minus 119873

min

119873max

minus 119873min

120587119903119888

119894119895= 119865119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

(22)


120587119889ℎ

119894119895= 119879119894119895(119873119894119895 119873119895119894 119873

conj119894119895

)

times sum

119896 = 119894119895

sum

119897 = 119894119895

(1 minus cos2120596119896119894119895119897) times 119908119905

119894119896(119903119894119896) 119908119905

119895119897(119903119895119897)Θ

times (sin (120579119895119894119896) minus 119904


119894119895119897) minus 119904

min)

cos120596119896119894119895119897

=

119903119895119894times 119903119894119896

10038161003816100381610038161003816119903119895119894times 119903119894119896

10038161003816100381610038161003816

sdot

119903119894119895times 119903119895119897

10038161003816100381610038161003816119903119894119895times 119903119895119897

10038161003816100381610038161003816

(23)


1199081015840

119905(119903119894119895) = 1198781015840(1199051015840

119888(119903119894119895)) (24)


1199051015840

119888(119903119894119895) =

119903119894119895minus 119903

min119894119895

119903max119894119895

minus 119903min119894119895

(25)


119864119871119869

119894119895= 119878 (119905

119903(119903119894119895)) 119878 (119905

119887(119887lowast

119894119895)) 119862119894119895119881119871119869

119894119895(119903119894119895)

+ [1 minus 119878 (119905119903(119903119894119895))] 119862119894119895119881119871119869

119894119895(119903119894119895)

(26)


119881119871119869

119894119895(119903119894119895) = 4120576

119894119895[(

120590119894119895

119903119894119895

)

12

minus (

120590119894119895

119903119894119895

)

6

] (27)








119905119903(119903119894119895) =

119903119894119895minus 119903119871119869min119894119895

119903119871119869max119894119895

minus 119903119871119869min119894119895

(29)



119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)


119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)


119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)





FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119887switch

119905119887(119887119894119895) =

119887119894119895minus 119887

min119894119895

119887max119894119895

minus 119887min119894119895

(30)


119862119894119895= 1 minusmax 119908

119894119895(119903119894119895) 119908119894119896(119903119894119896) 119908119896119895(119903119896119895)

forall119896119908119894119896(119903119894119896) 119908119896119897(119903119896119897) 119908119897119895(119903119897119895) forall119896 119897

(31)


119864tors119896119894119895119897

= 119908119896119897(119903119896119897) 119908119894119895(119903119894119895)119908119895119897(119903119895119897)119881

tors(120596119896119894119895119897) (32)

where

119881tors

(120596119896119894119895119897) =

256

405

120576119896119894119895119897

cos10 (120596119896119894119895119897

2

) minus

1

10

120576119896119894119895119897 (33)





FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




FF

(a)

(b)


Axial strain000 005 010 015 020

Pote

ntia

l ene

rgy

per a

tom

(eV

)



minus78

minus76

minus74

minus72

minus70

minus68

minus66

minus64













Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










Axial strain000 005 010 015 020 025

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120




(a)

Axial strain000 005 010 015 020 025 030

Tens

ile st

ress

(GPa

)0

20

40

60

80

100

120




(b)

00 01 02 03 04 050

20

40

60

80

100

120

140

160

180


Axial strain

Tens

ile st

ress

(GPa

)



(c)

00 01 02 03 040

40

80

120

160

200

240


Axial strain

Tens

ile st

ress

(GPa

)



(d)







Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








Axial strain000 005 010 015 020 025 030 035

Tens

ile st

ress

(GPa

)

0

20

40

60

80

100

120

140

160

180

Airebo potential

Cut-off onset154nm

Cut-off onset144nm

150nmCut-off onset





Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Axial strain00 01 02 03 04 05

Pote

ntia

l ene

rgy

per a

tom




minus80

minus75

minus70

minus65

minus60

minus55

minus50

LD = 91


(a)

Tota

l ene

rgy

per a

tom


Axial strain00 01 02 03 04 05



minus80

minus75

minus70

minus65

minus60

minus55

minus50


(b)






Liew et al [14]

Axial strain00 01 02 03 04 05

LD = 91

0

50

100

150

200

250

300

Tens

ile st

ress

(GPa

)



(a)

0

50

100

150

200

250

300



Axial strain00 01 02 03 04 05


LD = 451

Tens

ile st

ress

(GPa

)



(b)





0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






0

50

100

150

200

250



LD = 91

Tens

ile st

ress

(GPa

)


(a)



0

50

100

150

200

250


Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa


(b)









0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







0

50

100

150

200

Axial strain00 01 02 03 04 05

Tens

ile st

ress

(GPa

)

CNT LD = 91


(a)




0

50

100

150

250

200

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





0

50

100

150

200

250

Strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

20

40

60

80

100

120

140




0

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)


(b)





119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119910120590119906 GPa 120576

119906V


91


451



91


451


Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

Axial strain00 01 02 03 04 05

LD = 91

Tens

ile st

ress

(GPa

)


(a)

Four-walled 5-5 10-10 15-15 20-20 CNT



0

50

100

150

200

250

300

Axial strain00 01 02 03 04 05

LD = 451

Tens

ile st

ress

(GPa

)

(b)





119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119910120590119906 GPa 120576

119906


91


451



91


451


Four-walled 5-5 10-0 15-15 20-20 CNT

91


451







(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(a)

(b)

(c)









5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






5 Conclusions





References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article study of the nanomechanics of cnts under

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