Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 2, pp. 531-540, Warsaw 2011 MODELLING OF CARBON NANOTUBES BEHAVIOUR WITH THE USE OF A THIN SHELL THEORY Aleksander Muc Cracow University of Technology, Institute of Machine Design, Kraków, Poland e-mail: olekmuc@mech.pk.edu.pl In the paper, various problems connected with the possibility of mo- delling of single walled carbon nanotubes as thin cylindrical shells are discussed.The interatomicpotentials playing the fundamental role in the description are presented. Special attention is focused on various types of nonlinearities (geometrical andmaterial). The open problems arising in the modeling are also emphasised. Key words: carbon nanotubes, shells, interatomic potential 1. Introduction The investigations on carbon nanotubes behaviour have been mainly focu- sed on the experimental description andmolecular dynamics simulations such as classical molecular dynamics, tight-binding molecular dynamics and the ab initio method. However, the researchers have been seeking more efficient computational methods with which it is possible to analyse the large scale of CNTs in a more general manner. Yakobson et al. (1996) found that the continuum shell model could predict all changes of buckling patterns in ato- mistic molecular-dynamics simulations. The analogousness of the cylindrical shell model and CNTs leads to extensive application of the shell model for CNT structural analysis. Ru (2000a,b, 2001) used the CM approach and si- mulated the effect of van derWaals forces by applying a uniformly distributed pressure field on the wall, the pressure field was adjusted so as to give the same resultant force on each wall of the tube. It has verified that the me- chanical responses of CNTs can be efficiently and reasonably predicted by the shell model provided that the parameters, such as Young’s modulus and effective wall thickness, are judiciously adopted. Wang et al. (2006) studied buckling of double-walled carbon nanotubes under axial loads, by modeling 532 A. Muc CNTs using solid shell elements. Han et al. (2003, 2006) investigated torsional buckling of a DWNT and MWNT embedded in an elastic medium. Han et al. (2005) also studied bending instability of double-walled carbon nanotubes. Yao and Han (2006, 2007) analysed the thermal effect on axially compressed and torsional buckling of a multi-walled carbon nanotube. Some conclusions were drawn that at low and room temperature the critical load for infinite- simal buckling of a multi-walled carbon nanotube increases as the value of temperature change increases, while at high temperatures the critical load for infinitesimal buckling of a multi-walled carbon nanotube decreases as the value of temperature change increases. Nonlinear postbuckling behaviour of carbon nanotubes under large strain is significant to whoch great attention is paid by some researchers (Wang et al., 2005; Leung et al., 2006). The torsional postbuckling behaviour of single-walled ormulti-walled carbon nanotubeswas discussed in details by Yiao and Han (2008). The problems encountered in the numerical modelling of pristine and de- fective carbon nanotubes were demonstrated in details by Muc (2009, 2010). In the mentioned references, the linear and nonlinear (iterative) approaches were illustrated. A successfulwork has been also conductedwith continuummodelling such as dynamic studies. A comprehensive review of the literature dealing with the analysis of wave propagation in CNTs with the use of shell theories was presented by Liew and Wang (2007), although the authors focused the at- tention mainly on the application of thick shell theories. They pointed out that there was growing interest in the terahertz physics of nanoscale mate- rials and devices, which opened a new topic on phonon dispersion of CNTs, especially in the terahertz frequency range. Hu et al. (2008) proposed to use nonlocal shell theories in the analysis of elastic wave propagation in single- or double-walled carbon nanotubes. However, it is worth to emphasise that the vibrational characteristics of CNTs are studiedwith the use of both beam the- ories and different variants of shell theories – see e.g. Natsuki et al. (2008) and Ghorbanpourarani et al. (2010). Recently, thermal vibrations have attracted considerable attention – see e.g. Tylikowski (2008), where the dynamic sta- bility analysis was conducted with the use of stochastic methods. The cited above work demonstrates also another tendency in the dynamic analysis of multi-walled CNTs connected with the application of a multiple-elastic shell modelwhich assumes that each of the concentric tubes ofmultiwall carbonna- notubes is an individual elastic shell and coupledwith adjacent tubes through van der Waals interaction. The broader discussion of that class of problems was presented e.g. by Xu andWang (2007). Modelling of carbon nanotubes... 533 The aim of the present paper is to discuss various problems encountered in the modelling of single-walled carbon nanotubes as continuous cylindrical thin shells. 2. Inter-atomic relations At the beginning of our considerations, let us introduce basic relations descri- bing carbon-carbon (C-C) interactions. They characterise physical bahaviour of carbon nanotubes, being in fact atomistic structures, and are fundamental in the further transformation frommolecular dynamics relations to continuum (shell) mechanics. The structure of nanotubes is obtained by conformational mapping of a graphene sheet onto a cylindrical surface. The nanotube radius is estimated by using the relation R= r0 √ 3(m2+n2+mn) 2π (2.1) where r0 = 0.141nm is the carbon-carbon distance. The integers n and m denote the number of unit vectors a1 and a2 along two directions in the honeycomb crystal lattice of graphene. If m = 0, they are called ”zigzag” nanotubes; if n = m, they are called ”armchair” nanotubes. For any other values of n and m, the nanotubes are called ”chiral” because the chains of atoms spiral around the tube axis instead of closing around the circumference. To capture the essential feature of chemical bonding in graphite, Brenner (1990) established an interatomic potential (called as theREBOpotential) for carbon in the following form V (rij)=VR(rij)−BijVA(rij) (2.2) where rij is the distance between the atoms i and j, VR and VA are the repulsive and attractive pair terms (i.e., depending only on rij), and are given by VR(r)= D(e) S−1 exp[− √ 2Sβ(r−R)] VA(r)= D(e)S S−1 exp [ − √ 2 S β(r−R) ] (2.3) In the above expression, the cut-off function is assumed to be equal to 1 to avoid a dramatic increase in the interatomic force. 534 A. Muc Theparameter Bij inEq. (2.2) represents themultibody coupling between the bond from the atoms i and j and the local environment of the atom i, and is given by Bij = [ 1+ ∑ k(6=i,j) G(θijk) ]−δ (2.4) where θijk is the angle between bonds i−j and i−k, and the function G is given by G(θ)= a0 [ 1+ c20 d20 + c20 d20+(1+cosθ) 2 ] (2.5) and the term Bij is expressed in the symmetric form Bij = 1 2 (Bij +Bji) (2.6) The set of material parameters is adopted here as follows De =0.9612nNnm S=1.22 β=21nm −1 R=0.139nm δ=0.5 a0 =0.00020813 c0 =330 d0 =3.5 (2.7) In contrast to the REBO potential function, in which the bond stretch and bond angle are coupled in the potential, Belytschko et al. (2002) proposed the modifiedMorse potential function, which can be expressed as the sum of energies that are associated with the variance of the bond length Vstretch, and the bond angle Vangle, i.e. V =Vstretch +Vangle Vstretch =De{[1− exp(−β(r−r0))]2−1} (2.8) Vangle = 1 2 kθ(θ−θ0)2[1+ksextic(θ−θ0)4] Thematerial constants are following r0 =1.421 ·10−10m De =9 ·10−19Nm β=1.8 ·1010m−1 θ=120◦ kθ =0.9 ·10−18Nm/rad2 ksextic =0.754rad−4 (2.9) BydifferentiatingEq. (2.2) orEq. (2.8)1, the stretching force of atomicbonds is obtained. The force variationswith the bond length are almost the samewhile Modelling of carbon nanotubes... 535 the bond angle is kept constant as 2π/3. However, for the REBO potential, the force varieswith the bondangle variations, whereas for themodifiedMorse potential it is always constant. Thus, the inflection point (force peak) is not constant for the REBO potential. As it is reported, both bond lengths and bond angles vary asCNTs are stretched. Therefore, in our numericalmodel, it is necessary to consider two possible formulations of interatomic potentials to analyse and compare the influence of those effects on the nonlinear behaviour and fracture strain. Fromthe interatomic potentials that are shown inEqs. (2.2) and (2.8)1, the stretching force that results fromthebondelongation and the twistingmoment that results from the bond angle variation can be calculated as follows F(ri)= ∂V ∂ri M(θ)= ∂V ∂θ (2.10) Figure 1a compares the interatomic stretching force for the REBO and modified Morse potentials in the tensile regime, whereas Fig.1b shows the bond anglemoment for theREBOandmodifiedMorse potentials. Let us note that they present nonlinear behaviour. Fig. 1. Tensile (a) force and (b) moment distributions 3. Continuum shell model To describemechanical behaviour of carbon nanotubes in the form of classical shell relations, it is necessary to introduce the equivalent Young modulus E and the equivalent shell thickness h – Fig.2. 536 A. Muc Fig. 2. Continuummodel of carbon nanotube structure In general, three different approaches are possible: • to assume the above-mentioned values; for instance Yakobson et al. (1996) suggested that the effective Young modulus is 5.5TPa and the wall thickness of the carbonnanotube is 0.066nmbasedon their SWCNT buckling results obtained by MD simulation and the continuummecha- nics shell model • to fit the results to atomistic simulation results of tension rigidity EA=Eh/(1−ν2) and bending rigidity EI =Eh3/[12(1−ν2)], where ν is the Poisson ratio; this gives (Huang et al., 2006)) h= √ 12 EA EI =3 √ 2 ( ∂V ∂cosθijk ) 0 (∂2V ∂r2ij )−1 0 (3.1) • to determine analytically tensile, shear, bending and torsional rigidities directly from the interatomic potential, and therefore avoid any fitting not-well-defined elastic modulus and thickness (Wu et al.,2008). This atomistic-based shell theory gives the relation among the increments of second Piola-Kirchhoff stress T, moment M, Green strain E, and curvature K as T=L :E+H :K M=H :E+S :K (3.2) where L,H and S are the fourth-order rigidity tensors obtained analy- tically from the interatomic potential. Modelling of carbon nanotubes... 537 Using the above-mentioned approaches, it is possible to evaluate buckling loads of cylindrical shells, analyse their post-buckling behaviour, etc., just employing themethods of analysis well-known for shell problems. In this way, it is possible to describe the behaviour of equivalent carbon single-walled or multi-walled carbon nanotubes. However, there is always an open question dealing with the accuracy and correctness of such approaches. Some of those problems will be discussed in the next section. 4. Open problems Peng et al. (2008) determined the order of error for approximating single- walled carbon nanotubes by a thin shell. The ratio of atomic spacing r0 to the single-walled carbon nanotubes radius R (i.e. r0/R), is used to identify the order of error. They considered the structural response of single-walled carbon nanotubes subject to tension (or compression), torsion, bending and internal (or external) pressure. They proved that only for the order of error equal to 40%-(5.5) armchair single walled carbon nanotubes can be modelled as thin shells with constant thickness and isotropic mechanical properties. The extensions of theabove resultswerepresentedbyWu et al. (2008). The authors defined degrees of: anisotropy, nonlinearity and coupling, i.e., down to the radius of single-walled carbon nanotubes at which the tension/bending coupling becomes negligible in constitutive relation (3.2). Numerical modelling of single-walled carbon nanotubes constitutes a se- parate class of problems. Some of them were discussed by Kalmakarov et al. (2006). The cited work presents also the comparison of Young’s modulus and the equivalent thickness predicted with the use of various theories. Acknowledgement The Polish Research Foundation PB 1174/B/T02/2009/36 is gratefully acknow- ledged for financial support. References 1. Belytschko T., Xiao S.P., Schatz G.C., Ruogg R.S., 2002, Atomistic simulations of nanotube fracture,Physical Review B, 65, 235430 538 A. Muc 2. Brenner D.W., 1990, Empirical potential for hydrocarbons for use in simu- lating the chemical vapor deposition of diamond films,Physical Review B, 42, 9458-9471 3. Ghorbanpourarani A.,Mohammadimehr M., Arefmanesh A., Ghase- mi A., 2010, Transverse vibration of short carbon nanotubes using cylindrical shell and beammodels,Proc. I.Mech. Engrs – Part C: J.Mech. Eng. Sc.,224, 745-756 4. Han Q., Lu G.X., 2003, Torsional buckling of a double-walled carbon nano- tube embedded in an elastic medium,Eur. J. Mech. A Solids, 22, 875-883 5. Han Q., Lu G.X., Dai L.M., 2005, Bending instability of an embedded double-walled carbon nanotube based on Winkler and van der Waals models, Compos. Sci. Technol., 65, 1337-1346 6. Han Q., Yao X.H., Li L.F., 2006, Theoretical and numerical study of tor- sional buckling of multiwall carbon nanotubes, Mech. Adv. Mater. Structures, 13, 329-337 7. HuY.-G., LiewK.M.,WangQ.,HeX.Q.,YakobsonB.I., 2008,Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes, J. Mech. and Physics Sol., 56, 3475-3485 8. HuangY.,Wu J.,HwangK.C., 2006,Thickness of grapheneand single-wall carbon nanotubes,Physical Review B, 74, 245413 9. Kalamkarov A.L., Georgiades A.V., Rokkam S.K., Veedu V.P., Ghasemi-Nejhad M.N., 2006, Analytical and numerical techniques to pre- dict carbon nanotubes properties, International Journal of Solids and Structu- res, 43, 6832-6854 10. Leung A.Y.T., Quo X., He X.Q., et al., 2006, Postbuckling of carbon nanotubes by atomic-scale finite element, J. Appl. Phys., 99, 124308 11. Liew K.M.,Wang Q., 2007, Analysis of wave propagation in carbon nanotu- bes via elastic shell theories, Int. J. Eng. Sc., 45, 227-241 12. Muc A., 2009, Modeling of CNTs/nanocomposites deformations and tensile fracture,Proc. 17th International Conference on Composite Materials (ICCM- 17), EdinburghUK 13. Muc A., 2010, Design and identification methods of effective mechanical pro- perties for carbon nanotubes,Mat. Des., 31, 1671-1675 14. Natsuki T., Ni Q.-Q, Endo M., 2008, Analysis of the vibration characteri- stics of double-walled carbon nanotubes,Carbon, 46, 1570-1573 Modelling of carbon nanotubes... 539 15. Peng J.,Wu J., HwangK.C., Song J., HuangY., 2008,Can a single-wall carbon nanotube be modeled as a thin shell? Journal of the Mechanics and Physics of Solids, 56, 2213-2224 16. Ru C.Q., 2000a, Effect of van der Waals forces on axial buckling of a double- walled carbon nanotube, J. Appl. Phys., 87, 7227-7231 17. RuC.Q., 2000b,Effective bending stiffness of carbonnanotubes,Phys. Rev. B, 62, 9973-9976 18. RuC.Q., 2001,Axially compressedbucklingofadoublewalledcarbonnanotube embedded in an elastic medium, J. Mech. Phys. Solids, 49, 1265-1279 19. Tylikowski A., 2008, Instability of thermally induced vibrations of carbon nanotubes,Arch. Appl. Mech., 78, 49-60 20. Wang L.F., Hu H.Y., Guo Wl., 2005, Postbuckling of carbon nanotubes subjected to cyclic load,Acta Mech. Solida Sinica, 18, 123-129 21. Wang C.M., Ma Y.Q., Zhang Y.Y., Ang K.K., 2006, Buckling of double- walled carbon nanotubes modeled by solid shell elements, J. Appl. Phys., 99, 114317 22. WuJ.,HwangK.C.,HuangY., 2008,Anatomistic-basedfinite-deformation shell theory for single-wall carbon nanotubes, J. Mech. Phys. Solids, 56, 279-292 23. Wu J., Peng J., Hwang K.C., Song J., Huang Y., 2008, The intrinsic stiffness of single-wall carbonnanotubes,Mechanics ResearchCommunications, 35, 2-9 24. Xu C.L., Wang X., 2007,Matrix effects on the breathingmodes of multiwall carbon nanotubes,Composite Str., 80, 73-81 25. YakobsonB.I.,BrabecC.J.,BernholcJ., 1996.Nanomechanics of carbon tubes: instability beyond linear response,Phys. Rev. Lett., 76, 2511-2514 26. Yao X.H., Han Q., 2006, Buckling analysis of multiwalled carbon nanotu- bes under torsional load coupling with temperature change, J. Eng. Mater. Technol., 128, 419-427 27. Yao X.H., Han Q., 2007, Investigation of axially compressed buckling of a multi-walled carbon nanotube under temperature field,Compos. Sci. Technol., 67, 125-134 28. Yao X.H., Han O., 2008, A continuum mechanics nonlinear postbuckling analysis for single-walled carbon nanotubes under torque, Eur. J. Mech. A Solids, 27, 796-807 540 A. Muc Zastosowanie teorii powłok w zagadnieniach modelowania deformacji nanorurek węglowych Streszczenie W artykule przedstawiono problematykęmodelowania deformacji jednościennych nanorurek węglowych przy pomocy teorii cienkościennych powłok cylindrycznych. Wopisie zastosowano teorię potencjałów oddziaływańmiędzyatomowych. Szczególną uwagę zwrócono na kwestie opisu nieliniowości geometrycznych i fizycznych. Podkre- ślono także istnienie wielu nierozwiązanych do chwili obecnej zagadnień związanych zmodelowaniem deformacji nanorurek. Manuscript received April 26, 2010; accepted for print December 20, 2010