A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes

ec o s. In addi ion o he ex ac ion o elas ic ma e ial enso s, hese models ha e been used wi h he fini e elemen me hod o sol e bounda y alue p oblems, such as he nano-inden a ion o silicon (Tadmo e al., 1999). I he de o ma ion is no uni o m enough o he local heo y o hold, mixed con inuum a omis ic app oaches ha e been p o- posed o deal wi h inhomogenei ies o de ec s (Tadmo e al., 1996; Shenoy e al., 1999). The appeal o he app oach based on he Bo n ule s ems om he ac ha i gi es ise o fini e de o ma ion cons i u i e models based on he nano-scale physics a he han phenomenologi- cal ma e ial models. The in e -a omic po en ials, based on expe imen al da a and quan um me- chanical conside a ions o calcula ions, a e hus a he co e o he esul ing s ain ene gy densi y. A local quasicon inuum has also been de eloped based on he igh -binding me hod (Tadmo e al., 1999). This pape deals wi h he applica ion o such a heo y o he mechanics o ca bon nano ubes. Since he disco e y o hese c ys alline ubes in 1991, many s udies ha e ocused on hei unique mechanical p ope ies, h ough expe imen s (Yu e al., 2000; Chop a e al., 1995; Yu e al., 2001b), molecula dynamics (MD) and molecula me- chanics (MM) simula ions (Be nholc e al., 1998; Gao e al., 1998) and fi s -p inciples calcula ions (Zhou e al., 2001; Mai i, 2000). Al hough molec- ula simula ions seem well sui ed o s udy hese sys ems, hey a e no comple ely sa is ac o y. In- deed, hey a e e y demanding om he compu- a ional poin o iew. The leng h scales ( AA) and ime scales (ps) ha mus be esol ed a e o en well below he scales o p ac ical in e es o a pa ic- ula p oblem. In addi ion, al hough sys ems o o e one million nuclei a e cu en ly being ana- lyzed, one can always en ision la ge p oblems o which he compu ing capabili ies do no suffice. An app oach ha can alle ia e some o he d awbacks o molecula simula ions is he use o con inuum mechanics. The abili y o con inuum models o desc ibe he mechanics o nano ubes has been no ed by se e al au ho s. C oss-sec ion con- inuum models ha e been used o explain expe i- men al obse a ions on he ans e se s abili y o nano ubes (Chop a e al., 1995; Yu e al., 2001a). These ex emely simplified models b ing insigh s as well as quan i a i e in o ma ion on he physical phenomena ha go e n he s abili y o he ci cula and he collapsed configu a ions obse ed in na- no ubes. The ib a ional p ope ies o ca bon na- no ubes ha e been in es iga ed h ough linea elas ici y by Sohlbe g e al. (1998). The elas ic p ope ies o ca bon nano ubes as a con inuum, neglec ing all cu a u e effec s, ha e in es iga ed by Lu (1997). Zhong-Can e al. (1997) conside ed he nano ube o be an inex ensible memb ane, and ob ained an exp ession o he elas ic ene gy in e ms o he cu a u e o a amily o simple de- o ma ions. Yakobson e al. (1996) used he heo y o elas ic shells and linea ized bi u ca ion analysis o s udy he buckling pa e ns o comp essed ca - bon nano ubes obse ed in MD simula ions. Qian e al. (2001) used a 3D con inuum heo y com- bined wi h a mesh- ee app oxima ion o s udy C60 molecules inside nano ubes. Ne e heless, he p oposed models so a a e ei he o e -simplified, es ic ed o he linea e- gime, o o e y pa icula si ua ions, and do no cons i u e a sys ema ic con inuum app oach o he mechanics o nano ubes. This is pa icula ly ue wi h ega ds o he la ge de o ma ions. Indeed, expe imen s (Chop a e al., 1995; Fal o e al., 1997), MD/MM simula ions (Be nholc e al., 1998) and fi s -p inciples calcula ions (Mai i, 2000) show ha ca bon nano ubes unde go e y la ge de o ma ions, wi h highly non-linea beha - io and s ill emain elas ic in he sense ha he de o ma ions a e e e sible, wi h s able bonds and in ac bond opology. F om hese conside a ions, de eloping a fini e de o ma ion model based on he Bo n ule and on nano-scale physics applicable o nano ubes would be o g ea in e es and would fill in a gap in he p esen use o con inuum models o model c ys- alline films one a om hick. Apa om he physical insigh s ha a con inuum model b ings, such a desc ip ion could be he basis o an effi- cien nume ical simula ion me hodology, in con- as wi h he some imes oo de ailed molecula simula ions. Efficiency becomes an issue when nano- opes bundles o ens o hund eds o na- no ubes o mul i-walled nano ubes se e al mi- c ons long (Ruoff e al., 1993; Yu e al., 2000) a e 2 o be analyzed. Fu he mo e, a con inuum me- chanics heo y allows us o exploi he symme y o ce ain si ua ions explici ly, analogously o he plane s ain and plane s ess si ua ions in 3D elas ici y. Such educed models o he ans e se beha io o nano ubes a e one o he opics p e- sen ed in his pape . Un o una ely, he adi ional app oach based on he Bo n ule wo ks o bulk ma e ials bu ails o ex end di ec ly o he case o c ys alline films and opes one a om hick de o ming in highe di- mensional spaces, i.e. 3D in he case o films, and 2D o 3D in he case o opes. The p esen pape desc ibes some mechanical effec s ha a ise om a ecen ex ension o he Bo n ule o memb anes (A oyo and Bely schko, 2002). The ex ension is based on he diffe en ial geome y concep o he exponen ial map, and is he e called he exponen ial Bo n ule. An al e na i e app oach has been e- po ed by F iesecke and James (2000). The ou line o his pape is as ollows: we fi s p esen he Bo n ule o bulk ma e ials, in es i- ga e i s s uc u e and explain why i b eaks down o films (Sec ion 2). Then, a e some geome ic p elimina ies, Sec ion 3 in oduces he p oposed exponen ial Bo n ule in an abs ac and gene al way. This abs ac p esen a ion o he heo y is complemen ed by i s ealiza ion in he simples , ye comple e, si ua ion, i.e. an a omic chain de o ming in wo dimensions. The o mula ion o he equi - alen con inuum ope-like objec is de ailed in Sec ion 4, and a simple example illus a ing he effec i eness o his model in mimicking he a omic chain is p o ided. P e ious o he applica ion o he heo y o ca bon nano ubes, hei c ys alline s uc u e, as well as he ins ance o in e -a omic po en ial conside ed, a e desc ibed in Sec ion 5. Sec ion 6 desc ibes in de ail he model o he ans e se mechanics o ca bon nano ubes. The implemen a ion o he new heo y o he a bi a y de o ma ion o he con inuum memb ane in 3D is p esen ed in A oyo and Bely schko (2002). The B a ais mul i-la ice na u e o g aphene equi es he ea men o addi ional in e nal a iables, he so-called inne displacemen s. Addi ionally, he an de Waals in e ac ions a e also accoun ed o in he con inuum heo y, and he con inuum a ia ional s a emen o he p oblem as well as he Lag angian s ess measu es ha na u ally a ise a e desc ibed. Finally, a alida ion es compa ing he p oposed con inuum model disc e ized wi h fini e elemen s o molecula calcula ions is p o ided in Sec ion 7. Se e al simula ions highligh ing he ele ance o he an de Waals o ces in he ans e se configu a ions o ca bon nano ubes and nano- opes a e also included in his sec ion. 2. B eakdown o he Bo n ule o films The o mula ion o a fini e de o ma ion con- inuum model o space-filling de ec less c ys als based on he Bo n ule is ela i ely s aigh o - wa d. The Bo n ule links he a omis ic de o ma- ion o ha o he con inuum medium. Then, a ep esen a i e c ys alli e is conside ed, and, o a gi en con inuum de o ma ion, he con inuum s ain ene gy densi y is defined o be he ene gy o he c ys alli e subjec o he de o ma ion di ided by i s olume. The de ails o he p ocedu e a e p esen ed in se e al o he a icles e e enced in Sec ion 1, and will also be b iefly desc ibed la e in he p esen pape . The ocus o his sec ion is on he undamen al kinema ic assump ion ha links he con inuum and he a omic de o ma ions, i.e. he Bo n ule. The de ails o he a omic model a e delibe a ely omi ed. La e , an ins ance o an a omic model is adop ed. 2.1. The s anda d Bo n ule Assume o he momen ha we a e dealing wi h space-filling con inuum bodies, i.e. open subse s o he ambien Euclidean space. Le Ube he de o ma ion ha maps he unde o med body X0Rn, in o Rn,nbeing ei he 1, 2 o 3. I X deno es a poin in he unde o med body, i s image a e de o ma ion is x¼UðXÞ. The de o med body is deno ed as X¼UðX0Þand is an open se o Rn. The de o ma ion g adien is he de i a i e o he ec o - alued ec o ial unc ion U,F¼ DU¼oU=oX2Rnn. A each poin X, he de o - ma ion g adien is a linea ans o ma ion om Rn in o Rn, which maps ‘‘infini esimal’’ ma e ial ec- o s, dx¼FdX(see Mal e n, 1969, p. 156). 3 F om a diffe en ial geome y poin o iew, he de o ma ion g adien is called he angen map o U, and is deno ed as F¼TU. Le us call he in- fini esimal neighbo hoods o Xand x he angen spaces o he unde o med body and he de o med one, espec i ely deno ed as TXX0and TxX(see Fig. 1 o an illus a ion). Then, using his lan- guage, he de o ma ion map Umaps he unde- o med body in o he de o med one, and he angen map F¼TUmaps he angen space o he unde o med body in o he angen space o he de o med body. In he absence o slips, phase ansi ions and o he special c ys allog aphic phenomena, he Cauchy and Bo n hypo hesis o c ys als a e equi alen o homogeneous de o ma ions (E - icksen, 1984). Wha is e e ed o as he Bo n ule in some wo ks is simply called he me hod o ho- mogeneous de o ma ions in o he s (Ma in, 1975; Cousins, 1978). The Bo n hypo hesis consis s o assuming ha he la ice ec o s de o m as would ma e ial line elemen s in a homogeneous de o - ma ion: a¼FA;ð1Þ whe e Adeno es an unde o med la ice ec o and a he same ec o in he de o med c ys al. The geome y o he la ice ec o s, ha is hei leng h and he angles hey o m wi h o he la ice ec o s in he de o med c ys al, can he e o e be ex ac ed om he con inuum de o ma ion h ough he G een de o ma ion enso C¼FTFusing s anda d con inuum mechanics ela ions: kak¼ ACA pand cos h¼ACB kakkbk;ð2Þ whe e Band b ep esen ano he unde o med and de o med la ice ec o and his he angle a and b o m in he de o med c ys al. Once he geome y o he de o med la ice ec o s is linked o he con inuum de o ma ion, a cons i u i e model based on he a omic in e ac ions can be cons uc ed by iden i ying he con inuum s ain ene gy densi y wi h he po en ial ene gy o he a omic sys em o a ep esen a i e cell di ided by i s olume. One could a gue ha he ule exp essed by Eq. (1) is o mally inconsis en , because he la ice ec o s Aand a, each connec ing wo a omic po- si ions, a e physical en i ies ha lie in he unde- o med and de o med body espec i ely, while he angen map F¼TUmaps elemen s o he angen o he unde o med body in o elemen s o he an- gen o he de o med body. This inconsis ency can also be iewed om a mo e classic s andpoin : he la ice ec o s ha e fini e leng h while he de o - ma ion g adien maps ‘‘infini esimal’’ ma e ial ec o s, dx¼FdX. These objec ions a e ci cum- en ed by no ing ha o homogeneous de o ma- ions, Eq. (1) holds exac ly, e en o ma e ial ec o s o fini e leng h. The Bo n ule assumes ha , a leas ‘‘locally’’, i.e. in he scale o he la ice ec o s, he de o ma ion is homogeneous. 2.2. Why he case o films is mo e difficul Conside now he case in which we ha e a single a om hick c ys alline film (such as a g aphene shee ) de o ming a bi a ily in 3D. I is na u al in his case o ea he con inuum solid as a mem- b ane wi hou hickness. The shee is hen a wo- mani old embedded in R3(a su ace). I is assumed ha he a oms lie on he su ace (CauchyÕs hy- po hesis), and he e o e he la ice ec o s a e cho ds o he su ace. We would like o use he Bo n ule in o de o exp ess he geome y o he de o med la ice ec o s in e ms o he some con inuum a iable cha ac e izing he de o ma- ion o he su ace, such as he G een de o ma ion enso . Suppose ha he unde o med body is plana , like a plana g aphene shee . In his case, X0an open se in R2. The de o ma ion map ans o ms his o iginally plana body in o a cu ed mem- Fig. 1. De o ma ion map and i s angen map o space filling bodies. 4 F¼g1I1þg2I2;ð7Þ whe e I1;I2gis he dual basis o I1;I2g. Thus, a ec o W¼WIII2TXX0is ans o med by he de o ma ion g adien in o w¼FW ¼WIgI2 TUðXÞX. The e o e, he ma ix ep esen a ion o F in he Ca esian/con ec ed basis gIIJgis he 22 iden i y ma ix, he in o ma ion abou he de o ma ion being con ained in he con ec ed basis ec o s. We can also define he G een de- o ma ion enso C¼FTF. I s componen s in he Ca esian basis o X0coincide wi h hese o he me ic enso in he con ec ed basis p esen ed in Eq. (6). The enso Ccan be used o measu e leng h, angle and a ea changes due o he de o ma ion in e ms o unde o med body quan i ies, i.e. Cop- e a es in TX0. In pa icula he elemen o a ea o Xcan be w i en in e ms o he elemen o a ea o X0as dX¼JdX0whe e he Jacobian is J¼ de gIJ p. Also, he s e ch in he di ec ion o a uni ec o E2TXX0is K¼ECðXÞE p. 3.1.2. The second undamen al o m: he cu a u e The uni no mal o he su ace Xcan be defined as n¼g1g2 kg1g2k;ð8Þ whe e kkdeno es he Euclidean no m. The sec- ond undamen al o m o he de o med body kcan be exp essed in he basis gIgJgin e ms o i s componen s: kIJ ¼ngI;J;ð9Þ whe e gI;Jdeno es he de i a i e o gIwi h espec o XJ. The no mal cu a u e kna a poin xo he su ace Xand in a gi en di ec ion defined by he uni ec o ¼ IgI2TxX, is he minimum o he cu a u es o all he cu es o Xpassing h ough x angen o . I can be ob ained as: knðxÞ¼kIJ I J:ð10Þ Suppose he no mal cu a u e o he de o med body is o be compu ed a a poin x¼UðXÞ2Xin a gi en di ec ion V¼VIII2TXX0o he unde- o med body. This di ec ion co esponds in he de o med body o ¼FV ¼VIgI, whe e Eq. (7) has been used. The e o e, a e no malizing , he esul ing exp ession o he no mal cu a u e is: knðxÞ¼ kIJ VIVJ gMN VMVN p;ð11Þ whe e he denomina o co esponds o he Eu- clidean no m o . 3.1.3. The exponen ial map A simple defini ion o he exponen ial map is gi en in Mo gan (1993) o a mani old M: The exponen ial map exppa a poin pin M maps he angen space TpMin o Mby send- ing a ec o in TpM o he poin in Ma dis- ance j jalong he geodesic om pin he di ec ion . The exponen ial map is in e ible and diffe en- iable in a neighbo hood o each egula poin po he mani old. I can be defined because o he ex- is ence and uniqueness o geodesics a any poin gi en a di ec ion in he angen space. The expo- nen ial map is defined he e in abs ac e ms be- cause i s e alua ion equi es he knowledge o he geodesics. In gene al, ob aining he geodesics in- ol es sol ing he geodesic diffe en ial equa ions. These equa ions a e a sys em o non-linea o di- na y diffe en ial equa ions whose unknowns a e he pa ame ic coo dina es o he geodesic, and whose coefficien s a e he Ch is offel symbols o he su ace. Finding he geodesics, and hus he exponen ial map, is much simple in some pa ic- ula cases, as will be shown o he cylinde . Mo e de ails abou he exponen ial map o su aces can be ound in do Ca mo (1976). Fig. 3 p o ides an illus a ion o how he exponen ial map b ings a angen ec o o he su ace. Fig. 3. Illus a ion o he exponen ial Bo n ule; he geodesic a xin he di ec ion o wis ep esen ed by a dashed line. 6 3.2. Exponen ial Bo n ule In he p esen heo y, he con inuum solid equi alen o he o iginal single laye c ys alline film is a memb ane wi hou hickness. The nuclei o he a omis ic sys em lie on his su ace and consequen ly he la ice ec o s a e cho ds o he su ace. Le Adeno e an unde o med la ice ec- o . In he p esen se ing, since he unde o med body is plana , X0and TX0can be iden ified, and consequen ly Acan be ans o med h ough he de o ma ion g adien . The esul o he ans o - ma ion w¼FA is simply he de o med la ice ec o we would ob ain h ough he s anda d Bo n ule. Howe e he ec o wis angen o he de- o med su ace X, no a cho d. Conside he ol- lowing gene alized kinema ic ule, called he exponen ial Bo n ule in he ollowing: a:¼expUðXÞFA:ð12Þ Desc ibed in wo ds, his map akes a la ice ec o in he unde o med body Aemana ing om Xand ans o ms i in o a ec o win he angen o he de o med body Xa x¼UðXÞ. Then, his ec o is mapped om he angen space o he de o med su ace h ough he exponen ial map, which ‘‘b ings’’ he esul o he s anda d Bo n ule back o he su ace, hence defining a cho d (see Fig. 3 o an illus a ion o his p ocedu e). Thus, he exponen ial Bo n ule links he de o ma ion o he la ice ec o s o he de o ma ion o he con in- uum objec , since bo h he de o ma ion g adien and he exponen ial map a e defined in e ms o he de o ma ion map U. This ex ended kinema ic ule p o ides a heo- e ical amewo k o he applica ion o c ys al elas ici y o cu ed c ys als, by ec i ying he sho comings o he s anda d Bo n ule. No e howe e ha i s p ac ical implemen a ion is no s aigh o wa d, since he e alua ion o he expo- nen ial map equi es he de e mina ion o he geodesics, which in gene al en ails he in eg a ion o a sys em o wo non-linea diffe en ial equa- ions. This esul s in a compu a ionally e y complex me hod ha is necessa ily non-local. He e we p esen app oxima ions o he exponen ial map ha ende he model local and compu a ionally easible. 4. A omic chain in 2D In his sec ion, we illus a e he exponen ial Bo n ule o he simples case, an a omic chain de o ming in 2D. The esul ing con inuum model is a hype -elas ic ope whose s ain ene gy densi y depends on he s e ch and he cu a u e o he con inuum objec . This cons i u i e model is based exclusi ely on he a omis ic desc ip ion o he chain. In his case he exponen ial map is ap- p oxima ed a each poin by he exponen ial map o he ci cle, o which a closed- o m exp ession is s aigh o wa d. 4.1. A omic model The s ain ene gy o he a omic sys em is de- sc ibed by means o bond s e ch Vsand bond angle Vhpo en ials. The s ain po en ial ene gy o he a omic chain can be w i en as a unc ion o he nuclea posi ions xi: Pchainðx1;...;xnÞ¼X nB k1 VsðakÞþX mB l1 VhðhlÞ; ð13Þ whe e akdeno es he bond leng hs, hldeno es he angle ha adjacen bonds o m, and nBand mBa e he numbe o bonds and adjacen bonds, espec- i ely. This pa icula a omis ic model is chosen o simplici y, bu he app oach is no es ic ed o his s uc u e o he in e -a omic po en ial by any means. The exponen ial Bo n ule p o ides a link be ween he a omis ic and he con inuum de o - ma ions, and can be combined wi h any a omis ic model o choice, no es ic ed o closes -neighbo models. 4.2. Con inuum model As illus a ed in Fig. 4, he unde o med body is conside ed o be a 1D line segmen ha is allowed o de o m in 2D. The e o e, he de o ma ion map can be desc ibed as x¼UðXÞ¼U1ðXÞi1þU2ðXÞi2 wi h X2X0Rand i1;i2g he basis o R2. In his case, he componen s o he de o ma ion g adien a e ½F¼½U1 ;X;U2 ;XT, and he G een de o ma ion enso Cis a scala , whose squa e oo is he s e ch Ko he de o med ope: 7 K¼C p¼ ðU1 ;XÞ2þðU2 ;XÞ2 q:ð14Þ The no mal cu a u e kno he de o med ope can be w i en as: kn¼1 K3ðU2 ;XU1 ;XX U1 ;XU2 ;XX Þ;ð15Þ and can be in e p e ed geome ically as he in e se o he adius o cu a u e o he cu e. As we men ioned in Sec ion 3.2, in o de o ob ain a p ac ical me hod he exponen ial Bo n ule needs o be app oxima ed. I is desi able ha he app oxima ion o he exponen ial Bo n ule leads o a local model, i.e. one in which he s ain ene gy depends on he local de o ma ion o he ope. The s a egy ollowed o ob ain such an ap- p oxima ion is o pe o m he exponen ial map a each poin , no o he o iginal cu e, bu o a ci cle o adius ¼1=knwi h he same no mal as he o iginal cu e (see Fig. 4). Thus, locally, his ci cle eplaces he o iginal cu e. The exponen ial map o he ci cle is eadily a ailable in closed o m. The fi s pa o he exponen ial Bo n ule maps he la ice ec o Ao leng h Ain o a ec o an- gen o he cu e whose componen s a e ½w¼A½F. The e o e, i s leng h is w¼C pA:ð16Þ The exponen ial map o he ci cle is illus a ed in Fig. 5. The leng h o he angen ec o wis ‘‘walked’’ on he geodesic o ob ain expUðXÞw, and he e o e he cho d a. Since he geodesic o he ci cle is i ially he ci cle i sel , he leng h o he a c defined by he ends o ais w. Le hdeno e he angle o med by wo adjacen de o med la ice ec o s. Conside he iangle o med by he ends o aand he cen e o he ci cle. This iangle is isosceles, and i s equal angles a e h=2. The e o e i s hi d angle, he angle sub- ended by he a c o leng h w,isc¼ph. Con- sequen ly, we can ela e he leng h o he a c, w, o he adius o he ci cle ¼1=knand he angle c: w¼c ¼ðphÞ=kn:ð17Þ Since he leng h o he unequal side o he iangle can be easily compu ed as a¼kak¼2 sin c 2;ð18Þ i ollows ha a¼2 kn sin knw 2and h¼pknw:ð19Þ No e om Eqs. (16) and (19) ha he quan i ies a and h, which a e he a gumen s o he a omis ic ene gy (see Eq. (13)), a e exp essed in e ms o he con inuum de o ma ion. The nex s ep is o conside a ep esen a i e c ys alli e o he a omis ic sys em, which in his case is a cell o leng h Aincluding a single nucleus in he unde o med c ys al. In a homogeniza ion p ocess, he ene gy o his de o med cell con ain- ing one bond and one angle be ween adjacen bonds is iden ified o he s ain ene gy densi y o he con inuum mul iplied by he unde o med ol- ume o he cell: AWðUÞ¼VsðaÞþVhðhÞ. Since ou aim is o o mula e a hype -elas ic con inuum model, he elas ic po en ial WðUÞis a s ain ene gy pe unde o med olume, in his case unde o med Fig. 5. The exponen ial map o he ci cle defined a each poin o he cu e by he uni no mal and he no mal cu a u e. Fig. 4. Illus a ion o he con inuum ope like model o an a omic chain de o ming in 2D. 8 leng h. The con inuum s ain ene gy densi y de- pends on he de o ma ion map U h ough he local s ain measu es Cand kn. The e o e, he hype - elas ic po en ial o he con inuum ope can be w i en as: WðC;knÞ¼1 AVs2=knsinðknC pA=2Þ hinþVhp hknC pAio:ð20Þ The o al s ain ene gy o he con inuum sys em app oxima ing he a omis ic ene gy o Eq. (13) can hen be w i en as: P opeðUÞ¼ZX0 WðC;knÞdX0:ð21Þ By aking de i a i es o he hype -elas ic po en ial Wwi h espec o he s ain measu es, Lag angian s ess measu es a ise: he wo k conjuga e o Cis an axial s ess analogous o he second Piola Ki ch- hoff s ess enso , and he conjuga e o knis a bending momen -like s ess. Second de i a i es lead o he axial, bending and coupled axial- bending elas ic angen moduli. 4.3. Example and discussion Suppose an ini ially ec ilinea unde o med ope o leng h nA is ben in o a ci cle o adius wi h uni o m s e ch. The con inuum s e ch is K¼2p =ðnAÞ, and he cu a u e is kn¼1= . Since he a oms a e pos ula ed o lie on he con inuum su ace, he co esponding equispaced a omic chain con aining nbonds is de o med in o a egula polygon o nsides whose ci cumci cle has a adius . The bond leng h and angle p e- dic ed by he con inuum model (see Eqs. (16) and (19)) a e: a¼2 sin p nand h¼p2p=n:ð22Þ I is easy o see ha hese p edic ions coincide exac ly wi h he ac ual bond leng hs and angles o he a omic chain de o med in o a egula polygon. The e o e, he p edic ed ene ge ics o his fini e de o ma ion a e also exac . O cou se, o a gen- e al de o ma ion wi h non-cons an s e ch and cu a u e, he local app oxima ion o he expo- nen ial Bo n ule will lead o app oxima e ene - ge ics. The examples p esen ed la e demons a e, howe e , ha his app oxima ion is e y accu a e. The inadequacy o he s anda d Bo n ule can be illus a ed easily in he p esen example. The s anda d Bo n ule co esponds o aking a¼ w¼FA. Suppose ha ou ec ilinea 1D unde- o med body is de o med in o a ci cle wi hou s e ch, i.e. K¼C¼1. The applica ion o he s anda d Bo n ule leads o de o med la ice ec- o s ha a e angen o he ope. Consequen ly, wo la ice ec o s emana ing om he same nu- cleus emain collinea a e de o ma ion, so he angle hey o m is unchanged i espec i e o he bending o he ope. Fu he mo e, since he ope is ben wi hou s e ch, he leng h o he de o med la ice ec o s also emains unchanged (see Eq. (16)). The e o e, he ene gy o such a model will emain unchanged, and he esul ing ope has ze o bending s iffness. Howe e , he eal la ice ec o s do no emain coplana and hei leng h changes due o he cu - a u e e en i K¼C¼1, since om Eq. (22) i ollows ha o his isome ic de o ma ion a¼nA=psin p nand h¼p2p=n:ð23Þ The e o e, he ene gy o he a omic sys em will change when de o med in his ashion. Thus, a con inuum model based on he s anda d Bo n ule is blind o he ac ha he ope is being ben , and assigns ze o ene gy change o he de o ma ion, in sha p con as wi h he exponen ial Bo n ule, which p edic s he co ec ene ge ics. Al hough an in ui i e app oach would associa e he con inuum s e ch o he s e ch o he bonds, and he con inuum cu a u e o changes in bond angles, he p oposed model couples hese de o - ma ion modes. Indeed, he con inuum bond leng h a, which is he a gumen o he in e -a omic s e ch po en ial, depends bo h on Cand knin a non-linea ashion. The same applies o he con- inuum bond angle h. This ea u e is essen ial and makes he con inuum model exac o de o ma- ions ha map an ini ially s aigh chain in o a ci cula a c wi h cons an s e ch. Thus, as in he case o he s anda d Bo n ule o bulk c ys alline ma e ials, he esul ing con inuum model o he ope is exac o homogeneous de o ma ions. 9 The con inuum s a emen o he p oblem o finding s able equilib ium solu ions is hen gi en by: U¼a g in W2CPðWÞ  ;ð59Þ whe e Cis he app op ia e space o de o ma ions o ial unc ions accoun ing o essen ial bound- a y condi ions. Acco ding o he p inciple o s a- iona y ene gy, he equilib ium solu ions o he sys em a e s a iona y poin s o he po en ial en- e gy unc ional, and hey e i y he p inciple o i ual wo k: 0¼dPðUÞ ¼ZX0 ob WW oC:dCþob WW okn dkn!dX0 ZX0 BdUdX0þdPnb;ð60Þ whe e dUdeno es he i ual de o ma ion. The a ia ions o he non-bonded con inuum po en ial can be w i en as: dPnb ¼1 2 4 S2 0ZX0ZX0BX V0 nb kUðXÞUðYÞk ½UðXÞUðYÞ ½dUðXÞdUðYÞdX0YdX0X:ð61Þ Le us also define he s ess measu es, always e alua ed a he elaxed inne displacemen s ^ gg. Recalling he exp ession o he s ain ene gy den- si y in Eq. (48) and ollowing a simila a ionale o ha used o ob ain Eq. (52), we ob ain: S¼2ob WW oC¼2oW oC ¼2 S0X 3 l1 V0 s okalk oC "þ2X 3 k1 oVh oh ohk oC  þoVh o 1 okaik oCþoVh o 2 okajk oC#;ð62Þ and m¼ob WW okn¼oW okn ¼1 S0X 3 l1 V0 s okalk okn "þ2X 3 k1 oVh oh ohk okn  þoVh o 1 okaik oknþoVh o 2 okajk okn#:ð63Þ The in-plane s ess Sco esponds p ecisely o he Second Piola Ki chhoff s ess, while mis a mo- men -like s ess. No e ha , because o he special o m o C(see Eq. (33)), Shas only wo non-ze o componen s, which a e ela ed o he ac ions in he axial and he ci cum e en ial di ec ions. On he o he hand, mis he e a scala . The gene al heo y o a bi a y de o ma ions is gi en in A oyo and Bely schko (2002). No e ha , since he memb ane has no hickness, he uni s o Sa e o ce di ided by leng h, while mis exp essed in uni s o o ce (bending momen di ided by leng h). Using he G een s ain enso E¼1=2ðCIÞ, we can ew i e he p inciple o i ual wo k as: 0¼ZX0ðS:dEþmdknÞdX0 ZX0 BdUdX0þdPnb:ð64Þ Depending on he ea men o he axial s e ch K1 (see Eq. (30)) diffe en si ua ions can be s udied: Plane s ain: We can conside he si ua ion in which he alue o K1is p esc ibed. In his case, he unknowns o he a ia ional p oblem (64) a e U2and U3(see Eq. (30)). I K1¼1, a de o ma ion analogous o plane s ain condi ions is achie ed. This applies o e y long o axially cons ained nano ubes. K1can also be p esc ibed an a bi a y alue o s udy he ans e se beha io o s e ched o comp essed nano ubes; he beha io will change due o he non-linea i y o he model. In his si - ua ion, dK1¼0 and he axial componen o he memb ane s ess does no appea in he a ia ional p inciple. This means ha he axial s ess can be compu ed a pos e io i, bu does no play a ole in he solu ion o he p oblem. Plane s ess: Al e na i ely, he axial componen o he memb ane s ess can be p esc ibed, o in- 16 s ance, o be ze o. This would be he case o axially uncons ained nano ubes. In his case, in addi ion o U2and U3, he axial s e ch K1becomes an unknown o he p oblem. 7. Valida ion and ep esen a i e simula ions In his sec ion, nume ical simula ions o s able configu a ions o ca bon nano ubes in diffe en si ua ions a e epo ed. The educed con inuum model desc ibed in he p e ious sec ion is used and he a ia ional p inciple desc ibed in Eq. (64) is disc e ized by Gale kin fini e elemen s (FE). Thus, he o iginal disc e e molecula sys em is e- placed by a con inuum model which is subse- quen ly ans o med by he FE me hod in o ano he disc e e sys em. Howe e , in p inciple we a e ee o design he FE disc e iza ion so ha he FE model has ewe deg ees o eedom han he o iginal sys em. Fu he mo e, since he con- inuum model is 2D, while he ull a omis ic model is 3D, he compu a ional cos is u he educed. Fi s , he exponen ial Bo n ule-based con in- uum model is alida ed by compa ing FE simula- ions based on i wi h ull a omis ic calcula ions. In hese compa isons he in e -a omic po en ials used in he MM simula ions a e used o cons uc he con inuum cons i u i e equa ion, and analo- gous bounda y condi ions a e conside ed in bo h calcula ions. Since he con inuum model is in- ended o mimic he a omis ic sys em, which is iewed as ‘‘ ue’’, he e m e o should be un- de s ood as de ia ion o m he a omis ic model. Simula ions show ha he ag eemen is excellen wi h ega d o he ene ge ics as well as o he s able configu a ions. Simula ions o a model based on he s anda d Bo n ule a e also p o ided, illus- a ing he deficiencies o such a model. The ex- ponen ial Bo n ule simula ions also show ha , o he es ed si ua ions, he elaxa ion o he inne displacemen s g ea ly affec s he ene ge ics bu has e y li le impac on he minimum ene gy config- u a ions. The con inuum model is hen applied o simula e se e al si ua ions whe e he ans e se beha io o ca bon nano ubes and he effec o an de Waals in e ac ions a e impo an . A final ex- ample o he gene aliza ion o he model o h ee dimensions is p esen ed, wi h a wis ing es o a [10,10] nano ube beyond he poin o s uc u al ins abili ies. The in e -a omic po en ials all in o he gene al o m desc ibed in Eq. (28). The wo-body po en ial Vsis a Mo se po en ial while he h ee-body po- en ial depends only on he angle Vhand is ha - monic wi h a sex ic co ec ion. The pa ame e s a e aken om he MM2 model. The non-bonded in- e ac ions a e based on he classical Lenna d- Jones (6 12) po en ial. The a ia ional p inciple in Eq. (64) imposes es ic ions on he fini e elemen in e pola ion spaces. The i ual in e nal wo k e m in ol es a ia ions on he cu a u e o he es unc ions, and he e o e he fini e elemen space needs o be H2, i.e. ha e up o second squa e in eg able de- i a i es. This is why C1He mi e fini e elemen s a e chosen. No e ha he disc e iza ion o he configu a ion desc ibed in Eq. (30) equi es he app oxima ion o he scala unc ions U2ðÞ and U3ðÞ, i.e. he cu e in R2desc ibed by hese unc ions needs o be pa ame ized wi h espec o he fini e elemen deg ees o eedom. Each o hese unc ions is app oxima ed by piecewise C1 cubic polynomials, and he e o e, each node I ca ies ou deg ees o eedom: U2 I,U3 I,ðU2Þ0 Iand ðU3Þ0 I. The in e nal and ex e nal wo k e ms o he a ia ional p inciple a e in eg a ed using 3 Gauss poin s pe elemen , while he in eg a ion o he non-bonded in e ac ions e m may equi e mo e in eg a ion poin s depending on he size o he fi- ni e elemen s ela i e o he an de Waals equi- lib ium dis ance. Fou in eg a ion poin s a e equi ed o his e m in some o he simula ions. The BFGS quasi-New on echnique is used bo h in he elaxa ion o he inne displacemen s and in he global ene gy minimiza ion. This i e - a i e me hod only equi es g adien s o he ob- jec i e unc ion and app oxima es he in e se o i s Hessian using in o ma ion om he p e ious i e - a ions. Fo some o he la ge examples in ol ing mo e han one nano ube, and when he ini ial configu a ion is e y a om equilib ium, dy- namical elaxa ion is used o ob ain a good fi s guess which is u he efined wi h he BFGS minimiza ion algo i hm. 17 7.1. Valida ion es To alida e he p oposed educed con inuum model, a FE disc e ized e sion is compa ed o a MM model. A [32,0] zigzag ca bon nano ube ( he s anda d desc ip ion o ca bon nano ubes in e ms o wo in ege s is desc ibed by Sai o e al. (1992)) is conside ed (see Fig. 9(a)). The molecula model used in he compa ison has 384 nuclea posi ions, ha is 1152 deg ees o eedom, while he FE model has 20 nodes and consequen ly 80 deg ees o eedom. No e ha he disc e e FE model e- duces he compu a ional cos , no only because la ge elemen s ela i e o he c ys al cell size can be used, bu also because o i s educed dimensio- nali y. The fi s configu a ion s udied consis s o sim- ply olling a g aphene shee in o a ube in an iso- me ic ans o ma ion, wi hou any kind o elaxa ion. This configu a ion is called O iginal ube in Table 1. The able shows he excellen ag eemen be ween he ene gy ob ained wi h he molecula model and ha ob ained ia he con- inuum model and FE. Acco ding o he las e- ma k o Sec ion 6.2, he ene gy o he con inuum model should be exac in his si ua ion. No e howe e ha he con inuum memb ane is disc e - ized using an app oxima ion space ha does no ep oduce exac ly a ci cle, and hus in oduces disc e iza ion e o s. Then se e al ‘‘plane s ain’’ si ua ions a e con- side ed. This condi ion is en o ced in he molecu- la model by p esc ibing o ze o he nuclea displacemen s in he di ec ion o he axis o he ube a he nuclei loca ed a bo h ends o he ube. In he con inuum model, we simply en o ce K1¼1. Also, wo kinds o ene gy minimiza ion a e conside ed. The fi s one eezes he inne dis- placemen s o hose o he g aphene shee in equilib ium, i.e. in he con inuum model by p e- sc ibing g¼0. In his pa icula example, in oking symme y conside a ions, his cons ained mini- miza ion can be easily implemen ed in he molec- Fig. 9. (a) Ac ual molecula model used in compa ison, (b) compa ison o 20 elemen exponen ial Bo n ule con inuum model wi h MM and (c) esul s ob ained wi h a model cons uc ed om he s anda d Bo n ule. Table 1 Compa ison o 20 elemen model (CþFE) wi h MM: ene gy in J/mol g0 Relaxed g MM CþFE E o (%) MM CþFE E o (%) O iginal ube 14.58 14.69 0.81 Relaxed ube 10.26 10.28 0.22 6.338 6.324 0.22 Squeezed A 20.85 21.22 1.8 12.95 13.11 1.2 Squeezed B 48.56 49.17 1.3 30.68 30.46 0.75 18 ula model by p esc ibing o ze o he displace- men s o all he nuclei in he di ec ion o he ube axis. This incomple e analysis is pe o med o highligh he effec o he inne elaxa ion. The o he analysis is an uncons ained s uc u al op- imiza ion o all he nuclea posi ions. In he con inuum, he inne displacemen s a e elaxed in o de o calcula e b WW a each Gauss poin . The si ua ions conside ed a e: Relaxed ube: The O iginal ube is elaxed wi hou any cons ain o he han he plane s ain condi ions. Squeezed A: Displacemen s a he ends o one diame e o he ube a e p esc ibed so ha his diame e o is squeezed o 3=4 o i s o iginal size. Squeezed B: Displacemen s a he ends o one diame e o he ube a e p esc ibed so ha his diame e o is squeezed o 1=2 o i s o iginal size. Table 1 p esen s he equilib ium ene gies o bo h he MM and he Con inuum FE simula ions, as well as he ela i e e o o he FE calcula ion wi h espec o MM. A posi i e alue o e o means ha he MM ene gy is lowe han he FE ene gy. No e ha his e o includes con ibu ions no only om he modelling o he disc e e a omic sys em as a memb ane, bu also om he FE dis- c e iza ion. Fig. 9(b) compa es he equilib ium configu a- ions o he con inuum/FE model and he MM model in he Squeezed B si ua ion. Despi e he la ge de o ma ions o which he ube is subjec ed, he ag eemen is excellen . Table 1 shows ha he equilib ium ene gies ob ained wi h he con inuum model a e in all he cases e y accu a e app oxi- ma ions o he MM ene gies. The disc epancies a e in all he cases below 2%. The effec o he inne elaxa ion in he magni ude o he ene gies is e y impo an . In his able 20 fini e elemen s ha e been used, while 32 hexagonal cells span he same pe ime e in he MM model. The e o e, we expec he FE model o be mo e cons ained and he e- o e yield highe equilib ium ene gies. This can be no iced in he columns co esponding o ozen inne displacemen s. Howe e , when hose a e e- laxed, he FE model eaches lowe ene gies han he molecula model, s ill emaining e y accu a e. P obably he con inuum ea men o he inne displacemen s allows o his ex a elaxa ion. Al hough he effec o he inne elaxa ion in he equilib ium ene gies is e y impo an , in hese simula ions i s effec on he s able configu a ions is negligible. This can be explained by no ing ha he in-plane beha io o he model is e y s iff, while he flexu al beha io is e y complian . The e o e, a sligh pe u ba ion o in-plane de o ma ion ( he inne ea angemen s a e an in-plane effec ) has d ama ic influence on ene gy, bu no in hese flexu al-domina ed op imal de o ma ions. This sugges s ha in hese examples, he inne elax- a ion is nea ly uncoupled om he bending de- o ma ion. This is no he case o o he ypes o de o ma ion (A oyo and Bely schko, 2002). Table 2 shows he esul s ob ained wi h 36 fini e elemen s. The e o s ob ained a e smalle in all he cases excep in he Squeezed B si ua ion wi h inne elaxa ion. This indica es ha in gene al he iche disc e iza ion dec eases he o e all e o , bu also ha he fine mesh allows o he modeling e o s o mani es hemsel es. Indeed, he e o p obably inc eases in he las case because he con inuum model is mo e complian han he molecula one wi h ega d o he inne displacemen s. Howe e , simula ions ca ied ou wi h e en fine meshes in- dica e ha he esul s ‘‘con e ge’’ o a e y accu- a e esul . Thus, e en i he FE model is efined beyond he uni cell size, he con inuum model Table 2 Compa ison o 36 elemen model (CþFE) wi h MM: ene gy in J/mol g0 Relaxed g MM CþFE E o (%) MM CþFE E o (%) O iginal ube 14.58 14.60 0.14 Relaxed ube 10.26 10.28 0.21 6.338 6.324 0.22 Squeezed A 20.85 21.05 0.96 12.95 12.99 0.31 Squeezed B 48.56 49.01 0.93 30.68 30.34 1.1 19 appa en ly does no exhibi fine ea u es ha canno be p esen in he molecula model. This excellen beha io con as s wi h he si u- a ion encoun e ed when a con inuum model o he memb ane is di ec ly cons uc ed om he Bo n ule wi hou he p oposed exponen ial ex- ension. In his case he esul ing hype -elas ic po en ial is non-con ex. Indeed, as discussed in Sec ion 4.3, he ene gy o such a model is in a ian unde isome ic de o ma ions (bending wi hou s e ch), i.e. he model has ze o bending s iffness. This eflec s in a pa hological mesh dependency in he nume ical implemen a ion o such a model: since he disc e e FE space canno ep esen all isome ic de o ma ions, he disc e e p oblem can s ill be sol ed, bu as he mesh is efined, he nu- me ical me hod picks solu ions wi h inc easingly fine ea u es. Fig. 9(c) illus a es his ac , and sha pe kinks in he nume ical solu ion a e ob- se ed as he mesh is efined. The equilib ium en- e gy o he FE solu ions is almos ze o, which is no ealis ic. This is eminiscen o he si ua ion encoun e ed in o he ma e ials, o which he Fig. 13. Equilib ium configu a ion o a bundle o se en closely packed [22,0] nano ubes. Fig. 10. Which is mo e s able, ci cula o collapsed? (Answe : o he [20,0] and [26,0] ubes, ci cula , and o he [32,0] and [40,0] ubes, collapsed.) Fig. 12. Equilib ium configu a ions o pai s o nano ubes in an de Waals con ac . Fig. 11. T ans e se s abili y o a mul i walled nano ube. 20 s ain ene gy densi y is physically con-con ex, leading o non-unique solu ions wi h inc easingly fine ea u es, as epo ed by Daco ogna (1989, p. 276) and e e ences he ein. 7.2. T ans e se de o ma ion simula ions The nex simula ions illus a e he applica ion o he con inuum/FE model o he ans e se me- chanics o nano ubes in diffe en si ua ions. In hese applica ions, he compu a ional cos o analogous MM simula ions would be much highe han he cos o he p esen ed calcula ions. This is especially ue wi h ega ds o he non-bonded in e ac ions. The fi s example s udies he s abili y o he ci cula and he collapsed configu a ions o ca bon nano ubes. Because o he an de Waals a ac- ion po en ial, he ene gy o he sys em is educed when wo walls adhe e. On he o he hand, o he wall o a nano ube o come in con ac wi h i sel , significan elas ic ene gy is equi ed. This adeoff is p obably esponsible o he obse a ion by Gao e al. (1998) ha below a ce ain adius, only he ci cula configu a ion is s able. Fo g ea e adii, he collapsed configu a ion is a leas me a-s able. Subsequen ly, ano he h eshold adius sepa a es he nano ubes o which he ci cula configu a ion is ene ge ically a o able om hose in which he collapsed configu a ion is. Fig. 10 shows he sim- ula ions pe o med o se e al nano ubes. In his and subsequen figu es, he nodes shown a e nodes o he fini e elemen mesh; hey a e no a oms. In hese calcula ions, he ully elaxed ci cula con- figu a ion is de o med so ha he wall o he nano ube is b ough in con ac wi h i sel a he an de Waals equilib ium dis ance, and hen he ene gy is minimized. The sign o he diffe ence in ene gy be ween he ci cula configu a ion and he elaxed configu a ion is also epo ed, i.e. a posi i e diffe ence means ha he ene gy o he configu a ion p esen ed on he igh is lowe . In some cases, he nano ube goes back o he o iginal configu a ion ( his is he case o he [20,0] nano- ube). This implies ha he collapsed configu a ion is no s able. The collapsed configu a ion is s able o he [26,0] nano ube, bu his only cons i u es a local minimum o he ene gy since he ci cula configu a ion has lowe ene gy. Fo he [32,0] and [40,0] nano ubes, he collapsed configu a ion is he ene ge ically a o able s uc u e. This is expec ed because la ge nano ubes a e mo e flexible and ha e mo e wall a ea o gain adhesion ene gy. Fig. 11 displays a simila analysis o a mul i-walled nano ube o which he collapsed configu a ion yields lowe ene gy . A simila compe i ion o elas ic and adhesion ene gy occu s when wo nano ubes a e b ough o he an de Waals equilib ium dis ance. Fig. 12 shows he equilib ium configu a ions ob ained when his nume ical expe imen is pe o med wi h nano ubes o diffe en sizes. Again, he la ge na- no ubes ha e la ge po ions o fla ened walls. We also epo a simula ion o a bundle o na- no ubes unde plane s ain. Fig. 13 shows he equilib ium configu a ion o he sys em. A TEM image o such a nano ope has been epo ed by Sal e a e al. (1999). Ca bon nano ubes end o be closely packed in hexagonal la ices in he nano- opes and c ys als o nano ubes (Thess e al., 1996; Schli le e al., 2001). As can be seen om Fig. 13, he equilib ium configu a ion displays a fla ening o he nano ube walls, o pa ial polygonaliza ion. 7.3. Th ee dimensional simula ion The heo y p esen ed has been used o con- s uc a memb ane applicable in he gene al 3D Fig. 14. Twis ing o a [10,10] nano ube: de o med geome y o wis ing angles o 38,210and 360, and c oss sec ion o he de o med memb ane a he cen e o he ube o he abo e h ee configu a ions. 21 de o ma ion o ca bon nano ubes (A oyo and Bely schko, 2002). This mo e gene al memb ane can be disc e ized wi h subdi ision fini e elemen s, and he s uc u al ins abili ies epo ed in expe i- men s and a omis ic simula ions can be analyzed a e y low compu a ional cos . The analysis o wis ing a [10,10] nano ube is p o ided in Fig. 14, o a Te soff-B enne po en ial. No e ha , he de o med geome ies ha e been pos -p ocessed, and he compu a ional mesh has abou 18 ele- men s a ound he pe ime e . Each end o he nano ube is inc emen ally o a ed 360in opposi e o ien a ions. The fi s snapsho o he de o ma ion shows he configu a ion when he fi s ins abili y om a uni o m wis ing occu s, and he co esponding c oss-sec ion is shown a he bo om o he figu e. Fu he wis ing causes he wall o he nano ube o come in an de Waals con ac wi h i sel , as clea ly shown in he c oss-sec ion in he bo om o Fig. 14. Beyond 210, a seconda y ins abili y de- elops, and he ube olds on o i sel . F om he c oss-sec ion i is appa en ha he an de Waals in e ac ions a e esponsible o his buckled mo - phology. In he absence o hese long- ange o ces, he memb ane in e -pene a es and he seconda y s uc u e is no obse ed. This 3D memb ane has been shown o p o ide e y accu a e ene ge ics and de o med geome ies e en o e y la ge de- o ma ions (A oyo and Bely schko, 2002). 8. Conclusions We ha e u he explo ed a me hodology o cons uc con inuum models o one-a om hick c ys alline films. The p oposed model is a hype - elas ic memb ane whose elas ic po en ial ene gy is w i en in closed- o m exclusi ely in e ms o he in e -a omic po en ials ha cons i u e he molec- ula desc ip ion o he sys em. The analysis o he p esen wo k is based on he exponen ial he Bo n ule (A oyo and Bely schko, 2002), a ki- nema ic assump ion linking he a omic and he con inuum de o ma ions when he c ys al is a cu ed film. This ex ension is based on he ex- ponen ial map. An illus a i e example o an a omic chain de o ming in wo dimensions has been p esen ed. The esul ing simple ope-like con inuum model encompasses all o he unda- men al ideas. The gene al me hodology hen is pa icula ized o analyze he ans e se mechanics o ca bon nano ubes. This model explici ly exploi s he symme y o such a de o ma ion, and leads o a model o educed dimensionali y. The hype -elas- ic po en ial, as well as s ain and s ess measu es a e p o ided, and a con inuum o mula ion o he non-bonded in e ac ions is de i ed. The p oposed model is disc e ized using fini e elemen s, yielding an al e na i e simula ion me hod ha is as e han a omis ic calcula ions. Se e al simula ions highligh ing he ele ance o an de Waals in e ac ions in he ans e se mechanics o nano ubes a e epo ed. The esul s show ha he con inuum model based on he ex- ponen ial Bo n ule e y well app oxima es he s able configu a ions and ene gies o he co e- sponding MM model. Resul s ag ee wi h MM calcula ions wi hin 2% in he equilib ium ene gies. This sha ply con as s wi h he non-physical e- sul s ob ained om a model based on he s anda d Bo n ule. We also show he impo an effec o he inne ea angemen s o he c ys al s uc u e on he equilib ium ene gies. A ull 3D simula ion il- lus a es he applica ion o he p esen heo y o analyze he s uc u al ins abili ies o nano ubes obse ed in expe imen s and a omis ic calcula- ions. Acknowledgemen s The suppo o he ‘‘la Caixa’’ G adua e P o- g am o M. A oyo, and he Na ional Science Founda ion and he U.S. A my Resea ch Office is g a e ully acknowledged. Re e ences A oyo, M., Bely schko, T., 2002. An a omis ic based fini e de o ma ion memb ane o single laye c ys alline films. Jou nal o he Mechanics and Physics o Solids 50, 1941 1977. Be nholc, J., B abec, C.J., Na delli, M.B., Mai i, A., Roland, C., Yakobson, B.I., 1998. Theo y o g ow h and mechanical p ope ies o nano ubes. Applied Physics A 67, 39 46. 22 B enne , D.W., 1990. Empi ical po en ial o hyd oca bons o use in simula ing chemical apo deposi ion o diamond films. Physical Re iew B 42 (15), 9458 9471. Chop a, N.G., Benedic , L.X., C espi, V.H., Cohen, M.L., Louie, S.G., Ze l, A., 1995. Fully collapsed ca bon nano ubes. Na u e 377, 135 138. Cousins, C.S.G., 1978. Inne elas ici y. Jou nal o Physics C, 4867 4879. Daco ogna, B., 1989. Di ec me hods in he calculus o a ia ions. In: Applied Ma hema ical Sciences, ol. 78. Sp inge Ve lag, Be lin. do Ca mo, M.P., 1976. Diffe en ial geome y o cu es and su aces. P en ice Hall, Englewood Cliffs, NJ. E icksen, J.L., 1984. Phase ans o ma ions and ma e ial ins abili ies in solids. In: Gu in, M.E. (Ed.), The Cauchy and Bo n Hypo heses o C ys als. Academic P ess, Lon don, pp. 61 77. Fal o, M.R., Cla y, G.J., Taylo , R.M., Chi, V., B ooks, F.P., Washbu n, S., Supe fine, R., 1997. Bending and buckling o ca bon nano ubes unde la ge s ain. Na u e 389, 582 584. F iesecke, G., James, R.D., 2000. A scheme o he passage om a omic o con inuum heo y o hin films, nano ubes and nano ods. Jou nal o he Mechanics and Physics o Solids 48, 1519 1540. Gao, G., C ßa ggin, T., Godda d III, W., 1998. Ene ge ics, s uc u e, mechanical and ib a ional p ope ies o single walled ca bon nano ubes. Nano echnology 9, 184 191. Lu, J.P., 1997. Elas ic p ope ies o ca bon nano ubes and nano opes. Physical Re iew Le e s 79 (7), 1297 1300. Mai i, A., 2000. Mechanical de o ma ion in ca bon nano ubes ben ubes s ubes pushed by a omically sha p ips. Chemical Physical Le e s 331, 21 25. Mal e n, L.E., 1969. In oduc ion o he mechanics o a con inuous medium. P en ice Hall, Englewood Cliffs, NJ. Ma sden, J.E., Hughes, T.J., 1983. Ma hema ical ounda ions o elas ici y. P en ice Hall, Englewood Cliffs, NJ. Ma in, J.W., 1975. Many body o ces in me als and he b ug ge elas ic cons an s. Jou nal o Physics C 8, 2837 2857. Mo gan, F., 1993. Riemannian geome y, a beginne Õs guide. Jones and Ba le Publishe s. Ne ins, N., Chen, K., Allinge , N.L., 1996. Moleculas mechan ics (MM4) calcula ions on alkenes. Jou nal o Compu a ional Chemis y 17 (5 6), 669 694. Qian, D., Liu, W.K., Ruoff, R.S., 2001. Mechanics o C60 in nano ubes. Jou nal o Physical Chemis y B 105, 10753 10758. Ruoff, R.S., Te soff, J., Lo en s, D.C., Sub amoney, S., Chan, B., 1993. Radial de o ma ion o ca bon nano ubes by an de Waals o ces. Na u e 364, 514 516. Sai o, R., Fuji a, M., D esselhaus, G., D esselhaus, M.S., 1992. Elec onic s uc u e o chi al g aphene ubules. Applied Physics Le e s 60 (18), 2204 2206. Sal e a , J.P., B iggs, G.A.D., Bona d, J.M., Bacsa, R.R., Kulik, A.J., S oockli, T., Bu nham, N.A., Fo  oo, L., 1999. Elas ic and shea moduli o single walled ca bon nano ube opes. Physical Re iew Le e s 82 (5), 944 947. Schli le , R.R., Seo, J.W., Gimzewski, J.K., Du kan, C., Sai ullah, M.S.M., Welland, M.E., 2001. Single c ys als o single walled ca bon nano ubes o med by sel assembly. Science 292, 1136 1139. Shenoy, V.B., Mille , R., Tadmo , E.B., Rodney, D., Phillips, R., O iz, M., 1999. An adap i e fini e elemen app oach o a omic scale mechanics he quasicon inuum me hod. Jou nal o he Mechanics and Physics o Solids 47, 611 642. Sohlbe g, K., Sump e , B.G., Tuzun, R.E., Noid, D.W., 1998. Con inuum me hods o mechanics as a simplified app oach o s uc u al enginee ing o nanos uc u es. Nano echnol ogy 9, 30 36. Tadmo , E.B., O iz, M., Phillips, R., 1996. Quasicon inuum analysis o de ec s in solids. Philosophical Magazine A 73 (6), 1529 1563. Tadmo , E.B., Smi h, G.S., Be ns ein, N., Kaxi as, E., 1999. Mixed fini e elemen and a omis ic o mula ion o complex c ys als. Physical Re iew B 59 (1), 235 245. Thess, A., Lee, R., Nikolae , P., Dia, H., Pe i , P., Robe , J., Xu, C., Lee, Y.H., Kim, S.G., Rinzle , A.G., Colbe , D.T., Scuse ia, G.E., Tom aanek, D., Fische , J.E., Smalley, R.E., 1996. C ys alline opes o me allic ca bon nano ubes. Science 273, 483 487. Weine , J.H., 1983. S a is ical mechanics o elas ici y. Wiley, New Yo k. Yakobson, B.I., B abec, C.J., Be nholc, J., 1996. Nanome chanics o ca bon ubes: Ins abili ies beyond he linea esponse. Physical Re iew Le e s 76 (14), 2511 2514. Yu, M., Dye , M.J., Ruoff, R.S., 2001a. S uc u e and mechanical flexibili y o ca bon nano ube ibbons: An a omic o ce mic oscopy s udy. Jou nal o Applied Physics 89 (8), 4554 4557. Yu, M., Kowalewski, T., Ruoff, R.S., 2001b. S uc u al analysis o collapsed, and wis ed and collapsed, mul iwalled ca bon nano ubes by a omic o ce mic oscopy. Physical Re iew Le e s 86 (1), 87 90. Yu, M., Lou ie, O., Dye , M.J., Moloni, K., Kelly, T.F., Ruoff, R.S., 2000. S eng h and b eaking mechanism o mul i walled ca bon nano ubes unde ensile load. Science 287, 637 640. Zanzo o, G., 1996. The Cauchy Bo n hypo hesis, nonlinea elas ici y and mechanical winning in c ys als. Ac a C ys allog aphica A 52, 839 849. Zhong Can, O. Y., Su, Z. B., Wang, C. L., 1997. Coil o ma ion in mul ishell ca bon nano ubes: compe i ion be ween cu a u e elas ici y and in e laye adhesion. Physical Re iew Le e s 78 (21), 4055 4058. Zhou, G., Duan, W., Gu, B., 2001. Fi s p inciples s udy on mo phology and mechanical p ope ies o single walled ca bon nano ube. Chemical Physical Le e s 333, 344 349. 23