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.
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