July 2008
EPL, 83 (2008) 10003 www.epljou nal.o g
doi: 10.1209/0295-5075/83/10003
Molecula dynamics simula ions o homogeneous solids
using mul i-laye ed s uc u es
F. A. Gilabe (a) and A. Cas ellanos
Depa men o Elec onics and Elec omagne ism, Facul y o Physics, Uni e si y o Se ille
A da. Reina Me cedes s/n, 41012, Se ille, Spain, EU
ecei ed 19 Feb ua y 2008; accep ed in final o m 16 May 2008
published online 11 June 2008
PACS 02.70.Ns – Molecula dynamics and pa icle me hods
PACS 34.20.C – In e a omic po en ials and o ces
PACS 62.20.-x – Mechanical p ope ies o solids
Abs ac – The main goal o his wo k is o model a homogeneous compu e ma e ial wi h well-
defined mechanical p ope ies. To ca y ou he model ma e ial, an in e nal s uc u e a anged
in laye s wi h diffe en a om sizes is implemen ed using a simple in e a omic law o Lenna d-
Jones ype (LJ). We show ha imposing an app op ia e scaling law be ween he in e a omic
po en ials om diffe en laye s, we ob ain he same mechanical p ope ies as i he ma e ial was
homogeneous. Employing his scheme, gi en a fixed space olume o be occupied by he solid, his
s uc u al a angemen allows o dec ease d as ically (∼30–80%) he equi ed numbe o a oms as
compa ed wi h he case o a homogeneous solid, dec easing he compu a ional effo and speeding
up calcula ions. In ha espec , his p ocedu e is an analogous o mesh e inemen s me hodologies
usually applied in he con inuum app oaches.
Copy igh c
EPLA, 2008
In oduc ion. – Compu a ional designs o solids and
so ma e buil om simplified mic oscopic laws o in e -
ac ion ha e become an ex emely use ul ool o model
ma e ials wi h p esc ibed mac oscopic p ope ies. Many
o hem a e de o ed o unde s and phenomena such as
plas ici y, phase ansi ions, c eeps, la ge de o ma ions,
ac u e p opaga ion and so on. Among he ex ensi e
li e a u e on his a ea, he esea ch made in [1,2] is
wo h men ioning. O e he las yea s, all hese nume -
ical designs a e commonly used wi hin he amewo k o
s a is ical mechanics, since a di ec compa ison wi h he
heo e ical models o liquids and solids is possible [3,4].
Ne e heless, o ge a eliable s a is ical ep esen a ion,
all hese ealiza ions s ill demand high-pe o mance calcu-
la ions in iew o he la ge numbe o pa icles equi ed.
Fo such eason, hyb id models a e now e y popula in
ma e ial sciences hanks o he eno mous quan i y o sa ed
calcula ions. This enables esea che s o in oduce much
mo e de ailed physical aspec s in he model, and he e o e
i is no longe necessa y o sac ifice he physical desc ip-
ion by i ue o ge ing compu a ional imp o emen s. In
a hyb id nume ical design, a con inuum desc ip ion o he
ma e ial coexis s oge he wi h an a omis ic scale le el o
ep esen he in e es zone o he p oblem [5]. In ew wo ds,
we may say ha when he zone desc ibed by MD finishes,
(a)E-mail: [email p o ec ed]
i is immedia ely con inued by a FEM mesh coupled in
he in e ace. This mixed s a egy is now commonly used
in compu a ional es s o nano-inden a ion pe o med on
models o me allic ma e ials (among many o he s, see o
example [6]), whe e he egion suffe ing la ge de o ma ions
is desc ibed by he a omic ep esen a ion, and his is in
u n sus ained by he con inuum egion whe e he de o -
ma ions a e small and p edominan ly elas ic.
In his s udy, we p esen an al e na i e scena io,
wi hou abandoning he disc e e na u e o he in e nal
ep esen a ion o a solid, whose o igins, as a as we
know, s a ed back wi h he bidimensional esea ch made
in [7,8]. In ou de e minis ic desc ip ion, co espond-
ing o he mesoscopic le el, we assume ha pa icles
can be conside ed as “supe -a oms”, which a e ully
cha ac e ized by hei posi ions, eloci ies, in e pa icle
dis ances and o ces. Using his desc ip ion, we design
a mac oscopic solid in e nally s uc u ed in laye s. Each
laye is an amo phous assembly o supe -a oms ha
possess a con enien ly selec ed “size” o in e ac ion ange
obeying a simple scaling law. This gene a es a solid
bulk locally ea anged by amo phous a omic laye s wi h
diffe en hicknesses. The in e ac ion be ween a oms has
been modeled by means o LJ po en ial, which has been
used o model fluids (see pionee esea ch in e s. [9,10])
and solids such as me allic glasses, alloys and an de
Waals solids [11–13]. This po en ial is simple, well known
10003-p1
F. A. Gilabe and A. Cas ellanos
and widely used o s udy he mic oscopic o mac oscopic
c osso e in condensed ma e physics.
This pape is o ganized as ollows: Fi s ly, we p esen
some basic p elimina y defini ions, nomencla u e and he
undamen al physical meaning o he a iables employed
in his s udy. Subsequen ly, he mo i a ion o his al e na-
i e desc ip ion is commen ed, oge he wi h ou p oposal
o he in e ac ion and scaling laws. La e on, we shall
desc ibe he cen al aspec s o ou compu a ional imple-
men a ion, ha is, a de ailed desc ip ion o he laye ing
p ocedu e as well as he ad an ages ob ained om i .
Finally, he esul s ob ained om he mechanical es s by
means o uniaxial and iso opic comp essions a e exposed,
and compa ed wi h hose esul s ob ained om he
“classical” o homogenous solid bulk.
P elimina y issues. – Le us conside a pai o a oms
sepa a ed by a sepa a ion dis ance and subjec ed o
he ac ion o a cen al po en ial u( ) and in e ac ion
o ce ( )=u( ). No e ha he ollowing defini ions
a e independen om he po en ial o m excep o he
non-di ec ional and pai s-addi i i y ea u es. Le us use
designa ions σ, 0,and 1 o he dis ances a which
he po en ial and i s fi s and second de i a i es become
ze o, i.e.:u(σ)=0,u( 0)=0andu ( 1) = 0. Fo all usual
po en ials, σ<
0<
1. The physical meaning o hese
dis ances a e: σ—ha d sphe e adius, 0—equilib ium
dis ance and 1—b eak dis ance. We shall no use
he quan i y σ. The h ee mos impo an dimensional
cha ac e is ics o he in e ac ion a e ene gy, s eng h
and s iffness o he bond, gi en by D=−u( 0), max =
u( 1)andC=u( 0), espec i ely. All hese quan i ies
a e posi i e. In his s udy we use Dand max as he
uni s o ene gy and o ce, espec i ely. Fo he p essu e
we define he magni ude σ0= max/ 2
0, whose meaning
is he s ess needed o b eak a single in e a omic bond
(changing he sign o nega i e). The uni o ime is he
pe iod o oscilla ion o a single a om o mass mgi en by
T0=2πm/C. Finally, he c i ical dissipa ion coefficien
gi en by γc=√mC ep esen s he quan i y o mass ha
can be slowed down pe uni ime. The c i ical alue
co esponds o he iscosi y o an effec i e medium whe e a
pa icle is imme sed and whose alue p e en s he pa icle
o comple e an oscilla ion due o he ac ion o he po en ial
field. In ha espec , we ha e conside ed an a he mal
p oblem. The e o e he simples way o ep esen a global
dissipa i e mechanism is o fix a cons an dissipa ion
coefficien , in ou case γ=0.03γc. We a e awa e o his
delibe a e simplifica ion, since his choice does no allow
us o con ol he he mal p ope ies o he sys em. To
con ol he empe a u e o he ma e ial an addi ional
equa ion o γshould be included, e ol ing in ime and
eadjus ing he ene gy o he sys em [14]. Ne e heless, in
his wo k we a e in e es ed in calcula ing he mechanical
p ope ies unde e y low empe a u es.
To pe o m he mechanical es s, he ba os a ic p oce-
du e de ised by [15] has been implemen ed. I consis s o a
ma hema ical p ocedu e ha enables us o apply any kind
o ex e nal s essed/comp essed s a e on he solid, so ha
he in e nal s ess o he sys em, gi en by he exp ession
o he Cauchy s ess enso ,
σαβ =1
V
i<j
∂u( ij)
∂ α
ij
β
ij,(1)
can ma ch he ex e nally applied p essu e. In (1), he
cen al cha ac e o he po en ial has been assumed,
α
ij deno es he α-componen o he in e a omic dis ance
be ween a oms iand j,andVis he olume occupied by
he solid. P essu e changes a e accomplished by changing
he coo dina es o he pa icles and he size o he
compu a ional cell unde pe iodic bounda y condi ions.
Using his p ocedu e, we ha e ca ied ou uniaxial and
iso opic es s. F om hese es s, he elas ic moduli can be
accu a ely calcula ed using he in e nal s a e o s esses
(wi h eq. (1)) and de o ma ions ( ela i e changes o he
pe iodic cell gi en by αβ ), i.e.
E=2σ2
xx +σxx(σyy +σzz)−(σyy +σzz)2
xx(2σxx +σyy +σzz)−(yy +zz)(σyy +σzz),(2)
ν=σxx(yy +zz)−xx(σyy +σzz)
(yy +zz)(σyy +σzz)−xx(2σxx +σyy +σzz),(3)
B=1
3
σxx +σyy +σzz
xx +yy +zz
,(4)
whe e E,νand Ba e he Young modulus, he Poisson
coefficien and he bulk modulus, espec i ely. Fo mulae
(2)–(4) can be ob ained assuming an iso opic and homo-
geneous ma e ial (o a ma e ial wi h cubic symme y)
oge he wi h he cons i u i e equa ions o elas ici y [16].
Mo i a ion, in e ac ion and scaling laws. – The
mo i a ion is o c ea e a compu e solid bulk ma e ial
wi hou comp omising nei he a good knowledge o hei
mechanical p ope ies no he compu a ional effo associ-
a ed o an ele a ed numbe o pa icles. This imp o emen
is aimed a designing a compu e bulk ma e ial: i) wi h
well-defined mechanical beha io , ii) wi h simple bu
well-de e mined physics, iii) wi h an affo dable numbe o
pa icles and, i ) using an efficien and simple algo i hm.
We combine wo ideas: fi s , an inhomogeneous in e nal
s uc u e and secondly, an app op ia e scaling law o he
in e a omic po en ials.
We chose o build he e ogeneous a angemen s o a oms
wi h o de ing by laye s and wi h diffe en a om sizes.
This s uc u al ep esen a ion o he ma e ial is specially
sui able o pe o m uniaxial es s. Compu a ional effi-
ciency is no o iously imp o ed, since his o de ing allows
us o classi y a oms using echniques inspi ed by pa allel
compu ing using pa icle sys ems [17]. The simple idea
behind ou p oposal is ha using diffe en a om sizes
in a gi en olume, he numbe o a oms pe olume
dec eases, and he e o e he numbe o pai in e ac ions
dec eases as well. The esul is an ob ious speed-up
o he compu a ions: he final pu pose o his esea ch
is o ob ain he same mechanical beha io as in he
10003-p2
MD simula ions using mul i-laye ed s uc u es
Fig. 1: (Colou on-line) Le : de ail o he building ma e ial
in laye s. Black boxes a e i ually bounding he egion o
he sample co esponding wi h a laye , and he ed ones a e
enclosing he in e aces be ween a couple o laye s. Righ :
snapsho o he in e ace be ween wo laye s wi h diffe en
a om sizes. An app op ia e cu -off dis ance mus be se . Fo
an efficien calcula ion (see ex ), his cu -off will depend on
he scaling ac o used in he in e a omic po en ial o bo h
adjacen laye s.
homogeneous case bu educing he compu ing ime
significa i ely. A p oblem o which his me hodology
will be pa icula y sui able is he con ac be ween a
de o mable sphe e and a igid plane. As desc ibed in [18],
he sphe e can be ob ained om he cu o a block made
o his he e ogeneous ma e ial. Since he mos impo an
e en s will occu in he con ac zone, his con ac is mo e
p ecisely desc ibed by he laye ha con ains he smalles
a oms, whils he es o he egions a om he con ac
zone a e ep esen ed by la ge a om sizes.
The idea is o c ea e a block o he bulk ma e ial
a anged by laye s, as i is shown in fig. 1, whe e he
laye ing p ocess is indica ed by i ual boxes. Each laye
has a diffe en a om size. When we say “a om sizes” we
a e e e eing o equilib ium dis ances inside he amo phous
la ice. In ou simple elec ion, he ela ion be ween a om
sizes belonging o a couple o adjacen laye s scales as
l+1 =k
l(l⩾0), whe e lis he a om size o he laye l.
The fi s laye o a oms con ains he smalles a oms wi h
size 0defined p e iously. Thus, he a om size in he
n- h laye is n=kn 0. Taking in o accoun his scaling,
he po en ial ene gy be ween a oms om wo con iguous
laye s land mis
ulm( ij)=Dlm  lm
ij 12
−2 lm
ij 6
,(5)
whe e ij is he dis ance be ween a oms and lm is he
equilib ium dis ance equal o (kl 0+km 0)/2. This is a
long- ange po en ial bu in p ac ice one usually defines a
cu -off dis ance om which he po en ial ac ion becomes
negligible. In he case o a homogeneous s uc u e, we se
his cu -off dis ance o cu -off =2.1 0, ha co esponds o
a ypical dis ance beyond which o ces be ween pai s a e
essen ially ze o. Fo he scaled po en ial w i en in eq. (5),
we ha e es ablished an analogous cu -off dis ance gi en by
cu -off,lm =2.1 lm.
The elas ic moduli o a solid composed by wo laye s
indexed by le e s land m, espec i ely should be
in a iable i he po en ial pa ame e s sa is y he ollowing
scaling o mulae
lm =γlm 0,D
lm =γ3
lmD, γlm =1
2(kl+km).(6)
To ob ain hese ela ions, we ha e easoned as ollows.
Le us suppose a hypo he ic elas ic modulus o ou
homogeneous ma e ial deno ed by M. This modulus has
a dependency o he po en ial pa ame e s Dand 0gi en
by M=λD/ 3
0. The quan i y λis a numbe depending
i) on he spa ial dis ibu ion o he la ice nodes (i.e., he
in e nal s uc u e: amo phous, FCC, BCC, o wha e e )
and ii) on he in e ac ion ange be ween a oms (i.e.,
fi s , second o a he neighbo shells). Fo a wo-laye ed
compu e ma e ial, he same elas ic modulus should ha e a
dependency gi en by M=λDlm/ 3
lm. Then, we admi he
ollowing assump ion: bo h ypes o solids, homogeneous
and wo-laye ed, will ha e he same effec i e in e nal
s uc u e, i.e.,λ=λ. A e a s aigh o wa d ope a ion,
we ob ain he ela ions w i en in (6). The e o e, i his
assump ion is ue, a he e ogeneous ma e ial composed by
mo e han 2 laye s will keep hei elas ic moduli equal o
he homogenous ones i he scaling gi en by (6) is ulfilled
be ween each in e ace o med by wo con iguous laye s.
Gene al aspec s o he implemen a ion. – An
efficien MD ealiza ion equi es, a leas , he ollowing
ea u es: i) o admi a sui able cu -off dis ance o a oid he
unnecessa y coun ing o pa icles ha do no con ibu e,
ii) an efficien algo i hm o sea ching and so ing he
nea es neighbo s and iii) a con enien di ision o he
p oblem using echniques based on he domain decom-
posi ion p ocedu e [17]. In a polydispe si e size sys em, as
he one we a e dealing wi h he e, one can ackle he in e -
pa icle in e ac ion in wo ways: 1) o conside a mean
cu -off dis ance o all laye s using i o any pai wi h
non-equal a om sizes (see he igh pic u e in fig. 1), o 2)
o ake ad an age o he laye ed s uc u e o classi y a oms
acco ding o hei sizes and ecalcula ing he app op ia e
cu -off dis ance as a unc ion o he scaling ac o .
The fi s s a egy is e y easy o implemen bu
p oduces an inefficien handling o in e ac ing pai s and
an inaccu a e calcula ion o o ces and in e nal s esses.
In ou si ua ion, o compu e he o al o ce ac ing on one
a om we need o conside he adjacen a oms possessing
diffe en sizes. To conside an a e age cu -off dis ance
may lead o miss a oms beyond his ange and whose
con ibu ions o he in e ac ion o ce a e no negligible.
On he o he hand and a he same ime, since each
a om has a lis o po en ial neighbo s o in e ac wi h, o
assume his a e age cu -off dis ance may lead o include
a oms in he lis whose con ibu ions a e negligible. This
implies addi ional and expensi e ope a ions o inse ion,
calcula ion and ex ac ion o memo y alloca ions.
In ha espec , he second s a egy, al hough i is a bi
mo e elabo a ed o implemen , leads o a mo e efficien
10003-p3
F. A. Gilabe and A. Cas ellanos
Fig. 2: (Colou on-line) Snapsho showing how a con inuum
bulk ma e ial is ob ained once a laye ed sample wi h nl=5,
N= 21311, k=4/3 eaches he equilib ium. In his case,
he pe iodic bounda y condi ions a e only imposed in he
di ec ions o he plane pe pendicula o he column axis o he
sample.
handling o he in e ac ing pai s. This educes he uly
necessa y in e ac ing numbe o a oms compa ed wi h he
p e ious s a egy and wi h he homogeneous case. Also,
he in e a omic o ces and he s ess enso componen s
a e p ope ly calcula ed, since any in e ac ing pai is no
missed. The p oblem is s aigh o wa dly di ided in o
wo s ages: fi s , we pe o m he in e ac ions be ween
a oms wi h he same size, i.e. inside a laye , which is
compu a ionally enclosed wi hin a i ual bounding box,
as he black ones depic ed in he le pic u e o fig. 1, and
secondly, we pe o m he in e ac ions in he zone c ea ed
by he in e ace be ween wo adjacen laye s, as he
ed i ual box shown in he same pic u e. This simple
classifica ion o in e ac ions sa es many unnecessa y
memo y accesses (a oms inside i ual boxes a e loca ed
in con iguous blocks o memo y) and helps o con ol all
o ces calcula ed on each pa o he ma e ial.
Laye ing de ails. – This p ocedu e has been imple-
men ed in a C++ code as ollows. An amo phous and
mechanically equilib a ed block o monosized a oms
is c ea ed a low empe a u e (<0.1 K). This block is
eplica ed nl−1 imes, whe e nlis he numbe o laye s.
Each copy is placed on he p e ious block, bu scaling he
a om size acco ding o l+1 =k
l,i.e., he copied block
is “infla ed”. Due o he expansion o he scaled block,
he bounda y box has o be eadjus ed o keep he same
c oss-sec ion as he o iginal block. Thus, he a oms alling
ou side he eadjus ed box mus be emo ed. In each new
laye he mass densi y emains app oxima ely cons an
Fig. 3: A om numbe as a unc ion o he numbe o laye s in
a he e ogeneous-laye ed sys em in compa ison wi h a homoge-
neous sys em (solid line).
while he nume ical densi y dec eases d as ically. Using
his p ocedu e, i is easy o calcula e he numbe o a oms
in he laye l, gi en by Nl=N0/k2l, whe e N0is he a om
numbe in he o iginal block. Nex , he p og am de ec s
nlbounding boxes co esponding o he laye s con aining
monosized a oms and nl−1 bounding boxes co e-
sponding o in e aces con aining wo-sized a oms (wi h a
negligible nume ical cos ). Then, he key leading o an effi-
cien pe o mance is ha he in e ac ion ea men in each
laye becomes iden ical as in a homogeneous case, i.e.,
Dlm, lm wi h l=m, excep o he in e ace whe e l=m.
The esul is a complexi y linea wi h he o al pa icle
numbe , he e o e he compu a ion ime o his scheme is
educed acco ding o his numbe . Once a configu a ion
is ob ained, he las s ep consis s o ge ing a con inuum
bulk. Laye s mus s ick o each o he , as fig. 2 shows,
whils o ces, s esses and ene gy a e moni o ed. A e he
equilib ium, he new mul i-laye sys em is achie ed wi h
ze o s esses and low empe a u e. In fig. 3 we show he
ad an age o using his me hodology o wo g ow h a es
k. These esul s can be easily checked, since he o al
numbe o a oms in a configu a ion, as fig. 2 shows, can
be accu a ely calcula ed jus making he ollowing sum:
N=
nl−1

l=0
N0
k2l=N0
1−k−2nl
1−k−2.(7)
This exp ession is no hing mo e han he p edic ed a om
numbe in a mul ilaye ed specimen, co esponding o a
simple geome ical se ies wi h a e kand nl e ms. In
fig. 3 we ha e ep esen ed wi h dashed lines hose alues
acco ding o eq. (7), while symbols a e he esul s ob ained
a e he cons uc ion o he solid o diffe en laye
numbe s. The solid line ma ks he limi es ablished by
homogeneous-like cons uc ions. Figu e 4 shows he huge
ad an ages ob ained om he laye ing in e ms o he
sa ed numbe o a oms in a gi en olume: he numbe o
a oms needed o pe o m he model is d as ically educed,
be ween 30%–80%, depending on nland k.
Elas ic coefficien s and mechanical esponse. – A
a ie y o specimens has been c ea ed, a ying he a om
numbe (n), he laye numbe (nl) and he scaling size
10003-p4
MD simula ions using mul i-laye ed s uc u es
Table 1: Young modulus E, Poisson coefficien νand bulk modulus Bcalcula ed o diffe en configu a ions o he compu e
bulk laye ed ma e ial. Thei exp essions a e gi en by o mulae (2)–(4).
nlkn E/σ
0νB/σ
0
1 1.0 1000 24.5±0.80.370 ±0.131.5±0.2
1 1.0 8000 22.5±0.80.377 ±0.004 31.05 ±0.07
1 1.0 10648 22.1±0.40.379 ±0.002 31.2±0.02
2 1.40 1626 22.55 ±0.07 0.379 ±0.001 31.8±0.3
2 1.50 4375 22.45 ±0.07 0.3849 ±0.001 32.5±0.3
2 1.33 4737 22.48 ±0.02 0.380 ±0.002 32.4±0.4
2 1.50 5113 23.105 ±0.014 0.3765 ±0.0002 31.32 ±0.06
3 1.40 1432 24.4±0.70.372 ±0.008 32.2±0.5
3 1.33 2079 24.3±0.40.371 ±0.005 31.8±0.3
3 1.33 7924 23.5±0.10.375 ±0.001 31.78 ±0.12
4 1.33 8728 23.0±0.40.379 ±0.003 31.9±0.2
5 1.33 21311 24.0±0.50.373 ±0.005 31.4±0.3
Fig. 4: Imp o emen s ob ained using he laye ed model. The
educ ion in he numbe o pa icles is plo ed as a unc ion o
henumbe o laye s,nl.
ac o (k). Table 1 shows he esul s o mechanical es s
o hese laye ed ma e ials. Young modulus and Poisson
coefficien a e ob ained om uniaxial es s, while he
bulk modulus was calcula ed om iso opic comp essions.
No e ha he case nl= 1 co esponds o he homogeneous
ma e ial.
The esul s om able 1 sugges ha he elas ic moduli
seem o be cons an , despi e small diffe ences p obably
caused by a sligh size effec o he sample. These alues
we e ob ained applying s esses below 10−3σ0, p oducing
de o ma ions much smalle han 0.2%: we made su e
ha he samples ne e abandoned he elas ic egime.
Uniaxial s ess was applied in he di ec ion pe pendicula
o he laye ing, and he iso opic es in he h ee spa ial
di ec ions. In bo h cases pe iodic bounda y condi ions
we e conside ed. When an uniaxial comp ession imposing
a e y slow s ain a e is pe o med (˙T
0=10
−5), he
mechanical esponse o he compu e ma e ial is clea ly
linea , as fig. 5 shows. This diag am illus a es he s ess-
s ain cu e whe e he p essu e and s ain a e calcula ed
acco ding o he o iginal dimensions o he sample and
no he ins an aneous alues. We also checked ha he
same samples beha ed in a simila way i a es o uniaxial
ension was ca ied ou . In fig. 5, he con inuous line was
ob ained om a e aging o e fi e diffe en specimens. The
Fig. 5: S ess-s ain diag am o homogeneous and he e oge-
neous ma e ial. The ma e ial is de o med imposing a quasi-
s a ic–like bu con inuous cell sh inking on he pe iodic cell
ha con ains he ma e ial. The laye ed specimen has 3 laye s
wi h a g ow h a e k=4/3.
ba s ep esen he s anda d de ia ion om he a e age o
he he e ogeneous ma e ial (simila ones a e ob ained o
he homogeneous case). These he e ogeneous specimens
we e c ea ed using an iden ical p ocedu e bu changing
he ini ial dis ibu ions o posi ions and eloci ies o he
a oms du ing he s age o amo phous p epa a ion. Each
specimen consis ed o h ee laye s wi h k=4/3. In o de o
compa e wi h he homogeneous case, we ha e also plo ed
wi h dashed line he s ess-s ain cu e ob ained om
comp essing a homogeneous compu e ma e ial occupying
an equi alen olume. Bo h ypes o s uc u es esponded
wi h he same mechanical beha io . E en in he domain
o la ge de o ma ions, a e age alues we e always wi hin
he nume ical e o ba s. This si ua ion is also p esen in
expe imen al esul s o physical ma e ials.
Fo he sake o comple eness, we pe o med (on he
he e ogeneous ma e ial used in fig. 5) uniaxial es s
imposing he ex e nal p essu e by means o a s epwise
mechanism using he ba os a ic p ocedu e desc ibed in
e . [15] The leaps o p essu e we e se o 0.05σ0. Figu e 6
shows he esul an beha io . Two cycles consis ing o
loading and unloading we e ca ied ou . In he fi s cycle
he specimen was comp essed om 0 up o 0.25σ0(black
10003-p5

F. A. Gilabe and A. Cas ellanos
Fig. 6: (Colou on-line) S ess-s ain diag am o he he e oge-
neous ma e ial imposing small leaps o p essu e. Each symbol
ep esen s a mechanical equilib a ed s a e a low empe a u e.
The ma e ial defini ely b eaks once he p essu e exceeds he
alue 0.5σ0. Compa ison wi h he alues ob ained om homo-
geneous s uc u es ga e an excellen ag eemen , p o ing ha
effec s o he scaling law a e exac ly he expec ed ones.
ci cles). Once he equilib ium was eached, he specimen
was immedia ely unloaded up o ze o p essu e ( ed
ci cles). The second cycle s a ed eloading he p e ious
final s a e a ze o s ess, inc emen ing s ep by s ep up
o 0.5σ0(black c osses). Decomp ession p ocess was om
0.5σ0 e u ning o 0 ( ed c osses). To finish his es , he
las s a e o he second eload was newly comp essed up
o 0.65σ0(black squa es). In his case we obse ed ha
he ma e ial b oke once he alue 0.5σ0was exceeded
(de o ma ion jumped om 4% up o 12%).
Conce ning his b eakage, he e a e wo specific quan i-
ies ha comple e he cha ac e iza ion o he mechanical
beha io oge he wi h he elas ic moduli: he onse o
plas ici y σY, o yield s eng h (beyond which he ma e ial
flows plas ically) and he maximum s ess ha he ma e -
ial can sus ain σU, o ul ima e s eng h. F om calcula ions
a e aged om o e 20 diffe en samples o homogeneous
bulk ma e ials consis ing o 104a oms each, we ha e
ob ained ha hese quan i ies a e σY/σ0=0.29 ±0.12
and σu/σ0=0.51 ±0.08. In figs. 5 and 6 i can be seen ha
he he e ogenous ma e ial beha es iden ically wi hin his
ange o alues. In ac we ob ained simila alues o hese
wo quan i ies: σY/σ0=0.3±0.1andσu/σ0=0.50 ±0.1.
Conclusion. – Wi hin he amewo k o he MD
app oach, a simple compu a ional p ocedu e based on
he scaling o he in e a omic po en ial was used o
model he e ogenous amo phous solids ha possess he
same mechanical p ope ies as i hey had an in e nal
homogeneous s uc u e. The laye ed model was subjec ed
o a ious mechanical es s. F om he es s, Young and
bulk moduli, he Poisson coefficien and he s ess-s ain
diag ams (con olling he s ain a e o he applied p es-
su e) we e ob ained. The mechanical esponse showed by
laye ed s uc u es exhibi ed he same beha io as i hey
we e homogeneous. The combina ion be ween simple scal-
ing laws on he in e a omic po en ial and an app op ia e
laye ing makes i possible o educe up o 80% he nume i-
cal densi y o equi ed a oms o desc ibe he solid. This se
o ideas p esen ed in his pape a e sugges ed as he begin-
ning o a me hodology simila o he mesh efinemen s
app oach in FEM, al hough applied o disc e e simula ion
using pa icles. In ha sense, an immedia e applica ion
ha we ha e al eady s a ed is a de ailed s udy o he
elas ic-plas ic and ully plas ic egime in adhesi e nano-
con ac s [18]. This me hod will allow us o build a much
la ge sphe ical body o ca y ou he con ac wi h a igid
plane. We shall be in a posi ion o ob ain he JKR solu-
ion, commonly used o desc ibe he con ac be ween a
sphe ical mac oscopic so body and a igid fla su ace,
in he elas ic egime [19].
∗∗∗
The au ho s wish o hank P o . A. M. K i so o
his e y use ul con ac s and discussions. This wo k has
been suppo ed by he Minis e io de Educaci´on y Ciencia
o he Spanish Go e nmen unde p ojec FIS2006-03645
and by he Jun a de Andaluc´ıa unde p ojec FQM-421.
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