Molecular dynamics simulations of homogeneous solids using multi-layered structures

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. 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