Macroscopic probabilistic modeling of concrete cracking: First 3D results

Transcription

1 Fraur Mhani Conr Conr Sruur - Rn Advan in Fraur Mhani Conr - B. H. Oh, al.(d) 200 Kora Conr Iniu, Soul, ISBN Maroopi probabilii modling onr raking: Fir 3D rul J.-L. Tailhan, P. Roi & S. Dal Pon Laboraoir Cnral d Pon Chaué, Pari, Fran ABSTRACT: Th papr propo an original approah for imply modling h omplx raking pro mniiou ompoi ruur. Th bai ida h modl i o ak ino aoun h hrognou naur onr h prn iniial df a h main faor influning raking pro for a givn rd marial volum. In hi n, h modl onidr volum ff, rom diribuion mhanial propri rak loalizaion in i formulaion. Thrfor, h modl i abl o bridg h gap bn h loal dripion h mhanim a h marial lvl h global rpon a h ruural lvl. 2D fir 3D imulaion validaion ar prnd. INTRODUCTION Th dripion rak i ruial for prdiing h lif xpany onr ruur uh a dam, nular por plan vl, a (nular or no) orag ruur, unnl,. Th dvlopmn modl providing informaion on h hararii rak in onr (rak opning paing), for a givn nvironmn, loading limi ondiion i ill a hallnging ak. Morovr, a prinn modl hould alo ak ino aoun om hararii bing bhind raking pro for a givn volum onr: no only al ff, bu alo phnomna rlad o h hrognou naur onr uh a iniial df in h marial, raking nulaion propagaion. Th objiv hi papr i o provid a maroopi modl apabl bridging h gap bn h loal dripion h mhanim a h marial lvl h global rpon a h ruural lvl. In h propod modl, fini lmn ar laially onidrd a lmnary volum marial. A far a h onr i hrognou, mhanial paramr ar dfind via aiial diribuion (rquiring only o paramr) bad on a larg xprimnal ampaign hld a LCPC (Roi al. 994). Th modl i alo aimd o rprn rak iniiaion, propagaion loalizaion by a impl maroopi probabilii diipaiv mhanim (lai-prfly plai bril) dpnding on h lmn iz. Thi kind approah allo o obain a prinn, aiial global rpon,, imulanouly, loal informaion (uh a rak mouh opning diribuion) ha an b xploid, for xampl, in h oupld modling fluid ranfr. Thi modling ragy i dvlopd in 2D in 3D, rul ar ompard o original xprimnal (four poin bnding Brazilian ) prformd a LCPC. Th omparion ill b givn no only in rm h global anr bu alo on rak opning diribuion. 2 HETEROGENEITY OF CONCRETE AND CRACKING PROCESSES Conr i a porou mulipha marial hr h olid marix i formd by mn pa aggrga hr void ar filld ih liquid ga. In ohr ord, onr i, by naur, a hrognou marial, hih alay onain innr df uh a por rak (vn undr no xrnal load). Morovr, hrogniy i h main au onr aiial volum ff, hih influn al ff a h ruural lvl (Bazan 2000) via h raking pro. Hrogniy volum ff ar ap hih ar rily orrlad ha hould b pifially akn ino aoun hn daling ih onr modling. Roi (Roi al. 994), afr having prformd a hug xprimnal ampaign on nil bhavior onr, inrprd raking pro a h folloing: r dvlop in onr du o h xrnal load aggrga nd o onnra hm in hir nighborhood; on h ohr h, h mn pa alo h bound bn h pa h aggrga ar h pla hr df ar load, i.. ar pla lor rngh; raking ari hn a high r m a lo rngh, hn propaga hrough h mn pa; aggrga

2 vnually play h rol barrir or marial bridg in h propagaion rak. Volum ff ar h onqun h impa h hrogniy on h bhavior h rd volum. Th lor, o h al h rd volum, h al h loal r onnraion (gnrad by h hrogniy) i, h largr i i influn on h maroopi bhavior. Morovr, a far a hrognii df ar romly diribud in h marial, h mhanial propri ar rom paramr. In h xprimnal udy (Roi al. 994), auhor hod, for xampl, an inra h man valu h diprion h nil rngh v. a dra h marial volum for a am onr mix dign. Thi ff ha bn found for diffrn yp onr, bu ih diffrn magniud. An mpirial al ff la ha bn hn ablihd for h man nil rngh m( f ) h ard dviaion σ( f ) a funion aily maurabl quanii uh a h raio volum h pimn V ovr volum h oar grain h onr Vg (h raio V/Vg an b rlad o h iz h major hrogniy) h ard ompriv rngh onr f (onidrd hr a a good indiaor h mn pa qualiy). 3 NUMERICAL PROBABILISTIC MODELING OF CRACKING PROCESSES Th numrial modling ak pla in h gnral framork h fini-lmn mhod (FEM). Th undrlying, bai, ida i o onidr a fini lmn volum lik a marial volum o aum ha phyial mhanim influning h raking pro rmain h am havr h al obrvaion. Conidring h hrogniy h marial, mhanial propri ar romly diribud ovr h mh: h al la id in h prviou ion hav bn xrapolad o h volum h fini lmn ud a inpu daa in a numrial modling bad on a probabilii approah. Conidring h modling rak iniiaion, propagaion loalizaion o ragi ar hr prnd: a dir xplii modl an original oninuum bad approah. 3. A dir Approah Roi (Roi al. 992) originally prn a probabilii modl implmnd via a dir-xplii approah in hih inrfa lmn ar ud o drib h dioninuii. Th volum h maiv lmn hih ar adjan o h onidrd inrfa lmn, a a h rfrn (marial) volum. Th mhanial J D ( h, propri T ) h h inrfa lmn (nil rngh) h maiv lmn (young modulu) Th ar proporionaliy onidrd a fiin romly D(h,T) diribud variabl. moiur A prmabiliy id abov, diribuion i a nonlina hararii (man h valu rlaiv humidiy ard h dviaion) mpraur ar obaind from & an Najjar xrapolaion 972). Th moiur h mpirial ma balan formula ( ion ha h 2). variaion I hould in b im poind h ou ar ma ha h modl i volum aimd a xpliily onr rprning (ar onn lo-alizd rak parn divrgn in onr h aking moiur ino flux aoun J b q volum ff. Th modl i onidrd a probabilii, bu afr h rom diribuion mhanial propri ovr h mh, Jh ompuaion rmain drminii. I i hn nary o follo a Mon Carlo mhod hrfor Th ar o onn prform a an larg b numbr ompuaion xprd a h for vaporabl aiially ar validaing h (apillary a rul. Sal ff vapor, ar ffivly adorbd akn ar) ino aoun h modl h non- (hmially i auo-ohrn bound) in ar h n n (Mil ha daa a h loal Panazopoulo al ar ohrn & Mill ih 995). rul I i ra a h global al aum in a ha gnri h la vaporabl aking ino ar aoun volum ff i a fu rlaiv an dfin humidiy, onr h, dgr mhanial hydraion propri a ah dgr al. Alhough ilia fum loally raion, no nrgy, i.. i diipad (h failur ag-dpndn h lmnary orpion/dorpion volum rmain lai-prfly (Norling bril), Mjonll h 997). modl Undr allo hi o aum aiially rprning by ubiuing a global Equaion diipaion ino nrgy hrough inlai Equai obain ridual rain, ning bhavior. Aording o h loal probabilii harar h approah, h volum h h lmn ha o b + ( D h) & + & + uffiinly mall hn h ompard h o h volum h mhd ruur or o h zon iz hr r gradin an dvlop (i.. h fraur pro hr /h i h lop h orpion/ zon). Thi an lad o vry mall raio V/Vg iohrm (alo alld moiur apa hih fall ou h domain validiy uppord by govrning quaion (Equaion 3) mu b h xprimnal ampaign (Roi al. 994). An by appropria boundary iniial ondii invr analyi (Tailhan al. 2007) ha hn bn Th rlaion bn h amoun ud o drmin h xrapolaion h mpirial ar rlaiv humidiy i alld formula o h mall raio V/Vg domain. Th iohrm if maurd ih inraing original iz-ff la i hrfor updad ill humidiy dorpion iohrm in h b ud in h fini lmn analyi. a. Ngling hir diffrn (Xi al. Nvrhl, hi modling ragy ha, hovr, om horoming. Som quion ari in h folloing, orpion iohrm ill b rfrn o boh orpion dorpion h appliabiliy h dir-xplii modl. By h ay, if h hyri h Firly, h main riiim, hih an b mad, i iohrm ould b akn ino aoun, o ha rak inviably dpnd on h orinaion rlaion, vaporabl ar v rlaiv humi h ona lmn, vn if h rom diribuion b ud aording o h ign h varia h mhanial propri mpr hi ff. rlaiviy humidiy. Th hap h Sondly, for mor global approah, a h al iohrm for HPC i inflund by many p a hol ruur for xampl, uh modl lad o pially ho ha influn xn prohibiiv ompuaional o a h u ona hmial raion, in urn, drm lmn doubl h numbr nod. Thi i vn ruur por iz diribuion (arraio, mn hmial ompoiion, SF mor niiv in h a 3D modling. Th onidraion juify an nhanmn oard a oninuum bad approah. Suh a modl uring im mhod, mpraur, mix.). In h liraur variou formulaio m mor adqua in many iuaion in pariular hn daling ih ral ruur. If om- found o drib h orpion iohrm onr (Xi al. 994). Hovr, in h pard o a dir modl, a oninuum modl do papr h mi-mpirial xprion pro no rquir ona lmn, i.. no pr-orind Norling Mjornll (997) i adopd b rak (any rak dirion i favord). Proding FraMCoS-7, May 23-28, 200

3 3.2 J D ( A oninuum h, T ) h Approah () Th oninuum bad approah i dfind a a maroopi Th proporionaliy al hr r fiin rain D(h,T) a i ar alld dfind. moiur A hi prmabiliy al, i i horially i i a nonlinar poibl funion o ablih h rlaiv a oniuiv humidiy rlaionhip h mpraur bn r T (Bažan rain & Najjar dfining 972). h Th maroopi moiur ma bhavior balan rquir h marial. ha h Craking variaion pro in im an h b ar hn akn ma ino pr uni aoun volum by onidring onr (ar a diipaiv onn mhanim ) b qual a o h marial divrgn al. h In moiur hi maning, flux J rily paking, rak mu b onidrd a uffiinly mall (miro rak) diffud in h hol marial J (2) rprnaiv lmnary volum. To imporan fa hav o b poind ou: - Th Uually, ar onn h idnifiaion an b xprd h marial a h um bhavior h vaporabl i prformd ar on laboraory (apillary ampl ar, hih ar iz vapor, ha o b adorbd largr han ar) h Rprnaiv h non-vaporabl Elmnary (hmially Volum (REV) bound) in ar ordr o proprly n (Mill ak 966, ino aoun Panazopoulo h marial & Mill hrogniy. 995). I i Hovr, raonabl hn o daling aum ih ha onr h vaporabl hi iz ar i no i n a funion in aordan rlaiv ih humidiy, h iz h, h dgr fini lmn hydraion, ud in, h modling. dgr ilia I i hu fum nary raion, o prform, i.. an (h, xrapolaion ag-dpndn h idnifid orpion/dorpion xprimnal bhavior iohrm o al (Norling h Mjonll fini 997). lmn. Undr Thi hi rquir aumpion aking ino, ) aoun by ubiuing al hang, Equaion i.. volum ino Equaion ff mu 2 on b onidrd obain a hi ag. - Th loalizaion rak, gnrally ourring a h pak, ha o b arfully akn ino aoun. h Bfor loalizaion, + ( D h) marial & ingriy + & i + & qui n prrvd vn if h marial i vrly damagd. Af- (3) h h r loalizaion, marial ingriy fail uh ha i i impoibl hr /h o i onidr h lop h po-pak h orpion/dorpion ning bhavior iohrm a rprnaiv (alo alld moiur h bhavior apaiy). h marial. govrning In ohr quaion ord, (Equaion afr h 3) pak mu b hif ompld from a Th marial by appropria bhavior boundary o a ruural iniial ondiion. bhavior (Roi 998). Th Numrial rlaion bn ranlaion h amoun h problm vaporabl ar moly ar lading rlaiv o rong humidiy mh i niivii alld adorpion non objiv iohrm rpon if maurd (Bazan ih & Jirak inraing 2002). rlaiviy Addiional humidiy aumpion dorpion mu b iohrm don o olv in h hi oppoi problm. a. Ngling hir diffrn (Xi al. 994), in h Th folloing, modl ak orpion ino aoun iohrm a h ill fini b ud lmn ih lvl rfrn h o ap boh orpion a follo: dorpion ondiion. By - Firly, h ay, i i if aumd h hyri ha i poibl h o moiur dfin maroopi iohrm ould quanii b akn havr ino aoun, h iz o diffrn h fini rlaion, lmn, vaporabl hhr ar i i v marial rlaiv rprnaiv humidiy, mu or no. b ud I i aording hn uppod o h ha ign h mhanial h variaion bhavior h rlaiviy h fini humidiy. lmn dpnd Th hap on i iz h orpion poiion, iohrm i.. h for bhavior HPC i inflund ah fini by many lmn paramr, i pron o pially rom ho variaion, ha influn hu aking xn aoun ra h marial hmial hrogniy. raion, in urn, drmin por h ruur - Th mhanial por iz bhavior diribuion h (ar-o-mn fini lmn (pr- raio, mn po-loalizaion) hmial ompoiion, i rplad by SF an onn, quivaln uring marial im bhavior. mhod, Sin mpraur, i i onidrd mix addiiv, a a marial.). In bhavior, h liraur hi quivaln variou formulaion bhavior do an no b hav found a o ning drib branh h orpion afr h iohrm pak. A diipaiv normal mhanim onr (Xi i hon al. 994). o rprn Hovr, h in hol h prn raking papr pro, h mi-mpirial pr- po-loalizaion. xprion propod Th quivaln Norling bhavior Mjornll i dfind (997) via i an adopd quivaln bau in d- by i formaion xpliily aoun nrgy. I for an h b voluion argud ha hydraion h loal diipaiv raion mhanim SF onn. i no Thi rprnaiv orpion iohrm loal rad nrgy amoun rally diipad by h marial during raking. A h nd h raking pro, hn h oal amoun availabl nrgy i diipad, failur h fini lmn i aumd o b ( h,, ) G (, ) + bril. 0( g ) h Th diipaiv mhanim i rprnd via prf plaiiy. Thi hoi i juifid by h impli- (4) iy h approah oghr 0( g ih ) h h llablihd horial framork h robu K (, ) numrial implmnaion. Th prinipl h nrgy quivaln i dpid in Figur. Dail ar givn hr in h (Tailhan fir rm 2009). (gl iohrm) rprn h phyially bound (adorbd) ar h ond rm (apillary iohrm) rprn h apillary ar. Thi xprion i valid only for lo onn SF. Th fiin G rprn h amoun ar pr uni volum hld in h gl por a 00% rlaiv humidiy, i an b xprd (Norling Mjornll 997) a G (, ) k + k vg vg (5) hr k vg k vg ar marial paramr. From h maximum amoun ar pr uni volum ha an Figur. Prinipl h quivaln for a uniaxial nil bhavior. fill all por (boh apillary por gl por), on an alula K a on obain A far a h uniaxial bhavior dpnd on h rd volum marial prn 0 g om h romn, h 0 ara undr h urv i alo a rom G quaniy, inflund by volum ff. Conuivly, σ m h diipad nrgy (W (6) K (, ) 0 g h ) an b d onidrd a rom paramr h laiplai quivaln modl (alo inflund by volum ff). Th Th marial gnral paramr la, dfining k vg h k vg hararii g h probabilii diribuion for σ m W an b alibrad by fiing xprimnal daa rlvan o d v. fr h (vaporabl) paramr ar h modl onn (i.. f in onr V/Vg, a ion variou 2), ag hav (Di o Luzio b idnifid & Cuai via 2009b). an xprimnal ampaign or via a numrial ampaign uing h dir approah prnd abov. 2.2 Tmpraur voluion Th hoi a numrial uppor for h rprnaion No ha, a h arly rak ag, loalizaion in h hmial i imporan raion a i hould aoiad ombin ih mn h rlaiv hydraion impliiy SF raion implii modl ar xohrmi, (hih h ar mpraur pariularly fild uiabl i no for uniform bing ud for non-adiabai in h dripion ym vn larg if ruur) h nvironmnal oghr ih mpraur h apaiy i onan. giving Ha om onduion xra informaion an b nary dribd for in a onr, propr rak a la dripion. for mpraur Thr fini no lmn xding approah 00 C (Bažan hav bn & Kaplan d on 996), diffrn by onfiguraion Fourir la, hih in ordr rad o valua h vnual r loking mh dpndn: a Rahid-lik (Rahid q λ T968) modl (in hih lmn iffn (7) i rdud o zro a oon a an nrgy hrhold i rahd), hr q a i fixd h rak ha modl flux, (Droz T i 987) h abolu an mbddd mpraur, formulaion λ i (Alfaia h ha onduiviy; 2003). Aording hi o our rul, h Rahid-lik modl did no xhibi Proding FraMCoS-7, May 23-28, 200

4 r loking oghr ih h propod probabilii approah provd o b mh indpndn. For h raon, hi modl ha bn raind for h furhr probabilii analyi. 4 EXPERIMENTAL VALIDATION Th oninuum modling prnd in h prviou ion ha bn firly ompard o an original xprimnal prformd a LCPC. Th xprimn oni a four poin diplamn-onrolld bnding on a plain onr bam. Th bam gomry i a 70x20x5m prim. Th pan i 60 m long. Th onan momn zon i 20 m long. Th onr ud i an ordinary onr (E35GPa, f50mpa, f3mpa, valu xprimnally drmind). Diplamn (ar maurd on h fron fa via 6 LVDT, h bnding vrial diplamn i alo rordd. Th numrial probabilii approah i prformd aording h folloing p: - 30 ompuaion ar prformd ih h dir approah for imulaing h uniaxial nil bhavior h onr. A man bhavior i ddud from h rul. - An invr analyi i don on h man bhavior o drmin h paramr h oninuum approah - Th paramr ar ud o h modling h bnding bhavior. Again 30 ompuaion ar prformd. Th bam ha bn modld via T3 rgular lmn ( Figur 3); h Rahid-lik modl ha bn ud. Rul ar givn Figur 2. Figur 2. Global bhavior: xprimnal (bold), numrial anr (gry) man (irl). Th orrlaion bn h xprimnal rul h man urv i qui good a h xprimnal rul i onaind in h h numrial anr i vry lo o h man anr. Th maroopi modl provid no only a global anr bu alo om J D ( loal h, T ) informaion h on rak opning diribuion. Th Figur 3 ho a ypial rak parn Th proporionaliy h rak opning fiin urv D(h,T) ar prnd in Figur moiur 4. I prmabiliy i inring o i obrv i a nonlina ha no only a main h maro rlaiv rak humidiy i rprnd h mpraur bu alo h muli-raking & Najjar harar 972). Th h moiur global ma failur i rprnd. ha h variaion in im h ar ma balan volum onr (ar onn ) b q divrgn h moiur flux J J Th ar onn an b xprd a h vaporabl ar (apillary a vapor, adorbd ar) h non- (hmially bound) ar n (Mil Panazopoulo & Mill 995). I i ra aum ha h vaporabl ar i a fu rlaiv humidiy, h, dgr hydraion dgr ilia fum raion,, i.. ag-dpndn orpion/dorpion (Norling Mjonll 997). Undr hi aum by ubiuing Equaion ino Equai obain Figur 3. Typial rak parn a h nd h imulaion. h + ( D h) h h Figur 4. Crak opning (numrial -gry-, man anr - irl- xprimnal -bold-). Figur 5. Exprimnal dvi for h Brazilian. & + & + hr /h i h lop h orpion/ iohrm (alo alld moiur apa Th ond validaion govrning quaion onrn (Equaion h imulaion a Brazilian 3) mu b by, appropria prformd boundary a LCPC a iniial ll. ondii Th xprimn oni Th rlaion in applying bn a ompriv h amoun load on an m ar diamr rlaiv onr humidiy ylindr. i Th alld voluion h iohrm ylindr diamr if maurd i rordr ih inraing on boh fa h humidiy pimn, dorpion hir man iohrm valu i in h ud a h loading a. paramr Ngling hir h ing diffrn dvi (Xi al. avoiding, in hi ay, h folloing, loading inabilii. orpion iohrm ill b rfrn o boh orpion dorpion By h ay, if h hyri h iohrm ould b akn ino aoun, o rlaion, vaporabl ar v rlaiv humi b ud aording o h ign h varia rlaiviy humidiy. Th hap h iohrm for HPC i inflund by many p pially ho ha influn xn hmial raion, in urn, drm ruur por iz diribuion (arraio, mn hmial ompoiion, SF uring im mhod, mpraur, mix.). In h liraur variou formulaio found o drib h orpion iohrm onr (Xi al. 994). Hovr, in h papr h mi-mpirial xprion pro Norling Mjornll (997) i adopd b Proding FraMCoS-7, May 23-28, 200

5 J Th D ( onr h, T ) h ud i h am a dribd abov. () And h modling ragy i alo h am a h on Th prviouly proporionaliy dpid. fiin I i undrlind, D(h,T) i ha alld h ompuaion moiur prmabiliy i drivn in h i i am a nonlinar ay a funion h xprimn: h rlaiv h diamri humidiy h xpanion mpraur i alo T hr (Bažan h indir & Najjar loading 972). paramr. Th moiur ma balan rquir ha h variaion in im h ar ma pr uni volum onr (ar onn ) b qual o h divrgn h moiur flux J J (2) Th ar onn an b xprd a h um h vaporabl ar (apillary ar, ar vapor, adorbd ar) h non-vaporabl (hmially bound) ar n (Mill 966, Panazopoulo & Mill 995). I i raonabl o aum ha h vaporabl ar i a funion rlaiv humidiy, h, dgr hydraion,, dgr ilia fum raion,, i.. (h,, ) Figur ag-dpndn 6. Applid load v. orpion/dorpion diamri xpanion urv. iohrm Comparion (Norling bn Mjonll xprimnal 997). rul Undr (dod hi aumpion rd urv) imulaion (blu urv h magna urv rprning hir man valu). by ubiuing Equaion ino Equaion 2 on obain h + h ( D h) n (3) h & + & + hr /h i h lop h orpion/dorpion iohrm (alo alld moiur apaiy). Th govrning quaion (Equaion 3) mu b ompld by appropria boundary iniial ondiion. Th rlaion bn h amoun vaporabl ar rlaiv humidiy i alld adorpion iohrm if maurd ih inraing rlaiviy humidiy dorpion iohrm in h oppoi a. Ngling hir diffrn (Xi al. 994), in h folloing, orpion iohrm ill b ud ih rfrn o boh orpion dorpion ondiion. By h ay, if h hyri h moiur Figur iohrm 7. 3D ould rak parn b akn a h ino nd aoun, h imulaion. o diffrn rlaion, vaporabl ar v rlaiv humidiy, mu b ud aording o h ign h variaion h rlaiviy humidiy. Th hap h orpion iohrm for HPC i inflund by many paramr, pially ho ha influn xn ra h hmial raion, in urn, drmin por ruur por iz diribuion (ar-o-mn raio, mn hmial ompoiion, SF onn, uring im mhod, mpraur, mix addiiv,.). In h liraur variou formulaion an b found o drib h orpion iohrm normal onr (Xi al. 994). Hovr, in h prn papr h mi-mpirial xprion propod by Figur Norling 8. Diribuion Mjornll (997) horizonal i adopd diplamn bau on boh i fa h pimn. & xpliily Again, h aoun orrlaion for h bn voluion h xprimnal hydraion rul raion h imulaion SF onn. i qui Thi good, orpion a far iohrm a only on rad rul i hon hr (Figur 6 o Figur 8). Nvrhl, h imulaion larly ho h 3D harar h raking pro in h pimn, lading o a rong diymmry bn boh fa. Th ( h,, ) G (, ) + rak parn h horizonal diplamn ar 0( g ) h alo larly diymmri Figur 7 Figur (4) 8. Th fa hav bn alo xprimnally obrvd. 0( g ) h K (, ) 5 CONCLUSIONS In hr hi h papr, fir a rm modl (gl rprning iohrm) raking rprn pro phyially in onr bound (adorbd) hrough a robu ar oninuum h ond ap- h proah rm (apillary oupld ih iohrm) a impl rprn numrial h modling apillary dioninuii ar. Thi xprion i prnd. i valid 2D only for 3D lo validaion onn SF. ar Th dpid. fiin On Ghould rprn no forg h amoun ha h impliiy ar pr uni h volum numrial hld modling h gl por i maningful a 00% only rlaiv if h humidiy, approah i i oupld an b xprd ih h aiial (Norling diribuion Mjornll 997) propri a h givn al la. Thi oluion ragy allo o proprly ak ino aoun al ff h hrognou naur G (, ) k + k (5) onr, providing vg vga rliabl global anr a ll a loal informaion uh a rak parn rak hr opning. k Th nhanmn h modl oard maximum 3D i amoun a nary ar p pr o uni ak volum ino aoun ha an vg k vg ar marial paramr. From h h fill all omplx por (boh hr apillary dimnional por naur gl por), gomry on an rak alula Kgiving a on a aifaory obain dripion h loal-global bhavior a ruur. 0 g h G 0 REFERENCES (6) K (, ) Alfaia, J. Simon, 0A. g Sluy, h L.J Nonhomognou diplamn jump in rong mbddd dioninuii, In.J.Solid Sru., 40(2), Bazan, Th Z.P. marial Siz paramr Eff, In.J.Sol.Sru., k vg k vg 37 : g an Bazan, b alibrad Z. Jirak, by fiing M. xprimnal Nonloal daa ingral rlvan formulaion plaiiy damag: urvy progr, o fr (vaporabl) ar onn in onr a J.Eng.Mh., : Droz, variou P. ag 987. (Di Modèl Luzio numériqu & Cuai du 2009b). ompormn nonlinéar d'ouvrag maif n béon non armé, EPFL,Lauann. Rahid, 2.2 Tmpraur Y.R Analyi voluion prrd onr prur No vl, ha, Nul.Eng.D., a arly ag, 7, in h hmial raion Roi, P. & Wu, X Probabilii modl for marial bhaviour analyi appraimn onr ruur. aoiad ih mn hydraion SF raion ar Mag. xohrmi, Of Conr. R., h mpraur 44(6): fild i no uniform Roi, for non-adiabai P., Wu, X., L ym Maou, F. vn & Bllo, if h A. nvironmnal 994. Sal ff on onr i onan. in nion. Ha Marial onduion Sruur, an b 27 mpraur dribd (72): in onr, a la for mpraur no Roi, P., 998. Réflxion ur l'uiliaion d modél xding 00 C (Bažan & Kaplan 996), by oninu non linéair pour la pri n omp d la Fourir fiuraion la, d hih ruur rad n béon, BLPC, 27, Tailhan, J.-L., Roi, P., Lombar, J. & Gurrir, F From q a dioninuou λ T o a maroopi modlling h mod (7) I raking bhaviour mn-bad ompoi. FRAM- COS 6, Caania, Ialy. Tailhan, hr J.-L., q i Dal h Pon, ha S. & flux, Roi, T P i h From abolu loal o mpraur, global probabilii λ modling i h ha onr onduiviy; raking, ubmid o Annal olid ruural in hi mhani. Proding FraMCoS-7, May 23-28, 200