Simulation of tensile performance of fiber reinforced cementitious composite with fracture mechanics model

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1 Fratur Mhani Conrt Conrt Strutur - High Prorman, Fibr Rinord Conrt, Spial Loading Strutural Appliation- B. H. Oh, t al. (d) Kora Conrt Intitut, ISBN Simulation tnil prorman ibr rinord mntitiou ompoit ith ratur mhani modl J. Zhang Dpartmnt Civil Enginring, Tinghua Univrity, Bijing, China C. K.Y. Lung Dpartmnt Civil Environmntal Enginring, HKUST, Hong Kong, China Y. Gao Dpartmnt Civil Enginring, Tinghua Univrity, Bijing, China ABSTRACT: Uniaxial tnil prorman ibr rinord mntitiou ompoit i imulatd bad on ratur mhani ritria, ith pii objtiv tudy phnomna train-hardning multipl raking undr dirt tnion. In modl, intad dribing matrix ratur ritan by a ingl paramtr at rak tip, parat it in part, a rak tip ughn (KIC_M) a tnion tning urv rprnting intrloking t aggrgat. Th lattr i addd ibr bridging tr om up ith ovrall bridging tr v rak opning rlation or ratur analyi. To analyz rak propagation, a uprpoition mthod i mployd alulat tr intnity ar at rak tip rultd rom both applid load rak bridging tr. For a partiular rak iz, orrponding load i alulatd a valu hn KIC_M i rahd at rak tip. Uing modl, t variou matrial paramtr, inluding matrix ughn, initial lo iz, ibr ontnt pimn gomtry on tnil prorman ar invtigatd. Th rquirmnt or tnil train-hardning multipl raking ar analyzd poibl mthod or matrial prorman optimization ar diud. INTRODUCTION Conrt i a typial brittl matrial hr irt raking in tnion i aompanid by immdiat loalization dormation ollod by draing load. In normal rinord onrt trutur, a tr rah tnil trngth onrt undr mhanial /or nvironmntal load, a mall numbr idly pad dirt rak ill orm rak idth quikly opn a maroopially viibl lvl. Th ormation idly opnd rak allo atr or hmial agnt, uh a diing alt, go through ovr layr om in ontat ith rinormnt.. Th durability onrt trutur i n igniiantly atd. Many mthod hav bn propod improv durability onrt trutur in pat, but mot ou on tranport proprti un-rakd onrt, ith littl attntion paid ontrol rak. To prvnt rapid pntration atr orroiv hmial through rak, a undamntal approah rdu rak idth in onrt during it rvi tag ha b dvlopd (Lph t al. 6). In rnt yar, a la high prorman ibr rinord mntitiou ompoit, alld Enginrd Cmntitiou Compoit (ECC), ith an ultimat trngth highr than ir irt raking trngth ormation multipl raking during inlati dormation pro ha bn dvlopd (Li, ). Atr irt raking, tnil load-arrying apaity ontinu inra, rulting in trainhardning aompanid by multipl raking. For ah individual rak, rak idth irt inra tadily up a rtain lvl n tabiliz at a ontant valu. Furr inra in train apaity i rultd rom ormation additional rak until raking rah a aturatd tat ith rak paing limitd by tr tranr apability ibr. Atr that, a ingl rak loaliz load loly drop ith inrad dormation. Typially, train loalization our at a tnil train 3-5%, ith rak paing 3-6mm rak idth around 6µm (Li, ). Crak uh a mall idth ill hav littl t on atr prmability matrial (Wang, t al. 997). With littl dgradation in tranport proprti undr high dormation, durability trutur an b maintaind. Th dign ritrion or ECC i irt propod by Li Lung (Li & Lung, 99) urr dvl-

2 opd J = D ( in h, ubqunt T ) h invtigation (Li 993, Lung () 996, Ka t al. 999). To om up ith ondition Th or proportionality multipl raking iint b ahivd, D(h,T) i ratur alld analyi moitur i prmability prormd on a it rak i a nonlinar bridgd by untion ibr. Alo, in rlativ analyi, humidity h rak tmpratur i takn b T (Bažant propagating & Najjar in an 97). ininit Th pa. moitur In rality, ma balan hovr, rquir an air void that that variation i not bridgd in tim by ibr atr an ma rv pr a unit rak volum initiar. onrt In or (atr ord, ontnt ) rak b qual i initially not divrgn bridgd by ibr, moitur lux J bridging t bom mor mor igniiant a rak gro in iz. Alo, hn rak approah boundary = Jpimn, tr intnity du both () t applid load bridging tr ill vary igniiantly Th atr rom ontnt that in an b ininit xprd domain. a To tak um t vaporabl in onidration, atr a n ratur mhani modl i propod in thi papr or analy- (apillary atr, atr vapor, adorbd atr) non-vaporabl i raking in ibr rinord mntitiou ompoit. (hmially bound) atr n (Mill 966, Pantazopoulo & Mill 995). It i raonabl Th ild ratur mhani originatd in aum that vaporabl atr i a untion 9' ith A. A. Griith' ork on ratur brittl matrial uh a gla (Griith 9). It rlativ humidity, h, dgr hydration, α, mot dgr ilia um ration, α igniiant appliation, hovr,, i.. hav = bn (h,α or,α ) = ag-dpndnt orption/dorption iorm ontrol brittl ratur atigu ailur mtalli trutur uh a prur vl, airplan, (Norling Mjonll 997). Undr thi aumption by ubtituting Equation in Equation on hip t. In lat thirty yar, many attmpt hav obtain bn mad apply ratur mhani onpt mnt-bad ompoit, inluding mortar, onrt ibr rinord onrt (FRC). Prviou + & n (3) tudi on t thi la h matrial α hav hon that a rlativ larg miroraking zon, rrrd a ratur pro zon hr matrial bhav hr nonlinarly, / i lop orption/dorption xit adjant rak ront. Sin iorm (alo alld moitur apaity). Th iz zon i not mall in omparion govrning quation (Equation 3) mut b ompltd iz mmbr, tr ditribution ithin by appropriat boundary initial ondition. zon ha b onidrd xpliitly in analyi Th rlation btn amount vaporabl rak propagation. To major kind modl hav atr rlativ humidity i alld adorption bn dvlopd drib ratur pro zon, iorm i maurd ith inraing rlativity inluding () ititiou rak modl (FCM) propod by Hillrborg t al. (976) () rak humidity dorption iorm in oppoit a. Nglting ir dirn (Xi t al. 994), in b ory propod by Bazant t al. (983). Th olloing, orption iorm ill b ud ith ormr approah modl pro zon a part rrn both orption dorption ondition. rak but ith bridging tr govrnd by By ay, i hytri moitur rak opning. In or ord, bhavior iorm ould b takn in aount, dirnt pro zon ollo a rtain rak bridging la. rlation, vaporabl atr v rlativ humidity, mut Th lattr imagin pro zon xit ithin a b ud aording ign variation init b idth in hih mirorak ar uniormly ditributd bhavior atr raking an rlativity humidity. Th hap orption b iorm dribd or HPC by a tning i inlund tr-train by many rlationhip. paramtr, In pially litratur, tho that inlun modl ar xtnt omtim rat rrrd hmial a ohiv ration modl,, or ratur in turn, pro dtrmin modl. por In trutur prnt ork, por FCM iz ill ditribution b applid (atr--mnt aount or ratio, bhavior mnt hmial pro ompoition, zon. SF ontnt, uring In tim pat, mthod, rarhr tmpratur, had attmptd mix additiv, u laial t.). In LEFM litratur rak variou bridging ormulation la analyz an b rak ound propagation drib in matrial orption hih iorm xhibit normal rak bridging, onrt uh (Xi t a al. ibr 994). rinord Hovr, rami in prnt ibr rinord papr onrt mi-mpirial (Cox t xprion al. 99, Li propod t al. 986). by Sin Norling a harp Mjornll rak tip (997) i till i nviagd adoptd bau b prnt it at xpliitly lading aount dg or pro volution zon in hydration onrt, rinord ration rami, SF ontnt. or vn Thi rinord orption mtal, iorm it mor rad raliti aum bridging or ithin pro zon rdu nt tr intnity ar at rak tip, but a non-zro tr intnity till xit (Cox t al. 99, Li t al. 986). Th rak ( h, α, α ) = G ( α, α ) + propagating ritrion linar lati ratur mhani n rmain appliabl abov ( g α α ) h matrial, a long a ontribution pro zon (4) rak tip tr intnity ar ( g α i αxpliitly ) h inorporatd. K ( α, α ) In prnt papr, mod I rak propagation in ibr rinord mntitiou ompoit i imulatd bad hr on ratur irt trm mhani (gl iorm) ritria, ith rprnt pii objtiv phyially bound tudy (adorbd) ondition atr or multipl raking ond trm (apillary iorm) rprnt apillary our undr dirt tnion. Compard xiting atr. Thi xprion i valid only or lo ontnt modl on propagation bridgd rak in ibr SF. Th iint G rprnt amount matrial, atr pr r unit volum ar thr hld additional in gl onidration por at % in thi rlativ ork. humidity, Firtly, aum it an rak b xprd initiat (Norling rom intrnal Mjornll dt 997) (uh a a ntrappd air bubbl) hih ar not bridgd by ibr. Thror, ibr bridging t do not at rak initiation but om in G ( α, α ) = k α + k α play during vg propagation vg rak. Sondly, hn multipl raking our, rak opning i normally hr k vg ontrolld k vg ar matrial vry mall paramtr. valu. Intad From dribing maximum amount matrix atr ratur pr ritan unit volum by a that ingl an paramtr ill all por at (both rak apillary tip, por parat gl it por), in on part, an alulat rak K tip a on ughn obtain (K IC_M ) a tnion tning urv rprnting intrloking t aggrgat. Th lattr i addd g αibr α bridging.88α +.α tr om up ith G ovrall bridging tr (6) ith K ( α, αrak ) = opning rlation or ratur analyi. Thirdly, init idth g α αmmbr i onidrd in analyi. To analyz rak propagation, a uprpoition Th matrial mthod paramtr i mployd k alulat vg k vg g an tr b alibratd intnity by ar itting at xprimntal rak tip data rultd rlvant rom both r (vaporabl) applid load atr ontnt rak in bridging onrt tr. at For variou a partiular ag (Di rak Luzio iz, & Cuati orrponding 9b). load i alulatd a valu hn K IC_M i rahd at rak tip. Uing modl, t variou matrial paramtr, inluding matrix ughn (K IC_M ),. Tmpratur volution iz Not that, initial at unbridgd arly ag, la in ibr hmial ontnt ration tnil aoiatd prorman ith mnt ar hydration invtigatd. Th SF rquirmnt ration ar xormi, or tnil train-hardning tmpratur ild multipl i not uniorm raking ar analyzd poibl mthod or matrial or non-adiabati ytm vn i nvironmntal tmpratur i ontant. Hat ondution an b prorman optimization ar diud. dribd in onrt, at lat or tmpratur not xding C (Bažant & Kaplan 996), by Fourir la, hih rad PROBLEM FORMULATION q = λ T In prnt modl, in aggrgat in (7) ma trix ar vid a bridging lmnt a ratur hr q i hat lux, T i abolut ughn alulatd rom raking load or a tmpratur, λ i hat ondutivity; in thi givn pimn gomtry (uh a a ingl nod Proding FraMCoS-7, May 3-8,

3 bam undr bnding load) rv a ritria rak propagation. In or ord, rak propagation our hn: K tip = K IC () hr K IC i ratur ughn matrial K tip.i nt tr intnity rulting rom both applid load bridging tr. Thu, problm rdu obtaining rak tip tr intnity ar du xtrnal or rak bridging tr rptivly. Figur. Prinipl uprpoition in dgd pimn undr uniaxial tnil tr rak bridging tr. A an xampl, a ingl dg noth pimn undr uniaxial tnion load ill b onidrd. Figur ho a ingl dg noth pimn ith initial la iz (unbridgd rak), a, bridgd rak lngth, a xtrnal tnil load, a. Th bridging tr ating on rak ura along raking tion i b, hih i a untion rak opning. Bad on uprpoition hm hon in Figur, rak tip tr intnity ar an b obtaind by umming ontribution K a xtrnal load K b bridging or (Zhang & Li 4), i.. K = K + K () tip a b K a an b alulatd a tr intnity ar du prn ontant tr a along hol rak (inluding bridgd part). Undr tnil load P, a =P/bt, hr b t ar dpth idth pimn rptivly. Thn K a i alulatd by K a a = G( x, a, h) dx (3) a hr G(x,a,h) i ight untion rprnting ontribution a unit or on rak ura rak tip tr intnity ar i pii pimn gomtry rak oniguration (Tada 985). For a ingl dg noth pimn undr uniaxial tnion, it i givn by h ( Jx= / D ( a, ah, / T ) h h G( x, a, h) = (4) π / a( x / a ) hr Th proportionality iint D(h,T) moitur prmability it i a nonlina g( x / a, a / h) h ( x / a, a / h) = rlativ humidity h tmpratur 3 / ( & Najjar a / h) 97). Th moitur ma balan ith g(x/a, a/h) that dind variation by in tim atr ma 3 g( r, ) = g( ) + rg volum ( ) + r g3onrt ( ) + r g 4(atr ( ) ontnt ) b q 5 g ( ) = divrgn +.84( ) +.66 moitur ( lux ) J g( ) = 3.5 = J 3 g ( ) = t ( ).64 ( ) ( ) 3/ Th atr ontnt 3 an b xprd 3/ a g ( ) = ( ) 3 vaporabl atr (apillary a ( vapor, ) +.98 ( adorbd ) atr) non- Similar (hmially abov, ontribution bound) atr K b n (Mil bridging or Pantazopoulo rak tip tr & Mill intnity 995). ar It i ra an b givn by aum that vaporabl atr i a u rlativ humidity, h, dgr hydration a dgr ilia um ration, α Kb = G( x, a, h) b( ( x)) dx, i.. = (6) = ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum Th undamntal by matrial ubtituting proprty Equation rak in Equati bridging la b ((x)) obtain ill b givn a an matrial proprty input. Thu, or a givn gomtry, loading modl, rak oniguration rak bridging la, i rak pril, (x), x + (, ( a) D i hknon, ) = αk & tip + an α& + b alulatd by abov quation. Whn K tip ahiv K IC-M valu, rak tart propagat. No rmaining hr problm / i i ind lop rak pril or a givn rak iorm lngth. (alo Folloing alld moitur tard apa orption/ drivation outlind govrning Cox quation Marhall (Equation (99), 3) mut b rak opning pril by appropriat (x) an b boundary rlatd initial ap-onditplid tnil tr a Th rlation bridging btn tr b ((x)) amount a atr rlativ humidity i alld ' 8 a a iorm ' ' i maurd ' ' ' δ ( x) = G( x, a, h)[ a b( x )] dx G( x, a, h) da E x ith inraing (7) humidity dorption iorm in th a. Nglting ir dirn (Xi t al. Thu, or a givn olloing, rak lngth, orption a (a a iorm, a ill b initial unbridgd la rrn iz), olving both orption quation (), dorption () (7) numrially, By ritial ay, xtrnal i load hytri apaity a rak pril iorm (x) an ould b obtaind. b takn in aount, rlation, vaporabl atr v rlativ humi b ud aording ign varia 3 NUMERICAL rlativity METHOD humidity. Th hap iorm or HPC i inlund by many p Intad olving pially abov tho quation that inlun by intgration xtnt during itration ard hmial l-onitny ration, (x), in turn, dtrm problm an b olvd in matrix orm. With rrn ingl dg noth pimn hon in trutur por iz ditribution (atrratio, mnt hmial ompoition, SF Figur, a numbr nod ar ditributd along uring tim mthod, tmpratur, mix potntial ratur lin. Th opning loing t.). In litratur variou ormulatio tr ating on rak ura ar rplad by ound drib orption iorm nodal or that ar govrnd by xtrnal tnil onrt (Xi t al. 994). Hovr, in th load rak opning diplamnt aording papr mi-mpirial xprion pro trrak opning rlationhip rptivly. Norling Mjornll (997) i adoptd b Whn ratur ughn mnt matrix i Proding FraMCoS-7, May 3-8,

4 rahd J = D ( h, at T ) h rak tip, nod i pilt in () nod a pair oppoit or i impod on Th proportionality nod. Aum iint nod numbr D(h,T) i alld rak tip moitur m+, prmability quation it () i a nonlinar (7) an untion n b xprd rlativ in humidity matrix h orm tmpratur a: T (Bažant & Najjar 97). Th moitur ma balan rquir m that variation in tim atr ma pr unit Ktip = G( x j, a, h)( Pj Fj ) volum onrt (atr ontnt ) b qual i = divrgn moitur lux J (8) m = k jk ( Pj Fj ) j = = J () t m i 8 ' ' ' i Th = atr Gontnt ( xj, a, h ) Gan ( x, ab i, hxprd ) li ( Pj Fa j) um E i= i j= vaporabl atr (apillary atr, atr (9) m vapor, adorbd atr) non-vaporabl (hmially = k j( Pj bound) Fj ) atr n (Mill 966, j= Pantazopoulo & Mill 995). It i raonabl aum that vaporabl atr i a untion hr rlativ k ik humidity, k i ar h, dgr inluning hydration, ar α xtrnal dgr load /or ilia ititiou um ration, or on α, i.. rak =, tip (h,αtr ) intnity = ag-dpndnt ar rak orption/dorption opning rptivly. iorm F i an b (Norling rlatd Mjonll i by 997). Undr thi aumption by ubtituting Equation in Equation on / obtain i i Fi = B l p i + B l () i i + + ( D h) + & n (3) Dtaild dription abov rlation xplanation hr / i paramtr lop ill b orption/dorption givn in nxt tion. iorm For (alo a givn alld rak moitur lngth, a apaity). tr rak Th idth govrning rlation, quation by olving (Equation quation 3) mut (8), b (9) ompltd (), by appropriat ritial xtrnal boundary load apaity initial a ondition., ititiou or F i Th rak rlation pril btn (x) ar amount obtaind. vaporabl Th rak mouth atr opning rlativ diplamnt humidity (CMOD) i alld an adorption b alulatd iorm rom i quation maurd (9) a ith ll. inraing Th onvntional rlativity tnil humidity tr-dormation dorption diagram, iorm uh in a oppoit trrak a. Nglting lngth ir tr-rak dirn mouth (Xi t opning al. 994), diplamnt olloing, (CMOD) orption urv iorm an n ill b b obtaind ud ith by in rrn abov numrial both orption produr. dorption ondition. By ay, i hytri moitur iorm ould b takn in aount, dirnt 4 MATERIAL PARAMETERS FOR MODEL rlation, vaporabl atr v rlativ humidity, mut INPUT b ud aording ign variation rlativity humidity. Th hap orption Th paramtr ud in modl inlud ratur iorm or HPC i inlund by many paramtr, ughn matrix KIC_M, raking trngth, pially tho that inlun xtnt rat initial unbridgd la iz a rak bridging la hmial ration, in turn, dtrmin por (or tr-rak opning rlationhip). trutur por iz ditribution (atr--mnt () Fratur ughn raking trngth matrix. ratio, mnt hmial ompoition, SF ontnt, uring tim mthod, tmpratur, mix additiv, Th ughn mnt matrix i a ritial matrial proprty in imulation rak propagation. t.). In litratur variou ormulation an b ound drib orption iorm normal In pat, om tudi had bn arrid out onrt (Xi t al. 994). Hovr, in prnt dtrmin ratur ughn mnt pat a papr mi-mpirial xprion propod by ll a mortar onrt (Higgin t al. 976, Norling Mjornll (997) i adoptd bau it Nau t al. 969). In tudi, ontribution xpliitly aount or volution hydration ration SF ontnt. Thi orption iorm rad ( h, α, α ) = G ( α, α ) + ( g α α ) h ( g α α ) h K ( α, α ) (4) Figur hr. Finit irt lmnt trm nod (gl iorm) ititiou rprnt or on nod along phyially potntial bound ratur (adorbd) lin. atr ond trm (apillary iorm) rprnt apillary atr. Thi pro xprion zon i i inludd valid only in or alulating lo ontnt ratur SF. Th ughn, iint i.. G rprnt pak load in amount load- CMOD atr pr urv unit volum i ud a hld in ritial gl load por or at K IC % alulation. rlativ humidity, Thror it maurd an b xprd valu (Norling K IC i trongly Mjornll inlund 997) a by ontnt aggrgat i iz dpndnt. Hovr, i ontribution pro zon i onidrd in rak bridging la, G ( α, α ) = k α + k α n ratur vg ughn vg mnt pat or mortar a ll a onrt ill b a ontant, indpndnt hr pimn k iz. In thi a, ritial load at vg k vg ar matrial paramtr. From hih maximum rak amount tart propagat, atr pr unit i.. volum tarting that point an ill nonlinarity all por (both in apillary load-cmod por urv, gl por), hould on b ud an alulat or ratur K a ughn on obtain alulation (Zhang & Lung & Chung 6, Zhang & Lung & Xu 9a). g α α In addition, raking trngth dind a.88α +.α G tr lvl at hih initial rak tart propagat K ( α i, α ) = bginning point rak bridging la, (6) hih i rquird in gmodl α αalulation. Both raking trngth ratur ughn an b dtrmind dirtly rom xprimnt. For a prnod bam undr bnding vg k Th matrial paramtr k load, aording vg g an b alibratd by itting xprimntal data rlvant lati ory, CMOD xtrnal load (P) r (vaporabl) atr ontnt in onrt at oby a linar rlationhip bor initiation variou ag (Di Luzio & Cuati 9b). raking. Atr initial raking, linar rlationhip btn P CMOD no longr xit. Thu raking. Tmpratur load, P volution an b dtrmind by point hr P-CMOD urv dviat rom initial Not that, at arly ag, in hmial ration linar portion. Thi point i rgardd a tranition aoiatd ith mnt hydration SF ration point rom linar-lati tag nonlinarlati tag, in hih a ititiou rak tart d- ar xormi, tmpratur ild i not uniorm or non-adiabati ytm vn i nvironmntal vlop. Typial graph illutrating dtrmination tmpratur i ontant. Hat ondution an b raking load rom thr-point bnding tt on dribd in onrt, at lat or tmpratur not nod bam i hon in Figur 3. Bad on P xding C (Bažant & Kaplan 996), by valu, orrponding raking trngth,, i Fourir la, hih rad alulatd rom a init lmnt analyi that tak noth t in aount. At am tim, ratur ughn K IC_M i alulatd alo rom (7) P q = λ T through laial ratur mhani ory. Bad on hr tt rult q i providd hat in lux, (Zhang T t i al. 9b), abolut rlationhip tmpratur, btn λ i raking hat trngth ondutivity; ratur in thi ughn PVA ibr rinord mntitiou Proding FraMCoS-7, May 3-8,

5 ompoit an b xprd a: = 5.78 () K IC _ M hr i in unit MPa K IC i in unit MPam /. In prnt papr, t K IC-M /or raking trngth on ompoit tnil prorman i invtigatd by varying K IC-M rom.5mpam /.MPam /, orrponding hang rom.3mpa 5.MPa. Load (N) P CMOD (mm) Figur 3. Dtrmination raking load rom bnding tt. () Crak bridging la A a undamntal matrial proprty, rak bridging la mntitiou ompoit, uh a mortar, onrt ibr rinord mnt ompoit hav bn invtigatd both xprimntally ortially during rnt yar. Th xprimntal rult ho that hap tr-rak idth urv ibr rinord mnt ompoit i omplx gratly inlund by typ amount ibr ud (Stang t al. 99). A miromhani-bad modl or tr-rak idth rlationhip FRC matrial ha bn dvlopd by Li t al. (993). Th modl provid a bai undrting inlun miro-paramtr on hap tr-rak idth urv i pially uul or matrial dign. In thi ork, miromhani-bad modl dvlopd by Li t al. (993) i ud a rak bridging la (-), hih i givn by: ( ) = ( ) = L gτv L Load (N) CMOD (mm) ( ) τl or or L () π / =, =, g = ( + ) d ( + η) E d 4 + hr. In quation, d, L, V, E t or ibr diamtr, lngth, volum ration lati modulu rptivly. τ i ibr/matrix bond trngth. i nubbing iint that rlting t inlination an- J gl. η = V E / VmE= D ( m h, T ) V m, he m ar matrix volum ration matrix lati modulu. Typial PVA ibr ibr/matrix intraial Th proportionality paramtr iint ud in D(h,T) moitur prmability it i a nonlina modl ar litd in Tabl. rlativ humidity h tmpratur Apart rom ibr bridging, matrix itl alo xhibit bridging ation du prn in aggr- & Najjar 97). Th moitur ma balan that variation in tim atr ma gat. Du lak a good phyial modl, an volum onrt (atr ontnt ) b q mpirial modl propod by Stang t al. (99) i divrgn moitur lux J ud or matrix bridging alulation in prnt ork. In modl, matrix bridging tr m i xprd a a untion = J rak opning a: t m = Th atr ontnt an b xprd a p (3) vaporabl atr + (apillary a vapor, adorbd atr) non- (hmially bound) atr n (Mil Pantazopoulo & Mill 995). It i ra hr i matrix bridging tr at =, i.. matrix aum that vaporabl atr i a u raking trngth. Th paramtr p drib rlativ humidity, h, dgr hydration hap tning urv ith inra rak dgr ilia um ration, α opning. Th paramtr orrpond, i.. = = ag-dpndnt orption/dorption rak opning hn tr ha droppd hal (Norling Mjonll 997). Undr thi aum. For onrt, good itting xprimntal by ubtituting Equation in Equati rult an b obtaind ith p=. obtain =.5mm (Stang t al. 99). Bau prnt ork i oud on matrix ithout oar aggrgat, a mallr valu i ud + ( =.mm). Th tal rak h bridging ompoit an b obtaind by umming () (3) a hon in (). Typial tr-rak idth hr rlationhip PVA ibr / i lop orption/ rinord mnt ompoit ith parat omponnt i hon in iorm (alo alld moitur apa govrning quation (Equation 3) mut b Figur 4. by appropriat boundary initial onditi 8. Th rlation btn amount Total Bridgingatr rlativ humidity i alld iorm i maurd ith inraing 6. humidity dorption iorm in th Fibr Bridging a. Nglting ir dirn (Xi t al. 4. olloing, orption iorm ill b rrn both orption dorption By ay, i hytri. iorm ould b takn in aount, Matrix Bridging rlation, vaporabl atr v rlativ humi. b ud aording ign varia rlativity humidity. Th hap Crak idth (mm) Figur 4. Crak bridging iorm in ibr or rinord HPC i mntitiou inlund ompoit. pially tho that inlun xtnt by many p hmial ration, in turn, dtrm (3) Initial unbridgd trutur la iz por iz ditribution (atrratio, la mnt ar typially hmial ompoition, our SF In mnt matrix, hr rak ar uring initiatd. tim Mot la mthod, inhrnt tmpratur, rom mix mixing pro t.). hav In iz litratur blo -mm, variou ormulatio ir xitn may ound onidrably drib rdu orption raking iorm trngth (Zhang & onrt Li 4). (Xi In t al. modl, 994). Hovr, initial in th la i an quivalnt papr rak rultd mi-mpirial rom dt xprion at pro ura pimn. Norling Th Mjornll dt (997) might i rult adoptd b Str (MPa) Proding FraMCoS-7, May 3-8,

6 rom J = D ( air h, T ) void, h aggrgat/mnt pat intraial () rak or poibl damag in matrial (.g. Th hrinkag proportionality rak). In iint prnt D(h,T) tudy, i a alld t initial moitur unbridgd prmability la iz it a i ithin a nonlinar.5 untion.5 mm i rlativ ud in humidity modl h alulation tmpratur invtigat T (Bažant ir & Najjar t 97). on tnil Th moitur prorman. ma balan rquir that variation in tim atr ma pr unit 5 volum RESULTS onrt AND (atr DISCUSSION ontnt ) b qual divrgn moitur lux J In thi tion, tnil prorman a ingl dg pimn mad PVA ibr rinord mntitiou t ompoit i imulatd ith dvlopd = J () modl. Th inlun ratur ughn matrix KIC_M Th atr (ith ontnt t an b xprd alo involvd), a ibr um bridging vaporabl tr rltd atr by ibr (apillary ontnt atr, a ll atr a initial vapor, unbridgd adorbd la atr) iz a on tnil non-vaporabl prorman (hmially ompoit bound) ar atr prntd n (Mill diud. 966, Th Pantazopoulo rlatd matrial & Mill paramtr 995). It ud i raonabl in modl ar aum litd that in Tabl vaporabl hr ontant atr i ibr/matrix a untion intraial rlativ bond humidity, trngth h, dgr i aumd hydration, dpit α varia-, tion dgr in matrix ilia trngth um ration, ughn. α, i.. = (h,α,α ) = ag-dpndnt orption/dorption iorm Tabl (Norling. Paramtr Mjonll 997). matrix, Undr ibr thi ibr/matrix aumption intra. by Tnil ubtituting E trngth Equation d L in E m Equation τ on obtain (GPa) (mm) (mm) (GPa) (MPa) (MPa) & n (3) Et ratur ughn Figur 5a 5b ho t matrix ughn K hr / i lop orption/dorption IC_M on tnil bhavior, in trm tnil iorm (alo alld moitur apaity). Th tr vru rak lngth rak mouth opning govrning quation (Equation 3) mut b ompltd rlation rptivly. Th imulation rult ar or a by appropriat boundary initial ondition. ingl dgd pimn ith a ontant unbridgd initial noth.5mm. From igur, it an b Th rlation btn amount vaporabl atr rlativ humidity i alld adorption obrvd that global tnil load arrying apaity or a ixd rak bridging la inra ith iorm i maurd ith inraing rlativity humidity dorption iorm in oppoit inra K IC_M at initial tag rak dvlopmnt. Atr rak lngth ahiving about 8% a. Nglting ir dirn (Xi t al. 994), in olloing, orption iorm ill b ud ith tal pimn idth, all urv tart onvrg am loading apaity, hih i rrn both orption dorption ondition. By ay, i hytri moitur maximum ibr bridging apaity. With inra iorm ould b takn in aount, dirnt matrix ughn, tnil tr vru rak rlation, vaporabl atr v rlativ humidity, mut lngth rlation i gradually hangd rom a mononi inraing trnd an initially draing trnd b ud aording ign variation rlativity humidity. Th hap orption ollod by inraing tr. Thi bhavior an alo iorm or HPC i inlund by many paramtr, b ound in urv tnil tr vru rak pially tho that inlun xtnt rat mouth opning (Fig. 5b). Th rat tr dra hmial ration, in turn, dtrmin por at initial tag rak propagation i rdud trutur por iz ditribution (atr--mnt ith draing matrix ughn inally hang ratio, mnt hmial ompoition, SF ontnt, a trnd ith mononi inraing tr a matrix uring tim mthod, tmpratur, mix additiv, ughn i rdud a rtain valu, hih i t.). In litratur K IC_M =.5MPam / variou ormulation an b in thi a. Th inra tnil trngth ith rak lngth ho poibility ound drib orption iorm normal onrt (Xi t al. 994). Hovr, in prnt multipl-raking in matrial undr tnion, papr mi-mpirial xprion propod by hih grat intrt or matrial dutility improvmnt. I raking tr i blo Norling Mjornll (997) i adoptd bau it pak tnil xpliitly trngth aount or loal volution tr at maximum hydration rak ration opning SF along ontnt. rak Thi lngth orption i alo iorm lor than rad pak bridging tr, ompoit ha train-hardning potntial, i.. r xit a margin load arrying apaity or dvloping quntial multipl raking. ( h, α, α ) = G ( α, α ) + a/h ( g α α ) h (4) Tnil tr (MPa) ( g α α ) h K ( α, α ) K IC =.,.5,.,.5MPam / hr irt trm (gl iorm) rprnt phyially bound (adorbd) atr ond 4. trm (apillary iorm) rprnt apillary atr. Thi xprion i valid only or V. =% lo ontnt SF. Th iint G a =.5mm rprnt amount d=.5mm atr pr unit volum hld in gl por at %. rlativ humidity, 5 5 it an b 5 xprd 3 35(Norling 4 Crak lngth (mm) Mjornll 997) a (a). G ( α, α ) = k α + k α K IC =., vg.5,., vg.5mpam / 8. Tnil tr (MPa). g α α.88α +.α G V=%, a=.5mm. K ( α, α ) = Crak mouth g α opning α (mm) hr k vg k vg ar matrial paramtr. From maximum 6. amount atr pr unit volum that an ill all por (both apillary por gl por), on an alulat K a on obtain (6) (b) Figur 5. Tnil tr vru rak lngth (a) rak mouth Th opning matrial (b) paramtr urv dgd k pimn ith dirnt vg k vg g an matrix b alibratd ughn. by itting xprimntal data rlvant r (vaporabl) atr ontnt in onrt at variou Th rult ag (Di prnt Luzio in & Figur Cuati 5 9b). larly ho that lor matrix ratur ughn, highr liklihood multipl raking. Thi i onitnt ith. Tmpratur xprimntal volution rult that ECC ith highr matrix Not that, trngth at arly ha ag, a lor in tnil hmial train ration apaity (Zhang aoiatd t al. ith 9). mnt On hydration or h, SF ration abov rult ar xormi, alo indiat tmpratur that rdution ild i not uniorm matrix ughn or non-adiabati ill hlp ytm rdu vn i tr nvironmntal jump hih our tmpratur during i loading, ontant. hih Hat i ommonly ondution obrvd an b in dribd tnil tt in onrt, (Zhang t at al. lat 9). or tmpratur It nd not b notd that ibr/matrix intraial bond trngth xding C (Bažant & Kaplan 996), by bond rlatd paramtr ar aumd b ontant vn hn K IC_M i varying. Thi aumption Fourir la, hih rad may b raonabl bau prviou xprimntal rult hon that t matrix proportion (7) on q = λ T ibr/matrix bond trngth PVA ibr rinord hr mntitiou q i hat ompoit lux, i T limitd i du abolut trong tmpratur, hmial bond λ i btn hat PVA ondutivity; ibr mnt in thi matrix (Li t al. ). Proding FraMCoS-7, May 3-8,

7 Et ibr bridging Mot high prorman ibr rinord mntitiou ompoit ar rinord by hort diontinuou ibr. Crak bridging providd by ibr in ompoit i on mot important ar govrning maroopi mhanial proprti ompoit. Figur 6a 6b diplay t ibr volum ontnt (V ), hih govrn ibr bridging, on tnil tr vru rak lngth rak mouth opning (CMOD) diagram rptivly. Figur 7 prnt orrponding tr-rak opning rlationhip ompoit ith dirnt ibr ontnt. In alulation, K IC_M =.MPam / ibr paramtr litd in Tabl r ud. Fibr volum %,.%,.% r ud in imulation. Aording rult, or givn PVA ibr rlatd ibr/matrix intraial paramtr, a ibr ontnt rah %, ompoit tart ho train-hardning bhavior. By ontrat, a ibr ontnt i lor than %, train tning undr tnil load an b xptd. Clarly, rquirmnt on ibr bridging, rprntd by ibr ontnt, i ritial or train-hardning ahivmnt. For givn matrix, ibr ibr/matrix intraial paramtr, volum ibr ndd or trainhardning multipl raking prorman ompoit an b obtaind through modl imulation. A an xampl illutrat ibr bridging, rak pril orrponding tr ditribution along rak lngth at dirnt rak propagation tag (rltd by rak lngth) ar hon in Figur 8 9 or ibr ontnt % rptivly. In alulation, K IC_M =.MPam / initial la iz a =.5mm r aumd. Dirn in rak hap a ll a tr ditribution along rak an b obrvd or a. Whn r i no ibr rinormnt, rak opning inra mononially ith ditan rom rak tip, o maximum minimum rak opning our at rak mouth tip rptivly. Th tr ditribution along rak ho obviou tning ith inra rak opning. For ompoit ith % ibr addition, at initial tag rak groth, or rak lngth l than 3mm, rak opning along rak lngth inra mononially ith ditan rom rak tip. A rak lngth inra a rtain lvl, ay about 8% pimn idth, rak pril ho an ar hap ith maximum rak opning ourring nar ntr rak. Th orrpondd tr ditribution along rak lngth diplayd in Figur 9b i alo intrting. For rak lngth ovr 5mm, tr valu along rak bom almot ontant xpt or a mall zon nar rak tip tr inra ith inraing rak opning. Thi i imilar oalld tady-tat J raking = D ( h, T ) that h i bai rquirmnt or matrial train-hardning (Li & Lung, 99, Li 993, Lung Th 996). proportionality Hovr, tady-tat iint D(h,T) hr rr moitur almot ontant prmability tr ditribution it a nonlina along rak, rar than rlativ humidity ontant rak h opning along rak & Najjar lngth 97). dribd Th in moitur ormr mod- ma balan tmpratur l (Li & Lung that 99, Li variation 993, Lung in tim 996). Th atr ma prnt ork ho volum that an onrt ar hap (atr rak ontnt pril ) b q may b rultd vn divrgn undr tady-tat moitur lux trainhardning ondition. J. = J t Tnil tr (MPa) Tnil tr (MPa) Str (MPa ) V =.%,.%,% K IC =.MPam /, a =.5mm Crak lngth (mm) Th atr ontnt an b xprd a vaporabl atr (apillary a vapor, adorbd atr) non- (hmially bound) atr n (Mil Pantazopoulo & Mill 995). It i ra aum that vaporabl atr i a u rlativ humidity, h, dgr hydration dgr ilia um ration, α, i.. = = ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum by ubtituting (a) Equation in Equati obtain V =.%,.%,% = α & + & + govrning quation (Equation 3) mut b By ay, i hytri rlativity humidity. Th hap pially tho that inlun xtnt 4. hr iorm / i (alo lop alld orption/ moitur apa. by appropriat boundary initial onditi Th rlation K IC =.MPam btn /, a =.5mm amount. atr rlativ humidity i alld iorm.5. i maurd.5 ith.inraing Crak humidity mouth opning diplamnt dorption (mm) iorm in th (b) Figur 6. Tnil tr a. vru Nglting rak lngth ir dirn (a) rak (Xi t al. mouth opning (b) urv olloing, dgd orption pimn ith iorm dirnt ill b matrix ughn. rrn both orption dorption. iorm ould b takn in aount, V =%,.%,% rlation, vaporabl atr v rlativ humi 8. b ud aording ign varia 6. iorm or HPC i inlund by many p 4. hmial ration, in turn, dtrm trutur por iz ditribution (atrratio,. mnt hmial ompoition, SF uring tim mthod, tmpratur, mix t.). In litratur variou ormulatio.. ound. drib.3.4orption.5 iorm onrt Crak (Xi idth (mm) t al. 994). Hovr, in th Figur 7. Str rak papr opning rlationhip mi-mpirial ompoit xprion pro ith dirnt ibr ontnt. Norling Mjornll (997) i adoptd b α Proding FraMCoS-7, May 3-8,

8 J Crak mouth opning (mm) D.3 ( h, T h moitur. prmability a=5,.5,,7.5,5,.5mm it i a nonlinar untion Tnil tr (MPa). V =, K IC =.MPam /, a =.5mm = J x (mm) (a) 3. Th atr K IC =.MPam ontnt /, V =, an a =.5mm b xprd a um vapor, adorbd atr) non-vaporabl rlativ humidity, a=5,.5,,7.5,5,.5mm h, dgr hydration, α, by. ubtituting Equation in Equation on x (mm) obtain (b) Figur 8. Crak pril (a) tr ditribution (b) at dirnt rak lngth dgd pimn ith V =. + & n (3) Crak mouth opning (mm). V =%, K IC =.MPam /, a =.5mm a=36.5,35.5,35,33.75,3.5,3,5,,mm iorm.5 (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd by appropriat boundary initial ondition.. Th rlation btn amount vaporabl atr rlativ humidity i alld adorption iorm.5 i maurd ith inraing rlativity humidity dorption iorm in oppoit a. Nglting ir dirn (Xi t al. 994), in. olloing, orption iorm ill b ud ith rrn both orption x (mm) dorption ondition. By ay, i hytri (a) moitur. iorm ould b takn in aount, dirnt a=36.5,35.5,35,33.75,3.5,3,5,,mm rlation, 9. vaporabl atr v rlativ humidity, mut b ud aording ign variation 8. rlativity humidity. Th hap orption iorm 7. or HPC i inlund by many paramtr, pially 6. tho that inlun xtnt rat hmial ration, in turn, dtrmin por 5. trutur por iz ditribution (atr--mnt ratio, 4.mnt hmial ompoition, SF ontnt, uring 3. tim mthod, tmpratur, mix additiv, t.). In litratur variou ormulation an b. ound drib 5 5 orption iorm normal 4 onrt (Xi t al. 994). x Hovr, (mm) in prnt papr mi-mpirial (b) xprion propod by Figur 9. Crak pril (a) tr ditribution (b) at dirnt rak lngth dgd pimn ith V Norling Mjornll (997) i adoptd bau it =%. Tnil tr (MPa) = ) () Th proportionality iint D(h,T) i alld rlativ humidity h tmpratur T (Bažant & Najjar 97). Th moitur ma balan rquir that variation in tim atr ma pr unit volum onrt (atr ontnt ) b qual divrgn moitur lux J t () vaporabl atr (apillary atr, atr (hmially bound) atr n (Mill 966, Pantazopoulo & Mill 995). It i raonabl.5 aum that vaporabl atr i a untion dgr ilia um ration, α, i.. = (h,α,α ) = ag-dpndnt orption/dorption iorm (Norling Mjonll 997). Undr thi aumption hr / i lop orption/dorption xpliitly. aount or volution hydration ration V =%, SF K IC =.MPam ontnt. / Thi orption iorm 8. rad Tnil tr (MPa) 6. a =.5,.5,.5,.75,,.5mm ( h, α4., α ) = G ( α, α ) + ( g α α ) h. ( g α α ) h. K ( α, α ) Crak lngth (mm) Tnil tr (MPa). Mjornll. 997) a (a) atr. 6. Thi xprion i valid only or lo ontnt atr 4. pr unit volum hld in gl por at % V =%, K IC =.MPam /. G ( α, α. ) = k.5 α + k. α vg Crak vgmouth opning (mm) (b) Figur. Tnil tr vru rak lngth (a) rak mouth opning hr k(b) vg urv k vg ar dgd matrial pimn paramtr. ith dirnt From initial la maximum iz. amount atr pr unit volum that an ill all por (both apillary por gl por), on an alulat 6. K a on obtain a =.5,.5,.5,.75,.,.5mm g α α 4..88α +.α G K ( α, α ) = g α α. Tnil tr (MPa) (4) hr airt =.5,.5,.5,.75,,.5mm trm (gl iorm) rprnt phyially 8. bound (adorbd) atr ond trm (apillary iorm) rprnt apillary SF. Th iint G rprnt amount rlativ humidity, it an b xprd (Norling (6) Th matrial paramtr k vg k vg g an b alibratd. by itting xprimntal data rlvant r (vaporabl)..5atr. ontnt.5 in onrt. at Crak mouth opning (mm) Figur. variou ag Clo (Di up on Luzio tnil & Cuati tr-rak 9b). mouth opning urv dgd pimn ith dirnt initial la iz.. Tmpratur volution Et initial unbridgd la iz Not that, at arly ag, in hmial ration aoiatd ith mnt hydration SF ration Figur ar xormi, a b tmpratur ho t ild i not initial uniorm unbridgd or non-adiabati la iz ytm (a ) on vn tnil i nvironmntal tr vru rak tmpratur lngth i ontant. rak mouth Hat opning ondution (CMOD) an rlation b dribd rptivly. in onrt, Figur at lat ho or tmpratur lo up not xding tr- CMOD C urv (Bažant & Kaplan bginning 996), part by mor Fourir larly. la, hih In alulation, V =% K IC_M =.MPam / rad r ud. From urv, it an b n that or ixd K IC-M rak bridging q = λ T la, a trnd imilar (7) t K IC_M i alo ound or a hanging rom hr.5mm q i.5mm. hat lux, Th largr T i initial abolut la iz tmpratur, a, lor λ i irt hat raking ondutivity; load. Thi in indiat that nlargmnt initial thi unbridgd Proding FraMCoS-7, May 3-8,

9 la ill nhan train-hardning multipl raking potntial ompoit. Thi ida ha alrady bn ud in prodution high dutility ECC by Li t al. (6) Wang t al. (7), ho introdud mall plati bubbl in matrix inra tnil train apaity. Th numbr iz ditribution initial unbridgd la in matrix ontrol numbr rak that ill orm during train hardning prior rahing pak bridging tr. Thror, paramtr ill alo govrn ultimat tnil train apaity ompoit. Apparntly, unbridgd la in matrix play an important rol in multipl raking prorman ompoit. High dutility i loly aoiatd ith dnity multipl rak, aturatd multipl raking an only b rahd hn a uiint numbr la xit. Th inhrnt la in mntitiou matrix, uh a por ak boundari btn pha po a rom natur (Wang t al. 7). Th uniormity la ill inlun numbr rak ourring undr a rtain tnil load lvl. Whn mor la ar having am or imilar iz, highr ill b numbr rak ourring undr a giv load. Thu, ompoit dutility an b improvd by manually adding mall olid partil hih ha rlativ ak intraial bond trngth matrix. By inorporating partil imilar iz dvlop a vry narro ditribution initial la iz in matrix, mor rak ill orm undr loly inrad loading nhan tnil dutility. 6 CONCLUSIONS Thi artil prnt a ortial tudy on modl I rak propagation in mntitiou ompoit ith pii objtiv tudy phnomna train-hardning multipl raking undr dirt tnion. A ratur mhani bad modl or rak propagation imulation ibr rinord mntitiou ompoit undr dirt tnion i dvlopd. Th modl aum rak initiat rom intrnal dt hih ar not bridgd by ibr initially. In modl, intad dribing matrix ratur ritan by a ingl paramtr at rak tip, parat it in part, a rak tip ughn a tnion tning urv rprnting intrloking t aggrgat. Th lattr i addd ibr bridging tr om up ith ovrall bridging tr ith rak opning rlation or ratur analyi. Uing modl, t variou matrial paramtr, inluding matrix ughn, initial la iz in matrix ibr ontnt on tnil prorman ar invtigatd. With modl rult, tranition rom tnion tning train hardning multipl raking bhavior an b quantiid. Lor ratur ughn matrix, largr initial la iz improvd J = D ( rak h, T ) hbridging ith a highr ibr ontnt ill nhan poibility trainhardning multipl Th raking. proportionality Whil iint gnral D(h,T) onluion ar imilar moitur prmability tho rom onvntional it i a nonlina ori, prnt analyi rlativ i blivd humidity h provid tmpratur a mor aurat dription & Najjar 97). raking Th moitur pro ma in balan ibr rinord that mntitiou variation ompoit in tim giv atr ma bttr timat volum rang onrt miro-paramtr (atr ontnt or ) b q dign ompoit divrgn ith pudo-dutil moitur lux bhavior. J = J t ACKNOWLEDGEMENTS Th atr ontnt an b xprd a Support rom National vaporabl Sin atr Foundation (apillary a China (No.58789) vapor, Tinghua adorbd Univrity atr) ar non- gratully aknoldgd. (hmially bound) atr n (Mil Pantazopoulo & Mill 995). It i ra aum that vaporabl atr i a u REFERENCES rlativ humidity, h, dgr hydration dgr ilia um ration, α Bazant, Z.P. & Oh,B.H., 983. Crak b ory or ratur, i.. = onrt. Matrial = ag-dpndnt Strutur, 6: orption/dorption Cox, B.N. & Marhall, (Norling D.B. 99. Mjonll Stabl 997). untabl Undr olution or bridgd by rak ubtituting in variou pimn. Equation Ata Mtall. in Equati thi aum Matr., 39 (4): obtain Griith, A.A., 9. Th phnomna ruptur lo in olid. Phil. Tran. Roy. So., ri A: Higgin, D.D. & Baily, J.E Fratur maurmnt on mnt pat. Journal Matrial + ( DSin, h) = : α& + α& + Hillrborg, A., Modér, M. & Ptron, P-E Analyi rak ormation rak groth by man ratur mhani init hr lmnt, Cm Conr R, 6: / i lop orption/ Ka, T. & Li, V.C A N Miromhani Dign iorm (alo alld moitur apa Thory or Pudo Strain Hardning Cmntitiou Compoit, ASCE J. govrning Enginring quation Mhani, (Equation 5: ) mut b Lph, M.D., Li, V.C., by appropriat Lph, M.D. boundary & V.C. Li, 6. initial Long onditi Trm Durability Prorman Th rlation Enginrd btn Cmntitiou amount Compoit. Intrnational atr Journal rlativ or humidity Rration i alld Building Monumnt, iorm ():9-3. i maurd ith inraing Lung, C.K.Y., 996. Dign Critria or Pudo-dutil Fibr Compoit, ASCE humidity J. Enginring dorption Mhani, iorm :- in th 8. a. Nglting ir dirn (Xi t al. Li V.C., Stang, H. & Krnhl, olloing, H., 993. orption Miromhani iorm ill b rak bridging in rrn ibr rinord both onrt, orption Matrial dorption Strutur, 6: By ay, i hytri Li, V. C. & Wang, S. 6. Mirotrutur Variability iorm ould b takn in aount, Maroopi Compoit Proprti High Prorman Fibr Rinord rlation, Cmntitiou vaporabl Compoit. atr Probabiliti v rlativ humi Enginring Mhani, b ud aording (3):-6. ign varia Li, V. C., Wu, C., Wang, rlativity S., Ogaa, humidity. A. & Sai, Th T.. hap In-otra Tailoring iorm or Strain-hardning or HPC PVA-ECC. i inlund ACI by Ma-many p trial Journal, 99 pially : tho that inlun xtnt Li, V.C. & Lung, C.K.Y. 99. Stady Stat Multipl Craking Short hmial Rom Fibr ration Compoit",, ASCE in turn, J. dtrm Enginring Mhani, trutur 8: por iz ditribution (atrratio, 986. Fratur mnt pro hmial in onrt ompoition, SF Li, V.C. & Liang, E. ibr rinord mntitiou uring tim ompoit. mthod, Journal tmpratur, Enginring Mhani, t.). In (6): litratur variou ormulatio mix Li, V.C From ound Miromhani drib Strutural orption Enginring-- Dign Cmntiu Compoit or Civil En- iorm ginring Appliation. onrt JSCE (Xi J. t al. Strutural 994). Mhani Hovr, in th Earthquak Enginring, papr : mi-mpirial xprion pro Li, V.C.,. Advan Norling ECC Mjornll Rarh, (997) ACI Spial i Pub- adoptd b Proding FraMCoS-7, May 3-8,

10 J = liation D ( h, Ton ) Conrt: h Matrial Sin Appliation, () SP 6-3: Nau, D.J. & Lott, J.L., 969. Fratur ughn portl mnt Th proportionality onrt, Journal iint Amrian D(h,T) Conrt i Intitut, alld moitur prmability it i a nonlinar untion Stang, H. rlativ & Aarr, humidity T., 99. h Evaluation tmpratur rak T (Bažant idth in & Najjar FRC ith 97). onvntional Th moitur rinormnt. ma balan Cmnt rquir Conrt Compoit, 4 (): that variation in tim atr ma pr unit Tada, H., Pari, P.C. & Irin, G., 985. Th tr analyi volum rak hbook, onrt nd (atr dn. Pari ontnt Prod. ) In., b St qual Loui, Miouri. moitur lux J divrgn Wang, K. Wang, K., Jann, D., Shah, S. & Karr, A., 997. Prmability Study Crakd Conrt, Cmnt Conrt = Rarh, J 7: () Wang, t S. & Li, V. C., 7. Enginrd Cmntitiou Compoit ith High-volum Fly Ah. ACI Matrial Journal, 4 Th (3):33-4. atr ontnt an b xprd a um Zhang J., vaporabl Lung, C. K. atr Y. & Chung, (apillary Y. N., 6. atr, Flxural atr vapor, Prorman adorbd Layrd atr) ECC-Conrt Compoit non-vaporabl Bam. (hmially Compoit Sin bound) Thnology, atr 66 :5-5. Zhang J., Lung, C. K. Y. & Xu S., 9a. n (Mill 966, Evaluation Pantazopoulo Fratur Paramtr & Mill Conrt 995). rom It Bnding i raonabl Tt. Matrial that Strutur, vaporabl In pr. atr i a untion aum Zhang, rlativ J. & humidity, Li, V.C., 4. h, dgr Simulation hydration, mod I rak α propagation in ilia ibr um rinord ration, onrt α, dgr by, i.. ratur =mhani. (h,α,α ) = Cmnt ag-dpndnt Conrt Rarh, orption/dorption 34: iorm Zhang, J. Gong, C X., Ju, X.C. & Guo, Z L., 9b. Flxural (Norling Mjonll 997). Undr thi aumption prorman high Dutil ibr rinord mntitiou by ompoit. ubtituting Enginring Equation Mhani, in In pr, Equation (in Chin). on Zhang, obtain J. Gong, C X., Zhang, M H. & Guo, Z L., 9. Enginrd Cmntitiou Compoit ith Charatriti Lo Drying Shrinkag. Cmnt Conrt Rarh, 39: & n (3) hr / i lop orption/dorption iorm (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd by appropriat boundary initial ondition. Th rlation btn amount vaporabl atr rlativ humidity i alld adorption iorm i maurd ith inraing rlativity humidity dorption iorm in oppoit a. Nglting ir dirn (Xi t al. 994), in olloing, orption iorm ill b ud ith rrn both orption dorption ondition. By ay, i hytri moitur iorm ould b takn in aount, dirnt rlation, vaporabl atr v rlativ humidity, mut b ud aording ign variation rlativity humidity. Th hap orption iorm or HPC i inlund by many paramtr, pially tho that inlun xtnt rat hmial ration, in turn, dtrmin por trutur por iz ditribution (atr--mnt ratio, mnt hmial ompoition, SF ontnt, uring tim mthod, tmpratur, mix additiv, t.). In litratur variou ormulation an b ound drib orption iorm normal onrt (Xi t al. 994). Hovr, in prnt papr mi-mpirial xprion propod by Norling Mjornll (997) i adoptd bau it xpliitly aount or volution hydration ration SF ontnt. Thi orption iorm rad ( h, α, α ) = G ( α, α ) + ( g α α ) h ( g α α ) h K ( α, α ) (4) hr irt trm (gl iorm) rprnt phyially bound (adorbd) atr ond trm (apillary iorm) rprnt apillary atr. Thi xprion i valid only or lo ontnt SF. Th iint G rprnt amount atr pr unit volum hld in gl por at % rlativ humidity, it an b xprd (Norling Mjornll 997) a G ( α, α ) = k α + k α vg vg hr k vg k vg ar matrial paramtr. From maximum amount atr pr unit volum that an ill all por (both apillary por gl por), on an alulat K a on obtain α α K (, ) = g α α.88α +.α G g α α (6) Th matrial paramtr k vg k vg g an b alibratd by itting xprimntal data rlvant r (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 9b).. Tmpratur volution Not that, at arly ag, in hmial ration aoiatd ith mnt hydration SF ration ar xormi, tmpratur ild i not uniorm or non-adiabati ytm vn i nvironmntal tmpratur i ontant. Hat ondution an b dribd in onrt, at lat or tmpratur not xding C (Bažant & Kaplan 996), by Fourir la, hih rad q = λ T (7) hr q i hat lux, T i abolut tmpratur, λ i hat ondutivity; in thi Proding FraMCoS-7, May 3-8,