Static and fatigue failure simulation of concrete material by discrete analysis

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1 Fratur Mhani Conrt Conrt Strutur - Rnt Advan in Fratur Mhani Conrt - B. H. Oh, t al.(d) 2010 Kora Conrt Intitut, Soul, ISBN Stati fatigu failur imulation onrt matrial dirt analyi K. Nagai Th Univrity Tokyo, Tokyo, Japan K. Matumo Tokyo Intitut Thnology, Tokyo, Japan ABSTRACT: Thi papr drib an approah dvlop an analytial hm for a prdition lif pan onrt matrial. Failur imulation onrt at matrial al dirt analyi mthod, hih an onidr fft it omponnt inluding nvironmntal fft, i ondutd valuat onrt mhanial damag or fratur dirtly. A hronologial damag, a frot damag fft i introdud mo al fratur. Fatigu imulation i ondutd modling tim-dpnd phnomnon inluding lati, vio-lati, vio-plati raking omponnt. 1 INTRODUTION Conrt matrial i, in fat, onitd aggrgat mnt pat, rfor it bhavior i htrognou. Hovr, a tr-train modl onrt in ontinuou numrial mthod uh a finit lmnt analyi orrpond an avrag rpon rtain ara, onrt i tn tratd a homognou matrial in nginring fild. Whil uh homognou modling onrt i uful in dign, r ar om problm. Epially, modling rak i on mot diffiult point. For xampl, loalization ma diffuion in rakd onrt annot b imulatd dirtly mard rak modl in hih rak i avragly introdud ovr lmnt. On olution for problm i dirt numrial analy, in hih rak ar diontinuouly introdud. Th Rigid Body-Spring Modl (RBSM) (Kaai 1978), in hih mhanial bhavior objt i rprntd rigid body lmnt pring intronnt m, i a dirt numrial mthod ha bn applid for analyi onrt trutur matrial (Bolr & Sai 1997, 1998). In thi tudy, RBSM a applid for mortar onrt ubjtd tati fatigu in dimnion. Modling matrial i bad on mo-al onpt, in hih matrial failur at maro al i aumd b an aumulation tnil har fratur at miro al, rfor no fratur our in omprion rgion tr-train modl onntd pring. Th tnil har bhavior ar rprntd linr tning modl har lipping ritria hih i a funtion har tr, rptivly. For tati analy, aggrgat, mortar ir intrfa r paratly modld ontitutiv la a introdud. Bad on thi onpt, hronologial phnomnon i prditd in thi tudy ar dtrioration frot damag fatigu. To imulat mhanial haratriti damagd onrt frot-damag, zro trngth lmnt onpt moal plati tnil train ar introdud in RBSM pring onidr xprimntally obrvd raking plati dformation aud frot damag. For fatigu analy, tim-dpndnt modl hih inlud lati, vio-lati, vio-plati raking omponnt a dvlopd bad on tati modl n introdud in RBSM. 2 STATIC SIMULATION OF CONCRETE MATERIAL Th RBSM dvlopd Kaai (1978) i on dirt numrial analyi mthod. Analytial modl i dividd in polyhdron lmnt ho fa ar intronntd pring. Eah lmnt ha tranitional on rotational dgr frdom tho ar givn at om point inid lmnt. Th point ork a a nod ontrut a pring nrk. Normal har pring ar plad on fa (Fig. 1). Sin rak initiat propagat along boundary fa, mh arrangmnt may afft fratur dirtion. To avoid formation rak ith a rtain dirtion, a rom gomtry i introdud uing a Voronoi

2 diagram. Th Voronoi diagram i olltion Voronoi ll. Eah ll rprnt mortar or aggrgat lmnt in analyi. Contitutiv modl givn pring btn lmnt ar xplaind in mainly our prviou rarh (Nagai t al. 2004). Th pring in omprion zon alay at latially nvr ho brakag nor tning bhavior. Aftr it rah tnil trngth, tning bhavior govrnd rak idth i givn. For har pring, an la plati modl i applid. It man omprion failur pimn in maro al i prntd mo al tnil har failur btn lmnt(fig. 2). Spring btn aggrgat lmnt bhav latially only o a not hav fratur in thi tudy. Simulation failur onrt undr uniaxial omprion tnion ondition ar ondutd hr hap oar aggrgat in onrt i irular. Th analyzd pimn ( mm) hr numbr lmnt i 3,619 inluding 1,619 lmnt aggrgat i hon in Figur 3. Avrag lmnt iz i 2.60 mm 2. Aggrgat volum in pimn i 38%. In omprion tt, typ ondition ar imulatd hr p botm boundari pimn ar fixd not fixd in latral dirtion. In tnion tt, boundari in latral dirtion ar not fixd. Th prditd tr-train rlationhip dformation at failur ar prntd in Figur 3 4. In omprion tt, natur prditd urv tr-train in axial latral dirtion ar imilar xprimntal rult mntiond Sk t al. (1979). Th rult imulation hr boundary in latral dirtion i not fixd i prntd in graph, in hih a light rdution in maroopi trngth du limination frition on boundary i obrvd in analyi, imilarly xprimnt. Sin boundary pimn annot rtrit xpanion pimn in latral dirtion, pimn fail jut aftr rapid inraing latral train.dformation pimn at failur ho diffrnt maro rakingpattrn du boundary ondition in omprion a obrvd in uual xprimnt(fig. 3). In tnil analyi, imulatd tr-train urv dformation at failur, in hih loalizd rak i prditd, ar imilar obrvd in uual xprimnt. Th rlationhip btn ompriv tnil trngth onrt i xamind. Maroopi rpon imulation hang du loation aggrgat mh arrangmnt o uniaxial omprion tnion tt 10 pimn hr loation aggrgat lmnt mhing ar diffrnt r arrid out for thr targt trngth. Figur 5 ho obtaind trngth rlationhip ith propod Equation rlationhip JSCE (2002) hr analyi rprodu rlationhip fairly. Bad on analyi J D ( h, in T ) thi h haptr, dtrioration frot-damag fatigu modl ar intalld imulation Th proportionality i arrid out. offiint D(h,T) moitur prmability it i a nonlina rlativ humidity h tmpratur & Najjar 1972). Th moitur ma balan variation in tim atr ma volum onrt (atr ontnt ) b q divrgn moitur flux J t J Th atr ontnt an b xprd a Figur 1. Mhanial modl vaporabl RBSM. atr (apillary a vapor, adorbd atr) non- (hmially bound) atr n (Mil Pantazopoulo & Mill 1995). It i ra aum vaporabl atr i a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norling Mjonll 1997). Undr thi aum ubtituting Equation 1 in Equati obtain Figur 2. Modl pring. h h t h & + & + hr /h i lop orption/ iorm (alo alld moitur apa govrning quation (Equation 3) mut b appropriat boundary initial onditi Th rlation btn amount atr rlativ humidity i alld iorm if maurd ith inraing humidity dorption iorm in th a. Nglting ir diffrn (Xi t al. folloing, orption iorm ill b rfrn both orption dorption By ay, if hytri iorm ould b takn in aount, rlation, vaporabl atr v rlativ humi b ud aording ign varia rlativity humidity. Th hap iorm for HPC i inflund many p pially tho influn xtnt Figur 3. Conrt modl failur pattrn. hmial ration, in turn, dtrm trutur por iz ditribution (atrratio, mnt hmial ompoition, SF uring tim mthod, tmpratur, mix t.). In litratur variou formulatio found drib orption iorm onrt (Xi t al. 1994). Hovr, in th papr mi-mpirial xprion pro Figur 4. Etimatd tr-tain Norling urv. Mjornll (1997) i adoptd b Proding FraMCoS-7, May 23-28, 2010

3 J ) D ( h, T h Figur 5. Compriv-tnil trngth rlationhip. (1) Th proportionality offiint D(h,T) i alld moitur prmability it i a nonlinar funtion rlativ humidity h tmpratur T (Bažant & Najjar 1972). Th moitur ma balan rquir variation in tim atr ma pr unit volum onrt (atr ontnt ) b qual divrgn moitur flux J J (2) t 3 FROST DAMAGE INFLUENCE ON MECHANICAL Th atr ontnt PROPERTIES an b xprd OF CONCRETE a um vaporabl atr (apillary atr, atr Frot vapor, damag adorbd i on atr) typ nvironmntal non-vaporabl dtrioration (hmially our bound) in atr old rgion. Typial frot n (Mill 1966, damag Pantazopoulo onit & Mill urfa 1995). aling It i raonabl intrnal miroraking aum aud vaporabl xpanion atr i a funtion matrial (i.., rlativ mortar). humidity, Hr h, lattr dgr on i hydration, tratd. Moal, ontitutiv dgr ilia modl fum ration, frot-damagd onrt ar, i.. (h,, ) dvlopd ag-dpndnt in thi tudy orption/dorption through numrial iorm imulation (Norling uing Mjonll a -dimnional 1997). Undr RBSM. thi aumption Th aim imulation ubtituting i Equation prdit 1 in maro Equation bhavior 2 on frot-damagd obtain onrt ubjtd mhanial. Zro trngth lmnt onpt moal plati tnil train ar introdud in h RBSM pring onidr xprimntally h t obrvd h raking plati dformation & + & + & n (3) aud frot damag. To introdu frot damag fft, modifyation original modl i arrid out. Firtly, hr /h i lop orption/dorption iorm (alo alld moitur apaity). Th pring i aumd hav laplati govrning quation (Equation 3) mut b ompltd fratur bhavior. A a rult, un appropriat boundary initial ondition. r path do not pa through origin, but Th rlation btn amount vaporabl ha om platiity. Th platiity inra atr rlativ humidity i alld adorption tiffn dra a train at un iorm if maurd ith inraing rlativity point inra. To modl thi phnomnon, a linar humidity dorption iorm in oppoit un-r path follo nvlop a. Nglting ir diffrn (Xi t al. 1994), in tr-train urv in omprion at a train ε pa i folloing, orption iorm ill b ud ith adoptd, a hon in Figur 6. Sondly, frot rfrn both orption dorption ondition. ation au intrnal miroraking, hih lad By ay, if hytri moitur xpanion onrt. Thi xpanion an b iorm ould b takn in aount, diffrnt rprntd rmaining plati tnil train rlation, vaporabl atr v rlativ humidity, mut onrt ha bn ubjtd frot ation aftr b ud aording ign variation any rlativity mhanial humidity. Th ha hap bn rmovd. orption At moal, iorm for HPC rmaining i inflund tnil train many du paramtr, frot ation pially an tho b imulatd influn longation xtnt rat pring. hmial In ration thi papr,, thi in rmaining turn, dtrmin plati tnil por train trutur por iz pring ditribution i rfrrd (atr--mnt a moal plati ratio, mnt tnil train hmial ε pf. Du ompoition, rom SF ontnt, natur uring miroraking tim mthod, aud tmpratur, frot mix ation, additiv, moal t.). In plati litratur tnil variou train formulation ar ditributd an b romly found drib among orption pring. iorm In thi tudy, a trunatd onrt (Xi t al. ditribution 1994). Hovr, moal in prnt plati tnil papr train mi-mpirial i dribd a xprion hon in propod Figur 7. If Norling moal Mjornll plati (1997) tnil i train adoptd xd bau train it orrponding maximum rak idth, xpliitly pring aount annot for arry volution tr hydration any mor. ration Bid, SF ontnt. har pring Thi i unabl orption iorm bar har rad tr, ir. Elmnt ar no longr abl arry tr ar alld zro trngth lmnt (ZSE). ZSE rprnt lmnt hav bom tally fraturd du frot damag, 1 hih i ( h,, ) G (, ) 1 + dfind a funtion 1 moal plati tnil 10( g ) h train ( Fig. 7).Th dtail modifiation 1 (4) ar dribd in rfrn (Uda t al. 2009). 10( g ) h K (, ) hr firt trm (gl iorm) rprnt phyially bound (adorbd) atr ond trm (apillary iorm) rprnt apillary atr. Thi xprion i valid only for lo ontnt SF. Th offiint G 1 rprnt amount atr pr unit volum hld in gl por at 100% rlativ humidity, it an b xprd (Norling Mjornll 1997) a Figur 6. Contitutiv modl for pring frotdamagd onrt. G (, ) k + k 1 vg vg g G 1 Figur 7. Ditribution 0 moal 1 plati tnil train (ε pf ) dfinition zro trngth lmnt. (6) K (, ) 1 g 1 1 Comprion analyi a ondutd on fiv pimn ith diffrnt lvl avrag plati Th matrial paramtr k vg k vg g tnil train aumd aud diffrnt dgr 1 an b alibratd fitting xprimntal data rlvant frot damag, mathing tho maurd in a fr (vaporabl) atr ontnt in onrt at prviou xprimntal tudy (Haan t al. 2004). Th variou ag (Di Luzio & Cuati 2009b). plati tnil train givn ah pimn ar ditributd from zro a valu ti avrag plati 2.2 Tmpratur tnil train, volution or 0, 340, 700, 1,240 1,783 µ for fiv pimn, rptivly. Th modl Not, at arly ag, in hmial ration analyzd hav dimnion mm.str-train aoiatd ith mnt hydration SF ration urv analytial modl ar ompard ith ar xormi, tmpratur fild i not uniform xprimntal data from Haan xprimntal tudy in for non-adiabati ytm vn if nvironmntal Figur 8 hr tr i izd trngth tmpratur i ontant. Hat ondution an b non-damagd onrt rult. On anding dribd in onrt, at lat for tmpratur not branh, r i omparativly good orrlation xding 100 C (Bažant & Kaplan 1996), btn analytial xprimntal rult in Fourir la, hih rad tiffn trngth rdution du frot-damag. Pur tnil analyi a ondutd on am q λ T pimn a ud in omprion analyi. Figur (7) 9 ho obtaind tr-train urv. Although r hr a q no i xprimntal hat flux, data T ompar, i it abolut i lar tmpratur, tnil trngth λ i hat tiffn ondutivity; dra in ith thi inraing frot damag. To ompar rlationhip (5) hr k vg k vg ar matrial paramtr. From maximum amount atr pr unit volum an fill all por (both apillary por gl por), on an alulat K 1 a on obtain Proding FraMCoS-7, May 23-28, 2010

4 btn lo ompriv tnil trngth in frot-damagd onrt, analyi pimn ith moal plati tnil train 50, 100, 500, 1,000, 2,500µ a alo arrid out in addition analyi dribd abov. Figur 10 prnt rult omputation prviou xprimntal rult (Haan t al. 2002a,b, 2003, 2004,Matumura t al. 2003). Th good orrlation furr prov validity thi moal modl for prditing maroal bhavior frot-damagd onrt. Figur 8. Computd xprimntal tr-train urv frot damagd onrt in omprion. Figur 9. Computd tr-train urv frot damagd onrt in tnion. Figur 10. Strngth lo ratio rlationhip frot damag. Bnding analyi J on D ( nothd h, T ) h bam pimn frot-damagd onrt a arrid out for thr a moal plati tnil Th proportionality train: 0 (undamagd) offiint 340µ D(h,T) (modrat damag), moitur 2,000µ prmability (riou damag). it i a nonlina Th valu moal rlativ plati humidity tnil h train tmpratur r hon giv & Najjar am 1972). rlativ Th dynami moitur lati ma balan modulu a in xprimntal variation tudi in tim arrid out atr ma Haan t al. (2002b) volum Haan onrt (2003). (atr All pimn ontnt ) b q hav dimnion divrgn mm ith moitur a noth flux 50 J mm dp at ntr. Eah pimn i dividd in 5,200 polyhdron lmnt, a hon in Figur 11. Th pimn i upportd J impl upport t loadd at ntr undr diplamnt ontrol. Th load dfltion urv Th obtaind atr ontnt in thi analyi an b xprd ar a ompard ith tho obtaind vaporabl in atr xprimntal (apillary a tudi (Haan t vapor, al. 2002b, 2003) adorbd in Figur atr) 12. Th non- load on both xprimntal (hmially bound) analytial atr rult ar n (Mil izd ir Pantazopoulo orrponding & maximum Mill 1995). valu It at i ra pak point. aum It i obrvd vaporabl r i atr good i a fu agrmnt rlativ izd humidity, load-dfltion h, dgr urv hydration btn analytial dgr xprimntal ilia fum rult ration, xpt in, i.. tning branh ag-dpndnt pimn ith orption/dorption εpf2,000 µ, onfirming appliability (Norling Mjonll propod 1997). Undr modl thi for aum prditing obrvd ubtituting xprimntal Equation trnd. 1 in Th Equati analyi ll imulat obtain lo both arrying apaity tiffn du frot damag, a found in xprimnt. Maroraking initiat at noth hn maximum load i rahd. h Aftr rak tart, & + & + load gradually dra h tith inraing h rak idth rak propagat p fa pimn a hon in Figur 13. hr /h i lop orption/ iorm (alo alld moitur apa govrning quation (Equation 3) mut b appropriat boundary initial onditi Th rlation btn amount atr rlativ humidity i alld iorm if maurd ith inraing humidity dorption iorm in th a. Nglting ir diffrn (Xi t al. Figur 11. Bnding tt modl. folloing, orption iorm ill b rfrn both orption dorption By ay, if hytri iorm ould b takn in aount, rlation, vaporabl atr v rlativ humi b ud aording ign varia rlativity humidity. Th hap iorm for HPC i inflund many p pially tho influn xtnt hmial ration, in turn, dtrm trutur por iz ditribution (atrratio, mnt hmial ompoition, SF uring tim mthod, tmpratur, mix t.). In litratur variou formulatio found drib orption iorm onrt (Xi t al. 1994). Hovr, in th papr mi-mpirial xprion pro Norling Mjornll (1997) i adoptd b Figur 12. Load-dfltion urv bnding tt. Proding FraMCoS-7, May 23-28, 2010

5 J ) D ( h, T h (1) Th proportionality offiint D(h,T) i alld moitur prmability it i a nonlinar funtion rlativ humidity h tmpratur T (Bažant & Najjar 1972). Th moitur ma balan rquir Figur 13. Failur raking pattrn (ε pf 340µ at dfltion 1.5 mm). variation in tim atr ma pr unit volum onrt (atr ontnt ) b qual divrgn moitur flux J 4 FATIGUE SIMULATION OF MORTAR J (2) Fatigu t imulation mploy a mthodology am a tati analyi xpt ontitutiv la. Th timdpndnt Th atr ontitutiv ontnt modl an b xprd ud in a fatigu um analy vaporabl ar hmatially atr dran (apillary in Figur atr, 14. Bai atr onpt vapor, adorbd thi modl atr) i tal dformation non-vaporabl mortar (hmially i dividd bound) in lati, atr vio-lati, n (Mill vio- 1966, plati Pantazopoulo raking & Mill omponnt. 1995). It Eah i raonabl omponnt orrpond aum matrial vaporabl information atr i at a funtion inid mortar rlativ a humidity, hon in h, dgr figur. hydration, Thrfor, tnion, tning dgr ilia har fum tranfr ration, modl, i.. ar introdud (h,, ) in ag-dpndnt raking omponnt orption/dorption in iorm har dirtion, (Norling Mjonll rptivly. 1997). Matrial Undr thi ontant aumption ud in modl ubtituting ar dtrmind Equation from 1 in mix Equation proportion 2 on uring obtain priod mortar. Firt, matrial proprti uh a ompriv trngth, lati modulu rp offiint in maro lvl ar alulatd from h input. + Sond, ( D h ) matrial & + proprti & + & in maro n (3) h t h lvl ar onvrtd tho in mo lvl uing probability dnity funtion. Lat, modl ontant ar hr givn /h bad i on lop bai ory orption/dorption RBSM /or rult iorm paramtri (alo alld analy. moitur For dtail, apaity). Th author govrning prviou quation rarh (Equation (Matumo 3) mut b t al. ompltd 2008). appropriat boundary initial ondition. Th rlation btn amount vaporabl atr rlativ humidity i alld adorption iorm if maurd ith inraing rlativity humidity dorption iorm in oppoit a. Nglting ir diffrn (Xi t al. 1994), in folloing, orption iorm ill b ud ith rfrn both orption dorption ondition. By ay, if hytri moitur iorm ould b takn in aount, diffrnt rlation, vaporabl atr v rlativ humidity, mut b ud aording ign variation rlativity humidity. Th hap orption Figur 14. Tim-dpndnt ontitutiv modl for fatigu analy iorm for it HPC orrpondn i inflund mirotrutural many paramtr, fatur. pially tho influn xtnt rat hmial Figur ration 15 ho, analytial in turn, pimn dtrmin ud por in fatigu trutur analy. por Thi iz ditribution pimn do (atr--mnt not ontain oar ratio, mnt aggrgat hmial (mortar). ompoition, Spimn iz SF i ontnt, 75 mm x150 uring mm tim onit mthod, tmpratur, 1,800 Voronoi mix ll. additiv, Load i t.). applid In on litratur p urfa variou diplamnt formulation or an load b ontrolld found drib ondition. In orption thi ri, iorm latral dirtion onrt p (Xi urfa t al. i 1994). fixd. Hovr, Tabl 1 lit in analytial prnt a. papr Totally mi-mpirial 8 a r xprion ondutd propod in uni-axial omprion Norling Mjornll tnion. (1997) To i adoptd m bau ar undr it mononi xpliitly aount for obtain volution tati ompriv hydration tnil ration trngth. SF Uing ontnt. trngth, Thi orption applid iorm tr lvl rad in fatigu analyi r givn. In a, omputation a arrid out 3 tim for ah analytial a ith hanging hap lmnt variation matrial proprti 1 in mo lvl ( h,, ) G (, ) 1 + xamin variation 1 analytial rult. A a 10( g ) h rult mononi analy, ompriv 1 (4) trngth f m MPa tnil trngth f t 3.41 MPa r obtaind. To 10 dtrmin ( g failur ) h undr K (, ) 1 1 fatigu, 1a mthod dvlopd author (Matumo t al. 2008) a ud. hr firt trm (gl iorm) rprnt phyially bound (adorbd) atr ond trm (apillary iorm) rprnt apillary atr. Thi xprion i valid only for lo ontnt SF. Th offiint G 1 rprnt amount atr pr unit volum hld in gl por at 100% rlativ humidity, it an b xprd (Norling Mjornll 1997) a Figur G ( 15., Mortar ) k modl + ud k in 1 fatigu analy. vg vg (5) Tabl hr 1. kanalytial a for fatigu. vg k vg ar matrial paramtr. From maximum amount atr Loading pr ondition unit volum an fill Min No. all por (both apillary por Max gl tr por), on Typ Dirtion tr an alulat K 1 a on obtain ratio ratio CM Mononi - CF Comprion g CF80 Fatigu G CF (6) K TM (, ) Mononi - 1 TF80 g 0.8 Tnion 1 TF75 Fatigu TF Th matrial paramtr k vg k vg g 1 an b alibratd fitting xprimntal data rlvant fr (vaporabl) atr ontnt in onrt at variou ag (Di Luzio & Cuati 2009b). 2.2 Tmpratur volution Not, at arly ag, in hmial ration aoiatd ith mnt hydration SF ration ar xormi, tmpratur fild i not uniform Figur for non-adiabati 16. Str-train ytm rlationhip vn obtaind if in nvironmntal CF90. tmpratur i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not xding 100 C (Bažant & Kaplan 1996), Fourir la, hih rad q λ T (7) hr q i hat flux, T i abolut tmpratur, λ i hat ondutivity; in thi Figur 17. Str-train rlationhip obtaind in TF80. Proding FraMCoS-7, May 23-28, 2010

6 Figur 18. S-N urv in omprion. Figur 19. S-N urv in tnion. Figur 20. Crak pattrn in mononi fatigu analy in omprion (Dformation i multiplid 10 tim). Figur 16 J17 D ( ho h, T ) xampl h tr-train rlationhip obtaind in fatigu omprion tnion. With inra Th proportionality numbr offiint yl, D(h,T) lop hap moitur un-r prmability it i urv a nonlina bom gntlr onvx rlativ donard humidity a h obrvd tmpratur in prviou tudi & Najjar (Hatano 1972). t al. Th 1962, moitur Muguruma ma balan t al. 1970, Matuita t al. variation 1979). in tim atr ma Figur 18 volum 19 ho onrt rlationhip (atr ontnt btn ) b q applid tr ratio divrgn numbr moitur yl flux J up failur (S-N urv) in omprion tnion, rptivly. Th figur inlud xprimntal rult Morri t al. (1981) a J ll. Thy ould b t imulatd ll uing analytial modl propod in thi tudy. Th atr ontnt an b xprd a Figur 20 ho rak vaporabl pattrn atr at failur (apillary a obtaind in mononi vapor, fatigu adorbd atr) analy non- in omprion. Amount (hmially rak bound) at failur atr undr n (Mil mononi Pantazopoulo i l than & Mill undr 1995). fatigu It i ra bom aum gratr ith vaporabl mallr applid atr i a fu tr ratio. Thi rlativ i drivd humidity, from h, dgr mthod for hydration dtrmining failur dgr dvlopd ilia fum ration, author, i.. (Matumo t al. 2008). ag-dpndnt That i, orption/dorption in failur happn hn (Norling matrial Mjonll trngth 1997). bom Undr lor thi aum than applid tr, ubtituting trngth rdution Equation i 1 mor in Equati prominnt ith mallr obtain applid tr. Uing rult in CF90, fratur mhanim mortar undr fatigu a invtigatd. Point A, B C hon in Figur h 20 ar loatd hr & + & + maro rak propagatd. h Figur t 21 h ho tr-train bhavior onnt pring loatd at point A, B C. Th indiar for 1t load yl ar hr /h i lop orption/ pointing a loation f plot. Thi man iorm (alo alld moitur apa tr 1t yl ar in omprion. At govrning quation (Equation 3) mut b initial, pring appropriat at boundary point A ha initial bn onditi tnd alrady. Aftr Th rlation, tr btn at point amount A i rlad du atr un-r rlativ humidity pro. i To alld atify for quilibrium, iorm if maurd rlad ith tr inraing i rditributd humidity or ara. At dorption 5th iorm yl, in th pring at point a. B tartd Nglting b ir tnd diffrn bau (Xi t al. tr rditribution. folloing, orption Str rla iorm ill b rditribution from rfrn point both B promotd orption tning dorption or ara, By n ay, pring if at hytri point C tartd b tnd iorm at 23rd ould b takn yl. in Finally, aount, load arrying apaity rlation, vaporabl pimn atr bam v rlativ l humi than applid b for ud aording failur happnd. ign Thi varia bhavior orrpond rlativity fratur humidity. pro Th hap atual mortar a follo. iorm for HPC i inflund many p [1] At initial, pially miro tho rak influn happn at xtnt ak portion hmial matrial. ration, in turn, dtrm [2] During un-r trutur por pro, iz ditribution tr i (atrratio, rlad from miro mnt rak. hmial ompoition, SF [3] Th rlad uring tr tim i rditributd mthod, tmpratur, or mix ara. It indu t.). a n In rak litratur formation. variou formulatio [4] Rpating found abov pro drib [1], [2] orption [3], iorm rak dvlop onrt propagat (Xi t al. ovr 1994). tim. Hovr, in th [5] Whn rak papr gro mi-mpirial biggr xprion load pro arrying apaity Norling Mjornll matrial (1997) bom i l adoptd b Proding FraMCoS-7, May 23-28, 2010

7 J than D ( h, T happlid for, failur mortar (1) happn. ) Th proportionality offiint D(h,T) i alld moitur prmability it i a nonlinar funtion rlativ humidity h tmpratur T (Bažant & Najjar 1972). Th moitur ma balan rquir variation in tim atr ma pr unit volum onrt (atr ontnt ) b qual divrgn moitur flux J t J (2) Th atr ontnt an b xprd a um vaporabl atr (apillary atr, atr vapor, adorbd atr) non-vaporabl (hmially bound) atr n (Mill 1966, Pantazopoulo & Mill 1995). It i raonabl aum vaporabl atr i a funtion rlativ humidity, h, dgr hydration,, dgr ilia fum ration,, i.. (h,, ) ag-dpndnt orption/dorption iorm (Norling Mjonll 1997). Undr thi aumption ubtituting Equation 1 in Equation 2 on obtain h + h t ( D h) n (3) h & + & + hr /h i lop orption/dorption Figur 21. Str-train bhavior onntd pring loatd iorm (alo alld moitur apaity). Th at point A, B C. govrning quation (Equation 3) mut b ompltd appropriat boundary initial ondition. 5 CONCLUSIONS Th rlation btn amount vaporabl atr rlativ humidity i alld adorption Th iorm folloing if maurd onluion ith r inraing obtaind rlativity in thi tudy. humidity dorption iorm in oppoit 1) a. Numrial Nglting imulation ir diffrn failur (Xi t al. 1994), onrt in ubjtd folloing, orption uni-axial tati iorm ill a b ondutd ud ith rfrn RBSM. both Calulatd orption tr-train dorption urv ondition. ho a By imilar ay, hap if hytri in uual xprimntal moitur rult. iorm In omprion, ould b takn prditd in aount, tr-train urv diffrnt rlation, hang vaporabl in Poion atr ratio v rlativ ar imilar humidity, tho mut in b ud xprimnt. aording Diffrnt ign rak pattrn variation du rlativity diffrnt humidity. Th boundary hap ondition orption an b iorm imulatd for HPC raonably i inflund ll. In addition, many paramtr, analyi pially prdit tho ompriv influn xtnt tnil rat trngth hmial rlationhip ration onrt, fairly. in turn, dtrmin por trutur 2) It hould b por notd iz ditribution analy (atr--mnt in thi tudy ratio, ar mnt in dimnion hmial rfor ompoition, thr dimnional SF ontnt, uring fratur tim propagation mthod, tmpratur, along aggrgat mix additiv, urfa t.). an In not b litratur imulatd. variou For formulation pri an rprodution b found drib failur orption pro iorm htrognou onrt matrial, (Xi t al. thr 1994). dimnional Hovr, in analyi prnt i papr nary. mi-mpirial xprion propod Norling Mjornll (1997) i adoptd bau it & 3) xpliitly Simulation aount failur for frot-damagd volution hydration onrt ration a alo ondutd SF ontnt. uing Thi RBSM. orption Str-train iorm rad urv obtaind in analy ar imilar tho obtaind in xprimnt. Th analyi an imulat ll diffrn in dgradation trngth tiffn in tnion 1 omprion. ( h,, ) G (, ) 1 + Th rlationhip 1 btn rdution 10( g ) h ompriv tiffn trngth, rdution 1 (4) tnil trngth tiffn, rdution tnil ompriv 10( g trngth ) h obtaind in K (, ) 1 1 analyi 1agr ll ith tho obtaind in xprimnt. Load-dfltion urv bnding tt on nothd-bam pimn undamagd hr frot-damagd firt trm (gl onrt iorm) obtaind rprnt in phyially analyi bound ar almot (adorbd) imilar ith atr tho obtaind ond in trm (apillary xprimnt. iorm) Th dutility rprnt pimn apillary atr. ith Thi frot-damag xprion i i highr valid only than for lo ithout ontnt damag. SF. Th offiint G 1 rprnt amount 4) atr Simulation pr unit volum failur hld in mortar gl por undr at fatigu 100% rlativ humidity, a alo ondutd it an b xprd RBSM (Norling ith Mjornll introduing 1997) a tim-dpndnt ontitutiv modl a mthod for dtrmining failur dvlopd author. Str-train urv obtaind in G (, ) k + k (5) 1analy agr vg ll vg in trm tiffn rdution tranition un-r urv. In hr addition, k S-N rlationhip obtaind in vg k vg ar matrial paramtr. From maximum analyi amount r ompard atr pr ith unit volum xprimntal an fill rult. all por Th (both analyi apillary an por imulat gl ll por), fatigu on an liv alulat mortar K 1 a on in obtain omprion tnion. Amount rak at failur undr mononi bom largr than tho undr fatigu g. Thi i bau trngth rdution G matrial undr fatigu bom mor (6) K ( prominnt, ) than undr mononi. 1 Furrmor, fratur gmhanim 1 mortar undr 1 fatigu a invtigatd bad on tr-train bhavior onntd pring. Str Th matrial paramtr k vg k vg g 1 an i rlad around initial miro rak, lading b alibratd fitting xprimntal data rlvant tr rditribution anor ara. A a rult, fr (vaporabl) atr ontnt in onrt at rak propagat ovr tim. Thi pro variou ag (Di Luzio & Cuati 2009b). our for many miro rak finally au global failur. 2.2 Tmpratur volution Not, at arly ag, in hmial ration RERERENCES aoiatd ith mnt hydration SF ration ar xormi, tmpratur fild i not uniform Bolr, J.E. & Sai, S Dirt modling hortfor fibr non-adiabati rinformnt ytm in mntitiou vn if ompoit. nvironmntal Advand tmpratur Cmnt Bad i Matrial ontant. 6: Hat ondution an b Bolr, dribd J.E. in & onrt, Sai, S at lat Fratur for analyi tmpratur uing pring not xding nrk ith 100 C rom (Bažant gomtry. & Kaplan Enginring 1996), Fratur Fourir Mhani la, 61: hih rad Haan, M Modling tr-train rlationhip onrt damagd frzing thaing yl. Ph.D. q i, λ THokkaido Univ., Sapporo, Japan. (7) Haan, M., Nagai, K., Sa, Y., & Uda, T. 2002a. Str-train bhavior in tnion omprion onrt damagd hr q i hat flux, T i abolut frzing thaing yl. Pro., 2nd Int. Workhop tmpratur, on Frot Ritan λ i Conrt, hat ondutivity; M. J. Stzr in H.-J. thi Kk, d. RILEM: Proding FraMCoS-7, May 23-28, 2010

8 Haan, M., Nagai, K., Sa, Y., & Uda, T. 2002b. Tnil tr rak idth modl for plain onrt damagd frzing thaing ation. Pro. Japan Conrt Intitut 24(2): Haan, M., Okuyama, H., & Uda, T Th damag mhanim train indud in frot yl onrt. Pro. Japan Conrt Intitut 25(1): Haan, M., Okuyama, H., Sa, Y., & Uda, T Str-train modl onrt damagd frzing thaing yl. Journal Advand Conrt Thnology 2(1): Hatano, T Bhavior onrt yli omprion load. Proding JSCE 84: (in Japan) JSCE Stard pifiation for onrt trutur, Strutural prforman vrifiation. Tokyo: JSCE. Kaai, T N dirt modl ir appliation imi rpon analyi trutur. Nular Enginring Dign 48: Matumo, K., Sa, Y., Uda, T. Wang, L Moopi Analyi Mortar undr High-Str Crp Lo-Cyl Fatigu Loading. Journal Advand Conrt Thnology 6(2): Matumura, T., Katura, O. & Yohino, T Proprti frot damagd onrt timation dgr frot damag. J. Strut. Contr. Eng. 563: (in Japan) Matuhita, H. & Makizumi, J DT. ( h1979., T ) hdformational haratriti onrt undr rpatd omprion tr. Proding Japan Conrt Intitut 1: (in Japan) Morri, A. D. & Garrtt, Th G. G. proportionality A omparativ offiint tudy D(h,T) tati fatigu moitur bhaviour prmability plain tl it i fibr a nonlina rinford mortar in omprion rlativ humidity dirt h tnion, tmpratur Th Intrnational Journal & Najjar 1972). Cmnt Th Compoit moitur ma balan Lightight Conrt 3(2): variation in tim atr ma Muguruma, H. & Tominaga, M Str-train rlation onrt undr rpatd volum ovr-load. onrt Journal (atr ontnt Soity ) b q Matril Sin, divrgn Japan 19(200): moitur (in Japan) flux J Nagai, K., Sa, Y. & Uda, T Moopi imulation failur mortar onrt 2D RBSM. Journal Advand Conrt Thnology J 2(3): Sk A.F., Hannant, D.J. t & William, R.I.T Th fft aggrgat onntration upon trngth modulu latiity onrt. Th atr Magazin ontnt Conrt an Rarh b xprd a 31(109): vaporabl atr (apillary a Uda, T., Haan, M., vapor, Nagai, K., adorbd Sa, Y. & atr) Wang, L non- Moal imulation influn frot damag on (hmially bound) atr mhanial proprti onrt. Journal Matrial in n (Mil Civil Enginring Pantazopoulo 21(6): & Mill 1995). It i ra aum vaporabl atr i a fu rlativ humidity, h, dgr hydration dgr ilia fum ration,, i.. ag-dpndnt orption/dorption (Norling Mjonll 1997). Undr thi aum ubtituting Equation 1 in Equati obtain h h t h & + & + hr /h i lop orption/ iorm (alo alld moitur apa govrning quation (Equation 3) mut b appropriat boundary initial onditi Th rlation btn amount atr rlativ humidity i alld iorm if maurd ith inraing humidity dorption iorm in th a. Nglting ir diffrn (Xi t al. folloing, orption iorm ill b rfrn both orption dorption By ay, if hytri iorm ould b takn in aount, rlation, vaporabl atr v rlativ humi b ud aording ign varia rlativity humidity. Th hap iorm for HPC i inflund many p pially tho influn xtnt hmial ration, in turn, dtrm trutur por iz ditribution (atrratio, mnt hmial ompoition, SF uring tim mthod, tmpratur, mix t.). In litratur variou formulatio found drib orption iorm onrt (Xi t al. 1994). Hovr, in th papr mi-mpirial xprion pro Norling Mjornll (1997) i adoptd b Proding FraMCoS-7, May 23-28, 2010