An analytical study on the stress-strain relation of PVA-ECC under tensile fatigue

Transcription

1 Fratur Mhani Conrt Conrt Strutur - High Prforman, Fibr Rinford Conrt, Spial Loading Strutural Appliation- B. H. Oh, t al. (d) Kora Conrt Intitut, ISBN An analytial tudy on th tr-train rlation PVA-ECC undr tnil fatigu K. Kakuma, T. Matumoto. T. Hayahikawa & X. H Hokkaido Univrity, Sapporo, Japan ABSTRACT: Thi tudy propo th tr-train rlationhip Enginrd Cmntitiou Compoit rinford with polyvinyl alohol fibr (PVA-ECC) undr tnil fatigu. Th mhanim fatigu dgradation ECC i bridging tr dgradation on rak plan, th dgradation i modld in bridging law by introduing th hang miromhanial paramtr. In thi tudy, fibr fatigu ruptur i rgardd a a dgradation fator, th ritrion fibr fatigu ruptur dpndnt on numbr yl i applid. From th alulatd bridging tr-rak opning diplamnt rlationhip, it i hown that multipl raking ritrion an b atifid up to a rtain numbr fatigu yl. In ordr to obtain th tr-train rlationhip, th bridging tr-rak opning diplamnt rlationhip i introdud into finit lmnt analyi a dirt rak modl. Th timatd volution tr dgradation agr wll with th volution obtaind from uniaxial tnil fatigu tt PVA-ECC, howing th validity th urrnt tudy. INTRODUCTION Enginrd Cmntitiou Compoit (ECC) i on kind Dutil Fibr Rinford Cmntitiou Compoit, whih how pudo train hardning bhavior undr uniaxial tnion, ha high tnil train apaity. ECC alo how th fft rak opning diplamnt ontrol a th valu i dfind to b l than.mm (JSCE 7). In addition, th rdution fatigu trngth i mallr, th fatigu durability i highr than normal onrt or onvntional fibr rinford mntitiou ompoit, whih ar train tning typ matrial, du to multipl fin rak. From th haratriti, ECC i xptd to b applid to infratrutur a a rpair or rinformnt matrial uh a ovrlay or undrlay bridg lab, in whih fatigu dgradation aud by rpatd traffi load bom a ignifiant problm. In th matrial dign ECC, miromhani fratur mhani an xplain th tati proprty (Li 993). It i poibl to optimiz th matrial dign by a paramtri tudy. Fatigu dign matrial or trutur, howvr, i onidrd bad on tati load arrying apaity. Rntly, om tudi on fatigu hav bn prformd, th fatigu proprty ECC or trutur with ECC, uh a th mhanim fatigu dgradation ECC th fft rinformnt with ECC, ha bn rvald. Flxural fatigu tt ECC bam (Matumoto t al. 3) a whl truking tt ECCtl ompoit dk (Mitamura t al. 6) ar xampl thm. Howvr, mot tudi on fatigu ar bad on xprimnt, a fatigu modl ECC i not uffiintly dvlopd. Th mhanim fatigu dgradation ECC i th dgradation tr tranfrrd by fibr on rak plan. From th point dvlopmnt fatigu dign mthod, th propoition fatigu modl ECC onidring th dgradation mhanim fatigu lif prdition mthod ar nary. Thrfor, thi tudy dvlop th tr-train rlationhip ECC undr tnil fatigu. Th onpt fibr bridging tr dgradation i bad on miromhani, th hang miromhanial paramtr du to fatigu ar introdud. Th trtrain rlationhip i obtaind by applying th bridging tr dgradation modl to finit lmnt analyi. From th drivd rlationhip, th volution tr dgradation i diud by omparing with uniaxial fatigu tnion tt polyvinyl alohol fibr rinford ECC (PVA-ECC) ondutd by Matumoto t al.. BRIDGING STRESS DEGRADATION ECC i dignd bad on miromhani, whih i th mhanial modl formulating th bhavior th omponnt ompoit. Bridging law, th rlationhip btwn bridging tr rak opning diplamnt, i an important rlationhip, it an timat th tnil proprty ECC uh a tnil trngth, fratur nrgy, raking tat, t. Fatigu dgradation ECC i aud by th dgradation

2 bridging J = D ( h, tr, T ) h it alo an b timatd in bridging law. In thi haptr, bridging tr dgradation () i modld Th proportionality bad on bridging fiint law. D(h,T) i alld moitur prmability it i a nonlinar funtion. th Bridging rlativ law humidity h tmpratur T (Bažant & Najjar 97). Th moitur ma balan rquir Bridging that th variation law fibr in tim rinford th watr mntitiou ma pr ompoit volum i obtaind onrt (watr from miromhanial ontnt w) b qual param- to th unit tr divrgn about proprti th moitur fibr, flux J matrix fibrmatrix intrfa. Th typial rlationhip bfor prpak i hown in Figur. For xampl, maximum bridging = tr J th ara undr bridging tr () urv orrpond to tnil trngth fratur nrgy Th ompoit, watr ontnt rptivly. w an b xprd a th um Th th initiation vaporabl watr pudo w train hardning bhavior (apillary watr, watr vapor, ECC, th adorbd uniqu pot-raking watr) th bhavior, non-vaporabl i alo timatd (hmially by bridging bound) law watr a hown w in th following n (Mill 966, (Li. Pantazopoulo 993). & Mill 995). It i raonabl to aum that th vaporabl watr i a funtion rlativ σ pak humidity, (σ f ) i h, dgr hydration, α, () dgr ilia fum ration, α, i.. w =w (h,α,α ) = ag-dpndnt orption/dorption iothrm J' b / J tip (Norling Mjonll 997). Undr thi aumption () by ubtituting Equation into Equation on whr obtain σ pak =maximum bridging tr; (σ f ) i =rak trngth; J b =omplmntary nrgy th rlationhip btwn bridging tr rak opning diplamnt; J tip =fratur α& toughn + α& + w& matrix n (3) at = rak tip Equation i th ritrion for tr inra aftr rak whr initiation, / i th maning lop that th maximum orption/dorption bridging tr iothrm ompoit (alo alld mut moitur b largr apaity). than rak Th trngth govrning in quation ordr to (Equation obtain th 3) train mut hardning b ompltd bhavior. by appropriat boundary initial ondition. Equation Th rlation i btwn th ritrion th amount to obtain tady vaporabl tat raking, watr th rlativ uniqu humidity pro i alld rak propagation. adorption Undr iothrm tady if tat maurd raking, with rak inraing propagat rlativity with ontant humidity rak dorption opning diplamnt iothrm in undr th ontant oppoit loading, a. Nglting th initiation thir diffrn multipl (Xi t rak al. 994), i promotd th following, bau orption bridging iothrm tr rtaind will b ud although with in rak rfrn lngth to both bom orption larg. dorption ondition. By Bridging th way, law if in th thi hytri tudy a numrial th moitur modl drivd iothrm from would Fibr Pullout takn into Modl aount, (Li. 99), two diffrnt whih i rlation, th bai vaporabl mhanial watr modl v rlativ hort humidity, fibr rinford b ud ompoit. aording to In th th ign following th variation tion, th mut numrial rlativity modl humidity. i xplaind. Th hap th orption iothrm for HPC i inflund by many paramtr,.. pially Fibr tho Pullout that Modl influn xtnt rat th Fibr hmial Pullout ration Modl i, th mhanial in turn, dtrmin modl, whih por formulat trutur th por fibr iz pullout ditribution bhavior (watr-to-mnt from matrix drivd ratio, mnt bad on hmial th following ompoition, aumption. SF ontnt, - uring Fibr tim ar 3-D mthod, romly tmpratur, ditributd mix in additiv, loation t.). In orintation. th litratur variou formulation an b - found Dbonding to drib btwn th orption fibr iothrm matrix our normal at onrt th id (Xi t rak, al. 994). progr Howvr, to th in id th prnt mbdmnt. th mi-mpirial xprion propod by papr Norling Mjornll (997) i adoptd bau it xpliitly aount for th volution hydration σb ration SF ontnt. Thi orption iothrm rad σpak w ( h, σa α, α ) = G ( α, α ) + J b ( g α α ) h Jtip ( g α α ) h K ( α, α ) δa δpak Figur. Bridging tr-rak opning diplamnt rlationhip bfor pak. whr th firt trm (gl iothrm) rprnt th - phyially Th intrfaial bound (adorbd) bond btwn watr fibr th matrix ond i trm ubjtd (apillary to fritional iothrm) bond. rprnt th apillary - watr. Th Thi dformation xprion i matrix valid i only mall for nough low ontnt ompard Th with fiint th lip G SF. fibr rprnt o that th it an amount b ngltd. pr unit volum hld in th gl por at % watr - rlativ Th humidity, fft Poion it an b ratio xprd fibr (Norling i ngltd, 997) a th lati modulu fibr i on- Mjornll idrd to b ontant. - Fibr ruptur do G ( α, α ) = k α + k not α our bau fibr tnil (5) vg vg trngth i largr than axial tr. Th aumption ar for mntitiou ompoit whr rinford k vg kwith vg ar nylon matrial fibr paramtr. or polypropyln From th fibr. maximum Som amount thm hould watr pr b xpd, unit volum dpnding that an on fill fibr all por proprti (both apillary for rinformnt. por Hr, gl por), om on ar xpd an alulat by auming K on PVA obtain fibr... Intrfaial bond trngth Intrfaial wbond.88 btwn α +.α fibr G matrix i laifid into two kind: hmial bond fritional (6) K ( α, α ) = bond (Ka t al. 998a). Larg hmial bond trngth i th pifi haratriti PVA fibr. In thi tudy, th pro th hang intrfaial bond, Th dbond matrial paramtr hmial bond k th tranition to vg k vg g an fritional b alibratd bond, by i fitting dfind xprimntal a a funtion data rlvant rlativ to lip fr (vaporabl) fibr. watr ontnt in onrt at variou ag (Di Luzio & Cuati 9b). τ = τ [ { ( δ / δ τ )} ] for δ δ τ τ = τ i + ( τ τ i )Exp[ ατ { ( δ / δ τ )}] for δ τ < δ. Tmpratur volution (3) Not that, at arly ag, in th hmial ration whr τ=fibr-matrix intrfaial bond trngth; aoiatd with mnt hydration SF ration τ =hmial bond trngth; τ i =fritional bond ar xothrmi, th tmpratur fild i not uniform trngth; δ τ,α τ =paramtr δ τ i rlativ lip for non-adiabati ytm vn if th nvironmntal whn dbond hmial bond tart, α τ dtrmin th lop th tranition hmial bond to tmpratur i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not fritional bond. xding C (Bažant & Kaplan 996), by Fourir law, whih rad..3 Fibr ruptur For PVA-ECC, fibr ruptur mut b takn into aount bau fibr tnil trngth i rlativly (7) q = λ T mall ompard with intrfaial bond trngth btwn whr fibr q i th matrix. hat Th flux, bridging T i tr th i abolut alulatd tmpratur, by th ummation λ i th hat pullout ondutivity; tr tranfrrd in thi δ Proding FraMCoS-7, May 3-8,

3 by ah fibr with onidring th romn fibr poition inlining angl. σ = V b f π / ( L f / ) oφ φ = z= P A f p( φ) p( z) dzdφ whr σ b =bridging tr; P=pullout load tranfrrd by ah fibr; p(z)=probability about fibr loation; p(φ)=probability about fibr inlining angl; V f =fibr volum fration; A f =ro tion ara fibr- Fibr ruptur our whn axial tr fibr rah fibr tnil trngth, th rupturd fibr ar rmovd from Equation Fibr trngth rdution For fibr mbddd into matrix, th trngth dra du to th abraion fibr urfa bnding fft (Ka t al. 998a). Thi apparnt trngth rdution i xprd by th following quation. σ fu = σ n f ' φ fu (5) whr, σ fu =apparnt fibr trngth; σ n fu=nominal fibr trngth; f =rdution fator fibr trngth; φ=fibr inlining angl..5 Th bridging tr-rak opning diplamnt rlationhip Figur (numbr yl=) i th rlationhip btwn bridging tr rak opning diplamnt obtaind from th abov modl. Th rlationhip an rprodu th rlationhip drivd by Ka t al. (998b). Th diffrn btwn thi tudy modl Ka modl ar tratmnt intrfaial bond trngth lati modulu matrix. For th formr, paramtr in Equation 3 ar dtrmind bad on Ka modl. For th lattr, th matrix proprty an b ngltd bau th dformation matrix i uffiintly mall du to th ratio lati modulu btwn fibr matrix.. Dgradation rlation ingl fibr Fatigu dgradation ECC i aud by bridging tr dgradation, it i aud by fibr fatigu ruptur fibr pullout from matrix. Th dominant fator hang, dpnding on th proprty fibr for rinformnt (Matumoto t al. 3). Mor rupturd fibr ar obrvd than pullout fibr in th a PVA-ECC with rlativly low fibr tnil trngth ompard with intrfaial bond trngth. On th othr h, th numbr i oppoit in th a polythyln fibr rinford ECC with rlativly high fibr tnil trngth ompard with intrfaial bond trngth. Bridging tr (N/mm ) 4 3 J = ) D ( h, T h Th proportionality ( N=,,, 3, 4 fiint, 5 D(h,T) ) moitur prmability it i a nonlina th rlativ humidity h tmpratur & Najjar 97). Th moitur ma balan that th variation in tim th watr ma volum onrt (watr ontnt w) b q divrgn th moitur flux J. =.4 J.6.8. Crak opning diplamnt (mm) Figur. Bridging tr-rak opning diplamnt undr fatigu. Th watr ontnt w an b xprd a th vaporabl watr w (apillary wa In bridging law, dgradation vapor, fator adorbd an watr) b mod-ld th non- by th hang (hmially miromhanial bound) paramtr. watr win n (Mil miromhanial Pantazopoulo paramtr, th & Mill hang 995). fibr It i ra tnil trngth aum intrfaial that th bond vaporabl trngth xpr watr i a fu fibr fatigu ruptur rlativ humidity, fibr pullout h, dgr from matrix, hydration rptivly. In th dgr propod ilia modl, fum ration, only fibr α fatigu ruptur i onidrd = ag-dpndnt a a fatigu orption/dorption dgradation, i.. w =w fator, th (Norling hang Mjonll fibr 997). tnil Undr trngth, thi aum whih dpnd on by numbr ubtituting yl, Equation i introdud into Equati into bridging law, obtain auming PVA-ECC (Kakuma t al. 9a,b). Th ritrion fibr fatigu ruptur i rprntd by th following quation + ( D a h) = a funtion α& + α& + w numbr yl. i σ whr / i th lop th orption/ N = k f log( iothrm N) (alo alld moitur apa n σ fu govrning quation (Equation 3) (6) mut b by appropriat boundary initial onditi whr σ i N=fibr tnil Th rlation trngth btwn at Nth th yl; amount σ n fu=initial fibr watr tnil trngth; rlativ k f =paramtr; humidity i alld N=numbr yl iothrm if maurd with inraing Fatigu proprty humidity PVA fibr dorption ha not bn iothrm uffiintly rvald a. bau Nglting th fatigu thir tt diffrn ingl (Xi t al. in th fibr i diffiult. th Hr, following, th paramtr, orption k iothrm, dtr-wilmind a rfrring rfrn to th S-N to both diagram orption mtal dorption a b fibr ruptur at, By th yl way, undr if th th hytri tr ratio th.5. iothrm would b takn into aount, two rlation, vaporabl watr v rlativ humi b ud aording to th ign th varia.3 Th bridging rlativity tr-rak humidity. opning diplamnt Th hap th rlationhip undr iothrm fatigu for HPC tnion i inflund by many p Figur how pially th rlationhip tho that btwn influn bridging xtnt tr rak hmial opning diplamnt ration, undr in turn, variabl numbr yl, trutur N=,, por iz, ditribution 3, 4 (watr- dtrm 5. In th figur, ratio, bridging mnt tr hmial dtriorat ompoition, whn SF numbr yl uring inra. tim In th mthod, rlationhip tmpratur, bfor maximum bridging t.). In tr, th litratur bridging variou tr dg- formulatio mix radation i not hown. found to Figur drib i th a orption rlationhip iothrm whn only fibr onrt trngth i (Xi a variabl t al. 994). paramtr Howvr, in in th th ummation papr fibr pullout th mi-mpirial tr in Equation xprion 4, pro th numbr Norling rupturd Mjornll fibr do (997) not hang i adoptd at b Proding FraMCoS-7, May 3-8,

4 th J = rang D ( h, T ) mall h rak opning diplamnt with () th mall fibr pullout tr. Figur Th proportionality 3 i th rlationhip fiint btwn D(h,T) th i valu alld J moitur b /J tip prmability numbr yl, it i whih a nonlinar timat funtion th limit th rlativ howing humidity multipl h raking tmpratur proprty. T (Bažant In th alulation & Najjar 97). th Th figur, moitur th valu ma balan J tip, 3.kJ/m rquir, i that ontant th variation without in rgard tim to th numbr watr ma yl pr unit bau volum th hang onrt (watr fratur ontnt toughn w) b qual matrix to th i not divrgn applid. Th th moitur following flux about J tnil bhavior undr fatigu i uggtd whn paying attntion to th ritrion tady tat raking dfind in Equation =. Th J train hardning bhavior an b () ob- taind although bridging fft dtriorat du to th Th inra watr ontnt rupturd w an fibr b bau xprd Equation a th um i atifid th vaporabl bfor numbr watr wyl rah,. Epially (apillary watr, watr vapor, bfor adorbd yl, watr) multipl th rak non-vaporabl aturat bau (hmially th J b bound) /J tip valu watr i largr w than 3., th ritial Pantazopoulo valu to how & aturatd Mill 995). multipl It i raking. raonabl Whn to n (Mill 966, th aum J b /Jthat tip valu th vaporabl approah watr th i ritial a funtion valu, J rlativ b /J tip =, humidity, with inra h, dgr numbr hydration, yl, α it i, diffiult dgr to ilia obtain fum multipl ration, raking α proprty. Aftr, i.. w =w (h,α,α ) numbr = ag-dpndnt yl i ovr orption/dorption,, th loalization iothrm partiular (Norling Mjonll rak without 997). howing Undr thi multipl aumption raking probably by ubtituting our a Equation am a hown into in Equation gnral hort on fibr obtain mntitiou ompoit. 3 UNIAXIAL TENSILE FATIGUE TEST = α& + α& + w& n (3) In ordr to how th validity th tr-train rlationhip undr tnil fatigu propod in thi tudy, whr th analytial / i th lop th orption/dorption rult i ompard with uniaxial tnil fatigu tt PVA-ECC ondutd by Matu- iothrm (alo alld moitur apaity). Th govrning quation (Equation 3) mut b ompltd moto t al.. In thi haptr, th ummary by appropriat boundary initial ondition. th rult ar xplaind. Th rlation btwn th amount vaporabl watr rlativ humidity i alld adorption 3. iothrm Matrial if maurd tt pimn with inraing rlativity humidity dorption iothrm in th oppoit Stati fatigu tnil tt wr ondutd for a. Nglting thir diffrn (Xi t al. 994), in PVA-ECC. Th mix proportion th ECC th th following, orption iothrm will b ud with proprti PVA fibr ar hown in Tabl, rfrn to both orption dorption ondition. rptivly. By th way, if th hytri th moitur Th hap pimn i hown in Figur 4. iothrm would b takn into aount, two diffrnt Thr pimn nin pimn wr prpard rlation, vaporabl watr v rlativ humidity, mut for tati loading fatigu loading tt, rptivly. b ud aording to th ign th variation th rlativity humidity. Th hap th orption iothrm for HPC i inflund by many paramtr, 3. pially Loading tho produr that influn xtnt rat th hmial ration, in turn, dtrmin por Both tati fatigu loading wr ondutd undr diplamnt ontrol. Th uniaxial tnil tt trutur por iz ditribution (watr-to-mnt ratio, mnt hmial ompoition, SF ontnt, wr ondutd undr tati loading ondition bfor fatigu loading. Th tati tnil trngth uring tim mthod, tmpratur, mix additiv, t.). In th litratur variou formulation an b th train apaity bfor loalization ECC wr found to drib th orption iothrm normal dtrmind. Bad on th tnil train apaity onrt (Xi t al. 994). Howvr, in th prnt from th tati tt, thr lvl maximum tnil papr th mi-mpirial xprion propod by train lvl wr aignd for fatigu pimn. Norling Mjornll (997) i adoptd bau it xpliitly 4 aount for th volution hydration ration SF ontnt. Thi orption iothrm rad 3 w ( h, α, α ) = G ( α, α ) + ( g α α ) h J' b /J tip ( g α α ) h K ( α, α ) Numbr yl Figur 3. Th J b /J tip valu-numbr yl rlationhip. whr th firt trm (gl iothrm) rprnt th Tabl phyially. Mix bound proportion (adorbd) ECC. watr th ond Watr trm (apillary iothrm) rprnt th apillary Cmnt watr. Thi xprion i valid.3 only for low ontnt Fin aggrgat.4 SF. Th fiint G Supr platiizr rprnt th amount.3 Mthylllulo watr pr unit volum hld.7 in th gl por at % rlativ humidity, it an b xprd (Norling Tabl Mjornll. Proprti 997) a PVA fibr. Lngth (mm) Diamtr (µm) 37.7 Volum G ( α, α ) fration = k (%) α + k α. (5) vg Elati modulu (kn/mm vg ) 36.7 Fibr tnil trngth 6 (N/mm whr ) k vg k vg ar matrial paramtr. From th Intrfaial maximum bond amount trngth watr. pr unit volum that an (N/mm fill all ) por (both apillary por gl por), on an alulat K a on obtain mm mm α α K (, ) = w.88α +.α G 5 mm mm 5 mm Th matrial paramtr k vg k vg g an b alibratd by fitting 6 mmxprimntal data rlvant to fr (vaporabl) watr ontnt in onrt at variou ag (Di Luzio & Cuati 9b). mm. Tmpratur volution mm (6) Not that, at arly ag, in th hmial ration aoiatd with mnt hydration SF ration ar xothrmi, th tmpratur fild i not uniform Figur 4. Th iz tt pimn. for non-adiabati ytm vn if th nvironmntal tmpratur i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not Th xding ltd C maximum (Bažant tnil & Kaplan train lvl 996), wr by., Fourir.5 law, whih., rad thr pimn wr ondutd for ah tnil train lvl. q Th uniaxial tnil fatigu tt wr prformd = λ T undr train ontrol ondition. Spimn wr ubjtd to a 4Hz inuoidal yli loading. Th tt (7) wa whr ondutd q i th with hat ontant flux, T amplitud i th btwn abolut maximum tmpratur, tnil train λ i th hat minimum ondutivity; tnil train. thi Proding FraMCoS-7, May 3-8,

5 Maximum numbr yl in fatigu loading wa, yl. 3.3 Tt rult 3.3. Stati tt From th uniaxial tnil tt, th uniqu tnil bhavior ECC, tnil train inra with rpatd tr inra dra, wa obrvd. Aftr that, rak loalization ourrd whn tnil train qual to.4. Thi valu tnd to b mall ompard with othr uniaxial tnil tt PVA-ECC with imilar fibr proprty mix proportion. Howvr, it i rgardd that thi tt ha no problm a a ompard xprimnt bau multipl raking train hardning bhavior aftr initial raking wr obrvd Fatigu tt In th uniaxial tnil fatigu tt, tnil tr gradually rdud at low fatigu loading yl, th rat tr rdution inrad whn numbr yl inrad. Furthrmor, th tnil tr ECC tndd to b ontant or narly ontant whn th numbr yl wa in th rang btwn, 5, yl, dpnding on th maximum tnil train lvl. From th volution tnil tr aftr, yl, it wa uggtd that PVA fibr hav fatigu limit a hown in mtal. On th obrvd rak plan aftr fatigu loading, th numbr rupturd fibr wa largr than pulld out fibr, fibr fatigu ruptur wa th dominant mhanim fatigu dgradation ECC. 4 THE STRESS-STRAIN RELATIONSHIP UNDER TENSILE FATIGUE Thr ar two mthod to rprnt rak bhavior in finit lmnt analyi: dirt rak modl mard rak modl. Th formr an dirtly onidr th proprty rak uh a rak opning diplamnt rak lngth by introduing joint lmnt at rak loation. Th lattr trat rakd lmnt a ontinuou lmnt vn aftr raking by auming that rak ditribut a whol lmnt. Bridging tr dgradation rlationhip propod in th prviou haptr i th rlationhip on rak plan, maning that it i bad on th onpt dirt rak modl whn applid to finit lmnt analyi. Thrfor, th tr-train rlationhip undr fatigu obtaind from bridging tr dgradation i bnfiial to prform fatigu analyi ECC a mmbr or trutur. In thi haptr, th tr-train rlationhip undr tnil fatigu i drivd, th validity i diud by omparing with uniaxial tnil fatigu tt PVA-ECC hown in haptr 3. Th proportionality fiint D(h,T) moitur prmability it i a nonlina th rlativ humidity h tmpratur & Najjar 97). Th moitur ma balan that th variation in tim th watr ma 8mm volum onrt (watr ontnt w) b q Figur 5. Analytial modl. divrgn th moitur flux J 3 J = ) D ( h, T h diplamnt 4. Etimating produr = J Th tr-train rlationhip undr tnil fatigu PVA-ECC i timatd Th watr by applying ontnt finit w an lmnt b xprd a analyi. Crak ar th modld vaporabl bad watr on th w onpt (apillary wa dirt rak, vapor, th bridging adorbd tr-rak watr) opning diplamnt (hmially undr fatigu bound) drivd in watr haptr w n (Mil th non- i introdud by intrfa Pantazopoulo lmnt. & Mill 995). It i ra Finit lmnt aum modl i that hown th in vaporabl Figur 5, whih watr i a fu rfr to th mauring rlativ rang humidity, tnil h, train dgr uniaxial dirt tnil dgr tt (JSCE ilia fum 7). ration, Analyi α, i i.. w =w hydration ondutd by uing = thi ag-dpndnt modl du to orption/dorption th following raon although (Norling th iz Mjonll i diffrnt 997). from Undr th pimn uniaxial by tnil ubtituting fatigu tt Equation in Figur 4. into Equati thi aum - Crak wr obtain obrvd only at th mauring rang tnil train with mall ro tion ara in th xprimnt. - Th pro raking i not hangd undr un-iaxial tnion. = α& + α& + w Elmnt iz i dtrmind by rfrring to th intrval rak obrvd whr in / th i uniaxial th lop tnion th tt orption/ ondutd by Ka iothrm t al. (alo (998b), alld a minimum moitur apa lngth ah lmnt govrning i about quation.5mm. (Equation Load i 3) givn by diplamnt. by appropriat Crak initiat boundary whn axial initial tr onditi mut b rah rak trngth, Th rlation intrfa btwn lmnt th amount i inrtd into th whol watr raking rlativ ro tion, humidity aum- i alld ing tady tat raking. iothrm Thn, if maurd th romn with inraing rak trngth i onidrd humidity bad dorption on th probability iothrm in th dnity funtion ubjtd a. Nglting to Gau thir ditribution. diffrn (Xi t al. In th fatigu analyi, th following, th ontitutiv orption iothrm rlation will b intrfa lmnt, rfrn th rlationhip to both orption btwn bridging dorption tr rak By opning th way, diplamnt, if th hytri dtrio-oratd bad on th iothrm apparnt would numbr b takn yl. into aount, two th Alo, th rdution rlation, rak vaporabl trngth watr undr v fatigu rlativ humi i dfind in th following. b ud aording to th ign th varia rlativity humidity. Th hap th f N = kmlog( iothrm N) for HPC i inflund by many p f pially tho that influn xtnt (7) hmial ration, in turn, dtrm trutur por iz ditribution (watrratio, whr f N =rak trngth at N yl; f =initial rak trngth(=3n/mm mnt hmial ompoition, SF uring ); tim k m =paramtr mthod, tmpratur, mix Th valu k t.). m i /7, In th rfrring litratur to variou fatigu formulatio dign formulation found onrt to drib (JSCE ). th orption iothrm onrt (Xi t al. 994). Howvr, in th papr th mi-mpirial xprion pro Norling Mjornll (997) i adoptd b Proding FraMCoS-7, May 3-8,

6 4. J = D ( Etimatd h, T ) h rult () 4.. Stati analyi Th Th tnil proportionality tr-train fiint rlationhip D(h,T) undr i alld tati loading moitur i prmability hown in Figur it 6 i (numbr a nonlinar yl=. funtion Figur th rlativ 7 i th humidity rak ditribution h tmpratur whn tnil T (Bažant train qual & Najjar to 97).., Th.5 moitur.. ma balan From Figur rquir 6 that 7, th it variation i onfirmd in tim that multipl th watr raking ma proprty pr unit volum th pudo onrt train (watr hardning ontnt bhavior w) qual an b to xprd divrgn in analyi. th moitur Tnil flux train J at tnil trngth, th ultimat tnil train, rah about.5. Th imilar valu ultimat tnil train wa obtaind = J () from uniaxial tnil tt PVA-ECC with th imilar fibr proprti mix proportion. Thrfor, it i hown Th watr that th ontnt tnil w an bhavior xprd ECC a an th b um timatd th vaporabl by bridging watr law w (apillary dirt rak watr, modl. watr On vapor, th othr adorbd h, th watr) ultimat th tnil non-vaporabl train obrvd (hmially in th tt bound) xplaind watr in haptr w n (Mill 3 wa about 966,.4, Pantazopoulo th & alulatd Mill 995). rlationhip It i raonabl do not rprodu aum that ultimat th vaporabl tat. In xprimnt, watr i a funtion th tnil to train rlativ apaity humidity, an h, b dgr xtrmly hydration, poor in whih α, initial dgr dft ilia loal fum volum ration, fration α, i.. wnarby =w (h,α rak,α ) influn = ag-dpndnt th multipl orption/dorption raking proprty. In ontrat, iothrm fibr (Norling Mjonll matrix xri 997). Undr th prforman thi aumption idally in analyi. by ubtituting Thi probably Equation au into th diffrn Equation tnil obtain train apaity btwn analyi xprimnt. on 4.. Fatigu analyi Th tr-train + ( D rlationhip h) = α& + undr α& tnil + w& fatigu with variabl numbr yl,,, n (3), 3, 4 5, ar hown in Figur 6. Hr, tning bhavior whr i not / modld, i th lop o that th orption/dorption lop in tning iothrm aumd (alo bad alld on th moitur tati tt apaity). ondutd Th by Ka govrning t al. quation (998b).(Equation 3) mut b ompltd by In appropriat Figur 6, boundary tnil tr initial ondition. ultimat tnil train Th dra rlation btwn whn bridging th amount tr dtriorat vaporabl with watr th inra rlativ humidity numbr i yl. alld Bfor adorption, yl, iothrm if pudo maurd train with hardning inraing bhavior rlativity i obrvd humidity bau dorption maximum iothrm bridging tr in th i oppoit largr than a. rak Nglting trngth. thir Alo, diffrn th magnitud (Xi t al. 994), tnil in tr th following, rdution orption i am iothrm rat a th will rdution b ud with rak rfrn trngth to both bau orption rmarkabl dorption bridging ondition. tr dgradation By th way, i not if hown th hytri in pr-pak rlationhip th moitur in Figur iothrm. would Th inra b takn into th numbr aount, two rupturd diffrnt fibr rlation, with vaporabl th inra watr numbr v rlativ yl humidity, au mut th rdution b ud aording maximum to th bridging ign tr. th variation Aftr, th yl rlativity ECC humidity. how tning Th hap bhavior without th orption howing iothrm hardning for HPC bhavior i inflund bau by tnil many tr paramtr, rah- pially maximum tho bridging that influn tr at xtnt initial raking. rat Figur hmial 8 how ration th maximum, bridging in turn, tr-numbr dtrmin por th yl trutur rlationhip por iz th ditribution rak trngth-numbr (watr-to-mnt yl ratio, rlationhip. mnt hmial From thi ompoition, figur, th SF tr-train ontnt, rlationhip uring tim an b mthod, aily tmpratur, timatd. mix additiv, t.). Abov In th rult litratur approximatly variou formulation orrpond an to th b multipl found to raking drib th proprty orption timatd iothrm by bridging normal tr onrt dgradation (Xi t al. modl 994). in Howvr, haptr. in Critial th prnt numbr papr yl th mi-mpirial howing multipl xprion raking, propod timatd by Norling Mjornll (997) i adoptd bau it xpliitly 4 aount for th volution hydration ration SF ontnt. Thi orption iothrm rad 3 ( N=,,, 3, 4, 5 ) Tnil tr (N/mm ) w ( h, α, α ) = G ( α, α ) + ( g α α ) h ( g α α ) h K ( α, α ) 3 Tnil train (%) Figur 6. Tnil tr-train rlationhip. whr th firt trm (gl iothrm) rprnt th phyially bound (adorbd) watr th ond trm (apillary iothrm) rprnt th apillary watr. Thi xprion i valid only for low ontnt SF. Th fiint G rprnt th amount watr pr unit volum hld in th gl por at % rlativ humidity, it an b xprd (Norling Mjornll 997) a (a) ε t =. G ( α, α ) = k α + k α vg vg whr k vg k vg ar matrial paramtr. From th maximum amount watr (b) pr ε t =.5 unit volum that an fill all por (both apillary por gl por), on an alulat K a on obtain () ε t =. K ( α, α ) = Figur 7. Crak ditribution. 4 Th matrial paramtr k vg k vg g an b alibratd by fitting xprimntal data rlvant to 3 fr (vaporabl) watr ontnt in onrt at variou ag (Di Luzio & Cuati 9b). Str (N/mm ) w.88α +.α G g α. Tmpratur volution maximum bridging tr rak trngth Not that, at arly ag, in th hmial ration aoiatd with mnt hydration SF ration ar xothrmi, th tmpratur 3 fild i 4 not uniform 5 for non-adiabati ytm Numbr vn yl if th nvironmntal Figur tmpratur 8. Th volution i ontant. maximum Hat bridging ondution tr an rak b trngth. dribd in onrt, at lat for tmpratur not xding C (Bažant & Kaplan 996), by Fourir law, whih rad by th tnil tr-train rlationhip, i, yl, whil that timatd in haptr i, yl. q = Although λ T th limit howing multipl raking (7) i ovrtimatd ompard with in haptr du to th whr diffrn q i th analytial hat flux, objt, T i it i th aily abolut timatd tmpratur, by th rlationhip λ i th hat btwn ondutivity; rak trngth in thi maximum bridging tr. α (5) (6) Proding FraMCoS-7, May 3-8,

7 Th analytial rult i ompard with th uniaxial tnil fatigu tt about th ratio btwn th tnil tr at N yl, σ N, th valu at initial yl, σ, whn th tnil train qual to th foud valu,.,.5.. Figur 9 how th rlationhip btwn th σ N /σ valu numbr yl. In th figur, analytial rult agr wll with xprimntal on in all a, th ritrion fibr fatigu ruptur dfind in Equation 6 i propr rlationhip to timat th dgradation bridging tr PVA-ECC. Whn th volution th σ N /σ valu foud on th diffrn maximum tnil train i n, th lop tnd to bom tp with th inra tnil train in both analyi xprimnt. Thi i bau fibr fatigu ruptur i promotd by inra fibr pullout load whih i aud by xpanion rak opning diplamnt, orrponding to th inra tnil train. Alo, in all a, th lop bom larg whn numbr yl inra bau fibr trngth rdution promot fibr fatigu ruptur. In ral ompoit, th magnitud th rdution apparnt fibr tnil trngth hang dpnd on th amount dformation or lip fibr, although Equation 6 i imply dfind a a funtion numbr yl. In thi analyi, th rak opning diplamnt at foud tnil train ar imilar ah othr bau th diffrn th prpard train amplitud uniaxial tnil fatigu tt i omparativly mall, o that th ritrion an rprodu th xprimnt undr vral tnil train. Thrfor, additional xprimnt undr larg tnil train th r-vrifiation th propod modl may b nary. 5 CONCLUSIONS Thi tudy dvlopd th tnil tr-train rlationhip PVA-ECC undr uniaxial fatigu tnion. Th fatigu dgradation modl ECC wa bad on bridging tr dgradation, fibr fatigu ruptur wa onidrd a a dgradation fator. Th trtrain rlationhip wa obtaind by applying th bridging tr-rak opning diplamnt rlationhip into finit lmnt analyi a dirt rak modl. Th rult obtaind from thi tudy ar hown in th following. Th bridging tr-rak opning diplamnt rlationhip undr tnil fatigu wa obtaind by onidring th hang miromhanial paramtr in bridging law. A a rult, it wa uggtd that multipl raking bhavior ECC an b hown undr fatigu loading bfor numbr yl rah a rtain ritial valu. σ N /σ σ N /σ Th proportionality fiint D(h,T) moitur prmability it i a nonlina th rlativ humidity h tmpratur.5 No. & Najjar 97). Th moitur ma balan No. that th variation in tim th watr ma No.3 volum onrt (watr ontnt w) b q Analyi divrgn th moitur flux J Numbr yl = J (a) ε t =. Th watr ontnt w an b xprd a Pantazopoulo & Mill 995). It i ra th vaporabl watr w (apillary wa vapor, adorbd watr) th non- (hmially bound) watr w n (Mil.5 aum that th vaporabl watr i a fu No. rlativ humidity, h, dgr hydration No. No.3 dgr ilia fum ration, α, i.. w =w Analyi = ag-dpndnt orption/dorption (Norling Mjonll 997). Undr thi aum by ubtituting 3 Equation into Equati obtain Numbr yl (b) ε t =.5 = α& + α& + w whr / i th lop th orption/ iothrm (alo alld moitur apa govrning quation (Equation 3) mut b.5 No. by appropriat boundary initial onditi No.Th rlation btwn th amount No.3 Analyi watr rlativ humidity i alld iothrm if maurd with inraing humidity 3 dorption 4 5 iothrm 6 in th a. Numbr Nglting yl thir diffrn (Xi t al. th following, orption iothrm will b rfrn () εto t =. both orption dorption Figur 9. Th σ N /σ valu By th numbr way, if yl th rlationhip. hytri th iothrm would b takn into aount, two Th volution rlation, tr vaporabl dgradation watr obtaind v rlativ from humi th tr-train rlationhip b ud aording rprodud to th th ign rdution bridging tr rlativity obrvd humidity. in th Th uniaxial hap tn- th th varia il fatigu tt iothrm PVA-ECC. for HPC Thi i man inflund that by th many p tnil ontitutiv pially law for tho fatigu that analyi influn ECC xtnt a mmbr or trutur hmial i givn. ration, in turn, dtrm For th futur trutur tudi, th following por iz ditribution ar onidrdratio, mnt hmial ompoition, SF (watr- To xp th uring fatigu tim mod mthod, inluding tmpratur, anothr mix dgradation fator, t.). fibr In pullout, th litratur i nary variou from formulatio th point th dvlopmnt found to drib fatigu th modl orption indpndnt th fator onrt bridging (Xi t tr al. 994). dgradation. Howvr, in th iothrm papr th mi-mpirial xprion pro Norling Mjornll (997) i adoptd b σ N /σ J = ) D ( h, T h Proding FraMCoS-7, May 3-8,

8 J = To D ( timat h, T ) h th fatigu durability ECC () a mmbr or trutur by uing th fatigu modl propod Th proportionality in thi tudy. fiint D(h,T) i alld moitur prmability it i a nonlinar funtion th rlativ humidity h tmpratur T (Bažant REFERENCES & Najjar 97). Th moitur ma balan rquir that th variation in tim th watr ma pr unit Japan volum Soity onrt Civil Enginr. (watr ontnt. Stard w) b qual pifiation to th for onrt trutur Strutural prforman vrifiation. (in Japan) th moitur flux J divrgn Japan Soity Civil Enginr. 7. Rommndation for dign w ontrution High Prforman Fibr Rinford = Cmnt J Compoit with multipl fin rak () Ka, T. t al. 998a. Intrfa proprty apparnt trngth high-trngth hydrophili fibr in mnt matrix. Journal Th Matrial watr in ontnt Civil Enginring. w an b (): xprd 5-3. a th um Kakuma, th vaporabl K. t al. 9a. watr Etimation w (apillary tr-train watr, rlation watr vapor, ECC undr adorbd uniaxial tnil watr) fatigu. th Proding non-vaporabl Japan (hmially Conrt Intitut. bound) 3(): watr (in w n Japan) (Mill 966, Kakuma, Pantazopoulo K. t al. & 9b. Mill Flxural 995). fatigu It i analyi raonabl PVAto ECC bad on miromhani approah. Intrnational aum Confrn that on th Computational vaporabl Dign watr in i Enginring. a funtion Ka, rlativ T. humidity, t al. 998b. h, Matrial dgr dign hydration, dvlopmnt α, dgr high-dutility ilia ompoit fum ration, rinford α, i.. with w =w hort (h,α rom,α ) = polyvinyl ag-dpndnt alohol fibr. orption/dorption Proding Japan Conrt iothrm Intitut. (): Mjonll ). (in Japan) Undr thi aumption (Norling Li, V. C. 99. Pot-rak aling rlation for fibr-rinford by ubtituting Equation into Equation on mntitiou ompoit, Journal Matrial in Civil Enginring. 4(): obtain Li, V. C From miromhani to trutural nginr- w dign mntitiou ompoit for ivil ngi- ing-th nring appliation. Journal Strutural = α& + α& Mhani + w& n (3) Earthquak Enginring. h (): α Matumoto, T. t al. 3. Mhanim multipl raking fratur DFRCC undr fatigu flxur. Journal whr Advanφd / Conrt i th Thnology. lop th (3): orption/dorption Matumoto, iothrm T. (alo t al. alld 4. Efft moitur fibr apaity). fatigu ruptur Th on govrning bridging tr quation dgradation (Equation in fibr 3) rinford mut b mntitiou ompltd by ompoit. appropriat Proding boundary FRAMCOS-5 initial ondition. (): Mitamura, H. t al. 6. Invtigation for ovrlay rinformnt Th mthod rlation on btwn tl dk th utilizing amount Enginrd vaporabl Cmntitiou Compoit. rlativ Journal humidity Matrial, i alld Conrt adorption Stru- watr iothrm tur Pavmnt. if maurd 6(): with inraing (in Japan) rlativity humidity dorption iothrm in th oppoit a. Nglting thir diffrn (Xi t al. 994), in th following, orption iothrm will b ud with rfrn to both orption dorption ondition. By th way, if th hytri th moitur iothrm would b takn into aount, two diffrnt rlation, vaporabl watr v rlativ humidity, mut b ud aording to th ign th variation th rlativity humidity. Th hap th orption iothrm for HPC i inflund by many paramtr, pially tho that influn xtnt rat th hmial ration, in turn, dtrmin por trutur por iz ditribution (watr-to-mnt ratio, mnt hmial ompoition, SF ontnt, uring tim mthod, tmpratur, mix additiv, t.). In th litratur variou formulation an b found to drib th orption iothrm normal onrt (Xi t al. 994). Howvr, in th prnt papr th mi-mpirial xprion propod by Norling Mjornll (997) i adoptd bau it xpliitly aount for th volution hydration ration SF ontnt. Thi orption iothrm rad w ( h, α, α ) = G ( α, α ) + ( g α α ) h ( g α α ) h K ( α, α ) whr th firt trm (gl iothrm) rprnt th phyially bound (adorbd) watr th ond trm (apillary iothrm) rprnt th apillary watr. Thi xprion i valid only for low ontnt SF. Th fiint G rprnt th amount watr pr unit volum hld in th gl por at % rlativ humidity, it an b xprd (Norling Mjornll 997) a G ( α, α ) = k α + k α vg vg (5) whr k vg k vg ar matrial paramtr. From th maximum amount watr pr unit volum that an fill all por (both apillary por gl por), on an alulat K a on obtain α α K (, ) = w.88α +.α G (6) Th matrial paramtr k vg k vg g an b alibratd by fitting xprimntal data rlvant to fr (vaporabl) watr ontnt in onrt at variou ag (Di Luzio & Cuati 9b).. Tmpratur volution Not that, at arly ag, in th hmial ration aoiatd with mnt hydration SF ration ar xothrmi, th tmpratur fild i not uniform for non-adiabati ytm vn if th nvironmntal tmpratur i ontant. Hat ondution an b dribd in onrt, at lat for tmpratur not xding C (Bažant & Kaplan 996), by Fourir law, whih rad q = λ T (7) whr q i th hat flux, T i th abolut tmpratur, λ i th hat ondutivity; in thi Proding FraMCoS-7, May 3-8,