Fracture analysis of strain hardening cementitious composites by means of discrete modeling of short fibers

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1 Fraur Mhani Conr and Conr Sruur - Rn Advan in Fraur Mhani Conr - B. H. Oh, al.(d) 200 Kora Conr Iniu, Soul, ISBN Fraur analyi rain hardning mniiou ompoi by man dir modling hor fibr M. Kunida, K. Kozaa, N. Uda & H. Nakamura Dp. Civil Enginring, Nagoya Univriy, Nagoya, apan ABSTRACT: Thr ar a id variy hor fibr rinford mn ompoi, uh a rain hardning and mulipl fin raking in nion. Quaniaiv marial dign mhod onidring h propri marix, fibr and hir inrfa hould b ablihd. Thi papr inrodu numrial modl by man hrdimnional mo-al analyi (i.. marix, fibr and hir inrfa ar dirizd rpivly.), o imula h nil fraur inluding rain hardning and mulipl fin raking. Th fibr ih pifi lngh and diamr hav bn arrangd ihin h pimn, and loading in analyi hav bn alo arrid ou o ompar ih xprimnal rul. I a larifid ha h propod modl an b roughly imulad h nil failur high rngh Srain Hardning Cmniiou Compoi (SHCC) inlud rak parn. And h ff marix rngh, i variaion and fibr diribuion on h nil fraur r numrially invigad. INTRODUCTION Th prforman hor fibr rinford mniiou ompoi ha bn nhand in rn yar, ih a variy marial dign mhod bing propod o ahiv h rquird mhanial prforman. Typial marial i Srain Hardning Cmniiou Compoi (SHCC), uh a Enginrd Cmniiou Compoi (ECC) dvlopd by Li, V.C. (993). Th marial a dignd bad on miro mhani prinipl. Th ff fibr orinaion and hrogniy du o h fibr diribuion anno b, hovr, invigad in ordinary marial dign onp dirly. And a n marial uing hybrid fibr (i.. ombinaion diffrn fibr lngh or fibr yp) ha bn alo propod. Bolandr & Saio (997), for inan, ondud 2-D analyi in hih hor fibr r dirizd a bam lmn o xamin nil fraur fibr rinford onr paramrially, h ff h fibr diribuion on h mhanial prforman h ruling ompoi. Kunida al. (2008) propod hr-dimnional mo-al analyi for SHCC bad on a Rigid-Body-Spring Modl (RBSM) (Kaai 978) o luida h nil fraur ordinary SHCC (ECC) ih PVA fibr. And h propod modl an ll imula h nil fraur SHCC. Hovr, h appliabiliy h modl o ohr yp SHCC (.g. high rngh) hould b larifid. Thi papr prn h analyial rul high rngh SHCC by man dir modling hor fibr. In ordr o inrpr h fraur mhanim (rngh and rain apaiy, rak parn), paramri analy onrning variaion marix rngh and fibr diribuion a ondud. 2 OUTLINE OF PROPOSED MODEL 2. Oulin RBSM Figur ho an xampl a pair lmn (Voronoi ll in h Fig.) ompoing h lmn iffn marix in h RBSM (Bolandr & Bron 2004). Th lmn r aumd o b rigid bodi, hil ing 6 dgr frdom for ah rigid lmn. Six pring r plad on ah boundary plan ah ll, in h normal ( pring) and angnial (2 pring) o h boundari and roaional dirion (3 pring). Th marial propri (nil rngh, lai modulu, and fraur nrgy) h marix r adopd a h mhanial propri h normal and angnial pring, hra h mhanial propri h roaional pring r aumd bad on h liraur (Bolandr & Bron 2004). No ha fraur a valuad uing h rak band bad on h dian bn ah lmn h o minimiz h ff h dirizaion on h raking propri (Thomur al. 200).

2 D ( h, T h () Th proporionaliy fiin D(h,T) i alld moiur prmabiliy and i i a nonlinar funion h rlaiv humidiy h and mpraur T (Bažan & Najjar 972). Th moiur ma balan rquir ha h variaion in im h ar ma pr uni Figur volum. Elmn onr and dfind (ar dgr onn ) frdom. b qual o h divrgn h moiur flux = (2) Th ar onn an b xprd a h um h vaporabl ar (apillary ar, ar vapor, and adorbd ar) and h non-vaporabl Figur (hmially 2. Fibr loaion bound) and zro-iz ar pring. n (Mill 966, Panazopoulo & Mill 995). I i raonabl o aum ha h vaporabl ar i a funion 2.2 Modling fibr aion bfor raking rlaiv humidiy, h, dgr hydraion,, and mn marix dgr ilia fum raion,, i.. = (h,, ) Bfor = ag-dpndn raking mn orpion/dorpion marix, h r iohrm born by (Norling fibr Mjonll a alulad 997). bad Undr on hi h aumpion har lag hory by by ubiuing Cox (952). Equaion Cox xprd ino h Equaion nil 2 r on and diribuion obain aing on fibr in h marix a Equaion. Th auhor hrfor xprd h rinforing ff fibr bfor raking by alulaing h h nil + r ( D hfibr ) = uing & + Equaion & +. & n (3) h h oh β ( l / 2 x) σ hr f = E f ε m /h i h lop h orpion/dorpion () oh β ( l / 2) iohrm (alo alld moiur apaiy). Th govrning quaion (Equaion 3) mu b ompld hr by appropria σ f = nil boundary r and fibr iniial (MPa), ondiion. E f = lai modulu Th rlaion fibr bn (MPa), εh m = amoun rain marix vaporabl in h dirion ar and rlaiv h fibr, humidiy l = fibr i lngh alld (mm), adorpion x = dian iohrm from if h maurd nd fibr ih (mm), inraing and β = rlaiviy onan drmind humidiy and by dorpion marial onan iohrm marix in h and oppoi fibr (.0 a. in Ngling hi udy). hir diffrn (Xi al. 994), in h folloing, orpion iohrm ill b ud ih 2.3 rfrn Modling o boh fibr orpion aion and afr dorpion raking ondiion. By mn h marix ay, if h hyri h moiur iohrm ould b akn ino aoun, o diffrn 2.3. rlaion, Oulin vaporabl modling ar v rlaiv humidiy, mu Shor b ud fibr aording ih h o pifi h ign lngh h variaion r randomly h arrangd rlaiviy in humidiy. lmn auming Th hap h pimn h orpion iz o b iohrm analyzd, for HPC o a i o inflund aain h by pifid many paramr, fibr onn pially by volum. ho ha A hon influn Figur xn and 2, a ra zro-iz h pring hmial a raion plad on and, a poin, urn, hr drmin a fibr ro por o ruur lmn and por boundary, iz diribuion and h bonding (ar-o-mn r h raio, fibr mn a ad hmial on h ompoiion, pring. Th SF mbdmn onn, lngh uring im l and and h mhod, angl o mpraur, h normal mix dirion addiiv, o rak.). In plan h (orinaion liraur variou angl) formulaion φ r alo an alulad found for o ah drib fibr. h No orpion ha h iohrm orinaion normal angl b a onr drmind (Xi from al. 994). h innr Hovr, produ in h normal prn vor papr and h h mi-mpirial vor in h xprion dirion propod fibr orinaion. Norling Mjornll (997) i adopd bau by i xpliily aoun for h voluion hydraion raion and SF onn. Thi orpion iohrm rad ( h,, ) = G (, ) + 0( g ) h 0( g ) h K (, ) Figur 3. Pullou diplamn obaind from rak idh (bfor ning in pullou bhavior). (4) hr h fir rm (gl iohrm) rprn h phyially bound (adorbd) ar and h ond rm (apillary iohrm) rprn h apillary ar. Thi xprion i valid only for lo onn SF. Th fiin G rprn h amoun ar pr uni volum hld in h gl por a 00% rlaiv humidiy, and i an b xprd (Norling Mjornll 997) a G (, ) = k + k vg vg (5) hr k vg and k vg ar marial paramr. From h maximum amoun ar pr uni volum ha an fill all por (boh apillary por and gl por), on an alula K a on obain K (, ) = g G 0 g (6) Th marial paramr k vg and k vg and g an Figur 4. Produr for alulaion fibr bridging for. b alibrad by fiing xprimnal daa rlvan o fr (vaporabl) ar onn in onr a variou Afr ag raking, (Di Luzio h r & Cuai ranfr 2009b). by fibr aro a rak plan in h mn marix (hrafr rfrrd o a h marix ) a alulad by h p hon 2.2 Tmpraur in Figur 3 voluion and 4 (Kunida al. 2008). Half No h ha, alulad a arly rak ag, idh in W h 0 a hmial aumd raion o b h aoiad lngh ih fibr mn pull-ou hydraion diplamn and SF bfor raion ning ar xohrmi, bond r h mpraur lip rlaion. fild Afr i no h uniform ning, for non-adiabai i a alo aumd ym ha vn h if fibr h having nvironmnal horr mbdmn mpraur lngh i onan. a pulld Ha onduion ou. Pull-ou an diplamn b dribd in a onr, aumd a o la b qual for mpraur o oal rak no idh. xding Th hap 00 C (Bažan lip diribuion & Kaplan a drmind 996), by hrough Fourir h la, analyi hih rad ingl fibr pull-ou. If h hap lip diribuion i knon, hn h bond r q diribuion i drmind from h bond = λ T r-lip rlaionhip. Th bridging for fibr (7) aro rak an hn b alulad by ingraing h hr bond q r i h diribuion ha flux, in h T dirion i h abolu h fibr mpraur, axi. Uniaxial and λ bond i h r-lip ha onduiviy; urv a hon in hi in Figur 5 a idnifid hrough h rul ingl Proding FraMCoS-7, May 23-28, 200

3 fibr pull-ou (Kozaa al. 2008). No ha h ar o bindr raio (W/B) ud marix a 0.8, and high rngh Polyhyln (PE) fibr a ud in h. D ( h, T h Th proporionaliy fiin D(h,T) moiur prmabiliy and i i a nonlina h rlaiv humidiy h and mpraur & Najjar 972). Th moiur ma balan ha h variaion in im h ar ma volum onr (ar onn ) b q divrgn h moiur flux = Figur 5. Bond r-lip rlaion (PE fibr, W/B=0.8). Figur 6. Bridging inlind fibr o rak plan. Figur 7. Shap dumbbll-hapd pimn. Evnually, normal for o rak plan du o bridging fibr a modld. Hovr, h ff raking on ion har rian i no onidrd Inra bond rngh du o orinaion angl A hon in Figur 6, a fibr a no alay arrangd prpndiular o rak plan, ha man h fibr ha orinaion angl φ. Li al. (990) rvald h inra bond rngh in h inlind fibr pull-ou, and drivd h folloing Equaion; Figur 8. Tnil r-rain Th ar urv onn pimn an ih b diffrn fibr onn. h vaporabl ar (apillary a xprd a vapor, and adorbd ar) and h non- (hmially bound) ar n (Mil Panazopoulo & Mill 995). I i ra aum ha h vaporabl ar i a fu rlaiv humidiy, h, dgr hydraion dgr ilia fum raion,, i.. = = ag-dpndn orpion/dorpion (Norling Mjonll 997). Undr hi aum by ubiuing Equaion ino Equai obain h + ( D h) = h h & + & + Figur 9. Final rak parn afr nil. hr /h i h lop h orpion/ fφ iohrm (alo alld moiur apa F = F 0 govrning quaion (Equaion 3) (2) mu b by appropria boundary and iniial ondii hr F = pull-ou load Th (N), rlaion F 0 = bn pull-ou load h amoun a an orinaion angl ar 0 rad.(n), and rlaiv φ = orinaion humidiy angl i alld fibr (rad.), f = iohrm paramr if on maurd nubbing ff. ih inraing Rgarding h paramr humidiy f, 0.5 and a dorpion idnifid iohrm hrough in h numrial rul a. (Kozaa Ngling 2009). hir diffrn (Xi al. h folloing, orpion iohrm ill b Dra rfrn rupur rngh o boh orpion fibr and du dorpion o orinaion angl By h ay, if h hyri h Thr ar xprimnal iohrm finding ould b ha akn h ino rupur aoun, o rngh fibr rlaion, nd o vaporabl b lor han ar h v valu rlaiv humi givn in h manufaurr b ud aording aalogu o in h h ign a h varia PVA fibr (Kanda rlaiviy & Li 998). humidiy. Th orinaion Th hap angl h ah fibr o iohrm rak for HPC plan i aff inflund rupur by many p rngh fibr ilf, pially a dribd ho ha in influn h folloing xn and Equaion; hmial raion and, in urn, drm ruur and por iz diribuion (arraio, mn hmial ompoiion, (3) SF n f ' φ σ σ fu = fu uring im and mhod, mpraur, mix hr σ fu = rupur.). rngh In h liraur fibr (MPa), variou σ n formulaio fu = apparn rupur rngh found o a drib an orinaion h orpion angl iohrm 0 rad.(mpa), φ = orinaion onr (Xi angl al. fibr 994). (rad), Hovr, f = in h rngh rduion papr faor. h In mi-mpirial h prviou rarh xprion pro ih PVA fibr, Norling 0.3 a adopd Mjornll a a (997) f valu. i adopd b Proding FraMCoS-7, May 23-28, 200

4 D h T h (, () I ha alo bn xprimnally onfirmd ha h pull-ou Th proporionaliy fibr a obrvd fiin in h D(h,T) a i PE alld fibr having moiur ulra prmabiliy high rngh. and In i hi a udy, nonlinar h apparn funion fibr h rngh rlaiv valu humidiy a ud h and for mpraur h analyi. T (Bažan & Najjar 972). Th moiur ma balan rquir ha h variaion in im h ar ma pr uni volum onr (ar onn ) b qual o h divrgn h moiur flux = (2) Th ar onn an b xprd a h um h vaporabl ar (apillary ar, ar vapor, and adorbd ar) and h non-vaporabl (hmially bound) ar n (Mill 966, Panazopoulo & Mill 995). I i raonabl o Figur 0. Diriizd mn marix and fibr. aum ha h vaporabl ar i a funion rlaiv humidiy, h, dgr hydraion,, and dgr ilia fum raion,, i.. = (h,, ) 3 = ANALYSIS ag-dpndn FOR CRACKING orpion/dorpion OF iohrm (Norling COMPOSITES Mjonll WITH 997). STRAIN Undr hi HARDENING aumpion and by ubiuing Equaion ino Equaion 2 on 3. obain Oulin marial rpon for analyi Th arg hi analyi i Ulra High Prforman- Srain Hardning Cmniiou Compoi (UHP h SHCC) dvlopd + ( D by h) Kunida = & + al. (2007) & + and & n Kamal (3) h h al. (2008). Th marial xhibi high rngh ovr 8MPa in nion, high rain apaiy in nion mor han hr 2%, ih /h i xrmly h lop highr h durabiliy. orpion/dorpion Uniaxial nil iohrm (alo pimn alld onaining moiur 0.5%, apaiy)..0% Th and.5% govrning PE fibr quaion (6 mm (Equaion in lngh 3) and mu 0.02 b ompld mm in diamr) by appropria r arrid boundary ou. and A iniial hon ondiion. in Figur 7, dumbbll-hapd Th rlaion bn pimn h having amoun a ro-ion vaporabl 30 ar x 3 and mm rlaiv r ud humidiy in h i o alld maur adorpion h rain iohrm zon if maurd 00 mm in lngh. ih inraing rlaiviy humidiy and dorpion iohrm in h oppoi Tabl a.. Ngling Marial propri hir diffrn for analyi. (Xi al. 994), in h folloing, orpion iohrm ill b ud ih rfrn o boh orpion and dorpion ondiion. By h ay, if h hyri h moiur iohrm ould b akn ino aoun, o diffrn rlaion, vaporabl ar v rlaiv humidiy, mu b ud aording o h ign h variaion h rlaiviy humidiy. Th hap h orpion iohrm for HPC i inflund by many paramr, pially ho ha influn xn and ra h hmial raion and, in urn, drmin por ruur and por iz diribuion (ar-o-mn raio, mn hmial ompoiion, SF onn, uring im and mhod, mpraur, mix addiiv, 3.2.). Oulin In h liraur analyi variou formulaion an b found o drib h orpion iohrm normal Th onr analyi (Xi modl, al. 994). hih Hovr, ar dirizaion in h prn mn papr marix h mi-mpirial and fibr inluding xprion an inrfa, propod ar by hon Norling in Figur Mjornll 0. (997) Th i zon adopd mauring bau 30 x 3 i xpliily aoun for h voluion hydraion x raion 00 mm and a SF akn onn. ou Thi h orpion pimn iohrm for modling, rad and h rain a drmind in h ara 00 mm in lngh imilarly o h xprimn. Whn uing a RBSM, h rak parn, hih govrn h rain apaiy dirly in hi udy, i affd by ( h,, ) = G (, ) + h mh iz and hap. For impliiy, hovr, 0( g ) h h rangl-hapd lmn a ud and h mh (4) iz a mm in h dirion h longiudinal axi h pimn, hih 0( g i l han ) h xprimnal K (, ) rak paing, and 3mm in h dph dirion, o a o horn h alulaion im (a o-dimnional- lik diviion). Fibr r modld by randomly dipring hr h hm fir in rm h (gl pimn iohrm) a hon rprn in Figur h 0. phyially A for h bound boundary (adorbd) ondiion, ar ford and h horizonal rm diplamn (apillary a iohrm) givn o rprn h righ id h lmn, apillary ond hil ar. omplly Thi xprion fixing i h valid lf only id for lmn lo onn h pimn. SF. Th fiin G rprn h amoun ar Tabl pr uni giv volum h phyial hld in propri h gl por h a fibr, 00% marix, rlaiv and humidiy, fibr-marix and i inrfa an b xprd for h analyi. (Norling Mjornll 997) a 3.3 Trial analyi for pimn ih diffrn fibr Gonn (, ) = k + k vg vg (5) Figur ho h r-rain rlaionhip hn h hr nil k vg and rngh k vg ar h marial marix paramr. a aumd From o h b 7.5MPa. maximum Th amoun nil ar rngh pr uni a volum drmind ha an hrough fill all por h flxural (boh apillary rngh por and marix gl por), ilf, on in hih an alula h flxural K a on rngh obain a 7.5MPa. Th xprimnal rul ar alo rprnd in h figur. In h a 0.5%, h fir rak ourrd a g r 7.5MPa, and r inra up o G 3MPa 0 afr uddn drop in r. Finally, ning bhavior K ( a, ) (6) obrvd. = In hi a, oal rpon an b 0 ll imulad. g Thi modl alo prdi h ning bhavior in h Th a marial.0%, paramr alhough k xprimnal rul vg and k vg and g an ho lighly rain hardning up o.0% in rain. b alibrad by fiing xprimnal daa rlvan o fr Rgarding (vaporabl) h analyial ar onn rul for in.5% onr a, h a analyi variou ag rprodud (Di Luzio h & Cuai rpad 2009b). r gain and lo bginning afr h fir rak on and h larg ulima nil rain xding 5%, alhough h 2.2 Tmpraur rul hod voluion only rain apaiy ranging from 2.0% o 2.5%. No inra in h r up o h No ha, a arly ag, in h hmial raion poin aoiad 2.0% ih in mn rain hydraion r obrvd and SF bau raion raking ar xohrmi, rngh h mpraur 7.5MPa. Afr fild ha, i no i uniform xhibi rain hardning up o 5.2% in rain. Th oal rpon for non-adiabai ym vn if h nvironmnal mpraur anno i b onan. imulad Ha quaniaivly. onduion Figur an 2 b ho dribd analyial in onr, rak parn a la a for rain mpraur.0%, 2.0% no and 4.0%, rpivly. In rain hardning marial xding 00 C (Bažan & Kaplan 996), by ih Fourir mulipl la, fin hih rak, rad numbr rak rongly aff h rain apaiy in nion. A hon in rak parn hi analyi, mo inrfa lmn, hih r orrponding o rak ponially, (7) q = λ T r opnd, and i a indud onan rak paing hr in ah q rain i h lvl. ha I m flux, ha T hi bhavior h abolu gav muh mpraur, rain valu and λ in i h h analyi. ha onduiviy; in hi Proding FraMCoS-7, May 23-28, 200

5 D ( h, T h Th proporionaliy fiin D(h,T) moiur prmabiliy and i i a nonlina h rlaiv humidiy h and mpraur & Najjar 972). Th moiur ma balan ha h variaion in im h ar ma volum onr (ar onn ) b q divrgn h moiur flux = Figur 3. Eff marix rngh (.5% a). Th ar onn an b xprd a In ah a, r h valu vaporabl ju bfor ar loalizaion (apillary a (failur) in h xprimn vapor, and a adorbd roughly ar) imulad and by h non- h analyi. Thi (hmially man ha fibr bound) bridging ar rngh, n (Mil hih onrol h Panazopoulo rngh ompoi & Mill in 995). hi marial, i alo valuad aum in ha ah h ri, vaporabl and h ar mod- i a fu I i ra ling in hi udy rlaiv i appropria humidiy, in h, rm dgr fibr hydraion bridging rpon. dgr Hovr, ilia oal fum rpon raion, rprnd by r = rain ag-dpndn urv a no orpion/dorpion imulad, i.. = quaniaivly. On (Norling h raon Mjonll i 997). xiing Undr marial ha diribuion by ubiuing rngh Equaion marix and ino fibr Equai hi aum orinaion, obain h au qunial raking ihin a pimn. In h folloing ion, paramri udi rlad o diribuion marix h rngh and fibr diribuion + r ( D ondud. h) = & + & + h h Figur. Sr-rain urv pimn ih diffrn fibr onn. Figur 2. Analyial rak parn (.5% a). 4 PARAMETRIC hr ANALYSIS /h i h lop h orpion/ iohrm (alo alld moiur apa 4. Eff marix govrning rngh quaion (Equaion 3) mu b by appropria boundary and iniial ondii Figur 3 ho h analyial rul ih diffrn marix rngh, pially in h a fibr onn Th rlaion bn h amoun ar and rlaiv humidiy i alld.5%. Thr kind marix rngh (5.0MPa, 7.5MPa iohrm if maurd ih inraing and 0MPa) r adopd a a main paramr. A humidiy and dorpion iohrm in h hon in Figur 3, a h marix rngh bam a. Ngling hir diffrn (Xi al. lor, h rain apaiy bam highr ih rain h folloing, orpion iohrm ill b hardning bhavior. Bau h fibr bridging rngh rfrn o boh orpion and dorpion a l han 0MPa in h a.5%, i m ha By h ay, if h hyri h marix rngh lor han 0MPa i rquird o xhibi iohrm ould b akn ino aoun, o ignifian rain hardning bhavior. rlaion, vaporabl ar v rlaiv humi b ud aording o h ign h varia rlaiviy humidiy. Th hap h iohrm for HPC i inflund by many p pially ho ha influn xn and hmial raion and, in urn, drm ruur and por iz diribuion (arraio, mn hmial ompoiion, SF uring im and mhod, mpraur, mix.). In h liraur variou formulaio found o drib h orpion iohrm onr (Xi al. 994). Hovr, in h papr h mi-mpirial xprion pro Figur 4. Eff diribuion Norling Mjornll marix rngh (997) (.5% i a). adopd b Proding FraMCoS-7, May 23-28, 200

6 D ( h, T h () Th proporionaliy fiin D(h,T) i alld moiur prmabiliy and i i a nonlinar funion h rlaiv humidiy h and mpraur T (Bažan & Najjar 972). Th moiur ma balan rquir ha h variaion in im h ar ma pr uni volum onr (ar onn ) b qual o h divrgn h moiur flux = (2) Th ar onn an b xprd a h um h vaporabl ar (apillary ar, ar Figur 5. Changing rak idh up o final failur (.5% vapor, and adorbd ar) and h non-vaporabl a). (hmially bound) ar n (Mill 966, Panazopoulo & Mill 995). I i raonabl o 4.2 aum Eff ha h diribuion vaporabl marix ar rngh i a funion along rlaiv pimn humidiy, axi h, dgr hydraion,, and dgr ilia fum raion,, i.. = (h,, ) Figur = ag-dpndn 4 ho h orpion/dorpion analyial rul ih iohrm diffrn (Norling propry Mjonll marix. 997). Undr No ha hi aumpion diribuion and marix by ubiuing rngh Equaion along h pimn ino Equaion axi i 2 on aumd obain o b normal diribuion, and hr kind andard dviaion (S.D.=0MPa,.0MPa and 2.0MPa) r analyzd undr h avragd rngh 0.0MPa. h + ( D h) = & + & + & n (3) Th h rul ih h S.D. 0MPa hod only o rak, bau r rahd up o 0MPa i, a dribd in prviou. Afr ha, ning bhavior hr /h i h lop h orpion/dorpion a obrvd. Hovr, onidring andard dviaion (S.D.) hangd fraur pro dramaially iohrm (alo alld moiur apaiy). Th govrning quaion (Equaion 3) mu b ompld from rain ning o rain hardning, a hon by appropria boundary and iniial ondiion. in Figur 4. In a phyial bhavior, fir rak our a ak ro ion in a pimn, and - Th rlaion bn h amoun vaporabl ar and rlaiv humidiy i alld adorpion ond rak our a ondly ak ro ion. iohrm if maurd ih inraing rlaiviy Squnial raking along pimn axi m o b humidiy and dorpion iohrm in h oppoi rongly affd by diribuion marix rngh. a. Ngling hir diffrn (Xi al. 994), in Rgarding analyial rul, largr andard dviaion marix rngh produd highr rain apa- h folloing, orpion iohrm ill b ud ih rfrn o boh orpion and dorpion ondiion. iy in hi analyi, undr h aumpion onan By h ay, if h hyri h moiur bridging rngh fibr. iohrm ould b akn ino aoun, o diffrn rlaion, vaporabl ar v rlaiv humidiy, mu b ud aording o h ign h variaion h rlaiviy humidiy. Th hap h orpion iohrm for HPC i inflund by many paramr, pially ho ha influn xn and ra h hmial raion and, in urn, drmin por ruur and por iz diribuion (ar-o-mn raio, mn hmial ompoiion, SF onn, uring im and mhod, mpraur, mix addiiv,.). In h liraur variou formulaion an b found o drib h orpion iohrm normal onr (Xi al. 994). Hovr, in h prn papr h mi-mpirial xprion propod by Figur 6. Crak parn pimn onidring diribuion marix rngh (.5% a). Norling Mjornll (997) i adopd bau i xpliily aoun for h voluion hydraion raion and SF onn. Thi orpion iohrm rad ( h,, ) = G (, ) + 0( g ) h 0( g ) h K (, ) (4) hr h fir rm (gl iohrm) rprn h phyially bound (adorbd) ar and h ond rm (apillary iohrm) rprn h apillary ar. Thi xprion i valid only for lo onn SF. Th fiin G rprn h amoun ar pr uni volum hld in h gl por a 00% rlaiv humidiy, and i an b xprd (Norling Mjornll 997) a G (, ) = k + k vg vg (5) hr k vg and k vg ar marial paramr. From h maximum amoun ar pr uni volum ha an fill all por (boh apillary por and gl por), on an alula K a on obain Figur 7. Sr rain urv onidring andard dviaion marix rngh. g G 0 (6) KTablG (, 2. ) = Avragd numbr fibr in a ro ion and i andard dviaion. g Avragd numbr fibr Sandard dviaion in a ro ion Ca Th marial paramr k vg and k52 vg and g an b Ca2 alibrad by fiing xprimnal 565 daa rlvan o Ca fr (vaporabl) ar onn in onr a variou ag (Di Luzio & Cuai 2009b). 2.2 Tmpraur voluion No ha, a arly ag, in h hmial raion aoiad ih mn hydraion and SF raion ar xohrmi, h mpraur fild i no uniform for non-adiabai ym vn if h nvironmnal mpraur i onan. Ha onduion an b dribd in onr, a la for mpraur no xding 00 C (Bažan & Kaplan 996), by Fourir la, hih rad q = λ T (7) hr q i h ha flux, T i h abolu mpraur, and λ i h ha onduiviy; in hi Figur 8. Numbr fibr along pimn axi. Proding FraMCoS-7, May 23-28, 200

7 Figur 5 ho avragd rak idh during fraur pro in h a.5% a. Th analyial modl imula ady a raking bhavior ha man no ignifian inra rak idh up o final fraur, and hi bhavior i imilar o xprimnal rnd (Li 2003). Figur 6 ho rak parn in h a.5% ih S.D.=.0MPa. Alhough prviou analyi ihou onidring S.D. marix rngh indiad onan rak paing a hon in Figur 2, h analyi onidring S.D. marix rngh ho non uniform rak paing in ah p bau randomly diribud marix rngh along pimn axi. Bad on hi onp onidring diribuion marix rngh, diffrn fibr onn a an b alo imulad. Epially, lighly rain hardning hon in Figur 7(b) a rprodud, hil h prviou analyi imad ning bhavior.0% a, a hon in Figur (b). Figur 9. Sr ain urv pimn ih diffrn fibr diribuion. 4.3 Eff fibr diribuion along pimn axi In ordr o a h ff fibr diribuion in h pimn, analyi a ondud by hanging h andard dviaion numbr fibr aro h ah ro ion, hih i orrponding o ponially rak plan. In hi modl, ah fibr i randomly poiiond ihin h pimn, and avragd numbr fibr aro ah ro ion a abou 28,800, a abulad in Tabl 2. Thr kind analyi ih diffrn andard dviaion numbr fibr in ah ro ion (SD=0, 52, 565 and 396) r ondud, a hon in Figur 8. Figur 9 ho h r-rain rlaionhip h pimn having diffrn fibr diribuion. Th analyial rul ho ha largr diribuion fibr having ro ion ih l fibr ignifianly rdu rain apaiy. Sin l fibr impar draing bridging rngh a a ro ion o a ompoi, rduion no only rain apaiy = D ( h, T ) bu halo ulima rngh a obrvd a hon in Figur 9. Th proporionaliy fiin D(h,T) moiur prmabiliy and i i a nonlina 5 CONCLUSIONS h rlaiv humidiy h and mpraur & Najjar 972). Th moiur ma balan In hi udy, a hr-dimnional ha h variaion in analyi im mhod h ar ma ih dirizd volum hor fibr onr a ad (ar onn hrough ) b q h analyi high divrgn rngh rain h hardning moiur flux mniiou ompoi (UHP-SHCC). Th finding ob- aind inlud h folloing: = () By analyi ondud on a marial ih high rngh rain Th hardning ar onn and mulipl an b xprd fin a rak in nion (UHP-SHCC), h vaporabl hi ar modl roughly (apillary a imulad mhanial vapor, rpon and adorbd rprnd ar) by and r h non- rain urv and (hmially rak parn. bound) ar n (Mil (2) Paramri Panazopoulo analyi a & ondud Mill 995). o inrpr fraur mhanim aum ha h rain vaporabl hardning ar and i a fu I i ra mulipl fin rak rlaiv in UHP-SHCC. humidiy, h, Srain dgr apaiy hydraion a inrad du dgr o largr ilia diribuion fum raion, marix, i.. = rngh. Srain apaiy, = ag-dpndn hovr, drad orpion/dorpion ih inraing diribuion (Norling Mjonll fibr. 997). Undr hi aum by ubiuing Equaion ino Equai Th auhor innd obain o nhan h modl o a o b uful for h dvlopmn fibr rinforing marial. h + ( D h) = & + & + h h REFERENCES hr /h i h lop h orpion/ Bolandr,.E. & Saio, S. 997.Dir modling horfibr rinformn iohrm in mniiou (alo ompoi, alld moiur Advand apa Cmn Bad Marial, govrning 6, quaion (Equaion 3) mu b Bolandr,.E. & Bron, by appropria S Cohiv boundary Zon and Modling iniial ondii Fraur in Irrgular Th Lai, rlaion Fraur bn Mhani h amoun Conr Sruur, Pro. ar and FRAMCOS-5, rlaiv humidiy i alld Cox, H Th Elaiiy and Srngh Papr and Ohr iohrm if maurd ih inraing Fibrou Marial, Briih. Applid Phyi, 3. Kamal, A., Kunida, humidiy M., Uda, and N. dorpion & Nakamura, iohrm H in h Evaluaion Crak a. Opning Ngling Prforman hir diffrn a Rpair Marial ih Srain h Hardning folloing, Bhavior, orpion Cmn iohrm and Con-ill b (Xi al. r Compoi, rfrn 30(0), o boh orpion and dorpion Kanda, T. & Li, V. C Inrfa Propry and Apparn By h ay, if h hyri h Srngh High-Srngh Hydrophili Fibr in Cmn Marix, ournal iohrm Marial ould in Civil b Enginring, akn ino ASCE, aoun, o 0(), 5-3. rlaion, vaporabl ar v rlaiv humi Kaai, T., 978. N b Dir ud aording Modl and o Thir h ign Appliaion h varia o Simi Rpon rlaiviy Analyi humidiy. Sruur, Th Nular hap Engi-onring Dign, 48, iohrm for HPC i inflund by many p h Kozaa, K Sudy on Tnil Fraur Mhanim Ulra High Prforman-Srain Hardning Cmniiou pially ho ha influn xn and Compoi and hmial I Modling, raion Mar Thi and, in Nagoya urn, drm Univriy. (in apan) ruur and por iz diribuion (arraio, E. Brühilr, mn E hmial &Nakamura, ompoiion, H SF Kunida, M., Dnarié, Challng for Srain uring Hardning im and Cmniiou mhod, mpraur, Compoi mix Dformabiliy Vru Marix Dniy, Pro. h fifh.). In h liraur variou formulaio Inrnaional RILEM Workhop on HPFRCC, Kunida, M., Kozaa, found K., Uda, o drib N., Nakamura, h orpion H. & Ogura, iohrm H Thr-dimnional onr (Xi Mo-al al. Analyi 994). for Hovr, Srain in h Hardning Cmniiou papr Compoi, h mi-mpirial Crp, Shrinkag xprion and pro Durabiliy Mhani Norling Conr Mjornll and Conr (997) Sruur, i adopd b Pro. CONCREEP8, Proding FraMCoS-7, May 23-28, 200

8 Kozaa, = D ( hk.,, T ) Kunida, h M., Kanda, T. & Nakamura, H () Bond Propri Organi Fibr Embddd in Ulra High Srngh Marix, Proding h apan Conr Iniu, Th 30(), proporionaliy (in apan) fiin D(h,T) i alld Li, moiur V.C., Wang, prmabiliy Y. & Bakr, and S. i 990. i a Eff nonlinar Inlining funion Angl, h rlaiv Bundling, humidiy and Surfa h and Tramn mpraur on Synhi T (Bažan Fibr & Najjar Pull-ou 972). from a Th Cmn moiur Marix, ma Compoi, balan 2(2), rquir ha h variaion in im h ar ma pr uni Li, V.C From Miromhani o Sruural Enginring volum Th Dign onr Cmniiou (ar onn Compoi ) b for qual Civil o Engi- h divrgn h moiur flux = (2) Th ar onn an b xprd a h um h vaporabl ar (apillary ar, ar vapor, and adorbd ar) and h non-vaporabl (hmially bound) ar n (Mill 966, Panazopoulo & Mill 995). I i raonabl o aum ha h vaporabl ar i a funion rlaiv humidiy, h, dgr hydraion,, and dgr ilia fum raion,, i.. = (h,, ) = ag-dpndn orpion/dorpion iohrm (Norling Mjonll 997). Undr hi aumpion and by ubiuing Equaion ino Equaion 2 on obain h + h ( D h) = n (3) h & + & + hr /h i h lop h orpion/dorpion iohrm (alo alld moiur apaiy). Th govrning quaion (Equaion 3) mu b ompld by appropria boundary and iniial ondiion. Th rlaion bn h amoun vaporabl ar and rlaiv humidiy i alld adorpion iohrm if maurd ih inraing rlaiviy humidiy and dorpion iohrm in h oppoi a. Ngling hir diffrn (Xi al. 994), in h folloing, orpion iohrm ill b ud ih rfrn o boh orpion and dorpion ondiion. By h ay, if h hyri h moiur iohrm ould b akn ino aoun, o diffrn rlaion, vaporabl ar v rlaiv humidiy, mu b ud aording o h ign h variaion h rlaiviy humidiy. Th hap h orpion iohrm for HPC i inflund by many paramr, pially ho ha influn xn and ra h hmial raion and, in urn, drmin por ruur and por iz diribuion (ar-o-mn raio, mn hmial ompoiion, SF onn, uring im and mhod, mpraur, mix addiiv,.). In h liraur variou formulaion an b found o drib h orpion iohrm normal onr (Xi al. 994). Hovr, in h prn papr h mi-mpirial xprion propod by Norling Mjornll (997) i adopd bau i & xpliily nring Appliaion, aoun for ournal h voluion Sruural Mhani hydraion raion and Earhquak and SF Enginring, onn. Thi SCE, orpion 0(2), iohrm Li, rad V. C. 2003: On Enginrd Cmniiou Compoi(ECC) A Rvi h Marial and I Appliaion,. Advand Conr Thnology, (3), Thomur,. & Bolandr,.E. & Kunida, M Rduing Mh Bia on Fraur ihin Rigid-Body Spring Nork,,, ) = G ( ) ( h ournal Sruural, + Mhani and Earhquak Enginring, SCE, No.682/I-56, ( g ) h (4) 0( g ) h K (, ) hr h fir rm (gl iohrm) rprn h phyially bound (adorbd) ar and h ond rm (apillary iohrm) rprn h apillary ar. Thi xprion i valid only for lo onn SF. Th fiin G rprn h amoun ar pr uni volum hld in h gl por a 00% rlaiv humidiy, and i an b xprd (Norling Mjornll 997) a G (, ) = k + k vg vg (5) hr k vg and k vg ar marial paramr. From h maximum amoun ar pr uni volum ha an fill all por (boh apillary por and gl por), on an alula K a on obain K (, ) = g G 0 g (6) Th marial paramr k vg and k vg and g an b alibrad by fiing xprimnal daa rlvan o fr (vaporabl) ar onn in onr a variou ag (Di Luzio & Cuai 2009b). 2.2 Tmpraur voluion No ha, a arly ag, in h hmial raion aoiad ih mn hydraion and SF raion ar xohrmi, h mpraur fild i no uniform for non-adiabai ym vn if h nvironmnal mpraur i onan. Ha onduion an b dribd in onr, a la for mpraur no xding 00 C (Bažan & Kaplan 996), by Fourir la, hih rad q = λ T (7) hr q i h ha flux, T i h abolu mpraur, and λ i h ha onduiviy; in hi Proding FraMCoS-7, May 23-28, 200