Size-effect of fracture parameters for crack propagation in concrete: a comparative study

Transcription

1 Computr and Concrt, Vol. 9, No. 1 (2012) Tchnical Not Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy Shailndra Kumar 1,2 and S.V. Barai* 2 1 Dpartmnt of Civil Enginring, National Intitut of Tchnology, Jamhdpur , India 2 Dpartmnt of Civil Enginring, Indian Intitut of Tchnology, Kharagpur , India (Rcivd Sptmbr 1, 2009, Rvid Fbruary 18, 2011, Accptd April 6, 2011) Abtract. Th iz-ffct tudy of variou fractur paramtr obtaind from two paramtr fractur modl, ffctiv crack modl, doubl-k fractur modl and doubl-g fractur modl i prntd in th papr. Fictitiou crack modl (FCM) for thr-point bnd tt gomtry for crackd concrt bam of laboratory iz rang mm i dvlopd and th diffrnt fractur paramtr from iz ffct modl, ffctiv crack modl, doubl-k fractur modl and doubl-g fractur modl ar valuatd uing th input data obtaind from FCM. In addition, th fractur paramtr of two paramtr fractur modl ar obtaind uing th mathmatical cofficint availabl in litratur. From th tudy it i concludd that th fractur paramtr obtaind from variou nonlinar fractur modl including th doubl-k and doubl-g fractur modl ar influncd by th pcimn iz. Th fractur paramtr maintain om dft intrrlationhip dpnding upon th pcimn iz and rlativ iz of tial notch lngth. Kyword: concrt fractur; fractur proc zon; cohiv tr ditribution; nonlinar fractur modl; iz-ffct; thr-point bnding tt. 1. Introduction During , vral xprimntal and numrical invtigation provd that th claical form of linar latic fractur mchanic (LEFM) approach cannot b applid to normal iz concrt mmbr. Th inapplicability of LEFM wa du to th prnc of larg and variabl iz of fractur proc zon (FPZ) ahad of th crack-tip. From th pat tudi it bcam clar that th fractur mchanic can b a uful and powrful tool for th analyi of growth of ditributd cracking and it localization in concrt if th oftning bhavior of th matrial i takn into accot. Th actual application of tnion-oftning contitutiv law wa known til about mid Thn uing nonlinar fractur mchanic, Hillrborg and co-workr (1976) put forward a pionr work in which th dvlopmnt of fictitiou crack modl (FCM) or cohiv crack modl (CCM) for th crack propagation tudy of rinforcd concrt bam wa introducd. Thraftr, a numbr of nonlinar fractur modl hav bn propod and ud to prdict th nonlinar fractur bhavior of quaibrittl matrial lik concrt. Th ar: crack band modl (CBM) (Baz ant and Oh 1983), two paramtr fractur modl (TPFM) (Jnq and Shah 1985), iz ffct modl (SEM) (Baz ant t al. 1986), ffctiv crack modl (ECM) (Nallathambi and Karihaloo 1986), K R -curv mthod bad on cohiv forc ditribution in th FPZ (Xu and Rinhardt 1998, 1999a), doubl-k * Corrponding author, Ph. D., kbarai@civil.iitkgp.rnt.in

2 2 Shailndra Kumar and S.V. Barai fractur modl (DKFM) (Xu and Rinhardt 1999a,b,c) and doubl-g fractur modl (DGFM) (Xu and Zhang 2008). FCM and CBM ar bad on th numrical mthod whra TPFM, SEM, ECM, K R -curv mthod, DKFM and DGFM ar bad on th modifid LEFM concpt. A brif litratur urvy on th variou fractur paramtr obtaind from diffrnt nonlinar fractur modl i carrid out and prntd in th ubqunt ction. Th prnt contribution will xplor th bhavior of th diffrnt fractur paramtr with rpct to pcimn iz and rlativ iz of tial notch lngth. Th main objctiv of th papr i to how th iz-ffct bhavior of th diffrnt fractur paramtr obtaind uing TPFM, SEM, ECM, DKFM and DGFM in a rlativ mannr. Th intrrlationhip of th fractur paramtr i alo focud and analyzd. For thi purpo, FCM for thr-point bnd tt gomtry for crackd concrt bam of laboratory iz rang mm i dvlopd and th fractur paramtr from SEM, ECM, DKFM and DGFM ar valuatd uing th rquird input data obtaind from FCM. In addition, imilar rult of TPFM ar obtaind with th hlp of fractur pak load obtaind from FCM and th mathmatical cofficint rportd in th litratur (Plana and Elic 1990). 2. Litratur rviw Th cohiv crack modl i a impl mthod and i an idalizd approximation of a phyical localizd fractur zon. Th modl ha grat potntial to dcrib th nonlinar matrial bhavior in th victy of a crack and at th crack-tip. Th non-linarity i automatically introducd by uing cohiv tr-crack opning diplacmnt rlation (oftning fction) acro th crack fac nar th crack-tip, which lad tr intnity factor to b zro. Th cohiv crack mthod wa firt propod by Barnblatt (1962) and Dugdal (1960). Whil Barnblatt (1962) applid cohiv crack mthod to analyz th brittl fractur bhavior, Dugdal (1960) introducd it to modl ductil fractur bhavior of a matrial. Hillrborg t al. (1976) tially applid cohiv crack mthod (or fictitiou crack modl) to imulat th oftning damag of concrt tructur. Thr matrial proprti uch a modulu of laticity E, iaxial tnil trngth f t, and pcific fractur nrgy G F ar rquird to dcrib th cohiv crack modl. Th G F i dfind a th amot of nrgy ncary to crat on it of ara of a crack. In addition, th hap of oftning fction of concrt play an important rol on th rult prdictd by cohiv crack modl. RILEM Tchnical Committ 50-FMC (1985) propod a mthod uing thr point bnd tt (TPBT) bam to obtain th valu of G F. Although th RILEM rcommndation (1985) prntd xprimntal dtrmination of fractur nrgy uing thr-point bnd tt, it ha bn a mattr of dicuion in th pat bcau th valu of fractur nrgy obtaind from diffrnt xprimnt uing RILEM procdur wr affctd by th pcimn iz. Plana and co-workr (Guina t al. 1992, Plana t al. 1992, Elic t al. 1992, 1997) carrid out a carful analyi of th tt procdur for dtrmination of fractur nrgy. In th xtniv tudy, Plana and co-workr prntd that th apparnt iz ffct on th fractur nrgy could b rducd by nhancing th xprimntal mthod. A dtaild xplanation for xprimntal dtrmination of cohiv crack fractur paramtr uing TPBT uch a: tnil trngth, tial part of th oftning fction, fractur nrgy and bilinar oftning curv can b n in Sction 7.3 of th txt book (Baz ant and Plana 1998). Extniv litratur i availabl on th u of cohiv crack modl. Th rcnt tudi (Kim t al. 2004, Rolr t al. 2007, Park t al. 2008, Zhao t al. 2008, Elic t al. 2009) how th application of cohiv crack modl for charactrizing th oftning fction and prdicting th nonlinar fractur charactritic

3 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 3 of concrt uing variou tt configuration. Baz ant and Oh (1983) dvlopd th crack band modl in which th fractur proc zon i modld a a ytm of paralll crack that ar continuouly ditributd (mard) in th ft lmnt. Th mard or ditributd crack i jutifid du to prnc of random natur of microtructur. Th matrial fractur proprti ar charactrizd by thr paramtr uch a: f t, width of fractur proc zon ovr which th microcrack ar aumd to b iformly prad h c and G F (dfind a th product of th ara dr tr-train curv and h c ). Th matrial bhavior i charactrizd by th contitutiv tr-train rlationhip. Th two paramtr fractur modl wa dvlopd by Jnq and Shah (1985). In thi modl, th actual crack i rplacd by an quivalnt fictitiou crack. Th modl involv two valid fractur paramtr for cmntitiou matrial: th critical tr intnity factor at th tip of th quivalnt crack lngth at pak load and th corrponding valu of th crack-tip opning diplacmnt (CTOD) known a critical crack-tip opning diplacmnt CTOD c. Th loading and loading crack mouth opning diplacmnt (CMOD) complianc of tandard thr point-bnd pcimn wr ud to dtrmin th valu of critical ffctiv crack lngth a c. RILEM (1990a) procdur i followd for dtrmng th fractur paramtr and CTOD c from th tt rult carrid out on th TPBT pcimn. Baz ant and co-workr (Baz ant t al. 1986) introducd iz ffct modl, which dcrib th matrial fractur bhavior uing two paramtr: th fractur nrgy G f and critical ffctiv crack lngth xtnion c f at pak load for inftly larg tt pcimn. Th fractur paramtr ar dtrmind from th maximum load of gomtrically imilar notchd pcimn of diffrnt iz according to th RILEM (1990b) guidlin. Nallathambi and Karihaloo (1986) introducd ffctiv crack modl to valuat ffctiv crack xtnion a c bad on complianc calibration approach. Th baic principl of dtrmng th ffctiv crack xtnion i to obtain th mid-pan dflction of th tandard thr-point-bam tt uing cant complianc from a typical load-dflction plot up to th pak load P u and corrponding dflction i δ u. According to th ffctiv crack modl, th fractur in th ral tructur t in whn th tr intnity factor bcom critical at crack lngth qual to a. Th dtail of formulation and calculation procdur of th fractur paramtr of ffctiv crack modl can b n in Karihaloo and Nallathambi (1991). Xu and Rinhardt (1999a) prntd th thr tag of crack propagation in concrt: crack tiation, tabl crack propagation and tabl crack propagation bad on tt of th larg iz compact tnion (CT) pcimn and mall iz TPBT pcimn. Th analyi of th tt rult advocatd doubl-k fractur modl which can rprnt all th thr tag of cracking phnomna in th fractur proc of concrt. According to thi critrion, th two iz indpndnt paramtr can b ud to dcrib th fractur proc of concrt. Th firt paramtr i trmd a tial cracking toughn K KC which i dirctly calculatd by th tial cracking load and tial notch lngth uing LEFM formula. Th othr paramtr i known a tabl fractur toughn which can b obtaind by pak load and ffctiv crack lngth a c uing th am LEFM formula. From th availabl xprimntal rult, it wa alo hown that doubl-k fractur paramtr and ar not dpndnt on iz of th pcimn. Furthr, th valuatd valu of th critical crack-tip opning diplacmnt CTOD c, howd that thi valu appar to b iz dpndnt (Xu and Rinhardt 1999b). Th paramtr and computd from fractur tt on th mall iz wdg-plitting tt (WST) pcimn how that th ar indpndnt of th rlativ iz of tial notch lngth, lightly dpndnt on th iz and indpndnt of th thickn of th pcimn (Xu and

4 4 Shailndra Kumar and S.V. Barai Rinhardt 1999c). Th K R -curv mthod bad on cohiv tr ditribution in th FPZ introducd by Xu and Rinhardt (1998, 1999a) for complt fractur proc dcription of concrt diffr from th convntional mthod of th R-curv. Th ditribution of cohiv tr along th FPZ at diffrnt tag of loading condition i takn into accot in ordr to valuat th K R -curv which can analyz th complt fractur proc of concrt. In addition, th doubl-k fractur paramtr wr introducd in th tability analyi uing th K R -curv. Rcntly, Xu and Zhang (2008) propod th doubl-g fractur critrion bad on th concpt of nrgy rla rat coniting of two charactritic fractur paramtr: th tiation fractur nrgy rla G IC and th tabl fractur nrgy rla G IC. Th valu of G IC i dfind a th Griffith fractur urfac nrgy of concrt mix in which th matrix rmain till in latic tat dr th tial cracking load P and th tial crack lngth a o. Onc th load valu P on th tructur i incrad byond to th valu of P, a nw crack urfac (macro-cracking) i formd and th cohiv tr along th nw crack urfac tart to act. At th ont of tabl crack propagation, th total nrgy rla G IC conit of tiation fractur nrgy rla G IC and th C critical valu of th cohiv braking nrgy G IC. Th fractur modl bad on modifid LEFM (TPFM, SEM, ECM, DKFM, K R -curv aociatd with cohiv forc ditribution) ar bad on tr intnity factor (SIF) concpt xcpt for th doubl-g fractur modl which i bad on th nrgy approach. Thrby, th ductility proprty i alo aociatd with th nrgy approach bad fractur paramtr. Extniv tt rult uing two pcimn gomtri namly TPBT of iz rang mm and WST of iz rang mm on dtrmination of doubl-g fractur paramtr and doubl-k fractur paramtr wr prntd by Xu and Zhang (2008). Within crtain cattr rang in th tt rult it wa concludd that th doubl-g fractur paramtr wr iz indpndnt ovr th iz rang of 200 mm. Th doubl-g fractur paramtr wr convrtd to th ffctiv tiation toughn and ffctiv tabl fractur toughn quivalnt to th doubl-k fractur paramtr uing th rlationhip: K = EG. It wa fod that th valu of quivalnt fractur paramtr in trm of SIF at th ont of crack tiation and th ont of tabl fractur uing doubl-g fractur critrion and doubl-k fractur critrion ar in vry clo. It i wll known that th nonlinar fractur modl captur adquatly th tructural iz-ffct Fig. 1 Siz-ffct a a plot of nominal trngth v. iz on a bilogarithmic cal

5 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 5 ovr th uful rang of applicability. Th iz-ffct i th dcra in nominal trngth of gomtrically imilar tructur ubjctd to ymmtrical load whn th charactritic iz of th tructur i incrad. Thr ar two xtrm of iz-ffct law a hown in Fig. 1: (i) trngth critria and (ii) LEFM iz-ffct. Th formr yild no iz-ffct whra th lattr how th trongt iz-ffct i.. nominal trngth i invrly proportional to th quar root of th tructural dimnion. Karihaloo and Nallathambi (1989) ud th tt data of thr-point bnding pcimn for th comparion of improvd ECM and th TPFM. It wa fod that prdiction from both th modl ar in good agrmnt. From th variou ourc of xprimntal rult, Karihaloo and Nallathambi (1989) howd that fractur toughn valu obtaind from th ECM and th TPFM and alo from th ECM and th SEM ar in good agrmnt. A imilar prdiction btwn ECM and TPFM wa alo obrvd from th comparion of fractur paramtr uing diffrnt ourc of xprimntal rult (Karihaloo and Nallathambi 1991). It wa fod that irrpctiv of th concrt trngth, th fractur paramtr obtaind uing ECM ( and a c ) ar practically inditinguihabl from th corrponding paramtr ( and CTOD c ) dtrmind uing TPFM. Th iz-ffct rlationhip btwn FCM, SEM and TPFM wr dvlopd by Plana and Elic (1990) that prdictd almot th am fractur load for practical iz rang ( mm) of prcrackd concrt bam for TPBT gomtry. In addition, it wa obrvd from th iz-ffct tudy that fractur load prdictd by th SEM and th TPFM could divrg about 28% and 31% rpctivly for aymptotically larg iz (D ) bam. Latr, bad on th imilar approach (Plana and Elic 1990), a iz-ffct tudy btwn FCM and ECM wa prntd by Karihaloo and Nallathambi (1990). In th tudy, it wa hown that th prdictd fractur load from both th modl for th practical iz rang of TPBT configuration ar inditinguihabl and in th aymptotic limit (of inft iz), th prdiction diffrd by about 17%. It wa alo hown that th numrical rult of in TPFM and G f in SEM ar vry imilar (Tang t al. 1992, Baz ant t al. 1991, Baz ant 2002) and approximatly quivalnt throughout th whol iz rang and th cond paramtr of ach modl can b obtaind by CTOD c = 32G f c f whr E' i E/(1-ν 2 ) for plain train and i E for plain tr ca. Plana and co-workr (Plana and Elic 1990, 1991, 1992, Elic and Plana 1996) carrid out xtniv tudi on iz-ffct of concrt pcimn uing variou fractur modl including CCM, TPFM and SEM. For cohiv crack modl, a corrlation btwn fractur nrgy and th charactritic lngth a th baic paramtr wa drivd by Plana and Elic (1990, 1991) which dpnd on th hap of th oftning fction. Furthr, th pak load for th cohiv crack modl can b compltly dfind uing tial linar oftning for th normal xprimntal rang of pcimn iz. An lgant dcription for corrlation of cohiv crack modl with Baz ant SEM and Jnq-Shah TPFM ha bn prntd by Baz ant and Plana (1998) in Chaptr 7 of thir txt book. It ha bn pointd out in th book that a corrlation btwn th fractur paramtr of th variou modl can b tablihd uing iz ffct rult. Bad on th rult of Plana and Elic (1990, 1992), a rlationhip btwn cohiv crack fractur nrgy, tnil trngth and horizontal intrcpt from th tial linar oftning for quai-xponntial oftning fction ha prntd in th abov book. Th rlation can b furthr ud to dtrmin th cohiv crack charactritic lngth and th fractur paramtr of Baz ant SEM and Jnq-Shah TPFM. Th dtaild dcription and πe (1)

6 6 Shailndra Kumar and S.V. Barai rlationhip can b n in th txt book (Baz ant and Plana 1998). Ouyang t al. (1996) tablihd an quivalncy btwn TPFM and SEM bad on inftly larg iz pcimn. It wa fod that th rlationhip btwn CTOD c and c f thortically dpnd on both pcimn gomtry and tial crack lngth and both th fractur modl can raonably prdict fractur bhavior of quai-brittl matrial. Elic and Plana (1996) alo prntd a comprhniv rviw ovr th rlvanc to iz ffct prdiction bad on comparion of diffrnt modl of concrt fractur uing cohiv crack modl a th rfrnc. It wa fod that implr modl uch a: th quivalnt latic crack aociatd with R-curv approach, Baz ant' SEM and Jnq-Shah' TPFM fit inid thi chm and ar hirarchically rlatd. Xu t al. (2003) conductd concrt fractur xprimnt on both th thr-point bnding notchd bam and th wdg plitting pcimn with diffrnt rlativ tial crack lngth according to th xprimntal rquirmnt for dtrmng th fractur paramtr in th doubl-k fractur modl and th two paramtr fractur modl. Th comparativ rult howd that th critical crack lngth a c dtrmind uing th two diffrnt modl ar hardly diffrnt. Th valu of and CTOD c maurd for DKFM ar in good agrmnt with and CTOD c maurd for TPFM. Hanon and Ingraffa (2003) dvlopd th iz-ffct, two-paramtr, and fictitiou crack modl numrically to prdict crack growth in matrial for thr-point bnd tt. Th invtigation howd that if th thr modl mut prdict th am rpon for inftly larg tructur, thy do not alway prdict th am rpon on th laboratory iz pcimn. Howvr, th thr modl do agr at th laboratory iz pcimn for crtain rang of tnion oftning paramtr. It md that th rlativ iz of tnion oftning zon mut b l than approximatly 15% of th ligamnt lngth for th two-paramtr fractur modl to prdict imilar bhavior a of fictitiou crack modl. Furthr, it appard that th total rlativ iz of tnion oftning zon i not an indication for th iz-ffct modl to prdict th imilar rpon a of th fictitiou crack modl. Rolr t al. (2007) plottd th iz-ffct bhavior of xprimntal rult, numrical imulation uing cohiv crack modl, iz-ffct modl and two paramtr fractur modl for thr-pointbnd tt pcimn. From th analyi of rult it i fod that th iz-ffct bhavior calculatd from SEM and TPFM rmbl cloly. From th fractur tt (Xu and Zhang 2008), it i alo clar that th corrponding valu of doubl-k fractur paramtr and doubl-g fractur paramtr ar quivalnt at tial cracking load and tabl pak load. Cuati and Schaffrt (2009) prntd prci numrical imulation bad on cohiv crack modl for for computation of iz-ffct curv uing typical tt configuration. Th rult wr analyzd with rfrnc to SEM to invtigat th rlationhip btwn th iz-ffct curv and th iz ffct law. Th practical implication of th tudy wr alo dicud in rlation to th u of th iz-ffct curv or th iz ffct law for idntification of th oftning law paramtr through th iz ffct mthod. Exprimntal rult and analy availabl in th litratur (Xu and Rinhardt 1999a, 1999b, 1999c) how that th doubl-k fractur paramtr ar almot indpndnt of pcimn iz. Furthrmor, it i pointd that th principl of th dvlopmnt of fictitiou crack modl and doubl-k fractur modl ar contrary to ach othr. In th dvlopmnt of fictitiou crack modl, no ingularity i conidrd at th crack-tip whra, in th doubl-k fractur modl, th cohiv tr do not ncarily abolih th tr ingularity condition. To thi nd, th author (Kumar and Barai 2010) invtigatd th iz-ffct tudy btwn FCM and DKFM imilar to tho for

7 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 7 TPFM, SEM and ECM with rfrnc to FCM. From th tudy uing thr-point bnd tt pcimn, it wa fod that both th fractur modl (fictitiou crack modl and doubl-k fractur modl) yild almot th am valu of tabl fractur load and crack tiation load up to 400 mm dpth of th bam, byond thi th diffrnc in prdictd load may incra. Th prdiction in aymptotic bhavior of crack tiation load and tabl fractur load with rgard to fictitiou crack modl ar rlativly varid. Th prdiction ar mor conrvativ by about 20 and 22% rpctivly for aymptotic larg iz (D ). Th author (Kumar and Barai 2008, 2009a) ud TPBT and CT pcimn of iz rang 100 D 600 mm and 100 D 500 mm rpctivly to carry out numrical tudi on th doubl-k fractur paramtr. In both th tudi it wa dmontratd that th fractur paramtr and ar influncd by th pcimn iz. Th doubl-g fractur critrion i imilar to th doubl-k fractur modl that i th cohiv tr do not ncarily abolih th tr ingularity condition lik to th fictitiou crack modl. In th numrical tudy (Kumar and Barai 2009a), th input data obtaind from FCM wa ud to obtaind doubl-g fractur paramtr and it wa obrvd that th paramtr and ar influncd by th pcimn iz. From th prviou numrical tudi carrid out by diffrnt rarchr it i clar that mot of th fractur paramtr ar affctd by pcimn iz. Th rult hav bn rportd paratly and hnc it i difficult to mak a prci comparion among thm. Morovr, a comparativ tudy rgarding th othr paramtr (uch a CTOD c of TPFM, a of ECM, ac and CTOD c of DKFM or DGFM, c f of SEM) in ach of th fractur modl i not focud jointly in th litratur. Th prnt papr will addr a comparativ iz-ffct tudy uing fractur paramtr obtaind from TPFM, SEM, ECM, DKFM and DGFM with rfrnc to FCM. Sinc cohiv crack modl i widly ud to tudy th crack propagation phnomnon of concrt, th am modl i applid to obtain th input paramtr for th othr fractur modl. For prdicting th crack formation, it propagation and load-cmod rpon during fractur and fatigu in concrt, th rcnt tudi can alo b rfrrd to. Rcntly, Gar (2007) ud th dicrt crack-concpt to tudy th 3D propagation of tnil-dominatd failur in plain concrt. Th Partition of Unity Ft Elmnt Mthod (PUFEM) wa applid and th trong dicontinuity approach wa followd in th numrical modling. Th modl wa applid to tudy concrt failur during th PCT3D tt and th prdictd numrical rult wr compard to xprimntal data. Th P-CMOD rpon, th crack formation and th train fild wr compard to xprimntal data of th PCT3D tt. Th dvlopd numrical concpt provid a clar intrfac for contitutiv modl and allow an invtigation of thir impact on complx bhavior of concrt cracking dr 3D condition. Phillip (2009) dvlopd a nw modl uing modifid nrgy fctional to accot for molcular intraction in th victy of crack tip, rulting in Barnblatt cohiv forc, uch that th modl bcom fr of tr ingulariti. For th conitncy of th modl, th crack rvribility wa allowd and local mmizr of th nrgy fctional wr conidrd. Th modl wa olvd in it global a wll a in it local vrion for a impl on-dimnional xampl. It wa concludd that whil th global nrgy mmization ha a nonnical rult, prdicting failur dr any nonzro load, th local mmization corrctly prdict failur dr a critical poitiv load. Th modl alo corrctly prdict th location of crack formation. Alhoaibi (2010) prntd th numrical imulation of fatigu crack growth in arbitrary 2D gomtri dr contant amplitud loading by th uing a nw ft lmnt oftwar. In th imulation, an automatic adaptiv mh wa carrid out in th victy of th crack front nod and in th lmnt which rprntd th highr tr ditribution. Th fatigu crack dirction and th corrponding tr-

8 8 Shailndra Kumar and S.V. Barai intnity factor wr timatd at ach mall crack incrmnt by mploying th diplacmnt xtrapolation tchniqu dr facilitation of ingular crack tip lmnt. A conitnt tranfr algorithm and a crack rlaxation mthod wr propod and implmntd in th modl. Uing vral tt pcimn, th prdictd fatigu lif wa validatd with rlvant xprimntal data and numrical rult obtaind by othr rarchr. Th comparion of th rult how that th dvlopd numrical modl i capabl of dmontrating th fatigu lif prdiction rult a wll a th fatigu crack path atifactorily. 3. Matrial proprti and dtrmination of fractur paramtr Fictitiou crack modl or cohiv crack modl for tandard pcimn of thr-point bnding tt a hown in Fig. 2 i dvlopd in th prnt tudy. In thi mthod, th govrning quation (Ptron 1981, Carpintri 1989) of crack opning diplacmnt (COD) along th potntial fractur lin i writtn. Effct of lf-wight of th bam i alo conidrd in th numrical modl. Th influnc cofficint of th COD quation ar dtrmind uing linar latic ft lmnt mthod. Th COD vctor i partitiond according to th nhancd algorithm introducd by Plana and Elic (1991). Finally, th ytm of nonlinar imultanou quation i dvlopd and olvd uing Nwton-Raphon mthod. Svral commonly ud hap of oftning curv uch a bilinar, xponntial, nonlinar, quai-xponntial, tc. ar availabl in th litratur. Th dtaild xprion of th oftning curv can b fod in th litratur (Kumar and Barai 2009b). Any of th oftning curv lik bilinar or nonlinar curv can b conidrd for th iz-ffct tudy howvr, quai-xponntial oftning curv i lctd in th prnt tudy bcau om of th paramtr of iz ffct rult of TPFM drivd by Plana and Elic (1990) hav bn ud in ordr to obtain fractur paramtr for TPFM. Th paramtr of quai-xponntial fction ud in th tudy ar: A = and B = Sam concrt mix (Plana and Elic 1990) i takn in th prnt invtigation for which f t =3.21 MPa, E =30 GPa, and G F = 103 N/m. Th valu of ν i aumd to b For TPBT pcimn of notchd concrt bam with B = 100 mm, iz rang 100 D 400 mm and S/D = 4, th ft lmnt analyi i carrid out for dtrmng th fractur pak load and th corrponding CMOD uing fictitiou crack modl at tial crack lngth/dpth (a o /D) ratio ranging btwn Four nodd ioparamtric lmnt ar conidrd for ft lmnt calculation. Th half of th bam i Fig. 2 Thr point bnding tt (TPBT) pcimn gomtry

9 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 9 Fig. 3 Ft lmnt dicrtization of TPBT Tabl 1 Pak load and corrponding CMOD for tandard TPBT obtaind uing FCM for matrial proprti: f t =3.21 MPa, E = 30 GPa and G F =103N/m D (mm) P u (N) a o /D CMOD c (µm) P u (N) CMOD c (µm) P u (N) CMOD c (µm) P u (N) CMOD c (µm) dicrtizd a hown in Fig. 3 and 80 numbr of qual lmnt ar takn along th dpth of th bam. Th pak load P u and corrponding critical valu of CMOD (CMOD c ) ar gaind from th numrical modl uing FCM ar prntd in Tabl 1. 2 Th paramtr l ch = EG F / f t of cohiv crack modl i ud for comparion of numrical rult. In addition, th maximum iz of coar aggrgat d max i takn a 19 mm for all th ubqunt computation. Sinc loading and loading during tt of fractur pcimn i rquird to obtain th fractur paramtr of TPFM: and CTOD c (CTOD c of TPFM) according to th procdur outlind in RILEM Draft Rcommndation TC89-FMT (1990a), it i not poibl with th availabl rult obtaind uing FCM to dtrmin th fractur paramtr. Thrfor, th paramtr i prcily valuatd with th hlp of invr analyi uing th xprion and mathmatical cofficint prntd by Plana and Elic (1990) in which th author dtrmind th fractur paramtr of TPFM for th am TPBT pcimn and matrial proprti. Th critical ffctiv crack xtnion for inft iz a c i dtrmind uing Eq. (1) in which c f = a c, G f = G FS. Finally, th iz-ffct quation of TPFM i cat in th following form.

10 10 Shailndra Kumar and S.V. Barai EG FC G FC a (2) 2 K INu G c k αo l ch = FS l ch k( α o ) D Whr α = a/d, k'(α) i th 1 t drivativ of k(α) with rpct to α, G FC i qual G F and G FS i th quivalnt fractur nrgy obtaind uing TPFM. In Eq. (2) th mathmatical cofficint a c /l ch wa obtaind a (Plana and Elic 1990) for ach gomtry (a o /D) ranging btwn within an accuracy lvl of 3%. Th tr intnity factor K IN corrponding to nominal tr σ N i dtrmind uing LEFM formula givn in Tada t al. (1985). For thr-point-bnding tt gomtry, S = 4D, th following formula ar ud. K IN = σ N Dk( α) Whr k(α) i a gomtric factor and σ N i th nominal tr in th bam du to xtrnal load P and lf wight of th tructur which i givn by ( ) (3) 1.99 ( ) α α ( 1 α )( α + 2.7α2 ) = ( 1+ 2α) ( 1 α) 3 2 k α (4) 3S σ N = P [ + w g S] (5) 4BD 2 whr th w g i lf wight pr it lngth of th tructur. Th K INu of Eq. (2) can b obtaind uing Eq. (3) in which: K IN = K INu for σ N = σ Nu (whn P = P u ) and α = α o = a o /D. In th prnt tudy, th valu of K INu i dtrmind uing th valu of P u obtaind from FCM for a particular TPBT pcimn. Thn, for a givn gomtry and matrial proprti, th G FS i dtrmind uing Eq. (2). Finally, th CTOD c i valuatd uing Eq. (1) and th i calculatd uing th following LEFM formula. = EG FS Th computd valu of both th fractur paramtr and CTOD c ar givn in Tabl 2. For th givn pak load and tial notch lngth, th fractur paramtr of SEM, G f and c f ar dtrmind adopting th procdur givn in RILEM Draft Rcommndation TC89-FMT (1990b) for thr-point bnd tt pcimn a hown in Fig. 2. Furthr, th quivalnt critical tr intnity b factor i obtaind uing th tandard LEFM quation for comparion purpo. Th rult ar prntd in Tabl 2. Fractur paramtr and a of ECM ar obtaind uing th quation givn by Karihaloo and Nallathambi (1990). In thi mthod firt of all th a i obtaind by uing th rgrion quation (Karihaloo and Nallathambi 1990) for givn matrial and gomtrical proprti of a TPBT pcimn and thn th valu of i calculatd uing LEFM quation. Both th fractur paramtr dtrmind ar prntd in Tabl 2 for TPBT pcimn at a o /D ratio ranging btwn Th tiation toughn and tabl fractur toughn of th TPBT pcimn can b obtaind uing analytical mthod (Xu and Rinhardt 1999b) in which th numrical intgration for dtrmng th cohion toughn rquir pcializd numrical tchniqu bcau of ingularity problm at intgral bodary. To avoid thi difficulty, th author (Kumar and Barai 2008) put forward application of ivral wight fction which nabl on to calculat th cohion toughn in a clod form quation without compromiing in accuracy of rult. Hnc, in prnt tudy th doubl-k fractur paramtr ar dtrmind uing fiv trm wight fction mthod a (6)

11 Tabl 2 Comparion of variou fractur paramtr for th matrial and gomtrical proprti: = 3.21 MPa, E = 30 GPa, G F =103 N/m, d max = 19 mm, B = 100 mm, S/D = 4 D (mm) a o /D Fractur paramtr of SEM b (MPa-m 1/2 ) c f Fractur paramtr of TPFM (mm) (MPa-m 1/2 ) CTOD c (µm) a c Fractur paramtr of ECM (mm) (MPa-m 1/2 ) a /D Doubl-K fractur paramtr (MPa-m 1/2 ) (MPa-m 1/2 ) a c /D Doubl-G fractur paramtr CTOD c (µm) (MPa-m 1/2 ) (MPa-m 1/2 ) f t S i z - f c t o f f r a c t u r p a r a m t r f o r c r a c k p r o p a g a t i o n i n c o n c r t : a c o m p a r a t i v t u d y 1 1

12 12 Shailndra Kumar and S.V. Barai mntiond lwhr (Kumar and Barai 2008). Sinc th oftning rlation of concrt i alo rquird for dtrmng th paramtr of DKFM, modifid bilinar oftning fction of concrt (Xu and Rinhardt 1999b, Xu and Zhang 2008) i adoptd in prnt calculation. Th ffct of lf wight on th computation of ffctiv crack lngth and th fractur paramtr ar takn into conidration a mntiond by Kumar and Barai (2009a). Th rult of fractur paramtr and for th TPBT pcimn ar prntd in Tabl 2. Th analytical mthod (Xu and Zhang 2008, Kumar and Barai 2009a) i ud for dtrmng of doubl-g fractur paramtr. Thrfor, it i convnint to obtain th ffctiv doubl-k fractur paramtr i.. ffctiv tiation fractur toughn and ffctiv tabl fractur toughn in trm of quivalnt tr intnity factor uing doubl-g fractur modl. Modifid bilinar oftning fction of concrt i alo ud for dtrmng th fractur paramtr of DGFM in prnt calculation. Th computd valu of doubl-g fractur paramtr ar hown in Tabl 2. All calculation ar prformd with dvlopd computr program uing MATLAB (Vrion 7). 4. Siz-ffct tudy uing variou fractur modl 4.1 Siz-ffct of critical tr intnity factor b In Tabl 2, th dnot th quivalnt critical valu of SIF obtaind uing G f and LEFM quation. From th tabl it i clar that th fractur paramtr of SEM ar indpndnt of pcimn iz whra thy ar marginally dpndnt on gomtrical factor a o /D ratio. Th raon i obviou. In th SEM, th fractur nrgy G f by dftion i indpndnt of tt pcimn iz although thi i tru only approximatly inc th iz ffct law i not xact. Th G f i alo indpndnt of th pcimn hap. Thi bcom clar by ralizing that th fractur proc zon occupi a ngligibly mall fraction of th pcimn volum in an inftly larg pcimn. Thrfor, mot of th pcimn i latic, which impli that th fractur proc zon at it bodary i xpod to th aymptotic nar-tip latic tr and diplacmnt fild which ar known from LEFM and ar th am for any pcimn gomtry. Hr, th fractur proc zon mut b in th am tat rgardl of th pcimn hap. For thi raon, th computd fractur paramtr of TPFM, of ECM, and of DKFM and and of DGFM at a o / b D ratio ar cald down to and plottd in Fig. 4-7 rpctivly. Fig. 4 Siz-ffct bhavior of variou fractur paramtr at a o /D ratio = 0.2 Fig. 5 Siz-ffct bhavior of variou fractur paramtr at a o /D ratio = 0.3

13 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 13 Fig. 6 Siz-ffct bhavior of variou fractur paramtr at a o /D ratio = 0.4 Fig. 7 Siz-ffct bhavior of variou fractur paramtr at a o /D ratio = 0.5 From th figur it i obrvd that all th fractur paramtr ar influncd by pcimn iz hnc xhibit iz-ffct. Th fractur paramtr of variou fractur modl with rfrnc to b of SEM maintain crtain rlationhip with th non-dimnional paramtr l ch /D. From Fig. 4 it i obrvd that fractur paramtr at critical condition of ECM, of DKFM and of DGFM ar clo to ach othr and how imilar variation with rpct to th l ch /D. Siz-ffct bhavior of of TPFM at critical condition i imilar to that of th, and howvr, th magnitud of th i omwhat l than tho mntiond abov. Thi man that TPFM prdict th mot conrvativ rult of critical tr intnity factor at tabl failur. Th of DKFM and of DGFM ar fod to b vry clo at tial cracking load and how almot imilar iz-ffct bhavior. Fig. 5-7 alo how th am iz-ffct bhavior a dmontratd in Fig. 4 xcpt for th paramtr of of pcimn iz 100 mm at a o /D ratio of 0.5. Thi dviation rprntd in th figur i du to probably th limitation of th applicability of th rgrion formula for dtrmng th valu of a in ECM. It i obrvd that th ratio of critical valu of tr intnity factor prdictd by ECM, DKFM and DGFM to critical valu of tr intnity factor prdictd by SEM i clo to 1. Furthrmor, it i vidnt that th and ar l dpndnt on th pcimn iz conidrd in th prnt tudy. Thi bhavior wa alo obrvd in th prviou tudi (Kumar and Barai 2008, 2009a). In th numrical tudy (Kumar and Barai 2008), it wa hown that th paramtr i rlativly l dpndnt on th pcimn iz ranging btwn mm, howvr, byond th iz rang 400 mm, a dcra in th valu i obrvd. Similarly, th author (Kumar and Barai 2009a) rportd that th paramtr i almot indpndnt of th pcimn iz ranging btwn mm and byond th iz rang 300 mm, a harp dcra in th valu i obrvd. In addition, it wa obrvd that th dcra with th incra in th pcimn iz. Th dicrpancy fod in th rult particularly with th and may b poibly du to diffrnt oftning fction mployd in th calculation bcau th rult of K and IC ar omwhat dpndnt on th oftning fction of concrt. b b From Tabl 2 and Fig. 4, th ratio of th /, /, K / b b IC KIC, /, / and b / at a o /D ratio 0.2 ar fod to b 0.611, 0.943, 0.941, 0.900, and rpctivly for D = 100 mm and tho ar 0.830, 1.162, 1.091, 1.039, and rpctivly for D = 400 mm. On th othr xtrm, from Fig. 7, th ratio of th b b /, /, / b KIC, /,

14 14 Shailndra Kumar and S.V. Barai b b / and / at a o /D ratio 0.5 ar fod to b 0.799, 1.220, 0.936, 0.904, and rpctivly for D = 100 mm and tho ar 0.903, 1.166, 1.079, 1.047, and rpctivly for D = 400 mm. Th rult indicat that th TPFM prdict th mot conrvativ valu of th critical tr intnity factor whra clo rult ar prdictd by th ECM, DKFM and DGFM. Th obrvation i in conitnt with th aumption mad for th dvlopmnt of variou fractur modl. In TPFM, th LEFM quation ar applid for computation of diffrnt fractur paramtr in which only latic part of th total CMOD i conidrd for dtrmng th critical ffctiv crack lngth. Th loading and loading i prformd for th maurmnt of latic part of th total CMOD. Th inlatic part of that CMOD i nglctd in calculation which poibly rult in rlativly lowr valu of critical ffctiv crack lngth and. In ECM, th nonlinar P-δ (load-dflction) bhavior bfor attainmnt of pak load i conidrd. Similar to complianc calibration mthod, th pak load and corrponding mid pan dflction (cant modulu) i ud to valuat th valu of a whra th tial lop of th P-δ curv i ud to dtrmin th latic modulu of concrt mix. In DKFM or DGFM, th linar uprpoition aumption (Xu and Rinhardt 1999b) conidring P-CMOD plot i ud to obtain th critical ffctiv crack lngth a c. Thi aumption can b applid to dtrmin th fictitiou ffctiv crack xtnion for complt analyi of fractur proc in concrt. For critical condition, th ffctiv crack lngth i dtrmind uing cant CMOD complianc at pak load whra th latic modulu of concrt mix may b dtrmind uing tial complianc of P- CMOD plot. Hnc, th linar uprpoition aumption tak into accot th nonlinarity ffct in th P-CMOD curv bfor attainmnt of th tabl condition. Thi procdur m to b imilar to th mthod for calculating critical ffctiv crack xtnion in ECM. From th abov xplanation it i clar that th may b th lowt valu whra th fractur paramtr,, hould b in clo agrmnt. 4.2 Effct of pcimn iz on th CTOD c and CTOD c Th CTOD c obtaind uing TPFM and th CTOD c valuatd uing DKFM or DGFM ar plottd with rpct to th non-dimnional paramtr l ch /D in Fig. 8 and 9 rpctivly. It i obrvd from th figur that th CTOD c and CTOD c maintain a dft rlationhip with th pcimn iz for a givn valu of a o /D ratio and thy incra a th pcimn iz incra. It i alo obrvd from th figur that th CTOD c and CTOD c dpnd on th a o /D ratio for a givn Fig. 8 Siz-ffct bhavior of CTOD c obtaind uing TPFM Fig. 9 Siz-ffct bhavior of CTOD c obtaind uing DKFM

15 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 15 Fig. 10 Rlationhip of th CTOD c and CTOD c obtaind btwn uing TPFM and DKFM pcimn iz. Th valu of CTOD c ar mor cattrd particularly for mallr iz of pcimn whn compard among th diffrnt a o /D ratio whra tho valu of CTOD c ar mor clor and l cattrd and appar to b in a narrow band for iz-rang mm conidrd in th tudy. A rlationhip btwn CTOD c and CTOD c i prntd in Fig. 10 in which th ratio CTOD c / CTOD c i plottd with rpct to th paramtr l ch /D. It i n from th figur that th ratio CTOD c /CTOD c maintain a dft rlationhip with th pcimn iz and th ratio dcra a th pcimn iz incra. For a o /D ratio 0.2, th valu of CTOD c /CTOD c i 0.625, 0.561, and for pcimn iz of 100, 200, 300 and 400 mm rpctivly and th am for a o /D ratio 0.5 i 0.847, 0.686, and rpctivly. Nglcting th ffct of a o /D ratio, th man valu of CTOD c /CTOD c for pcimn iz rang 100 and 400 mm ar dtrmind and fod to b and rpctivly. It man that th prdictd CTOD at critical load uing TPFM i rlativly mor conrvativ than that prdictd by DKFM or DGFM. 4.3 Effct of pcimn iz on th a of ECM and a c of DKFM or DGFM Th critical ffctiv crack xtnion ratio a /D obtaind uing ECM and a c /D computd uing DKFM or DGFM ar plottd with l ch /D in Fig. 11 and 12 rpctivly. A imilar trnd on both th paramtr a /D and a c /D i obrvd from th figur. Th valu a / D and a c /D ratio ar dpndnt on a o /D ratio and pcimn iz. Th aumption for dtrmng both th paramtr a /D and a c /D ar diffrnt. Th cant complianc at critical load on P-δ curv Fig. 11 Siz-ffct bhavior of a /D obtaind uing ECM Fig. 12 Siz-ffct bhavior of a c /D obtaind uing DKFM

16 16 Shailndra Kumar and S.V. Barai Fig. 13 Rlationhip of th quivalnt critical crack xtnion obtaind btwn uing ECM and DKFM i ud for valuation of a /D raio whra th linar uprpoition aumption i applid on P- CMOD curv to dtrmin th a c /D valu. In prnt calculation, th rgrion quation (Karihaloo and Nallathmabi 1990) i ud for valuation of a /D ratio whil P-CMOD curv with linar uprpoition aumption i ud for dtrmng th a c /D ratio. Finally, an intrrlation btwn a /D and a c /D i plottd in Fig. 13. It i intrting to obrv th figur that rlationhip btwn a /D and a c /D ratio dpnd on th pcimn iz and gomtrical factor. Howvr, xcpt for D = 100 at a o /D = 0.5, th ratio a /a c i vry clo to 1 that i ffctiv crack xtnion at critical load obtaind uing ECM and DKFM or DGFM i almot quivalnt for th iz-rang conidrd in th tudy. 4.4 Rlation btwn c f of SEM and a c of TPFM From Tabl 2 it i n that th c f lightly vari with th a o /D ratio. For comparion purpo, th man valu of c f i obtaind a mm and th man valu of a c i fod a mm. Th ratio of a c /c f i which how that th ffctiv crack xtnion for inftly larg tructur prdictd by TPFM i mor conrvativ than th am prdictd uing SEM by about 38.64%. 5. Concluion In th prnt tudy th iz-ffct analyi of variou fractur paramtr obtaind from th important xiting fractur modl wa prntd. Th fractur paramtr wr dtrmind on thr-point bnd tt of iz-rang mm for which th input data wr obtaind from cohiv crack modl. A comparativ iz-ffct tudy wa carrid out uing th poibl fractur paramtr from TPFM, SEM, ECM, DKFM and DGFM. In gnral, it wa obrvd that all th fractur paramtr wr dpndnt on gomtrical factor and pcimn iz. From prnt numrical tudy th following rmark can b highlightd. Th fractur paramtr of all th fractur modl including doubl-k and doubl-g fractur paramtr xhibitd iz-ffct bhavior. Th critical tr intnity factor obtaind uing SEM, ECM, DKFM and DGFM appar to b clo to ach othr with an rror rang of ±20%. TPFM prdictd th mot conrvativ critical tr intnity factor. Th fractur paramtr of doubl-k and doubl-g fractur modl prdictd th rult vry

17 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 17 clo to ach othr at tial cracking and tabl cracking load. Th crack-tip opning diplacmnt at tabl fractur load prdictd uing TPFM wa mor conrvativ than that prdictd uing DKFM or DGFM by about in th rang of 27-47%. Thi valu wa obtaind on th bai of th man valu of crack-tip opning diplacmnt at tabl fractur load from TPFM and DKFM or DGFM for pcimn iz 100 and 400G mm rpctivly. Th critical ffctiv crack lngth obtaind uing ECM and DKFM or DGFM wa vry clo to ach othr. Th ffctiv crack xtnion for inftly larg tructur prdictd by TPFM wa mor conrvativ than th am prdictd uing SEM by about 39%. Rfrnc Alhoaibi, A.M. (2010), Ft lmnt procdur for th numrical imulation of fatigu crack propagation dr mixd mod loading, Struct. Eng. Mch., 35(3), Barnblatt, G.I. (1962), Th mathmatical thory of quilibrium crack in brittl fractur, Adv. Appl. Mch., 7(1), Baz ant, Z.P. (2002), Concrt fractur modl: tting and practic, Eng. Fract. Mch., 69, Baz ant, Z.P., Gttu, R. and Kazmi, M.T. (1991), Idntification of nonlinar fractur proprti from iz ffct tt and tructural analyi bad on gomtry-dpndnt R-curv, Int. J. Rock Mch. Min., 28(1), Baz ant, Z.P. and Oh, B.H. (1983), Crack band thory for fractur of concrt, Matr. Struct., 16(93), Baz ant, Z.P., Kim, J.K. and Pfiffr, P.A. (1986), Dtrmination of fractur proprti from iz ffct tt, J. Struct. Eng. - ASCE, 112(2), Baz ant, Z.P. and Plana, J. (1998), Fractur and iz ffct in concrt and othr quaibrittl matrial, Florida CRC Pr. Carpintri, A. (1989), Cup catatroph intrprtation of fractur intability, J. Mch. Phy. Solid, 37(5), Cuati, G. and Schauffrt, E.A. (2009), Cohiv crack analyi of iz ffct, Eng. Fract. Mch., 76, Dugdal, D.S. (1960), Yilding of tl ht contang lit, J. Mch. Phy. Solid, 8(2), Elic, M. and Plana, J. (1996), Fractur mchanic paramtr of concrt an ovrviw, Adv. Cm. Bad Matr., 4, Elic, M., Guina, G.V. and Plana, J. (1992), Maurmnt of th fractur nrgy uing thr-point bnd tt: Part 3- Influnc of cutting th P-δ tail, Matr. Struct., 25, Elic, M., Guina, G.V. and Plana, J. (1997), On th maurmnt of concrt fractur nrgy uing thrpoint bnd tt, Matr. Struct., 30, Elic, M., Rocco, C. and Rolló, C. (2009), Cohiv crack modling of a impl concrt: xprimntal and numrical rult, Eng. Fract. Mch., 76, Gar, T.C. (2007), Validation of 3D crack propagation in plain concrt. Part II: Computational modling and prdiction of th PCT3D tt, Comput. Concrt, 4(1), Guina, G.V., Plana, J. and Elic, M. (1992), Maurmnt of th fractur nrgy uing thr-point bnd tt: Part 1 - Influnc of xprimntal procdur, Matr. Struct., 25,, Hanon, J.H. and Ingraffa, A.R. (2003), Uing numrical imulation to compar th fractur toughn valu for concrt from th iz-ffct, two-paramtr and fictitiou crack modl, Eng. Fract. Mch., 70, Hillrborg, A., Modr, M. and Ptron, P.E. (1976), Analyi of crack formation and crack growth in concrt by man of fractur mchanic and ft lmnt, Cmnt Concrt R., 6, Jnq, Y.S. and Shah, S.P. (1985), Two paramtr fractur modl for concrt, J. Eng. Mch. - ASCE, 111(10),

18 18 Shailndra Kumar and S.V. Barai Karihaloo, B.L. and Nallathambi, P. (1989), An improvd ffctiv crack modl for th dtrmination of fractur toughn of concrt, Cmnt Concrt R., 19, Karihaloo, B.L. and Nallathambi, P. (1990), Siz-ffct prdiction from ffctiv crack modl for plain concrt, Matr. Struct., 23(3), Karihaloo, B.L. and Nallathambi, P. (1991), Notchd bam tt: mod I fractur toughn, Fractur Mchanic Tt mthod for concrt, Rport of RILEM Tchnical Committ 89-FMT (Editd by S.P. Shah and A. Carpintri), Chamman & Hall, London, Kim, J.K., L, Y. and Yi, S.T. (2004), Fractur charactritic of concrt at arly ag, Cmnt Concrt R., 34, Kumar, S. and Barai, S.V. (2008), Influnc of pcimn gomtry and iz-ffct on th K R -curv bad on th cohiv tr in concrt, Int. J. Fractur, 152, Kumar, S. and Barai, S.V. (2009a), Equivalnc btwn tr intnity factor and nrgy approach bad fractur paramtr of concrt, Eng. Fract. Mch., 76, Kumar, S. and Barai, S.V. (2009b), Effct of oftning fction on th cohiv crack fractur paramtr of concrt CT pcimn, Sadhana-Acad. P. Eng. S., 36(6), Kumar, S. and Barai, S.V. (2010), Siz-ffct prdiction from th doubl-k fractur modl for notchd concrt bam, Int. J. Damag Mch., 9, Kwon, S.H., Zhao, Z. and Shah, S.P. (2008), Effct of pcimn iz on fractur nrgy and oftning curv of concrt: Part II. Invr analyi and oftning curv, Cmnt Concrt R., 38, MATLAB, Vrion 7, Th MathWork, Inc., Copyright Nallathambi, P. and Karihaloo, B.L. (1986), Dtrmination of pcimn-iz indpndnt fractur toughn of plain concrt, Mag. Concrt R., 38(135), Ouyang, C., Tang, T. and Shah, S.P. (1996), Rlationhip btwn fractur paramtr from two paramtr fractur modl and from iz ffct modl, Matr. Struct., 29(2), Park, K., Paulino, G.H. and Rolr, J.R. (2008), Dtrmination of th kink point in th bilinar oftning modl for concrt, Eng. Fract. Mch., 7, Ptron, P.E. (1981), Crack growth and dvlopmnt of fractur zon in plain concrt and imilar matrial, Rport No. TVBM-100, Ld Intitut of Tchnology. Philip, P. (2009), A quaitatic crack propagation modl allowing for cohiv forc and crack rvribility, Intract. Multical Mch., 2(1), Plana, J. and Elic, M. (1990), Fractur critria for concrt: mathmatical validation and xprimntal validation, Eng. Fract. Mch., 35, Plana, J. and Elic, M. (1991), Nonlinar fractur of cohiv matrial, Int. J. Fractur, 51, Plana, J. and Elic, M. (1992), Shrinkag igntr and tructural iz-ffct, In Fractur Mchanic of Concrt Structur, Z.P. Baz ant, d., Elvir Applid Scinc, London, Plana, J., Elic, M. and Guina, G.V. (1992), Maurmnt of th fractur nrgy uing thr-point bnd tt: Part 2-Influnc of bulk nrgy diipation, Matr. Struct., 25, RILEM Draft Rcommndation (TC50-FMC) (1985), Dtrmination of fractur nrgy of mortar and concrt by man of thr-point bnd tt on notchd bam, Matr. Struct., 18(4), RILEM Draft Rcommndation (TC89-FMT) (1990a), Dtrmination of fractur paramtr ( and CTOD c ) of plain concrt uing thr-point bnd tt, Matr. Struct., 23(138), RILEM Draft Rcommndation (TC89-FMT) (1990b), Siz-ffct mthod for dtrmng fractur nrgy and proc zon iz of concrt, Matr. Struct., 23(138), Rolr, J., Paulino, G.H., Park, K. and Gadick, C. (2007), Concrt fractur prdiction uing bilinar oftning, Cmnt Concrt Compo., 29, Tada, H., Pari, P.C. and Irwin, G. (1985), Th tr analyi of crack handbook, Pari Production Incorporatd, St. Loui, Miouri, USA. Tang, T., Shah, S.P. and Ouyang, C. (1992), Fractur mchanic and iz ffct of concrt in tnion, J. Struct. Eng. - ASCE, 118(11), Xu, S. and Rinhardt, H.W. (1998), Crack xtnion ritanc and fractur proprti of quai-brittl matrial lik concrt bad on th complt proc of fractur, Int. J. Fractur, 92, Xu, S. and Rinhardt, H.W. (1999a), Dtrmination of doubl-k critrion for crack propagation in quai-brittl

19 Siz-ffct of fractur paramtr for crack propagation in concrt: a comparativ tudy 19 matrial, Part I: Exprimntal invtigation of crack propagation, Int. J. Fractur, 98, Xu, S. and Rinhardt, H.W. (1999b), Dtrmination of doubl-k critrion for crack propagation in quai-brittl matrial, Part II: Analytical valuating and practical mauring mthod for thr-point bnding notchd bam, Int. J. Fractur, 98, Xu, S. and Rinhardt, H.W. (1999c), Dtrmination of doubl-k critrion for crack propagation in quai-brittl matrial, Part III: compact tnion pcimn and wdg plitting pcimn, Int. J. Fractur, 98, Xu, S. and Zhang, X. (2008), Dtrmination of fractur paramtr for crack propagation in concrt uing an nrgy approach, Eng. Frac. Mch., 75, Xu, S., Rinhardt, H.W., Wu, Z. and Zhao, Y. (2003), Comparion btwn th doubl-k fractur modl and th two paramtr fractur modl, Otto-Graf J., 14, Zhao, Z., Kwon, S.H. and Shah, S.P. (2008), Effct of pcimn iz on fractur nrgy and oftning curv of concrt: Part I. Exprimnt and fractur nrgy, Cmnt Concrt R., 38, CC Abbrviation CBM CCM CMOD CMOD c COD CT CTOD CTOD c DGFM DKFM ECM FCM FPZ LEFM SEM SIF TPBT TPFM WST crack band modl cohiv crack modl crack mouth opning diplacmnt critical valu of crack mouth opning diplacmnt crack opning diplacmnt compact tnion crack-tip opning diplacmnt critical valu of crack-tip opning diplacmnt doubl-g fractur modl doubl-k fractur modl ffctiv crack modl fictitiou crack modl fractur proc zon linar latic fractur mchanic iz ffct modl tr intnity factor thr-point bnding tt two paramtr fractur modl wdg-plitting tt