Stress-compatible embedded cohesive crack in CST element

Transcription

1 Fratur Mhani of Conrt an Conrt Strutur - Rnt Avan in Fratur Mhani of Conrt - B. H. Oh, t al.() 2010 Kora Conrt Intitut, Soul, ISBN Str-ompatibl mb ohiv rak in CST lmnt J.F. Oln & P.N. Pouln Thnial Univrity of Dnmark, Dnmark ABSTRACT: A impl lmnt ith an mb trong iontinuity for moling ohiv raking of onrt i prnt. Th lmnt iffr from prviou lmnt of th mb typ, in that a onitnt tr fil i obtain by irt nformnt of tr ontinuity aro th rak. Th iplamnt iontinuity i mol in an XFEM fahion; hovr, th iontinuou iplamnt fil i pial, alloing for th irt nformnt of tr ontinuity. Thi in turn allo for limination of xtra gr of from nary for ribing th rak formation, thu th lmnt ha th am numbr of from a it ontinuou bai: CST. Th goo prforman of th lmnt i montrat by it ability to imulat thrpoint bning of a noth onrt bam. Th avantag of th lmnt i it impliity an th traightforar implmntation of it. Hanling ituation ith multipl rak ill not rquir furthr vlopmnt, although thi i not montrat hr. 1 INTRODUCTION Th 2D moling of onrt raking in th framork of FEM ha alray rah an avan lvl, an vlopmnt in thi fil ovr th pat a hav bn haratriz a XFEM, th Xtn Finit Elmnt Mtho, (Blythko & Blak 1999), (Moë t al. 1999) an (Moë & Blythko 2002). For rfinmnt of th mtho appli to ohiv raking alloing for partly rak lmnt, (Afrg t al. 2007) an (Mougaar t al. 2009). Although th mtho ar vry fftiv, th rabak, hovr, i that it tak a lot of bookkping, at th ytm lvl a ll a at th lmnt lvl, to nur onformity an to mol paration of th iontinuou part of an lmnt auratly, i.. in uh a ay that it allo for fully inpnnt formation of th to parat part. Th lmnt hih i propo hr i a 2D lmnt of th mb iontinuity typ ith a trong iontinuity; thi mb iontinuity typ a originally introu by (Dvorkin, E.N. t al. 1990). Furthr vlopmnt r prnt by (Olivr 1996); thi ork i ba on th aum nhan train approah an iffr from th prnt ork in that it o not nfor tr ontinuity irtly, but in a variational mannr. Th avantag of th mb lmnt typ i that it i impl to implmnt an that it o not ompliat th builing of th FE ytm Equation. Th rabak of it i that it lak th ability to rat ompltly inpnnt lmnt paration an that iplamnt ontinuity i violat on lmnt i ro by a rak; hovr, in many a th fiini might b onir to b of minor importan. Th prnt lmnt tak avantag of th ytmati ontrution of iplamnt fil in XFEM, xpt that th iontinuou hap funtion ar not th am a th ontinuou. Bi from bing of th mb typ th prnt lmnt i haratriz by having a onitnt tr fil hih i ontinuou aro th iontinuity. Enforing thi tr ontinuity in th lmnt obtain th poibility of liminating th xtra gr of from (DOF) nary for ribing th iontinuity; hrby an lmnt i raliz ith no xtra DOF ompar to th ontinuou vrion of th lmnt. Th prnt lmnt allo for ohiv raking ith a tr ritrion for rak initiation, i.. a rak initiat if th uni-axial tnil trngth i x by th firt prinipl tr an no tr intnity i aum at th rak tip, auming mooth rak lour. 2 KINEMATICS Th moling of a trong iontinuity, uh a a rak, i ba on th tratgy of th xtn finit lmnt mtho for th approximation of a iplamnt fil ith a iontinuity. Thu, th iplamnt fil approximation in an lmnt ith a iontinuity i tablih by ombining th iplamnt fil orrponing to th ontinuou lmnt ith th iplamnt fil orrponing

2 to th iontinuity. For th linar intrpolation of th ontinuou iplamnt fil onir a thr-no triangular lmnt, th CST. Th lmnt i givn th poibility of a trong iontinuity along a traight lin roing th lmnt. Th nformnt of tr ompatibility btn th tr in th ontinuou part of th lmnt an th tr briging th iontinuity lin i th ntial ia of th prnt ork. Str ompatibility i nur by maning that th orr of variation of th iontinuou iplamnt fil along th iontinuity lin mut math th orr of variation of th ontinuou tr fil, an that th iontinuou iplamnt fil prou qual tr on oppoit i of th iontinuity. In th a of a CST thi la to a iplamnt fil ith a ontant jump in th iplamnt along th iontinuity lin, hih ill prou ontant briging tr along th iontinuity mathing th ontant tr fil of th CST. Furthr, thi iplamnt fil mut prou qual graint on ithr i of th iontinuity. A hap funtion that allo for a ontant jump aro a traight lin an at th am tim only introu qual an ontant train on ithr i of th iontinuity lin i hon in Figur 1. Th hap funtion may b rittn in trm of th natural oorinat ζ aoiat ith th lmnt vrtx hih i alo a vrtx of th triangular ub-omain : ζ 1 in N (1) ζ in Thi typ of hap funtion ha prviouly bn appli by (Olivr 1996). Only to DOF ar n to rib th iontinuou formation. W ho th global omponnt of th iontinuity vtor an ollt th DOF in th vtor T V [ ΔV1, ΔV2]. Eah DOF i aoiat ith a iontinuou hap funtion a givn in (1), an th iontinuou iplamnt fil vtor u i introu a u N 0 NV, N 0 N (2) By ombining u ith th ontinuou iplamnt fil vtor u N V for th normal CST lmnt arriv at th total formation fil: u u u N V N V (3) hr N i th uual linar intrpolation matrix an V th uual of vtor. J ) D ( h, T h Th atr ontnt an b xpr a of th vaporabl atr (apillary a Figur 1. Top: Straight vapor, iontinuity an aorb lin fining atr) lmnt an ubomain, an th aoiat (hmially natural oorinat boun) ζ. atr n (Mil th non- Pantazopoulo & Mill 1995). It i ra Bottom: Illutration aum of that ontant th vaporabl iontinuity atr i a fu hap funtion. rlativ humiity, h, gr of hyration Aopting th Voigt gr notation of ilia an fum auming ration, a lin- ar train maur, th ag-pnnt train in th ontinuou orption/orption part, i.. of th lmnt may (Norling b rittn Mjonll a th 1997). vtor Unr thi aum by ubtituting Equation 1 into Equati ε BV BV obtain (4) hr B i th uual ontant train intrpolation matrix an B ( D h) i th ontant train intrpolation & & h matrix riv from th iontinuou hap fun-tion. Not that B i th am on both i of th iontinuity. hr / i th lop of th orption/ iothrm (alo all moitur apa govrning quation (Equation 3) mut b 3 VARIATIONAL by appropriat AND FEM bounary FORMULATIONS an initial oniti Th rlation btn th amount of Conir a linar atr lati an boy rlativ in a humiity tat of plan i all tr or plan train. iothrm Th boy if maur i ituat in ith a plan inraing Cartian oorinat humiity ytm an ( x orption 1, x 2). Aum iothrm that a in th rak ha form a. in th Nglting boy, Figur thir iffrn 2. Th unit (Xi t al. tangnt vtor to th th folloing, urvilinar orption rak path iothrm -ilnot by, an rfrn th normal to to both th orption rak path an i orption b not by n. At any By point th ay, along th if th rak hytri ( n, ) -ofin a loal right-han iothrm oul oorinat b takn ytm. into aount, Th to th ngativ i of rlation, th rak vaporabl i fin atr a v th rlativ i humi hr n oini b u ith aoring th outar to th normal ign to of th th varia rak fa. Th tr rlativity tranfr humiity. ovr th rak Th i hap givn of th a th normal an iothrm tangntial for HPC tration, i influn σ n an by τ n, many p rptivly. Th pially ar th tho gnraliz that influn tr xtnt in an th rak, an thir hmial ork-onjugat ration gnraliz an, in train turn, trm ar th normal opning trutur of an th por rak, iz Δ uitribution n un un,(atr- ratio, lip mnt of hmial th rak ompoition, fa, SF an th tangntial Δ u u u. Th uring gnraliz tim an tr mtho, an tmpratur, train mix in th rak ar ollt t.). In in th vtor litratur aoring variou to formulatio foun to rib th orption iothrm σ r n onrt u n, Δ (Xi t al. 1994). Hovr, in th σ papr τ u th mi-mpirial xprion (5) pro n Δu Norling Mjornll (1997) i aopt b Th proportionality offiint D(h,T) moitur prmability ζ an it i a nonlina of th rlativ humiity h an tmpratur & Najjar 1972). Th moitur ma balan that th variation in tim of th atr ma volum 1 of onrt (atr ontnt ) b q ivrgn of th moitur flux J t J Proing of FraMCoS-7, May 23-28, 2010

3 hr J D ( h, ut ) hignifi a jump in th iplamnt (1) fil u rlat to th loal ( n, ) oorinat ytm. Th proportionality offiint D(h,T) i all moitur Th tr prmability briging an th it rak i a ar nonlinar typially funtion funtion of th of rlativ th rak humiity opning h an an tmpratur liing a ll T (Bažant a of th & Najjar hitory 1972). of opning Th moitur an liing. ma At balan thi point rquir aum that th that variation th briging in tim tr of th atr may b ma rittn pr a unit a funtion volum of of onrt u : (atr ontnt ) b qual to th ivrgn of th moitur flux J r r σ σ ( u ) (6) J I (2) Th t intrnal virtual ork δw an th xtrnal E virtual ork δ W may b tat a Th atr ontnt an b xpr a th um of I T T r δw th vaporabl δεσ δ S atr u σ (apillary atr, atr (7) S vapor, an aorb atr) an th non-vaporabl E T T (hmially δw δu f boun) δ Γ u t atr (8) Γ n (Mill 1966, t Pantazopoulo & Mill 1995). It i raonabl to aum that th vaporabl atr i a funtion of hr σ an ε ar th ork onjugat tr an rlativ humiity, h, gr of hyration, train vtor, f i th omain loa vtor,, an t i gr of ilia fum ration, th prrib bounary tration, i.. vtor, an (h, th, prfix δ not th variation of th ubqunt fil. ) ag-pnnt orption/orption iothrm (Norling Mjonll 1997). Unr thi aumption an Th un-rak bulk matrial i aum to b linar lati an ontitutiv Equation i rittn in by ubtituting Equation 1 into Equation 2 on obtain th form σ D ε (9) ( D h) & & & n (3) h hr D i th appropriat matrial tiffn matrix. hr / i th lop of th orption/orption iothrm Str ompatibility (alo all aro moitur th iontinuity apaity). Th i r nur govrning by quation maning (Equation th tration 3) mut t b on omplt th rak fa by appropriat to qual th bounary briging an tr initial onition. Th rlation btn th amount of vaporabl r r r (10) T atr t an σ rlativ humiity i all aorption iothrm if maur ith inraing rlativity r hr humiity T an i orption a tranformation iothrm matrix in pning th oppoit on th a. orintation Nglting of thir rak. iffrn Th tration (Xi t al. may 1994), b trmin th folloing, from orption iothrm ill b u ith in rfrn to both orption an orption onition. By th ay, if n th 1 0 hytri n r 2 of th moitur iothrm t mσ, oul m (11) b takn into 0 n2 n aount, to iffrnt 1 rlation, vaporabl atr v rlativ humiity, mut b u aoring to th ign of th variation of th hr m i a matrix ba on th lmnt of th rlativity humiity. Th hap of th orption normal vtor of th rak fa n ( n1, n2). Introuing into (6), (10) an (11) th lmnt fil ap- iothrm for HPC i influn by many paramtr, pially tho that influn xtnt an rat of th proximation (4) an (9), an utilizing that hmial r ration an, in turn, trmin por u T V th tr ompatibility rquirmnt trutur an por iz itribution (atr-to-mnt may b xpr at th lmnt lvl a ratio, mnt hmial ompoition, SF ontnt, T uring r md tim B V an B mtho, V tmpratur, σ r ( T r V mix aitiv, t.). In th litratur variou formulation ) an (12) b foun to rib th orption iothrm of normal onrt Th natur (Xi t of al. thi 1994). Equation Hovr, pn in on th (6), prnt an in papr gnral th it mi-mpirial i nonlinar in xprion V. Hovr, propo it may b by olv Norling for Mjornll V at th (1997) lmnt i lvl, aopt thu bau alloing it for xpliitly th limination aount of for th th DOF volution ribing of hyration th iontinuity. ration an Th SF variation ontnt. of Thi V may orption b tablih iothrm by ra taking th variation of (12): 1 ( h,, ) G (, ) ( g ) h 1 Figur hr 2. th Global firt oorinat trm (gl ytm iothrm) ( x1, x2) an rprnt loal rak th oorinat phyially ytm boun ( n, )(aorb). atr an th on trm (apillary iothrm) rprnt th apillary atr. Thi xprion i vali r r δ δ σ only u for lo ontnt T md B V B V δv of SF. Th offiint G (13) 1 rprnt u V th amount of atr pr unit volum hl in th gl por at 100% rlativ r W introu humiity, D an a it an th b gnraliz xpr tangntial (Norling tiffn Mjornll of 1997) th rak a briging, i.. r r G (, σ) k k 1 vg vg D u (5) (14) hr k vg an k vg ar matrial paramtr. From th r maximum Ralizing amount that of uatr V pr T unit, th volum variation that an of V fill all may por b iolat (both apillary from (13) por in th an form gl por), on an alulat K 1 a on obtain δv ZδV (15) 10 g h G ( r ) r r 1 K r, 1 10 g h 1 1 hr 10( g ) h K (, ) (4) (6) Z D T T mdb T mdb (16) Th approximation to th intrnal virtual ork in an lmnt Th matrial may no paramtr b xpr k vg an by k vg an g 1 an b alibrat by fitting xprimntal ata rlvant to fr I T T T T T rt r δw δ V (vaporabl) [ ] S B Z B atr σ ontnt Z T in σ onrt (17) at S variou ag (Di Luzio & Cuati 2009b). hr all quantiti no rlat to an lmnt. Th tr 2.2 Tmpratur that ntr volution into thi xprion ar trmin Not that, a: at arly ag, in th hmial ration aoiat ith mnt hyration an SF ration σ ar DBV xothrmi, BV th tmpratur fil i not uniform (18) for r r r σ non-aiabati σ ( T V ytm vn if th nvironmntal ) (19) tmpratur i ontant. Hat onution an b rib in onrt, at lat for tmpratur not xing Equation 100 C (17) ontitut (Bažant th & lmnt Kaplan noal 1996), for by vtor Fourir q la, uh hih that ra may rit δw I δv T q. Th iffrntial form of th intrnal virtual ork ra q λ T (7) I T T r W δ δε σ δ S u σ (20) hr q i th hat S flux, T i th abolut tmpratur, an λ i th hat onutivity; in thi Proing of FraMCoS-7, May 23-28, 2010

4 In an lmnt thi may b xpr olly in trm of th uual (ontinuou) DOF vtor I T T T T W δ δv B Z B D B B Z T rt r r S ZT DT Z S V V : (21) Thi Equation ontitut th lmnt tiffn matrix kt uh that may rit δw I δv T ktv. Th tangnt tiffn matrix i only ymmtri if th r matrix D i ymmtri, hih i not narily r th a. Hovr, for itrativ purpo D may r rt b rpla by th ymmtri matrix ½ D D. 4 NUMERICS On th global lvl th virtual ork Equation furnih th irt quilibrium Equation: QV ( ) R (22) hr Q i th um of lmnt noal for tablih through Equation (17)-(19), V i th global DOF vtor an R i th global noal loa vtor trmin from (8). Equation (22) i a t of nonlinar Equation hih ar olv itrativly applying th linar inrmntal rlation K V R (23) T hr K T i th global tangnt tiffn matrix ambl from lmnt matri tablih through (21), V i th global inrmntal DOF vtor an R i global inrmntal loa vtor. Th lmnt prnt in th prviou tion i ba on th CST, hovr, it allo for th formation of a iplamnt iontinuity or a rak. Thu hav nam th lmnt CST. Th CST ha thr no an ix DOF, to at ah no ribing th iplamnt vtor. Th atual valu of th iontinuity vtor i alulat at th lmnt lvl an no global DOF ar n to rprnt th vtor lmnt. A rak i form if th prinipl tr in an lmnt x th uni-axial tnil trngth, an th normal to th iontinuity lin i paralll to th prinipl tr vtor. If a nighboring lmnt i in th rak tat, th rak in th atual lmnt i for to onnt to th nighboring rak; othri it i for to pa through th ntr of th lmnt. Th nonlinar quilibrium Equation may b olv by tanar FEM prour; hr hav appli th orthogonal riual algorithm (Krnk 1995). Convrgn i nur at vry loa tp in trm of an nrgy J ritrion D ( h, T ) rlat h to th lati nrgy of th initial lati loa tp. Th proportionality offiint D(h,T) moitur prmability an it i a nonlina 5 EXAMPLE of th rlativ humiity h an tmpratur & Najjar 1972). Th moitur ma balan Th apabiliti that of th th CST variation lmnt in tim ar of montrat through th volum moling of onrt of th (atr RILEM ontnt thr-) b q th atr ma point bning tt ivrgn (Vanrvall of th 2000) moitur for trmining fratur mhanial proprti of onrt, flux J hih ha bom a bnhmark tt alo for th numrial moling of ohiv J fratur propagation. t A i vi of th tt tup i hon in Figur 3. Th bam ha a quar Th atr ro-tion ontnt an an a 25 b xpr mm a p noth ut prpniular of th vaporabl to th atr bottom fa in (apillary a th mi-tion plan; vapor, it an i imply aorb upport atr) an at th th non- n an loa (hmially by a loa ating boun) vrtially atr onar at mi-pan. n (Mil Pantazopoulo & Mill 1995). It i ra aum that th vaporabl atr i a fu rlativ humiity, h, gr of hyration gr of ilia fum ration,, i.. ag-pnnt orption/orption (Norling Mjonll 1997). Unr thi aum by ubtituting Equation 1 into Equati obtain Figur 3. Gomtry of th RILEM tt ( D bam h) ith a & 25 mm & h noth. Th bam i aum hr to / b at i th in lop onrt of th ith orption/ haratriti aoring iothrm to Tabl (alo 1. all Th tnil moitur oftning of th onrt govrning i aum quation to (Equation oby a linar 3) mut b apa ohiv la a by pit appropriat in Figur bounary 4. Th an initial rak oniti propagation in thi tt Th i rlation ominat btn by th th opning amount of of th rak, hn atr th an mix rlativ mo humiity haratriti i all of th onrt ar iothrm not onir. if maur Hovr, ith liing inraing of th rak fa humiity mut prou an orption om tr iothrm in orr in th to avoi unrtrit a. Nglting rigi boy thir motion iffrn of th (Xi t al. rak lmnt th part folloing, hih i only orption attah iothrm to on ill b no. Thrfor, rfrn liing i to mol both orption latially an orption ith an infrior tiffn. By No th intration ay, if btn th hytri th to of th rak formation iothrm i onir. oul b takn into aount, to rlation, vaporabl atr v rlativ humi Tabl 1. Conrt matrial paramtr. b u aoring to th ign of th varia Paramtr rlativity humiity. Valu Th hap of th Young moulu, Eiothrm for HPC i 37.4 influn GPa by many p pially tho that influn xtnt an Poion ratio, ν 0.2 hmial ration an, in turn, trm Tnil trngth, f t trutur an por iz 3.5 MPa itribution (atrratio, mnt hmial 160 J/mompoition, 2 SF Fratur nrgy, G f uring tim an mtho, tmpratur, mix Th CST lmnt t.). i In tt th litratur againt th variou prforman of an intrfa foun lmnt to rib uppli th ith orption th om- iothrm formulatio mrially availabl onrt FEM (Xi o t al. DIANA 1994). Hovr, (Diana in th 2003). In th DIANA papr rfrn th mi-mpirial mol th xprion bulk of pro th bam i mol Norling ith Mjornll CST lmnt, (1997) an i intr- aopt b Proing of FraMCoS-7, May 23-28, 2010

5 fa J D ( lmnt h, T ) h ar pr-loat to mol a rak path (1) onfin to th mi-pan tion of th bam. Th proportionality offiint D(h,T) i all moitur prmability an it i a nonlinar funtion of th rlativ humiity h an tmpratur T (Bažant & Najjar 1972). Th moitur ma balan rquir that th variation in tim of th atr ma pr unit volum of onrt (atr ontnt ) b qual to th ivrgn of th moitur flux J Figur 4. Linar J tnion oftning urv. t (2) Th atr ontnt an b xpr a th um of th vaporabl atr (apillary atr, atr vapor, an aorb atr) an th non-vaporabl (hmially boun) atr n (Mill 1966, Pantazopoulo & Mill 1995). It i raonabl to aum that th vaporabl atr i a funtion of rlativ humiity, h, gr of hyration,, an gr of ilia fum ration,, i.. (h,, ) ag-pnnt orption/orption iothrm (Norling Mjonll 1997). Unr thi aumption an by ubtituting Equation 1 into Equation 2 on obtain Figur 5. Loa-fltion urv of th RILEM tt bam ith ( D h) & & & n (3) a 25 mm h noth. Comparion h btn a DIANA mol ith 48 intrfa lmnt ovr th ligamnt hight, an to CST mol ith untrutur mh having 14 an 28 lmnt ovr hr th ligamnt / i hight, th lop rptivly. of th orption/orption iothrm (alo all moitur apaity). Th govrning To mak quation a fair omparion (Equation 3) btn mut omplt th CST mol by appropriat an th bounary rfrn an mol, initial th onition. CST mol i allo Th rlation to vlop btn on rak th amount only, although of vaporabl it ha th atr apability an rlativ of moling humiity multi i rak all vlopmnt. aorption Th iothrm on-rak if maur rtrittion ith i obtain inraing by only rlativity alloing humiity an lmnt an orption to rak if it iothrm ha a nighbor in th hih oppoit i rak. a. Nglting Hovr, thir to gt iffrn tarting, (Xi th t firt al. 1994), lmnt in to th rak folloing, i xmpt orption from iothrm thi rul. ill b u ith rfrn Th rfrn to both orption bam a an mol orption in onition. a rgular mh By th ith ay, 48 lmnt if th ovr hytri th hight of th of th moitur bam. Untrutur iothrm oul mh b takn r into u aount, for to th iffrnt CSTmol rlation, ith vaporabl to niti atr v furnihing rlativ humiity, 14 an 28 mut lmnt b u ovr aoring th ligamnt to th hight, ign of rptivly. th variation of th rlativity Simulat humiity. loa-fltion Th hap urv of ar th hon orption in Figur iothrm 5. for Th HPC fltion i influn i maur by many at paramtr, mi-pan rlativ pially to th tho man that valu influn of th xtnt formation an rat of th bam hmial mi-plan ration at th an, upport. in turn, Rult trmin for both por mh trutur ompar an por nily iz itribution ith th rult (atr-to-mnt for th rfrn ratio, mol, mnt although hmial th CST ompoition, mol lightly SF ontnt, ovrtimat uring tim th an pak mtho, loa. tmpratur, It houl b mix mphaiz, aitiv, though, t.). In that th th litratur rak path variou ha formulation not bn prfin an b in foun th CST to rib mol. th In orption Figur 6 iothrm lmnt of mh normal lo onrt to th (Xi noth t al. an 1994). rak Hovr, path at in pak th loa prnt ar hon, papr th rptivly mi-mpirial for th xprion to mh niti propo appli. Norling Only Mjornll th mil (1997) part i of aopt th bam bau (±25mm by it xpliitly aount for th volution of hyration ration an SF ontnt. Thi orption iothrm ra 1 ( h,, ) G (, ) ( g ) h 1 10( g ) h K (, ) (4) hr th firt trm (gl iothrm) rprnt th phyially boun (aorb) atr an th on trm (apillary iothrm) rprnt th apillary atr. Thi xprion i vali only for lo ontnt of SF. Th offiint G 1 rprnt th amount of atr pr unit volum hl in th gl por at 100% rlativ humiity, an it an b xpr (Norling Mjornll 1997) a G (, ) k k 1 vg vg (5) hr k vg an k vg ar matrial paramtr. From th Figur maximum 6. Elmnt amount mh of for atr CST pr mol unit lo volum to th that noth, an an fill all rak por path (both at pak apillary loa. Th por mil an part gl of por), th bam on (±25mm an alulat from th K 1 mi-tion) a on obtain i hon for to mh niti. from th mi-tion) i hon. Th 10 gining h 1 of th CST rak 0 i of our G a 1 onqun of th oar (, ontant ) train fil of th lmnt. Hovr, th ovrall irtion 10 g of th h rak i in lin (6) K 1 1 ith th mi-tion plan. 1 Th matrial paramtr k vg an k vg an g 1 an 6 b CONCLUSION alibrat by fitting xprimntal ata rlvant to fr (vaporabl) atr ontnt in onrt at A variou impl ag lmnt (Di Luzio all & Cuati CST ith 2009b). an mb iontinuity for moling onrt raking ha bn 2.2 Tmpratur prnt. Th volution lmnt iffr from prviou trong iontinuity lmnt of th mb or XFEM Not that, typ, at arly in that ag, th in tr th fil hmial i onitnt ration ithin aoiat th lmnt ith mnt nuring hyration tr ontinuity an SF ration aro th ar rak. xothrmi, Th irt th tmpratur nformnt fil of tr i not ompatibility for non-aiabati ha allo ytm for th limination vn if th of nvironmntal xtra DOF uniform for tmpratur ribing i th ontant. rak opning Hat an onution liing, thu an th b CST rib ha th in am onrt, numbr at lat of DOF for tmpratur a th CST. not xing Th goo 100 C prforman (Bažant of & th Kaplan CST 1996), ha bn by montrat Fourir la, in hih th ra Mo I bnhmark tt: th thr-point bning of a noth onrt bam. q λ T Th avantag of th CST lmnt i th impliity of it. Implmntation i traightforar an (7) th hr hanling q i of th a multipl hat flux, rak T ituation i th ill abolut not rquir tmpratur, furthr an vlopmnt. λ i th hat onutivity; in thi Proing of FraMCoS-7, May 23-28, 2010

6 REFERENCES Afrg, J.L., Pouln, P.N. an Niln, L.O A onitnt partly rak XFEM lmnt for ohiv rak groth. Int. J. Numr. Mth. Engng. Vol. 72: Blythko, T. an Blak, T Elati rak groth in finit lmnt ith minimal rmhing. Int. J. Numr. Mth. Engng. Vol. 45: Diana Ur Manual DIANA Ur Manual, Elmnt Library, (Eition 8.1). TNO Builing an Contrution Rarh, Dlft, Th Nthrlan. Dvorkin, E.N., Cuitiño, A.M. an Gioia, G Finit lmnt ith iplamnt intrpolat mb loalization lin innitiv to mh iz an itortion. Comp. Mth. in Appl. Mh. an Engng. Vol. 90: Krnk, S An orthogonal riual prour for nonlinar finit lmnt quation. Int. J. Numr. Mth. Engng. Vol. 38: Moë, N., Dolbo, J. J an DBlythko, ( h, T ) h T A finit lmnt mtho for rak groth ithout rmhing. Int. J. Numr. Mth. Engng. Vol. 46: Moë, N., an Blythko, Th T. proportionality Extn finit offiint lmnt D(h,T) mtho for ohiv moitur rak prmability groth. Engng. an Frat. it i Mh. a nonlina Vol. 63: of th rlativ humiity h an tmpratur Mougaar, J.F., Pouln, & Najjar P.N. an 1972). Niln, Th L.O. moitur A partly ma balan an fully rak XFEM lmnt ba on highr orr that th variation in tim of th atr ma polynomial hap funtion for moling ohiv fratur. Submitt for publiation. volum of onrt (atr ontnt ) b q Olivr, J Molling ivrgn trong of iontinuiti th moitur in flux oli J mhani via train oftning ontitutiv quation. Part 1: Funamntal. Part 2: Numrial imulation. Int. J. Numr. Mth. Engng. Vol. 39: J Vanrvall, L Tt t an ign mtho for fibr rinfor onrt. Rommnation for bning tt. Matr. & Strut. Vol. 33: 3-5. Th atr ontnt an b xpr a of th vaporabl atr (apillary a vapor, an aorb atr) an th non- (hmially boun) atr n (Mil Pantazopoulo & Mill 1995). It i ra aum that th vaporabl atr i a fu rlativ humiity, h, gr of hyration gr of ilia fum ration,, i.. ag-pnnt orption/orption (Norling Mjonll 1997). Unr thi aum by ubtituting Equation 1 into Equati obtain ( D h) h & & hr / i th lop of th orption/ iothrm (alo all moitur apa govrning quation (Equation 3) mut b by appropriat bounary an initial oniti Th rlation btn th amount of atr an rlativ humiity i all iothrm if maur ith inraing humiity an orption iothrm in th a. Nglting thir iffrn (Xi t al. th folloing, orption iothrm ill b rfrn to both orption an orption By th ay, if th hytri of th iothrm oul b takn into aount, to rlation, vaporabl atr v rlativ humi b u aoring to th ign of th varia rlativity humiity. Th hap of th iothrm for HPC i influn by many p pially tho that influn xtnt an hmial ration an, in turn, trm trutur an por iz itribution (atrratio, mnt hmial ompoition, SF uring tim an mtho, tmpratur, mix t.). In th litratur variou formulatio foun to rib th orption iothrm onrt (Xi t al. 1994). Hovr, in th papr th mi-mpirial xprion pro Norling Mjornll (1997) i aopt b Proing of FraMCoS-7, May 23-28, 2010