A CRACK-TRACKING TECHNIQUE FOR LOCALIZED DAMAGE IN QUASI-BRITTLE MATERIALS

Transcription

1 A CRACK-TRACKING TECHNIQUE FOR LOCALIZED DAMAGE IN QUASI-BRITTLE MATERIALS Migul Crvra a, Luca Pla b *, Robrto Clmnt a, Pr Roca a a Intrnational Cntr for Numrical Mthods in Enginring (CIMNE), Tchnical Univrsity of Catalonia (UPC), Campus Nort, Jordi Girona 1-3, Barclona, Spain. b DISTART Dpartmnt, Univrsity of Bologna, Vial Risorgimnto 2, Bologna, Italy. Abstract - This work prsnts a procdur to simulat th growth and propagation of localizd tnsil cracks on quasi-brittl matrials. Th so-calld smard damag approach, which consists in standard finit lmnts and local nonlinar constitutiv laws, is rcovrd and improvd in ordr to rprsnt crack localization and avoid spurious msh-bias dpndnc in th discrt problm. This is achivd by mans of th implmntation of a local crack-tracking algorithm which can rproduc individual (discrt) cracks and nsur objctivity of th finit lmnt problm solution. Th prformanc of th localizd damag modl is strssd by mans of th analyss of structural cas-studis. Compard to th smard crack approach in its original form, th prsntd procdur shows clarly a bttr capacity to prdict ralistic collaps mchanisms. Th proposd tracking tchniqu is rlativly inxpnsiv. Kywords: Continuum Damag Modl, Strain Localization, Tnsil Cracking, Crack- Tracking Tchniqu, Quasi-Brittl Matrials. * Corrsponding author. addrsss: migul.crvra@upc.du (Migul Crvra), luca.pla@unibo.it (Luca Pla ), clmnt@cimn.upc.du (Robrto Clmnt), pr.roca.fabrgat@upc.du (Pr Roca)

2 Nomnclatur A C d D E f 0 G f h Finit lmnt ara Isotropic linar-lastic constitutiv tnsor Damag indx Spcific dissipatd nrgy Young s modulus Uniaxial tnsil strngth Tnsil fractur nrgy Avrag lmnt siz H dis Discrt softning paramtr H mat Matrial softning paramtr l dis Discrt crack charactristic width l Finit lmnt siz l mat Matrial charactristic lngth p i r Unit vctor associatd with i-th principal dirction Damag thrshold intrnal variabl r xcl Exclusion radius r nigh Radius of th nighbourhood whr V c is computd r 0 v V c Initial valu of th damag thrshold intrnal variabl Poisson s ratio Crack avrag dirction vctor V c, Vctor which forms an angl α with vctor V c V α Crack dirction vctor for th currnt lmnt Angl btwn V c and V α Maximum curvatur angl ε Φ Λ σ σ Strain tnsor Damag critrion Damag thrshold surfac shap tnsor Strss tnsor Effctiv strss tnsor - 2 -

3 + σ, σ Tnsil and comprssiv ffctiv strss tnsors σ i τ i-th principal ffctiv strss Equivalnt strss : Doubl contraction. Macaulay brackts Acronyms CMOD DCA E-FEM FE FEM SCA X-FEM Crack Mouth Opning Displacmnt Discrt Crack Approach Elmntal nrichmnt Finit Elmnt Mthod Finit Elmnt Finit Elmnt Mthod Smard Crack Approach Xtndd Finit Elmnt Mthod 1. Introduction Th numrical modlling of tnsil cracking phnomna on quasi-brittl matrials is on of th ky topics in Computational Failur Mchanics. Two main difficultis hav to b ovrcom in th discrt finit lmnt (FE) problm: th rprsntation of th opning crack, both in trms of displacmnt and/or strain filds, and th prdiction of th dirction for crack propagation. With rgard to th first issu, th most promising of th nwly proposd mthods ar nhancmnts of th classical Discrt Crack Approach (DCA) and Smard Crack Approach (SCA) [1]. In th Elmntal nrichmnt Finit Elmnt Mthod (E-FEM) [2-6] th displacmnt fild at th crack is assumd to b discontinuous and th strain fild is dcomposd into a rgular part, outsid th crack, and a singular part at th crack. This, togthr with th xplicit control on th nrgy dissipatd in th formation of th crack, rprsnts a link with th stablishd tradition of Fractur Mchanics. Nvrthlss, such approach dos not rally dpart from th usual continuum framwork. Th Xtndd Finit Elmnt Mthod (X-FEM) [7-10], whn combind with lvl sts [11], is abl to rprsnt th discontinuity gomtry, th displacmnt fild across th crack and th dvlopd singular fild at th crack tip in trms of nodal valus at th nods of th msh. Th accuracy and th vrsatility of th approach ar rmarkabl but a disadvantag lis in th spcial intgration ruls rquird insid th affctd finit - 3 -

4 lmnts to account for what happns at and outsid th discontinuity. An xhaustiv rviw about rcnt advancs in X-FEM is rportd in [12]. It has bn obsrvd in [13] that th computational cost in X-FEM is highr than in E- FEM and it incrass vn mor in cas of multipl cracks. Intgratd stratgis with transition from Continuum Damag to macro-cracks simulatd by mans of X-FEM [14] or E-FEM [15] hav bn also proposd in ordr to corrct initial misprdictions of th crack dirction during its volution. Othr options hav bn rcntly invstigatd. On on hand, in [16,17] th displacmnt discontinuity is rplacd with a rgularizd approximation within th framwork of th X-FEM. A mso-scal msh-siz indpndnt charactristic lngth rlatd to th procss zon width is introducd. Thrfor, th displacmnt and strain filds at th crack ar continuous with continuous drivativ, nabling on to adopt standard intgration procdurs instad of sub-lmntal ons. On th othr hand, in [18,19] a smard-mbddd continuum crack modl is prsntd which incorporats th ffct of th displacmnt jumps in th strain fild of th lmnts, rathr than th actual jumps thmslvs. Ncssary corrctions ar introducd in th modl to avoid msh-siz and msh-bias dpndncy. Th solution for th lattr drawback is found in th form of a msh corrctd crack modl, whr th structur of th inlastic strain tnsor is linkd to th gomtry of th crackd lmnt. Such approach can b considrd a particular cas of th classical SCA, implmntd at constitutiv lvl. Figurs 1-2 sktch th comparison btwn th diffrnt approachs dscribd, with rlation to th crack modlling in th discrt FE modl. Th diffrncs among thm ar subtl whn rfrrd to a 1D situation, but th diffrnt implications in 2D and 3D ar significant, sinc it is ncssary to stablish th functional rlationships btwn th filds of displacmnts (continuous and discontinuous) and strains (boundd and unboundd), and thn with thos of strsss. Th displacmnt and strain filds at th crack rprsntd by th strong discontinuity thoris of E-FEM [2] and X-FEM [9] ar dpictd in Figur 1a, whras thir rgularizd vrsions ar rportd rspctivly in Figurs 1b and 1c. Olivr t al. [4] us a small rgularization lngth to obtain a boundd strain fild at th discontinuity dfind in trms of th displacmnt jump, masurd as th diffrnc of th displacmnt fild at both sids of th band, s Figur 1b. This lngth may dcras during th dformation procss according to an volution law. Bnvnuti [16] introducs a rgularization lngth rlatd to th width of th procss zon that rmains constant during th dissipativ procss and insid which - 4 -

5 both th displacmnt and strain filds ar continuous, s Figur 1c. Such rgularizd vrsions of E-FEM/X-FEM rcall, in a crtain sns, th classical SCA of Rashid [1], dpictd in Figur 2a, vn if in this cas th charactristic lngth is qual to th rlvant dimnsion of th finit lmnt whr th crack is to b rprsntd. This similarity is xploitd in th mbddd SCA proposd by Crvra [18,19], s Figur 2b, whr concpts dvlopd for th E-FEM ar incorporatd in a SCA contxt. With rgard to th scond issu mntiond at th bginning of this introduction, i.. th prdiction of th dirction for crack propagation, it is dmd as th main difficulty to b ovrcom in th discrt FE problm. In th classical SCA it has always bn implicitly assumd that th critrion for th onst of cracking, which is stablishd in trms of strsss/strains, also must automatically dfin th dirction of propagation. This may b a natural assumption in th continuous problm, with propr valuation of strss and strain valus and dirctions. Howvr, in th discrt problm th strss and strain filds valuatd in th vicinity of th crack tip diffr gratly from bing xact. Thrfor, th computd damag distribution is incorrct, as it dpnds spuriously on th alignmnt of th finit lmnt msh. This rror must b ovrcom if rasonabl solutions ar to b obtaind with th SCA. On th othr hand, th application of E-FEM or X-FEM invariably nds th us of discontinuity tracking algorithms, in ordr to stablish which lmnts li in th crack path and nd to b nrichd [20-23]. Succssful applications of such tchniqus point to th potntial advantags of using a crack tracking algorithm in th discrt format of th crack propagation problm, also if continuous displacmnt filds ar usd in th intrpolation basis. Th crack tracking tchniqu marks th finit lmnts which can damag and prvnts th othrs from failing. This ssntial fatur minimizs th numbr of possibl solutions, so it hlps to idntify th xpctd on; lads to a bttr rprsntation of th xpctd solution for mixd-mods fractur problms, which ar oftn charactrizd by curvd cracks [24]; avoids (or limits) th mchanical dissipation outsid th crack track. A bttr dscription of th dissipativ phnomnon is achivd by forcing th crack to dvlop along a singl row of finit lmnts, sinc th lmntal softning paramtr is dirctly rlatd to th fractur nrgy of th matrial

6 Th crack tracking algorithm to b usd in crack propagation problms must b consistntly linkd to th cracking critrion, as this is th stablishd cracking mchanism at continuum lvl. For a Rankin critrion basd on th valu of th imum tnsil principal strss, it is consistnt to assum that th crack propagats in th plan orthogonal to th corrsponding first strss ignvctor. Non-local tracking stratgis dtrmin th crack path making rfrnc not to local variabls valus but to thir non-local countrparts, accounting for th influnc of th nighbourhood on a crtain matrial point [25,26]. Non-local avraging tchniqus ar rathr cumbrsom sinc th modular lmnt-wis natur of finit lmnt analyss is spoild. As a consqunc, th complxity of implmntation is rlativly high. A global crack tracking tchniqu has bn proposd by Olivr and co-workrs [5,6] within th framwork of E-FEM and combind with th SCA by Crvra & Chiumnti [24,27] aftrwards. Such a mthodology considrs th valuation of th propagation dirction as a sparat problm, indpndnt from th local valus of th discrt strss/strain filds. Th dirction of propagation is valuatd by solving a conductionlik problm which, by dfinition, is sufficintly wll-bhavd and dos not prsnt any singular point in th vicinity of th advancing crack. Howvr, th solution of such additional problm for ach mchanical loading stp involvs som programming complxity and additional computational cost. As a rmdy for ths drawbacks, a partial domain crack tracking algorithm has bn mployd [15,28] in which th scalar fild with isolins is constructd only for th domain which actually is or potntially will b affctd by th discontinuity. A practical comparison btwn diffrnt crack path tracking tchniqus is rportd in [29]. This work dals with an nhancd local crack-tracking algorithm, in which th propagation dirction is valuatd locally and corrctd opportunly dpnding on potntially damaging and alrady damagd lmnts on th crack [30]. Such innovativ tool, combind with a continuum damag constitutiv law with strain-softning, provids a numrical modl for th solution of problms involving tnsil cracking in quasi-brittl matrials, such as concrt and masonry [31]. Unlik with th us of th standard SCA, th rsulting damag in th ultimat conditions appars localizd in individual cracks, rprsntd via continuous paths. Morovr, th rsults do not suffr from spurious msh-siz or msh-bias dpndnc. Th proposd modl is much lss computationally intnsiv than th similar on usd in [24,27], which is basd on a global crack-tracking tchniqu

7 Th papr is organizd as follows: first, a brif rviw of th local continuum damag modl is rportd; thn, th proposd local crack-tracking tchniqu is discussd in dtail. Finally, th numrical tool is validatd via th FE analyss of rlvant structural xampls. 2. Local Continuum Damag Modl In this work a constitutiv modl basd on th Continuum Damag Mchanics thory is considrd. Sinc th adoptd formulation is mostly idntical to th on dscribd in [27], it will b brifly summarizd hrinaftr, rfrring th radr to th citd papr for any furthr dtail. 2.1 Constitutiv Modl Th constitutiv quation for th damag modl is dfind as ( d) ( d) σ = 1 σ = 1 C: ε (1) whr th ffctiv strss σ [32] can b computd in trms of th total strain tnsor ε, C is th usual (fourth ordr) isotropic linar-lastic constitutiv tnsor and d is th damag indx, i.. an intrnal-lik scalar variabl qual to zro whn th matrial is undamagd and qual to on whn it is compltly damagd. As our aim is to us a scalar damag modl snsitiv only to tnsil strsss contributions, a split of th ffctiv strss tnsor into tnsil and comprssiv componnts is carrid out according to [33]: 3 + = σ i i i and = + i= 1 σ p p σ σ σ (2) whr σ i dnots th i-th principal strss valu from tnsor σ, p i rprsnts th unit vctor associatd with its rspctiv principal dirction and th symbols. ar th Macaulay brackts ( x x, if x 0, x 0, if x 0) = = <. Th scalar positiv quantity, trmd as quivalnt strss τ, is dfind in ordr to idntify loading, unloading or rloading situations for a gnral 3D strss stat, according to: τ = σ : Λ: σ (3) - 7 -

8 Th shap of th damag thrshold surfac in th ffctiv strss spac is dfind by th non-dimnsional fourth ordr tnsor Λ, which is assumd in this work qual to Λ = p1 p1 p1 p 1 (4) lading to th wll-known Rankin critrion in tnsion. If on dsignats by r th gratst valus (hr trmd as thrshold) vr rachd until an instant t by th quivalnt strss τ, incrmnts of th damag variabl d during tim intrval [ t, t t] + ar assumd to occur only whn τ t+ t > r t (s [34] for dtails). Accordingly, th following damag critrion is introducd: ( τ r) Φ, = τ r 0 (5) Basd on Equations (3) and (5), th initial lastic domain is dfind by th condition τ = r 0, whr r 0 f 0 tnsion. =, bing f 0 th strss that dfins th onst of nonlinarity on 1D 2.2 Strain softning and discrt crack width rgularization Th damag indx d d( r) = is xplicitly dfind in trms of th corrsponding currnt valu of th damag thrshold r, so that it is a monotonically incrasing function such that 0 d 1. In this work, th softning law taks th following xponntial form [35] 1 r xp 2 r r = dis r r0 0 0 ( ) H d r (6) whr constant H dis 0 is th discrt softning paramtr. Sinc Bazant and Oh [36] it was rcognizd that th us of constitutiv laws with softning in th SCA lads to solutions strongly dpndnt on th msh rfinmnt. In ordr to nsur msh-siz objctiv rsults, th discrt softning law has to b modifid in such a way that th nrgy dissipatd ovr a compltly dgradd finit lmnt is qual to a givn valu, which dpnds on th matrial fractur nrgy and on th lmnt siz. Accordingly, th spcific dissipatd nrgy D is adjustd for ach FE in th crack band, so that th quation Dl = G (7) dis f - 8 -

9 applis, whr G f is th matrial tnsil fractur nrgy and l dis is th discrt crack charactristic width, i.. th computational width of th fractur zon. This can b drivd according to th consistnt mthodology suggstd by Olivr [37]. For th damag modl with th xponntial softning it can b provd that th spcific dissipatd nrgy is 1 D = 1+ H dis ( f ) E (8) and, thrfor, using (7) and (8) th discrt softning paramtr is dfind as H dis = l mat ldis l dis (9) whr th matrial charactristic lngth is l 1 2 mat = Hmat, with Hmat = ( f0 ) ( 2 EGf ) dpnding only on th matrial proprtis. It has to b rmarkd that this rgularization procdur dos not apply to th continuum format of th cracking problm. If discontinuous displacmnts ar considrd in th continuous problm, th nonlinar constitutiv bhaviour can b stablishd in trms of a traction-vrsus-jump law. If th discrt problm rproducs th displacmnt jumps, as it happns in th E-FEM/X-FEM, thn no rgularization procdur is ncssary (Figur 1a), although it is also usd in th rgularizd vrsions of ths formulations (Figurs 1b and 1c). Contrariwis, in th SCA, msh-siz rgularization is ncssary bcaus of th smaring of th displacmnt jump ovr th rsolution lngth which th FE msh is abl to achiv (Figur 2). Obviously, this only rfrs to th discrt problm, and it dpnds on th lmnt siz. It is important to not that whn th msh is rfind th continuous constitutiv modl is rcovrd as a limit cas. In th framwork of local modls and FE analysis, th stat variabls ar computd at th intgration points in trms of th local strain (and/or strss) history. Thrfor, th discrt crack charactristic width is rlatd to th volum (or ara) of ach finit lmnt [38]. For linar simplx lmnts, th discrt crack charactristic width can b takn as th rprsntativ siz of th lmnt, l dis = l. In this work, and assuming that 2 th lmnts ar quilatral, th siz of th lmnt will b computd as l = ( 4 3) for triangular lmnts. A - 9 -

10 3. Local Crack-Tracking Tchniqu Although th problm of msh-siz dpndnc in SCA has bn satisfactorily ovrcom, as mntiond in Sction 2.2, th issu concrning th msh-bias dpndnc of th FE solution still rmains unsolvd. It is wll-known that only if th spatial discrtization is dsignd in such way that an appropriat path for th advancing crack is availabl, th SCA solution rsults satisfactory. In this work, a solution to th problm is found by making us of a crack-tracking tchniqu. Th local crack-tracking algorithm dtcts th point whr a crack is originatd and thn it lts th crack propagat prpndicularly to th trajctory of th first principal tnsil strss. Th algorithm marks th finit lmnts prtaining to th crack path which can xprinc potntial damag. Th crack is forcd to dvlop along a singl row of finit lmnts. Th rgularization procdur using th matrial proprtis and th lmnt charactristic lngth nsurs that dissipation will b lmnt-siz indpndnt, s Equation (7). Th proposd mthod is applid at vry tim stp during th analysis, just bfor th strss valuation. Th mthod works with a flags systm, whr finit lmnts ar lablld to dlimit th zons whr cracks will appar or dvlop. Th critria usd to dfin ths zons dpnd on th magnitud and dirction of th principal strsss at ach lmnt. Th algorithm has bn implmntd for 2D problms using thr-nodd standard lmnts with continuous displacmnt fild. Dspit thir wll-known approximability limitations, constant strain-triangls nsur an fficint implmntation. Th procdur is dividd into two stps. First, nw cracks ar dtctd by chcking th strss valus at vry finit lmnt locatd on th boundary of th structur. Thn, th track of finit lmnts prtaining to th crack path is markd by th algorithm, in ordr to comput th crack propagation dirction. 3.1 Nw Crack Dtction Th input data of this first stag of th procdur ar (i) th principal tnsil strss valus of th lmnts locatd on th boundary of th msh and (ii) th list of th lmnts lablld as crack root, all rfrrd to th prvious tim stp. Thn, th following oprations ar carrid out: 1. Nw lmnts ar lablld as potntial crack roots. For this aim, w considr som critria:

11 Th crack starts onc th principal tnsil strss valu rachs th matrial strngth, according to th Rankin tnsil critrion. This chck is prformd only on finit lmnts locatd on th boundary of th msh. Thrfor, cracks ar assumd to start only from th bordr of th structur. Whn svral nighbour lmnts rach th tnsil strngth at th sam tim stp, th xclusion radius critrion is applid. This radius r xcl (dfind by th usr) is th minimum distanc imposd btwn two crack root lmnts, and it is usd to guarant th cration of sparatd discrt cracks. Among all th lmnts which hav rachd th tnsil strngth at th sam tim stp, and which ar containd into th xclusion radius, th on with th gratst principal tnsil strss is lablld as crack root. 2. Th spatial coordinats of th crack origin ar computd. Th midpoint of th lmnt sid locatd at th msh boundary is considrd. In cas of cornr lmnts, th cntroid is assumd, s Figur 3. Th output data of th nw cracks dtction procdur is th list of th potntial crack root lmnts, togthr with thir spatial coordinats. 3.2 Crack Propagation Th input data of this scond stag of th procdur ar (i) th list of th potntial crack root lmnts, togthr with thir spatial coordinats, (ii) th principal tnsil strss valus and dirctions of all th msh lmnts and (iii) th list of th lmnts blonging to consolidatd cracks rfrrd to th prvious tim stp. Thn, th following oprations ar carrid out: 1. Dtrmin th tip of th crack lmnt for ach xisting consolidatd crack. This is dfind as th damagd lmnt with only on nighbour damagd lmnt. 2. For ach tip of th crack lmnt, as wll as for th nw crack root lmnts, th following procsss ar xcutd: Dtrmin th xit point coordinats. A vctor is drawn from th ntry point coordinats (dfind blow), using th dirction prpndicular to th principal tnsil dirction of th lmnt. Th xit point is dfind as th intrsction of that vctor with th corrsponding fac of th lmnt, s Figur 4a

12 Dtrmin th nxt potntial lmnt on th crack for th currnt tim stp. This is th nighbouring lmnt whos fac in common with th currnt lmnt corrsponds with th fac whr its xit point is locatd. Th lmnt is markd as a potntial lmnt blonging to this crack. Dtrmin th ntry point coordinats. Th ntry point of th nw lmnt on th crack is locatd at th sam coordinats of th xit point of th prvious lmnt, s Figur 4b. Rpat th thr prvious stps, taking th nw potntial lmnt as th crack tip lmnt. For ach crack, th prvious procdur is rpatd until on of th following critria is satisfid: Strss thrshold critrion. Elmnt tracking and lablling is stoppd whn th principal tnsil strss is lowr than a thrshold dfind by th usr. Th xprinc has dmonstratd that 75% of tnsil strngth usually works wll. Crack mting critrion. Th procdur stops whn a prviously damagd lmnt, or an lmnt markd as a potntially cracking on, is found along th currnt crack. This mans that two cracks hav mt, and from thn on thy will b considrd as a singl on. Boundary critrion. Whn th xit point of an lmnt is on th boundary of th structur, th cracking procss finishs. 3. Onc any of th prvious critria is rachd, th currnt crack is considrd totally dvlopd and th nxt on is studid, by rstarting th cycl. Finally, aftr applying this procdur to all th cracks, ach lmnt will hav on of th thr following labls: Intact lmnt, not abl to damag (out of potntial crack track; it will kp lastic bhaviour during th currnt tim stp) Intact lmnt, abl to damag (in a potntial crack track; it will initiat inlastic bhaviour if th matrial strngth is rachd) Damagd lmnt (blonging to a crack consolidatd in prvious tim stps; it will dvlop inlastic bhaviour during th rst of th calculations) Th analysis procdur rcognizs ths labls and activats th corrsponding constitutiv law (lastic or damag) in ach lmnt for th currnt tim stp. Also, onc th strsss hav bn updatd and th damagd indxs ar known, th lmnts with potntial cracking that rally suffr damag ar rlablld as includd in a consolidatd

13 crack for th rst of calculations. Finally, lmnts potntially cracking that do not suffr damag ar rstord to thir original status, i.. unlablld. Th proposd algorithm, basd on th dirctions of th strsss in th prvious tim stp, is of first ordr accuracy (th rror is proportional to th siz of th tim stp). Mor accurat algorithms can b formulatd. For instanc, a scond ordr prdiction on th strsss at th currnt tim stp can b don in trms of th strsss at th two prvious tim stps. Such procdur would rduc th dpndnc of th computd crack track on th siz of th tim stp significantly, particularly in th cas of curv crack trajctoris. 3.3 Maximum Curvatur Critrion Th implmntation of th crack tracking algorithm in th form dscribd just in Sctions and lads to som problms in cas of bnding strss stats. Figur 5 shows th FE simulation of an advancing flxural crack in th middl of a 3-point loadd concrt bam. Th contour of tnsil damag is zoomd in th proximity of th nutral axs, as wll as th tnsil principal dirctions (rd arrows). As can b sn, sinc th algorithm assums th dirction of th crack to b prpndicular to th principal tnsil strss, th track should propagat from th crack tip (lmnt A) to lmnt B and thn to lmnt C. This is obviously rronous, bcaus th vrtical crack should go up to th lmnt D. Thrfor, th local crack tracking tchniqu nds a spcific dvic to ovrcom such a drawback. In this papr, th imum curvatur critrion is introducd in ordr to corrct spurious changs of propagation dirction. Th procdur consists in idntifying and corrcting th suddn chang of curvatur in th crack track, bfor marking ach potntial lmnt. Making rfrnc to Figur 6a, th following paramtrs ar considrd: Crack dirction vctor for th currnt lmnt V. Crack avrag dirction vctor V c. It is qual to th vctorial sum of th lmntal cracking vctors. Th lmnts considrd in th calculations ar thos potntial at th currnt tim stp and thos consolidatd at th prvious tim stps whos cntroids li insid a nighbourhood of radius r nigh. Such lngth is dfind by th usr and it is masurd from th cntroid of th currnt tip lmnt (s Figur 6b). Angl α btwn V c and V

14 Maximum curvatur angl α, dfind by th usr at th bginning of th calculations. Vctor V c, which forms an angl α with vctor V c. If it rsults that α α, th considrd lmnt is markd as potntial for th currnt tim stp with a corrsponding dirction V. Thn, th following lmnt of th crack is considrd. If it rsults that α > α, th crack dirction is dflcting sharply (s Figur 6a) and a corrction is carrid out, using V c instad of V. This mans to impos th crack th avrag dirction xhibitd until th stp considrd with rspct to a spcifid nighbourhood of radius r nigh (s Figur 6b). In th cas of problms involving straight or small curvatur cracks, th choic of r nigh obviously dos not affct particularly th crack dirction prdiction. In th cas of problms with curvd cracks, on th othr hand, vctor V c has to b computd making rfrnc to a limitd numbr of FEs in ordr to simulat corrctly th curvatur of th propagating crack. Thrfor, th choic of r nigh may influnc th rsults accuracy, as it will b shown in Sction 4.3. Onc th crack dirction corrction is carrid out, th standard procdur is followd, i.. th lmnt is markd as potntial and th nw on is considrd. It is worth mntioning that crack tracking algoritms ar snsitiv to th prcision achivd in th computd valus of th strsss (or strains) at th root of th crack. Point-wis convrgnc on th strss (or strain) valus for a givn FE formulation is a rsarch topic which dsrvs in-dpth invstigation. 4. Validation Exampls In this Sction, th proposd localizd damag modl basd on a crack-tracking tchniqu is validatd by mans of numrical structural xampls. First, a simpl bnchmark xampl consisting in a hold strip subjctd to uniaxial tnsion is analyzd. Th scond validation xampl is a thr point bnding tst on a concrt bam. Thn, a mor complicatd cas study is considrd, that is th simulation of a mixd-mod bnding tst on a concrt bam. Finally, th analysis of a masonry smicircular arch is prsntd, for th particular cas of asymmtrical vrtical load

15 Calculations ar prformd with an nhancd vrsion of th finit lmnt program COMET [39], dvlopd at th Intrnational Cntr for Numrical Mthods in Enginring (CIMNE, Barclona). Th problm is solvd incrmntally in a (psudo) tim stp-by-stp mannr. Within ach stp, a modifid Nwton Raphson mthod (using th scant stiffnss matrix), togthr with a lin-sarch procdur, is usd to solv th corrsponding non-linar systm of quations. Convrgnc of a tim stp is attaind whn th ratio btwn th norm of th itrativ rsidual forcs and th norm of th total xtrnal forcs is lowr than 1%. Pr- and post-procssing ar don with GiD [40], also dvlopd at CIMNE. 4.1 Hold Strip undr Uniaxial Traction Th numrical analysis of a hold strip subjctd to uniaxial strtching is considrd, in ordr to point out th diffrnc btwn a traditional smard damag modl and its nhancd vrsion improvd by a crack tracking algorithm. This xampl has bn alrady solvd in [27] making us of a global crack-tracking tchniqu. Th sam matrial data ar considrd: Young s modulus E = 30 GPa, Poisson s ratio v = 0.2, tnsil strngth f 0 = 2 MPa and mod I fractur nrgy G = 100 J m f 2. Th spcimn siz is 200 x 400 mm 2 and th radius of th prforation is qual to 10 mm. Axial vrtical displacmnts ar applid to both th strip nds. Sinc th problm is symmtrical, only th right half of th computational domain is considrd and discrtizd in two diffrnt unstructurd mshs with avrag msh sizs of h = 5 mm (2023 nods) and h = 2.5 mm (7648 nods). Th problm is analyzd assuming two-dimnsional plan strain conditions. First, th traditional smard damag modl is considrd in calculations. Figur 7 shows th computd dformd shaps and tnsil damag contours in th two diffrnt mshs for a (half)-imposd vrtical displacmnt of 0.1 mm. As shown, th crack grows from th prforation, thn it propagats horizontally and it suddnly dviats from th xpctd corrct path, following a lin of FEs inclind of about 30. This rsult is dfinitly spurious and strongly dpndnt on th considrd spatial discrtizations. Thn, th damag modl basd on a local crack-tracking tchniqu is adoptd in th analyss. Th choic of th tracking paramtrs is irrlvant in this xampl. In fact, r xcl dfinition is not ncssary sinc a uniqu crack is xpctd. No corrction of th

16 crack dirction is rquird bcaus th strss fild is uniformly uniaxial, so α and r nigh ar irrlvant. Figur 8 shows how th crack propagats corrctly along th horizontal axis of symmtry, without following any favourabl path givn by th considrd mshs. Figur 9 shows th (half)-load vs. (half)-imposd vrtical displacmnt curvs obtaind in th two analyss with local crack-tracking tchniqu. As shown, th rsults ar rmarkably msh-siz indpndnt. It is worth noticing that th rsults prsntd in Figurs 8-9 ar idntical to th ons obtaind in [27] by a global crack-tracking tchniqu bcaus th mshs usd ar th sam and both th algorithms prdict xactly a horizontal crack. Nvrthlss, th global tracking approach is mor computationally xpnsiv than th local on, sinc it is ncssary to solv a conduction-lik problm at ach stp of th analysis. Th local crack-tracking algorithm, on th othr hand, stands out bcaus of its fficincy and lowr computing tim. In fact, if th analysis with h = 2.5 mm is run in a standard PC quippd with a singl Pntium GHz, 2 GB Ram procssor, th absolut CPU tim rsults qual to sconds using th global tracking and qual to sconds using th local tracking. Sinc th computing tim for th classical SCA is qual to sconds, th additional calculation ffort for th local crack-tracking is only 24.5 % of that rquird for th global on point Bnding Bam Th scond xampl is a thr point bnding tst on a notchd concrt bam [41] whos dimnsions and boundary conditions ar givn in Figur 10. Th bhaviour of th proposd local crack-tracking algorithm in bnding problms is invstigatd, as wll as th influnc of th load stp siz. Th msh is constitutd by 2380 lmnts and 1248 nods. Th FEs avrag siz is h = 2 mm in th proximity of th cracking zon. Th two adoptd load incrmntation stratgis consist of 100 and 1000 load stps; accordingly, th vrtical displacmnt δ = 1 mm is applid in th midpoint through stps of magnitud mm and. Th matrial data ar: Young s modulus E = 20 GPa, Poisson s ratio mm v = 0.2, tnsil strngth f 0 = 2.4 MPa and mod I fractur nrgy G = 113 J m f 2. Th bam has thicknss qual to 100 mm and it is analyzd undr two-dimnsional plan strain assumptions.

17 As xplaind in Sction 3.3, th imum curvatur critrion is ncssary in th cas of flxur problms to avoid th crack about-turn phnomnon (Figur 5). Accordingly, svral FE analyss hav bn xcutd in ordr to assss th influnc of α on th rsults. With a stp magnitud of, valus α 50 prdict th corrct mm crack dirction, s Figur 11. For gratr valus of α, th prdictd crack suffrs from sharp changs of dirction during its propagation, lading to incorrct rsults. For a smallr stp magnitud of α 60., th aformntiond uppr limit incrass to mm Figur 12 shows th comparison btwn th xprimntal [41] and th numrical loaddisplacmnt curvs. Not that th curvs obtaind with th two diffrnt load stp sizs tally vry closly. Good agrmnt is achivd, both in trms of load pak valu and nrgy dissipation in th nonlinar rang. Dfinition of r xcl is not ncssary in this xampl sinc it prsnts a uniqu crack. Also, an adquat numbr of nighbour lmnts is slctd with r 10 mm in ordr to nigh rproduc th corrct straight dirction of th crack. 4.3 Mixd-Mod Bnding Bam Th third xampl is a plan-strain notchd bam subjctd to mixd-mod bnding tst. Th xprimntal vidnc rportd in [42] is simulatd by mans of th proposd modl. Figur 13 dpicts th gomtry of th problm and th boundary conditions applid to th concrt bam. In a first cas th stiffnss at th uppr lft support is assumd qual to zro ( K = 0, i.. a thr point bnding bam), whil in a scond cas is infinit ( K =, i.. a four point bnding bam). Th load P is applid by imposing vrtical displacmnts. Th following matrial proprtis ar assumd for concrt: Young s modulus mod I fractur nrgy E = 38 GPa, Poisson s ratio v = 0.2, tnsil strngth f 0 = 3 MPa and 2 G f = 69 J m. Th computational domain is discrtizd in triangular lmnts, with a total numbr of 7028 nods. Th avrag msh siz in th zon intrstd by th tnsil fractur is h = 2.5 mm. Figur 14 shows th computd dformd shaps at collaps obtaind assuming α = 20 for th spcimn 1 (thr point tst) and α = 32 for th spcimn 2 (four point tst). In th formr analysis it is assumd r = 5 mm whil in th lattr nigh r nigh = 10 mm. Th dfinition of r xcl is not ncssary in ths xampls sinc thy

18 prsnt a uniqu crack. Figur 15 shows th comparisons btwn th xprimntal and th numrical crack tracks, in which it is possibl to apprciat a rmarkabl agrmnt. Figur 16 prsnts th Load vs. CMOD curvs for spcimns 1 and 2. In th first cas (Figur 16a), th numrical curv is quit similar to th xprimntal ons, vn if th numrical modl slightly undrstimats th strngth in th last part of th calculus. In th scond tst (Figur 16b), th concordanc of numrical rsult with th xprimntal vidnc is clar, until a suddn collaps occurs for CMOD = 0.11 mm. For spcimn 2, a imum curvatur angl α = 32 has bn chosn in ordr to avoid sharp changs of th crack dirction. Sinc th crack is straight, th assumd valu r = 10 mm is irrlvant for th rsult shown in Figur 14b. nigh A snsitivity analysis rgarding th paramtrs usd for th tracking algorithm has bn carrid out for spcimn 1, bcaus it prsnts an intrsting cas of a crack with a crtain curvatur. Th first paramtr which has bn invstigatd is α bcaus it plays a ky rol in th crack dirction corrction through th imum curvatur critrion, as shown in Sction 4.2. Valus of α 55 ar ncssary to avoid sharp changs of th crack dirction. Blow this uppr-bound valu, som variations in th numrical crack track can b apprciatd for α = 40 and thn for α = 20. It is noticd that thr is an initial lngth of th crack of about 40 mm which is insnsitiv to th variation of th tracking paramtrs. It is at this distanc from th notch whr th algorithm coms into play. This is vidnt in Figur 17, whr th computd crack tracks (dpictd by th damagd lmnts) can b compard to th xprimntal nvlop. Onc th crack about-turn phnomnon is avoidd, thanks to th rstriction imposd by α, th ffct of th choic of r nigh is invstigatd. Rsults corrsponding to valus of r nigh = 20, 10 and 5 mm ar shown. Valus of r nigh > 20 mm yild th sam rsults as with r = 20 mm. On on hand, Figur 17 shows that for r = 20 mm th nigh nigh rsulting crack track is almost straight. Valus of r = 10 mm or r = 5 mm, on th nigh nigh othr hand, allow th crack to chang dirction during propagation. For α = 20 and r = 5 mm th numrical crack track lis ntirly insid th xprimntal nvlop. nigh 4.4 Asymmtrically Loadd Smicircular Arch Th fourth xampl consists in a smicircular arch with thicknss and width qual to 1 m and a radius of 5 m. Both th supports ar fixd and a vrtical load is applid

19 according to th schm rportd in Figur 18a. Th following matrial proprtis ar considrd for masonry: Young s modulus E = 5 GPa, Poisson s ratio v = 0.2, tnsil strngth f = 0.2 MPa and mod I fractur nrgy G = 10 J m f 2. All th numrical analyss ar carrid out undr th hypothsis of plan strain, with unitary thicknss assumd. Gravitational loads ar applid to th arch, bfor a vrtical load is monotonically incrasd by considring load stps of 3 kn in th cas of forc control or incrmnts of 0.05 mm undr displacmnt control. Th computational domain is discrtizd in triangular lmnts, with a total numbr of 5989 nods and a avrag msh siz h = 50 mm. Bfor th numrical analyss, a static limit analysis has bn carrid out in ordr to assss th ultimat load valu (about 150 kn) and th position of th plastic hings, dfind by lttrs A-D in Figur 18b. Such rsults hav bn also validatd through th RING softwar [43]. Thn, th masonry arch bhaviour is studid by mans of th traditional smard damag approach, considring both th load and th displacmnt control analyss. Figur 19 shows th curv of load vs. vrtical displacmnt in th application point of forc P. As can b sn, th ultimat load calculatd valu is quit similar to th on drivd by th limit analysis. Figur 20 prsnts th damag distribution at stags 1-5 (rfr to Figur 19) of th calculus carrid out undr forc control. Th attntion is focusd on th locations whr a hing dvlopmnt is xpctd, i.. th points A-D of Figur 18b. Th structural bhaviour rmains linar until P = 110 kn, which corrsponds to stag 1. Damag occurs first at points B and D, which ar subjctd to th highst valus of tnsil strss. Thn, a suddn xtnsion of th damagd zon ariss in such points, as shown at stag 3. At stag 4, cracks B and D still rmains radially orintd, whil at points A and C a quit distributd damag occurs at th xtrados. At stag 5, crack C progrsss downwards and sprads ovr a quit xtndd rgion, similar to a bulb. Aftr stag 5, as soon as crack A occurs, sctions B, C and D damag compltly and th arch collaps mchanism is activatd. No mor quilibrium points ar found undr load control. Figur 21 prsnts th rsults of th analysis undr displacmnt control. Compard to th prvious analysis, it is possibl to dtct th unloading occurring aftr th formation of cracks B and D (branch 1-2 of Figur 19). At stag 2, crack B is not as sprad as th quivalnt on obtaind undr load control. Thrfor, it is possibl to conclud that in th smard damag approach th solution dpnds on th particular loading path

20 followd. Anothr unloading occurs aftr th formation of cracks A and C (branch 4-6 of Figur 19). As shown in Figur 21, both th cracks follow a path givn by th spatial discrtization adoptd. Thrfor, it is vidnt th spurious msh-bias dpndnc of th solution. Aftr stag 6, th structur still prsnts a low stiffnss, until anothr unloading is discovrd (point 7 of Figur 19), which corrsponds to th formation of a hing in B (s Figur 21, in which crack B is shown bfor and aftr th jump). Finally, th arch is analyzd by considring th localizd damag modl with cracktracking tchniqu. Th following corrction paramtrs ar assumd for th imum curvatur critrion: α = 30 and r = 300 mm. Th formr has bn chosn in nigh ordr to avoid spurious changs of cracks dirction, whras th lattr is th minimum valu ncssary to rproduc corrctly th radial dirction of cracks. Th xclusion radius r xcl has bn assumd qual to 3000 mm, making rfrnc to th positions of th plastic hings drivd from th prvious limit analysis. Howvr, valus within th wid rang 1500 mm 3200 mm lad to a corrct prdiction of th hings r xcl positions. Figur 22 shows th curv of load vs. vrtical displacmnt in th application point of forc P. In th cas of displacmnt control, th modl is abl to captur th unloading occurring aftr th formation of th first crack, dnotd by branch 2-3. It is worth noticing that from point 4 th rspons of th analyss undr forc or displacmnt control ar almost th sam, as wll as th cracks growth (Figur 23a) and th collaps mchanism (Figur 23b). Morovr, th damag distribution computd thanks to th tracking tchniqu dos not follow th msh-bias, nsuring th corrct xpctd solution of radial cracks. 5. Conclusions This papr prsnts a local crack-tracking tchniqu for th numrical modlling of tnsil cracks in quasi-brittl matrials. Th classical smard crack approach is improvd by forcing th crack to dvlop along a singl row of finit lmnts, as a function of th dirction of th principal tnsil strss. Th algorithm works with a flag systm whr th finit lmnts which ar allowd to damag during th currnt tim stp ar appropriatly lablld. Th proposd modl has shown th following rmarkabl faturs: Th simulation of th damag distribution in ultimat conditions is mor ralistic than in th classical SCA. Th localizd cracks prdictd by th tracking modl

21 rproduc consistntly th ons usually xprincd by concrt or masonry structurs, which dvlop gradually and lad to th full collaps mchanism. Compard to th classical SCA, it dos not rquir a significant additional computation cost. Compard to th global crack-tracking tchniqus, th proposd algorithm rquirs a lss intnsiv computational ffort. Th dissipation is lmnt-siz indpndnt du to th rgularization procdur dpndnt on th matrial proprtis and th charactristic lngth. Th analysis rsults ar msh-bias indpndnt, nsuring th solution objctivity whn diffrnt orintations of th msh ar considrd in th discrt problm. Th crack track corrction, prformd according to a imum curvatur critrion, avoids th spurious changs of propagation dirction which ar usual in bnding dominatd problms. Th validation xampls hav shown th fficincy and th robustnss of th local crack-tracking tchniqu, which turns out to b suitabl for tnsion and mixd-mod cracking problms. Th snsitivity analyss with rspct to th tracking paramtrs hav assssd thir influnc on numrical solutions: α, r nigh and r xcl Th imum curvatur angl α is th most influntial paramtr, sinc it prvnts th crack from turning back in bnding problms. Th rlativ indpndnc of th rsults with rspct to th chosn valu for α has bn dmonstratd, vn for diffrnt load stps. Th dfinition of r nigh is usful to rproduc th corrct track of curvd cracks. It can b intrprtd as a rfinmnt paramtr for crack track corrction. Th xclusion radius r xcl is ncssary in multi-crack problms in ordr to rprsnt th corrct failur mchanism. Givn th snsitivity of th obtaind crack paths on th slctd paramtrs ( α in particular), snsitivity analyss similar to th ons conductd in th papr ar advisabl whn th proposd tracking algorithm is usd to prdict crack tracks without th hlp of known xprimntal rsults. This rmark is of cours applicabl to any of th xisting crack tracking algoritms. Snsitivity analyss ar almost mandatory in nonlinar computations in ordr to assss th rliability of th rsults obtaind

22 6. Acknowldgmnts Th studis prsntd hr hav bn dvlopd within th rsarch projcts BIA and SEDUREC (CSD ), fundd by DGE of th Spanish Ministry of Scinc and Tchnology, whos assistanc is gratfully acknowldgd. 7. Rfrncs [1] Rashid Y. Analysis of prstrssd concrt prssur vssls. Nucl Engrg Ds 1968; 7: [2] Simó J.C., Olivr J., Armro F. An analysis of strong discontinuitis inducd by strainsoftning in rat-indpndnt inlastic solids. Comput Mch 1993; 12: [3] Olivr J. Continuum modling of strong discontinuitis in solid mchanics using damag modls. Comput Mch 1995; 17: [4] Olivr J., Crvra M., Manzoli O. Strong discontinuitis and continuum plasticity modls: th strong discontinuity approach. Int J Plast 1999; 15: [5] Olivr J., Husp A.E., Samanigo E., Chavs W.V. Continuum approach to th numrical simulation of matrial failur in concrt. Int J Numr Anal Mthods Gomch 2004; 28: [6] Olivr J., Husp A.E. Continuum approach to matrial failur in strong discontinuity sttings. Comput Mthods Appl Mch Engrg 2004; 193: [7] Blytschko T., Black T. Elastic crack growth in finit lmnts with minimal rmshing. Comput Mthods Appl Mch Engrg 1999; 45(5): [8] Mös N., Dolbow J., Blytschko T. A finit lmnt mthod for crack growth without rmshing. Int J Numr Mthods Engng 1999; 46: [9] Mös N., Blytschko T. Extndd finit lmnt mthod for cohsiv crack growth. Engng Fract Mch 2002; 69: [10] Sukumar N., Mös N., Moran B., Blytschko T. Extndd finit lmnt mthod for thrdimnsional crack modlling. Int J Numr Mthods Engng 2000; 48: [11] Sthian J.A. Lvl St Mthods and Fast Marching Mthods. Cambridg Univrsity Prss, Cambridg, [12] Blytschko T., Graci R., Vntura G. A Rviw of Extndd/Gnralizd Finit Elmnt Mthods for Matrial Modlling. Modll Simul Matr Sci Eng 2009; 17(4):

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24 [27] Crvra M., Chiumnti M. Msh objctiv tnsil cracking via a local continuum damag modl and a crack tracking tchniqu. Comput Mthods Appl Mch Engrg 2006; 196: [28] Fist C., Hofstttr G. Msh-insnsitiv strong discontinuity approach for fractur simulations of concrt. In: Procdings of th 9th intrnational confrnc on numrical mthods in continuum mchanics, [29] Jägr P., Stinmann P., Kuhl E. Modling thr-dimnsional crack propagation - A comparison of crack path tracking stratgis. Int J Numr Mthods Engng 2008; 76: [30] Clmnt R. Análisis Estructural d Edificios Históricos mdiant Modlos Localizados d Fisuración (in Spanish). Ph.D. Thsis. Tchnical Univrsity of Catalonia, [31] Pla L. Continuum Damag Modl for Nonlinar Analysis of Masonry Structurs. Ph.D. Thsis. Tchnical Univrsity of Catalonia, Univrsity of Frrara, [32] Lmaitr J., Chaboch J.L. Aspcts phénoménologiqus d la ruptur par ndommagmnt. J Méc Appl 1978; 2: [33] Crvra M., Olivr J., Faria R.: Sismic valuation of concrt dams via continuum damag modls. Earthquak Eng Struct Dyn 1995; 24(9): [34] Simó J.C., Ju J.W. Strain- and strss-basd continuum damag modls I Formulation. Int J Solids Struct 1987; 23: [35] Crvra M. Viscolasticity and Rat-Dpndnt Continuum Damag Modls. Monography Nº-79. CIMNE, Tchnical Univrsity of Catalonia, [36] Bazant Z.P., Oh B.H. Crack band thory for fractur of concrt. Matr Struct 1983; 16: [37] Olivr J. A consistnt charactristic lngth for smard cracking modls. Int J Numr Mthods Engng 1989; 28: [38] Crvra M., Hinton E., Hassan O. Non linar analysis of rinforcd concrt plat and shll structurs using 20-nodd isoparamtric brick lmnts. Comput Struct 1987; 25(6): [39] Crvra M., Aglt d Saracibar C., Chiumnti M. COMET: COupld MEchanical and Thrmal analysis - Data Input Manual Vrsion 5.0. Tchnical rport IT-308. CIMNE, Tchnical Univrsity of Catalonia, [40] GiD: th prsonal pr and post-procssor. CIMNE, Tchnical Univrsity of Catalonia, Availabl from:

25 [41] Rots J.G. Computational modlling of concrt fractur. Ph.D. Thsis. Dlft Univrsity of Tchnology, [42] Gálvz J.C., Elics M., Guina G.V. and Planas J. Mixd mod fractur of concrt undr proportional and nonproportional loading. Int J Fract 1998; 94: [43] Gilbrt M. RING masonry arch analysis softwar [last updat: ]. Availabl from: 8. Figurs Captions Figur 1 Rcnt FEM approachs to crack modlling: a) X-FEM/E-FEM without rgularization; b) rgularizd E-FEM [4]; c) rgularizd X-FEM [16]. Figur 2 SCA: a) classical [1] and b) mbddd [18,19]. Figur 3 Location of cracks origin coordinats. Figur 4 Exit point of lmnt I (a); nw potntial lmnt J and its ntry point (b). Figur 5 Crack about-turn in th proximity of th nutral axis, undr bnding conditions. Figur 6 Maximum curvatur critrion: a) tip lmnt corrction and b) nighbour lmnts chosn to comput V c (with thicknd sids). Figur 7 Dformd msh (x100) and tnsil damag contour with a smard damag modl: msh with avrag siz (b). h = 5 mm (a) and msh with avrag siz h = 2.5 mm Figur 8 Dformd msh (x100) and tnsil damag contour with th localizd damag modl: msh with avrag siz h = 5 mm (a) and msh with avrag siz h = 2.5 mm (b). Figur 9 Load vs. displacmnt for hold strip with th localizd damag modl. Comparison among diffrnt msh sizs. Figur 10 Thr point bnding tst stup [41]. Figur 11 a) Damag in th bnding bam, b) dtail of crack and c) dformd msh (x35). Figur 12 Load vs. displacmnt for bnding bam. Comparison among diffrnt magnituds of th analysis stp. Figur 13 Mixd-mod bnding tst stup [42]. Figur 14 Numrical dformd shaps (x100 and x300) at collaps and dtails of cracks: a) spcimn 1 (thr point tst) and b) spcimn 2 (four point tst)

26 Figur 15 Comparison btwn th xprimntal and numrical crack tracks for th spcimns 1 and 2. Figur 16 Load vs. CMOD for th spcimn 1 (a) and th spcimn 2 (b). Figur 17 Spcimn 1: numrical crack track snsitivity to r nigh and α. Figur 18 Smicircular masonry arch: a) gomtry [m]; (b) thrust lin and plastic hings obtaind via limit analysis [43]. Figur 19 Comparison btwn Load vs. Vrtical Displacmnt curvs for limit analysis and smard damag modl undr load or displacmnt control. Figur 20 Smard damag modl (analysis undr forc control): cracks growth at diffrnt stags of th calculus and final collaps mchanism (amplification of msh dformation: x100). Figur 21 Smard damag modl (analysis undr displacmnt control): cracks growth at diffrnt stags of th calculus and final collaps mchanism (amplification of msh dformation: x100). Figur 22 Comparison btwn Load vs. Vrtical Displacmnt curvs for limit analysis and localizd damag modl undr load or displacmnt control. Figur 23 Localizd damag modl: a) dtail of cracks at collaps and b) collaps mchanism (amplification of msh dformation: x100)

27 Figur 1 Rcnt FEM approachs to crack modlling: a) X-FEM/E-FEM without rgularization; b) rgularizd E-FEM [4]; c) rgularizd X-FEM [16]. Figur 2 SCA: a) classical [1] and b) mbddd [18,19].

28 Figur 3 Location of cracks origin coordinats. Figur 4 Exit point of lmnt I (a); nw potntial lmnt J and its ntry point (b). Figur 5 Crack about-turn in th proximity of th nutral axis, undr bnding conditions.

29 Figur 6 Maximum curvatur critrion: a) tip lmnt corrction and b) nighbour lmnts chosn to comput V c (with thicknd sids). Figur 7 Dformd msh (x100) and tnsil damag contour with a smard damag modl: msh with avrag siz h = 5 mm (a) and msh with avrag siz h = 2.5 mm (b).

30 Figur 8 Dformd msh (x100) and tnsil damag contour with th localizd damag modl: msh with avrag siz h = 5 mm (a) and msh with avrag siz h = 2.5 mm (b). Figur 9 Load vs. displacmnt for hold strip with th localizd damag modl. Comparison among diffrnt msh sizs.

31 Figur 10 Thr point bnding tst stup [41]. Figur 11 a) Damag in th bnding bam, b) dtail of crack and c) dformd msh (x35). Figur 12 Load vs. displacmnt for bnding bam. Comparison among diffrnt magnituds of th analysis stp.

32 Figur 13 Mixd-mod bndig tst stup [42]. Figur 14 Numrical dformd shaps (x100 and x300) at collaps and dtails of cracks: a) spcimn 1 (thr points tst) and b) spcimn 2 (four points tst). Figur 15 Comparison btwn th xprimntal and numrical crack tracks for th spcimns 1 and 2.

33 Figur 16 Load vs. CMOD for th spcimn 1 (a) and th spcimn 2 (b). Figur 17 Spcimn 1: numrical crack track snsitivity to r nigh and α.

34 Figur 18 Smicircular masonry arch: a) gomtry [m]; (b) thrust lin and plastic hings obtaind via limit analysis [43]. Figur 19 Comparison btwn Load vs. Vrtical Displacmnt curvs for limit analysis and smard damag modl undr load or displacmnt control.

35 Figur 20 Smard damag modl (analysis undr forc control): cracks growth at diffrnt stags of th calculus and final collaps mchanism (amplification of msh dformation: x100).

36 Figur 21 Smard damag modl (analysis undr displacmnt control): cracks growth at diffrnt stags of th calculus and final collaps mchanism (amplification of msh dformation: x100).