Accurate simulation of frictionless and frictional cohesive crack growth in quasi-brittle materials using XFEM

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1 Aue smulo o oless ol ohesve k goh qus-le mels usg XFEM B.L. Khloo & Q.Z. Xo Shool o Egeeg, C Uves, Quee s Bulgs, The Pe, Nepo Ro, C CF AA, UK ABSTRACT: Ths ppe susses he k p smpo els o oless ol ohesve ks qus-le mels. Ths hs ee me possle e eomg he ohesve-l o spel u uvesl poloml. Fo ue smulo o k goh qus-le mels usg he eee/geelze e eleme meho (XFEM), he leg em ohe ue spleme smpo el s use s he ehme uo he p o ohesve k. The opeg ompoe ohe sme el s lso use s he l guess opeg pole o el eee ohesve segme. A sll mssle sess eove (SAR) ehque s use o eove he sesses he k p. Fll, pue moe I ohesve k polem s lse o emose he hess o glol esposes lol els oe umell he XFEM. INTRODUCTION The ohesve zoe (o k) moel o Hlleog e l. (976) hs ee eesvel use he su o lolso lue qus-le mels suues. Eles e l. () hve susse s vges lmos. I he mos el use s omulo ohe moel (see Fg. elo), s ssume h he sess s ehvou s soop le els, h he k s e po hee he mmum esle ppl sess ehes he esle segh ohe uke mel, h he k s oee oml o he eo o. A evoluo l s lso posule o he mooo moe I log so h he ohesve sess s uque uo ohe k opeg hh, o oee, eeses moooll log he ohesve zoe. The ohesve k popges he s p ehes. Alhough hs s omulo ohe ohesve k moel s hghl smple, s le o pue he essee ohe ue poess oee spemes suues (see Khloo 99). e Bos e l. () hve gve ose oveve ohe vous s o umell mpleme he ohesve zoe mehoolog. The olue h he eee/geelze e eleme meho (XFEM) (Moës e l. 999, Sououls e l., Khloo & Xo, Xo & Khloo ) poves pope epeseo ohe see he o ohesve zoe omulos vog mesh s. The XFEM ehes he s lol FE ppomos h ko omo ou he polem, suh s spleme sou oss k, he smpo soluo k p, o s sou oss ee, h he use ohe po o u (PU). I os h he FEM, vos he use o meshes oomg h he sou pve emeshg s he sou gos. I he XFEM omulo ohe ohesve zoe moel, Wells Slus () use he jump uo s ehme uo o he hole ohesve k, hee he ohesve k p ouhes he eleme ou. Moës Belshko () use he jump uo o he p ohe ohesve k o je o s p, h uo je o he p. Ths ppoh hle ses hh he ohesve k p oes o ouh he eleme ou. Hoeve, he ope h o-ee k () ohesve zoe K I Fgue. A el o-ee k emg ue poess (ohesve) zoe (FPZ) h esul sess se p () hose es lose smoohl e s p (K I ). The mel ouse he FPZ s le els, u h he FPZ s soeg.

2 uo oes o epese he ue smpo ue ohe spleme/sess el je o he ohesve k p. Hso Hso () moelle sog ek soues sol mehs oseg eleme vese sou s oule eleme eh hl o hh s use o he epolo o oe se ohe sou. Ths ppoh s ul o use he he sou es h eleme. Z Belshko () ehe ll ke le-oe o qu 6-oe gul elemes lug he elemes og he k p he sg uo. Ale e l. () emee spleme jumps hh o o ee o e homogeeous h eh FE. M Peego () oue s FE moel u spleme sou, oe o epoue he pl usp-lke shpe ohe poess zoe he p o ohesve k. Hoeve, hs u uo oes o epese he ue gul suo ohe spleme je o he p. The lk o ok o he smpo els he p o ohesve k s supsg oseg he espe use o ohesve k moels. Ths lk hs ee eel lle Xo Khloo (6) ho oe uvesl smpo epsos ohesve k p, logous o Wllms epsos o-ee k p. Coulom o o he ohesve k es s lso osee. The oees ohe epsos o ouse epe olel o he soeg-l he ou oos. These uvesl epsos e vl o oml oheso-sepo l (.e. soeg l) h e epesse spel poloml om. The emose h m ommol-use oheso-sepo ls, e.g. egul, le, le epoel, e epesse hs spel om. I hs ppe, e ll summze he eomule poloml soeg-l he k p smpo els o oless ol ohesve ks qus-le mels gve Xo Khloo (6). We ll use he leg em o he ue spleme smpo el he p o ohesve k s he ehme uo he XFEM. The opeg ompoe ohe sme el s lso use s he l guess opeg pole o el eee ohesve segme. The p ohe gog ohesve k oes o hve o eme o eleme ou. Ruse () hs sho h elvel smll eos he eemo ohe k ph eleo gle le o sg umulve evo ohe k ph ove e k legh. Theeoe elle lss o ohesve k popgo eques ue kolege ohe k p el. I oe o o ue gul suo ohe sess je o he k p, sll mssle sess eove (SAR) sheme (Xo & Khloo, 6, Xo e l. 7) ll e ope. SAR uses ss uos, hh mee he equlum equos h he om he lol o oos o he ou, movg les sques (MLS) o he sesses smplg (e.g., quue) pos oe he XFEM. The mos el use sess eove sheme (eoe s AVG) ll lso e use o ompso. AVG smpl veges he sess vlues eh oe evlue om je elemes -le epolo om he Guss pos, epoles he vege sesses usg shpe uos. We ll lse ohe leul speme me o qus-le mels h le l sho h he goh o ohesve k qus-le mel e uel pee h ve ose mesh hh s o possle h he FEM. We ll lso emose he hess o glol esposes lol els oe umell he XFEM. POLYNOMIAL COHESIVE LAW FOR QUASI-BRITTLE MATERIALS I oe o o he seple smpo el ohesve k p ( ems o uos, e lol pol ooes eee he p, see Fg. ) qus-le mels lke oee, Xo Khloo (6) eomule he soeg l o he ollog poloml o omlze om () ˆ ˆ ˆ () hee e he sess oml o he ohesve k e he ul esle segh, espevel; e he opeg spleme ohe ohesve k es, he l opeg spleme ohe pe-esg mok p he egs o go;,, e g pmees. Relo () o () epese e ve o soeg ls. Fo emple, he el use le soeg l () e epesee () h he ol ovshg oees -, -.,.e. ˆ ˆ ˆ ( ) () ˆ The oelo oee s. The le l () s ompe h () o () Fg. ; he o e sgushe o he sle ohe gue.

3 /. Le ohesve l () Poloml () o () /. Coelsse e l. (96) Poloml ().... / Fgue. Le eso-soeg l. / Fgue. Ble eso-soeg l. Smll, o he el use le esosoeg l (Fg. ) ( ˆ ) ˆ ˆ () ( ˆ ), ˆ < ˆ ˆ ( ˆ, ˆ ˆ ˆ ˆ, ˆ ), s o le ps e ee o o le ls s sho Fg.. The s p e e o () usg e eo o s (6) ˆ The seo p e e o () usg e eo o s ˆ (7) Coelsse e l. (96) oue he ollog epoel elo () () (), C C e () C C e g pmees. Fo oml oee (NC) h es 7 kg/m, ompessve segh 7 MP, Youg moulus E 9 GP,. MP, 6 μm, spe ue eeg G F J/m (e ue he eso-soeg uve), C C 6.9. Ths gm lso e e () h -7, -6.79, 67., -,.. The oelo oee s. The e ompe Fg., o e sgushe o he sle ohe gue. Weh (99) oue he ollog soeg eloshp m m ˆ ˆ (9).. / Fgue. Cohesve l (). /.. Cohesve l (9), m.7 Poloml () o ().. / Fgue. Cohesve l (9). m.7 o oee h ompessve segh MP. I he ge o ŵ, e (9) usg () h -6.99, 9.979, ,.67, -.. The oelo oee s. Whe ŵ, ˆ.; he ŵ >, ˆ s eglgl smll. As ompe Fg., he o e sgushe o he sle ohe gue. CRACK TIP ASYMPTOTIC FIELDS OF FRICTIONLESS AND FRICTIONAL COHESIVE CRACKS IN QUASI-BRITTLE MATERIALS Muskhelshvl (9) shoe h, o ple polems, he sesses splemes he Ces ooe ssem (see Fg. ) e epesse ems oo l uos φ(z) χ(z) ohe omple vle z e ' ' [ φ ( z) φ ( z)] τ '' '' [ zφ ( z) χ ( z)] ' ' μ( u v) κφ( z) zφ ( z) χ ( z) () hee pme eoes eeo h espe o z ove omple ojuge. I (), μ E [ ( ν )] s he she moulus; he Kolosov os s κ ν o ple s o κ ( ν ) ( ν ) o ple sess. Fo geel ple me moe I II polem, he omple uos φ(z) χ(z) e hose s sees o omple egevlue Gous uos (Sh & Leoz 96, Khloo & Xo ) φ( z) A z λ A e λ λ z () χ ( ) λ λ ( λ ) B z B e

4 hee he omple oees e A B. The egevlues λ oees,, e el. The smpo lss ohe k p els ees o sshe pope smme oos log he le o eeso ohe ohesve k, ou oos o he ohesve k es. I oml ohesve sepo pples o he k es, eloshp () ees o e sse ove he ohesve zoe. The sesses he ohesve k p e o-sgul (euse he sess es o K I ). Moeove, he ollog oos ee o e sse: () he Coulom o o he k e s osee, ± μ τ τ () hee μ equls he posve o egve vlue ohe oee o ke o, hh s ssume o e os, epeg o he elve slg eo ohe o k es: μ > he elve slg δ > μ < he δ <. Whe μ, he ohesve k es e oless. () he ohesve k es e pue moe I oo, τ τ, τ v () The legh ohe poess (ohesve) zoe s eeme he oo he omll opee p ohe k. Fo Coulom ol ohesve k h oml ohesve sepo (), he omplee smpo soluos e ompose oo ps. The s p oespos o ege egevlues λ, ( ) μ,,,, () ( )( ) ( ) os ± o ± ˆ () hee ( )( )( ) os, ( ) (6) se ± he. These soluos hve o-zeo τ log he ohesve k es, u zeo k opeg slg δ. The seo p ohe smpo soluos oespos o oege egevlues λ,,,,,, (7) κ μ s o ˆ μ κ s () Cose he ue N ems o ŵ (), eoe, ( > ) N ˆ (9) The epso o ŵ (9) se o he poe / s lso ue o N ems, se hese ems lue ol he ue N ems o ŵ. Hee N ˆ () hee ( ) ( )!, () N () ( ) ( ) eoes he h evve. The s ve oees e 9 6 () 9 I e hoose N, he e ssg he ohesve eloshp () e hve he ollog epessos o oees (6)

5 () Fo o-ege egevlues (7), he oees m e ege s epee, so h oees e lel epee o o. Fo ege egevlues (), oees m lso e ege s epee, so h oees o epe lel o,. Hoeve, he oees o ege egevlues ll epe oh lel o o ege egevlues olel o o o-ege egevlues v (6), (), (9), () (). The hee ole ue ohe polem s elee hese ole eloshps eee he oees ohe smpo els. Noe h he ove smpo soluo o e eue o pue moe I ohesve k (ompe () ()), se log he le o eeso o he k,, he she sess oes o vsh (τ ). Hoeve, he poeues e sml; he omplee smpo els hve ee gve Xo Khloo (6). The s em oespog o o-ege egevlue h gves oml spleme sou ove he ohesve k es s u κ os os μ κ s s μ v () SIMULATION OF COHESIVE CRACK GROWTH IN QUASI-BRITTLE MATERIALS USING XFEM To moel he ohesve ks he XFEM, s lol FE spleme ppomo ou he k s ehe h souous Hevse uos log he k es eh he k p lug he ope o-ee p, he k p smpo spleme els oes suoug he ohesve k p usg he PU. The ppomo o splemes o eleme e epesse he ollog om h u ( ) h v ( ) I u φ ( ) v u φm( ) v m M k I j J I ( p k ) m ( p k ) m φ j ( ) H ( ) j j () hee I s he se o ll oes he eleme, (u, v ) e he egul egees o eeom oe, φ s he FE shpe uo ssoe h oe, J s he suse o oes hose suppo s esee he k u o o ove ohesve k ps, he uo H() s he Hevse uo ee o he k sou, ( j, j ) e he oespog ol egees o eeom. M k s he suse o oes h e ehe ou he ohesve k p k h he smpo splemes u (p k) v (p k) ( ). u p k m v ( p k ) m e ehme uos ope oe m h oespog ol pmees epee o ohe oes. Cose om Ω og k Γ, s sho Fg. 6. The ou Γ s ompose ohe segmes Γ u, Γ, Γ. The p ohe k o hh soeg l s ve,.e. he FPZ, s eoe Γ oh. Pese splemes u e mpose o Γ u, hle os F e mpose o Γ. The k sue Γ elug Γ oh s ssume o e o-ee. The ohesve os, - e mpose o he uppe loe sues o Γ oh. The vul ok equo hou o oes s gve Wells Slus () Moës Belshko () s ( u) : vω Ω ( v) Γ Γ F vγ oh Γ hee s he Cuh sess, (6) s eoes he smme p ohe ge opeo. The splemes u mus elog o he spe U o kemll mssle spleme els u U { u U : u u o Γ, u souous o Γ } (7) u hee he spe U s ele o he egul ohe soluo. The es uo v mus elog o he Γ u F u Γ Γ - oh Ω, Γ oh Γ -, - Γ Fgue 6. Bo h k Γ volvg ohesve zoe Γ oh sujee o pese ou os F o Γ splemes u o Γ u.

6 spe V ee v V { v V : v o Γ, v souous o Γ } () u The ohesve os he sepo (v) v v e ele soeg l o Γ oh. Fo soeg l ompose o le segmes, equo (6) e solve usg se moulus eo sheme he he legh ohe ohesve zoe eel los e gve. The em se moulus hee ees o he h he sess m o XFEM oe om (6) eles el he ol splemes o los. Dsplemes () e use s he k p ehme uo () o moe I ohesve k. The uko oee epes o he soeg l, ou lo oos. I s osee s ol egees o eeom elev ehme oes XFEM. Dels ohe mplemeo o XFEM, SAR, smulo o ohesve k goh e ou Xo e l. (7). ou-o-ple eo). A le soeg l se o ple s oo e osee. The mel popees e E 6. GP, ν.,.9mp, G F N/m. The mesos o oe legh e N mm, espevel. / Fgue. Opeg pole ohe ohesve k. P ILLUSTRATION OF THE MODE I COHESIVE CRACK TIP FIELDS l I oe o vsulse he ohesve k p els eve Seo, s eess o eeme he uko oees g he umell ompue k p els h he heoell oe els. Ths eques sophse opmzo sheme. Fo he pese pupose o lluso, e ll solve moe I ohesve k polem sho Fg. 7 usg he oe smpo els he umell ompue opeg pole ohe ohesve k. We ose he suom oue oke les Fg. 7, ssume he legh ohe ohesve k s. mm. The opeg pole ohe ohesve k (Fg. ) ops he ompue esuls o hee po e em h ve smll l k o legh.mm he oom mpo ohe em (Fg. 9) he log sge h ol ohesve k.mm lo/( ) (Xo & Khloo 6). The geomel pmees e mm, l, ( s he speme hkess he P/ P/ Fgue 9. A hee-po e em (TPB). u.e-.e -.E- -.E Fgue 7. Illuso ohe ohesve k polem. Fgue. No-vshg splemes sesses log he le o eeso ohe ohesve k ( ).

7 u.e-6 6.E-6.E-6.E-6.E -.E-6 -.E E E-.E-.E-.E-..E-.E -.E- -.E- -.E- -.E- -.E v τ Fgue. Dsplemes sesses log he le.. The splemes sesses log he le o eeso ohe ohesve k he le h us. mm suoug he k p e ploe Fgs.. As epee, he sess he ohesve k p s equl o, o sess ohe loos ehes. 6 NUMERICAL EXAMPLE I hs seo, e ll lse ege ke ple ue ul eso (Fg. ). The geomel pmees e mm, l,. A se o ple s oo s osee. The l ege k hs legh o.mm. The popees E 6.9 GP, ν., le soeg l () h. MP,. MP,.79 mm,.7 mm, G F N/m, oe All Khloo () o el oml segh oee e use. To meshes, s sho Fg., e use he lss. The ose mesh osss o egul elemes, gvg ol o oes. The e mesh osss o egul elemes, gvg ol o 7 oes. Boh meshes e uoml ve -eo. Fo he ose mesh, he el les o elemes hve el hegh (-eo) o mm; he emg elemes hve el hegh o 9mm. Theeoe, elemes he el zoe e mm sques. Fo he e mesh, he el 6 les o elemes hve el hegh o mm; he emg elemes hve el hegh o 9mm. Theeoe elemes he el zoe e mm sques. We ll su glol esposes lug he lo eomo ehvou he evoluo ohe ohesve zoe, s ell s lol popees lug he opeg pole suo ohe ohesve sess he ohesve zoe, splemes sesses log he le.mm (he seo le o elemes he ose mesh, o he ouh le o elemes he e mesh) ou he k p log he le o eeso ohe k. The eo o usg o meshes s o su he mesh sze sesv ohe glol esposes s ell s ohe k p els. Fuhemoe, he esuls om he e mesh ll e use s eeee soluo, se o ll soluos e vlle. The oveol -oe le sopme Q elemes e use s kgou elemes. The poel ue lous oes h he speme s s o smme. The k s moelle ehg he oes o he k es h jump hou he oule oes h e use he ol FEM. The s le o oes suoug he ohesve k p e ehe h (). The opeg ompoe ohe sme el s lso use s he l opeg pole o el eee ohesve segme he smulo o ohesve k popgo. Sml o Moës Belshko (), - eo o oes h ooes (, ) (, 6mm) -eo ohe oe h ooes (mm, mm) e ose; he lo s sue ove legh o 6mm o he ose mesh mm o he e mesh (o elemes). Fgue. A ege-ke ple ue ul eso. l

8 () 6 6 log ege s sho Fg.. Ovousl, shos o mesh epee. The evoluo ohe sze ohe ohesve zoe oespog o he o hes ohe le soeg l () s sho Fg.. I lso evels ve ek mesh epeee. The opeg pole suo o ohesve sesses he ohesve zoe e sho Fg. 6, sho some ek mesh epeee. Novshg splemes sesses log he le.mm, he le o eeso ohe k log sges A e ploe Fgs 7. The splemes o.mm om he ose mesh gee que ell h he e mesh. The sesses o he ose mesh oe SAR e moe ue h he e eeo AVG, gee que ell h he e mesh. Alog he le o eeso ohe k, he spleme u - eo s ve sesve o he mesh sze, hoeve he emg o-vshg spleme sess ompoes e o. The geeme eee AVG, SAR he e mesh s geell ve goo, hoeve, SAR s ee h AVG lose o he ohesve k p. Noe h lose o he ohesve k p, he mmum vlues ohe m sesses ou o he ohesve k e. Ths eue ohe sess suo hs lso ee oe Pls e l. (). () Fgue. Cose () e () mesh o he le hl ohe speme. I he smulo, he s eme ohe ohesve k s.mm, he he ohesve k popges segme o legh mm e eh sep he ose mesh, hee segmes o legh mm eh he e mesh. The sesses he p ohe ohesve k eovee SAR e use o juge hehe o o he p ll popge. The sesses o he e mesh eovee AVG SAR e el log he le.mm he le o eeso ohe k. Theeoe, ll sess esuls pesee elo e o he ose mesh, uless ohese meoe. The mesos ohe spleme legh pmees e mm, hose ohe sesses MP. The vo ohe esle sess spleme v -eo he el po ohe / A B C Cose mesh Fe mesh.. v /l Fgue. The o-mesol esle sess-el eeso uves ohe ul sgle-ege ohe eso ple. FPZ / Cose mesh Fe mesh () A ()... Tol k legh / B C Fgue. Evoluo ohe ohesve zoe sze s he ohesve p vels hough he ple. () () oespo o he s seo hes ohe le soeg gm (Fg. ).

9 / / B C Cose mesh Fe mesh /.. / Cose mesh Fe mesh B.. / C / Fgue 6. Opeg pole suo o ohesve sesses he FPZ pl log sges (Fg. ). The use ohe ehme uo he XFEM hs ee oe o eeoe he oo ohe see ssem (Sououls e l. ). Ths oes o e glol esposes o he sesses he s eeg. Hoeve, m e sgl he splemes je o he p ohe ohesve k, so h e ompso ohe ose e mesh pes (Fg. ) ope he lss s o possle. Hee, e ompe se he splemes elve o he p ohe ohesve k. 7 DISCUSSION AND CONCLUSIONS The omplee smpo epsos o oless ol ohesve ks e logous o he Wllms epsos le sols. The e vl o m ommol use sepo ls, e.g. egul, le, le, epoel, e h o hou Coulom o o k es. Glol esposes lke lo eomo uves, evoluo ohe ohesve zoe, e o sesve o he sze ohe ope mesh he XFEM. Lol popees lke he opeg pole suo o he ohesve sesses he ohesve zoe, espell k p splemes sesses e moe sesve o he mesh sze. Alhough o sgul ess he p o ohesve k, he sesses oe e eeo ohe splemes e o ue, o e use o pe uel he goh o he p. The SAR, hh sses el he ohomogeeous ohesve os, gves moe ue esuls h he el use AVG je o he k p. I hgh u ohe lol els s eque (e.g. hese els e use o juge he goh (u-up) (v-vp).. -. Cose mesh Fe mesh Cose mesh Fe mesh τ De AVG SAR Fe mesh De AVG SAR Fe mesh De AVG SAR Fe mesh Fgue 7. No-vshg splemes sesses log he le.mm log sge A (Fg. ). ohe p), elvel e mesh ees o e use ogehe h ue sess eove mehos, suh s SAR. Fo moe I ohesve ks, he sesses eovee SAR e hghl ue log he le o eeso ohe ohesve k, e use el o pe he goh ohe k. Fo geel me-moe ks, uhe sues e eque o hehe he SAR sesses he p, o o smll le elosg e (e.g. oe o o) les elemes ou he p shoul e use o pe he goh ohe k.

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