XFEM and EFG Cohesive Fracture Analysis for Brittle and Semi-Brittle Materials

Transcription

1 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () XFEM and EFG Cohsiv Fracur Analysis for Bril and Smi-Bril Marials Yong Guo and C.. Wu Livrmor Sofwar chnology Corporaion 7374 Las Posias Road, Livrmor, CA 94551, USA yguo@lsc.com, cwu@lsc.com Absrac h fini lmn analysis of dynamic fracur in solids and srucurs is challnging du o h modling of arbirary crack growh in h coninuum domain. h wll-known msh siz and msh orinaion dpndncs add mor difficulis ino h analysis of his yp of problms. n his prsnaion, w ar going o inroduc wo numrical mhods in modling h dynamic fracur in bril marials for solid and srucurs in LS-DYNA. Boh mhods wr dvlopd by Blyschko and his group [1, 14] and wr basd on a srong disconinuiy approach combind wih cohsiv forcs for h crack iniiaion and propagaion. n EFG mhod, a visibiliy mhod is uilizd o dfin h cracks in h solids and a fas ransformaion mhod [18] is applid o handl h boundary condiions in h crackd mdia. h XFEM mhod is implmnd o modl h dynamic fracur in srucurs. h XFEM mhod can b viwd as a combinaion of lvl ss mhod and pariion of uniy mhod [15] in h dscripion of cracks. Boh acadmic bnchmarks and indusrial applicaions will b prsnd using hs wo mhods. Advanags and disadvanags will also b discussd. 1 nroducion Bril and smi-bril srucurs ofn dvlop complx fracur and fragmnaion parns during h failur procss and h dmand for analyzing h fracur parn and disribuion of fragmn siz has bn h focus in many scinific rsarchs such as hyprvlociy impac, crashworhinss, xplosiv drilling. Dspi a lo of work has bn don in h pas o sudy h physics of fracur and fragmnaion parns, a brak-hrough numrical chnology in h failur simulaion is sill missing. h main difficuly manas from h inhrn muli-scal naur of failur procss. For xampl, h crack iniiaion and propagaion ar affcd by h prsnc of flaws a h micro-scal and mulipl cracks occur hrough a complx communicaion procss of srss-wav inracions bwn hm. n liraurs, hr ar wo main ways o numrically modl h marial failur in bril and smi-bril marials. h firs on is accomplishd by assuming h formaion of discr cracks and zons of local damag in h coninuum sns. hs damag zons can b rprsnd in a numrical modl by mans of smard crack modls, whr h discr crack opning is rprsnd by srain concnraions. his approach is ofn rfrrd o as h wak-disconinuiy approach in h coninuum damag mchanics (CDM) framworks. As a rsul, i lads o h wll-known msh-dpndn problm in ra-indpndn marial and crain yp of localizaion limir is rquird o rmdy his numrical dfc [4]. n fac, h localizaion limir forcs h localizaion o occur in a givn volum insad of a surfac, mainaining h usfulnss of xisn volumric dissipaion modl. Mor rcnly, a srong-disconinuiy approach in conjuncion wih cohsiv modl is gaining 4-1

2 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc incrasing inrs in modling marial failur [9, 1]. his approach is dvisd o capur h physical disconinuiy, i.. fracurs, cracks c in spcific kinmaics wihin fini lmn or mshfr mhods. h cohsiv failur is inroducd o giv h xplici rprsnaion of cracks and h as o handl crack branching and fragmnaion. n h craion of nw failur surfacs, ach opning cohsiv zon dissipas a crain amoun of cohsiv nrgy and h oal nrgy dissipad in h cracking procss is hrfor rlad o h crack pah and hus minimiz h msh-dpndn problm as sn in CDM. Mshfr mhods ar h opic of rcn rsarch in many aras of compuaional scinc and nginring. On of h arly incnivs o dvlop mshfr mhod was is abiliy o handl crack propagaion problm. Ohr advanags of h mshfr mhods can also b found in many liraurs [3, 4, 7]. h firs papr uilizs h mshfr chnology in h cohsiv fracur analysis was givn by Klin al. [6]. Afr ha, svral mshfr formulaions wr proposd in h modling of fracur bhaviors. Exndd Elmn Fr Galrkin Mhod (XEFG) was proposd by Rabczuk al. [1] o modl hr-dimnsional cohsiv cracks in boh saics and dynamics. Zi al. [13] proposd a nw way for h crack closur nar h ip ha dos no rquir crack ip nrichmn. Park [9] proposd a modifid cohsiv modl in mshfr fracur analysis. On h ohr hand, XFEM is an applicaion of h srong disconinuiy approach of h mshfr fracur mhod o h radiional fini lmn mhod. n h xndd fini lmn mhod, h pariion of uniy is uilizd o incorpora h nrichmn funcions associad wih h crack ino h fini lmns so ha arbirary cracks can b modld wihou rmshing. Disconinuous pariions of uniy nrichmns wr firs usd o modl cracks by Blyschko and Black [19], who usd h disconinuous nar ip fild o modl h nir crack for laso-saic problms. n Mos al. [14] and Dolbow al. [], a sp funcion nrichmn was dvlopd for lmns complly cu by h crack. h approach was gnralizd o arbirary disconinuiis, including disconinuiis in drivaivs and angnial valus of displacmn in Blyschko al. [1], and was applid o 3-D saic problms by Mos and Gravouil [, 3]. n his mhod, h disconinuiis ar complly indpndn of h fini lmn msh: hy can cross lmns in any mannr. Across h disconinuiy, hy imposd a racion-displacmn law, i.. a cohsiv law. h nrgy dissipaion across h disconinuiy was chosn o mach h nrgy of fracur. n his papr, boh h mshfr fracur mhod for solid and h xndd fini lmn mhod for shll srucur fracur ar rviwd and implmnd in LS-DYNA. hy boh us h srong disconinuiy approach and h cohsiv law for h kinmaics of h disconinuous crack surfacs. Som bnchmarks and indusrial applicaions ar usd o dmonsra h advanags and disadvanags of h mhods. Rviw of Mshfr Cohsiv Fracur Approximaion n conras o cohsiv lmn mhod [8] in h fracur analysis whr h cohsiv surfac is dfind along h lmn dg, h rprsnaion of crack in h mshfr mhod is dpicd by h so-calld visibiliy cririon []. h mid-plan cohsiv surfac in mshfr domain is shown in Figur 1 and is givn by 4-

3 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () FEM 1 x( η) = Φ ( ) + Ψ ( ( )) + Ψ ( ( )) η X J X η u J J X η u J (1) = 1 + J Ω J Ω x η FEM ( η) Φ ( η) 1 Ψ ( X) Ψ ( X) X( η) = + J J X + u + η J uj () Ω X Ω X η = 1 J J whr domains on h uppr and lowr par of h crack ar dnod by + Ω and Ω, and ar dfind in h iniial configuraion. Eq. () is h Jacobian of h cohsiv surfac paramrizaion along h mid-plan. Figur 1: Cohsiv surfac is dfind by mshfr visibiliy. 3 Rviw of XFEM Fracur Approximaion n h xndd fini lmn fracur analysis, h cracks ar dfind by h lvl s mhod. h surfacs of disconinuiy Γ α ar dscribd by a signd disanc funcion f ( x) = min x x sign( n ( x x)) x Γ α (3) whr x is a poin on h surfac of disconinuiy Γ α and n is a uni normal o h surfac of disconinuiy. h poin x is h closs poin o x and h orhogonal projcion of x on Γ α. h disconinuiy corrsponds o f ( x) = and h wo aras wih diffrn signs of f (x) corrspond o wo domains across h disconinuiy, as shown in Figur. h approximaion clos o h disconinuiy (h shadd ara in Figur ) consiss of wo pars: h sandard fini lmn approximaion and h nrichmn, as in Eq. (4). Eq. (5) shows h nrichmn of a sp funcion for cracks cu hrough h lmn. u h FEM ( X) = Φ ( ξ ) u + Ψ ( X) q = 1 w (4) FEM ( X) = Φ ( )( H ( f ( X) ) H ( f ( X ))) Ψ ξ (5) 4-3

4 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc f > Γ α f < Figur : Cohsiv surfac is dfind by lvl s in XFEM. 4 niially-rigid Cohsiv Law Many cohsiv modls blong o h yp of iniially-lasic cohsiv law whr h ffciv Young s modulus is dpndn of modl rfinmn and h rsuls ar no convrgn. n h iniially-rigid cohsiv law, h cohsiv surfac is only inroducd as ndd. hrfor, corrc wav spds can b capurd bfor any crack occurs. n his sudy, w adopd a modifid cohsiv law [11] for h crack iniiaion and propagaion in boh EFG and XFEM mhod. A linar iniially-rigid cohsiv law is shown as in Figur 3. max 1 λ max 1 Figur 3: niially rigid cohsiv law. λ h displacmn jump λ is dfind by ) + β n + δ n ( δ + δ ) λ u = n u ( δ (6) whr un and u ar crack opning displacmns in normal and angnial dircions obaining from Eq. (1) in EFG and Eq. (4) in XFEM. h marial consans involvd in Eq. (6) includ: 4-4

5 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () δ n and δ n and δ n ar criical valus a which crack aks plac in normal and angnial dircions rspcivly, δ ar rgularizaion paramrs ha ar inroducd o prvn h imdisconinuiy and hus limina h numrically insabiliy [11]. α is h paramr coupling normal and shar racions and λ cr is h criical displacmn jump. h corrsponding normal racion and angnial racion ar obaind by h sandard cohsiv rlaionships by following β fs n + = α max (7) n = λ u λ u α and (8) 1 n max 1 max = λ δ n 1 λcr λ δ 1 λcr whr n is h normal racion, is h angnial racion and max is h maximum normal racion ha h crack surfac can bar bfor failur. No ha h cohsiv racions ar dfind on an un-dformd ara pr uni. Accordingly, h nodal forcs follow from h racions can b obaind by h surfac ingraion along h crack surfac. h implmnaion flow char of h iniially-rigid cohsiv law is shown in Figur 4. Figur 4: Flow char for crack iniiaion and propagaion using iniially-rigid cohsiv law. h final discr quaions can b drivd in a sandard fashion and givn by kin x coh = f in f f (9) f + 4-5

6 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc f kin = Ω ρ N NH f X dω u (( 1) ( )) (1) f in = Ω Ω B H (( 1) f ( X )) d σ (11) f x = N bh 1) f ( X )) dω + Ω Γ, ρ (( N H (( 1) f ( X )) dγ, (1) f coh c = ( 1) N n dγ Γ,, τ (13) 5 Numrical Exampls 5.1 Edg-crackd pla undr impulsiv loading Nx w simula an xprimn rpord by Kalhoff and Winklr [16] in which a pla wih wo iniial dg nochs is impacd by a projcil. h xprimn is shown in Figur 5. n h xprimn a low srain ra, bril failur wih a crack propagaion angl of abou 7 is obsrvd [16]. 1mm 75mm 5mm 1mm v 5mm y x 1mm Figur 5: Exprimnal s-up for dg-crackd pla undr impulsiv loading; only half of h pla is modld. Du o h wofold symmry of h configuraion, only h uppr half of h pla is modld: A h boom dg of h fini lmn modl, u y = and x =. h iniial impac vlociy is applid on h lf dg on h sgmn y 5mm. W assumd ha h projcil has h 4-6

7 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () sam lasic impdanc as h spcimn, so w applid on half of h projcil spd, 16.5m/s for h bril fracur mod, o h lf dg as an iniial condiion. h iniial noch was modld by including wo lins of nods sparad by.3mm. h marial is a maraging sl 3 18Ni19 and is marial propris ar ρ = 8kg/m, E=19GPa and ν =. 3 [17]. W usd a cnral diffrnc im ingraion schm wih a Couran numbr of.1. W found ha a low Couran numbr is ncssary for h lmns which conain a disconinuiy. 4 5 A cohsiv crack modl wih fracur nrgy G F = N/m and δ max = m and a linar cohsiv law was usd. For h crack iniiaion cririon, w usd h maximum nsil srss cririon. Numrical simulaion was mad wih a 5x5 msh shown in Figur 6. Boh h mshfr mhod and h XFEM ar usd o solv h problm. 1 v { symmry Figur 6: 5x5 msh wih pr-crack. h rsul of crack pah from XFEM is shown in Figur 7. h avrag angl from h iniial crack ip o h final crack ip is abou 6.5 and h iniial crack angl is abou 67.5 ; h crack pah is narly sraigh. his angl is smallr han h obsrvd angl [16] and h angl obaind by msh-fr mhods. h crack pah from EFG fracur mhod is shown in Figur 8. h avrag crack angl is abou 69., vry clos o h xprimn obsrvaion. h accura rsul is parially bcaus of h highr approximaion ordr of h mshfr mhod. 4-7

8 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc 6.5 Figur 7: Crack pah by XFEM. Figur 8: Final crack pah by EFG. 5. A nochd pla undr bnding s n his xampl, a bril fracur of hick pla undr bnding is simulad. his classical problm of a singl-dgd nochd pla suppord in wo poins is shown in Figur 9. h 4-8

9 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () problm is simulad using EFG mhod. h marial propris of h pla ar givn in nondimnsional uni, Young s modulus = , Poisson s raio =.3, dnsiy =.4-9, mod nrgy rlas ra =.5 and h criical displacmn jump =.1 wih h corrsponding maximum normal racion = 5. Figur 9: Singl-dgd nochd pla undr bnding. Figur 1 shows h final crack pach in a rsulan displacmn plo. h rsul dmonsras h capabiliy of h prsnd mhod o modl a curvd crack. 5.3 Cylindr shll undr pulling Figur 1: Final crack pah. h las xampl is a hin cylindr shll wih a pr-crack undr axial pulling. h righ nd of h shll is fixd and h lf nd is pulld wih a vlociy of V = 5mm/μs, as shown in Figur h marial is kinmaic plasic wih ρ = kg/mm, E=7GPa, ν =. 3 and σ y =.1GPa, E p =.GPa. h fracur nrgy rlas ra is 5kN/m and h cohsiv law paramrs ar: σ max =.GPa and δ c =.5mm. h problm is solvd wih wo mshs: a coars msh wih 186 lmns and a fin msh wih 744 lmns. h crack posiions a diffrn ims wih h fin msh ar shown in Figur 1. Figur 13 shows h comparison of h rsulan pulling forc obaind using h wo mshs. h fin msh yilds smoohr forc curv han h coars msh. 4-9

10 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc V Fixd Figur 11: Cylindr shll undr pulling. =4.us =5.us =6.us =6.5us Figur 1: Crack propagaion a diffrn ims. 4-3

11 11 h nrnaional LS-DYNA Usrs Confrnc Simulaion () Figur 13: Comparison of rsulan pulling forc using wo mshs. 6 Conclusion h mshfr fracur mhod for solids and h xndd fini lmn mhod for shll srucurs ar prsnd in his papr and implmnd in LS-DYNA. h wo mhods us h srong disconinuiy approach o modl h cracks and h cohsiv zon modl for h fracur kinmaics. Boh mhods show hir possibiliis o modl dynamic fracur wih arbirary cracks and wihou rmshing. Currnly hs mhods ar br fid o modl cracks in bril and smi-bril marials. Svral issus which ar no addrssd in his papr such as h ramn of mulipl cracks, rsponss from diffrn crack iniiaion and propagaion criria, h abiliy o modl conac bwn dbris and crack closur problms will b furhr invsigad. Rfrncs 1. Blyschko,. and abbara, M., Dynamic fracur using lmn-fr Galrkin mhods, nrnaional Journal for Numrical Mhods in Enginring, 39, , Blyschko,., Lu, Y. Y., and Gu, L., Elmn-fr Galrkin mhods, nrnaional Journal of Numrical Mhods in Enginring, 37(), 9-56, Blyschko,., Krongauz, Y., Organ, D., Flming, M. and Krysl, P., Mshlss mhods: An ovrviw and rcn dvlopmns, Compur Mhods in Applid Mchanics and Enginring, 139, 3-47, Chn, J. S., Pan, C., Wu, C.. and Liu, W. K., Rproducing krnl paricl mhods for larg dformaion analysis of non-linar srucurs, Compur Mhods in Applid Mchanics and Enginring, 139, 195-7, Chn, J.S., Wu, C.., Blyschko,., Rgularizaion of marial insabiliis by mshfr approximaion wih inrinsic lngh scals, nrnaional Journal of Numrical Mhods in Enginring, 47, ,. 6. Klin, P. A., Foulk, J. W., Chn, E. P., Wimmr, S. A. and Gao, H. J., Physical-basd modling of bril fracur: cohsiv formulaions and h applicaion of mshfr mhods, horical and Applid Fracur Mchanics, 37, , Li, S. and Liu, W. K., Mshfr and paricl mhods and hir applicaions, Applid Mchanics Rviws, 55, 1-34,. 8. Oriz, M. and Pandolfi, A., Fini-dformaion irrvrsibl cohsiv lmns for hr-dimnsional crackpropagaion analysis, nrnaional Journal for Numrical Mhods in Fluids, 44, ,

12 Simulaion () 11 h nrnaional LS-DYNA Usrs Confrnc 9. Park, C. K., h dvlopmn of a gnralizd mshfr approximaion for solid and fracur analysis Ph.D. hsis dissraion, h Gorg Washingon Univrsiy, U.S.A., Rabczuk,., Blyschko,., A hr-dimnsional larg dformaion mshfr mhod for arbirary volving cracks, Compur Mhods in Applid Mchanics and Enginring, 196, 777-7, Sam, C. H., Papoulia, K. D. and Vavasis, S.A., Obaining iniially-rigid cohsiv fini lmn modls ha ar mporally convrgn, Enginring Fracur Mchanics, 7, 47-67, Zavairi, P. D. and Espinosa, H. D., Grain lvl analysis of cramic microsrucurs subjcd o normal impac loading, Aca Marialia, 49(), , Zi, G, Rabczuk,. and Wall, W., Exndd mshfr mhods wihou branch nrichmn for cohsiv cracks, Compuaional Mchanics, 4, , Mos, N., Dolbow, J., Blyschko,., A fini lmn mhod for crack growh wihou rmshing, nrnaional Journal for Numrical Mhods in Enginring, 46: , Babuska,. and Mlnk, J.M., h pariion of uniy mhod, nrnaional Journal for Numrical Mhods in Enginring, 4:77-758, Kalhoff, J.F. and S. Winklr, S., Failur mod ransiion a high ras of shar loading, nrnaional Confrnc on mpac Loading and Dynamic Bhavior of Marials, 1: , Dckr, R.F., Sourc Book on Maraging Sls, Amrican Sociy for Mals, Wu, C.. and Lu, H.S., Pracical fas mshfr analysis, U.S. Pan, Blyschko,. and Black,., Elasic crack growh in fini lmns wih minimal rmshing, nrnaional Journal for Numrical Mhods in Enginring, 45:61 6, Dolbow, J., Mos, N., and Blyschko,., Disconinuous nrichmn in fini lmns wih a pariion of uniy mho, Fini Elmn Analysis and Dsign, 36(3):35 6,. 1. Blyschko,., Mo s, N., Usui, S. and Parimi, C., Arbirary disconinuiis in fini lmn. nrnaional Journal of Numrical Mhods in Enginring, 5(4): , 1.. Mo s, N., Gravouil, A. and Blyschko,., Non-planar 3d crack growh by h xndd fini lmn and lvl ss. par i: Mchanical modl, nrnaional Journal of Numrical Mhods in Enginring, 53: ,. 3. Gravouil, G., Mo s, N. and Blyschko,., Non-planar 3d crack growh by h xndd fini lmn and lvl ss. par ii: lvl s upda, nrnaional Journal of Numrical Mhods in Enginring, 53: ,. 4-3