Numerical modelling of quasi-brittle fracture with the rate-dependent multiple embedded discontinuity approach

Transcription

1 Ra Maa Joural of Srucural Mchacs Vol. 49 o pp rmsura..f/rmlh/ Th Auhors 206. Op accss ur CC BY-SA 4.0 lcs. umrcal mollg of quas-brl fracur wh h ra-p mulpl mb scouy approach Tmo Sasala Summary. Ths arcl als wh 2D umrcal mollg of fracur quas-brl marals. For hs a ra-p mulpl mb scouy mol s vlop o smula quas-brl fracur wh h f lms cox. I h prs mofcao of h mb scouy approach h scous ar pr-mb bfor h aalyss o ach f lm of h msh. Wh h chos cosa sra ragl lm hs rsuls hr scouy ls ach or paralll o h ss of h ragl lm. Each scouy l has s ow splacm ump a loag surfac. Th splacm umps a lmal srsss ar smulaously solv wh h mulsurfac plascy chqus. Ra-pcy s corpora wh h vscoplasc cosscy approach. Th global quaos of moo ar solv m by xplc m grao. Th mol prformac s mosra umrcal xampls whr h uaxal so a comprsso ss ar smula. Ky wors: brl fracur mulpl mb scous umrcal mollg FEM Rcv 22 Spmbr 205. Accp 4 Jauary 206. Publsh ol 9 Dcmbr 206. I mmory of Profssor Juha Kos Irouco umrcal mollg of quas-brl fracur has b gag crasg rs urg h las fw cas u o s mporac may fls of grg such as cocr srucurs a gochcal aalyss. May umrcal cos bas o h f lm a h scr lm mhos hav b vlop o smula h brl fracur. For a rvw o compuaoal mhos for fracur brl a quas-brl sols s []. Th maor challg such aalyss s h umrcal mollg of crac propagao. A promsg umrcal chqu o scrb crac propagao s h mb scouy approach [2-5]. Ths chqus allow hr a sra wa or splacm srog scouy s a f lm. Ths s achv by 25

2 hacm of h sra or splacm fl orr o capur h scouy. Sasala [6] succssfully appl hs mho for umrcal mollg of roc fracur. A clos rlav of hs mho s h x f lm mho XFEM whch rchs h lm rpolao oal bass by xplog h paro of uy propry [7]. Howvr h mb scouy approach s chos hs papr bcaus has a compuaoal avaag ovr h XFEM u o h complly local aur of h rqur hacms whl offrg h sam of accuracy a covrgc. Parcularly h aoal grs of from rprsg h splacm or sra ump ca b lma by sac cosao. Morovr h aoal grs of from ar global h XFEM a hus cras urg h propagao of scouy. I h usual applcao of h mb scouy mho for quas-sac aalyss sgl mb scouy a crac wh a ormal paralll o h frs prcpal rco of h srss sor s rouc urg h aalyss o a f lm wh h frs prcpal srss xcs h sl srgh of h maral. A vara mplmao bas o mulpl or rscg scous mb succssvly paralll o lm facs gs s propos [8]. Mulpl scous approach ffcvly allvas h srss locg a sprag problms ypcal for mb scouy approach as show [9 0]. Th mplmao of mulpl scous paralll o lm gs s smlar o h classcal rfac or cohsv zo lms mho bu cosrably smplr a compuaoally mor ffc as hr cohsv zo lms rqurg uplcao of msh os or coac rfacs ar upo closg cracs. I h prs papr a mofcao of h mulpl scous approach whr h scous ar mb bfor h aalyss paralll o h gs of h cosa sra lm CST s prs a appl o smulao of roc fracur. Each scouy has s ow ra p loag surfac so ha h compuaoal mulsurfac chqus ca b mploy solvg for h splacm umps a lm srsss. Ra-pcy s rouc for umrcal sably rasos wh h vscoplasc cosscy approach. Th mol prformac s mosra 2D umrcal xampls of roc comprsso a so ss solvg h quaos of moo xplcly m. Thory of h mol Th hory of h prs mol s prs hs sco. Frs h srog scouy macs s brfly scrb. Th h f lm mplmao of h mb scouy macs s prs. Fally h mulpl mb scouy mol s summarz a h soluo mho for h splacm umps s oul. Srog scouy macs A boy occupyg oma Ω R 2 wh a bouary Ω s Fgur s spl o wo so pars Ω a Ω by a srog scouy l f by s ormal a 26

3 ag m. Th splacm a sra fl assumg fsmal formao ca b compos as a fuco of h locao x Ω as s whr s h mrc par of h gra opraor u x rprss h splacm fl whou h scouy u x s h splacm ump a u o scouy a H x s h Havs fuco a h scouy. Fuco ϕx s f a suboma wh u x u x s ε x u x Ω ϕ Ωϕ Ωϕ so ha x 0 ϕ wh x Ω \ Ωϕ ϕ x x Ω \ Ωϕ wh \ sgfyg h s horc ffrc a s C 0 -coous bw 0 a wh x Ω. Th raso for usg h composo sa of ϕ H x x ϕ u x h aural o wh ϕ x 0 s ha boh ux a u x may b ozro a h bouary Ω whch mas ha hr ffc shoul b a o cosrao h f lm cox wh mposg h ssal bouary coos. I s mor cov o us composo as fuco M x apparg Equao rsrcs h ffc of h splacm ump o suboma M x u x ϕ x δ x u x Ω ϕ. Fgur. Doma cross by a scouy l. Morovr h rsul H δ o Drac la fuco δ a cosa splacm ump assumpo ylg s u 0 wr us h rvao of h sra. F lm mplmao of srog scouy macs As h boy Fgur s scrz wh h CST lms h mb scouy bcoms a sragh l s a lm as s llusra Fgur 2. Th FE vrso of Equao ras u u sol H sol ε u δ ε M 27 ε 2

4 whr h splacm ump s o by whl a u ar h saar lar wh CST rpolao fucos a oal splacms 23 wh summao o rpa cs rspcvly. I s o ha h pcy of h quas o h placm vcor x Equao a hcforh ar om for brvy of oaos. Fgur 2. CST lm wh a scouy l a fuco paralll o lm ss c. M b a hr scous Thus h f lm cox fuco ϕ Equao s f wh h rpolao fuco sol rla o h solary o s sol wh s suppor bg a sgl lm. Fally s o ha accorg o h hac assum sras EAS cocp h rms comprsg h scrz sra ca b f as h compabl par ε a h hac par ε [8]. Th FE scrz vrso of h wa form of h balac of lar momum ca b oba by saar argums a s o show hr. Th varao of h hac par of h sra s Equao 2 s bas o h EAS cocp whr h hac mos ar cosruc h sra spac orhogoal o h srss fl. Applyg h EAS cocp a h Prov-Galr formulao h varao of h hac sra a h L2-orhogoaly coo ca b wr as [8] δ ε β A Ω δ ε : σω 0 β l δ 3 whr β R 2 os a arbrary varao of h splacm ump A s h ara of a f lm a l s h lgh of a scouy. Subsuo of δ ε o h sco quao Equao 3 gvs afr som mapulaos A Ω \ σ Ω 0 l Wh h CST lm hs quao rprsg ssally a wa form of raco couy bcoms also h local srog form of raco couy.. σ 4 28

5 29 sc h gras Equao 4 ar cosas. Ths EAS cocp bas formulao provs a vry smpl mplmao whr hr h xplc poso of h scouy l wh h lm or s lgh s cssary o b ow. Ra p mulpl mb scouy mol A hr-surfac rscg scouy mol s prs hr. As mo Irouco h scous ar plac o ach lm of h msh paralll o lm ss as llusra Fgur 2c. Thrby h u ormal for ach scouy s calcula as whr s h rpolao fuco of o. Th sgl scouy macs ca b x a sraghforwar mar o h mulpl scouy cas sc ach scouy has s ow splacm ump. Thrby Equao 2 bcoms Wh hs xprsso for sra h raco vcor for ach scouy ca b wr as wh E bg h lascy sor. I shoul b o ha h srss sor σ 7 s commo o all scous h sam lm. Th loag fucos smlar o hos [9] sofg ruls a voluo laws for ach scouy ar f as whr x os h absolu valu of scalar x m os h u ag of a scouy ar h ral varabl a s ra rla o h sofg law for a scouy a σ a s ar h sl srgh a h vscosy of h maral. Morovr h s h sofg moulus of h xpoal sofg rul. Paramr g ha corols h al slop of h sofg curv s calbra by h mo I fracur 23 wh o summao o H u ε u u δ 6 ε E σ : : xp 3 Ic q G g g g h s h q q φ λ φ λ σ σ σ β φ x E m 8

6 rgy GIc. Fally β s a shar corol paramr ha govrs h raco bw h sl mo-i a h shar mo-ii compo of h raco. Th formal smlary of Equao 7 a 8 o plascy hory abls afr lmag h ra of h ral varabl λ h xploao of saar compuaoal mulsurfac plascy chqus solvg for h splacm umps a upag h ral varabls s [9]. I parcular h crms of h magus of h splacm umps δλ ca b solv grally as assumg φ > 0 for all 23 Gδλ F wh φ T G E x 0 0 y y x φ s δ h 0 23 whr marx oao s us a x ar y ar h compos of. Afr h crm δλ s solv h crms of h splacm umps raco vcors a h ral varabls ar upa accorg o Equao 8. Th w srss sa s h calcula accorg o h fo of srss Equao 7. I shoul b o ha h cas whr h magus of all hr splacm umps ar acv h of h srss grao... λ δλ > 0 23 os o acually ralz. Ths follows from h paro uy propry of h rpolao fucos gvg 3 0. Thus oly wo p g mos ar possbl 2D [8]. Th prs formulao rsuls a smpl compuaoal schm whr h global quaos of moo ar solv wh a xplc m graor a h local problm for srsss splacm umps a ral varabls wh h lasc-plasc opraor spl. Fally s mphasz ha h prs mol corporas loag ra ssvy so ha ca b appl yamc loag coos. Morovr o solvg for δλ h agoal rs of G mus rma posv. Ths may bcom crucal f h sofg curv s xrmly sp.. h absolu valu of h s grar ha h sffss rm. Thus h posvy s scur by h prsc of rm s/ whch s usually of svral orrs of magu. x y 0 y x F φ 9 umrcal xampls Prs mol prformac s s hs sco boh a h maral po lvl usg a sgl lm msh a a h srucural lvl whr h uaxal comprsso a so ss o roc l maral ar smula. Th maral proprs a from [0] val for Sasa gra a mol paramrs for smulaos ar gv Tabl. 30

7 Tabl. Maral proprs a mol paramrs Quay Symbol Valu U Youg s moulus E 67.3 GPa Elasc lm srss so σ 8.9 MPa Posso s rao ν 0.27 Maral sy ρ 266 g/m 3 Mo I fracur rgy G Ic /m Vscosy for scouy s 0.00 MPa s/m Shar corol paramr β Maral po lvl ss Bfor h laboraory sampl lvl smulaos h mol prcos ar mosra a h maral po lvl so. For hs h mol rspos s s wh h sgl CST lm mol a bouary coos show Fgur 3. Fgur 3. Sgl lm mol a bouary coos for maral po lvl smulaos. Th lm s lgh s s o 0 mm a h magu of cosa vlocy s 0.00 m/s ach loa cas. Th ffc of vscosy s s LC whl ohr loa cass h valu gv Tabl s us. Th smulao rsuls ar show Fgur 4. Srss [MPa] LC s 0.00 s 0. s s 0 MPas/m Srss [MPa] LC2 σ x σ y LC3 τ xy LC4 σ y LC4 τ xy Sra x 0-3 a Sra x 0-3 b Fgur 4. Th mol prcos a h maral po lvl smulaos: ffc of vscosy LC a a mol rspos LC2 LC3 a LC4 b. 3

8 Th ffc of h vscosy moulus o h mol rspos show Fgur 4a s ypcal for vscoplascy mols.. h hghr h moulus h hghr h pa srgh a h mor ucl h pos-pa bhavor. Oly crac 3 show Fgur 3 ops LC. I LC2 h mol prco s aurally cal boh x a y rco as h magus of crac 2 a 3 opg vlop qually. Wh h loag s mpos o o 3 oly x rco LC3 oly h shar srss of h lm ffrs from zro Fgur 3b. I hs loa cas crac a 2 boh op h bgg of h sofg procss. Howvr crac bcoms acv almos mmaly so ha oly crac 2 opg vlops urg h loag. Fally LC4 crac a 3 boh op frs upo rachg h sl srgh bu h almos mmaly afr oly crac 3 opg rmas acv so ha h of h sofg procss crac opg s oly 6.6E-5 mm whl ha of crac 3 s 0.03 mm. Laboraory sampl lvl ss Uaxal so a comprsso ss ar smula hr wh h mol show Fgur 5 orr o mosra h mol prformac a h laboraory sampl lvl. Th maral proprs a mol paramrs ar hos show Tabl. Fgur 5. Mol CST msh 4276 lms a bouary coos for uaxal so a comprsso smulaos. Th cosa vlocy bouary coo wh vy 0.02 m/s h so s smulaos a vy 0. m/s comprsso s appl hr. Th smulao rsuls wh wo ffr valus of h shar corol paramr ar show Fgur 6 a 7. 32

9 Fgur 6. Smulao rsuls for uaxal so s: Axal srss vs. sra curvs a fal falur mos wh β b a β 0.5 c. Th fal falur mos Fgur 6b a c ar prs rms of h magu of h sum of splacm ump vcors orms for ach lm. Th ffc of h shar paramr s obvous h rsuls: valu β las o a oubl crac sysm wh a cl r ca Fgur 6b by a ash l. Wh h shar paramr valu s cras o β 0.5 h macrocrac s approxmaly orhogoal o h loag rco whch s h obsrv cas h xprms for rocs a cocr. Wh hs smallr valu of β h prc maral rspos s mor brl as wll. Fally h uaxal comprsso s rsuls ar show Fgur 7 h gomchacs covo of rgarg comprssv srss a sra posv s aop hr. Fgur 7. Smulao rsuls for uaxal comprsso s: Axal laral a volumrc sra vs. axal srss curvs a fal falur mos wh β b a β 0.65 c. Th srss-sra curvs Fgur 7a xhb ypcal faurs obsrv h xprms of roc ur uaxal comprsso [0]. Volumrc sra s frs compaca bu chags h o laa wh ough cracs ar opg. Th ffc of shar paramr s subsaal hr: crasg s valu from o 0.65 oubls h pa srss. Thus wo xrm mos of mol bhavour wh rspc o hs paramr ca b f. 33

10 Th frs s gv by valu β 0 whch allows a f shar compo for h raco vcor rsulg a f comprssv srgh uaxal comprsso s. Th sco s gv by crasg β fly whch allows o shar a all rsulg a zro comprssv srgh uaxal comprsso s. As for h prc falur mos hy xhb h ypcal slghly cl localzao bas. Morovr h ruc shar corbuo rsuls cosrably wr localzao bas as ca b obsrv Fgur 7c. Th aur of hs falur mos s mx mo-i/ii falur. Dscusso a coclusos A ra-p mulpl mb scouy mol was vlop o smula quas-brl fracur wh h f lms cox. I h prs approach h scous wr pr-mb bfor h aalyss o ach f lm of h msh rsulg a formulao smlar cra rspcs o h classcal rfac or cohsv zo lms. Howvr h prs approach s smplr a mor ffc ha h cohsv zo approach sc hr uplcao of msh os or coac rfacs upo crac closur ar. Th prs mho s smpl havg rlavly small umbr 7 oal of mol paramrs. Th shar paramr corollg h raco bw h sl a shar mos of fracur has a subsaal ffc o h comprssv srgh a h falur mos prc wh h mol. By ausg h sl srgh a h shar corol paramr h mol ca prc h corrc sl a comprssv srghs as wll as ralsc falur mos of a quas-brl maral such as roc so a uaxal comprsso as was show h umrcal smulaos. As for h msh pcy of h rsuls prc wh h prs mho shoul b o ha hr ar wo localzao lmrs of ffr aur h prs mol. Th frs s prov by h mb scouy mho whch s ow o b msh p. Th sco s prov by h cluso of vscosy h mol. Thrfor h rsuls shoul b prcpl msh p. Howvr furhr vsgaos o hs ssu ar pospo o h fuur sus of h prs mho. Rfrcs [] T. Rabczu. Compuaoal Mhos for Fracur Brl a Quas-Brl Sols: Sa-of-h-Ar Rvw a Fuur Prspcvs. ISR Appl Mahmacs Arcl ID hp://x.o.org/0.55/203/84923 [2] J.C. Smo J. Olvr F. Armro. A aalyss of srog scous uc by sra-sofg ra-p lasc sols. Compuaoal Mchacs 2: [3] J.C. Smo J. Olvr. A w approach o h aalyss a smulao of sra sofg sols : Z.P. Baza al. Es. Fracur a Damag Quasbrl Srucurs E. a F.. Spo Loo 994 pp [4] J. Olvr. Mollg srog scous sol mchacs va sra sofg cosuv quaos. Pars : fuamals. Iraoal Joural for umrcal 34

11 Mhos Egrg 39: o:0.002/sici :2<3575::aid-me65>3.0.co;2-e [5] J. Olvr. Mollg srog scous sol mchacs va sra sofg cosuv quaos. Par 2: umrcal smulao. Iraoal Joural for umrcal Mhos Egrg; 39: o: 0.002/SICI :2<360::AID-ME64>3.0.CO;2-4 [6] T. Sasala. Ra-Dp Emb Dscouy Approach Icorporag Hrogy for umrcal Molg of Roc Fracur. Roc Mchacs a Roc Egrg 48: o:0.007/s [7]. Moës J. Dolbow T. Blyscho. A f lm mho for crac growh whou rmshg. Iraoal Joural for umrcal Mhos Egrg 46: o:0.002/sici :<3::aid- ME726>3.0.CO;2-J [8] R. Raulovc O.T. Bruhs J. Moslr. Effcv 3D falur smulaos by combg h avaags of mb Srog Dscouy Approachs a classcal rfac lms. Egrg Fracur Mchacs 78: o:0.06/.gfracmch [9] J. Moslr. O avac soluo srags o ovrcom locg ffcs srog scouy approachs. Iraoal Joural for umrcal Mhos Egrg 63: o: 0.002/m.329 [0] O.M. Mahaba. Ivsgag h fluc of mcro-scal hrogy a mcrosrucur o h falur a mchacal bhavour of gomarals. PhD Thss Graua Dparm of Cvl Egrg Uvrsy of Toroo 202. Tmo Sasala Tampr Uvrsy of Tchology Dparm of Cvl Egrg P.O. Box 600 FI-330 Tampr Fla mo.sasala@u.f 35