Theoretical and Numerical Modelling of Creep Crack Growth in a Carbon-Manganese steel

Transcription

1 Thortical ad Numrical Modllig o Crp Crack Growth i a Carbo-Magas stl Yatomi, M. 1, Nikbi, K. M. 2, O Dowd, N.P. 2 ad Wbstr, G. A. 2 1 Rsarch Laboratory, Ishikawajima-Harima Havy Idustris Co. Ltd, 1 Shiakahara-cho, Isogo-ku, Yokohama, Japa Abstract 2 Dpartmt o Mchaical Egirig, Imprial Collg Lodo South Ksigto Campus, Lodo SW7 2AZ, UK This papr prsts a umrical study o crp crack growth i a ractur mchaics spcim. Th matrial proprtis usd ar rprstativ o a carbo-magas stl at 360 o C ad th costitutiv bhaviour o th stl is dscribd by a powr law crp modl. A damag-basd approach is usd to prdict th crack propagatio rat i a compact tsio spcim. Elastic-crp ad lastic-plastic-crp aalyss ar prormd usig two dirt crack growth critria to prdict crack xtsio udr pla strss ad pla strai coditios. Th pla strai crack growth rat prdictd rom th umrical aalysis is oud to b lowr tha that prdictd rom ductility xhaustio pla strai modl (kow as th NSW modl), which uss th crp ractur mchaics paramtr C ad th dvlopmt o crp damag dirctly ahad o th crack tip to prdict crp crack growth rats udr pla strai/pla strss coditios. A modiid NSW modl (NSW-MOD) is prstd i which th ct o th damag agl at th crack tip is cosidrd i ordr to prdict this dirc. I th modl it is assumd that ractur occurs irst at th valu o th crack tip agl, at which th crp strai, rachs its maximum valu. It is oud that th w NSW-MOD givs a bttr prdictio o th pla strai uppr-boud o th xprimtal data. Kywords: crp, crack growth rat, ractur mchaics, C, high tmpratur tstig, Fiit lmt aalysis, damag mchaics, costrait. 1. Itroductio For dsig ad saty assssmt purposs it is ot cssary to stablish th sigiicac o dcts i compots subjctd to crp ad crp/atigu loadig. A umbr o assssmt procdurs,.g. [1] [5], ar availabl or this purpos. Wh such procdurs ar usd at th dsig stag, th sizs o postulatd dcts ar dtrmid by th rsolutio o o-dstructiv ispctio mthods. Othrwis th sizs o dcts dtctd i srvic ar usd to mak stimats o rmaiig litims. 1

2 I [6] a modl or crp crack growth (CCG) was itroducd, basd o th aalytical orm o th strss ad strai ilds i th viciity o a sharp, growig crack udr stady stat coditios, ad assumig that crack growth was du to th accumulatio o crp strai i a procss zo ahad o th growig crack i a crpig structur. Th modl (calld th NSW modl, hratr) dirctly rlats th CCG rat to th crp ractur mchaics paramtr C. Such a modl was oud to provid a sa uppr boud or crp crack growth i a rag o matrials. With rct advacs i iit lmt (FE) mthods, mor complx approachs ca b applid i th study o CCG, which may provid mor accurat prdictios tha rlativly simpl aalytical solutios. I particular, cotiuum damag mthods hav b widly usd to prdict ailur at high tmpratur,.g. [7] [11]. Grally, ths studis hav ocusd o th prdictio o th tim to crack iitiatio ad o th ruptur li o th compot ad hav ot xamid th CCG rgim. I this work, a ucoupld damag-basd approach is usd to simulat crack growth rom th iitial trasit stat to th stady stat rgim, withi a iit lmt ramwork. Th approach ollows that i [12] [14], whrby ods ar rlasd wh damag rachs a critical valu, simulatig th ormatio o a sharp crack. Th aalysis ocuss o th study o CCG i a carbo magas (CM) compact tsio (CT) spcim. Th prdictios obtaid rom th FE aalysis ar compard with thos rom th thortical NSW modl, a hacd vrsio o th NSW modl, which will b prstd i sctio 3, ad xprimtal CCG data or a CM stl. 2. Dormatio at High Tmpratur At high tmpraturs mtals xhibit rat dpdt (crp) dormatio udr costat load. For may matrials, udr stady stat (costat strai rat) coditios th crp strai rat, c, may b rlatd to th strss by a powr law, c (1) = Cσ whr C ad ar matrial costats (which may dpd o tmpratur). Mor grally, crp dormatio ca b cosidrd to b composd o thr rgims, amly primary, scodary ad trtiary crp rgims. Grally, Eq. (1) is usd to dscrib th stady stat or scodary crp rat. Altrativly, a avrag crp rat obtaid dirctly rom crp ruptur data ca b usd to accout or all thr stags o crp dormatio: σ A = = o = AAσ t r σ 0 A A, (2) 2

3 whr is th uiaxial ailur strai, t r is th tim to ruptur i a uiaxial crp tst ad o, σ o, A A ad A i Eq. A A A 0 σ 0 (2) ar matrial proprtis, rlatd i such a way that =. I this papr th crp xpots,, ad A, will b usd itrchagably ad rrs to th xpot o a crp law o th typ i Eq. (1) or (2). Ulss aothr ailur mchaism itrvs, crp dormatio will vtually lad to ruptur, which is grally associatd with th coalscc o voids alog grai boudaris. High valus o triaxial tsio ca hac void uclatio ad growth ladig to rducd crp ductility ad crp-brittl bhaviour. Svral modls (.g. [15], [16]) which accout or th iluc o stat o strss o th dormatio ad damag procsss hav b proposd, which ar basd o th assumptio that strai rat is govrd by th quivalt strss ad void growth ad iitiatio mchaisms by th ratio btw hydrostatic strss ad quivalt strss (triaxiality). I [16] a xprssio or multiaxial crp ductility,, basd o a mchaism o grai boudary cavitatio or a powr law crp matrial. From this modl th ratio o th multiaxial to uiaxial ailur strai, / is giv by = sih sih 2h , (3) whr h = σ m /σ is th ratio btw th ma (hydrostatic) strss ad quivalt (vo Miss) strss at a matrial poit. Udr uiaxial coditios, h = 1/3 ad Eq, (3) givs / = 1, as xpctd. 3. Fractur Mchaics at High Tmpratur Th thory bhid th corrlatio o high tmpratur crack growth data sstially ollows that o lastic-plastic ractur mchaics. Various aspcts o th charactrisatio o crp crack growth hav b rviwd i [17] ad [18]. For situatios whr liar lastic coditios prvail (short tims ad/or low loads) th liar lastic strss itsity actor, K, may b usd to prdict crp crack growth. Udr stady stat crp coditios, howvr, th crack tip strss ad strai rat ilds ar charactrisd by th paramtr C ad liar lasticity may o logr b applicabl. For a powr law crpig matrial with crp law o th orm o Eqs. (1) or (2), th strss ad strai rat i th viciity o a sharp crack tip ar giv by (s.g. [17]), σ ij C = σ 0 0σ 0 I r 1 ( + 1 ) ~ σ ( θ, ), (4) ij 3

4 ( + 1 ) C ~ ij = 0 ij ( θ, ) 0σ 0 I r, whr r ad θ masur distac ad agl rom th crack tip, rspctivly, I is a paramtr which dpds o th crp xpot,, ad out o pla strss stat (pla strss vs. pla strai) ad, ad ~σ ij ad ~ ij ar dimsiolss uctios o, θ, ad out o pla strss stat. Th paramtr C i Eq. (4) may b obtaid rom a path idpdt itgral ad is aalogous to th J itgral or o-liar lastic bhaviour [19]. C may also b itrprtd as a rgy rlas rat aalogous to th rgy diitio o J. Th C itgral has b widly usd as a paramtr or corrlatig CCG udr stady stat crp coditios. For th compact tsio (CT) spcim, ollowig ASTM E1457 [20], C may b calculatd rom crp load li displacmt rat, Δ c, usig th xprssio, c PΔ C = F, (5) B b whr P is th applid load, b is th rmaiig ligamt ahad o th crack, B is th t thickss btw sid-groovs wh usd ad F is a actor which dpds o crack lgth, spcim gomtry ad crp strss idx,. 3.1 Two Paramtr Crack Tip Fild I [21] th sigl paramtr charactrisatio o Eq. (4) was xtdd by icludig highr ordr trms i th crack tip ilds. Th crack tip strss ild is rprstd by a quatio o th orm: σ σ ij 0 C = 0σ 0I r 1/( + 1) ~ σ ( θ, ) + Q ˆ σ ( θ, ). (6) ij ij Th irst trm i Eq. (6) is th HRR ild (as bor). Th crack tip ilds ar thus dscribd by two paramtrs: C, which cotrols th amplitud o th crack tip sigularity, ad Q, which masurs th costrait o th body through th dviatio o th ull ild solutio rom th HRR solutio. From lastic powr-law plastic [22] ad lastic powr law crp [23], aalyss th dviatio o th ull ild solutio rom th HRR solutio has b s to corrspod closly to a uiorm hydrostatic strss stat, i.. ˆ σ ij δ, whr δ ij is th Krockr dlta. 3.2 Modls o Stady Stat Crp Crack Growth Basd o th orm o th crack tip ilds i Eq. (4) ad usig a ductility xhaustio argumt it was show i [6], that th crp crack growth rat, a, may b writt as ij 4

5 1 ( + 1) C C = = rc c I σ 0 0 I ( Ar ) + 1 a, whr r c is th siz o th crp procss zo ad (7) is th appropriat multiaxial crack tip ductility. Th modl is kow as th NSW modl. I [25] it is rcommdd that udr pla strss coditios th multiaxial ductility,, b tak as th uiaxial ailur strai,, ad /30 udr pla strai coditios. Th pla strss ad pla strai NSW lis should th spa th xprimtal CCG data, with Eq. 7 prdictig that th crack growth rat udr pla strai coditios is approximatly 30 tims highr tha that udr pla strss coditios at th sam valu o C (additioal stat o strss cts o CCG rat tr through I i Eq. 7). I th NSW modl it is implicitly assumd that ractur occurs irst at th valu o th crack tip agl,θ, at which th quivalt crp strai, quatiid by ~ ( θ, ) i Eq. (4), rachs its maximum valu. A mor gral xprssio ca b obtaid, which cosidrs th dpdc o ~ ( θ, ) ad o agl, θ. For this situatio, th NSW modl may b xtdd to giv a modiid crack growth rat, (hratr rrrd to as th NSW-MOD modl ): 0 C 1/(+ 1) a r ~ NSW MOD ( 1) c ( θ, ) ( θ, ) 0σ 0I = +. (8) /( + 1) Th orm o Eq. 8 is th sam as that o Eq. 7, but th dpdc o ~ ad agl θ ad is icludd. Sic hydrostatic ad Miss strss dpd o θ ad, th dpdc o o strss stat may b valuatd by substitutig h = σ ( θ, )/ σ ( θ, ) or h i Eq. (3) or ay othr appropriat modl which dscribs th strss-stat dpdc o th crp ductility. Solutios or th crack tip distributios ~ σ (, θ ) ad ~ σ (, m θ ) ar tabulatd i [26] or svral valus o. Figur 1 shows th variatio o [ ~ σ ] ( θ, ) or = 5, 10 ad 20 udr pla strss ad pla strai coditios (ot that ~ (, ) [ ~ (, ) ] θ = σ θ ). It is s i Figur 1 that th maximum valu o [ ~ σ ] ( θ, ) is uity at θ 0 ad 90 udr pla strss ad pla strai coditios, rspctivly. At θ = 0 th dirc i th valu o [ ~ σ ( θ, ) ] udr pla strss ad pla strai coditios ca b up to a actor o 100 dpdig o th valu o. Figur 2 shows th valu o th ailur strai m o (ormalisd by uiaxial ailur strai, ) or th sam valus o, usig Eq. (3) i cojuctio with th solutio or h i [26]. It is s that th valu o icrass with agl, θ, both udr pla strss ad pla strai coditios with a much strogr dpdc udr 5

6 pla strai coditios (s Figur 2b). It may also b s that at θ = 0 th multiaxial ductility ca b up to 500 tims lowr udr pla strai tha udr pla strss coditios (dpdig o th valu o ). Sic both multiaxial ailur strai,, ad crp strai, ~, dpd o agl θ, th maximum valu o th CCG rat will occur at th agl whr th ratio ~ is a maximum. Figur 3 shows th agl givig th maximum valu o CCG rat, θˆ, udr pla strss ad pla strai coditios. It is s that udr pla strss coditios ˆ θ = 0 (dirctly ahad o th iitial crack pla), whil udr pla strai coditios th valu o θˆ dpds o, ˆ θ 0 or = 5, θˆ o 90 or = 20. Figur 4 shows th ratio o CCG rat udr pla strss ad pla strai coditios or dirt valus o at th sam valu o C obtaid rom th NSW-MOD modl. From Figur 4(a), th maximum valu o CCG rat udr pla strai coditios is about 3 7 tims gratr tha that udr pla strss, although th ratio dpds o th valu o. I w assum that crack growth occurs at a agl ˆ θ = 0 ( Figur 4b), th CCG rat udr pla strai coditios is up to 7 tims astr tha that udr pla strss coditios, dpdig o th valus o (th ratio dcrass with icrasig ). Not that or = 20, th prdictd crack growth rat or θˆ = 0 is lowr udr pla strai coditios tha udr pla strss coditios. This is du to th lowr lvl o crp strai rat udr pla strai coditios or this valu o (s Figur 1) Ect o costrait o crp crack growth I [27] a modl or prdictig th stady stat crack growth rat at dirt lvls o costrait, giv by th Q strss, was proposd. Equatio (6) is usd to rprst th CCG rat (rathr tha Eq. (4) as i th NSW or NSW-MOD modl), ad th CCG rat is th giv as a a g, (9) = NSW MOD whr d ~ 1 g = 1 ~ dh θ, ( ) Q C 2 ~ (, ) σ θ I σ ( + 1) 1 ( + 1) c r, (10) As bor, ~ xampl. So that is th multiaxial ailur strai valuatd rom ~ h = ~ σ / ~ σ usig Eq. (3) or m d = 2 tah 2h dh (11) 6

7 With th Cocks ad Ashby modl, th crp xpot,, ad th HRR triaxiality strss ratio, h, ar s to iluc th crack growth rat through th uctio g as wll as through th valu or ~ usd i th NSW or NSW-MOD modl. I [27] th dpdc a NSW o crp strai ad triaxiality was ot cosidrd ad was usd i Eq. (9) rathr tha a NSW MOD (Is that corrct?)(i [27], th dpdc o crp strai ad triaxiality is cosidrd ad th agl is ot cosidrd. a is usd i Eq. (9).) 4. Exprimtal Data NSW MOD I this work crp crack growth at 360 C i a carbo magas stl (C-M) is xamid. Th matrial proprtis or th carbo magas (C-M) stl wr obtaid rom uiaxial tsil tsts ad crp tsts. Th rsults ar rom a sigl batch o matrial ad prormd withi a collaborativ programm, [28]. Th rlvat mchaical proprtis ar giv i Tabl 1 [14]. Figur 5 shows th stady stat CCG rat or th C-M stl. Ths data hav b obtaid rom a rag o spcim sizs (spcim width rags rom 7.5 mm to 50 mm) ad both dp crackd compact tsio ad sigl dg otch bd (thr poit bd) spcims hav b tstd. Th valu o C has b obtaid rom th load-poit displacmt rat ollowig Eq. (5) ad th crack lgth dtrmid usig th pottial drop tchiqu accordig to ASTM E1457 [20]. Oly data valid accordig to [20] ar icludd (.g. thos data collctd i th trasitio rgio or or amouts o crack growth lss tha 0.2 mm ar xcludd). It is s that th data all withi a rlativly arrow scattr bad ad th dpdc o crack growth rat, da/dt, o C ollows a ar liar trd o a log-log scal. Also show i Fig. 5(a) ad 5(b), rspctivly, ar th NSW prdictios (Eq. 7) ad th NSW-MOD prdictios (Eq. 8) or pla strss ad pla strai usig th matrial proprtis i Tabl 1. Th uiaxial ailur strai,, usd i both modl is 18%, which is th masurd valu rom uiaxial crp tsts [14]. Th additioal paramtrs dd or th NSW ad NSW-MOD modl ar listd i Tabl 2. Th valu tak or th critical distac r c is 15 μm, which th avrag grai siz or th matrial (ot that th prdictios ar ot vry ssitiv to th valu o r c ). Figur 5(a) shows that th pla strss NSW modl is clos to th ma o th masurd crack growth rat ad th pla strai NSW modl sigiicatly ovrstimats th crack growth rat (by almost a ordr o magitud). Two lis ar icludd or th pla strai NSW-MOD aalysis i Fig. 5(b), th li corrspodig to th agl, θˆ, at which ~ is maximum (dirctio o maximum crp crack growth) ad th NSW-MOD prdictio takig θˆ = 0, i.. th dirctio dirctly ahad 7

8 o th currt crack tip. Th lattr prdictio will b compard with iit lmt prdictios i Sctio 5. It may b s i Figur 5(b) that th li or θˆ = 0 is somwhat blow th li o maximum prdictd crack growth rat (th lis ar coicidt or pla strss coditios). Th NSW-MOD modl or ithr pla strss or pla strai coditios li quit clos to th xprimtal data ad th pla strai NSW-MOD modl or θˆ = 0 givs a uppr boud to th xprimtal data. Ths rsults suggst that th NSW- MOD pla strai modl provids a mor accurat uppr boud stimat o CCG rat tha th NSW pla strai modl or ths data. Furthr isight ito th crp crack growth bhaviour ca b obtaid through a iit lmt aalysis as will b discussd i th xt sctio. 5. Fiit Elmt Modllig A virtual ractur tstig procdur usig th iit lmt (FE) mthod [12 14] has b dvlopd to prdict crp crack growth rats i Carbo Magas CT spcims. It has b oud that ucoupld cotiuum damag mthods ca b usd to prdict crp crack growth withi a umrical ramwork [14]. I this sctio, a bri rviw o th approach is prstd. 5.1 Damag Accumulatio Modl Followig th discussio o Sctio 3, it is assumd that CCG occurs by a ductil mchaism. A damag paramtr, ω, is did such that th rat o damag accumulatio is rlatd to th quivalt crp strai rat, c, by, c ω =. (12) Th damag, ω, accumulats with tim du to th accumulatio o crp strai, rom ω = 0 at t = 0, ad ailur occurs at a matrial poit wh ω = 1. I this work th volutio o damag is ot coupld to th dormatio, as i.g. [7], so that th crp rat o th matrial is ot hacd du to th accumulatio o damag. 5.2 Elastic, plastic ad crp strais Calculatios hav b prormd usig lastic-crp ad lastic-plastic-crp matrial dscriptios. I th lattr cas th plastic strais ar udrstood to b idpdt o strai rat givig th total strai as l pl cr = +, (13) + whr l, pl ad cr ar lastic, plastic ad crp strais rspctivly. As discussd i Sctio 2 th crp rspos is dscribd by a scodary crp law usig th avrag 8

9 crp proprtis. Th yild strgth o th stl at 360 o C is 240 MPa, which is rlativly low. Thror th ct o plasticity may b importat or this matrial. Th plastic rspos is assumd to b govrd by a Vo-Miss low rul with isotropic strai hardig ad was obtaid by ittig to uiaxial tsil tst data at 360 C. Th post-yild strai hardig rspos is tratd as pic-wis liar up to th UTS (= 570 MPa) byod which o strai hardig was tak to occur. For a lastic-crp aalysis or durig uloadig th plastic strai rat is zro. 5.3 Fiit Elmt Modl A typical iit lmt (FE) msh is illustratd i Figur 6. A compact tsio (CT) spcim is aalysd with spcim width W = 25 mm ad iitial crack lgth to spcim width, a/w = Th load is applid at poit P, as idicatd i Figur 6. All FE aalyss wr coductd usig ABAQUS 5.8 [29] ad th msh cotais approx our odd two dimsioal lmts. Small strai thory has b usd i th aalysis to b cosistt with th assumptios o th NSW ad NSW-MOD modls. Both pla strss ad pla strai aalyss hav b carrid out. Crack growth was modlld usig a odal-rlas tchiqu [14] wh damag, ω, rachs uity ahad o th crack tip, th od at th crack tip is rlasd. Th odal rlas is implmtd through th MPC subrouti i ABAQUS [29] ad ods ar rlasd ovr a sigl icrmt oc th ailur coditio is satisid i th adjact lmt. Rgular squar lmts wr usd i th viciity o th crack tip (s ist to Figur 6) so th crack grows through a rgio o uiorm lmts. Th msh siz at th crack tip is approximatly 15 μm (which is also th critical distac usd i th NSW ad NSW-MOD modls). It is assumd i th FE aalysis that th crack grows i th pla o th iitial crack rot, i.. alog th symmtry pla. Calculatios hav b prormd usig lastic-crp ad lastic-plastic-crp bhaviour. I th lattr cas th plastic strais ar udrstood to b idpdt o strai rat. Th crp rspos is dscribd by a scodary crp law usig th avrag crp proprtis. 6. Fiit Elmt Rsults Rsults ar irst prstd or crp ductility, = 50%. This valu is ot rprstativ o th masurd crp ductility o th C-M stl, but has b xamid i ordr to provid a rlativly slow crack growth rat i th FE aalysis so that trasit cts ar rlativly small (i.. crack growth occurs udr prdomiatly stady stat coditios) ad dirct compariso may b mad with th NSW ad NSW-MOD modls. I Sctio 6.3, dirct compariso with th xprimtal crack growth data ad th FE prdictios will 9

10 b mad, by takig = 18%, which is th masurd uiaxial crp ductility o th C-M stl. 6.1 Fiit lmt prdictio o crp crack growth rat Figur 7 shows th CCG rat prdictd rom th FE aalysis plottd agaist C or th pla strss/strai aalyss with = 50% ad all othr matrial proprtis tak rom Tabl 1. Th trasit parts o th crack growth curvs (th tails ) hav b rmovd or ach aalysis. Thus oly rsults udr (global) stady stat coditios ar prstd. (How, prcisly wr ths tails dtrmid? Did you bas it o tim > trasitio tim or just judg it by y rom th igur?) (Δa > 0.2 mm) Th paramtr C was calculatd rom th FE load li displacmt rat (s Eq. (5)) ad good agrmt btw th li itgral C valu ad that obtaid rom Eq. (5) has b obsrvd ovr th rgio o itrst, [14]. Th crack growth rat, da/dt, is dtrmid dirctly rom th FE aalysis, with th currt crack tip positio is tak to b at th positio o th most rctly rlasd od. I prvious work [12], [13], FE rsults o th typ show i Figur 7 hav b compard with th NSW modl. Hr, th prdictios rom th NSW-MOD modl ar also cosidrd. Th NSW ad NSW-MOD prdictios ar th sam as thos i Figur 5 xcpt that th ailur strai,, has b tak to b 50% so th CCG rat is rducd by a actor o approx. 3. I th NSW-MOD modl, th crack growth dirctio is tak to b i th pla o th crack ( θˆ = 0 ) to b cosistt with th FE aalysis. It may b s i Figur 7(a) that udr pla strss coditios th CCG rats prdictd rom th lasticcrp, th lastic-plastic crp FE aalysis ad th NSW-MOD modl ar i good agrmt ad slightly abov th origial NSW modl. Sic strss lvls ar somwhat lowr i pla strss, th ct o icorporatig plastic dormatio is ot sigiicat. Udr pla strai coditios (Figur 7b) th NSW-MOD modl prdicts a lowr CCG rat ovr th rag o C tha th NSW modl. At low lvls o C th lastic ad lasticplastic crp FE prdictios ar similar (as xpctd) ad clos to th NSW-MOD modl. At high valus o C (C > 1 J/m 2 h) th FE prdictio rom th lastic-crp aalysis is cosidrably highr tha th NSW-MOD modl ad appars to approach th NSW li. Th lastic-plastic FE aalysis prdicts a CCG rat somwhat highr tha th NSW-MOD modl at low valus o C ad alls slightly blow th NSW-MOD li at highr valus o C. Th high crack growth rats s i th lastic-crp aalysis ar blivd to b du to ustady crack growth or th lastic-crp matrial at high valus o C. This will b discussd urthr i Sctio

11 Not that or th NSW-MOD modl th CCG rat udr pla strai coditios is about 1.5 tims highr tha that udr pla strss coditios (s Figur 7 ad also Figur 4) at th sam valu o C. I gral it is s that th lastic-plastic-crp FE rsults ar cosistt with th NSW-MOD modl udr pla strss ad pla strai coditios. 6.2 Ratio o crp displacmt rat ad load-li displacmt rat ASTM E1457 [20] spciis bouds udr which C is applicabl i trms o th loadli crp displacmt rat dividd by th load-li total displacmt rat ( Δ c / Δ ). Th c crp displacmt rat, Δ, is obtaid by subtractig th istataous displacmt i rat, Δ, (du to crack growth) rom th total displacmt rat, i.. Δ c i = Δ Δ. (14) For covic, th istataous displacmt rat, plastic trms, i.. Δ i, is ot split ito lastic ad i i, i, p Δ = Δ + Δ. (15) c t For covic th otatio Δ = Δ Δ is usd i this sctio. I [20] it is statd that i Δ > 0.5, th crack growth rat may b charactrisd by C. Valus o Δ < 0. 5, imply that that th istataous displacmt rat is a sigiicat ractio o th total displacmt rat, idicatig that rapid o-stady ( crp-brittl ) crack growth is occurrig, which caot b charactrisd by th stady stat crp paramtr, C. Figur 8 shows th valu o Δ as a uctio o C obtaid rom th FE aalysis udr pla strss ad pla strai coditios or th lastic-crp ad lastic-plastic crp aalysis. Δ is calculatd i th sam way as would b do i a xprimt: Δ t is obtaid dirctly rom th load poit displacmt rat ad th istataous i displacmt rat Δ is calculatd by [20], 2 Δ i ab 2K =, (16) P E whr K is th liar lastic strss itsity actor ad E is th ctiv lastic modulus (E/(1 - v 2 ) or pla strai ad E or pla strss). Not that th plastic displacmt rat, p Δ, has b tak to b zro. It is s i Figur 8 that udr pla strss coditios th valu o Δ or both th lastic crp aalysis ad lastic-plastic crp aalysis is wll abov 0.5 or th rag o C valus cosidrd. Thus crack growth occurs udr stady stat ( crp-ductil ) coditios ad ca b charactrisd usig C. Udr pla strai coditios th valu o 11

12 Δ or th lastic crp aalysis dcrass with icrasig C (though rmais abov 0.5). This is du to th icras i th valu o th lastic displacmt rat, Δ, as th crack growth rat icrass. For th lastic-plastic crp aalysis th valu o Δ or lastic-plastic crp aalysis icrass slightly with th valu o C ad rmais about 0.5. This suggsts that or th lastic-plastic crp aalysis th crack growth occurs udr stady crp coditios. Figur 7 idicats that although th valu o Δ rmais abov 0.5 or th lastic-crp aalysis, th crack growth data ar ot dscribd by th stady stat NSW-MOD modl ad such a dscriptio would b o-cosrvativ. (Th additioal cosrvatisms ihrt i th NSW modl sur that or this cas th FE crack growth rats ar boudd by th pla strai NSW li, though this is ot a gral rsult.) Th umrical aalysis thus suggsts that th coditio that Δ > 0.5 dos ot guarat that stady stat coditios prvail ad th us o C is a appropriat paramtr. It may b s rom Figur 7 ad Figur 8, that or Δ < th CCG rat prdictd rom th FE aalysis is highr tha th NSW-MOD modl ad or Δ > th CCG rat prdictd rom th FE aalysis is clos to or lowr tha th NSW-MOD modl (i.. All C or pla strss ad C > 2 J/m 2 h or pla strai). Thror th umrical aalysis suggsts that i Δ > 0. 85, strss coditios ar prdomiatly udr crp-ductil stady stat coditios ad th crack growth may b charactrisd usig C ad th NSW-MOD modl. 6.3 Prdictd iluc o costrait, Q, o crp crack growth Th compact tsio spcim is a high costrait spcim ad Q i Eq. (6) is xpctd to b clos to zro ovr th rag o coditios xamid. Figur 9 shows th variatio o Q with C durig crack growth, or a lastic-crp ad a lastic-plasticcrp aalysis udr pla strss/strai coditios. (A valu o σ 0 = 600 MPa is usd to valuat Q rom Eq. 6, so that Q = 1 implis that th crack tip strss is 600 MPa lss tha that prdictd by th HRR solutio.) Not that Q is did rlativ to th appropriat pla strss or pla strai HRR distributio so th valus i Figur 9(a) ad (b) caot b compard dirctly. It ca b s rom Figur 9 (a) that udr pla strss coditios th valu o Q is clos to zro ad almost idpdt o crack tip plasticity up to C valus o about 100 J/m 2 h. Th ct o plasticity is mor vidt or th pla strai cas, illustratd i Figur 9(b). For valus o C > about 0.01 J/m 2 h, th valu o Q, is lowr or a lastic-plastic aalysis tha or a lastic aalysis. Th largst dviatio o Q rom zro i Figur 9 is s or th lastic-plastic pla strai aalysis. Figur 10 shows th compariso o th CCG rat udr pla strai 12

13 coditios prdictd rom th FE aalysis i cojuctio with th Budd ad Aisworth modl (s Eq. (16)), i which th cotributio o Q to th crp crack growth rat is cosidrd. (Is this actually a Budd ad Aisworth-MOD modl or dos it tak ito accout th agular dpdc o strai ad triaxiality?) (th agl is ot cosidrd but th triaxiality is cosidrd) Th two lis ar almost idistiguishabl idicatig that i this cas icorporatio o Q dos ot sigiicatly act th prdictd crack growth rat. 6.4 Compariso with Exprimtal Data or C-M Th prvious aalyss wr or = 50% to sur stady stat crp coditios. I this sctio th rsults obtaid rom th modl takig = 18% ar compard with th NSW ad NSW-MOD modl ad with th xprimtal data or C-M. (I [13] ad [15] xprimtal data or th C-M stl hav b compard with th NSW modl.) Figur 11 shows th compariso btw th FE prdictio, th NSW modl (Eq.(7)) ad th NSW-MOD modl (Eq. (8)) with = 18%. (Hav ay tails b rmovd i th FE aalysis? I so, o what basis?)(δa > 0.2 mm) Elastic-plastic-crp FE aalyss wr coductd to prdict th CCG rat ad, as bor, it is assumd that crack growth is i th pla o th crack, θˆ = 0. Ovr th rag whr th xprimtal data ar availabl, it is s that th pla strss ad pla strai FE prdictios giv similar crack growth rats (cosistt with Figur 7) ad all withi th scattr bad o th data. Howvr, it may b s that or low valus o C (outsid th rag o th xprimtal data) th pla strai FE aalysis prdicts a cosidrably highr crack growth rat tha ithr th pla strss FE prdictio or th NSW-MOD modl ad is v abov th pla strai NSW modl (Figur 11a). Figur 12 shows th valu o Δ vrsus C or th FE aalysis. It is s that udr pla strss coditios th valus o Δ is abov 0.5 ovr th rag o C valus cosidrd. Udr pla strai coditios, Δ icrass with icrasig C ad at low valus o C, th valus o Δ is wll blow 0.5, idicatig that wh is suicitly low, crack growth ca occur i th arly trasit crp rgim, whr Δ i, is a sigiicat ractio o Δ i. Udr ths coditios, th us o th NSW or NSW-MOD modl may b qustioabl. It should b poitd out that th FE aalysis dos ot rly o a stady stat assumptio. Thus th prdictio o Figur 11 is ot icorrct; th rsult simply idicats that or this matrial ad at low valus o C th crack growth rat caot b charactrisd by C ad th NSW or NSW-MOD modls. Thus, th prdictd hacd crack growth rat at low C valus may b a ral ct ad should b cosidrd wh xtrapolatig rom short 13

14 trm tsts (high C) to log trm tsts (low C), th lattr big mor rprstativ o coditios i actual idustrial compots. It may b s i Figur 11 that or pla strss coditios thr is good agrmt ovr almost th ull rag o C btw th FE prdictios ad th NSW-MOD modl (th sam trd was i th prvious sctio or = 50%). I cosidrig th valu o Δ i Figur 12, it is s that providd th valu o Δ is abov 0.65, th NSW-MOD pla strai modl provids a cosrvativ stimat o th CCG rat, compard to th FE prdictio. Th NSW pla strai modl givs cosrvativ stimats providd Δ > Isuicit xprimtal data or th C-M stl ar availabl to coirm this rsult, though data or a stailss stl tstd at high tmpratur ad at low C lads to a highr prdictd CCG rat compard to thos at highr lvls o C. 7. Coclusio A iit lmt study o crp crack growth (CCG) usig a damag variabl withi a iit lmt ramwork to quatiy tim dpdt crack tip dgradatio has b carrid out. Th matrial xamid is carbo-magas stl tstd at 360 C (i th crp rgim). A powr law crp modl is usd to dscrib th crp bhaviour o th stl ad both pla strss ad pla strai coditios ar xamid. Th prdictd CCG rat is corrlatd usig th crp paramtr C dtrmid rom th load-li displacmt rat. A modiid ductility xhaustio modl NSW-MOD which is drivd rom th NSW modl is prstd which taks ito accout th ct o th th maximum damag agl at th crack tip to prdict CCG rats. Th modl compars avourably with th xprimtal CCG data orm Carbo- Magas tsts, FE prdictios. Th NSW-MOD modl udr pla strai coditios givs lss cosrvativ prdictios tha th NSW modl udr pla strai coditios ad uppr bouds o xprimtal data. For Δ < th CCG rat prdictd rom FE aalysis is highr tha th NSW-MOD modl. Th covrgc o th pla strss ad pla strai prdictios at high valus o C is blivd to b du to th rductio o Q ad high valu o Δ (i.. stady stat). Rrcs [1] BS7910, Guid o mthods or assssig th accptability o laws i usio wldd structurs, Lodo, BSI, [2] R5, Dct assssmt cod o practic or high tmpratur mtallic compots, British Ergy Gratio Ltd., 2000 [3] ASME Boilr ad prssur vssl cod, sctio XI: Ruls or i-srvic ispctio o uclar powr plat compots, Amrica Socity o Mchaical Egirs

15 [4] R6, Assssmt o th itgrity o structurs cotaiig dcts, Rvisio 3, British Ergy Gratio Ltd., [5] AFCEN, Dsig ad costructio ruls or mchaical compots o FBR uclar islads, RCC-MR, Appdix A16, AFCEN, Paris, 1985 [6] Nikbi, K. M., Smith, D. J. ad Wbstr, G. A., Iluc o Crp Ductility ad Stat o Strss o Crp Crack Growth, i Advacs i Li Prdictio Mthods at Elvatd Tmpraturs, 1983, [7] Hayhurst, D.R., Dimmr, P.R. ad Morriso, C.J., Dvlopmt o cotiuum damag i th crp ruptur o otchd bars, Phil. Tras. R. Soc. Lod. A 1984; 311, [8] Hyd, T.H., Su, W. ad Bckr, A.A., Crp crack growth i wlds: a damag mchaics approach to prdictig iitiatio ad growth o circumrtial cracks, It. J. Prssur Vssls Pipig 2001; 78, [9] Bllgr, E. ad Bussy, P., Phomological modllig ad umrical simulatio o dirt mods o crp damag volutio, It. J. Sol. Struct. 2001; 38, [10] Prri, I.J., ad Hayhurst, D.R., Cotiuum damag mchaics aalyss o typ IV crp ailur i rritic stl cross-wld spcims, It. J. Prssur Vssls Pipig, 1999; 76, [11] Bassai, J.L. ad Hawk, D.E., Iluc o damag o crack-tip ilds udr smallcrp coditios, It. J. Fractur 1990; 42, [12] Yatomi, M., O Dowd, N. P., Nikbi, K. M. Modllig o damag dvlopmt ad ailur i otchd bar multiaxial crp tsts, Fatigu Fractur Eg. Matls. Struct., 2004; 27, [13] Yatomi, M, O'Dowd, N. P ad Nikbi, K. M., Computatioal modllig o high tmpratur stady stat crack growth usig a damag-basd approach, i PVP- Vol. 462, Applicatio o Fractur Mchaics i Failur Assssmt Computatioal Fractur Mchaics, ASME 2003, Ed. P.-S. Lam, ASME Nw York, NY 10016, 2003, [14] Yatomi, M., Nikbi, K. M., O Dowd, N. P., Crp Crack Growth Prdictio Usig a Crp Damag Basd Approach, It. J. Prssur Vssls Pipig, 2003; 80, [15] Ric, J.R. ad Tracy, D.M., O th Ductil Elargmt o Voids i Triaxial Strss Filds, J. Mch. Phys. Solids, 1969; 17, [16] Cocks, A.C.F. ad Ashby, M.F., Itrgraular ractur durig powr-law crp udr multiaxial strss, Mtal Scic, 1980; 14, [17] Wbstr, G.A. ad Aisworth, R.A. 1994, High Tmpratur Compot Li Assssmt, Chapma ad Hall. 15

16 [18] Ridl, H, 1987, Fractur at High Tmpraturs, Sprigr-Vrlag Brli, Hidlbrg. [19] Ric, J.R., 1968, Mathmatical aalysis i th mchaics o ractur, i Tratis o Fractur (d. H. Libowitz), vol. 2, Acadmic Prss, Nw York. [20] ASTM, E : Stadard Tst Mthod or Masurmt o Crp Crack Growth Rats i Mtals, Aual Book o ASTM Stadards, 2001, 3, [21] Shih, C.F., O Dowd, N. P. ad Kirk, M.T., 1993, A ramwork or quatiig crack tip costrait. Costrait Ects i Fractur (Editd by E.M. Hacktt, K.H. Schwalb ad R.H. Dodds), Amrica Socity or Tstig ad Matrials, Philadlphia, 2 20 [22] O'Dowd N.P., Shih, C.F., Family o Crack Tip Filds Charactrizd by a Triaxiality Paramtr I. Structur o Filds, J. Mch. Phys. Solids, 1991; 39, [23] Bttiso A.D., O'Dowd N.P., Nikbi K. ad Wbstr G.A., Two paramtr charactrisatio o crack tip ilds udr crp coditios, IUTAM Symposium o Crp i Structurs, S. Murkami ad N. Oho Eds., , Kluwr Acadmic Publishrs, Nthrlads, [24] O'Dowd, N.P., Applicatios o two paramtr approachs i lastic-plastic ractur mchaics, Egirig Fractur Mchaics, 52, 1995, [25] Ta, M., Clard, N.J.C., Nikbi, K.M. ad Wbstr, G.A. Compariso o crp crack iitiatio ad growth i our stls tstd i HIDA, It. J. Prs. Vss. Pipig., 2001; 78, [26] Shih, C.F., Tabls o Hutchiso-Ric-Rosgr sigular ild quatitis, Brow Uivrsity Rport MRL E-147, Providc, RI [27] Budd P.J. ad Aisworth R.A. Th Ect o Costrait o Crp Fractur Assssmts. It. J. Fractur, 1999; 97, [28] Rsults rom th Crp Crack Growth i Carbo-Magas Stls at C, Brit/Euram 7463 Collaborativ Projct, Europa Commissio [29] ABAQUS vrsio 5.8, Hibbitt, Karlsso & Sors, Ic. [30] C. M. Davis, F. Mullr, K. M. Nikbi, N. P. O Dowd ad G. A. Wbstr, Aalysis o Crp Crack Iitiatio ad Growth i Dirt Gomtris or 316H ad Carbo Magas Stls, submittd or publicatio,

17 Tabl 1: Matrial costats or th high itrog C-M stl at 360 C (or A A ad A, strss is i MPa ad tim i hours). Tmpratur Youg's modulus σ y A A A 360 C 190 GPa 240 MPa % Tabl 2: Matrial proprtis usd i NSW modl ad NSW-MOD modl i Figur 5 Strss coditios I / (0, 10) ( = 0, = 10) Pla strss μm Pla strai μm ~ θ r c 17

18 (a) 1 ~ σ θ ( ), 0.1 = 5 = 10 = 20 (b) Crack Pla strss θ θ ( ) ~ σ θ, 0.1 = 5 = 10 = 20 Crack Pla strai θ Figur 1: Dpdc o quivalt strss uctio ~ σ i HRR distributios o agl θ ad ; (a) pla strss ad (b) pla strai θ 18

19 (a) = 5 = 10 = 20 / 0.01 (b) Pla strss θ 1 = = 10 / 0.01 = 20 Pla strai θ Figur 2: Dpdc o ormalisd multiaxial ailur strai,, ormalisd by uiaxial ailur strai,, o agl θ ad ; (a) pla strss ad (b) pla strai. 19

20 90 θ or a max. θˆ 45 Pla strai Pla strss Figur 3: Dpdc o prdictd agl o maximum crp crack growth rat, crp xpot,, udr pla strss ad pla strai coditios. θˆ, o 20

21 (a) 10 8 a PE max /aps max (b) a PE /a PS or θ = Figur 4: Ratio btw prdictd crack growth rat udr pla strai ad pla strss coditios at th sam valu o C; (a) crack grows i th dirctio o th maximum valu max o CCG rat ( a ) ad (b) crack grows at a agl o 0. 21

22 (a) 10-1 NSW modl (pla strai) 10-2 da/dt, (mm/h) NSW modl (pla strss) data C, (J/m 2 h) (b) da/dt, (mm/h) NSW-MOD modl (pla strai) NSW-MOD modl (pla NSW-MOD strai, ˆ θ modl = 0 ) (pla strai) NSW-MOD modl (pla strss) data C, (J/m 2 h) Figur 5: Stady stat crp crack growth rat, da/dt, vrsus C or th C-M stl at 360 C; (a) xprimtal data ar compard to NSW modl (b) xprimtal data ar compard to NSW-MOD modl; ( = 18% ad = 10 ad othr proprtis ar i Tabl 1 ad Tabl 2) 22

23 P a W Figur 6: Fiit lmt msh or crp crack growth aalysis o a CT spcim. Loadig is applid at poit P, idicatd i th igur. 23

24 (a) = 50 % da/dt, (mm/h) NSW-MOD modl(θ = 0) Pla strss NSW modl Pla strss Elastic Elastic-plastic C, (J/m 2 h) (b) = 50 % NSW modl Pla strai da/dt, (mm/h) NSW-MOD modl(θ = 0) Pla strai Elastic Elastic-plastic C, (J/m 2 h) Figur 7: CCG rat prdictd rom FE aalysis vrsus C with NSW ad NSW-MOD modl udr (a) pla strss ad (b) pla strai coditios. ( = 50% ad = 10 ad othr proprtis ar i Tabl 1 ad Tabl 2) 24

25 (a) Δ Elastic plastic Pla strss Elastic C, (J/m 2 h) (b) Δ Figur 8: Elastic plastic Pla strai Elastic C, (J/m 2 h) Δ c Δ = Δ plottd agaist C or (a) pla strss ad (b) pla strai. 25

26 (a) 0.20 σ 0 = 600MPa 0.10 Q Pla strss = 50 % Pla strss,lastic plastic Pla strss,lastic C, (J/m 2 h) (b) 0.20 σ 0 = 600MPa 0.10 Q Pla strai = 50% Pla strai, lastic plastic Pla strai, lastic C, (J/m 2 h) Figur 9: Variatio o Q with C (a) pla strss ad (b) pla strai 26

27 Budd & Aisworth modl da/dt, (mm/h) NSW modl Pla strai NSW-MOD modl(θ = 0) Pla strai 10-7 F.E. aalysis C, (J/m 2 h) Figur 10: Compariso o CCG rat udr pla strai btw FE aalysis CCG modls or lastic-plastic-crp. ( = 50%, = 10 ad othr modl proprtis ar i Tabl 1 ad Tabl 2)) 27

28 (a) FE, FEM, Pla strss FE, FEM, Pla strai NSW, Pla strss NSW, Pla strai da/dt, (mm/h) Δ < C, (J/m 2 h) Bad or Exprimtal CT data = 18 % (b) FEM, FE, Pla strss FEM, FE, Pla strai NSW-MOD, xt_nsw, Pla strss(θ (θ = = 0) 0) NSW-MOD, xt_nsw, Pla strai(θ (θ = = 0) 0) da/dt, (mm/h) Δ < 0.5 Bad or Exprimtal CT data C, (J/m 2 h) = 18 % Figur 11: Compariso btw FE prdictio, xprimtal data ad aalytical modls or C-M stl; (a) NSW modl ad (b) NSW-MOD modl ( θˆ = 0 ). ( = 18%; = 10) 28

29 Δ Figur 12: Pla strss Pla strai C, (J/m 2 h) Δ c Δ = Δ plottd agaist C or FE prdictio ( = 18%; = 10) 29