J. Liu, Z. L. Zhang and C. Thaulow, A dynamic void growth model, Proceedings of the 11 th Int. Congress on Fracture (ICF11), Turin, Italy, March

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1 J. Lu, Z. L. Zhag ad C. Thaulow, A dyamc vod growth modl, Procdgs of th th It. Cogrss o Fractur (ICF), Tur, Italy, March -5, 5

2 A YNAMIC OI GROWTH MOEL J. Lu, Z. L. Zhag ad C. Thaulow INTEF Matrals ad Chmstry ah Norwga Uvrsty of cc ad Tchology, Trodhm, Norway ABTRACT I ths ar, th roblm of th dyamc growth of a sgl shrodal vod a owr law vsco-lastc matrx matral has b studd ad a w vod growth modl whch s caabl of dscrbg th dformato ad fractur of ductl matrals udr dyamc loadg codtos s rstd. Partcular attto s ao rtal ffct, ratsstvty ffct ad vod sha.. INTROUCTION Mcrovod uclato, growth ad coalscc ar th domatg mchasms of ductl fractur. For statc loadg, vod growth roblms hav b studd by McCltock [] ad Rc ad Tracy []. Gurso [6] xtdh vod growth modl by Rc ad Tracy to a rssur ddt costtutv quato whr ot oly th vod growth but also th ffct of vod growth o th lastc flow of a orous matral has b tak to cosdrato. Modfcatos to brg th Gurso modl to ralstc rdctos hav b mad by Ndlma ad Tvrgaard []. Zhag t al. [8] hav studh mcrovod coalscc roblm ad fouhat th lastc lmt load modl by Thomaso [] works vry wll as a coalscc crtro for th Gurso modl. A so-calld comlt Gurso modl whr o mrcal crtcal vod volum fracto at vod coalscc s dd has b troducd by Zhag t al. [9]. Wth th succssful alcato of ductl fractur modls, th d for dyamc ductl fractur modls caabl of dscrbg dformato ad fractur of structurs udr dyamc loadg codtos has b stadly crasg, scally th last t yars wh fractur mchacs was troduco th auto dustry. Although a larg body of work xsts o quas-statc mcrovod growth, vod growth udr dyamc loadg has draw rlatvly modst attto sc th org work of Carroll ad Holt [7]. Th ffct of dyamc loadg o th vod growth aturally volvs thr ascts: thrmal ffct, stra-rat sstvty ad rta ffct. arous studs hav show (Tog ad Ravchadra, Corts) that most cass th thrmal ffct s lss sgfcat comard wth th ffct of rta ad stra-rat sstvty. I ths ar w focus th roblm of dyamc growth of a sgl shrodal vod a owr law vsco-lastc cll lmt, whch has a cofocal shrodal sha. Partcular attto s ao rtal ffct, rat-sstvty ffct ad vod sha.. REPREENTATIE OLUME ELEMENT AN MATERIAL PROPERTIE I ths study, a shrodal (axsymmtrc) cavty wth sm-axs a (alog x 3 ) ad b (alog x ad x ), mbddd a cofocal shrodal rrstatv volum lmt wth sm-axs A (alog x 3 ) ad B (alog x ad x ) has b cosdrd, Fgur. I th rrstatv volum lmt, c a b A B, dots th focal dstac, ad th cctrcts of th r ad outr shrods, = c a, = c A. Th followg two gomtrcal aramtrs hav b usd, th vod volum fracto f = ab / AB ad vod asct rato

3 w = a / b. Th r ad outr cctrcts ca b calculatd trms of ths aramtrs. Hr w oly cosdr th cas wth rolat vod ( A B ). Fgur : Th rrstatv volum lmt modl I orthogoal shrodal coordats, th so - λ surfacs ar cofocal shrods wth sm-axs ad cctrcty dotd a, b ad rsctvly: a = c C o s h λ b = c h λ = c a = C o s h λ () I artcular, th surfac of th vod ah xtral boudary ar so- λ surfacs corrsodg to som valu λ ad λ rsctvly. Th ozro mtrc coffcts for ths systm of coordats ar gv by (Moo ad cr [3]): ( C o s h λ C o s h θ ) ( h λ h θ ) g λ λ = g θ θ = c = c g ϕ ϕ = c h λ h θ () Th xrsso of th lmtary volum shrodal coordat s: d v = c h λ g θ d λ d θ d ϕ λ λ ( ) 3 = c h λ h θ h λ h θ d λ d θ d ϕ (3) Throughout th aalyss, w assum that th matrx surroudg th vod rsods to mootoc strssg as a vsco-lastc sold wth flow rul: d ( s ) σ s σ N σ N N 3 s φ σ σ ε σ = ε =, ε = (4) ε, φ ( s ) = whr, σ s th strss tsor, s th dvatorc strsss tsor, d s th rat of dformato tsor, σ ad ε th ffctv Mss strss ad stra, rsctvly, σ s a flow strss, ad ε ar matral costats, N s th rat sstvty aramtr, N. Th macroscoc rat of dformato of th rrstatv volum lmt s dfd trms of th vlocty fld, v, o th surfac of th cll lmt, = ( v v ) d (5)

4 whr s th volum of th cll lmt, s ts outr surfac ad s th ut outward ormal o. Th avrag rat of work : d of th cll lmt s dfd as: : d = σ d d (6) Usg th rcl of vrtual work ad glctg th body forcs, th abov quato bcoms : d d v = σ v d ρ v d (7) Followg Molar ad Mrcr [5] ad usg th dfto of th dyamc macroscoc strss d v Σ = σ ρ x, whr s mass dsty, th rlato btw th mcroscoc ad macroscoc strsss of th volum lmt modl rads, : = : d d v ρ (8) Th macroscoc lastc dssato Φ ( ) s dfd by : Φ ( ) = I f φ ( d ) d v ρ (9) For ay vlocty fld, v, satsfyg codtos of homogous boudary stra rat, o ca comutr th ovrall dssato Φ ( ) corrsodg to th vlocty fld cosdrhrough umrcal tgrato ovr th volum. Wh Φ ( ) s obtad, th macroscoc yld locus ca b obtad by Φ = th quato ( ). 3. ELOCITY FIEL Followg Gåråu [3], th homogous stra rat tsor o th outsd surfac of th cll lmt has th followg form = 3 3, = () A two-tral vlocty fld has b trd currt study. ( ) ( ) v = v v () ( ) ( ) ( ) Whr both v ad v ar comrssbl flds, v satsfs a codto of homogous boudary stra rat o th outr surfac of th cll lmt wth a stra rat tsor T as follow, 3

5 ξ ( ) v = T x T = ξ B ξ A () ad ( ) v corrsods to a homogous stra o th tr doma, as follow ( ) ( ) v = T x (3) whr ξ = B A 3 3 (4) ffrt to Gologau s [4] xaso vlocty fld, ths vlocty fld v oly satsfs th homogous stra rat boudary codto o th outsd surfac of th cll lmt. Ths vlocty fld wll rduc to th classcal comrssbl xaso fld usd by Gurso th shrcal ad cyldrcal cass wh th llsod sha bcoms shrcal or cyldrcal rsctvly. 4. A YNAMIC OI GROWTH MOEL Basd o q (8), th lastc dssato fucto ca b wrtt as a summato of a quas-statc ad a dyamc art: ( ) ( ) ( ) Φ = Φ Φ ( : d ) = d d v ρ d (5) Usg th fld q () q (5) arasformg to shrodal coordats, w obta ( ) ( : d ) σ m ( m ) ε 4π Φ = 3 λ π π m 3 = ε ( sh sh ) sh sh c λ θ λ θ d λdθ dϕ AB λ d (6) 3 λ π π d v Φ ( ) = ρ 4π AB λ dt 3 c ( sh λ sh θ ) sh λ shθ dλdθ dϕ (7) whr ε s a fucto of vlocty fld v. Followg Gåråu s work, th stmat of th volum avrag q (6) rads (Gåråu [3]): m σ 4ξ Φ ( ) 4 F m ξ ( m f ) ε η d η (8) whr ξ ad ξ ar two costats, as follows: 4

6 ( ) ( ) ξ = ( ), ξ = 3 3 =, = 3 3 a d m = N (9) Aftr a lgthy drvato wth th hl of Mathmatca rogram th dyamc lastc ottal fucto ca b wrtt: 3 3 ( ) ρc ( F F F3 F4 F5 F6 ) Φ = () F = F F () F = F F () Whr F (=,, 3, 5, 4, 4, 6, 6) ar aalytcal fuctos of cctrcts of th r ad outr shrods, ad (Lu, []). Assumg th vod rmas shrodal durg th dformato, wh th vlocty fld v s scalzo q (), w obta th followg dffrtal quato: ( 3 3 ( )) f w = 3 f (3) ( ) ( 3 3 )( ) w ( 3 ) f Wth th sam rocdur, w obta th voluto quato of aramtr a as follow: a = ( ) ( ) ( ) ( ( )) ( ) ( 3 ) a (4) Basd o th comrssblty of th matrx matral, th voluto quato of vod volum fracto ca b drvd as follow: ( ) ( ) f = f T r (5) 5. ICUION AN CONCLUION Th vod growth modl s comard wth th FEM rsults (Lu, []). A xaml s show fgur. Th agrmt of th rst modl wth FEM rsults s qut satsfactory o a wd rag of stra rat xct that th voluto of vod sha ca ot b rdctd rcsly. Comaro othr modls th ltratur, ths modl ot oly cludh stra rat sstvty ad vod sha ffct as som rvous work, but also cludd quattatv trms for rtal ffct whch rlatd wth th vod sha ad sz. 5

7 Fgur. Comarso wth FEM rsults. f =., =.6, =-3 (/s), 33 =9 (/s). st: statc calculato; dy: dyamc calculato. 6. REFERENCE. J. Lu, Modlg of dyamc ductl fractur, Ph thss, Th Norwga Uvrsty of cc achology, 4.. J.R. Rc ad.m. Tracy, O th ductl largmt of vods traxal strss flds, J. Mch. Phys. olds, 7, -7, M. Gåråu, J.C. Mchl, P. uqut, A mcromchacal aroach of damag vscolastc matrals by voluto sz, sha ad dstrbuto of vods, Comut. Mthods Al. Mch. Egrg., 83, 3-46,. 4. Gologau M., Lblod J. B. ad vaux J., Aroxmat modls for ductl mtals cotag o-shrcal vods cas of axsymmtrc rolat llsodal cavts, J. Mch. Phys. olds, 4, , A. Molar ad. Mrcr, Mcromchacal modllg of orous matrals udr dyamc loadg, J. Mch. Phys. olds, 49, , 6. A.L. Gurso, Cotum thory of ductl rutur by vod uclato ad growth: Part I ---- Yld crtra ad flow ruls for orous ductl mda, J. Eg. Matr. ad Tch., Jauary, -5, M.M. Carroll ad A.C. Holt, tatc ad yamc Por-Collas Rlatos for uctl Porous Mtrals, J. Al. Phys., ol. 43, Arl, , Z. L. Zhag ad E. Nm, A w falur crtro for th Gurso- Tvrgaard dlatatoal costtutv modl, It. J. Fractur, 7, 3-334, Z. L. Zhag, C. Thaulow ad J. Ødgård, A comlt Gurso Modl basd aroach for ductl fractur, Eg. Fractur Mch., 67, 55-68,.. Thomaso PF, uctl fractur of mtals, Prgamo Prss, Oxford, 99.. A. Ndlma ad. Tvrgaard, A aalyss of ductl rutur otchd bars, J. Mchacs ad Physcs of olds, ol. 3, 46-49, F. A. McCltock, A crtro for ductl fractur by th growth of hols, J. al. Mch. 35,363, Moo P. ad cr. E., Fld Thory Hadbook, rgr-rlag, Brl, 97. 6