THEORETICAL AND NUMERICAL ISSUES ON DUCTILE FAILURE PREDICTION AN OVERVIEW

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1 Informatka w Tchnologii Matriałów Publihing Hou AKAPIT Vol. 0, 00, No. 4 THEORETICAL AN NUMERICAL ISSUES ON UCTILE FAILURE PREICTION AN OVERVIEW JOSE MANUEL E ALMEIA CESAR E SA*, FILIPE XAVIER COSTA ANRAE, FRANCISCO MANUEL ANRAE PIRES artmnt of Mchanical Enginring EMc Facult of Enginring, Univrit of Porto FEUP Rua r. Robrto Fria Porto, Portugal *Corronding author: cara@f.u.t Abtract Th main goal of thi ar i to giv a gnral ovrviw of om of th rcnt advanc accomlihd in th dcrition of ductil damag, both from a thortical and numrical oint of viw. To tart with, th claical local thor with rgard to th thrmodnamic of irrvribl roc i rviwd whr a gnral lato-latic damag modl i tablihd. It i alo highlightd th aumtion and limitation bhind th claical thor whn th contitutiv quation ar obtaind from th olution of a contraind maximiation roblm. Rcnt advanc on th non-local modlling of ductil damag ar alo addrd whr w hd om light on th rincil and conqunc of non-localit in lato-latic damag modl. Th iu rgarding th fficint numrical imlmntation of both local and non-local thori ar alo dicud whr cial attntion i dvotd to th imlmntation of non-local modl. In articular, a novl comutational tratg, uitabl for imlmntation in commrcial rogram, i rntd for th xlicit finit lmnt cod LS-YNA in dtail. A FORTRAN cod xcrt i givn in which th main t for th imlmntation of th modl ar chmaticall dictd. Th ffctivn of th non-local modl i ad through th imulation of an aximmtric cimn and a ht mtal forming roc. It i hown that in both ca th non-local numrical tratg i abl to diminih th athological mh dndnc inhrntl rnt in local lato-latic damag modl. K word: ductil failur, damag, lato-laticit, non-local modl, LS-YNA ur-dfind ma. INTROUCTION Mtal forming roc ar gnrall charactrizd b th fact that th involv ignificant chang in ha of a art, in th olid tat, through th ma flow undr larg latic dformation, controlld b contact and friction b man of tool. Thir indu imortanc com from th larg varit of tructural art manufacturd b th roc in a larg divrit of indu ctor (automotiv, aronautic, conumtion good, tc.). Th tat and gomtr of th final art dnd on variou factor: load condition, gomtr of matric, tool, lubrication of contact zon, gomtr of r-form and ma forming limit, to nam onl th mot imortant on. Thrfor, th oibilit of matring all th factor i dciiv for dvlomnt and otimization of mtal forming roc bad on larg latic dformation of mtal. In thi act numrical tool bad mainl on th Finit Elmnt Mthod hav bcom indinabl. Th lat twnt ar hav witnd a grat dvlomnt and volution on both thortical and numrical abiliti to dal with th hnomna. To a larg xtnt it i oibl toda to rdict dformation mchanim, tr, forc, ma ror ISSN

2 ti chang, th influnc of tool gomtr, lubrication condition, tc. Th grat dvlomnt in comuting faciliti mad wa for imortant thortical advanc on th tratmnt of larg dformation of inlatic ma which could not b ad bfor, which in turn romtd th dvlomnt of nw and fatr algorithm to olv incraingl comlx roblm. Nowada thr xit owrful commrcial cod which ar abl to imulat comlx forming roc and giv vr uful information for thir imrovmnt and otimization. Howvr, on act that nd till om rogr and in which commrcial cod commonl fail to giv th adquat ron i th rdiction of ma formabilit, undr comlx loading ath. Nvrthl, thi i a dciiv fatur in ordr to b abl to rdict dfctiv art in roc lik forging or to dcrib roc in which fractur i a art of th roc itlf a in ht blanking or mtal cutting. In larg dformation of mtal, whn latic dformation rach a thrhold lvl, which ma dnd on th loading, th fatigu limit and th ultimat tr, a ductil damag roc ma occur concomitantl with th latic dformation du to th nuclation, growth and coalcnc of micro-void. Somtim tho cod includ a otriori fractur indicator that ar not alwa uitabl for all dformation ath. In th imulation of bulk forming roc, it i common to utili fractur critria bad on th comutational valuation of function of om tat variabl and that dnd on th dformation tor. Tho critria ma b, broadl, claifid in two grou: on bad on micromchanic which includ a rimr tat variabl th total latic work (Frudnthal, 950), th maximum latic har work or th quivalnt latic train (atko, 966); and anothr bad on th growth of dfct which includ gomtric act (McClintock, 968; Ric & Trac, 969), growth mchanim, dndnt on rincial tr (Cockcroft & Latham, 968) or hdrotatic rur (Norri t al., 978; Atkin, 98), or ma bhaviour couling (Oan t al., 978; Tai & Yang, 987; Lmaitr, 985). In fact, in man ca tho critria do not tak into account th fact that th damag localization it ma b awa from th zon whr th maximum quivalnt latic dformation i locatd or that damag volution ma b diffrnt for comrion or traction tr tat or diffrnt triaxialit tr tat. For th roc, modl bad on Continuou amag Mchanic (CM) (Lmaitr, 996) ma offr a bttr undrtanding of th hical hnomnon and bcom an imortant tool in th formabilit and fractur rdiction. Mor rcntl, a lot of rarch i bing focud on multical modl which ma giv an imortant inight on how th damag mchanim at lowr cal ma manift at th macro lvl and affct th dfinition of th hnomnological law which ar uuall adotd at thi lvl. Nvrthl, du to th hug amount of comutr caabiliti ndd, in mot ca nonxitnt nowada, th alication of th multi-cal modl in ractical trm ar till far from bing a ralit and for th momnt indutr will hav to rl on th alication of hnomnological modl at th macro cal. Th oftning inducd b th tandard imlmntation of tho damag modl in finit lmnt olution lad to mh and orintation dndnc a th localization ffct ar not corrctl dalt with b mh rfinmnt. On of th olution for thi roblm i th u non-local modl (Pijaudir- Cabot & Bažant, 987; Bort & Mühlhau, 99; Vr t al., 995; Strömbrg & Ritinmaa, 996; Polizzotto t al., 998; Borino t al., 999; Jirák & Rolhovn, 003; Car d Sa t al., 006; Car d Sa & Zhng, 007; Jirák, 007; Andrad t al., 009a). Non-local modl includ om lngth cal information, rlatd with localization ffct du to microtructur htrognit, in ordr to avrag an intrnal variabl aociatd with th diiativ roc. Two t of modl ar uuall aumd for thi uro: intgral and gradint modl. In thi work, an intgral non-local modl i adotd, in which th intrnal diiativ non-local variabl i th damag. Th modl i imlmntd, b man of ur-dfind ubroutin, in th commrcial cod LS-YNA and om xaml ar ud to a it rformanc.. LOCAL CONSTITUTIVE MOEL On of th mor wll tablihd hnomnological modl for ductil damag in mtal i bad on th work of Lmaitr (Lmaitr, 996) in which th damag variabl,, i a maur of th dicontinuiti dnit r unit ara in ach urfac of a rrntativ volum lmnt at th mo-cal. Thrfor, damag i a calar variabl and th modl i onl valid for iotroic damag in which a homognou ditribution of micro-caviti i aumd. In ordr to avoid a formulation aociatd to ach t of dfct or damag growth mchanim, th rin- 80

3 cil of train quivalnc i otulatd b which i aumd that th contitutiv law of th damagd ma i th am a th on aumd for th undamagd ma, but in which th tr tnor i ubtitutd b th ffctiv tr tnor dfind a: ~ () whr i th Cauch tr tnor. Lmaitr modl i bad on th thor of CM and thrmodnamic of irrvribl roc which trongl coul lato-laticit and damag at th contitutiv lvl. In thi framwork, two otntial ar introducd: a tat otntial, th Hlmholtz fr nrg otntial, which i a function of th tat variabl and from which i oibl to dfin thir aociatd variabl; and a diiation otntial aociatd to th volution of th tat variabl charactriing th diiativ roc. A th two diiativ roc aociatd to damag and laticit rult from diffrnt hnomna at th micro-cal, it i aumd that th tat otntial can b lit in two trm a,r ε () In thi xrion, whr kinmatic hardning and tmratur ffct ar nglctd, rrnt th cific latic otntial of th damagd ma and i th cific latic otntial aociatd to hardning. In quation (), ε i th latic train tnor and R i th iotroic hardning variabl From th otntial, it i oibl to driv th tat variabl ; ; Y R (3) whr i th thrmodnamic forc aociatd with th iotroic hardning variabl, Y i thrmodnamic forc aociatd with damag and i th dnit of th ma. A a conqunc of th train quivalnc rincil, th cific latic otntial of th damagd ma i dfind b whr (, ) : : i th fourth ordr laticit tnor (4) Th Cauch tr ha thn th following xrion in accordanc to quation () and th train quivalnc rincil : (5) Th aociatd variabl to damag i thn writtn a : : (6) Y or, quivalntl, a q Y (7) 6G K whr i th hdrotatic tr, q i th undamagd quivalnt von Mi tr and G and K ar rctivl th har and th bulk moduli... Potntial of diiation for laticit and damag For aociatd laticit, th otntial of diiation i rlatd to th articular ild (limit) function adotd. For th ca of th von Mi law, could with th rincil of train quivalnc, th otntial of latic diiation tak th following form: F q R 3 : R (8) whr σ i th ild tr function and th dviatoric tr tnor. Likwi, it i oibl to dfin a otntial of diiation for ductil damag in th form of th limit function: r Y Fd ( Y ) H acc d (9) r whr r i a ma aramtr rrnting th nrg trngth of damag and i a nondimnional ma aramtr which i a function of th tmratur. Th Haviid function H of th quivalnt latic train acc, who incluion i jutifid in th nxt ction, i dfind from a critical valu d which triggr th ont of damag. 8

4 .. Stat variabl volution From th cond rincil of th thrmodnamic of irrvribl roc, th Claui-uhm inqualit imli that th nrg of th diiativ roc of laticit and damag mut b qual or gratr than zro:,, Y ε R Y 0 (0) : Th volution of th intrnal variabl ma b obtaind from th gnraliation of th rincil of maximum latic diiation for could laticit and damag. Conquntl, admitting that, for a crtain lvl of latic dformation ε and a crtain damag valu, among all th oibl tat tr that atif both th latic and damag limit function, th olution will b obtaind b maximiing that nrg. In Lmaitr modl, it i imlid that th rtriction ar imod, rorting to onl on Lagrang multilir. Thi i a imlification of th modl that conquntl do not allow th two diiativ roc, laticit and damag, to volv aratl. Nvrthl, thi ma b a valid aumtion for ductil damag whr thi i almot alwa th ca. That i alo th raon wh th Haviid function i includd in th damag limit function in ordr to rvnt damag volution rior to a thrhold valu of latic dformation. Th Lagrangian aociatd to thi rincil i thn tatd a: L (,, Y, ) : R Y F (, ) F ( Y ) () whr i th Lagrang multilir. Th volution law for th tat intrnal variabl ma thn b obtaind a: L F 3 0 () q d L F 0 R R (3) L Fd 0 ( Y ) ( Y ) Y H ( acc d ) (4) r Rcurring to th rviou quation and, for th ca of th von Mi law, it i oibl to xr th quivalnt latic train in th following form: : 3 acc 3 : q (5) which rmit to rtat th rviou quation that dfin th volution of intrnal variabl, for th ca of could laticit and damag, a: 3 acc (6) q R (7) acc Y acc H acc d (8) r 3. NON-LOCAL MOELLING OF UCTILE AMAGE At thi oint, it i imortant to rmark that th contitutiv framwork rntd in th lat ction fall within th o-calld local aroach. It i widl known that th tandard local thor uffr from a athological dndnc on atial dicrtiation whn undr oftning rgim. Within a tical finit lmnt framwork, thi corrond to having uriou dndnc on th mh rfinmnt ud to imulat th fracturing roc. Thi lad to unralitic numrical olution inc th fracturing zon tnd to bcom infinitl mall a th mh i rfind. Aiming to ovrcom thi iu, th non-local aroach ha bn addrd b vral author in th litratur (Pijaudir-Cabot & Bažant, 987; Bort & Mühlhau, 99; Vr t al., 995; Strömbrg & Ritinmaa, 996; Polizzotto t al., 998; Borino t al., 999; Jirák & Rolhovn, 003; Car d Sa t al., 006; Car d Sa & Zhng, 007; Jirák, 007; Andrad t al., 009a; among othr). In th nonlocal thor, th contitutiv bhaviour i no longr indndnt of nighbour ma oint. In fact, an intrinic lngth i incororatd in th continuum thor, ithr b man of an intgral or a gradint aroach. A a conqunc, a diffuiv ffct i introducd into th modl, rading th fracturing zon ovr a finit ara. Ovr th at two and half dcad, vral author hav addrd non-localit a an ffctiv tool to rvnt uriou localiation and to liminat athological mh dndnc. Aftr th ionring work of Pijaudir-Cabot and Bažant (987), who adotd a non-local formulation of intgral-t for 8

5 th mh-innitiv dcrition of quai-brittl ma, vral rlvant contribution on th ubjct hav mrgd. Initiall, mot of thm hav concntratd on th modlling of ma lik concrt and oil (Pijaudir-Cabot & Bažant, 987; Vr t al., 995; Prling t al., 996; Bnvnuti & Borino, 00; Borino t al., 003; among othr). Howvr, ductil ma hav alo, mor rcntl, bn focu of attntion in man ar (Gr t al., 003; Car d Sa t al., 006; Mdiavilla t al., 006; Ricci & Brünig, 007; Andrad t al., 009b; Andrad t al., 00). In thi ction, w brifl rvi th gnral conct of non-local modlling b adoting th intgral aroach. Som of th rcnt advanc accomlihd within th framwork of non-local modl of intgral-t ar alo highlightd and dicud. Finall, a gnral non-local modl for lato-latic ductil damag i rntd. 3.. Gnral act of non-localit In a gnral n, a non-local variabl can b obtaind b avraging th aociatd local quantit through th following intgral g ( ( x, ) g( ) dv ( ) (9) V whr ( x, ) i an avraging orator givn b ( x, ) ( x, ) (0) ( x, ) dv ( ) V whr ( x, ) i a wight function, hr, conidrd to b x ( x, ) () Th intgral of Equation (9) incororat a diffuiv ffct in th contitutiv modl, which rvnt th chon non-local variabl to urioul locali into a narrow band a th atial dicrtiation bcom finr. Thrfor, a dmontratd b Andrad t al. (009b) and Car d Sa t al. (00), onl th variabl that influnc th oftning rgim hould b rgularid. On imortant act of th non-local thor i th aumtion of an avraging orator which i indndnt of th hitor of dformation. A dicud b Andrad t al. (00), uch hothi i articularl imortant from a comutational oint of r viw inc th numrical imlmntation of th nonlocal modl i much imlr if comard to th ca whr ( x, ) i a function of dformation. A a dirct conqunc, th rat of th avraging orator vanih, i.., ( x, ) ( x, ) 0 t () Thu, th rat of th chon non-local variabl i iml givn b g ( ( x, ) g ( ) dv ( ) (3) V At thi oint, om intrrtation and concluion can b withdrawn from th abov tatmnt. For intanc, if th hitor of dformation ha no influnc on th non-local avraging orator, it man that th intrinic lngth rmain contant a th bod undrgo dformation. Thu, th iz of th fracturing ara i inhrntl aumd to b contant and dictatd b r. Howvr, thr i no xrimntal vidnc uorting thi tatmnt. Aarntl, a mor ralitic modlling would conidr an volving intrinic lngth, for which r would b a function of othr variabl that ar known to ignificantl influnc th diiativ fracturing roc,.g., th latic train, th tr tat, th intrnal dgradation or th hitor of dformation itlf. On th othr hand, th conidration of a noncontant intrinic lngth a wll a an avraging orator which i dndnt on th hitor of dformation i till vr challnging, dmanding furthr dvlomnt both on thortical and comutational oint of viw of th non-local thor. Nonthl, th rnt non-local thor ha alrad bn uccfull mlod in th tak of rducing athological mh dndnc and rovid mor ound rult than th tandard local thor. It i alo imortant to mntion that th aumtion of a dformation-indndnt avraging orator facilitat th xtnion of th non-local thor to finit train. Thi oint ha bn dicud in mor dtail b Andrad t al. (00), whr diffrnt avraging tratgi at finit train hav bn rntd. Thrfor, in th following, w will limit ourlv to how th thor in mall train in ordr to k notation iml. Howvr, th xtnion to th finit train domain can b traightforwardl don b mloing th am rocdur adotd b Andrad t al. (00). 83

6 3.. Non-local lato-latic damag modl In th work of Andrad t al. (009b) and Car d Sa t al. (00), diffrnt non-local variabl hav bn chon to nhanc th local continuum. Th rult hav hown that damag i a good candidat, ciall du to th fact that it highl influnc th oftning rgim. Thrfor, w will conidr hrin th choic of damag a th non-local variabl for th dfinition of th non-local modl. Th latic-damag contitutiv rlation can thn b xrd a ( ( : ( ) ( x (4) Th ild function, th latic flow rul and th volution of iotroic hardning ar alo rwrittn, rctivl givn b q( F ( ( R( ) (5) ( and 3 ( ( ( (6) ( q( R ( ( (7) In contrat to th modl of Sction, all contitutiv variabl hav bn xlicitl writtn a function of thir atial location, gnricall rrntd b x. Thi tm from th fact that th non-local formulation ild on an nrichd continuum thor for which th oition of th ma oint in th bod i ntial for th dtrmination of it contitutiv bhaviour. Morovr, inc non-local damag, x, i now th actual tat variabl, it rat hould la th rol of damag volution. From quation () and (3), w know that whr x ( x, ξ) ( ξ) dv ( ξ) (8) V Y ( ξ) ( ξ) ( ξ) (9) ( ξ) r whr ξ corrond to th global coordinat of a givn urrounding oint in th vicinit of x. It i worth mntioning that th Haviid function ha not bn conidrd in quation (9). Howvr, it incluion in th contitutiv modl i rathr traightforward. Finall, th Kuhn-Tuckr condition ( ( 0, F ( 0 and ( F ( 0 ) mut b fulfilld, lading to th final contitutiv modl, ummarid in tabl. It i imortant to rmark that, inc vr ma oint dnd on it nighbourhood, th Kuhn-Tuckr condition mut, in ractic, imultanoul hold for all oint of th whol bod. A a mattr of fact, th intgral charactr of th nonlocal modl ild on a ubtantiall mor comlicatd contitutiv roblm for which analtical olution ar difficult to obtain. A tratg that numricall olv th non-local ma roblm through a global intgration algorithm ha bn addrd b Andrad t al. (00). In Sction 4, w will rnt an altrnativ imlmntation tratg uitabl for xlicit formulation. 4. NUMERICAL IMPLEMENTATION 4.. Local damag modl On of th oibl modification for Lmaitr damag modl i achivd b dirgarding th ffct of kinmatic hardning. A hown b d Souza Nto (00), th conidration onl of iotroic hardning lad to a rmarkabl iml and fficint numrical algorithm. Th algorithm for th numrical intgration of th contitutiv modl tart with th dfinition of th tical latic tat. Notic that th contitutiv bhaviour i mant to b locall comutd at vr ma oint within a gnric tim intrval t, t n n, whr th contitutiv variabl n, n, R n and n ar known a riori. Th goal of th algorithm i to find th udatd valu of n, n, R n and n for a givn train incrmnt within t, t n n. In a tical finit lmnt framwork, th ma oint corrond to th Gau intgration oint. Th latic train tnor i givn b n n (30) from which it i oibl to comut th latic tr tnor, writtn a. ~ σε n : n (3) 84

7 Tabl. Claical non-local damag modl with (i) Strain tnor additiv lit ( ( ( (ii) Elatic law ( ( ( : ( q( (iii) Yild critrion F ( ( R( ) ( (iv) Platic flow (v) Evolution of iotroic hardning (vi) Evolution of damag (vii) Kuhn-Tuckr condition ( 3 R ( ( ( ( ( q( V whr Y ( ξ) ( ξ) ( ξ) ( ξ) r x ( x, ξ) ( ξ) dv ( ξ) ( 0 ; F ( 0 ; ( F ( 0 Conidring now th dviatoric/hdrotatic lit of th tr tnor, th latic tr tnor i altrnativl writtn a ~ ~ ~ n n σn I (3) whr ~ n and ~ n ar th ffctiv dviatoric and hdrotatic tr, rctivl givn b whr ~ n d n ; d n d n d ; n v n G ε K, (33) vn vn v, (34) in which th train dviator and th volumtric train hav bn dnotd, rctivl, b d and v. Finall, with quation (33) it i oibl to comut th latic von Mi quivalnt tr from th ffctiv dviatoric tr a q~ 3 ~ n n (35) whr q n i ncar for th ror valuation of th ild function and to chck whthr th udoincrmnt i latic or latic. Following tandard rocdur of latic rdictor/rturn maing chm, Lmaitr contitutiv modl can b writtn in it (udo-)tim-dicrtid vrion b th following tm of quation n n 3 n n q n R n R n Y n n n n r q n R n 0 n (36) whr n, R n, and n ar th unknown of th incrmntal initial boundar valu contitutiv roblm. It i imortant to rmark that th lat quation of th tm i th conitnc condition which, in ractic, act a a contraint for th contitutiv roblm. Th abov tm of quation i unattractiv from th numrical oint of viw du to th high comutational burdn if comard to imlr latolatic modl (.g. von Mi iotroic laticit). Howvr, b rforming om rlativl traightforward oration, it i oibl to rduc th tm of quation (36) to a ingl calar non-linar quation, which can b writtn a F( )( ~ ) 3G ( n q n ) ( q ~ ) 3G n Y ( ) r (37) whr i now th onl unknown. In fact, th calar quation abov i much imlr to b workd out, ignificantl rducing th comutational cot r Gau oint. Th drivation of quation (37) i omittd hr for convninc and th radr i rfrrd to d Souza Nto (00) for furthr dtail. Th comlt tr udat algorithm for th fficint numrical intgration of Lmaitr imlifid modl b man of a full imlicit latic rdictor/rturn maing chm (Simo & Hugh, 998) i ummarid in tabl in udo-cod format. 4.. Non-local damag modl Th numrical imlmntation of non-local modl of intgral-t ha bn uuall conidrd a on of th main diadvantag of thi kind of rgularid modl. Th main raon for that i th rnc of th intgral in th contitutiv volution quation that rvnt th modl to hav th coni- 85

8 tnc condition fulfilld locall. Som imlicit gradint-dndnt lato-latic modl rood in th litratur hav tackld th roblm b adding th non-local variabl in th global tructural roblm and thn atifing th conitnc condition onl in a wak n (Prling t al., 996; Gr t al., 003). Adoting th intgral-t thor, othr author hav mlod a olution chm whr all intgration oint ar olvd imultanoul (Strömbrg & Ritinmaa, 996; Andrad t al., 009a, 009c, 00; Car d Sa, 00). In th articular ca of lato-latic damag modl, th work of Car d Sa t al. (00) and Andrad t al. (009a, 009c, 00) hav rntd dtaild algorithm for th numrical imlmntation of non-local modl of intgral-t. Th tratg wa bad on a global vrion of th tical rturn-maing chm, for which all th intgration oint ar intgratd imultanoul. Thi aur th Tabl. Str udat algorithm for Lmaitr imlifid (local) modl. Howvr, for it imlmntation, on nd to hav acc to all intgration oint of th finit lmnt mh in ordr to rocd with th global intgration chm. In gnral, thi fact do not rrnt a roblm if on ha full acc to th ourc cod of th finit lmnt rogram, or, at lat, acc to th algorithm that ambl th intrnal forc vctor. Unfortunatl, mot commrcial FE cod do not allow th ur to do th ncar modification inc th rtrict th ur-dfind contitutiv modl to b comutd for a gnric Gau oint onl. Th global intgration aroach can b avoidd b ubtituting it b altrnativ formulation that onl aroximat th non-local thor. For xaml, Tvrgaard and Ndlman (995) hav rood a non-local modl intndd for th dcrition of ductil ma whr th hav aroximatd th rat of th non-local variabl (in thir ca, th oroit f ) b f tn tn K f nl (38) (i) fin latic tat n n ; ~ n G ε d n R n R n ; ~ n K ; q~ 3 ~ v n n n (ii) Chck latic admiibilit IF ~ qn Rn 0 THEN SET n and EXIT n ENIF (iii) Solv th ingl ridual quation ( q ~ n ) Y F G n )( q ~ ( ) 3 ( n ) 3G r for with th Nwton-Rahon mthod whr R ~ n n Y 6G K (iv) Udat tr and intrnal variabl 3G n q n R n qn ~ n ; n ~ qn n n I ; n n ~ n ; n n q n n R G n 3 vn I n ;. (v) EXIT conitnc condition to b fulfilld at all ma oint at th nd of a givn dformation t. Within an lato-latic damag framwork, thi algorithm ha rovd vr fficint and rlativl iml to imlmnt in xiting finit lmnt cod. whr f and f ar, rctivl, th local and nl non-local rat of th oroit and K i a nalt factor givn b K nl tn f (39) tn f tn whr f n and f t ar th local and non-local rat of th oroit at th lat convrgd incrmntal t. A a mattr of fact, thi mthodolog can b intrrtd a an "xlicit non-local intgration" inc onl information of th rviou t i ud for th comutation of th non-local quantit. Th main advantag of thi tratg i that th tat udat rocdur k it tical local format. On th othr hand, a major rtriction ari inc th timt mut b kt mall nough to avoid intabiliti in th olution. In xlicit cod, howvr, th timt iz ha ncaril to b mall in ordr to rndr tabl olution whr a critical tim-t i dictatd b th iz of th mallt lmnt of th mh. Thu, in thi ca, th u of an aroximat xrion lik th on in quation (38) i vr attractiv from a comutational oint of viw inc th trudat can b don locall and th tim-t will b ncaril kt mall ithr wa. Following th ida, w hav imlmntd a imilar non-local tratg in th commrcial FE rogram LS-YNA. 86

9 Th imlmntation i dcribd in dtail in th following. To tart with, an xrion quivalnt to th on in quation (38) can b writtn for th non-local modl dcribd in Sction 3. Thrfor, non-local damag i now givn b t n K nl tn t t n n tn (40) It i imortant to rmark that th rlation abov mut b conitntl introducd in th local tr udat algorithm. Whn th volution of th chon non-local variabl i xlicitl givn in th numrical imlmntation of th local contitutiv modl, th modification i rathr traightforward. Howvr, whn algbraic maniulation hav bn rvioul don, on mut tak cial car in ordr to corrctl add th nalt factor in th ridual quation. For th ca of th local damag algorithm rntd in Sction 4., th ingl ridual quation mut b modifid to F( )( ~ ) 3G ( n q n ) K nl ( q ~ 3G n ) Y r (4) In addition, th rat of damag mut b tord aftr convrgnc ha bn achivd with th local ma intgration algorithm. No furthr modification ar ncar in th tr udat algorithm. Howvr, on till nd to comut th nonlocal countrart of th damag rat, which rquir having th valu of th local damag rat of th urrounding oint. In fact, thi t ma till b vr comlx dnding on th commrcial FE rogram ud. In th ca of LS-YNA, th acc to information of othr lmnt i availabl, ciall bcau th oftwar allow th ur to crat vctorial imlmntation. Thrfor, th arra containing th tr, latic train, and th hitor variabl of th lmnt of th mh ar radil availabl in th ur ubroutin urmathn (in th ca of olid) and hav th dimnion of nlq, which i a aramtr dndnt on th machin architctur. Whnvr th oftwar ntr th ubroutin urmathn, th information of a givn grou of lmnt (th iz of th grou dnd on th aramtr nlq) i ulid. Th ur ubroutin i thn calld a man tim a ncar until th ma intgration algorithm of all lmnt of that articular grou ha bn carrid out. For intanc, if nlq i dfind to b 96 and th mh ha 000 olid lmnt, th ubroutin urmathn will b accd tim b LS-YNA, whr at ach tim th arra igx, and hv will tor, rctivl, th tr, th latic train and th hitor variabl of a diffrnt grou of 96 lmnt. Thi cod tructur can b convnintl ud for th imlmntation of th non-local tratg inc w nd th rat of th urrounding oint onl at th rviou convrgd tim. Thrfor, a gnral rocdur can b adotd a follow. In th firt incrmntal t, all th information ncar (nodal coordinat, connctiviti, tc.) for th comutation of th non-local factor, ij, can b rtrivd and tord in ur-dfind arra. Manwhil, th ma intgration i don locall for all lmnt of th mh and, aftr convrgnc of th local ma algorithm, th rat of chon non-local variabl (in th rnt ca, damag) i tord in a ur-dfind arra. Thi information will b ncar in th ubqunt t whn th nonlocal avrag for th comutation of th nalt factor, K nl, will b udatd. In turn, th nalt factor i inut a an argumnt of th local ma ur routin. In ordr to avoid numrical intabiliti, th nalt factor i t to.0 if th local rat of damag i l than a givn rcribd valu (.g. 0 ). A critical t in th rocdur dcribd abov i th corrct torag of th ur-dfind arra. To rvnt data lo and to aur that th information comutd in th rviou t will b availabl in th nxt on, th FORTRAN77 av tatmnt hould b iud at th bginning of th ubroutin urmathn for th ur-dfind arra. In ordr to know th currnt t (or ccl, a it i calld in LS-YNA), th following common block nd to b includd at th bginning of th ubroutin urmathn: common/bk06/idmm,iadd,ifil, maxiz,nccl,tim(,30) Th variabl nccl i an intgr that tor th numbr of th currnt t. For th dtrmination of th non-local avraging factor, a wll a th lmnt connctiviti and th nodal coordinat, it i in gnral air to handl with th xtrnal I, which can b ail rtrivd through th function lqfinv(id_intrnal,it). 87

10 Tabl 3. FORTRAN cod xcrt for th imlmntation of th non-local modl in LS-YNA for olid lmnt ubroutin urmathn (...) common/bk06/idmm,iadd,ifil,maxiz, nccl, tim(,30) dimnion btaij(...), dot_nlv(mxlm,) dimnion connct(mxlm,8), coord(mxnod,3) C Sav data in arra for th nxt t av btaij, dot_nlv, connct, coord if(nccl.q..and.nnm.q.0)thn C Initiali rat for nalt factor calculation do ilm=,nlm dot_nlv(ilm,)=0.0! local rat dot_nlv(ilm,)=0.0! non-local rat nddo ndif if(nccl.q.)thn C Gt connctiviti (.g. linar hxahdron) do i=lft,llt ilm=lqfinv(nnm+i,) connct(ilm,)=lqfinv(ix(i),) connct(ilm,)=lqfinv(ix(i),) connct(ilm,3)=lqfinv(ix3(i),) connct(ilm,4)=lqfinv(ix4(i),) connct(ilm,5)=lqfinv(ix5(i),) connct(ilm,6)=lqfinv(ix6(i),) connct(ilm,7)=lqfinv(ix7(i),) connct(ilm,8)=lqfinv(ix8(i),) nddo C Gt coordinat call gt_coordinat (..., coord, r_mm(dm_,...) ndif if(nccl.q..and.nnm.q.0)thn call comut_non-local_factor (..., btaij, connct, coord,...) ndif if(nccl.g..and.nnm.q.0)thn call comut_non-local_rat (..., btaij, dot_nlv,...) ndif do 90 i=lft,llt C Gt global lmnt I ilm=lqfinv(nnm+i,) C Comut nalt factor nalt_factor=dot_nlv(ilm,)/ dot_nlv(ilm,) C Call local ur ma routin 4 call umat4 (..., nalt_factor, rat_local,...) C Stor local rat of damag dot_nlv(ilm,)=rat_local 90 continu nd ubroutin urmathn A FORTRAN cod xcrt chmaticall dicting th imlmntation rocdur i givn in tabl 3 for rfrnc. Not that onl th main t for olid lmnt hav bn rntd. Nvrthl, th am conct can b traightforwardl xtndd for hll lmnt (in ubroutin urmat). 5. NUMERICAL EXAMPLES 5.. Aximmtric cimn In thi firt xaml, th anali of an aximmtric cimn ( figur ) i carrid out in ordr to a th rgulariing ffct of th rood numrical imlmntation. A imilar cimn ha bn invtigatd b th author in rviou contribution (Andrad t al., 009c, 00) uing an imlicit in-hou finit lmnt cod. Thr mh rfinmnt uing linar quadrilatral lmnt with rducd intgration ( figur ) hav bn conidrd in ordr to catur th athological mh dndnc. Th ma rorti adotd ar givn in tabl 4. Tabl 4. Ma rorti for th aximmtric xaml Prort Young' modulu Poion' ratio amag xonnt amag dnominator Initial ild tr Yild hardning curv Non-local charactritic lngth Valu E = 0000 MPa ν = 0.3 =.0 r =.5 MPa σ o = MPa R MPa o mm It i worth mntioning that th rgulariing ffct of th non-local thor can onl b achivd if th charactritic lngth i larg nough to an at lat om lmnt. Thrfor, in th rnt ca, th valu of r. 0 mm ha bn chon aiming to fulfil thi condition rathr than bad on xrimntal maurmnt. Nonthl, thi conidration i raonabl nough to a th non-local algorithm. r 88

11 INFORMATYKA W TECHNOLOGII MATERIAŁÓW Th cimn i loadd at it nding dg with a low vlocit that grow linarl with tim, thr- for, minimiing th ffct of inrtia. Figur how th damag contour for th local ca. W notic that damag tnd to concntrat at th criti- cal lmnt manwhil it numrical valu alo tnd to incra. Convrl, th non-local figur 3, it i oibl to notic that both th damaging ara and th damag valu ar kt narl contant uon mh rfinmnt. olution rovid a diffrnt bhaviour. Obrving Fig.. amag contour for th local ca. Fig. 3. amag contour for th non-local ca. Fig.. Gomtr and diffrnt mh rfinmnt for th axi- mmtric cimn. Fig. 4. Evolution of damag and nalt factor at th critical oint of th fint mh (non-local volution of th damag variabl and of th non-local nalt factor, K nl, ar lottd at th critical oint of th fint mh. At th initial tag, whn damag i tilll low, th nalt factor i kt contant and qual to th unit. Howvr, olution). In figur 4, th 89

12 Thi xaml i bad on th anali carrid out b d Souza Nto t al. (994). It conit of th imulation of a ht mtal forming roc ( figur 5). Onl on quartr of th gomtr ha bn imulatd. it th fact that thi kind of roc uuall rquir aniotroic contitutiv law for an accurat rdiction, th u of an iotroic modl i hlful and ufficint to dmontrat how th ro- od non-local th athological mh dndnc iu. Nonthl, it i worth mntioning that th am non-local mthodolog can b xtndd to account for aniotro. Th ht i conidrd to b mad of an aluminium allo, whil th unch and th di ar rgardd a rfct rigid ma. All th ma rorti mlod ar givn in tabl 5. iffrnt mh rfinmnt uing linar hll lmnt with 5 imlmntation can ignificantl allviat intgration oint hav bn ud to modl th ht a hown in figur 6. A gnral urfac-to-urfac contact algorithm availabl in LS-YNA ha bn mlod to conidr th frictional contact btwn th art. Both tatic and dnamic friction coffi- cint wr aumd to b qual to Th d- namic ffct hav bn minimid b aling a linarl growing low vlocit on th unch. Similar to th lat xaml, a charactritic lngth thatt i ufficintl larg to gt th influnc of nough ur- rounding lmnt ha bn adotd. Tabl 5. Ma rorti for th ht forming xaml Prort Valu Young' E = MPa modulu whn damag bgin to volv raidl, th nalt factor dcra accordingl, which low down th rat of damag volution. A a conqunc, uri- ou localiation i avoidd Sht forming Poion' ratio amag xonnt amag dnominator Initial ild tr Yild hardning curv Non-local charactritic lngth o ν = 0..3 =..0 r =.5 MPa σ o = MPa r R mmm MPa Fig. 5. Simulation of th ht forming roc: gomtr and finit lmnt modl for th coart mh (th modl ha bn mirrord in rct to th XZ lan). Fig. 6. Mh rfinmnt of th ht for th mtal forming imulation (to viw). In figur 7, th damag contour for th local o- lution ar lottd for th am unch dilacmnt. Clarl, damag tnd to locali vr raidl uon mh rfinmnt. On th othr hand, whn th non- 90

13 Fig. 8. amag contour for th non-local ca (to viw). 6. CONCLUSIONS Fig. 7. amag contour for th local ca (to viw). A gnral ovrviw of th gnral framwork for th dcrition of ductil damag ha bn addrd in thi ar. Th tandard local thor ha bn rvid whr om imortant intrrtation hav bn highlightd. Th gnral act of th nonlocal thor incororatd into a continuum damag 9 local olution i adotd, th localiation ffct ar ignificantl attnuatd inc th damag valu do not var a much a in th local ca ( figur 8). Furthrmor, th diffrnc on th contour ar alo mallr, ciall on th zon whr damag concntrat.

14 framwork hav alo bn focu of dicuion whr om of th main aumtion hav bn rviwd and clarifid. Th numrical imlmntation of an altrnativ non-local formulation ha bn rntd in dtail for th finit lmnt cod LS-YNA. Th rult of two aml ca hav hown that th mthodolog wa abl to allviat th athological mh dndnc obrvd du to th oftning rgim caud b th damag modl. Howvr, inc th rood non-local formulation i onl an aroximation of th tru non-local thor, it i till not clar if th mthodolog will rorl rvnt localiation for a widr numbr of ca. Thrfor, a dr amnt of th rntd non-local tratg, togthr with lmnt roion tchniqu undr vral circumtanc, hould b focu of furthr invtigation. ACKNOWLEGEMENTS F.X.C. Andrad wa uortd b th Programm AlBan, th Euroan Union Programm of High Lvl Scholarhi for Latin Amrica, cholarhi no. E074075BR. REFERENCES Atkin, A. G., 98, Poibl xlanation for unxctd dartur in hdrotatic tnion-fractur train rlation, Mtal Scinc, 5, Andrad, F.X.C., Andrad Pir, F.M., Car d Sa, J.M.A., Malchr, L., 009a., Nonlocal intgral formulation for a laticit-inducd damag modl, Comutr Mthod in Ma Scinc, 9(), Andrad, F.X.C., Car d Sa, J.M.A., Andrad Pir, F.M., Malchr, L., 009b, Nonlocal formulation for Lmaitr ductil damag modl, Procding of X Intrnational Confrnc on Comutational Platicit, Barclona, Sain. Andrad, F.X.C., Andrad Pir, F.M., Car d Sa, J.M.A., Malchr, L. 009c, Imrovmnt of th numrical rdiction of ductil failur with an intgral nonlocal damag modl, Intrnational Journal of Ma Forming,, Andrad, F.X.C., Car d Sa, J.M.A., Andrad Pir, F.M., 00, A ductil damag nonlocal modl of intgral-t at finit train: formulation and numrical iu, Intrnational Journal of amag Mchanic, (In rint). Bnvnuti, E., Borino, G., 00, A thrmodnamicall conitnt nonlocal formulation for damaging ma, Euroan Journal of Mchanic A/Solid,, Borino, G., Failla, B., Polizzotto, C., 003, A mmtric nonlocal damag thor, Intrnational Journal of Solid and Structur, 40, Borino, G., Fuchi, P., Polizzotto, C., 999, A thrmodnamic aroach to nonlocal laticit and rlatd variational rincil, Journal of Alid Mchanic, 66, Car d Sa, J.M.A., Aria, P.M.A., Zhng, C., 006, amag modlling in mtal forming roblm uing an imlicit non-local gradint modl, Comutr Mthod in Alid Mchanic and Enginring, 95, Car d Sa, J.M.A., Zhng, C., 007, A comarion of mhl and finit lmnt aroach to ductil damag in forming roc, Comutr Mthod in Ma Scinc, 7(), Car d Sa, J.M.A., Andrad Pir, F.M., Andrad, F.X.C., 00, Local and nonlocal modling of ductil damag, Chatr in book Advancd Comutational Ma Modling: From Claical to Multi-cal Tchniqu, Wil-VCH Vrlag GmbH, Winhim (Grman). Cockcroft, M. G., Latham,. J., 968, uctilit and workabilit of mtal, Journal of th Intitut of Mtal, 96, atko, J., 966, Ma Prorti and Manufacturing Proc, John Wil & Son, Nw York. Bort, R., Mühlhau, H., 99, Gradint-dndnt laticit: formulation and algorithmic act, Intrnational Journal for Numrical Mthod in Enginring, 35, Souza Nto, E.A., 00, A fat, on-quation intgration algorithm for th Lmaitr ductil damag modl, Communication in Numrical Mthod in Enginring, 8, Souza Nto, E.A., Prić,., Own,.R.J., 994, A modl for latolatic damag at finit train: algorithm iu and alication, Enginring Comutation,, Vr, J.H.P., Brklman, W.A.M., van Gil, M.A.J., 995, Comarion of nonlocal aroach in continuum damag mchanic, Comutr & Structur, 4, Frudnthal, A.M., 950, Th Inlatic Bhaviour of Enginring Ma and Structur, John Wil & Son, Nw York. Gr, M.G.., Ubach, R.L.J.M., Engln, R.A.B., 003, Strongl non-local gradint-nhancd finit train latolaticit, Intrnational Journal for Numrical Mthod in Enginring, 56, Jirák, M., Rolhovn, S., 003, Comarion of intgral-t nonlocal laticit modl for train-oftning ma, Intrnational Journal of Enginring Scinc, 4, Jirák, M., 007, Nonlocal damag mchanic, Rvu Euroén d Géni Civil,, Lmaitr, J., 996, A Cour on amag Mchanic, Sringr, Nw York. Lmaitr, J., 985, A continuou damag mchanic modl for ductil fractur, Journal of Enginring Ma and Tchnolog, 07, McClintock, F.A., 968, A critrion for ductil fractur b growth of hol, Journal of Alid Mchanic, 35, Mdiavilla, J., Prling, R., Gr, M.G.., 006, A nonlocal triaxialit-dndnt ductil damag modl for finit train laticit, Comutr Mthod in Alid Mchanic and Enginring, 95, Norri,.M., Raugh, J.E., Moran, B., Quiñon,.F., 978, A latic-train, man-tr critrion for ductil fractur, Journal of Enginring Ma and Tchnolog, Tranaction ASME, 00, Oan, M., Shima, S., Tabata, T., 978, Conidration of baic quation and thir alication in th forming of mtal owdr and orou mtal, Journal of Mchanical Working Tchnolog,, Prling, R.H., Bort, R., Brkmal, W.A., Vr, J.H., 996, Gradint-nhancd damag for quai-brittl ma, Intrnational Journal for Numrical Mthod in Enginring, 39, Pijaudir-Cabot, G., Bažant, Z.P., 987, Nonlocal damag thor, Journal of Enginring Mchanic, 3(0),

15 Polizzotto, C., Borino, G., Fuchi, P., 998, A thrmodnamic conitnt formulation of nonlocal and gradint laticit, Mchanic Rarch Communication, 5(), Ric, J. R., Trac,. M., 969, On th ductil nlargmnt of void in triaxial tr fild, Journal of th Mchanic and Phic of Solid, 7, 0-7. Ricci, S., Brünig, M., 007, Numrical anali of nonlocal aniotroic continuum damag, Intrnational Journal of amag Mchanic, 6, Strömbrg, L., Ritinmaa, M., 996, FE-formulation of a nonlocal laticit thor, Comutr Mthod in Alid Mchanic and Enginring, 36, Simo, J.C., Hugh, T.J.R., 998, Comutational Inlaticit, Sringr, Nw York. Tai, W., Yang, B.X., 987, A nw damag mchanic critrion for ductil fractur, Enginring Fractur Mchanic, 7, Tvrgaard, V., Ndlman, A., 995, Effct of nonlocal damag in orou latic olid, Intrnational Journal of Solid and Structur, 3(8/9), PROBLEMY TEORETYCZNEJ I NUMERYCZNEJ ANALIZY PLASTYCZNEGO PĘKANIA WYBRANE ZAGANIENIA Strzczni Clm rac jt rzdtawini rzglądu najnowzch oiągnięć w zakri oiu latczngo ękania, zarówno od tron tortcznj jak i numrcznj. Na wtęi omówiono klaczną lokalną torię odnozącą ię do trmodnamiki niodwracalnch roców, wkorztwaną do oiu ogólngo modlu rężto-latczngo ękania. Naświtlono równiż założnia i ogranicznia klacznj torii ojawiając ię wtd, kid równania konttutwn ą uzkiwan z rozwiązania roblmu minimalizacji z ograniczniami. Omówiono tż otatni oiągnięcia w zakri ni lokalngo modlowania rężto-latczngo ękania, okazując zaad i konkwncj wnikając z ni lokalngo traktowania tgo rocu. Zagadninia związan z fktwnm numrcznm oim lokalnj i nilokalnj torii ą równiż omówion w rac, a cjalną uwagę oświęcono imlmntacji modli ni lokalnch. Nowa tratgia obliczniowa, umożliwiająca imlmntację tch modli w komrcjnch rogramach mulacjnch, zotała zczgółowo rzdtawiona na rzkładzi LS-YNA. Fragmnt kodu w jęzku FORTRAN, w którch okazano główn kroki imlmntacji, ą rztoczon w rac. Efktwność ni lokalngo modlu jt ocniona na odtawi mulacji odkztałcania oiowomtrcznch róbk oraz roców tłocznia. Wkazano, ż w obdwóch analizowanch rzadkach nilokalna tratgia numrczna ozwala na obniżni wrażliwości rozwiązania na rozmiar iatki, któr jt nirozrwalni związan z rężto-latcznmi modlami ękania. Rcivd: Jul 3, 00 Rcivd in a rvid form: Octobr 0, 00 Acctd: Novmbr 4, 00 93