Ths atcle aeaed n a jounal ublshed by Elseve. The attached coy s funshed to the autho fo ntenal non-commecal eseach and educaton use, ncludng fo nstucton at the authos nsttuton and shang wth colleagues. Othe uses, ncludng eoducton and dstbuton, o sellng o lcensng coes, o ostng to esonal, nsttutonal o thd aty webstes ae ohbted. In most cases authos ae emtted to ost the veson of the atcle (e.g. n Wod o Tex fom) to the esonal webste o nsttutonal eostoy. Authos equng futhe nfomaton egadng Elseve s achvng and manusct olces ae encouaged to vst: htt://www.elseve.com/coyght
Jounal of the Mechancs and Physcs of Solds 57 (9) 34 368 Contents lsts avalable at ScenceDect Jounal of the Mechancs and Physcs of Solds jounal homeage: www.elseve.com/locate/jms Laws of cack moton and hase-feld models of factue Vncent Hakm a,, Alan Kama b a Laboatoe de Physque Statstque, CNRS-UMR855 assocé aux unvestés Pas VI et VII, Ecole Nomale Suéeue, 4 ue Lhomond, 753 Pas, Fance b Physcs Deatment and Cente fo Intedsclnay Reseach on Comlex Systems, Notheasten Unvesty, Boston, MA 5, USA atcle nfo Atcle hstoy: Receved 4 June 8 Receved n evsed fom 8 Octobe 8 Acceted Octobe 8 PACS: 6..Mk 46.5þa 46.5. x Keywods: Factue Phase feld Ansotoy Eshelby tenso Heng toque abstact Recently oosed hase-feld models offe self-consstent desctons of bttle factue. Hee, we analyze these theoes n the quasstatc egme of cack oagaton. We show how to deve the laws of cack moton ethe by usng solvablty condtons n a etubatve teatment fo slght deatue fom the Gffth theshold o by genealzng the Eshelby tenso to hase-feld models. The analyss ovdes a smle hyscal nteetaton of the second comonent of the classc Eshelby ntegal n the lmt of vanshng cack oagaton velocty: t gves the elastc toque on the cack t that s needed to balance the Heng toque asng fom the ansotoc suface enegy. Ths foce-balance condton can be nteeted hyscally based on enegetc consdeatons n the tadtonal famewok of contnuum factue mechancs, n suot of ts geneal valdty fo eal systems beyond the scoe of hase-feld models. The obtaned law of cack moton educes n the quasstatc lmt to the ncle of local symmety n sotoc meda and to the ncle of maxmum enegy-elease-ate fo smooth cuvlnea cacks n ansotoc meda. Analytcal edctons of cack aths n ansotoc meda ae valdated by numecal smulatons. Inteestngly, fo knked cacks n ansotoc meda, foce-balance gves sgnfcantly dffeent edctons fom the ncle of maxmum enegy-elease-ate and the dffeence between the two ctea can be numecally tested. Smulatons also show that edctons obtaned fom foce-balance hold even f the hase-feld dynamcs s modfed to make the falue ocess evesble. Fnally, the ole of dssatve foces on the ocess zone scale as well as the extenson of the esults to moton of lana cacks unde ue antlane shea ae dscussed. & 8 Elseve Ltd. All ghts eseved.. Intoducton The edcton of the ath chosen by a cack as t oagates nto a bttle mateal s a fundamental oblem of factue mechancs. It has classcally been addessed n a theoetcal famewok whee the equatons of lnea elastcty ae solved wth zeo tacton bounday condtons on cack sufaces that extend to a sha t (Bobeg, 999). In ths descton, the stess dstbutons nea the cack t have the unvesal dvegent foms (Wllam, 957; Iwn, 957) s m ð; YÞ ¼ Km ffffffffffffffff f m ðyþ, () Coesondng autho. E-mal addess: hakm@ls.ens.f (V. Hakm). -596/$ - see font matte & 8 Elseve Ltd. All ghts eseved. do:.6/j.jms.8..
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 343 whee K m ae the stess ntensty factos (SIF) fo the thee standad modes I, II, o III of factue (m ¼ ; o 3), Y s the angle between the adal vecto of magntude wth ogn at the cack t and the local cack decton and the exlct exessons of the f s ae ecalled n Aendx C (see Eq. (C.4)). The alled enegy-elease-ate (o cack extenson foce) eads, fo lane stan, G ¼ aðk þ K ÞþK 3 =ðmþ, () whee n denotes Posson s ato, m s the shea modulus, and a ð nþ=ðmþ. Followng Gffth (9), Iwn (957) ostulated that fo the cack to oagate, G must exceed some mateal-deendent theshold G c that s theoetcally equal to twce the suface enegy (G c ¼ g), but often lage n actce. Gffth s theoy ovde one cteon fo cack oagaton but s nsuffcent to detemne cuvlnea cack aths o cack knkng and banchng angles. Lke othe oblems n factue, the edcton of the cack decton of oagaton was fst examned (Baenblatt and Cheeanov, 96) fo mode III whch s smle because the antlane comonent of the dslacement vecto u 3 s a uely scala Lalacan feld. In ths case, the stess dstbuton nea the t, can be exanded as s 3Y m qu 3 qy ¼ K 3 ffffffffffffffff cos Y ma sn Y þ. (3) The domnant dvegent contbuton s always symmetcal about the cack decton. As a consequence, the knowledge of K 3 alone cannot edct any othe ath than a staght one. To avod ths masse, Baenblatt and Cheeanov (96) etaned the subdomnant sn Y tem, whch beaks ths symmety. They hyotheszed that a cuvlnea cack oagates along a decton whee A ¼, when the stess dstbuton s symmetcal about the cack decton. In subsequent extensons of ths wok, seveal ctea have been oosed fo lane loadng, fo whch the tensoal natue of the stess felds makes t ossble to edct non-tval cack aths uely fom the knowledge of the SIFs (Goldsten and Salgank, 974; Cotteell and Rce, 98). The geneally acceted condton K ¼ assumes that the cack oagates n a ue oenng mode wth a symmetcal stess dstbuton about ts local axs (Goldsten and Salgank, 974) and s the dect analog fo lane stan ðu 3 ¼ Þ of the condton A ¼ fo mode III. Ths ncle of local symmety has been atonalzed usng lausble aguments (Cotteell and Rce, 98) n the classcal famewok of factue mechancs. It has also been agued to follow fom contnuty of the chemcal otental at the factue t, n a model that ncooates suface dffuson (Bene and Machenko, 998). A full devaton eques an exlct descton of the ocess zone, whee elastc stan enegy s both dssated and tansfomed nonlnealy nto new factue sufaces. As a esult, how to extend ths ncle to ansotoc mateals, whee symmety consdeatons have no obvous genealzaton, s not clea (Made, 4a). Ths s also the case fo cuved thee-dmensonal factues although ths aeas lttle-noted n the lteatue. In addton, ath edcton emans lagely unexloed fo mode III even fo sotoc mateals. Contnuum models of bttle factue that descbe both shot scale falue and macoscoc lnea elastcty wthn a self-consstent set of equatons have ecently been oosed (Aanson et al., ; Kama et al., ; Eastgate et al., ; Wang et al., ; Macon and Jagla, 5; Satschek et al., 6). These models have aleady shown the usefulness n vaous numecal smulatons. Fo both antlane (Kama and Lobkovsky, 4) and lane (Heny and Levne, 4) loadng, they have oven caable to eoduce the onset of cack oagaton at Gffth theshold as well as dynamcal banchng (Kama and Lobkovsky, 4; Heny, 8) and oscllatoy (Heny and Levne, 4) nstabltes. In a quasstatc settng, ths contnuous meda aoach dffes n st but has nonetheless much n common wth a vaatonal aoach to bttle factue (Fancfot and Mago, 998) oosed to ovecome lmtatons of Gffth theoy. Ths s esecally aaent when the latte s mlemented numecally (Boudn et al., ), usng deas (Amboso and Totoell, 99) ntally develoed fo mage segmentaton (Mumfod and Shah, 989). In ths atcle, we analyze these self-consstent theoes of bttle factue fo mode I/II cacks as well as fo cacks movng unde ue antlane shea (mode III) and show how to deve laws of moton fo the cack t. Ths ovdes, n atcula, elatons whch genealze the ncle of local symmety fo an ansotoc mateal. Futhemoe, we valdate these elatons by hase-feld smulatons. Ths valdaton s caed out both fo the tadtonal vaatonal fomulaton of the hase-feld model wth a so-called gadent dynamcs, whch guaantees that the total enegy of the system,.e. the sum of the elastc and cohesve eneges, deceases monotonously n tme, and fo a smle modfcaton of ths dynamcs that makes the falue ocess evesble. We fnd that both fomulatons yeld essentally dentcal cack aths that ae well edcted by the laws of cack moton deved fom the hase-feld model. In sotoc meda, the ncle of local symmety and the ncle of maxmum enegy-elease-ate gves dentcal edcton fo smooth cuvlnea cacks and vey small dffeences fo knked angles (see e.g. Cotteell and Rce, 98; Hutchnson and Suo, 99; Amestoy and Leblond, 99). Ths makes t dffcult to numecally dstngush these two ctea. Thus, fo knked cacks, t has emaned somewhat unclea what haens fo stesses n the ga above the maxmum enegy-elease-ate theshold stess (whee enegetc consdeatons alone aea to eque cack oagaton) but below the mnmal stess whee oagaton wth local symmety/foce-balance s ossble. Inteestngly, we fnd that the foce-balance and maxmum enegy-elease-ate ncles can gve sgnfcantly dstnct edctons n ansotoc meda fo some choces of suface enegy ansotoy and loadng condtons. Numecal smulatons, whch ae at the lmt of what s comutatonally feasble wth a fnte-dffeence mlementaton of the hase-feld equatons on a egula mesh, show that knk cacks emege fom the man cack t at an ntal angle, whch aeas close to the one edcted by
344 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 maxmum enegy-elease-ate, and tun on the ocess zone length scale towads the angle edcted by foce-balance fo staght knks. Fo claty of exoston, the elatons deved fom the hase-feld model ae summazed fst n Secton and nteeted hyscally n the context of evous esults fom the factue communty. Ths secton stesses why the second comonent of the Eshelby confguatonal foce eendcula to the cack axs s both hyscally meanngful and motant fo the detemnaton of cack aths. Ou aoach s alcable to a lage class of dffuse nteface desctons of bttle factue. Howeve, fo claty of exoston, we base ou devaton on the hase-feld model ntoduced by Kama et al. (). As ecalled n Secton 3, n ths descton, the dslacement feld s couled to a sngle scala ode aamete o hase-feld f, whch descbes a smooth tanston n sace between unboken ðf ¼ Þ and boken states ðf ¼ Þ of the mateal. We focus on quasstatc factue n a macoscocally sotoc elastc medum wth neglgble netal effects. Mateal ansotoy s smly ncluded by makng the suface enegy gðyþ, deendent on the oentaton y of the cack decton wth esect to some undelyng cystal axs. In Secton 4, we analyze the quasstatc moton of a cack oagatng n mode I/II, etubatvely fo small deatue fom Gffth theshold ðjg G c j=g c 5Þ and small ansotoy. The cack laws of moton ae shown to be detemned n a usual manne by solvablty condtons, comng fom tanslaton nvaance aallel and eendcula to the cack t axs. A dffeent devaton s ovded n Secton 5 by genealzng Eshelby (975) tenso to hase-feld theoes. The elaton and dffeences between foce-balance and maxmum enegy-elease-ate ctea ae then consdeed. The atcula case of moton unde ue antlane shea s dscussed n Secton 6. Ou analytcal edctons ae comaed wth numecal hase-feld smulatons n Secton 7 whee we also examne the senstvty of the esults to the evesblty of the falue ocess. Ou conclusons and some futhe esectves of ths wok ae then esented n Secton 8. Futhe nfomaton on the hase-feld model of Kama et al. () s ovded n Aendx A n the smle context of a stetched one-dmensonal band. Detals of some of ou calculatons ae ovded n Aendces B and C. A shot veson of ths wok has been ublshed n Hakm and Kama (5).. An ovevew of the hyscal ctue and man esults n the classcal factue fomalsm In the fomalsm of contnuum factue mechancs, cack oagaton has been tadtonally analyzed by consdeng the cack extenson foce G defned by Eq. (). Ths s a uely confguatonal foce that onts along the cack axs n the decton of oagaton whee Gdl s the amount elastc enegy eleased when the cack advances nfntesmally along ths axs by a dstance dl. When consdeng the oagaton of a geneal cuvlnea cack, howeve, t s necessay to consde the extenson of a cack at some small nfntesmally angle dy wth esect to ts cuent axs as dected schematcally n Fg.. Physcally, one would exect a confguatonal foce, dstnct fom G, to be assocated wth the exta amount of elastc enegy that s eleased f the cack oagates by dl along ths new decton, denoted hee by ^t, as oosed to oagatng the same dstance along ts cuent axs, denoted by ^x. Ths addtonal foce on the cack t was consdeed by Eshelby (975). It can be nteeted hyscally as oducng a toque on the cack t that changes the cack oagaton decton so as to maxmze the elastc enegy eleased. The foce that oduces ths toque must act eendculaly to the cack oagaton decton and ts magntude s smly dg G y lm dy! dy, (4) whee dg s the dffeence between the cack extenson foce along the new decton and the old decton,.e. along ^t and ^x n Fg.. Ths toque s analogous to the well-known Heng (95,. 43) toque actng on the juncton of thee cystal gans of dffeent oentatons n a olycystallne mateal, wth the man dffeence that G y s a confguatonal foce n the esent factue context whle the Heng toque s oduced by a hyscal foce assocated wth the gan bounday enegy, g gb ðyþ, whch s geneally ansotoc. Ths foce acts eendculaly to each gan bounday segment at the juncton of thee gans wth a magntude dg gb =dy. Fg.. Schematc eesentaton of an nfntesmal extenson P P of the cack of length dl at and angle dy measued wth esect to the cack axs. The aows ontng eendcula to the cack denote the two analogs G y and G cy of the Heng toque assocated wth the dectonal deendence of the cack extenson foce and the factue enegy aound the cack axs, esectvely.
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 345 Ths analogy suggests that thee should geneally be two toques actng on the cack t. The fst, aleady mentoned, s Eshelby s confguatonal elastc toque G y assocated wth the dectonal deendence of the cack extenson foce n efeence to the local cack axs. The second s the hyscal toque assocated wth the dectonal deendence of the factue enegy, defned hee by G c ðyþ, whch should have a magntude dg c ðyþ=dy G cy by dect tanslaton of Heng s esult fo factue. It follows that the balance of foces at the cack t should yeld two condtons. The fst s the standad condton of the classcal factue fomalsm assocated wth the balance of foces along the cack axs, G ¼ G c. The second s the condton G y ¼ G cy ¼ g y, (5) whch coesonds hyscally to the balance of the two afoementoned toques actng on the cack t. Whle G y ulls the cack n a decton that tends to maxmze the elease of elastc enegy, G cy ulls the cack n a decton that mnmzes the enegy cost of ceatng new factue sufaces. The second equalty on the ght-hand sde (.h.s.) of Eq. (5) only holds n some deal bttle lmt whee the factue enegy s equal to twce the suface enegy, defned hee by gðyþ, and g y dgðyþ=dy. We note that ths deal bttle lmt s exact fo the class of hase-feld models analyzed n ths ae but at best only aoxmate even fo a stongly bttle mateal such as glass. The ssue of the quanttatve evaluaton of G cy, howeve, should be ket seaate fom ts ole n cack ath edcton that s ou man focus n ths ae. To see how ths toque balance condton ovdes an exlct edcton fo the cack ath, t s useful to deve an exesson fo G y by elementay means, dectly fom the defnton of Eq. (4), nstead of by evaluatng an Eshelby Rce tye ntegal aound the cack t (Rce, 968; Eshelby, 975), as done late n ths ae (see Secton 5 and Aendx C); whle both methods yeld the same answe, the fome s moe hyscally tansaent. Fo ths uose, we use the known exessons fo the new SIFs K and K at the t (coesondng to P n Fg. ) of an nfntesmally small knk extenson of length dl of a sem-nfnte cack (Amestoy and Leblond, 99). In the lmt of vanshng knk angle, these exessons ae gven by K ¼ K 3K dy= þ, (6) K ¼ K þ K dy= þ to lnea ode n dy ndeendently of dl, whee K and K ae the SIFs at the t (coesondng to P n Fg. ) of the ognal staght cack. Usng Eq. () wth these new SIFs to defne GðdyÞ, we obtan at once that dg ¼ GðdyÞ GðÞ ¼ ak K dy, and hence usng Eq. (4), that G y ¼ ak K. Substtutng ths exesson fo G y n the toque balance condton (5), we obtan the condton K ¼ G cy ¼ g y, (8) ak ak whee second equalty only holds n the deal bttle lmt as befoe. In the sotoc lmt whee G cy vanshes, ths condton educes to the ncle of local symmety whch assumes that the cack oagates n a ue oenng mode ðk ¼ Þ. In contast, fo an ansotoc mateal, K s fnte wth a magntude that deends both on K and the local cack oagaton decton,.e. G cy deends on the decton of the cack wth esect to some fxed cystal axs n a cystallne mateal. Fo smlcty, we have estcted ou devaton to a stuaton whee lnea elastcty s sotoc (e.g. hexagonal symmety n two dmensons), but Eq. (8) could staghtfowadly be extended to a moe geneal stuaton whee lnea elastcty s also ansotoc. The ecognton that the toque balance condton (5) can be used to detemne the geneal ath of a cack n a bttle mateal s the cental esult of ths ae. Ths condton sheds lght on the hyscal ogn of the ncle of local symmety n the sotoc lmt and shows how t can be genealzed quanttatvely to ansotoc mateals. Although the confguatonal foce eendcula to the cack t was consdeed exlctly by Eshelby (975) t has been lagely gnoed untl ecently. Ths s ehas because the dslacement of a small segment of cack eendcula to tself, whch one mght navely exect to esult fom such a foce, would aea unhyscal and uneconclable wth the evesblty of the factue ocess. Whle such a moton s unhyscal, t should be clea fom the esent consdeatons that all the toques actng on the cack t, both the elastc confguatonal toque G y and the hyscal toque G cy lnked to factue enegy ansotoy, have been obtaned solely fom the consdeaton of an nfntesmal, hyscally admssble, extenson of the cack at a small angle fom ts axs. In equatng these two toques at the cack t, the man assumton made s that the dynamcs on the ocess zone scale s able to samle dffeent ossble mcoscoc states so as to emt local elaxaton to mechancal equlbum. Thee have been moe ecent attemts to ncooate the Eshelby elastc toque n the classcal factue fomalsm, whee factue sufaces ae teated as mathematcally sha boundaes extendng to the cack t (Adda-Beda et al., 999; Oleaga, ; Made, 4a). In atcula, based on enegetc consdeatons, Made (4b) has oosed an equaton of moton that educes to Eq. (8) fo smooth cuvlnea cacks, albet not fo knked cacks, n the lmt when a henomenologcally ntoduced elaxaton length scale (Hodgdon and Sethna, 993) vanshes. In the context of these evous studes, the hase-feld aoach has the motant advantage of emovng many of the ambgutes that ase when consdeng the moton of the cack t n the classcal factue fomalsm. In atcula, t makes t ossble to (7)
346 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 goously deve a foce (toque) balance condton fom the condton fo the exstence of a oagatng cack soluton that s satally dffuse on the nne scale of the ocess zone and that must match smoothly to the standad soluton of lnea elastcty on the oute scale of the samle sze. Ths condton educes to Eq. (5) (o Eq. (8) fo sotoc elastcty) n the lmt of vanshng cack velocty. Futhemoe, t holds geneally fo both smooth and knked cacks and edcts knk angles that dffe fom those edcted by maxmum enegy-elease-ate. Futhemoe, ths foce-balance condton contans addtonal contbutons fo fnte cack velocty assocated wth dssatve foces on the ocess zone scale. Inteestngly, the esults of the hase-feld analyss show that the comonent of the dssatve foce eendcula to the cack t vanshes fo oagaton n sotoc meda because both the stess dstbuton and the hase feld ae symmetcal about the cack axs n ths case. Consequently, wthn the hase-feld famewok, dssatve foces do not change the condton K ¼ fo cack oagaton n sotoc meda. Fo oagaton n ansotoc meda, n contast, small velocty-deendent coecton to the toque balance condton (8) ase because ths symmety s boken. 3. The KKL hase-feld model of factue Factue s geneally descbed n dffuse nteface models (Aanson et al., ; Kama et al., ; Eastgate et al., ; Wang et al., ; Macon and Jagla, 5) as a softenng of the elastc modul at lage stans. Ths can be done uely n tem of the stan tenso but t oduces feld equatons wth devatve of hgh ode (Macon and Jagla, 5). Hee, we adot the altenatve aoach of ntoducng a sulementay feld f, a scala ode aamete o hase feld, that descbes the state of the mateal and smoothly nteolates between ntact ðf ¼ Þ and fully boken ðf ¼ Þ states. Fo defnteness, we base ou devaton on the secfc model oosed by Kama et al. () wth enegy densty E, E ¼ E f ðfq j fgþ þ gðfþðe stan E c ÞþE c, (9) whee q j q=qx j denotes the atal devatve wth esect to the catesan coodnate x j (j ¼ ; ; 3) and E stan s the elastc enegy of the ntact mateal. The equatons of moton ae deved vaatonally fom the total enegy of the system that s the satal ntegal E ¼ d 3 x E () of the enegy densty. In the quasstatc case, these ae ¼ de du ¼ q qe qe k j q½q j u k Š qu, () k w q t f ¼ de df ¼ q qe j q½q j fš qe qf. () The thee Eule Lagange equaton () fo the catesan comonents u k of the dslacement vecto ðk ¼ ; ; 3Þ ae smly the statc equlbum condtons that the sum of all foces on any mateal element vansh. The fouth equaton () fo f s the standad Gnzbug Landau fom that govens the hase-feld evoluton, wth w a knetc coeffcent that contols the ate of enegy dssaton n the ocess zone,.e. t follows fom Eqs. () and () that de dt ¼ w d 3 x de. (3) df In the smlest case of an sotoc elastc medum and sotoc f, the hase feld and stan enegy ae smly E f ðfq j fgþ ¼ k ðfþ, (4) E stan ðfu gþ ¼ l ðu Þ þ mu u, (5) whee u ¼ðq u j þ q j u Þ= s the usual stan tenso of lnea elastcty. No asymmety between dlaton and comesson s ncluded snce ths s not necessay fo ou esent uoses. The boken state of the mateal becomes enegetcally favoed when E stan exceeds the theshold E c and gðfþ s a monotoncally nceasng functon of f that descbes the softenng of the elastc enegy at lage stan ðgðþ¼þ and oduces the usual elastc behavo fo the ntact mateal ðgðþ ¼; g ðþ ¼Þ. In addton, the elease of bulk stess by a cack eques the functon gðfþ to vansh faste than f fo small f, as ecalled n Aendx A. We theefoe choose gðfþ ¼4f 3 3f 4,asnKama et al. (), Kama and Lobkovsky (4), and Heny and Levne (4). Wth these choces, the sotoc suface enegy s equal to g ¼ ffffffffffffffffffffff ke c df ffffffffffffffffffffffffffffffffffff gðfþ :765 ffffffffffffffffffffff ke c, (6) as shown n Aendx A (Eq. (A.)), by eeatng the analyss of Kama et al. ().
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 347 In the esent ae, we analyze the case of a hase-feld enegy E f ðfq j fgþ wthout otatonal symmety whch gves an ansotoc suface enegy. A smle examle used fo conceteness and fo the numecal smulatons s ovded by a smle two-fold ansotoy n the hase-feld enegy E f ¼ k ðjfj þ q fq fþ. (7) The suface enegy of a staght factue nteface oented at an angle y wth the x-axs ases fom the vaaton of the elastc and hase felds n a decton tansvese to the factue, namely wth fðx; yþ ¼f½ x snðyþþycosðyþš. Theefoe, the only dffeence between Eqs. (4) and (7) n ths one-dmensonal calculaton of the suface enegy (Aendx A) s the elacement of k by k½ ð=þsn yš n the ansotoc case. The alled ansotoc suface enegy thus follows dectly fom the sotoc exesson (6) and eads qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff gðyþ ¼g ð=þsn y. (8) It educes of couse to the sotoc suface enegy g of Eq. (6) n the! lmt. Wth the secfc eneges of Eqs. (5) and (7), the vaatonal hase-feld equatons ead q j ½s gðfþš ¼, k½ f þ q xy fš g ðfþðe stan E c Þ¼ w q tf, (9) whee s s the usual stess tenso fo an sotoc medum s ¼ lu kk d ;j þ mu. () Ou am n the followng sectons s to analyze the laws that goven the moton of a cack t n ths self-consstent descton. 4. Laws of cack moton as solvablty condtons 4.. The t nne oblem In the hase-feld descton, the obtenton of laws of moton fo a cack t can be vewed as an nne oute matchng oblem. The hase-feld equaton (9) ntoduce an ntnsc ocess zone scale x ¼ k=ðe ffffffffffffffffffffffffffffffffff c Þ. The nne oblem conssts n the detemnaton of a soluton of Eq. (9) at the ocess zone scale x. The bounday condtons on ths nne oblem ae mosed at a dstance fom the cack t much geate than the ocess zone scale ðbxþ and much smalle than any macoscoc length. They should concde wth the shot-dstance asymtotcs of the oute oblem, namely the usual detemnaton of the elastc feld fo the cack unde consdeaton. Theefoe, the mosed bounday condtons on Eq. (9) ae () that the mateal s ntact ðf! Þ away fom the cack tself and () that fo mxed mode I/II condtons, as consdeed n ths secton and the next one, the asymtotc behavo of the dslacement feld s u ð; YÞ ffffffffffff ½K 4m d I ðy; nþþk d II ðy; nþš, () whee m s the shea modulus and the functons d m ae dectly elated to the unvesal dvegent foms of the stess (Eq. ()) and ae exlctly gven (n ola coodnates) by Eqs. (C.6) and (C.7) of Aendx C. The values of K and K ae mosed by the bounday condtons at the macoscoc scale and do not sgnfcantly vay when the cack t advances by a dstance of ode x. In the fame of the cack t movng at velocty v, Eqs. (9) thus ead q j ½s gðfþš ¼, k f g ðfþðe stan E c Þ¼ v w q xf kq xy f. () 4.. Petubatve fomalsm and solvablty condtons Ou fst aoach fo obtanng the laws of cack t moton conssts n analyzng the slowly movng solutons of Eq. () wth bounday condtons () etubatvely aound an mmoble Gffth cack. Fo sotoc elastc and hase-feld eneges and a ue oenng mode, ths Gffth cack coesonds to the statonay soluton that exsts fo aðk c Þ ¼ G c. Accodngly, we consde, fo a small deatue fom Gffth theshold, dk ¼jK K c j=kc 5 and Note that wth coodnates x ; y otated by =4 wth esect to the x; y-axes, the hase-feld enegy eads, E f ¼ðk=Þð þ =Þðq x fþ þðk=þ ð =Þðq y fþ.
348 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 fo a small K 5K c, a slowly movng cack wth a small (two-fold) ansotoy n f-enegy (Eq. (8)). Ou am s to fnd the elatons between K and the ansotoy, as well as between K ; K and the velocty v, equed fo the soluton exstence. Lneazaton of Eqs. () aound the sotoc Gffth cack u ðþ ; f ðþ wth the substtutons u ¼ u ðþ f ðþ þ f ðþ, gves q j ½s ðþ gðf ðþ ÞŠ þ q j ½s ðþ g ðf ðþ Þf ðþ Š¼, k f ðþ g ðf ðþ Þs ðþ u ðþ g ðf ðþ Þf ðþ ½E stan E c Š¼ v w q xf ðþ kq xy f ðþ. (3) Ths can symbolcally be wtten as u ðþ LB @ u ðþ C A ¼ v B C B @ A k C @ A, (4) w q x f ðþ q xy f ðþ f ðþ whee L s the lnea oeato on the left-hand sde (l.h.s.) of Eq. (3). The bounday condtons at nfnty ae that f ðþ vanshes and that u ðþ behaves asymtotcally as n Eq. () but wth K elaced by dk, the small deatue fom Gffth theshold, and K s also assumed to be small. The lnea oeato L ossesses two ght zeo-modes, that ase fom the nvaance of the zeoth-ode oblem unde x and y tanslatons, and can be exlctly obtaned by nfntesmal tanslaton of the mmoble Gffth cack. Fo a geneal lnea oeato, the detemnaton of the left zeo-modes would nonetheless be a dffcult oblem. Howeve, the vaatonal chaacte of the equatons of moton moses qute geneally that L s self-adjont (see Aendx B) and that left zeo-modes ae dentcal to ght zeo-modes. Thus, takng the scala oduct of the two sdes of Eq. (4) wth the two tanslaton zeo-modes ovdes two exlct solvablty condtons fo Eq. (4). The scala oduct wth a left zeo-mode ðu L ; ul ; fl Þ can geneally be wtten as dx dy ðu L ; ul ; fl ÞLB @ u ðþ u ðþ f ðþ C A ¼ dx dy f L þ u ðþ ; f ¼ v w q xf ðþ þ kq xy f ðþ. (5) Snce the left vecto s a zeo-mode of L, the only contbuton to the l.h.s. of Eq. (5) comes fom bounday tems, dx dy ðu L ; ul ; fl ÞLB @ u ðþ u ðþ f ðþ I C A ¼ dsn j f½u L s ðþ u ðþ s L Š gðfðþ Þ þ½u L fðþ u ðþ f L Šg ðf ðþ Þs ðþ þ k½f L q f ðþ f ðþ q f L Šg, (6) whee n s the outwad contou nomal and the contou ntegal s taken counteclockwse along a ccle (of adus ) centeed on the factue t. 4.3. Tanslatons along x and cack velocty The zeo-mode coesondng to tanslatons along x s ðq x u ðþ ; q xu ðþ ; q xf ðþ Þ. On the.h.s. of Eq. (5), the tem ootonal to the ansotoy vanshes (by symmety o exlct ntegaton). The.h.s. of Eq. (6) can be smlfed snce on a ccle of a lage enough adus, gðf ðþ Þ equals unty eveywhee excet n the egon whee the ccle cuts the factue ls. Ths egon of non-constant f s fa away fom the cack t whee the cack s to a vey good aoxmaton nvaant by tanslaton along x and q x f q x u. Theefoe, dx dyðq x u ðþ ; q xu ðþ ; q xf ðþ ÞL B @ u ðþ u ðþ f ðþ I C A ¼ dsn ½u ðx;iþ s ðþ u ðþ s ðx;iþ Š ¼ K dk ð nþ, (7) m whee the exlct fomulas (C.6), (C.7) fo the elastc dslacements aound a staght cack, have been used to obtan the last equalty as detaled n Aendx C (see Eq. (C.4)). Comason between Eqs. (7) and (5) fnally ovdes the natual
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 349 esult that the cack velocty s ootonal to the deatue fom Gffth theshold, v dx dy½q w x f ðþ Š ¼ K dk ð nþ ¼dG. (8) m 4.4. Tanslatons along y and cack decton A second condton on cack moton ases fom the zeo-mode coesondng to tanslatons along y, ðq y u ðþ ; q yu ðþ ; q yf ðþ Þ. In ths case, only the tem ootonal to the ansotoy contbutes to the l.h.s. of Eq. (6). dx dy q y f ðþþ q xy f ðþþ ¼ dy½q y f ðþþ j x¼ Š. (9) Smlaly to Eq. (7), the.h.s. of Eq. (6) smlfes when the ntegaton contou s a lage enough ccle u ðþ I dx dyðq y u ðþ ; q yu ðþ ; q yf ðþ ÞLB u ðþ @ C A ¼ dsn ½u ðy;iþ s ðþ u ðþ s ðy;iþ Š f ðþ ¼ K K ð nþ, (3) m whee agan the exlct evaluaton n the last equalty s detaled n Aendx C (see Eq. (C.)). Thus, the second elaton of cack moton eads þ K K m ð nþ ¼k dy½q y f ðþþ j x¼ Š. (3) Eq. (3) educes to the ncle of local symmety (.e. K ¼ ) fo an sotoc medum and ovdes the aoate genealzaton fo the consdeed ansotoy. Befoe futhe dscussng these esults and the hyscal consequences, we esent a dffeent devaton n the next secton. 5. Genealzed Eshelby Rce ntegals The second aoach, whch we usue hee, dectly exlots the vaatonal chaacte of the equatons of moton and the nvaance unde tanslaton. It yelds dentcal solvablty condtons as the aoach of Secton 4 when G G c and symmety beakng etubatons ae small, but t s moe geneal snce t does not eque these quanttes to be small. 5.. The genealzed Eshelby tenso As shown by Noethe (98) n he classc wok, to each contnuous symmety of vaatonal equatons s assocated a conseved quantty (chage) and an alled dvegenceless cuent. Sace (and tme) tanslaton nvaance ae well known to gve the dvegenceless enegy momentum tenso n feld theoes (Landau and Lfshtz, 975). Eshelby (95) and followng authos (Rce, 968; Eshelby, 975; Gutn and Podo-Gudugl, 998; Adda-Beda et al., 999; Oleaga, ) have shown the usefulness of the analogous tenso fo classcal elastcty theoy. Hee, we consde the genealzed enegy momentum (GEM) tenso whch extends the Eshelby (975) tenso fo lnea elastc felds by ncooatng shotscale hyscs though ts addtonal deendence on the hase-feld f. We fnd t convenent to defne the fou-dmensonal vecto feld c a ¼ u a fo a3 and c a ¼ f fo a ¼ 4, whee u a ae the comonents of the standad dslacement feld. The nne oblem Eq. () can then be ewtten n the condensed fom d a;4 v w qe q f ¼ q j q½q j c a Š qe a ; a ¼ ;...; 4, (3) qc whee hee and n the followng summaton s mled on eeated ndces (fom to 3 on oman ndces and fom to 4 on geek ones). Chan ule dffeentaton ovdes the smle equalty, q E ¼ qe qc a q c a þ qe q½q j c a Š q jq c a. (33) Usng Eq. (3) to elmnate qe=qc a fom the.h.s. of Eq. (33), we obtan q j T ¼ v w q fq f fo ¼ ;, (34)
35 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 whee the GEM tenso T eads T Ed qe q½q j c a Š q c a. (35) The GEM tenso T s the sought extenson of the classcal Eshelby (95) tenso T E of classcal lnea elastcty T E ¼ E stand s jk q u k. (36) The GEM tenso T educes dentcally to T E n the ntact mateal whee the hase-feld s constant (f ¼ ). Both tensos ae non-symmetc n the two ndces. The dvegence of the GEM tenso taken on ts second ndce vanshes n the zeovelocty lmt, when dssaton n the ocess zone also vanshes. 5.. Laws of cack moton In ode to take advantage of Eq. (34), we ntegate the dvegence of the GEM tenso ove a lage dsk O centeed on the cack t (see Fg. ), followng Eshelby (975) comutaton of the confguatonal foce on the cack t teated as a defect n a lnea elastc feld and subsequent attemts to deve ctea fo cack oagaton and stablty (Gutn and Podo- Gudugl, 998; Adda-Beda et al., 999; Oleaga, ). The motant dffeence wth these evous comutatons s that, hee, the GEM tenso (35) s well defned eveywhee, so that the cack tself s ncluded n the doman of ntegaton. The ntegal of the dvegence of the GEM tenso can be wtten as a contou ntegal ove the lage ccle qo boundng the dsk O, F ¼ dst n j þ dst n j v d~x q C A!B B!A w fq f ¼. (37) O We have decomosed the ccle qo nto: () a lage loo C A!B aound the t n the unboken mateal, whee A (B) sata heght h below (above) the cack axs that s much lage than the ocess zone sze but much smalle than the adus R of the contou, x5h5r and () the segment ðb! AÞ that taveses the cack fom B to A behnd the t, as llustated n Fg.. In both ntegals, ds s the contou aclength element and n j the comonents of ts outwad nomal. Eq. (37) ovdes an altenatve bass to edct the cack seed and ts ath fo quasstatc factue. The F s can be nteeted as the aallel ð ¼ Þ and eendcula ð ¼ Þ comonents wth esect to the cack decton, of the sum of all foces actng on the cack t. In Eq. (37), the thee ntegals tems fom left to ght, esectvely, eesent confguatonal, cohesve, and dssatve foces. We examne them n tun. 5... Confguatonal foces and Eshelby toque We take A and B fa back fom the t and close to the cack on a macoscoc scale but wth the dstance h between A and B much lage than the ocess zone scale. Namely, we consde the mathematcal lmt h!þ; R!þ wth h=r! whee R s the dstance fom A and B to the cack t. In ths lmt, the fst ntegal n Eq. (37) s taken on a ath that s entely n the unboken mateal whee f s constant and equal to unty. Thus, the tenso T educes to the classcal Eshelby tenso T E Þ (Eq. (36)) the fst ntegal n Eq. (37) yelds the two comonents of the usual confguatonal foces Fðconf, F ðconf Þ ¼ dst E n j. (38) C A!B The fst comonent, F ðconf Þ, s the cack extenson foce and also Rce s (968) J ntegal, F ðconf Þ ¼ dst E j n j. (39) C A!B n B A x =+h x = h x R Ω x Fg.. Satally dffuse cack t egon wth f ¼ contou seaatng boken and unboken mateal (thck sold lne).
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 35 B A C F D θ E Fg. 3. Sketch showng the contou ntegal decomoston n Eq. (4). The cack wth ts vtual extenson at an angle y s dected by the thck bold lne. The ntegal contou follows the geat ccle fom A to B; t contnues along the ue l of the cack fom B to C and then along the ue l of the vtual extenson fom C to D; t then enccles the extended cack t followng the small ccle fom D to E; fnally t comes back to A along the lowe cack ls va F. Wth the known foms of the elastc dslacement felds nea the cack t, as detaled n Aendx C (see Eq. (C.8)), one obtans the well-known exesson () of the cack extenson foce, F ðconf Þ ¼ G ¼ aðk þ K Þ. (4) The second comonent F ðconf Þ can be comuted n an analogous way fom the elastc dslacement felds nea the cack t, (Eq. (C.9)) and one obtans F ðconf Þ ¼ ak K. (4) As dscussed eale, F ðconf Þ s the Eshelby (975) toque that can be nteeted hyscally f one magne extendng the cack t by a small amount at a small angle y fom the man t axs. Then F ðconf Þ s equal to the angula devatve of the cack extenson foce GðyÞ at y ¼. Ths equalty can be seen n two ways. Fst, we can use the geneal oetes of the Eshelby tenso. We denote wth a tlde the elastc quanttes coesondng to the cack wth the small extenson of length s at an angle y. Snce the cack extenson s along the decton ðcosðyþ; snðyþþ, we consde the alled vecto obtaned fom the Eshelby tenso, T E yj cosðyþ T E j þ snðyþ T E j. The flux of ths vecto vanshes when taken though the contou that goes along the geat ccle fom A to B and then contnues n a classcal way along the ls of the extended cack, as dawn n Fg. 3. C A!B þ B!C þ þ þ þ ds T E yj n j ¼. (4) C!D C D!E E!F F!A The two ntegals on the factue ls fom C to D and E to F do not contbute snce the ntegand vanshes: the y decton s along the ath and the nomal stesses vansh on the factue ls. The same agument shows that the ntegand s smly equal to Ẽstan snðyþ fo the ntegals fom B to C and F to A along the ls of the ognal factue. The ntegal on the small ccle aound the extended cack t C D!E s equal to GðyÞ whee GðyÞ s the enegy-elease-ate at the end of the small cack extenson. Eq. (4) thus educes to ½cosðyÞ T E j þ snðyþ T E j Š n j ¼ GðyÞþ dx snðyþ½ẽþ stan ðxþ Ẽ stanðxþš, (43) C A!B R whee Ẽþ stan and Ẽ stan, esectvely, denote the elastc stan enegy denstes on the ue and lowe factue ls. The equed dentty between F ðconf Þ and the angula devatve of d G=dyj y¼ follows fom dffeentaton of Eq. (43) wth esect to y at y ¼. When ths s efomed, thee ae two knds of tems. Tems comng fom the dffeentaton of the exlct tgonometc functons n Eq. (43) and tems comng fom the mlct deendence uon y of tlde quanttes. Howeve, n the ntegal on the l.h.s of Eq. (43), these mlct tems vansh as the length s of the extenson s taken to zeo, and n the ntegal on the.h.s. they ae multled by a vanshng sne functon. Moeove, fo the staght factue at y ¼, only s xx s non-zeo on the factue ls and t s of ooste sgn on the ue and lowe factue ls (Eq. (C.4)). The stan enegy denstes whch ae quadatc n the stess s xx ae theefoe equal on the ue and lowe factue ls and afte dffeentaton, the contbuton of ntegal tem on the.h.s. of Eq. (43) vanshes at zeo. Fnally, n the lmt of a vanshng extenson length ðs! Þ tlde quantty tend towad the (non-tlde) values on the ognal factue and one obtans F ðconf Þ ¼ T E j n d GðyÞ j ¼ lm C A!B s! dy G y ðþ. (44) y¼ Subdomnant tems, comng fo nstance fom a macoscoc cuvatue of the cack, could be dffeent on the two cack ls but note that the ntegal ange s on a length scale that s vanshngly small on a macoscoc scale.
35 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 Ths elaton between the second comonent F ðconf Þ and the angula devatve of GðyÞ can also be obtaned by comang the exlct exessons n tem of the SIF K and K. As t s well known, the SIF K and K at the end of a small extenson can be exessed as lnea combnaton of K and K (Amestoy and Leblond, 99) K ¼ F ðyþk þ F ðyþk, K ¼ F ðyþk þ F ðyþk, (45) wth clealy F ðþ ¼F ðþ ¼ and F ðþ ¼F ðþ ¼ and the devatve at y ¼, F ðþ ¼F ðyþ ¼, as aleady mentoned n Secton (Eqs. (6) and (7)). A detaled comutaton (Amestoy and Leblond, 99) ovdes the othe two devatves F ðþ ¼ 3 and F ðþ ¼. Theefoe, one obtans d GðyÞ lm s! dy ¼ ak K ½F ðþþf ðþš ¼ ak K. (46) y¼ Ths s ndeed dentcal to the exesson of F ðconf Þ obtaned by a dect comutaton (Eq. (4)) and t ovdes a second devaton of Eq. (44). 5... Cohesve foces An motant new ngedent n Eq. (37) s the second oton of the lne ntegal ð R B!AÞ of the GEM tenso that taveses the cack. Ths ntegal eesents hyscally the contbutons of cohesve foces nsde the ocess zone. To see ths, we fst note that the ofles of the hase feld and the thee comonents of the dslacement can be made to deend only on x ovded that the contou s chosen much lage than the ocess zone sze and to tavese the cack eendculaly fom B to A. Wth ths choce, we have that n ¼, n ¼, along ths contou and theefoe that, fo ¼ þh F ðcohþ ¼ dst j n j ¼ dx T. (47) B!A h The satal gadents aallel to the cack decton ðq c k Þ gve vanshngly small contbutons n the lmt h=x!þand R=x!þwth h=r!. Thus, the ntegand on the.h.s. of Eq. (47) educes to the enegy of a one-dmensonal cack whch, as ecalled n Aendx A (Eq. (A.)), s tself ndeendent of the stan and can be dentfed to twce the suface enegy g þh F ðcohþ ¼ dx Eðf; q f; q u Þ¼ g. (48) h Ths yelds the exected esult that cohesve foces along the cack decton exet a foce ooste to the cack extenson foce wth a magntude equal to twce the suface enegy. One smlaly obtans fo ¼, n the same lmt x5h5r, the othe comonent F ðcohþ of the foce eendcula to the cack decton þh þh F ðcohþ qe þh qe ¼ dst j n j ¼ dx T ¼ dx q B!A h h qq c c f a ¼ dx a h qq f q f. (49) The last equalty comes fom the fact that the only consdeed ansotoy s n the hase-feld at E f of the enegy densty and that, as above, gadents aallel to the cack decton gve neglgble contbutons fa behnd the cack t. F ðcohþ can be exessed as the angula devatve of the suface enegy at the cack t decton y ¼. Fo a mateal boken along a lne lyng at a decton y wth the x-axs, the dslacement and hase felds only deend on the nomal coodnate ¼ x snðyþþx cosðyþ. The local enegy densty E½f; q f; q f; q u Š s theefoe equal to E½f; snðyþq f; cosðyþq f; q u Š. The alled suface enegy eads gðyþ ¼ þ d E½ snðyþq f; cosðyþq f; q u Š. (5) Dffeentaton wth esect to y bngs on the.h.s. of Eq. (5) tems comng fom the exlct deendence of the ntegand on y as well as tems comng fom the mlct deendence of the felds on the beakng angle (fo an ansotoc mateal). Howeve, the contbuton of the mlct tems vanshes snce fo any gven angle the felds mnmze the total enegy and no feld vaaton leads to an enegy change at lnea ode. Theefoe, one obtans d þ dy g qe þ ¼ dx y¼ qq f ð q qe f fþ ¼ dx qq f ð q fþ, (5) snce educes to x fo y ¼. Comason of Eqs. (49) and (5) shows that F ðcohþ ¼ d dy g y¼ (5) as announced.
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 353 Of couse, elaton (5) can also be checked by dect comutaton fo any exlct fom of the hase-feld enegy. Fo nstance, n the smle case of Eq. (7), one obtans fom Eq. (49), F ðcohþ ¼ qe f dx qq f q f ¼ dx kðq fþ ¼ g, (53) whee, fo the last equalty, t should be noted (see Aendx A) that the second ntegal n Eq. (53) s equal to the enegy (by unt length) of the cacked mateal whch s tself equal to g. The esult of Eq. (53) ndeed agees wth Eq. (5), g ¼ g y ðþ, snce the suface enegy n the decton y s gven by Eq. (8). The foce F ðcohþ s the dect analog of the Heng (95,. 43) toque g y ¼ dg=dy on gan boundaes. Ths toque tends to tun the cack nto a decton that mnmzes the suface enegy. 5..3. Dssatve foces The last tem n Eq. (37) gves the two comonents of the dssatve foce F ðdsþ ¼ vw þ þ dx dx q fq f. (54) The lmt whee the dsk aea O tends to nfnty has been taken snce the ntegand vanshes outsde the ocess zone. In contast to the confguatonal and cohesve foces, the dssatve foce clealy deends on the detal of the undelyng dffuse nteface model. 5..4. Foce-balance and ansotoc genealzaton of the ncle of local symmety Substtutng the esults of Eqs. (39) (54) nto Eq. (37), the two condtons of Eq. (37) can be ewtten n the comact fom F ¼ G G c F ðdsþ ¼, (55) F ¼ G y ðþ G cy ðþ F ðdsþ ¼, (56) whee we have used the fact that G cy ¼ g y. Eq. (55) togethe wth Eq. (54) edcts the cack seed fo G close to G c v w R dx dx ðq f Þ ðg G cþ, (57) whee f s the hase-feld ofle fo a statonay cack (Kama and Lobkovsky, 4), and thus the ntegal n the denomnato above s just a constant of ode unty. Eq. (56), n tun, edcts the cack ath by mosng K at the cack t, K ¼ ðg cy ðþþf ðdsþ Þ=ðaK Þ. (58) The comonent F ðdsþ of the dssatve foce vanshes wth the cack velocty n the quasstatc lmt. So, n ths lmt, the mcoscoc detals of the ocess zone do not lay a ole and the cack decton s unquely detemned by the dectonal ansotoy of the mateal though the smlfed condton K ¼ G cy ðþ=ðak Þ. (59) Eq. (59) elaces the ncle of local symmety fo a mateal wth an ansotoc suface tenson enegy. It educes of couse to the ncle of local symmety n an sotoc mateal, snce then G cy vanshes. One can also note that qute emakably, Eq. (59) only contans macoscocally defned aametes and s ndeendent of the detaled hyscs of the ocess zone. Outsde the quasstatc lmt, K ¼ should contnue to hold fo an sotoc mateal snce F ðdsþ vanshes even fo a fnte cack seed. The latte follows fom the symmety of the nne hase-feld soluton fo a oagatng cack wth K ¼, fðx ; x Þ¼fðx ; x Þ, whch mles that the oduct q fq f n Eq. (54) s ant-symmetc and that the satal ntegal of ths oduct vanshes. In an ansotoc mateal, howeve, f s geneally not symmetcal about the local cack axs and F ðdsþ should geneally be non-zeo. The cack decton should then become deendent on the detals of the enegy dssaton n the ocess zone. A small velocty exesson fo the dssatve foce eendcula to the cack axs can be obtaned by consdeng the hase-feld ofle that coesonds to a statonay Gffth cack n an ansotoc mateal. Hee, f A, s unquely defned as the statonay hase-feld ofle that exsts fo a unque a of values of K and K that satsfy the condtons of equlbum aallel and eendcula to the cack axs, aðk þ K Þ¼G cðþ and ak K ¼ G cy ðþ, esectvely. Fo small velocty, Eq. (54) must theefoe educe to F ðdsþ ¼ vw IðÞ whee the ntegal IðÞ dx dx q f A q f A (6)
354 V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 s a dmensonless constant that, lke G c and G cy, deends geneally on the local oentaton of the cack wth esect to some fxed efeence axs chosen hee as y ¼. Fo small velocty, Eq. (58) theefoe becomes K ¼ ðg cy ðþþvw IðÞÞ=ðaK Þ, (6) whee IðÞ vanshes n the sotoc lmt snce f A aoaches f and hence becomes symmetcal about the cack axs n that lmt. 5.3. Comason wth the maxmum enegy-elease-ate cteon In sotoc mateals, the ncle of local symmety and the maxmum-enegy-elease-ate cteon concde fo smooth cacks but gve slghtly dffeent esults n geneal. A well-studed case s the gowth of a knk extenson at the t of a staght cack, as dscussed below. Fo smooth cacks, t should be noted that foce-balance, the esent ansotoc genealzaton (Eq. (59)) of the ncle of local symmety, also concdes wth a genealzaton of the maxmum enegy-elease-ate cteon to ansotoc mateals. One way to defne the latte s to eque cack gowth to take lace n the decton that maxmzes GðyÞ gðyþ (Leblond, 5) whee as befoe GðyÞ s the enegy-elease-ate fo an nfntesmal extenson at the cack t at an angle y (and y ¼ s the decton of the unextended cack). Fo a smooth cack, the condton that ths quantty be maxmal n the cack decton yelds d dy ½ GðyÞ gðyþš y¼ ¼. (6) Wth the hel of Eq. (46), ths s seen to be dentcal to Eq. (59) as stated. Fo non-smooth cack aths, foce-balance (Eq. (59)) and the maxmum enegy-elease-ate cteon gve dffeent edctons. The gowth of a knked extenson at the t of a staght cack subject to mxed-mode I/II loadng has been studed as a test case by many authos (see e.g. Amestoy and Leblond, 99; Hutchnson and Suo, 99) and efeences theen). Although dstnct, the edctons fo the theshold enegy-elease-ates and the knk angles n an sotoc medum ae vey close fo the two ctea. Thus fo sotoc meda, the dffeence between the two ctea s nsgnfcant n actce but nonetheless leads to some questons of theoetcal nteest. 3 Fo nstance, t has emaned unclea what the fate of a cack would be f loaded n the naow ga n whch knk oagaton s allowed by the maxmum enegy-eleaseate cteon but fobdden by the ncle of local symmety. Inteestngly, the stuaton s dffeent fo a knked cack n an ansotoc medum. It s not dffcult to fnd cases whee thee s a sgnfcant dffeence between the knk angles edcted by the maxmum enegy-elease-ate cteon and the foce-balance condton (Eq. (59)) whch genealzes the ncle of local symmety. As a smle llustaton, we consde the gowth of a staght cack wth a load ceatng a mxed-mode sngula stess dstbuton at the cack t, wth SIF K and K (Eq. ()). The SIFs K ðyþ and K ðyþ at the end of a small extenson, at an angle y wth the decton of the man cack, ae gven as lnea combnatons of K and K as descbed by Eq. (45). Wth these notatons, the enegy-elease-ate at the t of the small extenson eads GðyÞ ¼a½K ðyþ þ K ðyþ Š. (63) Foce-balance, Eq. (59), edcts that above the Gffth theshold G fb, a knked extenson wll gow at an angle y fb such that G fb ¼ Gðy fb Þ¼gðy fb Þ, (64) ak ðy fb ÞK ðy fb Þ dg dy ¼. (65) y¼yfb Maxmum enegy- elease-ate edcts the dffeent Gffth theshold G me and extenson angle y me that satsfy G me ¼ Gðy me Þ¼gðy me Þ, (66) d G dy dg y¼yme dy ¼. (67) y¼yme We have numecally nvestgated the dffeence between these two edctons fo a smle two-fold ansotoy qffffffffffffffffffffffffffffffffffffffffffffffffffffff gðyþ ¼g cosðyþ, (68) whee, fo convenence, the ansotoy axes have been otated by =4 comaed to ou evous choce (Eq. (8)) and we have uosely chosen a lage value of ¼ to oduce an examle whee the knk angles edcted by the ncle of maxmum enegy-elease-ate and the foce-balance condton dffe makedly. As befoe, the decton of the man cack stands at y ¼. 3 We ae ndebted to J.J. Mago fo emhaszng ths ont to us.
V. Hakm, A. Kama / J. Mech. Phys. Solds 57 (9) 34 368 355.. -. -. Gffth Foce balance Maxmum enegy-elease-ate 4 6 8 θ (deg) 7. 6 5 θ fb.5 θ (deg) 4 3 θ me Λ fb /Λ me..5-5 -4-3 - - -5-4 K /K -3 - - K /K Fg. 4. Comason of foce-balance and maxmum enegy-elease-ate cteon edctons fo a knk gowng at the t of a man cack unde mxedmode loadng n an nfnte medum wth ansotoc suface enegy gven by Eq. (68). (A) The angles edcted by the two ctea ae shown fo the atcula SIF ato K =K ¼ 4 and two dffeent loads, one (thn lnes) just above the theshold load fo the maxmum enegy-elease-ate and the othe (thck lnes) just above the theshold load fo the foce-balance cteon. Fo the maxmum enegy-elease-ate theshold load, the admensoned dffeence between the enegy-elease-ate at the small extenson t and the ansotoc nteface enegy ½ GðyÞ gðyþš=ðg Þ (thn sold lne) s negatve eveywhee excet nea the maxmum y me 34. The devatve of ths cuve (thn dot-dashed lne) ntesects zeo at y me, the angle edcted by the maxmum enegy-elease-ate cteon (Eq. (67)). The angle y fb edcted by foce-balance, y fb 57:5, s gven by the vanshng of the foce-balance condton (thn dashed lne, Eq. (65) lotted hee n an admensoned fom afte dvson by g ) but t does not coesond to a gowng extenson snce ½ Gðy fb Þ gðy fb ÞŠ=ðg Þ s negatve fo ths load. The thck lnes show analogous cuves fo the slghtly lage load that coesonds to the theshold of oagaton fo the foce-balance condton. The ntesecton of the foce-balance condton (thck dashed lne) wth zeo gves y fb 58 at theshold. (B) Knk angle as a functon of K =K fo foce-balance (y fb, sold lne) and maxmum enegy-elease-ate (y me dashed lne) at the Gffth theshold fo cack gowth at that atcula angle. (C) ato of the theshold loads fo gowth edcted by the two ctea as measued by the ato L fb =L me of the theshold SIFs fo the aent cack (see man text fo detals). The comutaton of the enegy-elease-ate at the t of the small extenson GðyÞ and of ts devatve eques the evaluaton of the functons F ðyþ (Eq. (45)) and of the devatves fo fnte angles. Ths has been efomed by usng the sees exansons of the F s (u to ode y ) as gven by Amestoy and Leblond (99). 4 We magne that the load on the aent cack as measued by the SIFs at ts t s gven by K ¼ Lðg =aþ = ; K ¼ L k ðg =aþ =, whee the facto ðg =aþ = s ntoduced fo convenence to make L dmensonless (L ¼ =4 coesonds to the onset of oagaton of a staght cack fo K ¼ fo the ansotoy (68)). As shown n Fg. 4A, when L s nceased fom zeo, keeng a constant ato K ¼ K =K of the mode I and II SIFs, a fst ctcal value L me s attaned fo whch the maxmum enegyelease-ate cteon allows knk gowth at angle y me. When L s nceased futhe a second ctcal value, L fb, s eached fo whch the knked extenson can gow at an angle y fb wth foce-balance holdng at ts t. The obtaned values of the extenson angles and of the ato L fb =L me ae dslayed n Fg. 4B and C, as a functon of the ato K ¼ K =K. As n the sotoc case, the load dffeence between the two ctea emans small (of the ode of % n the esent case, as seen n Fg. 4C) but, qute emakably, t s seen n Fg. 4B that the esence of ansotoy (Eq. (68)) endes the angle dffeence between the two ctea much moe onounced than n an sotoc medum. The lage angle dffeence oens u the ossblty to dscmnate between the two ctea n full numecal smulatons. Also, even though the fobdden ga between the theshold loads fo oagaton gven by the maxmum enegy-elease-ate and 4 We have checked that these sees gve a vey ecse estmaton of the F ðyþ s u to y ¼ 9, by comason wth the full numecal evaluatons of Katzav et al. (7).