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2 Engineering Frture Mehnis 76 (29) Contents lists ville t SieneDiret Engineering Frture Mehnis journl homepge: Anlysis of frture n elmintion in lmintes using 3D numeril moelling Mrel V. Ci Alfro, Akke S.J. Suiker, *, René e Borst, Joris J.C. Remmers Chir of Engineering Mehnis, Fulty of Aerospe Engineering, Delft University of Tehnology, Kluyverweg 1, 2629 HS Delft, The Netherlns Chir of Numeril Methos in Engineering, Deprtment of Mehnil Engineering, Einhoven University of Tehnology, Einhoven, The Netherlns rtile info strt Artile history: Reeive 31 Jnury 28 Reeive in revise form 17 July 28 Aepte 1 Septemer 28 Aville online 18 Septemer 28 Keywors: Interfe mge moel Douly-eflete rk Crk tunneling Centre-rke tensile speimen Fire metl lmintes GLARE The stti filure ehviour of the fire metl lminte GLARE is exmine using 3D finite element simultions. The onfigurtion nlyse is entre-rke tensile speimen ompose of two luminium lyers snwihing ross-plie, fire-epoxy lyer. The rk n elmintion growths re simulte y mens of interfe elements equippe with mixe-moe mge moel. The moe-mixity is erive from n energy riterion typilly use in liner elsti frture mehnis stuies. The mge kineti lw is rte-epenent, in orer to simulte rte effets uring interfil elmintion n to voi numeril onvergene prolems ue to rk ifurtions. The numeril implementtion of the interfe mge moel is se on kwr Euler pproh. In the ounry vlue prolem stuie, the filure responses of GLARE speimens ontining elsti luminium lyers n elsto-plsti luminium lyers re ompre. The evelopment of plsti eformtions in the luminium lyers stilizes the effetive filure response, n inreses the resiul strength of the lminte. For qusi-rittle GLARE speimen with elsti luminium lyers, the resiul strength is governe y the toughness for interfil elmintion, n is in lose orresponene with the resiul strength otine from lose-form expression erive from energy onsiertions. Conversely, for utile GLARE speimen with elsto-plsti luminium lyers, the resiul strength is lso etermine y the reltion etween the frture strength n the yiel strength of the luminium. The mount of onstrint y the horizontl isplements t the vertil speimen eges hs moerte to smll influene on the resiul strength. Furthermore, the ultimte lminte strength is lower for lrger initil rk length, n shows to e in goo orresponene with experimentl vlues. Ó 28 Elsevier Lt. All rights reserve. 1. Introution Filure uner tensile loing is esign limiting hrteristi of fire-reinfore, lyere omposites. Experimentl stuies hve shown tht the effetive tensile response of these mterils is oune y the evelopment of vrious filure mehnisms t lower sle, suh s trnsverse mtrix rking, fire eohesion n frture, n interfil elmintion, see for exmple [1 7] n referenes therein. In ition, moelling stuies hve emonstrte tht the rking n elmintion ptterns typilly oserve in lyere omposites epen upon lol geometril n mteril properties, suh s the numer, lotion n size of initil flws, the stking sequene, the fire volume frtion, the toughness n stiffness hrteristis of the iniviul plies, the interfil elmintion toughness, n the presene of resiul stresses [8 16]. * Corresponing uthor. Tel.: ; fx: E-mil ress: A.S.J.Suiker@tuelft.nl (A.S.J. Suiker) /$ - see front mtter Ó 28 Elsevier Lt. All rights reserve. oi:1.116/j.engfrmeh

3 762 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) Tunneling iretion Tunneling iretion Tunneling iretion Moe I rk without elmintion H shpe rk with onstnt elmintion length Unstle elmintion growth in ll iretions Mehnism 1 Mehnism 2 Mehnism 3 Fig. 1. Three possile filure senrios for rittle lminte ompose of two issimilr, isotropilly elsti mterils sujete to unixil tension (tken from Suiker n Flek [17]). Mehnism 1: Tunneling of stle moe I rk with elmintion sent. Mehnism 2: Tunneling of stle H-shpe rk with onstnt elmintion length. Mehnism 3: A tunneling rk with unstle elmintion eveloping in ll iretions. Reently, Suiker n Flek [17] stuie the ompetition of three possile filure senrios for lminte ompose of two issimilr isotropilly elsti lyers, sujete to unixil tension, see Fig. 1. These filure senrios re ssume to hve grown from lrge pre-existing flw in the mi-lyer (mteril #1), where mehnism 1 reflets the tunneling of stle moe I rk in the mi-lyer with elmintion sent, mehnism 2 represents the tunneling of stle H-shpe rk with onstnt elmintion length n mehnism 3 reltes to tunneling rk with unstle elmintion growth in ll iretions. It ws foun tht the opertive filure mehnism is strongly etermine y the reltive toughness of lyer n interfe, n to lesser extent y the stiffness mismth of the lyers, the lotion(s) of the initil flws(s), n the numer of plies. The results presente in [17] were etermine omining 2D liner elsti frture mehnis solutions for the prolems of (i) plne-strin elmintion of n H-shpe rk n (ii) stey-stte tunneling of n H-shpe rk. In line with this pproh, the rk nuletion phse ws ignore, s well s the presene of plsti eformtions in the iniviul lyers. These effets, however, my ontriute to the stti filure ehviour of lmintes ontining utile lyers, suh s the fire metl lmintes ARALL n GLARE. 1 In the present pper, the effets of plstiity n rk nuletion on the tensile filure response of GLARE re exmine y mens of 3D finite element nlyses. The onfigurtion stuie is ompose of two luminium lloy sheets snwihing ross-plie, fire-epoxy lyer, where horizontl initil rk is ple t the entre of the speimen, ross the thikness of the luminium sheets, i.e., entre-rke tensile speimen is stuie. After imposing the tensile loing, the entre rk strts to tunnel in the luminium lyers n inues elmintion t the interfes etween the luminium n fire-epoxy lyers, effets tht re simulte y mens of interfe elements equippe with mixe-moe mge moel, see lso [18 24]. For the interfe mge moel use in the present stuy, the formultion of the moe-mixity is se upon n energy riterion regulrly pplie in liner elsti frture mehnis stuies [12,25], using erivtion proeure similr to the one propose reently y Turon et l. [26]. In ition, the kineti lw esriing the evolution of the mge proess is tken s rte-epenent. This is one to ount for rte effets generte uring interfil elmintion, n to voi numeril onvergene prolems inue y rk ifurtions. The inorportion of these two fetures in the interfe mge formultion istinguishes the present moel from most other moels presente in the literture. The iniviul luminium n fire-epoxy lyers of the lminte re moelle y soli-like shell elements [22,24,27]. These elements llow for liner strin fiel in thikness iretion, whih vois the effet of Poisson-thikness loking tht ppers in onventionl volume elements with high spet rtio in sptil imensions, i.e., slener volume elements [28]. The isotropi, elsto-plsti ehviour of the luminium lyer is simulte using J 2 -plstiity moel with n exponentilly sturting hrening lw. The ross-plie, fire-epoxy lyer is moelle s isotropilly elsti. This simplifition, whih is resonle if the elsti mismth etween fires n mtrix is moerte, llows prt of the numeril results to e quntittively ompre to those presente in Suiker n Flek [17]. Furthermore, the frture mehnisms oserve in the present stuy n then e equtely vlite ginst the filure senrios epite in Fig. 1. The pper is orgnise s follows: Setion 2 ontins the formultion of the interfe mge moel use for the simultion of frture within the lminte lyers, n mixe-moe elmintion long the interfes etween the lyers. The 1 ARALL n GLARE re lmintes ompose of n lternting stk of luminium sheets n fire-epoxy (prepreg) lyers, where ARALL is reinfore y rmi fires n GLARE is reinfore y glss fires, see [5] for more etils on the hrteristi properties, mnufturing proeures n pplitions of these mterils.

4 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) trtion seprtion lw n the rte-epenent kineti lw for interfil mge re speifie, where the moe-mixity for mge growth is erive from liner elsti frture mehnis onepts. In Setion 3, the time isretistion of the moel is isusse, whih is se on kwr Euler pproh. In Setion 4, the interfe mge moel is inorporte into three-imensionl finite element moel for entre-rke GLARE speimen sujete to unixil tension. The geometry n ounry onitions re speifie, followe y isussion of the finite element isretistion n the mteril properties. The numeril results otine with the moel re isusse in Setion 5, strting with the response of qusi-rittle GLARE lminte with elsti luminium lyers (where the results re ompre to those presente in [17]), followe y the response of utile GLARE lminte with elsto-plsti luminium lyers. The filure response is ompute for two ifferent types of ounry onitions t the vertil eges of the speimen. The setion ens with stuy of the effet of the initil rk length on the lminte filure strength, where the results re ompre with experiments of e Vries [6]. In Setion 6, the min nlysis results re summrise. 2. Formultion of the interfe mge moel In the urrent setion the governing equtions of the interfe mge moel re presente. The rrngement n tretment of these equtions within onseutive lgorithmi frmework n e foun in Setion 3. For ohesive zone moels use in 3D soli mehnis nlyses, the trtions t i t the interfe moelling the ohesive zone n the reltive isplements v i ross the interfe onsist of three omponents: i 2f1; 2; 3g, with the numers enoting the norml iretion n the two tngentil iretions t the interfe, respetively. For onveniene, the tngentil iretions in the plne of rk propgtion re tken prllel (inex 2 ) n perpeniulr (inex 3 ) to the iretion of rk. The trtions n reltive isplements re relte y mens of onstitutive formultion, whih, in the present stuy on interfil mge, hs the form t i ¼ð1 ÞC ij v j C ij 1j h v 1 i where i; j 2f1; 2; 3g; with the mge prmeter oune s Here, ¼ orrespons to the initil, unmge stte, n ¼ 1 to the stte t whih the integrity of the interfil mteril point is fully lost. Further, C ij is the elsti stiffness tensor, given y C ij ¼ K ij ; with K stiffness prmeter n ij the Kroneker elt symol. Oserve from the lst term in Eq. (1) tht rk penetrtion of two opposite rk fes is voie, y presriing these fes to intert elstilly in the norml iretion of the interfe uring ontt, with the elsti ontt stiffness eing equl to K. The ft tht rk fe ontt is hrterise y negtive vlue of the norml rk fe isplement v 1 is ounte for y the Muley rkets hi, whih re efine s hxi ¼ 1 ðx þjxjþ. 2 During loing proess, the mge in n interfil mteril point evolves with eformtion, s formlly expresse y ¼ ^ðjþ with j eformtion history vrile tht is monotonilly inresing (sine mge is n irreversile proess). The speifi form of ^ðjþ orrespons to the shpe of the softening urve of the trtion seprtion lw. In the present stuy, liner softening lw is opte, s shemtize in Fig. 2, where the onset of mge reltes to j ¼ v (orresponing to ¼ ) n the ompletion of mge is reflete y j ¼ v u (orresponing to ¼ 1), with v n v u the equivlent rk fe isplements t whih mge is onsiere to e initite n omplete, respetively. Although the softening lw my hve other forms (e.g., multi-liner, exponentilly eying), the shpe of the softening lw ommonly hs minor influene on the hrteristis of the frture proess, espeilly in the se of utile frture [29 31]; the frture proess is minly etermine y the ultimte trtion t u n the frture toughness G (whih equls the re uner the trtion seprtion urve, see Fig. 2). As further shown in Fig. 2, for speifi vlue of j the equivlent trtion is equl to Kv ðv u jþ=ðv u v Þ, ð1þ ð2þ Onset of mge proess Effetive trtion t u =Kv Kv (v u -κ) (v u -v ) K G (1-)K v κ v u Effetive reltive isplement Completion of mge proess Fig. 2. Trtion seprtion lw.

5 764 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) or, lterntively, in terms of the mge prmeter, equl to ð1 ÞKj. Equting these two expressions for the trtion les to the following expression for the mge prmeter: ¼ ^ðjþ ¼ vu ðj v Þ jðv u v Þ : In ontrst to rte-inepenent mge proesses, for rte-epenent mge proesses the evolution of the mge prmeter is not set y the eformtion only, ut lso y its rte, s esrie y speifi kineti lw. The tul vlue of the orresponing history vrile j is then otine y the inverte form of Eq. (3). In the present stuy, the following rte-epenent kineti lw is propose: 8 >< ^Fðk; jþ for k P j n v _ ¼ g 6 j < v u ; ð4þ >: for 6 k < j or j ¼ v u ; where g is relxtion prmeter (with imension of time) n ^Fðk; jþ is the mge loing funtion. In Eq. (4), the upper expression reflets the rte of mge when the effetive eformtion k exees the threshol j, wheres the lower expression sets the rte of mge equl to zero when (i) the threshol vlue hs not (yet) een rehe, (ii) the interfil mteril point is in stte of unloing, or (iii) the mge proess hs omplete. qthe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eformtion mesure k is tken here s the Eulien qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi norm of the vetor of reltive rk fe isplements, k ¼jvj ¼ v 2 1 þ v2 sh, with v sh the totl sher isplement, v sh ¼ v 2 2 þ v2 3. Furthermore, the loing funtion hs the form ^Fðk; jþ ¼^f ðkþ ^ðjþ ¼ vu ðk v Þ kðv u v Þ vu ðj v Þ jðv u v Þ ; where the right expression is otine y sustituting Eq. (3) for ^ðjþ, n hoosing the form of ^f ðkþ to e similr s ^ðjþ. The speifi form of the kineti lw, Eq. (4), is nlogous to the form often use in viso-plstiity moelling (see e.g. [32]), with the equivlent plsti strin rte eing reple y the mge rte n the stti yiel funtion y the mge loing funtion. In the limit of the relxtion prmeter going to zero, g!, the kineti lw, Eq. (4), turns into the rte-inepenent loing onition, ^Fðk; jþ ¼, whih, s n e oserve from Eq. (5), is ientil to k ¼ j. Uner these irumstnes, the loing unloing onitions re represente y the Kuhn Tuker reltions ðk jþ _j ¼ ; k j 6 ; _j P : ð6þ Hene, the present interfe mge moel n e use for esriing oth (virtully) rte-inepenent frture in the metl lyers of GLARE n rte-epenent elmintion t the metl-prepreg interfes, y setting the relxtion prmeter g oringly. In mixe-moe frture proesses, the equivlent rk fe isplements v n v u ppering in Eq. (5) re epenent on the reltion etween the norml n sher isplements t the interfe. As reently propose y Turon et l. [26], this reltion my e pture y the following moe-mixity prmeter: v sh ¼ v sh þhv 1 i : ð7þ In this efinition, pure moe I loing is reflete y v sh ¼, n thus y ¼, wheres pure sher loing reltes to v 1 ¼, n thus to ¼ 1. Turon et l. [26] emonstrte tht the funtions v ¼ ^v ðþ n v u ¼ ^v u ðþ n e ompute opting speifi mixe-moe filure riterion from liner elsti frture mehnis. The mixe-moe riterion opte in the present stuy is moel regulrly use to hrterise mixe-moe toughness t for rittle interfil frture [12,25], i.e. ð3þ ð5þ G I G I; þ G II G II; þ G III G III; ¼ 1; ð8þ where G I, G II n G III re the moe I, moe II n moe III energy relese rtes, n G I;, G II; n G III; re the toughnesses uner pure moe I, pure moe II n pure moe III loing onitions. For simpliity, the moe II n moe III frture toughnesses re ssume to hve ommon vlue, G sh; ¼ G II; ¼ G III;, s result of whih the riterion (8) reues to G I G I; þ G sh G sh; ¼ 1; ð9þ with G sh ¼ G II þ G III. Essentilly, the ove frture riterion is n extension of the well-known Griffiths riterion, G ¼ G, where G is the totl energy relese rte mesure t the rk tip n G is the effetive frture toughness (whih thus epens on the moe-mixity of the loing). As lrey mentione, the frture toughness G is represente y the re uner the trtion seprtion urve in Fig. 2, n thus n e ompute s G ¼ G ðþ ¼ 1 2 K^v ðþ^v u ðþ: ð1þ In the limits of pure moe I loing ( ¼ ) n pure sher loing ( ¼ 1), the frture toughness in Eq. (1), respetively reues to:

6 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) G I; ¼ ^G ð ¼ Þ ¼ 1 2 Kv 1 vu 1 ; G sh; ¼ ^G ð ¼ 1Þ ¼ 1 2 Kv sh vu sh ; ð11þ where v 1 ¼ tu 1 =K is the isplement t whih mge is initite uner pure moe I loing, with tu 1 eing the ultimte norml trtion, v sh ¼ tu sh =K is the isplement relte to mge initition uner pure sher loing, n vu 1 n vu sh re the orresponing ultimte isplements t whih mge qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is omplete, with the ultimte sher trtion t u sh eing ompute from the two tngentil trtion omponents s t u sh ¼ ðt u 2 Þ2 þðt u 3 Þ2. In orer to fin expressions for v n v u in terms of the moe-mixity, Griffiths riterion, G ¼ G, is inorporte into Eq. (9), whih les to G I G I; þ G sh G sh; ¼ G G : ð12þ This expression n e further evelope y writing the totl energy relese rte s 2 G ¼ G I þ G sh n sustituting this form into the right-hn sie of Eq. (12). In line with this eomposition, the iniviul rk moe omponents of the energy relese rte re expresse in terms of the reltive rk fe isplements s G I ¼ v 2 1 n G sh ¼ v 2 sh, with proportionlity ftor (with imension of fore length -3 ) tht epens on the stiffness n geometry properties of the onfigurtion uner onsiertion n on the tul position long the rk fes t whih the reltive rk fe isplements re evlute (ommonly mesure with respet to the origin of the rk tip, see [12]). Invoking Eq. (1), n using the efinition of the moe-mixity prmeter, Eq. (7), to express the reltive norml isplement in terms of the reltive sher isplement s v 1 ¼ v sh ð1 Þ=, Eq. (12) n e elorte into n expression for v u : "!!# v u ¼ ^v u ðþ ¼ 2ð1 þ Þ ð1 Þ þ 2 ; ð13þ Kv G I; G sh; with G I; n G sh; given y Eq. (11). An expliit expression for the prmeter v n e foun y sustituting Eq. (11) 1,2 into Eq. (13), repling v u, v u 1 n vu sh y the orresponing initil vlues v, v 1 n v sh, n solving for v. This results in vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u v ¼ ^v ðþ ¼v 1 þ v t sh v 2 1 þ ð1 Þv 2 : ð14þ sh Note from Eqs. (13) n (14) tht the moe-mixity is inee the only vrile in the expressions for v u n v (sine the other prmeters represent interfil frture t). 3. Time isretistion of the interfe mge moel In orer to perform finite element simultions with the interfe mge moel presente in the previous setion, the moel formultion nees to e isretise in time. Using kwr Euler pproh, for eh isrete time intervl ½t n ; t nþ1 Š the moel vriles in n interfil integrtion point re evlute t time t nþ1 (¼ t n þ Dt nþ1 ) ssuming the orresponing vlues t the previous time step t n re known. In isplement-se finite element metho, the inrementl upte t integrtion point level is governe y the reltive isplements v nþ1 ross the interfe. These isplements re provie s input from the glol itertive proeure t the system level. With the reltive isplements, the eformtion mesure k is upte s k nþ1 ¼ ^kðv nþ1 Þ¼kv nþ1 k. This vlue is ompre ginst the history prmeter j n ompute t the previous time step, where for k nþ1 > j n mge is ssume to our, n for k nþ1 6 j n the response is onsiere to e elsti (i.e., no mge hs ourre (yet), or the integrtion point is sujete to elsti unloing). The history prmeter is initilise s j n ¼ v nþ1 ¼ ^v ð nþ1 Þ, using Eq. (14) with nþ1 ¼ v sh;nþ1 =ðv sh;nþ1 þhv 1;nþ1 iþ. Susequently, the mge inrement D nþ1 is ompute y omining Eqs. (4) n (5) (with ll prmeters evlute t t nþ1 ), with the time isretistion of the mge rte in orne with D nþ1 _ nþ1 Dt nþ1. This les to 8 < ðf nþ1 n ÞDt nþ1 for k D nþ1 ¼ g nþ1 > j n ; þ Dt nþ1 : for 6 k nþ1 6 j n : ð15þ 2 An itive eomposition of the energy relese rte into its iniviul rk moe omponents is llowe if the stiffness properties of the two elsti ulk mterils seprte y propgting, rittle interfil rk re in greement with the seon Dunur s stiffness mismth prmeter eing equl to zero, see [12]. This onition is not met stritly for most of the elsti frture onfigurtions stuie in this work. Moreover, the present interfe rk moel is use in ounry vlue prolems where the ulk mterils experiene plsti yieling. However, se on heuristi resoning it my e ssume tht the effet of this isrepny on the omputtionl results remins reltively smll.

7 766 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) where f nþ1 is given y f nþ1 ¼ vu nþ1 k nþ1 v nþ1 k nþ1 v u nþ1 ; ð16þ v nþ1 n v u nþ1 ¼ ^vu ð nþ1 Þ is lulte from Eq. (13). Formlly, the mge proess is omplete if the mge prmeter rehes unity. Corresponingly, in the present lgorithm ompletion of mge is heke y mens of the onition: n þ D nþ1 > 1 e, where e is smll positive vlue ( < e 1), introue here to improve the glol numeril onvergene ehviour when lolly mge hs omplete. If the ove onition hols, the mge proess is onsiere to hve finishe n, oringly, the mge prmeter is then set equl to the mximum vlue, nþ1 ¼ 1 e. Hene, the upte of the mge prmeter n e onisely formulte s nþ1 ¼ min ð n þ D nþ1 ; 1 eþ: ð17þ For ny mge vlue within the rnge 6 nþ1 6 1 e, the trtion vetor, t nþ1 ¼ ^tðv nþ1 ; nþ1 Þ, is ompute y sustituting the reltive isplement vetor, v nþ1 together with the upte mge vrile, nþ1 ¼ ^ðv nþ1 Þ, into Eq. (1), whih results in t i;nþ1 ¼ð1 nþ1 ÞKv i;nþ1 nþ1 K 1i h v 1;nþ1 i with i 2f1; 2; 3g; where use hs een me of Eq. (2). In ition, the history prmeter j is upte using the inverte form of Eq. (3), i.e. v nþ1 vu nþ1 j nþ1 ¼ v u nþ1 ðvu nþ1 v nþ1 Þ : ð19þ nþ1 From the ove expression, it n e onfirme tht ompletion of mge orrespons to j nþ1 ¼ v u nþ1 (when ignoring the ontriution y the smll vlue e in Eq. (17)). Note tht the present time integrtion proeure n e performe without ny itertions, ue to the use of the speifi mge loing funtion, Eq. (5), in the kineti lw, Eq. (4). Alterntive, more omplex forms of the mge loing funtion require n itertive time integrtion proeure t the integrtion point level, whih oviously is omputtionlly more expensive. The tngent opertor neessry for onstruting the stiffness mtrix t the system level follows from ð18þ ^t nþ1 v nþ1 ¼ o^t nþ1 ov nþ1 þ o^t nþ1 o nþ1 o^ nþ1 ov nþ1 ; ð2þ whih, with Eq. (18), les to ^t i;nþ1 ¼ K ij nþ1 K v j;nþ1 h v 1;nþ1 i ij þ 1i 1j v 1;nþ1 o^ nþ1 K v i;nþ1 þ 1i h v 1;nþ1 i : ð21þ ov j;nþ1 Here, the erivtive o^ nþ1 =ov nþ1 n e etermine from Eqs. (15) to (17) s 8 v nþ1 vu nþ1 Dt nþ1 v g j;nþ1 þ Dt nþ1 o^ nþ1 ov j;nþ1 ¼ >< k 3 nþ1 ðvu nþ1 v nþ1 Þ 1 þ h v 1;nþ1i 1j v j;nþ1 for k nþ1 > j n n < nþ1 < 1 e; >: else; ð22þ where the multiplition ftor ð1 þ 1j hv 1;nþ1 i=v j;nþ1 Þ hs een e in the upper expression to ount for the ft tht the erivtive in the norml iretion of the interfe, o^ nþ1 =ov 1;nþ1, is zero uring (elsti) rk fe ontt. For resons of simpliity, the extensive terms relte to the erivtives o^v nþ1 =ov j;nþ1 n o^v u nþ1 =ov j;nþ1 re ignore in Eq. (22). This simplifition is eptle if the moe-mixity vries rther slowly with eformtion, using these erivtives to remin reltively smll. However, for ounry vlue prolems where the moe of frture hnges ruptly with eformtion, itionl omputtions not presente here hve shown tht the tngent opertor given y Eq. (22) my inue rsti ollpse in the glol onvergene rte (i.e., trnsition from qurti to qusi-liner onvergene rte). Uner these irumstnes it my e preferentil to ompute the tngent opertor numerilly (see [33] for the se of smll eformtions n [34,35] for the se of finite eformtions), in orer to preserve the qurti onvergene rte t the system level n to voi the nlytil etermintion of the extensive erivtives o^v nþ1 =ov j;nþ1 n o^v u nþ1 =ov j;nþ1. 4. Numeril moel of entre-rke 2/1 GLARE lminte sujete to unixil tension 4.1. Geometry n ounry onitions The interfe mge moel presente in the previous setion is use to stuy moe I frture n interfil elmintion in entre-rke 2/1 lminte sujete to unixil tension, see Fig. 3. The 2/1 ly-up refers to lminte ompose of two luminium lyers snwihing single fire-epoxy lyer. Lmintes reinfore y glss fires re ommerilly known

8 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) u 2/1 GLARE lminte 2 L = 6 mm u W = 4 mm Length entre rk: 2 = 4.8 mm Fig. 3. Dimensions of entre-rke tensile speimen me of 2/1 GLARE. s GLARE. For this mteril the luminium n fire-epoxy lyers ommonly hve thiknesses of.3 mm n.25 mm, respetively (see e.g. [6]), whih les to totl speimen thikness of.85 mm. The speimen length is tken s L ¼ 6 mm n its with equls W = 4 mm. The initil pre-rk t the entre of the speimen is ple ross the omplete thikness of the two luminium lyers, n hs length of 2 ¼ :12W ¼ 4:8 mm. The tensile loing is impose y presriing remote norml isplement u 1 t the top n ottom eges of the lminte, whih inues uniform remote tensile stress r 1 in the luminium lyer t short istne from the top n ottom eges. Qusi-stti loing onitions re wrrnte y pplying reltively smll nominl strin rte _u 1 =L ¼ 1 5 s 1. As emonstrte y experimentl n moelling stuies on the stti n ftigue filure ehviour of entre-rke lmintes [5,6,17,36 38], uner the pplie remote tensile loing the initil pre-rk will strt to tunnel s moe I rk through the luminium sheet, therey inititing elmintion long the interfe etween the luminium (mteril #1) n fire-epoxy (mteril #2) lyers. The omplete frture mehnism hs een skethe in Fig. 4 (Left), n n e enote s tunneling, ouly-eflete rk [17]. Sine the ouly-eflete rks in the two outer lyers of the 2/1 speimen tken together resemle n H-shpe rk, this frture mehnism is omprle to the filure senrios epite in Fig. 1. u u #1 #2 #1 #1 #2 #1 Delmintion length Tunneling iretion y z x u u Fig. 4. Left: ouly-eflete rk in the two outer lyers of 2/1 GLARE lminte sujete to unixil tension (impose y remote isplement u 1 ). The lminte is ompose of two luminium (mteril #1) lyers snwihing single, ross-plie, fire-epoxy (mteril #2) lyer. Right: One symmetril otnt of the entre-rke tensile speimen, use in the numeril simultions.

9 768 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) In orer to exmine how plstiity in the luminium lyer ffets the filure hrteristis of the speimen, simultions in whih the luminium lyer ehves isotropilly elsti re ompre to simultions where the luminium is moelle s isotropilly elsto-plsti. In oth ses the fire-epoxy lyer is moelle s isotropilly elsti. This is n eptle simplifition if the internl struture of the fire-epoxy lyer is ross-plie, i.e., me of two sulyers of.125 mm thikness with the uni-iretionl fires oriente uner n 9, respetively, see Fig. 4 (Left), n the elsti mismth etween the fires n the epoxy is moerte. The ssumption of n isotropi fire-epoxy lyer inreses the egree of symmetry of the prolem, suh tht only one otnt of the tul geometry my e moelle, see Fig. 4 (Right). Trivilly, this signifintly reues the omputtionl time of the nlysis. Furthermore, this simplifition llows the present omputtionl results to e ompre to results reporte y Suiker n Flek [17], whih relte to frture n elmintion proesses in lyups ompose of rittle, isotropi elsti lyers. The symmetry of the onfigurtion epite in Fig. 4 (right) is wrrnte y fixe n roller supports, n y using ustomize interfe element tht reltes the interfe trtions of the moe I rk in the luminium (propgting long the horizontl x y symmetry plne hlf-wy the speimen) to hlf of the reltive rk fe isplements ross the interfe. The effet of the ounry onitions t the right ege of the speimen on the filure response is stuie y onsiering two ses. The first se, referre to s BC1, reflets the unonstrine sitution, where the norml trtion t the right vertil ege of the speimen (i.e., the norml trtion in the x-iretion) is presrie s zero. The seon se, referre to s BC2, reflets onstrine sitution, where the norml isplement t the right vertil ege of the speimen (i.e., the isplement in x-iretion) is set to zero. For oth ses the sher trtions t ll speimen ounries re presrie to e zero Finite element isretistion The numeril nlyses re performe within lrge-isplement, smll-strin frmework, where the luminium n fire-epoxy lyers re meshe with 16-noe, iso-prmetri soli-like shell elements, with Guss qurture [22,24,27]. As illustrte in Fig. 5, these elements hve eight noes t their element orners (inite y the soli irles 1 8 ) n eight noes hlf-wy eh element sie (inite y the open irles 9 16 ). In ition, they hve four internl noes (inite y the open squres ), whih re positione t the four orners of the mi-surfe tht is lote hlf-wy the element thikness. The internl noes re use for onstruting liner strin fiel in the thikness iretion (i.e., the f-iretion) of the element, whih remeies the prolem of Poisson-thikness loking tht hrterises onventionl volume elements with high length-to-thikness rtio see [22,24,27] for more etils. At the plne t whih interfil elmintion etween the luminium lyer n the fire-epoxy lyer is expete, the top (or ottom) surfe of the soli-like shell element, whih is spnne y eight noes in the n g plne, is onnete to sixteennoe interfe element, see Detil A in Fig. 6. This interfe element hs eight noes per surfe, n is equippe with 3 3 Newton Cotes qurture. The numeril formultion of the geometrilly non-liner interfe element is se on [18,19], see lso [22]. At the plne long whih moe I frture in the luminium lyer evelops, the sie surfe of the soli-like shell element, whih onsists of six noes in the g f (or n f) plne, is tthe to 12-noe interfe element (whih hs six noes per surfe), equippe with 3 2 Newton Cotes qurture. The totl numer of soli-like shell elements use for meshing the luminium lyer n the fire-epoxy lyer is 2 7 = 14, see Fig. 6. Corresponingly, the numer of sixteen-noe interfe elements use for meshing the plne long whih elmintion is expete equls 7. In ition, η 2 ζ ξ Geometril orner noe Geometril mi sie noe Internl noe Fig. 5. Geometry of 16-noe, iso-prmetri soli-like shell element. The element is ompose of 16 geometril noes (inite y the irles 1 16 ) n four internl noes (inite y the squres ).

10 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) BC 1 u y sls elements (fire-epoxy) DETAIL A sls elements (luminium) vertil speimen ege 3 y z x interfe elements BC 2 DETAIL A z x Initil rk 2.4 Point A Tunneling iretion Interfe elements { Symmetry plne.425 Symmetry plne ( All mesures re in [mm] ) Symmetry plne z x Fig. 6. Finite element mesh n ounry onitions for the GLARE 2/1 speimen. The se BC1 refers to speimen where the horizontl isplement t the vertil speimen ege is unonstrine, i.e., trtion-free ounry, wheres for the se BC2, epite in the lower right inset, these horizontl isplements re fully onstrine. The upper right inset shows the etils of the onnetion etween the soli-like shell (sls) elements n the interfe elements, lose to the initil pre-rk. the numer of 12-noe interfe elements use for moelling the plne long whih the moe I rk in the luminium evelops is 22. This results into totl numer of 2122 elements. As further illustrte in Fig. 6, the finite element mesh is reltively fine lose to the plne long whih the moe I rk propgtes, n eomes orser towrs the top ege of the speimen t whih the loing is pplie. The reltively smll interfe elements pturing the moe I rk growth n the initil prt of the interfil elmintion proess re pproximtely squre-shpe, hving size D. For the frture prolems stuie in [39], onvergene of the numeril results upon mesh refinement ws oserve when the vlue of D ws hosen smller thn 4 times the ultimte seprtion v u use in the trtion seprtion lw. Aoringly, in the present stuy the size of the interfe elements t the moe I rk tip n t the elmintion tip is hosen suh tht D=v u is pproximtely equl to 1.5 n 2.5, respetively Mteril properties The mteril t use in the numeril moel is liste in Tle 1. The elsto-plsti response of the luminium lyers is simulte using stnr J 2 -flow theory (Von Mises plstiity), where the yiel strength r y evolves in orne with n exponentilly-sturting hrening lw: ð1 exp ð nj p ÞÞ: ð23þ r y ¼ ^r y ðj p Þ ¼r y þ ru y r y The evolution from the initil yiel strength r y (= 35 MP) to the ultimte yiels strength ru y (= 43 MP) (representing the sturtion stte) is in orresponene with unixil stress strin t reporte in [4] for the luminium lloy 224-T3 typilly use in GLARE. The yieling sturtion stte of this type of luminium is rehe t n equivlent plsti strin j p of out 15%, whih is pture y setting the hrening rte prmeter in Eq. (23) s n ¼ 27[ ]. The effetive Young s moulus n the Poisson s rtio of the fire-epoxy lyer re ompute y verging the stiffness vlues n Poisson s rtios in the fire-iretion n perpeniulr to the fire-iretion, s reporte in [4]. The ummy elsti stiffness K of the interfe elements moelling the moe I rk n the elminting rk is tken reltively

11 77 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) Tle 1 Mteril properties of GLARE lminte Prmeter(s) Vlue(s) Aluminium lyer Young s moulus E ¼ 72 [GP] Poisson s rtio m ¼ :33 [ ] Initil yiel strength r y ¼ 35 [MP] Ultimte yiel strength r u y ¼ 43 [MP] Hrening rte prmeter n ¼ 27 [ ] Fire-epoxy lyer Young s moulus E ¼ 32 [GP] Poisson s rtio m ¼ :2 [ ] Moe I rk in luminium Elsti stiffness K ¼ [N/mm 3 ] Ultimte norml trtion (3 ses) t u 1 ¼ 43 [MP] (= 1.4r ), 61 [MP] (= 2.r ), 915 [MP] (= 3.r ), with r ¼ 35 [MP]) Frture toughness G I; ¼ 112 [N/mm] Relxtion prmeter g ¼ [s] Numeril offset t mge ompletion e ¼ 1 8 [ ] Interfil elmintion Elsti stiffness K ¼ [N/mm 3 ] Ultimte norml trtion t u 1 ¼ 5 [MP] Ultimte sher trtion t u 2 ¼ tu 3 ¼ 25 [MP] Frture toughnesses G I; ¼ G II; ¼ G III; ¼ 4 [N/mm] Relxtion prmeter g ¼ [s] Numeril offset t mge ompletion e ¼ 1 6 [-] high, suh tht in the elsti regime the interfil eformtions remin negligily smll. Oserve from Tle 1 tht the stiffness vlue for the elminting rk is tken little lower thn tht of the moe I rk in the luminium, whih hs een one in orer to optimise the onvergene ehviour of the numeril simultions. The vlue of the moe I p frture toughness, G I;, is etermine opting the vlue for the ritil moe I stress intensity ftor K I; ¼ 89:6 MP ffiffiffiffiffi m reporte in Hshgen [22] (whih ws otine y lirting the rk mouth opening isplement n the effetive rk length in entre-rke luminium sheet sujete to unixil tension), n sustituting this vlue into Irwin s reltion, G I; ¼ K 2 I; =E, where E is the Young s moulus of the luminium. Vlues for the ultimte norml trtion of the moe I rk in the luminium, t u 1, hve not een wiely reporte in the literture. In Hshgen [22], the ultimte norml trtion ws tken pproximtely 1.3 times higher thn the initil yiel strength of the luminium lyer. Furthermore, the stuies of Tvergr n Huthinson [29] n Chen et l. [39] showe tht the rtio etween the ultimte norml trtion of the rk n the initil yiel strength of the surrouning elsto-plsti ulk mteril my hve signifint influene on the effetive rk growth resistne. Corresponingly, in the present nlysis the ultimte trtion is vrie, onsiering the following three vlues: t u 1 ¼ 1:4r,2:r n 3:r, with the referene stress tken s r ¼ 35 MP (whih equls the initil yiel strength r y in the elsto-plsti moel for the luminium). The moe I frture toughness of elminting rk is otine from test t for 2/1 GLARE speimen sujete to moe I elmintion [41]. For simpliity, the moe II n III elmintion toughnesses re hosen to hve the sme vlue s the moe I toughness, so tht the effetive elmintion toughness G eomes inepenent of the moe-mixity. The ultimte trtions for the elminting rk, t u 1, tu 2 n tu 3, re tken equl to the vlues reporte in Hshgen [22], whih were etermine from sher n tensile tests on prepreg mteril typilly use in GLARE. The relxtion prmeter g for the moe I rk is tken very smll suh tht uner qusi-stti loing onitions the rk vnement ours virtully rte-inepenently. For the elminting rk, the relxtion prmeter g is hosen slightly lrger in orer to voi onvergene prolems ue to rk ifurtions; itionl simultions not presente here hve shown tht in the limit of truely rte-inepenent frture ehviour (i.e., g! ) the onvergene of the present numeril simultions nnot e wrrnte. A etile nlysis on rte effets uring interfil elmintion is onsiere to e topi for future stuies. Finlly, oth for the moe I rk n the elminting rk the prmeter e, whih represents the (smll) numeril offset t mge ompletion, see Eq. (17), is tken equl to the inverse vlue of the orresponing elsti stiffness K of the interfe. 5. Moelling results In Tle 2 n overview of the numeril simultions is given. With the present seletion of simultions, the influene of the following four spets on the effetive filure response is stuie: (i) The vlue of the ultimte trtion of the moe I rk in the luminium, (ii) the genertion of plstiity in the luminium lyers, (iii) the type of ounry onitions t the vertil speimen ege, n (iv) the initil rk length.

12 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) /1 ly-up with elsti luminium lyers For the se where the luminium lyer ehves elstilly, Fig. 7 epits the remote stress in the luminium lyer (evlute t the top of the lminte, ner the horizontl ege t whih the loing is pplie) plotte ginst the norml opening of the moe I rk in the luminium (mesure t the tip position of the initil pre-rk, s represente y point A in Fig. 6) for three ifferent vlues of the ultimte norml trtion, t u 1, of the moe I rk (s liste in Tle 1). The hrters f, whih inite ifferent eformtion stges, orrespon to the elmintion profiles in Fig. 7 n for the ses where the ultimte trtion of the moe I rk is t u 1 ¼ 1:4r n t u 1 ¼ 2:r, respetively. Sine the elmintion evolution for the se with t u 1 ¼ 3:r looks similr to tht of the se with t u 1 ¼ 2:r (with the only ifferene eing somewht stronger elmintion evelopment uring the initil loing stge), the former se hs not een visulize. From Fig. 7 it n e oserve tht the remote stress initilly inreses, susequently rehes mximum vlue, then ereses (thus representing n unstle, softening response) n finlly symptotes to (resiul) stress level tht remins pproximtely onstnt uner ontinuing rk opening. Clerly, the vlue of the mximum remote stress is higher for higher vlue of the frture strength t u 1 of the luminium, i.e., r1 mx ¼ 72 MP, 76 MP n 913 MP for tu 1 ¼ 1:4r,2:r n 3:r, respetively. The ft tht the lol ultimte trtion influenes the remote stress inites tht the size of the frture proess zone of the tunneling rk nnot e ignore with respet to its length n/or the totl speimen with. If this woul e the se, the rk woul e of the Griffith type, where the remote stress only epens on the lol toughness hrteristis of the rk [12]. However, ue to the rk riging ehviour y the fire-epoxy lyer, the size of the proess zone of the tunneling rk is signifint. A more etile isussion on the size of the proess zone of the luminium rk will follow lter in this setion. The elmintion profiles in Fig. 7 n re otine y grphilly onneting the integrtion points in whih the mge hs just exeee the vlue of.99 (whih thus is lose to the mximum mge vlue ¼ 1). The x- n z-oorintes use for enoting the lotion of the profiles re in orresponene with the oorinte system epite in Fig. 6, with the origin tken t the entre of the initil pre-rk. It n e oserve tht the ifferenes in elmintion growth for t u 1 ¼ 1:4r n t u 1 ¼ 2:r re smll. In oth ses, the elmintion profile initilly evelops with inresing length of the moe I rk in the luminium (stges n ), where the mximum elmintion tkes ple t the speimen entre. When the moe I rk rehes the vertil speimen ege, the elmintion strts to evelop minly in the vertil iretion. After stge the elmintion profile hs eome lmost uniform ross the speimen with (only ner the free ege the elmintion is slightly lrger), where the growth proess losely pproximtes the onition of stey-stte elmintion. In Suiker n Flek [17] lose-form expression hs een presente for the remote filure stress for stey-stte elmintion tht propgtes long the two outermost plies of unixilly-loe, rittle lminte. This expression is erive from the energy ifferene upstrem n ownstrem of the elmintion front, n res sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 ss ¼ 4E 1 G ðn 2Þw 1E1 þðn 1Þw 2E2 2w 1 nw 1E1 þðn 1Þw 2E2 : ð24þ Here, r 1 ss is the remote stress in the luminium lyer, n is the numer of mteril #1 lyers, G is the elmintion toughness, n w 1 n w 2 re the thiknesses of the mteril #1 n #2 lyers. Further, the stiffness E i is efine y ( E i ¼ E i=ð1 m 2 i Þ for plne strin; ð25þ for plne stress; E i where E i n m i re the Young s mouli n Poisson s rtios of mterils i ¼ 1 (luminium) n 2 (fire-epoxy). During uniform, stey-stte elmintion the lminte pprohes plne-stress onition, sine the right ege of the speimen is trtion-free, in orne with the ounry onitions of se BC1 epite in Fig. 6. Sustitution of the mteril prmeters liste in Tle 1 into Eq. (24), with the elmintion toughness tken s G ¼ 4 N/mm (rell tht the effetive elmintion toughness is inepenent of the moe-mixity n thus equls the toughness vlue of the iniviul frture moes), n the numer of luminium lyers s n ¼ 2 (in orresponene with 2/1 ly-up), for plne-stress onition les to remote stress of r 1 ss ¼ 547 MP. This stress vlue, whih is represente in Fig. 7 y horizontl, she line, is in lose greement with the resiul strength following from the numeril nlyses. Tle 2 Overview of the simultions, whih re hrterise y vritions of the ounry onitions, BC1 n BC2 (see Fig. 6), ifferent vlues of the ultimte moe I trtion t u 1 of the moe I rk in the luminium, ifferent lmelle properties, n ifferent reltive initil rk length 2 =W t u 1 ¼ 1:4r t u 1 ¼ 2:r t u 1 ¼ 3:r Aluminium elsti n BC1 & BC2 BC1 BC1 fire-epoxy elsti 2 =W ¼ :12 2 =W ¼ :12 2 =W ¼ :12 Aluminium elsto-plsti BC1 & BC2 BC1 BC1 n fire-epoxy elsti 2 =W ¼ :12,.28,.48 2 =W ¼ :12 2 =W ¼ :12

13 772 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) Remote stress σ [MP] e σ ss = 547 [MP] t u 1 = 1.4σ * t u 1 = 2.σ * t u 1 = 3.σ * f Norml rk opening v 1 [mm] z-oorinte [mm] z-oorinte [mm] e f x-oorinte [mm] e f x-oorinte [mm] Fig. 7. Filure response for 2/1 ly-up with elsti luminium lyers. () Norml opening v 1 of the moe I rk in the luminium (mesure t point A, see Fig. 6) versus the remote stress r 1 in the luminium lyer, for three ifferent vlues of the ultimte norml trtion t u 1 of the moe I rk in the luminium. The horizontl, she line inites the stey-stte remote stress r 1 ss ¼ 547 MP ompute with Eq. (24), n the hrters f inite ifferent eformtion stges uring the loing proess. () Delmintion profiles for eformtion stges f (see ()), for the se where the ultimte trtion of the moe I rk in the luminium is t u 1 ¼ 1:4r (= 43 MP). The x- n z-oorintes relte to the oorinte system in Fig. 6. () Delmintion profiles for eformtion stges f (see ()), for the se where the ultimte trtion of the moe I rk in the luminium is t u 1 ¼ 2:r (= 61 MP). The x- n z-oorintes relte to the oorinte system in Fig. 6. As illustrte y Eq. (24), the resiul strength uring stey-stte elmintion minly epens on the elmintion toughness G, n is inepenent of the luminium frture strength t u 1. The ltter spet n lso e oserve from Fig. 7, n, whih illustrte tht t stges e n f oth the r 1 1 response n the elmintion profiles re virtully

14 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) ientil for the ifferent frture strengths t u 1. The rittle, lmost rte-inepenent ehviour of the elmintion proess is further onfirme y the pperne of reltively smll proess zone t the elmintion tip, whih tully n e ignore with respet to the speimen length L. The moe-mixity uring stey-stte elmintion my e evlute in orne with the efinition given y Eq. (7). An lterntive mesure for the moe-mixity, typilly use in liner elsti frture mehnis [12], is w ¼ tn 1 r zy ðrþ r yy ðrþ r¼l ; ð26þ whih is se on the sher (r zy ) n norml (r yy ) stresses evlute t speifie horizontl istne r ¼ l he of the rk (or elmintion) tip. Note tht the efinition, Eq. (26), ssumes tht the rk propgtes in the z-iretion long plne seprting two (possily issimilr) ulk mterils, with the norml to the rking plne pointing in the y-iretion, where pure moe I onitions orrespon to w ¼, n pure moe II onitions relte to w ¼ 9. In ohesive zone moelling there is no stnr proeure to ientify the rk tip from trtion seprtion lw. However, efinition regulrly use is se on relting the rk tip to the point t whih the interfe trtion rehes the pek vlue t u [42], whih is onsistent with the experimentl oservtion tht miromehnil proesses re not only tive he of the rk tip, ut lso in the wke of the rk tip [43]. With this efinition, the present numeril results show tht uring stey-stte elmintion the moe-mixity, Eq. (26), for the se t u 1 ¼ 1:4r vries etween minimum vlue w ¼ 31 mesure t smll istne he of the rk tip, n mximum vlue w ¼ 9 mesure extly t the rk tip. The mximum vlue reflets purely moe II onition. This onition is rehe sine the rk fes t the rk tip re in ontt, s result of whih the moe I ontriution vnishes. In Suiker n Flek [17], the moe-mixity for the se of stey-stte elmintion in rittle, unixilly-loe 2/1 lminte with stiffness mismth of E 2 = E 1 :4 (whih is in orresponene with the elsti prmeters of the luminium n fire-epoxy lyers presente in Tle 1) hs een mesure s w 54, where the istne l hs een ritrrily speifie s the semi-with of the moe I rk in the luminium, l ¼ :15 mm. Oviously, this vlue flls within the rnge 31 < w < 9 following from the present numeril nlyses. In the wke of the rk tip, the moe-mixity in the present simultions vries etween vlue w ¼ 9 mesure extly t the rk tip n vlue w ¼ 85 mesure when the mge is lose to unity, 1, n the herene etween the rk fes is virtully lost. Hene, it my e onlue tht in the wke of the rk tip the filure proess is moe II ominte. For the ses with higher mximum trtion, t u 1 ¼ 2:r n t u 1 ¼ 3:r, the moe-mixities he n ehin the rk tip hve similr rnges, with the min ifferene eing tht the minimum vlue he of the rk tip is somewht lrger, i.e., w ¼ 38 n w ¼ 4 for t u 1 ¼ 2:r n t u 1 ¼ 3:r, respetively. The trtion profile of the moe I rk in the luminium is plotte in Fig. 8 (t u 1 ¼ 1:4r ) n Fig. 8 (t u 1 ¼ 2:r ) ross the omplete speimen with, uring vrious eformtion stges of the loing proess. The urve inite y n sterisk,, represents the elsti response of the interfe (plotte y she line) prior to rking (i.e., efore the ultimte trtion t u 1 is rehe t the tip of the initil pre-rk), while the urves, whih relte to the eformtion stges inite in Fig. 7, orrespon to moe I rk eveloping in the luminium. At eformtion stge, the integrtion points next to the initil pre-rk hve just entere the softening regime, while the integrtion points t lrger istne from the pre-rk still hve not rehe the ultimte trtion t u 1 n therefore respon elstilly. During eformtion stges n ll integrtion points ross the speimen with hve rehe the softening regime. Note tht uring softening the trtions vry only wekly in the x-iretion, suh tht the size of the proess zone of the luminium rk sustntilly exees the speimen with. As expete, lose to the pre-rk the vrition in the softening profile is slightly lrger for lrger ultimte trtion t u 1, initing tht the frture proess ours little less spre out. Finlly, t stge ll integrtion points ross the speimen with hve rehe the en of the softening regime, n the rk fes of the luminium rk hve eome fully seprte. During the finl phse of the rk tunneling proess (i.e., etween stges n ), signifint elmintion hs evelope, see Figs. 7 n, where the effetive response of the lminte hs eome unstle, see Fig. 7. In the stuy of Suiker n Flek [17] on lmintes sujete to unixil tension, this unstle tunneling mehnism ws nme mehnism 3, see Fig. 1. Although the stuy in [17] reltes to n ielly rittle lminte of infinite size, where the size of the proess zone of the tunneling rk is equl to zero, the typil fetures of mehnism 3 thus lso re oserve in qusi-rittle lminte where the tunneling rk is hrterise y proess zone of sustntil length. Essentilly, the pperne of mehnism 3 n e expline from the speifi mteril prmeters tht hrterise the 2/1 GLARE lminte, i.e., the elsti stiffness of the fire-epoxy lyer, whih, s lrey mentione, is out.4 times smller thn the elsti stiffness of the luminium lyer, n the rtio etween the elmintion toughness n the moe I frture toughness, whih hs rther low vlue of.4, see Tle 1. The filure mehnism mp presente in Fig. 18 of Suiker n Flek [17] onfirms tht for these mteril prmeters mehnism 3 is inee the opertive filure mehnism uring rk tunneling in 2/1 lminte. This filure mehnism mp further inites tht the stiffness mismth etween the fire-epoxy lyer n the luminium lyer hs moerte influene on the pperne of this speifi frture mehnism, n tht it is minly etermine y the reltively low toughness for interfil elmintion. In ft, it is shown tht the elmintion uring rk tunneling remins sent if the elmintion toughness eomes lrger thn the toughness for moe I frture in the luminium. For suh reltively high elmintion toughness the tunneling rk eomes stle n of the pure moe I type (i.e., mehnism 1 in Fig. 1), in orresponene with remote, stey-stte tunneling stress tht is out three to four times lrger thn tht otine for

15 774 M.V. Ci Alfro et l. / Engineering Frture Mehnis 76 (29) Norml trtion t 1 [MP] Norml trtion t 1 [MP] * Elsti response prior to moe I rking * x-oorinte [mm] * Elsti response prior to moe I rking * x-oorinte [mm] Fig. 8. Norml trtion t 1 of the moe I rk tunneling in the elsti luminium lyer, evlute ross the speimen with, t vrious eformtion stges. The urve * refers to n elsti interfil response n the urves (whih orrespon to the eformtion stges inite in Fig. 7) reflet the frture proess. () Cse where the ultimte trtion of the moe I rk is t u 1 ¼ 1:4r. () Cse where the ultimte trtion of the moe I rk is t u 1 ¼ 2:r (=61 MP). the present elmintion toughness. Oviously, this onlusion hols s long s the luminium lyers o not eform plstilly. The influene of plstiity in the luminium on the effetive filure response of the lminte is investigte in the susequent setion /1 ly-up with elsto-plsti luminium lyers In Fig. 9 the remote stress in the luminium lyers is plotte versus the norml opening of the moe I rk in the elstoplsti luminium lyers. It n e oserve tht higher frture strength t u 1 of the luminium les to higher remote stress, i.e., t rk opening v 1 ¼ 1:9 mm the remote stress in the luminium equls r 1 ¼ 557 MP, 611 MP n 642 MP for t u 1 ¼ 1:4r, 2:r n 3:r, respetively. For the se with the highest luminium frture strength, t u 1 ¼ 3:r,tv mm the ontinuum elements lose to the rk tip hve eome strongly istorte, whih is ue to the genertion of sustntil plsti eformtions in these elements. Consequently, t this stge the numeril proeure fils to onverge. Although it is expete tht this prolem n e overome y further eresing the size of the elements ner the rk tip, this strtegy hs not een explore in more etil ue to the lrey lrge omputtionl times of the present simultions (i.e., the soli-like shell elements use in the numeril moel re omputtionlly expensive). Note tht the urves epite in Fig. 9 o not show softening rnh, whih is in ontrst to the effetive response of ly-up with elsti luminium lyers, see Fig. 7. Hene, it my e onlue tht the energy issiption in the plsti zone he of the rk in the luminium stilizes the effetive filure response of the lminte. For lrger frture strength t u 1 of the luminium, the plsti zone, n thus the energy issiption in this zone, eomes lrger, s result of whih the filure response eomes more stle, see Fig. 7. The elmintion profiles for the eformtion stges f inite in Fig. 9 re plotte in Fig. 9 (t u 1 ¼ 1:4r ) n 9 (t u 1 ¼ 2:r ). Apprently, uring rk tunneling the elmintion front for speimen with elsto-plsti luminium lyers