C5-539 (1) JOURNAL DE PHYSIQUE Colloque CS, supplement au n010, Tome 49, octobre 1988 ON THE INTERFACE CRACK MODELS

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1 JOURNAL DE PHYSIQUE Colloqu CS, supplmnt au n010, Tom 49, octobr 1988 C5-539 ON THE INTERFACE CRACK MODELS T. SUGA(l), S. SCHMAUDER and G. ELSSNER Max-Planck-Institut für Mtallforschung, Institut für Wrkstoffwissnschaftn, V-7000 Stuttgart, F.R.G. Rsum - Ls modls du continu pour un fissur dans l'intrfac sont discuts. Ls incompatiblits logiqus prsnts dans l champ d tnsion du modl convntionnl, comm par xmpl ls singularits d'oscillation t ls intrpntrations d la surfac d fissur, sont lmins n modifiant ls conditions d joints a la point d fissur. Un modl d la point d fissur st prsnt t discut par rapport a la disprsion inlastiqu d l'nrgi t l manqu d'intrfacs d matriaux htrogns. Abstract - Th continuum modls for an intrfac crack ar rviwd. Th logical inconsistncis prsnt in th local strss fild of th convntional modl, such as th oscillatory singularitis and th intrpntration of th crack surfacs, ar rmovd by modifying th boundary conditions at th crack tip. An intrprtation of th crack tip modls is givn and discussd with rspct to inlastic nrgy dissipation during intrfacial failur of htrognous matrials. I - INTRODUCfION Smibrittl fractur along intrfacs of htrognous matrials (such as solid stat bondd cramic-mtal joints or composits) can occur whn th nrgy stord in th spcimn by xtrnal work bcoms sufficint to supply th fractur nrgy ndd for crating nw surfacs. Th trm "smibrittl" mans that th rgion in which th nrgy is consumd by th failur procss is limitd to th crack tip. Th idal fractur nrgy of an intrfac corrsponds to th thrmodynamic work of adhsion W. which is th diffrnc btwn th sum of surfac nrgis and th intrfacial nrgy of th matrials bondd: W'~M+~-~M. Any failur procss 1S accompanid by irrvrsibl inlastic nrgy dissipation. Thrfor, dnoting its contribution to th fractur nrgy by Wp, th thrmodynamic critrion for an intrfac fractur is writtn as /1/: whr G is th critical valu of th gnralizd strain nrgy rlas rat G. This quation is a rational gnralization of th Griffith-Orowan formula for th smibrittl failur of homognous matrials. If th inlastic dformation of th spcimn is 1imi td to a narrow rgion in th vicini ty of th crack, th gnral izd strain nrgy rlas rat G corrsponds to Irwin's lastic strain nrgy rlas rat Gwhich can b calculatd by an lastic analysis of th crackd matrial. If th load-dformation rlationship of th spcimn is not linar, G can also b obtaind by modifying th lastic strain nrgy rlas rat G by th xprimnta11y stimatd nonlinarity cofficint /2/. Exprimntal vrification of th nrgy critrion for intrfacial fractur is givn in arly works by Malyshv and Salganik /3/, Mulvill /4/, and Saxna /5/, followd by th xtnsiv xprimnts by Suga and Elssnr /1,6,7/. Th most important rsult in /1/ is that th dissipatd work Wp is dpndnt on th rvrsibl work W. Thr- (1) (1how at: Th Univrsity of Tokyo, Faculty of Enginring, Dpartmnt of Prcision Enginring, 113 Tokyo, Japan.

2 C5-540 JOURNAL DE PHYSIQUE for quation (1) should b writtn: or. mor gnrally: G G = W. + Wp (~M. ~. ~M) (3) In /1/ it was also shown that th total intrfacial fractur nrgy is strongly dpndnt on th crystallographic orintation of th bondd matrials. and thus is dirctly affctd by th work of adhsion W. vn though th masurd valu of W. is much smallr than that of WP This apparntly contradictory rsult is xplainabl by Wp bing a function of W. In ordr to rprsnt th function Wp xplicitly. an adquat modlling of th nrgy dissipation procss is ncssary. Th simpl modl givn in /1/ which can stimat th fraction of th rvrsibl fractur nrgy and th nrgy dissipatd in th vicinity of an intrfacial crack suggsts that Wp is a monotonic function of W. Thrfor. quation (3) can b writtn: 1_ W. 1 - a (4) whr a ~ 1. A mor comprhnsiv analysis must b don in ordr to undrstand th mchanism of intrfacial failur rlatd to th nuclation of microcracks in an intrfac or th intraction of dislocations with an intrfac crack. Such an analysis rquirs mor dtaild knowldg of th strss stat in th vicinity of an intrfac crack. which can b dscribd only by an adquat intrfac crack modl. In contrast with th wil stablishd crack modl in singl phas matrials. intrfac crack modls hav not bn succssfully applid to intrfacial fractur phnomna. In th prsnt papr w rviw th prvious attmpts of th modlling of an intrfac crack. Th problms of intrfac crack modls sm to lad to th qustion of how th intrfac crack in an actual intrfac can b simplifid and idalizd by a modl with a sharp and prfct intrfac within th framwork of continuum mchanics. (2) II - CXlNVENTIONAL INTERFACE CRACK MODEL Th problm of th dtrmination of th strss fild in th vicinity of an intrfac crack in lastically dissimilar matrials has bn studid by Williams /8/. England /9/. Erdogan /10/. Ric and Sih /11/. and Lowngrub and Snddon /12/ among othrs. A comprhnsiv summary of ths works is givn in /13/. Th displacmnt fild u, and u. and th strss fild o and Tr in th vicinity of th intrfac crack tip ar found to possss th following dominant trms: x (u - iur) z (5) X-I z (o - (6) itr) in polar coordinats with z = r xp(i). Th xponnt X of th abov trms is a complx numbr of th form 1/2 + ie.. whr E. is th bimatrial constant givn by E.= 1/2~ In«I+ß)/(I-ß» and ß is on of th Dundurs' composit paramtrs /1/. For singl phas homognous matrials. E. is rducd to zro. and thrfor th strss componnts ar invrsly proportional to th squar root of th distanc r from th crack tip. In contrast with this wll-known squar root singulari ty. th non-zro imaginary part of th xponnt X of th intrfac crack lads to undsirabl rsults from th physical point of viw: th strss componnts oscillat and th uppr and lowr surfacs of th crack wrinkl and ovrlap in th crack tip rgion. Indd. if on dfins a complx strss intnsity factor K = Kr + Krr to indicat th strngth of th singular ity. th strss fild at th intrfac nar th crack tip and th opning of th crack surfacs can b writtn as (7)

3 C5-541 Au (u - iur)=rr - (u - iur)=-rr -4~ I EX " (Kr -iklr)/~" r1/2 - i/(1/2 - i) (8) whr EX is th ffctiv lastic modulus of th dissimilar matrial dfind in 11/. As sn in th abov quations th trm r -i = cos (lnr) - i sin (lnr) (9) lads to th oscillatory singularity. If on introducs a charactristic nw dimnsion of lngth, ro into quation (7): ~ -1/2 - i (o - itr)=o = (Kr -. ik, r)/'1 21T " (r/ro) lngth or a thn th componnts of th strss intnsity factor, Kr and Krr, ar changd altrnatly du to th trm (ro)1/2 + i. Th priod of th oscillation is givn by ro = xp (21Tll1), and vn for th strongst dissimilarity corrsponding to ß = 0.5 and c = it taks valus at vry larg intrvals: xl0-32, 2.5xl0-15, Ox1015, 1.6xl tc. It should b mphasizd, howvr, that th sign of th ral part of quation (9) is changd at th points of r = 1.25xl0-4 and 7.99xl03 nar th rfrnc lngth r = 1. For a cntral crack at th intrfac of two lastically dissimilar bondd plans loadd at infinity with a uniform strss fild. 00 and To. th strss intnsity factor is: K = Kr - ik'1 = (00 - ito) ~" (2a)ic " (1-2i) (11) whr 2a is th crack lngth. It is notd that th trm (2a)i givs an oscillatory charactr also to th strss intnsity factor. (10) III - MODELS FüR THE CRACK HP REGION Th convntional intrfac crack modl dscribd abov has bn criticizd by many fractur mchanists, bcaus it lads both to th oscillatory bhaviour of th strss fild and to th ovrlapping of th crack surfacs in th small rgion around th crack tip whr th intrfacial failur will b initiatd. Also th strss intnsity factor has an ambiguous dpndnc on an arbitrary lngth paramtr, and thrfor it is not possibl to rlat th componnts of th strss intnsity factor K" Kr, to th typ of loading mod in th vicinity of th crack tip. Ths undsirabl faturs rsul t from th traction-fr condi tion assumd on th whol crack surfacs of th convntional intrfac crack modl. Th charactr of th local strss fild of th crack tip rgion obviously dpnds on th mannr in which th conditions chang from th intrfac bond to th crack surfacs. Thrfor, by changing th boundary conditions a mor ralistic charactr of th local strss fild can b obtaind. Such attmpts hav bn mad by svral authors. Comninou 114/ has shown that th strss singularitis loos th oscillatory bhaviour by assuming that th crack is not compltly opn and that its surfacs ar in contact with ach othr nar th crack tip (contact zon modl). An altrnativ modl for th crack tip was introducd by Mak t al. 115/, which assurns that thr is no rlativ slip btwn th crack surfacs in th crack tip rgion (intrlock modl). Although both modls liminat th imaginary part of th strss singularity, an undsirabl aspct rmains concrning th strss intnsi ty factar. in that on of its componnts (Kr for th contact zon modl, Krr for th intrlock modl) is always liminatd. In othr words. no normal strss xists in th intrfac ahad of th crack tip in th contact zon modl, rgardlss of th actual loads applid to th bimatrial. Similarly no shar strss xists in th intr lock modl. This dilmma was rsolvd by th "transition zon modl", introducd by Suga and Eissnr 116,17/. In this modl an intraction btwn th crack surfacs is assumd. Th transitions acting on th crack surfacs ar xprssd as: (12) (O - itr)=±1t = -ßEx/4 (I-ß2)"8(Aur + iau)18r Th strsss in th intrfac ar charactrizd by both strss intnsity factors Kr and Krr as givn in th following xprssion: (o - itr)=o = (Kr - ikr 1)/~" r-1/2 (13)

4 C5-542 JOURNAL DE PHYSIQUE This is xactly th sam xprssion as that of homognous matrials. Th crack opning is xprssd by: Au = -8(l-ß2)/Ex (Kr - ikrr)/~ r1/2 (14) Th lastic strain nrgy rlas rat G is calculatd by th following quation: giving 15 G = -1/2 1im 1/0 f (u(r) Au(o-r) + T (r) Au (o-r»dr (15) o-'ij 0 r r th rlationship Th strss functions of this modl ar constructd by a combination of thos of th contact zon modl and th intrlock modl. In Figs. 1 and 2 slctd strss distributions around th crack tip ar compard with thos of th convntional crack tip. Full dscriptions for th complx strss functions and th strss and displacmnt fild for th modl ar givn in /17/. (16) IV - INTERPRETATION OF INTERFACE CRACK MODELS Som fundamntal qustions ar: how is th siz of th crack tip rgion dtrmind. and how can th assumd boundary conditions for th crack tip rgion b justifid? Ths qustions can b rducd to a mor fundamntal qustion: why must an intrfac crack modl b usd to charactriz th fractur bhaviour of an intrfac? If on assums an atomic structur of th intrfac crack tip and a propr intratomic potntial, th strss fild at th crack tip or th idal cohsion nrgy of th intrfac can b calculatd undr crtain boundary conditions such as displacmnts of th atoms on th crack surfacs or strsss acting on th assmbly of atoms. Howvr, how can such displacmnts or strsss for th small assmbly of atoms b dtrmind from macroscopic masurmnts of th appl id forc and th dformation of th spcimn? In this cas a modl dscribing th local strss fild of th crack tip is rquird. Th prsnt crack tip modl lads. howvr, to a singular crack tip strss fild. which is not physically ralistic. Th fractur mchanics argumnt is not that th strngth of th strss fild itslf but rathr th critical nrgy rlas rat G should b slctd as a physical paramtr which rprsnts th atomistic cohsiv nrgy indpndnt of th crack modl. and that th strss intnsi ty factors drivd from th crack modl ar usful only bcaus of thir possibility to charactriz th loading mod in th vicinity of th crack tip. Thus th ambiguity rlating to th siz and th conditions of th crack tip rgion can b allowd. Th slctd modl should b justifid not physically but by fractur toughnss tsts giving a practical vrification of strss intnsity factors. Th crack tip zon siz has bn invstigatd analytically only for an intrfac crack with th contact zon or an intr lock zon in a dissimilar infinit plan undr uniform tnsil or shar strss /14, /. Th zon siz was dtrmind undr th assumption of th xistnc of comprssiv strss in th contact zon or a critical crack opning displacmnt in th intrlock zon. rspctivly. Howvr. from th forgoing viwpoint, a mor gnral mthod should b dvlopd to dtrmin th strss intnsity factors for a givn tip zon siz and for any configuration of intrfac cracks. rathr than dtrmining th tip zon siz. Othr attmpts to avoid th oscillatory crack tip strss fild ar also givn in th litratur: Sinclair /21/ has proposd a crack tip modl with a wdg form. Th opning angl of th crack tip is dtrmind such that th strss fi1d has th usual squar root singu1arity. Atkinson /22/ has shown that a crack at th intrfac rgion with th lastic moduli continuously varying across th intrfac can rmov th anomalis of th strss singularity. This modl may allow tratmnt of a diffus intrfac, although it may b difficult to dscrib th local strss fild xplicitly. Atkinson also dmonstratd that a crack in a thin intrfac layr can likwis rmov strss anomalis /22/.

5 C5-543 o 180' -180 o 180' Fig. 1 Normalizd tnsil strss a for KI = 0 at th crack tips of th convntional intrfac crack modl (lft) and th transition modl (right) ,0 ",of KnIK['O (Kn,OJ -1,0-1SO" o 180' o 180' Fig. 2 Normalizd shar strss Tr for KII = 0 at th crack tips of th convntional intrfac crack modl (lft) and th transition modl (right).

6 C5-544 JOURNAL DE PHYSIQUE v - CüNCLUDING REMARKS Th anomal is of th local strss fild can b rmovd by introducing an adquat crack tip modl. What rmains is th dvlopmnt of a gnral mthod to dtrmin th local strss intnsity factors. Th convntional intrfac crack modl offrs no rational local strss fild. Howvr. th solution of th modl is still valid for th dscription of th asymptotic bhaviour of th strss fild far from th crack tip, as vrifid xprimntally /23.24/. With rgard to th nrgy dissipation in th intrfacial failur procss as mntiond in th introduction, it is still unknown whthr dislocations intract with an intrfac crack in its nar-tip local strss fild or in th far-from-crack-tip strss fild. Furthr fforts should focus on th micromchanics approach to th intrfacial fractur problm from both xprimntal and analytical points of viw. REFERENCES /1/ G. Elssnr. T. Suga and M. Turwitt, J. d Physiqu 46 (1985) C4-597 /2/ H. Libowitz and J. Eftis. Engng. Fract. Mch. ~ (1971) 267 /3/ B.M. Malyshv and R.L. Salganik, Int. J. Fract. Mch. 1 (1965) 114 /4/ D.R. Mulvill. P.W. Mast and R.N. Vaishnav. Engng. Fract. Mch. ~ (1976) 555 /5/ A. Saxna. Fibr Si. Tchnology 12 (1979) 111 /6/ T. Suga and G. Elssnr. Z. Wrkstofftch. 16 (1985) 122 /7/ T. Suga. I. Kvrns and G. Elssnr, Z. Wrkstofftch. 15 (1984) 371 /8/ M.L. Williams, Bull. Sismol. Soc. Amr. 49 (1959) 199 /9/ A.H. Eng land, J. Appl. Mch. E32 (1965) 400 /10/ F. Erdogan. J. Appl. Mch. E32 (1965) 403 /11/ J.R. Ric and G.C. Sih. J. Appl. Mch. E32 (1965) 418 /12/ M. Lowngrub and I.N. Snddon. Int. J. Engng. Si. 11 (1973) 1025 /13/ A. Piva and E. Viola. Engng. Frac. Mch. 13 (1980) 143 /14/ M. Comninou. J. Appl. Mch. E44 (1977) 631 /15/ A.F. Mak. L.M. Kr, S.H. hn and J.L. Lwis. J. Appl. Mch. 47 (1980) 347 /16/ T. Suga and.g. Elssnr. J. d Physiqu 46 (1985) C4-657 /17/ T. Suga. PhD. Thsis, Univrsity of Stuttgart (1983) /18/ M. Comninou and D. Schmusr. J. Appl. Mch. 46 (1979) 345 /19/ C. Atkinson. Int. J. Fractur 19 (1982) 131 /20/ A.K. Gautsn and J. Dundurs. J. Appl. Mch. 54 (1987) 93 /21/ G.B. Sinclair. Int. J. Fractur 16 (1980) 111 /22/ C. Atkinson. Int. J. Fractur 13 (1977) 807 /23/ E.E. Gdoutos and G. Papakaliatakis. Engng. Fract. Mch. 16 (1982) 177 /24/ P.S. Thocaris. Acta Mchanica 24 (1976) 99