SIMULATION OF DELAMINATION GROWTH UNDER HIGH CYCLE FATIGUE USING COHESIVE ZONE MODELS

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1 Pro P. Camanho t al. Simulation of lamination SIMULTIO OF DELMITIO ROWTH UDER HIH CYCLE FTIUE USI COHESIVE ZOE MODELS PEDRO P. CMHO *, LBERT TURO **, JOSEP COST ** * DEMEI, Facula Engnharia, Univrsia o Porto Rua Dr. Robrto Frias, Porto, Portugal pcamanho@f.up.pt ** MDE, Polytchnic School, Univrsity of irona Campus Montilivi s/n, 1771 irona, Spain albrt.turon@ug.s, josp.costa@ug.s BSTRCT: cohsiv zon mol is propos for th simulation of lamination growth in composit matrials unr high-cycl fatigu loaing. Th basis for th formulation is an intrfacial constitutiv law that links fractur mchanics an amag mchanics rlating th volution of th amag variabl,, with th crack growth rat, a/. Th cohsiv zon mol is implmnt in BQUS finit lmnt co an us in th simulation of carbon-poxy tst spcimns cyclically loa in mo I, mo II an mix-mo I an II. Th accuracy of th mol is assss by comparing th prictions with xprimntal ata. Kywors: Composit matrials; lamination; fatigu; cohsiv mols. RESUMO: Est artigo aprsnta um molo cosivo para simular a laminagm matriais compósitos sob faiga altos ciclos. O molo constitutivo proposto rlaciona a volução a variávl ano,, com a vlocia crscimnto a laminagm, a/. O molo cosivo é implmntao no cóigo lmntos finitos BQUS é utilizao na simulação provts fabricaos m carbono-poxy carrgaos ciclicamnt m moo I, moo II moo misto I II. O molo é valiao comparano as suas prvisõs com rsultaos xprimntais. Palavras chav: Matriais compósitos; laminagm; faiga; molos cosivos. 1. ITRODUCTIO High-cycl fatigu is a common caus of failur of arospac structurs. In laminat composit matrials, th fatigu procss involvs svral amag mchanisms that rsult in th graation of th structur. On of th most important fatigu amag mchanisms is intrlaminar amag (lamination). In a graation procss involving high cycl-fatigu, amag volution can b obtain as th sum of th amag caus by static or quasi-static ovrloas an th amag that rsult from th cyclic loas. Th amag volution prouc by cyclic loas is usually rprsnt as a function of th numbr of cycls an strains (or isplacmnt jumps) [1-], whr a amag volution law xprss in trms of th numbr of cycls is stablish a priori. Howvr, th amag volution law must b xprss as a function of svral paramtrs that hav to b ajust through a trialan-rror calibration of th whol numrical mol. In this papr, an altrnativ approach is propos whrby th volution of amag is bas on linking Fractur Mchanics an Damag Mchanics, an rlating th volution of th amag variabl,, with th crack growth rat, a/. Th prsnt mol is implmnt by mans of a usrwrittn lmnt in BQUS [4] by aing th amag volution law formulat in th cohsiv lmnt prviously vlop by th authors [5].. COSTITUTIVE MODEL FOR HIH-CYCLE FTIUE Th amag volution in a graation procss involving high-cycl fatigu can b consir as th sum of th amag caus by th quasi-static loas an th amag caus by th cyclic loas: t static cyclic Th volution law is formulat using Fractur Mchanics an crating a link btwn Fractur Mchanics an Damag Mchanics to rlat th amag variabl,, to th crack growth rat, a/. Th volution of th amag variabl is rlat to th volution of th crack surfac as follows: (1) 4 Ciência Tcnologia os Matriais, Vol., n.º 1/, 8

2 Simulation of lamination Pro P. Camanho t al. whr is th amag ara, an is th growth rat of th amag ara. Whil th scon trm in th right han si of quation () must b charactriz xprimntally, th first trm,, can b obtain from ithr a Damag Mchanics approach or a Fractur Mchanics approach. In th framwork of th Fractur Mchanics, th fraction of th amag ara,, with rspct to th total ara,, can b writtn as a function of th issipat nrgy: Ξ () c whr Ξ is th fraction of th nrgy pr unit surfac issipat uring th amag procss, i.., th ara unr th cohsiv law for th currnt amag thrshol, an c is th critical nrgy rlas rat. ssuming no chang btwn mos, c is constant, whil Ξ is a function of th cohsiv law us an th currnt amag thrshol. Using this approach, th rivativ of th amag variabl with rspct to th amag ara can b writtn as: () c (4) Ξ Th rivativ of th nrgy issipation with rspct to th amag variabl is obtain from th quations of th constitutiv law us. Using a bilinar constitutiv law: ( ) Ξ Δ ( c Using quation (5), quation (4) can b writtn as: 1 Δ ( ( ) For a spcimn with just on crack front, th crack growth rat is qual to th sum of th amag surfac growth rats of all lmnts in th cohsiv zon. In th othr rgions of th spcimn, thr is no possibility of nw surfac gnration. (5) (6) (7) Using th simplification that is constant ovr th cohsiv zon, th prvious quation can b writtn as: C (8) whr th ratio rprsnts th numbr of aras in which th cohsiv zon has bn ivi. In a finit lmnt nvironmnt, this ratio rprsnts th numbr of lmnts that span th cohsiv zon. Rarranging trms in quation (8), th surfac amag growth rat can b writtn as: Using quations (4) an (9) in (): c b c a Ξ Ξ (9) (1) whr c pns on th matrial us an loaing mo, an pns on th cohsiv law us in th Ξ formulation of th surfac traction-isplacmnt jump rlation. Th ara of th cohsiv zon can b comput using: E C η b (11) whr b is th with of th spcimn, is th nrgy rlas rat, E is th Young s moulus of th bulk matrial in th o irction prpnicular to th crack plan, an τ is th intrfacial strngth. Th paramtr η pns on th softning law an gomtry of th spcimn, bing only a matrial proprty qual to π for vry larg gomtris. 4. CRCK ROWTH RTE aa Th crack growth rat unr fatigu loaing,, is a loa an matrial-pnnt charactristic that has bn wily stui. Th crack growth rat of brittl composits may b xprss using Paris law: m a C (1) c whr C an m ar paramtrs that pn on th mo ratio an must b trmin xprimntally. Δ, is th cyclic variation in th nrgy rlas rat, which can b comput using th constitutiv law of th intrfac: Ciência Tcnologia os Matriais, Vol., n.º 1/, 8 41

3 Pro P. Camanho t al. Simulation of lamination Δ )Δ (1) 1.5 Dfining th rvrsibility factor, R, th rlation btwn th minimum an th maximum isplacmnt is: 1.5 min max Δ Δ R (14) Using (1) an (14): max f max 1+ R Δ Δ KΔΔ Δ ( Δ Δ 4. MODEL VLIDTIO (15) f Crack growt h, a-a a (mm) x1 4 4.x1 4 6.x1 4 8.x1 4 1.x1 5 Cycls Th propos mol was implmnt in BQUS (4) as a usr-fin lmnt. Th prliminary valiation of th mol was prform by comparing its prictions with publish xprimntal ata. Th Doubl-Cantilvr Bam spcimn tst by sp [6] was simulat to analyz mo I fatigu lamination. Th rlvant matrial proprtis us in th mol ar rport by sp [6]. Th tst spcimn is mm wi, 15mm long, with two 1.55mm thick arms with an initial crack of 5mm. Th spcimn is loa by constant momnts to nsur a constant valu of th nrgy rlas rat, rgarlss of th crack lngth []. This mol is rquir to obtain th crack growth rats for iffrnt valus of th appli nrgy rlas rat. pplying constant momnts to th spcimn, th nrgy rlas rat is rlat to th momnt as: M I (16) bei whr E is th longituinal flxural Young s moulus, an I is th scon momnt of ara of th spcimn s arm. Fig. 1. Prict rlation btwn th crack lngth an th numbr of cycls. Svral simulations corrsponing to iffrnt lvls of th appli nrgy rlas rat for four iffrnt loa ratios wr conuct to simulat th crack growth unr mo I loaing. Th rsults obtain an th xprimntal ata obtain by sp t al. [6] ar compar in Figur. It is obsrv that th constitutiv mol accounts for all thr rgions of fatigu crack growth. In rgion II, whr crack growth rats follow th Paris Law, a goo agrmnt btwn th prictions an th xprimntal ata is obtain. In rgion I thr is ngligibl crack growth rat for small valus of th normaliz nrgy rlas rat an th numrical ata follows th trn of th xprimntal ata. iffrnc btwn th numrical an th xprimntal ata is obsrv in rgion III. On of th rasons for this iffrnc is that th crack growth rats prsnt in rgion III ar vry high. Thrfor, a low-cycl insta of a high-cycl fatigu mol is likly to b mor appropriat for this rgion. Howvr, in spit of this iffrnc, th mol can also prict rgion III crack growth rat, whr th Paris law quation is not vali. Th application of constant momnts to th arms of th DCB spcimn rsults in a linar rlation btwn th crack lngth an th numbr of cycls. For xampl, th rlation btwn th crack growth an th numbr of cycls is shown in Figur 1 for a ratio of 4% btwn th appli nrgy rlas rat an th fractur toughnss. Crack growth rat, a/ (mm/cycl),5 5E-4 5E-5 5E-6 5E-7 5E-8 5E-9 5E-1 5E-11 5E-1 Exprimntal (R) umrical (R.) umrical (R.) umrical (R.5) umrical (R.7),,,4,5,6,7,8,9 1 ormaliz Enrgy Rlas Rat, Imax / Ic Fig.. Comparison of th xprimntal ata with th prict crack growth rat; snsitivity to th loa ratio unr mo I loaing. 4 Ciência Tcnologia os Matriais, Vol., n.º 1/, 8

4 Simulation of lamination Svral tsts wr conuct to simulat th crack growth rat unr mo II loaing for iffrnt rangs of th nrgy rlas rat. Th xprimntal ata on fatigu rivn lamination growth rport in sp t al. [6] was slct for comparison. Th imnsions an th matrial of th spcimn ar th sam us for th DCB spcimn prviously scrib. To impos pur mo II loaing conitions, th four-point n notch flxur (4EF) tst spcimn was us. In th 4EF spcimn, th nrgy rlas rat is rlat to cp th appli momnt,, as []: II cp 4 bei (17) Th finit lmnt mol us is similar to that us in th simulation of th mo I tst, th only iffrncs bing th bounary conitions an loas. Th matrial proprtis us in th simulations ar givn by sp [6]. Th crack growth rats obtain from th iffrnt simulations an th xprimntal ata slct for comparison ar shown in Figur. goo corrlation btwn th xprimntal ata an th numrical prictions is obsrv. Crack g rowth rat, a/ (mm),5 5E-4 5E-5 5E-6 5E-7 Exprimntal ata umrical priction,6,8,1,,4,6,8 1 ormaliz nrgy rlas rat, ( IImax / IIc ) Fig.. Comparison of th xprimntal ata with th crack growth rat obtain from th numrical simulation for a mo II 4EF tst. umrical mols wr vlop to simulat th crack growth rat unr mix-mo loaing with I II for iffrnt nrgy rlas rats. Th xprimntal ata on fatigu rivn lamination growth rport in [6] was slct for comparison. Th imnsions an th matrial of th spcimn ar th sam us for th DCB spcimn prviously scrib. Pro P. Camanho t al. iffrnt momnts appli to th arms of th spcimn. Th nrgy rlas rat is rlat to th appli momnt, M, as []: I II M bei (18) Th finit lmnt mol us was similar to that us in th simulation of th DCB tst, whr th momnts appli to th arms of th spcimn wr moifi to impos mixmo loaing. Th loa is appli in two stps an th matrial proprtis us in th simulations ar givn [6]. Th rsults obtain from th simulations an th xprimntal ata ar shown in Figur 4. Lik in th xampls of spcimns loa unr pur mo I an mo II, a goo corrlation prictions an xprimnts is obsrv. Crack rowth Rat, a/ (mm/cycl),1 1E- 1E-4 1E-5 1E-6 umrical Exprimntal 1E-7,1,,,4,5,6,7 ormaliz Enrgy Rlas Rat, max / c Fig. 4. Comparison of th xprimntal ata with th crack growth rat obtain from th numrical simulation for a mix-mo tst with I II. 5. COCLUSIOS thrmoynamically consistnt amag mol for highcycl fatigu lamination was vlop. Th volution of th amag variabl was riv by linking Fractur Mchanics an Damag Mchanics to rlat amag volution to crack growth rats. Th amag volution laws for cyclic fatigu wr combin with th law of amag volution for quasi-static loas within a cohsiv lmnt prviously vlop by th authors. Th mol was valiat by comparing its prictions with publish xprimntal ata. Th mol was abl to rprouc th tst ata without th n of aitional ajustmnt paramtrs that ar typically us in othr fatigu growth mols. In th stanar mix-mo bning tst, th appli nrgy rlas rat changs with th crack lngth. To obtain an nrgy rlas rat inpnnt of th crack lngth, th bounary conitions of th FEM wr introuc using Ciência Tcnologia os Matriais, Vol., n.º 1/, 8 4

5 Pro P. Camanho t al. CKOWLEDEMETS Th financial support of th Portugus Founation for Scinc an Tchnology (FCT) unr th projct PDCTE/554/EME/ is acknowlg. 6. REFERECES [1] RH.H.J. Prlings, W..M. Brlmans, R. Borst, an M..D rs, Intrnational Journal of umrical Mthos in Enginring, 9 () Simulation of lamination [] V.K. oyal an E.R. Johnson, 44th I/SME/SCE/HS Structurs, Structural Dynamics an Matrials Confrnc, () orfolk, Virginia. [4] BQUS Usr's Manual, Vrsion 6.5, Hibbitt, Karlsson & Sornsn, Pawtuckt, RI, 5. [5]. Turon, P.P. Camanho, J. Costa, C.. Dávila, Mchanics of Matrials, 8 (6) 17. [6] L.E. sp,. Sjgrn an E.S. rnhalgh, J. Comp. Tchnol. Rs., (1) 55. [] P. Robinson, U. alvantto, D. Tumino an. Bllucci, Intrnational Journal for umrical Mthos in Enginring, (7). 44 Ciência Tcnologia os Matriais, Vol., n.º 1/, 8

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