MULTILEVEL ANALYSIS OF DELAMINATION INITIATED NEAR THE EDGES OF COMPOSITE STRUCTURES N. Carrere 1, T. Vadellos 1, E. Marti 1 ONERA, 9 av. de la Divisio Leler, 930 Châtillo, Frae LCTS, 3 Allée de la Boétie, Domaie Uiversitaire, 33600 Pessa, Frae iolas.arrere@oera.fr, thomas.vadellos@oera.fr, marti@lts.u-bordeaux1.fr SUMMARY The preset study is aimed at developig a method to desribe delamiatio iitiated ear the edges o omposite strutures. I this study, two aspets are uder ivestigatio : (i) the modelig of the sigular stress fields ear the edges ad (ii) two omplemetary methods to predit the oset of delamiatio. Keywords: edge effets, ohesive zoe models, frature mehais, delamiatio, multilevel strategy INTRODUCTION Due to their high mehaial properties, omposite materials are more ad more employed i may aerospae appliatios. However, the preset simulatio tools used durig desig ad oeptio of ew strutures do ot take ito aout the omplexity of damage/rupture mehaisms ad the multisale ature of omposites. Moreover, some key problems remai very sesitive. The first oe oers the preditio of the stregth of high stress gradiet parts of the strutures. The seod oe oers the modelig of delamiatio whih ould ot be, otrary to i-ply damage, desribed by a otiuous damage model. Obviously, these two problems arise i lamiate omposites due to a mismath i elasti properties betwee plies. I suh a ase, sigular iterlamiar stresses are ideed reated leadig to iterply debodig. The aim of this paper is to propose a robust methodology to aalyze the delamiatio (oset threshold ad if eessary propagatio) iitiated ear the edges ad the possible iteratio with damage iside the plies. This paper is devided i x setios. The first oe is devoted to the strategy developed to predit the delamiatio iitiatio ear the edges. I this setio, the FE method to alulate the sigular stress field ear the edges is preseted. Two omplemetary approahes are developed to predit the oset of delamiatio. The seod setio is devoted to the idetifiatio ad the ompariso with experimetal results. Fially, the last setio is devoted to disussio ad olusio. STRATEGY FOR THE PREDICTION OF DELAMINATION INITIATED NEAR THE EDGES Calulatio of sigular stress field ear edges
Due to elasti mismath betwee the plies, stress field ear the edges are sigular. Various approahes have bee proposed i the literature i order to ivestigate the stress field ear the edges of a omposite lamiate based o liear assumptios [ 1]. However, the aim of this paper is to propose a methodology that ould be applied to predit the oset of iter-ply damage (i.e. delamiatio) ad itra-ply damage (matrix rakig). This is the reaso why, the elasti assumptio is o more valid. A Fiite Elemet modelig is thus eessary to alulate the stress fields. I order to apture the sigularities, it is eessary to use very fie meshes leadig to high omputatioal ost. Two approahes are developed: The first oe is a D approah, based o a variatioal approah iitially proposed i [ ]. It osists i a.5d model that supposes a uiform strai o the legth of the lamiate. Fores, whih deped o the elasti properties of the plies, are applied to the boudary of the mesh (see Figure 1). Figure 1 : Priiple of the D approah proposed i [ ]. The seod approah osists i usig 3D volume elemets. The boudary oditios ad a mesh are preseted i Figure. Figure : Boudary oditios ad mesh for the 3D FE modellig The results obtaied by these two methods are ompared with those obtaied i the literature for istae with CLEOPS [ 3]. See Figure 3 for a ompariso betwee the 3D FE approah ad the exat results provided by CLEOPS. The two approahes
preseted i this paper lead to very similar results. However, the omputatioal ost of the D approah is very low as ompared with the 3D oe. O the other had, the 3D approah is more geeral ad ould be applied to every geometry (for istae edges of a hole) ad loadigs while the D model is restrited to plae plates subjeted to i-plae loadigs. 0.30 0.7 0.5 0.6 0.0 0.5 0.15 0.4 0.3 0.10 0. 0.05 0.1 0.00 0.0-0.05 0 1 3 4 5 6 7 8 9 10 Szz Y (legth) Syz Sxz (a) -0.1 0 1 3 4 5 6 7 8 9 10 Szz Y (legth) Syz Sxz (b) Figure 3 : Compariso betwee a 3D FE model (dot i the figure) ad the exat soltui provided by CLEOPS [ 3] for (a) [+/-10 ]s (b) [+/-30 ]s lamiates. Preditio of the oset of delamiatio The stress fields beig sigular ear the edges it is ot possible to apply a simple stress or strai based riterio. Differet methods ould be foud i the literature to overome this problem for istae by averagig, over a legth L, the shear stress at the iterfae [3]. However, this approah requires to idetify the parameter L whih has o physial sese. This is the reaso why, i the preset study, two omplemetary approahes are proposed: the first oe is based o a mixed stress ad eergy riterio ad the seod oe ivolves ohesive zoe models. Mixed riterio to desribe the oset of damage The mixed riterio is based o a eergeti oditio ad a stress riterio. The eergeti riterio ompared the hage of the potetial eergy W ad the material toughess G. W is expressed as a variatio betwee the fial state (with a rak of legth d) ad the iitial state (without a rak). This eergeti oditio is give by the followig equatio: G i (d) W(0) W(d) d = = A(l, d)reε G (1) where W is the potetial eergy for a ostat exteral loadig ε, G is the iterfaial toughess. A (l,d) is a dimesioless parameter whih depeds oly o the geometry ( l ) ad of the legth rak (d). R is the thikess of the plies. This relatio ivolves a
iremetal eergy release rate G i (d) sie the lassial ifiitesimal iremet of rak is replaed here by a fiite iremet of rak. I the preset study, the stress riterio is a simple imum riterio. It implies that, prior to the rupture, a state of stress σ greater tha the stregth of the iterfae σ takes plae o a distae at least equal to the legth of the iitiated rak: σ ( l, y) k(l, y)eε σ () where k (l, y) is a dimesioless parameter whih depeds oly o the geometry ( l ) ad of the oordiates alog the iterfae (y). The proedure to alulate the dimesioless parameters ould be foud i [ 4]. Charateristi evolutios of the eergy release rate (represeted by A (l,d), see eq 1) ad the stress itesity fator ( k (l, y) ) i the ase of a delamiatio iitiated from the edge are give i Figure 4a ad b. A 8 x 10-3 6 4 d 0 0 0.5 1 1.5.5 d (a) k(l,y) 0.14 0.1 0.1 0.08 0.06 0.04 0.0 0 1 y 3 4 (b) Figure 4 : A(l,d) as a futio of the rak legth (a) ad k (l, y) as a futio of the y absissa (b). Figure 4a shows a irease of the eergy release rate as a futio of the rak legth d. A (l,d) is imal for a rak legth oted d. For a mootoi loadig, the eergy riterio is satisfied for A (l,d )RE = ε G (3) The iitiatio of the rak with a legth d is possible if : k(l,d ) E (3) ad (4) lead to ε = σ (4) EG A(l,d ) = L (5) L R = σ k (l,d )
Two ases must be osidered: 1. L L tha legth. I this ase, the stress riterio (4) is satisfied, o a legth greater d, before the eergy riterio (3). I this ase, the rak is iitiated o a * d =. L < L d, ad the ritial loadig is o more a futio of. I this ase, the ritial loadig is a futio of is iitiated o a legth A(l,d k (l,d * d whih orrespods to the solutio of σ σ, G ad the rak ) Lr = (6) ) R Fially, the ritial loadig is alulated : ε ε 0 = 1 A(l,d * ) with G ε 0 = (7) RE Cohesive zoe models to desribe de delamiatio Cohesive zoe elemets are used to simulate the iterfaes betwee the plies. The behaviour of these iterfaes is elasti with softeig damage rules i ormal tesio ad shear; moreover, fritio is take ito aout i shear. The mehaial behaviour of the iterfae is desribed by the relatios betwee the ormal ad tagetial relative displaemets (U, U t ) of these odes, ad their respetive ormal ad tagetial tratios (T, T t ). The damage evolutio is take ito aout by the damage variable λ whih ombies the tesio ad the shear damages as follows [ 5]: λ = U δ + U t + δ t The o-liear relatios betwee ( U, U ) ad ( T, T t t ) have the form: U T = F( λ) δ U t Tt = α F( λ) δ t 7 ad F(λ) is hose as F( λ ) = σ (1 λ) for 0 λ 1, where σ ad ασ are 4 respetively the imum values of T ad T t i pure modes. The damage parameter λ varies otiuously from 0 (loally boded ase) to 1 (loally deboded ase). The omplete separatio betwee two orrespodig odes ours for λ=1 (F(λ)=0) so that δ ad δ t are the imum values of the relative displaemets U ad U t i pure ormal ad pure shear modes respetively. I the preset study, o distitio is made betwee ormal ad tagetial loadigs, it meas that α = 1 ad δ t = δ. Two parameters have to be idetified: σ ad δ.
A very importat poit oers the defiitio of the iitiatio of the rak ad of its legth. I the preset paper, it has bee supposed that a rak is reated as soo as the damage variable λ is equal to 1 i (at least) oe Gauss Poit, the rak tip is loated at the last damage Gauss Poit. IDENTIFICATION AND COMPARISON WITH EXPERIMENTAL RESULTS The aim of this setio is to idetify ad validate these two approahes agaist experimetal results provided i [ 6]. The material uder ivestigatio is a arbo/epoxy G947/M18 lamiate. The mehaial properties of the ply are give i Table 1. The ply thikess is equal to 0.190 mm. Table 1 : Mehaial properties of the G947/M18 ply material E 11 (GPa) E (GPa) E 33 (GPa) G 1 (GPa) G 3 (GPa) G 13 (GPa) 1 3 13 97.6 8 8 3.1.7 3.1 0.37 0.5 0.37 [+/-θ ] s lamiates are ivestigated i the preset paper. The results obtaied i [ 6] are reported i Table. Two importat poits are evideed i this table (i) a irease i thikess of the speime (it meas ireases) leads to a derease of the ritial stress, ad (ii) a irease i θ leads to a derease of the ritial load. Table : Experimetal results from [ 6] for [+/-θ]s lamiates Stakig sequee Mode of delamiatio Critial stress (MPa) Std deviatio [+/-10] s III 81 10 [+/-10 ] s III 73 13 [+/-10 3 ] s III 690 10 [+/-10 4 ] s III 67 16 [+/-0 ] s Quasi pure III 481 10 [+/-0 4 ] s Quasi pure III 453 8 [+/-30] s Quasi pure III 391 8 [+/-30 ] s Quasi pure III 351 5 [+/-30 3 ] s Quasi pure III 339 5 [+/-30 4 ] s Quasi pure III 38 8 Idetifiatio of the models
The idetifiatio of the mixed riterio model eessitates, first of all, to perform FE alulatios to alulate A (l,d) ad k (l, y). I order to idetify the ouples (G,σ ) that orrespod to the experimetal ritial load ε leadig to the iitiatio of the delamiatio, G (ad the ε 0 * ) is fixed i order to alulate d (see eq. 5 ad 6). Fially, L (σ ) is alulated usig eq. 7. It is worth metioig that oly a few elasti FE alulatios are eessary to alulate A (l,d) ad k (l, y). Usig these two parameters, the idetifiatio of all the ouples (G,σ ) leadig to the orret experimetal ritial load, ivolved oly some aalytial treatmets. Table 3 shows the idetifiatio for the [+/-10]s ad the [+/-0]s lamiates. Table 3: Idetifiatio of the best ouples (G,σ ) for the [+/-10 ] s ad the [+/-0 ] s lamiates Stakig sequee ± 10 ± 0 σ (MPa) 344 406 G (Jm - ) 86.5 60.7 Error betwee experimetal ad umerial ritial load (%) 1.18 3.44 a*(µm) 1-0.5-1.5 The idetifiatio of the ohesive zoe models, ivolvig o-liear FE alulatios, is more omplex. Ideed, (G,σ ) ouples are iteratively hose thaks to a optimizatio algorithm. A fiite elemet alulatio is performed util the oset of damage. The ritial umerial load is ompared to the experimetal oe. The iterative proess is performed util the overgee betwee the umerial ad the experimetal load. This approah ould lead to a high omputatioal time ad provides oly oe ouple (G,σ ) leadig to the experimetal ritial load. I the preset paper, a strategy, based o the use of a surrogate model (also alled respose surfae), is used. A surrogate model osists i buildig a aalytial relatio (most geerally polyomial) betwee some iputs (i the preset ase G,σ ) ad the outputs (i the preset ase the ritial load) usig a limited umber of rus of the referee modellig (see Figure 5). Fially, this aalytial approximatio is used to determie the values of the outputs as a futio of the variatio of the iputs. I this study, the approximatio is performed thaks to a sparse polyomial expasio. It is worth metioig that the aalytial model is just a approximatio ad some tools have bee developed i order to irease the ofidee i this approximatio [ 7].
1000 800 Critial stress 600 400 00 0 400 00 G 0 0 100 00 300 400 sigma Figure 5 : Priiple of the respose surfae method (RSM) to assess the evolutio of the ritial stress as a futio of the iterfaial parameters G,σ. The blak dots orrespod to the idetifiatio poits. The surfae orrespods to the results obtaied by the RSM. Figure 6a ad Figure 6b show the ouples (G,σ ) leadig to a ritial load with a error of 10% as ompared with the experimetal oes for the [+/-10]s ad [+/-10 ]s lamiates. It is worth metioig that a large umber of ouples (G,σ ) permits to predit the experimetal results for the two lamiates. The value idetified from the mixed riterio is lose to (but lower) some values give by this aalysis o ohesive zoe models. (a) (b) Figure 6 : Idetifiatio of the ouples (G,σ ) orrespodig to the experimetal ritial load for a [+/-10] s (a) ad a [+/-10 ] s (b) lamiate. The blak dots orrespod to (G,σ ) parameters leadig to a error lower tha 10%.
The evolutio of the ritial load as a futio of the thikess is show i Figure 7. The model permits to reprodue i a orret maer the mai experimetal evidees i.e. a derease of the ritial load if the thikess ad/or the agle of the plies irease. 1000 900 [+/-10 ] s PM10 G=8.4 Jm - S=359 MPa 800 700 G=86.5 Jm - S=344 MPa sigma 600 [+/-0 PM0 ] s G=60.7 Jm - S=406 MPa 500 400 G=8.4 Jm - S=359 MPa 300 0 1 3 4 5 Figure 7 : Evolutio of the ritial load as a futio of the umber of plies for the [+/-10 ] s ad [+/-0 ] s lamiates. The dashed urves are predited with the (G,σ ) provided i Table 3. The otiuous lie orrespods to the best fit for the two lamiates. DISCUSSION AND CONCLUSION A strategy has bee preseted to predit the oset of delamiatio iitiated from the edges. Two approahes have bee proposed to attai this goal. The first is based o a mixed riterio that eessitates oly elasti FE alulatios to be applied. The seod approah used 3D FE alulatios with ohesive zoe elemets. As ompared with the lassial methods used i the literature, these two approahes do ot eessitate the idetifiatio of a ritial legth. Oly the ritial eergy release rate G ad the stregth σ of the iterfae eessitate to be idetified. These two approahes have permitted to reprodue experimetal evidees suh as a irease i thikess of the speime leadig to a derease of the ritial stress, ad (ii) a irease i θ leadig to a derease of the ritial load. However, it has bee show that :
I order to use these methods, very fie meshes are eeded. Ideed, the rak legth is very small (from 0.5 to µm). This is the reaso why, i order to apply this approah to a omplex struture, a omputatioal strategy is eeded (for istae a global->loal approah). A D FE approah has also bee preseted i order to redue drastially the omputatioal ost. The mixed riterio approah ad the ohesive zoe models are omplemetary. Ideed, the mixed riterio dimesioless parameters are oly a futio of the elasti properties ad of the geometry. These parameters are alulated usig simple elasti modeligs. Thaks to these parameters, aalytial equatios permit to assess the effet of (G,σ ) o the ritial loadig leadig to the oset of delamiatio. O the other had, ohesive zoe models eessitate expesive o-liear FE alulatios whih are a futio of the elasti properties, the geometry ad the iterfaial parameters (G,σ ). I order to assess the effet of the iterfaial parameters, a strategy based o surrogate model is eessary. However, otrary to the mixed riterio, ohesive zoe models permit to take ito aout all the o-liearities (visosity, i-plae damage i the plies) ad the ouplig betwee itra-ply damage ad delamiatio. I order to reprodue the experimetal results, is eessary to idetify the iterfaial parameters of eah stakig sequee. This problem is probably due to the evolutio of the mixed-mode delamiatio (see Table ). Ideed, it is well kow that the ritial eergy release rate is a futio of this mixed-mode delamiatio, ad this poit must be itrodued i the preset aalysis for both the eergy release rate ad the ritial stregth of the iterfae. Referees 1. Mittelstedt C. ad Beker W., Free-edge effets i omposite lamiates, Applied Mehais Reviews, Vol. 60, pp 17-45, 007.. Pipes R.B. ad Pagao N.J., Iterlamiar stresses i omposite lamiates uder uiform axial extesio, Joural of Composite Materials 4. 538 548,1970. 3. Medysys, Cleops referee maual, 008. 4. Marti E., Peters P.W.M., Leuillo D., Queisset J.M., Coditios for matrix rak defletio at a iterfae i erami matrix omposites, Mater. Si. Eg. A., 50, pp. 91-30, 1998. 5. Tvergaard V., "Effet of fibre debodig i a whisker-reifored metal", Materials Siee ad Egieerig, 03-13, 1990. 6. Lorriot T., Mario G., Harry R., Wargier H., Oset of free-edge delamiatio i omposite lamiates uder tesile loadig, Composites: Part B 34, 459 471, 003. 7. Carrere N., Rollet Y., Leory F.-H., Maire J.-F, Effiiet strutural omputatios with parameters uertaity for omposite appliatio, aepted for publiatio i Composite Siee ad Tehology, Available o lie 4 Otober 008.