Study of transversal and delamination cracks in [0/90] symmetric laminates by means of the linear elastic brittle interface formulation

ESTUDIO DE GRIETAS TRANSVERSALES Y DE DELAMINACI ´ ON EN LAMINADOS SIM´ ETRICOS [0/90] DE MATERIAL COMPUESTO USANDO LA FORMULACI ´ ON DE INTERFASE EL ´ ASTICA LINEAL-FR ´ AGIL STUDY OF TRANSVERSAL AND DELAMINATION CRACKS IN [0/90] SYMMETRIC LAMINATES BY MEANS OF THE LINEAR ELASTIC BRITTLE INTERFACE FORMULATION L. T´ a a a, V. Man iˇ c, A. Bl´ azquez, E. G aciani, F. Pa ´ ıs G upo de Elas icidad y Resis encia de Ma e iales, Escuela T´ecnica Supe io de Ingenie os, Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092 Se illa, Espa˜na. E-mail: l a[email p o ec ed], m[email p o ec ed], [email protected], g[email p o ec ed], p[email p o ec ed] RESUMEN En el p esen e abajo se es udia el inicio y c ecimien ode g ie as ans e sales y de delaminaci´onen laminados sim´e icos [0/90], bajo ca gas de acci´on en la di ecci´on de las ib as en las l´aminas ex e nas. El p oblema se esuel e con un c´odigo basado en el M´e odo de los Elemen os de Con o no (MEC) y usando una nue a o mulaci´on de in e ase el´as ica lineal- ´agil (IELF). Es a o mulaci´on incluye una ley de compo amien o el´as ica lineal- ´agil en la in e ase que pe mi e el uso de un c i e io de allo en modo mix o. La o mulaci´on IELF hademos adoque puedese usada pa a es udia y ca ac e iza el compo amien o de las di e en es g ie as de in e ase que apa ecen en es e p oblema. El sis ema de ma e iales usado en es e p oblema es un laminado de ib a de ca bono con ma iz epoxi (AS4/8552 Hexcel) [03/903]S. Los esul ados num´e icos mues an que la o mulaci´on IELF implemen ada en el c´odigo MEC es una he amien a ´u il y e icien e pa a desc ibi el compo amien o de las g ie as ans e sales y de delaminaci´on en laminados sim´e icos [0/90]. ABSTRACT This pape s udies ans e sal and delamina ion c ack onse and g ow h in [0/90] symme ic lamina es unde ac ion loads in he ib e di ec ion o he ex e nal laminas. The p oblem is sol ed by means o a Bounda y Elemen Me hod (BEM) code and using a new Linea Elas ic B i le In e ace (LEBI) o mula ion. This o mula ion includes a linea elas ic-b i le cons i u i e law in he in e ace ha allows he use o a mixed mode ailu e c i e ion. The LEBI o mula ion has p o ed o be capable o s udy and cha ac e ize he beha iou o he di e en in e ace c acks in ol ed in his p oblem. The ma e ial sys em used is ca bon ib e ein o ced polyme lamina e wi h epoxy ma ix (AS4/8552 Hexcel) [03/903]S. Nume ical esul s show ha he LEBI o mula ion implemen ed in a BEM code is a use ul and e icien ool able o desc ibe he ans e sal and delamina ion c acks beha iou in [0/90] symme ic lamina es. PALABRAS CLAVE: Composi es, Delamina ion, Mixed mode ac u e, In e ace c ack 1. INTRODUCTION Composi es a e expe iencing a massi e use in p ima y s uc u es o comme cial ae oplanes. The associa ed cu- en demand o diminishingweigh equi esabe e know- ledge o mechanisms o ailu e. An essen ial s ep is o gene a e physically based ailu e o damage c i e ia. All his leads o he necessi y o e isi ing classical p oblems o composi e lamina es such as he mechanism o damage in [0/90]Slamina es [1]. In he p esen wo k he linea elas ic-b i le in e ace (LE- BI) model de eloped in T´a a a e al. [2, 3] has been en- hanced, esul ing in a linea elas ic-b i le cons i u i e law ha akes in o accoun he a ia ion o he ac u e ough- ness wi h he ac u e mode mixi y. The cons i u i e law conside s he possibili y o linea elas ic ic ionless con- ac be ween ib e and ma ix once a po ion o he in e - ace is b oken. The LEBI model has been implemen ed in a 2D colloca ional BEM code. The p esen pape is a s ep o wa d owa ds he cha ac- e iza ion o he mechanisms o ailu e o a [0/90]Slami- na e. I s aim is o cha ac e ize he beha iou o he wo ypes o c ack in ol ed in he p oblem ( ans e se and delamina ion) by means o he LEBI o mula ion, ying o connec p edic ions wi h he obse ed damage o spe- cimens, as shown in [4]. 2. DESCRIPTION OF THE PROBLEM The p oblem analyzed is shown in Fig. 1. I ep esen s a [0/90]Slamina e unde ensile loading in he di ec ion o he 0◦ ib es, ha was sol ed p e iously by means o he BEM and using he Vi ual C ack Closu e Technique (VCCT) o calcula e he Ene gy Release Ra e (ERR) by Bl´azquez e . al. [5, 6] and by Pa ´ıs e . al. [1]. The i s damage in his lamina e is expec ed o be he nu- clea ion and g ow h o c acks in he 90◦ply ans e se o he load. When one o hese c acks app oaches he in e - ace wi h he 0◦ply,i is accep ed ha i s ops. New ans- e se c acks appea in he 90◦ply wi h inc easing load un il he c ack densi y eaches a c i ical alue. T ans e - se ma ix c acking in 90◦ply leads o a load edis ibu ion in he adjacen 0◦plies and induces local s ess concen- a ions a he neighbo hood o he c ack ips ha , when he ips a e nea o he in e ace, can in ol e signi ican in e lamina delamina ion [1, 7]. Fo he case analyzed in his wo k, hema e ial sys ems is a ca bon–epoxy (AS4/8552 Hexcel) lamina e [03/903]S, di ec ion 1 being conside ed he ib e di ec ion: E11 = 45.6 GPa, E22 =E33 =16.2 GPa, ν12 =ν13 =0.278, ν23 = 0.4, G12 =G13 =5.83 GPa, G23 =5.786 GPa. The hal - hickness o he se o 90◦ply, , and he hickness o each o he se o 0◦plies is 0.55 mm. The a e age sepa a ion be ween ans e se c acks is aken o be 2L=4 mm. Figu e 1: T ans e se and delamina ion c acks in [0/90]S lamina e, aken om [1]. 3. MODEL OF THE PROBLEM The p e iously desc ibed p oblem ha will be s udied, has some usually accep ed damagepa e ns[1], see Fig. 2. One ha ing only a ans e se c ack Fig. 2(a) and (b), he second ha ing a ans e se c ack ha has eached he in- e ace wi h he 0◦lamina and has de lec ed, s a ing a symme ic delamina ion Fig. 2(c). The hi d damage pa - e n shown Fig. 2(d) indica es ha delamina ion will s a o appea be o e he ans e se c ack eaches he in e - ace. This scheme ep esen ed in Fig. 2(d) is known in F ac u e Mechanics as he Cook-Go don mechanism [8]. As men ioned be o e, he LEBI model is used o s udy he p esen p oblem. The cases shown in Fig. 2 exhibi sym- me y (wi h espec o he ho izon al middle plane in he igu e), his ac allows us o s udy he delamina ionc ack g ow h ei he using he con igu a ion shown in Fig. 3(a) o Fig. 3(b). Ne e heless, i we wan o s udy he on- se and g ow h o he ans e sal c ack (modeled using he LEBI model), i is necessa y he use o he con igu a- ion p esen ed in Fig. 3(a) because o he use o in e ace elemen s ha , in he p esen implemen a ion, needs he p esence o bo h solids adjacen o he in e ace. In he BEM model used o simula e he geome y shown in Fig. 3(a) he uni o m bounda yelemen mesh has 3860 (a) (b) (c) (d) Figu e 2: Damage con igu a ions conside ed wi h a ans e se c ack: (a) no eaching he in e ace, (b) e - mina ed a he in e ace, (c) de lec ed a he in e ace, and (d) app oaching a damaged in e ace (mechanism o Cook-Go don). Sligh ly modi ied e sion o a pic u e om [1]. (a) (b) Figu e 3: Geome y and bounda y condi ions o he de- lamina ion p oblem in [0/90] symme ic lamina es. linea elemen s, while in he o he case (Fig. 3(b)) he mesh is o med by 2040 linea elemen s. In bo h cases he cons an elemen size is 5µm. 4. NUMERICAL RESULTS FORTRANSVERSE AND DELAMINATION CRACKS 4.1. T ans e se c acks As men ionedbe o e, o s udy he onse and g ow h o he ans e se c ack, he geome y shown in Fig. 3(a) is used. The LEBI elemen s ha e been included a he in e ace be ween he 0◦ply and 90◦ply, and also in he assumed c ack pa h o he ans e se c ack ha could appea in he 90◦ply (shown in Fig. 3(a) wi h a dashed line). Thep ope ieso he in e acesa he woposi ions ( ans- e sal o delamina ion) we e conside ed o be di e en , see Table 1. Table 1: Conside ed combina ions o he in e ace p o- pe ies in he delamina ion p oblem o [0/90] symme ic lamina es. Posi ion GIc(Jm−2) ¯σc(MPa) kn(MPa/µm) ans e sal 75 61 24.807 delamina ion 75 90 54 No ice ha he ac u e oughness in mode I, GIc, is con- side ed o be he same in bo h posi ions, bu he c i ical s ess, ¯σc, a ies leading o di e en alues o kn. Finally, he a io kn/k =3 is conside ed. Figu e 4: Applied s ain e sus he leng h o he ans e - se c ack a. Figu e 5: P edic ions o he ERR (G), o a ixed applied s ain o 1%, e sus he leng h o he ans e se c ack a. In Fig. 4 he ac ual applied a e age (longi udinal) s ain, ε=¯u/L, e sus he leng h o he ans e se c ack ais plo ed. I can be seen ha once a c i ical s ain is applied he c ack g owsin an uns able manne (less applied s ain is needed) un il i eaches he in e ace be ween he 0◦ply and 90◦ply. Al hough i is no shown in he abo e men ioned igu e he ac u e ene gyGcwasalways equal oGIc =75Jm−2, due o he mode I beha io o he ans e sal c ack and acco ding o Table 1. Fig. 5 ep esen s he dis ibu ion o he alues o he ERR a he c ack ip o he g owing ans e se c ack e sus he leng h a, o a ixed applied s ain o 1%. 4.2. Delamina ion c acks In he ollowing, he onse and g ow h o he delamina- ion c ack is s udied. In pa icula he e ec s o he a ia- ion o he geome y o be used and he conside a ion o no o an elas ic con ac algo i hm a e elucida ed in he ollowing sec ion. 4.2.1. Di e en geome ies To s udy he onse and g ow h o he delamina ion c ack, he geome ies shown in Fig. 3(a) and (b) a e used. The p ope ies co espond o he ones shown in Table 1. The con ac algo i hm is also ac i a ed. In Fig. 6, a compa ison o he ac ual applied s ain e - sus he leng h o he delamina ion c ack dis plo ed, o he wo geome ies shown in Fig. 3. In ac , he alue o d ep esen s, hal o he o al leng h o he delamina ion c ack. Fo he geome y shown in Fig. 3(b) is d=d1, while o Fig. 3(a) dis de ined as d=(d1+d2)/2. The eason o his de ini ion is ha d1is no necessa ily equal o d2, as some imes in modelinga c ack g ow hi is possi- ble o ob ain non-symme ic solu ions o o iginally sym- me ic p oblems. The g ow h o he delamina ion c ack is an example o his beha io . Fig. 7 ep esen s a com- pa ison o he a ia ions o he ac u e ene gy neces- sa y o cause he c ack g ow h e sus he leng h d; while Fig. 8 shows a compa ison o he a ia ions o he alues o he ERR o he ans e se c ack e sus he leng h a, o a ixed applied s ain o 1%, o he wo geome ies shown in Fig. 3. In Figs. 6-8 compa isons o he esul s Figu e 6: Compa ison o he applied s ain e sus he leng h o he delamina ion c ack d, o he geome ies shown in Fig. 3. o he onse and g ow h o he delamina ion c ack ob- ained by using he wo geome ies shown in Fig. 3 a e Figu e 7: Compa ison o he dis ibu ion o he ac u e oughness, ene gy necessa y o cause he c ack g ow h, e sus he leng h d, o he geome ies shown in Fig. 3. Figu e 8: Compa ison o he p edic ions o he ERR (G), o a ixed applied s ain o 1%, e sus he leng h o he delamina ion c ack d, o he geome ies shown in Fig. 3. p esen ed. As can be seen om hese igu es, he esul s o geome y shown Fig. 3(a) p esen s some spu ious os- cilla ions. This ac is caused because al hough he p o- blem is symme ic, he geome y shown in Fig. 3(a) and he sequen ially linea analysis used [2], allow o a non- symme ic c ack g ow h, while in he geome y shown in Fig. 3(b) a symme ic c ack g ow h is assumed. In his manne , once one c ack b anch ( op o bo om) ini ia es an uns able g ow h (less applied s ain is necessa y o c ack g owing) he c ack g ows only ough his b anch, leading o d1,d2. Be o e a con inuos uns able g ow h ini ia es in he second b anch, some spu ious beha io is ob ained caused because he c ack g ows in bo h b an- ches al e na ing small ad ances in bo h c ack b anches, see Figs. 7-8. In pa icula , picks and alleys ob ained in Fig. 6 o he geome y shown in Fig. 3(a) a e ela- ed o he uns able g ow h o one b anch ollowed by he uns able g ow h o he o he b anch. Ne e heless, his beha io is obse ed along he uns able g ow h zone only, and once he c ack g ow h becomes s able he sym- me y o he p oblem is eco e ed. Thus, in he ollowing o a oid his spu ious beha io (wi h no clea physically meaning), wi h he aim o educe compu ing ime and be- cause he ob ained esul s a e e y simila , only he geo- me y shown in Fig. 3(b) will be used in he ollowing. In any case he obse ed al e na ing c ack g ow h will e- qui e a u he and mo e ex ensi e s udy in he u u e. 4.2.2. Con ac in luence The delamina ion c ack s a s being an open c ack and, a e a ce ain c ack leng h size is eached, i s a s o be pa ially in con ac . Thus, he in luence o con ac in his speci ic p oblem is s udied. In one case, once he LEBI elemen is b oken i may en e in ic ionless con ac [9]. In he o he case in e pene a ion is allowed wi hou cau- sing any con ac s esses. Again he p ope ies aken whe- e he ones de ined in Table 1 and he geome y used is he one shown in Fig. 3(b). Figu e 9: Compa ison o he applied s ain e sus he leng h o he delamina ion c ack d, conside ing con ac o no . Figu e 10: Mixi y angle ψσob ained a he c ack ip e - sus he leng h o he delamina ion c ack d, conside ing con ac o no . In Fig. 9 a compa ison o he ac ual applied s ain e sus he leng h o he delamina ion c ack dis plo ed, consi- de ing con ac o no . No ice ha when con ac is consi- de ed he delamina ion c ack ini ially beha es in a s able manne , hen a ela i ely la ge uns able g ow h appea s whe e an ins abili y phenomenon called snap- h ough a- kes place, and inally a s able g ow h is eached again. I con ac is no conside ed, allowing o e lapping o dela- mina ion c ack aces, he c ack g ow h is always s able. Fig. 10 ep esen s a compa ison o he mixi y angle ψσ, (ψσ=τ/σ) [9], ob ained a hec ack ip e sus he leng h o he delamina ion c ack d. In a i s s age a s able c ack Figu e 11: Compa ison o he dis ibu ion o he ac u- e oughness, ene gy equi ed o cause he delamina ion c ack g ow h, e sus he leng h d, conside ing con ac o no . Figu e 12: Compa ison o he p edic ions o he ERR (G), o a ixed applied s ain o 1%, e sus he leng h o he delamina ion c ack d, conside ing con ac o no . g ow h in mixed mode (ψσ≤90◦) is obse ed o wo cases. A e his s age a g ea di e ence o ψσ alues is ob ained. When con ac is conside ed, a small s age o c ack g ow h wi h he c ack ip closed appea s, and a - e his small s age he c ack g ows in mixed mode again, wi h he c ack ip opened o ming a “bubble” nea he c ack ip. On he o he hand, when con ac is no consi- de ed, he s age o c ack g ow h wi h he c ack ip closed is e y la ge. Fig. 11 ep esen s a compa ison o he dis ibu ion o he ac u e ene gy necessa y o cause he c ack g ow h e - sus he leng h d. In his case i is no ewo hy ha when con ac is conside ed he ac u eene gynecessa y o g ow h is less han when con ac is no conside ed. Thus, con ac makes easie c ack p opaga ion. This ac could be ex- plained because o he ailu e c i e ia used. As de ailed in [9], when he c ack ip is in comp ession i needs mo e ene gy (g ea e angen ial s esses which con ol he ai- lu e in his p oblem) o g ow h. Fig. 12 shows a compa ison o he dis ibu ion o he alues o he ERR o he delamina ion c ack e sus he leng h d, o a ixed applied s ain o 1%. I can be seen ha , he di e ences o he alues o he ERR o he dela- mina ion c ack a e almos negligible i con ac is consi- de ed o no . Figu e 13: No mal s esses ob ained a he in e ace be - ween 0◦and 90◦plies o he di e en load s eps shown in Fig. 9. Figu e 14: Tangen ial s esses ob ained a he in e a- ce be ween 0◦and 90◦plies o he di e en load s eps shown in Fig. 9. In Fig. 13 and Fig. 14 he no mal and angen ial s es- ses a he in e ace be ween 0◦and 90◦plies a e plo ed o di e en load s eps shown in Fig. 9 when elas ic con- ac is conside ed. As can be seen om hese igu es he angen ial s esses a e much la ge han he no mal s es- ses du ing he delamina ion c ack g ow h. In pa icula in Fig. 13, he “bubble” o med nea he c ack ip can be obse ed (σ=0). In Fig. 15 he de o med shapes o he same load s eps conside ed in Fig. 13 and Fig. 14 a e shown. These de o med shapes a e mul iplied by 20 only in he x-di ec ion. 5. CONCLUSIONS As shown by he nume ical esul s p esen ed in his wo k, he LEBI o mula ion seems o be a p omising ool o desc ibe he beha io o he ans e sal and delamina ion c acks in [0/90] symme ic lamina es. As shown by he ealis ic esul s p esen ed, he BEM ool de eloped can be conside ed a use ul ool, due o he ac ha he c ack p opaga ion can be modeled by using he same uni o m mesh. While o he VCCT [1, 10] a e ined mesh is ne- cessa y nea he c ack ip, hus o di e en load s eps, (a) (b) (c) (d) (e) Figu e 15: De o med shapes ob ained o he di e en load s eps shown in Fig. 9, mul iplied by 20 in he x- di ec ion only. di e en meshes a e needed. I is ema kable ha he on- se and uns able g ow h o he ans e sal c ack equi es a signi ica i ely lowe load (applied s ain) han he one needed o he onse o he delamina ion c ack. In all he analyzed cases, he ans e sal c ack eached he in e a- ce be ween 0◦and90◦plies be o e he delamina ionc ack onse . Once he delamina ion c ack s a s o g ow h h ee s ages can be clea ly iden i ied: (i) ini ial s age - ini ially a ela i ely sho s able g ow h o he delamina ion c ack akes place wi h open ac ion- ee c ack aces; (ii) in e - media e s age - he delamina ion c ack g ow h becomes uns able and he e is con ac be ween he c ack aces, ex- cep o he zone close o he c ack ip whe e a kind o bubble appea s; (iii) inal s age - he delamina ion c ack g ow h becomes s able again. I is in e es ing o no ice ha al hough he p oblem con igu a ion (geome y, ma- e ial and bounda y condi ions) is symme ic i s solu ion including a ans e sal and delamina ion c ack may be non-symme ic. This ac was obse ed in he second s a- ge o he delamina ion c ack g ow h (uns able), whe e an non-symme ic g ow h can be obse ed in he wo b an- ches o he delamina ion c ack. 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