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.
ACKNOWLEDGEMENTS
The wo k was suppo ed by he Jun a de Andaluc´ıa (P o-
jec s o Excellence TEP-1207, TEP 02045 and P08-TEP
04051), he Spanish Minis y o Educa ion and Science
h oughP ojec TRA2006-08077andMAT2009- 140022.
REFERENCES
[1] F. Pa ´ıs, A. Bl´azquez, L. N. McCa ney,and V. Man-
iˇc. Cha ac e iza ion and e olu ion o ma ix and
in e ace ela ed damage in [0/90]Slamina es unde
ension. Pa I: Nume ical p edic ions. Composi es
Science and Technology, 70:1168–1175,2010.
[2] L. T´a a a, V. Man iˇc, E. G aciani, J. Ca˜nas, and
F. Pa ´ıs. Analysis o a c ack in a hin adhesi e la-
ye be ween o ho opic ma e ials. An applica ion
o composi e in e lamina ac u e oughness es .
Compu e Modeling in Enginee ing and Sciences,
58(3):247–270,2010.
[3] L. T´a a a, V. Man iˇc, E. G aciani, and F. Pa ´ıs.
BEM analysis o c ack onse and p opaga ion along
ibe -ma ixin e ace unde ans e se ension using
a linea elas ic-b i le in e ace model. Enginee-
ing Analysis wi h Bounda y Elemen s, 35:207–
222, 2011.
[4] F. Pa ´ıs, A. Bl´azquez, L. N. McCa ney, and A. Ba-
oso. Cha ac e iza ion and e olu ion o ma ix and
in e ace ela ed damage in [0/90]Slamina es unde
ension. Pa II: Expe imen ale idence. Composi es
Science and Technology, 70:1176–1183,2010.
[5] A. Bl´azquez, V. Man iˇc, F. Pa ´ıs, and L. N. Mc-
Ca ney. S ess s a e cha ac e iza ion o delami-
na ion c acks in [0/90] symme ic lamina es by
BEM. In e na ional Jou nal o Solids and S uc-
u es, 45:1632–1662,2008.
[6] A. Bl´azquez, V. Man iˇc, F. Pa ´ıs, and L. N. McCa -
ney. BEM analysis o damage p og ess in 0/90 la-
mina es. Enginee ing Analysis wi h Bounda y Ele-
men s, 33:762–769, 2009.
[7] JM. Be helo . T ans e se c acking and delamina-
ion in c oss-ply glass- ibe and ca bon- ibe ein-
o ced plas ic lamina es: s a ic and a igue loading.
Applied Mechanics Re iews, 56:111–147, 2003.
[8] J. Cook, J. E. Go don, C.C. E ans, and D.M. Ma sh.
A Mechanism o he Con ol o C ack P opaga ion
in All-B i le Sys ems. P oceedings o he Royal So-
cie y o London. Se ies A, Ma hema ical and Physi-
cal Sciences, 282:508–520,1964.
[9] L. T´a a a. Damage ini ia ion and p opaga ion
in composi e ma e ials. Bounda y elemen analy-
sis using weak in e ace and cohesi e zone models.
PhD Thesis. Uni e sidad de Se illa: Se illa, 2010.
[10] A. Bl´azquez. T ans e sal and delamina ion c ack
onse and g ow h in [0/90]Slamina es. GERM P i-
a e Comunica ion, 2010.