C ack onse and g ow h a he ib e-ma ix in e ace unde a emo e
biaxial ans e se load. Applica ion o a coupled s ess and ene gy
c i e ion
V. Man iˇc, I.G. Ga c´ıa
G oup o Elas ici y and S eng h o Ma e ials, School o Enginee ing, Uni e si y o Se ille
Camino de los Descub imienos s/n, 41092 Se ille, Spain
Abs ac
A heo e ical model o p edic ion o he c i ical load gene a ing a c ack onse a he ib e-ma ix in e ace
unde a emo e biaxial ans e se load is p esen ed. In pa icula , his wo k is ocused on he ension
domina ed ailu e. A e an ab up onse he c ack g ows uns ably up o achie ing an a es leng h.
A simple plane s ain model o a single ci cula inclusion su ounded by an unbounded ma ix allows
o ob ain conclusions app oxima ely alid o a dilu e ib e packing. Linea iso opic elas ic beha iou
is assumed o bo h inclusion and ma ix. Two classical elas ic solu ions o bo h pe ec ly bonded
and pa ially debonded ci cula inclusions a e used oge he wi h a coupled s ess and ene gy c i e ion,
p oposed ecen ly in he amewo k o Fini e F ac u e Mechanics, and a phenomenological law o ac u e
oughness o in e ace c acks g owing in ac u e mixed mode. The ob ained analy ical and semi-analy ical
exp essions make easy o s udy he in luence o all he dimensionless pa ame e s go e ning he ib e-
ma ix sys em beha iou : Dundu s elas ic bima e ial cons an s αand β, he in e ace b i leness numbe
γ, he load biaxiali y pa ame e η, and he ac u e mode-sensi i i y pa ame e λ. A size e ec o he
inclusion adius on he c i ical load is p edic ed, smalle inclusions being s onge and less dependen
on he seconda y load. Finally, an expe imen al p ocedu e o measu emen o he ib e-ma ix in e ace
ac u e and s eng h p ope ies is p oposed.
Keywo ds: composi es, ci cula inhomogenei y, c ack ini ia ion, in e ace debond, size e ec , ini e
ac u e mechanics, ac u e houghness, damage mechanism, ailu e c i e ia, b i leness numbe
1. In oduc ion
Composi es ein o ced by long ib es a e commonly used as a s uc u al ma e ial in ligh weigh s uc-
u es a p esen . In ae ospace applica ions, whe e ligh weigh is a key aspec o he design, hei le el
o s uc u al esponsibili y has signi ican ly inc eased as hey a e massi ely used in p ima y s uc u es.
Howe e , ou unde s anding o he damage mechanisms occu ing in hese composi es on di e en scales
is s ill insu icien . Thus, i is necessa y o gene a e mo e knowledge abou hese mechanisms in o de o
a oid he p esen high le el o unce ain y in he ailu e loads p edic ed in he design.
One o he mos complex ailu e mechanisms on mic o scale in hese composi es is associa ed o he
ma ix ailu e, also called in e - ib e ailu e. In pa icula , he ension domina ed mechanism ollows a
well desc ibed sequence o s ages, see Hull and Clyne (1996) and Pa ´ıs e al. (2007): i) ailu e is ini ia ed
a he ib e-ma ix in e ace as small debonds, ii) he in e ace c acks g ow along he in e ace un il a
ce ain a es angle and hen iii) kink ou he in e ace owa ds he ma ix, i ) coalescence o g owing
ma ix c acks gene a es a mac oc ack which may cause he ailu e o he unidi ec ional lamina.
The p esen wo k is ocused on he wo i s s eps o his ailu e mechanism: c ack ini ia ion and
g ow h a he ib e-ma ix in e ace. P oblem o a single ib e embedded in a ma ix including a pa ial
debond has been in ensi ely s udied by many au ho s o a long ime, see Pa ´ıs e al. (2007) and Man iˇc
Email add esses: [email p o ec ed] (V. Man iˇc), [email p o ec ed] (I.G. Ga c´ıa)
P ep in submi ed o In e na ional Jou nal o Solids and S uc u es
(2009) o comp ehensi e e iews. Ne e heless, he debond onse has no a ac ed su icien a en ion
up o he las decade. Resul s p esen ed in bibliog aphy ha e usually been ob ained by compu a ional
me hods as cohesi e zone o weak in e ace models, see o ins ance Ca pin e i e al. (2005), Xie and
Le y (2007), and T´a a a e al. (2011)). Man iˇc (2009) p oposed a heo e ical model o p edic he c ack
ini ia ion along he ib e-ma ix in e ace unde an uniaxial emo e ension. This heo e ical model is
based on he coupled c i e ion in oduced by Leguillon (2002) in he amewo k o he Fini e F ac u e
Mechanics, see also Co ne i e al. (2006) and o a e iew Taylo (2007). This model p oposes o apply
bo h he s ess and ene gy c i e ia simul aneously as a su icien condi ion o he in e ace c ack onse .
In he p esen wo k, an ex ension o he heo e ical p ocedu e de eloped by Man iˇc (2009) o he case
o ension domina ed emo e biaxial ans e se loads is de eloped. No e ha he e ec o he seconda y
comp ession in addi ion o a domina ing ension has been demons a ed o be impo an by Pa ´ıs e al.
(2003) and Co ea (2008). This analysis is ca ied ou o dilu e ib e packing whe e he in luence o
adjacen inclusions is almos negligible, whe eas o dense ib e paking, a nume ical me hod should be
employed based on o he solu ions, e.g. Kushch e al. (2010). In addi ion, se e al models de eloped unde
he assump ion o dilu e ib e packing ha e demons a ed being use ul o explain expe imen al esul s
e en o densely packed composi es, see Co ea (2008). A e a e ision o he solu ion o he elas ic
inclusion pe ec ly bonded o he ma ix in Sec ion 2, he elas ic p oblem o an inclusion wi h a debond is
analysed in Sec ion 3, whe e a gene al analy ical solu ion is simpli ied o his p oblem. Coupled c i e ion
is de eloped and applied in Sec ion 4 and he esul s o he c i ical c ack leng h and emo e load a
he onse a e ob ained. Sec ion 5 desc ibes he in luence o he emo e seconda y load and he inclusion
size on he esul s ob ained. Finally a new expe imen al p ocedu e o an indi ec measu emen o he
s eng h and ac u e in e ace p ope ies is p esen ed in Sec ion 6.
2. S esses in a single inclusion unde a emo e biaxial ans e se load
S ess based ailu e c i e ia usually conside he s ess s a e p io o he damage appea ance. Hence
he aim o his sec ion is o s udy he ac ions along he ib e-ma ix in e ace unde emo e biaxial load
ans e se o he ib e axis. Assuming ce ain hypo hesis, a classical elas ic solu ion is pa icula ized o
his p oblem p o iding closed o m exp essions o ac ions. Finally, he ac ion dependence on he key
p oblem pa ame e s is discussed.
Conside a ci cula cylind ical inclusion o adius aembedded in an in ini e ma ix and pe ec ly
bonded along i s la e al in e ace. Le (x, y, z) and ( , θ, z) be sui ably de ined ca esian and cylind ical
coo dina e sys ems, he z-axis being coinciden wi h he inclusion (longi udinal) axis. Remo e uni o m
biaxial ans e se load (σ∞
x, σ∞
y) is applied pa allel o he he wo axes, xand y, ans e se o he
cylind ical inclusion axis, see Figu e 1.
Figu e 1: The inclusion p oblem con igu a ion.
An analy ic solu ion o s esses in his p oblem was deduced by Goodie (1933). As was shown by
2
Ha diman (1954) he s esses inside he inclusion a e cons an . Following Man iˇc (2009), hese inclusion
s esses a e ew i en in e ms o he Dundu s bima e ial cons an s αand β(Dundu s (1967, 1969)) as
ollows:
σ(1)
xσ(1)
xy
σ(1)
xy σ(1)
y!=σ∞
xk0
0k−m+σ∞
yk−m0
0k,(1)
whe e
k(α, β) = 1
2
1 + α
1 + β
2 + α−β
1 + α−2βand m(α, β) = 1 + α
1 + β,(2)
and he Dundu s bima e ial cons an s a e de ined as
α=µ1(κ2+ 1) −µ2(κ1+ 1)
µ1(κ2+ 1) + µ2(κ1+ 1) and β=µ1(κ2−1) −µ2(κ1−1)
µ1(κ2+ 1) + µ2(κ1+ 1),(3)
wi h µk=Ek/(2(1 + νk)) and κk= 3 −4νk,Ekand νkdeno ing he Young’s modulus and Poisson’s
a io, espec i ely. I can be shown ha 0 ≤k≤5/3 and 0 ≤m≤2.
Le a dimensionless load-biaxiali y pa ame e ηbe de ined as he a io o emo e s esses
η=σ∞
y
σ∞
x
.(4)
Then, no mal and angen ial ac ions, σand τ, ac ing along he in e ace ( =a) can be exp essed as a
unc ion o he pola angle θ(see Figu e 1) and he pa ame e ηas
σ(θ)
σ∞
x
=σ (θ)
σ∞
x
=k+ (k−m)η−(1 −η)msin2θ, (5a)
τ(θ)
σ∞
x
=−σ θ(θ)
σ∞
x
= (1 −η) sin θcos θ. (5b)
In iew o he p oblem symme y only angles 0◦≤θ≤90◦will be conside ed o he sake o simplici y.
In he ollowing and wi hou loss o gene ali y i will be assumed ha σ∞
x>0 and σ∞
x≥σ∞
y, i.e.
η≤1. The de i a i e o σ(θ) e alua ed om (5a),
∂σ(θ)
σ∞
x
∂θ = (η−1)msin (2θ).(6)
This shows ha no mal ac ion is a dec easing unc ion o θ o θ∈[0◦,90◦] and any η≤1. Acco ding
o his and he exp ession in (5a), σ(θ) achie es i s maximum alue a θ= 0◦:
σmax =σ(θ= 0◦) = σ∞
x·k+σ∞
y·(k−m).(7)
No e ha kand k−m ep esen , espec i ely, he ela i e con ibu ion o emo e s esses σ∞
xand σ∞
y
o his maximum alue o σ(θ).
Following (5a) and (7), he in luence o σ∞
yon he no mal s esses σ(θ) is de e mined by he a io
k/m. In pa icula , ension σ(θ= 0◦)>0 is gene a ed by a emo e comp ession σ∞
y<0 ( ension σ∞
y>0)
o k/m < 1 (k/m > 1), assuming :::::
smallσ∞
x&0.
Recalling ha σ∞
x>0, he semiangle θ0 o which he in e ace no mal ac ions anish is gi en om
(5a) by
θ0(η;α, β) = a csin sk
m+k
m−1η
1−η.(8)
Acco ding o his exp ession and he analysis in Appendix A, he angle θ0∈[0◦,90◦] does no exis o
3
Table 1: Examples o iso opic bima e ials cons an s (1, inclusion; 2, ma ix)
Bima e ial E1(GPa) ν1E2(GPa) ν2
Glass/epoxy 70.8 0.22 2.79 0.33
Ca bon/epoxy 13.0 0.20 2.79 0.33
α β ε E∗(GPa)
Glass/epoxy 0.919 0.229 -0.074 6.01
Ca bon/epoxy 0.624 0.136 -0.044 5.09
k m k/m θη(◦)η0
Glass/epoxy 1.44 1.56 0.9205 16.3 0.086
Ca bon/epoxy 1.32 1.43 0.9200 16.4 0.087
all he conside ed alues o η∈(−∞,1] and all admissible alues o k
m∈[3/4,+∞), see Man iˇc (2009).
The condi ion o anishing de i a i e o (5a) wi h espec o ηgi es he angle θη o which he in e ace
no mal ac ion is independen o η,
θη(α, β) = a ccos k
m= 90◦−θ0(η= 0).(9)
This exp ession makes sense only i k≤m. Then, θηdi ides he in e ace sec o 0 ≤θ≤90◦in o wo
egions. In he i s egion (θ < θη) whe e ∂σ
∂η <0, σdec eases wi h inc easing he emo e seconda y load
σ∞
y. On he con a y, in he second egion (θ > θη) whe e ∂σ
∂η >0, σinc eases wi h inc easing he emo e
seconda y load σ∞
y. Hence, θηis a key angle o e alua e he in luence o load biaxiali y.
The alues o he abo e de ined and used cons an s cha ac e izing he in e ace ac ion dis ibu ion
o wo ypical ib e ein o ced composi es a e p esen ed in Table 1. The glass/epoxy bima e ial will be
used as an example in he p esen wo k.
Plo s o no mal ac ions dis ibu ions along he in e ace ob ained om (5a) o glass/epoxy a e
shown in Figu e 2.
Η=1
Η=0.5
Η=0
ΘΗ
Η=-2
Η=-1.5
Η=-1
Η=-0.5
Θ0HΗ=-1L
Θ0HΗ=0L
0
20
40
60
80
-3
-2
-1
0
1
2
ΘHºL
Σ
Σx
¥
Figu e 2: Dis ibu ion o he no mal ac ions along he in e ace o se e al alues o ηand glass/epoxy.
3. The solu ion o a c ack a he in e ace o a single inclusion unde a emo e biaxial
ans e se load
Ene gy based ac u e c i e ia conside a c acked con igu a ion. Hence he aim o his sec ion is o
analyse he p oblem o a pa ial debond a he ib e-ma ix in e ace. Unde ce ain assump ions, a
classical elas ic solu ion pa icula ized o he p esen p oblem p o ides closed o m exp essions o a
ac u e mode mixi y and he ene gy elease a e, hei dependence on he key p oblem pa ame e s being
poin ed ou .
4
Conside now he p oblem con igu a ion om he p e ious sec ion al e ed by he p esence o a c ack
a he in e ace. In iew o he ac ha he maximum o σ(θ) is achie ed a θ= 0◦, see (6), i will be
assumed ha his c ack is symme ically si ua ed wi h espec o he x-axis, wi h a semidebond angle
θd≥0 and an in ini e leng h in he z-axis di ec ion, see Figu e 3.
Figu e 3: The in e ace c ack p oblem con igu a ion.
A mo e gene al p oblem has been analy ically s udied by Toya (1974) and se e al o he au ho s using
he open model o in e ace c acks. As will be seen la e on, he alidi y o he analy ic solu ions based
on he open model is limi ed and, hus, compu a ional me hods should some imes be used employing he
con ac model o in e ace c acks, see Pa ´ıs e al. (2007) o a e iew.
The in e ace ac ions a a poin placed ahead o he c ack ip a he pola angle θ=θd+θl,θl>0,
see Figu e 3, can be exp essed by pa icula izing Toya’s solu ion o s esses1and ew i ing i in e ms
o he Dundu s pa ame e s,
σ(θ, θd, η, β)−iτ(θ, θd, η, β) = −σ∞
x
2
1−α
1−βχ(θ, θd, β)p(θ, θd, η, β),(10)
whe e χ(θ, θd, β) and p(θ, θd, η, β) a e de ined in Appendix B). I should be no iced ha he ac ions
along he in e ace a e independen o he inclusion adius a. Ra io o he in e ace shea and no mal
ac ions ahead he c ack ip a a small e e ence leng h (ei he geome y o ma e ial based) gi es a
measu e o ac u e mode mixi y o an in e ace c ack. Thus, he angle ψa a e e ence angle θl, measu ed
om he c ack ip (see Man iˇc (2009) o he discussion abou his e e ence angle o a simila p oblem),
is de ined as:
an ψ(θd;θl, η, ε) = τ(θd+θl)
σ(θd+θl).(11)
This angle will be used as a sui able measu e o he ac u e mode mixi y. Figu e 4 shows he e olu ion
o ψ(θd) o di e en alues o he load-biaxiali y pa ame e η.
The ERR o he in e ace c ack p opaga ing a i s uppe c ack ip a an angle θdcan be exp essed,
ew i ing Toya’s exp ession as p e iously, by
G(θd;σ∞
x, σ∞
y;a;E∗, α, β) = (σ∞
x)2a
E∗ˆ
G(θd;η;α, β),(12)
1The ollowing alues o he pa ame e s used by Toya: φ= 0 and ε∞= 0 a e aken.
5

Ψ = 90º
Η=-2
Η=-1.5
Η=-1
Η=-0.5
Η=0
Η=0.5
Η=1
0
20
40
60
80
0
20
40
60
80
100
120
ΘdHºL
ΨHºL
Figu e 4: Examples o he e olu ion o he ac u e mode mixi y angle ψ(ob ained om Toya’s solu ion o he open model
o in e acial c acks) aking θl= 0.1◦, o di e en alues o ηand glass/epoxy.
whe e E∗is he ha monic mean o he e ec i e elas ici y moduli
1
E∗=1
21−ν2
1
E1
+1−ν2
2
E2(13)
and ˆ
Gis a dimensionless no malized ERR whose exp ession is p esen ed in Appendix C. Acco ding o
exp ession (12), he ERR a ies linea ly wi h he a io a/E∗and quad a ically wi h he emo e load σ∞
x.
Figu e 5 shows he e olu ion o he no malized ERR ˆ
G(θd) , and also o i s asymp o es o θd≈0◦
gi en by (D.1), o di e en alues o he load-biaxiali y pa ame e η, see Appendix D. Validi y o hese
plo s is limi ed by he alidi y o he open model o in e ace c acks. No ice, in ela ion o Figu e 4, ha
comp essions ahead o he c ack ip co espond o |ψ|>90◦and can become ele an o η < 0. These
comp essions may ha e associa ed a ele an o e lapping o c ack aces close o he c ack ip. This is no
physically admissible, so i may in alida e Toya’s solu ion o some alues o θdand η.
Due o he abo e men ioned o e lapping, an addi ional ic i ious e m, co esponding in some sense
o Mode I, appea s in he compu a ion o he ERR, causing some o e es ima ion o ˆ
G. This o e es ima-
ion can be s udied by using he ela ion be ween he ERR based ac u e mode mixi y and he s ess
based ac u e mixi y acco ding o Man iˇc and Pa ´ıs (2004). This ela ion allows pa i ioning ˆ
Gin o
wo componen s ˆ
G(θd) = ˆ
GI(θd, δθ) + ˆ
GII (θd, δθ), o a gi en i ual-c ack-s ep angle δθ, as shown in
Appendix E, leading o,
ˆ
GI,II (θd, δθ) = 1
2ˆ
G(θd)(1 ±F(ε) cos(2(ψ(θd, θl) + ψ0(δθ/θl, ε))),(14)
whe e θlis he e e ence angle o ψ(11) and he oscilla ion index εis gi en in e ms o βin (B.4).
Figu e 6 shows he indi idual componen s o he ERR co esponding o a small i ual-c ack-s ep
angle δθ = 0.5◦. These plo s allow o cla i y he ange o alidi y o Toya’s exp ession o ERR o la ge
alues o θdand di e en alues o η. Dec easing alues o ηdec eases he ange o he alues o θd, whe e
Toya’s exp ession o ERR is alid. This igu e also con i ms ha he cause o he s ongly inc ease o ˆ
G
o la ge alues o θdand η < 0 is associa ed o a ic i ious con ibu ion o ˆ
GIdue o a la ge o e lapping.
In o de o cla i y he in luence o he emo e seconda y load σ∞
yon he alues o ˆ
G, i is use ul o
s udy he a ia ion o he de i a i e o ˆ
G(D.1) wi h espec o he load-biaxiali y pa ame e ηa θd∼
=0◦,
d2ˆ
G
dηdθdθd=0
=2π(k−m) (k+ (k−m)η)1+4ε2
cosh (πε).(15)
This exp ession shows again he impo ance o he pa ame e k/m. In ac , he sign o he a ia ion o
6
Η=-2
Η=-1
Η=0
Η=1
asymp o es
0
20
40
60
80
0
2
4
6
8
10
ΘdHºL
G
`
Figu e 5: Examples o he no malized ERR (ob ained om Toya’s solu ion o he open model o in e acial c acks) and i s
asymp o es o di e en alues o ηand glass/epoxy.
he asymp o ic slope o ˆ
Ga θd∼
=0◦wi h ηis di ec ly cha ac e ized by
sign d2ˆ
G
dηdθdθd=0
= sign k
m−1k
m+k
m−1η.(16)
Acco ding o (16), a change o mono onici y o he asymp o ic slope o ˆ
Ga θd∼
=0◦occu s o (see
Appendix A)
η=
k
m
1−k
m
=1
η0
.(17)
Thus, he sign o de i a i e in (15), c . (16), o k
m>1 is posi i e o 1/η0< η ≤1 and nega i e o
η < 1/η0, whe eas o k
m<1 i is nega i e o all he alues o η≤1. Consequen ly, o bima e ials wi h
k
m<1, a emo e seconda y ension σ∞
y>0 will hinde he c ack onse om he ene ge ic app oach poin
o iew, while a seconda y comp ession σ∞
y<0 will acili a e i .
4. In e ace c ack onse a a single inclusion unde a emo e biaxial ans e se load
The app oach de eloped by Leguillon (2002) in he amewo k o he Fini e F ac u e Mechanics will
be used o p edic he c ack onse , in a simila way as done in Man iˇc (2009) o emo e uniaxial load.
The key idea o his app oach is o combine a s ess and an ene gy c i e ion o p edic he c i ical load
o igina ing c ack onse and he c ack leng h a he onse . The eason o applying he coupled c i e ion
is ha i is no possible o ob ain a solu ion o he c ack onse p oblem by applying each c i e ion
indi idually wi hou some ex a assump ions due o he ollowing easons:
7
(a) η=−2
G
`
G
`
II
ic i ious
G
`I
0
20
40
60
80
0
5
10
15
20
ΘdHºL
G
`
(b) η=−1
G
`
G
`I
G
`
II
ic i ious
0
20
40
60
80
0
2
4
6
8
10
ΘdHºL
G
`
(c) η= 0
G
`
G
`
II
ic i ious
G
`I
0
20
40
60
80
0
1
2
3
4
5
ΘdHºL
G
`
(d) η= 1
G
`
G
`
II
G
`I
0
20
40
60
80
0
1
2
3
4
5
ΘdHºL
G
`
Figu e 6: Plo s o he indi idual componen s o he ERR associa ed o δθ = 0.5◦ o di e en alues o he load-biaxiali y
pa ame e ηand glass/epoxy. (a) η=−2. (b) η=−1. (c) η= 0. (d) η= 1.
•A s ess c i e ion can de e mine he minimal load bu i canno de e mine he size o he c ack
o igina ed a he onse .
•An applica ion o he in ini esimal G i i h c i e ion equi es an exis ing c ack o ob ain a alue o
Gdi e en om 0 in o de o ul ill he condi ion G≥Gc>0.
Howe e , coupling bo h c i e ia and eleasing he G i i h condi ion o in ini esimal c ack g ow h by
allowing ini e inc emen s o c ack ad ancing, pe mi s o ob ain he leng h o he c ack o igina ed and
he c i ical load equi ed o he onse .
In iew o he emo e ension σ∞
xdomina ing he c ack onse and he p oblem symme y wi h espec
o he x-axis, onse o a c ack si ua ed as shown in Figu e 3 will be assumed in ag eemen wi h expe i-
men al obse a ions. Reasons why, in spi e o he symme y wi h espec o he y-axis, only one c ack
appea s, esul ing in an asymme ic con igu a ion a e he c ack onse will be discussed in a o hcoming
pape by Ga c´ıa e al. (2012).
Fi s , in Subsec ion 4.1, he s ess c i e ion is p esen ed and applied o he s ess s a e analysed in
Sec ion 2. Second, a condi ion imposed by he inc emen al ene gy c i e ion is ob ained in Subsec ion 4.2
wi h he aid o he analysis in oduced in Sec ion 3. Then, bo h condi ions a e combined in Subsec ion 4.3
leading o he p edic ion o he c i ical load and semiangle. Finally, he pos -c ack-onse e olu ion and
he applicabili y o he open model o in e ace c acks in he p esen p oblem a e discussed in Subsec ions
4.4 and 4.5, espec i ely.
8
4.1. S ess c i e ion
A s ess c i e ion is usually in oked i no c ack exis s a p io i. The p esen s ess c i e ion is based
on he idea o he exis ence o an in e ace ensile-s eng h σc, de ined as he maximum ension ha he
in e ace can sus ain. Thus, in he p esen p oblem, he inclusion-ma ix in e ace can b eak a he poin s
de ined by an angle θwhe e,
σ(θ)≥σc,(18)
de ining a ensile c i e ion. Acco ding o Figu e A.1 and Sec ion 2, his c i e ion canno be ul illed o
η≤1/η0and k/m > 1 because he whole in e ace is unde comp ession and no c ack onse can be
p edic ed ollowing he s ess c i e ion. Hence, in he ollowing analysis, i will be assumed ha ei he
η > 1/η0o k/m ≤1.
Then, combining (18) and (5a), he s ess c i e ion can be exp essed as
σ∞
x
σc≥1
k+ (k−m)η−(1 −η)msin2θ=s(θ, η, α, β).(19)
Assuming a su icien ly la ge emo e loading, gi en by (19) o θ= 0◦,
σ∞
x
σc≥min
θs(θ, η) = s(0◦, η) = 1
k+ (k−m)η>0,(20)
an angle θσ
c∈[0◦,90◦] can be de ined by σ(θσ
c) = σc. Then, due o he dec easing cha ac e o σ(θ) (see
(6) and discussion in Sec ion 2), condi ion (18) is e i ied o all θ∈[0◦, θσ
c],
θσ
c= a csin sk+ (k−m)η−σc
σ∞
x
(1 −η)m,(21)
Acco ding o a discussion in Sec ion 2, o a gi en alue o ηan angle θ0(8) may exis whe e he ac ion
is ze o. Then, condi ion (19) leads o an in ini e load o θ=θ0, which is an uppe limi o he alues
o θσ
c,
θσ
c< θ0(η;α, β) (22)
Then, combining all he condi ions ela ed o he s ess c i e ion, he maximum angle o a debond and
he unc ion sa e de ined in a igo ous manne sui able o compu a ional p oposes in Appendix F.
Figu e 7 shows a ep esen a ion o he s ess c i e ion o glass/epoxy as de ined in Table 1 o di e en
alues o η. As p edic ed, all he cu es o he s ess c i e ion a e inc easing. Thus, o a load ( alues o
σ∞
x/σcand η) wo zones can be de ined in his diag am i θσ
cexis s, a zone whe e a debond is possible
[0◦, θσ
c], and ano he whe e i is no possible (θσ
c,180◦].
No e ha an angle θη(see Figu e 7) can be de ined whe e he s ess c i e ion is independen o he
emo e seconda y load σ∞
yas demons a ed in Sec ion 2. This semiangle sepa a es he in e ace in o wo
egions, a egion (θ < θη) whe e a seconda y comp ession σ∞
y<0 acili a es a debond onse and ano he
egion (θ > θη) whe e i hinde s a debond.
Finally i should be no iced ha he s ess c i e ion is no su icien o uniquely cha ac e ize he
debond onse , as i p o ides only one inequali y o wo unknowns, he c i ical emo e load and debond
angle a e he onse .
4.2. Inc emen al Ene gy c i e ion
An inc emen al G i i h c i e ion is used he e wi h he aid o exp essions de eloped in Sec ion 3. Fi s ,
an ene gy balance o he onse o an in e ace c ack o a ini e leng h is in oduced and i s di e en e ms
a e pa icula ized o his p oblem and analysed. Finally, a condi ion o he minimum load o igina ing
an ene ge ically allowed ib e-ma ix debond is deduced by means o a dimensionless unc ion o he c ack
leng h ep esen ing he a io o he dissipa ed o he eleased ene gy.
9
o γ( ecall ha k/m < 1 and η0>0 o glass/epoxy). Howe e , a non-mono onic bounda y cu e o he
sa e egion is obse ed o g ea e alues o γin Figu e 12. In ac , i is obse ed ha he cu e which
joins he poin s wi h he maximum c i ical emo e load σ∞
cx shows ha he maximum is si ua ed a η= 1
jus o γ→0+. Fo mode a e alues o γ, he maximum c i ical emo e load σ∞
cx co esponds o η < 0.
The eason o his beha iou is cla i ied in Figu e 13 whe e wo si ua ions a e explained.
Γ=0.2
Γ=0.4
Γ=0.6
Γ=0.8
Γ=1
Γ=1.2
Γ=1.4
Γ=1.6
Γ=1.8
Γ = 2
SAFE
Scena io A
Scena io B
FAILURE
Scena io A
Th eshold cu e
Maximum
c i ical load
Asymp o e o Γ ® 0+:
Hk-mLΣcy +kΣcx = Σc
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Σcx
¥
Σc
Σcy
¥
Σc
Figu e 12: C i ical biaxial loads o igina ing a c ack o di e en alues o γ, aking λ= 0.3, θl= 0.1◦and glass/epoxy.
In he i s case, he alue o γis ela i ely small, Figu e 13(a), which is associa ed o small alues o
θc, as p e iously demons a ed. Fo small alues o θc, he e ec o he seconda y load σ∞
yis he same
o bo h he s ess c i e ion and ene gy c i e ion cu es, bo h cu es descending when η educes. In he
second case, he alue o γis la ge , Figu e 13(b), and he alues o θca e g ea e han θη, see (9) and
discussion below. Then, o alues o γo igina ing θc> θη, an inc ease o he emo e seconda y load σ∞
y
makes less es ic i e he s ess c i e ion and mo e es ic i e he ene gy c i e ion. Thus, he mono ony
o he unc ion σ∞
cx(σ∞
cy ) can be b oken down as obse ed in Figu e 12 o mode a e alues o γ.
The s aigh line de ined by (44) ep esen s, acco ding o Figu e 12, a limi o ailu e en elope cu es
o γ→0+. No e ha he ailu e en elope cu es o γ→0+in Figu e 12 show he mos ele an
in luence o he seconda y load σ∞
yon he alue o he c i ical load σ∞
cx.
Figu e 12 also shows he ” h eshold cu e” which sepa a es scena ios A and B. I is in e es ing o
ema k ha g ea e alues o γco espond o a la ge ange o ailu e beha iou go e ned by scena io B.
On he con a y, he p esence o a emo e seconda y comp ession σ∞
y<0 leads o scena io A, o small
16

(a)
Η = 0.9
S ess c i e ion
Ene gy c i e ion
Η = -1
Η = 0.9
Η = 0
Η = -1
Η = 0
Η = 0.9
Γ=0.4
0
10
20
30
40
50
60
70
0.0
0.5
1.0
1.5
2.0
DΘHºL
Σcx
¥
Σc
(b)
Η = -1
Η = 0
Η = 0.9
S ess c i e ion
S ess c i e ion
S ess c i e ion
Η = 0.9
Η = -1
Η = 0
Ene gy c i e ion
Η = 0
Η = -1
Η = 0.9
Γ=1
0
10
20
30
40
50
60
70
0.0
0.5
1.0
1.5
2.0
DΘHºL
Σcx
¥
Σc
Figu e 13: S ess and ene gy c i e ia cu es, aking λ= 0.3, θl= 0.1◦and glass/epoxy o wo di e en alues o γ. (a)
γ= 0.4, (b) γ= 1
and mode a e alues o γ.
Figu e 14 s udies he in luence o αand β alues on he biaxial sa e egion o γ→0+, o selec ed
heo e ical (bu possible) bima e ials and also o usual composi es. F om Dundu s’ α−βpa allelog am
i is seen ha he mos common bima e ials ha e e y simila p ope ies in he debond onse p oblem. A
mo e ex ensi e lis o α−β alues o eal bima e ials can be ound in Suga e al. (1988) and Schmaude
and Meye (1992).
The sa e egion in he limi case γ→0+is de ined by he in e sec ion o he semiplanes including
he o igin o coo dina es and limi ed by he s aigh line de ined by (44) and he symme ic one wi h
espec o he bisec o o he coo dina e axes. F om (44), he posi ion o he co ne poin o he sa e
egion (η= 1) is gi en by
σ∞
cx
σc
=σ∞
cy
σc
=1
2k−m.(45)
17
(a)
1
3
-1
0.6
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
2
¥
0.7
AA
BB
DD
EE
FF
0.5
Ca bon/epoxy
Bo o/epoxy
A amid/epoxy
Glass/polyes e
Glass/epoxy
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
C
-1.0
-0.5
0.0
0.5
1.0
-0.4
-0.2
0.0
0.2
0.4
Α
Β
(b)
A
B
C
SAFE
FAILURE
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Σcx
¥Σc
Σcy
¥
Σc
SAFE
FAILURE
D
E
F
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Σcx
¥Σc
Σcy
¥
Σc
FAILURE
SAFE
Glass/epoxy
Bo o/epoxy
A amid/epoxy
Glass/polyes e
Ca bon/epoxy
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Σcx
¥Σc
Σcy
¥
Σc
Figu e 14: (a) α−βdiag am o bima e ials in plane s ain wi h iso alue cu es o γ→0+co esponding o: solid
lines wi h alues o σ∞
cx/σc=σ∞
cy /σc, and dashed lines wi h alues o ∂σ∞
cx/∂σ∞
cy . (b) Some biaxial ailu e en elopes o
γ→0+ o selec ed poin s in he α−βdiag am and usual composi es. A: k/m = 0.75, B: k/m = 1, C: k/m →+∞, D:
k/m = 1.25, E: k/m = 1.25, F: k/m = 1.25, Ca bon/epoxy: k/m = 0.9200, Glass/polyes e : k/m = 0.9201, Glass/epoxy:
k/m = 0.9205, A amid/epoxy: k/m = 0.9217, Bo o/epoxy: k/m = 0.9204
The slope o he linea ela ion in (44) cha ac e izes he in luence o he seconda y load σ∞
yon he c i ical
load σ∞
cx. This slope can be exp essed as:
lim
γ→0+∂σ∞
cx
∂σ∞
cy =η0(46)
whe e η0is de ined in (A.1) and i s ange in (A.2), see also Figu e A.1. Hence, his slope is only dependen
on he elas ic bima e ial p ope ies and does no depend on he in e ace p ope ies. In iew o he ange
o possible alues o he slope (A.2), he sa e egion is always con ex, c . Figu e 14(b).
Fo k/m < 1 he slope (46) is posi i e and an inc ease in he seconda y load σ∞
yinc eases he c i ical
load σ∞
cx necessa y o o igina e a debond, see Figu e 14. This is he case o he wo bima e ials de ined
in Table 1, i.e. glass/epoxy and ca bon/epoxy. Howe e , an opposi e e ec is p edic ed o k/m > 1, see
Figu e 14. In ac , his dependence ma ches up wi h he e ec o he seconda y load σ∞
yp edic ed by
18
Goodie ’s solu ion o he in e ace poin θ= 0◦, see (1), which shows ha a comp ession o ension is
expec ed a θ= 0◦when a seconda y load σ∞
y>0 is applied o k/m < 1 o k/m > 1, espec i ely.
The p esen esul s show ha he in luence o he seconda y load σ∞
yon he c i ical load σ∞
cx is a mos
mode a e in usual composi es. Taking in o accoun ha in he case γ→0+, analysed in Figu e 14, he
alues o σ∞
cx a e he mos sensi i e o he alues o σ∞
y, in gene al o k/m < 1 he in luence o σ∞
yon he
alue o σ∞
cx exis s bu i is small o a mos mode a e. Fo bima e ials as glass/epoxy and ca bon/epoxy
wi h k
m<1, hese esul s ag ee wi h he hypo hesis in oduced by Pa ´ıs e al. (2003). Acco ding o his
hypo hesis a seconda y comp ession σ∞
ymakes easie he debond onse o hese bima e ials and educes
he c i ical load σ∞
cx as shown in Figu e 14(b). Howe e , o bima e ials wi h k
m>1 he e ec is opposi e,
a seconda y comp ession σ∞
y<0 inc eases he c i ical load σ∞
cx.
4.4. Pos -onse e olu ion
A e he onse o a new c ack, an uns able g ow h o he c ack is possible depending on he ela ion
be ween G(θd) and Gc(θd) o θd≥θcand acco ding o he c i e ion o he classical (in ini esimal)
in e ace ac u e mechanics (see, e.g. Pa ´ıs e al. (2007); Man iˇc e al. (2006)). Thus, he condi ion o
he u he c ack g ow h will be
G(θd, η)≥Gc(ψ(θd, η)), θd≥θc.(47)
The c ack will s op g owing a an a es angle θa≥θc e i ying G(θa, η) = Gc(ψ((θa, η)) i o angles
θd&θac i e ion (47) is no ul illed. The s abili y o he pos -onse g ow h o he c ack is di e en o
wo scena ios A and B sepa a ed by γ h wo pos -onse scena ios being possible:
•Fo γ < γ h,G(θc, η)> Gc(ψ(θc, η)) and he c ack is expec ed o con inue g owing in an uns able
manne up o an a es angle θa> θE
min, which can be shown simila ly as in Man iˇc (2009).
•Fo γ≥γ h,θc=θE
min,G(θc, η) = Gc(ψ(θc, η)) and dG/dθd|θd=θE
min ≤dGc/dθd|θd=θE
min , see (31)
and ela ed discussion in Subsec ion 4.2. The e o e, assuming s ic inequali y in (31) (which, in
ac , has been e i ied in all p esen calcula ions), no uns able c ack g ow h is usually expec ed
a e he c ack onse and θa=θE
min.
Figu es 11(a) and 15 we e compu ed by implemen ing he abo e ideas. The alues o Gand Gcand
hei de i a i es a e compa ed o θd≥θcin o de o ind he a es semiangle θa. Acco ding o hese
igu es, a long uns able c ack g ow h a e he c ack onse is p edic ed o small alues o γ(b i le
con igu a ions), whe eas sho (o ze o) uns able c ack g ow h is p edic ed o la ge ales o γ( ough
con igu a ions).
4.5. Applicabili y o he open model o in e ace c acks
The applicabili y o he heo e ical model de eloped is limi ed by i s assump ions, pe haps he mos
es ic i e being he usage o he open model o in e ace c acks. Toya’s (1974) solu ion assumes negligible
o e lapping o ac ion- ee c ack aces. The angle o he o e lapping zone a he c ack ip can be
es ima ed by he o mula deduced by Hills and Ba be (1993) and gene alized by G aciani e al. (2007),
which ew i en o he p esen case is de ined as he la ges alue o
θI(θd, η) = θl·exp [((2n−1/2) π−ψ(θd, η, θl)signε+ a c an(2 |ε|)) /|ε|],(48)
lowe han he semidebond angle θd, wi h nbeing an in ege .
Figu e 16 shows he e olu ions o ˆ
G o glass/epoxy and di e en alues o he load biaxiali y pa am-
e e η. Addi ionally, o γ= 1.5, co esponding o ela i ely ough con igu a ions, he alues o θc,θE
min
and θacompu ed by he p esen model a e indica ed. Finally, angles θI,1% o which he o e lapping
zone ep esen 1% o he c ack leng h, i.e. θI/2θd= 0.01, p o iding a easonable limi o alidi y o he
open model, a e also p esen ed in Figu e 16. This igu e shows ha all he alues o θca e lowe han he
e e ence limi θI,1%, he e o e he open model is accep able o he e alua ion o θcand he c i ical load
σ∞
cx. Howe e , i migh no be ully accep able when compu ing θa o la ge nega i e alues o η. No e
ha , he abo e discussed limi on he semiangle θdis mainly due o somewha inaccu a e e alua ion o
19
Γ=0.2
Γ=0.6
Γ=1
Γ=1.4
Γ = 2
Γ = 2
Γ=1.4
Γ=1
Γ=0.6
Γ=0.2
A es semiangle
C i ical semiangle
Θmin
EHindependen o ΓL
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
0
20
40
60
80
Η
ΘcHºL
Θmin
EHºL
ΘaHºL
Figu e 15: Semiangles θc,θE
min and θaas unc ions o he biaxiali y pa ame e η o di e en alues o γ, aking θl= 0.1◦,
λ= 0.3 and glass/epoxy.
ˆ
Gbecause o a la ge o e lapping zone a he c ack ip, see also Figu e 6 and he ela ed discussion. A
co ec p ocedu e o he e alua ion o ˆ
Gis such cases would equi e employing he con ac model o
in e ace c acks as in Pa ´ıs e al. (2007) and Co ea (2008).
5. Size e ec o he inclusion adius aon he c ack onse and i s a ia ions wi h he biaxiali y
A size e ec in he p esen debond onse p oblem can be unde s ood as a dependence o he c i ical
emo e load (σ∞
cx,σ∞
cy ) and c i ical semiangle θcon he only geome ical pa ame e in he p esen p oblem,
he inclusion adius a. As will be seen, his size e ec is di ec ly ela ed o he b i le- o- ough ansi ion
go e ned by he b i leness numbe γ, see Figu e 11.
Le a bima e ial cha ac e is ic leng h a0be de ined in e ms o he in e ace p ope ies σcand G1c
and he ha monic mean o e ec i e Young moduli E∗,
a0=G1cE∗
σ2
c
.(49)
Then, he no malized adius a
a0and γa e ela ed by
γ= a0
a.(50)
Analogously o he h eshold alue γ h (41), a h eshold alue o ais de ined as
a h =a0
γ2
h
.(51)
In ac , all he analysis p esen ed abo e aking as e e ence γcan be exp essed as a dependence on a
in an analogous manne . Fo su icien ly la ge alues o a, which co espond o small alues o γ, he
20
Θc
Θmin
E
Θ0
Θa
Η = -2
Η = 1
Θdwi h ΘÈ
2Θd
=0.01
Η = 0
Η = -1
inc easing Ηwi h DΗ = 0.2
0
20
40
60
80
0
2
4
6
8
10
12
ΘdHºL
G
`
Figu e 16: Plo s o e olu ion o alues o dimensionless ERR limi ed by he semiangles which es ima e he alidi y o he
model and ep esen a ion o impo an alues o semiangles o he esul s, aking λ= 0.3, θl= 0.1◦,γ= 1.5 o glass/epoxy.
c i ical semiangle θcand he c i ical emo e load σ∞
cx can be app oxima ed by he ollowing exp essions,
see Sec ion 4,
θc∼
=2cosh2(πε)
π(1 + 4ε2)
a0
aand σ∞
cx
σc∼
=1
k+ (k−m)η,(52)
whe eas o a≤a h,
θc=θE
min and σ∞
cx
σc
=qg(θE
min, η) a0
a.(53)
As ollows om (52) and (53), he c i ical c ack-semileng h aθcis cons an and independen o aand
η o la ge a, whe eas o small ai is linea ly p opo ional a.
The abo e desc ibed asymp o ic beha iou o θcand σ∞
cx can be easily iden i ied in Figu e 17 whe e
he a ia ions o θc,θaand σ∞
cx as unc ions o aa e plo ed. In pa icula , in Figu e 17(a) and (b) i is
seen ha θc=θa=θE
min and σ∞
cx ≈1/√a, espec i ely, o a≤a h. Thus, σ∞
cx inc eases d as ically o
small inclusions while o la ge inclusions i ends o a cons an alue gi en by he s ess c i e ion applied
a θ= 0◦. As can be obse ed in Figu e 17(b), he size e ec on σ∞
cx is simila o di e en alues o η,
being qui e independen o he combina ion o emo e ans e se loads.
6. Expe imen al p ocedu e o he measu emen o he b i leness numbe γ, in e acial
ensile s eng h σcand ac u e oughness G1c
F ac u e p ope ies o he ib e-ma ix in e aces a e e y impo an o he mac oscopic beha iou
o ib e ein o ced composi es. Howe e he expe imen al measu emen o hese is e y di icul o be
ca ied ou . An indi ec expe imen al p ocedu e is p oposed he e o ob aining i s he alue o γ, and
subsequen ly he alues o σcand G1c, o a bima e ial. Elas ic p ope ies o he bima e ial (E∗, α, β) a e
21

(a)
C i ical semiangle
A es semiangle
Η=1
Η=0
Η=-1
Θmin
E
0
1
2
3
4
5
6
0
20
40
60
80
aa0
ΘcHºL
Θmin
EHºL
ΘaHºL
(b)
1
k+Hk-mLΗ
Η=1
Η=0
Η=-1
0
1
2
3
4
5
6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
aa0
Σcx
¥
Σc
Figu e 17: (a) C i ical semiangle θcand (b) C i ical emo e ension in σ∞
cx as a unc ion o he inclusion adius a, aking
λ= 0.3, θl= 0.1◦and glass/epoxy.
assumed o be known, as hey can be measu ed by ca ying ou s anda d ma e ial es s o each ma e ial
sepa a ely.
To de e mine he in e ace p ope ies, he sole measu e o he c i ical s ess o he case o emo e
uniaxial ension (η= 0) is no su icien . The eason is ha he c i ical s ess depends no only on γbu
also on σc, see he no maliza ion used in (38). Ne e heless, i he c i ical s ess is also measu ed o a
biaxial load (η6= 0), hen, he a io o hese c i ical s esses depends only on he alue o γdue o he
in luence o γ alue on he solu ion θco (37). This is he key idea behind he expe imen al p ocedu e
p oposed. This p ocedu e employs plo s o he a io o c i ical s esses σ∞
cx(η6=0)
σ∞
cx(η=0) as a unc ion o γo η.
As an example, Figu e 18 shows alues o σ∞
cx(η6=0)
σ∞
cx(η=0) o glass/epoxy, aking θl= 0.1◦and λ= 0.3.
The s eps o he expe imen al p ocedu e a e b ie ly explained in he ollowing:
1. De e mine he c i ical s ess σ∞
cx in he uniaxial ension es (η= 0). This is a ela i ely easy es ,
hus a good accu acy is expec ed.
2. De e mine ano he c i ical s ess σ∞
cx in a biaxial es o η6= 0. Combining he plo s in Figu e 18(a)
and a ough a p io i es ima ion o he γ alue, choose he mos sui able alue o η o es by looking
22
(a)
Η = -2
Η = -1.75
Η = -1.5
Η = -1.25
Η = -1
Γ hHΗ= 0.95 L
Γ hHΗ= -0.75 L
Γ hHΗ= -0.5L
Γ hHΗ= -0.25 L
Γ hHΗ= 0L
Γ hHΗ= 0.25 L
Γ hHΗ= 0.5L
Γ hHΗ= 0.75 L
Η = -0.75
Η = -0.5
Η = -0.25
Η = 0
Η = 0.25
Η = 0.5
Η = 0.75
Η = 0.95
0
1
2
3
4
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Γ
Σcx
¥
Σcx
¥HΗ = 0L
(b)
Γ=0
Γ=0.2
Γ=0.4
Γ=0.6
Γ=0.8
Γ=1
Γ=1.4
Γ=1.8
Γ = 2
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Η
Σcx
¥
Σcx
¥HΗ = 0L
Figu e 18: G aphs o de e mina ion o γby applying he expe imen al p ocedu e p oposed. P edic ions o σ∞
cx(η)
σ∞
cx(η=0) as a
unc ion o (a) γand (b) η o glass/epoxy, θl= 0.1◦and λ= 0.3.
o an in e ible segmen o he pe inen unc ion plo ed in Figu e 18(a) and o i s maximum
slope. Capabili ies o he es ing machine may ep esen an addi ional cons ain .
3. E alua e he a io o he measu ed c i ical s ess σ∞
cx(η6=0)
σ∞
cx(η=0) . Then, es ima e a alue o ˜γ om he
measu ed a io o he c i ical s esses o he chosen alue o η om Figu e 18(a).
4. Es ima e a alue o he c i ical semiangle θcby sol ing he ollowing nonlinea equa ion, employing
˜γ, see (37):
˜γqg(˜
θc, η) = s(˜
θc, η) (54)
5. Es ima e he in e acial ensile s eng h σc om (38),
˜σc=˜σ∞
cx(η)
˜γqg(˜
θc, η)
,(55)
whe e ˜σ∞
cx is one o he wo alues measu ed, ei he o η= 0 o η6= 0.
23
6. Es ima e he in e acial ac u e oughness G1c om (36)
˜
G1c=(˜σc˜γ)2a
E∗.(56)
A ew commen s ollow wi h e e ence o a possible s umbling block ackled in he 2nd s ep o he
abo e p ocedu e.
As can be obse ed om Figu e 18(a), he a io σ∞
cx(η6=0)
σ∞
cx(η=0) o a gi en ηis no an injec i e (one- o-one)
unc ion o γ o he whole ange o γconside ed. Ne e heless, his a io may become an injec i e
unc ion o γwhen es ic ed o a sui able in e al o γ, e.g. o small alues o γ oughly in he ange
0< γ .1. In gene al, an a p io i es ima e o γwill be e y use ul in choosing a sui able η6= 0 o he
biaxial es and a pe inen in e al o γwhe e he abo e a io is an injec i e unc ion.
Recall ha o γ > γ h he c i ical semiangle is cons an (θc=θE
min) and he ene gy c i e ion de e -
mines he c i ical emo e ension, which is di ec ly p opo ional o γ, see (40). Then, o a gi en alue
o η, he alue o pg(θE
min(η), η) is ixed. Thus, he a io shown in Figu e 18(a) o γ≥γ h(η) and
γ≥γ h(η= 0) is
σ∞
cx(η)
σ∞
cx(η= 0) =pg(θmin(η), η)
pg(θmin(η= 0), η = 0).(57)
Hence, his a io is a cons an independen o γ, as can be obse ed in Figu e 18(a). Ob iously o hese
a he la ge alues o γ, he p oposed expe imen al p ocedu e (including he biaxial es o only one
alue o η) is no di ec ly applicable.
Ne e heless, epea ing he biaxial es s o se e al adequa ely chosen alues o ηand applying leas
squa e i ing o unc ions plo ed in Figu e 18(b) could p o ide a good es ima ion o γ, and subsequen ly
o σcby (55) and G1cby (56) as well.
7. Concluding Rema ks
1. The p oblem o he onse o a debond o a ini e leng h a he ini ially undamaged in e ace o a
ci cula cylind ical inclusion embedded in an in ini e ma ix subjec ed o a emo e biaxial ans e se
load has been s udied. A heo e ical model has been de eloped assuming linea elas ic plane s ain
s a es be o e and a e he debond onse and su icien ly dilu e packing. The c i ical biaxial load
leading o he onse o a debond symme ically si ua ed wi h espec o he domina ing emo e
ension is p edic ed oge he wi h he debond size. The p esen model is based on he Fini e
F ac u e Mechanics app oach in oduced by Leguillon (2002) combining a poin wise no mal ension
c i e ion wi h an inc emen al ene gy c i e ion.
I is expec ed ha he p esen wo k will con ibu e o he knowledge o he go e ning pa ame-
e s and o o e all unde s anding o he ailu e mechanism in he ibe composi es unde ension
domina ed ans e se loads. A special a en ion has been gi en o he in luence o a seconda y
comp ession/ ension on he alue o he c i ical (p ima y) ension. Al hough he p esen wo k
is ocused on a s i inclusion embedded in a complian ma ix (glass/epoxy composi e has been
used as a ep esen a i e example), mos esul s a e gene ally alid o any combina ion o elas ic
bima e ial pa ame e s.
Fo he sake o simplici y a emo e biaxial s ess s a e (σ∞
x, σ∞
x) wi h σ∞
xy = 0 is assumed. Ne -
e heless, he p esen model and esul s may be easily adap ed o a gene al emo e in-plane s ess
s a e wi h σ∞
xy 6= 0 by wo king in i s p incipal coo dina e sys em and assuming ha a leas one
p incipal s ess is ension.
2. The p edic ions o he p esen model a e go e ned by he dimensionless b i leness numbe γin o-
duced o in e ace c acks in Man iˇc (2009). A consequence o his ac is ha a size e ec on he
c i ical emo e load and on he size o he debond a onse is p edic ed by he p esen model.
I may be use ul o ealize ha an al e na i e b i leness numbe gi en in e ms o he c i ical
S ess In ensi y Fac o (SIF) in ac u e Mode I K1c, ins ead o he c i ical ERR G1c, can be
p oposed. Taking in o accoun he ela ion be ween he complex S ess In ensi y Fac o Kand
24
Ene gy Release Ra e Gin in e acial ac u e mechanics (see Malyshe and Salganik (1965)) G=
|K|2/(E∗·cosh2(πε)), his al e na i e b i leness numbe is exp essed as
γK=γ·cosh (πε) = K1c
σc
1
√a.(58)
Recall ha he exp ession o γKin e ms o K1c eminds he classical de ini ion o he b i leness
numbe sin homogeneous ma e ials by Ca pin e i (1981). No e ha , ε= 0 o a c ack in a
homogeneous ma e ial, hus γK=γin his case.
3. The way how he emo e seconda y load σ∞
yin luence he c i ical alue o emo e ension σ∞
cx
depends on he alue o he a io k/m =1
2
2+α−β
1+α−2β, de ined in e ms o he Dundu s elas ic bima e ial
pa ame e s αand β. In pa icula , o γ1, a emo e seconda y comp ession σ∞
ydec eases o
inc eases σ∞
cx i k/m < 1 o k/m > 1, espec i ely. This esul , o k/m < 1, is cohe en wi h he
hypo hesis p oposed and expe imen ally e i ied by Pa ´ıs e al. (2003) o a pa icula ca bon/epoxy
composi e.
Fo mode a e o la ge alues o γ,γ&1, his model p edic s an almos negligible in luence o he
seconda y comp ession σ∞
yon he c i ical ension σ∞
cx o he glass/epoxy bima e ial s udied, ha ing
a somewha la maximum o η.0 (see Figu e 12). This obse a ion is ela ed o he ac ha
he c i ical semidebond angles θcp edic ed o hese alues o γa e su icien ly la ge o make he
in luence o he seconda y comp ession σ∞
ymo e complex. Howe e , he la e conclusions should
be accep ed wi h a cau ion in iew o he ange o model applicabili y, which appea s o be e y
sui able o b i le con igu a ions bu o a lesse ex en o ough ones.
4. In addi ion o he inclusion-ma ix debond onse mechanism s udied in he p esen wo k, o he
ailu e mechanisms can occu in he inclusion-ma ix sys em unde emo e ans e se loads. This
is, o example, he case o he domina ing comp essi e load s udied by he coupled s ess and
ene gy c i e ion in Quesada e al. (2009), whe e pa allel c acks in he inclusion and he ma ix a e
p edic ed. Ano he example o inclusion-ma ix debond con igu a ions no allowed by he p esen
assump ion o he debond symme ically si ua ed wi h espec o he p incipal di ec ions o he
emo e load we e s udied in Co ea e al. (2008). Expe imen al es s o specimens subjec ed o
emo e ans e se comp essions show debonds o igina ing a in e ace posi ions wi h la ge shea
s esses. Thus, in o de o comple e he pic u e o ailu e en elopes shown in Figu e 12, such
con igu a ions should be s udied in a simila way as done in he p esen wo k. In o de o ake
in o accoun he in luence o in e ace shea s esses, he s ess c i e ion should be e ised. The
Moh -Coulomb c i e ion (used in b i le ma e ials and soil mechanics, see Ca pin e i (1986) o a
e iew) appea s o be a sui able candida e as i shows a good ag eemen wi h he expe imen s
ca ied ou by Toda e al. (2001) and Ogiha a and Koyanagi (2010) abou ailu es a in e aces.
Acknowledgemen s
The au ho s hank o P o . F. Pa ´ıs o his mo i a ion and con inuous suppo o he p esen wo k.
The au ho s also hank o D . E. Co ea o he Ma hema ica code o Toya’s solu ion used o checking
p oposes. This wo k was suppo ed by he Jun a de Andaluc´ıa and he Spanish Minis y o Science
and Inno a ion, h ough he P ojec s TEP4051 and MAT2009-14022, espec i ely. I.G. Ga c´ıa also
acknowledges he suppo by he Spanish Minis y o Educa ion h ough he FPU G an 2009/3968.
Appendix A. Domain o exis ence on he angle θ0whe e no mal ac ion anishes
A h eshold pa ame e η0can be de ined as
η0(α, β) = k
m−1
−1.(A.1)
25