- 1 -
Equi alen ene gy elease a e and c ack s abili y in he End No ched Flexu e
wi h inse ed olle mixed mode I/II es
A. Boyano*a, J. De G aciaa, A. A eseb, F. Mujikab
a,b Ma e ials + Technologies G oup, Depa men o Mechanical Enginee ing
a Facul y o Enginee ing o Vi o ia-Gas eiz, Uni e si y o he Basque Coun y (UPV/EHU)
Nie es Cano, 12, 01006 Vi o ia-Gas eiz, Spain
*e-mail: [email p o ec ed] // Tel: +34 945 013933
b Facul y o Enginee ing o Gipuzkoa - Uni e si y o he Basque Coun y (UPV/EHU)
Plaza de Eu opa, 1, 20018 San Sebas ian, Spain
Abs ac
The c ack p opaga ion pe o mance o he End No ched Flexu e wi h inse ed olle es is analyzed. An
equi alen ene gy elease a e is p oposed aking in o accoun he in e ac ion o he modes I and II,
based on he linea ailu e c i e ion. Expe imen al esul s ob ained wi h specimens o F593/T300
ca bon/epoxy unidi ec ional composi e show a good ag eemen wi h he p oposed app oach. The
s abili y condi ion is heo e ically de eloped based on he de i a i e o he equi alen ene gy elease
a e, unde ixed load condi ion and ixed displacemen condi ion. Expe imen al es s ha e been ca ied
ou o assess he p oposed equi alen ene gy elease a e and o e alua e he c ack s abili y condi ion.
An analysis o he in luence o compliance measu emen on he c ack leng h and on he equi alen
ene gy elease a e is included. Based on he esul s ob ained, es condi ions wi h ini ial mode a ios
be ween 65% and 75% a e ecommended.
Keywo ds: Composi es; Delamina ion; Mixed-mode; Ene gy Release Ra e; C ack S abili y.
This is he accep manusc ip o he ollowing a icle ha appea ed in inal o m in
Theo e ical and Applied F ac u e Mechanics 87: 99-109 (2017),
which has been published in inal o m a
h ps://doi.o g/10.1016/j. a mec.2016.11.001.
© 2016 Else ie L d. unde CC BY-NC-ND licence (h ps://c ea i ecommons.o g/
licenses/by-nc-nd/4.0/)
- 2 -
1 INTRODUCTION
One o he main objec i es in ac u e mechanics is o measu e he ac u e oughness o ma e ials. I is
supposed ha a s able c ack g ow h is equi ed o eliable ene gy elease a e cu es de e mina ion.
Di e en ype o es con igu a ions ha e been used o in es iga e he mixed-mode I/II ac u e o
di e en ma e ials, such as composi e lamina es [1-3], wood [4-6], o adhesi ely bonded join s [7-11].
Howe e , he s abili y condi ion o he mixed-mode es is no de ined in he s anda d. The only
men ion done is ha in he high mode II egime, he delamina ion g ow h is o en uns able, p ecluding
p opaga ion oughness alues om being de e mined. I is also s a ed ha he use o longe ini ial
delamina ions inc eases he endency o s able delamina ion g ow h [12]. Wi h espec o pu e modes I
and II, he Double Can ile e Beam es (DCB) is conside ed always s able [13], and he End No ched
Flexu e es (ENF) is s able i a/L>0.7 [14,15].
Phillips and Wells [16], s udied he s abili y o ans e se c acks in composi es. They assumed as s abili y
limi he de i a i e o s ess wi h espec o he c ack leng h ob aining a c i ical alue. I he ini ial c ack
is less han he c i ical one, he c ack p opaga es uns ably a a s ess alue de e mined by he ini ial
ac u e ene gy and i g ows spon aneously unde dec easing load. On he con a y, c acks which a e
ini ially la ge han he c i ical size a e s able and only g ow i he load inc eases.
Allix and Co igliano [17] deal wi h he p oblem o simula ing he mixed-mode I/II c ack p opaga ion.
They compa ed he analy ical esul s om he Linea Elas ic F ac u e Mechanics (LEFM) hypo heses and
nume ical esul s conce ning c ack s abili y o he Asymme ic End Loaded Spli (AELS) specimen. The
analy ical esul s s a ed ha unde load con ol i was always uns able and unde displacemen con ol
he s abili y condi ion was a/L>0.42. By means o he nonlinea nume ical model p oposed, he limi o
s able c ack p opaga ion unde displacemen con ol was a/L =0.435 which is e y close o he analy ical
esul .
Szek enyes [18], e i ied he adi ional compliance based c i e ion, based on he de i a i e o he
ene gy elease a e (ERR) by expe imen al obse a ions. He used anspa en ma e ial o isually
measu e c ack leng h and c ack p opaga ion. The ansi ion om s abili y o ins abili y was de ined as
he poin jus be o e a c ack jump. He ound ha he s abili y o he sys em depends on he de i a i e o
he c i ical displacemen de ined as he displacemen o he load applica ion poin a c ack ini ia ion.
The ela ionship be ween he c i ical displacemen and he c ack leng h was de e mined by expe imen s
in many specimens o di e en ini ial c ack leng hs, o di e en es me hods. The c i ical alue o c ack
leng h was de e mined by di e en ia ion a e cu e i ing o expe imen al da a. He deduced ha he
poin o ins abili y was always whe e he c i ical displacemen eached i s minimum alue.
- 3 -
In he p esen wo k he c ack p opaga ion pe o mance is analyzed in a mixed mode I/II End No ched
Flexu e specimen wi h inse ed olle (ENFR), ecen ly p oposed [19,20]. Taking in o accoun he
in e ac ion be ween he wo modes, an equi alen ene gy elease (ERR) concep has been de eloped
and he s abili y condi ion has been de e mined by means o he de i a i e o ha equi alen ene gy
elease a e. An e o analysis has been ca ied ou in o de o de e mine he in luence o compliance
measu emen on he c ack leng h and on he equi alen ERR. Analyzing he expe imen al da a du ing
he p opaga ion, he p oposed equi alen ERR has been assessed and he c ack s abili y has been
e alua ed. Op imum es condi ions o es s pe o med unde displacemen con ol ha e been
p oposed.
- 4 -
NOMENCLATURE
a,
a
c ack leng h and c ack inc emen , espec i ely
ci
dis ances om he suppo o he posi ion o he olle
b,2h
wid h and hickness o he specimen, espec i ely
Cspec, Cs, Cexp,
compliance o he specimen, o he sys em and expe imen al compliance,
espec i ely
kS
s i ness o he sys em
0, P0
ini ial displacemen and ini ial load, espec i ely
displacemen o he middle poin o he specimen
spec,
exp
calcula ed and expe imen al displacemen o he middle poin o he specimen,
espec i ely
E
lexu al modulus
GLT, GLT’
in-plane shea modulus and ou o plane shea modulus, espec i ely
E11, E22, G12
longi udinal elas ic modulus, ans e sal elas ic modulus and shea elas ic modulus
espec i ely
GI,GII , G
ene gy elease a es o mode I, mode II, and o al, espec i ely
GIc,GIIc , Gc
c i ical ene gy elease a e o mode I, mode II, and o al, espec i ely
Geq,Geqc
equi alen ene gy elease a e and c i ical equi alen ene gy elease a e,
espec i ely
KCa, KaG
e o coe icien s ha co espond o he in luence o he measu emen o
compliance on he c ack leng h, and he in luence o he de e mina ion o c ack
leng h on equi alen ene gy elease a e, espec i ely.
L
hal span o he es
P
applied Load
Y
o ce exe ed by he olle
R
olle adius
W, U, U*
wo k done by ex e nal o ces, s ain ene gy, and complemen a y s ain ene gy,
espec i ely
- 5 -
2 ANALYTICAL BACKGROUND
2.1 Compliance o ENFR es
The End No ched Flexu e wi h inse ed Rolle (ENFR) es con igu a ion has been p oposed [19] and
expe imen ally assesed [20] ecen ly. In o de o ge mixed mode, a olle is in oduced be ween he
wo su aces o he c ack and he specimen is es ed in ENF con igu a ion. The mode II is p o ided by he
ex e nal load and he mode I is ob ained by he opening o he c ack due o he inse ion o he olle as
shown in Fig. 1. The olle can be loca ed a he ou e side o a he inne side o he suppo .
Fig. 1 ENFR Tes con igu a ion wi h he olle loca ed a he inne side
Being Y he o ce exe ed by he olle and being i he o ce ha gene a es he displacemen , aking
in o accoun only bending e ec s, when he olle is loca ed a he inne side o he suppo , he Y o ce
is [19]:
(1)
3
3
(2 )
4 ( ) 8( )
bh
R P
a ac
a
Ec
Yc
Whe e R is he adius o he olle ; c is he dis ance om he suppo whe e i is loca ed; a is he c ack
leng h; P is he applied load; E he lexu al modulus; b he wid h o he specimen; and 2h he o al
hickness o he specimen.
The displacemen o he load applica ion poin aking in o accoun only bending e ec s is [19]:
(2)
3 3 3 2
3
(2 )
(3 2 3 )
8 4 ( )
P R a c
a L c ac
bh aE c
- 6 -
Acco ding o Eq.(2), due o he olle in oduced be ween he specimen a ms, he e is an ini ial nega i e
displacemen wi hou ex e nal load applica ion. Fu he mo e, when he displacemen is null, he e is a
posi i e load. The ini ial condi ions can be calcula ed eplacing P=0 and
=0 in Eq.(2) and lead o [19]:
(3)
0
3 3 3 2
0
3
(2 )
4 ( )
2 (2 )
( )(3 2 3 )
R a c
a c
R bh a c
a c a c ac
E
L
P
The heo e ical load-displacemen cu e be o e c ack p opaga ion occu s is depic ed in Fig. 2.
P0
P
P
Fig. 2 Load-displacemen cu e o ENFR es be o e c ack p opaga ion
Acco ding o Fig. 2 he compliance o he ENFR es is [19]:
(4)
3 3 3 2
0
3
0 0
3 2 3
8
CP
a L c a
bhP E
c
P
2.2 Ene gy elease a e in he ENFR es
Acco ding o G i i h [21] and I win [22], he ene gy balance in a small c ack ad ance can be exp essed
as:
(5)
dW dU Gbda
Whe e dW is he wo k done by ex e nal o ces; dU is he change in s ain ene gy; G is he ene gy needed
o he c ack ad ance pe uni a ea; b is he wid h o he c ack; and da is he di e en ial c ack ad ance.
The di e en ial wo k done by he ex e nal o ces Fi in hei espec i e displacemen s
i, assuming he
epea ed index con en ion is . The complemen a y s ain ene gy U* is de ined as:
i i
dW Fd
(6)
*
i i
U F U
Di e en ia ing Eq.(6), and eplacing in he ene gy balance equa ion o Eq.(5) i esul s:
(7)
*
i i
dU dF Gbda
- 7 -
In spi e o he c ack ad ance is an i e e sible p ocess, i is assumed ha an elemen al a ia ion he
complemen a y s ain ene gy is an exac di e en ial. Since he s a e a iables a e Fi and a, hus:
(8)
* *
*
i
i
U U
dU dF da
F a
Iden i ying he i s e ms in Eq.(7) and Eq.(8) i esul s he heo em o Engesse -Cas igliano. Iden i ying
he second summands i esul s:
(9)
*
1
i
F c e
U
Gb a
In he ENFR es he wo k is done by wo o ces, Y and P. The wo k ca ied ou by Y is ela ed o he
ini e displacemen imposed by he olle , and he wo k ca ied ou by P is ela ed o he applica ion o
he load. Acco ding o Eq.(9) he complemen a y s ain ene gy mus be used o de e mining G i o ces
a e used as s a e a iables.
When he olle is posi ioned a he inne side o he c ack ip, aking in o accoun only bending e ec s,
he ene gy elease due o each ac u e mode can be exp essed as ollows [19]:
(10)
2 3 2 2
4 2 2 3
2 2
2 3
33 3
4( ) 4 ( ) 16
9
16
I
II
R E h PRc P c
Ga c b a c E b h
P a
GE b h
The alues o GI and GII o Eq. (10) ag ee wi h hose ob ained by William´s pa i ion me hod [23]. In he
case ha c=0 and aking in o accoun only bending e ec s, hey ag ee wi h hose ob ained by
Szek enyes [24] o his es con igu a ion.
In many cases G is de e mined based on he compliance, using he I win-Kies app oach [25]. In he
p esen case, he calcula ion o G based on he de i a i e o he compliance is:
(11)
2 2 2 2 2
2 3 2 3
9 3
2 16 16
P dC P a P c
Gb da E b h E b h
In Eq. (11) wo summands co esponding mode I o Eq. (10) a e no included. Then, in he p esen case
he app oach based on he compliance is no alid. I can be concluded ha i is also alid in cases whe e
he ex e nal wo k is ca ied ou by a unique o ce.
- 8 -
3 EQUIVALENT ENERGY RELEASE RATE
C ack p opaga ion only occu s i he ene gy a ailable o a c ack ex ension, G, is enough o p o ide all
he ene gy ha is equi ed o c ack g ow h [26]. The ene gy equi ed o c ack g ow h is called c i ical
ene gy elease a e Gc. Fo pu e modes, he c i ical condi ion o c ack p opaga ion is:
(12)
I Ic
II IIc
G G
G G
Howe e , when mode I and mode II a e in ol ed, he ene gy elease a e is . Then, he
I II
G G G
c i ical alue o he ene gy elease a e Gc when c ack p opaga es depend on he mode a io GII/G. Fo
ins ance, in a mixed mode es wi h a g ea mode a io, Gc mus be close o GIIc and when he mode a io
is low Gc mus be close o GIc. Thus, in spi e o GIc and GIIc being ma e ial p ope ies, Gc is no .
In he p e ious s udy conce ning he expe imen al assessmen o ENFR [20], i was ound ha he linea
c i e ion is sui able o ep esen ing he c ack p opaga ion in an ENFR es , whe e mode a io a ies
du ing he es . The e o e, he condi ion o he c ack p opaga ion can be de ined as:
(13)
1
I II
Ic IIc
G G
G G
Exp essing Eq.(13) in simila o m o Eq.(12), he c ack p opaga ion condi ion leads o:
(14)
I IIc II Ic Ic IIc
G G G G G G
The le membe o Eq. (12) is he ene gy a ailable o a c ack ex ension and he igh membe is he
c i ical alue o c ack ad ance. Compa ing Eq.(12) and Eq.(14) an equi alen ene gy elease a e Geq and
an equi alen c i ical alue Geqc a e de ined as:
(15)
1/2
1/2
eq I IIc II Ic
eqc Ic IIc
G G G G G
G G G
The e o e, i can be s a ed hen ha c ack p opaga es when:
(16)
eq eqc
G G
To he bes knowledge o he au ho s, equi alen alues o ene gy elease a e gi en in Eq. (15) ha e no
been p e iously de ined.
- 9 -
4 ANALYSIS OF CRACK STABILITY
4.1 S abili y de ini ion
A gene al de ini ion o s abili y is ha a sys em is s able i a ini e change in he inpu pa ame e s does
no cause an in ini e change in he ou pu alues [27]. Rega ding ac u e es s ca ied ou in uni e sal
es ing machines, he inpu pa ame e s a e he load o he displacemen applied by he es ing machine
and he ou pu alue is he c ack leng h. The e o e, a ac u e es is s able du ing c ack g ow h when an
in ini esimal change o he load o he displacemen does no cause an in ini e change in he c ack
leng h. Acco ding o Schwalbe e al [28], a c ack ex ension unde displacemen con ol is s able when
he c ack s ops when he applied displacemen is held cons an . In he p esen s udy, he s abili y
condi ion is applied o he equi alen ene gy elease a e Geq:
(17)
eq eqc
eq eqc
dG dG and G G
da da
Assuming ha Geqc is cons an du ing c ack g ow h, he s abili y c i e ion om Eq.(17) simpli ies o:
(18)
0
eq
dG
da
Replacing Eq. (15) in he condi ion gi en in Eq.(18), he condi ion o s abili y can be exp essed as
ollows:
(19)
1/2
10
2
eq I II
I IIc II Ic IIc Ic
dG dG dG
G G G G G G
da da da
I he esul o Eq.(19) is posi i e, hen he c ack g ow h is uns able, because he ene gy eleased is mo e
han ha needed o c ea e a new su ace a ea. I i is nega i e ex e nal wo k mus be done o keep he
c ack mo ing.
I is wo h no ing ha he equi alen ene gy elease a e p oposed is based on he ul illmen o he
linea c i e ion. I ha c i e ion is no sui able he p esen s abili y analisis is no alid.
4.2 Fixed load condi ion
Conside ing ha GI = GI (a,P)and GII = GII (a,P), he o al de i a i e o he ERR due o each mode ac u e
wi h espec o he c ack leng h, can be exp essed as ollows:
- 16 -
0
0.5
1
1.5
2
20 30 40 50 60
a (mm)
KaG
ENF
R0.5-c0
R0.5-c10
R1.5-c0
Fig. 8 In luence o c ack leng h ela i e e o on he ela i e e o o Geq
Fig. 8 shows he absolu e alue o he coe icien KaG o di e en es condi ions. In he ENF es , KaG =1
o any c ack leng h since in Eq.(28) GI=0. In he case o R0.5-c0, when he c ack leng h a>24mm he
absolu e alue o KaG is below 1. Mo eo e , he e is a minimum alue when a=30mm. In he case o
R0.5-c10, he cu e has a simila shape bu he c ack leng h should be a>39mm o ge a alue o KaG
below 1, and he minimum alue co esponds o a=45mm. In he case o R1.5-c10 when a>43mm i
gi es an absolu e alue o KaG below 1, being he minimum alue when a=52mm. The e o e when R o c
inc ease, he minimum c ack leng h o ge KaG<1 inc eases.
Acco ding o Eq.(28), i is possible o analyze di ec ly he in luence o he ela i e inpu e o o he
compliance in he ela i e ou pu e o o Geq as i ollows
(29)
eq
aG Ca
a
eq a
GC
K K
G C
Fig. 9 In luence o compliance ela i e e o on he ela i e e o o Geq
- 17 -
Fig. 9 shows he absolu e alue o he p oduc o wo coe icien s KaG KCa. The absolu e alues o he
p oduc KaG KCa. lowe han 1 ake place in simila c ack leng hs o hose ob ained in he analysis o KaG.
In o de o de e mine an op imum es condi ion, he p oduc o KaG KCa should be aken in o accoun .
Fig. 10 Absolu e alue o KaG KCa e sus mode a io GII/G
Fig. 10 shows he absolu e alue o KaG KCa e sus mode a io GII/G. When he mode a io is g ea e ,
which is ela ed o a longe c ack leng h o ixed aues o c, R, he absolu e alue o KaG KCa is lowe .
Thus i means ha he ela i e e o o Geq is educed wi h espec o he ela i e e o o he
expe imen al compliance.
Table 2 summa izes he op imum c ack leng hs depending on he di e en c i e ia.
Table 2 Op imum c ack leng hs in mm.
STABILITY CONDITION
ERROR ANALYSIS
TEST
NOMENCLATURE
Load
Con ol
Displacemen
con ol
KCa<1
KaG<1
KaG KCa<1
ENF
No s able
a > 42
a > 42
KaG=1
a > 42
R0.5-c0
a < 30
s able
a > 42
a > 24
a > 27
R0.5-c10
a < 45
s able
a > 42
a > 38
a > 39
R1.5-c0
a < 51
s able
a > 42
a > 43
a > 43
In o de o de ine op imum es condi ions, he c ack leng h has o sa is y he s abili y condi ion and o
minimize he ela i e e o in he de e mina ion o Geq.
- 18 -
6 EXPERIMENTAL
6.1 Ma e ials and es appa a us
T6T/F593 p ep egs p o ided by Hexcel Composi es wi h a 55% olume-con en o ib e we e used o
p oduce lamina es. The pla es we e manu ac u ed by ho p ess molding. Six een-laye ed unidi ec ional
lamina es, [0]16, we e made wi h a Te lon ilm in oduced cen e ed du ing he piling up p ocess in o de
o make he ini ial c ack. The specimens we e cu wi h a diamond disc saw, being he nominal hickness
and wid h 3 mm and 15 mm, espec i ely. The edges o he lamina e we e disca ded o he p epa a ion
o he specimens. Tes s we e pe o med on an MTS-Insigh 100 elec omechanical es ing machine
equipped wi h a 5kN load cell, ope a ing in a displacemen con olled mode. In o de o a oid he
in luence o he esin ich a ea he specimens we e p ec acked in mode II by a ENF es , inc easing he
c acked leng h a ound 5 mm.
All he specimens we e es ed using a p ocedu e based on h ee-poin bending es s a i e di e en
spans p oposed by Mujika [29], in o de o ob ain he lexu al modulus, E and he ou o plane shea
modulus GLT’, which is equal o he in-o -plane shea modulus GLT assuming ha he ma e ial is
ans e sely iso opic. The mean alues co esponding o i e specimens we e:
E = 107.4 (±1.4) GPa
GLT’ = 4.3 (±0.4) GPa
6.2 Expe imen al es condi ions
Se e al mixed mode es s ha e been pe o med in o de o compa e he esul s wi h he ones ob ained
applying he heo e ical s abili y condi ion. The expe imen al displacemen is de ined as 0 when he
con ac be ween he load nose and he specimen wi hou olle occu s, as i is shown in Fig. 11. A e
inse ing he olle , he e is an ini ial nega i e displacemen
0 o he ze o load condi ion, as seen in Fig.
2. Ac ually, he con ac in he es ing machine has been de ined when he load was 0.5 N.
- 19 -
Fig. 11 Ini ial Condi ions in Load-Displacemen cu e
The de e mina ion o he c ack leng h a e e y poin whe e P and
a e measu ed, is based on he
a ia ion o he compliance du ing he c ack ad ance, based on he Beam Theo y including Bending
Ro a ion e ec s (BTBR) me hod de eloped o he ENF es by A ese e al [30]. Fo he expe imen al
analysis bending o a ion e ec s ha e no been included because he suppo olle adii a e 2.5mm and
hus he in luence is negligible [30]. In a p e ious wo k [20], i was e i ied ha in he ENFR es he
c ack leng h can be de e mined wi hou op ical me hods. In spi e o only bending e ec s ha e been
p esen ed o simplici y in he analy ical backg ound, he calcula ions conce ning he expe imen al pa
ha e been ca ied ou including also shea e ec s.
Since he con ibu ion o each ac u e mode a ies du ing he c ack p opaga ion, he mode a io a he
ini ia ion poin is he pa ame e chosen o de ine he ype o mixed mode es . To de e mine ha ini ial
mode a io, he alue o ene gy elease a e o each mode con ibu ion has been de e mined
subs i u ing he alue o he c ack leng h in Eq.(10), when a=0.25mm, conside ing i as nonlinea i y
poin (NL). The NL poin is one o he de ini ions o c ack ini ia ion p esen ed in he s anda d [12].
The nomencla u e ha has been used in o de o iden i y each es condi ion is ai-Rj-ck. ai is he nominal
ini ial c ack leng h; Rj is he inse ed olle adius; and ck is he alue o he c dis ance ha de ines he
posi ion o he olle . Fo ins ance, a40-R1-c8 is he es wi h ini ial nominal c ack leng h o 40 mm, an
inse ed olle o 1mm adius and posi ioned a 8mm a he inne side o he suppo .
The nominal geome ic dimensions a e span 2L=120mm, wid h b=15mm and hickness 2h=3mm.
- 20 -
All he es condi ions a e summa ized in Table 3. Fo simplici y, a numbe will be used in he legends o
he g aphics.
Table 3 Summa y o expe imen al es condi ions
ID
NOMENCLATURE
INITIAL GII/G
(%)
Tes 1
a40-R0.5-c0
96
Tes 2
a31-R0.5-c0
90
Tes 3
a45-R1.5-c0
74
Tes 4
a43-R0.9-c8
66
Tes 5
a38-R0.7-c8
59
6.3 Resul s and discussion
6.3.1 Ene gy elease a e cu es
The de e mina ion o he c ack leng h a any poin o he es whe e P and a e measu ed, allows he
calcula ion o he ene gy elease a e a any poin du ing he c ack p opaga ion. The expe imen al
alues o GI and GII o he es s p esen ed in Table 3, ha e been ob ained subs i u ing he alue o he
c ack leng h, in Eq.(10). In Fig. 12 he expe imen al ene gy elease a e cu es due o each mode can be
seen. In he legend o he igu es besides he iden i ie o he es he ini ial mode a io is included in
b acke s.
0 2 4 6 8 10 12
200
400
600
800
1000
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
C ack ad ance (mm)
GI o GII (J/m2)
Uppe Cu es GII
Lowe Cu es GI
- 21 -
Fig. 12 Expe imen al Ene gy Release Cu es
As i can be seen in Fig. 12, he uppe cu es co espond o GII, and he lowe ones o GI. The cu e o GI
and he cu e o GII ha co espond o he same es condi ion a e d awn wi h he same colo . When
c ack p opaga es, he cu es o GI dec ease slowly. A he same ime, he cu es o GII inc ease. This
means ha he con ibu ion o each mode is changing du ing c ack p opaga ion.
In Fig. 13 he o al ene gy elease a e G= GI + GII is depic ed.
0 2 4 6 8 10 12
0
200
400
600
800
1000
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
C ack ad ance (mm)
G=GI+GII (J/m2)
Fig. 13 Expe imen al o al Ene gy Release Ra e
When he mode II is p edominan he o al ene gy elease a e ends o a cons an alue. When he
mode I is mo e impo an he o al G inc eases wi h c ack ad ance. Those ends ag ee quali a i ely wi h
he esul s ob ained in R-cu es o he pu e modes conce ning ENF and DCB es s, espec i ely: a
pla eau in ENF es s [30] and he inc ease o G in DCB es s o he same ma e ial [31].
6.3.2 Linea c i e ion, no malized mode a ios and equi alen ene gy elease a e cu es
In he p e ious s udy conce ning he expe imen al assessmen o ENFR [20], i was ound ha he linea
c i e ion o Eq.(23) is sui able o ep esen ing he c ack p opaga ion in an ENFR es , whe e mode a io
a ies du ing he es .
The alue o GIIc is ou imes GIc in he case o he ma e ial used in his s udy as i can be seen in Table 1.
The e o e, in o de o analyze he di ec con ibu ion o each mode o ailu e, ins ead o GI and GII,
no malized mode a ios ha e been de ined [20]:
- 22 -
(30)
0 0
I II
I II
Ic IIc
G G
G G
G G
These no malized mode a ios o Eq.(30) a e he summands o he linea c i e ion. The ini ial alues o
he o al and he no malized mode a ios gi en in Eq.(30) a e shown in Table 4.
Table 4 Ini ial mode a io and ini ial no malized mode a io
ID
INITIAL MODE RATIO
(GII/G) (%)
INITIAL NORMALIZED MODE
RATIO (GII0) (%)
Tes 1
96
51
Tes 2
90
45
Tes 3
74
37
Tes 4
66
30
Tes 5
59
22
The es wi h he highes ini ial mode a io and he one wi h he lowes mode a io ha e been plo ed in
Fig. 14 in e ms o he no malized mode a ios.
0246810
0
0.2
0.4
0.6
0.8
1
GI0 Tes 1 (96%)
GII0 Tes 1 (96%)
GI0 Tes 5 (59%)
GII0 Tes 5 (59%)
C ack ad ance(mm)
GI0 and GII0
Fig. 14 Expe imen al no malized mode a ios GI0and GII0
When he mode II is clea ly p edominan he GII0 cu e is abo e he GI0 cu e, as in Fig. 12. Ne e heless,
when he ini ial mode a io is 59%, he cu e o GI0 is abo e he cu e o GII0 du ing almos all he
p opaga ion, in con as o wha happened wi h GII and GI.
As linea c i e ion is adop ed, he sum o he wo no malized mode a ios GI0+GII0 depic ed in Fig. 15,
should be close o 1.
- 23 -
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
C ack ad ance(mm)
GI0 + GII0
Fig. 15 Sum o expe imen al no malized mode a ios
The mean alue o he cu es du ing p opaga ion is always highe han 0.85, and in he case o es s 3
and 4 is e y close o 1. The maximum de ia ion occu s in he es s wi h mode a io nea o pu e mode
II, es s 1 and 2, and he maximum ela i e e o is 15% wi h espec o he linea c i e ion.
0 2 4 6 8 10
100
200
300
400
500
600
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
C ack ad ance (mm)
Geq (J/m2)
Fig. 16 Equi alen ERR, expe imen al Geq
The p oposed equi alen ene gy elease a e o Eq. (15) has been applied o he expe imen al da a and
he esul s a e p esen ed in Fig. 16. In all he cases he equi alen ERR ends o a cons an alue. The
mean alues o he cu es du ing p opaga ion a e be ween 505 and 555 (J/m2). The alue o he pla eau
ag ees wi h he heo e ical alue o GCeq=550 (J/m2) o Table 1. The maximum de ia ion co esponds o
he es s 1 and 2 and in hose cases he maximum ela i e e o is 8.5% wi h espec o he heo e ical
- 24 -
alue o GCeq.Consequen ly, acco ding o he esul s shown in Fig. 16, he p oposed equi alen ERR
app oach can be conside ed sui able o de ining c ack p opaga ion condi ion o he ENFR es .
6.3.3 S abili y e alua ion
Du ing he es ing phase, i has been assumed ha uns able c ack g ow h occu s when he e is a jump in
he load-displacemen cu e. In Fig. 17 he load-displacemen cu es o he es s p esen ed in Table 3
a e p esen ed. The ci cula ma ke included in igu es indica es he c ack ini i a ion poin o he NL
poin .
0 1 2 3 4 5
0
50
100
150
200
250
300
350
Displacemen (mm)
Load (N)
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
Fig. 17 Expe imen al load-displacemen cu es
When he ini ial mode a io is close o pu e mode II, es s 1 and 2, when he c ack s a s o p opaga e
he load con inues inc easing, which means ha i is beha ing in a s able manne . A e he load eaches
i s maximum alue i dec eases suddenly. Besides, he e a e jumps in bo h cu es. The e o e he es s 1
and 2 can be conside ed s able du ing he mos pa o he p opaga ion, and a he inal s age hey
become uns able. In he es o he cases, he e a e no jumps o sudden load d ops du ing c ack
p opaga ion. I can be concluded hen, ha hey a e all s able.
The de ini ion o s abili y o sec ion 4.1, which de ines a es as s able i an in ini esimal change o
displacemen , does no cause an in ini e change in he c ack leng h, has been aken in o accoun o
analyze he expe imen al da a. In Fig. 18 he ela ionship be ween c ack leng h and he displacemen is
depic ed.
- 25 -
012345
25
30
35
40
45
50
55
60
Displacemen (mm)
a (mm)
Tes 1 (96%)
Tes 2 (90%)
Tes 3 (74%)
Tes 4 (66%)
Tes 5 (59%)
Fig. 18 C ack leng h e sus displacemen
In he cu es o es s 1 and 2 a he i s s age o he p opaga ion, a ini e change in displacemen causes
a ini e change in he c ack leng h and hus hey can be conside ed s able. Howe e , a he inal s age o
he p opaga ion a small change in displacemen causes a much g ea e change in c ack leng h, and he e
a e e en jumps in he cu e. The e o e, hey become uns able.
Fo he es o he es s, he c ack p opaga ion s a s ea lie , and he slope o he cu e is simila , which
means ha a ini e change in he displacemen is needed o ha e a ini e change in he c ack leng h.
Consequen ly, hey a e all s able acco ding o his de ini ion.
This is he same conclusion d awn by analyzing P-
cu es.
Since he expe imen al es s ha e been ca ied ou a a cons an displacemen a e which means unde
displacemen con ol, heo e ical s abili y condi ion based on he de i a i e o Geq,o Eq. (26) is applied
o he expe imen al da a.
