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Fa igue delamina ion damage analysis in composi e ma e ials h ough a ule
o mix u es app oach
Ali eza Tahe zadeh-Fa da,b,∗, Se gio Jiméneza,b, Alejand o Co nejoa,b, Eugenio Oña ea,b,
Lucia G a iela Ba bua,b
aPoly echnic Uni e si y o Ca alonia (UPC), Campus No d, 08034 Ba celona, Spain
bIn e na ional Cen e o Nume ical Me hods in Enginee ing (CIMNE), Campus No d UPC, 08034 Ba celona, Spain
ARTICLE INFO
Keywo ds:
Rule o mix u es
Homogeniza ion
Fa igue
Delamina ion
Composi es
Fini e elemen me hod
ABSTRACT
The p esen s udy in es iga es delamina ion damage ini ia ion and p opaga ion wi hin a homogeniza ion heo y
o mix u es, using he concep o i ual laye s and i ual in e aces. I elimina es spa ial disc e iza ion o
laye s, in oducing a esul an damage a iable o cap u e s uc u e’s bulk esponse unde bo h mono onic
and cyclic loads. Fa igue-induced de e io a ion is classi ied in o sub-c i ical, c i ical, and o e -c i ical s ages
based on in e acial s esses. Calib a ion is conduc ed employing he widely-a ailable Wöhle cu es o each
loading mode independen ly. An ad ance-in- ime s a egy is included in he model o enhance he simula ion
speed. The eliabili y o he app oach is assessed o c ack ini ia ion and p opaga ion sepa a ely h ough
s anda d es coupons, showing good co ela ion wi h expe imen al da a in mode I, mode II, and mixed-
mode loading condi ions. Depending on he calib a ion p ocedu e adop ed, he model is applicable o a wide
ange o s ess a ios. In addi ion, i could be in eg a ed in o any s anda d ini e elemen amewo k using
he desi ed numbe o elemen s h ough he hickness ega dless o he physical amoun o laye s. This allows
easy modi ica ion o s acking sequences o he numbe o laye s wi hin he cons i u i e law wi hou mesh
s uc u e changes, acili a ing simula ion o la ge-scale composi e lamina es wi h minimal accu acy loss and
educed compu a ional cos s.
1. In oduc ion
O e he pas ecen yea s, composi es ha e a ac ed much a en-
ion as he po en ial ma e ials o be u ilized in di e en indus ies
including ae ospace, au omo i e, ma ine, e c. [1,2], due o hei su-
pe io mechanical p ope ies such as co osion and a igue esis ance,
ligh weigh , high s eng h and ease o anspo a ion [3–6]. The mul i-
ma e ial na u e o composi e ma e ials gi es mo e eedom o enginee s
o designing pu poses, howe e , his would lead o mo e damage
mechanisms being engaged once hese ma e ials a e unde s uc u al
loading as well as mo e unce ain y in hei bulk esponse due o manu-
ac u ing complexi ies [7–9]. The e o e, mo e ad anced me hodologies
a e equi ed in s udying he beha io o composi es.
Damage mechanisms in composi es a e mainly comp ised o in a-
lamina and in e -lamina modes [10]. Acco ding o he complex na u e
o hese damages and hei in e ac ions wi hin he s uc u e, simu-
la ion schemes ha e encoun e ed challenges o comp omise be ween
accu acy and compu a ional cos [11]. Among all deg ada ion modes,
in e ace ailu e o delamina ion has been o signi ican impo ance o
∗Co esponding au ho a : Poly echnic Uni e si y o Ca alonia (UPC), Campus No d, 08034 Ba celona, Spain.
E-mail add esses: [email p o ec ed],[email p o ec ed] (A. Tahe zadeh-Fa d).
i s majo e ec on he s eng h and s i ness o he composi e [12–
15]. Indeed, lamina ed composi es a e ab ica ed h ough building up
he in-plane ibe - ein o ced laminae and a e mainly expec ed o bea
loading in he ibe di ec ion. Howe e , gene al loading condi ions
usually impose some h ough- he- hickness s esses o he composi e,
and can lead o ailu e o he ma ix and ini ia ion o he delamina ion
phenomenon [16,17]. Debonding in he adjacen laye s may no be
ca as ophic wi hin he in-plane loading, none heless, o composi es
unde ou -o -plane loading o bending condi ions, i could be a al [18,
19]. P edic ion o his ini ia ion and consequen p opaga ion o he
delamina ion c ack has been associa ed wi h expe imen al ull-scale
es ing p ocedu es owing o he lack o enough knowledge and accu a e
nume ical models, which, in u n, has made he design p ocedu es o
be highly expensi e in e ms o ime and cos [20].
Fa igue delamina ion damage is mainly composed o h ee dis inc
s ages o ini ia ion, onse and p opaga ion, and each o hem has been
cha ac e ized by di e en nume ical app oaches [21]. The ini ia ion
phase e e s o he o ma ion o he c ack om an in ac ma e ial
and commonly o igina es om he impe ec ions o med du ing he
h ps://doi.o g/10.1016/j.comps uc .2024.118613
Recei ed 11 June 2024; Recei ed in e ised o m 8 Augus 2024; Accep ed 23 Sep embe 2024
Composi e S uc u es 351 (2025) 118613
A ailable online 27 Sep embe 2024
0263-8223/© 2024 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
A. Tahe zadeh-Fa d e al.
manu ac u ing p ocess [11]. A common me hod o accoun o his
s age is de ining an ini ia ion zone whe e damage could s a appea ing
acco ding o he models based on S-N cu es [22,23]. In his way, S-
N cu es a e o mula ed using expe imen ally-de ined pa ame e s and
each mode is calib a ed independen ly. Delamina ion onse , howe e ,
is de ined as a isible inc emen in he c ack h ough he medium
om a c ack s a e . Al hough he e is an uns able delamina ion c ack
p opaga ion a e c ack ini ia ion, c ack onse is ollowed by a s able
condi ion o c ack ex en ion [24]. C ack onse could be analyzed in
a simila manne as ini ia ion by conside ing i ing pa ame e s o
he expe imen al da a [25], howe e his me hod lacks he abili y o
conside he mixed-mode and s ess a ios.
Linea elas ic ac u e mechanics (LEFM) is a widely-used app oach
in simula ing he a igue c ack in he p opaga ion phase [26,27].
These models mainly employ any o ms o he Pa is law o make
a connec ion be ween he c ack g ow h and he ene gy elease a e
(ERR) [11]. The e o e, o he me hods should also be inco po a ed
wi hin his o mula ion o calcula e he ene gy elease a e o he s ess
in ensi y ac o such as Vi ual C ack Closu e Technique (VCCT) [8,28–
30], i ual c ack ex ension [31,32], o c ack su ace displacemen
ex apola ion me hods [33]. Despi e his, LEFM based models su e
om se e al p oblems such as he inabili y o be used in hyb id
composi es [34] o ailu e o ini ia e he a igue c ack wi hin a p is ine
ma e ial [24].
Cyclic Cohesi e Zone Me hod (CCZM) can also be used o simula e
a igue c ack in composi es which is based on de ining a ac ion-
sepa a ion law o damage e olu ion wi hin he in e ace [23,35–39].
Based on how he cyclic load e ec s a e aken in o accoun , wo
di e en me hods o hys e esis loop and en elope load damage models
ha e been conside ed [40]. Hys e esis loop damage models deg ade he
s eng h and he s i ness wi hin a cycle-by-cycle algo i hm and a e well
sui ed o low-cycle a igue egime [41]. Ne e heless, calib a ion o
he pa ame e s is conside ed o be a highly challenging ask [40]. En e-
lope load damage models, on he o he side, u ilize jumping s a egies
combined wi h he Pa is law o e ol e delamina ion damage wi hin he
in e ace. This o mula ion could be bene icial in he high-cycle a igue
egime and is capable o conside ing di e en ac o s such as mode mix
and s ess a ios. Howe e , implemen ing Pa is law wi hin he cohesi e
amewo k can be complex [42], and linking he damage a iable o
he ac ion-sepa a ion o mula ions is no s aigh o wa d [20]. In ad-
di ion, i is no clea i he Pa is law should be conside ed as a ma e ial
p ope y o a he a p oblem dependen a iable. Consequen ly, a igue
delamina ion damage models ha a e no based on he Pa is law a e
highly aluable.
Mo eo e , cohesi e zone me hods ha e se e al sho comings.
Mainly since hey a e dependen on he de ini ion o an in e ace
medium, c ack pa h should be known a p io i [43]. Addi ionally, inco -
po a ing in e ace elemen s wi hin all in e aces would be a challenging
ask especially in composi es wi h la ge numbe o laye s. Since CCZMs
a e dependen on he de ini ion o a ac ion-sepa a ion law wi h an
ini ial linea pa , hey could manipula e he bulk esponse o he
s uc u e in i s elas ic egion when he e is no damage e ol ed wi hin
he in e ace [44]. Selec ion o he slope in CCZM linea pa is also
ad-hoc and equi es a uning p ocess [45,46].
Recen ly, a con inuum mechanics based a igue model has been
p oposed by uni ying di e en phenomena such as damage, plas ici y,
iscosi y and empe a u e e ec s [47], and has been imp o ed in o he
applica ions as well [48,49]. Indeed, he o mula ion igge s s eng h
and s i ness de e io a ion once a speci ic s ess c i e ion is sa is ied.
The deg ada ion is accoun ed o by a damage- ype cons i u i e law,
while being sensi i e o he cyclic loads h ough a a igue educ ion
ac o [50]. The ac o also a ec s he ac u e ene gy so ha an ene gy
dissipa ion is in oduced o he model as a esul o he cyclic loading.
Model pa ame e s a e calib a ed by using cu e i ing o he common
S-N cu es. The model is expec ed o be applicable in a wide spec um
o a igue scena ios while being capable o cap u ing bo h damage
ini ia ion and p opaga ion o e di e en amoun o s ess a ios.
In he cu en s udy, a homogeniza ion heo y o mix u es has
been p esen ed o model a igue delamina ion damage esponse in
lamina ed composi es. The wo k is based on a p e ious esea ch by he
au ho s [51] whe e loading condi ions we e limi ed o he mono onic
egime. Ra he han conside ing cohesi e elemen s a each in e ace,
he p oposed model employs a homogenized medium whe e he e is no
need o explici ly disc e ize composi e laye s, which could signi ican ly
acili a e he p e-p ocessing s age as well as he op imiza ion o he
design p ocess, since no e-meshing is equi ed when he in e nal
con igu a ion o he composi e changes. This is e en mo e p onounced
when modeling composi es wi h la ge numbe o laye s by accoun ing
he numbe o elemen s needed h ough he hickness i espec i e o
he ac ual numbe o laye s. The cyclic loading e ec is in oduced o
he o mula ion by de ining a a igue educ ion unc ion which could
be calib a ed acco ding o he common S-N cu es o mode I and
mode II, independen ly. Delamina ion damage e ec , in bo h con ex s
o c ack ini ia ion and p opaga ion, is e lec ed o he adjacen i ual
bulk laye s so ha he bulk esponse o he s uc u e is accu a ely
ep oduced, while no imposing any limi a ions on he cons i u i e law
used o each laye . The e o e, in addi ion o he in e -laye damages,
he e is he possibili y o in e a-laye mechanical phenomena such
as damage and plas ici y. Since he ad-hoc in e ace penal y s i ness
concep is elimina ed om he o mula ion, s uc u al esponse will no
longe be manipula ed in he elas ic egime. The p oposed model is
compa ible wi h any ini e elemen code and is conside ed o be well-
sui ed o simula ing a igue delamina ion in la ge-scale composi es
while main aining he accu acy wi hin an accep able ange.
2. Ma hema ical o mula ion
2.1. P elimina y concep s
In his sec ion, some in oduc o y in o ma ion ega ding he unde -
lying ule o mix u es me hod and in e acial s esses is p o ided. The
eade is e e ed o he p e ious wo k [51] o a comple e desc ip ion
and undamen al hypo heses.
Composi e lamina es a e mainly composed o a speci ic numbe o
ibe ein o ced laminae o ien ed a di e en di ec ions. T usdell and
Toupin [52] i s ly p esen ed a me hod o uni y all laye s’ beha io
h ough a homogeniza ion heo y. La e on, he me hod has been
de eloped by o he esea che s as well [53,54]. I is mainly based on
he assump ion ha s ains expe ienced by all laminae a e he same
whils he s ess o he composi e is ob ained by aking a supe -posi ion
o he s ess in each laminae acco ding o i s olume ic pa icipa ion,
i.e.:
𝜺1=𝜺2=⋯=𝜺𝑛(1)
𝝈=
𝑛
∑
𝑖=1
(𝑘𝑖⋅𝝈𝑖)(2)
whe e 𝜺𝑖,𝑘𝑖and 𝝈𝑖a e he s ain enso , olume ic pa icipa ion and
he s ess enso o he 𝑖 h laye , espec i ely. F om he homogeniza ion
poin o iew, he composi e is made up o i ual bulk laye s and
i ual in e aces, as p esen ed in Fig. 1, in a way ha he esponse
o he s uc u e is ep oduced co ec ly in he mac o scale.
The i s s ep in o mula ing he delamina ion is o p epa e a oun-
da ion o calcula e in e acial s esses. Since ee edge e ec s a e no
accoun ed o in he homogenized medium, a s aigh o wa d a e aging
scheme is employed o ob ain s esses a he in e aces, as ollows:
𝝈𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑖=
𝝈𝑏𝑢𝑙𝑘
𝑖+𝝈𝑏𝑢𝑙𝑘
𝑖−1
2(3)
whe e 𝝈𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑖is he in e acial s ess enso a he 𝑖 h in e ace while
𝝈𝑏𝑢𝑙𝑘
𝑖is he bulk laye s ess enso a he 𝑖 h laye .
Composi e S uc u es 351 (2025) 118613
2
A. Tahe zadeh-Fa d e al.
Fig. 1. Vi ual bulk laye s and i ual in e aces h ough he homogenized medium [51].
Acco ding o Balzani e al. [55], delamina ion is d i en by a no mal
s ess componen ac ing pe pendicula o he delamina ion plane and
wo shea s ess componen s ac ing on he delamina ion plane. Consid-
e ing he coo dina e sys em p esen ed in Fig. 1, wo equi alen s esses
a e calcula ed o be ed in o he delamina ion damage model o mode
I and mode II loadings:
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑖 =⟨𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑧𝑧,𝑖 ⟩(4)
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑖 =√(𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑥𝑧,𝑖 )2+ (𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑦𝑧,𝑖 )2(5)
whe e 𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑖 and 𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑖 a e he in e acial equi alen no mal and
shea s esses a he 𝑖 h in e ace. ⟨∙⟩a e he Macaulay b acke s o
p e en damage e olu ion in no mal comp ession loads [51].
2.2. Fa igue delamina ion damage e olu ion
In o de o accumula e he a igue e ec in o he cu en delam-
ina ion o mula ion, a simila scheme o ha desc ibed in [49] is
adop ed he e. To his end, he ailu e indica o s 𝐹𝑛,𝑖 and 𝐹𝑠,𝑖 p e iously
de eloped a each in e ace 𝑖in [51] a e modi ied by a a igue educ ion
unc ion (𝑓𝑟𝑒𝑑 ) as below:
𝐹𝑛,𝑖 =
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑖
𝑓𝑟𝑒𝑑
𝑛,𝑖 (𝑁𝑐
𝑛,𝑖, 𝑅𝑛,𝑖, 𝑆𝑚𝑎𝑥
𝑛,𝑖 )−𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑡ℎ,𝑖 >0(6)
𝐹𝑠,𝑖 =
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑖
𝑓𝑟𝑒𝑑
𝑠,𝑖 (𝑁𝑐
𝑠,𝑖, 𝑅𝑠,𝑖, 𝑆𝑚𝑎𝑥
𝑠,𝑖 )−𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑡ℎ,𝑖 >0(7)
whe e subsc ip ions n and s e e o no mal mode I and shea mode
II loadings, espec i ely. 𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑡ℎ,𝑖 and 𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑡ℎ,𝑖 a e his o ical a iables
o he in e ace h eshold s ess in mode I and mode II, and hei
ini ial alue would conside o be no mal and shea in e ace s eng hs,
espec i ely. These alues will be upda ed a each s ep o he maximum
his o ical no mal (𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑛,𝑖 ) and shea (𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑠,𝑖 ) uni-axial s esses
ob ained p e iously. Once inequali ies (6) and (7) a e sa is ied, he
c ack is pe mi ed o g ow.
𝑓𝑟𝑒𝑑 (𝑁𝑐
𝑚,𝑖, 𝑅𝑚,𝑖, 𝑆𝑚𝑎𝑥
𝑚,𝑖 )is in oduced o he ailu e indica o s o ake
he cyclic load e ec s in o accoun by ampli ying he s ess s a e. I s
alue anges om 1, indica ing no cyclic load e ec , down o 0 a
he asymp o e. Acco ding o he a igue c ack p opaga ion h ough he
in e ace, which is shown in Fig. 2, h ee di e en s ages could be
conside ed a each in eg a ion poin , whils each s age equi es speci ic
o mula ion o 𝑓𝑟𝑒𝑑 , as desc ibed in he ollowing.
2.2.1. Sub-c i ical a igue s age
Once a igue c ack ini ia es and s a s p opaga ing, he e a e some
a eas in he a - ield whe e he maximum cyclic s ess a he in e ace
(𝑆𝑚𝑎𝑥) is below he a igue limi , 𝑆𝑡ℎ, a a speci ic s ess a io, 𝑅=
Fig. 2. Di e en s ages expe ienced by a band o elemen s in (a) Sub-c i ical (blue),
(b) C i ical (g ay), and (c) O e -c i ical ( ed) egions as a igue c ack p opaga es in he
di ec ion illus a ed by a ows.
𝑆𝑚𝑖𝑛∕𝑆𝑚𝑎𝑥, ma ching he blue a ea in Fig. 2(a). An elemen loca ed in
his a ea (black dashed s ip) should no expe ience any cyclic load
e ec s, and hence, he a igue ela ed quan i ies will be main ained in
hei ini ial alues, as below:
𝑓𝑟𝑒𝑑
𝑚,𝑖 = 1 (8)
𝑁𝑐
𝑚,𝑖 = 0 (9)
whe e 𝑁𝑐
𝑚,𝑖 is he numbe o cycles a he 𝑖 h in e ace and 𝑚s ands
o 𝑛o 𝑠in no mal and shea loadings, espec i ely. As he c ack
Composi e S uc u es 351 (2025) 118613
3
A. Tahe zadeh-Fa d e al.
p opaga es, he black dashed a ea becomes close o he c ack ip and
he maximum cyclic s ess le els will inc ease, which necessi a es a
di e en desc ip ion o he 𝑓𝑟𝑒𝑑 .
2.2.2. C i ical a igue s age
When a speci ic elemen expe iences s esses g ea e han he a-
igue limi and below he s a ic s eng h, he cyclic load e ec s should
be conside ed h ough 𝑓𝑟𝑒𝑑 . This si ua ion is obse ed by he elemen s
wi hin he shaded a ea in he g ay egion in Fig. 2(b), whe e a ia ion
o 𝑓𝑟𝑒𝑑 could igge wo kinds o non-linea i y. The i s one is he
g adual ampli ying o he in e acial s esses (o li e ally, he g adual
dec ease o he s eng h) as a esul o he 𝑓𝑟𝑒𝑑 e olu ion when 𝑁𝑐is
g owing. The second one, which is a consequence o he i s one, is
he apid p opaga ion o he c ack once 𝑁𝑐 eaches he c i ical numbe
o cycles (𝑁𝑓) and inequali ies (6) and (7) a e me .
The de ini ion o he 𝑓𝑟𝑒𝑑 unc ion can be done based on ei he G-
N (ene gy elease a e s. numbe o cycles) o S-N (In e acial s ess
s. numbe o cycles) diag ams. Indeed, he model in his s udy has
o igina ed om he high-cycle a igue (HCF) o mula ion p esen ed by
Olle e al. [47], which has been de eloped o me als based on S-
N cu es. Success ul implemen a ion and sa is ac o y esul s o his
o mula ion in HCF egime [49,56] igge ed he idea o applying
he same app oach in composi e ma e ials [57]. The e o e, al hough
calib a ing he model based on G-N cu es seems o be mo e applicable
as o mo e a ailabili y o hese cu es o composi e ma e ials, p esen
o mula ion has been es ablished based on he s ess cu es. Suppo ing
his app oach, he e a e ecen ly-published pape s whe e S-N cu es
ha e been employed, and a calib a ion p ocess has been conduc ed
acco dingly o model he delamina ion p opaga ion in lamina ed com-
posi es [58,59]. This may indica e he po en ial o he S-N cu es in his
con ex . Mo eo e , acco ding o [42], some in e -connec ions could be
es ablished be ween he common Pa is law (da/dN s. G) and he S-N
cu e in composi es. This would be an asse o he cu en app oach in
his s udy, since he e is he po en ial o eed he Pa is law o he model
and make calib a ions acco dingly. This issue has no been add essed
in cu en s udy, and would be a ma e o u u e in es iga ions.
De ini ion o 𝑓𝑟𝑒𝑑 o he in e ace is conduc ed h ough a i ing
p ocedu e wi h espec o he common S-N su aces o he ype [49]:
𝑆𝑚,𝑖(𝑅𝑚,𝑖, 𝑁𝑐
𝑚,𝑖) =𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖) + (𝑆𝑢
𝑚,𝑖 −𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖))⋅exp{−𝛼𝑡
𝑚,𝑖(𝑅𝑚,𝑖)
⋅(log 𝑁𝑐
𝑚,𝑖)𝛽𝑓
𝑚,𝑖 }(10)
𝑅𝑚,𝑖 and 𝑆𝑢
𝑚,𝑖 being he s ess a io and s a ic s eng h o mode 𝑚a
he 𝑖 h in e ace, espec i ely. 𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖)is he a igue limi a he 𝑖 h
in e ace:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
i |𝑅|≤1⇒𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖) = 𝑆𝑒
𝑚,𝑖 +(𝑆𝑢
𝑚,𝑖 −𝑆𝑒
𝑚,𝑖)⋅(1 + 𝑅𝑚,𝑖
2)𝑆𝑅1
𝑚,𝑖
i |𝑅|>1⇒𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖) = 𝑆𝑒
𝑚,𝑖 +(𝑆𝑢
𝑚,𝑖 −𝑆𝑒
𝑚,𝑖)⋅(1 + 𝑅𝑚,𝑖
2𝑅𝑚,𝑖 )𝑆𝑅2
𝑚,𝑖
(11)
whe e 𝑆𝑒
𝑚,𝑖 is he a igue limi a 𝑅𝑚,𝑖 = −1.𝛼𝑡
𝑚,𝑖(𝑅𝑚,𝑖)is a pa ame e
which depends on he s a e o loading as desc ibed below:
⎧
⎪
⎨
⎪
⎩
i |𝑅|≤1⇒𝛼𝑡
𝑚,𝑖(𝑅𝑚,𝑖) = 𝛼𝑓
𝑚,𝑖 +(1 + 𝑅𝑚,𝑖
2)⋅𝐴𝑈𝑋𝑅1
𝑚,𝑖
i |𝑅|>1⇒𝛼𝑡
𝑚,𝑖(𝑅𝑚,𝑖) = 𝛼𝑓
𝑚,𝑖 −(1 + 𝑅𝑚,𝑖
2𝑅𝑚,𝑖 )⋅𝐴𝑈𝑋𝑅2
𝑚,𝑖
(12)
𝑆𝑅1
𝑚,𝑖 ,𝑆𝑅2
𝑚,𝑖 ,𝛼𝑓
𝑚,𝑖,𝛽𝑓
𝑚,𝑖,𝐴𝑈𝑋𝑅1
𝑚,𝑖 and 𝐴𝑈𝑋𝑅2
𝑚,𝑖 a e ma e ial p ope ies and
could be calib a ed a each in e ace 𝑖 o each mode o loading 𝑚. The
mo e Wöhle diag ams in di e en s ess a ios a e conside ed in he
calib a ion, he mo e accu a e desc ip ion o he S-N su ace will be
ob ained in a wide spec um o s ess a ios.
Once S-N su ace has been de ined and o mula ed, an exp ession
could be p oposed o a igue educ ion unc ion in he o m
𝑓𝑟𝑒𝑑
𝑚,𝑖 (𝑁𝑐
𝑚,𝑖, 𝑅𝑚,𝑖, 𝑆𝑚𝑎𝑥
𝑚,𝑖 ) = exp ⎡⎢⎢⎢⎢⎣
ln (𝑆𝑚𝑎𝑥
𝑚,𝑖 ∕𝑆𝑚,𝑢,𝑖)
(log 𝑁𝑓
𝑚,𝑖)(𝛽𝑓
𝑚,𝑖)2(log 𝑁𝑐
𝑚,𝑖)(𝛽𝑓
𝑚,𝑖)2⎤⎥⎥⎥⎥⎦
.(13)
Acco ding o Fig. 3, when a cyclic load a a speci ic 𝑆𝑚𝑎𝑥
𝑚,𝑖 is applied
wi hin he c i ical a igue s age, a unique 𝑓𝑟𝑒𝑑
𝑚,𝑖 unc ion could be ound
which in e sec s he no malized S-N cu e a 𝑁𝑓
𝑚,𝑖.𝑁𝑓
𝑚,𝑖 is a c i ical
quan i y which will be used in he nex sessions o he jumping
s a egy. I s alue could be ob ained by se ing 𝑁𝑐
𝑚,𝑖 =𝑁𝑓
𝑚,𝑖 in Eq. (13),
hen equa ing he ou come o he esul om Eq. (10):
𝑁𝑓
𝑚,𝑖(𝑅𝑚,𝑖, 𝑆𝑚𝑎𝑥
𝑚,𝑖 ) = 10
⎡⎢⎢⎢⎢⎢⎢⎣
⎡⎢⎢⎢⎣
−1
𝛼𝑡
𝑚,𝑖(𝑅𝑚,𝑖)⋅ln⎛⎜⎜⎜⎝
𝑆𝑚𝑎𝑥
𝑚,𝑖 −𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖)
𝑆𝑢
𝑚,𝑖 −𝑆𝑡ℎ
𝑚,𝑖(𝑅𝑚,𝑖)⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦
1
𝛽𝑓
𝑚,𝑖 ⎤⎥⎥⎥⎥⎥⎥⎦(14)
Gene ally, as long as 𝑁𝑐
𝑚,𝑖 < 𝑁𝑓
𝑚,𝑖, he non-linea i ies would be s eng h
de e io a ion and ac u e ene gy dissipa ion, acco ding o he 𝑓𝑟𝑒𝑑
𝑚,𝑖
e olu ion. Once 𝑁𝑐
𝑚,𝑖 ≥𝑁𝑓
𝑚,𝑖, in addi ion o he s eng h, he s i ness
will be deg ading as well, which co esponds o a apid ailu e p ocess
in he in eg a ion poin le el acco ding o inequali ies (6) and (7).
Ma lab so wa e [60] is employed in he i ing p ocess and o
ind p ope alues o he engaged pa ame e s. To his end, he mo e
expe imen al da a is p o ided, he wide he eliable egion will be in
e ms o s ess a io, 𝑅, and maximum s ess, 𝑆𝑚𝑎𝑥. In o he wo ds, he
model should be well-p epa ed p io o simula ions o be able o ea
as close as possible o he p o ided expe imen al Wöhle su ace. In his
way, he o mula ion could be applicable in di e en loading scena ios.
2.2.3. O e -c i ical a igue s age
O e -c i ical a igue s age e e s o he egion whe e p edic i e
s esses a e g ea e han he s a ic s eng h o he ma e ial. This si -
ua ion happens as he c ack p opaga es h ough he in e ace and he
c ack opening inc eases cycle-by-cycle, as is he case o he elemen s
loca ed wi hin he ed egion in Fig. 2(c). The i e e sible deg ada ion
p ocess o he in e ace has o be e lec ed in he 𝑓𝑟𝑒𝑑
𝑚,𝑖 quan i y. How-
e e , a di e en o mula ion han he one o he c i ical egion should
be employed. Consequen ly, a simila app oach o [61] is adop ed by
conside ing a a igue-induced deg ada ion as:
d𝑓𝑟𝑒𝑑
𝑚,𝑖
d𝑁𝑐
𝑚,𝑖
= − 1
𝛾𝑚,𝑖 (𝑁𝑐
𝑚,𝑖)−𝜂𝑚,𝑖 (15)
whe e 𝛾𝑚,𝑖 and 𝜂𝑚,𝑖 a e in e ace pa ame e s desc ibing he a e o deg a-
da ion, and hei alue could be ob ained acco ding o he expe imen al
campaign.
2.3. Fa igue delamina ion damage e ec
Once he ailu e c i e ia in Eqs. (6) and/o (7) a e sa is ied – ei he
by e olu ion o 𝑓𝑟𝑒𝑑
𝑚,𝑖 o inc easing he 𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑚,𝑖 – he damage a iable
o each mode 𝑚and in e ace 𝑖is calcula ed independen ly acco ding
o an exponen ial so ening law [51,62]:
𝑑𝑚,𝑖 = 1 −
𝜎0,𝑡ℎ
𝑚,𝑖
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑚,𝑖
exp [𝐴𝑚,𝑖(1 −
𝜎𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒
𝑚,𝑖
𝜎0,𝑡ℎ
𝑚,𝑖 )](16)
whe e
𝐴𝑚,𝑖 =1
𝐶𝑚,𝑖𝐺𝑚,𝑖
(𝜎0,𝑡ℎ
𝑚,𝑖 )2𝑙𝑐
−1
2
(17)
Composi e S uc u es 351 (2025) 118613
4
A. Tahe zadeh-Fa d e al.
Fig. 3. No malized Wöhle and a igue educ ion unc ion cu es in a gene al loading s a e.
𝜎0,𝑡ℎ
𝑚,𝑖 ,𝐶𝑚,𝑖,𝐺𝑚,𝑖, and 𝑙𝑐being ei he mode I o mode II in e acial
s eng h, modulus, ac u e oughness and cha ac e is ic leng h, e-
spec i ely. As explained in p e ious wo k [51], ans e se Young’s
and ans e se shea moduli o he bulk laye s a e conside ed as he
in e acial no mal and shea moduli, whils hal o he elemen leng h
is ega ded as he cha ac e is ic leng h.
Since he e is no dedica ed medium o he in e ace laye , de-
lamina ion damage e ec s should be p ope ly e lec ed in he i ual
bulk laye s so ha he beha io o he s uc u e a mac o-scale is well-
cap u ed. To his end, each bulk laye is a ec ed by wo maximum
no mal (𝑑𝑛) and shea (𝑑𝑛) damages as below [51]:
𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑧𝑧,𝑖 = (1 − max[𝑑𝑛,𝑖, 𝑑𝑛,𝑖−1])𝜎𝑏𝑢𝑙𝑘
𝑧𝑧,𝑖 (18)
𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑥𝑧,𝑖 = (1 − max[𝑑𝑠,𝑖, 𝑑𝑠,𝑖−1])(1 − max[𝑑𝑛,𝑖, 𝑑𝑛,𝑖−1])𝜎𝑏𝑢𝑙𝑘
𝑥𝑧,𝑖 (19)
𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑦𝑧,𝑖 = (1 − max[𝑑𝑠,𝑖, 𝑑𝑠,𝑖−1])(1 − max[𝑑𝑛,𝑖, 𝑑𝑛,𝑖−1])𝜎𝑏𝑢𝑙𝑘
𝑦𝑧,𝑖 (20)
whe e (𝜎𝑏𝑢𝑙𝑘
𝑧𝑧,𝑖 , 𝜎𝑏𝑢𝑙𝑘
𝑥𝑧,𝑖 , 𝜎𝑏𝑢𝑙𝑘
𝑦𝑧,𝑖 )a e p edic i e s esses ac ing on he delamina-
ion plane. They would be conside ed as he in eg a ed s esses (𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑧𝑧,𝑖 ,
𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑥𝑧,𝑖 and 𝜎𝑏𝑢𝑙𝑘, 𝑑
𝑦𝑧,𝑖 ) once a ec ed by he delamina ion damage. All
o he s ess componen s emain una ec ed in e ms o he delamina ion
phenomenon [55].
To illus a e he pe o mance o he p esen ed o mula ion in di -
e en scena ios, wo ypes o loading condi ions a e schema ically
p esen ed in Fig. 4. In Fig. 4(a), a cyclic load is applied wi hin he
c i ical a igue s age ollowed by a mono onic loading pa . The model
is expec ed o ac i a e he i s non-linea i y as he esul o he 𝑓𝑟𝑒𝑑
𝑚,𝑖
e olu ion and dec ease he in e ace s eng h acco dingly. Howe e , in
Fig. 4(b), a mono onic load is applied ollowed by a cyclic load wi hin
he o e -c i ical a igue egion. As i is ob ious, he esponse ollows
he same pa h as in he mono onic loading egime in he i s cycle,
whils a p og essi e deg ada ion in bo h s eng h and s i ness happens
once he load is swi ched o a cyclic one.
2.4. Ad ance-in- ime s a egy
Acco ding o Eqs. (13) and (15), he a igue delamina ion me hod
de eloped in he p esen s udy employs h ee independen quan i ies
o 𝑁𝑐
𝑚,𝑖,𝑅𝑚,𝑖 and 𝑆𝑚𝑎𝑥
𝑚,𝑖 in he c i ical egime, and one quan i y o 𝑁𝑐
𝑚,𝑖
in he o e -c i ical egion o iden i y he s a e o he sample wi hin he
cyclic loading. Al hough 𝑅𝑚,𝑖 and 𝑆𝑚𝑎𝑥
𝑚,𝑖 a e compu ed a each s ep and
a e comple ely dependen on he exe ed ex e nal loads, 𝑁𝑐
𝑚,𝑖 is a con-
inuous a iable in he ime domain and equi es p og essi e g ow h
wi hin he simula ion. This could be p oblema ic, speci ically in high-
cycle a igue si ua ions due o he high compu a ional cos associa ed.
The e o e, a p ocedu e should be conside ed o skip a numbe o cycles
while compensa ing o i s e ec s wi hin he o mula ion o educe he
compu a ional cos s.
To his end, an ad ance-in- ime (AIT) s a egy is conside ed once
s able condi ions a e ob ained in e ms o he cyclic loading exe ed.
The me hod is based on he jump s a egy p oposed in [49,50,57,63],
howe e i is enhanced o accoun o wo di e en loading mode con-
di ions. Indeed, p e ious e sion o he AIT me hod has been p esen ed
o iso opic ma e ials whe e he dis inc ion be ween mode I o mode
II loadings was no necessa y. In he case o he nume ical amewo k
p oposed in his s udy, ha is no longe he case, a me hod is de eloped
o iden i y o mos c i ical mode o loading, and he jump is se up
acco dingly.
Le us conside ha mode I and mode II loadings a e ac i a ed in
he c i ical a igue egion, as shown schema ically in Fig. 5. To main ain
gene ali y, loading modes a e e e ed o as modes 𝑘and 𝑙. Depending
on he ime when ei he mode has eached he c i ical a igue egion,
he numbe o cycles be o e jump (𝑁𝑐
𝑖) could be di e en , since in he
sub-c i ical a igue s age cycles a e no coun ed acco ding o Eq. (9).
On he o he hand, he equi ed numbe o cycles o each he damage
ini ia ion condi ions could be ob ained using Eq. (14) o each loading
mode, independen ly. I would be easonable o employ he mos
c i ical loading mode in e ms o he dis ance o he damage ini ia ion
condi ions, so ha he o he mode emains in he allowable S-N a ea
a e he jump. As a esul , he loading mode wi h he minimum amoun
o 𝑁𝑓−𝑁𝑐will be ed in o he AIT s a egy.
O he combina ions o loading modes a e s ill possible, whe e he
selec ion o he mode o he AIT s a egy is qui e s aigh o wa d.
Wi hin all o he possibili ies, loading mode in he o e -c i ical a igue
s age would be selec ed o AIT s a egy compa ed o he one in he
c i ical a igue s age, while c i ical a igue s age will be p io i ized o e
he sub-c i ical a igue egion.
The jump is expec ed o occu once s able condi ions a e eached
in he s ess s a e. While he c ack is p opaga ing, he maximum cyclic
s ess is con inuously e ol ing a each in eg a ion poin . In his way,
s abili y condi ion means a pla eau in he damage and s ess e olu ion
which could be moni o ed employing wo indica o s:
𝜑𝑚,𝑖 =|||||
𝑆𝑚𝑎𝑥
𝑚,𝑖,𝑗+1 −𝑆𝑚𝑎𝑥
𝑚,𝑖,𝑗
𝑆𝑚𝑎𝑥
𝑚,𝑖,𝑗+1 |||||< 𝑡𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 (21)
𝜓𝑚,𝑖 =|||||
𝑅𝑚,𝑖,𝑗+1 −𝑅𝑚,𝑖,𝑗
𝑅𝑚,𝑖,𝑗+1 |||||< 𝑡𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 (22)
Composi e S uc u es 351 (2025) 118613
5
A. Tahe zadeh-Fa d e al.
Fig. 4. Model pe o mance in wo di e en loading scena ios: (a) cyclic load ollowed by a mono onic ension wi hin he c i ical a ea and (b) mono onic ension ollowed by a
cyclic loading wi hin he o e -c i ical egion.
Fig. 5. C i icali y le el o wo di e en loading modes in e ms o he numbe o emaining cycles o ailu e.
𝜑𝑚,𝑖 and 𝜓𝑚,𝑖 being maximum s ess and s ess a io s abiliza ion no ms,
while 𝑆𝑚𝑎𝑥
𝑚,𝑖,𝑗 and 𝑅𝑚,𝑖,𝑗 being he maximum s ess and he s ess a io
o mode 𝑚, in e ace 𝑖and ime s ep 𝑗, espec i ely. Once s abili y
condi ions in Eqs. (21) and (22) a e sa is ied, he p ocess ime as well
as he numbe o cycles a each in eg a ion poin will be upda ed, and
he 𝑓𝑟𝑒𝑑
𝑚,𝑖 is calcula ed based on he newly-ob ained numbe o cycles.
3. Nume ical examples and esul s
The po en ial o he p oposed model is s udied in his sec ion
h ough di e en loading scena ios in bo h damage ini ia ion and dam-
age p opaga ion egimes. Unidi ec ional ca bon ibe /epoxy p ep eg
(IM7/8552) p ope ies ha e been employed, and he alida ions a e
conduc ed on his composi e in all a igue c ack e olu ion assessmen s.
I should be no ed ha , in he cu en s udy, he e ec o ibe
b idging has no been conside ed in he simula ions. Indeed, he p e-
sen ed a igue model ac s in an in eg a ion-poin -wise app oach, whe e
all he calcula ions a e conduc ed a each Gauss-poin wi hin he ini e
elemen model. This implies ha he basis o he a igue model in
cu en assessmen is a local amewo k. Howe e , aking in o accoun
he b idging e ec necessi a es addi ional in o ma ion ega ding he
ex en o delamina ion damage c ack, since he R-cu e phenomenon
depends upon he leng h o he c ack p opaga ed. To his end, non-local
me hods should be aken in o accoun o e icien ly conside he ibe
b idging e ec s. Igno ance o he R-cu e e ec would no impose any
p oblems o he damage ini ia ion s age, howe e , damage p opaga-
ion would be a ec ed by his assump ion. I will be demons a ed ha
in he absence o he ibe b idging e ec s, he esul s s ill all wi hin
an accep able ange o he expe imen al da a.
The p esen ed a igue o mula ion is implemen ed h ough he
open-sou ce code K a os Mul i-physics [64,65]. Geome y and mesh
gene a ion has been conduc ed u ilizing p e and pos p ocessing so -
wa e GiD [66].
3.1. Cons i u i e law pa ame e s calib a ion
As i has been men ioned in Sec ion 2.2.2, a Ma lab i ing algo i hm
is employed o calib a e he a igue pa ame e s o he model. To
his end, Wöhle diag ams o he in e ace in each loading mode is
equi ed o e ec i ely cha ac e ize he a igue delamina ion beha io .
T ans e se ension [18,67] and shea [22,68] a igue es s ha e been
conduc ed on he IM7/8552 ca bon/epoxy composi e a di e en s ess
a ios (𝑅𝑚,𝑖) and s ess le els (𝑆𝑚𝑎𝑥
𝑚,𝑖 ) un il he c ack is ini ia ed wi hin
he in e ace. The esul ed expe imen al S-N diag ams a e u ilized o
i a cu e in he o m o Eq. (10) o each mode I and mode II. The
expe imen al Wöhle cu es along wi h he calib a ed model esponses
a e p esen ed in Fig. 6.
Tables 1 and 2p esen he ma e ial p ope ies as well as he
ob ained alues o he calib a ed in e acial pa ame e s, espec i ely.
These p ope ies will be used wi hin he simula ions in he ollowing
sec ions o bo h a igue damage ini ia ion and p opaga ion egimes.
Since all he simula ions a e conduc ed a 𝑅𝑚,𝑖 <1, he e is no need o
calib a e 𝑆𝑅2
𝑚,𝑖 and 𝐴𝑈𝑋𝑅2
𝑚,𝑖 pa ame e s.
Composi e S uc u es 351 (2025) 118613
6
A. Tahe zadeh-Fa d e al.
Fig. 6. Fa igue ma e ial model calib a ion in (a) ans e se no mal (mode I) and (b) shea (mode II) con igu a ions compa ed o he expe imen al da a [18,68].
Table 1
Mechanical p ope ies o he IM7/8552 ca bon/epoxy p ep eg [22].
Mechanical p ope y Value
𝐸1[GPa] 161
𝐸2=𝐸3[GPa] 11.4
𝐺12 =𝐺13 [GPa] 5.17
𝐺23 [GPa] 3.98
𝜈12 =𝜈13 0.32
𝜈23 0.43
3.2. Fa igue delamina ion damage ini ia ion assessmen
In o de o in es iga e he model capabili y o ini ia e a igue c ack
in an undamaged medium, wo s anda d es s ha e been employed in
each loading mode, as illus a ed in Fig. 7. Th ee-poin bending es is
u ilized o ini ia e he a igue c ack in mode I loading. To his end, a
cen al displacemen is cycled wi h a equency o 1 Hz while he wo
ends o he can ile e beam a e se o be ixed h ough he whole wid h
(𝑤= 6.35 mm), as shown in Fig. 7(a).
Cyclic displacemen domains a e chosen in a way ha hey p oduce
maximum no mal s esses o 90,93,97,101, and 105.5 MPa wi hin he
cen al egion. Ca bon ibe s a e aligned in he loading di ec ion and
Table 2
Calib a ed in e acial a igue pa ame e s o modes I and II [68].
In e ace p ope y No mal load Shea load
(Mode I) (Mode II)
Ene gy elease a e (𝐺) [J/m2] 212 600
In e acial s eng h (𝜎0
𝑡ℎ) [MPa] 111 85
In e acial modulus (𝐶) [GPa] 11.4 5.17
𝑆𝑒[MPa] 87.7 39.1
𝑆𝑅1[MPa] 48.88 55.81
𝛼𝑓2.14e−3 1.69e−4
𝛽𝑓4.49 6.71
𝐴𝑈𝑋𝑅1−2.80e−3−2.58e−4
𝛾25 30
𝜂0.72 0.72
in 90◦angle wi h espec o he longi udinal di ec ion. Hexahed on
elemen s wi h he size o 0.1 mm a e employed in he cen al egion,
whe e he c ack is expec ed o ini ia e, and coa se meshes a e op ed o
he a - ield a ea. The cycle is de ined o induce 0.1 s ess a io s a es
acco ding o he expe imen al p ocedu e, and he esul s a e p esen ed
in Fig. 8 compa ed agains he expe imen al da a. Wi hin he expe i-
men al es s, specimens in bo h s anda d es s ail ca as ophically once
damage has ini ia ed, which allows simple de ec ion o he damage
Composi e S uc u es 351 (2025) 118613
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A. Tahe zadeh-Fa d e al.
Fig. 7. Schema ic iew o (a) h ee-poin bending and (b) double-no ched shea es s unde cyclic load (dimensions in millime e s).
Fig. 8. Ini ia ion o he a igue c ack in he h ee-poin bending con igu a ion unde mode I loading compa ed wi h he expe imen al da a a 𝑅= 0.1[18,23].
onse . In he nume ical amewo k, howe e , he necessa y numbe o
cycles o ini ia e he delamina ion damage is calcula ed acco ding o
Eq. (14). This alue has been conside ed as he c ack ini ia ion poin
wi hin he simula ions.
As i can be obse ed, he model cap u es he c ack ini ia ion
cycle ai ly well a each s ess le el, and he nume ical ou pu lies
wi hin he expe imen al sca e egion. In he simula ions, a igue c ack
p opaga es ca as ophically once i has been ini ia ed, which aligns
wi h he expe imen al indings [18,23]. In Fig. 9, he c ack ini ia ion
s ep and i s comple e p opaga ion is p esen ed a he s ess le el o
97 MPa. Th ough he simula ion, a deac i a ion algo i hm is employed
o e ase ully damaged elemen s so ha he model con e ges mo e
easily.
Valida ing he pe o mance o he model in mode I loading, i would
be aluable o check i s applicabili y in shea mode as well. The e o e,
expe imen al double-no ched shea es esul s a e u ilized o unning
pu e mode II simula ions [68]. The con igu a ion o he es is shown in
Fig. 7(b) whe e ca bon ibe s a e aligned in he longi udinal di ec ion.
A cyclic axial displacemen is applied o one end h ough he whole
wid h (𝑤= 12.7 mm) so ha i p oduces shea s esses a he cen al
pa . F om he nume ical poin o iew, conside ing solely he cen al
pa would be enough o un he simula ions. Two s ess a ios o 0.1
and 0.3 a e used along wi h a cyclic load o 1 Hz. The elemen size and
ype a e he same as in he h ee-poin bending es . Fig. 10 summa izes
he nume ical and expe imen al esul s.
The model ou pu o mode II loading shows a good ag eemen wi h
he expe imen al esul s. This indica es ha he p oposed o mula ion
would be able o deal wi h he c ack ini ia ion phenomenon in bo h
mode I and mode II loading condi ions in an accu a e manne . The
nex s age is assessing he pe o mance o he model in he p opaga ion
egime, which will be discussed in he ollowing sec ion.
Composi e S uc u es 351 (2025) 118613
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A. Tahe zadeh-Fa d e al.
Fig. 9. Th ee-poin bending a igue simula ions in he s ess le el o 97 MPa in (a) ini ia ion s age a 𝑁𝑐= 218 964 and (b) comple e ailu e a 𝑁𝑐= 218 986.
Fig. 10. Ini ia ion o he a igue c ack in double-no ch shea con igu a ion in (a) 𝑅= 0.1and (b) 𝑅= 0.3compa ed wi h he expe imen al da a [68].
Composi e S uc u es 351 (2025) 118613
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