L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
65
Focused on C ack Tip Fields
Mul i-pa ame e ac u e mechanics:
c ack pa h in a mixed-mode specimen
L. Malíko á, H. Šimono á, B. Kucha czyko á
B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ins i u e o S uc u al Mechanics, Ve eří 331/95, 602 00 B no,
Czech Republic
[email p o ec ed], h p://o cid.o g/0000-0001-5868-5717
[email p o ec ed], h p://o cid.o g/0000-0003-1537-6388
[email p o ec ed], h p://o cid.o g/0000-0002-7123-5099
P. Mia ka
Academy o Sciences o he Czech Republic, Ins i u e o Physics o Ma e ials, . . i., Žižko a 22, 616 62 B no, Czech Republic
and B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ins i u e o S uc u al Mechanics, Ve eří 331/95, 602 00 B no,
Czech Republic
[email p o ec ed], h p://o cid.o g/0000-0002-4953-4324
ABSTRACT. A mixed-mode geome y has been chosen o in es iga e a c ack
p opaga ion using he mul i-pa ame e ac u e mechanics concep . The so-
called Williams’ se ies expansion is used o he c ack- ip s ess ield
app oxima ion. I has been shown ha applica ion o he gene alized ac u e
mechanics concep can be c ucial o ma e ials wi h speci ic ac u e
beha iou , such as elas ic-plas ic o quasi-b i le one, when ac u e occu s no
only in he e y icini y o he c ack ip, bu also in a mo e dis an su ounding.
Then, conside ing he highe -o de e ms o he Williams’ expansion in
ac u e c i e ia (desc ibing he c ack s abili y and/o c ack p opaga ion
di ec ion) can b ing mo e p ecise esul s. The coe icien s o he Williams’
expansion mus be calcula ed nume ically ( o ins ance by means o he o e -
de e minis ic me hod in his wo k) o each c acked con igu a ion, which is
e y ime-consuming, and he analysis is e y ex ensi e e en o a ew basic
c acked specimen con igu a ions. On he o he hand, a sui able choice o he
geome ical con igu a ion o he c acked disc enables pe o ming expe imen s
only on he specimens ha could p o e he heo y abou he impo ance o
using he highe -o de e ms.
KEYWORDS. C ack- ip ield; Mul i-pa ame e ac u e mechanics; Fini e
elemen analysis; Mixed-mode; Williams’ expansion.
Ci a ion: Malíko á, L., Šimono á, H.,
Kucha czyko á, B., Mia ka, P., Mul i-
pa ame e ac u e mechanics: c ack pa h in a
mixed-mode specimen, F a u a ed In eg i à
S u u ale, 49 (2019) 65-73.
Recei ed: 21.04.2019
Accep ed: 25.04.2019
Published: 01.07.2019
Copy igh : © 2019 This is an open access
a icle unde he e ms o he CC-BY 4.0,
which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided he
o iginal au ho and sou ce a e c edi ed.
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
66
INTRODUCTION
l hough ac u e mechanics has been s udied o app oxima ely one cen u y, p ope assessmen o c ack beha iou
is s ill unde in es iga ion o many esea che s. One o he cu en asks is mo e p ecise desc ip ion o he c ack-
ip s ess/displacemen ield ha is undamen al in o ma ion necessa y o addi ional mo e complex ac u e
analyses. In he las yea s, he mul i-pa ame e concep seems o be help ul, when some o he ac u e mechanics issues a e
sol ed [1-3]. Con a y o he mos common single-pa ame e linea elas ic ac u e mechanics app oach wo king mainly
only wi h he s ess in ensi y ac o ( i s e m o he Williams’ expansion [4]), he gene alized ac u e mechanics akes in o
accoun also he second (non-singula ) e m o he WE, he so-called T-s ess, as well as he e ms o highe o de s.
Signi icance o he unca ed o m o he Williams’ se ies has been in es iga ed by he au ho s’ collec i e o ins ance in [5-
8]. Mo eo e , gene alized ac u e c i e ia a e aking in o accoun mo e e ms o he Williams’ expansion s a o appea in
scien i ic wo ks. Fo example in [9, 10] he gene alized maximum angen ial s ess c i e ion is in oduced, which conside s
in he ac u e c i e ion no only he s ess in ensi y ac o bu also he T-s ess.
In his wo k, a mixed-mode c acked con igu a ion was chosen and he c ack p opaga ion angle was in es iga ed nume ically
by means o ini e elemen s. The s udy p esen ed ep esen s a pilo analysis ha should help o selec a sui able con igu a ion
o he semi-ci cula disc, pa icula ly he ini ial c ack leng h and he c ack inclina ion angle. This specimen will be conside ed
in planned bending ac u e expe imen s on specimens made o ine-g ained composi es based on he alkali-ac i a ed ma ix
(AAM). Se e al con igu a ions a e sea ched, whe e i can be expec ed ha he desc ip ion o he c ack- ip s ess ield ia
mul i-pa ame e ac u e mechanics b ings much be e esul s o he c ack pa h han he classical one-pa ame e ac u e
mechanics concep .
Cu en ly, iden i ica ion o he basic ma e ial cha ac e is ics o he specially selec ed ma e ial a e aking place. Alkali ac i a ed
aluminosilica e ma e ials ep esen an al e na i e o o dina y Po land-cemen -based ma e ials, educing he en i onmen al
impac o building indus y and o e ing new imp o ed p ope ies. These binde s a e made h ough he mixing o some
non- o poo ly-c ys alline aluminosilica e-based ma e ial, such as blas u nace slag o ly ash, wi h an alkaline ac i a o
(hyd oxides, ca bona es o he mos p e e ably silica es) and wa e [11, 12]. Type and dosage o he ac i a o as well as he
way o cu ing ha e a signi ican e ec on he hyd a ion cou se and inal mechanical p ope ies [13]. The choice o he
composi e wi h AAM o he esea ch co esponds o he lack o in o ma ion abou i s ac u e beha iou .
THEORETICAL BACKGROUND
Mul i-Pa ame e F ac u e Mechanics (MPFM)
he mul i-pa ame e ac u e mechanics concep assumes ha he c ack- ip s ess/displacemen ield is desc ibed by
means o he Williams’ expansion [4] ha was de i ed o a homogeneous elas ic iso opic c acked body wi h an
a bi a y emo e loading. The in ini e powe se ies o he c ack- ip s ess ield can be w i en in he o m:
𝜎 ∑𝐴


 𝑟
𝑓󰇛𝑛,𝜃󰇜∑𝐵


 𝑟
𝑔󰇛𝑚,𝜃󰇜 , whe e 𝑖,𝑗 ∈󰇝𝑥,𝑦󰇞 ; (1)
simila ly, he in ini e powe se ies o he c ack- ip displacemen ield can be w i en in he o m:
u∑A

 
 󰇛n,θ,E,ν󰇜∑B

 
g󰇛m,θ,E,ν󰇜 , whe e i ∈ 󰇝x,y󰇞 . (2)
These a e he wo basic equa ions and hei unca ed o m is used in he esea ch in oduced in his pape . Rega ding he
symbols used, i holds ollowing:

ij and ui ep esen he s ess enso and displacemen ec o componen s, espec i ely; ( ,

 symbolize he pola coo dina es (conside ing he cen e o he coo dina e sys em a he c ack ip and he c ack aces lying
on he nega i e x-axis); ij ,gij and i , gi s and o he known unc ions co esponding o loading mode I ( ) and loading mode
II (g) and hei exp essions can be ound in classical ex books on ac u e mechanics; E and

ep esen Young’s modulus
and Poisson’s a io, espec i ely. The only unknown symbols a e he coe icien s An and Bm o he indi idual e ms o he
se ies – hei alues depend on he ela i e c ack leng h, load, geome y o gene ally, on he c acked specimen con igu a ion.
The e o e, i is necessa y o calcula e hese pa ame e s nume ically and he me hod used wi hin his wo k is desc ibed in he
ollowing ex .
A
T
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
67
O e -De e minis ic Me hod (ODM)
O e -de e minis ic me hod (ODM) was chosen as he p ocedu e o calcula ion o he coe icien s o he indi idual e ms
o he Williams’ expansion. These e ms a e usually neglec ed and only he i s (singula ) e m co esponding o he well-
known s ess in ensi y ac o is aken in o accoun . ODM enables de e mina ion o an a bi a y numbe o he coe icien s
o he se ies in oduced in Eqns. 1 and 2 when pa icula condi ions a e sa is ied.
The p inciple o he me hod consis s in di ec applica ion o Eqn. 2 o displacemen ec o componen s. When k nodes
a e selec ed a ound he c ack ip, a sys em o 2k algeb aic equa ions o he a iables An and Bm can be de ined, when
displacemen s u and as well as he nodes coo dina es and

a e known ( om a ini e elemen model o om an
expe imen ). In o de o ul il he basic idea o he me hod (sol ing o an o e -de e mined sys em o equa ions), i mus
hold ha 2k is la ge han N+M+2, whe e N and M a e he numbe s o he mode I and mode II Williams’ expansion e ms
in he i s unca ed o m, espec i ely. Mo e de ails, ecommenda ions and/o es ic ions ega ding he applica ion o he
me hod can be ound o ins ance in [14, 15]. No e ha he ODM was chosen wi h ega d o i s low equi emen s on he
so wa e and ma hema ical de ini ions/p ocedu es in con as o o he me hod de i ed in li e a u e, such as Hyb id C ack
Elemen (HCE) me hod, Bounda y Colloca ion Me hod (BCM) e c. [16-18].
Maximum Tangen ial S ess c i e ion (MTS)
When a c ack is loca ed a bi a ily o he loading di ec ion, he mixed-mode loading is spoken abou . In o de o desc ibe
he c ack beha iou , pa icula ly i s ajec o y h ough he specimen, se e al ac u e c i e ia ha e been de i ed, see o
ins ance [19]. In his wo k, he Maximum Tangen ial S ess (MTS) c i e ion and minimum S ain Ene gy Densi y (SED)
c i e ion we e chosen. Bo h a e common and o en used when he c ack pa h shall be in es iga ed. The MTS c i e ion is a
s ess-based c i e ion and is independen on he plane s ess o plane s ain condi ions. I s basic idea consis s in he
assump ion ha a c ack will p opaga e in he di ec ion whe e he angen ial s ess


eaches i s maximum [20]. W i en
ma hema ically:

 0 and 
0 . (3)
When he c i e ion is used in his esea ch, he angen ial s ess componen is exp essed ia Williams’ expansion simila ly o
Eqn. 1 and he maximum o ha unc ion is sea ched nume ically in Wol am Ma hema ica code [21].
Minimum S ain Ene gy Densi y c i e ion (SED)
Simila ly, he kink angle o he c ack can be ound by means o he minimum S ain Ene gy Densi y (SED) c i e ion [22][23],
bu he minimum o he unc ion S is sea ched. This condi ion can be again w i en by means o he de i a i es:

 0 and 
0 , whe e 𝑆
 󰇣
󰇛𝜎 𝜎
󰇜𝜎
𝜎 𝜎

󰇤 . (4)
The s ess enso componen s a e app oxima ed by means o he Williams’ expansion conside ing a ious numbe s o he
ini ial e ms and a p ocedu e o inding he minimum o he s ain ene gy densi y ac o S is p og ammed.
The basic di e ence be ween he mul i-pa ame e (gene alized) c i e ia and he classical (one-pa ame e ) ones is he
exis ence o a leng h pa ame e whe e he mul i-pa ame e c i e ion is applied. Because he e does no exis any
unambiguous ecommenda ion how o choose his adial dis ance (only some heo ies ha i should be ela ed o ma e ial
cha ac e is ics [24-26]), se e al a ious alues we e es ed wi hin his s udy.
NUMERICAL MODEL
he analysis on he c ack beha iou and impo ance o he MPFM has been pe o med on a semi-ci cula disc loaded
in bending (SCB), see e.g. [27]. One o he ad an ages o his kind o specimens is he ini ia ion o he ension c ack
e en unde he comp essi e loading, see Fig. 1. Also, he inclina ion o he c ack enables o es specimens unde
a ious mixed mode (I+II) condi ions.
The pa ame e s o he nume ical model we e as ollows: adius o he disc R = 50 mm, span be ween suppo s S = 80 mm,
he c ack leng h a ied be ween a = 10 ÷ 35 mm and he c ack inclina ion angle a ied be ween

= 10° ÷ 50°. Thus, hi y
a ious con igu a ions we e in es iga ed wi h mixed-mode le el be ween KI/KII = 1.5 ÷ 8.
T
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
68
In o de o ge basic ac u e mechanical p ope ies o he alkali-ac i a ed ma e ial unde s udy, se e al es s ha e been
pe o med on p isms wi h dimensions 40  40  160 mm. Young’s modulus o 35 GPa and Poisson’s a io o 0.23 we e
hen applied in he nume ical simula ions based on he esul s o 3-poin bending es and esonance me hod, espec i ely.
Figu e 1: Schema and geome y o he semi-ci cula disc loaded in bending used o he ac u e analysis in his esea ch.
a) b)
Figu e 2: Expe imen al campaign on 40 × 40 × 160 mm c acked p isms unde 3-poin -bending: a) expe imen al se up; b) de ail o he
c ack pa h h ough he specimen.
The nume ical model has been c ea ed as a wo-dimensional (2D), comme cial ini e elemen so wa e ANSYS [28] has
been used. The c ack de lec ion has been s udied unde he comp essi e o ceP = 1 kN conside ing plane s ain bounda y
condi ions. A quad ila e al elemen ype (PLANE183) wi h quad a ic basis unc ions ha e been used o modelling o he
semi-ci cula disc. The FE mesh a he c ack ip has been e ined in o de o ake in o accoun c ack ip singula i y. Fo
applica ion o he ODM, nodes a he adial dis ance o 4 mm ha e been selec ed and hei displacemen s used o e alua ion
o he coe icien s o highe -o de e ms. S ess componen s necessa y o applica ion o men ioned ac u e c i e ia ha e
been exp essed ia Williams’ expansion aking in o accoun up o 10 ini ial e ms o he powe se ies. The c i e ia ha e been
applied a c i ical dis ances C o 0.1, 0.5 and 1 mm. Resul s ob ained a e p esen ed and discussed in he ollowing sec ion.
RESULTS & DISCUSSION
n he ollowing igu es, he dependences o he c ack de lec ion angle ob ained o a ious c acked con igu a ions and
conside ing a ious pa ame e s a e p esen ed. Le ’s summa ize wha pa ame e s ha e been a ied:
 Ini ial ela i e c ack leng h,

= a/R;
 C ack inclina ion angle,

;
I
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
69
 C i ical dis ance, whe e he mul i-pa ame e ac u e c i e ia a e applied, c;
 Numbe o ini ial e ms o he Williams’ expansion conside ed o c ack- ip s ess ield app oxima ion.
In Figs. 3 o 6 only selec ed esul s a e in oduced, because he ex en o he analysis is oo la ge. In luence o he abo e-
men ioned pa ame e s on he c ack de lec ion angles ob ained has been in es iga ed in o de o ind he app op ia e
con igu a ions o subsequen expe imen al campaign ha should p o e he signi icance o conside ing he highe -o de
e ms when he c ack- ip s ess ield is econs uc ed.
Compa ison be ween mul i-pa ame e MTS and SED c i e ion and hei applica ion a a ious dis ances can be obse ed
o sho c acks wi h

= 0.2 in Fig. 3 and o longe c acks wi h

= 0.5 in Fig. 4, espec i ely.
a) MTS,

= a/R = 0.2, c = 0.1 mm b) MTS,

= a/R = 0.2, c = 1.0 mm
c) SED,

= a/R = 0.2, c = 0.1 mm d) SED,

= a/R = 0.2, c = 1.0 mm
Figu e 3: C ack de lec ion angles

ob ained on he SCB specimens o sho c acks (a/R = 0.2) ia mul i-pa ame e MTS and SED
c i e ion applied a c i ical dis ances o 0.1 mm and 1.0 mm o a ious c ack inclina ion angles (

= 10, 20, 30, 40 and 50°) and
conside ing a ious numbe s o ini ial e ms o Williams’s expansion (N, M = 1, 2, 3, 6 and 10).
Resul s p esen ed in Figs. 5 and 6 ep esen s compa ison o he c ack de lec ion angles ob ained ia mul i-pa ame e MTS
and SED c i e ion a a ious c i ical dis ances in dependence on he choice o he numbe o ini ial e ms o he Williams’
expansion o wo inclina ion angles (

= 10° and

= 50°) o sho c acks wi h

= 0.2 (Fig. 5) and o longe c acks wi h

= 0.5 (Fig. 6), espec i ely.

L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
70
a) MTS,

= a/R = 0.5, c = 0.1 mm b) MTS,

= a/R = 0.5, c = 1.0 mm
c) SED,

= a/R = 0.5, c = 0.1 mm d) SED,

= a/R = 0.5, c = 1.0 mm
Figu e 4: C ack de lec ion angles

ob ained on he SCB specimens o longe c acks (a/R = 0.5) ia mul i-pa ame e MTS and SED
c i e ion applied a c i ical dis ances o 0.1 mm and 1.0 mm o a ious c ack inclina ion angles (

= 10, 20, 30, 40 and 50°) and
conside ing a ious numbe s o ini ial e ms o Williams’s expansion (N, M = 1, 2, 3, 6 and 10).
Based on he g aphs p esen ed abo e, ollowing conclusions can be s a ed:
 E en a small c i ical dis ances om he c ack ip ( c = 0.1 mm) i has been p o ed ha highe -o de e ms o he
Williams’ expansion could imp o e he accu acy o he es ima ed c ack de lec ion angle.
 This e ec is s onge o longe c acks and when la ge dis ances om he c ack ip a e used o applica ion o he
gene alized ac u e c i e ia.
 The SED c i e ion is less sensi i e o he choice o he numbe o he highe -o de e ms han he MTS c i e ion.
 Gene alized ac u e c i e ia a e mo e impo an o c acks unde mixed mode I+II, han o cases when loading
mode I p e ails (i.e. in ou case o la ge

angles).
In connec ion wi h he expe imen al esea ch in ended by he au ho s, ollowing ecommenda ions can be concluded: The
expe imen al campaign should be pe o med (as much as possible) on SCB specimens wi h la ge c acks and la ge ini ial
de lec ion angles

in o de o be able o in es iga e he e ec o he highe -o de e ms conside ed in he gene alized
ac u e c i e ia on es ima ion o u he c ack p opaga ion angles.
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
71
a) MTS,

= a/R = 0.2,

= 10° b) MTS,

= a/R = 0.2,

= 50°
c) SED,

= a/R = 0.2,

= 10° d) SED,

= a/R = 0.2,

= 50°
Figu e 5: C ack de lec ion angles

ob ained on he SCB specimens o sho c acks (a/R = 0.2) ia mul i-pa ame e MTS and SED
c i e ion applied a c i ical dis ances o 0.1, 0.5 and 1.0 mm o c ack inclina ion angles 10° and 50° and conside ing up o 10 ini ial
e ms o Williams’s expansion.
I he sugges ed app oach is no able o desc ibe he c ack pa h in he ine-g ained composi e based on he alkali-ac i a ed
slag as a quasi-b i le ma e ial, o he ac u e c i e ia ( ha a e mo e sui able o non-b i le ac u e and mixed-mode
condi ions) should be applied, such as maximum a e age angen ial s ess (MATS) c i e ion [29] o o he s. Ano he
possibili y is o use so-called Equi alen Ma e ial Concep (EMC) sugges ed ecen ly by To abi [30] and conside a i ual
b i le ma e ial exhibi ing linea elas ic beha iou ins ead o he eal quasi-b i le one; de ails on his concep can be ound
in he e e ed li e a u e.
CONCLUSIONS
ape deals wi h a pa ame ic s udy on a mixed-mode I+II specimen in o de o ind a sui able geome ical
con igu a ion o a SCB disc o p o ing impo ance o conside ing he highe -o de e ms o he Williams’ expansion
du ing app oxima ion o c ack- ip s ess ield. The analysis shall be pe o med on a no el ma e ial: ine-g ained
composi e based on he alkali-ac i a ed slag (an al e na i e o o dina y po land-cemen -based ma e ials, educing he
en i onmen al impac o building indus y). The c ack de lec ion angle is s udied ia mul i-pa ame e MTS and SED c i e ia
o a ious geome ical con igu a ions and se e al ecommenda ions a e concluded based on he esul s ob ained. The
la ges di e ences be ween he classical single-pa ame e ac u e mechanics and he mul i-pa ame e concep a e obse ed
in SCB specimens wi h la ge c acks and la ge ini ial de lec ion angles

. Thus, he eal ine-g ained composi e specimens
will be p oduced wi h hese pa ame e s.
P
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
72
a) MTS,

= a/R = 0.5,

= 10° b) MTS,

= a/R = 0.5,

= 50°
c) SED,

= a/R = 0.5,

= 10° d) SED,

= a/R = 0.5,

= 50°
Figu e 6: C ack de lec ion angles

ob ained on he SCB specimens o la ge c acks (a/R = 0.5) ia mul i-pa ame e MTS and SED
c i e ion applied a c i ical dis ances o 0.1, 0.5 and 1.0 mm o c ack inclina ion angles 10° and 50° and conside ing up o 10 ini ial
e ms o Williams’s expansion.
ACKNOWLEDGEMENT
inancial suppo om he Czech Science Founda ion (p ojec No. 18-12289Y) and om he Facul y o Ci il
Enginee ing, B no Uni e si y o Technology (p ojec No. FAST-S-19-5896) is g a e ully acknowledged.
REFERENCES
[1] Be o, F., Lazza in, P. (2010). On highe o de e ms in he c ack ip s ess ield, In . J. F ac ., 161, pp. 221–226.
DOI: 10.1007/s10704-010-9443-3.
[2] Du, Z.Z., Hancock, J.W. (1991). The e ec o non-singula s esses on c ack ip cons ain , J. Mech. Phys. Solids, 39,
pp. 555–567. DOI: 10.1016/0022-5096(91)90041-L.
[3] Ka ihaloo, B.L. (1999). Size e ec in shallow and deep no ched quasi-b i le s uc u es, In . J. F ac ., 95, pp. 379–390.
DOI: 10.1023/A:1018633208621.
[4] Williams, M.L. (1957). On he s ess dis ibu ion a he base o a s a iona y c ack, J. Appl. Mech. (ASME), 24, pp. 109–
114.
[5] Malíko á, L. (2015). Mul i-pa ame e ac u e c i e ia o es ima ion o c ack p opaga ion di ec ion applied o a mixed-
mode geome y, Engng. F ac . Mech., 143, pp. 32–46. DOI: 10.1016/j.eng acmech.2015.06.029.
F
L.. Malíko á e alii, F a u a ed In eg i à S u u ale, 49 (2019) 65-73; DOI: 10.3221/IGF-ESIS.49.07
73
[6] Malíko á, L., Veselý, V. (2014). The in luence o highe -o de e ms o Williams se ies on a mo e accu a e desc ip ion
o s ess ields a ound he c ack ip, Fa igue F ac . Engng Ma e . S uc ., 38, pp. 91–103. DOI: 10.1111/ e.12221.
[7] Malíko á, L., Veselý, V. (2017). In luence o he elas ic misma ch on c ack p opaga ion in a silica e-based composi e,
Theo . Appl. F ac . Mech., 91, pp. 25–30. DOI: 10.1016/j. a mec.2017.03.004.
[8] Veselý, V., F an ík, P., Sobek, J., Malíko á (Šes áko á), L., Sei l, S. (2014). Mul i-pa ame e c ack ip s ess s a e
desc ip ion o es ima ion o nonlinea zone wid h in silica e composi e specimens in componen spli ing/bending
es geome y, Fa igue F ac . Engng. Ma e . S uc ., 7(25), pp. 69–78. DOI: 10.1111/ e.12170.
[9] Smi h, D. J., Aya ollahi, M. R., Pa ie , M. J. (2001). The ole o T-s ess in b i le ac u e o linea elas ic ma e ials
unde mixed-mode loading, Fa igue F ac . Engng Ma e . S uc ., 24(2), pp. 137–150.
DOI: 10.1046/j.1460-2695.2001.00377.x.
[10] Hou, C., Z. Wang, Liang, W., Li, J. (2016). De e mina ion o ac u e pa ame e s in cen e c acked ci cula discs o
conc e e unde diame al loading: A nume ical analysis and expe imen al esul s. Theo . Appl. F ac . Mech., 85,
pp. 355–366. DOI: 10.1016/j. a mec.2016.04.006.
[11] P o is, J.L., an De en e , J.S. (eds) (2014). Alkali ac i a ed ma e ials: s a e-o - he-a epo , RILEM TC 224-AAM.
Do d ech : Sp inge .
[12] Shi, C., K i enko, P. V., Roy, D. (2006). Alkali-Ac i a ed Cemen s and Conc e es. Oxon: Taylo & F ancis.
[13] Shi, C., Day, R. L. (1996). Some ac o s a ec ing ea ly hyd a ion cha ac e is ics o alkali-slag cemen s, Cemen Conc e e
Res., 26 (3), pp. 439–447. DOI: 10.1016/S0008-8846(96)85031-9.
[14] Aya ollahi, M.R, Neja i, M. (2010). An o e -de e minis ic me hod o calcula ion o coe icien s o c ack ip asymp o ic
ield om ini e elemen analysis, Fa igue F ac . Engng Ma e . S uc ., 34, pp. 159–176.
DOI: 10.1111/j.1460-2695.2010.01504.x.
[15] Šes áko á, L. (2013). How o enhance e iciency and accu acy o he o e -de e minis ic me hod used o de e mina ion
o he coe icien s o he highe -o de e ms in Williams expansion, Appl. Mech. Ma e ., 245, pp. 120–125.
DOI: 10.4028/www.scien i ic.ne /AMM.245.120.
[16] Ka ihaloo, B.L., Xiao, Q. Z. (2001). Accu a e de e mina ion o he coe icien s o elas ic c ack ip asymp o ic ield by
a hyb id c ack elemen wi h p-adap i i y, Engng. F ac . Mech., 68(15), pp. 1609–1630.
DOI: 10.1016/S0013-7944(01)00063-7
[17] Tong, P., Pian, T. H. H., Las y, S. J. (1973). A hyb id-elemen app oach o c ack p oblems in plane elas ici y, In . J.
Nume . Me hods Eng., 7(3), pp. 297–308. DOI: 10.1002/nme.1620070307
[18] Xiao, Q. Z., Ka ihaloo, B. L., Liu, X. Y. (2004). Di ec de e mina ion o SIF and highe o de e ms o mixed mode
c acks by a hyb id c ack elemen , In . J. F ac ., 125(3), pp. 207–225. DOI: 10.1023/B:FRAC.0000022229.54422.13.
[19] Qian, J., Fa emi, A. (1996). Mixed mode a igue c ack g ow h: a li e a u e su ey, Engng. F ac . Mech., 55(6), pp. 969–
990. DOI: 10.1016/S0013-7944(96)00071-9.
[20] E dogan, F., Sih, G.C. (1963). On he c ack ex ension in pla es unde plane loading and ans e se shea , J. Basic
Engng., 85, pp. 519–525. DOI:10.1115/1.3656897.
[21] Wol am Ma hema ica Documen a ion Cen e . Wol am Resea ch, Inc., Champaign (2018).
[22] Sih, G.C. (1973). Some basic p oblems in ac u e mechanics and new concep s, Engng. F ac . Mech., 5, pp. 365–377.
[23] Sih, G.C. (1974). S ain ene gy densi y ac o applied o mixed mode c ack p oblems, In . J. F ac . Mech., 10, pp. 305–
321.
[24] Sewe yn, A., Lukaszewicz, A. (2002). Ve i ica ion o b i le ac u e c i e ia o elemen s wi h V-shaped no ches, Engng.
F ac . Mech., 69, pp. 1487–1510. DOI: 10.1016/S0013-7944(01)00138-2.
[25] Sih, G.C., Ho, J.W. (1991). Sha p no ch ac u e s eng h cha ac e ized by c i ical ene gy densi y, J. Theo . Appl. F ac .
Mech., 16, pp. 179–214. DOI: 10.1016/0167-8442(91)90044-K.
[26] Susmel, L., Taylo , D. (2008). The heo y o c i ical dis ances o p edic s a ic s eng h o no ched b i le componen s
subjec ed o mixed-mode loading, Engng. F ac . Mech., 75, pp. 534–550. DOI: 10.1016/j.eng acmech.2007.03.035
[27] Fakh i, M., Amoosol ani, E., Aliha, M.R.M. (2017). C ack beha io analysis o olle compac ed conc e e mix u es
con aining eclaimed asphal pa emen and c umb ubbe , Engng. F ac . Mech., 180, pp. 43–59.
DOI: 10.1016/j.eng acmech.2017.05.011.
[28] ANSYS P og am Documen a ion (2019). Use ’s manual e sion 19.0. Swanson Analysis Sys em, Inc., Hous on.
[29] Ma ienko, Y.G., Semeno a, M.M. (2015). The concep o he a e age s ess in he F ac u e P ocess Zone o he
sea ch o he c ack pa h. F a . In . S u ., 34, pp. 276–281. DOI: 10.3221/IGF-ESIS.34.29.
[30] To abi, A.R. (2012). Es ima ion o ensile load-bea ing capaci y o duc ile me allic ma e ials weakened by a V-no ch:
The equi alen ma e ial concep . Ma . Sci. Engng., A 536, pp.249–255. DOI: 10.1016/j.msea.2012.01.007.