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Pee - e iew unde esponsibili y o he scien i ic commi ee o he 23 Eu opean Con e ence on F ac u e – ECF23
10.1016/j.p os .2022.12.126
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Pee - e iew unde esponsibili y o he scien i ic commi ee o he 23 Eu opean Con e ence on F ac u e – ECF23
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Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
Nume ical s udy o specimen wi h s eel inclusion:
In luence o in e acial ansi ion zone
Michal Vyhlídala,*, Jan Klusákb
aB no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 331/95, 602 00 B no, Czech Republic
bCzech Academy o Sciences, Ins i u e o Physics o Ma e ials, Žižko a 22, 616 00 B no, Czech Republic
Abs ac
In his pape , he in luence o he in e acial ansi ion zone (ITZ) be ween s eel inclusion and ine-g ained cemen -based composi e
on ac u e beha iou is in es iga ed. Specimens o he nominal dimensions 40 × 40 × 160 mm wi h he s eel inclusion o he
p isma ic shape wi h nominal dimensions 8 × 8 × 40 mm we e p o ided wi h an ini ial cen al edge no ch wi h a dep h 12 mm.
The aim o his pape is o analyse he beha iou o such specimen by means o nume ical modelling by ini e elemen me hod
(Ansys so wa e). A simpli ied 2D model (plane s ain) based on he ac u e es con igu a ion was c ea ed. The c ack p opaga ion
assessmen was based on he c i e ion o he a e age alue o angen ial s ess 𝜎𝜎θθ calcula ed o e ce ain dis ance d in dependence
on he pola coo dina e θ. I is assumed ha he c ack is ini ia ed when he a e age s ess 𝜎𝜎θθ(𝜃𝜃0) eaches i s c i ical alue
𝜎𝜎θθ,c(𝜃𝜃0), which depends on he ac u e oughness 𝐾𝐾Ic o he ma e ial and on he dis ance d.
F om he de ailed nume ical analysis o he desc ibed ac u e es , we concluded ha he c ack p opaga ion pa h depends s ongly
on he ac u e p ope ies o ITZ. The cen al inclusion wo ks as an obs acle o c ack p opaga ion only in he ini ial s age o ailu e.
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: : S eel inclusion, ac u e es , nume ical simula ion, c ack p opaga ion pa h, he c i e ion o a e age angen ial s ess.
* Co esponding au ho .
E-mail add ess: Michal.Vyhlidal@ u b .cz
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
Nume ical s udy o specimen wi h s eel inclusion:
In luence o in e acial ansi ion zone
Michal Vyhlídala,*, Jan Klusákb
aB no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 331/95, 602 00 B no, Czech Republic
bCzech Academy o Sciences, Ins i u e o Physics o Ma e ials, Žižko a 22, 616 00 B no, Czech Republic
Abs ac
In his pape , he in luence o he in e acial ansi ion zone (ITZ) be ween s eel inclusion and ine-g ained cemen -based composi e
on ac u e beha iou is in es iga ed. Specimens o he nominal dimensions 40 × 40 × 160 mm wi h he s eel inclusion o he
p isma ic shape wi h nominal dimensions 8 × 8 × 40 mm we e p o ided wi h an ini ial cen al edge no ch wi h a dep h 12 mm.
The aim o his pape is o analyse he beha iou o such specimen by means o nume ical modelling by ini e elemen me hod
(Ansys so wa e). A simpli ied 2D model (plane s ain) based on he ac u e es con igu a ion was c ea ed. The c ack p opaga ion
assessmen was based on he c i e ion o he a e age alue o angen ial s ess 𝜎𝜎θθ calcula ed o e ce ain dis ance d in dependence
on he pola coo dina e θ. I is assumed ha he c ack is ini ia ed when he a e age s ess 𝜎𝜎θθ(𝜃𝜃0) eaches i s c i ical alue
𝜎𝜎θθ,c(𝜃𝜃0), which depends on he ac u e oughness 𝐾𝐾Ic o he ma e ial and on he dis ance d.
F om he de ailed nume ical analysis o he desc ibed ac u e es , we concluded ha he c ack p opaga ion pa h depends s ongly
on he ac u e p ope ies o ITZ. The cen al inclusion wo ks as an obs acle o c ack p opaga ion only in he ini ial s age o ailu e.
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: : S eel inclusion, ac u e es , nume ical simula ion, c ack p opaga ion pa h, he c i e ion o a e age angen ial s ess.
* Co esponding au ho .
E-mail add ess: Michal.Vyhlidal@ u b .cz
Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007 1001
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
Nume ical s udy o specimen wi h s eel inclusion:
In luence o in e acial ansi ion zone
Michal Vyhlídala,*, Jan Klusákb
aB no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 331/95, 602 00 B no, Czech Republic
bCzech Academy o Sciences, Ins i u e o Physics o Ma e ials, Žižko a 22, 616 00 B no, Czech Republic
Abs ac
In his pape , he in luence o he in e acial ansi ion zone (ITZ) be ween s eel inclusion and ine-g ained cemen -based composi e
on ac u e beha iou is in es iga ed. Specimens o he nominal dimensions 40 × 40 × 160 mm wi h he s eel inclusion o he
p isma ic shape wi h nominal dimensions 8 × 8 × 40 mm we e p o ided wi h an ini ial cen al edge no ch wi h a dep h 12 mm.
The aim o his pape is o analyse he beha iou o such specimen by means o nume ical modelling by ini e elemen me hod
(Ansys so wa e). A simpli ied 2D model (plane s ain) based on he ac u e es con igu a ion was c ea ed. The c ack p opaga ion
assessmen was based on he c i e ion o he a e age alue o angen ial s ess 𝜎𝜎θθ calcula ed o e ce ain dis ance d in dependence
on he pola coo dina e θ. I is assumed ha he c ack is ini ia ed when he a e age s ess 𝜎𝜎θθ(𝜃𝜃0) eaches i s c i ical alue
𝜎𝜎θθ,c(𝜃𝜃0), which depends on he ac u e oughness 𝐾𝐾Ic o he ma e ial and on he dis ance d.
F om he de ailed nume ical analysis o he desc ibed ac u e es , we concluded ha he c ack p opaga ion pa h depends s ongly
on he ac u e p ope ies o ITZ. The cen al inclusion wo ks as an obs acle o c ack p opaga ion only in he ini ial s age o ailu e.
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: : S eel inclusion, ac u e es , nume ical simula ion, c ack p opaga ion pa h, he c i e ion o a e age angen ial s ess.
* Co esponding au ho .
E-mail add ess: Michal.Vyhlidal@ u b .cz
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
Nume ical s udy o specimen wi h s eel inclusion:
In luence o in e acial ansi ion zone
Michal Vyhlídala,*, Jan Klusákb
aB no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 331/95, 602 00 B no, Czech Republic
bCzech Academy o Sciences, Ins i u e o Physics o Ma e ials, Žižko a 22, 616 00 B no, Czech Republic
Abs ac
In his pape , he in luence o he in e acial ansi ion zone (ITZ) be ween s eel inclusion and ine-g ained cemen -based composi e
on ac u e beha iou is in es iga ed. Specimens o he nominal dimensions 40 × 40 × 160 mm wi h he s eel inclusion o he
p isma ic shape wi h nominal dimensions 8 × 8 × 40 mm we e p o ided wi h an ini ial cen al edge no ch wi h a dep h 12 mm.
The aim o his pape is o analyse he beha iou o such specimen by means o nume ical modelling by ini e elemen me hod
(Ansys so wa e). A simpli ied 2D model (plane s ain) based on he ac u e es con igu a ion was c ea ed. The c ack p opaga ion
assessmen was based on he c i e ion o he a e age alue o angen ial s ess 𝜎𝜎θθ calcula ed o e ce ain dis ance d in dependence
on he pola coo dina e θ. I is assumed ha he c ack is ini ia ed when he a e age s ess 𝜎𝜎θθ(𝜃𝜃0) eaches i s c i ical alue
𝜎𝜎θθ,c(𝜃𝜃0), which depends on he ac u e oughness 𝐾𝐾Ic o he ma e ial and on he dis ance d.
F om he de ailed nume ical analysis o he desc ibed ac u e es , we concluded ha he c ack p opaga ion pa h depends s ongly
on he ac u e p ope ies o ITZ. The cen al inclusion wo ks as an obs acle o c ack p opaga ion only in he ini ial s age o ailu e.
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: : S eel inclusion, ac u e es , nume ical simula ion, c ack p opaga ion pa h, he c i e ion o a e age angen ial s ess.
* Co esponding au ho .
E-mail add ess: Michal.Vyhlidal@ u b .cz
2 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
1. In oduc ion
Conc e e is a composi e, which is o med, in a esh s a e, om coa se agg ega e, ine agg ega e, binde (mos ly
Po land cemen ) and wa e , e en ually admix u es. Due o he hyd a ion p ocess conc e e ob ain i s aluable
p ope ies – high s eng h (mainly comp essi e), du abili y, i e esis ance e c. Du ing he hyd a ion and ha dening,
he sh inkage and o he p ocesses occu and lead o he o ma ion o mic o and mac o-de ec s (c acks, po es, e c.).
Ne e heless, conc e e s uc u es a e designed using p ocedu es men ioned in s anda ds – e. g. EN 1992-1-1 (2011)
o ecommenda ions which do no conside he exis ence o de ec s, such as c acks, po es, inclusions, ansi ion zones,
e c. Howe e , hese discon inui ies can ac as po en ial weak elemen s, which in luences he ac u e beha io
o conc e e s uc u es. In he mo e complex design o hese s uc u es (e. g. complex geome y), i is ad isable o apply
he p inciples o ac u e mechanics.
In his pape , ma e ial discon inui y is o med by special s eel inclusion placed in he middle o he es specimen.
Such a specimen made o ine-g ained cemen -based composi e was es ed in h ee-poin bending con igu a ion.
The aim o his pape is o iden i y he in luence o he in e ace be ween s eel inclusion and ma ix which is o med
by he in e acial ansi ion zone (ITZ). In he pape , esul s o nume ical assessmen o c ack p opaga ion s ages a e
p esen ed.
Nomencla u e
d a e aging dis ance
E Young’s modulus
EITZ Young’s modulus o he in e acial ansi ion zone
Fappl applied loading o ce
Fc i c i ical alue he applied o ce
𝐹𝐹𝑖𝑖𝑖𝑖(𝑛𝑛,𝜃𝜃) shape unc ions
GMTS gene alized maximum angen ial s ess
H1 gene alized s ess in ensi y ac o
ITZ in e acial ansi ion zone
KI s ess in ensi y ac o in loading mode I
KIc ac u e oughness (unde pu e mode I)
KIc, ITZ ac u e oughness (unde pu e mode I) o he in e acial ansi ion zone
KIc, MTX ac u e oughness (unde pu e mode I) o he ma ix
MTX ma ix – ine-g ained cemen -based composi e
p1 s ess singula i y exponen
𝑟𝑟,𝜃𝜃 pola coo dina es
ν Poisson’s a io
𝜎𝜎
ij
s ess enso componen
𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃) angen ial s ess
𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃) a e age angen ial s ess
𝜎𝜎𝜃𝜃𝜃𝜃,c c i ical alue o he a e age angen ial s ess
2. Theo e ical backg ound
2.1. The In e acial T ansi ion Zone
The exis ence o he ITZ be ween agg ega e and cemen pas e was i s desc ibed in he 1950s by Fa an (1956).
The p ope ies o he ITZ and i s impac on he beha io o cemen -based composi es ha e been s udied om many
poin s o iew since ha . The ITZ is a egion o abou 50 µm in size, and i s mic os uc u e is o med mainly
by e ingi e needles and po landi e pla es (Diamond and Huang, 1998). The signi ican ea u e o he ITZ is mainly
1002 Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 3
i s highe po osi y compa ed o he bulk ma ix which is ela ed o he highe local wa e o cemen a io
(Sc i ene e al., 2004). The local inc ease in po osi y leads o poo e bonds be ween he componen s, and hus
o lowe alues o he mechanical ac u e pa ame e s o he ITZ, see e. g. Zacha da e al. (2018)
o Vyhlídal and Klusák (2020). The p ope ies o ITZ in luence he o e all ac u e beha io o conc e e.
2.2. Linea elas ic ac u e mechanics and i s gene alized o m
F ac u e mechanics is a widely used ool o an assessmen o c ack beha io in ma e ials. Linea elas ic ac u e
mechanics supposes c acks in homogenous, iso opic, and linea ly elas ic ma e ial, see Ande son (2005). Williams
(1957) p esen ed a uni e sal solu ion o he s ess componen s a ound a c ack ip in he o m o in ini e se ies. Nea
c ack ip (𝑟𝑟 → 0), only he i s (singula ) e m is used o desc ibe he s ess ield, while he highe o de e ms
o he in ini e se ies can some imes be neglec ed. Componen s o he s ess enso a e gi en by he supe posi ion
o h ee basic ailu e modes de ined by I win (1957), whe e he opening mode I is p edominan ailu e mode.
Gene alized linea elas ic ac u e mechanics deals wi h an assessmen o gene al singula s ess concen a o s,
such as sha p no ches, bi-ma e ial no ches, bi-ma e ial inclusions, in e ace c acks e c. – see Klusák e al. (2013),
De Co e e al. (2017), o Klusák e al. (2016). In mos cases, he loading modes canno be assigned o he se ies e ms,
so he e ms a e labeled by A abic subsc ip s. The dependence o s ess on pola coo dina e is exp essed by a gene al
s ess singula i y exponen p1:
𝜎𝜎
ij
=𝐻𝐻1
√2𝜋𝜋 𝑟𝑟−𝑝𝑝1𝐹𝐹𝑖𝑖𝑖𝑖
1(

)
  
ሺͳሻ
whe e σij a e he s ess enso componen s o i, j = ,

. Gene alized s ess in ensi y ac o H1 ha e he uni s
[MPa𝑚𝑚𝑝𝑝1], and Fij a e he shape unc ions. Fo c acks in homogeneous media, he s ess in ensi y ac o s can be
de e mined di ec ly in he FEM so wa e, e.g. Ansys Inc. so wa e (2021), while o he gene al singula s ess
concen a o s, a ious di ec o in eg a ion me hods a e used, see Ping e al. (2008), Klusák e al. (2008)
o P o an e al. (2008). The alues H1 a e asce ained om a nume ical solu ion o he s udied geome y, ma e ials,
and bounda y condi ions.
2.3. C i e ion o s abili y based on a e age s ess ahead o he c ack ip
In he pape , he c ack p opaga ion condi ion is de e mined om he gene alized maximum angen ial s ess
(GMTS) c i e ion. This s abili y condi ion is based on wo well-known ac u e mechanics s abili y condi ions – (a)
c ack ini ia ion occu s i he s ess in ensi y ac o KI eaches i s c i ical alue KIc (also known as ac u e oughness),
see e.g. Ande son (2005), and (b) c ack will p opaga e in he di ec ion whe e he angen ial s ess σθθ is maximal, see
E dogan and Sih (1963). The GMTS s abili y c i e ion employs he a e age s ess σθθ(𝜃𝜃) calcula ed ac oss a dis ance
d ahead o he c ack ip. The dis ance d is usually chosen in dependence on he mechanism o a up u e (dimension
o a plas ic zone o me als, dimension o p ocess zone o quasi-b i le ma e ials, ma e ial g ain size, o ini e supposed
c ack ini ia ion/p opaga ion inc emen ). The a e age s ess σθθ(𝜃𝜃) ahead o he c ack ip is gi en by he ollowing
equa ion.
𝜎𝜎𝜃𝜃𝜃𝜃 =1
𝑑𝑑∫𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃)𝑑𝑑𝑟𝑟
𝑑𝑑
0 (2)
The c ack p opaga ion di ec ion is supposed in he di ec ion o maximum o he a e age angen ial s ess esul ing
om nume ical analysis. The c i ical alue o angen ial s ess σθθ(𝜃𝜃) co esponding o he c ack ini ia ion is
de e mined om he knowledge o c ack p opaga ion condi ions o a c ack in homogeneous ma e ial unde mode I,
whe e ac u e oughness KIc is employed:
𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 =2𝐾𝐾𝐼𝐼𝐼𝐼
√2𝜋𝜋𝑑𝑑 (3)
Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007 1003
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 3
i s highe po osi y compa ed o he bulk ma ix which is ela ed o he highe local wa e o cemen a io
(Sc i ene e al., 2004). The local inc ease in po osi y leads o poo e bonds be ween he componen s, and hus
o lowe alues o he mechanical ac u e pa ame e s o he ITZ, see e. g. Zacha da e al. (2018)
o Vyhlídal and Klusák (2020). The p ope ies o ITZ in luence he o e all ac u e beha io o conc e e.
2.2. Linea elas ic ac u e mechanics and i s gene alized o m
F ac u e mechanics is a widely used ool o an assessmen o c ack beha io in ma e ials. Linea elas ic ac u e
mechanics supposes c acks in homogenous, iso opic, and linea ly elas ic ma e ial, see Ande son (2005). Williams
(1957) p esen ed a uni e sal solu ion o he s ess componen s a ound a c ack ip in he o m o in ini e se ies. Nea
c ack ip (𝑟𝑟 → 0), only he i s (singula ) e m is used o desc ibe he s ess ield, while he highe o de e ms
o he in ini e se ies can some imes be neglec ed. Componen s o he s ess enso a e gi en by he supe posi ion
o h ee basic ailu e modes de ined by I win (1957), whe e he opening mode I is p edominan ailu e mode.
Gene alized linea elas ic ac u e mechanics deals wi h an assessmen o gene al singula s ess concen a o s,
such as sha p no ches, bi-ma e ial no ches, bi-ma e ial inclusions, in e ace c acks e c. – see Klusák e al. (2013),
De Co e e al. (2017), o Klusák e al. (2016). In mos cases, he loading modes canno be assigned o he se ies e ms,
so he e ms a e labeled by A abic subsc ip s. The dependence o s ess on pola coo dina e is exp essed by a gene al
s ess singula i y exponen p1:
𝜎𝜎
ij
=𝐻𝐻1
√2𝜋𝜋 𝑟𝑟−𝑝𝑝1𝐹𝐹𝑖𝑖𝑖𝑖
1(

)
  
ሺͳሻ
whe e σij a e he s ess enso componen s o i, j = ,

. Gene alized s ess in ensi y ac o H1 ha e he uni s
[MPa𝑚𝑚𝑝𝑝1], and Fij a e he shape unc ions. Fo c acks in homogeneous media, he s ess in ensi y ac o s can be
de e mined di ec ly in he FEM so wa e, e.g. Ansys Inc. so wa e (2021), while o he gene al singula s ess
concen a o s, a ious di ec o in eg a ion me hods a e used, see Ping e al. (2008), Klusák e al. (2008)
o P o an e al. (2008). The alues H1 a e asce ained om a nume ical solu ion o he s udied geome y, ma e ials,
and bounda y condi ions.
2.3. C i e ion o s abili y based on a e age s ess ahead o he c ack ip
In he pape , he c ack p opaga ion condi ion is de e mined om he gene alized maximum angen ial s ess
(GMTS) c i e ion. This s abili y condi ion is based on wo well-known ac u e mechanics s abili y condi ions – (a)
c ack ini ia ion occu s i he s ess in ensi y ac o KI eaches i s c i ical alue KIc (also known as ac u e oughness),
see e.g. Ande son (2005), and (b) c ack will p opaga e in he di ec ion whe e he angen ial s ess σθθ is maximal, see
E dogan and Sih (1963). The GMTS s abili y c i e ion employs he a e age s ess σθθ(𝜃𝜃) calcula ed ac oss a dis ance
d ahead o he c ack ip. The dis ance d is usually chosen in dependence on he mechanism o a up u e (dimension
o a plas ic zone o me als, dimension o p ocess zone o quasi-b i le ma e ials, ma e ial g ain size, o ini e supposed
c ack ini ia ion/p opaga ion inc emen ). The a e age s ess σθθ(𝜃𝜃) ahead o he c ack ip is gi en by he ollowing
equa ion.
𝜎𝜎𝜃𝜃𝜃𝜃 =1
𝑑𝑑∫𝜎𝜎𝜃𝜃𝜃𝜃(𝑟𝑟,𝜃𝜃)𝑑𝑑𝑟𝑟
𝑑𝑑
0 (2)
The c ack p opaga ion di ec ion is supposed in he di ec ion o maximum o he a e age angen ial s ess esul ing
om nume ical analysis. The c i ical alue o angen ial s ess σθθ(𝜃𝜃) co esponding o he c ack ini ia ion is
de e mined om he knowledge o c ack p opaga ion condi ions o a c ack in homogeneous ma e ial unde mode I,
whe e ac u e oughness KIc is employed:
𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 =2𝐾𝐾𝐼𝐼𝐼𝐼
√2𝜋𝜋𝑑𝑑 (3)
4 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
The c ack p opaga es when he mean alue o angen ial s ess
σθθ(𝜃𝜃0)
exceeds i s c i ical alue
𝜎𝜎𝜃𝜃𝜃𝜃,c
.
𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃0)≥ 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 (4)
In he case o bi-ma e ial combina ions, he c ack p opaga es in o di ec ion θ0 and ma e ial m co esponding
o he lowes alue o c i ical applied o ce Fc i ,m. The c i ical applied o ce Fc i ,m depends on ac u e oughness
o pa icula ma e ial KIc,m and on he a io o he c i ical alue 𝜎𝜎𝜃𝜃𝜃𝜃,𝑐𝑐 and he a e age angen ial s ess co esponding
o he applied o ce Fappl.
𝐹𝐹c i ,𝑚𝑚 = 𝐹𝐹appl 𝜎𝜎
𝜃𝜃𝜃𝜃,c(𝐾𝐾Ic,𝑚𝑚)
𝜎𝜎𝜃𝜃𝜃𝜃(𝜃𝜃0,𝑚𝑚) (5)
3. Nume ical s udy
3.1. Nume ical model
A simpli ied 2D model o he specimen was c ea ed in Ansys Inc. so wa e (2021). The dimensions o he specimen
a e in Fig. 1. The specimen con ains a s eel inclusion wi h dimensions 8×8×40 mm placed in he middle o he span.
The specimens include an ini ial no ch wi h a dep h o 12 mm.
Fig. 1. Designed specimen geome y (dimensions in mm).
Ma e ials we e modelled as linea , elas ic, and iso opic and hey we e ep esen ed by he elas ic cons an s,
i.e. Young’s modulus E, and Poisson’s a io ν. F ac u e p ope ies we e de ined by he ac u e oughness KIc (Table 1).
Table 1. O e iew o he ma e ial’s pa ame e s used in he nume ical model.
Ma e ial
E
[GPa]
ν
[–]
K
Ic
[MPa∙m1/2]
Ma ix (MTX) – ine-g ained cemen -based composi e
39.3
0.20
0.58
ITZ
14.7
0.20
0.7K
Ic, MTX
Inclusion – s eel S235
210
0.30
i ele an
Young’s modulus o elas ici y o he ITZ EITZ was es ima ed acco ding o app oaches in Zacha da e al. (2018).
F ac u e oughness o he ITZ was chosen as 0.7KIc, MTX.
The c ack was modelled as ideally sha p. Because o high s ess g adien s su ounding c ack ips and edges, highly
e ined meshes we e equi ed nea he c ack ip, bo om, le and uppe co ne o s eel inclusion, see Fig. 2.
In he ollowing s udy o c ack p opaga ion, he esul ing c i ical o ces will be compa ed o a e e ence specimen.
The e e ence specimen has he same geome y bu is made o homogeneous ma e ial wi h he p ope ies o ma ix.
In he g aphs o esul s, he c i ical o ces co esponding o he specimens wi h s eel inclusion a e labeled as STE,
while Fc i o he e e ence specimens a e labeled as REF.
1004 Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007
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Fig. 2. Radial mesh a ound – c ack ip (le ); le co ne o s eel inclusion (middle); op co ne o s eel inclusion ( igh ).
3.2. C ack ini ia ion om he no ch
Dis ance d was conside ed as he leng h be ween he c ack ip and bo om co ne o he inclusion, which a e bo h
po en ial s ess concen a o s. Acco ding o he heo ies o ini e ac u e mechanics, see Taylo e al. (2005),
Co ne i e al. (2006), Taylo (2017), we suppose, ha his dis ance d co esponds o cu en c ack inc emen o inal
leng h. We suppose ha he b idge be ween he ini ial c ack (edge no ch) and he inclusion b eaks a once. The s ess
concen a ion ahead o he c ack ip and he inclusion is shown in Fig. 3.
Fig. 3. Isosu aces o he i s p incipal s ess o he specimen wi h he s eel inclusion (le ) and he homogeneous e e ence specimen ( igh ).
Va ious no ch dep hs would lead o di e en s ess dis ibu ions as shown in Vyhlídal and Klusák (2019), whe e i
was ound ha he diamond blade o he saw can damage he specimen mo e han expec ed. He e in he s udy
o he no ch wi h he dep h 12 mm, he c i ical applied o ce was de e mined as c ack p opaga ion in he ma ix:
Fc i (STE) = 1.36 kN. Simila ly, he c i ical o ces we e calcula ed o he e e ence specimens: Fc i (REF) = 1.14 kN.
3.3. C ack p opaga ion om he bo om co ne o he inclusion
In his s age, we analyzed he c ack p opaga ion in he bo om co ne o he inclusion. The dis ibu ion o he i s
p incipal s ess can be seen in Fig. 4 (le ), while a g aph desc ibing he de elopmen o he c i ical o ce Fc i
in dependence on he dis ance d is shown in Fig. 4 ( igh ). I should be no ed ha he c i ical o ce Fc i was de e mined
o he c ack p opaga ion along he in e ace, whose ac u e oughness was equal o 70 % o he ac u e oughness
o he ma ix in he nume ical model. The c i ical o ces o e e ence specimen a e also shown in Fig. 4 ( igh ).
Al hough he c ack p opaga ion along he in e ace is no na u al, in con as o he e ical c ack p opaga ion
in he case o e e ence specimens, he Fc i (STE) eaches lowe alues in compa ison o he Fc i (REF). I is caused
by he lowe ac u e oughness o he in e ace KIc, ITZ compa ed o he KIc, MTX.

Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007 1005
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 5
Fig. 2. Radial mesh a ound – c ack ip (le ); le co ne o s eel inclusion (middle); op co ne o s eel inclusion ( igh ).
3.2. C ack ini ia ion om he no ch
Dis ance d was conside ed as he leng h be ween he c ack ip and bo om co ne o he inclusion, which a e bo h
po en ial s ess concen a o s. Acco ding o he heo ies o ini e ac u e mechanics, see Taylo e al. (2005),
Co ne i e al. (2006), Taylo (2017), we suppose, ha his dis ance d co esponds o cu en c ack inc emen o inal
leng h. We suppose ha he b idge be ween he ini ial c ack (edge no ch) and he inclusion b eaks a once. The s ess
concen a ion ahead o he c ack ip and he inclusion is shown in Fig. 3.
Fig. 3. Isosu aces o he i s p incipal s ess o he specimen wi h he s eel inclusion (le ) and he homogeneous e e ence specimen ( igh ).
Va ious no ch dep hs would lead o di e en s ess dis ibu ions as shown in Vyhlídal and Klusák (2019), whe e i
was ound ha he diamond blade o he saw can damage he specimen mo e han expec ed. He e in he s udy
o he no ch wi h he dep h 12 mm, he c i ical applied o ce was de e mined as c ack p opaga ion in he ma ix:
Fc i (STE) = 1.36 kN. Simila ly, he c i ical o ces we e calcula ed o he e e ence specimens: Fc i (REF) = 1.14 kN.
3.3. C ack p opaga ion om he bo om co ne o he inclusion
In his s age, we analyzed he c ack p opaga ion in he bo om co ne o he inclusion. The dis ibu ion o he i s
p incipal s ess can be seen in Fig. 4 (le ), while a g aph desc ibing he de elopmen o he c i ical o ce Fc i
in dependence on he dis ance d is shown in Fig. 4 ( igh ). I should be no ed ha he c i ical o ce Fc i was de e mined
o he c ack p opaga ion along he in e ace, whose ac u e oughness was equal o 70 % o he ac u e oughness
o he ma ix in he nume ical model. The c i ical o ces o e e ence specimen a e also shown in Fig. 4 ( igh ).
Al hough he c ack p opaga ion along he in e ace is no na u al, in con as o he e ical c ack p opaga ion
in he case o e e ence specimens, he Fc i (STE) eaches lowe alues in compa ison o he Fc i (REF). I is caused
by he lowe ac u e oughness o he in e ace KIc, ITZ compa ed o he KIc, MTX.
6 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
Fig. 4. Isosu aces o he i s p incipal s ess (le ), and a g aph desc ibing he dependence o he Fc i on he alue o d ( igh ).
3.4. C ack p opaga ion in he le co ne o he inclusion
The dis ibu ion o a e age angen ial s ess 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on he pola coo dina e

and he dis ance d,
o he c ack wi h i s ip a he le co ne o s eel inclusion can be seen in Fig. 5 ( igh ). The pola coo dina e

be ween
0 ° and 90 ° ep esen s he a ea o he s eel inclusion which means, ha he di ec ion

= 0° is o ien ed in he edge
o he inclusion (c ack ace) and posi i e alues a e coun ed coun e clockwise. The g aph clea ly desc ibes
he maximum 𝜎𝜎𝜃𝜃𝜃𝜃 in he di ec ion abou 135°, which goes o he ma ix (almos e ically). This is also
in he ag eemen wi h he dis ibu ion o he i s p incipal s ess (Fig. 5, le ).
Fig. 5. Isosu aces o he i s p incipal s ess (le ), A e age opening s ess 𝜎𝜎𝜃𝜃𝜃𝜃 ( igh ).
Ne e heless, o de e mine he c ack p opaga ion di ec ion, c i ical applied o ces mus be de e mined sepa a ely
o each ma e ial egion. O cou se, p opaga ion o he s eel inclusion is no possible, and only he ma ix o he ITZ
can be conside ed as po en ial c ack p opaga ion egions. These wo egions a e cha ac e ized by he alues o ac u e
oughness KIc,MTX, and KIc, ITZ. The g aph o Fc i is shown in Fig. 6 in dependence on he ma e ial egions
and on he dis ance d. We can see ha he minimum alue o c i ical o ce is in he ITZ, because we ha e chosen
he ac u e oughness o ITZ as 0.7 × KIc,MTX. I ac u e oughness o ITZ is g ea e , c ack p opaga es e ically
o he ma ix, i i is less han o equal o 0.7  KIc,MTX, c ack p opaga es along he in e ace be ween s eel inclusion
and he ma ix. No e ha also he dis ance d plays a ole in he compe i ion be ween he c ack p opaga ion di ec ion
in o MTX o ITZ.
The compa ison o c i ical o ces o specimen wi h he inclusion and he e e ence specimen is no easy he e,
because he c ack p opaga es in a s aigh line in homogeneous ma e ial. Fo his eason, he alues o Fc i (REF) a e
calcula ed o he c ack ip in he cen e o he specimen. Due o he lowe alues o he ac u e oughness o he ITZ
KIc, ITZ, he Fc i (STE) also eaches lowe alues in compa ison o he Fc i (REF).
1006 Michal Vyhlídal e al. / P ocedia S uc u al In eg i y 42 (2022) 1000–1007
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 7
Fig. 6. Dis ibu ion o c i ical o ce Fc i in dependence on he pola coo dina e

, and in dependence on he dis ance d (on he le );
minimum alue o c i ical o ce Fc i in dependence on he dis ance d (on he igh ).
3.5. C ack p opaga ion om he op co ne o he inclusion
The g aph in Fig. 7 shows he dis ibu ion o a e age angen ial s ess o he c ack wi h i s ip a he op co ne
o s eel inclusion in dependence on he pola coo dina e and dis ance d. C ack p opaga ion in he almos e ical
di ec ion (abou 215°) is expec ed. Fu he , he c ack will u n in e ical di ec ion. The dependences o c i ical o ces
Fc i on he dis ance d a e shown in Fig. 7 ( igh ) o he specimens bo h wi h and wi hou he inclusion. Simila ly
o he sec ion 3.4, he alues o Fc i (REF) a e calcula ed o he c ack ip in he posi ion which co esponds o he op
co ne o he inclusion. The di e ences be ween Fc i (REF) and Fc i (STE) a e due o he di e en di ec ions o c ack
p opaga ion o bo h specimens (STE and REF) bu a e negligible in his case.
Fig. 7. Dis ibu ion o a e age alues o angen ial s ess 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on pola angle (po en ial c ack p opaga ion di ec ion), c i ical o ce
Fc i in dependence on he dis ance d, espec i ely.
4. Conclusions
The nume ical s udy in he p e ious pa ag aphs es ima ed po en ial c ack p opaga ion pa hs and he c i ical applied
o ces du ing c ack p opaga ion s ages: c ack ini ia ion in he cen al edge no ch, and p opaga ion in he bo om, le ,
and uppe co ne s o he inclusion. The Fig. 8 shows he e olu ion o he c i ical o ce Fc i in he c ack s ages.
The esul s o c i ical o ces o c ack p opaga ion a ound he inclusion a e compa ed o he c ack p opaga ion
in homogeneous ma e ial ( he e e ence specimen).
Al hough he dependence on he dis ance d is o en discussed, i is appa en , ha i is e y weak. C ack p opaga ion
pa h depends mainly on he ac u e p ope ies o ITZ, see he sec ion 3.4. The compa ison o he specimens
wi h and wi hou he inclusion shows ha he cen al inclusion wo ks as an obs acle o c ack p opaga ion only
in he beginning s age – he c ack ini ia ion in he no ch ip, whe e Fc i (STE) is highe han Fc i (REF).
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens,
as we epo ed in p e ious pape s, see e.g. Vyhlídal e al. (2022), Vyhlídal and Klusák (2020) o Zacha da e al. (2018).
8 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
Fig. 8. E olu ion o he c i ical applied o ce Fc i o he c ack p opaga ion s ages in dependence on he dis ance d.
Acknowledgemen s
This ou come has been achie ed wi h he inancial suppo o he B no Uni e si y o Technology unde p ojec
No. FAST-J-22-8038. Au ho s a e g a e ul o suppo om he Czech Science Founda ions, p ojec No. 21-08772S.
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E dogan, F., Sih, G. C., 1963. On he C ack Ex ension in Pla es Unde Plane Loading and T ans e se Shea . Jou nal o Basic Eng. 85, 519–525.
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no ches, Key Eng. Ma e . 385–387, 409–412.
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Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 7
Fig. 6. Dis ibu ion o c i ical o ce Fc i in dependence on he pola coo dina e

, and in dependence on he dis ance d (on he le );
minimum alue o c i ical o ce Fc i in dependence on he dis ance d (on he igh ).
3.5. C ack p opaga ion om he op co ne o he inclusion
The g aph in Fig. 7 shows he dis ibu ion o a e age angen ial s ess o he c ack wi h i s ip a he op co ne
o s eel inclusion in dependence on he pola coo dina e and dis ance d. C ack p opaga ion in he almos e ical
di ec ion (abou 215°) is expec ed. Fu he , he c ack will u n in e ical di ec ion. The dependences o c i ical o ces
Fc i on he dis ance d a e shown in Fig. 7 ( igh ) o he specimens bo h wi h and wi hou he inclusion. Simila ly
o he sec ion 3.4, he alues o Fc i (REF) a e calcula ed o he c ack ip in he posi ion which co esponds o he op
co ne o he inclusion. The di e ences be ween Fc i (REF) and Fc i (STE) a e due o he di e en di ec ions o c ack
p opaga ion o bo h specimens (STE and REF) bu a e negligible in his case.
Fig. 7. Dis ibu ion o a e age alues o angen ial s ess 𝜎𝜎𝜃𝜃𝜃𝜃 in dependence on pola angle (po en ial c ack p opaga ion di ec ion), c i ical o ce
Fc i in dependence on he dis ance d, espec i ely.
4. Conclusions
The nume ical s udy in he p e ious pa ag aphs es ima ed po en ial c ack p opaga ion pa hs and he c i ical applied
o ces du ing c ack p opaga ion s ages: c ack ini ia ion in he cen al edge no ch, and p opaga ion in he bo om, le ,
and uppe co ne s o he inclusion. The Fig. 8 shows he e olu ion o he c i ical o ce Fc i in he c ack s ages.
The esul s o c i ical o ces o c ack p opaga ion a ound he inclusion a e compa ed o he c ack p opaga ion
in homogeneous ma e ial ( he e e ence specimen).
Al hough he dependence on he dis ance d is o en discussed, i is appa en , ha i is e y weak. C ack p opaga ion
pa h depends mainly on he ac u e p ope ies o ITZ, see he sec ion 3.4. The compa ison o he specimens
wi h and wi hou he inclusion shows ha he cen al inclusion wo ks as an obs acle o c ack p opaga ion only
in he beginning s age – he c ack ini ia ion in he no ch ip, whe e Fc i (STE) is highe han Fc i (REF).
The mechanical ac u e pa ame e s o he ITZ s ongly in luence he o e all ac u e beha io o es specimens,
as we epo ed in p e ious pape s, see e.g. Vyhlídal e al. (2022), Vyhlídal and Klusák (2020) o Zacha da e al. (2018).
8 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
Fig. 8. E olu ion o he c i ical applied o ce Fc i o he c ack p opaga ion s ages in dependence on he dis ance d.
Acknowledgemen s
This ou come has been achie ed wi h he inancial suppo o he B no Uni e si y o Technology unde p ojec
No. FAST-J-22-8038. Au ho s a e g a e ul o suppo om he Czech Science Founda ions, p ojec No. 21-08772S.
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