Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
A ailable online 13 Decembe 2022
0955-2219/© 2022 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/).
P edic ion o edge and unnelling c ack o ma ion in laye ed ce amics using
a s ess-ene gy ac u e c i e ion
Roman Papˇ
sík
a
, Oldˇ
ich ˇ
Se eˇ
cek
b
, Anna-Ka ha ina Ho e
a
, I ina K ale a
a
, Jose K ei h
a
,
Raúl Be mejo
a
,
*
a
Mon anuni e si ¨
a Leoben, Depa men o Ma e ials Science, F anz Jose S aße 18, 8700 Leoben, Aus ia
b
B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Technick´
a 2896/2, 616 69 B no,
Czech Republic
ARTICLE INFO
Keywo ds:
Laye ed ce amics
Coupled c i e ion
Fini e ac u e mechanics
Residual s esses
Edge c acks
Tunnelling c acks
ABSTRACT
A coupled s ess-ene gy c i e ion is u ilized o p edic ini ia ion o bo h edge and unnelling c acks in laye ed
ce amics con aining he mal esidual s esses. Edge (su ace) c acks may o igina e in laye s ha ing high
comp essi e in-plane s esses while unnelling (in e nal) c acks may o m in laye s wi h high ensile in-plane
s esses. This wo k in es iga es he in luence o bo h he esidual s esses magni ude and laye hickness on
he o ma ion o su ace c acks and p o ides a design map de ining sa e egions whe e no c acks will be p esen
in he sin e ed mul ilaye a chi ec u e upon eaching he oom empe a u e. Necessa y s ess and ene gy inpu s
o e alua e he coupled c i e ion a e calcula ed using he ini e elemen me hod. Simula ion esul s a e alida ed
wi h expe imen al obse a ions on sample a chi ec u es ab ica ed wi h laye s o a ious hicknesses and in-
plane he mal esidual s esses. The good ag eemen demons a es he po en ial o he s ess-ene gy coupled
c i e ion o designing c ack- ee mul i-laye ed ce amic a chi ec u es.
1. In oduc ion
Ce amic ma e ials a e used in s uc u al applica ions equi ing high
ha dness, empe a u e s abili y, esis ance o oxida ion, co osion o
wea . Despi e he ad an ageous p ope ies o ce amics, hei use is
usually limi ed in applica ions whe e high sa e y and eliabili y a e
equi ed. The main p oblem is he inhe en b i leness due o he low
ac u e oughness and he signi ican s eng h a iance caused by he
p esence o laws inside he mic os uc u e. Flaws a e mos ly induced
upon p ocessing (po es, inclusions, e c.) o du ing inal machining
(no ches o sc a ches). Such laws may hen, unde c i ical condi ions,
be esponsible o he o ma ion and p opaga ion o c acks, causing o al
componen ailu e.
One app oach o p o ec ing componen s agains ca as ophic ailu e
is embed "p o ec i e" laye s o a es he p opaga ion o su ace c acks
[1]. This migh be done, o example, by induc ion o comp essi e e-
sidual s esses in he c i ical loca ion/laye , dec easing he s ess in-
ensi y ac o a he c ack ip. The mal esidual s esses a e induced
du ing he cooling down p ocess om sin e ing empe a u e and a e
p ima ily caused by a he mal s ain misma ch be ween indi idual
ma e ials. This di e en ial s ain may be associa ed wi h di e ences in
he mal expansion (CTE), phase ans o ma ions o chemical eac ions
occu ing in pa icula laye s. Tailo ing he loca ion o in-plane
comp essi e esidual s esses in he laye ed a chi ec u e may inc ease
he damage ole ance o he ce amic componen [2]. Concu en ly,
ensile esidual s esses a e simul aneously induced in he composi e and
ha e an opposi e e ec , i.e., hey migh p omo e c ack p opaga ion.
The e o e, he igh balance be ween comp essi e and ensile s esses
and o he geome ical and ma e ial pa ame e s mus always be ound.
Two ypical c ack ypes associa ed wi h esidual s esses can be
obse ed in ce amic lamina es, i.e. “edge c acks” (Fig. 1a) and
“ unnelling c acks” (Fig. 1b) [2]. Edge c acks a e associa ed wi h
ou -o -plane ensile s esses a he su ace o he comp essi e laye . In
his case, he comp essi e in-plane esidual s ess in he bulk anishes a
he su ace, and he ensile ou -o -plane s ess componen appea s a his
loca ion ins ead. ˇ
Se eˇ
cek e al. [3] ound ha he magni ude o ensile
ou -o -plane s esses and he magni ude o he in-plane comp essi e
s esses a e o simila o de o magni ude. On he one hand, i he
ou -o -plane ensile s ess componen ac i a es a law, an edge c ack
usually de elops along he whole pe ime e o he laye and ex ends in o
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (R. Be mejo).
Con en s lis s a ailable a ScienceDi ec
Jou nal o he Eu opean Ce amic Socie y
jou nal homepage: www.else ie .com/loca e/jeu ce amsoc
h ps://doi.o g/10.1016/j.jeu ce amsoc.2022.12.022
Recei ed 1 July 2022; Recei ed in e ised o m 9 Decembe 2022; Accep ed 12 Decembe 2022
Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2929
he dep h pe pendicula ly o he specimen/componen su ace. On he
o he hand, unnelling c acks appea p ima ily in laye s con aining
in-plane ensile esidual s esses. Tensile bi-axial in-plane esidual s ess
ac ing inside he ensile laye may ac i a e a law, which usually esul s
in a c ack o ma ion and u he p opaga ion un il i appea s a he ee
su ace pe pendicula ly o he in e aces wi h o he laye s.
Edge and unnelling c acks in b i le laye s a e pa icula c acks
obse ed and s udied o almos h ee decades [4,5]. These c acks a e
no limi ed o mul i-ma e ial ce amic componen s bu can also be
obse ed o ins ance in glass ib e epoxy lamina es [6]. An ex e nal
mechanical load migh also induce his ype o c acks wi hou he
con ibu ion o esidual s esses.
Empi ical obse a ions [5] show ha unde he same magni ude o
in e nal esidual s esses edge c acks only ini ia e in laye s hicke han a
ce ain alue (c i ical hickness). The e a e wo common app oaches o
explain his: (i) he law s a is ics app oach based on he Weibull heo y
sugges s ha , due o smalle e ec i e olume, he appa en s eng h
inc eases; (ii) he ene gy app oach s a es ha a c ack in he hin laye
canno p opaga e since he s eady-s a e ene gy elease a e is oo low.
These explana ions a e based on he heo y o linea elas ic ac u e
mechanics (LEFM), which always assumes he p esence o a c ack. Such
c ack shall p opaga e as long as G i i h’s ene gy c i e ion is ul illed, i.e.
he ene gy elease a e G o e comes he ac u e ene gy G
c
(gi en by he
ac u e oughness o he co esponding ma e ial). Howe e , when he
ini ial c ack does no exis , LEFM canno p edic i s onse . Ins ead, he
Fini e F ac u e Mechanics (FFM) app oach which employs a coupled
s ess-ene gy c i e ion (CC) needs be implemen ed.
FFM conside s he o ma ion o a c ack as a ac u e e en [7] and
does no ega d he his o y o i s o ma ion. The momen o c ack
ini ia ion is de e mined by he “coupled c i e ion”, which equi es ha
bo h a s ess and ene gy c i e ia mus be ul illed simul aneously o a
c ack o nuclea e [8]. The size, shape o loca ion o a c i ical law (a
c ack) is usually no known p io o ac u e; hence, i needs o be
assumed. I was demons a ed by Leguillon e al. [9] and ˇ
Se eˇ
cek e al.
[3,10] o edge c acks and by Ga cia e al. [11,12] o unnelling c acks
ha he coupled c i e ion go e ns c ack ini ia ion and can p edic he
size e ec . Howe e , in bo h cases, sys ema ic alida ion o he used
models was lacking.
In his wo k, ini e elemen models o p edic ing edge c ack and
unnelling c ack o ma ion in laye ed ce amics was de eloped, based on
a coupled s ess-ene gy c i e ion. Two ma e ials wi h di e en he mo-
elas ic cons an s (i.e. Young’s modulus and coe icien o he mal
expansion (CTE)), a e used o induce esidual s esses in dissimila
adjacen laye s. The edge c ack model exploi s he axial symme y and
he edge c ack ini ia ion is assumed o always occu in only one laye .
The model o p edic ion o he unnelling c ack onse was c ea ed as a
3D model, bu due o assump ions abou he ini ial c ack shape, a 3- old
symme y could be exploi ed. In bo h cases, a pa ame ic analysis
conside ing a ious hicknesses o c acked laye s and di e en
magni ude o esidual s esses in he laye s was pe o med o de e mine
he c i ical condi ions necessa y o he onse o bo h ypes o c acks. To
e i y he p edic ions o he employed models, ce amic lamina ed
samples wi h dis inc ma e ials and a ious hicknesses o pa icula
laye s we e manu ac u ed and analysed o he p esence o edge and
unnelling c acks.
2. Expe imen al
2.1. Fab ica ion o samples
Samples we e manu ac u ed by he ape cas ing echnology. Fig. 2
depic s wo symme ic a chi ec u es o specimens manu ac u ed o
in es iga ion o edge c acks (Fig. 2a) o unnelling c acks (Fig. 2b).
Using he ape cas ing echnique, d ied apes o squa e shape and
40 ×40 mm size we e s acked and wa m-p essed by 20 MPa a 75 ◦C o
15 min, ollowed by he iso-s a ic lamina ion a 20 MPa/75 ◦C o
30 min, and by binde bu n-ou a 600 ◦C o 2 h. A e wa ds, he
s acked pla es we e cold-isos a ic p essed wi h 100 MPa o 15 min and
sin e ed a 1550 ◦C (hea ing a e 10 ◦C/min) o 2 h o achie e high
ela i e densi y.
Th ee di e en ma e ials we e used o manu ac u ing he samples:
(i) pu e alumina (A0), (ii) alumina wi h 50 ol% o (s abilized) e ag-
onal zi conia (A50TZ) and (iii) alumina wi h 15 ol% o (non-s abilized)
monoclinic zi conia (A15MZ). The ma e ial p ope ies o used ce amics
a e summa ized in Table 1. The choice o A15MZ ce amic was based on
p e ious expe imen al obse a ions o samples con aining laye s wi h
ei he 20 ol% o 10 ol% o monoclinic zi conia (A20MZ and A10MZ,
espec i ely), whe e c acks we e obse ed in all A0 laye s (combined
wi h A20MZ), and no c acks we e ound in any A0 laye (combined wi h
A10MZ).
To in es iga e edge c acks, he A0 laye s we e embedded be ween
he ZTA50 laye s (Fig. 2a) o induce ensile ou -o -plane esidual s esses
a he ee su ace o he A0 laye s (alumina laye s). The choice o laye
hicknesses was based on p elimina y analy ical calcula ions o s esses.
Fo in es iga ion o unnelling c acks, he A0 laye was embedded be-
ween A15MZ laye s (Fig. 2b) o induce ensile in-plane esidual s ess in
he A0 alumina laye s.
Young’s modulus E o indi idual laye s was de e mined on bulk
specimens om loading cu es o displacemen con olled 3-poin
bending es s. Uni e sal es ing machine (Messphysik, Mic os ain,
Fü s en eld, Aus ia) wi h a 100 N load cell and a ix u e wi h 30 mm
ou e span was used. Expe imen ollowed he EN 843–2 s anda d [13]
wi h he c osshead speed 0.5 mm/min. Ambien condi ions we e 24 ◦C
empe a u e and 42% humidi y. A 1 N p eload and maximum load o
35 N we e selec ed o al e na e loading/unloading o h ee samples pe
ma e ial. Common alues o Poisson’s a ios
ν
we e assumed.
Secan coe icien s o he mal expansion
α
(CTE) we e de e mined
om dila a ion cu es o monoli hic p isma ic ba s. Specimens wi h
Fig. 1. Examples o (a) he ci cum e en ial edge c ack; (b) he unnelling c ack [2].
R. Papˇ
sík e al.
Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2930
s anda dized leng h o 25 mm we e hea ed om 30 ◦C up o 900 ◦C
using a dila ome e (Ne zsch 402E, 95100 Selb, Ge many). The change
o leng h was egis e ed du ing hea ing wi h 1 h long holding segmen s
a e e y 100 ◦C wi h 5 ◦C/min hea ing a e in be ween. The CTE was
calcula ed be ween he oom empe a u e 25 ◦C and an es ima ed s ess-
ee empe a u e 1470 ◦C [14] om an ex apola ion o leng h a
holding segmen s. Du ing he cooling om sin e ing, ypically a ound
1100 ◦C, zi conia in A15MZ ma e ial unde goes a phase ans o ma ion
[15], om he e agonal o he monoclinic phase, wi h an associa ed
olume inc ease. This phase change canno be obse ed and aken in o
accoun when dila a ion is measu ed on al eady sin e ed specimen in a
empe a u e ange below 900 ◦C.
The ac u e oughness K
Ic
was de e mined using he single edge V-
no ched beam me hod (SEVNB) acco ding o he ISO 23146 s anda d
[16]. A uni e sal es ing machine (Zwick 010, Zwick/Roell Ulm, Ge -
many) wi h a 200 N load cell was used a ambien condi ions o 24 ◦C
empe a u e and 32% humidi y. Specimens we e es ed using a 4-poin
bending ix u e wi h a 40 mm ou e span and a 20 mm inne span wi h
applied displacemen a e o 0.5 mm/min.
Su aces o specimens we e spu e ed wi h a gold using he Ag a
Spu e Coa e and in es iga ed using a SEM (JEOL JCM-6000Plus,
Neoscope, JEOL L d., Tokyo, Japan). Edge and unnelling c ack we e
sough in all laye s.
The s eng h o A0 was measu ed by 4-poin -bending me hod using a
uni e sal es ing machine Zwick Z010 (Zwick/Roell, Ulm, Ge many).
Fo s a is ical signi icance, 26 samples we e used. Ambien condi ions
we e 23 ◦C empe a u e and 37% ela i e humidi y. A c oss beam speed
o 1.5 mm/min and a p e-load o ce o 10 N was chosen. The s eng h o
A50TZ and A15MZ was no measu ed, since i is no needed o he
calcula ions; c acking is expec ed only in he A0 laye s.
3. Theo y and calcula ions
In his sec ion, he compu a ional model o an edge c ack and a
unnelling c ack is desc ibed. The ini e ac u e mechanics is b ie ly
in oduced as a complemen a y app oach o he linea elas ic ac u e
mechanics and he me hodology o p edic ion o c ack ini ia ion by he
coupled s ess-ene gy c i e ion (CC) is summa ized.
3.1. Residual s esses in lamina es
Edge and unnelling c acks in ce amic lamina es o igina e solely due
o a p esence o esidual s esses caused by a misma ch in he mal s ains
be ween laye s o di e en ma e ials. No ex e nal mechanical load is
equi ed o he o ma ion o hese ypes o c acks. The magni ude o
ensile and comp essi e esidual s esses in he co esponding laye s can
be con olled by changing he olume a io o ma e ials [17]. The ol-
ume a io 
Vi o he i- h ma e ial in a composi e o N ma e ials is de ined
as:

Vi=Vi
∑N
i=1Vi
(1)
whe e Vi is he olume o he i- h ma e ial.
Figs. 3a and 3c illus a e he manu ac u ed specimens in c oss-
sec ions exploi ing he 3- old symme y. The esidual s esses a e
depic ed along pa hs A and B in he inne mos laye om he ee su ace
o he bulk. No mal s esses
σ
along pa hs A and B a e plo ed in Figs. 3b
and 3c, espec i ely. Pa h leng hs a a e no malised by co esponding
laye hickness inne
A0 and s esses a e no malised by he co esponding in-
plane (x-y) esidual s esses
σ
in. In he case o pa h A, he ou -o -plane
s ess
σ
zz can be as high as he in-plane s ess
σ
xx o
σ
yy and hus p o-
mo es c ack ini ia ion a he su ace. In he case o pa h B, he in-plane
s ess
σ
yy is homogeneous and cons an in he bulk bu dec eases owa ds
he su ace. Thus, we in e ha he mos a ou able loca ion o ini ia-
ion o unnelling c acks is no a he su ace, bu below he su ace.
3.2. Compu a ional model o edge c acking
The edge c acking was simula ed on a ci cula i e-laye disc (Fig. 4).
Since he edge c ack is o med all along he whole ci cum e ence o he
disc, his ask can be sol ed as an axisymme ic p oblem. Al hough he
manu ac u ed specimen was a pla e, simila condi ions o c ack o -
ma ion a he edges exis on he side o he disc, wi h less demanding
compu a ional ime. The inne laye and wo ou e laye s ha e cha ac-
e is ics o ma e ial A0 (see Table 1), which is a ma e ial wi h smalle
CTE han ha o he o he ma e ial o he disc, esul ing in comp essi e
in-plane esidual s esses in A0 laye s. Remaining laye s we e made o
A50TZ ma e ial, whe e ensile esidual in-plane s esses we e induced.
A he ee su ace o A0 laye s he in-plane comp essi e esidual s esses
ans o m in o he ou -o -plane ensile s esses and anish, which migh
be (upon ce ain condi ions) esponsible o he o ma ion o ci cum-
e en ial edge c acks [3].
The geome y o he disc is depic ed in Fig. 4a. The disc adius was
ixed a R =20 mm, he inne laye hickness was a iable in he ange
om 10 µm o 500 µm, he A50TZ laye s we e a leas 400 µm hick and
hey we e adjus ed oge he wi h ou e mos laye s o each he desi ed
le el o esidual s esses. The c ack dep h a was simula ed om 0 µm up
o he dep h whe e ensile ou -o -plane s esses anished.
The me idian c oss-sec ion o he disc was disc e ized by quad a ic
2D elemen s (PLANE183) ha ing he axisymme ic op ion ac i a ed.
The mesh was coa se in gene al (elemen edge leng h was se app oxi-
ma ely o 5 µm), bu a e ined mesh o app oxima ely 50 nm (Fig. 4b)
was used in he c ack icini y. The c ack was inc emen ally opened by
disconnec ing coinciden nodes on opposi e aces along he expec ed
c ack pa h. The mesh did no change du ing he c ack g ow h o a oid
Fig. 2. A chi ec u es o specimens manu ac u ed o analyses o (a) edge c ack o ma ion and; (b) unnelling c ack o ma ion.
Table 1
Ma e ial p ope ies o ce amics used o manu ac u ing o samples.
Ma e ial p ope y Ma e ial
A0 A50TZ A15MZ
E [GPa] 398 ±3 292 ±3 354 ±4
ν
[–] 0.23 0.23 0.23
α
[K
−1
] 8.3⋅10
−6
9.6⋅10
−6
7.5⋅10
−6
σ
0
[MPa] ~350 n/a n/a
K
Ic
[MPa⋅m
1/2
] 3.40 ±0.4 4.8 ±0.3 4.3 ±0.1
G
c
[J⋅m
−2
] 27.9 ±6.5 74.7 ±9.4 48.3 ±1.3
R. Papˇ
sík e al.
Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2931
spu ious a ia ions o he ene gy. On he symme y axis displacemen s
in he adial di ec ion we e ixed (u
( =0) =0) and addi ionally in one
node o he symme y axis (a z=0) a ze o displacemen in he axial
di ec ion was p esc ibed o a oid igid mo ion upon he simula ion.
Co ec ness o he axisymme ic simpli ica ion and co esponding
alues o he calcula ed ene gy elease a es we e e i ied by a com-
pa ison wi h a ull 3D disc model, whe e a pe ec ma ch was eached.
3.3. Compu a ional model o unnelling c acking
Ini ia ion o a unnelling c ack was simula ed on a symme ic pla e
composed o h ee laye s. The ou e laye s we e made o ma e ial wi h
smalle CTE (A15MZ), o induce comp essi e s esses and he inne laye
wi h la ge CTE (A0) esul ing in a o ma ion o ensile esidual s esses
(Table 1). The hickness o ou e laye s was kep a 500 µm and he
hickness o he inne laye a ied be ween 20 µm and 300 µm. Leng h
and wid h we e adjus ed o a oid in luence o edges on he laye cen e.
By exploi ing he h ee old symme y, only 1/8 o he ac ual geome y
was su icien o he modelling o he p oblem. I was assumed ha he
c ack has a ci cula o ellip ic shape ( he so called “penny-shape c ack”),
loca ed in he middle o he inne laye wi h aces o ien ed in pa allel
wi h he x-y c oss-sec ion (Fig. 5a). The wo axes desc ibing he c ack
size a e a
x
and a
z
. The loca ion o he c ack was in he egion o ho-
mogeneous ensile s ess a om he ee su ace, whe e he ensile
s ess sligh ly dec eased (bu no anished).
The geome y was disc e ized by 3D hexahed al quad a ic elemen s
(SOLID186). The icini y o he c ack on was swep by elemen s wi h
shi ed nodes close o he c ack on o be e cap u e he singula i y
(Fig. 5b). Iden ical mesh opology o models wi h and wi hou c ack
was kep again o a oid spu ious ene gy a ia ions. Besides he em-
pe a u e change only displacemen bounda y condi ions on he sym-
me y planes we e p esc ibed. Namely, displacemen s on all nodes on
each symme y plane we e ixed in he no mal di ec ion (excep hose
which a e inside he ellip ical c ack in y-z plane o enable opening o he
c ack).
Fig. 3. (a, c) Schema ics o he mul ilaye designs. (b, d) Residual s ess p o iles along pa hs A and B in manu ac u ed specimens (b,d) o size L×W×H. S esses
σ
a e no malised by he in-plane esidual s ess
σ
in and he pa h leng hs a a e no malised by he associa ed laye hickness inne
A0.
Fig. 4. (a) C oss-sec ion o he specimen wi h adius R in he plane o c ack wi h leng h a; (b) de ail o he ini e elemen mesh wi h a de ail o he ensile ( ed) ou -o -
plane s ess
σ
zz a he ee su ace o inne laye .
R. Papˇ
sík e al.
Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2932
3.4. Fini e ac u e mechanics and coupled c i e ion
To ind condi ions upon which he c ack will ini ia e, he coupled
s ess-ene gy c i e ion was employed [18]. Fi s , he s ess condi ion
mus be ul illed all along he p ospec i e c ack pa h (s esses no mal o
he c ack plane mus be highe han he ensile s eng h
σ
c) and subse-
quen ly he inc emen al ene gy elease a e Ginc mus be highe han i s
c i ical alue G
c
. The inc emen al ene gy elease a e is gene ally
calcula ed as he di e ence be ween he po en ial ene gy Π o he body
wi h and wi hou a c ack o size/su ace A, espec i e, di ided by his
su ace as s a ed in he ollowing ela ion:
Ginc := − Π(A) − Π(0)
A(2)
In he case o ci cum e en ial edge c acks he inc emen al ene gy
elease a e was calcula ed as ollows:
Ginc = − Π(a) − Π(0)
π
R2−
π
(R−a)2(3)
whe e Π(a)and Π(0)a e he po en ial ene gies [in J] o he disc wi h and
wi hou a c ack, espec i ely. The a ea o he c ack (gi en in he de-
nomina o o equa ion (3)) is ha o an annulus. Fo each combina ion
o he olume a io and he laye hickness, he coupled c i e ion was
e alua ed acco ding o a g aphical demons a ion shown in Fig. 6. As he
empe a u e dec eases a e he sin e ing p ocess, esidual s ess in he
ou -o -plane di ec ion
σ
zz a ise on he ee edges o he lamina e. In
Fig. 6a,
σ
zz and Ginc a e plo ed o wo empe a u e di e ences. A
−700 ◦C, he s ess ep esen ed by he (g een) dashed cu e does no
o e come he s eng h, no does he Ginc ep esen ed by ( iole ) do ed
cu e o e come GC. The coupled c i e ion is ul illed only a e signi i-
can dec ease o empe a u e. In Fig. 6b, he Ginc a e plo ed o c ack
pa hs no malised by he laye hickness in which he c ack is loca ed and
p opaga es. Al hough he
σ
zz s esses along he p ospec i e c ack pa h
a e he same in all 3 laye s, only in he hickes (pu ple) laye he ene gy
condi ion is a ou able o he c ack onse since i s alues exceed G
c
(o
1 in he no malized g aph) o he same c ack leng h as he s ess con-
di ion exceeds
σ
c alue.
Since, o his pa icula case, i is no impo an a which empe a-
u e a c ack ini ia es, bu only i i ini ia es, we can e alua e he coupled
c i e ion only o he wo s -case s a e eached wi h he maximum em-
pe a u e di e ence ΔTmax = − 1450℃.
In he case o unnelling c ack, he inc emen al ene gy elease a e
was calcula ed as ollows:
Ginc = − Π(A) − Π(0)
π
axaz
(4)
whe e Π(A)and Π(0)a e he ene gies [in J] o he whole body (mul i-
plied om he eigh olume o he 3- old symme ic model) wi h and
wi hou a c ack espec i ely. The exp ession in he denomina o o
equa ion (4) is he a ea o an ellip ic c ack wi h majo and mino axis a
x
and a
z
, espec i e.
Fig. 7 illus a es he e alua ion o he coupled c i e ion o wo laye s
wi h hickness 100 µm (Fig. 7a) and 300 µm (Fig. 7b), espec i ely. The
s ess no malised by s eng h is ep esen ed by he g een su ace and he
ed and iole su aces ep esen s no malised Ginc. In Fig. 7a he s ess
c i e ion is ul illed in he hicke laye a a empe a u e di e ence ha
is smalle han which can be achie ed by cooling om sin e ing and a
he same ime he ene gy c i e ion is ul illed also (bo h su aces a e
abo e alue 1) and hus an ellip ic c ack may o m. In Fig. 7b he
s eng h c i e ion has been ul illed in he hinne laye a maximal
achie able empe a u e bu despi e ha , Ginc is unable o each Gc ( i-
ole su ace o e coming alue 1) e en in an ex eme case, when he
Fig. 5. (a) A c oss-sec ion ske ch o geome y wi h a po en ial ellip ic c ack. (b) De ail o mesh a ound he c ack icini y exploi ing 3- old symme y.
Fig. 6. E alua ion o he coupled c i e ion o edge c ack: (a) in laye o cons an hickness and cons an olume a io o inc easing empe a u e and (b) in laye s o
di e en hicknesses a cons an olume a io and maximal achie able empe a u e di e ence.
R. Papˇ
sík e al.

Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2933
ini ial ellip ic c ack would ex end ac oss he whole laye hickness and
had ex eme aspec a io. Thus, o ma ion o unnelling c acks is
p e en ed.
Necessa y inpu s in o he coupled c i e ion, such as s esses along he
p ospec i e c ack pa h o he inc emen al ene gy elease a e, as a
unc ion o he c ack leng h, we e calcula ed using he ini e elemen
simula ion in Ansys Mechanical [19]. All essen ial esul s we e expo ed
o he subsequen pos -p ocessing in he ma hema ical so wa e Ma lab
[20], whe e he ul ilmen o CC was e alua ed.
4. Resul s and discussion
4.1. Edge c acks
Resul s o he pe o med pa ame ic s udy a e shown in Fig. 8a. I
shows a cha o wo egions sepa a ed by a black solid cu e de ining
he c i ical combina ion o laye hickness and le el o esidual
comp essi e s esses leading o ini ia ion o edge c acking. Posi ion and
shape o his cu e depend p ima ily on he alues o ma e ial ac u e
mechanics cha ac e is ics – namely o he ensile s eng h and ac u e
oughness – see also [3]. In he uppe -le egion, he coupled c i e ion is
ul illed and condi ions o he edge c ack o ma ion a e hus a ou able.
In he lowe - igh egion, nei he s ess o ene gy c i e ion a e ul illed,
hence, edge c acks should no ini ia e.
The a chi ec u e in Fig. 8b was chosen such ha laye hicknesses
co esponded o ull symbols (colou ed ma ke s) in Fig. 8a. One can see
ha edge c acks indeed ini ia e in hick laye s ( ep esen ed by a ed
iangle) – as he coupled c i e ion is ul illed and no edge c acks we e
obse ed in he hin laye ( ep esen ed by he g een ci cle) e en hough
he comp essi e s ess magni udes in bo h laye s we e he same. In he
laye wi h an in e media e hickness (depic ed by a yellow squa e) no
c acks we e obse ed, since he ensile s eng h o he laye was no
eached. This c ack ex ended along he whole ee su ace and in o a
ce ain dep h which is o he o de o he laye hickness, as expe i-
men ally e idenced by subsequen ly polishing he specimens om he
side.
4.2. Tunnelling c acks
The esul o pe o med simula ions analysing condi ions o he
unnelling c ack o ma ion is depic ed in Fig. 9a. The g aph a ea is
di ided by a e ical line co esponding o 63 d quan ile o he
measu ed s eng h (63% p obabili y o ailu e). In-plane s esses le
om his line a e so high ha i is almos gua an eed ha he s ess
c i e ion be ul illed. Righ om hese lines, he s esses a e so low ha i
is e y unlikely ha he s ess c i e ion be sa is ied. The cha is u he
spli by a hick black cu e, abo e which he ene gy c i e ion is ul illed.
A c oss-sec ion showing he ee su ace o he manu ac u ed spec-
imen is shown in Fig. 9b. The unnelling c ack is clea ly isible in he
hickes laye (depic ed by a ed ci cle), while he o he (depic ed by
yellow squa es and blue iangles) emain wi hou c acks. Su ace laye s
(depic ed by g een s a s) we e also ac u ed, howe e p edic ion o
hese su ace c acks was no modelled in his wo k.
By compa ing Figs. 9a and 9b a good ag eemen be ween p edic ions
and empi ical obse a ions was ound, which demons a es ha he
coupled c i e ion is a powe ul and applicable ool o designing c ack-
Fig. 7. E alua ion o he coupled c i e ion o he unnelling c ack (a) in a 300 µm hick laye a lowe han maximal achie able empe a u e; (b) in a 100 µm hick
laye a maximal achie able empe a u e.
Fig. 8. (a) Regions o ul ilmen /non- ul ilmen o CC o edge c ack, (b) SEM images o he specimen wi h e idence o edge c acking. Symbols o di e en shape and
colou s ep esen di e en laye hicknesses.
R. Papˇ
sík e al.
Jou nal o he Eu opean Ce amic Socie y 43 (2023) 2928–2934
2934
ee ce amic componen s. We cau ion he eade ha he model o
unnelling c acks de eloped he e does no s udy c ack o ma ion in
su ace laye s, whe e s ess dec eases signi ican ly; his will be
add essed in he u u e wo k.
5. Conclusion
This wo k demons a es he abili y o he coupled s ess-ene gy c i-
e ion o p edic he ini ia ion o edge o unnelling c acks in a bi-
ma e ial laye ed ce amic a chi ec u e. The size e ec ( hickness) in in-
di idual laye s is go e ned by he ul ilmen o he coupled s ess-ene gy
c i e ion, no only by he s ess o he ene gy c i e ion alone. Fo bo h
edge and unnelling c acks, he e exis s a egion whe e c ack may no
ini ia e as a consequence o he ene gy c i e ion no being ul illed e en
i he s ess eached he s eng h ha co esponds o 99% p obabili y o
ailu e (99 h quan ile o he Weibull s eng h dis ibu ion). An ad an-
age o he coupled c i e ion is ha i only equi es he ac u e ough-
ness, he ensile s eng h and he elas ic ma e ial p ope ies o laye s,
whe e hese c acks a e in es iga ed. Resul s he ein can be used as a
guide o designing componen s ha ing no p ocessing c acks induced
upon he cooling down p ocess om he sin e ing empe a u e. Expe -
imen al obse a ions showed a good ag eemen wi h he p esen ed nu-
me ical models and con i m he abili y o he coupled s ess ene gy
c i e ion in p edic ing c ack o ma ion in laye ed ce amics designed
wi h esidual s esses.
Decla a ion o Compe ing In e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgemen s
Funding o his esea ch was p o ided by he Eu opean Resea ch
Council (ERC) excellen science g an “CERATEXT” h ough he Ho izon
2020 p og am unde con ac 817615.
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Fig. 9. (a) Cha illus a ing egions o ul ilmen o CC o unnelling c acks. (b) SEM images o he specimen wi h e idence o unnelling c acks (ci cled). Symbols o
di e en shapes and colou s ep esen di e en laye hicknesses.
R. Papˇ
sík e al.