Prediction of edge and tunnelling crack formation in layered ceramics using a stress-energy fracture criterion

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. Re e ences [1] R. Be mejo, Z. Chlup, L. ˇ Ses ´ ako ´ a, O. ˇ Se eˇ cek a R. Danze , S a egies o op imize he s eng h and ac u e esis ance o ce amic lamina es,“ Mechanical P ope ies and Pe o mance o Enginee ing Ce amics and Composi es VII, 2012. [2] R. Be mejo, Towa d seashells unde s ess, J. Eu . Ce am. Soc. ol. 37 (2017) 3823–3839. [3] O. ˇ Se eˇ cek, M. Ko oul, D. Leguillon, ´ E. Ma in a R. Be mejo, Modelling o edge c ack o ma ion and p opaga ion in ce amic lamina es using he s ess-ene gy coupled c i e ion, Eng. F ac . Mech., S . 167 (2016) 45–55. [4] S. Ho a Z. Suo, Tunneling c acks in cons ained laye s, J. Appl. Mech., S . 60 (1993) 890–894. [5] S. Ho, C. Hillman, F.F. Lange a Z. Suo, Su ace c acking in laye s unde biaxial, esidual comp essi e s ess, J. Am. Ce am. Soc., S . 78 (1995) 2353–2359. [6] A. Pa izi, K.W. Ga e , aJ.E. Bailey, Cons ained c acking in glass ib e- ein o ced epoxy c oss-ply lamina es, J. Appl. Mech., S . 13 (1978) 195–201. [7] Z. Hashin, Fini e he moelas ic ac u e c i e ion wi h applica ion o lamina e c acking analysis, J. Mech. Phys. Solids, S . 44 (1996) 1129–1145. [8] D. Leguillon, S eng h o oughness?, 13, Eu o. J. . Mech. – A/Solids, S . 21 (2002) 61–72, 13. [9] D. Leguillon, O. ˇ Se eˇ cek, ´ E. Ma in a R. Be mejo, Edge c acking due o a comp essi e esidual s ess in ce amic lamina es, Comp es Rendus M´ ecanique, S . 343 (2015) 192–198. [10] O. ˇ Se eˇ cek, M. Ko oul, D. Leguillon, ´ E. Ma in a R. Be mejo, Unde s anding he edge c ack phenomenon in ce amic lamina es, F a . Ed. In eg i a S u ., S . 9 (2015) 362–370. [11] I. Ga cía, V. Man iˇ c, A. Bl´ azquez, F. Pa ís, T ans e se c ack onse and g ow h in c oss-ply lamina es unde ension, In . J. Solids S uc . ol. 51 (2014) 3844–3856. [12] I. Ga cía, B.J. Ca e , A.R. Ing a ea, aV. Man iˇ c, A nume ical s udy o ans e se c acking in c oss-ply lamina es by 3D ini e ac u e mechanics, Compos. Pa B: Eng., S . 95 (2016) 475–487. [13] EN 843–1: Ad anced echnical ce amics – Monoli hic ce amics – Mechanical p ope ies a oom empe a u e – Pa 1: De e mina ion o lexu al s eng h, B ussel: Eu opean Commi ee o S anda diza ion, 2008. [14] Z. Chlup, H. Had aba, D. D dlík, K. Maca, I. Dlouhý a R. 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[19] Ansys Mechanical, Release 2020 R2. [20] MATLAB. The Ma hwo ks, Inc, Na ick, Massachuse s, 2021. 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.