Modeling complex crack paths in ceramic laminates: A novel variational framework combining the phase field method of fracture and the cohesive zone model

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 O iginal A icle Modeling complex c ack pa hs in ce amic lamina es: A no el a ia ional amewo k combining he phase field me hod o ac u e and he cohesi e zone model V. Ca ollo a,⁎ , J. Reinoso b,a , M. Paggi a a IMT School o Ad anced S udies Lucca, Piazza San F ancesco 19, 55100 Lucca, I aly b Elas ici y and S eng h o Ma e ials G oup, School o Enginee ing, Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092 Se ille, Spain ARTICLE INFO Keywo ds: Phase field model o ac u e Cohesi e zone model C ack deflec ion C ack b anching Lamina es ABSTRACT The compe i ion be ween c ack pene a ion in he laye s and cohesi e delamina ion along in e aces is he ein in es iga ed in e e ence o lamina e ce amics, wi h special a en ion o he occu ence o c ack deflec ion and c ack b anching. These phenomena a e simula ed acco ding o a ecen a ia ional app oach coupling he phase field model o b i le ac u e in he laminae and he cohesi e zone model o quasi-b i le in e aces. I is shown ha he p oposed a ia ional app oach is pa icula ly sui able o he p edic ion o complex c ack pa hs in- ol ing c ack b anching, c ack deflec ion and cohesi e delamina ion. The effec o diffe en in e ace p ope ies on he p edic ed c ack pa h o uosi y is in es iga ed and he abili y o he me hod o simula e ac u e in laye ed ce amics is p o en in ela ion o expe imen al da a aken om he li e a u e. 1. In oduc ion Ce amic ma e ials a e la gely used in echnological applica ions, especially wi h he aim o achie ing a desi ed esis ance o se e e wea o co osi e phenomena a high empe a u es. Howe e , he main d awback o ce amics ega ds hei b i leness and, o inc ease hei oughness, lamina es a e o en used al e na ing ce amic and me allic laye s. Fo ins ance, in [1,2], Al/SiC and Al/TiN lamina es ha e been explo ed and es ed. The me allic laye s make he composi e able o wi hs and highe de o ma ions by means o he de elopmen o plas- ici y a se e al loca ions wi hin he specimen, and he e o e inc easing he o e all oughness o he lamina e. The same oughening p ocess has been achie ed by al e na ing ce amic laye s wi h polyme ic laye s in [3]. The main d awback o hese solu ions is ha me als and polyme s loose hei mechanical p ope ies a high empe a u es and ha e a low wea esis ance. A possible way o enhance he oughness o ce amics is o in oduce quasi-b i le in e aces [4,5]. Then, a s ack o ce amic laye s al e na ed by hin laye s o a e y b i le ce amic is a possible effec i e echnological solu ion. Such b i le laye s ac as a quasi-b i le in e aces which make he c ack pa h e y complex, hus inc easing he o e all ma e ial oughness by ac ing on he c ack o uosi y. This me- chanism has been fi s ly heo ized in he amewo k o linea elas ic ac u e mechanics (LEFM) by he so called Cook–Go don mechanism [6]. I is he esul o c ack b anching and c ack deflec ion ypical o he compe i ion be ween c ack pene a ion in he laye s and delamina ion along he exis ing in e aces. Ano he app oach o os e complex c ack pa hs is o in oduce po ous laye s be ween he ce amic ones [7]. Howe e , he d awback o po ous ma e ials is hei low wea esis ance. Al e na i ely, ough in e aces wi h p eexis ing de ec s could be in- se ed among he laye s [8]. The ough in e ace can be made o a ce amic ma e ial which gua an ees he esis ance o wea and co o- sion, also a high empe a u es. The de ec s in he ough in e aces gua an ee he de elopmen o c ack deflec ion wi h a consequen oughening o he ma e ial. Conside ing he echnological s a egies he ein desc ibed, he in- oduc ion o an in e ace wi h ailo ed p ope ies clea ly eme ges as a s a egy o c ea e a complex c ack pa e n and consequen ly enhance he appa en ma e ial oughness. In his a icle, we in es iga e hese possibili ies by examining and modelling he in e ac ion be ween c ack pene a ion in he laye s and cohesi e delamina ion a he in e aces be ween ma e ials wi h diffe en elas ic and ac u e p ope ies. This p oblem has been ma hema ically add essed in many publica ions using diffe en me hodologies, c . [9–11]. Diffe ing om p e ious in es iga- ions, his p oblem is he ein analysed by means o he nume ical me hod published in [12], which has been ecen ly ex ended o a 3D fini e de o ma ion amewo k in [13] and applied o ac u e o ani- so opic polyc ys alline Silicon in [14]. The no el y o he cu en ap- p oach elies upon he inno a i e a ia ional amewo k combining he h ps://doi.o g/10.1016/j.jeu ce amsoc.2018.01.035 Recei ed 30 No embe 2017; Recei ed in e ised o m 21 Janua y 2018; Accep ed 22 Janua y 2018 ⁎ Co esponding au ho . E-mail add ess: [email p o ec ed] (V. Ca ollo). Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 A ailable online 02 Feb ua y 2018 0955-2219/ © 2018 The Au ho (s). Published by Else ie L d. 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/). T phase field me hod o ac u e and he cohesi e zone model. In pa i- cula , he ole played by he in e nal cha ac e is ic leng h scales o he wo app oaches is igo ously analysed in o de o unde s and hei e - ec on he esul ing c ack pa h and i s o uosi y as a way o enhance he o e all composi e oughness. The manusc ip is o ganized as ollows. Sec ion 2summa izes he p incipal ea u es o he a ia ional amewo k he ein employed. In Sec ion 3, he nume ical me hod is applied o p edic he c ack pa h in lamina es. Special a en ion is de o ed o examining he effec o ough and quasi-b i le in e aces and ep oducing also expe imen al esul s ela ed o a ce amic lamina e aken om li e a u e. Finally, he main conclusions o he cu en in es iga ion a e d awn in Sec ion 4. 2. Va ia ional model In his sec ion, he coupled phase field and cohesi e zone model o mula ion de eloped in [12] is esumed. We fi s p esen he unda- men al hypo hesis o he cu en coupling app oach in Sec ion 2.1. La e , he phase field me hod o b i le ac u e o mula ion and he cohesi e zone model compa ible wi h he phase field model a e ou - lined in Sec ions 2.2 and 2.3, espec i ely. Finally, he co esponding fini e elemen o mula ion wi hin he infini esimal de o ma ion se ing is de i ed in Sec ion 2.4. 2.1. Fundamen al hypo hesis The o mula ion he ein p esen ed is de eloped in he gene al Euclidean space o dimension n dim unde infini esimal de o ma ion se ing. Le us conside a body ∈Ωℝ ndi m wi h a gene ic shape, whe e he bounda ies o he body a e deno ed by ∂ ∈ − Ωℝ n 1 dim (Fig. 1). Ki- nema ic and ac ion bounda y condi ions can be espec i ely p e- sc ibed on he disjoin ed pa s o he bounda ies ∂Ω u and ∂Ω (wi h ∂Ω ∪∂Ω u =∂Ωand ∂Ω ∩∂Ω u =∅). Then, he p esc ibed displace- men s and ac ions a e deno ed by: =∂ = ∂ u u σnon Ω and · on Ω , u (1) whe e nis he ou wa d no mal uni ec o o he body, and σis he Cauchy s ess enso . The body o ces a e ep esen ed by he unc ion → :Ω ℝ ndi m . The composi e is cha ac e ized by quasi-b i le in e - aces, Γ i , and a c ack in he laye ep esen ed as an in e nal dis- con inui y, Γ b (see Fig. 1(a)). A gene ic poin in he bulk o he body is deno ed by he ec o o i s Ca esian coo dina es x, while a gene ic poin on he in e ace Γ i is deno ed by he ec o x c . The ee ene gy unc ional which go e ns he mechanics o he body Ωis defined as [15,16]: ∫∫ =+= + ∖ψuu ε Π (,Γ) Π(,Γ) Π(Γ) ()dΩ dΓ, ecΩΓ ΩΓ Γ G (2) whe e ψ e (ε) is he elas ic ene gy densi y, εis he s ain field, and c G is he ac u e ene gy. The main idea o couple he phase field app oach o b i le ac u e and he cohesi e zone model is o spli he ac u e ene gy unc ion c G in wo pa s. One pa ( c b G ) desc ibes ac u e in he laye s and i is modelled by he phase field app oach. The second pa ( i G ) desc ibes he cohesi e ac u e o he in e aces and i is modelled by he cohesi e zone app oach. Then, he ee ene gy unc ional in Eq. (2) can be e- w i en as: ∫∫ ∫ =++= + +∖ψuεu g Π( , Γ , Γ) Π Π Π ( )dΩ ( , )d Γ (,, )dΓ, bi ec b i ΩΓ ΓΩΓ Γ Γ bib i G G d hd (3) whe e gdeno es he ec o o displacemen discon inui ies a he in- e ace, h is an his o y pa ame e as in [17] o a oid e-healing o he ma e ial and he non uniqueness o he solu ion, and d is he phase field deg ada ion a iable which will be u he de ailed in he nex sec ion. 2.2. Phase field app oach o b i le ac u e The phase field app oach o b i le ac u e [15,18] is a a ia ional app oach which conside s a c ack as a diffuse damage ins ead o a sha p discon inui y (Fig. 1(a)). Wi hin his amewo k, he po en ial ene gy o he bulk is o mula ed as ollows: ∫∫ =+∇ψγuε Π (, ) (, )dΩ (, )dΩ, bc bx ΩΩ G dd dd (4) Fig. 1. Schema ic ep esen a ion o an a bi a y body wi h a discon inui y in he domain and an in e ace: (a) Le : disc e e discon inui y in he domain. Righ : smea ed discon inui y in he domain based on he phase field concep . (b) Diffusi e c ack modeling solu ion o he one-dimensional c ack p oblem. V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 2995 whe e ψ ε(, ) dis he elas ic ene gy s o ed in he bulk, he symbol ∇ x • deno es he spa ial g adien ope a o , and ∇γ(, ) x dd is he so called c ack densi y unc ional which eads [15]: ∇= +∇γl l (, ) 1 22 || xx 22 dd d d (5) whe e lis he phase field in e nal leng h pa ame e go e ning he sha pness o he c ack acco ding o he equa ion p esen ed in Fig. 1(b) o he mono-dimensional case. The elas ic ene gy in Eq. (4) akes he ollowing o m: =+ +− ψ ψψεεε(, ) () () () , ee dgd (6a) =+ +++ ψ λμεε ε() 2( [] ) [ ] , e22 (6b) =+ −−− ψ λμεε ε() 2( [] ) [ ] , e22 (6c) whe e λand μa e he Lamé cons an s, [•] deno es he ace ope a o , ϵ + and ϵ − deno e he posi i e and nega i e coun e pa o he s ain enso , espec i ely, and g()dis a deg ada ion unc ion: =− +() (1 ) . 2 K gd d (7) In Eq. (6a), he elas ic ene gy has been spli in i s posi i e and ne- ga i e coun e pa s acco ding o he o mula ion in [19,20]. The posi- i e coun e pa o he elas ic ene gy is p oduced by he ensile s esses while he nega i e coun e pa is p oduced by comp ession. Posi i e and nega i e s esses can be compu ed by he de i a i e o he elas ic ene gy wi h espec o he s ain enso . Then, he in oduc ion o he posi i e and nega i e spli o he s ain enso leads o: =∂ ∂=+ = + +− ± ±± ψλμσεσσ σ ε1 ε:ˆ() ; wi h ( [] ) 2 , gd (8) Then, since he deg ada ion unc ion in Eq. (7) affec s only he posi i e coun e pa , damage can only de elop when he ma e ial is unde ension, a oiding damage g ow h in comp ession. 2.3. Cohesi e zone model compa ible wi h he phase field In his sec ion, he classical linea cohesi e zone model wi h ension cu -off[21] is pa icula ized in o de o ake in o accoun he effec o he bulk damage d . Fi s o all, he cohesi e coun e pa o he ac u e ene gy in Eq. (3) is decomposed in he sum o he Mode I and Mode II ac u e ene gies, I G and II G , espec i ely. Based on he o mula ion ou lined in [12], he c i ical c ack opening displacemen (g c ) depends on he bulk damage d acco ding o he linea ela ion =− +ggg() (1 ) ccc,0 , 1 ddd , whe e ==gg(0 ) cc,0 d and ==gg(1 ) cc,1 d. Then, he cohesi e ac ion s. ela i e displacemen laws o Mode I and Mode II ake he o m shown in Fig. 2, and a e desc ibed by he ollowing equa ions: =⎧ ⎨ ⎩ << ≥=⎧ ⎨ ⎩ << ≥ σ k τ k,i 0 1; 0, i 1, ,i 0 1; 0, i 1. n g g g g g g g g g g g g nn n nc nc nc c c c (9) whe e σand τa e he Mode I and Mode II ac ions, espec i ely, gis he ela i e displacemen , and he subsc ip nand e e s o opening and sliding, espec i ely. The s iffness o he cohesi e ela ion, k, depends on damage d acco ding o he o mulae: == () () kk kk,. nn g g g g ,0 2 ,0 2 nc,0 nc c,0 c (10) whe e k 0 and g 0 a e, espec i ely, he s iffness and c i ical ela i e displacemen s a =0 d . In his o mula ion, among he diffe en po en ial al e na i es, we ha e adop ed he hypo hesis o keeping cons an he c i ical ene gy elease a e o he in e ace, c i G , o many easons. Fi s ly, he model is o mula ed s a ing om he G iffi h ene gy balance c i e ion, hen in Eq.(3) we se up a clea and explici spli be ween he dissipa ed ene gy due o he bulk ac u e and o he in e ace delamina ion. Co e- spondingly, c i G becomes he ma e ial pa ame e ha go e ns he in- e ace ailu e. This choice endows a clea cha ac e iza ion o he in- e ace ac u e ene gy. In ac , among he ou pa ame e s en e ing he cohesi e law, c i G can be ob ained in a s aigh o wa d manne om expe imen al es s [22]. Finally, he mixed mode ailu e c i e ion p oposed in [23] is he e- wi h conside ed o igge he in e ace ailu e: ⎜ ⎟⎜⎟ ⎛ ⎝ ⎞ ⎠+⎛ ⎝⎞ ⎠=1 , I i i i i IC 2 II IIC 2 G G G G (11) whe e I i G and i II G a e he dissipa ed ac u e ene gies which ake he o m: == −+ −+ ng kg() , () . I i n g gg i g gg 1 2,0 2 [(1 ) ] II 1 2,0 2 [(1 ) ] nc,0 2 nc,0 nc,1 2 c,0 2 c,0 c,1 2 GG dd dd dd (12) The c i ical ac u e ene gies i IC G and i IIC G a e: ==gk gk,. ini IC 1 2nc,0 2,0 IIC 1 2 c,0 2,0 GG (13) 2.4. Fini e elemen o mula ion The fini e elemen o mula ion o he p e ious ac u e mechanics models is he ein de i ed. Fi s o all, he weak o m o he ee ene gy unc ional in Eq. (3) is deduced using he s anda d Gale kin p ocedu e. Then, he a ia ion o he bulk ene gy unc ional (Eq. (4)) wi h espec o he displacemen s uand he phase field a iable d akes he o m: Fig. 2. Schema ic ep esen a ion o he cohesi e zone model coupled wi h he phase field a iable o b i le ac u e in he bulk. (a) Mode I CZM ac ion σ s. g n . (b) Mode II CZM ac ion τ s. g . V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 2996 ∫∫ ∫ =−− + ⎡ ⎣+∇ ∇ ⎤ ⎦ +∀ + ψ δδ uu σε ε uu u δΠ ( , δ , d, δd) :δ dΩ 2(1 d)δd ()dΩ l·()dΩ δΠ ( , δ ) δ , δd, be c b l b xx ΩΩ Ω 1 ,ex 2 G dd d d (14) whe e ∈= = ∂ ∈δδuuuuu{| onΩ, } uu1 H V is he ec o o he dis- placemen es unc ion, and ∈= = ∈δδδ{| 0onΓ, } b0 H dV dd d dis he damage es unc ion. The con ibu ion o he ex e nal o ces in he a ia ion o he bulk ene gy unc ional is defined as ollow: ∫∫ =∂+ ∀ ∂ δδ δ δδδuu u u uΠ(,) ·dΩ ·dΩ , . b ,ex ΩΩ d (15) Finally, he a ia ion o he in e ace ene gy unc ional Π Γ iin Eq. (3) is defined as: ∫ ⎜⎟ =⎛ ⎝∂∂+∂∂⎞ ⎠∀δδδ δ δ δδuu u uuuuΠ(, , , ) (, ) (, ) dΓ , . ii ΓΓ ii GG dd dd ddd (16) The phase field model has been implemen ed wi hin a 4-node iso- pa ame ic fini e elemen . The cohesi e zone model has been im- plemen ed using a 4-node in e ace fini e elemen . The de ailed fini e elemen implemen a ion in he so wa e FEAP [24] and all he ela ed ope a o s can be ound in [12] and a e omi ed he e o he sake o b e i y. 3. Simula ion o complex c ack pa hs in laye ed ce amics The p edic ion o he c ack pa h in ce amic lamina es is a e y challenging issue. Va ious ac o s such as he elas ic misma ch o he cons i uen ma e ials and he p ope ies o he in e aces [23,25] make he c ack p opaga ing along o uous pa hs which a e e y difficul o be simula ed using p e ious nume ical me hods. The p esen app oach esol es mos o hei d awbacks p ima ily associa ed o emeshing o co ne -case p oblems. 3.1. Complex c ack pa hs o lamina es unde ensile loading: analysis o ailu e pa e ns wi h quasi-b i le o ough in e aces In his sec ion we s udy he effec o he in e ace oughness on he esul ing c ack pa h. The case o a single-edge no ched bi-ma e ial la- mina e unde ension (Fig. 3) is he ein conside ed. The lamina e has been modelled using he phase field fini e elemen s o he bulk and he in e ace fini e elemen s compa ible wi h he phase field be ween each laye . The ma e ials which compose he lamina e a e: a so ma e ial 1 wi h high ac u e oughness; a s iffma e ial 2 wi h low ac u e oughness. The ma e ial pa ame e s a e collec ed in Table 1. Th ee cases ha e been examined: (1) lamina e wi h pe ec ly bonded laye s; (2) lamina e wi h ough in e aces; (3) lamina e wi h b i le in e aces. No e ha he oughness o he in e aces is always la ge han he oughness o he ma e ials composing he laminae, o simula e configu a ions consis en wi h echnological solu ions o ce amics lamina es. In he fi s simula ion, no in e ace elemen s a e in oduced in o de o simula e ully bonded laye s. The phase field in e nal leng h lis se e y small o bo h laye s (see Table 1), o ep oduce LEFM p edic ions as discussed in [26]. The esul s o he simula ions a e shown in Fig. 4. The fi s laye ha shows c ack nuclea ion is he second one (ma e ial 2). In such a laye , wo pa allel c acks a e p edic ed o p opaga e si- mul aneously (Fig. 4(a)) and can be conside ed as wo b anches o he ini ial no ch in he laye 1. A e inc easing he applied load, each c ack in he second laye u he b anches in he nex b i le laye (Fig. 4(b)). The same p ocess con inues o he nex b i le laye bu only wo b anches a e now de eloped (Fig. 4(c)). A his s age, he c acks in he b i le laye s s a connec ing h ough he ma e ial 1. Finally, ailu e o he specimen is achie ed (Fig. 4(d)) and i is he esul o a complex c ack pa h mos ly localized in he mid-span c oss-sec ion o he specimen. In he second simula ion, we in oduce in e ace elemen s be ween each laye . The pa ame e s used o he in e ace a e σ c,0 =τ c,0 = 100 MPa and k 0 = 2000 MPa/mm o model a s iffquasi- b i le in e ace. As expec ed, he e olu ion o he p edic ed c ack pa h is e y diffe en om ha o he p e ious simula ion. The fi s laye whe e c acks nuclea e is he second one (Fig. 5(a)) wi h he same pa e n as in he fi s simula ion. Immedia ely a e ha , delamina ion along he in e ace be ween he fi s and he second laye akes place (Fig. 5(b)). Then, he fi s laye is c acked by he p opaga ion om he Fig. 3. Specimen geome y. Table 1 Geome y and ma e ial/in e ace pa ame e s. Geome y pa ame e s L6 mm Specimen leng h l y 0.25 mm Laye hickness h0.005 mm In e ace hickness Mechanical pa ame e s ma e ial 1 E 1 70,000 MPa Ma e ial 1 Young modulus 1 0.34 Ma e ial 1 Poisson a io 1 G 0.025 N/mm Ma e ial 1 ac u e ene gy l 1 0.0075 mm Ma e ial 1 phase field leng h scale pa ame e Mechanical pa ame e s ma e ial 2 E 2 300,000 MPa Ma e ial 2 Young modulus 2 0.14 Ma e ial 2 Poisson a io 2 G 0.005 N/mm Ma e ial 2 ac u e ene gy l 2 0.0075 mm Ma e ial 2 phase field leng h scale pa ame e Mechanical pa ame e s in e ace k 0 2000 MPa/mm Ini ial s iffness o in e ace σ c,0 ,τ c,0 100 MPa Ini ial peak s ess o he ough in e ace c i G 2.5 N/mm C i ical ene gy elease a e o ough in e ace σ c,0 ,τ c,0 1 MPa Ini ial peak s ess o he b i le in e ace c i G 0.025 N/mm C i ical ene gy elease a e o b i le in e ace σ c,0 /σ c ,τ c,0 / τ c 1 Ra io be ween he ini ial and final alue o he peak s ess o he in e ace V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 2997 no ch (Fig. 5(c)). Con inuing wi h he simula ion, c acking p oceeds in he ma e ial 2 laye s oge he wi h he de elopmen o delamina ion a in e aces (Fig. 5(d)). A ailu e, delamina ion makes he c ack pa e n dis ibu ed along he whole specimen, as a p ima y diffe ence om he esul s o he fi s simula ion. Ano he impo an aspec is ha he majo i y o he ma e ial 1 laye s a e no c acked, apa om he fi s laye con aining he no ch. In he hi d simula ion we in oduce a mo e b i le in e ace, se ing σ c,0 =τ c,0 = 1 MPa and k 0 = 2000 MPa/mm. The e olu ion o he c ack pa h is again qui e diffe en om he p e ious cases (Fig. 6). Thus, in he cu en case, fi s , delamina ion is p edic ed o occu be ween he fi s and he second laye (Fig. 6(a)). Then, he c ack s a s p opaga ing om he no ch un il i impinges on o he delamina ed in e ace (Fig. 6(b)). Subsequen ly, b anching is p edic ed o ake place in he second laye (Fig. 6(c)), and each b anched c ack is de eloped s a ing om he poin s whe e delamina ion was a es ed. Fu he mo e, c ack Fig. 4. C ack e olu ion in he simula ion wi h ully bonded laye s. Fig. 5. C ack e olu ion in he simula ion wi h ough in e ace (σ c,0 = 100 MPa). V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 2998 pene a ion igge s delamina ion along he second in e ace. Finally, he sample ailu e is he esul o a sudden delamina ion along all he in e aces and c acks in he b i le laye s (Fig. 6(d)). Again, he ma e ial 1 laye s ha e been p ese ed om c acking. The o ce–displacemen cu es o he abo e h ee simula ions a e shown in Fig. 7. All o hem p esen an ini ial nonlinea i y due o he o ma ion o damage in he bulk. A e ha , he cu e o he fi s si- mula ion s a s loosing he load-bea ing capaci y slowly wi h a smoo h so ening. This is due o he g adual c ack p opaga ion wi hin he specimen. The cu es co esponding o he second and hi d simula- ions show a comple ely diffe en c ack pa e n and he nonlinea e - ec s a e much mo e p onounced in he load–displacemen cu e due o he occu ence o delamina ion. The de elopmen o c acking and de- lamina ion e en s cause mul iple d ops in he esis an o ce. The co espondence be ween c ack/delamina ion e en s and he d ops in he o ce can be examined closely in Fig. 8. In hese g aphs, he o ce–displacemen cu es a e plo ed oge he wi h o he wo quan- i ies, he o al c ack p opaga ion leng h and he po ion o c ack p o- paga ion leng h in jus he bulk. Bo h quan i ies a e no malized wi h espec o he laye hickness, l y . These plo s show ha , in gene al, delamina ion and c acking occu simul aneously. This is in ag eemen wi h he pa e ns in Figs. 5 and 6, whe e in some cases delamina ion igge s c ack o ma ion in he laye s o ice- e sa. Ano he impo an conside a ion is ha he delamina ion leng h in he simula ion wi h quasi-b i le in e aces is bigge han he in simula ion wi h oughe in e aces. Mo eo e , we compa e he cu en p edic ions wi h he heo e ical esul s p esen ed in [6], which complied wi h he so-called Cook–- Go don mechanism. Acco ding o ha heo y, he in oduc ion o a b i le in e ace should inc ease he appa en ma e ial s eng h. The esul s epo ed in Fig. 7 seem o challenge his heo y, since a b i le in e ace educes he appa en ma e ial s eng h as compa ed a oughe one. This con adic ion can be explained by no ing ha he assump ions o he Cook–Go don model a e no ully sa isfied in he p esen com- pu a ional se ing. One o he mos ele an diffe ences ega ds he ac ha in he Cook–Go don model he ma e ial is conside ed homo- geneous and linea elas ic un il he condi ion o ac u e o he in e ace is achie ed. A e delamina ion, discon inui ies a e conside ed along he in e ace and, consequen ly, he in e ace s a s in e e ing wi h he linea elas ic ac u e mechanics c ack ip s ess dis ibu ion. In he p esen model, on he o he hand, in e aces a e in oduced as an ad- di ional complian ma e ial om he e y beginning o he simula ion. In [12] i has been shown ha he beha iou o such a sys em depends on he a io be ween he in e ace p ocess zone size and he bulk ene gy dissipa ion zone size (l CZM /l). Depending on his a io, he appa en s eng h o he o e all ma e ial anges be ween he s eng hs uled by he ollowing limi models: a model wi h p e ec bonded in e aces and fini e l; a model wi h elas ic bulk ma e ial and fini e l CZM . In ou si- mula ions, he a io l CZM /lis kep cons an . In ac , he bulk p ope ies do no change ( hen lis cons an ), and l CZM is also cons an since kis fixed. This la e aspec is a consequence o dimensional analysis con- side a ions in [12] om which we deduced ha ∝=lEσEk//(2) c i CZM max 2 G . The consequence o ha ing a cons an a io l CZM /lis ha delamina ion is igge ed ea lie in he p esence o a b i le in e ace. Due o his, we ha e also longe delamina ion pa hs o a b i le in e ace. These conclusions lead o he second impo an diffe ence wi h e- spec o he Cook–Go don model, whe e he load is supposed o be applied as a emo e ensile s ess a infini y. The assump ion o infini e plane plays an impo an ole, since he e is no cons ain on he size o he delamina ion pa h. As a esul , a b i le in e ace could lead o a longe delamina ion wi h a consequen delayed c ack p opaga ion, in- c easing he elas ic ene gy elease a e. The specimens ha we simu- la ed a e no long enough o be conside ed as infini e. Due o ha , when he in e ace is b i le, delamina ion eaches he bounda ies o he specimen causing he ailu e o he whole in e ace. This is also wha accele a es he final ailu e o he specimen. As a esul , o a b i le in e ace, he appa en s eng h o he specimen esul s lowe han o a mo e duc ile one. Finally, he e a e o he assump ions made in he Cook–Go don model ha a e no in line wi h he hypo heses o he p esen app oach. Fo ins ance, Cook and Go don neglec ed he effec o he e ical s ess σ y in hei in e ace ac u e c i e ion. They made his simplifica ion Fig. 6. C ack e olu ion in he simula ion wi h b i le in e ace (σ c,0 = 1 MPa). V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 2999 because hei sys em is an homogeneous ma e ial wi h an in e ace, while he e we ha e a lamina e wi h diffe en elas ic p ope ies o he laminae. This means ha σ y is no cons an ac oss he laminae and he e a e jumps in co espondence o he in e aces. We canno say quan i- a i ely how much is he effec o he a ia ion o σ y on he in e ace delamina ion and on he o e all s eng h. Ne e heless, we ha e good mo i a ions o belie e ha σ y canno be neglec ed and could be ano he impo an sou ce o possible disc epancies wi h espec o he Cook–Go don model. 3.2. C ack p opaga ion in Si 3 N 4 /BN mic o-lamina e In his sec ion, we ep oduce he expe imen al esul s in [4] con- ce ning he 4-poin bending es o a silicon ni ide/bo on ni ide (Si 3 N 4 /BN) mic o-lamina e. Bo h cons i uen ma e ials a e b i le ce amics. The lamina e is s uc u ed wi h laye s o Si 3 N 4 o hickness be ween 40 μm and 60 μm, al e na ed by laye s o BN o a iable hickness be ween 2 μm and 10 μm(Fig. 9(a)). The BN laye s ac as quasi-b i le in e ace be ween he Si 3 N 4 laye s. The 4-poin bending es geome y is shown in Fig. 9(b). The di- mensions o he specimen a e: o al span L= 5 mm, hickness T= 3 mm, wid h W= 4 mm. The ou e and inne span o he 4-poin bending es a e, espec i ely, S 1 = 4 mm and S 2 = 2 mm. This geo- me y is disc e ized wi h phase field fini e elemen s o he Si 3 N 4 laye s, ep esen ing he bulk ma e ial, and cohesi e in e ace fini e elemen s compa ible wi h phase field o he BN laye s. Due o he a iable hickness o he Si 3 N 4 laye s, in ou simula ion he hickness associa ed o he laye s o his ma e ial is se equal o 40, 50 o 60 μm. The as- signmen o he Si 3 N 4 laye hickness is andomly chosen acco ding o a uni o m dis ibu ion. The in e ace hickness, on he o he hand, is se Fig. 7. Fo ce–displacemen cu e o he h ee simula ions. Fig. 8. Fo ce–displacemen cu e compa ed wi h he c ack and delamina ion leng h o he case o σ max = 100 MPa (le ) and σ max = 1 MPa ( igh ) no malized by he laye hickness l y . The le e e e o he co esponding image in Fig. 5 o he case o σ max = 100 MPa and o Fig. 6 o he case o σ max = 1 MPa. Fig. 9. (a) Si 3 N 4 /BN s uc u e o he mic olamina e; (b) 4-poin bending expe imen al es geome y. Image (a) ep in s om [4]. V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 3000 Fig. 10. Nume ically p edic ed s. expe imen al o ce–displacemen cu es. Fig. 11. F ac u e p opaga ion and delamina ion esul ing om he nume ical simula ion. Fo each subfigu e he e is he damage con ou plo on op and he x-displacemen con ou plo on bo om. V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 3001 cons an and equal o 5 μm. The ma e ial and ac u e pa ame e s o he bulk a e: Young modulus E = 310 GPa, Poisson a io ν= 0.27, ac u e ene gy s G =9μN/μm and phase field in e nal leng h scale l=2μm. The in e ace pa ame e s a e: σ c,0 =τ c,0 = 32 MPa and k 0 = 70 MPa/ μm. The expe imen al o ce–displacemen cu e (Fig. 10) shows an in- i ial linea beha iou un il he peak load o 475 N is eached. A e his poin , he load-ca ying capaci y o he specimen d ops down o a ound 40% o he peak load. Then, he load con inue inc easing un il a second d op is obse ed when he alue o 240 N is eached. A e his second d op, he load-ca ying capaci y is educed o 10% o ha a peak load. The specimen main ains his le el un il final ailu e. Fig. 11 shows he e olu ion o he p edic ed c ack pa e n. Due o symme y o he geome y and o he bounda y condi ions, only hal o he domain has been simula ed, using a ound 380.000 fini e elemen nodes and a fine mesh in he egion whe e he c ack is expec ed o p opaga e. The simula ion shows an ini ial linea beha iou un il he peak load o 475N is eached (Fig. 11(a)). Then, an in e ace si ua ed in he middle o he specimen hickness ails due o delamina ion (Fig. 11(b)). A e his fi s delamina ion e en , he specimen con inues gaining load-ca ying capaci y un il he Si 3 N 4 laye s s a ailing. Ac- co ding o he nume ical p edic ions, he laye s ha fi s ail a e ha a he in ados and he one immedia ely nex o he delamina ed in e ace (Fig. 11(c)). The c ack is p edic ed o con inue i s p opaga ion owa ds he ex ados o he specimen un il final ailu e (Fig. 11(d)). The esul o he simula ion in e ms o o ce–displacemen cu e shows a good ag eemen wi h he expe imen al one (Fig. 10). The ini ial pa o he cu e, he peak load and he d op in load-ca ying capaci y a e e y well p edic ed, cap u ing he c i ical load a which damage e en s occu . The final pa o he simula ion shows a mo e p og essi e ailu e e olu ion un il collapse. This is no in pe ec ag eemen wi h expe i- men s, since we suppose ha he second d op in he o ce no iced in expe imen s is caused by a second se e e delamina ion e en which we we e no able o ep oduce nume ically. One possible sou ce o his misma ch can be a ibu ed o he in e ace pa ame e s, since no ex- pe imen al cha ac e iza ion was a ailable. Ano he impo an esul o he simula ion is he e y sa is ac o y p edic ion o he c ack pa e n ea u es con o ming o he expe imen al e idences. The expe imen al image in Fig. 12(a) shows he s ong c ack deflec ion and b anching de eloped due o delamina ion. The same beha iou has been ep oduced by he nume ical simula ion (Fig. 12(b)) whe e bo h phenomena a e p esen in he simula ed c ack pa e n. 4. Conclusions In his wo k, a no el a ia ional amewo k combining he phase field me hod o ac u e and he cohesi e zone model app oach has been applied o modelling complex c ack pa hs in ce amic lamina es wi hin he con ex o he fini e elemen me hod. In pa icula , he p edic i e capabili ies o he p oposed modelling ools ha e been exploi ed in ela ion o ce amic lamina es. The effec o ough o quasi-b i le in e aces has been quan ified and compa ed o he case o ully bonded laye s. Based on he nume ical p edic ions, we ha e nume ically quan ified ha in oducing in e aces is a way o inc ease he o uosi y o he c ack pa h inc easing he ene gy dis- sipa ed be o e ailu e. I has been also iden ified ha he peak cohesi e ac ions go e n he de elopmen o c ack pene a ion, c ack b anching and c ack deflec ion phenomena. Mo eo e , he applicabili y o he cu en compu a ional me hod has been examined h ough i s applica ion o eal lamina es s uc u es composed by Si 3 N 4 /BN laye s. The nume ical p edic ions showed ha he p oposed amewo k enabled ep oducing he key ea u es o he c ack pa e n and he o ce–displacemen cu e obse ed in he ex- pe imen s, p o ing he capabili ies o he p oposed app oach o be used as a ool o he design and cha ac e iza ion o ce amic lamina es. In ligh o he p e ious a gumen s, his wo k is expec ed o p o ide a sui able modelling amewo k o mo e applica ion-based p oblems conce ning c ack pa e n p edic ions in ce amic composi es, among o he al e na i e mul ilaye o he e ogeneous ma e ials. Mo eo e , as a esea ch pe spec i e, a compa ison o he p oposed app oach wi h he me hod p o ided in he heo y o s uc u ed de- o ma ions [27–29] could be o in e es o u he in es iga ions, o p o ide a e-in e p e a ion o he phase field app oach o ac u e in e ms o s uc u ed de o ma ions. Acknowledgmen s The au ho s acknowledge unding ecei ed om he Eu opean Resea ch Council unde he Eu opean Union's H2020 P og amme ERC G an Ag eemen No. 737447 (ERC P oo o Concep 2016 “Pho o ol aic wi h supe io c ack esis ance”–PHYSIC). JR acknowl- edges he suppo o he p ojec unded by he Spanish Minis y o Economy and Compe i i eness/FEDERMAT2015-71036-P and he Andalusian Go e nmen P ojec o Excellence No. P12-TEP-1050. The au ho s would like o hank he anonymous e iewe o his use ul sugges ions. Re e ences [1] M. Sebas iani, K.-E. Johanns, E.-G. He be , F. Ca assi i, G.-M. Pha , A no el pilla inden a ion spli ing es o measu ing ac u e oughness o hin ce amic coa ings, Philos. Mag. 95 (16–18) (2015) 1928–1944. [2] N. Li, H. Wang, A. Mis a, J. Wang, In si u nanoinden a ion s udy o plas ic co- de o ma ion in Al-TiN nanocomposi es, Sci. Rep. 4 (2014). Fig. 12. (a) Expe imen al c ack pa e n; (b) nume ical p edic ions o he c ack pa e n (magnified). Image (a) adap ed om [4]. V. Ca ollo e al. Jou nal o he Eu opean Ce amic Socie y 38 (2018) 2994–3003 3002