Strength and reliability of WC-Co cemented carbides: Understanding microstructural effects on the basis of R-curve behavior and fractography

ACCEPTED MANUSCRIPT S eng h and eliabili y o WC-Co cemen ed ca bides: Unde s anding mic os uc u al e ec s on he basis o R-cu e beha io and ac og aphy J. M. Ta agó1,2,ϕ, D. Cou eaux1,3, Y. To es4, E. Jiménez-Piqué1,2, L. Schneide 5, J. Fai 6 and L. Llanes1,2 1CIEFMA, Depa men o Ma e ials Science and Me allu gical Enginee ing, EEBE, Uni e si a Poli ècnica de Ca alunya, 08019 Ba celona, Spain 2Ba celona Resea ch Cen e in Mul iscale Science and Enginee ing, Uni e si a Poli ècnica de Ca alunya, 08019 Ba celona, Spain 3Facul ad de Ingenie ía Mecánica, Uni e sidad de O ien e, 90500 San iago de Cuba, Cuba. 4Depa amen o de Ingenie ía y Ciencia de los Ma e iales y del T anspo e, Uni e sidad de Se illa, 41092 Se illa, Spain 5Sand ik Machining Solu ions, Co en y CV4 0XG, UK 6Sand ik Hype ion, Co en y CV4 0XG, UK Φ Cu en add ess: Sand ik Hype ion, 08107 Ma o elles, Spain ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Abs ac S eng h and eliabili y o WC-Co cemen ed ca bides (ha dme als) a e dependen on e ec i e ac u e oughness as well as on na u e, size and dis ibu ion o p ocessing laws. Rega ding oughness, hey exhibi a c ack g ow h esis ance (R-cu e) beha io , de i ed om he de elopmen o a mul iligamen b idging zone a he c ack wake. Acco dingly, success ul implemen a ion o ac u e mechanics equi es conside a ion o angency c i e ion, be ween applied s ess in ensi y ac o and R-cu e, in addi ion o ac og aphic inspec ion. I is he aim o his s udy o e alua e he s eng h beha io o a se ies o expe imen al WC-Co g ades on he basis o R-cu e ailu e c i e ia. Resul s indica e ha mic os uc u al e ec s on he s eng h o ha dme als may be sa is ac o ily a ionalized ollowing he e e ed c i e ion. The analysis includes conside a ion o na u e and dis ibu ion o ac u e o igins, ound o be mo e di e se and wide espec i ely o he ha de ine-g ained g ades. This expe imen al e idence, oge he wi h he ac ha hese ha dme als exhibi s eepe ising R-cu es han oughe coa se-g ained ones, leads o lowe eliabili y o he o me . This in es iga ion documen s and alida es he g ea ele ance o R-cu e beha io o op imizing he mechanical pe o mance o WC-Co cemen ed ca bides on he basis o mic os uc u al design. Keywo ds WC-Co cemen ed ca bides, s eng h, eliabili y, ac og aphy, R-cu e beha io ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT In oduc ion B i le ac u e is one o he mos common ailu e mechanisms in he applica ion o cemen ed ca bides as o ming ools and s uc u al componen s, pa icula ly unde he combined ac ion o mechanical and con ac loads [1]. In a la ge numbe o cases ac u e s ems om he p opaga ion o p e-exis ing laws ei he in insic o he manu ac u ing and/o pos - ea men p ocesses (e.g. Re . [2–6]) o induced unde se ice condi ions (e.g. Re s. [7–9]). Simila o o he b i le ma e ials, ensile s eng h o cemen ed ca bides depends on he size o he majo law and exhibi s wide sca e due o he a iabili y associa ed wi h de ec dis ibu ion. Acco dingly, s eng h is size-dependen as he p obabili y o inding a majo law inc eases wi h specimen size. Wi hin his con ex , Weibull s a is ics [10] ha e been equen ly ecalled as a sui able app oach o a ionalize he up u e s eng h and eliabili y o ha dme als (e.g. Re s. [11,12]). An ex ensi e numbe o s udies ha e been de o ed o analyze he in luence o he mic os uc u e on he ac u e s eng h o ha dme als (e.g. Re . [1,4,13–18]). Gu land and Ba dzil documen ed ha he s eng h ini ially inc eases wi h he binde mean ee pa h (λCo) un il i eaches a maximum a λCo alues a ound 0.3-0.6 μm, and hen i s a s dec easing [13]. This beha io was associa ed wi h a b i le-duc ile ansi ion. Fo small λCo alues, ac u e mechanisms a e go e ned by he b i le ac u e o he WC con inuous phase. The e o e, he dec ease o he lexu al s eng h wi h dec easing λCo was ela ed o la ge WC amoun s and o highe con igui y alues o he ca bide phase. On he opposi e side o he cu e, he s eng h was domina ed by he ex ensi e plas ic de o ma ion o he binde phase. Thus, ac u e s eng h was in e sely p opo ional o λCo due o a dec ease o he cons ain deg ee exe ed by he ca bide phase on he binde , when inc easing λCo alue. This mic os uc u e-s eng h pe spec i e was sha ed by Fang [17], who plo ed ac u e s eng h agains ha dness and documen ed ha he o me goes h ough a peak a in e media e alues o he la e . In line wi h Gu land’s ea lie discussion, Fang pos ula es ha s eng h o ha dme als is based on he compe i i e ac ion played be ween ha dness and ac u e oughness. Howe e , Gu land’s heo y could no explain he high ac u e s eng h dispe sion measu ed in cemen ed ca bides. This pe cep ion changed wi h he implemen a ion o linea elas ic ac u e mechanics (LEFM) o a ionalize he b i le beha io o ha dme als [2,3,14]. Wi hin his ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT con ex , he ac u e oughness is he ma e ial p ope y ha , combined wi h he c i ical p e-exis ing law sizes and he applied s ess, de e mines he ac u e s eng h o cemen ed ca bides. LEFM app oach conside s a ully b i le beha io wi h a ailu e p ocess gi en by ex ension o c i ical laws, wi hou s able subc i ical g ow h be o e ac u e [1]. Howe e , as widely epo ed in li e a u e, cemen ed ca bides exhibi subc i ical c ack g ow h unde mono onic loads. I yields (and esul s om) he de elopmen o a c ack-b idging mul iligamen mechanism a he c ack wake [19,20]. This mechanism en ails he de elopmen o a ising c ack g ow h esis ance cu e [21]. The au ho s o his pape ecen ly epo ed a ele an in luence o mic os uc u e on he R-cu e cha ac e is ics o ha dme als [22]. I is he pu pose o his in es iga ion o e alua e he mic os uc u e-s eng h ela ions o a se ies o expe imen al WC-Co g ades on he basis o R-cu e ailu e c i e ia. In doing so, special a en ion is paid o explaining he impac o R-cu e beha io on he s eng h and eliabili y o cemen ed ca bides. Ma e ials and expe imen al aspec s The in es iga ed ma e ials we e a se o expe imen al WC-Co cemen ed ca bide g ades supplied by Sand ik Hype ion. Eigh g ades co esponding o di e en combina ions o binde con en and ca bide mean g ain size we e s udied. Specimen codes and key mic os uc u al pa ame e s - binde con en (Vw %), mean g ain size (dWC), ca bide con igui y (CWC), and binde mean ee pa h (λCo) - a e lis ed in Table I. In he ine g ades, conc e ely 3UF, 6UF, 10UF, 11M and 15UF, a small amoun o C 3C2 was added as g ain g ow h inhibi o . Mean g ain size was measu ed ollowing he linea in e cep me hod, using ield emission scanning elec on mic oscopy (FE-SEM) mic og aphs aken in a JEOL-7001F uni . Ca bide con igui y and binde mean ee pa h we e deduced acco ding o empi ical ela ions om open li e a u e [5,23,24]. Mechanical cha ac e iza ion includes ha dness (HV30), lexu al s eng h (σR) and ac u e oughness (KIc). Ha dness was measu ed using a Vicke s diamond py amidal inden e and applying a load o 294N. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT In all he o he s cases, es ing was conduc ed using a ou -poin bending ully a icula ed es jig wi h inne and ou e spans o 20 and 40 mm, espec i ely. Flexu al s eng h es s we e pe o med on an Ins on 8511 se ohyd aulic machine, and 10 o 15 specimens o 45x4x3 mm3 we e es ed pe g ade. The su ace subjec ed o he maximum ensile loads was polished o mi o -like inish and he edges we e cham e ed o educe hei e ec as s ess aise s. Flexu al s eng h esul s we e analyzed using Weibull s a is ics. F ac u e oughness was de e mined using 45x10x5 mm3 single edge p e-c acked beam (SEPB) specimens, wi h a no ch leng h- o-specimen wid h a io o 0.3, ollowing he p ocedu e p oposed by To es e al. [25]. A de ailed ac og aphic analysis on b oken specimens was ca ied ou by means o FE-SEM in o de o de ec c i ical laws ha p omo ed ailu e. Subsequen ly, he size o iden i ied c i ical laws was measu ed using he image analysis ee-a ailable so wa e ImajeJ. Resul s and discussion In luence o he mic os uc u e on he lexu al s eng h Ha dness, ac u e oughness, ac u e s eng h and Weibull cha ac e is ic s eng h (σ0) and modulus (m) alues o he in es iga ed cemen ed ca bides a e gi en in Table II. The Weibull plo s o s udied ma e ials a e shown in Fig. 1. Ve y in e es ing, he ha de g ades (i.e. 3UF, 6UF and 10UF) exhibi a bi- modal s eng h dis ibu ion. The e o e, he Weibull modulus o hese g ades is no epo ed. On he o he hand, he o he ma e ials exhibi a clea ly uni-modal dis ibu ion. A ac og aphic analysis by means o FESEM was conduc ed in o de o iden i y he c i ical laws ha p omo ed ailu e. Some examples o he di e en ypes o c i ical laws e idenced o s udied ha dme als a e shown in Fig. 2. The la ge sca e measu ed o he ha de g ades can be di ec ly associa ed wi h wo di e en de ec popula ions, consis ing o po es and inclusions o he low s eng h alues, and abno mal coa se WC g ains o he op pa o he Weibull dis ibu ion. I is also in e es ing o poin ou ha he highe s eng h de e mined alues co espond o he specimens wi h he smalle de ec sizes, in line wi h he heo e ical amewo k es ablished by LEFM [3,26]. In addi ion, ga he ed in o ma ion om Fig. 1 and 2 indica es ha unde he conside a ion o de ec s o di e en na u e bu simila equi alen size, de ec s such as po es, ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT agglome a es o inclusions, imply a g ea e se e i y han in insic mic os uc u al he e ogenei ies (i.e. abno mal coa se g ains). This is in ag eemen wi h ideas gi en in li e a u e o o he ce amic ma e ials [27]. Fu he mo e, i he Weibull analysis o 6UF g ade is pe o med uniquely in he specimens ac u ing om mic os uc u al he e ogenei ies (i.e. he uppe pa o he Weibull dis ibu ion), he Weibull modulus would ise om 7 up o 23. This is a clea indica ion ha an exhaus i e con ol o c i ical laws na u e esul s in ele an s eng h and eliabili y enhancemen s in cemen ed ca bides. On he o he hand, c i ical de ec popula ion o he oughe g ades is cha ac e ized by an absence o po es and inclusions, oge he wi h a endency o simply consis on mic os uc u al he e ogenei ies such as abno mally coa se WC pa icles. In Fig. 3, ac u e s eng h is plo ed as a unc ion o binde mean ee pa h. In acco dance wi h p e ious in es iga ions [13,28], lexu al s eng h inc eases wi h λCo un il i eaches a maximum a alues a ound 0.25 μm and hen i dec eases o la ge λCo alues. I can he e o e be concluded ha he ma e ial wi h he highes s eng h is nei he he oughes no he ha des , concep s ha end o me ge in indus ial a eas. The e o e, a comp omise be ween ac u e oughness and ha dness is equi ed o achie e an op imum pe o mance o he ma e ial. I is also in e es ing o ema k ha s eng h dispe sion e idenced o in es iga ed cemen ed ca bides g ades wi h λCo o e 0.25 μm is signi ican ly lowe han ha obse ed o g ades wi h smalle binde mean ee pa hs. Two main ac o s may explain his signi ican di e ence in s eng h sca e . Fi s , as p e iously indica ed, and p obably he main eason, he Weibull cu e o he s udied ha dme als wi h a sho λCo exhibi a bi-modal beha io (Fig. 1). Meanwhile Weibull plo s o he o he s udied ha dme als a e clea ly uni-modal. Second, as p e iously commen ed in he in oduc ion, cemen ed ca bides exhibi a ising c ack g ow h esis ance beha io ha impa s eliabili y and damage ole ance o ha dme als ools and componen s [29–32]. The e ec i e eliabili y enhancemen de i ed om he c ack b idging mechanism is a unc ion o bo h ac u e oughness and ac u e s eng h o he composi e ma e ial [22]. S eng h eliabili y inc eases wi h he mic os uc u al size, de ined as dWC + λCo, un il i eaches a pla eau ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT o dWC + λCo leng hs o a ound 1 μm [22]. Consequen ly, i is easonable o expec lowe Weibull modulus o ha dme als wi h sho binde mean pa hs han o coa se ma e ials. Es ima ion o he c i ical law size The c i ical law sizes o in es iga ed ma e ials we e es ima ed acco ding o LEFM and R-cu e angency c i e ia by assuming hem o be in e nal ci cula laws ( o embedded ci cula c acks, Y has a alue o 1.12 [33]). On he one hand, LEFM ailu e c i e ion ela es he ac u e s eng h (σR) wi h he c i ical law size (ac ) acco ding o he exp ession [34]: (1) whe e Y is an un-dimensional c ack geome ic ac o . On he o he hand, when conside ing an R-cu e beha io , he angency c i e ia o uns able c ack p opaga ion p e ails. This c i e ia de ines ha he ma e ial ca as ophically ails when he ollowing wo condi ions a e sa is ied [32]: (2) (4) whe e Kapp is he applied s ess in ensi y ac o , KR is he c ack- ip esis ance s ess in ensi y ac o and a is he c ack leng h. Kapp and a may be ela ed o he applied s eng h (σapp) acco ding o LEFM basic equa ion [34]: (3) The c ack g ow h esis ance cu e o in es iga ed ha dme als can be es ima ed om an equa ion ecen ly published by he au ho s o desc ibe R-cu e beha io o cemen ed ca bides [21,22]: (4) ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT whe e K is he c i ical c ack- ip s ess in ensi y ac o equi ed o c ack ini ia ion, Δa is he subc i ical c ack g ow h leng h and is a c ack leng h no malizing pa ame e . The e o e, in acco dance wi h ins abili y c i e ion o R-cu e ma e ials, he size o he c i ical laws can be de e mined as ollows: (5) Acco ding o angency c i e ion, ising R-cu e ma e ials ac u ing om he p opaga ion o “sho ” c acks a e no able o de elop he whole oughening capabili y, and ac u e a e ec i e oughness le els (Ke ) lowe han he in insic ac u e oughness measu ed o long c acks [21,22,32,35,36]. Fu he mo e, R- cu e ma e ials also exhibi s able subc i ical c ack g ow h ex ension be o e ca as ophic ailu e. Reached e ec i e oughness le els and subc i ical c ack g ow h leng hs can be de e mined acco ding o angency c i e ion and hey a e a unc ion o bo h R-cu e shape and ini ial law [32,36]. This p ocess is exempli ied in Fig. 4 o he 6UF and 15C in es iga ed cemen ed ca bides. The a ea o he iden i ied de ec s du ing he ac og aphic analysis was expe imen ally measu ed and compa ed o he es ima ed ones. In doing so, de ec s we e conside ed o be ci cula shaped and he e o e, an equi alen c i ical adius (ac ) is epo ed. C i ical law sizes expe imen ally measu ed and es ima ed using he LEFM ( ull black squa es) and angency (whi e ci cles) c i e ia a e plo ed in Fig. 5 agains he binde mean ee pa h. No e ha in he case o he expe imen ally measu ed c i ical de ec sizes, a leng h ange is gi en whe e he bo om and op alues co espond o he smalle and la ge iden i ied laws, espec i ely. No de ec s we e iden i ied du ing he ac og aphic examina ion o he ha dme al g ade wi h he longes binde mean ee pa h (22M) and consequen ly hey a e no shown in he g aph. Absolu e di e ences be ween he es ima ed law sizes es ima ed ollowing he LEFM and he angency ailu e c i e ia a e a he small o he ha de g ades, bu hey a e ai ly la ge o he oughe cemen ed ca bides and inc easing wi h binde mean ee pa h. As i is clea ly shown in Fig. 5, a eally good ag eemen be ween he c i ical de ec sizes expe imen ally measu ed and hose es ima ed ollowing LEFM and R-cu e app oaches is a ained o he ha de g ades, e en hough ob ained ac ,R-cu e leng hs a e sligh ly lowe . Howe e , o he oughe g ades, ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT es ima ed ac ,R-cu e alues clea ly ma ch wi h he sizes expe imen ally measu ed in he iden i ied c i ical laws, whe eas he applica ion o he LEFM heo y esul s in a law size o e es ima ion. This ac e lec s he need o aking in o conside a ion he subc i ical c ack g ow h ex ension and he e ec i e de eloped oughness o a ionalizing he ac u e beha io o cemen ed ca bides [22]. On he one hand, low Ke alues imply la ge es ima ed size di e ences be ween bo h c i e ia. On he o he hand, la ge subc i ical c ack g ow h alues lead o an o e es ima ion o c i ical law sizes by LEFM. Mic os uc u al e ec s on ailu e p obabili y As p e iously commen ed, he ac u e beha io o cemen ed ca bides is o s ochas ic na u e [2– 4,11,12,14]. The e o e, p obabilis ic ailu e mechanics a e essen ial o p ope designing wi h his b i le- like ma e ial [37,38]. Acco ding o Weibull dis ibu ion s anda ds [39], he ange o ailu e p obabili ies co e ed by he expe imen al assessmen o he Weibull cu e inc eases wi h he dimension o he sample (i.e. he numbe o es ed specimens). Howe e , due o he cos s o he specimens he numbe o es ed specimens is usually a he small [40]. In his in es iga ion, he sample size was limi ed be ween 10 o 15 specimens, and he e o e he ange o conside ed p obabili ies o ailu e was kep om a 5% o a 95%. Ne e heless, o designing pu poses o ools and componen s, he measu ed cha ac e is ic s eng h usually needs o be ex apola ed o eally low p obabili ies o ailu e and o di e en e ec i e loaded olumes [40]. In cemen ed ca bides his ex apola ion is no s aigh o wa d, as hey exhibi a R-cu e beha io ha may imply a de ia ion om Weibull s a is ics [11,22,40]. Mo eo e , i has been obse ed in his in es iga ion, in some cases he ac u e o cemen ed ca bides exhibi a mul i- modal law popula ion whe e a Weibull dis ibu ion is no expec ed o occu [40]. The e o e, a deep knowledge o he ac u e p ocess o cemen ed ca bides is demanded o he p ope design o hea ily s essed ools and componen s. Acco ding o Weibull heo y [10,41], he s eng h o an indi idual specimen (σi) o a sample can be deduced o a gi en ailu e p obabili y (P ) by using he ollowing equa ion: ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT [34] G.R. I win, F ac u e, in: Handbuch De Physics, Sp inge -Ve lag, Be lin, 1958. [35] J.J. K uzic, R.L. Sa e , M.J. Ho mann, R.M. Cannon, R.O. Ri chie, The u ili y o R-cu es o unde s anding ac u e oughness-s eng h ela ions in b idging ce amics, J. Am. Ce am. Soc. 91 (2008) 1986–1994. [36] M.E. Launey, R.O. Ri chie, On he ac u e oughness o ad anced ma e ials, Ad . Ma e . 21 (2009) 2103–2110. [37] D. Rubes, B. Smoljan, R. Danze , Main ea u es o designing wi h b i le ma e ials, J. Ma e . Eng. Pe o m. 12 (2003) 220–228. [38] R. Danze , T. Lube, P. Supancic, R. Damani, F ac u e o ce amics, Ad . Eng. Ma e . 10 (2008) 275–298. [39] ASTM C 1239-00: S anda d p ac ice o epo ing uniaxial s eng h da a and es ima ing Weibull dis ibu ion pa ame e s o ad anced ce amics, USA, 2000. [40] R. Danze , P. Supancic, J. Pascual, T. Lube, F ac u e s a is ics o ce amics – Weibull s a is ics and de ia ions om Weibull s a is ics, Eng. F ac . Mech. 74 (2007) 2919–2932. [41] J.B. Wach man, Mechanical p ope ies o ce amics, 2nd ed., John Wiley & Sons, New Yo k, 1996. [42] J.M. Ta agó, G. Fa gas, L. Ise n, S. Do lo, E. Ta es, C.M. Mülle , E. Jiménez-Piqué, L. Llanes, Mic os uc u al in luence on ole ance o co osion-induced damage in ha dme als, Ma e . Des. 111 (2016) 36–43. [43] J.M. Ta agó, S. Do lo, J. Es e e, L. Llanes, In luence o he mic os uc u e on he he mal shock beha io o cemen ed ca bides, Ce am. In . 42 (2016) 12701–12708. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Lis o ables Table I. Mic os uc u al pa ame e s o in es iga ed cemen ed ca bides. Table II. Ha dness, lexu al s eng h, Weibull cha ac e is ic s eng h, Weibull modulus and ac u e oughness alues o in es iga ed cemen ed ca bides. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Lis o igu es Figu e 1: F ac u e s eng h alues in he o m o Weibull dis ibu ion o s udied ha dme al g ades. The in es iga ed cemen ed ca bide g ades wi h he lowe binde mean ee pa h exhibi s a: (a) bi-modal Weibull dis ibu ion, whe eas he o he g ades ha e (b) uni-modal Weibull cu es. Figu e 2: FESEM mic og aphs o ailu e ini ia ion si es; namely: (a) ca bide agglome a e in he 3UF g ade, (b) po e in he 10UF g ade, (c) abno mal coa se g ain in he 10C g ade and, (d) binde less agglome a e o coa se WC pa icles in he 15C g ade. Figu e 3: F ac u e s eng h as a unc ion o he binde mean ee pa h. Figu e 4: Es ima ed c ack esis ance cu es o he 6UF and 15C in es iga ed ha dme als ( ull lines). Uns able c i e ion is eached when angency condi ions be ween he R-cu e and he applied s ess in ensi y ac o (dash-do lines) a e ul illed. Dashed lines indica e he subc i ical c ack g ow hs leng hs and e ec i e oughness le els eached when angency c i e ia is accomplished (i.e. uns able c ack g ow h). F ac u e is es ima ed o he expe imen ally measu ed lexu al s eng h alues o 3287MPa and 2570MPa o 6UF and 15C, espec i ely. Figu e 5: Es ima ed and expe imen ally de e mined c i ical laws sizes. Figu e 6: F ac u e s eng h as a unc ion o ac u e oughness (coo dina e axis) and p obabili y o ailu e (o dina e axis). ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT FIGURES (a) (b) Figu e 1: F ac u e s eng h alues in he o m o Weibull dis ibu ion o s udied ha dme al g ades. The in es iga ed cemen ed ca bide wi h he lowe binde mean ee pa h (a) exhibi s a bi-modal Weibull dis ibu ion whe eas he o he g ades (b) ha e uni-modal Weibull cu es. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT (a) (b) (c) (d) Figu e 2: FESEM mic og aphs o ailu e ini ia ion si es; namely: (a) ca bide agglome a e in he 3UF g ade, (b) po e in he 10UF g ade, (c) abno mal coa se g ain in he 10C g ade and, (d) binde less agglome a e o coa se WC pa icles in he 15C g ade. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Figu e 3: F ac u e s eng h as a unc ion o he binde mean ee pa h. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Figu e 4: Es ima ed c ack esis ance cu es o he 6UF and 15C in es iga ed ha dme als ( ull lines). Uns able c i e ion is eached when angency condi ions be ween he R-cu e and he applied s ess in ensi y ac o (dash-do lines) a e ul illed. Dashed lines indica e he subc i ical c ack g ow hs leng hs and e ec i e oughness le els eached when angency c i e ia is accomplished (i.e. uns able c ack g ow h). F ac u e is es ima ed o he expe imen ally measu ed lexu al s eng h alues o 3287MPa and 2570MPa o 6UF and 15C, espec i ely. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Figu e 5: Es ima ed and expe imen ally de e mined c i ical laws sizes. ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Figu e 6: F ac u e s eng h as a unc ion o ac u e oughness (coo dina e axis) and p obabili y o ailu e (o dina e axis). ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT TABLES Vw .% dWC (MPa) Cwc (MPa) λCo (μm) 3UF 3 0.37 ± 0.22 0.74 ± 0.42 0.09 ± 0.05 6UF 6 0.40 ± 0.21 0.61 ± 0.12 0.12 ± 0.06 10UF 10 0.39 ± 0.19 0.46 ± 0.06 0.16 ± 0.06 11M 11 1.12 ± 0.71 0.38 ± 0.07 0.42 ± 0.28 10C 10 2.33 ± 1.38 0.31 ± 0.11 0.68 ± 0.48 15UF 15 0.47 ± 0.22 0.36 ± 0.02 0.24 ± 0.11 15C 15 1.70 ± 1.08 0.27 ± 0.07 0.77 ± 0.54 22M 22 1.64 ± 0.75 0.19 ± 0.04 1.13 ± 0.56 Table I. Mic os uc u al pa ame e s o in es iga ed cemen ed ca bides. ACCEPTED MANUSCRIPT