Crack propagation in a brittle DCB specimen assessed by means of the Williams’ power expansion

L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 34 Focussed on “C ack Pa hs” C ack p opaga ion in a b i le DCB specimen assessed by means o he Williams’ powe expansion Lucie Malíko á Ins i u e o S uc u al Mechanics, B no Uni e si y o Technology (BUT), B no, Czech Republic [email p o ec ed], h p://o cid.o g/0000-0001-5868-5717 Seyed Mohammad Ja ad Raza i, Filippo Be o Depa men o Mechanical and Indus ial Enginee ing, No wegian Uni e si y o Science and Technology (NTNU), T ondheim, No way [email p o ec ed], h p://o cid.o g/0000-0002-2574-065X [email p o ec ed], h p://o cid.o g/0000-0002-4207-0109 ABSTRACT. A double can ile e beam geome y has been chosen in o de o in es iga e he impo ance o he highe -o de e ms o he Williams’ powe expansion o he c ack pa h es ima ion. The c ack p opaga ion has been es ed expe imen ally on a b i le polyme hylme hac yla e (PMMA) specimen and al hough he mode I loading condi ions we e applied, he c ack kinked om i s o iginal plane immedia ely and p opaga ed owa ds he bo om side o he specimen. I has been shown ha his phenomenon is connec ed o he magni ude and sign o he T-s ess and o he le el o he cons ain gene ally. In his wo k, he in luence o he hi d and highe e ms o he Williams’ se ies on he c ack p opaga ion is in es iga ed. The gene alized o m o he well-known maximum angen ial s ess ac u e c i e ion o de e mina ion o he c ack p opaga ion angle has been es ed and discussed. The obse ed di e ences in he c ack ajec o y o di e en specimens ha e been ound o be ela ed o he magni ude o he highe -o de e ms o he s ess enso componen s a he c ack ip. KEYWORDS. C ack de lec ion; DCB specimen; Gene alized MTS c i e ion; Geome y e ec ; Williams’ powe expansion. Ci a ion: Malíko á, L., Raza i, S.M.J., Be o, F., C ack p opaga ion in a b i le DCB specimen assessed by means o he Williams’ powe expansion, F a u a ed In eg i à S u u ale, 48 (2019) 34-41. Recei ed: 28.11.2018 Accep ed: 07.01.2019 Published: 01.04.2019 Copy igh : © 2019 This is an open access a icle unde he e ms o he CC-BY 4.0, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho and sou ce a e c edi ed. INTRODUCTION s one o he impo an ypes o ma e ial ailu e, b i le ac u e has been widely aken in o conside a ion by esea che s, ying o p opose p ecise ailu e p edic ion c i e ia o di e en b i le and quasi-b i le ma e ials such as ocks, conc e es, ce amics and polyme s. Dealing wi h mode I loading condi ion o a componen wi h symme ic geome y wi h espec o he c ack line which is unde he symme ic loading condi ion, he c ack p opaga ion om he c ack ip is expec ed o be along he ini ial plane o he c ack. Howe e , p e ious esea ches showed ha o he A L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 35 case o specimens wi h high geome y cons ain , c ack can de ia e om he symme y line o he specimen. In his case he ac u e load is expec ed o be di e en om he specimens wi h lowe cons ain e ec . I has been shown in p e ious esea ches ha di e en ac u e oughness alues can be ob ained o an iden ical ma e ial when using a ious specimens’ geome ies [1-9]. This pape shall show di e ences be ween he well-known one-pa ame e ac u e mechanics concep and he mul i- pa ame e one. Whe eas he o me one uses he s ess in ensi y ac o (SIF) as he single-con olling pa ame e o assessmen o he ac u e esponse o he specimen/s uc u e [10], he la e one is based on he app oxima ion o he s ess/displacemen c ack- ip ield by means o he Williams’ expansion (WE) [11], i.e. an in ini e powe se ies o iginally de i ed o a homogeneous elas ic iso opic c acked body subjec ed o an a bi a y emo e loading. The mul i-pa ame e concep is e y o en connec ed o ac u e analyses pe o med on elas ic-plas ic o quasi-b i le ma e ials, see e.g. in [12- 17]. The mul i-pa ame e app oach seems o be help ul and mo e accu a e when ac u e p ocesses occu in a mo e dis an su ounding a ound he c ack ip. Fo ins ance, he in luence o he second (non-singula ) e m o he WE on he ac u e beha iou o b i le/quasi-b i le ma e ials has been in es iga ed in se e al wo ks [18-24]. Aya ollahi e al. [22] p oposed wo ac u e c i e ia based on s ain ene gy densi y o ake in o accoun he e ec o i s non-singula e m o s ess in WE. Acco ding o hei esul s highe accu acy o he ac u e load p edic ion can be ob ained by use o wo pa ame e ac u e c i e ia. Addi ionally, unlike he o me single pa ame e ac u e c i e ia, he new o mula ions we e able o success ully p edic he cu ilinea c ack g ow h pa h unde he in luence o geome y cons ain s. Raza i e al. [24] e alua ed he mode I ac u e beha io o i e di e en geome ies o p e-c acked specimens made o PMMA and h ee di e en ocks using an ene gy-based c i e ion namely A e age S ain Ene gy Densi y (ASED). They epo ed ha o speci ic ca ego ies o ma e ials such as ocks, he e ec o geome y cons ain is no negligible. Among he s udied geome ies in hei esea ch, applica ion o only he i s s ess e m in Williams’ se ies expansion o ac u e p edic ion o Tape ed Double Can ile e Beam (TDCB) specimens made o Ha sin ma ble ock esul ed in 47% di e ence wi h he ASED esul s ob ained by conside ing all s ess e ms in a con ol olume a ound he c ack ip o he same specimen. The basic ask, when he WE shall be used o he s ess/displacemen ield app oxima ion, is o de e mine he coe icien s co esponding o he indi idual e ms o he Williams’ se ies. In his pape , he o e -de e minis ic me hod (ODM) is used [25]. This wo k is de o ed o an in es iga ion o he ini ial c ack p opaga ion angle in se e al double can ile e beam con igu a ions. The kink angle is es ima ed by means o he maximum angen ial s ess c i e ion [26]. I s common as well as gene alized o m is applied, and a pa ame ic s udy is pe o med in o de o desc ibe he e ec o he specimen wid h, he numbe o e ms o he WE aken in o accoun du ing he analysis and he adial dis ance om he c ack ip whe e he c i e ion is applied. The esul s ob ained om he nume ical analysis a e compa ed o he expe imen al ones. METHODOLOGY AND BASIC TERMS C ack- ip ields app oxima ion illiams [11] showed ha he c ack- ip displacemen /s ess ield can be desc ibed as ollows:    ,,,,,, 0 2 1 2EmgB En A u ii u M m m m u N n n n i   whe e i  {x,y}. (1)   mgB m n A n ijij M m m m N n n n ij , 2 , 21 1 2 1 1 2        whe e i, j  {x,y}. (2) Eqs. 1 and 2 ep esen he unca ed o m (N and M a e he numbe s o he WE e ms co esponding o he loading modes I and II, espec i ely) o he in ini e powe se ies enabling he app oxima ion o he c ack- ip ields. Pa icula ly, ui and σij deno e he displacemen ec o and s ess enso componen s, espec i ely. The powe se ies is de ined in he pola coo dina e sys em ( ,θ) wi h i s cen e a he c ack ip. The meaning o he o he symbols is as ollows: u, gu … known dimensionless displacemen s unc ions co esponding o mode I and II, espec i ely, ha can be ound in li e a u e; W L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 36 σ, gσ … known dimensionless s ess unc ions co esponding o mode I and II, espec i ely, ha can be ound in li e a u e; E,  … Young’s modulus and Poisson’s a io; An, Bm … he only coe icien s ha need o be calcula ed nume ically o each speci ic geome ic con igu a ion ep esen ing he well-known highe -o de e ms coe icien s o mode I and II, espec i ely. I should be no ed ha he i s e ms o he se ies, A1 and B1, co espond o he classical s ess in ensi y ac o s, KI and KII, as hey a e known in he classical one-pa ame e linea elas ic ac u e mechanics concep ; i holds ha KI = A12 and KII = –B12. Iden i ica ion o he coe icien s o he Williams’ expansion I has been men ioned ha a e y impo an ask is o de e mine he coe icien s o he WE e ms co ec ly. This is mos ly pe o med nume ically by means o he ini e elemen (FE) me hod. Se e al me hods o es ima ion o he coe icien s we e de i ed, es ed and applied in he pas such as hyb id c ack elemen me hod, bounda y colloca ion me hod e c. [27- 31], bu mos o hem we e complica ed and demanding special elemen s o di icul FE o mula ions. In his wo k, an o e -de e minis ic me hod (ODM) [25] is applied because o i s simplici y. This p ocedu e is based on he o mula ion o linea leas -squa es and i equi es he basic nodal solu ion o he ac u e mechanics ask. The e o e, an a bi a y egula FE code can be u ilized o i s applica ion. The use o he me hod consis s in de ini ion o he displacemen ield a ound he c ack ip, see Eq. 1. When he nume ical analysis on he c acked specimen/s uc u e is ca ied ou , a se o nodes a ound he c ack ip is selec ed and hei displacemen s oge he wi h hei pola coo dina es a e aken as inpu s o Eq. 1. The only a iables An and Bm can be hen calcula ed om he sys em o equa ions. The p inciple o he ODM lies in he ela ion be ween he numbe o he equa ions and numbe o he coe icien s ha shall be de e mined: a minimum o (N + M)/2 + 1 nodes need o be conside ed in o de o de e mine N + M coe icien s. Mo e in es iga ions on he accu acy, con e gence, mesh sensi i i y, in luence o he ounding o numbe s e c. can be ound in [32-35]. Maximum Tangen ial S ess (MTS) c i e ion MTS c i e ion p edic s ha a c ack p opaga es in he di ec ion whe e he angen ial s ess,   , eaches i s maximum [26]. In he classical one-pa ame e ac u e mechanics concep an explici ela ion o he kink angle has been de i ed: II 22 IIII 2 2a c an 8 K KKK    (3) Ne e heless, because he mul i-pa ame e concep is used in his wo k, he angen ial s ess alues mus be app oxima ed ia Williams’ powe expansion conside ing a ious numbe s o he ini ial WE e ms and hen he ini ial c ack p opaga ion angle has been es ima ed. No e ha a new dependence a ises du ing his p ocedu e: he ini ial kink angle depends on he dis ance whe e he c i e ion is applied and he e o e a ious dis ances om he c ack ip a e conside ed in he analysis, see he ollowing sec ions. SPECIMEN GEOMETRY/NUMERICAL MODEL he Double Can ile e Beam (DCB) ype o he c acked specimen was chosen o he s udy p esen ed aking in o accoun a ious c ack wid h W (30, 90 and 150 mm), see Fig. 1. The majo ad an age o choosing his ype o specimens is ha hey a e cha ac e ized by simple shapes and loading condi ions, while hey can p o ide a wide ange o highe -o de e ms o s ess in WE. The es specimens we e cu om 10 mm hick PMMA pla e and es ed unde s a ic loading a oom empe a u e wi h a displacemen a e o 0.1 mm/min. As is epo ed in e . [36], he specimens we e all ac u ed suddenly om he c ack ip and wi h linea load-displacemen cu es con i ming he b i le ac u e beha io o he es ed PMMA samples. The expe imen al se up is desc ibed in de ail in e . [36]. Fo c ea ing he c acks, i s a e y hin s ip saw blade o hickness 0.2 mm was used o c ea e a no ch wi h an ini ial dep h sligh ly less han a/W = 0.5. Then, a sha p c ack was c ea ed by p essing a azo blade ca e ully o make he inal c ack leng h o each specimen equal o a/W = 0.5. The ex e nal load was applied h ough wo pin holes de ised on each specimen. T L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 37 Figu e 1: Schema o he Double Can ile e Beam (DCB) specimen wi h i s sizes: wid h W, heigh H, c ack leng h a and loading o ces F. The nume ical analysis (as well as expe imen s) was pe o med on he specimens wi h ollowing dimensions and pa ame e s: specimen heigh H = 30 mm, specimen wid h W = 30, 90 and 150 mm, specimen hickness B = 10 mm, c ack leng h a = W/2. The alue o he loading o ce was chosen wi h ega d o he ac u e o ce leading o c ack p opaga ion obse ed du ing he expe imen al esea ch p esen ed in [36], F = 319, 141 and 87 N (co esponding o he specimen wid h 30, 90 and 150 mm). In o de o ge he nodal displacemen s ha a e necessa y o ODM applica ion, a nume ical model o he c acked specimen was c ea ed in ANSYS comme cial so wa e [37] wi h he ollowing ea u es: 2D, linea elas ic, meshed wi h quad a ic PLANE183 elemen s, e ined mesh a ound he c ack ip (wi h shi ed mid-side nodes emphasizing he s ess singula i y), plane s ain condi ions, Young’s modulus E = 2900 MPa and Poisson's a ion  = 0.35. Subsequen ly, he ODM p ocedu e was p og ammed in Wol am Ma hema ica so wa e [38]. Fo es ima ion o he coe icien s o he WE, he nodes wi h hei coo dina es and displacemen s a he dis ance o 1 mm om he c ack ip we e u ilized. A ypical mesh pa e n used o ini e elemen analysis is illus a ed in Fig. 2. As can be obse ed om Fig. 2, highe densi y o elemen s has been used o he a ea nea he c ack ip o ge accu a e esul s in he egion wi h singula s ess alues. Figu e 2: Typical mesh pa e n used o ini e elemen analysis o he c acked specimen. The maximum angen ial s ess c i e ion was applied in he ollowing manne . The angen ial s ess a ound he c ack ip a a ious adial dis ances c in he ange om 0.1 o 1 mm was econs uc ed by means o he Williams’ powe se ies. The angle θ, whe e he σθθ eaches i s maximum alue was selec ed as he di ec ion o he u he c ack p opaga ion. Du ing applica ion o he mul i-pa ame e /gene alized ac u e c i e ion, a ious numbe s o he WE e ms we e assumed (up o 10 ini ial e ms). L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 38 RESULTS OF THE PARAMETRIC STUDY AND THEIR DISCUSSION n he ollowing esul s, only he dependences o he DCB specimens wi h he wid h 90 and 150 mm a e p esen ed. The eason is ha when he c ack p opaga ion in he DCB specimen wi h W = 30 mm was in es iga ed, no de ia ion om he o iginal c ack plane was obse ed, i.e. he c ack p opaga es in he o iginal di ec ion ega dless he dis ance o he c ack ip whe e he c i e ion is applied as well as he numbe o he WE e ms conside ed. In Figs. 3 and 4 he dependences o he ini ial c ack di ec ion on he numbe o he WE e ms conside ed and c i ical dis ance om he c ack ip whe e he MTS c i e ion is applied a e p esen ed. In he igu es only he da a ob ained om nume ical analysis a e plo ed. The alues o he ini ial c ack p opaga ion angle obse ed du ing expe imen s a e 0, 9 and 56 deg ees o W = 30, 90 and 150 mm, espec i ely [36], o compa ison see he black lines in Fig. 4. a) b) Figu e 3: Dependence o he ini ial c ack p opaga ion angle es ima ed by means o he mul i-pa ame e maximum angen ial s ess c i e ion on he numbe o he WE e ms conside ed when a ious c i ical dis ances om he c ack ip a e assumed: a) DCB wi h W = 90 mm; b) DCB wi h W = 150 mm. a) b) Figu e 4: Dependence o he ini ial c ack p opaga ion angle es ima ed by means o he mul i-pa ame e maximum angen ial s ess c i e ion on he adial dis ance om he c ack ip when a ious numbe s o he WE e ms a e conside ed: a) DCB wi h W = 90 mm; b) DCB wi h W = 150 mm. The gene alized s ain ene gy densi y (GSED) c i e ion is mo eo e used o es ima ion o he ini ial kink angle in [36] and he esul s co espond well wi h he expe imen (conside ing he c i ical dis ance o 0.1 mm). Un o una ely, in o ma ion abou he dependence o he esul s on he c i ical dis ance alue is missing. Bu he GSED seems o be a use ul ool o calcula ion o he c ack p opaga ion angle. Applica ion o he gene alized MTS c i e ion in oduced in his pape depends s ongly on he choice o he c i ical dis ance and he e o e a en ion should be de o ed o his phenomenon. I L. Malíko á e alii, F a u a ed In eg i à S u u ale, 48 (2018) 34-41; DOI: 10.3221/IGF-ESIS.48.05 39 I should be also men ioned ha he ac u e load o a b i le ma e ial is commonly ob ained using single pa ame e ac u e c i e ia which a e o mula ed based on he c i ical alues o s ess in ensi y ac o co esponding o he ac u e load o he ma e ial. Al hough he main ocus o he cu en esea ch is on c ack kinking angle, howe e , i is expec ed o ge mo e accu a e esul s o ac u e load by conside ing highe -o de e ms o WE in calcula ions. Acco ding o he esul s p esen ed in his pape , conside ing he highe -o de e ms o WE in calcula ion o ini ial c ack p opaga ion angle leads o mo e accu a e c ack ajec o y p edic ions. Hence, he single pa ame e ac u e c i e ia can p o ide eliable ac u e p edic ions only o limi ed ange o geome ies. Based on he esul s in oduced, se e al s a emen s can be o mula ed:  P obably he mos impo an conclusion is ha he classical (one-pa ame e ) MTS c i e ion is no able o desc ibe he c ack de lec ion when DCB specimens wi h la ge wid h (i.e. highe geome ic cons ain s) a e in es iga ed. The classical MTS c i e ion p edic s ha he c ack will p opaga e in i s o iginal di ec ion wi hou any kinking o all he DCB co igu a ions conside ed. Ne e heless, he expe imen s show ha o specimens wi h he wid h 90 and 150 mm he c ack de ia es om i s o iginal plane.  The esul s also show ha when he MTS c i e ion is applied a dis ances e y close o he c ack ip, he singula i y o he i s WE e ms p e ails and he c i e ion does no desc ibe he c ack kinking.  The ini ial inc ease o he es ima ed ini ial kink angle in Fig. 3 is p obably connec ed o a ce ain egion a ound he c ack ip, whe e he second e m o he WE (non-singula T-s ess) becomes impo an . This phenomenon can be seen also in Fig. 4, whe e he cu e ep esen ing he esul s co esponding o he app oxima ion by means o ini ial wo e ms lies highe han o he cu es in a ce ain egion o dis ances om he c ack ip. The heo e ical e ec o he la ge T-s ess on he c ack p opaga ion unde mode I is desc ibed in de ail in [39].  A good ag eemen be ween he esul s o he gene alized MTS c i e ion and he expe iman al campaign can be ound o he c i ical dis ance o 0.5 mm in he case o DCB specimens wi h W = 150 mm, and o c i ical dis ance be ween 0.3 and 0.5 mm in he case o DCB specimens wi h W = 90 mm.  The ini ial kink angle calcula ed by means o he mul i-pa ame e MTS c i e ion conside ing 3 WE and mo e han 3 WE e ms does no change signi ican ly when mo e han 3 WE e ms a e coside ed, see he s eadying o he cu es in Fig. 3 and o e laping o he cu es in Fig. 4.  I has been epo ed by Aya ollahi e al. [16, 30] ha o ma e ials wi h la ge ac u e p ocess zone adius, e.g. ock ma e ials, he highe -o de e ms o WE becomes no longe negligible in o e al ac u e beha iou o he componen . Hence, he highe -o de e ms o s ess can play an impo an ole in ac u e s eng h and c ack ajec o y o ma e ials possessing la ge ac u e p ocess zone. In ha case, he impo ance o using a ac u e c i e ion, which coun s o highe -o de e ms, would be e en mo e impo an .  Al hough he cu en me hodology is p esen ed o a speci ic b i le polyme , (i.e. PMMA), howe e , he same me hod can be employed o p edic he ac u e ajec o y o o he c acked and no ched componen s made o b i le and quasi-b i le ma e ials. No e ha he signi icance o he T-s ess has been in es iga ed in ensi ely in [40]. whe e a mixed-mode loading con igu a ion was analysed.  The c ack kinking angles o di e en es specimens a e p edic ed in he cu en esea ch by conside ing he highe -o de e ms o WE se ies. The same me hodology can be used o ge he ull c ack pa h by use o he inc emen al me hod in ini e elemen so wa e and p edic ing he c ack g ow h angle o each c ack inc emen . CONCLUSIONS a ious DCB specimen con igu a ions ha e been in es iga ed by means o nume ical me hods in o de o es ima e he angle o he ini ial c ack p opaga ion. Al hough he c ack in he DCB specimen is loaded in he mode I, i is expe imen ally obse ed, ha o DCB specimens wi h la ge wid h (90 and 150 mm), he c ack de lec s om i s o iginal di ec ion. In his pa ame ical s udy his phenomenon is desc ibed by means o he gene alized o m o he well- known MTS c i e ion because he classical (one-pa ame e ) o m o he c i e ion is no able o desc ibe he c ack de lec ion. I can be ecommended o DCB specimens wi h highe cons ain e ec s o use he mul i-pa ame e o m o he MTS c i e ion wi h a leas h ee ini ial WE e ms o ge be e esul s o he kink angle. I is also shown ha he choice o he app op ia e c i ical dis ance whe e he ac u e c i e ion is applied is c ucial. Based on his s udy, he c i ical dis ance close o 0.5 mm seems o be op imal o he conside ed ma e ial. V L. 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