A unified brittle fracture criterion for structures with sharp V-notches under mixed mode loading

Transcription

1 Jural f Mechaical Sciece ad Techlgy Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 A uified brittle fracture criteri fr structures with sharp V-tches uder mixed mde ladig Ji Kwag Kim ad Sag Bg Ch * Divisi f Mechaical Egieerig ad Autmati, Kyugam Uiversity, Masa 63-70, Krea (Mauscript Received December 3, 2007; Revised February 7, 2008; Accepted March 23, 2008) Abstract A uified brittle fracture criteri fr cracks ad V-tches uder mixed mde ladig is prpsed by extedig the maximum circumferetial stress criteri ad Nvzhilv s criteri. The mixed mde fracture tughess ad crack rietati f PMMA plates with a sharp V-tch are predicted by the prpsed criteri. Tests were als carried ut i rder t ivestigate the mixed mde fracture f the PMMA plates. The fracture criteri is validated by cmparis t experimetal results. Keywrds: Sharp V-tch; Maximum circumferetial stress criteri; Uified brittle fracture criteri; Stress itesity factr; RWCM trducti The stress fields arud the tips f cracks r sharp V-tches, as shw i Fig., are geerally sigular. As a result, crack iitiati ccurs at these sharp crers. The effrts f may egieerig ivestigatrs have bee fcused develpig a uified methd t evaluate the fracture stregth f cracks ad sharp V- tches. Therefre, a iterestig prblem i fracture mechaics is t establish a simple ad accurate fracture criteri that eables the predicti f crack iitiati ad crack rietati ear the tip f sharp V- tches uder mixed mde ladig. Several fracture criteria t evaluate the fracture tughess ad fracture agle, such as the maximum circumferetial stress criteri [], the miimum strai eergy desity criteri [2], ad the maximum eergy release rate criteri [3], have bee prpsed. The maximum circumferetial stress criteri is simple ad the mst cmmly used. This criteri states the fllwig: () A crack prpagates frm the V- tch crack tip alg the directi θ fr which the * Crrespdig authr. Tel.: , Fax.: address: sbch@kyugam.ac.kr DO 0.007/s y circumferetial stress is maximum, ad (2) a fracture ccurs whe the maximum circumferetial stress σ θθ reaches a critical stress value σ c, which is equal t the fracture stress uder the uiaxial tesi. Sewery [4] prpsed a fracture criteri fr sharp V-tches i brittle materials uder Mde ladig; it is based Nvzhilv s criteri ad the maximum circumferetial stress criteri. Nvzhilv s prpsal [5] states that a fracture may ccur whe the mea stress alg a small distace ahead f the tip f Fig.. Cfigurati f a V-tch with tracti-free faces.

2 270 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 a V-tch reaches the critical breakig stress. Several researchers [6-] have als preseted theretical ad experimetal results fr V-tch prblems. their studies, the results fr either Mde r Mde are preseted, but there are t may results fr mixed mde ladig. Sewery et al. [2] studied the mixed mde ladig prblems. The criteri was frmulated by cmbiig the rmal stress i perpedicular directi ad the shear stress i parallel directi t crack prpagati. Greestedt et al. [3] carried ut a similar research cracks ad V-tches i a expaded plyviyl chlride (PVC) frm. Hallstrőm ad Greestedt [4] als studied mixed mde prblems f cracks ad V-tches. As they idicated i their wrks, the fracture directi θ varies with the distace r fr all V-tches except cracks uder the mixed mde ladig. Due t this feature, it is difficult t apply the maximum circumferetial stress criteri ad Nvzhilv s criteri t mixed mde prblems. this paper, the fllwig extesis t the maximum circumferetial stress criteri ad Nvzhilv s criteri are assumed: () A crack prpagates alg θ fr which the circumferetial stress is maximum at a distace f r = d frm the V-tch r the crack tip, ad (2) a fracture ccurs whe the mea circumferetial stress σ reaches the critical stress σ c, which is equal t the fracture stress i uiaxial tesi. The exteded maximum circumferetial stress criteri ad Nvzhilv s criteri are applied t predict the fracture tughess ad crack rietati fr V-tches uder a mixed lad. The effectiveess ad validati f the prpsed criteri are discussed by cmparis f the experimetal results f plymethyl methacrylate (PMMA) material with the predicted results. rder t determie the critical stress itesity factrs frm the fracture lads btaied by the fracture test f a PMMA plate with arbitrary wedge agles uder mixed mde ladig, the reciprcal wrk ctur itegral methd (RWCM) [5-8] based Betti s reciprcal wrk therem ad fiite elemet aalysis is used. The frmulati f the RWCM is give i Ref. [9]. The verificati f the RWCM will be discussed fr the mixed mde cditi. 2. Stress ad displacemet fields arud the tip f a sharp V-tch We aalyze a sharp V-tch i a liear twdimesial elastic bdy, as shw i Fig.. The budary cditi fr bth the tch surfaces with arbitrary wedge agles θ = α ad α 2 is tractifree. this study, the wedge agles are csidered t be equal, that is, α = α 2 = α. Williams eigefucti expasi series [20], it is well kw that the eigevalues f Mde ca be btaied frm λ si(2 α) + si(2 αλ) = 0 ad thse f Mde ca be calculated frm λ si(2 α) + cs(2 αλ) = 0. Subsequetly, the eigevalues f the peig ad slidig mde are deted as ϕ ad η, respectively. The stress ad displacemet fields fr structures with a sharp tch are described i terms f plar crdiates ( r, θ ) cetered at the tip f a V-tch as fllws [8]: ϕ η r A ( ) (, ) R r A u () j r θ = gj θ + gj() θ, 2µ 2µ = () ( j= r, θ) σ ( ) (, ) A R A () ij r θ f (), ij f ϕ ij r θ η r θ = + = (2) ( i, j = r, θ ) where gr ( θ) ( κ ϕ ) C[( ϕ ) θ] + C[( ϕ + ) θ = g ( ) ( ) S[( ) ] S[ ( ) θ θ κ + ϕ ϕ θ + ϕ + θ +, 2 ϕα] + ϕc[ ( ϕ + ) θ + 2 α] 2 ϕα] + ϕs[ ( ϕ + ) θ + 2 α] gr ( θ) ( κ η ) S[( η ) θ] S[( η + ) θ = g ( ) ( ) C[( ) ] C[( ) θ θ κ + η η θ η + θ, 2 ηα ] ηs[ ( η + ) θ+ 2 α] 2 ηα ] + ηc[ ( η + ) θ+ 2 α] frr( θ ) ϕ{ (3 ϕ) C[( ϕ ) θ] + C[( ϕ + ) θ fθθ ( θ) = ϕ {( ϕ + ) C[( ϕ ) θ] C[( ϕ + ) θ f ( ) { ( ) S[( ) ] S[ ( ) rθ θ ϕ ϕ ϕ θ + ϕ + θ +, 2 ϕα ] + ϕc[ ( ϕ + ) θ+ 2 α] } 2 ϕα ] ϕc[ ( ϕ + ) θ+ 2 α] } 2 ϕα ] + ϕs[ ( ϕ + ) θ+ 2 α] } ad frr( θ) η{ (3 η) S[( η ) θ] S[( η + ) θ = fθθ ( θ) η {( η + ) S[( η ) θ] + S[( η + ) θ f ( ) { ( ) C[( ) ] C[ ( ) rθ θ η η η θ + η + θ +. 2 ηα ] ηs[ ( η + ) θ+ 2 α] 2 ηα ] + ηs[ ( η + ) θ+ 2 α] 2 ηα ] ηc[ ( η + ) θ+ 2α ]

3 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~ Here, κ = (3 4 ν ) fr a plae strai ad κ = (3 4 ν) /( + ν) fr a plae stress, C = csie, ad S = sie. µ ad ν are the shear mdulus ad Piss s rati, respectively. A R ad A are udetermied eigevectr cefficiets. The variati f the sigular eigevalues with respect t the wedge agle is shw i Fig. 2. this research, the Mde stress itesity factr K, which is related t the sigular Mde eigevalue ϕ ad the udetermied eigevectr cefficiet AR( ϕ ), is defied by the fllwig Eq. (5). Mrever, the Mde stress itesity factr K ca be described by usig η ad A ( η ), as shw i Eq. (6). η is the sigular eigevalue f the slidig mde ad A ( η ) is the udetermied eigevectr cefficiet assciated with η. By usig ly tw sigular terms with ϕ ad η i the eigefucti expasi series, the stress ad displacemet fields arud the V-tch tip ca be represeted as fllws: Fig. 2. Variati f eigevalues with the wedge agle α. ϕ η r K (, ) r K u () j r θ = gj θ + gj( θ), 2µ 2π 2µ 2π (3) ( j = r, θ ) σ (, ) K K () ij r θ = fij f (), ϕ ij 2 r θ + η π 2 π r θ (4) ( i, j = r, θ) where K lim 2 ϕ σ = θθ π = r 0 fθθ(0) 2 π A R, (5) η σ K lim 2 r = r θ π = r 0 frθ(0) 2π A. (6) 3. Brittle fracture criteri fr V-tches uder mixed mde ladig As stated i itrducti, the directi f the maximum circumferetial stress varies with the distace r fr all V-tches except cracks uder mixed mde ladig. Fr this reas, it is difficult t apply the maximum circumferetial stress criteri ad Nvzhilv s criteri t mixed mde prblems. The fllwig extesis t the maximum circumferetial stress criteri ad Nvzhilv s criteri are assumed: () A crack prpagates alg the directi θ fr which the circumferetial stress is maximum at a small distace r = d frm the V- tch r crack tip ad (2) a fracture ccurs whe the value f σ reaches that f σ c, which is equal t the fracture stress i uiaxial tesi. σ ca be expressed as fllws: σ = σθθ (, r θ) dr d, (7) 0 d where d is assumed t be a material cstat ad the crack rietati θ is the agle prducig [ σθθ ( d, θ )] max the circumferece f radius, r = d as shw i Fig. 3. By substitutig the circumferetial stress cmpet σ θθ f Eq. (4) it Eq. (7), the fllwig expressi is btaied: Fig. 3. The mea circumferetial stress ver the legth ear the tip f a V-tch. d σ = K fθθ( θ) + ϕ 2 π ϕ d. (8) K fθθ( θ) η η d

4 272 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 σ is related t the breakig tesile stress σ c which ca be btaied frm the tesile test f a utched specime: σ σ c. (9) The brittle fracture f a egieerig structure will ccur whe the value f σ reaches that f σ c. The fllwig fracture criteri is btaied by Eqs. (8) ad (9): K cr f θθ( θ) + 2 π ϕ ϕ d σ c, (0) K cr f θθ( θ) = η η d σ c cr where K cr ad K are the critical values f K ad K uder the mixed mde ladig cditi, respectively. Sice K becmes zer i the case f Mde, Eq. (0) ca be represeted as fllws: d f (0) K cr = θθ 2π ϕ σ c /( ϕ ). () the case f crack prblems, the sigular eigevalue fr Mde is btaied as ϕ = 0.5 ad K is deted as K fr easy idetificati. A fracture ccurs whe the value f K reaches that f the fracture tughess K C. Therefre, d ca be btaied frm K C ad σ c by emplyig experimetal results [4]: d 2 2KC = 2π σ. (2) c As stated abve, θ ca be determied by usig the cditi f σθθ ( r, θ) / θ = 0 at r = d frm Eq. (3). cr ϕ d cr η d K ( σθθ )' = ( fθθ( θ) )' 2 π, (3) K + ( fθθ( θ ))' = 0 2 π where the prime symbl represets differetiati. Frm Eqs. (0) ad (3), the critical values f K ad K fr the mixed mde cditi ca be btaied as fllws: K cr ϕ 2 π d fθθ fθθ( θ) fθθ ( θ) fθθ = ( ( θ) ) ' ϕ η ( ( θ) ) ' θ= θ K cr η 2 π d fθθ fθθ( θ) fθθ( θ) fθθ = ( ( θ) )' η ϕ ( ( θ) )' θ= θ σc, (4) σc. (5) A uified brittle fracture criteri fr cracks ad V- tches uder mixed mde ladig was btaied by extedig the maximum circumferetial stress criteri ad Nvzhilv s criteri. The mixed mde fracture tughess ad crack rietati f PMMA plates with a sharp V-tch ca be predicted by this criteri. 4. Experimets ad discussis 4. Specimes ad the ladig fixture Tests were carried ut t measure the critical lads at fracture ad the fracture agles f plates with a sharp V-tch. The phtgraph ad cfigurati f the ladig fixture with a specime are shw i Fig. 4. The ladig fixture is f Richard s type [2], ad it ca be used fr varius mixed ladig mdes by chagig the ladig agle γ. The specime material fr the LEFM test is PMMA, which is brittle. The specime dimesis are as fllws: height H = 60 mm, width W = 40mm, tch legth a = 20mm ad thickess t = 5mm. The tches i the specimes were shaped by a vertical millig machie ad were cstraied by tw steel plates with a tch agle f 2β = 50º. The ttal umber f specimes prepared was 90: 8 specimes fr each tch agle f 2β = 5º, 30º, 60º, 75º, ad 90º. The ladig was perfrmed by varyig the ladig agle frmed by the lgitudial axis f the PMMA plate with respect t the ladig directi, as shw i Fig. 4(b). The test ladig agles were γ = 0º (Mde ), 30º, 45º, 60º, 75º ad 90º (Mde ). The specime ladig was carried ut i a NSTRON-448 serv-ctrlled testig machie. The tests were cducted i air at rm temperature ad the crsshead speed was 0.5mm/mi. The critical lads at fracture are tabulated i Table. Sice sme specimes ruptured at the fixig hles,

5 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~ Table. Fracture lad f V-tched PMMA specimes. Fracture lad [kn] Ladig Ntch agle(γ) agle(2β) Table 2. plates. θ measured fr the brke V-tched PMMA Ladig Ntch agle(γ) agle(2β) Crack rietati Table 2. t was bserved that θ icreases with the ladig agle, whereas θ decreases as the tch agle icreases. σ c was als btaied by usig the test results f seve tesile specimes. The test results give σ c = 6.7MPa with a stadard deviati f 5. MPa. All the tesile fractures ccurred at the maximum lad withut eckig. 4.2 Determiati f the stress itesity factrs (a) Fig. 4. Phtgraph ad cfigurati f the ladig fixture with a V-tch specime. several fracture lads were t recrded i Table. The experimetal data i this research were similar i tred t thse f Sewery et al. [2]. As shw i Table, the fracture lad icreases with the tch agle ad ladig agle. Furthermre, the fracture lads i Mde fr γ = 90º are remarkably larger tha thse uder Mde fr γ = 0º. The critical values fr γ = 30 suddely icrease i cmparis with thse fr γ = 0º ad are prduced the aalgus value i all the tch agles. The measured value f θ fr the brke specimes is preseted i (b) T calculate the stress itesity factrs f a V-tch, the RWCM was used. K ad K assciated with A R ad A, respectively, ca be calculated by usig the RWCM. The stresses ad displacemets required fr the calculati are btaied by a fiite elemet aalysis, which uses the cmmercial fiite elemet aalysis cde ANSYS [22]. Fr verifyig the validity ad accuracy f the RWCM prgram i the mixed mde cditi, the stresses σ rr fr the FEA ad RWCM the ctur f radius r =0.2 a with respect t the V-tch tip i Fig. 4(b) are cmpared. Three cases are examied. The first case is 2β =20º, aw=0.5, / H / W =.5, ad γ =45º; the secd case is 2β =60 º ad γ =90º; ad the third case is 2β =90º ad γ =60º, as shw i Fig. 4(b). The FE mdel fr the

6 274 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 Table 3. Eigevectr cefficiet assciated with each eigevalue. Case 2 3 Eigevalue Assciated eigevectr cefficiet φ A R (φ ) φ A R2 (φ 2 ) φ A R3 (φ 3 ) η A (η ) η 2 A 2 (η 2 ).505 η A 3 (η 3 ) η A 4 (η 4 ) φ A R (φ ) η A (η ) η 2 A 2 (η 2 ) φ A R (φ ) η A (η ) η 2 A 2 (η 2 ) Fig. 5. FE-mdel fr the first case with 2β =20º, a/ W =0.5, H / W =.5, ad γ =45º. first case is shw i Fig. 5. Ladig pit ad fixed hles were cected via multipit cstrait equati (MPC) [22] t distribute the frce applied at the ladig pit t des arud the hles, takig it accut the gemetry f the fixed hles. The eigevectr cefficiets were evaluated uder the plae stress cditi. Similar aalyses have bee carried ut fr all the ther cases, ad the results f thse fr each case are preseted i Table 3. The secd eigevalue η 2 = f Mde is a eigevalue related t the rtatial cmpet f rigid bdy mti [9]. Therefre, the eigevalue η 2 = cat affect the stress distributi ear the tch tip. the first case, ( σ rr ) FEA btaied by usig the fiite elemet aalysis ad ( σ rr ) RWCM calculated by substitutig the eigevectr cefficiets determied frm the RWCM prgram it the radial stress cmpet σ rr f Eq. (2) are shw i Fig. 6; these stresses are rmalized by tracti T i (= F/( t W) ) with respect t tesile ladig. t is shw that there is a csiderable differece i the values betwee ( σ rr ) FEA ad ( σ rr ) RWCM btaied by usig ly the terms f the sigular eigevalues ϕ ad η. By addig the terms f the sigular eigevalues t σ rr f Eq. (2), ( σ rr ) RWCM appraches ( σ rr ) FEA, as shw i Fig. 6. The secd case is subjected t Mde ladig. Hece, the eigevectr cefficiets assciated with Mde becme zer, as shw i Table 3. Fig. 7 shws that ( σ rr ) RWCM determied by the Mde sigular eigevalue η is csistet with ( σ rr ) FEA. the third case, Fig. 8 shws that the value f ( σ rr ) RWCM calculated by usig ly the sigular terms is i gd agreemet with that f ( σ rr ) FEA. Fig. 6. Cmparis f ( σ rr ) FEA ad ( σ rr ) RWCM accrdig t the umber f eigevalues the ctur f radius r =0.2 a, whe a/ W =0.5, H / W =.5, 2β =20º, ad γ =45º. Fig. 7. Cmparis f ( σ rr ) FEA ad ( σ rr ) RWCM accrdig t the umber f eigevalues the ctur f radius r =0.2a, whe a/ W =0.5, H / W =.5, 2β =60º, ad γ =90º.

7 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~ Table 4. Material prperty d ad K c fr PMMA. Material cstat d [m] Fracture tughess K c [MPa m 0.5 ] Value Fig. 8. Cmparis f ( σ rr ) FEA ad ( σ rr ) RWCM accrdig t the umber f eigevalues the ctur f radius r =0.2 a, whe a/ W =0.5, H / W =.5, 2β =90º, ad γ =60º. 4.3 Verificati f the prpsed fracture criteri The critical lads at fracture were measured t calculate AR( ϕ ) ad A ( η ) thrugh umerical aalysis by usig the RWCM prgram. The critical stress itesity factrs were determied frm the RWCM prgram by csiderig the stresses ad displacemets cmputed frm the fiite elemet aalysis usig the critical lads at fracture. the fiite elemet aalysis, the plae stress cditi was used ad the elastic prperties f the PMMA material are Yug s mdulus E = 339MPa ad Piss s rati ν =0.36. T verify the prpsed fracture criteri, the critical stress itesity factrs determied by the fracture lad will be cmpared with the fracture curve predicted by the criteri f Eqs. (4) ad (5). The crack rietatis measured fr the brke specimes will als be cmpared with the results predicted by the criteri f Eq. (3). The value f d shuld be determied i rder t predict the crack iitiati. Hwever, it is difficult that PMMA specimes with the cfigurati f a crack are fabricated exactly. Therefre, istead f determiig d frm Eq. (2) by usig K C, it ca be btaied by substitutig the K cr fr each V- tch agle f 2β = 5º, 30º, 60º, 75º, ad 90º it Eq. () because d is a material cstat. this paper, the mea value f all the results btaied frm the abve substituti is used as the material cstat d ad the fracture tughess K C ca be als calculated frm Eq. (2) by usig the mea value d. These values are preseted i Table 4. Fig. 9 shws K cr as predicted by Eq. () usig the mea value d ad the experimetal results. Fig. 9. K cr predicted by d uder Mde ladig. Figs. 0(b)-(f) shw a cmparis f the predicted curves f the critical stress itesity factrs f Eqs. (4) ad (5) fr each V-tch agle f 2β =5º, 30º, 60º, 75º, ad 90º with the experimetal results i the mixed mde cditi. Fig. 0(a) shws ly the predicted curves f the critical stress itesity factrs fr 2β =0º withut the experimetal results. The crack rietatis predicted by Eq. (3) are als shw with the experimetal results i Figs. (b)-(f). Fig. (a) shws ly the predicted crack rietatis fr 2β =0º withut the experimetal results. t is well kw that θ becmes zer fr all tch agles uder Mde ladig. Hwever, uder Mde ladig, θ varies frm θ = º i the case f 2β =0º t º i the case f 2β =90º as shw i Fig. 2. The results shw that all the fracture curves may prvide a cservative estimate f the fracture stregth. This fact is due t sharp V-tch machiig difficulties. We kw that the sharpess f the tip f a specime is imprtat. But, i V-tch machiig, it is uavidable that V-tches have the tch rt radii. The small ruded shape f the tch tips will icrease the fracture lad f the specimes. Hece, it ca be stated that the predicted ad experimetal results are i early gd agreemet. We examied ce mre usig the experimetal data fr 2β =20º, 40º, 60º, ad 80º give by Sewery et al. [2] fr PMMA. The material prperties f PMMA [23] are E =3300MPa ad ν =0.35. The plae stress cditi is used i the fiite elemet

8 276 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 (a) 2β = 0 (b) 2β = 5 (c) 2β = 30 (d) 2β = 60 (e) 2β = 75 (f) 2β = 90 Fig. 0. Predicted fracture curves ad experimetal critical stress itesity factrs uder mixed mde ladig. (a) 2β = 0 (b) 2β = 5 (c) 2β = 30 (d) 2β = 60 (e) 2β = 75 (f) 2β = 90 Fig.. Predicted ad experimetal crack rietati uder mixed mde ladig. aalysis. d was determied frm Eq. () i the same maer. ts mea value was f d = m ad the value f K C was.346mpa m 0.5. Figs. 3(a)-(d) shw the fracture curves as predicted by the criteri ad the critical stress itesity factrs btaied frm the experimets. The value f θ predicted frm Eq. (3) ad the measured data f Sewery et al. [2] are shw i Figs. 4(a)-(d).

9 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~ t ca be stated that the predicted ad experimetal results are als i early gd agreemet. 5. Cclusis Fig. 2. Crack rietati fr differet tch agles uder Mde ladig. The purpse f this paper is t prpse a uified brittle fracture criteri fr cracks ad V-tches uder mixed mde ladig by extedig the maximum circumferetial stress criteri ad Nvzhilv s criteri. (a) 2β = 20 (b) 2β = 40 (c) 2β = 60 (d) 2β = 80 Fig. 3. Predicted fracture curves ad Sewery s experimetal critical stress itesity factrs uder mixed mde ladig. (a) 2β = 20 (b) 2β = 40 (c) 2β = 60 (d) 2β = 80 Fig. 4. Predicted ad Sewery s experimetal crack rietati uder mixed mde ladig.

10 278 J. K. Kim ad S. B. Ch / Jural f Mechaical Sciece ad Techlgy 22 (2008) 269~278 The crack iitiatis ad rietatis predicted by the prpsed criteri agree early with ur experimetal results ad Sewery s tests. This study shws that the prpsed criteri is csiderably simpler ad ca easily predict the crack iitiati ad rietati fr cracks ad V-tches uder mixed mde ladig. Ackwledgmet This wrk was supprted by the Kyugam Uiversity Fudati Grat Refereces [] F. Erdga ad G. C. Sih, O the crack extesi i plates uder plae ladig ad trasverse shear ladig, Jural f Basic Egieerig, Trasactis f ASME, 85 (963) [2] G. C. Sih, Strai eergy desity factr applied t mixed mde crack prblems, teratial Jural f Fracture, 0 (974) [3] M. A. Hussai, S. L. Pu ad J. Uderwd, Strai eergy release rate fr a crack uder cmbied mde ad mde, Fracture Aalysis, ASTM STP 560, ASTM, Philadelphia, (974) [4] A. Sewery, Brittle fracture criteri fr structures with sharp tches, Egieerig Fracture Mechaics, 47 (994) [5] V. V. Nvzhilv, O ecessary ad sufficiet criteri f brittle fracture, Prikladaja Matematika, 33 (969) [6] J. M. Whitey ad R. J. Nuismer, Stress fracture criteria fr lamiated cmpsites ctaiig stress ccetratis, Jural f Cmpsite Material, 8 (974) [7] M. L. Du ad W. Suwit, Cuigham S. May CW. Fracture iitiati at sharp tches mde, mde, ad mild mixed mde ladig, teratial Jural f Fracture, 84 (997) [8] L. O. Jerkvist, Fracture f wd uder mixed mde ladig. Experimetal ivestigati f Picea abies, Egieerig Fracture Mechaics, 68 (200) [9] M. Stradberg, Fracture at V-tches with ctaied plasticity, Egieerig Fracture Mechaics, 69 (2002) [0] F. J. GÓmez ad M. Elices, Fracture f cmpets with V-shaped tches, Egieerig Fracture Mechaics, 70 (2003) [] F. J. GÓmez ad M. Elices, A fracture criteri fr bluted V-tched samples, teratial Jural f Fracture, 27 (2004) [2] A. Sewery, S. Pskrbk ad Z. MrÓz, Brittle fracture i plae elemets with sharp tches uder mixed-mde ladig, Jural f Egieerig Mechaics, 23 (997) [3] J. L. Greestedt, S. HallstrŐm ad J. Kuttekeuler, O cracks emaatig frm wedges i expaded PVC fam, Egieerig Fracture Mechaics, 54 (996) [4] S. HallstrŐm ad J. L. Greestedt, Mixed mde fracture f cracks ad wedge shaped tches i expaded PVC frm, teratial Jural f Fracture, 88 (997) [5] M. Ster, E. B. Becker ad R. S. Duham, A ctur itegral cmputati f mixed-mde stress itesity factrs, teratial Jural f Fracture, 2 (976) [6] W. C. Carpeter, A imprved techique fr determiig higher rder eigevectr cefficiets, teratial Jural f Fracture, 37 (988) [7] W. C. Carpeter, Cmmets the eigevalue frmulati f prblems with cracks, V-tched cracks, ad crers, teratial Jural f Fracture, 68 (994) [8] W. C. Carpeter, sesitivity f the reciprcal wrk ctur itegral methd t higher rder eigevectrs, teratial Jural f Fracture, 73 (995) [9] J. K. Kim ad S. B. Ch, A Aalysis f Eigevalues ad Eigevectrs fr V-tched Cracks i Pseud-istrpic Dissimilar Materials, teratial Jural f Precisi Egieerig ad Maufacturig, 3 (2002) [20] M. L. Williams, Stress sigularities resultig frm varius budary cditis i agular crers f plates i extesi, Jural f Applied Mechaics, 9 (952) [2] H. A. Richard, A ew cmpact shear specime, teratial Jural f Fracture, 7 (98) R05-07 [22] ANSYS User s ad Thery Mauals, Revisi 0.0. [23] A. Sewery ad A. Lukaszewicz, Verificati f brittle fracture criteria fr elemets with V-shaped tches, Egieerig Fracture Mechaics, 69 (2002)