NUMERICAL VERIFICATION OF THE RELATIONSHIP BETWEEN THE INPLANE GEOMETRIC CONSTRAINTS USED IN FRACTURE MECHANICS PROBLEMS.


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1 Marci Graba Numerical Verificatio of the Relatioship Betwee he IPlae Geometric CoStraits used i Fracture Mechaics Problems NUMERICAL VERIFICAION OF HE RELAIONSHIP BEWEEN HE INPLANE GEOMERIC CONSRAINS USED IN FRACURE MECHANICS PROBLEMS Marci GRABA * * ielce Uiversity of echology, Faculty of Mechatroics ad Machie Desig, Al. 1lecia PP 7, ielce, Polad Abstract: I the paper, umerical verificatio ad catalogue of the umerical solutios based o Modify Boudary Layer Approach to determie the relatioship betwee stress ad stress are preseted. Based o the method proposed by Larsso ad Carlsso, the stress value are calculated for some elasticplastic materials for differet value of stress ad exteral load expressed by Jitegral. he ifluece of the exteral load, stress value ad material properties o stress value were tested. Such catalogue may be useful durig solvig the egieerig problems, especially while is eeded to determie real fracture toughess with icludig the geometric costraits, what was proposed i FINE procedures. ey words: Fracture, Crack, Stress Fields, HRR, MBLA, iplae Costraits, Stress, Stress. 1. DESCRIPION OF HE SRESS FIELD NEAR CRAC IP HEOREICAL BACGROUNDS For mode I of loadig, stress field ahead of a crack tip i elastic liear isotropic material ca be give by (Williams, 1957): θ θ 3θ 11 = = I xx cos 1 si si 2πr 2 2 2, θ θ 3θ 22 = = I yy cos 1 + si si 2πr 2 2 2, θ θ 3θ 12 = τ = I xy cos si cos 2πr 2 2 2, 33 = zz = for plae stress, ( ) + = zz= ν for plae strai, 33 xx yy where ij is the stress tesor, r ad θ are as defied i Fig. 1, ν is Poisso s ratio, I is the Stress Itesity Factor (SIF). crack y Fig. 1. Defiitio of the coordiate axis ahead of a crack tip he z directio is ormal to the page (ow Fig. based o (Williams, 1957) r τ xy θ yy τ yx xx (1) x he SIF defies the amplitude of the crack tip sigularity. hat is, stresses ear the crack tip icrease i proportio to. Moreover, the stress itesity factor completely defies the crack tip coditios; if is kow, it is possible to solve for all compoets of stress, strai, ad displacemet as a fuctio of r ad θ. his sigleparameter descriptio of crack tip coditios turs out to be oe of the most importat cocepts i fracture mechaics (Williams, 1957). I 1968, Hutchiso (1968) ad Rice ad Rosegre (1968) derived the sigular stressstrai fields at a crack tip i a powerlaw hardeig material (which called the HRRfield). he hardeig law used by Hutchiso ad Rice ad Rosegre is give by: ε = α, (2) ε where is a referece stress value that is usually equal to the yield stregth, ε= /E, α is a dimesioless costat, ad is the strai hardeig expoet. Assumig the RambergOsgood material, the results obtaied by Hutchiso ad Rice ad Rosegre for plae strai ad for plae stress may be expressed i the followig form: 1 J 1+ ij = ~ ij(, ) α ε Ir θ, (3) J 1+ εij = αε ~ εij(, ) α ε Ir θ, (4) where J is the Jitegral, I is a itegratio costat that depeds o, (, ) ad (, ) are dimesioless fuctios of ad. hese parameters also deped o the stress state (i.e. plae stress or plae strai). All this fuctio may be determied usig the algorithm ad computer program preseted by Gałkiewicz ad Graba (26). 38
2 acta mechaica et automatica, vol.6 o.2 (212) he Jitegral defies the amplitude of the HRR sigularity, just as the stress itesity factor characterizes the amplitude of the liear elastic sigularity. hus, Jitegral completely describes the coditios withi the plastic zoe. A structure i smallscale yieldig has two sigularitydomiated zoes: oe i the elastic regio, where stress varies as 1/ ad oe i the plastic zoe where stress varies as (1/ ) /( ). he latter ofte persists log after the liear elastic sigularity zoe has bee destroyed by crack tip plasticity. he HRR sigularity cotais the same apparet aomaly as the liear elastic fracture mechaics sigularity; amely, both predict ifiite stresses as r. he sigular field does ot persist all the way to the crack tip, however. he large strais at the crack tip cause the crack to blut, which reduces the stress triaxiality locally. he bluted crack tip is a free surface; thus xx must vaish at r =. he aalysis that leads to the HRR sigularity does t cosider the effect of the bluted crack tip o the stress fields, or does it take accout of the large strais that are preset ear the crack tip. his aalysis is based o small strai theory, which is the multiaxial equivalet of egieerig strai i a tesile test. Small strai theory breaks dow whe strais are greater tha about 1%. directio parallel to the crack advace directio. he mode I stress field becomes: = ij I 2πr f ij ( θ) +, (5) ν here is a uiform stress i the x directio (which iduces a stress ν i the z directio i plae strai). 22 / J = 116kN/m SEN(B) plae strai a/w =.5 W = 4mm = 315MPa = 5 E = 26MPa ν =.3 ε = /E =.153 large strai (FEM) small strai (FEM) HRR solutio ψ = x 22 /J Fig. 2. he stress distributio ear a crack tip (curves were obtaied usig the FEM for small ad fiite strai ad HRR formula) McMeekig ad Parks (1979) performed crack tip fiite elemet aalyses that icorporated large strai theory ad fiite geometry chages. Some of their results are show i Fig. 2, which is a plot of stress ormal to the crack plae versus distace. he HRR sigularity is also show o this plot. Note that both axes are dimesioless i such a way that both curves are ivariat, as log as the plastic zoe is small compared to specime dimesios. he solid curve i Fig. 2 reaches a peak whe the ratio x /J is approximately uity, ad decreases as x. his distace correspods approximately to twice the Crack ip Opeig Displacemet (COD). he HRR sigularity is ivalid withi this regio, where the stresses are iflueced by large strais ad crack blutig. Preseted above solutios for stress (Eq. (1) for liear fracture mechaics ad Eq. (3) for oliear fracture mechaics) oly describe the ear tip field ad cosider oly the first term of aylor expasio. I the liear case, the secod term i the aylor expasio correspods to the so called stress which acts i the Fig. 3. Modified boudary layer aalysis. he first two terms of the Williams series are applied as boudary coditios (ow Fig. based o Aderso, (1995)) he more commo form of Eq. (9) is the followig etry: = ( θ) δ1 iδ, (6) 1j 2πr I ij fij + where δij is the roecker delta. We ca assess the ifluece of the stress by costructig a circular model that cotais a crack, as illustrated i Fig. 3. O the boudary of this model, let us apply iplae tractios that correspod to Eq. (5). A plastic zoe develops at the crack tip, but its size must be small relative to the size of the model i order to esure the validity of the boudary coditios, which are iferred from a elastic solutio. his cofiguratio, ofte referred to as a modified boudary layer aalysis, simulates the ear tip coditios i a arbitrary geometry, provided the plasticity is well cotaied withi the body. Fig. 4 is a plot of fiite elemet results from a modified boudary layer aalysis (Neimitz et al., 27) that show the effect of the stress o stresses deep iside the plastic zoe, obtaied for large strai assumptio. he special case of = correspods to the smallscale yieldig limit, where the plastic zoe is a egligible fractio of the crack legth ad size of the body, ad the sigular term uiquely defies the eartip fields. he sigleparameter descriptio is rigorously correct oly for =. Note that egative values cause a sigificat dowward shift i the stress fields. Positive values shift the stresses to above the smallscale yieldig limit, but the effect is much less proouced tha it is for egative stress. Note that the HRR solutio does ot match the = case. he stresses deep iside the plastic zoe ca be represeted by a power series, where the HRR solutio is the leadig term. Fig. 4 idicates that the higher order plastic terms are ot egligible whe =. 39
3 Marci Graba Numerical Verificatio of the Relatioship Betwee he IPlae Geometric CoStraits used i Fracture Mechaics Problems Modified Boudry Layer Approach = 315 MPa = 5 ν =.3 E = 26 MPa 22 / Fig. 4. he stress distributio i frot of the crack, computed usig modified boudary layer approach for costat SIF, I, ad chagig stress. Properties of material like i the Sumpter ad Forbes paper (Sumpter ad Forbes, 1992) I a cracked body subject to Mode I loadig, the stress, like I, scales with the applied load. he biaxiality ratio relates to stress itesity: πa β =, (7) I where a is the crack legth. For a throughthickess crack i a ifiite plate subject to a remote ormal stress β = 1. hus a remote stress iduces a stress of  i the x directio. I the oliear fracture mechaics, the factor was itroduced by O Dowd ad Shih (1991, 1992) to accout for differece betwee the HRR field ad fiite elemet results. he factor (also called stress) correspods to a additioal hydrostatic pressure. he modified by O Dowd ad Shih stress field is obtaied as: ( ij) + δij =, (8) ij HRR where (ij)hrr is give by Eq. (3). For small scale yieldig, Eq. (8) ca be writte i the followig form: ( ij) δij = = + ij. (9) he parameter ca be iferred by subtractig the stress field for the = referece state from the stress field of iterest. O'Dowd ad Shih ad most subsequet researchers defied as follows: ( ) ( yy) yy = = for θ= ad r = 2. (1) J J = 3 kn/m / = 1,16 / = 1, / = ,75 / = ,5 / = ,25 / = / =,25 / =,5 / = 1, ψ = x 22 /J Referrig to Fig. 4, we see that is egative whe is egative. For the modified boudary layer solutio, ad are uiquely related. Fig. 5 is a plot of versus for a two work hardeig expoets. A relatio betwee ad stress, based o umerical calculatios, usig large strai theory ad icremetal strai plasticity is give by O Dowd ad Shih (1991) as: 2 ( ) + a ( ) a ( ) 3 a + a +, (11) where coefficiets a, a1, a2 ad a3 were give oly for two work hardeig expoets: for =5: a=.1, a1=.76, a2=.32, a3=.1 ad for =1: a=.1; a1=.76, a2=.52, a3=. Proposed by Eq. (11) relatioship is the result of the matchig mathematical formula to umerical results. I 1995 O Dowd (O Dowd, 1995) was proposed, to describe the relatioship betwee ad stress by liear formula, as = for for <. (12) > Equatio (11) for =5 Equatio (11) for =1 Equatio (12) / Fig. 5. Relatioship betwee ad stress, based o Equatios (11) ad (12) (based o O Dowd ad Shih (1991)) Fig. 5 shows graphically the mutual relatioship betwee the ad stress, based o formulas (11) ad (12). he use of both depedecies is relatively simple. I the case of formula (11) must kow the values of the coefficiets a, a1, a2 ad a3. Seems to be easier to use formula (12), because it is required oly kowledge level of stress, which for various geometries ca be foud i the literature (Sherry et al., 1991), (Leevers ad Rado, 1983). Whe Equatio (12) will be used i aalysis of the real structural elemet, as ca be observed, this formula did ot take ito accout the geometry characterizatio, for example material properties, exteral load, kid of specime. Formula (12), takes ito accout oly crack legth, because, stress deped o relative crack legth (Leevers ad Rado, 1983). However, thaks to its simplicity, Equatio (12) has foud use i the solvig egieerig problems ad it was recommeded by FINE procedures (FINE, 26). It should be oted that both Equatios  (11) ad (12) were based o aalysis of small scale yieldig, based o Modify Boudary Layer Approach (MBLA). It should be oted also that both Equatios geerally do ot iclude exteral load, which i the case of plae strai strogly affects the stress level. 4
4 acta mechaica et automatica, vol.6 o.2 (212) 2. UILIZAION OF HE CONSRAIN PARAMEERS IN HE EVALUAION OF FRACURE OUGHNESS Both costrait parameters, the ad stress were foud applicatio i Europea Egieerig Programs, like SINAP (SINAP, 1999) ad FINE (FINE, 26). he stress or stress is applied uder costructio of the fracture criterio ad to assessmet the fracture toughess of the structural compoet. Real fracture toughess, may be evaluated usig the formula proposed by Aisworth ad O Dowd (1994). Aisworth ad O Dowd have show that the icrease i fracture i both the brittle ad ductile regimes may be represeted by a expressio of the form: C mat = mat mat k [ 1+ α( βl ) ] r β Lr > (13) βl < where mat is the fracture toughess for plae strai coditio obtaied usig FINE procedures, ad β is the parameter calculated usig followig formula: ( L ) r β =, Lr for elastic materials, for elasticplastic materials, r (14) where Lr is the ratio of the actual exteral load P ad the limit load P (or the referece stress), which may be calculated usig FINE procedures (FINE, 26). he costats α ad k which are occurrig i Eq. (13), are material ad temperature depedet (ab. 1). Sherry ad et al., (25a, b) proposed the desigatio procedures to calculate the costats α ad k. hus J ad  theories have practical applicatio i egieerig issues. ab 1. Some values of the α ad k parameters, which are occurrig i Eq. (13) (SINAP, 1999; FINE, 26) material temperature fracture mode α k A533B (steel) 75 C cleavage A533B (steel) 9 C cleavage A533B (steel) 45 C cleavage Low Carbo Steel 5 C cleavage A515 (steel) +2 C cleavage ASM 71 Grade A +2 C ductile he reciprocal relatioship betwee iplae costrait parameters, for what ca be cosidered the stress ad stress may be very useful i practical egieerig problems, to determiatio of the real fracture toughess or i failure assessmet diagrams (FAD) aalysis whe correctio of the FAD curve usig costrait parameter is doe (SINAP, 1999), (FI NE, 26). hus, i this paper, catalogue of the  trajectories obtaied usig MBLA aalysis will be preseted. he ifluece of the material properties will be tested. 3. DEAILS OF HE NUMERICAL ANALYSIS I the umerical aalysis, ADINA System (ADINA, 28a, b) was used. Computatios were performed for plae strai usig small strai optio ad the Modify Boudary Layer Approach (MBLA) model. he MBLA model cosists of big circle, which radius aroud the crack tip where the boudary coditios are modeled is 1 meters log. Due to the symmetry, oly a half of the circle was modeled (see Fig. 6). he boudary coditios are modeled usig the followig relatioship: u1 = + u2 = ν E ( 1+ ν) 2 ( 1 ν ) r cos( θ) E E ( 1+ ν) E ( 1+ ν) r si( θ) κ 1+ r θ cos + 2 θ 2π si 2, (15) κ + 1+ r θ si + 2 θ 2π 2 2 cos 2 where is the stress itesity factor (SIF) calculated form Jitegral value usig formula = /(1 ) i preseted i the paper umerical program the followig values of the Jitegral were tested: J={1, 25, 5, 1, 25, 5}kN/m; r ad θ are polar coordiates; ν is the Poisso s ratio; E is Youg s modulus; is the stress expressed i stress uit ( ) i preseted i the paper umerical program, the followig values of the parameter were tested: ={.5,.25,, .25, .5, 1}; κ=34ν for plae strai; κ=(3ν)/(1+ν) for plae stress. he radius of the crack frot was equal to rw=5 16 m. he crack tip was modeled as half of arc. he crack tip regio about to m was divide ito 5 semicircles. he first of them, was at least 2 times smaller the the last oe. he fiite elemet mesh was filled with the 9ode plae strai elemets. he size of the fiite elemets i the radial directio was decreasig towards the crack tip, while i the agular directio the size of each elemet was kept costat. It varied from θ=π/19 to θ=π/3 for various cases tested. he whole MBLA model was modeled usig 2584 fiite elemets ad 1647 odes. he example fiite elemet model for MBLA aalysis is preseted i Fig. 6. I the FEM simulatio, the deformatio theory of plasticity ad the vo Misses yield criterio were adopted. I the model the stress strai curve was approximated by the relatio: ε for =, where α=1. (16) ε α( ) for > he tesile properties for the materials which were used i the umerical aalysis are preseted below i ab. 2. I the FEM aalysis, calculatios were doe for sixtee material cofiguratios, which were differed by yield stress ad the work hardeig expoet. 41
5 Marci Graba Numerical Verificatio of the Relatioship Betwee he IPlae Geometric CoStraits used i Fracture Mechaics Problems a) a) b) b) c) Fig. 6. he MBLA model which was used i the umerical program a); Crack tip regio usig i the MBLA model b) he Jitegral were calculated usig two methods. he first method, called the virtual shift method, uses cocept of the virtual crack growth to compute the virtual eergy chage. he secod method is based o the J itegral defiitio: J = [wdx 2 t ( u x1 )ds ], (17) C where w is the strai eergy desity, t is the stress vector actig o the cotour C draw aroud the crack tip, u deotes displacemet vector ad ds is the ifiitesimal segmet of cotour C. ab. 2. he mechaical properties of the materials used i umerical aalysis ad the HRR parameters for plae strai [MPa] E [MPa] 26 ν ε=/e α ~θθ (θ = ) I RESULS OF HE NUMERICAL ANALYSIS Fig. 7 presets the ifluece of the stress parameter value o the shape ad size (deoted i Fig. as rp) of the plastic zoe ear crack tip. For smaller value of the stress parameter, the greater plastic zoe is observed, if the same level of the Jitegral was used to calculate the boudary coditios ad the same material characteristic was established. he bigger plastic zoe is observed for MBLA model characterized by smaller yield stress (see Fig. 8). 42 Fig. 7. Ifluece of the stress parameter o shape ad size of the plastic zoe for MBLA model characterized by =315MPa, =1, J=1kN/m: a) =.5, rp=.29m; b) =, rp=.33m; c) =.5, rp=.9m (brighter regio is the plastic zoe) For smaller values of the stress parameter the, smaller value of the stress are observed. he ifluece of the work hardeig expoet o =() trajectories should be cosidered for a umber of cofiguratios. For the cases characterized by yield stress 5MPa ad for Jitegral values betwee 1 ad 25kN/m (which are used to determie the boudary coditios), the lower values of the stress are observed for weakly stregthe materials (see Fig. 9). For theses cases almost parallel arragemet of the =() trajectories is observed. he highest o the chart are situated curves for strogly stregthe material (=3). I other cases, whe the Jitegral value is equal to or greater tha 5kN/m, it ca be see the itersectig of the =() curves for differet values of the work hardeig expoet (see Fig. 1). For small values of the stress parameter (which meas that we are dealig with a case of highlevel of the flat geometric costraits), the lower values of parameter are observed for weakly hardeig materials. Icrease value of the stress parameter makes cuttig curves ad the reversal of the tred o the chart  the a smaller values of the parameter are observed for the case of the strogly hardeig materials. Numerical aalysis show, that the ifluece of the yield stress o =() trajectories is quite complex. For small values of the Jitegral (J=1kN/m or J=25kN/m), it ca be cocluded, that =() curves characterized by regularity of arragemet, especially for strog hardeig materials (=3 ad =5). he lowest o the charts are arraged the =() curves, described by small value of the yield stress. his meas that for the materials characterized by higher yield stress, the higher values of the parameter are obtaied (see Fig. 11).
6 acta mechaica et automatica, vol.6 o.2 (212) a) .4 b) c) .8 =315MPa J=5kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 1. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =315MPa, J=5kN/m.5 Fig. 8. Ifluece of the yield stress o shape ad size of the plastic zoe for MBLA model characterized by =, =1, J=1kN/m: a) =5MPa, rp=.13m; b) =1MPa, rp=.3m; c) =15MPa, rp=.1m (brighter regio is the plastic zoe) =5MPa J=1kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 9. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =5MPa, J=1kN/m he ifluece of the yield stress o =() trajectories is sigificat for the low level of the Jitegral which was used to determiatio of the boudary coditios. Sometimes the ifluece of the yield stress o stress value is egligible, whe material characterized by big work hardeig expoet ad the level of the Jitegral is quite large (J 5kN/m). For weakly hardeig materials, =() trajectories are parallel, whe J itegral characterized by very high level (for example J=25kN/m or J=5kN/m) =3 J=25kN/m E=26GPa ν=.3 =315MPa =5MPa =1MPa =15MPa Fig. 11. he ifluece of the yield stress o the =() trajectories for MBLA characterized by =3, J=25kN/m he most importat coclusio cocers the ifluece of the Jitegral (which is used to determiatio of the boudary coditios for MBLA model) o stress value. As show by umerical calculatios, the value of the parameter as a fuctio of the stress parameter i a very small extet depeds o the Jitegral value adopted to determie the boudary coditios at MBLA issue. A very little impact (hardly isigificat), or the lack of impact is characterized for MBLA models, for which the Jitegral level used to determiatio of the boudary coditios was equal to or greater tha 5kN/m (see Fig. 12). Fig. 13 presets the ifluece of the work hardeig expoet o stress value for differet level of the stress parameter, which may be cosidered as a measure of the iplae costrait parameter. For the case of low costraits (low value of the stress, equal to .5 or 1.), stress value decreases whe the value of the work hardeig expoet icreases. Whe the value of the stress parameter is greater tha .25, it ca be see that stress value is costat or slightly icreases if value of the work hardeig expoet icreases. 43
7 Marci Graba Numerical Verificatio of the Relatioship Betwee he IPlae Geometric CoStraits used i Fracture Mechaics Problems a) a).4 J=5kN/m =315MPa E=26GPa ν= b) J [kn/m] =3 =315MPa E=26GPa ν=.3 =.5 =.25 = =.25 =.5 =1. b) J=1kN/m =5MPa E=26GPa ν=.3 =.5 =.25 = =.25 =.5 =1. c) J [kn/m] =5 =315MPa E=26GPa ν=.3 =.5 =.25 = =.25 =.5 =1. c) J=25kN/m =1MPa E=26GPa ν=.3 =.5 =.25 = =.25 =.5 = Fig. 12. he ifluece of the Jitegral value which was used to determiatio of the boudary coditios, o stress value for differet stress parameter: a) =3, =315MPa; b) =5, =315MPa; b) =1, =5MPa 5. CONCLUSIONS J [kn/m] =1 =5MPa E=26GPa ν=.3 =.5 =.25 = =.25 =.5 =1. I the paper, catalogue of the umerical solutios based o Modify Boudary Layer Approach to determie the relatioship betwee stress ad stress were preseted. Based o method proposed by Larsso ad Carlsso, the stress value were calculated for sixtee elasticplastic materials for differet value of stress ad exteral load expressed by Jitegral both parameter were used to determie the boudary coditios, which are ecessary to carry out the MBLA aalysis. he ifluece of the stress parameter ad material properties o stress value were tested =.5 =.25 = =.25 =.5 =1. Fig. 13. he ifluece of the work hardeig expoet o stress value for differet level of the stress parameter: a) J=5kN/m, =315MPa; b) J=1kN/m, =5MPa; c) J=25kN/m, =1MPa Obtaied umerical results lead to followig coclusios: for smaller value of the stress parameter, the smaller value of the stress are observed; the ifluece of yield stress o the stress value is sigificat for the case characterized by low level of the Jitegral, adopted to determie the boudary coditios at MBLA issue; i case of a higher level of the Jitegral, it ca be observed a little impact of the yield stress o the stress value, sometimes this effect is egligible; for differet stress value, stress value depeds o work hardeig expoet ; this ifluece should be discussed for each MBLA models separately; the stress parameter very weak depeds (or i geeral does ot deped) o the level of the Jitegral adopted 44
8 acta mechaica et automatica, vol.6 o.2 (212) to determie the boudary coditios i MBLA issue, especially for the case whe the level of the Jitegral is equal to or greater tha 5kN/m. Preseted i the paper such catalogue may be useful durig solvig the egieerig problems, especially while is eeded to determie real fracture toughess with icludig the geometric costraits, what was proposed i FINE procedures (FINE, 26). REFERENCES 1. ADINA (28a), ADINA 8.5.4: ADINA: heory ad Modelig Guide  Volume I: ADINA, Report ARD 87, ADINA R&D, Ic., ADINA (28b), ADINA 8.5.4: ADINA: User Iterface Commad Referece Maual  Volume I: ADINA Solids & Structures Model Defiitio, Report ARD 86, ADINA R&D, Ic., Aisworth R.A., O'Dowd N.P. (1994), A Framework of Icludig Costrait Effects i the Failure Assessmet Diagram Approach for Fracture Assessmet, ASME Pressure Vessels ad Pipig Coferece, PVPVol 287/MDVol47, ASME. 4. Betego, C. ad Hacock, J.W. (1991), wo Parameter Characterizatio of ElasticPlastic Crack ip Fields, Joural of Applied Mechaics, Vol. 58, Bilby, B.A., Cardew, G.E., Goldthorpe, M.R., Howard, I.C. (1986), A Fiite Elemet Ivestigatio of the Effects of Specime Geometry o the Fields of Stress ad Strai at the ips of Statioary Cracks, Size Effects i Fracture, Istitute of Mechaical Egieers, Lodo, FINE Fitess for Serwice Procedure Fial Draft (26), Edited by M. oçak, S. Webster, JJ. Jaosh, RA. Aisworth, R. oers. 7. Gałkiewicz J., Graba M. (26), Algorithm for Determiatio of (, ), (, ), (, ), ( ) ad ( ) Fuctios i HutchisoRiceRosegre Solutio ad its 3d Geeralizatio, Joural of heoretical ad Applied Mechaics, Vol. 44, No. 1, Hutchiso J. W. (1968), Sigular Behaviour at the Ed of a esile Crack i a Hardeig Material, Joural of the Mechaics ad Physics of Solids, 16, Irwi, G.R. (1957), Aalysis of Stresses ad Strais ear the Ed of a Crack raversig a Plate, Joural of Applied Mechaics, Vol. 24, Leevers P.S., Rado J.C. (1983), Iheret Stress Biaxiality i Various Fracture Specime Geometries, Iteratioal Joural of Fracture, 19, McMeekig, R.M. ad Parks, D.M. (1979), O Criteria for JDomiace of Crack ip Fields i LargeScale Yieldig, ASM SP 668, America Society for estig ad Materials, Philadelphia, Neimitz A., Graba M., Gałkiewicz J. (27), A Alterative Formulatio of the RitchieottRice Local Fracture Criterio, Egieerig Fracture Mechaics, Vol. 74, O Dowd N. P. (1995), Applicatios of two parameter approaches i elasticplastic fracture mechaics, Egieerig Fracture Mechaics, Vol. 52, No. 3, O Dowd N. P., Shih C. F. (1992), Family of Crackip Fields Characterized by a riaxiality Parameter II. Fracture Applicatios, J. Mech. Phys. Solids, Vol. 4, No. 5, O Dowd N. P., Shih C.F. (1991), Family of Crackip Fields Characterized by a riaxiality Parameter I. Structure of Fields, J. Mech. Phys. Solids, Vol. 39, No. 8, Rice J. R., Rosegre G. F. (1968), Plae Strai Deformatio Near a Crack ip i a Powerlaw Hardeig Material, Joural of the Mechaics ad Physics of Solids, 16, Rice, J.R. (1968), A Path Idepedet Itegral ad the Approximate Aalysis of Strai Cocetratio by Notches ad Cracks, Joural of Applied Mechaics, Vol. 35, pp Sherry A.H., Hooto D.G., Beardsmore D.W., Lidbury D.P.G. (25), Material costrait parameters for the assessmet of shallow defects i structural compoets Part II: costrait based assessmet of shallow cracks, Egieerig Fracture Mechaics, 72, Sherry A.H.,Frace C.C., Goldthorpe M.R. (1995), Compedium of stress solutios for two ad three dimesioal cracked geometries, Fatigue & Fracture of Egieerig Materials & Structures, Vol. 18, No. 1, Sherry, A.H., Wilkes M.A., Beardsmore D.W., Lidbury D.P.G. (25), Material costrait parameters for the assessmet of shallow defects i structural compoets Part I: Parameter solutios, Egieerig Fracture Mechaics, 72, SINAP (1999), SINAP: Structural Itegrity Assessmet Procedures for Europea Idustry. Fial Procedure, BriteEuram Project No BE Rotherham: British Steel. 22. Seddo, I.N. (1946), he Distributio of Stress i the Neighbourhood of a Crack i a Elastic Solid, Proceedigs, Royal Society of Lodo, Vol. A187, Sumpter J. G. D., Forbes A.. (1992), Costrait based aalysis of shallow cracks i mild steels, Proceedigs of WI/EWI/IS It. Cof o Shallow Crack Fracture Mechaics, oughess ests ad Applicatios, Paper 7, Cambridge U Westergaard, H.M. (1939), Bearig Pressures ad Cracks, Joural of Applied Mechaics, Vol. 6, Williams, M.L. (1957), O the Stress Distributio at the Base of a Statioary Crack, Joural of Applied Mechaics, Vol. 24, Ackowledgmets: he support of the ielce Uiversity of echology Faculty of Mechatroics ad Machie Desig through grat No 1.22/7.14 is ackowledged by the author of the paper. ANNEX  NUMERICAL RESULS OBAINED FOR ALL MBLA MODELS ab. A.1. Results for MBLA models, characterized by J=1kN/m J=1kN/m =315MPa J=1kN/m =5MPa
9 Marci Graba Numerical Verificatio of the Relatioship Betwee he IPlae Geometric CoStraits used i Fracture Mechaics Problems J=1kN/m =1MPa J=1kN/m =15MPa ab. A.2. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.3. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa ab. A.4. Results for MBLA models, characterized by J=1kN/m J=1kN/m =315MPa
10 acta mechaica et automatica, vol.6 o.2 (212) J=1kN/m =5MPa J=1kN/m =1MPa J=1kN/m =15MPa ab. A.5. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.6. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa
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