Microsoft Word - numero 14 articolo 3 A C C. Dep Sh Res Dh s.k AB init pla by mi res wit KE IN inv esti Fra min stat dep spe frac the ana und line pre crit me P Analysis ompact M. Sharan epartment of M hashidhar K search Center, harwad-58000 kudari@rediffm BSTRACT. Th tiation direc astic zone si analytical a xed mode (I sults are com th reference EYWORDS. P NTRODUCTIO rediction life pred depends vestigators [5- imates of cra acture Mecha nimum plasti tes that the pending on t ecimen [8]. H cture specim e plastic zone alysis are mis der mixed m early with re esent work is terion for cra echanics (LEF P of crack t Tensil naprabhu Mechanical Eng K. Kudari Department o 02, India fmail.com he minimum ction under m ze (PZS). In and the elast I/II) loading mpared to a to the loadi Plastic zone; ON n of crack ini diction of eng on the loadin -7] have prop ack-tip plastic anics (LEFM ic zone radius crack initiate the loading d Hence, this ki en estimated e under mixed ssing in their mode loading spect effectiv to validate th ack initiation FM) regime fo C.M. Sh k-tip pla le Shear ngineering, Bap of Mechanical m plastic zon mixed mode n this invest tic finite ele g according analyze the m ing angle and Mixed mod itiation and o gineering mat ng angle [4], posed fractur c zone size in M) approach. s (MPZR) th es in the dire direction. It is ind of study by numerica d mode loadi investigation condition us ve stress inte he finite elem angle with re or plane stres haranaprabhu et astic zo r (CTS) puji Institute of l Engineering S ne radius (M e loading. T tigation, the ement comp to von Mise minimum p d stress inten de I/II; Mini orientation wi terials [1-3]. I and the loadi re criteria for n mixed mod Recently, Bi eory for crac ection where s well known needs detaile al method suc ing by the fin n. Sharanapr sing finite ele ensity factor ent results wi eference to t ss condition. t alii, Frattura ed one in a Specim of Engineering SDM, College MPZR) criteri The MPZR t shape and s putations in es yield crite lastic zone nsity factor. imum plastic with its propag In mixed mo ing angle alter r the predictio de fracture. Th ian and Kim ck initiation an the radius o n that plastic ed informatio ch as the fini nite element m rabhu and Ku ement analysi and is indep ith the theore the loading an d Integrità Strutt men g & Technology e of Engineerin ion is one of theory is bas size of crack a Compact eria. The the radius (MPZ c zone radiu gation path u ode fracture, rs the shape a on of the cra he analytical m [4] and Kh ngle in mixed f plastic zon zone size (P on about the ite element m method, but t udari [10] ha is. The autho pendent of lo etical results t ngle and stre turale, 14 (2010) gy, Davanager ng and Techno f the recent sed on the t k-tip plastic t Tensile Sh eoretical and ZR) criterio s; Finite elem under mixed- it is known t and size of th ack-initiation estimates we han and Kh d mode mono ne takes eithe PZS) depend crack-tip pla method. Benr the details of ave studied si ors have dem oading angle. to analyze the ss intensity f ) 27-35; DOI: 10 re 577004, In ology, criterions to theoretical c zones have hear (CTS) s d the finite e n for crack- ment analysi mode loading that the crack he crack-tip pl angle based ere obtained b raisheh [7] h otonic loadin er a local or s on the typ astic zone sha rahou et al. [9 f the mixed m ize and shape monstrated th Hence, the e minimum pl factor in linea 0.3221/IGF-ESIS. ndia o estimate cr omputation been estima specimen un element anal -initiation an is. g is desirable k initiation a lastic zone. Sev on the analy by Linear Ela have propose ng. MPZR the global minim e of the frac ape and size 9] have estim mode plastic z e of plastic z hat MPZR va objective of lastic zone ra ar elastic frac 14.03 27 rack s of ated nder lysis ngle e for ngle veral ytical astic ed a eory mum cture in a ated zone zone aries f the adius cture http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true mailto: s.kudari@rediffmail.com C.M 28 AN Th K K F M. Sharanaprabh NALYTICAL E or a sin Cartesian xx  yy  xy  zz  0zz  he stress inten I p K w t   II p K w t   F hu et alii, Frattu ESTIMATION gular elastic n co-ordinate 1 co 2 IK r    1 c 2 II K r    1 s 2 IK r    ( )xx yy   0 Pla nsity factor fo co s 1 b bt w      sin 1 b bt w      ura ed Integrità S N OF CRACK field of an i es (x, y) shown Figure 1: s 1 sin 2 2     cos 1 si 2     in cos 2 2   2 ) K 2 I r    ne stress r CTS specim 1 0 b w     1 0 b w     Strutturale, 14 (2 K INITIATIO infinite plate n in Fig.1 are Crack tip stres 3 sin 2 2 K      3 in sin 2 2     3 cos 2 K   cos 2 IIK   men of interes 0 .2 6 0 .5 5 b w     0 .23 0 .67 b w      2010) 27-35; DO ON ANGLE with a crack e given as [4]. sses expressed sin 2 2IIK     sin 2IIK    cos 1 2IIK     sin 2    st is given by  2 .6 5 0 .0 w b b        1 .40 2 .0 w b       OI: 10.3221/IGF-E k, the stress in Cartesian co cos cos 2   cos cos 2 2   sin sin 2   Plane Strain [11].  0 8 b w b b w b        08 b w b b w b       ESIS.14.03 near the cra o-ordinates. 3 2    3 s 2     3 2    2 b    2    ack tip expres ssed in term (1) (2) (3) (4) (5) (6) (7) s of http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true wh spe MI Wh foll wh sin exp Th FIN sho Bo 90° loa is a bel A s geo Th et a I T here p is the a ecimen and  INIMUM PLA t is assume radius evalu r          here r is the r lowing form. ( xx  here y is the y gular stress f pressed in the  ,Ir K I K K he crack initiat NITE ELEM he gene specime referred own in Fig.2, rrego et al. [1 ° (pure Mode ading jig [10] a ascertained b low, in the sim 1 FF 2 FF  3 FF series of elast ometry due to he loading and al. [12] and ar I T applied force,  is the loadin ASTIC ZONE ed in this stud uated from th 0 o      radius of plas 2) (yy yy  yield stress in field of equa e following fo  1 , 4IIK    52 sin 2 16II K    tion angle is g ENT ANALY eral-purpose f en [10] under d to as the C , the dimensi 12]. The loadi e –I) to study along the six by applying u milar manner    cos 2 1 6 FF sin5 FF     cos 2 1 4 FF tic finite elem o the lack of l d displacemen re clearly sho C.M. Sh b is the crack ng angle. E RADIUS CR dy that the di he von Mises 2 2 r       stic zone. Th 2) (yy zz  n uniaxial tens ations 1 to 3 orm. 2 2 1 3 1 4I y K      6 sin 3 16   given by Eq. YSIS finite elemen mixed mode ompact Mixe ions of the s ing of the sp the plastic d holes as show uniaxial point r to that demo   sins b c   sins b c ment calculatio loading symm nt boundary c own in Fig.4. haranaprabhu et k depth of th RITERION irection of cr yield criterio 0 o        he von Mises 2( )zz xx   sion. As an ap in the abov 1 cos 2  2sin    8, the relative nt (FE) code e loading has ed Mode (CM specimen con ecimen is app eformation ah wn in Fig. 3. I loads F1 to onstrated by R       ons have bee metry. A typic conditions us Two-dimens t alii, Frattura ed he specimen, rack initiation n. The crack 0 yield criteria 2 26( xy yz   pproximate d e yield criter 22cos 2     e minimum m ABAQUS is been conside MM) specime nsidered in th plied at variou head of the c In the presen F6 as shown Richard [11]. n made on th cal two-dimen ed in this ana ional elastic F d Integrità Strutt w is the width n coincides w initiation can for the three 2 ) 2z zx   determination rion and solv 2 15 2 4IIK    must have pos s used in thi ered in the pr en [4]. The s he analysis ar us angles (), crack-tip. The nt FE analysis n in the Fig. he CTS speci nsional FE m alysis are simi FE calculatio turale, 14 (2010) h of the spec with the direct n be determin e dimensiona 2 y n of the plasti ve for the pla 2 9 2sin 2 4   sitive circumf is study. A C resent study. specimen geo re similar to , 0° (pure Mo e load is appli s the specimen 4 and estima men (Fig. 2) mesh used in t ilar to the one ons were perf ) 27-35; DOI: 10 imen, t is the tion of minim ned by minimi al object can c zone, one c astic zone ra cos 2   ferential stres Compact Ten This kind of ometry used i the one used ode-II), 18°, 3 ied at various n loading at v ated by Eqns considering t the analysis is e used in the formed using 0.3221/IGF-ESIS. e thickness of mum plastic z izing r [4]: (8) be written in (9) can substitute dius r. It can (10) s. nsile Shear (C f specimen is in the analys d in the work 36°, 54°, 72° s angles  usin various angles s. 11-13 as gi (11) (12) (13) the full specim s shown in Fi work of Borr eight noded 14.03 29 f the zone n the e the n be CTS) also sis is k of and ng a s () iven men g. 4. rego iso- http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true C.M 30 par 284 thir has MP RE und the Th Sha Eq loa D M. Sharanaprabh rametric quad 44. The maxi rd of yield st s been consid Pa, Poisson’s Figure 2: S (all di ESULTS AND ifferen zone s compu der mixed mo e similar appli his nature of v aranaprabhu qns. (6) and (7 ad 4kN and 1 D hu et alii, Frattu drilateral elem imum load ap tress to keep dered to be lin ratio () of 0 Specimen confi imensions in m F D DISCUSSIO nt load steps shape and siz uted for vari ode loading i ied load, for variation of K [13]. The ma 7). The estima 0 kN. This p ura ed Integrità S ments conside pplied is 10k the analysis d near elastic ty 0.3 and elastic iguration used mm), thickness Figure 4: FE m N were applied ze ahead of th ious load step .e variation o a loading ang KII vs. KI is in agnitudes of K ated theoretic plot indicates Strutturale, 14 (2 ering plane str kN and the co domain appro ype pertaining c modulus (E) in the analysis = 3 mm. mesh, loading a d to the spec he crack-tip. T ps and loadin of KII vs. KI f gle below 70° n good agree KI and KII ha cal values of s that there ex 2010) 27-35; DO ress condition orresponding oximately und g to interstitia ) of 197 GPa s and boundary c imen to estim The stress int ng angles () for various lo ° the stress in ement with th ave also been stress intensit ists some dis OI: 10.3221/IGF-E n. The numb g applied stre der LEFM. I al free steel (IF a [8]. conditions used mate the stre tensity factors using the AB oading is depi ntensity facto he results sho computed by ty factors hav crepancy in t ESIS.14.03 er of element ss is 62 MPa In these calcu F) possessing Figure 3: Lo d in FE analysi ss intensity fa s in mixed mo BAQUS post icted in Fig.5. or in mode-I i own by Benra y analytical fo ve been superi the estimation ts used in the a, which is ap ulations, the m g the yield stre oading Jig. is. factor and to ode loading ( t processor. T . This figure i is more than ahou et al. [9] ormulations c imposed in F n of stress int e FE analysis pproximately material beha ength (y) of study the pla (KI and KII) w The failure lo indicates that that of mod ] and Kudari cited in the ab Fig. 5 typically tensity factor was one avior f 155 astic were ocus t for e-II. and bove y for rs by http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true ana res FE com Th () res Th dep iso con Th pur ana = pla are Fig alytical formu pectively. Th E analysis thr mputed using effK  Figure 5: Vari he computed m for various a ults [9, 13]. he plastic zon pendent on lo -contours of nstitutive mo his is different rposes of the alysis. The se =36o (Mixed m astic zone con e set to zero. gure 7: The seq analyti ulation and pr his discrepanc rough loadin g the relation 2 2 I IIK K iation of KII vs magnitudes o applied loads ne shapes obt oading angle. the effective del is used fo t from the ac e present stud equential deve mode-I and I ntour obtaine quential develo ical Eq. (10) fo C.M. Sh resent FE res cy in estimate ng Jig. The e [9]: s. KI for differe of Keff estimat s in Fig.6. Th tained by the The shape o stress, which or the materia ctual plastic zo dy the consid elopment of II) is shown i ed in each loa opment of plas or various loadi haranaprabhu et sults. It is fou ed magnitude effective stre ent applied loa ted by analyti e nature of v Eq. 10 is sh of the plastic h causes yield al, contour plo one that is af deration is fo crack-tip pla in Fig. 8. The ad step. For s stic zone obtain ing angles. t alii, Frattura ed und that ther es of stress in ess intensity ads. F ical and the f variation of K own in Fig.7 zone ahead o ding according otting of the ffected by str ocused on th astic zone for e plastic zone simplicity the ned by Fig ap d Integrità Strutt re is 10.5% an ntensity facto factors (Keff) Figure 6: Variat finite element Keff vs.  (Fig.6 . This figure of a crack-tip g to von Mis von Mises str ess redistribu e plastic zon r various app e contours in displacemen gure 8: Sequen pplied loads. N ap turale, 14 (2010) nd 2.4% erro r attributed t ) under mixe tion of Keff vs.  t analyses are 6) is in good indicates tha p by FEA has es criterion [1 ress correspo ution during p ne, due to the plied loads an n Fig. 8 are ob nt scaling of th ntial developme Number 1,2,3,4 pplied load 4, 6 ) 27-35; DOI: 10 or in estimatio to varied load ed mode loa  for different e plotted again agreement w at the shape o s been ascerta 14]. Since, in onds to the pl plastic flow. H e considerabl nd typically fo btained by su he specimens ent of plastic z indicates the p 6, 8 and 10 kN 0.3221/IGF-ESIS. on of KI and ding conditio ading have b (14) t applied loads. nst loading a with similar ea of plastic zon ained by plot this study ela lastic zone sh However, for ly greater eas or loading an uperimposing s shown in Fi zone for variou plastic zone fo N. 14.03 31 d KII on in been . angle arlier ne is tting astic hape. r the e of ngles g the ig. 8 us or http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true C.M 32 To the loa var var enc step (iii) zon Th Fig ma fac elem esti app this and wh par Th Fig elem fou ind (mo Th for of sm loa the val zon M. Sharanaprabh illustrate the e contours are ading angles ( rious  remai rious loading claves shown ps and loadin ) angle at wh ne radius (MP Figure 9: Ty he variation of g.11, Fig.12 a agnitudes of r ctors. It must ment analysis imation of b proximation. s figure that, d it is highest hich is 3.33 tim rticular value he variation o g. 12. It is al ment analysis und that ther dicates that th ode-II). hese results (rp r the similar m the material maller plastic z ading is consi e FEA values lues of θ with ne size occur hu et alii, Frattu e developmen e superimpos (), the plastic ins almost sim angles is in in Fig. 7, Fig ng angle (β), hich the maxi PZR) and (v) ypical shapes o =0o - 90o un f the plastic z and Fig.13 re rp estimated b t be noted th s is slightly l both the an Fig. 11 illust for a particul t for =0o (m mes that of M of Keff, mode f the analytic lso noted tha s is slightly la e is less than he magnitude p and (rp)max) magnitude of in Mode-I is zone area ah dered to be m of θ with the h the loading a s () changes ura ed Integrità S nt of the crack sed and show c zone under milar, with so good agreem g. 8 and Fig. such as: (i) p imum extent angle at whic of crack-tip pla nder 10kN load zone characte espectively. F by analytical hat for variou larger than th nalytical and trates that the lar magnitude mode-II). The Mode-I. Thes e-I loading ca cal and the fin at for various arger than tha n 2% of error e of (rp)max for indicate clea f Keff. This an much lower head of the c more dangero e loading angl angle, β is ob s from 0o to 9 Strutturale, 14 (2 k-tip plastic z wn in Fig. 9. F -goes a rotati ome change in ment with the 9, several pla plastic zone si of plastic zo ch MPZR occ astic zones for d. erizing param Fig. 11 show Eq. 10 and F us loading an hat of the th FEA results e plastic zone e of Keff (for e e difference in se results infe an lead to ma nite element s loading ang at of the analy r in estimatio r a particular arly that the a nalysis infers t r than under crack-tip, the ous than mod le, β is depict bserved. This 90o as  is va 2010) 27-35; DO zones for vari From this figu ion. It is also n the size and e theoretical r astic-zone cha ize along the one occurs, , curs, o. Thes meters rp and ( ws the variati FEA are com ngles (β), the heoretical valu s. This discr e size ahead o example, 500 n magnitudes er that, due to terial fracture values of (rp) gles (β), the ytical values f on of both th value of Kef area of plastic that for the si Mode-II load mode-I load de-II. The var ed in Fig. 13. plot indicates ried from 0o OI: 10.3221/IGF-E ious loading a ure one can f interesting to d orientation results as sho aracterizing p crack plane, , measured fr se parameters Figure 10: Sc (rp)max vs. Keff ion of rp vs. mpared for pa e plastic zone ues. It is fou repancy may of the crack-t MPa mm1/2) s of rp for = o minimum p e earlier than )max vs. Keff fo maximum pl for a particula he analytical a ff is least for  c zone in mo imilar magnit ding. One ca ding leads to riation of the A small disc s that the ang to 90o both f ESIS.14.03 angles, =0o t find that for s o note that th . The nature own in Fig. 7 arameters can rp, (ii) maxim rom the crack s are schemat chematic repre characterizing f and  vs.  f Keff for var articular value e size along t und that there y be due to tip increases w ) the value of =0o and 90o f plastic zone r any mixed m or various loa lastic zone si ar effective st and finite ele =90o (mode de-I is much tude of Keff th n conclude f early fractur e theoretical v crepancy in th gle at which t for the analyt to 90o and ap similar applie he shape of th of growth of 7. From the r n be estimate mum plastic z k plane, (iv) tically illustrat esentation of p parameters. for CTS speci rious loading es of effectiv the crack pla e is less than o the finite with Keff. It i f rp is least for for Keff = 500 radius ahead o mode or mode ading angles ( ize (rp)max est tress intensity ement results. e-I) and it is smaller than he energy abs from this ana re, hence in values of θ ar he analytical a the maximum tical and the F pplied load 10 ed load at var he plastic zon f plastic zone results for pla ed at various l zone size, (rp minimum pla ted in Fig. 10. lastic zone imen is show angles (). ve stress inten ane for the fi n 2% of erro element ana is also clear f r =90o (mod 0 is about 5 m of crack-tip f e-II loading. () is depicte timated by fi y factor. It is . This figure highest for  n that of mod sorption capa alysis that, du fracture, Mo re compared w and finite elem m extent of pla FEA results. 0kN, rious ne at e for astic load p)max, astic . wn in The nsity finite or in alysis from de-I) mm, for a ed in finite also also =0o de-II acity ue to de-I with ment astic It is http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true inte a c for a si Th Th min dis fou loa dec the loa Mo elem inc eresting to fin considerable a r all various a imilar angle  Figure 1 Figure 1 he variation o he estimated nimum plasti crepancy in t und that there ading angle r creases from e loading con ading angle is ode-II loading ment values. crease in the nd that = amount of de applied loads.  for a consta 11: Variation o 3: Variation of f minimum p theoretical v ic zone radiu the estimation e is 10.5% an espectively. T loading angle nditions in fi s changed fro g is about 1. It is observ applied load. C.M. Sh for mode-I a eviation betw From Fig. 13 ant loading an of rp vs. Keff for f θ vs.  for va plastic zone ra values of min us in Fig.14 n of minimu nd 2.4% error The percenta e 900 to 00. Th inite element om 0o (Mode .987 times le ved from Fig. . In the case haranaprabhu et nd mode-II l ween  and  3 one can un ngle, . r various  arious loads adius (MPZR nimum plasti typically for um plastic zon r in estimatio age error for his discrepanc t analysis. Th e-II) to 90o ( ss than that . 14 that the of applied lo t alii, Frattura ed loading condi is observed. nderstand that R) vs. loading ic zone radiu load 4kN an ne by analytic n of minimum the theoreti cy in estimate his figure ind (Mode-I). It i of Mode-I fo ratio of MP oad of 10 kN d Integrità Strutt itions only. F The nature o t the extensio Figure 12: V Figure 14: Va angle,, for v us have been nd 10 kN. T cal formulatio m plastic zon ical and FEA ed magnitude dicates that th is interesting for applied lo PZR between N, the ratio i turale, 14 (2010) For the mixed of the variatio on of the plas Variation of (rp) ariation of MPZ various applie n superimpos This plot ind on and prese ne for mode-I A values of m es of MPZR a he magnitud to note that oad 4kN both n Mode-I and is found to b ) 27-35; DOI: 10 d mode (=36 on of  vs.  stic zone take )max vs. Keff for ZR vs.  for va ed loads is de sed with the dicates that th ent finite elem I (β=900) and minimum pla attributed to a de of MPZR t the magnitu h for the ana d Mode-II de be 2.0 almost 0.3221/IGF-ESIS. 6o to 72o) the is almost sim es place almos r various  arious loads epicted in Fig e FEA value here exists so ment results. d mode-II (β= astic zone ra approximatio increases as ude of MPZR alytical and fi ecreases with t same as tha 14.03 33 re is milar st in g.14. s of ome It is =00) adius on in the R in finite the at of http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true C.M 34 app Mo be CT Th plo res of pro equ is 0 ind Th Th or v In ana loa dec com ana thr MP nat of spe init M. Sharanaprabh plied load 4k ode-II loading used as inpu TS specimen. he variation o ot of MPZR pectively. It i loading angle oportional to uation to all t 0.0066 for an dependent of eff MPZR K eff MPZR K he Eqns. 15 an he Eqns. 15 an vice versa. T this investiga alytical and th ads along with creases from mputed for v alytical o co rough loading PZR in a CTS ture of the va Bian and Ki ecimen under tiates almost hu et alii, Frattu kN. These res g only. The re uts for minim f the minimu vs. Keff for v is interesting es. These figu the effective the MPZR da nalytical resul loading angle 0.0066 R m M  0.0068 R m M  nd 16 show th Figure nd 16 can be The proposed ation the angl he finite elem h the similar 82.5° to 0° a various applie ompared with g jig. Bian an S specimen. T ariation of o im [4], as th r Mode-I loa perpendicula ura ed Integrità S sults clearly d esults of MPZ mum plastic zo um plastic zo various loadin to note from ures infer tha e stress intens ata. Such a lin lts and 0.0068 e can be expr 1 2mm MPa 1 2mm MPa hat there is an 15: Variation o used to estim Eqns. 15 and le at which M ment analysis. results by Bia as  changes ed loads and h the FEA  d Kim [4] ha These investig vs.  obtained he theoretical ading crack i ar to the ligam Strutturale, 14 (2 demonstrate ZR estimated one radius (M one radius (M ng angles est m these figure t for any load sity factor. Th near fit is show 8 for FEA re essed as follo (Analytical) (FEA) n insignifican of MPZR vs. K mate MPZR in d 16 can be o MPZR occurs The variation an and Kim [ from 0° to 9 a particular o for various ave considere gators have u d in present a results in [4 initiates along ment. In this 2010) 27-35; DO that the spec d using the th MPZR) criteri MPZR) vs. eff timated by an es that the var ding angle (fo he proportion wn in Fig.15a esults. From ows: nt difference i Keff for various n a CTS spec of great use in (o) is also st n of o vs. loa [4] is depicted 90°. The resu  is almost s s  can poss ed that the cr used the magn analysis show 4] are obtain g the ligame investigation OI: 10.3221/IGF-E cimen experie heoretical and ion for crack fective stress nalytical and riation of of M or the analytic nality constan a and Fig.15b these results, n the results s  (a) analytica cimen indepen n MPZR crite tudied with re ading angle o d in Fig.16. T ults shown in similar. A sm sibly be attrib rack in mixed nitude of o wn in Fig. 16 ed for plane nt, and for a n a validation ESIS.14.03 ences minimu the finite ele initiation [4] intensity fact FEA are dep MPZR vs. K cal and the F nt can be eval b. The slope , the relation obtained by F al and (b) FEA ndent of the l eria. espect to the obtained in th This figure sh n Fig. 16 indic mall dissimilar buted to varie d mode loadin for defining t slightly diffe strain analy a specimen u n of minimum um plastic zo ement in this in mixed mo tor (Keff) is al picted in Fig Keff is linear an FEA), the gro luated by fitti of the estima between the FEA and ana A results. loading angle loading angle his analysis fo hows that the cate that the rity observed ed loading co ng initiates in the crack init ers from the t sis. Fig. 16 s under mode- m plastic zone one radius un investigation ode loading f lso studied. .15a and Fig. nd is independ owth of MPZ ing a straight ated linear fit e MPZR and (15) (16) alytical metho e if Keff is kno e () both for or various app magnitude o magnitude o in the estim ondition in F n the direction tiation angle. theoretical res shows that fo -II loading c e radius (MP nder can for a The .15b dent ZR is line line Keff ds. own, r the plied of o of o ated FEA n of The sults for a rack PZR) http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true crit tip CO AC RE [1] [2] [3] [4] [5] [6] [7] [8] [9] [10 [11 [12 [13 [14 T A teria and crac plastic zones ONCLUSION he analy For the Finite el CKNOWLEDG uthors Engine EFERENCES L. Nobile, T C. M. Sonsi X. Pitoiset, L. C. Bian, K K. Golos, B B. Wasiluk, S. M. A. Ka S. K. Kudar K. H. Benr 595. 0] C. M. Shara Oxford, En 1] HA. Richar Düsseldorf: 2] L. P. Borreg 3] S. K. Kudar 4] E. Gdoutos T A ck initiation a s and its vario NS ytical results a mixed mode lement analys GMENT gratefully ack eering & Tech S Theoretical an ino, Internatio I. Rychlik, an K. S. Kim, In B. Wasiluk, In K. Golos, Fa ahan, M. K. K ri, B. Maiti, K rahou, M. Ben anaprabhu, S. ngland, (2009) rd, Bruchvorh : VDI-Verlag go, F. V. Antu ri, C. M. Shar s, G. Papakali C.M. Sh angle (θ0) is m ous parameter Fi are in excelle analysis of C sis. knowledge th hnology, Hub nd Applied F onal Journal o nd A. Preumo nternational J nternational J atigue and Fr Khraisheh, Int K. K. Ray, Jou nguediab, M. . K. Kudari, A ) 13. hersagen bei g, 631(1985) 1 unes, J. M. C ranaprabhu, I itakis, Interna haranaprabhu et made by consi rs under mixe gure 16: Variat ent agreement CTS specimen he computati bli-580 031, In racture Mech of Fatigue, 23 ont, Fatigue a ournal of Fat Journal of Fra acture of Eng ternational Jo urnal of Strain Belhouri, N American In überlagerter . osta, J. M. Fe nternational J ational Journa t alii, Frattura ed idering a deta ed mode load tion of ovs.  t with the FE n one can use ional facilities ndia. hanics, 33 (20 3 (2001) 159. and Fracture o tigue, 26 (200 acture, 102 (2 gineering Mat ournal of Plas n Analysis, 42 ait-Abdelaziz stitute of Phy normal- und erreira, Intern Journal of En al of Fracture d Integrità Strutt ail compariso ding. for various loa EA results fo e analytical fo s provided b 00) 107. of Engineerin 04) 1169. 2000) 341. terials and Str sticity, 20 (200 2 (2007) 125. z, A. Imad, C ysics, USA E d schubbeans national Journ ngineering, Sc e, 32 (1987) 14 turale, 14 (2010) on of the anal ads. or CTS specim ormulations ra y the Resear ng Materials a ructures, 23 ( 04) 55. Computationa Editor: Profes pruchung vo nal of Fatigue cience and Te 43. ) 27-35; DOI: 10 lytical and the men under LE ather than go ch Centre, B and Structure (2000) 381. al Materials Sc sor Alexande on risen VDI , 28 (2006) 61 echnology, 2 0.3221/IGF-ESIS. e FEA for cr EFM conditi oing for exten B V B Colleg s, 24 (2001) 7 cience, 38 (20 er M. Korsun Forschungsh 18. (2010) 13. 14.03 35 rack- ons. nsive ge of 715. 007) nsky, heft. http://www.gruppofrattura.it http://dx.medra.org/10.3221/IGF-ESIS.14.03&auth=true << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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