Microsoft Word - numero 3 art 1 M. Zakeri et al., Frattura ed Integrità Strutturale, 3 (2008) 2 - 10; DOI: 10.3221/IGF-ESIS.03.01 2 1 INTRODUCTION Many structural materials are subjected to crack forming and propagation during their service life. These cracks in- fluence the stress distribution in the component and can result in significant decrease of its strength. Because of the importance of safety and reliability, the crack prob- lem has been of interest to a large number of researchers. Elastic stress field around a crack tip is usually written as a set of infinite series expansions as [1]: ( ) I xx 1/2II K θ θ 3θ = cos 1 - sin sin 2 2 22πr K -θ θ 3θ + sin 2 + cos cos + T + O r 2 2 22πr σ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ( ) I yy 1/2II K θ θ 3θ = cos 1 + sin sin 2 2 22πr K θ θ 3θ + sin cos cos + O r 2 2 22πr σ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (1) ( ) I xy 1/2II K θ θ 3θ = cos sin sin 2 2 22πr K θ θ 3θ + cos 1 - sin sin + O r 2 2 22πr σ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ On the influence of T-Stress on photoelastic analysis under pure mode II loading Mahnaz Zakeri Mechanical Eng. Dep., Iran University of Science and Technology, Narmak, Tehran 16844, Iran e-mail: m_zakeri@iust.ac.ir Chiara Colombo Politecnico di Milano, Dipartimento di Meccanica, Via la Masa 34, 21058 Milano e-mail: chiara.colombo@mecc.polimi.it RIASSUNTO: Dalla definizione classica dello stato di sollecitazione elastica in prossimità dell’apice di una cricca, il termine T costante nello sviluppo in serie del fattore di intensificazione degli sforzi esiste solo in presenza del modo I di carico. Tuttavia, recenti studi mostrano che il T-stress può esistere anche in condi- zione di modo II, e modificare significativamente il campo di sforzi elastici presenti nell’intorno dell’apice della cricca. Questi effetti possono essere visualizzati e testati sperimentalmente col metodo della fotoelasti- cità. In questo lavoro è proposto uno studio sull’influenza del T-stress in cricche sollecitate secondo il modo II e i suoi effetti sul campo di frange visibili sperimentalmente. I provini utilizzati sono dischi, chiamati Bra- zilian disks, al cui interno sono contenute cricche centrali da analizzare: i risultati sperimentali indicano che questi tipi di provini contengono valori negativi di T-stress. I valori ottenuti sperimentalmente sono poi con- frontati con i risultati di simulazioni numeriche. Per meglio interpretare le differenze tra valori sperimentali e numerici, sono inoltre state eseguite analisi FEM 3D: i risultati mostrano l’influenza della reale geometria del fronte sui valori dei fattori di intensificazione degli sforzi. ABSTRACT. According to the classical definition for in-plane modes of crack deformation, the constant stress term T exists only in the presence of mode I. However, recent studies show that this term can exist in mode II conditions as well, and significantly affect the elastic stress field around the crack tip. These effects can be visualized using the experimental method of photoelasticity. Based on the analytical studies, presence of the T-stress in mode II cracks transforms the isochromatic fringe patterns from symmetric closed loops to asymmetric and discontinuous shapes. In this paper, presence of the T-stress in mode II cracks and its effects on the fringe patterns is experimentally investigated. The test specimens are Brazilian disks containing very sharp central cracks: experimental results indicate that these specimens contain negative values of T-stress. Experimental values are then compared to numerical results. To better understand the differences between experimental and numerical values, a thee dimensional analysis is performed with the finite element method: results show the influence of the real geometry of the crack front on the stress intensity factors. KEYWORDS: sharp crack generation, curved crack tip, Brazilian disk specimen, T-stress, mode II loading. http://dx.medra.org/10.3221/IGF-ESIS.03.01&auth=true http://www.gruppofrattura.it http://www.gruppofrattura.it wher crack are t mode term, ent o this terms domi F Base mode tions cond analy can a effec mode The c the c an im mate als or of T- ture leads is po This lahi a fect fractu affec that T crack II sp aroun T af stress nifica in m study Cons the c ordin xx =σ re r and θ a k tip (Fig. 1) the singular e II stress in , often called of the distanc expansion, r s which are inated zone. Figure 1. Crac d on the cl es [1], the T- of mode I dition. Howe ytical and nu also exist in ct can introdu e II brittle fra constant stre rack tip. The mportant effe rials, whethe r elastic-plas -stress influe path. Presen s the crack to ositive, the c effect is not and Abbasi considerably ure path in m cts the mode T is the mos k tip constrai ecimens exh nd the crack ffects the siz ses inside th antly by a re mode II can ying of mode sidering this rack tip in m nate system a IIK -θ= sin 22πr ⎛ ⎜ ⎝ are the polar ). The two fi stresses, dep ntensity fact d the T-stres ce r from the represented e usually n ck tip coordina assical defin -stress exists and II, and ever, some p umerical rese mode II prob uce significa acture. ess term T ac e amounts of ect on the br er in predom stic cases. It ences the sta nce of the n o grow along crack deviate t restricted to [6] have sho y the angle mode II as w II fracture to st important int in constra hibiting smal tip, Ayatolla ze and shap he plastic zo emote T-stres introduce c e II brittle fra point, the el mode II can b as: θ θ 2 + cos 2 2 ⎡⎞ ⎛ ⎟ ⎜⎢ ⎠ ⎝⎣ M. Zakeri et r coordinate first terms in pending on tors KI and ss, is constan e crack tip. T by O(r1/2), a neglected in ates and stress nition of cra s only in mod it vanishes published re arches indica blems [2-4], ant inaccurac cts over a larg f this stress a rittle fracture minantly linea has been sho ability and d negative T-s g its plane, w es from its i o mode I con own that the between the ell. Presence oughness. It parameter fo ained yieldin ll to moderat ahi et al. [8] pe of the pl one are also ss. Thus, ign considerable acture. lastic stress c be expressed θ 3θ cos + 2 2 ⎤⎞ ⎛ ⎞ ⎟ ⎜ ⎟⎥ ⎠ ⎝ ⎠⎦ t al., Frattura s centred at n each expan the mode I KII. The sec nt and indepe The next term are higher o the singula s components. ack deforma de I or comb in pure mod esults of sev ate that this t and ignorin cies in predic ge distance f and its sign h e of enginee ar elastic ma own that the direction of f stress in mo while when th initial plane nditions. Aya T-stress can e crack line e of T-stress has been sh or describing ng [7]. For m te scale yield have shown lastic zone. influenced oring the T-t inaccuracie components in Cartesian ( )1/2+ T + O r ed Integrità S 3 t the nsion and cond end- ms of order arity . ation bina- de II veral term g its cting from have ering ateri- sign frac- de I he T [5]. atol- n af- and also own g the mode ding that The sig- term es in near n co- yyσ xyσ The dete peri met calc spe bee from Usi men gen thre ods whi vide fiel four frin num New squ Nur ana Alth high prob pur sug mod hav [13 con due spe The pres the usin foll is p peri ing ate buc use cen The num are (FE agre Strutturale, 3 II y K = sin 2πr IIK= cos 2πr ⎛ ⎜ ⎝ e crack param ermined usin imental meth thod of phot culating the cimens [e.g. en suggested m photoelast ing the stress ntal optic equ neral, a non- ee unknown have been ich, the over e a more ac d method th r or more nges. The re merically, an wton-Raphso uares. Anoth rse and Pat alysis which i hough the fu h degree of blems, some e mode II. O ggest that the de II condit ve been obse ]. As describ nsistency bet e to neglectin cimens. e main obje sence of T i isochromati ng the exper lowing, a bri presented. Th imental prog a mechanica sheets (thic ckling in the d to make B ntral cracks, l e observed f merical predi validated by EM). Althou eement with (2008) 2 - 10 θ θ n cos 2 2 ⎛ ⎞ ⎛ ⎜ ⎟ ⎜ ⎝ ⎠ ⎝ θ θ 1 - sin 2 2 ⎡⎛ ⎞ ⎛ ⎟ ⎜⎢ ⎝ ⎠ ⎝⎣ meters KII an ng different hods. Among toelasticity h e crack par 9,10]. More d and utilized tic fringe pat s series expa uations for a -linear equa parameters K suggested t r- determini curate analy hat can use t arbitrary po esultant non nd the fittin on method her full-field tterson [12], is more comp full-field met accuracy for etimes the re On the other e fringe patte tions [11]. H rved in some bed by previ tween the th ng the effect ctive of thi in a mode II ic fringe pat rimental met ief review on hen, differen gram are des al shock, in ckness of 10 following lo Brazilian disk loaded in mo fringe pattern ictions. Also y using result ugh the expe h FEM, the 3θ cos + 2 ⎞ ⎛ ⎞ ⎟ ⎜ ⎟ ⎠ ⎝ ⎠ θ 3θ sin 2 2 ⎤⎞ ⎛ ⎞ ⎟ ⎜ ⎟⎥ ⎠ ⎝ ⎠⎦ nd T in these analytical, n g the experim has been fre rameters in eover, severa d to determi tterns. ansion (Eq. an isochroma ation is obta KI, KII, and T to solve this stic techniqu ysis [11]. Be the coordina oints on giv n-linear equa ng process i and the m d technique , based on plicated. thods genera r mode I and esults are no r hand, the t erns are alw However, as e of the prev ious research heory and ex t of T-stress s paper is t I specimen, tterns around thod of phot n the analyti nt steps of th cribed: crack sufficiently 0 mm) to av oading phas k specimens ode II conditi ns are finall o, calculated ts from finite erimental re ere were so ( )1/2+ O r ( )1/2+ O r e equations c numerical, an mental techni equently use various cra al procedures ine KI, KII a 1) and the f atic fringe [1 ained in term T. Different m s equation a ue is able to ecause it is a ates r and θ ven isochro ations are s nvolves bot method of is propose complex Fo ate solutions d mixed mod ot satisfactor theoretical r ways symmet symmetric fr vious experim hes [2,13], th xperiments c in some mo to investigat and its effec d the crack t toelasticity. I ical relations he performe ks are create thick polyca void problem e. This proc s containing ions. ly compared crack param e element an sults had a ome minor e (2) can be nd ex- iques, ed for acked s have and T funda- 11], in ms of meth- among o pro- a full- from omatic solved th the least ed by ourier s with de I/II ry for results tric in ringes ments his in- an be ode II te the cts on tip by In the s [13] ed ex- d, us- arbon- ms of cess is sharp d with meters nalysis good errors whic front also i crack singu apply node 2 M IS Base of an press m2τ wher the f and h shear nents m(2τ Subs math aroun prese param S = ⎛⎜ ⎝ In wh h could be r t through the investigated k front is ass ular element ying the qua e elements. MATHEMA SOCHROM d on the cla n isochroma sed as [11]: Nf = h re τm is max fringe order h is the thic r stress τm is s with this eq 2 m xx) = (σ - σ tituting stres hematical eq nd a mode I ented in ref. meters: 2 Nf hT ⎛ ⎞ ⎟ ⎝ ⎠ , B = hich a is the related to th e specimen th by using a 3 sumed to be s around the arter point ATICAL RE MATIC FRIN assical conce atic fringe a imum in-pla and materia kness of spe s related to quation [11]: 2 2 yy xyσ ) + 4σ ss terms from quation for II crack tip [13] and d II T πa = K , r′ e crack lengt Figure 3 M. Zakeri et he curved sha hickness. Th 3D finite elem e in a circula e crack tip a technique o ELATIONS NGES epts of photo around the c ane shear str al fringe valu ecimen. Also the Cartesia m Eqs. (2,3) a fringe l is written in defining thre = r/2a th for edge c 3. Created sem t al., Frattura ape of the c his latter effe ment model. ar arc form; are generated on quadratic OF oelasticity, lo crack tip is ( ress. N and f ue, respectiv o, the maxim an stress com ( ) in Eq. (4), oop develop n a simple f ee dimension ( cracks and se mi-natural cra ed Integrità S 4 rack ect is The and d by 20- ocus ex- (3) f are vely, mum mpo- (4) , the ping form nless (5) emi- crac equ isoc as: r′ whe Thi con whi mat clos 90° [11 2.b. r = Figu tip: acks: a) front v Strutturale, 3 ck length fo uation is obta chromatic fri 2bb +± =′ ere: ⎜ ⎝ ⎛ = sinb is equation p ntinuous alon ile, in the ca tic fringes is sed loops, sy , similar to , 14]. A typi . 2 IIhK1= 2π Nf ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ure 2. Typical a) T≠0, b) T= view, b) throu (2008) 2 - 10 or central c ained. Solvin inges in pres )1(2 34)(1( SB S − −−+ + 2 2 3 cosn θ θ predicts asym ng the crack ase of zero T s obtained as ymmetric ab the earlier a cal scheme o ( ) 2 24 - 3sin θ l isochromatic =0. gh the sheet th cracks, a qu ng this equa ence of T-str )sin3 2 θ ⎟ ⎠ ⎞ 2 sin2 θ mmetric fring k edges (see T-stress, the s Eq.(7) that out direction analytical re of these loop ) c fringes arou hickness. uadratic alge ation, the loc ress is determ ges which ar Fig. 2-a). M locus of iso t suggests a ns θ = 0° an sults present ps is shown in und a mode II ebraic cus of mined (6) re not Mean- ochro- set of nd θ = ted in n Fig. (7) I crack 3 E 3.1 For e elasti utiliz of th first jet te gen o come by ap crack with tal cr the s R=66 2), re in th ness, speci Since all re ting p accor heati 145° ture in th check the d 3.2 As a fring [N/(m to he EXPERIME Specimen Pr experimental ic stress fie zed. The disk ickness t =1 an initial sm echnology. T of -196°C tem e completely pplying a me k obtained in sharp tips (F rack lengths sheet was cu 6.5mm (spec espectively. I his way are n and there is ially in the ca e the specim esidual stress process were rding to Fig ng rate is C, which is for polycarb he furnace ked in the po disks are almo Determinati an optical p ge value for mm·fringe)] eat treatmen ENTAL PRO reparation l investigatio ld, two Bra ks were made 0mm. For cr mall notch w The notched s mperature fo y brittle, and echanical sho n this way is Fig. 3-a). For were 2a=58 ut in the fo cimen N-1) It is notable not perfectly s a curvature ase of thick s mens should b ses induced d e removed b g. 4. It is se decreased in the minimu bonate [15]. for 59 hour olariscope m ost stress-fre ion of Materi property of an intact p [11]. Since t nt, a materia Figure M. Zakeri et OCEDURE on of T-stres azilian disk e from a poly reation of th was made by sheet was pu or 15 minute d then the cr ock on the no very close to r the perform 8.8mm and 2 orm of two and R=60mm that the crac y straight thr which may sheets (Fig. 3 be stress-free during the cr by using a th een from the n temperatu um glass tran The specim rs and then machine. It w ee. ial Fringe Va photoelastic polycarbonate the specimen al calibratio e 4. Applied th t al., Frattura E ss effects on specimens w ycarbonate s he central cra using the w ut in liquid n es in order to rack was cre otched zone. o a natural c med tests, the 2a=60mm. T disks of ra m (specimen ck tips gener rough the th affect the res 3.b). e before load racking and hermal treatm e figure that ures higher nsition temp ens were pla n, stresses w was observed Value c materials, e is about f ns were expo n test was hermal cycle t ed Integrità S 5 n the were sheet acks, water nitro- o be- eated The rack e to- Then adius n N- rated hick- sults ding, cut- ment t the than pera- aced were that the f =7 osed per- form ess. sam spe test Tes unlo was isoc the cen 3.3 The fram plie forc P=3 sho [4], tion pos for o remove resi Strutturale, 3 med to deter . For this pu me material w cimens. This t under diago st was condu oading, and s obtained as chromatic fri two fringes ntre of disk. Figure 5. I Photoelast e cracked B me as shown ed by using t ce amount. T 367.5N for s uld be ment , the angle α n (see Fig. 6) ed to mode N-1 (a/R=0. idual stresses f (2008) 2 - 10 rmine the fri urpose, a disk was put in th s disk was th onal compre ucted in two considering s f =6.9 [N/(m inge patterns of order N= Isochromatic f tic Tests Brazilian disk n in Fig. 6. the loading s The loads w specimens N tioned that u between the ) was such s II condition 44), and α=2 from test spec inge value af k of diamete e furnace alo hen employed essive load a o steps inclu g both cases, mm·fringe)] s in the calib =8 are joinin fringes in cali ks were site Compressiv screw and th were selected N-1 and N-2 using the ea e crack line a elected that n. These ang 23.2° for N-2 cimens. fter thermal er 50mm from ong with the d for a calibr according to uding loading , the fringe . Fig. 5 show bration disk ng together i ibration disk. ed in the lo ve loads wer he gage show d as P=525N 2, respective arlier FEM r and loading d the crack wa gles were α= 2 (a/R=0.5). F proc- m the main ration [16]. g and value ws the when in the oading re ap- ws the N and ely. It results direc- as ex- =24.5° Fig. 7 show tips f 4 E As s aroun obser with tence term calcu this a using and t lect s ders. gram deter tant n fittin F ws the resulta for the two d EXPERIME shown in Fi nd the crack rved discont the theoret e of the T-st can be quan ulating the c aim, the obta g a compute the Image Pr some data p These data m prepared fo rministic tech non-linear eq ng process in Figure 6. Load ant isochrom disks. ENTAL RE ig. 7, the ob k tips are asy tinuous loop ical predicti tress in mod ntified by usi crack tip par ained isochro er code prepa rocessing To points from f were utilize or a full field hnique [11]. quations are nvolves both ding frame em Figure M. Zakeri et matic fringes a SULTS btained pho ymmetric in ps are in go ions, and co de II conditi ing the comm rameters, KI omatic fringe ared in MAT oolbox was e fringe loops d in another d analysis ba In this tech numerically h the Newto mployed to app e 7. Isochroma t al., Frattura around the c toelastic frin both cases. ood consiste onfirm the e ions. This st mon methods , KII, and T es were analy TLAB softw employed to of different r MATLAB ased on the o nique, the re y solved, and on-Raphson ply compressi atic fringes aro ed Integrità S 6 rack nges The ency exis- tress s for T.For yzed ware, col- t or- pro- over- esul- d the me- thod kno sent It s spe the Tab disk Exp the of T thes ive load on th ound the crack Strutturale, 3 d and the m own paramet ted in Tab. 1 hould be me ct to KII in b crack tips w Specime N-1 N-2 N-1 N-2 ble 1. Experi ks, compared w perimental fi investigated T-stress in pu se results, th e disks and de k tips of the d (2008) 2 - 10 method of le ters KII, and 1. entioned tha both cases. H was subjected en Experim 4.34 4.82 -0.38 -0.26 imental result with FEM res indings pres d Brazilian d ure mode II hey are com efinition of the disks: a) N-1, b east squares. d T were c at KI was ve Hence it could d to mode II c KII [MPa ment FEM 4 4.7 2 5.13 T-Stress [M 89 0.40 62 0.27 ts obtained f ults [4]. sented in Ta disks contain condition. In mpared with e characteristi b) N-2. . Finally, th calculated as ery small wi d be assume conditions. m m ] M Error 1 7.9% 3 6.0% MPa] 0 2.7% 7 2.9% from the Bra b. 1 indicate n negative v n order to va numerical r ic dimensions he un- s pre- th re- d that azilian e that values alidate results . [4] o that t two m cially curve was n mode by de that i 5 IN C In ex possi mens use o mens Figu line) obtained from there is a go methods, tho y in the case ed crack fron not taken int els. Hence, t eveloping a is explained NFLUENCE CRACK FRO xperimental ible to analy s considering of polycarbo s allows to i ure 9. Finite e ). m FEM anal ood agreeme ough there a e of KII. This nt through th to considerat the role of c three dimen in the next s E OF THE ONT studies usin yze the stress g them as 2 onate sheet to nvestigate o element mode M. Zakeri et lysis (see Ta ent between are some mi s problem m he specimen tion in the 2D rack tip curv nsional finite ection. CURVATU ng the phot s condition i 2D-models. I o prepare th nly the pres Figure 8. M el with the ap t al., Frattura ab. 1). It is s the results f inor errors e may be due to thickness w D finite elem vature is stu e element m URE OF TH toelasticity, inside the sp In this way, e cracked sp ence of mod Meshing conf pplied loads an ed Integrità S 7 seen from espe- o the hich ments died model HE it is peci- , the peci- des I and pola fact the as w out- III. crac thre elem The spe mm in c nes as s spo the figuration arou nd displacem Strutturale, 3 d II condition ariscope and tors values, stress condi well. In this -of-plane str With the ai ck front on th ee dimension ment softwar e model sch cimens used m and thickne corresponden s and has a shown in Fi ndent to the curvature e und the curved ments (crack p (2008) 2 - 10 ns. However d utilized to are an integ ition along t way, the cra resses that le im to study he stress inte nal model is res PATRAN hematizes a d in experim ess h=10 mm nce of a diam curved front ig. 8. Even e real situatio effect though d crack tip. osition is ind r, the fringe calculate th gral of the lig the thickness ack tip can a ead to the p the influenc ensity factor developed b N/ABAQUS. Brazilian di ental part, w m. The centr meter, cross t in the form if it is not c ons, the circ h the thickn dicated in surf es observed a he stress inte ght that desc s of the spec lso be impos presence of e of a curvi s determinat by using the . isk similar t with radius R ral crack is p es the disk t m of a circula completely c ular arc incr ness of the face with a th at the ensity cribes cimen sed to mode linear tion, a finite to the R=120 placed thick- ar arc, corre- reases crack hicker M. Zakeri et al., Frattura ed Integrità Strutturale, 3 (2008) 2 - 10 8 front and enable to get an indication about the stress in- tensity factors trend along a non-straight crack front. Crack curvature radius through the thickness is 10 mm, and the maximum extension of the crack is 2a=96 mm, indicated with a black thicker line in Fig. 9. The angle α between the direction of application of the compressive force (F=375N) and the crack line is 25.4°. This angle is chosen according to [4] in order to obtain pure mode II on the crack, considering the problem in 2D plane stress state. Displacements of the nodes in which the force is applied, are forced to be in line with the loading direc- tion. Since the results in terms of KII are equivalent consider- ing both the crack tips, only for one of them the mesh has been refined in the circumferential direction. In this way, it is possible to reduce the analysis run time, without loosing accuracy in the final result. The material of the disk in numerical model is the poly- carbonate, with elastic modulus of E=2480MPa and Pois- son’s ratio ν=0.38, according to [11]. Solid elements used for the modeling have a shape function of the second or- der, with a midside node in each edge. This choice allows having more nodes despite a not excessively refined mesh. Moreover, the use of quadratic element is neces- sary to use the quarter point technique [17, 18], that is to move the midside nodes next to the tip to ¼ of the edge length, which results in a better stress gradient in this area with singularity in the crack tip. Since good results are achievable with these elements even if the singularity is not well modeled on lines other than elements edges [19, 20], no collapsed element is used. It should be mentioned that to get better results in J- integral evaluation and consequently on stress intensity factors assessment, mesh directions should always be perpendicular to the crack front [21], avoiding distorted elements. However, the circular shape of the crack front causes a particular pattern for the mesh through the spe- cimen thickness. As shown in Fig. 8, in the upper part form point A to B, the mesh is more regular and the ele- ments of this region describe the radial directions per- pendicular to the crack front. In the lower part, the arc geometry makes it impossible to draw a regular mesh, and the normal to the crack front is not coincident with the mesh direction. Numerical results are obtained starting from node 1 cor- responding to point A to node 33 that is point C in Fig. 8. Convergence of J-integral and stress intensity results is obtained at the third contour. The trend of stress intensity factors can be graphically observed in Fig. 10 in function of the node distance from the surface. However, the stress intensity factor values obtained near to point C should not be taken into consideration, since elements present a high level of distortion producing low accuracy in the results. Values of the first three nodes are moreover invalid in the discussion, since the third contour integral cannot be cal- culated and results are infected by the presence of the surface border. 6 DISCUSSION The semi-natural cracks created with a mechanical shock after making brittle the polycarbonate in the liquid nitro- gen, have a nonlinear curved tip through the thickness. When the specimen containing such a crack is subjected to mode II loading condition, the global deformation of the crack front is in-plane sliding in X-direction. Howev- er, considering local coordinate systems n-t moving along the crack tip curve (see Fig. 11), the global displacement of the crack tip points will have two components. The normal component in n-direction leads to mode II; and the tangential component in t-direction implies that there is also mode III deformations in local view. In order to find the effect of specimen thickness on the numerical results, they can be compared with the pre- vious results [4] obtained from 2D finite element model- ing. For this aim, a new parameter KIIeq is defined as: 2 2= +IIeq II IIIK K K (8) Figure 10. Stress intensity factors along the crack front in function of the depth. (-●- KI, -♦- KII, -▲- KIII, -x- KIIeq, __ KII-2DFEM [10]). -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 0 2 4 6 8 10 Distance from the surface [mm] K [M P a√ m m ] M. Zakeri et al., Frattura ed Integrità Strutturale, 3 (2008) 2 - 10 9 which presents the equivalent mode II stress intensity factor in X-direction of the global coordinate system. It can be noticed from Fig. 10 that KI is negligible with respect to KII and KIII for all the considered nodes. Also, KIII is initially less than KII. Increasing the curvature that is going toward points B and C, KIII values are increasing and finally becoming more than KII values. However, values of KIIeq remain about constant, except from surface nodes which are not valid as described before. The ob- served difference between KIIeq and the result of 2D mod- el [4] shows that the thickness of specimen affects the ideal plane stress conditions and leads to some errors in the photoelastic experiment results. 7 CONCLUSION In this research, presence of the T–stress and its effects on the elastic stress field around a mode II crack tip were experimentally studied. Very sharp cracks were created in polycarbonate sheets by using a new method with dif- ferent steps. The cracks obtained in this way are com- pletely sharp, but the crack tip has a curved shape through the thickness of the specimen. Specimens were cut in the form of centrally cracked Brazilian disk speci- mens. Photoelastic experiments were conducted on these specimens subjected to mode II loading conditions, to de- termine from the isochromatic fringe patterns the crack parameters KI, KII, and T by using computer codes devel- oped with the MATLAB software. Experimental results revealed that the specimens had negative T–stresses in mode II condition. The experimental results were consistent very well with numerical bidimensional predictions in that the T-stress significantly affects the symmetric shape of the fringe loops, and causes the loops to become asymmetric and discontinuous along the crack edges. However, there were some minor errors which could be related to the curved shape of the crack front through the specimen thickness. The effect of crack tip curvature on the crack parameters was also investigated by developing a 3D finite element model. The crack front was assumed to be in a circular arc form and, even if it is not com- pletely correspondent to the real situations, aim of this model is to get an indication about the stress intensity factors trend along a non-straight crack front. The numerical results show that though the global defor- mation of the crack is in-plane sliding (mode II), in local coordinates there are two shear components which are parallel and perpendicular to the crack front. That is, the crack tip points are subjected to a combination of mode II and mode III deformations. This local mixed mode condi- tion can lead to some errors in the experimental results, which can be a source of difference of experimental re- sults compared to the values of the finite element model. 8 REFERENCES [1] M.L. Williams, “On the Stress Distribution at the Base of a Stationary”, Journal of Applied Mechanics, (1957) 109-114. [2] M.R. Ayatollahi, M. Zakeri, M.M. 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