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1 86-81!"# ( 6 1-./ ' '1 '!,&' () 89/. :;6 7!/ < - -./ -./ 1., - -./ -./ 1., - -./ -./ 1., 9= >9? 8@A.6 B,9. 9 D >. -,, (8 >?;< =89 : 67 ' > 6 I<. >H > ;. > E N A F ' Q A F ABAQUS J P Y W6: X H KV9 TU A Y Y., <-Q U >?;< [H J< F< Z7 ' '. A \ 8@6 Q ' ' :,E' omparison of the onventional and Extended Finite Element ethods in Fracture Prediction of Keyhole Notch Specimen under ixed ode oading A. Nasrnia Department of echanical Engineering, University of Isfahan, Isfahan, Iran F. Haji Aboutalebi Department of echanical Engineering, University of Isfahan, Isfahan, Iran. Poursina Department of echanical Engineering, University of Isfahan, Isfahan, Iran Abstract Due to the noticeable stiffness and also very low deformability, brittle materials have vast application in the various industries such as automobile, navy, and aero. Therefore, investigation of failure behaviour and mechanical fracture of these materials is one of the most important challenges of the engineers and researchers. The main aim of this paper is to compare the performance of the extended finite element method (XFE) with the conventional finite element method (FE) in prediction of crack initiation and propagation in a brittle material. For this purpose first, applying the damage models based on the FE and XFE available in ABAQUS finite element software, crack onset and growth in a graphite specimen with keyhole notch is simulated. Then, values of crack initiation angle and maximum load capacity in the sample under different mixed mode loading conditions are predicted, compared with experimental results and validated. omparison of the numerical results obtained from the FE and XFE simulations with the empirical results reveals that the XFE in addition to decreasing the computational costs and also reducing sensitivity to the element size in the mesh has higher accuracy and more performance in prediction of crack initiation and propagation in the brittle materials, compared with the FE. Keywords: onventional Finite Elements ethod (FE), Extended Finite Element ethod (XFE), Fracture of Keyhole Notch Specimen, ixed ode oading..q c E Y - A VE d N >.[],. J K \ A? A = @<H. P\ KV9 - A - Q c -Q< g 6< <F<h T - R.[] KV9.[7 6], J Q O i j?.? ) A 6U - B F: A A 6U Q I ^. J? _7 E W6: [H Y 1.[-1], VE \ N - \ 1< VE \ ohesive Zone odel (Z) 1 inear Elastic Fracture echanics (EF) f.hajiaboutalebi@eng.ui.ac.ir : 9/1/9 :! 9//1 :' (!

2 1-./ @ P\ A ' P1<. V 1 A 6U < - 8 H > sv = A Z7., U / A A 1 '., <E 6t H FJ _7 ( 1 W 6 u6 > ' >.[8] Q [ A i Q< <B >. / < d [. F = - 1, A - Q i i u6 [H R.[8] Q I n < [ Q Q c -Q< R - >?;< φ B., /ψ φ K q - A ' / P1< Z7 B A < A fv Q7 fv <-ψ B A fv,- - A.= K B > Bn \ W K. A fv Q7 T.= n 1 8 sv.q - A.= Q7 T.= m -1 /.= wv T.= ψ. 'Z [H > ;. ; - 9 -Q [8], c IVK ^ /6: I< N <.[11-9] I n = F J Q., / U n A fh^ N /6: A n o <' ' > /. EA 8 8 8U pk 1 < q? J I n F B < q P\ N >, F [KV. >Q: J = < E - n <- P [KV9 ' N X 1? Q K c - > >. K >?;< [1] -Q< >\ X.[9] X Q F ' N iu F<h, KV9 A = PJ = / ABAQUS J / RS., 9 KV Y. F J KV9. X F ' A 9 Y Y. E \ U =89 F< >Qi ' Q H -Q Y >?;<. E [1]ψ ' φ 7K?J6/ '6I9/ 8H9-/ -1< Enrichment functions 1 Extended Finite Element ethod (XFE) onventional Finite Element ethod (FE) inimum Strain Energy Density (SED) aximum Tangential Stress (TS) 8

3 ^ i 67 =U < n, Q7 ( 8 q ) r E U r E (, H/ (,. 8 Q7 <- - Q H 8.. < - = 6. FJ [1] 91 T(B96 -PQ -6 ', -< 69N,. 8 p 6U 8 O E K < VE \ [ H - Q F F V. F F - R / / -. p - Q /., 8 O :? K! NH E p <F p J! F<. F< x. Q yi N K <F O ABAQUS J P K 8 < >. J= ' F F 6 F 1 6 F < K >. 6 F B F B > - W K 6 F 7 K.[1] z/ -Q E 1 B (PE PS) K. < K - s V [ > A F 9 6 F 6 F A < K > NE NS) < K. T U X r : 8 9 A.= (QNE QNS.,> K < [1] I9/ 'O.! -< PQ R O 8 - J= K 9 H Q F - W6 ABAQUS J P 8QH } F - > P -@ 6, [ KV9 A > Y J= (, I <?;< 8 >?;< J= Y. [1] - Q< > \ X 1 -./ '! S./ O 8-1- >?;< 8QH 8 9 } F N s V Q < 6, <o p K Q >., JP 8, N a <o J > 6 ρ. q. 8 s V J θ (, :; 6 (H@ < 6 ( ' (Y X 6. -< ~ P\. P 8 q ( = ), Q7 Q H 1 (W X@ N < Q F N ' /, 8 9 F: K J= aximum Principle Stress (PS) aximum Principle Strain (PE) aximum Nominal Stress (NS) aximum Nominal Strain (NE) Quadratic Nominal Stress (QNS) 6 Quadratic Nominal Strain (QNE) 8 8

4 7!/ < 6. I9/ <B9-6< 89/. :;6 '1'! S./ O8 -- -;< ' N 1? 89U Q F TU N X >?., W K 1 = sv ve N> X >?;<.[1], H = sv [1] B= = X 8 / 6 1/6 /8. Q7 J P [1] [.N-1'. ( Pam ) [^: (Pa) \ (Pa) - O i (Pa). PH (Pa) u. PH. ( Pam ) 1? 1? [1] T\,- O. [.N- ' [^:... / 9]B F z/ K > sv TU J ^: X.[1] PH J P 1N r: < = F /., W K [1] N X >? ;< 1N key K I = πρσ max key K II = limr ρ { :, H F [ O i πρ σ rθ ( r,) [1 1.6( ρ / r).7( ρ / r) 1.87( ρ / r) ] key, ρ Keff Fapp = key, ρ K F eff, c cr J H 6 key,ρ K eff Z7 } ( 1) ( ) ( ) <- O σ rθ σ max ρ r. F PH pk K F [ O i,- O F cr F app key,ρ K eff, c x K F [ O i 1N - @<H H O i O i V / >?;< X ~ 8U., F 8QH } sv., = TU Q F A A 6U. 6 /71 /7 /7 9/7 /898 /19 /8 (Pa) 6 F 6 F F (Pa) F F. ( Pam ( Pam < 9 J? A F A F 8U.< FQ ve H 8@6 7!/ < 6. I9/ <B9-< X@N. :;6 8

5 8. n =U < ^ i - K F 9 (PE PS) K.,F A <K > sv [ > A. TU 6 F 6 F A.= (QNEQNSNENS) sv., >K X r: 89 9 ' < 8@6 [H/ A F. E -&' S./ 6 7( 'O _1- ' 11 /11 / /6 [ 1] 69/ _1' '1 '! θ SED /9 7 / 67 / / 1 /8 7 /6 θ TS 8 / /97 9 /1 θ Exp [ 1] 6 /1 9 /6 6 /7 -&' S./ 6 7( 'O _1- ' 6 /1 /8 11 / [1] 69/ _1' 1-./ '! θ PE /7 7 / 7 /98 11 /11 / /6 θ PS /9 7 / 67 / θ Exp [ 1] 6 /1 9 /6 6 /7 6 6 O J Q Q F ) ' ' K F >?;< 1? 6 F 6 F ) H ( F B F B F F [1] 8U Y - K < F J } 8U Y } ' <?,, U 8@6 [H/ 8QH >Q: O \ 9 > >Qi ;7 ' > [Z F< _7 F > E PJ., \ H H 1 A / ' < ;. A 6U? <,.= -Q< A > I ^. J ', R9 <QB 1? z/ K > sv. H I n ^: ;U W K V sv ve / H > σ u Wc = E PH :[1] ( ) <- O W c E σ u \ X 89 Q7 J P 1 N Q r: ;U 1? : H 89 () K / J ve (1 ν )( 8ν ) K R = Ic π σ u - O i 1? <- re 8Z< 9 8H ( ) O ν K Ic.. R r: - 9 q - ;U U.. TU 6 F.= W Fapp = W cr Fcr :, V / ( 1? <- O F cr F app W cr W A K > 1N-,-Q<.[1] TU 6 F A 6U - J 89 -Q< A > P. <E J A N X., >K J A <@ P R., 8U X F Q Q F ) ' ' (6 F K >?;< 1? 6 F ) H [1] 8U Y - K < F >?;<.,,-Q<. } 8

6 [1] 69/ _1' 1-./ '! &',- S./ 6 7( '9_1-6 ' 1/ 6/78 F QNE (KN) /7 /1 6/77 / F QNS (KN) /7 /91 1/78 7/1 F NE / / 7/ /8 F NS /71 /9 1/8 / F PE / / 1/7 /7 F PS /9 /9 F Exp [1] /99 8/ 6/8 /6 /6 7/ 6/7 /8 /8 1/6 6/ /7 /7 6/ /11 /8 /9 1/86 1/ /6 / 6/1 /6 /86 /96 /1 /11 6 J9-6 [1] riffith A.A., The phenomenon of rupture and flow in solids, philosophical transactions, Vol. 1, pp , 19. [] Irwin.R., Analysis of stress and strains near the end of a crack traversing a plate, Applied echanics, Vol., pp , 197. [] Rice J.R., Path-independent integral and the approximate analysis of strain concentration by notches and crack, Applied echanics Transactions Vol., pp , [] Barenblatt., The matematical theory of equilibrium cracks in brittle fracture, Advances in Applied echechanics, Vol. 7, pp. -19, 196. [] Hillerborg A., odeer., Petersson P., Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, ement and oncrete Research, Vol. 6, pp , [6] arter B.J., ajtai E.Z., Ayari.., riteria for brittle fracture in compression, Engineering Fracture echanics, Vol. 7, pp. 9-7, 199. [7] Fischer K.F., Review of brittle fracture criteria in case of static and cyclic mixed mode loading, Theoretical and Applied Fracture echanics, Vol. 1, pp , 198. [8] Belytschko T., Black T., Elastic crack growth in finite elements with minimal remeshing, Numerical ethod in Engineering, Vol., pp. 61-6, [9] Torabi A.R., Pirhadi E., Stress-based criteria for brittle fracture in key-hole notches under mixed mode loading, European Journal of echanics and Solids, Vol. 9, pp. 1-1, 1. [1] Berto F., ampagnolo A., Ayatollahi., V-notches subjected to combined tension and torsion loadings: the application of the fictitious notch rounding concept", Engineering Fracture echanics, Vol. 18, pp. 8-96, 1. [11] ampagnolo A., Berto F., eguillon D., agoda T., ode II loading in sharp V-notched components: a comparison among some recent criteria for brittle fracture assessment, Procedia Structural Integrity, Vol., pp , 16. [1] azzarin P., Berto F., Ayatollahi A.R., Brittle failure of inclined key-hole notches in isostatic graphite under in-plane mixed mode loading, Fatigue and Fracture of Engineering aterials and Structures, Vol. 6, pp. 9-9, 1. [1] oes N., Dolbow J., Belytschko T., A finite element method for crack growth without remeshing, Numerical ethod in Engineering, Vol. 19, pp. 11-1, [1] ABAQUS 6.1 Help Documentation. ABAQUS Theory anual. A F 6U 8 <E, N. F X <K, \ 9 (QNS NSPS) F. E (QNE s6k H VE > Q 1/6 VE > >?;< 6 F F K x 6Q7 J (SED TS) >1 /6 /6 <- -&' S./ 6 7( '9_1- ' /6 8 /7 /61 /68 [ 1] 69/ _1' '1 '! F SED / /8 /1 / 1 / /1 /91 /16 F TS /9 / / /98 F Exp [ 1] /99 /1 /1 /11 6 9;1- 8U Y F<h > ' 9 >? ;< E (SEDTS) (QNE 6Q7 9 <- Z7. ', H U H F< N X > '... F 86