3-D SQCE Model and Its Application in Fracture Mechanics *

Transcription

1 3-D SQCE Modl and Its Application in Fractur Mchanics * Zhichao Wang Sr. ad Enginr Applid Mchanics Dpt., Emrson Climat Tchnology, USA Tribikram Kundu - Profssor Enginring Mchanics Dpt.,Th Univrsity of Arizona, USA Zonji Cao - Profssor ShangHai JiaoTong Univrsity, China Abstract In this papr, th 3-D SQCE crack lmnt modl was dvlopd. A numbr of 3-D crack xampl problms wr solvd and compard with analytical and publishd numrical solutions. Th rsults showd that th SQCE modl provids a consistnt and accurat solution tchniqu to calculat strss intnsity factor (SIF) and solv SIF controlld fatigu problms. Compard to most of th singular lmnt modls, th thr-dimnsional SQCE modl givs a gnralizd mthodology to combin local analytical solution with finit lmnt intrpolation functions to crat spcific lmnts and is rlativly simpl in programming. This formulation procdur can b usd to comput strss intnsity factors of 2-D and 3-D crack problms, strss concntration problms as wll as othr nginring problms with singularity or high gradint of fild variabls. For crack problms, only th local solutions of Mod I, II and III ar ndd to obtain SIFs of 3-D cracks with any shap. Introduction Sinc th 1950 s, th strss intnsity factor (SIF) has playd an important rol in th application of Fractur Mchanics for nginring dsign and analysis. With th dvlopmnt of matrials tchnology, such as composit matrials and nw lctronic packaging matrials, th mthods for accurat calculation of SIFs ar ndd to valuat th rsidual strngth and crack growth of crackd componnts. Th approachs usd to dtrmin SIFs includ th analytical, xprimntal, and numrical mthods. Concrning th numrical mthods, FEM and BEM ar th most widly usd in Computational Fractur Mchanics, spcially th FEM. FEM mthods in th calculation of 3-D SIFs may b classifid into thr catgoris, which ar in- Spring lmnt, FEM altrnating, and various spcific crack lmnt mthods. in-spring modl [2, 3], proposd by Ric-vy [2], is mainly suitabl to solv 3-D surfac crack problms, particularly Mod-I long surfac crack problms. In othr words th crack shap is limitd to prsrv th accuracy. in-spring crack lmnt has th advantags that only 2-D analytical crack solutions ar ndd to obtain 3-D SIFs. Thrfor it is rlativly simpl to implmnt. And th solutions of in-spring modl ar consistnt compar to thos obtaind using various hybrid singular lmnts, which may b th rason that this modl has bn usd by many commrcial FEM softwar. in-spring modl producs poor rsults nar th intrsction of fr surfac and th crack. Th FEM altrnating mthod [4-5] has no limitation on crack shaps, which combin analytical and ordinary FEM solutions to comput SIFs through an altrnating itration procdur. An analytical solution of th similar crack in an infinit body is rquird to construct th solution. This approach is th most tim consuming on bcaus of th itration procdur involvd to obtain convrgnt rsults. This mthod is frquntly usd to calculat 3-D SIFs bcaus it dos not rquir spcial singular lmnts. Compar to in-spring modl and th FEM altrnating tchniqu, spcial singular lmnt mthods [6-10] had bcom most popular during 1980s. Howvr only a fw typs of spcific singular lmnt modls ar suitabl for solving 3-D crack problms, such as Nwman s polynomial singular lmnt [11], Aturi s hybrid strss/strain singular * This projct was sponsord by NSF, China

2 lmnt [8] and th Quartr-Point lmnt [12-14]. Th singular lmnts cratd basd on various variation principls such as hybrid displacmnt singular lmnt, hybrid strss singular lmnt and hybrid/mixd singular lmnt can b usd to obtain SIFs dirctly with high prcision. Howvr th rsults of hybrid singular lmnts ar gnrally snsitiv to lmnt siz and boundary conditions and not always consistnt. This papr prsnts a nw 3-D singular quasi-compatibl lmnt (SQCE) formulation tchniqu, which is a gnralizd mthod to combin local analytical solution with finit lmnt intrpolation functions to crat spcific lmnts. Th SQCE modl has svral advantags: (1) simpl to implmnt, (2) only th local solutions of Mod I, II, and III ar ndd to obtain thr dimnsional crack SIFs with various crack front shaps undr any complx loads, (3) its rsults ar consistnt and not snsitiv to th siz of th singular lmnts, (4) accurat and saving computr tim, (5) it is ffctiv to solv strss concntration and othr fild problms with singularity or high strss gradint. ocal Solution of 3-D Crack Problms As shown in Figur 1, arc SS is a sgmnt of th crack front with an arbitrary shap. Th local strss fild at any point P along th crack front can b xprssd as 1 T σ = α Sλ α K (1) 4 2r whr α is th transformation matrix from th local coordinat systm (n,t,z) to th global coordinat K = K K K systm (x,y,z). { } T local strss givn as I II III is th SIF vctor at point P and S λ is th angl function matrix of θ 5θ θ 5θ 3cos + cos 7sin sin θ θ 8νcos 8νcos θ θ θ θ 5cos cos sin + sin 0 S λ = (2) θ 0 0 4sin 2 θ 0 0 4sin 2 θ 5θ θ 5θ sin + sin 3cos + cos

3 Th local displacmnt filds can b writtn as Figur 1. Th SQCE Elmnt on a Crack in 1 r u = α Mλ K (3) 4µ 2 whr µ is th shar modulus and M λ is th angl matrix of local displacmnts givn as θ 3θ θ 3θ (5 8 ν) cos cos (9 8 ν) sin sin θ Mλ = 0 0 8sin (4) θ θ θ θ (7 8 ν)sin sin (3 8 ν)cos cos Substitution of Eq. (1) in th quations of 3-D lastic constitutiv rlation lads to th local strain filds 1 1 T 1 ε = D α Sλ α K = BK (5) 8µ 2r 8µ 2r in which D is th 3-D lastic matrix and B is th local strain matrix. Displacmnt Modl of 3-D SQCE Elmnt (1). ocal Elmnt Solutions Similar to displacmnt fild, gnrally SIFs is a continuous function but vary along th crack front lin. Tak lmnt as an xampl shown in Figur 1. Whr arc SS is th crack front lin of th lmnt. Using isotropic intrpolation function on SS, th SIF vctor K at any point P along th crack front lin SS can b xprssd as, K = λ (6) whr is on-dimnsional shap function and givn as λ is th lmnt nodal SIF vctor along th crack lin

4 in which K { } T λ = 1 2 i n i KK K K (7) is th SIF vctor of nod i givn as { } T Ki = KI KII K III (8) i Substitution of Eq. (6) into Eq. (1), (3), and (5) lads to lmnt local strss, strain, and displacmnt solutions 1 T σ = α Sλ α λ 4 2r (9) 1 ε = B λ 8µ 2r (10) 1 r 4µ 2 u = α Mλ λ (11) (2). Drivation of th Elmnt Displacmnt, Strain, and Strss Filds For FEM displacmnt mthod, th most important tchniqu is th formulation of lmnt displacmnt filds. Basd on th ida of th prvious work [1], th gnralizd 3-D SQCE displacmnt filds could b xprssd as u = Nq + (u N q) (12) Rgarding to th spcific singular lmnt formulation, most rsarchrs dirctly combin th local displacmnt fild ( u ) with rgular lmnt displacmnt fild ( Nq ). Thus th displacmnt cross th boundaris btwn rgular and singular lmnts is not continuous that crats convrgnc and accuracy problms. Th ida of Equation (12) is subtracting th displacmnt ( Nq) from local analytical solution ( u ) which can b xprssd by rgular lmnt displacmnt fild. This, in turn, assurs th compatibility btwn singular and rgular lmnts. In Equation (12) q is th lmnt nodal displacmnt vctor of local solutions givn by, { } T q = u1 u2 ui u n (13) in which u i is th local displacmnt vctor of nod i. Displacmnt modl (12) can b usd to simulat various fild problms that contain singularity or high gradint of fild variabls. For linar lastic 3-D crack problms, substitution of Eq. (11) into Eq. (12) lads to th SQCE lmnt displacmnt fild. u = Nq + (M N M) λ (14) t us assum that th total numbr of singular lmnts is m, thn th ntir filds of displacmnts and strains can b xprssd as Nq (M N M) u j m j = + λ (15) Nq j > m

5 + λ Bq ε = Bq Bk j m j j > m whr B k is th strain matrix drivd from Eq. (15), givn as Bk = E( )(M NM) (17) in which E( ) DBq DBk σ j m j = + λ (18) DBq j > m (16) is th drivativ oprator. Th SQCE lmnt strss filds can thn b xprssd as Elmnt Stiffnss and Global Equations With th lmnt strss, strain, and displacmnt filds availabl abov, w can obtain th global quations of SQCE finit lmnt systm. Substitution of Eq. (15), (16), and (18) into th systm potntial nrgy quation lads to, whr n b xprssd as 1 π= ε ε + Γ NE T T T ( D dv ( u f dv u Td ) V V 2 j 1 2 Γ = NE NE T 1 T T 1 T T nλ nλ λ j= 1 2 j= 1 2 = qk λ + λ k λ λ F + qkq qf (19) k,k,and F λ λ λ ar crack dpndnt stiffnss matrics and nodal load vctor, rspctivly that can T knλ = B D B V kdv T kλ = B V k D Bk dv (20) F = λ (M V NM)f dv + (M NM)T d Γ Γ 2 t us introduc global displacmnt and SIF vctor and stiffnss matrics, thn th systm potntial nrgy can b writtn as 1 T T 1 T T T π= QKQ+ QK Λ NΛΛ+ Λ KΛΛ QF Λ F Λ (21) 2 2 Q Λ,K NΛ,K Λ,and, FΛ ar global matrics and vctors corrsponding to Eq. (19). Whras Λ is whr global nodal strss intnsity factor vctor. Using stationary principl and considring th variation with rspct to QandΛ hav th global displacmnt and SIF quations as KQ = F 1 T T Λ= KΛ KN ΛQΛ + KN Λ F Λ in which K andf ar th global stiffnss matrix and load vctor, rspctivly, givn as of Eq. (21), w now (22)

6 1 T K = K KN K K Λ Λ NΛ 1 (23) F = F K NΛK Λ F Λ whr th scond trms on th right hand sid ar th contribution of singular lmnts to th FEM systm. Comparison of Solutions To valuat th nw 3-D SQCE modl, a numbr of xampl problms, such as smi-lliptical and lliptical 3-D crack problms hav bn solvd using SQCE lmnts. In this papr rsults of normalizd strss intnsity factors ar prsntd for a finit smi-lliptical surfac crack shown in Figur 2. Figur 3 showd on FEM msh of ths xampls of a smi-lliptical surfac crack subjctd to a uniform tnsion. Only a quartr of th crackd body is considrd bcaus of th symmtry of th problm. Gnrally 150 to 200 lmnts including 8 to 10 SQCE lmnts ar ndd along and on ach sid of th * * K crack front lin. Th normalizd strss intnsity factors K ( K = I ) of ths 3-D crack problms hav bn computd and compard with both analytical and publishd numrical solutions. Tabl 1 and Figur 4 showd th comparison of SQCE s rsults (K I * ) for diffrnt ratios of a/c with Grn-Snddon s analytical solutions [15]. In Tabl 2 only th maximum K I * of SQCE solutions for th ratio a/c varying from 0.2 to 1.0 ar compard with analytical solutions. Tabl 3 and Figur 5 show th comparisons of SQCE rsults with Atluri s and Nwman s numrical rsults. I I S t a π Q Figur 2. 3-D Crack Body with a Smi-Elliptical Surfac Crack

7 Figur 3. 3-D Crack FEM Msh Figur 4. Comparisons of SQCE Rsults with Analytical Solutions

8 Figur 5. Comparisons of SQCE Rsults with Numrical Solutions

9 Conclusions A nw 3-D SQCE modl and its finit lmnt cod hav bn implmntd for computing SIFs along a smooth crack lin with any shap undr complx loads. Basd on th modling thory and a larg numbr of computations, th following conclusions can b mad. (1) Th SQCE modl has no limitation on loading and crack shaps. In othr words, it can b usd to find SIFs of 3-D cracks of any shap with a smooth crack front lin subjctd to any typ of loads. (2) Th SQCE modl producd rliabl and consistnt rsults. It is not snsitiv to lmnt siz and a quasi-compatibl condition btwn singular lmnts and non-singular lmnts ar guarantd. (3) As th ralistic local solution filds ar wll simulatd by SQCE lmnt displacmnt modl, th siz of singular lmnts can b larg. As a rsult, fwr singular lmnts ar ndd for a 3-D crack body. For xampl, for a quartr of th crackd body, 8 to 10 SQCE lmnts ar gnrally nough to obtain accptabl numrical rsults. (4) To nsur intr-lmnt compatibility btwn SQCE lmnts and standard lmnts, it is rcommndd that for 2-D problms us 8 nod SQCE lmnts and for thr dimnsional problms us 24 nod SQCE lmnts. (5) Th SQCE modl provids a gnralizd formulation tchniqu to combin any analytical solutions with isotropic finit lmnt displacmnt filds. It can b usd to simulat any nginring fild problms with singularitis or high gradints. It has th potntial to solv strss concntration problms with complicatd gomtry and loading, such as th strss concntration and crack propagation problms of composit matrials, turbin ngins or lctronic packags. Rfrncs 1. Zhichao Wang, isu Zhang, and Chun-Tu iu, A Nw Formulation Mthod of Qausi-Compatibl Finit Elmnt and Its Application in Fractur Mchanics, Enginring Fractur Mchanics, Vol. 37, No. 6, , J. R. Ric and N. vy, Th Part-Through Surfac Cracks in an Elastic Plat, ASME J. Appl. Mch. 39, , F. Dalal and F. Erdogan, Application of th in-spring Modl to a Cylindrical Shll Containing a Circumfrntial or Axial Part-through Crack, ASME J. Appl. Mch. 49, , S. N. Atluri, Analysis of Embddd and Surfac Flaws in Transvrsly Isotropic oading by FE Altrnating Mthod, ASME J. Appl. Mch. Vol. 58, n2, , Jun, F. W. Smith, A. S. Kobayshi, and A. F. Emry, Strss Intnsity Factors for Pnny-shapd Cracks, Part 1 Infinit Solid, ASME J. Appl. Mch., 34, , D. M. Tracy and T. S. Cook, Analysis of Powr Typ Singularitis Using Finit Elmnt, Int. j. Numrical Mth. Engng, Vol. 11, , T. H. H. Pian, Crack Elmnt, Proc. World Congrss on Finit Elmnt Mthods in Structural Mchanics, Editd by J. Robinson and Assoc., Dorst, England, S. N. Atluri, A. S. kobyashi, and M. Nakagaki, An Assumd Displacmnt Hybrid Finit Elmnt Mthod for Fractur Mchanics, Int. J. Fractur 11, , D. R. J. Own and A. J. Fawks, Enginring Fractur Mchanics, Numrical Mthod and Applications, Pinridg Prss td, Swansa, UK, 1983.

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