Transient analysis of collinear cracks under anti-plane dynamic loading

Transcription

1 valable onlne at Proceda Engneerng () ICM Transent analyss of collnear cracks under ant-plane dynamc loadng K. C. Wu * and J. C. Chen Insttute of ppled Mechancs, atonal Tawan Unversty, Tape, Tawan bstract The problem of a homogeneous lnear elastc body contanng multple collnear cracks under ant-plane dynamc load s consdered n ths work. The cracks are smulated by dstrbutons of dslocatons and an ntegral equaton relatng tractons on the crack planes and the dslocaton denstes s derved. The ntegral equaton n the Laplace transform doman s solved by Gaussan-Chebyshev ntegraton quadrature. The dynamc stress ntensty factor assocated wth each crack tp s calculated by a numercal nverse Laplace scheme. The proposed method was appled to calculate the stress ntensty factors for M equally spaced cracks of dentcal length subject to mpact loadng wth M =4. Comparson of the numercal result for a sngle crack wth the analytc soluton shows that the present method s hghly accurate. Publshed by Elsever Ltd. Open access under CC BY-C-D lcense. Selecton and peer-revew under responsblty of ICM Keywords: collnear cracks; dynamc stress ntensty factor.. Introducton When a lnear elastc sold contanng cracks s subject to appled loads, the stress feld near the crack tps ehbts a square-root sngularty. The ampltudes of the sngular stress feld are characterzed by the stress ntensty factors, whch are used as the parameters governng crack propagaton. The stress ntensty factors depend on appled loadng, geometry, and appled loadng. Because of the dffcultes n satsfyng * Correspondng author. Tel.: ; fa: E-mal address: wukc@webmal.am.ntu.edu.tw Publshed by Elsever Ltd. Open access under CC BY-C-D lcense. Selecton and peer-revew under responsblty of ICM do:.6/j.proeng..4.5

2 K. C. Wu and J. C. Chen / Proceda Engneerng () the boundary condtons, only a lmted number of closed-form solutons of stress ntensty factor est. For most problems numercal methods must be employed to determne the stress ntensty factors. One of the popular methods for solvng crack problems wth statc loadng s the dslocaton method. nalyss of the problem by ths method may be broken down nto stages: frst the tractons arsng along the lne of the crack n the uncracked body are determned. The crack s then nserted and the unsatsfed tractons cancelled by nsertng a contnuously varyng densty of dsplacement dscontnutes, or dslocatons, along the lne of the crack. Ths formulaton leads to an ntegral equaton whch may be dscretzed usng Gaussan-Chebyshev quadrature to the desred degree of refnement []. Once the resultng smultaneous lnear equatons are solved the stress ntensty factors can be readly determned from the dslocaton denstes at the crack tps. The dslocaton method, however, has rarely been appled to the problems nvolvng dynamc loadng. For sotropc materals Cochard and Madaraga [] has derved an ntegral equaton for dynamc antplane shear (mode III) loadng. The ntegral equaton contans a space-tme convoluton ntegral, whch s Cauchy sngular n space as n the statc case. ddtonally there s another term whch s only present n the dynamc case and s assocated wth radaton dampng by wave emsson. The ntegral equaton has been numercally mplemented by Morrssey and Geubelle [3] wth a spectral scheme, n whch an ntegral n tme obtaned by applyng the Fourer transform n space s solved. In the spectral formulaton the Cauchy sngularty of the ntegral s removed and the Gauss ntegraton quadrature cannot be utlzed. Chen and Tang [4] obtaned an ntegral equaton as that n [] but dd not correctly nclude the radaton term. evertheless they showed that f the Laplace transform n tme s appled to the ntegral, the Cauchy sngularty s preserved and the same ntegraton quadrature for statc loadng may be employed. It s the man objectve of ths work to etend the method to study collnear cracks under dynamc loadng. The ntegral equatons n [] wll be derved based on the fundamental soluton of a dslocaton. The equaton wll be solved frst n the Laplace transform doman usng Gauss ntegraton quadrature and then nverted to calculate the stress ntensty factors n the tme doman. The proposed method was appled to calculate the stress ntensty factors for M equally spaced cracks of dentcal length subject to mpact loadng wth M =4.. Basc Equatons The equaton of moton n ant-plane shear deformaton s,, u, () where =,, u s the dsplacement n the 3 drecton, t s tme and s the mass densty, a comma n the subscrpt denotes partal dfferentaton, and an overhead dot stands for tme dervatve. The stress-stran relatons for sotropc materals are u,,,, () where sthe shear modulus. The equaton of moton n terms of the dsplacement s gven by substtutng () nto () as u u u (3),,. The general soluton of (3) for > can be represented as [5,6]

3 96 K. C. Wu and J. C. Chen / Proceda Engneerng () u Re[ f( w)] H t / c, (4) where Re stands for real part, H s the Heavsde step functon, c / s the shear wave speed, and w s Here y = t, =, and c y y y/ c w y / y y y. From Eq. (), can be epressed as w Re f '( ) Re[ f( )] ct, where s the Drac delta functon. 3. Fundamental Soluton Consder a screw dslocaton of Burgers vector, whch suddenly appears at t = at the orgn n an nfnte body that s ntally at rest and stress-free. The slp plane s assumed to concde wth the negatve -as. From the jump condton for u and contnuty condton for at =, (4) and (6) yeld, for t >,. ( f ' f ' ) ( f ' f ' ) y, (7) ( f ' f ' ) ( f ' f ' ), (8) where the superscrpts denote the lmts as. The soluton of f (w) may be shown to be Substtuton of (9) nto (4) and (6) gves From (6) at = s gven as where f( w) lnw. (9) 4 u Im[ln w] Ht / c. () L y H t, () c / / L y y c H y c. () (5) (6) 4. Collnear Cracks Consder collnear cracks located at b a, M and = n an nfnte body. The cracks may be smulated as a dstrbuton of dslocaton t, as

4 K. C. Wu and J. C. Chen / Proceda Engneerng () , t t, From () and (3) the stress at = may be epressed as dd. (3) M b a t, t L, dd c t t Equaton (4) recovers the result obtaned n [] usng a dfferent approach. Takng Laplace transform of (4) yelds the followng Cauchy type sngular ntegral equaton: c b a b a. (4) M ˆ b a ˆ, s s, s ˆ U s/ c ˆ, sd, (5) where ˆf denotes Laplace transform of f, III. z U z z K z K d (6) and K n s the modfed Bessel functon of order n. Equaton (5) may be solved for, s subsequently, by nverse Laplace transform, t. 5. umercal Methods Let b a,, the ntegral n (5) may be epressed as, and b a ˆ ˆ U s/ c ˆ, s d U s/ c ˆ, sad b a. (7) Moreover to ncorporate the square-root sngularty of dslocaton densty at the crack tps, let gˆ, s ˆ, s, (8) a where ĝ s a regular functon, and (7) becomes b a ˆ ˆ, ˆ g s U s/ c ˆ, s d U s/ c d ba. (9) Snce as, U, (9) s an ntegral of Chauchy's type and can be evaluated usng the followng Gauss-Chebyshev ntegraton formula [] b a b a ˆ U s/ c ˆ, sd Uˆ / ˆ j s c gj, s, () j where j b a cos j, j,,. ˆ, s Consder net n (6), whch s related to ˆ, s for b a, polynomals, ˆ, smay be epressed as by, s ˆ, 3 b 3 a j ˆ s d, (). By substtutng (8) nto () and appromatng ĝ by Chebyshev

5 98 K. C. Wu and J. C. Chen / Proceda Engneerng () Tk j sn ˆ j, () ˆ 3 k g j k k where cos. Wth () and (), (5) can be dscretzed as M, s C, ˆ j j g j, s ˆ. (3) j k Equaton (3) can be solved for gˆ j, sby settng b a cos,,, M, k,,. ddtonal equatons are obtaned by the crack closure condtons ˆ j b a, s g,,, M. (4) The stress ntensty factors n the Laplace transform doman s determned by where r s the dstance from the crack tp. Usng (8), (5) yelds j ˆ r K lm ˆ III s r, s, (5) r j K, / cot ˆ III b a s a g j j 4 j j, / tan ˆ III j j 4 j, (6) K b a s a g. (7) The stress ntensty factors n the tme doman may be calculated usng several numercal schemes for Laplace nverson. In ths work, the method proposed by Mller and Guy [7] s adopted. 6. Eamples Consder M equally spaced cracks of length a subject to H t, where s a constant. The dstance between the centers of the cracks s assumed to be 3a. The stress ntensty factors for M =4, were computed and the numercal results are shown, respectvely, n Fg. Fg B.. a) /.8 a) /.8.4 tp.4 tp B tp

6 K. C. Wu and J. C. Chen / Proceda Engneerng () Fg.. Stress ntensty factors for M =. Fg.. Stress ntensty factors for M =..6 B C.6 B C D.. a) /.8 a) /.8.4 tp C tp B tp.4 tp D tp C tp B tp Fg. 3. Stress ntensty factors for M = 3. Fg. 4. Stress ntensty factors for M = 4. For M =, the computed mamum normalzed KIII was.4, whch agrees closely wth the analytc value 4 /.7 [8]. For M =, the mamum normalzed KIII occurred at tp B and reached a hgher value.33 than that for a sngle crack. For M = 3, 4, the mamum normalzed KIII occurred at tp C and D, respectvely, but no further enhancement was observed. It appears that as far as the dynamc overshoot of the stress ntensty factors s concerned, t s suffcent to consder only two cracks. It was also checked that the correspondng statc values were approached as the tme ncreased n all cases consdered. References [] Erdogan F, Gupta GD, Cook TS. umercal soluton of sngular ntegral eqautons. In: Sh GC, edtor. Methods of analyss and solutons of crack problems, Leyden: oordhoff; 974, p [] Cochard, Madaraga R. Dynamc faultng under rate-dependent frcton. Pure and ppl Geophys 994;4: [3] Morrssey JW, Geubelle PH. umercal Scheme for Mode III Dynamc Fracture problems. Int J. umer Meth Engng, 997;4:8 96. [4] Chen W, Tang R. Cauchy Sngular Integral Equaton Method for Transent ntplane Dynamc Problems. Engng Fracture Mech996; 54: [5] Wlls JR. Self-smlar problems n elastodynamcs. Phl Trans R Soc Lond 973;443: [6] Wu KC. Dynamc Green s Functons for nsotropc materals under nt-plane Deformaton. J Mech999; 5: 6. [7] Mller Ma K, Guy Jr T. umercal nverson of the laplace transform by use of jacob polynomals. SIM J umer nal 966; 3: [8] Thau S, Lu TH. Dffracton of transent horzontal shear waves by a fnte crack and a fnte rgd rbbon. Int J Engng Sc 97;8: