SEMI-ANALYTICAL BEM APPLIED TO EVALUATE DYNAMIC STRESS INTENSITY FACTORS OF FRACTURED BODIES

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1 ISSN SEMI-ANALYTICAL BEM APPLIED TO EVALUATE DYNAMIC STRESS INTENSITY FACTORS OF FRACTURED BODIES Danel Nelson Macel & Humberto Breves Coda Abstract The Boundary element method s recognzed nowadays as a very powerful tool to deal wth fracture mechancs problems. The most of the wors carred out so far uses the socalled crac elements to model the dsplacement feld n the vcnty of the crac tp. These elements are usually adopted together wth the well-nown dual fracture mechancs methods (hypersngular formulatons), sub-regon technques (sngular formulatons) and other procedures to smulate the crac exstence,.e. tracton free nternal surfaces. In ths wor, smple quadratc boundary elements assumed as external boundary approxmatons are taen to model the crac and also to compute the dsplacements feld n the crac tp vcnty. The exstence of the crac s smulated by an actual hole defned nsde the doman. For the approach, closed expressons for all ntegral terms along the elements are analytcally obtaned and then the stress ntensty factors computed accurately by usng the stress feld around the tp. Easy extenson to non-lnear fracture models (plastcty or mpact on fractured surfaces) s an mportant characterstc of the proposed approach. Numercal examples are shown n order to demonstrate the accuracy of ths smple approach. Keywords: fracture mechancs; boundary elements; analytc ntegrals. INTRODUCTION It s well now that nowadays the BEM s the most powerful numercal method to analyse general fracture mechanc problems. Ths s due to ts ablty on representng the stress felds nsde the analysed doman. Followng the hstorcal aspects of BEM appled to fracture mechancs, the frst attemptng to calculate stress ntensty factors (SIFs) were due to Cruse & Van-Buren (97) and Cruse (97). In those wors SIFs were calculated by ordnary Boundary Element Method formulaton usng smple numercal ntegraton procedures to perform the boundary ntegrals and errors about 4% were acheved. After that Snyder & Cruse developed Green functons consderng a crac n an nfnte doman to be used as the weghtng functons, generatng BEM formulatons Mestre em Engenhara de Estruturas - EESC-USP, dnmacel@sc.usp.br Professor do Departamento de Engenhara de Estruturas da EESC-USP, hbcoda@sc.usp.br Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

2 64 Danel Nelson Macel & Humberto Breves Coda that naturally consdered the crac presence. It maes possble to compute wth an enormous precson SIFs n fnte geometry. Some authors that followed ths approach as Mews and Telles & Gumarães may be mentoned. Some dffcultes as, for example, generatng curved cracs of even mult-craced bodes are present. To overcome these dffcultes the sub-regon technque had bee adopted, together wth the well nown quarter pont boundary element as for example n Blandford et all. Ths formulaton present some dffcultes when the growth of a fracture should be consdered, a new mesh (for the couplng) defnton s needed each tme the openng s expected. Nowadays, the dual boundary element for fracture mechancs s one of the most appled formulaton to SIFs determnaton and t s used together quarter pont type fracture elements and hysngularty are properly treated by the sngularty subtracton technque. In dynamcs smlar procedures are employed and one can menton the wors of Chrno et all, Gallego & Granadosl, Domngues & Gallego and Albuquerque et all. There s the formulaton presented by Ventrn 3 that uses stress resduals to reproduce crac behavour. In ths study, the frst experence of Cruse s reproduced. Ths tme the numercal ntegrals over ordnary boundary elements are performed analytcally consderng straght elements wth quadratc varable approxmatons. The analytcal ntegrals are developed n a local co-ordnate system for source ponts belongng or not to the ntegrated element. The result s rotated to the global co-ordnate system and then added to the global BEM matrces. As n Cruse wor the crac s dscretzed as a small hole nsde the doman and the stress ntensty factor s calculated va stress dstrbuton near the crac tp. No specal (quarter pont type) elements are employed near the tp dscretzaton, loong forward to non-lnear applcatons. The Kelvn fundamental soluton together wth Houbolt scheme and trangular cells are used to run dynamc cases. Some examples are shown n order to demonstrate the applcablty of ths smple methodology. DEVELOPMENTS As mentoned before, the formulaton presented here s almost the same of ordnary BEM for D elastc analyss. So the dsplacement ntegral equaton s: C + Ω u (s) = u p (s, q)b (q)dω(q) (s, q)u (q)d(q) + u (s, q)p (q)d(q) + () and the dsplacement gradent ntegral equaton for nternal ponts s: u (s) = p (s, q)u (q)d(q) u (s, q)p (q)d(q) +, Ω u,, (s,q) λ (q)dω(q), () where the Kelvn fundamental soluton values are: Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

3 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 65 u ( s,q ) = [( 3 4υ )ln( r ) δ r, 8π ( υ )G r, ] (3) r p ( s,q ) = {[( υ ) δ + r, r, ] ( υ )( r, η r, 4π ( υ )r n η )} (4) u, [(3 4ν)r, δ + r, r, r, ( δ r, + δ r, )] (s,q) = (5) 8πG( ν)r p, (s,q) = 4π( ν)r + ( ν) [ ( ν) δ r, + 8r, r, r, ( δ r, +δ r, )] ( η r, r, η r, r, δ η + δ η δ η ) r, r, η } r + n (6) It s mportant to note that the dsplacement gradent s calculated here loong forward to J ntegral calculatons. After applyng spatal approxmatons, choosng sources ponts equal to the nodal ponts and performng, analytcally, the boundary ntegrals see next secton) results: { }, ( s) [ h] s, { U} [ g] s, { P} [ b] s, { Ψ U = } (7) where ψ s a doman term that by the mposton of D Alambert prncple turns nto: HU (t) + CU(t) & + MU(t) & = GP(t) + Bb(t) (8) for dynamc applcatons. The dsplacement gradent statc equaton s: U = H U G P B Ψ (9) In t s dynamc verson one has: U (t) = H U(t) G P(t) M U(t) & C U(t) & + B b (0) Houbolt scheme together wth trangular cells have bee used to approach nertal terms. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

4 66 Danel Nelson Macel & Humberto Breves Coda 3 ANALYTIC INTEGRAL DEVELOPMENT In ths secton only the man steps requred to develop the referred ntegrals are descrbed,.e., local axes, varables lmts and transformaton acoban. Full expressons can be found n Macel. For a general source pont the local system of co-ordnated s depcted n fgure. θ s a X r(s,q) X X α η X Fgure - Local co-ordnate system and general source pont poston. The ntegrals to obtan the sub-matrces [h], [g], [h] e [g] can be wrtten as: f (r, θ, α) d () has: The fundamental soluton s also wrtten n the local co-ordnate system, so one f (r, θ) d () It s easy to note that: d = dx One can transform equaton () nto: f (r, θ)d = θ θ a f ( θ) dθ (3) sen θ Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

5 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 67 where the followng relatons have been used: a a = rsen θ r = (4) sen θ cosθ X = X(s) + a (5) sen θ dx a = dθ sen θ (6) The quadratc shape functons can be wrttten, n local co-ordnate, as: cosθ 3 cosθ φ ( θ ) = X( s ) + a X( s ) + a + (7) L senθ L senθ 4 cosθ cosθ φ ( θ ) = X( s ) + a X( s ) + a (8) L senθ L senθ cosθ cosθ φ3 ( θ ) = X( s ) + a X( s ) + a (9) L L senθ senθ where L s the element length. gven: To fnsh the descrpton the followng example to calculate the term g s θ g θ K senθ cosθ a u φ d = = X( s ) a dθ + L senθ (0) sen θ where: K = () 8π( ν)g Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

6 68 Danel Nelson Macel & Humberto Breves Coda For smplcty φ n equaton (0) s the frst lnear approxmaton functon, not the quadratc ndcated n expresson (7), now one can develops the ndcated ntegrla and fnd the desred matrx value. Complet terms for both lnear and quadratc elements can be found n Macel. 4 GEOMETRY AND SIFS EXTRACTION STRATEGY The geometry of a pre-exstent crac and the ponts locaton to extract the stress ntensty factos for the analysed examples are shown n fgure. h a y θ = 0 T x T a a a/8 a a/7 X X a/6 Fgure - a) Geometry b) Extracton ponts. The numercal soluton for the stress ntensty factor begns to approach the analytcal soluton for a closed crac when h < 0 a, n ths wor t has been assumed 3 the relaton h < 0 a for the analysed examples, because t seemed near enough to the closed crac and of a reasonable sze for numercal computatons. To valdate the extracton ponts poston a analyss smlar to the one done by Pars & Cañes 6 has been done. Ths analyss leads to a relable correlaton between the soluton presented by the logarthm curve extracton (very expensve) and the one obtaned by usng the proposed extracton poston. To save space ths analyss s not shown here. Examples: 4. Mode I statc analyss Ths example s dvded nto two cases; the frst s related to a plane stress problem. The second s related to a plane stran problem, table shows the physcal and geometrcal characterstcs (see fgure 3) for both problems. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

7 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 69 Table - Geometryc and physcal propertes Plane stress Plane stran σ 000 gf/cm.0 Mpa E. x 0 6 gf/cm 00 Mpa ν a.0 cm.4 mm W 0 cm 0 mm L 40 cm 40 mm h.0 x 0-3 a.0 x 0-3 a σ a W σ L Fgure 3 - General geometry, mode I statc analyss. For ths case, the analytcal soluton s: K I = 3 ( + 0,56( a W ),5( a W ) +,00( a W ) ) σ πa Table presents the results and errors for ths example: Table - Results for plane stress and plane stran mode SIF calculaton Plane stress ( K I ) analtc 79, 74 g Plane stran 3 cm 3 / 89848,07 Pa m ( I ) MEC K 3 / 89,4 gf cm 907,90 Pa m 3 / Error.47% 0.36% Obvously that the one does not expect equal results n ths case because the analysed problems are very near, but not the same. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

8 70 Danel Nelson Macel & Humberto Breves Coda 4. Mxed modes I and II, statc case In ths example a mxed mode s studyed for the problem depcted n fgure 4. It s a plane stra problem and the followng characterstcs were adopted (see fgure 4): σ =,0 Mpa, E = 00 Gpa, ν = 0,3, a = 7,07 mm, W = 30 mm, L = 60 mm, α = 45º. h = x 0-3 a a σ W σ α L Fgure 4 - Plate wth nclned crac. In ths example two dscretzatons have been adopted; the frst, used 60 boundary elements for the exteror boundary and 00 along the crac surface. The second used 60 boundary elements along the exteror boundary but only 0 elements along the crac surface. Results are depcted n table 3, and one may note the good stablty of the results regardng the crac dscretzaton. Table 3 - SIFs varaton regardng crac mesh densty Crac dscretzaton KI KII Varaton.4% 4.0% In fgure 5 one can see the deformed shape of the proposed example. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

9 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 7 Fgure 5 - Deformed confguraton. 4.3 Dynamc KI determnaton In ths example the same geometry and physcal propertes of example (plane stran) are adopted. The mass densty s ρ = 5000 Kg/m 3 and the load, σ =,0 Mpa, s suddenly appled at nstant t=0s. In fgure 6 the normalzed dynamc K I s depcted together wth other results extracted from lterature. 3.5 MEC - MMBEM Dual TDBEM KI Normalzado KI/Ko Tempo t [µ s] Fgure 6 - Stress ntensty factor K I for a crac n the center of a plate. The adopted dscretzaton s 80 boundary elements to the total crac dscretzatons and 350 mass cells. In ths case the cells were consdered passng over the crac, as t s done n MMBEM formulatons applyng dual recprocty and closed cracs. The results are compared wth the ones obtaned by FEDELINSKI et al that employed a TDBEM formulaton and a Dual recprocty analyss wth 3 boundary elements and 5 nternal ponts together wth specal crac tp elements. The results are n good agreement. The non-zero values obtaned before the longtudnal wave reach the crac s due to the strong non-causalty of MMBEM caused by the use of Kelvn fundamental soluton that presents non-zero values over all the doman for any nstant. Ths same geometry has been adopted for the plane stress case wth ρ = 0,00 Kg/cm 3 and t = 0, x 0-4 s usng 60 boundary elements (00 along the crac) and 050 cells notng passng over the crac. The acheved path for the stress ntensty factor s depcted n fgure 7 and the crac openng behavour s shown n fgure 8. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

10 7 Danel Nelson Macel & Humberto Breves Coda 3 MEC - MMBEM.5 KI Normalzado KI/Ko Tempo t [ 0-4 s] Fgure 7 - Stress ntensty factor for a open crac. The path s smlar to the one obtaned by consderng the crac closed, but the behavor presents more modes. X (cm) npt=0 npt=5 npt=50 npt=75 npt=00 npt=5 npt=50 npt=75 npt=00 npt=5 npt= X (cm) Fgure 8 - Crac openng behavour durng analyss (npt=number of tme steps). From fgure 7 and 8 one concludes that negatve K I s related to crac closng behavour and that durng the man duraton of the analyss the closng relatve dsplacement s about h/8 and then there s no necessty for dong collson analyss for opened cracs. However f the crac s really closed more research s requred to analyse the mportance of the colldng fenomenum. 4.4 Inclned crac, dynamc analyss The geometry of the problem s depcted n fgure 4. The adopted physcal propertes are the same of the plane stran case of the prevous example. The load, σ =,0 Mpa, s suddenly appled (t=0), as n fgure 4. In fgure 9 and 0 the values of K I and K II are compared wth the obtaned by FEDELINSKI et. al. and DOMINGUEZ & GALLEGO. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

11 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 73.6 MEC - MMBEM TDBEM. Dual KI Normalzado KI/Ko Tempo t [µ s] Fgure 9 - Dynamc ntensty factor K I for a nclne crac 0 boundary elements along the crac and 60 nternal cells (passng over the crac) have been adopted for ths analyss. The dual recprocty analyss used 0 boundary elements and 0 nternal ponts together wth specal crac elements. The adopted tme step s t=0, µs.,6 MEC Analítco - MMBEM TDBEM Dual K II Normalzado KII/Ko, 0,8 0, ,4 Tempo t [µs] Fgure 0 - Dynamc K II for nclned crac. The results are n good agreement n spte of the lost of causalty manly n the K I mode for the MMBEM. 5 ACKNOWLEDGEMENTS The authors would le to tans CAPES Brazl for supportng ths research. 6 CONCLUSIONS It has been shown that by means of analytcal boundary ntegrals, the statc and dynamc fracture mechanc problem can be properly treated by ordnary boundary Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

12 74 Danel Nelson Macel & Humberto Breves Coda elements, as proposed by T.A. Cruse n poneer wors on Boundary Elements. The MMBEM needs more mass dscretzaton than the dual recprocty, so the mplementaton of DBEM n ths nd of analyss s greatly recommended. The necessty of specal crac tps elements should be consdered, but for further non-lnear applcaton the unty partton must be consdered. It s mportant to note that from ths formulaton hydraulc fracture mechancs holds naturally and plastc problems can be solved wthout much modfcatons. 7 REFERENCES ALBUQUERQUE, E. L.; SOLLERO, P., ALIABADI, M. H. (00). The boundary element method appled to tme dependent problems n ansotropc materals. Int. J. Sol. Struc., v.39, n.5, p ALIABADI, M. H.; ROOKE, D. P. (99). Numercal Fracture Mechancs. The Netherlands: Kluwer Academc Publshers. BLANDFORD, G. E.; INGRAFFEA, A. R.; LIGGET, J. A. (98). Two-dmentonal stress ntensfy factor computatons usng the Boundary Element Method. Int. J. Num. Meth. Engn., v.7, p CODA, H. B. (00). Dynamc and statc non-lnear analyss of renforced meda: a BEM/FEM couplng approach. Comp. Struc., v.79, p CRUSE, T. A. (97). Numercal evaluaton of elastc stress ntensfy factor by the boundary-ntegral equaton method. In: SWEDLON, J. L. (Ed.). The surface crac: physcal problems and computatonal solutons. New Yor. CRUSE, T. A.; VAN BUREN, W. (97). Dmensonal elastc stress analyss of a fracture specmen wth an edge crac. Int. J. Num. Mech., v.7, p.-6. CHIRINO, F.; GALLEGO, R.; SAEZ, A.; DOMÍNGUEZ, J. (994). A comparatvestudy of 3 boundary-element approaches to transent dynamc crac problems. Engn. Anal. Bound. Elem., v.33, n.3, p DOMÍNGUEZ, J.; GALLEGO, R. (99). Tme boundary element method for dynamc stress ntensfy factor computatons. Int. J. Num. Meth. Engn., v.33, n.3, p FEDELINSKI, P; ALIABADI, M. H.; ROOKE, D. P. (995). Boundary element formulaton for the dynamc analyss of craced structures. In: ALIABADI, M. H. (Ed.). Dynamc Fracture Mechancs. Computatonal Mechancs Publcatons. GALLEGO, R.; GRANADOS, J. J. (00). Transent crac propagaton usng boundary elements. In: ASCE ENG. MECH. CONFERENCE. 5., New Yor: Columba Unversty. MACIEL, D. N. (003). Determnação dos fatores de ntensdade de tensão estátcos e dnâmcos va MEC com ntegração analítca em coordenadas locas. São Carlos. Dssertação (Mestrado) Escola de Engenhara de São Carlos Unversdade de São Paulo. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006

13 Sem-analytcal BEM appled to evaluate dynamc stress ntensty factors of fractures bodes 75 MEWS, H. (987). Calculaton of stress ntensfy factors for varous crac problems wth the Boundary Element Method. In: BREBBIA, C. A. et. al. (Ed.). Boundary Elements IX. PARÍS, F.; CAÑAS, J. (997). Boundary Element Method fundamentals and applcatons. Oxford: Oxford Unversty Press. PORTELA, A.; ALIABADI, M. H.; ROOKE, D. P. (99). Dual boundary element method: Effcent mplementaton for craced problems. Int. J. Num. Meth. Engn., v.33, p SNYDER, M. D.; CRUSE, T. A. (975). Boundary-ntegral equaton analyss of craced ansotropc plates. Int. J. Fracture., n., p TELLES, J. C. F.; GUIMARAES, S. (000). Green s funton: A numercal generaton for fracture mechancs problems va Boundary Elements. Comput. Meth. Appl. Mech., v.88, n.4, p VENTURINI, W. S. (994). A new boundary element formulaton for crac analyss. In: BREBBIA, C. A. (Ed.). Boundary Element Method XVI. Cadernos de Engenhara de Estruturas, São Carlos, v. 8, n. 3, p , 006