3d dual bem_cisilino

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Three-dimensional BEM analysis for fatiguecrack growth in welded cbmponents -

A. P. Cisilino & M, H. Aliabadi*Wessa imtitntr of Technology. Askurst Lodge, Ashurst, Sot~thnrrrpton SO40 7AA. UK

(Received 10 April 1996:accepted8 May 1996)

In this paper a general procedure for the analysis of three-dimensionalmultiple fatigue crack growth is presented.The crack propagation s simulatedusing an incremental crack extension analysis based on the strain energydensity criterion and the Paris law. For each crack extension the dualboundary element method is used o perform a single region analysisof thecracked component. Stress ntensity factors are computed along the crackfronts usinga displacement asedmethod. Crack extensionsare automaticallymodelled with the introduction of new boundary elements along the crackfronts and a localized rediscretization n the area where the cracks ntersect hefree surfaces.The capability of the procedure is demonstratedby solving anumber of multiple edge-crack examples.Copyright 0 1996Elsevier ScienceLtd.

1 INTRODUCTfONGeometric discontinuities are created in all fabricationprocesses. These discontinuities act as clusters ofsurface breaking defects. introducing a high prob-ability of crack initiation which is frequently followedby a subcritical crack growth period that eventuallyleads to failure, limiting the components life.In welded components the initial defects areassociated with the fusion and solidification process(cracking, metallurgical transformations, residualstresses, inclusions etc.) and they are located in a zonewhich usually has a high level of stresses induced bythe geometric discontinuity of the weld toe. Due tothe periodicity in the geometry of the weld toe(specially for automatic welding) crack initiationpoints are regularly distributed along the weld toe,resulting in the formation of similar periodic arrays ofcracks. Recent experimental and theoretical resultsdemonstrate that the rate at which small cracksinitiate and propagate is strongly dependent on crackinteraction, microstructural characteristics and re-sidual stresses.-5The different ways in which the cracks interactdepend primarily on their spatial distribution, appliedstresses and the problem geometry. Although therehave been some advances in the fracture theory ofcrack interaction, it appears that there is stillGAuthor to whom correspondence hould be addresed.

insufficient knowledge to be able to treat this problemwith confidence.Most of the research in the field of crack interactionhas been limited to coplanar cracks. Murakami andNemat-Nasser6. have applied the body force methodto obtain stress intensity factor solutions for twodissimilar interacting surface cracks. ODonoghue,using the alternating method in conjunction with thefinite element method and an analytical solution for asingle cracks has determined the stress intensityfactors for interacting semi-elliptical surface cracks incylindrical pressure vessels and embedded ellipticalcracks in infinite solids. Health has also used thealternating method, but to solve the problem ofmultiple corner cracks along a hole bore in a plate;and McComb has carried out an experimental studyof fatigue crack coalescence and interaction incentred-hole remote-loaded polycarbonate specimens.In the assessment of non-coplanar multiple andirregularly oriented cracks the knowledge is verylimited and not many results have been reported.Through the use of the current code methods (ASMESection XI, BSI PD6493, etc.), cracks are usuallyrecharacterized as a single crack with a boundingcurve as the new crack shape. These recharacteriza-tion procedures follow mostly empirical ruies whichcan vary significantly among the different codes, as is

illustrated by Tu. Jiang using finite elements hasdeveloped an empirical formula to evaluate the stressintensity factors of two parallel cracks.


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Fatigue crack growth in welded components 137for the dual boundary element method when therelative displacements between the crack surfaces areintroduced as new unknowns. This allows the numberof unknowns on the crack to be halved, which, as willbe shown later. is the region of the model with thelargest number of unknowns.The general modelling strategy used in this work issimilar to that reported in Mi and Aliabadi and canbe summarized as follows:

* Only one of the crack surfaces is discretized andthe traction boundary eqn (6) is applied forcollocation. The discretization is done usingdiscontinuous 9-noded quadratic elements. Dis-continuous elements are used to fulfil thecontinuity requirements of the field variables forthe existence of Cauchy and Hadamard principalvalue integrals. Special elements developed byMi and Aliabad? are placed on the crack front toreproduce the variation fi in the displacementfield at the crack tip.

l Continuous 9-noded quadratic elements are usedover the remaining boundary of the model?except at the intersection of a crack and aboundary surface. In these regions edge discon-tinuous elements are employed to avoid acommon node at the intersection. The displace-ment integral eqn (5) is used to collocate in bothcases.This simple strategy is robust and allows the dual

boundary element method to effectively modelgeneral edge crack problems. Crack tips, crack edgecorners and crack kinks do not require specialtreatment, since they are not located at nodal pointswhere the collocation is carried out.

3 CRACK EXTENSION ANALYSISThe numerical simulation of crack growth involves anincremental crack extension analysis. In the first stagethe initial cracks are defined and the boundaryelement method is applied to perform a stress analysisand the stress intensity factors are evaluated along thecrack fronts. The incremental direction along thecrack front is evaluated by the minimum strain energycriterium and the incremental size by the Paris law.The incremental extension of the cracks is modelledby adding new elements at the crack front. Aremeshing strategy is employed to redefine thediscretization at the regions where the cracks intersectthe free surface (noted here as tip areas). A newboundary element solution is then carried out for thenew configuration and the incremental procedure isrepeated. The above incremental analysis is per-formed until the predefined crack length is reached.The relative displacements of the crack surfaces AUare calculated by the boundary element analysis and

used in the near crack tip stress field equations toobtain the local mixed mode stress intensity factors.For any point P located on the crack surface there is asection plane orthogonal to the plane of the crack andnormal to the tangent to the crack front at some point(say Q). When the one point formula is employed,stress intensity factors at the point Q are evaluated as:

KF= E ;4(1 - v) J SAli::

K;= E4(1- v) d n Auf;2rEK;,=- 4(1 + v)

(7)

where E is the Youngs modulus and v the Poissonsratio. The terms ALL:, AU{ and AU: are projections ofALL, the displacement evaluated at point P, on thelocal coordinate directions (i.e. normal, binormal andtangential) at the crack front.Several criteria have been proposed to describe thelocal direction of mixed-mode crack growth. Amongthem, the most popular are the maximum principalstress, the maximum energy release rate and theminimum strain energy density. In this work theminimum strain energy criterion formulated by Sib isadopted. The criterion states that the direction ofcrack propagation at any point along the crack front istoward the region with the minimum value of strainenergy density factor S as compared with otherregions on the same spherical surface surrounding thepoint. Referring to the coordinate system (n, b, t), thestrain energy factor S for a volume elementdv = dn db . dt on the crack front can be expressed interms of the stress intensity factors K,, Kr, and K,,, asfollows:

S = aI, Kf + 2alzKIKII + a,,Kf, + a33KfII (8)where CI,,, 012, a2* and a33 are trigonometricpolynomials of sin(e) and cos(8), and 8 is thedirection of the crack extension in the local plane. Theangle 8 at each point along the fronts is obtained bysolving:

In this work eqn (9) is solved using the bisectionmethod.During fatigue crack growth, the relationshipbetween incremental size and the number of cycles isrepresented by the Paris law, which states that

where daldN is the rate of change of crack length withrespect to the loading cycles: C and nz are constantsthat depend upon material, load frequency, environ-


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ment and mean load: and AK,rr is the cyclic value ofthe effective stress intensity factor given byAK eff = Km.? - f(+cl1 efi (11-l

Combining the expression of Kc, proposed byGerstle, the resultant expression for KeR is:

.4K:,T = (AK1 + B {AK,# + 2AK:t (WFinally the discrete amount Aa corresponding to acrack front point where bK,% occurs is given fron eqn(lo), in an approximate form, by:

Ao = CaK,,,g . AN (13)In this work three sets of stress intensity factors arecomputed on each element along the crack front, byusing the relative displacements Au of the secondrow collocation nodes (Fig. 1). Then by using

expressions (8) and (9) the propagation angle 8 is alsocomputed at the points Q. The resultant propagationdirection can be referred to the global system ofcoordinates and expressed as a propagation vector Vwith components (u.$, up, uf).The incremental crack extension analysis assumes apiece-wise linear discretization of the crack path. Newspecial crack tip elements are placed along the crackfront for each increment of the crack extension; aprocedure that requires the stress intensity factorvalues and propagation directions at the geometricnodes along the crack front ((2 in Fig. 1). Aninterpolation K(s), where s is the position along thecrack front of the hK values, is used to obtain thestress intensity factors at the geometric nodes AKa.For the special case where these geometric nodes areplaced at the crack tips, AK are calculated byextrapolating K(s) to the free surface. The effectivestress intensity factors at the geometric nodes AK$are computed using expression (12). The procedure tocompute the propagation vector at these nodes 5varies according to the location of Q:l for corner nodes shared by two crack front elements

K=f(s)Resultant

a Coilocation nodes0 Geom etrical nodes

on the crack front(Q )x x Pants of SIF evaluationfQ)

Fig. 1. Crack front propagation vectors.

the propagation vector 0 is taken as the average ofthe propagation vectors of the two closestneighbouring points Q,l for midside element nodes the propagation vector istaken equal to that of closest point Q, and* for nodes located at the crack tips, the value ofthe components of the propagation vector(& lf, uf) are taken from the extrapolatedvalues of the components of the propagation vectorsof the interior points Q and then projected on tothe free surface plane.Finally the crack extensions at the geometric nodesA3n are computed using expression (13).The incremental analysis also requires the modifica-tion of the boundary element mesh at the tip areas.The remeshing strategy developed for these areas isshown in Fig. 2. For each crack extension only a fewelements located around the previous and new cracktips are removed and the tip area is remeshed with6-node triangular discontinuous elements using aDelauny-based triangularization algorithm.~~ Theremoved elements are those intersected by a circlecentred at the new crack tip (point B in Fig. 2) andwith a radius equal to a certain fraction of the crackextension. The triangulation points consist of A (theold crack tip) and B (the new crack tip) and thegeometric corner nodes C located on the boundary ofthe undiscretized patch. This strategy which accom-modates the required local mesh modificationsminimizes the extra computation necessary to solve

I BEx

//

I / I /(4

(b)

Fig. 2. Crack tip remeshing procedure.


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Frrtigue crack growth in welded components I.19

n hc de f a

a

Fig. 3. Schematic representation of system matrixassembling.

the new configuration and is suitable for the automaticsimulation of multiple cracks approaching each other.After each rediscretization the entries in theboundary element system matrix corresponding onlyto the altered sections of the mesh have to beupdated. These entries correspond to the dashed areain Fig. 3, according to the following rules:(a) nodes belonging to the quadrilateral elements thatare unaffected by the rediscretization process:(b) nodes shared by the removed quadrilateral

elements and elements still present in the newdiscretization;(c) nodes on the previous crack surfaces, except thoselocated on the crack front elements;

(d) nodes located on the elements on the crack front(although the geometry of these elements remainsunaffected they are now standard elements, notspecial crack tip ones);(e) new crack front element nodes:(f) nodes belonging to the triangular elements thatare unaffected by the rediscretization process:(g) nodes belonging to the new triangular elementsintroduced to the model.

4 EXAMPLES4.1 Two equal coplanar semi-circular cracksThe first example is a prismatic bar containing twoidentical and symmetrical coplanar semi-circularcracks of radius II. The bar is subjected to a remotetensile stress (T at its ends. Its dimensions scaled to theoriginal crack radius CI are shown in Fig. 4. The initialdistance between the two adjacent cracks tips (B, andA2 in Fig. 4) is equal to 0.4~7. The following Paris lawis employed to estimate the crack growth:

with a mean load ratio R = 0.The evolution of the crack shapes is shown in Fig. 4for seven propagation increments. The crack profilesin Fig. 4 are such that the same number of loadingcycles is taken to develop from one contour to thenext. The reference number of cycles AN,, is fixed at750 cycles. Crack coalescence takes place between thefirst and second propagation increments and thetransition from two cracks to one crack was assumed

Fig. 4. A prismatic bar with two equal semicircular coplanar cracks under remote tension. Model geometry and predicted crackprofiles.


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A. P. Cisilino. M. H. Aliabadi

Initial geometry

Propagation increment #5

Propagation increment #6

Propagation increment #3 Propagation increment #7

Fig. 5. Crack discretization of the first example.

to occur when the cracks overlap. Once the crackscoalesce, the resultant crack tends rapidly to asemi-elliptical shape.Figure 5 shows the crack discretization for the initialconfiguration and the seven propagation increments.Due to dramatic changes in the crack shape during thecoalescence process it was only possible, with theexisting code, to use the automatic remeshing strategy,previously described, in a few propagation increments.The rest of them were done manually.Crack growth curves for the outer distance andmaximum depth are compared with those obtained byKishimoto et al. using finite elements in Fig. 6. Thegeneral evolution of the crack profile are generally ingood agreement but they differ for the outer distanceduring the last propagation increments.

' 1 1 ' 00 1 2 3 4 5 6 7 8N/No

Fig. 6. Crack growth curves predicted by the finite andboundary element methods.

xx

070

1.00 0 75 0 50 025 0.00 025 050 075 1 00CR% ,m e,/z

Fig. 7. Variation of the mode I stress ntensity factors alongthe crack fronts for eachpropagation increment.

The stress intensity factors AK, for the growingcracks are shown in Fig. 7. For each crack thenormalized value of the stress intensity factors areplotted as a function of the angle between thehorizontal axis and the radial line from the centre ofthe initial crack. It can be seen that the value of thestress intensity factor at adjacent crack tips increase asthe cracks approach each other. It rapidly increases atthe contact zone in the early coalescence; it stays highwhile the single crack shape is sharply concave, andfinally starts decreasing as the crack adopts a regularcrack front. The evolution of the ratio AK,,,/AK,i, isplotted in Fig. 8 together with results from Kishimotoet a1.7 This ratio also reaches its maximum valueduring coalescence, after which it starts decreasingand tends towards one.

0.50 I I I I 0 1 2 3 4 5 6 7N/No

Fig. 8. Changeof the ratio K,,,/K ,,,,,,along the crack frontduring crack development predicted by the finite andboundary element methods.


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Fatigue crack growth in welded components 141

Fig. 9. Predicted crack profiles for the second example.

4.2 Two dissimilar and coplanar semi-circular cracks

In this example it is assumed that the specimen of theprevious example has now two dissimilar coplanarcracks. The radii of the cracks are such that the radiusof the smaller (a,) is half of the larger one (a1 = 2a,),to which all dimensions of the example are referred.The initial distance between the closest crack tips (B,and A2 in Fig. 9) is half the init ial radius of the largercrack. Al l material properties and propagation lawsare the same as for the first example. Figure 9illustrates the evolution of the crack profiles for eightpropagation increments. The evolution of the mode Istress intensity factor a long the crack fronts ispresented in Fig. 10. The general behaviour in the

evolution of the crack shape and stress intensityfactors is similar to that of the first example.4.3 Two offset semi-circular paralle l cracksThe third example is a prismatic bar containing twoidentical offset semi-circular paralle l planar cracks.The dimensions of the bar as well as the relativepositions of the cracks scaled to the orig inal crackradius a are shown in Fig. 11. The bar is subjected to aremote tensile stress u at its ends. Crack growth isestimated using the same Paris law as in the previousexamples.

Figure 12 is a rear view of the specimen where someof the boundary elements on its lateral face have beenremoved to show the cracks more clearly. It illustrates

070

0 60

0 50

0401 I , I , I , I , 11 00 0 75 050 025 000 025 050 0 75 1 00

W-8, VT o2 /n

Fig. 10. Varia tion of the mode I stress intensity factorsalong the crack fronts for each propagation increment. Fig. 11. A prismatic bar with two equal semicircularout-of-plane parallel cracksunder remote tension.


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142 A. P. Cisilino, M. H. Aliahadi

\

Fig. 12. Predicted crack profiles for the third example.

the evolution of the crack shapes for 5 propagationincrements. In this example the reference number ofcycles AN,, is 3000 cycles and the propagationincrements are not constant. Also shown in Fig. 12 arethe discontinuous triangular elements introduced tothe model during the rediscretization process aftereach crack extension. The subfigure in the top righthand corner shows the crack propagation on the freesurface.The evolution of the stress intensity factorcomponents AK,, AKII and AK,,, are plotted for bothcracks in Figs 13, 14 and 15. Since the cracks nowpropagate out of plane it is no longer suitable torepresent position on the crack front as a function of theangle 19as before. In this example, the position on thecrack front is represented by the normalized distancegiven by the ratio of the distance s, measured from theA, crack tips (see Fig. 11) over the total crack frontlength C The behaviour of AK, is almost unaffected bythe presence of the second crack for the two first crack

109 0.75 0 50 025 0.00 0.25 0.50 073 1 00(f,-sp sz/t.Fig. 14. Variation of the mode II stress ntensity factorsalong the crack fronts for each propagation ncrement.profiles when the adjacent crack tips do not pass overeach other. However, this is not the case after thethird increment, since a shielding effect takes placeand AK, values dramatically decrease for the adjacenttips. In contrast to what happens to AK,, the values ofAK,, are earlier influenced by the presence of thesecond crack. Their absolute values achieve amaximum to start with and decrease after a secondcrack increment. The asymmetric evolution in thevalues of AK,,, monotonically increases throughoutthe propagation process. However, these values aresmall compared to AK, and hence not significant.

5 CONCLUSIONSA dual boundary element method (DBEM) procedurefor the numerical simulation of three-dimensionalmultiple fatigue crack growth and interaction has been

040 / I > I * I I100 0 73 0 30 023 0.00 025 050 073 100L -s, )/I s,le

mode I stress ntensity factors along the crack fronts for eachig. 13. Variation of the propagation ncrement.


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143

Fig. 15. Variation of the mode III stress ntensity factors along the crack fronts for each propagation ncrement.

presented in this paper. The DBEM single regionanalysis is shown to be particularly suitable for solvingmultiple crack problems, facilitating the constructionof the model and its remeshing after each crackextension. The localized remeshing strategy whichaccommodates the changes in original system matricesas well as the use of the displacement discontinuity inthe DBEM formulation allow important savings in thecomputing time and memory requirements. Theprocedure has been successfully employed to modelthe interaction and coalescence of coplanar andnon-coplanar surface fatigue cracks.

ACKNOWLEDGEMENTSThis first author wishes to express his thanks forfunding provided by the Consejo National deInvestigaciones Cientificas y Tecnicas de la ReptiblicaArgentina (CONICET) and the Foreign and Com-monwealth Office of the United Kingdom.

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3d dual bem_cisilino

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