8pcf_geometries

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8as JORNADAS DE FRACTURA - 2002

LIFE PREDICTION USING FINITEELEMENTS IN COMPLEX GEOMETRIES

R.A. CLUDIO*, R. BAPTISTA*, V. INFANTE**, C.M. BRANCO **

* Department of Mechanical Engineering, EST/Instituto Politcnico de SetbalCampus do IPS, Estefanilha, 2914-508 Setbal

** Department of Mechanical Engineering, Instituto Superior TcnicoAv. Rovisco Pais, 1096 Lisboa Codex

Abstract. One of the major problems in numerical determination of crack propagation parameters isthe mesh generation around the crack tip. Most finite element programs dont have, yet, tools for crackmeshing using collapsed quarter point elements. This paper reports some of the principal difficultiesthat can be found modelling 3D cracks in complex geometries. The most frequent tasks, like finiteelement program choosing, elements that should be used, the mesh generation process and nonlinearcalculus are described. Some tricks and a list of complete solutions are given for the most frequentproblems. Finally are presented references where the presented instructions were applied successfully.

Resumo. Um dos maiores problemas na determinao numrica dos parmetros da Mecnica daFractura a gerao de malha junto fenda. A maioria dos programas de elementos finitos no temcapacidade para gerao de malhas usando elementos colapsados e singulares. Este artigo fazreferncia s principais dificuldades que se encontraram em modelao 3D de fendas, em geometriascomplexas. So descritas as principais etapas a seguir, tais como a escolha do programa de elementosfinitos, elementos a usar, processo de gerao de malhas e clculo no linear, apontando algumas pistaspara a resoluo da maioria dos problemas encontrados. Finalmente so apresentadas referncias paratrabalhos onde estas instrues foram aplicadas com sucesso.

1 INTRODUCTIONThe Finite Element Method (FEM) is a verypowerful technique, frequently used in designto predict stress distributions. However,sometimes a simple stress analysis is notsufficient. Low cycle fatigue crack propagationpredictions are an essential part in the lifeassessment of critical components working inhard conditions and when a fail can causecatastrophic loses. Testing real sizecomponents or event test components insimulated conditions can be very expensiveand time consuming. Predicting operating life

can be an option, but to do that is essential todetermine fracture mechanic parameters, likefor example K. Most components in servicehave geometries where K solutions arentderived. Numerical procedures like the FEtechnique are commonly used to solve thesesolutions. One of the most difficult tasks in Kevaluation using FEM is modelling cracks ingeometries that in most situations are complex.Some commercial programs are well suited for2D were isnt difficult to introduce a crack in ashape using collapsed quarter point elements.Generating a 3D fracture model is considerably

more involved than a 2D model.

Fatigue material properties can be obtainedfrom small laboratory specimens at low cost.

This paper describes the procedures used tofind K solution for two cases. In both cases 3Dcracks were modelled. Due to the geometrycomplexity with the cracks and becausesingular collapsed elements should be used, toget better solutions, sometimes problemsappeared. The majority of these problems arehere reported, including solutions that hadbeen taken.

First of all this paper describes the 3D

elements normally used in fracture mechanicalproblems and the procedures used to modifythese elements to improve solution quality.The next step was to choose FE code. In thiscase ANSYS and ABAQUS were used and thereasons why are reported.

Mesh generation was when more time wasspent. Its presented some features used toimprove mesh quality and reduce time, like thevolume subdivision and the use of parametersto define geometry and mesh. It is alsoexplained how conversion from ANSYS to

ABAQUSwas done.


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Some problems with processing are alsoreported including how convergence in someelastic-plastic problems was obtained.

The technique used to get K parameters was toobtain J integral directly from FE post-

processor. With K solutions calculated by FEmethod, together with appropriate fatigue dataand crack propagation laws, its possible toobtain component life.

All that work resulted from two cases studiedby the authors a Gas Turbines disc and a TJoint improveded by Hammer Peening. Acomplete description and results for the twocases are published in ref. [1],[2] and[3].

2 FINITE ELEMENT PROGRAMSUSED

One thing to do when working with FE is tochoose the program. Part of the FE programsavailable arent able to calculate fracturemechanic parameters. Some programs could begood in some modules but bad in others. Thebest thing to do is to take a look for theprogram manuals and for the samples that aregiven, to verify if that program can really dowhat is wanted without too much effort. Inboth cases that were modelled it hasntpossible to find any program that was good

enough, in all modules, for fracture mechanicsanalyses. The authors only had available foruseABAQUSandANSYS.

ABAQUS is a very powerful program, which isone of the most used software by scientificcommunity. In our work group we already hadsome experience withABAQUS. ABAQUSpre-processor, for us, is the horst part of theprogram. At two years ago, when this workwas started, ABAQUS had a module calledABAQUS/Pre for pre-processing purposes.ABAQUS/Pre was a graphical interface for

problem definition that generates a text file forABAQUS/Standard module. TheABAQUS/Premodule had some tools for mesh generationaround the crack tip but this tools didnt workwell for 3D purposes, as Miranda[4] observed.Now instead of ABAQUS/Pre, the ABAQUSprogram has a new product calledABAQUS/CAE. This is a complete ABAQUSenvironment that provides interface forcreating, submitting, monitoring, andevaluating results fromABAQUS/StandardandABAQUS/Explicit simulations. ButABAQUS/CAE hasnt yet any tools for mesh

generation around crack tip, which makes thisprogram not appropriated for fracture

mechanic problems. Its also possible to writethe input file by hand without any graphicalinterface, but is a very difficult task when its acomplex geometry.ABAQUS/Standard is a module that reads aninput file, prepares the problem for solving and

solves the equations. This module has alsosome post-processor capacities includingsolving the most common fracture mechanicsparameters.With ABAQUS is possible to solve theproblem, but without a good pre-processor isdifficult to model a complex geometry with acrack. Then, it was necessary to search foranother program.The next step was to verify what ANSYS coulddo. ANSYS is able to calculate FractureMechanical parameters like stress intensityfactors, the J-integral, and the energy release

rate. In meshing capacities was found thatANSYS have only tools for mesh generationaround crack tip in 2D geometries. After, sometests made, it was verified that if a volume wasgenerated from a 2D mesh in one area andusing the sweep function, its possible toobtain a 3D crack tip mesh. After that wasverified that ANSYS has a very good graphicalinterface with powerful tools for meshgeneration. With ANSYS its also possibleparameterise the problem, very important whenthe crack size has to be updated.When the authors tried to solve the problem, a

lot of errors with the mesh appeared withoutapparent solution.The option taken was to model the problemusing ANSYS, then exporting the data toABAQUS and calculating the solution withABAQUS.

For summarise, the FE programs used were:- Pre-processing, ANSYS 5.5 Due to the

complex geometry, ANSYS code was usedinstead ofABAQUS. Some programs had tobe made to convert node and element datato theABAQUScode.

- Processing, ABAQUS 5.8.14 code, with theSTANDARD module.

- Post-processing, ABAQUS 5.8.14 code,with thePOST module.

3 ELEMENTS USED

3.1 Isoparametric Singular ElementsThe isoparametric singular elements are

obtained from the classical 2D 8 node and 3D20 node isoparametric elements, positioningthe intermediate nodes near to the crack at


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quarter-point positions. Special crack tipelements are unnecessary. Crack tip elementscan be obtained from standard isoparametricelements, changing the node positions toquarter-point[5].

Henshell et al[5] and Barsoum[6] found thatin an 8 node quadratic isoparametric elementthe strain field becomes singular at the cornernode if the mid-side nodes are placed at thequarter points of the sides emanating from thiscorner node.

The change in the positioning of the mid-sidenodes in isoparametric elements for achievingthe desired singularity was extended to 3D.The region of square root singularity behaviourexists in every cross sectional plane orthogonalto the crack edge.

3.2 Collapsed ElementsThe collapse of one side in the 8 node 2Delement generates, at the collapsed side, thestrain needed. In the 20 node 3D brick element,the singularity is obtained at the crack frontwhen a face is collapsed. Once the cornernodes have been positioned, only the mid-sidenode on crack front doesnt have its positionfixed.

In the majority of the problems, the mostefficient mesh design at the crack tip hasproven to be the spider web configuration(fig. 1), which consists of concentric rings ofelements that are focused toward the crackfront [7]. The inner-most ring of elements aredegenerated to triangle prisms. The spider webdesign facilities a smooth transition from a finemesh at the tip to a coarser mesh remote fromit. Elastic analysis of K can be accomplishedwith relatively coarse meshes, since modernmethods eliminate the need to solve local cracktip fields accurately.

Figure 1 Spider Web configuration.

In the linear elastic case, G can be expressed asthe scalar product of the J integral vector and a

vector in the plane of the crack and normal tothe crack front [8]. So, for a 2D plane straintensile crack, with the crack advancing

uniformly through the thickness in its plane, Jand G are equivalent[9], so K can be obtainedfrom J using equation (1).

21 =

EGK (1)

A universal optimum size does not exist and itis difficult to give general guidelines for properuse of singular elements for all cases of crackproblems. But, using appropriate transitionelement sizes, a universal optimum size lieswithin a 15-25% crack length for a 5% errorbound. A convergence study is recommendedto identify the lower bound of the error, whenaccurate results are searched. If the results arebeing used to calculate the stress intensityfactor and if the method for calculating thisparameter is being on the far-field FE solution,the results can be insensitive to the singular

element size.

Murti et al[10] concluded that the maximumspan angle for each singular triangle is /2 andhence acute singular elements should be used.So, it is required that the minimum number ofsingular elements enclosing the crack tip isfive (preferably six).

The non-uniformity of the singular elementscan be measured by the ratio of maximum tominimum side length. This ratio should be asclose as possible to unity for best results [10].

This behaviour is partly due to inherentproperties of the isoparametric finite elements(FE) itself. With large aspect ratio anddistortion, there is a considerable increase inthe degree of non-linearity of isoparametricmapping, which leads to a decrease inaccuracy. More importantly, the radialvariation within the element is a function ofelement size. Thus, when non-uniform singularelements are used, the local radial variationfrom one element to another varies, leading toan improper modelling of crack-tip behaviour,causing a decrease in accuracy [10].

4 MESH GENERATIONMesh generation is one of the most difficulttasks when modelling a FE problem. Infracture mechanics problems, the mostimportant region to model is around the edgeof the crack. Generating a 3-D fracture modelis considerably more involved than a 2-Dmodel, especially when the geometry is notregular.

The use of hexahedral elements instead oftetrahedral provides more accurate solutions


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[11] and [12], but was consumed more timeand promote more problems in meshgeneration. With tetrahedral elements the meshgeneration in ANSY S is totally automatic,including in irregular volumes. The use ofhexahedral elements requires regular volumes

and the mesh generation must be done withsimpler functions like extrude and sweep. In allsimulations made hexahedral elements werealways used.

The first step was to define the geometry. Thesoftware used was the MECHANICALDESKTOP (a CAD program), because is aparametric based program with superior 3Dfunctions for geometry definition. Thisprogram was also chosen because is possible tocan be use the explode function. With thisAUTOCAD function its possible to change

from, parametric to nom parametric, volumesto surfaces and surfaces to lines easily. This isvery important, if a volume has to besubdivided in small regular volumes or tomade some repairs.

Geometry subdivision is a very important taskif the objective is to generate good qualitymeshes. To use hexahedral elements inANSYSthe region has to be divided in regular volumes(fig. 2).

Figure 2 Subdivisions made to help thegeneration of regular meshes.

Regular volumes are those witch can begenerated using a base area that is extruded or

sweeped. A regular volume has only 6 areas(fig. 3). The source and final area must have

the same topology and the volume cant haveany discontinuities in the middle. As saidbefore were used some typical AUTOCADfunctions to subdivide the exploded geometryfrom MECHANICAL DESKTOP. To exportthe geometry for ANSYS, for crack definition

and mesh purposes, the IGES standardinterface was used.

Figure 3 The sweep process in mesh

generation.

Careful must be taken dividing the volume;sometimes small volumes needed to be creatednear larger ones resulting in meshes toodistorted.

The cracks were defined parametrically, inANSYS, as function of crack depth and halfcrack length at the surface, a andc. Parametricmodelling is available in most common FEcodes. In the cases studied a base model wasalready meshed without the crack tip region

defined. When the problem was loaded inANSYS two dialogs appear to enter the cracksize. With crack size defined, the geometry andmesh were updated for the size entered. Theparameterisation was one of the most timeconsumption in all process.

Some problems appeared when meshing withcrack tip elements. ANSYS generated well thecollapsed quarter point elements in areas, butwhen the sweep function was used, the nodes changed gradually to position untilthe mesh reached the target area. To correct

this problem a macro was made, with ANSYSfunctions, to change the nodes that were inwrong position to position.

The elements, in cracks with a circular frontshape, have the same appearance along thefront. But when modelling cracks withelliptical shape the elements along the fronthave different aspect ratios resulting in moredistorted meshes, making the a/c relation alimitation for crack generation process.

Another thing that users should be carefulabout is with the total number degrees offreedom (DOF). With the available hardware it


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was only possible to solve problems with lessthan 200.000 DOF. The hardware used was aPC, Pentium III 500MHz processor with320MB of RAM and 10GB. Our limitation insolving large problems was only the size of thedisk.

For the most refined meshes, submodellingwas carried out in the area near the crack,using the medium mesh solution to establishthe boundary conditions (fig. 4).

Figure 4 Sub-modelling technique

Another problem that sometimes appears inmesh generation is when the crack sizebecomes near an edge (fig. 5). In that case itsimpossible to do the mesh. The solution that

was taken was to extrapolate the K results forthe crack sizes that couldnt be done.

Figure 5 A large centre crack near anedge.

5 EXPORTING DATA FROM ANSYSTO ABAQUS

As said before ABAQUS program was used tosolve the problem and ANSYS to meshed thegeometry. One additional task was to do aprogram to export data from ANSYS toABAQUS because ANSYS doesnt have anyexport function. What was done was

something similar to what the old PRE-ABAQUS program did, that was writing a nodeand element file and then adding someinformation like loads, boundary conditions,material properties and some commands fordata requesting. This operation consists in

building the INP (input) file. A program wasmade in VISUAL FORTRAN to change the listformat, of nodes and elements, to ABAQUSinput file format. The nodes and elements areused for mesh definition.

6 NONLINEAR SOLUTI ONSThere are three basic notions to regard whenexecuting a FE analysis:

Analysis Step: the basic division ofanalysis, consisting on a specific loading

situation, boundary conditions, type ofanalysis and output variables. A fullycompleted analysis is composed ofseveral steps, which are executed by theprocessor following a previous definedorder.

Step Increment: if necessary each step isdivided into several increments. Whenexecuting a nonlinear analysis theprocessor has to follow a law, whichimplies the total load division intosmaller portions, in order to simulate thenonlinear behaviour. The initial value of

the step increment may be suggested bythe user, or calculated by the processor,but influences tremendously the finalresult.

Iteration: it is an attempt to obtain asolution for the equilibrium equations, ineach increment. When the processordoesnt reach the equilibrium for a givenincrement in a single iteration, then itbegins a new iteration until theequilibrium is found. If the equilibrium isnot obtained in a predefined number ofiterations, the size is reduced to the

original increment.

6.1 Iterations and convergence of anincrement

In a nonlinear analysis the structure reply canbe represented in the fig. 6. FE processorattempts to solve each increment using the

initial structure stiffnessdu

dPK =0 , based upon

the structure initial configuration u0 andupdated load increment P, to calculate an

initial displacement correction .0uuc aa =


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Figure 6- Nonlinear load-displacementcurve

Based on these results the processor candetermine if the solution has reached theequilibrium. In order to do this a new structurestiffness is calculated, based on the newconfiguration ua, which allows it to obtain theinternal forces for this iteration Ia. Thedifference between the total applied load andthese internal forces are named as the residualforce (fig. 7):

aa IPR = (2)

IfRa is null at every degree of freedom in themodel, point a in the next figure would lie onthe load-deflection curve, and the structurewould be in equilibrium.

Figure 7 First iteration in an increment.

In a nonlinear problem this value is alwaysdifferent of zero, therefore the processor as tocompare with a defined tolerance in a way toverify the iteration equilibrium. By definitionthis tolerance is 0.5 % of the averaged force

applied to all the structure DOF along the time.The processor throughout the simulationcomputes this average. If the residual force isless then the current tolerance value, theprocessor accepts ua as an equilibriumconfiguration, but not before verifying theincrement convergence. In order to do this theprocessor calculates the incrementaldisplacement value:

0uuu aa = (3)

and verifies ifca is not superior to 1 % of this

value. I f so, the processor executes another

iteration, repeating the process until thenecessary convergence is obtained.

Figure 8 Second iteration.

On the second iteration the processor uses anew structure stiffness, already calculated, andthe same load increment value, in order to

calculate a new configuration ub. Thedetermination of the new value for the residualforce Rb, a new displacement correction

abb uuc = and a new incremental

displacement 0uuu bb = is allowed (fig.

8). All these values are evaluated in order toverify the iteration equilibrium and incrementconvergence. The process is repeated untilthese two are verified. This way, in eachiteration is necessary to form a model stiffnessmatrix and to solve the equilibrium equations.Each iteration is equivalent, in computational

effort, to conduct a complete linear analysis,which is solved in a single increment anditeration. The computational effort to solve anonlinear problem is several times superiorthan for analysing a linear one, without theguarantee of the solution convergence.

The initial increment value of each step isessential. For ABAQUS if this value is notgiven to the processor, itll begin from thelargest possible reducing the increment valueto 25 %. If after 16 iterations the solutionhavent reach the equilibrium, this process is

repeated until the convergence is obtained.From this point on, the processor will adaptthis increment in the running simulation. Beingso all initial effort of searching the bestincrement value may be speared if the usersuggests a good estimation.

7 POST PROCESSINGNear the free boundaries, when K solution isderived from FE, the solution isnt good. Thishappens because the stress field singularity

changes from 1/ r to , where is aparameter that can change from 1 to 0.5,

K.r


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depending on the Possion coefficient and anglebetween crack and surface [13], [14], [15] e[16]. Since the elements are only well suited

for 1/ r singularity some errors appearbecause the singularity is different. When theangle between the crack and surface is 90 (themost common situation) Benthen [17] foundan analytical solution for the singularity. In ref.

[15] and [16] its possible to find solutions

for some angles when . Someinvestigators solve this problem using finemeshes and extrapolating the solution frominside the body to the surface[16] and[18].

0.3 =

8 VALIDATION OF RESULTSSometimes it s possible to find in literature K

solutions for geometries and load case similarsto the situation that is being studied. One of themost complete references in this area isMurakami [19] and [20] and the PD6493document [21]. The most frequent situation isthat there is nothing in literature similar to thecase that is being studied. So directcomparation cant be made. The best thing todo is to be cautious generating the mesh,applying the right boundary conditions andanalyse carefully the finite element solution ina post-processor.

A very good indicator of mesh quality is Jintegral. J integral is an energy parameter,independent from contour to contour. In thefinite element codes, J integral is a quantitythat can be computed along different contours.If the result is not the same in differentscontours this is an indicator that mesh qualityis not good. This indicates a need for meshrefinement near the crack tip. Numerical testssuggest that the estimate from the first ring ofelements abutting the crack front does notprovide a high accuracy result, so at least twocontours are recommended. As well contours

far from the crack tip normally giveunsatisfactory results.

Only convergence in different contours is notsufficient to guarantee that the solution isgood. The correct thing to do, is test alsodifferent meshes to ensure that the solutionremains constant.

Another way to check solution is usingPommier solutions [22]. These equations applyto engineering problems, and it is possible toobtain K knowing only the stress field near the

crack zone. They are very useful to use with afinite element post-processor code. These

equations have some limitations because theycan only apply to semi-elliptical crack in semi-infinite bodies. The stress field must beinterpolated by a 3rd degree polynomial. In [2]and [1] is published a case where the Pommierequations were used to check K solutions.

9 THE STATE OF ARTEach time a new version of a finite elementprogram comes to market new things appearrelated with fracture mechanics parametersevaluation. So probably soon are ableprograms to mesh in an automatic waygeometries with crack tip elements.

There are some programs like ZENCRACK(the most known) that are able to model

automatically cracks in general shapes. Thesesprograms operate on an existing finite elementmesh and introduce the crack by replacingsome of the existing brick elements by a crackblock. The only thing that the user has to do isto mesh the geometry without any crack. Thecrack shape can be fully arbitrary and can beused in mixed mode propagation. Since thecrack front can be automatically updated,applying a propagation law with a simplealgorithm, is possible to predict componentlifetime and crack geometry. Dhondt [23] and[24] presents some works where he uses this

methodology successfully.

10 CONCLUSIONS1. Finite element analysis is a powerful

method to get fracture mechanicparameters. Can be used to replace part ofthe expensive tests in real components byfatigue tests in small laboratory test pieces.

2. Mesh generation still continues one of themost difficult tasks to perform calculus

with finite elements when a crack has to besimulated. This is because finite elementprograms arent yet well suited to generatecrack tip elements.

3. Parameterisation of the crack front size isrecommended. Its difficult to do but as thefirst crack is modelled the next ones aremuch easier to do. This is because thecrack size must be updated a lot of times toget a reasonable curveKvsa.

4. Determining the initial value for the loadincrement is fundamental when performinga nonlinear simulation. This value has a


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large influence in the final results and alsomakes a difference in the simulationrunning time. Therefore using the correctvalue provides good results with lowcomputational effort.

5. Programs like ZENCRACK seams to beone of the most practical alternatives thatexist today for meshing cracks in complexgeometries.

REFERENCES

[1] Cludio, R. A.., "Previso de vidafadiga/fluncia em discos de turbina deturborreactores", MSc thesis, TechnicalUniversity of Lisbon, Instituto Superior

Tcnico, April 2001. Also available inelectronic version athttp://ltodi.est.ips.pt/rclaudio/ (inPortuguese).

[2] Claudio, R., Branco, C. M., Gomes, E.,Byrne, J ., Life Prediction of a GasTurbine Disc Using the Finite ElementMethod, 8th Portuguese Conference onFracture, Vila Real, 20-22 February 2002.

[3] Infante, V., Branco, C. M., Baptista, R.,Gomes, E., Fracture Mechanics Analysis

of Welded Joints Repaired by HammerPeening, 8th Portuguese Conference onFracture, Vila Real, 20-22 February 2002.

[4] Miranda, Rui M. A.; Anlise de Fissurasde Canto pelo Mtodo dos ElementosFinitos Linear Elstico e Elastoplstico;Tese de Mestrado, Instituto SuperiorTcnico, Universidade Tcnica deLisboa, Dezembro de 1996.

[5] Henshell, R.D., Shaw, K.G., Crack TipFinite Elements are Unnecessary, Int.Journal for Num. Methods in Eng., Vol.9, pp. 495-507, 1975.

[6] Barsoum, R.S., On the Use ofIsoparametric Finite Elements in LinearFracture Mechanics,. Int. Journal forNum. Methods in Eng., Vol. 10, pp. 25-37, 1976.

[7] Antunes, Fernando J. V., Propagao deFendas por Fadiga a Alta Temperaturaem Inconel 718. Tese de Doutoramento,

Faculdade de Cincias e Tecnologia daUniversidade de Coimbra, 1999. (inPortuguese)

[8] DeLorenzi, H.G., Energy Release RateCalculations by the Finite ElementMethod, Eng. Fracture Mechanics, Vol.21, n 1, pp. 129-143, 1985.

[9] Li, F.Z., Shih, C.F., Needleman, , AComparison of Methods for CalculatingEnergy Release Rates, Eng. FractureMechanics, Vol. 21, n 2, pp. 405-421,1985.

[10]Murti, V., Valliappan, S., A UniversalOptimum Quarter Point Element, Eng.Fracture Mechanics, Vol. 25, n 2, pp.237-258, 1986.

[11]ABAQUS/Standard Users Manual, VolI, II e III; ABAQUS/Theory Manual;ABAQUS/Standard Example Problems

Manual, Vol I e II, ABAQUS/StandardVerification Manual; ABAQUS/PostUsers Manual, Version 5.8; Hibbitt,Karlsson & Sorensen, Inc; 1999.

[12]ANSYS manuals, Version 5.5, September1998.

[13]Dhondt, G., Analysis of the BoundaryLayer at the Free Surface of a HalfCircular Crack, Eng. Fracture Mech.,Vol. 60, n 3, pp 273-290, 1998.

[14]Fonte, Manuel A., Anlise dapropagao de fendas semi-elpticas emveios sob flexo e toro, Tese deDoutoramento em Eng. Mecnica,Universidade Tcnica de Lisboa, InstitutoSuperior Tcnico, 1997.

[15]Pook, L. P.; Crack Profiles and CornerPoint Singularities; Fatigue FractureEng. Mater. Struct, 23, pp.141-150,Setembro 1999.

[16]Bazant, Z. P.; Estenssoro, L. F.; SurfaceSingularity and Crack Propagation, Int.J. Solids and Structures, vol. 15, pp.1411-1426, 1979.

[17]Benthen, J . P., State of Stress at theVertex of a Quarter-Infinite Crack in aHalf-Space, Int. J. Solid and Structures,Vol. 13, pp. 479-492, 1977.

[18]Carpinteri, A., Elliptical-Arc SurfaceCracks in Round Bars, Fatigue Eng.Mater. Struc. Vol. 15 n 11, pp. 1141-

1153, 1992.
http://ltodi.est.ips.pt/rclaudio/http://ltodi.est.ips.pt/rclaudio/


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[19]Murakami, Y .; Stress IntensityHandbook, Vols. 1 e 2, Pergamon Press,Oxford, 1987.

[20]Murakami, Y .; Stress IntensityHandbook, Vol. 3, The Society of

Materials Science, Japan, 1992.

[21]PD6493 Document. British Standard BS7910. Guide on Methods for Assessingthe Acceptability of Flaws in Structures,1997.

[22]Pommier, S.; Sakae, C.; Murakami, Y .;An Empirical Stress Intensity Factor Setof Equations For a Semi-Elliptical Crack

in a Semi-Infinite Body Subjected to aPolynomial Stress Distribution; Int. J.of Fatigue, Vol. 21, pp. 243-251, 1999.

[23]Dhondt, G.; Automatic Three-Dimensional Cyclic Carck Propagation

Predictions with Finite Elements at theDesign Stage of an Aircraft Engine,NATO RTO-MP-8, May 1998.

[24]Dhondt, G., Automatic 3-D mode Icrack propagation calculations with finiteelements, Int. J. Numer. Meth. Engng.,vol. 41, 739-757. 1998.

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