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A Bell&Hc:weI tntormabcnCompany 3XlNortrIZeebRoao.AttnAlbor.Ml &8151346lJSA

313.'761-4700 eoo:521..o&Xl

HIERARCmCAL FINITE ELEMENTS FOR STRESS CONCENTRAnON AND STRESS SINGULARITY PROBLEMS

by CHarvindel' S. Sidhu, B.E.

Athesissubmitted totheSchool01 Graduate Studiesinpartialfulfilmeot of the

requirm1ents for thedegreeof Master ofEngineering

Faculty of Engineering and AppUed Science Memorial University of Newfoundland

August, 1995

St.Jobo's NewfoundlaDd Canada

.+.

Acquisitionsand Bibliographic 5ervices 395W~Slrwl oa-aON K1AllN4

c..-

Biblothequenationale duCanada Acquisitionset S8mces bibliographiques 385."WIIIingIcn oaa-ClN K1AllN4

.,...

The author has granted a non- exclusive licence allowing the National LIbrary of Canada to reproduce.loan,.distnbute or sell copies of this thesis in microfor:m, paper or electronic formats.

The author retains ownetS!rip of the copyrightinthis thesis. Neither the thesisnorsubstantialextracts fromit may beprintedor otherwise reproduced without the author'5 permission.

L'autem a accordenne licence Don exclusive permettant

a

la Bibliotheque nationale du Canada de reproduire. peeter, distnbuer au vendre des copiesde cettethesesous la forme de microfiche/film, de reproduction sur papier au sur format electronique.

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these.

Nila thesenides extraits sohstaotiels de ceUe-<:i ne doivent

etre impnmes

au autremcnt reproduits sans son autorisation.

0-£12-25885-8

To myparents and Anjana

ABSTRACT

The p-version finite elemenr:method.offers a distinct advantage of savings in computational timeandeffonincomparison with lIle conventional. h-version finite element method. The h-version uses low order e1emenlS and convergence studies are doneby successive refinements ofthemesh that imply analyzingtheproblem afresh.In contrast, the: p-version uses a fixed discretization ofthedomainandhigher order elements are successively employed during convergence studies. Thecomputational advantage resultS fromtheusc of hierarchical shape fu.octions. coarsemeshesandfaster raleS of convergence withdecreasednumberof degrees of freedom. Consequently.the use of hierarchical finite elements can be especially advantageous for stress concemration andstress singularity problems that require very refined meshesinlhe h-version.

TIleaccuracy of hierarchical finite elememsis demonstrated. withtheuse of coarse meshesfor beamandT-plate weld joint problems. A 20Menrichedhierarchical-[mite elementisdeveloped for^stressintensity factor evaluationlhatembodiestheinverse square rootstresssingularity by iocludiDginits formulationthestress intensity factors as additional degrees of freedom. Stress intensity factors for cracked specimens are numerically evaluated quite accurately using very coarse meshes involving fewer degrees of freedomincomparison with conventional analyses. This conceptisthen extended(0

3D crack: problems and favourable results are obtained.

iii

ACKNOWLEDGEMENTS

I would like⁽⁰express my sincere thanks£0Dr. K. Munaswamy. my research supervisor, for his invaluable guidance, encouragement and financial support during the pursuitof my Master of Engineering degree.

1am.thankful toDr. R. Seshadri. Dean. Faculty of EngineeringandApplied Science for his continued interestandadviceduringmygraduateprogram.Igr.ltcfully acknowledge the valuable help received fromDr. 1.1. Sharp, Associate Dean. Faculty of Engineering andApplied Science for making my stayatMemorial a pleasant one.

Iam alsogratefultotheSchool of Graduate Studiesand theFaculty ofEngineeringand AppliedScience for providing financial assis£aDCe wil.hout which this work. would not have bttn possible.

Profound thankstomy dear wife Anjana for ber love, understaDdingandconstant moral support.

iv

CONTENTS

ABSTRACT ACKNOWLEDGEMENrS LIST OF FIGURES LISTOF TABLES

SYMBOLS AND ABBREVIATIONS 1 INTRODUCTION

1.1 Finite Elemem Method 1.2p-version FEM

1.3 Compumtional Fracture Mechanics 1.4 Objectives of the Thesis 1.S Layout oftheThesis

2 BACKGROUND AND SCOPE OF WORK 2.1 Literawre Review

2.1.1p-versionFEM

2.1.2 Compuwioaalfracturemechanics 2.2Scope ofthe SlUdy

2.2.1 Formulation of 20 P-elem.ent 2.2.2 Formulation of 2D EH-elemem 2.2.3 Formulation of 3D P-&EH-elements 2.2.4 Solution stta[C:gy

2.2.5 Numerical verificationaDdresults

iii iv viii

15 19 19 20 20 20 21

3 FINITE ELEMENT FORMULATION 3.1 GeoeralizedFE Formulation 3.2p-versionFE Formulation 3.3 20 Ell-Element 3.4 3Dp.&EH-Elements

3.4.1 Enriched 3D P-element 4 COMPUTER. IMPLEMENTATION

4.1 Computer Program 4.1.1 Input

4.1.2 Elemental stiffness matrix evaluation 4.1.3 Assemblyandsolution of global stiffness matrix 4.2 CompulationalEase

S NUMERICAL STUDIES AND DISCUSSIONS 5.1 Numerical Application

5.2 Tests for 20 P-Element 5.2.1 Soon cantilever beam. problem 5.2.2 Cook's membrane problem.

5.3 Tests for 20 EH-Element

5.3.1 Double edge cracked tensile specimen 5.3.2 Slam cracked teosiJe specimen 5.4 Analysis ofT-plate Weld Joint

5.4.1 Stress coocenb'1.tion factors . 5.4.2 Stress intensity factors 5.5 3D Ell-Element

5.5.1 Double edge cracked specimen 5.5.2 Compact tension fracture specimen 5.5.3 Semi-circular surface crackina plate

vi

Z2 22 21 31 35 31 43 43 43 44 44 45 48 48 49 49 52 55 55 51 59 60 10 14 14 16 19

6 CONCLUSIONS AND RECOMMENDATIONS 6.1 Conclusions

6.1.1 P-elemeDlS 6.l.2 EH-elemeots 6.2 Recommendations REFERENCES APPENDIX A

vii

83 83 84 84 86 87

90

LIST OF FIGURES

Figure Page

3.1 Quadrilateralmasterelement 22

3.2 Coordinatesystem. at cracktip 31

3.3 3Dmasterelement 35

3.4 Coordinate system atlhecrack front (30) 3g

5.1 Cantileverbeamproblem 49

5.2 Meshes for cantileverbeam 50

5.3 Cook's membrane problem 52

5.4 MeshesforCook's membrane 53

5.5 Double edge cracked tensile specimen 55

5.6 MeshesandresWlSfor double edge cracked specimen 56 5.7 Mesh of distoned serendipity elements (Barsoum) 56

5.8 Slaol cracked tensile specimen 57

5.9 Meshesandresults for slam cracked tensilespec~n 58

5.10 Gromeoy ofT·plale weld joint 59

5.1l Meshes foc SCF evaluation 61

5.12 StressdistributiODthroughthickness 62

t=78mm. tIT-I.O, 45° weld profile. bending

5.13 Stress distribution through thicmess 63

t=78mm, tIT=1.0, AWS weld profile.bending

5.14 Stress dislribution through thickness 64

t=78mm.tIT=1.0. 45° weld profLle. tension

viii

5.15 Sttess distributionthroughthickness 65 t=78mm..lfI'=l.O, AWS weld profile. lension

5.16 Sttess distribution lhrough thickness 66

t-78mm. lfI'=O.75. 45° weld profile. beading

5.17 Scress dismbution lbrough lbickness 61

t=78mm. tIT=0.75. AWS weld profile. bending

5.18 Sttess distributionlbroughlbiclmess 68

t=78mm. tIT=O.7's. 45° weld profile. lension

5.19 Stress distributionthroughthickness 69

t=78mm. tIT=O.7,S. AWS weld profile. lension

5.20 Meshes for SIF for 45° weld profile. tIT=I.O 11

5.21 SIF vs crack: depth 12

5.22 Fatigue crack:growth(Bell) 13

5.23 Fatigue crackgrowthusing EH-elements 13

5.24 Mesbes for 3D double edge cracked specimen 15

5.25 Compact tension fracture specimen 11

5.26 Meshes for compact tension fracture specimen 18

5.27 Semi-<:ircular crackina plate 80

5.28 Meshes for semi-<:in:ular crack 81

5.29 SIF results alongthecr.tCk front 82

ix

LIST OF TABLES

Table Page

5.1 Cantilever beam problem (reference values) SO

5.2 Cantilever beam problem(resu.lts) 51

5.3 Cook's membrane problem (referellCe values) 52

5A Cook's membrane problem(results) 53

5.5a SCF.4S0weld profile.bendingload 60

5.5b SCF, AWS weld profile. bending load 60

5.6a SCF. 45° weld profLIe, tensile load. 61

5.6b SCF, AWS weld proftle, tensile load 61

5.7 SIFvscrack depth 71

5.8 KlKwfor double edge cracked specimen. 4 elemem mesh 75 5.9

K/Kw

for double edge cracked specimen. 24 element mesh _ 76 5.10 KIK:!I)for compact tension specimen.4 elements

n

5.11 KJK10for compact teasion specimen. 32 elements 78

<>

{}

[I II' G"

[nJ 1("K",K", w.r.t.

20,30 AWS CPU OOF EH- FE FEM LEFM MUNSID NDOF p- RI SCF SIF

SYMBOLS AND ABBREVIATIONS

Row vector Column vector Matrix

Shapefunction of orderp Geomeaic shape function of orderp Reference numberf1

MooeI,Mode II, Mode ITStreSSintensity factors withrespect to

Two dimensional.three dimensional American Welding Society Central Processing Unit Degrees of Freedom Enriched Hierarchical- Finite Element Finite Element Melh<X1 LiJleatElastic Fracture Mecbanics Finite e1emem analysisprogramdeveloped Number of DOF

Hierarchical- Reduced Integration Slress Concentration Factor StreSS Intensity Factor

xi

Chapter I INTRODUCTION

1.1 Finite Element Method

Finite element melhod isthe most popular and widely used teChnique forthe computational analysis of engineering problems. The procedures involvedinthefinite elementmethod are : The formulation oftheproblemindisplacemeDl:. variational or weighted residual form.thefinite element discretization and.theeffective solution ofthe resulting system of equations. 1bese basic steps arethesame foralmostall kindsof problems andthefinalresultisa complete numerical process implemented onthe digital computer. 1becomputational timeandeffortisinvolvedinthefollowing :The formulationandthenumerical integration ofthefIniteelemelll manices, the assemblage ofthe element matricesintothecomplete fmi(e elemem system matricesand the numerical solution ofthesystem equilibrium equatiODS.

Finite elements are employed to represent, depending on thetypeof problem. beam.

lruSS members. regions ofpla.ne stress, plane strain. andaxisymmetric. three- dimensional.plateor shell behaviour. A suitable division oftheproblem domainis

essentialtoany analysis.inaddition. it becomes important for the analyst to refine[he meshof elementsinsome suitable mannertoenhance anddetermine

me

accuracy ofthe analysisdlroughconvergence stUdies. For

me

analysis of complex problems. where complexityisdueto

me

geometty orstressraiserslike streSS concentration regionsand cracks.a detailed meshisrequired. Thisintroducesa large number ofunJcnown variables which increases the computational effoItrequiredinthemunerical process.

The procedureisrepeated by remes.hing andreanajyLiu~theproblem for convergence studies.Complex systemSalsoinvolve considerable human effort for design. verification andrefinementsofthemesh. Theoverall time and effortrequiredfor a computer analysis increasesinproportion to tile number of unknown variables and complexity of therequiredmesh. For any analysis it is desirable to reduceboththebumanandthe computer timeandeffoIt astheseresources are always limited.

Convergence studies. wbicb are essentialtoany analysis. can be undertakenintwO ways.

'Theconventional method isthesub-division of the mesbintoa finer mesh, wbich decreasesthesize and increasesthenumber of finite elemenB that represemthedomain.

especially at areas of^stresscOncentratiODS and streSS singularities. Thisiscalled h·

r¢n.emenJ.or simply~hreji.noun.c.wberehrefers rothesize oftheelementandthe methodis referred to as the h·version finite element method (FEM). Thesecond uncommon but powerful methodisto increase the polynomial order of interpolation functions for the unknown variables associated with the elements. selectively or over the entire domain.This is Icnown as p-rejinement or simply tnl!sh enn·chment. where prefers to the order oftheelement and the analysis is said toemploy p-version FEM.

Convergenceissaid to be achieved ash ...0 (h-convergence) or asp ...CXl (p- convergence),

1.2 p-version FEM

Inp-version FEM.p.the order of the element impliestheorder of the interpolation function fortheunknownvariable overtheelemental domain. Mesh enrichmenl can furtherbeperformed in twO ways. viz., i)withelementsthatuse regular (noD- hierarchical) interpolation fiJ.octioQS of higber order,anditlwithelements that use higher order"hierarchical~interpolation functions. Hierarcbical implies that the lower order interpolation functions form. a subset oftheinterpolation functiODS of higher orders. The use of higber order elements withIlOn~hierarchicinterpolation functions does DOt reduce thecomputational time and effort appreciably as additional data is required for the new unkoownsor degrees of freedomand theentire system of higher order matricesneeds tobere-<:omputedandre-solved. Withtheuse of hierarchical elements onlythemauiees that correspond to the·additiona.l~degrees of freedomDeedtobeevaluatedand assembled, thus reducingthecomputational efron.Inaddition. less effoltisrequiredin thesolution stage during convergence studies as lower order system maaices form. a subset of higber order system mauices. Theuse of p-version FEM results in fewer degrees of freedom for an analysis due to improved convergeoce rates that varyina geometric progression compared with algebraic progression for h-version.Inthiswork.

p-version referstotheuse of hierarchical finite elements unless otherwise mentioned.

[ :

Hierarchical elements thus employ hierarcbical basis or interpolation functions forthe unknowns.The:advantages of reduction in[becomputational effortand timeare dueto thefoUowing: ConsttUCtion of coarsemeshessufficient to modelthegeometry of

me

problem; lower order matricesneednotbere-evaluated. re-assembledanduiangula.rized as theyremainunchanged duetothehierarchical naru.reandformpanofthehigher order maaices during convergeoce srodies. Also remesbiog can be avoidedtoa large extent during convergence studies by selecting a suitable mesh fortheinitialanalysisand increasingtheorder oftheelements keepingthesize ftxed. Dueto hieran:hical forms theresulting matrices have improved conditioning that allows effective iterative solution techniques tobeadopted-The:p-venlon FEMalsooffers ease of error estimation which incotpOrated suitablyintothenumerical analysis can leadtoself adaptive selective mesh enrichment.

Special problems of interest thatfux1immense use ofthe ftniteelement method are those that involvestresscoocentrations due to geometryandstress singularities due to cracks.

etc. For such problems h-version FEM commonly employs a detailed mesh of quadratic serendipity elements introducing a large number ofdegreesof freedom. The p-version FEM can thereforebeeffectively employed to reduce the required effort significantly.

Forstresscoocentration problems. a detailedmeshofstandardserendipity elements over the domainand.a very detailed mesh inthestress concentration region is replaced by a coarse mesh of hierarchical elements with a slight ref"mement atthestress concentration region. The convergence ofthesolution is then achieved simply by, perhaps selective.

mesh enrichment. Stress singularity problems however are DOt so easytodeal with.

, 4

1.3 Computational Fracture Mechanics

Common problems ofstresssingularity behaviour are those comainingDmcbes. cracks.

sharpchangesinthegeometry asinan L-shaped domain. etc. FractW'e mechanics deals withsuch problemsanditsimponanceintheanalysis of scrucru.res with cracks bas increasedmanifold over the recent years.Inlinearelastic stress analysis the stresses and deflectionsinthe cracktipregion depend on a single parameter. known as the stress intensity factor(Sm, K.Themagnitude ofK depends00the:strueruralgeometry and theloading system.IfK exceedsthefracnue toughness ofthematerial crack: instability occurs. Thus evaluation of K for any crack eonfigwatiODisaUthatisrequiredto assess whether failurewilloc:curorDOt. Inmostinstances. closed form solutions are DOt available and K must bede[erminednumerically. Special finite elements I'hat exhibit inve~square root stress singularityatme cracktipimprove the accuracy and reduce theneedfor a high degree of mesb refinement at the cracktip. A lot of effortbasgOne in10thedevelopment of special crack: tip elemelUS todealwithtbesingularitiesinh- ve~iODFEM. Theseelementsstill~a large amount of computational effortand.

theevaluation ofstressinteosity (actors for complex geometries. especiallyinthree dimensions.isquitecumbersome. 'Thep-version FEM can reducetheeffort and therefore can be applied effectivelytofracture mechanics problems wilh development of an efficient crack tip elementinsome suitable fashion.

1.4 Objectives of the Thesis

A lotof reseatth advocatingtheadvanr.ages of p-versioo FEM has been published.

however00commercial p-venionFEsoftwareisyet available. For fracOJ.re mechanics problemsthecommonlyusedelements such as the<listenedquadratic sererxlipity elements require a very detailed mesh especiallyinthe crack tip region and thus are not compulatiooallyefficient. Moreover.intheknowledge oftheaulhor. no special crack tipelement for efficient useinp-version FEMisavailableinliterao.ue.

Theobjective ofthiswork.is. thus.toobtain apracticalinsightintOthenumerical accuracy of p-version finite element method for plane elasticity problems aDdtodevelop an efficient hierarchical crack tip element for lineae elastic fracture mechanics problems.

Duetothe aaractive features of a 20 hierarchical crack tip element. itisalso decided toextend it to a3Dhierarchical cracktipelement. An efficient 3D cracktipelementis requiredasrea.Ilife fraetuR: mechanics problems are essentiallythreedimensioaalin natureandinvolve a large number ofdegreesof freedom. Thereduction of degrees of freedom mainlythroughthe use of coarsemeshes isa boon for any3D analysis where usually eventhevisual inspection ofthemesbisa uemendouswk..

[nordertoachievetheobjectives. a ftnite element analysis programisdeveloped using hierarchicalandhierarchical crack tip elementsinconjunction with a suitable solution strategy. The elemeots are first tested for commonly analyzed problems having analytical

resultsor refereoce values and men used to solve practical problems. Asthep-version isknowntobecompuwionally advantageousincomparison withtheh-version FEM.in thisworle. me emphasisisontheperformance oftheelemems in lerms of aumerica1

1.5 Layout of the Thesis

ChapterOMgives an introduction^[Qtheh-andp-versioDS of the finite element method.

Chaptertwoprovides a delailed literanue review ontherelevant topicsanddefmes the scope of this study. The formulation ofthep-version fmite element method is given in thechapter three along wimtheformulation ofthespecial cracktipelement used for fracture mechanics problems for both two-dimensional plane elasticityandthree- dimensional problems. TheimplemeOIatioo of the fonnulationisdiscussed in chapter four vis-a-visthedevelopment of a fmite e1eme01 program MUNSID. Italso includes a brief discussion ofthecomputational effort forthep-version FE program. Choptu five presemsthenumerical results obtained fromtheanalysis oftestproblems forboth2D and3D analyses. The conclusions ofthisstudy and recommendations are presentedin chapter sU:.

Chapter II BACKGROUND AND SCOPE OF WORK

2.1 Literature Review

Thefinite element melbodingeneral includes three basic concepts. The coocepr. of matrixandlinearalgebra(0provide foundatiOD for a thorough understanding ofthefinite element procedures. The formuJation of the finite element method andthenumerical procedurestoevaluatetheelement matticesandthemauices of the complete elemem assemblage. Lastly. methods of efficicm solution ofthefiniteelementequilibrium equations.Thesefundamentals ace amply illustratedinme^(eXtbooks by many authors.

namely, Zienkiewicz [11, Irons (2], Bathe [3].etC. The following sections presentthe relevant Uteramre review ofthep-version FEM andtheconventional techniques for evaluation of fracture mechanics parameters.

2.1.1 p-version FEM

ODeofthefIrSt works on hierarc.b:ica.l p-version FFMisby Peano (4]. New families of

co

andC^tinterpolation functionsare presented complete up to an arbitrary polynomial degreep.1beseinterpolation functions are formulated for triangular elements in area coordinates. 'Thefundamental cbaracteristics ofthisfamily of finite elementsis :th~

shope functions corresponding to an inIerpoJarion oforrkr pCOnstilUl~a subset ofth~ s~t of shape functions corruponding to an inurpoJarion oford~rp+1andth~refor~ rh~

snffnus matrix ofth~ ~Jem~ntoford~rpis a sub·matrix of the stiffness matrix ofth~

~/~m~ntoford~rp+I.The nodal variables corresponding totheconstant strain triangle (element of orderI)arethefunctional valuesatthevenices. For the linearstrain triangle (element of order2)theadditionaloodalvariablesarethesecond derivatives of theapproximating fuoction evaluated. alongme sides atthemid-side points. Theshape functions corresponding tothemid-side nodes ofthelinearstrainaiangle vanish atthe vertices ofthetriangle. Thesehierarchical finite elements are noted as indispensable tools for realizing convergence with respect to increasing polynomial orders due to the hierarchicalnature oftheresultingstiffness matrix. where the triangularization effort is saved by utilizingthemady decomposed lower order maaices. With respect tothe problem of using different order elements inthesame mesh,thehigher order derivatives associated with edgesincommon with a lower order elementareset tozero. Peaoo proposedtheselection oftbeshape functionsas closely onbogonal to one another as possible.

RossowandKatz[5] presented. two dimensionalC'hierarchic triangular fInite elements of arbitral)' polynomial order. It is shownthatelemental arrays for high polynomial order may be efficiently computed by using hierarcbal elements together with

n,9

precomputed arrays. For polynomial oroerp

=

1.thenodalvariables arethevalue ofthe functional variable ateachvertex. For p?:.2.thenodalvariables aretheones for p-lin additionto thepthorder tangential derivatives atthemid·sides oftheelement. For a complete polynomial orderp. with(p+l)(p+2)12coefficients. (p-l)(p-2)n additiooal nodalvariables are defined forp2:3. 1beseadditiooaloodaIvariables are taken as mixedpartial derivatives ofthefunction evaluated at the origin. When transformed to the global coordinates. these variables do DOt equate across inrer-element boundaries.

hence have no effect externaltotheelementandarecalledinrernalnodalvariables. [n amixedmesh. higher order elements are demotedtolower order elements alongthe common edges by equatingthecommonoodaIvariables and eliminatingthehigher order variables. Relative efficieocy ofh-andp<onvergence with respect tothecomputational timeiscomparedandp-convergeoceisobservedbothgloballyandat particular points.

The technique of employing hierarchical elements. precomputed arrays. andp.

convergence appeared competitiveinterms of computational efficiency with the conventioaal finite element approach. Also. the use of large elements simplified the presentation of input data and interpretation of results.

Babuska,SzaboandKatz [6]discuss concepts of p-<:oavergeoce ofthefinite element method.thesingularity problem, numerical examplesandcomputer implememation of the p-version. The shape functions chosen for a one dimensionalbarproblem are the integrals of the Legendre's polynomials which form. an orthogonal family with respect totheenergy inner pnxluct. For the computer implementation precomputed arrays based on bieran:hic families are utilized.It isnoted that asthenumber of degrees of freedom

II 10

increaseprogressively. major computational effort occursinthesolution phase. For the asympwticrateof convergence withrespect

w

theNDOF: forSTnf)()(h solutiofU, Jr convergenceisnot limited by an upperboundpolynom.ia.l degree as inh~version;andin case ofnon-smooth solutions. when a singularityiscaused by comers.therateofJr convergenceisalmosttwicethatof h<onvergeo::e. Asp-version can beusedin conjunction with optimally designed meshes.the meshdesign seemed much less critical forthep-versionthanfortheh-version. Observations include: Reductioninvolume of input data; less critical nature of roundoff problemsinp-version; no major differencein thesolution times for p-version and h-version for the same NDOF; and adaptivityin p- version seems simpler by proposing mesh grading on a prior basis. either manually or with standard mesh generamrs.andthen making adaptive changes by adjustingp.

Zienkiewicz. Gago and Kelly[7] discussthemerits of hierarchical formsinutilizing previous solutions and computation as well as pcrmining a simple iteration when attempting a refinement. Thebienrc:hical degrees of freedom appear as perw.rbations ontheoriginal solution ratherI.ha..nits substiwte andtheresultingmatrices have a more dominantly diagonal form thanthatobtainable from identical number of non-hierarchic degrees of freedom. Thisensures an improved conditioning ofthematrix and a faster rate of ileration convergence than would be possiblewithnon-hieran:b.ic forms. The penurbation narure of hierarchic forms also provides an immediate estimate oftheerror inthe solution. The paper discusses the error estimatesandaddresses the issue of adaptive refinement. Itis recommended to limittheorderpto a maximum of 4 and start from a reasonable mesh notcallingforp>4,sincetheuse of very high polynomial

n

11

orders could lead to local oscillation.

Szabo [8] presents generalguidelinesfor prior design of mesbes. It isDOted that with theuse of properly designed mesbes.theperformance of p-cnrichmemisvery close to the bestperformance attainable bythe finiteelement method. lbeestimation and conttol of the errorinrmite element analysis are based onh-or p-reflnementsandp- refinement makes it convenient and inexpensive to obtain information aboutthequality ofthefinite element solutions. Theproposed mesh designis thecoarsest possible mesh controUed entirely by the geomeuy ofthedomain wbentheexact solution is smooth.

Theonly restriction on themeshisthesmooth mapping oftheelements.Incase of non- smoothexactsolutions. the points of singularityandareas of.stressconcentration need to be isolated by one or more layers ofsmallelemeDlsandthemesbisgraded sucb that the element sizes areingeomeaic progression withthesmallestelement(s) located where the stress gradients arethelargest. The geomeaic progression with a common factor of about 0.15isfound suitable. Inthe case ofstructureswith many singularities it is found not oecessary to refine the mesh in the neighbourhood of e"'eI}' singular point with care being taken that overall and local equilibrium conditions are satisfied. Babuska[9]also gives referencetooptimal design of meshes.

WiebergandMoller [10] describe hierarchical FE-formulationandadaptive procedures for staticanddynamic problems. The sequence of nested equation systems that resultS from a hierarchical finite element formulationisexamined. Due tothespecialstructure ofthematrices formulated using hierarchic basis functionsandease of developing error

n

12

indicators for funber refinements, iterative solution procedures areusedwhen an increasingnumber of variables areintroducedintotheapproximations. New possibilities are presented for iterative solution ofnestedequation systems for elastic static problems resultingfrom hierarchicbasisfunctions. Out ofthefour algorithms outlined,threeare basedonthepre-conditiooed conjugate gradiem method andthefounhisa two-level multigrid method.

Babuska,Griebel and Pitkaranta [11] address the question of optimal selection of the shape functions for p-type finite elements and discusses the effectiveness oftheconjugate gradient and multilevel iteration method for solving che corresponding linear system. A unit squ.a.re master element is considered and the three groups of shap.. functions associated wim me element are, namely,thenodalshape functions, theside shape functionsandtheinternal shape functions. Anodalshape functionisassociated with a venex oftheelementandiszero ontheopposite sides ofthevenex itisassociated with.

A sideshapefunctionis associated with a sideandiszero onallthreeother sides ofthe element.Aninternalshapefunctioniszero onallfour sides and bastheclwacter of a 'bubble' function. Various sets oftheseshape functions are considered along wiLh the oon·hierarchic shape functions and bigonomeaic shape functions. Thesesets are then compared for optimal selection of shape functions through computational analysis.Only two dimensional cases are addressed in the paper.

Morris, Tsuji and CamevaH [121 proposedandimplemented a solution strategy for taking advantage ofthehierarchical structure of linear equation sets. Thekey novelty of the

II 13

algorithmistheabilitytochoose dynamically between iterative and direct solution techniques based on a set of heuristics. The iterative solverisbased ontheconjugate gradientmethodandmecombination ofthedirectanditerative solvers allow foran efficient solution path. havingtherobustnessof adirect:solution algorithm. andthe efficiency ofaniterative solver in utilization ofbothCPU and storage.

Avaluable discussion onthestate oftheartofthep-version ofthefinite element method isgiveninBabuska [9]. Varioustheorems arepresentedtodefinethe p-andhp-version ofthe FEM. Anumerical example illustratestheperformance ofp-version with different number of layers at the stress singularity point (graded mesh) and compares itco a uniform mesh. For implementationthethreekinds of shape functions discussed are aodal.sideandinternalshapefunctions. Forthecomputation of the local stiffness mattixtherule for number ofGausspointsis2[inleger(pn»)+2. Thep- andhp- versions for two-dimensional problems were implemented inthecommercial software PROBEby Noetic Tech.• St.Louiswiththefirst releasein1985 andthesecoDd in 1986 andit wastestedand usedextensivelyintheindustry.The fe.awres considered imporwu.

bytheengineering users wereincreasedlevel of confidence inthecomputation. lower human time requirement due to simplicity and flexibility ofinput,rapid convergence and flexibility includingth~use of large aspect ratios. flexibility of mesh design. easy learningandrobust perfonnance.

D 14

2.1.2 Computational fracture mechanics

Chan. TubaandWilson (13] usedmefiniteelement method for the evaluation of stress intensity factors. Thedisplacement FEMisusedandfirst order displacement functions for triaDgular elemeots are assumed,inwhich.thedisplacemems vary lioearly overthe elemem resultinginconstamstrainsandstressesintheelemenL A verydetailedmesh isconsiden:d. wbere

me

smallestelements are difficull to see withtheoakedeye. For thecomputation of^stressintensity factors. thIec: approaches outlined arethedisplacement method.stresSmethodand lhelineintegral(eaergy) method.Thestress intensity factOrs in thedisplacement method are determined from the coaelation of lhefinite element nodal point displacements withthewell known cracktipdisplacement equations. The stress methodissimilar tothedisplacement method andthenodalpoint streSses are correlatedwith theknown cracktipstressequatioDS.In theline integral method,theline integralJgiven by Rice [14]isnumerically integrated along an arbitrary contour surroundingthecrack tipandthestressinIecsity factorisevaluaced as a function ofJ.

modulw of elasticity and the Poisson's raCo. The elemenrs howeverdoIl()(represellt me singularDeartipdeformationand theanalyses included a very large number of degrees of freedom. They proposed potential improvemeotsby including higher order displacement functions.

Tracey (15] introduced a new type of two-dimensional finite element which incorporates the inverse square root singularityinstresses near a crack tipinan elastic medium.

Triangular ·singular· elements that areproposedembody the singular stresS fields coaespondingtothecracktip. The crack:tipserves asthecentre of a whole ring of

II 15

triangular elements and joinedintheradial direction with quadriiareraJ isoparametric elements of order ODe. 'Thesetriangular elemems are fournIXIe quadrilaterals with two oodes coincidentatthecrack tip but distiDguishable by their angular rotation. Tracey (16]alsodescnbed a special three-dimensional element aoaJogous 10thetwo-dimensional element for evaluation of suess intensity factors. Inthisformulation a sixnodewedge shaped singularity element with special displacement interpolation functionsisfocussed around the crack frontand issurrounded by eightnodedisoparametric brick elements.

Aneight point Gaussian quadratureisusedto evaluatethestiffness matrix.

Hellen [17] proposed a virtualcrackextension method wheretheeoergy release rates are computed usingvinua.lcrack extensions.lbestressintensity factors are evaluated using theirknownrelationship with eoergy release rate for a virtual amount of crackgrowth.

Analgorithmisdevised to calculate the energy difference for two crack positions. close together. using only one mesh.andaltering the tip stiffnesses to assess the energy difference. To increasetheaccuracy of results. quadratic special crack tip elements developed by Blackburn [20] areusedinconjunction withtheproposed method. The special elements are triangularintwO dimensions with,-Yo displacement variation radiating fromthecrack tip. wilereristheradial distance fromthecracktip.

Heosbelland Shaw [18] inttoduced the requiredsingularity at the comer of a quadrilateral isoparametric ftnite element by moving the mid-side nodes to the quarter point towards the crack tip. For determination of stress intensity factorsincracked bodies itis suggested to usethe standardquadratic isoparametric elements withthe

IT 16

distorted isoparamet:ric elements atthecrack tip. Howevertheresu1tsobtainedby displacement melhods depended onthepoint chosen for displacemenr: evaluation and reliable results wereobtainedby using displacementsatpointsalongthefree surface of

!heend.

Barsoum[19] individually developed siogular elementsbasedonthesametechniqueused by HensheU and Shaw[l81 fortheevaluation ofsaessintensity factors. Eight ooded quadrilateral. six-ooded aiangular elementsandthree dimensional twenry noded cubic andprismelementsaredistorted by moving mid-sidenodesclosest tothecrack at the quaner points. This results in a

,.'h

stress singularity at the cracktipcorresponding[0 the I:beoreticalstress singularity of linear fracture mechanics. The elememsaresbown to haverigidbodymotionandCODStanl strain modes and satisfythepalChleSt.10 both cases[18.19) a sizeablenumberof elementsarerequired.10modelthecrack tipaswell astherestoftheproblem domain.

Blackburn [201p~nteda Iriangular element with vertexandmid-edge nodes with special displacemeOl functiOQS [() representthesingularity behaviourintwo-dimensional problems. Blackburn and HeUen [21] extendedthespecialtw<Kiimensional element10 three-dimensional problems as a 15 noded wedge shaped special crack tip elemem. The stress intensity factors are calculated by displacement methCKl.,lineintegral methodand also bythe method of vinual crack ex.tensions.

Benzley [22] developed a special crack tip element from a linear isoparametric element, IT 11

wilhoneCOrDercorrespooding to thecrack:tip. and enrichingtheconventional displacement assumptionwith thedisplacement field correspooding tolhesingulac portion of the elasticity solution. Inadditionto DOdaIdisplacements.~andKgbecome basic unknownsinsucb elementsandan: c:al0llatM direct1yin thefinite element analysis. A linear zeroing functionis usedfor the emicbcd elementStomake themcompatible with theadjacenr: oon-cnriched elemenIS.

Gifford and Hilton [23] enriched a conventional quadratic isoparamet:ric element as suggested by Beazley. Howevertheelement formulationisof the non-conformingtype andno zeroing functionis considered. 8x8 numerical quadratureisrequired forthe accurate numeric:al integration oftheenriched element.

bjuandNewman [241 presentedstressintemity factors for sballowanddeep semi- elliptical surface cracks in plateS subjectm to uniform temion. 1bestressintemity factors are evaluated usingnodalfora: methodand thefinile elementmodelscoDSUUcted using singularandisoparameaic elements involve a vo=ry large number ofdegreesof freedom. Theresults presented inlhispaper are widelyusedfor refen:oce as well as for comparison studies.

HepplerandHansen[2S]utilized GiffordandHillon's twelve node enriched serendipity element for the calculation oflinear elastic planar stress inlemity faclors for rectilinear anisotropic materials subjected lO biaxial loading.Inaddition. conforming formulatiom usingthreedifferent zeroing functions, linear. super-ellipse and polynomial. are studied

n 18

andcompared with non-conforming formulatioos. Forthenon-confanning casehigh accuracyisachieved wilh a few elementsandthey lX>teda substantial stiffening ofthe finite element model fortheconforming cases. Amongthethreezeroing functiol1S.the polynomialfunction which reinforcestheexpected singular behaviourDearthe cnck tip performedthebest.

2.2 Scope of the Study

Asiatic p-version fInite element analysis program that employs quadrilateral hierarchical finite elementsisdeveloped for plane elasticity problems. The interpolation functioos for P..elements are enrichedtoderive special crack tip elements (enriched hierarchical elements) for use in linear elastic fracture mechanics problems. All analyses srudyp- convergence thereby eliminatingtheneed for redesign of mesh for convergence srudies.

2.2.1 Formulation of 2D-bierarcbical element

The shape functions for quadrilateralhienucb..ica.l elements are derived from LegeDdre's polynomials.Theinterpolation functions are incomplete polynomials in a fashion similar to that of standard serendipity elements. For an elemeOl of orderpthere are4pnodes each having two degrees of freedom. The nodal variables corresponding to lhe comer nodes arethephysical degrees of freedom,thedisplacementsuandv.intheglobal cartesian coordinates.x and y respectively. The higher order nodal variables forthepm order element arethe pthderivatives ofthedisplacements along the side atthemid-side

IT 19

node. For evaluation ofthestiffness matrices.P+1gauss quadratuttisadopted foran element of orderp.

2.2.2 Formulation of 2D..,nriched hierarchical element

Theenrichment ofthehic:rarcbica1elementisbased on Benz1ey's[22] principle due to simplicity of the formulation andas the stress intensity factors are presented directly in theoutput. For the compatibility of adjacent P-andEH--e(ements azeroing function is required for the EH-element. bowever it basbeen sbownin(25)thatthezeroing function stiffens the system. 1bereforeDOzeroing function fortheEH--elementisconsidered,the formulationis anon-conforming one andis similar(0thatusedby Gifford and Hilton [23]fora serendipity element. 1becrack: element stiffness matrices are numerically integrated by using 8 point gauss quadrature.

2.2.3 Formulation of 3D P-

and

EH..,lements

TIleshape functionsusedfortheP--elementandtheprinciple for enricbmenttothecrack tip element for a two-dimensional caseare simply extended to three-dimensions.

2.2.4 Solution strategy

The elemental stiffness malricesare assembled intotheglobal stiffness matrix in a skyline fonnat. The global stiffness matrix for high order elements is panitione<i, the block diagonal matrices aretriaogularizedbygaussian elimination and a combination of directanditerative techniquesisusedfor the solution.

n

20

2.2.5 Numerical verification

and

results

Thetestproblems selected for verification of 2D P..elemcot are the: commonlyusedshan cantilever beam problemandtheCook's membrane problem. Thetestsfor 2D EH- elements are a double edge cracked tensile specimen and a slant cracked tensile specimen withmixedmodeSIFsolution.

Thepractical problem selectedisaT-plate weld jointanalyzedby Bell [26] forstress cooceotrationandsttessintensity factors. TheSCF's are evaluated for 45⁰andAWS weld profiles and the fatigue Life of a 45⁰weld jointis also estimated. The results are then compared with those obtained by Bell.

The test problems for the 3D EH-elemeDlS are double edge cracked tensileandcompact tension specimens. Thedouble edge crack tensilespecimenisan extension oftbc:20 problem.amtheselected compact tension specimenis lbat soLvedbyTracey [16]. A semi-drcular surface crack:ina plate subjected to teosionisthenanalyzed andresultsare compared with those givenbyRajuandNewman (24}.

II , 21

Chapter ill FINITE ELEMENT FORMULATION

3.1 Generalized FE Formulation

The displacement finite element approach used in this study is derived from the principles of virtual work. 'Thisformulation can alsobederived using the variational principle or GaJerkin's weighted residual approach. For (wo dimensional plane elasticity problems the unknowns are the displacements u and v in the global Xand Y coordinate system.

Thedisplacements arefirst interpolated over lhe domain of an element represented by a local coordinate system.

Thestandardprocedures givenin[1.2.31 are thenusedtodetermine the shape functions. The quadrilateral mastc:r elementinlocal ~ and'I coordinate systemis representedinFig. 3.1. The values of local coordinates for the element lie between±1. Theinterpolation of

Figure3.1 Quadrilateral master element

displacementsuaDdvinteans of nodal variables. by appropriate shape functions N.in any region ofthelocal elementisgiven by :

.0

⁼(N(~.~»)(uOI v·=(N(~.~»)(v·1

.• (3.1)

where {ue} aDd{V"}are£benodalvalues forthelocal elemenL Thestrainsin

me

element are given by :

(cOl'

tJ

^{= • [Blldol •• (3.2)}

where(8)is a matrix conWning derivatives of shape functionsinthe global coordinates and{de} isthenodaldisplacement vecmr given by :

UtdeDOtesu at node I,aDdPismeorder of the imerpolation function.

. . (3.3)

The mapping ofthelocal elementtotheglobal elementrequiredfor the evaluation of global sttessesandstrains is done bytheJacobianI, that depends on the geomeuy of the global element. The geometry oftheelementisapproximatedinterms of the nodal coordinates(X;,yJby suitable geometric shape functions, G :

ill 23

x=~GI~

..

y=

.. EG,

^y,

i-I

• • (3.4)

1be formulationissub-.iso-or super-parametricifthe order of G (given as pin

Eqn.(3.4» is lessthan.equalto or greaterthanthe order of N respectively. The JacobianJisgivenby :

Using Eqn.(3.4)wehave:

.• (3.5)

· . (3.6)

1be£B)maai:t is partitionedintosub-matrices correspondingtoeach node.andisgiven by:

lB]

= (•••

18,1. . -)

lB,]

=

0 ay

aN, aNI aNI

aya;:

ill 24

• • (3.7)

where the sub-matrix

[BJ

correspoDdstothe ith node. Since the displacement shape functions are fonnulatedinlocal coordinates. to determinetheelements of matrix [BJ.

thefoUowing relatioosbipis used :

Theelemeotalstressesace obtained by the constiwtive law:

\o'} •

tJ

⁼/Dllo'} • /D1[Blld'}

where [01. the constitutive matrix given for plane strain (pe) is :

•. (3.8)

•. (3.9)

1;']

^{•• (3.10)}

aDdfor planestress(po)is :

[ 1 • [D]:~vl

per I-v² 0 0

.• (3.11)

Thestiffness matrixandthe load vector (considering only surface forces) for an elemen[

are given by :

ID,25

[K '] •

f .

[BIT (OJ [BJ dO [F']'[(N)Tf,dl'

• • (3.12)

whereQrepresents the domain ofdieelementand asurface loadC.actsontheboundary r. Thestiffnessmatrixisformulatedinthelocal coordinate system.beDceitiswritten

, ,

[K '] • [, [,[BJT(OJ[BJ d(d~^{111 · • (3.13)}

where

I

J

I

is the determinant of the Jacobian. Gauss quadrature ruleis applied to numerically integrate the element stiffness matrix giveninEqn.(3.13). Gauss imegration usingNGxNG pointsisgiven as :

NO NO

[K'] •

LL

[B((,,";lJT (OJ [B((,,";lJ 11«(,,";l1 w, w; . .(3.14)

I-'J-'

Wiandwjaretheweights correspoodingtoithand

fth

gausspointsrespectively.

Oncethestiffness matrices ofall theelements representingthedomain are evaluated as descnDed above.theglobal stiffnessandload maaices are obtained by the summation or assembly oftheelements of local matricestothecorrespoDding elements of global matrices :

[K"]⁼

L

[K'l [F"] •

L

[F']

· . (3.15)

1besystem of equationsin thedisplacement formulation oftheFFMisgiven as : ill 26

[K"I(dl • {Fol where {d}istheglobalvector ofnodaldisplacements.

•. (3.16)

Thedisplacementshapefunctions. N.inp-version FEM are hierarchicalinaacureand ared.iscu.ssedinthe DeXt.section. Thegeometricshapefunctions,QP.are chosen asthe linearandquadratic sereodipityshapefunctions correspollding to a quadrilateral master element for order(P)oaeand two respectively :

i=1,2,3,4

.• (3.17)

• ~(l-<'J<h~~),

=

~(l'«;XI-~'l,

i=5,7 i=6,8

The geometric shape functions giveninEqn.(3.17)attalsoused as displacement shape functions forh-type elementsintheh-versionFEMandare an entirely different set for different orders.

3.2 p-version FE Formulation

This section presents lhe hierarchical shape functions usedinthis work. For the ilIusttationpurposesthederivation for a second order hierarchical elementisdiscussed

m :21

in particular. Inaddition onlythederivatioDS for displacementuare provided aod are obtained for vinan analogous manner.

1be displacement interpolation foru overthelocalquadri1aleraIelementisgiven by :

. . (3.18)

• (P)(ol

wbere {a}isthecoefficient vector.Thenodes correspondingtoa hierarchical elemen[

arethefour corner ocx1es forthe firstorderandfor eacb incremen[ in order a se[ of four hierarchical nodesisplaced. one8[each of me (our mid-sides. Thus. for a second order P-elemem there are four comer nodesandfour hierarchical mid-side nodes. Evaluating u in Eqn.(3.18) a[ me four corner nodes we have:

•• (3.19)

~degree of freedomat:

me

mid-side nodeisthehier.ln:.hicaldegreeof fn:edom defmed aslbc:double derivative ofthedisplacemen[ alongmeside on whichtheoodelies.aDd isevaIuared a[

mal

node. Thwtllispseudo -disptacemem- a[

me

mid-side oode 5 is given by:

where1,isthelength oftheside on whichoode5 lies. Similarlythehierarchical

ill 28

variables attheoodes 6, 7and8are evaluated. The coefficient macrix {a}isobtained intermsofthenodalvariablesandis usedinEqn.(3.18)toob<a.inthedisplacementsin termsofoodalvariablesviashape functiom.1beoperationisgivenas :

· . (3.21)

u"=[PJ[q-l[u"1 • (N)[u"[

Theshape functionsN obtainedinthis fashion are hierarc.b.icalinnature. The hierarchicalshape functions however are not unique. Thefirst order P-shape functions correspoodtothe corner nodesandareequalto thefirst order sereooipity shape functions.QI,giveninEqn.(3.17). Toreducecoupling between successive solutionsthe higher order shape functions are derived from Legendre's polynomials that are orthogonalinthe-I to+1 range. A one dimemional polynomial term. LP, for the pth order shape function for a line element corresponding to a side of Lhe master element having two comer nodesandone mid-side node is obtained as :

· . (3.22) whereP,istheLegendre's polynomial of orderp. The polynomial expression forthe second order.U.is :

L'⁼«'-1) ^{· . (3.23)}

andpolynomial expression fortheshapefunction for the mid-side oode 5 (along '7--1) isobtained as :

ill 29

· . (3.24)

For me secoDd order hierarchical variablethedouble derivative ofthis shapefunction.

Eqn.<J.24) mustequalunityatDade5. Thus

· . (3.25)

andthecompleu: set of 2nd order hierarchical shape functions for a quadratic P-element correspondingtomefour mid-side nodesis :

Hi

⁼

~(l"U('l2-1{~r

NJ • i«'-l)(l+"{~r N; .. ~(1-~)(111_l)(~r

· . (3.26)

For athirdorder element out of atotalof (Welve shape functions.thefour corner shape functiODSremainthesame Le.equaltoG1,andfourmid-side shape functions arethe same as giveninEqo.(3.26).Theremaining four additional shape functions correspond to the new hierarchical degrees of freedom (thethirdderivative of the displacement along theside)andare similarly derived.[athisstudy onlynodaland side shape functions are comidc:red thus an element of orderp has4pnodes(4 cornerand4p4 mid-side)and4p

ill 30

correspoDding shape fwJctions.

3.3 20 EH-Element

For abodycontaining a crackthelocal coordinate syStem atthecrack tip isillUStrated inFig. 3.2.

x'

aDdy'

are

thelocal

axes cemredatlhecrack tip with x' axisoriented alongthecrack plane. r and

e

arethepolar coordinates of a pointinthecrack tip region w.r.L the loc:alx'.y'axes. 4>istheorientation ofthecrack tip w.r.t.theglobal X axis. mea.sured +ve counter·

clockwise. Thedisplacementsinthe

vicinity ofthecnck tip given by

rtgUre

3.2 Coordinate system at cracktip GiffordandHilton [23J are :

.. (3.27)

where fl. g10 etc. are functions that depend onr,

e

andq,inthe crack tip coordinate system. Gtheshear modulus andIIthe Poisson's ratio. ~and Kn are the mode I (tension)andmode II (shear) stress intensity factors. The functionsflo etc.alegiven as

ill 31

who«

f₁•

-.!-

4G~l2;"J

rp..!.-)l~

^[(2y-l)

"""~

2^-

cos~J

2 - sinq[(2y.1)

siu-i - siu-TII

g, •

-.!-

_4G~lbJ

rp..!.-)l~

^[(2y.3)

siu~

₂

• sin~J

₂

•

sin~

^[(2y-3)

COS-i •"""-Til

f, •

-.!-

_4G~lhJ

rp..!.-)

^(sinq[(2y-l)

"""~

₂ ^-

cos~J

₂

•

~

^[(2y.1)

siu-i - siu-TII

g, •

-.!-

_4G~lhJ

rp..!.-) (sin~

^{[(2y .3)}

sin~

₂^•

sin~J

₂

-

~

^[(2y-3)

"""-i •"""-Til

• • (3.28)

· . (3.29)

{ 3-4.

y" 3-v I ••

plane strain

planestt= • . (3.30)

Thehierarchical element displacemem assumption. Eqn.(3.18),isenriched by me crack tipdisplacementas :

u~'"u· .. u"

• • (3.31)

Proceeding ina Standard sequence, Eqn.(3.18) to Eqn.(3.21). for evaluating the

m :32

displacements in anelementin rerms of the nodal variables. the displacement imerpolation ofthe"eruicbed~second order P-clementisgivenby :

u,;. •

(N)(u"- K,

If, -

tN,

f,,-

tN,

f{j

i-I i_5

• (N)(u" -K,P, -KgQ,

. .

v';' • (N)(v")+ K,If,-

EN,

f,;-

EN, r;{J

I-I i-5

• (N)(v'l •K,P, •KgQ,

• • (3.32)

· • (3.33)

f([ is similar tothedefinition of hierarchicnodalvariables and is givenby :

• • (3.34)

andis evaluated attheithnode. The displacement interpolation u for an enricbed hieran:bica1 element of arbitrary orderp is givenby :

. . .

u,;.

(N)(u"l- K,

[f,- EN, f

_u-

EENt f{,'J

I-I p-1j-1

. . .

• K.[g,-

EN, g,,- EEN,' g{,'J

I-I p_1j_1

where i andjcorrespondtothecomer and mid-side nodes respectively.

ill J3

• • (3.35)

TIlestrains ina second order P-elemem ace given by :

" •

_• ~("I _ax

^{+ . .}

_-';ax

^{aI',+ . .}""1I

ilQ, ax

"

_{' a y}

=

~(v"1

^+K

_lay

^{aI',+ . .}

_{.... ay} ilQ,

y' • _'" (~lv'l+ ~lu"l _{ax ay}

(aP, aP,) (ilQ, ilQ,)

+It,

ay+ a;: +

^K.

ay + a;:

whicbinmatrix form arc :

(,'I

=

lB",llcS,;.1 1cS,;.I

^T•(u,

v, ., v, ...

u,

v, K, K,,)

· . (3.36)

· . (3.37)

In thisformulation Kland

Ku

ace included as additional degrees of freedomandthe corner node corresponding to local coordinates of

e

^=-1and1'1=-1is located at the crack tip. For dle evaluation of functions fl'etc.,thevalue oftbfor the nodes lying on one surface ofthecrack faceis+180° while for

me

node ontheother surfacethevalueis _180°. The(BJmatrix is partitionedandgiven as :

[B!=(B, .. B, .. B,B,.)

[B,.!=

· • (3.38)

m

l4

where[B.JisgiveninEqn.(3.7) and[BJcorresponds tothedegrees of freedom

0Ci. Ka>

ofthe·pseudo· node atthecrack tip. The evaluation of the localandglobalstiffness marrixisthesame as descnOedinsection 3.2.

3.4 3D P- & EH-Elements

The <levelopmenr: ofthe3D hieruchi.c e1emenlisasimple extension ofthe 2DP~lemeD1.lbemasterelememis definedinthelocal~.71.

r

coordinates as shown in Fig. 3.3. The displacement shape functions for a 2nd order 3D hierarchic element are giveninappendix A. Thegeometric shape functions. G. for order 1and. 2 arethe serendipity shape functions

Figure 3.3 3D master elemem

associated with an 8-noded and a 2Q..nCKied serendipity brick element respectively. The Jacobian for the 3DP~lementis given by :

[1] •

ax '!:L

Ur.

a. a, a.

~'!:L~

a., a., a.,

ax

iJy Ur.

a.acac

ill 35

•• (3.39)

Thederivatives of shape functionsinglobal coordinates for the evaluation of(B)matrix are givenby :

•• (3.40)

The(B]matrix for a second order 3D P-elemeatispartitioned into 20 sub-matrices (corresponding to 8 comer aodes and 12 mid-side hierarchic oodes)andalong with a panitionec1 matrix is giveninEqn.(3.41). Eqn.(3.42) depicts the constitutive relationship which relates the stressesandstrains. The steps for fonnulation and evaluation of the stiffness matrix are the same as for a 2D element.

[B] : (.

18.1

.)

aN, 0

: 1

ax ay-

aN,

aN,

•• (3.41) [BJ :

ax

aN, aN,

ay ax

aN, aN,

~

ay-

aN, 0 ^aN,

ax a;;-

m

36

la'( : [011.'(

1-.

a a

I'

1-.

. ^{a a}

~:

a, 1-.

a a

.• (3.42)

a, : _ _E_ _ 1-2v E,

2

"

(1-tvXl-2v)

"

,.

1-2v2

,.

symm 1-2v

2

3.4.1 Enriched 3D P-..Lement

Asshown by Sib andLiebowitz[27],thedisplacements

u.:.

v: and

w;

neartheedge ofthecrackindirectioosx',ylandZl(Fig.3.4) parallel to the normal. me binormal and the wrgenttothecrack froot areexp~as :

u.: '"

KJf₁ Ka^&1

v: '"

~^C:z Kg&:1

w;:

^K",f,

wheref_{l ,}etc. are given as :

m

37

•• (3.43)

z

~x

)L,..-J'---,.

f'ieure3.4 Coordinate system atlhecrack front (3D)

l+VJ¥ 6 36

f • - - [(5-8v_--

cos-I

I 4E 1: 2 2

g '"

~ ~

[(9-8v)sin!.

sin~]

I 4E~-; 2 2

t, •

~;, ~ ~ [(7-8V)siD~- smT1

g, • -~ ~ [(3-8v_~. cos~l

_{4E~• 2 2}

(, .. 2(l+v)

~ sin~

E ~^{-; 2}

.. (3.44)

Thedisplacements u:, etc. areinthe local CfclCk coordinate system (x',y~Z/). For enrichment ofthedisplacement assumption oftheP-element. lhese are transformed intO displacementsintheglobal coordinate system,Ue. etc. The transformation matrix. T. is

m 38

• • (3.45) a 3x3matrixof direction cosines formed betweenthelocalandglobal coordinate axes.

[11 •

[~;: ~;: ~;:j

T z'.. Tz'y Tz'l;

where Til=T •.•= thecosiDeof the angIebetweenthelocal xIandglobal X axes.

1bedisplacemeDlS\1,;.etc.intheglobal coordinateS ace determined fromtherelationship

andaregiven as :

Uc '"K1 (flit+T2tiz)-+

Kn

(Tugl+T21Ct)+

Km

[T'tt;}

• K_tP_t +

Ka

Q_t+

Km

R₁

• . (3.46)

• • (3.47)

TIledispl.acemeot interpolation of the enriched 3DP~lementisthengiven by :

• • (3.48)

Usingthe standardprocedure.thesecond order displacement interpolationin theelement interms ofthe nodalvariablesisgiven by :

ill 39

. .

u.; •

^(N)lu'}+K,lP,-_~N,Pu-_~N;

p:tJ

... Ku[Ql-

t

^{N;Qu -}

t

^N;

^Q~l

+K,.[R,-

tN,

^R.,;-

tN, R~~

i-I I·'

•. (3.49)

Inthe20 enrichment.theadditional nodal degrees of freedom are mode Iand mode II mess imc:nsity factolS.Inthe3Dcase there arethree nodesalongthecrack front.[wo comerand one mid-side node. Eachnode isassociatedwiththree unknowns. mode I.

modeII and modeill~intensil:y facrors. thus there are9 additional DOF for a 3D enriched elemem:.TheSIP's areassumedtohave a quadratic variation alongthecrack frontand this variationisgiven as :

K" •

M,K,,+ M, K,,+ M,

K"

K,. •M,

K"

+ M, K,,+ M,

K"

",he",

M, •

~';~

M,= I-~'

M,'~

.. (3.50)

•. (3.51)

~in Eqn.(3.S0) denotes '"at.nodei.Thelocal axesfJis equaltothemaster element axis

r

andliesalongthecrack frontandme z' axis of the cracktipcoordinate system.

Combining equations (3.49)and(3.50)thedispLtcemem interpOlation for me enriched elementiswrittenas :

>K"A.,. >K"A." >K"A.,.

""KU~7 +Kn~.+KnAu

· . (3.52)

Forthederivation of the[B}matrixlet the displacement interpolation forYandwbe given by :

Y~^{... (N){VC) ...K}ll~I^...~l

Au ...

^K,I

An

> K"

A,.

>K" Au > K"

A,.

> K" A." , K" Au ' K" A"

w'; ...

(N)(WC) ....Kll A,l ....~1An~1',1

An

> K"

^{A,. ,}

K" Au ' K"

A"

, K" A." > K" A" , K" A"

Using equations (3.52) to (3.54),thestrainsare evaluated as :

• • (3.53)

• • (3.54)

{,'I=[BJ(d"1 .. (3.55)

IdC)T ..(~VI WI . . .11m_{v20 Wzo Kn}~1IC:lI K12~Kn_{K u}

KzJ

K.,3)

m ;41

andthe[8] matrix is given by :

ilA" aA

_u

aA"

~

ax aA,

aA" aA" aA"

ay ay aA,

aA" aA" aA"

[B,.,J •

ax ax aA,

aA

21 +

aA

_u

aA". aA

u

aA". aA"

ax ^ay ax ^ay ax ^ay aA". aA" aA". aA" aA".

ilA"

ax ^ay ax ^ay az ay

dA.,1+ dA,,1 a~+aAu

aA". aA"

ax ax ax ax ax ax

•. (3.56)

The stiffness matrix for the second order 3D EH-elementis then evaluated using the standard procedure describedinprevious sections.

ill 42

Chapter IV COMPUTER IMPLEMENTATION

4.1 Computer Program

A.staticp-venion finite element analysis program MUNSIDisdevelopedinC language.

The capabilities include analysis of plane elasticitYandLEFM problemsintwoandthree dimensions. One ofthe primaryaimsofthisstudyis[0developanddeterminethe numerical accuracy oftheenriched hierarchical elements. The fmile elements are quadrilateral shaped so a quadrilateral mesh generatorwithautomaticnodenumbering andelemental connectivitydata generationisalso developed (or ease of input. The program utilizes p-cnricbmemandmemeshDeednotberefmed for convergence stUdies.

ThethreemainsegmentSoftheprogram are : Elemental stiffness matrix formulation.

global assembly oftheelemental matricesand the solution routine. Various aspects of MUNSIDarediscussedin thefoUowiog sections.

4.1.1

Input

BesidesthesrandaId input parameters such as geometricalandpbysical properties ofthe problem.theorder oftheelements, the order of geometric shape fu.octions, plane suess

or planestrainanalysis option.thenumber of ileratioasaDdan option forcracktip analysis are alsorequiredasinpUl. For20 crack problemsthevalue of!p. orieruation ofthecraclcplaneW.r.l.theglobalcoordinaleaxes.aDdfor3D problemstheVecoor normal 00the crack:planeare additiooalinputparameters required.

4.1.2 Elemental stiffness matrix evaluation

For most oftheanalysesthesub-parametric element formulationis utilized. Theorder of the elementisalmost always greaterthanunicyaspissuccessively increased while the element edges are generally straightandthereforetheorder ofthegeometric shape functionsisone. A maximum geometric order of cwoisusedfor problems having curved boundaries. The elements are notdefinedasiso-or sub-parametricinthe strictest sense asthegeomeaic shape fu.nl:tioos are DOt hierarchical butstandardserendipity shape functions. Gaussian integrationisemployed fortheevaluation ofthestiffness matrices.

Tbe(p+1)Gauss integrationruJeisusedtointegrate a partitioned sub-matrix_~.where p isthehigher order of the matrix Le.p

=

rnax(iJ). To avoid zero eigenvalues inthe stiffness matricesthe reducedintegration ruleisDOtused.however. reduced integration forthesecond orderisfound valid as long astheelemen[isDOl enriched further.

4.1.3 Assembly and solution of global stiffness matrix

Since me elements oftheglobal stiffness matrices lie predominantly along lhe diagonal.

toavoid Storing a large number of zero elements. each global partitioned matrix is assembledina skyline format. TheGaussian elimination isused todecompose the diagonal partitioned matricesinto theLDL^Tform (Bathe[3} Chap. 8).andtheoff-

IV 44