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Approximate Methods for Evaluating Mode Mixity in Delaminated Composites

Permanent Link: http://ufdc.ufl.edu/UFE0024103/00001

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Title: Approximate Methods for Evaluating Mode Mixity in Delaminated Composites
Physical Description: 1 online resource (66 p.)
Language: english
Creator: Vigroux, Nicolas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: ctfm, delamination, fea, fracture, mutifidelity, vcct
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In laminated materials, such as composites, interlaminar damage is one of the major modes of failure. Due to the increasing incorporation of composites into aircraft structures, this kind of insidious damage, also called delamination, has become an air safety concern. Therefore, two main research fields have emerged to answer the safety requirements of the aviation industry: the first focuses on detection of delaminations, whereas the second develops delamination models to understand what constraints control the interface crack growth. Currently sophisticated structural health monitoring tools can provide information on damage of a structure in real time. However, the information will be most useful if one can perform a stress analysis of the damaged structure in real-time allowing prognosis of the life of the structure. Thus, the aim of the study is to implement high-fidelity delamination finite element models and compare them with speedy analytical methods to allow a gain in computational speed without losing much accuracy. Fracture Mechanics principles are used in the analysis of 2D and 3D finite element models that were conducted on a basic interface problem: the double cantilever beam fracture test. Furthermore, numerical methods, such as the Virtual Crack Closure Technique, are implemented in those models to assess the total energy release rate along with the mode mixity, which permits to fully characterize the delamination at the crack tip. Finally, in a multifidelity approach, 2D analytical Crack Tip Force Method that provides inexpensive computing solutions is compared to the high-fidelity 3D Finite Element Analysis to improve its prediction capability of the criticality of the damage of the structure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nicolas Vigroux.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Sankar, Bhavani V.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024103:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024103/00001

Material Information

Title: Approximate Methods for Evaluating Mode Mixity in Delaminated Composites
Physical Description: 1 online resource (66 p.)
Language: english
Creator: Vigroux, Nicolas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: ctfm, delamination, fea, fracture, mutifidelity, vcct
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In laminated materials, such as composites, interlaminar damage is one of the major modes of failure. Due to the increasing incorporation of composites into aircraft structures, this kind of insidious damage, also called delamination, has become an air safety concern. Therefore, two main research fields have emerged to answer the safety requirements of the aviation industry: the first focuses on detection of delaminations, whereas the second develops delamination models to understand what constraints control the interface crack growth. Currently sophisticated structural health monitoring tools can provide information on damage of a structure in real time. However, the information will be most useful if one can perform a stress analysis of the damaged structure in real-time allowing prognosis of the life of the structure. Thus, the aim of the study is to implement high-fidelity delamination finite element models and compare them with speedy analytical methods to allow a gain in computational speed without losing much accuracy. Fracture Mechanics principles are used in the analysis of 2D and 3D finite element models that were conducted on a basic interface problem: the double cantilever beam fracture test. Furthermore, numerical methods, such as the Virtual Crack Closure Technique, are implemented in those models to assess the total energy release rate along with the mode mixity, which permits to fully characterize the delamination at the crack tip. Finally, in a multifidelity approach, 2D analytical Crack Tip Force Method that provides inexpensive computing solutions is compared to the high-fidelity 3D Finite Element Analysis to improve its prediction capability of the criticality of the damage of the structure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nicolas Vigroux.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Sankar, Bhavani V.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024103:00001


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APPROXIMATEMETHODSFOREVALUATINGMODEMIXITYINDELAMINATED COMPOSITES By NICOLASVIGROUX ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2009 1

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c 2009NicolasVigroux 2

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ToMelissa 3

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ACKNOWLEDGMENTS Ioweadebtofgratitudetomyadvisor,sponsor,andfriend,DrBhavaniV.Sankar. Igratefullyacknowledgehissupportandguidance.IalsothankDrYoupingChenand DrAshokV.Kumarforbeingpartofmycommitteeandfortheirvaluablefeedback. Furthermore,Iwouldliketoacknowledgeallmyfamily,inparticularmyparentsClaude andJean-PaulandmysiblingsVincentandMariefortheirunconditionalsupportand love.Withoutthem,thisacademicandresearchexperiencewouldnothavebeenpossible. Finally,IamgreatlythankfultomycolleaguesoftheCenterforStudiesofAdvanced StructuralCompositesandmyclassmatesinthedepartmentofMechanicalandAerospace Engineering.Theirassistanceandfriendshiphavebeeninvaluable. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................12 1.1DenitionofComposites............................12 1.2DamageinDelaminatedComposites.....................12 1.3ScopeoftheStudy...............................14 2LINEARELASTICFRACTUREMECHANICS..................15 2.1StressIntensityApproach...........................15 2.2EnergyCriterion................................17 2.3TheJContourIntegral.............................18 3THEDOUBLECANTILEVERBEAMFRACTURETEST...........20 3.1TheoreticalApproachtoModeMixityAssessment.............20 3.1.1ModeIStrainEnergyReleaseRate..................20 3.1.2Zero-Volume J integral aroundtheCrackTip..............23 3.1.3ModeMixityCalculations.......................25 3.2AlgorithmEssentialtoFEADataProcessing:VirtualCrackClosureTechnique26 3.2.1EnergyReleaseRateDerivation....................26 3.2.2FiniteElementProcess.........................28 3.3AnalyticalApproachtoModeMixityComputation:CrackTipForceMethod31 3.3.1LaminatedBeamEquations......................31 3.3.2ModesofFractureComponents....................34 3.3.3ValidationofCTFM..........................36 4FINITEELEMENTANALYSIS..........................37 4.1The2DDoubleCantileverBeamFiniteElementAnalysis..........37 4.1.1FiniteElementModel..........................37 4.1.2ResultsandDiscussion.........................38 4.1.3DiscrepancyInherentto2DAnalysis.................41 5

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4.2DoubleCantileverBeamPlateElementModel................44 4.2.1FiniteElementProcedure........................44 4.2.2ResultsandDiscussion.........................47 4.3The3DAnalysisofaSymmetricDoubleCantileverBeam.........47 4.3.1Descriptionofthe3DSDCBModel..................47 4.3.2Convergenceof G upontheRenementoftheMesh.........48 4.3.3DelaminationFrontCurveforanArbitraryModeMixity......50 4.3.4PerfectCorrelationamongTheoretical,Analytical,andNumerical Methods.................................52 5MULTI-FIDELITYRESPONSESURFACEFORASYMETRICDOUBLECANTILEVER BEAM.........................................55 5.1ComputerDesignExperiments.........................56 5.1.1The3DFEModeling..........................56 5.1.2The2DAnalyticalMethod.......................57 5.2LowFidelityAnalysiswithHighQualitySurrogates.............57 5.2.1CrackTipForceMethodPolynomialResponseSurface.......57 5.2.2EnergyReleaseRateCorrectionResponseSurface..........59 5.2.3PhaseAngleCorrectionResponseSurface...............61 6CONCLUSIONS...................................63 REFERENCES.......................................65 BIOGRAPHICALSKETCH................................66 6

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LISTOFTABLES Table page 4-1ABAQUSinput....................................38 4-2Beamtheoryand2DFEA G I computationin N:mm )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 .............41 4-3AccuracyofABAQUSdeectionoutput......................42 4-4Outputof2DbimaterialDCBFEanalysis.....................44 5-1Predictioncapabilitiesofenergyreleaseratecorrectionresponsesurfaces....59 5-2Predictioncapabilitiesofphaseanglecorrectionresponsesurface.........62 7

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LISTOFFIGURES Figure page 1-1AA587'stailnrecoveredfromJamaicaBay,CourtesyofNationalTransportation Safetyboard......................................13 2-1Domainofvalidityofmajorfailuremechanisms..................15 2-2Stressesnearthecracktipinanelasticmaterial..................16 2-3Basicfracturemechanicsmodes...........................16 2-4Arbitrarycontouraroundacracktip........................18 3-1Crackedplatesubmittedtoxedloads.......................21 3-2Load-displacementdiagram.............................22 3-3Additivityofenergyreleaserate...........................25 3-4Cracktipvicnityina2Dniteelementmodelbeforethecrackclosure.....26 3-5Virtualcrackclosureina2Dniteelementmodel................27 3-6Crackfrontregionin2Dand3D..........................28 3-7Sublaminatesindelaminatedbeam.........................31 3-8Interfacesforcesatthecracktip...........................33 3-9Crack-tipforceactingonthetopsub-laminates..................33 3-10EnergyReleaseRatewithrespecttomodemixity.................36 4-1The2Dniteelementmodel.............................38 4-2Meshingsystemnearthecracktip.........................39 4-3Stresseldsingularityin 1 p r nearthecracktip...................39 4-4 J integral underplanestress..............................40 4-5 J integral underplanestrain..............................40 4-6FiniteElementanalysisofacantileveredbeamdeection.............42 4-7Deectionofsublaminatesinopeningmode....................42 4-8Meshrenementaroundthecracktip........................43 4-9A2DplateFEmodel.................................45 4-10A2Dplatemodelwithspringsasconnectors....................46 8

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4-11Descriptionofabeamelement............................47 4-12Comparisonof2Dplateand3Dresultsinanopeningmodefracturetest....48 4-13SDCBFiniteElementModel.............................48 4-14AnalysisofthestateofstressinModeI......................49 4-15Convergenceoftheenergyreleaserate.......................49 4-16DelaminationfrontinModeI............................51 4-17StateofstressinModeII..............................51 4-18DelaminationfrontinModeII............................52 4-19ApplicationofasingledeadloadinaFESDCB..................52 4-20DelaminationfrontinanarbitraryMixMode...................53 4-21Correlationoftotal G .................................53 4-22Correlationofthemodemixity...........................54 5-1Assymetricdoublecantileverbeam.........................55 5-2ApplyingcouplestotheADCBlegs.........................57 5-3Polynomialresponsesurfaceoftheenergyreleaserate..............58 5-4Correctionresponsesurface.............................60 5-5Phaseanglecorrectionresponsesurface.......................61 9

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience APPROXIMATEMETHODSFOREVALUATINGMODEMIXITYINDELAMINATED COMPOSITES By NicolasVigroux May2009 Chair:BhavaniV.Sankar Major:MechanicalEngineering Inlaminatedmaterials,suchascomposites,interlaminardamageisoneofthe majormodesoffailure.Duetotheincreasingincorporationofcompositesintoaircraft structures,thiskindofinsidiousdamage,alsocalleddelamination,hasbecomeanair safetyconcern. Therefore,twomainresearcheldshaveemergedtoanswerthesafetyrequirements oftheaviationindustry:therstfocusontothedetectionofdelaminations,whereasthe seconddevelopdelaminationmodelstounderstandwhatconstraintscontroltheinterface crackgrowth. Currentlysophisticatedstructuralhealthmonitoringtoolscanprovideinformationon damageofastructureinrealtime.However,theinformationwillbemostusefulifonecan performastressanalysisofthedamagedstructureinreal-timeallowingprognosisofthe lifeofthestructure. Thus,theaimofthestudyistoimplementhigh-delitydelaminationnite elementmodelsandcomparethemwithspeedyanalyticalmethodstoallowagainin computationalspeedwithoutlosingmuchaccuracy. FractureMechanicsprinciplesareusedintheanalysisof2Dand3Dniteelement modelsthatwereconductedonabasicinterfaceproblem:thedoublecantileverbeam fracturetest.Furthermore,numericalmethods,suchastheVirtualCrackClosure Technique,areimplementedinthosemodelstoassessthetotalenergyreleaserate 10

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alongwiththemodemixity,whichpermitstofullycharacterizethedelaminationatthe cracktip.Finally,inamultidelityapproach,2DanalyticalCrackTipForceMethod thatprovidesinexpensivecomputingsolutionsiscomparedtothehigh-delity3DFinite ElementAnalysistoimproveitspredictioncapabilityofthecriticalityofthedamageof thestructure. 11

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CHAPTER1 INTRODUCTION 1.1DenitionofComposites Acompositematerialisacombinationofdierentmaterialsthatresultsinasingle structurewithadistinguishableinterface.Sincemanyindustrialapplicationse.g., aerospaceandautomotivestructuresrequireanincreasinglevelofperformanceandat thesametimeweightreduction,compositematerialswithnovelberperformsandtough epoxyresinarebeingdeveloped. Thosepropertiesdependontheconstituentmaterialsbutalsoontheinterface. Therefore,theoptimizationofthecompositematerialsselectionandmanufacturing processeshasledtoawiderangeofapplications.Forinstance,theycanbeextremely valuableeitherasacorematerial,highstrengthandstinessskinsorouterprotective layers. Becauseofvariouscharacteristicstheycanoer,theycanbeassociatedtoforma laminatestructureinwhichlaminaearejoinedtogetherwithprecured,prepregorwet lay-upcongurations.Amongthoseprocesses,theprepreglay-uponeisthemostcommon intheaerospaceindustry,principallybecauseitallowstheproductionofhighbervolume fractioncompositepartse.g.,largeglass/epoxy/honeycombsandwichfairingsforthe Airbus330/340aptracks.However,duringthismaturemanufacturingprocess,even thoughsqueezingrollersareusedtoremoveairentrappedbetweenprepregsheets,the fabricationofdefect-freepartsremainsachallenge.Hence,insuchaircraftstructures,due totheexistenceofthosemicrovoidslocatedatthelaminarinterface,stressconcentration eectscanleadtodebondingdamage:amajorfailuremodeinlaminatedcomposites. 1.2DamageinDelaminatedComposites Interlaminardamage,alsocalleddelamination,representsactuallyanairsafety concernbecausethisinsidiouskindoffailureisapotentialsourceofhumantragedy. Indeed,dierentfactorsasvariousascyclicstresses,lowvelocityimpacts,manufacturing 12

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defects,lightningorbirdstrikes,cancontributetothepropagationofinterfacecracks. Undernomonitoringassistanceofcrackextension,thismodeoffailurecanthenbecome catastrophic.Forinstance,followingthecrashofacommercialairplaneinQueensN.Y., onNovember12 th 2001,investigatorshavefoundthatthekillingofall260peopleinthe planeandvepeopleontheground,canbetracedtosuchadefectinthecarbon/epoxy compositetailsectionofAmericanAirlinesight587.Thefailureoftheattachmentpoints thatheldtheverticalstabilizertothefuselageisobviousinFigure1-1. Figure1-1.AA587'stailnrecoveredfromJamaicaBay,CourtesyofNational TransportationSafetyboard Therefore,tofacethatchallenge,alotofinterestisfocusedintothedetection andcharacterizationmethodofdelamination.Asthismodeoffailureisnotvisible, non-destructivetechniques,suchasultrasonicinspection,radiographyorvibrationand thermalmethodsareunderconsideration.Theultimategoalofthosestudiesistogivean accuratestrainmappingdistributionofthewholestructure.Thosedatacanthenbeused bycomputationalcodesrunningmodelsofdelaminationbehaviortoassessthecriticality ofeachdefectofthestructures.Consequently,aremotemonitoringassistanceofaircrafts isonlypossibleifthosecomputationscanberealizedinrealtime:thepilotusuallyneeds toreactquickly! 13

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1.3ScopeoftheStudy Thepurposeoftheresearchistoestablisharelation,albeitempirical,between thehigh-delitydelaminationmodelsandfastanalyticalmodelstogainspeedinthe calculationofthecriticalityofinterlaminardefectswithoutlosingaccuracy.Toassessthe criticalityofthosestaticpre-existingcracksoraws,aLinearElasticFractureMechanics LEFMapproachisfollowed.Itsprincipleswillpermitustoanalyzethedelamination modelsthatweareimplementing. Thechapter2ofthisthesisisdevotedtoLEFMfundamentalprinciples:anapproach basedonenergybalance,thatisshowntobeequivalenttoclassicalstressanalysisof cracks,introducesglobalfractureparametersusefulinthesafetyassessmentofany crackedstructure.Energycriterionispreferredthroughoutthestudyasabettermeans tocharacterizemodesoffracture.Asaresult,thoseprinciplesareheavilyusedinchapter 3,throughthederivationoftheoretical,analyticalandnumericalmethodsthatallowthe evaluationofmodemixityinabasicinterfaceproblem:adoublecantileverbeamfracture test. Inchapter4,theVirtualCrackClosureTechniqueisimplementedintodierentnite elementmodelstosimulatethemechanicalbehaviorofacrackedbody.Theirrespective accuracywastestedbycomparingtheglobalfractureparametersobtainedtotheoretical ones.Here,limitationsassociatedto2DFEmodelingwerehighlightedwhereas3D FEanalysiswasprovedtoaccuratelypredictthedelaminationbehavior,butatahigh computationalcost. Suchaccuratepredictionisusedinchapter5toimprovethepredictioncapability ofthespeedyanalyticalCrackTipForceMethod.Theintroductionofmulti-delity correctionresponsesurfacespermitstogainincomputationalspeedwithoutlosingmuch accuracy. 14

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CHAPTER2 LINEARELASTICFRACTUREMECHANICS ThefundamentalconceptsofLEFMonlyappliedtomaterialsthatobeyHooke'slaw. Indeed,thefracturetoughnessofcompositesisusuallylowenoughtoallowthecollapseof thestructurebeforeanyonsetorappearanceofplasticityeects.Figure2-1showsthatin thislineardomain,suchstructureexperiencesabrittlefracture. Figure2-1.Domainofvalidityofmajorfailuremechanisms Moregenerally,inlayeredstructureanalysis,theLinearElasticFractureMechanics remainsapplicablewhenthefailurezoneismuchsmallerthanthesmallestdimensionof thespecimen,whichisusuallythelaminate'sthickness.Twoequivalentapproachesto LEFMprevailed:theenergyandstressintensityapproach,whicharedescribednext,to providenecessarybackground. 2.1StressIntensityApproach Whenanylinearelasticcrackedbodyissubjectedtoexternalforces,astressanalysis showsthatinacoordinateaxis,asdenedinFigure2-2,thestresseldcanbeexpressed asaseriesin 1 p r n .Therefore,inthevicinityofthecracktip,asthehigherorderterms vanish,arst-orderasymptoticapproximationEq.2{1isvalid:thestresseldsolely 15

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Figure2-2.Stressesnearthecracktipinanelasticmaterial dependson 1 p r 8 > > > > < > > > > : x y z 9 > > > > = > > > > ; = K I p 2 r 8 > > > > < > > > > : f 1 f 2 f 3 9 > > > > = > > > > ; + K II p 2 r 8 > > > > < > > > > : g 1 g 2 g 3 9 > > > > = > > > > ; {1 Thelocalfractureparameters K i introducedinEq.2{1arecalledstressintensityfactors. Thesubscripts i denotethefracturemodesassociatedtospecicloadingconditions,as describedinFigure2-3. f and g arefunctionsthatdependonthecrackconguration.An exhaustivelistofpossiblecongurations,alongwithcorrespondingfunctionsaredetailed inthe"stressanalysisofcrackshandbook"[1]. Figure2-3.Basicfracturemechanicsmodes 16

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Accordingtotheprincipleofsuperposition,anyloadcongurationcanbesplitinto thosethreedierentmodes.Thelatterarecalledmodesoffracturebecausefailureofthe structurecanoccurifthestressintensityfactor K i associatedtoaspecicmode i exceeds alimit K ic ,thatdenesthefracturetoughnessofthematerialinaspecicmode.Thus, thestructureremainssafeundertheconditionsofEq.2{2. i 2f 1 ; 2 ; 3 g f K I K Ic ; K II K IIc < 1{2 wherefunctionfdependsontheparticularfracturecriterionused.Hence,todetermine whichmodeoffractureispredominant,amodemixityparameterhasbeenintroduced:the phaseangleEq.2{3,whichisthemeasureofmodeIItomodeIloading[2]. =tan )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K II K I {3 Ananalogparametercanalsobedenedwiththeout-of-planeshearingmode.The knowledgeofmodemixityalongwithtotalstressintensityfactorfullycharacterizesthe stressbehavioratthecracktip. 2.2EnergyCriterion Theenergycriterionforfracture,rstproposedbyGrith[3],statesthatifthe energyavailableforacrackgrowthisgreaterthanthematerialresistance,thenfracture occurs.Furthermore,whenthecrackextends,newsurfacesarecreatedintheelasticbody: theenergythathasbeenbroughttothesystembytheapplicationofasetofexternal forcesisreleased.Thatrateofchangeinpotentialenergywithrespecttothecrackareais theStrainEnergyReleaseRateSERR G G = )]TJ/F21 11.9552 Tf 10.494 8.087 Td [(d dA {4 Here,toguarantythesafetyofthestructuretheglobalparameter G mustsatisfyan analogcondition: i 2f 1 ; 2 ; 3 gG i < G ic {5 17

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where G ic representsanalternativedenitionofthefracturetoughness.Duetothe orthogonalityofthefracturemodes,underlinearelasticassumption, G canbebroken downintoitscomponentsineachmodeoffracturebythelinearrelationshipderivedin Eq.2{6. G = G I + G II + G III {6 Furthermore,Irwincracksclosureanalysis[4]establishedarelationshipbetweenthosetwo approachesEq.2{7. G = K 2 I E + K 2 II E + K 2 III 2 {7 inwhich E = E forplanestress, E = E 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 and istheshearelasticmodulus.Eq.2{7 isderivedundertheassumptionofaselfsimilarcrackgrowth.Inotherwords,theshape ofthecrackmustremainthesamealongitsstraightaheadpropagation.Thathypothesis isusuallynotmetformixedmodefracture,inwhichcrackkinkingcanoccur.However,in quasi-staticstate,suchconsiderationisirrelevant.Therefore,energyandstressintensity approachesarefullyequivalentforlinearelasticmaterials. 2.3TheJContourIntegral The J integral onanarbitrarycontour)-326(aroundthetipofthecrack,asshowninFigure 2-4,hasbeenpresentedbyRice[5]asthepath-independentintegralofEq.2{8. Figure2-4.Arbitrarycontouraroundacracktip J = Z )]TJ/F15 11.9552 Tf 7.779 10.793 Td [( wdy )]TJ/F21 11.9552 Tf 11.955 0 Td [(T i @u i @x ds {8 18

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where w isthestrainenergydensity )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(w = R ij 0 ij d ij T thetractionvector T i = ij n j and u thedisplacementvector.Fornonlinearandelasticmaterials,itispossibletoshow underquasi-staticconditionsthattheJcontourintegralisequivalenttothetotalenergy releaserate )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J = )]TJ/F22 7.9701 Tf 10.494 4.708 Td [(d dA = G ;anelegantproofisderivedbyAnderson[6]. 19

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CHAPTER3 THEDOUBLECANTILEVERBEAMFRACTURETEST Thisfracturetestisabasicinterfacialfractureproblemthatdescribesprettywell thegeneralbehaviorofacompositeinterlaminardamagebecauseeachplyofacomposite cantypicallybemodeledashomogeneous.Furthermore,specicgeometricconditionson thecrackandligamentlengthofthatstructure,respectivelydenotedby a and W )]TJ/F21 11.9552 Tf 12.35 0 Td [(a in Figure3-1,canreducethesizeoftheplasticzone,thatisknowntobeproportionalto K Ic Ys 2 [6].Thus,ifthosetwoparametersarebigenough,alargezoneaheadofthecrack tipisK-controlled,whichmeansthatthestressintensityfactor K totallycharacterizes stressesnearthecracktip.Inaddition,theexternalforcesappliedareselectedtoavoid reachingthemaximumloadapplicabletothestructure.Consequently,thecrackdoesnot propagatefastandthecrackedbeamcanthenbestudiedunderquasi-staticassumption. Therefore,theabovefracturetestwillbedescribedundertheassumptionofalinear elasticmaterialunderquasi-staticbehavior,hypothesisthatallowstheequivalency betweentheenergyreleaserate G andtheJcontourintegral J integral 3.1TheoreticalApproachtoModeMixityAssessment 3.1.1ModeIStrainEnergyReleaseRate InFigure3-1,thegeometricparametersofthefacturetestinwhichforcesareapplied toeachlegofthebeam,aredetailed.Inthissection,thetopandbottomforceswill beassumedtohaveequalnormsandoppositedirections.Asseenearlier,thatloading conditioniscalledMode I oropeningmode. Irwin[7]denedstrainenergyreleaserate G astheenergyrequiredtoextendacrack byasmallincrementEq.3{1. G = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(d dA {1 inwhichisthepotentialenergyand A theareacreatedbytheextensionofthecrack. GrithEnergybalance[3]statesthatcrackextensionoccurswhen G reachesacritical value,alsocalledfracturetoughnessofthematerial.Indeed,thisenergybalanceEq.3{2 20

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Figure3-1.Crackedplatesubmittedtoxedloads showsthattheenergyrequiredtocreatenewsurfacesequalsthestrainenergythatis released. dE dA = d dA + dW s dA =0{2 where E isthetotalenergyand W s theworkneededtocreatenewsurfaces.Therefore, thecriticalstrainenergyreleaserate G c canbeexpressedas G c = dW s dA =2 s {3 where s istheenergyneededtocreateanewsurface.Asthecrackextends,twonew surfacesaredefactocreatedonbothsidesofthecracktip,whichexplainedthepresence ofthenumber2intheaboveequation. AsshowninFigure3-2,theforceappliedtobothligamentsofthedoublecantilever beamisxed:theplateisthenload-controlled.Thus,thecrackpropagatesbya smallincrementataxedloadbeforeunloadingthestructure,asdepictedinthe load-displacementdiagraminFigure3-2.ThepotentialenergyisdenedinEq.3{4. = U )]TJ/F21 11.9552 Tf 11.955 0 Td [(W F {4 21

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where W F istheworkdonebyexternalforceand U isthestrainenergystoredinthe body.Inthisspecicdoublecantileverbeamtest, W F = F and U = R 0 Fd = F 2 Consequently,= )]TJ/F21 11.9552 Tf 9.298 0 Td [(U .SubstitutingthatexpressionintoEq.3{1,weobtainEq.3{5. G = dU dA F {5 Figure3-2.Load-displacementdiagram Figure3-2showsthattheincreaseofthestrainenergyduetotheextensionofthe crackcanbederivedas dU F = F 2 + Fd )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(+ d F 2 = Fd 2 {6 Accordingtobeamtheory,arelationshipisestablishedbetweentheforce F andthe displacementEq.3{7. = 2 Fa 3 3 EI {7 22

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where I = Bh 3 12 and E istheYoungModulusofthehomogeneousmaterial.Substituting thatresultintotheexpressionofGderivedinEq.3{5givesEq.3{8. G = F 2 B d da F = F 2 a 2 BEI {8 Finally,substitutingthemomentofinertiaI,asderivedabove,intoEq.3{8permits toreachanexpressionoftheModeIstrainenergyreleaserate G I infunctionofthe loadingconditions,thematerialandgeometricpropertiesEq.3{9. G I = 12 F 2 a 2 B 2 h 3 E {9 3.1.2Zero-Volume J integral aroundtheCrackTip The J integral ispath-independent.Therefore,thecontouronwhichthelineintegral isdenedcanbemovedveryclosetothecracktip.Let'sthenconsiderthreevertical pathsAB,CDandEFrespectivelynumbered1,2and3,nexttothecracktipasshownin Figure3-1.Thosepathsformedaltogetheraclosedlinealongthecracktip.As n x ds = dy and T i = ijn j ,thelineintegraldenedinEq.2{8canbewrittenas J = Z )]TJ/F15 11.9552 Tf 7.779 10.793 Td [( wn x )]TJ/F21 11.9552 Tf 11.955 0 Td [( ij n j @u i @x ds {10 Alongthepath1,usingthedenitionofthestrainenergy,wederiveEq.3{11. J = Z B A 1 2 ij ij n x )]TJ/F21 11.9552 Tf 11.955 0 Td [( ij n j u i;x ds {11 Forthatparticularpath, n x = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1and n y =0.Substitutingthenormalvectorcomponents givesEq.3{12. J = Z B A )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 xx x + xy xy + xx u ;x + xy v ;x ds {12 Usingstrainsdenitions x = u ;x and xy = 1 2 u ;y + v ;x givesEq.3{13. J = Z B A 1 2 xx u ;x + 3 4 xy v ;x )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 4 xy u ;y ds {13 23

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whichcanberearrangedEq.3{14tomakeappearedthestrainenergydensity w andthe rotation t = 1 2 u ;y )]TJ/F21 11.9552 Tf 11.955 0 Td [(v ;x ofthebeamcrosssectionatthecracktip. J = Z B A 1 2 xx x + xy xy )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 xy u y )]TJ/F21 11.9552 Tf 11.955 0 Td [(v ;x ds {14 Next,aseparationoftheintegralisrealizedEq.3{15. J = Z B A wds )]TJ/F15 11.9552 Tf 11.955 0 Td [( t Z B A xy ds {15 Atthisstepofthecomputation,weidentifytheshearforce V .Hence,theexpressionof J integral alongthepath1issimpliedEq.3{16. J = w L )]TJ/F15 11.9552 Tf 11.956 0 Td [( t V 1 {16 where w L representsthestrainenergyperunitlengthalongthepath1.Byanalogy, alongthepath2,thesameformisobtainedEq.3{17. J = w L )]TJ/F15 11.9552 Tf 11.955 0 Td [( t V 2 {17 Forthelastpath, n x =1,whichresultsinachangeofsignEq.3{18. J = )]TJ/F21 11.9552 Tf 9.298 0 Td [(w L + t V 1 {18 Asthepathslieinthevicinityofthecracktip,theshearforceresultantsmustsatisfy, byanargumentofcontinuity,theequilibriumcondition V 1 + V 2 = V 3 .Addingthe threeintegralspreviouslyderivedtheshearforcesvanishfromtheresultingequation.An equationisconsequentlyderivedbetweenthe J integral andthesolestrainenergydensities, expressedhereperunitlength. J = J + J + J = w L + w L )]TJ/F21 11.9552 Tf 11.955 0 Td [(w L {19 Therefore,forelasticmaterialsunderquasi-staticassumptions,theequivalency between J integral and G ,permitstoreachaconvenientwaytocomputethestrainenergy 24

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releaserateinadoublecantileverbeamfracturetest:thelatteristhedierencebetween thestrainenergydensitiesperunitlengthjustbehindandaheadofthecracktip. 3.1.3ModeMixityCalculations Sincewearedealingwithlinearelasticfracture,principleofsuperpositionis applicable.Thus,consideringasetofforcesasdescribedinFigure3-3,allmodemixities canbeassessedthroughthecomputationofthestrainenergydensityalongeachcross section,behindandaheadofthecracktipinadoublecantileverbeamfractureanalysis. Figure3-3.Additivityofenergyreleaserate Behindthecracktip,thestrainenergydensityperunitlengthineachcrosssection ofthebeamisidenticalinbothpuremodes: w L = w L = M 2 2 EIB ,whereasthatquantity is,aheadofthecracktip,eithernullinopeningmodeorequalto w L = M 2 4 EIB inshearing mode.SubstitutingintoEq.3{19theaboveexpressionofthestrainenergydensitiesper unitlengthgivesusanexpressionforpuremodesstrainenergyreleaseratesEq.3{20. G I = J I = a 2 )]TJ/F22 7.9701 Tf 6.675 -4.871 Td [(F 1 + F 2 2 2 EIB ; G II = J II = 3 a 2 )]TJ/F22 7.9701 Tf 6.675 -4.871 Td [(F 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(F 2 2 2 4 EIB {20 Asamatteroffact,theaboveresultisthesamethantheonepreviouslyderivedinpure Mode1usingbeamtheoryEq.3{8.Inanarbitrarymodeoffracture,calledmodemix, inwhich F 1 and F 2 arerespectivelyappliedonthetopandbottomhalf,asshownin Figure3-3,theenergyreleaseiscomputingbyaddingitsmodecomponentsEq.3{21. G MixMode = a 2 8 EIB 7 2 F 2 1 + F 2 2 + F 1 F 2 {21 25

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3.2AlgorithmEssentialtoFEADataProcessing:VirtualCrackClosure Technique ThevirtualCrackClosureTechniqueVCCTpermitstoassesswithasimple algorithmthefractureparametersforallthreefracturemodes.Wewilldiscusslaterthe accuracyofsuchtechniqueinaniteelementanalysisofa3Ddoublecantileverbeam modelimplementedonABAQUS.Atthispoint,wewillfocusontotheprincipleson whichthetechniqueisbasedbeforedetailingthevariousstepsofthealgorithmandits implementationinthecommercialsoftwareMATLAB. 3.2.1EnergyReleaseRateDerivation Accordingtoenergyconservationprinciples,Irwin[4]assumesthattheenergy releasedbyanincrementalcrackextension a shouldbeequalthattheamountofenergy necessarytoclosethecracktoitsinitialstate.Inatwodimensionalconguration,suchas theoneshowninFigure3-4,thisworkcanbecomputedas W = 1 2 Z a 0 u r r )]TJ/F15 11.9552 Tf 11.955 0 Td [( a dr {22 where isthestressand u therelativedisplacementatadistance r behindthecracktip. Figure3-4.Cracktipvicnityina2Dniteelementmodelbeforethecrackclosure 26

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Figure3-5.Virtualcrackclosureina2Dniteelementmodel Therefore,inquasi-staticstate,theenergyreleaserateis G =lim a 0 W a =lim a 0 1 2 Z a 0 u r r )]TJ/F15 11.9552 Tf 11.955 0 Td [( a dr {23 RybickiandKanninen[8]showedthattheexpressioninEq.3{23canbecomputed numericallythroughniteelementanalysis.Inniteelement,sincenoforcesare transferredthroughelementedges,forcesneedtobeappliedonsharenodes,asillustrated inFigure3-4inordertoclosethecrackandreachthestatedescribedinFigure3-5.The correspondingwork W neededis W = 1 2 Fu {24 where F istheforcerequiredtomaintaintogethernodes i and i and u thedistance betweenthosenodesbeforetheclosure.Thisforcecanbeapproximatedbytheforceat node j oftheelement2.Thisinternalforcewhichisequalwithanoppositedirectionthan theforceatnode j oftheelement4canbecomputedinthesamestepanalysisfromthe stressintheelement.Consequently,theanalysisoftheniteelementwiththecrackclosed Figure3-5isnotneeded:thisisoneofthemainadvantageofthattechnique[9]. 27

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Iftheniteelementmeshisrenedenoughtoallowtheconvergenceofthelimitin Eq.3{23,thestrainenergyreleaseratecanbecomputedfortheModesIandII,only modesavailablein2D.ThiscalculationisledusingEq.3{25. G i = 1 2 B a F i u i {25 inwhichthesubscript i indicatesthetypeofloadingconditions.Inopeningmodei=I, theforceisnormaltothecrackplanewhereasinshearingmodei=II,theforceis parallel.Inbothcases,thedirectionofthedisplacementiscollineartothecorresponding forceapplied. Shivakumarextendedtherangeofapplicationfrom2Dto3D[10],whichallowsnot onlyagaininaccuracybutalsoaModeIIIassessment.Theintegralformulationtakesthe followingform G i =lim a 0 1 2 w i a Z i +1 i )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 Z a 0 u r;s r )]TJ/F15 11.9552 Tf 11.955 0 Td [( a;s drds {26 wherethecouples s w i and r a arethedistanceandtheelementlengthrespectively alongandnormaltothecrackfront; isthestressdistributionaheadofthecrackfront and u therelativedisplacementbehindthecrackfront. 3.2.2FiniteElementProcess Figure3-6.Crackfrontregionin2Dand3D 28

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Ina3Dsolidniteelementmodelcomprising20-nodesbrickelementCRD20R,we considertheassociationoftwoelementslocatedabovethecrackplane,onbothsidesof thecracktip.ThenodesofthecracktipinterfaceelementsareshowninFigure3-6.The applicationofEq.3{26insuchstructuregivesEq.3{27. G = 1 2 2 3 X i =1 F c i a i )]TJ/F21 11.9552 Tf 11.955 0 Td [(A i + 2 X i =1 F d i b i )]TJ/F21 11.9552 Tf 11.955 0 Td [(B i {27 whereisboththewidthandthelengthofthequadraticelement. a i A i b i and B i are thepositionvectorsofthebrick-elementnodes,asdepictedinFigure3-6.Therefore,the dierencescomputedinEq.3{27permitstoassesstheopeningandshearingdistancesat adistanceand 2 fromthecracktip,respectivelyalongthedirectionzforopeningand alongxandyforin-planeandout-of-planeshearing.Thosedistancesaremultipliedby theforces F c i and F d i calculatedatnodes c i and d i respectivelyinordertoevaluatethe workneededtoextendthecrackbythelengthoftheelement.Eachforceiscomputed usingABAQUSNFORCoutputthatisthenodalforcecausedbythestressinaspecic element.Whenanodeislocatedonthecracktip,theassociatedforcepresentedinEq. 3{27isthesumofthenodalforcefrombothtopelementsaheadandbehindthecracktip. Anotherwaytodothecalculationistoconsidertwohalvesofelementsonbothsides ofthecracktip.Theoriginofthatnewreferenceelementisthusshiftedby 2 inthe directionofthecracktip.InFigure3-6,itsmiddleisthenode c 3 anditslengthisthe distance= c 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 2 .Withthisnewreference,theenergyreleaserateisgivenby G = 1 2 2 F c 3 a 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A 3 + F d 2 b 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(B 2 + 1 2 F c 2 a 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A 2 + F c 4 a 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A 4 {28 Inthatcase,asthereferenceelementdoesnotcoincidewiththeelementsofthe niteelementmodel,theforce F c 2 iscomputedaddingthenodalforcesofthe4FE elements,locatedonthetopofthecrackplane,thatsurroundthisnode.Duetothe fundamentalFEequation KU = F inwhichinternalforcesareunloadedtosatisfy theequilibrium,suchnodalforceisoppositetotheonethatonecouldcomputetaking 29

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intoconsiderationthesetof4elementslocatedbelowthecracktipplane.Thosetwo dierentapproximationsgivesensiblythesameresults,especiallyifanemeshhasbeen implemented. Inordertosplittheenergyreleaserateintothethreemodesoffracture,itisrequired toderiveeachofthemusingtherespectiveforceanddisplacementcomponentsofeach directionofthecoordinateaxisdenedinFigure3-6.Suchapproximationoftheenergy releaseratecomponentsisexpressedinEq.3{29[11]. G I = F z w 2 B a G II = F x u 2 B a G III = F y v 2 B a {29 ImplementationoftheVCCTinMATLAB TheimplementationoftheVirtualCrackClosureTechniquehasbeendoneusing thecommercialsoftwareMATLAB.Theinputnecessarytorunsuchcodearetheoutput oftheniteelementmodel,i.e.thenodalforcesandthedisplacementsthatwehave denedforeachunitcell.ThoseABAQUSoutputsareaccessiblethroughaHistory outputrequest.Then,itispossiblefromABAQUStostoreallforcesinavectorcolumn, whereeachlineisassociatedtothenodeofanelement.Understandinginwhichorder ABAQUSlistthe20nodesofeachelementallowstheextractionoftheforcecomponents thatappearinEq.3{27andEq.3{28. Tocomputethedisplacementcomponents,ananalogousprocedureisfollowed:the positionofeachnodethatisusedtocomputedisplacementcomponentsareextracted beforebeingrearrangedinaconvenientway.Finally,thecomputationoftheenergy releaserateisdoneusingthedatastoredforeachelementalongthecracktip.This algorithmpermitsustoobtainthedelaminationfrontcurve,i.e.thestrainenergyrelease rateinfunctionofthewidthofthebeam. 30

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3.3AnalyticalApproachtoModeMixityComputation:CrackTipForce Method Inthissection,wedescribesimpleandclosed-formsolution,rstproposedbySankar [12],ofthemodeIandmodeIIenergyreleaseratesforthegeneraldelaminatedbeam problem,asdepictedinFigure3-7. Figure3-7.Sublaminatesindelaminatedbeam Theenergyreleaseratesareexpressedintermsofallaxialloads,bendingmoments andshearforcesappliedatthecrosssectionofthecracktip. 3.3.1LaminatedBeamEquations Wearehereconsidering,inthederivationofthesetofequationsthatpermitsto computeenergyreleaseratesandmodemixity,thexy-planeasthereferenceplanefor bothtopandbottomlayers;thisassumptionpermitsusnottointroduceanyosetvalues alongthecomputation.Accordingtotheclassicalbeamtheory,thedisplacementeldfor laminatedcompositebeamisoftheform u x;z = u 0 x + z x w x;z = w 0 x {30 Intheabovein-planedisplacementequation, u 0 x istheaxialdisplacementof pointsinthereferenceplaneand, x therotationaboutthey-axis.Furthermore, thetransversedisplacementissupposedtostayconstantthroughthethickness.Strain 31

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relationsderivedfromthepreviouslydescribeddisplacementeldare xx u ;x = u 0 ;x + z ;x {31 where ;x denotespartialdierentiationwithrespectto x Foreachorthotropiclayeroflaminatedcompositebeam,thestress-strainrelationsare xx = Q 11 xx xz = Q 55 xz {32 inwhich Q 11 and Q 55 arestinessterms.Forplanestrainwithaberinthesame orientationthanthedirectionofthebeam,thosetermsaredenedby Q 11 = E 11 )]TJ/F22 7.9701 Tf 6.587 0 Td [( 12 21 and Q 55 = G 13 Forthetoplayer,theforceandmomentresultantsaredenedas F T = b P;M;V c = b Z h 1 0 b xx ;z xx ; zx c dz {33 inwhich P M and V arerespectivelytheaxial,bendingandshearforceresultants; b and h 1 arethewidthandthethicknessofthetoplayer.Thestressresultantsarerelatedto strainsvia F =[ S ] e wherethecompliancematrix[ S ]isgiveninEq.3{34. S = 2 6 6 6 6 4 a 11 b 11 0 b 11 d 11 0 00 a 55 3 7 7 7 7 5 {34 e T = x 0 x xz {35 Thestinesstermsappearingabovearedenedas b a 11 ;b 11 ;d 11 ;a 55 c = Z h 1 0 Q 11 ;z Q 11 ;z 2 Q 11 ; Q 55 dz {36 x and arerespectivelythebeamcurvatureandtheshearcorrectionfactorthatcan beassumedas5/6accordingtoSankaretal[13].Forthebottomlayer,thoseequations 32

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arestillvalid,exceptfortheboundaryoftheintegralthatneedstotakeintoaccountthe dierentthickness. Figure3-8.Interfacesforcesatthecracktip Nearthecracktip,afreebodydiagramanalysispermitstowritetheequilibrium betweenforceandmomentresultants F 3 + F 4 = F 1 + F 2 {37 Sincethesublaminates3and4areintactaheadofthecracktip,thestrainsareidentical. ThisequivalenceisstatedinEq.3{38. e 3 = e 4 =[ C t ] F 3 =[ C b ] F 4 {38 InEq.3{38,[ C t ]and[ C b ]arerespectivelythestinessofthetopandbottomlayers andtheyareequalto[ S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 t ]and[ S )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b ],respectively. Solvingfor F 4 ,weobtainEq.3{39. F 4 =[ C t ] )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 [ C b ]+ I )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 F 1 + F 2 {39 Figure3-9.Crack-tipforceactingonthetopsub-laminates 33

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FromthefreebodydiagramequilibriumofFigure3-9,anexpressionofthecracktip forceisderivedinEq.3{40. F ct = F 1 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F 3 {40 UsingtherelationshipsderivedinEq.3{40andEq.3{37,anewexpressionofthe cracktipforce F ct isreachedEq.3{41.Thatnewformisadvantageousbecausetheonly variablesarethematerialpropertiesexpressedinthestinessterms C t and C b andtheset ofexternalloadsappliedtothebody F 1 and F 2 F ct = F 4 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F 2 =[ C t ] )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 [ C b ]+ I )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 F 1 + F 2 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F 2 {41 3.3.2ModesofFractureComponents Assumingquasi-staticconditionsandelasticpropertiesforthematerialconsidered, theenergyreleaserate G isequalto J integral .Therefore,usingtheexpressionofthe zero-volume J integral aroundaclosedcontourinthevicinityofthecracktipforadouble cantileverbeam,statedinEq.3{19,anewrelationEq.3{42isestablishedforthetotal energyreleaserate. G = w L + w L )]TJ/F21 11.9552 Tf 11.956 0 Td [(w L )]TJ/F21 11.9552 Tf 11.955 0 Td [(w L {42 where w i L isthestrainenergydensityperunitlengthforaspecicsub-laminate,as denedinEq.3{43. w L = 1 2 F T [ C ] F {43 inwhich F isthevectorforceasdescribedinEq.3{33and[ C ]isthestinessofthe sublamintes.[ C ]iscomputedtakingtheinverseofthecompliancematrix[ S ]Eq.3{34, whichdependsofthesublaminate.Substitutingthatexpressionofthestrainenergy densityintoEq.3{42givesEq.3{44. G = 1 2 )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(F T 1 [ C t ] F 1 + F T 2 [ C b ] F 2 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F T 3 [ C t ] F 3 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F T 4 [ C b ] F 4 {44 34

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TheuseofthestrainidentityrelationsipderivedinEq.3{38andequilibriumbetween forcespresentedinEq.3{37allowsustoestablishanewrelation3{45. G = 1 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( F 3 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F 1 T [ C t + C b ] F 3 )]TJ/F37 11.9552 Tf 11.955 0 Td [(F 1 {45 SubstitutingEq.3{40intoEq.3{45,weobtainamorecompactformEq.3{46. G = 1 2 F T ct C t + C b F ct {46 Iftheforceatthecrack-tipisrepresentedbythecolumnvector F T ct = F x CF z ,the energyreleaseratecanbedecomposedintodierentfracturemodes. G = 1 2 F x CF z 2 6 6 6 6 4 c 11 c 12 0 c 12 c 22 0 00 c 66 3 7 7 7 7 5 8 > > > > < > > > > : F x C F z 9 > > > > = > > > > ; = 1 2 c 11 F 2 x + c 12 F x C + 1 2 c 22 C 2 + c 66 F 2 z {47 Thecomponentsoftheenergyreleaseratecanbesplitintothetwofracturemodes. Themode III doesn'tinterveneherebecauseourmodelisplanar.Theforce F x provokes thetwolaminatesofthedoublecantileverbeamtoslideoneachother:ithasthena shearingeect.Moreover,thecouple C andtheforce F z areresponsiblefortheopening ofthecrack:thosecomponentsbelongtoModeIeect.Therefore,thecomponentsofthe energyreleaseratecanbeexpressedasinEq.3{48. G I = 1 2 c 22 C 2 + c 66 F 2 z ; G II = 1 2 c 11 F 2 x + c 12 F x C {48 Finally,fromtheenergyreleaseratemodecomponents,themodemixitycanbe computedundertheassumptionofalinearelasticmaterialEq.3{49. =tan )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K II K I = tan )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 r G II G I {49 35

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Figure3-10.EnergyReleaseRatewithrespecttomodemixity 3.3.3ValidationofCTFM TheenergyreleaseratecomponentsstatedinEq.3{48havebeencomputedusing Matlabfordierentloadcongurations.Tochecktheaccuracyoftheresultsobtained,the samecalculationhasbeenledusinganotheranalyticalmethoddetailedbyHutchinson andSuointheirreviewon"Mixedmodecrackinginlayeredmaterials"[2]. Thetotalstrainenergyreleaseobtainedbybothmethodsisidentical:thestandard deviationofthosetwodistributions,whichareplottedingure3-10,isindeedaround0.2. Moreover,anevenbetterequivalencyisreachedforthephaseangle:thestandard deviationisinthiscaseclosedto3 : 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 36

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CHAPTER4 FINITEELEMENTANALYSIS Aninterlaminardamagestudywasconductedinadoublecantileverbeam.The purposeoftheniteelementanalysis,realizedusingtheversion6.7ofthesoftware ABAQUS,istoassesswithhighaccuracythemodemixityfordierenttypesofloading conditions,geometricandmaterialproperties.First,weproposed2Dniteelement modelsinordertoshowtheinherentdiscrepancyofeitherplanestressorplanestrain 2-dimensionnalanalysis.Then,a2Dplatemodelisintroduced.Theapplicationofthe VCCTalgorithmonthatmodelpermitstoobtaintheenergyreleaseratecomponents alongthewidth.Finally,a3Dniteelementmodelofanhomogeneous,isotropic, symmetricdoublecantileverbeamiscreated.Themodemixityisthencomputedthrough theimplementationofthesamenumericalmethod:theVirtualCrackClosureTechnique. Thathighdelityanalysiswillbecomparedtoanalyticaltechniquessuchasthe CrackTipForceMethodwhosemainadvantageisbasedonitslowcomputationcost.As amatteroffact,combininghighdelityof3Dniteelementanalysiswithexpensiveness analyticalmethodswillallowustobenetfrombothcomputations. 4.1The2DDoubleCantileverBeamFiniteElementAnalysis Thepurposeofthe2Danalysisistoassesstheeectsoftheloadingforcesona crackedplateandtheaccuracyofmodemixitycalculationinmodelingincludingmesh elementsthatallowalownumberofcomputationstepsincomparisonwithcumbersome 3Dniteelementmodels. 4.1.1FiniteElementModel First,letusconsiderasymmetricdoublecantileverbeamtestinopeningmode. Forthatcase,thesamesurfacetractionisappliedonbothlegsofthatbeaminwhicha crackhasbeencreatedinthemid-plane.Moreover,weareassumingastraightextension ofthecrack.Intheimplementationonaniteelementcode,thatassumptionpermits ustomodelonlyhalfofthebeamandthenreducetheCPUtimeapproximatelyby 37

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afactoroftwo.Thismodelreductionispossiblethankstotheapplicationonthe remainingstructureofasetofboundaryconditions,asshownontheniteelementmodel ofFigure4-1,whichalsointroducesalltheparametersofthisfracturetest: a and W are respectivelythecrackandligamentlength, B isthewidthand F isashearforceapplied oneachlegofthedoublecantileverbeamwhoseheightis h Figure4-1.The2Dniteelementmodel Thematerialchosenforthattestisanhighstrengthaluminumalloythatisassumed tobehomogeneous,linear,elasticandisotropic.Thus,onlytwoengineeringconstants arerequiredtocharacterizeitsmechanicalproperties:theYoung'smodulus E andthe Poisson'sratio .ThoseinputtotheniteelementcodeimplementedonABAQUS,along withgeometricpropertiesandloadconditionsaredetailedinTable4-1. Table4-1.ABAQUSinput loadgeometricpropertiesmaterialproperties F=100Na=10mm,W=20mm,h=1mm E =70 : 10 3 MPa = : 35 Inordertoreachthemostaccurateresultpossible,aspecialattentionhasbeen devotedtothetypeofelementsandtothemeshingsystem.Therefore,aroundthecrack tip,asshowninFigure4-2,themeshhasbeenrenedandthesweepalgorithmhasbeen usedtolinkthesurfacemeshingschemeadoptedforthecracktipareawiththestructured quadrilateralelementsthatareusedtomeshtherestofthebeam. 4.1.2ResultsandDiscussion Theassessmentofthecracktipstresseldishererealizedusingapathconstituted ofnodesthatlieonthemid-planeofthedoublecantileverbeam,wheretheinterlaminar 38

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Figure4-2.Meshingsystemnearthecracktip damageoccurs.PlottinginlogscaleFigure4-3thestress y normaltothecrack propagationdirectionwithrespecttothedistancefromthecracktiphighlightsthefact thatthestresseldissubmittedtoasingularityin 1 p r inalargelinearelasticzonenear thecracktip. Figure4-3.Stresseldsingularityin 1 p r nearthecracktip Asaresult,thelinearelasticfracturemechanicscanbeapplied.Duetotheelastic behaviorofthematerial,thestrainenergyreleaserate G isequalto J integral whichis directlycomputedbyanABAQUSplug-inintheversion6.7-1usedtorunthesimulation. Thus,arequestofthathistoryoutputpermitstoaccessthatdata.Theanalysishasbeen madeforplanestressandplanestrainelementsandtheassociated J integral resultsalong dierentcontoursnearthecracktipareshowninFigure4-4andFigure4-5. 39

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Figure4-4. J integral underplanestress Figure4-5. J integral underplanestrain Thosegraphdemonstratethat,exceptfortherstcontourwhichprobalyliesinthe plasticzone,the J integral ispath-independent.Toassesstheaccuracyofthecomputation ofthestrainenergyreleaseratevia2DFiniteElementanalysis,wecompute,usingthe sameparametersthantheoneinputtedintheFEmodel,thetheoreticalexpressionof G I derivedinanearliersectionandstatedinEq.4{1. G I = 12 F 2 a 2 B 2 h 3 E {1 where E = E forplanestressand E = E 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 forplanestrain.Table4-2showsthe existenceofanonnegligiblediscrepancy. 40

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Table4-2.Beamtheoryand2DFEA G I computationin N:mm )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 Beamtheory2DFEAnalysisdierential% planestress G I =171 : 43 G I =195 : 3213.93 planestrain G I =150 : 43 G I =171 : 3913.93 Consequently,this2Danalysisallowedustoassessitsinaccuracyeventhough themeshingsysteminvolvedthousandsofelementsinthevicinityofthecracktip. Furthermore,atthatpoint,otheranalysisestablishedthatreningthemeshdoesnot conveybetterresults:theinaccuracyinthecomputationoftheenergyreleaserateina crackedplateisprobablyinherenttothe2dimensionalaspectandtothetypeofelements used.But,anotheroriginofthatdiscrepancy,whichisexactlythesameforbothplane stressandplanestrainconditions,couldalsolieintheapproximationdonebythenite elementsoftwareABAQUSwhileimplementingthismodel.Toremovethissuspicion regardingtheaccuracyofthatcommercialsoftware,theenergyreleaseratehasbeen computedusingonlyABAQUSdeectionoutput,whoseaccuracyhasbeenprovedtobe excellentcomparedtobeamtheory. 4.1.3DiscrepancyInherentto2DAnalysis AccuracyofABAQUSdeectioncalculation Inordertochecktheaccuracyoftheniteelementsoftwareused,wehavecompared theABAQUSoutputofthemaximumdeectionofasimplecantileveredbeamFigure 4-6withthetheoreticalresultderivedfrombeamtheoryEq.4{2. = Fl 3 3 E I {2 inwhich F istheforceappliedattheedgeofthebeam, l thelengthofthebeamand E = E forplanestressand E = E 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 forplanestrain,where E istheYoung'smodulus and thePoisson'sratio.TheFEAsimulationhasbeenexecutedunderbothplanestress andplanestrainconditions.Moreover,theshearingforce F hasbeenmodeledasasurface 41

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Figure4-6.FiniteElementanalysisofacantileveredbeamdeection traction.ThecomparisonoftheresultsTable4-3showsanexcellentcorrelation:the ABAQUSdeectionoutputarethusaccurate. Table4-3.AccuracyofABAQUSdeectionoutput planestressplanestrain FE analytical 1.00151.0003 Discretizationof G Figure4-7.Deectionofsublaminatesinopeningmode 42

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Inadoublecantileverbeam,thestrainenergydensityofanysublaminatescanbe expressedthroughitsdeectionfromthemid-plane. U a = 1 2 F 1 a 1 + F 2 a 2 {3 Suchexpression,statedinEq.4{3,canbeinsertedintoEq.4{4toobtaintheenergy releaseratefromanitedierencederivation,whichisvalidunderfrictionlessassumption oftheinterface[14]. G = @U @A crack = U B a = 1 B U a + a )]TJ/F21 11.9552 Tf 11.955 0 Td [(U a a {4 Thedeectionoutputrequiredhavebeenobtainedthroughaplanestressanalysisofa bi-materialdoublecantileverbeamwith2linearisotropicmaterials:SteelandAluminum. TherespectiveYoung'smodulus,necessarytodescribethelinearelasticbehaviorofthe structureare E Fe =200 GPa and E Al =70 GPa .Furthermoreastraightaheadcrack hasbeeninsertedandaseamhasbeendenedtoallowthecrosssectionbehindthecrack tiptoslideoneachother:thestressisconcentratedatthecrack.Moreover,quadratic eight-nodalelementshavebeenused,andarenementofthemeshwasdonedening asweepregionnearthecracktip,asshowninFigure4-8.Elsewhere,theelementsare structured. Figure4-8.Meshrenementaroundthecracktip 43

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Table4-4.Outputof2DbimaterialDCBFEanalysis a =10 mm a + a =10 : 5 mm 1 m 2.5982.972 2 m 6.6677.663 UJ:mm 3 463.24531.79 J int N:mm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 128.55142.12 G th N:mm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 115.71127.58 error%11.111.4 Throughanhistoryoutputrequest,thedeectionofthebeambutalsothe J integral havebeenobtainedfortwodistinctcongurations:thecracklengthhasbeenincreasedin thesecondonefrom10mmto10.5mm.Table4-4showsthetheoreticalvalueof G was derivedusingEq.4{5. G th = F 2 a 2 2 BI 1 E Fe + 1 E Al {5 SubstitutingthevaluesofthestrainenergydensitiesintoEq.4{4,weobtainavalue closedby1.3%fromthearithmeticalmeanofthetwo J integral output.Therefore, ABAQUS J integral outputareingoodcorrelationwithacomputationbasedondeection results,whichhavebeenprovedtobeaccurate.Hence,asthetotalenergyreleaserate can'tbecorrectlyassessedthroughthattoosimplisticdoublecantileverbeammodeling, anotherFEmodelinvolvingshellelementshavebeenimplementedinABAQUSandis describedinthenextsection. 4.2DoubleCantileverBeamPlateElementModel 4.2.1FiniteElementProcedure Inthismodel,wearestilltryingtobenetfromthespeedycomputingabilityoftwo dimensionalanalysis,buthereabetterdescriptionofthebondingbetweenlaminatesis reached.Indeed,thetwosub-laminatesaregoingtobemodeledbytwodistinctplatesand thecrackisnotgoingtobeimplementedusinganypre-conguredABAQUSplug-in. Behindofthecracktip,thereisnoconnectionbetweenthetwoplates,whereasahead ofthecracktip,thereisacontinuouselasticbodythatneedstobemodeled.Therefore,as wecanuseonlydiscreteconnectorstoestablishalinkbetweenthetwoplates,anemesh 44

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isimplementedonthepartofeachplatethatneedstobeconnectedtogether.Foreach pairofnodes,aconnector,suchasaspringpermitstobondthestructure. Furthermore,thisstructureissolid,thecohesionbetweenmoleculesisstrong;thus itisnecessarytomodelthespringswithahugestiness.Moreover,anosetvaluehas beenincorporatedinthedecriptionoftheshellelementsinordertodisposethenodes ofbothtopandbottomplatesonthemid-plane.Thisosetintroductionpermitsto getconnectorsdenedwithadistancenull.Analternativeapproachtobondtheplates altogetheristoconsiderbeamelements,whosekinematicscharacteristicsaredetailedin Figure4-11.Tomodelthousandsofspringsorbeams,asshowninFigure4-9,aMATLAB codehasbeenincorporatedtotheABAQUSpythonscript. Figure4-9.A2DplateFEmodel ExecutionoftheVCCTinthespringmodel Betweeneachpairofnodesaheadofthecracktip,threespringshavebeenmodeled: oneineachdirectionoftheCartesiancoordinates.Eachofthosepairsdenesaunitcell intheVCCTalgorithm.Furthermore,thespringsaredenedbyastiness K .Hence, theonlyoutputrequiredfromABAQUS,oncethesimulationcomplemented,arethe displacementsbetweenthenodeswherethespringshavebeentiedupandjustbehind. Thestrainenergyreleaseratescanbecomputedineachunitcellalongthecracktip throughtheexpressionderivedinEq.4{6. G i = 1 2 B a F i u i a )]TJ/F15 11.9552 Tf 11.955 0 Td [( a {6 inwhich F i = Ku i a istheforceinthesprings. a and B arerespectivelythedistance betweentworowsofsprings,andtwocolumnsinthatmatrixlledwithmorethanthree 45

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thousandssprings.Moreover, i representsthedirectionassociatedtoaspecicmode offracture.Anexampleofsuchsimulation,whichhasbeenruninanopeningmode,is presentedinFigure4-10. Figure4-10.A2Dplatemodelwithspringsasconnectors Themaindicultyinthedataprocessingliesinthechoiceofthestiness.Indeed, thebestapproximationwouldbetodenestinessasbigastheniteelementsoftware allows,butinthatcase,astheforceremainsconstant,thedisplacementbetweeneach pairofnodesatthecracktiparesosmallthatitisnotalwayspossibletoquantifythem throughanyhistoryoutputrequest.Indeed,thenumberofdigitsneededoverowsthe limitnumberpossibletoallocate.Toovercomethathurdle,anothermodelhasbeensetup witharigidconnectioninstead. ImplementationoftheVCCTinthebeamelementmodel Themodelingofarigidconnectionthroughbeamelement,whichABAQUS descriptionispresentedinFigure4-11,ispossiblebecausesuchconnectorallowsthe evaluationoftheforcesandmomentsthatareappliedatthenodeswhichdeneits edges.Indeed,thenodesofthetopplateareinvirtualcontactwiththoseofthe bottomonethankstotheosetoftheplanesonwhichthenodeshavebeendened:a zero-displacementconditionmustthenbesatised. Finally,substitutingtheforcesandmomentsoutputintoanexpressionanalogtoEq. 4{6permitstoreachanapproximationoftheenergyreleaserate. 46

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Figure4-11.Descriptionofabeamelement 4.2.2ResultsandDiscussion Inthismodelwherethesublaminatesofthatsymmetricdoublecantileverbeam fracturetesthavebeendenedby2Dshellelementsandtheconnectorsusedtojointhe structurethathasnotyetbeensplitbythegrowthofthecrackarebeamelements,the necessaryABAQUSoutputareconnectorforcesandmomentsalongwithdisplacements behindthecrack.Thefractureanalysishasbeenconductedindierentmixitymodesbut theassessmentofthetotalenergyreleaseratethroughthattechniquehasrevealedtohave asignicantdiscrepancy.Forinstance,inmodeIloadingconditionsandunderthesame geometricparmeters,Figure4-12showsanonnegligiblegapbetweentheresulting2D beamelementand3Dsolidelementmodelsintheenergyreleaseratedistributionalong thewidth. 4.3The3DAnalysisofaSymmetricDoubleCantileverBeam 4.3.1Descriptionofthe3DSDCBModel First,a3Dsoliddeformableparthasbeencreatedfromtheextrusionofasketch deningonelateralfaceofthebeam.Thisparthasbeenpartitionedinordertodene aspeciccellforthesublaminatesinterface.Furthermore,asolidhomogeneoussection whosematerialisanisotropicaluminumalloycharacterizedbyasetoftwoengineering 47

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Figure4-12.Comparisonof2Dplateand3Dresultsinanopeningmodefracturetest Figure4-13.SDCBFiniteElementModel constantsYoung'smodulus E =70 : 10 3 MPa andPoisson'sratio =0 : 35hasbeen assignedtothatpart.Then,20-nodequadraticbrickCRD20Relements,whichbelongto thequadratic3Dstressfamily,havebeenusedtomeshthehexagonalstructuredsystem withareducedintegrationelementcontrol.Beforemeshingtheinstance,theglobalsize oftheseedhasbeenrened.Moreover,acontourintegralcrackhasbeenimplemented deningitsfrontandextensiondirection ~q .Aseamhasalsobeenassignedtothesurface betweenthetwosublaminates,behindthecracktip.Finally,astaticstephasbeen created.Inthelatter,boundaryconditionsandloadshavebeencongured,asshownin Figure4-13. 4.3.2Convergenceof G upontheRenementoftheMesh InamodeIconguration,theABAQUSniteelementmodeltypicallygivesthe characteristicsateofstressshowninFigure4-14.Dependingontheseedglobalsize 48

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Figure4-14.AnalysisofthestateofstressinModeI selected,the J integral valueextractedforeachunitcellalongthecracktipthroughan historyoutputrequestedvaries,asFigure4-15illustrates. Figure4-15.Convergenceoftheenergyreleaserate Indeed,thesegraphsshowthatthecurveddelaminationfrontalongthewidthis smootherwhenthenumberofelements,whichisinverselyproportionaltotheseed selected,increased.Testingourmodelwithanermesh,wasincompatiblewiththe computermemoryallocationpossibilities.But,thecurveobtainedwitharenedseedof 0.25,givesaccurateenoughresultssinceitappearsinFigure4-15thatthelattercurve seemstohavealreadyconverged. Itisinterestingtonotethatinordertoreachtheconvergenceoftheenergyrelease rate;itisnotrequiredtorenethemeshspecicallyinthecracktipareacomparedto 49

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therestofthemodelbecausewemadeastaticassumptionforthecrackanalysis.Aner meshintheregionsurroundingthecracktipisespeciallyneededindynamicmodelsto allowthepropagationofthecrackbyasmallerincrement. AsdisplayedinFigure4-15,the J integral reachedaminimumattheedgeofthebeam andamaximumatthecenter.InmodeI,theenergycomesprincipallyfromlongitudinal bendingandthenonuniformtrendiscausedbytheanticlasticcurvaturethatoccursin eachlegofthedoublecantileverbeam.Duringthedelaminationgrowth,achangeinthe specimenbendingrigidityoccurs,astheconstraintconditionsvaryfromplanestrainfor shortdelaminationtoplanestressforlongone.Thisvariationofconstraintscontrolsthe shapeofthedistributionoftheenergyreleaserate.Furthermore,aquanticationofthat variationinthespecimenbendingrigiditycanbeassessedthrougharatio,introducedby Davidson[15],involvingDCBproperties. 4.3.3DelaminationFrontCurveforanArbitraryModeMixity VCCTreferenceelementselection IntheVCCTdescriptionofthechapter3,twodierentreferenceelementswere introducedinordertocomputethefrontdelaminationcurveofastraightaheadcrack. ImplementingbothoftheminMatlabgivessensiblythesameresult:thecontinuityof thestress,allowedbythenemeshselected,permitsthosetwoapproximationstobe equivalent.However,aslightlydierenceisnoticeableattheedgeofthedelamination frontbecause,fortheunitcellthathasbeenosetbyhalfofthedistanceseparatingtwo consecutivenodesalongthecrackfront,theresultiserroneousatbothends.Indeed,a mid-sidenodeofthequadrilateralmeshingsystemusedismissing. Anexcellentmatchingwith J integral ToassesstheaccuracyoftheVirtualCrackClosureTechniqueimplementedin the3DSDCB,theresultingtotalenergyreleaseratehasbeencomparedtoABAQUS computationof J integral .Suchcomparisonisallowedbytheelasticbehaviorofthe material. 50

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Figure4-16.DelaminationfrontinModeI FromtheanalysisofseveralSDCBniteelementmodelscoveringawiderange ofmodemixitypossibilitiesanexcellentmatchinghavebeennoticedeitherformodeI Figure4-16,modeIIFigure4-18,oranarbitrarymixmodeFigure4-20. InFigure4-17,areshownthestateofstressoftheniteelementmodelwhenthe sameforcehasbeenappliedinthesamedirectiontobothlegsofthesymmetricdouble cantileverbeam.Inthatcase,thelegsslideoneachotherduetothefrictionlesscontact allowedbytheimplementationoftheseaminthemodel. Figure4-17.StateofstressinModeII ApplyingtheVCCTalgorithmtothesetofforcesanddisplacementsdescribedin Chapter3,highlightsagaintheexcellentcorrelationof G total with J integral butalsoconveys thepresenceofmodeIIIcomponentsattheedgeofthebeamFigure4-18.Therefore, theshearingforcesappliedthroughasurfacetractionontheedgeofthecrackedbeam legs,createcomponentsslightlyoutofthereferenceplane.Asthisedgeeectisminor, 51

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itwillbeneglectedinthecomparisonwiththeothertheoreticalandanalyticalmethods thathavebeenderivedinchapter3fora2Dconguration.Furthermore,suchagraph demonstratesthattheshearingforcesareresponsibleforthemodeIIfracturecomponent, whichisalmostuniformacrossthewholewidthofthebeam. Figure4-18.DelaminationfrontinModeII Inamixmodeconguration,wherethebottomhalfoftheSDCBhasbeensubmitted toadeadload,theeectsnoticedinbothpuremodesarecombinedFigure4-20. Figure4-19.ApplicationofasingledeadloadinaFESDCB 4.3.4PerfectCorrelationamongTheoretical,Analytical,andNumerical Methods Foranyarbitrarymixmode,thetotalenergyreleaserate,obtainedthroughthe VirtualCrackClosureTechnique,matchnotonlywiththeJcontourintegralbutalsowith theresultingequationsofbothderivationof J integral fromstrainenergydensitiesofeach 52

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Figure4-20.DelaminationfrontinanarbitraryMixMode crosssectionofthebeamEq.3{21andCrackTipForceMethodEq.3{48.Indeed,the threedistributionsofthetotalenergyreleaserateoverlayFigure4-21. Figure4-21.Correlationoftotal G Inaddition,thephaseanglecomputation,whichisameasureofthemodemixity, givessensiblythesameresultsineithermethod.ThesmallgapFigure4-22,between theexpressionofthephaseangle,computedthroughtheVCCT,infunctionoftheone 53

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Figure4-22.Correlationofthemodemixity computedinthesameloadcongurationsbutwiththeCTFMcanbepartiallyimputedto thepresenceofashearingcomponentoutoftheplaneofdelamination. 54

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CHAPTER5 MULTI-FIDELITYRESPONSESURFACEFORASYMETRICDOUBLE CANTILEVERBEAM Aspreviouslyshown,thestudyofasymmetricdoublecantileverbeamfracturetest demonstratesthatthelow-delityCrackTipForceMethodcanaccuratelypredictthe globalfractureparametersassociatedwithcrackpropagation.Nevertheless,achangein theselecteddesignspace,forinstanceavariationinthethicknessratio h 1 h 2 ,asintroduced inFigure5-1,revealsthattherearediscrepanciesbetweenthetwomethods. Figure5-1.Assymetricdoublecantileverbeam InthisChapter,ourgoalistocombinethehigh-delity3DFEmodeltothefast CrackTipForceMethodtoobtainhighaccuracyatlowcomputationalcostinthe determinationoftheglobalfractureparameters G andassociatedwiththeasymetric doublecantileverbeamfracturetestforanygivenloadconguration.First,acorrection surfacetechniqueispresentedtocouplethetwomethodsofanalysisatvariouspoints ofinterestinthedesignspace.Then,thelatterisimplementedtotthevaluesofthe ratiooftheenergyreleaseratesandthedierenceofthephaseanglesoftheanalytical andnumericalmethodsatspecicdesignpoints.Finally,thepredictivecapabilityof theresponsesurfaceisassessed.Moreover,particularattentionisgiventothedesignof experimentsbydetailingtherespectiveloadcongurationsthatareimplementedinthe3D FEand2Danalyticalmodelsinordertocapturetheeectsoftheopeningandin-plane 55

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shearingmodes.Theeectsoftheout-of-planeshearingmodescanbeneglectedasthe3D FEmodelrevealsthattheenergyreleaserateinthisfracturemodeismuchsmallerthan thetwoothermodecomponents. 5.1ComputerDesignExperiments Giventheloadcongurationappliedtobotharmsofthedoublecantileverbeam,as depictedinFigure5-1,theprincipleofsuperpositionpermitstosplitthisdoublecantilever beamfracturetestinto3distincttests.Ineachofthelatter,onlyasinglekindofforceis applied:axialloads,couplesorshearingforces. Thosefracturetestsallowsustogeneratecomputerdesignexperimentsinboth3D FEand2Danalyticalmodelbyselectingdierentthicknessratio. 5.1.1The3DFEModeling ToimplementthethreepreviouslydescribedfracturetestsintotheABAQUS niteelementsoftware,the3Dmodelintroducedinchapter4isused.Inthelatter, thethicknessratiowillbechangedwhilebuildingthemodel.Thethicknessratiosselected are1,1.5,2and3.Wedon'tfocusonextremethicknessratiosbecauseacrackthat wouldpropagateatthesurfaceofthebodyisnotlikelytocausethefailureofthewhole structure. Thethreesetsofloadingconditionsarecreatedbyapplyingasurfacetractioninthe caseofashearingforceandbyapplyingapressureotherwise.Thepressureisuniformly appliedtotheedgesurfacetocreatetheaxialforcewhereastocreateacoupleonone arm,thepressureisappliedaccordingtoalineardistributionwhosecenterispositioned atthemiddleoftheleg:asetofforceshavingoppositedirectionsisthereforecreated,as depictedinFigure5-2. Whenapplyingacoupleononeligament,the3DFEmodelneedstohavethesame numberofrowsofelementsateachsideoftheligament'smiddle.Otherwise,thesetof forcescan'tcorrectlycreatethedesiredcouple.Furthermore,aviolationofthecontact boundariesisalsoobservedwhenthepreviousconditionisnotsatised. 56

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Figure5-2.ApplyingcouplestotheADCBlegs 5.1.2The2DAnalyticalMethod ThecrackTipForceMethoddescribedinchapter3isimplementedtocompute atlowcomputationalcosttheglobalfractureparametersdenoted G CTFM and CTFM foragivensetofloads P;M;V appliedtoeachligamentofthedoublecantilever beam.Exceptinthecaseoftheapplicationofmomentstothedoublecantileverbeam arms,whichissimilarinhigh-delity3DFEmodelandlow-delity2Danalyticalone, theapplicationofshearingforcesandaxialloadspresentssomeproblems.Indeed,to produceanequivalentloadingconditionasinthe2DFEmodelwhereshearingforces areappliedattheedgeofeacharm,itisnecessarytotranslatetheshearingforceapplied intoamoment.Therefore,bothshearingforceandmomentsneedtobeinputtedin theanalyticalmethod.Furthermore,theaxialloadscannotbedirectlyappliedinthe 2Danalyticalmodel:amomentisnecessarytotakeintoaccountthefactthatinthe analyticalmodeltheaxialload P isnotappliedatthemiddleofeachligamentbutatthe delaminationplane. 5.2LowFidelityAnalysiswithHighQualitySurrogates 5.2.1CrackTipForceMethodPolynomialResponseSurface DuetothelowcomputationalcostoftheCrackTipForceMethod,largenumber oflowdelityanalysiscouldbeperformedtogeneratehigh-orderpolynomialresponse surfaces.Inthecaseofapplicationofmoments,afth-orderpolynomialresponsesurface oftheenergyreleaseratehasbeengeneratedinFigure5-3usingthedataof64design pointswhosevariablesarethethicknessratio h 1 h 2 andthephaseangle CTFM ,which characterizestheratioofmodeIItomodeIenergyreleaserate.Thedesignvariablesare 57

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Figure5-3.Polynomialresponsesurfaceoftheenergyreleaserate normalizedtoimprovethestabilityoftheMATLABcodeusedtogeneratethepolynomial responsesurface. Theperformanceofthattisevaluatedthroughthecomputationof R 2 ,that measuresthefractionofthevariationinthedatacapturedbytheresponsesurface. Suchpredictorisdenedastheratioofthevariation SS r oftheresponsesurface^ z from theaverageofthedatapoints z andthevariation SS z ofthedatafromitsaverage. SS r SS z and R 2 arerespectivelypresentedinEq.5{1,Eq.5{2andEq.5{3. SS r = n z X i =1 ^ z i )]TJ/F15 11.9552 Tf 12.665 0 Td [( z 2 {1 SS z = n z X i =1 z i )]TJ/F15 11.9552 Tf 12.664 0 Td [( z 2 {2 where n z isthenumberofdatapoints R 2 = SS r SS y {3 58

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FortheresponsesurfaceofFigure5-3,thisratioisveryclosedto1: R 2 =0 : 998465. Furthermore,sincetheadjustedformof R 2 ,denedinEq.5{4isalsofoundtobemuch closedto1 R 2 a =0 : 998209,wearesureaboutthepredictioncapabilityofourmodel. R 2 a =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 2 n z )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n z )]TJ/F21 11.9552 Tf 11.955 0 Td [(n {4 where n z and n arerespectivelythenumberofdatapointsandthenumberofcoecients ofthepolynomialresponsesurface.Forafth-order,thiscoecientisequalto21. 5.2.2EnergyReleaseRateCorrectionResponseSurface Thehigh-delityanalysisgivesafewnumberofaccurateenergyreleaseratevalues thataregoingtobeusedtocorrectthelowdelityresponse.Foreachthicknessratio h 1 h 2 =1 ; 1 : 5 ; 2 ; 3,4valuesof G areobtainedforvariousphaseangles.Then,aquadratic correctionresponsesurfaceisplottedinFigure5-4usingthose16designvaluesbytting thevalueoftheratioofthehigh-delityenergyreleaserate G 3 D andtheapproximate energyreleaseratevaluesthatarecomputedforthesamephaseangleandthickness ratiousingthe5 th orderpolynomialresponsesurface. Thequadraticorderhasbeenchosenforthecorrectionresponsesurfacebetween thehigh-delityandlowdelityanalysisbecauseahigherpolynomialorderdoesnot improvethepredictioncapabilityofthecorrectionfactor.Indeed, R 2 a decreasesbetween thequadraticandcubicpolynomialorder,asshowninTable5-1. Table5-1.Predictioncapabilitiesofenergyreleaseratecorrectionresponsesurfaces R 2 R 2 a quadraticCRS0.69.53 cubicCRS0.78.47 Themulti-delityenergyreleaserate G multifidelity atagivendesignpointisderivedin Eq.5{5. G multifidelity = G 2 D :CRS {5 59

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Figure5-4.Correctionresponsesurface Theprocedureistondthe2Dapproximatedvalue G 2 D usingthe5 th order polynomialCTFMresponsesurface,thenmultiplythelatterbythevalueofthecorrection responsesurfaceatthesamedesignpoints.Tovalidatethismulti-delityapproach, thispreviousprocedurehasbeenimplementedataround20designpointsthathavenot beenusedintheconstructionofthecorrectionresponsesurface.Atthosepoints,the G multifidelity valuescanbecomparedtothehigh-delityvaluesresultingfromthe3D FiniteElementAnalysis.Thiscomparisonhasshownthattheintroductionofthehigh qualitysurrogatesinthelow-delityanalysispermitstoreducetheaverageerror,as denedinEq.5{6. e av = 1 n z n z X i =1 j z i )]TJ/F15 11.9552 Tf 13.439 0 Td [(^ z i j {6 where n z isthenumberofdesignpoints, z i isthehigh-delityenergyreleaseratevalueat designiand^ z i isthe2DCTFMapproximatedvalue. Usingthe5 th orderpolynomialresponsesurfacetogetthe2Danlyticalapproximation oftheenergyreleaserate,theaverageerrorwiththe3Dhigh-delity23designpointsis 60

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around11%.Themulti-delydesignreducedthataverageerrorto6.1%.Consequently, thismulti-delitydesignapproachoftheasymetricdoublecantileverbeamimprovesthe predictioncapabilityofthetotalstrainenergyreleaserateusingonlyafewnumberof highqualitysurrogates3DFEdesignpoints. 5.2.3PhaseAngleCorrectionResponseSurface Implementingthesameapproachtothephaseangle,acorrectionresponsesurface isshowninFigure5-5.Thedierenceofthehigh-delityandlow-delityphaseangles ishereusedtopredictthe3DFEvalue VCCT usingalargenumberofanalytical phaseanglesderivedinfunctionoftwodesignvariables:thethicknessratioandthe 2Danalyticalphaseanglethatresultsfromtheapplicationofacoupleatthetopligament ofthedoublecantileverbeam.Thecoupleappliedatthebottomligamentischosentobe constantandequalto1inboth2Danalyticaland3DFEcomputerexperimentsrun. Figure5-5.Phaseanglecorrectionresponsesurface Table5-2showsanexcellentpredictioncapabilityofthepolynomialresponse surfaceofthephaseangledierenceexpressedinradianswithrespecttothenormalized 61

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analyticalphaseangle.Changingthemodemixityboundsfrom[0 ; 2 ]to[ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1]permitsa stabilizationoftheoptimizationprocess. Table5-2.Predictioncapabilitiesofphaseanglecorrectionresponsesurface R 2 R 2 a quarticCRS0.97.93 Asaresult,usingtheenergyreleaserateandphaseanglecorrectionresponsesurface, itispossible,givenathicknessratioandasetofcouplesappliedatbothligamentsof theasymetricdoublecantileverbeam,toapproximatethoseglobalfractureparameters inrealtime.Indeed,thecomputationcostinvolvedinthecalculationofthelow-delity fractureparametersandintheapplicationofthecorrectionfactorscomputedforthe specicdesignselectedislow.Furthermore,thoseglobalfractureparametersareassessed withoutlosingmuchaccuracyaswedemonstratethataquadraticcorrectionresponse surfacecanalreadyimprovethepredictivecapabilityofthe2Danalyticalapproximation byaconsiderableamount. 62

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CHAPTER6 CONCLUSIONS Ourstudyshowstheassessmentofthetotalenergyreleaserate G total alongwiththe modemixity,whichmeasuresthemodeIIshearingtomodeIopeningloading,via thesuccessfulimplementationofanalgorithm,theVirtualCrackClosureTechnique,in 2Dplate,and3Dsolidniteelementmodelsofadoublecantileverbeamfracturetest. Underquasi-staticandlinearelasticbehavior,thedelaminationfrontcurveobtainedisin excellentagreementwiththe J integral curvecomputedbyABAQUS.Thelatterhasalso beenfoundinagreementwithanexpressionoftheenergyreleaserateobtainedthrougha nitedierenceofitsdenitionitselfi.e.,thechangeinpotentialenergywithrespectto theareacreatedbyanextensionofthecrack. Furthermore,theCrackTipForceMethod,rstproposedbySankar[13]hasbeen validatedfromtheexcellentcorrelationshownbyitscomparisonwiththegeneralstandard solution,describedinHutchinsonandSuo'sreview[2].Then,thisanalyticalmethod hasbeencomparedtothehighdelity3DFiniteElementAnalysisofahomogeneous, isotropicandsymmetricdoublecantileverbeam:thedierencenoticedwassmall.In addition,theresultsobtainedwithbothmethodswereveryclosetotheoreticalresults, whichidentifythestrainenergyreleaserateinadoublecantileverbeamasthedierence betweenthestrainenergydensitiesofeachcrosssectionofthelegsthatarelocatedjust behindandaheadofthecracktip. Astudyofthevariationofthicknessratiosshowsthatthereisnolongeranagreement betweentheresultsderivedusingtheCrackTipForceMethodandthe3DFiniteElement Analysis,whenthedoublecantileverbeamisnotsymmetric.Yet,inamulti-delity approach,itisshownthatalargenumberoflow-delity2Danalyticalvaluesofthe fractureparameterscombinedwithafewnumberofhigh-delitysurrogate3DFEvalues canpredictatlowcomputationalcostthefractureparametersforaspecicdesignwithout losingmuchaccuracy. 63

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Finally,itwouldbeinterestingtostudytheeectofthevariationofthematerial properties.Thelogicalnextstepwouldbetoconsiderorthotropicmaterialsbefore consideringplieswithdierentorientations.Astheenergyreleaserateisalsoafunctionof thecracklengthwhenthelatterbecomessmall,itwouldalsobeadvantageoustocomplete ourcorrectionresponsebetweenhigh-delityandlow-delityanalysestointegratethe cracklengthasadesignvariableintheexpressionofthedelaminationfrontcurve. 64

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REFERENCES [1]H.Tada,P.C.Paris,G.R.Irwin,Thestressanalysisofcrackshandbook,2nd edition,Parisproduction,Saint-Louis,MO,1985. [2]J.W.Hutchinson,Z.Suo,Mixedmodecrackinginlayeredelasticmaterials,Adv. Appl.Mech.2963{191. [3]A.A.Grith,Thephenomenaofruptureandowinsolids,Philos.T.Roy.Soc.A 221163{198. [4]G.R.Irwin,Analysisofstressesandstrainsneartheendofacracktraversingaplate, J.Appl.Mech.24361{364. [5]J.R.Rice,Apathindependentintegralandtheapproxiamteanalysisofstrain concentrationbynotchesandcracks,J.Appl.Mech.35379{386. [6]T.L.Anderson,FractureMechanics,FundamentalsandApplications,CRCPress, 2005. [7]G.R.Irwin,Onsetoffastcrackpropagationinhighstrengthsteelandaluminium alloys,Sagamoreresearchconferenceproceedings2289{305. [8]E.F.Rybicki,M.F.Kanninen,Aniteelementcalculationofstressintensityfactors byamodiedcrackclosureintegral,Eng.Fract.Mech.9931{938. [9]A.Leski,Implementationofthevirtualcrackclosuretechniqueinengineeringfe calculations,FiniteElem.AnalDes.43261{268. [10]K.N.Shivakumar,P.W.Tan,J.C.Newman,Avirtualcrack-closuretechniquefor calculatingstressintensityfactorsforcrackedthreedimensionalbodies,Int.J.Fract. 3643{50. [11]D.Xie,S.B.Biggers,Progressivecrackgrowthanalysisusinginterfaceelementbased onthevirtualcrackclosuretechnique,FiniteElem.AnalDes.42977{984. [12]B.V.Sankar,Aniteelementformodelingdelaminationsincompositebeams, Comput.Struct.38239{246. [13]O.Park,B.V.Sankar,Crack-tipforcemethodforcomputingenergyreleaseratein delaminatedplates,Compos.Struct.55429{434. [14]X.Sun,B.D.Davidson,Adirectenergybalancefordeterminingenergyrelease ratesinthreeandfourpointbendendnotchedexuretests,Int.J.Fract.135 51{72. [15]B.D.Davidson,R.A.Schapery,Eectofanitewidthondeectionandenergy releaserateofanorthotropicdoublecantileverspecimen,J.Compos.Mater.22 640{656. 65

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BIOGRAPHICALSKETCH NicolasVigrouxwasborninParis,France,in1985.Heattendedpublicschoolsin Chamalieres,agatewaytothemarvelousCha^nedesPuysoftheMassifCentralinFrance. AftergraduatingfromhighschoolwithhonorsinJuly2003,hesuccessivelywentto LyceeLouisLeGrand,ParisandLyceeLakanal,Sceaux,wherehereceivedanintensive trainingprincipallyinmathematics,physicsandengineeringsciencetocompeteinthe Frenchnationwideselectiveexam.InSeptember2005,hewasadmittedtooneofthemost prestigiousGrandesEcoles:theEcoleNationaleSuperieuredesMinesdeSaint-Etienne fromwhichhereceivedin2008thetitleofIngenieurCivildesMinesmaster'sdegree withthehighesthonorsMentionTresBien.Alongsidetheengineeringandmanagerial formationheexperiencedthere,hisinterestineconomicsdrovehimtoattendclasses deliveredbytheEconomic,AdministrationandAccountingSuperiorInstituteofthe UniversityJeanMonnetinSaint-Etienne,fromwhichhegraduated,in2007,witha BachelorofScienceinEconomics.Insummer2007,heperformednanoprecipitates characterizationbytransmissionelectronmicroscopyinnewaeronauticalalloysatthe NationalResearchCenterinToulouse.InAugust2007,hejoinedtheDepartmentof MechanicalandAerospaceEngineeringattheUniversityofFloridawherehehasbeen pursuinghisMasterofScienceonfracturemechanics. 66