Investigations of the thermomechanical behavior of a coarse-grained aluminum multicrystal using Constrained full-field measurements methods

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Investigations of the thermomechanical behavior of a coarse-grained aluminum multicrystal using Constrained

full-field measurements methods

Li Li, Félix Latourte, Jean Michel Muracciole, Laurent Waltz, Laurent Sabatier, Bertrand Wattrisse

To cite this version:

Li Li, Félix Latourte, Jean Michel Muracciole, Laurent Waltz, Laurent Sabatier, et al.. Investigations of the thermomechanical behavior of a coarse-grained aluminum multicrystal using Constrained full- field measurements methods. Optics and Lasers in Engineering, Elsevier, 2019, 112, pp.182-195.

�10.1016/j.optlaseng.2018.08.003�. �hal-01885346�

Investigations ofthe thermomechanicalbehavior ofa coarse-grained

aluminum multicrystal usingConstrained full-field measurements methods

L. Li^{a, ∗}, F.Latourte^b, J.-M. Muracciole^{a, c}, L.Waltz^{a, c}, L.Sabatier^{a, d}, B.Wattrisse^{a, c}

aLMGC, Univ. Montpellier, CNRS, Montpellier, France

bEDF R&D, MMC Department, Moret-sur-Loing, France

cLaboratoire de Micromécanique et d’Intégrité des Structures (MIST), IRSN-CNRS-Université de Montpellier, Montpellier, France

dAix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

Keywords:

Digital image correlation (DIC) Constrained DIC

InfraRed Thermography (IRT) Constrained IRT

EBSD analysis

Displacement and strain mapping Crystal plasticity

a b s t r a ct

Withtheintentionofachievinganexperimentalgrainscaleenergybalanceatfinitestrainandatthegrainscale, amechanicaltestonacoarse-grainedaluminiumispresentedinthispaperusingtwocomplementaryimaging techniquesbasedonvisibleandinfraredlight.SpecificimageprocessingmethodsreferredtoasConstrainedDig- italImageCorrelation(ConstrainedDIC)andConstrainedInfraRedThermography(ConstrainedIRT)areapplied toinvestigatethethermomechanicalbehavioratthemicrostructuralscale.ConstrainedDICisusedtoobtain displacementandstrainfieldsduringthetest,whileConstrainedIRTprovidesanestimateoftemperatureand heatsourcefieldsinducedbythemechanicalloading.Theproposed“constrained” methodsallowtoenforcean adjustablelevelofconstraintsonameasuredfield(displacementortemperature)withoutreferringtoaspecific finite-elementdescription.Inthatmanner,itispossibletodecouplethemeasurementmodelandtheinterpre- tationmodelwhilekeepingregularizingconstraints(suchascontinuityofthefields).Inthispaper,wemainly focusonthekinematicanalysisoftheexperimentaltest.ElectronBackscatterDiffraction(EBSD)isalsousedin thiscasetoexperimentallycharacterizethemicrostuctureofa3mmthickspecimenwithcentimetricgrainsize.

1. Introduction

Polycrystallinemetalsusuallypossessamicrostructurecomposedof anaggregationofcrystallinegrainswithvaryingsize,morphologyand orientation.Duringamacroscopictensileloading,thediversityofgrain orientationsandtheintrinsicanisotropyofcrystal plasticityleadsto strongheterogeneitiesinthematerialplasticresponse,andconsequently toaninhomogeneousthermaldistributionduetothermomechanicalef- fects.

Recently, heterogeneous phenomena on mechanical andthermal fieldshavebeenstudiedinmetallicmaterialsatthegranularscale[1–

6].Alltheseworkshaveshownthevarietyofmicromechanicalmod- ellingissuesthatcanbeaddressed usingclassicalDIC(DigitalImage Correlation)&IRT(InfraRedThermography)method.Hereafter,a“Con- strained” surfaceDICorIRTmethodisproposedtoenrichthekinematic orthermaltransformationofneighbouringelements(orgrains)byim- posingcontinuity(ordiscontinuity)conditionsonthedisplacement(or thedisplacementgradientcomponent) oronthetemperature(orthe temperaturegradient).

Performingstrainfieldandheatsourcemeasurementsultimatelyal- lowstoaccesstotheevolutionofthemechanicalandcalorimetricen-

ergiesinvolvedinthetransformation.Thisassessmentcontributestoa betterknowledgeofthelocalthermomechanicalsignatureofthemate- rialdeformationmechanisms.

As mentioned, two data processing methods (Constrained DIC [7,8]andConstrainedIRT[9])arerequiredtoperformkinematicand thermalmeasurementsthatarebothneededtoconductalocalenergy balancewithineachgrainduringamechanically-loadedtest.Inthelight ofthisgeneralobjective,wemainlyfocusinthispaperonthekinematic aspectoftheaforementionedgeneralmethodology.

First,theprincipleofConstrainedDICmethodwillbeintroduced.

Then,thenumericalvalidationofConstrainedDICmethodwillbeper- formedonnumericalexampleassociatedtocrackedpolycrystallineag- gregates.Afterwards,thisnovelmethodwillbeappliedtorealexperi- mentalimages.

Infact,surfacedisplacementfieldmeasurementsofmaterialssub- jectedtovariousloadings(e.g.mechanicalloadingorthermalloading) areanimportanttaskforexperimentalistsaddressingchallengesinthe fieldofsolidmechanics.

Inrecentyears,anincreasingnumberofspectaculardevelopments inopticalfull-fieldmeasurementtechniqueshasbeenwitnessed[10], includingbothinterferometrictechniquesandnon-interferometrictech- niques.However,theinterferometrictechniquesinvolvedelicateproce-

dureswhicharenotalwayseasilytransferabletoconventionaltesting laboratories.Consequently,theDigitalImageCorrelation(DIC)method widelyconsideredasarepresentativenon-interferometricopticaltech- nique,hasbeenlargelyacceptedandcommonlyusedasapowerfuland flexibletoolforsurfacedisplacementandstrainmeasurementintheex- perimentalsolidmechanicsfield[2,4,6,11–14].

Thesemeasurementsareparticularlyvaluableinthesensethatthey allowtheinterpretation ofcomplex tests atdifferent scalesandthat theyarenaturallyadaptedtoscaletransitions.Forthesereasons,they havebeenlargelyusedtocharacterisethedeformationmechanismsorto proposeandvalidatemicromechanicalmodelsorscaletransitionlaws.

Fromamicrostructuralviewpoint, polycrystallinematerials area discretestructurethatarecomposedofjointedgrainswithvaryingsizes andorientations.Thecharacterisationandmeasurementofgrainstruc- turesisofgreatinteresttoMaterialsScientistsbecausetheyaredirectly relatedtothephysicalpropertiesofmatter[15,16].

Ourobjectivehereistheunderstandingoftherelationshipbetween themicrostructuralparametersandthemechanical behaviourof the heterogeneousmaterialsatthemacroscopicscale,inparticularatthe granularlengthscale[17–19].

Usingtheclassicallocalapproaches,thematerialmicrostructureis notaccountedforinthekinematiccomputation:

• Firstly,theintroducedsubsets(forDIC)areindependentlydefined fromthemicrostructure

• Secondly,asthetransformationofneighbouringsubsetsaresepa- ratelyprocessed,sosubsetsmayoverlap.

Thisisaninherentdisadvantageoftheselocalmethodswhendealing withheterogeneousstructureproblems.

Nevertheless,classicallocalDICmethodshavebeenwidelyusedto highlighttheheterogeneityinkinematicfields[2,6,12,20],inalarge rangeofsituations dealingforinstance,with thefracturemechanics (intergranularorintragranular)problems.

2. PrincipleofconstrainedDICmethod

Global DIC methods were proposed to determine the displace- mentandstrainfieldsonthewholeimage.Thesemethodsproposeto parametrizethekinematicfieldsusingalimitedsetofdegreesoffree- domwhichtendstoregularizetheDICproblem.Thesemethodswere firstlyintroducedtoimposethecontinuityofmeasureddisplacement onafinite-elementmesh[21,22]orusingB-splines[23,24].GobalDIC methodswereafterwardsextendedtoallowsomediscontinuitiesinthe displacementfieldstoaccountforcrackdevelopment[25,26].

TheConstrainedDICmethodproposed herecorrespondstoanal- ternativetoglobalDICmethods.Itreliesonameshthatrespectsthe materialmicrostructureanditintroducesshapefunctionsthatareex- pressedintherealspaceandnotontheassociatedreferenceelement (asinclassicalfiniteelements).Theshapefunctionscanbeanykind (wegenerallyuselinear,bi-linear,quadratic,bi-quadraticpolynomial functions),andtheshapefunctionchoiceisindependentoftheshapeof theelement.Themostsignificantdifferencewithglobal(finite-element based)DICmethodsreliesinthefactthatthelevelofrestrictionbetween twoadjacentelementscanbemodifiedbychoosingthenumber(andthe location)ofpointswheretoenforcethecontinuityconditionsontheel- ementboundary.Itisalsointerestingtonotethattheproposedmethod allowstohandleinthesameframeworkclassicallocalDICmethods (whichcorresponds toaregularrectangular meshwithnocontinuity conditionbetweeneachelements)toglobalfinite-elementbasedmeth- odsonregularmeshes(byimposingcontinuityconditionsontheends ofeachelementboundary).

AsclassicalDICapproaches(whetherlocalorglobal),theproposed methodalsorelies ontheBrightnessConservationequation[27]mo- tivatingtheuseofapatternrecognitionalgorithmforthedetectionof changesinthegreyleveldistributionoftargetedsurfaceduringloading.

Indeed,themainstepsofConstrainedDICmethodarethefollowing:

Fig.1. Spatialdescriptionofthegeometryofapolycrystallineaggregation.The grainboundariesareinmagentaandtheelementcontoursareinblack.Andthe threereddotsareforspatialmatchingprocedure[6].(Forinterpretationofthe referencestocolorinthisfigurelegend,thereaderisreferredtothewebversion ofthisarticle.)

• Spatialdiscretizationofthegeometry

ThroughanEBSDanalysis,atwo-dimensionalarrayofdataasso- ciatedwith themicrostructureis providedbymicroscopic devise [28,29].Afterwards,thismicrostructuralmap(Fig.1a)canbeused toperformaspatialdiscretization(FiniteElementtype)inorderto respectasmuchaspossibletherealmicrostructure.Theobtained meshisusedforsubsequentprocessingofthekinematicresponse.

Inordertooptimizethemeshingprocedure,therealgrainbound- aries(whitecontoursinFig.1a)aresimplifiedandpolygonizedso astokeepthelargegrainsandregroupthesmallestones,asshown inFig.1binmagenta.Byconstruction,thelevelofmicrostructural simplificationhastobeadjusteddependingonthespatialresolution associatedwiththekinematicand/orthermalmeasurement.Thein- troduceduncertaintyduringthegrainboundaryextractionoperation isnotquantified,whichissupposedtobenegligibleinthispaper.

Afterwards,anunstructuredmeshiscarefullyappliedonthe“simpli- fied” geometry(representingthemicrostructure)withineachgrain inordertokeeptherepresentationofphysicalgrainboundaries,as showninFig.1c.Insideeachgrain,thesmallestmeshunitiscalled an“element”,whichisequivalentofthecorrelationsubsetforclas- sicalDICmethods.Theelementcontoursareaccuratelydetermined.

Thecomputationalmeshunderlying themicrostructureisdefined intheinitialconfiguration.Thekinematicvariablesassociatedwith eachelementdescribingthephysicaltransformationofthematerial willbeintroducedinthenextsection.

• Descriptionofthephysics

Thespecimen might be subjected todifferent loadings (traction, compression,shearorrotation).Dependingonthemechanicalsitu- ationunderconsideration,thedisplacementfieldcanbecontinuous (continuousmedium)ordiscontinuous (granularmedium orfrac- ture).ThemethoddevelopedhereproposestoenrichtheDICformu- lationinordertointroduceconstraintsintheDICalgorithmscom- patiblewiththecontinuityordiscontinuityofdisplacementfield.

Inordertodescribethekinematicphysics,apolynomialshapefunc- tionisassignedtoeachelementeofthemeshtorepresentthelocal displacementvariations(Eq.(1)).

𝐮^𝐞_𝐱(𝑋𝐶𝐶𝐷,𝑌𝐶𝐶𝐷,𝑝^𝐞_𝑋)=

𝑑𝑘∑𝑋 𝑘=0

𝑑𝑙𝑋

∑

𝑙=0

𝑎^𝐞_𝑘𝑙𝑋_𝐶^𝑘_𝐶𝐷𝑌_𝐶^𝑙_𝐶𝐷

𝐮^𝐞_𝐲(𝑋𝐶𝐶𝐷,𝑌𝐶𝐶𝐷,𝑝^𝐞_𝑌)=

𝑑𝑘∑𝑌 𝑘=0

𝑑𝑙𝑌

∑

𝑙=0

𝑏^𝐞_𝑘𝑙𝑋_𝐶^𝑘_𝐶𝐷𝑌_𝐶^𝑙_𝐶𝐷

(1)

where,𝐮^𝐞_𝐱and𝐮^𝐞_𝐲arethecomponentsoflocaldisplacementfield(u^e) forelementeinthedirections𝑋⃗and⃗𝑌,and(X_CCD,Y_CCD)represents theLagrangiancoordinatesinpixels.Vector𝐩^𝐞=(𝑝^𝐞_𝑋,𝑝^𝐞_𝑌)=(𝑎^𝐞_𝑘𝑙,𝑏^𝐞_𝑘𝑙) gathersthe kinematicshape function parameters. The order and thetype of the kinematic shape function can be chosen accord- ingtorequirements.Thedisplacementfields ineach elementare thusdescribedintherealspace,andnotthrougha“referenceele- ment”,whichisthecaseforclassicalFiniteElement(FE)descriptions [25,30].

• Introductionoftherestriction

Intheprevioussection,kinematictransformationsforeachindivid- ualelementhavebeenintroduced.Wewillnowdetailthepossible relationshipsbetweenkinematictransformationofneighbouringel- ements.

Threesituationscanbeencountered:

• Thetransformations between neighbouringelementsarecom- pletelyindependent.

• Atleastonecomponentofthedisplacementvectoriscontinuous throughthecommonboundary.

• Adisplacementjumpisallowedonthecommonboundary.

ThefirstsituationcorrespondstoclassicallocalDICmethods.When imposingcontinuityconditionsonbothdisplacementcomponents, thesecondsituationisanalogoustoglobalDICmethods[31].Here, thecontinuitycondition canbe enforcedon thewholeboundary (exactrestriction)orinalimitednumberofnodes(partialrestric- tion).Inthesamespirit,continuityconditionscanalsobeintroduced atboundaryonthedisplacementderivatives.Thethirdonecorre- spondstounilateralcondition(crackopening).Thiskindofrestric- tionswillnotdiscussedinthispaper.

Afterthedisplacementshapefunctionhasbeenchosenforeachel- ement,restrictionscanbeintroducedbetweenthekinematicfields associatedwitheachpairofneighbouringelements.Fig.2schemat- icallyillustratesthesituationforagivenpairofadjacentelements (elementiandelementj)ofthekinematicmesh.Asproposed,the degreesofthepolynomialsforkinematicdescriptionin elementi andelementjarenotnecessarilyidentical.

Theboundary𝑙_𝑢^𝑖𝑗 betweenelementsiandjis modeledasalinear relationshipbetweenXandY,whosecoefficientsdependonlyonthe meshgeometry.Forarelatively“horizontal” boundary,asshownin Fig.2,theboundary𝑙^𝑖𝑗_𝑢 isexpressedas

𝑌_𝐶𝐶𝐷=𝛼_𝑢𝑋_𝐶𝐶𝐷+𝛽_𝑢 (2)

Naturally,thecaseofa“vertical” boundaryisdeducedbyinverting theroleofXandY,expressedas𝑋_𝐶𝐶𝐷=𝛼_𝑢^′𝑌_𝐶𝐶𝐷+𝛽_𝑢^′.

Fig.2. Descriptionoftheboundary𝑙_𝑢^𝑖𝑗betweentwoadjacentelementsiandj anddefinitionofthelocalNormal-Tangentialcoordinatesystem(𝑁⃗^𝑖𝑗,⃗𝑇^𝑖𝑗)ofthe boundary𝑙^𝑖𝑗𝑢.

Byconstruction,thecoefficients{𝛼u,𝛽u}or{𝛼^′_𝑢,𝛽_𝑢^′}oftheboundary expressionaredefined onlybythegeometricalmesh.Thecompo- nentsofthenormalandtangentialvectorsoftheboundary,𝑁⃗^𝑖𝑗and

⃗𝑇^𝑖𝑗,areexpressedfrom𝛼uor𝛼_𝑢^′asfollows:

• fortherelatively”horizontal” boundary:

𝑁⃗^𝑖𝑗= ( −𝛼_𝑢

1 )

and⃗𝑇^𝑖𝑗= ( 1

𝛼𝑢

)

(3)

• fortherelatively”vertical” boundary:

𝑁⃗^𝑖𝑗= ( 1

−𝛼^′_𝑢 )

and⃗𝑇^𝑖𝑗= ( 𝛼_𝑢^′

1 )

(4) The restriction conditions are introduced along the element boundary𝑙_𝑢^𝑖𝑗 usingthelocalNormal-Tangentialcoordinatesystem (𝑁⃗^𝑖𝑗,⃗𝑇^𝑖𝑗) oftheboundary.Asmentionedabove,differentkindof restrictionscanbeimposed:

• continuityrestriction:equalityofthevariable(oritsderivative)on bothsidesoftheboundary

• jumprestriction:inequalityofthevariable(oritsderivative)on bothsidesoftheboundary

Inthispaper,weonlyfocusedondescribingcontinuityrestrictions.

They correspondtotheintroductionof linearequations between theparametersdescribingkinematicfieldsoftwoadjacentelements.

Thecontinuityofthedisplacementfield(u^e)isimposedinthelocal Normal-Tangentialcoordinatesystemoftheboundary,inorderto imposeeitheranormaloratangentialdisplacementcontinuity(or bothsimultaneously).

Furthermore,restrictionconditionscanalsobeimposedonthedis- placementgradientontheelementboundaries.

Finally,takingintoaccountthesedifferentrestrictionsleadstoim- posethecorrespondinglinearequationsbetweenthetwoadjacent elementsiandj,thatcanbeexpressedasalinearsystem:

[𝐀^𝐢𝐣_𝐔]{

𝐏^𝐢𝐣_𝐔}

={𝟎} (5)

where[ 𝐀^𝐢𝐣_𝐔]

isthekinematicelementaryrestrictionmatrixbetween element i and j, and {

𝐏^𝐢𝐣_𝐔}

is the elementary vector containing alltheunknownkinematicparameters(𝐩^𝐢,𝐩^𝐣)=(𝑎^𝑖_𝑘𝑙,𝑏^𝑖_𝑘𝑙,𝑎^𝑗_𝑘𝑙,𝑏^𝑗_𝑘𝑙)for thesetwoadjacentelements.Thelinearrelationsbetweenpⁱandp^j (Eq.(5))allowstodecreasethenumberofindependentparameters tobedeterminedbycorrelationforelementiandj.Theintroduction ofthislinearrelationsreducesthenumberofDegreesOfFreedom (DOFs)requiredtodescribethekinematicfield.

Byiteratingthisoperationforallboundariesonwhichcontinuity restrictionsareapplied,aglobalkinematicrestrictionmatrixA_Uis builtforthemesh,aswellasaglobalvectorP_Ucontainingallthe kinematicparameters.

[𝐀_𝐔]{

𝐏_𝐔}

={𝟎} (6)