USINGCHEMO-THERMO-MECHANICALLYCOUPLEDCRYSTALPLASTICITYSIMULATIONS
TOINVESTIGATETHEPROCESSOFWHISKERFORMATIONIN
¯
¡
SNTHINFILMS
By
AritraChakraborty
ADISSERTATION
Submittedto
MichiganStateUniversity
inpartialfumentoftherequirements
forthedegreeof
ChemicalEngineeringŒDoctorofPhilosophy
2019
ABSTRACT
USINGCHEMO-THERMO-MECHANICALLYCOUPLEDCRYSTALPLASTICITYSIMULATIONS
TOINVESTIGATETHEPROCESSOFWHISKERFORMATIONIN
¯
¡
SNTHINFILMS
By
AritraChakraborty
¯
-Sn(Sn)whiskersgrowingfromtincoatingsinelectronicdeviceshavebeenaseriousconcern
toitsreliabilityastheynucleateinunpredictablelocations,cangrowtoalengthofseveralmil-
limeters,andhence,areabletoshort-circuitdifferentpartsofanelectronicassembly.Itwas
foundthatalloyingSnwithlead(Pb)mitigatedwhiskerformation.However,withthedriveto-
wardslead-freeelectronicsfromearly2000,duetothehazardousnatureofPb,reliabilityprob-
lemsduetotinwhiskersresurfaced,thereby,rekindlinganinteresttounderstandtheirgov-
erningmechanismsinordertoproposecientmitigationstrategies.Withsuchadrive,sig-
niantresearchhasbeenpublishedhighlightingdifferentwhiskercharacteristics(morphol-
ogy,propensity,andkinetics).However,todate,thereisnotheorythatcouldexplain
theformationofthesetousstructures,withsometimesevencontradictingobservations
betweendifferentresearchgroups.Despitesomecontradictions,itiswidelyacceptedthattin
whiskersarearesultofstress-drivendiffusioninthewithdiffusionoccurringprimarily
throughthegrainboundarynetwork.Stressesinthesecanoriginatefromvarioussources,
suchas,upondeposition,indentation,thermalexpansionmismatch,andbyformationofvolu-
minousCu
6
Sn
5
intermetalliccompoundsattheinterfacebetweenacoppersubstrateandthe
Amongthesesources,thermalstraininghasbeenusedtodeliberatelycontrolthestresses
indepositedonaSilicon(Si)substrate(havingalowerthermalexpansioncoecientthan
¯
-Sn)toanalyzetheeffectofappliedstressonwhiskerpropensity.
Inthepresentworksuchexperimentalconditionsofthermallystraining
¯
-Snismim-
ickedusingsimulaitons,andtheresultingmechanicsandkineticsareinvestigatedusingfully
coupledchemo-thermo-mechanicalsimulationsinacrystalplasticitycontinuummechanical
framework.Thegoaloftheistoidentifycrystallographicandgeometricfactorsthatmodu-
latethestress-driventransportofatomsalongthegrainboundarynetwork.Inthatregard,
athermo-mechanicalstudyisdonethathighlightsthedominatinginenceofglobal
urecomparedtograingeometryandgrain-sizedistribution.Itisalsoestablishedthat
whiskernucleationisindeedalocalphenomenonasnolong-rangestressgradientsarefound.
Moreinvolvedchemo-thermo-mechanicalsimulationsareperformedtounderstandthekine-
maticsandkineticsofatomredistributioninthesethermallystrainedfortwoloadingcon-
ditions.Basedonthesesimulationsitisinferredthatplasticrelaxationplaysadominantrole
instressevolutioncomparedtodiffusionkinetics.Thedominantroleoflm-textureondeter-
miningwhiskernucleationsites(lowcompressivelocationsinthebiaxiallystrainedm)can
beattributedtothehighanisotropyofthebody-centeredtetragonal
¯
-Sncrystalstructurethat
involvesanelasticanisotropy,athermalanisotropy,andaverycomplexplasticanisotropy.The
complexplasticanisotropyarisesduetotheavailabilityofmultipleslipfamilieswithasmall
numberofslipsystemsperfamily.PlasticityreadilyhappensinSnevenatroomtemperature,
whichcorrespondstoabout0.6
T
m
,and,therefore,requirestheestablishmentofanaccurate
constitutivedescription.
Inthatregard,thesecondhalfofthisworkfocusesonestablishingthereliabilityof
InverseIndentationAnalysis(
IIA
)asameanstoidentifysuchconstitutiveplasticparameters
fordifferentcrystalstructures.Inthismethod,theerrorinexperimentalandsimulatedinden-
tationresponse(bothinloadŒdisplacement
and
surfacetopography)isminimizedtoobtain
theconstitutivematerialparameters.Thereliabilityofthis
IIA
methodisestablishedfor
face-centeredcubic(fcc)materials,whereitisfoundthatanycrystalorientationisadequately
sensitivefortheoptimizationtoidentifytheparameters.Suchisnotthecaseforlesssymmetric
hexagonalmaterials,wherethecacyofthe
IIA
methodreliesheavilyontheselectedcrystal
orientation,anditisproposedtoselectcrystalorientationsthataresensitivetoallslipfamilies
consideredintheconstitutivedescription.Furthereffortswillapplytheproposed
IIA
method
to
¯
-Sn.
ACKNOWLEDGEMENTS
Iexpressmydeepestgratitudetomyadvisor,Dr.PhilipEisenlohr,forhisimmensepatience,
support,andcontinuousencouragementtostriveforexcellenceduringmygraduateresearch.
IthasbeenarealpleasuretoworkandlearnfromDr.Eisenlohr.Hisabilitytoexplaincom-
plicatedthingsthroughsimpleexamples;investigatingascientiproblemsystematically;and
alwayshavinganopenmindtonewideas,arethingsIwillalwayslookuptofortherestofmy
life.Withhisconstantguidanceandsupport(bothprofessionalandpersonal),Iwasableto
contributemypartintheofcrystalplasticitymulti-physicsmodeling,aftercomingfroma
backgroundinchemicalengineering.
IwouldliketothankDr.ThomasBieler,whosewillingnesstohelpmeandanswermynu-
merousquestionswithenthusiasm,issomethingIcherishedthoroughlyduringmygraduate
study.Myapproachtoresearch,andtheassociatedtechnicalchallenges,havebeensig
cantlyedbytheseveraldiscussionswhichIhadwithDr.Bieler.Iwouldalsoliketo
thankDr.KalyanmoyDeb,forhisinsightsregardingfundamentalsofoptimizationalgorithms.
Hiscourseonoptimizationforengineeringapplicationsintroducesavastarrayofdifferental-
gorithmsaswellastheirphysicalandmathematicalbasis,thathelpedmeformulatingmyown
optimizationmodule.Ialsowanttothankmyothercommitteemember,Dr.DonaldMorelli,for
providingseveralinterestingperspectivesregardingmyresearch.Iextendmysincerethanksto
Dr.MartinCrimp,Dr.CarlBoehlert,andalloftheirstudents(theentiremetalsresearchgroup),
aswellasthemembersoftheComputationalMaterialsMechanicsresearchgroup,forthestim-
ulatingdiscussionsandthevaluablesuggestionsthathelpedmeshapemygraduateresearch
work.IalsothankDr.GregSwain,intheDepartmentofChemistry,forallowingmetoperform
electro-depositionexperimentsinhislab,andhisstudentKirtiBhardwajforherimmensehelp
withtheexperiments.
AmassivethankstoDr.PraveenKumarfromIndianInstituteofScience,Bangalore,India,
andDr.PiyushJagtap,nowinBrownUniversity,forintroducingmetotheworldoftinwhiskers.
iv
IthasbeenapleasuretogettoknowPiyush,whohasbeenagreatfriend,andalsofamiliarized
mewiththeexperimentalintricaciesrelatedtowhiskers.MysincerethankstoDr.Pratheek
ShanthrajandDr.MartinDiehlinMax-Planck-InstitutfürEisenforschung,Düsseldorf,Ger-
many,fortheirconstanthelpandguidanceindevelopingthemulti-physicssolverinDAMASK
andmakingitopensource.Theyhavebeenimmenselypatientandalwayswillingtohelpme
withanyqueriesIhad.IalsothankDr.LikChuanLeeandDr.SeungikBaekfromtheDe-
partmentofMechanicalEngineering,foralwaysbeingthereforanyqueriesonmechanicsand
numericalmethods.Itisgreattohaveinteractedandlearnedfromthem.
Mygraduatestudywouldnothavebeensosmoothwithouttheconstantsupportofallof
myfriendsinMichiganStateUniversitywhohavebecomemyfamilyhere,especiallyŠSapDa,
Preetam,Sabya,Kanchan,Kirti,Sayli,Tarang,Yashesh,Oishi,Chetan,Indraroop,Chen,Tridip
Da,Tias,Atri,Natalia,David,Alex,Eric,Joe,Mariana,Jialin,HongKang(andsurelymanymore).
Lastbutnottheleast,Iamgratefulformyparents,friends,andfamilymembersinIndiafor
alwayslovingandsupportingme,andencouragingmetochasemyambition.
PartsofthisresearchworkwasfundedbyNSFgrantDMR-1411102.Foranyerrorsorinad-
equaciesthatmayremaininthiswork,ofcourse,theresponsibilityisentirelymyown.
TABLEOFCONTENTS
LISTOFTABLES
............................................
ix
LISTOFFIGURES
...........................................
x
CHAPTER1GENERALINTRODUCTION
...........................
1
1.1Motivation
..........................................1
1.2Overview
...........................................5
CHAPTER2LITERATUREREVIEW
...............................
12
2.1Background
..........................................12
2.2Materialpropertiesandanisotropyoftin
........................12
2.3Currentunderstandingoftinwhiskerformation
....................15
2.3.1Whiskergeometryandcrystallography
.....................15
2.3.2Tinmplatingandresultingmicrostructure
.................18
2.3.3Causesofwhiskers
.................................20
2.3.4Whiskermitigationstrategies
...........................22
2.3.5Growthmodesofwhiskers
............................23
2.3.6Modelsofwhiskerformation
...........................26
2.4Modelingframework
....................................33
2.4.1Continuummechanics
...............................34
2.4.2CrystalPlasticityMulti-physicsframework
...................38
2.4.2.1Numericalschemes
...........................41
2.5Summary
...........................................46
CHAPTER3THERMOMECHANICALCRYSTALPLASTICITYMODELINGOFTHER-
MALLYSTRAINED
¯
¡
SNFILMS
.........................
49
3.1Background
..........................................49
3.2Simulationdetails
......................................50
3.2.1Geometrydiscretization,texture,andboundaryconditions
......50
3.2.2Continuummechanicalframework
.......................51
3.2.3Constitutivedescription
..............................53
3.3Simulationresults
......................................54
3.3.1Spatialvariabilityofgrainboundaryhydrostaticstress
............55
3.3.2Effectoftexture
................................56
3.3.3Effectofgrainsizedistribution
..........................59
3.3.4Effectofobliquesurfacegrains
..........................60
3.3.5Effectof
¯
-Sncrystalanisotropy
.........................62
3.3.6Establishmentoflocalcrystallographiceffect
.................62
3.4Discussion
..........................................67
CHAPTER4FULLYCOUPLEDCHEMO-THERMO-MECHANICALMODELINGOF
THERMALLYSTRAINED
¯
¡
SNFILMS
.....................
71
vi
4.1Introduction
.........................................71
4.2Modeldevelopment
.....................................72
4.2.1Constitutiveequations
...............................73
4.2.2Residualformulation
................................77
4.3Simulationdetails
......................................78
4.3.1Simulationgeometryandboundaryconditions
................78
4.4ResultsandDiscussion
...................................80
4.4.1Kinematicconsequence
..............................81
4.4.2Comparisonofdifferentstresses
.........................83
4.4.3Kineticsofthetransport
..............................86
4.5Summary
...........................................88
CHAPTER5INVERSEINDENTATIONANALYSISANDITSAPPLICATIONINFACE-
CENTEREDCUBICMATERIALS
.........................
91
5.1Motivation
..........................................91
5.2Background
..........................................92
5.3Introduction
.........................................93
5.4Simulationset-up
......................................97
5.4.1Finiteelementdiscretization
...........................97
5.4.2Phenomenologicalmaterialpointmodel
....................97
5.5Globalsensitivityanalysis
.................................99
5.6OptimizationMethodology
.................................100
5.6.1Universaloptimizationmodule
.........................100
5.6.2NelderŒMeadsimplexoptimizationalgorithm
.................100
5.6.3Objectivefunctionforoptimization
.......................104
5.6.4Determinationofmostsensitiveconstitutiveparameters
..........105
5.7Results
.............................................108
5.7.1Iuenceofiteelementmeshsize
......................108
5.7.2Iuenceofobjectivefunction
..........................108
5.7.3Reproducibilityofparameteridenticationfordifferentobjectivefunctions
109
5.7.4Robustnessofparameteridenticationfordifferentindentationorien-
tations
........................................113
5.7.5Dual-orientationobjectivefunction
.......................115
5.7.6Iuenceofthereferencepoint
.........................115
5.7.7Predictionofslipsystemactivity
.........................117
5.8Discussion
..........................................119
5.9Summary
...........................................121
CHAPTER6RELIABILITYOFINVERSEINDENTATIONANALYSESINHEXAGONAL
MATERIALS
.....................................
122
6.1Background
..........................................122
6.2Introduction
.........................................123
6.3CRSSvaluesfromsurfacetopographyandinstrumentedindentation
........125
6.4Virtualexperiments
.....................................126
6.4.1Constitutivematerialmodel
...........................127
6.4.2Virtualindentation
.................................128
6.5Hybridoptimizationalgorithm
..............................129
6.6Reliabilityofinverseindentationanalysis
........................132
6.6.1SelectingorientationsforperformingtheInverseIndentationAnalysis
..132
6.6.2Results
........................................134
6.7Discussion
..........................................139
6.8Summary
...........................................141
CHAPTER7CONCLUSIONSANDFUTUREWORK
......................
143
7.1Conclusions
..........................................143
7.2Futurework
..........................................145
BIBLIOGRAPHY
............................................
148
viii
LISTOFTABLES
Table2.1:Slipfamiliesin
¯
-Sncrystalstructure[
Leeetal.
,
2015
].
...............14
Table3.1:Materialpropertiesfordilatationalair,body-centeredtetragonal
¯
-Sn,and
isotropicsubstrate.
....................................70
Table4.1:Materialpropertiesfordilatationalair,body-centeredtetragonal
¯
-Sn,and
isotropicsubstrate.
....................................90
Table5.1:Resultsofglobalsensitivityanalysisforthefouradjustableparametersus-
ingeachofthethreeobjectivefunctions
²
LD
,
²
topo
,and
²
combo
andtwoval-
ues(
n
Æ
25and
n
Æ
5)forthestressexponentinthekineticsequationde-
scribedinSection
5.4.2
.Tabulatedvaluesrepresent
¹
¤
asdescribed,with
largernumbersindicatingahigherrelativeeoftherespectivepa-
rameter.CrystalorientationsarelabeledinaccordancewithFig.
5.4
.
......107
Table5.2:Iuenceoftargetpointinparameterspaceontheoptimizedparameterval-
uesusing
²
combo
astheobjectivefunctionand
²
tol
=0.005.Thereferenceval-
uesforeachoftheparametersareshowninparenthesesandtherespective
boundschosenfortheoptimizationareshowninbrackets.
............116
Table6.1:Resultsofthe
IIA
analysisperformedforsixdifferentcrystalorientationsla-
belledinFig.
6.4
fortwodifferentreference
input
CRSS.
..............138
ix
LISTOFFIGURES
Figure1.1:ImageshowingdevicefailureduetoshortcircuitbyformationofSn
whiskersjoiningtwopinsinanintegratedcircuitchipinaprintedcircuit
board.
NASAwhiskerweb-site
............................2
Figure1.2:Otherdevicefailuresduetowhiskerformation-left:potentiometerinbrake
peddleinCorolla2003;right:cardguidesusedinspaceshuttlestohold
printedcircuitboards(2008).
NASAwhiskerweb-site
...............3
Figure1.3:NumberofreportedcasesofmedicaldevicefailureduetoSnwhiskerfor-
mationfrom2006-2016.[
Alagarsamyetal.
,
2018
]
.................4
Figure2.1:Deformationresistanceofbulktin(opensymbols,[
Mohamedetal.
,
1973
,
Mathewetal.
,
2005
,
WeertmanandBreen
,
1956
,
McCabeandFine
,
2002
,
Adevaetal.
,
1995
])andthin(dots,[
Boettingeretal.
,
2005
,
Chason
etal.
,
2014
])undervariousloadingconditionsandhomologoustempera-
tures(shadesofred).
..................................13
Figure2.2:Examplesoftinwhiskers,hillocks,andsmallerprotrusions.Topleftimage
byPeterBush(SUNYBuffalo)[
BoguslavskyandBush
,
2003
],bottomleftim-
agefromNASAmetalwhiskerphotogallery,centerimagesbyPiyushJagtap
andPraveenKumar(IIScBangalore,unpublished),crosssectionscutbyfo-
cussedionbeamandcorrespondingsketchesonrightby[
Chasonetal.
,
2013
].
16
Figure2.3:fiObliquegrainflwithshallowgrainboundariesatthebaseofatypical
whisker[
Saroboletal.
,
2013
].
.............................16
Figure2.4:Nucleationandgrowthofawhisker14h(a),14.3h(b),14.6h(c),17h(d),
21h(e)afterdeposition(ontoCusubstrate)[
Jadhavetal.
,
2010a
].
........24
Figure2.5:Growthofahillockwithextensiverotationimaged6h(a),8h(b),12h(c),
18h(d),50h(e)afterdeposition[
Jadhavetal.
,
2010a
].
..............24
Figure2.6:Sequenceofcomplexhillockgrowthwithvaryingcross-sectionandpartial
bending6h(a),12h(b),18h(c),32h(d),44h(e),56h(f),76h(g),138h
(h)afterdeposition.Insetsshowschematicofshapeevolutionhighlighting
rotationoftheoriginalsurface.Arrowspointtograinboundaryfeaturesin
(d)thatarevisibleasridgesonsideofthehillockin(h).From[
Jadhavetal.
,
2010a
].
..........................................25
Figure2.7:Conceptualunderstandingofwhiskerandhillockformationmechanisms
[
Jadhavetal.
,
2010a
].
..................................27
x
Figure2.8:Kinematicsofacontinuumbodyundergoingdeformation[
Diehl
,
2016
].
...35
Figure2.9:Illustrationoftheintermediateconationsresultingfromthemulti-
plicativedecompositionofthedeformationgradient[
Rotersetal.
,
2019
].
Selectingthecrystalorientationasinitialvalueof
F
p
(t
Æ
0)
Æ
O
0
guarantees
thatthelatticecoordinatesystemintheplasticgurationalwayscoin-
cideswiththelabcoordinatesystem[
Maetal.
,
2006
].
..............39
Figure2.10:Solutionstrategyforthevariouskinematicquantities,[
Rotersetal.
,
2019
].
..41
Figure3.1:Discretizedgeometryofatinbetweenarigidisotropicsubstrate(or-
ange,partlyshowingvoxelsizeinthecornerofleftimage)andsoftdi-
latationalfiairfl(faintblue).Boundaryregionbetweencolumnartingrains
(shadesofyellow)ishighlighted(green)andconstitutesthe(sub)volumeof
themforwhichhydrostaticstressgradientsareassumedtodriveatom
diffusion.Selectedgrainsofthecolumnarshownatleftaretrans-
formedintoanobliquesurfacegrain(grayvolumeinthemiddleandright
views).
ChakrabortyandEisenlohr
[
2018
]
......................50
Figure3.2:Forcedtemperaturechangeoftinmsonrigidsubstrate(black)andre-
sultingcompressivevonMisesstressinthesimulatedfordifferent
textures(coloredlines)comparedtomeasuredstressevolutiononaSi
substrate(circlesfrom
Peietal.
,
2017
).Pointofmaximumloadisreached
after20min(atendofheating)andservesasreferenceforallsubsequently
reportedstressvalues.
ChakrabortyandEisenlohr
[
2018
]
............55
Figure3.3:Spatialvariabilityofgrainboundary(top)andbulkgrain(bottom)hydro-
staticstress
p
formaximumload(reachedafter20min,seeFig.
3.2
)illus-
tratedatthreedifferentdepthsthroughthe(surfacetosubstratefrom
lefttoright).Substrateisnotshownwhilepartofthecolumnargrainstruc-
tureofthetinisillustratedandcoloredaccordingtothecrystallo-
graphicdirectionthatisparalleltothesurfacenormal(closeto
h
100]inthis
example,seecolorcodeinstandardstereographictriangle).
Chakraborty
andEisenlohr
[
2018
]
..................................56
Figure3.4:Distributionsofgrainboundaryhydrostaticstressatthreedifferentdepths
withintinofeithergreen,blue,orredtexture(
h
100],
h
110],and
h
001],fromlefttoright).Light,medium,anddarklineshadescorrespond
tothelayerdepthsshowninFig.
3.3
left,center,andright,i.e.,atthe
surface,middle,andsubstrateinterface.Inversepoleuremapsof
normalandonein-planedirectionillustratethethreedifferenttexturesas-
signedtootherwiseidenticalgrainstructure.
ChakrabortyandEisenlohr
[
2018
]
...........................................57
xi
Figure3.5:Comparisonofgrainboundaryhydrostaticstressdistributionsbetween
withanisotropicandisotropicthermalexpansion(solidanddashed)
havingeither
h
100],
h
110],or
h
001]texture(green,blue,andredfrom
lefttoright).Hydrostaticstressmapsatthesubstrateinterfacerevealno
qualitativechangeinthehighdegreeofspatialvariability(leftandright
halves,bottomrow).
ChakrabortyandEisenlohr
[
2018
]
..............58
Figure3.6:Variationofgrainareadistribution(left)forthreedifferentstructures
(labelledfiafl,fibfl,andficfl)andtheirassociateddistributionsofhydrostatic
stressonthegrainboundarynetwork(middle)formshaving
h
100]
texture(seeinversepoleecoloringofpartlyshowngrainstructure).
Spatialmaps(rightfia"tofic")showthelayerdirectlyatopthe(invisible)
substrateandillustratetherapidoscillationof
p
consistentlyobservedfor
eachofthethreestructures.
ChakrabortyandEisenlohr
[
2018
]
.........59
Figure3.7:Distributionofchangeinhydrostaticstress
p
acrossthegrainboundary
networkduetotransformationofone(fiafltoficfl)ormultiple(fidfl)colum-
nargrainsintoobliqueones.Thesegeometricalalterationsaffectonlya
smallfractionofthegrainboundarynetwork,asdemonstratedbythenar-
rowoveralldistributions(left).Spatialmapsconmthelocalizedinence
ofobliquegrainsontherelative(top)andabsolute(bottom)changein
p
for
thelayerclosesttothesubstrate(termedfibottomflinFig.
3.3
)withthe
transformed(formerlycolumnar)grainshowninorange.
Chakrabortyand
Eisenlohr
[
2018
]
.....................................61
Figure3.8:Distributionofhydrostaticstress
p
acrossthegrainboundarynetwork(and
atmaximumload)forthreedifferentglobaltextures,toprow:
h
100];
middlerow:
h
110];andbottomrow:
h
001].Ineachoftheplots,lighter
curveslabelled`I'correspondstothe
p
distributionforasimulationhaving
bothmechanicalandthermalanisotropywhichactasareferencedistri-
bution.Thisreferencedistributioniscomparedtothe
p
distributionsob-
taineduponmaking
¯
-Snthermallyisotropicbutmechanicallyanisotropic
(darkercurves,labelled`II',inleftcolumn);thermallyanisotropicbutelas-
ticallyisotropic(darkercurves,labelled`III',inmiddlecolumn);andboth
thermallyandelasticallyisotropic(darkercurves,labelled`IV',inrightcol-
umn),i.e.,keepingonlyplasticanisotropy.
.....................63
Figure3.9:Toprowshowsthereferencetexture(right:outofplane;left:in-plane)
andgrainmorphologyforthe
h
100]lmusedearlier.Inthesubsequent
imagesthetextureandgrainmorphologyofthepartinsidetheblackcircle
remainsunchangedwhileoutsidethegrainsarealongwithashuf-
ofthetexture.Toptobottomshowsagradualdecreaseintheareaof
unchangedpart(radiusoftheblackcircle)fromabout5grainsizes(second
row)to
¼
oneadjacentgrainfromthereference(central)grain.
.........64
xii
Figure3.10:Distancedistributionoftheunchangedpart,approximatedistancesbe-
tweenunchangedvoxelsandthecentralvoxeloftheblackcirclesinFig.
3.9
withfia",fib",fic",andfid"representingrows2to5.
................65
Figure3.11:Differencesinhydrostaticstressesalongthegrainboundarynetworkfor
thedifferentradiiofunchangedtextureandgrainmorphology(rows2to
5inFig.
3.9
)withthatofthereferencecase(row1inFig.
3.9
)atthelm-
substrateinterface(left)andsurface(right).Thisisa2-Dhistogram
plotwheretheshadeofgrayrepresentsthevaluemeaningdarkerblackcir-
clescorrespondtozerochangeinthestressvalues(y-axis),andthelocation
ofthecirclesindicatethedistancefromthecentralgrain'svoxel.Theblack
line,ineachcase,indicatestheapproximatedistancebetweenthechanged
andunchangedpartinthefromthecentralgrain(radiusoftheblack
circleinFig.
3.9
)
.....................................66
Figure4.1:Arepresentativevoxelwithamaterialpointshowingthexdirection
(alongx-axis)andgrainboundarynormaldirection,
n
,alongy-axis.The
grainboundaryxperunitlinecrosssectionalongz-direction,foran
atomicdistanceisgivenby(

1/3
J
GB
).Subsequently,therateofmono-
layeradditionisrepresentedby

m
whichcorrespondstothedivergenceof
thisthroughanotheratomicdistanceof

1/3
.
.................77
Figure4.2:Discretizedgeometryofatinbetweenarigidisotropicsubstrate(or-
ange,showingvoxelsize)andsoftdilatationalfiairfl(faintgray).Boundaries
(lightblue)betweenthecolumnargrains(green)havedifferentchemical
(diffusion)propertiesbutsharethermalandmechanicalpropertieswith
thebulk.Partofthegeometryalsorevealstheglobal
h
100]lmtexture.
...............................................79
Figure4.3:AveragevonMisesstressevolutioninthelmforthecoupledchemo-
thermo-mechanicalmodel(blue)tothatofthecaseofwithoutanydiffu-
sionalw(red),foratemperaturechangeof20K(left)and40K(right)in
10and20minrespectively(shownbytheblackcurve),followedbyarelax-
ationofabout60min.Thedasheddarkgraylinerepresentsthepointof
maximumload/straininthesystem,whichisusedasareferencelocation
forfurthercomparisoninthestudy.
.........................81
Figure4.4:Kinematicconsequenceofatomredistributionisshownbycomparingdis-
tributionofdeterminantoftheeigenstraindeformationgradients
F
i
along
thegrainboundarynetwork,atmaximumloadforthecoupledchemo-
thermo-mechanicalmodel(blue)tothecaseofwithoutanydiffusionalw
(red),foratemperaturechangeof20K(left)and40K(right).
..........82
xiii
Figure4.5:Spatialvariationofthepressure(toprow)andtheconcentrationchangein
numberofatoms(bottomrow),pergrainboundaryarea,forthetwoload-
ingconditionsof20K(leftcolumn)and40K(rightcolumn)atthemaxi-
mumloadandatthelmŒsubstrateinterface.Asexpected,locationsoflow
compression(fired"regions)inthepressureplots(toprow)correspondto
locationsofatomgains(fired"regions)intheconcentrationplots(bottom
row).
............................................83
Figure4.6:Normal(alongnormaldirection)andshear(intheboundaryplane)
tractioncomponents(right)derivedateachpointofthegrainbound-
arybasedonthelocalboundaryplanenormal(middle)ofanexemplary
(oblique)grainisolatedfromtheentire(left,onlygrainboundarynet-
workshown).
.......................................84
Figure4.7:Probabilitydistributionplotsforpressure(left),ashearcomponentofthe
traction(middle),andthetractioncomponentactingalongthemnor-
maldirection(right,foronlytheobliquegrainboundary)alongthegrain
boundarynetwork,allcomputedatmaximumstraincondition.Thesolid
anddashedlinescorrespondtothedistributionofthesevaluesatthe
surfaceandbstrateinterfacerespectively.
.................85
Figure4.8:Arealconcentration(pergrainboundarysurfaceinm
2
)distributionwithin
thegrainboundarynetworkattherateinterface(dashedcurves)
andatthesurface(solidcurves)foraloadof20K(left)and40K(right)
atthemaximumstraincondition.
..........................86
Figure4.9:Averageconcentrationevolutionoftheexemplaryobliquegrainboundary
(highlightedinFig.
4.6
)forthetwodifferentthermalloadingconditionsof
20Kand40K.
......................................87
Figure5.1:Finiteelementmodelincludingindenterandsinglecrystallinesubstrate.
Indentationissimulatedbyprescribingdownwardfollowedbyupwardmo-
tionoftherigidindentersurfaceatconstantvelocity.Nodaldisplacements
ofsubstratearefullyrestrictedonthebottomandsidefaces.
...........96
Figure5.2:Iterativeoptimizationsetuptoidentifyadjustableparametersofacrystal
plasticityconstitutivelaw.Basedonaselectedstrategy,theoptimizerad-
juststheparameters,whicharethenfedasinputintotheCPFEsimulation
usingthematerialpointmodelDAMASK.Objectivefunctionvalueisob-
tainedastheloadŒdisplacement(
²
LD
)and/orsurfacetopography(
²
topo
)
deviationfromagivenreference.
...........................101
xiv
Figure5.3:Two-dimensionalillustrationofthesequenceofoperationsperformedby
theNMsimplexalgorithmasimplementedhere.Bluetrianglemarksthe
initialsimplex,redtriangletheonethatresultsfromthespoperation
mentionedontheconnectingarrow.Verticesarelabelledbynotationsas
describedinthetext.Thestrategytonavigateawayfromaninfeasiblever-
texisdisplayedbythegraysimplexinthebottomleft.
..............103
Figure5.4:Crystallographicorientationsoftheindentationaxisconsideredinthe
presentstudy.
.......................................105
Figure5.5:Iuenceofmeshsizeonthestabilityoftheimplementedinverseanalysis.
Thehorizontallinein(
a
)denotesthetolerancecriterion;verticallinesin
(
b
)showtheparameterboundsfortheoptimization.
...............109
Figure5.6:ComparisonofloadŒdisplacementandsurfacetopographydeviations(ab-
solutevaluesbetween0to
Å
10nm)fromreferenceresponsewithparam-
etersoptimizedtowithin
²
tol
Æ
0.005oftherespectiveobjectivefunction
(fromlefttoright,
²
LD
,
²
topo
,and
²
combo
Æ
(
²
LD
Å
²
topo
)/2)foranexemplary
indentationalongdirectionfif"inFig.
5.4
.ThereferenceloadŒdisplacement
curveisshownbyblack.
................................110
Figure5.7:Optimizedparameterset(
¿
0
,
¿
sat
,and
h
0
)usingdifferentobjectivefunc-
tions(
²
LD
,
²
topo
,and
²
combo
,lefttoright)andtolerancevalues
²
tol
(topto
bottom)resultingfromverandominitialsimplicesforonecrystal-
lographicindentationdirection.Thehorizontalgraylinemarksthetarget
value(
Æ
1),verticallinesspantheboundsofeachparameter.Theverti-
calbartotherightofeachurerepresentsthestandarddeviationofthe
overallparameterset.
..................................111
Figure5.8:Toprowcorrespondstotheevolutionofobjectivefunctionvalue(forthe
bestpointinthesimplex)withnumberoffunctionevaluations(cost)for
differentobjectivefunctions(lefttoright:
²
LD
,
²
topo
,
²
combo
)thatdidnot
reach
²
tol
Æ
0.005(horizontalgrayline).Therelativedeviationbetweenre-
sultingparametersetsandtargetvaluesisshowninthebottomrow.Termi-
nationbeforethemaximumallowablenumberoffunctionevaluations(45)
resultedduetothedegeneracyofthesimplexatalocalminimum.
.......112
Figure5.9:Bottomrowrepresentsparameterestimationstabilityfordifferentin-
dentationorientationsanddifferentessfunctions(lefttoright:
²
LD
,
²
topo
,
²
combo
and
²
dual
).Toprowgivescorrespondingobjectivefunc-
tionvaluevs.cost,withtolerance
²
tol
Æ
0.005indicatedbythegrayhorizon-
talline.Theanalysiswasperformedfor4differentcrystalorientationsfor
²
LD
and
²
topo
,7differentcrystalorientationsfor
²
combo
,and6differentori-
entationpairsfor
²
dual
.Greenandbluecurvesin
c
indicate
²
LD
and
²
topo
comprisingtheblack(exemplary)
²
combo
evolution.
...............114
xv
Figure5.10:Relationbetweenaccuracyofslipsystemresponseandqualityofconsti-
tutiveparameteridention.Sevensetsofoptimizedparameters(top),
eachfung
²
combo
·
0.005,sortedbydecreasingaccuracy,andresulting
forthesevencrystalorientationsshowninFig.
5.4
.Deviationinaccumu-
latedshearbetweensimulationsusingthetarget(
°
ref
)andtheidenti
parameters(
°
opt
)contrastedtotheaccumulatedshearvaluesobservedin
thereferencevaluesimulation(bottom,excludingany
°
ref
Ç
10
¡
4
).
......118
Figure6.1:Schematicrepresentationoftheinverseindentationanalysis.
..........125
Figure6.2:GeometrygeneratedinMarc-Mentattoperformtheindentationsimula-
tions.
...........................................129
Figure6.3:Illustrationofthestochasticparticleswarmoptimizationalgorithmbeing
usedinthisstudyfor
IIA
.
................................130
Figure6.4:Differentcrystalorientationsusedtoperformthesensitivityanalysisin
hexagonalcp-Ti.
.....................................133
Figure6.5:Sensitivityofdifferentcrystalorientationstodeformationondifferentslip
familiesasrepresentedbytheradiusofthecircles.
................135
Figure6.6:ResultingCRSSvaluesforallthesixsinglecrystalindentationsforthetwo
differentreferenceCRSS.
................................140
Figure7.1:Proposedstructureforidentifyingwhatconditionsinthetinmmi-
crostructurecauseagraintoformawhisker.
....................145
xvi
CHAPTER1
GENERALINTRODUCTION
1.1Motivation
Whiskersaregenerallyreferredtoasousstructuresthatshowahighaspectratioin
theirlength(height)comparedtotheircross-sectionarea(width).Thegrowthoftheselamen-
tarymetalstructuresonmetalswasobservedin1945fromCd-electroplatedcondenser
thatledtotheshort-circuitfailureoftheelectronicequipment[
Cobb
,
1946
].Threeyearslater
thegrowthof(tin)SntsfromSncoatedconnectorswasalsoidentwhichcauseda
short-circuitfailureoftelephonetransmissionlinesoftheBellTelephoneCorporation[
Comp-
tonetal.
,
1951
].Othermetalwithlowmeltingpoints,suchasZn,Ag,andCu,werealso
foundtoexhibitsimilarwhiskeringbehaviors[
Colemanetal.
,
1957
,
Brenner
,
1957
],howeverSn
whiskershavebeenthemostextensivelyresearchedamongthembecauseoftheirsignt
useinelectronicdevices.
Sncoatingsarefrequentlyappliedonelectroniccomponentsduetotheirfavorableprop-
ertiessuchasexcellentsolderability,ductility,electricalconductivity,andcorrosionresistance
[
LeeandLee
,
1998
].Thus,thefunctionaldegradationofelectronicdevices,andsometimes
theircompletefailureduetotheformationofSnwhiskersisamajorreliabilityconcern.It
wasdiscoveredempiricallyin1960sthatadditionoflead(Pb)topureSnpreventedwhisker
formation[
Arnold
,
1966
],whichledtothesuccessfulemploymentofwhisker-resistantsolu-
tionsforsolderingandcoatingproblemsinelectronicdevices.SincePballoyingmitigatedthe
long-standingproblemofwhiskerformationinelectronicdevices,itreducedtheattentionto
understandthemechanismscausingsuchwhiskersandtheirgrowthkinetics.However,after
thelegislativedirectivetobanPbfromelectronicgoodsduetoitshazardousnature[
Parliament
andoftheEuropeanUnion
,
2003
]astrongmovetowardsPb-freeelectronicsresulted,causing
arenewedinterestinthis70yearoldproblemofwhiskerformationthatledtoapoolofex-
1
Figure1.1:ImageshowingdevicefailureduetoshortcircuitbyformationofSnwhiskersjoining
twopinsinanintegratedcircuitchipinaprintedcircuitboard.
NASAwhiskerweb-site
perimentalandtheoreticalinvestigationsoverthepastcoupleofdecades.Suchanexponential
drivetowardsunderstandingcriticalfactorsgoverningtheformationofSnwhiskersfromelec-
troplatedcoatingswasfurthermotivatedbythefactthatpureSnorSn-basedalloyshavebeen
consideredasthemostsuitablecandidatestoreplaceeffectivelyandeconomicallyPbSnalloys
[
Chenetal.
,
2007
,
AbtewandSelvaduray
,
2000
].
EventhoughtheelectronicindustrywasoneofthemostaffectedduetoformationofSn
whiskerscausingshort-circuitbybridgingtwocomponentsoftheelectronicassembly,shown
schematicallyinFig.
1.1
,othersectorsthathavesufferedduetowhiskersincludeautomobiles,
spaceshuttles,nuclearpowerplants,andmedicaldevices.Figure
1.2
showsawhiskerjoining
twocomponentsofapotentiometer(left)presentinthebrakepeddleinaToyotaCamry,mak-
2
Figure1.2:Otherdevicefailuresduetowhiskerformation-left:potentiometerinbrakepeddle
inCorolla2003;right:cardguidesusedinspaceshuttlestoholdprintedcircuitboards(2008).
NASAwhiskerweb-site
ingthemrecallalltheir2003models,and(right)thetwocomponentsofcardguidesusedto
holdprintedcircuitboardsinspaceshuttlesthatareconnectedbywhisker.
Huangetal.
[
2018
]
highlightstheriskofSnwhiskergrowthinnuclearpowerplantsandtheneedformanagements
atalllevelstocometogetherintakingstepsinconductingriskassessmentandqualityassur-
ancestudies.
Amorerecentarticle[
Alagarsamyetal.
,
2018
]focusesontheimportanceofwhiskermitiga-
tionstrategiesforproducingreliabledevicesandthehighstakesassociatedwithfailureofsuch
devices.FromFig.
1.3
itappearsthatevenin2016therehavebeenconsiderableamountofde-
vicefailurecasesthatareattributedtoformationoftinwhiskers,therebyfurthermotivatingthe
3
Figure1.3:NumberofreportedcasesofmedicaldevicefailureduetoSnwhiskerformation
from2006-2016.[
Alagarsamyetal.
,
2018
]
needtoproperlyunderstandthefactorscausingsuchtaryprotrusions.Furthercasesof
failuresduetowhiskerformationiswelldocumentedin
NASAwhiskerweb-site
.
Therefore,withtherekindleddrivetounderstandmechanismscausingwhiskerformation,
multitudeofexperimentalobservationsandafewnumericalstudiesoverthepastcoupleof
decadeshavebeenperformedinordertoidentifytheirrootcausesandproposeientand
reliablemitigationstrategies.However,eventodaytherelacksacomprehensiveunderstanding
ofthisphenomena[
Zhangetal.
,
2015b
]thatlimitstheconpredictionof(i)theappar-
entrandomwhiskerlocations,(ii)theirobservedspatialdensities,(iii)theirwidearrayofmor-
phologies,andlastbutnotleast(iv)theirvariableincubationperiodswhichpreventsformula-
tionofaconsistentwhiskergrowthmodel.Inthenextsectionabriefoverviewofthecurrent
4
understandingislaidoutfollowedbytheobjectivesofthepresentwork.
1.2Overview
Tinwhiskersarebelievedtobefiperfect"singlecrystals[
Fisheretal.
,
1954
,
LeBretandNor-
ton
,
2003
]emanatingfromsinglegrainsonly[
Choietal.
,
2003
,
GalyonandPalmer
,
2005
,
Boet-
tingeretal.
,
2005
,
Zhaoetal.
,
2006
,
Tuetal.
,
2007
,
Nakadairaetal.
,
2008
,
Wangetal.
,
2014
,
Steinetal.
,
2015b
],withdiametersapproximately1-2
&
m(comparabletothegrainsize)and
withatomsbeingaddedfromthebottom[
KoonceandArnold
,
1953
].Theirtypicalgrowth
rateshavebeenreportedtobearound1Ås
¡
1
[
Ellisetal.
,
1958
,
GlazunovaandKudryavtsev
,
1963
,
FurutaandHamamura
,
1969
]whichmightvaryinpresenceofexternallyappliedstresses
[
Fisheretal.
,
1954
].Startingfromtheirinitialobservationtherehavebeenmultipletheoriesthat
areproposedtodescribetheprocessofwhiskerformation,suchasdislocation-basedmecha-
nisms[
Eshelby
,
1953
,
Frank
,
1953
,
Lindborg
,
1976
,
LeeandLee
,
1998
],whiskerbeingaresult
ofdynamicrecrystallizationandabnormalgraingrowth[
BoguslavskyandBush
,
2003
,
Vianco
andRejent
,
2009
,
Etienneetal.
,
2012
,
Viancoetal.
,
2015
],andwhiskersprovingtobeastress-
relaxationmeansfortheandgrowingfromgrainshavinginclinedgrainboundaries[
Tuand
Li
,
2005
,
Smetana
,
2007
,
Galyon
,
2011
,
Chasonetal.
,
2013
,
HeandIvey
,
2015
].Amongtheafore-
mentionedtheories,thestressrelaxationtheoryisthemostdiscussedandrelevantonedueto
theimmenseexperimentalsupport[
Zhangetal.
,
2015b
],andhencewouldbethefocusofthis
work.Evenwiththisgeneralagreementofstressbeingthecauseofwhiskerformation,there
existsseveraldiscrepanciesinliteratureintermsofitsnature(compressive,tensile,orboth),
theextentofthegradient,effectofplasticdeformation,roleofsurfaceoxidelayer,andmost
importantlytheeffectofcrystalstructureandorientation.Inthisworkwetrytoaddresssome
oftheopenquestionsandaimtohighlighttheimpactof
¯
-Sncrystalanisotropyingoverning
theoverallprocess.
Stressin
¯
-Snlmscanarisefromanumberofcausessuchasindentation[
Yangand
Li
,
2008
],bending[
Crandalletal.
,
2011
],thermalexpansionmismatchbetweentheand
5
itssubstrate[
Peietal.
,
2017
],or,themostcommonlyobserved,byformationofvoluminous
Cu
6
Sn
5
intermetalliccompoundsattheinterfacebetweenaCusubstrateandtheSn[
Cha-
sonetal.
,
2013
].Especiallyforthelattertwostressorigins,agradientofdecreasinghydrostatic
stressformsbetweenthemŒsubstrateinterfaceandthelmsurface[
Hektoretal.
,
2018
].Mea-
surementsofstressevolution(quantiedfromn-substratecurvaturemeasurements)
indicatesthatplasticrelaxationplaysanimportantroleandnotablymodulatesthewhisker
propensity[
Chasonetal.
,
2008
].Sinceamongthesestressinthestressesinduced
bythermalexpansionmismatchcanbecarefullycontrolledanditseffectcanbequantied
throughexperiments,soourfocusinthisworkistoanalyzethestresseldsandthesubse-
quentstress-drivendiffusiongeneratedbythermallystrainingthesystem.Moreover,itisour
hypothesisthatinadditiontomacroscopicgradientsinducedbytheexternalloadingcondi-
tions,thepolycrystallinenatureandtheinherentanisotropyof
¯
-Sncanbeexpectedtofurther
modulatethehydrostaticstress(anditsgradients)withintheInconsequence,aspatially
varyingchemicalpotentialofSnatomswouldresult,whichthenactsasthedrivingforcefor
atomicredistribution.
Themassredistributionislikelytobeoccurringsolelyviagrainboundariessinceatthetyp-
icaloperatingtemperaturestheself-diffusivityofSnisordersofmagnitudeslowerinthebulk
thanonagrainboundary(GB)[
Jagtapetal.
,
2017
]causingustofocusourattentiontostresses
generatedalongthegrainboundarynetworkoftheBasedontheoverallunderstanding
andcurrentknowledge,thekeyfactscanbesummarizedas:
Ł
Whiskerformationisastressreliefmechanismthatoccursinwithpredominantly
columnargrainstructurebyatomattachmentatsubsurfaceboundariesofobliquesur-
facegrainsthatdonotextendthroughthecompletelmthickness.Whethersuchoblique
grainsoccurnaturallyduringthedepositionprocess,orwhethertheyaretriggeredby
stress,inserviceconditions,isnotyetestablished.
Ł
Theassociatedatomxisdrivenbyapositivegradientofhydrostaticstressthatlow-
6
ersthechemicalpotentialofatomsonthewhiskergrainboundariesrelativetoother
grainboundaries.Typically,stressgradientsdevelopbetweenthebaseandsurfaceof
aSninresponsetoanumberofpossiblecauses,forinstance,thegrowthofinter-
metalliccompounds(typicallyCu
6
Sn
5
forCusubstrates),thermalexpansionmismatch
ofthesubstrate,orlocalindentation.Nevertheless,whiskerformationisalsotriggeredby
macroscopicallyhomogeneouscompressionortension[
Crandalletal.
,
2011
].
Ł
Sincebulkself-diffusivityofSnisroughlytenordersofmagnitudeslowerthangrain
boundary(GB)diffusionattypicaloperatingtemperatures,theatomisexclusively
supportedontheGBnetworkandthetransportismodulatedbythegrainboundaryhy-
drostaticstress.
Ł
Incubationperiodsoflargelyvaryingdurationareobserved,dependingonthemechan-
icalloadeitheralreadypresentand/ordevelopingwithinthem.Typically,thewhisker
nucleationrate,aswellastheindividualwhiskerelongationrate,decreaseovertime,par-
ticularlyfordecreasingstress.
Ł
Thestressevolution(quanfromm-on-substratecurvaturemeasurements)
indicatesthatplasticrelaxationplaysanimportantroleandcouldnotablymodulatethe
whiskerpropensity.
Fromtheincidenceofthedrivetodevelopeffectivewhiskermitigationstrategiesfromearly
2000,therehasbeenawiderangeofexperimentalobservations,however,therehasbeenlim-
itedprogressindevelopingaccurateandpracticalmodelscapableofcapturingthiscomplex
multi-physicssystem.Someofthemodelingeffortsarehighlightednext,whileamoredetailed
reviewonrelevantliteratureispresentedinChapter
2
.
Amajorshortcomingoftheexistingmodelsofwhiskergrowth(transportofatomstowards
aparticularfiwhiskergrain")isthefactthattheygenerallytendtosidestepthequestionofthe
nucleationeventandpostulateaconstant,non-zerolevelofstressbeingmaintainedatthe
whiskerrootgrainboundariesoncethenucleationhashappened[
TuandLi
,
2005
,
Sarobol
7
etal.
,
2013
,
Peietal.
,
2017
].Aroundthislocationofstress,aradiallysymmetricstress
gradientemergeswhenbalancingtheunderlyingsourceofcompressivelmstress,plasticre-
laxation,anddiffusivemassredistribution.Subsequentadhoctingofanobservedmstress
evolutionwaspossiblewiththestresslevelatthewhiskerrootbeingcomparabletothew
stressofSnatambienttemperatures.Suchaantresistanceagainstatominxatthe
whiskerrootwasqualitativelyrationalizedbasedonafrictionalcomponentexertedbythegrain
boundariesagainstout-of-planewhiskergrainmotion[
Saroboletal.
,
2013
],aresistancethat
increaseswithdecreasingobliqueness(inclinationfromthesurfacenormal).Nevertheless,it
remainselusivewhyplasticrelaxationofthewhiskergrainwouldnotcontinuouslydecrease
itscompressivestress.Furthermore,allcurrentmodels[
TuandLi
,
2005
,
Buchoveckyetal.
,
2009a
,
Saroboletal.
,
2013
,
Peietal.
,
2017
]effectivelysimplifythegeometryintoonedimen-
sion,partlyconsiderplasticrelaxationmechanismstovariousdegreesofsophistication,and
prescribeastressconditionof
the
whiskergrain,whichcanthenbettedtoreproduce
observedwhiskergrowthrates.Yet,noneofthosemodelscanberegardedastruly
predictive
,
sincemanyopenandcriticalaspectsofwhiskerformationarenotexplicitlyaddressedbythem.
Firstandforemost,theseventstriggeringparticulargrainstogrowawhiskerareignored
andcannotbededucedfromthesimulationresults,posingthequestionabouttherarityofsuch
aneventasobservedfromthewhiskerdensitymeasurements.Secondly,thelong-rangestress
gradient(extendingabout50
&
m,[
Peietal.
,
2017
])thatispredictedtodeveloparoundawhisker
grainisculttomeasureandhasyettobeconmedexperimentally,whilesomecontrary
observationshavebeenreportedaboutthedominantinnceofverticalstressgradients[
So-
biechetal.
,
2008
].Moreover,geometricallymodelsdisregardtheeffectofelasticand
plasticanisotropyofSn,andhence,cannotaccountforthemeasuredinoftheglobal
textureonwhiskerpropensity.Sincewhiskernucleationisultimatelya
localphenomenon
[
Buchovecky
,
2010
,
Sobiechetal.
,
2011
],specicsofthewhiskergrainneighborhoodinterms
ofgeometryandcrystalorientationsareverylikelystrongdecisivefactors.
Therefore,animportantgoaloftheproposedresearchistoenhancepastmodelingand
8
simulationeffortsbyemployingthree-dimensional,full-eld,andchemomechanicallycou-
pledcrystalplasticitysimulationsofstressedtinforwhichactualwhiskernucleationsites
areknown.Themajorquestiontobeaddressedwiththissimulationframeworkiswhetheran
oblique(surface)grainisnotonlyanecessarybutalsoa
sucient
conditiontotriggerwhisker
formation,i.e.,would
any
obliquegraininanotherwisecolumnarexperienceconditions
thatleadtoperpetualaccretionofatomstothisgrain?Inthisstudy,aresidualstressevolu-
tionstudyisperformedforthermallystressedtoanalyzethespatialvariationofsuchstress
.Theideaistoseewhethersucientatomscanbetransportedtowhiskergrainrootpro-
ducingshearthatishighenoughtobreakthenativeoxidelayerandpushoutthewhisker.From
theinitialthermo-mechanicalmodel,theroleofmicrostructureandmaterialanisotropyisalso
analyzedtoidentifythecriticalfactor(s)thatsigntlyinencetheatomredistribution.
Following,thethermo-mechanicalmodel,afullycoupledchemo-thermo-mechanicalmodel
isdevelopedtostudythekinematicconsequenceofsuchatomredistribution,intermsofthe
grainboundary(GB)hydrostaticstress,thenormaltraction,andthesheartraction.Moreover,
fromsuchspatiallyresolvedfull-simulationsofatomredistributionunderthermomechan-
icaldrivingforces,theroleoftheGBnetworkinmodulatingtheatomuxanditsnettransport
capacitycanberelatedtotheapparentlyshallowstressgradientthatisofteninferredfromex-
perimentalobservations.
Additionally,the
¯
-Snplasticityisyettobefullyestablishedinliteratureduetothecomplex
tetragonalcrystalstructure.Inthatregard,anecientmethodologyusinginverseanalysisis
developedtoidentifycriticalplasticityparameters(suchastheinitialowstress,alsoreferredas
criticalresolvedshearstress(CRSS))foreachoftheslipfamiliesinlow-symmetrymaterialsby
minimizingtheerrorbetweentheexperimentalandsimulatedsinglecrystalnanoindentation
data.Developmentofthisinverseanalysisframework,alongwithitsperformancefordifferent
crystalstructureswillalsobehighlightedinthiswork,usingvirtualexperimentaldata.
Overallthethesischapterscanbesummarizedas:
Ł
Chapter
2
containsaliteraturereviewaboutthecurrentunderstandingaboutthewhisker
9
growthmechanisms,existingmodelsandtheirdrawbacks,effectofprocessparameters,
andexistenceofcontradictingobservations,therebyfurthermotivatingtheneedoffull-
threedimensionalanalysisforsuchproblems.
Ł
Chapter
3
providesthebackgroundandformulationoffuldthermomechanicalsim-
ulationsinacrystalplasticityframeworkalongwithsignicantresultsabouttheeffects
ofglobaltexture,graingeometry,grainsizedistribution,andcrystalanisotropyon
grainboundaryhydrostaticstresswhichmodulatestheatomtransportinthe
Ł
Chapter
4
providesdetailsaboutthedevelopmentofthefullycoupledmulti-physics
chemo-thermo-mechanicalmodel.Thepreliminaryresultsobtainedfromthecoupled
simulationsarediscussedfortwodifferentboundaryconditionsintermsoftheirkinet-
ics,averageess,andtheshearcontributions.
Ł
Chapter
5
highlightstheInverseIndentationAnalysis(
IIA
)frameworkthatisdeveloped
toidentifycriticalplasticityconstitutiveparameters,alongwiththemeanstoperforma
highsensitiveparameterselection,anditsperformanceforhighsymmetricface-centered
cubicmaterials.
Ł
Chapter
6
discussesthestrategytousethe
IIA
frameworkforlowsymmetryhexagonal
materials,highlightstheimportanceofcrystalorientationselectionformaterialshaving
multipleslipfamilies,andthereliabilityofthe
IIA
methodforhexagonalmaterials.
Ł
Chapter
7
providesanoverviewaboutthekeyunderstandinggainedthroughthiswork
alongwithsuggestionsaboutthepossiblepathforwardindevelopingecientcrystallog-
raphybasedwhiskermitigationstrategies.
Apartfromthedevelopmentsinthecurrentoftinwhiskerresearch,withthedevel-
opmentofsuchmulti-physicsmodelstostudystress-drivendiffusionwouldopenmanynew
doorstoanalyzetransportmechanismsofcondensedmatterincomplex(non-hydrostatic)me-
chanicalstresselds,atopicoflong-lastinginterest[
LarchéandCahn
,
1973
,
Ostrovskyand
10
Bokstein
,
2001
,
Chakrabortyetal.
,
2008
,
Brenner
,
1956
].Moreover,understandinghowstresses
etheformationofmetallicwirestructureswouldhelpincientlydesigningfunc-
tionalnanostructures(suchasnanowires)andnanocomponentassemblies,especially,asthese
haveuniqueelectrical,mechanical,optical,andmagneticproperties.Hence,byunderstand-
ingmechanismscausingSnwhiskers,suchaknowledgecouldthenbecientlytransferredto
fabricateothermetallicnanowiresbysuitablycontrollingthestressescausingthem[
Cuietal.
,
2009
,
ErdemAlaca
,
2009
,
Wangetal.
,
2008
,
Karimetal.
,
2006
,
Sakaetal.
,
2007
,
Shimetal.
,
2009
,
LuandSaka
,
2009
].
11
CHAPTER2
LITERATUREREVIEW
2.1Background
Thischapterexploresrelevantliteratureusedforthecurrentwork,andhasbeendivided
intothreesections.Inthesection,theanisotropyof
¯
-Sncrystalstructureisdiscussed
alongwithacomparisonbetweenitscreepbehaviorasabulkmaterialtothatofausing
existingdatafromliterature.Subsequently,adetailedreviewisdoneontherelevantaspectsof
tinwhiskermechanismsalongwithdifferentfactorsmodulatingthekineticsoftheformation
ofsuchlamentousstructuresfromms.Finally,areviewabouttheitestraincontinuum
mechanicalframeworkthatisusedtoperformthecrystalplasticitymulti-physicssimulations
isdiscussedalongwithsomeofthecriticalnumericalchallenges.
2.2Materialpropertiesandanisotropyoftin
Tinmeltsat
T
m
Æ
505K(232
±
C),thus,ambienttemperaturecorrespondstoahomologous
temperature
T
/
T
m
Æ
0.6.Thelow-temperatureallotrope
®
ŒSn(graytin)isnon-metallicwitha
diamondcubiclatticestructure,andisthermodynamicallystableonlybelow13
±
C.
¯
-Sn(white
tin)hasabody-centeredtetragonal(bct)crystalstructurewith
c
/
a
Æ
0.545,andisgenerally
stabletowellbelowroomtemperaturesincethe
¯
to
®
transformationiskineticallyhindered.
(Inthistextfi
¯
-Snfl,fiSnfl,andfitinflareusedsynonymouslyfor
¯
-Sn.)
Thebctstructureof
¯
-Sncauseshighanisotropywithregardtotheelasticmodulus,thermal
expansioncient,andplasticbehavior,thusrenderingitsbehaviorhighlysensitivetocrys-
talorientation.Forexample,theYoung'smodulusandthermalexpansioncocientof
¯
-Snin
h
001
i
istwiceasthatin
h
100
i
direction[
Leeetal.
,
2015
].Plasticdeformationoftinunderambi-
entandelevatedtemperaturesisgenerallycomparabletoothermetallicmaterials,i.e.,follows
atypicalpower-lawrelationbetween(stationary)deformationrateandapplied(unidirectional)
12
Figure2.1:Deformationresistanceofbulktin(opensymbols,[
Mohamedetal.
,
1973
,
Mathew
etal.
,
2005
,
WeertmanandBreen
,
1956
,
McCabeandFine
,
2002
,
Adevaetal.
,
1995
])andthin
(dots,[
Boettingeretal.
,
2005
,
Chasonetal.
,
2014
])undervariousloadingconditionsand
homologoustemperatures(shadesofred).
13
stress(Fig.
2.1
).However,areviewdoneonplasticdeformationof
¯
-Snby
YangandLi
[
2007
]re-
vealstheincompleteunderstandingofitsslipactivity.Thisisduetothedifferencesinselecting
governingfactorsforplasticdeformation,suchasresistancetoforestdislocation[
Fujiwaraand
Hirokawa
,
1987
]or`effectiveyieldstrength'[
SidhuandChawla
,
2008
].Theorientationdepen-
denceofactiveslipsystemswasalsoinvestigatedbyanalyzingidealsheardeformationof
¯
-Sn
singlecrystalusinginciplesdensityfunctionaltheory(DFT)[
Kinoshitaetal.
,
2012
]fora
perfectcrystal,whichsuggestedthatthemostactiveslipsystemsareon{110}planes.Onthe
otherhand,indentationstudiesby
Fujiwara
[
1997
]on
¯
-Snrevealedtheslipactivityof{100}
and{110}planestobethemostlikely.Thoughtherehasbeensomeconsensusontheavailable
slipsystems(seeTable
2.1
),theirrelativeactivityhasyettobequantiedbyanyconstitutive
model[
Bieleretal.
,
2011
].
Table2.1:Slipfamiliesin
¯
-Sncrystalstructure[
Leeetal.
,
2015
].
ModeSlipfamilyMultiplicity
1{100)
h
001]2
2{110)
h
001]2
3{100)
h
010]2
4{110)
h
1
¯
11]4
5{110)
h
1
¯
10]2
6{100)
h
011]4
7{001)
h
010]2
8{001)
h
110]2
9{011)
h
0
¯
11]4
10{011)
h
100]4
11{011)
h
1
¯
11]8
12{211)
h
0
¯
11]8
13{211)
h
¯
111]8
Theself-diffusivityoftinexhibitsalargedisparitybetweenbulkandgrainboundary,with
thelatterdiffusionientbeingordersofmagnitudehigher[
MeakinandKlokholm
,
1960
,
CostonandNachtrieb
,
1964
,
HuangandHuntington
,
1974
,
LangeandBergner
,
1962
].Con-
sequently,masstransportfortypicalin-servicetemperaturesisexclusivelysupportedbythe
grainboundarynetwork.Moreover,tinformsatenaciousoxidethateffectivelysuppressessur-
14
facediffusionevenunderrelativelysmallpartialpressuresofoxygen,andwhosepresencehave
beenreportedtouencewhiskergrowthintermsofmaintainingcompressivestressesinthe
by
Tuetal.
[
2007
].
2.3Currentunderstandingoftinwhiskerformation
Inthissectionrelevantbackgroundregardingwhiskermorphologyandproposedgrowth
mechanismsisdiscussed.Someoftheprominentmodelstodescribetheprocessofwhisker
formationishighlightedalongwiththeirunderlyinglimitations,therebyrequiringfurthereffort
inadvancedcomputationaleffortstopredictwhiskernucleationandmotivatingthepresent
work.Despitethelargebodyofresearchbeendedicatedtodeterminingthemechanismsof
whiskerformation[
Galyon
,
2005
]foroversevendecades,thewholeprocessisstillnotwell
understoodandanacceptedwhiskermitigationtechniqueinplaceofPbadditionisyettobe
determined.Althoughitisbynowgenerallyacceptedthatstressisthedrivingforceforwhisker
growth[
Galyon
,
2005
,
Chasonetal.
,
2008
,
Jadhavetal.
,
2010b
,
Tuetal.
,
2007
,
Sobiechetal.
,
2008
,
LeeandLee
,
1998
,
Smetana
,
2007
,
ViancoandRejent
,
2009
],thisknowledgealonedoes
notexplainthesubsequentstepsleadingtotheiractualformation.
2.3.1Whiskergeometryandcrystallography
Figure
2.2
highlightsexemplarywhiskermorphologiesoutofthevastgeometriesandmi-
crostructuresreportedintheexpansivebodyofwhiskerinvestigationsoversevendecades(see
[
Galyon
,
2005
]foranoverview).Whiskerdiameteriscomparabletothegrainsizeofthem
fromwhichitgrowsandremainsconsistentduringgrowthwithsometimessharpbends[
Bu-
chovecky
,
2010
].Time-seriesevolutionofgrowingwhiskersconmedthattheatomattach-
menthappensatthebaseofthewhiskerandnotatthetoporside,alongwiththefactthat
theyoriginatefromasinglegrain[
Jadhavetal.
,
2010a
,
Susanetal.
,
2013
].Focusedionbeam
(FIB)sectionsthroughthebaseofwhiskersrevealthatthesewhiskerstypicallyoriginatefrom
anon-columnarsub-surfacegrainthatdoesnotextendthroughthemthickness[
Boettinger
15
Figure2.2:Examplesoftinwhiskers,hillocks,andsmallerprotrusions.TopleftimagebyPe-
terBush(SUNYBuffalo)[
BoguslavskyandBush
,
2003
],bottomleftimagefromNASAmetal
whiskerphotogallery,centerimagesbyPiyushJagtapandPraveenKumar(IIScBangalore,un-
published),crosssectionscutbyfocussedionbeamandcorrespondingsketchesonrightby
[
Chasonetal.
,
2013
].
Figure2.3:fiObliquegrainflwithshallowgrainboundariesatthebaseofatypicalwhisker
[
Saroboletal.
,
2013
].
etal.
,
2005
,
Smetana
,
2007
,
Jagtapetal.
,
2017
],andisfrequentlytermedanfiobliquegrainflor
fichevrongrainfl(Fig.
2.3
)inliterature(andhenceforthinthistext).
Whiskerscancontainmultiplecrystalorientations
FortierandKovacevic
[
2012
](asin
Fig.
2.2
thirdcolumn),butwhiskersmostlyconsistofsinglecrystals[
Galyon
,
2005
,
Susan
etal.
,
2013
,
Tuetal.
,
2007
,
Osenbachetal.
,
2005
,
LeBretandNorton
,
2003
]thatareoftentimes
straightwithoccasionalabruptchanges(kinks)ormoregradualchangesingrowthdirection
(seeFig.
2.2
andsecondcolumn).Extensivetimelapseobservationsweremadetocapture
theeffectofthesekinksonthekineticsofwhiskergrowthby
Susanetal.
[
2013
].Inthesame
16
study,twotypesofkinkswereobservedduringthegrowthofawhisker.Inonecase,referred
inthetextastypeA,sharpkinksappearedatthebaseofthewhiskerwiththemorphologyand
orientationremainingunchangedfortheremainderofthegrowthperiod;whileincaseoftype
Bkinks,orientationandmorphologicalchangeswereobservedduringthewhiskergrowthpro-
cess.Moreover,thetypeAkinksdidnotuencethegrowthrateofthewhisker,whilewithtype
Bkinks,asigntdecreaseorcompletestoppageofwhiskergrowthwasobserved.Sucha
caseofintermittentgrowthhavealsobeenreportedinotherstudies[
LeBretandNorton
,
2003
,
JiangandXian
,
2006
].Thisobservedmorphologicalandkineticschange(increaseanddecrease
inwhiskerdiameterasshowninFig.
2.2
secondcolumn)duringtheprocessofwhiskergrowth
wasattributedtothedynamicanderraticbehaviorofthegrainboundariesoftheunderlying
whiskergrain.Apartfromtheselongandmostlystraightwhiskers,asecondprotrudedmor-
phologyisobservedthathasamuchloweraspectratios(Fig.
2.2
lastcolumn)andistermed
fihillock".Hillocksarecharacterizedbytheirmuchwiderbasethanthetypicalgrainsizeofthe
andinvolvesigntgrainboundarymigration[
Chaudhari
,
1974
,
Boettingeretal.
,
2005
,
Osenbachetal.
,
2007
].Oftentimesawhiskerhavebeenobservedwithsuchawiderbaseatthe
bottom[
LinandLin
,
2008
],indicatingthatthegrowthmode(whiskerorhillock)canvaryover
time.
WithcarefulelectronbackscatterdiffractionmeasurementsofTypeAkinkedwhiskersitwas
observedthatthesewhiskersaresinglecrystals,inthatthecrystallatticeorientationremains
unchangedacrossthekinkswiththekinkanglerepresentingtheanglebetweenthetwogrowth
directions[
Susanetal.
,
2013
].
Ithasbeenobservedexperimentallyby
Steinetal.
[
2015b
]thattheactivegrowthdirectionof
whiskerscorrespondstolowindexdirections,
h
001
i
,
h
101
i
,and
h
111
i
withoccasionalpresence
oflowindex
h
201
i
and
h
102
i
directions.Suchafavorabilityofgrowthdirectionstobelow-index
directionswerealsoobservedbyothers[
Buchovecky
,
2010
].However,whetherthereexistsany
correlationbetweentheunderlyingwhiskergraintothegrowthdirection(mechanism)isstill
underactivedebate[
Susanetal.
,
2013
,
Chasonetal.
,
2013
].Suchalackofanyapparentunique-
17
nessforaplausiblewhiskergrainwasalsoobservedthroughrecentfull-multiphysicssim-
ulations[
ChakrabortyandEisenlohr
,
2018
]aswellasdiffractionstudies[
Jagtapetal.
,
2017
,
Pei
etal.
,
2016
].Reportedwhiskerdensitiesrangefrom1000cm
¡
2
to10000cm
¡
2
afteronetotwo
monthsofaging[
Tu
,
1973
,
LeeandLee
,
1998
]witharandomdistributionacrossthemsur-
face,withtwowhiskersoccasionallyeruptingmuchcloserthanthemeanspacing,thusempha-
sizingtheimportanceoflocalneighborhoodofthewhisker.Itisalsoimportanttonoticethat
whiskerdensitydependsontheplating(processing)conditionstodeposittheselms,however
thereexistscoupleofordersofmagnitudedifferencebetweenthenumberofwhiskernucle-
ationspots(whereagrain"pops"outbutdoesnotgrowlong)tothatofthenumberoflong
whiskers[
Susanetal.
,
2013
].
2.3.2Tinmplatingandresultingmicrostructure
Themostcommonroutetosynthesizelmsisviaelectrochemicalplating,wherethesubstrate
actsastheanodewithaSnbasedelectrolyte,andpureSnactingasanode.Basedontheelec-
trodepositionconditionstheresultingpropertiesvarygreatly,whichinturnaffectsthe
whiskerpropensity.Thedepositionconditionsgenerallyincludedepositiontime,bathtem-
perature,electrolytecomposition,andcurrentdensitywhichsubsequentlythenaffectsm
globalpropertiessuchasthickness,grainstructure,andcoatingcompositionaswellaslocal
propertiessuchascrystallineorientationdistributionlmtexture).Variousstudieshavebeen
performedtocorrelatesucheffects,likeforexampleaccordingto[
JagtapandKumar
,
2015
],
changingtheelectrodepositionbathtemperaturefrom30
±
Cto60
±
C(atcurrentdensity
of30mAcm
¡
2
)alteredthelmgrainstructurefromcolumnartomostlyequiaxedandthusre-
sultedinareductionofwhiskerpropensityby4ordersofmagnitude.Theeffectofelectrolyte
compositionontheresultingtexture(crystallographicdirectionalongthenormal)
alongwiththesubsequentwhiskerpropensityhavealsobeenstudied[
LalandMoyer
,
2005
,
Saroboletal.
,
2010
,
Sobiechetal.
,
2011
].
Sobiechetal.
[
2011
]concludedthatcolumnarSnde-
positedoncopper(Cu)substrateexhibited{321}bertexture,whileelectrolytecomposition
18
ofSn90Pb10(wt.%)onCuexhibited{110}mtexture.
LalandMoyer
[
2005
]cmedthat
with
h
220
i
werelesswhiskerpronecomparedto
h
321
i
texture,whichdepended
ontheamountoforganicadditivesintheelectrolyte.Basedontheeffectofminoralloyingof
CuandPb,
Saroboletal.
[
2010
]generatedadefect-phasediagramtoidentifythosecombina-
tionsofcompositionsthatshowedtheleastwhiskerpropensity.Filmtexture(crystallographic
directionalongthelmnormal)alsodependedonthesubstratecompositionasstudiedby
[
Jagtapetal.
,
2018
]whoreportedadominanttextureof{100}whenitwasdepositedon
Cusubstrateorbrass(Cu-35wt%),ascomparedtoa{110}texturewhenthesubstratehad
anickel(Ni)underlayer.Awidearrayofliteratureexiststhatcomparedifferentlmproper-
tiesonwhiskergrowth(andpropensity)suchasmthickness[
Pinoletal.
,
2011
,
Jadhavetal.
,
2010b
,
Chasonetal.
,
2011
,
Chengetal.
,
2011
,
Jadhavetal.
,
2012a
,
LuandHsieh
,
2007
],grain
size[
Jadhavetal.
,
2010b
,
Chasonetal.
,
2011
,
Jadhavetal.
,
2011
,
Peietal.
,
2012
],platingcondi-
tions[
Pinoletal.
,
2011
,
Yenetal.
,
2011
,
Sobiechetal.
,
2010
,
Vicenzoetal.
,
2010
],microstructure
[
Jadhavetal.
,
2011
,
Boettingeretal.
,
2005
,
Yuetal.
,
2010
,
Mathewetal.
,
2009
,
Okamotoetal.
,
2009
]andcomposition[
Jadhavetal.
,
2011
,
Boettingeretal.
,
2005
,
Kaoetal.
,
2006
,
Moonetal.
,
2010
].However,adirectcomparisontoisolatethegoverningfactorsisultfromsuchstud-
iesduetothehugevariabilityintheprocessingconditions.Forexample,
Jagtapetal.
[
2018
]
reportedahigherwhiskerpropensityforlmsdepositedonbrass(Cu-35wt%)substratethat
had{100}textureascomparedtothosedepositedonpureCuusingstannoussulfateasthe
electrolyte,whiledepositedby[
Steinetal.
,
2015b
]onanominallysimilarsubstrateand
having{001}textureshowedlesserwhiskerpropensityascomparedtothoseonpureCu
usingmethanesulfonicacidwithorganicadditivesastheelectrolyte.Thiscomparisonbrings
forwardtheexistingcontradictionintinwhiskerliterature,alongwiththesignicantuence
oflmtexture,aswellastheneedforasystematicstudyforaccuratelyidentifyingthegoverning
mechanismsforwhiskergrowth.
19
2.3.3Causesofwhiskers
Toaccuratelyidentifytheunderlyingmechanismsleadingtowhiskerformationishighlycom-
plicatedasitinvolvesamultitudeofconcurrentprocesses,suchasinterdiffusion,phasetrans-
formation,stressgenerationandrelaxation.Itisageneralconsensusinthetinwhiskerresearch
communitythatstressisrequiredtogeneratesuchprotrusions.Inmostobservationswhiskers
areobservedundercompressivestressesinthem,howevertherehavebeenreportsofwhisker
undertensilestressesaswell[
Crandalletal.
,
2011
]whilesomeobservationshaveemphasized
moreontheimportanceofthestressgradientratherthanthesignofthestressitself[
Sobiech
etal.
,
2008
].Moreover,eventhoughwhiskersarearesultofstressesinthetheyarenot
theonlystressrelaxationmechanism,andothermechanismssuchasplasticity(creep)dictate
theamountofstressineachlayer[
ChasonandJadhav
,
2016
].Thiswasdemonstratedby
[
Fisheretal.
,
1954
]inthe1950s,whoshowedthatwhiskergrowthratesscalewithexternally
applied(compressive)stress.Apartfromexternalloadingimpositionssuchasindentationor
bending[
YangandLi
,
2008
,
Crandalletal.
,
2011
],compressivestresscanalsoarisefrom,for
instance,residualstressduringthedepositionprocess,thermalexpansionmismatchofm
andsubstrate,electromigration,ordisplacivephasetransformationssuchasthoseresulting
fromcorrosionortheformationofvoluminousintermetallics(Cu
6
Sn
5
)inconnectionwithCu
substrates[
LeeandLee
,
1998
].Sndepositedonsilicon(Si)substratesdevelopcompres-
sivebiaxialstressesuponheatingduetothethermalexpansionmismatchbetweenthelmand
thesubstrate[
Peietal.
,
2016
].Itwasalsoobservedthatincreasedcompressivestressesdueto
electromigrationcansigntlyincreasethegrowthrateofSnwhiskersfrombyupto
8Ås
¡
1
[
Liuetal.
,
2004
].Thesuggestedmechanismisthatthedirectionelectronwinduces
Snmigrationfromcathodetotheanodesidetherebyleadingtothebuild-upofcompressive
stressesandthusformationofSnwhiskersontheanodeside.Amongtheaforementioned
causesofstressgeneration,themostcommonlyobservedisduetotheformationofCu
6
Sn
5
resultingfrominterstitialdiffusionofCufromthesubstrateintotheandmostlyattheSn
grainboundaries.
Sobiechetal.
[
2011
]observedadirectcorrelationbetweentheamountof
20
Cu
6
Sn
5
formationtothewhiskernucleationpropensity,andproposedthataregular(homoge-
nous)Cu
6
Sn
5
morphologyreduceswhiskernucleationpropensity,while
Chasonetal.
[
2013
]
foundnosuchcorrelation.Asmentionedearlier,thesestressesprimarilyrelaxedbyplasticity
andthediffusivetransportofSnatomsfromaregionofhighcompression(lowtension)toa
regionlowcompression(hightension)viathegrainboundaries.
TheSnfeaturesthatareobservedtohaveaprominenteffectonwhiskerpropensity
areitsthicknessandthegraincharacteristics(size,structure,andorientation)[
WalshandLow
,
2016
,
Ashworthetal.
,
2015
].Increasingthethicknessincreasesthewhiskerincubationpe-
riodtherebyreducingthewhiskerpropensity,sinceathickerlayerismoreeffectiveinrelaxing
theimposedstressesascomparedtoathiner[
IllésandHorváth
,
2013
].
Chasonetal.
[
2013
]
establishedthatthemaximumcompressivestressinthe(causedbycontinuousprecipita-
tionofCu
6
Sn
5
atthesubstrateinterface)decreaseswithincreasingthicknessbyetching
thesameelectrodepositedtinonaCusubstrate,thusindicatingthatthickerareplas-
ticallysofterandcanrelaxmoreiently.Withthelowerlevelofnetcompressivestressinthe
thewhiskerdensityandrateofwhiskerformationalsodecreased.Othermeasurements
ofstressrelaxationkineticsontwosimilardepositedonSisubstrateandhavingdifferent
thicknessalsormedthatthickerrelaxstressesmoreeasilyascomparedtothinerlms
[
ShinandChason
,
2009b
].Inthisstudy,itwasalsofoundthattheplasticdeformationkinetics
oftinmswereconsistent(intermsofactivationenergyandstressexponent)with
¯
-Snbulk
behaviorandthatthepresenceofnativesurfaceoxidelayerontopimpedesplasticrelaxation.
Eventhough[
LeeandLee
,
1998
]foundnosystematiceffectofgrainstructureonwhisker-
ring,recentstudieshaveclearlydemonstratedthatwithequiaxedgrainmorphologyis
moreientinstressrelaxationascomparedtothosewithcolumnarstructurewhichshow
higherwhiskerpropensity[
Sauteretal.
,
2010
,
Jadhavetal.
,
2012b
,
Sobiechetal.
,
2011
].
Theeffectofgrainorientationmtexture)havealsobeenattributedasoneofthedecisive
factorsforwhiskerorhillockpropensitywithsometexturesbeingmorewhiskerproponentthan
others[
Jagtapetal.
,
2017
,
2018
,
Saroboletal.
,
2010
].However,thereisnoconclusivestudythat
21
clearlydemonstratestheeffectoflmtexture.
Theroleofthenativetinoxidelayeronwhiskerpropensityisalsonotfullyunderstood.The
presenceofnativesurfaceSn-oxidelayerisconceivedbymanyauthorsasanimportantpa-
rameterforwhiskerformation[
Tuetal.
,
2007
,
Kumaretal.
,
2008
,
LeeandLee
,
1998
,
TuandLi
,
2005
,
Chasonetal.
,
2008
,
Buchoveckyetal.
,
2009a
,
b
,
Boettingeretal.
,
2005
].Experimental[
Ku-
maretal.
,
2008
,
Chasonetal.
,
2008
]andtheoretical[
Buchoveckyetal.
,
2009a
,
b
]investigations
indicatethatwithoutthepresenceofa(passivating)oxidelayerthecompressivestressgener-
atedbyCu
6
Sn
5
formationcouldrelaxuniformlyviaCoblecreep[
Coble
,
1963
]alongtheentire
layersurface,i.e.Snatomscoulddiffusefromthecolumnargrainboundariestothefreesurface
ofSn[
ShinandChason
,
2009a
].However,
Moonetal.
[
2005
]concludedthatthesurfaceoxide
canonlyhaveaminimaleffectonwhiskergrowthbycomparing(underultrahighvacuum)the
Ar
+
sputter-cleanedpartofanelectrodepositedSn-1.5wt%Culmtothepartstillcontaining
thenativeoxidelm.Thisnotionwasalsosupportedbyarecentinvestigationthatobserved
whiskerformationonacompressedAui.e.foramaterialwithoutanysurfaceoxide[
Cran-
dalletal.
,
2012
].Nevertheless,systematicremovalofthesurfaceoxidefromafractionofthe
surfaceinpatchescomprisingmany
¯
-Sngrainsresultedinwhiskerformationexclusively
fromthoseclearedareas[
Suetal.
,
2011
].However,verylocaloxideremovalontopofonlya
fewgrainsdidnottriggerwhiskerformationfromanyoneofthoseuncoveredgrains,whilea
whiskeroriginatedfromaclose-bygrainthatwasstillcoveredbysurfaceoxide[
Jadhavetal.
,
2010a
].
2.3.4Whiskermitigationstrategies
WiththemovetowardsPbfreeelectronicsafteritwasaccidentallydiscoveredtoinhibit
whiskers[
Arnold
,
1966
],therehavebeensomewhiskermitigationstrategies,butunfortunately
noneofthemareaseffectiveasPballoyingwasknowntobe.Thesestrategiestominimize,pre-
vent,orinhibitwhiskergrowthinclude(i)alloyingtinwitheitherbismuth(Bi)[
Sandnesetal.
,
2007
,
Baatedetal.
,
2011a
,
Jadhavetal.
,
2012b
]orAg[
Baatedetal.
,
2011a
]whichproducesa
22
moreequiaxedgrainstructurethatismoreientinplasticallyrelaxingthegenerated
stressesthancolumnarstructureswould,andtherebyreducingthedrivingforceforwhisker
formation;(ii)annealing,orfipost-baketreatment",toreducetheresidualstress[
Dittesetal.
,
2003
]andpromoteformingamoremorphologicallyhomogeneouslayerofintermetalliccom-
pounds[
Sobiechetal.
,
2008
]ascomparedtotheirgenerallyobservedirregularity;and(iii)us-
ingofanickel(Ni)underlayertopreventtheformationofSn-Cuintermetallic[
Dittesetal.
,
2003
,
Baatedetal.
,
2011b
].Thealloyingstrategieshavesomeassociatedchallenges,suchas
reductionoffatiguelifeuponaddingBi,whileanundesireddendriticgrowthisoftenassoci-
atedwithAgaddition.Theapplicationofconformal(polymericormetallic)coatingsisalso
effectiveinreducingthewhiskernucleationdensityandtheirgrowth,butnotnecessarilyelim-
inatedevicefailureduetothem,assometimeswhiskershavebeenreportedtopiercethrough
thecoatings[
HuntandWickham
,
2010
,
Mahanetal.
,
2014
,
Doudricketal.
,
2015
].Thus,these
solutionsappeartobemoretemporalandthereforetohaveaconclusivemitigationstrategy,
amorecomprehensiveunderstandingaboutwhiskernucleationandgrowthisrequired.Itis
ourbelievethatperformingfullhightysimulationswouldhelpindesigningcrys-
tallographicbasedmitigationstrategieswhichwouldbemuchmoreeffectivethantheabove
mentionedphenomenologicalsolutions.
2.3.5Growthmodesofwhiskers
Criticallyexaminationofthegrowthmorphologiesofwhiskers/hillockcanprovidesignt
insightsintothemechanismcontrollingtheirgrowth,sincetheirshapeandorientationis
closelyrelatedtothewayinwhichatomsareaccreditedintothem.Togetaholdofthecomplete
picture,onemustconsiderhowthestressinthegetgenerated,followedbythemech-
anismbywhichthisstressgovernsthematerialredistributioninthebytransportingSn
atomstothewhiskeringgrain,andnallyhowthistransportedmaterialgetsincorporatedinto
thewhisker.Inthissectionthesomeoftheexperimentalobservationsofwhiskersfromthe
immensestudiesperformedbyDr.EricChason'sresearchgroupatBrownUniversityiscon-
23
Figure2.4:Nucleationandgrowthofawhisker14h(a),14.3h(b),14.6h(c),17h(d),21h(e)
afterdeposition(ontoCusubstrate)[
Jadhavetal.
,
2010a
].
Figure2.5:Growthofahillockwithextensiverotationimaged6h(a),8h(b),12h(c),18h(d),
50h(e)afterdeposition[
Jadhavetal.
,
2010a
].
sideredinanattempttohighlighttherelationbetweenwhiskergrowthmechanismandtheir
subsequentmorphology.
Thetimeseriesevolutionofwhiskernucleationandgrowthasobservedby
Jadhavetal.
[
2010a
]isshowninFig.
2.4
.Byexaminingthesequencefrom(a)to(c),itisapparentthatthere
existsarapidchangeinthesurfacemorphologyaroundthelocationfromwherethewhisker
willgrowbylifting(breakingthrough)theoxidelayeronthesurface.Thedetailsaboutthe
breakageoftheoxidelayeris,however,notclearfromtheimages.Withtheemergingwhiskers
thereexistsnoothersurfacecontaminationorotherdefectseitheronthewhiskergrainorany
othergrainsurroundingit.Thusthemicrostructureandgrainmorphologyofthewhiskergrain
anditsneighborhoodremainsprettymuchunchangedbothduringwhiskernucleationaswell
asduringitsgrowth.Moreover,oncetheoxide-layerwascracked(whiskernucleationstage),
thewhiskerswereobservedtogrowinanearlyconstantdirectionandhadashortphaseoffast
initialgrowththatquicklydecaystoanearlyuniformrate.Whileinotherexperimentswhiskers
wereobservedtogrowintermittentlywithpausesduringtheirgrowthorhadsharpbendsor
kinks[
Galyon
,
2005
,
Ellisetal.
,
1958
,
Tuetal.
,
2007
,
Sobiechetal.
,
2008
].Suchvariabilityin
24
Figure2.6:Sequenceofcomplexhillockgrowthwithvaryingcross-sectionandpartialbending
6h(a),12h(b),18h(c),32h(d),44h(e),56h(f),76h(g),138h(h)afterdeposition.Insets
showschematicofshapeevolutionhighlightingrotationoftheoriginalsurface.Arrowspoint
tograinboundaryfeaturesin(d)thatarevisibleasridgesonsideofthehillockin(h).From
[
Jadhavetal.
,
2010a
].
whiskergrowthmechanismsandtheirsubsequentmorphologieswereattributedtothediffer-
entenvironmentalconditionsby
Jadhavetal.
[
2010a
].Forexample,thedifferenceduetothe
samplesbeingkeptinairormeasuredinanscanningelectronmicroscope(SEM)instrument
withapoorer(
È
4
£
10
¡
4
Pa)qualitybasevacuum,andthussuggestingthatthepresenceof
oxygen,watervapor,orothergasesmayplayaroleinthenon-uniformgrowthofwhiskers.
Figure
2.5
showsa(short)whiskerwhereonesidelookstoremainattachedtothesurface.It
appearsthatthegrowthofthiswhiskerprogressesina180°curvedfashionuntilthewhiskertip
hitsthesurfaceofaneighboringgrain.Therotationseemstohappenduetoonesidegrowing
outwardatamuchfasterratecomparedtotheoppositeside.Sincethereappearstobenomor-
phologicalchangesofmaterialabovethesurface,materialextrusionoccursclearlybyaddition
ofSnatomsatthebaseofthewhisker.
Theprimarydifferencebetweenthegrowthmechanismsleadingtowhiskerorhillockfor-
mationisthefactthatwithhillocksthereisbelievedtobelateralgraingrowthattributedby
25
thehighdegreeofgrainboundarymobility,whileincaseofwhiskersthereexistslimitedlateral
grainboundarymovementwhichcompelsthewhiskertosimplygrowupward.Thehillock
growthsequence,asdepictedinFig.
2.6
,alsoappearstostartatasinglegrainsimilartoa
whiskerinitiation.Thetopsurfaceofthishillockhoweverrotates(similartoFig.
2.5
)asitgrows
untilitisorientedapproximatelyperpendiculartothesurface.Duringthisgrowthofthe
hillockwhereitpushesupward,itsbasestartstowidensimultaneously(Fig.
2.6
dŒh),indicative
oftheextensivelateralgraingrowthbythehillockgrainassuggestedearlier.Asthehillockcon-
sumesadjacentgrains,thehorizontalgraingrowthisroughlyconstrainedbythegrainbound-
ariesonthesurfacewhichreducesthekineticsoftheprocesswithfurthergrowthofthehillock
grain.Thisinterfacebetweenthegrowinghillockgrainandtheneighboringunchanged(sta-
tionary)grainsdeterminethehorizontalhillockboundary.Thesequenceofthisgrowthoften
proceedsinastep-wisemanner,withanincrementinhorizontalgrain-growthfollowedbyan
incrementintheverticalgrowth,therebyleadingtostriationmarks(horizontalsteps)onthe
sidesurfacesofthehillock.Thesestriationscanbecomprehendedtocorrespondtothesizeof
thebaseofthehillockwhenitwaspushedoutofthesurfacewhichcanbeutilizedtorecreate
thehistoryofthehillockmorphology.
2.3.6Modelsofwhiskerformation
Jadhavetal.
[
2010a
]proposedaconceptuallong-rangetransportmodelforwhiskergrowth,as
illustratedinFig.
2.7
,basedontheobservedgrowthmorphologiesandthesmallchangesin
grainvolumesintheimmediatevicinityofthewhiskersandhillocks.Suchanotionoflong-
rangetransporti.e.,accretionofatomstothewhiskergrainfromalargeareahasalsobeen
experimentallyobservedby
Reinboldetal.
[
2009
].Foralmthatisundercompressivestress,
thechemicalpotentialdifferencebetweentheinclinedandtheverticalgrainboundarieswould
supportthemasstransportofatomstowardstheseoblique(inclined)surfacegrains.Suchan
effectofgraingeometryinmaintainingthechemicalpotentialwasalsoproposedby
Smetana
[
2007
]whoalsosuggestedafollowinggrainboundaryslidingmotiontoejectthewhiskerup-
26
Figure2.7:Conceptualunderstandingofwhiskerandhillockformationmechanisms[
Jadhav
etal.
,
2010a
].
ward.Subsequently,verticalgrowth(topleft)resultswhenthemassx(asindicatedbythick
arrowsinFig.
2.7
)intothewhiskergrainoccursevenlyinallthegrainboundariesofthewhisker
grainandinaspeciwaywhichresultsinaconsistentandparticularcrystaldirection(asin-
dicatedbythinarrows).Thereasonforsuchacoordinatedaccretionofatomsisyettobeun-
coveredinliteratureasitappearstoencompassahighdegreeofcomplexity.Verticalwhisker
growthfromcolumnargrainswouldthenrequireastrongmechanicalbiastomaintainastress
gradientandhenceasubsequentchemicaltothewhiskergrainboundaries.Suchame-
chanicalbiasforthewhiskergrainisoftenattributedtoplasticity,i.e.,thewhiskergrainun-
dergoesahighdegreeofplasticdeformationwhichalsoreducesitsstrainenergydensity[
Sun
etal.
,
2011
].Thedislocationslipcausingthisplasticitywouldalsohelpinredistributingvolume
toabovethesurface,however,therationalizationofthestrictcrystallographicgrowthasaresult
ofconservativedislocationglideisdiculttocomprehend.Curvedwhiskermorphologies(top
rightinFig.
2.7
)hasbeenrationalizedbyasizableimbalanceofmassinbetweensomegrain
boundaryfacesofthewhiskergrainascomparedtotherest,andthesubsequentlackoffast
enoughredistributionoftheaccreditedmassacrossallthegrainboundarysurfaces.Hence,the
surface(s)receivingthemostuxgrowsthemost,therebygeneratingthecurvedmorphology.
However,suchacurvedwhiskergeometryhasbeenrationalizedbasedonadifferentfrictional
27
resistanceargumentby
Saroboletal.
[
2013
].Asmentionedearlierandshownagaininbottom
rowinFig.
2.7
,thegrowthofhillocksisconceptuallysimilartothatofthewhiskerwiththe
distinctivefeaturethatthegrainboundaries(ofthehillockgrain)are(intermittently)moving
concurrentlyduringtheverticalgrowthofthe(hillock)grain.
Someresearchershave,however,laidmoreemphasisonthestressgradientandnotonthe
natureofthestressitselfŒmeaningtheyhaveproposedwhiskergrowtheveninatensilestress
state,buthavingasuitablestressgradientsthatallowforcoordinatedtransportofatomstothe
fiwhiskergrain".Inthisregard,
Sobiechetal.
[
2011
]emphasizedontheimportanceofboth
in-planeandthroughthicknessstressgradientaswell.Accordingtohim,formationofCu
6
Sn
5
alongtheSngrainboundariesattheCu/Sninterfacesinducesin-planecompressivestressin
theSnlayer,particularlyinthedepthrangeoftheSncoatingwhereCu
6
Sn
5
formationpro-
ceeds.Following,closetothesurfaceoftheSnlayerresultsinin-planetensilestresscondition
thatismostpronouncedatthosesurfacelocationswhereformationofCu
6
Sn
5
ismostdistinc-
tive,asobservedunderneaththewhisker.Asaconsequence,bothnegativeout-of-planeresid-
ualstress-depthgradients,inthedirectionofincreasingdepth,andnegativein-planeresidual
stressgradientsinthedirectionofwhiskernucleationsitetowardswhiskersurroundingsoccur.
ThisthenprovidesthedrivingforcesforthetransportofSnatomstothewhiskernucleation
site.
Inthefollowing,someoftheprominenttheoriesregardingmechanismsofwhiskergrowth
andtheirshortcomingsarebrihighlighted.
Thersttheoriesonwhiskergrowthwasbasedonatomicmovementthroughdislocations,
wherein
Peach
[
1952
]proposedthatwhiskersgrowasameanstoreducethedislocationenergy
broughtaboutbyscrewdislocationspresentinthecenterofthewhiskergrain.
Eshelby
[
1953
],
Frank
[
1953
]alsotriedtoexplaintheextrudedwhiskergrowthbytransferofatomsfromthe
baseofthewhiskerbroughtaboutbyadislocationfromeitheraFrank-Readsourceatthebot-
tomorduetothestress-gradientcausedbysurfaceoxidationrespectively.Otherdislocation
basedtheoriesofwhiskergrowthsuchasatwo-stagediffusion-dislocationbasedmodelpro-
28
posedby
Lindborg
[
1976
],whereadislocationloopexpandsbyclimbtilltheforceisbalanced
followingadislocationglidetoprotrudeaswhiskers.
LeeandLee
[
1998
]alsoproposedgrowth
ofwhiskersfromaprismaticloopundertheactionofaPeach-Koehlerforceduetothestress
thatispresentintheduetotheformationofintermetalliccompounds(IMCs)atthe
substrate-interface.However,suchdislocationbasedtheorieshaveseldombeenaccepted
inthewhiskerresearchcommunitybasedonthefactsthatwhiskerswereobservedtogrowfrom
thebaseandnotfromthetip
KoonceandArnold
[
1953
];thelackofdislocationsbeingobserved
inthewhiskergrain;andthefactthatforsuchtheoriestoholdtruethewhiskergrowthaxis
shouldalwaysbeparalleltoBurgersvector(slipdirectionorficlose-packeddirection")whichis
notobservedinthetransmissionelectronmicroscopy(TEM)studies
LeBretandNorton
[
2003
].
Sincetinisathighhomologoustemperatureunderambientconditions,itreadilydeforms
plasticallywhenloadedandrecrystallizesaround30
±
C.Thesefactsmotivatedwhiskerforma-
tiontheoriesbasedonrecrystallization.Earlyproponentsofthefirecrystallizationtheoryfor
whiskergrowth"include[
Ellisetal.
,
1958
,
GlazunovaandKudryavtsev
,
1963
,
FurutaandHama-
mura
,
1969
]whoproposedthatwhiskersgrewasameanstolowerthestrainenergyinthe
Ellisetal.
[
1958
]evenfamouslystatedthatfiwhiskergrowthisbutaspecialcaseofrecrystal-
lizationandgrowthinvolvingmasstransportfl.Since
¯
-Snhasalowmeltingpointthatletsit
exhibithightemperatureplasticityevenatroomtemperature,thegrainsundergoinglargede-
formation(andhencewithhigherstrainenergy)servedtobethenucleationsiteswhichthen
ledtowhiskergrowthbyrecrystallization.
Kakeshitaetal.
[
1982
]alsosupportedthenotion
thatrecrystallizationisanecessarystepforwhiskerformationbasedonhisexperimentalstud-
iesbetweentwotypesofms:onehavinglargewell-polygonizedgrainmorphologywhilethe
otherhavingsmall,non-polygonizedgrainstructure.Heattributedthescenariotobea
situationinwhichrecrystallizationhadalreadyhappenedtherebyrequiringnofurtherneed
togrowwhiskers,whilethelatterbeingthecasewhichdidnotundergorecrystallizationand
henceloweredtheenergybywhiskergrowth.Basedonthistheorysmall,sub-micron,grains
recrystallizedtoformgrainswithasizeofabout1
&
mto2
&
m,whichthenbecomeprobable
29
whiskergrains.Following,
BoguslavskyandBush
[
2003
]proposedthespontaneouswhisker
formationasanfiabnormalgraingrowth"mechanismwherenewatomsareaddedtothelattice
ofthewhiskergrainbytheprocessofrecrystallizationasameanstominimizethestrainen-
ergy(storedasdislocations)inthewhiskergrain.Suchanotionisalsoguidedbythefactthat
therecrystallizationtemperatureoftinisaround30
±
Candthusenablingroomtemperature
recrystallization.Itwasalsoproposedthatthewhiskernucleationsiteswerethoseareasthat
underwentsevereplasticdeformationaswellasonesthatarehighlymisorientedwithrespect
toitsneighborsfiwhichgivestheneededgrowthmobility"forthenewgrainstogrow[
Vianco
andRejent
,
2009
,
Viancoetal.
,
2015
].Eventhoughtheabovefirecrystallizationbasedtheories"
ofwhiskergrowthwerequiteprevalent,therehavebeencertaindirectcontradictionssuchas
thefactthatthepropensityofwhiskergrowthislargerinlargecrystallizedgrainsascompared
tosmall-graindeposits,[
BoguslavskyandBush
,
2003
]whichisindirectcontradictionwiththe
dingsof
FurutaandHamamura
[
1969
].Moreover,basedontheexperimentalobservations
by
Jadhavetal.
[
2010a
],
Peietal.
[
2012
],
Jagtapetal.
[
2017
]thewhiskergrainwasfoundto
existfromthebeginningandnotformedlaterasaresultofrecrystallization.Furthermore,
throughouttheliteratureithasbeenvaguelyproposedthatthereexistssomelong-rangedif-
fusionofSnatomsintothewhiskergrainandthecrystalorientationofthewhiskergrainandits
neighborhoodplaysasigniantroleinthegrowthofthewhiskers.However,withthe
¹
-X-Ray
diffraction(
¹
-XRD)studiesby
Peietal.
[
2016
]itwasobservedthattherelacksanypeculiarity
inthestressandstraindsatthevicinityofthewhiskergrain.Suchalackofuniquenesswas
alsoobservedinthefractionofhigh-anglegrainboundariesbetweenthewhiskergrainandits
neighbors.Moreover,thisunderlyingassumptionthatrecrystallizationbyitselfisalong-range
masstransportmechanismisinherentlyculttocomprehendasatomswouldonlyundergo
slightdisplacements(oflessthaninteratomicdistances)whensweptbythereorientationfront
thattransfersthemfromtheirinitiallyoccupiedcrystallatticetothenewone.
Theoriginandmagnitudeofmasstransferbylong-rangediffusionwasformulatedby
Tu
[
1994
].Inhismodel,hethegrowthkineticsasaradiallysymmetricdiffusion
30
problemwherethetinatomiscollectedfromthemeasuredaverageareaperwhisker.The
stressgradientsupportingthisuxvariedfromtheaverage(compressive)stressatafar
awaypointeldoftransport)toazerostressconditionatthecentrallylocatedwhiskergrain.
Thezero-stressboundaryconditionforthewhiskergrainwasrationalizedbythepresenceofa
fiweakoxidefllayerontopinthat,oncethesurfaceoxideontophasbeenfracturedbytheverti-
calexpansionofthewhiskergrainitcouldeasilyaccommodateanynewatomsbeingaddedinto
thewhiskergrainboundary,asopposedtoanintactoxidelayerthatwouldprovideamechan-
icalconstraint.Themajorshortcomingofthismodelbeing,inordertotheexperimentally
observedwhiskergrowthrates,therequiredcompressivestressesintheneedstobemuch
highercomparedtothoseobtainedtypicallyinexperiments.Thismodelwaslaterimprovedby
Hutchinsonetal.
[
2004
],wherethecollectioncrosssectionwasincreasedfromamonoatomic
layertothefullheight.Morerecently,[
Saroboletal.
,
2013
]redtheassumptionofzero
stressboundaryconditionatthewhiskerroot(proposedbyTu)byconsideringadditionalfric-
tionalresistancetoatomaccretionontotheboundariesofanobliquewhiskergrain.Inthis
tion,accretionofatomsisrestrictedtohappenatthefractionofthewhiskergrain
boundaryareathatareconsideredashorizontalfacetswhileaslidingfrictionstressispostu-
latedtoactontheverticalpartsofthegrainboundary.Thisfrictionalforceopposestheout-
wardmotionofthewhiskerwhosemagnitudedependsontheinclinationofthewhiskergrain
boundaries.Thisledtotherationalizationoftheexperimentallyobservedpreferenceforshal-
lowwhiskergrains.Furthermore,bymakingtheslidingresistancedependontheslidingveloc-
ity,theirproposednon-steadystategrowthmodelcouldpredictsimilarfistickŒslip"growththat
isobservedexperimentally[
Eshelby
,
1953
,
Chasonetal.
,
2008
].Inthismodel,theeffectofgrain
geometryandsize,stress,andoxidelayerthicknesswashighlighted,howevertheoriginof
afavorablestressgradientwasnotsp
Themostadvancedimprovementoftheradiallysymmetriclong-rangetransportdiffusion
modelcenteredonapredeterminedwhiskergrainhasbeenreportedby
Peietal.
[
2017
]build-
ingonformerworksfromthesamegroupatBrownUniversity[
Buchoveckyetal.
,
2009b
,
Bu-
31
chovecky
,
2010
].Inthisnascentformulation,theconditionsdictatingthestressevolutionin
theandthesubsequentdiffusiontothewhiskerbaseemergeasanaturaloutcomefrom
thebalancebetweenthestraingeneration,throughCu
6
Sn
5
formationorviathermalexpan-
sionmismatch,andthesubsequentstressrelaxation,viarate-dependentplasticityanddiffusive
masstransportleadingtowhiskers,inthechemomechanicallycoupledsystem.Diffusivetrans-
portisexplicitlytriggeredbyreducingthecompressivestressboundaryconditionatthecentral
whiskergraintoapredeterminedvaluewithrespecttoitssurroundings.Thisvalueismostoften
attributedtotheyieldstressofSnandtheunderlyingreasoningbeingthiswhiskergrainhasun-
dergonemoreplasticitycomparedtoitsneighbors.Inthesesimulations,astresspredevel-
opsaroundthewhiskergrainsuchthattheamountofmassdiffusingintothewhiskergrainand
thesubsequentwhiskergrowthvelocityisdeterminedbytheappliedstrainrateratherthanthe
actualwhiskerspacing.Modelvalidationisdonebycomparingtotheexperimentallyobtained
stressprandwhiskergrowthrates,therebyleadingthemtoconcludethatthelocalmi-
crostructureisoflessimportance,asthesimpleisotropicmodelwithoutanyhardeningcould
capturetheimportantphysicalprocessesdeterminingthewhiskeringandtherate-dependent
plasticity.However,suchaninformationofthelocalmicrostructuremightbecrucialiniso-
latinggrainsthataremoreprobabletowhiskerformation,andhenceaddressingthecritical
questionofwhiskernucleation.Thetwoideaswhichexistregardingagrainformingwhiskers
areeitheranoblique(inclined)grainboundarywhichareeitherpreexistingorlaterformedvia
lateralgraingrowth[
Chasonetal.
,
2014
]orweakgrainsthatundergoantamountofplas-
ticdeformation.Noneofthetwonotionsisconclusiveascontradictingobservationshavebeen
reported,anditisourbelievethatthesequestionscouldbeansweredviathree-dimensional
fullchemo-thremo-mechanicalsimulationsinacrystalplasticityframeworkwhichtakes
intoaccounttheeffectofcrystalanisotropy.
32
2.4Modelingframework
Crystalplasticity(CP)materialsolverscanaccommodatetheeffectofcrystalanisotropy
andneighborhoodtextureindeterminingtheoverallresponseofthesystemunderapplied
boundaryconditions.Inthesamecontext,fifud"simulationscorrespondtothefactthat
thesimulationgeometryincorporatesspatialgraininformationobtainedfromexperiments.In
thisworktheCPsimulationsareperformedusingtheopensourcefiDüsseldorfAdvancedMa-
terialSimulationToolkitfl(DAMASK)[
Eisenlohretal.
,
2014
]developedinMax-Planck-Instiut
fürEisenforschungGmbH(MPIE)[
Roters
,
2011
,
Rotersetal.
,
2012
,
2019
].Variousconstitutive
modelsandhomogenizationschemesforcrystalplasticitysimulationsatdifferentlengthscales
areavailableintheDAMASKsoftware,whichalsoprovidesaexibleinterfacewithcommercial
iteelement(FE)packagessuchasABAQUSandMSC.Marc.Apartfromitbeingthematerial
pointsolver(tosolveconstitutiveeldequations),DAMASKalsohasitsownintegratedbound-
aryvaluesolverthatusesspectralmethodsthatreducescomputationtimesigantlycom-
paredtoite-elementbasedsolvers[
Eisenlohretal.
,
2013
,
Shanthrajetal.
,
2015
,
Diehl
,
2016
].
Thecrystalplasticity(CP)modelisbasedonacontinuummechanicalestrainframework
thatdescribesthemechanicalbehaviorofthebodyunderconsideration,andalsoneedsthe
fimicrostructure"orgraininformation.Materialresponseisthenobtainedbysolvingforglobal
equilibriumconditionsalongwithinter-graincompatibility(achievedthroughtheprescribed
homogenizationscheme)thatalsoensuresthefuofthelocalcompatibilityandequi-
libriumconditionateachdiscretizationpoint,anapproachmuchdifferentfromtheearlyplas-
ticitymodelsthatenforceduniformstrain
Taylor
[
1938
]orresolvedshearstress
Sachs
[
1929
]
distributionwithineachgrain.
Inthissection,abriefbackgroundaboutcontinuummechanicsisprovidedtointro-
ducethedifferentcontinuummechanicalexpressionsandtheirrelevance.Subsequently,a
briefoverviewisgivenabouttheconceptsofgeneralcrystalplasticitytheoryaswellasthenu-
mericalmethodsusedtosolvetherespectivedequations.However,theconstitutivemodels
usedtorunthesimulationsarenotdescribedinthissectionandwillbeexplainedseparatelyin
33
subsequentchapterswherethesimulationresultsarediscussed.
2.4.1Continuummechanics
Intheofcontinuummechanicsthemechanicalbehaviorofabodyismodeledasahypo-
theticalcontinuousmassratherthandiscreteparticles.Basedonthisassumption,anyobject
wouldthencompletelythespaceitoccupies.Thishastheadvantagethatthedeformation
behaviorofthematerialcouldbedescribedusingcontinuousmathematicalfunctions,while
thedisadvantagebeingdiscontinuitiesinthebody,suchascracks,areneglected.Tohavea
consistentsolutionforaprobleminacontinuummechanicalframework,thefollowingsetsof
equationshastobesupplied:
Ł
deformationcompatibility
Ł
mechanicalequilibriumorbalanceofforces
Ł
constitutivedescription
Incontinuummechanicsthekinematicsofdeformationisdescribedthroughvariousstages
inthedeformationhistoryofthebodyorconations.Eventhoughthebodyoccupiesdif-
ferentgurations,therestillexistscontinuityduringthedeformationsuchas:
Ł
Thematerialpointsformingaclosedcurveatanyinstantwillalwaysformaclosedcurve
atanysubsequenttime.
Ł
Thematerialpointsformingaclosedsurfaceatanyinstantwillalwaysformaclosedsur-
faceatanysubsequenttimeandthematterwithintheclosedsurfacewillalwaysremain
within.
Inthissense,thereference(undeformedortime-independent)conationoccupyinga
region
B
0
inspace(withitenumberofmaterialpoints)undercertain(time-dependent)
loadingwouldattainthecurrent(deformedortime-dependent)curationoccupyingare-
gion
B
t
inspace.Thelocationofthematerialpointsing
B
0
isgivenby
x
2
B
0
,andthatin
34
Figure2.8:Kinematicsofacontinuumbodyundergoingdeformation[
Diehl
,
2016
].
B
t
isgivenby
y
2
B
t
,asillustratedinFig.
2.8
.Eachconationcanhaveitsownbasisvec-
tors,howeverusingageneralorthonormalCartesiancoordinatesystemforbothconurations
avoidsthecomplexityofrepresentingvectorswithdifferentbasisfordifferentconations.
Ingeneralthereexiststwowaystodescribethemotionofabody,i.e.,eitherasLagrangian(ma-
terial)descriptionorEulerian(spatial)description.IntheLagrangiandescriptionthereference
conationisalsotheundeformedconurationandthepositionandphysicalproperties
oftheparticlesaredescribedintermsofthereferenceconurationcoordinatesandtime.It
canbethoughtasifanobserverstandingintheframeofreferenceof
B
0
observedhowthe
bodychangesovertimetoreach
B
t
.Since,thechoiceofreferenceurationisarbitraryin
general,theresultsobtainedinLagrangiandescriptionareindependentofthechoiceofinitial
timeandreferencecuration.WhileEulerianorspatialdescriptionfocusesonthecurrent
conationandisakintoobservingwhathappenstoapointinspace.Inthiswork,like
35
inmostcasesofsolidmechanics,themotionisdescribedusingtheLagrangiandescription,
whereadeformationmap
Â
(
x
):
x
2
B
0
!
y
2
B
t
isusedtomappoints
x
inthereferencecon-
urationtopoints
y
inthecurrentconuration.Thedisplacement
u
ofamaterialpointata
deformationstate
t
)isthengivenby:
u
(
x
)
Æ
Â
(
x
)
¡
x
(2.1)
Alinesegmentd
x
inaninitesimalneighborhoodofamaterialpoint
x
inthereference
conationisthenpushedforwardintothecurrentcurationby:
y
Å
d
y
Æ
y
Å
@
y
@
x
¢
d
x
Å
O
(d
x
2
).(2.2)
Neglectingtermsofhigherorder,d
y
canbeexpressedas:
d
y
Æ
@
y
@
x
¢
d
x
Æ
Grad
Â
|
{z
}
Æ
:
F
(
x
)
¢
d
x
,
(2.3)
where
F
(
x
)isthesecondordertensorcalledthedeformationgradient.
F
(
x
)mapsthenites-
imallinesegmentd
x
inthereferencegurationtod
y
inthecurrentconation
Roters
etal.
[
2019
].AsillustratedinFig.
2.8
thedeformationgradienttensor
F
(theargument
x
orthe
spatialdependenceisassumedimplicitly)isa2-pointtensor,meaningithasonebasisinthe
referenceurationandtheotheroneinthecurrentconuration.Since
F
isamapping
functionitsinverse,
F
¡
1
,existsaswellwhichmapselementsfromcurrentconurationback
tothereferenceconation.Thecommonlyobservedtermsfipushforward"andfipullback"
onaquantityreferstotheoperationby
F
or
F
¡
1
onthatquantitytoeithermapintocurrentcon-
uration(fromreferenceconuration)orbackintoreferencecuration(fromcurrent
conation)respectively.
Thematerialvelocitydisdenedas:
d
v
Æ
d
u
(
x
)
d
t
Æ

u
,
Æ

Â
,
(2.4)
36
where
u
(
x
)isthematerialdisplacementwhichvarieswithtime(deformation).Sincethe
referenceconurationisassumedtobetime-independent,i.e.,d
x
/d
t
Æ
0,therelation

u
Æ

Â
holds.Thevelocitygradient
L
isthenthespatialgradientofthevelocity
@
v
/
@
y
andcan
alsobeexpressedas:
L
Æ

FF
¡
1
(2.5)
Thedeformationgradient,ingeneral,isassociatedwitharotationandastretch,andsince
itisinvertibleitcanalsobedecomposeduniquelybyafipolardecomposition":
F
Æ
VR
Æ
RU
,
(2.6)
where
R
istherotationtensorand
U
and
V
representstherightandleftstretchtensorsrespec-
tively.Dependingontheurationsvariousstrainandstressmeasuresexist,forexample
ifwewanttoexpressthestraincompletelyinreferenceconurationthenitisexpressedas
G
REEN
ŒL
AGRANGE
straintensoredas(
F
T
F
¡
I
)/2;whileastrainmeasurecompletelyin
currentconationistheE
ULER
ŒA
LMANSI
straintensoras(
I
¡
F
¡
T
F
¡
1
)/2.Thus,the
pushforwardforG
REEN
ŒL
AGRANGE
straintensoristheE
ULER
ŒA
LMANSI
straintensor.
Similartotheabovementionedstrainmeasures,thestressmeasuresofthebodyundergo-
ingdeformationcanbeexpressedindifferentconations.TheC
AUCHY
stresstensor
¾
is
expressedcompletelyinthecurrent(ordeformed)gurationsinceitisastheforce
expressedinthecurrentgurationactingonanitesimalareaalsointhecurrentg-
urationandhenceisasymmetrictensor.Similarly,thesecondP
IOLA
ŒK
IRCHHOFF
stress-tensor
isasymmetrictensorrepresentedcompletelyinthereference(stress-free)conation.In
staticequilibriumwithoutanybodyforcestheequilibriumcondition:
¾
ij
,
j
Æ
0,(2.7)
whichisconvenientlywrittenintheitestraintheoryas,
0
Æ
Div
P
Ær¢
P
(2.8)
37
where
P
representsthenon-symmetricP
IOLA
ŒK
IRCHHOFF
stresstensor.
Withthisbriefcontinuummechanicaloverview,thecrystalplasticityframeworkwherethe
simulationsareperformedinthisworkisexplainednext.
2.4.2CrystalPlasticityMulti-physicsframework
Crystalplasticity(CP)modelsarewell-establishedlawsthattakeintoaccounttheaffectof
materialanisotropyinresponsetoanexternalloadandhenceisapowerfultoolincompu-
tationalmaterialssciencetoinvestigatestructureŒpropertyrelationships.TheCPmodeling
techniquehasbeensuccessfullyimplementedtostudyawiderangeofmicromechanicalphe-
nomenarangingfromevolutionofslipresistanceandstrainhardeninginsinglecrystalstome-
chanicalresponseinapolycrystallineaggregate[
Rotersetal.
,
2010b
,
2019
].Eventhoughthe
formulationofthismodelingframeworkoriginatedtoinvestigatetheplasticdeformationof
amaterialbycrystallographicslipondiscreteslipsystems,withincreasingvariabilityinthe
environmentalconditionsofapplicationsleadingtodrasticmicrostructuralchangesinthesys-
tem(suchasphasetransformation,cracknucleation,hightemperaturedeformation),thisCP
frameworkhasrecentlybeenextendedtoincludethecomplexcouplingofotherconstitutive
equations(damage,reaction-diffusion,thermal,electromigration)toformulateamore
unandgeneralmulti-physicstoolinordertohaveabetterrepresentationofthephysical
systemandhencehigheraccuracyinprediction[
Shanthrajetal.
,
2016
,
2019
].Inthiswork,the
opensourcecrystal-plasticitymultiphysicsframeworkDAMASKwasmoditoincludethe
stress-drivendiffusionkineticsinordertosimulatethemasstransportintin.Integration
ofthenewconstitutivelawalongwiththesubsequentnumericalstrategyintotheexistingmod-
elingframeworkisrelativelystraightforwardduetothemodularstructureofDAMASK[
Roters
etal.
,
2019
],andsomeofitsrelevantfeatureswillbediscussedinthesubsequentparagraphsin
thecontextofsolvingcoupleddequationsinthisframework.
InDAMASK,theconstitutiveequationsareprescribedinthesinglecrystallevelwhichare
thenhomogenizedbyappropriatehomogenizationschemestogettheoverallsystemresponse.
38
Figure2.9:Illustrationoftheintermediateconurationsresultingfromthemultiplicativede-
compositionofthedeformationgradient[
Rotersetal.
,
2019
].Selectingthecrystalorientation
asinitialvalueof
F
p
(t
Æ
0)
Æ
O
0
guaranteesthatthelatticecoordinatesystemintheplasticcon-
urationalwayscoincideswiththelabcoordinatesystem[
Maetal.
,
2006
].
Inthisworkthefull-simulationsareperformedwhereeachgraininthepolycrystallineag-
gregateisspatiallyresolvedintoalargenumberofmaterialpoints/voxelsandtheeldequa-
tionsaresolvedateachofthesematerialpointshencenohomogenizationschemeisneeded.
Moreover,asmentionedearlier,thecurrentworkincludestheeffectofstress-drivenmasstrans-
porthenceastrongcouplednumericalframeworkisrequiredtointimatelyincludethekine-
maticconsequencesofsuchtransportŒbothataconstitutive(governingtransportequation)
andatthecontinuum(subsequentkinematicsduetotransport)levels.
Inthiscontext,thenumericalframeworkinDAMASKisbasedontheedeformation
theorywherethedeformationgradient,
F
,governingthemotionofthebodyismultiplicatively
decomposedintoalatticepreservingplasticdeformationgradient,
F
p
,followedbythein-
termediatelatticedistortingorfieigenstrain"deformationgradient,
F
i
,andllytheelastic
39
deformationgradient,
F
e
asillustratedinFig.
2.9
.
F
Æ
F
e
F
i
F
p
(2.9)
Theplasticdeformationgradient
F
p
mapstheundeformed(initial)conationintoavol-
umeconservingplasticallydeformedguration,while
F
i
mapsthesubsequentvolumenon-
conservingeigenstrainconationthataccommodatesstress-freestrains(fieigenstrains")
suchasthermalstrainsorchemicalstrains.Thedeformedconationmapping(that
involvesbothrotationandstretch)isobtainedbytheelasticdeformationgradient,
F
e
.More-
over,toavoidunnecessaryrotationsofthecontinuumquantitiesduetothedifferenceinrefer-
enceframesoftheindividualgrainsandtheglobal(lab)frame,thedeformationmap(
F
p
)
isinitializedwiththeorientationmatrixcorrespondingtotheinitialcrystalorientation
O
0
,i.e.
F
p
(
t
Æ
0)
Æ
F
e
T
(t
Æ
0)
Æ
O
0
suchthattheplasticcurationofeachcrystalcorrespondstoa
commoncubeorientation[
Maetal.
,
2006
].Followingthis,thecurrentorientationmatrix(
O
)
canthenbecalculatedfrom
F
e
throughthepolardecomposition[
Rotersetal.
,
2019
].
ThesecondP
IOLA
ŒK
IRCHHOFF
stresstensor
S
dependsontheG
REEN
ŒL
AGRANGE
strainten-
soras:
S
Æ
C
¢
E
e
,(2.10)
where
C
isthestiffnesstensor.Thestress
S
,alongwiththemicrostructuralstate(givenbythe
correspondingconstitutiveequation(s)),drivestheplasticvelocitygradient
L
p
aswellasthe
eigenstrainvelocitygradient
L
i
(summationofboththethermalandchemicalvelocitygradi-
ents).Thevelocitygradientsaregivenby:
L
Æ

FF
¡
1
(2.11)
ThisinterdependencyamongthevariouskinematicquantitiesisillustratedinFig.
2.10
whichalsoshowstheself-consistentnumericalstrategy.Inthat,theplasticvelocitygradi-
ent(
L
p
)issolvedbyapredictor-correctorschemeunderanassumed
state
andtheeigenstrain
(intermediate)velocitygradient
L
i
.Withaobtained
L
p
andthepreviously
state
,thefol-
lowingeigenstrainvelocitygradient(s)aresolvedsequentially(dependingontheproblemthere
40
Figure2.10:Solutionstrategyforthevariouskinematicquantities,[
Rotersetal.
,
2019
].
mightbemultiplesourcesof
L
i
).Finally,acheckisperformedtoseewhethertheobtainedve-
locitygradientsisconsistentwiththeassumed
state
ornot[
Shanthrajetal.
,
2019
,
Rotersetal.
,
2019
].Ifthedifferencebetweentheassumedandthepredicted
state
isbelowtheprescribed
tolerance,thesolutionisconsideredtobeconvergedotherwisethenextiterationstarts.The
numericalstrategiestosolvethedifferentialequationsaswellasthesubsequentintegrationis
mentionedbriinthesubsequentsection.
2.4.2.1Numericalschemes
Thesolutionoftheboundaryvalueproblem,i.e.,staticequilibriumisobtainednumerically
usingoneofthemanyexistingtechniques.Themostcommonmethodsofsolvingsuchstatic
41
equilibriumproblemsiseithervianiteelementmethod(FEM,)[
Courant
,
1943
,
Zienkiewicz
,
1967
,
Bathe
,
2014
,
Zienkiewiczetal.
,
2013
]orthroughspectralmethodsusingfastF
OURIER
transform(FFT)introducedby[
MoulinecandSuquet
,
1994
]andlaterby[
Leben-
sohn
,
2001
,
Kaßbohmetal.
,
2006
,
Spahnetal.
,
2014
,
Eisenlohretal.
,
2013
,
Willot
,
2015
,
DeGeus
etal.
,
2017
].ThecriticaldifferencebetweenFEMandFFTmethodbeing,inFEMthesolution
oftheequationisapproximatedbylocalshapefunctions(lower-orderpolynomials)that
arenon-zeroinaitedomain(fielement")andarethensummedovertogetthetotalapprox-
imate(weakform)solution.WhilethespectralFFTmethodsapproximatethewholedomain
byonelargesetofbasisfunctions[
Shanthrajetal.
,
2019
].Whenthesegloballynon-zeroansatz
functions(exceptattheirroots)aretrigonometricthemethodusesthefastF
OURIER
transform
tosolveforthem,andhencereferredasFFT.TheFEMmethodhasbeenthemostpopularin
solvingmicromechanicalequationsduetothepossibilityofinvestigatingsystemshaving
complexgeometries,whilethespectralFFTbasedmethodsinherentlyinvestigatesaperiodic
system.
Asmentionedearlier,theopensourcemulti-eldmaterialpointsolver,DAMASK,usedin
thisstudyhasitsinherentspectralFFTboundaryvaluesolverbutcouldalsobeusedinconjunc-
tionwithcommerciallyavailableFEMsolversasausermaterialsubroutinesuchasMSC-Marc
andABAQUS.InChapter
5
,theuseofthematerialsolverDAMASKalongwithcommercialFEM
solverMSC-Marcisdiscussedwithrespecttonanoindentationsimulations,whilethecoupled
thinmicromechanicalsimulationsinChapters
3
and
4
areperformedusingtheFFTbased
spectralsolverofDAMASK.
Thekeythingforanymaterialsolveristoprovideameasureofstress(
S
,
P
,or
¾
depend-
ingonthesolver)foraninputmeasureofdeformationgradient,
F
.Thefollowingoutlinesthe
stepsundertakenbythematerialsolverinDAMASKtoevaluateaconsistentvalueofstressfor
agivenvalueof
F
ateachmaterialpoint.Asmentionedearlier,thepartitioningoftheglobal
F
receivedfromtheboundaryvaluesolverinto
F
ateachmaterialpointandthesubse-
quenthomogenizationofthestressateachmaterialpointintoaglobalstressishandled
42
bythefipartitioningandhomogenization"moduleinDAMASKbasedontheprovidedscheme,
andisexcludedinthissection(see[
Rotersetal.
,
2019
]forfurtherdetails).Thus,ateachtime
stepandwithaknown
F
ateachmaterialpointthefollowingsystemofequationsissolvedina
self-consistentway:
F
Æ
F
e
F
i
F
p
(2.12a)

F
Æ
LF
,forboththeinelasticdeformationgradients,(2.12b)
L
Æ
X
n
f
n
(
S
,...),forboththeinelasticvelocitygradients,(2.12c)
wherethesecondP
IOLA
ŒK
IRCHHOFF
stress,whichistheworkconjugateoftheG
REEN
Œ
L
AGRANGE
straintensor,isobtainedfromtheelasticconstitutiveequation,
S
Æ
f
(
E
e
,...
)
.These
equationsaresolvedinanimplicitmannerwiththevelocitygradientintegralsbeingapproxi-
matedas(withamaterialstate)foratimeinterval
¢
t
:
F
p
(
t
)
¡
F
p
(
t
0
)
¢
t
Æ
L
p
(
t
)
F
p
(
t
)and(2.13a)
F
i
(
t
)
¡
F
i
(
t
0
)
¢
t
Æ
L
i
(
t
)
F
i
(
t
),(2.13b)
whichresultsintheinelasticdeformationgradientsattheendofthetimeincrementbeing
F
p
(
t
)
Æ
¡
I
¡
¢
t
L
p
(
t
)
¢
¡
1
F
p
(
t
0
)and(2.14a)
F
i
(
t
)
Æ
(
I
¡
¢
t
L
i
(
t
)
)
¡
1
F
i
(
t
0
),(2.14b)
fromwhich
F
e
canbecalculatedusingEq.(
2.12a
).Theunknownsfortheabovesystemofcou-
plednon-linearalgebraicequations,
F
e
,
F
i
,and
F
p
,whosesolutionwouldthenprovideastress
statesatisfyingthemultiplicativedecompositionaswellasfungtheconstitutivelawsfor
therespectivevelocitygradients,representedhereasgenericfunctionsEq.(
2.12c
).
Since,thedifferentelds(givenbyeither
L
i
forthermalandchemicalstressesor
L
p
for
mechanicalstresses)arecompletelycoupled,theirsolutionisobtainedusingatwo-level
43
predictorŒcorrectorscheme(asshowninFig.
2.10
)bymyminimizingthefollowingresiduals:
R
p
¡
f
L
p
,
e
L
i
¢
Æ
f
L
p
¡
L
p
¡
M
p
¡
f
L
p
,
e
L
i
¢¢
and(2.15a)
R
i
¡
f
L
p
,
e
L
i
¢
Æ
e
L
i
¡
L
i
¡
M
i
¡
f
L
p
,
e
L
i
¢¢
,(2.15b)
with
f
L
p
and
e
L
i
denotingthepredictedvaluesof
L
p
and
L
i
.TheseresidualsEqs.(
2.15a
)
and(
2.15b
)arethenminimizedusingaN
EWTON
ŒR
APHSON
schemewithavariable
steplengthuntilaconsistentsolutionisobtained,withinastaggerediterativeloop[
Shanthraj
etal.
,
2019
,
Rotersetal.
,
2019
].ThesolutionalgorithmisoutlinedinAlgorithm
1
.Convergence
oftheN
EWTON
ŒR
APHSON
schemeisachievedwhentheresidualdropsbelowagiventolerance
²
p
Æ
max
¡
²
a
,
²
r
¯
¯
¯
¯
L
p
¯
¯
¯
¯
2
,
²
r
¯
¯
¯
¯
f
L
p
¯
¯
¯
¯
2
¢
(2.16a)
²
i
Æ
max
¡
²
a
,
²
r
jj
L
i
jj
2
,
²
r
¯
¯
¯
¯
e
L
i
¯
¯
¯
¯
2
¢
(2.16b)
wherethevaluesofabsoluteandrelativeerrors(
²
a
and
²
r
)areprescribed.
Theadvantageofformulatingtheproblemasaresidualminimizationistheuseofopen
sourcesolverssuchasPETSc(PortableExtensibleToolkitforScienComputation),devel-
opedby[
Balayetal.
,
2013
]andPETScteaminArgonneNationalLab,tothesolution.
PETSchasvariousnumericalalgorithmstosolveasystemoflinearornon-linearequationsin
themostientwayandisafundamentalpackagefortheusageofDAMASK.Thespatialgra-
dientsofthemechanicalquantitiesarecalculatedusingtheforward-backwarditedif-
ferencevariantintroducedby
Willot
[
2015
],
Schneideretal.
[
2015
]thatreducesthefrequently
observeductuationsintheresults(alsoknownasthefiGibbsphenomenonfl).Thegradientand
divergencecalculationsforthechemicalquantitiesareperformedinrealspace.Themi-
crostructuralstateintegrationisdoneusinganimplicitxedpointiterativescheme[
Kalidindi
etal.
,
1992
].Forthespectralsolverusedinthiswork,thestaticequilibrium(i.e.theboundary
valueproblem)equationissolvedusingthedirectvariationalformulationoriginallyintroduced
by
Eisenlohretal.
[
2013
]andlaterby
Shanthrajetal.
[
2015
]usingthefibasicflscheme
whichisacollocationbasedapproachatthegridpoints[
Eisenlohretal.
,
2013
].
44
Algorithm1:
Self-consistentintegrationofkinematicquantitiesataxedinternalmate-
rialstate[
Rotersetal.
,
2019
].
Data:
[
F
]
t
n
,[
F
p
]
t
n
¡
1
,[
F
i
]
t
n
¡
1
Result:
[
F
p
]
t
n
,[
F
i
]
t
n
,[
F
e
]
t
n
,[
S
]
t
n
1
Initialisation:
[
f
L
p
]
0
t
n
Æ
[
L
p
]
t
n
¡
1
,
[
e
L
i
]
0
t
n
Æ
[
L
i
]
t
n
¡
1
,
j
=1
2
L
i
loop:
3
while
k
R
i
k
2
¸
²
i
do
4
[
F
i
]
t
n
Æ
³
I
¡4
t
[
e
L
i
]
j
¡
1
t
n
´
¡
1
[
F
i
]
t
n
¡
1
5
k
=1
6
L
p
loop:
7
while
k
R
p
k
2
¸
²
p
do
8
[
F
p
]
t
n
Æ
³
I
¡4
t
[
f
L
p
]
k
¡
1
t
n
´
¡
1
[
F
p
]
t
n
¡
1
9
[
F
e
]
t
n
Æ
[
F
]
t
n
[
F
p
¡
1
]
t
n
[
F
i
¡
1
]
t
n
10
[
S
]
t
n
Æ
f
³
[
F
e
]
t
n
,[
F
i
]
t
n
´
11
R
p
Æ
[
f
L
p
]
k
¡
1
t
n
¡
L
p
³
[
S
]
t
n
,[
F
i
]
t
n
´
12
[
f
L
p
]
k
t
n
Æ
[
f
L
p
]
k
¡
1
t
n
¡
®
p
³
@
f
L
p
R
p
´
¡
1
R
p
13
k
=
k
+1
14
end
15
R
i
Æ
[
e
L
i
]
j
¡
1
t
n
¡
L
i
³
[
S
]
t
n
,[
F
i
]
t
n
´
16
[
e
L
i
]
j
t
n
Æ
[
e
L
i
]
j
¡
1
t
n
¡
®
i
³
@
e
L
i
R
i
´
¡
1
R
i
17
j
=
j
+1
18
end
45
Apartfromtheabove(relativelydetailed)overviewofthecomputationalframework,the
individualconstitutivelawswillbediscussedasapartofsimulationdetailsineachchapter,
alongwithafewlinesaboutthecontinuummechanicalitestrainframeworktoallowfora
coherentreading.
2.5Summary
Thecouplednumericalframeworkdescribedinthischapterprovidesaconvenienttoolto
investigatethekinematicsassociatedwiththestressassistedtransportofSnatomsalongthe
grainboundarynetworkintinmsalongwiththeassociatedeffectofthestressontheirkinet-
ics.Moreover,since
¯
-Snhasaveryhighhomologoustemperatureevenatroomtemperature,
theaddedcouplingofthemechanical(plasticandelastic)effectsontheoverallkinematicsof
thesystemcannotbeneglected.Thefullcalculationsperformedinthisworktakeintoac-
countboththelong-rangeandtheshort-rangegraininteractionstherebyincludingtheeffects
ofgrainneighborhoodsindeterminingtheoverallstrthatinturndictatethetrans-
port,[
Liuetal.
,
2010
].Thehighmaterialanisotropyof
¯
-Snthatcansigtlyaffectthe
overallsystemresponseoftheunderloadwasincorporatedeasilyinthiscrystalplasticity
framework.Themostcriticaladvantageofthecurrentframeworkistheabilitytodirectlyincor-
porateexperimentallyobserveddata(grainsizedistribution,andcrystalorientation)intothe
simulationssoastohaveadirectcomparisonbetweentheexperimentalandsimulatedresults,
therebyensuringthetyofthesimulatedresults.
InChapter
3
,fuldthermo-mechanicalsimulationsoftinareperformedbymim-
ickingtheexperimentsof
Peietal.
[
2016
].Theconstitutiveequationsusedforthedifferent
alongwiththesimulatedgeometryaredescribed.Criticaloutcomesfromsuchsimula-
tionshelpedinproposingahypothesisforthewhiskernucleationprocess(anditssubsequent
growth)thatisnotincontradictionwiththeexistingstress-controlledwhiskerformationtheo-
riesinliterature,alongwithhighlightingtheeffectsofglobaltexture,localgraingeometry,
andgrainsizedistribution.
46
Following,thethermo-mechanicalmodelwasimprovedtoaddthestress-driventransport
equationalongwithitskinematicconsequenceinChapter
4
.Thedetailsoftheconstitutive
equationsalongwiththenumericalsolutionstrategythatwasdevelopedtosolvethex
equationswithcomponentshavinghighpropertycontrastisalsoexplained.Thegoalofthe
simulationsdiscussedinthischapteristohighlightthecontributionofwhiskerformationon
theoverallstress-relaxationofthesystem.Moreover,thekinematicconsequenceofsuchmass
redistributionisalsoaddressed.
Asmentionedpreviously,
¯
-Snhasaveryhighplasticanisotropywith13slipfamiliesthat
areavailabletoaccommodatetheplasticshapechange.Moreover,sinceitshowshightemper-
atureplasticityevenatroomtemperature(300K
¼
0.6
T
m
)theplasticcontributiontooverallde-
formationbecomescrucialthusrequiringanaccuratedescriptionofitsplasticity.Eventhough
thecoupledchemo-thermo-mechanicalsimulationsareperformedusingavailabledatafor
¯
-
Snplasticity[
Leeetal.
,
2015
],whicharetheinitialowstressvalues(alsoreferredtoascriti-
calresolvedshearstresses(CRSS)ofthedifferentslipfamilies),thesevaluesareyettobees-
tablisheditively.Thus,inordertoestimatereliablevaluesoftheseinitialwstresses,
amodiedInverseIndentationAnalysis(
IIA
)methodologyisproposed,originallydeveloped
by
Zambaldietal.
[
2012
],wheretheplasticityparametersareidentiedbycomparingexper-
imentalandsimulatedsinglecrystalnanoindentations.InChapter
5
,thedetailsofthemod-
methodologyareprovidedalongwithitsreproducibilityandrobustnessforsymmetric
face-centeredcubicmaterials(withsingleslipfamily)usingvirtualexperimentalreferences.
Following,thereliabilityofthismethodologyisstudiedforlowersymmetrichexagonalcom-
merciallypuretitanium(assumingthreeslipfamilies),alsousingvirtualexperimentalrefer-
ences,inChapter
6
.Havingestablishedthereliability,thenextstepwouldbetoimplement
suchamethodologyforevenmoreanisotropic
¯
-Sn(having13slipfamilies)andtocompare
theidentiparameterswiththoseexistinginliterature.Thiswouldfacilitateinperforming
higtycoupledmulti-materialsimulationswithanaccuratematerialdescription.
Apartfromthegeneralprogresstowardsdevelopingthisrobustmultimaterialsimula-
47
tionsystemwithhighphasecontrastbetweencomponents(suchasthedifferenceindiffusivity
ofSnatomsinthebulkandthegrainboundaries),thegoalofthisworkistoanalyzethecritical
processofwhiskernucleationanditsgoverningfactors.Onceweareabletodeterminewhich
grains(orwhatarethedictatingconditions)couldleadtowhiskers,bettercrystallographicmit-
igationstrategiescouldbeproposed(suchasglobalorlocallmtexturecontrol)andtherelia-
bilityissuesduetowhiskerscouldbeavoided.
48
CHAPTER3
THERMOMECHANICALCRYSTALPLASTICITYMODELINGOFTHERMALLYSTRAINED
¯
¡
SnFILMS
3.1Background
Inthischapter,three-dimensionalfuldthermo-mechanicallycoupledcrystalplasticity
simulationsofthermallystrained
¯
-Snthinareshownusingthenumericalframeworkdis-
cussedinthepreviouschapterinSection
2.4
.Theobjectiveofthisworkistobetterunderstand
theconditionsgoverningthegenerallyrarenucleationofwhiskersfromtheselmsthrough
thesimulatedresultsaftervalidatingthemwithexperimentalresults.Criticalaspectsofthis
studyincludedeterminationofcrystallographicandgeometricfactorsthatsigntlyaffect
thehydrostaticstressalongthegrainboundarynetworkoftheandhencewoulde
thesubsequentstress-drivenmasstransportalongtheseboundariespriortotheonsetofany
whiskerformation.Inthiswork,theisapproximatedasaperiodicstructurewithapprox-
imatelyhundredcolumnargrainsonanrigidisotropicsubstratemimickingtheexperimental
conditionsof[
Peietal.
,
2017
],whereacompressivebiaxialstressisimposedintheupon
heatingthebstratesystem,duetodifferencesintheircientofthermalexpansion.
Adilatationallayerakintoairwithnegligiblestiffnessandstrengthcomparedtotheisalso
addedonthetopoftheThisdilatationallayeravoidstheneedofhavingperiodicbound-
aries,
MaitiandEisenlohr
[
2018
],aswellas,couldlaterbeadjustedtomimicthemechanical
effectofanysurfaceoxidelayer.Criticalobservationsfromthesesimulationsincludethespa-
tialvariabilityofthehydrostaticstressesalongthegrainboundarynetworkwhichalsovaries
basedontheglobaltexture;stress-gradientsnotexceedingoneortwograinsizes;and
negligibleeffectofgraingeometryorgrainsizedistributionascomparedtotexture(local
crystallographicneighborhood),rmingthenotionthatwhiskerformationisindeedalocal
phenomenaasreportedinliterature
Buchovecky
[
2010
],
Sobiechetal.
[
2011
].Fromthesesim-
49
Figure3.1:Discretizedgeometryofatinlmbetweenarigidisotropicsubstrate(orange,partly
showingvoxelsizeinthecornerofleftimage)andsoftdilatationalfiairfl(faintblue).Boundary
regionbetweencolumnartingrains(shadesofyellow)ishighlighted(green)andconstitutes
the(sub)volumeoftheforwhichhydrostaticstressgradientsareassumedtodriveatom
diffusion.Selectedgrainsofthecolumnarshownatleftaretransformedintoanoblique
surfacegrain(grayvolumeinthemiddleandrightviews).
ChakrabortyandEisenlohr
[
2018
]
ulationsitisalsoinferredthat
¯
-Snhavingatextureandwiththeiraxisparallel
to
h
001]islesswhiskerpronecomparedtothehavingtheireraxisparallelto
h
100].In
thefollowing,Section
3.2
,thedetailsaboutthesimulationgeometry,boundaryconditions,and
inputtextureinformationisdiscussedfollowedbyabriefaccountofthecontinuummechan-
icalframeworkandconstitutivedequations.AfterhighlightingtheresultsinSection
3.3
a
hypothesisaboutaplausiblewhiskerformationmechanismislaidinSection
3.4
alongwitha
summaryoftheresultsfromthestudy.Partsoftheworkinthischapterisalreadypublishedin
ChakrabortyandEisenlohr
[
2018
].
3.2Simulationdetails
Inthissectionthedetailsaboutthecoupledthermo-mechanicalsimulationsmimicking
oneoftheexperimentalconditionsin
Peietal.
[
2017
]isoutlined.
3.2.1Geometrydiscretization,texture,andboundaryconditions
Thesimulatedgeometryconsistsofa1
&
mthinofbody-centeredtetragonal(bct)
¯
-Snwith
103columnargrainsabovea1.4
&
mrigidisotropicsubstrate,whichhasasmallercientof
thermalexpansion,anda0.8
&
mlayerofdilatationalmaterialmimickingfreesurfacecondition.
Theentiregeometryisdiscretizedbyaregulargridof128
£
148
£
32
Æ
606208voxels,eachof
50
volume0.1
£
0.1
£
0.1
&
m
3
asillustratedinFig.
4.2
(left).Toanalyzetheeffectofobliqueshaped
surfacegrains,insomesimulationsone(orfew)columnargrain(s)aremadeoblique(inclined
grainboundaries)asinFig.
4.2
(right).
PoissonŒVoronoitessellationofthetinvolume(using103seedpoints)resultsinarela-
tivelynarrowgrainsizedistribution.Toarriveatwiderdistributions,graingrowthissimulated
throughcurvature-driven(andin-planeonly)motionofthegrainboundaries
Eisenlohretal.
[
2017
].
Crystallographicorientationsofeachgrainareselectedsuchthataparticularcrystallo-
graphicdirectionispredominantlyalignedwiththenormaldirection,resultinginaber
textureforthelm.Thethreetexturesconsideredinthisworkare
h
100],
h
110],and
h
001]
(crystallographicnotationsadoptedfrom
Leeetal.
[
2015
]),whicharereferredtoasfigreenfl,
fibluefl,andfiredflinaccordancewiththestandardinversepoleecolormapping.
Allthreelayers(substrate,andair)areinitiallystress-free,heatedby40Kin20min,and
thenkeptatconstanttemperaturetoobservestressrelaxationforanother160min(blackcurve
inFig.
3.2
),whichissimilartotheexperimentalconditionsof
Peietal.
[
2017
]
Theresultingthermo-mechanicallycoupledsystemwassolvedunderperiodicdisplace-
mentboundaryconditionsusingthefibasicflschemeofthespectralsolver
Shanthrajetal.
[
2015
],
Eisenlohretal.
[
2013
]thatispartoftheopen-sourceDüsseldorfAdvancedMaterialSim-
ulationKit(DAMASK).
3.2.2Continuummechanicalframework
Thecontinuummechanicalframeworkonwhichthethermo-mechanicalsimulationsareper-
formedisbasedontheestraindeformationtheoryasdescribedindetailintheprevious
chapterSection
2.4.1
.Averybriefhighlightisalsomentionedinhereforthesakeofeasein
readingthroughthesubsequentsectionsdescribingtheconstitutiveequations.
Thedeformationgradientelddescribesthetranslationofmaterialpointsoriginallylocated
at
x
intheundeformedconationtoanewlocation
y
inthecurrent(ordeformed)cu-
51
rationbythedeformationgradienttensorgivenas:
F
Æ
@
x
@
y
.(3.1)
Inthepresentcontext,suchadeformationisbroughtaboutbycontributionofdifferent
processessuchthatthetotaldeformationgradientcanbedecomposed
Meggyes
[
2001
]as
F
Æ
F
e
F
th
F
p
(3.2)
intoalattice-preservinginelasticdeformationgradient
F
p
,alattice-distortinginelasticdefor-
mationgradient
F
th
,andanelasticdeformationgradient
F
e
.
F
p
mapstheundeformedcon-
urationtothefiplasticflguration.
F
th
accountsforthethermalexpansionandmapsthe
plastictothefiintermediateeigenstrainflcuration.Lastly,thisintermediateeigenstrain
conationismappedtothenalfideformedflconationby
F
e
.
Itisimportanttonotethatforthesimulationsshowninthischaptertheintermediateeigen-
strainconationisaccountedforsolelybythethermalstrain,whileinthenextchapter,
Chapter
4
,thiseigenstrainwillalsoaccountforthekinematicsduetoatomtransportviaan
additivedecomposition.
Thetimeevolutionofbothinelasticdeformationgradients
F
p
and
F
th
isgivenintermsof
theirrespectivevelocitygradients
L
p
and
L
th
followingthewrules

F
p
Æ
L
p
(
M
p
,...)
F
p
(3.3)

F
th
Æ
L
th
(
M
th
,...)
F
th
.(3.4)
Bothvelocitygradientsaredrivenbytheirrespectivework-conjugateMandelstresses
M
p
Æ
(
F
e
F
th
)
T
F
e
F
th
S
¼
S
(3.5)
M
th
Æ
(det
F
th
¡
1
)
F
th
SF
th
T
,(3.6)
where
S
Æ
C
:(
F
e
T
F
e
¡
I
)/2(usingHooke'slaw)isthesecondPiolaŒKirchhoffstress,
C
thefourth-
orderstiffnesstensor,and
I
thesecond-orderidentitytensor.Inaddition,thevelocitygradients
alsodependonthematerialstateandassociatedconstitutiveparameters.
52
Staticequilibriumforthiscoupledsystemofequationsisfoundbyastaggerediteration
schemethatsolves
F
p
under
F
th
andmaterialstate,thenfor
F
th
atstillmaterialstate,
andeventuallyforconsistentmaterialstatewithineachtimeincrementasdescribedpreviously
inSection
2.4.2.1
andcanalsobefoundin
Shanthrajetal.
[
2019
].
3.2.3Constitutivedescription
Theemployedcrystalplasticityframeworknaturallyallowstoincorporatetheanisotropyin
elasticity,plasticity,andthermalexpansion.Thermalexpansionduetoachangeintemper-
ature
T
givesrisetotheeigenstrainvelocitygradient
L
th
Æ

T
®
ii
1
Å
®
ii
(
T
¡
T
0
)
i
Æ
1,2,3,(3.7)
whichisusedinthedescriptionofbody-centeredtetragonal(bct)
¯
-Snandtheisotropicsub-
strate.Thematrixof(anisotropic)thermalexpansioncientsisdenotedby
®
andtheinitial
temperatureby
T
0
.
Theplasticvelocitygradientisadditivelycomposedfromsliprates
L
p
Æ
X
®

°
®
s
®

n
®
,(3.8)
wheretheunitvectors
s
®
and
n
®
indicatetheslipdirectionandslipplanenormalforeachslip
system(indexedby
®
)ofthedifferentslipfamiliesin
¯
-Sn.Theresolvedshearstress
¿
®
Æ
M
p
¢
¡
s
®

n
®
¢
(3.9)
isthedrivingforceforslipatrates

°
®
Æ

°
0
¯
¯
¯
¯
¿
®
g
®
¯
¯
¯
¯
n
sgn
¡
¿
®
¢
,(3.10)
withmaterialparameters

°
0
and
n
.Thedislocationdefectstructureisparameterizedphe-
nomenologically
Peirceetal.
[
1982
],i.e.,sliponeachsystem
®
facesaresistance
g
®
thatevolves
duetosliponallsystems(indexedby
¯
)as

g
®
Æ
X
¯
h
0
¯
¯
¯
¯
¯
1
¡
g
¯
g
1
¯
¯
¯
¯
¯
a
sgn
Ã
1
¡
g
¯
g
1
!
¯
¯
¯

°
¯
¯
¯
¯
,(3.11)
53
withinitialhardeningslope
h
0
,hardeningexponent
a
,andsaturationslipresistance
g
1
Hutchinson
[
1976
].Thisphenomenologicalcrystalplasticitymodelisusedtodescribetheme-
chanicalbehaviorofthetinThesubstrate,inexperiment,istypicallyanorderofmagni-
tudethickerthanthelmandisthereforeassumedisotropicallyelasticwithstiffnessselected
largeenoughsothatonitsfurtherincreasethesystemresponsedidnotchange.Thelayerof
softdilatationalmaterialontopoftheensuresafreesurfaceconditionandusestheconsti-
tutivedescriptionproposedby
MaitiandEisenlohr
[
2018
].Thecomponentsoftheanisotropic
thermalexpansion,elasticstiffnesstensor,andconstitutiveparametersfor
¯
-Sncrystalplastic-
ityareadoptedfrom
Leeetal.
[
2015
]andarelistedinTable
3.1
alongwithcorrespondingvalues
forthesubstrateanddilatationalfiairfl.
3.3Simulationresults
Theselectedconstitutiveparametersresultinthestressrelaxationbehaviorshownin
Fig.
3.2
whenthermallystrainingthebstrateassembly(illustratedinFig.
4.2
).The
solidcoloredlinesrthreedifferentcrystallographictextures,characterizedbythecrys-
tallographicdirection(either
h
100],
h
110],or
h
001])thathasthehighestprobabilityofbeing
normaltothelmsurface.Thesimulatedstressresponsesofatinforallthethreetextures
andwithagrainstructureseemtogenerallyagreecloselywiththeexperimentallymea-
suredrelaxationbehavior
Peietal.
[
2017
]forwhichtheactualmtexturewasnotreported.
Thedashedverticallineat20mininFig.
3.2
indicatesthetimeofmaximumlmstressandis
selectedastheinstancefordataanalysis.SincethediffusionofSnatomsislocalizedonthe
grainboundarynetworkanddrivenbyagradientinhydrostaticstress,thequantityofinterest
isthegrainboundaryhydrostaticstressofmagnitude
p
,locallyobtainedbyaveragingpairsof
voxelsseparatedbyagrainboundary.
54
Figure3.2:Forcedtemperaturechangeoftinonrigidsubstrate(black)andresultingcom-
pressivevonMisesstressinthesimulatedfordifferentlmtextures(coloredlines)
comparedtomeasuredstressevolutiononaSisubstrate(circlesfrom
Peietal.
,
2017
).Pointof
maximumloadisreachedafter20min(atendofheating)andservesasreferenceforallsubse-
quentlyreportedstressvalues.
ChakrabortyandEisenlohr
[
2018
]
3.3.1Spatialvariabilityofgrainboundaryhydrostaticstress
Thespatialvariabilityofthegrainboundaryhydrostaticstressdevelopinginthetinminre-
sponsetothethermomechanicalloadingisshowninFig.
3.3
.Themagnitudeofcompressive
stressrangesfromabout0MPato40MPa(closetothesubstrateandatmaximumload,see
Fig.
3.2
)andexhibitssubstantialheterogeneitythatrstheheterogeneityofgrainmor-
phologyandlatticeorientationinthisgrainpatch.Inthecaseof
h
100]asdominantcrystallo-
graphicdirectionalignedwiththenormal,i.e.thefigreenfltextureinFig.
3.3
,thevariability
ofcompressivehydrostaticgrainboundarystressmarkedlydecreasesfromtheate
interfacetoamorehomogenousrangeof20MPato30MPaatthelmsurface(righttoleft
inFig.
3.3
),duetothereducedmechanicalconstrainttowardsthelmsurface.Additionally,
55
Figure3.3:Spatialvariabilityofgrainboundary(top)andbulkgrain(bottom)hydrostaticstress
p
formaximumload(reachedafter20min,seeFig.
3.2
)illustratedatthreedifferentdepths
throughthe(surfacetosubstratefromlefttoright).Substrateisnotshownwhilepartof
thecolumnargrainstructureofthetinisillustratedandcoloredaccordingtothecrystallo-
graphicdirectionthatisparalleltothesurfacenormal(closeto
h
100]inthisexample,seecolor
codeinstandardstereographictriangle).
ChakrabortyandEisenlohr
[
2018
]
withinthelimitedspatialextentofthepolycrystallinestructurediscretizedinthisinvestiga-
tion,thereisnoindicationofcorrelatedlong-rangestressgradients(spanningmultiplegrains),
whichisconsistentwithreportedexperimentalobservations
Peietal.
[
2016
].
3.3.2Effectoftexture
¯
-Snexhibitsstronganisotropy,therefore,crystallographictextureislikelyaantin
enceinthedevelopmentofstressmagnitudeanditsheterogeneity.Thedistributionsof
hydrostaticpressure
p
asfunctionoftexture(green,blue,andred)anddepthwithinthe
areplottedinFig.
3.4
atthesamethreedepthsasshowninFig.
3.3
.Similartothecase
ofagreenbertexturethatisspatiallymappedinFig.
3.3
,bothothertextures(blueand
red)showadecreaseintherangeof
p
fromtherateinterfacetothesurface(dark
tolightcolorshadesinFig.
3.4
).Theoverallwidthofthehydrostaticstressdistributionsisno-
56
Figure3.4:Distributionsofgrainboundaryhydrostaticstressatthreedifferentdepthswithintin
ofeithergreen,blue,orredbertexture(
h
100],
h
110],and
h
001],fromlefttoright).Light,
medium,anddarklineshadescorrespondtothelayerdepthsshowninFig.
3.3
left,center,and
right,i.e.,atthesurface,middle,andsubstrateinterface.Inversepoleuremapsofm
normalandonein-planedirectionillustratethethreedifferenttexturesassignedtootherwise
identicalgrainstructure.
ChakrabortyandEisenlohr
[
2018
]
tablywideratalldepthsinthethathaveeithergreenorbluetextureincomparison
tothewitharedbertexture.Thisstarkcontrastbetweengreen/bluecomparedtothered
texturemaybeconnectedtothefactthatonlytheredbertexturehasatransverselyisotropic
thermalexpansion.
Toanalyzetherelativeinencesofthermalandmechanicalanisotropy,theactualther-
malexpansionanisotropyof
¯
-Snischangedfrom40
£
10
¡
6
K
¡
1
Æ
®
[001]
6Æ
®
[010]
Æ
®
[100]
Æ
20
£
10
¡
6
K
¡
1
toanisotropiccasewith
®
[001]
Æ
®
[010]
Æ
®
[100]
Æ
20
£
10
¡
6
K
¡
1
.The
p
distribu-
tionsresultingforthesetwocasesinofgrainstructureandwiththethreedifferent
57
Figure3.5:Comparisonofgrainboundaryhydrostaticstressdistributionsbetweenwith
anisotropicandisotropicthermalexpansion(solidanddashed)havingeither
h
100],
h
110],or
h
001]bertexture(green,blue,andredfromlefttoright).Hydrostaticstressmapsatthesub-
strateinterfacerevealnoqualitativechangeinthehighdegreeofspatialvariability(leftand
righthalves,bottomrow).
ChakrabortyandEisenlohr
[
2018
]
textures
h
100],
h
110],and
h
001]areshowninFig.
3.5
.Thereisalmostnochangeinthe
stressdistributionfortheredtexture(Fig.
3.5
right),whichremainsverynarrow.Incon-
trast,thestressdistributionsforboththegreenandbluetexturesreduceappreciablyinwidth
andslightlyshifttolowercompressivestressvalues.Thelattereffectbeingcausedbythere-
ducedeffectivemagnitudeofthermalexpansionwhenlowering
®
[001]
from40to20
£
10
¡
6
K
¡
1
.
Theobservedhighdegreeofspatialvariabilityin
p
doesnotqualitativelychangeforanyofthe
threetexturesevenafterchangingtoisotropicthermalexpansion(Fig.
3.5
bottom).Conse-
quently,thestillappreciabledifferenceinthespreadofhydrostaticstressonthegrainbound-
arynetworkforthegreen/bluebertexturecomparedtotheredonehastobeattributedtothe
mechanical(elasticandplastic)anisotropyof
¯
-Sn.
58
Figure3.6:Variationofgrainareadistribution(left)forthreedifferentstructures(labelled
fiafl,fibfl,andficfl)andtheirassociateddistributionsofhydrostaticstressonthegrainboundary
network(middle)forhaving
h
100]texture(seeinversepolegurecoloringofpartly
showngrainstructure).Spatialmaps(rightfia"tofic")showthelmlayerdirectlyatopthe
(invisible)substrateandillustratetherapidoscillationof
p
consistentlyobservedforeachof
thethreestructures.
ChakrabortyandEisenlohr
[
2018
]
3.3.3Effectofgrainsizedistribution
Afurtherpotentialeonthe
p
distributionisthegrainsizedistribution.Toanalyze
itsrole,threedifferentgrainstructuresarecompared:thegrainstructurementioned
before(herelabelledfiafl),aversionofitafterbeingsubjectedtoarticialgraingrowthandre-
sultinginawiderdistribution(labelledfibfl),andanexperimentallymeasuredpatch(la-
belledficfl)
Jagtapetal.
[
2017
],allhavingaboutahundredgrains.Allthreegrainstructuresare
againsubjectedtothethermalboundaryconditionsshowninFig.
3.2
(blackcurve).Despite
thesignicantvariationintheirgrainsizedistribution(Fig.
3.6
left),the
p
distributionresulting
foreachmstructureremainsessentiallyidentical(Fig.
3.6
right).WhilenotshowninFig.
3.6
,
theobservedinvariancein
p
despitesigniantchangesingrainsizedistributionissimilarly
obtainedformswith
h
110]and
h
001](blueandred)textures.Apparently,theuence
ofgrainmorphologyonthegrainboundaryhydrostaticstressdistributionisminutecompared
tothatofthetexture.
59
3.3.4Effectofobliquesurfacegrains
Focusedionbeampreparedcross-sectionalviewsoftinwhiskersrevealthattheygrowfrom
obliquegrains,i.e.,grainsthatfeatureinclinedgrainboundaries.Suchashapeisgeometrically
necessarywhenwhiskergrowthsisbeingfacilitatedbyaccretionofatomsattheseboundaries.
However,itisnotclearwhetherthepresenceofsuchagraingeometryinitselfentailsalocally
relativelylowcompressivestressstatefavorableforinitialatomadditiontotherootofa
po-
tential
whiskergrain.Togaugehowstronglythesheerpresenceofobliquegrainsaltersthe
p
distributionintheand,inparticular,alongthegrainboundarynetwork,one(orseveral)
randomlyselectedgrain(s)inthecolumnar(subscriptficolfl)arechangedtoanobliquege-
ometry.Theresultingchange
p
¡
p
col
intheoverallaswellasthespatialdistributionofgrain
boundaryhydrostaticstressisanalyzedfortheexemplarycaseof
h
100](green)lmtex-
ture.
Figure
3.7
(left)presentsthedistributionofchangesinhydrostaticstressontheentiregrain
boundarynetworkforthreecaseswithanisolatedtransformationofonegrain(fiafltoficfl)and
oneadditionalcasewithmultipleobliquegrains(fidfl).Theobservedchangeforthemajorityof
affectedlocationsislimitedtoabout5MPa(correspondingtoarelativechangeofabout10%).
Thischangeisstronglylocalizedtotheimmediatevicinityoftheaffectedgrain(s),asexempli-
bythespatialmapsofthechangein
p
col
inthelayerclosesttothesubstrate(Fig.
3.7
bottom).Thelimitedrangeofinence,whichextendsonlytotheboundariesofthedirectly
neighboringgrains,isevenmoreapparentinthemapsofrelativestresschanges(Fig.
3.7
top).
60
Figure3.7:Distributionofchangeinhydrostaticstress
p
acrossthegrainboundarynetworkduetotransformationofone(fiafltoficfl)
ormultiple(fidfl)columnargrainsintoobliqueones.Thesegeometricalalterationsaffectonlyasmallfractionofthegrainboundary
network,asdemonstratedbythenarrowoveralldistributions(left).Spatialmapsmthelocalizeduenceofobliquegrains
ontherelative(top)andabsolute(bottom)changein
p
forthelayerclosesttothesubstrate(termedfibottomflinFig.
3.3
)with
thetransformed(formerlycolumnar)grainshowninorange.
ChakrabortyandEisenlohr
[
2018
]
61
3.3.5Effectof
¯
-Sncrystalanisotropy
¯
-Sncrystalishighlyanisotropicduetoitsbody-centeredtetragonalcrystalstructure.The
anisotropyisexhibitedbothinmechanical(elasticandplastic)aswellasthermalanisotropy,
andhencewouldhaveasignicantimpactonthestressevolutioninthelm.Theeffectofcrys-
talanisotropyisinvestigatedbycomparingsimulatedgrainboundary
p
distributionsobtained
uponmakingthe
¯
-Sncrystalisotropicinitsthermalbehavioronly(leftcolumninFig.
3.8
),
elasticbehavioronly(middlecolumninFig.
3.8
),andbothelasticandthermalbehavior(right
columninFig.
3.8
),andcomparingeachofthemwiththereferencecasewherethecrystalis
havingbothmechanicalandthermalanisotropies(seeFig.
3.8
).ItappearsfromFig.
3.8
(com-
paringleftandmiddlecolumns)thatboththethermalandelasticanisotropyof
¯
-Snishaving
equallystrongeonthe
p
distributionalongthegrainboundarynetwork.Moreover,by
comparingthedarkercurves(labelled`IV')intherightcolumninFig.
3.8
,whereonlytheplastic
anisotropyof
¯
-Snisinncingthe
p
distribution,thedominantroleof
¯
-Snplasticitycanbe
inferred,andthiseffectappearstobedifferentfordifferenttextures.
Thisstudyhighlightstheinadequaciesofsimplisticisotropicmodelsbeingpreviouslyused
todescribethematerialbehaviorof
¯
-Sn,aswellasmotivatesustoidentifystrategiestoestab-
lishaccurateplasticparametersof
¯
-Sn.
3.3.6Establishmentoflocalcrystallographiceffect
Inthissectionthefilocalizedeffect"oftextureingoverninghydrostaticstressalongthegrain
boundarynetworkisstudiedbyconsideringanexemplarytextured
h
100],asshownintop
rowinFig.
3.9
.Theaimofthisinvestigationistoverifywhetherasubpartofentiregeometry
couldbestudiedtogetanideaaboutthedifferentstresses(shearandnormal)generatedalong
thegrainboundarystatisticallybychanginggrainmorphologyandorientationneighborhood
butkeepingtheglobaltexturesame.Hence,itismorelogicaltoperformsuchanalysisfora
referencegraininthelmandthenvaryingit'sneighborhoodafteracertaingrainsizeseach
timeandanalyzethechangesintheresultinghydrostaticstressdistributionalongthegrain
62
Figure3.8:Distributionofhydrostaticstress
p
acrossthegrainboundarynetwork(andatmax-
imumload)forthreedifferentgloballmtextures,toprow:
h
100];middlerow:
h
110];and
bottomrow:
h
001].Ineachoftheplots,lightercurveslabelled`I'correspondstothe
p
distri-
butionforasimulationhavingbothmechanicalandthermalanisotropywhichactasarefer-
encedistribution.Thisreferencedistributioniscomparedtothe
p
distributionsobtainedupon
making
¯
-Snthermallyisotropicbutmechanicallyanisotropic(darkercurves,labelled`II',in
leftcolumn);thermallyanisotropicbutelasticallyisotropic(darkercurves,labelled`III',inmid-
dlecolumn);andboththermallyandelasticallyisotropic(darkercurves,labelled`IV',inright
column),i.e.,keepingonlyplasticanisotropy.
63
Figure3.9:Toprowshowsthereferencetexture(right:outofplane;left:in-plane)andgrain
morphologyforthe
h
100]lmusedearlier.Inthesubsequentimagesthetextureandgrain
morphologyofthepartinsidetheblackcircleremainsunchangedwhileoutsidethegrainsare
shuedalongwithagofthetexture.Toptobottomshowsagradualdecreaseinthe
areaofunchangedpart(radiusoftheblackcircle)fromabout5grainsizes(secondrow)to
¼
oneadjacentgrainfromthereference(central)grain.
64
Figure3.10:Distancedistributionoftheunchangedpart,approximatedistancesbetweenun-
changedvoxelsandthecentralvoxeloftheblackcirclesinFig.
3.9
withfia",fib",fic",andfid"
representingrows2to5.
boundarynetworkforsuchachange.Rows2-5inFig.
3.9
illustratesthisexample,wherethe
unchangedareawithrespectthecentralreferencegrain(centralgrainintheblackcircle)is
graduallydecreased,seeFig.
3.9
forthedistancedistributionoftheunchangedpart.
Subsequently,thedifferencebetweenthehydrostaticstressesalongthegrainboundarynet-
workascomparedatmaximumloadforthedifferentcases(rows2to5inFig.
3.9
)tothatofthe
reference(row1inFig.
3.9
)isshowninFig.
3.11
forthesubstrateinterface(left)andatthe
surface(right).
AsevidentfromFig.
3.11
presenceofdifferenceinstressesfromthereferencecasecorrelates
wellwiththedemarcationbetweenthechangedandtheunchangedpart(blackverticalline
inFig.
3.11
).Thus,ifthegrainofinterestisthecentralgrainintheblackcirclesinFig.
3.9
thenitsstressalongthegrainboundarynetworkleadingtoitdependsontheextremelylocal
crystallographicneighborhood,andhenceasubpartoftheentiregeometry(leadingtooneor
twograindiameters)couldbeusedtostudytheinenceofcrystallographicneighborhoodon
thestresses.
65
Figure3.11:Differencesinhydrostaticstressesalongthegrainboundarynetworkforthediffer-
entradiiofunchangedtextureandgrainmorphology(rows2to5inFig.
3.9
)withthatofthe
referencecase(row1inFig.
3.9
)atthem-substrateinterface(left)andlmsurface(right).
Thisisa2-Dhistogramplotwheretheshadeofgrayrepresentsthevaluemeaningdarkerblack
circlescorrespondtozerochangeinthestressvalues(y-axis),andthelocationofthecircles
indicatethedistancefromthecentralgrain'svoxel.Theblackline,ineachcase,indicatesthe
approximatedistancebetweenthechangedandunchangedpartinthelmfromthecentral
grain(radiusoftheblackcircleinFig.
3.9
)
66
3.4Discussion
Hydrostaticstressaltersthechemicalpotentialofatomsand,thus,aspatialvariationin
p
willcauseadrivingforceforatommigrationfromareasofhighpressuretothoseoflowpres-
sure,resultinginanoppositexofvacancies.Potentiallocationsforwhiskerformationare
thenassociatedwithareasoflowhydrostaticstressandwouldattractatomsfromareasexpe-
riencingmorecompressivestress.Anoverallwiderrangeof
p
isintensifyingthisimbalance
ofchemicalpotential,whichmightbeattributedtoahigherpropensityforwhiskerformation
(everythingelseremainingunchanged).Underthispremise,atinwith
h
001](red)er
textureshouldexhibitalesserdegreeofwhiskerformationthanlmshavingeither
h
110]or
h
100](blueorgreen)texture,sincethesimulationresultsinthisstudyrevealtheformer
texturetoresultinamuchnarrower
p
distributioncomparedtothelattertwo.Inenceof
texturecouldthenbeonepossibleexplanationforthecontradictingobservationsreportedin
literatureaboutthepropensityofwhiskerformationfortinmsonbrasssubstrates.Forin-
stance,withnominallysimilarbrasssubstratecompositions,
Steinetal.
[
2015a
]reportedlower
whiskerpropensityfora
h
001]while
Jagtapetal.
[
2018
]foundhigherpropensitybutona
h
100]lm.Hence,amoredirectexperimentalvericationofthehypothesisthatthewidthof
thehydrostaticstressdistributionintinlmsisofprimeimportancetounderstandingwhisker
formationwouldrequirethermally-strainedtinlmsvaryingonlyintheirtexture.
Thechangesinthedistributionof
p
resultingfromthearticialcaseofisotropicthermal
expansionshowninFig.
3.5
indicatethatthe(unaltered)mechanicalanisotropyplaysatleast
asimportantaroleasthethermalanisotropyincausinghydrostaticstressthatdevelopsinther-
mallystressed
¯
-Sn.Hence,foraccuratepredictionsof
¯
-Snbehavioritisquintessen-
tialtoincludethe
¯
-Sncrystalanisotropyinthemodelingframework.
AccordingtothedingsreportedinFig.
3.6
andFig.
3.7
,neitherthegrainsizedistribu-
tionnorthereplacementofcolumnarbyobliquegraingeometrydrasticallyaltersthelocaland
global
p
distribution.Thus,theobliqueshapeinitselfcanonlybea
necessary
conditionfor
agraintogrowawhisker
HeandIvey
[
2015
],
Galyon
[
2011
],
Smetana
[
2007
]but
cannotbea
67
sucientcondition
sincethisshapedoesnotintrinsicallygeneratefavorablehydrostaticstress
gradientsforinxofSnatomsthatwouldberequiredforwhiskergrowth.
Basedonthestudyinvestigatingtheroleofcrystalanisotropy,Fig.
3.8
,itappearsthatboth
mechanicalandthermalanisotropiesareequallyuencingthe
p
distributionalongthegrain
boundarynetwork,andhenceneedstobeaccountedfortohaveaccuratepredictions.
Thepresentworkinvestigatedtheuenceofcrystallographictexture,grainsizedistribu-
tion,andpresenceofobliquegrainsonthedevelopmentofhydrostaticstress
p
inthermally
strainedcolumnartinonarigidsubstratewhileexcludinganyvacancydiffusion.The
maingscanbesummarizedasfollows.
Ł
Tinlmswith
h
001]textureshowacomparablysmallspreadinoverallhydrostaticstress.
Incontrast,fortheothertwoinvestigatedtextures(
h
100]and
h
110])largeoverallstress
variabilityisobserved.Hence,
h
100]lmsareexpectedtobemorepronetowhisker
growthcomparedto
h
001].
Ł
Forallthreeertextures,nosmoothlong-rangestressgradientsareobserved,as
p
spa-
tiallyoscillatesoverdistancescomparabletothegrainsize,i.e.,
p
isessentiallyuncorre-
latedafteradistanceofonetotwograins.
Ł
Neithergrainsizedistributionnorthepresenceofobliquesurfacegrainsnotablyalters
the
p
distribution.
Ł
¯
-Sncrystalanisotropyhasasigntroleinthestressevolutionintheandhence
simplisticisotropicmodelswouldbeinadequateinprovidinghighdelitysolutions.
Ł
Alocalsubstructurecouldbeusedtostudytheinenceofcrystallographicneighbor-
hoodontheresultinggrainboundarystresses,andhencethesubsequentatomredistri-
bution.
Theresultsareconsistentwiththefollowinghypothesizedchainofeventsleadingtowhisker
formation:Thewhiskernucleationsitewillbealocationthatexhibitslowcompressivestress
68
relativetoitsimmediategrainneighborhood.Theoccurrenceofsuchneighborhoodsisex-
pectedtobepredominantlydeterminedbythecrystallographicorientationdistributionofthe
Duetothelocallyhighchemicalpotentialofvacancies,atomswillmigratetothislocation
alongthegrainboundarynetwork.Providedthatthegraininquestionisanobliquesurface
grain,theuxofatoms(i)increasestheshearforcesalongitssurfacegrainboundarytraces
and(ii)decreasestheimbalanceofchemicalpotential,whichreducesfurtherinx.Ifthepres-
surethatbuildsupduetotheinwingatomsexceedstheresistingforceofanoxidelayeror
othersurfacecoating
before
theequilibrationofchemicalpotentialshutsdownfurtheratom
acircumferentialcrackinthesurfacecoverdevelops.Surfacecrackingremovestheformer
kinematicconstraintontheinclinedgrainboundariessuchthatthechemicalpotentialofva-
canciesattheboundariesofthisgrainbecomeshigherthananyoftheremaininggrainsinthe
excludingotherfiwhiskeringflgrainsinsimilarlyrelaxedconditions.Thewhiskercanthen
growfromthisgrainbydrainingatomsfromanowprogressivelyenlargingareaofin.
Suchgrowthcancontinueaslongasthehydrostaticstressinthecollectionvolumeexceedsthe
energyrequiredtocreatefreshwhiskersurface(basedonabalanceofsurfaceandlatticestrain
energy[
Choietal.
,
2003
],ahydrostaticstressofabout1MPaissucienttosupportawhisker
of2
&
mdiameter).
69
Table3.1:Materialpropertiesfordilatationalair,body-centeredtetragonal
¯
-Sn,andisotropic
substrate.
ValueUnit
Air
C
11
10.0GPa
C
12
0.0GPa
g
0
:
g
1
0.333:0.667MPa
h
0
1.0MPa
a
2
M
3
n
20

°
0
10
¡
3
s
¡
1
¯
-Sn
C
11
72.3GPa
C
12
59.4GPa
C
13
35.8GPa
C
33
88.4GPa
C
44
22.0GPa
C
66
24.0GPa
g
0
:
g
1
{100)
h
001]8.5:11.0MPa
g
0
:
g
1
{110)
h
001]4.3:9.0MPa
g
0
:
g
1
{100)
h
010]10.4:11.0MPa
g
0
:
g
1
{110)
h
1
¯
11]4.5:9.0MPa
g
0
:
g
1
{110)
h
1
¯
10]5.6:10.0MPa
g
0
:
g
1
{100)
h
011]5.1:10.0MPa
g
0
:
g
1
{001)
h
010]7.4:10.0MPa
g
0
:
g
1
{001)
h
110]15.0:10.0MPa
g
0
:
g
1
{011)
h
0
¯
11]6.6:9.0MPa
g
0
:
g
1
{211)
h
0
¯
11]12.0:13.0MPa
h
0
20MPa
a
2.0
n
6

°
0
2.6
£
10
¡
8
s
¡
1
®
h
100]
20
£
10
¡
6
K
¡
1
®
h
001]
40
£
10
¡
6
K
¡
1
Substrate
C
11
270.0GPa
C
12
90.0GPa
®
h
100
i
1
£
10
¡
6
K
¡
1
70
CHAPTER4
FULLYCOUPLEDCHEMO-THERMO-MECHANICALMODELINGOFTHERMALLY
STRAINED
¯
¡
SnFILMS
4.1Introduction
Inthischapter,fullycoupledchemo-thermo-mechanicalsimulationsoftinareper-
formedtoanalyzethestress-drivendiffusionkineticsintheseprecedingwhiskernucle-
ationandgrowth,aswellastounderstandtheextentoftheirkinematicconsequence.Inthis
regard,thetransportismostlikelytooccursolelyalongthegrainboundarynetworksincethe
grainboundarydiffusivityofSnisordersofmagnitudehigherthantheSnbulkdiffusivity[
Jag-
tapetal.
,
2017
].Likeinthepreviouschaptershowingthermo-mechanicalsimulations,Chap-
ter
3
,inthischapteraswell,thecoupledsimulationsmimictheexperimentalconditionsof
Pei
etal.
[
2017
]whereatindepositedonarigidsiliconsubstrateisunderbiaxialcompressive
stresswhenthesystemisheated(withaheatrate),duetothedifferencesinthethermal
expansionientofthelmandthesubstrate.Thechoiceofmimickingsuchasimula-
tionisbecauseofthelargedegreeofcontrolontheappliedstressonthesystem,ascompared
toothersourcesofstresssuchasformationofCu
6
Sn
5
intermetalliccompounds,indentation
[
YangandLi
,
2008
],orbending[
Crandalletal.
,
2011
].
ThesimulationsareperformedusingtheopensourcesimulationsoftwareDAMASK
whereintheeffectofcrystalanisotropy(elastic,plastic,andthermal)couldeasilybeincorpo-
ratedtoaccountforthelmtextureeffects.Withsuccessfulfuldthree-dimensionalfully
coupledsimulations,performedherein,severalcomplexprocessescausingaparticulargrain
tonucleateawhisker(andsubsequentlygrow)canbede-convolutedandinvestigatedsequen-
tiallytodeterminethemostcriticalfactorsandhenceallowustoproposereliablemitigation
strategiestoavoidwhiskers.ThenumericalframeworkandthevariousaspectsofDAMASK,
suchasformulationofresidualsforequations,calculationofspatialgradientsusinga
71
forward-backwardscheme,andsolutionoftheboundaryvalueproblemusingtheinherent
spectralbasedFFTsolver,havealreadybeenoutlinedinrelativedetailinSection
2.4
andcan
beobtainedinfurtherdetailin[
Rotersetal.
,
2019
,
Eisenlohretal.
,
2014
].However,forthe
currentfullycoupledsimulationstheresidualformulationforthethermalandstress-driven
transportequationshavebeenrevisitedfromtheiroriginalformulation[
Shanthrajetal.
,
2019
,
Svendsenetal.
,
2017
]tobetteraccountforthestarkphasecontrastinthediffusivityofthe
transportedspeciesbetweenthegrainboundariesandthegrainbulk.
Thechapterisstructuredasfollows:inSection
4.2
,rsttheconstitutiveequationsforthe
coupledchemo-thermo-mechanicalmodelisoutlinedinSection
4.2.1
,followedbythenumeri-
calmethodtoformulatetheresidualforthethermalandchemicaldequationsSection
4.2.2
.
SubsequentlyinSection
4.3
thedetailsofsimulationgeometryandmeanstoextractrelevant
informationisprovidedwhichisfollowedbytheresultsanddiscussionSection
4.4
.
4.2Modeldevelopment
ThecontinuummechanicalsolutionframeworkofDAMASKisformulatedbasedonthe
ite-strainframeworkinvolvingamultiplicativedecompositionofthetotaldeformationgra-
dient
F
Æ
F
e
F
in
F
p
(4.1)
intoalattice-preservingplasticdeformationgradient
F
p
thatmapstotheplasticcuration,
alattice-distortinginelasticdeformationgradient
F
in
mappingfurthertotheeigenstraincon-
uration,andanelasticdeformationgradient
F
e
thatmapstheeigenstrainconationto
thenaldeformedguration.AsexplainedinChapter
2
,theintermediateeigenstrain(or
stress-freestrain)curationsaccountforboththethermalstrainduetodifferenceinthe
thermalexpansionientsaswellasthekinematicsmodulatedbythetransportofatoms.
Thetimeevolutionoftheplasticandintermediatedeformationgradients
F
p
and
F
in
isgivenin
72
termsoftheirrespectivevelocitygradients
L
p
and
L
in
followingthewrules

F
p
Æ
L
p
(
M
p
,...)
F
p
(4.2)

F
in
Æ
L
in
(
M
in
,...)
F
in
.(4.3)
Bothvelocitygradientsaredrivenbytherespectivework-conjugateMandelstresses
M
p
Æ
(
F
e
F
in
)
T
F
e
F
in
S
¼
F
in
T
F
in
S
(4.4)
M
in
Æ
(det
F
in
¡
1
)
F
in
SF
in
T
,(4.5)
where
S
Æ
C
:(
F
e
T
F
e
¡
I
)/2isthesecondPiolaŒKirchhoffstresswithfourth-orderstiffnessten-
sor
C
andsecond-orderidentitytensor
I
,andfurtherdependonthematerialstateandasso-
ciatedconstitutiveparameters.Theintermediateplasticvelocitygradient
L
in
isadditivelyde-
composedintoitsconstituents,whichinthisstudyarethethermalvelocitygradient
L
th
and
chemicalvelocitygradient
L
ch
L
in
Æ
L
th
Å
L
ch
(4.6)
andhavetheirindividualconstitutiveequationsmentionedinthenextsection.
Theinterdependencyofthevelocitygradientsmakesthesystemfullycoupledandissolved
basedonthestaggeredsolutionstrategyoutlinedinSection
2.4
andexplainedindetailin[
Shan-
thrajetal.
,
2019
].Theboundaryvalueproblemissolvedusingthefibasic"schemeofthespec-
tralbasedFFTformulationalsooutlinedinSection
2.4
.
4.2.1Constitutiveequations
SimilartothepreviousChapter
3
,theemployedcrystalplasticityframeworkincorporatesthe
anisotropyinelasticity,plasticity,andthermalexpansionaswellasdifferentbulkandgrain
boundarydiffusivity.Thegoverningequationsdictatingthekinematicsaregivenbytheircor-
respondingvelocitygradients.Theplasticvelocitygradientisadditivelycomposedfromslip
rates
L
p
Æ
X
®

°
®
s
®

n
®
,(4.7)
73
withtheunitvectors
s
®
and
n
®
indicatingtheslipdirectionandslipplanenormalforeachslip
system(indexedby
®
)ofthedifferentslipfamiliesin
¯
-Sn.Theresolvedshearstress
¿
®
Æ
M
p
¢
¡
s
®

n
®
¢
(4.8)
drivestheslipatrates

°
®
Æ

°
0
¯
¯
¯
¯
¿
®
g
®
¯
¯
¯
¯
n
sgn
¡
¿
®
¢
,(4.9)
withmaterialparameters

°
0
and
n
.Thephenomenologicalplasticitylawof[
Peirceetal.
,
1982
]
isusedinthisstudytodescribethemechanicalbehaviorofthewheredislocationslipon
eachslipsystemisafunctionofacriticalresistance
g
®
,whichevolvesduringslipas

g
®
Æ
X
¯
h
0
¯
¯
¯
¯
¯
1
¡
g
¯
g
1
¯
¯
¯
¯
¯
a
sgn
Ã
1
¡
g
¯
g
1
!
¯
¯
¯

°
¯
¯
¯
¯
,(4.10)
withinitialhardeningslope
h
0
,hardeningexponent
a
,andsaturationslipresistance
g
1
[
Hutchinson
,
1976
].
Thethermalvelocitygradient(accountingforthethermalstraininthesystem)
L
th
Æ

T
®
ii
1
Å
®
ii
(
T
¡
T
0
)
i
Æ
1,2,3,(4.11)
isconstitutivelyrelatedtothethermalexpansionandthechangeintemperature
T
forboththe
body-centeredtetragonal(bct)
¯
-Snandtherigidisotropicsubstrate,
®
beingthecoecient
matrixofthermalexpansion,and
T
0
indicatestheinitialtemperature.Theconstitutiveequa-
tionresponsibleforthetemperaturechangeisgivenby:
½
Cp
@
T
@
t
Æ¡r¢
f
T
Å

q
,(4.12)
where

q
representstherateatwhichthebstratesystemisheated.
f
T
istheheatuxand
½
and
Cp
representingthedensityandspeciheatofthesystem.
Transportofanyspeciesisdeterminedbythedifferencesintheir,whichdependson
theirchemicalpotential.Ingeneral,thechemicalpotentialhasaconcentrationtermandan
energyterm,chapter2in
PorterandEasterling
[
1992
].However,inthecurrentformulation
74
wherethediffusivewisconsideredonlyalongthegrainboundaries,theconcentrationterm
intheequationisneglectedmakingthethechemicalpotentialasolefunctionofthegrain
boundarynormalstress.
Thusthexequationforsuchasystemthatisrelativelydilute(suchthatthethermody-
namicfactorrelatingthediffusivityandthemobilitycanbeneglected)isthengivenas:
J
Æ¡
DC
kT
r
¹
(4.13)
with
D
beingthediffusivityofthespecies(incurrentstudyitsvaluecorrespondstotingrain
boundarydiffusivity),
C
representstheconcentrationofthediffusingspecies,
r
isthespatial
gradientoperator,and
k
and
T
representstheBoltzmannconstantandtemperature.Thechem-
icalpotential
¹
isafunctionofatomicvolume

andgrainboundarynormalstress[
Buchovecky
etal.
,
2009b
],andisgivenas
¹
Æ¡

¾
n
,(4.14)
¾
n
Æ
¾
¢
(
n

n
)(4.15)
where
n
representsthegrainboundarynormal.Withsinglediffusingspeciestherighthand
side
C
inEq.(
4.13
)isinverseoftheatomicvolume,
C
Æ
1/

[
Tuetal.
,
2007
].Theatom
duetothein-planegradientinchemicalpotential,relevantforthepresentstudyconcerning
grain-boundarydiffusion,isthengivenby
J
Æ¡
DC
kT
(
I
¡
n

n
)
r
¹
(4.16)
Æ¡
D
kT
(
I
¡
n

n
)
r
¾
:(
n

n
)(4.17)
wherethediffusionalongthegrainboundarynormaldirectioniseliminatedbythemodied
diffusivityexpression,
D
(
I
¡
n

n
)
.However,inthepresentstudythegrainboundary,normal
isnotyetincorporatedinthemodelleadingtotheapproximationoftheaboveEq.(
4.17
)by
J
Æ¡
D
kT
(
I
¡
n

n
)
r
¾
:(
n

n
)(4.18)
¼¡
D
kT
r
¾
h
(4.19)
75
with
¾
h
Æ
1
3
Tr
M
in
(4.20)
Sincetheuxislimitedtothethickness
±
GB
ofthegrainboundary,thexperlinearcross-
sectiongrainboundary(i.e.,forawidthof
±
GB
andperlengthalongtheotherareadirection)
followsas
J
GB
Æ¡
D
GB
±
GB
kT
r
¾
h
(4.21)
Thespatialgradientinthegrainboundary
J
GB
entailsarateofchange

N
Ær¢
J
GB
(4.22)
ofatomsonthegrainboundaryarea,whichtranslatesintoarateofmonolayeraccumulation
(orloss)givenby:

m
Æ

1/3
r¢
³

1/3
J
GB
´
,(4.23)
where
¡

1/3
J
GB
¢
correspondstotheforanatomiclength,anditsdivergenceforanatomic
distance

1/3
givesthemonolayeratomaddition.Theabovetransportofatomsduetothe
accumulationEq.(
4.23
)isschematicallyshownforagrainboundaryvoxelinthesimulation
geometryinFig.
4.1
,wherethevector
J
GB
representsthegrainboundaryforagrain
boundaryofwidth
±
GB
perunitlengthalongtheotherareadirection(correspondingtoz-
directioninFig.
4.1
).
Finally,thekinematicconsequence(chemicalstrain)permonolayerisexpressedas
L
ch
Æ
²

m

1/3
d
(
n

n
)
|
{z
}
¼
I
(4.24)
where
²
isthestrainientwhichisnegative(reducingstrainuponaddition)forvacancy
transportandpositive(increasingstrainuponaddition)foratomtransport.
Thisfullycoupledsystemofpartialdifferentialequationsissolvednumericallyusingastag-
geredsolutionscheme,whiletheindividualconstitutiveequationsaresolvedbyformulatinga
residual,andminimizingitusingtheopensourcelibraryPETSctogettherequisitesolution.
76
Figure4.1:Arepresentativevoxelwithamaterialpointshowingthedirection(alongx-axis)
andgrainboundarynormaldirection,
n
,alongy-axis.Thegrainboundaryuxperunitline
crosssectionalongz-direction,foranatomicdistanceisgivenby(

1/3
J
GB
).Subsequently,
therateofmonolayeradditionisrepresentedby

m
whichcorrespondstothedivergenceofthis
throughanotheratomicdistanceof

1/3
.
4.2.2Residualformulation
Thesolutionofthetransportequationforstress-drivenmasstransferasshownbyequations
Eqs.(
4.21
)and(
4.22
)thatgivesthenumberofatomsaccumulatedperunitheightofthem
perunittimeissolvedbyformulatingasaresidualminimizationproblem.Theadvantageof
formulatingsucharesidualistheabilitytouseoneoftheseveralin-builtnumericalmethods
intheopensourcepackagePETSctosolvethisresidualinthemostientway.Thecritical
aspectofsolvingtheequationasad(andnotaconstantoutofthedivergencetermin
Eq.(
4.23
))isduetothepresenceofthestrongcontrastinthediffusivityoftheSnatomsinthe
bulkcomparedtothegrainboundaries.Itisnotedthatthegrainboundariesinthecurrent
simulationarerepresentedby
¼
2voxelsfromeachofthegrainsnexttotheboundary(similar
77
tothegeometryusedinChapter
3
).Thus,fortheinterfacebetweentwograins,theboundary
comprisesofatotalof
¼
4voxels,2fromeachgrainwiththeorientationoftheirparentgrain.
Toproperlyaccountforthiscontrastinbulkandgrainboundarydiffusivitiesforeachma-
terialpointtheiscalculatedusingtheminimumdiffusivitybetweenitselfanditsneighbor.
Thisisjusinthesensethatifanatomistryingtodiffusefromthebulktotheboundary
itwouldnotbeabletodiffuseduetotheverylowbulkdiffusivity,whileifanatomistryingto
diffusefromtheboundarytothebulkitwillalsoberestrictedsinceitcannotdisplaceanatom
inthebulk(againduetolowbulkdiffusivity).Thesituationissimilartoaseriesofchemical
reactionswheretheratecontrollingstepistheslowestreaction.
4.3Simulationdetails
4.3.1Simulationgeometryandboundaryconditions
Areducedmicrostructurewith
¼
30grainsfromtheoriginalmicrostructureconsistingof
¼
100
grains(whichwasusedforthethermo-mechanicalanalysisinChapter
3
)isusedasatestcaseto
analyzethekinematicconsequenceofthefullycoupledchemo-thermo-mechanicalprocesses.
Moreover,oneofthe30grainsinthemicrostructureismadefioblique"(inclinedgrainbound-
aries)andisconsideredasthereferencegrainforperforminganygraincomparisonstudy.The
reducedgeometryisrepresentativeofthecrystallographiceffectsforthisreferencegrain,since
theneighborhoodeffectsarehighlylocalized(asestablishedinSection
3.3.6
).
Thereducedlayeredgeometryconsistsofa3.0
&
mthinwith30grainsofbody-centered
tetragonal(bct)
¯
-Sn(generatedusingPoisson-Voronoitessellationwith30seedpoints)be-
tweena1.2
&
mthickelasticrigidisotropicsubstrateatthebottomanda0.6
&
msoftdilatational
materialontopasshowninFig.
4.2
.Thislayerofsoftdilatationalmaterialontopofthem
ensuresafreesurfacecondition,thusmimickingfiair",andusestheconstitutivedescription
proposedby
MaitiandEisenlohr
[
2018
].Thesubstratethicknessismuchlowerthanthatob-
servedinrealitytoreducethecomputationcost,however,itsstiffnessissignicantlyincreased
(around32timeshigherYoung'smoduluscomparedtoreal
¯
-Sn)tomakeitrigid.Thisvaluefor
78
Figure4.2:Discretizedgeometryofatinbetweenarigidisotropicsubstrate(orange,show-
ingvoxelsize)andsoftdilatationalfiairfl(faintgray).Boundaries(lightblue)betweenthecolum-
nargrains(green)havedifferentchemical(diffusion)propertiesbutsharethermalandme-
chanicalpropertieswiththebulk.Partofthegeometryalsorevealstheglobal
h
100]lmtexture.
thesubstratestiffnessishighenoughtoensureitsrigidityinthatfurtherincreasingthevalue
didnotaffectthesimulationresults.Theentiregeometryisdiscretizedbyaregulargridof
48
£
96
£
96
Æ
442368voxels,eachofvolume0.1
£
0.1
£
0.1
&
m
3
,alsoshowninFig.
4.2
.Thegrain
orientationsareselectedtorepresentatexturewiththeaxis(thenormaldirec-
tion)beingalignedwithaparticularcrystaldirection,whichinthiscaseischosenas
h
100],as
shownpartlyinFig.
4.2
.Biaxialcompressivestressesgeneratedintheuponheatingdueto
thedifferentthermalexpansionof
¯
-Snandthesubstrate(thehavinganorderofmagni-
tudehigherexpansioncientthanthesubstrate).Theresultingstressesrelaxbyplasticity
andchemicaltransport(stress-drivendiffusion).Diffusionalowoccursonlythroughthegrain
boundaries(highlightedinshadeofblueinFig.
4.2
)duetomuchhigher(tenordersofmagni-
tude)grainboundarydiffusivitycomparedtograinbulkdiffusivity[
Jagtapetal.
,
2017
].The
presentstudysolvesthiscoupledchemical-thermo-mechanicalprobleminacrystalplastic-
ityframework,therebyincorporatingtheeffectofcrystalorientationonthemicro-mechanical
response.
79
Inthiswork,theinitiallystress-freelayeredgeometry(substrate,andair)isheatedto
twodifferentconditionsŒi)20K;andii)40Kin10and20minrespectively,withasubsequent
relaxationofabout60min.Sincediffusionoftinatomsisconsideredinthisstudy,thew
isallowedonlyinthetin(sllyalongthegrainboundaries).Thechemicalxat
thelmsurfaceisrestrictedbyspecifyingazerodiffusivityvalueforthedilatationalairlayer,
thusmimickingthepresenceofapassivatingoxidelayerthatpreventsvacanciestodiffuseinto
thelmfromthesurface.Theseboundaryconditionsgiverisetoacoupledchemical-thermo-
mechanicalsystemofpartialdifferentialequations,describedinSection
4.2.1
whicharethen
solvedusingthefibasicflschemeofthespectralsolver[
Shanthrajetal.
,
2015
,
Eisenlohretal.
,
2013
]thatispartoftheopen-sourceDüsseldorfAdvancedMaterialSimulationKit(DAMASK)
[
Rotersetal.
,
2019
].
The
¯
-Snconstitutiveparametersareobtainedfrom[
Leeetal.
,
2015
]listedinTable
4.1
along
withtheconstitutiveparametersusedforthesubstrateandair.Thethermo-mechanicalmodel
usingtheseconstitutiveparametershavealreadybeenvalidatedusingexperimentaldatainthe
previouschapterSection
3.3
.
4.4ResultsandDiscussion
Todeterminethemacroscopicconsequenceofthestress-drivendiffusionkinematics,
Fig.
4.3
comparestheaveragevonMisesstressevolutionofthelmforafullycoupledsimu-
lation(bluecurves)tothatofasolelythermo-mechanicalsimulation(redcurves)forthetwo
loadingconditionsofa20Kchangeoftemperaturein10min(blackcurveinleftgure)and
thatofa40Kchangeoftemperaturein20min(blackcurveinrighte)followedbyarelax-
ationofabout60minforboth.Basedontheresults,thereappearstobenegligibleeof
thediffusionalkinematicsontheaveragestressevolutionoftheHowever,itiscriticalto
pointoutthatconsideringsuchakinematicconsequencelowers(byasmallamount)theaver-
agestressinthemasthebluecurvesareclosertozerostresscomparedtotheredcurvesin
Fig.
4.3
,whichisexpectedfromsuchaprocess.Also,theaboveobservationismoreprominent
80
Figure4.3:AveragevonMisesstressevolutioninthelmforthecoupledchemo-thermo-
mechanicalmodel(blue)tothatofthecaseofwithoutanydiffusionalw(red),foratem-
peraturechangeof20K(left)and40K(right)in10and20minrespectively(shownbytheblack
curve),followedbyarelaxationofabout60min.Thedasheddarkgraylinerepresentsthepoint
ofmaximumload/straininthesystem,whichisusedasareferencelocationforfurthercom-
parisoninthestudy.
inthelaterstagesofloadingandthesucceedingrelaxationstages.
Inthenextsection,thekinematiceffectofthestress-drivendiffusionisanalyzedforthe
grainboundarynetworkthatservesasthesoleconduitforthetransportofSnatoms.
4.4.1Kinematicconsequence
Inthissection,theaimistoanalyzetheeffectofthekinematicconsequenceoftheatomre-
distributioninthe(alongthegrainboundarynetwork)bycomparingtheresultsobtained
fiwithdiffusional"wtothatofthecasewherethefinodiffusional"wisconsidered(purely
thermo-mechanicalsimulations).
Thekinematiceffectofthediffusionalwisintegratedinthemodelthroughtheeigen-
strainvelocitygradient
L
i
,asmentionedinSection
4.2.1
,alongwiththethermalstrain(refer
toSection
2.4
forfurtherdetails).Figure
4.4
showsthiseffectbycomparingthedeterminant
oftheeigenstraindeformationgradients(representingtherelativevolumetricchange)along
thegrainboundarynetworkforthetwotemperatureboundaryconditions(left:20K,andright:
81
Figure4.4:Kinematicconsequenceofatomredistributionisshownbycomparingdistribution
ofdeterminantoftheeigenstraindeformationgradients
F
i
alongthegrainboundarynetwork,
atmaximumloadforthecoupledchemo-thermo-mechanicalmodel(blue)tothecaseofwith-
outanydiffusionalow(red),foratemperaturechangeof20K(left)and40K(right).
40K).Theredistributionofmassincreasesthevolumeatlocationswhereatomsgetaddedand
decreasesinareasofatomdepletion,ascanbeseenbythesolidbluecurve.Theredcurve
correspondstotheappliedthermalstrainwhichisuniformacrossthem.
Thepressurealongthegrainboundarynetworkmodulatestheatomredistributionandis
shownatthemaximumstrainconditionfortheateinterface(wherethestressesare
maximum)underthetwoloadingconditionsintheleftandrightcolumnsinFig.
4.5
respec-
tively.Theveracityoftheequationsandtheoverallmodelisconmedbythefactthatthe
transportofatomsoccursfromregionsofhighcompression(fiblue")toregionsoflowercom-
pression(fired")asseenbycomparingpressureandconcentrationplotsinFig.
4.5
.Moreover,
thehighdegreeofspatialvariationinpressuredistributionalongthegrainboundarynetwork
isriveofthegloballmtextureof
h
100],similartothepressurevariationsshowninSec-
tion
3.3
,whichalsosupportsthefactthatthereducedgeometry(with30grains)isrepresenta-
tiveoftheoverallgeometry(with100grains)whenthetextureisthesame.
82
20Kchange40Kchange
pressure
0MPa
-30MPa
concentration
change
1
12
atoms
¡
1
12
atoms
Figure4.5:Spatialvariationofthepressure(toprow)andtheconcentrationchangeinnum-
berofatoms(bottomrow),pergrainboundaryarea,forthetwoloadingconditionsof20K(left
column)and40K(rightcolumn)atthemaximumloadandatthelmŒsubstrateinterface.As
expected,locationsoflowcompression(fired"regions)inthepressureplots(toprow)corre-
spondtolocationsofatomgains(fired"regions)intheconcentrationplots(bottomrow).
4.4.2Comparisonofdifferentstresses
Whiskersareverylikelytogrowbyadditionofatomsatthebaseofthewhiskergrain(oblique
surfacegrains)andthenbybreakingthetinoxidelayerontop(thatformsinstantaneously).
Similartotheworkof
ChakrabortyandEisenlohr
[
2019
],thedifferenttractions(shearand
normal)actingalongthegrainboundariesareanalyzedtogetanideaabouttheapproximate
ranges,withanotionthattheymighthaveamechanisticaffect.Anexemplaryvisualization
withanobliquegrainboundaryisshowninFig.
4.6
,wheremostofthenormalsareidentied
properlywiththeexceptionofafewscatterednormals.Thissmallscatterisunlikelytoaffect
83
Figure4.6:Normal(alonglmnormaldirection)andshear(intheboundaryplane)traction
components(right)derivedateachpointofthegrainboundarybasedonthelocalboundary
planenormal(middle)ofanexemplary(oblique)grainisolatedfromtheentire(left,only
grainboundarynetworkshown).
theoveralldistributionofthevalues.
Togaugetheeffectofthekinematicsofatomredistributiononthesetractionvalues,dis-
tributionplotsofthepressurevariation,normaltractions(i.e.,tractiononthegrainboundary
planeactingalongnormaldirection)andasheartraction(i.e.,tractionactingintheplane
oftheboundary)areshownforthetwoloadingconditions(of20K,toprow,and40K,bottom
row)attherateinterface(solidline)andattheface(dashedline)inFig.
4.7
.
Foralltheconditions,themagnitudeofthevalueatthelmŒsurfaceismuchsmallerthanthat
atthelmŒsubstrateinterfaceduetothefreesurfacecondition.Also,forbothloadingcondi-
tionsinvestigatedhere,thekinematicconsequenceofthediffusionalwreducedthepressure
distribution(bluecurvesbeingnarrowerthanredcurvesinleftcolumninFig.
4.7
),whilenegli-
gibleeffectofsuchkinematicscouldbeseenforthetractionvalues.Thismightindicatethatjust
byadditionofatomsalongthegrainboundarynetworkdoesnotkinematicallyalterthestresses
ientlytobreaktheoxidelayerontop.However,suchacasemightalsobestothe
presentlyconsideredmicrostructure,amorereliableestimationcanbemadebyperforminga
statisticalestimationofthevariationofthesestressesunderdifferentinitialmicrostructures.
84
Figure4.7:Probabilitydistributionplotsforpressure(left),ashearcomponentofthetraction(middle),andthetractioncomponent
actingalongthenormaldirection(right,foronlytheobliquegrainboundary)alongthegrainboundarynetwork,allcomputed
atmaximumstraincondition.Thesolidanddashedlinescorrespondtothedistributionofthesevaluesatthefaceand
ateinterfacerespectively.
85
Figure4.8:Arealconcentration(pergrainboundarysurfaceinm
2
)distributionwithinthegrain
boundarynetworkattherateinterface(dashedcurves)andatthemsurface(solid
curves)foraloadof20K(left)and40K(right)atthemaximumstraincondition.
Itwouldalsobeinterestingtoseefromamechanisticpointofview,howthetractionsacting
alongthegrainboundarynormalaffecttheextrusionofmaterialoutfromthefacilitated
bytheobliquegeometryofthegrain.Anothercriticaldrawbackofthecurrentlyconsidered
transportmodelisthepossibilityofuxalongthegrainboundarynormaldirectionwhichis
unlikelytohappeninthecaseofthegrainboundarydiffusion.
Thearealconcentration(pergrainboundarysurfaceinm
2
)variationatthebstrate
interface(dashedlines)andatthelmsurface(solidlines)forthetwoloadingconditions(left:
20Kandright:40K)isshowninFig.
4.9
.Qualitatively,theconcentrationvariationlookssimilar
betweenthelmŒsubstrateinterfaceandtheface,whichisriveofthestronger
dominanceofthein-planegradientascomparedtotheverticalpotentialgradient.
4.4.3Kineticsofthetransport
Toanalyzethekineticsofthediffusionprocess,theevolutionofconcentrationofanexemplary
obliquegrainboundaryovertimeisshowninFig.
4.9
forthetwoloadingconditionsconsid-
eredinthisstudy,witheachmonolayercorrespondingtoathicknesschangeof0.3nm.For
themicrostructure(grainmorphologyandtexture)consideredinthecurrentstudy,thereisa
86
Figure4.9:Averageconcentrationevolutionoftheexemplaryobliquegrainboundary(high-
lightedinFig.
4.6
)forthetwodifferentthermalloadingconditionsof20Kand40K.
netlossofatomsfromtheobliquegrainindicatingthatitactsasasourceofatomsduetothe
stresspredevelopingintheforthe(randomly)chosengrainarrangement.Thiswould
alsoindicatethattheconsideredobliquegrainwouldmostlikelynotbeawhiskernucleation
site.Therateoflossofatoms(slopeofthecurvesinFig.
4.9
)ismorepronouncedduringload-
ingandreducesmarkedlyduringtherelaxationstages.Sucharesultcanbeattributedtothe
dominanteffectofplasticrelaxationascomparedtostressrelaxationbyatomredistribution,
sincethekinematicconsequencesduetoatomredistributionappearedtobeinsigntfor
thepresentcase.Itisalsocriticaltonotethatthegrainboundarydiffusivityconsideredinthe
currentsimulationsisthreeordersofmagnitudeslowerthanthoseusedin[
Buchoveckyetal.
,
2009b
],thatusesactuallymeasuredvaluesofdiffusivity,duetonumericalconvergence.Hence,
thekineticsresultsshowninthisstudyisanunderestimationoftheactualphenomenon.How-
ever,theconclusionsdrawn(insignicanteofthediffusionalwontheshearforces
andthemaveragestressevolutionascomparedtoplasticity)wouldremainunchangedsince
thekinematicconsequenceofatomredistributionisverysmall,asalsoreportedby
Buchovecky
87
[
2010
]whereachangeincoupleofordersofmagnitudeindiffusivitydidnotchangethestress
results.
4.5Summary
Ageneralizedfullycoupledchemo-thermo-mechanicalmodelinthecrystalplasticity
frameworkisoutlinedinthischaptertostudythestress-drivendiffusionintinthinms.By
comparingtheaveragelmstressevolutionwithandwithoutthediffusionalkinematics,itis
concludedthattheatomredistributionhadlimitedinenceoverthestresswhichcor-
respondstothebulkstressesmeasuredexperimentallyviacurvaturemeasurements.Suchan
observationiscontradictorytothereportedresultof
Buchoveckyetal.
[
2009a
],wherethere-
laxedstressis10MPahigherinthefortheelasto-plasticsimulationascomparedtothatof
includinggrainboundarydiffusioninthesimulation.Similarresultswerealsoreportedin
Pei
etal.
[
2017
]wherethestrainduetowhiskerrelaxationwasabout30%ofthetotalstrain.Inour
casethekinematicconsequenceappearstobemuchlowerthanthat.Aplausiblereasonfor
suchanoutcomemightbethelowervalueofdiffusivityusedinthisstudycomparedtothose
usedinthemodelof
Buchoveckyetal.
[
2009a
],
Peietal.
[
2017
]wheretheyhaveusedexperimen-
tallymeasureddiffusivityvalues.Anotherreasonforthedifferencemightbethedifferencesin
thedescriptionof
¯
-Snplasticitybetweenthepresentstudyusingaphenomenologicalpower
lawmodel,versusthatofanisotropicvonMisesdescriptionusedin
Buchoveckyetal.
[
2009a
],
Peietal.
[
2017
].Thisfurthermotivatestheneedfortheaccuratedescriptionof
¯
-Snplastic-
ity,whichisyettobeestablishedinliterature,andamethodologytoaccuratelypredictsuch
plasticityparameterswouldbetakenupinthefollowingcoupleofchapters(Chapters
5
and
6
).
Thekinematiceffectofatomredistributiononthehydrostaticstressandthedifferenttrac-
tioncomponents(shearandnormal)inthegrainboundarynetworkarealsostudied,anditis
concludedthatthekinematicsdueatomredistributionconsiderablyaffectedthegrainbound-
aryhydrostaticstressascomparedtothedifferenttractioncomponents,Fig.
4.7
.Thismightbe
astandaloneresultforthestudiedmicrostructure.Furtherinsightcanbegainedviaastatistical
88
analysisofthesetractionvaluesforvariousmicrostructures(i.e.,crystalneighborhoodand/or
texture).Also,forthecurrentlysimulation,theobliquegrainboundaryappearstocontinu-
ouslyloseatomswhichwouldmakeitanatomsourceandpreventittonucleateawhisker.This
suggeststhat,contrarytotheobservationoftheobliquegrainseenhere,therecouldbeasitua-
tionwheretheobliquegrainboundarywouldgainatomsandnucleateawhisker.However,the
locationofsuchgrainswouldstillbegovernedbythelocalcrystallographicneighborhood.
Thepreliminaryconclusionsmadefromthepresentstudyneedtobefurtherestablishedby
moredetailedanalysis.Asmentioned,themodelproposedhereindoesnoteliminatexalong
thegrainboundarynormaldirection,andhenceasuitableationtothecurrentformu-
lationwouldbetoincorporategrainboundarynormaltocalculatethestressactingalongthe
grainboundarynormaldirectionanduseonlythatquantityasameasureofchemicalpotential
insteadofthehydrostaticstressconsideredpresently.Also,thekineticsreportedinthisworkis
anunderestimation,asthegrainboundarydiffusivityvalueusedinhereisaroundthreeorders
ofmagnitudeslowerthanreality.Havingthecapabilityofparallelcomputationwouldenableto
userealisticdiffusivityvalues,astheyrequireasmallertimestepthanthatusedforthepresent
simulations.Performingarigorousstatisticalstudyusingaccuratemodeldescriptionandwith
differentgrainmicrostructuresandtexturedistributions(i.e.,initialconditions)wouldprovide
ameasureoftherangeofthevariousstressquantities(shearandnormaltractions)andhence
increaseourinsightonwhethertheyareientinbreakingthetinoxidelayerontop.More-
over,suchastatisticalstudywouldalsoprovideadistributionofcaseswheretheobliquegrain
actsasanatomsink(therebybeingapotentialwhiskernucleus),comparedtoitbeinganatom
source.Suchadistributionofgrainsactingatomsinks,canthenbecomparedtoexperimentally
observedwhiskerdensityvalues.
89
Table4.1:Materialpropertiesfordilatationalair,body-centeredtetragonal
¯
-Sn,andisotropic
substrate.
ValueUnit
Air
C
11
10.0GPa
C
12
0.0GPa
g
0
:
g
1
0.333:0.667MPa
h
0
1.0MPa
a
2
M
3
n
20

°
0
10
¡
3
s
¡
1
¯
-Sn
C
11
72.3GPa
C
12
59.4GPa
C
13
35.8GPa
C
33
88.4GPa
C
44
22.0GPa
C
66
24.0GPa
g
0
:
g
1
{100)
h
001]8.5:11.0MPa
g
0
:
g
1
{110)
h
001]4.3:9.0MPa
g
0
:
g
1
{100)
h
010]10.4:11.0MPa
g
0
:
g
1
{110)
h
1
¯
11]4.5:9.0MPa
g
0
:
g
1
{110)
h
1
¯
10]5.6:10.0MPa
g
0
:
g
1
{100)
h
011]5.1:10.0MPa
g
0
:
g
1
{001)
h
010]7.4:10.0MPa
g
0
:
g
1
{001)
h
110]15.0:10.0MPa
g
0
:
g
1
{011)
h
0
¯
11]6.6:9.0MPa
g
0
:
g
1
{211)
h
0
¯
11]12.0:13.0MPa
h
0
20MPa
a
2.0
n
6

°
0
2.6
£
10
¡
8
s
¡
1
®
h
100]
20
£
10
¡
6
K
¡
1
®
h
001]
40
£
10
¡
6
K
¡
1
D
GB
1
£
10
¡
16
m
2
s
¡
1
D
bulk
2
£
10
¡
28
m
2
s
¡
1
±
GB
0.5nm

2.7
£
10
¡
29
m
3
²
1.0
c
0
0numberofatoms
Substrate
C
11
1500.0GPa
C
12
500.0GPa
®
1
£
10
¡
6
K
¡
1
90
CHAPTER5
INVERSEINDENTATIONANALYSISANDITSAPPLICATIONINFACE-CENTEREDCUBIC
MATERIALS
5.1Motivation
Theincompletematerialdescriptionof
¯
-Snarisesduetothelackofestablishedplasticity
parameters,theinitialvalueof
g
(sometimesalsoreferredtoascriticalresolvedshearstress,
CRSS),intheliterature,especiallywhenusingthephenomenologicalmaterialmodel.Thisis
importantsinceplasticityof
¯
-Snisknowntoaffectthewhiskerformation[
Chasonetal.
,
2008
].
Plasticdeformationinthehighlyanisotropic
¯
-Sn,becauseofthebody-centeredtetragonal
crystalstructure,canbeaccommodatedbythethirteenslipfamiliesthatareavailable.Even
thoughthepresentsimulationsusereportedvaluesfortheseparameters,areliableparameter
estimationstrategyforcrystalplasticityconstitutivemodelsislacking.Aninverseapproachis
proposedinthisworktoidentifyconstitutiveparametersthatreliesonminimizingtheerrorbe-
tweenthesimulatedandexperimentalsinglecrystalnanoindentationresponses.Themethod
wasoriginallyproposedby
Zambaldietal.
[
2012
],however,astudyregardingitsecacyhas
beenlacking.Inthischapter,themethodologyusingInverseIndentationAnalysis(
IIA
)isim-
plementedforface-centeredcubic(fcc)materialstodetermineitseffectivenessinsuchahighly
symmetricsystemwithonlyasingleslipfamilyandusingvirtualexperimentalreferencedata
withaprioriknownparametervalues.Someionstotheoriginalmethodarealsopro-
posedthatresultedinincreasedeffectivenessofthemethodology.Criticalaspectsforaneffec-
tiveoptimization,suchassensitivityanalysisofeachparameters,dimensionalreduction,and
effectofinitialguess,isalsodiscussed.Finally,afterestablishingthetyofthe
IIA
forfccmaterials,itsreliabilityistestedformoreasymmetrichexagonalmaterials(againwith
virtualexperiments)havingmultipleslipfamiliesinChapter
6
.Havingestablishedthereliabil-
ityoftheproposedmethodologyforcubicandhexagonalmaterials,thenextstepwouldbeto
91
implementitforevenlowersymmetrictetragonal
¯
-Sn,andestablishitsplasticbehavior.
Thecontentsofthischapterisadaptedfromthealreadypublishedworkin
Chakrabortyand
Eisenlohr
[
2017
].
5.2Background
Thefeasibilitytodeterminetheadjustableparametersofsinglecrystalplasticityconstitu-
tivelawsbyaninverseapproachthatminimizesthedeviationbetweenthemeasuredandsim-
ulatedindentationresponseofindividualgrains(ofapolycrystallinesample)isanalyzedin
thischapterforthecaseofface-centeredcubic(fcc)crystalstructure.Optimizationusesthe
NelderŒMead(NM)simplexalgorithm,thatwastocircumventregionsinparameter
spacewheretheevaluationoftheobjectivefunctionfails.Aphenomenologicalpower-lawis
usedastheconstitutivedescriptionforcrystalplasticity.Simulatedcasesofindentationwith
prescribedconstitutiveparametervaluesserveasthevirtualreference,oftenreferredtointhe
chapterasvirtualexperiments.Asensitivityanalysisstudyisalsodiscussedinordertogauge
therelativeinofdifferentadjustableparametersontheoverallmaterialresponseunder
indentation.Followingthesensitivityanalysis,thereliabilityofthemethodologyisassessed
basedonit'sreproducibilityandrobustnesswithdifferentobjectivefunctionsinvolvingthe
loadŒdisplacementresponseandresidualsurfacetopographyfordifferentindentationcrystal
orientations.ItappearsthatusingbothinformationfromloadŒdisplacementandresidualsur-
facetopographyprovidesamoreconvexobjectivefunctionsurfacefortheoptimizertoidentify
thetargetvalues(globalminimum)withoutbeingsigntlyedbytheselectedcrys-
talorientation.
Thechapterisstructuredasfollows:afteraliteraturereviewaboutthedevelopmentofin-
dentationasatechniquetoidentifymaterialparametersinSection
5.3
,thesimulationdetails
alongwiththeconstitutivematerialmodeldescriptionarepresentedinSection
5.4
followedby
anexplanationaboutthestructureoftheinverseoptimizationstrategyinSection
5.6
.Critical
resultsobtainedfromthecomprehensiveinvestigationarehighlightedinSection
5.7
followed
92
byadiscussioninSection
5.8
andtheconclusiveremarksinSection
5.9
.
5.3Introduction
Thepredictionofthemechanicalresponseandinternalstructuralevolutionofcrystalline
solidischallengingduetotheinherentanisotropyoftheelasticandplasticpropertiesofsingle
crystals.Integrationofanisotropyintocrystalplasticity(CP)modelsofdeformationhasbeen
quitesuccessful(see[
Rotersetal.
,
2010b
]forarecentreview)andismostrelevantformetals
thatexhibitstrongcrystallographictextureresultingfromprocessingorwhenoneormoredi-
mensionsofanengineeringcomponentapproachtheinternalgrainsizeofthematerial.While
thesophisticationoftheunderlyingconstitutivedescriptionofdeformationandstructureki-
neticsvariesamongdifferentCPapproaches,forallofthemtheaccuracyofpredictionlargely
dependsonthevaluesselectedfortheadjustableconstitutiveparameters.
Correctidentionofconstitutiveparametervaluesisanongoingareaofsigniant
researchinterestasitinvolvestheinverseproblemofmatchingsimulationoutcomestoex-
perimentalreference.Onewaywouldbetouseexperimentalreferencefrompolycrystalline
(macroscopic)deformationunderunidirectionaltensionorcompressioninconnectionwith
eitheranisotropicplasticdescription[
MahnkenandStein
,
1996
,
KajbergandLindkvist
,
2004
]
oraphenomenologicalcrystalplasticitymaterialmodel[
Herrera-Solazetal.
,
2014
].In[
Herrera-
Solazetal.
,
2014
],singlecrystalmaterialparameterswereextractedusingagradient-based
LevenbergŒMarquardtoptimizationalgorithmthatmatchedstressŒstrainresponsesofe
elementsimulationsofrepresentativevolumeelementstocorrespondingexperiments.Are-
producibilitystudyforthisapproachrevealedhighreproducibilityforsomeparametersas
comparedtoothers[
Herrera-Solazetal.
,
2015
],indicatingeitherthepresenceofmultiplelo-
calminimaintheobjectivefunctionsurfaceorlackofsensitivityforcertainparametersover
others,andthus,callingforfurtherinvestigation.Analternativeandmoredirectwayistouse
singlecrystal(microscopic)deformationsinceitexcludestheproblematicconvolutioncaused
bymultiplegrainsdeformingandinteractinginparallelandappears,hence,tobeamore
93
promisingvenue.Instrumentednano-indentationexperimentshavebeenanecientwayto
generatemicroscopicdeformationresponsefrommultipleorientationsintheformofloadŒ
displacementcurvesandsurfacetopographyafterindentation.
Indentationexperimentsasatooltoidentifymaterialparameters,wasrstproposedby
HuberandTsakmakis
[
1999
]and
Huberetal.
[
2001
]byusinganarialneuralnetwork(ANN)
andonlyindentationloadŒdisplacementresponse.Finiteelement(FE)simulationswereper-
formedbasedonanisotropicplasticityconstitutivemodelincludingisotropicandkinematic
hardeningtogeneratesimulatedloadŒdisplacementcurves.Inputtothenetworkincludedthe
simulated(reference)loadŒdisplacementcurveswhiletheoutputcorrespondedtotheparam-
etervalues.MultipleFEsimulationswereperformedtotrainthenetwork.Theeffectivenessof
thisFE-ANNmethodologywastestedbyimplementingittoidentifythematerialparametersfor
differentmaterialsusingexperimentalloadŒdisplacementcurvesby
Klötzeretal.
[
2006
].Itwas
observedthatthemethodologycouldonlycapturetheelasticresponsewithcertainty,while
thereexistedahighscatteramongtheidentplasticparameters.AmodicationtotheFE-
ANNmethodologywasproposedby
Hajalietal.
[
2008
]byconsideringonlytheloadingpartof
theloadŒdisplacementresponseandusingdimensionlessmaterialparametersfortheiridenti-
ation.However,sigtdeviationfromexperimentalresponsetothatobtainedfromFE-
ANNparameterswereobtainedespeciallyintheplasticregime.Adifferentapproach,butstill
usingonlytheindentationloadŒdisplacementresponse,wasadoptedby
Nakamuraetal.
[
2000
]
whereinheincorporatedthesequentialstochasticKalmanlteralgorithm,usedmostlyinsig-
nalprocessing,toestimateYoung'smodulusandPoisson'sratioforfunctionallygradedma-
terials.Thestrategywasalsoextendedtoidentifyplasticparametersfortransverselyisotropic
materials[
NakamuraandGu
,
2007
,
Yonezuetal.
,
2009
].Subsequently,withtheadventandease
ofaccessinmeasuringsurfacetopographies,suchasbyatomicforcemicroscopyorwhitelight
interferometry,
Bolzonetal.
[
2004
]proposedtoincludetheresidualimprintafterindentation
forestimatingtheconstitutiveparameters.Theyassumedanisotropicelasto-plasticmaterial
modelandobservedthatinclusionoftopographydeviationintheobjectivefunctiongenerally
94
improvedtheaccuracyofparameterestimationinthegradient-baseddeterministicoptimiza-
tionalgorithmthatwasused.
Bocciarellietal.
[
2005
],
BocciarelliandMaier
[
2007
]extended
thescopetoelastic-perfectlyplasticmaterialmodelswithorthotropicsymmetry(usingayield
surfaceapproach[
Hill
,
1948
])fororthotropicmaterialsagainstbotharticialreferencedataand
realexperimentaldata.Thegeneralshortcomingsassociatedwithgradient-basedoptimization
andstrongsimplicationsintheconstitutivedescriptionintermsofthealoptimizationsolu-
tiondependingontheinitialparameterswasrmed.Nevertheless,withaprioriknowledge
ofviableplasticparameterrangessuchanapproachgavesatisfactoryresultsformulti-layered
systems[
Moyetal.
,
2011a
]andaluminumalloys[
Moyetal.
,
2011b
].Toreducethecomputation
costofevaluatingthegradientnumerically,severalstudiesfocussedonestimatinghigh-quality
approximationsofthesolutiondfromalimitednumberofnumericalresultsusingproper
orthogonaldecomposition(POD)withradialbasisfunctions[
BuljakandMaier
,
2011
,
Bolzon
andTalassi
,
2013
,
Bocciarellietal.
,
2014
,
HamimandSingh
,
2017
,
ArizziandRizzi
,
2014
].Other
ionstoreducecomputationtimeofcalculatingthegradientforgradient-basedopti-
mizationalgorithmshavealsobeeninvestigatedusingapproachessuchasparametercoupling
[
RauchsandBardon
,
2011
],solvingtheadjointproblem[
ConstantinescuandTardieu
,
2001
]
andmakingdimensionlessequationsbasedonaprioriFEMsimulations[
Yonezuetal.
,
2010
,
Wangetal.
,
2010
].
Inallabovestudies,deformationbehaviorhasbeenapproximatedbysimpliedmaterial
modelsthathardlyconsiderplasticanisotropyofthematerial.
Sánchez-Martínetal.
[
2014
]
postulatedvaluesforcrystalplasticityconstitutiveparameters(ofaparticularMgalloy,MN11)
bycomparinghardnessvaluesfromgrainindentationsimulationsusingsetsofparameters
reportedforsimilarmaterialsintheliterature,andchoosingthatonegivingtheclosestto
theindentationexperiments.Comparingresponsesbetweenexperimentandsimulationofax-
isymmetricinstrumentednano-indentationintoindividualgrainsbyincorporationofacrystal
plasticitymodelwasrstperformedby
ZambaldiandRaabe
[
2010
]onface-centeredtetragonal
°
-TiAl.Theirstudyledtotheinversemethodologyofidentifyingcrystalplasticityparameters
95
Figure5.1:Finiteelementmodelincludingindenterandsinglecrystallinesubstrate.Inden-
tationissimulatedbyprescribingdownwardfollowedbyupwardmotionoftherigidindenter
surfaceatconstantvelocity.Nodaldisplacementsofsubstratearefullyrestrictedonthebottom
andsidefaces.
usinggradient-freeNelderŒMead(NM)simplexoptimizationbasedonmatchingthesimulated
loadŒdisplacementandsurfacetopography[
Zambaldietal.
,
2012
]tomeasureddataforhexag-
onalcommerciallypuretitanium(cp-Ti).However,theconstitutiveparametersobtainedfor
cp-Tidifferedsigntlyfromotherreports(see[
Lietal.
,
2013
]forsummary).
Inthischapterthecacyoftheinversemethodologysuggestedby
Zambaldietal.
[
2012
]
isanalyzedonface-centeredcubic(fcc)crystalsystemsusingfipseudo-experimental"reference
datatodeterminetherobustnessofsuchanapproachwithmaterialorientation.
96
5.4Simulationset-up
5.4.1Finiteelementdiscretization
Similartothepreviouswork,[
Zambaldietal.
,
2012
],athree-dimensionaliteelement(FE)
modelofconosphericalindentation(seeFig.
5.1
)isgeneratedusingtheopensourcetoolkit
fiSTABiXfl[
MercierandZambaldi
,
2014
].Thesinglecrystallinecylindricalsubstrateofheight
4
&
manddiameter8
&
misdiscretizedby4596linearhexahedralelementsbasedonameshsen-
sitivitystudy(Section
5.7.1
).Nodaldisplacementsarexedonthebottomandoutersurfaces.
Theindenterofradius1
&
mandconeangleof90°isassumedrigidwithaCoulombfriction
cientof0.3.Itsverticaldisplacementislinearlyvariedfromzeroto0.5
&
mandbackin
10s,whicharediscretizedinto1334equalincrements.Allthesimulationsareperformedus-
ingthecommercialiteelementsoftwareMarc2013.1(MSCSoftwareCorporation,Newport
Beach,CA)oncomputeserversmaintainedbytheDivisionofEngineeringComputingServices
atMichiganStateUniversity.Toreducethecomputationtime,thegeometryisalsodecom-
posedintofourequaldomains(i.e.,sectorsof90°each)usingthein-builtfunctionalityinMarc
2013.1.
5.4.2Phenomenologicalmaterialpointmodel
Tosolvethemechanicalboundaryvalueproblemposedbyconosphericalindentation,aphe-
nomenologicalconstitutivemodel[
Peirceetal.
,
1982
]widelyusedincrystalplasticityandim-
plementedaspartoftheopensourcesimulationtoolkitfiDüsseldorfAdvancedMaterialSimu-
lationToolkitfl(DAMASK)[
Rotersetal.
,
2012
]isinterfacedtoMarc2013.1throughthematerial
subroutinefihypela2fl.Abriefoverviewoftheconstitutivedescriptionismentionedbelow,fur-
therdetailscanbefoundin
Rotersetal.
[
2010a
].Thecurrentmodelisbasedonthemultiplica-
tivedecompositionofthetotaldeformationgradient
F
intoplasticandelasticcomponentsfor
97
describingthekinematicsunderalargestrainframework[
Lee
,
1969
]
F
Æ
F
e
F
p
(5.1)
with
F
p
beingtheplasticdeformationgradientand
F
e
relatingtotheelasticstretchand
rotation.
Therelationbetweenplasticdeformationgradient
F
p
andtheplasticvelocitygradient
L
p
is
givenbythewrule:

F
p
Æ
L
p
F
p
(5.2)
Thedislocationdefectstructureisparameterizedphenomenologicallyintermsofslipresis-
tanceoneachofthetwelve{111}
h
110
i
slipsystems.Theplasticvelocitygradientisadditively
composedfromsliprates
L
p
:
Æ
X
®

°
®
s
®

n
®
(5.3)
wheretheunitvectors
s
®
and
n
®
indicatetheslipdirectionandslipplanenormalforslipsys-
tem
®
Æ
1,...,12.Theresolvedshearstress
¿
®
Æ
C
³
F
e
T
F
e
¡
I
´
/2:
s
®

n
®
(5.4)
isthedrivingforceforslipatrates

°
®
Æ

°
0
¯
¯
¯
¯
¿
®
¿
®
crss
¯
¯
¯
¯
n
sgn
¡
¿
®
¢
,(5.5)
withmaterialparameters

°
0
(referenceshearrate)and
n
(stressexponent).Componentsofthe
anisotropicelasticstiffnesstensor
C
wereselectedas
C
11
Æ
107GPa,
C
12
Æ
61GPa,and
C
44
Æ
29GPa.Thecriticalresolvedshearstress
¿
®
crss
evolvesinthecourseofslipas[
Hutchinson
,
1976
]

¿
®
crss
Æ
X
¯
q
®¯
h
0
¯
¯
¯
¯
¯
1
¡
¿
¯
crss
¿
sat
¯
¯
¯
¯
¯
a
sgn
Ã
1
¡
¿
¯
crss
¿
sat
!
¯
¯
¯

°
¯
¯
¯
¯
,(5.6)
98
withadjustableparameters
h
0
,
a
,
¿
sat
,and
q
®¯
rectingthesixtypesofdislocationinterac-
tionsobservedinfccsystems.Inadditiontothesaturationvalue
¿
sat
,theinitialvalueof
¿
crss
,
denotedas
¿
0
,isalsofreetoadjust.
Intotal,theabovephenomenologicaldescriptioncomprises12adjustableparametersin
thecaseoffcclatticestructure,thosebeing

°
0
,
n
,
¿
0
,
¿
sat
,
a
,
h
0
,andsixcientsfordis-
locationinteractions,i.e.,self,coplanar,collinear,Hirthlock,LomerŒCottrelllock,andglissile
junction.Sincethevaluesofthematerialparameters

°
0
and
n
mostlyuencenumericalsta-
bilityandonlymarginallythematerialmicromechanics,theyarenotconsideredasadjustable
parametersintheoptimization[
Zambaldietal.
,
2012
].Moreover,thesixinteractionparame-
tersaregenerallyobtainedfromalowerscaledislocationdynamicssimulationandhenceare
alsoconsideredtobeconstant.
5.5Globalsensitivityanalysis
Theglobalsensitivityanalysisisperformedbasedonthefielementaryeffectsmethod"pro-
posedby
Morris
[
1991
]toeconomicallycollectpartialderivativesinahigh-dimensionalunit
domain.Partialderivatives
@
f
/
@
x
i
arecalculatedbasedonpairsofpointsthatonlydifferby
¢
Æ
0.5
p
/(
p
¡
1)alongthecoordinateaxis
i
.Thosepairsareselectedfromagridthathastwo
pointsperparameterspacedimensionandwhichispopulatedby
r
timesstartingapathat
theoriginandtakingone
¢
stepalongeachdimensioninrandomorder.Theabsolutevalues
ofthemultiplepartialderivativesresultingalongeachdimensionareaveragedfollowingthe
suggestionof
Campolongoetal.
[
2007
].
¹
¤
i
:
Æh
¯
¯
@
f
/
@
x
i
¯
¯
i
,(5.7)
wherelargervaluesof
¹
¤
i
indicatealargerofparameter
i
on
f
.Forthecurrentstudy,
p
Æ
r
Æ
4werechosen.
99
5.6OptimizationMethodology
5.6.1Universaloptimizationmodule
Figure
5.2
illustratestheoptimizationstrategythatiterativelyadjuststheconstitutiveparame-
ters,performscrystalplasticityeelementsimulation(s)ofsinglecrystalindentation,and
comparestheresultingloadŒdisplacementdataand/orsurfacetopographytotheirreference
untilthedeviationmeetsagiventolerance.Theoptimizerusedinthisstudyisimplemented
asageneralPythonclassthatcanbeequippedwithdifferentstochasticanddeterministicop-
timizationalgorithmssuchasParticleSwarmOptimization[
KennedyandEberhart
,
1995
]or
NelderŒMeadsimplex[
NelderandMead
,
1965
].Itsuniversalityresultsfromthepossibilityto
besubclassedwithanarbitraryevaluationoftness,whichinthepresentcasetriggerstheCPFE
simulation(s),thepost-processingofitsresultstoextractloadŒdisplacementdataandsurface
topography,andthecalculationofthecorrespondingdeviationsfromaknownreference(ob-
jectivefunctionvalue).
SincethesystemofconstitutiveequationsselectedinSection
5.4.2
hasfouradjustablepa-
rameters
¿
0
,
¿
sat
,
h
0
,and
a
,theassociatedoptimizationproblemisfour-dimensional.Hence,
forsuchrelativelylow-dimensionalproblem,thedeterministicNelderŒMeadstrategyischosen
asasuitableoptionforitssolution.
5.6.2NelderŒMeadsimplexoptimizationalgorithm
TheNelderŒMead(NM)simplexalgorithm,originallyproposedin
1965
[
NelderandMead
,
1965
],hasbeenoneofthebestknownalgorithmstosolveawideclassofunconstrainedopti-
mizationproblemsduetoitssimplicityandeaseofimplementation.ThestandardNMsimplex
algorithmconstructsasimplexhaving
N
Å
1verticesanditerativelyrelocatestheworstvertex
untilthesimplexhasmovedandshrunktoanoptimumlocationinthe
N
-dimensionalparam-
eterspace.SincetheNMalgorithmdoesnotdependongradientinformation,itisverysuitable
forthepresentclassofinverseproblems[
HamimandSingh
,
2017
].Asimple,yethighlyeffec-
100
Figure5.2:Iterativeoptimizationsetuptoidentifyadjustableparametersofacrystalplasticity
constitutivelaw.Basedonaselectedstrategy,theoptimizeradjuststheparameters,whichare
thenfedasinputintotheCPFEsimulationusingthematerialpointmodelDAMASK.Objective
functionvalueisobtainedastheloadŒdisplacement(
²
LD
)and/orsurfacetopography(
²
topo
)
deviationfromagivenreference.
tive,ionisproposedtothealgorithmtomoveitawayfromregionswherenoviable
solutiontothenessfunctioncanbefound(whentheFEsimulationdoesnotconverge).A
secondationistomapthedomainofparameterspacethatisassumedtocontainthe
optimumsolutionontoaunitdomainsuchthatdistancesareindependentoftheparticular
directioninparameterspace.
Thealgorithmtondtheminimumofanobjectivefunction
f
reads:
Ł
Generate
N
Å
1randompoints
p
intheremappedparameterspaceasverticesoftheinitial
simplex,
N
beingthenumberofconstitutiveparameters,i.e.,thedimensionofparameter
space.
Ł
Evaluatetheobjectivefunction
f
foreachvertextodeterminethebest(subscriptfibfl),
second-worst(subscriptfisfl),andworst(subscriptfiwfl)one.
101
Ł
Excludetheworstvertex
p
w
anddeterminethecentroid(subscriptfig")oftheremaining
simplex
p
g
Æh
p
i¡
p
w
/
N
.(5.8)
Ł
fiRtion"of
p
w
aboutthecentroid
p
g
toyield
p
r
Æ
p
g
Å
®
¡
p
g
¡
p
w
¢
.(5.9)
Ł
Propagatethesimplexdependingonthefollowingconditions:
1.
If
f
r
isgloballybest,checkafurtherfiexpandedflpoint
p
e
Æ
p
g
Å
°
¡
p
r
¡
p
g
¢
(5.10)
andreplace
p
w
withthebetterof
p
r
and
p
e
.
2.
If
f
r
isbetterthanthecurrentsecondworst,replace
p
w
by
p
r
.
3.
Otherwisecontractthebetterof
p
w
or
p
r
towards
p
c
Æ
p
g
Å
¯
¡
p
w
j
r
¡
p
g
¢
.(5.11)
Replace
p
w
by
p
c
if
f
c
improvesover
f
w
,otherwiseshrinkallverticesaccordingto
p
j
Æ
p
b
Å
±
¡
p
j
¡
p
b
¢
(5.12)
Ł
Anypointforwhichtheobjectivefunctioncannotbeevaluatedisrelocatedtofallsome-
wherebetweenthecentroidoftheoverallsimplexandtherionoftheinfeasible
pointwithrespecttothatcentroid
p
Æh
p
iÅ
»
¡
h
p
i¡
p
¢
,(5.13)
with
»
auniformrandomnumberbetween0and1.Therandomnessavoidssituations
whenthegeneratedpointturnsouttobeinfeasibleaswell.Previousworksonvisco-
plasticparameteridenationusingNMsimplexalgorithm[
KajbergandWikman
,
2007
,
KajbergandLindkvist
,
2004
,
Zambaldietal.
,
2012
]didnotincludeastrategytoavertin-
feasiblepointsinparameterspace.
102
Figure5.3:Two-dimensionalillustrationofthesequenceofoperationsperformedbytheNM
simplexalgorithmasimplementedhere.Bluetrianglemarkstheinitialsimplex,redtrianglethe
onethatresultsfromthespoperationmentionedontheconnectingarrow.Verticesare
labelledbynotationsasdescribedinthetext.Thestrategytonavigateawayfromaninfeasible
vertexisdisplayedbythegraysimplexinthebottomleft.
Ł
Moreover,eachpointisforcedtoremainwithintheunitdomain.
Sincetheobjectivefunctionevaluationrequiresaiteelementsimulationandiscom-
putationallyveryexpensive,theientsoffirection",fiexpansion",ficontraction",and
fishrink",whicharetheadjustableparametersfortheNMsimplexalgorithm[
SingerandNelder
,
2009
],arechosentobe
®
Æ
1,
°
Æ
0.5,
¯
Æ
2,and
±
Æ
0.5,respectively,asassignedinmostimple-
mentationsofNMsimplexalgorithm.Figure
5.3
illustratesthedifferentoperationsoftheNM
simplexalgorithmimplementedinthisstudyalongwiththestrategytogenerateanewpointof
evaluationwhenaninfeasiblepointisencounteredduringtheoptimization.
103
5.6.3Objectivefunctionforoptimization
Possibleinputfortheobjectivefunctionare
²
LD
and
²
topo
,i.e.thedifferencesbetweenmea-
suredandsimulatedindentationsintermsofloadŒdisplacementresponse
F
(
h
)andsurface
topography
z
(
x
,
y
),respectively.Thereisclearagreementthattheformerisessentialfora
quantitativelycorrect,buttheofincludingthesurfacetopographydeviationisnot
yetultimatelysettled[
Bolzonetal.
,
2004
,
RauchsandBardon
,
2011
,
Mengetal.
,
2016
].Toclar-
ifythebof
²
topo
,threedifferentalternativesfortheobjectivefunction(being
²
LD
,
²
topo
,
andtheiraverage)aretestedwithregardtoreproducibilityandrobustness.Reproducibility
correspondstotheconsistencyoftheidentconstitutiveparametersasafunctionofthe
initialsimplexchoice,whilerobustnessmeanstheconsistencyofoptimizedparametervalues
betweendifferentindentedcrystalorientations.
Theerror
²
LD
isobtainedbyintegratingtheabsolutedifferenceoftheloadingpartofthe
loadŒdisplacementcurvebetweenthereference(ref)andoptimizedsimulation(sim)overeach
timestep,
i
,andnormalizedbytheintegralofthereferenceloadŒdisplacementresponse.
¢
F
i
:
Æ
¯
¯
F
sim,
i
¡
F
ref,
i
¯
¯
(5.14)
²
LD
:
Æ
P
i
(
¢
F
i
Å
1
Å
¢
F
i
)(
h
i
Å
1
¡
h
i
)
P
i
(
F
ref,
i
Å
1
Å
F
ref,
i
)(
h
i
Å
1
¡
h
i
)
(5.15)
Asimilarstrategyisalsousedtocalculatetheerrorinsurfacetopography
²
topo
:Afterbothto-
pographiesareadjustedsuchthattheelevationfarfromtheindentisatzeroandinterpolated
ontoaregulargrid(256
£
256),theabsolutedifferenceinelevationwasintegratedoverallgrid
points
j
andnormalizedbytheintegralofthereferencetopographyheight.Whencomparing
withrealexperimentswhereaccurateinformationoftipgeometryisulttoacquireand
maynotmatchthesimulatedtip,asystematicerrorarisesbetweenexperimentalandCPFEto-
pographyofthetipimpression.Hence,onlypositiveheights,correspondingtofipileup"around
theindent,wereconsideredforcalculating
²
topo
.
²
topo
:
Æ
P
j
¯
¯
¯
z
j
,sim
¡
z
j
,ref
¯
¯
¯
P
j
z
j
,ref
(5.16)
Thecombined(average)functionvalueofbothapproachesistermed
²
combo
Æ
(
²
LD
Å
²
topo
)/2.
104
Figure5.4:Crystallographicorientationsoftheindentationaxisconsideredinthepresentstudy.
5.6.4Determinationofmostsensitiveconstitutiveparameters
Theglobalsensitivityanalysis,followingthefielementaryeffectsmethodflintroducedby
Mor-
ris
[
1991
]anddiscussedinSection
5.5
,isusedtodeterminetherelativeeofthefour
adjustableparameters,
¿
0
,
¿
sat
,
h
0
and
a
,ontheloadŒdisplacementandsurfacetopography
responsefordifferentcrystalorientations.Theinnceofeachofthefourparametersare
qualitativelyrankedbasedon
¹
¤
,whichisthemeanoftheirabsolutepartialderivatives
@
f
/
@
x
where
x
representsapointinthefourdimensionalparameterspace(see
5.6.4
forexplanation).
Threedifferentcasesofdetermining
¹
¤
wereconsideredwherein
@
f
betweentwopointswas
calculatedbasedoneitherthedeviationinloadŒdisplacementresponse(similarto
²
LD
)orde-
viationinthesurfacetopographyresponse(similarto
²
topo
)orastheaveragedeviationofboth
loadŒdisplacementandsurfacetopographyresponse(similarto
²
combo
).
Forthepresentanalysistheboundsofthefourparametersareselectedtobe
¿
0
2
[1,100]MPa,
¿
sat
2
[1,100]MPa,
a
2
[2,5]and
h
0
2
[5,550]MPa,basedonthealreadyestablished
crystalplasticityconstitutiveparametersforfccaluminum[
KalidindiandSchoenfeld
,
2000
].As
105
mentionedearlier,thevalueofreferenceshearrate

°
0
andstrain-ratesensitivityparameter
n
areto10
¡
3
s
¡
1
and25,respectively.Table
5.1
summarizesthe
¹
¤
valuesofthefourpa-
rametersinvestigatedforthethreedifferentcasesandforsevendifferentcrystalorientations
labelledinFig.
5.4
.Itistobenotedthatforthecurrentstudytherewereafewsamplepoints
wheretheCPFEsimulationsfailedtoconvergeandavalid
¹
¤
forthatparametercouldnotbe
obtained(indicatedbyfiŒflinTable
5.1
).
FromTable
5.1
itisclearthat
¿
0
and
¿
sat
arethemostialparametersforallthreeob-
jectivefunctions,followedbythehardeningslope
h
0
whilethehardeningexponent
a
turnsout
tobetheleastsensitiveforallthevecrystalorientations.Thesensitivityanalysisisrepeated
withadifferentvalueofthestressexponent(
n
Æ
5)fortwodifferentcrystalorientations(fia"
andfib"inFig.
5.4
)andsimilarresultsof
¿
0
and
¿
sat
beingthemostinentialparametersfol-
lowedby
h
0
and
a
areobtained,asshowninthebottompartofTable
5.1
.Thisunderpinswhy
somestudies(e.g.[
Herrera-Solazetal.
,
2014
,
Zambaldietal.
,
2012
])didonlychoose
¿
0
and
¿
sat
asdesignvariablesintheoptimization.Lowersensitivityofhardeningparametersakintothe
presentstudyarealsoobservedby
Alcalaetal.
[
2008
].
Thus,theglobalsensitivityanalysishelpstoqualitativelyrankthemostinentialparame-
tersinaconstitutivelawandisverycialforthecurrentcomputationallyexpensivestudy
ofcrystalplasticityparameterestimation.
106
Table5.1:Resultsofglobalsensitivityanalysisforthefouradjustableparametersusingeachofthethreeobjectivefunctions
²
LD
,
²
topo
,and
²
combo
andtwovalues(
n
Æ
25and
n
Æ
5)forthestressexponentinthekineticsequationdescribedinSection
5.4.2
.
Tabulatedvaluesrepresent
¹
¤
asdescribed,withlargernumbersindicatingahigherrelativeoftherespectiveparameter.
CrystalorientationsarelabeledinaccordancewithFig.
5.4
.
Orientation
²
LD
²
topo
²
combo
¿
0
¿
sat
h
0
a
¿
0
¿
sat
h
0
a
¿
0
¿
sat
h
0
a
n
Æ
25
a1.480.500.1180.0540.660.5890.0910.0621.070.540.1040.058
b1.340.490.3450.0970.770.6700.2850.1301.060.580.3150.113
c1.470.500.1410.0930.660.6450.1170.0911.070.570.1300.092
d1.500.520.1510.1030.630.6730.1050.0911.070.600.1280.097
e1.120.510.1500.0360.850.7500.1200.0151.000.630.1340.025
f1.42Œ0.1220.0700.55Œ0.0300.0501.00Œ0.0700.060
g1.42Œ0.1200.0700.56Œ0.0800.0701.00Œ0.1000.070
n
Æ
5
a1.440.420.1110.050.740.5770.1190.0701.090.500.1150.06
b1.380.420.1100.050.750.6100.1250.0751.070.510.1170.06
107
5.7Results
Thepseudo-experimentalreferenceindentationsareobtainedwithaparametersetchosen
as
¿
0
Æ
31MPa,
¿
sat
Æ
63MPa,
h
0
Æ
400MPa,

°
0
Æ
10
¡
3
s
¡
1
,
n
Æ
25,and
a
Æ
2.25,basedonthose
ofaluminum[
KalidindiandSchoenfeld
,
2000
].Thesixdislocationinteractionparametersare
basedon[
Kubinetal.
,
2008
]withvaluesofself=1.4,coplanar=1.4,collinear=3.0,Hirthlock
=1.0,glissilejunction=1.4,Lomerlock=1.4.Inresponsetotheperformedsensitivityanalysis,
thethreemostialparametersareselectedasdesignvariablesfortheoptimizationwith
associatedsearchdomainsof[20,40]MPafor
¿
0
,[40,70]MPafor
¿
sat
,and[300,450]MPafor
h
0
.
5.7.1Ieofelementmeshsize
Todeterminewhetherthemeshsizeintheparametervalueidenation,thesame
optimizationisperformedwiththreedifferentdiscretizations(2532,4596,and6696hexahedral
elementswithlinearshapefunctions).Theeofthemeshresolutionisstudiedforan
exemplarycrystalorientation(fia"inFig.
5.4
)using
²
combo
astheobjectivefunction.EachNM
optimizationwasstartedwithidenticalinitialsimplex,andanerrortolerance
²
tol
Æ
0.005was
setastheterminationcriterion.
Figure
5.5
demonstratesthenegligibleuenceofmeshsizeontheidenilityofthe
optimizedparameters.Theessevolution(Fig.
5.5a
)issimilarforallthreediscretizations
andleadstostablenalparameters(Fig.
5.5b
).Basedonthesedings,ameshwith4596ite
elementsisselectedforthesubsequentstudies.
5.7.2Iofobjectivefunction
Figure
5.6
comparesforanexemplaryindentationdirection(fifflinFig.
5.4
)andforallthree
selectedobjectivefunctions(
²
LD
,
²
topo
,and
²
combo
fromlefttoright)theloadŒdisplacement
andsurfacetopographydeviationsafterparameteroptimizationtowithin
²
tol
Æ
0.005.Itis
observedthatwhenusingeither
²
LD
or
²
topo
astheobjectivefunctionforparameterestima-
108
(a)nessevolution
(b)optimizedparameters
Figure5.5:Iofmeshsizeonthestabilityoftheimplementedinverseanalysis.The
horizontallinein(
a
)denotesthetolerancecriterion;verticallinesin(
b
)showtheparameter
boundsfortheoptimization.
tion,therespectiveerrorinthenon-includedresponse(surfacetopographyfor
²
LD
andloadŒ
displacementincaseof
²
topo
)remainsrelativelylarge(graynumbersinFig.
5.6
).Incontrast,
theuseof
²
combo
minimizestheerrorinbothloadŒdisplacementandsurfacetopographyre-
sponsessimultaneously.
5.7.3Reproducibilityofparameteridentiationfordifferentobjectivefunctions
Toestablishthereproducibility,parameteroptimizationwiththeNMalgorithmisstartedfrom
edifferent(random)initialsimplicesforeachobjectivefunctionwhilekeepingtheindented
crystalorientation(fif"inFig.
5.4
)constant.Figure
5.7
showstherelativedeviationsoftheop-
timizedparameterscollectedfromtheerunsandcomparesthemamongthethreeobjective
functions(
²
LD
,
²
topo
,and
²
combo
)witherrortolerancesof
²
tol
Æ
0.2,0.01,and0.005.Theav-
eragestandarddeviationsfromthetargetsolutionforeachoftheninecasesshowninFig.
5.7
isrepresentedbyaverticalbaradjacenttothee.Clearly,withstrictertolerancethestan-
darddeviationamongmultipleoptimizationrunsisreduced,asthelengthoftheverticalbar
decreasesfromtoptobottomineachcolumnofFig.
5.7
.Itistobenotedthateventhough
theaveragestandarddeviationsarecomparablefor
²
LD
and
²
combo
,therelativescatterinthe
109
²
LD
²
topo
²
combo
²
tol
Æ
0.005
²
LD
Æ
0.003
0.091
0.002
²
topo
Æ
0.031
0.00460.007
0
10nm
Figure5.6:ComparisonofloadŒdisplacementandsurfacetopographydeviations(absolute
valuesbetween0to
Å
10nm)fromreferenceresponsewithparametersoptimizedtowithin
²
tol
Æ
0.005oftherespectiveobjectivefunction(fromlefttoright,
²
LD
,
²
topo
,and
²
combo
Æ
(
²
LD
Å
²
topo
)/2)foranexemplaryindentationalongdirectionfif"inFig.
5.4
.ThereferenceloadŒ
displacementcurveisshownbyblack.
optimizedvaluesfor
¿
0
and
¿
sat
isnegligiblewhen
²
combo
wasusedastheobjectivefunction
ascomparedto
²
LD
.Theobjectivefunction
²
topo
(basedonlyonsurfacetopography)exhibited
thepoorestreproducibility,whereascomparablereproducibilitieswereobtainedwith
²
LD
and
²
combo
.Thelargescatterin
h
0
isexpectedsinceithastheleastuenceonthenessamong
thethreedesignvariables(asobservedinTable
5.1
),rendering
h
0
relativelydiculttoidentify
accurately.Itisinterestingtonote(andpresentlyunclear)that
h
0
exhibitsacomparablyhigh
degreeofreproducibilityforthetwootherobjectivefunctions,
²
LD
and
²
topo
.
Itisnotedthatwiththemoststricttoleranceof0.005onlythreeoutofverunsgenerated
110
²
LD
²
topo
²
combo
0.2
0.01
²
tol
Æ
0.005
Figure5.7:Optimizedparameterset(
¿
0
,
¿
sat
,and
h
0
)usingdifferentobjectivefunctions(
²
LD
,
²
topo
,and
²
combo
,lefttoright)andtolerancevalues
²
tol
(toptobottom)resultingfrome
randominitialsimplicesforonecrystallographicindentationdirection.Thehorizontal
graylinemarksthetargetvalue(
Æ
1),verticallinesspantheboundsofeachparameter.The
verticalbartotherightofeacherepresentsthestandarddeviationoftheoverallparameter
set.
111
Figure5.8:Toprowcorrespondstotheevolutionofobjectivefunctionvalue(forthebestpoint
inthesimplex)withnumberoffunctionevaluations(cost)fordifferentobjectivefunctions(left
toright:
²
LD
,
²
topo
,
²
combo
)thatdidnotreach
²
tol
Æ
0.005(horizontalgrayline).Therelative
deviationbetweenresultingparametersetsandtargetvaluesisshowninthebottomrow.Ter-
minationbeforethemaximumallowablenumberoffunctionevaluations(45)resulteddueto
thedegeneracyofthesimplexatalocalminimum.
parametersthatmetthistolerance.Theoptimizedparametersetforthesenon-convergedruns
alongwiththeevolutionoftheirness(i.e.theobjectivefunctionvalueforthebestpointin
thesimplex)withnumberoffunctionevaluationsisshowninFig.
5.8
.Fromthenessplotsin
Fig.
5.8
itisevidentthatineachofthesixcasesthesimplexgotstuckinalocalminimumwith
negligibleimprovementoversubsequentgenerations.Despiteallsixidentiedparametersets
beingcomparablyfaroffthetarget,itisinterestingtonotethattheobjectivefunctionvaluesfor
²
combo
remainnotablylargerthanthosefor
²
LD
and
²
topo
.
112
5.7.4Robustnessofparameteridentiationfordifferentindentationorientations
Toestablishrobustness,parameteroptimizationusingeachofthethreeobjectivefunctionsis
repeatedforsevendifferentcrystalorientations(seeFig.
5.4
)startingfromrandominitialsim-
plices.Thetoleranceforminimizingeachobjectivefunction(
²
LD
,
²
topo
,and
²
combo
)issetto
²
tol
Æ
0.005,consistentwiththebottomrowinFig.
5.7
,andthenumberofobjectivefunction
evaluationsiscappedat45toavoidoverlyhighcomputationalcostincurredbytheCPFEsim-
ulations.
Figure
5.9
showsthetrajectoryoftheobjectivefunction(toprow)andthealdeviations
(bottomrow)betweentargetandoptimizedparametersetsuponeitherreachingthespecied
tolerance(graylineat0.005)orexceedingtheallowednumberoffunctionevaluations.For
²
LD
asobjectivefunction(bluecurves,Fig.
5.9e
),allthreeoptimizedparametersexhibitsomescat-
terofabout10%aroundtheirtargetvalues.When
²
topo
isusedasobjectivefunction(green
curves,Fig.
5.9f
),thescatterin
¿
0
and
¿
sat
ismuchlargerthanobservedfor
²
LD
but
h
0
isidenti-
relativelyaccurately.Despitethisapparentlylargescatter,theratio
¿
sat
/
¿
0
afteroptimiza-
tionisfoundtobefairlystableandveryclosetothecorrectone,i.e.therelativedeviationfrom
theirtargetisvirtuallythesamefor
¿
0
and
¿
sat
.Finally,with
²
combo
asobjectivefunction(red
curves,Fig.
5.9g
),thescatterof
¿
0
and
¿
sat
aroundthetargetisthesmallestamongallthree
objectivefunctions.However,acomparablylargescatterof
h
0
isobserved.
AsevidentinFig.
5.9b
,thetolerance
²
topo
·
0.005wasnotmetwithinthelimitednumberof
objectivefunctionevaluationsfortwooutofthefourselectedcrystalorientations,whichindi-
catesaweakoverallgradientin
²
topo
towardstheglobalminimum.Incontrast,allseven(four)
testedcrystalorientationsmetthistolerancewithin45iterationswhenusing
²
combo
(
²
LD
).
Figure
5.9c
illustratesforoneexemplarycase(blackline)outoftheseventestedindenta-
tionorientations(redlines)thetypicallyobservedtrendofanopposingevolutioninthecom-
ponents
²
LD
(blueline)and
²
topo
(greenline)thataverageto
²
combo
(blackline).
113
(a)
(b)
(c)
(d)
(e)
²
LD
(f)
²
topo
(g)
²
combo
(h)
²
dual
Figure5.9:Bottomrowrepresentsparameterestimationstabilityfordifferentindentationorientationsanddifferentessfunc-
tions(lefttoright:
²
LD
,
²
topo
,
²
combo
and
²
dual
).Toprowgivescorrespondingobjectivefunctionvaluevs.cost,withtolerance
²
tol
Æ
0.005indicatedbythegrayhorizontalline.Theanalysiswasperformedfor4differentcrystalorientationsfor
²
LD
and
²
topo
,7
differentcrystalorientationsfor
²
combo
,and6differentorientationpairsfor
²
dual
.Greenandbluecurvesin
c
indicate
²
LD
and
²
topo
comprisingtheblack(exemplary)
²
combo
evolution.
114
5.7.5Dual-orientationobjectivefunction
Theconcurrentuseofmultipleorientationsintheobjectivefunctionmightimprovethepa-
rameterestimationquality,asreported,forinstance,by
Herrera-Solazetal.
[
2014
].Inorderto
testforthispossibility,anobjectivefunction
²
dual
Æ
(
²
combo
(
a
)
Å
²
combo
(
b
))/2thataveragesthe
individualessesobtainedfromtwoorientations
a
and
b
isevaluatedforsixdifferentorien-
tationpairschosenfromthesevenindividualorientationsofFig.
5.4
.Randominitialsimplex
verticesandthetighttolerancevalueof
²
tol
Æ
0.005areselectedasbeforeandamaximumof
45objectivefunctionevaluationswereallowed.Thesimultaneoususeoftwoorientationsin
theobjectivefunction
²
dual
didnotnoticeablyimprovethequalityoftheoptimizedconstitu-
tiveparametersoverthesingleorientationcase
²
combo
(compareFig.
5.9h
toFig.
5.9g
).The
twomostsensitiveparameters,
¿
0
and
¿
sat
,couldagainbeidentiedtowithinafewpercentof
theirrespectivetargetvalues,with
h
0
againexhibitingnotablylargerdeviations.However,due
tothedoubledeffortperobjectivefunctionevaluation,theoverallcomputationalcosttoreach
asptolerance(0.005inthepresentcase)was,onaverage,abouttwiceaslargeasinthe
singleorientationcase(compareFig.
5.9d
toFig.
5.9c
).
5.7.6Iofthereferencepoint
ThestabilityoftheNMsimplexoptimizationalgorithmisinvestigatedbyselectingthreedif-
ferenttargetpoints(optima)inthreedifferentparameterbounds,aslistedinTable
5.2
.The
optimizationisperformedforanexemplaryorientation(fia"inFig.
5.4
)using
²
combo
asthe
objectivefunctionwith
²
tol
=0.005andstartingwitharandomsimplex.
ThedifferenttestcasesshowninTable
5.2
notonlyhighlightstheindifferenceofthesimplex
algorithmtoreferenceparametervaluesbutalsoindicatesthatevenifthereferenceparameters
areclosetotheboundaryofthedomainchosenfortheoptimization,theusedNMsimplex
algorithmwith
²
combo
astheobjectivefunctionisabletosuccessfullyidentifythem.
115
Table5.2:Inenceoftargetpointinparameterspaceontheoptimizedparametervaluesusing
²
combo
astheobjectivefunction
and
²
tol
=0.005.Thereferencevaluesforeachoftheparametersareshowninparenthesesandtherespectiveboundschosenforthe
optimizationareshowninbrackets.
¿
0
/MPa
¿
sat
/MPa
h
0
/MPa
²
combo
optimized(reference);[bounds]optimized(reference);[bounds]optimized(reference);[bounds]
19.98(20);[10,40]63.40(63);[40,70]382.0(400);[300,450]0.0038
49.60(50);[20,60]62.30(63);[40,70]416.8(400);[300,450]0.0045
30.76(31);[20,40]62.50(63);[40,70]151.5(150);[60,300]0.0040
116
5.7.7Predictionofslipsystemactivity
Todeterminehowaccuratelytheslipactivitycouldbecaptured,thedeviationofaccumulated
shearbetweenthereferenceparametersimulation(
°
ref
)andthesimulationwithoptimizedpa-
rameters(
°
opt
)iscollectedfromallFEintegrationpointswherebothaccumulatedshearvalues
exceeded1
£
10
¡
4
.Thisanalysisisperformedpereachofthetwelvefccslipsystemsforseven
differentoptimizedparametersetsasshowninFig.
5.10
.Theleftcolumnpresentstheseseven
constitutiveparametersets,whiletherightcolumncomparesthedistributionofthe(absolute)
point-wisedeviationofaccumulatedsheartotheaccumulatedshearvaluesobservedintheref-
erenceparametersimulation(excludingpointswhere
°
ref
Ç
10
¡
4
).Graycurvescorrespondto
dataonindividualslipsystems,whiletheblackandredcurvesrepresenttheoverallpopulation
acrossallslipsystems.
FromFig.
5.10
itcanbeinferredthattheclosertheoptimizedvaluesaretothetargetforall
threeadjustableparametersthebetteristheslipactivitycaptured.Thechosenobjectivefunc-
tionbasedonloadŒdisplacementandsurfacetopographyisratherinsensitivetothehardening
slope
h
0
,i.e.thetoleranceismetforallsevenparametersets,but
h
0
notablyinencesthe
slipactivities.Thus,includingslipmechanicsinformationintheobjectivefunctionmayim-
provetheidenticationof
h
0
,however,quantifyingindividualslipsystemactivityisextremely
cultfromrealexperiments.
117
Figure5.10:Relationbetweenaccuracyofslipsystemresponseandqualityofconstitutiveparameteridentication.Sevensetsof
optimizedparameters(top),eachfuing
²
combo
·
0.005,sortedbydecreasingaccuracy,andresultingforthesevencrystalorien-
tationsshowninFig.
5.4
.Deviationinaccumulatedshearbetweensimulationsusingthetarget(
°
ref
)andtheidenparameters
(
°
opt
)contrastedtotheaccumulatedshearvaluesobservedinthereferencevaluesimulation(bottom,excludingany
°
ref
Ç
10
¡
4
).
118
5.8Discussion
Theanalysisofreproducibilityandrobustnessfordifferentobjectivefunctionsrevealsafew
noteworthypointsthatareaddressedinthefollowing.
Thecombinationofbothoriginalobjectivefunctions,i.e.
²
LD
Å
²
topo
Æ
2
²
combo
,appearsto
increasetheoverallsteepnessoftheresultingnesshypersurface,particularlyinthevicinityof
thecorrectparameterset(target).Inotherwords,
²
combo
increasesfasterwithfidistanceflfrom
thetargetcomparedtoeither
²
LD
or
²
topo
.Thiscanbeinferredfrom(i)thegenerallycloser
matchbetweentheoptimizedandtargetvaluesof
¿
0
and
¿
sat
observedfor
²
combo
(Fig.
5.7
right)comparedtoeither
²
LD
or
²
topo
(Fig.
5.7
leftandcenter),inparticularatthetightesttol-
erance(bottomrow),and(ii)theconsistentlylargervaluesof
²
combo
comparedtoeither
²
LD
or
²
topo
for(non-converged)solutionsthatareallcomparablyfarfromthetarget(Fig.
5.8
).The
latterentailsthatobjectivefunctionvaluesof
²
LD
and
²
topo
inlocalminimaarecomparable
tothosefoundclosetothetarget.Hence,theywouldllaxtolerancesdespitebeingsub-
stantiallydifferentthanthetargetsolution.Correspondingly,
²
combo
appearstoexhibitamore
consistentgradientclosetothetargetpoint.Moreover,theoverallsmoothnessoftheobjective
functionisalsolikelyhigherfor
²
combo
asitbalancestheoftentimescomplementaryresponse
oftheindividualobjectivefunctions
²
LD
and
²
topo
.
Anotherwaytointerpretthesuperiorperformanceof
²
combo
canbebasedontheobserva-
tionsmadeinFigs.
5.9e
and
5.9f
where,ontheonehand,
²
topo
leadtoproperidentiationof
theratio
¿
sat
/
¿
0
,and
²
LD
resultedinamoreconsistentidentiationof
¿
0
.Hence,merging
²
LD
and
²
topo
canbeexpectedtocombineboththeseialattributes,asseemstobeindeed
thecase.
Inthecourseofexploringtheparameterspace,infeasiblepoints(wheretheobjectivefunc-
tioncouldnotbeevaluated,i.e.,theindentationsimulationdidnotconverge)areoftentimes
encountered.Nevertheless,thesearchneverterminatedduetotheseobstacles,sincethepro-
posedstrategyforadjustingthesimplexturnedouttobesuccessfultocontinuethesearch.
TheproposedinversemethodologywiththemodiNMsimplexoptimizationalgorithm
119
appearedtobeinsensitivetotherelativepositionofthetargetintheparameterspace.Thiswas
demonstratedbyselectingdifferentvaluesfor
¿
0
,
¿
sat
,and
h
0
andperformingmultipleopti-
mizationrunsthatallresultedinveryaccurateidenticationof
¿
0
and
¿
sat
,while
h
0
exhibited
relativelylargerscatter.
Thechoiceofusingasimulatedindentationbasedonthesameconstitutivelawasusedlater
onfortheparameteridenticationmeansthat,inprinciple,evenaninltolerance
(
²
tol
!
0)canbereached.Figure
5.9c
showsexamplesofstableratesofconvergence.However,
duetotheparticulartopographyoftheobjectivefunction
²
combo
,i.e.amuchhighercurvature
withrespectto
¿
0
and
¿
sat
comparedto
h
0
(afeaturecomparabletothefiRosenbrockflfunction),
itwasnotuncommonforthesimplextoshrinkinsizemuchquickerthanapproachingthe
optimumlocation.Sucharelativelysmallsimplexthentookmanystepstomovetowardsthat
optimum,hence,therateoffurtherconvergencewasnotablyreduced(notshowninFig.
5.9c
).
Inthisstudy,areasonabletolerancevalueandnumberofobjectivefunctionevaluations
areselectedowingtotheveryhighcomputationcost.Whencomparingthecurrentficomputa-
tionalexperiment"withrealexperimentalresults,thetightesttoleranceachievedinthisstudy
(
²
tol
Æ
5
£
10
¡
3
)couldbeextremelychallengingsincethechosenconstitutivedescriptionmight
notbeabletocaptureallaspectsthatinthemeasuredloadŒdisplacementorsurface
topographycharacteristics.Ontheotherhand,amorerealisticphysics-basedconstitutivede-
scriptionmightenableaclosermatchbetweensimulationandexperimentalreference,and,
thus,smallattainabletolerances.
Concurrentlyevaluatingtheresponsesoftwocrystallographicindentationdirectionsdid
notnotablyreducethescatterobservedintheresultingoptimizedparameters,inparticular
withregardto
h
0
.Thisisunderstandable,sincevirtuallyanysetof
¿
0
,
¿
sat
,and
h
0
ident
withonegivenindentationdirectionwillalsofuthesameobjectivefunctiontolerancefor
anotherindentationdirection(seeFig.
5.9g
).Hence,theinclusionofthissecondevaluation
doesnotaddsucientlyindependentinformation,butplainlyincreasestheevaluationcost.
Incaseofmaterialswithcrystalstructureshavingmultipleslipfamilies,suchashexagonalor
120
tetragonalsystems,multipleindentationdirectionsaremorelikelytoyieldindependentde-
formationresponsessincetherelativeactivityamongthedifferentslipfamiliestypicallyvaries
withindentationdirection,andhenceforthosesystemschoiceofcrystalorientation(s)would
playamoresigntrolethanthepresentcaseoffccmaterials.
5.9Summary
Basedonthecomprehensiveanalysisperformedinthisstudy,itcanbeconcludedthatin-
verseindentationanalysisusingsinglecrystalnanoindentationisareliableavenuetowards
successfulidentionofmaterialparametersinfccmaterials.Thesensitivitiesofthefour
adjustableparametersofthephenomenologicalcrystalplasticityconstitutivelawused,
¿
0
,
¿
sat
,
h
0
,and
a
,wereindifferenttobothcrystalorientationandthestressexponent
n
thatcharacter-
izesthematerialwbehavior.Aglobalsensitivityanalysisqualitativelyrevealedtheuence
ofthedifferentconstitutiveparameters,whichthenallowstoselectthemostinentialad-
justableparametersandthushelpstoreducethedimensionalityoftheoptimizationproblem.
IncorporatingbothloadŒdisplacementandsurfacetopographyinformationintoanobjective
functionincreasedtheeffectivenessoftheNelderŒMead(NM)simplexalgorithmtoaccurately
adjusttheconstitutiveparameters.ThemodiationtotheNMsimplexalgorithmproposedin
thisstudyappearstobequiteeffectiveasitallowedthesimplextoovercomeinfeasibleregions
intheparameterspace.Finally,theindentationresponsefromonecrystalorientationturned
outtobeientinpredictingtheconstitutiveparameters
¿
0
and
¿
sat
withhighcone,
while
h
0
hadslightlylowerconce.Usingtheseconclusionsformoreanisotropiccrystal
structuresthathavemultipleslipfamilies,suchashexagonalcommerciallyavailabletitanium,
thereliabilityofthe
IIA
methodologywillbetakenupinthenextchapterChapter
6
.
121
CHAPTER6
RELIABILITYOFINVERSEINDENTATIONANALYSESINHEXAGONALMATERIALS
6.1Background
AfterestablishingthereliabilityoftheInverseIndentationAnalysis(
IIA
)forface-centered
cubic(fcc)materialswithsingleslipfamilyinChapter
5
,inthischapterthismethodologyisex-
tendedtolesssymmetrichexagonalmaterialswheremultipleslipfamiliescanco-activateand
inturncanaffecttheecacyofthisparameterestimationmethod.Forsingleslipfccmaterials
theselectionoforientationdidnothavemuchinenceontheoutputparameters,whilefor
hexagonalmaterialswithdifferentslipfamilyiscoulddominate(ornot)foraparticulartexture.
Hence,itisnotunusualtopresumethatthechoiceofthecrystalorientationtoperformthissin-
glecrystalinverseidenationplaysanimportantrole.Moreover,forfccmaterials,sincethe
objectivefunctionforoptimizationwasrelativelysmooth(convex)themodideterministic
NelderŒMeadsimplexoptimizationalgorithmturnedouttobeient,whileduetothecom-
plexityintheparameterspaceforhexagonalmaterials(becauseofexistenceofmultipleslip
familieswithstarkcontrastintheslipactivities)ahybridstochastic-followed-by-deterministic
optimizationalgorithmmightbebettersuitabletoeliminateinfeasiblelocationsintheparam-
eterspace.
InthischapterthisInverseIndentationAnalysisistestedforexemplaryhexagonalmaterial
ofcommerciallyavailabletitanium(cp-Ti)byconsideringvirtualexperimentalresultsasrefer-
ence.LikeforcubicfccmaterialsinChapter
5
,suchvirtualexperimentsprovidecientmeans
toexclusivelyinvestigatethereliabilityofthemethodology(sincetheresultsareknownapri-
ori)whilewithrealexperimentsthereexistsadditionaleffects(relatedtoexperimentalerroror
shortcomingoftheusedmodel)thatmightaffecttheresultanddecreasetheciencyofthe
methodology.ThroughoutliteraturetherehavebeenmultipleCRSSvaluesofcp-Tiwhichhave
beensometimescontradictory
Lietal.
[
2013
],thusmakingitaninterestingmaterialtostudy.
122
Thechapterstartswithabriefbackgroundaboutidentionofslipparametersinhexagonal
materials,Section
6.2
,followedbyadescriptionofthematerialmodelbeingusedtoperform
thesinglecrystalindentationsimulations.Finally,theresultsofthereliabilitystudyisdiscussed
inSection
6.6
.Itiscriticaltonotethattheiencyofthereliabilityofthe
IIA
methodologyis
crucialinimplementinginevenlowersymmetricbody-centeredtetragonal
¯
-Snthathaseven
highernumberofslipfamiliesaccommodatingadeformation(13availableslipfamilies).
6.2Introduction
Crystallographicslipinhexagonalmetalsinvolvesanumberofgeometricallydistinctslip
familiescharacterizedbytheirslipdirectionandslipplane,suchasbasal,prismatic,orpyra-
midalamongwhichthebasalandtheprismslipfamiliesrequiremuchlowershearstressesto
slipcomparedtopyramidalslipsystems[
Hutchinson
,
1977
].Duetotheparticularhexagonal
latticesymmetry,manyofthosefamilieshaveonlyfewsymmetricallyequivalentslipsystems
(familymembers).Furthermore,differentslipfamiliesgenerallybecomeactiveatdifferentre-
solvedshearstresses.Theseintrinsiccharacteristicsrenderthemechanicalprocessing(rolling,
stamping,extrusion)ofhexagonalmetalstechnologicallymuchmorechallengingthanforcu-
bicmetalssuchasaluminumorsteels.Consequently,substantialefforthasandisbeende-
votedtosimulatethecrystalplasticityofhexagonalmetals(andtheiralloys)withthegoalto
understandandpredictthebulkmechanicalresponseaswellasitsgrain-scalemicromechan-
ics[
MayeurandMcDowell
,
2007
,
Bieleretal.
,
2009
,
Zhangetal.
,
2010
,
Yangetal.
,
2011
,
Wang
etal.
,
2011
,
ChengandGhosh
,
2015
,
Choietal.
,
2012
,
Hémeryetal.
,
2017
,
Zhangetal.
,
2015a
,
Baudoinetal.
,
2018
].Thus,theabilitytoquantifytheintrinsicslipresistancesbecomescrucial
innotonlytheaccuratepredictionofthesesinglephasematerialsbutalsotodeterminethe
effectofalloyadditionontheirplasticity.
Anumberofmethodshavebeensuggested,bothexperimentallyandviasimulation,to
quantifytheso-calledcriticalresolvedshearstress(CRSS)atwhichthedifferenthexagonalslip
familiesbecomeactive.Adetailedlistofsuchmethodscanbefoundin
Lietal.
[
2013
]thatalso
123
introducedanewmethodologytoidentifytheCRSSvaluesofhexagonalsystemsexperimen-
tallybyrelatingthefrequencyofsurfacesliptraceobservationstotheirtheoreticalpropensity.
ZambaldiandcoworkerspioneeredtheideaofiterativelyadjustingtheCRSSvaluesinacrystal
plasticitysimulationofsinglecrystalindentationwithaconosphericaltipuntilthesimulated
loadŒdisplacementresponseandremnantsurfacetopographymatchthecorrespondingexper-
iment[
ZambaldiandRaabe
,
2010
,
Zambaldietal.
,
2012
].
Herrera-Solazetal.
[
2014
]obtained
singlecrystalmaterialparametersusingagradient-basedLevenbergŒMarquardtoptimization
algorithmthatmatchedstressŒstrainresponsesofniteelementsimulationsofarepresentative
volumeelementstocorrespondingexperimentswithmultipleloadingconditions.Theabove
twomethodologiesfallundertheclassofinverseproblemswheretheproperidentionof
theCRSSvaluesminimizestheerrorbetweenthesimulatedandexperimentalresponse.Ina
morerecentstudy,theplasticityyieldparameters(CRSS),inhexagonalTi-7Al,alongwiththe
correspondingevolutionoftheirplasticbehaviorwasinvestigatedbothatroomtemperature
andhightemperatureusingfarhighenergyX-raydiffractionmethods(ffŒHEDM)
Pagan
etal.
[
2017
,
2018
].Withsuchdiversetechniquesofparameteridenationthereexistsawide
discrepancyinthesubsequentreportedvaluesofCRSSforthesamematerial,[
Lietal.
,
2013
,
Wangetal.
,
2017
,
Dastidaretal.
,
2015
],therebydecreasingthereliabilityofsuchpublishedval-
ues.Inthischapterthe
IIA
methodology,thatuses
singlecrystal
responsetoidentifytheCRSS
valuestherebyeliminatingtheauxiliaryeffectsassociatedwithgrainboundaries,precipitates,
andcrystallographicneighborhood,isusedtostudyit'scacyforhexagonalmaterials.The
originalmethodologyproposedby
Zambaldietal.
[
2012
]andlatermodiedandstudiedin
detailin[
ChakrabortyandEisenlohr
,
2017
]whereitsreliabilitywasalsoestablishedforfaceŒ
centeredcubicmaterialsChapter
5
.Asamodelhexagonalmaterial,commerciallypuretita-
nium(cp-Ti)isanalyzedandthepossibilityofitstwinningisneglectedtofocusexclusivelyon
itsslipbehavior.Moreover,sincecomparingactualexperimentalindentationresponsewith
simulationincludesthelimitationsoftheunderlyingcrystalplasticitymodelŒthereliabilityof
thismethodologyistestedagainstvirtualexperimentswheretheexpectedCRSSvaluesforthe
124
Figure6.1:Schematicrepresentationoftheinverseindentationanalysis.
slipfamiliesareknownapriori.
Thischapterstartswithadetaileddescriptionofthemethods,Section
6.3
followedbyintro-
ducingtheconceptofvirtualexperimentsinSection
6.4
withthedescriptionoftheconstitutive
crystalplasticitymodelinSection
6.4.1
.Thereliabilitystudyofthismethodincp-Tiisdetailed
inSection
6.6
followedbydiscussionaboutsomeoftheconcernsregardingthemethodology.
6.3CRSSvaluesfromsurfacetopographyandinstrumentedindentation
TheCRSSvaluesfordifferentslipfamiliesareinverselypredictedbycomparingtheexperi-
mentalandcrystalplasticityiteelement(CPFE)simulatedsinglecrystalnanoindentation(s)
responseswithaspheroconicalindenter.ThisInverseIndentationAnalysis(
IIA
)methodology
(shownschematicallyinFig.
6.1
)isassociatedwithanobjectivefunction,wheretheCRSSvalues
aretheadjustableparametersforoptimization,andanoptimizationalgorithmthatminimizes
theobjectivefunctionwiththeoptimumpointbeingthepredictedsolutionfortheCRSSval-
ueswiththismethodology.Generally,fromindentationteststwomajoroutputsareobtainedŒ
theloadŒdisplacementresponseandthesubsequentsurfacetopography.Ithasbeenestab-
125
lishedinChapter
5
andin
ChakrabortyandEisenlohr
[
2017
],thatcombiningboththeloadŒ
displacementresultswiththesurfacetopographyprovidedabetterchanceofaccuratelyiden-
tifyingtheparametersfortheoptimizer.Hence,inthisstudyaswell,aweightedsummation
oftheerrorintheloadŒdisplacement
²
LD
andsurfacetopography
²
topo
betweenthesimulated
andthereferencedataisusedtotheobjectivefunction.Thechosenweightsbeing0.3
for
²
LD
and0.7for
²
topo
.Thechoiceofgivingmoreweightagetotheerrorinsurfacetopogra-
phyisbasedontheassumptionthattheslipbehaviorisredmoresigntly(andnon-
uniquely)inthesurfacepileŒuptopographyascomparedtotheloadŒdisplacement.Moreover,
inthisworkahybridoptimizationalgorithmisusedtominimizetheobjectivefunctioncon-
stitutingatastochasticparticleswarmoptimization(pso)followedbythedeterministic
NelderŒMeadsimplex(NM)tominimizetheobjectivefunction.Theideabehindsuchase-
lectionistoensureproperexplorationandexploitationoftheparameterspace,inthat,the
stochasticalgorithmensuresahigherexplorationwhilethedeterministicalgorithmisabetter
exploiter.
6.4Virtualexperiments
Inthecontextofthispaper,virtualexperimentsrelatetotheprocedureofgeneratingex-
perimentaldatawithsimulationswithaprioriknownparameters.Hereinweemploycrystal
plasticitysimulations,thattakeintoaccountthetextureeofthematerial,tosimulate
thematerialresponseaftersinglecrystalnanoindentation(s).Performingsuchavirtualexper-
imentalanalysisisbenecialindeterminingthereliabilityofamethodologysincethesolution
isknownaprioriwhichleadstoproperquantionoftheerrorbetweentheactualsolution
andthepredictionbyexclusiveconsiderationofmethodologywithouttheinnceofother
factors(suchasprecipitates,grainboundaries,inclusions,etc.).Thecrystalplasticitysimula-
tionsareperformedusingthematerialsolverinDAMASKandusingthecommerciallyavailable
iteelementboundaryvalueproblemsolverMSCMarcŒMentat.Thenumericsofthecrystal
plasticityiteelementsimulationisthesameasthatdiscussedinthepreviouschapterChap-
126
ter
5
.Theconstitutivematerialdescriptionandthegeometryusedinthisstudyisdiscussed
next.
6.4.1Constitutivematerialmodel
Themodelframeworkisbasedontheitestraindeformationtheorywheretotaldeformation
gradient,
F
,
F
Æ
F
e
F
p
(6.1)
ismultiplicativelydecomposedintoplasticandelasticcomponents(
F
e
and
F
p
,[
Lee
,
1969
]).
Therelationbetweenplasticdeformationgradient
F
p
andtheplasticvelocitygradient
L
p
is
givenbythewrule:

F
p
Æ
L
p
F
p
(6.2)
Theplasticvelocitygradient
L
p
Æ
X
®

°
®
s
®

n
®
(6.3)
isadditivelycomposedfromcrystallographicslipatrates

°
®
.Theunitvectors
s
®
and
n
®
align
withtheslipdirectionandslipplanenormalofeachslipsystem
®
Æ
1,...,
N
,with
N
indicating
thenumberoftotalslipsystemsacrossallconsideredslipfamilies.Theresolvedshearstress
¿
®
Æ
C
³
F
e
T
F
e
¡
I
´
/2:
s
®

n
®
(6.4)
isthedrivingforceforslip,where
C
denotesthe(fourth-order)tensorofelasticity.
Crystallographicslipiscommonlymodeledbasedonthephenomenologicalconstitutive
descriptionpioneeredby
Peirceetal.
[
1982
].Inthatmodel,whichisalsousedforthisstudy,the
dislocationdefectstructureisparameterizedintermsofastress
g
®
thatresistssliponsystem
®
.Thiscriticalresolvedshearstressevolvesinthecourseofslipas[
Hutchinson
,
1976
]

g
®
Æ
X
¯
q
®¯
h
0
¯
¯
¯
¯
¯
1
¡
g
¯
g
1
¯
¯
¯
¯
¯
a
sgn
Ã
1
¡
g
¯
g
1
!
¯
¯
¯

°
¯
¯
¯
¯
,(6.5)
127
withinitialhardeningslope
h
0
Æ
200MPa,hardeningexponent
a
Æ
2.0,andsaturationslipre-
sistance
g
1
Æ
300,200,and300MPaforbasal,prism,andpyramidal
h
c
Å
a
i
slipfamiliesre-
spectively,neglectingthesliponthepyramidal
h
a
i
planes(aswasalsodonein
Zambaldietal.
[
2012
]).Forthecurrentvirtualruns
q
®¯
isselectedas1forbothselfhardening(
®
Æ
¯
)andlatent
hardening(
®
6Æ
¯
).Crystallographicslipoccursatrates

°
®
Æ

°
0
¯
¯
¯
¯
¿
®
g
®
¯
¯
¯
¯
n
sgn
¡
¿
®
¢
,(6.6)
withreferenceshearrate

°
0
Æ
0.001s
¡
1
andstressexponent
n
Æ
20.Thekinematicsofthemate-
rialsolveroutlinedaboveareimplementedaspartoftheopensourcesimulationtoolkitfiDüs-
seldorfAdvancedMaterialSimulationToolkitfl(DAMASK)[
Rotersetal.
,
2012
]whichiscou-
pledwiththecommercialiteelementsolverMSCMarctosolvethesinglecrystalindentation
boundaryvalueproblem.
Forthisinverseanalysistheinitialvalueofthe
g
forthethreeslipfamilies,basal,prism,and
pyramidal
h
c
Å
a
i
(whicharethemostobservedslipfamiliesincp-Ti),areconsideredasad-
justableparametersforoptimizationwithathree-dimensionalparameterspace.Pyramidal
h
a
i
slipandtwinisneglectedtoreducethedimensionalityoftheproblem,whileotheradjustable
parametersofthemodelsuchas
g
1
ofeachfamily,initialhardeningslope
h
0
,exponents
n
,
and
a
areallxedduetotheirlowersensitivitytotheobjectivefunctionasdemonstratedinthe
previouschapterinSection
5.6.4
.
6.4.2Virtualindentation
Thecrystalplasticityiteelement(CPFE)indentationsimulationsareperformedusingcom-
mercialsoftwareMarc2013.1(MSCSoftwareCorporation,NewportBeach,CA)andusingthe
materialmodelinDAMASKthatisinterfacedthroughthematerialsubroutinefihypela2flon
computeserversmaintainedbytheDivisionofEngineeringComputingServicesatMichigan
StateUniversity.Themodelgeometry,showninFig.
6.2
,consistsof19980hexahedralelements
and22693nodeswithdimensionsof8
&
mx8
&
mx4
&
m.Furthermore,theregionclosetothein-
128
Figure6.2:GeometrygeneratedinMarc-Mentattoperformtheindentationsimulations.
denterisdiscretizedsinceitexperiencesthemostdeformation.Thenodaldisplacements
areonthebottomandoutersurfaces.Indentationisperformedtoadepthof0.3
&
mwitha
rigidspheroconicalindenteroftipradius1
&
mwithaconeangleof90°andnocontactfriction
isconsideredbetweentheindenterandthesubstrate.Theverticaldisplacementoftherigid
indenterto0.3
&
misdonein10secondsdiscretizedinto800equalincrements.Additionally,
toreducethecomputationtime,thegeometryisdecomposedintofourequaldomains(i.e.,
sectorsof90°each)withinMarcandthensimulated.
6.5Hybridoptimizationalgorithm
Asmentionedpreviouslytheinverseanalysisusesahybridoptimizationalgorithmthatis
writteninPython.Thetwoalgorithmsbeingusedaretheinitialstochasticparticleswarmop-
timization(pso)algorithmdescribedin[
EberhartandShi
]andaversionofthede-
terministicNelderŒMead(NM)simplexalgorithmasusedin
ChakrabortyandEisenlohr
[
2017
]
129
Figure6.3:Illustrationofthestochasticparticleswarmoptimizationalgorithmbeingusedin
thisstudyfor
IIA
.
andinitiallyintroducedby
NelderandMead
[
1965
].
TheNMsimplexmethodhavealsobeenintroducedindetailinpreviouschapterChapter
5
,
andthesameisusedinthisstudyaswell(samevalueofthealgorithmparameters).Thereason
forchoosingthisdeterministicNMsimplexmethodistwofoldŒrstitiseasytoimplement,
andsecondlyitisagradientfreemethodwhichishighlybsincecalculationofgradi-
entsforsuchanon-linearFEproblemiscomputationallyextremelyexpensive.Thestochastic
psoalgorithmusedinthisstudyusessetofalgorithmparametersassuggestedby
Eberhartand
Shi
.Inthisalgorithmthetrajectoryofaparticleinthepopulationisgovernedbytwoattractors
Œi)globalbestpositionoftheparticleinthepopulation;andii)particle'sownbestpositionas
illustratedinFig.
6.3
.Thisallowsecientexplorationoftheparameterspacebeforeclosing
towardsthe(global)optimumsolution.Forthe
i
th
particleina
D
-dimensionalspacerepre-
sentedas
X
i
Æ
(
x
i
1
,
x
i
2
,...,
x
iD
)andwiththebestpreviousposition(positionwithit'sbest-
ness)givenby
P
i
Æ
(
p
i
1
,
p
i
2
,...,
p
iD
),andwiththeglobalbestposition(positionoftheglobal
130
bestess)being
P
g
,thevelocityoftheparticle
i
,
V
i
,andtheirsubsequentposition
X
i
,
new
arethengivenas:
V
i
Æ
w
¤
V
i
Å
c
1
¤
rand()
¤
(
P
i
¡
X
i
)
Å
c
2
¤
rand()
¤
(
P
g
¡
X
i
)(6.7)
X
i
,new
Æ
X
i
Å
V
i
(6.8)
(6.9)
c
1and
c
2areaccelerationandaresetto1.2inthisstudywith
w
beingtheinertiaweightthatwas
setto0.3.therand()arefunctionsgeneratingrandomnumbersbetween(0,1).Themaximum
velocityineachdirectionwasalsorestrictedto75%inthatdimension.Itisimportanttonote
thatthechoiceoftheseparametervaluesarebasedonageneralthumbrule,sinceduetothe
highcomputationcostofeachCPFEcalculationaparametersensitivitystudyisimpractical,
andhenceavoided.Moreover,duetothishighcomputationcostthetotalnumberofparticlesin
thepopulationisrestrictedto10(forthecurrentthreedimensionalproblem),whichislowand
henceanexhaustiveexplorationisnotpossible,however,havingthesubsequentdeterministic
rundidachieveresultsclosetotheglobalminimumaswillbeshownlater.
Anothercriticalaspectofthisinverseproblemisthepresentofinfeasibleregionsinparam-
eterspace,orcombinationsofparametervaluesforwhichtheCPFEsimulationfailstocon-
verge.Forthepsoalgorithm,wheneversuchaninfeasiblepointisobtainedaveryhighpenalty
of1
£
10
10
isreturnedtherebycausingalargepenaltyinthatlocation.Thisispossiblesincethe
stochasticpsodoesnotrelyontheexactvalueofthetnesstoproceed.Whileforthedetermin-
isticNMsimplexalgorithmwheretheexactvalueofnessisneededtoproceed,thealgorithm
avertsanyinfeasiblepointbyringitaboutthecenterofgravityofthecurrentsimplexas
explainedinSection
5.6.2
andalsopublishedin
ChakrabortyandEisenlohr
[
2017
].
Thetolerancefordecidingtheoptimumissetto2
£
10
¡
2
alongwithanadditionaltermi-
nationcriterialofamaximumof100FEevaluationsforeachoftheoptimizationalgorithm.
Thetoleranceof2
£
10
¡
2
isdecidedsuchthatitisstrictenoughtocapturetherelativeactivity
ofthedifferentslipfamiliestogiveauniquesolutionoftheoptimization(withrespecttothe
131
reference),whilelaxenoughtonotrequireveryhighFEruns(morethan100).Theinitialpop-
ulationof10particlesforthepsoalgorithmisselectedrandomlyfromadomainof10MPato
400MPaforallthethreeslipfamilies,whiletheinitialsimplexforthedeterministicNMsim-
plexalgorithmwithfourpoints(correspondingtotheverticesofthefourdimensionalsimplex
inthecurrentproblem)isconstructedbyarandomselectionfromtheoptimizedal)pso
population.RandomselectionforgeneratingtheNMsimplexissuitablesincetheoptimized
psopopulationislessscatteredandmoreconcentratedinthefeasibleregionintheparame-
terspace.Furthermore,similarto
ChakrabortyandEisenlohr
[
2017
],theparameterdomainis
convertedtoaunitdomaintoexcludeinenceofindividualvaluesoftheparameters.
6.6Reliabilityofinverseindentationanalysis
Theadvantageofusingvirtualexperimentstopredictthereliabilitiesofthedifferent
methodologiesofpredictionofCRSSvaluesistheaprioriknowledgeoftheexpectationŠhence
afaircomparisoncanbedonebetweenthepredictedCRSSandthetarget/expected/input
CRSS.Thereliabilityofthe
IIA
methodologyisassessedbasedonsixdifferentsinglecrystal
orientationsandfortwodifferentCRSSinputvalues.
6.6.1SelectingorientationsforperformingtheInverseIndentationAnalysis
Akeyaspectforhexagonal(orotherlowsymmetrymaterials)isthepresenceofmultipleslip
familiestoaccommodateadeformation,hence,any(randomly)selectedorientationmaynot
besucientincapturingtheactivityfromalltheslipfamilies,therebymakingsomeofthe
slipfamilieslesssensitivetodeformation.Thus,aneffectiveselectionofcrystalorientationto
performthe
IIA
inhexgonalcp-Tiiscrucialtoenabletheoptimizertoaccuratelyidentifytheslip
parametersforalltheslipfamilies.Previously,suchaconsequencewasavertedbyconsidering
allorientationstogetherin[
Zambaldietal.
,
2012
],however,thisdependencewasnotexplicitly
addressed.
Inthisstudy,therelativesensitivitiesofdifferentslipfamilies(i.e.basal,prism,andpyrami-
132
Figure6.4:Differentcrystalorientationsusedtoperformthesensitivityanalysisinhexagonal
cp-Ti.
dal
h
c
Å
a
i
)fordifferentcrystalorientationsselectedfromthestandardstereographictriangle
ofhexagonalcp-Ti,showninFig.
6.4
,isqualitativelyanalyzedbycalculatingthechangeinthe
combinedloadŒdisplacementandsurfacetopographyresponsesforachangeintheCRSSval-
uesforthedifferentslipfamiliesforthedifferentcrystalorientations.
Thissensitivityanalysisisperformedusingamethodofelementaryeffectsorig-
inallyproposedby
Morris
[
1991
]andby[
ChakrabortyandEisenlohr
,
2017
].The
methodisalsooutlinedinthepreviouschapterinSection
5.6.4
,whereitwasusedtoidentify
themostsensitiveadjustableparametersofthephenomenologicalpowerlawbeingusedtode-
scribethematerialbehavior.Thismethodessentiallyprovidesaientwayofcalculatingthe
derivativeoftheoutputfunction(loadŒdisplacementandtopographyresponse)withrespectto
achangeininputparameter(CRSS)inahighdimensionalspace.Theciencyisinselecting
relevantpointsintheparameterspace.Thesensitivityofacrystalorientationtodifferentslip
families'CRSS(parameters)isthencalculatedbytakingthemeanofabsolutesensitivitiesfor
thedifferentparametersforthecrystalorientation.Itisimportanttonotethatsuchamethod
isaqualitativeestimationoftherelativesensitivitiesandprovidesaguidanceinmakinganef-
133
fectivechoiceofcrystalorientationstoperformthe
IIA
.Inthisstudythesensitivityanalysisis
performedforalltheorientationsshowninFig.
6.4
forthebasal,prism,andpyramidal
h
c
Å
a
i
slipfamilieswithintheboundsof[50Œ500MPa]forallthethreefamilies.Theresultingsensi-
tivitiesisshownqualitativelyinFig.
6.5
forthethreedifferentfamilies(blueŒbasal;redŒprism;
andorangepyramidal
h
c
Å
a
i
).Thelocationofthecirclescorrespondtothecrystalorientation
forwhichtheanalysisisdone,whilethediameterofthecirclerepresentsthesensitivityofthe
particularslipfamily.Thus,foracrystalorientationhavingahighersensitivityforthebasalslip
family(meaningbasalslipsystemsaredominantindeterminingthedeformationresponsefor
thatcrystalorientation)itwouldbehavingalargercirclecomparedtothoseoftheotherslip
familiesatthesamelocation.Theimportanceofperformingsuchananalysisistheunderlying
assumptionthatorientationforwhichaparticularslipfamilyishighlysensitive(dominantslip
mode;largercircle),theoptimizationalgorithmwillhaveabetterchanceofaccuratelypredict-
ingtheCRSSvalueofthatfamilywhentheparticularorientationisselectedforoptimization.
Basedontheaboveassumption,ahypothesiscanbeproposedthatstatesthatthecrystalori-
entationswherealltheslipfamiliesarehighlysensitivetoallthethreeslipfamiliesshouldbe
selectedtoperformthe
IIA
foraccuratepredictionsoftheadjustableparameters(CRSS)with
theleastcomputationtime.
6.6.2Results
Totestthishypothesis,six(arbitrary)differentcrystalorientationsareselected,andthere-
liabilityofthe
IIA
forpredictingCRSSvaluesforthethreedifferentslipfamiliesconsidered
(i.e.basal
h
a
i
,prism
h
a
i
,andpyramidal
h
c
Å
a
i
)isevaluatedfortwodifferentsetsofin-
putCRSSvalues.TheresultsthusobtainedarelistedinTable
6.1
.Asmentionedearlier,
theInverseIndentationAnalysisinthisstudyusesahybridoptimizationalgorithmwith
astochasticparticleswarmoptimization(pso)followedbythedeterministicNMsimplex
method.InTable
6.1
thebestparameters(globalbest)obtainedaftertheinitialpsorunis
shownincolumns6,7,8,whilethealpredictedparameters,CRSS,oftheoptimizerafter
134
(a)basal
h
a
i
(b)prism
h
a
i
(c)pyramidal
h
c
Å
a
i
Figure6.5:Sensitivityofdifferentcrystalorientationstodeformationondifferentslipfamilies
asrepresentedbytheradiusofthecircles.
135
thesubsequentNMsimplexalgorithmisshownincolumns10,11,and12,foreachofthethree
slipfamiliesofbasal,prism,andpyramidal
h
c
Å
a
i
respectivelylabelledasfibasal",fiprism"and
fipyrCA"inTable
6.1
.Rows2,3,and4showsthesensitivityvaluesforthebasal,prism,and
pyramidal
h
c
Å
a
i
slipfamiliesrespectivelybasedonthesensitivityanalysisdiscussedinthe
previoussectionandillustratedinFig.
6.5
.InFig.
3.11
,thecolumnshowsthesixsingle
crystalorientationsforwhichthereliabilitystudywasdone,moreover,thetableisdividedinto
twosectionsŒthetopsixrowsshowsthe
IIA
resultswithareferenceparameterof[80,110,180]
MPacorrespondingtobasal,prism,andpyramidal
h
c
Å
a
i
slipfamiliesrespectively;whilefor
thebottomsixrowstheoptimizationresultswithareferenceparameterof[120,60,180]MPa
forthethreeslipfamilies.Thereasonforchoosingsuchreferencesisbasedonthefactthatin
theformercasethetwomostactiveslipfamiliesincp-Ti,basalandprism,havetheirCRSSval-
uesveryclosetoeachother(i.e.,80and110MPa)whileforthelattercasethesevaluesarefar
apart(i.e.,120and60MPa).Hence,itcanbepresumedthatfortheformerreferenceCRSScase
wherethebasalandprismslipfamilieshavecomparablevalues,thetaskofaccuratelyidenti-
fyingthebasalandprismCRSSvaluesismorecultsincebothbasalandprismsystemscan
leadtothenaldeformationresponseatsimilarglobalstressstate(duetotheircloseactivation
values)ascomparedtothelatterreferenceCRSScase.Moreover,thetworeferenceCRSScon-
ditionsalsorepresentstwoclassesofhexagonalmaterialsŒone(likecp-Ti,
c
/
a
ratiolessthan
1.6333)wheretheprismismoreactivethanbasal;andothers(likeTi525,
c
/
a
rationgreaterthan
1.6333)wherebasalismoreactivethanprismslip,therebyfurtherhighlightingtheacyof
the
IIA
methodology.
ItcanalsobenoticedfromTable
6.1
thatforbothCRSSreference,thealoptimizedpa-
rametersareclosetotheexpectedreferenceformostofthesixorientationsexceptorientation
fig".Also,inalmostallruns(forbothreferenceCRSSreferences)theoptimizationproceededto
thesubsequentNMsimplexrunaftertheinitialpso,exceptforcrystalorientationfid"forCRSS
reference[120,60,180]MPa,wherethetoleranceof2
£
10
¡
2
isachievedinthepsostep.As
canbeeseenfromtheresults,sucharelativelylaxtoleranceselectedforthecurrenthexagonal
136
materialstudy,ascomparedtothatsetinfccmaterials,isstrictenoughforthepredictedpa-
rameterstobeuniqueandclosetotheglobalminimum(referenceCRSSvalueshighlightedin
boldinTable
6.1
).ForthereferenceinputCRSSof[80,110,180]MPacorrespondingtobasal,
prism,andpyramidalslipfamilies,thetoleranceof2
£
10
¡
2
isnotsatfororientationsfia",
fic",andfiq"(apartfromfig").Amongthethree,fororientationsfia"andfic"theNMsimplex
optimizationreachedthemaximumallowableFEcalculationsof100therebyleadingtotheter-
minationoftheoptimization,whilefororientationfiq"thesimplexconvergedprematurelyto
alocalminimum.ForthereferenceinputCRSSvaluesof[120,60,180]MPacorrespondingto
basal,prism,andpyramidalslipfamilies,thetoleranceisnotachievedfororientationfic"(apart
fromfig")becausethesimplexconvergedprematurelytoalocalminimum.
137
Table6.1:Resultsofthe
IIA
analysisperformedforsixdifferentcrystalorientationslabelledinFig.
6.4
fortwodifferentreference
input
CRSS.
OrientationsensitivityPSOresultsNMresults
basalprismpyrCAnessbasalprismpyrCAessbasalprismpyrCA
80
,
110
,
180
a0.430.440.8720.1909165.9768.47260.560.04783.06104.55179.22
c0.410.470.8640.1736168.0466.7238.70.03481.14118.1196.1
d0.490.2840.8650.255542.7172.7330.80.00781.9110.3179.3
g0.2470.0890.660.1533400.0132.6110.560.1533400.0132.6110.56
k0.460.480.850.153113.6793.15282.40.016480.0114.17187.63
q0.450.480.900.1161100.3100.4219.40.023381.31116.7192.1
120
,
60
,
180
a0.430.440.8720.09168.3132.91300.670.019129.757.1185.0
c0.410.470.8640.106144.544.8394.50.098154.738.6399.0
d0.490.2840.8650.004120.360.6181.1xxxx
g0.2470.0890.660.1441370.068.2117.10.1441370.068.2117.1
k0.460.480.850.0325127.853.7200.20.019129.456.0187.0
q0.450.480.900.1932236.729.8269.10.012113.861.1173.6
138
6.7Discussion
Theintermediateoptimumobtainedafterthestochasticpsoalgorithm(shownincolumns
6,7,8inTable
6.1
)showsthelimitationofthepsoalgorithminachievingthenesstolerance
(of2
£
10
¡
2
)formostorientations.However,forbothreferenceCRSSruns,theinverseanalysis
isabletoapproachthetargetafterthedeterministicoptimizationalgorithmformostofthe
crystalorientationsstudied(shownincolumns10,11,12inTable
6.1
).Thisisnotunexpected,
since,thepsoalgorithmisusedtoientlyexploretheparameterspaceinordertoidentifya
feasibleregion.Moreover,thepopulationsizeisalsoquitesmall(i.e.,10particles)whichalso
reducesthecacyofthepsoalgorithm.
Inbothcases,theoptimizationsolutionissigantlydifferentfromthetargetforthecrys-
talorientationlabelledfig"thatalsohastheleastsensitivityforallthreeslipfamiliesascom-
paredtothesensitivitiesoftheslipfamiliesforalltheothereorientations,illustratedand
labelledinFig.
6.6
.
Asmentionedintheprevioussection,forsomeoftheorientationsinbothreferenceCRSS
runs,thetolerancewasnotmet(apartfromorientationfig")inthenumberofFEruns.
However,therelativeinenceofeachoftheslipfamiliesisqualitativelycapturedforallof
theseorientationsthathadhighsensitivitiesforalltheslipfamiliescomparedtoorientation
fig"(wheretheoptimizationfailed).Itisalsointerestingtonotethatfortheorientationswhere
theobjectivefunctiontoleranceof2
£
10
¡
2
ismet,theoptimizedvaluesarewithin7%errorof
thetargetvalues,indicatingtheeffectivenessofthechosentolerance.Thus,theinitialhypoth-
esisaboutselectingcrystalorientationswhere
all
thethreeslipfamiliesarehighlysensitive
forperformingthisinverseanalysiswiththeproposedhybridoptimizationalgorithmappears
tobeagoodoneinaccurateandreliableestimationofthoseCRSSvalues.Therefore,sucha
methodologycannowbeusedinevenlowersymmetricbody-centeredtetragonalSntoiden-
tifyitsplasticityparameters.
139
Figure6.6:ResultingCRSSvaluesforallthesixsinglecrystalindentationsforthetwodifferentreferenceCRSS.
140
6.8Summary
AnimportantchallengeforidenationofCRSSvaluesforthedifferentslipfamiliesus-
ingsinglecrystalorientationsistheselectionofthecrystalorientation(s).Mostoften,abunch
oforientationsareselectedandusedtominimizetheerrorbetweenthesimulatedandrefer-
enceresponses(loadŒdisplacementand/ortopography),however,forthemtheinitialguessfor
theoptimizerand/ortheparameterboundsareselectedbasedonsomeaprioriknowledge,
[
Zambaldietal.
,
2012
].Inthisstudyanattemptwasmadetodesignamethodologythatwould
notrequireanyaprioriinformationabouttheCRSSvaluesandcouldreachthetargetsolution
withaantlylargeparameterbounds,closetoablackboxoptimization.Toestablishthe
reliabilityofsuchanapproach,virtualsimulationswereusedasexperimentalreference(with
knownCRSSvalues)andtheinverseanalysiswasemployedtoseewhethertheinputCRSScan
beobtainedafteralminimization.
Thechallengeofselectingproperorientationistackledbyselectinganorientationwhose
indentationresponseishighlysensitiveforalltheslipfamilies.Fromtheresultsoftheinverse
indentationanalysisperformedinthisstudyforsixdifferentcrystalorientations,Table
6.1
,it
isobservedthatfortheorientationthatisleastsensitive(fig")forallthethreeslipfamilies,the
optimizationalgorithmfailedtoconvergetoafeasibleregion(regionclosetothetarget)inthe
parameterspace.Thisisexpectedsincelessersensitivityimplieslesserinenceoftheparam-
etersontheselectedobjectivefunctionvalue.Fortheothereorientationswithrelatively
highersensitivities,theoptimizedsolutionisveryclosetothetarget.However,itisimportant
tonotethatwiththeselectedparametervaluesforthe
IIA
framework(sptolerance,op-
timizationparameters,andmaximumallowablefunctionevaluations)thesimplexsometimes
convergestoalocalminimaevenforhighsensitiveorientationsandthetoleranceisnotmet
(fia",fic",fiq").However,evenforsuchcasestherelativeeofeachoftheslipfamilies
(CRSSvalues)isqualitativelycapturedanditcanbepredictedthattheselocalminimaareina
feasibleregionintheparameterspaceandisclosetotheglobalminimum(thetargetsolution).
Also,itappearsthatinsteadofconsideringmultiplecrystalorientationsinparallelformini-
141
mization,usingafewhighsensitivesingleorientationsseparately,issucienttoconclusively
predicttherelativeslipsystemresistanceofthedifferentslipfamiliesforhexagonalmaterials.
Thus,havingestablishedthereliabilityoftheproposedInverseIndentationAnalysisforfcc
andhexagonalmaterialsbasedonorientationselectionafterperformingasensitivityanalysis,
thisproposedmethodologycannowbeusedtoidentifyplasticityparametersfor
¯
-Snandpro-
videevenhigheritysimulationresults.
142
CHAPTER7
CONCLUSIONSANDFUTUREWORK
7.1Conclusions
Inthisthesisamulti-physicsfullycoupledchemo-thermo-mechanicalmodelinacrystal
plasticitycontinuummechanicalframeworkisestablishedtoinvestigatethestress-drivendif-
fusionofSnatomsintinundergoingthermalstraining,inordertounderstandthegov-
erningfactorsfortheprocessofwhiskernucleationandtheirsubsequentkinetics.Through
theinitialthermo-mechanicalmodeloutlinedinChapter
3
,thedominantroleoftexture
onthegrainboundaryhydrostaticstressdistribution(thatsolelymodulatestheatomtrans-
portintheisestablishedascomparedtograingeometryandgrainsizedistribution.Itis
alsopredictedfromthethermo-mechanicalsimulationsthattherelacksanylongrangespatial
gradientsinstressesthatwouldpromotelong-rangediffusiontherebyre-enforcingthenotion
thatthewhiskernucleationprocessisindeedalocalphenomenon.Duetothelocalizednature
ofthecurrentproblem,itispossibletoinvestigateareducedgeometryforthemoreinvolved
chemo-thermo-mechanicalmodeldiscussedinChapter
4
.Withthepreliminaryresultsfrom
thefullycoupledsimulations,itcanbeinferredthatthekinematicconsequenceofatomre-
distributionhasanegligibleeffectonthestressrelaxationascomparedtotheplasticity.
Moreover,suchakinematicconsequenceofatomsdiffusingfromaregionofhighcompression
toaregionoflowcompressionalongthegrainboundarynetworkismostpronouncedinthehy-
drostaticstressvaluesandhasminimaleffectontheshearandnormaltractioncomponentsfor
thestudiedmicrostructure.However,addingthekinematicsduetotransportindeedreduces
thestresses,ascomparedtothecasewherenodiffusionalkinematicsisconsidered.Finally,it
isalsoestablishedthattheplasticanisotropyof
¯
-Snplaysanimportantroleinthemechanics,
therebymakingtheuseofsimpleisotropicmodelscientandrequiringtheestablishment
ofaccuratedescriptionofthecomplexplasticbehaviorobservedin
¯
-Sn.
143
Inthatregard,anInverseIndentationAnalysisbasedmethodologyisinvestigatedinChap-
ter
5
toidentifysuchconstitutiveparametersforcrystalplasticitymaterialmodelsbyminimiz-
ingtheerrorbetweenexperimentalandsimulatedsinglecrystalnanoindentationresponse.For
suchaninversemethodologyitisestablishedthatusingcombinedloadŒdisplacementandsur-
facetopographyerrorprovedtobeareliableobjectivefunctionforminimizationwithamod-
NelderŒMeadsimplexoptimizationalgorithm.Thereliabilityofthismethodologyis
establishedforface-centeredcubicmaterialswhereitisconcludedthattheinitialwstress
couldbeaccuratelyidenwithnegligibleerror,followedbythesaturationstress,andthe
hardeningslope,whichisalsoriveoftheirsensitivitiestotheobjectivefunction.More-
over,forfccmaterialsselectinganyrandomcrystalorientationfortheinverseanalysisproved
tobeeffectiveintheparameterestimation.Suchalityofselectinganyrandomorienta-
tionsisnotpossibleforlowsymmetrycrystalsstructuressuchashexagonalorbody-centered
tetragonal.Henceamoreintelligentchoiceoforientationselectionforlowsymmetrymateri-
alsisproposedinChapter
6
,whereasensitivityanalysisisperformedfordifferentregions
intheorientationspaceforthedifferentslipfamilies,andthenthoseorientationsareselected
wherealltheconsideredslipfamiliesaresensitivetotheindentationloadingcondition.The
reliabilityofthisproposedsensitivitybasedInverseIndentationAnalysisisestablishedforex-
emplaryhexagonalcommerciallypuretitaniumwithahybridoptimizationalgorithm.Using
thehybridalgorithmallowedanexplorationoftheparameterspace,byusingstochastic
particleswarmoptimization,aswellasanalexploitationofafeasibleregionbydeterminis-
ticNelderŒMeadsimplexalgorithm.Havingestablishedthereliabilityofthissensitivitybased
IIA
forhexagonalmaterialswithtwodifferentreferenceparameters,thismethodcanthenbe
employedtoestimatetheplasticparameters(initialwstressvalues)of
¯
-Sn.
Additionally,fromatechnicalstandpoint,themulti-physicsmodeldevelopedinthiswork
couldbeeasilytranslatedtostudyotherprocessesinvolvingstress-drivendiffusionwithad-
ditionaltermsintheeldequationŒsuchasstress-corrosioncracking.Moreover,theinverse
optimizationcodedevelopedinthisworkcouldbeusedasageneraltoolkitforanyoptimiza-
144
Figure7.1:Proposedstructureforidentifyingwhatconditionsinthetinmmicrostructure
causeagraintoformawhisker.
tionproblemduetoitsmodularstructure[
Maitietal.
,
2018
].
7.2Futurework
Theultimategoalofthisworkistoanswerthequestionofwhichgrainsinthetinmi-
crostructurewouldnucleatewhiskerssothatwecanproposeaneffectivewayofwhiskermit-
igationandpreventdevicefailureduetotheirformation.Figure
7.1
outlinestheoverallwork-
wforidentionofsuchgrains,whereinputofthemicrostructureinformationforthe
usingexperiments,eitherelectronbackscatterdiffraction(EBSD)dataforgrainstruc-
tureorX-raydiffractiondataforoveralltexture,andfeedintothecontinuummechanicalcou-
145
pledchemo-thermo-mechanicalmodelinthecrystalplasticityframeworktosimulatetheki-
neticsandstressvariation.Subsequently,aone-to-onecomparisonbetweenthegrainswhere
whiskersareobservedtogrowexperimentallywiththeircorrespondingstress-stateandplas-
ticaccommodationfromthefull-simulationswillhelpinnarrowingthecrystallographic
governingfactorsmodulatingthewhiskernucleationprocess.Inanidealscenariothecompar-
isonshouldbedoneagainstthecaseswherewhiskersgrowduetoCu
6
Sn
5
formation(Sn
depositedoncopperorbrasssubstrate)sincetheyarethemostcommon.However,dueto
thecomplexstateofstresses(andloadingcondition)itbecomesverytrickytodrawdent
conclusionsbycomparingexperimentalobservationstosimulatedresultsfromCu
6
Sn
5
growth
boundaryconditions.Incontrastthermalstrainingprovidesadirectcomparisonbetweenthe
experimentandsimulation,andmorepracticalconclusionscanbedrawn.
Acriticalstepforhavingsimulationdataconsistentwiththerealityistohaveanaccurate
materialdescriptionofthesystem.Asigantfoundationtoachievethishasalreadybeen
laidinthisthesis,alongwithsomeimmediatefuturework.Firstandforemost,animportant
andnecessaryadvancementinthechemo-thermo-mechanicalmodelistheinclusionofgrain
boundarynormalstressasthechemicalpotentialforstress-drivendiffusioninsteadofgrain
boundaryhydrostaticstress(pressure).Secondly,theacrossthegrainboundarywidth
(alongthegrainboundarynormal)isnotzeroleadingtoatomtransportinthenormaldirection,
whichisnotthecaseinreality.Toavoidthis,onewaywouldbetosubtractthediffusivitycom-
ponentalongthegrainboundarynormaldirection.Therefore,toachieveboththeabovecon-
ditionsinsimulation,thegrainboundarynormalateachvoxelshouldbeincorporatedintothe
model.Also,inthisworkthegrainboundaryshearforcesandkineticsareinvestigatedforasin-
glemmicrostructure.Hence,togetabetterquantitativeunderstandingoftheseshearforces,
astatisticalanalysisneedstobedonefordifferentmicrostructures(havingthesameglobal
texturebutdifferentlocalneighborhoodfortheobliquegrainand/orwithdifferentglobaltex-
ture,aswell)todeterminewhethertheseaverageshearforcesarecapableofbreakingthetin
oxidelayer(ofagiventhickness)ontop.Thisanalysisisimportantsincewhiskersareoften
146
observedtogrowafterbreakingtheoxidelayerontop.Additionally,thematerialdescription
wouldbeincompletewithoutanaccuratematerialmodelforSnplasticity.Inthisthesis,the
proposedInverseIndentationAnalysisframeworkofplasticityparameterestimationisalready
establishedforface-centeredcubicandhexagonalmaterials.Hence,theimmediatenextstep
wouldbetoidentifysuchparameters(i.e.theinitialwstressorthecriticalresolvedshear
stressforthedifferentslipfamilies)for
¯
-Sn.Asmentionedearlier,theidentionof
¯
-Sn
plasticityparametersisimportantduetoitshighhomologoustemperatureatoperatingcondi-
tionstherebyshowingsigantplasticityorhightemperaturecreepbehavior.
Apartfromthesimulationworks,someexperimentalstudiescanalsobedonetoverifysome
oftheconclusionsmadeinthisworkbasedonsimulateddataŒsuchastheroleoftheglobalm
texture.Toachievethis,thindifferingonlyintextureshouldbesynthesizedandthensub-
jectedtothermalstrainconditions,withouttheuenceofCu
6
Sn
5
tohaveafaircomparison.
Subsequently,itcanalsobestudiedwhetherthelmtexturecanbemodulatedbycontrolling
theprocessingconditionsinordertoobtainatexturethatismoreresistanttowhiskerforma-
tion(e.g.
h
001],asobservedinthisstudy).Anotheropenquestionwhichcanbeanswered
throughcarefulexperimentationistherelationshipbetweentheobliquesurfacegrainstoover-
allwhiskerdensity.Forachievingsuch,multiplefocus-ionbeamsectioningcouldbedoneon
randomlyselectedareasintheandthencountingthenumberofgrainshavinginclined
boundariesandcomparethemtotheexperimentallyobservedwhiskerdensityvalues.
Thus,withsuccessfulcompletionoftheaboveproposedworkwewouldbeabletounder-
standthecenturyoldproblemofwhiskernucleationfromthinandsubsequentlyprevent
thembymodifyingprocessvariablesthatleadtomicrostructuresthatarelessproneforming
whiskers.
147
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