Modeling slip gradients and internal stresses in crystalline microstructures with distributed defects

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MODELINGSLIPGRADIENTSANDINTERNAL
STRESSESINCRYSTALLINEMICROSTRUCTURES
WITHDISTRIBUTEDDEFECTS
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RAMINAGHABABAEI
B.S.(Hons.),UNIVERSITYOFTEHRAN,2006
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ATHESISSUBMITTED
FORTHEDEGREEOFDOCTOROFPHILOSOPHY
DEPARTMENTOFMECHANICALENGINEERING
NATIONALUNIVERSITYOFSINGAPORE
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201
1
I

DEDICATION

Tomydearparents
MitraandAmir
 whohavesupportedandencouragedmefrombirth

Tomybelovedwife
Marjan
whohasofferedmeunconditionalloveandhappiness

II

ACKNOWLEDGEMENTS
Thisdissertationwouldnothavebeenpossiblewithouttheguidanceandthe
supportofseveralindividualswhohelpedmewiththeirvaluableassistanceinthe
preparationandcompletionofthisstudy.
Firstandforemost,IwouldlikeexpressmydeepgratitudetomysupervisorDr.
ShailendraP.JoshiforhissoundadviceandcarefulguidanceduringmyPh.D.The
innumerablediscussionsIhadwithhimprovidedmeagoodunderstandingofthe
mechanicsandphysicstogether.Withouthissupport,thisworkwouldneverhavebeen
accomplished.
IwouldliketowarmlythankProfessorJ.N.Reddyforhissupportandintroducing
metothefieldofnonlocaltheories.Hisprofoundunderstandingofthecontinuum
mechanicsandfiniteelementtheorieshelpedmealotincompletingthiswork.
Inaddition,IwouldliketothankProfessorR.NarasimhanfromtheIndianinstitute
ofScienceforfruitfuldiscussionsIhadwithhim.Amongmypeers,Igreatlyvaluethe
friendshipIsharewithHamidrezaMirkhani.Iappreciatethehelpheextendedduring
myPhDandmanyusefuldiscussionswehadonthetopicsinmechanicsofmaterials.I
alsothankmyfriendsandcolleaguesDr. JingZhang andA.S.Abhilashfortheir
commentsandsuggestionsaboutmyworks.Ialsogratefullyacknowledgetheresearch
scholarshipprovidedtomebyNationalUniversityofSingapore.
IowemyspecialthankstomylovelywifeMarjanwhohaschosentospendherlife
withmeasmysoulmate.Finally,thisundertakingcouldneverhavebeenachieved
withouttheencouragementofmywonderfulfather,motherandsisterwhohave
supportedmefrombirth.
III

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TABLE OF CONTENTS
DEDICATION I
ACKNOWLEDGEMENTS II
TABLEOFCONTENTS III
SUMMARY VI
LISTOFTABLES VII
LISTOFFIGURES VIII
LISTOFSYMBOLS XII
1 INTRODUCTION 1
1.1 Length‐scaleeffectsinresponseofmaterials 1
1.2 Length‐scaleEffectsinCrystallineMicrostructures 3
1.2.1PlasticDeformationatDifferentLength‐scales 4
1.2.2ABriefOverviewofExperimentalObservationsofLength‐scale
EffectsinPlasticity: 10

1.2.3ContinuumdescriptionsofDislocation‐mediatedCrystalPlasticity
 13
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1.2.3.1 Classicalcrystalplasticity 13
1.2.3.2 ContinuumcrystalplasticitywithGNDs 15
1.3 ScopeandObjectivesoftheThesis 18
2 AMechanism‐BasedGradientCrystalPlasticityInvestigationofMetalMatrix
Composites 20
2.1 Introduction 20
2.2 ComputationalImplementationofMSGCPTheory 24
2.2.1Slipgradientcalculation 27
2.2.2Timeintegrationscheme 28
2.3 Length‐scaledependentMMCresponseinducedbythermalresidual
stresses 29

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2.3.1Computationalresultsforsinglecrystalswithinclusions 32
2.3.2CrystalorientationandinclusionsizeeffectsonthermalGND
densitydistribution 34
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IV
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2.3.3
Size‐dependentstress‐strainresponsewithpre‐existingthermal
GNDdensity 43
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2.3.4Inclusionshapeeffectonstress‐strainresponsesinthepresenceof
thermalGNDdensity 47
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2.3.5ThermalGNDdensitydistributioninpolycrystallineMMCunder
thermalloading 52
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2.4 Grainsize‐inclusionsizesinteractioninMMCatmoderatestrainusing
MSGCP 54
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2.4.1ModelMicrostructures 58
2.4.2Length‐scaledependentpolycrystallineresponse 61
2.4.3Length‐scaleDependentMMCResponse 63
2.4.4Grainorientationandmeshsizeeffects 64
2.4.5Grainsize‐inclusionSizeInteractionstrengthening 66
2.4.6AnalyticalModelforInteractionStrengthening 70
2.5 SummaryandOutlook 75
3 Length‐scaleDependentContinuumCrystalPlasticitywithInternalStresses
 77 
3.1 Introduction 77

3.2 Background 80
3.3 KinematicsofCompatibleandIncompatibleDeformations 84
3.3.1CompatibilityofLatticeCurvature: 85
3.3.2RelationbetweenIncompatibleElasticStrainTensorandtheGND
DensityTensor: 87
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3.4 InternalStressTensor:StressFunctionApproach 88
3.4.1InternalStressunderPlaneStrainCondition:IsotropicElasticity92
3.4.2InternalStresswithElasticAnisotropy 95
3.5 ThermodynamicallyConsistentVisco‐plasticConstitutiveLaw 96
3.5.1Firstlawofthermodynamics:PowerBalance 97
3.5.2Secondlawofthermodynamics:Powerimbalance 98
3.6 ResultsandDiscussion 101
3.6.1TaperedSingleCrystalSpecimenSubjectedtoUniaxialLoading101
3.6.2SingleCrystalLamellaSubjectedtoSimpleShear 110
3.7 Summary 115
4 ACrystalPlasticityAnalysisofLength‐scaleDependentInternalStresseswith
ImageEffects 117
4.1 Introduction 117
V

4.2
 NonlocalContinuumTheorywithInternalStressandImageFields120
4.3 SingleCrystalSpecimenunderPlane‐StrainPureBending:RoleofFree
Surfaces 125
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4.4 Length‐scaleDependentPureBendingResponseofSingleCrystals139
4.4.1Monotonicresponse 143
4.4.2ComparisonwithExperiment 146
4.4.3Length‐scaleDependentBauschingerEffect 155

4.5 SummaryandOutlook 161
5 SummaryandRecommendations 163
5.1 Summary 163
5.2 Recommendationsforfuturework 166
6 ListofPublication 169
7 Bibliography. 170
AppendixA. ANoteonContinuumDescriptionsofGNDDensityTensor 189
AppendixB. Kernelfunctions 194
AppendixC. Numericalintegrationconvergencestudy 200
VI

SUMMARY
This thesis addresses a formulation, computational implementation and
investigationoflength‐scaleeffectsinthepresenceofheterogeneitiesandinternal
stressesincontinuumcrystalplasticity(CCP).First,weimplementagradientcrystal
plasticitytheoryinafiniteelementframework.Usingthis,weinvestigatethecrystal
orientation‐dependentsizeeffectsduetothermalstressesontheoverallmechanical
behaviorofcomposites.Then,throughsystematicsimulations,wedemonstrate
additionalHall‐Petchtypecouplingresultingfrominclusionsize‐grainsizeinteraction
andproposeananalyticalmodelforthesame.Sincethecontinuumcrystalplasticity
augmentedbyshortrangeinteractionofdislocationsfailstopredictlength‐dependent
strengtheningatyieldingpoint,athree‐dimensionalconstitutivetheoryaccountingfor
length‐scaledependentinternalresidualstressesisdeveloped.Thesecond‐order
internalstresstensorisderivedusingtheBeltramistressfunctiontensorthatisrelated
totheNyedislocationdensitytensor.Oneofthecommonsourcesoftheseinternal
residualstressesisthepresenceofensemblesofexcess(GN)dislocationswhich
sometimesreferredtoasamesoscopiccontinuumscale.Theresultinginternalstressis
discussedintermsofthelong‐rangedislocation‐dislocationanddislocation‐boundaries
elasticinteractionsandphysicalandmathematicaloriginsofcorrespondinglength
scalesareargued.Itwillshowthatinternalstressisafunctionofspatialvariationof

GNDdensityinabsenceoffiniteboundarieswhereinternalstressarisesfromGND–
GNDlongrangeelasticinteractions.Howeverinpresenceoffiniteboundariessuchas
freesurfacesorinterfaces,additionalsourceofinternalstressispresentduetolong
rangeinteractionbetweenGNDandboundaries.Usingtheseapproaches,weinvestigate
severalimportantexamplesthatmimicrealproblemswhereinternalstressesplayan
importantroleinmediatingtheoverallresponseundermonotonicandcyclicloading.
VII

LIST OF TABLES
Tables      Page
Table 2‐2.Activatedslipsystemsfortwolimitingcrystalorientations 37
Table 2‐3.MicrostructuralsizecombinationsforMMCsimulations 66
Table 2‐4.MicrostructuralsizecombinationsforMMCsimulations 74
Table 3‐1.Summaryofgoverningequations 100
Table 3‐2.Summaryofconstitutiveequations 101
Table 3‐3.Summaryofunknownvariablesandavailableequations 101
Table 4‐1.Parametersusedintheanalyticalmodelforinternalstressandpredictionof
beambehaviorresponse. 143
Table 4‐2.LocalandglobalcoordinatesofactiveslipsystemaccordingtoMotzetal.,
(2005)singlecrystalbendingexperiment 147
VIII

LIST OF FIGURES
Figures      Page
Figure 1.1.Plasticdeformationandappropriateunitprocessesformodelingat
differentscales 7
Figure 1.2.Dislocationinteractionsatdifferentlength‐scales 9
Figure 1.3.Schematicofgeometricallynecessarydislocations(GNDs)pileupat
grainboundaryinordertoaccommodatecompatibleplasticdeformation. 11
Figure 1.4.FormationofGNDinpresenceofstraingradientin(a)bendingofsingle

crystal(b)nano/microindentation(c)metalmatrixcompositecontainsnano/micro
inclusions. 12
Figure 2.1.Kinematicsofsinglecrystaldeformation 24
Figure 2.2.(a)AnEight‐nodeplanestrainFEwithfourGPsand(b)alinearpseudo‐
elementconstructedfromtheGPsoftheactualFEwhereandarethelocal
isoparametriccoordinates.Theslipandnormaldirectionsandofatypicalslip
systemarealsoshown(b). 27
Figure 2.3.Metalmatrixcomposite(MMC)withuniformarrangementofinclusions
andunitcellcomprisingsinglecrystalmatrixandsquareinclusion. 33
Figure 2.4.Crystalorientationandinclusionsizedependentdistributionof
effectiveGNDdensity


|Δ|500,

1 35
Figure 2.5.(a)DistributionofeffectiveGNDdensity


alongthediagonallineas
showninembeddedfigure.
|
Δ
|
500(b)evolutionofaverageGNDdensity



duringcoolingprocess(


1. 36
Figure 2.6.Distributionofnormalstress


underthermalloadingfordifferent
crystalorientationofmatrix(

1). 38
Figure 2.7.(a)EffectiveGNDdensitydistributionfordifferentinclusionsizes,
(b)averagethermalGNDdensity


evolutionduringthermalcoolingfordifferent
inclusionsizes,(c)InverserelationofaveragethermalGNDdensity


andinclusion
size
|Δ|500,45

. 41
Figure 2.8.Contributionsofindividualmismatchcomponentsunderthermal
loading(

1. 42
Figure 2.9.Truestress‐truestrainresponseforMMCmodelsunderthermo
mechanicalloading.BulkbehaviorispredictedbyCCPwhilesizedependentbehavioris
modeledusingMSGCPforinclusionsize

1,45°. 44

Figure 2.10.Influenceofthepriorthermalloadingon(a)truestress‐truestrain
responseand(b)hardeningrate.(

1,45°), obtained from MSGCP
calculations. 45
IX

Figure 2.11.AverageGNDdensityevolutionunderconsequentthermal‐mechanical
loading.(

1,45°) 47
Figure 2.12.DistributionofthermalGNDdensityaroundsquareandcircular
inclusionsembeddedinsinglecrystalwith(a)0°and(b)45°. 48
Figure 2.13.Truestress‐truestrainresponseforMMCmodelscomprisingtwo
differentinclusionshapes.0°. 49
Figure 2.14.InfluenceofinclusionshapeonthermalresidualstressesinMMCbased
on(a)CCPand(b)MSGCP.0° 51
Figure 2.15.Schematicindicatinganinteractionbetweeninclusionshapeandsize
effectsatthelocationsofstressconcentrations. 51
Figure 2.16.EffectiveGNDdensitydistributioninpolycrystallineMMCwithrandom
grainorientationfordifferentgrainsize(a)

0.5μmand(b)

0.25μm.


1,
|
Δ

|
500 53
Figure 2.17.AverageGNDdensitydistributionevolutioninsinglecrystallineand
polycrystallineMMC 54
Figure 2.18.MMCwithmicron‐sizedinclusionsembeddedinananocrystalline
matrix(JoshiandRamesh,2007) 55
Figure 2.19.Representativemodelsfor(a,c)polyXand(b,d)MMCarchitectures..59
Figure 2.20.Truestress‐truestrainresponsesforpolyXmodelswithdifferentgrain
sizes. 62
Figure 2.21.Normalizedgrainsizedependentflowstressat2%forpolyXwith
identicalgrainorientations.TheplotalsoincludestheempiricalHall‐Petch

.
and
inversegrainsize


fits. 62
Figure 2.22.Grain‐sizedependenttruestress‐truestraincurvesforMMC(solid
lines)with

2.ThecorrespondingpolyXresponses(Figure 2.20)arealso
includedforcomparison. 64
Figure 2.23.StandarddeviationinΔarisingforagivencomputationalmodel
withfixed

butdifferentrealizationsofgrainorientations.Asexpected,thevariationis
smallerforfiner

. 65

Figure 2.24.Meshconvergenceforthestress‐straincurvesofMMC

2,


1withdifferentmeshsizes. 65
Figure 2.25.Flowstress2%normalizedbybulkpolyXyieldstressvariation
ofMMCsasafunctionofgrainsize. 67
Figure 2.26.Inclusionsizeeffectonthenormalizedflowstress(normalizedbybulk
polyXyieldstress)forlargegrainsizes,

3

(negligiblegrainsizeeffect). 68
Figure 2.27.DistributionoftheeffectiveGNDdensity/alongpatha‐
b

2fordifferentgrainsizes. 69
X

Figure 2.28.Schematicofaninclusionembeddedinapolycrystallinemassoffiner
grains. 71
Figure 2.29.Variationoftheinteractionstrengtheningwiththeproduct



. 74
Figure 3.1.ExamplesillustratingthecontributionsofGNDdensitytoenhanced
hardeningin(a)purebeambending‐dissipativehardening,(b)non‐uniformbending‐
dissipativeandenergetichardening. 82

Figure 3.2.Schematicillustratingthenon‐localityarisingfromthepresenceofGND
densityatacontinuumpointandthedistributionoftheGNDdensityaroundthatpoint.
 83
Figure 3.3.VariationofatypicalcomponentofthethirdgradientoftheGreen
functioninEq( 3.31) 91
Figure 3.4.Ataperedbarunderuniaxialloading.Dashedtaperededgesindicatethat
theyaresufficientlyawayfromthecenterlineofthespecimen 102
Figure 3.5.Plasticslipalongbaraxisyforvariousratioof/

fortapered
specimenundermonotonictension 105
Figure 3.6.Resolvedshearstressversusplasticslipatfortaperedbarunder
monotonictensionforvariousratios(a)/

,and(b)/

. 106
Figure 3.7.Distributionofnormalizedinternalshearstress 
∗
/alongthetapered
specimenundermonotonictensionfor(a)2.86°,(b)5.71°.50 107
Figure 3.8.Resolvedshearstressversusplasticslipatfortaperedbarunder
cyclicloading(a)100,(b)50. 108
Figure 3.9.Resolvedshearstressversusplasticslipaty=Lforvarioustaperedangle
undercyclicloading(=100)(a)2.86°,(b)5.71°. 109
Figure 3.10.Asinglelamellawithinanano‐twinnedcrystalundersimpleshear..110
Figure 3.11.(a)Normalizedresolvedshearstress/

versusaverageplasticslipas
afunctionoffor90°,(b)Normalizedresolvedshearstress/


versusnormalized
lamellathicknessat0.2%. 112
Figure 3.12.(a)Distributionofplasticsliponaslipsystemasafunctionoffor
90°versusdistancenormalizedbylamellathickness(b)Normalizedinternal
resolvedshearstress
∗
/alongthelamellathicknessasafunctionoffor90°,and
(c)Normalizedinternalresolvedshearstress
∗
/versusnormalizedlamellathickness.
 114
Figure 4.1.Decompositionoftheinternalstressproblemforaspecimenhostinga
generalGNDdensitydistribution.Seetextfordiscussion. 123
Figure 4.2.SchematicshowingeffectiveGNDarrangementinaspecimenunder
uniformcurvature.Thespecimenthicknessis2andtheGNDdensityisdescribed
bytheglobal,andlocal,coordinates. 126
Figure 4.3.Internalstresscomponentsvariationacrossthicknessfor0.2
5

. 128
XI

Figure 4.4.Variationofnormalizedinternalstress

alongthenormalized
specimenthicknessfordifferentvaluesofnormalizedinternallength‐scale. 129
Figure 4.5.Variationofnon‐dimensionalstressesindirection(



and


)over
beamthicknessforagivennormalizedinternallength‐scale10(Eq.4.8a,b).Note
thatthecomponentsareequalandoppositeresultinginoverall

∗
0. 132
Figure 4.6.Variationof


withYandL.(SeeEq.4.10a) 133
Figure 4.7.Variationof


respectto(a)YatL=10and(b)LatY=1.(SeeEq.4.10b)
 134
Figure 4.8.Variationofthenormalizedtotalinternalstress

withnormalized
internallength‐scaleatspecimensurface(1). 135
Figure 4.9.a)Normalizedstressvariationacrossnormalizedspecimenthickness
/at


0.05,b)Stress‐straincurvesatspecimensurfaces1fordifferent
valuesof

/. 145

Figure 4.10.ContributionofshortrangeGNDinteractionversus/

andlongrange
GNDinteractionsversus./

onflowstressat5%surfacestrain. 146
Figure 4.11.Schematicofsinglecrystalspecimenunderpurebending,crystal
orientationandcorrespondingactiveslipsystems. 148
Figure 4.12.Comparisonoftheanalyticalresults(Eq.4.17)fordifferentvaluesof


withtheexperimentalresultsofMotz,etal(2005). 150
Figure 4.13.TypicalGNDarrangementindoublesymmetricslipdeformationunder
purebending. 152
Figure 4.14.Bending‐straighteningcyclicresponseofsinglecrystallinespecimen
orientedfordoublesymmetricslip 156
Figure 4.15.Overallstressvariationacrossspecimenthicknessatdifferentstrain
showninfigure4.14. 158
Figure 4.16. Length‐scale dependent dissipative (isotropic) and energetic
(kinematic)hardeningcomponentsofpurebendingresponsesfortwodifferent
specimenthickness 159



XII

LIST OF SYMBOLS
Inthisdissertation,thefollowingdefinitionsareusedandaCartesiancoordinate
systemwithunitvectorbase


,

,

applies.
Quantities Notation
Scalar ,,


Vector ,
Secondandhigherordertensor ,
Kroneckerdelta 


Permutationtensor 



Operators Notation
Innerproduct ∙




Crossproduc
t










Tensorproduc
t
⨂








Trace 







Vectordifferentialoperator 




Gradien

t


.



.


Divergence 

.

∙.
Curl 

.

.
Incompatibility


.



.








XIII

Nomenclature Notation
Deformationgradient 
Displacementgradient 
Velocitygradient 
Compatible/Incompatiblestrain 

,


Latticecurvature 
Rotationvector 
Spintensor 
Incompatibilitytensor 
GNDdensitytensor A
Slipdirectionof

slipsystem



Normaldirection 

EffectiveGNDdensity 




Plasticslip 


Plasticsliprate 


Referenceplasticslip 



Appliedstresstensor 


Internalstresstensor 
∗

Internalstressduetodislocation‐dislocation
interaction



Internalstressduetodislocation‐boundary
interaction(Imagestress)



Appliedresolvedshearstress




Internalresolvedshearstress

∗


Beltramistressfunctiontensor 
Slipresistance 


Hardeningmodulus


Elasticmodulus/Compliancetensor ,
Displacement 
Bodyforce 
Tractionforce 







1

1 INTRODUCTION
1.1 Length-scale effects in response of materials

Naturereliesonengineeringitscreationsinahierarchicalmannerinorderto
impartimpressivepropertiesforarangeofapplications(Endy,2005;Fratzl,2007;Gao
etal.,2003).Intriguingexamplesofnaturalstructuralsystemssuchasspider’ssilk
(Vollrath,2000)andnacreinabaloneshells(Meyers,2008)indicateimpressive
strengthsresultingfromstrong,hierarchicalarchitecturesatsmalllength‐scalescoupled
withrobustfailureresistancemechanisms.Oursingularquesttomimicnaturehas
spawnedtremendousexcitementinsynthesizingmaterialsandconstructingstructures
thatareaimedatusingsomeofthenaturalprinciples.Thenotionofthestatement
SmallerisStrongerhasfar‐reachingimplicationsinengineeringthematerialsthatpush
thelimitsofstructuralperformance.
Length‐scaleeffectsonmaterialproperties,oftentermedassizeeffects,areofgreat
importanceincurrentengineeringandscientificapplicationsthatrangefromlarge‐scale
structuresthatdemandhighstrengthatlowerweight(e.g.automotive,aerospace
systems)tominiaturizedmicroandnano‐scaledsystemsthatarebeingadoptedin
biomedicalandelectronicsapplications.Incrystallinemetals,size‐effectsarereported
inavarietyofmaterialpropertiesincludingelasticity(Agrawaletal.,2008;Wuetal.,
2005),plasticity(Dehm,2009;GreerandHosson,2011),thermal(Rohetal.,2010)and
electricalconductivities(Boukaietal.,2008),asspecimendimensionsand/or
microstructuralfeatures(e.g.diameterinnanowire,grainsizeincrystallinemetals)are
reduced.Anunderstandingoftheseeffectsisespeciallyimportantasourabilityto
designandmanufacturestructuresatminiaturizedlength‐scalesandwithnano‐scaled
2

internalstructurescontinuestoacquirehigherlevelsofsophistication(ZhuandLi,
2010).
Inmetallicmicrostructures,ageneraltrendreportedinartificialsystemsisthat
microstructureswithsmallerfeaturesexhibitstrongerbehaviorsthanthosewith
coarserfeatures(GreerandHosson,2011).Forexample,theyieldstrengthof
nanocrystallinepurealuminumwithanaveragegrainsizeof40nmisnearly10times
morethanthatofacoarse‐grainedpurealuminum(Gianolaetal.,2006).Nanotwinned

copperwithtwinthicknessof~35nmisnearly7timesstrongerthancoarse‐grained
purecopper(Luetal.,2009).Forafixedinclusionvolumefractiontheyieldstrengthofa
metalmatrixcomposite(MMC)increasesdramaticallywithdecreasinginclusionsize
(Lloyd,1994).Myriadexamplespertainingtothinfilms(HaqueandSaif,2003),
miniaturizedbeams(Motzetal.,2005),pillars(GreerandNix,2006),rods(Wongetal.,
1997)unequivocallyendorsethesmallerisstrongerphenomenon.Inotherwords,with
allotherpropertiesheldconstant,thesmallerthegeometricalormicrostructuralsize
thestrongeramaterialisexpectedtobe.Seenslightlydifferently,theseexamples
suggestthattheelasticandplasticpropertiesofmaterialsceasetobepurelymaterial
parameters asthespecimen dimensions ormicrostructural features approach
characteristicmicrostructurallength‐scale(GreerandHosson,2011).Allofthese
observationshaveacommonmessage: smaller is stronger.Inabroadsense,thesize‐
dependentbehaviorsofmicroandnano‐scaledstructuresareassociatedwiththehigh
surface(orinterface)areatovolumeratio.Thisisin‐turnbasedontheideathatthe
atomicinteractionsatboundariestendtobedifferentfromthoseinthebulkofa
material.
Rapidincreaseincomputationalpowerintherecentdecadeshasenabled
performing computational simulations that supplement, or at times enable,
experimentalinvestigationsintothephysicsandmechanicsatsmalllength‐scales.An
3

importantquestionthatarisesisthatofthechoiceofspatialandtemporalresolutions.
Atomistic provide a virtual experimental paradigm to capture the prevailing
mechanismsatveryhighspatio‐temporalresolution,butmaybecomecomputationally
prohibitiveatlargerstructurallength‐scale(evenbeyondafewhundrednm).Atthe
otherextreme,continuummechanicsprovidesastrongtheoreticalconstructthatcanbe
extremelyusefulifappropriatelyendowedwithanabilitytopredictsize‐effects,albeit
atthelossofsub‐scaledetails.Athirdpossibilityisjudiciouslycombiningtheatomistics
andcontinuummechanicstoprovideaconcurrentmulti‐scalemodelingapproach.The
choiceofanapproachisdictatedbythedetailsweareinterestedinandthescalesthat

needtobebridgedwiththeavailablecomputationalpower.
Inthiswork,ourfocusisonasmallsubsetwithinthevastexpanseoflength‐scale
dependentbehaviors.Weareinterestedinsomeofthesize‐effectsthatprevailinthe
mechanicalbehaviorofcrystallinemetals.Aparticularcategoryofsize‐effectscovered
inthisthesispertainstocrystallineplasticitythatarisesfrominteractingeffects
betweendislocationsandtheirambience.Forexample,dislocationsgetstoppedbyhard
boundariesandgetannihilatedbyfreesurfaces.Inanotherscenario,dislocationstalkto
otherdislocationsintheirneighborhood.Alltheseeventsresultinlength‐scale
dependentmacroscopicplasticresponsesthatmanifestasstrengtheningofamaterial.
Weprobesomeoftheseeffectsinheterogeneouscrystallinemicrostructuresofcurrent
interestthroughanalyticalandcomputationalapproaches.
Tosetthestagefortherestofthethesis,webrieflydiscussdislocationplasticityin
crystallinemetalsasitcanbedescribedatvariouslength‐scales.
1.2 Length-scale Effects in Crystalline Microstructures
Duringthelastcoupleofdecades,crystallinemetallicmaterialsespeciallyFace‐
Centered‐Cubic(FCC)metalsarevastlyusedasthenano/microstructuresfornumerous
4

applications.Therefore,itiscriticallyimportanttoobtainfundamentalinsightintotheir
length‐scaledependentmechanicalbehavioratmicroandnanoscales.Theexperimental
andtheoreticalaspectsoftheselength‐scaledependentbehaviorsarediscussedinthe
followingsections.
1.2.1 Plastic Deformation at Different Length-scales
Incrystallinematerials,theunitprocessesthataredeemedrelevanttodescribe
plasticitymustbeidentifiedbasedonthelengthandtime‐scalesofinterest.Froma
thermodynamicviewpoint,movementofthedislocationsduringplasticdeformationis
mediatedbycrystallatticeresistance.Thiscrystallatticeresistancecanorneedstobe
definedatdifferentscales.Atthefinestlength‐scale(atomistic),itisaninherently
dynamicalprocessofatomicmotions.Inthedevelopmentofanincrementallycoarse‐
grainedapproach,someofthemicrostructuraldetailsatthefinerscalearesmearedout

bymakingcertainassumptionswithregardsthelength‐andtime‐scalesatthesub‐scale
vis‐à‐visthecurrentscalesofinterest.Thisoftenprovidesamotivationtodefineamore
relevantunitprocessatthecoarserlength‐scalebycoarseningthesub‐scaledefect
dynamics.ThereviewarticlebyZaiserandSeeger(2002)servesasausefulreference.A
possiblecascadingflowofsuchamulti‐scalingprocess(Fig.1.1)thatisdeemeduseful
forthisthesisisbrieflydiscussedhere:
Atomicscale–describestheindividualatomintermsofitsfinercomponentssuch
aselectrons.Density functionaltheory(DFT)isthemostpopularmethodtoinvestigate
thetotalground‐levelenergyandpropertiesofasystemofinteractingelectronsin
particularatomsandmolecules(ShollandSteckel,2009).Itusesthefunctionalofthe
electrondensity,whichprovidesthepotentialfunctionasabasisformoleculardynamic
simulations.
5

Nanoscopic scale– Atthisscale,theindividualatomsandmoleculesareresolved
wheretheinformationfromtheatomicscalethatiscoarse‐grainedistheinteratomic
interaction.Moleculardynamics(MD)isapowerfultooltocomputationallysimulatethe
physicalmotionsofatomsandmoleculesunderexternalstimuli.InMDsimulations,the
Newton’sequationsofmotionforasystemofinteractingparticlesarenumerically
solvedwhereintermolecularinteractionsaredescribedbyapotentialfunctionprovided
bytheatomicscale.Areasonablylargeensembleofatomsismodeled,andtheelastic
andplasticpropertiesemergenaturallythroughinteratomicinteractions.Atthisscale,
theunitprocessthatdescribesplasticdeformationisthenucleationandmobilityof
individualdislocationswithinacrystallinelattice.Giventheinherentdynamicsof
atomicmotions,typicalMDcalculationsneedhightemporalresolutionintheorderof
femtotopicoseconds.Theinteractivelong‐andshort‐rangeinteractionsbetween
dislocationsarenaturallyresolvedatthisscaleandprovidetheessentialphysicsthat
canberationalizedasconstitutivedescriptionsatcoarserscales.Nanoscopiclattice
resistanceisreferredtoasthePeierlsstress.Itdependsstronglyonthestrainrateand
canbethermallyactivated;hence,itisreferredtoasthethermallatticeresistance.

Microscopic scale –Atthislength‐scale,theatomisticresolutionissmearedout
renderinganelasticcontinuum,butthediscretenessofdislocationsisretained.Theyare
modeledaslinesingularitieswithinanelasticcontinuumandtheirevolutionis
describedthroughasetofconstitutiverulesthatareformulatedbasedonthesubscale
observations.Thecrystallatticeinformationisretainedintheformofanisotropicelastic
stiffnesstensorandslipsystemsonwhichdislocationsglide.Thecorresponding
mathematicalconstructandnumericalimplementationiscommonlyreferredtoas
Discrete Dislocation Dynamics (DDD),ifinertialtermsareretained(CazacuandFivel,
2010).Internalstressesaroundindividualdislocationsareaccountedforatthislength‐
scaleandareinherentlynon‐local,renderingalength‐scaledependentpseudo‐
continuumframework.WhileDDD(anditsstaticcounterpartignoringinertia)can
6

modelrelativelybiggercomputationaldomainscomparedtoMDwhileaccountingfor
short‐andlong‐rangedislocationinteractions,thephysicaldimensionsarestill
restrictivetoafewmicronsmakingitsomewhatdifficulttoapplytolargerscale
calculationsthatspanseveralto.
Mesoscopic scale –Atthisscale,thephysicalpropertiesofamaterialare
representedascontinuousvariables(continuum).Asinthemicroscopicscale,the
directionalelasticityatthecrystallatticelevelisincorporatedthroughanisotropic
elasticity.However,insteadoftrackingplasticactivitythroughmotionofdiscrete
dislocations,equivalentconstitutivelawsforplasticsliponindividualslipplanesare
writtenintermsofdislocationdensitiesonthoseslipplanes(Asaro,1983;Maetal.,
2005).Initsconventionalform,length‐scaleeffects(Burger’svectorinformation)in
crystalplasticityarelostduetohomogenizationfromdiscretedislocationsto
dislocationdensity.However,someoftheseeffectscanbeincorporatedbyappealingto
non‐localfieldtheories(Eversetal.,2004;Gurtin,2002;Hanetal.,2005a).Thisscale
canbeconsideredasabridgebetweenthemicroscopicandmacroscopicscalewherethe
mechanicsatfinerlength‐scalesisaccountedforusingappropriateconstitutive
relations.

Mesoscopic(andmicroscopic)internalstressesareusuallyreferredtoasathermal
lattice resistancetodislocationmotion,whichareindependentoftemperatureand
strainrateexceptforitstemperaturedependencethroughtheshearmodulus(Hulland
Bacon,2001;ZaiserandSeeger,2002).
Macroscopic scale–Bulkscaleresponsesdevoidofsize‐effectsarewell‐described
atthisscaleusingclassicalcontinuumplasticity(KhanandHuang,1995).Traditionally,
theelasticandplasticbehaviorsaredescribedbydeterministicconstitutivelaws
resultingfromaveragingthemicro‐structural information (e.g.dislocationcell
structuresanddislocationspacing)atfinerscalesoverarepresentativevolumethat
7
comprises sufficientnumber of crystal orientations to render ahomogenized
continuum.Suchaveragingproceduresnaturallysmearoutmuchofthemicrostructural
informationandmoreimportantly,theinherentmicrostructuralfeatures,givinglength‐
scaleindependentframeworks.Again,thisapproachworkswellinmanycases,butfails
tocapturesize‐effectsthatarisefrommicrostructuraldifferences.Forexample,suchan
approachessentiallypredictsthesame(size‐independent)yieldstrengthandhardening
responseforananocrystallinematerialandacoarse‐grainedmaterial.Recentattempts
admitlength‐scaleeffectsinsuchamacroscopictheorywithoutresortingtocrystallevel
slipdetails(AbuAl‐RubandVoyiadjis,2006;FleckandHutchinson,1997;NixandGao,
1998;VoyiadjisandAl‐Rub,2005).
Figure

1.1.Plasticdeformationandappropriateunitprocessesformodelingatdifferent
scales
Atsmalllength‐scales,dislocationmechanismsareenrichedbythepresenceof
boundaries.Forexample,short‐rangeinteractionssuchasdislocationnucleation,
8

annihilation, and multiplication mechanisms and long‐range interaction elastic
interactionbetweendislocationsmaybeinfluencedbyinterfacessuchasgrainortwin

boundaries, and/or free surfaces. Therefore, additional interactions between
dislocationsandboundariesshouldbetakenintoaccountfornano/micro‐scale
structureswherehighsurface(orinterface)areatovolumeratioiscommon.Insingle
crystalsunderuniformloadingconditions,length‐scaledependentyieldandflow
strengthsareobservedwithdecreasingspecimendimensionsandtheunderlying
mechanismsareassociatedwithdislocationactivitiesthataremodulatedbyfree
surfaces(GreerandNix,2006);(Shanetal.,2007).Innanostructuredpolycrystalline
metalssuchasnanograinedandnanotwinnedmetals(Haque,2004;Luetal.,2009),a
Hall‐Petchbehaviorarisesfromdislocationinteractionwithgrainandtwinboundaries
intheformofdislocationpile‐up.
Atcontinuumscales,dislocationinducedplasticitymaybebroadlyclassifiedinto
twogroupsbasedonthewaytheyaccumulateinduringplasticdeformation.Statistically
storeddislocations(SSD)accumulatebystatisticaltrappingofthedislocationsto
accommodateplasticslip(Ashby,1970).Atanatomisticscale,individualdislocations
produceinternalstressesintheirvicinity,butatlargerscales(mesoandabove),these
arecanceledintheprocessofaveragingout,sinceSSDsbydefinitionarerandomly
distributed.Anothertypeofdislocationsarisesfromthenecessitytoaccommodatelocal
latticecurvaturesthatariseduetonon‐uniformplasticdeformation(Nye,1953;Ashby,
1970).Ashby(1970)referredtotheseastheGeometricallyNecessaryDislocations
(GNDs).GNDsactasadditionalobstaclestothemotionofSSDs,butthemselvesdonot
contributetoplasticstrain(GaoandHuang,2003).IncorporatingGNDswithin
continuumframeworksendowthemwithanabilitytopredictalength‐scaledependent
macroscopicresponseundernon‐uniformplasticdeformation(AcharyaandBassani,
2000;Ashby,1970;Flecketal.,2003;NixandGao,1998).
9
ThefollowingGNDrelatedmechanismscouldbeidentifiedintermsofstressesor
resistancemechanismsatdifferentscales(Figure 1.2):
 Short‐rangeinteractionsofGNDswithSSDsasanadditionalthermal
latticeresistancewhichoccursinnanoscopicscale(Acharyaand
Bassani,2000;NixandGao,1998).

 Long‐rangeelasticGND‐GNDinteractiondescribedatthemesoscopic
scaleasathermalinternalstressesthatinfluencedislocationmobility
(Kröner,1967).
 Long‐rangeelasticinteractionbetweenGNDsandboundariessuchas
freesurfacesmanifestingasathermallatticeresistance,whichare
describedasimagestressfieldsatthemesoscopiccontinuumscales
Figure

1.2.Dislocationinteractionsatdifferentlength‐scales
10

Figure 1.2alsogivessomeexamplesofeachoftheinteractions.Thefocusthiswork
ismodelingtheplasticdeformationincrystallinematerialsaccountingforthelength‐
scaleeffectsthatpersistatthemesoscopicscale.Whiletheseeffectsaremainlyascribed
tothepresenceofGNDsthatarein‐turnrelatedtostraingradients,somedislocation
mechanismsproducesize‐effectsevenintheabsenceofstraingradientsandarebriefly
mentionedlaterinthischapter,forclarity.Eachofthesemaypossessanassociated
length‐scalethatmustbecomparedwiththelength‐scalesofinterest.Manyatimes,the
length‐scaleareproblem‐dependentandmaybedeterminedbystructuregeometry,
deformationprofile,materialmicrostructure,physicalpropertiesofboundariesandso
on(VoyiadjisandAl‐Rub,2005).
1.2.2 A Brief Overview of Experimental Observations of Length-
scale Effects in Plasticity:
Severalsimilarobservationsarereportedinmicro‐scaledspecimensinavarietyof
heterogeneousdeformationconditionsincludingbendingofsingle‐andpoly‐crystalline
beamsandthinfilms(HaqueandSaif,2003;Huberetal.,2002;Motzetal.,2005;Stolken
andEvans,1998).Specifically,theobservedtrendisthattheflowstressincreasesasthe
specimenthicknessreduces.Furthermore,thissizeeffectisenhancedinpresenceof
substratewhichcausesadditionalpile‐upofdislocationsatthefilm‐substrateinterface.
Similarbehaviorisobservedinmicroandnanoindentation,whichexhibitlength‐scale

dependenthardness(MaandClarke,1995;McElhaneyetal.,1998;NixandGao,1998).
Inmetalmatrixcomposites(MMCs),highermacroscopicstrengthandhardeningis
reportedwithdecreasinginclusionsizewhilekeepingitsvolumefractionconstant.
11


Figure 1.3.Schematicofgeometricallynecessarydislocations(GNDs)pileupatgrain
boundaryinordertoaccommodatecompatibleplasticdeformation.
Inalltheabove‐mentionedandsimilarscenarios,thelength‐scaleeffectsare
attributedtothepresenceofGNDsthataccumulateinadditiontoSSDsinorderto
compensateincompatibilitiesintheplasticdeformation(Figs. 1.3andFigure 1.4)arising
duetorelevantreasons(e.g.elasto‐plasticandthermalexpansionmismatchbetween
theinclusionandmetalmatrixinMMCsorincompatibleplasticdeformation,(Ashby,
1970;Flecketal.,1994).
ItisusefultomentionherethatalthoughmechanicsapproachesrelyingGND‐
inducedstrengtheninghavegainedpopularityandisalsothemaintopicofthisthesis,
thesemaynotbetheonlyorthemostrelevantmechanismsinstrengthening.



Incompatibledeformation
atGB
GNDpileupatGBtoaccommodate
compatibledeformation
F
F