MODELINGSLIPGRADIENTSANDINTERNALSTRESSESINCRYSTALLINEMICROSTRUCTURES

WITHDISTRIBUTEDDEFECTS

RAMINAGHABABAEI

B.S.(Hons.),UNIVERSITYOFTEHRAN,2006

ATHESISSUBMITTED

FORTHEDEGREEOFDOCTOROFPHILOSOPHY

DEPARTMENTOFMECHANICALENGINEERING

NATIONALUNIVERSITYOFSINGAPORE

2011

I

DEDICATION

Tomydearparents

MitraandAmir

whohavesupportedandencouragedmefrombirth

Tomybelovedwife

Marjan

whohasofferedmeunconditionalloveandhappiness

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

preparationandcompletionofthisstudy.

First and foremost,I would like expressmy deep gratitude tomy 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

mechanicsandphysicstogether.Withouthissupport,thisworkwouldneverhavebeen

accomplished.

IwouldliketowarmlythankProfessorJ.N.Reddyforhissupportandintroducing

me to the field of nonlocal theories. His profound understanding of the continuum

mechanicsandfiniteelementtheorieshelpedmealotincompletingthiswork.

Inaddition,IwouldliketothankProfessorR.NarasimhanfromtheIndianinstitute

ofScience for fruitfuldiscussions Ihadwithhim.Amongmypeers, Igreatlyvalue the

friendship IsharewithHamidrezaMirkhani. Iappreciate thehelpheextendedduring

myPhDandmanyusefuldiscussionswehadonthetopicsinmechanicsofmaterials.I

also thank my friends and colleagues Dr. Jing Zhang and A.S. Abhilash for their

commentsandsuggestionsaboutmyworks.Ialsogratefullyacknowledgetheresearch

scholarshipprovidedtomebyNationalUniversityofSingapore.

IowemyspecialthankstomylovelywifeMarjanwhohaschosentospendherlife

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

supportedmefrombirth.

III

TABLE OF CONTENTS

DEDICATION....................................................................................................................................I 

ACKNOWLEDGEMENTS................................................................................................................II 

TABLEOFCONTENTS.................................................................................................................III 

SUMMARY.......................................................................................................................................VI 

LISTOFTABLES..........................................................................................................................VII 

LISTOFFIGURES........................................................................................................................VIII 

LISTOFSYMBOLS.......................................................................................................................XII 

1  INTRODUCTION....................................................................................................................1 

1.1  Length‐scaleeffectsinresponseofmaterials..................................................1 

1.2  Length‐scaleEffectsinCrystallineMicrostructures......................................3 

1.2.1 PlasticDeformationatDifferentLength‐scales........................................4 

1.2.2 ABriefOverviewofExperimentalObservationsofLength‐scaleEffectsinPlasticity:............................................................................................10 

1.2.3 ContinuumdescriptionsofDislocation‐mediatedCrystalPlasticity....................................................................................................................................13 

1.2.3.1  Classicalcrystalplasticity.............................................................................13 

1.2.3.2  ContinuumcrystalplasticitywithGNDs................................................15 

1.3  ScopeandObjectivesoftheThesis....................................................................18 

2  AMechanism‐BasedGradientCrystalPlasticityInvestigationofMetalMatrixComposites.................................................................................................................20 

2.1  Introduction.................................................................................................................20 

2.2  ComputationalImplementationofMSGCPTheory.....................................24 

2.2.1 Slipgradientcalculation..................................................................................27 

2.2.2 Timeintegrationscheme.................................................................................28 

2.3  Length‐scaledependentMMCresponseinducedbythermalresidualstresses...........................................................................................................................29 

2.3.1 Computationalresultsforsinglecrystalswithinclusions................32 

2.3.2 CrystalorientationandinclusionsizeeffectsonthermalGNDdensitydistribution...........................................................................................34 

IV

2.3.3 Size‐dependentstress‐strainresponsewithpre‐existingthermalGNDdensity..........................................................................................................43 

2.3.4 Inclusionshapeeffectonstress‐strainresponsesinthepresenceofthermalGNDdensity.........................................................................................47 

2.3.5 ThermalGNDdensitydistributioninpolycrystallineMMCunderthermalloading...................................................................................................52 

2.4  Grainsize‐inclusionsizesinteractioninMMCatmoderatestrainusingMSGCP.............................................................................................................................54 

2.4.1 ModelMicrostructures.....................................................................................58 

2.4.2 Length‐scaledependentpolycrystallineresponse..............................61 

2.4.3 Length‐scaleDependentMMCResponse.................................................63 

2.4.4 Grainorientationandmeshsizeeffects...................................................64 

2.4.5 Grainsize‐inclusionSizeInteractionstrengthening...........................66 

2.4.6 AnalyticalModelforInteractionStrengthening....................................70 

2.5  SummaryandOutlook.............................................................................................75 

3  Length‐scaleDependentContinuumCrystalPlasticitywithInternalStresses.......................................................................................................................................77 

3.1  Introduction.................................................................................................................77 

3.2  Background..................................................................................................................80 

3.3  KinematicsofCompatibleandIncompatibleDeformations...................84 

3.3.1 CompatibilityofLatticeCurvature:............................................................85 

3.3.2 RelationbetweenIncompatibleElasticStrainTensorandtheGNDDensityTensor:...................................................................................................87 

3.4  InternalStressTensor:StressFunctionApproach.....................................88 

3.4.1 InternalStressunderPlaneStrainCondition:IsotropicElasticity92 

3.4.2 InternalStresswithElasticAnisotropy....................................................95 

3.5  ThermodynamicallyConsistentVisco‐plasticConstitutiveLaw...........96 

3.5.1 Firstlawofthermodynamics:PowerBalance.......................................97 

3.5.2 Secondlawofthermodynamics:Powerimbalance.............................98 

3.6  ResultsandDiscussion..........................................................................................101 

3.6.1 TaperedSingleCrystalSpecimenSubjectedtoUniaxialLoading101 

3.6.2 SingleCrystalLamellaSubjectedtoSimpleShear.............................110 

3.7  Summary......................................................................................................................115 

4  ACrystalPlasticityAnalysisofLength‐scaleDependentInternalStresseswithImageEffects..........................................................................................................117 

4.1  Introduction...............................................................................................................117 

V

4.2  NonlocalContinuumTheorywithInternalStressandImageFields120 

4.3  SingleCrystalSpecimenunderPlane‐StrainPureBending:RoleofFreeSurfaces........................................................................................................................125 

4.4  Length‐scaleDependentPureBendingResponseofSingleCrystals139 

4.4.1 Monotonicresponse........................................................................................143 

4.4.2 ComparisonwithExperiment.....................................................................146 

4.4.3 Length‐scaleDependentBauschingerEffect........................................155 

4.5  SummaryandOutlook...........................................................................................161 

5  SummaryandRecommendations..............................................................................163 

5.1  Summary......................................................................................................................163 

5.2  Recommendationsforfuturework..................................................................166 

6  ListofPublication...........................................................................................................169 

7  Bibliography.....................................................................................................................170 

AppendixA.  ANoteonContinuumDescriptionsofGNDDensityTensor...........189 

AppendixB.  Kernelfunctions...........................................................................................194 

AppendixC.  Numericalintegrationconvergencestudy...........................................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

additionalHall‐Petch typecoupling resulting from inclusion size‐grain size interaction

and propose an analytical model for the same. Since the continuum crystal plasticity

augmentedbyshortrange interactionofdislocations fails topredict length‐dependent

strengtheningatyieldingpoint,athree‐dimensionalconstitutivetheoryaccounting for

length‐scale dependent internal residual stresses is developed. The second‐order

internalstresstensorisderivedusingtheBeltramistressfunctiontensorthatisrelated

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

sometimesreferredtoasamesoscopiccontinuumscale.Theresultinginternalstressis

discussedintermsofthelong‐rangedislocation‐dislocationanddislocation‐boundaries

elastic interactions and physical and mathematical origins of corresponding length

scales are argued. Itwill show that internal stress is a function of spatial variation of

GND density in absence of finite boundarieswhere internal stress arises fromGND –

GND long rangeelastic interactions.However inpresenceof finiteboundaries suchas

free surfaces or interfaces, additional source of internal stress is present due to long

rangeinteractionbetweenGNDandboundaries.Usingtheseapproaches,weinvestigate

several important examples thatmimic real problemswhere internal stressesplay an

importantroleinmediatingtheoverallresponseundermonotonicandcyclicloading.

VII

LIST OF TABLES

Tables Page

Table 2‐2.Activatedslipsystemsfortwolimitingcrystalorientations..................................37 

Table 2‐3.MicrostructuralsizecombinationsforMMCsimulations........................................66 

Table 2‐4.MicrostructuralsizecombinationsforMMCsimulations........................................74 

Table 3‐1.Summaryofgoverningequations.....................................................................................100 

Table 3‐2.Summaryofconstitutiveequations.................................................................................101 

Table 3‐3.Summaryofunknownvariablesandavailableequations....................................101 

Table 4‐1.Parametersusedintheanalyticalmodelforinternalstressandpredictionofbeambehaviorresponse.......................................................................................................143 

Table 4‐2.LocalandglobalcoordinatesofactiveslipsystemaccordingtoMotzetal.,(2005)singlecrystalbendingexperiment...................................................................147 

VIII

LIST OF FIGURES

Figures Page

Figure 1.1. Plastic deformation and appropriate unit processes for modeling atdifferentscales.......................................................................................................................................................7 

Figure 1.2.Dislocationinteractionsatdifferentlength‐scales..............................................9 

Figure 1.3. Schematic of geometrically necessary dislocations (GNDs) pile up atgrainboundaryinordertoaccommodatecompatibleplasticdeformation...........................11 

Figure 1.4.FormationofGNDinpresenceofstraingradientin(a)bendingofsinglecrystal (b) nano/micro indentation (c) metal matrix composite contains nano/microinclusions...............................................................................................................................................................12 

Figure 2.1.Kinematicsofsinglecrystaldeformation...............................................................24 

Figure 2.2.(a)AnEight‐nodeplanestrainFEwithfourGPsand(b)alinearpseudo‐element constructed from the GPs of the actual FE where and are the localisoparametric coordinates. The slip and normal directions and of a typical slipsystem arealsoshown(b).........................................................................................................................27 

Figure 2.3.Metalmatrixcomposite(MMC)withuniformarrangementofinclusionsandunitcellcomprisingsinglecrystalmatrixandsquareinclusion.........................................33 

Figure 2.4. Crystal orientation and inclusion size dependent distribution ofeffectiveGNDdensity |Δ | 500, 1 ........................................................................35 

Figure 2.5.(a)DistributionofeffectiveGNDdensity alongthediagonallineasshown in embedded figure. |Δ | 500 (b) evolution of average GND density duringcoolingprocess( 1 ..........................................................................................................36 

Figure 2.6. Distribution of normal stress under thermal loading for differentcrystalorientationofmatrix( 1 )...............................................................................................38 

Figure 2.7.(a)EffectiveGNDdensity distributionfordifferentinclusionsizes,(b) average thermal GND density evolution during thermal cooling for differentinclusion sizes, (c) Inverse relationof average thermalGNDdensity and inclusionsize |Δ | 500, 45 ...........................................................................................................................41 

Figure 2.8. Contributions of individual mismatch components under thermalloading( 1 ..........................................................................................................................................42 

Figure 2.9. True stress‐true strain response for MMC models under thermomechanicalloading.BulkbehaviorispredictedbyCCPwhilesizedependentbehaviorismodeledusingMSGCPforinclusionsize 1 , 45°.......................................................44 

Figure 2.10. Influence of the prior thermal loading on (a) true stress‐true strainresponse and (b) hardening rate. ( 1 , 45°), obtained from MSGCPcalculations............................................................................................................................................................45 

IX

Figure 2.11.AverageGNDdensityevolutionunderconsequentthermal‐mechanicalloading.( 1 , 45°)......................................................................................................................47 

Figure 2.12. Distribution of thermal GND density around square and circularinclusionsembeddedinsinglecrystalwith(a) 0°and(b) 45°..................................48 

Figure 2.13. True stress‐true strain response for MMC models comprising twodifferentinclusionshapes. 0°.............................................................................................................49 

Figure 2.14.InfluenceofinclusionshapeonthermalresidualstressesinMMCbasedon(a)CCPand(b)MSGCP. 0° ..........................................................................................................51 

Figure 2.15.Schematic indicatingan interactionbetween inclusionshapeandsizeeffectsatthelocationsofstressconcentrations..................................................................................51 

Figure 2.16.EffectiveGNDdensitydistributioninpolycrystallineMMCwithrandomgrain orientation for different grain size (a) 0.5μm and (b) 0.25μm.

1 , |Δ | 500 ...............................................................................................................................53 

Figure 2.17. Average GND density distribution evolution in single crystalline andpolycrystallineMMC.........................................................................................................................................54 

Figure 2.18. MMC with micron‐sized inclusions embedded in a nanocrystallinematrix(JoshiandRamesh,2007)...............................................................................................................55 

Figure 2.19.Representativemodelsfor(a,c)polyXand(b,d)MMCarchitectures..59 

Figure 2.20.Truestress‐truestrainresponsesforpolyXmodelswithdifferentgrainsizes...........................................................................................................................................................................62 

Figure 2.21.Normalizedgrainsizedependentflowstressat 2%forpolyXwithidenticalgrainorientations.TheplotalsoincludestheempiricalHall‐Petch . andinversegrainsize fits..........................................................................................................................62 

Figure 2.22. Grain‐size dependent true stress‐true strain curves for MMC (solidlines) with 2 . The corresponding polyX responses (Figure 2.20) are alsoincludedforcomparison.................................................................................................................................64 

Figure 2.23. Standard deviation in Δ arising for a given computationalmodelwithfixed butdifferentrealizationsofgrainorientations.Asexpected,thevariationissmallerforfiner ............................................................................................................................................65 

Figure 2.24.Meshconvergenceforthestress‐straincurvesofMMC 2 ,1 withdifferentmeshsizes .................................................................................................................65 

Figure 2.25.Flowstress 2%normalizedbybulkpolyXyieldstressvariationofMMCsasafunctionofgrainsize............................................................................................................67 

Figure 2.26.Inclusionsizeeffectonthenormalizedflowstress(normalizedbybulkpolyXyieldstress)forlargegrainsizes, 3 (negligiblegrainsizeeffect)..................68 

Figure 2.27. Distribution of the effective GND density / along path a‐b 2 fordifferentgrainsizes.........................................................................................................69 

X

Figure 2.28.Schematicofaninclusionembeddedinapolycrystallinemassoffinergrains........................................................................................................................................................................71 

Figure 2.29.Variationoftheinteractionstrengtheningwiththeproduct ....74 

Figure 3.1. Examples illustrating the contributions of GND density to enhancedhardeningin(a)purebeambending‐dissipativehardening,(b)non‐uniformbending‐dissipativeandenergetichardening.........................................................................................................82 

Figure 3.2.Schematicillustratingthenon‐localityarisingfromthepresenceofGNDdensityatacontinuumpointandthedistributionoftheGNDdensityaroundthatpoint......................................................................................................................................................................................83 

Figure 3.3. Variation of a typical component of the third gradient of the GreenfunctioninEq( 3.31).........................................................................................................................................91 

Figure 3.4.Ataperedbarunderuniaxialloading.Dashedtaperededgesindicatethattheyaresufficientlyawayfromthecenterlineofthespecimen...............................................102 

Figure 3.5.Plastic slip alongbaraxisy forvarious ratioof / for taperedspecimenundermonotonictension.......................................................................................................105 

Figure 3.6.Resolvedshearstressversusplasticslipat fortaperedbarundermonotonictensionforvariousratios(a) / ,and(b) / ....................................106 

Figure 3.7.Distributionofnormalizedinternalshearstress ∗/ alongthetaperedspecimenundermonotonictensionfor(a) 2.86°,(b) 5.71°. 50....................107 

Figure 3.8.Resolvedshearstressversusplasticslipat fortaperedbarundercyclicloading(a) 100,(b) 50...................................................................................................108 

Figure 3.9.Resolvedshearstressversusplasticslipaty=Lforvarioustaperedangleundercyclicloading( =100)(a) 2.86°,(b) 5.71°..........................................................109 

Figure 3.10.Asinglelamellawithinanano‐twinnedcrystalundersimpleshear..110 

Figure 3.11.(a)Normalizedresolvedshearstress / versusaverageplasticslipasafunctionof for 90°,(b)Normalizedresolvedshearstress / versusnormalizedlamellathicknessat 0.2%..................................................................................................................112 

Figure 3.12.(a)Distributionofplasticslip onaslipsystemasafunctionof for90° versus distance normalized by lamella thickness (b) Normalized internal

resolvedshearstress ∗/ alongthelamellathicknessasafunctionof for 90°,and(c)Normalizedinternalresolvedshearstress ∗/ versusnormalizedlamellathickness...................................................................................................................................................................................114 

Figure 4.1.Decompositionof the internalstressproblemforaspecimenhostingageneralGNDdensitydistribution.Seetextfordiscussion...........................................................123 

Figure 4.2. Schematic showing effective GND arrangement in a specimen underuniformcurvature.Thespecimenthickness is 2 andtheGNDdensity isdescribedbytheglobal , andlocal , coordinates........................................................................................126 

Figure 4.3. Internal stress components variation across thickness for 0.25 ..............................................................................................................................................................128 

XI

Figure 4.4. Variation of normalized internal stress along the normalizedspecimenthickness fordifferentvaluesofnormalizedinternallength‐scale ............129 

Figure 4.5.Variationofnon‐dimensionalstressesin direction( and )overbeamthicknessforagivennormalized internal length‐scale 10 (Eq.4.8a,b).Notethatthecomponentsareequalandoppositeresultinginoverall ∗ 0...........................132 

Figure 4.6.Variationof withYandL.(SeeEq.4.10a)..................................................133 

Figure 4.7.Variationof respectto(a)YatL=10and(b)LatY=1.(SeeEq.4.10b)..................................................................................................................................................................................134 

Figure 4.8. Variation of the normalized total internal stress with normalizedinternallength‐scale atspecimensurface( 1)......................................................................135 

Figure 4.9. a) Normalized stress variation across normalized specimen thickness/ at 0.05,b)Stress‐straincurvesatspecimensurfaces 1 fordifferent

valuesof / ............................................................................................................................................145 

Figure 4.10.ContributionofshortrangeGNDinteractionversus / andlongrangeGNDinteractionsversus. / onflowstressat5%surfacestrain..........................................146 

Figure 4.11. Schematic of single crystal specimen under pure bending, crystalorientationandcorrespondingactiveslipsystems........................................................................148 

Figure 4.12.Comparisonoftheanalyticalresults(Eq.4.17)fordifferentvaluesof withtheexperimentalresultsofMotz,etal(2005)........................................................................150 

Figure 4.13.TypicalGNDarrangementindoublesymmetricslipdeformationunderpurebending......................................................................................................................................................152 

Figure 4.14. Bending‐straightening cyclic response of single crystalline specimenorientedfordoublesymmetricslip........................................................................................................156 

Figure 4.15.Overall stress variation across specimen thickness at different strainshowninfigure4.14.......................................................................................................................................158 

Figure 4.16. Length‐scale dependent dissipative (isotropic) and energetic(kinematic) hardening components of pure bending responses for two differentspecimenthickness.........................................................................................................................................159 

XII

LIST OF SYMBOLS

Inthisdissertation,thefollowingdefinitionsareusedandaCartesiancoordinate

systemwithunitvectorbase , , applies.

Quantities Notation

Scalar , ,

Vector ,

Secondandhigherordertensor ,

Kroneckerdelta

Permutationtensor

Operators Notation

Innerproduct ∙

Crossproduct

Tensorproduct ⨂

Trace

Vectordifferentialoperator

Gradient . .

Divergence . ∙ .

Curl . .

Incompatibility . .

XIII

Nomenclature Notation

Deformationgradient

Displacementgradient

Velocitygradient

Compatible/Incompatiblestrain ,

Latticecurvature

Rotationvector

Spintensor

Incompatibilitytensor

GNDdensitytensor A

Slipdirectionof slipsystem

Normaldirection

EffectiveGNDdensity

Plasticslip

Plasticsliprate

Referenceplasticslip

Appliedstresstensor

Internalstresstensor ∗

Internalstressduetodislocation‐dislocation

interaction

Internalstressduetodislocation‐boundary

interaction(Imagestress)

Appliedresolvedshearstress

Internalresolvedshearstress ∗

Beltramistressfunctiontensor

Slipresistance

Hardeningmodulus

Elasticmodulus/Compliancetensor ,

Displacement

Bodyforce

Tractionforce

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

impartimpressivepropertiesforarangeofapplications(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

strengthsresultingfromstrong,hierarchicalarchitecturesatsmalllength‐scalescoupled

with robust failure resistance mechanisms. Our singular quest to mimic nature has

spawnedtremendousexcitementinsynthesizingmaterialsandconstructingstructures

that are aimed at using some of the natural principles. The notion of the statement

SmallerisStrongerhasfar‐reachingimplicationsinengineeringthematerialsthatpush

thelimitsofstructuralperformance.

Length‐scaleeffectsonmaterialproperties,oftentermedassizeeffects,areofgreat

importanceincurrentengineeringandscientificapplicationsthatrangefromlarge‐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

biomedicalandelectronicsapplications. Incrystallinemetals, size‐effects are reported

in a varietyofmaterial properties including elasticity (Agrawal et al., 2008;Wuet al.,

2005),plasticity(Dehm,2009;GreerandHosson,2011),thermal(Rohetal.,2010)and

electrical conductivities (Boukai et al., 2008), as specimen dimensions and/ or

microstructuralfeatures(e.g.diameterinnanowire,grainsizeincrystallinemetals)are

reduced. An understanding of these effects is especially important as our ability to

designandmanufacturestructuresatminiaturized length‐scalesandwithnano‐scaled

2

internal structures continues to acquire higher levels of sophistication (Zhu and Li,

2010).

Inmetallic 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

nanocrystallinepurealuminumwithanaveragegrainsizeof40nmisnearly10times

morethanthatofacoarse‐grainedpurealuminum(Gianolaetal.,2006).Nanotwinned

copperwithtwinthicknessof~35nmisnearly7timesstrongerthancoarse‐grained

purecopper(Luetal.,2009).Forafixedinclusionvolumefractiontheyieldstrengthofa

metal matrix composite (MMC) increases dramatically with decreasing inclusion size

(Lloyd, 1994). Myriad examples pertaining to thin films (Haque and Saif, 2003),

miniaturizedbeams(Motzetal.,2005),pillars(GreerandNix,2006),rods(Wongetal.,

1997)unequivocallyendorsethesmallerisstrongerphenomenon.Inotherwords,with

all other propertiesheld constant, the smaller the geometrical ormicrostructural size

the stronger a material is expected to be. Seen slightly differently, these examples

suggest that theelasticandplasticpropertiesofmaterials cease tobepurelymaterial

parameters as the specimen dimensions or microstructural features approach

characteristic microstructural length‐scale (Greer and Hosson, 2011). All of these

observations have a commonmessage: smaller is stronger. In a broad sense, the size‐

dependentbehaviorsofmicroandnano‐scaledstructuresareassociatedwiththehigh

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 andmechanics at small length‐scales. An

3

importantquestionthatarisesisthatofthechoiceofspatialandtemporalresolutions.

Atomistic provide a virtual experimental paradigm to capture the prevailing

mechanismsatveryhighspatio‐temporalresolution,butmaybecomecomputationally

prohibitive at larger structural length‐scale (even beyond a few hundred nm). At the

otherextreme,continuummechanicsprovidesastrongtheoreticalconstructthatcanbe

extremelyuseful ifappropriatelyendowedwithanabilitytopredictsize‐effects,albeit

atthelossofsub‐scaledetails.Athirdpossibilityisjudiciouslycombiningtheatomistics

andcontinuummechanicstoprovideaconcurrentmulti‐scalemodelingapproach.The

choiceofanapproachisdictatedbythedetailsweareinterestedinandthescalesthat

needtobebridgedwiththeavailablecomputationalpower.

Inthiswork,ourfocusisonasmallsubsetwithinthevastexpanseoflength‐scale

dependent behaviors.Weare interested in some of the size‐effects thatprevail in the

mechanicalbehaviorofcrystallinemetals.Aparticularcategoryofsize‐effectscovered

in this thesis pertains to crystalline plasticity that arises from interacting effects

betweendislocationsandtheirambience.Forexample,dislocationsgetstoppedbyhard

boundariesandgetannihilatedbyfreesurfaces.Inanotherscenario,dislocationstalkto

other dislocations in their neighborhood. All these events result in length‐scale

dependentmacroscopicplasticresponsesthatmanifestasstrengtheningofamaterial.

Weprobesomeoftheseeffectsinheterogeneouscrystallinemicrostructuresofcurrent

interestthroughanalyticalandcomputationalapproaches.

Tosetthestagefortherestofthethesis,webrieflydiscussdislocationplasticityin

crystallinemetalsasitcanbedescribedatvariouslength‐scales.

1.2 Length-scale Effects in Crystalline Microstructures

During the last couple of decades, crystalline metallic materials especially Face‐

Centered‐Cubic(FCC)metalsarevastlyusedasthenano/microstructuresfornumerous

4

applications.Therefore,itiscriticallyimportanttoobtainfundamentalinsightintotheir

length‐scaledependentmechanicalbehavioratmicroandnanoscales.Theexperimental

andtheoreticalaspectsof these length‐scaledependentbehaviorsarediscussedinthe

followingsections.

1.2.1 Plastic Deformation at Different Length-scales

In crystallinematerials, the unit processes that are deemed relevant to describe

plasticity must be identified based on the length and time‐scales of interest. From a

thermodynamicviewpoint,movementofthedislocationsduringplasticdeformation is

mediatedbycrystal latticeresistance.Thiscrystal latticeresistancecanorneedstobe

defined at different scales. At the finest length‐scale (atomistic), it is an inherently

dynamicalprocessof atomicmotions. In thedevelopmentof an incrementally coarse‐

grainedapproach,someofthemicrostructuraldetailsatthefinerscalearesmearedout

bymakingcertainassumptionswithregardsthelength‐andtime‐scalesatthesub‐scale

vis‐à‐visthecurrentscalesofinterest.Thisoftenprovidesamotivationtodefineamore

relevant unit process at the coarser length‐scale by coarsening the sub‐scale defect

dynamics.ThereviewarticlebyZaiserandSeeger(2002)servesasausefulreference.A

possiblecascadingflowofsuchamulti‐scalingprocess(Fig.1.1)thatisdeemeduseful

forthisthesisisbrieflydiscussedhere:

Atomicscale–describestheindividualatomintermsofitsfinercomponentssuch

aselectrons.Density functionaltheory(DFT)isthemostpopularmethodtoinvestigate

the total ground‐level energy and properties of a system of interacting electrons in

particular atomsandmolecules (Sholl andSteckel, 2009). Ituses the functionalof the

electrondensity,whichprovidesthepotentialfunctionasabasisformoleculardynamic

simulations.

5

Nanoscopicscale–Atthisscale, the individualatomsandmoleculesareresolved

where the information from the atomic scale that is coarse‐grained is the interatomic

interaction.Moleculardynamics(MD)isapowerfultooltocomputationallysimulatethe

physicalmotionsofatomsandmoleculesunderexternalstimuli.InMDsimulations,the

Newton’s equations of motion for a system of interacting particles are numerically

solvedwhereintermolecularinteractionsaredescribedbyapotentialfunctionprovided

by theatomic scale.A reasonably largeensembleof atoms ismodeled, and theelastic

andplasticpropertiesemergenaturallythroughinteratomicinteractions.Atthisscale,

the unit process that describes plastic deformation is the nucleation andmobility of

individual dislocations within a crystalline lattice. Given the inherent dynamics of

atomicmotions, typicalMDcalculationsneedhigh temporal resolution in theorderof

femto to pico seconds. The interactive long‐ and short‐range interactions between

dislocations arenaturally resolvedat this scale andprovide the essential physics that

can be rationalized as constitutive descriptions at coarser scales. Nanoscopic lattice

resistanceisreferredtoasthePeierlsstress.Itdependsstronglyonthestrainrateand

canbethermallyactivated;hence,itisreferredtoasthethermallatticeresistance.

Microscopic scale – At this length‐scale, the atomistic resolution is smeared out

renderinganelasticcontinuum,butthediscretenessofdislocationsisretained.Theyare

modeled as line singularities within an elastic continuum and their evolution is

describedthroughasetofconstitutiverulesthatareformulatedbasedonthesubscale

observations.Thecrystallatticeinformationisretainedintheformofanisotropicelastic

stiffness tensor and slip systems on which dislocations glide. The corresponding

mathematical construct and numerical implementation is commonly referred to as

DiscreteDislocationDynamics (DDD), if inertial terms are retained (Cazacu and Fivel,

2010).Internalstressesaroundindividualdislocationsareaccountedforatthislength‐

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 toMDwhile 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

calculationsthatspanseveral 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

dislocationdensity.However,someoftheseeffectscanbeincorporatedbyappealingto

non‐local field theories (Eversetal.,2004;Gurtin,2002;Hanetal.,2005a).Thisscale

canbeconsideredasabridgebetweenthemicroscopicandmacroscopicscalewherethe

mechanics at finer length‐scales is accounted for using appropriate constitutive

relations.

Mesoscopic(andmicroscopic)internalstressesareusuallyreferredtoasathermal

lattice resistance to dislocation motion, which are independent of temperature and

strainrateexceptforitstemperaturedependencethroughtheshearmodulus(Hulland

Bacon,2001;ZaiserandSeeger,2002).

Macroscopicscale–Bulkscaleresponsesdevoidofsize‐effectsarewell‐described

atthisscaleusingclassicalcontinuumplasticity(KhanandHuang,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.Suchaveragingproceduresnaturallysmearoutmuchofthemicrostructural

informationandmoreimportantly,theinherentmicrostructuralfeatures,givinglength‐

scaleindependentframeworks.Again,thisapproachworkswellinmanycases,butfails

tocapturesize‐effectsthatarisefrommicrostructuraldifferences.Forexample,suchan

approachessentiallypredictsthesame(size‐independent)yieldstrengthandhardening

responseforananocrystallinematerialandacoarse‐grainedmaterial.Recentattempts

admitlength‐scaleeffectsinsuchamacroscopictheorywithoutresortingtocrystallevel

slipdetails(AbuAl‐RubandVoyiadjis,2006;FleckandHutchinson,1997;NixandGao,

1998;VoyiadjisandAl‐Rub,2005).

Figure 1.1. Plastic deformation and appropriate unit processes for modeling at differentscales

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

interactionbetweendislocationsmaybeinfluencedbyinterfacessuchasgrainortwin

boundaries, and/or free surfaces. Therefore, additional interactions between

dislocations and boundaries should be taken into account for nano/micro‐scale

structureswherehighsurface(or interface)areatovolumeratio iscommon.Insingle

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 andNix, 2006); (Shan et al., 2007). In nanostructured polycrystalline

metalssuchasnanograinedandnanotwinnedmetals (Haque,2004;Luetal.,2009),a

Hall‐Petchbehaviorarisesfromdislocationinteractionwithgrainandtwinboundaries

intheformofdislocationpile‐up.

At continuum scales, dislocation inducedplasticitymay be broadly classified into

twogroupsbasedonthewaytheyaccumulateinduringplasticdeformation.Statistically

stored dislocations (SSD) accumulate by statistical trapping of the dislocations to

accommodate plastic slip (Ashby, 1970). At an atomistic scale, individual dislocations

produceinternalstresses intheirvicinity,butat largerscales(mesoandabove), these

are canceled in the process of averaging out, since SSDs by definition are randomly

distributed.Anothertypeofdislocationsarisesfromthenecessitytoaccommodatelocal

latticecurvaturesthatariseduetonon‐uniformplasticdeformation(Nye,1953;Ashby,

1970). Ashby (1970) referred to these as the Geometrically Necessary Dislocations

(GNDs).GNDsactasadditionalobstaclestothemotionofSSDs,butthemselvesdonot

contribute to plastic strain (Gao and Huang, 2003). Incorporating GNDs within

continuumframeworksendowthemwithanabilitytopredictalength‐scaledependent

macroscopic response under non‐uniform plastic deformation (Acharya and Bassani,

2000;Ashby,1970;Flecketal.,2003;NixandGao,1998).

9

ThefollowingGNDrelatedmechanismscouldbe identified intermsofstressesor

resistancemechanismsatdifferentscales(Figure 1.2):

Short‐rangeinteractionsofGNDswithSSDsasanadditionalthermal

lattice resistance which occurs in nanoscopic scale (Acharya and

Bassani,2000;NixandGao,1998).

Long‐rangeelasticGND‐GNDinteractiondescribedatthemesoscopic

scaleasathermalinternalstressesthatinfluencedislocationmobility

(Kröner,1967).

Long‐rangeelasticinteractionbetweenGNDsandboundariessuchas

free surfaces manifesting as athermal lattice resistance, which are

describedasimagestressfieldsatthemesoscopiccontinuumscales

Figure 1.2.Dislocationinteractionsat differentlength‐scales

10

Figure 1.2alsogivessomeexamplesofeachoftheinteractions.Thefocusthiswork

ismodeling theplastic deformation in crystallinematerials accounting for the length‐

scaleeffectsthatpersistatthemesoscopicscale.Whiletheseeffectsaremainlyascribed

to the presence of GNDs that are in‐turn related to strain gradients, somedislocation

mechanismsproducesize‐effectsevenintheabsenceofstraingradientsandarebriefly

mentioned later in this chapter, for clarity. Each of these may possess an associated

length‐scalethatmustbecomparedwiththelength‐scalesofinterest.Manyatimes,the

length‐scale are problem‐dependent and may be determined by structure geometry,

deformationprofile,materialmicrostructure,physicalpropertiesofboundariesandso

on(VoyiadjisandAl‐Rub,2005).

1.2.2 A Brief Overview of Experimental Observations of Length-

scale Effects in Plasticity:

Severalsimilarobservationsarereportedinmicro‐scaledspecimensinavarietyof

heterogeneousdeformationconditionsincludingbendingofsingle‐andpoly‐crystalline

beamsandthinfilms(HaqueandSaif,2003;Huberetal.,2002;Motzetal.,2005;Stolken

andEvans,1998).Specifically,theobservedtrendisthattheflowstressincreasesasthe

specimen thickness reduces. Furthermore, this size effect is enhanced in presence of

substratewhichcausesadditionalpile‐upofdislocationsatthefilm‐substrateinterface.

Similarbehaviorisobservedinmicroandnanoindentation,whichexhibit length‐scale

dependenthardness(MaandClarke,1995;McElhaneyetal.,1998;NixandGao,1998).

In metal matrix composites (MMCs), higher macroscopic strength and hardening is

reportedwithdecreasinginclusionsizewhilekeepingitsvolumefractionconstant.

11

Figure 1.3. Schematic of geometrically necessary dislocations (GNDs) pile up at grainboundaryinordertoaccommodatecompatibleplasticdeformation.

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

compensateincompatibilitiesintheplasticdeformation(Figs. 1.3andFigure 1.4)arising

due to relevant reasons (e.g. elasto‐plastic and thermal expansionmismatch between

the inclusion andmetalmatrix inMMCs or incompatible plastic deformation, (Ashby,

1970;Flecketal.,1994).

It is useful to mention here that although mechanics approaches relying GND‐

inducedstrengtheninghavegainedpopularityand isalsothemaintopicofthisthesis,

thesemaynotbetheonlyorthemostrelevantmechanismsinstrengthening.

IncompatibledeformationatGB

GNDpileupatGBtoaccommodatecompatibledeformation

F

F

12

(a)

(b) (c)

Figure 1.4.FormationofGNDinpresenceofstraingradientin(a)bendingofsinglecrystal(b)nano/microindentation(c)metalmatrixcompositecontainsnano/microinclusions.

A somewhat disconnected result is the recently observed size‐dependent

strengtheningofsinglecrystallinematerialsundernominallyuniformdeformations(e.g.

uniaxialtensionorcompression)atstructuralscalesbelowafewmicrons(Uchicetal.,

2004;Uchicetal.,2009).TheGNDmechanism isnotexpected tobeoperativeorbea

dominantmechanism in thesecasesdue to theabsenceof latticecurvatures.This is a

relativelynascentareaofresearchandseveralpostulateshavebeenrecentlyadvocated.

Theseincludethedislocationstarvationmodel(Dehm,2009;GreerandNix,2006;Nixet

al., 2007)which suggests that in a smaller specimen,dislocations readily escape from

the free surfaces (aided by image stresses) in comparison to the rate of dislocation

nucleation andmultiplication, or the source‐limiteddislocationplasticity (Dehm, 2009;

Uchicetal.,2009),whichsuggeststhatfewerdislocationsourcesinthecaseofsmaller

specimenscomparedtolargerspecimensisalsolikelytoproduceasimilarsize‐effect.In

general, many of the aforementioned mechanisms may operate in tandem and

13

contribute synergistically or compete with each other to produce overall plastic

responses.

Thespatialresolutionthatwefocusoninthisthesisisthesinglecrystal.Inthenext

section,webrieflysummarizesomeoftheproposedlength‐scaledependentcontinuum

approachesthataccountforsomeoftheGNDeffectsdescribedinFig.1.2.

1.2.3 Continuum descriptions of Dislocation-mediated Crystal

Plasticity

1.2.3.1 Classical crystal plasticity

Classical continuum plasticity theories are generally based on macroscopic

behaviors of materials in plastic region where materials are considered as a

homogenizedcontinuumbody.Theanisotropicplasticbehaviorofcrystallinematerials

was pioneered by works of Taylor and coworkers (Taylor, 1934; Taylor and Elam,

1923), andSchmid, (1924)whoproposed themovementof thedislocations in crystal

latticeas amajor sourceofplasticdeformation.Basedon theseobservations,Hill and

Rice(1972)andAsaroandRice(1977)developedarobustframeworkforsinglecrystal

plasticity.AcomprehensivereviewofsinglecrystalplasticityhasbeengivenbyAsaro

(1983).These theoriesexplicitlyaccount foranisotropicplasticity throughslipsystem

informationinthattheplasticslipcanoccurincertaindirections,theslipdirectionsand

oncertainatomicplanes,theslipplanes.Thediscretenessofatomisticsissmearedout.

Phenomenologicalhardeninglawsareprescribedthatattempttoadheretothephysics

of the hardening processes (Bassani and Wu, 1991; Peirce et al., 1983). The Taylor

hardeningmodel typically serves as a standard expression to describe the hardening

induced by myriad short‐range dislocation‐dislocation interactions , for example, a

generalizedmodelproposedbyFranciosi(1980)

14

( 1.1)

where isthecriticalresolvedshearstress(CRSS)on slipsystem,and and are

theshearmodulusandBurgersvector,respectively.Thecoefficients apportionthe

hardening components that account for both, self and latent hardening and is a

continuum field variable describing the SSD density on slip system. These

coefficients implicitly accounted for macroscopic isotropic hardening behavior arises

from short range dislocation interaction mechanisms in nanoscopic scale such as

multiplication,annihilation,joganddipoleformationandcrossslip.

In generalized dislocation based crystal plasticity individual dislocation

mechanismsandtheirevolutionlawsincorporatedintocontinuumframeworkinterms

of continuum microstructural field variables (Prinz and Argon, 1984; Roters et al.,

2000). Roters et. al. (2000) have proposed a dislocation based crystal plasticity for

polycrystalline materials, which is mainly concern about SSD density while GND

contributions are neglected. In their approach, plastic deformation is introduced in

termsofthreeinternalstatevariablesasmobileandimmobiledislocationdensityinthe

cellinteriorsandimmobiledislocationdensityinthecellwallsandtheirevolutionlaws.

The kinematic hardening in macroscopic continuum scale is addressed by

ArmstrongandFredrick(1966;2007)intermsofbackstresstensor.Later,ithasbeen

extended into conventional crystal plasticity framework (Cailletaud, 1992). The

evolutionlawforbackstresstensorincrystalplasticityframeworkissometimeswritten

as(VoyiadjisandHuang,1996;XuandJiang,2004)

| | ( 1.2)

15

where is the plastic slip rate on slip system, and and are coefficients

obtainedfromexperiments.Inmicroscopicscale,thebackstressarisesfromlongrange

elasticinteractionbetweendislocationsincellstructureandareresponsibleforclassical

Bauschingereffects (Mughrabi, 1983). In conventional crystallinematerialswith large

grainsizes,thecellstructureandaveragedislocationspacingarenearlyindependentof

the specimen sizes and consequently internal stress is only function of plastic strain.

However, as microstructural or specimen dimensions decrease, the dislocation

arrangementsandtheirinteractionsmaybesignificantlyaffected.

1.2.3.2 Continuum crystal plasticity with GNDs

Withincreasingquesttowardstrongandductilematerialsatlowoverallweightfor

large‐scale structures on the one hand and the rapid development of miniaturized

structures small scale devices on the other, predictive modeling of length‐scale

dependentmaterial behavior has assumed a central role to analyze and design novel

materials and structures. However, a robust understanding of length‐scale dependent

mechanismsisachallengingproblem.Although,classical(i.e.length‐scaleindependent)

crystalplasticitytheoriescapturethebehaviorsofbulkcrystallinematerialswithgood

accuracy, they fail to predict length‐scale effects since no explicit microstructural

informationisincluded.Furthermore,performingMDsimulationsonrealistictimeand

length‐scalesfornano/microstructuresareverycostly.Alogicalrecourseistodevelop,

continuumcrystalplasticitytheoriesthatareendowedwithGNDinformationwithin.

Alongside the SSD interactions, the GND‐SSD and GND‐boundary interactions

become importantat small length‐scales.NixandGao(1998)proposed that theGNDs

act as the obstacles formovement of other dislocations and provide additional short‐

range interactionwith other dislocations. Since thenatureof these interactions is the

same forbothSSDsandGNDs, theyreformulated theTaylorhardeningmodelwithan

16

additional term that arises from thepresenceofGNDs,which in‐turn is related to the

straingradient.Theassociatedlength‐scaleisrelatedtotheBurgersvectorthatisscaled

byelasticshearmodulusandbasicmaterialstrength.Thisapproachhasbeenextended

intocrystalplasticityframework(MSG‐CP)by(Hanetal.,2005a).AcharyaandBassani

(2000)applied thesameconceptby introducingahardeningmodulusasa functionof

both,strainandstraingradienttoaccountforbothSSDsandGNDsinteractions.Since,

these theories do not include higher‐order stresses and boundary conditions, the

generallyreferredtoasthelower‐ordergradienttheories.Thesetheorieshavecapability

tocapturesizedependentflowstressatmoderatestrainwhereflowstressisdominated

by short range interaction of dislocations (Acharya, 2003; Schwarz et al., 2008).

However, they fail topredict size‐dependentyieldstrengthat initial stageofplasticity

becausetheyignorethelong‐rangeelasticinteractioneffects.

This latteraspect that isrelatedtosmallstrainscanbemodeledby incorporating

theinternalstressesthatariseduetotheGNDs(EvansandHutchinson,2009;Fleckand

Hutchinson,1997).UnliketheSSDdensity,anaverageGNDdensityoveramesoscopic

volume result in net internal residual stresses through long‐range elastic interactions

between the GNDs. Kröner (1967) incorporated the long‐range interaction of

dislocations into continuum mechanics through the nonlocal constitutive equations

using integral formulation. LaterAifantis (1984; 1987) accounted for this effect using

constitutiveequations that includeplasticstraingradient terms.FleckandHutchinson

(2001; 1993) proposed higher‐order phenomenological strain gradient plasticity

theoriesusingreformulationoftheyieldfunctionthatincludedgradienttermsandthat

introduce additional boundary conditions. Gurtin and coworkers (Anand et al., 2005;

Gurtin, 2002, 2010; Gurtin and Anand, 2005) generalized this theory using

thermodynamicframeworkbyproposinganadditionaldefectenergyduetodefectslike

dislocations.Thisadditionalenergyiswork‐conjugatetothehigher‐orderstressesthat

are related to the second gradients of plastic strain, requiring higher‐order boundary

17

conditions. With the same concept, different approaches have been advocated to

develop nonlocal theories (Abu Al‐Rub et al., 2007; Anand et al., 2005; Gudmundson,

2004; Polizzotto, 2009; Voyiadjis andDeliktas, 2009). In all of these theories, length‐

scales enter into the continuum equations to be mathematically consistent, but their

physicaloriginandconnectionwithmaterialmicrostructuresareunclear.

Tobetterunderstandthelength‐scaledependentbehavior,underlyingmechanisms

andoriginoflengthscaleparameters,thedefectenergyandcorrespondinghigher‐order

stress and boundary conditions need to be interpreted in terms of micro structural

information.Recently,thelong‐rangeelasticinteractionofGNDsatmesoscopicscaleis

modeled into continuum plasticity using dislocation theory of infinitemediumwhere

length‐scales are defined in termsof thedislocation correlationdistance (Evers et al.,

2004;GerkenandDawson,2008;Mesarovic,2005).Thiscorrelationdistancerelatesto

thecollectivebehaviorofdislocationsstatisticalmechanicsapproachwhichexplainthe

originofstraingradienttermsinsizedependentcontinuumtheories.

Summarizing, there are two main groups of strain gradient theories mostly

accountingforshort‐andlong‐rangeinteractionsbetweendislocations:thelower‐order

and higher‐order strain gradient theories. It has been shown that the short‐range

interaction isamajorsourceof sizedependencyatmoderatestrainwheredislocation

density is large enough (Acharya, 2003; Schwarz et al., 2008). However, the higher‐

orderstraingradienttheoriesaresuccessfulinexplainingthesize‐dependentresponse

atyieldandtheytieittothelong‐rangeinteractionbetweenGNDs(Borg,2007;Evans

andHutchinson,2009;Niordson,2003a).Thedifficultywithhigher‐orderb.c.’sisthatit

may not be always easy to identify appropriate descriptions for general interfaces

(Voyiadjis and Deliktas, 2009) and typically, the computational effort is significantly

large.

18

1.3 Scope and Objectives of the Thesis

In this dissertation, we investigate the length‐scale dependent behaviors of

microstructures due to the presence and non‐homogeneous distribution of the GNDs.

The formulation focuses on face‐centered‐cubic (FCC) materials and their size

dependentbehaviorsundernon‐uniformplasticdeformation.Abroadobjectivehereis

to physically incorporate the GND related mechanisms into a continuum framework

throughtheconceptofkinematicincompatibilityoftheunderlyinglattice.

InChapter2,we focusourattentionon the length‐scaledependentbehavior that

arise from short‐range interactions between the SSD and GND densities, which

manifests as enhanced flow hardening atmoderate strains. At such strains, the long‐

rangeelasticeffectsduetoGNDsareexpectedtobenegligible(Acharya,2003;Schwarz

etal.,2008).ThisGNDinducedhardeningmodeledthroughTaylorhardening(Nixand

Gao,1998)asextendedtocrystalplasticity(Hanetal.,2005a).Theresultingmechanism

based strain gradient crystal plasticity is implemented within ABAQUS® via user‐

material subroutine (UMAT). First,we investigate the gradient‐induced size‐effects in

singlecrystalswithembeddedinclusionsunderthermo‐mechanicalloading.Theroleof

internal stresses due to prior thermal loading is probed as a function of crystal

orientation,and inclusionshapeandsize. Then,wefocusourattentiononthe length‐

scale dependent interaction effects in polycrystalline MMC due to the grain size and

inclusionsizes.Weproposeasimpleanalyticalmodelforthisinteractioneffect.

Chapter3presentsconcernstheroleofGNDsinproducinglong‐rangeinteractions

that manifest as internal stresses. We develop a nonlocal crystal plasticity theory

accounting for these long‐range GND interactions using stress functions approach as

appliedtoelasticallyisotropicmaterials.Wesystematicallyshowthatnonlocalinternal

stressesdevelopduetonon‐homogeneousspatialdistributionoftheGNDdensity.Using

19

thermodynamic framework these internal stresses are incorporated into continuum

crystal plasticity as an additional irreversible stored energy (defect energy). The

internalstressesappearasadditionalresolvedshearstressinthecrystallographicvisco‐

plastic constitutive law for individual slip systems. Using this formulation, we

investigate boundary value problems involving isotropic single crystals subjected to

monotonic and cyclic loading. The resulting length‐scale dependent isotropic and

kinematichardeningbehaviorsareinvestigatedintermsofshort‐rangeandlong‐range

GND interactions. Finally, we close the chapter by discussing the extension of this

approachtocrystallinematerialswithelasticanisotropy.

In the theory presented in Chapter 3 ignores the long‐range elastic interactions

between the GND density and boundaries, the so‐called image stresses. These image

stresses may have significant effects in miniaturized specimens and are therefore

important. In Chapter 4, this additional long‐range interaction is incorporated by

augmenting the formulation in Chapter 3 with another kernel (Green) function that

accounts for traction‐free surfaces. The resulting additional internal stresses are

introducedintermsofGNDdensity‐surfaceelasticinteraction.Whilethebasicapproach

isgeneral,wechoose thin filmunderpurebendingasamodelproblem to investigate

the length‐scaledependentbehavior. We show that these additional internal stresses

produce a length‐scale dependent macroscopic response even in the case of such a

system that comprises a nominally uniformdistribution of GND density.We compare

our resultswith experiments and provide a physical interpretation of the underlying

length‐scale.

Finally,Chapter5summarizestheaccomplishmentsofthisPhDthesisandprovides

recommendationsforfuturework.

20

2 A Mechanism-Based Gradient Crystal

Plasticity Investigation of Metal Matrix

Composites

2.1 Introduction

Theadventofnanostructuring techniqueshas led toanunprecedentedgrowth in

the area of synthesizing metal matrix composites (MMC) with exceedingly superior

strengths.ItispossibletosignificantlyenhancethestrengthofMMCsoverthatachieved

byconventionalstrengtheningfromloadtransfer,bysynthesizingmicrostructureswith

nanocrystalline matrices, incorporating small sized reinforcing inclusions, or a

combinationofboth(Lloyd,1994;NanandClarke,1996;SekineandChent,1995).Grain

boundaries(gb’s)createstrongbarrierstodislocationsprovidinghigherbaselinematrix

strength that canbe further improvedby theadditionof reinforcing inclusionsMMCs

through a load‐transfermechanism.Thus, onemay rely on synthesizinghigh‐strength

MMCs solely by using nanocrystalline matrices. Alternatively, the length‐scale

dependent strengthening from micron or sub‐micron sized inclusions attributed to

interaction of the geometrically necessary dislocations (GNDs) with matrix‐inclusion

interfacesmayalsoprovideanotherpathtostrengthenhancement.However,boththe

strengtheningstrategieshavetodealwithonecommoncaveat–theenhancementinthe

strength usually comes at the cost of precipitous reduction in the ductility. The latter

alternative might be attractive, because it allows using smaller inclusion volume

fractions(v.f.)thatmayhelpmitigatethestrength‐ductilitydichotomytosomeextent.

Recent experimental and analytical efforts have aimed at understanding the size‐

effectsinMMCs(e.g.(Balint,2005;Daietal.,1999;JoshiandRamesh,2007;Kiseretal.,

1996; Lloyd, 1994; Nan and Clarke, 1996))and have led to the development of novel

21

compositemicro‐architectures (Habibi et al., 2010; Joshi andRamesh, 2007; Ye et al.,

2005). These investigations indicate that one has to judiciously choose appropriate

values for the microstructural design degrees of freedom in imparting optimal

functionalcharacteristicstoanMMC.Analyticalandcomputational investigationshave

focused on implementing length‐scales in the conventional plasticity theory based on

theGNDargumentasappliedtoMMCs(e.g.(Cleveringaetal.,1997;JoshiandRamesh,

2007;NanandClarke,1996;Niordson,2003b;Xueetal.,2002;Zhouetal.,2010).From

a mechanistic viewpoint there are several challenging aspects that need to be

understood in the length‐scaledependentMMC response. For example, thephysics of

plasticeventsattheinclusion‐matrixinterfaces(i‐m)andatgb’s(andtriplejunctions)

duetothermalandmechanicalloading,communicationbetweenthei‐minterfacesand

gb’s,grainorientationeffects,inclusionandgrainsizedistributions,thermalandelastic

mismatchbetweenphasesandseveralmore.Whileitmaybeimportanttoincorporate

these mechanisms, a single mechanistic framework that is capable of resolving the

microstructural details and concurrently also embeds appropriate physics for all the

interfacialmechanismsisdifficulttoconceiveatthemoment.Acomparativelytractable

settingispossibleifonechoosestosimplifyand/orignoresomeoftheaspects.Crystal

plasticity enriched with length‐scale features can effectively handle the kind of

resolutionnecessaryfortheproblem.

Inthischapter,wefocusourattentiononthe length‐scaleeffects inMMCsarising

fromshort‐range interactionbetweenSSDsandGNDs,which isdominantatmoderate

strains where dislocation density is high (Acharya, 2003; Schwarz et al., 2008). To

account for these interactions within a continuum framework, we resort to the

Mechanism‐based Slip Gradient Crystal Plasticity (MSG‐CP) developed by Han, et al.

(2005a)thathasitsrootsinthepioneeringworkoftheNixandGao(1999;1998).The

MSGCPframeworkaccountsforlength‐scaleeffectsintheslipsystemconstitutivelaws

by including slip gradients on individual slip systems that are related to their GND

22

densities. Given that in the present work the grains and inclusions are explicitly

resolved,slipgradientsnaturallyariseatgb’sandi‐minterfacesduetheirelasto‐plastic

and thermal mismatch. However, the MSGCP approach is a lower‐order theory

compared to a higher‐order framework1, because it does not invoke additional

boundary conditions (b.c.’s) at interfaces that are related to the gradient of the GND

density, i.e. Laplacian of the plastic slip (Abu Al‐Rub, 2009; Borg, 2007; Geers et al.,

2007;Gurtinetal.,2007;KurodaandTvergaard,2006;KurodaandTvergaard,2008a,b;

McDowell, 2008; Voyiadjis and Deliktas, 2009). Consequently, lower‐order CP

approaches cannot model some of the enhanced interactions between interfaces and

dislocations that higher‐order CP approaches are capable of handling. For example,

(Borg, 2007) introduced a higher‐order CP theory that includes a material

parameter to tune the inter‐granular interaction at gb’swith impingingdislocations.

Usingthis,heinvestigatedtheroleofgrainboundariesonthemacroscopicbehaviorsof

simulated polycrystals and demonstrated that0 ∞ determines the amount of

strengthening at yield. Notably, the 0 case (gb’s fully transparent to dislocations)

degeneratestoalower‐ordertheory.Asindicatedby(Borg,2007)theseb.c.’stogether

with the choice of interface material parameters may have a profound effect on the

nature of polycrystalline strengthening and hardening predicted by these theories.

Althoughahigher‐ordertheorywouldbesuitedforthepresentproblem(Fredrikssonet

al., 2009), the difficulty with higher‐order b.c.’s is that it may not be always easy to

identifyappropriatedescriptions forgeneral interfaces (VoyiadjisandDeliktas,2009).

1Lower‐ordergradienttheoriesintroducelength‐scalethroughfirstgradientofplasticslip

thatrelatesonlytothepresenceoftheGNDdensity.Ontheotherhand,higher‐ordergradient

theoriesincorporatetheGNDdensitydistributioneffecttooandrelatetothemtothesecond

gradientofplastic slip. Thisleadstoaconstitutivelawintheformofapartialdifferential

equationthatnecessitateshigher‐orderb.c.’s.

23

Moreover,thecomputationaleffortforhigher‐orderCPissignificantlylargerthantheir

lower‐ordercounterparts.Ontheotherhand,duetotheinherentinabilityoftheMSGCP

in handling enhanced long‐range interactions between dislocations and interface the

length‐scale effect appears only in the flow behavior rather than at yield (Evans and

Hutchinson, 2009). However, despite some of its limitations, we choose the MSGCP

theorykeepinginviewitssimplicityinthenumericalimplementationwithinexistingCP

framework, computational expense for the present work and a relatively established

physicalunderstandingofthelength–scaleparameters.

In the following section, we first give a brief outline of the computational

implementation ofMSGCP (Han et al., 2005a) as user‐material subroutine (UMAT) in

ABAQUS/STANDARD®finiteelementsoftware.Usingtheimplementedformulation,we

first investigate size‐effects in single crystalMMCsdue to thermo‐mechanical loading.

ThisisaclassicsourceofGNDexistencethatarisesduetothermalresidualstressesthat

pre‐exist in anMMCmicrostructure due to themismatches in the thermal expansion

coefficients (CTE) of the matrix and the inclusions together with elastic and plastic

mismatches. The corresponding GND density is referred to here as the thermal GND

densitytodistinguishitfromtheGNDdensitythatarisesduringmechanicalloading.We

simulate the roleofpre‐existing thermalGNDdensity on the subsequentmacroscopic

and microscopic behaviors under mechanical loading as a two‐step process. These

thermo‐mechanicalsimulationsessentiallyrestricttheirattentiontosinglecrystalMMC

inabidtounderstandthelocalmicroscopicdetailsthatarisearoundtheinclusionsthat

are embedded within large grains. In section 2.4, we take a step further and model

polycrystalline MMCs that include both, grain and inclusion size‐effects under

mechanical loading.Theobjective is toquantify thenatureof the interactionbetween

thesetwomicrostructuralsizesontheoverallresponse.Throughthesepolycrystalline

simulations, we propose a simple analytical model that can be easily integrated into

24

homogenized continuum calculations such as the Mori‐Tanaka approach (Joshi and

Ramesh,2007).

2.2 Computational Implementation of MSGCP Theory

ThekinematicsandkineticsofMSGCPapproachimplementedinthisworkclosely

follow the conventional continuum crystal plasticity framework of Asaro and co‐

workers(Asaro,1983;Peirceetal.,1983),exceptthatalength‐scaleeffectisintroduced

intheslipsystemhardening.

Figure 2.1.Kinematicsofsinglecrystaldeformation

Based on the multiplicative decomposition of deformation gradient proposed by

Lee(1966),incaseoffinitedeformation,thetotaldeformationgradient is

( 2.1)

where and representtheelasticandplasticpartsofthedeformationgradient,

respectively (Figure 2.1). The spatial velocity gradient in the current state is (Asaro,

1983)

25

( 2.2)

.

. ( 2.3)

where and are the rateof deformationand spin tensors, respectively.The super

scriptseandpsignifytheelasticandplasticparts,respectively.Weassumethatasingle

crystal deforms plastically solely by crystalline slip and the elastic behavior of the

crystal isconstantduringplasticdeformation.The latticeorientation isaffectedsolely

byelasticpartofthetotaldeformationgradient.

The elastic constitutive equation for a single crystal proposed by Hill and Rice

(1972)isadoptedinthiscode.Theconstitutivelawforplasticsliprate isassumedas

. . ( 2.4)

where and are the reference plastic slip rate and resolved shear stress on slip

system and istheoverallhardeningoflatticeduetobothSSDandGNDdensities.

TheevolutionlawfortheSSDinducedhardeningformultipleslipdeformationisgiven

by

( 2.5)

where is a matrix representing self and latent hardening coefficients given by

(Asaro,1983),

26

(nosumon )

,

( 2.6)

In Eq. (2.6), is initial hardening modulus, is the saturation value for the

resolved shear stress, is the critical resolved shear stress, ∑ is the total

cumulative shear strainonall slip systemsand (~1‐2) accounts for the interaction

betweendifferentslipsystems.IntheMSGCPapproach,theGNDdensity on slip

system is assumed to contribute to its overall hardening via Taylor hardeningmodel.

Consequently, is(Hanetal.,2005a)

( 2.7)

where the internalmaterial length‐scale ,with as themagnitudeof

Burgersvector, astheoverallshearmodulusand asanempiricalmaterialconstant

ranging between 0.1‐0.5. In Eq. ( 2.7), is an effective scalar measure of the GND

densitytensorontheslipsystem

| | ( 2.8)

where and arerespectively, theslipdirectionandslip‐planenormal for slip

system. The effect of slip gradient is related to the GND density in each slip system

via ⁄ .

27

2.2.1 Slip gradient calculation

Inimplementingthislength‐scalefeaturewithinaUMAT,oneneedstocalculatethe

slipgradientsateachGausspoint(GP).Weexploittheconceptofshapefunctionsthatis

atthecoreofatypicalfiniteelement(FE)formulationforevaluatingtheslipgradients

corresponding to each slip systems. For illustration purposes, we present the

formulationapplicableforan8‐nodeplanestrainFE,buttheapproachcanbeextended

todifferenttypesofFE’s.Asiswidelyknown,thenumberofGP’sinaFEdeterminesthe

orderofintegration.The8‐nodeplanestrainelementthatweadopthere(CPE8R)usesa

reduced integration procedure in order tominimize the effects due to shear locking.

Therefore,foreachFEallthestatevariables(i.e.individualandtotalslip,slipgradients,

etc.) and stresses are calculated at these GP’s. To calculate the plastic slip gradients

withinanFE,weapplytheapproachsimilartotheoneisusedincalculatingstrainsfrom

displacementsinaconventionalFEformulation(Reddy,2006).WithinaFE,weconsider

a 4‐node pseudo‐element (Figure 2.2) constructed by joining the GP’s describable by

linearshapefunctions 1 4 .

(a) (b)

Figure 2.2.(a)AnEight‐nodeplanestrainFEwithfourGPsand(b)alinearpseudo‐elementconstructedfromtheGPsoftheactualFEwhere and arethelocalisoparametriccoordinates.Theslipandnormaldirections and ofatypicalslipsystem arealsoshown(b).

x

y

ξ

ηsm

28

Thelocalisoparametriccoordinates and ofthepseudo‐elementarerelatedto

theglobalcoordinates and viathedeterminant oftheJacobianmatrix.

( 2.9)

Theslipgradientvector intheslipdirection withineachelementisobtained

usingthechainruleofpartialdifferentiation

( 2.10)

andtheCartesianslipgradientsarerelatedtothepseudo‐elementshapefunctionsby

; ( 2.11)

where istheplasticslipat pseudo‐node(i.e.GPoftheactualFE)andon slip

system2.TheCartesianderivativesof (Eq.2.8)arecalculatedusingEq.( 2.9).

2.2.2 Time integration scheme

Thetime integrationused intheUMATisbasedonthe implementationbyHuang

(1991),butcontainsaugmented informationabout theGNDeffects.Forcompleteness,

wesummarizethemethodhere.Theincrementalslipon slipsystemis

2ThisissimilartotheoneadoptedinABAQUStocalculatestrainfromnodal

displacements ,e.g., ∑ .

29

∆ ∆ 1 ∆ ( 2.12)

wheretheparameter introducesalinearinterpolationbetweenvalueofsliprate at

thebeginningandendofthetimeincrement(Peirceetal.,1983).The 0degenerates

toEulerforwardtimeintegrationscheme,buttherecommendedvalueis0.5.Eq.( 2.12)

canbesolvedusingNewton‐Raphsontechnique

∆ ∆ 1 ∆ ∆ 0 ( 2.13)

Then,theplasticsliprateiscomputedas

∆ .∆∆

. ∆ ( 2.14)

Here, thevaluesof stressandsolutiondependent statevariablesareevaluatedat

the end of each time increment and this allows using larger time increment. Further

details on the basic implementation of the user subroutine UMAT and incremental

formulationscanbefoundinthereportbyHuang(1991).

2.3 Length-scale dependent MMC response induced by

thermal residual stresses

Synthesis ofmetalmatrix composites (MMCs) typically involvemoderate to high

temperatureprocessing followedbycoolingdowntoroomtemperature.Such thermal

processes cause the internal residual stresses in MMCs due to high CTE and elastic

mismatches between the matrix and inclusion. It is well‐established that for

conventional coarse‐grainedMMCs, the inclusion andmatrix properties togetherwith

the volume fraction (v.f.), shape and arrangement of inclusions govern the overall

stress‐strain behavior (Christman et al., 1989; Corbin andWilkinsona, 1994; Qiu and

Weng,1991;Shenetal.,1995).ArsenaultandTaya(1987)experimentallyinvestigated

30

the effects of thermal residual stress on the overall strengthening and hardening

behaviorofMMCsundermonotonictensileandcompressiveloading.Theyexplainedthe

tension‐compression asymmetry through an analytical model based on Eshelby’s

equivalentinclusionapproach.ThestrengtheningandhighhardeningbehaviorofMMC

occur due to high triaxiality in the stress state exists within thematrix region at the

inclusion‐matrix(i‐m)interface(Christmanetal.,1989;LiandRamesh,1998;Llorcaet

al.,1991;Shenetal.,1995).Mostoftheseworksemployinclusionsthatareseveraltens

ofmicronsinsize.Lloyd(1994)observedthatforafixedinclusionv.f.thestrengthofthe

MMCincreasedwithdecreasing inclusionsizes(intherangeof fewmicrons). Inother

words, the MMC response becomes length‐scale dependent‐ an effect that has been

explainedintermsofGNDs(Ashby,1970;Nye,1953).Theseareadditionaldislocations

that arise due to the thermo‐elastic mismatch between the inclusion and the matrix

(ArsenaultandShi,1986;BarlowandHansen,1995;Daietal.,1999;Daietal.,2001a;

Dunand and Mortensen, 1990, 1991; Joshi and Ramesh, 2007). Several experimental

observationsindicatehighdislocationdensityatreinforcement‐matrixinterfacedueto

thermo‐elastic mismatch between reinforcement material and matrix (Arsenault and

Shi,1986;BarlowandHansen,1995;DunandandMortensen,1991).

Alength‐scaledependentmetalplasticityframeworkbecomesnecessaryinorderto

correctly predict size‐effects in MMCs, including failure (Dai et al., 1999; Dai et al.,

2001a).Recently,suchalength‐scaledependentbehaviorhasbeenaccountedforwithin

finite element (FE)basedcomputationalworksby introducingplastic straingradients

within the homogenized constitutive laws for continuum plasticity (Xue et al., 2002;

Zhang et al., 2007; Zhou et al., 2011). Ohashi (2004) has explored the distribution of

GNDdensityaroundcuboidalandspherical inclusionembedded in theFCCcrystalline

matrixunderuniaxialloadingemployingdislocationbasedcrystalplasticityframework.

However, thesenumericalapproacheshavenotaccountedfor theeffectsdue to initial

GND density due to thermal processing. It is imperative to accounting for the (size‐

31

dependent) thermal residual stresses within these frameworks, because they have

importantcontributionstokinematichardening.Analyticalmodelshavebeenproposed

based on the idea of dislocation punching (Arsenault and Shi, 1986; Dai et al., 1999;

Dunand and Mortensen, 1990, Taya et al., 1991; Qu et al., 2005) incorporated the

contribution from the thermal GNDdensitywithinFE‐based strain gradientplasticity,

butinarathersimplisticmannerasanadditionaluniformbackgroundmatrixstrength.

In comparison, underscoring the fact that these thermal GNDsmay not be uniformly

distributed in thematrix (Arsenault and Shi, 1986;Mukherjee et al., 1995; Suh et al.,

2009). Suh et al., (2009) proposed an FE‐based discrete punched zone approach that

includesanadditionalregionsurroundingan inclusionwhosestrength isenhancedby

the presence of thermal GNDs and investigated the length‐scale dependent

strengtheningandinterfacialfailureofMMCs.

Common to all the aforementioned modeling approaches is the assumption of

homogenizedmatrixplasticityinthattheyignorespecificcrystallographicorientations,

which are important in discerning the local deformation fields that affect global

composite response (Schmitt et al., 1997; Barlow and Liu, 1998). One may envisage

scenarioswhere crystallographic orientation effects are important in determining the

inclusion‐induced size effects (Barlow andHansen, 1995; Shu, 2000). For example, in

polycrystallineMMCswheninclusionsaremuchsmallerthanthesurroundinggrainsthe

local crystallography would be expected to decide the GND distribution. Likewise, in

highly textured MMC architectures the overall crystallographic orientation would be

expected to produce strong plastic anisotropy, which may also influence the GND

induced size effect. Single crystal plasticity based approaches are valuable in such

scenarios. Someworks have been carried out to investigate the size‐dependentMMC

behaviorsusingcrystalplasticityframeworks(Cleveringaetal.,1997;ShuandBarlow,

2000), but they ignore the presence of preexisting heterogeneously distributed

thermally‐inducedGNDdensity.

32

Inthefollowing,weaddressboththeseissuesbyadoptingaMechanism‐basedSlip‐

gradientCrystalPlasticity(MSGCP)approachandperformingsimulationsofmodelMMC

architectures under thermal and mechanical loads. We first elucidate the role of

crystallographicorientationon the inclusionsizedependent thermalGNDdensity.We

quantify the individual contributions therein from the thermal and elasto‐plastic

constituent mismatches. Then, for a given crystal with embedded inclusion, this

crystallography mediated non‐uniform thermal GND density is retained as a starting

motif under subequentmechanical loading,which gets superposedon themechanical

GNDdensity.Thesimulationsnotonlyresolvethe inclusionsizeandshape‐dependent

distributionoftheGNDdensity,theyalsoprovideusefulinformationaboutevolutionof

their averagedmeasures as a function of strain (Dai et al., 2001).We alsomodel and

discuss the size‐dependent dependent tension‐compression asymmetry under

mechanical loadingproducedduetothepre‐existingthermalGNDdensity.Mostofthe

results and discussions presented here pertain to single crystalmatrices, butwe also

demonstratetheapplicabilityofsuchanapproachtopolycrystalMMCarchitectures.The

detailsoftheslip‐gradientcrystalplasticitytheorycanbefoundinHanetal.(2005) and

its computational implementation as a user‐material subroutine within ABAQUS/

STANDARDasdiscussedinprevioussection.

2.3.1 Computational results for single crystals with inclusions

To enable consistent comparison of the size‐effect across different parametric

models, first we consider MMC unit cells comprising square inclusions embedded in

singlecrystalmatrixwithplane‐strainconditionintheout‐of‐planedirection(Fig.2.3).

33

Figure 2.3.Metalmatrixcomposite(MMC)withuniformarrangementofinclusionsandunitcellcomprisingsinglecrystalmatrixandsquareinclusion.

Asshowninthefigure2.3thecrystalorientationθforanFCCstructureisdefined

as the anglemadeby the [100] crystaldirectionwith the global loadingdirection

and[001]crystaldirectionistakentocoincidewiththeglobal direction.Forallthe

cases, the left and bottom edges are constrained along the and directions,

respectively; the top edge is allowed to move vertically, but constrained to remain

straight during deformation. Table 2.1 gives the material properties used in the

simulations, which are representative of pure Al (matrix) and SiC (inclusions). For

simplicity, we assume isotropic elastic properties for the matrix, but anisotropic

elasticitycanbeeasilyimplemented.

Table 2‐1.ConstituentparametersusedinsinglecrystalMMCsimulations

ParameterElastic

Modulus

Poisson’s

ratio

Burgers

vector

Initial

hardening

Modulus

CRSSSaturation

stress

Coefficientof

thermal

expansion

(GPa) (nm) /

Matrix(m) 70 0.33 0.25 510 60 109 23.6e‐06

Inclusion(i) 190 0.19 ‐ ‐ ‐ ‐ 4.3e‐06

34

The thermo‐mechanical loading condition is simulated as a two‐stepprocess that

mimics the real scenario in that the thermal GND density arises due to thermal

quenching as a part of processing, while the mechanical GND density accumulates

during subsequent mechanical loading. A typical two‐step thermo‐mechanical

simulationusingMSGCP isperformedas follows: in the first step,weapplyauniform

temperaturereductionΔ 500 totheentireunitcell.Thisquenchingeffectresultsin

a thermal GND density that is heterogeneously distributed within the matrix. The

secondstepconstitutesusingthispre‐existingGNDdensitydistributionasabackground

motif on the same starting microstructure and performing a new calculation under

actual mechanical loading of interest. Thus, following the thermal loading step, a

uniform displacement b.c. in the x1-direction is applied to the right edge (Fig. 2.3)

producinganominalstrainrate 1 10 .Weconsiderseveralcrystalorientations

0 , 10 , 30 45 and inclusion sizes 1, 2, 5 10 to elucidate the

crystallographicallymediatedsize‐effects.Inallthesimulations,theinclusionv.f.iskept

constant( 0.05).

Underthermalandmechanicalloads,size‐dependentresponsesariseasaresultof

theplasticdeformationcarriedbytheGNDsduetotheinterfacialstressesarisingfrom

thethermo‐elasto‐plasticmismatchbetweentheinclusionandthematrix.Theeffective

GND density under thermal/ mechanical loading condition is calculated using the

constitutivelawsprovidedinprevioussection(Eq.2.8).

2.3.2 Crystal orientation and inclusion size effects on thermal GND

density distribution

Figure2.4a‐d exhibit several interesting featurespertaining to thedistributionof

the thermal GND density for two 1 and 10 as a function of different

35

crystalorientations ,priortothemechanicalloading.Foremost,itcanbevalidatedthat

forafixed thedistributionof issimilar fordifferent inclusionsizes;asexpected,

themagnitudeismuchlargerforthefinerinclusion(Daietal.,2001b).

Figure 2.4. Crystalorientationand inclusionsizedependentdistributionofeffectiveGNDdensity (|Δ | 500, 1 .

However, a key result is that the simulations quantify the heterogeneity in the

distribution of , which underscores the limitations that may arise from the

assumption of uniform enhancement of matrix strength due to quenching (Dai et al.,

2001b; Taya et al., 1991). Expectantly, the resulting matrix strengthening cannot be

uniformandmaybeexacerbatedbytheheterogeneityofinclusiondistribution.

36

(a)

(b)

Figure 2.5.(a)DistributionofeffectiveGNDdensity alongthediagonallineasshowninembeddedfigure. |Δ | 500 (b)evolutionofaverageGNDdensity duringcoolingprocess( 1 .

Distancefrominclusioncorner,x(

Eff.GNDdensity

(

0.0 0.2 0.4 0.6 0.8 1.0 1.20

5

10

15

20

25

30

35

40

45

0 10 30 45

37

This becomes especially critical when working with composites tend to host

relativelydilutev.f.'sofmicronorsub‐micronsizedreinforcements(ZhangandWang,

2008) compared to conventionalMMCswhichusuallyhosthighv.f. of inclusionswith

sizes in the range of tens ofmicrons. The GND density ismainly concentrated at the

sharp inclusion corners to accommodate the strong lattice incompatibility that causes

stress concentration. Table 2.2 shows the slip systems that contribute to the GND

distribution for different crystal orientations. The distribution is determined by the

numberofactiveslipsystems,whichisfunctionof .Figure2.5ashowsthemagnitude

ofeffectivethermalGNDdensity alongthelineAB(seeinset)withdistancefromthe

inclusion corner for different crystal orientation and |Δ | 500, 1 . It can be

seen that the local GND density near the matrix‐inclusion interface is very high

(~10 )whileitsmagnituderapidlydecreasesawayfromtheinterface.Thelocal

GND density magnitude at the interface is much higher for 45 where fewer slip

systemsactivelyparticipate(Table2.2)andtheyareperpendiculartomatrix/inclusion

interface which cause the maximum thermo‐elasto‐plastic incompatibility at

matrix/inclusioninterface.

Table 2‐2.Activatedslipsystemsfortwolimitingcrystalorientations

SlipNormal SlipDirection 0 45 (1,1,1) (0,‐1,1) ‐ X(1,1,1) (‐1,0,1) X X(1,1,1) (‐1,1,0) X ‐(‐1,1,1) (0,‐1,1) ‐ ‐(‐1,1,1) (1,0,1) X ‐(‐1,1,1) (1,1,0) X ‐(1,‐1,1) (0,1,1) ‐ ‐(1,‐1,1) (‐1,0,1) X ‐(1,‐1,1) (1,1,0) X ‐(1,1,‐1) (0,1,1) ‐ ‐(1,1,‐1) (1,0,1) X X(1,1,‐1) (‐1,1,0) X X

38

Figure 2.5b shows the average thermal GND density in the

matrix volume for different ′ and fixed inclusion size ( 1 . Beyond the

initialelasticdeformationstage, changeslinearlywith|Δ |,whichisconsistentwith

the analyticalmodels (Arsenault andTaya, 1987;Dai et al., 1999;Dai andBai, 2001).

Notably, isabout:10 at|Δ | 500,whichisinthesamerangeasevaluated

in experiments (Barlow and Liu, 1998) and predicted from the dislocation punching

modelof(ArsenaultandShi,1986).However,theadditionalinformationthatweobtain

fromthefigureistheeffectofcrystalorientationon .Evenwithisotropicelasticityas

assumed here, higher lattice incompatibility produces higher GND density especially

withincreasing|Δ |.

0 10

30 45

Figure 2.6. Distribution of normal stress under thermal loading for different crystalorientationofmatrix( 1 ).

39

Figure2.6depictsthedistributionofthenormalresidualstress aroundthe

inclusionfordifferent .Itistensileinthevicinityofthetopandbottominterfaces.Note

thatthemaximumtensilestressislargerinthecrystalswithlargerincompatibilityand

willbelargerforsmallerinclusions.Asimilartrendisalsoobservedfor (notshown),

whichistensilealongtheverticalfacesoftheinclusion.Althoughnotexploredhere,the

likelyimplicationonthematrixfailurenucleationarisingfromthenormalstressesdue

to thermal excursion is of interest. The results indicate that the with decreasing

inclusion size and increasing incompatibility, the matrix may become susceptible to

nucleating voids near the interface. Such a possibility may eventually lead to failure

eitherthroughvoidgrowthinthematrixoronethatculminatesintointerfacefailure.

Figure2.7apresentsthermalGNDdensitydistributionfordifferentinclusionsizes

forhighestincompatiblecase( 45 .Itcanbeseenthatforsmallerinclusionsize,the

magnitude of thermal GND density and the area its distributed over are larger.

Consistentwithanalyticalmodels,figure2.7band2.7cconfirmsthelineardependence

of on|Δ |anditsinversedependenceontheinclusionsizeforgiven|Δ |.Whilethe

trends broadly corroborate with analytical models, the simulations provide a deeper

insight into the apportioning of to the thermal mismatch and elasto‐plastic

mismatch,whichisalsoinfluencedbycrystalorientation.

40

1 2

5 10

(a)

(b)

0 100 200 300 400 5000.0

0.1

0.2

0.3

0.4

0.5

1 m inclusion 2 m inclusion 5 m inclusion 10 m inclusion

Magniture of  

A 

41

(c)

Figure 2.7.(a)EffectiveGNDdensity distributionfordifferentinclusionsizes,(b)averagethermalGNDdensity evolutionduringthermalcoolingfordifferentinclusionsizes,(c)InverserelationofaveragethermalGNDdensity andinclusionsize |Δ | 500, 0 .

Notethatinadditiontothethermalmismatchthatiscommonlyaccountedfor(e.g.

(Arsenault andShi, 1986;Dai et al., 1999;DunandandMortensen,1990;Dai andBai,

2001)),theelasto‐plasticmismatchbetweenthematrixandinclusionalsocontributesto

theGNDdensityunderthermalloading.Thislattercontributionisnotapparentinmost

of the analytical and numerical models. For example, Dai et al. (2001) provide an

expression for the GND density that develops due to elastic mismatch under

mechanicallyappliedstrain,butitsapplicationtotheelasticmismatchstraindeveloped

duringthermalloadingisnotaccountedfor.Toisolatetheseindividualcontributionsto

thermal GND density, we performed additional simulations in that elasto‐plastic

mismatchissuppressed,butthethermalmismatchisretained.Ascanbeseeninfigure

2.8 for theparticular caseof 0 and 1 , isnearly30%higherwhen the

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

Simulation results

Y=ax-1

Inclusionsize, (

Ave.GNDdensity

(

Simulation results

42

elasto‐plasticmismatch is accounted for. This difference is accentuated by increasing

latticeincompatibility.

Figure 2.8.Contributionsofindividualmismatchcomponentsunderthermalloading( 1 .

Figures 2.5b and 2.7c together indicate an interesting interplay between the

inclusionsizeandcrystalorientation.Weobserve thatan increase in theGNDdensity

due to a smaller inclusion could be compensated (at least partially) if the lattice is

oriented in amanner thatproducesweak incompatibility. To first order, these effects

due to thermal excursion may be accounted for in a manner similar to that derived

analytically(DunandandMortensen,1991;Daietal.,2001)

Δ Δ ( 2.15)

where isapre‐factorthatembedstheinformationoftheeffectsofcrystallographically

defined elastio‐plastic properties of the matrix and the inclusion shape. For square

inclusions,comparingEq.2.15withthesimulationresults(Fig.2.5b), 30 40,with

0 100 200 300 400 5000.0

0.1

0.2

0.3

0.4

0.5

0.6

Thermal+Elastoplastic GND Thermal GND

Thermalmismatch

Elasto‐plasticmismatch

Magniture of  

A 

Thermo-elasto-plastic mismatch Thermal mismatch

43

the lower bound delimited by the lowest lattice incompatibility (e.g. 0 in our

simulations) and theupperboundby thehighest lattice incompatibility (e.g. 45 )

withrespect to the loadingaxis.Note that thepre‐factor isnearly thrice thatof the

analyticallyobtainedcoefficient(Daietal.,2001).

2.3.3 Size-dependent stress-strain response with pre-existing

thermal GND density

Followingthethermalprocessingthatproduceresidualstresses,theMMCbehavior

under compressive and tensile mechanical loads could differ significantly.For

conventional coarse‐grainedMMCswith large inclusions,the thermal residual stresses

provide a basis for the tension‐compression asymmetry in the stress‐strain

response.This isalsoexpected inMMCswith fine‐scaled inclusionswithanadditional

complexityarisingfromtheirsize‐dependency.Asnotedintheintroductorypartofthis

section,we subjected the same MMC unit cells to monotonic tension and

compression(Fig.2.9).The solid curves in figure2.9are the average true stress‐true

strainMMCresponses intensionandcompressioninthepresenceof locked‐in length‐

scale dependent thermal residual stresses 1 , 0.05, 45° .The

classicalcrystalplasticity(CCP)results(i.e.ignoringtheslipgradients)areshownbythe

dashedcurves for thesame and .Notethat thestress‐straincurvesareplottedup

to1%strainonly,inorder tohighlight theeffectofpriorthermal residualstresseson

theinitialstrengtheningandhardening.Inbothcasesthethermalresidualstresseslead

toanasymmetry in the tensileandcompressiveresponses,indicating theirubiquitous

role irrespective of whether size‐effects are accounted for or not.Specifically,the

compressive response is stronger compared to the tensile response,because the

44

thermal residualstressesare tensile innature(ArsenaultandTaya ,1987;Duttaetal.

,1993).

Figure 2.9.Truestress‐truestrainresponseforMMCmodelsunderthermomechanicalloading.BulkbehaviorispredictedbyCCPwhilesizedependentbehaviorismodeledusingMSGCPforinclusionsize 1 , 45°.

Lookingatdifferentinclusionsizesandcrystalorientationrevealsthatthethermal

residual stress is tensile in the vicinity of the top and bottom interfaces and its

magnitudeislargerforsmaller forall .Likewise,forafixed themagnitudeofthe

normal stress increases with increasing incompatibility, . .

.AcomparisonoftheCCPandSGCPresponsesin

Fig.2.9reveals that the thermalGNDcontribution in compression(differencebetween

bluesolidanddashedlines)ishigherthanintension(differencebetweenredsolidand

dashed lines).On theotherwords,under compressive loading,a higheroverall load is

required toovercome the initial tensile thermal residual stressand thiscauseshigher

overall stress compared to the tensile loading case where the initial tensile thermal

residualstressaugmentstheappliedtensioncausingyieldingatasmalleroverallload.

0.000 0.005 0.0100

100

200

300

Compression (MSGCP) Compression (CCP) Tension (MSGCP) Tension (CCP)

Tru

e S

tres

s (M

Pa)

True Strain

Compression (MSGCP) Compression (CCP) Tension (MSGCP) Tension (CCP)

45

(a) (b)

Figure 2.10.Influenceofthepriorthermalloadingon(a)truestress‐truestrainresponseand(b)hardeningrate.( 1 , 45°),obtainedfromMSGCPcalculations.

Figure2.10 indicates that the hardening rate in tension is higher than that in

compression,atleastintheinitialstagesofplasticity.Tofurtherstudytheevolutionof

hardeningduetothermo‐mechanicallyinducedGNDeffects,wecomparethehardening

rates of the two‐step thermo‐mechanical loading for monotonic tension and

compressionwiththatofthemechanical‐onlyloadingcase.Notethatinthemechanical‐

onlycasetheresponseisthesameundertensionandcompression.Figure2.10ashows

influence of prior thermal loading on the subsequent stress‐strain response under

tensileandcompressivemechanical loadingwhencomparedwith themechanical‐only

loadingcaseandfigure2.10bshowsthecorrespondinghardeningrates / .Itcanbe

seen that in the presence of initial thermal loading the subsequent tensile yield

strength(measured at0.2%strain)is lower,but the overall hardening rate is higher

compared to the mechanical‐only loading over the strain range considered.On the

contrary,thecompressiveyieldstrengthishigherandthehardeningrateisalsohighin

theincipientstage,butwithincreasingstrainthelatterquicklydropsbelowthetensile

response and asymptotes toward the hardening rate of the mechanical‐only loading

condition.These observations may be rationalized as follows:at the initial stages of

0.000 0.002 0.004 0.006 0.008 0.0100

50

100

150

200

250

Tru

e S

tres

s (M

Pa

)

True Strain

Thermo-Mechanical (compression) Thermo-Mechanical (tension) Mechanical

0.002 0.004 0.006 0.008 0.0100

20000

40000 Thermo-Mechanical (compression) Thermo-Mechanical (tension) Mechanical

Ha

rden

ing

rate

(M

Pa)

True Strain

Thermo-Mechanical (compression) Thermo-Mechanical (tension) Mechanical

46

compressive loading,macroscopic compressive stresses at the inclusion‐matrix

interfaces are compensated by the initial tensile thermal residual stresses generated

duringthethermalloadingattheinterface(figure2.11).Naturally,thenetlocalstresses

at the interface are lowered and therefore,the applied stress has to be increased for

macroscopic yielding(Fig.2.10a).Then,for the same strain the stress is higher in the

thermal plus compressive loading case compared to themechanical‐only loading case

and consequently,the initial hardening rate in the former is higher.The rapiddrop in

the compressive hardening rate indicates that the strengthening effect due to prior

thermal residual stresses decreases with increasing strain.This may be construed as

some of the initial thermal GND density being annihilated by the mechanical GND

densityaccumulatedduringthesubsequentmechanicalloading.

Figure2.11showstheevolutionoftotal(thermal+mechanical)GNDdensityforthe

tensileandmechanicalloadingcases.IntheinitialstagesthetotalGNDdensityisnearly

the same as that developed during prior thermal history. However, as deformation

progressesweobserveadropintheGNDdensityundercompressiveloadingindicating

partial annihilation of thermal GNDs; in comparison, under tensile loading the GND

densityincreasesmuchmorerapidly.Inotherwords,thelatticecurvaturegeneratedat

theinclusion‐matrixinterfaceduetothermalloadingiscompensatedbythatgenerated

due to the mechanical loading. As the loading continues, the mechanically‐induced

latticecurvature(mechanicalGNDdensity)prevailsovertheinitialthermalGNDdensity

andgoverns themacroscopichardening.Asmentioned in theprecedingparagraph, in

thecaseoftension,theexternaltensilestressarisingfromthemechanicalloadingadds

totheinitialtensilethermalresidualstresses.ThisnaturallycausestheMMCtoyieldat

alowerappliedload,butasdeformationprogressesmechanicalGNDdensityaddstothe

initialthermalGNDdensityproducinganenhancedhardeningrate.

47

Figure 2.11.AverageGNDdensityevolutionunderconsequentthermal‐mechanicalloading.( 1 , 45°)

2.3.4 Inclusion shape effect on stress-strain responses in the

presence of thermal GND density

TheoreticalstudiesonMMCreveal thatoverallMMCresponse isstronglyaffected

by inclusion shapes. (Meijer, 2000) used a cubic unit cell to investigate the residual

thermalstress/straininparticulatereinforcedmetalmatrixcomposites.Theyfoundthat

the sharp corners and edges of the cube shaped particles result in stress/strain

localization and lead to a much larger initial hardening behavior than the spherical

inclusions. (Chen et al., 1999) extensively studied effect of inclusion shape and its

morphology on the MMC response. They showed that the plastic strain near the

inclusion‐matrix interfacevarieswith inclusionshape.Whenthecurvatureatacorner

increasestheplasticstraingetsmoreconcentratedarounditandhasasignificanteffect

onthethermalresidualstressandstraindistributions.(Xueetal.,2002)observedthat

thereisarelativelyweakinteractionbetweeninclusionsizeandshapeforinclusionsize

largerthan7.5 whenthethermalresidualstressesareignored.Here,weinvestigate

48

the coupling between the inclusion size and shape in the presence of initial thermal

loading.Todoso,weperformthetwo‐stepthermo‐mechanicalsimulations forsquare

and circular inclusions, with 0.05, which gives 1 and

1.13 .

Figure2.12showsthecrystalorientation‐dependentthermalGNDdistributionfor

thetwoinclusionshapesandcorrespondingactiveslipdirections.Forboththeshapes,

large incompatible deformations result in high GND density in the vicinity of i‐m

interface,whichisdistributedinthedirectionofslipsystems.Quantitatively,thespatial

extentoftheGNDaffectedzonesaresimilarforboththeinclusionshapes;however,the

GNDdensityismoreconcentratednearthesharpcornersforthesquareinclusion.

(a)

(b)

Figure 2.12.DistributionofthermalGNDdensityaroundsquareandcircularinclusionsembeddedinsinglecrystalwith(a) 0°and(b) 45°.

49

Studieson theeffectof inclusion shapeusingcontinuumplasticityhave indicated

thathigher triaxialityof the stress‐stateexists ata sharpcornerof a square inclusion

comparedtothestressstateinthecloseproximityofacircularinclusion.Thisraisesthe

overallstress‐strainbehavioroftheMMCswithsharp‐corneredinclusionscomparedto

thosewithmoreroundedinclusions(Meijer,2000).Interestingly,wefindthesituation

to be somewhat different when the GND effects are included.Figure2.13shows the

MSGCPtruestress‐truestrainresponsesofMMCunderthermo‐mechanical loadingfor

the square and circular inclusions embedded in single crystal with 0.05and

0° .For comparison,the CCP results are also included in the same plot with same

simulation parameters.The CCP results show that the shape of inclusion has an

influence on the overall stress‐strain behavior with the square inclusion providing a

strongerresponsethanitscircularcounterpart.Thisisqualitativelyconsistentwiththe

trendindicatedby(Meijer,2000).However,theshapeeffectdramaticallyreducesinthe

presenceofthethermalGNDdensity.

Figure 2.13.Truestress‐truestrainresponseforMMCmodelscomprisingtwodifferentinclusionshapes 0° .

0.000 0.005 0.010 0.015 0.020 0.0250

50

100

150

200

250

300

350

Tru

e st

ress

(M

Pa)

True strain

CCP- Square inclusion CCP- Circular inclusion MSGCP- Square inclusion MSGCP- Circular inclusion

50

The diminished shape effect in the presence of thermal GND density can be

explainedintermsoftheinteractionbetweentheGNDaffectedzoneati‐minterfaceand

the high triaxiality stress at a sharp corner.Figure2.14shows the von‐Mises stress

distributions around the square and circular inclusions as obtained from theCCPand

MSGCP calculations.The CCP simulation, being length‐scale independent, does not

accountforthethermalGNDdensityandisakintoalargeinclusioncasewheretheGND

effectsarenegligible.Note that thesimulationresultsare for 0°,whichpossesses

thelowestincompatibilityamongstthecasesevaluatedhere(seeforexample,Fig.2.5a),

whichmeansthatthethermalGNDdensitywouldalsobethelowest.Eveninthiscase,

astheinclusionsizedecreasesthestressintheGNDaffectedzonetendstooverlaythe

regionsofhightriaxiality(sharpcorners)sothattheregionaroundthesquareinclusion

appearstohavemoredistributedstressarounditcomparedtothecasewhereeitherthe

thermally induced GND density is not present or is negligible. Figure 2.15 further

elaboratesthisideaschematically,indicatingthatthestressconcentrationsatthesharp

corners tend to be engulfed by overall high stress around a square inclusion as the

inclusionsizedecreases.Thistemperstheeffectofinclusionshapeonthemacroscopic

behavior,becausethestressstatearoundasharp inclusionappearstobequalitatively

closer to that of an inclusionwith rounded corners, the limiting case being a circular

inclusion. We note in passing that we observe qualitatively similar trends for the

45°andarenotrepeated forbrevity. Inconclusion, thesimulationsreveal that in

the presence of initial thermal GND density the inclusion shape effect on the overall

MMC response diminishes at smaller inclusion sizes. In addition to this, although the

stress around the inclusion is higher it is relatively more uniformly distributed

comparedtothecasewheretheGNDeffectisabsentandthisisexpectedtoshieldsucha

materialagainstfracturearisingfromlocallyhighstresses.

51

(a)

(b)

Figure 2.14.InfluenceofinclusionshapeonthermalresidualstressesinMMCbasedon(a)CCPand(b)MSGCP. 0°

Figure 2.15.Schematicindicatinganinteractionbetweeninclusionshapeandsizeeffectsatthelocationsofstressconcentrations.

GND affected zone

Stress concentration

 

 

>      

52

2.3.5 Thermal GND density distribution in polycrystalline MMC

under thermal loading

In the previous sections, the role of inclusion size and crystal orientation are

investigated using unit cell model comprising a single inclusion embedded inside a

singlecrystalmatrix.Inthissection,weexploretheapplicabilityofthemodelonMMC

with polycrystallinematrix where grain sizes are smaller than inclusion sizes. In the

next section we will investigated the role of grain size, inclusion size and their

interaction in MMC overall response undermechanical loading via mechanism based

straingradienttheory.Here,weadoptunitcellapproachincludingboththethermaland

theelasto‐plasticmismatchesbetweenthegrainsaswellastheinclusionandthegrains

surrounding it (figure 2.16). The crystallographic elastic properties for the grains are

168.4, 121.4and 75.4 .Thecolorsforindividualgrainsrepresent

their orientation as specified by the numerals within the grains surrounding the

inclusioninFig.2.16.ThecontourplotsinFigs......aandbrespectivelyshowthethermal

GNDdensity distribution grain sizes 0.5 and0.25 . Inboth case, 0.05 ,

and 1 .

As a comparison, Fig. 2.17 shows the evolution of the average thermal GND

densities inthepolycrystallineMMCanditssinglecrystalcounterparts.First, itcanbe

seenthattheaverageGNDdensityishigherinthecaseofpolycrystallineMMCsasthe

grain boundaries act as additional sources of elasto‐plastic incompatibility.

Interestingly, this enhancement is only weakly dependent on the grain size. This

indicates that the elastic anisotropy in polycrystalline matrices do not produce a

significantcouplingintothegrainsize‐dependenceofthethermalGNDdensity.Second,

weobserveisthatthefinergrainsizeproducesamoreuniformdistributionoftheGND

density, which is also important because it can help induce uniform strengthening

aroundtheinclusion.Finally,notethatthesinglecrystalcasesgivealowerGNDdensity,

53

eventhatwiththelargestincompatibility 45 .Fromthis,wemayconjecturethat

for highly textured polycrystalline cases (a limiting case being a single crystal) the

thermal GND density may be somewhat lower than a poylcrystal MMC with random

texture.Inotherwords,texturedpolycrystalsmayproduceweakerinitialstrengthening

duetothermalhistorycomparedtoarandomlytexturedpolycrystalMMC.

(a)

(b)

Figure 2.16.EffectiveGNDdensitydistributioninpolycrystallineMMCwithrandomgrainorientationfordifferentgrainsize(a) 0.5 μmand(b) 0.25 μm. 1 , |Δ | 500

54

Figure 2.17.AverageGNDdensitydistributionevolutioninsinglecrystallineandpolycrystallineMMC

2.4 Grain size-inclusion sizes interaction in MMC at

moderate strain using MSGCP

Mostof thepreviousefforts onMMCbehaviors andunderlyingmechanismshave

beenconcentratedontheunitcellapproacheswhereasingleinclusionrepresentingits

v.f., is embedded in a single crystal or a homogenized matrix that is endowed with

enrichedplasticitydescriptions.However,itisimportanttonotethelimitationsofthese

models in terms of the microstructural characteristics: an inclusion embedded in a

singlecrystalresemblesapolycrystallinemasswhosegrainsaremuchbiggerthanthe

particles(Cleveringaetal.,1997)sothatthegb’sdonotinterfereinthestrengthening

response (e.g. a sub‐micron sized inclusion embedded within a large grain of a

polycrystal). The other extreme is the assumption of a homogenized matrix with

discrete inclusions (Nan and Clarke, 1996; Suh et al., 2009; Xue et al., 2002), which

0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

PolyX, dg=0.25 m

PolyX, dg=0.5 m

SingleX, = 45o

SingleX, = 0o

Magniture of  

A 

55

resembles a polycrystalline mass with grains that are much finer (allowing

homogenization of the matrix) than the inclusions. In practice, one may encounter

important intermediate cases in addition to these two extremes, especially for

nanostructured composites. For example, the trimodal Al‐alloy composites (Joshi and

Ramesh,2007;Zhangetal.,2008)possessgrainsizesthatareinthesamerangeasthose

ofthereinforcingparticles(Figure 2.18).Insuchsituationsitmaynotbeappropriateto

assume either a homogenizedmatrixmodel or a single crystal approximation. Rather

one has to explicitly resolve both inclusion and its surrounding grains within the

microstructuretocapturethelength‐scaledependenciesontheoverallresponse.

Figure 2.18.MMCwithmicron‐sizedinclusionsembeddedinananocrystallinematrix(JoshiandRamesh,2007)

Thisobservationposesinterestingquestions–whatrolesdothegrainsizeandthe

inclusionsizeplayintheoveralllength‐scaledependentresponseofanMMC?Howdoes

one account for or model the interaction between these microstructural features? Is

there a range of grain size‐inclusion size combinations that produces significant

synergistic contributions? Is itpossible toquantify this interaction, forexample, asan

additional hardening contribution? To our knowledge, these questions have not been

addressedviaeitheranalyticalorcomputationalmodelingatanylength‐scale.

56

From amechanistic viewpoint this is a challenging problem as there are several

aspectsthatonehastounderstand,forexample,thephysicsoftheplasticeventsatthe

inclusion‐matrix (i‐m) interfaces and at gb’s (and triple junctions), communication

between the i‐m interfacesandgb’s, grainorientation effects, inclusionandgrain size

distributions, and several more. While it may be important to incorporate these

mechanisms, a single mechanistic framework that is capable of resolving the

microstructural details and concurrently also embeds appropriate physics for all the

interfacialmechanismsisdifficulttoconceiveatthemoment.Acomparativelytractable

settingispossibleifonechoosestosimplifyand/orignoresomeoftheaspects.Crystal

plasticity enriched with length‐scale features can effectively handle the kind of

resolutionnecessaryfortheproblem.Initssimplestversion,itispossibletomodelMMC

microstructures using CP by explicitly resolving the grains and inclusions and

accounting for some of the size‐dependent mechanisms, but ignoring some of the

intricate details such as size and spatial distributions of grains and inclusions, gb

deformationprocessesandfailure3.

Withthisnotion,wedemonstrateacomputationalapproachbasedonlength‐scale

dependent crystal plasticity (CP) to answer the questions posed in the preceding

paragraph.Specifically, thisworkresorts tothemechanism‐basedslipgradientcrystal

plasticity (MSGCP) (Han et al., 2005a) theory. MSGCP accounts for size‐effects by

incorporatingslipgradientsthatarerelatedtotheGNDdensitieswithintheconstitutive

description of individual slip systems. Given that both grains and inclusions are

explicitly resolved in this approach, slip gradients naturally arise at gb’s and i‐m

3ThesedetailscanbeincludedwithinCP,buttheycomplicatetheproblembyintroducing

severaladditionalvariablesandunderstandingtheireffectsontheoverallbehaviorwould

requiresignificantcomputationaleffort.

57

interfacesduetheirelasto‐plasticmismatchandareaccountedforintheMSGCPtheory.

However,this approachisessentiallyalower‐ordertheorycomparedtoahigher‐order

framework, because it does not invoke additional boundary conditions (b.c.’s) at

interfaces. Consequently, the lower‐order CP approaches cannot model some of the

enhanced interactions between interfaces and dislocations that the higher‐order CP

approachesarecapableofhandling.Althoughahigher‐ordertheorywouldbesuitedfor

the present problem (Fredriksson et al., 2009), (Bardella and Giacomini, 2008), the

difficulty with higher‐order b.c.’s is that it may not be always easy to identify

appropriate descriptions for general interfaces (Voyiadjis and Deliktas, 2009).

Moreover,thecomputationaleffortforhigher‐orderCPissignificantlylargerthantheir

lower‐ordercounterparts.Ontheotherhand,duetotheinherentinabilityoftheMSGCP

inhandlingenhancedinteractionsbetweeninterfacesanddislocationsthelength‐scale

effect appears only in the flow behavior rather than at yield (Evans and Hutchinson,

2009).However,despitesomeofitslimitations,wechoosetheMSGCPtheorykeepingin

view its simplicity in the numerical implementation within existing CP framework,

computational expense for the present work and a relatively established physical

understandingofthelength–scaleparameters.Inthisregard,theresultspresentedhere

on the grain size‐inclusion size interaction are applicable in the flow regime, i.e at

moderatestrains,ratherthanatyield(AcharyaandBeaudoin,2000).However,wealso

notetheexperimentalobservationof(KouzeliandMortensen,2002)thatthesizeeffect

intheflowregimeofMMCsfollowssimilartrendsasatyieldandreturntothisaspectin

theclosingsectionofthischapter.

Inthenextsection,wedescribethemodelmicrostructuresadoptedinthepresent

work and the procedure to isolate the individual length‐scale effects arising from the

grainsize,inclusionsizeeffectsandthegrainsize‐inclusionsizeinteractions.

58

2.4.1 Model Microstructures

Toenableconsistentcomparisonacrossdifferentparametricmodels,weconsider

highlyidealizedMMCmicrostructurescomprisingsquaregrainsandinclusions.Wealso

assume that the inclusions are regularly arranged, and the gb’s and interfaces remain

intactthroughoutthedeformation.Figure 2.19showscanonicalpolycrystal(Figure 2.19

a, c) andMMC (Figure 2.19 b, d) microstructures amongst several considered in the

presentwork.Oneextremecaseiswheretheinclusionismuchsmallerthanthegrainso

that iteffectivelyresideswithinthegrain(Figure 2.19b),andtheothercase iswhere

theinclusionismuchbiggerthanthegrains(Figure 2.19d)sothatmultiplegrainsshare

aninclusioninterface.Agrainorientation(Figure 2.19a)forthisFCCcrystalstructureis

definedhere as the anglemadeby the [100] crystal directionwith the global loading

direction and [001] crystal direction is considered to coincidewith the global

direction. The associated color for each grain acts as a reference for the other

microstructures4. Within each MMC configuration the grain size and inclusion

size areconstant.Thisenablesorganizingthemicrostructuralarrangementsintotwo

broadcategories:(a) (Figure 2.19c),and(b) (Figure 2.19d).

4 Insection2.4.6webrieflydiscussthestatisticaleffectofthenumberofgrainswith

randomgrainorientationsonthestress‐strainresponses.

59

(a) (b)

(c) (d)

Figure 2.19.Representativemodelsfor(a,c)polyXand(b,d)MMCarchitectures.

Forcase (a)weconstructa36grainpolycrystalwithrandomcrystalorientations

witheachgrainembeddingoneinclusion.Forcase(b)asingleparticleissurroundedby

randomly oriented grains. Note that only when ≫ would a computational cell

asymptote to a unit cell approximation that is commonly adopted (Dai et al., 2001a;

KouzeliandMortensen,2002;NanandClarke,1996;Zhangetal.,2007)however,most

worksdonotstatetheassumptionsonthematrixmicrostructuraldetailsexplicitly. In

such cases, it is not obvious how the matrix strengthening due to grain size would

couplewiththecontributionfrominclusionsize.Toquantifythegrainsizeandparticle

sizeeffects:

47  38  78  5  19  42 

85  83  38  55  22  21 

64  7  40  37  15  58 

47  34  67  72  58  6 

68  16  67  14  66  76 

75  40  3  34  58  15 

6gd m

 

 

2id m

60

(i)First,wemodelpolycrystallinemasses comprisinga fixednumberof grainsof

size ,without inclusions (c.f. Figure 2.19 a, c). These simulations are performed for

microstructureswithdifferentgrainsizes,butkeepingtheinitialorientationsbetween

thedifferentmicrostructuresunchanged.

(ii) The samemicrostructures in (i) are again simulated with inclusions of fixed

size andvolumefraction (e.g.Figure 2.19bandd).

Steps(i)and(ii)areappliedtodifferentinclusionsizeswithfixed ,Basedon(i),

theflowstress ofabarepolycrystallinemassatafixedstrainis

∆ ( 2.16)

where isthesize‐independentflowstressofthepolycrystallinemasswithlargegrain

sizesforagivensetofcrystallographicorientation,andΔ istheadditionalgrainsize‐

dependentflowstressderivedfromtheslipgradientsatgb’s.Likewise,fromstep(ii)the

flowstressforanMMC maybewrittenas

Δ

∆ Δ

Δ

Δ Δ Δ

( 2.17)

whereΔ isthesize‐independentflowstresspurelyduetotheinclusionv.f.,Δ isthe

contributionduetoinclusionsizeeffectarisingfromtheslipgradients(GNDs)atthei‐m

interface andΔ is an additional contribution thatmayexist due to the synergistic

effectsbetween and .NotethatthegrainsizecontributioniscommontoEqs.( 2.16)

and( 2.17).

Forall the cases, the left andbottomedgesare respectively constrained along

and directions, the top edge is allowed to move vertically, but remain straight. A

uniform velocity b.c. is applied to the right edge producing a nominal strain

61

rate 1 10 . Inwhat follows,we refer to thepolycrystallinemicrostructures

sans inclusionsasPolyX and thosewith inclusionsasMMC.Table1gives thematerial

propertiesusedinthesimulations.ThesepropertiesarerepresentativeofpureAlasthe

matrixandSiCasthe inclusions.Forsimplicity,weassumeisotropicelasticproperties

fortheAl,butanisotropicelasticitycanbeeasilyimplemented.

2.4.2 Length-scale dependent polycrystalline response

Figure 2.20showsthesize‐dependentpolyXtruestress‐truestrainresponseswith

differentgrainsizes.Thereddashedcurveisthebaselinecalculationwithoutgradient

effectsthatrepresentsapolyXwithlargegrains.Asnotedearlier,thenatureoftheslip

gradientmodelimplementedhereissuchthatthelength‐scaleeffectmanifestsitselfin

the hardening response rather than at yield (Evans andHutchinson, 2009;Han et al.,

2005a). Therefore, we measure the average flow stress at 2% nominal strain to

demonstratethesizeeffects.Figure 2.21showsthestrongdependenceoftheflowstress

on .Theplotalso includes thepopularHall‐Petch typeempirical fit ~ . to the

simulation results alongside the inverse grain size correlation . (Acharya and

Beaudoin, 2000) applied their version of the length‐scale dependent CP theory to

investigate grain size effects in polycrystals and obtained corroborations with

experiments at moderate strains. Experimental evidences (Hommel and Kraft, 2001;

Nix, 1989; Venkatraman and Bravman, 1992) and theoretical models (Ohno and

Okumura,2007;Sinclairetal.,2006;Zhangetal.,2007;Zhouetal.,2011)makecasesfor

boththetypesofdependencies,butforconsistentcomparisonhereweadopttheHall‐

Petchrelationforsubsequentdiscussions.

62

Figure 2.20.Truestress‐truestrainresponsesforpolyXmodelswithdifferentgrainsizes.

Figure 2.21.Normalizedgrainsizedependentflowstressat 2% forpolyXwithidenticalgrainorientations.TheplotalsoincludestheempiricalHall‐Petch . andinversegrainsize fits.

Decreasinggrainsize

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05

CCP

dg= 6 m

dg= 1 m

dg= 500 nm

dg= 300 nm

Tru

e S

tre

ss (

MP

a)

True strain

0

2

4

6

8

10

12

0.1 1 10

g /

Y )

*10

0

Grain size, dg

( m)

Simulation results

Hall-Petch behaviour

Inverse grain size relation

Simulation results

Hall-Petch fit

Inverse grain size fit

63

2.4.3 Length-scale Dependent MMC Response

We now discuss the results obtained from the MMC simulations. For clarity, we

focusinitiallyonthecasewithfixed 2 anddifferent ′ 6 and1 ,but

subsequentlyalsodiscusstheeffectofinclusionsize.Figure 2.22showstheresponseof

MMCs(solidcurves)fordifferentgrainsizes.Forcomparison,thepolyXresultsforthe

samegrain sizesandorientationsarealso included in the figure (dashedcurves).The

red dashed and solid curves respectively denote the response of the polyX andMMC

without the gradient effects (i.e. conventional crystal plasticity). As expected,

irrespectiveofwhethergradientsareincludedornottheMMCflowstressishigherthan

itspolyX counterpartdue to thepresenceof inclusions in the former Δ . Theblue

solid curve is theresponseofanMMCwith inclusions thataremuchsmaller than the

grains(e.g.Figure 2.19b),whereasthegreensolidcurveisforthecasewherethegrains

are smaller than the inclusions (e.g. Figure 2.19 d). Interestingly, in the presence of

gradients,thelatterexhibitsahigherhardeningrateoveritspolyX(dashedgreencurve)

counterpartcomparedtotheformer(solidanddashedbluecurves).Thissuggeststhat

thereexistsaninteractionbetweenthegb’sandthei‐minterfaceswhenthegrainsizes

are comparable to or smaller than the inclusion sizes. In the following section, we

quantify this interaction through systematic simulations with different grain and

inclusion sizes. In the next section, we briefly discuss the mesh convergence studies

performed on one combination.

64

Figure 2.22.Grain‐sizedependenttruestress‐truestraincurvesforMMC(solidlines)with 2 .ThecorrespondingpolyXresponses(Figure 2.20)arealsoincludedforcomparison.

2.4.4 Grain orientation and mesh size effects

Sincethefocusofthisworkistocapturetheinteractioneffects,weinvestigatethe

influence of random grain orientations onΔ in regime for different for

fixed 0.12and 2 .Note that in thepresent2D investigation thenumberof

grains inanRVEisthenequalto ⁄ .Thisindicatesthatbyreducingthegrain

size,moregrainswithrandomorientationsaremodeledintheRVEandthisshouldhelp

reducethestatisticalvariationduetograinorientation.Figure 2.23showsthevariation

inΔ ,shownbytheerrorbars, forfivedifferentrealizationspergrainsize.Indeed,

the standard deviation in Δ arising from random choice of grain orientations

reduces with decreasing grain size. We also investigated mesh dependency and

convergencefora limitednumberofMMCsimulationsandpresentonesuchresult for

thecasewith 2 and 1 .

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05

Conv. (PolyX)

dg=6 m (PolyX)

dg=1 m (PolyX)

Conv. (MMC)

dg=6 m (MMC)

dg=1 m (MMC)

Tru

e S

tre

ss (

MP

a)

True Strain

∆ ∆ ∆  

∆  ∆  

65

Figure 2.23.StandarddeviationinΔ arisingforagivencomputationalmodelwithfixed butdifferentrealizationsofgrainorientations.Asexpected,thevariationissmallerforfiner .

Figure 2.24ashowsthatthestress‐straincurvesconvergewithfinermeshsize. In

addition, the flow stress at a true strain of 0.04 is depicted versus total number of

elementsusedinthemodelinFigure 2.24b.

(a) (b)

Figure 2.24.Meshconvergenceforthestress‐straincurvesofMMC 2 ,1 withdifferentmeshsizes .

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5

it/ 2

2

(MP

a)

4 6 8 10 12

Grain size, dg (m)

in

t/

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05

d = 240 nm d = 120 nm

d = 60 nm d = 30 nm

300

305

310

315

320

325

330

0.03 0.035 0.04 0.045 0.05

Finer mesh

True strain

Tru

e st

ress

(M

Pa)

311

312

313

314

315

316

317

318

319

0 1E+4 2E +4 3E+4 4E+4

Flo

w s

tre

ss a

t 4

% s

trai

n (

MP

a)

No. Elem ent

66

2.4.5 Grain size-inclusion Size Interaction strengthening

Tosystematicallydiscerntheinteractioneffectthatexistswhentheinclusionsizeis

inthesamerangeorsmallerthanthegrainsize,weperformedFEsimulationsofMMCs

with various grain size‐inclusion size combinations (see Table 2‐3). The procedure

adopted is discussed here brieflywithin the context of a fixed , and . First, two

simulations are performed for the MMC with both SGCP and conventional crystal

plasticity(CCP).Thealgebraicdifferencebetweentheoverallstress‐strainbehaviorsof

these two gives the totalMMC strengthening Δ Δ Δ due to

thegrainsize,inclusionsizeandinteractionterms(Eq.( 2.17).ThegrainsizeeffectΔ

is obtained as the difference between the polyX‐SGCP and polyX‐CCP response that

possess the same grain sizes and orientations as the MMC. Subtracting the 1

portion of the grain size effectΔ from the total MMC strengtheningΔ , the

combined inclusion size and interaction effects are isolated, i.e. Δ Δ .

Thisprocedureisperformedfordifferentgrainsizesandinclusionsizeswithfixed .

Table 2‐3.MicrostructuralsizecombinationsforMMCsimulations

InclusionSize

5 2 1

GrainSize

0.83,1.6,5,15 0.33,0.66,1,2,6,15 0.33,0.5,1,3,15

Figure 2.25 shows the normalizedΔ as a function of at 2% nominal strain

for 1 , 2 and5 .Notethatstrengtheningbehaviorcanbesplitupintotwo

distinct regions. The first region is , characterized byΔ that is larger for

smallerinclusionsizes.Inthisregime,thecurvesremainhorizontalandparalleltoeach

other over the range,meaning that the grain size does not play anymajor role in

contributingtotheoverallMMCstrengthening.Inotherwords,for theinclusion

67

strengthening is grain‐size independent and only inclusion size‐effect prevails. To

extracttheinclusionsizeeffect,weconsidercaseswith 3 wheretheinteraction

effectisnegligible.

Figure 2.25.Flowstress 2% normalizedbybulkpolyXyieldstressvariationofMMCsasafunctionofgrainsize.

ItcanbeseenthatinFigure 2.26thattheflowstressvariesas . ,whichcanbe

explainedbyTaylorhardeningdescription that is embedded in theMSGCP (Dai et al.,

2001a)

Δ √3 √36

( 2.18)

where istheTaylorfactorand isthematrixshearmodulusand istheapplied

strain.

68

Figure 2.26.Inclusionsizeeffectonthenormalizedflowstress(normalizedbybulkpolyXyieldstress)forlargegrainsizes, 3 (negligiblegrainsizeeffect).

The second regime in Figure 2.25 corresponds to where a dramatic

increaseinstrengtheningisobserved,whichmustbeduetotheinteractionbetweenthe

i‐minterfacesandgb’s.Inthisregime,theinteractioneffectforagiven issimplythe

deviationofthecurvefromitsbaselineinclusionstrengtheningatlargegrainsizes,i.e.

Δ Δ Δ . Note that for a fixed the interaction effect is larger for

smaller . Further, for smaller inclusions the interaction effect kicks in at

correspondingly smaller grain sizes. In other words, the inclusions do not feel their

neighboring grains unless the characteristicmicrostructuralwavelengths of the latter

arecomparableorsmallerthantheformer.

ThemannerinwhichtheGNDdensitycomponentofthetotaldislocationdensityis

distributed depends strongly on the grain size and the inclusion size. As an example,

Figure 2.27showstheGNDdensitydistributionsalonganodalsegmentstaringfroman

i‐m interface traversing through the matrix for three cases with different ′ and

fixed .Clearly,thepresenceofmultiplegb’s(smallergrains)aroundaninclusionleads

0

1

2

3

4

5

0 1 2 3 4 5 6

i/ y )

1

00

Particle size, di (m)

Simulation results

Equation 3

69

toahigherGNDdensityatthei‐minterfaceaswellasGNDaccumulationatthegb’s.This

effectisfurtherenhancedforsmallerinclusionsizes(notshownhere).

Figure 2.27.DistributionoftheeffectiveGNDdensity / alongpatha‐b 2 fordifferentgrainsizes.

Wepositthattheintersectionofagbandaninclusioninterfacecanbeconsidered

asanadditionalsourceofdislocationactivitythatleadsincreasingdislocationdensityin

itsvicinityandcontributestotheoverallhardeningasaninteractioneffect,Δ .Inthe

nextsection,weproposeananalyticalmodelbasedonthishypothesistoaccountforthe

dependenceofΔ onboththegrainsizeandtheinclusionsize.

Interaction effects discussed in the context of MMCs have also been observed in

polycrystalline thin films on substrates. There, strong interactions exist between the

gb’s and the relatively rigid substrates. Although these effects have been addressed

based on the grain sizes and film thickness, they have mostly been accounted for

separately rather than as an interactive effect (Hommel and Kraft, 2001; Nix, 1989;

Venkatraman andBravman, 1992). Interestingly, (Hommel andKraft, 2001) indicated

λ

0.0E+00

1.0E+03

2.0E+03

3.0E+03

4.0E+03

5.0E+03

6.0E+03

7.0E+03

8.0E+03

9.0E+03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dg=0.66um dg=1um dg=6um

a b

x/λ

GBs

μm μm μmE

ffec

tive

GN

D d

ensi

ty -η/

b (1

/μm

²)

70

that the dislocation density measured in their film‐substrate experiments was larger

than the computed total dislocationdensity,which is summation of the SSDandGND

densities. Furthermore, (Choi and Suresh, 2002; Nicola et al., 2005) pointed out that

grain size and film thickness are coupled and not independent. Hence, a linear

combination of grain size and film thickness may not adequately capture the overall

size‐dependent behavior of thin film structures, similar to the present scenario. The

MMC architectures considered here bear microstructural resemblance with

polycrystallinethin filmsonsubstrates,and itwouldbe interestingtoperformsimilar

studiesonthesearchitectures.

2.4.6 Analytical Model for Interaction Strengthening

From Figure 2.27, we note that the GND density distribution arising from the

kinematicincompatibilitieswithinanMMCarchitectureisstronglyaffectedbyboththe

grain and the inclusion size. As shown, these in turn affect its flow stress. However,

currenthomogenizedmicromechanicalmodelsdonotaccountfortheeffectduetothis

synergisticinteractionanditisusefultodevelopasimpleanalyticaldescriptionforthe

same.Basedonthenotionthatintersectionsofgb’swithaninclusionserveaspotential

regionsof enhanceddislocationgeneration,weproposeaphenomenologicalmodel to

quantify the dependence of the interaction effect on the grain size and inclusion size.

The idea of intersections serving as dislocation sources has been recently laid out by

(Forest and Aifantis, 2010) in the case of nano‐twinned materials where gb‐twin

boundaryintersectionsmaynucleatedislocations.However,itisimportanttoascertain

if indeedagb‐interfacejunctioninanMMCcouldserveasadislocationsource.Toour

knowledgetherearenoexplicitmicroscopicexperimentalevidencesonMMCstofortify

this hypothesis. However, as discussed in the closing paragraphs of the preceding

section the thin film‐rigid substrate systems do exhibit similar coupling andwe seek

someguidancefromexperimentalinvestigationsonthem.Indeed,thereareevidencesof

71

dislocationsemanatingfromsubstrate‐gbintersections(e.g.(Legrosetal.,2009)),which

indicates that such intersections canbepotential sources.Weuse these experimental

evidencestoputforthourmodelfortheMMCproblem.

Figure 2.28 shows a computational cell of size considered for

developingtheanalyticalmodel.Thiscellcomprisesaninclusionofsize and

grainsofsize .

Figure 2.28.Schematicof aninclusionembeddedinapolycrystallinemassoffinergrains.

Thenumberofspecialdislocationsourcesalongalineformedbytheintersectionof

thegbandinclusionfaceis

/ ( 2.19)

where is a factor introduced to account for the fact that only a certain fraction of

atomic positions may contribute as dislocation sources (Forest and Aifantis, 2010).

Then,thetotalnumberofintersectionlinesalonganinclusionsurfaceis

72

( 2.20)

where is a geometrical factor that depending on the dimensionality of the problem

and cross‐sectional shape of the inclusion. From Eqs. ( 2.19) and ( 2.20) the number

densityofdislocationsourcesmaybewrittenas

/ ∙ ⁄ ∙ ⁄ ( 2.21)

where is a representative volume (Figure 2.28). Noting that for a given

RVE, ⁄ ,weobtain

( 2.22)

Equation ( 2.22) indicates that the dislocation source density depends linearly on the

inclusion v.f. and inversely on the inclusion and grain sizes. We propose that an

additionaldislocationdensity emanatesfromthesesourcesandcanbequantifiedas

/

( 2.23)

where is the average length of the nucleated dislocations and is the total

number of the dislocation which is nucleated from one source. The plastic strain

accommodatedbythese dislocationsmaybewrittenas(vonBlanckenhagenet

al.,2004)

/

( 2.24)

FromEq.( 2.19),wemaywrite .Assumingthat canbeexpressedas

afractionofthetotalplasticstrain ,thedislocationsemanatedfromeachsourceis

73

( 2.25)

where is a ratio of the total plastic strain to . Substituting Eq. ( 2.25) into Eq.

( 2.24)weobtain

( 2.26)

PuttingEq.( 2.22)inEq.( 2.27)andusingTaylorhardeningmodel,wewrite

∆| | .

( 2.27)

where √3 and .ThroughEq.( 2.27),theinteractioneffectexhibitsa

Hall‐Petch type relation with both the grain size and the inclusion size. Figure 2.29

shows theΔ versus relationship obtained from all the FE simulations

performed in this work for different combinations. Notably, with appropriate

parameters (Table 2‐4) the trend from Eq. ( 2.27) corroborates well with the FE

simulation result, indicating the precise nature of the interaction effect. Thus, in the

analytical modeling of MMCs with size effects, one may account for the grain size‐

inclusionsizeinteractionthroughEq.( 2.27).

74

Figure 2.29.Variationoftheinteractionstrengtheningwiththeproduct .

Therefore,atmoderatestrains,theoverallMMCflowstrengthremainsindependent

ofthegrainsizeanddependsonlyontheinclusionsizeforthe cases,butshows

a strong coupling between them for the cases. The transition from an

uncoupled to a coupled (interaction effect) behavior occurs at . Our detailed

simulations enable isolating this interaction effect as a function of the two

microstructuralfeatures.

Table 2‐4.MicrostructuralsizecombinationsforMMCsimulationsParameter Value Unit

Taylorfactor( 0.3 ‐

Shearmodulusofmatrix( 27 10

Burgersvector( ) 0.25

Strainfactor( ~ 5 10 ‐

Simulation results

Analytical solution (Eq. 15)

0

2

4

6

8

10

0 5 10 15 20 25 30

(

int/

Y )

100

di.d

g (m2)

Simulation results

Equation 15Equation 2.27

75

2.5 Summary and Outlook

Inthischapter,wedevelopedan in‐houseUMATforABAQUS/STANDARD®finite

element code that implements the MSGCP theory to investigate the length‐scale

dependentresponsesofMMCarchitecturesunderthermalandmechanicalloading.

We investigated the role of inclusion size and shape using unit cell model

comprising a single inclusion embedded inside a single crystalmatrix. The simulation

results showed the length‐scale dependent asymmetric responses under monotonic

tensionand compression loading,which are related to theprior thermalGNDdensity

due to thermo‐elastic mismatch at the i‐m interfaces. The pre‐existing thermal GND

densitywasshowntoenhancetheoverallMMChardeningbehaviorinbothtensileand

compressivemechanical loading; however, the increase in hardening is higher in the

caseoftension.

Systematic computational simulations on bare polycrystalline and MMC

architectures were performed in order to isolate the contributions due to grain size,

inclusion size and the interaction thereof. We showed that at moderate strains, the

overallMMCflowstrengthremainsindependentofthegrainsizeanddependsonlyon

theinclusionsizeforthe cases,butexhibitsastrongcouplingbetweenthemfor

the cases. The transition from an uncoupled to a coupled (interaction effect)

behavioroccursat .Basedonthenotionofenhanceddislocationsourcedensity,

weproposedaphenomenologicalmodel thatquantifies their relationship as adouble

Hall‐Petchtypebehavior.Suchaninteractiontermcouldbeincorporatedwithinlength‐

scaledependenthomogenizedapproachestoaccountfortheinteractioneffect.

In this chapter,wemainly focusedon the short‐range interaction as arising from

the thermal and mechanical GNDs. The length‐scale dependent response at yield

observedherewasbecauseofthelength‐scaledependentpriorthermalresidualstress

76

thatwasaccountedforinthesimulations.Inthenextchapter,wediscussanothertype

oflength‐scaledependentinternalresidualstressthatarisesfromthelong‐rangeelastic

interactions between the GNDs. We present a theoretical formulation based on

kinematicincompatibilityandastressfunctionapproachforthislong‐rangeeffectthat

isincorporatedintoacontinuumcrystalplasticityframework.

77

3 Length-scale Dependent Continuum Crystal

Plasticity with Internal Stresses

3.1 Introduction

Conventionalcontinuumplasticity theoriesaresize‐independentandtreat the

plasticbehaviorofcrystallinemetalsasamaterialresponsethatdoesnotdependon

geometric or microstructural length‐scales. However, there are compelling

experimentalevidencesof strengthening innanostructuredmaterials compared to

their coarse‐grained counterparts. Experiments on miniaturized specimens also

suggest that the yield strength ceases to be a purely material parameter as the

specimendimensionsapproachcharacteristicmicrostructurallength‐scalessuchas

grain size, cell‐wall spacing, dislocation spacing etc. At these length‐scales, the

mechanisms of plasticity may be significantly altered giving rise to macroscopic

phenomena such as strong strengthening and modified hardening that are

intimatelytiedtothemicrostructuralandmacrostructuraldetails.Toexplainsome

of the experimentally observed length‐scale effects, traditional continuum

mechanicsofplasticdeformationisaugmentedwithavarietyofmechanismssuchas

strain gradients (Fleck and Hutchinson, 1993), dislocation starvation, limited

dislocation sources (Dehm, 2009; Uchic et al., 2009) and so on. In practice, such

effectsmayoperate intandemandmaycontributesynergisticallyorcompetewith

eachother toproduceoverallplastic responses.Of thedifferentmechanism‐based

length‐scaledependentplasticitytheories,nonlocalapproachesincorporatingstrain

gradients have gained popularity. Such approaches invoke the existence of excess

dislocations which are commonly referred to as the Geometrically Necessary

Dislocations (GNDs), (Nye, 1953) and are necessary to maintain geometric

78

compatibilityduringplasticdeformation.Thereareseveralversionsofthegradient

plasticitytheoriesavailableinliterature,butthecoreconceptistheassumptionthat

thelocalkinematicsandkineticsofdeformationatacontinuumpointaremodulated

byitssurroundingpoints.Theintroductionofagradienttermintroducesalength‐

scaleintotheconventionalplasticityandendowsitwithanabilitytopredictlength‐

scale dependent plastic behavior. In thiswork,we formulate a nonlocal approach

basedoncontinuumdislocationtheorythataugmentstheclassicalcrystalplasticity

theory with length‐scale dependent internal residual stresses. First, we briefly

summarize some of the strain gradient plasticity theories with reference to their

salientfeaturesincorporatingfirstandhighergradientsofstrain.

Fleck andHutchinson (1993) introduced higher‐order stresses corresponding

to the first gradient of plastic strain in the classical plasticity theory tomodel the

length‐scale dependent responses in micro‐beam bending, torsion of micro‐wires

and micro‐indentation. Gao and co‐workers (Gao, 2001; Nix and Gao, 1998)

providedaphysicalbasis for themicrostructural length‐scale in theirMechanism‐

Based Strain Gradient (MSG) plasticity theory that was based on the Taylor

hardening model. Han and co‐workers (Han et al., 2005a, b) extended the MSG

theory to crystal plasticity. These approaches, requiring higher‐order boundary

conditions, have been further refined to include thermodynamically consistent

descriptionsofthedislocationdensity(e.g.(AbuAl‐Rubetal.,2007)).Ontheother

hand, lower order theories (Acharya and Bassani, 2000; Huang et al., 2004; Shu,

2001)avoidthecomplicatingfeaturesofthehigher‐ordertheoriesbyneglectingthe

higher‐order stresses in thegoverningequations.Recently,EvansandHutchinson,

(2009) compared the lower‐order andhigher‐ordergradient theories and showed

that even in the case of first‐gradient theories the nature of the length‐scale

dependentformulationmayexhibiteitherstrengtheningatyieldoranenhancement

inhardeningafteryield.

79

Recent approaches basedon second gradients of plastic strains formulate the

length‐scale dependent plasticity in a thermodynamically consistent manner

(Bardella, 2006, 2008; Gurtin, 2000, 2002; Gurtin et al., 2007) ascribing their

presence to the distribution of defects. These approaches predict enhanced

strengthening, hardening and the internal stress (aka back‐stress) induced

asymmetryinthetension‐compressioncyclicresponse(theBauschingereffect)asa

function of microstructural parameters. Acharya and Roy (2006) developed a

phenomenologicalmesoscopic fielddislocationmechanicsapproach(PMFDM) that

accountsforGNDsindissipativeandenergeticaspectsbasedonincompatibleelastic

distortions.Ertürk, etal.,(2009);Evers,etal., (2004);GerkenandDawson, (2008),

andKurodaandTvergaard(2008a,b)developedphysicallybased,crystalplasticity

theoriesinthattheback‐stressthatdeterminestheeffective(i.e.appliedstressplus

thesize‐dependentbackstress)shearstressforplasticsliponaslipplaneisderived

usingtheVolteradislocation theory.Concurrently,Yefimov,etal., (2004b)derived

similarexpressionsfortheeffectiveshearstressonaslipplanecorrespondingtothe

edge dislocation density using a statistical‐mechanics approach. These different

approaches provide a similar computational construct and may be interpreted in

termsofeachother(KurodaandTvergaard,2006).

Inthischapter,weformulateastressfunctionbasedapproachtoderivelength‐

scale dependent three‐dimensional (3D) internal residual stress tensor arise from

long range interaction among GNDS in the non‐homogeneous spatial GND

distribution density using continuum dislocation theory. Invoking the Beltrami

stress function tensor Sadd, (2005), we systematically relate the length‐scale

dependent internal residual stress tensor ∗to thegradientof theNyedislocation

densitytensor viatheconstitutiveequationsforabulk,linearelasticsolid.This3D

internalstresstensorthatautomaticallyincludestheeffectsofbothedgeandscrew

GNDs is described in terms of the incompatible parts of the continuumkinematic

80

variables,namelytheelasticstrainandcurvaturetensors.Thedefinitionof ∗via

ensures its equilibrium and can naturally blend into the conventional equilibrium

equation. The higher order derivatives of the Green function in ∗decay rapidly

withdistanceintroducingalength‐scalethathastoberelatedtothemicrostructural

detailsofaboundaryvalueproblem(bvp).Thislength‐scaledependentcontinuum

frameworkisthenextendedtothecrystalplasticitytheoryusingthekinematicsand

kinetics of crystallographic slip, expressed in a thermodynamically consistent

manner(Gurtin,2002;KurodaandTvergaard,2008b).

Weconsidertwobvpinvolvingsinglecrystalsunderplanestrainconditionwith

symmetric double slip, namely, (i) a tapered specimen under uniaxial loading

(ignoring the free surface effects), and (ii) constrained simple shear of a single

lamellawithimpenetrableboundaries,fromalayeredmicrostructurethatmimicsa

nano‐twinnedgrain.The importanceof thesize‐dependenthardeningmechanisms

ishighlightedthroughtheseexamples.Theresultsarediscussedwithinthecontext

oftheexperimental/computationalinvestigationsreportedintheliteratureonthe

length‐scaledependentsinglecrystalplasticityundermonotonicandcyclicloading.

3.2 Background

As an illustration to distinguish between the length‐scale dependent

mechanisms due to the GND density, we consider two examples of crystalline

lattices subjected to curvatures. Here, we do not account for the free surfaces by

implicitlyassumingthatthecrystallatticeisembeddedinanelasticregionofsame

elasticproperties(Mesarovicetal.,2010).Figure 3.1ashowsthesurroundingregion

ofacontinuumpointwhereinthecrystal latticeisunderpurebendingresulting in

constant lattice curvature (Fleck and Hutchinson, 1993). Noting the continuum

description of GND used by Gurtin and coworkers (Cermellia and Gurtin, 2000;

81

Gurtin, 2002) andGao and coworkers (Han et al., 2005a;Nix andGao, 1998) (see

appendix A) the non‐uniform strain along the axis results in a non‐zero GND

density component that is proportional to the curvature (Nye, 1953)5.

However,atanysectionalongthe directionthecurvatureisaconstant and

therefore,theGNDdensitycomponentisalsohomogeneouslydistributedalongthe

axes.Consequently,at anycontinuumpoint theaveragestress fieldsdue to the

presence of the GNDs cancel out. In this problem, the size‐dependent hardening

mechanismisrelatedtothepresenceofGNDdensityandcorrespondingshortrange

interactionbetweenSSGsandGNDswhichisthedissipativehardeningmechanism

and corresponds to the first gradient of plastic strain (e. g.(Acharya and Bassani,

2000;NixandGao,1998)).

However, in thesecondcase(Figure 3.1b) the latticecurvaturevaries linearly

alongthe directionandcorrespondingly,theGNDdensityalsovarieslinearly(i.e.

,where and areconstants characterizedby theappliedstimulus

andmaterialcompliance).Asdemonstratedlater,thisleadstotwocontributionsto

hardening, one purely due to the presence of the GND density similar to the first

illustration(shortrangedislocationinteraction)andanadditionaltermduetoanet

internal stress that exists owing to its non‐homogeneous distribution, as their

averagestressfieldsatacontinuumpointmaynotcancelout(longrangeinteraction

among GNDs). The resulting hardening is sometimes referred to as energetic

hardening. Therefore, this internal residual stress due to the non‐zero gradient of

theGNDdensityaswellasdissipativehardeningmechanismmustbeaccountedfor

topredictboththesize‐dependenthardeningmechanisms.

5BasedoncontinuumdescriptionofGNDdensitybyNye(1953),non‐uniformstrainalong

the axisresultsinanon‐zeroGNDdensitycomponent .

82

(a) (b)

Figure 3.1. Examples illustrating the contributions of GND density to enhancedhardening in (a) pure beam bending ‐ dissipative hardening, (b) non‐uniform bending ‐dissipativeandenergetichardening.

In discrete dislocation plasticity, the internal stress enters the formulation

through the Peach‐Koehler force acting on a dislocation (e.g.(Giessen and

Needleman,1995))

∗ ( 3.1)

where is the Peach‐Koehler force vector on ith dislocation, is the Burgers

vectorforthatdislocation, isitsunittangentvectorand istheappliedstress.

∗istheinternalstressfieldfromthe jthontheithdislocation,whichissuperposed

overalldislocations.While these length‐scaledependent internalstressesmaynot

play a big role in the response of conventional bulk crystalline materials,

experiments show that it cannot be ignored in the regimes where the

microstructurallength‐scalesmediatethemacroscopicresponse.Forexample,thin‐

f-f

No residual stress

MM

x

y

Residual stress

F

2f-fx

y

83

film tension and cyclic bending experiments by Xiang and Vlassak, (2006) exhibit

length‐scale mechanics of strengthening and the Bauschinger effect in passivated

specimenscomparedtotheunpassivatedones.Inthiscasethepassivationlayersact

ashardboundariesthatobstructdislocationsescapingthroughthesurfaces,leading

to the accumulation of dislocations necessary to accommodate geometric

incompatibilities. Very recently,Kiener, et al., (2010) experimentally observed the

BauschingereffectinsinglecrystalCumicro‐beamsundercantileverbending.Even

undernominallyhomogeneous loadingconditionssuchasuniaxialcompressionor

tension the strain and curvature gradients may be induced in miniaturized

specimensbyvirtueofgeometricimperfectionsorfabricationdefectsthatmayplay

vitalrolesinthestrengtheningandhardeningofsinglecrystalspecimens(El‐Awady

etal.,2009a;El‐Awadyetal.,2009b;Fricketal.,2008).

Unlikethediscretedislocationmodelingwherethesuperpositionofthestress

fieldsduetoindividualdislocationsproducesaninherentlynonlocaltheory(Giessen

andNeedleman,1995),inhomogenizedapproachesthatsmearoutthediscreteness

ofdislocationswemustadoptalength‐scaledependentapproachtoaccountforthe

internalstressesduetodislocationarrangement(Figure 3.2).

Figure 3.2. Schematic illustrating the non‐locality arising from the presence of GNDdensityatacontinuumpointandthedistributionoftheGNDdensityaroundthatpoint.

Continuum scale

Dislocation

Micro-scale

Net GND

Integration area

G

G

G

GbLG

G

G

G G

G

bl

G

G

84

3.3 Kinematics of Compatible and Incompatible

Deformations

Incontinuummechanicsthedeformationgradienttensor is

( 3.2)

where is the total displacement gradient tensor.Under small deformation

and small strain assumptions, the total displacement gradient tensor may be

additivelydecomposedintotheelasticandplasticparts

( 3.3)6

Further,wehave

( 3.4)

where istheinfinitesimalstraintensorand istheinfinitesimalrotationtensor.

Further, we assume additive decomposition of the elastic and plastic strains into

theircompatibleandincompatibleparts7

( 3.5)

6Incrystalplasticity,whichwelaterrelateto, ∑ ⊗ where isthe

plasticsliponslipsystem comprisingslipdirection andslipnormal .7Recently,Mesarovicetal.(2010)establishedtheproofthatforanystrainfield,there

existsauniqueorthogonaldecompositionintocompatibleandincompatiblestrainfields.

85

and, ( 3.6)8

Theincompatiblepartoftheelasticstraintensor maybeobtainedfromthe

strain compatibility conditions. In the presence of internal defects such as

dislocations the requirement of compatibility of the total strain introduces an

incompatibilitytensor (Kröner,1959)

( 3.7)

where ∙ , ≝ ∙ . InEq. ( 3.7) the incompatibility tensor,obtained

as the secondgradientof the elastic (orplastic) strain tensor, is ameasureof the

deviationof the elastic (or theplastic) strains from their compatible counterparts

duetothepresenceofexcessdislocationsthatresultininternalstressesinaddition

tothoseduetotheappliedloads.

3.3.1 Compatibility of Lattice Curvature:

Under general loading conditions not only will the strains be non‐uniform

(leading to a strain gradient), but the curvatures (i.e. first gradient of strain)may

alsovarybetween twomaterialpoints.Thecorresponding tensor is referred toas

the Nye tensor (Nye, 1953). In what follows, we systematically relate the

incompatibility tensor to the first gradient of the elastic part of the lattice

curvaturetensor.

8Henceforth,thesubscriptsCandIindicatethecompatibleandincompatibleparts,

respectively,ofthekinematicquantities.

86

The total lattice curvature tensor is given as the gradient of the rotation

vector

12 ,

12 , ( 3.8)

Notingthat , 0,weobtain

12 , . , ( 3.9)

Forconvenience,weset . . , .ThenEq.( 3.9)mayberewrittenas

( 3.10)

Again, in the presence of internal defects the total lattice curvature is still

compatible; however, its elastic and plastic parts may individually be

incompatible.Theincompatiblepartof is

( 3.11)9

Thecompatibilityofthetotallatticecurvaturetensorthengives10

( 3.12)

Thecompatiblepartofthecurvaturetensor maybeconsideredastheelastic

lattice curvature due to the externally applied non‐uniform stress, while the

incompatiblepartof thecurvature is theadditional latticecurvatureduetothe

9NotethatEq.( 3.11)maybeequivalentlywrittenintermsoftheincompatibleparts

oftheplasticstrain andcurvature tensors.

10Thecompatibilityconditionforthecurvaturetensoris . Then,

⇒ .

87

atomicmisfitinthepresenceofGNDs.Further, maybeconceptuallydecomposed

intotheirplastic andelastic parts.Theplasticpartisthelatticecurvature

arising due to the presence of GND density (Nye, 1953) while the incompatible

elasticcurvaturetensor istheadditionallatticecurvaturethatcorrespondstothe

internal residual stress field due to the surrounding excess dislocation density.

Thesedifferentpartsofthetotallatticecurvaturemaybeexplainedbyresortingto

thetwoillustrationsinsection3.2.Inthepurebendingcase, and represent

theelastic latticecurvatureandadditional latticecurvaturedue to thepresenceof

theGNDdensity, respectively,while the vanishes.However, in thenon‐uniform

curvatureexamplethelatticecurvatureduetotheatomicmisfit includesboththe

elasticandplasticparts.Takingthecurlof ,weobtain

( 3.13)

Equation ( 3.13) establishes that the gradient of the incompatible elastic

curvature tensor is non‐zero if anonlinear strain (or stress) field existsdue to an

inhomogeneousGNDdensitydistributioninagivenregion.

3.3.2 Relation between Incompatible Elastic Strain Tensor and

the GND Density Tensor:

Nye(1953)definedtheGNDdensitytensor whosecomponentsarerelatedto

theplasticpartoftheincompatiblelatticecurvaturetensor(SeeappendixA)

12

( 3.14)

where . Applying the curl operator to Eq. ( 3.14), we

obtain

0

88

( 3.15)

Noting the compatibility conditions for the curvature [see Eq. ( 3.12) and Eq.

( 3.13)], we obtain the relation between the incompatible elastic strain and GND

densitytensors

( 3.16)

Since,therighthandsideofEq.( 3.16)issymmetricwerewritethisequationby

consideringthesymmetricpartofthelefthandsideaswell

( 3.17)

i.e. ,

12 , , ( 3.18)

Summarizing,wehaveestablishedtherelationbetweenthegradientoftheGND

densitytensorandthesecondgradientoftheincompatibleelasticstraintensor[Eq.

( 3.17)]inacontinuumsense.Thisequationiscentraltoderivingtheexpressionsfor

theinternalresidualstresstensor,whichisdiscussedinthenextsection.

3.4 Internal Stress Tensor: Stress Function Approach

In the preceding section,we introduced an incompatible elastic strain tensor

thatisrelatedtotheGNDdensitytensor[Eq.( 3.17)].Correspondingtothisstrain

tensor,weintroduceawork‐conjugateinternalstresstensor ∗viaHooke’slaw

∗ ∶ ( 3.19)

where isthefourth‐orderelasticstiffnesstensorforthebulkmaterial.Then,using

superposition,thetotalstressis

89

∶ ∶ ∶ ∶∗

( 3.20)

InvertingEq.( 3.19)werewriteEq.( 3.17)as

∗,

12 , , ( 3.21)

where isthefourth‐ordercompliancetensor.

IntroduceasymmetricBeltramistressfunctiontensor (Sadd,2005)tosolve

Eq.( 3.21),suchthat

∗, ( 3.22)11

FromEqs.( 3.7)and( 3.22)

∶ ( 3.23)

Foranisotropicmedium, dependsonlyontheshearmodulus andPoisson’s

ratio .Then,Eq.( 3.23)simplifiesto(Kröner,1959)

( 3.24)

where 12 1 2

( 3.25)

A fully three‐dimensional solution of Eq. ( 3.24) for an infinite medium is

obtainedusingGreen’sfunction (Kröner,1959)

| | ′3 ( 3.26)

11Eq. ( 3.22)satisfiesstressequilibriumequationbecause ∙ 0

90

| |

| |

8 ( 3.27)

where is theGreen function thatdependson thedimensionalityof theproblem

andtheelasticstiffnessofthematerial(i.e.isotropicoranisotropic).Substituting

fromEq.( 3.17),componentsof are

12

| | ,′3

12

| | ,′3

( 3.28)

UsingtheGreen‐Gausstheoremandsettingthesurfacetermatinfinityequalto

zero,Eq.( 3.28)canberewrittenintheform

12

| | ,′3

12

| | ,′3

( 3.29)

Weassumeappropriateboundaryconditions (Groma,2003;Mesarovic,2005)

whensolvingthebvp’ssothatthesurfaceeffectsduetoimagedislocationfieldsmay

beneglected.

InvertingEq.( 3.25)andsubstituteitinEq.( 3.22),weobtain

∗ 21

∗ 2 , 1 ,

( 3.30)

where

91

12 ,

12 ,

( 3.31)

Equation( 3.30)isthe3Dconstitutivelawfortheinternalstresses,whichcanbe

solvedanalyticallyornumericallyoncethedistributionofA isknown.IfEq.( 3.30)

weretobeintegratedexactlyoverthewholecontinuumdomain,itwouldmeanthat

the stress field due to theGNDdensity at each point influences the stress field at

every other point in the body. However, it can be seen that Eq. ( 3.31), and

consequently,theinternalstressconstitutiverelation(Eq.( 3.30)arefunctionsofthe

thirdgradientof ,whichrapidlydecaystozero(Figure 3.3).

Figure 3.3.VariationofatypicalcomponentofthethirdgradientoftheGreenfunctioninEq( 3.31)

Therefore,wemayconsiderasmall,butfiniteregion aroundacontinuum

point wherein the GND density distribution is accounted for (Evers et al., 2004;

Gerken and Dawson, 2008; Groma, 1997; Mesarovic, 2005). Using the Taylor

92

expansionof aroundthepointr in theregion ′andassuming thatonly the

firstgradientofthisseriesisimportant(Groma,2003),weobtain

,12 , ∙ | | ,

12 , ∙ | | ,

( 3.32)

Theonlyparameterwhichremainstobechosenisthe integrationvolume, ,

whichdefinesa length scale in theproblem that givesnonlocal ∗(Eq.3.32).Note

that ∗doesnotdependon atacontinuumpoint,butonlyon itsgradient.Using

the crystallographic definition of ∑ ⊗ , the resolved

components of ∗on a slip system are obtained as the Laplacian of the plastic

slip .Foraslipsystem thecontributionsfromotherslipsystemstoitsinternal

stressautomaticallyenterstheformulation.

3.4.1 Internal Stress under Plane Strain Condition: Isotropic

Elasticity

Althoughtheresultobtainedintheprecedingsectiongivesa3Dinternalstress

tensor[Eq.( 3.31)]weexplicitlywriteitscomponentsforthesimplercaseofplane

strain. Assuming a plane strain condition in the ‐direction, the only non‐zero

components of the GND density tensor (i.e. containing dislocation lines in the z‐

direction)are , , .Then,thenon‐zerocomponentsofincompatibilitytensor

are[seeEq.( 3.17)]

, , ;

12 , ;

12 ,

( 3.33)

93

FromEq.( 3.22)thein‐planestressescanbeobtainedfromthestressfunction

tensor

∗ , ; ∗, ; ∗

, ( 3.34)

where issimilartoanAirystressfunction.Theout‐of‐planestressesare

∗, – , ; ∗

, , ( 3.35)

Equation ( 3.24) can be solved inwhich 1 /2 and then, the

internalstresscomponentsare

∗ 21 , . , , . ,

, . , , . ,

∗ 21 , . , , . ,

, . , , . ,

∗ 21 , . , , . ,

, . , , . ,

( 3.36a‐c)

where is the integration area that defines a length‐scale. For the plane strain

conditiontheappropriateGreenfunctionis(Kröner,1959)

| |

8ln | | ( 3.37)

94

choosingasquareregion astheintegrationarea,weobtain

∗ 21

0.068 , 0.25 ,

∗ 21

0.25 , 0.068 ,

∗ 21

0.068 , 0.068 ,

( 3.38a‐d)

where ∗ , ∗ , ∗ ∗ are the internal stress components due to edge

dislocations,whilethoseduetothescrewcomponents ∗ ∗ arezero.When

described in terms of the crystal plasticity framework the resolved internal shear

stress ∗ due to Eqs. ( 3.38 a‐d) on slip system is ∗ ∙ ∗ . These

internal stresses bear close resemblance with those derived in the recent works

(Geers et al., 2007; Gerken and Dawson, 2008; Yefimov et al., 2004b). Note that

gradientsinthedislocationdensitiesmayprevailinsinglecrystalspecimensdueto

a variety of reasons including geometric imperfections (Uchic et al., 2009), small

misorientations, fabrication‐induced defects (El‐Awady et al., 2009b)etc. In

polycrystallinematerials,changesincrystalorientationsacrossgrainboundariesor

twinboundariesmayalso setup regionswithhighGNDdensitygradients in their

vicinity.WehighlightsomeoftheseaspectsthroughtheexamplesintheResultsand

Discussionsection.

Thelength‐scaleinthistheoryismathematicallynecessary,butitmustalsobe

physicallymeaningful.Recently,Mesarovicetal (2010)showed thata length‐scale

emergesfromthethermodynamiccoarseningerroroftheenergiescorrespondingto

thecontinuousandsemi‐discreterepresentationsofstackedpile‐ups,whichisofthe

orderofaverageslipplanespacing ~100 foreachslipsystem.Dependingonthe

95

specific problem, microstructural length‐scales may be related to, for example,

averagespacingofobstaclestodislocationmotion inthe formofgrainboundaries

(polycrystals), second‐phase particles (heterogeneous alloys and composites),

dislocationandcell‐wallarrangements(singlecrystals). Inotherwordsthe length‐

scale has to be determined by the microstructural details and may be problem‐

dependent.One interpretationof the length‐scaleemerges fromthecomparisonof

Eqs.( 3.38a‐d)withthatofYefimov,etal.(2004)andrelatestotheaveragespacing

ofdislocations,i.e. ~ (alsosee,Groma,etal.,2003andBayleyetal,2006).With

thisinterpretationtheinternallength‐scalemayrangebetweenfewtensofnm(very

high dislocation density, e.g. Dao, et al., 2006; Lu, et al., 2009) to a few (low

dislocationdensity,e.g.miniaturizedsinglecrystals)andthelength‐scaleitselfmay

evolve with deformation. In the next section, we briefly discuss the extension of

currentstressfunctionapproachtoaccountforelasticanisotropy.

3.4.2 Internal Stress with Elastic Anisotropy

In section 3.4, we derived the internal residual stress using Beltrami stress

functionforanelasticallyisotropicmaterial.Here,theextensionforanisotropiccase

ispresentedbasedonpreviousworkbyKröner,(1955).Fortheanisotropiccasethe

incompatibilityequationiswrittenas(compareEq.( 3.24))

( 3.39)

where is fourth‐order stress function tensor and is a scalar sixth‐order

differentialoperatorwhichisgivenby

∙ ∙ ( 3.40)

96

InEq.(3.40) ∙ , isasecondordertensoroperatorwith asthe

elasticconstant.TheGreenfunctionsolutionofequation(3.39)isgivenby(compare

Eq.( 3.26))

| | ( 3.41)

where is the appropriate Green function for the anisotropic case. For cubic

symmetry,theGreenfunctionis(Burger,1939;Kröner,1953)

| | ∙ | | , /96 ( 3.42)

FurtherstudiesfortheanisotropiccasescanbefoundintheworksofLeutzand

Bauer,(1976)andSteedsandWillis(1979).Then,theinternalstress ∗isgivenby

Eq.( 3.22)wheretherelationbetweenBeltramistressfunctiontensor andfourth‐

orderstressfunctiontensor isdefinedusingasecond‐orderdifferentialoperator

as

( 3.43)

Theexplicitformulationfor operatorforcubicsymmetrymediahasbeen

derivedbyKröner,(1955)andextendedtofullyanisotropiccasebyMichelitschand

Wunderlin(1996).

3.5 Thermodynamically Consistent Visco-plastic

Constitutive Law

Inthissection,wederivetheequilibriumandconstitutiveequationsforcrystal

plasticity including the internal stress using the purelymechanical version of the

thermodynamiclaws.

97

3.5.1 First law of thermodynamics: Power Balance

Givenavirtualdisplacementfield ,thevirtualexternalpowerofanysub‐body

ofvolume boundedbysurface is

∙ ∙ ( 3.44)

where isthetractionvectoronaplanewhoseunitnormalis andand isthe

bodyforcevector.Thevirtual internalpower inthe includingthe internalresidual

stressis

∗ ∙ ( 3.45)

Foranyvirtualdisplacementfield, the internalandexternalpowersshouldbe

balanced,sothat

∙ ∙ ∗ ∙ ( 3.46)

Usingthedivergencetheorem,weobtain

∗ ∙ ∗ ∙

0

( 3.47)

Sincethisequationshouldbevalidforallsub‐body andanyarbitraryvirtual

displacement ,thenonlocaltractionconditionis

∗ ( 3.48)

and,thenonlocalforcebalanceis

∗ 0 ( 3.49)

NotethatfromEq.( 3.22), ∗isalwaysequaltozero.Then,Eq.( 3.49)yields

the classical force balance equation. Writing the plastic part of the total virtual

98

displacement gradient vector in terms of the crystal plasticity framework, in the

absenceofanymacroscopicmotion,wehave

⨂ ( 3.50)

andtheprincipleofvirtualpower[seeEq.( 3.46)]yields

∗ ∗ ∙

∗ ∙ ∙

( 3.51)

where isthetotalshearstressonslipsystem .UsingEq.( 3.50),weobtain

∗ 0 ( 3.52)12

∗ ∗ ∙ ( 3.53)

where ∙ is the resolved shear stress due to external loads and

∗ isthemicroscopictractionvector.

3.5.2 Second law of thermodynamics: Power imbalance

To derive the constitutive equation in the presence of the internal residual

stress, we rewrite the second law of thermodynamics within the framework of

crystalplasticity.Theclassicalformofsecondlawforisothermalconditionis

12Equation( 3.52)isthesameasthemicro‐forcebalanceequationofGurtin(2002)

where ∗ where isthemicro‐stressvector.Inthepresentapproachtheinternal

stressiswork‐conjugatetotheincompatibleelasticstraintensor,akintoGurtin’sdefect

stress(Gurtin,2002)thatwork‐conjugateswiththeGNDdensitytensor.

99

∶ 0 ( 3.54)

where isthefreeenergy.Notingtheorthogonaldecompositionofthetotalstrain

tensor(Mesarovic,etal.,2010),thetotalfreeenergymaybedecomposedas

( 3.55)

where isthestandardelasticstrainenergycorrespondingtothecompatiblepart

of the elastic strain tensor and is the defect energy that corresponds to the

incompatiblepartoftheelasticstraintensor.SubstitutingEq.( 3.55)inEq.( 3.54)we

obtain

∙ ∗ ∙ ∗ 0 ( 3.56)

Thisinequalityshouldholdforallchoicesof , and ;thelinearityofthis

inequality in and respectively provides the sufficient conditions for

macroscopicandmicroscopicenergeticconstitutiveequations

( 3.57)

∗ ( 3.58)

andfromtheinequality,weobtain

∗ 0 ( 3.59)

A visco‐plastic constitutive law satisfying the inequality in Eq. (3.56) can be

writtenas

∗∗ ( 3.60)

where isthetotalcrystallographicslipresistanceduetotheSSDdensityandthe

presenceoftheGNDdensity(Hanetal.,2005a).Writingthetotalinternalpowerand

100

comparing with the theory of Gurtin, et al. (Gurtin et al., 2007) we identify the

energeticanddissipativehardeningterms

:

∗:

∗ ( 3.61)

The first term in Eq. ( 3.61) represents the length‐scale independent stress

power (reversible stored power) associated with externally applied loads. The

second term is referred to as the length‐scale dependent energetic power

(irreversiblestoredpower)as it isassociatedwiththe internalresidualstressand

incompatible elastic strain that will tend to reorganize the GND density from an

energetically efficient configuration. The third term in Eq. ( 3.61) is the plastic

dissipation due to the SSD (length‐scale independent) and GND (length‐scale

dependent)densities.

Table )3‐1)‐)3‐3) summarize key expressions developed in the present

approach. Note that the internal stress tensor is blended into the continuum

frameworkthroughordinaryequationsofforcebalanceandtractioncondition,and

additionalgoverningequationsarenotrequired.

Table 3‐1.Summaryofgoverningequations

Straindecomposition

Kinematicrelation ,12 , ,

Localforcebalance 0

Non‐localforcebalance ∗ 0

101

Table 3‐2.Summaryofconstitutiveequations

Localelasticconstitutivelaw

∶

Nonlocalinternalstressconstitutivelaw

∗ ∶ 2 , 1 ,

Nonlocalvisco‐plasticconstitutivelaw

∗

∗

Table 3‐3.Summaryofunknownvariablesandavailableequations

Unknownparameters #unknowns Governingequations #equations 6 ∗ 0 3 6 ∗ ∶ 6 6 , ∗, … 6 3 6∗ 6 ∗ ∗ , … 6

Total# 27 Total# 27

3.6 Results and Discussion

Inthissection,weinvestigatetwoproblemsinvolvingsinglecrystalspecimens

usingthenonlocalvisco‐plasticconstitutiverelation(Eq.( 3.60)).Forsimplicitywe

consideratwo‐dimensionalplanestrainsetupwithcrystalsorientedforsymmetric

doubleslipwithrespecttotheloadingdirection.

3.6.1 Tapered Single Crystal Specimen Subjected to Uniaxial

Loading

Figure 3.4showsa taperedsinglecrystalof lengthL inplanestraincondition

that is constrainedagainst slipatoneendandsubjected to anaxial force at the

102

otherend.Thecrystal isassumedtodeformundersymmetricdoubleslip.Theslip

systemsareorientedatanangle giving

sin , cos , 0 , cos , sin , 0 , sin , cos , 0 , cos , sin , 0

Figure 3.4. A tapered bar under uniaxial loading. Dashed tapered edges indicate thattheyaresufficientlyawayfromthecenterlineofthespecimen

Althoughthisgeometryismotivatedbytherecentmicro‐pillarexperimentson

single crystals, there are important differences that are discussed briefly before

proceedingwiththesolution.First,theactualproblemisessentially3D,whereaswe

assumeaplanestraincondition.Further,asmentionedearlier,thepresentapproach

does not account for free surfaces thatmay give rise to image stresses and cause

othermechanismsofstrengthening.Tocircumvent thecomplexityassociatedwith

thefreesurfaceeffects,weimplicitlyassumeaquasi‐1Dsituationinthatthetapered

boundariesareconsideredtobesufficientlyawayfromthecrystalcenterandtheir

103

presenceisaccountedforonlythroughthestressvariationalongitslengthfroman

appliedforce13.

We apply a uniaxial force F at the top and assume that at the base of the

specimen 0 , 0 (specifically, zero, in this example) and

γα 0 0sothatthedislocationsarefreetomoveintothebase,akintoamicro‐

pillar.Theonlynon‐vanishingstresscomponentisthen ⁄ ,where

is thewidthofthecrystalatsection thatchanges linearlyfrom at the loaded

edge to at the constrained edge. Then, the resolved shear stress on each slip

system is sin cos and the corresponding plastic slip is

.Withtheplasticstraintensor ∑ ⊗ theplastic

slipgradientis 0, , , 0 .InthecrystallographictermstheGNDdensitytensor

is ∑ ⊗ ;therefore,weobtain

0 0 00 0 0

2 , 0 0

wherec andsdenotecos andsin , respectively.For thisproblemtheonlynon‐

zeroGNDdensitycomponentistheonewiththeBurgersvectorinthe direction

andthedislocationlineinthe direction.Forthetotalsliphardeningweadoptthe

SSDandGNDdependenthardeningoftheform(Hanetal.,2005b)

13Alternatively,onemayassumethatthetaperededgesarecoated(i.e.nofreesurface

forthedislocationtoexitthespecimen)withasufficientlythickmaterialofelasticproperties

sameasthatofthecrystalsothattheydonotpileupalongthoseedges.Theseareobviously

highlyidealizedassumptions,butenableustoconsiderasimplersystemtoprovidesemi‐

analyticalsolutions.

104

1 | | ,

/

( 3.62)

Thenon‐zerocomponentsoftheinternalstresstensor[seeEq.( 3.38a‐d)]are

∗1

∙ ,

∗ 0.271

∙ ,

∗ ∗ ∗

∗ 0

( 3.63)

andthecorrespondingresolvedinternalresidualshearstressoneachslipsystemis

∗ ∗, , ( 3.64)

where.

. Substituting Eqs. ( 3.62) and ( 3.64) into Eq. ( 3.60) and integrating

withrespecttotime,weobtain

,

1 | | ,

( 3.65)

Equation ( 3.65) is solved using the fourth‐order Runge‐Kutta method. The

material and geometric parameters used are 0.01, 5⁄ , 1000 ,

45°, 0.2 , 0.4 , 10. The results for monotonic and cyclic

loadingarediscussednext.

a.Monotonicloading:

To begin with, we investigate the influence of by setting 0. Figure 3.5

showsthevariationofthemagnitudeofplasticslipalongthelengthofthespecimen

fordifferentratios.Asexpected,themagnitudeoftheplasticslipdecreases

withdecreasing ,thatisforasmallerspecimensizeorlarger .

105

Figure 3.5. Plastic slip along bar axis y for various ratio of / for taperedspecimenundermonotonictension

With decreasing the internal stress term inEq. ( 3.65) becomes increasingly

dominant and provides a strong resistance to plastic slip. The increasing internal

stresswithdecreasing tendstohomogenizetheplasticslipasobservedfromthe

trendoftheplasticslipvariationwithdecreasing .

Next, we highlight the relative influence of the two length‐scale dependent

dissipativehardeningmechanisms,i.e.thedissipativehardeningduetothepresence

of GND (corresponding to ) and the one due to GND density gradient

(correspondingto )ontheoverallresponseofthecrystal.Figures 3.6aandbshow

the normalized resolved shear stress on a slip system versus the magnitude of

plasticslipat fordifferentvaluesof and ratios.Figure 3.6ashows

that the length‐scaledependentdissipativehardeningdue to thepresenceofGND

influencesthepost‐yieldresponse,whichhasbeenpreviouslyreportedbyHan,etal.

(2005b)inthecontextofMSG‐CPtheory.

106

(a)

(b)

Figure 3.6. Resolved shear stress versus plastic slip at for tapered bar undermonotonictensionforvariousratios(a) ⁄ ,and(b) ⁄ .

However, this enhanced hardening effect is discernable onlywhen the slip is

appreciably large,well beyond the initial yield. In comparison, Figure 3.6b shows

thattheinternalstresssignificantlyinfluencesboththeresponseatincipientslipas

well as at relatively larger slip. That is, the length‐scale dependence due to the

gradientoftheGNDdensityhasastrongerinfluenceonboththestrengtheningand

0 0.005 0.01 0.015 0.02

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

g

L

l

β = 1000β = 10β = 2β = 1β = 0.5

0g

Decreasing β

0 0.005 0.01 0.015 0.02

b

L

l

0g

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

η= 1000η = 100η = 50η = 25η = 15η =10

Decreasing η

107

hardeningbehaviorofa crystal compared to that arising from thepresenceof the

GNDdensity.Thisisfurtherexacerbatedgiventhattherangeof ratiosconsidered

here is relatively small compared to the range of ratios. Figure 3.7 signifies the

influenceofgeometric imperfectiononthedistributionofthe internalstressalong

thespecimenlength.Thelargerthe initialtaperthemorenon‐homogeneousisthe

GND density distribution that causes higher resolved internal shear stresses on

individual slip systems. Consequently, the plastic slip on the slip systems would

becomehardergivinganoverallplasticallystrongerresponse.

The strong strengthening and hardening observed in this example is

qualitatively similar to the specimen length‐scale dependent strengthening

behaviors reported in some of the recent experiments on miniaturized single

crystals(e.g.(Fricketal.,2008))thatindicatepresenceoftheGNDdensity.Whilethe

actual mechanisms of strengthening in such miniaturized experiments have not

beenfullyunraveled,theresultsfromthepresentworkcorrelatequalitativelywith

the experimentally observed size‐dependent plasticity in the presence of strong

gradients(Maaßetal.,2009).

Figure 3.7. Distribution of normalized internal shear stress ∗ along the tapered

specimenundermonotonictensionfor(a) 2.86°,(b) 5.71°. 50.

AlthoughinthepresentcasethegradientintheGNDdensityisduetospecimen

taper,presenceoffillets,lowangleboundaries(Uchicetal.,2009)orsurfacedamage

layersdue to fabricationmayalsoproducesignificantgradientsat small specimen

108

sizes(El‐Awadyetal.,2009b).Thus,thenon‐gradientbasedsize‐effects(e.g.source‐

limiteddislocationplasticity,dislocationstarvation)postulatedinsuchexperiments

may be augmented by those due the internal stresses arising from the non‐

homogeneousdistributionoftheGNDdensity.

b.Cyclicloading:

Wenowinvestigatetheresponseofthetaperedsinglecrystalspecimenundera

singletension‐compressioncycle.Manymetalsexhibitthewell‐knownBauschinger

effectunder cyclic loading. In thepresentwork, the internal stress tensorderived

from the inhomogeneous GND density distribution produces a size‐dependent

Bauschingereffect.Forsimplicity,wesuppressthecontributionfromthedissipative

hardeningbysetting 0.

(a) (b)

Figure 3.8.Resolvedshearstressversusplasticslipat fortaperedbarundercyclicloading(a) 100,(b) 50.

Figure 3.8 shows the resolved shear stress versus plastic slip curves at

plotted for two different values of . We also include the response of the same

specimens under monotonic loading. For fixed the monotonic compressive

responseismuchstrongerthanifthespecimenwereloadedunderasingletension‐

Cyclic loading

Monotonic compression

Cyclic loading

Monotonic compression

0g

0 0.01 0.02 0.03 0.04 0.050 0.01 0.02 0.03 0.04 0.05

1

0.5

0

-0.5

-1

1

0.5

0

-0.5

-1

0g

109

compression cycle. The disparity between the monotonic and cyclic responses

increases with decreasing giving a length‐scale dependent Bauschinger effect

(Kiener et al., 2010; XiangandVlassak, 2006).However,upon reverse loading the

directionoftheresolvedshearstressduetoexternal loadreverses,butthatofthe

internal stress does not as the GND arrangement is unaffected. This causes the

specimentoyieldatasmallerloadinthereverseloading.Thehardeningbehavioris

alsoweaker in the reverse loading compared to the initial forward response. In a

realistic scenario with more than two slip systems, one may observe stronger

hardening due to latent hardening that may accentuate the Bauschinger effect

(Bayley et al., 2006). Figure 3.9 shows that geometric imperfections strongly

influence the Bauschinger effect and it increases with increasing degree of

imperfection.Suchanasymmetricresponsecannotbepredictedsolelybya theory

thatdoesnotaccountfortheeffectofdistributionofthedislocationdensity.Thisis

true irrespective of the particular nature of the strain gradient theory (Xiang and

Vlassak,2006).

(a) (b)

Figure 3.9. Resolved shear stress versus plastic slip at y=L for various tapered angleundercyclicloading( =100)(a) 2.86°,(b) 5.71°.

110

3.6.2 Single Crystal Lamella Subjected to Simple Shear

In the previous problem, the internal stress appeared because of non‐

homogeneous distribution of the stress due to geometric imperfections. Here, we

considerthe internalstress inaspecimenwithnogeometricnon‐uniformities,but

due to the pile‐up of dislocations at impenetrable boundaries. Consider a layered

crystal as shown in Figure 3.10. We isolate a single layer from this crystal and

assumeittobeasemi‐infinitelamellaofthickness2 withsymmetricdoubleplanar

slipsubjected tosimpleshear.Thisgeometry is reflectiveofa typical twin lamella

withinagrainofanano‐twinnedpolycrystal(Lietal.,2010).

Figure 3.10.Asinglelamellawithinanano‐twinnedcrystalundersimpleshear.

Weconsiderhardboundaryconditionsonthelamellaboundariessuchthat

0 0, 2 0; 1,2 ( 3.66)

These conditions ensure that noplastic slip occurs along a slip system at the

boundaries causingdislocations to pileup there. Theonlynon‐zero componentof

macroscopicstress in thisproblemis .Thecorrespondingresolvedshearstress

111

andplasticslipduetotheexternalloadare,respectively, cos 2

and ,where istheorientationoftheslipsystemswithrespecttothe

loading direction. The plastic slip gradient is 0, , , 0 and the continuum

dislocationdensitytensor forthiscaseis

0 0 00 0 00 2 , 0

FromEq.( 3.38a‐d)andusingSchmidlaw,theresolvedinternalshearstresson

eachslipsystemis

∗ ∗ ∗0.1361 , cos 2 ( 3.67)

andthecorrespondingplasticslipis

0.1361 , 2

1 | | ,

( 3.68)

The material parameters are the same as in the previous example, except ,

whichissetequalto90°providingthehighestplasticincompatibility.Figure 3.11a

showsthenormalizedshearstress‐averageplasticslipresponsefordifferent .

Asexpected,strongstrengtheningoccurswithdecreasing .Figure 3.11bshowsthe

normalizedshearstressat 0.002asafunctionofnormalizedlamellathickness.

It is interesting to note that for the range of values shown in the figure the

strengtheningtrendcompareswellwithHall‐Petchbehavior.

For a given applied stress, Eq. ( 3.68) plastic slip variation along the lamella

thicknesscanbeobtained,subjecttotheboundaryconditionsinEq.( 3.66).Forfixed

appliedloading( 1.5 ),Figures 3.12aandbrespectivelyshowthevariationof

plasticslipandnormalizedinternalresolvedshearstressonaslipsystemalongthe

112

normalizedthicknessordinatefordifferentvaluesof .Asshowninfigures3.12,for

fixed (i.e.samematerial),whenthelamellathicknessismuchlargercomparedto

the internal length‐scale ≫ 1 , only a very narrow region is affected by the

boundaryandawayfromittheeffectdecaysrapidly.

(a)

(b)

Figure 3.11.(a)Normalizedresolvedshearstress ⁄ versusaverageplasticslipasafunctionof for 90°, (b)Normalized resolvedshear stress ⁄ versusnormalizedlamellathicknessat 0.2%.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

η = 1

η =1.3

η = 2

η = 3.3

η = 5

η = 10

η = 20

0g

avg

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

0g

0.5

0

1.73bg l

2 0.92R

bl

113

The plastic slip away from the boundary reaches a constant value

(Figure 3.12a),which corresponds to the absence of internal stress in that region

(Figure 3.12b).However, as the lamella thickness approaches the internal length–

scale → 1 theboundaryaffectedzone(b.a.z.)occupiesasignificantportionofthe

lamella.Themagnitudeofinternalstressoverthelamellathicknessincreasesandat

thesametimeitbecomesmorediffuse,i.e.itextendsovertheentirelamellarregion.

Correspondingly,itbecomesincreasinglydifficulttoproduceplasticslip.Witheven

further decrease in the lamella thickness 1 the internal stress distribution

within the lamella becomes nearly uniform (except at the boundary) and its

magnitude tends to saturate. Figure 3.12c captures this aspect clearly in that it

showsaninitialstrongincreaseinthenormalizedinternalresidualshearstressas

decreases, but a tendency to saturate at very small . Although not shown in

Figure 3.11b, the corresponding slip system strengthening also tends to saturate

withthesaturationoftheinternalresidualshearstress.

(a)

0 0.5 1 1.5 2y

0.1

0.08

0.06

0.04

0.02

0

η = 0.2η = 1η = 5η = 10η = 20η = 50η = 100

114

(b)

(c)

Figure 3.12. (a)Distribution of plastic slip on a slip systemas a function of for90° versus distance normalized by lamella thickness (b) Normalized internal

resolvedshearstress ∗ ⁄ alongthelamellathicknessasafunctionof for 90°,and(c)Normalizedinternalresolvedshearstress ∗ ⁄ versusnormalizedlamellathickness.

η = 0.01η = 0.02η = 0.2η = 1η = 5η = 10η = 20η = 50η = 100

2

4

6

8

y 0 0.4 0.8 1.2 1.6 2

*

1

0.8

0.6

0.4

0.2

0

bl

*

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1 10 100

115

3.7 Summary

Inthischapter,wedevelopedanonlocalcrystalplasticityapproachenrichedby

internal residual stresses that arise due to the non‐homogeneous distribution of

GND densities. The salient feature of this work is the analytical derivation of the

length‐scaledependent3Dinternalstresstensorusingthestressfunctionapproach.

This second order internal stress tensor blends into the conventional equilibrium

equationsandboundaryconditions.Inthecrystalplasticityframework,theinternal

stressesappearasadditionalresolvedshearstressesoneachslipsystemalongside

those due to the externally applied loads. The visco‐plastic constitutive law for

crystallographic slip that includes this effect is presented in a thermodynamically

consistent manner. The connections between the continuum and crystallographic

variablesinthisformulationrenderitusefulforthedevelopmentofcomputational

frameworkforthe deformationtheoryaswellassmallstraincrystalplasticity

theory.

Theanalytical exampleshighlight the importanceof the internal stresson the

size‐dependent strengthening and hardening in single crystals. Geometric

imperfections can cause strong gradients in the GND density and lead to a

strengtheningof theoverallstress‐strainresponse inspecimensthataresubjected

tonominallyuniaxialmacroscopic loads.Asevident from the secondexample, the

internal stress‐induced strengthening of a slip system is akin to the Hall‐Petch

behavior,buttendstosaturateatsmallmicrostructuralsizes.

Asaclosure,webrieflycomparethepresentapproachwithsomeoftheexisting

nonlocaltheoriesthatincorporatetheinternalresidualstresses.Thekeyequations

derived here for internal stresses results bear close resemblance with the

pioneering efforts of Groma and co‐workers that use a statistical approach

116

mimicking the collective behavior of dislocations (Groma, 2003; Groma andBakó,

1998;Zaiseretal.,2001).RecentversionsofthisapproachbyYefimovetal(2004)

have been developed for edge dislocations. The approach of Gerken and Dawson

(2008)againrestrictsitsfocusontheinternalstressesduetoedgedislocations.In

theirtheory,theaverageinternalstressfieldsderivedfromVolterradislocationsare

simplified assuming a bilinear variation of the GND density, but the motivation

behind this choice is not obvious. Recently, Ertürk, et al (2009) presented a

sophisticatedcrystalplasticityapproachwithbackstressesaccountingforthelatent

hardening effects due to both screw and edge GNDs. In comparison to these

approaches, the present approach is based on a continuum theory of kinematic

coarseningandalsoincludescontributionsfromedgeandscrewcomponentsofthe

GNDs. Theprojection of the gradient of theGNDdensity tensoron to a given slip

system leads to the contribution from other slip systems providing a latent

hardening effect. Further, it should be possible to extend this approach to

anisotropicelasticcasesusingappropriatestressfunctions(Seesection3.4.2).This

aspectseemsnotwell‐outlinedinmosttheoriesinthatthedevelopmentisrestricted

toelasticallyisotropiccases.

OurfocusinthischapterwasoninteractionofGNDsininfinitemediumwhere

GNDinteractionwithboundariessuchasfreesurfacesandinterfacesareneglected.

It showed that internal stresses only present when GNDs are distributed non‐

uniformlyandGNDdensitygradientpresent.Inthenextchapter,wewillaccountfor

GNDs‐boundaries interaction inthe finiteregion. Itshowsthateven inpresenceof

uniform GND density distribution, internal stresses arise due to GND‐boundary

interactionswhichaffectoverallbehaviorofsmallscalestructures.

117

4 A Crystal Plasticity Analysis of Length-scale

Dependent Internal Stresses with Image

Effects

4.1 Introduction

Internal stresses arise in crystalline metals due to ensembles of geometrically

necessarydislocations(GNDs)thataccommodatelatticeincompatibilities.Animportant

macroscopic consequence of these internal stresses is that they produce length‐scale

dependentstrengtheningunder forwardloadingandkinematichardeningundercyclic

loading with decreasing microstructural and/ or specimen sizes (Kiener et al., 2010;

Motzetal.,2005).Theseinternalstressesappearbecauseofthelong‐rangedislocation‐

dislocation and dislocation‐boundary interactions. Of particular interest are the long‐

rangeGND‐GNDandGND‐freesurfaceinteractions.Whilemoleculardynamics(MD)and

discretedislocationdynamics(DDD)accountfortheseinternalstressesasalength‐scale

dependententity(e.g.Fiveletal.1996;Yanetal.2004),coarserrealizationsbasedon

continuum approaches, e.g. crystal plasticity, rely on augmenting the traditional

kinematics and kineticswith additional length‐scale dependent features. A continuum

crystal plasticity description of the internal stress due to long‐range GND‐GND

interactionappearsasthe firstgradientoftheGNDdensity withrespecttoaslip

direction (Evers et al., 2004; Gerken and Dawson, 2008; Gurtin, 2002; Kuroda and

Tvergaard, 2008b). A natural requirement of this result is that the GND density

shouldbespatiallynon‐uniform.Indeedinmanycases, variesalongslipdirection

due to a variety of situations including geometric non‐uniformity (see chapter 3), or

deformationmappingleadingtogradients,e.g.simpleshear,(Eversetal.,2004;Gurtin

etal.,2007;Yefimovetal.,2004b).However,incaseswhere doesnotvaryspatially

118

suchanexpositionpredictszerointernalstress.Forinstance,auniformcurvatureinthe

case of pure bending results in also being constant (Han et al., 2005b). From a

physical viewpoint however, internal stresses should exist even under uniform

curvature conditionsor evenunderhomogeneous loading (Guruprasad andBenzerga,

2008),becauseof theadditional long‐rangeGND‐surface interactions.Theseenhanced

interactions are automatically resolved in ahigh resolution approach suchasMDand

arealsomodeledinDD frameworksthroughappropriatecorrectivetractionboundary

conditions (Cleveringa et al., 1999; Hou et al., 2008;Motz et al., 2008; Yefimov et al.,

2004a). To our knowledge most length‐scale dependent continuum crystal plasticity

frameworkswith internal stresses donot explicitly discuss image stress fields arising

fromthelong‐rangeelasticinteractionsbetweentheGNDsandfreesurfaces.(Bayleyet

al., 2006; Evers et al., 2004; Gerken and Dawson, 2008; Gurtin, 2002; Kuroda and

Tvergaard,2008).Recently,VinogradovandWillis(2008)andCherednichenko(2010)

derived a continuum crystal plasticity framework incorporating image stress fields

using statistical mechanics based approach (Groma, 1997). They provided explicit

solutionsforimagestressesinastripundersimpleshearduetothepresenceofahard

boundary (causing dislocation pile up) rather than a traction‐free boundary.

Thermodynamically‐based frameworks developed by Gurtin (2002) and Mesarovic

(2005) provide pathways to introduce these additional effects. Mesarovic (2005)

proposedathermodynamicframeworkaddressingthe long‐rangeGND‐GNDandGND‐

boundaryinteractionsanddiscusseditsapplicabilityinthecontextofimpenetrableand

penetrableboundaries.Gurtin(2002)providedabasistoaccountfortheimagestresses

through higher‐order traction b.c. that can be adopted for traction‐free surfaces. This

micro‐traction b.c. is sometimes reinterpreted in an equivalent null edge and screw

dislocation densities at the free surfaces (Ertürk et al., 2009; Hayashi et al., 2011;

KurodaandTvergaard,2009;Yefimovetal.,2004a).However,suchanequivalentnull

119

GNDprescriptionmaynotfullyaccountforthelong‐rangeimageeffectsproducedbya

traction‐freesurface.

SimilartotheDDapproaches(e.g.(Fiveletal.,1996;Lubardaetal.,1993;Vander

Giessen andNeedleman, 1995; Yan et al., 2004)), superposition of image stress fields

due to dislocation ensembles described by a continuum densitymeasure would be a

natural way to satisfy the b.c.’s at free surfaces. There are some classic studies on

obtaining imagestress fieldsarising fromasingledislocationhosted inasemi‐infinite

medium(Jagannadham and Marcinkowski, 1979; Lubarda and Kouris, 1996b), in the

proximityofabi‐materialinterface(Chouetal.,1975;JagannadhamandMarcinkowski,

1980;LubardaandKouris,1996a)andinathinstripwithtwofreesurfaces(Fotuhiand

Fariborz, 2008; Hartmaier et al., 1999; Ting, 2008). (Saada, 2008) provided a brief

reviewontheimagefieldsarisingfromplanardislocationarraysandtheircontribution

to theplasticdeformation. (Khanikaretal.,2011;Lubarda,2006;WeinbergerandCai,

2007)investigatedimageeffectsingeometriesmimickingmicro‐scalespecimens.

Inthischapter,wepresentacoarse‐grainedapproachthataccountsforimagefields

withincontinuumcrystalplasticityarisingfromthelong‐rangeelasticinteraction(LRI)

betweenaGNDdensity field andbounding free surfacesof the specimen thathosts

thisGNDfield.Theapproachexpandsonourpreviouswork(chapter3)byintroducinga

generalized stress function that now incorporates appropriate boundary corrections

through image fields. As a model system, we analyze a thin specimen experiencing

uniformcurvatureundertheactionofanexternalbendingmoment.Theanalysisshows

thateveninthecaseofauniformGNDdensitydistribution,internalstressesoccurfrom

twosources

I. LRIarising from finite spatial extentof theGNDdensity (embedded in an

infinitemedium).

II. LRIbetweentheGNDdensityandfreesurfacesappearingasimagefields.

120

Thestressfunctioncorrespondingto(i)isobtainedviaappropriateinfinitemedium

Green’s function (chapter 3) and the resulting stress fields are akin to the Volterra

solutionasappliedtofinitespatialextentoftheGNDdensityinaninfinitemedium.The

solution to (ii) is incorporated by writing an available stress function for a single

dislocation in a finite medium derived using complex Fourier transform approach

(FotuhiandFariborz,2008),equivalentlyintermsoftheGNDdensity.Theformulation

isappliedtoinvestigatelength‐scaledependentresponsestriggeredbyinternalstresses

–(a)strengtheningundermonotonicpurebendingasafunctionofdecreasingspecimen

thicknessand(b)kinematichardeningundercyclicpurebending.Wealsocompareour

results with experiments and DD simulations and propose a likely origin of variable

internallength‐scaleforinternalstresses.

Inthefollowingsection,wefirstprovidekeyequationspertainingtoourprevious

workbasedonstressfunctions(Chapter3)andthenextendtheapproachbyincluding

additionalstressfunctionsprovidingimagefields.

4.2 Nonlocal Continuum Theory with Internal Stress and

Image Fields

AsdiscussedintheprecedingChapter,thetotalstress comprisesthestressdueto

externallyappliedloads andinternalstress ∗,givenby

∶ ∶ ∶ ∶∗

( 4.1)

where isthetotalelasticstraintensor, isthecompatibleelasticstrainarisingfrom

lattice stretching due to external loading, is the incompatible elastic strain tensor

arisingfrominternalstressduetodistributeddefectsand isthe fourth‐orderelastic

stiffness tensor. Internal stresses arise from the presence of defects that could span

several orders of length‐scales. One of the common sources of these internal residual

121

stresses is the presence of ensembles of excess dislocations (GNDs) and the relevant

resolution is sometimes referred to as amesoscopic continuum (Zaiser and Seeger,

2002). At this length‐scale, it is appropriate to describe GNDs by equivalent density

fieldsthatcouldvaryspatially.Ithasbeenwell‐establishedthattheseinternalstresses

occur due to the long‐range dislocation‐dislocation and dislocation‐interface elastic

interactions(HullandBacon,2001;Mughrabi,1983;ZaiserandSeeger,2002).Aspecial

caseof the latter is thedislocation‐ free surface interaction,which is the focusof this

work.

The incompatible part of elastic strain is given by the incompatibility

condition(chapter3),

( 4.2)

whereN is incompatibility tensor (Kröner, 1959) andA is the GND density tensor14.

Then,fromEq.(4.1)and(4.2)canbewrittenas

∶ ∗ ( 4.3)

A solution to Eq. (4.3) in terms of the internal stresses arising from may be

obtained by introducing a second‐order Beltrami stress function tensor (Kröner,

1959)

∗ 21

( 4.4a)

14TheGNDdensitytensorcanbewrittenas where isplasticdisplacement

gradient(Gurtin,2002).ThenegativeandtransposeofGNDdensitytensor isoftenreferredto

asNye’stensor.(Nye,1953;ArsenlisandPark,1999)

122

where isthesecondorderidentitytensorand isaGreen’sfunction.Thecomponent

formofEq.(4.4)is

∗ 2 , 1 ,

12 ,

12 ,

(4.4b)

The elastic Green’s function G in Eq. (4.4) depends on the dimensionality and

geometryof theproblemandmay includeappropriatetermsaccounting forparticular

boundaryconditions. Kröner (1959)proposeda solutionbasedonaGreen’s function

foranelasticinfinitemedium,whichgivesinternalstressfieldsduetoGNDdensitythat

do not account for image effects. However, in systems with finite boundaries these

internal stresses need a correction in order to properly account for the long range

dislocation‐boundaryinteraction.Itisthislattercorrectiontermedasimagestressthat

isafocusofthiswork.

To incorporate the image stresses due to dislocation‐boundary interaction using

GND density fields, the internal stress tensor in Eq. (4.1) may be conceptually

decomposedas

∗ ( 4.5)

Figure4.1illustratesthisconceptuallyforabodywhoseboundaryisafreesurface.

The full problem (Fig. 4.1a) comprises a body bounded by a finite boundary

subjectedtoexternaltractions .Itisassumedthat hostsasmoothlyvaryingGND

density field where is the position vector. The problem may be conceptually

decomposedintotwoauxiliaryproblems:

123

(i)externalstressfieldsin subjectedto (Fig.4.1b),and

(ii)internalstressfieldsin inthepresenceof (Fig.4.1c).Thisproblemmaybe

further considered to be a superposition of two sub‐problems: (ii‐a) internal stress

field inadislocatedbody embeddedwithinaninfinitemedium thatgivesriseto

a spurious tractions (Fig. 4.1d), and (ii‐b) internal stress fields produced by

application of equal and opposite tractions at the boundary to eliminate (Fig.

4.1e).

Figure 4.1.Decompositionof the internalstressproblemforaspecimenhostingageneralGNDdensitydistribution.Seetextfordiscussion.

   

   

 

 

 

 

 

 

 

 

 

 

     

   

     

   

124

In accordance with the foregoing decomposition, here is calculated using the

elastic Green’s function in the presence of defects (chapter 3) providing non‐local

stress fields due to GND density in an infinitemedium, while the additional stress

arising from the GND‐boundary interaction needs an appropriate corrective kernel

function .Itisusefultomentionthatwhile isafunctionofonlythespatiallocationof

apointofinterestinthemediumfromthedislocation, isafunctionofboth,thespatial

positionofapointfromthedislocationanditsproximitytothefiniteboundary.Further,

the nature of is expected to depend on the details of the boundary, e.g. a

rigid/deformableinterfaces,freesurfaceandstraightorarbitraryboundaries.Although

aGreen’s functionbasedtreatmentprovidesanelegantapproachtosuchproblems,at

timesitmaybedifficulttoobtainanappropriatekernelfunctionfor ,e.g.anarbitrarily

curvedboundary.However,kernel functionshavebeenderived forsome fundamental

cases accounting for finiteness of domains. Some of the examples involving single

dislocations or dislocation arrays include geometries such as an elastic half‐space

(Head,1953; JagannadhamandMarcinkowski,1978;LeeandDundurs,1973;Lubarda

and Kouris, 1996b; Ma and Lin, 2001) an infinitely long strip with two parallel free

surfaces (Fotuhi and Fariborz, 2008; Moss and Hoover, 1978; Nabarro, 1978) , a

straight, rigid interface separating two dissimilar half‐spaces (Chou et al., 1975;

Jagannadham and Marcinkowski, 1980). These kernel functions aim at providing

fundamental solutions based on discrete dislocations to the problems of image fields,

but may differ based on the conceptual appeal (e.g. image dislocation versus surface

dislocations).Inwritingappropriatecorrectivestressesbasedonacontinuumanalogof

dislocations (i.e.dislocationdensity), it ispossible touse these fundamental solutions.

Themodelproblemdiscussedinthisworkisthatofpurebendingofathinfilm,whichis

represented by a constant GND density tensor and the corrective stress field is

obtained by extending the basic construct developed by Fotuhi and Fariborz (2008)

where the stress function is directly obtained using complex Fourier transformation

125

approach.Althoughthediscussionispresentedindetailforthemodelbendingproblem,

itshouldbepossibletoconstructsimilarsolutionsforotherinterfacesinteractingwith

edgeandscrewGNDdensitiesusingabove‐mentionedfundamentalkernelfunctionsor

similarapproaches(Cherednichenko,2010;VinogradovandWillis,2008).

4.3 Single Crystal Specimen under Plane-Strain Pure

Bending: Role of Free Surfaces

This sectionpresents ananalytical formulation for thenon‐local internal stresses

arisingfromGNDdensitydistributionincludingtheeffectofboundingsurfacesthrough

imagestressfields.Asmentionedintheprecedingsection,thisisachievedbyusingthe

kernel functions derived by Fariborz and Fotuhi (2008) for a single dislocation in a

boundedisotropicmedium(AppendixA).Weillustratetheproblemasfollows:consider

aninfinitedomainofacrystalwithauniformGNDdensityfield .Adiscreteequivalent

of thisproblemisan infiniteregioncontaining infinitely longequallyspacedarraysof

dislocationswithidenticalBurgersvector.Itcanbeshownthatforthisarrangementthe

averagestressfield(integrationofthestressfieldsarisingfromindividualdislocations)

overagivenregionthatdefinesamesoscopiclength‐scaleiszero.Thisisbecauseeach

individualdislocationappears tobeat the centerof theGNDarray and the individual

stressfieldscanceleachotherwhenintegrated(Eq.(4.4)).Ontheotherhand,ifapartof

such a defective (dislocated) region is embedded in a pristine (i.e. dislocation‐free)

material of same or different elastic properties, or is simply removed from the host

material, a net internal stress must exist due to (a) finite spatial extent of the GND

density, and (b) additionalLRIbetween theGNDsand theboundaries. In theextreme

situationoftheregionboundedbyfreesurfaces,suchastructurewouldrepresentathin

film thathasbeen subjected toauniformplastic curvature (Fig. 4.2).The extentover

which the internal stress is felt away from the free boundaries should depend on the

126

internal length‐scale corresponding to the long‐range dislocation stress fields and the

specimendimensions.

Figure 4.2. Schematic showing effective GND arrangement in a specimen under uniformcurvature. The specimen thickness is 2 and the GND density is described by the global, andlocal , coordinates.

Foraplanestrainconditioninthe direction,Eq.(4.5)canbewrittenas(Fotuhi

andFariborz,2008)inthecomponentformas(seeAppendixA)

∗ χ χ χ χ

∗ χ χ χ χ

∗ χ, χ χ χ

( 4.6)

where and arethecomponentsof representingdislocationswithlinelength

inthe directionandBurgersvectoralong and directions,respectively.Theχandχ

are thekernel functions representing the stress fieldsof anedgedislocationwithunit

Burgersvectorinaninfinitemediumanditscorrespondingcorrectionduetotraction‐

free boundaries, respectively (given in Appendix A). In Eq. (4.6), while the field is

translationally invariant, i.e. it does not carry the information about the absolute

2h

2l

  

 

 

127

positionofapointfromthefreesurface, thisunaccountedfeatureis incorporatedinto

theoverallexpressionbythe fieldwhichembedstherelevantspatialinformation.

Without losing the essential features in Eq. (4.6), we consider a thin film with

uniform GND density subjected to external bending moment under plane strain

condition as a model problem. Consider a structure of length 2 ( direction) and

thickness 2 ( direction) with a plane‐strain condition in the z –direction.

Assume that the structure is subjected to pure bending in the plane (Fig. 4.2).

The axis lies at the neutral plane and the coordinate is measured from this

neutral plane. A typical point in the continuum is located at , from the neutral

plane and carries a local coordinate system , with it. Further, the structure is

assumedtobesufficientlylongsothatthe endfaces(i.e.loadedfaces)andtheout‐of‐

plane faces do not contribute to image stress fields. The end faces constitute the

thickness of the structure. Since the structure is under pure bending, the in‐plane

internalshearstress ∗ mustbezeroandonlythenormalcomponentsoftheinternal

stresstensor(Eq.(4.6))exist.Assumingthattheradiusofcurvatureofthestructureis

much larger than its thickness, theGNDdensitycanbeconsideredtobeuniformover

theentirethickness.

NotethatinEq.(4.6)theintegrationisperformedoveranarea,whichimpliesthat

theinternalstressatagivenpoint,say , ,isinfluencedbytheGNDdensitiesand

their stress‐fields in its finite neighborhood. Such a non‐local representation allows

introducinganinternallength‐scale intotheproblem(Eversetal.,2004;Gerkenand

Dawson,2008).This length‐scale isproblem‐dependentandrequiressomediscussion.

Thislength‐scaleisproblem‐dependentandrequiressomediscussion.Wereturntothe

issueof themicroscopicunderpinningsofsuchan internal length‐scale laterwhenwe

compareourresultswithexperiments,butbrieflydiscuss itherewithinthecontextof

thegeometryunderconsideration.If ismuchlargerthan ,theninthepresenceofa

128

uniform GND density the surfaces would influence regions only in its proximity|

| | | | |.15Therefore,inpresenceofuniformdistributionofGNDs, onlyfeltin

athinsurfacelayerwiththicknessof duetothefinitenessofdislocationdistribution.

Similarly significantlyinfluencetheinternalstressinthesurfacelayerduetoimage

effect. Therefore, overall the internal stress variation remain zero away the neutral

plane before becoming non‐zero closer to the surfaces. Note that image field has

small contribution in region| | | |. ). Each component of internal stresses is

depictedinFigure 4.3for 0.2.

Figure 4.3.Internalstresscomponentsvariationacrossthicknessfor 0.2 5 .

15BasedontheDDsimulations,fromadislocation‐surfaceinteractionviewpoint,perhapsit

wouldbemoreappropriatetoassumeadislocation‐freezone(DFZ)ofafixedthicknessinthe

proximityofasurface,irrespectiveofthespecimenthickness(Cleveringaetal,.1999).Suchan

assumptionwouldintroduceanadditionallength‐scaleintotheformulation.However,wedonot

considerthisaspectinthecurrentwork.

-0.2 -0.1 0.0 0.1 0.2-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

xx

xx

 

Non-dimensional stress

 

2h  1 

2 3 

y 

1 

2 

3 

129

Theoverall internalstressvariationacrossbeamthickness isshowninFigure 4.4

fordifferentvalueof . It canbe seen that internal stress decreasewithdecreasing

whichmeans increasingspecimenthicknesswhere isconstant. In thecurrentwork,

the internal stress arises from interaction of GNDs among themselves and with free

surfaceswhichprovide internal stresseseven in caseofuniformdistributionofGNDs.

Justrecently,(Hayashietal.,2011)presentedsimilarinternalstressprofilesinrelatively

thick(comparedtotheirchoseninternallength‐scale)beamsunderpurebending.They

employedahigher‐orderslipgradienttheorythatinducesGNDdensitygradientsowing

toanullGNDdensityb.c.atfreesurfaces.

Figure 4.4. Variation of normalized internal stress along the normalized specimenthickness fordifferentvaluesofnormalizedinternallength‐scale .

However,theydidnotexplicitlyaccountfortheGND‐surfaceLRIthatcauseimage

effects.However,thistheoryaccountforinternalstressarisefromGNDinteractioninan

infinitemediuminpresenceofnon‐uniformGNDdistributionandconsequently image

effectsduetofreesurfacesarenottakenintoaccount.Inaddition,finiteboundariesare

not explicitly considered in this group of higher order theories and so they are not

-0.4 -0.2 0.0 0.2 0.4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

L=0.2 L=0.1 L=0.03

(y) 

 

130

capabletopredicttheinternalstressforareanearthefiniteboundaries.Therefore,for

specimenwhichthicknessiscomparableorsmallerthanlengthscaleparameter, finite

boundaries effect and corresponding image effect have to be considered.As such, the

effect of the internal stress on the overall response may be negligibly small. Very

recently, (Hayashi et al., 2011) presented similar internal stress profiles in relatively

thick(comparedtotheirchoseninternallength‐scale)beamsunderpurebending.They

employedahigher‐orderslipgradienttheorythatinducesGNDdensitygradientsowing

toanullGNDdensityb.c.atthefreesurfaces.However,theydidnotexplicitlyaccount

fortheGND‐surfaceLRIthatcauseimageeffects.

Forcaseswith theentirethicknessmayparticipateindeterminingthestress

fieldateverypointalongthethickness.Itisthislatterscenariothatexhibitsinteresting

length‐scaledependentcharacteristicsandisthemainfocusofthiswork.Consequently,

weperformthe integration (Eq. (4.6)) in the directionover theentire thickness .As

forthe direction,atleastinthepresentscenarioof ≫ ,itwouldbereasonableto

assumethat ≪ .Therefore,inthe direction,wemayrestrictourattentionovera

distance oneithersidesofatypicalpoint (fig4.2).Asshownlater, mayberelated

to the correlation distance between dislocations in a dislocated network (Zaiser and

Seeger(2002)andMughrabi(1975,2004)).Notethatifthestructurehasafinitelength

suchthat theeffectsdueto endfacesmustalsobetakenintoaccount;thiscase

isnotconsideredhere.

Returning to the case of pure bending about ‐axis, is the only non‐zero

componentoftheGNDdensitytensoranditisuniformoverentirestructure.TheDDD

simulationsofYefimovetal.(2004a)andMotzetal.(2008)alsoindicateanear‐uniform

distributionofdislocationsovermuchofthebeamthickness,exceptintheproximityof

thesurfaceswherethespacingbetweenindividualdislocationsonaslip‐planeslightly

increases. In this paper, the GND density is assumed to be uniform over entire beam

131

structure. SpecializingEq. (4.6) forpurebending,weobtainafter re‐arrangementand

normalization

∗

∗

( 4.7)

where ⁄ , ⁄ , ⁄ and ⁄ .

Since and are odd functions of (see Appendix A), ∗ vanishes

automatically and consistently satisfies the shear traction boundary condition

at .Asimilarconditionfor ∗ requiresthatitmustalsovanishat .From

Eqs.(4.7b),the and componentsof ∗ are

12 1

, , , , ,

( 4.8)

where and arethenon‐dimensionalstressesthatdependon and ,and isa

non‐dimensional spatial frequency coefficient that appears in the complex Fourier

transform solution of the . stress function (Fotuhi andFariborz, 2008).AppendixA

givesdetailedaccountofthefunctionsinvolvedinEqs.4.8band4.9b.Figure4.5shows

thevariationof and overthebeamthickness.Itcanbeseenthroughtheseplots

thatnotonlydoes ∗ vanishattheboundaries, italsovanishesovertheentirebeam

thickness as the magnitude as the distribution of exactly equal to and opposite

of ,independentof .

132

Therefore,theonlynon‐zerointernalstressis ∗ thatreflectsthecurvaturedueto

the uniform GND density distribution. From Eqs. (4.8), the and components

of ∗ readasfollows

12 1

3

, , , , ,

( 4.9)

InEqs.(4.9)thenon‐zeroGNDdensitycomponent popsoutoftheintegralasit

is independentof thespatialcoordinates for thepresentproblem. Ingeneral, theGND

densitycomponentsshouldberetainedinsidetheintegraliftheyvaryspatially.Inthat

case, the internal stresses would also be induced by smooth spatial GND density

gradients.

Figure 4.5. Variation ofnon‐dimensional stresses in direction ( and ) overbeamthickness for a given normalized internal length‐scale 10 (Eq. 4.8a,b). Note that thecomponentsareequalandoppositeresultinginoverall ∗ 0.

-1.0 -0.5 0.0 0.5 1.0-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

No

n-d

ime

nsi

ona

l str

ess

Y

sigma yy-im sigma yy-f 

 

133

ItisinformativetonotethatEq.(4.8a)and(4.9a)aretheaveragedsolutionsofthe

Volterra‐based result for a two‐dimensional array of GNDs distributed over 2

embeddedinanelasticallyisotropicmedium.Thus, internalstressexistsinthisregion

that isdelineatedby the finiteextentof theGNDdensity fieldembedded inamedium

(Eq. 4.9a). It is furthermodulated by the image stress fields due to dislocation ‐ free

surfaceinteractionsbytheadditionalterm(Eq.4.9b).Equation(4.9a)canbeintegrated

analyticallyandFig.4.6showsthevariationof with and .Forafixed , varies

nonlinearly with .Specifically, → 0as → 0and asymptotes to a constant value

for ≫ 1.

Figure 4.6.Variationof withYandL.(SeeEq.4.10a)

Ontheotherhand,foragiven , varieslinearlyoverthethicknessandchanges

fromcompressivetotensileabouttheneutralaxisconsistentwiththenotionofbending

stresses.ThislattervariationcompareswellqualitativelyandquantitativelywiththeDD

 

0

2

4

6

8

10

-3

-2

-1

0

1

2

3

-1.0

-0.5

0.0

0.5

1.

YL

134

simulation results of (Cleveringa et al., 1999) for a single crystal under pure bending,

whichalsoshowsalinearvariationofthebaselinestressfield(i.e.beforecorrectingfor

the image fields) across the beam thickness16. Unlike the calculation, the

variation(Eq.4.9b)overthethicknessrequiresnumericalintegrationandisperformed

using Gauss‐Laguerre quadrature (Press et al., 1992) 17. Appendix B provides the

detailed procedure of the numerical integration and related convergence study. As

showninFig.4.7aandb, variationwith and isqualitativelysimilartothatof .

(a) (b)

Figure 4.7.Variationof respectto(a)YatL=10and(b)LatY=1.(SeeEq.4.10b)

FromEqs.(4.9a,b),theoverallinternalstress ∗ canbewrittenas

∗ ( 4.10)

16Fig7inCleveringa’swork(1999)showsthatthe stressmagnitudelinearlyvariesfrom

0MPa(neutralaxis)to 600MPa forabeamwith 4 and 12 with

27 subjectedtoaplasticrotationof0.013radians.Weobtainthesamevaluesby

substitutingtheseparametersinEq.(4.9a).

17Apartfromthis,inEq.(4.9a,b),thePoisson’sratio cannotbeisolatedfrom .

components.Weset 0.33forallthecalculations.

-1.0 -0.5 0.0 0.5 1.0-3

-2

-1

0

1

2

3

 

 

 

0 5 10 15 200.0

0.5

1.0

1.5

2.0

2.5

3.0

 

 

 

135

where (Fig4.8).Thevalueof asymptotesto~1 3⁄ as ⁄ tends

to infinity, which indicates that the internal stress saturates in the cases where the

specimenthickness ismuchsmallercomparedtotheinternal length‐scale.Notethat

could vary because of changes in , , or both. An important question is: should

change with , and if so, how? In other words, should remain fixed, increase or

decrease with ? If changes linearly with then constant and therefore,

constant(Fig.4.8).Insuchascenario,althoughtheinternalstressisnon‐zeroitisnot

length‐scale dependent. On the other hand, decreasing (increasing) with increasing

(decreasing) indicatesthattheinternalstresswillalsodecrease(increase).Thisisalso

true if constant. Thus, except for the constant case the remaining possibilities

providethetrendexpectedfromalength‐scaledependentinternalstresstheory.Aswe

showlaterinthepaper,comparingthepredictedresultswithexperimentssuggeststhat

thepossibilityof increasingwithdecreasing isaplausiblescenario.

Figure 4.8. Variation of the normalized total internal stress with normalized internallength‐scale atspecimensurface( 1).

0 20 40 60 80 1000.0

0.1

0.2

0.3

0 5 10 15 200

1

2

3

 

 

 

  

 

‐ 

136

Equation (4.10) brings to fore a noteworthy feature in that the length‐scale

dependent internal stress shows an explicit dependence on the magnitude of GND

density.Thisisincontrasttomostcontinuumformulationsforlength‐scaledependent

internal stresses that predict a non‐zero internal stress only if GND density gradient,

ratherthananon‐zeroGNDdensity,ispresent(Eversetal.,2004;GerkenandDawson,

2008; Gurtin, 2002). Thus, the length‐scale dependent internal stressesmay exist not

onlybecauseofanon‐uniformGNDdensitybutalsobecauseoftheimagefieldsandthe

lattermaycauseinternalstressesevenif theGNDdensity isnominallyuniformovera

given finite region. From this viewpoint, the finite spatial extent of a uniform GND

densityfielddelineatedbytwofreesurfacesconsideredinthisworkservesasasimple,

yetinsightfulexampletofurtherprobetheinternalstresscharacteristicsinthepresence

ofimagefields.

Beforediscussing the results of theparticularmodelproblem, it is useful tonote

that the decomposition of the real problem in Fig. 4.1 and the treatment explicitly

presented for the bending problem bears some semblance with the superposition

techniqueusedinDDPapproaches(VanderGiessenandNeedleman(1995))(although

therearesomeimportantdifferences,discussedasfollowing).Usingthisbroadanalogy,

theoveralltraction atafreeboundarywithunitnormal maybewrittenas(Fig.

4.1)

( 4.11)

with beingthemicro‐tractioncorrespondingtotheinfinitemediumsolution(Fig.

4.1d)and beingthecorrectivetractionappliedtoremovethespurioustraction

at the free surfaces arising from fields (Fig. 4.1e). Thus, from a superposition

viewpointthefollowingtractionsb.c.’sconditionsshouldbesatisfied

137

∗ ( 4.12)

Forageneralgeometrywithprescribedmacroscopickinematicand tractionb.c.’s,

thedistributionof canbeobtainedfromslipgradients.Thetractions arisingfrom

the infinite medium assumption (Fig. 4.1d) may be calculated either analytically or

numerically. Then, the stress fields arising from corrective image tractions

(∵ ∗ applied at the free boundaries may be obtained by solving the

correspondingboundary‐valueproblemusingFEmethod.Thesuperpositionofthetwo

solutions (Fig. 4.1c) together with the stress fields from prescribed b.c.’s (Fig. 4.1b)

wouldthenprovidetheoverallsolution.

Asshowninchapter3, the internalstress ∗maybecomparedtoGurtin’smicro‐

stress that appears as an additional higher‐order quantity on a slip system that

needstobeprescribedattheboundaries.Onatraction‐freeboundary(Gurtin,2002)

∙ ( 4.13)

which indicates that the component of the micro‐stress along the normal to the

boundaryshouldbezero.Basedonthepresentwork,Eq.(4.13)maybeexpandedas

∙ ∙ ( 4.14)

where ∙ and ∙ fields represent the appropriate micro‐stress fields. The scalar

components arising from the dot products ∙ and ∙ are the

projectionsofthemicro‐tractionactingnormaltotheboundary,sothatwemayobtain

fromallslipsystems

( 4.15)

Note that Eq. (4.15) is very similar to the one arising from Eq. (4.12). Thus, the

micro‐stress construed as net Peach‐Koehler (P‐K) force density (Gurtin, 2002)

138

manifestsas image stress ata traction‐freeboundary.This canalsobeascertainedby

notingthesimilaritybetweentheinternalstressduetoaGNDdensityandtheP‐Kforce

actingonadiscretedislocationinthepresenceofexternaltractions,otherdislocations

andboundaries(Lubarda,2006)

( 4.16)

where isthePeach‐Koehlerforcevectoronithdislocation, istheBurgersvector,

is itsunit tangentvectorand is theappliedstress. is the internalstress tensor

from the jth on the ith dislocation (dislocation‐dislocation interaction) in an infinite

medium and is the image field contribution arising from the dislocation‐boundary

interactionofthe dislocationtotheP‐Kforceonthe dislocation.Inouropinion,

displacement‐based formulationswith slip gradients (orGNDdensities) as degrees of

freedom(e.g.KurodaandTvergaard,2009;Hayashi,etal,2011)donotclearlyconnect

the null GND density with the higher‐order natural b.c.’s at free surfaces. Their

equivalentrepresentationasnullGNDdensityseemstoaccountonlyforthe . termin

Eq. (4.14), because the internal stress expressions are based on infinite medium

assumptionthatdiscountthe . contribution.

Although,theforegoingexpositionisconceptuallyanalogoustoDDPapproaches,it

isusefultonoteatleastafewimportantdifferencesbetweenthetwoapproaches.First,

unlike theDDP approach thatmodels dislocations as discrete elastic singularities, the

presentsetuptreatsthemasacontinuousdensityfield.Consequently,thestressfields

areexpectedtobesmoothcomparedtotheDDPresultsastheysmearoutfluctuations

atafiner‐scale.Second,byvirtueofintegration,onlytheinformationregardingthenet

Burgersvectorisretained.Thisresultsinaccountingonlyforthecontributionsfromthe

GNDdensityandtheinfluenceduetoSSDsis lost. Incomparison,theDDPapproaches

explicitly account for both LRI and SRI contributions arising from SSDs and GNDs

139

(Cleveringaetal.,1999;GuruprasadandBenzerga,2008;YefimovandVanderGiessen,

2005). The SRI interactions from SSDs must be incorporated via phenomenological

constitutiveprescription.Notwithstanding thesedifferences, thepresenthomogenized

approach still enablesmaking connectionswith its counterpart in the DDP approach.

For example, it provides a possible description of the length‐scale adopted in coarse‐

grainedcontinuumapproaches.Likewise,as justdiscussed italsoenables interpreting

thehigher‐orderboundaryconditionsthatappearincoarse‐grainedapproaches(Bayley

etal.,2006;Gurtin,2002;Hayashietal.,2011).

With this background, the model is extended to crystal plasticity framework to

investigate the pure bending problem with reference to length‐scale effects under

monotonic and cyclic loading. The stresses due to the externally applied bending

momentsuperposewiththeinternalstressesinducedbytheuniformGNDdensitywhile

accounting for the image stress fields arising from the GND‐surface interaction to

produceanoveralllength‐scaledependentresponse.Theresultsarecomparedwiththe

micro‐beambendingexperimentsandDDsimulations(Cleveringaetal.,1999;Motzet

al.,2005;Motzetal.,2008)andindoingso,anattemptismadetoconnecttheinternal

material length‐scale with microstructural underpinnings that define plastic

deformation.

4.4 Length-scale Dependent Pure Bending Response of Single

Crystals

In this section, we analyze the length‐scale dependent behavior of an elastically

isotropic single crystal beam under pure bending. We adopt a nonlocal plastic

constitutivedescriptionforeachslipsystemwithinthesinglecrystalthatisaugmented

bytheGNDeffects.TheGND‐inducedstrengtheningmaybecategorizedasthatarising

from:(i)dislocation‐dislocationintersections,termedasshort‐rangeinteractions(SRI),

140

andisrepresentedviaTaylorhardeningmodelfortheslipsystemstrengthening(Han

etal.,2005a)and(ii)dislocation‐dislocationanddislocation‐boundaryLRIthatmanifest

asinternalstresses,describedintheprecedingsections.Forsimplicity,weassumethat

thesinglecrystalinthisplanestrainsetupisorientedforsymmetricdoubleslipwithan

angle respect to beam axis (Figure 4.2) and is subjected to total curvature . As

mentionedearlier,wealsoassume ⁄ ≫ 1andtherefore,ignoretheimageeffectsfrom

the end faces in the direction. For a single crystal that is elastically isotropic and

plasticallyincompressible,therelevantplasticstrainsunderplane‐strainpurebending

maybewrittenusingtheclassicalKirchhoffbeamtheory(Hanetal.,2005b)

, 0. ( 4.17)

where is mean plastic beam curvature and is the elastic curvature

obtainedasthecurrentstressdividedbytheelasticmodulus.Incrystallographicterms,

theplasticstrain tensor ∑ ⊗ ,where is theplasticslipon

slipsystemdefinedbytheslipdirection andslip‐planenormal .Forthesymmetric

doubleslip,theplasticsliponeachslipsystemis

2

( 4.18)

where sin istheSchmidfactor.ThecrystallographicdescriptionoftheGND

density tensor is ∑ ⊗ where . . (Han et al., 2005b).

Therefore,GNDdensitytensorisobtainedas18

18ThetransposeofGNDdensitytensor isoftenreferredtoasNye’stensor.(Nye,1953;

ArsenlisandPark,1999)

141

0 0 00 0 0

0 0 ( 4.19)

It can be seen that the only non‐zero GND density component is , which

representsthedensityofGNDswithBurgersvectorinthe directionanddislocation

lineinthe direction.Intherate‐independentlimitthefollowingconditionresultsin

plasticflow

∗ ( 4.20)

where and ∗ 2 areresolvedshearcomponentsof theexternallyapplied

and internally developed stresses, respectively and is the current total slip system

hardness that develops through dislocation‐dislocation SRI. Proposals to account for

SSD and GND induced slip system hardening include Taylor model (Nix and Gao,

1998)and its variant e.g. ~ ⁄ (Fleck et al., 1994; Evans and

Hutchinson, 2009)which has amore generalized form ~ ⁄ (Abu

Al‐Rub,2004;NixandGao,1998;VoyiadjisandAl‐Rub,2005)where isequivalently

thetotalplasticstrain, isanappropriate length‐scaleparameterand , and are fit

parameters. The hardening function embeds information regarding the

conventionalsize‐independentstrainhardeningduetoSSDs.Althoughnotthefocusof

thispaper,anotablepointisthatthereseemstobenoconsensusinthepreciserange

of ; it appears to depend on the details of the underlying formulation (Evans and

Hutchinson,2009;Husseinetal.,2008;NixandGao,1998;VoyiadjisandAl‐Rub,2005).

Note thatwith appropriate choice of parameters , and (with 2and 1

oneretrievestheTaylorhardeningmodel(NixandGao,1998).Nonetheless,asourfocus

isprimarilyontheinternalstressesarisingfromimageeffectsduetofreesurfaces,we

deferfurtherthediscussiononthismattertofutureinvestigationsandchooseasimple

Taylor‐likehardeningapproximationhere

142

1 | | ( 4.21)

where 0 is the initial slip resistance for slip system , is the ratio of hardening

modulusover the initial resistance and |. |indicates themagnitudeof plastic slip. The

secondterminsidethesquarerootmodelsthelength‐scaledependentSRIthroughslip

gradients onindividualslipsystemsthatareassociatedaneffectivemeasureofthe

GNDdensity on slip system , | ⊗ | (Han et al, 2005a). Substituting

Eq.(4.21)intoEq.(4.20)andapplyingappropriatetransformation,weobtain

1 2

| |

2

| cos |

( 4.22)

Equation ( 4.22) gives the macroscopic stress‐plastic strain relationship for the

problemthataccountsforlength‐scaleeffectsowingto(a)SRIthroughsliphardening,

and(b)LRIthroughtheinternalstress.Notably,theinternalstress(theLRIterminEq.

( 4.22))nowincorporatestheimageeffectsarisingfromthetwoboundingsurfaces.The

corresponding length‐scale is introduced through the non‐dimensional internal

stress .

Inthesubsequentsections,wepresentquantitativeresultsanddiscussthelength‐

scaledependentstrengtheningarisingfromtheLRandSRinteractionsinthepresence

of free surfaces under monotonic and cyclic pure bending. We also compare the

predicted results with recent micro‐beam bending experiments. Through this

comparison,we postulate that the characteristic internal length‐scale associatedwith

internalstressesmaynotbeafixedparameterbutcouldberelatedtoamicrostructural

parameterthatmayitselfvarywiththecharacteristicstructuraldimension.Finally,we

143

discuss length‐scale dependent Bauschinger effect in the presence of free surfaces.

Unlessotherwisementioned,weset 0.

4.4.1 Monotonic response

Tobeginwith,weinvestigatethevariationoflength‐scaledependentinternalstress

(Eq. ( 4.22)). To highlight its role in the overall response, we suppress both the SRI

contributions,bysetting 0(noSSDhardening)and 0(noGND‐inducedTaylor

hardening),implyinganon‐hardeningtypematerial.WiththeparametersinTable 4‐1,

and symmetricdouble slip systemswith angles= 45°Figure 4.9a showsnormalized

overall stress variation across the normalized specimen thickness ⁄ for

differentvaluesofnormalizedinternallength‐scale ⁄ .

Table 4‐1. Parameters used in the analyticalmodel for internal stress and prediction of beam

behaviorresponse.

Parameter Value Unit

Taylorfactor( 0.3 ‐

Shearmodulus( 47 10

Burgersvector( ) 0.255

Criticalslipresistance( 60

Thebenchmarksolution(bluedashedcurve)is forthesameproblembutwithout

the length‐scale dependent internal stress (classical crystal plasticity – CCP). As

expected the CCP response results in a constant stress over the specimen thickness

becausetheonlycontributioncomesfrom ,whichisconstantoverthethickness(Eq.

( 4.22)).However,withLRIincluded(Eq.( 4.22))theoverallstressvarieslinearlyalong

thethickness.Further,thislinearvariationbecomesstrongerwithincreasing ,i.e.with

increasing and/ordecreasing .Animportantaspectthatshouldbehighlightedisthe

144

distribution of the internal stress over the specimen thickness. In Figure 4.9a, for a

given the internal stress issimply thedifferencebetweenthe inclined line for that

andthedashedline(Eq.4.22).Theplotindicatesthattheinternalstressexistsoverthe

entire specimen thickness for the values considered in this example. This is a direct

consequence of the assumption in writing the integration limits to obtain the

internalstress(Eq.4.7).Inthecasewhere ,onemayexpecttheinternalstressto

decaytozerosomedistanceawayfromthespecimensurfaces.For ≫ ,theinternal

stresswillexistonlyasaboundarylayereffectandbezeroovermuchofthespecimen

thickness as shown in previous sections. Very recently, (Hayashi et al., 2011) showed

that non‐zero internal stresses exist only close to the surface in their specimenswith

thicknesses in the range of 25‐50 , whereas away from them the internal stresses

decaytozero.Thisisexpectedgiventhatthelength‐scalecorrespondingtotheinternal

stressesispostulatedtobeintherangeoffewmicrons.

(a)

 

 

-10 -8 -6 -4 -2 0 2 4 6 8 10-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Classical Plasticity L=1 L=5 L=20

145

(b)

Figure 4.9.a)Normalizedstressvariationacrossnormalizedspecimenthickness /at 0.05, b) Stress‐strain curves at specimen surfaces 1 for different valuesof / .

Since the overall stress varies across the specimen thickness, we consider the

stress‐strainrelationshipatthebeamsurface (Eq.( 4.22))todescribetheoverall

response. Figure 4.9b depicts the length‐scale dependent relationship

where isthetotalsurfacestrainand 2 1⁄ isthe

surface elastic strain. For a fixed the internal stress induces increasingly stronger

hardeningasspecimenthickness decreasesevenunderpurecurvatureconditions.

Conversely, fora fixed , the internal stress increaseswith increasing .Fromthe

figure4.10itcanbeseenthattheLRIcontributionincreasesas~1/ ,whichisanatural

outcomeofthepresentinternalstressformulation.

 

 

0.00 0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

CCP L=1 L=5 L=20

146

Figure 4.10. Contributionof short rangeGND interaction versus / and long rangeGNDinteractionsversus. / onflowstressat5%surfacestrain.

Havinginvestigatedtheroleofimagefieldsonthelength‐scaledependentinternal

stresses,inthefollowingsectionwemakeattempttoconnectthepresentresultswitha

recent experimental result. In the process, we also discuss the nature of the internal

length‐scale .

4.4.2 Comparison with Experiment

(Motz et al., 2005) performed bending tests on copper single crystal specimens

withan<110>{111}orientationwherethebeamaxisiscollinearwiththecrystal<110>

direction and neutral plane is along the {111} plane (Fig. 4.11). For this crystal

orientation,therearefouractiveslipsystemswithSchmidfactor0.408,whileitiszero

on the remaining slip systems (Table 4.2). Their experiments show that the flow

stress relationship formonotonicbendingof single crystal coppermicro‐beams

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

LRI strengthening Conventional CP

 

147

roughlyfollows ∝ .Wecompareourmodelpredictionswith theseexperimental

observations19.

Table 4‐2.LocalandglobalcoordinatesofactiveslipsystemaccordingtoMotzetal.,(2005)

singlecrystalbendingexperiment

SlipsystemNo.

Slipdirection NormaldirectionLocal

coordinateGlobal

coordinateLocal

coordinateGlobal

coordinate1 (0,‐1,1) (0.5,0,‐0.86) (‐1,1,1) (‐0.81,0.33,‐0.47)2 (‐1,0,1) (‐0.5,0,‐0.86) (1,‐1,1) (0.81,0.33,‐0.47)3 (1,0,1) (0.5,0.81,‐0.28) (‐1,1,1) (‐0.81,0.33,‐0.47)4 (0,1,1) (‐0.5,0.81,‐0.28) (1,‐1,1) (0.81,0.33,‐0.47)

For this,we consider the specimen geometry to be the same as in the preceding

section, butmodel two sets of conjugate slip systems as observed in theMotz et al’s

experiments(SeeFigure 4.11).

Noting Eq. ( 4.17) and the crystallographic description of plastic strain

∑ ⊗ weobtain

3 3 4

( 4.23)

19Theexperimentalsetup(Motzetal.,(2005))isthatofcantileverbendingsubjectedtoa

concentratedforceatthefreeend.Therefore,thecurvatureandGNDdensityvaryalongbeam

axis.However,mostoftheplasticdeformationisconcentratednearthecantileverroot,andin

thiswork,weassumeauniformcurvatureconditiontomimicthisregion(seefig.7ofMotzetal.,

(2005)).

148

Figure 4.11. Schematic of single crystal specimenunder pure bending, crystal orientationandcorrespondingactiveslipsystems.

The crystal orientation in the experimental set up satisfies the plane strain

conditions for plasticity, which is in agreement with our previous assumption in

derivinginternalstressconstitutiveequation.

Figure 4.12 compares the experimental variation (green curve and circles)

together with the theoretical predictions (dashed curves and symbols) with the

parameter values provided in Table 4‐120. The different dashed curves signify the

predicted variation for different values of . The blue dashed curve is the

predicted variationaccountingforonlytheGND‐SRIterminEq.( 4.22).Overthe

entire range of beam thicknesses investigated byMotz et al (2005) the predicted SRI

contribution to the overall strengthening falls short of the experimentally reported

strengthening indicating that there should be an additional contribution to

strengthening. Further, the discrepancy is accentuated with decreasing . That this

20Forcomparisonwiththeexperiment,weset 4soastoproducethesameamountof

SSDhardening(~70MPaat5%totalstrain)inordertoobtainthesize‐independentlimitofthe

macroscopicflowstressforbulkspecimens(Motzet.al,2005).

149

additionalcontributionshouldarisefrominternalstressesissubstantiatedbybending

experiments(Kieneretal.,2010)and(DemirandRaabe,2010)atsimilarlength‐scales,

but under cyclic loading,which exhibit pronounced length‐scale dependent kinematic

hardening(Bauschingereffect)duringloadreversal,alsopredictedby(Houetal.,2008)

in the DDP analysis of pure bending ofminiaturized beams. An interesting source of

deviation from the classic Taylor hardening model was recently highlighted by

Guruprasad and Benzerga (2008). They indicated that even under nominally

homogeneousloading(tensionorcompression),presenceoflocalGNDstructuresatthe

micro‐scale may cause additional hardening at small length‐scales, especially over

moderate to large strains. They also proposed an augmented hardening model to

accountforsuchhigher‐ordereffects.

Notethat for thevaluesof consideredinFigure 4.12,eachcurvecompareswell

with theexperimentonly inaparticular regimeofbeam thickness. Forexample,with

1 (red curve) thepredicted corroborateswellwithexperimental values for

large beam thickness 3.5 . However, for thinner beams the predicted

strengthening is much lower than the actual value. Interestingly, for beams with

3.5 oneobservesbettercorroborationwiththeexperimentforlargervaluesof

such that at the smallest 1 the predictions with 10 corroborate very

well with the experiments. Based on this comparison, it may be postulated that the

evolution of internal length‐scale changes with decreasing specimen thickness. It

becomes imperative to seek a plausible explanation for such a dependency, which is

discussedinthesubsequentparagraphs.

150

Figure 4.12.Comparisonoftheanalyticalresults(Eq.4.17)fordifferentvaluesof withtheexperimentalresultsofMotz,etal(2005).

Basedonthiscomparison,itmaybepostulatedthattheinternallength‐scaleitself

increaseswithdecreasingspecimenthickness.Itbecomesimperativetoseekaplausible

explanation for such a dependency,which is discussed in the subsequent paragraphs.

Onephysicalinterpretationoftheinternallength‐scale thatgovernsinternalstresses

in amesoscopic continuum is a correlation length‐scale emerging from the collective

behavior of dislocations21. (Groma, 1997, 2003; Groma and Bako, 2000; Groma and

21ZaiserandSeeger(2002)introducedmesoscopicinternalstressesarisingfromlong‐range

interactionsofdislocationensembles,whichvaryonthecharacteristicscaleofthedislocation

densityvariationincomparisontomicroscopicinternalstressesinthevicinityofasingle

dislocation.Basedonthisdefinition,theinternallength‐scaleassociatedwiththemesoscopic

Specimen thickness

 

1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

1000 Experimental results (Motz, et. al. 2005) l

c=10 m

lc=3 m

lc=1 m

MSG-CP theory (Han, et. al. 2005) Conventional CP

151

Bakó, 1998; Zaiser and Aifantis, 2003; Zaiser et al., 2001; Zaiser and Seeger, 2002).

(HahnerandZaiser,1997)proposedascalinglawforcorrelationlength,whichgivesa

correlation length around 1 for deformed Cu in stage‐II hardening at room

temperature. For a given specimen, the initial may be defined by the starting

dislocation substructure or the initial dislocation sourcedistribution. Itmay evolve as

thedislocationsubstructureevolvesduringdeformation.Basedontheseideas,wemay

consider the relevant length‐scale to be the correlation distance over which

dislocationensemblesinteractthroughtheirstress‐fields.Conventionally isdescribed

intermsofmultiplesoftheaveragedislocationspacing , i.e.~ ⁄ ,where isthe

overall dislocation density (Weiss and Montagnat, 2007; Zaiser and Seeger, 2002).

Mughrabi(1975,1983,2001)suggestedthat shouldbe intherangeof10‐100times

the average dislocation spacing . On a given slip‐planewith dislocations arranged in

parallel (Figure 4.13), the average on that particular slip‐plane can be obtained

directly from their arrangement. However, normal to the slip‐plane the dislocation

spacing is set by another length‐scale , which is the distance between two adjacent

slip‐planes. The existenceof such aneffective slip‐plane spacing stems from the fact

thatnot all theplanes in aparticular slip systemareactivatedduring initial stagesof

deformation. Indeed, this natural length‐scale emerges from activation of dislocation

sources(e.g.Frank‐Readsources)onafewofthepotentialslip‐planes–aprocessthatis

statisticalinnature(Yefimovetal.,2004b).

continuumwassuggestedtobeoftheorderoffewmicrons(alsoseeWeissandMontagnat

(2007)).

152

Figure 4.13. Typical GND arrangement in double symmetric slip deformation under purebending.

Assumingforourcurrentproblem(Figure 4.13)thatthereexist dislocationsin

anarea where istheprojectedslip‐planespacing,theGNDdensitycomponent

canbewritteninadiscretesenseas

( 4.24)

where ⁄ on a given slip plane. For crystalline materials, (or ) is in the

rangeoftensofnanometers22(Deshpandeetal.,2005).Mesarovicetal(2010)showed

that appearsasanaturalinternallength‐scaleinthecontinuumdescriptionofinternal

stress by invoking the thermodynamic coarsening error between the discrete and

22 / cos istheprojectionoftheactualslip‐planespacing ,whichis~100 .

153

continuum descriptions of a GND density field. From a kinematic coarsening

perspective, if we smear out the dislocation arrangement along the slip‐plane by

adopting the notion of continuously distributed dislocations, but retain the discrete

nature of distribution across it, then the correlation length‐scalemaybedescribed as

(Mughrabi,(1975),ZaiserandSeeger(2002)andMesarovic,etal(2010))

( 4.25)

where ~10 100(Mugharbi, 1975) may be viewed as the number of slip planes

around a given point that influence the stress field at that point. For example,

if 30 , we obtain ~300 3 from Eq. ( 4.25), which is in the range

suggestedbyvariouscontinuumandDDPapproaches(Deshpandeetal.,2005;Hahner

and Zaiser, 1997; Weiss and Montagnat, 2007; Zaiser et al., 2001). Note that the

assumption of discreteness of slip‐planes may be relevant in a scenario where a

specimen initially hosts sparsely distributed dislocation sources, which would also

indicate activation of fewer slip‐planes giving a larger .Recent explorations in small

scalecrystalplasticityhaveresultedinproposalsforscale‐dependentplasticitythatrely

onnovelmechanismssuchasdislocationstarvation(GreerandNix,2006;Greeretal.,

2005),exhaustionhardening(Benzerga,2009)andsourcetruncation(El‐Awadyetal.,

2011; Kiener and Minor, 2011; Parthasarathya et al., 2007). It is likely that these

mechanismsmaycouple intothedescriptionof internalstressesthroughthenotionof

internal length‐scale. Although we do not explicitly account for such mechanisms,

comparison of our calculations with experiments provides an interesting perspective

fromtheviewpointofinternalstressesthatmayconnecttotheseproposals.

Now, based on our postulate that increases with decreasing (Figure 4.12), a

larger for smaller specimen means that either is larger for fixed ,or that more

numberof slip‐planesparticipate indetermining the internal stress at apoint. Three‐

dimensionalDDDsimulationsofsinglecrystallinebeamunderbendingforbothsingle

154

anddouble‐symmetricslip(Motz,et.al.(2008))showthatforsmallerbeamthicknesses,

fewerdislocationsourcesareactivatedgivingalowerinitialdislocationsourcedensity

thanforathickerbeam.Inthesamesimulations,plasticslipinthinnerbeamsoccursin

localizedslipbandsthatarespacedmuchwiderapartcomparedtothoseinthethicker

beamswhereslipbandsarecloseenoughtoresemblearelativelyhomogeneousplastic

slip. Asmentioned in the preceding paragraph, from the viewpoint of Eq. ( 4.24) and

Figure 4.12, this suggests that below a certain value of , for a fixed GND density a

reduction in theaveragedensityofdislocationsourcesshould indeed lead to theslip‐

planes being spaced wider apart, i.e. larger with smaller . This observation is also

consistentwiththenotionofsource‐limitedplasticityingeometricallyconfinedsystems

(El‐Awadyetal.,2009a;Espinosa,2005;Parthasarathyetal.,2007);Shi,et.al.(2004)).

FromFigure 4.12, (andtherefore, ) remainsnearlyconstantabove ~4 ,but

increasesdramaticallybelowthatvalue.

It is interesting to consider an allied length‐scale dependency from the

perspective introduced by Chakravarthy and Curtin (2010). For specimen under

uniaxial tension, their DD and continuum analysis showed that the amount of

strengthening increases with increasing ratio of the obstacle to dislocation source

density ⁄ forafixedspecimenthickness.Inthepresentcontext,thismaybe

viewed as follows: For a fixed , Eq. ( 4.24) requires that for a fixed (i.e. fixed

curvature) thenumberof dislocations in thepile‐upover the region must

increaseforalarger .Inotherwords,thereshouldbemoredislocationsperslip‐plane

to accommodate the same curvature. This is also akin to saying that the density of

obstacles to dislocation nucleation or motion per slip‐plane is higher, because

each new dislocationmust overcome the back stress produced by existing obstacles.

Further, higher means more widely spaced slip‐planes concomitant to fewer

dislocation sources, i.e. lower .With this, it can be postulated that the argument of

higher producing larger strengthening due to enhanced internal stress is also

155

consistent with the results of Chakravarthy and Curtin (2010). In the context of the

present problem,we obtain that for two specimens experiencing same curvature, the

specimenwiththesmallerthicknessexhibitshigheroverallinternalstressduetowider

slip‐planespacingthatcoupleswithmorenumberofdislocationsperslip‐planeacting

asobstaclestodislocationmotion.

In summary,we posit that the proposed origin of the inverse dependence of the

internal length‐scale corresponding to the internal stresses on the specimen size

stems from thepaucity of active slipplanes (arising from lackof sufficientnumberof

sourcesand/ortruncatedsources)thatdeterminetheiraveragespacing .Wenotein

passing that from Figure 4.8, this also means that the internal stress increases non‐

linearlywithdecreasing .

4.4.3 Length-scale Dependent Bauschinger Effect

Finally, we investigate the cyclic pure bending behavior of the single crystal

specimenusingdevelopedcontinuumframeworkintheprevioussection.Inparticular,

we mimic a single cycle comprising forward bending leading a prescribed plastic

curvature followedbystraightening tooriginalundeformedgeometry,whichhasbeen

experimentallyreportedtoexhibita length‐scaledependentBauschingereffect(Demir

andRaabe,2010).InEq.( 4.22)weset 0duringtheforwardloading.

Figure 4.14showsbending‐strengtheningresponsesofthesinglecrystalspecimens

underdoublesymmetricslipconditionfortwothicknesses, 1 and10 .There

areseveral interestingfeaturesthatcanbeextractedfromthisfigure.Firstly, itcanbe

seen that the length‐scaledependent internal stressescausemoresevereBauschinger

effect (yield asymmetry) for the thinner specimen than for the thicker specimen

(Kiener,(2010);DemirandRaabe(2010);Hayashi,et.al.(2011);Houet.al(2008)and

KurodaandTvergaard(2008)).Duringforwardloading,theinternalstressactsasback‐

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stress that resists plastic deformation and manifests as macroscopic strengthening,

whileduringstraighteningthesameinternalstressassistsplasticdeformationcausinga

decreaseintheyieldstrength.DemirandRaabe(2010)referredtothisasamechanical

Bauschingereffect,whichisduetointernalstressesthatarisefrompolarizeddislocation

(GNDs) arrangement during forward bending. Secondly, the hardening rate in the

reverse plasticity is higher for thinner specimen. Finally, for a given thickness the

hardening rate in reverseplasticity is lower than in its forward counterpart. This last

aspectcanbeexplainedasaconsequenceoftheidealizedscenarioofnoSSDhardening

andvanishingGNDdensitywithdecreasingcurvatureasassumedinouranalysis.

Figure 4.14. Bending‐straightening cyclic response of single crystalline specimenorientedfordoublesymmetricslip

CorrespondingtoFig.4.14,theGND‐inducedSRIandLRItermsarenonzeroatthe

point of reverse yield and are equal to those at the strain atwhich unloading begins.

0.000 0.005 0.010 0.015 0.020

-6

-4

-2

0

2

4

6

lc=10 m, t=1 m

lc=1 m, t=10 m

CCP

 

 

157

These contributions gradually decrease with decreasing curvature until they vanish

when the specimen attains its original shape. Consequently, the overall stress also

becomesequal to theyieldstressdeterminedby (Eq.4.22). Inrealityhowever, the

presence of SSD density and the remobilization of GNDs as SSDs due to decreasing

curvature (Demir and Raabe, 2010) may produce a higher hardening rate in the

straighteningstage,whichisnotcapturedhere.Inadditiontheyobservedanenhanced

hardening during straighteningwith decreasing specimen thickness. This likely arises

because the GND density that evolves to accommodate increasing curvature during

forward loading progressively decreases during the straightening phase and this

accumulateddensitymanifestsitselfasincreasedmobiledislocationdensitythatmayin

turn enhance the overall hardening in the flow regime. In Figure 4.14, this effect is

accountedfor inthestraighteningphasewithintheTaylorhardeningterminasimple

waybyassumingthatthefractionalreductionintheGNDdensityisthesameasthatof

the fractional increase in the SSD density. The GND‐induced SRI and internal stress

termsarenonzeroatinitialstagesofunloading,butgraduallydecreasewithdecreasing

curvature until it reaches zero when the specimen attains its original shape.

Concurrently,thehardeningduetoremobilizedGNDdensitytermiszeroatinitialstage

ofunloading,butevolveswithstressreversal.

Figure 4.15 shows the overall stress variation across specimen thickness at

different levels of surface strain indicated by the open circles in Fig. 4.14.While CCP

(Figure 4.15a)exhibitsidenticalvariationoverthethicknessunderforwardandreverse

loading(noBauschingereffect),thelength‐scaledependentresults(Figure 4.15bandc)

showincreasinglystrongerasymmetrywithdecreasedspecimenthickness.Asexpected,

for the latter cases the stress distribution under forward loading shows increasingly

highersurfacestresswithdecreasingthicknessandalowersurfacestressunderreverse

loading.

158

(a)

(b)

(c)

Figure 4.15.Overallstressvariationacrossspecimenthicknessatdifferentstrainshowninfigure4.14.

Further,thestressdistributionoverthespecimenthicknessatthepointofreverse

yielding strongly depends on and (blue curves). Note that at the point of reverse

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

 

 

 

 

 

 

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

 

 

 

-6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

159

yielding, theCCPresult showsa linear tension to compression transitionbetween the

neutral axis and the free surface. With decreasing and concurrently higher the

inflection point moves toward the surface to the extent that it may even disappear.

Figure 4.14and4.15revealthatsurfacesmaydeformplasticallyuponreversalloading

even with a positive stress. Figure 4.16 displays the contributions from length‐scale

dependentisotropic(dissipative)andkinematic(energetic)hardeningmechanismsfor

twodifferentspecimenthicknessesincyclicbending‐straightening.

Figure 4.16. Length‐scale dependent dissipative (isotropic) and energetic (kinematic)hardeningcomponentsofpurebendingresponsesfortwodifferentspecimenthickness

Since the SSD hardening is ignored here, the length‐scale independent plastic

dissipationisonlyduetoperfectplasticity.Therefore,thehardeningbehaviorobserved

in the figure is fully ascribed to the length‐scale dependent dissipative and energetic

components of GNDs. Inset in Fig. 4.14 shows that the isotropic hardening can be

obtained by deducting the initial yield stress (OA) from subsequent yield stress

0.000 0.005 0.010 0.015 0.020

-6

-4

-2

0

2

4

6

 

 

                 

LengthscaledependentplasticdissipationLengthscaleindependentplasticdissipationLengthscaledependentIrreversiblestoredenergyLengthscaleindependentreversiblestoredenergy

 

 

 

 

 

 

Isotropichardening

Kinematichardening

160

(BC/2).Thekinematichardening contribution isobtainedbydeducting from the

stressatthepointwheretheloadisreversed(pointBonthecurve).

Fromathermodynamicviewpoint,writingthetotalinternalenergy(seechapter3)

identifythedifferentcomponentsofenergyinvolvedinplasticdeformationas

:

∗:

∗

( 4.26)

The first term in Eq. ( 4.26) represents the length‐scale independent reversible

storedenergyassociatedwithexternallyappliedloads.Thesecondtermisreferredtoas

the length‐scale dependent irreversible stored energy as it is associated with the

internalresidualstressand incompatibleelasticstrainthatwill tendtoreorganizethe

GNDdensityfromanenergeticallyefficientconfiguration.ThethirdterminEq.( 4.26)is

theplasticdissipationduetotheSSD23(length‐scale independent isotropichardening)

andGND(length‐scaledependentisotropichardening)densities.

It can be seen than regardless of beam thickness, the contribution of energetic

hardening is higher than the dissipative onewhich is depicted in figure 4.10 aswell.

Furthermore, higher irreversible energy store and dissipate in thinner beam due to

polarizedGNDdistributionwhichprovide the length‐scaledependent responses.Note

that length scale dissipative energy arises from short‐range interaction of GNDswith

SSD and causes isotropic hardening, while long‐range elastic GND‐GND and GND‐

23Intheresultspresentedinfigure4.14and4.15,theSSDhardeningisneglectedbysetting

C=0inEq.(4.17)andplasticenergyisdissipatedonlybecauseofshortrangeinteractionofGNDs

withSSDs

161

boundaryinteractionscauseirreversibleenergy,sometimesreferredtoasdefectenergy

(Gurtin,2002)andcausekinematichardening(Bauschingereffect).

4.5 Summary and Outlook

Thischapterpresentsanelasticitybasedapproachtoaccountforimageeffectsdue

to GND‐free surface interaction producing length‐scale dependent internal stresses in

crystalplasticity.The approachdevelopedhere isgeneric in that the internal stresses

fromGNDdensityinaninfinitemediumiscorrectedbyanadditionaltermthataccounts

forGND‐boundarylong‐rangeinteraction.Thelatterexplicitlydependsonthenatureof

theboundaryandtheparticularcasetreatedhereisthatofafreesurface.Itshouldbe

possible to apply the concept to other continuum analogs based on fundamental

solutionssuchaselastichalf‐spacesandinterfacesbetweendissimilarmaterialsunder

homogeneousornon‐homogeneousdeformations.

The pure bending example shows that finite spatial extent of the GND density

produces net long‐range elastic interactionwhen image fields are taken into account

thatmanifestas internalstressesevenwhentheGNDdensityisuniformlydistributed.

Theresultsshowthattheseeffectsbecomeincreasinglyimportantasthecharacteristic

specimen size approaches the internal material length‐scale. If the specimen size is

much larger than the internal length‐scale the internal stresses are still non‐zero, but

onlyintheproximityofthefreesurfaces.Aproposalthatstemsfromthepresentwork

is thenatureof the internal length‐scale contributing toward internal stresses.This is

expressed in terms of the average slip‐plane spacing through the dislocation density

argument. A comparison of the model predictions with experimental results on

monotonic micro‐beam bending suggests that the initial internal length‐scale should

increasewithdecreasingcharacteristic specimensize.Thisproposal is rationalizedby

appealingtorecentexperimentalandcomputationalresultsonminiaturizedspecimens

162

subjected to homogeneous or non‐homogeneous loading. Cyclic responses of the

specimens under pure bending and straightening are also explored and the

experimentallyobserved length‐scaledependentBauschingereffect iscaptured.Again,

it is important to note that this effect stems from the GND‐free surface long‐range

interaction. The resulting contributions of the isotropic and kinematic hardening are

quantifiedinthestress‐strainresponses.

In closing,wewould like tomention the recentworkbyCherednichenko (2010)

that introduces the idea of continuum representation of image fields, although the

underlying mathematical representation is different from ours. In comparison to the

presentwork thatrelieson the theoryofelasticityandkinematic incompatibility, that

workconstructsimagefieldsthroughstatistical‐mechanicsbasedensembleaveragingof

dislocations (Groma, 1997; Groma and Balogh, 1999; Yefimov et al., 2004), similar to

that of Vinogradov and Willis (2008). The resulting elegant formulation embeds an

enriched nonlocal constitutive law that tracks plasticity through dislocation density

evolution includingnucleation,whileaccounting for imageeffects.Although the image

fieldsconstructedthereare foradifferentboundaryvalueproblem(simpleshearofa

constrainedthinstriparisingwheredislocationspile‐upatthehardboundary),itwould

beinterestingtodrawbroadcorrelationsbetweenthesetwoapproaches.

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5 Summary and Recommendations

In thischapter,wesummarize thekeycontributionsof this thesisandprovidean

outlookintothefuturedevelopmentsalongthechosendirection.

5.1 Summary

Classical continuum crystal plasticity theories are successful to predict inelastic

behaviorofmaterialinmacroscopicscalewhileitfailstocapturesizedependencyofthe

material when sub structural counterparts or specimen sizes are comparable with

microstructuralcharacteristic length.Therefore, toexplainsomeof theexperimentally

observed length‐scale responses in crystalline materials, traditional continuum

mechanics of plastic deformation is augmented with a variety of dislocation related

mechanisms. In this thesis, we explore the length‐scale dependent behaviors of FCC

crystallinematerials in terms of GND interactionmechanisms at different scales. The

threeGNDrelatedmechanismsinvestigatedinthisthesisaresummarizedasfollows:

(i) Short‐range interactionbetweenGNDs and SSDs is considered in chapter 2 in

termsoflength‐scaledependentTaylorhardeningmodel.Thisphenomenonismodeled

using the Mechanism‐based Slip Gradient Crystal Plasticity (MSG‐CP), which is

implemented by the author as a user‐material subroutine (UMAT) within

ABAQUS/STANDARD®.Usingthisimplementation,thelength‐scaledependentbehavior

ofmodel single crystalMMC architectures under thermal andmechanical loading are

investigated.Specifically,roleofinitialthermalGNDdensityonthesubsequentresponse

undermechanical loading is investigated for certain crystal orientations. The length‐

scale dependent thermo‐mechanical response is modeled as a two‐step procedure

withinfiniteelementanalysisthatcloselyfollowstheactualsynthesisroute.Theresults

164

exhibit the characteristic tension‐compression asymmetry arising from pre‐existing

internal(thermally‐induced)residualstressesthatisnowlength‐scaledependent.

(ii)AnotherimportantresultthatemanatesfromtheMSGCP‐MMCinvestigationsis

the negligible effect of inclusion shape (whether round or sharp‐cornered) at small

sizes.Thisisbecausethehightriaxialitythatexistsinthematrixsurroundingthesharp

corners issmearedoutbythehardenedGNDzonearoundthe inclusions that isabout

thesamespatialextentasthestressconcentrationatcorners.

(iii) The MSGCP implementation is also used in systematic simulations of

polycrystalline MMC architecture to delineate the interaction strengthening

contributionasarisingfromthegrainsizeandtheinclusionsizeinteraction.Thiseffect

hasbeentraditionallyignoredinMMCsimulationsto‐date.Oursimulationspredictthat

theinteractiontermappearsasHall‐Petchtypecontribution.Basedonthisobservation,

weproposeananalyticalsolution for this interactioneffectbasedon the ideathat the

inclusion‐grainintersectionsactasspecialsourcesthatemitadditionalGNDsandthese

GNDspile‐upatgrainboundariesresultinginadditionalstrengthening. .Theproposed

analyticalmodelisdeemedamenabletohomogenizedapproaches.

(iv) The thesis then expands its scope to address the length‐scale dependent

internal stress arising from long‐range GND‐GND interaction. A nonlocal continuum

crystalplasticityisdevelopedtoaccountfortheseresidualstressesarisingfromanon‐

homogeneous distribution of GND density. The thesis proposes a stress functions

approach to analytically derive the length‐scale dependent 3D internal stress tensor.

This internal stress tensor is incorporated into continuum framework using

thermodynamiclaws.

(v) The model examples treated semi‐analytically with the aforementioned

formulation highlight the importance of the internal stress on the length‐scale

dependent strengthening and hardening in single crystals. The tapered single crystal

165

example that approximately mimics a micro‐pillar subjected uniaxial loading

substantiatesthatofgeometricimperfectionsinsuchstructurescausestronggradients

intheGNDdensityand leadtoastrengtheningoftheoverallstress‐strainresponsein

specimensthataresubjectedtonominallyuniaxialmacroscopicloads.Thesimpleshear

of a single crystal lamellamimics a nano‐twinwithina grain thatundergoes shearing

deformationandtheinternalstress‐inducedstrengtheningofaslipsystemexhibitsHall‐

Petch typebehavior. Interestingly, at small structural sizes the strengthening tends to

saturate.

(vi) Finally, the thesis addresses the long‐range interaction between the GND

densityandtraction‐freeboundaries.Buildinguponthestressfunctionsapproach,the

internal stress formulation incorporates additional image stresses due to this

contributionthroughanadditionalstressfunction.Theresulting internalstresstensor

representing image fields is length‐scale dependent. In analyzing a model micro‐

architecturecomprisingasinglecrystal thin filmunderpurebending, it isshownthat

internalstressesalsooccurundernominallyuniformcurvatures.Theformalismreveals

thatthemicro‐tractionboundaryconditionsintheformofPeach‐Koehlerforcedensity

due toGNDdensity invoked in thework of Gurtin (2002) can be construed as image

stresses imposed due to GND‐surface long‐range interaction. A comparison with the

experimental results suggests that internal length‐scale is a functionof structural and

microstructural dimensions and may increase with decreasing specimen dimensions.

The structurally dependent internal length‐scale is explained in terms of the recent

proposals of paucity of dislocation sources and enhancement of obstacle densitywith

decreasingstructuraldimensions.

166

5.2 Recommendations for future work

The work compiled in this thesis may serve to provide a basis to variety of

problems in themechanics and physics of length‐scale dependent plasticity. Some of

theseareidentifiedanddiscussedhere:

(i)IntheMMCmodeling,wehaveassumedhighlyidealizedmicrostructuresinthe

form of grain shapes and inclusion arrangement. Further, it may be useful to mimic

microstructures that are closer to the real microstructures. A simple, yet plausible

approximation could be that of hexagonally‐shaped grains rather than the square‐

shaped approximations chosen in this work. The former would be a better

approximation, because typically one finds triple junctions in real microstructures

unlikethequad‐junctionsinthemicrostructurewithsquaregrains.

(ii) The length‐scale dependent MMC behaviors including thermo‐mechanical

loading conditions may be extended to polycrystalline microstructures. This would

require invoking the elastic anisotropy of the crystals to render differential stresses

acrossgrainboundaries,whichisalreadypresent intheMSGCPUMATdevelopedhere

butwasnotusedforsimplicity.

(iii) The area of hierarchical MMCs (Joshi and Ramesh, 2007) is an exciting

direction to use the MSGCP approach. Several local and global features of such

composite‐within‐compositearchitecturescanbeinvestigatedwithhighresolutionasa

functionofjudiciouslyarrangedtopologieswithvaryinggrainsizes.

(iv) Our MMC investigations do not invoke higher‐order boundary conditions,

whichwould be necessary to capture enhanced interactions between the dislocations

and interfaces. Therefore, it would be worthwhile to endow the current crystal

plasticity code with higher‐order gradients. Within ABAQUS, this would entail

developing a full‐fledged user‐element (UEL) that would enable calculating second

167

gradients of plastic slip (i.e. first gradient of the GND density), of which the current

UMATwouldbeapart.

(v)Thestress functionsapproachtoaccount for internalstresseswhileattractive

and insightful, has limitations that explicit solutions are possible only for a limited

numberofcasessuchasthosewithregulargeometry,idealboundaryconditions,elastic

isotropy.Although,wehaveprovidedabriefoutlineoftheextensionofthisapproachto

elastically anisotropic materials, a complete solution to a boundary value problem is

expected to be quite complicated. Hence, it may be useful, or perhaps necessary, to

developafiniteelementapproachthatcalculatestheseinternalstresses.

(vi) In thecontextofstress functionapproachwhichhasdeveloped inthis thesis,

some fundamental caseswhich are interesting for largemechanics community canbe

investigated where appropriate Green functions are available or can be derived. For

instance,

‐ Torsion of single crystalline micro wire comprising uniform distribution of

screwdislocation.

‐ MMC unit cell comprising circular inclusion and crystalline matrix where

internal stresses develop at the inclusion/matrix interface due to dislocation

pilesup.

‐ Microcompressionoftaperedpillaraccountingforimageeffects.

(viii) Alternatively, the surface dislocationmodel proposed by Jagannadham and

Marcinkowski (1978) can be incorporated to account for image effect while simple

infinitemediumgreenfunctionsareused.

(viii) In the case of complicated topologies and non‐ideal higher‐order traction

boundary conditions, the stress function approach is not a very useful approach to

168

obtain image fields. A brief discussion and a possible approach are outlined in the

closingsectionofChapter4,whichshouldprovideadirectionforfuturework.

169

6 List of Publication

ArticlespublishedinJournals:Aghababaei,RandJoshi,SP(2011)Grainsize–inclusionsizeinteractionin

metal matrix composites using mechanism‐based gradient crystal plasticity.International Journal of Solids and Structures, 48 (18) 2585‐2594.

Aghababaei,R,Joshi,SPandReddy,JN(2011)Nonlocalcontinuumcrystalplasticitywithinternalresidualstresses.JournaloftheMechanicsandPhysicsofSolids,59,713–731.

Aghababaei, R and Reddy, JN (2009), Nonlocal Third‐Order Shear

DeformationPlateTheorywithApplication toBendingandVibrationofPlates.JournalofSoundandVibration,326,277‐289.

Articlesinpreparation/submission:

Aghababaei,RandJoshi,SP.ACrystalPlasticityAnalysisofLength‐scale Dependent Internal Stresses with Image Effects (under review inJournaloftheMechanicsandPhysicsofSolids).

Aghababaei, R and Joshi, SP. Length‐scale dependent compositeresponse induced by thermal residual stresses (manuscript inpreparation)

Conferencepresentations:

Aghababaei,RandJoshi,SP(2011)GrainSize‐InclusionSizeInteractioninMetal Matrix Composites at Moderate Strains. International Conference onMaterials forAdvancedTechnologies, ICMAT, (June26‐July 1, 2010), Singapore.Aghababaei, R, Joshi, SP and Reddy, JN (2010) A Nonlocal ContinuumTheoryAccounting for SizeDependentBauschingerEffect.9thWorldCongresson ComputationalMechanics and 4th Asian Pacific Congress on ComputationalMechanics,WCCM/APCOM2010 (19 – 23 July 2010), Sydney, AustraliaAghababaei, R and Joshi, SP (2010) A Nonlocal Continuum TheoryAccountingforSizeDependentBauschingerEffect.16thUSNationalCongressonTheoreticalandAppliedMechanics,USNCTAM(June27‐July2,2010),PennStateUniversity, Pennsylvania, USAAghababaei R, Joshi, SP and Zhang, J (2010) Length‐Scale DependentResponseofHierarchicalCompositesusingEnrichedPolycrystalPlasticity.16thUSNationalConCongressonTheoreticalandAppliedMechanics(June27‐July2,2010),PennStateUniversity,Pennsylvania,USA.

170

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189

Appendix A. A Note on Continuum Descriptions of

GND Density Tensor

Strain gradient (nonlocal) theories invoke the existence of excess dislocations

commonly referred to as the geometrically necessary dislocations (GNDs) that are

necessarytomaintaingeometriccompatibilityduringplasticdeformation(Ashby,1970;

Nye, 1953). Hence a continuum description of dislocations is necessary to explicitly

account their effects into continuum theories. Here, wemake a comparison between

different continuum descriptions of GNDs corresponding to the different basis which

researchersusedinthisfield.

(Nye, 1953) first presented the tensorial form of the GND density in continuum

framework, now commonly referred to as Nye dislocation density tensor. This

dislocationdensitytensoriswrittenintermsofscalardislocationdensityas

⊗ α⨀

α ⊗ α (A.1)

where the subscriptN indicates Nye’s definition. and ⨀ are the scalar edge and

screwGNDdensityonslipsystem respectively, istheBurgersvectormagnitudeand

αand αareunitvectorsinthedirectionofBurgersvectoranddislocationlineforslip

system , respectively.Note that thisexpansiondependson the choiceofbasisaswe

showlater,butforanyprescribedbasisthescalardensitiesareunique.

Sincethecontinuumdescriptionof thescrewdislocations is thesame indifferent

conventions,withoutlossofgenerality,weonlyconsideredgeGNDsinremainingpartof

this article for only one slip system. Clearly, the discussion can be generalized for

multipleslipaswell.

190

Assuminga local coordinateonadislocationshown inFigureA.1associatedwith

slipsystemwithslipin directionandnormalin direction.

FigureA.1EdgedislocationinlocalandglobalcoordinatesbasedonNyedefinition

For a dislocation shown in Figure A.1, Eq. (A.1) iswritten in terms of the global

coordinatesas

⊗ (A.2) 24

wherebistheBurgersmagnitudes.Theonlynon‐zerocomponentofGNDdensitytensor

isobtainedas .

Nye’s definition of positive and negative edge dislocations under plane strain

assumptionwheredislocationlinesareinplaneareshowninFig.A.2(Nye,1953)

24Forasingleedgedislocation, , where , istwodimensionalDiracdelta

function.

191

FigureA.2.PositiveandnegativeedgedislocationsaccordingtoNye’sdefinition(Nye,1953)

Based on this definition of dislocation density tensor, the relation between the

dislocationdensityandlatticecurvatureisobtainedas

12

(A.3)

where istheincompatibleplasticlatticecurvature.Notethatelasticlatticecurvature

isnotconsideredinEq.(A.3)asinitiallyderivedby(Nye,1953).

Inchapter3wedecomposedthetotallatticecurvatureintermsofcompatibleand

incompatible terms.Forexample, considerplanestrainbendingofacrystalline lattice

showninFig.A3.Nyerelationiswrittenas (Eq.(A.3))whereboth and

are negative continuum quantities while corresponding Burgers vector is

positive(FigureA.2).

FigureA.3.Nye’sdefinitionofedgedislocationunderplanestrainassumption

x

y

z

b (1,0,0)

b(-1,0,0)

R

192

TheGNDdensitytensorgenerallycanberelatedtothestraingradient.Considera

smooth surface bounded by a closed curve . The net Burger’s vector of the

dislocationspiercingthrough isdefinedby(usingStokesformula)

∙ (A.4)

where isunitnormalto and iselasticdeformationgradient.UsingStokesformula,

theGNDdislocationdensitytensorisobtainedas

(A.5)

Inthecaseofsmalldeformationtheory,theelasticdeformationgradientiswritten

intermsofdisplacementgradientas .Then,Eq.(A.5)maybeapproximated

by

(A.6)

ThisconventionforGNDdensitydescriptionisusedby(Ashby,1970)and(Forest,

2008)aswell.ArsenlisandPark(1999)adoptedasimilarnotation,butwiththesmall

difference that the dislocation line is defined in the opposite sense to that of theNye

definition.TheyrewroteEq.(A.2)as ⊗ ,whichresultsinthenon‐zero

componentsgivenas .Similarly,latticecurvature‐GNDdensityrelation

whereelasticlatticecurvatureisneglectedisrewrittenas

12

(A.7)

andtheGNDdensitytensorisdefinedas

(A.8)

Inthesmalldeformationtheory,weobtain

193

(A.9)

Inshort,thecontinuumdescriptionsofGNDdensitytensorprovidedbyNye(1953)

andArsenlisandParks(1999)canberelatedas

(A.10)

wherethenegativesignissimplyduetothedifferenceinthedirectionsofdislocation

lineinthesetwodescription(FigureA.2andFigureA.3).

Another continuum description of the GND density tensor is used extensively by

Gurtin and coworkers (Cermellia and Gurtin, 2000; Gurtin, 2002) and Gao and

coworkers(Gao,2001;Hanetal.,2005b;NixandGao,1998)

α ⊗ α⨀

α ⊗ α (A.11)

which is the transpose of Nye’s definition of the GND density tensor. Then, for

dislocationshowninFigureA.1,theonlynonzerocomponentof is .

Usingthisnotation,thelatticecurvature‐GNDdensityrelationiswrittenas

12

(A.12)

Asasummary,weobtain

(A.13)

Thisdissertation follows theGNDdensitydescription thathasbeenpromotedby

Gurtin,Gaoandcoworkers.

194

Appendix B. Kernel functions

As shown in chapter 3, internal residual stress for elastically isotropic medium

underplanestrainconditioncanbewrittenas

∗ , , , , , , (B.1)

where is thekernel function,whichdependson thedimensionality, geometryof the

problemandtheelasticpropertiesofthematerial.Notethatfirstindicesin associated

with the Burger vector direction while last two indices prescribes the stress

components.

B‐1Elasticallyisotropicinfinitemediumsolution

Thekernel functioncorresponding to theelastically isotropic infinitemediumcan

beobtainedfrominfiniteGreenfunctionsolutionas

, (B.2)

where the infiniteGreen function for infinitemediumhas been proposed by (Kröner,

1959)as

2ln (B.3)

where , is the local Cartesian coordinate system and 2 1⁄ is the

effective stiffnesswith shearmodulus andPoisson’s ratio . SubstitutingEq. (A.3) in

Eq. (A.2),weobtainconventionalkernel functionsakintotheVolterrasolution,which

representsthestressfieldofanedgedislocationwithunitBurgersvectorinaninfinite

medium(HirthandLothe,1982)as

195

3,

,3

(B.4)

where 2 1⁄ is the effective stiffness with shear modulus and Poisson’s

ratio and2histhespecimenthickness.

B‐2Elasticallyisotropicfinitemediumsolution

Generallytheexplicitformulationsfordislocationkernelfunctionsinthepresence

offiniteboundariesaresignificantlycomplicated.Fortheproblemstudiedinthiswork,

stress function based approach using complex Fourier transform is adapted from

(FotuhiandFariborz,2008).Forcompleteness,wedescribethesefinitekernelfunctions

forastructurewithedgedislocationsofunitBurgersvectorsin and directions,but

with a somewhatmodifiednotations.We introducea local coordinate system , at

the point where the internal stress is required, while the origin for the global

coordinates , isplacedattheneutralaxes(fig.2).Sinceinfinityassumptionismade

inxdirection,thekernelfunctionareindependentofglobalxcoordinate.

, , , , , , , ,

, , ,

(B.5)

where is a transform variable in the complex Fourier transformation approach

representingnon‐dimensionalspatialfrequencyand

, , , 0 1

, , , 1 0

, , , 0 1

, , , 1 0

(B.6)

196

2 ∙ cos , 1 ∙ sin

2 ∙ cos , 1 ∙ sin

∙ cos , 1 ∙ sin

∙ cos , 1 ∙ sin

1 ∙ sin , ∙ cos

1 ∙ sin , ∙ cos

(B.7)

4Δ

A ∙ cosA ∙ sin

1 1

1 1

C ∙ cosC ∙ sin

2 1

2 1

4Δ

A ∙ cosA ∙ sin

1 1

1 1

C ∙ cosC ∙ sin

2 1

2 1

4Δ

A ∙ cosA ∙ sin

1 1

1 1

C ∙ cosC ∙ sin

1 1

(B.8)

197

4Δ

A ∙ cosA ∙ sin

1 1

1 1

C ∙ cosC ∙ sin

1 1

4Δ

A ∙ sinA ∙ cos

1 1

C ∙ sinC ∙ cos

1 1

1 1

4Δ

A ∙ sinA ∙ cos

1 1

C ∙ sinC ∙ cos

1 1 1

1

Thecoefficientsaredefinedas

,

,

,

,

Δ

(B.9)

198

1 2 2 1 1 1

2 2 1 1

1 2 1 2

2 4 1

2 4 1

1 2 2 1 1 1 2

2 1 1 1 2

1 2

2 2 1 1

2 2 1 1

2

2 4 2 1

2 4 2 1

4 4 2 1

4 4 2 1

2 4 2 1

2 4 2 1

4 4 2 1

4 4 2 1

4 4 2 1

4 4 2 1

(B.10)

199

2 4 2 1

2 4 2 1

4 4 2 1

4 4 2 1

2 4 2 1

2 4 2 1

Noting the components of kernel function associatedwith finitemedium, it

canbeseenthatinterchangingtheindicesdonotchangethefunctionalityofthekernel

function (e.g. ). This suggests that itmay alsobepossible towrite a finite

kernelfunctionintermsofacorrespondingGreen’sfunction suchthat , .

200

Appendix C. Numerical integration convergence

study

ToevaluateinternalstressarisingfrominteractionbetweenGNDsandfreesurfaces

(SeeEqs.4.8and4.9),weneedtonumericallyintegratethekernelfunctionforisotropic

finitemediumprovidedinappendixB.Inthisappendix,webrieflystudytheintegration

procedureandconvergence.WeonlyshowtheintegrationprocedureforEq.4.9while

similarprocedurewithsameresultshasbeendoneforEq.4.9whichdoesnotpresent

here.

Forsetuptheproblem,firstwerewriteEq.4.9as

, , , , , (C.1)

Tobeginwith,weinvestigatethevariationofP1with andnormalizedvariables

M / ,N / andY / .FigureC.1showsindetailthevariationoftheterm

P1withrespectto and fordifferentvaluesof and .Generallyitcanbeseenthat

independent of other parameters, P1 is a continuous and decay function of which

ensureconvergenceofinfiniteintegrationinexpressionP2.Notethatdecayingdistance

on doesnotchangewithvariationofMhoweveritincreaseswhenYapproachesto1.

To perform numerical integration in P2,we used Gauss‐Laguerre quadraturemethod

(Pressetal.,1992)where

(C.2)

201

where isthei‐throotofLaguerrepolynomial ,nisnumberofintegrationpoints

and .

Y=0

(a) (b) (c)

Y=0.5

(d) (e) (f)

Y=0.9

(g) (h) (i)

M=20 M=5 M=0

FigureC.1.VariationofP1expressionwithrespecttothe andNfordifferentvalueofMandY.

202

(a) (b)

FigureC.2.(a)variationofP1versus (b)P2integrationconvergencebyincreasingthenumberofintegrationpoints(Y=0.9,M=5,N=‐0.1)

TheconvergenceofexpressionP2 is investigated inFig.C2aandbwithvariable

usedincase(h)infigureC.1and =‐0.1.Itcanbeseenthatforthecertainvalueused,

expressionP2convergesfor 600.NotethatexpressionP2givesusnondimensional

stressexertedon thepointatposition byadislocationwithdistanceof( , ) from

thepoint(figureC.3a).

(a)

(b) (c) (d)FigureC.3.(a)Illustrationofdislocationinthinfilm,Normalizedimagestressexertedat

point(b)Y=0,(b)Y=0.5,(c)Y=0.9byadislocationatposition(M,N)fromthatpointcalculatedformexpressionP2.

0 5 10 15 20 25 30 35 40-0.2

-0.1

0.0

0.1

0.2

P1

0 200 400 600 800 1000

-0.010

-0.005

0.000

0.005

0.010

P2

n (No. integration points)

  

 

-1.0 -0.5 0.0 0.5 1.0-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

P2

Dislocation distance form the point (N)

Y=0 M=5

-1.5 -1.0 -0.5 0.0 0.5-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

P2

Dislocation distance form the point (N)

Y=0.5 M=5

Y=0.9 M=5

-1.5 -1.0 -0.5 0.0-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

P2

Dislocation distance form the point (N)

203

InFig.C3b‐d, thevariationofexpressionP2 isshownasa functionof thevertical

distance ofthedislocationfromthepointkeepingthehorizontaldistance fixedfor

differentpointsacrossthebeamthickness.ItcanbeseenthatP2isalinearfunctionof

.

IntegrationofexpressionP2overentirebeamthickness(expressionP3)provides

the non‐dimensional image stress at a point Y by a continuous finite dislocationwall

withdistanceofMfromthepoint(Fig.C4a).Thisnon‐dimensionalstressisequaltoarea

underthesurfacesshowninFig.C1.Thenon‐dimensionalimagestressfieldofafinite

dislocationwallinathinfilmcapturesinFig.C4batdifferentpositionof .Itconcluded

thatafinitedislocationwallinathinfilmgeneratealinearimagestressacrossthefilm

thicknesswhich is negative at top and positive at the bottomof the film. In addition,

Variation of expression P3 at the film surface ( 1) versus distance and its

convergencearedrawn inFig.C4c It canbe seen that the stressofdislocationwall is

rapidly decay to zero by increasing the distance . This observation is in agreement

withsaturationof withincreasing (seefigure4.7b)whereintegrationisperformed

overM.TheareaunderthecurveshowninFig.C4from0to givesthe forspecified

.SimplecompositeSimpson’sruleisusedtoperformnumericalintegrationforsecond

andthirdintegralinEq.(C.1)usingMAPLEsoftware.FigureC.4cdisplaysconvergence

oftheP3withthenumberofintegrationsegments.

204

(a)

(b)

(c)

FigureC.4.(a)Illustrationoffinitedislocationwallinthinfilm,(b)VariationofP3acrossfilmthicknessfor 5,(c)variationofP3atfilmsurface( 1)withrespecttothedistance anditsconvergence.

  

-1.0 -0.5 0.0 0.5 1.0-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

P3

Y

M=5

0 20 40 60 80 100

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

P3

M

0 10 20 30 40 50-0.0475

-0.0470

-0.0465

-0.0460

-0.0455

-0.0450

-0.0445

-0.0440

P3

No. segments