Table Of ContentOXFORD SERIES ON MATERIALS MODELLING
SeriesEditors
AdrianP.Sutton,FRS
DepartmentofPhysics,ImperialCollegeLondon
RobertE.Rudd
LawrenceLivermoreNationalLaboratory
Oxford Series on Materials Modelling
Materialsmodellingisoneofthefastestgrowingareasinthescienceandengineer-
ing of materials, both in academe and in industry. It is a very wide field covering
materialsphenomenaandprocessesthatspantenordersofmagnitudeinlengthand
morethantwentyintime.Abroadrangeofmodelsandcomputationaltechniques
hasbeendevelopedtomodelseparatelyatomistic,microstructuralandcontinuum
processes. A new field of multi-scale modelling has also emerged in which two
or more length scales are modelled sequentially or concurrently. The aim of this
series is to provide a pedagogical set of texts spanning the atomistic and micro-
structural scales of materials modelling, written by acknowledged experts. Each
bookwillassumeatmostarudimentaryknowledgeofthefielditcoversanditwill
bringthereadertothefrontiersofcurrentresearch.Itishopedthattheserieswill
beusefulforteachingmaterialsmodellingatthepostgraduatelevel.
APS,London
RER,Livermore,California
1. M.W.Finnis:Interatomicforcesincondensedmatter
2. K.Bhattacharya:Microstructureofmartensite—Whyitformsandhowitgives
risetotheshape-memoryeffects
3. V.V.Bulatov,W.Cai:Computersimulationsofdislocations
4. A.S.Argon:Strengtheningmechanismsincrystalplasticity
5. L.P.Kubin:Dislocations,mesoscalesimulationsandplasticflow
Forthcoming:
T.N.Todorov:Electricalconductioninnanoscalesystems
D.N.Theodorou,V.Mavrantzas:Multiscalemodellingofpolymers
Dislocations, Mesoscale
Simulations and Plastic Flow
Ladislas P. Kubin
Laboratoired’ÉtudedesMicrostructures,
CNRS-ONERA,Châtillon,France
3
3
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PREFACE
Predicting the mechanical response of materials is the ultimate objective of dis-
location theory. Some may even believe that this goal will never be reached. The
challenge is, indeed, formidable, for several reasons that are stated in the intro-
ductory chapter of this book. After the period of euphoria of the years 1950–70,
dislocationtheorywasconfrontedwithseveralunsolvableproblems.Thequestion
of bridging the huge range of time and length scales between electronic structure
calculations of dislocation cores and the mechanical response of bulk materials
reappeared by the end of the twentieth century under the name of multiscale
modelling. The body of knowledge accumulated by physical metallurgy was ren-
ovatedbytheimportationofnewconceptsfromnon-lineardynamicsandstatistical
mechanics, the expansion of ab initio and atomistic simulations and the devel-
opment of sophisticated experimental methods for investigating materials at all
scales. Mesoscale simulations emerged in the same period with the objective of
fillingastrategicgapbetweenatomic-scalestudiesandthecontinuummechanical
framework.
Like others in this series on materials modelling, the present book on
Dislocations, Mesoscale Simulations and Plastic Flow is intended for graduate
students and researchers in materials science and mechanical engineering. It dis-
cussesmesoscopicaspectsofdislocationsandcrystalplasticityinelementalmodel
materials, as well as the necessary background for establishing connections with
otherscales.Sincethisisahugedomain,theemphasisisontopicsthatrequirean
updateorasynthesis—areasofpotentialexpansionandearlyseminalstudiesthat
are still of primordial importance. Abundant references are provided for further
reading. Two books in this series are complementary to the present one and can
alsobeconsulted.Thecomputationalaspectsofmesoscalesimulationsaretreated
in a chapter of Computer Simulations of Dislocations by Vassily V. Bulatov and
WeiCai.Alloyingeffects,onwhichlittleissaidinthepresentbook,arediscussed
intwochaptersofStrengtheningMechanismsinCrystalPlasticitybyAliS.Argon.
The content of this book includes the individual and collective properties of
dislocations in crystals and their mechanical response in the domain of temperat-
ures where dislocation glide and do not climb. The approach adopted consists in
goingfromearlyexperimentalandmodellingstudiestothepresentstateoftheart.
GeneralaspectsarerecalledinChapter1,whereasChapters2and3discussdislo-
cationsandplasticityinmaterialswhereplasticflowisgoverned,respectively,by
interactionswithobstaclesandthelatticeresistance.
vi PREFACE
There are few three-dimensional codes for dislocation dynamics simulations,
but most of them are now public. For this reason, an objective of the present
book is to present a comprehensive treatment of such simulations, their basic
principles, limitations, achievements and potential developments. A guide to dis-
location dynamics simulations and to the available existing codes is presented in
Chapter 4. Chapter 5 illustrates the power of the method through a synthesis of
representativeapplications,whichrangefromnanomaterialstostructuralmaterials
andfrommodeltolarge-scalesimulations.
Synthetizingthestateoftheartinamovingfieldisalwaysaperilousexercise.
Nevertheless, some rules for conducting simulations are not bound to change. It
is important to draw a clear demarcation line between questions that we believe
we understand, others that we do not understand well, or not at all, and others
for which the answers are speculative. Performing simulations requires having in
mindaspecificproblemtosolve,acriticalviewoftherelevantpublishedliterature
and clear ideas about the input needed for this purpose. Mesoscale simulations
reproducetosomeextenttherealbehaviourofdislocationsandprovideadynamic
visionofplasticity,whichisoftenabsentfrommodels,aswellasnumericalvalues
that cannot always be accessed by other means. However, reproducing does not
necessarily mean understanding. To become valuable, simulated outputs have to
be connected to the world of real materials through comparison with experiment
and,wheneverpossible,modelling.
Acknowledgements
Iamgratefultomanycolleaguesforclarifyingdiscussionsandcriticalcomments
onsomepartsofthebookandforprovidingusefulmaterial.Iwouldliketothank
in particular Joel Bonneville, Daniel Caillard, Marc Fivel, Volker Mohles, Satish
Rao, Klaus W. Schwarz, the group of Clain’s Silicon Valley, Sandrine Brochard,
JulienGodet,LaurentPizzagalliandJacquesRabier,aswellasauthorswhokindly
suppliedmewithoriginalmaterialandgavemetheirauthorizationforreproducing
theirpublishedfigures.IalsowouldliketothanktheserieseditorsRobertE.Rudd
and Adrian P. Sutton for their corrections and suggestions on several chapters of
this book. Exchanges with Sönke Adlung, Jessica White and her predecessors at
OxfordUniversityPresswerenotonlyasourceofstimulation,butalsoofanswers
toinnumerablequestions.MyspecialgratitudegoestoBenoitDevincreforalong-
standingandfruitfulcollaborationandhisconstanthelpduringthepreparationof
thisbook.IamalsogreatlyindebtedtoRonanMadecforprovidingmewithmany
original figures. By the end of the 1980s, Fred Kocks suggested to Gilles Canova
and myself that we should do some work together. This is how the first three-
dimensionaldislocationdynamicssimulationwasinitiated.Thisbookisdedicated
tothememoryofGillesCanova.
CONTENTS
FigurePermissions xi
1 BackgroundandDefinitions 1
1.1 Introduction 1
1.2 Dislocationcoreproperties 3
1.2.1 Coreenergyandstructure 3
1.2.2 Cross-slipandthelatticeresistance 5
1.3 Elasticpropertiesofdislocations 8
1.3.1 Strainenergyofastraightdislocation 8
1.3.2 Forceonadislocation 10
1.3.3 Linetension 12
1.3.4 Linetensionstrengthening 15
1.4 Dislocationvelocity 16
1.4.1 Effectivestress 16
1.4.2 Governingmechanisms 17
1.4.3 Orowan’slaw 19
1.5 Multiscalemodelling 21
1.6 Introductionto3DDDsimulations 23
1.6.1 Briefhistoricalsketch 23
1.6.2 Furtherimplementation 25
2 Obstacle-controlledPlasticFlow 27
2.1 Outline 27
2.2 Free-flightvelocity 28
2.2.1 ThePeierlsstressinfccmetals 28
2.2.2 Phonondrag 29
2.3 Dislocation–dislocationinteractions 34
2.3.1 Short-rangeinteractionsinfcccrystals 35
2.3.2 Junctionformationanddestruction 38
2.3.3 Jogs 41
2.4 Cross-slipinfcccrystals 42
2.4.1 Modelsforcompactcross-slip 42
2.4.2 TheFriedel–Escaigmechanism 43
2.4.3 Theactivationenergyforcross-slip 44
viii CONTENTS
2.4.4 Escaig’seffectandEscaig’sbarrier 47
2.4.5 Experimentalchecks 49
2.4.6 Stress-freeconstrictionenergies 50
2.4.7 Atomisticstudiesofcross-slip 51
2.4.8 Themultiplerolesofcross-slip 55
2.5 Flowstressanddislocationdensities 56
2.5.1 Dislocationstrengthening 57
2.5.2 Foreststrengthening 58
2.5.3 Jogstrengthening 61
2.5.4 Generalizeddislocationstrengthening 63
2.6 Mechanicalresponseandmicrostructures 65
2.6.1 Resolvedstress–straincurves 65
2.6.2 StageI 67
2.6.3 StageII 68
2.6.4 StageIII 69
2.6.5 StageIV 70
2.6.6 Similitudeandself-similarity 73
2.6.7 Thestorage–recoverymodel 77
2.7 Collectivedislocationbehaviour 81
2.7.1 Themodellingofdislocationpatterns 81
2.7.2 Dislocationavalanches 83
3 Lattice-controlledPlasticFlow 90
3.1 Outline 90
3.2 Thelatticeresistanceinbccmetals 91
3.2.1 Deformationpropertiesofbccmetals 91
3.2.2 Corestructureofscrewdislocations 96
3.2.3 Non-SchmideffectsandPeierlsstresses 100
3.2.4 Kink-pairmechanismsandmodels 102
3.2.5 Strengtheningandsofteninginbccmetals 113
3.3 Prismaticslipinhcpmetals 118
3.3.1 Slipsystemsandscrewdislocationcores 118
3.3.2 ThePeierlsstressinTiandZr 121
3.3.3 Locking–unlockinginhcpmetals 123
3.4 Dislocationsinsilicon 124
3.4.1 Introduction 124
3.4.2 Dislocationsinthediamondcubiclattice 125
3.4.3 Dislocationcoresintheglideset 127
3.4.4 Experimentalmethods 129
3.4.5 Themultiplicationyieldpointofsilicon 131
3.4.6 Velocitiesinthekink-diffusionmodel 133
3.4.7 Dislocationvelocitiesandactivationenergies 134
CONTENTS ix
3.4.8 Thelength-independentregime 137
3.4.9 Dislocationsathighstress 139
4 AGuideto3DDDSimulations 145
4.1 Introduction 145
4.2 Elasticproperties 146
4.2.1 Outline 146
4.2.2 Discretizationofdislocationlines 147
4.2.3 Localproceduresandoptimization 148
4.2.4 Corefields 149
4.2.5 Theself-stress 151
4.2.6 Fromself-stresstoeffectivestress 155
4.2.7 Furtheroptimization 156
4.2.8 Elasticanisotropy 159
4.2.9 Dissociateddislocations 161
4.3 Localrules 162
4.3.1 Outline 162
4.3.2 Dislocationmobilityandvelocity 162
4.3.3 Dislocationcross-slip 166
4.3.4 Otherlocalrules 170
4.4 Boundaryconditions 170
4.4.1 Periodicboundaryconditions 171
4.4.2 Finiteboundaryconditions 175
4.4.3 Othermethodsforfinitesizes 177
4.5 Current3DDDsimulations 178
5 ApplicationsofDDSimulations 181
5.1 Outline 181
5.2 Dislocationintersections 181
5.2.1 Intersectionsandreactions 182
5.2.2 Theinteractioncoefficients 187
5.3 Atomic-scaledefects,precipitationstrengthening 193
5.3.1 Dislocationsandsoluteatoms 193
5.3.2 Dislocationsandirradiationdefects 194
5.3.3 Dislocationclimb 196
5.3.4 Precipitationstrengthening 197
5.4 Collectivedislocationprocesses 199
5.4.1 Intermittencyandavalanches 199
5.4.2 Fromintermittenttocontinuousflow 202
5.4.3 Dislocationpatterns 205
5.4.4 Patterningincyclicdeformation 207
5.4.5 Shockloading,highstrainrates 208
