Crystals, Defects and Microstructures. Modeling Across Scales

808 Pages·2001·21.72 MB·English

by  Rob Phillips

This page intentionally left blank
Crystals,DefectsandMicrostructures
Materialssciencehasemergedasoneofthecentralpillarsofthemodernphysical
sciencesandengineering,andisnowevenbeginningtoclaimaroleinthebiolog-
ical sciences. A central tenet in the analysis of materials is the structure–property
paradigm, which proposes a direct connection between the geometric structures
withinamaterialanditsproperties.
The increasing power of high-speed computation has had a major impact on
theoretical materials science and has permitted the systematic examination of
this connection between structure and properties. In this textbook, Rob Phillips
examinesthevariousmethodsthathavebeenusedinthestudyofcrystals,defects
and microstructures and that have made such computations possible. The author
presentsmanyofthekeygeneralprinciplesusedinthemodelingofmaterials,and
punctuates the text with real case studies drawn from recent research. A second
key theme is the presentation of recent efforts that have been developed to treat
problemsinvolvingeithermultiplespatialortemporalscalessimultaneously.
This text is intended for graduate students and researchers in science and
engineering with an interest in the theoretical constructs that have been devised
toundertakethestudyofmaterials.

Crystals, Defects and Microstructures
Modeling Across Scales
ROBPHILLIPS
BrownUniversity,ProvidenceRI
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
©Robert B. Phillips 2004
First published in printed format 2001
ISBN 0-511-04108-X eBook(netLibrary)
ISBN 0-521-79005-0 hardback
ISBN 0-521-79357-2 paperback
DedicatedtoSonic’saccomplices

Contents
Preface pagexv
Acknowledgements xxi
NotesonUnits,ScalesandConventions xxiv
Partone: ThinkingAbouttheMaterialWorld 1
1 IdealizingMaterialResponse 3
1.1 AMaterialWorld 3
1.1.1 Materials: ADatabookPerspective 3
1.1.2 TheStructure–PropertiesParadigm 8
1.1.3 ControllingStructure: TheWorldofHeatandBeat 12
1.2 ModelingofMaterials 14
1.2.1 TheCaseforModeling 14
1.2.2 ModelingDefined: ContrastingPerspectives 15
1.2.3 CaseStudiesinModeling 18
1.2.4 ModelingandtheComputer: NumericalAnalysisvsSimulation 25
1.3 FurtherReading 26
2 ContinuumMechanicsRevisited 29
2.1 ContinuumMechanicsasanEffectiveTheory 29
2.2 Kinematics: TheGeometryofDeformation 31
2.2.1 DeformationMappingsandStrain 32
2.2.2 GeometryofRigidDeformation 35
2.2.3 GeometryofSlipandTwinning 36
2.2.4 GeometryofStructuralTransformations 37
2.3 ForcesandBalanceLaws 39
2.3.1 ForcesWithinContinua: StressTensors 39
2.3.2 EquationsofContinuumDynamics 41
2.3.3 ConfigurationalForcesandtheDynamicsofDefects 44
2.4 ContinuumDescriptionsofDeformationandFailure 51
2.4.1 ConstitutiveModeling 51
vii
viii Contents
2.4.2 LinearElasticResponseofMaterials 51
2.4.3 PlasticResponseofCrystalsandPolycrystals 54
2.4.4 ContinuumPictureofFracture 60
2.5 BoundaryValueProblemsandModeling 64
2.5.1 Principle of Minimum Potential Energy and Reciprocal
Theorem 64
2.5.2 ElasticGreenFunction 66
2.5.3 MethodofEigenstrains 69
2.5.4 NumericalSolutions: FiniteElementMethod 72
2.6 DifficultieswiththeContinuumApproach 75
2.7 FurtherReading 76
2.8 Problems 78
3 QuantumandStatisticalMechanicsRevisited 81
3.1 Background 81
3.2 QuantumMechanics 82
3.2.1 BackgroundandFormalism 82
3.2.2 CatalogofImportantSolutions 87
3.2.3 FiniteElementsandSchro¨dinger 94
3.2.4 QuantumCorrals: AFiniteElementAnalysis 101
3.2.5 MetalsandtheElectronGas 103
3.2.6 QuantumMechanicsofBonding 109
3.3 StatisticalMechanics 115
3.3.1 Background 115
3.3.2 EntropyofMixing 119
3.3.3 TheCanonicalDistribution 122
3.3.4 InformationTheoreticApproachtoStatisticalMechanics 126
3.3.5 StatisticalMechanicsModelsforMaterials 129
3.3.6 BoundsandInequalities: TheBogoliubovInequality 135
3.3.7 CorrelationFunctions: TheKinematicsofOrder 137
3.3.8 ComputationalStatisticalMechanics 139
3.4 FurtherReading 142
3.5 Problems 144
Parttwo: EnergeticsofCrystallineSolids 147
4 EnergeticDescriptionofCohesioninSolids 149
4.1 TheRoleoftheTotalEnergyinModelingMaterials 149
4.2 ConceptualBackdropforCharacterizingtheTotalEnergy 152
4.2.1 AtomisticandContinuumDescriptionsContrasted 152