Physical Foundations of Continuum Mechanics

438 Pages·2012·2.166 MB·English

by  Murdoch A.I.

PHYSICAL FOUNDATIONS OF
CONTINUUM MECHANICS
A.IanMurdoch’sPhysicalFoundationsofContinuumMechanicswill
interestengineers,mathematicians,andphysicistswhostudythemacro-
scopicbehaviourofsolidsandfluidsorengageinmoleculardynamical
simulations. In contrast to standard works on the subject, Murdoch’s
bookexaminesphysicalassumptionsimplicitincontinuummodelling
fromamolecularperspective. Insodoing, physicalinterpretationsof
conceptsandfieldsareclarifiedbyemphasisingboththeirmicroscopic
origin and sensitivity to scales of length and time. Murdoch expertly
applies this approach to theories of mixtures, generalised continua,
fluid flow through porous media, and systems whose molecular con-
tentchangeswithtime.Elementsofstatisticalmechanicsareincluded,
forcomparison,andtwoextensiveappendicesaddressrelevantmath-
ematical concepts and results. This unique and thorough work is an
authoritativereferenceforbothstudentsandexpertsinthefield.
A. Ian Murdoch is Professor Emeritus of Mathematics at the Uni-
versity of Strathclyde, Glasgow. His work on continuum mechanics
has been widely published in such journals as the Archive for Ratio-
nalMechanicsandAnalysis,ProceedingsoftheRoyalSociety,Journal
ofElasticity,InternationalJournalofEngineeringScience,Continuum
MechanicsandThermodynamics,andtheQuarterlyJournalofMechan-
ics&AppliedMathematics.Heistheco-editoroftwobooks:Modelling
MacroscopicPhenomenaatLiquidBoundariesandModellingCoupled
Phenomena in Saturated Porous Materials, and author of published
lecturenotes,FoundationsofContinuumModelling.DrMurdochhas
taughtandlecturedatmanydistinguishedmathematicsandengineering
schoolsaroundtheworld.

Physical Foundations of
Continuum Mechanics
A. Ian Murdoch
UniversityofStrathclyde
CAMBRIDGE UNIVERSITY PRESS
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Murdoch,A.I.
Physicalfoundationsofcontinuummechanics/A.IanMurdoch.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-0-521-76558-9
1. Continuummechanics. 2. Fluidmechanics. I. Title.
QC155.7.M87 2013
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Contents
Preface page xiii
1 Introduction 1
1.1 Motivation 1
1.2 Contents 2
2 SomeElementsofContinuumMechanics 6
2.1 Preamble 6
2.2 MatterandItsDistribution 6
2.3 MotionofMatter:KinematicsandMaterialPoints 7
2.4 TheFormal(Axiomatic)ApproachtoMatterandMaterialPoints 9
2.5 MassConservation 11
2.6 DynamicsI:GlobalRelations 14
2.6.1 Introduction 14
2.6.2 LinearMomentumBalance 14
2.6.3 RotationalMomentumBalance 16
2.6.4 RigidBodyDynamics 17
2.7 DynamicsII:LocalRelations 24
2.8 Thermomechanics 29
2.8.1 GlobalBalanceofEnergy 29
2.8.2 AsideontheSpinVectorFieldwandPowerExpendedby
Couples 29
2.8.3 LocalBalanceofEnergy 30
3 MotivationforSeekingaMolecularScale-DependentPerspectiveon
ContinuumModelling 33
3.1 Preamble 33
3.2 TheNaturalContinuumPrejudice 33
3.3 TheContinuumViewpointonMassDensityρ 34
3.4 BoundariesandtheScaleDependenceofρ 34
3.5 Continuityofρ andtheDiscreteNatureofMatter 36
3.6 Velocity 37
3.7 ThePressureinaGas 41
vii
viii Contents
3.8 Reproducibility 42
3.9 SummaryofConceptualProblems 42
3.10 MotivationforSpace-TimeAveragingofMolecularQuantities 43
4 SpatialLocalisation,MassConservation,andBoundaries 44
4.1 Preamble 44
4.2 WeightedAveragesandtheContinuityEquation 44
4.3 TheSimplestChoicew(cid:3) ofWeightingFunction 48
4.3.1 Definitionofw(cid:3) 48
4.3.2 TheBoundaryCorrespondingtow(cid:3) 48
4.3.3 Integrationofρ(cid:3) andp(cid:3) overaRegion 52
4.3.4 AWrinkletoBeResolved:UseofaMollifier 53
4.3.5 FurtherMollificationConsiderations 56
4.3.6 RegularityofMollifiedFields:PolynomialMollifiers 58
4.3.7 MollificationasaNaturalConsequenceofSpatial
Imprecision 60
4.4 OtherChoicesofWeightingFunction 61
4.4.1 CellularAveraging 61
4.4.2 ChoicesAssociatedwithRepeatedAveraging 62
4.4.3 OtherChoices 67
4.5 TemporalFluctuations 69
4.6 Summary 70
5 Motions,MaterialPoints,andLinearMomentumBalance 71
5.1 Preamble 71
5.2 MotionsandMaterialPoints 72
5.3 MotionsandMaterialPointsforNon-ReactingBinaryMixtures 74
5.4 LinearMomentumBalancePreliminaries:IntermolecularForces 76
5.5 LinearMomentumBalance 80
5.5.1 DerivationoftheBalanceRelation 80
5.5.2 TheThermalNatureofD 82
w
5.5.3 ComparisonofContributionsT−andD toT 83
w w w
5.6 DeterminationofCandidateInteractionStressTensors 84
5.6.1 Preamble 84
5.6.2 SimpleForm 84
5.6.3 FormforPairwise-BalancedInteractions 85
5.6.4 SimpleChoiceofb forPairwise-BalancedInteractions 85
ij
5.6.5 Hardy-TypeChoiceofb forPairwise-BalancedInteractions 86
ij
5.6.6 Noll-TypeChoiceofb forPairwise-BalancedInteractions 87
ij
5.6.7 Conclusions 87
5.7 CalculationofInteractionStressesfortheSimplestFormof
WeightingFunctionw(cid:3) 88
− −
5.7.1 Determinationofa andCalculationof T and T 88
i s w(cid:3) sb w(cid:3)
5.7.2 DeterminationofbˆH andCalculationof T− 90
ij H w(cid:3)
5.7.3 TheGeometricalComplexityofbN 91
ij
5.8 ComparisonofInteractionStressTensorsfortheSimplestFormof
WeightingFunctionw(cid:3) 91
Contents ix
5.8.1 ValuesforTwoSimpleGeometries 91
5.8.2 IntegrationoverPlanarSurfaces 92
5.9 IntegralsofGeneralInteractionStressTensorsoverthe
BoundariesofRegularRegions 95
5.9.1 ResultsforaGeneralChoiceofWeightingFunction 95
5.9.2 ResultsforChoicew=w(cid:3) 98
5.9.3 FurtherRemarksforChoicew=w(cid:3) 100
6 BalanceofEnergy 102
6.1 Preamble 102
6.2 DerivationofEnergyBalances 102
6.3 ASubatomicPerspective 110
7 Fine-ScaleConsiderations:Moments,CoupleStress,Inhomogeneity,
andEnergetics 115
7.1 Preamble 115
7.2 GeneralisedMomentofMomentumBalance 115
7.3 InhomogeneityandMomentofMassConservation 121
7.4 Fine-ScaleEnergetics 123
7.5 SummaryandDiscussion 128
8 TimeAveragingandSystemswithChangingMaterialContent 130
8.1 Preamble 130
8.2 Motivation 130
8.3 TimeAveraging 132
8.4 TheTime-AveragedContinuityEquation 134
8.5 Time-AveragedFormsofLinearMomentumBalance 135
8.6 Time-AveragedFormsofEnergyBalance 137
8.7 SystemswithChangingMaterialContentI:General
GlobalConsiderations 139
8.8 SystemswithChangingMaterialContentII:SpecificGlobal
Examples 146
8.8.1 Rocketry 146
8.8.2 JetPropulsion 148
8.8.3 FallingRaindrop 150
8.9 SystemswithChangingMaterialContentIII:LocalEvolution
EquationsatSpecificScalesofLengthandTime 150
8.9.1 MassBalance 150
8.9.2 LinearMomentumBalance 153
8.9.3 EnergyBalance 157
8.9.4 ConcludingRemarks 164
8.10 Summary 165
9 ElementsofMixtureTheory 167
9.1 Preamble 167
9.2 MassConservationandMaterialPointsfora
Non-ReactingMixtureConstituent 167