Evolution of Phase Transitions: A Continuum Theory

259 Pages·2006·1.924 MB·English

by  Abeyaratne R., Knowles J.K.

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EVOLUTION OF PHASE TRANSITIONS
This work began with the authors’ exploration of the applicability of the
finite deformation theory of elasticity when various standard assumptions
suchasconvexityoftheenergyorellipticityofthefieldequationsofequi-
librium are relinquished. The finite deformation theory of elasticity turns
outtobeanaturalvehicleforthestudyofphasetransitionsinsolidswhere
thermal effects can be neglected. This is a valuable work for those inter-
estedinthedevelopmentandapplicationofcontinuum-mechanicalmodels
thatdescribethemacroscopicresponseofmaterialscapableofundergoing
stress- or temperature-induced transitions between two solid phases. The
focusisontheevolutionofphasetransitions,whichmaybeeitherdynamic
orquasi-static,controlledbyakineticrelationthatintheframeworkofclas-
sicalthermomechanicsrepresentsinformationthatissupplementarytothe
usualbalanceprinciplesandconstitutivelawsofconventionaltheory.The
book should be of interest to mechanicians, material scientists, geophysi-
cists,andappliedmathematicians.
Rohan Abeyaratne is the Quentin Berg Professor of Mechanics and Head
of the Department of Mechanical Engineering at MIT. He received his
bachelor’sdegreefromtheUniversityofCeylonandhisdoctoratefromthe
CaliforniaInstituteofTechnology.AmonghishonorsaretheE.O.E.Pereira
GoldMedal(1975),DenHartogDistinguishedEducator(1995),MacVicar
Fellowship (2000), Fellow, American Academy of Mechanics (1996) and
Fellow, American Society of Mechanical Engineers (1998). His primary
researchinterestisinnonlinearphenomenainmechanics.
JamesK.KnowlesistheWilliamR.KenanProfessorofAppliedMechanics,
Emeritus,attheCaliforniaInstituteofTechnology.HereceivedhisS.B.and
Ph.D.degreesfromMIT,andheholdsanhonorarySc.D.degreefromthe
National University of Ireland. He is a Fellow of the American Academy
of Mechanics (AAM), the American Association for the Advancement of
ScienceandtheAmericanSocietyofMechanicalEngineers(ASME).Heis
apastpresidentofAAM,andheisarecipientofMIT’sGoodwinMedalfor
teaching,theEringenMedaloftheSocietyofEngineeringScienceandthe
KoiterMedaloftheASME.Hisprimaryresearchinterestsareinnonlinear
phenomena in continuum mechanics, and in analytical issues in fracture
mechanicsandthetheoryofelasticity.
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EVOLUTION OF PHASE
TRANSITIONS
A Continuum Theory
ROHAN ABEYARATNE
MassachusettsInstituteofTechnology
JAMES K. KNOWLES
CaliforniaInstituteofTechnology
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cambridge university press
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Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521661478
© Cambridge University Press 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-54713-3 OCeISBN
isbn-13 978-0-521-66147-8 hardback
isbn-10 0-521-66147-1 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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TotheC7:Gina,Kenny,Kevin,Kristen,Liam,Linus,&Nina;
andtheJ4:Jackie,John,Jeff,&Jamey.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pagexiii
PartI Introduction
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
1.1 Whatthismonographisabout 3
1.2 Someexperiments 7
1.3 Continuummechanics 9
1.4 Quasilinearsystems 10
1.5 Outlineofmonograph 11
PartII PurelyMechanicalTheory
2 Two-WellPotentials,GoverningEquations
andEnergetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Introduction 19
2.2 Two-phasenonlinearlyelasticmaterials 20
2.3 Fieldequationsandjumpconditions 25
2.4 Energeticsofmotion,drivingforceanddissipation
inequality 27
3 EquilibriumPhaseMixturesandQuasistatic
Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
3.1 Introduction 32
3.2 Equilibriumstates 33
3.3 Variationaltheoryofequilibriummixtures
ofphases 37
3.4 Quasistaticprocesses 42
3.5 Nucleationandkinetics 44
3.6 Constantelongationrateprocesses 47
3.7 Hysteresis 53
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viii CONTENTS
4 Impact-InducedTransitionsinTwo-Phase
ElasticMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Introduction 59
4.2 Theimpactproblemfortrilineartwo-phase
materials 61
4.2.1 Theconstitutivelaw 61
4.2.2 Theimpactproblem 64
4.3 Scale-invariantsolutionsoftheimpactproblem 66
4.3.1 Solutionswithoutaphasetransition 66
4.3.2 Solutionswithaphasetransition:Thetwo-wavecase 67
4.3.3 Solutionswithaphasetransition:Theone-wavecase 68
4.3.4 Thetotalityofsolutions 69
4.4 Nucleationandkinetics 71
4.5 Comparisonwithexperiment 74
4.6 Othertypesofkineticrelations 77
4.7 Relatedwork 77
PartIII ThermomechanicalTheory
5 Multiple-WellFreeEnergyPotentials . . . . . . . . . . . . . . 85
5.1 Introduction 85
5.2 Helmholtzfreeenergypotential 86
5.3 Potentialenergyfunctionandtheeffectofstress 88
5.4 Example1:ThevanderWaalsFluid 90
5.5 Example2:Two-phasemartensiticmaterial
withcubicandtetragonalphases 95
6 TheContinuumTheoryofDrivingForce. . . . . . . . . . .105
6.1 Introduction 105
6.2 Balancelaws,fieldequationsandjumpconditions 106
6.2.1 Balancesofmomentumandenergyin
integralform 106
6.2.2 Localizationofthebalancelaws 106
6.3 Thesecondlawofthermodynamicsand
thedrivingforce 108
6.3.1 Entropyproductionrate 108
6.3.2 Drivingforceandthesecondlaw 110
6.3.3 Drivingforceinthecaseofmechanical
equilibrium 111
7 ThermoelasticMaterials . . . . . . . . . . . . . . . . . . . . . 113
7.1 Introduction 113
7.2 Thethermoelasticconstitutivelaw 113
7.2.1 Relationsamongstress,deformationgradient,
temperatureandspecificentropy 113
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CONTENTS ix
7.2.2 Theheatconductionlaw 116
7.2.3 Thepartialdifferentialequationsofnonlinear
thermoelasticity 116
7.2.4 Thermomechanicalequilibrium 117
7.3 Stabilityofathermoelasticmaterial 118
7.4 Aone-dimensionalspecialcase:uniaxialstrain 120
8 KineticsandNucleation. . . . . . . . . . . . . . . . . . . . . .124
8.1 Introduction 124
8.2 Nonequilibriumprocesses,thermodynamicfluxes
andforces,kineticrelation 124
8.3 Phenomenologicalexamplesofkinetic
relations 127
8.4 Micromechanicallybasedexamples
ofkineticrelations 128
8.4.1 Viscosity-straingradientmodel 130
8.4.2 Thermalactivationmodel 131
8.4.3 Propagationthrougharowof
imperfections 133
8.4.4 Kineticsfromatomisticconsiderations 134
8.4.5 Frenkel-Kontorowamodel 136
8.5 Nucleation 139
PartIV One-DimensionalThermoelasticTheory
andProblems
9 ModelsforTwo-PhaseThermoelasticMaterials
inOneDimension . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.1 Preliminaries 149
9.2 MaterialsofMie-Gru¨neisentype 151
9.3 Two-phaseMie-Gru¨neisenmaterials 153
9.3.1 Thetrilinearmaterial 153
9.3.2 Stabilityofphasesofthetrilinearmaterial 156
9.3.3 Othertwo-phasematerialsofMie-Gru¨neisen
type 159
10 QuasistaticHysteresisinTwo-PhaseThermoelastic
TensileBars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1 Preliminaries 163
10.2 Thermomechanicalequilibriumstates
foratwo-phasematerial 164
10.3 Quasistaticprocesses 166
10.4 Trilinearthermoelasticmaterial 167
10.5 Stresscyclesatconstanttemperature 169
10.6 Temperaturecyclesatconstantstress 173