Configurational Forces as Basic Concepts of Continuum Physics

245 Pages·2000·0.838 MB·English

by  Morton E. Gurtin (auth.)

Configurational Forces
as Basic Concepts of
Continuum Physics
Morton E. Gurtin
Springer

FormygrandchildrenKatie,Grant,andLiza
Contents
1. Introduction 1
a. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
b. Variationaldefinitionofconfigurationalforces . . . . . . . . . 2
c. Interfacialenergy.Afurtherargumentforaconfigurational
forcebalance . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
d. Configurationalforcesasbasicobjects . . . . . . . . . . . . . 7
e. Thenatureofconfigurationalforces . . . . . . . . . . . . . . . 9
f. Configurational stress and residual stress.
Internalconfigurationalforces . . . . . . . . . . . . . . . . . . 10
g. Configurationalforcesandindeterminacy . . . . . . . . . . . . 11
h. Scopeofthebook . . . . . . . . . . . . . . . . . . . . . . . . 12
i. Onoperationaldefinitionsandmathematics. . . . . . . . . . . 12
j. Generalnotation.Tensoranalysis . . . . . . . . . . . . . . . . 13
j1. Ondirectnotation . . . . . . . . . . . . . . . . . . . . 13
j2. Vectorsandtensors.Fields . . . . . . . . . . . . . . . 13
j3. Third-ordertensors(3-tensors).TheoperationT :(cid:1). . 15
j4. Functionsoftensors . . . . . . . . . . . . . . . . . . . 16
A. Configurationalforceswithinaclassicalcontext 19
2. Kinematics 21
a. Referencebody.Materialpoints.Motions. . . . . . . . . . . . 21
b. Materialandspatialvectors.ThesetsE andE . . . . . 22
space matter
c. Materialandspatialobservers . . . . . . . . . . . . . . . . . . 23
d. Consistencyrequirement.Objectivefields . . . . . . . . . . . 23
viii Contents
3. Standardforces.Working 25
a. Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
b. Working.Standardforceandmomentbalancesasconsequences
ofinvarianceunderchangesinspatialobserver . . . . . . . . . 26
4. Migrating control volumes. Stationary and time-dependent
changesinreferenceconfiguration 29
a. MigratingcontrolvolumesP (cid:1)P(t).Velocityfieldsfor∂P(t)
¯
and∂P(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
b. Changeinreferenceconfiguration . . . . . . . . . . . . . . . . 31
b1. Stationarychangeinreferenceconfiguration . . . . . . 31
b2. Time-dependentchangeinreferenceconfiguration . . . 32
5. Configurationalforces 34
a. Configurationalforces . . . . . . . . . . . . . . . . . . . . . . 34
b. Workingrevisited . . . . . . . . . . . . . . . . . . . . . . . . 35
c. Configurationalforcebalanceasaconsequenceofinvariance
underchangesinmaterialobserver . . . . . . . . . . . . . . . 36
d. Invariance under changes in velocity field for ∂P(t).
Configurationalstressrelation . . . . . . . . . . . . . . . . . . 37
e. Invariance under time-dependent changes in reference.
Externalandinternalforcerelations . . . . . . . . . . . . . . . 38
f. Standard and configurational forms of the working.
Powerbalance . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6. Thermodynamics. Relation between bulk tension and energy.
Eshelbyidentity 41
a. Mechanicalversionofthesecondlaw . . . . . . . . . . . . . . 41
b. Eshelbyrelationasaconsequenceofthesecondlaw . . . . . . 42
c. Thermomechanicaltheory . . . . . . . . . . . . . . . . . . . . 44
d. Fluids.Currentconfigurationasreference. . . . . . . . . . . . 45
7. Inertiaandkineticenergy.Alternativeversionsofthesecondlaw 46
a. Inertiaandkineticenergy . . . . . . . . . . . . . . . . . . . . 46
b. Alternativeformsofthesecondlaw . . . . . . . . . . . . . . . 47
c. Pseudomomentum . . . . . . . . . . . . . . . . . . . . . . . . 47
d. Lyapunovrelations. . . . . . . . . . . . . . . . . . . . . . . . 48
8. Changeinreferenceconfiguration 50
a. Transformationlawsforfreeenergyandstandardforce . . . . 50
b. Transformationlawsforconfigurationalforce . . . . . . . . . 51
9. Elasticandthermoelasticmaterials 53
a. Mechanicaltheory . . . . . . . . . . . . . . . . . . . . . . . . 54
a1. Basicequations . . . . . . . . . . . . . . . . . . . . . 54
Contents ix
a2. Constitutivetheory . . . . . . . . . . . . . . . . . . . 54
b. Thermomechanicaltheory . . . . . . . . . . . . . . . . . . . . 56
b1. Basicequations . . . . . . . . . . . . . . . . . . . . . 56
b2. Constitutivetheory . . . . . . . . . . . . . . . . . . . 57
B. The use of configurational forces to characterize
coherentphaseinterfaces 61
10.Interfacekinematics 63
11.Interfaceforces.Secondlaw 66
a. Interfaceforces . . . . . . . . . . . . . . . . . . . . . . . . . 66
b. Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
c. Standardandconfigurationalforcebalancesattheinterface . . 68
d. InvarianceunderchangesinvelocityfieldforS (t).Normal
configurationalbalance . . . . . . . . . . . . . . . . . . . . . 69
e. Powerbalance.Internalworking . . . . . . . . . . . . . . . . 70
f. Secondlaw.Internaldissipationinequalityfortheinterface . . 71
g. Localizationsusingapillboxargument . . . . . . . . . . . . . 72
12.Inertia.Basicequationsfortheinterface 74
a. Relativekineticenergy . . . . . . . . . . . . . . . . . . . . . 74
b. DeterminationofbS andeS . . . . . . . . . . . . . . . . . . 75
c. Standardandconfigurationalbalanceswithinertia . . . . . . . 77
d. Constitutiveequationfortheinterface. . . . . . . . . . . . . . 78
e. Summaryofbasicequations . . . . . . . . . . . . . . . . . . . 79
f. Globalenergyinequality.Lyapunovrelations . . . . . . . . . . 80
C. An equivalent formulation of the theory.
Infinitesimaldeformations 81
13.Formulationwithinaclassicalcontext 83
a. Background. Reason for an alternative formulation
intermsofdisplacements . . . . . . . . . . . . . . . . . . . . 83
b. Finitedeformations.ModifiedEshelbyrelation . . . . . . . . . 84
c. Infinitesimaldeformations . . . . . . . . . . . . . . . . . . . . 86
14.Coherentphaseinterfaces 88
a. Generaltheory . . . . . . . . . . . . . . . . . . . . . . . . . . 88
b. Infinitesimaltheorywithlinearstress-strainrelationsinbulk. . 89
x Contents
D. Evolvinginterfacesneglectingbulkbehavior 91
15.Evolvingsurfaces 93
a. Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
a1. Background.Superficialstress . . . . . . . . . . . . . 93
a2. Superficialtensorfields . . . . . . . . . . . . . . . . . 94
b. Smoothlyevolvingsurfaces . . . . . . . . . . . . . . . . . . . 97
b1. TimederivativefollowingS.Normaltimederivative. . 97
b2. Velocityfieldsfortheboundarycurve∂G ofasmoothly
evolvingsubsurfaceofS .Transporttheorem . . . . 99
b3. Transformationlaws . . . . . . . . . . . . . . . . . . 100
16.Configurationalforcesystem.Working 101
a. Configurationalforces.Working. . . . . . . . . . . . . . . . . 101
b. Configurationalforcebalanceasaconsequenceofinvariance
underchangesinmaterialobserver . . . . . . . . . . . . . . . 102
c. Invarianceunderchangesinvelocityfields.Surfacetension.
Surfaceshear. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
d. Normalforcebalance.Intrinsicformfortheworking. . . . . . 104
e. Powerbalance.Internalworking . . . . . . . . . . . . . . . . 105
17.Secondlaw 108
18.Constitutiveequations 110
a. Functionsoforientation . . . . . . . . . . . . . . . . . . . . . 110
b. Constitutiveequations . . . . . . . . . . . . . . . . . . . . . . 111
c. Evolutionequationfortheinterface . . . . . . . . . . . . . . . 113
d. Lyapunovrelations. . . . . . . . . . . . . . . . . . . . . . . . 114
19.Two-dimensionaltheory 115
a. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
b. Configurationalforces.Working.Secondlaw . . . . . . . . . . 116
c. Constitutivetheory. . . . . . . . . . . . . . . . . . . . . . . . 118
d. Evolutionequationfortheinterface . . . . . . . . . . . . . . . 119
e. Corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
f. Angle-convexity.TheFrankdiagram . . . . . . . . . . . . . . 120
g. Convexity of the interfacial energy and evolution
oftheinterface . . . . . . . . . . . . . . . . . . . . . . . . . . 124
E. Coherent phase interfaces with interfacial energy
anddeformation 127
20.Theoryneglectingstandardinterfacialstress 129
a. Standardandconfigurationalforces.Working . . . . . . . . . 129
Contents xi
b. Powerbalance.Internalworking . . . . . . . . . . . . . . . . 131
c. Secondlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
c1. Secondlaw.Interfacialdissipationinequality. . . . . . 132
c2. Derivationoftheinterfacialdissipationinequality
usingapillboxargument . . . . . . . . . . . . . . . . 132
d. Constitutiveequations . . . . . . . . . . . . . . . . . . . . . . 133
e. Construction of the process used in restricting
theconstitutiveequations . . . . . . . . . . . . . . . . . . . . 135
f. Basicequationswithinertialexternalforces . . . . . . . . . . 135
f1. Standardandconfigurationalbalances . . . . . . . . . 135
f2. Summaryofbasicequations . . . . . . . . . . . . . . 136
g. Globalenergyinequality.Lyapunovrelations . . . . . . . . . . 137
21.General theory with standard and configurational stress
withintheinterface 138
a. Kinematics.Tangentialdeformationgradient . . . . . . . . . . 138
b. Standardandconfigurationalforces.Working . . . . . . . . . 139
c. Powerbalance.Internalworking . . . . . . . . . . . . . . . . 142
d. Secondlaw.Interfacialdissipationinequality . . . . . . . . . . 144
e. Constitutiveequations . . . . . . . . . . . . . . . . . . . . . . 145
f. Basicequationswithinertialexternalforces . . . . . . . . . . 147
g. Lyapunovrelations. . . . . . . . . . . . . . . . . . . . . . . . 147
22.Two-dimensionaltheorywithstandardandconfigurationalstress
withintheinterface 149
a. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
b. Forces.Working . . . . . . . . . . . . . . . . . . . . . . . . . 150
c. Powerbalance.Internalworking.Secondlaw . . . . . . . . . . 152
d. Constitutiveequations . . . . . . . . . . . . . . . . . . . . . . 155
e. Evolutionequationsfortheinterface . . . . . . . . . . . . . . 156
F. Solidification 157
23.Solidification. The Stefan condition as a consequence of the
configurationalforcebalance 159
a. Single-phasetheory . . . . . . . . . . . . . . . . . . . . . . . 159
b. Theclassicaltwo-phasetheoryrevisited.TheStefancondition
asaconsequenceoftheconfigurationalbalance . . . . . . . . 160
24.Solidificationwithinterfacialenergyandentropy 163
a. Generaltheory . . . . . . . . . . . . . . . . . . . . . . . . . . 163
b. Approximate theory. The Gibbs-Thomson condition as a
consequenceoftheconfigurationalbalance . . . . . . . . . . . 166
c. Free-boundary problems for the approximate theory.
Growththeorems. . . . . . . . . . . . . . . . . . . . . . . . . 167