Nonlinear mesoscopic elasticity : the complex behaviour of granular media including rocks and soil

411 Pages·2009·15.05 MB·English

by  Robert A Guyer, Paul A Johnson, Prof

RobertA.GuyerandPaulA.Johnson
NonlinearMesoscopicElasticity

RobertA. Guyer and Paul A. Johnson
Nonlinear Mesoscopic Elasticity
The Complex Behaviour of Granular Media
including Rocks and Soil
WILEY-VCH Verlag GmbH & Co. KGaA
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TheDeutscheNationalbibliothekliststhis
T.J.Ulrich,P.JohnsonandR.A.Guyer,
publicationintheDeutscheNationalbib-
InvestigatingInteractionDynamicsofElastic
liografie;detailedbibliographicdataare
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availableontheInternetat
ApplyingtheTimeReversalMirror,Phys.Rev.
<http://dnb.d-nb.de>.
Lett.,98.10430(2007)
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ISBN 978-3-527-40703-3
V
Contents
Preface XI
Acknowledgements XIII
1 Introduction 1
1.1 Systems 1
1.2 ExamplesofPhenomena 4
1.3 TheDomainofExploration 6
1.4 Outline 7
References 8
2 Microscopic/Macroscopic Formulation of the Traditional Theory
ofLinearandNonlinearElasticity 11
2.1 PrefatoryRemarks 12
2.2 FromMicroscopictoContinuum 13
2.2.1 AMicroscopicDescription 13
2.2.1.1 MicroscopicEnergyandMicroscopicStrain 13
2.2.1.2 Phonons 16
2.2.2 MicroscopicDescriptionandThermodynamics 20
2.2.2.1 TheQuasiharmonicApproximation,inPrinciple 20
2.2.2.2 TheQuasiharmonicApproximationtoF 22
2.2.3 FromMicroscopicModeltoContinuumElasticity 24
2.3 ContinuumElasticityandMacroscopicPhenomenology 29
2.3.1 Displacement,Strain,andStress 29
2.3.2 DynamicsoftheDisplacementField 31
2.3.3 CouplingContinuumElasticitytoAuxiliaryFields 32
2.3.4 InhomogeneousElasticSystems 32
2.4 Thermodynamics 33
2.4.1 ThermodynamicDerivatives 33
2.4.2 SeriesExpansionforE 33
S
2.4.3 SeriesExpansionforE 34
Z
2.4.4 SeriesExpansionforF 35
T
2.4.5 AssemblethePieces 36
2.5 EnergyScales 37
NonlinearMesoscopicElasticity:TheComplexBehaviourofGranularMedia
includingRocksandSoil.RobertA.GuyerandPaulA.Johnson
Copyright©2009WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40703-3
VI Contents
References 38
3 TraditionalTheoryofNonlinearElasticity,Results 39
3.1 QuasistaticResponse;LinearandNonlinear 41
3.1.1 QuasistaticResponse;Linear 41
3.1.2 QuasistaticResponse;Nonlinear 42
3.2 DynamicResponse;Linear 44
3.3 Quasistatic/DynamicResponse;Nonlinear 45
3.4 DynamicResponse;Nonlinear 47
3.4.1 BasicEquations 47
3.4.2 WavePropagation 48
3.4.2.1 l+l →l 48
3.4.2.2 t+t →l andl+t →t 51
3.4.2.3 l+l+l →l,l+2l →l andmore 52
3.4.3 ResonantBar 55
3.4.3.1 l+l+l →l 55
3.4.3.2 l+t →t 57
3.5 ExoticResponse;Nonlinear 59
3.6 GreenFunctions 64
3.6.1 GreenFunction,FreeSpace 64
3.6.2 GreenFunction,ResonantBar 65
References 67
4 Mesoscopic Elastic Elements
andMacroscopicEquationsofState 69
4.1 Background 69
4.2 ElasticElements 70
4.2.1 Hertz–MindlinContacts 70
4.2.2 HystereticHertzianContacts 72
4.2.3 HertzianAsperities 72
4.2.4 VanderWaalsSurfaces 73
4.2.5 Other 75
4.3 EffectiveMediumTheory 76
4.4 EquationsofState;Examples 83
4.4.1 HertzianContacts 83
4.4.2 VanderWaalsSurfaces 87
4.4.3 GeneralizationandCaveats 94
References 95
5 AuxiliaryFields 97
5.1 Temperature 97
5.2 Saturation 102
5.2.1 Saturation/StrainCoupling 102
5.2.2 Saturation/StrainResponse 108
5.3 TheConditioningField,X 112
References 112
Contents VII
6 HystereticElasticElements 113
6.1 FiniteDisplacementElasticElements;QuasistaticResponse 113
6.1.1 FiniteDisplacementElasticElements:TheModel 113
6.1.2 FiniteDisplacementElasticElement:ImplementingtheModel 116
6.2 FiniteDisplacementElasticElements:Inversion 122
6.3 FiniteDisplacementElasticElements:DynamicResponse 122
6.3.1 FiniteDisplacementElasticElement:ResonantBar 123
6.3.1.1 LumpedElementModel 123
6.3.1.2 Stress-StrainforFiniteDisplacementElasticElement 125
6.3.1.3 ResonantBarResponse 126
6.3.2 FiniteDisplacementElasticElement:WaveMixing 130
6.4 ModelswithHysteresis 136
6.5 Summary 136
6.6 ModelswithHysteresis,Detail 137
6.6.1 HertzianContacts 137
6.6.1.1 TheMindlinModel;ShearContactwithFriction 137
6.6.1.2 AnfccLatticeofHertz–MindlinContacts 138
6.6.2 TheMasingRules 139
6.6.3 TheEndochronicFormalism 141
References 144
7 TheDynamicsofElasticSystems;FastandSlow 145
7.1 Fast/SlowLinearDynamics 145
7.1.1 QuasistaticResponse 146
7.1.2 ACResponse 149
7.2 FastNonlinearDynamics 153
7.3 AuxiliaryFieldsandSlowDynamics 158
7.3.1 X =TheConditioningField 159
7.3.2 X =Temperature 162
7.4 Summary 163
References 165
8 QandIssuesofDataModeling/Analysis 167
8.1 AttenuationinLinearElasticSystems 168
8.1.1 WaveVectorDispersion 168
8.1.2 ExtractingElasticEnergy 168
8.1.3 Other 169
8.2 NonlinearAttenuation 170
8.2.1 NonlinearDampling:TraditionalTheory 170
8.2.2 NonlinearDamping:HystereticElasticity 171
8.3 WhyMeasureQ? 171
8.4 HowtoMeasureQ 172
8.4.1 MeasurementofQ inaLinearSystem 172
8.4.2 MeasurementofQ inaNonlinearSystem 174
8.5 ResonantBarRevisited 176
8.5.1 ModelingaResonantBar 176
VIII Contents
8.5.2 DataProcessing 178
8.5.2.1 ConstantFieldAnalysis,CFA 180
8.5.2.2 TemplateAnalysis 186
References 197
9 ElasticStateSpectroscopiesandElasticStateTomographies 199
9.1 Spectroscopies 199
9.1.1 Linear,Homogeneous 200
9.1.2 Nonlinear,Homogeneous 200
9.2 Tomographies,Linear,Inhomogeneous 202
9.2.1 TimeofFlight 202
9.2.2 NormalMode 203
9.2.3 NormalMode,aNumericalExample 206
9.2.4 TimeReversal 209
9.2.5 TimeReversal,aNumericalExample 215
9.3 Tomographies,Nonlinear,Inhomogeneous 218
9.3.1 TimeofFlight 218
9.3.2 NonlinearNormal-ModeTomography 220
9.3.3 NonlinearTime-ReversalTomography 223
References 225
10 QuasistaticMeasurements 227
10.1 SomeBasicObservations 227
10.2 QuasistaticStress-StrainData;Hysteresis 231
10.3 CouplingtoAuxiliaryFields 236
10.3.1 Saturation 237
10.3.2 Temperature 242
10.4 Inversion 247
10.4.1 Simpleσ–εProtocolandMinimalistModel 248
10.4.2 Elaborateσ–εProtocolandMinimalistModel 251
10.4.3 TheRelationshipofσ–εDatatoDynamics 254
10.4.3.1 ApproximateTreatmentofσ–εData 254
10.4.3.2 Dynamics 256
10.4.3.3 QuasistaticDynamics 257
References 258
11 DynamicMeasurements 261
11.1 Quasistatic-Dynamic 261
11.1.1 Pressure-Dynamic 261
11.1.2 Temperature-Dynamic 264
11.1.3 Saturation-Dynamic 265
11.2 Dynamic–Dynamic 267
11.2.1 Dynamic–Dynamic:WaveMixing 268
11.2.1.1 NoncollinearWaveMixing 268
11.2.1.2 CollinearWaveMixing 270