Computational Modeling of Multiphase Geomaterial

GEOTECHNICAL ENGINEERING

Computational Modeling of Multiphase Geomaterials discusses how numerical methods play a very important role in geotechnical engineering and in the related activity of computational geotechnics. It shows how numerical methods and constitutive modeling can help predict the behavior of geomaterials such as soil and rock.

After presenting the fundamentals of continuum mechanics, the book explores recent advances in the use of modeling and numerical methods for multiphase geomaterial applications. The authors describe the constitutive modeling of soils for rate-dependent behavior, strain localization, multiphase theory, and applications in the context of large deformations. They also emphasize viscoplasticity and watersoil coupling.

Features

Explains how to predict the behavior of geomaterials

Contains the governing equations for multiphase geomaterials

Discusses the constitutive modeling of multiphase geomaterials, including elastoplastic and elastoviscoplastic models

Presents numerical methods, such as the finite element method, for analyzing geomaterials

Covers the latest developments in geomechanics, including the deformation-seepage flow coupled analysis of an unsaturated river embankment

Drawing on the authors well-regarded work in the field, this book provides you with the knowledge and tools to tackle problems in geomechanics. It gives you a comprehensive understanding of how to apply continuum mechanics, constitutive modeling, finite element analysis, and numerical methods to predict the behavior of soil and rock.

ISBN: 978-0-415-80927-6

9 780415 809276

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FUSAO OKASAYURI KIMOTO

COMPUTATIONALMODELING OFMULTIPHASEGEOMATERIALS

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A S P O N P R E S S B O O K

Y132935 cvr mech.indd 1 6/7/12 10:11 AM

COMPUTATIONALMODELING OFMULTIPHASE

GEOMATERIALS

A SPON PRESS BOOK

COMPUTATIONALMODELING OFMULTIPHASE

GEOMATERIALS

FUSAO OKASAYURI KIMOTO

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2013 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government worksVersion Date: 20120619

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v

Contents

Preface xvAcknowledgments xvii

1 Fundamentalsincontinuummechanics 1

1.1 Motion 11.2 Strainandstrainrate 2

1.2.1 Straintensor 21.2.2 Compatibilityrelationofstrain 41.2.3 Shearstrainanddeviatoricstrain 51.2.4 Volumetricstrain 6

1.3 Changesinarea 71.4 Deformationratetensor 81.5 Stressandstressrate 10

1.5.1 Stresstensor 101.5.2 Principalstressesandthe

invariantsofthestresstensor 121.5.3 Stressratetensorandobjectivity 16

1.6 Conservationofmass 191.7 Balanceoflinearmomentum 201.8 Balanceofangularmomentumandthe

symmetryofthestresstensor 221.9 Balanceofenergy 231.10 EntropyproductionandClausiusDuheminequality 241.11 Constitutiveequationandobjectivity 26

1.11.1 Principleofobjectivityandconstitutivemodel 271.11.2 Timeshift 281.11.3 Translationalmotion 281.11.4 Rotationalmotion 28

References 29

vi Contents

2 Governingequationsformultiphasegeomaterials 31

2.1 Governingequationsforfluidsolidtwo-phasematerials 312.1.1 Introduction 312.1.2 Generalsetting 322.1.3 Densityofmixture 332.1.4 Definitionoftheeffectiveandpartial

stressesofthefluidsolidmixturetheory 342.1.5 Displacementstrainrelation 342.1.6 Constitutivemodel 352.1.7 Conservationofmass 352.1.8 Balanceoflinearmomentum 352.1.9 Balanceequationsforthemixture 382.1.10 Continuityequation 38

2.2 Governingequationsforgaswatersolidthree-phasematerials 412.2.1 Introduction 412.2.2 Generalsetting 412.2.3 Partialstresses 422.2.4 Conservationofmass 432.2.5 Balanceofmomentum 452.2.6 Balanceofenergy 46

2.3 Governingequationsforunsaturatedsoil 462.3.1 Partialstressesforthemixture 472.3.2 Conservationofmass 482.3.3 Balanceoflinearmomentumforthethreephases 492.3.4 Continuityequations 51

References 52

3 Fundamentalconstitutiveequations 55

3.1 ElasticBody 553.2 Newtonianviscousfluid 573.3 Binghambodyandviscoplasticbody 583.4 vonMisesplasticbody 593.5 Viscoelasticconstitutivemodels 59

3.5.1 Maxwellviscoelasticmodel 603.5.2 KelvinVoigtmodel 613.5.3 Characteristictime 62

Contents vii

3.6 ElastoplasticModel 633.6.1 Yieldconditions 643.6.2 Additivityofthestrain 653.6.3 Loadingconditions 663.6.4 Stabilityofelastoplasticmaterial 673.6.5 Maximumworktheorem 693.6.6 Flowruleandnormality(evolutional

equationofplasticstrain) 713.6.7 Consistencyconditions 73

3.7 Overstresstypeofelastoviscoplasticity 763.7.1 Perzynasmodel 763.7.2 DuvautandLionsmodel 773.7.3 PhillipsandWusmodel 80

3.8 Elastoviscoplasticmodelbasedonstresshistorytensor 803.8.1 Stresshistorytensorandkernelfunction 813.8.2 Flowruleandyieldfunction 81

3.9 Otherviscoplasticandviscoelasticplastictheories 833.10 Cyclicplasticityandviscoplasticity 833.11 Dissipationandtheyieldfunctions 86References 89

4 FailureconditionsandtheCam-claymodel 91

4.1 Introduction 914.2 Failurecriteriaforsoils 92

4.2.1 FailurecriterionbyCoulomb 924.2.2 FailurecriterionbyTresca 934.2.3 FailurecriterionbyvonMises 934.2.4 FailurecriterionbyMohr 944.2.5 MohrCoulombfailurecriterion 944.2.6 MatsuokaNakaifailurecriterion 944.2.7 Ladefailurecriterion 954.2.8 Failurecriteriononplane 954.2.9 LodeangleandMohrCoulombfailurecondition 98

4.3 Cam-claymodel 1024.3.1 OriginalCam-claymodel 1024.3.2 Ohtastheory 1084.3.3 ModifiedCam-claymodel 1104.3.4 Stressdilatancyrelations 112

References 113

viii Contents

5 Elastoviscoplasticmodelingofsoil 115

5.1 Rate-dependentandtime-dependentbehaviorofsoil 1155.1.1 Strainrate-dependentbehaviorofclayeysoil 1155.1.2 Creepdeformationandfailure 1175.1.3 Stressrelaxationbehavior 1205.1.4 Strainrate-dependentcompression 1205.1.5 Isotaches 123

5.2 Viscoelasticconstitutivemodels 1255.3 Elastoviscoplasticconstitutivemodels 126

5.3.1 Overstressmodels 1265.3.2 Time-dependentmodel 1275.3.3 Viscoplasticmodelsbasedon

thestresshistorytensor 1275.4 Microrheologymodelsforclay 1285.5 AdachiandOkasviscoplasticmodel 128

5.5.1 Strainrateeffect 1375.5.2 SimulationbytheAdachiandOkasmodel 138

5.5.2.1 Effectofsecondaryconsolidation 1385.5.2.2 Isotropicstressrelaxation 140

5.5.3 Constitutivemodelforanisotropicconsolidatedclay 141

5.6 Extendedviscoplasticmodelconsideringstressratio-dependentsoftening 141

5.7 Elastoviscoplasticmodelforcohesivesoilconsideringdegradation 1425.7.1 Elastoviscoplasticmodelconsideringdegradation 1425.7.2 Determinationofthematerialparameters 1485.7.3 Strain-dependentelasticshearmodulus 149

5.8 Applicationtonaturalclay 1505.8.1 OsakaPleistoceneclay 1505.8.2 OsakaHoloceneclay 1525.8.3 Elastoviscoplasticmodelbasedon

modifiedCam-claymodel 1535.9 Cyclicelastoviscoplasticmodel 156

5.9.1 Cyclicelastoviscoplasticmodelbasedonnonlinearkinematicalhardeningrule 157

5.9.2 Cyclicelastoviscoplasticmodelconsideringstructuraldegradation 1585.9.2.1 Staticyieldfunction 158

Contents ix

5.9.2.2 Viscoplasticpotentialfunction 1595.9.2.3 Kinematichardeningrules 1595.9.2.4 Strain-dependentshearmodulus 1615.9.2.5 Viscoplasticflowrule 162

References 164

6 Virtualworktheoremandfiniteelementmethod 171

6.1 Virtualworktheorem 1716.1.1 Boundaryvalueproblem 1716.1.2 Virtualworktheorem 173

6.2 Finiteelementmethod 1756.2.1 Discretizationofequilibriumequation 1756.2.2 Discretizationofcontinuityequation 1796.2.3 Interpolationfunction 1806.2.4 Triangularelement 1816.2.5 Isoparametricelements 183

6.3 DynamicProblem 1906.3.1 Timediscretizationmethod 191

6.3.1.1 LinearaccelerationmethodandWilsonmethod 191

6.3.1.2 Newmarkmethod 1926.3.1.3 Centralfinitedifferencescheme 193

6.3.2 Massmatrix 1936.4 Dynamicanalysisofwater-saturatedsoil 193

6.4.1 Equationofmotion 1946.4.2 Continuityequation 201

6.4.2.1 Galerkinmethod 2016.4.2.2 Finitevolumemethod 202

6.4.3 Timediscretization 2066.4.3.1 Equationofmotion 2076.4.3.2 Continuityequation 207

6.5 Finitedeformationanalysisforfluidsolidtwo-phasemixtures 2106.5.1 Effectivestressandfluidsolidmixturetheory 2106.5.2 Equilibriumequation 2116.5.3 Continuityequation 2146.5.4 Discretizationoftheweakforms

fortheequilibriumequationandthecontinuityequation 216

x Contents

6.5.4.1 Discretizationoftheweakformsfortheequilibriumequation 216

6.5.4.2 Discretizationoftheweakformforthecontinuityequation 219

References 220

7 Consolidationanalysis 223

7.1 Consolidationbehaviorofclays 2237.2 Consolidationanalysis:smallstrainanalysis 225

7.2.1 One-dimensionalconsolidationproblem 2257.2.2 Two-dimensionalconsolidationproblem 2307.2.3 Summary 234

7.3 Consolidationanalysiswithamodelconsideringstructuraldegradation 2347.3.1 Effectofsamplethickness 2357.3.2 SimulationofAboshisexperimentalresults 238

7.3.2.1 Determinationofmaterialparameters 2387.3.2.2 Elasticparameters 2387.3.2.3 Viscoplasticparameters 2387.3.2.4 Consolidationanalysis 239

7.3.3 Effectofdegradation 2417.4 Consolidationanalysisofclayfoundation 244

7.4.1 Introduction 2447.4.2 Consolidationanalysisofsoftclay

beneaththeembankment 2447.4.2.1 Soilparameters 2447.4.2.2 Soilresponsebeneathembankment 245

7.5 Consolidationanalysisconsideringconstructionoftheembankment 2497.5.1 Numericalexample 251

References 255

8 Strainlocalization 259

8.1 Strainlocalizationproblemsingeomechanics 2598.1.1 Angleofshearband 260

8.2 Localizationanalysis 2618.3 Instabilityofgeomaterials 2648.4 Noncoaxiality 2708.5 Currentstress-dependentcharacteristicsandanisotropy 271

Contents xi

8.6 RegularizationofIll-posedness 2718.6.1 Nonlocalformulationofconstitutivemodels 2728.6.2 Fluidsolidtwo-phaseformulation 2738.6.3 Viscoplasticregularization 2738.6.4 Dynamicformulation 2738.6.5 Discretemodelandfiniteelement

analysiswithstrongdiscontinuity 2748.7 Instabilityandeffectsofthetransportofporewater 274

8.7.1 Extendedviscoplasticmodelsforclay 2768.7.2 Instabilityanalysisoffluid-

saturatedviscoplasticmodels 2788.7.2.1 Instabilityunderlocally

undrainedconditions 2788.7.2.2 Instabilityanalysisconsidering

theporewaterflow 2818.8 Two-dimensionalfiniteelementanalysis

usingelastoviscoplasticmodel 2828.8.1 Effectsofpermeability 2828.8.2 Strainlocalizationanalysisbythegradient-

dependentelastoviscoplasticmodel 2868.8.2.1 Finiteelementformulation

ofthegradient-dependentelastoviscoplasticmodel 286

8.8.2.2 Effectofthestraingradientparameter 2878.8.2.3 Effectoftheheterogeneity

ofthesoilproperties 2888.8.2.4 Mesh-sizedependency 290

8.9 Three-dimensionalstrainlocalizationanalysisofwater-saturatedclay 2918.9.1 Undrainedtriaxialcompressiontests

forclayusingrectangularspecimens 2928.9.1.1 Claysamplesandthetestingprogram 2928.9.1.2 Imageanalysis 293

8.9.2 Three-dimensionalsoilwatercoupledfiniteelementanalysismethod 294

8.9.3 Numericalsimulationoftriaxialtestsforrectangularspecimens 2958.9.3.1 Determinationofthe

materialparameters 2958.9.3.2 Boundaryconditions 296

xii Contents

8.9.3.3 Comparisonbetweenexperimentalandsimulationresults 297

8.9.3.4 Three-dimensionalshearbands 3018.9.3.5 Effectsofthestrainrates 302

8.10 Applicationtobearingcapacityandearthpressureproblems 305

8.11 Summary 306References 307

9 Liquefactionanalysisofsandyground 317

9.1 Introduction 3179.2 Cyclicconstitutivemodels 3179.3 Cyclicelastoplasticmodelforsand

withageneralizedflowrule 3199.3.1 Basicassumptions 3199.3.2 Overconsolidationboundarysurface 3199.3.3 Fadingmemoryoftheinitialanisotropy 3219.3.4 Yieldfunction 3229.3.5 Plasticstraindependenceoftheshearmodulus 324

9.3.5.1 Method1 3249.3.5.2 Method2 3249.3.5.3 Method3 3259.3.5.4 Method4 325

9.3.6 Plasticpotentialfunction 3269.3.7 Stressstrainrelation 328

9.4 Performanceofthecyclicmodel 3299.4.1 Determinationofmaterialparameters 329

9.5 Liquefactionanalysisofaliquefiableground 3329.5.1 VerticalarrayrecordsonPortIsland 3349.5.2 Numericalmodels 3349.5.3 Commonparameters 3359.5.4 Parametersforelastoplasticitymodel 3389.5.5 Parametersforelastoviscoplasticitymodel 3389.5.6 ParametersforRambergOsgoodmodel 3399.5.7 Finiteelementmodelandnumericalparameters 3409.5.8 Numericalresults 340

Contents xiii

9.6 Numericalanalysisofthedynamicbehaviorofapilefoundationconsideringliquefaction 3419.6.1 Simulationmethods 3449.6.2 Resultsanddiscussions 345

References 349

10 Recentadvancesincomputationalgeomechanics 353

10.1 Thermo-hydro-mechanicalcoupledfiniteelementmethod 35310.1.1 Temperature-dependentviscoplasticparameter 35410.1.2 Elasticandtemperature-dependentstretching 35710.1.3 Weakformoftheequilibrium

equationforwatersoilmixture 35810.1.4 Continuityequation 36010.1.5 Balanceofenergy 36310.1.6 Simulationofthermalconsolidation 365

10.2 Seepagedeformationcoupledanalysisofunsaturatedriverembankmentusingmultiphaseelastoviscoplastictheory 37010.2.1 Introduction 37010.2.2 Governingequationsandanalysismethod 37110.2.3 Constitutivemodelforunsaturatedsoil 371

10.2.3.1 Overconsolidationboundarysurface 37110.2.3.2 Staticyieldfunction 37210.2.3.3 Viscoplasticpotentialfunction 37310.2.3.4 Viscoplasticflowrule 37310.2.3.5 Constitutivemodelforporewater:

soilwatercharacteristiccurve 37410.2.4 Simulationofthebehaviorofunsaturated

soilbyelastoviscoplasticmodel 37510.2.5 Numericalanalysisofseepage

deformationbehaviorofalevee 37610.2.5.1 Analysismethod 37610.2.5.2 Deformationduringtheseepageflow 377

References 385

xv

Preface

Overthelastthreedecades,studiesonconstitutivemodelsandnumeri-cal analysis methods have been well developed. Nowadays, numericalmethodsplayaveryimportantroleingeotechnicalengineeringandinarelatedactivity calledcomputationalgeotechnics.Thisbookdealswiththeconstitutivemodelingofmultiphasegeomaterialsandnumericalmeth-odsforpredictingthebehaviorofgeomaterialssuchassoilandrock.Thebook provides fundamental knowledge of continuum mechanics, con-stitutivemodeling,numericalmethodsformultiphasegeomaterials,andtheirapplications. Inaddition, themonograph includesrecentadvancesinthisarea,namely,theconstitutivemodelingofsoilsforrate-dependentbehavior,strainlocalization,themultiphasetheory,andtheirapplicationsinthecontextoflargedeformations.Thepresentationisself-contained.Muchattentionhasbeenpaidtoviscoplasticity,watersoilcoupling,andstrainlocalization.

Chapter1presents the fundamental conceptandresults incontinuummechanics,suchasmotion,deformation,andstress,whicharenecessaryforunderstandingthefollowingchapters.Thischapterhelpsreadersmakeaself-consistentstudyofthecontentsofthisbook.

Chapter2dealswiththegoverningequationsformultiphasegeomaterialsbased on the theory of porous media, such as water-saturated and airwatersoilmultiphasesoilsincludingsoilwatercharacteristiccurves.Thischapterisessentialforthestudyofcomputationalgeomechanics.

Chapter3startswiththeelasticconstitutivemodelandreviewsthefun-damentalconstitutivemodelsincludingplasticityandviscoplasticity.Fortheplasticitytheory,thestabilityconceptinthesenseofLyapunovisdiscussed.Attheendofthechapter,cyclicplasticityandviscoplasticitymodelsarepresentedwithkinematicalhardeningrules.

InChapter4,failurecriteriaandtheCam-claymodelarereviewed.Forthefailurecriteria,manywell-knowncriteriahavebeenproposedinthischapter,fromCoulombscriteriontoMatsuokaNakaiscriterion.Then,theCam-claymodel is reviewedsince themodel includesadescriptionof thebasicpropertiesofsoilbehaviorsuchasdilatancyandthecriticalstateconcept.

xvi Preface

Chapter 5 is devoted to the rate- and time-dependent behavior andmodelingofsoils.Atfirst,typicalrate-andtime-dependentbehaviorsofsoilsarereviewedbasedontheexperimentalmeasurements.Severalrate-dependentmodelsarediscussedandelastoviscoplasticmodelsbasedontheCam-claymodelandPerzynasviscoplasticity theoryarepresented.AdachiandOkasmodelisfirstdescribedandthenanelastoviscoplasticmodelconsideringstructuraldegradationisintroduced.Thechapterendswiththecalibrationofthesemodelsusingtheexperimentalresults.

InChapter6,thevirtualworktheoremispresentedandthenthefiniteelementmethodfortwo-phasematerials isdescribedforquasi-staticanddynamicproblemswithintheframeworkoftheinfinitesimalstraintheory.

Chapter7dealswithatypicalmultiphasephenomenonofsoils;namely,theconsolidationproblem.Inparticular,theeffectsofsamplethicknessonconsolidation,usingAboshiswell-knowndata,andtheanomalousbehav-iorofporewaterdevelopmentintheclayfoundationbeneaththeembank-ment,duringloadingandaftertheendofconstructionembankment,arenumericallyanalyzed.

Chapter8 startswith a reviewof the studyon the strain localizationbehaviorofsoils.Severalissuesrelatedtothestrainlocalizationarethendiscussed for rate-independent and rate-dependent models. Finally, anumericalanalysisofthestrainlocalizationofwater-saturatedclayispre-sentedfortriaxialtestsandpracticalproblems.

InChapter9,aliquefactionanalysismethodispresentedwithacyclicelastoplasticmodelusingthetwo-phasetheorypresentedinChapter2forwater-saturatedsoils.Applicationsoftheliquefactionbehaviortoaman-madeislandduringanearthquakeandofthesoilpilestructureinterac-tionareshown.

Chapter10dealswithrecentadvancesingeomechanics.Itincludesthetemperature-dependentbehaviorofsoilssuchasconsolidationduetothechangeintemperature,andthenumericalanalysisofairwatersoilcou-pledproblems;namely,thedeformationseepageflowcoupledanalysisofanunsaturatedriverembankmentispresented.

xvii

Acknowledgments

During the writing and preparation of this book, the authors becameindebtedtomanyresearchersandstudents.Inparticular,weexpressoursincere thanks to Dr. K. Akai and Dr. T. Adachi, Emeritus Professorsof Kyoto University; Dr. H. Aboshi, Emeritus Professor of HiroshimaUniversity,forgivingusdataonconsolidation;Dr.S.Leroueil,ProfessorofLavalUniversity;Dr.A.YashimaofGifuUniversity;Dr.T.KodakaofMeijo University; Dr. R. Uzuoka of Tokushima University; Dr. Y. Higoof Kyoto University; Dr. F. Zhang of Nagoya Institute of Technology;Dr.K.Sekiguchi;Dr.A.Tateishi;Dr.Y.Taguchi;Dr.S.Sunami;Dr.M.Kato;Dr.M.J.Jiang,Dr.C.-W.Lu;Y.-S.Kim;Dr.Garcia;Dr.MojtabaMirjalili;Dr.R.Kato;Dr.YoungSeokKim;Dr.A.W.Karnawardena;Dr.H.Feng;Dr.Md.R.Karim;Dr.B.Siribumrungwong;Mr.T.Takyu;Mr.T.Satomura;Mr.N.Nishimatsu;Ms.T.Ichinose;Mr.Takada;andthegraduatestudentsoftheGeomechanicsLaboratoryofKyotoUniversityfortheircontributionsanddiscussions.WethankMs.ChikakoItohforherdailyassistance;Mr.ShahbodaghKhanBabak,aPhDstudentofKyotoUniversity,forhisassistanceinpreparingthefigures;andMs.H.GriswoldforherEnglishcorrections.Finally,wededicatethisbooktoourfamilies,inparticular,O.KeikoandK.Keiko.

Manythanksarealsoduetothefollowingorganizationsandtheresearchersforpermissionforusetheindicatedfigures:ProfessorH.Aboshi,Figure5.8;Professor T. Adachi, Figure 5.7a,b; Professor Liam Finn, Figure 5.3a,band Figure 5.5; Gihodo Syuppan Co. Ltd. (Dr. M. Saito), Figure 5.6;Professor G. Sllfore, Figure 5.9; American Society of Civil Engineers(ASCE),Figure10.1andFigure10.3;ASTM,Figure5.2andFigure5.12a,b;InstitutionofCivilEngineers(Gotechnique),Figure5.1a,bandFigure5.11;andNationalResearchCouncilofCanada(NRC)(CanadianGeotechnicalJournal),Figure5.10.

1

Chapter1

Fundamentals in continuum mechanics

In this book, we use vectors and tensors in components, and the directnotationsforthesevectorsandtensorsaregivenwithoutfurtherexplana-tion.Adotdenotesacontractionoftheinnerindices,forexample,a bi i

a bsothatA Bij ij A:B.

1.1 MOTION

The position of the material point X ii( 1 2 3)= , , of a body at time t isexpressedby

x x X ti i j ( )= , (1.1)

Material pointXi canbe givenby thepositionof xi at a time t= 0.Equation (1.1) expresses the motion of the material point of the body.TherectangularCartesiancoordinatesusedinthisbookaredescribedbyo e e e( )1 2 3 , , , withoriginoandunitbasevectorei .

Therearetwomethodsfordescribingthemotionofaparticle.Oneisthematerialdescription,inwhichthemotionisexpressedbymaterialpointXi ,andtheotheristhespatialdescription,inwhichthemotionisexpressedbyspatialcoordinatesxi .ThematerialdescriptioniscalledtheLagrangiandescriptionandthespatialdescriptioniscalledtheEuleriandescription.

Thevelocityvectorofaparticleisgivenby

v

x X tt

ii j( )=

,

(1.2)

In thematerial description, the acceleration of a particle in a body isexpressedby

a

v X tt

ii j( )=

,

(1.3)

2 Computationalmodelingofmultiphasegeomaterials

Inthespatialdescription,ontheotherhand,theaccelerationofaparticleisgivenby

a

v x tt

vv x tx

ii j

ki j

k

( ) ( )= ,

+

,

(1.4)

1.2 STRAIN AND STRAIN RATE

1.2.1 Strain tensor

Strainisthechangeinshapeorthechangeinvolumeofabodyduringtheapplicationof forcetothebody.Weneedanobjectivemeasureofstrainthatcanbederivedthroughchangesinthevariationofthelineelement.

Letusconsider themotionof thebodyshown inFigure1.1.Materialpoints P and Q have moved to points P and Q after the deformation.PointsQandQarethepointslocatedinthevicinityofpointsPandP.

Distance,dS,betweenpointsPandQ,isgivenby

dS dX dXa a2 = (1.5)

and thedistancebetweenpointsP andQ after thedeformation,ds, isgivenby

ds dx dxb b2 = (1.6)

wherethesummationconventionisusedfora,b= 1,2,3.

ui + dui

ui

dxi

Xi xi

PdXi

Q

Q

P

x1, X1

x2, X2

x3, X3

Figure 1.1 Motion.

Fundamentalsincontinuummechanics 3

Displacementvectoru ii( 1 2 3)= , , isgivenby

x X ui i i= + (1.7)

TakingthedifferencebetweenEquations(1.5)and(1.6),wehave

ds dS dx dx dX dX F F dX dX

xX

xX

dX dX E dX dX

k k k k ki kj ij i j

k

i

k

jij i j ij i j

( )

2

2 2 = =

=

= (1.8)

where Fij xXij

= is the deformation gradient and i j i jij (1 0 ) = , = ; , isKroneckersdelta.Eij inEquation(1.8)iscalledtheGreenstraintensor.

SubstitutingEquation(1.7)intoEquation(1.8),weget

E

uX

uX

uX

uX

iji

j

j

i

k

i

k

j

12

=

+

+

(1.9)

Forthecaseofinfinitesimalstrain,thatis, uXi

j1| |


where ij isgivenby

ux

ux

iji

j

j

i

12

=

(1.13)

andrepresentstherotationofasmallelement,forexample,therotationisnotzerofortherigidbodymotion.

1.2.2 Compatibility relation of strain

Thestraintensorhassixcomponents,althoughthedisplacementvectorhasonlythreecomponents.Thisindicatesthatweneedthreeindepen-dentequationstoobtainthedisplacementvectorfromthestraintensor.However, six compatibility equations exist among the strain compo-nents. As for the compatibility equations, three of them are indepen-dent,andcompatibility equationsarenecessaryandprovide sufficientconditions for single-value displacements in a simple connected body(seeMalvern1969).

As for the differentiation of the displacementstrain relations withrespecttocoordinates,weobtainthecompatibilityequationsas

y x x yxx yy xy

2

2

2

2

2

+

=

(1.14)

z y y zyy zz yz

2

2

2

2

2

+

=

(1.15)

x z z xzz xx zx

2

2

2

2

2

+

=

(1.16)

y z x x y z

xx yz xz xy22

=

+

+

(1.17)

x z y x y z

yy yz xz xy22

=

+

(1.18)

x y z x y z

zz yz xz xy22

=

+

(1.19)


Equations(1.14)to(1.19)canbeexpressedbyatensornotationas

ij kl kl ij ik jl jl ik0 + =, , , , (1.20)

wherei,j,k,l= 1,2,3.

1.2.3 Shear strain and deviatoric strain

Let us consider the deformation shown in Figure 1.2. The displacementvectorisgivenbyu c y u c x c cx y 01 2 1 2= , = , , > .

Then,

uy

ux

c cxy xyx y2 ( )1 2 = =

+

= + (1.21)

andthestraincomponents, xx yy , ,arezero.Assuming a small deformation gradient, u

yx

is given by 1 and ux

y

isgivenby 2 ,namely,

uy

ux

x y1 2

+

= + (1.22)

This indicatesthat xy expresseschanges intheangle, inotherwords,changesintheshape,thatis,shearingdeformation.

eij ij ij kk

13

= (1.23)

isdefinedasthedeviatoricstraintensor.

xO2

1

ux

ux

y

Figure 1.2 Sheardeformation.


1.2.4 Volumetric strain

SettingVasthevolumeafterthedeformationandV0asthevolumebeforethedeformation,volumetricstrain v isexpressedby V V Vv kk ( )0 0 = = / .

IfweexpressthevolumebeforethedeformationbyV dX dX dX0 1 2 3= ,

dX dX dX dX dX dX dX dX dX

o

v [(1 )(1 )(1 ) ]

( )

11 22 33 1 2 3 1 2 3 1 2 3

11 22 33

= + + + /

= + + + (1.24)

whereo()isthehigher-ordersmallterm.Wecandisregardthehigher-ordertermforthesmalldeformationcase.

Next,wewillconsiderthechangesinvolumeforthefinitedeformationcase.Thevolumeofthesmallhexahedronafterthedeformation,dV,isgivenby

dV d d d dx dx dxijk i j kx x x( )= = (1.25)

where ijk isapermutation(oralternating)symbol.The volume of the small hexahedron before the deformation, dV0, is

givenby

dV d d d dX dX dXpqr p q rX X X( )0 = = (1.26)

Usingthedeformationgradient,Fij xXij

= ,weget

J det F F F Fmn ijk pqr ip jq kr( )

16

= (1.27)

Usingthefollowingrelation:

ijk pqr

ip iq ir

jp jq jr

kp kq kr

=

(1.28)

weobtain

det F F F Fpqr mn ijk ip jq kr( ) = (1.29)

As for Equation (1.29), it is worth noting that det F F F Fmn ijk i j k( ) 1 2 3= followingtheexpansionofthedeterminantdet Fmn( ).


Ifpqrisanevenpermutationof1,2,3,wehavethedeterminantandifpqrisoddwehavethenegativeone.

Consequently,wehave

dV F F F dX dX dX

det F dX dX dX JdV dV

ijk ip jq kr p q r

pqr mn p q r( ) 00

0

=

= = =

(1.30)

where 0 andaretheinitialmassdensityandthecurrentmassdensity,respectively.

Disregardingthehigher-ordertermleadsto

J det F

uX

iji

i

( ) 1= +

(1.31)

Therefore,weobtainthefollowingrelationconsistenttoEquation(1.24)as

dV dVdV

ii v0

0

= = (1.32)

1.3 CHANGES IN AREA

ThechangesinareahavebeenestimatedbyNansonsformulaincontin-uummechanics(Malvern1969)andaregivenby

ds J dSTn F N 0= (1.33)

wherenistheunitnormaltoareadsinthecurrentconfiguration,dsisanareainthecurrentconfiguration,Nistheunitnormaltotheinitialcon-figuration,F 1 istheinverseofthedeformationgradient,anddS0isanareaintheinitialconfiguration.

Surface vector dSN 0 at point X in the referential configuration isexpressedby

dS d dN X X0 = (1.34)

wheredXisaninfinitesimalvectoratpointX.Surfacevector dsn atpointxinthecurrentconfigurationisexpressedby

ds d d n ds dx dxs spq p qn x x , ( )= = (1.35)

wheredx isaninfinitesimalvectoratpointx.


FromEquation(1.34),weobtain

N dS

Xx

Xx

dx dxi ijkj

p

k

qp q0 =

(1.36)

Then, multiplying both sides of Equation (1.36) by Xxis

and usingEquations(1.28)and(1.35),weget

JXx

N dS n dsis

i s0

= (1.37)

inwhichtherelation JXx

Xx

Xx

spq ijki

s

j

p

k

q

1 =

Consequently, we get Equation (1.33), called Nansons theorem, sincecomponentsof F 1 and TF are

F

Xx

FXx

iji

jijT j

i

,1 =

=

1.4 DEFORMATION RATE TENSOR

Whenwedealwiththelargedeformationofabody,thematerialconfigurationchangeseachtime,andthedeformationratetensorisusefulfortheanalysis.

TakingatimederivativeofEquation(1.8)leadsto

ddt

ds dS dxddt

dxk k( ) 2 ( )2 2 = (1.38)

ddt

dxddt

xX

dXxX

ddtdXk

k

mm

k

mm( ) =

+

vX

dXkm

m=

dv L dxk km m= = (1.39)

L

vx

iji

j

=

(1.40)

whereLij iscalledthevelocitygradienttensor.


Thevelocitygradient tensor canbe separated into symmetricpartDij andantisymmetricpartWij as

L D Wij ij ij= + (1.41)

D

vx

vx

iji

j

j

i

12

=

+

(1.42)

W

vx

vx

iji

j

j

i

12

=

(1.43)

SubstitutingtheprecedingequationsintoEquation(1.39)gives

ddt

ds dS dxddt

dx dx v dx dx L dx

dx D dx dxW dx

i i i i m m i im m

i im m i im m

( ) 2 ( ) 2 2

2 2

2 2 = = =

= +

, (1.44)

SinceWij isskewsymmetric, dxW dxi im m2 0= .Hence,wehave

ddt

ds dSddt

ds dx D dxi im m( ) ( ) 22 2 2 = = (1.45)

Subsequently,Dij isusedtoexpressthemeasureofthedeformationrateatthecurrentconfiguration,whichiscalledtherate-of-deformationtensororthestretchingtensor.Incontrast,Wij denotestherateofrotationandiscalledthespin.

In a small deformation field, we do not distinguish the deformationbetweenthecurrentandthereferenceconfigurations.Hence,wecanusestrainrate ij insteadofthedeformationratetensoras:

ux

ux

iji

j

j

i

12

=

+

(1.46)


1.5 STRESS AND STRESS RATE

1.5.1 Stress tensor

Theforcesactingonabodycanbeclassifiedintotwoforces:thebodyforceandthesurfaceforce.Thebodyforceistheforceactingonthebodyremotely,suchasthegravitationalforce,whichisproportionaltothemassvolume.Thesurfaceforceistheforceactingonthebodythroughthesurface,whichisproportionaltotheareaofthesurfaceandiscalledthestressvector.

Letusconsiderthesurfaceforce t dst x n( , , ) ,showninFigure1.3(a),actingonthesmallsurfaceelementdsofthecrosssectionofthebodyatapositionx.tisasurfacetractionvectorperunitareaactingonsideIIfromsideI.

Incontrast,theforceactingonsideIfromsideIIhasthesamemagnitudeofforceasthatfromsideItosideII,onlyinanoppositedirection.

t ds t dst x n t x n( , ) ( , ), = , (1.47)

inwhichnistheunitnormalvectortothesurface.Thesurfaceforceperunitareatiscalledthestressvectororthetractionvector.

Considerthestressstateofatetrahedron,whichisinequilibriumunderthesurface,andthebodyforcesshowninFigure1.3(b).

TheareaofABCisS,theareaofOBCis S1,theareaofOCAisS2 ,andtheareaofOABis S3.Then,

S Sn n n ni i n ( )1 2 3= , = , , (1.48)

F3F1

F2

IIt(n)

t(n)

Pn

n

I

t(e1)

t(e2)

t(e3)

t(n)

e1

132

e2

e3

C

Pn

x2O B

x1

x3

A

(a) (b)

Figure 1.3 (a)Forcebalance.(b)Tractionvectorsinequilibrium.


The reason is as follows:Whenwe set the intersectionpointbetweentheperpendicularlinefrompointOandABCaspointP,andOP= h,thevolumeofthetetrahedronisSh/3.

Hence,

Sh S AO S BO S CO1 2 3= = = (1.49)

Then,

S S h AO ncos1 1 1/ = / = (1.50)

where ni isthedirectioncosine cos i .Fromtheequilibriumoftheforcesactingonthetetrahedron,wehave

S S S S Sht n t e t e t e F a( ) ( ) ( ) ( ) ( ) 3 01 1 2 2 3 3+ + + + / = (1.51)

whereFisthegravitationalforceandaistheinertialforce.Ash0,Equation(1.51)becomes

n n nt n t e t e t e( ) ( ) ( ) ( ) 01 1 2 2 3 3+ + + = (1.52)

Equation(1.52)canberewrittenas

ni

i it n t e( ) ( ) 01

3

+ ==

(1.53)

Then,ifweusethefollowingexpression

nk

k kn e1

3

==

(1.54)

thetractionvectorbecomes

n n nk k k k k kt e t e t e( ) ( ) ( )= = (1.55)

whereEinsteinssummationconvention n nk

k k k ke e1

3 =

isused.


Thestressvectorcanbedefinedas

m

k

mk kt e e( )1

3

=

(1.56)

FromEquations(1.55)and(1.56),wegetCauchysfundamentaltheoremofstressvectoras

nm k

mk k mt n e( )1

3

= , =

(1.57)

where mk iscalledthestresstensor.Thekthcomponentof t isgivenby tk = nmk m . mk denotes thecom-

ponentofthestressvectorinthe xkdirectionactingontheperpendicularplanetothexm axis.

Whenwedisregardthecouplestress(seeSection1.8),thestresstensorbecomessymmetricfromtheequilibriumofthemomentas

ij ji = (1.58)

TheCauchystresstensorisexpressedbyboth ij andTij inthisbook.

1.5.2 Principal stresses and the invariants of the stress tensor

Ingeneral,thestressvectorisnotparalleltothenormalvectorofthesec-tion,asshowninFigure1.3.Inacertaindirection,however,thestressvec-torisparalleltothedirectionofnormalvector,ni ,inwhichdirectionstressvector,ti ,canbeexpressedby

t n ni ji j i= = (1.59)

whereisthemagnitudeofthestressvector.Since ij issymmetric,wehave

n nji ij j ij ij i( ) ( ) 0 = = (1.60)

Equation(1.60)isasetoflinearhomogeneousequationsforniandhasanontrivialsolution,thatis,ni 0 ifandonlyifthefollowingrelationholds:

det ij ij 0| |= (1.61)


Equation (1.61) is an eigenvalue equation. When the stress tensor issymmetric,Equation(1.61)hasthreerealroots(realeigenvalues).Thesethreeeigenvalues, 1 2 3 , , ,arecalledprincipalstresses.Thedirectionofni ,whichsatisfiesEquation(1.60),iscalledtheprincipalstressdirection.

Equation(1.61)canbewrittenas

I I I 03

12

2 3 + = (1.62)

I1 11 22 33= + + (1.63)

I ( )2 11 22 22 33 33 11 122

232

312= + + + + (1.64)

I 2 ( )3 11 22 33 12 23 31 11 232

22 312

33 122= + + + (1.65)

SinceEquation(1.61)holdsfortheprincipalstressconditions,wehave

I I I ( )( )( ) 03

12

2 3 1 2 3 + = = (1.66)

Consequently,fromtherelationbetweenrootsandcoefficients, I I I1 2 3, , canbeexpressedas

I1 1 2 3= + + (1.67)

I2 1 2 2 3 3 1= + + (1.68)

I3 1 2 3= (1.69)

Since I I I1 2 3, , areinvariantsundertherotationofthecoordinates,theyarecalledthefirst,thesecond,andthethirdinvariants,respectively.

Alternatively,thesethreeinvariantscanbeexpressedbyI I I1 2 3, , as

I ii1 = (1.70)

I ij ij

12

2 = (1.71)

I ij jk ki

13

3 = (1.72)


Thedifferencebetweenthestresstensorandthemeanvalueofstressten-sor, m ,iscalledthedeviatoricstresstensor,sij ,as

sij ij m ij= (1.73)

m

13( )11 22 33 = + + (1.74)

Forthedeviatoricstresstensor,threestressinvariantsexist, J J J1 2 3, , ,as

J J s s J s s sij ij ij jk ki0

12

13

1 2 3= , = , = (1.75)

The angle of the coordinates for specifying the principal stresses isobtainedbysetting xy 0 = inEquation(1.77)as

xy

xx yy

tan22

=

(1.76)

xy xy

xx yycos2( )

2sin2 =

(1.77)

where xy isacomponentofthestresstensor,whichistransformedwithrespecttotherotationofthecoordinates,andistheanglebetweenthereferencecoordinatesandcorrespondstotheprincipalstressdirections.

PROBLEMShowtheprincipalstressesandtheirdirectionsforthefollowingstresstensor:

[ ] ij =4 1 11 2 11 1 2

Answer:Ifwesetastheprincipalstress,

det4 1 11 2 11 1 2

0

=


Thisyields ( ) ( ) ( ) ( )2 4 2 4 2 2 02 + = .Then,theprincipalstressesare1 5= ,2 2= ,and 3 1= .

From Equation (1.60), the direction corresponding to 51 = is givenas n n n n n n n n n0 3 0 3 01 2 3 1 2 3 1 2 3 + + = , + = , + = . Then, : :n n n1 2 3= : :2 1 1, where n n n ni ( )1 2 3= , , are the components of the unit principaldirectionvector.Similarly,theprincipaldirectionfor 22 = isobtainedas

n n n n n n n2 0 0 01 2 3 1 3 1 2+ + = , + = , + = . Then, n n n 1 1 11 2 3: : = : : . Theprincipaldirection for 13 = is n n n n n n3 0 01 2 3 1 2 3+ + = , + + = ,

and thenn n n 0 1 11 2 3: : = : : .

LetusconsiderthecurrentforcevectorbythenominalstressvectorthatisthestressvectorwithrespecttosurfaceareadS0inthereferenceconfigu-rationas

t T ni ji j= (1.78)

s Ni ji j= (1.79)

where ij isthenominalstresstensororthefirstPiolaKirchhoffstresstensor.Sincetheforceisinequilibrium,

t ds s dSi i 0= (1.80)

Nansonstheorem,Equation(1.37),gives

JN dS

xX

n dsjs

js0 =

(1.81)

BysubstitutingEquations(1.78)and(1.79)forEquation(1.80)andusingEquation(1.81),weobtain

JT

xX

kik

jji=

(1.82)

Then,

= =

J JXxTij

i

kkjF T or

1 (1.83)


1.5.3 Stress rate tensor and objectivity

When we consider the stress rate, we have to examine the objectivityof it.Theobjectivity isdefinedastheindependenceofthemotionwithrespect to theobservers.Since thephysical lawhas tobeobjective, thephysicallawincludingtheconstitutiveequationofmaterialsshouldsat-isfytheobjectivity.Here,wewilldiscusstheobjectivityofthestressandthestressratetensor.Theobjectivityfortheconstitutiveequationswillbediscussedinthenextsection.Herein,theobjectivityfollowsmostlythatbyMalvern(1969).

Theobserverforaneventiscalledareferenceframe.Forthedifferentobservers,thereexistsatransformationamongthem,whichisexpressedbyaEuclidtransformation.

The Euclid transformation between two frames, tx( , ) and tx( , )* * , isgivenby

t tx Q x c( ) ( )= + (1.84)

t t t0= (1.85)

where tQ( )denotesanorthogonaltensorthatexpressestherotationbetweentwoframes, tc( )istherelativemotionoftheorigin,and t0expressesthetimedifference.

Forthechangeinreferenceframefromxto x*byEquations(1.84)and(1.85),scalarC,vectoru,andtensorEaretransformedas

C C= (1.86)

u Qu= (1.87)

E QEQT= (1.88)

Forphysicallawstobeobjective,theyhavetobedescribedbytensorquan-tities.ThedifferentiationofEquation(1.84),withrespecttotime,provides

t t t tv c Q v Q x( ) ( ) ( ) ( )= + +

t t t t tTc Q v Q Q x c( ) ( ) ( ) ( )( ( )) = + + (1.89)

wherethesuperimposeddot( )indicatestimedifferentiation.


Ifweset

TA QQ= (1.90)

Aistheangularvelocitytensoroftheunstarredframetothestarredframe.Inthefollowing,wewillexaminethedeformationgradient(F x XiJ i J= / ),

velocity gradient (L v xij i j L FF 1= / , = ), rate of deformation (stretching)tensorD,spintensorW,andCauchystresstensorT.

Ifboththenewandtheoldframesinthereferencestatearethesame,fromdx Qdx QFdX= = ,weobtain

F QF= (1.91)

Differentiatingtheprecedingequation,

F QF QF = + (1.92)

F F F F Q QFF Q AT T1 1 1 = = + (1.93)

Then,bysettingL F F 1= ,velocitygradienttensorLbecomes

L QLQ AT= + (1.94)

Consequently,Lisnotobjective.Sincethestretchingtensor(orthedeformationratetensor),D L LT( ),12= +

satisfiesthetransformationas

D QDQT= (1.95)

Disobjective.Moreover,thespintensor,W L LT( )12= ,transformsas

W QWQ AT= + (1.96)

Hence,Wisnotobjective.Wisacontinuumspintensoranddoesnotrep-resentrigidbodymotion.

TheCauchystresstensorfollowsthetransformationas

T QTQT= (1.97)


TakingatimederivativeofEquation(1.97)gives

TQTQ QTQ QTQT T

T = + + (1.98)

Then,thetimederivativeofthestresstensorisnotobjective.Letuscon-siderthequantityasT WT TW + .Since

Q AQ = (1.99)

QQ IT = (1.100)

T

Q Q AT = (1.101)

wehave

T W T T W Q T WT TW QT[ ] + = + (1.102)

Hence,Tisobjective.

T T WT TW = + (1.103)

whereTiscalledtheJaumannstressratetensor.Theotherstressrate,theJaumannderivativeofKirchhoffstress( JT),

o

T

,isobjective.

T T T L L

( ), ( )o

iitr tr L= + = (1.104)

DifferentiatingEquation(1.82)withrespecttotime,

vX

xX

JT JTkj

jik

jji ki ki

+

= + (1.105)

Hence,

xX

JT JTvX

J TJJT

vxT

J T L T L T

k

jji ki ki

k

jji

ki kik

ppi

ki pp ki kp pi

( )

= +

= +

= +

(1.106)


MultiplyingEquation(1.106)by

X

xq

k,wehave

Xx

xX

JXx

T L T L Tqk

k

jji

q

kki pp ki kp pi

( )

=

+ (1.107)

Subsequently,

JXx

Sjij

kki

=

(1.108)

S T L T L Tki ki pp ki kp pi + (1.109)

whereSij isthenominalstressratewithrespecttothecurrentconfiguration.

1.6 CONSERVATION OF MASS

TheconservationofmasscontinuousmediumVisexpressedasthebal-anceofmassofvolumeVholdsifthereisnomassinflowintovolumeVandnomassisproducedinV.Inthiscase,thebalanceofmassforanarbitraryvolumeVisexpressedby

DDt

dvV

0 = (1.110)

whereD/Dtdenotesthematerialtimederivative.Equation(1.110)gives

+ =

DDt

vx

dvV

i

i

0 (1.111)

Whentheintegrandisacontinuousfunction,thelocalformforthebal-anceofmassis

DDt

vxi

i

0+

= (1.112)

Incontrast,usingEquation(1.29),themassconservationlawisexpressedby

dV dV J= =0 0 0or (1.113)


where is the mass density after the deformation and 0 is the densitybeforethedeformation.

Equation (1.112) is an Eulerian description and Equation (1.113) is aLagrangiandescriptionofthemassconservationlaw.

1.7 BALANCE OF LINEAR MOMENTUM

ThebalanceoflinearmomentumisgivenbythestatementthechangeinthelinearmomentumofthebodyoccupyingregionR(=volumeV+theboundaryS)isproportionaltotheforceactingonthebody.Thebalanceofmomentumisexpressedas

DDt

v dv t ds b dvV

iS

iV

i = + (1.114)

where DDt

isthematerialtimederivative,vi isthevelocityvector, ti isthestressvector,bi isthebodyforce,andisthemassdensity.

Theleft-handsideofEquation(1.114)indicatesthetimechangeofthelinearmomentum,andthefirstandsecondtermsontheright-handsideexpressthesurfaceforceandthebodyforce,respectively.

UsingCauchystheorem(Equation1.57, t ni ji j= )andtheGausstheo-rem,andconsideringthebalanceofmass,wehave

V

i iji

j

a bx

dv

=

0 (1.115)

wherea Dv Dti i= / istheaccelerationterm.IfEquation(1.115)holdslocally,

xb aji

ji i

+ = (1.116)

Whendisregardingtheaccelerationterm,thatis,forthequasi-staticcase,Equation(1.116)iscalledtheequilibriumequation.

Equation (1.116) can be expressed in component form for a two-dimensionalproblemasx x x y x z,1 2 3 , ,x,ycomponents.

x y zbxx yx zx x 0

+

+

+ = (1.117)


x y zbxy yy zy y 0

+

+

+ = (1.118)

x y zbxz yz zz z 0

+

+

+ = (1.119)

wereb b bx y z,, arecomponentsofthebodyforcevector.Thebalanceoflinearmomentuminthereferenceconfigurationisgivenby

a dV dV b dViVV

ji j i

V

0 0 , 0 0 0 = + (1.120)

where ai istheaccelerationvectorand ij isthenominalstresstensorinEquation(1.83).

Taking a time derivative of the first term on the right-hand side ofEquation(1.120)andusingNansonstheoremand J 0 = ,wefind

DDt

N dS JSXxN dS

JSXx J

xX

n ds

S n ds

ji

S

j kij

kS

j

kij

kS

p

jp

ki

S

k

1

0 0

=

=

=

(1.121)

Hence,aratetypeofbalanceoflinearmomentum,withrespecttothecurrentconfiguration,isobtainedas

a dv S n ds b dvi ki kSV

i

V

= + (1.122)

Underthestaticconditionswithconstantbodyforce,theprecedingequa-tionbecomes

S dvki kV

0, = (1.123)

TheaboverateequilibriumequationwillbeusedfortheupdatedLagrangianformulationoftheboundaryvalueproblem.


1.8 BALANCE OF ANGULAR MOMENTUM AND THE SYMMETRY OF THE STRESS TENSOR

Fromthebalanceofangularmomentum,thesumofthemomentumiszerointhecaseofazerotimeratefortheangularmomentum.Then,

s ds dv dv s ds

v vt n r b r a r M( ) 0 + + = (1.124)

wheredenotesthevectorproduct, t n( )isthestressvector,bisthebodyforcevector,aistheinertiaforcevector,risthepositionvector,andMisthecouplestressvector.Couplestressiscalledmomentstressandcannotbe disregarded for materials with a significant rotation of the particles,suchasgranularmaterials.

Equation(1.124)canbewrittenincomponentformas

t x ds b x dv a x dv M ds

sijk k j

Vijk k j

Vijk k j

si 0 + + = (1.125)

where ijk isthepermutationsymbol.UsingCauchystheoremandthedivergencetheorem,wehave

sijk k j

sijk j mk m

vijk j

mk

mjt x ds x n ds x x = =

+

kk dv (1.126)

Consideringcouplestresstensorij,weobtainM ni ji j ;= andEquation(1.125)becomes

V

ijk jmk

mk k

Vijk jkx x

b a dv

+ +

++

=ji

jxdv 0 (1.127)

Upon substitutingEquation (1.116), thefirst termofEquation (1.127)becomeszero.

Hence,

x

dvV

ijk jkji

j

0 +

= (1.28)


Whenthestressdistributioniscontinuous,thelocalformforEquation(1.128)is

xijk jk

ji

j

0 +

= (1.129)

whencouplestresstensorjiiszero.Fori= 1, 123 23 132 32 23 32 23 320+ = = =, .Ingeneral,

ij ji = (1.130)

Consequently, thestresstensor issymmetricwhencouplestresstensorjiiszero.

1.9 BALANCE OF ENERGY

Theenergyconservationlawiscalledthefirstlawofthermodynamics,anditisdescribedasfollows:Thetimerateoftotalenergyofthemasssystemisequaltothesumoftheexternalmechanicalworkratedonebythebodyforceandthesurfaceforce,heatinflowthroughthesurfaceofthebodyandtheotherenergysupply.

K E F Q + = + (1.131)

K

DDt

v v dvV

i i12

= (1.132)

F b v dv t v ds

Vi i

Si i = + (1.133)

E edv

V

= (1.134)

Q hdv q n ds

V Si i = (1.135)

whereK istherateofthemechanicalenergy,Eistheinternalenergy,eistheinternalenergydensity,Fistheexternalworkrate,Qistheheatinflowandothersuppliesofenergy,histheenergysupplydensitysuchasradia-tion,andqiistheheatflowvector.


Fromtheconservationoflinearmomentum,wehave

F

DDt

v v dv D dvV

i iV

ij ij12 = + (1.136)

Then,

E D dv hdv q n ds

Vij ij

V Si i

= + (1.137)

ThelocalformforEquation(1.131)becomes

e D h qij ij i i = + , (1.138)

1.10 ENTROPY PRODUCTION AND CLAUSIUSDUHEM INEQUALITY

Thesecondlawofthermodynamicsisdescribedasfollows:Thetimerateoftheentropyofthebodyisnotlessthanthechangeinentropyassociatedwiththeheatinflowandtheothersuppliesofenergy.

Inotherwords,theentropyproductionduringthemotionofabodyisnotalwaysnegative.

N H (1.139)

N

DDt

dvV

= (1.140)

whereistheentropydensity.

H

hdv

qn ds

V S

ii

= + (1.141)

whereisthetemperature.

DDt

h qi

i

,

(1.142)

ThelocalformforEquation(1.139),ClausiusDuhemInequality,is


Ifweset D Dt = / ,usingthefirstlawofthermodynamics,thepreced-ingequationbecomes

D e q

xij ij i

i

10 +

(1.143)

If we set Helmholtzs free energy function as = e, Equation(1.143)becomes

D q

xij ij i

i

( )1

0 + +

(1.144)

Letusconsiderthecasewithasmallchangeindensityandthefollow-ingrelations:

ije ( ) = , (1.145)

Dij ije ij

vpij ij = + , = (1.146)

where ije and ijvp are the elastic strain rate and the inelastic strain rate,

respectively.FromEquation(1.144),wehave

+

+

ije ij

eij ij

1qq

xi i

0 (1.147)

ij

ije =

, =

(1.148)

Hence,

q

xij ij

vpi

i

10

(1.149)

Equation(1.149)indicatesthattheinternalentropyproductionoccursduetotheinelasticstrainandtheheatflow.


TruesdellandNoll(1965)definedinternalentropyproductionas

D e q

xij ij i

i

1 1 102 = +

(1.150)

ThestrongsufficientconditionsforEquation(1.150)tobetruearegivenbythefollowingtwoinequalities:

D eij ij

1 10 = +

(1.151)

q

xi

i

102

(1.152)

WhenFourierslawofheatflowisexpressedas

qi i,= :heatconductioncoefficient (1.153)

Equation(1.152)becomes

xi

102

2

(1.154)

Consequently,

0 (1.155)

Namely,isnonnegative.An important thermodynamical framework for the plasticity theory

hasbeenstudiedbyCollinsandHoulsby(1997)basedontheZieglerstheory of dissipation function. The related results will be presented inChapter3.

1.11 CONSTITUTIVE EQUATION AND OBJECTIVITY

Aswasbeenmentionedearlier, therearenine fundamental laws incon-tinuummechanics,exceptforelectromagneticlaws.Theseincludethemassconservation law (1), theconservation lawsof linearmomentum(3), theconservationlawsofangularmomentum(3),theconservationofenergy(1),andtheentropyproductioninequality(1,constraintcondition).


Ontheotherhand,therearenineteenvariablescontainedinthelaws,namely,themassdensity(1),velocitycomponents(3),thecomponentsofthestresstensor(9),temperature(1),thecomponentsoftheheatflowvector(3),internalenergy(1),andentropy(1).

Hence,elevenmoreequationsarerequired todescribe theresponseofmaterials.Theseelevenequationsarecalledconstitutiveequationsinordertospecifytheresponsecharacteristicsofmaterials.Thenumberofequa-tionsiseleven,thatis,sixforstressstrainrelations,threeforheatflux,oneforinternalenergy,andoneforentropy.

Constitutive equations are not given a priori but are derived basedon experiments satisfying the fundamental laws and objectivity. Thewell-known Hookes law is a typical constitutive equation for elasticmaterials.

1.11.1 Principle of objectivity and constitutive model

Theresponseofamaterialtoexternalactionisindependentoftheobserver.Itindicatesthattheconstitutiveequationshouldbeindifferenttochangesinthecoordinateframe.Inthefollowing,wewilldiscusstheobjectivityoftheconstitutiveequationsandhowtheprincipleofobjectivityprescribesconstitutiveequations.

Constitutiveequationsdescribethematerialsinherentresponsetoexter-nalactionandareexpressedbyafunctionalofthehistoryofmotionanddeformation.Thisfunctionaliscalledtheconstitutivefunctional.

Forexample,whenthestresstensorTatamaterialpointXisdeterminedbythemotionofmaterialpoint X inthevicinityofX,constitutivefunc-tionalGisgivenby

t tT G x X X X[ ( ) ( ) ]= , , , , (1.156)

ConstitutivetensorfunctionalGhastobeindifferentwithrespecttotherigidrotation,thetranslationalmotion,andthetimeshiftsothatconstitu-tivefunctionalGsatisfiestheprincipleofobjectivityas

t t t tx X Q x X c( ) ( ) ( ) ( ), = , + (1.157)

t t t0= (1.158)

t t t t t t tG x X X X G x X X X[ ( ) ( ) ] [ ( ) ( ) ]0, , , , , = , , , , , (1.159)

whereX isapointinthevicinityofpointX.


Intheabove,weassumethatfunctionalGdependsonthemotionofthematerialpointsinthevicinityofpointXand,ingeneral,theconstitutivefunctionaldependsonthepasthistory( t t < ).Herein,however,wedisregarditforthesakeofsimplicity.

1.11.2 Time shift

Whenweset t IQ( ) = , tc( ) 0= ,t t0 = ,and = , = t t tx X x X( , ) ( , ) 0.Then,forexample,thestresstensorTcanbeexpressedbyfunctionalGas

T G x X X X[ ( 0) ( 0) 0]= , , , , , (1.160)

This indicates that the functionaldoesnotexplicitlydependon time. Inotherwords,theresponsefunctional,whichexplicitlydependsontime,isnotobjective.

1.11.3 Translational motion

Whenweset t IQ( ) = , t tc x X( ) ( )= , ,andt 00 = ,

t t t t tx X x X x X( ) ( ) ( ), = , , , = (1.161)

t t t tT G x X x X X X[ ( ) ( ) ( ) ]= , , , , , , (1.162)

1.11.4 Rotational motion

When tQ( )issettobearbitraryand tc( ) 0= andt 00 = ,

t t t t t t tTQ T X Q G Q x X X X T( ) ( ) ( ) [ ( ) ( ) ( ) ], = , , , , , = (1.163)

FromEquation(1.162),weget

t t t t t t t tTQ G x X x X X X Q G Q x X x X X( ) [ ( ) ( ) ( ) ] ( ) [ ( )( ( ) ) ( )) ], , , , , = , , ,,

(1.164)

TakingtheTaylorseriesof tx X( ), around tx X( ), ,assumingthecontinu-ityofthefunctional,wehave

t t t dx X x X F X X( ) ( ) ( ), , = , (1.165)


Hence,

t t t tTQ TQ G QF X X X( ) ( ) [ ( ) ( ) ]= , , , , (1.166)

Thedependenceof the functionalon the relativeposition leads to thedependenceonthedeformationgradient,Fi j xX

ij

=, ,andthedependenceonthestrain.

Forexample,considerthecaseinwhichthedeformationratetensor,D,dependsonthestress,namely,

D G T( )= (1.167)

TheobjectivefunctionalGsatisfies

QDQ G QTQT T( )= (1.168)

forarotationQ.Fromaphysicalpointofview,itcanbeseenthatthedeformationrate

tensorrotatesaccordingtotherotationoftheloadingsystem.Hence,theprincipleofobjectivitycanbeviewed theprincipleof the space isotropy(Figure1.4).

REFERENCES

Belytschko, T., Liu, W.K., and Moran, B. 2000. Nonlinear Finite Elements forContinuaandStructures,JohnWiley&Sons,NewYork.

Boehler, J.P. 1987. Applications of Tensor Functions in Solid Mechanics, CISMCoursesandLectures,No.292,Springer-Verlag,NewYork.

y T

TD ij

Dij

x

xO

y

Figure 1.4 Changeofreferenceframe.


Collins, I.F., and Houlsby, G.T. 1997. Application of thermomechanical prin-ciples to themodelingofgeotechnicalmaterials,Proc.Roy.Soc.LondonA,453:19752001.

Eringen,A.C.1967.MechanicsofContinua,NewYork:JohnWiley&Sons.Fung, Y.C., and Tong, P. 2001. Classical and Computational Solid Mechanics,

WorldScientific.Gurtin, M.E. 1982. An Introduction to Continuum Mechanics, New York:

AcademicPress.Malvern,L.E.1969.IntroductiontotheMechanicsofaContinuousMedia,New

York:Prentice-Hall.Maugin,G.A.1992.TheThermomechanicsofPlasticityandFracture,Cambridge

UniversityPress.Spencer, A.J.M. 1988. Continuum Mechanics, Longman Scientific and Technical,

NewYork.Truesdell, C., and Noll, W. 1965. The non-linear field theories of mechanics, In

EncyclopediaofPhysics,Vol.III,Part3,ed.byS.Flgge,Berlin:Springer-Verlag.Ziegler,H.1983.AnIntroductiontoThermomechanics,2nded.,ElsevierScience,

North-Holland,Amsterdam.

31

Chapter2

Governing equations for multiphase geomaterials

Oneoftheimportantcharacteristicsofgeomaterialsisthatthematerialis composed of solid, liquid, and gas in general. In this chapter, gov-erning equations for the analysis includingbalance lawsand constitu-tiveequationsarepresentedbasedonthetheoryofporousmedia,thatis,animmisciblemixtureofsolidandfluids.First,governingequationsforfluidsolidmaterialsareshown,theninSection2.2, thegoverningequations for gasliquidsolid three-phase materials are formulated.In Section 2.3 the equations for the unsaturated saturated soils arepresented.

2.1 GOVERNING EQUATIONS FOR FLUIDSOLID TWO-PHASE MATERIALS

2.1.1 Introduction

Thegoverningequationsforporewatersoilcoupledproblemscanbederived fromBiots theoryofwater saturatedporousmedia,which isbased on continuum mechanics (Biot 1941, 1955, 1956, 1962; Atkinand Craine 1976; Bowen 1976). Before Biots work, Fillunger (1913)proposedatheoryofporousmediafilledwithwater.Historicaldevel-opment of the theory of porous media has been well documented bydeBoer(2000a,b).VariousmethodsareproposedforBiotstwo-phasemixture theory depending on the method of approximation and thechoiceofunknownvariables(Coussy1995;LewisandSchrefler1998;Zienkiewicz et al. 1999; Ehlers, Graf, and Ammann 2004). In manycomputer programs for liquefaction and consolidation analyses, a u-pformulation is adopted in which the displacement (u) of the solid andporewaterpressure(p)areusedastheunknownvariables,becausewecanreducethedegreeoffreedomalthoughtheu-pformulationprovidesadifferentsolutioninthehighfrequencyrangeforthehigherpermeability


(Zienkiewiczetal.,1980,LIQCARes.DevelopmentGroup,2005).Thisu-p formulation can be easily applied to the consolidation problemssincethedisplacementofporewaterisexplicitlyintroduced.

2.1.2 General setting

Thefollowingassumptionsareadoptedintheu-pformulation:

1.Aninfinitesimalstrainisused. 2.Therelativeaccelerationofthefluidphasetothatofthesolidphaseis

muchsmallerthantheaccelerationofthesolidphase. 3.Thegrainparticlesinthesoilareincompressible. 4.Theeffectofthetemperatureisdisregarded.

Themotionofthemixtureforthemultiphasemediumisdescribedbythesuperpositionofmultiphases inthecontextofcontinuumtheoryasshowninFigure2.1.Forthefluidsolidtwo-phasemixture,weassumethat each point within the mixture is occupied simultaneously by twoconstituents and described by the rectangular Cartesian coordinates(Figure2.2).

Inthefollowings,thematerialtimederivativeisgivenby

DDt t

vx

a

ia

ia

( )

=+

(2.1)

wheresuperscriptaindicatesphaseaand via isthevelocityofthematerial

inphasea,xia isthepositionofparticleofaphase.

Water

Soil particleSolid fluid mixture

Solid phase Fluid phase

Figure 2.1 Superpositionofsolidandfluidphases.

Governingequationsformultiphasegeomaterials 33

Atcurrentstateattimet, x xis

if( ) ( )= fortwophases;(s)standsforsolid

and(f)forfluidphasesinFigure2.2.Hence,materialtimederivativeforthemultiphasescanbedescribedinthespatialcoordinatexi andthesuper-scriptscanbeneglectedas:

DDt t

vx

ia

i

=+

(2.2)

2.1.3 Density of mixture

Thedensitiesofthesolidphase,s

,andthefluidphase,f

,aredefinedas

s s f fn n = =( ) ,1 (2.3)

wherenistheporosity,s isthedensityofthesolid,andf isthedensityofthefluid.

ThedensityofthemixtureisdescribedusingEquations(2.1)and(2.2)as

= + = +s f s fn n( )1 (2.4)

InBiotstheory,thewater-saturatedsoilisdescribedbythesuperpositionofthesolidphaseandthefluidphaseasshowninFigure2.1.

x3

x2

x1

at time t

at current time t

xi Xi( f )

xi(s)Xi(s) ( f )= xi = xi

Figure 2.2 Geometricarrangement.


2.1.4 Definition of the effective and partial stresses of the fluidsolid mixture theory

The total stress is givenby the sumof thepartial stressesactingon thephasesas

ij ij

sijf= + (2.5)

where ijs isthepartialstresstensorofthesolidphaseand ij

f isthepartialstresstensorofthefluidphase.

Thepartialstressesforthefluidphaseandthesolidphasearegivenby

ijf

ijnp= (2.6)

n pij

sij ij(1 ) = (2.7)

where ij isthetotalstresstensor, ijistheeffectivestresstensor,pistheporewaterpressure,nistheporosity,and ij istheKroneckersdelta.Inthederivation,tensionispositivebuttheporewaterpressure ispositiveforcompression.

The total stress isdescribedby theeffective stressand theporewaterpressureas

= pij ij ij (2.8)

2.1.5 Displacementstrain relation

FromAssumption1,thedisplacementstrainrelationsforthesolidandthefluidphasesaredefinedas

ijs i

s

j

js

iijf i

f

j

jux

u

xux

u=

+

=

+1

212

,ff

ix (2.9)

whereijs isthestraintensorofthesolidphaseandui

sisthedisplacementvectorofthesolidphase,ij

f isthestraintensorofthefluidphase,anduif

isthedisplacementvectorofthefluidphase.Thestrainratesaregivenbythetimedifferentiationofstrainsas

ijs i

s

j

js

i

ux

ux

=

+

12

,

ijf i

f

j

jf

i

ux

ux

=

+

12 (2.10)

where()denotesthetimedifferentiation.


2.1.6 Constitutive model

The constitutive relations of the solid phase are given by the relationsbetweentheincrementalstrainsandeffectivestressincrementsas

= Dij ijkl kl

s (2.11)

where ijistheeffectivestressincrementtensor,Dijkl isthemodulusten-sor,and kl

s isthestrainincrementtensorofthesolidphase.Inthecaseoftheelastoplasticmodel,itbecomes

D Dijkl ijklep= (2.12)

Whendisregardingtheviscousresistanceoffluidphase,theconstitutiveequationisgivenby

p Kf

iif= (2.13)

whereKf istheelasticvolumetricmodulusoftheporefluid.

2.1.7 Conservation of mass

Themassconservationlawsforthesolidandthefluidphasesaregivenby

+

=

s sis

itux

( ) 0 (2.14)

+

=

f fif

itux

( ) 0 (2.15)

2.1.8 Balance of linear momentum

Thelinearmomentumconservationlawsforthetwophasesaregivenas

sis

iijs

j

siu R xb

=

+ (2.16)

fif

iijf

j

fiu R xb

+ =

+ (2.17)


wherebi isthebodyforcevector,andRi isthetermexpressingtheenergydissipationduetotherelativemotionbetweenthesolidandthefluidphases(Biot1956).

R n

kwi

wi=

(2.18)

i if

is

w n u u = ( ) (2.19)

wherekisthepermeabilitycoefficient(assumedtobescalarbecauseofisot-ropy),and iw istherelativevelocityvectorofthefluidphasetothesolidphase. w

f g= istheunitweightoftheporewaterwiththegravitationalaccelerationg.

WhenwedescribeRibyEquation(2.18),itiseasilyshownthattheequa-tionofmotionforthefluidphaseisageneraldescriptionofDarcyslaw.

ConsideringEquation(2.19)andthattheporosityisconstant,weobtainthe following equation from the equation of motion for the fluid phaseEquation(2.17),aftermanipulation:

fis

f

i iijf

j

fiu n

w Rx

b

+ + =

+ (2.20)

Iftherelativeaccelerationisalmostzero,

u wis

i (2.21)

Equation(2.20)canbeapproximatedconsideringEquation(2.21):

fis

iijf

j

fiu R xb

+ =

+ (2.22)

Substituting Equations (2.2), (2.5), and (2.18) into Equation (2.22),weobtain

n u n

kw

npx

n bf is w

ii

fi

+ =

+ (2.23)

Whenweassumethatthespatialgradientoftheporosityissufficientlysmall,thefollowingequationholds:

=nxi

0 (2.24)


SubstitutingEquation(2.24)intoEquation(2.23),wehave

f i

s wi

i

fiu k

wpx

b + =

+ (2.25)

Then,whenthesecondtermontheright-handsideofEquation(2.25)isabodyforceduetothegravitationalforce,andwedisregardthedynamicterm,thefollowingequationholds:

+ =px

bi

fi 0 (2.26)

Hence,p gxf= 1inwhichx1isacoordinateinthedirectionofthegravita-tionalforceandgisthegravitationalacceleration;thenpisthencalledthehydrostaticpressure.

FromEquation(2.25),wehavethefollowingequation:

i

w

fis

i

fiw

ku

px

b = +

(2.27)

Then,disregardingtheaccelerationtermandsettingthedirectionof x1forthedirectionofgravitationalforcegandb g1 = ,wehave

1

1 11w

k px

g kx

px

w

f

w

=

=

=

khx1

(2.28)

inwhich wf g= andhisthetotalhead.

Thetotalheadisexpressedby

h

px

pz

w w

= = + 1

(2.29)

wherez x= 1,zistheelevationhead,andp

w isthepressurehead.It is seen thatEquation (2.28) isDarcys law. It shouldbenoted that

theconservationlawofthelinearmomentumofthefluidphaseEquation(2.17)isageneraldescriptionofDarcyslaw.Inthepreceding,thefunda-mental equationsof governing equationshavebeendescribed.Next,wewillderivethebalanceequationofthewholemixtureandthecontinuityequationfromthefundamentalequations.


2.1.9 Balance equations for the mixture

ByaddingEquation(2.16)andEquation(2.17),weobtainthebalanceoflinearmomentumofthemixtureas

sis f

if ij

s

j

ijf

j

si

fu u

x xb

+ =

+

+ + bbi (2.30)

Upon substitution of Equations (2.3), (2.4), and (2.19) into Equation(2.30),thefollowingequationisderived:

i

s fi

ij

jiu w xb + =

+ (2.31)

FromAssumption2,namely,inthecasethattherelativeaccelerationcanbeneglectedasEquation(2.21),Equation(2.31)becomesthebalanceoflinearmomentumforthemixtureas

i

s ij

jiu xb =

+ (2.32)

Forthebalanceofangularmomentumofthemultiphasematerials,weassumethebalanceofangularmomentumforthephasesaswellasforthewholemixture.

2.1.10 Continuity equation

SubstitutionofEquation(2.1)intothemassconservationequationofthesolidphaseEquation(2.14)leadstothefollowingequation:

( )

( ) {( ) }( )1

1 11

+

+

+ n

tn

tn ux

ns

s s is

iis ux

s

i

=

0 (2.33)

Inasimilarway,substitutingEquation(2.2)intothemassbalanceequa-tionofthefluidphaseEquation(2.15)gives

n

tnt

nux

nux

ff f i

f

iif

f

i

+

+

+

=

( )

0 (2.34)


MultiplyingEquation(2.33)by f s/ andaddingtheresultandEquation(2.34),wehave

f f if

is

i

f int

nt

n u ux

+

+

+

( ) { ( )}1 ss

i

f

if

f

i

f

s

s

ux

nt

ux

nt

+

+

+

+

( )1 ii

ss

iu

x

=

0

(2.35)

InEquation(2.35),thefirsttermisequaltozero.TakingintoaccountAssumption1andsubstitutingEquation(2.10)andEquation(2.19) intoEquation(2.35),wehave

+ +

+

+i

iiis

f

f

if

f

is

wx

nt

ux

n

( )1

+

= s

is

s

itu

x 0 (2.36)

where iis isthevolumetricstrainrateofthesolidphase.

Considering the material time derivative, Equation (2.36) can beexpressedas

+ + +

=ii

iis

f

f

s

swx

n n

( )1

0 (2.37)

Herein,whenweassume the incompressibilityof the soil constituentssuchassoilparticles(Assumption3),thefollowingequationholds:

s = 0 (2.38)

SubstitutionofEquation(2.38)intoEquation(2.37)leadstothefollow-ingequation:

+ + =ii

iis

f

fwx

n

0 (2.39)

Theprecedingequationisthecontinuityequationforthecaseofincom-pressibilityofthesolidconstituent.Thethirdtermoftheleft-handsideofEquation(2.39)denotesacompressibilityoftheporewater.


Neglecting the effect of temperature, the following equation is givensincethetimederivativeofmassoftheporefluid(f fV )iszero:

D VDt

f f( )= 0 (2.40)

Afterthemanipulationofthisequationwehave

f

f

f

f iifV

V

= = (2.41)

Substitutingtheconstitutiveequationofthefluid,Equation(2.13),intoEquation(2.41)gives

f

f f

p

K

= (2.42)

UponsubstitutionofEquation(2.42)intoEquation(2.39),thecontinuityequationbecomes

+ + =ii

iis

fwx

n

Kp 0 (2.43)

InEquation(2.43),thisequationincludestherelativevelocityofthefluidphasetothesolidphase.

IfAssumption2oftheu-pformulationisadopted,wecanexpresswibytheaccelerationofthesolidphaseandtheporewaterpressureashasbeenshowninEquation(2.27)andcanbeeliminatedinEquation(2.43).

SubstitutingEquation(2.27)intoEquation(2.43),weget

+

x

ku

px

bi w

fis

i

fi

+ + =iis

f

n

Kp 0 (2.44)

Ifthebodyforcebiisconstant,andthespatialgradientsofpermeabilityandthedensityofthefluidaresufficientlysmall,consideringEquation(2.8),thefinalformofthecontinuityisgivenas

k p

x

n

Kp

w

fiis

iiis

f

+ + =

2

20 (2.45)


2.2 GOVERNING EQUATIONS FOR GASWATERSOLID THREE-PHASE MATERIALS

2.2.1 Introduction

Geomaterials generally fall into the category of multiphase materials.Theyarebasicallycomposedofsoilparticles,water,andair.Thebehaviorofmultiphasematerialscanbedescribedwithintheframeworkofamac-roscopiccontinuummechanicalapproachthroughtheuseofthetheoryofporousmedia(deBoer2000b).Thetheoryisconsideredtobeagener-alizationofBiotstwo-phaseporoustheoryforsaturatedsoil(Biot1941,1955,1956).

Proceeding fromthegeneralgeometricallynonlinear formulation, thegoverningbalancerelationsformultiphasematerialscanbeobtained(deBoer2000b;LoretandKhalili2000;LewisandSchrefler1998;EhlersandGraf,2003;Ehlersetal.,2004).Massconservationlawsforthegasphaseaswellasfortheliquidphaseareconsideredinthoseanalyses.Inthefieldofgeotechnics,airpressureisassumedtobezeroinmanyresearchworks(Shengetal.2003),sincegeomaterialsusuallyexistinanunsaturatedstatenearthesurfaceofthegroundandwehavenotenoughdataonadevelop-mentofairpressure.Consideringgashydratedissociationintheseabedground,however,wehavetodealwiththehighlevelofgaspressurethatexistsdeepintheground(Kimotoetal.2007),thismeansthatthemassbalanceforthreephasesmustbeconsidered.Okaetal.(2006)proposedanairwatersoilcoupledfiniteelementmodelinwhichtheskeletonstressis used as a stress variable, and the suction effect is introduced in theconstitutiveequationforsoil.Furthermore,theconservationofenergyisrequiredwhen there isa considerable change in temperatureduring thedeformation process. Vardoulakis (2002) showed that the temperatureof saturated clay rises with plastic deformation. Oka et al. (2004) andKimotoetal.(2007)numericallysimulatedthethermalconsolidationpro-cess,whichwillbeshowninChapter10.

2.2.2 General setting

The material to be modeled is composed of three phases, namely, solid(S),water(W),andgas(G),whicharecontinuouslydistributedthroughoutspace.Totalvolume(V)isobtainedfromthesumofthepartialvolumesoftheconstituents,namely,

V V S W G

= =( , , ) (2.46)


Thevolumeofvoid,Vv ,whichiscomposedofwaterandgas,isgivenasfollows:

V V W Gv

= =( , ) (2.47)

Volumefraction, n,isdefinedasthelocalratioofthevolumeelementwithrespecttothetotalvolume,namely,

n

VV

= (2.48)

n S W G

= =1 ( , , ) (2.49)

Thevolumefractionofthevoid,thatis,porosity,n,iswrittenas

n nVV

V VV

n W Gv S

S= = =

= =

1 ( , ) (2.50)

Thevolumefractionofthefluid,nF ,isgivenby

n n W GF = = ( , ) (2.51)

Thevolumefractionconcepthasbeenadoptedtoconstructthetheoryofmixture(Mills1967;Morland1972).ThehistoricaldevelopmentofthevolumefractiontheoryhasbeenwelldiscussedbydeBoer(2000b).

Inaddition,thewatersaturationisrequiredinthemodel,namely,

s

V

V V

n

n n

n

nr

W

W G

W

W G

W

F=

+=

+= (2.52)

2.2.3 Partial stresses

Byanalogytothewater-saturatedsoil,weassumethat

= n PijS

ijS F

ij (2.53)

ijW W W

ijn P= (2.54)

ijG G G

ijn P= (2.55)


wherePF istheaveragepressureofthefluidssurroundingthesolidskeleton(Bolzonetal.1996)givenby

P s P s PF

rW

rG= + ( )1 (2.56)

and ij isaskeletonstress.Theskeletonstress,whichwillbeexplainedinthefollowing,isreason-

abletodescribethebehaviorofsolidskeletonintheconstitutiverelation.Totalstresstensor, ij ,isobtainedfromthesumofthepartialstresses,

ij ,namely,

= =

S W Gij ij ( , , ) (2.57)

and

Pij ijF

ij = + (2.58)


Theconservationofmassforthesolid,water,andgasphases,(=S,W,G),isgiveninthefollowingequation:

( ) = + =( )t n q m S W GMi i

, , , (2.59)

inwhich isthematerialdensity,qMi isthefluxvector,and misthemass

changerateofphaseperunitvolume.Thefluxvectorisexpressedintermsofthevelocityoftheflowas

q n v S W GMi i = =( , , ) (2.60)

where vi isthevelocityofphase.

Therelativevelocityoftheflow,Vi ,withrespecttothesolidphaseis

V n v v W Gi i iS = =( ) ( , ) (2.61)


TheconservationlawsinEquation(2.59)forthesolid,water,andgasphasesareexpressedwithwatersaturation, sr ,andthevolumefractionofvoid,nF,as

+ = s F F s i i

s sn n v m ( ) ,1 (2.62)

n s s n n s ns v mW F

rW

rF F

rW

rW

i iW W, + + + = (2.63)

n s n s n s n s vF r

G Fr

G Fr

G Fr

Gi iG( ) ( ) ( ) ,1 1 1 + + == m

G (2.64)

whereweassumetheincompressibilityofsoilparticles, S = 0,andn nF = istheporosity.

Assumingthatthespatialgradientofthevolumefractionsarezero,weobtainfollowingrelationsfromEquations(2.62)to(2.64)as

s n s n V s v s

m mr

FW

W rF

i iW

r i iS

r

s

s

W

W

+ + + =, , 0 (2.65)

s n s n s v V s

m mr

FG

G rF

r i iS

i iG

r

s

s

G

G(1 ) (1 ) (1 ) 0, ,

+ +

= (2.66)

Asfordescribingchangesinthegasdensity,theequationofidealgasescanbeused,thatis,

G

G GM PR

= (2.67)

G

G G GMR

P P=

2 (2.68)

inwhichMGisthemolecularweightofgas,Risthegasconstant,isthetemperature,andtensionispositiveintheequation.

DividingEquation(2.68)byEquation(2.67)yields

G

G

G

G

P

P= (2.69)


2.2.5 Balance of momentum

Momentumbalanceisrequiredforeachphase,namely,

n v n F P S W Gi ji j i i = + =, ( , , ) (2.70)

inwhich Fi is thegravity forceand Pi is related to the interaction term

givenin

P D v v D D S W Gi i i

= = = ( ), ( , , , ) (2.71)

where Dareparameters thatdescribe the interactionwitheachphase.Themomentumbalanceequationforeachphaseisobtainedwiththefol-lowingequationswhentheaccelerationisdisregarded:

n P n F D v v D v vji jS F

iS S

iSW

iS

iW SG

iS

iG' ( ) ( ) ( ) 0, , + = (2.72)

n P n F D v v D v vW W

iW W

iWS

iW

iS WG

iW

iG( ) ( ) ( ) 0, + = (2.73)

n P n F D v v D v vG G

iG G

iGS

iG

iS GW

iG

iW( ) ( ) ( ) 0, + = (2.74)

D W G ( , , )= aregivenas

D

n g

kD

n g

kWS

W W

WGS

G G

G= =

( ),

( )2 2 (2.75)

inwhich kWand kGarethepermeabilitycoefficients for thewaterphase

and thegasphase, respectively.Weassume that the interactionbetweenwaterandgasphasesDGW andDWGiszero.

When the spacederivativeofvolume fraction n i, isnegligible,Darcys

lawforthewaterphaseandthegasphaseisobtainedfromEquations(2.73)and(2.74),respectively,as

V n v v

kgP Fi

W WiW

iS

W

W iW W

i,( ) ( )= =

(2.76)

V n v v

kgP Fi

G GiG

iS

G

G iG G

i,( ) ( )= =

(2.77)


ThesumofEquations(2.72)to(2.74)leadsto

ji jE

iEF n S W G, , ( , , )+ = = =0 (2.78)

It is worth noting that Darcy-type laws such as Equations (2.76) and(2.77)arenotobjectivesincetheyincludevelocitybutareagoodapprox-imation (Eringen 2003). In addition, as has been printed out in Section2.2.1.9, it is assumed that the angular momentum is balanced for thephasesforthewholemixture.

2.2.6 Balance of energy

Thefollowingenergyconservationequationisappliedinordertoconsidertheheatconductivity:

c D h Q

Eijvp

ij i i, ( ) = + (2.79)

c n c S W GE( ) = = ( , , ) (2.80)

wherecisthespecificheat,isthetemperatureforallthephases,Dijvpis

theviscoplasticstretchingtensor, hi istheheatfluxvector,and Q istheheatsource.

Heatflux,hi ,isgivenby

hi

Ei= , (2.81)

E n=

( =S,W,G) (2.82)

inwhich isthethermalconductivity.

2.3 GOVERNING EQUATIONS FOR UNSATURATED SOIL

In the theoryofporousmedia, theconceptof theeffective stress tensorisrelatedtothedeformationofthesoilskeletonandplaysanimportantrole.The effective stress tensorhasbeendefinedbyTerzaghi (1943) for


water-saturated soil. However, the effective stress needs to be redefinedif the fluid is made of compressible materials. In the present study, theskeletonstresstensor, ij ,isdefinedandthenusedforthestressvariablein the constitutive relation for the soil skeleton (Okaetal.2006,2008,2010),whichhasbeencalledaverageskeletonstressbyJommi(2000)andGallipolietal.(2003).

2.3.1 Partial stresses for the mixture

Thetotalstress tensor isassumedtobecomposedofthreepartialstressvaluesforeachphase:

ij ij

sijf

ija= + + (2.83)

where ij is the total stress tensor,and ijs , ij

f ,and ija are thepartial

stresstensorsforsolid,liquid,andair,respectively.Consideringthevolumefraction(Ehlersetal.2004),thepartialstress

tensorsforunsaturatedsoilcanbegivenby

ijf

rf

ijnS p= (2.84)

ija

ra

ijn S p= ( )1 (2.85)

= n Pijs

ijF

ij(1 ) (2.86)

P S p S pF

rf

ra= + ( )1 (2.87)

where pf and pa are theporewaterpressureand theporeairpressure,respectively,nistheporosity,Sr isthedegreeofsaturation, ij istheskel-etonstress,andPF istheaverageporepressure.Thetensionispositiveinthischapter.

Fortheunsaturatedsoil,wewillusetheskeletonstress(Okaetal.2006,2008;Kimotoetal.2010)asthebasicstressvariableinthemodelalongwithsuction.TheskeletonstressonlyappliestothesoilskeletonandEquation(2.86)comesfromtheanalogytotheeffectivestressforthewater-saturatedsoil.ByaddingEquations(2.84),(2.85),and(2.86)wehaveskeletonstressas

Pij ijF

ij = + (2.88)


TheskeletonstresswasfirstadvocatedbyJommi(2000)astheaverageskeletonstress,whichwasdefinedasthedifferencebetweenthetotalstressandtheaveragefluidpressure.Ehlersetal.(2004)callediteffectivestress.However,hereinwecalleditskeletonstresstoavoidconfusingitwithmeanvalueoftheskeletonstress.

Adoptingtheskeletonstressprovidesanaturalapplicationofthemix-turetheorytounsaturatedsoil.ThedefinitioninEquation(2.88)issimilartoBishopsdefinitionfortheeffectivestressofunsaturatedsoil.InadditiontoEquation(2.88),theeffectofsuctionontheconstitutivemodelshouldalwaysbetakenintoaccount.Thisassumptionleadstoareasonablecon-sideration of the collapse behavior of unsaturated soil, which has beenknownasabehavior thatcannotbedescribedbyBishopsdefinition fortheeffectivestressofunsaturatedsoil.Introducingsuctionintothemodel,however,makesitpossibletoformulateamodelforunsaturatedsoil,start-ingfromamodelforsaturatedsoil,byusingtheskeletonstressinsteadoftheeffectivestress.


Themassconservationlawforthethreephasesisgivenby

+

=

J JiJ

itux

( ) 0 (2.89)

where J istheaveragedensityfortheJphaseand uiJ isthevelocityvector

fortheJphase.

s sn = ( )1 (2.90)

f

rfnS = (2.91)

a

ran S = ( )1 (2.92)

whereJ=s, f,anda, inwhich thesuperscriptss, f,anda indicate thesolid,theliquid,andtheairphases,respectively;nistheporosity;andSr isthesaturation.

J isthemassbulkdensityofthesolid,theliquid,andthegas.


2.3.3 Balance of linear momentum for the three phases

The conservation laws of linear momentum for the three phases aregivenby

i

ssi i

ijs

j

siu Q R xb

=

+ (2.93)

fif

iijf

j

fiu R xb

+ =

+ (2.94)

aia

iija

j

aiu Q xb

+ =

+ (2.95)

where u J a f siJ ( , , )= aretheaccelerationvectorsforthethreephases,bi is

thebodyforce,Qi denotes the interactionbetweenthesolidandtheairphases, and Ri denotes the interaction between the solid and the liquidphases.Theseinteractionterms,QiandRi ,canbedescribedas

R nS

kwi r

wf i

f=

(2.96)

Q n S

g

kwi r

a

a ia= ( )1

(2.97)

wherekf isthewaterpermeabilitycoefficient,ka istheairpermeability, if

w istheaveragerelativevelocityvectorofwaterwithrespecttothesolidskel-eton,and i

aw istheaveragerelativevelocityvectorofairtothesolidskeleton.

Therelativevelocityvectorsaredefinedby

if

r if

is

w nS u u = ( ) (2.98)

ia

r ia

is

w n S u u = ( )( )1 (2.99)


UsingEquation(2.98),Equation(2.94)becomes

fis

rif

iijf

j

fiu nS

w Rx

b

+ + =

+

1 (2.100)

We deal with the behavior of soil in which the difference betweenaccelerationsofthesoilskeletonandporefluidissufficientlysmall.Thisassumptionisreasonableexceptforthehighfrequencyproblemandveryhighpermeability(Zienkiewiczetal.1980).Forthisreasonweassumethatwif 0,inthiscase,usingEquations(2.84),(2.91),and(2.96),Equation

(2.100)becomes

nS u nS

kw nS

px

nS brf

is

rwf i

fr

f

ir

fi

+ =

+ (2.101)

inwhichweassumethatthespatialgradientsofporosityandsaturationaresufficientlysmall.Thesameassumptionwillbetakeninthefollowingderivationsofthegoverningequations.

Aftermanipulation,theaveragerelativevelocityvectorofwatertothesolid skeleton and the average relative velocity vector of air to the solidskeletonareshownas

if

f

w

f

i

fis f

iwk p

xu b =

+

(2.102)

ia

a

a

a

i

ais a

iwk

g

px

u b =

+

(2.103)

inwhich wia 0isassumedduetothereasonmentionedearlier.

Based on the aforementioned fundamental conservation laws, we canderiveequationsofmotionforthewholemixture.SubstitutingEquations(2.90), (2.91), and (2.92) into the given equation and adding Equations(2.93)to(2.95),wehave

( ) ( ) + + =

+u nS u u n S u ux

bis rf

if

is

ra

ia

is ij

ji(1 ) (2.104)

whereisthemassdensityofthemixtureas = + +f a s ,and ij isthetotalstresstensor.


Fromthefollowingassumptions,

is

if

is

u u u >> ( ) (2.105)

is

ia

is

u u u >> ( ) (2.106)

equationsofmotionforthewholemixturearederivedas

i

s ij

jiu xb =

+ (2.107)

2.3.4 Continuity equations

Using the mass conservation law for the solid and the liquid phases,Equation(2.89)(J=s,f)andEquations(2.90)and(2.91),andassumingtheincompressibilityofsoilparticles,weobtain

{ }

+ + + =nS u u

xS nS nS

r if

is

ir ii

sr

f

f r

( )

0 (2.108)

Incorporating Equation (2.102) and p Kf iif= (Kf: volumetric elastic

coefficient) into the previous equation leads to the following continuityequationfortheliquidphase:

+

x

ku

px

bi

f

w

fis

f

i

fi

+ + + =S nS nSp

Kr ii

sr r

f

f 0 (2.109)

Similarly, we can derive the continuity equation for the air phase byassuming that the spatial gradients of porosity and saturation are suffi-cientlysmall:

+

x

ku

px

bi w

ais

a

i

ai

a

+ + =( ) ( )1 1 0S nS n Sr ii

sr r

a

a

(2.110)

Forthesaturationwewilluseaconstitutiveequationcalledwaterchar-acteristicrelationorwaterretentionrelation.


Sincesaturationisafunctionofsuction,thatis,thepressurehead,thetimerateforsaturationisgivenby

nS n

dSd

dd

ddp

pdd

prr

cc

w

c1 =

=

(2.111)

where = VVw is the volumetric water content, pc is the matric suction

(p p pc a f= ( )), = pc w/ is thepressurehead forsuction,andC dd=

isthespecificwatercontent.

Andweneedtheconstitutiveequationforairphasesuchasidealgas.

REFERENCES

Atkin,R.J.,andCraine,R.E.1976.Continuumtheoriesofmixtures:basictheoryandhistoricaldevelopments,Q.J.

Computational Modeling of Multiphase Geomaterial

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