Computational Modeling of Multiphase Geomaterial
Transcript of Computational Modeling of Multiphase Geomaterial
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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
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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
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COMPUTATIONALMODELING OFMULTIPHASE
GEOMATERIALS
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A SPON PRESS BOOK
COMPUTATIONALMODELING OFMULTIPHASE
GEOMATERIALS
FUSAO OKASAYURI KIMOTO
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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
International Standard Book Number-13: 978-1-4665-7064-1 (eBook - PDF)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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)
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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.
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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| |
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4 Computationalmodelingofmultiphasegeomaterials
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)
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Fundamentalsincontinuummechanics 5
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.
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6 Computationalmodelingofmultiphasegeomaterials
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( ).
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Fundamentalsincontinuummechanics 7
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.
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8 Computationalmodelingofmultiphasegeomaterials
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.
-
Fundamentalsincontinuummechanics 9
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)
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10 Computationalmodelingofmultiphasegeomaterials
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.
-
Fundamentalsincontinuummechanics 11
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.
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12 Computationalmodelingofmultiphasegeomaterials
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)
-
Fundamentalsincontinuummechanics 13
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)
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14 Computationalmodelingofmultiphasegeomaterials
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
=
-
Fundamentalsincontinuummechanics 15
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)
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16 Computationalmodelingofmultiphasegeomaterials
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.
-
Fundamentalsincontinuummechanics 17
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)
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18 Computationalmodelingofmultiphasegeomaterials
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)
-
Fundamentalsincontinuummechanics 19
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)
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20 Computationalmodelingofmultiphasegeomaterials
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)
-
Fundamentalsincontinuummechanics 21
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.
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22 Computationalmodelingofmultiphasegeomaterials
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)
-
Fundamentalsincontinuummechanics 23
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.
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24 Computationalmodelingofmultiphasegeomaterials
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
-
Fundamentalsincontinuummechanics 25
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.
-
26 Computationalmodelingofmultiphasegeomaterials
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).
-
Fundamentalsincontinuummechanics 27
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.
-
28 Computationalmodelingofmultiphasegeomaterials
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)
-
Fundamentalsincontinuummechanics 29
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.
-
30 Computationalmodelingofmultiphasegeomaterials
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
-
32 Computationalmodelingofmultiphasegeomaterials
(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.
-
34 Computationalmodelingofmultiphasegeomaterials
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.
-
Governingequationsformultiphasegeomaterials 35
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)
-
36 Computationalmodelingofmultiphasegeomaterials
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)
-
Governingequationsformultiphasegeomaterials 37
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.
-
38 Computationalmodelingofmultiphasegeomaterials
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)
-
Governingequationsformultiphasegeomaterials 39
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.
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40 Computationalmodelingofmultiphasegeomaterials
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)
-
Governingequationsformultiphasegeomaterials 41
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)
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42 Computationalmodelingofmultiphasegeomaterials
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)
-
Governingequationsformultiphasegeomaterials 43
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)
2.2.4 Conservation of mass
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)
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44 Computationalmodelingofmultiphasegeomaterials
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)
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Governingequationsformultiphasegeomaterials 45
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)
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46 Computationalmodelingofmultiphasegeomaterials
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
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Governingequationsformultiphasegeomaterials 47
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)
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48 Computationalmodelingofmultiphasegeomaterials
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.
2.3.2 Conservation of mass
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.
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Governingequationsformultiphasegeomaterials 49
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)
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50 Computationalmodelingofmultiphasegeomaterials
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.
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Governingequationsformultiphasegeomaterials 51
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.
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52 Computationalmodelingofmultiphasegeomaterials
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.