Table Of ContentAPrimeronTheoreticalSoilMechanics
APrimeronTheoreticalSoilMechanicsisaboutadaptingcontinuummechanicsto
granularmaterials.Thefieldofcontinuummechanicsoffersmanyfruitfulconcepts
andmethods,howeverthereisdeclininginterestinthefieldduetoitscomplexand
fragmentednature.Thisbook’spurposeisthereforetofacilitatetheunderstandingof
thetheoreticalprinciplesofsoilmechanics,aswellasintroducingthenewtheoryof
barodesy.Thistitlearguesforbarodesyasasimplealternativetotheplasticitytheory
usedcurrentlyandprovidesasystematicinsightintothisnewconstitutivemodelfor
granularmaterials.Thisbookthereforeintroducesacomplexfieldfromafreshand
innovative perspective using simple concepts, succinct equations and explanatory
sketches. Intended for advanced undergraduates, graduates and PhD students, this
titleisalsoaptforresearchersseekingadvancedtrainingonfundamentaltopics.
DimitriosKolymbasisEmeritusProfessorattheUniversityofInnsbruck,Austria,where
he worked from 1994 to 2017 as Professor at the Unit of Geotechnical and Tunnel
Engineering. He is the author of several books and many papers, and invented the
newconstitutivetheoriesofhypoplasticityandbarodesy.
Published online by Cambridge University Press
Published online by Cambridge University Press
A Primer on Theoretical
Soil Mechanics
DIMITRIOS KOLYMBAS
UniversityofInnsbruck
Published online by Cambridge University Press
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Informationonthistitle:www.cambridge.org/9781009210331
DOI:10.1017/9781009210348
©DimitriosKolymbas2022
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Firstpublished2022
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Contents
Preface pageix
1 GranularMaterialsasSoftSolids 1
1.1 SoilandGeotechnicalEngineering 1
1.2 GranulatesinChemicalEngineering 2
1.3 CanWeConsiderGranularMediaasContinua? 2
1.4 DifferencesbetweenGranulatesandOtherSolids 3
2 MechanicalBehaviourofSoil:ExperimentalResults 5
2.1 TheMeaningofMechanicalBehaviour 5
2.2 ElementTests 5
2.3 TypicalLaboratoryTests 6
2.4 OedometricTest 6
2.5 DrainedTriaxialTest 7
2.6 UndrainedTriaxialTests 12
2.7 CyclicTests 14
2.8 TrueTriaxialApparatus 16
2.9 SimpleShear 18
2.10 Strain-versusStress-Control 19
2.11 RoleofTime 20
2.12 AccuracyofTestResults 22
2.13 LookingintotheSamples 23
3 MechanicalBehaviourofSoil:Intuitively 25
3.1 EquationsversusIntuition 25
3.2 ProportionalPathsforGranularMaterials 25
3.3 RelationbetweenStrainPathsandStressPaths 25
3.4 ProportionalStrainingStartingatT(cid:2)0 26
3.5 TriaxialTests 27
4 VectorsandTensors 31
4.1 PurposeofThisChapter 31
4.2 Vectors 31
4.3 Tensors 32
5 Fields 35
5.1 FieldsinContinuumMechanics 35
5.2 Coordinates 35
v 5.3 VectorFields 36
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vi Contents
5.4 ContinuousFieldsandDiscontinuities 37
6 Deformation 38
6.1 DeformationandGrainRearrangement 38
6.2 HowtoDescribeDeformation? 38
6.3 EulerandLagrangeApproaches 38
6.4 DeformationGradient 39
6.5 Rotation 40
6.6 DisplacementGradient 41
6.7 TimeandSpatialDerivatives 43
6.8 Example:SimpleShear 44
6.9 EquationsofCompatibility 45
7 Stress 47
7.1 WhatIstheStressTensor? 47
7.2 MohrCircle 48
7.3 PrincipalStressSpace 48
7.4 StressTensorinCylindricalCoordinates 49
7.5 InvariantsandEigenvalues 50
7.6 Example:StressinaShearBox 51
8 ConservationLaws(BalanceEquations) 52
8.1 IntegralsofMotion 52
8.2 ConservationLawsasFieldEquations 52
8.3 WeakSolutionoftheEquilibriumEquation 55
8.4 JumpRelations 56
8.5 IntegralRepresentationsofConservationEquations 57
8.6 StressandIntergranularForces 57
8.7 StressFields 58
9 InternalFrictionandShearStrength 62
9.1 MeaningofStrength 62
9.2 DryFrictioninContinuumMechanics 62
9.3 FrictionAngle 64
9.4 Cohesion 65
9.5 RockasFrictionalMaterial 67
10 Collapse 69
10.1 ImportanceofCollapseinSoilMechanics 69
10.2 ThePhenomenonofCollapse 69
10.3 PlastifiedZones 70
10.4 CollapseMechanisms 75
10.5 Safety 78
10.6 ImminentCollapse 81
11 ConstitutiveEquations 82
11.1 ConstitutiveEquationversusConstitutiveLaw 82
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vii Contents
11.2 WhyDoWeNeedConstitutiveEquations? 82
11.3 AreConstitutiveEquationsDispensableinView
ofArtificialIntelligence? 83
11.4 MaterialConstants 83
11.5 Calibration 84
11.6 ResponseEnvelopes 84
11.7 ProportionalStrainPaths 85
11.8 LargeDeformations 86
11.9 RoleofThermodynamics 86
11.10 ComparisonofConstitutiveEquations 87
12 Elasticity 88
12.1 DefinitionofElasticity 88
12.2 LinearElasticity 88
12.3 ModificationsofElasticity 89
12.4 ElasticityinSoilMechanics 90
13 ElasticWaves 92
13.1 PurposeofThisChapter 92
13.2 WhatAreWaves? 92
13.3 KinematicWaves 93
13.4 ElasticWavesinOne-dimensionalContinua 94
13.5 WavesinBodiesofFiniteDimensions 98
13.6 BodyWaves 100
13.7 RayleighWaves 101
13.8 ImpairmentDuetoWaves 102
14 PlasticityTheory 103
14.1 RelevanceofPlasticityTheory 103
14.2 One-dimensionalOriginofPlasticity 103
14.3 YieldFunction,Loading–Unloading 103
14.4 NormalityRule 104
14.5 CollapseorLimitLoadTheorems 105
14.6 ElastoplasticRelationsforSoil 106
14.7 CriticismofPlasticityTheoryinSoilMechanics 107
15 Hypoplasticity 109
15.1 HypoplasticityasanAlternativetoElastoplasticity 109
15.2 Non-linearRateEquations 109
15.3 Notation 110
15.4 IncrementalNon-linearity 110
15.5 MathematicalDescriptionofIrreversibility 110
15.6 EmergenceofHypoplasticity 111
15.7 LinkstoElastoplasticity 112
15.8 ImprovingMemorybyMeansofIntergranularStrain 112
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viii Contents
16 Barodesy 113
16.1 Introduction 113
16.2 Notation 113
16.3 DerivationoftheConstitutiveEquation 114
16.4 TheEquationsofBarodesy 115
16.5 CriticalStatesRevisited 116
16.6 TheR-function 119
16.7 Calibration 122
16.8 SimulationofElementTests 123
16.9 BarodesyforSand 124
16.10 BarodesyforClay 129
16.11 ReflectinguponBarodesy 132
16.12 NumericalSimulationofElementTests 133
17 Uniqueness 136
17.1 MeaningofUniquenessandRelatedNotions 136
17.2 UniquenessinElementTests 136
17.3 ShearBandsandFaults 137
18 Symmetry 141
18.1 GeneralRemarks 141
18.2 PrincipleofMaterialFrameIndifference 141
18.3 IsotropicMaterials 146
18.4 Scaling 147
18.5 MechanicalSimilarity 150
19 InteractionwithWater 154
19.1 WaterinSoil 154
19.2 MultiphaseMaterials 154
19.3 EffectiveStressinWater-saturatedSoil 156
19.4 Darcy’sLaw 157
19.5 BalanceEquations 158
19.6 Consolidation 160
19.7 GroundwaterFlow 160
19.8 UnsaturatedSoil:AnExkurstoPhysicalChemistry 164
20 ComputinginSoilMechanics 170
20.1 PitfallsofComputing 170
20.2 ProblemswithGeotechnicalEngineeringComputations 171
20.3 LimitsofContinuumMechanics 173
20.4 QualityofNumericalResults 173
20.5 CopingwithUncertainPredictions 174
21 Outlook 177
21.1 OpenQuestions 177
References 178
Index 185
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Preface
Socratestaughtustodistinguishbetweenwhatweunderstandandwhatwedo
notunderstand.
Theoretical soil mechanics is continuum mechanics adapted to soil, a granular
material. Continuum mechanics is a complicated, subtle and difficult science. It
suffersfromfragmentation;itsnotationisnotunique,andthedegreeofmathematical
abstractionvariesgreatly.Someresearchersgetlostinit.Themanydiversenumerical
applicationsaddtotheconfusion;theircontinuummechanicalfoundationsareoften
hidden.
Theseaspectsimplyadeclininginterestincontinuummechanicsforsoils,which
is also enhanced by the emergence of computer-based micromechanical methods.
Insteadofthecontinuum,agglomeratesofindividualparticlesareconsidered.This
approach has its merits but still does not provide a global understanding for the
behaviourofsoil.Certainly,thecontinuumisanabstraction,justlikethemasspoint
ofclassicalmechanics.Thediscontinuousstructureofmatter(suchassand)canbe
reconciled with the assumption of the continuum by considering the continuously
distributedfieldquantitiesasexpectedvaluesofdiscontinuousdistributions.
Themainpurposeofthisbookistoawakenthereader’sinteresttowardstheelegant
field of continuum mechanics and its applications in soil mechanics. An important
part therein is the constitutive law of soils, now in a completely new frame – the
theoryofbarodesy.
Every science starts from its fundamentals and proceeds to ever higher peaks of
increasingcomplexityandrefinement.Thus,theauthorofatextbookthatisintended
tobeclearandconcisefacesthequestionsof‘wheretostart?’and‘wheretostop?’.
Clearly, the answers can only be subjective in the hope that the reader can retrieve
themissinginformationinbothdirections.Agoodbookshouldrathermotivatethe
readertowardsasearchinliteratureandinone’sownresearch.
Deeply convinced that ‘less is more’, I should like to present the reader with a
guidetounderstanding,ratherthanarepositoryofequations.Ihavealsoattempted
tofindnewandsimpleraccesstoseveralitems.IndoingsoIhavefollowedWalter
Noll’ssuggestionsfortheroleoftheprofessor:
‘The professor’s focus is on understanding, gaining insight into, judging the
significance of, and organizing old knowledge. He is disturbed by the pile-up of
undigestedandill-understoodnewresults.Heisnothappyuntilhehasbeenable
tofittheseresultsintoalargercontext.Heishappyifhecanfindanewconceptual
frameworkwithwhichtounifyandsimplifytheresultsthathavebeenfoundbythe
researcher.’
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