Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime, and Size Effect

320 Pages·2017·12.907 MB·English

by  Zdenek P. Bazant, Jia-Liang Le

ProbabilisticMechanicsofQuasibrittleStructures
Strength,Lifetime,andSizeEffect
Quasibrittle materials are becoming increasingly important for modern engineering.
They include concretes, rocks, fiber composites, tough ceramics, sea ice, bone, wood,
stiffsoils,rigidfoams,glass,dentalandbiomaterials,aswellasallbrittlematerialson
themicroornanoscale.Theirsalientfeatureisthatthefractureprocesszonesizeisnon-
negligible compared to the structural dimensions. This causes intricate energetic and
statistical size effects and leads to size-dependent probability distribution of strength,
transitionalbetweenGaussianandWeibullian.
ProbabilisticMechanicsofQuasibrittleStructuresdiscussestheensuingdifficultchal-
lengesforsafedesign.
Drawinguponyearsofpracticalexperienceandusingnumerousexamplesandillustra-
tiveapplications,theauthorscover:
(cid:129) Rigorous theory with detailed derivations yet no superfluous mathematical sophisti-
cation.
(cid:129) Extensiveexperimentalverificationsandrealisticapproximationsfordesign.
(cid:129) Fracturekineticsanditssizeeffect.
(cid:129) Multiscaleanalyticaltransitiontothematerialscale.
(cid:129) Statisticsofstructuralstrengthandlifetime,sizeeffectandreliabilityindices.
(cid:129) Ramificationtogatedielectricsbreakdownwithanalogousmathematicalformulation.
Born and educated in Prague (Ph.D. 1963), Zdeneˇk Bažant joined Northwestern Uni-
versity in 1969, where he has been W.P. Murphy Professor since 1990 and simultane-
ouslyMcCormickInstituteProfessorsince2002,aswellasDirectorofCenterforGeo-
materials (1981–1987). He was inducted to NAS, NAE, American Academy of Arts
& Sciences, Royal Society London; to the academies of Italy, Austria, Spain, Czech
Republic, Greece and Lombardy; and to Academia Europaea and European Academy
of Sciences & Arts. He is an honorary member of ASCE, ASME, ACI, and RILEM;
receivedsevenhonorarydoctoratesaswellasthevonKarman,Timoshenko,Pragerand
Newmarkmedalsamongmanyhonors;wasawardedtheAustrianCrossforScienceand
Art1stClassfrompresidentofAustria;andwaspresidentofSES,IA-FraMCoSandIA-
ConCreep.Hehasauthoredsixbooksandover600papers.In2015,ASCEestablished
ZP Bažant Medal for Failure and Damage Prevention, and ZP Bažant Prize for Engi-
neeringMechanicswascreatedinCzechRepublic.Heisoneoftheoriginaltop100ISI
HighlyCitedScientistsinEngineering(www.ISIhighlycited.com).
Dr. Jia-Liang Le is currently Associate Professor of Civil, Environmental, and Geo-
EngineeringattheUniversityofMinnesota.HeobtainedhisPh.D.instructuralmechan-
icsfromNorthwesternUniversityin2010.HereceivedtheBestPaperAwardofthe48th
U.S.RockMechanics/GeomechanicsSymposium,the2015YoungInvestigatorAward
from the U.S. Army Research Office, and the 2017 ASCE Leonardo da Vinci Award.
Hisresearchinterestsincludefracturemechanics,probabilisticmechanics,scaling,and
structuralreliability.

Probabilistic Mechanics of
Quasibrittle Structures
Strength, Lifetime, and Size Effect
ZDENEˇK P. BAŽANT
NorthwesternUniversity
JIA-LIANG LE
UniversityofMinnesota
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Informationonthistitle:www.cambridge.org/9781107151703
©ZdeneˇkP.BažantandJia-LiangLe2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2017
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Bažant,Z.P.|Le,Jia-Liang,1980–
Title:Probabilisticmechanicsofquasibrittlestructures:strength,lifetime,andsizeeffect/
ZdeneˇkP.Bažant,NorthwesternUniversity,Jia-LiangLe,UniversityofMinnesota.
Othertitles:Probabilisticmechanicsofquasibrittlestructures
Description:Cambridge:CambridgeUniversityPress,[2017]|Includes
bibliographicalreferencesandauthorindex.
Identifiers:LCCN2016041636|ISBN9781107151703(hardback)
Subjects:LCSH:Fracturemechanics.|Brittleness.|Elasticanalysis
(Engineering)|Structuralanalysis(Engineering)
Classification:LCCTA409.B392017|DDC624.1/76–dc23
LCrecordavailableathttps://lccn.loc.gov/2016041636
ISBN978-1-107-15170-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
DedicatedtothememoryofAlfredMartinFreudenthal(1906–1977),widely
regardedasthefounderofthefieldofstructuralsafetyandreliability,
encompassingboththetheoriesofprobabilityandstatisticsand
thetheoriesofmechanicsofmaterialsandstructures.

Contents
Foreword pagexiii
Preface xv
1 Introduction 1
1.1 TheProblemofTailofProbabilityDistribution 1
1.2 HistoryinBrief 3
1.2.1 ClassicalHistory 3
1.2.2 RecentDevelopments 6
1.3 SafetySpecificationsinConcreteDesignCodesandEmbedded
ObstaclestoProbabilisticAnalysis 9
1.4 ImportanceofSizeEffectforStrengthStatistics 10
1.5 Power-LawScalingintheAbsenceofCharacteristicLength 12
1.5.1 NominalStrengthofStructureandSizeEffect 13
1.6 StatisticalandDeterministicSizeEffects 14
1.7 SimpleModelsforDeterministicSizeEffects 14
1.7.1 Type1SizeEffectforFailuresatCrackInitiation 15
1.7.2 Type2SizeEffectforStructureswithDeepCracks
orNotches 16
1.8 ProbabilityDistributionsofStrengthofDuctileandBrittle
Structures 19
2 ReviewofClassicalStatisticalTheoryofStructuralStrengthandStructural
Safety,andofStatisticsFundamentals 22
2.1 Weakest-LinkModel 22
2.2 WeibullTheory 23
2.3 ScalingofWeibullTheoryandPureStatisticalSizeEffect 24
2.4 EquivalentNumberofElements 26
2.5 StabilityPostulateofExtremeValueStatistics 27
2.6 DistributionsEnsuingfromStabilityPostulate 28
2.7 CentralLimitTheoremandStrengthDistributionof
DuctileStructures 30
2.8 FailureProbabilityWhenBoththeStrengthandLoadAreRandom,
andFreudenthalIntegral 32
viii Contents
3 ReviewofFractureMechanicsandDeterministicSizeEffectin
QuasibrittleStructures 35
3.1 LinearElasticFractureMechanics 35
3.2 CohesiveCrackModel 37
3.3 CrackBandModel 40
3.4 NonlocalDamageModelsandLattice-ParticleModel 44
3.5 OvercomingInstabilityofTestsofPost-PeakSoftening
ofFiber–PolymerComposites 46
3.6 DimensionalAnalysisofAsymptoticSizeEffects 47
3.7 Second-OrderAsymptoticPropertiesofCohesiveCrackorCrack
BandModels 50
3.8 TypesofSizeEffectDistinguishedbyAsymptoticProperties 51
3.9 DerivationofQuasibrittleDeterministicSizeEffectfrom
EquivalentLEFM 52
3.9.1 Type2SizeEffect 53
3.9.2 Type1SizeEffect 54
3.10 NonlocalWeibullTheoryforMeanResponse 56
3.11 CombinedEnergetic-StatisticalSizeEffectLawandBridgingof
Type1and2SizeEffects 57
4 FailureStatisticsofNanoscaleStructures 59
4.1 BackgroundofModelingofNanoscaleFracture 59
4.2 Stress-DrivenFractureofNanoscaleStructures 60
4.3 ProbabilityDistributionofFatigueStrengthatNanoscale 65
4.4 RandomWalkAspectofFailureofNanoscaleStructures 66
5 Nano–MacroscaleBridgingofProbabilityDistributionsofStatic
andFatigueStrengths 71
5.1 ChainModel 72
5.2 Fiber-BundleModelforStaticStrength 73
5.2.1 BrittleBundle 74
5.2.2 PlasticBundle 79
5.2.3 SofteningBundlewithLinearSofteningBehavior 81
5.2.4 BundlewithGeneralSofteningBehaviorandNonlocal
Interaction 84
5.3 Fiber-BundleModelforFatigueStrength 88
5.4 HierarchicalModelforStaticStrength 92
5.5 HierarchicalModelforFatigueStrength 97
6 MultiscaleModelingofFractureKineticsandSizeEffectunderStatic
andCyclicFatigue 100
6.1 PreviousStudiesofFractureKinetics 100
6.2 FractureKineticsatNanoscale 102
Contents ix
6.3 MultiscaleTransitionofFractureKineticsforStaticFatigue 103
6.4 SizeEffectonFractureKineticsunderStaticFatigue 106
6.5 MultiscaleTransitionofFractureKineticsunderCyclicFatigue 108
6.6 SizeEffectonFatigueCrackGrowthRateandExperimental
Evidence 112
6.7 MicroplaneModelforSizeEffectonFatigueKineticsunder
GeneralLoading 117
7 SizeEffectonProbabilityDistributionsofStrengthand
LifetimeofQuasibrittleStructures 119
7.1 ProbabilityDistributionofStructuralStrength 119
7.2 ProbabilityDistributionofStructuralLifetime 122
7.2.1 CreepLifetime 122
7.2.2 FatigueLifetime 127
7.3 SizeEffectonMeanStructuralStrength 129
7.4 SizeEffectsonMeanStructuralLifetimesandStress-LifeCurves 133
7.5 EffectofTemperatureonStrengthandLifetimeDistributions 136
8 ComputationofProbabilityDistributionsofStructuralStrengthandLifetime 139
8.1 NonlocalBoundaryLayerModelforStrengthandLifetime
Distributions 139
8.2 ComputationbyPseudo-randomPlacingofRVEs 144
8.3 ApproximateClosed-FormExpressionforStrength
andLifetimeDistributions 146
8.4 AnalysisofStrengthStatisticsofBeamsunderFlexuralLoading 152
8.5 OptimumFitsofStrengthandLifetimeHistograms 154
8.5.1 OptimumFitsofStrengthHistograms 154
8.5.2 OptimumFitsofHistogramsofCreepLifetime 157
8.5.3 OptimumFitsofHistogramsofFatigueLifetime 159
9 IndirectDeterminationofStrengthStatisticsofQuasibrittleStructures 161
9.1 RelationbetweenMeanSizeEffectCurveandProbability
DistributionofRVEStrength 161
9.2 ExperimentalVerification 164
9.2.1 DescriptionofExperiments 164
9.2.2 AnalysisofTestResults 166
9.3 DeterminationofLarge-SizeAsymptoticPropertiesof
theSizeEffectCurve 169
9.4 ComparisonwiththeHistogramTestingMethod 170
9.5 ProblemswiththeThree-ParameterWeibullDistributionofStrength 171
9.5.1 TheoreticalArgument 171
9.5.2 EvidencefromHistogramTesting 172
9.5.3 MeanSizeEffectAnalysis 173