Fracture mechanics

238 Pages·2011·2.862 MB·English

by  Alan T Zehnder

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Alan T. Zehnder
Fracture Mechanics
AlanT.Zehnder
SibleySchoolofMechanicaland
AerospaceEngineering
CornellUniversity
Ithaca
USA
[email protected]
ISSN1613-7736 e-ISSN1860-0816
LectureNotesinAppliedandComputationalMechanics
ISBN978-94-007-2594-2 e-ISBN978-94-007-2595-9
DOI10.1007/978-94-007-2595-9
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Preface
Fracture mechanics is a large and always growing field. A search of the Cornell
Libraryinwinter2006uncoveredover181entriescontaining“fracturemechanics”
inthesubjectheadingand10,000entriesinarelevancekeywordsearch.Thisbook
iswrittenforstudentswhowanttobegintounderstand,applyandcontributetothis
important field. It is assumed that the reader is familiar with the theory of linear
elasticity,vectorcalculus,linearalgebraandindicialnotation.
There are many approaches to teaching fracture. Here the emphasis is on con-
tinuummechanicsmodelsforcracktipfieldsandenergyflows.Abriefdiscussion
of computational fracture, fracture toughness testing and fracture criteria is given.
Theycontainverylittleonfractureatthemicromechanicalleveloronapplications.
Boththemechanicsandthematerialssidesoffractureshouldbestudiedinorderto
obtain a balanced, complete picture of the field. So, if you start with fracture me-
chanics, keep going, study the physical aspects of fracture across a broad class of
materialsandreaduponfracturecasestudies[1]tolearnaboutapplications.
I use these notes in a one-semester graduate level course at Cornell. Although
thesenotesgrowoutofmyexperienceteaching,theyalsoowemuchtoAresRosakis
fromwhomItookfracturemechanicsatCaltechandtoHutchinson’snotesonnon-
linearfracture[2].TextbooksconsultedincludeLawn’sbookonthefractureofbrit-
tlematerials[3],Sureshonfatigue[4]andJanssen[5],Anderson[6],Sanford[7],
Hellan[8]andBroberg[9].
I would like to thank Prof. E.K. Tschegg for generously hosting me during my
2004sabbaticalleaveinVienna,duringwhichIstartedthesenotes.Thanksalsoto
mystudentswhoencouragedmetowriteand,inparticular,toformerstudentsMike
CzabajandJakeHochhalterwhoeachcontributedsections.
References
1. ASTM,CaseHistoriesInvolvingFatigueandFracture,STP918(ASTMInternational,West
Cohshohocken,1986)
2. J.W.Hutchinson,ACourseonNonlinearFractureMechanics(DepartmentofSolidMechanics,
TheTechnicalUniversityofDenmark,1979)
v
vi Preface
3. B.Lawn,FractureofBrittleSolids,2ndedn.(CambridgeUniversityPress,Cambridge,1993)
4. S.Suresh,FatigueofMaterials,2ndedn.(CambridgeUniversityPress,Cambridge,1998)
5. M.Janssen,J.Zuidema,R.Wanhill,FractureMechanics,2ndedn.(SponPress,London,2004)
6. T.L. Anderson, Fracture Mechanics Fundamentals and Applications, 2nd edn. (CRC Press,
BocaRaton,1995)
7. R.J.Sanford,PrinciplesofFractureMechanics(PrenticeHall,NewYork,2003)
8. K.Hellan,IntroductiontoFractureMechanics(McGraw-Hill,NewYork,1984)
9. K.B.Broberg,CracksandFracture(AcademicPress,SanDiego,1999)
Ithaca,USA Alan T.Zehnder
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 NotableFractures . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 BasicFractureMechanicsConcepts . . . . . . . . . . . . . . . . . 3
1.2.1 SmallScaleYieldingModel . . . . . . . . . . . . . . . . . 4
1.2.2 FractureCriteria . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 FractureUnitConversions . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 LinearElasticStressAnalysisof2DCracks . . . . . . . . . . . . . . 7
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 ModesofFracture . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 ModeIIIField . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.1 AsymptoticModeIIIField . . . . . . . . . . . . . . . . . 9
2.4.2 FullFieldforFiniteCrackinanInfiniteBody . . . . . . . 13
2.5 ModeIandModeIIFields . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 ReviewofPlaneStressandPlaneStrainFieldEquations . . 16
2.5.2 AsymptoticModeIField . . . . . . . . . . . . . . . . . . 17
2.5.3 AsymptoticModeIIField . . . . . . . . . . . . . . . . . . 21
2.6 ComplexVariablesMethodforModeIandModeIICracks . . . . 21
2.6.1 WestergaardApproachforMode-I . . . . . . . . . . . . . 22
2.6.2 WestergaardApproachforMode-II . . . . . . . . . . . . . 22
2.6.3 GeneralSolutionforInternalCrackwithAppliedTractions 22
2.6.4 FullStressFieldforMode-ICrackinanInfinitePlate . . . 23
2.6.5 StressIntensityFactorUnderRemoteShearLoading. . . . 25
2.6.6 StressIntensityFactorsforCracksLoadedwithTractions . 26
2.6.7 AsymptoticModeIFieldDerivedfromFullFieldSolution 26
2.6.8 AsymptoticModeIIFieldDerivedfromFullFieldSolution 28
2.6.9 StressIntensityFactorsforSemi-infiniteCrack . . . . . . . 28
2.7 SomeComments . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
viii Contents
2.7.1 Three-DimensionalCracks . . . . . . . . . . . . . . . . . 29
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 EnergyFlowsinElasticFracture . . . . . . . . . . . . . . . . . . . . 33
3.1 GeneralizedForceandDisplacement . . . . . . . . . . . . . . . . 33
3.1.1 PrescribedLoads. . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 PrescribedDisplacements . . . . . . . . . . . . . . . . . . 34
3.2 ElasticStrainEnergy. . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 EnergyReleaseRate,G . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 PrescribedDisplacement . . . . . . . . . . . . . . . . . . 36
3.3.2 PrescribedLoads. . . . . . . . . . . . . . . . . . . . . . . 37
3.3.3 GeneralLoading . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 InterpretationofGfromLoad-DisplacementRecords . . . . . . . 38
3.4.1 MultipleSpecimenMethodforNonlinearMaterials . . . . 38
3.4.2 ComplianceMethodforLinearlyElasticMaterials . . . . . 41
3.4.3 ApplicationsoftheComplianceMethod . . . . . . . . . . 42
3.5 CrackClosureIntegralforG . . . . . . . . . . . . . . . . . . . . 43
3.6 GinTermsofK ,K ,K for2DCracksThatGrowStraight
I II III
Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.1 Mode-IIILoading . . . . . . . . . . . . . . . . . . . . . . 47
3.6.2 ModeILoading . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.3 ModeIILoading . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.4 GeneralLoading(2DCrack) . . . . . . . . . . . . . . . . 48
3.7 ContourIntegralforG(J-Integral) . . . . . . . . . . . . . . . . . 49
3.7.1 TwoDimensionalProblems . . . . . . . . . . . . . . . . . 49
3.7.2 Three-DimensionalProblems . . . . . . . . . . . . . . . . 51
3.7.3 ExampleApplicationofJ-Integral . . . . . . . . . . . . . 51
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 CriteriaforElasticFracture . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 InitiationUnderMode-ILoading . . . . . . . . . . . . . . . . . . 55
4.3 CrackGrowthStabilityandResistanceCurve. . . . . . . . . . . . 58
4.3.1 LoadingbyCompliantSystem . . . . . . . . . . . . . . . 60
4.3.2 ResistanceCurve . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Mixed-ModeFractureInitiationandGrowth . . . . . . . . . . . . 63
4.4.1 MaximumHoopStressTheory . . . . . . . . . . . . . . . 63
4.4.2 MaximumEnergyReleaseRateCriterion . . . . . . . . . . 65
4.4.3 CrackPathStabilityUnderPureMode-ILoading . . . . . 66
4.4.4 SecondOrderTheoryforCrackKinkingandTurning . . . 69
4.5 CriteriaforFractureinAnisotropicMaterials . . . . . . . . . . . . 70
4.6 CrackGrowthUnderFatigueLoading. . . . . . . . . . . . . . . . 71
4.7 StressCorrosionCracking . . . . . . . . . . . . . . . . . . . . . . 74
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Contents ix
5 DeterminingK andG . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 AnalyticalMethods . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1 ElasticityTheory. . . . . . . . . . . . . . . . . . . . . . . 77
5.1.2 EnergyandComplianceMethods . . . . . . . . . . . . . . 79
5.2 StressIntensityHandbooksandSoftware . . . . . . . . . . . . . . 80
5.3 BoundaryCollocation . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 ComputationalMethods:APrimer . . . . . . . . . . . . . . . . . 84
5.4.1 StressandDisplacementCorrelation . . . . . . . . . . . . 84
5.4.2 GlobalEnergyandCompliance . . . . . . . . . . . . . . . 85
5.4.3 CrackClosureIntegrals . . . . . . . . . . . . . . . . . . . 86
5.4.4 DomainIntegral . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.5 CrackTipSingularElements . . . . . . . . . . . . . . . . 90
5.4.6 ExampleCalculations . . . . . . . . . . . . . . . . . . . . 94
5.5 ExperimentalMethods . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5.1 StrainGaugeMethod . . . . . . . . . . . . . . . . . . . . 98
5.5.2 Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5.3 DigitalImageCorrelation . . . . . . . . . . . . . . . . . . 101
5.5.4 ThermoelasticMethod . . . . . . . . . . . . . . . . . . . . 103
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 FractureToughnessTests . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 ASTMStandardFractureTest . . . . . . . . . . . . . . . . . . . . 110
6.2.1 TestSamples . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.3 TestProcedureandDataReduction . . . . . . . . . . . . . 112
6.3 InterlaminarFractureToughnessTests . . . . . . . . . . . . . . . 113
6.3.1 TheDoubleCantileverBeamTest . . . . . . . . . . . . . . 113
6.3.2 TheEndNotchFlexureTest . . . . . . . . . . . . . . . . . 117
6.3.3 SingleLegBendingTest . . . . . . . . . . . . . . . . . . . 118
6.4 IndentationMethod . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Chevron-NotchMethod . . . . . . . . . . . . . . . . . . . . . . . 122
6.5.1 K Measurement . . . . . . . . . . . . . . . . . . . . . 123
IVM
6.5.2 K Measurement . . . . . . . . . . . . . . . . . . . . . . 124
IV
6.5.3 WorkofFractureApproach . . . . . . . . . . . . . . . . . 125
6.6 WedgeSplittingMethod . . . . . . . . . . . . . . . . . . . . . . . 127
6.7 K–RCurveDetermination . . . . . . . . . . . . . . . . . . . . . 130
6.7.1 Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.7.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.7.3 TestProcedureandDataReduction . . . . . . . . . . . . . 133
6.7.4 SampleK–Rcurve . . . . . . . . . . . . . . . . . . . . . 134
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135