Fracture Mechanics Fracture Mechanics C. T. Sun School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana Z.-H. Jin Department of Mechanical Engineering The University of Maine Orono, Maine AMSTERDAM•BOSTON•HEIDELBERG•LONDON NEWYORK•OXFORD•PARIS•SANDIEGO SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO AcademicPressisanimprintofElsevier AcademicPressisanimprintofElsevier 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,Oxford,OX51GB,UK (cid:13)c 2012ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem, withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuch astheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. 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ISBN:978-0-12-385001-0 ForinformationonallAcademicPresspublications visitourWebsiteatwww.elsevierdirect.com PrintedintheUnitedStates 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1 Tomywife,Iris andmychildren,Edna,Clifford,andLeslie C.T.Sun Tomywife,Zhen Z.-H.Jin Contents Preface............................................................................... xiii AbouttheAuthors................................................................... xvii CHAPTER 1 Introduction ....................................................... 1 1.1 FailureofSolids .................................................. 1 1.2 FractureMechanicsConcepts.................................... 2 1.3 HistoryofFractureMechanics................................... 5 1.3.1 GriffithTheoryofFracture................................ 5 1.3.2 FractureMechanicsasanEngineeringScience.......... 6 1.3.3 RecentDevelopmentsinFractureMechanics Research ................................................... 7 References.............................................................. 8 CHAPTER 2 GriffithTheoryofFracture........................................ 11 2.1 TheoreticalStrength.............................................. 11 2.1.1 AnAtomisticModel....................................... 11 2.1.2 TheEnergyConsideration................................. 13 2.2 TheGriffithTheoryofFracture.................................. 14 2.3 ARelationamongEnergies...................................... 17 References.............................................................. 22 Problems ............................................................... 22 CHAPTER 3 TheElasticStressFieldaroundaCrackTip.................... 25 3.1 BasicModesofFractureandStressIntensityFactor ........... 25 3.2 MethodofComplexPotentialforPlaneElasticity (TheKolosov-MuskhelishviliFormulas)........................ 27 3.2.1 BasicEquationsofPlaneElasticityandAiry StressFunction ............................................ 27 3.2.2 AnalyticFunctionsandCauchy-Riemann Equations .................................................. 29 3.2.3 ComplexPotentialRepresentationoftheAiry StressFunction ............................................ 30 3.2.4 StressandDisplacement................................... 32 vii viii Contents 3.3 WestergaardFunctionMethod ................................... 34 3.3.1 SymmetricProblems(ModeI)............................ 34 3.3.2 Skew-SymmetricProblems(ModeII).................... 36 3.4 SolutionsbytheWestergaardFunctionMethod................. 38 3.4.1 ModeICrack............................................... 38 3.4.2 ModeIICrack.............................................. 43 3.4.3 ModeIIICrack............................................. 46 3.4.4 ComplexRepresentationofStressIntensityFactor...... 48 3.5 FundamentalSolutionsofStressIntensityFactor............... 50 3.5.1 AFiniteCrackinanInfinitePlate ........................ 51 3.5.2 StressIntensityFactorsforaCrackSubjected toArbitraryCrackFaceLoads............................ 53 3.5.3 ASemi-infiniteCrackinanInfiniteMedium............. 54 3.6 FiniteSpecimenSizeEffects..................................... 55 3.7 Williams’CrackTipFields....................................... 56 3.7.1 Williams’CrackTipStressandDisplacement Fields:ModeIandII...................................... 57 3.7.2 Williams’CrackTipStressandDisplacement Fields:ModeIII ........................................... 63 3.8 K-Dominance..................................................... 66 3.9 Irwin’sK-BasedFractureCriterion.............................. 68 References.............................................................. 71 Problems ............................................................... 72 CHAPTER 4 EnergyReleaseRate.............................................. 77 4.1 TheConceptofEnergyReleaseRate............................ 77 4.2 TheRelationsbetweenGandK bytheCrackClosure Method............................................................ 78 4.3 TheJ-Integral..................................................... 82 4.3.1 JasEnergyReleaseRate.................................. 83 4.3.2 Path-Independence......................................... 86 4.3.3 RelationbetweenJandK ................................. 87 4.3.4 Examples................................................... 89 4.4 StressIntensityFactorCalculationsUsingtheFinite ElementMethod.................................................. 92 4.4.1 DirectMethod.............................................. 92 4.4.2 ModifiedCrackClosureTechnique....................... 93 Contents ix 4.5 Three-DimensionalFieldnearCrackFront ..................... 94 4.5.1 DistributionofStressIntensityFactorover Thickness .................................................. 95 4.5.2 PlaneStrainZoneattheCrackFront ..................... 99 References.............................................................. 101 Problems ............................................................... 102 CHAPTER 5 MixedModeFracture............................................. 105 5.1 ASimpleEllipticalModel ....................................... 105 5.2 MaximumTensileStressCriterion(MS-Criterion) ............. 108 5.3 StrainEnergyDensityCriterion(S-Criterion)................... 111 5.4 MaximumEnergyReleaseRateCriterion (ME-Criterion).................................................... 115 5.5 ExperimentalVerifications ....................................... 117 References.............................................................. 119 Problems ............................................................... 120 CHAPTER 6 CrackTipPlasticity............................................... 123 6.1 YieldCriteria...................................................... 124 6.1.1 TrescaYieldCriterion ..................................... 124 6.1.2 vonMisesYieldCriterion................................. 125 6.2 ConstitutiveRelationshipsinPlasticity.......................... 125 6.2.1 FlowTheoryofPlasticity ................................. 126 6.2.2 DeformationTheoryofPlasticity......................... 128 6.3 Irwin’sModelforModeIFracture .............................. 130 6.3.1 PlasticZoneSize........................................... 130 6.3.2 EffectiveCrackLengthandAdjustedStress IntensityFactor............................................ 133 6.3.3 CrackTipOpeningDisplacement......................... 134 6.4 TheDugdaleModel .............................................. 134 6.4.1 Small-ScaleYielding ...................................... 135 6.4.2 ACrackinanInfinitePlate................................ 137 6.5 PlasticZoneShapeEstimateAccordingtotheElastic Solution........................................................... 140 6.5.1 PrincipalStresses .......................................... 140 6.5.2 PlaneStressCase .......................................... 141