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Elementary engineering fracturemechanics DAVIDBROEK 1982MARTINUSNIJHOFF PUBLISHERS~ . amemberoftheKLUWERACADEMICPUBLISHERSGROUPII. BOSTON/THEHAGUE/DORDRECHT/LANCASTER.... Distributors for theUnitedStatesand Canada:KluwerBoston,Inc.,190OldDerbyStreet, Hingham,MA02043,USA for 01/ other countries:Kluwer Academic Publishers Group,Distribution Center, P.O.Box 322,3300 AHDordrecht,The Netherlands LibraryofCongress CatalogueCardNumber 82-45135 1stEdition1974 2ndrevisededition1978 3rdrevisededition1982 3rd printing1984 ISBN90-247-2580-1 Copyright 1984byMartinusNijhoff Publishers,The Hague. Allrightsreserved.Nopartof thispublicationmaybereproduced,storedina retrievalsystem,ortransmittedinanyformorbyanymeans,mechanical, photocopying,recording,orotherwise,withoutthepriorwrittenpermissionof thepublishers, Martinus Nijhoff Publishers,P.O.Box566,2501CN The Hague, The Netherlands. PRINTEDIN THE NETHERLANDS Contents Prefacetothefirstedition Prefacetothesecond edition Prefaceto thethird edition PartIPRINCIPLES ChapterI Summary of basicproblems and concepts l.lIntroduction 1.2Acrackinastructure 1.3Thestress atacracktip 1.4TheGriffithcriterion 1.5The crack opening displacement criterion 1.6Crackpropagation 1.7Closure Chapter 2 Mechanisms of fractureand crackgrowth 2.1Introduction 2.2Cleavagefracture 2.3Ductile fracture 2.4Fatigue cracking 2.5Environmentassistedcracking 2.6Service failureanalysis XI XIII XIV 3 3 6 8 15 17 18 22 24 24 31 38 48 59 62 v Contents Chapter 3 The elastic crack-tip stress field67 3.1The Airystressfunction67 3.2Complex stressfunctions68 3.3Solutiontocrackproblems69 3.4The effectof finitesize73 3.5Special cases77 3.6Ellipticalcracks80 3.7Some usefulexpressions86 Chapter 4 The crack tip plastic zone91 4.1The Irwinplastic zone correction91 4.2The Dugdale approach94 4.3The shape of the plastic zone 96 4.4Plane stress versus plane strain101 4.5Plastic constraintfactor105 4.6Thethickness effect107 Chapter 5 The energyprinciple115 5.1The energyreleaserate115 5.2The criterionforcrack growth119 5.3Thecrackresistance (R curve)122 5.4Compliance127 5.5The Jintegral131 5.6Tearing modulus 136 5.7Stability137 VT Chapter 6 Dynamics and crackarrest 6.1Crackspeedandkinetic energy 6.2The dynamic stressintensityand elastic energyreleaserate 6.3Crackbranching 6.4The principlesof crackarrest 6.5Crack arrestinpractice 6.6Dynamic fracturetoughness Chapter 7 Plane strainfracturetoughness 7.1The standardtest 7.2Sizerequirements 7.3Non-linearity 7.4Applicability Chapter 8 Plane stress andtransitionalbehaviour 8.1Introduction 8.2An engineering concept of plane stress 8.3TheRcurveconcept 8.4Thethicknesseffect 8.5Plane stresstesting 8.6Closure Contents 142 142 147 ISO ISS 162 165 170 170 174 177 181 185 185 185 193 199 208 216 VII Contents Chapter 9 Elastic-plastic fracture 9.1Fracturebeyondgeneralyield 9.2The crack tip opening displacement 9.3The possibleuseof the CTOD criterion 9.4Experimental determinationof CTOd 9.5Parameters affecting thecritical CTOD 9.6Limitations,fractureat generalyield 9.7Useof the Jintegral 9.8Limitations of the Jintegral 9.9Measurement of JIeand J R 9.10Closure Chapter10 Fatigue crack propagation 10.1Introduction 10.2Crack growthandthe stress intensity factor 10.3Factors affecting crackpropagation 10.4Variable amplitude serviceloading 10.5Retardation models 10.6Similitude 10.7Smallcracks 10.8Closure Chapter11 Fracture resistance of materials 11.1Fracture criteria 11.2Fatigue cracking criteria 11.3The effect of alloying and secondphase particles 11.4Effectof processing, anisotropy 11.5Effectof temperature 11.6Closure '"'' 219 219 221 224 225 228 231 235 237 240 245 250 250 250 256 262 266 271 278 282 288 295 295 297 304 309 311 Contents PartIIAPPLICATIONS Chapter12 Fail-safety and damagetolerance317 12.1Introduction317 12.2Meansto providefail-safety318 12.3Requiredinformationforfracturemechanicsapproach323 12.4Closure326 Chapter13 Determination of stressintensityfactors 13.1Introduction 13.2Analytical andnumericalmethods 13.3Finite elementmethods 13.4Experimentalmethods Chapter14 Practicalproblems 14.1Introduction 14.2Through cracks emanating fromholes 14.3Corner cracks atholes 14.4Cracks approaching holes 14.5Combinedloading 14.6Fatigue crackgrowthunder mixedmodeloading 14.7Biaxialloading 14.8Fracturetoughness of weldmenfs 14.9Servicefailureanalysis 328 328 330 330 338 347 347 347 352 356 359 366 369 371 374 Contents Chapter15 Fracture of structures 15.1Introduction 15.2Pressure vesselsandpipelines 15.3"Leak-before-break" criterion 15.4Material selection 15.5The useof the Jintegral forstructural analysis 15.6Collapse analysis 15.7Accuracy of fracturecalculations Chapter16 377 377 378 388 392 396 399 404 Stiffened sheet structures408 16.1Introduction408 16.2Analysis409 16.3Fatigue crack propagution413 16.4Residual strength415 16.5TheRcurve andtheresidual strengthof stiffenedpanels422 16.6Other analysismethods425 16.7Crack arres t 16.8Closure Chapter17 Prediction of fatiguecrack growth 17.1Introduction 17.2The loadspectrum 17.3Approximation of thestressspectrum 17.4Generation of astresshistory 17.5Crack growthintegration 17.6Accuracy of predictions 17.7Safetyfactors Author index Subject index 427 431 434 434 435 437 439 441 447 452 455 462 Preface to the first edition When askedtostartteaching acourseonengineeringfracturemechanics, Irealizedthataconcisetextbook,givingageneraloversightof thefield, didnotexist.Theexplanationisundoubtedlythatthesubjectisstillina stageof earlydevelopment,andthatthemethodologieshavestillavery limited applicability.It isnotpossibletogiverulesforgeneralapplication of fracturemechanicsconcepts.Yetourcomprehensionof crackingand fracture behaviour of materials and structures is steadily increasing. Further developmentsmaybeexpectedinthenottoodistantfuture,enabling usefulprediction of fracturesafety and fracture onthebasis of advancedfracturemechanicsprocedures.Theuserofsuchadvanced proceduresmusthave ageneralunderstanding of theelementaryconcepts, which areprovidedbythisvolume. Emphasis was placed on the practical applicationof fracturemechanics, butitwasaimedtotreatthesubjectinawaythatmayinterestboth metallurgistsandengineers.Forthelaller,somegeneralknowledgeof fracturemechanisms and fracturecriteriaisindispensableforanapprecia-tion of the limitations of fracturemechanics. Therefore ageneraldiscussion isprovidedonfracturemechanisms,fracturecriteria,andothermetal-lurgicalaspects,withoutgoingintomuchdetail.Numerousreferencesare providedtoenableamoredetailedstudyof thesesubjectswhicharestill inastageofspeculativetreatment.Metallurgistsandmeehanicistsneed toknowtheparticularproblemsofindustryinordertobeableto respondtothedemandsof engineers.Therefore,somepertinentpractical problems arediscussed. Noallemptwasmadetopresentadetailedreviewofeveryaspectof Preface theprinciplesandapplicationoffracturemechanics.Suchatreatment hasalreadybeengiveninthesevenvolumes"Fracture,Anadvanced treatise",editedbyLiebowitz.Instead,asthetitleindicates,anattempt wasmadetodealwiththeelementariesandwiththoseengineering applicationsthathavefoundsomeacceptance.Thetextmayserveasan introductiontotheliteratureandasabasisfortheunderstandingof forthcomingdevelopment.AlthoughIhavetriedtoarriveatabalanced presentationof thevarioussubjects,thetreatmentof certainsubjectsmay betraymypersonalinterests. Inviewofthestageofdevelopmentoffracturemechanics,acertain degree of speculation could notbe avoided.Where appropriate,speculative discussionsarespecifiedassuch.Attentionisfocussedonsubjectsthat arepromisingforquantitativeuseindesign.Qualitativefractureanalysis proceduresandtesting,suchastheuseof Charpytestdata,Robertson testsanddynamicteartests,arenotconsidered,becausethesehavebeen amplytreatedelsewhere. The textisconsideredsuitableforadvancedundergraduate orfirstyear graduate students.Butitmayalsoserveasageneralintroductiontothis relativelynewdisciplineforengineersandmetallurgistswhohavenot beenconfrontedearlierwithfracturemechanics. IamindebtedtomyformercolleaguesattheNa tionalAerospace LaboratoryN,L.R.,inwhoseenthusiasticfracturemcchanicsgroupI participatedduringtwelveyears.WithouttheirencouragementIwould nothavedevelopedtoastagewherewritingthistcxtwouldhavebeen possible.Inparticular,IwanttothankDr.R.J.H.Wanhill,who scrutinizedthetextandmademanyusefulsuggestionsforimprovement. I am gratefultoMissLucyLoomansforherassistanceinthepreparation ofthemanuscript,toMr.L.vandeEijkelfortheartwork,andto Ir.B.Pennekampforhishelpincorrectingtheproofs.Finally,Iwant toapologizetoallwhohavesufferedfrommylimitedinterestinother mattersduringpreparationof themanuscript. Delft,January1974DavidBroek XII Preface to the second edition Sincethepublicationofthefirsteditionthepredictionoffatiguecrack propagationhasbecomecommonpracticeinseveralareasofindustry. Thereforeitseemedappropriatetoaddanextrachapter17dealingwith crack growthprediction.Apartfromthatthissecondeditionisessentially thesameasthefirst,althoughafewareas,suchasdynamicfractureand mixedmodeloading,wereexpanded. I want to emphasize thatthe objective of thisbookisto dealinparticular with those areas of fract ure mechanics that have found practical application. Thebackground.assumptionsandlimitationsarepresentedandthe problems of engineering application are pointed out. The texttouchesupon matterspresentlyunderdevelopment,butIdonotbelievethatmore extciisivetreatmentisappropriateinatextbookonengineeringfracture mechanics. IamgratefulforthemanyappreciativecommentsIreceivedupon publicationof thefirstedition. Columbus.May1978DavidBroek XIII Prefacetothethird edition Duringthelastfiveyearsaconsiderableamountof researchonelastic-plasticfracturemechanicshasbeenconducted.Althoughelastic-plast}c. fracture concepts are still mostly inthe stage of paper and laboratory studies, sometechnicalapplicationsbegintoemerge.Thereforeitseemedappro-priate to expand thistextto cover elastic-plastic fracture mechanics inmore detail. To this end,the J integral has beengivenmore coverage inchapter 5 asanenergyconceptandinchapter9asastressfieldparameteranda fracture criterion, whereas chapter 15 has been extended with a discussion of thepractical aspects of fracturepredictions instructures of hightoughness materials. Intheareaof fatiguecrackpropagation,amoredetaileddiscussionof similituderequirementswasfeltnecessary.Thisresultedinan extension of chapter10. It istheaimof thisbooktoshowtheuseandapplicationof fracture mechanicstopractical problems. Thenumerous compromises that have to bemadeinpracticalapplicationshavetobebasedonsoundengineering judgement.Forthisreasonshortcomingsandlimitationsof thevarious fracturemechanicsconceptsarestronglyemphasized,andontheother hand,approximative andsimplifiedconceptsareamplydiscussedbecause theyare oftenmoreusefulthanthe"formal"methods. Iamgratefulthatthereisstillsufficientinterestinthistext,sothatthe publisher wouldentertainathirdedition. Columbus, January1981DavidBroek v,,, Part I Principles 1 Summary of basic problems and concepts 1.1Introduction Throughtheagestheapplicationofmaterialsinengineeringdesignhas poseddifficultproblemstomankind.IntheStoneAgetheproblems weremainlyintheshapingofthematerial.Intheearlydaysofthe BronzeAgeandtheIronAgethedifficultieswerebothinproduction andshaping.Formanycenturiesmetal-workingwaslaboriousand extremelycostly.Estimatesgothattheequipmentof aknightandhorse inthethirteenthcenturywasoftheequivalentpriceof aCenturiontank inWorldWarII. Withtheimprovingskillofmetalworking.applicationsofmetalsin structuresincreasedprogressively.Thenitwasexperiencedthatstructures huiltofthesematerialsdidnotalwayshehavesatisractorily,andun-expectedI;tilurcsortenoccurred.Detaileddescriptionsofcastingsand forgingsproducedIntheMiddleAgesexist.Whenjudgedwithpresent dayknowledge,theseproductionmethodsmusthaveheenliabletohuild importanttechnicald d i c i e n c e ~intothcstructure.Thismusthavemadc gunnerspray -whenignitingthccharge- thattheprojectilewouldbe properlydeliveredandtheharrelnothlownup... The\i,lstlyincreasinguscofmetalsinthenineteenthcenturyca used thenumberof,Iccidcntsandcasualtiestoreachunknownlevels.The numberofpeoplekilledinrailwayaccidentsinGreatBritainwasinthe orderoftwohundredperyearduringthedecade186U-1870.Mostof theaccidentswerearesultofderailingcausedbyrraeturesofwheels, axlesorrails.Andersonll]hasrecentlymadeaninterestingcompilation ofaccidentrcportsfrol11thelasttwohundredyears.Afewquotations follow: 3 JSUllllllary of hllsicprohlc/IlslIlIdconccpts "Onthe19thofMarchIXJOabout700personsassul11bledonthe Montrose bridgetowitnessaboatrace,whenoneofthe mainehainsgaveway ...andcausedconsiderablelossof life." "Onthe22ndof JanuaryIX66.aportionof theroof of theManches((;r railway stationIdl.causing deaths of twomen.The accidcntwas causcd byfailureof cast-ironstrutsconnectcd ... " "The failureofaImge gastankinNewYorkoccurredonDecemberD. 1898. killing and injuring anumber of people and destroying considcrable surroundingproperty." "AhighpressurewatermainburstatBoston.JanuaryJ.ll)!.'and floodedthe district. .. " "Engineering.February1866.Withsome P/iy tosixt.\'boilerexplosions annuallyintheUnitedKingdom attended asthey arcwithlossof many livesanddestructionorproperty.isitnottimethattheGoverment should appointacommissiontoinquireintothesubject T "The most seriousrailroad accidentof theweek occurredApril20 ( I XX7) andwascausedbythebreakingofadrawbar.Threewerekilledand twofatallyinjured." "The most railroad accidentof thewcck occurrcdMay 27(IXX7). Theburstingof awheelcausedthedeathsof sixpeople." "The mostseriousrailroad accident of the \I'cckoccurredJune23(1887) andwascausedbyabrokenrail.Onemanwaskilled." "ThemostseriousrailroadaccidentoftheweekoccurredonJuly2 (1887)andwascausedbythcbreaking of an axle." Someof theseacddentswcrecertainlyduetoapoordesign,butitwas gradually discoveredthatmaterialdeficienciesintheformof pre-existing flawscouldinitiatecracksandfractures.Preventionof suchflawswould improve structuralperformance.Betterproductionmethodstogetherwith increasingknowledgeandcomprehensionof materialpropertiesledtoa stage wherethenumber of failureswasreducedtomoreacceptablelen:ls. Aneweraof accidentpronestructuresstartedwiththeintroductionof all-weldeddesigns.Outof 2500LibertyshipsbuiltduringWorldWarII. 145brokeintwoandalmost700experiencedseriousfailures.Thesame disasterstruckmanybridgesandotherstructures.Informationonthese failures was also given by Anderson [I] and more specifica lIy e.g..byBiggs [2]. Thefailuresoftenoccurredunderconditionsoflowstresses(several shipsfailedsuddenlywhileintheharbour)whichmadethemseemingly 4 1.1in/rodll('/ion inexplicable.Asaresultextensiveinvestigationswereinitiatedinmany countries and especially inlheUnited States or America. This workrevealed thathereagain.flaws,lIldstressconcentrations(,lIldtoacertainextent illternalwere1'01'1;lilure. Therr,lcturesweretrulyhrittle:theywereaccompalliedhy littlc plasticddorm'ltion.Itturnedoutth,ltthehrittlerractureorsteelwas promotedhylowtemperaturesandhyconditionsortriaxi,dstresssuch asmayexistat,Isharpnotchoral1'IW.Underthesecircumstances structuralsteelcallfracturebycle'l\agc(clupter2)withoutnoticeahle dcrormation.Aboveacertaintemperature.calledthetrallsitioll temper'lture.thesteclhehavesinaductilemanner.Thetransitiontem-peraturemaygoupasaresultortheheatcycleduringthe\\clding process. Atpresent.brittlerracturesor \\(.'Idedstructureshuiltoutllr lowstrength stl'llctural canbes,ltislilctllrilyprevented.Ithast()heensuredthat thematerialisprlHlucedtohave,Ilowtransitiontemperatureandthat the welding process docs notraisetheductile-hrittletransition.Largl'stress concentrationsshouldhe,1\ oidedandtheweldsshouldbecheckedtohe \ irtuallyrreeof defects. ArterWorldWarIItheuscorhighstrengthmaterialshasincreased consider'lbly.Thesematerialsarcoftenselectedtorcalileweightsavings. Simultaneously.stressanalysismethodsweredevelopedwhichenahlea morereliabledeterminationor localstresses.Thispermittedsard)ractors tohereducedresultinginfurther\\eightsavings.Consequently.structures designedinhighstrengthmaterialshaveonlylowmarginsor saret).This meansthatsen'icestresses(sometimeswiththeaidor.3). Manyotherequationshavebeenproposed.Theyareanalysedina concisepaperbyPelloux[181.Furtherworktoderiveanequationwith asoundphysicall1asisilicertainlyneeded:itmustbeanticipatedthat thisIInalequationwillbeacomplicatedoneifitistohaveageneral validity.For thetechnicalproblemof fatiguecrackpropagationthesimple knowledgethatda/dNisafunctionof thestressintensityfactorwilloften besufficient.aswillbecomeapparentinthischapter. Aliltiguecycleisdefinedbyafrequencyandtwostressparameters. ThesecanbethemeanstressSOlandthestressamplitudeSa.the minimumstressinacycle(Sm;n = Sill- Sa)andthemaximumstress (Sm ... =Sm+ Sa).orothercombinationsoftwoofthesefourparameters. Aslongasthecyclera tio(R =SnllnSill;")equalszeroonecanspeak aboutthestressintensityfactorofthefatiguecycle.since Sill."=2Sa =Thehypothesisthattherateofcrackpropagationis a functionor thestress intensityfactorpresentsnodifficulties.WhenR -# 0 therangeof thestress intensity = 2.)'a \/IW isaninsufficientdescription ofthestressenvironmentorthecracktip.Thequestionariseswhether da/dNwillnowbeafunctionoforof themaximumstressintensity inacycle(K 01;"=SI1"" ,/ TW).or of both. Itappears[19.20]thattherateof crackpropagationisafunctionof both andK mwThiscanbeappreciatedfromfigure10.3.Itcanbe concludedthat dl/ dN ( 1004) Severalinvestigatorshavetriedtoestablishempiricalrelat.ionswhich attempttoincorporatetheeffectofthecycleratiosuchthatalldata couldbecondensedtoasinglecurve.BrockandSchijve[19]proposed a complicatedrelation.butalsothefollowingmoresimpleone: dll,. 2 d--= . 'N ( 10.5) 251

70 60 K(kg/mm3/2)-j-maxI - j- -+-H,'cW'"",c7"'bl'4 I 50 f---+-+--H-++i --Figure10.3.Hfcctorcycleratioontherclati'HlhetweenClackgCl"'thrateandstress intensityfactor/I'IJ.AsimilarequationwasgivenbyErdogan[20].Walkcr[21,22]usedthe more generalequation whichhemodifiedbyintroducingancrfectiveS,i\,yielding da=C 1'11\"whereI'lKisdefinedasifK =Snlax(I-R)"'J;-;;. dN ( 10.6) ( 10.7) Formanetal.[23]arguedthatda/dNshouldbecomeinfinitewhenthe crackreachesacriticalsize_i.e.whenK maXreachesK Ie'Theyarrivedat doC 11K"C 11K" dN=(1::/ffKI-':-I1-K= CI-R)(Ktc-Kma.) (HU\) whichcanberearrangedtogive: da -= dN C I'lKm K md\ Klc-Kmax ( 10.9) The differencesamongtheseexpressionsarcnotlarge,andnoneof them has ageneralapplicability.Eachonemaybe foundreasonablysatisfactory inalimitedregionorforlimitedsetsof data. Thequestionariseswhethercq(1004)stillholdsforR


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