https://ebookmass.com/product/a-practical-approach-tofracture-mechanics-1-edition-jorge-luis-gonzalez-velazquez/
Instant digital products (PDF, ePub, MOBI) ready for you
Download now and discover formats that fit your needs...
Introduction to Fracture Mechanics Robert O. Ritchie
https://ebookmass.com/product/introduction-to-fracture-mechanicsrobert-o-ritchie/
ebookmass.com
Fracture Mechanics: Fundamentals and Applications, Fourth Edition 4th Edition – Ebook PDF Version
https://ebookmass.com/product/fracture-mechanics-fundamentals-andapplications-fourth-edition-4th-edition-ebook-pdf-version/
ebookmass.com
Pathophysiology: A Practical Approach 3rd Edition
https://ebookmass.com/product/pathophysiology-a-practicalapproach-3rd-edition/
ebookmass.com
Bystander Society: Conformity and Complicity in Nazi Germany and the Holocaust Mary Fulbrook
https://ebookmass.com/product/bystander-society-conformity-andcomplicity-in-nazi-germany-and-the-holocaust-mary-fulbrook/
ebookmass.com
https://ebookmass.com/product/get-it-done-surprising-lessons-from-thescience-of-motivation-ayelet-fishbach/
ebookmass.com
Programming with Microsoft Visual Basic 2017 8th Edition
Diane Zak
https://ebookmass.com/product/programming-with-microsoft-visualbasic-2017-8th-edition-diane-zak/
ebookmass.com
The Virtues in Psychiatric Practice John R. Peteet
https://ebookmass.com/product/the-virtues-in-psychiatric-practicejohn-r-peteet/
ebookmass.com
Spatiotemporal changes of drought characteristics and their dynamic drivers in Canada Yang Yang & Thian Yew Gan & Xuezhi Tan
https://ebookmass.com/product/spatiotemporal-changes-of-droughtcharacteristics-and-their-dynamic-drivers-in-canada-yang-yang-thianyew-gan-xuezhi-tan/
ebookmass.com
Chemical Analysis and Material Characterization by Spectrophotometry Bhim Prasad Ka■e
https://ebookmass.com/product/chemical-analysis-and-materialcharacterization-by-spectrophotometry-bhim-prasad-ka%ef%ac%82e/
ebookmass.com
Adversarial Robustness for Machine Learning Chen
https://ebookmass.com/product/adversarial-robustness-for-machinelearning-chen/
ebookmass.com
APRACTICALAPPROACHTO FRACTUREMECHANICS
Thispageintentionallyleftblank
APRACTICALAPPROACHTO FRACTUREMECHANICS
JORGELUISGONZÁLEZ-VELÁZQUEZ
MetallurgicalandMaterialsEngineeringDeparmentof InstitutoPolitécnicoNacional, MexicoCity,Mexico
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
Copyright©2021ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorage andretrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowto seekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandour arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyright LicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions .
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightby thePublisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthis fieldareconstantlychanging.Asnewresearchand experiencebroadenourunderstanding,changesinresearchmethods,professional practices,ormedicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribed herein.Inusingsuchinformationormethodstheyshouldbemindfuloftheirownsafety andthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,or editors,assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasa matterofproductsliability,negligenceorotherwise,orfromanyuseoroperationofany methods,products,instructions,orideascontainedinthematerialherein.
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN:978-0-12-823020-6
ForinformationonallElsevierpublicationsvisitourwebsite at https://www.elsevier.com/books-and-journals
Publisher: MatthewDeans
AcquisitionsEditor: DennisMcGonagle
EditorialProjectManager: RachelPomery
ProductionProjectManager: PremKumarKaliamoorthi
CoverDesigner: VictoriaPearson
TypesetbyTNQTechnologies
1.Generalconceptsofmechanicalbehaviorandfracture1
1.1 Fracturemechanics fieldofapplication1
1.2 Definitionofstressandstrain3
1.3 Mechanicalbehaviorundertension6
1.4 Thestresstensor13
1.5 TheMohr’scircle17
1.6 Yieldcriteria21
1.7 Stressconcentration24
1.8 Definitionsandbasicconceptsoffracture27
1.9 Objectand fieldofapplicationoffracturemechanics31
2.Linearelasticfracturemechanics35
2.1 Cohesivestrength35
2.2 TheGriffithcriterion37
2.3 Thestressintensityfactor(Irwin’sanalysis)40
2.4 Solutionsofthestressintensityfactor48
2.5 Experimentaldeterminationofthestressintensityfactor56
2.6 Determinationofthestressintensityfactorbythe finiteelementmethod62
2.7 Theplasticzone66
2.8 Thecracktipopeningdisplacement72
3.Theenergycriterionandfracturetoughness75
3.1 Theenergycriterion75
3.2 TheR-curve81
3.3 Planestrainfracturetoughness86
3.4 Planestrainfracturetoughnesstesting(KIC)87
3.5 Effectofsizeonfracturetoughness94
3.6 Charpyimpactenergyfracturetoughnesscorrelations96
3.7 Dynamicfractureandcrackarrest98
4.Elastic-plasticfracturemechanics107
4.1 Elastic-plasticfractureandthe J-integral107
4.2 JIC testing113
4.3 Useofthe J-integralasafractureparameter118
4.4 Thecrack-tipopeningdisplacementasfractureparameter125
4.5 Thetwo-parametercriterion133
5.Fractureresistanceofengineeringmaterials145
5.1 Remainingstrength145
5.2 Materialsselectionforfractureresistance153
5.3 Materialpropertiescharts163
5.4 Failureanalysisusingfracturemechanics165
5.5 Reinforcementofcrackedstructures168
5.6 Theleak-before-breakcondition172
6.Fatigueandenvironmentallyassistedcrackpropagation177
6.1 FatiguecrackgrowthandParis’slaw177
6.2 EffectoftheloadratioontheFCGrate183
6.3 Fatiguecrackclosure184
6.4 Effectoftheenvironmentonfatiguecrackgrowth190
6.5 Effectofvariableloadsonfatiguecrackgrowth192
6.6 Effectofasingleoverloadonfatiguecrackgrowth195
6.7 Fatiguecracksemanatingfromnotchesandholes200
6.8 Stress-corrosioncracking204
6.9 Creepcrackgrowth208
6.10 Crackgrowthbyabsorbedhydrogen213
7.Structuralintegrity219
7.1 In-servicedamageofstructuralcomponents219
7.2 Generalaspectsofstructuralintegrity227
7.3 Remaininglifeofcrackedcomponents235
7.4 Amethodologyfortheestimationofremaininglife241
7.5 Structuralintegrityassessmentprocedure250
7.6 Exampleofastructuralintegrityassessment256
Preface
Rightafter finishingmyPhDinmetallurgyattheUniversityofConnecticutin1990,undertheguidanceofProfessorArthurJ.McEvily,a renownedpioneeronthestudyoffatiguecrackpropagationunderthe fracturemechanicsapproach,Ibecameafull-timeprofessorattheMetallurgyDepartmentoftheInstitutoPolitecnicoNacional(IPN),thesecondlargesthighereducationinstitutioninMexico,withmorethan200,000 students.Asayoungteacherandresearcher,Ihadthefullintention ofapplyingmyknowledgetothesolutionofthetechnologicalchallenges ofmycountryandperhapsmakeasignificantcontributiontothe fieldsof metallurgyandfracturemechanics.Asaresult,Iintroducedthevery first graduate-levelcourseonFractureMechanicsinMexicoin1990,and 2yearslater,IpersuadedPemex,Mexico’sstate-ownedoilcompany,to financeanacademy-industryresearchprojecttostudypipelinefractures.
After4yearsofintenseworkbothat fieldandlaboratoryIfundedthe GrupodeAnálisisdeIntegridaddeDuctos(PipelineIntegrityAssessment Group,GAID)atIPN,anorganizationcomposedbyprofessors,undergraduate,andgraduateengineeringstudentsandprofessionalsintendedto performfracturemechanicsresearch,technicalassistanceonmaintenance andfailureanalysis,andmostimportant,structuralassessmentofpipelines andhydrocarbonprocessingandstoragefacilities.Bythe2000sGAIDhad morethan500collaboratorstoprovidefracturemechanicsrelatedservices inmorethan60,000kmofpipelines,morethan144marineplatforms,six largeoilrefineries,andmorethan50hydrocarbonstorageanddistribution plants.ThemostsignificantcontributionofGAIDwastointroducethe structuralintegrityapproachintothemanagementofmaintenanceattheoil andgasindustryinMexico,culminatingwiththeissuingofthenational standardsonpipelineintegritymanagement,nondestructiveinspection,and manyothertechnicalspecificationsandrecommendedpractices,allofthem aimedtothepracticalapplicationsoffracturemechanics.
Thetopicoffracturemechanicswasararityinthe1990sMexico,since therewereonlyafewprofessionalsworkingonitbythattime.Evenafter submittingaprojectproposaltotheindustry,somehigh-levelexecutives toldmethat “fracturemechanicsisatheoreticalcuriositywithoutpractical applicationbeyondexplainingsomefailures.” Thisexperienceencouraged metowritemy firstbook,backin1998.ThebookwaswritteninSpanish
becauseIrealizedthatmanystudentsandindustryprofessionalsdidnot haveenoughproficiencyinEnglishtofullyunderstandandmoreimportantlytoapplythetheoreticalfoundationsofFractureMechanicstothe solutionofreal-lifeproblems.MySpanishbookonfracturemechanicshad asecondeditionin2004,andinbetweenIwroteabookonmechanical metallurgy,alsoinSpanish.Yearslater,encouragedbyAshokSaxenaand otherbrilliantcolleaguesIpublishedmybookson FractographyandFailure Analysis and MechanicalBehaviorandFractureofEngineeringMaterials in English.Bothbookshadagoodreceptionamongengineeringstudentsand professionals,butIstillhesitatedtopublishabookonfracturemechanicsin English,untilElsevierkindlyinvitedmetodoso.Afterconsideringthe excellent buthighlytheoreticalandfullofcomplexmathematics existingbooksonfracturemechanics,Idecidedtotakeadifferentapproach andtowriteatextintendedtomakethissubjectaccessibletostudentsthat enterforthe firsttimeintothistopicandprofessionalssearchingforquick andpracticalanswerstothefractureproblemstheyfaceintheir fieldof practice.Thatishow APracticalApproachtoFractureMechanics becamea reality.
Thisbookplacesemphasisonthepracticalapplicationsoffracture mechanics,avoidingheavymathematicaldemonstrations,withafewexceptions,andfocusinginsteadonmakingthephysicalconceptsclearand simple,providingexamplesfrommyreal-lifeexperience,butyetatalevel thatcanbeeasilyunderstoodandappliedbybothengineeringstudentsand practicingengineersthathavetheneedtolearnaboutitbuthaveneither thetimenorthebackgroundtounderstandahigh-leveltextbookor researchpaper.
Myintentionistointroducethereaderintothefracturemechanics field inalogicalandchronologicalsequence,asthefracturemechanicsconcepts weredevelopedalongthehistory;therefore,thebookbeginswithbrief introductionoftheimportanceofthestudyoffracturemechanicsand presentsthebasicdefinitionsofstressandstrainitssignificancefor mechanicaldesignandmaterialsselection.Chapters2and3introduce Griffith’sanalysisasthebackgroundforIrwin’sanalysisthatledtothe introductionoflinearelasticfracturemechanics.Theanalyticaland experimentalmethodstodeterminethestressintensityfactorandthe fracturetoughnessofengineeringmaterialsaredescribed,alongwiththe implicationsoflinearelasticfracturemechanicsandtheenergycriterionon thebehaviorofaloadedbodycontainingcracks.
Chapter4isdedicatedtoelastic-plasticfracturebyRice’sJintegraland thetwo-parametercriterionwhicharenowadaysthemostwidelyused methodtoassesscracklike flawsinstructuralintegrity.InChapter5,the conceptsoflinear-elasticandelastic-plasticfracturemechanicsareappliedto thedeterminationofremainingstrength,materialsselection,andtodescribe theuseoffracturemechanicsinfailureanalysisandthereinforcementof crackedstructures.Oncethestaticfractureproblemisunderstood,thenext developmentistheapplicationoffracturemechanicsconcepttotheunderstandingofthegradualorslowcrackpropagationphenomenainengineeringmaterialsunderserviceconditions.ThisisdiscussedinChapter6, beginningwithfatiguecrackgrowthandcoveringstress-corrosion-induced crackpropagation,creepcrackgrowth,andcrackgrowthbyabsorbed hydrogen.
Thebook finisheswiththelatestandmostimportantapplicationof fracturemechanicstoday,whichisthestructuralintegrityassessmentof crackedcomponents.Itpresentsabriefintroductiontothemaindamage mechanismsofin-servicecomponentstothenpresentthegeneralconcepts ofstructuralintegrity,includingtheestimationoftheremaininglife.Since themainattemptofthistextistoserveasapracticalguidancetopracticing engineers,acompleteexampleoftheassessmentofareal-lifecomponentis presented.
Thisbookisconsideredsuitableforengineeringstudents,designengineers,inspectionandmaintenanceengineersperformingFitness-For-Service (FFS)assessmentsinallindustries(oilandgas,powergeneration,construction,chemicalandpetrochemical,transportation,etc.)aswellasfor professionalsworkingatresearchlaboratories,engineering firms,insurance, andaslossadjusters.Thebookcouldbeusedinfracturemechanicscourses thatarebeingtaughtinmostmajoruniversitiesandhighereducation institutions,atbothundergraduateandgraduatelevels.
IamindebtedtomycolleaguesattheIPN,whomIdonotlisthereto avoidinvoluntaryomissions,andtomanyPemexengineers,speciallyto Ing.FranciscoFernandezLagos( ),Ing.CarlosMoralesGil,Ing.Javier HinojosaPuebla,andIng.MiguelTameDominguezfortheirsupportand encouragementtoallowmetofulfillmylifeprojectofapplyingscientific knowledgeintothesolutionofstrategicproblemsintheindustry.Iwishto dedicatethisbooktothememoryofProfessorArthurJ.McEvily,notonly forhisguidanceandsupportwhenIwasadoctoralstudent,butalsoforhis
x Preface
friendshipandinspirationformorethan30years.Finally,Iamgratefulto mydaughterCarolinaforhermanysuggestionsfortheimprovementofthis text,andtomybrotherJuanManuelforhishelpinthepreparationofthe manuscriptinEnglishlanguage.
April2020
JorgeLuisGonzález-Velázquez
CHAPTER1
Generalconceptsofmechanical behaviorandfracture
1.1Fracturemechanics fieldofapplication
Fractureisaphenomenonthathasreceivedconstantattention,practically sincemachinesandstructuresfounduseinbothwartimeandpeacetime. Particularly,theuseofmechanicalandstructuralcomponentssuchas beams,columns,shafts,pressurevessels,cables,gears,andsoforthhave alwayscomealongwiththeriskoffracture.Frequently,thefractureofa structuralcomponentisaccompaniedwithgreatmaterial,economicand humanlosses.Itisalsocommonthat,althoughfailuresmayoccuronceina lifetime,asinglefailurecanmeanagreatcatastrophe,whichisthecasefor airplanecrashes,explosionsofgaspipelines,ornuclearreactorfailures.The lossesarenotonlylimitedtohumanandeconomicones,butalsothereare additionallosses,suchasdelaysinproduction,environmentaldamage,and thedetrimentofthecompany’spublicperceptionandimage.Premature fractureofsmallcomponents,suchasscrewsandbolts,isalsoaninsidious problem,sinceconsumersassociateitwithpoorquality,whichresultsin salesreduction.Insummary,itwouldbeimpossibletoquantifythe magnitudeoflossescausedbyfracture-relatedfailures,butwhatiscertainis thatfracturemaybethelimitingfactorforthesuccessofindustriesand entireeconomies.
Fracturemechanicsisthedisciplinethatprovidesthebasisandmethodologyforthedesignandassessmentofcrackedcomponentsinorderto determinetheeffectofthepresenceofacrack.Itisalsoappliedtodevelop structuresandmaterialsmoreresistanttofracture.Throughtime,ithasbeen demonstratedthatthetraditionalcriteriaofstructuredesignunderthe assumptionofabsenceof flaws,andfurthercompensatingitseffectby meansofsafetyfactorsareriskyandoftenlackanytechnicalfoundation. Thefactisthat flaws,especiallycracks,inevitablyappearinbothmechanical andstructuralcomponentsduetopoormanufacturing,inadequate
APracticalApproachtoFractureMechanics
ISBN978-0-12-823020-6
https://doi.org/10.1016/B978-0-12-823020-6.00001-3
construction,orintroducedduringservice,sotheengineershavetodeal withthem,andthebestwayisbyanalyzingtheireffectonthemechanical behavior.
Theproblemoffracturehaskeptscientistsandengineersbusysince theerasofthegreatLeonardoDaVinciandGalileo.However,itwas untilthebeginningofthetwentiethcenturythatGrif fi thwasableto calculatethefracturestrengthofbrittlematerials,butevenso,theoretical andexperimentaldif fi cultieshinderedthedevelopmentoffracture mechanicsuntil1956,whenGeorgeR.Irwinintroducedtheconceptof stressintensityfactorandfracturetoughness,givingbirthtomodern fracturemechanics.Nowadays,thestudyoffracturemechanicsisa fundamentalpartofmechanical,materialsandmetallurgicalengineering. Asigni fi cantfactisthatover40%ofthearticlespublishedinengineering andmaterialssciencejournalsarerelated,directlyorindirectlytomechanicalbehaviorandfracture.Withintheindustry,fracturemechanics isextensivelyusedintheaeronautic,aerospace,chemicalprocessing,oil re fi ning,andnuclearindustries,andithasbeguntobeusedmore frequentlyintheautomobileindustry,pipelinehydrocarbontransport, andconstructionindustries.
Althoughtheeconomicalandsafeoperationofengineeringcomponentsandindustrialfacilitiesrequiresadesignresistanttocrackingand fracture,fracturemechanicsisofgreatusefulnessincomponentsthathave alreadybeeninservice.Ithelpstosetthecriteriafortheacceptanceor rejectionof flaws,establishingthefrequencyofinspectionsandsafeoperationallimitsofprocessequipmentandmachinery;thesestudiesareknown as StructuralIntegrity or Fitness-For-Service.Fracturemechanicsisalsousefulin failedcomponents,sinceitprovidestheanalyticaltoolstodetermineifthe causeoffailurewasanoverloadorthecomponenthaddefectsoutofthe acceptancelimits.
Theapplicationoffracturemechanicsinallstagesofthelifecycleof structuralandmechanicalcomponentsyieldsgreatwinsinsafetyand economysinceitreducesthefrequencyoffailuresandextendsthelifespan. Allofwhichallowengineerstopaymoreattentiontootherfundamental issuessuchasthedevelopmentofnewmaterialsandtheimprovementof designswhichresultinfurthertechnologicaladvance.
The fieldoffracturemechanicsisdividedintotwobroad fields,as illustratedin Fig.1.1.Atthemicroscopicscale,fracturemechanicsispartof thematerialsscience field.Theaimistostudytherelationshipbetween microstructureandfracturemechanisms,includingplasticityand
fractographicexamination.Atthemacroscopicscale,fracturemechanicsisa branchofmaterialsandmechanicalengineering,focusedonapplications, suchaslaboratorytestingtodeterminethefractureproperties,andsolving practicalproblems,suchasdefectassessment,crackarrest,materialsselection,andfailureanalysis,amongothers.
1.2Definitionofstressandstrain
TheFrenchmathematicianandscientistAugustinLouisCauch^ yintroduced theconceptofstressin1833.HeusedthemovementlawsbyEuler,and Newton’smechanics,todeterminethedisplacementsproducedonastatic solidbodysubjecttosurfaceloads.BasedonNewton’ssecondlaw,which states, “toeveryaction,thereisacorrespondingreaction,” Cauch^ y figured outthatwhenanexternalforceisappliedonastaticbody,aninternal reactionforcebalancingtheexternalforceisinstantaneouslyproduced.The magnitudeofsuchreactionisdirectlyproportionaltothemagnitudeofthe appliedforceandinverselyproportionaltothesizeofthecross-sectionarea, thisreactionbeing stress
UnderCauch^ y ’sprinciple,themechanicalbehaviorofsolidmaterials maybesummarizedasfollows:Loadsproducestresses,thestressescause strain,andstrainleadstofracture;thereforetheaimistodeterminethe stressesandstrainsproducedinaloadedsolidbodyanddeterminethe material’sstrengthtowithstandsuchstresseswithoutneitherexcessively strainingnorfracturing.
Tofacilitatetheanalysisofthemechanicalbehaviorofsolids,itis necessarytosimplifythesystem,becausematerialsarecomplexarraysof atoms,crystallinedefects,secondphases,andmicrostructuralheterogeneities.
Figure1.1 Fieldofstudyoffracturemechanics,accordingtothescalesize.
Thebasicassumptionsfortheinitialapproachofmechanicalbehavior includethefollowing:
• Thematerialisacontinuum:Thismeansthatmatter fillsinthetotal volumeandtherearenovoidsnorinterruptions.Underthisassumption,itcanbeestablishedthattherewillbeaninfinitesimalvolume(a volumethattendstozero,butitneveriszero),wheretheforcesand areasbedefined,sothestressexistsinapoint.Thisassumptionisalso thereasonwhythestudyofmechanicsofmaterialsiscalled continuum mechanics.
• Thematerialishomogeneous:Thewholevolumeis filledinwiththe sametypeofmatter.
• Thematerialisisotropic:Thepropertiesarethesameinanydirection. Basedontheseassumptions,astaticsolidundertheactionofanapplied externalforce FA remainsstatic,ifandonlyifthisforceisbalancedbyan internalforce Fi, ofthesamemagnitudeandoppositedirectionto FA.The forcecausesaninternalreactioninthesolidthatisdirectlyproportionalto themagnitudeoftheappliedforceandthenumberofparticlesresistingthe actionofsuchforce,beingproportionaltothecross-sectionarea A.The magnitudeoftheinternalreactionisthestress,representedbythesymbol s, andcanbedefinedas:
s ¼ F=A
Theinternalforceisavector,sinceithasmagnitudeanddirection, thereforeitcanbesplitintotwocomponents:Oneperpendiculartothe cross-sectionarea(Fn)andtheotherparallel(ortangential)tothecrosssectionarea(Ft),asshownin Fig.1.2.
Thestressproducedby Fn iscalled normalstress andisexpressedby
n ¼ Fn
External applied force, FA Internal area, A
Figure1.2 Internalreactionforcetoanappliedforcethatgivesorigintothestress conceptanditsdecompositionintonormal(Fn)andtangential(Ft)components.
Normalstressesaredividedintotwotypes;whentheinternalforces tendtoelongatethebody,theyarecalled tension stressesanditssignis positive(þ)andwhentheyshortenthebody,arecalled compression stresses andareofnegativesign( ).Ontheotherhand,thetangentialforceswill produce shear stress,representedbythesymbol s,beingcalculatedas
Thephysicaleffectsofthenormalandshearstressesonthebodyare quitedifferentandtherefore,theyhavetobetreatedseparately.Thetypical stressunitsaregivenin Table1.1
Cauch^ yalsoincludedtheconceptofstraininhisanalysisofthemechanicalbehaviorofsolidmaterials.Insimpleterms,strainisthechangeof shapeinabodyduetotheactionofstresses,anditisdividedintotwotypes:
Elongationstrain,identifiedbythesymbol ε,anddefinedasthechange oflength(lf l0)overtheinitiallength(l0)
Shearstrain,identifiedbythesymbol g,anddefinedasthechangeof straightangle q ofacubicvolumeelement,soitiscalculatedas
Where i isthedirectionofdisplacementand j istheoriginaldirectionofthe displacedside. Fig.1.3 schematicallyillustratesthetwotypesofstrain.
Thestressandstrainarerelatedbytheso-calledconstitutiveequations, whichappearineverybookofmechanicalbehaviororstrengthofmaterials.Thereaderisencouragedtobecomeacquaintedwiththem,sincethey arefundamentalfortheanalysisofthemechanicalbehaviorofengineering materials.
Table1.1 Typicalstressunits.
SystemUnitsCommonmultiple
InternationalPascal(Pa ¼ Nw/m2)MPa ¼ 106 Pa Englishpsi(psi ¼ lbf/plg2)ksi ¼ 1000psi Mkskg/mm2 kg/cm2 ¼ 100kg/mm2
Conversionfactors: 1ksi ¼ 6.895MPa 1kg/cm2 ¼ 14.23psi 1MPa ¼ 10.5kg/cm2
1.3Mechanicalbehaviorundertension
Thesimplestwaytoobservethemechanicalbehaviorofasolidisby applyingatensionforceonabodyofregularcross-sectionandrecordthe loadandelongationproducedinthetestspecimen.Thetypical Load versus Elongation recordofanengineeringmaterialisshownin Fig.1.4.Byusing thedefinitionofstressas s ¼ F/A0,where A0 istheinitialcross-sectionarea ofthetestspecimen,anddefiningtheelongationstrainas ε ¼ Dl/l0,where Dl istheelongationand l0 istheinitiallength,the stress-strain curvein tensioncanbereconstructedfromtheoriginalLoad-Elongationcurve. Since A0 and l0 areconstants,theshapeofthecurvedoesnotchange,only thescaleismodified,thereforethestress-straincurveinuniaxialtensionhas thesameshapeoftheLoad-Elongationcurve.
Asitcanbeobservedonthestress-straincurve,attheonsetofloading, theelongationisproportionaltothestress,butiftheloadisremoved,the bodyrecoversitsinitialshapeandlength;thisbehavioristermedas elastic.In mostmaterials,thestrainislinearlyproportionaltothestrainandthe constantofproportionalityiscalledthe Young’smodulus, representedbythe
Figure1.3 Schematicillustrationofengineeringstrain.
Figure1.4 TypicalLoad-ElongationorStress-Straincurveinuniaxialtensionofan engineeringmaterial.
symbol E.Whenthestresssurpassesalimitvaluethedeformationbecomes permanent,aconditiontermedas plasticstrain,thestressatthispointis termedasthe yieldstrength,anditusuallyrepresentedbythesymbol s0.On furtherloading,theloadhastobeincreasedtosustaintheplasticdeformation,thisbehavioriscalled strainhardening anditmakesthecurvetotake aparabolic-likeshape,whichmaximumisreferredas ultimatetensilestrength, representedbythesymbol suts.Justaftertheultimatetensilestrengthis reached,thematerialexperiencesalocalcontractionknownas necking,so thecross-sectionareaisrapidlyreducedandtheloaddrops,leadingtothe finalrupture.Themaximumelongationintensionloadingisreferredas ductility,identifiedbythesymbol εf.Theseessentialfeaturesofthestressstraincurveinuniaxialtensionallowdeterminingthefundamentalmechanicalpropertiesofengineeringmaterials,whichare:
• Young’smodulus(E):Proportionalityconstantbetweenstrainandstress intheelasticregime.
• Yieldstrength(s0):Tensionstressattheonsetofplasticdeformation.
• Ultimatetensilestrength(suts):Maximumtensionstressthatamaterial canwithstandbeforefailure.
• Ruptureelongationorductility(εf):Maximumplasticelongation measuredafterruptureintension.
Atthispoint,itisimportanttodiscussaboutthehardness,whichis generallyregardedasamechanicalproperty.Hardnessistheresistanceofa materialtobescratchedorindented,anditsvalue,regardlessthetesting method,involvesseveralmechanicalprocessessuchaselasticstrain,plastic strain,andstrainhardening.Additionally,itisaffectedbyotherfactorssuch asfriction,thicknessandsurface finish;therefore,hardnessisnotafundamentalmechanicalproperty.However,sinceitstestingissimple,fastand economical,ithasbecomeverypopularasaparametertocharacterizethe mechanicalbehaviorofengineeringmaterials.Thebasicruleisthatthe harderamaterialis,thehigherloadbearingcapability,butthedisadvantage isthathardmaterialsareusuallybrittle.Hardnessismeasuredinseveral scales,beingthemostcommoninengineeringmaterials:Brinell,Vickers, Rockwell,andShore(forplastics).
Theuniaxialtensiontestnotonlyprovidesthefundamentalmechanical propertiesofmaterials,butalsoprovidesinformationontheoverallperformanceofmaterialsunderstresses.Thisinformationisquiteusefulintheanalysis ofthemechanicalandfracturebehaviorofmaterials,allowingcomparisons formaterialsselection,materialsdevelopment,designandqualitycontrol, amongothers.Typicalmaterialbehaviorcategoriesaccordingtothestressstraincurveinuniaxialtensionisschematicallydepictedin Fig.1.5
Thematerialcategoriesaccordingtothiscriterionare:
Hardandbrittle: Highyieldstrength,lowstrainhardeningandpoor ductility.TheYoungmodulusmaybeveryhigh.Thesematerialshavehigh hardnessandstiffness,buttheyarebrittle,sotheydonotresistimpacts, strainandthermalshock.Theyareusedincuttingtoolsandpartsrequiring highabrasivewearanderosionresistance.
Softandductile: Lowyieldandtensilestrength,lowstrainhardeningand highductility,sotheyareductileandeasytoshapebymechanicalprocessing.TheYoungmodulusmaybelow,butnotnecessarily.Theyare usedtofabricatecomponentsofintricateforms,butlimitedlowload bearingapplications.
Highstrength: Highyieldandtensilestrength,highstrainhardeningand moderatetohighductility.TheYoungmodulusisusuallyhigh.These materialsresistheavyloads,strongimpactsandrequirehighenergyinputto causefracture,sotheyarewidelyusedforhighloadandimpactconditions, suchasbuildings,bridges,machinery,processequipment,allkindsof transportvehicles.
Weak: Theyfeaturelowmechanicalstrength,sotheyareusedinlow loadbearing,wear,andimpactconditionsandlowdurabilityitems,suchas disposablecupsandpackingstuffing.
Thetypicaltensilepropertiesofcommonengineeringmaterialsare shownin Table1.2.Itisobservedthatthestrongestmaterialsarethe metallicmaterials,whichmakesthemanexcellentchoiceforstructuraland mechanicalapplications.Polymershavethelowestmechanicalstrength levels,althoughtheirlowdensity,chemicalstabilityandeasyofmanufacturemakesthemidealforlowloadrequirements.Ceramicsandglassare highlyresistanttocompressionandexhibithighhardness,buttheyare
Figure1.5 TypicalmaterialbehaviorcategoriesaccordingtotheStress-Straincurvein uniaxialtension.
Table1.2 Mechanicalpropertiesofcommonengineeringmaterials.
Carbonsteel200 212250 1200340 1600120 80100 1100
Lowalloysteel200 212400 1100460 120014 100120 350
Stainlesssteel200 212170 1000480 224062 180140 650
Castiron200 212215 790350 100010 35100 290
Aluminumalloys70.330 50060 55022 3518 160
Cooperalloys13030 500100 55030 9029 160
Nickelalloys20070 1100340 120080 110100 350
Titaniumalloys90 110250 1250300 160014 12086 460
Zincalloys70e8080e450130e52010e10038e150
Elastomers0.01 0.62 9025 500.2 0.47 15
Polymers8 10018 7020 701 66 20 Foams0.02e
Brick15 2050-140a 7 141 2N/A Concrete15 3032 60a 2 60.35 0.45N/A Rocks7 2034-298a 5 170.7 1.5N/A Wood5 2030 7060 1005 9N/A Leather0.1 0.55 1020 263 5N/A
brittle,soitsapplicationislimitedtocompressiveloadingandnonimpact uses.
Accordingtothecontinuummechanicsprinciplesandtakinginto considerationthestress-strainbehaviorinuniaxialtension,themechanical behavioranalysisofloadedbodiesistypicallycarriedoutbythefollowing procedure:
(1) Definitionofthecomponentgeometry.
(2) Definitionofloads,magnitude,direction,andpointofapplication.
(3) Calculationofstresses.
(4) Calculationofstrainsanddisplacementsresultingofthestresses.
Thedesignbasedoncontinuummechanicsaimstodeterminetheloads and/orthesizeofthecomponentandtoselectamaterialofspecificmechanicalpropertiescapableofwithstandandtransmittheappliedloadsand/ ordisplacements.Todothis,thecross-sectionareaandtheappliedloads shouldbesuchthatthestressesarelowerthanthematerial’sstrength.This processcanbeexplainedbythemathematicaldefinitionofstress:
s ¼ F A
Where s isthestress, F istheappliedloadorexternalforceand A isthesize ofthecross-sectionarea.Thegeneraldesigncriterionisusually:
If s Limitstress,thecomponentfails.
Thelimitstressisusuallytheyieldstrengthmultipliedbyasafetyfactor (SF).Basedontheearlier-mentionedequation,thestress-basedmechanical designiscarriedoutundertheprocedureshownin Fig.1.6.First,theshape andsizeofthecomponentaredefined,todeterminethecross-sectionarea size(inmanydesigncodesthecross-sectionareasizeisdeterminedbythe thickness),then,theloadsareassignedandamaterialwithenoughmechanicalstrengthtowithstandthecalculatedstressesisselected.
Inengineeringpractice,thesafetyfactorisanadditionalmaterial strength,additionalthicknessand/oralimitload,allofthemestablishedin ordertocompensatethepresenceofadditionalorhigherstresses,eitherby unexpectedaccidentalloadsorbyabuse.Thesafetyfactorisalsointroduced tocompensateforlowerstrengthcausedbythepresenceof flawsand defectsinthematerial.Inmostmechanicaldesigncodes,themaximum allowablestressisthecalculatedstressmultipliedbythesafetyfactor, definingthe designstress,insuchcaseFSismorethanone,butifthesafety factorisappliedtothematerial’sstrength,thenFSislessthanone.
Themoreuncertaintyofin-serviceloads,thepoorerqualityofmaterial, andthemoreseverefailureconsequences,thehighersafetyfactor.
Regardlesstowhatvariablethesafetyfactorisapplied,itshouldgivea reasonablylargegapbetweenthematerial’sstrengthandthereaction stressesadschematicallyillustratedin Fig.1.7.Asitmaybeobserved FS givesthedesignsafetymargin,whichisthedifferencebetweentheminimumspecifiedmaterial’sstrengthandthelimitdesignload.Becausemost vendorssupplymaterialswithstrengthsomewhatgreaterthantheminimumspecified,thedesignsafetymarginbecomesaminimumexpected value;however,thisshouldnotbeconsideredasawarrantyofasaferdesign andtheusersshouldnotrelayonthefreeexcessstrength.Furthermore, moststructuralormechanicalcomponentsarenotregularlyoperatedat theirdesignlimit,andinmanycases,theoperationalloadsarefarbelowthe designlimit,suchashappenswithautomobiles,handtools,cranes,and manyothers,andadditionaloperationalsafetymarginisintroduced, allowingtheimplementationofdamagetolerancestrategies,asitwillbe furtherdiscussedinthisbook.
Formostengineeringmaterials,theelasticpartofthestress-straincurve linear,thereforeYoung’smodulusisdefinedas:
Figure1.6 Flowchartofstructuralcomponentsdesignbycontinuummechanics.
Figure1.7 Stresslevelsinthedesignofstructuralcomponentsresultingoftheuseof thesafetyfactor.
Bearinginmind,that ε ¼
0 and s ¼ F/A0,itcanbestatedthat
Theterm(AE/l0)isknownastheelasticcoefficient,highvaluesofthe elasticcoefficientresultinrigidstructures,thatislesslikelytoelastically deformunderload;whereaslowvaluesoftheelasticcoefficientindicate thatthestructureis flexible,anditwilleasilydeformunderload.Theelastic coefficientiswidelyappliedinthedesignofmechanicalandstructural components,sinceitallowstosetthephysicaldimensionsandselectthe appropriatematerialtocontroltheelasticstrain.
The firstcaseofelasticdesignare flexibleorelasticcomponents:those arethatshouldfeaturefairlylargeelasticstrainsor flexions,butnotasmuch astoreachyield,asitisthecaseofhelicalandleafsprings.Inotherwords, flexiblecomponentsaredesignedtohavelargecontrolledelasticstrains undertheappliedloads.Accordingtotheelasticcoefficient,thiscanbe achievedbylonglengths,smallcross-sectionareasandmaterialswithalow Young’smodulus.The firsttwoconditionsmake flexiblecomponentslong andslender.
Theothertypeofelasticdesignisrigidcomponents,whicharethosewhose excessiveelasticstrainisadversefortheirperformance,suchasbuilding
structures,supports,gears,andmachineparts.Inthesecomponents,the magnitudeoftheelasticstrainmustbelimitedtoaminimum,sotheelastic coefficientmustbehigh.Thiscanbeattainedbywideningthecross-section, shorteningthelengthandbyselectinghighYoungmodulusmaterials.The first twocharacteristicsmakerigidcomponentsshortandthick.
Thefollowingexampleillustratestheuseoftheelasticcoefficient. Ahollowrectanglebarof5cmexternalheightand4cmexternalwidth,1mm thickness,and50cmlong,tobeusedaspartofaframeisrequiredtosupportaload F ¼ 5000N.Themaximumallowabletensilestrainis Dl/l0 ¼ 2.0 10 4.The originaldesignusessteel,butanovelengineersuggestssubstitutingsteelbyaluminum tosaveweightandcost.Isthisagoodidea?Assumethatthecostofaluminumisthree timesthecostofsteel.ESteel ¼ 200 109 N/m2;EAluminum ¼ 70.3 109 N/m2
Solution:
Accordingtotheforce-elongationelasticformula:(Dl/l0) ¼ F/(AE)
A ¼ 1.96cm2 ¼ 0.000,196m2.
Substitutingvaluesforsteel: (Dl/l0)Steel ¼ (5000N)/(0.000196m2 200 109 N/m2) ¼
Substitutingvaluesforaluminum:
Noticethatthealuminumbardoesnotmeetthespecificationforallowablestrain. Iftheframehastobemadeofaluminum,inordertomeettheallowablestrain specification,itsthicknessneedtobechangedas
A ¼ F/[(
Iftheexternaldimensionshavetobethesame,thethicknessofthealuminum frameshallbe0.184cm ¼ 1.84mm,buttheimportantfactoristheweight.
Thealuminumframeweights1.6timeslessthanthesteelframe,butitwillcost 1.872timesmorethansteel,sointermsofweightsaving,aluminumisabetter choice,butintermsofcoststeelisbetter.
1.4Thestresstensor
AccordingtoCauch^ y ’sstresstheory,thestateofstressatonepointis describedbythestresstensorwhichisdeterminedbythestresscomponents
Stresscomponentsinaunitaryvolumeelementofasolidbody.
actingonthefacesofanelementofadifferentialvolume(acube)locatedat theoriginofaCartesiancoordinatesystemof x,y, and z axes,asshownin Fig.1.8.
Thisresultsinninestresscomponents,threenormalandsixshear.An indexnotationidentifieseachstresscomponent:
sij
Where i isthecube’sfacewherethestressisacting,and j isthedirection ofthestress.
Theninestresscomponentswritteninmatrixform,becomethe stress tensor,sincetheyarevectors,andithastheform:
Duetobalanceofmomentum,itcanbeeasilydemonstratedthatthe shearcomponentsaresymmetric,thatmeansthat sij ¼ sji,sothatreduces thestresstensortosixindependentcomponents,threenormalandthreeof shear.Theotherthanzerocomponentsofthestresstensordefinethe stateof stresses andrepresentthereactionsthatoccurinsideofastaticsolidbody subjecttoexternalloads.Themostcommonstatesofstressarepresentedin Table1.3.
Themethodologytodeterminethestateofstresses(qualitatively)ofa solidofregularshapeunderafewknownloadsisthefollowing: 1. Drawafreebodydiagramofthecomponent,indicatingtheapplied loadsintheformofvectors(arrowswithmagnitude,direction,and applicationpoint)aswellasthepointsofsupportand/orrestrictions todisplacement.
Figure1.8
Table1.3 Commonstatesofstress.
NameStresstensorExample
Deadweighthangingofabar
Thinwallcylinderunder internalpressure
Transmissionshaft
Wiredrawing
Thinsheetandfreesurfaces
Generalcase
Note:Thecommonpracticeistoeliminaterowsandcolumnswhichvaluesarezero.
2. Placeanorthogonalcoordinatesystem x, y, z,withitscenteronthe body’scenterofgravityorthegeometricalcentroid,preferablywith oneoftheaxesalignedwiththebody’ssymmetryaxisorparallelto thedirectionofthemainappliedload.
3. Drawacubeattheoriginofthecoordinatesystem,withitssidesaligned paralleltothecoordinatesystemaxes.
4. Identifytheinternalreactionloadsoneachofthecube’sfaces,asvectors ofthesamemagnitudeandinoppositedirectiontotheexternalforces.
5. Identifynormalandshearstresscomponentsgeneratedonthecubefaces byapplyingtheindexnotation,bearinginmindthatthe firstsubindexis thefacewheretheforceisactingon,andthesecondsubindexisthe directionofthereactionforce.Usetheruleofsignstodeterminethe
