常用应力强度因子计算方法比较.docx

常用应力强度因子计算方法比较.docx

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常用应力强度因子计算方法比较

27thICAFSymposium–Jerusalem,5–7June2013

ThePursuitofK:

ReflectionsontheCurrentStateoftheArtin

StressIntensityFactorSolutionsfor

PracticalAerospaceApplications

R.CraigMcClung,1Yi-DerLee,1JosephW.Cardinal,1andYajunGuo21SouthwestResearchInstitute,SanAntonio,Texas,USA

2JacobsESCG,Houston,Texas,USA

Abstract:

Thestressintensityfactor（Kisthefoundationof

fracturemechanicsanalysisforaircraftstructures.Thispaper

providesseveralreflectionsonthecurrentstateoftheartinK

solutionmethodsusedforpracticalaerospaceapplications,

includingabriefhistoricalperspective,descriptionsofsome

recentandongoingadvances,andcommentsonsomeremaining

challenges.Examplesareselectivelydrawnfromtherecent

literature,fromrecentenhancementsintheNASGROand

DARWINsoftware,andfromnewresearch,emphasizing

integratedapproachesthatcombinedifferentmethodstocreate

engineeringtoolsforreal-worldanalysis.Verificationand

validationchallengesarehighlighted.

INTRODUCTION

Thestressintensityfactor（commonlydenotedKisthefoundationoffracturemechanics（FManalysisforaircraftstructures.Thisparameterdescribesthefirst-ordereffectsofstressmagnitudeanddistributionaswellasthegeometryofbothstructure/componentandcrack.Hence,thecalculationofKisoftenthemostsignificantstepinfatiguecrackgrowth（FCGlifeanalysis.ThispaperprovidesseveralreflectionsonthecurrentstateoftheartinKsolutionmethodsusedforpracticalaerospaceapplications,includingabriefhistoricalperspective,descriptionsofsomerecentandongoingadvances,andcommentsonsomeremainingchallenges.Noattemptismadetobeexhaustiveinthisreview—thatwouldbeadauntingtask—butkeycitationsarewovenintothepracticalexperiencesoftheauthors.

HISTORICALSURVEY

Handbooks

TheearlycompilationsofKsolutionsinhandbooksbyTada,Paris,andIrwin[1]andRookeandCartwright[2]wereinvaluablecontributions.ThosecompilerscollectedmanydifferentpublishedKsolutionsavailableatthetimewhilealso

R.C.McClung,Y.-D.Lee,J.W.Cardinal,andY.Guo2

contributingnewsolutionsfromtheirownresearch.However,thesehandbookshadsomepracticallimitations.Firstofall,manyofthesolutionswerepresentedingraphicalformbasedoncomplexnumericalcomputations（withoutanycorrespondingclosed-formequations,andsotheycouldnotbeimmediatelyincorporatedintoengineeringsoftwareforproductionuse.Second,manyofthesolutionswereforconfigurationswithoutimmediatepracticalvalue:

infinitebodies,pointloads,out-of-planeloadingmodes,etc.Whilemanyofthesesolutionsunquestionablyprovidedessentialfoundationsforlaterwork,theywereoftennotaccessibletotheeverydaypractitionerwhoneededtoanalyzerealcracksinrealstructures,anddosoquickly.

Closed-FormEquationsDerivedFromFiniteElementResults

NewmanandRaju（NRmadesignificantearlycontributionstopracticalstructuralanalysisbydevelopingclosed-formKequationsforsurfaceandcornercracksinsimplifiedfinitegeometries,oftenbasedonempiricalfitsoffiniteelement（FEsolutions.Forexample,alandmarkRaju-Newman（RNpaper[3]summarizedtheFEcalculationofKvaluesforsemi-ellipticalsurfacecracksinfinite-thicknessrectangularplatesunderuniformtension.Stretchingthestateoftheartofthattime,theyemployedFEmodelsusingupto6900degreesoffreedom（!

.Notingthatpreviouslypublishedsolutionsforthesamegeometryexhibitedlargevariations,theycarefullyverifiedtheirmodelingtechniquesandcomparedtheirresultsagainstotherreliablesourceswhereavailable.Theirkeyresultwasasummarytableofcorrectionfactorsforasimplematrixofnormalizedcrackshapesandcrackdepthsatvariousangularpositionsaroundthecrackfront.Newman-Raju[4]thenusedthesediscretedatatoderiveanempiricalequationthatcouldbeusedtocalculateKquicklyandaccuratelyforanycrackshapeandnormalizedsize（relativetoplatedimensionswithinthescopeoftheoriginalFEmatrix,whichnowincludedbothtensionandbending.Thissimpleequationhasremainedinwidespreadusetothepresentday.

Alaterpaper[5]summarizedadditionalworkbyRNtodevelopKequationsforellipticalembedded,surface,andcornercracksinplates,andsurfaceandcornercracksatholesunderuniformremotetension,againbuildingonpreviouslypublishedtabulationsofFEresults.Insomecases,theKequationsincludedadditionalcorrectionfactorsforfinitegeometryeffects（finitewidth,onecrackvs.twosymmetriccracksbasedontheoreticalconsiderationsorotherpublishedwork.NRlaterpublishedslightlymodifiedversionsoftheseequations[6].

TheNRsolutionsbegantobemorewidelydisseminatedandusedaftertheywereincorporatedintoearlyversionsoftheNASA/FLAGROcomputercode,whichwasoriginallydevelopedtosupportfracturecontrolforthespaceshuttleorbiterandotherspacestructures[7].TheFLAGROteamalsobegantoexpandandamendtheoriginalNRsolutionstoaccommodateotherloadingmodesandgeometricvariations,aswellastoaddresssomeperceivedaccuracyissues.Otherresearchersandcomputercodeshavesubsequentlycontributedtheirownderivativesofthese

ThePursuitofK3solutions.Itisaremarkabletestimonialtothesesolutionsthattheyarestillinwidespreaduseoverthirtyyearslater,eventhoughtheoriginalFEmodelswereverycoarsebycurrentstandards.

RecentFiniteElementMethods

Computationalpowerhasincreaseddramaticallyoverthelastthirtyyears,ofcourse,andsotheprospectofusingthispowertogenerateimprovedKsolutionshasgrownmoreandmoreattractive.TheidealsituationwouldbetobeabletogeneratetheexactKsolutionforeachproblemofinterestusingafaithfulFEmodeloftheactualconfigurationofinterest（andtoupdatethecrackmodelasthecrackgrows.Itiscertainlypossibletodothistoday,andinfactseveralcommercialcomputercodesofferthecapability.Thiscanbeanattractiveoptionforsolvingveryspecificproblems（suchasacriticalfieldcrackingissue,buttheresourcerequirements（includingthecomputationtimeitselfstillrenderthisapproachimpracticalasageneraldesigntoolforcomplexstructureswithmanyfracture-criticallocations.

However,theincreasedcomputationalpowerisbeingusedtoupdatetheolderengineeringapproaches（e.g.,NR.Attheleast,theoriginalmatricesofFEsolutionscanberevisitedwithfinermeshes,orexpandedtowidergeometrylimits.Furthermore,thenewpowercanbecoupledwithautomatedmeshconstructionandsolutionmethodstogeneratemuchlargenumbersofsolutionsforamuchwiderrangeofparameters.TherecentleadersinthisareahavebeenFawazandAndersson（FA,whohaveusedthep-versionFEmethod.Theybeganbyrevisitingthebasiccorner-crack-at-holegeometry,movingontodevelopalargedatabaseofsolutionsfortwounequalcornercracksatholes[8].“Large”issomethingofanunderstatementinthiscase:

theirsingle（orsymmetriccorner-crack-at-holedatabasecontained7150combinationsofthenon-dimensionalratiosR/t,a/t,anda/c.Thedifferentcombinationsofdiametrically-opposedunequalcornercracks（undertension,bending,orbearingloadpushedthetotalnumberofsolutionsoverfivemillion（nottomentionthateachKsolutionwasobtainedatalargenumberofpositionsaroundtheperimeterofthecrack.

Thiswealthofinformationcouldbeusedinseveraldifferentways.Initially,FAusedittoevaluatethelegacyRNsolutionsandNRequations.Theyfoundthattheoldersolutionswereremarkablygoodinmanyplacesandnotsogoodinafewothers.Thispromptedsomeeffortstodevelopempiricalcorrections（basedonthenewFEdatabasetotheoriginalempiricalfitstotheoriginalFEdatabase.Unfortunately,theproductoftwonecessarilyinaccurateempiricalfitsisitselfnecessarilystillinaccurate.

However,theavailabilityofsuchadensematrixofreliablenewFEsolutionssuggestedthatitmightbepreferabletousethismatrixdirectlyasamastertablefromwhichlocalinterpolationcouldbeperformed,henceeliminatinganyinaccuraciesofempiricalmulti-variablefits.Unfortunately,thesizeofthematrices

R.C.McClung,Y.-D.Lee,J.W.Cardinal,andY.Guo4

canresultinverysubstantialpenaltiesforcomputermemory（gigabytesofstorageforonlyonefamilyofcrackgeometriesand,toalesserextent,forprocessingtimeaswell.FAhavecontinuedtogenerateevenmoresolutions（literallymillionsandmillionsforothergeometryfamiliesinworkthatislargelyunpublishedatthiswriting.Whilethepromiseofsuchbountyisexciting,itisnotyetclearhowtheinformationcouldbestbeputtopracticaluse,giventhememoryburden.

This“FEdatabase”approachtoKsolutionsalsoretainstwootherdisadvantages.Firstofall,eventhedensematricesofautomatically-generatedFEsolutionscannotcaptureallofthefinitegeometryeffects,suchastheinfluenceofnarrowplatewidthorholeoffset/shortedgedistance（neitherofwhichwasaddressedinthefivemillionunequalcornercracksolutions.Therefore,additionalcorrectionfactors（ofperhapsquestionableaccuracyorgeneralityarestillrequiredforpracticaluse.Second,thehugenumbersofresultsmakethetaskofverificationdifficult,ifnotimpossible.Howdowereallyknowthatallofthosesolutionsareactuallycorrect?

Thenumericalmethoditselfmayclaimthatnumericalconvergenceisaguaranteeofsuccess,butacarefulinspectionoftheoriginalFawaz-Anderssondatabasebythecurrentauthorsfoundsomezerosorevennegative（!

valueswheretheyshouldnotoccur.Further