1、mO=gIFmEmNATIONALADVISORYCOMMITTEEFORAERONAUTICS -e=90 fortwistingonly)shearstressatpointofhigheststressoctahedralshearstressjoiningpointofspecimen;seeforbendinginspecimensfatiguefatigueappliedbendinglimitstrengthinbendingstrengthintorsionbendingmomentmomentwhenmaximumstrainisatproportionalslopeofdi
2、mensionlessstress-straincurveaboveproportionallimitformattheresultingbendingortwistingstressescausedbyanisotropywillbelinearflmctionsofthenominalbendingortwistingstresses,respectively;andtheresultingbendingandtwistingstresseswillbethesamefunctionsofthenominalstressesregardlessofthecombinationofbendi
3、ngandtwistingemployed.Undertheseconditionsonecanthenreexaminealloftherationaltheoriesofinelasticaction(whichrequirespecificvaluesoftheratiob/t)bymultiplyingthenominaltwistingstressT byacorrec-tionconstantforeachmaterialsuchthattheresultingequationsatisfiestheconditionsforpurebendingandpuretwisting.A
4、 correc-tionconstantcouldhavebeenappliedtothebendingstressinsteadoftothetwistingstressortobothstresses;theresultwouldhavebeenthesame.Forexample,thetheoryofalimitingprincipalshearingstressmaybewrittenforcombinedbendingandtorsionasfollows:Thisequationrequiresthatb/t=2, asisshownbytionforthecasesofpure
5、bendingandpuretorsion.accordancewiththeabovesuggestionsonecancorrectmultiplyingeveryvaluewherebl and tl areof T introducedinequation.(5)solvingtheequa-Thereforeinforanisotropyby- “- -(5)by bl/2tlthemeasuredvalues of thefatiguestrengthin .Provided by IHSNot for ResaleNo reproduction or networking per
6、mitted without license from IHS-,-,-:Q NACATN2924 25. bendingandintorsion,respectively.Thenuniberofcyclesforwhichhland tl aredeterminedarethesameasthenuniberwhichereexpectedtoproducefailureunderthegivenvaluesof u and T. Iftheadjusted. valueT1=b17/2t1issubstitutedinequationresults:(5)thefollowingequa
7、tion()b12T2=t (6)Thisequationmayberationalizedandrewrittenas(7)theellipse-quadrantequation(seetableIII)bqwhichisidenticalwith. sinceforpuretwistinga71fromwhich._Thustheellipsequadrantmaybeconsideredtobearationalizedequationforthelimitingprincipalshesringstresswithacorrectionforanisotropyapplicablefo
8、rthesyecialtypeoftestingandorienta-tionofspecimensconsidered.Ofcourse,proofoftheeffectofsni-sotropymustbesoughtbeforethevalidityofthea%oveexplanationmaybedemonstrated.Inasimilarwaythiscorrectionmaybeappliedtotheotherrationaltheoriesoffailure.Thevaluesof b/t requiredbythesetheoriesandtheresultingequa
9、tionsaregivenintableIII.Itissignificanttonotethattheequationsforsixofthesetheories,whenrationalized,sreidenticaltotheempiricalequationproposedbyGough-theellipse-quadrantequation.Thesixtheoriesare:.(1)Principalshearstress(2)Principalshearstrain(3) tiergyofdistortion(4)OctahedralshearstressProvided by
10、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-26 NACATN2924(5)Totalenergyofdeformation.:(6)Magnitudeofstate-of-stressvectorThetheorieswhoserationalizedequations(withcorrectionforanisotropy)differfromtheellipsequadrantare:(1)Principalstress(2)PrincipalstrainEF
11、FECTOFMEANSTRESSANDSTATEOFSTISSCONSIDERINGANISOTROPYMeanStressSincesixofthetheoriesareidenticalinform(withinthestatesofstresscoveredbycombinedbendingandfbrsion)whencorrectedforanisotropythecurvesforonlyoneofthesixneedtobeplotted.Thetheoryofalimitingprincipalshearstresswasselectedbecauseofitspossibil
12、itiesasshownbythereviewoftheliteratureandbecauseoftheconsiderationsdiscussedinthenextsection.Theeffectofmeanstressisshowninfigures32to34fortheprincipal-shear-stresstheorywithcorrectionforanisotropy.Thevaluesof b and t usedincorrectingforbothalternatingandmeanstresseswerethevaluesofthebendingandtwist
13、ingfatiguestrehgths,respectively,atzeromeanstressandthegivennumberof-cycles.Thereisnoreasontothinkthatthevaluesof b and t areanymoreaccuratethanthefatiguestrengthatanycombinationofbendingandtwisting.Thusacertainlatitudecommensuratewiththescatterofthedatamightbeallowedinthevalueof b/t usedincorrectin
14、gforanisotropyifindicatedbythetrendofdataforcombinedbendingandtorsion. -.Thedisagreementwhichexistsinfigures32and34betweentheeldstrengthandthemaximumstressbeyondwhichthefatiguestrength - decreasesmarkedlymaybetheresultofcorrectingboththefatigue-strengthandmean-stressvaluesforanisotropy.Anexcellentco
15、rrelationwasobservedbetweentheyieldstrengthsfromstatictestsandresultspredictedbytheprincipal-sheartheoryasmentionedearlier.Thissuggeststhattheoriginalmaterialmayhavebeennemlyisotropicasfarasgeneralyieldingisconcernedandthatthematerialmayhavebeenaaisotropiconlyatthelevelofthelocalizedactionwhichiniti
16、atesfatiguefailure.Ifthisisthecasethenthecorrectionforanisotropyshouldhavebeenappliedonlytothealternatingcomponentsofthestress.Thisrevisiondoesnotaffectthediagramforbending,figure32,butdoesaffectfigures33and34. ThereviseddiagrsmfortorsionisshowninProvided by IHSNot for ResaleNo reproduction or netwo
17、rking permitted without license from IHS-,-,-NACATN2924 27. figure35. Theapparenteffectofyieldingonthefatiguestrengthresultingfromahighmeanstressisinbetteragreementwithstaticyieldinginfigure35.ThesamereasoningwhenappliedtotheGuesttheoryhadasimilarresultalthoughtheagreementwasnotsofavorable.Themagnit
18、udeofthecorrectionforyieldinganditseffectonthediagramofsmplitudeofstressagainstmeanstressareillustratedinfigures32snd34 in whichbothcorrectedanduncorrectedvaluesareplotted.StateofStressatDifferentMeanStressesFromordinatesatfourvaluesofmeanstressindiagramssuchasfigure35andfromthedataatotherstatesofst
19、ressforzeromeanstress,diagramsshowingtheeffectofstateofstresswereconstructedasshowninfigure36forthetheoryofa limitingvalueofprincipalshearstress(withacorrectionforanisotrophyappliedtothealter-natingstress).Anexaminationoffigure36r”Tedthatthistheoryisacloserrepresentationofthetestdatathan therconside
20、red,withthepossibleexceptionofGuesttslaw.Itshouldberecalled,however,thatthealternatingvaluesarethesame,exceptforaconstantfactor,forsixofthetheories-notjusttheshear-stresstheory.Thedia-gramsformeanstressesabovezeroare,however,dependentontheshear-stresstheory(orshear-straintheory,whichisidentical)sinc
21、ethemeanstressdoesnotcontainacorrectionforanisotropy.Theprincipal-stresstheoryandprincipal-straintheorywithcorrec-tionforanisotropywerealsoconsideredforzeromeanstress.Theagreementwiththetestdatawasnotnearlysogoodasfortheprincipal-shesr-stresstheorycorrectedforanisotropy.Theeffectofthecorrectionforyi
22、eldingisindicatedinfigure36byshowingtheordinatestoboththecorrectedanduncorrectedcurvesofalternatingagainstmeanstress.Whenthecorrectionsforanisotrophyandthedataforallstatesofconibinedstress,allvaluesofmeanstress,andallcyclestofailurewereconsidered,thetheoriesoffatiguefailurewerefoundtohavethefol-lowi
23、ngorderofmerit(wherethefirsttheoryrepresentsthetestdatatheclosest):(1)(2)(3)(4)(5)Principalshearstress(principalshearstrain,etc.)correctedforanisotrophyCompleteGuestlawPrincipalstraincorrectedforanisotrophyMagnitudeofstate-of-stressvectorTotalstrainenergyProvided by IHSNot for ResaleNo reproduction
24、or networking permitted without license from IHS-,-,-28 NACATN2924ENERGYANDOTHERCONSIDERATIONS sOfthesixtheorieswhoseequations_orendg.andtorsionbecome widenticalwhencorrectedforisotropytwoarebasedonanenergycon-cept.Also,evenwithoutcorrectionforenisotropy,energytheoriesrankhighintheorderoftherational
25、theories(seesectionentitled .-“EffectofStateofStress”).EnergyasaScalarThereareanumberofobservationswhichseemincompatiblewithenergyconcepts.Energyisascalarqusmt”itysothatthecharacteristicsofstrainenergyareindependentoforientationoftheprincipalstressesrelativetothematerial.Thusanisotropyinthematerials
26、houldnotaffecttheresultsoffatiguetestsofspecimenshavingdifferentorien-tationsifenergyisthecriterionoffatiguefailure.However,datareferredtointhesectionentitled“Data-onIsotropyinFatigueofMetals”indicatethatorientationprobablyaffectstheresultsoffatiguetests.Microscopicstudieshaveindicatedthatslipbandsa
27、representinthevicinityoffatiguecracksandthattheyprobablyprecedeactual .-.-fatiguecracks.Thisbeingthecase,theorientationofstress(orforces)wouldseemtobeanimportantfactorintheoriginoffatigue wcracking.Thusonemightquestionwhetherascalarquantitysuchasenergycouldbeacontrollingfactor.IthasbeensuggestedbyFo
28、wler(inaprivatediscussion)thatenergytheoriesmightbetestedifonecoulddeviseameansofmaintainingthestateandintensityofstressconstant-inaspecimen(nofluctuationinstress)sndfluctuatetheorientationoftheprincipalstressaxeswithrespecttothespecimen.Fatiguefracturemightormightnotoccurathighstresses,accordingtow
29、hetherorientakio-io-fs-tressingisorisnotafactorinthefatiguephenomena.It.seemstothewriterthatfatiguewouldveryprobablyoccur.Suchatestmightbeaccomplishedbyusingadiskasaspecimen.Thediskshouldhaveaheavyrimandbedishedonbothsidessothatwhenloadeddiametrallytherewouldbeasectionofuniformhighstressinthecenterw
30、hosestateofstresswasbiaxial(anunequaltensionandcompression).Ifthediskisloadedthroughrollerssndrotatedunderloadthestrainenergyinthehigh-stressportionofthediskwouldremainconstantbuttheprincipalstressdirectionswouldrotatewithrespecttothedisk.Insuchatestifeitherthestateofstressorthematerialis .isotropic
31、nothingshouldhappen,butifbothereanisotropicitshouldbepossibletoproducefatiguefracture.Thisshouldbetrueeven .thoughthematerialis statisticallyisotropicsincefatiguefractureoriginatesfromphenomenaoccurring in individualgrains,whoseproper-tiesme knowntobeanisotropic.Provided by IHSNot for ResaleNo repro
32、duction or networking permitted without license from IHS-,-,-NACATN292 29. Inviewoftheaboveitwouldseemthat correctionofanisotropictheorysuchasanenergytheoryforanisotropyismeaningless.This. leadsonetosuspectthatenergytheoriesarenotapplicabletothisproblem,orthattheconceptofenergyasascalarmustbemodifie
33、d.StrainEnerandMeanStressAnotherdifficultyarisesintryingtoapplytheenergytheorytotheproblemwhenthemeanstressisnotzero.Thevariationofstrainenergywithtimewhencalculatdfromstressesmustbebasedonthevariationofthetotalstress,notonthevariationofcomponentssuchasalternatingandmeanstress.Residualstressesmustal
34、sobeaddedtotheappliedstressesbeforecalculatingtheenergy.Ifthisisnotdonelsrgeerrorsinenergywillresultsincestressesaresquaredincomputingthestrainenergy.Thusdiagramssuchasfigures26to28inwhichthealternatingandmeantotalenergiesarecomputeddirectlyfromthealternatingandmeanstresses,respectively,donotpresent
35、thecor-rectrelationbetweenthecomponentsofenergy.Anattempttopresentamoreaccuratepictureoftheenergyrelation. distlosesthefollowingdifficulties:(1) Whentheminimumstressofthecycleisofthesamesignasthemaximumstressthereisnodifficultyotherthanthattheenergycycleisnotsinusoidalasthestresscyclewas.Themaximume
36、nergyiscalcu-latedfromthemaximumstressandtheminimumenergyfromtheminimumstress.Thealternatingandmeanenergiesarecalculatedashalfthedifferenceandhalfthesumofthesemaximumsandminimums.(2)Sinceenergyisascalsxandalwayspositiveacompletelyreversedsinusoidalcycleofstresswillproducenotacompletelyreversedcycleo
37、fstrainenergybutacycleofenergyvaryingfromzero(atzerostress)toamaximum(atbothmaximumandminimumstress)whichisnotsinusoidalandhasafrequencytwicethefrequencyofthestresscycle.Thusthetotalnumberofenergycyclessustainedbeforefractureistwicethetotalnuniberofstresscyclessustained.Thesefactshavenotbeendiscusse
38、d(asfarasislmown)inpreviousstudiesofenergytheoriesoffailureappliedtofatigue.(3)men thestressduringthecycleispartlyofonesignbutnmstlyoftheoppositesign,theenergycycleconsistsoftwoalternatepulsesofthesamesignbutdifferentmagnitude.Thefrequencyofthelargerpulseisthesameasthatofthestresscycle.Whentheabovef
39、actorsaretakenintoaccountanewdiagramrepre- ._sentingtheactualalternatingenergyagainstactualmeanenergymaybeconstructedasshowninfigure37.Thisisaccomplishedby(a)computingthealternatingandmeanenergiesasdescribedinitem(1)above,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
