03co /’-r-oo

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J.

PLASTIC

NATIONALADVISORYCOMMITTEE[FORAERONAUTICS

STRESS-STRAIN

TECHNICALNOTE2425

SUBJECTEDTO

By JosephMarin,

—.

FOR 75S-T6ALUNIHTUMALLOY

BIAXIAL TENSILE STRESSES

B. H. Ulrich, andW. P. Hughes

The PennsylvaniaStateCol.lege

.

Washington

August195i

‘TBIwMH!LWwNfM%2&Nl

. .----- -—.--——.-.——- -. -.= —-———. ..—-.-— -— .-,... . .. ... . . . ______ .__- ._._ ___ . . . .

TECHLIBRARYKAFB,NM

1

c

NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALNOTE2425

PIASTTCSTRESS-STRAINRELATIONSFOR75S-T6AWMINUMALLOY

SUBJECTEDTOBIAXIALTENSILESTRESSES

~ JosephMarin,B.H. U1.righ,andW. P.Hughes

Inthisinvestigation,thematerialtestedwasa 75S-T6aluminumalloyandthestresseswereessentiallybiaxialandtensile.Thebiaxialtensilestresseswereproducedina speciallydesignedtestingmachineby subjectinga thin-walLedtubularspecimentotial tensionandinternalpressure.Plasticstress-strainrelationsforvariousbiaxialstressconditionswereobtainedusinga clip-typeSR-4straingage.

Threetypesoftestsweremade: Constant-stress-ratiotests,variable-stress-ratiotests,andspecialtests.Theconstant-stress-ratiotestresultseve contioldataandshowedtheinfluenceofbiaxialstressesontheyield,fracture,andultimatestrengthofthematerial.Bymeam ofthevariable-stress-ratiotests,it ispossibleto determinewhetherthereisanysignificantdifferencebetweentheflowanddeformationme of theory.Finally,specialtestswereconductedto checkspecificassumptionsmadeinthetheoriesofplasticflow.

Thecorwtant-stress-ratiotestsshowthatthedeformationtheorybasedontheoctahedral,effective,or significantstress-strainrelationsisinapproximateagreement.withthetestresults.Thevariable-stress-ratiotestsshowthatboththedeformationandflowtheoryareinequallygoodagreementwiththetestresults.

INTRODUCTION

Machineandstructuralpartsmaybe subjectedto stressesbeyondtheyieldstrengthofthematerial.Oftenthesestressesarenotsimplestressesactinginonedirection,butarecombinedstressesactinginmorethanonedirection.To adequatelydeterminethefactorsof safetyina particularmember,it isnecessarytoknowtheplasticstress-strainrelations.Furthermore,inpartswhichme subjectedto initialresidual

——..— -—— —- — ..—.— -——— —-— “-

2 NACATN 2k25

stresses,suchashigh-pressurevessels,informationontheplasticstress-strainrelationsisimportant.AnothervaluableuseOf theplasticstress-strainrelationinmetalsisinthestudyandimprove-mentofformingoperations.

Inrecentyesrs,manytheorieshavebeenproposedfordefifingtheplasticcombinedstress-strainrelationsformetalsbasedonthesimple-tensionstress-strainrelations.Thesetheoriesareneededforthesolutionoftheengineeringproblemsmentionedintheforegoingpsragraph.However,forengineeringdesignpurposes,it isdesirableto lmowwbichoftheavailabletheories,ifam, Wee ~th thetestresultsforthevariouspossiblestressconditions.Inthepast,mostinvestigationshavebeenmadeforbiaxialtensionstressesandfortheconditioninwhichtheratiooftheprincipalstressesremainsconstantduringloading.Constant-stress-ratiotestsdonotdistinguishbetweentheflow-anddeformation-typetheorieslanditwasforthisreasonthatemphasisinthisreportisplacedonvariable-stress-ratiotests.Constant-stress-ratiotestsarealsoreportedinordertoprovidebasicinformationonthestrengthpropertiesofthematerialtested.Thepresentinvestigationisrestrictedto a 75s-T6aluminumalloysubjectedtobiaxialtensilestressesonly.

TheresearchhereinreportedwasconductedinthePlasticityLaboratoryofthePennsylvaniaStateCollegeunderthesponsorshipandwiththefinancialassistanceoftheNationalAdvisoryCommitteeforAeronautics.Dr.SsmBatdorfandhisassociatesatLangleyFieldgavevaluedsuggestionsintheplanningoftheresearchreportedherein.Messrs.B. H.Ulrich,W. P.Hughes,andL.W. Hu,researchassistants,conductedthetestsandcomputedthetestdata.PartsofthetestingmachineandthespecialstraingagewerebuiltbyMessrs.S.S.Eckl.ey,H. Johnson,andI.B@lme. Theforegoingindividualsinmakingappreciated.

assistancegivenby theNACAandthepossiblethisinvestigationisgreatly

SYMBOLS

d originalinternaldiameteroftubulsr

d internaldiameteroftubularspecimenP inches

specimen,inches

inplasticrange,

% thisr@port,whenreferenceismadeto-theflowanddeformationtheories,thesimpletheoriesbasedontheoctahedralshesrstressandstrainsreintended.

———

NACATN 2k25 3

.

..

k

F

n

P

P

t

‘P

XYY

a

aY

au

‘r

al~a2

a3

aleJa2e

alu~a2u

Young’smodulusofelasticity,psi

longitudinalandlateralnominalstrainsinplasticrange,respectively,inchesperinch

strengthcoefficientforsimpletension

Poisson’sratio

strain-hardeningcoefficientforsimpletension

internalpressure,psi

axialtewionload,pounds

originalwallthicknessoftubularspecimen,inches

wall.thickness.oftubularspecimeninplasticrange,inches

principal-stressratios

truestressinsimpletension,psi

yieldstressin simpletension,psi

nominalultimatestressinlongitudinaltension,psi

truerupturestressinlongitudinaltension,psi

truelongitudinalandrespectively,psi

trueradialprincipal

lateralprincipal

stresses,psi

stresses,

elasticlongitudinalandlateralprincipalstresses,respectively,psi

yieldlongitudinalandlateralprincipalstresses,respectively,psi

nominalultimatelongitudinalstresses,respectively,psi

truelongitudinalandlateralrupture,respectively,psi

andlateralprincipal

principalstressesat

.—— ——- _—.——

4 NAC!ATN 2425 ‘

significantstress,psi

truestraininstipletension,inchesperinch

trueprincipalstrains,inchesperinch

significantstrain,inchesperinch

totalprincipalstrains,inchesperinch

incrementinplasticflow (F(5)t@

TESTPROCEDURE

MaterialTestedandSpectien

Thematerialtestedinthisi~estigationwasa fullyheat-treatedaluminumalloydesignatedas75s-T6. Thematerialwassuppliedintubularextrudedforminlengthsof16 feetwithan internaldiameterof2 inchesanda wallthicknessof1/4inch.Thenominalchemicalcanposition,inadditiontoaluminumandnormalimpurities,consistsof 1.6 percentcopper,2.5percentmagnesium,andtracesofmanganeseandchromium.Nominalmechanicalpropertiesintensionasfurnishedby themanufacturerare: Ultimatestrength,88)000psi;yieldstrength(O.2 percentoffset),80,000PSi;m.od~usofelasticity,10.6X 106pSi;percentelongationin2 inches,10percent;andp~sson’sratioj0.33.

Thedimensionsofthemachinedspecimensareshowninfigure5 ofference1. Thespecimenusedhadan over-al-1lengthof I-6inches,

k.kha- tntermediatelengthofXl inchesofreducedwall.thicknessequalto about0.100* 0.002-inches.Theinternalsurfacewasleftintheextrudedform.Thewallthicknessofthetubularspecimenwasmeasuredusingtheapparatusdescribedinreference1. Theratioofthewallthicknessto diameterofthespecimenwas0.05,sothatthebiaxialstressesthroughoutthewallwereessentiallyconstant.The ‘ratioof diameterto lengthforthespecimenwasabout0.18,sothatasufficientlylongsectionofthespecimenwasavailablefreefrombendingstressesproducedby endrestraints.

,t

,

—.

NACATN 2&5 5

TestingMachine

Themachineusedforthetestsreportedinreference1 wasmodifiedforthepresentinvestigation.Changesinmethodsofapplyingtheinternalpressureandaxialloadsanda new-typeclipgagewereneces-saryinthepresentinvestigationtoobtainmoreaccuratelythestress-strainrelationsfortheinitialpartoftheplasticrange.‘Figure1showsfrontandsideviewsofthebiaxial-stressmachine.TheaxialtensileloadisappliedtothespectienS bymeansofa hydraulicjackJ,a vertical.rodR, anda leverL. The.axialloadismeasuredbya dynamometerD usingSR-4gages.TheleverL transmitstheloadtothespectienthroughsphericalseatsSt to insureaxialityof loading.ThefulcrumF oftheleverandtheendsoftheleveraresuppliedwithbearingsto e13minateerrorsdueto flriction.ThepullingrodR isprovidedwitha sphericalseatanda bearingto eltiinatebending.ApumpunitP wasusedto applythetiternalpressure.A 10W automotiveoilwith175SSUviscosityat100°F wasthefluldusedforapplyingtheinternalpressure.TheoilwassuppliedtothespecimenS by apmp P througha high-pressurepipelineto thelowerpullingheadH.Therateofpressureapplicationwascontrolledbymeansofa releasevalveV whichdischargedsurplusoilintotheoil-supplyreservoir.Theoilpressurewasmeasuredby a 10,000-poundU.S.EourdongageG.

.

.

Theaxialityoftheloadwascheckedas describedinreference1.Themachinewascalibratedforaxialloadingby usingacalibratingrodwithSR-4gagesinplaceofthespectienS andrecordtigthereadingsona calibratedmechanicaltypedynamometeratD. Theaxialloadonthespectiencouldbemeasuredwithin100pounds.Thepressuregagewascalibratedbeforetestingandwasfoundtohavea maximumerrorofabout2 percent.

MethodofMeasuringStrains

Theelasticstratisweremeasuredfora 13/16-iuch gage lengthby usingSR~ electricstraingages.TwoSR-4gages,onelongitudinalandonelateral,were’attachedto thespecimenatmidlengthandwereusedtomeasuretheelasticstrains(fig.2(a)).TheSR-4gageswerecementedto thespecimensticompliancewiththeprocedureprescribedby themanufacturer.Thestraingageswerecomectedthrougha switchboxB sothateachgagecouldbe successivelyswitchedintothecircuitwiththestrainindicatorI. Thestra’inindicatorI recordsthestraindirectlyinmicroinchesperinch.

Theforegoingmethodofmeasurtigstrainsislimitedtoamaxinnmstrainvalueofabout0.015inchpertich.Inordertomeasuretheplasticstrainsitwasnecessarytoprovidesomeotherldndof stratigage.A clip-typegageas showninfigures2 and3 wasusedtomeasure

-.—. ————- —— — –—— .—— —————. -— --

6 NACATN2425.

thelongitudinalandlateralplasticstratis.A CEP gage consistsofa rectangular-shapedframewiththecrossmembermadeofa phosphor-bronzestriptowhichSR-4electricgagesareattachedtotheupperandlowersurfaces.By thisarrangement,anadditionaltemperature-compensatinggageisnotrequiredandticreasedsensitivityisobtained.Bymeansoftheseclipgagesa largestratiatthepivotpointsoftheclipisreducedtoa smalJ-measurablestratiatthebridgeoftheclip.Thelongitudinalandlateralclipgagesmeasurestrainsto 0.00005inchperinch.Theclipgageinfigure2 madeitpossibletomeasureboththelongitudinalandlateralplasticstrainsontwogagelengths.Thegageswerecalibratedusingthedeviceshowninfigure2(b).A steppedplateC withnotchesalongtheedgesoftheplatespacedat fixedMowndistancesprovidesthestandardforCalibrattigtheclipgages.Thedistancesbetweenthenotcheswereaccuratelymeasuredbymicrometercalipersreatigto 0.0001inch.Withtheclipgageattachedtoa pairofnotches,theS@+ indicatorreadtigisrecorded.By useofthesuccessivenotchesandby observingthecorrespondingSR-4indicatorreadtigs,a calibrationoftheclipgagesismadepossible.

FinalStrainsatruptureweremeasuredto 0.01tichby useofdividersand-ascale.

MethodofTesting

Priortotesttng,SR-4gagesweregluedto a tubularspectien.Afteradjustingtheclipgagesandconnectingallstraingagestotheswitchingboxandstratiindicator,a zerosetof stratireadingsontheunloadedspechenwasrecorded.Oilwasthenpumpedthroughthespecimentoremoveanyairthatmightbe trappedinthespectien.Thedischargeoutletinthepulltigheadofthetestingmachinewasthensealedanda protectionshieldwasplacedaroundthespechen. Internalpressureoraxialloadsorbothtypesofloahg werethenappliedaccorhg topredete-ed values.ThemannerandmagnitudeoftheloadsappliednaturaIlydependeduponthespecifictypeoftest.Atselectedintervalsof loador strainthevaluesoftheloadsandstrainswererecorded.lbactureloadswerenotedandpemanentstrainsafterfractureweremeasured.

Priortotesttig,allspecimensweresubjectedtoa permanentprestrainof0.2percent,firstinthelongitudinaldirectionandtheninthelateraldtiection.Thisprocedurewasrecommendedby theNACAcommitteeforthisproject.Thepurposeoftheprestraintigistoreducetheamountofanisotropypresentintheetirudedtubularspectien.Influenceof suchprestra~tigisdescribedina paperby Templti(reference2).

..

. “

—. _.—. —— — .

NACATN 2425 7’

CONSTANT-STRESS-RATIOTESTS \

.Plasticstress-strainrelationsforvariousconstantbiaxial

stressratiosaretheusualtypeobtained.Toprovidethisstandardinformationandto obtaincontroldata,constant-stress-ratiotestswerealsoconductedaspartofthepresentinvestigation.It shouldbenotedthatconstant-stress-ratiotestsgivealsoinformationontheinfluenceofthecombinedstressratiouponthestrengthandductilityofthematerial.

Conventional

Theaveragecurveshowingstressandstrainforboththe

Stress-StrainResults

therelationsbetweentheconventionallongitudinalandhteralstressesis

showninfigures4 and5. Oneachstress~straincurvetheratio/

(Y2al ofthelateralto longitudinalstressisgiven.Thestrati

valuesplottedinfigures4 and5 were~asuredbytheSR-4gRgescementedtothespectiens.Formoststressratiosthreespectienswereused,butforallratiosat leasttwospecimensweretested.

.

Theequationsusedforbteralstressesplottedin

calculattigthefigures4 and5

pd2+ 4:ale =

4t(d+ t)

nominallongitudinalandwere,respectively,

(1)

(2)

(seereference1). Eqyation(2)forthelateralstressisthatbasedonaseumingthatthewallthicknessislsrge.Itwasnecessarytoconsiderthelateralstreps

since,forthevalue t/d=

5 percentgreaterthanthatthin-walledtube.

basedonthetheoryofthethick-walltube

0.05 used,cr2e= 1.05~ minusa value

obtainedby considertigthetheoryofthe

Thenominalor conventionalstrainvaluesplottedinfigures4and5 weredeterminedfromthevaluesoftheSR-4indicatorreadtig.Theindicatorreadingswerecorrectedforlateralsensitivityandthe“combined-stresseffect”sincethemanufacturersconstantsarebased

—. —.-

8 N/QTM2425

ona calibrationustiga steelBpecimenwitha Poisson~sratioof0.285.EquationsforobtatitigthecorrectedstrainusingtheindicatorreadingsaregiveninappendixB ofreference1.

Yield-strengthvaluesforaxialtension(as@ven intable1 for “stressratioequaltoO)werebasedonoffsetstrainof0.002inchperinch,as showninfigure4. Forthecombined-stresstestsanequivalentoffsetstrainwasused. Thedeterminationofthisequivalentoffsetstrainisexplainedin appendixB.

PlasticStress-StrainResults

Therelationbetweenthetruestressesandstratisfortheentirerangeof stressandforthevariou,prticipalstressratiosaregiveninfigures6 and7. Thesestress-stratirehtionsdifferfromtheconventionaldiagramsstic”etheyconsidera chan@ng.gagelengthandchangtigdimensionsofthespecimen.Thecurvesshowninfigures6and7 arebasedontheaveragenominalstress-strainrelationforatleasttwospecimens. ,

Thetrueplasticstrainsweredeterndnedfromtheclip-gagereadingsgivenby theSR-4indicator.Theconversionofthereadingto stratiininchesperinchisexphinedinreference1.

Itlateralstrains

csnbe shown(reference1)thatthetruelongitudinalandstratish termsofthenominallongitudinalandlateralel and -ePare

‘1 = lo& (1

$2= lo% (1

Thestressesintheplasticrange

}’+e-J(3)

+ e2)

mustbe determinedustigtheWensionsattheparticularloadvalues,sincethechangesindimensionsdurtigplasticflowareappreciable.Thetruelongitudinalandlateralstressescanbe obtainedby equations(1)and(2)providedtheinitialdismeterd andwallthicknesst arereplacedby theiractual “values~ and tp attheparticularloadsconsidered.Thatis,the

stressesintheplasticrangesxe

p&f+4;‘+ ktp(dp+~) (4)

Q

2

Thevaluesofthedimensions~ and tp canbe showntobe

?P=(l+e~+e2)

9

(5) “

(6)

‘P= (d+ 2t)(l+e~ - 2tp (7)

Thetruestress-stratidiagramsrepresentedinfigures6 and7 arebasedon stressesandstrainsas calculatedby equations(3),(4-),and(5).Thefracturepointsshowntifigures6 and7 werebasedonthestratisafterrupturecorrectedfortheelasticstratisjustpriortorupture.

Fromthedatagiven.infigures4 and5,valuesofthenominalultimatestrengthsforthevariousbiaxialstressratiosweredetermined.Thesevaluesarelistedintable2. Table3 showsthetruefracturestressesforvariousbiaxialstressratios,as determinedfromfig–urea6 and7. Table4 givesductilityvaluesby listingthenominalandtruestratisat fractureforvariousbiaxialstressratios.

A&lysisandDiscussion

Yieldstrength.-Yield-streng’thvaluesforvariousbiadalstressratios(appendixB) arecomparedwiththetheoreticalvaluesinfig-ure8 andtable1. Thecomparisonshowninfigure8 isbdsedupontheuniaxialstren~hinthelongitudinaldirection.Figure8 showsthatthemaximum-she%or stresstheoriesareinapproximateageementwiththetestresults.

Plasticstress-stratirelations.-Plasticstress-strainrelationsarecomparedwiththedeformationtheoryby plottingrelationsbetweenthesignificantstressandstrati(reference1)andcomparingtheserelationswiththetrueuniaxialstress-stratirelations(figs.9and10). Thevaluesofthesignificantstressandstrainwerecomputedbytheequations

~=\; ~.1- .2)2+(.2- .3)2+(.l - .3)3 (8)

—. .—- .— -—— —.— ————

10 NACATN 2425

‘=w==) (9)

Valuesof = and F arealsoreferredtoasthe“effectivestressandstrainlfandtheyareequivalenttothe“octahedralshearstressandstrati”exceptfora nmericalconstant.A studyoffigure100showsthatthedeformationor flowtheoriescanhe usedtoapproximatelypredictplasticstress-stratirelations.Thisconclusionisbasedontheagreementbetweenthevarioussignificantstress-strainrektionsandthetrueuniaxklstress-strainrelationas showninfigure10.

A comparisonofthetruestress-strainrelationsforeachprincipalstressandthevaluespredictedby theflowanddeformationtheoriesisgivenh figures6 and7. Thedeterminationofthetheo-reticalstress-strainrektionsby theflowanddeformationtheoriesisexpktiedinappendixA. Forconstantstressratiostheflowanddefamationtheoriescoincide.Formall stratisthetwotheoriesgivethesameresultswithinthepossibleaccuracyofthecalctitions.Figure10swws tit there iB good age~ent be~ef= the act~ stress-strainrelationsandthevaluespredictedbyboththeflowandthedeformationtheories.

BiaxialnmdnalUlt-te strength.-ValuesOfb~ n~ultimatestrengthas givenintable2 arecomparedinfigureI.lwithvaluespredictedby themaximum-stressor sheartheoryof failure.Figure11 showsthatthemaximum-stressor shesrtheoriesmaybe usedto approximatelypredictthenominalultimatebiaxialtensilestrengthsforAlcoa75S-T6aluminumalloy.

Biaxialtruefracturestrength.-Valuesofbiaxial.truefracturestrengthas listedh table3 arecomparedinfigure12withvaluesgivenby themaximum-stresstheory.An examinationof figure1.2showsthemaxtium-stressor sheartheoriesgiveanapproximatepredictionoffracturestrength.~ viewoftheneckingdownofthespechenbeyondtheulthateloadsandthesubsequent,changesh thestateof stressduetonecldng,thecomparisonbetweentheoriesandtestresultsisconsideredbetterthanmightbe expected.

Ductility.-Ductilityvaluesbaseduponboththetiitialandchangtiggagelengthsaregivenintable4 forvariousbkxialstresses.Boththenominalandtrueductilityvaluesintable4 showthattheductilitydecreaseswithincreaseiubiaxialityoftheprincipalstressratio cr21UlfromO to 1. Theinfluenceofbiaxialstressesonthe

ductilitycannotbe definitelydeterminedbecauseoftheeffectofanisotropyofthematerial.Theinitialprestressingofthematerialdidnothavethedesfiedinfluenceonthesnisotropyofthematerial.Thedirectionaleffectsinthespecimenarealsoindicatedby the

“

———

NACATN 2425 11

-.

differenceinthetruetensilestress-stratidiagrsmsforthelongi-tudinalandlateraldirectionsas showninfigure13. Thedifferenceinthetensilepropertiesinthetwodirectionsisalsoshownby thedifferenceinvaluesof k and n asobtainedfromfigure13andlistedintable5. Valuesof k and n are,respectively,thestrengthcoefficientandstrain-hardeningexponentintheequationa = k#, where u and G arethetruetensilestressandstrain,respectively.

VARIABLE-STRESS-RATIOTESTS

Theconstant-stress-ratiotestsdiscussedintheforegoingsectiondonotmakeitpossibleto distinmshbetweentheflowanddeformationtheoriessinceforconstantbiaxialstressratiosthetheoriescoincide.Variable-stress-ratiotestswereconductedinthisinvesti~tiontianattemptto showwhichofthetwotheoriesagreedbestwiththetestresults.

Variable-stress-ratiotestswereconductediness=tiallythessmemanner”astheconstant-stress-ratiotests,exceptthattheinternalpressurewasfirstappliedupto selectedvaluesandaxialtensileloadswerethenappliedto fracture.Thevalueoftheinternalpressurewasmaintatiedineachcasewhiletheaxialloadwasapplied.Themannerofloadingisindicatedinfigures14and15whichshowthenominalstress-strainrelationsforboththelongitudinalandlateralstresseswhenvariousloadingconditionswereused.Thenominal.stressesusedinplottingfigures14and15werecalculatedbyequhtions(1)and(2)andthestrainsweredeterminedasexplainedinreference1. usillgequations(3),(4),and(5)andtheaveragevaluesrepresentedby thecurvesinfigures14and15,thetruestressesandstratiswerecal-culatedandforeachloadingconditionthevaluesoftruestress-strainrelationswereplottedforboththelongitudinalandlateraldirections.Figures16and17showthesetruestress-strainrelations.Valuesofthetruestress-strainrehtionsas determinedby theflowanddefor-mationtheorieswerecomputedasex@@ned inappendixA. Thesevaluesarebasedonthetruetensionstress-strainrelationsaspreviomlynoted.A comparisonisshowninfigure16betweenthetestresultsandthevaluesofthestress-strainrektionspredictedby theflowanddeformationtheories.An examinationof figure16 showsthatboththeflowanddeformationtheoriesme inapproximateagreementwiththetestresultsandthatonecannotbe recommendedinpreferencetotheother.

To comparethedeformationtheoryandtestresults,significantstress-strainrelationswereplottedforthevariable-stress-ratiotestsas showninfigure18. Figure19 showsthesignificantstress-strain

—. —— ——— .— .—

. .

12 NACATM 24z5

relationsplottedwitha commonorigtiaewellasthetfueuniaxialtensilestress-strainrelation.An examtitionofthesignificantstress-straincurves,tifigure10 forconstantstressratiosindicatesthatsomeofthedifferencesbetweenthesignificantstress-strainrelationsh figure19aredueto anisotropy.Theanisotropyisshownby thedifferencebetweensignificantstress-strainrelationsinfig-ure10fortheuniaxialhteralandlongitudinalstresses- thatis,forprincipalstressratiosO and~.

u

.

SPECIALTESTS

Testson IsotropicYielding

It is assumedintheisotropicltiearflow’theoriest~t ~itialprestrainingwillnotproduceanisotropy.Thatis,itisassumedthatthereisisotropicyieldtig.To determtieexpertiental.lythevalidityofthisassumptionthefollowingtestsweremade. OnespectienwasloadedinlongituiUnaluniaxialtensionto a strainofabout5 percent.Thespectienwasunloadedandthenloadedunderuniaxiallateraltensionto failure.A secondspectienwasloadedinlongituti uniaxialtensiontoabout5-percentstrain,unloaded,andthenreloadedunderuniaxiallongitudinaltensionto failure.Iftheisotropic-yieldingassmnptionisvalidthesignificantstress-straincurvesforthesetwotestswouldcoincide.A plotofthesignificantstress-strainrelations .

showedthatthecurveswereinaboutas closeagreementasthesignifi-cantstress-stratirelationsforlongitudinaltensionandlateraltensioninfigure10. Furthermore,thelackof ductilityinthelateral.directiongavea smallover-allrangeof strain,makingthecomparisonofthesignificantqtress-strainplotsnotentirelyconclusive.Thatis,the3nitialanisotropyofthematerialmadeitdifficultto deteminewhetherisotropicyieldingoccurred.

TestsonCoticidenceofPrincipalStressandStrainAxes

Inthetheoriesofplasticity,itisassumedthatthedirectionoftheprincipalstressesandstrainsremainsthesameintheplasticrange.To checkthisassumption,a strainrosettewasplacedona tubularspecimenh ordertoprovidea meansof determiningtheprincipalstraindirections.Thespectienwasthensubjectedtoan internal.pressureandvaluesof strainsforthethreestrain-rosettedirectionsweremeasured “uptoa strainofabout1.5percent.ThepressurewasthenremovedandthepermanentP~sticstra~sweremeasured”‘rm ‘he‘trati-rosettereadingsthedtiectionsoftheprticipalplasticstrainsweredetermined. ,A comparisonofthedirectionsoftheprticipalplasticstressesandstrainsas showninfigure20 showsthatforpracticalpurposesthedirectionoftheseaxescoincideasassumedinthetheory.

,

NilCATN 2k25

CONCLUSIONS

13

Forthe7x-T6alumtiumalloytested,thefollowtigconclusionssremadeonthebasisoftheforegotigbiaxialtensiontests:

1.Thebiaxialyieldstrengthsmaybe safelypredictedby themaxtium-shesror stresstheories.

2.Thenominalbiaxialultimatestrengthsandthetruebiaxialfracturestrengthsareinapproximateagreementwithboththemaximum-stressandmaximnn-sheartheories.Forallthreekindsof strengbh,thelinesdefiningthetheoriesarenotdefinitelyfixedsincethetesttigofmorespectiensforuniaxialstressesmayhaveshiftedthelocationofthelinesdefiningthetheories.

3. Althoughthetestresultstidicatea decreaseinductilitywithbiaxialtensioncomparedwithuniaxialtension,theductilityvaluesmayhavebeeninfluencedby theanisotropyofthematerial.

4.QForconstantprincipalstressratios,theoctahedraldefo~tiontheorygivesa goodengineertigapproximationfordefiningtheplasticbiaxialstress-strainrelations.

5.Fortheparticularloadpathandprincipalstressesusedthevariable-stresstestresultsshowthatboththedeformationandflowtheoriesgivea goodapproximationto theactualstress-strainrelations. “

6. Forlargeplasticstrains,theassmnptionof isotropicyieldingmadeintheplasticitytheoriesish generalagreementwiththetestresults.

7. Forthetestsof constantprincipalstressratioitwasshownthattheprincipalaxesof stressandstraincoticidewithintheMnitsofpossibleexperimentalerror.Thisconclusionindicatesthatanyinitialanisotropyofthematerialdoesnotinfluencethetheoreticalvaluesas givenbythestipledeformationor flowtheories.

.ThePennsylvaniaStateCollege

StateCollege,Pa.,May27” 1950

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14 NACATN2k25

APPENDIXA

D~ON OFTHEORETICALSTRESS-STRAINRELATIONS

BYDEFORMATIONANDFLOWTHEORIES

Inthetiterpretationoftestresultsonplasticcombinedstress-strainrelations,thedeformation-andflow-typetheoriesareusuallybasedon distortion-ener~or octahedral-shear-stresscriterionsofflow. Thedeterdnationofthestress-stratirelationbasedon theuniaxialsimple-tensionstress-strainrelationforboththeorieswillbe outlinedinthefollowingsections.

Stress-StrainRelationsby theDeformationTheory

Onthebasisoftheassmnptionsthatthesmnoftheprincipalplasticstrainsis zeroandthattheratiosoftheprficip~sh~~stressesandstrainsareproportional,itcanbe shown(reference1)thattheprincipalplasticstrainsintermsoftheprincipalstressesare

~,= (:)~1 -:!2 + “3]]

(Al)

b equations(Al)ju and c arethetruestressandplasticstrainforstipletension.

Squaringbothsidesofequations” andaddingtheresultingequationsyield

— —- .—

(A2)

NACATN 2425

where

“

(A3) “

and

;=&\ (3-42+ ~,- .3)2+(U3- UJ’ (A4)

and~and~ arecalJedthesignificantoreffectivestressandstrain.

It isnowpossiblebymeansof equations(Al),(A2),(A3),and(A4)andthesimple-tensionstress-straincurvetoobtaintheprticipalplasticstrains.Thatis:

the

theof

(1)Forgivenvaluesoftheprincipalstressesu1, U2,and U3,valueofthesignificantstress~ canbe determinedlyequation(A4)

(2)Fromtheshple-tensionstress-plastic-strainrelationusingvalueof u = 6 obtainedinstep1,correspondingvaluesG = % are found

(3)stresses~2,and

(4)

With T and ~ knuwn,forgivenvaluesofequations(Al)canbe usedto determinethe

‘3Forothervaluesoftheprincipalstresses,

theprincipalplasticstrains~1,

theabovestepsmaybe repeated

To obtainthepredictedstress-straincurvesforeachofthe.principalstresses,itisfirstnecessaryto addtheplasticstrain

. valuestotheelastich orderto obtainthetotalstrains.Thatis,thetotalstrainsare

(A5)

.— ————— — -— -- — —— —

16 NACATN 2h25

By equations(A5) thetotalprincipalstrains~1’> ~2’>ad ~3’ cm

be determinedandthetheoreticalstress-strainrelationsbasedonthedeformationtheoryplotted.h figures6, 7, 16, and17 stress-strainrelationsbasedontheforegoingprocedureareshown.

Stress-StratiRelationsby theFlowTheory

Theflow-typetheoryforpredictingplasticstress-strainrelationsdiffersfrcmthedeformation-typetheoryby ass-g thattheincrementalchangestiprincipalshe= stressesareproportionalto theticrementalchangesh theprincipalshearstratis.Theproceduredevelopedinthefollowingdiscussionforthetubesubjectedto titernalpressureandaxialtensionisadaptedfromthegeneraltheorygivenby Shepherdinreference3.

.

Whenticrementsofprticipalshearstressandstrainareassumedtobeproportional,thenequations(A5)arereplacedby incrementsof

wheretheincrementsofplasticstratiare7

(A6)

/

(A7) ‘

Fromequations(A6) and(A7)thetotal-strainincrements,equaltothesumoftheelasticandplasticstrainincrements,become

.

.

.

\

NACATN 2h25 17

.

Thebeginningofplasticflowisdefinedby thedistortion-ener~theoryortheequivalentoctahedralshearstress.Thatis,if cryisthe

yieldstressinshpletensiontherelationbetweenthestresscomponentsforplasticflowis

‘Y2= ’12‘-U22+Cf2-fl~D2-02a3 -3

a3ul (A9)

It isthenassumedthatthefunctiongivenby equation(A9)whichdefinesbeginntigofplasticflowisa functiondefinm thesubsequentplasticflow.Thatis, 8B tiequtions(A7)and(A8)isassumedtotobe a functionF(G)b~ of ~ where ~ isdefinedby

(Ale)

Itwillbe assumedfurthermorethat~ (1)For 56<0,

5B=0 (All)

andtheincrementof strainiselastic.(2)For 55>0,

andtheticrementof strainiselasticandplastic.To determinetheprincipalstress-strainrel&ions,itisnecessaryto determinetheincrementof strainsfromequations(A8).To obtainthesestratiincrmentsthevaluesof 5B mustbe knownfora givensetof stresses.To determinebB equations(AIO)and(AH)wilJ.be used,togetherwiththestiple-tensionstress-straindiagram.Forsimpletension~byequations(A8),sinceal= a and U2= U3= O,

,

——— . —.—. __ -—.— -— .... . . —. —.-—.

.

18 Iw.fl TN 2k25

.

By equation(AIO)forsimpletensioncrl=a, a2= a3= O, and F= u,andequation(A13)canbewrittena6 .

Substitutingthevalueof bB fromequation(AK?)h equation(A14),

(=’-~+ 5F(5)5F

Fora ftiiteamountof strainingby sumingup

(A15)

thestratis,

(JU6)

Sincetheleft-handsideof equation(LL6)re~resentstheplasticstrainC,by equation(u6)

G =z5F(G)55 (A17)

Fromthetensiontestcurve,valuesof 6 = et - a/E canbe obtained

forgivenvaluesof a = 6. xSinceby equation(A17)e = a (G)55,

a graphcanbeplottedshowingtherelationbetween~~(~)~~ md ~.

Fromthisgraphandby graphicalintegrationvaluesof ~(~) canbeobtatied foreachvalueof =. Then,dividing~(6) valuesby thecorresponding3 values,the F(6) canbe determinedforeachz stress.It isthenpossibletoplota curveshowingtherelationbetwkenF(5) and F. WiththerelationbetweenF(=) and 6 knownfromthetensiontestresults,itisnowpossibletoobtainthetheo-reticalstress-stratirelationsforthetubesubjectedto internalpressureandaxialloading.To dothis,thefollowingstepsareinvolved: .

(1)Forvariousvaluesor U1 thevaluesofby equation(A1O)listedina tablecontainingthe

,

7 aredeterminedfollowingheadings:

.

— ~— ——— -—— —— -—- .—

MICATN 2423 19

(2)Fromthe F(3)- F curveobtatiedby usingthestress-straticurveforsimpletension,valuesof F(6) canbe foundforeachvalueandtheirmagnitudesplacedh thecorrectcolumnabove.

(3)me products

inthetable.

(4)Therelation

(5)Fm theplot

(F(B)al-&2)

~ arethencomputedand

77

listed

( )‘2. ~ isthenplotteda@nst b.F(6)al-z

obtainedinstep(4)thevalues

xF@(”l-%-2)canbe obtatiedsincethesevaluesaretheareasunderthecurvefortheparticularvalueof G. Thesevaluesaretheplasticstrainssinceby equations(A8)and(A12)thephsticstrains

q ,=>

561 -

— - 1

(6) Thenby addingtheplasticstrainsfromstep(5)totheelasticstrains,thetotalstrainsbecome

!I’’lmtis;by equations(M.8)thetheoreticalprincipalstress-stratirelationscanbe obtabed.Thecurvesdesignatedby theflowtheoryinfigures6,7, 16,and17wereplottedUstigequations(~8) ad theforegotigprocedure.

—— —— .- —

. —.—.

20 NACATN 2425

APPENDIXB ‘.

D~IOI? OFEQUIVALENTOFFSEH!STRAINFOR.

D~ION OFBIAXIALYIELDSTRESSES1

Formaterialswithstrain-hsrdening,,itiscomnonpracticetodeterminetheyieldstressby theuseoftheoffsetmethodas illustratedinfigure4 forshnpletension.Forstatesof combinedstressestheprocedureforthedeterminationofyieldstresseshasnotbeenstandard-izedandvariouEmethodshavebeenused. Themethoddevelopedinthefollowingd.iscuasionforthedeterminationofyieldstressappearstohe themostlogical.b thismethod,theyieldstressisbasedonanoffsetstrain- an equivalentoffsetstrati- a valuewhichtakesintoaccountthetifluenceof cmbinedstressesanda valuewhichisbasedontheoffsetstrainusedforshpletension.Thedeterminationofthisequivalentoffsetstratiisbasedonthedeformationtheory.

By thedeformationtheory,stnce u = 6, theprincipalstrainsgland ‘2 canbe obtainedfitermsoftheuniaxialstratie andtheprticipalstressesby substituttig= for a as givenby equation(A4)inequations(Al).Thatis,

where R istheprincipal

2~1-R+R’

(Bl)

stressratio G2/ul.

stratie = ~, theequivalentoffsetprin-Foran offsetplasticcipalstratisG1 and 62 are,by eq~tions(Bl),

o 0 7(2- R)

’10= 2-’0

G2 =

0 *\F%’”

(IQ)

.%his procedurewassuggestedbyMr.L.W.Hu,ResesrchAssistant,

ThePennsylvaniaStateCollege.

— — —— —— .

.

NACATN 2425 21

ForvariousvaluesoftheprincipalstressratioR = u2/al,

equations(B2)definetheequivalentoffsetstratisasusedinfigureskand5.

I

.

.

——.————. ——— — —.-

22 NACATN 2k25

1.Marin,Joseph,Faupel,J.H.,Dutton,V. L.,andBrossman,M. W.:BiaxialPlasticStress-StrainRelationsfor24&T AluminumAlloy.NACATN1536,1948.

2.Templiu,R.L.,andStumn,R. G.: SomeStress-StrainStudiesofMetals.Jour.Aero.Sci.,vol.7,no.5XMarch1940,pp.189-1980

3. shepherd,W.M.: PlasticStress-St~”inRelations.Proc.InstitutionMech.Eng.lvol.159,1948,pp.95-99,dis~sion,pp.99-I-14.(FomnerlyWarRnergencyIssueNo.39.)

.

.

—.—

NAcATN 2425 23

.

.

.

TABLE1

YIELDSTRESSESFORVARIOUSRATIOSOFBIAXIALSTRESSES

hngitudinalBtiid

Lateral

b

Stressratiosyield yield

stress stress,ratio,

s-teas,~ly

~ly@l % x=—

(psi) (psi) ‘Y

0 67.5 x 103” ----------- 0.9472.1 ----------- 1.OO76.6 ----------- 1.06

a72.o ------ ----- al. 00

0.5 73.5 39.7 x 103 1.0274.574.0

38.0 1004a38.8 al. 03

L o 72.5 70.5 1.0171.0 73.0 .98-74.0

?~*51003

%? al. 01

2.0

t

37.0 76.2 I 0.5138.0 76.0

● 53a37.5 a76.~ a.52

m ----------- 68.6 ------------ -—- m 5 ------ ----------- a70.1 -----

aAveragevalue.

--------------------

0.55.53

a.%

0.981.01

● 99a.99

L 061.06

aL 06

o.~● 99

a.97

-—. . —-— —— —-- —

24 NACATN 24a

TABLE2

NOMINALULTIMATESTRESSESFORV&UOUSBIAXIALSTRESSRATIOS

LongitudinalLateral ‘Biaxial Stressnominal nominalstress ultimate tit imate ratios

Spectienratio, stress, stress,

‘2/”1 - % ‘2U(psi) (psi) x = ‘Idauy = ‘~/”u

(k::h:ny 21 85.7X 103 0 1004 029 83.2 0 1.01 034 78.6 0 .95 0

a82*5 ao al.00 %

o..~ A2 90.0 45.0x 103 1.09 0.55A3 87.5 44.8 L 06 .5418 86.5 43.2 1005 .52

a88.o ‘%4.0 al. 07 a.53

1.0 10 88.0 < @.2 1.07 1.07I-1 77.4 77.5 .% ● 94E! 78.8 78.8 .96 .96

a8L4 a81.5 a.99 a.gg

2.0 37 40.4 80.4 0.49 0.9825 40.2 81.6 .49 ● 99

a40.3 a81.O a.49 a.98

3a o 72.5 0 0.88(Tr-mverse Al 73.1 0 .89tension) $ a72.8 a. a.~

aAveragemlue. -

—— — —.—— —

4 NACATN 2425

TABLE3

TRUE*FRACTURESTRESSESFORVARIOUSBIAXIALSTRESSRATIOS

Biaxialstressratio,

a2/ul

~ 0.5,,

1.0

2.0

(Lat~raltension)

212934

A2A318

10U.12

3725

38Al

LongitudinalIILateraltrue true Strem3fracture fracture ratiosstress,

a’ -

stress,al.r(psi) (psi)

97.0 x 103 0 1004 094.2 0 1.01 089.0 0

● 95 0a93.4 ao al.00 ao

25

.

95.8 48.1 X 103 1.03 0.5293.2 ;;.; 1.00 ●5192.1 .99 .49a93.7 a47:o al.01 a.51

95.1 93.0 L 02 1.0083.6 81.6 .90 .8885.3 83.1 .89a88.o a85.9 a:z a.92

41.2

I

80.6 0.44 0.8641.0 81.8 .44 .88a41.1 81.2 a.44 a.87

o 73.7 0 0,790 74.4 0 .81ao a74.o ao a.80

aAveragevalue. -

-.—. —---- .——-——— --—-— — —

26 N/WAm?2425

TABLE4

NOMINALANDTRUEDUCTllJITYVALUESFORVARIOUS.

BIAXIALSTRESSRATIOS

Biaxial Nominal Truestressratio, Specimen ductility ductility‘

/Crzal (h./in.) (h./in.)

21 I-2.1x 10-2 11.3x 10-2(Long~tudinal 29 13.5 u.6

tension) 34 13.0 12.2a12.8 au. o

0.5 A2 7.2A3 ;:2 - 6.718 6.0 5.8 c

a7.o a6.7

1.0 10 4.0 3*911 3.0w 3.5 ::;

a3.5 a3.5

2.0 37 2.0 1.925 2.0 1.9

a2.o al.9

38 2.5 2.1(Lat&l Al 1.5 1.3tension) a2.o al.7

.

.

aAveragevalue.

.

,

— —.—

I

.

I

I

I1

I

I

Loadingdirection

Longitudinal

T&anaverae

T- 5

mm sms—smAIN RIZATIONSFORDNIAXIALTEHSION5TS

-1-Nminal

specimenult hrmte

atretls

(psi)

21 85.7X 103

29I83.2

34I78.6

Ia&.538

I72.5

AlI73.1

a72. 8

Truefracture

BtreBB

(psi)

97.0x 103

94.2

89.0

‘993.4

73.7

74.4

‘a74.o

True Con6tantductIllty

constant

(h./in. ) (p:l)n

11.3 x 10-2 --------- — -.---

12.6 ----------- -----

L?.2 ---.—----- -----

*12.o $%09 x 1 aO.08

2.1 —- —--- --- -----

L 3 ---- —-- --- -----

*1. 7 %. 0!3 %.04

‘ aAverage Talue.

!I

~, ..-“ -~ n ‘–1

.,

(a)l%mrltview.

.

(b) Side view.

mgwe l.- Biaxl.al-6treEEtesting m&hine.

L ..—. . . . ___

(a) Clip gage attached to .9pecink3n. (b) Device for calibmtlmgclip gage.

NM&l

1 JE@re 2.- Photograph of clip gage.

1

I

I , Tubular specimen

&T1--

kI

1

—

—

—

—

—

—

SR-4 gages ~“ I \Phosphor bronze

7“SR-4 g

L

I

Figure 3.- DrawingOf dip I+=.

—

NACA‘IT?2425 31

.

9X1

B

7

6

.-mn

g“5a)$u)=.2 4

2.

3

2

I

o

kd in./in. Nominalstrain,iniin

Figure 4.- Longitudinalnominalstress-straindiagramsforconstantstressratios.

-—. ———.—..—— _— — __— — .

.

7

6

3

0

,u A

1 1 1 1 1 11 1

I 11t

Figure5.- Iateral nmlnal stress-straindlagrarnsfor cmmtant stremratios.

.1

I

Cn

. .

womo mfim True strain, bh.

Figure 6.- CcmparisOD of longitol true@33ticity theories for conatit stresepoint .

stress-8t* diagram withratios . R demotes mpt~

i

,

I

u0.005 True strain, inJimirdin

Figure 7.- Compa15Bon of lateral true stress-straindiagama withplasticity theorieB for constant stress ratios. R denotes rupturepoint.

w4=

. r

NACATN 2425 35

I

1.2

1.0

0’ .6.-

~

.2

0

6—-

● Test points ~—

o Averagetest pointsTheory I

Distortionenergy— — Maximumshear JJLL

o .2 .4 .6 .8 Lo 1.2~ly

Stress ratio, x=—Cy

Figure8.- Comparisqpofyield’strengthswiththeoriesoffailure.

.-.

—— —— -–———- —

i

I

9R

A

~

8r

/. - ~

A R

R

7 ,{ Y

(

s

2

, %/W 0.5 1.0 Lo 03

0 I

solo Significant .strarn, ire/in.in/in.

Figure 9.. Significant stress-drain relations for wiou.s condamtprincipal stress ratioa . R d.9110t.IsE ?mPture point. ~

I

I

.

I I I I I I I I I I I I c. ..,... .A:- 1 I I I I I I I I I I I I I I I I I I I I 1I I I I I I I I I IGltwwIullu

I I I I Illll>lf 11111111 II

91104 @-i’~~~ lm.LOI 1 1 1 1 1 1 1,

I I I I o! I I I I L

I I i

!-’/ I E s 4 8 6 70m5

0 An-

tnh Sigdflcmt strain, in./in.

Figure lo. - Compe.risenof significant wlmsss-strain relations withU.Oiaxialtme stress-strain value6 for constant stres6 mtios.

NACA!IT?242-5

.

1.2

Lo

.8

.6

.4

.2

00

●

�

. Test values 1—

o Averagetest valuesTheory r $“

—-— Maximumstress—II—Maximumshear

Im 1x

.2 .4 .6 .8 Lo 1.2

‘1uStress ratio, x= ~

Figure11.- Comparisonofnominalbiaxialultimatestresseswithvaluesfr~ msdnum-stresstheory.

.

— — ..—

NAC!ATN 2425 39

Ibab’

o-.-

1.2

Lo

.8

.6

.4

.2

0

—11 T●

0

— .——11—

I

“11

8

—110

.0

Averagetest valuesTheory

MaximumMaximum

stressshear +

●

~N?T

o .2 .4 .6 .8 Lo 1.2

%Stress ratio, x=~

Figure12.- Comparisonoftruebiaxialfracturestresseswithvaluesfrommaximuwstresstheory.

.— —.——

lox9

8

AT.—Q

6Lc-

f5+O-Jg4

e

3

2.006 .008 ,010 .015 ..020 ma .040 .050 .10 .20

True strain, in./in.

(a) Tubas tith u~u~ = 0.

Figure 13.. Tme stress-strain relations for”tension tests. R denotesrupture point.

I

I

I

41

9

7

6.-U)Q

4

3

2

+ -

- —

—

—

I

.010 .015

True strain, Min.

(p)robestith u2/ul= W.

Figure13.- Concluded.

.020 .030

..— -——— .——. —

u0.010in/in.

c.——

R

m3 ratio

onstmt I

Nominal strain, in/in.

Figure 14.- Nominallongitudinal stress-straindiagrams for variablestress ratios. R denotes rupture point.

I

8 ,4

=m

=&-R

u.wzuty,t

Ed/in,,

Ro-

?

Nornhol stro!n, in./in.

1 1 1 I

t

/

{4

/

Stress ratio

—— Constomt

Fiv Yj.- Ncminal lateral stress-strain relations for variableratios. R denotes rupture point.

Btresa

ii

&-

I ,4

R

J? ‘f

()

%’ ‘m ‘%Yauue,

h

c1?

I,

I

--l0.010InAn.

R OR

~ ‘-

/ y--- ---.$ 4---

f < F

I

c

%!Y

9 Stress ratio

-o- Variable

J-3 -C- Cmstont

Theory

I----- flow—-— Deformotkm

True strain; Win.

E

Es’

I

IHgure 16.- Ccmparison of true stress-straindiagrams with plasticity Q!theories for variable stress ratios. R denotee rupture point;* denotes point where unitial plastic straina equal.elastic gStraim .

.

alfin,

Figure 17. - !hue

True strain, in./in.

laterd stress-strain diagram forR d3L10tes zqture point.

variable stress ratios.

I

.,%

Figure l.a. - 61gnificant stress-atrd.ndiagrams for varialilestres6ratios. R denotes rupture point.

‘

I

I

I

I

10

9

e

7

I

.

4Xta

(

0

Figure 19. -Uniaxbl

3 4 5. 8 7 e * &

SignWicant strai~ (n./in .

Comparison of significant stress-strain relations withtrue stress-strain valuea for variable stress ratios.

o●

o0 0 6

0 0●

‘AEl

&o

---i-

0

6 0Orientation of strains

o 2 4

Later~l

Figure 2U.- Comparkon of

8

.

Specimen01● 2

10 14

nominal strain, 63, in,/in.

tiredioll Of IIOW St~ill with diTeCtiOll Of

ncminal stress -s. OA Is longitudinalaxle of specimen.

16X10-3