astmc1239-2013

1、Designation:C1239?13StardPracticefReptingUniaxialStrengthDataEstimatingWeibullDistributionParametersfAdvancedCeramics1ThisstardisissuedunderthefixeddesignationC1239thenumberimmediatelyfollowingthedesignationindicatesthey

2、earofiginaladoptioninthecaseofrevisiontheyearoflastrevision.Anumberinparenthesesindicatestheyearoflastreapproval.Asuperepsilon()indicatesaneditialchangesincethelastrevisionreapproval.1.Scope1.1Thispracticecoverstheevalua

3、tionreptingofuniaxialstrengthdatatheestimationofWeibullprobabilitydistributionparametersfadvancedceramicsthatfailinabrittlefashion(seeFig.1).TheestimatedWeibulldistributionparametersareusedfstatisticalcomparisonoftherela

4、tivequalityoftwometestdatasetsfthepredictionoftheprobabilityoffailure(alternativelythefracturestrength)fastructureofinterest.Inadditionthispracticeencouragestheintegrationofmechanicalpropertydatafractographicanalysis.1.2

5、Thefailurestrengthofadvancedceramicsistreatedasacontinuousromvariabledeterminedbytheflawpopulation.Typicallyanumberoftestspecimenswithwelldefinedgeometryarefailedunderisothermalwelldefineddisplacementfceapplicationcondit

6、ions.Thefceatwhicheachtestspecimenfailsisrecded.TheresultingfailurestressdataareusedtoobtainWeibullparameterestimatesassociatedwiththeunderlyingflawpopulationdistribution.1.3Thispracticeisrestrictedtotheassumptionthatthe

7、distributionunderlyingthefailurestrengthsisthetwoparameterWeibulldistributionwithsizescaling.Furthermethispracticeisrestrictedtotestspecimens(tensileflexuralpressurizedringetc.)thatareprimarilysubjectedtouniaxialstressst

8、ates.Thepracticealsoassumesthattheflawpopulationisstablewithtimethatnoslowcrackgrowthisoccurring.1.4ThepracticeoutlinesmethodstocrectfbiaserrsintheestimatedWeibullparameterstocalculateconfidenceboundsonthoseestimatesfrom

9、datasetswhereallfailuresiginatefromasingleflawpopulation(thatisasinglefailuremode).Insampleswherefailuresiginatefrommultipleindependentflawpopulations(fexamplecompetingfailuremodes)themethodsoutlinedinSection9fbiascrecti

10、onconfidenceboundsarenotapplicable.1.5Thispracticeincludesthefollowing:SectionScope1ReferencedDocuments2Terminology3SummaryofPractice4SignificanceUse5Interferences6OutlyingObservations7MaximumLikelihoodParameterEstimatsf

11、CompetingFlawDistributions8UnbiasingFactsConfidenceBounds9Fractography10Examples11Keywds12ComputerAlgithmMAXLAppendixX1TestSpecimenswithUnidentifiedFractureiginsAppendixX21.6ThevaluesstatedinSIunitsaretoberegardedasthest

12、ardperIEEEASTMSI10.2.ReferencedDocuments2.1ASTMStards:2C1145TerminologyofAdvancedCeramicsC1322PracticefFractographyacterizationofFractureiginsinAdvancedCeramicsE6TerminologyRelatingtoMethodsofMechanicalTestingE178Practic

13、efDealingWithOutlyingObservationsE456TerminologyRelatingtoQualityStatisticsIEEEASTMSI10AmericanNationalStardfUseoftheInternationalSystemofUnits(SI):TheModernMetricSystem3.Terminology3.1Properuseofthefollowingtermsequatio

14、nswillalleviatemisunderstinginthepresentationofdatainthecalculationofstrengthdistributionparameters.3.1.1censedstrengthdata—strengthmeasurements(thatisasample)containingsuspendedobservationssuchasthatproducedbymultipleco

15、mpetingconcurrentflawpopulations.1ThispracticeisunderthejurisdictionofASTMCommitteeC28onAdvancedCeramicsisthedirectresponsibilityofSubcommitteeC28.01onMechanicalPropertiesPerfmance.CurrenteditionapprovedAug.12013.Publish

16、edSeptember2013.iginallyapprovedin1993.Lastpreviouseditionapprovedin2007asC1239–07.DOI:10.1520C123913.2FreferencedASTMstardsvisittheASTMwebsitewww.astm.gcontactASTMCustomerServiceat.FAnnualBookofASTMStardsvolumeinfmation

17、refertothestard’sDocumentSummarypageontheASTMwebsite.Copyright?ASTMInternational100BarrHarbDrivePOBoxC700WestConshohockenPA194282959.UnitedStates1CopyrightASTMInternationalProvidedbyIHSunderlicensewithASTMNotfResaleNorep

18、roductionwkingpermittedwithoutlicensefromIHS``````````````3.2.13probabilitydensityfunction—functionf(x)isaprobabilitydensityfunctionfthecontinuousromvariableXif:f~x!$0(1)2``f~x!dx51(2)TheprobabilitythattheromvariableXass

19、umesavaluebetweenabisgivenbythefollowingequation:Pr~aXb!5abf~x!dx(3)3.2.14sample—collectionofmeasurementsobservationstakenfromaspecifiedpopulation.3.2.15skewness—termrelatingtotheasymmetryofaprobabilitydensityfunction.Th

20、edistributionoffailurestrengthfadvancedceramicsisnotsymmetricwithrespecttothemaximumvalueofthedistributionfunctionbuthasonetaillongerthantheother.3.2.16statisticalbias—inherenttomostestimatesthisisatypeofconsistentnumeri

21、caloffsetinanestimaterelativetothetrueunderlyingvalue.Themagnitudeofthebiaserrtypicallydecreasesasthesamplesizeincreases.3.2.17unbiasedestimat—estimatthathasbeencrectedfstatisticalbiaserr.3.2.18Weibulldistribution—contin

22、uousromvariableXhasatwoparameterWeibulldistributioniftheprobabilitydensityfunctionisgivenbythefollowingequations:f~x!5SmβDSxβDm21expF2SxβDmGx.0(4)f~x!50x#0(5)thecumulativedistributionfunctionisgivenbythefollowingequation

23、s:F~x!512expF2SxβDmGx.0(6)F~x!50x#0(7)wherem=Weibullmodulus(theshapeparameter)(0)β=scaleparameter(0).3.2.19Theromvariablerepresentinguniaxialtensilestrengthofanadvancedceramicwillassumeonlypositivevaluesthedistributionis

24、asymmetricalaboutthemean.Theseacteristicsruleouttheuseofthenmaldistribution(aswellasothers)pointtotheuseoftheWeibullsimilarskeweddistributions.Iftheromvariablerepresentinguniaxialtensilestrengthofanadvancedceramicisacter

25、izedbyEq47thentheprobabilitythatthisadvancedceramicwillfailunderanapplieduniaxialtensilestressσisgivenbythecumulativedistributionfunctionasfollows:Pf512expF2SσσθDmGσ.0(8)Pf50σ#0(9)where:Pf=probabilityoffailureσθ=Weibulla

26、cteristicstrength.NotethattheWeibullacteristicstrengthisdependentontheuniaxialtestspecimen(tensileflexuralpressurizedring)willchangewithtestspecimensizegeometry.InadditiontheWeibullacteristicstrengthhasunitsofstressshoul

27、dbereptedusingunitsofmegapalsgigapascals.3.2.20Analternativeexpressionftheprobabilityoffailureisgivenbythefollowingequation:Pf512expF2vSσσ0DmdVGσ.0(10)Pf50σ#0(11)Theintegrationintheexponentialisperfmedoveralltensileregio

28、nsofthetestspecimenvolumeifthestrengthcontrollingflawsareromlydistributedthroughthevolumeofthematerialoveralltensileregionsofthetestspecimenareaifflawsarerestrictedtothetestspecimensurface.Theintegrationissometimescarrie

29、doutoveraneffectivegagesectioninsteadofoverthetotalvolumearea.InEq10σ0istheWeibullmaterialscaleparameter.TheparameterisamaterialpropertyifthetwoparameterWeibulldistributionproperlydescribesthestrengthbehaviofthematerial.

30、InadditiontheWeibullmaterialscaleparametercanbedescribedastheWeibullacteristicstrengthofatestspecimenwithunitvolumeareafcedinunifmuniaxialtension.TheWeibullmaterialscaleparameterhasunitsofstress(volume)1mshouldbereptedus

31、ingunitsofMPa(m)3mGPa(m)3mifthestrengthcontrollingflawsaredistributedthroughthevolumeofthematerial.IfthestrengthcontrollingflawsarerestrictedtothesurfaceofthetestspecimensinasamplethentheWeibullmaterialscaleparametershou

32、ldbereptedusingunitsofMPa(m)2mGPa(m)2m.FagiventestspecimengeometryEq8Eq10canbeequatedwhichyieldsanexpressionrelatingσ0σθ.Furtherdiscussionrelatedtothisissuecanbefoundin7.6.3.3Fdefinitionsofotherstatisticaltermstermsrelat

33、edtomechanicaltestingtermsrelatedtoadvancedceramicsusedinthispracticerefertoTerminologiesE456C1145E6toappropriatetextbooksonstatistics(25).3.4Symbols:A=testspecimenarea(areaofeffectivegagesectionifused).b=gagesectiondime

34、nsionbaseofbendtestspecimen.d=gagesectiondimensiondepthofbendtestspecimen.F(x)=cumulativedistributionfunction.f(x)=probabilitydensityfunction.Li=lengthoftheinnerspanfabendtestspecimen.Lo=lengthoftheouterspanfabendtestspe

35、cimen.=likelihoodfunction.m=Weibullmodulus.m?=estimateoftheWeibullmodulus.m?U=unbiasedestimateoftheWeibullmodulus.N=numberoftestspecimensinasample.Pf=probabilityoffailure.C1239?133CopyrightASTMInternationalProvidedbyIHSu