1、Designation: D7846 16 An American National StandardStandard Practice forReporting Uniaxial Strength Data and Estimating WeibullDistribution Parameters for Advanced Graphites1This standard is issued under the fixed designation D7846; the number immediately following the designation indicates the year
2、 oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope*1.1 This practice covers the reporting of uniaxial strength
3、data for graphite and the estimation of probability distributionparameters for both censored and uncensored data. The failurestrength of graphite materials is treated as a continuous randomvariable. Typically, a number of test specimens are failed inaccordance with the following standards: Test Meth
4、ods C565,C651, C695, C749, Practice C781 or Guide D7775. The load atwhich each specimen fails is recorded. The resulting failurestresses are used to obtain parameter estimates associated withthe underlying population distribution. This practice is limitedto failure strengths that can be characterize
5、d by the two-parameter Weibull distribution. Furthermore, this practice isrestricted to test specimens (primarily tensile and flexural) thatare primarily subjected to uniaxial stress states.1.2 Measurements of the strength at failure are taken forvarious reasons: a comparison of the relative quality
6、 of twomaterials, the prediction of the probability of failure for astructure of interest, or to establish limit loads in an applica-tion. This practice provides a procedure for estimating thedistribution parameters that are needed for estimating loadlimits for a particular level of probability of f
7、ailure.2. Referenced Documents2.1 ASTM Standards:2C565 Test Methods for Tension Testing of Carbon andGraphite Mechanical MaterialsC651 Test Method for Flexural Strength of ManufacturedCarbon and GraphiteArticles Using Four-Point Loading atRoom TemperatureC695 Test Method for Compressive Strength of
8、Carbon andGraphiteC749 Test Method for Tensile Stress-Strain of Carbon andGraphiteC781 Practice for Testing Graphite and Boronated GraphiteMaterials for High-Temperature Gas-Cooled Nuclear Re-actor ComponentsD4175 Terminology Relating to Petroleum Products, LiquidFuels, and LubricantsD7775 Guide for
9、 Measurements on Small Graphite Speci-mensE6 Terminology Relating to Methods of Mechanical TestingE178 Practice for Dealing With Outlying ObservationsE456 Terminology Relating to Quality and Statistics3. Terminology3.1 Proper use of the following terms and equations willalleviate misunderstanding in
10、 the presentation of data and inthe calculation of strength distribution parameters.3.2 Definitions:3.2.1 estimator, na well-defined function that is dependenton the observations in a sample. The resulting value for a givensample may be an estimate of a distribution parameter (a pointestimate) assoc
11、iated with the underlying population. The arith-metic average of a sample is, for example, an estimator of thedistribution mean.3.2.2 population, nthe totality of valid observations (per-formed in a manner that is compliant with the appropriate teststandards) about which inferences are made.3.2.3 po
12、pulation mean, nthe average of all potentialmeasurements in a given population weighted by their relativefrequencies in the population.3.2.4 probability density function, nthe function f(x) is aprobability density function for the continuous random variableX if:fx! $0 (1)and*2fx! dx 5 1 (2)The proba
13、bility that the random variable X assumes avalue between a and b is given by:1This practice is under the jurisdiction of ASTM Committee D02 on PetroleumProducts, Liquid Fuels, and Lubricants and is the direct responsibility of Subcom-mittee D02.F0 on Manufactured Carbon and Graphite Products.Current
14、 edition approved Jan. 1, 2016. Published February 2016. Originallyapproved in 2012. Last previous edition approved in 2012 as D7846 12. DOI:10.1520/D7846-16.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of A
15、STMStandards volume information, refer to the standards Document Summary page onthe ASTM website.*A Summary of Changes section appears at the end of this standardCopyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1Pra , X , b! 5 *abfx! dx
16、 (3)3.2.5 sample, na collection of measurements or observa-tions taken from a specified population.3.2.6 skewness, na term relating to the asymmetry of aprobability density function. The distribution of failurestrength for graphite is not symmetric with respect to themaximum value of the distributio
17、n function; one tail is longerthan the other.3.2.7 statistical bias, ninherent to most estimates, this is atype of consistent numerical offset in an estimate relative to thetrue underlying value. The magnitude of the bias error typicallydecreases as the sample size increases.3.2.8 unbiased estimator
18、, nan estimator that has beencorrected for statistical bias error.3.2.9 Weibull distribution, nthe continuous random vari-able X has a two-parameter Weibull distribution if the prob-ability density function is given by:fx! 5SmDSxDm21expF2SxDmGx.0 (4)fx! 5 0 x #0 (5)and the cumulative distribution fu
19、nction is given by:Fx! 5 1 2 expF2SxDmGx.0 (6)orFx! 5 0 x #0 (7)where:m = Weibull modulus (or the shape parameter) ( 0), and = scale parameter ( 0).3.2.9.1 DiscussionThe random variable representing uni-axial tensile strength of graphite will assume only positivevalues, and the distribution is asymm
20、etrical about the popula-tion mean. These characteristics rule out the use of the normaldistribution (as well as others) and favor the use of the Weibulland similar skewed distributions. If the random variablerepresenting uniaxial tensile strength of a graphite is charac-terized by Eq 4, Eq 5, Eq 6,
21、 and Eq 7, then the probability thatthe tested graphite will fail under an applied uniaxial tensilestress, , is given by the cumulative distribution function:Pf5 1 2 expF2SDmGfor .0 (8)andPf5 0 for #0 (9)where:Pf= the probability of failure, and= the Weibull characteristic strength.3.2.9.2 Discussio
22、nThe Weibull characteristic strength de-pends on the uniaxial test specimen (tensile, compression andflexural) and may change with specimen geometry. In addition,the Weibull characteristic strength has units of stress andshould be reported using units of MPa or GPa.3.3 For definitions of other stati
23、stical terms, terms related tomechanical testing, and terms related to graphite used in thispractice, refer to Terminologies D4175, E6, and E456,ortoappropriate textbooks on statistics (1-5).33.4 Nomenclature:F(x) = cumulative distribution functionf(x) = probability density function+ = likelihood fu
24、nctionm = Weibull modulusm = estimate of the Weibull modulusmU= unbiased estimate of the Weibull modulusN = number of specimens in a samplePf= probability of failuret = intermediate quantity used in calculation of confi-dence boundsX = random variablex = realization of a random variable X = Weibull
25、scale parameter = estimate of mean strength = uniaxial tensile stressi= maximum stress in the Ith test specimen at failure= Weibull characteristic strength (associated with a testspecimen)= estimate of the Weibull characteristic strength4. Summary of Practice4.1 This practice provides a procedure to
26、 estimate Weibulldistribution parameters from failure data for graphite datatested in accordance with applicable ASTM test standards. Theprocedure consists of computing estimates of the biasedWeibull modulus and Weibull characteristic strength. Ifnecessary, compute an estimate of the mean strength.
27、If thesample of failure strength data is uncensored, compute anunbiased estimate of the Weibull modulus, and computeconfidence bounds for both the estimated Weibull modulus andthe estimated Weibull characteristic strength. Finally, prepare agraphical representation of the failure data along with a t
28、estreport.5. Significance and Use5.1 Two- and three-parameter formulations exist for theWeibull distribution. This practice is restricted to the two-parameter formulation.An objective of this practice is to obtainpoint estimates of the unknown Weibull distribution param-eters by using well-defined f
29、unctions that incorporate thefailure data. These functions are referred to as estimators. It isdesirable that an estimator be consistent and efficient. Inaddition, the estimator should produce unique, unbiased esti-mates of the distribution parameters (6). Different types ofestimators exist, includi
30、ng moment estimators, least-squaresestimators, and maximum likelihood estimators. This practicedetails the use of maximum likelihood estimators.5.2 Tensile and flexural specimens are the most commonlyused test configurations for graphite. The observed strengthvalues depend on specimen size and test
31、geometry. Tensile and3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.D7846 162flexural test specimen failure data for a nearly isotropicgraphite (7) is depicted in Fig. 1. Since the failure data for agraphite material can be dependent on the test speci
32、mengeometry, Weibull distribution parameter estimates (m , )shall be computed for a given specimen geometry.5.3 Many factors affect the estimates of the distributionparameters. The total number of test specimens plays asignificant role. Initially, the uncertainty associated with pa-rameter estimates
33、 decreases significantly as the number of testspecimens increases. However, a point of diminishing returnsis reached where the cost of performing additional strengthtests may not be justified. This suggests a limit to the numberof test specimens for determining Weibull parameters to obtaina desired
34、level of confidence associated with a parameterestimate. The number of specimens needed depends on theprecision required in the resulting parameter estimate or in theresulting confidence bounds. Details relating to the computa-tion of confidence bounds (directly related to the precision ofthe estima
35、te) are presented in 8.3 and 8.4.6. Outlying Observations6.1 Before computing the parameter estimates, the datashould be screened for outlying observations (outliers). Pro-vided the experimentalist has followed the prescribed experi-mental procedure, all test results must be included in thecomputati
36、on of the parameter estimates. Given the experimen-talist has followed the prescribed experimental procedure, thedata may include apparent outliers. However, apparent outliersmust be retained and treated as any other observation in thefailure sample. In this context, an outlying observation is oneth
37、at deviates significantly from other observations in thesample and is an extreme manifestation of the variability of thestrength due to non-homogeneity of graphite material, or largedisparate flaws, given the prescribed experimental procedurehas been followed. Only where the outlying observation is
38、theresult of a known gross deviation from the prescribed experi-mental procedure, or a known error in calculating or recordingthe numerical value of the data point in question, may theoutlying observation be censored. In such a case, the test reportshould record the justification. If a test specimen
39、 is deemedunsuitable either for testing, or fails before the prescribedexperimental procedure has commenced, then this should notbe regarded as a test result. However, the null test should befully documented in the test report. The procedures for dealingwith outlying observations are detailed in Pra
40、ctice E178.7. Maximum Likelihood Parameter Estimators7.1 The likelihood function for the two-parameter Weibulldistribution of a censored sample is defined by the expression(8):+ 5Hi51rSmDSiDm 21expF2SiDmGJj5r11NexpF2SjDmG(10)7.1.1 For graphite material, this expression is applied to asample where ou
41、tlying observations are identified under theconditions given in Section 6. When Eq 10 is used to estimatethe parameters associated with a strength distribution contain-ing outliers, then r is the number of data points retained in thesample, that is, data points not considered outliers, and i is thea
42、ssociated index in the first product. In this practice, the secondproduct is carried out for the outlying observations. Thereforethe second product is carried out from (j = r +1)toN (the totalnumber of specimens) where j is the index in the secondsummation. Accordingly, iis the maximum stress in the
43、 ithtest specimen at failure. The parameter estimates (the Weibullmodulus m and the characteristic strength ) are determinedFIG. 1 Failure Strengths for Tensile Test Specimens (left) and Flexural Test Specimens (right) for a Nearly Isotropic Graphite (7)D7846 163by taking the partial derivatives of
44、the logarithm of thelikelihood function with respect to m and then equating theresulting expressions to zero. Finally, the likelihood functionfor the two-parameter Weibull distribution for a sample free ofoutlying observations is defined by the expression:+ 5 i51NSmDSiDm 21expF2SiDmG(11)where r was
45、taken equal to N in Eq 10.7.2 The system of equations obtained by differentiating thelog likelihood function for a censored sample is given by (9):and5FSi51Ni!mD1rG1m(13)where:r = the total number of observations (N) minus the numberof outlying observations in a censored sample.7.3 For a censored sa
46、mple Eq 12 is solved first for m .Subsequently, is computed from Eq 13. Obtaining a closedform solution of Eq 12 for m is not possible. This expressionmust be solved numerically.7.4 When a sample does not require censoring Eq 11 is usedfor the likelihood function. For uncensored data, the parametere
47、stimates (the Weibull modulus m and the characteristicstrength ) are determined by taking the partial derivatives ofthe logarithm of the likelihood function given by Eq 11 withrespect to m and then equating the resulting expressions tozero. The system of equations obtained is given by (9):and5FSi51N
48、i!mD1NG1m(15)For an uncensored sample Eq 14 is solved first for m .Subsequently is computed from Eq 15. Obtaining a closedform solution of Eq 14 for m is not possible. This expressionmust be solved numerically.7.5 An objective of this practice is the consistent statisticalrepresentation of strength
49、data. To this end, the followingprocedure is the recommended graphical representation ofstrength data. Begin by ranking the strength data obtained fromlaboratory testing in ascending order, and assign to each aranked probability of failure Pfaccording to the estimator:Pfi! 5i 2 0.5N(16)where:N = number of specimens, andi = the ith datum.Compute the natural logarithm of the ith failure stress, andthe natural logarithm of the natural logarit
