1、Designation: D7846 12 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. Scope1.1 This practice covers the reporting of uniaxial strengthd
3、ata 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 Metho
4、ds 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 characterized
5、 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 fa
7、ilure.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 C
8、arbon andGraphiteC709 Terminology Relating to Manufactured 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 Pet
9、roleum, PetroleumProducts, and LubricantsD7775 Guide for 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 followi
10、ng terms and equations willalleviate misunderstanding in 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 esti
11、mate of a distribution parameter (a pointestimate) associated 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
12、 teststandards) about which inferences are made.3.2.3 population 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 ra
13、ndom variableX if:f(x) $0 (1)and*2f(x) dx 5 1 (2)The probability 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 and Lubricants and is the direct responsibility of Subcommittee D02.F0 onManufact
14、ured Carbon and Graphite Products.Current edition approved Dec. 1, 2012. Published August 2013. DOI: 10.1520/D7846-12.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer t
15、o the standards Document Summary page onthe ASTM website.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1Pr(a,X,b) 5 *abf(x) dx (3)3.2.5 sample, na collection of measurements or observa-tions taken from a specified population.3.2.6 sk
16、ewness, 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 distribution function; one tail is longerthan the other.3.2.7 statistical bias, ninherent to most estimates, this is a
17、type 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, nan estimator that has beencorrected for statistical bias error.3.2.9 Weibull distribution, nthe continuo
18、us random vari-able X has a two-parameter Weibull distribution if the prob-ability density function is given by:f(x) 5SmDSxDm21expF2SxDmGx.0 (4)f(x) 5 0 x #0 (5)and the cumulative distribution function is given by:F(x) 5 1 2 expF2SxDmGx.0 (6)orF(x) 5 0 x #0 (7)where:m = Weibull modulus (or the shape
19、 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 asymmetrical about the popula-tion mean. These characteristics rule out the use of the normaldistribution (a
20、s 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, and Eq 7, then the probability thatthe tested graphite will fail under an applied uniaxial tensilestre
21、ss, , 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 DiscussionThe Weibull characteristic strength de-pends on the uniaxial test specimen (tensile, compression andfl
22、exural) 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 statistical terms, terms related tomechanical testing, and terms related to graphite used in thispractice, r
23、efer to Terminologies C709, D4175, E6, and E456,orto appropriate textbooks on statistics (1-5).33.4 Nomenclature:F(x) = cumulative distribution functionf(x) = probability density function+ = likelihood functionm = Weibull modulusm = estimate of the Weibull modulusmU= unbiased estimate of the Weibull
24、 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 scale parameter = estimate of mean strength = uniaxial tensile stressi= maximum stress in the It
25、h 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 estimate Weibulldistribution parameters from failure data for graphite datatested in accordance
26、 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. If thesample of failure strength data is uncensored, compute anunbiased estimate of the Weibull
27、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 testreport.5. Significance and Use5.1 Two- and three-parameter formulations exist for theWeibull
28、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 functions that incorporate thefailure data. These functions are referred to as estimators. It isd
29、esirable 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, including moment estimators, least-squaresestimators, and maximum likelihood estimators. This practiced
30、etails 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 geometry. Tensile and3The boldface numbers in parentheses refer to the list of references at the
31、 end ofthis standard.D7846 122flexural 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 specimengeometry, Weibull distribution parameter estimates (m , )shall be computed for a given specim
32、en 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 decreases significantly as the number of testspecimens increases. However, a point of diminishi
33、ng 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 level of confidence associated with a parameterestimate. The number of specimens needed depends
34、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 estimate) are presented in 8.3 and 8.4.6. Outlying Observations6.1 Before computing the parameter esti
35、mates, 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 thecomputation of the parameter estimates. Given the experimen-talist has followed the prescribed experiment
36、al 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 onethat deviates significantly from other observations in thesample and is an extreme manifestation o
37、f 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 theresult of a known gross deviation from the prescribed experi-mental procedure, or a known err
38、or 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 is deemedunsuitable either for testing, or fails before the prescribedexperimental procedure ha
39、s 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 Practice E178.7. Maximum Likelihood Parameter Estimators7.1 The likelihood function for the two-par
40、ameter Weibulldistribution of a censored sample is defined by the expression(8):+ 5Hi51rSmD SiDm 21expF2SiDmGJj5r11NexpF2SjDmG(10)7.1.1 For graphite material, this expression is applied to asample where outlying observations are identified under theconditions given in Section 6. When Eq 10 is used t
41、o 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 theassociated index in the first product. In this practice, the secondproduct is carried out for th
42、e 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 ithFIG. 1 Failure Strengths for Tensile Test Specimens (left) and Flexural Test Specimens (rig
43、ht) for a Nearly Isotropic Graphite (7)D7846 123test specimen at failure. The parameter estimates (the Weibullmodulus m and the characteristic strength ) are determinedby taking the partial derivatives of the logarithm of thelikelihood function with respect to m and then equating theresulting expres
44、sions 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 taken equal to N in Eq 10.7.2 The system of equations obtained by differentiating thelog likeli
45、hood function for a censored sample is given by (9):i51N(i)mln(i)i51N(i)m21ri51rln(i) 21m5 0 (12)and5FSi51N(i)mD1rG1m(13)where:r = the total number of observations (N) minus the numberof outlying observations in a censored sample.7.3 For a censored sample Eq 12 is solved first for m .Subsequently, i
46、s 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 parameterestimates (the Weibull modulus m and the character
47、isticstrength ) 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):i51N(i)mln(i)i51N(i)m21Ni51Nln(i) 21m5 0 (14)and5FSi51N(i)m
48、D1NG1m(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 dat
49、a. 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 lo
