Designation: C1239 − 13Standard Practice forReporting Uniaxial Strength Data and Estimating WeibullDistribution Parameters for Advanced Ceramics1This standard is issued under the fixed designation C1239; the number immediately following the designation indicates the year 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 evaluation and reporting ofuniaxial strength data and the estimation of Weibull probabilitydistribution parameters for advanced ceramics that fail in abrittle fashion (see Fig. 1). The estimated Weibull distributionparameters are used for statistical comparison of the relativequality of two or more test data sets and for the prediction ofthe probability of failure (or, alternatively, the fracturestrength) for a structure of interest. In addition, this practiceencourages the integration of mechanical property data andfractographic analysis.1.2 The failure strength of advanced ceramics is treated as acontinuous random variable determined by the flaw population.Typically, a number of test specimens with well-definedgeometry are failed under isothermal, well-defined displace-ment and/or force-application conditions. The force at whicheach test specimen fails is recorded. The resulting failure stressdata are used to obtain Weibull parameter estimates associatedwith the underlying flaw population distribution.1.3 This practice is restricted to the assumption that thedistribution underlying the failure strengths is the two-parameter Weibull distribution with size scaling. Furthermore,this practice is restricted to test specimens (tensile, flexural,pressurized ring, etc.) that are primarily subjected to uniaxialstress states. The practice also assumes that the flaw populationis stable with time and that no slow crack growth is occurring.1.4 The practice outlines methods to correct for bias errorsin the estimated Weibull parameters and to calculate confi-dence bounds on those estimates from data sets where allfailures originate from a single flaw population (that is, a singlefailure mode). In samples where failures originate from mul-tiple independent flaw populations (for example, competingfailure modes), the methods outlined in Section 9 for biascorrection and confidence bounds are not applicable.1.5 This practice includes the following:SectionScope 1Referenced Documents 2Terminology 3Summary of Practice 4Significance and Use 5Interferences 6Outlying Observations 7Maximum Likelihood Parameter Estimators forCompeting Flaw Distributions8Unbiasing Factors and Confidence Bounds 9Fractography 10Examples 11Keywords 12Computer Algorithm MAXL AppendixX1Test Specimens with Unidentified FractureOriginsAppendixX21.6 The values stated in SI units are to be regarded as thestandard per IEEE/ASTM SI 10.2. Referenced Documents2.1 ASTM Standards:2C1145 Terminology of Advanced CeramicsC1322 Practice for Fractography and Characterization ofFracture Origins in Advanced CeramicsE6 Terminology Relating to Methods of Mechanical TestingE178 Practice for Dealing With Outlying ObservationsE456 Terminology Relating to Quality and StatisticsIEEE/ASTM SI 10 American National Standard for Use ofthe International System of Units (SI): The Modern MetricSystem3. Terminology3.1 Proper use of the following terms and equations willalleviate misunderstanding in the presentation of data and inthe calculation of strength distribution parameters.3.1.1 censored strength data—strength measurements (thatis, a sample) containing suspended observations such as thatproduced by multiple competing or concurrent flaw popula-tions.1This practice is under the jurisdiction of ASTM Committee C28 on AdvancedCeramics and is the direct responsibility of Subcommittee C28.01 on MechanicalProperties and Performance.Current edition approved Aug. 1, 2013. Published September 2013. Originallyapproved in 1993. Last previous edition approved in 2007 as C1239 – 07. DOI:10.1520/C1239-13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.1.1 Consider a sample where fractography clearly estab-lished the existence of three concurrent flaw distributions(although this discussion is applicable to a sample with anynumber of concurrent flaw distributions). The three concurrentflaw distributions are referred to here as distributions A, B, andC. Based on fractographic analyses, each test specimenstrength is assigned to a flaw distribution that initiated failure.In estimating parameters that characterize the strength distri-bution associated with flaw distribution A, all test specimens(and not just those that failed from Type A flaws) must beincorporated in the analysis to ensure efficiency and accuracyof the resulting parameter estimates. The strength of a testspecimen that failed by a Type B (or Type C) flaw is treated asa right censored observation relative to the A flaw distribution.Failure due to a Type B (or Type C) flaw restricts, or censors,the information concerning Type A flaws in a test specimen bysuspending the test before failure occurred by a Type A flaw(1).3The strength from the most severe Type A flaw in thosetest specimens that failed from Type B (or Type C) flaws ishigher than (and thus to the right of) the observed strength.However, no information is provided regarding the magnitudeof that difference. Censored data analysis techniques incorpo-rated in this practice utilize this incomplete information toprovide efficient and relatively unbiased estimates of thedistribution parameters.3.2 Definitions:3.2.1 competing failure modes—distinguishably differenttypes of fracture initiation events that result from concurrent(competing) flaw distributions.3.2.2 compound flaw distributions—any form of multipleflaw distribution that is neither pure concurrent nor pureexclusive. A simple example is where every test specimencontains the flaw distribution A, while some fraction of the testspecimens also contains a second independent flaw distributionB.3.2.3 concurrent flaw distributions—type of multiple flawdistribution in a homogeneous material where every testspecimen of that material contains representative flaws fromeach independent flaw population. Within a given testspecimen, all flaw populations are then present concurrentlyand are competing with each other to cause failure. This termis synonymous with “competing flaw distributions.”3.2.4 effective gage section—that portion of the test speci-men geometry that has been included within the limits ofintegration (volume, area, or edge length) of the Weibulldistribution function. In tensile test specimens, the integrationmay be restricted to the uniformly stressed central gagesection, or it may be extended to include transition and shankregions.3.2.5 estimator—well-defined function that is dependent onthe observations in a sample. The resulting value for a givensample may be an estimate 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.3.2.6 exclusive flaw distributions—type of multiple flawdistribution created by mixing and randomizing test specimensfrom two or more versions of a material where each versioncontains a different single flaw population. Thus, each testspecimen contains flaws exclusively from a single distribution,but the total data set reflects more than one type of strength-controlling flaw. This term is synonymous with “mixtures offlaw distributions.”3.2.7 extraneous flaws—strength-controlling flaws observedin some fraction of test specimens that cannot be present in thecomponent being designed. An example is machining flaws inground bend test specimens that will not be present inas-sintered components of the same material.3.2.8 fractography—analysis and characterization of pat-terns generated on the fracture surface of a test specimen.Fractography can be used to determine the nature and locationof the critical fracture origin causing catastrophic failure in anadvanced ceramic test specimen or component.3.2.9 fracture origin—the source from which brittle fracturecommences (Terminology C1145).3.2.10 multiple flaw distributions—strength controllingflaws observed by fractography where distinguishably differentflaw types are identified as the failure initiation site withindifferent test specimens of the data set and where the flaw typesare known or expected to originate from independent causes.3.2.10.1 Discussion—An example of multiple flaw distribu-tions would be carbon inclusions and large voids which mayboth have been observed as strength controlling flaws within adata set and where there is no reason to believe that thefrequency or distribution of carbon inclusions created duringfabrication was in any way dependent on the frequency ordistribution of voids (or vice-versa).3.2.11 population—totality of potential observations aboutwhich inferences are made.3.2.12 population mean—average of all potential measure-ments in a given population weighted by their relative frequen-cies in the population.3The boldface numbers in parentheses refer to the list of references at the end ofthis practice.FIG. 1 Example of Weibull Plot of Strength DataC1239 − 1323.2.13 probability density function—function f(x) is a prob-ability density function for the continuous random variable Xif:f~x! $0 (1)and*2``f~ x! dx 5 1 (2)The probability that the random variable X assumes avalue between a and b is given by the following equation:Pr~a,X,b! 5 *abf~x! dx (3)3.2.14 sample—collection of measurements or observationstaken from a specified population.3.2.15 skewness—term relating to the asymmetry of a prob-ability density function. The distribution of failure strength foradvanced ceramics is not symmetric with respect to themaximum value of the distribution function but has one taillonger than the other.3.2.16 statistical bias—inherent 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.3.2.17 unbiased estimator—estimator that has been cor-rected for statistical bias error.3.2.18 Weibull distribution—continuous random variable Xhas a two-parameter Weibull distribution if the probabilitydensity function is given by the following equations:f~x! 5SmβDSxβDm21expF2SxβDmGx.0 (4)f~x! 5 0 x #0 (5)and the cumulative distribution function is given by thefollowing equations:F~x! 5 1 2 expF2SxβDmGx.0 (6)orF~x! 5 0 x #0 (7)where:m = Weibull modulus (or the shape parameter) (0), andβ = scale parameter (0).3.2.19 The random variable representing uniaxial tensilestrength of an advanced ceramic will assume only positivevalues, and the distribution is asymmetrical about the mean.These characteristics rule out the use of the normal distribution(as well as others) and point to the use of the Weibull andsimilar skewed distributions. If the random variable represent-ing uniaxial tensile strength of an advanced ceramic is char-acterized by Eq 4-7, then the probability that this advancedceramic will fail under an applied uniaxial tensile stress σ isgiven by the cumulative distribution function as follows:Pf5 1 2 expF2SσσθDmGσ.0 (8)Pf5 0 σ#0 (9)where:Pf= probability of failure, andσθ= Weibull characteristic strength.Note that the Weibull characteristic strength is dependent onthe uniaxial test specimen (tensile, flexural, or pressurized ring)and will change with test specimen size and geometry. Inaddition, the Weibull characteristic strength has units of stressand should be reported using units of megapascals or gigapas-cals.3.2.20 An alternative expression for the probability offailure is given by the following equation:Pf5 1 2 expF2*vSσσ0DmdVGσ.0 (10)Pf5 0 σ#0 (11)The integration in the exponential is performed over alltensile regions of the test specimen volume if the strength-controlling flaws are randomly distributed through the volumeof the material, or over all tensile regions of the test specimenarea if flaws are restricted to the test specimen surface. Theintegration is sometimes carried out over an effective gagesection instead of over the total volume or area. In Eq 10, σ0isthe Weibull material scale parameter. The parameter is amaterial property if the two-parameter Weibull distributionproperly describes the strength behavior of the material. Inaddition, the Weibull material scale parameter can be describedas the Weibull characteristic strength of a test specimen withunit volume or area forced in uniform uniaxial tension. TheWeibull material scale parameter has units of stress·(volume)1/mand should be reported using units of MPa·(m)3/morGPa·(m)3/mif the strength-controlling flaws are distributedthrough the volume of the material. If the strength-controllingflaws are restricted to the surface of the test specimens in asample, then the Weibull material scale parameter should bereported using units of MPa·(m)2/mor GPa·(m)2/m. For a giventest specimen geometry, Eq 8 and Eq 10 can be equated, whichyields an expression relating σ0and σθ. Further discussionrelated to this issue can be found in 7.6.3.3 For definitions of other statistical terms, terms related tomechanical testing, and terms related to advanced ceramicsused in this practice, refer to Terminologies E456, C1145, andE6 or to appropriate textbooks on statistics (2-5).3.4 Symbols:A = test specimen area (or area of effective gage section,if used).b = gage section dimension, base of bend test specimen.d = gage section dimension, depth of bend test specimen.F(x) = cumulative distribution function.f(x) = probability density function.Li= length of the inner span for a bend test specimen.Lo= length of the outer span for a bend test specimen.+ = likelihood function.m = Weibull modulus.mˆ = estimate of the Weibull modulus.mˆU= unbiased estimate of the Weibull modulus.N = number of test specimens in a sample.Pf= probability of failure.C1239 − 133r = number of test specimens that failed from the flawpopulation for which the Weibull estimators are beingcalculated.t = intermediate quantity defined by Eq 27, used incalculation of confidence bounds.V = test specimen volume (or volume of effective gagesection, if used).X = random variable.x = realization of a random variable X.β = Weibull scale parameter.ε = stopping tolerance in the computer algorithm MAXL.µˆ = estimate of mean strength.σ = uniaxial tensile stress.σi= maximum stress in the ith test specimen at failure.σj= maximum stress in the jth test specimen at failure.σ