ASTM C1239-00
(Practice)Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics
Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics
SCOPE
1.1 This practice covers the evaluation and subsequent reporting of uniaxial strength data and the estimation of probability distribution parameters for advanced ceramics that fail in a brittle fashion. The failure strength of advanced ceramics is treated as a continuous random variable. Typically, a number of test specimens with well-defined geometry are failed under well-defined isothermal loading conditions. The load at which each specimen fails is recorded. The resulting failure stresses are used to obtain parameter estimates associated with the underlying population distribution. This practice is restricted to the assumption that the distribution 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 uniaxial stress states. Section 8 outlines methods to correct for bias errors in the estimated Weibull parameters and to calculate confidence bounds on those estimates from data sets where all failures originate from a single flaw population (that is, a single failure mode). In samples where failures originate from multiple independent flaw populations (for example, competing failure modes), the methods outlined in Section 8 for bias correction and confidence bounds are not applicable.
1.2 Measurements of the strength at failure are taken for one of two reasons: either for a comparison of the relative quality of two materials, or the prediction of the probability of failure (or, alternatively, the fracture strength) for a structure of interest. This practice will permit estimates of the distribution parameters that are needed for either. In addition, this practice encourages the integration of mechanical property data and fractographic analysis.
1.3 This practice includes the following:Section Scope 1 Referenced Documents 2 Terminology 3 Summary of Practice 4 Significance and Use 5 Outlying Observations 6 Maximum Likelihood Parameter Estimators for Competing Flaw Distributions7 Unbiasing Factors and Confidence Bounds 8 Fractography 9 Examples 10 Keywords 11 Computer Algorithm MAXL X1 Test Specimens with Unidentified Fracture OriginsX2
1.4 The values stated in SI units are to be regarded as the standard.
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Standards Content (Sample)
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Designation: C 1239 – 00
Standard Practice for
Reporting Uniaxial Strength Data and Estimating Weibull
Distribution Parameters for Advanced Ceramics
This standard is issued under the fixed designation C 1239; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope
Significance and Use 5
Outlying Observations 6
1.1 This practice covers the evaluation and subsequent
Maximum Likelihood Parameter Estimators for Competing 7
reporting of uniaxial strength data and the estimation of
Flaw Distributions
Unbiasing Factors and Confidence Bounds 8
probability distribution parameters for advanced ceramics that
Fractography 9
fail in a brittle fashion. The failure strength of advanced
Examples 10
ceramics is treated as a continuous random variable.Typically,
Keywords 11
ComputerAlgorithm MAXL X1
a number of test specimens with well-defined geometry are
Test Specimens with Unidentified Fracture Origins X2
failed under well-defined isothermal loading conditions. The
1.4 The values stated in SI units are to be regarded as the
load at which each specimen fails is recorded. The resulting
standard.
failure stresses are used to obtain parameter estimates associ-
ated with the underlying population distribution. This practice
2. Referenced Documents
is restricted to the assumption that the distribution underlying
2.1 ASTM Standards:
the failure strengths is the two-parameter Weibull distribution
C1145 Terminology of Advanced Ceramics
with size scaling. Furthermore, this practice is restricted to test
C1322 Practice for Fractography and Characterization of
specimens (tensile, flexural, pressurized ring, etc.) that are
Fracture Origins in Advanced Ceramics
primarily subjected to uniaxial stress states. Section 8 outlines
D4392 Terminology for Statistically Related Terms
methods to correct for bias errors in the estimated Weibull
E6 Terminology Relating to Methods of Mechanical Test-
parameters and to calculate confidence bounds on those esti-
ing
mates from data sets where all failures originate from a single
E178 Practice for Dealing With Outlying Observations
flaw population (that is, a single failure mode). In samples
E456 Terminology Relating to Quality and Statistics
where failures originate from multiple independent flaw popu-
2.2 Military Handbook:
lations (for example, competing failure modes), the methods
MIL-HDBK-790 Fractography and Characterization of
outlinedinSection8forbiascorrectionandconfidencebounds
Fracture Origins in Advanced Structural Ceramics
are not applicable.
1.2 Measurementsofthestrengthatfailurearetakenforone
3. Terminology
of two reasons: either for a comparison of the relative quality
3.1 Proper use of the following terms and equations will
of two materials, or the prediction of the probability of failure
alleviate misunderstanding in the presentation of data and in
(or, alternatively, the fracture strength) for a structure of
the calculation of strength distribution parameters.
interest. This practice will permit estimates of the distribution
3.1.1 censored strength data—strength measurements (that
parameters that are needed for either. In addition, this practice
is, a sample) containing suspended observations such as that
encourages the integration of mechanical property data and
produced by multiple competing or concurrent flaw popula-
fractographic analysis.
tions.
1.3 This practice includes the following:
3.1.1.1 Considerasamplewherefractographyclearlyestab-
Section
Scope 1 lished the existence of three concurrent flaw distributions
Referenced Documents 2
(although this discussion is applicable to a sample with any
Terminology 3
Summary of Practice 4
Annual Book of ASTM Standards, Vol 15.01.
1 3
This practice is under the jurisdiction ofASTM Committee C28 onAdvanced Discontinued—see 1992 Annual Book of ASTM Standards, Vol 07.02.
Ceramicsand is the direct responsibility of Subcommittee C28.02 on Design and Annual Book of ASTM Standards, Vol 03.01.
Evaluation. Annual Book of ASTM Standards, Vol 14.02.
Current edition approved Oct. 10, 2000. Published January 2001. Originally AvailablefromStandardizationDocumentsOrderDesk,Bldg.4SectionD,700
published as C1239–93. Last previous edition C1239–95. Robbins Ave., Philadelphia, PA 19111-5094, Attn: NPODS.
Copyright ©ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA19428-2959, United States.
C 1239 – 00
number of concurrent flaw distributions). The three concurrent the total data set reflects more than one type of strength-
flaw distributions are referred to here as distributions A, B, and controlling flaw. This term is synonymous with 88mixtures of
C. Based on fractographic analyses, each specimen strength is flaw distributions.”
assigned to a flaw distribution that initiated failure. In estimat- 3.2.7 extraneous flaws—strength-controlling flaws ob-
ing parameters that characterize the strength distribution asso- servedinsomefractionoftestspecimensthatcannotbepresent
ciated with flaw distribution A, all specimens (and not just in the component being designed. An example is machining
those that failed from Type A flaws) must be incorporated in flaws in ground bend specimens that will not be present in
the analysis to ensure efficiency and accuracy of the resulting as-sintered components of the same material.
parameter estimates. The strength of a specimen that failed by 3.2.8 fractography—the analysis and characterization of
a Type B (or Type C) flaw is treated as a right censored patterns generated on the fracture surface of a test specimen.
observation relative to the A flaw distribution. Failure due to a Fractography can be used to determine the nature and location
Type B (or Type C) flaw restricts, or censors, the information of the critical fracture origin causing catastrophic failure in an
concerning TypeAflaws in a specimen by suspending the test advanced ceramic test specimen or component.
before failure occurred by a Type A flaw (1). The strength 3.2.9 multiple flaw distributions—strength controlling flaws
from the most severe Type A flaw in those specimens that observed by fractography where distinguishably different flaw
failed from Type B (or Type C) flaws is higher than (and thus types are identified as the failure initiation site within different
tothe rightof)theobservedstrength.However,noinformation specimens of the data set and where the flaw types are known
is provided regarding the magnitude of that difference. Cen- or expected to originate from independent causes.
sored data analysis techniques incorporated in this practice 3.2.9.1 Discussion—An example of multiple flaw distribu-
utilize this incomplete information to provide efficient and tions would be carbon inclusions and large voids which may
relatively unbiased estimates of the distribution parameters. both have been observed as strength controlling flaws within a
data set and where there is no reason to believe that the
3.2 Definitions:
frequency or distrubution of carbon inclusions created during
3.2.1 competing failure modes—distinguishably different
fabrication was in any way dependent on the frequency or
types of fracture initiation events that result from concurrent
distribution of voids (or vice-versa).
(competing) flaw distributions.
3.2.10 population—the totality of potential observations
3.2.2 compound flaw distributions—any form of multiple
about which inferences are made.
flaw distribution that is neither pure concurrent nor pure
3.2.11 population mean—the average of all potential mea-
exclusive.Asimple example is where every specimen contains
surements in a given population weighted by their relative
the flaw distribution A, while some fraction of the specimens
frequencies in the population.
also contains a second independent flaw distribution B.
3.2.12 probability density function—the function f (x)isa
3.2.3 concurrent flaw distributions—a type of multiple flaw
probabilitydensityfunctionforthecontinuousrandomvariable
distribution in a homogeneous material where every specimen
X if:
of that material contains representative flaws from each inde-
pendent flaw population. Within a given specimen, all flaw
f ~x!$0
populations are then present concurrently and are competing (1)
with each other to cause failure.This term is synonymous with
and
88competing flaw distributions.”
`
3.2.4 effective gage section—that portion of the test speci- f ~ x! dx 51 (2)
*
2`
men geometry that has been included within the limits of
integration (volume, area, or edge length) of the Weibull
The probability that the random variable X assumes a value
distribution function. In tensile specimens, the integration may
between a and b is given by the following equation:
be restricted to the uniformly stressed central gage section, or
b
it may be extended to include transition and shank regions.
Pr~a , X , b! 5 f~x! dx
*
a
3.2.5 estimator—a well-defined function that is dependent
(3)
ontheobservationsinasample.Theresultingvalueforagiven
sample may be an estimate of a distribution parameter (a point
3.2.13 sample—a collection of measurements or observa-
estimate) associated with the underlying population.The arith-
tions taken from a specified population.
metic average of a sample is, for example, an estimator of the
3.2.14 skewness—a term relating to the asymmetry of a
distribution mean.
probability density function. The distribution of failure
3.2.6 exclusive flaw distributions—a type of multiple flaw
strength for advanced ceramics is not symmetric with respect
distribution created by mixing and randomizing specimens
to the maximum value of the distribution function but has one
from two or more versions of a material where each version
tail longer than the other.
contains a different single flaw population. Thus, each speci-
3.2.15 statistical bias—inherent to most estimates, this is a
men contains flaws exclusively from a single distribution, but
typeofconsistentnumericaloffsetinanestimaterelativetothe
trueunderlyingvalue.Themagnitudeofthebiaserrortypically
decreases as the sample size increases.
3.2.16 unbiased estimator—an estimator that has been cor-
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
this practice. rected for statistical bias error.
C 1239 – 00
3.2.17 Weibull distribution—the continuous random vari- material scale parameter can be described as the Weibull
able X has a two-parameter Weibull distribution if the prob- characteristic strength of a specimen with unit volume or area
ability density function is given by the following equations: loaded in uniform uniaxial tension. The Weibull material scale
1/m
parameter has units of stress·(volume) and should be re-
m21 m
m x x
3/m 3/m
f~x!5 exp — x.0 (4)
S DS D F S D G
ported using units of MPa·(m) or GPa·(m) if the
b b b
strength-controlling flaws are distributed through the volume
of the material. If the strength-controlling flaws are restricted
f~x!50 x#0 (5)
to the surface of the specimens in a sample, then the Weibull
and the cumulative distribution function is given by the
material scale parameter should be reported using units of
2/m
following equations: 2/m
MPa·(m) orGPa·(m) .Foragivenspecimengeometry,Eq
m
x
8 and Eq 10 can be equated, which yields an expression
F~x!51 2 exp 2 x .0 (6)
F S D G
b
relating s and s . Further discussion related to this issue can
0 u
be found in 7.6.
or
3.3 Fordefinitionsofotherstatisticalterms,termsrelatedto
F~x!50 x#0 (7)
mechanical testing, and terms related to advanced ceramics
used in this practice, refer to Terminologies D4392, E456,
where
m = Weibull modulus (or the shape parameter) (>0), and
C1145,andE6ortoappropriatetextbooksonstatistics(2345)
b = scale parameter (>0).
.
3.2.18 The random variable representing uniaxial tensile
3.4 Symbols:
strength of an advanced ceramic will assume only positive
values, and the distribution is asymmetrical about the mean.
A = specimen area (or area of effective gage section, if
Thesecharacteristicsruleouttheuseofthenormaldistribution
used).
(as well as others) and point to the use of the Weibull and
b = gage section dimension, base of bend test specimen.
similar skewed distributions. If the random variable represent-
d = gage section dimension, depth of bend test speci-
ing uniaxial tensile strength of an advanced ceramic is char-
men.
acterized by Eq 4-7, then the probability that this advanced
F(x) = cumulative distribution function.
ceramic will fail under an applied uniaxial tensile stress s is
f(x) = probability density function.
given by the cumulative distribution function as follows:
L = length of the inner load span for a bend test
i
s
specimen.
m
P 51– exp — s.0 (8)
F S D G
f
s
u L = length of the outer load span for a bend test
o
specimen.
P 50 s#0 (9) + = likelihood function.
f
m = Weibull modulus.
where:
mˆ = estimate of the Weibull modulus.
P = probability of failure, and
f mˆ = unbiased estimate of the Weibull modulus.
U
s = Weibull characteristic strength.
u N = number of specimens in a sample.
Note that theWeibull characteristic strength is dependent on
P = probability of failure.
f
theuniaxialtestspecimen(tensile,flexural,orpressurizedring) r = number of specimens that failed from the flaw
and will change with specimen size and geometry. In addition, population for which the Weibull estimators are
the Weibull characteristic strength has units of stress and being calculated.
t = intermediate quantity defined by Eq 27, used in
should be reported using units of megapascals or gigapascals.
3.2.19 An alternative expression for the probability of calculation of confidence bounds.
V = specimen volume (or volume of effective gage
failure is given by the following equation:
section, if used).
s
m
X = random variable.
P 51– exp — dV s.0 (10)
f F *S D G
s
v
x = realization of a random variable X.
b = Weibull scale parameter.
P 50 s#0(11)
f
e = stopping tolerance in the computer algorithm
The integration in the exponential is performed over all MAXL.
tensile regions of the specimen volume if the strength- µˆ = estimate of mean strength.
s = uniaxial tensile stress.
controlling flaws are randomly distributed through the volume
s = maximum stress in the ith test specimen at failure.
of the material, or over all tensile regions of the specimen area
i
s = maximum stress in the jth test specimen at failure.
j
if flaws are restricted to the specimen surface. The integration
s = Weibull material scale parameter (strength relative
O
is sometimes carried out over an effective gage section instead
to unit size) defined in Eq 10.
...
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