ASTM C1239-13(2018)
(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
SIGNIFICANCE AND USE
5.1 Advanced ceramics usually display a linear stress-strain behavior to failure. Lack of ductility combined with flaws that have various sizes and orientations leads to scatter in failure strength. Strength is not a deterministic property, but instead reflects an intrinsic fracture toughness and a distribution (size and orientation) of flaws present in the material. This practice is applicable to brittle monolithic ceramics that fail as a result of catastrophic propagation of flaws present in the material. This practice is also applicable to composite ceramics that do not exhibit any appreciable bilinear or nonlinear deformation behavior. In addition, the composite must contain a sufficient quantity of uniformly distributed reinforcements such that the material is effectively homogeneous. Whisker-toughened ceramic composites may be representative of this type of material.
5.2 Two- and three-parameter formulations exist for the Weibull distribution. This practice is restricted to the two-parameter formulation. An objective of this practice is to obtain point estimates of the unknown parameters by using well-defined functions that incorporate the failure data. These functions are referred to as “estimators.” It is desirable that an estimator be consistent and efficient. In addition, the estimator should produce unique, unbiased estimates of the distribution parameters (6). Different types of estimators exist, including moment estimators, least-squares estimators, and maximum likelihood estimators. This practice details the use of maximum likelihood estimators due to the efficiency and the ease of application when censored failure populations are encountered.
5.3 Tensile and flexural test specimens are the most commonly used test configurations for advanced ceramics. The observed strength values are dependent on test specimen size and geometry. Parameter estimates can be computed for a given test specimen geometry ( m^, ^σθ), but it is suggested that the paramet...
SCOPE
1.1 This practice covers the evaluation and reporting of uniaxial strength data and the estimation of Weibull probability distribution parameters for advanced ceramics that fail in a brittle fashion (see Fig. 1). The estimated Weibull distribution parameters are used for statistical comparison of the relative quality of two or more test data sets and for the prediction of the probability of failure (or, alternatively, the fracture strength) for a structure of interest. In addition, this practice encourages the integration of mechanical property data and fractographic analysis.
1.6 The values stated in SI units are to be regarded as the standard per IEEE/ASTM SI 10.
1.7 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
General Information
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Standards Content (Sample)
This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: C1239 − 13 (Reapproved 2018)
Standard Practice for
Reporting Uniaxial Strength Data and Estimating Weibull
Distribution Parameters for Advanced Ceramics
This standard is issued under the fixed designation C1239; 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 (´) indicates an editorial change since the last revision or reapproval.
1. Scope
Section
Scope 1
1.1 This practice covers the evaluation and reporting of
Referenced Documents 2
uniaxialstrengthdataandtheestimationofWeibullprobability Terminology 3
Summary of Practice 4
distribution parameters for advanced ceramics that fail in a
Significance and Use 5
brittle fashion (see Fig. 1). The estimated Weibull distribution
Interferences 6
parameters are used for statistical comparison of the relative Outlying Observations 7
Maximum Likelihood Parameter Estimators 8
quality of two or more test data sets and for the prediction of
for Competing Flaw Distributions
the probability of failure (or, alternatively, the fracture
Unbiasing Factors and Confidence Bounds 9
Fractography 10
strength) for a structure of interest. In addition, this practice
Examples 11
encourages the integration of mechanical property data and
Keywords 12
fractographic analysis.
ComputerAlgorithm MAXL Appendix X1
Test Specimens with Unidentified Fracture Appendix X2
1.2 Thefailurestrengthofadvancedceramicsistreatedasa
Origins
continuousrandomvariabledeterminedbytheflawpopulation.
1.6 The values stated in SI units are to be regarded as the
Typically, a number of test specimens with well-defined
standard per IEEE/ASTMSI10.
geometry are failed under isothermal, well-defined displace-
1.7 This international standard was developed in accor-
ment and/or force-application conditions. The force at which
dance with internationally recognized principles on standard-
eachtestspecimenfailsisrecorded.Theresultingfailurestress
ization established in the Decision on Principles for the
data are used to obtain Weibull parameter estimates associated
Development of International Standards, Guides and Recom-
with the underlying flaw population distribution.
mendations issued by the World Trade Organization Technical
1.3 This practice is restricted to the assumption that the
Barriers to Trade (TBT) Committee.
distribution underlying the failure strengths is the two-
parameter Weibull distribution with size scaling. Furthermore, 2. Referenced Documents
this practice is restricted to test specimens (tensile, flexural,
2.1 ASTM Standards:
pressurized ring, etc.) that are primarily subjected to uniaxial
C1145Terminology of Advanced Ceramics
stressstates.Thepracticealsoassumesthattheflawpopulation
C1322Practice for Fractography and Characterization of
is stable with time and that no slow crack growth is occurring.
Fracture Origins in Advanced Ceramics
1.4 The practice outlines methods to correct for bias errors
E6Terminology Relating to Methods of Mechanical Testing
in the estimated Weibull parameters and to calculate confi- E178Practice for Dealing With Outlying Observations
dence bounds on those estimates from data sets where all
E456Terminology Relating to Quality and Statistics
failuresoriginatefromasingleflawpopulation(thatis,asingle IEEE/ASTMSI10American National Standard for Use of
failure mode). In samples where failures originate from mul-
theInternationalSystemofUnits(SI):TheModernMetric
tiple independent flaw populations (for example, competing System
failure modes), the methods outlined in Section 9 for bias
3. Terminology
correction and confidence bounds are not applicable.
3.1 Proper use of the following terms and equations will
1.5 This practice includes the following:
alleviate misunderstanding in the presentation of data and in
the calculation of strength distribution parameters.
This practice is under the jurisdiction ofASTM Committee C28 on Advanced
Ceramics and is the direct responsibility of Subcommittee C28.01 on Mechanical
Properties and Performance. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
CurrenteditionapprovedJuly1,2018.PublishedJuly2018.Originallyapproved contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
in 1993. Last previous edition approved in 2013 as C1239–13. DOI: 10.1520/ Standards volume information, refer to the standard’s Document Summary page on
C1239-13R18. the ASTM website.
Copyright ©ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA19428-2959. United States
C1239 − 13 (2018)
3.2.2 compound flaw distributions—any form of multiple
flaw distribution that is neither pure concurrent nor pure
exclusive. A simple example is where every test specimen
contains the flaw distribution A, while some fraction of the test
specimensalsocontainsasecondindependentflawdistribution
B.
3.2.3 concurrent flaw distributions—type of multiple flaw
distribution in a homogeneous material where every test
specimen of that material contains representative flaws from
each independent flaw population. Within a given test
specimen, all flaw populations are then present concurrently
and are competing with each other to cause failure. This term
is 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 of
integration (volume, area, or edge length) of the Weibull
distribution function. In tensile test specimens, the integration
may be restricted to the uniformly stressed central gage
FIG. 1 Example of Weibull Plot of Strength Data
section, or it may be extended to include transition and shank
regions.
3.2.5 estimator—well-defined function that is dependent on
3.1.1 censored strength data—strength measurements (that
the observations in a sample. The resulting value for a given
is, a sample) containing suspended observations such as those
sample may be an estimate of a distribution parameter (a point
produced by multiple competing or concurrent flaw popula-
estimate) associated with the underlying population.The arith-
tions.
metic average of a sample is, for example, an estimator of the
3.1.1.1 Considerasamplewherefractographyclearlyestab-
distribution mean.
lished the existence of three concurrent flaw distributions
3.2.6 exclusive flaw distributions—type of multiple flaw
(although this discussion is applicable to a sample with any
distribution created by mixing and randomizing test specimens
number of concurrent flaw distributions). The three concurrent
from two or more versions of a material where each version
flaw distributions are referred to here as distributions A, B, and
contains a different single flaw population. Thus, each test
C. Based on fractographic analyses, each test specimen
specimencontainsflawsexclusivelyfromasingledistribution,
strength is assigned to a flaw distribution that initiated failure.
but the total data set reflects more than one type of strength-
In estimating parameters that characterize the strength distri-
controlling flaw. This term is synonymous with “mixtures of
bution associated with flaw distribution A, all test specimens
flaw distributions.”
(and not just those that failed from Type A flaws) must be
incorporated in the analysis to ensure efficiency and accuracy
3.2.7 extraneous flaws—strength-controllingflawsobserved
of the resulting parameter estimates. The strength of a test
in some fraction of test specimens that cannot be present in the
specimen that failed by aType B (orType C) flaw is treated as
component being designed.An example is machining flaws in
a right censored observation relative to the A flaw distribution.
ground bend test specimens that will not be present in
Failure due to a Type B (or Type C) flaw restricts, or censors,
as-sintered components of the same material.
the information concerningTypeAflaws in a test specimen by
3.2.8 fractography—analysis and characterization of pat-
suspending the test before failure occurred by a Type A flaw
terns generated on the fracture surface of a test specimen.
(1). The strength from the most severe Type A flaw in those
Fractography can be used to determine the nature and location
test specimens that failed from Type B (or Type C) flaws is
of the critical fracture origin causing catastrophic failure in an
higher than (and thus to the right of) the observed strength.
advanced ceramic test specimen or component.
However, no information is provided regarding the magnitude
3.2.9 fracture origin—thesourcefromwhichbrittlefracture
of that difference. Censored data analysis techniques incorpo-
commences (Terminology C1145).
rated in this practice utilize this incomplete information to
provide efficient and relatively unbiased estimates of the
3.2.10 multiple flaw distributions—strength-controlling
distribution parameters.
flawsobservedbyfractographywheredistinguishablydifferent
flaw types are identified as the failure initiation site within
3.2 Definitions:
differenttestspecimensofthedatasetandwheretheflawtypes
3.2.1 competing failure modes—distinguishably different
are known or expected to originate from independent causes.
types of fracture initiation events that result from concurrent
3.2.10.1 Discussion—An example of multiple flaw distribu-
(competing) flaw distributions.
tions would be carbon inclusions and large voids which may
both have been observed as strength-controlling flaws within a
data set and where there is no reason to believe that the
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
this practice. frequency or distribution of carbon inclusions created during
C1239 − 13 (2018)
fabrication was in any way dependent on the frequency or ing uniaxial tensile strength of an advanced ceramic is char-
distribution of voids (or vice-versa). acterized by Eq 4-7, then the probability that this advanced
ceramic will fail under an applied uniaxial tensile stress σ is
3.2.11 population—totality of potential observations about
given by the cumulative distribution function as follows:
which inferences are made.
σ
3.2.12 population mean—average of all potential measure-
m
P 5 1 2 exp 2 σ.0 (8)
F S D G
f
σ
θ
mentsinagivenpopulationweightedbytheirrelativefrequen-
cies in the population. P 5 0 σ#0 (9)
f
3.2.13 probability density function—function f(x) is a prob-
where:
ability density function for the continuous random variable X
P = probability of failure, and
f
if:
σ = Weibull characteristic strength.
θ
f x $0 (1)
~ !
Note that theWeibull characteristic strength is dependent on
theuniaxialtestspecimen(tensile,flexural,orpressurizedring)
and
and will change with test specimen size and geometry. In
`
f x dx 51 (2)
* ~ !
addition, the Weibull characteristic strength has units of stress
2`
and should be reported using units of megapascals or gigapas-
The probability that the random variable X assumes a
cals.
value between a and b is given by the following equation:
3.2.20 An alternative expression for the probability of
b
failure is given by the following equation:
Pr~a,X,b! 5 f~x! dx (3)
*
a
σ
m
3.2.14 sample—collection of measurements or observations
P 5 1 2 exp 2* dV σ.0 (10)
F S D G
f
v σ
taken from a specified population.
P 5 0 σ#0 (11)
f
3.2.15 skewness—term relating to the asymmetry of a prob-
The integration in the exponential is performed over all
ability density function. The distribution of failure strength for
advanced ceramics is not symmetric with respect to the tensile regions of the test specimen volume if the strength-
controlling flaws are randomly distributed through the volume
maximum value of the distribution function, but has one tail
longer than the other. of the material, or over all tensile regions of the test specimen
area if flaws are restricted to the test specimen surface. The
3.2.16 statistical bias—inherent to most estimates, this is a
integration is sometimes carried out over an effective gage
typeofconsistentnumericaloffsetinanestimaterelativetothe
section instead of over the total volume or area. In Eq 10, σ is
trueunderlyingvalue.Themagnitudeofthebiaserrortypically
the Weibull material scale parameter. The parameter is a
decreases as the sample size increases.
material property if the two-parameter Weibull distribution
3.2.17 unbiased estimator—estimator that has been cor-
properly describes the strength behavior of the material. In
rected for statistical bias error.
addition,theWeibullmaterialscaleparametercanbedescribed
3.2.18 Weibull distribution—continuous random variable X
as the Weibull characteristic strength of a test specimen with
has a two-parameter Weibull distribution if the probability unit volume or area forced in uniform uniaxial tension. The
density function is given by the following equations: Weibull material scale parameter has units of
1/m
m21 m stress·(volume) and should be reported using units of
m x x
3/m 3/m
f x 5 exp 2 x.0 (4)
~ ! S DS D F S D G MPa·(m) or GPa·(m) if the strength-controlling flaws are
β β β
distributed through the volume of the material. If the strength-
f~x! 5 0 x#0 (5)
controlling flaws are restricted to the surface of the test
specimens in a sample, then the Weibull material scale param-
and the cumulative distribution function is given by the
2/m
eter should be reported using units of MPa·(m) or
following equations:
2/m
GPa·(m) . For a given test specimen geometry, Eq 8 and Eq
m
x
F x 5 1 2 exp 2 x.0 (6) 10 can be equated, which yields an expression relating σ and
~ ! F S D G
β
σ . Further discussion related to this issue can be found in 8.6.
θ
or
3.3 Fordefinitionsofotherstatisticalterms,termsrelatedto
mechanical testing, and terms related to advanced ceramics
F x 5 0 x#0 (7)
~ !
used in this practice, refer to Terminologies E456, C1145, and
where:
E6 or to appropriate textbooks on statistics (2-5).
m = Weibull modulus (or the shape parameter) (>0), and
3.4 Symbols:
β = scale parameter (>0).
A = test specimen area (or area of effective gage section,
3.2.19 The random variable representing uniaxial tensile
if used).
strength of an advanced ceramic will assume only positive
b = gage section dimension, base of bend test specimen.
values, and the distribution is asymmetrical about the mean.
d = gage section dimension, depth of bend test specimen.
Thesecharacteristicsruleouttheuseofthenormaldistribution
F(x) = cumulative distribution function.
(as well as others) and point to the use of the Weibull and
f(x) = probability density function.
similar skewed distributions. If the rando
...
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