ASTM C1239-07
(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 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.
1.2 The failure strength of advanced ceramics is treated as a continuous random variable determined by the flaw population. Typically, a number of test specimens with well-defined geometry are failed under isothermal, well-defined displacement and/or force-application conditions. The force at which each test specimen fails is recorded. The resulting failure stress data are used to obtain Weibull parameter estimates associated with the underlying flaw population distribution.
1.3 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. The practice also assumes that the flaw population is stable with time and that no slow crack growth is occurring.
1.4 The practice 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 9 for bias correction and confidence bounds are not applicable.
1.5 This practice includes the following:
1.6 The values stated in SI units are to be regarded as the standard per IEEE/ASTM SI 10.
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Designation: C1239 − 07
StandardPractice for
Reporting Uniaxial Strength Data and Estimating Weibull
1
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 for 8
quality of two or more test data sets and for the prediction of
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
1.2 Thefailurestrengthofadvancedceramicsistreatedasa
Test Specimens with Unidentified Fracture Appendix
continuousrandomvariabledeterminedbytheflawpopulation. Origins X2
Typically, a number of test specimens with well-defined
1.6 The values stated in SI units are to be regarded as the
geometry are failed under isothermal, well-defined displace-
standard per IEEE/ASTMSI10.
ment and/or force-application conditions. The force at which
eachtestspecimenfailsisrecorded.Theresultingfailurestress
2. Referenced Documents
data are used to obtain Weibull parameter estimates associated
2
2.1 ASTM Standards:
with the underlying flaw population distribution.
C1145Terminology of Advanced Ceramics
1.3 This practice is restricted to the assumption that the
C1322Practice for Fractography and Characterization of
distribution underlying the failure strengths is the two-
Fracture Origins in Advanced Ceramics
parameter Weibull distribution with size scaling. Furthermore,
E6Terminology Relating to Methods of Mechanical Testing
this practice is restricted to test specimens (tensile, flexural,
E178Practice for Dealing With Outlying Observations
pressurized ring, etc.) that are primarily subjected to uniaxial
E456Terminology Relating to Quality and Statistics
stressstates.Thepracticealsoassumesthattheflawpopulation
IEEE/ASTMSI10American National Standard for Use of
is stable with time and that no slow crack growth is occurring.
theInternationalSystemofUnits(SI):TheModernMetric
System
1.4 The practice outlines methods to correct for bias errors
in the estimated Weibull parameters and to calculate confi-
3. Terminology
dence bounds on those estimates from data sets where all
failuresoriginatefromasingleflawpopulation(thatis,asingle
3.1 Proper use of the following terms and equations will
failure mode). In samples where failures originate from mul-
alleviate misunderstanding in the presentation of data and in
tiple independent flaw populations (for example, competing
the calculation of strength distribution parameters.
failure modes), the methods outlined in Section 9 for bias
3.1.1 censored strength data—strength measurements (that
correction and confidence bounds are not applicable.
is, a sample) containing suspended observations such as that
produced by multiple competing or concurrent flaw popula-
1.5 This practice includes the following:
tions.
1
This practice is under the jurisdiction ofASTM Committee C28 on Advanced
Ceramicsand is the direct responsibility of Subcommittee C28.01 on Mechanical
2
Properties and Performance. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved Feb. 1, 2007. Published February 2007. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 1993. Last previous edition approved in 2006 as C1239–06a. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/C1239-07. the ASTM website.
Copyright ©ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA19428-2959. United States
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C1239 − 07
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
...
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