ASTM C1239-13
(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^, σ^θ), but it is suggested that the parameter...
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 confid...
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Designation: C1239 − 13
Standard Practice 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
Ceramics and 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 Aug. 1, 2013. Published September 2013. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 1993. Last previous edition approved in 2007 as C1239–07. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/C1239-13. the ASTM website.
Copyright ©ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA19428-2959. United States
1
---------------------- Page: 1 ----------------------
C1239 − 13
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 concurrentl
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: C1239 − 07 C1239 − 13
Standard Practice 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. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. 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:
Section
Scope 1
Referenced Documents 2
Terminology 3
Summary of Practice 4
Significance and Use 5
Interferences 6
Outlying Observations 7
Maximum Likelihood Parameter Estimators for 8
Competing Flaw Distributions
Unbiasing Factors and Confidence Bounds 9
Fractography 10
Examples 11
Keywords 12
Computer Algorithm MAXL Appendix
X1
Test Specimens with Unidentified Fracture Appendix
Origins X2
1.6 The values stated in SI units are to be regarded as the standard per IEEE/ASTM SI 10.
1
This practice is under the jurisdiction of ASTM Committee C28 on Advanced Ceramicsand is the direct responsibility of Subcommittee C28.01 on Mechanical Properties
and Performance.
Current edition approved Feb. 1, 2007Aug. 1, 2013. Published February 2007September 2013. Originally approved in 1993. Last previous edition approved in 20062007
as C1239 – 06a.C1239 – 07. DOI: 10.1520/C1239-07.10.1520/C1239-13.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
C1239 − 13
FIG. 1 Example of Weibull Plot of Strength Data
2. Referenced Documents
2
2.1 ASTM Standards:
C1145 Terminology of Advanced Ceramics
C1322 Practice for Fractography and Characterization of Fracture Origins in Advanced Ceramics
E6 Terminology Relating to Methods of Mechanical Testing
E178 Practice for Dealing With Outlying Observations
E456 Terminology Relating to Quality and Statistics
IEEE/ASTM SI 10 American National Standard for Use of the International System of Units (SI): The Modern Metric System
3. Terminology
3.1 Proper use of the following terms and equations will alleviate misunderstanding in the presentation of data and in the
calculation of strength distribution parameters.
3.1.1 censored strength data—strength measurements (that is, a sample) containing suspended observations such as that
produced by multiple competing or concurrent flaw populations.
3.1.1.1 Consider a sample where fractography clearly established
...
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