ASTM D7846-21
(Practice)Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Graphites
Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Graphites
SIGNIFICANCE AND USE
5.1 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 Weibull distribution 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, such as moment estimators, least-squares estimators, and maximum likelihood estimators. This practice details the use of maximum likelihood estimators.
5.2 Tensile and flexural specimens are the most commonly used test configurations for graphite. The observed strength values depend on specimen size and test geometry. Tensile and flexural test specimen failure data for a nearly isotropic graphite (7) is depicted in Fig. 1. Since the failure data for a graphite material can be dependent on the test specimen geometry, Weibull distribution parameter estimates (m, Sc) shall be computed for a given specimen geometry.
FIG. 1 Failure Strengths for Tensile Test Specimens (left) and Flexural Test Specimens (right) for a Nearly Isotropic Graphite (7)
5.3 The bias and uncertainty of Weibull parameters depend on the total number of test specimens. Variability in parameter estimates decreases exponentially as more specimens are collected. However, a point of diminishing returns is reached where the cost of performing additional strength tests may not be justified. This suggests a limit to the number of test specimens for determining Weibull parameters to obtain a desired level of confidence associated with a parameter estimate. The number of specimens needed depends on the precision required in the resulting parameter estimate or in the resulting confidence bounds. Details relating to the computation of confidence bo...
SCOPE
1.1 This practice covers the reporting of uniaxial strength data for graphite and the estimation of probability distribution parameters for both censored and uncensored data. The failure strength of graphite materials is treated as a continuous random variable. Typically, a number of test specimens are failed in accordance with the following standards: Test Methods C565, C651, C695, C749, Practice C781 or Guide D7775. 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 limited to failure strengths that can be characterized by the two-parameter Weibull distribution. Furthermore, this practice is restricted to test specimens (primarily tensile and flexural) that are primarily subjected to uniaxial stress states.
1.2 Measurements of the strength at failure are taken for various reasons: a comparison of the relative quality of two materials, the prediction of the probability of failure for a structure of interest, or to establish limit loads in an application. This practice provides a procedure for estimating the distribution parameters that are needed for estimating load limits for a particular level of probability of failure.
1.3 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.
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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: D7846 − 21
Standard Practice for
Reporting Uniaxial Strength Data and Estimating Weibull
1
Distribution Parameters for Advanced Graphites
This standard is issued under the fixed designation D7846; 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* Graphite Mechanical Materials
C651 Test Method for Flexural Strength of Manufactured
1.1 This practice covers the reporting of uniaxial strength
CarbonandGraphiteArticlesUsingFour-PointLoadingat
data for graphite and the estimation of probability distribution
Room Temperature
parameters for both censored and uncensored data. The failure
C695 Test Method for Compressive Strength of Carbon and
strengthofgraphitematerialsistreatedasacontinuousrandom
Graphite
variable. Typically, a number of test specimens are failed in
C749 Test Method for Tensile Stress-Strain of Carbon and
accordance with the following standards: Test Methods C565,
Graphite
C651,C695,C749,PracticeC781orGuideD7775.Theloadat
C781 PracticeforTestingGraphiteMaterialsforGas-Cooled
which each specimen fails is recorded. The resulting failure
Nuclear Reactor Components
stresses are used to obtain parameter estimates associated with
D4175 Terminology Relating to Petroleum Products, Liquid
the underlying population distribution. This practice is limited
Fuels, and Lubricants
to failure strengths that can be characterized by the two-
D7775 Guide for Measurements on Small Graphite Speci-
parameter Weibull distribution. Furthermore, this practice is
mens
restricted to test specimens (primarily tensile and flexural) that
E6 Terminology Relating to Methods of Mechanical Testing
are primarily subjected to uniaxial stress states.
E178 Practice for Dealing With Outlying Observations
1.2 Measurements of the strength at failure are taken for
E456 Terminology Relating to Quality and Statistics
various reasons: a comparison of the relative quality of two
materials, the prediction of the probability of failure for a
3. Terminology
structure of interest, or to establish limit loads in an applica-
3.1 Proper use of the following terms and equations will
tion. This practice provides a procedure for estimating the
alleviate misunderstanding in the presentation of data and in
distribution parameters that are needed for estimating load
the calculation of strength distribution parameters.
limits for a particular level of probability of failure.
3.2 Definitions:
1.3 This international standard was developed in accor-
dance with internationally recognized principles on standard- 3.2.1 estimator, n—awell-definedfunctionthatisdependent
ization established in the Decision on Principles for the
ontheobservationsinasample.Theresultingvalueforagiven
Development of International Standards, Guides and Recom- sample may be an estimate of a distribution parameter (a point
mendations issued by the World Trade Organization Technical
estimate) associated with the underlying population. The arith-
Barriers to Trade (TBT) Committee. metic average of a sample is, for example, an estimator of the
distribution mean.
2. Referenced Documents
3.2.2 population, n—the totality of valid observations (per-
2
2.1 ASTM Standards:
formed in a manner that is compliant with the appropriate test
C565 Test Methods for Tension Testing of Carbon and
standards) about which inferences are made.
3.2.3 population mean, n—the average of all potential
1
measurements in a given population weighted by their relative
This practice is under the jurisdiction ofASTM Committee D02 on Petroleum
Products, Liquid Fuels, and Lubricants and is the direct responsibility of Subcom-
frequencies in the population.
mittee D02.F0 on Manufactured Carbon and Graphite Products.
3.2.4 probability density function, n—the function f(x) is a
Current edition approved Dec. 1, 2021. Published February 2022. Originally
approved in 2012. Last previous edition approved in 2016 as D7846 – 16. DOI:
probabilitydensityfunctionforthecontinuousrandomvariable
10.1520/D7846-21.
X if:
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
f x $0 (1)
~ !
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. and
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
...
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: D7846 − 16 D7846 − 21
Standard Practice for
Reporting Uniaxial Strength Data and Estimating Weibull
1
Distribution Parameters for Advanced Graphites
This standard is issued under the fixed designation D7846; 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 reporting of uniaxial strength data for graphite and the estimation of probability distribution
parameters for both censored and uncensored data. The failure strength of graphite materials is treated as a continuous random
variable. Typically, a number of test specimens are failed in accordance with the following standards: Test Methods C565, C651,
C695, C749, Practice C781 or Guide D7775. 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 limited to failure
strengths that can be characterized by the two-parameter Weibull distribution. Furthermore, this practice is restricted to test
specimens (primarily tensile and flexural) that are primarily subjected to uniaxial stress states.
1.2 Measurements of the strength at failure are taken for various reasons: a comparison of the relative quality of two materials,
the prediction of the probability of failure for a structure of interest, or to establish limit loads in an application. This practice
provides a procedure for estimating the distribution parameters that are needed for estimating load limits for a particular level of
probability of failure.
1.3 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.
2. Referenced Documents
2
2.1 ASTM Standards:
C565 Test Methods for Tension Testing of Carbon and Graphite Mechanical Materials
C651 Test Method for Flexural Strength of Manufactured Carbon and Graphite Articles Using Four-Point Loading at Room
Temperature
C695 Test Method for Compressive Strength of Carbon and Graphite
C749 Test Method for Tensile Stress-Strain of Carbon and Graphite
C781 Practice for Testing Graphite Materials for Gas-Cooled Nuclear Reactor Components
D4175 Terminology Relating to Petroleum Products, Liquid Fuels, and Lubricants
D7775 Guide for Measurements on Small Graphite Specimens
E6 Terminology Relating to Methods of Mechanical Testing
E178 Practice for Dealing With Outlying Observations
E456 Terminology Relating to Quality and Statistics
1
This practice is under the jurisdiction of ASTM Committee D02 on Petroleum Products, Liquid Fuels, and Lubricants and is the direct responsibility of Subcommittee
D02.F0 on Manufactured Carbon and Graphite Products.
Current edition approved Jan. 1, 2016Dec. 1, 2021. Published February 2016February 2022. Originally approved in 2012. Last previous edition approved in 20122016
as D7846 – 12.D7846 – 16. DOI: 10.1520/D7846-16.10.1520/D7846-21.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
D7846 − 21
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.
3.2 Definitions:
3.2.1 estimator, n—a well-defined function that is dependent on the observations in a sample. The resulting value for a given
sample may be an estimate of a distribution parameter (a point estimate) associated with the underlying population. The arithmetic
average of a sample is, for example, an estimator of the distribution mean.
3.2.2 population, n—the totality of valid observations (performed in a manner that is compliant with the appropriate test standards)
about which inferences are made.
3
...
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