Fine ceramics (advanced ceramics, advanced technical ceramics) - Weibull statistics for strength data (ISO 20501:2019)

This document covers the reporting of uniaxial strength data and the estimation of probability distribution parameters for advanced ceramics which 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 brought to failure 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 document is restricted to the assumption that the distribution underlying the failure strengths is the two-parameter Weibull distribution with size scaling. Furthermore, this document is restricted to test specimens (tensile, flexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress states. Subclauses 6.4 and 6.5 outline methods of correcting 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 (i.e. a single failure mode). In samples where failures originate from multiple independent flaw populations (e.g. competing failure modes), the methods outlined in 6.4 and 6.5 for bias correction and confidence bounds are not applicable.

Hochleistungskeramik - Weibullstatistik von Festigkeitswerten (ISO 20501:2019)

Céramiques techniques - Analyse statistique de Weibull des données de résistance à la rupture (ISO 20501:2019)

Fina keramika (sodobna keramika, sodobna tehnična keramika) - Weibullova statistika za podatke o trdnosti (ISO 20501:2019)

General Information

Status
Published
Publication Date
20-Dec-2022
Current Stage
6060 - Definitive text made available (DAV) - Publishing
Start Date
21-Dec-2022
Due Date
03-Oct-2024
Completion Date
21-Dec-2022

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SLOVENSKI STANDARD
SIST EN ISO 20501:2023
01-marec-2023
Nadomešča:
SIST EN 843-5:2007
Fina keramika (sodobna keramika, sodobna tehnična keramika) - Weibullova
statistika za podatke o trdnosti (ISO 20501:2019)

Fine ceramics (advanced ceramics, advanced technical ceramics) - Weibull statistics for

strength data (ISO 20501:2019)
Hochleistungskeramik - Weibullstatistik von Festigkeitswerten (ISO 20501:2019)

Céramiques techniques - Analyse statistique de Weibull des données de résistance à la

rupture (ISO 20501:2019)
Ta slovenski standard je istoveten z: EN ISO 20501:2022
ICS:
81.060.30 Sodobna keramika Advanced ceramics
SIST EN ISO 20501:2023 en,fr,de

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
SIST EN ISO 20501:2023
---------------------- Page: 2 ----------------------
SIST EN ISO 20501:2023
EN ISO 20501
EUROPEAN STANDARD
NORME EUROPÉENNE
December 2022
EUROPÄISCHE NORM
ICS 81.060.30 Supersedes EN 843-5:2006
English Version
Fine ceramics (advanced ceramics, advanced technical
ceramics) - Weibull statistics for strength data (ISO
20501:2019)

Céramiques techniques - Analyse statistique de Hochleistungskeramik - Weibullstatistik von

Weibull des données de résistance à la rupture (ISO Festigkeitswerten (ISO 20501:2019)

20501:2019)
This European Standard was approved by CEN on 20 December 2022.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this

European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references

concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN

member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by

translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management

Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,

Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,

Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and

United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels

© 2022 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 20501:2022 E

worldwide for CEN national Members.
---------------------- Page: 3 ----------------------
SIST EN ISO 20501:2023
EN ISO 20501:2022 (E)
Contents Page

European foreword ....................................................................................................................................................... 3

---------------------- Page: 4 ----------------------
SIST EN ISO 20501:2023
EN ISO 20501:2022 (E)
European foreword

The text of ISO 20501:2019 has been prepared by Technical Committee ISO/TC 206 "Fine ceramics” of

the International Organization for Standardization (ISO) and has been taken over as EN ISO 20501:2022

by Technical Committee CEN/TC 184 “Advanced technical ceramics” the secretariat of which is held by

DIN.

This European Standard shall be given the status of a national standard, either by publication of an

identical text or by endorsement, at the latest by June 2023, and conflicting national standards shall be

withdrawn at the latest by June 2023.

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. CEN shall not be held responsible for identifying any or all such patent rights.

This document supersedes EN 843-5:2006.

Any feedback and questions on this document should be directed to the users’ national standards body.

A complete listing of these bodies can be found on the CEN website.

According to the CEN-CENELEC Internal Regulations, the national standards organizations of the

following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,

Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,

Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of

North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and the

United Kingdom.
Endorsement notice

The text of ISO 20501:2019 has been approved by CEN as EN ISO 20501:2022 without any modification.

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SIST EN ISO 20501:2023
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SIST EN ISO 20501:2023
INTERNATIONAL ISO
STANDARD 20501
Second edition
2019-03
Fine ceramics (advanced ceramics,
advanced technical ceramics) —
Weibull statistics for strength data
Céramiques techniques — Analyse statistique de Weibull des données
de résistance à la rupture
Reference number
ISO 20501:2019(E)
ISO 2019
---------------------- Page: 7 ----------------------
SIST EN ISO 20501:2023
ISO 20501:2019(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2019

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2019 – All rights reserved
---------------------- Page: 8 ----------------------
SIST EN ISO 20501:2023
ISO 20501:2019(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

3.1 Defect populations ............................................................................................................................................................................... 1

3.2 Mechanical testing ............................................................................................................................................................................... 3

3.3 Statistical terms ...................................................................................................................................................................................... 3

3.4 Weibull distributions ......................................................................................................................................................................... 4

4 Symbols .......................................................................................................................................................................................................................... 5

5 Significance and use .......................................................................................................................................................................................... 6

6 Method A: maximum likelihood parameter estimators for single flaw populations ....................7

6.1 General ........................................................................................................................................................................................................... 7

6.2 Censored data .......................................................................................................................................................................................... 8

6.3 Likelihood functions ........................................................................................................................................................................... 8

6.4 Bias correction ........................................................................................................................................................................................ 8

6.5 Confidence intervals ........................................................................................................................................................................10

7 Method B: maximum likelihood parameter estimators for competing flaw populations ....13

7.1 General ........................................................................................................................................................................................................13

7.2 Censored data .......................................................................................................................................................................................14

7.3 Likelihood functions ........................................................................................................................................................................14

8 Procedure..................................................................................................................................................................................................................15

8.1 Outlying observations ....................................................................................................................................................................15

8.2 Fractography ..........................................................................................................................................................................................15

8.3 Graphical representation ............................................................................................................................................................16

9 Test report ................................................................................................................................................................................................................18

Annex A (informative) Converting to material-specific strength distribution parameters .....................19

Annex B (informative) Illustrative examples ............................................................................................................................................21

Annex C (informative) Test specimens with unidentified fracture origin ..................................................................28

Annex D (informative) Fortran program .......................................................................................................................................................31

Bibliography .............................................................................................................................................................................................................................36

© ISO 2019 – All rights reserved iii
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SIST EN ISO 20501:2023
ISO 20501:2019(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso

.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 206, Fine ceramics.

This second edition cancels and replaces the first edition (ISO 20501:2003), which has been technically

revised. It also incorporates the Technical Corrigendum ISO 20501:2003/Cor.1:2009.

The main changes compared to the previous edition are as follows:
— the terms and definitions in Clause 3 have been updated and modified;

— a method to treat a higher number of specimens (N > 120) has been introduced for method A:

maximum likelihood parameter estimators for single flaw populations;

— in Annex D, example codes have been added for calculating the maximum likelihood parameters of

the Weibull distribution with modern analysis software.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/members .html.
iv © ISO 2019 – All rights reserved
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SIST EN ISO 20501:2023
ISO 20501:2019(E)
Introduction

Measurements of the strength at failure are taken for one of two reasons: either for a comparison of

the relative quality of two materials regarding fracture strength, or the prediction of the probability

of failure for a structure of interest. This document permits estimates of the distribution parameters

which are needed for either. In addition, this document encourages the integration of mechanical

property data and fractographic analysis.
© ISO 2019 – All rights reserved v
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SIST EN ISO 20501:2023
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SIST EN ISO 20501:2023
INTERNATIONAL STANDARD ISO 20501:2019(E)
Fine ceramics (advanced ceramics, advanced technical
ceramics) — Weibull statistics for strength data
1 Scope

This document covers the reporting of uniaxial strength data and the estimation of probability

distribution parameters for advanced ceramics which 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 brought to failure 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 document is restricted to the assumption that the distribution underlying the failure strengths is

the two-parameter Weibull distribution with size scaling. Furthermore, this document is restricted to

test specimens (tensile, flexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress

states. Subclauses 6.4 and 6.5 outline methods of correcting 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 (i.e. a single failure mode). In samples where failures originate

from multiple independent flaw populations (e.g. competing failure modes), the methods outlined in 6.4

and 6.5 for bias correction and confidence bounds are not applicable.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
NOTE See also Reference [1].
3.1 Defect populations
3.1.1
flaw

inhomogeneity, discontinuity or (defect) feature in a material, which acts as stress concentrator due to

a mechanical load and has therefore a certain risk of mechanical failure

Note 1 to entry: The flaw becomes critical if it acts as fracture origin in a failed specimen.

© ISO 2019 – All rights reserved 1
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SIST EN ISO 20501:2023
ISO 20501:2019(E)
3.1.2
censored data

strength measurements (i.e. a sample) containing suspended observations such as that produced by

multiple competing or concurrent flaw populations

Note 1 to entry: Consider a sample where fractography clearly established the existence of three concurrent

flaw distributions (although this discussion is applicable to a sample with any number of concurrent flaw

distributions). The three concurrent flaw distributions are referred to here as distributions A, B, and C. Based

on fractographic analyses, each specimen strength is assigned to a flaw distribution that initiated failure. In

estimating parameters that characterize the strength distribution associated with flaw distribution A, all

specimens (and not just those that failed from type-A flaws) shall be incorporated in the analysis to ensure

efficiency and accuracy of the resulting parameter estimates. The strength of a specimen that failed by a

type-B (or type-C) flaw is treated as a right censored observation relative to the A flaw distribution. Failure

due to a type-B (or type-C) flaw restricts, or censors, the information concerning type-A flaws in a specimen

[2]

by suspending the test before failure occurs by a type-A flaw . The strength from the most severe type-A

flaw in those specimens that failed from type-B (or type-C) flaws is higher than (and thus to the right of) the

observed strength. However, no information is provided regarding the magnitude of that difference. Censored

data analysis techniques incorporated in this document utilize this incomplete information to provide efficient

and relatively unbiased estimates of the distribution parameters.
3.1.3
competing failure modes

distinguishably different types of fracture initiation events that result from concurrent (competing)

flaw distributions
3.1.4
compound flaw distribution

any form of multiple flaw distribution that is neither pure concurrent, nor pure exclusive

Note 1 to entry: A simple example is where every specimen contains the flaw distribution A, while some fraction

of the specimens also contains a second independent flaw distribution B.
3.1.5
concurrent flaw distribution
competing flaw distribution

type of multiple flaw distribution in a homogeneous material where every specimen of that material

contains representative flaws from each independent flaw population

Note 1 to entry: Within a given specimen, all flaw populations are then present concurrently and are competing

to each other to cause failure.
3.1.6
exclusive flaw distribution
mixture flaw distribution

type of multiple flaw distribution created by mixing and randomizing specimens from two or more

versions of a material where each version contains a different single flaw population

Note 1 to entry: Thus, each specimen contains flaws exclusively from a single distribution, but the total data set

reflects more than one type of strength-controlling flaw.
3.1.7
extraneous flaw

strength-controlling flaw observed in some fraction of test specimens that cannot be present in the

component being designed

Note 1 to entry: An example is machining flaws in ground bend specimens that will not be present in as-sintered

components of the same material.
2 © ISO 2019 – All rights reserved
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SIST EN ISO 20501:2023
ISO 20501:2019(E)
3.2 Mechanical testing
3.2.1
effective gauge section

that portion of the test specimen geometry included within the limits of integration (volume, area or

edge length) of the Weibull distribution function

Note 1 to entry: In tensile specimens, the integration may be restricted to the uniformly stressed central gauge

section, or it may be extended to include transition and shank regions.
3.2.2
fractography

analysis and characterization of patterns generated on the fracture surface of a test specimen

Note 1 to entry: Fractography can be used to determine the nature and location of the critical fracture origin

causing catastrophic failure in an advanced ceramic test specimen or component.
3.3 Statistical terms
3.3.1
confidence interval
interval within which one would expect to find the true population parameter

Note 1 to entry: Confidence intervals are functionally dependent on the type of estimator utilized and the sample

size. The level of expectation is associated with a given confidence level. When confidence bounds are compared

to the parameter estimate one can quantify the uncertainty associated with a point estimate of a population

parameter.
3.3.2
confidence level

probability that the true population parameter falls within a specified confidence interval

3.3.3
estimator
function for calculating an estimate of a given quantity based on observed data

Note 1 to entry: The resulting value for a given sample may be an estimate of a distribution parameter (a point

estimate) associated with the underlying population, e.g. the arithmetic average of a sample is an estimator of

the distribution mean.
3.3.4
population
collection of data or items under consideration
3.3.5
probability density function
pdf
function f (x) for the continuous random variable X if
fx ≥ 0 (1)
and
fx dx = 1 (2)

Note 1 to entry: The probability that the random variable X assumes a value between a and b is given by

Pr aX< () ()
© ISO 2019 – All rights reserved 3
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SIST EN ISO 20501:2023
ISO 20501:2019(E)
3.3.6
cumulative distribution function

function F (x) describing the probability that a continuous random variable X takes a value less than or

equal to a number x

Note 1 to entry: Therefore, the cumulative distribution function (cdf) is related to the probability density

function f (x) by
Fx =−Pr ∞< Xx< = fx´´dx (4)
() () ()

Differentiating Formula (4) with respect to x shows that the pdf is simple the derivative of the cdf:

dF x
fx = (5)

Note 2 to entry: According to 3.3.5, F (x) is a monotonically increasing function in the range between 0 and 1.

3.3.7
ranking estimator

function that estimates the probability of failure to a particular strength measurement within a

ranked sample
3.3.8
sample
collection of measurements or observations taken from a specified population
3.3.9
statistical bias

type of consistent numerical offset in an estimate relative to the true underlying value, inherent to most

estimates
3.3.10
unbiased estimator
estimator that has been corrected for statistical bias error
3.4 Weibull distributions
3.4.1
Weibull distribution

continuous distribution function which can be used to describe empirical data from measurements

where continuous random variable x has a two-parameter Weibull distribution if the probability

density function is given by
mm−1
 
mx   x 
 
fx = expw− henx ≥0 (6)
    
ββ β
 
    
 
fx =<00when x (7)
and the cumulative distribution function is given by
 
 
 
Fx =−10exp − whenx ≥ (8)
 
 
 
 
4 © ISO 2019 – All rights reserved
---------------------- Page: 16 ----------------------
SIST EN ISO 20501:2023
ISO 20501:2019(E)
Fx()=<00whenx (9)
where
m is the Weibull modulus (or the shape parameter) (>0);
β is the Weibull scale parameter (>0).

Note 1 to entry: The random variable representing uniaxial tensile strength of an advanced ceramic will assume

only positive values. If the random variable representing uniaxial tensile strength of an advanced ceramic is

characterized by Formulae (6) to (9), then the probability that this advanced ceramic will fail under an applied

uniaxial tensile stress σ is given by the cumulative distribution function.
 
 
 
P =−10exp − whenσ ≥ (10)
 
 σ 
 q 
 
P =<00whenσ (11)
where
P is the probability of failure;
σ is the Weibull characteristic strength.

Note 2 to entry: The Weibull characteristic strength is dependent on the uniaxial test specimen (tensile, flexural,

or pressurized ring) and will change with specimen geometry. In addition, the Weibull characteristic strength

has units of stress, and has to be reported using SI-units of Pa, or adequately in MPa or GPa.

Note 3 to entry: An alternative expression for the probability of failure is given by

 
 
1 σ
 
P =−1 exp − dV whenσ >0 (12)
 
 V 
V σ
00 
 
P =≤00whenσ (13)

The integration in the exponential is performed over all tensile regions of the specimen volume (V) if the

strength-controlling flaws are randomly distributed through the volume of the material, or over all tensile

regions of the specimen area if flaws are restricted to the specimen surface. The integration is sometimes carried

out over an effective gauge section instead of over the total volume or area. In Formula (12), σ is the Weibull

material scale parameter and can be described as the Weibull characteristic strength of a specimen with unit

volume or area loaded in uniform uniaxial tension. For a given specimen geometry, Formulae (10) and (12) can be

1/m

combined, to yield an expression relating σ and σ (this means: σσV = ). Further discussion related to

0 θ
q 0
this issue can be found in Annex A.
4 Symbols
A specimen area
b gauge section dimension, base of bend test specimen
d gauge section dimension, depth of bend test specimen
f (x) probability density function
© ISO 2019 – All rights reserved 5
---------------------- Page: 17 ----------------------
SIST EN ISO 20501:2023
ISO 20501:2019(E)
F(x) cumulative distribution function
L likelihood function
L length of the inner load span for a bend test specimen
L length of the outer load span for a bend test specimen
m Weibull modulus
estimate of the Weibull modulus
unbiased estimate of the Weibull modulus
N number of specimens in a sample
P probability of failure

q intermediate quantity defined in 6.5.1, used in calculation of confidence bounds

r number of specimens that failed from the flaw population for which the Weibull estimators

are being calculated

t intermediate quantity defined by Formula (22), used in calculation of confidence bounds

UF unbiasing factor
V tensile loaded region of specimen volume
V unit size volume
V effective volume
eff
x realization of a random variable X
X random variable
β Weibull scale parameter
σ uniaxial tensile stress
estimate of mean strength
σ maximum stress in the j th test specimen at failure

σ Weibull material scale parameter (strength relative to unit size) defined in Formula (12)

estimate of the WeibuII material scale parameter

σ Weibull characteristic strength (associated with a test specimen) defined in Formula (10)

estimate of the Weibull characteristic strength
5 Significance and use

5.1 This document enables the experimentalist to estimate Weibull distribution parameters from

failure data. These parameters permit a description of the statistical nature of fracture of fine ceramic

materials for a variety of purposes, particularly as a measure of reliability as it relates to strength data

6 © ISO 2019 – All rights reserved
---------------------- Page: 18 ----------------------
SIST EN ISO 20501:2023
ISO 20501:2019(E)

utilized for mechanical design purposes. The observed strength values are dependent on specimen size

 

and geometry. Parameter estimates can be computed for a given specimen geometry (,m σ ) but it is

suggested that the parameter estimates be t
...

SLOVENSKI STANDARD
oSIST prEN ISO 20501:2022
01-oktober-2022
Fina keramika (sodobna keramika, sodobna tehnična keramika) - Weibullova
statistika za podatke o trdnosti (ISO 20501:2019)

Fine ceramics (advanced ceramics, advanced technical ceramics) - Weibull statistics for

strength data (ISO 20501:2019)
Hochleistungskeramik - Weibullstatistik von Festigkeitswerten (ISO 20501:2019)

Céramiques techniques - Analyse statistique de Weibull des données de résistance à la

rupture (ISO 20501:2019)
Ta slovenski standard je istoveten z: prEN ISO 20501
ICS:
81.060.30 Sodobna keramika Advanced ceramics
oSIST prEN ISO 20501:2022 en,fr,de

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
oSIST prEN ISO 20501:2022
---------------------- Page: 2 ----------------------
oSIST prEN ISO 20501:2022
INTERNATIONAL ISO
STANDARD 20501
Second edition
2019-03
Fine ceramics (advanced ceramics,
advanced technical ceramics) —
Weibull statistics for strength data
Céramiques techniques — Analyse statistique de Weibull des données
de résistance à la rupture
Reference number
ISO 20501:2019(E)
ISO 2019
---------------------- Page: 3 ----------------------
oSIST prEN ISO 20501:2022
ISO 20501:2019(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2019

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2019 – All rights reserved
---------------------- Page: 4 ----------------------
oSIST prEN ISO 20501:2022
ISO 20501:2019(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

3.1 Defect populations ............................................................................................................................................................................... 1

3.2 Mechanical testing ............................................................................................................................................................................... 3

3.3 Statistical terms ...................................................................................................................................................................................... 3

3.4 Weibull distributions ......................................................................................................................................................................... 4

4 Symbols .......................................................................................................................................................................................................................... 5

5 Significance and use .......................................................................................................................................................................................... 6

6 Method A: maximum likelihood parameter estimators for single flaw populations ....................7

6.1 General ........................................................................................................................................................................................................... 7

6.2 Censored data .......................................................................................................................................................................................... 8

6.3 Likelihood functions ........................................................................................................................................................................... 8

6.4 Bias correction ........................................................................................................................................................................................ 8

6.5 Confidence intervals ........................................................................................................................................................................10

7 Method B: maximum likelihood parameter estimators for competing flaw populations ....13

7.1 General ........................................................................................................................................................................................................13

7.2 Censored data .......................................................................................................................................................................................14

7.3 Likelihood functions ........................................................................................................................................................................14

8 Procedure..................................................................................................................................................................................................................15

8.1 Outlying observations ....................................................................................................................................................................15

8.2 Fractography ..........................................................................................................................................................................................15

8.3 Graphical representation ............................................................................................................................................................16

9 Test report ................................................................................................................................................................................................................18

Annex A (informative) Converting to material-specific strength distribution parameters .....................19

Annex B (informative) Illustrative examples ............................................................................................................................................21

Annex C (informative) Test specimens with unidentified fracture origin ..................................................................28

Annex D (informative) Fortran program .......................................................................................................................................................31

Bibliography .............................................................................................................................................................................................................................36

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Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso

.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 206, Fine ceramics.

This second edition cancels and replaces the first edition (ISO 20501:2003), which has been technically

revised. It also incorporates the Technical Corrigendum ISO 20501:2003/Cor.1:2009.

The main changes compared to the previous edition are as follows:
— the terms and definitions in Clause 3 have been updated and modified;

— a method to treat a higher number of specimens (N > 120) has been introduced for method A:

maximum likelihood parameter estimators for single flaw populations;

— in Annex D, example codes have been added for calculating the maximum likelihood parameters of

the Weibull distribution with modern analysis software.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/members .html.
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Introduction

Measurements of the strength at failure are taken for one of two reasons: either for a comparison of

the relative quality of two materials regarding fracture strength, or the prediction of the probability

of failure for a structure of interest. This document permits estimates of the distribution parameters

which are needed for either. In addition, this document encourages the integration of mechanical

property data and fractographic analysis.
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INTERNATIONAL STANDARD ISO 20501:2019(E)
Fine ceramics (advanced ceramics, advanced technical
ceramics) — Weibull statistics for strength data
1 Scope

This document covers the reporting of uniaxial strength data and the estimation of probability

distribution parameters for advanced ceramics which 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 brought to failure 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 document is restricted to the assumption that the distribution underlying the failure strengths is

the two-parameter Weibull distribution with size scaling. Furthermore, this document is restricted to

test specimens (tensile, flexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress

states. Subclauses 6.4 and 6.5 outline methods of correcting 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 (i.e. a single failure mode). In samples where failures originate

from multiple independent flaw populations (e.g. competing failure modes), the methods outlined in 6.4

and 6.5 for bias correction and confidence bounds are not applicable.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
NOTE See also Reference [1].
3.1 Defect populations
3.1.1
flaw

inhomogeneity, discontinuity or (defect) feature in a material, which acts as stress concentrator due to

a mechanical load and has therefore a certain risk of mechanical failure

Note 1 to entry: The flaw becomes critical if it acts as fracture origin in a failed specimen.

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3.1.2
censored data

strength measurements (i.e. a sample) containing suspended observations such as that produced by

multiple competing or concurrent flaw populations

Note 1 to entry: Consider a sample where fractography clearly established the existence of three concurrent

flaw distributions (although this discussion is applicable to a sample with any number of concurrent flaw

distributions). The three concurrent flaw distributions are referred to here as distributions A, B, and C. Based

on fractographic analyses, each specimen strength is assigned to a flaw distribution that initiated failure. In

estimating parameters that characterize the strength distribution associated with flaw distribution A, all

specimens (and not just those that failed from type-A flaws) shall be incorporated in the analysis to ensure

efficiency and accuracy of the resulting parameter estimates. The strength of a specimen that failed by a

type-B (or type-C) flaw is treated as a right censored observation relative to the A flaw distribution. Failure

due to a type-B (or type-C) flaw restricts, or censors, the information concerning type-A flaws in a specimen

[2]

by suspending the test before failure occurs by a type-A flaw . The strength from the most severe type-A

flaw in those specimens that failed from type-B (or type-C) flaws is higher than (and thus to the right of) the

observed strength. However, no information is provided regarding the magnitude of that difference. Censored

data analysis techniques incorporated in this document utilize this incomplete information to provide efficient

and relatively unbiased estimates of the distribution parameters.
3.1.3
competing failure modes

distinguishably different types of fracture initiation events that result from concurrent (competing)

flaw distributions
3.1.4
compound flaw distribution

any form of multiple flaw distribution that is neither pure concurrent, nor pure exclusive

Note 1 to entry: A simple example is where every specimen contains the flaw distribution A, while some fraction

of the specimens also contains a second independent flaw distribution B.
3.1.5
concurrent flaw distribution
competing flaw distribution

type of multiple flaw distribution in a homogeneous material where every specimen of that material

contains representative flaws from each independent flaw population

Note 1 to entry: Within a given specimen, all flaw populations are then present concurrently and are competing

to each other to cause failure.
3.1.6
exclusive flaw distribution
mixture flaw distribution

type of multiple flaw distribution created by mixing and randomizing specimens from two or more

versions of a material where each version contains a different single flaw population

Note 1 to entry: Thus, each specimen contains flaws exclusively from a single distribution, but the total data set

reflects more than one type of strength-controlling flaw.
3.1.7
extraneous flaw

strength-controlling flaw observed in some fraction of test specimens that cannot be present in the

component being designed

Note 1 to entry: An example is machining flaws in ground bend specimens that will not be present in as-sintered

components of the same material.
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3.2 Mechanical testing
3.2.1
effective gauge section

that portion of the test specimen geometry included within the limits of integration (volume, area or

edge length) of the Weibull distribution function

Note 1 to entry: In tensile specimens, the integration may be restricted to the uniformly stressed central gauge

section, or it may be extended to include transition and shank regions.
3.2.2
fractography

analysis and characterization of patterns generated on the fracture surface of a test specimen

Note 1 to entry: Fractography can be used to determine the nature and location of the critical fracture origin

causing catastrophic failure in an advanced ceramic test specimen or component.
3.3 Statistical terms
3.3.1
confidence interval
interval within which one would expect to find the true population parameter

Note 1 to entry: Confidence intervals are functionally dependent on the type of estimator utilized and the sample

size. The level of expectation is associated with a given confidence level. When confidence bounds are compared

to the parameter estimate one can quantify the uncertainty associated with a point estimate of a population

parameter.
3.3.2
confidence level

probability that the true population parameter falls within a specified confidence interval

3.3.3
estimator
function for calculating an estimate of a given quantity based on observed data

Note 1 to entry: The resulting value for a given sample may be an estimate of a distribution parameter (a point

estimate) associated with the underlying population, e.g. the arithmetic average of a sample is an estimator of

the distribution mean.
3.3.4
population
collection of data or items under consideration
3.3.5
probability density function
pdf
function f (x) for the continuous random variable X if
fx ≥ 0 (1)
and
fx dx = 1 (2)

Note 1 to entry: The probability that the random variable X assumes a value between a and b is given by

Pr aX< () ()
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3.3.6
cumulative distribution function

function F (x) describing the probability that a continuous random variable X takes a value less than or

equal to a number x

Note 1 to entry: Therefore, the cumulative distribution function (cdf) is related to the probability density

function f (x) by
Fx =−Pr ∞< Xx< = fx´´dx (4)
() () ()

Differentiating Formula (4) with respect to x shows that the pdf is simple the derivative of the cdf:

dF x
fx = (5)

Note 2 to entry: According to 3.3.5, F (x) is a monotonically increasing function in the range between 0 and 1.

3.3.7
ranking estimator

function that estimates the probability of failure to a particular strength measurement within a

ranked sample
3.3.8
sample
collection of measurements or observations taken from a specified population
3.3.9
statistical bias

type of consistent numerical offset in an estimate relative to the true underlying value, inherent to most

estimates
3.3.10
unbiased estimator
estimator that has been corrected for statistical bias error
3.4 Weibull distributions
3.4.1
Weibull distribution

continuous distribution function which can be used to describe empirical data from measurements

where continuous random variable x has a two-parameter Weibull distribution if the probability

density function is given by
mm−1
 
mx   x 
 
fx = expw− henx ≥0 (6)
    
ββ β
 
    
 
fx =<00when x (7)
and the cumulative distribution function is given by
 
 
 
Fx =−10exp − whenx ≥ (8)
 
 
 
 
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Fx()=<00whenx (9)
where
m is the Weibull modulus (or the shape parameter) (>0);
β is the Weibull scale parameter (>0).

Note 1 to entry: The random variable representing uniaxial tensile strength of an advanced ceramic will assume

only positive values. If the random variable representing uniaxial tensile strength of an advanced ceramic is

characterized by Formulae (6) to (9), then the probability that this advanced ceramic will fail under an applied

uniaxial tensile stress σ is given by the cumulative distribution function.
 
 
 
P =−10exp − whenσ ≥ (10)
 
 σ 
 q 
 
P =<00whenσ (11)
where
P is the probability of failure;
σ is the Weibull characteristic strength.

Note 2 to entry: The Weibull characteristic strength is dependent on the uniaxial test specimen (tensile, flexural,

or pressurized ring) and will change with specimen geometry. In addition, the Weibull characteristic strength

has units of stress, and has to be reported using SI-units of Pa, or adequately in MPa or GPa.

Note 3 to entry: An alternative expression for the probability of failure is given by

 
 
1 σ
 
P =−1 exp − dV whenσ >0 (12)
 
 V 
V σ
00 
 
P =≤00whenσ (13)

The integration in the exponential is performed over all tensile regions of the specimen volume (V) if the

strength-controlling flaws are randomly distributed through the volume of the material, or over all tensile

regions of the specimen area if flaws are restricted to the specimen surface. The integration is sometimes carried

out over an effective gauge section instead of over the total volume or area. In Formula (12), σ is the Weibull

material scale parameter and can be described as the Weibull characteristic strength of a specimen with unit

volume or area loaded in uniform uniaxial tension. For a given specimen geometry, Formulae (10) and (12) can be

1/m

combined, to yield an expression relating σ and σ (this means: σσV = ). Further discussion related to

0 θ
q 0
this issue can be found in Annex A.
4 Symbols
A specimen area
b gauge section dimension, base of bend test specimen
d gauge section dimension, depth of bend test specimen
f (x) probability density function
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F(x) cumulative distribution function
L likelihood function
L length of the inner load span for a bend test specimen
L length of the outer load span for a bend test specimen
m Weibull modulus
estimate of the Weibull modulus
unbiased estimate of the Weibull modulus
N number of specimens in a sample
P probability of failure

q intermediate quantity defined in 6.5.1, used in calculation of confidence bounds

r number of specimens that failed from the flaw population for which the Weibull estimators

are being calculated

t intermediate quantity defined by Formula (22), used in calculation of confidence bounds

UF unbiasing factor
V tensile loaded region of specimen volume
V unit size volume
V effective volume
eff
x realization of a random variable X
X random variable
β Weibull scale parameter
σ uniaxial tensile stress
estimate of mean strength
σ maximum stress in the j th test specimen at failure

σ Weibull material scale parameter (strength relative to unit size) defined in Formula (12)

estimate of the WeibuII material scale parameter

σ Weibull characteristic strength (associated with a test specimen) defined in Formula (10)

estimate of the Weibull characteristic strength
5 Significance and use

5.1 This document enables the experimentalist to estimate Weibull distribution parameters from

failure data. These parameters permit a description of the statistical nature of fracture of fine ceramic

materials for a variety of purposes, particularly as a measure of reliability as it relates to strength data

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utilized for mechanical design purposes. The observed strength values are dependent on specimen size

 

and geometry. Parameter estimates can be computed for a given specimen geometry (,m σ ) but it is

suggested that the parameter estimates be transformed and reported as material-specific parameters

 

(,m σ ). In addition, different flaw distributions (e.g. failures due to inclusions or machining damage)

may be observed, and each will have its own strength distribution parameters. The procedure for

transforming parameter estimates for typical specimen geometries and flaw distributions is outlined in

Annex A.

5.2 This document provides two approaches, method A and method B, which are appropriate for

different purposes.

Method A provides a simple analysis for circumstances in which the nature of strength-defining flaws

is either known or assumed to be from a single population. Fractography to identify and group test

items with given flaw types is thus not required. This method is suitable for use for simple material

screening.

Method B provides an analysis for the general case in which competing flaw populations exist. This

method is appropriate for final component design and analysis. The method requires that fractography

be undertaken to identify the nature of strength-limiting flaws and assign failure data to given flaw

population types.

5.3 In method A, a strength data set can be analysed and values of the Weibull modulus and

 

characteristic strength (,m σ ) are produced, together with confidence bounds on these parameters. If

necessary, the estimate of the mean strength can be computed. Finally, a graphical representation of the

failure data along with a test report can be prepared. It should be noted that the confidence bounds are

frequently widely spaced, which indicates that the results of the analysis should not be used to extrapolate

far beyond the existing bounds of probability of failure. A necessary assumption for a valid extrapolation

(with respect to the tested effective volume V and/or small probabilities of failure) is that the flaw

eff
populations in all considered strength test pieces are of the same type.

5.4 In method B, begin by performing a fractographic examination of each failed specimen in order

to characterize fracture origins. Screen the data associated with each flaw distribution for outliers. If all

failures originate from a single flaw distribution compute an unbiased estimate of the Weibull modulus,

and compute confidence bounds for both the estimated Weibull modulus and the estimated Weibull

characteristic strength. If the failures originate from more than one flaw type, separate the data sets

associated with each flaw type, and subject these individually to the censored analysis. Finally, prepare a

graphical representation of the failure data along with a test report. When using the results of the analysis

for design purposes it should be noted that there is an implicit assumption that the flaw populations in

the strength test pieces and the components are of the same types.
6 Method A: maximum likelihood parameter estimators for single flaw
populations
6.1 General

This document outlines the application of parameter estimation methods based on the maximum

likelihood technique (see also References [13], [14], [20] and [21]). This technique has certain

advantages. The parameter estimates obtained using the maximum likelihood technique are unique

(for a two-parameter Weibull distribution), and as the size of the sample increases, the estimates

statistically approach the true values of the population more efficiently than other parameter

estimation techniques.
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6.2 Censored data

The application of the techniques presented in this document can be complicated by the presence of

test specimens that fail from extraneous flaws, fractures that originate outside the effective gauge

section, and unidentified fracture origins. If these complications arise, the strength data from these

specimens should generally not be discarded. Strength data from specimens with fracture origins

[3]

outside the effective gauge section and from specimens with fractures that originate from extraneous

flaws should b
...

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