1、 Reference number ISO 20501:2003(E) ISO 2003INTERNATIONAL STANDARD ISO 20501 First edition 2003-12-01 Fine ceramics (advanced ceramics, advanced technical ceramics) Weibull statistics for strength data Cramiques techniques Statistiques Weibull des donnes de rsistance ISO 20501:2003(E) PDF disclaimer
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6、quester. ISO copyright office Case postale 56 CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii ISO 2003 All rights reservedISO 20501:2003(E) ISO 2003 All rights reserved iiiContents Page Foreword iv 1 Scope 1 2 Terms a
7、nd definitions. 1 2.1 Defect populations 1 2.2 Mechanical testing 2 2.3 Statistical terms 3 2.4 Weibull distributions. 4 3 Symbols . 5 4 Significance and use 6 5 Method A: maximum likelihood parameter estimators for single flaw populations 7 5.1 General. 7 5.2 Censored data . 7 5.3 Likelihood functi
8、ons . 7 5.4 Bias correction 8 5.5 Confidence intervals. 9 6 Method B: maximum likelihood parameter estimators for competing flaw populations 11 6.1 General. 11 6.2 Censored data . 12 6.3 Likelihood functions . 12 7 Procedure. 13 7.1 Outlying observations 13 7.2 Fractography . 13 7.3 Graphical repres
9、entation 13 8 Test report 16 Annex A (informative) Converting to material-specific strength distribution parameters 17 Annex B (informative) Illustrative examples 19 Annex C (informative) Test specimens with unidentified fracture origin . 26 Annex D (informative) Fortran program . 29 Bibliography .
10、33 ISO 20501:2003(E) iv ISO 2003 All rights reservedForeword 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. Ea
11、ch 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 Internatio
12、nal Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standar
13、ds adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this document may be the subject
14、of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO 20501 was prepared by Technical Committee ISO/TC 206, Fine ceramics. INTERNATIONAL STANDARD ISO 20501:2003(E) ISO 2003 All rights reserved 1Fine ceramics (advanced ceramics, advanced technical cera
15、mics) Weibull statistics for strength data 1 Scope This International Standard 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 con
16、tinuous 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
17、the underlying population distribution. This International Standard is restricted to the assumption that the distribution underlying the failure strengths is the two-parameter Weibull distribution with size scaling. Furthermore, this International Standard is restricted to test specimens (tensile, f
18、lexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress states. Subclauses 5.4 and 5.5 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
19、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 5.4 and 5.5 for bias correction and confidence bounds are not applicable. Measurements of the strength at fail
20、ure are taken for one of two reasons: either for a comparison of the relative quality of two materials, or the prediction of the probability of failure (or alternatively the fracture strength) for a structure of interest. This International Standard permits estimates of the distribution parameters w
21、hich are needed for either. In addition, this International Standard encourages the integration of mechanical property data and fractographic analysis. 2 Terms and definitions For the purposes of this document, the following terms and definitions apply. 2.1 Defect populations 2.1.1 censored strength
22、 data strength measurements (i.e., a sample) containing suspended observations such as that produced by multiple competing or concurrent flaw populations NOTE Consider a sample where fractography clearly established the existence of three concurrent flaw distributions (although this discussion is ap
23、plicable 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 paramete
24、rs that characterize the strength distribution associated with flaw distribution A, all specimens (and not just those that failed from type-A flaws) must be incorporated in the analysis to assure efficiency and accuracy of the resulting parameter estimates. The strength of a specimen that failed by
25、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 by suspending the test before failure occurs by a type-A flaw 2. The strengt
26、h 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 Inte
27、rnational Standard utilize this incomplete information to provide efficient and relatively unbiased estimates of the distribution parameters. ISO 20501:2003(E) 2 ISO 2003 All rights reserved2.1.2 competing failure modes distinguishably different types of fracture initiation events that result from c
28、oncurrent (competing) flaw distributions 2.1.3 compound flaw distributions any form of multiple flaw distribution that is neither pure concurrent, nor pure exclusive NOTE A simple example is where every specimen contains the flaw distribution A, while some fraction of the specimens also contains a s
29、econd independent flaw distribution B. 2.1.4 concurrent flaw distributions a type of multiple flaw distribution in a homogeneous material where every specimen of that material contains representative flaws from each independent flaw population NOTE Within a given specimen, all flaw populations are t
30、hen present concurrently and are competing with each other to cause failure. This term is synonymous with “competing flaw distributions”. 2.1.5 exclusive flaw distributions a type of multiple flaw distribution created by mixing and randomizing specimens from two or more versions of a material where
31、each version contains a different single flaw population NOTE Thus, each specimen contains flaws exclusively from a single distribution, but the total data set reflects more than one type of strength-controlling flaw. This term is synonymous with “mixture flaw distributions”. 2.1.6 extraneous flaws
32、strength-controlling flaws observed in some fraction of test specimens that cannot be present in the component being designed NOTE An example is machining flaws in ground bend specimens that will not be present in as-sintered components of the same material. 2.2 Mechanical testing 2.2.1 effective ga
33、uge 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 In tensile specimens, the integration may be restricted to the uniformly stressed central gauge section, or it may be extended to i
34、nclude transition and shank regions. 2.2.2 fractography the analysis and characterization of patterns generated on the fracture surface of a test specimen NOTE Fractography can be used to determine the nature and location of the critical fracture origin causing catastrophic failure in an advanced ce
35、ramic test specimen or component. 2.2.3 proof testing applying a predetermined load to every test specimen (or component) in a batch or a lot over a short period of time to ascertain if the specimen fails due to a serious strength limiting defect NOTE This procedure, when applied to all specimens in
36、 the sample, removes potentially weak specimens and modifies the statistical characteristics of the surviving samples. ISO 20501:2003(E) ISO 2003 All rights reserved 32.3 Statistical terms 2.3.1 confidence interval interval within which one would expect to find the true population parameter NOTE Con
37、fidence 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
38、 a population parameter. 2.3.2 confidence level probability that the true population parameter falls within a specified confidence interval 2.3.3 estimator well-defined function that is dependent on the observations in a sample NOTE The resulting value for a given sample may be an estimate of a dist
39、ribution parameter (a point estimate) associated with the underlying population. The arithmetic average of a sample is, e.g., an estimator of the distribution mean. 2.3.4 population totality of potential observations about which inferences are made 2.3.5 population mean the average of all potential
40、measurements in a given population weighted by their relative frequencies in the population 2.3.6 probability density function function f (x) is a probability density function for the continuous random variable X if f (x) W 0 (1) and () 1 fxd x = (2) NOTE The probability that the random variable X a
41、ssumes a value between a and b is given by ()( ) b a Pr a X b f x dx = (3) 2.3.7 ranking estimator function that estimates the probability of failure to a particular strength measurement within a ranked sample 2.3.8 sample collection of measurements or observations taken from a specified population
42、2.3.9 skewness term relating to the asymmetry of a probability density function NOTE The distribution of failure strength for advanced ceramics is not symmetric with respect to the maximum value of the distribution function but has one tail longer than the other. ISO 20501:2003(E) 4 ISO 2003 All rig
43、hts reserved2.3.10 statistical bias inherent to most estimates, this is a type of consistent numerical offset in an estimate relative to the true underlying value NOTE The magnitude of the bias error typically decreases as the sample size increases. 2.3.11 unbiased estimator estimator that has been
44、corrected for statistical bias error 2.4 Weibull distributions 2.4.1 Weibull distribution continuous random variable X has a two-parameter Weibull distribution if the probability density function is given by () 1 exp when 0 mm mx x fx x = (4) or f (x) = 0 when x u 0 (5) and the cumulative distributi
45、on function is given by ()1e x p w h e n 0 m x Fx x = (6) or F(x) = 0 when x u 0 (7) where m is the Weibull modulus (or the shape parameter) ( 0); is the Weibull scale parameter ( 0) NOTE 1 The random variable representing uniaxial tensile strength of an advanced ceramic will assume only positive va
46、lues, and the distribution is asymmetrical about the mean. These characteristics rule out the use of the normal distribution (as well as others) and point to the use of the Weibull and similar skewed distributions. If the random variable representing uniaxial tensile strength of an advanced ceramic
47、is characterized by Equations 4 to 7, then the probability that this advanced ceramic will fail under an applied uniaxial tensile stress is given by the cumulative distribution function f 1e x p w h e n 0 m P = (8) P f= 0 when u 0 (9) where P fis the probability of failure; is the Weibull characteri
48、stic strength. ISO 20501:2003(E) ISO 2003 All rights reserved 5NOTE 2 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,
49、and should be reported using units of MPa or GPa. NOTE 3 An alternative expression for the probability of failure is given by f 0 1e x p w h e n 0 m V Pd V = (10) P f= 0 when u 0 (11) The integration in the exponential is performed over all tensile regions of the specimen volume if the strength-controlling flaws are randomly distributed through the volume of the material, or over all tens
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