1、Designation: E 2283 03Standard Practice forExtreme Value Analysis of Nonmetallic Inclusions in Steeland Other Microstructural Features1This standard is issued under the fixed designation E 2283; the number immediately following the designation indicates the year oforiginal adoption or, in the case o
2、f revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice describes a methodology to statisticallycharacterize the distribution of the la
3、rgest indigenous nonme-tallic inclusions in steel specimens based upon quantitativemetallographic measurements. The practice is not suitable forassessing exogenous inclusions.1.2 Based upon the statistical analysis, the nonmetalliccontent of different lots of steels can be compared.1.3 This practice
4、 deals only with the recommended testmethods and nothing in it should be construed as defining orestablishing limits of acceptability.1.4 The measured values are stated in SI units. For mea-surements obtained from light microscopy, linear feature pa-rameters shall be reported as micrometers, and fea
5、ture areasshall be reported as micrometers.1.5 The methodology can be extended to other materials andto other microstructural features.1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to est
6、ablish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E 3 Methods of Preparation of Metallographic SpecimensE 7 Terminology Relating to MetallographyE 45 Test Methods for Determining the Incl
7、usion Contentof SteelE 178 Practice for Dealing with Outlying ObservationsE 456 Terminology Relating to Quality and StatisticsE 768 Practice for Preparing and Evaluating Specimens forAutomatic Inclusion Assessment of SteelE 883 Guide for Reflected-Light PhotomicrographyE 1122 Practice for Obtaining
8、JK Inclusion Ratings UsingAutomatic Image AnalysisE 1245 Practice for Determining the Inclusion Content orSecond-Phase Constituent of Metals by Automatic ImageAnalysis3. Terminology3.1 DefinitionsFor definitions of metallographic termsused in this practice, refer to Terminology, E 7; for statistical
9、terms, refer to Terminology E 456.3.2 Definitions of Terms Specific to This Standard:3.2.1 Afthe area of each field of view used by the ImageAnalysis system in performing the measurements.3.2.2 Aocontrol area; total area observed on one specimenper polishing plane for the analysis. Aois assumed to b
10、e 150mm2unless otherwise noted.3.2.3 Nsnumber of specimens used for the evaluation. Nsis generally six.3.2.4 Npnumber of planes of polish used for the evalua-tion, generally four.3.2.5 Nfnumber of fields observed per specimen plane ofpolish.Nf5AoAf(1)3.2.6 Ntotal number of inclusion lengths used for
11、 theanalysis, generally 24.N 5 Ns Np(2)3.2.7 extreme value distributionThe statistical distribu-tion that is created based upon only measuring the largestfeature in a given control area or volume (1,2).3The continu-ous random variable x has a two parameter (Gumbel) ExtremeValue Distribution if the p
12、robability density function is givenby the following equation:fx! 51dFexpS2x 2ldDG3 expF2expS2x 2ldDG(3)and the cumulative distribution is given by the followingequation:1This practice is under the jurisdiction of ASTM Committee E04 on Metallog-raphy and is the direct responsibility of Subcommittee
13、E04.09 on Inclusions.Current edition approved Nov. 1. 2003. Published December 2003.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary p
14、age onthe ASTM website.3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.Fx! 5 exp2exp2x 2l! / d! (4)As applied to this practice, x, repres
15、ents the maximum feretdiameter, Length, of the largest inclusion in each control area,Ao, letting:y 5x 2ld(5)it follows that:Fy! 5 exp2exp2y! (6)andx 5dy 1l (7)3.2.8 lthe location parameter of the extreme value distri-bution function.3.2.9 dthe scale parameter of the extreme value distribu-tion func
16、tion.3.2.10 reduced variateThe variable y is called the reducedvariate. As indicated in Eq 6, y is related to the probabilitydensity function. That is y = F (P), then from Eq 6, it followsthat:y 52ln2lnFy! 52ln2lnP! (8)3.2.11 plotting positionEach of the N measured inclusionlengths can be represente
17、d as xi, where 1 # i # N. The datapoints are arranged in increasing order such that:x1# x2# x3# x4# x5.# xNThen the cumulative probability plotting position for datapoint xiis given by the relationship:Pi5iN 1 1(9)The fraction ( i /(N + 1) is the cumulative probability. F (yi)in Eq 8 corresponds to
18、data point xi.3.2.12 mean longest inclusion lengthLis the arithmeticaverage of the set of N maximum feret diameters of themeasured longest inclusions.L51N(i51i5NLi(10)3.2.13 standard deviation of longest inclusion lengthsSdev is the standard deviation of the set of N maximum feretdiameters of the me
19、asured longest inclusions.Sdev 5 (i51NLi2 L!2/ N 2 1!#0.5(11)3.2.14 return periodthe number of areas that must beobserved in order to find an inclusion equal to or larger than aspecified maximum inclusion length. Statistically, the returnperiod is defined as:T 511 2 P(12)3.2.15 reference area, Areft
20、he arbitrarily selected area of150 000 mm2. Arefin conjunction with the parameters of theextreme value distribution is used to calculate the size of thelargest inclusion reported by this standard. As applied to thisanalysis, the largest inclusion in each control area Aoismeasured. The Return Period,
21、 T, is used to predict how large aninclusion could be expected to be found if an area Areflargerthan Aowere to be evaluated. For this standard, Arefis 1000times larger than Ao. Thus, T is equal to 1000. By use of Eq 12it would be found that this corresponds to a probability valueof 0.999, (99.9 %).
22、Similarly by using Eq 6 and 7, the length ofan inclusion corresponding to the 99.99 % probability valuecould be calculated. Mathematically, another expression for thereturn period is:T 5ArefAo(13)3.2.16 predicted maximum inclusion length, Lmaxthelongest inclusion expected to be found in area Arefbas
23、ed uponthe extreme value distribution analysis.4. Summary of Practice4.1 This practice enables the experimenter to estimate theextreme value distribution of inclusions in steels.4.2 Generally, the largest oxide inclusions within the speci-mens are measured. However, the practice can be used tomeasur
24、e other microstructural features such as graphite nod-ules in ductile iron, or carbides in tool steels and bearing steels.The practice is based upon using the specimens described inTest Method E 45. Six specimens will be required for theanalysis. For inclusion analysis, an area of 150 mm2should beev
25、aluated for each specimen.4.3 After obtaining the specimens, it is recommended thatthey be prepared by following the procedures described inMethods E 3 and Practice E 768.4.4 The polished specimens are then evaluated by using theguidelines for completing image analysis described in PracticesE 1122 a
26、nd E 1245. For this analysis, feature specific measure-ments are required. The measured inclusion lengths shall bebased on a minimum of eight feret diameter measurements.4.5 For each specimen, the maximum feret diameter of eachinclusion is measured. After performing the analysis for eachspecimen, th
27、e largest maximum feret diameter of the measuredinclusions is recorded. This will result in six lengths. Theprocedure is repeated three more times. This will result in atotal of 24 inclusion lengths.4.6 The 24 measurements are used to estimate the values ofd and l for the extreme value distribution
28、for the particularmaterial being evaluated. The largest inclusion Lmaxexpectedto be in the reference area Arefis calculated, and a graphicalrepresentation of the data and test report are then prepared.4.7 The reference area used for this standard is 150 000mm2. Based upon specific producer, purchase
29、r requirements,other reference areas may be used in conjunction with thisstandard.4.8 When required, the procedure can be repeated to evalu-ate more than one type of inclusion population in a given set ofspecimens. For example, oxides and sulfides or titanium-carbonitrides could be evaluated from th
30、e same set of speci-mens.5. Significance and Use5.1 This practice is used to assess the indigenous inclusionsor second-phase constituents in metals using extreme valuestatistics.5.2 It is well known that failures of mechanical components,such as gears and bearings, are often caused by the presence o
31、fE2283032large nonmetallic oxide inclusions. Failure of a component canoften be traced to the presence of a large inclusion. Predictionsrelated to component fatigue life are not possible with theevaluations provided by standards such as Test Methods E 45,Practice E 1122, or Practice E 1245. The use
32、of extreme valuestatistics has been related to component life and inclusion sizedistributions by several different investigators (3-8). The pur-pose of this practice is to create a standardized method ofperforming this analysis.5.3 This practice is not suitable for assessing the exogenousinclusions
33、in steels and other metals because of the unpredict-able nature of the distribution of exogenous inclusions. Othermethods involving complete inspection such as ultrasonicsmust be used to locate their presence.6. Procedure6.1 Test specimens are obtained and prepared in accordancewith E 3, E 45 and E
34、768.6.2 The microstructural analysis is to be performed usingthe types of equipment and image analysis procedures de-scribed in E 1122 and E 1245.6.3 Determine the appropriate magnification to use for theanalysis. For accurate measurements, the largest inclusionmeasured should be a minimum of 20 pix
35、els in length. Forspecimens containing relatively large inclusions, objective lenshaving magnifications ranging from 10 to 203 will be ad-equate. Generally, for specimens with small inclusions, anobjective lens of 32 to 803 will be required. The samemagnification shall be used for all the specimens
36、to beanalyzed.6.4 Using the appropriate calibration factors, calculate thearea of the field of view observed by the image analysissystem, Af. For each specimen, an area of 150 mm2shall beevaluated. Using Eq 1, the number of fields of view required toperform the analysis is Nf= Ao/ Af= 150 / Af. Nfsh
37、ould berounded up to the next highest integer value; that is, if Nfiscalculated to be 632.31, then 633 fields of view shall beexamined.6.5 Image Analysis Measurements:6.5.1 In this practice, feature specific parameters are mea-sured for each individual inclusion. The measured inclusionlengths shall
38、be based on a minimum of eight feret diameters.6.5.2 For each field of view, focus the image either manu-ally or automatically, and measure the maximum feret diameterof each detected oxide inclusion. The measured feret diametersare stored in the computers memory for further analysis. Thisprocedure i
39、s repeated until an area of 150 mm2is analyzed.6.5.3 In situations where only a very few inclusions arecontained within the inspected area, the specimen can first beobserved at low magnification, and the location of the inclu-sions noted. The observed inclusions can then be remeasured athigh magnifi
40、cation.6.5.4 After the specimen is analyzed, using the accumulateddata, the maximum feret diameter of the largest measuredinclusion in the 150 mm2area is recorded. This procedure isrepeated for each of the other five specimens.6.5.5 The specimens are then repolished and the procedureis repeated unti
41、l each specimen has been evaluated four times.This will result in a set of 24 maximum feret diameters. Foreach repolishing step, it is recommended that at least 0.3 mmof material be removed in order to create a new plane ofobservation.6.5.6 The mean length, L, is then calculated using Eq 10.6.5.7 Th
42、e standard deviation, Sdev, is calculated using Eq11.6.6 The 24 measured inclusion lengths are sorted in ascend-ing order. An example of the calculations is contained inAppendix X1. The inclusions are then given a ranking. Thesmallest inclusion is ranked number 1, the second smallest isranked number
43、 2 etc.6.7 The probability plotting position for each inclusion isbased upon the rank. The probabilities are determined using Eq9: Pi= i /(N + 1). Where 1 # i # 24, and N = 24.6.8 A graph is created to represent the data. Plottingpositions for the ordinate are calculated from Eq 8: yi=ln(ln(Pi). The
44、 variable y in this analysis is referred to as theReduced Variate (Red. Var.). Typically the ordinate scaleranges from 2 through +7. This corresponds to a probabilityrange of inclusion lengths from 0.87 through 99.9 %. Theordinate axis is labeled as Red. Var. It is also possible toinclude the Probab
45、ility values on the ordinate. In this case, theordinate can be labeled Probability (%). The abscissa is labeledas Inclusion Length (mm); the units of inclusion length shall bemicrometers.6.9 Estimation of the Extreme Value Distribution Param-eters:6.9.1 Several methods can be used to estimate the pa
46、ram-eters of the extreme value distribution. Using linear regressionto fit a straight line to the plot of the Reduced Variate as afunction of inclusion length is the easiest method; however, itis the least precise. This is because the larger values of theinclusion lengths are more heavily weighted t
47、han the smallerinclusion lengths. Two other methods for estimating theparameters are the method of moments (mom), and the methodof maximum likelihood (ML). The method of moments is veryeasy to calculate, but the method of maximum likelihood givesestimates that are more precise. While both methods wi
48、ll bedescribed, the maximum likelihood method shall be used tocalculate the reported values of d and l for this standard.(Since the ML solution is obtained by numerical analysis, thevalues of d and l obtained by the method of moments are goodguesses for starting the ML analysis.)6.9.2 Moments Method
49、It has been shown that the param-eters for the Gumbel distribution, can be represented by:dmom5Sdev =6p(14)andlmom5 L2 0.5772 dmom(15)where the subscript mom indicates the estimates are basedon the moment method.6.9.3 Maximum Likelihood MethodThis method is basedon the approach that the best values for the parameters d and lare those estimates that maximize the likelihood of obtainingthe measured set of inclusion lengths. Since the extreme valuedistribution is based on a double exponential function, theE2283033maximization process i
