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1、JIS JAPANESE INDUSTRLAL STANDARD Weibull statistics of strength data for fine ceramics JIS R 1625-l9% Translated and Published bY Japanese Standards Association UDC 666.5:620.17 Printed in Japan 7s Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/
2、1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-JIS R Lb25-ENGL L77b 9733b08 053b1107 8TT W In the event of any doubt arising, the original Standard in Japanese is to be final authority Errata for JE (Eng
3、lish edition) are printed i n Standardization Journal, published monthly by the Japanese Standards Association. Errata will be provided upon request, please contact: Business Department, Japanese Standards Association 4-1-24, Akasaka, Minato-ku, Tokyo, JAPAN 107 TEL. 03-3583-8002 FAX. 03-3583-0462 J
4、 Errata are also provided to subscribers of JIS (English edition) in Monthly Infomation. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted
5、without license from IHS -,-,- STD*JIS R Lb25-ENGL L99b Li733b08 053b1i08 73b D UDC 666.5:620.17 JAPANESE INDUSTRIAL STANDARD J I S Weibull statistics of strength data R 1625-1996 for fine ceramics 1. two population parameter Weibull statistics which is a method for obtaining the form population par
6、ameter (Weibull coefficient) and scale population parameter of the instant break strength data obtained from the bending strength test and ten- sile strength test of fine ceramics at room temperature and elevated temperature. Scope This Japanese Industrial Standard specifies the single mode and Rema
7、rks: The following standards are cited in this Standard: JIS R 1600 JIS R 1601 JIS R 1604 JIS R 1606 JIS 2 8401 Glossary of terms relating to fine ceramics Testing method for flexural strength (modulus of rupture) of fine ceramics Testing method for flexural strength (modulus of rupture) of fine cer
8、amics at elevated temperature Testing methods for tensile strength of fine ceram- ics at room and elevated temperature Rules for rounding off of numerical values 2. R 1600 and the following definitions apply: Definitions For the purpose of this Standard, the definitions given in JIS (1) single mode
9、Weibull distribution Weibull distribution for brittle fracture due to only one kind of breaking cause. determines the form of probability density function in Weibull distribution and is an assessed value in practice. The larger this value the smaller the extension of strength distribution. scale pop
10、ulation parameter (oo) Strength at which the cumulative break- ing probability becomes 63.2% in two population parameter Weibull distri- bution and is an assessed value in practice. (2) form population parameter (Weibull coefficient: m) A parameter which (3) (4) two population parameter Weibull dist
11、ribution The Weibull distribution in which the location population parameter, out of three parameters of form population parameter, scale population parameter and location population parameter, is zero. (5) maximum likelihood method A method for estimating population param- eter from the strength da
12、ta. The method to determine the population pa- rameter in such a way that the probability (likelihood) of the obtained observing value to be realized is made the maximum. (6) Weibull plot graduated with the vertical axis In ln(1- F) -l and the horizontal axis lna (F: cumulative breaking probability,
13、 o: strength). ranking method of data probability from an order number of strength or breaking stress data, To plot the strength data on Weibull probability paper (7) A method for calculating the cumulative breaking Copyright Japanese Standards Association Provided by IHS under license with JSALicen
14、see=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- STD*JIS R Lb25-ENGL 177b q733b08 053b407 b72 2 R 1625-1996 3. For the analysis of strength data, the single mode and two population parameter
15、Weibull distribu- tion function shown by the following formula is used: Weibull distribution function used for statistical analysis F(a) = i-exp- - (:)“I where, F (o) : single mode and two population parameter Weibull distribution function m : form population parameter CT : strength or breaking stre
16、ss (Pa or N/mm2) ao : scale population parameter (Pa or N/mm2) 4. Statistical analvzine method 4.1 bending strength test data and tensile strength test data, and their test methods shall, as a rule, be as follows: (1) Bending strength test method shall be as specified in JIS R 1601 or JIS R 1604. St
17、rength data The data to be the object for statistical analysis shall be (2) (3) Tensile strength test method shall be as specified in JIS R 1606. The strength test should preferably be carried out on 30 test pieces or more in either case. 4.2 parameter Obtain the form population parameter m and the
18、scale population parameter ao, which are the unknown population parameters of the Weibull dis- tribution function, by means of the maximum likelihood method, and make the Weibull plot diagram. Estimating method of form population parameter and scale population The procedure shall be as follows: (1)
19、Calculate a tentative assessed value m, of the form population parameter r n by the following formula obtained by applying the maximum likelihood method to the Weibull distribution function or its probability density func- tion. i=i where, ai (i = 1 to n) : strength data of i-th sample n : number of
20、 strength data (2) Calculate the scale population parameter ao by the following formula and round it off to three significant figures in accordance with the specification in JIS Z 8401. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001,
21、User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- 8 3 R 1625-1996 where, rn, : tentative assessed value of form population pa- rameter (3) Calculate the form population parameter rn by the following formula and round it of
22、f to three significant digits in accordance with the specification in JIS Z 8401. m = B,. mt where, Bn : correction factor Bn is the function of the number of test pieces n, and is as follows: in case of n 5120, B, = (2.04-n-,+ l ) - in case of n 120, B, = 1 (4) Weibull plot of strength data The pro
23、cedure of Weibull plot shall be as follows: (a) Change the position of the strength data of n pieces to be statistically analyzed in the order from smaller strength. Calculate the cumulative breaking factor Fi using the following for- mula (median - rank method) for the data of each ranking i (i = 1
24、 to n) and make the corresponding table between i, Di and Fi. (b) i-0.3 n+0.4 F. =- where, n : number of strength data (total number of test pieces) i : ranking after changing to the order from smaller strength Fi : i-th cumulative breaking probability (c) According to the ranking i of the strength
25、in (b), plot the sets of oi and Fi on a Weibull probability paper graduated by In ln(1- F)-l in vertical axis and by lno in horizontal axis. Substitute the form population parameter rn and the scale population parameter ao obtained in (2) and (3) into the following formula and draw the regression st
26、raight line of Weibull plot. (d) 1 1- F In In- = rn In o - m In a , where, F : cumulative breaking probability rn : form population parameter obtained in (3) CT : strength or breaking stress (Pa or N/mm2) ao : scale population parameter obtained in (2) (Pa or N/mm2) Copyright Japanese Standards Asso
27、ciation Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- 4 R 1625-1996 5. Record The following information shall be recorded: (i) Test methods for s
28、trength (three-point bending test, four-point bending test and tensile test, and identification of elevated temperature or room tem- perature) Shape and dimension of test piece (2) (3) Number of test pieces (4) Strength data (5) Average value of strength (6) Result of analysis (a) Form population pa
29、rameter rn (b) Scale population parameter o , (c) Weibull plot diagram J Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license
30、from IHS -,-,- STD-JIS R Lb25-ENGL L77b i933b08 053bliL2 Lb7 5 R 1625-1996 Informative reference Example of analysis for estimation of population parameter using maximum likelihood method Preface estimation of population parameter in the single mode and two population param- eter Weibull statistics
31、of strength data for fine ceramics and does not form a part of the Standard. This Informative reference intends to show an example of analysis for 1. example of analysis are shown in Informative reference Attached Table 1. Used data The examples of the measured data which are used in this Procedure
32、The procedure shall be as follows: Carry out numerical analysis on the strength data in Informative reference Attached Table 1 by means of a computer using the formula shown in 4.2 (i) in the body and calculate the tentative assessed value rnt of the form population parameter rn. In this case, nt=25
33、.50. Substitute mt obtained in (i) and the strength data in Informative reference Attached Table 1 into the formula shown in 4.2 (2) of the body to calculate the scale population parameter ao. If rn, calculated in (i) is used, o . = 975.7. Because rn, obtained in (i) includes the deviation depending
34、 on the number of test pieces peculiar to the maximum likelihood method, calculate the correction factor B, in compliance with the number of test pieces using the formula shown in 4.2 (3) of the body and, calculate the form population parameter m by multiplying the tentative assessed value rn, by th
35、is correc- tion factor. When n = 30, B, = 0.9538, therefore if mt calculated in (i) is used, rn = 24.32. Make the Weibull plot diagram in accordance with the following procedure: Change the arrangement of the strength data shown in Informative reference Attached Table 1 in the order from smaller one
36、, and give them rank i. The results are shown in three columns on the left side of Informative reference Attached Table 2. By using the formula shown in 4.2 (4) (b) of the body, calculate the cumulative breaking probability Fi corresponding to rank i and, in addition, calculate Y; = In ln(1- Fi)-1,
37、Xj = Inai. The results are shown in three columns on the right side of Informa- tive reference Attached Table 2. Plot the set ofXi and Y i on Weibull probability paper. The example when using the data in Informative reference Attached Table 2 is shown in Informative reference Attached Fig. 1. Copyri
38、ght Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- 6 R 1625-1996 (d) After substituting the scale population parame
39、ter a , , and the form popu- lation parameter m obtained in (2) and (3) respectively into the formula shown in 4.2 (4) (cl of the body, calculate the respective strength a for two arbitrary cumulative breaking probability F, and connect these two points (o, FI by a straight line to make it the regre
40、ssion straight line. An example of regression straight line is shown in Informative refer- ence Attached Fig. 1. (5) An example of the program based on FORTRAN which analyzes numerically the procedures in (i) to (3) by the Newton-Raphson method and carries out the correcting calculation, and an exam
41、ple of the output of calculation re- sult when the strength data in Informative reference Attached Table 1 are inputted in this program are shown below: Remarks: In the numerical analysis by Newton-Raphson method, the m coordinate mi+l of the intersection of tangential line at g (mi) and M axis, for
42、 the tentative setting value mi of rn in the graph graduated on the longitudinal axis g (m) and the lateral axis m, is obtained and mi is replaced by it. By repeating this calcula- tion, it is possible to calculate the m value precise enough to obtain three significant digits. In this case, the rela
43、tion between mi and mi+l is shown by the following formula. For the initialized value of mi, if M calculated roughly, for instance, from the maximum value and the minimum value of strength data is used, the convergence is fast. In addition, oim in the formula has a possibility to overflow according
44、to computers, but it is possible to avoid by arranging the formula by obtaining the average value a , of the strength data and replacing Cai“ with O,“Z(OJCT)“. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for
45、Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-JIS R Lb25-ENGL L99b M LI933b08 053bLILLi T I T 7 R 1625-1996 a O . . E v M . 8 .e .d m o a . e E M .FI 8 U U ho I E U r( + E C u O h O I Fi * * n ea * * I d n a n * * * v h n h l-l n I % . 0 O
46、GI s O Fi O B O o * E 9 * m s E h U w + m m i - d a h O O O O rl n h ea # I w z v - + II 4 n i II R ri II U O rn l-l I O 9 II O 2 n O o II Ba“ H R II 2 5 O n O0 O O A F i * rn Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing,
47、 Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- STD.JIS R lb25-ENGL l99b q933b08 053b1i15 97b H 8 R 1625-1996 h a2 Y i E m a .d 6“ Y O Li O O E 8 .C Y te 7 e E m m a Y o .d .d m E .- 2 .d 2 a Y Y Cu O a: 8 2 2 . F I Y V o 3 .C Y a
48、 1 a M a2 M o fn cci O Cu O h a2 Q s C E 3 m Cu O Cu O 8 8 .M Y 8 Cu O .la 7 a 7 O .la Cu O Y 1 1 O ,a Y 1 7 ,a O c, 7 ,a 8 $ 8 r n Q Q a + , + a + + I S g II II II II Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/14/2007 02:58:04 MDTNo reproduction or networking permitted without license from IHS -,-,- STD-JIS R lb25-ENGL