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Determination of the Weibull Parameters from the Mean Value and the Coefficient of Variation of the Measured Strength for Brittle Ceramics

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Determination of the Weibull Parameters from the Mean Value and the Coefficient of Variation of the Measured Strength for Brittle Ceramics Journal of Advanced Ceramics 2017, 6(2): 149–156 ISSN 2226-4108 DOI: 10.1007/s40145-017-0227-3 CN 10-1154/TQ Research Article Determination of the Weibull parameters from the mean value and the coefficient of variation of the measured strength for brittle ceramics Bin DENGa, Danyu JIANGb,* aDepartment of the Prosthodontics, Chinese PLA General Hospital, Beijing 100853, China bAnalysis and Testing Center for Inorganic Materials, State Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Shanghai 200050, China Received: January 11, 2017; Accepted: April 7, 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract: Accurate estimation of Weibull parameters is an important issue for the characterization of the strength variability of brittle ceramics with Weibull statistics. In this paper, a simple method was established for the determination of the Weibull parameters, Weibull modulus m and scale parameter 0 , based on Monte Carlo simulation. It was shown that an unbiased estimation for Weibull modulus can be yielded directly from the coefficient of variation of the considered strength sample. Furthermore, using the yielded Weibull estimator and the mean value of the strength in the considered sample, the scale parameter 0 can also be estimated accurately. Keywords: Weibull distribution; Weibull parameters; strength variability; unbiased estimation; coefficient of variation 1 Introduction likelihood (ML) method. Many studies [9–18] based on Monte Carlo simulation have shown that each of these Weibull statistics [1] has been widely employed to methods has its benefits and drawbacks. model the variability in the fracture strength of brittle According to the theory of statistics, the expected ceramics [2–8]. In general, a two-parameter form of the value, E( ) , and the variance, D( ) , of the two- Weibull function is adopted to give the cumulative parameter Weibull distribution can be expressed as [19]: failure probability, P, of a component or specimen at a 1 E()0 1 (2) given applied stress, , as [2]: m m 22 21 P 1exp (1) D()0 1 1 (3) mm 0 Thus, the coefficient of variation for this distribution, where m is called the Weibull modulus and 0 is the ()cvw, is scale parameter with the same dimension as . 1/2 Different methods have been used to estimate the 212 11 D() mm Weibull parameters, m and 0 , the most popular being ()cvw (4) the least-square (LS) method and the maximum- E() 1 1 m Equation (4) shows that ()c is dependent only on * Corresponding author. vw E-mail: [email protected] the Weibull modulus, m, and decreases as m increases www.springer.com/journal/40145 150 J Adv Ceram 2017, 6(2): 149–156 when m > 1. Based on this relation, Gong et al. [20,21] the strength value, i , with the prescribed mtr : proposed an empirical equation for determining the 1/mtr 1 Weibull modulus, mes , from the coefficient of variation, i ln (6) 1RND i ()cvf, of measured fracture strength: Thus, a sample containing N strength values, , m (5) 1 es ()cvf 2, …, N, is obtained. where and are both constants dependent on the (ii) The yielded strength values are ordered from the sample size, and were determined and formulated by lowest to the highest and the ith result in the set of N analyzing the data produced by Monte Carlo simulation strength data, , is assigned a cumulative fracture in the original works of Gong et al. [20,21]. i probability, Pi , which is calculated with [22–24]: It should be noted that Gong et al.’s method is only i 0.5 P (7) for the determination of Weibull modulus, and the other i N Weibull parameter, the scale parameter 0 , should be The strengths and the probabilities are then analyzed, determined still with the conventional methods, i.e., the by using a simple, least-square regression method, LS method or the ML method. Undoubtedly, this according to the alternative form of Eq. (1): drawback would impede the application of this simple 1 method. ln lnmmes lnes ln( 0 ) es (8) 1 Pi The aim of this paper is to contribute supplementary to give the estimated values of the Weibull parameters, information towards a more accurate determination of m and () . the Weibull modulus based on Eq. (5) and then establish es 0es an approach for the direct determination of the scale (iii) The mean values, f , and the standard deviation, parameter 0 . Considering its versatility and simplicity, Sf , of the N strength data are calculated as the LS method was employed in the present work to 1 N yield the referenced Weibull parameters for the analyses f i (9) N i1 and comparison. N 221 Sff()i (10) N i1 2 Monte Carlo simulation procedure Then coefficient of variation, ()cvf, is given by Sf ()cvf (11) The basic idea of the Monte Carlo simulation can be f described as: suppose we have a material whose To study the effect of sample size, N is increased fracture strength variation follows a two-parameter progressively from 10 to 200. Similarly, mtr is distribution of known parameters, mtr and () 0tr (for increased progressively from 5 to 25 to study the effect the sake of simplicity, 0 is set to be unity throughout of Weibull modulus. For each set of given sample size, the whole study); thus we can choose N random N, and the prescribed Weibull modulus, mtr , the specimens of this material to yield a sample, “measure” above-mentioned procedure is repeated for 100 000 the fracture strength of each specimen by assigning a times in order to ensure statistical convergence of the random number in the internal of 0–1 as its results. corresponding fracture probability, P, and then estimate the Weibull parameters. In continuation of this idea, a computer program is 3 Results and discussion written to establish the relationships between the 3. 1 Refinement of Gong et al.’s method estimated Weibull parameters, mes and () 0es, the mean values, f , and the coefficient of variation, ()cvf, In the original works of Gong et al. [20,21], by of the “measured” fracture strengths. repeating the above-mentioned Monte Carlo simulation (i) Firstly, a sample of random number, RND (i = 1, i procedure 4000 times for a given N and mtr = 2, 3, 4, …, 2, …, N) in the interval of 0–1 is produced to calculate 25, 26, 100 000 (= 4000 25) samples of strength data www.springer.com/journal/40145 J Adv Ceram 2017, 6(2): 149–156 151 were produced. Then, a least-square regression analysis was applied with all the resultant 100 000 estimated parameters, mes and (),cvf to yield the parameters and in Eq. (5). Such a treatment, however, seems to be questioned. Figure 1 shows, as an example, some data produced with N = 30 and mtr = 2.5, 10, 20. It can be seen from the insert, which shows the enlargement of the small mtr region, that the data points produced with mtr= 2.5 show large scatter, making it unreliable to fit these data to a straight line. The dashed line in Fig. 1 is obtained by least-square regression analysis of the data produced with mtr 20. Clearly, most of the data points corresponding to mtr 2.5 locate above the dashed line. If all the data shown in Fig. 1 were fitted according to Eq. (5), however, most of the data points Fig. 2 Best-fit value of as a function of the prescribed mtr . corresponding to mtr 2.5 locate below the best-fit line (solid line in Fig. 1). of statistics, fitting the data produced with different mtr To get a clearer understanding on the experimental to a single straight line would result in some phenomena shown in Fig. 1, 100 000 samples of uncertainties in the best-fit values of and . strength data were produced with given N and mtr . As mentioned above, in the original works of Gong et Then the resultant data were analyzed according to Eq. al. [20,21], the regression analyses for determining (5) to yield and . Figure 2 shows the best-fit value of and were conducted within mtr ranging from 2 to 26 the parameter as a function of the prescribed mtr for and only 4000 data produced with each prescribed mtr some selected N. As can be seen, strongly depends on were used. Based on the above discussion, such a the prescribed mtr value, especially when the sample simulation procedure seems to be insufficient to ensure size N is smaller. On the other hand, the best-fit value of the statistical convergence and warrant a further the parameter is found to be very small, less than improvement. 5 510 in each case. These results imply that, each set of Therefore, three key issues were considered here to the mes – ()cvf data points produced with a given mtr improve Gong et al.’s simulation procedure: would fall along a straight line starting from the origin (i) For each given set of N and mtr , 100 000 samples and with a slope dependent of mtr . From the viewpoint were analyzed to ensure the statistical convergence. (ii) Noting that the Weibull modulus has typical values ranging 5 to 20 for brittle ceramics, the smallest and the largest mtr were set to be 5 and 25, respectively. (iii) For a given N, the mean values of mes and ()cvf corresponding to each prescribed mtr , were employed for the regression analyses for determining the parameters and . The relationships between the mean values of mes and ()cvf for different sample size, N, are now shown in Fig.
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