Determination of the Weibull parameters from the mean value and the coefficient of variation of the measured strength for brittle ceramics

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 σ0, 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 σ0 can also be estimated accurately.


Introduction
Weibull statistics [1] has been widely employed to model the variability in the fracture strength of brittle ceramics [2][3][4][5][6][7][8]. In general, a two-parameter form of the Weibull function is adopted to give the cumulative failure probability, P, of a component or specimen at a given applied stress, , as [2]: where m is called the Weibull modulus and 0  is the scale parameter with the same dimension as .
Different methods have been used to estimate the Weibull parameters, m and 0  , the most popular being the least-square (LS) method and the maximum-likelihood (ML) method. Many studies [9][10][11][12][13][14][15][16][17][18] based on Monte Carlo simulation have shown that each of these methods has its benefits and drawbacks.
According to the theory of statistics, the expected value, ( ) E  , and the variance, ( ) D  , of the twoparameter Weibull distribution can be expressed as [19]: Thus, the coefficient of variation for this distribution, v w ( ) c , is Equation (4) shows that v w ( ) c is dependent only on the Weibull modulus, m, and decreases as m increases  when m > 1. Based on this relation, Gong et al. [20,21] proposed an empirical equation for determining the Weibull modulus, es m , from the coefficient of variation, v f ( ) c , of measured fracture strength: where  and  are both constants dependent on the sample size, and were determined and formulated by analyzing the data produced by Monte Carlo simulation in the original works of Gong et al. [20,21]. It should be noted that Gong et al.'s method is only for the determination of Weibull modulus, and the other Weibull parameter, the scale parameter 0  , should be determined still with the conventional methods, i.e., the LS method or the ML method. Undoubtedly, this drawback would impede the application of this simple method.
The aim of this paper is to contribute supplementary information towards a more accurate determination of the Weibull modulus based on Eq. (5) and then establish an approach for the direct determination of the scale parameter 0  . Considering its versatility and simplicity, the LS method was employed in the present work to yield the referenced Weibull parameters for the analyses and comparison.

Monte Carlo simulation procedure
The basic idea of the Monte Carlo simulation can be described as: suppose we have a material whose fracture strength variation follows a two-parameter distribution of known parameters, tr m and 0 tr ( )  (for the sake of simplicity, 0  is set to be unity throughout the whole study); thus we can choose N random specimens of this material to yield a sample, "measure" the fracture strength of each specimen by assigning a random number in the internal of 0-1 as its corresponding fracture probability, P, and then estimate the Weibull parameters. In continuation of this idea, a computer program is written to establish the relationships between the estimated Weibull parameters, es m and Thus, a sample containing N strength values, 1  , 2  , …, N  , is obtained.
(ii) The yielded strength values are ordered from the lowest to the highest and the ith result in the set of N strength data, i  , is assigned a cumulative fracture probability, i P , which is calculated with [22][23][24]: The strengths and the probabilities are then analyzed, by using a simple, least-square regression method, according to the alternative form of Eq. (1): Then coefficient of variation, v f ( ) c , is given by To study the effect of sample size, N is increased progressively from 10 to 200. Similarly, tr m is increased progressively from 5 to 25 to study the effect of Weibull modulus. For each set of given sample size, N, and the prescribed Weibull modulus, tr , m the above-mentioned procedure is repeated for 100 000 times in order to ensure statistical convergence of the results.

1 Refinement of Gong et al.'s method
In the original works of Gong et al. [20,21], by repeating the above-mentioned Monte Carlo simulation procedure 4000 times for a given N and tr m = 2, 3, 4, …, 25, 26, 100 000 (= 4000  25) samples of strength data were produced. Then, a least-square regression analysis was applied with all the resultant 100 000 estimated parameters, es m and v f ( ) , c 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 tr m = 2.5, 10, 20. It can be seen from the insert, which shows the enlargement of the small tr m region, that the data points produced with tr m = 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 tr 20. m  Clearly, most of the data points corresponding to tr 2.5 m  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 corresponding to tr 2.5 m  locate below the best-fit line (solid line in Fig. 1).
To get a clearer understanding on the experimental phenomena shown in Fig. 1, 100 000 samples of strength data were produced with given N and tr . m Then the resultant data were analyzed according to Eq.
(5) to yield  and . Figure 2 shows the best-fit value of the parameter  as a function of the prescribed tr m for some selected N. As can be seen,  strongly depends on the prescribed tr m value, especially when the sample size N is smaller. On the other hand, the best-fit value of the parameter  is found to be very small, less than 510 5 in each case. These results imply that, each set of the es mv f ( ) c data points produced with a given tr m would fall along a straight line starting from the origin and with a slope dependent of tr m . From the viewpoint of statistics, fitting the data produced with different tr m to a single straight line would result in some uncertainties in the best-fit values of  and .
As mentioned above, in the original works of Gong et al. [20,21], the regression analyses for determining  and  were conducted within tr m ranging from 2 to 26 and only 4000 data produced with each prescribed tr m were used. Based on the above discussion, such a simulation procedure seems to be insufficient to ensure the statistical convergence and warrant a further improvement.
Therefore, three key issues were considered here to improve Gong Fig. 3. Also shown in Fig. 3 are the best-fit lines obtained by analyzing the data according to Eq. (5). It can be seen that, for each given sample size, N, an excellent linear relationship between es m and v f ( ) c is obtained and, in each case, the regression analysis  The best-fit values of the parameters  and  are shown in Fig. 4. Similar to the results obtained in the original work of Gong and Li [21], the sample-size dependence of  and  can be respectively described with the following empirical equations: 0.54 0.38exp 0.14 exp 8.98 48.25 The results given in Eqs. (12) and (13) are only slightly different from those reported by Gong and Li [21]. But it is clear, based on the above discussion, that Eqs. (12) and (13) would yield more accurate results for the determination of the parameters  and .

2 Statistical comparison of the estimated Weibull parameters
It is necessary to make a statistical comparison between the two estimators, one calculated directly from Eq. (5) (denoted as 1 m ) and the other estimated by the conventional LS method (denoted as 2 m ).    that estimated with the conventional LS method. Note that the standard deviation of 1 m is slightly smaller than that of 2 m . One can expect that the statistical distribution of 1 m would be somewhat narrower than that of 2 m . Figure 7 shows as examples the empirical probability density functions of es tr / m m estimated with the simple method and the LS method. In constructing these plots, the resultant 100 000 es tr / m m with tr m = 10 were classified into groups that fell into definite intervals. The number of estimators that fell into each interval was countered and normalized through division by 100 000 to produce the relative frequency of occurrence, the y-value, and the x-value is simply the midpoint of the given interval. Clearly, there is no distinct statistical difference between 1 t r / m m and 2 t r / m m , indicating that the simple method proposed in the present study is suitable for the determination of Weibull modulus.

3 Determination of the scale parameter
In the original works of Gong et al. [20,21], no discussion was made for the determination of the scale parameter, 0 .
 Even the Weibull modulus can be determined directly with the coefficient of variation of the strength sample, another method, e.g., LS method or ML method, is still required to estimate the scale parameter. This is one of the main drawbacks which may impede the application of this simple method. Now we try to establish an approach for the direct determination of the scale parameter 0 .
 Similar to the determination of Weibull modulus, we start with the expression of the expected value of the two-parameter Weibull modulus, Eq. (2).
As shown in Fig. 8, for x ranging from 1.0 to 1.4, ( ) x  can be approximately expressed as a polynomial form: where f  is the mean value of the measured strength and es m is calculated with Eq. (11). Once es m is obtained from Eq. (5), the  function in Eq. (15) can be calculated with Eq. (14).
Similarly, hereafter the 0 es ( )  determined with Eqs. (14) and (15)    Although Monte Carlo simulation has been widely used to evaluate the statistical properties of Weibull modulus estimated by different methods [9][10][11][12][13][14][15][16][17][18], only a little effort [10,12] has been devoted to the statistical properties of the estimated scale parameter 0  . This is because that, for each conventional point-estimation method, no significant variation occurs in determining 0  [12]. Figure 10 shows the empirical probability density functions for  Fig. 9, indicate that, for the conventional point-estimation methods such as LS method, 0  can be estimated with a higher degree of accuracy than m. Figure 11 compares the empirical probability density curves of

4 Examples of application
It is interesting to examine the joint effect of the    Table 2 of Ref. [25] in which the strength of a sintered silicon nitride was measured in three-point bending with specimens of 3 mm in thickness and 4 mm in width. Dataset 2 was selected from Fig. 1  there exists a good agreement between the two sets of estimated parameters, giving a sound support for the above discussion. Using these estimated values, the Weibull lines for each dataset can be drawn directly with Eq. (1). As can be seen in Fig. 12, the Weibull lines drawn based on the simple method are overlapped with those based on the conventional LS method.

Conclusions
Based on Monte Carlo simulation, a simple method was proposed in the present study for the determination of the Weibull parameters, m and 0  . This simple method can be summarized as (i) An unbiased Weibull modulus, es , m can be calculated directly with Eq. (5) using the coefficient of variation, v f ( ) c , of the strength sample. The parameters included in Eq. (5) are shown in Fig. 4 and can also be calculated with Eqs. (12) and (13).
(ii) Using es m calculated with Eq. (5) and the mean value of the strength in the considered sample, Detailed analyses confirmed that both m and 0  obtained by this simple method are statistically identical with those estimated with the conventional LS method. Especially, the standard deviations of both m and 0  obtained by this simple method are slightly smaller than those estimated with the conventional LS method, implying that the simple method would yield a more accurate estimation of the Weibull parameters.