Estimator type and population size for estimating the weibull modulus in ceramics

doi:10.1016/S0955-2219(97)00165-9

Journal of the European Ceramic Society

Volume 18, Issue 5, May 1998, Pages 451-455

https://doi.org/10.1016/S0955-2219(97)00165-9 Get rights and content

Abstract

Analysis of fracture data in ceramics can be made by applying probabilistic theories. In order the fracture probability to be expressed an estimator must be used. For economic reasons it is also necessary, the population size for estimating the shape ceramic parameter, to be found. In the present work a number of estimators and sets of different sample populations of kaolin ceramics are tested against the stability of the weibull shape parameter. The 2-parameter and 3-parameter Weibull functions and the Neville function were compared for best fitting of results. It seems that modulus remains almost constant in the range of the specimen numbers tested and a minimum number of about 30 specimens is acceptable for reliability predictions at least for the examined type of ceramic.

Introduction

The statistical function most commonly used in analysis of strength data of ceramics is the cumulative distribution function proposed by Weibull.[1] using the theorem of the product of independent event probabilities (the probability of an event comprising a number of independent events is equal to the product of the probabilities of the individual events considered separately).

Generally, the Weibull function, is a three parameter equation and it has the following form: $P_{f} =1−e^{−∫xx−x_{u}x_{u}^{m}dy} for x≻x_{u} P_{f} =0 for x≤x_{m}$

where:

The distribution of extreme values is influenced by the type of the function used to manipulate the data. Three types of asymptotic Extreme Value distributions have been developed.[2] The type I is unbounded in the direction of extreme value, the type II do not possess finite moments, and the type III is bounded in the direction of extreme value. The Weibull function use extreme value statistics and is a type III asymptotic extreme value distribution also known as Fisher–Tipper Type III distribution of smaller values and as the third asymptotic distribution of smallest extreme value.[3] Kittl and Diaz[4] proposed five mathematical ways for deduction of this function.

Another statistical function based in WLT theory where initiation of the failure takes place in the weakest part of the failure-prone volume is an extreme value function recently developed by Neville,5, 6, 7, 8 who tried to give a physical meaning to his function by considering the stress and strain near the crack tip. He considered that in a piece of material containing many microcracks, the failure-prone volume will increase in direct proportion to load. The failure-prone volume near the tip of the short length of crack front, δl, is linearly proportional to K ⁴, (K=stress intensity factor), or P ⁴ (P=load), and he expressed it as sampling S [S=P ⁴×(volume of piece), when all the pieces in a set have the same sizes S=P ⁴]. Neville considered a cumulative failure probability depending on the size of the surviving population and on some function g(S). He supposed that his function has the following form: $g S = S B_{p}^{D_{p}}$

He developed the following expression as a simple statistical criterion for the failure of pieces containing microcracks, supposing that failure will take place when sampling is sufficiently reached. This sufficiency of sampling is statistically distributed according to the function: $P_{f} = S B_{p}^{D_{p}} 1+ S B_{p}^{D_{p}}$

where: P_f is the cumulative probability of failure, S is the sampling, always linearly proportional to the actual amount of material loaded at greater than a given stress or strain. (It is defined as Ki ⁴, Ji ⁴, COD², and its effect, g(S) is equal to (S/B_p)^Dp, B_p is a scale parameter which can change with temperature and which normalises S (it is actually the median value of S for all values of D_p), D_p is a shape parameter which allows the shape to change with temperature. Different D_p values for a given material are also expected for different distributions of stresses.

In order for the fracture probability to be expressed an estimator must be used. The expected values of the probability of failure P_f can be obtained from the two estimators used by Trustrum and Jayatilaka:[9] $P_{f} = i N+1$

and $P_{f} + i−0·5 N$

where i is the ith (ranked) order of failure; and N is the total number of specimens experimentally tested.

They demonstrated that when less than 50 specimens are used for statistical analysis, the first of the above estimators give a more biased m. Other proposed estimators, (used also by Neville in his calculations[10]), follow the expressions:11, 12, 13 $P_{f} = i−0·3 N+0·4$

and $P_{f} = i− 38 N+ 14$

The first estimator corresponds to the median probability of failure. Examination of the statistical properties of the four estimators using Monte-Carlo simulation techniqu,[13] have shown that the popular $P_{f} = i N+1$ estimator gives the more biased m. This means that the value of m is lower and gives a higher and so, more conservative probability of failure and from engineering point of view is the best choice in reliability prediction[13] and is used in many statistical applications.[14] The formula $P_{f} = i+0·5 /N$ gives for N more than 20, the less biased m values and probably is the most preferable from a materials science point of view. Asloun et al.,[15] using the above estimator, have shown that the estimator and the sample size when it is higher than about 20 do not influence the results.

Results from flexural and tensile strength measurements on a carbon fibre–borosilicate composite[16] showed a poor fit of Weibull statistics, which was attributed to the small number (12) of tested specimens. Research has shown11, 17 that a sample size of about 30 specimens is acceptable for estimating Weibull parameters in ceramics and generally brittle materials.

Section snippets

Experimental Details

A series of brittle ceramic materials such as Remblend China clay from ECC International,[18] a pottery mixture and a brick clay were used for manufacturing all the test specimens. Details on fabrication, sintering programs, chemical, XRD analysis and AE analysis are given elsewhere.[19] All the batches of the 29, 63, 94 and 144 brick-clay samples were produced from the same raw material and under the same manufacturing and sintering conditions.

Results and Discussion

The effect of estimator type on Weibull Modulus was examined in several formulations and sintering temperatures and as an example the results on the K_IC values of kaolin notched specimens sintered at 1100°C and tested at 3-point bending, and are presented in Table 1. Graphical representation of the results are also shown in Fig. 1. The effect of sample size (population) was studied on 29, 63, 94 and 144 brick-clay samples fired at 900°C for 24 h and is presented in Table 2. The Weibull

Conclusions

In most of the cases, literature survey shows results from simulated data using the Monte Carlo technique. In the present work, ceramic fracture data show that the $P_{f} = i−0·5 N$ estimator gives the least bias to the m Weibull modulus, and the $P_{f} =pt i −0·3 n +0·4$ estimator is very close to the average value of the four examined estimators. The $P_{f} = i N+1$ estimator gives the more conservative m and reliability predictions using this modulus, give lower reliability values and it this sense Weibull function

References (19)

Weibull, W., J. Applied Mech., 1951, 18,...
Bury, K. V., Statistical Models in Applied Science. J. Wiley and Sons, New York,...
Kapur, K. C. and Lamperson, L. R., Reliability in Engineering Design. J. Wiley & Sons, New York,...
Kittl, P. and Diaz, G., Eng. Frac. Mech, 1990, 36(5),...
Neville, D. J., Proc. R. Soc. Lond., 1987, A. 410,...
Neville, D. J., Int. J. of Fracture, 1987, 34,...
Neville, D. J., Int. J. of Fracture, 1990, 36,...
Neville, D. J., Inter. J. Fracture, 1990, 44,...
Trustrum, K. and Jayatilaka, A. De. S., J. Mat. Sci., 1979, 14,...

There are more references available in the full text version of this article.

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