Fracture statistics of brittle materials: Weibull or normal distribution

Abstract

Brittle materials, such as ceramics, rock, and concrete, etc., have been widely used in engineering for their excellent resistance to heat, corrosion, and wear. But brittle materials also break easily, and their strength, i.e., the maximum stress they can withstand, varies unpredictably from component to component even if a set of nominally identical specimens are tested under the same conditions. Therefore, the strength of a brittle material is not a well defined quantity and has to be described with respect to fracture statistics ͓1-3͔. Furthermore, the assessment of reliability of brittle materials also requires a probability approach.As is well known, the Weibull distribution with a flexible two-parameter analytic formula has been found to successfully describe a large body of fracture or fragmentation data, especially for brittle materials, which is therefore suggested to be considered first. As Weibull mentioned in his pioneering papers, however, the Weibull distribution should be considered as an empirical one on an equal footing with other distribution functions ͓4͔. The normal or Gaussian distribution is another widely used function. Other possible candidates, which can be used for probability density function of failure, involve the lognormal, power law, and Type I extreme value distributions, etc. ͓5,6͔. In general, we attempt to identify an appropriate model for the data using the so-called goodness-of-fit tests. However, for small sample sizes, it is difficult to distinguish between the Weibull and normal distributions. In this paper, we will propose a simple quantitative method, which can be used to highlight the difference between the Weibull and normal or other favorite distributions, and find out which model is better.It is often supposed that a small volume in a brittle material is like a chain of many links, and if any link breaks, then the whole material will fail. Based on this weakest-link principle and an empirical function, the cumulative probability of failure of a brittle material subjected to a load , i.e., the Weibull strength distribution can be represented as F() ϭ1Ϫexp(Ϫ͓(Ϫ th )/ 0 ͔ m ), where 0 is a normalized material strength, th is the threshold stress ͑below which no failure will occur͒, and m is the Weibull modulus ͓4͔. Here, the Weibull modulus is a measure of the degree of strength dispersion, and is also called the shape factor. Then, the probability density function of the three-parameter Weibull distribution f ()ϭdF()/d is given byIn most applications, th is usually taken as zero.On the other hand, if brittle materials are manufactured and handled without special care, their strengths usually exhibit more or less symmetrical distributions, so the normal distribution could be a natural one to apply to these data ͓5͔. For the normal distribution, its probability density function iswhere and ␣ are the mean and standard deviation, respectively. In order to find the unknown parameters in a distribution function, the usual way is the linear regression ͑least-squares͒ procedur...

Is Weibull distribution the most appropriate statistical strength distribution for brittle materials?

“…In view of the presented statistical analysis as well as that of Lu et al 7 , it is important to revisit the basic theory of Weibull, which links the statistical probability of fracture to the probability of finding a critical crack size in the tested sample. Further investigation should focus on rationalizing/ justifying other strength distribution function from the perspective of the probabilistic theory of brittle fracture.…”

“…It is observed that based on the Akaike Information Criterion (AIC), two-parameter Weibull or normal distributions fit better than the three-parameter Weibull distribution. Although two-parameter Weibull distribution has been widely used in practice to model strength data, Lu et al 7 questioned the uncritical use of Weibull distribution in general.…”

Fracture statistics of dental ceramics: Discrimination of strength distributions

“…There are also some cases, however, where the Weibull distribution fails to fit strength data of ceramics [12][13][14][15][16]. Danzer [12] reported that the M a n u s c r i p t 3 Weibull theory is inconsistent in estimating the strength of small specimens because their effective volumes become less than the fracture origin.…”

The influence of microstructure on the fracture statistics of polycrystalline diamond and polycrystalline cubic boron nitride

“…However, recent evidence has shown that this distribution may not always be the most accurate description of experimentally measured strength data for brittle materials. Lu et al [5,6] compared the strength data for three different ceramics and analysed them using the Weibull and normal distributions. They found that for ZnO, the strength data was best characterized by the normal rather than the Weibull distribution.…”