Weibull Statistical Distribution

Despite the use of identical specimens, the mechanical properties of ceramics show considerable scatter in the measured results. The main reason for the scatter in the values measured is a consequence of the presence, size and distribution of cracks in ceramics. A mean value must be determined via statistical evaluation. The most commonly used statistical approach for describing experimental data is Gaussian normal distribution. In ceramics, however, the use of the Weibull distribution is preferable, reviewed below. 

Weibull distribution assumes that each elemental part of a bulk material has an individual property, i.e., a local strength. The fracture or failure probability, Pf, of each such element is integrated over the entire test piece, giving  

In cases of fracture with a fracture strength (or stress), 7f, the first term, under the integral in the nominator in the above relations, may be replaced to obtain  

The higher is m and the lower represents strength variability. Ve is the effective volume of the specimen and may be expressed as  

Vg is the first term of the integral being equal to V, since, in the nominator, a = Of for brittle materials  

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Two necessary conditions are required by the Weibull statistical distribution all samples should have the same size and distribution of tensile stress all samples contain a unified and uniform distribution of defects. The derivation of Weibull s distribution is as follows. 

W. A. Weibull, Statistical Distribution Function of Wide Applicability, J. Appl. Mech. 18(3), 293 297 (1951). 

Weibull, W. A Statistical Distribution Function of Wide Application. 7. Appl. Mech., Vol. 18, 1951, pp. 293. 

W. Weibull. A statistical distribution of wide applicability. J. Appl. Meehan. 1951, 38, 293-297. 

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

Weibull. W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293-297. Wells, J.K. and Beaumont, P.W.R. (1985). Debonding and pull-out processes in fibrous composites. J. Mater. Sci. 20, 1275-1284. 

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

The first step in the evaluation of an air quality standard is to select the statistical distribution that supposedly best fits the data. We will assume that the frequency distribution that best fits hourly averaged ozone concentration data is the Weibull distribution. Since the standards are expressed in terms of expected events during a 1-year period of 1-h average concentrations, we will always use the number of trials m equal to the number of hours in a year, 8760. We would use m < 8760 only to evaluate the parameters of the distribution if some of the 8760 hourly values are missing from the data set. 

All the models are based upon standard statistical distributions. The Multi-Hit model is based upon a Poisson distribution. The Probit model is based upon a normal distribution, and the One Hit, Multistage and Weibull models rely upon linear probabilities. Such distributions have proven applicability in dealing with substantial percentages of the population (up to 1 in 20). However, the models lose precision when they are pushed to extremes such as 1 in 10, such as encountered in risk assessment. Furthermore, homeostatic mechanisms such as DMA repair and immunological survellance may be poorly evaluated in risk assessment. Such high doses are administered to achieve the siaximum tolerated dose, that these mechanisms are surely overwhelmed in the animal studies. Additionally it should be noted that a risk of one in a million does not mean one tumor in the lifetime of a million people, it means that each individual has one chance in a million of developing a tumor in a lifetime. 

WEI39] Weibull A (1939) A Statistical Distribution Function of Wide Applicability. Ingenibrs Vetenskap Akademien, Handlingar No 154... 

Where (s) = is infinitesimal strength statistical distribution function, taking the form of Weibull distribution, namely ... 

The Weibull model cannot describe the volume dependence of strength data [22], although aWeibull modulus (m) can be extracted from the statistical distribution of strengths m is in the range 20-29. This value provides an evaluation of the scatter in strength data. 

The DA equation (4.2-5) is obtained by assuming the temperature invariance of the adsorption potential at constant loading and a choice of the Weibull s distribution to describe the filling of micropore over the differential molar work of adsorption. It can be shown to be a special case of an isotherm equation derived from the statistical mechanical principles when the loading is appreciable (Chen and Yang, 1994). They derived the following isotherm... 

Another feature of this material is its narrow distribution of the fracture strength. When the strength distribution was expressed in Weibull statistics, the Weibull modulus was 46 - substantially higher than the value of 26 obtained for a conventional, self-reinforced silicon nitride. Thus, the seeded and tape-cast silicon nitride showed a synergistic improvement in all of the important fracture attributes, such as fracture strength, fracture toughness, and strength stability (Weibull modulus). 

Weibull W, A statistical distribution function of wide apphcability, J Appl Mech, 18, 293, 1951. Tibbetts GG, Doll GL, Gorkiewicz DW, Moleski JJ, Perry TA, Physical properties of vapor-grown carbon-fibers. Carbon, 31(7), 1039 1047, 1993. 

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