Weibull measurement

K. Fredriksson, I. Lindgren, S. Svanberg, G. Weibull Measurements of the emission from industrial smoke stacks using laser radar techniques. Goteborg Institute of Physics Reports GIPR-121 (CTH, Goteborg 1976)... 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

Tlie life of an automobile seal lias a Weibull distribution with failure rate Z(t) =, where t is measured in years. What is tlie probability that the... 

Three ranges of values of n were considered, >1, 0.7—1.0 and <0.7. When n> 1, and particularly when 3 < n < 4, the Weibull distribution readily reduces to a normal distribution if the Erofe ev function is symmetrical about a = 0.5. [The Weibull distribution is symmetrical for n = 3.26, i.e. (1 — In 2)-1, and the inflection point varies only slowly with n.] Thus, under these conditions (3 < n < 4 and symmetry about a = 0.5), we may derive the parameters of the corresponding normal distribution (where p defines the half-life of the reaction and the dispersion parameter, a, is a measure of the lack of homogeneity of the surface centres), viz. 

According to Table 1, semi-invariants of higher order characterize the shape of the profile in terms of variance, skewness, and kurtosis. The outstanding merit of the Weibull distribution is that its shape parameter a provides a summarizing measure for this property. For other distributions, the characterization of the shape is less obvious. 

Gulino, R. and Phoenix, L. (1991). Weibull strength statistics for graphite fibers measured from the break progression in a model graphite/glass/cpoxy microcomposites. J. Mater. Sci. 26, 3107-3118. 

Figure 10.15 The decay of the transverse magnetisation (points) for ethylene-octene copolymer at different temperatures [136]. The decay was measured using the solid-echo pulse sequence. The solid lines represent the result of a least-squares adjustment of the decay using a linear combination of Weibull and exponential functions. The dotted lines represent the relaxation component with a long decay time. In the experiments the sample was heated from room temperature to 343 K (70 °C)...

This measure of heterogeneity generalizes the notion of heterogeneity as a departure from the classical first-order model initially introduced [121] for the specific case of the Weibull function. In addition, the above equation can also be used for comparison between two experimentally obtained dissolution profiles [131]. 

Coppard et al (1989) measured the breakdown field of polyethylene plaques loaded with metallic particles in very small quantities. Their results concern the range of low p. One of the purposes of this study was to determine the distribution probability F E y) to distinguish between the Gumbel and the Weibull distributions. But it appears to be very difficult a task experimentally, because a large number of samples are needed for the study. However, they were able to verify that the average of the breakdown fields varies with p in accordance with (2.68). 

Mean values in particle dimension, thickness, length and slenderness ratio (length/thickness), used in this test are shown in Table 1. These mean values were based on two hundred measurements of each furnish type and calculated using the Weibull distribution function. 

All these quantities are readily determined. However, the determination of lc and Of (/c) necessitates a statistical analysis using the Weibull model. Gf (/c) cannot be measured directly, since /c is usually less than 0.5 mm. Therefore, it is determined from the tensile strength Gf (/) at higher gauge lengths using Equation 20... 

The expected number of exceedances Nx(m) of a given concentration level in m measurements is given by (26.63), which, in the case of the Weibull distribution,... 

The Weibull distribution is the state of the art statistics in the mechanical design process of ceramic components [1 - 3]. Strength testing and data evaluation are standardised. A sample of at least 30 specimens has to be tested. The range of measured failure probabilities increases with the sample size [3, 13] and is - for a sample of 30 specimens - very limited (it is between 1/60 and 59/60). To determine the design stress, the measured data have to be extrapolated with respect to the volume and to the tolerated failure probability. This often results in a very large extrapolation span [3]. 

It should be noted that on the basis of a small sample size, e.g. only 30 specimens, it is not possible to differentiate between a Weibull, a Gaussian, or any other similar distribution functions, as shown by Lu et al. [14] using statistical measures or by Danzer et al. [12] using Monte Carlo simulations. This is caused by the inherent scatter of the data and the difference between sample and true population. The ultimate test for the existence of a Weibull distribution is to test a material on different levels of (effective) volumes. 

A more effective representation of the strength data is possible by using Weibull plots where the measured strength data are reported as a function of failure probability. Here, failure probability was calculated as ... 

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