Gumbel statistics/distribution

As we will see in the appropriate sections of the next two chapters, the precise ranges of the validity of the Weibull or Gumbel distributions for the breakdown strength of disordered solids are not well established yet. However, analysis of the results of detailed experimental and numerical studies of breakdown in disordered solids suggests that the fluctuations of the extreme statistics dominate for the entire range of disorder, even very close to the percolation point. 

We have studied the the fracture properties of such elastic networks, under large stresses, with initial random voids or cracks of different shapes and sizes given by the percolation statistics. In particular, we have studied the cumulative failure distribution F a) of such a solid and found that it is given by the Gumbel or the Weibull form (3.18), similar to the electrical breakdown cases discussed in the previous chapter. Extensive numerical and experimental studies, as discussed in Section 3.4.2, support the theoretical expectations. Again, similar to the case of electrical breakdown, the nature of the competition between the percolation and extreme statistics (competition between the Lifshitz length scale and the percolation correlation length) is not very clear yet near the percolation threshold of disorder. 

The distributions and pdf s, as well as certain statistical properties of these random variables, can be determined by applying the methods and results of order statistics (or statistics of extremes) (Gumbel 1958 Sarhan and Greenberg 1962 David 1981). 

It is shown that for thermal actions on bridges the Weibull distribution is often fitting well as the skewness a of the probabilistic distribution based on evaluated statistical data is considerably less (0,1 to 0,6) than the skewness of Gumbel distribution. 

It appears that despite the Eurocodes recommend the apphcation of the Gumbel distribution for modelling of thermal actions, the temperature components in bridges may be better represented by Weibull distribution. The skewness of the statistically evaluated data of temperatures is in a range fi om 0,1 to 0,6 what is considerably less than the skewness of the Gumbel distribution. 

A number of statistical transformations have been proposed to quantify the distributions in pitting variables. Gumbel is given the credit for the original development of extreme value statistics (EVS) for the characterization of pit depth distribution [13]. The EVS procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. The Gumbel distribution expressed in Eq 1, where X and a are the location and scale parameters, respectively, can then be used to characterize the dataset and estimate the extreme pit depth that possibly can affect the system from which the data was initially produced. 

Skewed distributions have been seldom reported or even noticed. It is more common to report results in terms of Gumbel or Weibull distributions, which take account of weak-link aspects of voltage breakdown. Standard statistical texts can be consulted for the details of such analysis. An extreme-value distribution for the breakdown voltage of two films is shown in Figure 17. 

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