Statistics extreme

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

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. 

Of course, in case it is given by /max (>> f e extreme statistics will always dominate over percolation statistics. We will discuss the experimental observations later. 

We will discuss these fracture properties of disordered solids, modelled by the random percolation models, and concentrate on their statistics, given by the cumulative failure strength distribution F a) under stress a, and the most probable fracture strength erf of such samples. We will discuss separately the cases for weak disorder p 1) and strong disorder p Pc)-The scaling properties of <7f near p pc and the nature of the competition between the percolation and extreme statistics here, will be discussed in detail. 

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. 

Consider the m random unordered variates c(fi), c(f2), , c tm), which are members of the stochastic process c(f,) that generates the time series of available air quality data. If we arrange the data points by order of magnitude, then a new random sequence of ordered variates c -m > c2 m > > cmm is formed. We call ci m the z th highest-order statistic or z th extreme statistic of this random sequence of size m. 

N. Mori, M. Onorato, P. A. E. M. Janssen, A. R. Osborne and M. Serio, On the extreme statistics of long crested deep water waves Theory and experiments, J. Geophys. Res., Ocean 112, C09011 (2007), doi 10.1029/2006JC004024. 

Suyuthi, A., Leira, B.J. and Riska, 2012a Short term Extreme Statistics of Local Ice Loads on Sip Hulls , Cold Regions Science and Technology, Elsevier, Vol. 82, pp. 130-143. 

Before the data can be apphed in design formulae and/or numerical or physical modelling tools, the data will require proper quality control and dedicated analyses (e.g., to determine trends and extreme statistics). The data analysis may also show that the available data sources are unreliable, incomplete or insufficient. If so, it may be decided to collect additional data by means of field surveys and/ or data purehase in combination with numerical modelling. Numerical modelling can also be applied to determine the operational site conditions for use in planning the reclamation works. 

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