A brief introduction to some theoretical ideas and models

2 A brief introduction to some theoretical ideas and models 

We give here a brief discussion of the various theoretical models and concepts or ideas, employed later for the understanding of the failure proper- 

In the first reading of this book, a detailed study of this section is not essential, although a general familiarity with the concepts and the notations is necessary. 

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A brief introduction to some theoretical ideas and models Table 1.3 Fractal dimensions of the percolation clusters... 

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. 

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