Fracture In Brittle Disordered Media

This thesis consists of three main chapters, an introduction, and an appendix. The introduction (chapter 1) gives a general historical introduction to the problem of brittle fracture in disordered media. Chapters 2 and 4 are concerned with various aspects of fracture in disordered fuse networks. Chapter 2 investigates the asymptotic properties of fracture strength distributions, and explores their relation with extreme value statistics. Chapter 4 deals with critical phenomena in brittle fracture. This chapter introduces the concept of finite-sized criticality as a means to explain how fracture can have mixed properties of abrupt and continuous phase transitions. Chapter 3 describes the collective dynamics at the non equilibrium metal insulator transition. The phenomenon of dielectric breakdown at the metal insulator transition shares several characteristics with fracture, and provides a suitable build up to the development presented in chapter 4. The first three parts of the appendix provide an introduction to the various mathematical tools required in order to better appreciate the content of this thesis. Appendix A.1 discusses the basics of extreme value theory, while A.2 and A.3 provide a light introduction to linear elastic fracture mechanics. Appendix A.4 summaries some results on crack propagation in graphene that are not sufficiently well developed to merit a chapter, and yet are developed enough to merit a mention