Spatial Fragmentations

When something big gets broken up into smaller pieces - for example rocks in an earthquake - how exactly the pieces break might depend on a whole host of parameters, such as their size, shape and structure. For example, long, thin pieces of rock are likely to break more quickly than compact blocks. A mathematical concept called a fragmentation process also describes a collection of objects that break up randomly into smaller and smaller pieces as time passes. However, the rules of how the breaking up happens can be quite restrictive: how a piece breaks, and how quickly it breaks, is only allowed to depend on its size. This allows us to give nice mathematical results about these processes but is not very useful for understanding real objects. In this project we propose to study a simple model of fragmentation where how quickly blocks break does depend on their shape as well as their size. We begin with a square, which after some time breaks up into two randomly-sized rectangles. These two rectangles then do the same: each of them waits some time and then splits into two. But the time they wait before doing so depends on the ratio of their height to their width. In fact, exactly as in the rocks example, long thin pieces are likely to break more quickly than roughly square pieces. The aim is that this work will be a first step towards understanding more complex fragmentation systems, where the break-up rules might be very complicated. But even the relatively simple object that we plan to study has been used as a model for real-life processes called martensitic avalanches. These describe the way in which impurities form in crystalline atomic structures - such as when metals are melted and then cooled quickly in order to increase their strength.