RS Fellow - EPSRC grant (2016): Spatial fragmentations

RS 研究员 - EPSRC 拨款 (2016)：空间碎片

• 批准号: EP/R005249/1
• 负责人: Matthew Roberts
• 金额: $ 25.94万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR005249%2F1
• 关键词: RS; Fellow; EPSRC; grant; 2016

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.

当大物体破碎成小块时（例如地震中的岩石），这些碎片的破碎程度可能取决于一系列参数，例如它们的大小、形状和结构。例如，长而薄的岩石可能比致密的岩石更快破裂。称为碎片过程的数学概念也描述了随着时间的推移随机分解成越来越小的碎片的对象集合。然而，破碎如何发生的规则可能非常严格：碎片如何破碎以及破碎的速度仅取决于其大小。这使我们能够给出有关这些过程的很好的数学结果，但对于理解真实物体并不是很有用。在这个项目中，我们建议研究一个简单的碎片模型，其中块破碎的速度取决于它们的形状和大小。我们从一个正方形开始，一段时间后它分成两个随机大小的矩形。然后，这两个矩形执行相同的操作：每个矩形等待一段时间，然后分成两部分。但他们在这样做之前等待的时间取决于他们的高度与宽度的比率。事实上，正如岩石的例子一样，长而薄的碎片可能比大致方形的碎片更快破裂。目的是这项工作将成为理解更复杂的碎片系统的第一步，这些系统的分解规则可能非常复杂。但即使是我们计划研究的相对简单的物体也被用作现实生活中称为马氏体雪崩过程的模型。这些描述了杂质在晶体原子结构中形成的方式，例如当金属熔化然后快速冷却以增加其强度时。