Dislocation patterns beyond optimality

金属，特别是钢铁，在我们的日常生活中扮演着基本的角色，并被用于无数的应用，从建筑行业到交通、能源、包装和家用电器。然而，不同的应用需要不同的金属材料性能，这些材料需要设计成满足要求。材料设计反过来需要对金属的机械性能与其化学成分和微观结构的依赖关系有深刻的了解。如果不能很好地理解位错，这是不可能的。位错是金属原子结构中的缺陷，这些缺陷在宏观尺度上共同决定了金属如何变形。因此，任何以预测能力为目标的宏观模型都必须考虑到这些缺陷的存在。然而，由于即使是很小的金属样本中的典型位错数量也非常高，因此建立一个跟踪每个单个位错的模型是不可行的，除非是非常小规模的问题。虽然在原子晶格水平上已经有了很好的位错模型，但如何在宏观的工程规模上将它们的存在和运动的影响合并到一个模型中还没有被很好地理解。在过去的十年里，这一挑战已经成为深入研究的焦点--在工程界，因为需要良好的宏观模型来开发新的金属和合金，在数学界，它在理解复杂的多尺度系统中发挥着核心作用。多年来，在不同的社区，以几种方式应对这一挑战。严格方法的最大优势是它是准确的；然而，要付出的代价是对可以分析的配置的强烈限制。在大多数数学文献中，位错被建模为直线和平行线，而它们实际上是三维曲线。这一假设极大地降低了理论的复杂性，尽管在这种理想化的背景下，数学挑战仍然数不胜数。近年来受到极大关注的一种特殊构型是垂直周期位错壁，类似于小角度晶界。它们如此受欢迎的原因之一是人们普遍认为它们代表了位错的最小能量排列。这项拟议的研究提出了一个更基本的问题：位错的能量有利构型是什么？另外，当模型中引入一个小但非零的温度时，低能位错结构是如何变化的？低能位错结构(LED)是金属微观结构的主要特征之一，如壁、团簇和晶胞。这些高密度配置增加了材料的抗塑性滑移，从而产生更坚固的材料。确定LED的特性，从而开发和优化LED是材料设计中的关键一步：这将使设计师能够构建轻便的结构，但仍具有很高的抗变形能力，从而产生更安全、更省油的汽车。因此，我们研究的每一项进展都与机械工程和工业有关。然而，这个项目的研究超出了金属缺陷的具体例子。位错是复杂粒子系统的一个典型例子，这里发展的分析将适用于各种问题，涉及大量个体代理人集体行为的推导，例如人群和交通动力学、蜂拥而至的网络。

Metals, and steel in particular, play a fundamental role in our everyday life and are used in countless applications, from the construction industry to transport, energy, packaging, and house appliances. Different applications, however, require different material properties of the metals, which need to be designed to meet the requirements. Materials design in its turn requires a deep understanding of the dependence of the mechanical behaviour of the metal on its chemical composition and microstructure. This is not possible without a good understanding of dislocations.Dislocations are defects in the atomic structure of metals which collectively, at the macroscopic scale, determine how metals deform. For this reason, any macroscopic model that aims for a predictive power has to take into account the presence of these defects. However, since the typical number of dislocations in even a small sample of metal is very high, formulating a model that keeps track of every single dislocation is unfeasible except for very small-scale problems. Although good models for dislocations are available at the level of the atomic lattice, it is not yet well-understood how to incorporate the effect of their presence and motion in a model at the macroscopic, engineering scale. This challenge has become the focus of intensive research in the last decade - in the engineering community because of the need of good macroscopic models for the development of new metals and alloys, and in the mathematical community for the central role it plays in understanding complex multiscale systems. Over the years, and in different communities, this challenge has been approached in several ways. The great advantage of a rigorous approach is that it is exact; the price to pay, however, is a strong limitation on the configurations that can be analysed. In the majority of the mathematical literature, dislocations are modelled as straight and parallel lines, while they are in fact three-dimensional curves. This assumption reduces the complexity of the theory enormously, although the mathematical challenges in this idealised setting are still countless. A special configuration that has received great attention in recent years is that of vertically periodic dislocation walls, similar to low-angle grain boundaries. One of the reasons why they are so popular is the general belief that they represent minimum energy arrangements for dislocations. The proposed research poses a more fundamental question: what are the energetically favourable configurations of dislocations? And, also, how do low-energy dislocation structures vary when a small but non-zero temperature is introduced in the model?Low-energy dislocation structures (LEDS) like walls, clusters and cells are one of the main features of the microstructure in metals. These high-density configurations increase the resistance of the material against plastic slip, thus leading to a stronger material. Characterising and hence exploiting and optimising LEDS is a key step in materials design: it would allow designers to construct lightweight structures which nevertheless have a high resistance to deformation, resulting, e.g., in safer and more fuel-efficient cars. Therefore, every advance in our research is relevant to mechanical engineering and industry.The research in this project however goes beyond the specific example of defects in metals. Dislocations are a paradigmatic example of a complex particle system and the analysis developed here would be applicable to a variety of problems dealing with the derivation of the collective behaviour of a large number of individual agents, e.g., crowd and traffic dynamics, swarming, networks.