Variational convergence for nonlinear high-contrast homogenisation problems

拟议的研究涉及在研究所谓的复合材料的行为时出现的数学问题，复合材料是两种或两种以上组成介质的非均质混合物。这门学科的实际重要性在于，一方面，目前工业中使用的绝大多数材料都是复合材料，另一方面，我们周围世界中的几乎所有物体（如木材或岩石等天然材料，生物种群等）都是异质介质，也可以被视为许多成分的“混合物”。在更广泛的背景下，所提出的工作的结果将直接关系到我们对一般称为“复杂系统”的知识的发展，它可以被粗略地定义为具有单个属性不同于整体系统属性的组件的系统。为了定量测量“混合”的影响，建模者通常会考虑与之相关的长度尺度，比如一块木板中纤维的平均厚度。为了更接近现实，可能需要考虑多个长度尺度，例如在胶合板中还需要考虑单个层的平均厚度的长度尺度。另一方面，在定性层面上，人们通常会说到与复合介质相关的“微观结构”，粗略地说，这意味着组成部分的几何排列。例如，这可以是球形夹杂物周期性分布的形式，或者根据其他一些规则。如果复合材料中存在的长度尺度之间的比率足够小，这对应于微观结构“足够细”，则介质的物理性质预计将“接近”某些均匀的“有效”材料的性质。要使这一观察结果变得严格，是一项有趣的数学任务，目前已经对性质在某种意义上“差别不大”的材料混合物进行了检验。然而，如果在其他均质介质（“矩阵”）中夹杂物的性质与矩阵的性质以某种“临界”方式缩放，则“有效”介质可能具有许多非标准特征，这些特征在应用中是有用的。这是由于不同长度尺度之间存在某种“耦合”，可以说是彼此“相互作用”。例如，在电磁波传播的情况下，这种精细混合物的频谱显示出具有“禁止”频率的“空白”（或“间隙”），这些“空白”被微观结构捕获。对这一事实的严格证明是几年前才发明的一种相当简洁的数学方法。相关的物理特性正在制造新一代光传输器件的过程中积极实现。开发先进的数学工具来捕捉这种长度尺度的相互作用是该项目的广泛目标。更具体地说，我们将扩展数学家中已知的伽马收敛技术，以研究矩阵和内含物性质之间高度对比的材料。使用这个扩展，我们将给出相应有效介质的描述。由于材料特性的高度对比，它将包含一个“耦合”不同长度尺度上的行为的方程系统。然后，我们将研究固体力学中的一个或两个模型，使用已发展的理论，并对其实际实施提出建议。

The proposed research relates to mathematical problems arising in the study of the behaviour of the so-called composite materials, which are heterogeneous mixtures of two or more constituent media. The practical importance of this subject lies in the facts that, on the one hand, a vast majority of materials currently used in industry are composites, and on the other hand, virtually all objects in the world around us (natural materials such as wood or rock, biological populations etc.) are heterogeneous media that can also be perceived as ``mixtures'' of a number of components. In a wider context, the results of the proposed work will be directly relevant to the development of our knowledge of what is generically called ``complex systems'', which can be loosely defined as systems with components whose individual properties differ from the properties of the system as a whole. In order to give a quantitative measure of the effect of ``mixing'', a modeller usually thinks of a length-scale associated with it, such as the average thickness of fibres in a plank of timber. It may happen that in order to get a closer match to reality, more than one length-scale has to be taken into account, for example in ply-wood there is also a length-scale of the average thickness of the individual layers. On the other hand, at the qualitative level one often speaks of a ``microstructure'' associated with a composite medium, meaning, roughly, the geometrical arrangement of the components. For example, this could be in the form of spherical inclusions distributed periodically or according to some other rule. If the ratio between the length-scales present in the composite is sufficiently small, which corresponds to the microstructure being ``sufficiently fine'', the physical properties of the medium are expected to be ``close'' to the properties of some homogeneous ``effective'' material. It is an interesting mathematical task to make this observation rigorous, which by now has been carried out for mixtures of materials whose properties ``do not differ too much'' in some sense. However, if the properties of inclusions in an otherwise homogeneous medium (``matrix'') are scaled in a certain ``critical'' way with the properties of the matrix, then the ``effective'' medium may possess a number of non-standard features, which are useful in applications. This is due to some sort of ``coupling'' between the different length-scales present, which may be said to ``interact'' with each other. For example, in the context of electromagnetic wave propagation, the spectrum of a fine mixture of this kind is shown to have ``lacunae'' (or ``gaps'') of ``forbidden'' frequencies, which get trapped by the microstructure. The rigorous proof of this fact is a rather neat piece of mathematics, which was devised only a few years ago. The related physical property is in the process of active implementation in manufacturing a new generation of optical transmission devices. The development of advanced mathematical tools that could capture the length-scale interactions of this kind is the wide aim of the project. More specifically, we will extend the techniques known among mathematicians as the Gamma-convergence in order to study materials with high contrast between the properties of the matrix and the inclusions. Using this extension we will give a description of the corresponding effective medium. Due to the high contrast in material properties, it will contain a system of equations that ``couple'' the behaviour at different lengthscales. We will then investigate one or two models in solid mechanics, using the developed theory, and make suggestions on their practical implementation.