Variational convergence for nonlinear high-contrast homogenisation problems

拟议中的研究涉及到在研究所谓的复合材料的行为中出现的数学问题，复合材料是两种或多种成分介质的非均匀混合物。这一主题的实际重要性在于，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，我们周围世界的几乎所有物体（天然材料，如木材或岩石，生物种群等）都是复合材料。是异质介质，也可以被视为许多成分的“混合物”。在更广泛的范围内，拟议工作的结果将直接关系到我们对一般所谓的"复杂系统“的知识的发展，复杂系统可以粗略地定义为具有其个别性质不同于整个系统性质的组成部分的系统。为了定量地衡量“混合”的效果，模型制作者通常会考虑与之相关的长度尺度，例如一块木板中纤维的平均厚度。为了更接近实际，可能需要考虑多个长度尺度，例如，在软木中，也存在单个层的平均厚度的长度尺度。另一方面，在定性层面上，人们经常谈到与复合介质相关的“微观结构”，大致意思是组分的几何排列。例如，这可以是周期性分布或根据某些其他规则分布的球形夹杂物的形式。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"有效”材料的性质。使这种观察严格化是一项有趣的数学任务，到目前为止，已经对性质在某种意义上“差别不大”的材料的混合物进行了观察。然而，如果在一种均匀介质（"基质“）中夹杂物的性质以某种"临界”方式与基质的性质成比例，那么“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。这是由于存在的不同长度尺度之间的某种"耦合“，可以说它们彼此”相互作用“。例如，在电磁波传播的背景下，这种精细混合物的频谱被证明具有“禁止”频率的“空隙”（或“间隙”），这些频率被微观结构捕获。这一事实的严格证明是一个相当简洁的数学，这是几年前设计的。相关的物理特性正在新一代光传输器件的制造中积极实现。开发能够捕捉这种长度-尺度相互作用的先进数学工具是该项目的广泛目标。更具体地说，我们将扩展数学家称为伽玛收敛的技术，以研究矩阵和夹杂物的属性之间具有高度对比的材料。利用这个推广，我们将给出相应的有效介质的描述。由于材料特性的高对比度，它将包含一个方程组，该方程组将不同长度尺度下的行为“耦合”起来。然后，我们将研究固体力学中的一个或两个模型，使用已开发的理论，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.