Variational convergence for nonlinear high-contrast homogenisation problems

非线性高对比度均匀化问题的变分收敛

• 批准号: EP/F03797X/1
• 负责人: Kirill Cherednichenko
• 金额: $ 18.72万
• 依托单位: CARDIFF UNIVERSITY
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FF03797X%2F1
• 关键词: Variational; convergence; nonlinear; contrast; homogenisation

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.

拟议的研究涉及所谓复合材料行为研究中出现的数学问题，复合材料是两种或多种组成介质的异质混合物。该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，我们周围世界上几乎所有的物体（天然材料，例如木材或岩石、生物种群等）是异质介质，也可以被视为多种成分的“混合物”。在更广泛的背景下，所提出的工作的结果将与我们对一般所谓的“复杂系统”的知识的发展直接相关，复杂系统可以被松散地定义为具有其各个属性不同于系统属性的组件的系统。整个系统。为了定量测量“混合”的效果，建模者通常会考虑与之相关的长度尺度，例如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，还存在各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”，大致意思是组件的几何排列。例如，这可以是周期性分布的球形内含物的形式或根据一些其他规则。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”某些均质材料的性质“有效”的材料。使这种观察变得严格是一项有趣的数学任务，到目前为止，已经针对其性质在某种意义上“相差不大”的材料混合物进行了这项工作。然而，如果均匀介质（“基质”）中夹杂物的特性以某种“关键”方式与基质的特性成比例，那么“有效”介质可能拥有许多非标准功能，在应用程序中很有用。这是由于存在的不同长度尺度之间存在某种“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显示出“禁止”频率的“缺陷”（或“间隙”），这些频率被微观结构。这个事实的严格证明是一个相当简洁的数学部分，它是几年前才设计出来的。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。该项目的广泛目标是开发能够捕获这种长度尺度相互作用的先进数学工具。更具体地说，我们将扩展数学家所熟知的伽玛收敛技术，以研究基体和内含物特性之间具有高对比度的材料。使用这个扩展，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，“耦合”不同长度尺度的行为。然后，我们将利用所发展的理论研究固体力学中的一两个模型，并就其实际实施提出建议。

项目成果

High contrast homogenisation in nonlinear elasticity under small loads

小载荷下非线性弹性的高对比度均匀化

• DOI: http://dx.10.3233/asy-171430
• 发表时间: 2017
• 期刊: Asymptotic Analysis
• 影响因子: 1.4
• 作者: Cherdantsev M

High contrast homogenisation in nonlinear elasticity under small loads

小载荷下非线性弹性的高对比度均匀化

• DOI: http://dx.10.48550/arxiv.1303.1224
• 发表时间: 2013
• 期刊: arXiv e-prints
• 作者: Cherdantsev Mikhail

On the existence of waves guided by a cavity in an elastic film

关于弹性薄膜中空腔引导的波的存在性

• DOI: http://dx.10.1093/qjmam/hbp007
• 发表时间: 2009
• 期刊: The Quarterly Journal of Mechanics and Applied Mathematics
• 作者: Cherednichenko K

Two-Scale G-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites

积分泛函的两尺度 G 收敛及其在非线性高对比度周期性复合材料均匀化中的应用

• DOI: http://dx.10.1007/s00205-011-0481-4
• 发表时间: 2011
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Cherdantsev M

Bending of thin periodic plates

薄周期板的弯曲

• DOI: http://dx.10.1007/s00526-015-0932-0
• 发表时间: 2015
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Cherdantsev M