Variational convergence for nonlinear high-contrast homogenisation problems

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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.