Elastic, acoustic and water wave propagation through inhomogeneous media

非均匀介质（即具有空间相关材料特性的材料）在我们周围的世界中无处不在。它们通过优化策略在我们自己的身体（例如骨骼和软组织）中自然发展，工程界在工业环境中大量使用它们（例如航空航天工业中使用的轻质复合材料，设计既轻又强）。通常，它们表现出复杂的行为，这不是其单独均匀（同质）成分的特征。它们在众多应用中的使用越来越多，这促使人们需要更好地了解它们的行为方式。为了更好地了解非均质材料（如骨和复合材料）的性能，需要进行静态和动态测试，但这些实验通常证明成本过高。分析模型可以帮助很大，因为首先它避免了这些实验的财务影响，其次它可以帮助提供洞察到的基本物理，支配这些材料的行为所涉及的主要机制。为了更好地理解非均匀材料，所使用的基本分析数学技术是均匀化理论、微观力学和多次散射理论。在本研究计划中，主要研究者打算发展上述数学理论，并使用它们来模拟固体和流体力学领域中有关波通过非均匀介质传播的各种问题。固体力学领域的一个例子是将该理论应用于波传播（超声波）通过骨骼的问题-这是可以在老年患者中检测骨质疏松症等疾病的主要机制。数学模型通过描述不同生物学和材料长度尺度对骨中有效波速的影响来辅助这一过程。正在研究的问题之一将是评估微裂纹对波传播的影响。在流体力学领域，待研究的问题的一个例子是（表面）水波与障碍物和涡流的多重相互作用。这种相互作用有许多应用，例如波浪对近海结构稳定性的影响以及湍流的研究。这项研究将在位于法国巴黎的三个研究所内进行，那里的研究人员都是上述领域的专家。

Inhomogeneous media (i.e. materials with spatially dependent material properties) are ubiquitous in the world around us. They develop naturally via optimization strategies in our own body (e.g. bone and soft tissues) and the engineering community has made great use of them in an industrial context (e.g. lightweight composite materials used in the aerospace industry designed to be both light but also strong). Frequently, they exhibit complex behaviour which is not characteristic of their individual uniform (homogeneous) constituents. Their increased use in a multitude of applications motivates the need to better understand the way that they behave. Both static and dynamic testing of inhomogeneous materials such as bone and composites takes place in order to better understand their properties but these experiments often prove prohibitively expensive. Analytical modelling can help enormously since firstly it avoids the financial implications of these experiments and secondly it can help to provide insight into the main mechanisms involved in the underlying physics which governs the behaviour of these materials. The fundamental analytical mathematical techniques used in order to better understand inhomogeneous materials are the theories of homogenization, micromechanics and multiple scattering.In this research proposal the principal investigator intends to develop the mathematical theories described above and use them to model a number of diverse problems regarding wave propagation through inhomogeneous media in the fields of solid and fluid mechanics. One example in the area of solid mechanics is the application of the theory to the problem of wave propagation (ultrasound) through bone - this is the principal mechanism by which disease such as osteoporosis can be detected in elderly patients. Mathematical models assist this process by describing the influence of the differing biological and material lengthscales on the effective wavespeed in bone. One of the problems under study will be to assess the effect that microcracks have on wave propagation. In the area of fluid mechanics, one example of a problem to be studied is that of the multiple interaction of (surface) water waves with obstacles and vortices. This interaction has a number of applications such as the effect of waves on off-shore structure stability and also the study of turbulence.This research will be carried out within three research institutes located in Paris, France, where the host researchers are experts in the areas described above.