Mathematical Sciences: Analysis and Optimization of the Effective Properties of Inhomogeneous Materials

本研究将分析的有效性质， 广泛的非均匀材料，出现在许多 技术和科学上的重要应用。 一 这项工作的主要目标将是获得严格的 有效属性的界限，如电导率和 弹性 调查人员将考虑综合 材料，特别是由三个或更多个 在应用中特别重要的组件，以及 其中有许多具有挑战性的数学问题， 保持开放。 也要研究的是媒体的行为， 组件具有高度对比的特性， 渗流效应在分析中起主导作用。 需要进行这种分析的例子包括 半导体、金属陶瓷、多孔介质、海冰和断裂 金属。 %%% 复合材料的使用量急剧增加， 和非均匀材料， 在过去的二十年里，应用。 的关键问题之一 有效设计和使用这些材料的根本问题是， 各种部件的布置决定了 有效的性能，如强度或导电性， 合成的 拟议研究的总体目标是 发展数学技术， 理解混合物几何形状和 有效的属性。

This research will analyze the effective properties of a broad range of inhomogeneous materials that arise in many technologically and scientifically important applications. One of the main thrusts of this work will be to obtain rigorous bounds on effective properties, such as conductivity and elasticity. The investigators will consider general composite materials, particularly those composed of three or more components, which are particularly important in applications, and for which there are many challenging mathematical questions that remain open. Also to be studied is the behavior of media whose components have highly contrasting properties and where percolation effects play the dominant role in the analysis. Examples where such an analysis is necessary include semiconductors, cermets, porous media, sea ice, and the fracture of metals. %%% There has been a dramatic increase in the use of composite and inhomogeneous materials for technological and scientific applications over the last two decades. One of the key questions underlying the efficient design and use of such materials is how the arrangement of the various components determines the effective properties, such as strength or conductivity, of the composite. The overall goal of the proposed research is to develop mathematical techniques which provide a quantitative understanding of this connection between mixture geometry and effective properties.