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

本研究将分析在许多技术和科学重要应用中出现的广泛的非均质材料的有效特性。这项工作的主要重点之一将是获得有效性质的严格界限，例如电导率和弹性。研究人员将考虑一般复合材料，特别是由三种或更多组分组成的复合材料，这在应用中特别重要，并且有许多具有挑战性的数学问题仍然开放。还需要研究的是那些成分具有高度对比性质的介质的行为，其中渗透效应在分析中起主导作用。需要这种分析的例子包括半导体、金属陶瓷、多孔介质、海冰和金属断裂。在过去的二十年里，复合材料和非均质材料在技术和科学应用中的使用急剧增加。有效设计和使用这种材料的关键问题之一是各种成分的排列如何决定复合材料的有效性能，如强度或导电性。提出的研究的总体目标是发展数学技术，为混合物几何形状和有效性质之间的这种联系提供定量的理解。

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.