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

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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.