Recent advances in the analysis of microstructure is providing models and methods for treating the kinds of optimization problems that arise in the study of microstructure. The main advance has been the development of theory and methods for treating the case in which arbitrary microstructures compete for the minimum (or maximum). This contrasts for example with micromechanics in which the geometry of the microstructure is assumed, or assumed up to the choice of a few parameters, and then the optimization or stress analysis is carried out under severe geometric restrictions. Micromechanics is effective in dealing with a particular experimentally observed microstructure, but not for understanding microstructures that might be optimal in a certain sense. Much of this recent research has been fueled by critical discussions among engineering scientists, mathematicians and electron microscopists. The intent of this paper is first to summarize, in terms accessible to a broad audience, the nature of this research and then to describe applications to the improvement of shape-memory and magnetostrictive materials. The general part of the lecture will focus on three areas, effective properties of materials, optimal design of materials and phase transformation and active materials. A central role is played by the question “How does one meaningfully average a quantity whose values vary rapidly on a microstructural scale?” A second recurring theme is that the optimal microstructure is predicted to have fine structure. The latter is closely related to the failure of conditions of material stability.

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