Microstructures minimizing the energy of a two phase elastic composite in two space dimensions.

Abstract: "For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. S. Vigdergauz has shown the existence of particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions. This 'Vigdergauz microstructure' consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization."