Optimal three-material wheel assemblage of conducting and elastic composites

We describe a new type of three material microstructures which we call wheel assemblages, that correspond to extremal conductivity and extremal bulk modulus for a composite made of two materials and an ideal material. The exact lower bounds for effective conductivity and matching laminates was found in Cherkaev (2009) and for anisotropic composites, in Cherkaev and Zhang (2011). Here, we show different optimal structures that generalize the classical Hashin–Shtrikman coated spheres (circles). They consist of circular inclusions which contain a solid central circle (hub) and radial spikes in a surrounding annulus, and (for larger volume fractions of the best material) an annulus filled with it. The same wheel assemblages are optimal for the pair of dual problems of minimal conductivity (resistivity) of a composite made from two materials and an ideal conductor (insulator), in the problem of maximal effective bulk modulus of elastic composites made from two linear elastic material and void, and the dual minimum problem.