  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, 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.
