We describe a new type of three material microstructures which we call
wheel assemblage, that correspond to extremal conductivity and
extremal bulk modulus for a composite made of two materials and an ideal
material. The exact lower bound for effective conductivity and matching
laminates were found in (Cherkaev, 2009) and for anisotropic composites, in
(Cherkaev, Zhang, 2011). Here, we show different optimal structures that
generalize of the classical Hashin-Shtrikman coated spheres (circles). They
consists 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 couple 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 linearly elastic material and
void, and the dual one.
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