  The set $GU_f$ of possible effective elastic tensors of composites built from
two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the
set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly
characterized. The material with tensor $\BC_2=0$ corresponds to a material
which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero
and we take the limit $\BC_2\to 0$). Specifically, recalling that $GU_f$ is
completely characterized through minimums of sums of energies, involving a set
of applied strains, and complementary energies, involving a set of applied
stresses, we provide descriptions of microgeometries that in appropriate limits
achieve the minimums in many cases. In these cases the calculation of the
minimum is reduced to a finite dimensional minimization problem that can be
done numerically. Each microgeometry consists of a union of walls in
appropriate directions, where the material in the wall is an appropriate
$p$-mode material, that is easily compliant to $p\leq 5$ independent applied
strains, yet supports any stress in the orthogonal space. Thus the material can
easily slip in certain directions along the walls. The region outside the walls
contains "complementary Avellaneda material" which is a hierarchical laminate
which minimizes the sum of complementary energies.
