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

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials