  The set $GU_f$ of possible effective elastic tensors of composites built from
two materials with positive definite elasticity tensors $\BC_1$ and
$\BC_2=\Gd\BC_0$ comprising the set $U=\{\BC_1,\Gd\BC_0\}$ and mixed in
proportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$.
The material with tensor $\BC_2$ corresponds to a material which (for technical
reasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and the
underlying microgeometries, have many aspects in common with the companion
paper "On the possible effective elasticity tensors of 2-dimensional printed
materials". The chief difference is that one has a different algebraic problem
to solve: determining the subspaces of stress fields for which the thin walled
structures can be rigid, rather than determining, as in the companion paper,
the subspaces of strain fields for which the thin walled structure is
compliant. 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 almost rigid
to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in
the orthogonal space. Thus the walls, by themselves, can support stress with
almost no deformation. The region outside the walls contains "Avellaneda
material" that is a hierarchical laminate which minimizes an appropriate sum of
elastic energies.
