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The set $GU_f$ of possible effective elastic tensors of composites built fromtwo 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 inproportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$.The material with tensor $\BC_2$ corresponds to a material which (for technicalreasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and theunderlying microgeometries, have many aspects in common with the companionpaper "On the possible effective elasticity tensors of 2-dimensional printedmaterials". The chief difference is that one has a different algebraic problemto solve: determining the subspaces of stress fields for which the thin walledstructures can be rigid, rather than determining, as in the companion paper,the subspaces of strain fields for which the thin walled structure iscompliant. Recalling that $GU_f$ is completely characterized through minimumsof sums of energies, involving a set of applied strains, and complementaryenergies, involving a set of applied stresses, we provide descriptions ofmicrogeometries that in appropriate limits achieve the minimums in many cases.In these cases the calculation of the minimum is reduced to a finitedimensional minimization problem that can be done numerically. Eachmicrogeometry consists of a union of walls in appropriate directions, where thematerial in the wall is an appropriate $p$-mode material, that is almost rigidto $6-p\leq 5$ independent applied stresses, yet is compliant to any strain inthe orthogonal space. Thus the walls, by themselves, can support stress withalmost no deformation. The region outside the walls contains "Avellanedamaterial" that is a hierarchical laminate which minimizes an appropriate sum ofelastic energies.

Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase