Which Elasticity Tensors are Realizable?

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.