WHICH ELASTICITY TENSORS ARE REALIZABLE

It is shown that any given positive definite fourth order tensor satis fying the usual symmetries of elasticity tensors can be realized as th e effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotro pic phase configured in an suitable microstructure. The building block s for constructing this composite are what we call extremal materials. These are composites ef 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 tenser.