For the efficient use and development of advanced materials it is necessary to develop simple yet accurate analytic relationships linking their average physical response to as large a set as possible of microstructural quantifiers. With this objective in mind, the theory of the effective elastic response of biphase composite materials, with arbitrary second-phase geometry, concentration, and orientation distribution, is examined here. A formulation is presented, which employs an equivalent poly-inclusion methodology, in the sense that its governing variable is the average inclusion strain concentrator for an entirely uniform body undergoing a uniform eigenstrain in a non-dilute family of morphologically identical internal regions (inclusions). Admissibility criteria for approximate inclusion strain concentrators are formulated. The relation of the present approach with the traditional one, based on the inhomogenity average strain concentrator, is investigated, and some deficiencies of current methods based on the latter approach are discussed. A family of fully admissible poly-inclusion concentrators is proposed, which differs by a single scalar-valued function of the volume fraction—the so-called “interaction function”. A specific interaction function is finally identified, which satisfies all admissibility criteria, is consistent with the known analytic solutions for biphase composites, and reproduces literature data to within experimental error.