| We consider the long-standing problem of characterizing the set of all possible effective tensors of two- and three-dimensional composites made of several isotropic linearly conducting phases in prescribed volume fractions. For more than two phases, a complete characterization of this set, the G-closure, is not known. The results presented in this dissertation follow from and expand upon the work of a number of people including Hashin and Shtrikman; Lurie and Cherkaev; Tartar; Murat and Tartar; Milton; Milton and Kohn; and Gibiansky and Sigmund.; The "translation bound," a generalization of the Hashin-Shtrikman bound, consists of several inequalities that the effective tensors of any composite must satisfy. The bound depends only on the G-closure parameters---the conductivities of the phases and the relative volume fractions. It is independent of the layout of the phases in the composite. The inequalities are known to be optimal for certain parameters. It is also known that there exist parameters for which they are not optimal. However, a complete characterization of the parameter set for which the bound is optimal is still missing.; We use a systematic approach based on "field optimality conditions". Using this approach, we consider the general, anisotropic bound and prove that it is optimal for a much larger range of parameters than was previously known. We do this for three or more phases and in two or three dimensions by constructing laminate composites that saturate one of the inequality bounds.; We illustrate a number of applications of this approach, including finite-rank iterated laminates, special "block structures" resembling those of Gibiansky and Sigmund, and infinite-rank laminates produced via a differential scheme. Finally, we discuss a numerical algorithm that gives an approximation of the G-closure in regions where the translation bounds may not be optimal. |
