Structure of Fields in Extremal 2D Conducting Multimaterial Composites

The paper discusses a long-standing problem of exact bounds for the effective properties of multimaterial composites. The suggested bounds refine Hashin-Shtrikman’s bounds in the region of parameters where the last ones are loose. They are derived by the description of gradient fields in the components of optimal structures. We show that the gradient fields vary in restricted domains; taking this into account, we modify the Translation method to obtain new exact bounds. These bounds are multifaced: the active or passive inequality constraints lead to algebraically different expressions and topologically different optimal structures. The bounds for the energy and optimal structures are explicitly computed for three-material conducting and elastic composites (one material is void). However, a unified approach to deriving a set of constraints in the supporting fields is yet to be developed.