Bounds for effective properties of multimaterial two-dimensional conducting composites

The paper suggests exact bounds for the effective conductivity of an isotropic multimaterial composite, which depend only on isotropic conductivities of the mixed materials and their volume fractions. These bounds refine Hashin–Shtrikman and Nesi bounds in the region of parameters where they are loose. The bounds by polyconvex envelope are modifies by taking into account the range of fields in optimal structures. The bounds are a solution of a formulated finite-dimensional constrained optimization problem. For three-material composites, bounds for effective conductivity are found in an explicit form. Three-material isotropic microstructures of extremal conductivity are found. It is shown that they realize the bounds for all values of conductivities and volume fractions. Optimal structures are laminates of a finite rank. They vary with the volume fractions and experience two topological transitions: For large values of m1m1, the domain of material with minimal conductivity is connected, for intermediate values of m1m1, no material forms a connected domain, and for small values of m1m1, the domain intermediate material is connected.