Optimal multimaterial composites: Bounds and structures

Andrej Cherkaev, cherk@math.utah.edu
Optimal multimaterial composites: Bounds and structures
The paper suggests a method for funding exact bounds for the effective conductivity moduli of multimaterial
composites. These bounds expand and refine Hashin–Shtrikman and Nesi bounds. We prove that the fields in
the materials within optimal structures vary in restricted domains and take this into account, obtaining more
restricted bounds. The new bounds are solutions of a formulated relaxed finite-dimensional constrained optimization problem. For two-dimensional conducting three-material composites, bounds for effective conductivity are explicitly computed. These bounds are exact: Three-material isotropic microstructures of extremal
conductivity are found that realize the bounds for all values of parameters. The optimal structures are laminates
of a finite rank, their parameters vary with the volume fractions and they experience two topological transitions:
For large values of material of minimal conductivity, its subdomain percolates (is connected), for intermediate
values of that fraction, no material forms a connected domain, and for small values of that fraction, the domain
of intermediate material percolates. Another type of isotropic optimal three–material structures is the “wheel
assemblages” that replaces the Hashin–Shtrikman coated circles.