Phase-interchange relations for the elastic moduli of two-phase composites

For Isotropic two-phase composites, we derive phase-interchange inequalities for the bulk and shear moduli in two and three space dimensions. We find optimal microstructures that realize part of the bulk and shear moduli bounds in two dimensions. Geometrical-parameter bounds and the translation method are used to prove the results. The phase-interchange relations are applied to composites with cavities or a perfectly rigid phase, composites near the percolation threshold, incompressible composites, composites with equal bulk or shear phase moduli, and symmetric composites.