Transport properties of densely packed composites. Effect of shapes and spacings of inclusions

We analyse transport properties of fluid/solid and solid/solid composites containing finite arrays of closely spaced rigid inclusions when a host medium is either an elastic matrix or an incompressible fluid. The appropriate choice of the number of inclusions and the symmetry of a periodicity cell allows us to introduce simple, yet physically relevant models so that effective characteristics of homogenized media can be investigated analytically. For various applied loads and shapes of (polydisperse) inclusions we demonstrate the spatial non-uniformity of geometric configurations corresponding to either lowest dissipation rate (for fluid/solid composites) or to minimal stiffness (for solid/solid composites). In order to find the optimal configurations, we use a unified framework based on asymptotic expansions in terms of inter-inclusion distances. Furthermore, we compare effective transport properties of composite materials containing inclusions with either flat or curved boundaries.