Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions

We demonstrate the bounding of the effective properties of random multiscale microstructures by means of essential and natural boundary conditions. The proposed method involves moderate sized lattices, not modified in the boundary zone, thereby allowing much faster calculations than the method of periodic boundary conditions. In case of a random two-phase lattice, scaling laws have been found for a wide range of contrasts. In the case of a disk-inclusion composite having circular inclusions with graded interphases, the presence of a graded interphase dramatically changes the effective conductivity compared to that of a composite with perfect interfaces.