Certain hierarchical laminates with a wide separation of length scales are known theoretically to have optimal transport and mechanical properties. We derive analytical expressions for the $n$-point probability functions that statistically characterize the microstructure for more general hierarchical laminates with an arbitrary number of stages and a finite separation of length scales. Using two-point probability information, we rigorously bound the effective conductivity (or dielectric constant) tensor for macroscopically anisotropic laminates and study how the separation of length scales affects the effective properties.

• Received 30 October 1995

DOI:https://doi.org/10.1103/PhysRevE.53.4368