Morphology and effective properties of multi-scale random sets: A review

Complex microstructures in materials often involve multi-scale heterogeneous textures, modeled by random sets derived from Mathematical Morphology. Our approach starts from 2D or 3D images; a complete morphological characterization by image analysis is performed, and used for the identification of a model of random structure. Morphological models enter into the prediction of effective properties by estimation, bounds, or from numerical simulations. Simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (electric, elastic, …) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields. Our approach is illustrated by various examples of multi-scale models: Boolean random sets based on Cox point processes and various random grains (spheres, cylinders), showing a very low percolation threshold, and therefore a high conductivity or high elastic moduli for a low volume fraction of a second phase. Multi-scale iterations of random media provide another source of morphologies with interesting overall properties.