The classical homogenization methods cannot capture entirely the complexity of the microstructures encountered in material engineering and design. This paper presents three different types of approach, which all tend towards a refined description of the pertinent microstructural characteristics. The first one is a step-wise homogenization based on a hierarchy of heterogeneity levels. It is applied to the prediction of elastic moduli of composites. The second one aims to better estimate the stress and strain distributions in and around inclusions with complex shapes. Semi-analytical methods (for non-ellipsoidal shapes) and numerical methods (for non-convex ones) are developed. In the third case, the microstructure cannot be treated as an usual continuum. Discrete numerical methods are then derived for predicting the properties of fiber networks or cellular materials.
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