Asymptotic expansion homogenization for heterogeneous media: computational issues and applications

Developments in asymptotic expansion homogenization (AEH) are overviewed in the context of engineering multi-scale problems. The problems of multi-scales presently considered are those linking continuum level descriptions at two different length scales. Concurrent research in the literature is first described. A recipe of the AEH approach is then presented that can be used for future developments in many areas of material and geometric non-linear continuum mechanics. Then, a derivation is outlined using the finite element method that is useful for engineering applications that leads to coupled hierarchical partial differential equations in elasticity. The approach provides causal relationships between macro and micro scales wherein procedures for homogenization of properties and localization of small-scale response are built-in. A brief discussion of a physical paradox is introduced in the estimation of micro-stresses that tends to be a barrier in the understanding of the method. Computational issues are highlighted and illustrative applications in linear elasticity are then presented for composites containing microstructures with complex geometries.