Three scales asymptotic homogenization and its application to layered hierarchical hard tissues

In the present work a novel multiple scales asymptotic homogenization approach is proposed to study the effective properties of hierarchical composites with periodic structure at different length scales. The method is exemplified by solving a linear elastic problem for a composite material with layered hierarchical structure. We recover classical results of two-scale and reiterated homogenization as particular cases of our formulation. The analytical effective coefficients for two phase layered composites with two structural levels of hierarchy are also derived. The method is finally applied to investigate the effective mechanical properties of a single osteon, revealing its practical applicability in the context of biomechanical and engineering applications.