Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study

Asymptotic homogenization is employed assuming a sharp length scale separation between the periodic structure (fine scale) and the whole composite (coarse scale). A classical approach yields the linear elastic-type coarse scale model, where the effective elastic coefficients are computed solving fine scale periodic cell problems. We generalize the existing results by considering an arbitrary number of subphases and general periodic cell shapes. We focus on the stress jump conditions arising in the cell problems and explicitly compute the corresponding interface loads. The latter represent a key driving force to obtain nontrivial cell problems solutions whenever discontinuities of the coefficients between the host medium (matrix) and the subphases occur. The numerical simulations illustrate the geometrically induced anisotropy and foster the comparison between asymptotic homogenization and well established Eshelby based techniques. We show that the method can be routinely implemented in three dimensions and should be applied to hierarchical hard tissues whenever the precise shape and arrangement of the subphases cannot be ignored. Our numerical results are benchmarked exploiting the semi-analytical solution which holds for cylindrical aligned fibers.