A top-down approach for the prediction of hardness and toughness of hierarchical materials

Many natural and man-made materials exhibit structure over more than one length scale. In this paper, we deal with hierarchical grained composite materials that have recently been designed to achieve superior hardness and toughness as compared to their traditional counterparts. Their nested structure, where meso-grains are recursively composed of smaller and smaller micro-grains at the different scales with a fractal-like topology, is herein studied from a hierarchical perspective. Considering a top-down approach, i.e. from the largest to the smallest scale, we propose a recursive micromechanical model coupled with a generalized fractal mixture rule for the prediction of hardness and toughness of a grained material with n hierarchical levels. A relationship between hardness and toughness is also derived and the analytical predictions are compared with experimental data.