Materials with structural hierarchy over nanometer to millimeter length scales are found throughout Kingdoms Plantaei> and Animalia. The idea of using structural hierarchy in engineering structures and materials goes back at least to Eiffel's Garabit Viaduct and then Tower. Incorporating hierarchy into honeycomb lattice structures has been the focus of a number of studies and has significance with regard to the application of honeycombs in impact energy absorption and structural protection, thermal isolation and as the structural core of sandwich panels. Here we explore the mechanical properties of two kind of cellular structures:​ hierarchical honeycombs and trabecular bone.

Hexagonal honeycomb structures are known for their high strength and low weight. We construct a new class of fractal-appearing cellular metamaterials by replacing each three-edge vertex of a base hexagonal network with a smaller hexagon and iterating this process. The mechanical properties of the structure after different orders of the iteration are optimized. We find that the optimal structure (with highest in-plane stiffness for a given weight ratio) is self-similar but requires higher order hierarchy as the density vanishes. These results offer insights into how incorporating hierarchy in the material structure can create low-density metamaterials with desired properties and function.

The second aim of this study was to explore the hierarchical arrangement of structural properties in cortical and trabecular bone and to determine a mathematical model that accurately predicts the tissue's mechanical properties as a function of these indices. By using a variety of analytical techniques, we were able to characterize the structural and compositional properties of cortical and trabecular bones, as well as to determine the suitable mathematical model to predict the tissue's mechanical properties using a continuum micromechanics approach. Our hierarchical analysis demonstrated that the differences between cortical and trabecular bone reside mainly at the micro- and ultrastructural levels. By gaining a better appreciation of the similarities and differences between the two bone types, we would be able to provide a better assessment and understanding of their individual roles, as well as their contribution to bone health overall.