Optimal Fractal-Like Hierarchical Honeycombs

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.

Figure 1

(a) Unit cell of regular (i.e., zeroth) to fourth order hierarchical honeycombs fabricated using 3D printing. The physical thickness of the structures is constant, $t_n=2\mathrm{mm}$, because of the limitations of the 3D printing. To maintain the structure density, therefore, the size of this unit cell increases as the order of the hierarchy increases. (b) Unit cell of the hierarchical honeycombs with regular structure (left) and with first order hierarchy (right). Here $F$ is an arbitrary concentrated force and $N_1$ and $N_2$ are the reaction forces at the midline.

Figure 2

Maximum achievable elastic modulus (elastic modulus limit) of hierarchical honeycombs for different relative densities $\overline{\rho }$ and different hierarchical orders $n$, normalized by the elastic modulus of a regular honeycomb of the same density. The numerically computed elastic modulus of hierarchical honeycombs with bending only is shown by the dashed curve. The elastic modulus limit found from our scaling analysis [Eq. (5)] is shown by (triangle) points. The inset shows comparison with the experimental results for hierarchical honeycombs with a density of $\overline{\rho }=0.054$.

Figure 3

Topology of the stiffest hierarchical honeycombs at different relative densities. The results show the values of $\gamma$ corresponding to the optimum structure of the hierarchical honeycombs at different relative densities. Maximum achievable hierarchical order and selected topologies of the stiffest hierarchical honeycombs in the specified relative density range are also shown at the top.

Figure 4

(a) The order of hierarchy that yields the stiffest hierarchical honeycomb as a function of the relative density. The results of numerical analysis (solid line) are shown together with scaling analysis results [Eq. (8)] (dashed line). (b) The limiting elastic modulus of the hierarchical honeycomb versus the relative density. The results of numerical analysis (circle markers) are shown together with scaling analysis results [Eq. (9)] (dashed curve).

Figure 5

Elastic modulus range for different order of hierarchy $n$ versus relative density. Dashed curve shows the limiting elastic modulus of the hierarchical honeycomb for specified relative density [Eq. (9)].