Honeycombs with hierarchical organization*

We investigated the mechanical behavior of two-dimensional hierarchical honeycomb structures using analytical, numerical and experimental methods. Hierarchical honeycombs were constructed by replacing every three-edge vertex of a regular hexagonal lattice with a smaller hexagon. Repeating this process builds a fractal-appearing structure. The resulting isotropic in-plane elastic properties (effective elastic modulus and Poisson's ratio) of this structure are controlled by the dimension ratios for different hierarchical orders. Hierarchical honeycombs of first and second order can be up to 2.0 and 3.5 times stiffer than regular honeycomb at the same mass (i.e., same overall average density). The Poisson's ratio varies from nearly 1.0 (when planar ``bulk'' modulus is considerably greater than Young's modulus, so the structure acts ``incompressible'' for most loadings) to 0.28, depending on the dimension ratios. The work provides insight into the role of structural organization in regulating the mechanical behavior of materials, and new opportunities for developing low-weight cellular structures with tailorable properties.