We investigate the elastic behavior of periodic hexagonal honeycombs over a wide range of relative densities and cell geometries, using both analytical and numerical approaches. Previous modeling approaches are reviewed and their limitations identified. More accurate estimates of all nine elastic constants are obtained by modifying the analysis of Gibson and Ashby (1997) to account for the nodes at the intersection of the vertical and inclined members. The effect of the nodes becomes significant at high relative densities. We then compare the new analytical equations with previous analytical models, with a numerical analysis based on a computational homogenization technique and with data for rubber honeycombs over a wide range of relative densities and cell geometries. The comparisons show that both the new analytical equations and numerical solutions give a remarkably good description of the data. The results provide new insights into understanding the mechanics of honeycombs and designing new cellular materials in the future

Additive manufacturing technologies offer new ways to fabricate cellular materials with composite cell walls, mimicking the structure and mechanical properties of woods. However, materials limitations and a lack of design tools have confined the usefulness of 3D printed cellular materials. We develop new carbon fiber reinforced, epoxy inks for 3D printing which result in printed materials with longitudinal Young's modulus up to 57 GPa (exceeding the longitudinal modulus of wood cell wall material). To guide the design of hierarchical cellular materials, we developed a parameterized, multi-scale, finite element model. Computational homogenization based on finite element simulations at multiple length scales is employed to obtain the elastic properties of the material at multiple length scales. Parameters affecting the elastic response of cellular composites, such as the volume fraction, orientation distribution, and aspect ratio of fibers within the cell walls as well as the cell geometry and relative density are included in the model. To validate the model, experiments are conducted on both solid carbon fiber/epoxy composites and cellular structures made from them, showing excellent agreement with computational multi-scale model predictions, both at the cell-wall and at the cellular-structure levels. Using the model, cellular structures are designed and experimentally shown to achieve a specific stiffness nearly as high as that observed in balsa wood. The good agreement between the multi-scale model predictions and experimental data provides confidence in the practical utility of this model as a tool for designing novel 3D cellular composites with unprecedented specific elastic properties.