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Density gradation has been analytically and experimentally proven to enhance the load-bearing and energy absorption efficiency of cellular solids. This research focuses on the analytical optimization (by virtual experiments) of polymeric honeycomb structures made from flexible thermoplastics to achieve density-graded structures with desired mechanical properties. The global stress-strain curves of single-density honeycomb structures are used as input to an analytical model that enables the characterization of the constitutive response of density-graded hexagonal honeycombs with discrete and continuous gradations and for various gradients. The stress-strain outputs are used to calculate the specific energy absorption, efficiency, and ideality metrics for all density-graded structures. The analytical results are shown to be in good agreement with previous experimental measurements. The findings of this research suggest that the choice of an optimal gradient depends on the specific application and design criteria. For example, graded structures wherein low-density layers are dominant are shown to outperform high density uniform honeycombs in terms of specific energy absorption capacity while possessing higher strength compared with low density uniform structures.

Honeycomb structures are widely used in engineering applications mainly due to their high strength to weight ratio. By changing the base material and geometry of the repeating unit cell structure, target effective properties can be achieved. Hierarchical honeycomb structures are known to have enhanced mechanical properties when compared to regular honeycomb structures. Therefore, it is important to understand and quantify the mechanical properties and the variation of these properties with the presence of hierarchy. This investigation builds upon prior work and considers the mechanical properties of two dimensional hierarchical honeycomb structures. Previous research of hierarchical honeycomb structures studied replacing the homogeneous cell walls with truss lattices, or by replacing the cell walls by composite layers. Another hierarchy was examined by replacing the vertices of hexagon by smaller hexagons. However, in contrast to these previous studies, reiterated hierarchy is studied in this work, where a first order hierarchy structure is created by placing smaller honeycombs inside the conventional honeycombs such that midpoints of edges of the base level-0 honeycomb are shared vertices of the smaller level-1 honeycomb. In this work, the in-plane effective mechanical properties of these reiterated hierarchical honeycomb structures are studied with both regular and auxetic honeycombs. Effective elastic moduli and Poisson's ratio properties are determined and compared for a range of different cell wall thickness ratios between the base level-0 and smaller level-1 hierarchy. For comparisons, the mass was kept constant in all cases. Given the total mass and thickness ratio of the level-0 to level-1 hierarchy, the mass distribution is varied. The mechanical properties are determined from finite element analysis of a patch of honeycombs in both uni-axial tension and shear loading conditions. By changing the thickness ratio of level-0 to level-1 hierarchy, a nonlinear variation in mechanical properties is observed showing maximum and minimum values at specific ratios. From the results of first order regular hierarchical honeycomb structures, it can be said that for the same mass, the effective Young's modulus for thickness ratio of 0.1 between level-0 divided by level-1 is maximum and is about 1.45 times that of the zeroth order. Maximum effective shear modulus occurs for the special case with thickness ratio of zero, corresponding to a special level-1 honeycomb structure with the level-0 structure removed, and is 1.57 times that of the zeroth order. From the results of first order auxetic hierarchical honeycomb structures, it can be said that the effective relative Young's modulus, and shear modulus of first order is higher for any thickness ratio than that of the zeroth order auxetic honeycomb structure of the same mass. The maximum effective Young's modulus occurs for thickness ratio 9 and is about 2.8 times that of the zeroth order. The maximum effective shear modulus of first order structure is maximum at ratio 0.1 and is 2.6 times that of the zeroth order.