Hierarchical honeycomb material design and optimization: Beyond linearized behavior

This paper explores the importance of nonlinear material properties in the design of hierarchical honeycomb materials. The recent literature on the design and optimization of linear material properties for hierarchical honeycombs is reviewed. Then a full nonlinear post-bifurcation numerical analysis is performed for five representative hierarchical honeycomb structures. Particular attention is paid to the following four nonlinear material properties: the critical load λ_c at which the structure first experiences an instability; the plastic critical load λ_p at which the onset of plasticity would occur (if no elastic instability occurred); the stable post-bifurcated structure of the honeycomb; and the purely elastic resilience of the nonlinear material. It is found that although the honeycomb's linear Young's modulus is optimally maximized at a hierarchy ratio of γ₁ ≈ 30%, the critical load is reduced by a factor of two (relative to the standard honeycomb) at this ratio. Further, the critical load displays a monotone decreasing trend with increasing hierarchy ratio. A similar trend is found for the plastic critical load. A non-monotone trend for the resilience is discovered and explained by a qualitative change in the stable post-bifurcated structure for the hierarchical honeycombs which occurs as the hierarchy ratio is increased. The observed loss of strength (decreased critical load) is significant and may negate any advantages of the increased Young's modulus. This result demonstrates the importance of considering nonlinear properties and their implications in the design and optimization of hierarchical materials.