Mechanics of mutable hierarchical composite cellular materials

Cellular structures having the internal volumes of the cells filled with fluids, fibres or other bulk materials are very common in nature. A remarkable example of composite solution is the hygroscopic keel tissue of the ice plant Delosperma nakurense. This tissue, specialized in promoting the mechanism for seed dispersal, reveals a cellular structure composed by elongated cells filled with a cellulosic swelling material. Upon hydrating, the filler adsorbs large amounts of water leading to a change in the cells' shape and effective stiffness. This paper, inspired by the configuration of the aforementioned hygroscopic keel tissue, deals with the analysis of a two-dimensional honeycomb made of elongated hexagonal cells filled with an elastic material. The system is treated as a sequence of Euler Bernoulli beams on Winkler foundation, whose displacements are derived by introducing the classical shape functions of the Finite Element Method. The assumption of the Born rule, in conjunction with an energy-based approach, provide the constitutive model in the continuum form. It emerges a strong influence of the infill's stiffness and cell walls' inclination on the macroscopic elastic constants. In particular, parametric analysis reveals the system isotropy only in the particular case of regular hexagonal microstructure. Even though a rigorous analysis of the keel tissue is well beyond our aim, the application of the theoretical model to estimate the effective stiffness of such biological system leads to results that are in good agreement with the published data, where the keel tissue is represented as an internally pressurized honeycomb. Specifically, an energetic equivalence gives an explicit relation between the inner pressure and the filler's stiffness. Optimal values of pressure and cell walls' inclination also emerge. Finally, the theory is extended to the hierarchical configuration and a closed form expression for the macroscopic elastic moduli is provided. It emerges a synergy of hierarchy and material heterogeneity in obtaining a stiffer material, in addition to an optimal number of hierarchical levels.