The in-plane elastic properties of hierarchical composite cellular materials: Synergy of hierarchy, material heterogeneity and cell topologies at different levels

The hierarchical organization of many biological materials plays a key role in their exceptional mechanical properties. Existing studies investigate how hierarchy affects the mechanical behavior of cellular materials and the vast majority of them assume empty cells. In reality, in numerous natural systems the cells are filled with fluids, fibers or other bulk materials to better resist external stimuli. Inspired by the highly efficiency of nature, this paper investigates the effects of adding hierarchy into a composite cellular material. Initially, the analytical expressions for the effective elastic moduli derived in the case of self-similarity reveal the system isotropy as for the not filled configuration. Then, from parametric analysis emerges a strong influence of the microstructure on the overall properties. We discovered that adding hierarchical levels to a filled cellular material can lead to a higher material specific stiffness only if the filler is stiffer than a critical value. Thus for classical cellular materials hierarchy is detrimental for the specific stiffness. In spite of this, for composite cellular solids an optimal number of hierarchical levels naturally emerges. In addition, numerical homogenization validates the analytical approach. Finally, the example of a hierarchical composite cellular material having different levels with different cell topologies is also considered. The present analysis provides an insight into the role of structural hierarchy on the in-plane elastic properties of composite cellular materials, as well as some possible ways to improve low-weight cellular structures by mixing different materials and varying the cell topology.