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<p class="indent">Many natural and man-made materials exhibit self-similar hierarchical microstructures on several length scales. The effective macroscopic mechanical properties of such materials or composites are affected by the number of hierarchical levels and the topology of microstructures. Although the effective mechanical properties can be determined numerically using homogenization techniques, the computational costs can become prohibitively high as the level of hierarchy increases. This paper proposes an analytical approach to predicting the effective stiffness of a class of materials and structures with self-similar hierarchical microstructures. For each microstructural configuration, a simple relationship between the effective stiffness and the hierarchical level is established and verified against results of finite element analysis or data in the literature. It is found that the simple relationships we have developed provide quite accurate stiffness predictions of various hierarchical materials and composites including the Menger sponge. For composites, the predicted effective stiffness is accurate even when one of the phases is near its incompressibility limit, with its Poisson ratio close to 0.5. Inspired by the Menger sponge and informed by our topology optimization result, we propose a lighter yet stiffer “cross sponge”.</p>

Predicting the effective stiffness of cellular and composite materials with self-similar hierarchical microstructures