The cellular nature of many biological materials, providing them with low density, high strength and high toughness, have fascinated many researchers in the field of botany and structural biology since at least one century. Bamboo, sponges, trabecular bone, tooth and honeybee combs are only few examples of natural materials with cellular architecture. It has been widely recognised that the geometric and mechanical characteristics of the microscopic building blocks play a fundamental role on the behavior observed at the macroscale. Up to date, many efforts have been devoted to the analysis of cellular materials with empty cells to predict the structure-property relations that link the macroscopic properties to the mechanics of their underlying microstructure. Surprisingly, notwithstanding the great advantages of the composite solutions in nature, in the literature a limited number of investigations concern cellular structures having the internal volumes of the cells filled with fluids, fibers or other bulk materials as commonly happens in biology. In particular, a continuum model has not been derived and explicit formulas for the effective elastic constants and constitutive relations are currently not available. To provide a contribution in this limitedly explored research area, this thesis describes the mathematical formulation and modelling technique leading to explicit expressions for the macroscopic elastic constants and stress-strain relations of biologically inspired composite cellular materials. Two examples are included. The first deals with a regular hexagonal architecture inspired by the biological parenchyma tissue. The second concerns a mutable cellular structure, composed by mutable elongated hexagonal cells, inspired by the hygroscopic keel tissue of the ice plant Delosperma nakurense. In both cases, the predicted results are found to be in very good agreement with the available data in the literature. Then, by taking into account the benefits offered by the complex hierarchical organisation of many natural systems, the attention is focused on the potential value of adding structural hierarchy into two-dimensional composite cellular materials having a self-similar hierarchical architecture, in the first case, and different levels with different cell topologies, in the second. In contrast to the traditional cellular materials with empty cells, the analysis reveals that, in the cell-filled configuration, introducing levels of hierarchy leads to an improvement in the specific stiffness. Finally, to offer concrete and relevant tools to engineers for developing future generations of materials with enhanced performance and unusual functionalities, a novel strategy to obtain a honeycomb with mutable cells is proposed. The technique, based on the ancient Japanese art of kirigami, consists in creating a pattern of cuts into a flat sheet of starting material, which is then stretched to give a honeycomb architecture. It emerges a vast range of effective constants that the so-called kirigami honeycomb structures can be designed with, just by changing the value of the applied stretch.

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