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All the five independent elastic properties/constants of micro- and nano-structured hierarchical and self-similar random irregular honeycombs with different degrees of cell regularity are obtained by analysis and computer simulation in this paper. Cell wall bending, stretching, and transverse shearing are the main deformation mechanisms of hierarchical honeycombs. The strain gradient effects at the micro-meter scale, and the surface elasticity and initial stress effects at the nano-meter scale are incorporated into all the deformation mechanisms in the analysis and finite element simulations. The results show that the elastic properties of hierarchical random irregular honeycombs strongly depend on the thickness of the first-order cell walls if it is at the micro-meter scale, and that if the thickness of the first-order cell walls is at the nano-meter scale, the elastic properties of hierarchical random irregular honeycombs are not only size-dependent, but are also tunable and controllable over large ranges. In addition, the geometrical properties of nano-structured hierarchical random irregular honeycombs are also tunable and controllable.

This paper investigates the effects of cell wall thickness, initial stress/strain, and cell regularity on the high strain compressive responses of micro- and nano-sized low density random irregular honeycombs. The strain gradient effects at the micrometer scale, and the surface elasticity and initial stress effects at the nanometer scale are incorporated into the dominant deformation mechanisms in finite element simulations. It is found that the dimensionless compressive stress strain relation strongly depends on the thickness of the cell walls at the micron scale, and at the nano-meter scale, this relation is not only size-dependent, but are also tunable and controllable over a large range. It is also found that under high strain compression, the Poisson's ratios of micro- and nano-sized low density random irregular honeycombs strongly depend on the cell regularity, but are almost independent of the cell wall thickness and the amplitudes of the initial stress or strain.

The mechanical and geometrical properties of hierarchically nano-structured random irregular honeycombs are obtained in this paper. All the possible deformation mechanisms: bending, stretching, transverse shear and hinging, are considered in the deformation analysis. The results show that if the size of the first order cell walls is at the nano-meter scale, the elastic and geometrical properties of a hierarchical and random irregular honeycomb are not only size-dependent, but also tunable over a large range.