Homogenization and equivalent in-plane properties of two-dimensional periodic lattices

The equivalent in-plane properties for hexagonal and re-entrant (auxetic) lattices are investigated through the analysis of partial differential equations associated with their homogenized continuum models. The adopted homogenization technique interprets the discrete lattice equations according to a finite differences formalism, and it is applied in conjunction with the finite element description of the lattice unit cell. It therefore allows handling of structures with different levels of complexity and internal geometry within a general and compact framework, which can be easily implemented. The estimation of the mechanical properties is carried out through a comparison between the derived differential equations and appropriate elasticity models. Equivalent Young's moduli, Poisson's ratios and relative density are estimated and compared with analytical formulae available in the literature. In-plane wave propagation characteristics of honeycombs are also investigated to evaluate phase velocity variation in terms of frequency and direction of propagation. Comparisons are performed with the values obtained through the application of Bloch theorem for two-dimensional periodic structures, to show the accuracy of the technique and highlight limitations introduced by the long wavelength approximation associated with the homogenization technique.