| Approaches to obtain analytical closed-form expressions for the macroscopic elastic, plastic collapse, and buckling response of various two-dimensional cellular structures are presented. First, we will provide analytical models to estimate the effective elastic modulus and Poisson's ratio of hierarchical honeycombs using the concepts of mechanics of materials and compare the analytical results with finite element simulations and experiments. For plastic collapse, we present a numerical minimization procedure to determine the macroscopic `plastic collapse strength' of a tessellated cellular structure under a general stress state. The method is illustrated with sample cellular structures of regular and hierarchical honeycombs. Based on the deformation modes found by minimization of plastic dissipation, closed-form expressions for strength are derived. The work generalizes previous studies on plastic collapse analysis of lattice structures, which are limited to very simple loading conditions. Finally, the method for calculation of buckling strength is based on classical beam-column end-moment behavior expressed in a matrix form. It is applied to regular, chiral, and hierarchical honeycombs with square, triangular, and hexagonal unit cells to determine their buckling strength under a general macroscopic stress state. The results were verified using finite element eigenvalue analysis. |
