Buckling of regular and auxetic honeycombs under a general macroscopic stress state in symplectic system

A symplectic stiffness method, based on Hamiltonian variational principle and symplectic eigen expansion, is presented to investigate the buckling of honeycomb structures under a general macroscopic stress state. The analytical closed-form expressions of critical forces are obtained for the possible buckling modes, which satisfy the deformation compatibility conditions for the cell beams, and the lowest value in the stability surface corresponds to the actual buckling force for the honeycomb. The results show that the buckling strengths and modes for the regular and auxetic honeycomb structures depend strongly on the biaxial stress state but weakly on the relative density. For some complex honeycombs, e.g., the triangular and re-entrant cell, the buckling modes change with the increasement of the orders. Considerable attention is focused on the physical mechanism of phase transformation. There is no possibility of a swaying buckling mode when the beam buckles symmetrically, and the re-entrant geometry has an important influence on the auxetic honeycomb buckling patterns. The present symplectic stiffness method is believed to be beneficial for the stability analysis of some frontier engineering fields such as flexible electronics and soft robots.