Nonlinear shear modulus of re-entrant hexagonal honeycombs under large deformation

The nonlinear, in-plane, shear modulus of re-entrant hexagonal honeycombs under large deformation is analytically derived by studying the mechanical behavior of cell structures, which is later verified by numerical simulations. A nonlinear, modified factor is proposed to characterize the difference of the honeycomb's shear modulus under large and small deformation, revealing its independence from the honeycomb's relative density. The effects of both strain and cell geometry on the honeycomb's shear modulus are investigated, exhibiting that the honeycomb's shear modulus increases with shear strain but decreases with the cell-wall-length ratio. For the effect of cell-wall angle, the re-entrant honeycomb's shear modulus decreases gradually with the cell-wall angle until reaching a minimum and then increases, which is highly different from the monotonically increasing relationship of conventional hexagonal honeycombs. When keeping the honeycomb's relative density constant, the re-entrant honeycomb's shear modulus monotonously increases with the cell-wall angle and reaches a maximum at h/l ≈ 3.25. Finally, the shear modulus of the re-entrant honeycombs is compared with that of conventional honeycombs. In contrast to the predictions of the classical continuum theory, the present study shows that the shear modulus of the re-entrant honeycomb with a negative Poisson's ratio is not always higher than that of the conventional honeycomb with a positive Poisson's ratio, which is dominated by the geometry of the cell structure.