The applicability of the effective medium theory to the dynamics of cellular beams

The applicability and the limitations of the effective medium assumption for the dynamics of cellular beams are studied. Beams made of uniform triangular and regular hexagonal cells are analysed. The natural frequencies and modal distributions as calculated using the detailed finite element model of the cellular networks are compared with those predicted based on equivalent homogeneous media of the same overall size and shape. It is observed that, for low mode number, a cellular beam behaves as a continuum, provided the cell size is significantly smaller than the external dimensions of the beam. Due to different deformation mechanisms, triangular cells show frequencies independent of area fraction whereas hexagonal cells show this dependence clearly. As the wavelength starts to become of the order of the heterogeneity, the continuum behaviour begins to break down. With the increase in mode number, cellular beams exhibit inherent flexibility with a progressive increase in their modal densities as compared to those of a homogeneous continuum. The modal density increases further when the cell walls start to resonate. During resonance, an abrupt rise in the modal density is observed for the triangular cells as the cell walls start deforming in the flexural mode instead of the axial mode. In contrast, for hexagonal cells, the predominant mode of cell wall deformation is always flexural.