Bending Cutoff Governs Phononic Band Gaps in Elastic Beam Lattices

Phononic band gaps have been widely studied in beam lattices, but the physical principle that determines the location of these gaps has remained unclear. Here, we first develop a 1D analytical model to derive axial and bending cutoff frequencies at the Brillouin zone boundary. The derivation shows that the bending cutoff frequency is determined by the beam slenderness parameter and the lattice periodicity (through the maximum wavevector). We then analyze 2D periodic beam lattices by computing dispersion relations and density of states (DOS). The results show that the bending cutoff frequency derived from the 1D analytical model governs the location of phononic band gaps in 2D periodic beam lattices. We further show that the same principle applies to 2D disordered beam lattices generated from point configurations including packings of hard disks, stealthy hyperuniform systems, and Poisson distributions. To this end, we develop a finite-element supercell model with periodic boundary conditions and then perform eigenfrequency analysis to obtain the DOS, thereby identifying the onset of a pseudogap (a frequency regime where the DOS is strongly suppressed without a complete band gap). Taken together, these results demonstrate that while lattice topology and the type of disorder can influence the details of the DOS, they play only a minor role compared to this universal cutoff mechanism. These findings establish a general design principle for phononic band gaps in elastic beam lattices.