Bounds on resonant bandgap limits in a branched 1D lattice modeled by Bloch’s theorem

Elastic metamaterials exhibit unique properties associated with local resonance band gaps. A 1D lattice unit cell comprising a monatomic chain connected to a branch is examined leading to mathematical relations relating the vibration behavior of the independent branch to the dispersion of the entire unit cell. This perspective is distinct from studying the resonances of the full system and relating them to the local-resonance band gaps. The closed-form relation, based on Bloch’s theorem, determines the dependence of the upper and lower band-gap limits on the resonances and antiresonances of the frequency response of the separate branch. This offers a formal approach for identifying bounds for the location of band-gap edges. Moreover, it demonstrates that local resonance band gaps form as a result of a balance between the inertia and restoring forces of the main chain and the branch effective restoring force. This framework is further employed to study a special case where the branch is constructed out of a finite number of repeating diatomic units where the periodic branch’s Bragg band gaps are exploited. Conditions are derived for a global unit-cell dispersion exhibiting super-wide local resonance band gaps or pass bands, super narrow pass bands, and tailored fano-resonances.