Double bandgap formation in a locally resonant metamaterial micromorphic beam

The present study investigates bandgap formation in micromorphic metamaterial Euler-Bernoulli beams, with the problem also modeled using a nonlocal strain gradient framework. Both models are mathematically well-posed, sharing identical governing equations but differing in the definiteness of their potential energy. Multiple oscillators, arranged in two distinct configurations according to the local resonance principle, are employed to realize a double bandgap structure. Application of Hamilton’s principle yields the non-dimensional governing equations of motion, expressed as a sixth-order system with corresponding boundary conditions. Bandgap edge frequencies are determined from wave dispersion analysis in an infinitely long beam by means of periodic unit cell modeling. Dispersion relations derived through homogenization and transfer matrix methods are consistent with each other and with previously established results. Two homogenization approaches are examined: a one-field displacement-based formulation, which neglects nonlocal oscillator inertia and fails to guarantee asymptotic consistency when the nonlocal and strain gradient parameters are equal and large; and a two-field formulation, which resolves this limitation and preserves consistency. Dispersion curves obtained from both homogenization schemes are compared with those from the transfer matrix method to assess accuracy. The dispersion characteristics derived from the infinite medium formulation are further validated by frequency response functions of a finite-length nanobeam, demonstrating agreement between spectral predictions and bounded system dynamics. Finally, a parametric study explores the influence of key parameters, including the nonlocal and strain gradient coefficients, unit cell length, and resonator-to-unit-cell mass ratio, on bandgap formation in metamaterial beams.