On the band gap formation in locally-resonant metamaterial thin-walled beams

This paper is concerned with flexural and torsional waves propagation in locally-resonant thin-walled beams with open cross section. A beam with monosymmetric cross section hosting a periodic array of translational and torsional resonators is considered. The resonators are designed to protect the beam from both vertical and lateral dynamic loads; while vertical loads acting in the symmetry plane induce flexural waves only, vertical loads acting out of the symmetry plane and lateral loads induce coupled flexural-torsional waves due to the monosymmetry of the beam cross section. Aim of the paper is to investigate the dispersive properties of the locally-resonant beam under consideration, focusing on coupled flexural-torsional waves. For this purpose, a homogenization approach is adopted to derive a simplified dispersion equation of the infinite locally-resonant beam. Starting from the observation that the latter equation is equivalent to the dispersion equation of a uniform infinite beam with homogenized inertial properties, it is possible to identify the location and the amplitude of band gaps; remarkably, it is found that band gaps for coupled flexural-torsional waves are located in the frequency region where the matrix of the homogenized inertial properties is negative definite. Further, it is demonstrated that band gaps arise when translational and torsional resonators are simultaneously connected to the beam and are tuned to the same frequency; on the other hand, band gaps are not guaranteed if only translational resonators or only rotational resonators are connected to the beam. Alternative exact methods based on the transfer matrix approach and the theory of generalized functions are employed to validate and substantiate the obtained results.