Band structure calculation of 3d ultrawide elastic metamaterials with embedded inertial amplification mechanisms

In this study, phononic band structure of three dimensional (3D) ultrawide elastic metamaterials with embedded inertial amplification mechanisms are obtained. In order to achieve that, inertial amplification mechanisms with different sizes and geometries are considered by applying periodic boundary conditions, also known as Bloch's boundary conditions, to the unit cells. First, typical wave propagation problems in one dimensional, two dimensional, and three dimensional periodic structures studied in the literature are investigated and benchmark studies are performed by COMSOL Multiphysics and ABAQUS/MATLAB programs. In this way, these models are tested and verified so that the phonon band structure of the 3D elastic metamaterials with embedded inertial amplification mechanisms can be calculated accurately. Inertial amplification mechanisms have complex geometries and their computational costs can be very high. Thus, analyses are done by using both COMSOL Multiphysics and ABAQUS/MATLAB programs. Also, the comparison of the results obtained using these programs with the FRF results of the 3×2 octahedron array enables the determination of the most accurate model. It is very likely to encounter many problems when applying Bloch's theorem to a complex system such as a 3D elastic metamaterial with embedded inertial amplification mechanisms. Among many problems, four possible problems are explained and their solutions are presented. Thus, it is shown that the band structure of any geometry can be easily obtained, regardless of the complexity of the geometry. To sum up, in the literature, the widest band gap in 3D is achieved by this method, and the band gap is found to be in between 6.37 - 90.26 Hz, with a ratio of the upper limit to the lower limit of 14.17. Hence, it is demonstrated that the 3D elastic metamaterial with embedded inertial amplification mechanisms impedes waves coming from all directions in a very wide frequency range.