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A magic square is defined as a square array of distinct positive numbers arranged such that the sum along the horizontal, vertical, and main diagonal directions are the same, which is called a magic number. Previously, we leveraged the magic square concept to create a new class of phononic structures whose dispersion behavior can be tuned without altering their global mass and stiffness. In this study, we study the dispersion behavior of a thin plate with non-structural mass arranged in a magic square pattern. The plate unit cell is partitioned into nine portions and individual point masses are embedded at the center of each portion. Our results show that the magic square mass-embedded plate provides a low-frequency out-of-plane bandgap whose widths can be controlled while maintaining the global mass of the plate structure. Additionally, we investigate a magic plate concept by applying different densities at each portion associated with the magic square patterns. Our preliminary results show that the magic density distribution results in the emergence of polarized bandgaps and provides a possibility of altering the plate modal frequency without altering the mode shape or the global mass.

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In this talk, we examine the elastic wave propagation behavior of infinite two-dimensional self-similar and non-self-similar hierarchical square periodic lattices. Using finite element analysis, we calculate the dispersion curves of different hierarchical lattice structures along their irreducible Brillouin zone (IBZ) paths. The effects of similarity, geometrical symmetry, and material symmetry on bandgap generation are studied. The mechanisms responsible for bandgap generation are explained by studying the mode shapes for different characteristic length ratios. We show that material symmetry can help induce wave attenuation bandgaps at low frequencies. Further, we classify the P- and S-wave propagation modes and study the effect of similarity on the generation of polarized bandgaps. We demonstrate that controlling the hierarchical similarity provides a robust method for tailoring the dispersion characteristics of periodic lattice structures.

Recent studies show that structures with triply periodic minimal surfaces (TPMS) provide enhanced mechanical, acoustical, and energy abortion performance. Previously, we have shown that breaking the symmetry of the gyroid lattice—one of the most used TPMS geometry—results in the creation of directional and polarized bandgaps. Here, we focus on the effect of breaking symmetry on the effective wave speeds of the gyroid structure. We analyze the wave speeds of different asymmetric gyroid lattices using the finite element analysis approach. Our analysis shows that certain gyroid asymmetries result in the transverse waves propagating faster than the longitudinal waves in particular direction. Our research shows that breaking the symmetry leads to previously unobserved anomalous polarization of elastic waves in asymmetric gyroids.