In this study, the wave propagation properties of lattice metamaterials with Koch fractal structures are investigated in terms of band structures and directional wave propagation. The analytical models of lattice metamaterials are established using the finite element method, and the dispersion relation is solved using the Bloch’s theorem. The band structures of the lattice metamaterials with different numbers of iterations are studied, and the group velocities at a selected frequency are calculated to analyze the directional wave propagation characteristics. Furthermore, dynamic responses of the finite structures are calculated using commercial finite element software to verify the band gaps and directional wave propagation behaviors in the lattice metamaterials. The results show that multiple and low band gaps are present in the lattice materials with various geometric parameters of the Koch fractal, and the position of the lowest band gap decreases as the number of iterations increases. The results indicate the potential applications of lattice metamaterials with Koch fractals for vibration isolation and multi-functional design.

The wave propagation in the tetrachiral and hexachiral lattices with local resonators are investigated and the wave behaviors with different geometrical parameters are analyzed. In this study, the tetrachiral and hexachiral lattices are assembled with the repeat unit cells and the unit cell contains a ring and a number of massless slender elastic ligaments. The ligaments are rigidly connected to the ring and the ring contains a heavy disk with its surrounding soft elastic annulus, which acts as the local resonators. The governing equations of the lattices are established by energy variation principle and the wave behaviors of the lattices are calculated by solving the eigenvalue problem with the Bloch’s theorem. The effects of chiral angles on the distributions of band gaps including the width and position are studied to investigate the effects of the chirality and local resonators on the formation of low-frequency band gaps. The first mode of the phase and group velocities are calculated to analyze the effects of the geometrical parameters on the directional frequency-dependent energy flows in the anisotropic structures. We also use the commercial finite element software COMSOL to simulate and verify the directional wave behaviors in the structure. We find that the first mode of the elastic wave spread only along certain specific directions in the tetrachiral structure. As the chiral angle increases, the speed of wave propagation is reduced, and the direction of wave propagation rotates clockwise. The hexachiral lattice exhibits effective isotropic property in the low frequency range, and the wave propagation velocity gradually decreases with the increase of the frequency of the elastic waves.