In this paper, star-shaped honeycombs are analyzed in terms of their equivalent mechanical behaviors and band gap properties. Firstly, by applying Castigliano’s second theorem, the effective Young’s modulus and Poisson’s ratio are derived by an analytical method used in structural mechanics. On the basis of Bloch’s theorem, the dispersion characteristics are then analyzed by the dynamic matrix in conjunction with the Wittrick–Williams (W–W) algorithm. It should be noted that the presented method can form a more simple stiffness and mass matrices of the proposed structures, compared with the traditional finite element (FE) method. Thereafter, the effects of the geometrical parameters on the effective constants and band gaps are investigated and discussed. Numerical results demonstrate that the negative Poisson’s ratio provides an enhanced effective Young’s modulus of the considered honeycombs. Furthermore, the band gap exists in a much lower frequency region with an unchanged summing band gap width when the Poisson’s ratio is in negative values. In general, the work can serve as a guide for the optimal design of cellular structures.

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.