Band gap analysis of star-shaped honeycombs with varied Poisson's ratio

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.