Pade-Based Strain Gradient Modeling of Bandgaps in Two-Dimensional Acoustic Lattice Metamaterials

A strain gradient (SG) continuum theory of two-dimensional (2D) lattice metamaterials based on Pade approximation has been proposed, called PSGM hereafter, to predict their acoustic dispersion characteristics. Square and triangular elastic lattices have been investigated for a demonstrating purpose. By applying Pade approximation to the Taylor expansion of displacement field, the fourth- and eighth-order SG continuum models are established. The dispersion relations obtained by the proposed model are examined by comparing with the results by discrete analyses as well as existing SG theories. It is confirmed that the proposed theory is always more accurate than the existing SG counterpart with the same SG order. Furthermore, the present formulation is free of any unrealistic instability issue in dispersion, which has challenged the existing SG theories. The results also show that the accuracy of PSGM will be enhanced with increasing SG orders. Within the present parameter settings, it can be found that the eighth-order PSGM can successfully capture the dispersive properties of both lattices throughout the first irreducible Brillouin zone.