Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials

A variational method is introduced to study the propagation of elastic waves in two-dimensional periodic systems. First, it has been applied to a binary system consisting of an array of high-density cylinders in an epoxy background, where the advantages of the method are pointing out in regard with the well-known plane-wave expansion formalism. Second, a comprehensive study is performed for the two-dimensional counterpart of the ternary systems recently studied by Liu et al. @Z. Liu et al., Science 289, 1734 ~2000!#. Numerical simulations predict that subfrequency gaps also appear because of the soft polymer that coats the cylinders. A simple mechanical model is introduced to give a physical insight of the phenomenon responsible for our findings.