LAUSR

Abstract Periodic structures are a type of metamaterial in which physical properties depend not only on the constituents of the unit cell but also on how the unit cells are arranged and interact with each other. Such structures have interesting wave propagation properties, making them suitable materials for acoustic filters and wave beaming devices. To analyze wave propagation in these structures, the Bloch theorem is commonly used. Bloch analysis was first developed to study electron wave propagation in crystals in which the interatomic potential function gives rise to significant interactions extending beyond nearest neighbors. In virtually all periodic engineering structures, however, each unit cell is interacting with adjacent cells through elastic contacts and forces. Methods developed for vibrational and wave propagation analysis of periodic engineering structures are generally concerned with forces exerted only by the nearest neighbor. As the complexity of a metamaterial depends largely on the interactions between unit cells, more interactions give rise to more complex behavior. Such complex behavior may, for instance, be manifest in a more complex band structure. In order to model structures in which interactions are not limited to nearest neighbors, it is necessary to derive equations of motion, get rid of forces by applying Bloch analysis, then obtain an eigenvalue problem. This procedure of modeling forces and then eliminating them in the appropriate way is the subject of the present paper. If the procedure is not implemented correctly, the result is an erroneous eigenvalue problem. The present analysis lays the foundation for vibrational analysis of structures in which interactions are not restricted to the nearest neighbor.