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.

中文翻译：

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周期性结构中最近邻之外的相互作用：力分析

摘要 周期结构是一种超材料，其物理性质不仅取决于晶胞的组成，还取决于晶胞的排列方式和相互作用。这种结构具有有趣的波传播特性，使其成为声学滤波器和波束装置的合适材料。为了分析这些结构中的波传播，通常使用布洛赫定理。Bloch 分析最初是为了研究晶体中电子波的传播而开发的，其中原子间势函数会引起超出最近邻的显着相互作用。然而，在几乎所有周期性工程结构中，每个单元格都通过弹性接触和力与相邻单元格相互作用。为周期性工程结构的振动和波传播分析开发的方法通常涉及仅由最近邻施加的力。由于超材料的复杂性很大程度上取决于晶胞之间的相互作用，更多的相互作用会产生更复杂的行为。例如，这种复杂的行为可能表现在更复杂的能带结构中。为了对相互作用不限于最近邻的结构进行建模，有必要推导出运动方程，通过应用布洛赫分析去除力，然后获得特征值问题。这种对力进行建模然后以适当的方式消除它们的过程是本文的主题。如果该过程没有正确执行，结果是一个错误的特征值问题。