This paper investigates methods of band gap formation in two-dimensional locally resonant elastic metamaterials and proposes a technique for computing such gaps as algebraic functions of integrated Bloch modes and metamaterial parameters. Mode shapes associated with the deformation of unit cells consisting of a hard matrix, soft filler, and hard resonator are investigated near the lower and upper bounds of the first local resonance band gap. Similarities reveal that Bloch modes at these bounds have the same non-dimensional geometries and locations within the irreducible Brillouin Zone irrespective of material properties, matrix and resonator thicknesses, and unit cell size. A theoretical model for the unit cell incorporating Mindlin plate theory is employed to compute the frequency of these modes in transverse vibration using conservation of energy. A least squares system identification algorithm is then described to generate surface equations for non-dimensional Bloch mode shapes computed by finite element analysis, employing a computation-saving Kronecker factorization and yielding continuous surface equations for the non-dimensional Bloch modes when rectangular cell components are utilized. Nine sample metamaterials are analyzed with this approach generating reliable predictions of the first band gap bounds. The presented framework offers insights into band gap formation, opens avenues in inverse unit cell design, and provides versatile algebraic relationships between band gap frequencies and unit cell parameters in such resonant metamaterials.

本文研究了二维局部共振弹性超材料中带隙形成的方法，并提出了一种计算带隙的技术，如积分布洛赫模式和超材料参数的代数函数。在第一个局部共振带隙的下限和上限附近研究了与由硬基体、软填料和硬谐振器组成的晶胞变形相关的模式形状。相似性表明，无论材料特性、矩阵和谐振器厚度以及晶胞尺寸如何，这些边界处的布洛赫模式在不可约布里渊区内都具有相同的无量纲几何形状和位置。使用包含 Mindlin 板理论的晶胞的理论模型来计算这些模式在使用能量守恒的横向振动中的频率。然后描述最小二乘系统识别算法以生成由有限元分析计算的无量纲 Bloch 模式形状的表面方程，使用节省计算的 Kronecker 分解并在矩形单元分量为无量纲 Bloch 模式时产生连续表面方程利用。用这种方法分析了九个样本超材料，生成了第一带隙界限的可靠预测。所提出的框架提供了对带隙形成的见解，为逆晶胞设计开辟了道路，