Fig. 1. Schematic of the MRE acoustic metamaterial cells and the irreducible first Brillouin zone: (a) Single-layer MRE acoustic metamaterial; (b) double-layer MRE acoustic metamaterial; (c) the irreducible first Brillouin zone of square lattices.MRE包覆层声学超材料元胞结构和不可约布里渊区　(a) 单包覆层MRE声学超材料; (b) 双包覆层MRE声学超材料; (c) 正方晶格第一不可约布里渊区

Fig. 2. The band gap structures of single-layer MRE acoustic metamaterial: (a) Magnetic field intensity of H = 0; (b) magnetic field intensity of H = 6 kOe. 单包覆层MRE声学超材料带隙结构　(a) 磁场强度H = 0; (b) 磁场强度H = 6 kOe

Fig. 3. Displacement field of the band gap boundaries of the single-layer MRE acoustic metamaterial: (a) The lower boundary of the band gap; (b) the upper boundary of the band gap.单包覆层MRE声学超材料带隙边界位移场　(a) 带隙下边界; (b) 带隙上边界

Fig. 4. The mass-spring model of single-layer MRE acoustic metamaterial: (a) Lower boundary of band gap; (b) upper boundary of band gap.单包覆层MRE声学超材料的质量-弹簧模型　(a) 带隙下边界; (b) 带隙上边界

Fig. 5. Dependence of the band gap boundaries on the applied magnetic field of single-layer MRE acoustic metamaterial and a comparison of the FEM and the mass-spring model for (a) magnetic field intensity of H = 0–10 kOe, (b) comparison of the FEM and mass-spring model. 单包覆层MRE声学超材料带隙随磁场强度变化及有限元法与质量-弹簧模型对比　(a) 磁场强度H = 0—10 kOe; (b) 有限元法与质量弹簧-模型对比

Fig. 6. The band gap of single-layer MRE acoustic metamaterial changes with (a) the core mass and (b) the filling rate.单包覆层MRE声学超材料带隙随材料和结构参数变化　(a) 随芯体质量增加变化; (b) 随填充率增加变化

Fig. 7. The band gap structures of the double-layer MRE acoustic metamaterial: (a) H = 0; (b) H = 6 kOe. 双包覆层MRE声学超材料带隙结构　(a) H = 0; (b) H = 6 kOe

Fig. 8. Displacement field of the band gap boundaries of the double-layer MRE acoustic metamaterial: (a) The lower boundary and (b) the upper boundary of the first band gaps; (c) the lower boundary and (d) the upper boundary of the second band gap; (e) point E in Fig.7 (a). 双包覆层MRE声学超材料带隙边界处位移向量场　(a) 第一阶带隙下边界; (b) 第一阶带隙上边界; (c) 第二阶带隙下边界; (d) 第二阶带隙上边界; (e) 图7(a)中E点

Fig. 9. The mass-spring model of the double-layer MRE acoustic metamaterial: (a) Lower boundary of the first and second band gap; (b) upper boundary of the first and second band gap.双包覆层MRE声学超材料质量-弹簧模型　(a) 第一阶与第二阶带隙下边界; (b) 第一阶与第二阶带隙上边界

Fig. 10. Dependence of the band gap boundaries on the applied magnetic field of double-layer MRE acoustic metamaterial and a comparison of the FEM and mass-spring model for (a) magnetic field intensity of H = 0–10 kOe, (b) comparison of the FEM and mass-spring model. 双包覆层MRE声学超材料带隙随磁场强度变化及有限元法与质量-弹簧模型对比　(a) 磁场强度H = 0—10 kOe; (b) 有限元法与质量-弹簧模型对比

Fig. 11. The band gap of double-layer MRE acoustic metamaterial changes with (a) the core mass, (b) the shell mass, (c) the filling rate.双包覆层MRE声学超材料带隙随材料和结构参数变化　(a) 芯体质量增加; (b) 壳体质量增加; (c) 填充率增加