Low frequency bandgap characteristics and application of a novel two-dimensional three-component cement-based phononic-like crystal composite material
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摘要: 为了拓宽混凝土超材料的弹性波带隙宽度和数量，本文基于局域共振理论设计了一种新型二维三组元水泥基拟声子晶体。首先，采用有限元方法计算和研究了该新型二维三组元水泥基拟声子晶体的能带结构、振动模态、位移场和衰减特性。其次，分析了带隙形成机制和影响因素，并根据质量-弹簧系统模型推导了带隙范围的理论估计式。最后，将该水泥基拟声子晶体应用到地铁道床上，分析了水泥基拟声子晶体地铁道床的减振性能。结果表明：该新型二维三组元水泥基拟声子晶体在200 Hz频段内打开了5条低频带隙，在带隙频率范围内，衰减值大多都在10 dB以上，衰减效果较好；带隙的打开与各原胞的振动特征呈现出对应关系，因特定原胞的平移振动触发，由特定原胞与基体的耦合作用的强度所控制；散射体材料的密度、包裹层材料的弹性模量及厚度是影响其带隙的主要因素；由新型二维三组元水泥基拟声子晶体组成的水泥基拟声子晶体地铁道床在1～200 Hz频段内的振动加速度均小于普通混凝土地铁道床，最大插入损失为10.22 dB，插入损失平均值为8.76 dB，具有显著的减振性能。Abstract: In order to widen the width and number of elastic bandgap of concrete metamaterials, a novel two-dimensional three-component cement-based phononic-like crystal was designed based on local resonance theory. Firstly, the finite element method was used to calculate and study the energy band structure, vibration mode, displacement field and attenuation characteristics of the novel two-dimensional three-component cement-based phononic-like crystal. Secondly, the formation mechanism and influencing factors of the bandgap were analyzed, and the theoretical estimation of the bandgap range was derived according to the mass-spring system model. Finally, the cement-based phononic-like crystal was applied to the subway track bed, and the vibration reduction performance of the cement-based phononic-like crystal subway track bed was analyzed. The results show that the novel two-dimensional three-component cement-based phononic-like crystal opens 5 low-frequency bandgaps in the 200 Hz frequency range, and the attenuation values are mostly above 10 dB within the bandgap frequency range, and the attenuation effect is good. The opening of the bandgap corresponds to the vibration characteristics of each primitive cell, which is triggered by the translational vibration of a specific primitive cell and controlled by the strength of the coupling between the specific primitive cell and the matrix. The density of scatterer material, elastic modulus and thickness of cladding material are the main factors affecting the bandgap. In the 1-200 Hz frequency band, the vibration acceleration of the cement-based phononic-like crystal subway track bed composed of the novel two-dimensional three-component cement-based phononic-like crystal is lower than that of the ordinary concrete subway track bed, and the maximum insertion loss is 10.22 dB and the average insertion loss is 8.76 dB, which has remarkable vibration reduction performance.
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图 1 水泥基拟声子晶体原胞结构及第一布里渊区
Figure 1. Cement-based phononic-like crystal primitive cell and first Brillouin region
图 3 传统水泥基声子晶体能带结构
Figure 3. Band structure of traditional cement-based phononic crystal
图 4 新型二维三组元水泥基拟声子晶体能带结构
Figure 4. Band structure of novel two-dimensional three-component cement-based phononic-like crystal
图 15 新型二维三组元水泥基拟声子晶体的频率响应函数计算模型
Figure 15. Frequency response function calculation model of novel two-dimensional three-component cement-based phononic-like crystal
图 16 新型二维三组元水泥基拟声子晶体的频率响应函数
Figure 16. Frequency response function of novel two-dimensional three-component cement-based phononic-like crystal
图 17 新型二维三组元水泥基拟声子晶体的散射体材料参数对带隙的影响
Figure 17. Effect of scatterer material parameters on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 18 新型二维三组元水泥基拟声子晶体的散射体结构参数对带隙的影响
Figure 18. Effect of scatterer structure parameters on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 19 新型二维三组元水泥基拟声子晶体的包裹层材料参数对带隙的影响
Figure 19. Effect of material parameters of wrapping layer on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 20 新型二维三组元水泥基拟声子晶体的包裹层结构参数对带隙的影响
Figure 20. Effect of structure parameters of wrapping layer on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 21 新型二维三组元水泥基拟声子晶体的水泥砂浆基体材料参数对带隙的影响
Figure 21. Effect of material parameters of cement mortar matrix on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 22 新型二维三组元水泥基拟声子晶体的水泥砂浆基体结构参数对带隙的影响
Figure 22. Effect of structure parameters of cement mortar matrix on bandgap of novel two-dimensional three-component cement-based phononic-like crystal
图 23 不同散射体形状的新型二维三组元水泥基拟声子晶体的计算模型
Figure 23. Calculation models of novel two-dimensional three-component cement-based phononic-like crystal with different scatterer shapes
图 24 不同形状散射体的新型二维三组元水泥基拟声子晶体的能带结构
Figure 24. Band structure of novel two-dimensional three-component cement-based phononic-like crystal with different scatterer shapes
图 25 新型二维三组元水泥基拟声子晶体内部散射体的多形状耦合的计算模型
Figure 25. Calculation models of multi-shape coupling of internal scatterers in novel two-dimensional three-component cement-based phononic-like crystal
图 26 新型二维三组元水泥基拟声子晶体原胞内部散射体多形状耦合的能带结构
Figure 26. Band structure of multi-shape coupling of internal scatterers in novel two-dimensional three-component cement-based phononic-like crystal
图 27 单振子和多振子局域共振声子晶体原胞的弹簧-质量系统等效模型
Figure 27. spring-mass system equivalent models of single-oscillator and multi-oscillator local resonance phononic crystal primitive cell
图 28 多振子局域共振声子晶体的无限质量-质量晶格结构
Figure 28. The infnite mass-in-mass lattice structure of local resonance phononic crystal with multiple oscillators
图 30 水泥基拟声子晶体地铁道床和普通混凝土地铁道床计算模型
Figure 30. Calculation models of cement-based phononic-like crystal subway track bed and ordinary concrete subway track bed
图 31 水泥基拟声子晶体地铁道床和普通混凝土地铁道床时域和频域曲线
Figure 31. Time domain and frequency domain curves of cement-based phononic-like crystal subway track bed and ordinary concrete subway track bed
图 32 水泥基拟声子晶体地铁道床和普通混凝土地铁道床三分之一倍频程及插入损失
Figure 32. One-third octave and insertion loss of cement-based phononic-like crystal subway track bed and ordinary concrete subway track bed
表 1 结构参数
Table 1. Structure parameters
Scatterer i Scatterer inner diameter Rsi/m Wrapping layer outer diameter Rci/m Lattice
constant a/m1 0.0275 0.033 0.1 2 0.018 0.0216 3 0.013 0.0156 4 0.01 0.012 5 0.007 0.0084 
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表 2 材料参数
Table 2. Material parameters
Components Density
ρ /(kg·m−3)Elastic modulus
E /PaShear modulus
G /PaScatterer 3300 1.2×1011 4.8×1010 Wrapping layer 12 1.34×105 5.93×104 Matrix 2000 3.45×1010 1.44×1010
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表 3 新型二维三组元水泥基拟声子晶体的带隙影响因素及其水平
Table 3. Bandgap influencing factors and their levels of novel two-dimensional three-component cement-based phononic-like crystal
Influence factors Level 1 Level 2 Level 3 Level 4 Level 5 Material parameters Scatterer E/Pa 1.2×109 1.2×1010 1.2×1011 1.2×1012 1.2×1013 ρ/(kg·m−3) 2300 2800 3300 3800 4300 v 0.15 0.20 0.25 0.30 0.35 Wrapping layer E/Pa 1.34×103 1.34×104 1.34×105 1.34×106 1.34×107 ρ/(kg·m−3) 6 9 12 15 18 v 0.09 0.11 0.13 0.15 0.17 Matrix E/Pa 3.45×108 3.45×109 3.45×1010 3.45×1011 3.45×1012 ρ/(kg·m−3) 1000 1500 2000 2500 3000 v 0.10 0.15 0.20 0.25 0.30 Structure parameters Scatterer Rs/m 0.8R1 0.9R1 R1 - - Wrapping layer Rc/m 0.8R2 0.9R2 R2 - - Matrix a/m - - 0.1 0.12 0.14 Notes: R1－Inner diameter of scatterer; R2－Outer diameter of wrapping layer.
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