Dispersion and Dissipation Relations of One-Dimensional Viscoelastic Phononic Crystals
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摘要: 基于运动方程和广义麦克斯韦本构，推导了一维黏弹性局域共振型和布拉格散射型声子晶体的色散与耗散关系。结果表明：对于时间谐波传播问题，这两种声子晶体的色散关系中均不存在禁带，波的衰减完全依赖于黏性耗散和周期性调制，且周期性调制会增强这种耗散作用；相反，对于自由波传播问题，两种声子晶体的色散关系中均存在禁带，但在禁带之外，波的衰减仍需借助于黏性耗散和周期性调制。研究结果对于由高分子材料构成的层状复合材料中的应力波传播研究具有一定的意义。Abstract: Based on equations of motion and the generalized Maxwell constitutive model, this paper derives the dispersion and dissipation relations of one-dimensional viscoelastic local resonance and Bragg scattering type phononic crystals. The results show that, for the time-harmonic propagation, band gap does not exist in the dispersion relation and the attenuation of wave solely relies on viscous dissipation and periodic modulation, which will enhance the dissipation; on the contrary, for the free wave propagation, there is a band gap in the dispersion relation, but beyond the band gap, the attenuation of wave is still dependent on viscous dissipation and periodic modulation. These results are valuable for the study on stress wave propagation in layered composite materials made of polymers.
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图 1 离散形式的一维黏弹性声子晶体
Figure 1. One dimensional viscoelastic phononic crystals in discrete form
图 2 不同
$\tau $ 下LR和BS模型的色散关系Figure 2. Dispersion relation of LR and BS model with different
$\tau $ 
图 3 N = 1和N = 3时BS模型的耗散关系
Figure 3. Dissipation relation of BS model when N = 1 and N = 3
图 4 N = 1和N = 3时LR模型的耗散关系
Figure 4. Dissipation relation of LR model when N = 1 and N = 3
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