Characteristics of frequency response for linear viscoelastic spherical divergent stress waves
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摘要: 基于线黏弹性球面波Laplace域的理论解, 得到了不同传播距离处粒子速度、粒子位移、应力、应变等力学量的传递函数。以标准线性固体模型为例, 重点讨论了粒子速度频率响应函数的传播特征, 指出随着传播距离的增加, 粒子速度幅频响应函数的高频响应会低于低频响应, 而在理想弹性条件下, 粒子速度幅频响应函数的高频响应一直高于低频响应。以弹性半径为0.025 m的空腔爆炸为例, 采用Laplace数值逆变换方法对粒子速度波形的演化进行了分析, 给出了粒子速度强间断幅值及粒子速度峰值随传播距离变化的衰减规律曲线, 指出黏弹性介质中粒子速度幅值的衰减曲线介于理想弹性介质中粒子速度幅值衰减曲线和黏弹性介质中粒子速度强间断幅值衰减曲线之间。Abstract: In this study the transfer functions of the mechanical quantities (i.e. the particle velocity, the particle displacement, the stress and the strain, etc.) at different propagation distances were analytically presented based on the solutions of the linear viscoelastic spherical stress wave in the Laplace domain, The propagating characteristics of the frequency response function for the particle velocity were examined with the standard linear solid model taken as an example. The results reveal that the high-frequency response of the frequency response function for the particle velocity in viscoelastic medium is less than that of the low-frequency response with the increase of the propagation distance; in an ideal elastic medium, however, it is always greater than that of the low-frequency response. With the cavity explosion with the elastic radius of 0.025 m taken as an example, the evolution of the wave form of the particle velocity was calculated using the numerical method of the inverse Laplace transform. The results reveal that the attenuation curve of the peak value for the particle velocity in viscoelastic medium falls in between the attenuation curve of the the peak value for the particle velocity in an ideal elastic medium and the attenuation curve of the amplitude of the strong discontinuity for the particle velocity in viscoelastic medium.
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Key words:
- spherical stress waves /
- frequency response /
- Laplace inversion transform
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图 3 幅频响应函数Hvr(r, r0, 2πfi)随频率f的变化
Figure 3. Amplitude-frequency response function Hvr(r, r0, 2πfi) vs. frequency f
图 4 卓越频率f0和上限频率f1随传播距离r的变化
Figure 4. Predominant frequency f0 and upper limit frequency f1 varying with propagation distance r
图 6 粒子速度峰值(强间断幅值)随传播距离r变化
Figure 6. Peak values of particle velocity vs. propagation distance r
表 1 黄土材料标准线性固体模型参数
Table 1. Parameters of the standard linear solid model for loess
密度ρ/(kg·m3) 弹性模量E0/GPa 弹性模量E1/GPa 松弛时间θ1/μs 泊松比μ 1 800 1.60 0.33 21.0 0.25 
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表 2 卓越频率f0和上限频率f1的变化
Table 2. Variations of predominant frequency f0 and upper limit frequency f1
弹性半径r0/m 0.5 km处的卓越频率f0/Hz 0.5 km处的上限频率f1/Hz 1 000 km处的卓越频率f0/Hz 1 000 km处的上限频率f1/Hz 0.025 110.76 419.65 2.48 11.71 0.25 109.28 414.41 2.48 11.71 2.5 73.82 361.39 2.47 11.70 25.0 26.92 278.37 2.33 11.32 250.0 9.43 184.49 1.19 10.09
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