Transient Response of One-Dimensional Viscoelastic Double-Layer Unsaturated Soil Columns Under Pot Cover Effect
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摘要:
为研究锅盖效应的存在对非饱和路基瞬态响应影响，将非饱和路基划分为锅盖效应影响层和非锅盖效应影响层，覆盖层简化为不透水不透气的边界条件；并考虑锅盖效应和土体的黏弹性，基于Biot形式的波动方程建立一维双层非饱和路基动力分析模型；继而根据黏弹性对应原理，在Laplace域内采用状态空间法和传递矩阵法给出瞬态响应问题的解析解答，再通过Laplace逆变换技术获得问题在时域内的解答；最后，通过退化算例，分析了黏滞阻尼系数、锅盖层饱和度和路基初始饱和度对瞬态响应的影响，验证本文解答方法的正确性. 结果表明：随着黏滞系数的增加，压缩波的波速呈增大趋势；近饱和状态时，黏弹性和弹性情况下锅盖层中孔压幅值分别可达无锅盖效应现象时的15.4倍和14.5倍；随着初始饱和度的增加，锅盖层和非锅盖层中的孔隙水压力分别呈现减小和增加趋势. 在工程实践中，应重视锅盖效应的动力致灾问题.
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关键词:
- 锅盖效应 /
- 非饱和路基 /
- 黏弹性 /
- 瞬态响应 /
- Laplace变换法
Abstract:To investigate the influence of the pot cover effect on the transient response of unsaturated subgrades, the unsaturated subgrade is divided into the pot-cover-effect-affected layer and the non-pot-cover-affected layer, and the cover layer is simplified as an impermeable and airtight boundary condition. Considering the pot cover effect and the viscoelasticity of the soil, a one-dimensional double-layer dynamic analysis model for unsaturated subgrades is established based on the Biot-type wave equations. According to the viscoelastic correspondence principle, the analytical solution of the problem is derived in the Laplace domain using the state-space method and the transfer matrix method. The solution in the time domain is then obtained through the Laplace inverse transform technique. The correctness of the proposed solution method is verified by degenerate examples. Finally, the effects of the viscous damping coefficient, the saturation of the pot cover layer, and the initial saturation of the subgrade on the transient response are investigated. The results indicate that the compression wave velocity tends to increase with the viscous coefficient. Under near-saturation conditions, the pore pressure amplitude in the pot cover layer is 15.4 and 14.5 times that of the scenario without the pot cover effect, for the viscoelastic and elastic cases, respectively. As the initial saturation increases, the pore water pressure in the pot cover layer decreases, whereas that in the non-pot-cover-affected layer exhibits an increasing trend. Therefore, in engineering practice, the dynamic disaster issues related to the pot cover effect should be given due attention.
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图 7 锅盖效应对非饱和路基中孔压影响结果
Figure 7. Influence of pot cover effect on pore pressure in unsaturated subgrade
图 9 不同初始饱和度下非饱和路基中孔压结果
Figure 9. Pore pressure with different initial saturation in unsaturated subgrade
图 10 不同初始饱和度下非饱和路基中固相位移结果
Figure 10. Solid displacement with different initial saturation in unsaturated subgrade
表 1 各符号含义
Table 1. Meaning of symbols
符号 意义 符号 意义 $ {\sigma }_{i} $ 总应力 $ {k}_{i\mathrm{w}} $ 孔隙水动力渗透系数 $ {p}_{i\mathrm{w}} $ 孔隙水压力 $ {k}_{i\mathrm{n}} $ 孔隙气的动力渗透系数 $ {p}_{i\mathrm{n}} $ 孔隙气压力 $ {S} _{i\mathrm{w}} $ 孔隙水饱和度 $ {u}_{i} $ 土骨架绝对位移 $ {S}_{i\mathrm{n}} $ 孔隙气饱和度 $ {w}_{i} $ 孔隙水相对位移 $ n $ 孔隙率 $ {v}_{i} $ 孔隙气相对位移 $ {\mu }_{i} $ 动剪切模量 $ {\rho }_{i} $ 土骨架密度 $ {K}_{\mathrm{b}} $ 土骨架的体积模量 $ {\rho }_{\mathrm{s}} $ 土颗粒密度 $ {K}_{\mathrm{s}} $ 土颗粒的体积模量 $ {\rho }_{i\mathrm{w}} $ 孔隙水密度 $ {K}_{\mathrm{w}} $ 孔隙水的体积模量 $ {\rho }_{i\mathrm{n}} $ 孔隙气密度 $ {K}_{\mathrm{n}} $ 孔隙气的体积模量 $ {\chi }_{i} $ 有效应力系数 $ {S} _{\mathrm{w}0} $ 束缚饱和度 $ a $ V-G 模型参数 $ d $ V-G 模型参数 $ {\lambda }_{i} $ Lame 参数 $ m $ V-G 模型参数 注：$ {\rho }_{{\mathrm{s}}} $为土颗粒密度，用于计算$ {\rho }_{{{i}}} $. 
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表 2 材料参数
Table 2. Material parameters
三相 参数 数值 数据来源 气相 气相体积模量（$ {K}_{\mathrm{n}} $）/kPa 145 文献[14] 气相动力渗透系数（$ {k}_{\mathrm{n}} $）/
$ ({\mathrm{m}}^{4}{{•}}(\mathrm{N}{{•}} \mathrm{s})^{-1}) $Sn × 10−6 文献[22] 气相密度（$ {\rho }_{\mathrm{n}} $）/(kg•m−3) 1.28 文献[14] 液相 液相体积模量（$ {K}_{\mathrm{w}} $）/GPa 2.25 文献[14] 液相动力渗透系数（$ {k}_{\mathrm{w}} $）/
$ ({\mathrm{m}}^{4}{{•}}(\mathrm{N}{{•}} \mathrm{s})^{-1}) $Sw × 10−6 文献[22] 液相密度（$ {\rho }_{\mathrm{w}} $）/(kg•m−3) 1000 文献[14] 固相 固相体积模量（$ {K}_{\mathrm{s}} $）/GPa 35 文献[14] 固相密度（$ {\rho }_{\mathrm{s}} $）/(kg•m−3) 2650 土骨架 体积模量（$ {K}_{\mathrm{b}} $）/GPa 8.33 文献[14] 动剪切模量（$ {\mu }_{\mathrm{s}} $）/MPa 3.85 孔隙率（$ n $） 0.45 饱和度（$ {S} _{\mathrm{w}} $） 0.40～0.95 束缚饱和度（$ {S}_{\mathrm{w}0} $） 0.05 有效内摩擦角（$ {\phi }^{\prime} $）/(°) 27 V-G 参数（$ a $）/Pa−1 1.0 × 10−4 V-G 参数（$ m $） 0.5 V-G 参数（$ d $） 2
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