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Analysis of one-dimensional rheological consolidation of double-layered soil with fractional derivative Merchant model and non-Darcian flow described by non-Newtonian index

基于非牛顿指数渗流和分数阶Merchant 模型的双层地基一维流变固结分析

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Abstract

To further investigate the one-dimensional (1D) rheological consolidation mechanism of double-layered soil, the fractional derivative Merchant model (FDMM) and the non-Darcian flow model with the non-Newtonian index are respectively introduced to describe the deformation of viscoelastic soil and the flow of pore water in the process of consolidation. Accordingly, an 1D rheological consolidation equation of double-layered soil is obtained, and its numerical analysis is performed by the implicit finite difference method. In order to verify its validity, the numerical solutions by the present method for some simplified cases are compared with the results in the related literature. Then, the influence of the revelent parameters on the rheological consolidation of double-layered soil are investigated. Numerical results indicate that the parameters of non-Darcian flow and FDMM of the first soil layer greatly influence the consolidation rate of double-layered soil. As the decrease of relative compressibility or the increase of relative permeability between the lower soil and the upper soil, the dissipation rate of excess pore water pressure and the settlement rate of the ground will be accelerated. Increasing the relative thickness of soil layer with high permeability or low compressibility will also accelerate the consolidation rate of double-layered soil.

摘要

为了进一步深入探讨饱和黏土双层地基一维固结机理, 引入Koeller 定义的弹壶元件修正 Merchant 模型描述饱和黏土的黏弹性变形, 引入非牛顿指数渗流模型描述固结过程中的非Darcy 渗流, 推导出一个新的饱和黏性土双层地基一维流变固结方程, 并利用隐式差分格式进行数值求解。通过文 献对比, 验证了本文数值算法的有效性。探讨了非牛顿指数渗流模型参数以及分数阶Merchant 流变 模型参数对双层地基一维流变固结过程的影响。结果表明, 在单面排水条件下, 上层中渗流参数和模 型参数对固结速率的影响起决定作用, 且随着下层土与上层土的相对压缩性的降低或相对渗透性的增 加, 地基孔压的消散速率和沉降速率都会加快;增大渗透性高或压缩性低的土层相对厚度, 也会加快 双层地基的固结速率。另外, 非牛顿指数的增大会延缓双层地基中的孔压消散, 使得沉降发展变慢。

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Abbreviations

b, c :

Ratio of permeability coefficient and thickness of the lower soil layer to the upper soil layer, respectively

C v :

Coefficient of consolidation

E :

Elastic modulus

E 0β, E 1β :

Modulus of independent spring and the series spring of the fractional kelvin body of the first soil layer, respectively, β=l, 2

E α(x):

Mittag-Leffler function

F β :

Viscosity coefficient

H :

Total thickness of double-layered soil

h β :

Thickness of the upper and lower soil layer when β=1 or 2, respectively

i’, j :

Time node and spatial node, respectively

i 0β, k uβ :

Non-Newtonian index and permeability coefficient, respectively

I 0β, V 0β, T, Dimensionless variables of i 0, F, t, u :

and

U, Z z :

respectively, as shown in Eq. (1)

J :

Creep compliance

P 0 :

Vertical distributed load

s :

Variable of Laplace transformation

U P, U S :

Degree of consolidation in terms of pore pressure and deformation, respectively

R 1 :

Ratio of independent spring modulus of the upper soil layer to lower soil layer

R 2 :

Ratio of the series spring of the fractional kelvin body of the upper soil layer to lower soil layer

v :

Flow velocity

α β :

Fractional order

η :

Viscous time

γ w :

Unit weight of water

ε :

Vertical strain

\(\bar \varepsilon \) :

Laplace transformation of ε

τ′ :

Effective stress

τ :

Vertical stress

\(\bar \sigma \) :

Laplace transformation of τ

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Authors and Affiliations

  1. School of Civil Engineering, Zhengzhou University, Zhengzhou, 450001, China

    Peng-lu Cui  (鹏陆), Zhong-yu Liu  (刘忠玉), Jia-chao Zhang  (张家超) & Zhi-cheng Fan  (范智铖)

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  1. Peng-lu Cui  (鹏陆)
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Contributions

The overarching research goals were developed by CUI Peng-lu, LIU Zhong-yu, ZHANG Jia-chao, FAN Zhi-cheng, LIU Zhong-yu contributed to the conception of the paper. ZHANG Jia-chao, FAN Zhi-cheng participated in the analysis of relevant calculated results. The initial draft of the manuscript was written by CUI Peng-lu, LIU Zhong-yu. All authors replied to reviewers’ comments and revised the final version.

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Correspondence to Zhong-yu Liu  (刘忠玉).

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Conflict of interest

CUI Peng-lu, LIU Zhong-yu, ZHANG Jia-chao, FAN Zhi-cheng declare that they have no conflict of interest.

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Project(51578511) supported by the National Natural Science Foundation of China

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Cui, Pl., Liu, Zy., Zhang, Jc. et al. Analysis of one-dimensional rheological consolidation of double-layered soil with fractional derivative Merchant model and non-Darcian flow described by non-Newtonian index. J. Cent. South Univ. 28, 284–296 (2021). https://doi.org/10.1007/s11771-021-4602-4

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