﻿ 基于非线性渗流的大变形固结有限元分析
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 哈尔滨工程大学学报  2017, Vol. 38 Issue (8): 1231-1237  DOI: 10.11990/jheu.201607027 0

### 引用本文

LI Gang, ZHANG Jinli, YANG Qing, et al. Finite element analysis of large-strain consolidation with nonlinear flow[J]. Journal of Harbin Engineering University, 2017, 38(8), 1231-1237. DOI: 10.11990/jheu.201607027.

### 文章历史

1. 西京学院 土木工程学院, 西安 710123;
2. 大连理工大学 建设工程学部土木工程学院 岩土工程研究所, 大连 116024;
3. 大连理工大学 海岸和近海工程国家重点实验室, 大连 116024;
4. 同济大学 地下建筑与工程系, 上海 200092

Finite element analysis of large-strain consolidation with nonlinear flow
LI Gang1,2, ZHANG Jinli2,3, YANG Qing2,3, JIANG Mingjing4
1. School of Civil Engineering, Xijing University, Xi'an 710123, China;
2. Institute of Geotechnical Engineering, School of Civil Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China;
3. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China;
4. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
Abstract: To study consolidation behaviors under non-Darcy flow, finite element equations and governing equations of plane consolidation were deduced with nonlinear flow (coupled non-Darcy flow with variable permeability coefficient) based on large-strain consolidation theory. The consolidation process of clays was studied by using a program that was developed based on finite element equations. The influence of non-Darcy flow and variable permeability coefficient on consolidation was analyzed by changing the parameters. Calculation results indicate that compared with the circumstances that consider Darcy flow, the settlement is smaller and the dissipation of excess pore water pressures is slower when non-Darcy flow is considered. The influence of parameter m (test constant of non-Darcy flow) on consolidation is greater than iL (initial hydraulic gradient of linear flow). The influence of permeability coefficient is similar to that of non-Darcy flow, and the influence of αc (ratio between compression index and permeability coefficient index) is remarkable. The settlement decreased obviously when considering non-Darcy flow and variable permeability coefficient. The dissipation of excess pore water pressure becomes slow, and the delay effects are more evident.
Key words: large strain    non Darcy flow    variable permeability coefficient    finite element analysis    excess pore water pressure    consolidation    settlement    compression index

1 渗透系数变化与非线性渗流下的固结控制方程 1.1 考虑渗透系数变化的非线性渗流问题描述

Hansbo[14-15]基于试验给出渗流速度v与水力梯度i之间的关系式为

 $v = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {c{i^m}}\\ {k\left( {i - {i_0}} \right)} \end{array}}&{\begin{array}{*{20}{c}} {i < {i_L}}\\ {i \ge {i_L}} \end{array}} \end{array}} \right.$ (1)

 $e = {e_0} - {C_c}\lg \left( {\frac{{p'}}{{{{p'}_0}}}} \right)$ (2)

Taylor[16]基于试验结果，给出渗透系数k与孔隙比e的关系为

 $\lg k = \lg {k_0} - \frac{{{e_0} - e}}{{{C_k}}}$ (3)

 $k = {k_0}{\left[ {\frac{{p'}}{{{{p'}_0}}}} \right]^{ - {\alpha _c}}}$ (4)

 $\left\{ \begin{array}{l} v = {k_0}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}{i^m}/\left( {mi_L^{m - 1}} \right),\;\;\;\;\;\;\;\left| i \right| < {i_L}\\ v = {k_0}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}\left( {i - \frac{{m - 1}}{m}{i_L}} \right),\;\;\;\;\;\left| i \right| \ge {i_L} \end{array} \right.$ (5)
1.2 平衡方程

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{L}}^{\rm{T}}}\mathit{\boldsymbol{S}} + {f_0} = 0\\ \mathit{\boldsymbol{S}} = \mathit{\boldsymbol{S' + T}}{\mathit{\boldsymbol{p}}_{ow}}\\ \mathit{\boldsymbol{S' = DE}}\\ \mathit{\boldsymbol{E}} = {\mathit{\boldsymbol{E}}_L} + {\mathit{\boldsymbol{E}}_N} \end{array} \right.$ (6)

1.3 连续方程

 $\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_y}}}{{\partial y}} = \frac{\partial }{{\partial t}}\left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}}} \right)$ (7)

 $\left\{ \begin{array}{l} {v_\psi } = {S_\psi }{k_{\psi 0}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}{\left| {{i_\psi }} \right|^m}/\left( {mi_L^{m - 1}} \right),\;\;\;\;\;\;\left| {{i_\psi }} \right| < {i_L}\\ {v_\psi } = {S_\psi }{k_{\psi 0}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}\left( {\left| {{i_\psi }} \right| - \frac{{m - 1}}{m}{i_L}} \right),\;\;\;\;\left| {{i_\psi }} \right| \ge {i_L} \end{array} \right.$ (8)

 $\left\{ \begin{array}{l} \frac{{\partial {v_\psi }}}{{\partial \psi }} = \left( {A + B} \right)\frac{{{k_{{\psi _0}}}}}{{{\gamma _w}}}\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}},\;\;\;\;\;\;\;\;\;\;\;\;\left| {{i_\psi }} \right| < {i_L}\\ \frac{{\partial {v_\psi }}}{{\partial \psi }} = \left( {C - D + E} \right)\frac{{{k_{{\psi _0}}}}}{{{\gamma _w}}}\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}},\;\;\;\;\;\;\left| {{i_\psi }} \right| \ge {i_L} \end{array} \right.$ (9)

 $A = \frac{{{\alpha _c}}}{{mi_L^{m - 1}\gamma _w^{m - 1}p'\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}}}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}{\left| {\frac{{\partial {p_{{\rm{ow}}}}}}{{\partial \psi }}} \right|^{m + 1}}$
 $B = \frac{1}{{i_L^{m - 1}\gamma _w^{m - 1}}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}{\left| {\frac{{\partial {p_{{\rm{ow}}}}}}{{\partial \psi }}} \right|^{m - 1}}$
 $C = \frac{{{\alpha _c}}}{{p'\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}}}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}{\left| {\frac{{\partial {p_{{\rm{ow}}}}}}{{\partial \psi }}} \right|^2}$
 $D = \frac{{\left( {m - 1} \right){i_L}{\gamma _w}{\alpha _c}}}{{mp'\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}}}}{\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}\left| {\frac{{\partial {p_{{\rm{ow}}}}}}{{\partial \psi }}} \right|$
 $E = {\left( {\frac{{p'}}{{{{p'}_0}}}} \right)^{ - {\alpha _c}}}$

 $\frac{{\partial {v_\psi }}}{{\partial \psi }} = {H_\psi }\frac{{{k_{{\psi _0}}}}}{{{\gamma _w}}}\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}}$ (10)

 ${H_\psi } = \left\{ \begin{array}{l} A + B,\;\;\;\;\;\;\;\;\;\;\left| {{i_\psi }} \right| < {i_L}\\ C - D + E,\;\;\;\;\left| {{i_\psi }} \right| \ge {i_L} \end{array} \right.$ (11)

 ${H_x}\frac{{{k_{{\psi _0}}}}}{{{\gamma _w}}}\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {\psi ^2}}} + {H_y}\frac{{{k_{{y_0}}}}}{{{\gamma _w}}}\frac{{{\partial ^2}{p_{{\rm{ow}}}}}}{{\partial {y^2}}} = \frac{\partial }{{\partial t}}\left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}}} \right)$ (12)

 ${\mathit{\boldsymbol{T}}^{\rm{T}}}\mathit{\boldsymbol{LH}}{\mathit{\boldsymbol{k}}_0}{\mathit{\boldsymbol{L}}^{\rm{T}}}\mathit{\boldsymbol{T}}{p_{ow}} - \frac{\partial }{{\partial t}}{\mathit{\boldsymbol{T}}^{\rm{T}}}\mathit{\boldsymbol{E}} = 0$ (13)

2 渗透系数变化与非线性渗流下的固结有限元方程 2.1 位移与孔压模式选取

 $\left\{ \begin{array}{l} \tilde u = \sum\limits_{i = 1}^8 {{N_i}{u_i}} \\ \tilde v = \sum\limits_{i = 1}^8 {{N_i}{v_i}} \\ \tilde p = \sum\limits_{j = 1}^8 {{N_j}{p_j}} \end{array} \right.$ (14)
 图 1 平面八结点等参单元 Fig.1 Plane eight-node isoparametric element

 $\left\{ \begin{array}{l} {N_{i\left( {1,3,5,7} \right)}} = \frac{1}{4}\left( {1 + {\xi _0}} \right)\left( {1 + {\eta _0}} \right)\left( {{\xi _0} + {\eta _0} - 1} \right)\\ {N_{i\left( {2,6} \right)}} = \frac{1}{2}\left( {1 - {\xi ^2}} \right)\left( {1 + {\eta _0}} \right)\\ {N_{i\left( {4,8} \right)}} = \frac{1}{2}\left( {1 - {\eta ^2}} \right)\left( {1 + {\xi _0}} \right) \end{array} \right.$ (15)
 ${N_{j\left( {1,2,3,4} \right)}} = \frac{1}{4}\left( {1 + {\xi _0}} \right)\left( {1 + {\eta _0}} \right)$ (16)

2.2 平衡方程离散

 ${\mathit{\boldsymbol{K}}_{DS}}\Delta {a_e} + {\mathit{\boldsymbol{K}}_C}\Delta p_{ow}^e = - \mathit{\boldsymbol{\bar R}} - {\mathit{\boldsymbol{R}}_{SP}}$ (17)

 ${\mathit{\boldsymbol{K}}_{DS}} = {\mathit{\boldsymbol{K}}_D} + {\mathit{\boldsymbol{K}}_S},{\mathit{\boldsymbol{K}}_D} = \int_{{V_0}} {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{DB}}{\rm{d}}{V_0}} ,$
 ${\mathit{\boldsymbol{K}}_S} = \int_{{V_0}} {{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{MG}}{\rm{d}}{V_0}} ,{\mathit{\boldsymbol{K}}_C} = \int_{{V_0}} {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{T\bar N}}{\rm{d}}{V_0}} ,$
 $\mathit{\boldsymbol{\bar R}} = \int_{{V_o}} {{\mathit{\boldsymbol{N}}^{\rm{T}}}{\mathit{\boldsymbol{f}}_0}{\rm{d}}{V_0}} = \int_{{A_o}} {{\mathit{\boldsymbol{N}}^{\rm{T}}}{\mathit{\boldsymbol{q}}_0}{\rm{d}}{A_0}} ,$
 ${\mathit{\boldsymbol{P}}_{SP}} = \int_{{V_0}} {{\mathit{\boldsymbol{B}}^{\rm{T}}}\left( {\mathit{\boldsymbol{S' + T}}{p_{ow}}} \right){\rm{d}}{V_0}} .$
2.3 连续方程离散

 $\mathit{\boldsymbol{K}}_C^{\rm{T}}\Delta {a_e} - \theta \Delta t{\mathit{\boldsymbol{K}}_{SP}}\Delta p_{ow}^e = {\mathit{\boldsymbol{R}}_{PQ}}$ (18)

 $\begin{array}{l} {\mathit{\boldsymbol{K}}_{SP}} = \int_{{V_0}} {\mathit{\boldsymbol{B}}_s^{\rm{T}}\mathit{\boldsymbol{Hk}}{\mathit{\boldsymbol{B}}_s}{\rm{d}}{V_0}} \\ {\mathit{\boldsymbol{R}}_{PQ}} = \Delta t\left( {{\mathit{\boldsymbol{K}}_{SP}}\Delta p_{ow}^e + {\mathit{\boldsymbol{R}}_Q}} \right) \end{array}$
3 非线性渗流的计算方法

 $\left\{ \begin{array}{l} i_x^{j + 1} = \frac{1}{{{\gamma _w}}}\frac{{\partial p_{ow}^j}}{{\partial x}} = \frac{{\left( {p_{1\# }^j + p_{4\# }^j - p_{2\# }^j - p_{3\# }^j} \right)}}{{4{\gamma _w}a}}\\ i_y^{j + 1} = \frac{1}{{{\gamma _w}}}\frac{{\partial p_{ow}^j}}{{\partial y}} = \frac{{\left( {p_{1\# }^j + p_{2\# }^j - p_{3\# }^j - p_{4\# }^j} \right)}}{{4{\gamma _w}b}} \end{array} \right.$ (19)

 $\left\{ \begin{array}{l} \frac{{{\partial ^2}p_{{\rm{ow}}}^{j + 1}}}{{\partial {x^2}}} = \left( {\frac{{\partial p_{{\rm{ow}}}^{j + 1}}}{{\partial x}} - \frac{{\partial p_{{\rm{ow}}}^j}}{{\partial x}}} \right)/\Delta x\\ \frac{{{\partial ^2}p_{{\rm{ow}}}^{j + 1}}}{{\partial {y^2}}} = \left( {\frac{{\partial p_{{\rm{ow}}}^{j + 1}}}{{\partial y}} - \frac{{\partial p_{{\rm{ow}}}^j}}{{\partial y}}} \right)/\Delta y \end{array} \right.$ (20)
4 计算结果与讨论 4.1 数值解与近似解析解比较

 图 2 数值解与近似解析解的比较 Fig.2 Comparison between numerical solutions and approximate analytic solutions
4.2 非Darcy渗流对固结的影响分析

 图 3 网格剖分 Fig.3 Mesh divided

 图 4 参数iL对沉降的影响 Fig.4 Influence of iL on settlement
 图 5 参数iL对超孔压的影响(500 d) Fig.5 Influence of iL on excess pore water pressure (500 d)
 图 6 参数m对沉降的影响 Fig.6 Influence of m on settlement
 图 7 参数m对超孔压的影响(500 d) Fig.7 Influence of m on excess pore water pressure (500 d)

4.3 渗透系数变化对固结的影响分析

 图 8 参数αc对沉降的影响 Fig.8 Influence of αc on settlement
 图 9 参数αc对超孔压的影响(500 d) Fig.9 Influence of αc on excess pore water pressure (500 d)
4.4 非Darcy渗流与渗透系数变化对固结的影响

 图 10 非Darcy渗流与变渗透系数对沉降的影响 Fig.10 Influence of non-Darcy flow and variable permeability coefficient on settlement
 图 11 非Darcy渗流与变渗透系数对超孔压的影响(500 d) Fig.11 Influence of non-Darcy flow and variable permeability coefficient on excess pore water pressure (500 d)
5 结论

1) 在100 d之后，与Darcy渗流相比，非Darcy渗流下的沉降较小，超孔压消散缓慢，且随固结时间的增加，差异越为明显，非Darcy渗流所引起的延迟效应也越来越显著，其中参数m的影响较大，而参数iL的影响较小，可忽略。原因在于土体固结过程中，土颗粒越来越密实，孔隙越来越小，导致渗透性降低；而随着超孔压的消散，水力梯度随之减小，导致渗流速度降低，由此导致上述规律。

2) 参数αc对固结过程具有重要影响，且随αc的增加，沉降逐渐减小，孔压消散趋于缓慢，延迟效应较为明显。

3) 参数mαc对黏土固结过程的影响较大，当m=1.5，αc=1.5时最为显著。建议在进行黏性土固结分析时，应考虑非Darcy渗流与渗透系数随固结变化的耦合作用。

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