饱和土一维有限应变固结控制方程辨析

王玮瑜; 雷国辉; 赵鑫; 戴传杰; 谷遇溪

doi:10.11779/CJGE20240077

饱和土一维有限应变固结控制方程辨析 English Version

王玮瑜^{1, 2,},
雷国辉^{1, 2, ,},
赵鑫^{1, 2},
戴传杰^{1, 2},
谷遇溪^{1, 2}

1.
河海大学岩土力学与堤坝工程教育部重点实验室, 江苏南京 210024
2.
河海大学江苏省岩土工程技术工程研究中心, 江苏南京 210024

基金项目:

国家自然科学基金项目 52178326

江苏省研究生科研与实践创新计划项目 KYCX23_0697

详细信息

作者简介:
王玮瑜(1996—)，男，博士研究生，主要从事软土固结理论的研究。E-mail: wangweiyu@hhu.edu.cn

通讯作者:
雷国辉, E-mail: leiguohui@hhu.edu.cn

中图分类号: TU43
计量
- 文章访问数: 0
- HTML全文浏览量: 0
- PDF下载量: 0
出版历程
- 收稿日期: 2024-01-23
- 网络出版日期: 2024-06-05
- 刊出日期: 2025-03-31

Understanding governing equations for one-dimensional finite strain consolidation of saturated soils

WANG Weiyu^{1, 2,},
LEI Guohui^{1, 2, ,},
ZHAO Xin^{1, 2},
DAI Chuanjie^{1, 2},
GU Yuxi^{1, 2}

1.
Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210024, China
2.
Jiangsu Research Center for Geotechnical Engineering Technology, Hohai University, Nanjing 210024, China

摘要

摘要: 饱和土的一维有限应变固结控制方程存在多种表达形式，有Eulerian描述和Lagrangian描述，各有不同的控制变量，包括孔隙率、孔隙比、应变、固结比和超静孔压。为明辨其适用条件，考虑固相的压缩性，建立了两种描述方法间基于孔隙率、孔隙比和位移的坐标转换关系，以及控制变量的时间导数间的转换关系，分析了两种描述方法便于求解的固结问题，并基于Eulerian描述，考虑固相和液相的压缩性和惯性，推导了一维有限应变固结控制方程组，包括连续性方程、动量平衡方程和达西渗流定律。忽略固相和液相的压缩性和惯性后，该方程组简化为一个具有单一控制变量的固结微分方程，通过坐标转换和时间导数转换，也得到了Lagrangian描述下的固结微分方程。将其退化为现有不同形式的有限应变固结控制方程，依据退化过程中涉及的基本假设，明确了这些控制方程的适用条件。
- 固相和液相 /
- 压缩性 /
- Eulerian和Lagrangian空间和时间转换 /
- 自然应变 /
- 固相和液相惯性 /
- 达西定律
Abstract: Various forms of governing equations are available for one-dimensional finite strain consolidation of saturated soils. They are expressed using either Eulerian or Lagrangian description, each with different dependent variables, including porosity, void ratio, strain, consolidation ratio and excess pore pressure. To better understand the applicability of these equations, the coordinate transformation relationship between the two descriptions is established in terms of porosity, void ratio and displacement considering the compressibility of solid. The transformation relationship between the time derivatives of dependent variable in the two descriptions is also established. The applicability of the two descriptions to solving consolidation problems is analyzed. Considering the compressibility and inertia of solid and liquid, a governing equation system for one-dimensional finite strain consolidation in Eulerian description is derived, including continuity equation, momentum balance equation and Darcy's law. After neglecting the compressibility and inertia of solid and liquid, the system of equations is simplified into a differential equation with a single dependent variable. Through the coordinate transformation and time derivative transformation, the consolidation differential equation with Lagrangian description is also obtained. The differential equations are degenerated into various existing forms of the finite strain consolidation governing equations, and the applicability of these governing equations is clarified through the basic assumptions involved in the degenerating process.
- solid and liquid /
- compressibility /
- spatial-temporal transformation between Eulerian and Lagrangian descriptions /
- natural strain /
- inertia of solid and liquid /
- Darcy's law

HTML全文

图 1 Lagrangian坐标系Oz_L和坐标z_L

Figure 1. Lagrangian coordinate system Oz_L and coordinate z_L

下载: 全尺寸图片幻灯片

图 2 Eulerian坐标系Oz_E和坐标z_E

Figure 2. Eulerian coordinate system Oz_E and coordinate z_E

下载: 全尺寸图片幻灯片

参考文献(37)

[1]	三笠正人. 軟弱黏土の压密–新压密理論とその応用[M]. 東京: 鹿島出版会, 1963. MIKASA M. The Consolidation of soft Clay–a New Consolidation Theory and its Application[M]. Tokyo: Kajima Institute Publishing Co., Ltd., 1963. (in Japanese)
[2]	MIKASA M, TAKADA N, OSHIMA A, et al. Nonlinear consolidation theory for nonhomogeneous clay layers and its application[J]. Soils and Foundations, 1998, 38(4): 205-212. doi: 10.3208/sandf.38.4_205
[3]	LEE K. An analytical and experimental Study of Large Strain Soil Consolidation[D]. Oxford: University of Oxford, 1979.
[4]	LEE K, SILLS G C. A moving boundary approach to large strain consolidation of a thin soil layer[C]// Proceedings of the Third International Conference on Numerical Methods in Geomechanics, Rotterdam, 1979.
[5]	MCVAY M, TOWNSEND F, BLOOMQUIST D. Quiescent consolidation of phosphatic waste clays[J]. Journal of Geotechnical Engineering, 1986, 112(11): 1033-1049. doi: 10.1061/(ASCE)0733-9410(1986)112:11(1033)
[6]	MCVAY M C, TOWNSEND F C, BLOOMQUIST D G. One-dimensional Lagrangian consolidation[J]. Journal of Geotechnical Engineering, 1989, 115(6): 893–898. doi: 10.1061/(ASCE)0733-9410(1989)115:6(893)
[7]	TAN T S, SCOTT R F. Finite strain consolidation—a study of convection[J]. Soils and Foundations, 1988, 28(3): 64-74. doi: 10.3208/sandf1972.28.3_64
[8]	SCHIFFMAN R L, VICK S G, GIBSON R E. Behavior and properties of hydraulic fills [J]. Journal of Geotechnical Engineering, 1989, 115(2): 280.
[9]	LANCELLOTTA R, PREZIOSI L. A general nonlinear mathematical model for soil consolidation problems[J]. International Journal of Engineering Science, 1997, 35(10/11): 1045-1063.
[10]	GIBSON R E, ENGLAND G L, HUSSEY M J L. The theory of one-dimensional consolidation of saturated clays[J]. Géotechnique, 1967, 17(3): 261-273. doi: 10.1680/geot.1967.17.3.261
[11]	GIBSON R E, SCHIFFMAN R L, CARGILL K W. The theory of one-dimensional consolidation of saturated clays. Ⅱ. Finite nonlinear consolidation of thick homogeneous layers[J]. Canadian Geotechnical Journal, 1981, 18(2): 280-293. doi: 10.1139/t81-030
[12]	KOPPULA S D. The Consolidation of Soil in Two Dimensions and with Moving Boundaries[D]. Edmonton: University of Alberta, 1970.
[13]	KOPPULA S D, MORGENSTERN N R. On the consolidation of sedimenting clays[J]. Canadian Geotechnical Journal, 1982, 19(3): 260-268. doi: 10.1139/t82-033
[14]	WALLIS G B. One-dimensional Two-Phase Flow[M]. New York: McGraw-Hill, 1969.
[15]	BEAR J. Dynamics of Fluids in Porous Media[M]. New York: American Elsevier Pub Co, 1988.
[16]	COUSSY O. Poromechanics[M]. Chichester: John Wiley & Sons, 2004.
[17]	AMBROSI D. Infiltration through deformable porous media[J]. ZAMM, 2002, 82(2): 115-124. doi: 10.1002/1521-4001(200202)82:2<115::AID-ZAMM115>3.0.CO;2-4
[18]	EL TANI M. Hydrostatic paradox of saturated media[J]. Géotechnique, 2007, 57(9): 773-777. doi: 10.1680/geot.2007.57.9.773
[19]	DE BOER R, EHLERS W. The development of the concept of effective stresses[J]. Acta Mechanica, 1990, 83(1): 77-92.
[20]	LADE P V, DE BOER R. The concept of effective stress for soil, concrete and rock[J]. Géotechnique, 1997, 47(1): 61-78. doi: 10.1680/geot.1997.47.1.61
[21]	陈晶晶, 雷国辉. 决定饱和岩土材料变形的有效应力及孔压系数[J]. 岩土力学, 2012, 33(12): 3696-3703. CHEN Jingjing, LEI Guohui. Effective stress and pore pressure coefficient controlling the deformation of saturated geomaterials[J]. Rock and Soil Mechanics, 2012, 33(12): 3696-3703. (in Chinese)
[22]	PANE V, SCHIFFMAN R L. A note on sedimentation and consolidation[J]. Géotechnique, 1985, 35(1): 69-72. doi: 10.1680/geot.1985.35.1.69
[23]	WYCKOFF R D, BOTSET H G, MUSKAT M, et al. The measurement of the permeability of porous media for homogeneous fluids[J]. Review of Scientific Instruments, 1933, 4(7): 394-405. doi: 10.1063/1.1749155
[24]	MUSKAT M, WYCKOFF R D. The Flow of Homogeneous Fluids Through Porous Media[M]. New York: McGraw-Hill Book Company, Inc., 1937.
[25]	HARR M E. Groundwater and Seepage[M]. New York: McGraw-Hill Book Company, 1962: 10-15.
[26]	YIH C S. Stratified flows[M]. 2d ed. New York: Academic Press, 1980.
[27]	Kovács G. Seepage Hydraulics[M]. Amsterdam: Elsevier Scientific Publishing Company, 1981: 476-482.
[28]	NADER J J. Darcy's law and the differential equation of motion[J]. Géotechnique, 2009, 59(6): 551-552. doi: 10.1680/geot.2008.T.014
[29]	孔祥言. 高等渗流力学[M]. 3版. 合肥: 中国科学技术大学出版社, 2020. KONG Xiangyan. Advanced Seepage Mechanics[M]. 3rd ed. Hefei: University of Science and Technology of China Press, 2020. (in Chinese)
[30]	ZIENKIEWICZ O C, CHANG C T, BETTESS P. Drained, undrained, consolidating and dynamic behaviour assumptions in soils[J]. Géotechnique, 1980, 30(4): 385-395. doi: 10.1680/geot.1980.30.4.385
[31]	韩昌瑞. 有限变形理论及其在岩土工程中的应用[D]. 武汉: 中国科学院研究生院(武汉岩土力学研究所), 2009. HAN Changrui. Finite Deformation Theory and its Application in geotechnical Engineering[D]. Wuhan: Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, 2009. (in Chinese)
[32]	陈明祥. 大变形弹塑性理论-上册, Ⅰ[M]. 北京: 科学出版社, 2022. CHEN Mingxiang. Elasticity and Plasticity of Large Deformation[M]. Beijing: Science Press, 2022. (in Chinese)
[33]	吴健. 饱和软土复杂非线性大变形固结特性及应用研究[D]. 杭州: 浙江大学, 2008. WU Jian. Study on Complex Nonlinear Large Deformation Consolidation Characteristics and Application of Saturated Soft Soil[D]. Hangzhou: Zhejiang University, 2008. (in Chinese)
[34]	丁洲祥, 朱合华, 丁文其. 大变形固结理论连续性条件的严格表述[J]. 同济大学学报(自然科学版), 2009, 37(4): 471-474. DING Zhouxiang, ZHU Hehua, DING Wenqi. A strict form of continuity condition in large-strain consolidation theory[J]. Journal of Tongji University (Natural Science), 2009, 37(4): 471-474. (in Chinese)
[35]	ORTENBLAD A. Mathematical theory of the process of consolidation of mud deposits[J]. Journal of Mathematics and Physics, 1930, 9(1/2/3/4): 73-149.
[36]	MCNABB A. A mathematical treatment of one-dimensional soil consolidation[J]. Quarterly of Applied Mathematics, 1960, 17(4): 337-347. doi: 10.1090/qam/113405
[37]	LEHNER F K. On the consolidation of thick layers[J]. Géotechnique, 1984, 34(2): 259-262. doi: 10.1680/geot.1984.34.2.259