EI、Scopus 收录
Volume 55 Issue 2
Feb.  2023
Turn off MathJax
Article Contents
Li Xikui, Zhang Songge, Chu Xihua. Effective pressure and generalized effective Biot stress of porous continuum in unsaturated granular materials. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 369-380 doi: 10.6052/0459-1879-22-407
Citation: Li Xikui, Zhang Songge, Chu Xihua. Effective pressure and generalized effective Biot stress of porous continuum in unsaturated granular materials. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 369-380 doi: 10.6052/0459-1879-22-407


doi: 10.6052/0459-1879-22-407
  • Received Date: 2022-09-02
  • Accepted Date: 2022-12-28
  • Available Online: 2022-12-29
  • Publish Date: 2023-02-18
  • The definition of effective pressure with associated formula of the Bishop parameter for unsaturated porous medium proposed in the frame of the theory of macroscopic porous continuum has been controversial for a long time. This also affects the correct prediction of directly related generalized Biot effective stress. Based on the Voronoi cell model described with the discrete system composed of solid particles, binary bond liquid bridges and liquid films, the present paper presents the definitions of effective internal state variables at local material points in unsaturated porous continua with low saturation, i.e. effective pressure and generalized Biot effective stress. Using the proposed Voronoi cell model, their expressions are formulated with the information of hydro-mechanical meso-structure and moso-response evolved with incremental loading process exerted on the representative volume element (RVE) of unsaturated granular material. With the derived effective pressure formula, it is demonstrated that the effective pressure tensor of unsaturated porous continuum is anisotropic. It has not only an anisotropic effect on hydrostatic components, but also an effect on shear stress components, of generalized Biot effective stress tensor. It is demonstrated that the fundamental defect of both the generalized Biot theory and the so-called bivariate theory lies in that it is assumed that effective pore pressure tensor representing the hydro-mechanical effect of two immiscible pore fluids on the solid skeleton of unsaturated porous continua is isotropic. In addition, the Bishop parameter introduced as the weighted factor to define the isotropic effective pore pressure tensor is assumed not related to the matrix suction with very important effect on the hydro-mechanical response occurring at local material points over unsaturated porous continua. The derived formulae of both generalized Biot effective stress and effective pressure (including effective Bishop parameter reflecting the isotropic effect of effective pressure) can be upscaled to a local material point, where the RVE is assigned, in macroscopic unsaturated porous continua, for computational multi-scale methods represented by the concurrent computational homogenization method for unsaturated granular materials.


  • loading
  • [1]
    Bishop AW. The principle of effective stress. Teknisk Ukeblad, 1959, 106(39): 859-863
    Skempton AW. Effective stress in soils, concrete and rock//Proc. Conf. on Pore Pressure and Suction in Soils, Butterworth, 1960: 4-16
    Bishop AW, Blight GE. Some aspects of effective stress in saturated and partly saturated soils. Geotechnique, 1963, 13: 177-197 doi: 10.1680/geot.1963.13.3.177
    Li XK, Zienkiewicz OC. Multiphase flow in deforming porous media and finite element solutions. Comput. Struct., 1992, 45: 211-227 doi: 10.1016/0045-7949(92)90405-O
    Zienkiewicz OC, Chan AHC, Pastor M, et al. Computational Geomechanics with Special Reference to Earthquake Engineering. Chichester: Wiley, 1999
    Gray WG, Schrefler BA. Thermodynamic approach to effective stress in partially saturated porous media. Eur. J. Mech. A: Solids, 2001, 20: 521-538 doi: 10.1016/S0997-7538(01)01158-5
    Gray WG, Schrefler BA. Analysis of the solid phase stress tensor in multiphase porous media. Int. J. Numer. Anal. Meth. Geomech., 2007, 31: 541-581
    Gray WG, Miller TC. Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems. Switzerland: Springer, 2014
    邵龙潭, 郭晓霞. 有效应力新解. 北京: 中国水利水电出版社, 2014

    Shao Longtan, Guo Xiaoxia. New Insight on the Effective Stress. Beijing: China Water & Power Press, 2014 (in Chinese)
    Auriault JL, Sanchez-Palencia E. Etude du comportement macroscopique d’un milieu poreux sature deformable. Journal de Mecanique, 1977, 16: 575-603
    Auriault JL. nonsaturated deformable porous media: quasistatics. Transport in Porous Media, 1987, 2: 45-64
    Nemat-Nasser S, Hori M. Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier: Amsterdam, 1999
    Li XS. Effective stress in unsaturated soil: a microstructural analysis. Geotechnique, 2003, 53: 273-277 doi: 10.1680/geot.2003.53.2.273
    Lu N, Likos WJ. Suction stress characteristic curve for unsaturated soil. J. Geotech. Geoenviron. Eng., 2006, 132: 131-142 doi: 10.1061/(ASCE)1090-0241(2006)132:2(131)
    Hicher PY, Chang CS. A microstructural elastoplastic model for unsaturated granular materials. Int. J. Solids. Struct., 2007, 44: 2304-2323 doi: 10.1016/j.ijsolstr.2006.07.007
    Hicher PY, Chang CS. Elastic model for partially saturated granular materials. J. Eng. Mech., 2008, 134: 505-513
    Scholtes L, Hicher PY, Nicot F, et al. On the capillary stress tensor in wet granular materials. Int. J. Numer. Anal. Meth. Geomech., 2009, 33: 1289-1313 doi: 10.1002/nag.767
    Pierrat P, Caram HS. Tensile strength of wet granula materials. Powder Technology, 1997, 91: 83-93 doi: 10.1016/S0032-5910(96)03179-8
    Richefeu V, El Youssoufi MS, Peyroux R, et al. A model of capillary cohesion for numerical simulations of 3D polydisperse granular media. Int. J. Numer. Anal. Meth. Geomech., 2008, 32: 1365-1383 doi: 10.1002/nag.674
    Than VD, Khamseh S, Tang AM, et al. Basic mechanical properties of wet granular materials: A DEM study. Journal of Engineering Mechanics, 2017, 143: C4016001
    Urso MED, Lawrence CJ, Adams MJ. Pendular funicular and capillary bridges: results for two dimensions. J. Colloid Interface Sci., 1999, 220: 42-56 doi: 10.1006/jcis.1999.6512
    El Shamy U, Groger T. Micromechanical aspects of the shear strength of wet granular soils. Int. J. Numer. Anal. Meth. Geomech., 2008, 32: 1763-1790
    杜友耀, 李锡夔. 二维液桥计算模型及湿颗粒材料离散元模拟. 计算力学学报, 2015, 32: 496-502 (Du Youyao, Li Xikui. 2D computational model of liquid bridge and DEM simulation of wet granular materials. Chinese Journal of Computational Mechanics, 2015, 32: 496-502 (in Chinese)
    李锡夔, 杜友耀, 段庆林. 基于介观结构的饱和与非饱和多孔介质有效应力. 力学学报, 2016, 48: 29-39 (Li Xikui, Du Youyao, Duan Qinglin. Meso-structure informed effective stresses in saturated and unsaturated porous media. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48: 29-39 (in Chinese)
    Li XK, Du YY, Zhang SG, et al. Meso-hydro-mechanically informed effective stresses and effective pressures for saturated and unsaturated porous media. European Journal of Mechanics A/Solids, 2016, 59: 24-36 doi: 10.1016/j.euromechsol.2016.03.005
    Oda M, Iwashita K. Mechanics of Granular Materials: An Introduction. Rotterdam: Balkema, 1999
    Walsh SDC, Tordesillas A, Peters JF. Development of micromechanical models for granular media: the projection problem. Granul Matter, 2007, 9: 337-352 doi: 10.1007/s10035-007-0043-5
    Li XK, Du YY, Duan QL. Micromechanically informed constitutive model and anisotropic damage characterization of cosserat continuum for granular materials. Int. J. Damage. Mech., 2013, 22: 643-682 doi: 10.1177/1056789512462427
    Zhang SG, Li XK, Du YY. A numerical model of discrete element-liquid bridge–liquid thin film system for wet deforming granular medium at low saturation. Powder Technology, 2022, 399: 117217 doi: 10.1016/j.powtec.2022.117217
    Souli´e, F, Cherblanc F, El Youssoufi MS, et al. Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials. Int. J. Numer. Anal. Meth. Geomech., 2006, 30: 213-228
    Li XK, Zhang SG, Duan QL. Second-order concurrent computational homogenization method and multi-scale hydro-mechanical modeling for saturated granular materials. Int. J. for Multiscale Computational Engineering, 2020, 18: 199-240 doi: 10.1615/IntJMultCompEng.2020031587
    Li XK, Zhang X, Zhang JB. A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials. Computer Methods in Applied Mechanics and Engineering, 2010, 199: 3137-3152 doi: 10.1016/j.cma.2010.06.016
    Li XK, Zienkiewicz OC, Xie YM. A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution. Int. J. for Numerical Methods in Eng., 1990, 30: 1195-1212 doi: 10.1002/nme.1620300608
    Schrefler BA, Zhan XY. A fully coupled model for water flow and airflow in deformable porous media. Water Resour Res., 1993, 29: 155-167 doi: 10.1029/92WR01737
    Lewis RW, Schrefler BA. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Second Edition. England: John Wiley & Sons Ltd, 1998: 1-492
    Fredlund DG, Rahardjo H. Soil Mechanics for Unsaturated Soils. New York: John Wiley & Sons, 1993: 1-517
    沈珠江. 理论土力学. 北京: 中国水利水电出版社, 2000

    Shen Zhujiang. Theoretical Soil Mechanics. Beijing: China Water Power Press, 2000 (in Chinese)
    Zienkiewicz OC, Shiomi T. Dynamic behavior of saturated porous media:the generalized Boit formulation and its numerical solution. Int. J. Numer. Anal. Meth. Geomech., 1984, 8: 71-96
  • 加载中


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索


    Article Metrics

    Article views (246) PDF downloads(70) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint