Constitutive equations and finite element implementation of strain localization in sand deformation

Abstract

A recently proposed model coupling with the solid-fluid of the saturated sand was utilized to study the deformation band. Based on the critical state plasticity model by Borja and Andrade, the hydraulic conductivity tensor was naturally treated as a function of the spatial discretization matrix about the displacement and the stress field, allowing a more realistic representation of the physical phenomenon. The fully Lagrangian form of the Darcy law was resolved by Piola algorithm, and then the flow law was gained, leading to the implementation of a modified model of the saturated sand. Then the criterion for the onset of localization was derived and utilized to detect instability. The constitutive model was implemented in a finite element program coded by FORTRAN, which was used to predict the formation and development of shear bands in plane strain compression of saturated sand. At last, the formation mechanism of the shear band was discussed. It is shown that the model works well, and the simulation sample bifurcates at 1.18% axial strain, which is in a good qualitative agreement with the experiment. The pore pressure greatly affects the onset and development of the deformation band, and it obviously increases around the localization-prone regions with the direction toward the outer side of the normal of the shear band, while the pore stress flows nearly horizontally and is distributed equally far away the shear band region.

Keywords

constitutive equations / deformation band / finite elements / hydraulic conductivity / sand

References

[1] AlsinyA., VardoulakisI. G., DrescherA.. Deformation localization in cavity inaction experiments on dry sand [J]. Géotechnique, 1992, 42(3): 395-410

[2] ShiC.-h., HuangL.-chong.. Calculation of soil deformation in area of disturbance due to pipe-jacking construction [J]. Journal of Central South University: Science and Technology, 2005, 36(2): 323-328

[3] ZhaoM.-h., LiuD.-p., ZhangL., JiangC.. 3D finite element analysis on pile-soil interaction of passive pile group [J]. Journal of Central South University of Technology, 2008, 15(1): 75-80

[4] AnandL.. Plane deformations of ideal granular materials [J]. Journal of the Mechanics and Physics of Solids, 1983, 31(2): 105-122

[5] RoscoeK. H., BurlandJ. H.. On the generalized stress-strain behavior of ‘wet’ clay [C]. Engineering Plasticity, 1968, Cambridge, Cambridge University Press: 535-609

[6] LandisE. N., NagyE. N., KeaneD. T.. Microstructure and fracture in three dimensions [J]. Engineering Fracture Mechanics, 2003, 70(7): 911-925

[7] OtaniJ., MukunokiT., ObaraY.. Characterization of failure in sand under triaxial compression using an industrial X-ray scanner [J]. International Journal of Physical Modelling in Geotechnics, 2002, 34(1): 15-22

[8] MeesF., SwennenV., GeetM., JacobsP.Applications of X-ray computed tomography in geosciences [M], 2003, London, Geological Society, Special Publications

[9] LenoirN., BornertM., DesruesJ., BesuelleP., ViggianiG.. 3D digital image correlation applied to X-ray micro tomography images from triaxial compression tests on argillaceous rock [J]. Strain, 2007, 43(3): 193-205

[10] DesruesJ., ChambonR., MokniM., MazerolleF.. Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography [J]. Géotechnique, 1996, 46(3): 529-546

[11] KetchamR. A., CarlsonW. D.. Acquisition, optimization and interpretation of X-ray computed homographic imagery: Applications to the geosciences [J]. Computers and Geosciences, 2001, 27(4): 381-400

[12] RICE J R. The localization of plastic deformation [C]// KOITER W T. Theoretical and Applied Mechanics. Amsterdam, 1976: 207–220.

[13] QianJ.-g., HuangM.-song.. An analytical solution for criterion of onset of strain localization of soils [J]. Rock and Soil Mechanics, 2005, 26(3): 432-436

[14] HuangM.-s., QianJ.-g., WuS.-ming.. A homogenization approach to localized deformation in saturated soils [J]. Chinese Journal of Geotechnical Engineering, 2002, 24(1): 24-28

[15] WangL. B., FrostJ. D., LaiJ. S.. Three-dimensional digital representation of granular material microstructure from X-ray tomography imaging [J]. Journal of Computing in Civil Engineering, 2004, 18(1): 28-35

[16] BelytschkoT., LiuW. K., MoranB.Nonlinear finite elements for continua and structures [M], 2000, West Sussex, John Wiley and Sons Ltd

[17] BorjaR. I., AndradeJ. E.. Critical state plasticity. Part VI: Meso-scale finite element simulation of strain localization in discrete granular materials [J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(37/40): 5115-5140

[18] BorjaR. I., TamagniniC.. Cam-clay plasticity. Part III: Extension of the infinitesimal model to include finite strains [J]. Computer Methods in Applied Mechanics and Engineering, 1998, 155(1/2): 73-95

[19] DafaliasY. F.. Plastic spin: Necessity or redundancy [J]. International Journal of Plasticity, 1998, 14(9): 909-931

[20] BearJ.Dynamics of fluids in porous media [M], 1972, New York, Elsevier Publishing Company Inc

[21] AndradeJ. E., BorjaR. I.. Capturing strain localization in dense sands with random density [J]. International Journal for Numerical Methods in Engineering, 2006, 67(11): 1531-1564

[22] JefferiesM. G.. Nor-sand: A simple critical state model for sand [J]. Géotechnique, 1993, 43(1): 91-103

[23] ChangC. S., MatsushimaT., LeeX.. Heterogeneous strain and bonded granular structure change in triaxial specimen studied by computer tomography [J]. Journal of Engineering Mechanics, 2003, 129(11): 1295-1307

[24] XuL.-m., WangX.-ran.. Numerical simulation of shear band in clayey soils using finite deformation theory [J]. Chinese Journal of Geotechnical Engineering, 2004, 26(2): 225-228