Multiscale and Multiphysics Modeling of Soils

This chapter addresses the multiscale and multiphysics modeling of soils explicitly. The presentation revolves around three main paradigms: continuum, discrete, and multiscale. The advantages and disadvantages of each of the paradigms are discussed and their particular developments are addressed. We show that continuum models are the backbone of current analysis at the engineering scale and furnish an appropriate framework to implement multiphysics couplings including thermal, hydraulic, mechanical, and chemical (THMC) effects. On the other side of the spectrum, discrete models have emerged and they are capable of capturing explicitly the discrete nature of granular soils. Progress has been made to make discrete models accurate. More recent developments include multiscale methods connecting continuum and discrete approaches. Multiscale methods show much promise and have been able to reproduce material behavior in the laboratory. We anticipate that future applications will demand more multi-scale and analysis of geologic materials.

[1]  J. Andrade,et al.  Flow Liquefaction Instability as a Mechanism for Lower End of Liquefaction Charts , 2017 .

[2]  I. F. Collins,et al.  A thermomechanical analysis of a family of soil models , 2002 .

[3]  Gaël Combe,et al.  FEM × DEM modelling of cohesive granular materials: Numerical homogenisation and multi-scale simulations , 2014, Acta Geophysica.

[4]  A. Anandarajah,et al.  Three-Dimensional Discrete Element Method of Analysis of Clays , 2003 .

[5]  He Huang,et al.  A Simple Multiscale Model for Granular Soils with Geosynthetic Inclusion , 2016 .

[6]  Mohammad Hossein Abbaspour-Fard,et al.  Modeling nonspherical particles using multisphere discrete elements , 2001 .

[7]  Farhang Radjai,et al.  BIMODAL CHARACTER OF STRESS TRANSMISSION IN GRANULAR PACKINGS , 1998 .

[8]  G. McDowell,et al.  Discrete element modelling of under sleeper pads using a box test , 2018, Granular Matter.

[9]  Yang Liu,et al.  A nonlocal multiscale discrete‐continuum model for predicting mechanical behavior of granular materials , 2016 .

[10]  Alexei A. Efros,et al.  Image quilting for texture synthesis and transfer , 2001, SIGGRAPH.

[11]  Francesco Calvetti,et al.  Micromechanical approach to slope stability analysis , 2004 .

[12]  Kristian Krabbenhoft,et al.  A contact dynamics approach to the Granular Element Method , 2014 .

[13]  Félix Darve,et al.  INCREMENTAL CONSTITUTIVE LAW FOR SANDS AND CLAYS: SIMULATIONS OF MONOTONIC AND CYCLIC TESTS , 1982 .

[14]  FengXiating,et al.  NUMERICAL MODELING FOR COUPLED THERMO-HYDRO-MECHANICAL AND CHEMICAL PROCESSES (THMC) OF GEOLOGICAL MEDIA——INTERNATIONAL AND CHINESE EXPERIENCES , 2003 .

[15]  B. Teppen,et al.  Molecular Dynamics Modeling of Clay Minerals. 1. Gibbsite, Kaolinite, Pyrophyllite, and Beidellite , 1997 .

[16]  T. Belytschko,et al.  A finite element with embedded localization zones , 1988 .

[17]  M. Blunt,et al.  Pore space reconstruction using multiple-point statistics , 2005 .

[18]  Mike Jefferies,et al.  NOR-SAND: A SIMPLE CRITICAL STATE MODEL FOR SAND , 1993 .

[19]  James K. Mitchell,et al.  Effects of Sample Preparation on Sand Liquefaction , 1977 .

[20]  R. L. Kondner Hyperbolic Stress-Strain Response: Cohesive Soils , 1963 .

[21]  J. Santamarina,et al.  Closure of "Particle Shape Effects on Packing Density, Stiffness, and Strength: Natural and Crushed Sands" , 2006 .

[22]  Xuxin Tu,et al.  Multiscale framework for behavior prediction in granular media , 2009 .

[23]  L. Laloui,et al.  Constitutive modeling of unsaturated aggregated soils , 2010 .

[24]  Hai-Sui Yu,et al.  CASM: a unified state parameter model for clay and sand , 1998 .

[25]  M. Gunaratne,et al.  Analysis of water seepage in a pavement system using the particulate approach , 2009 .

[26]  Antonio Gens Solé,et al.  Fully Coupled Thermo-Hydro-Mechanical Double-Porosity Formulation for Unsaturated Soils , 2016 .

[27]  Dimitrios Kolymbas,et al.  An outline of hypoplasticity , 1991, Archive of Applied Mechanics.

[28]  Ning Guo,et al.  3D multiscale modeling of strain localization in granular media , 2016 .

[29]  Atsushi Yashima,et al.  FEM-FDM coupled liquefaction analysis of a porous soil using an elasto-plastic model , 1994 .

[30]  Yannis F. Dafalias,et al.  A critical state sand plasticity model accounting for fabric evolution , 2014 .

[31]  Jidong Zhao,et al.  The signature of shear-induced anisotropy in granular media , 2013 .

[32]  Assessment of structural evolution of aggregated soil using neutron tomography , 2008 .

[33]  A. Whittle,et al.  Nanoscale elastic properties of montmorillonite upon water adsorption. , 2012, Langmuir : the ACS journal of surfaces and colloids.

[34]  Liange Zheng,et al.  A coupled THMC model of a heating and hydration laboratory experiment in unsaturated compacted FEBEX bentonite , 2010 .

[35]  L. Y. Hu,et al.  Multiple‐point geostatistics for modeling subsurface heterogeneity: A comprehensive review , 2008 .

[36]  G. Gudehus A COMPREHENSIVE CONSTITUTIVE EQUATION FOR GRANULAR MATERIALS , 1996 .

[37]  Félix Darve,et al.  Failure in geomaterials: continuous and discrete analyses , 2004 .

[38]  A. Whittle,et al.  Mesoscale properties of clay aggregates from potential of mean force representation of interactions between nanoplatelets , 2014 .

[39]  Giovanni Grasselli,et al.  Characterization of the effect of normal load on the discontinuity morphology in direct shear specimens using X-ray micro-CT , 2015 .

[40]  J. Andrade,et al.  Level set discrete element method for three-dimensional computations with triaxial case study , 2016 .

[41]  Xuxin Tu,et al.  Return mapping for nonsmooth and multiscale elastoplasticity , 2009 .

[42]  Richard A. Regueiro,et al.  Three‐dimensional ellipsoidal discrete element modeling of granular materials and its coupling with finite element facets , 2010 .

[43]  Christoph Goniva,et al.  LIGGGHTS – Open Source Discrete Element Simulations of Granular Materials Based on Lammps , 2011 .

[44]  A. Elgamal,et al.  Computational modeling of cyclic mobility and post-liquefaction site response , 2002 .

[45]  Jean-Luc Guermond,et al.  Nonlinear corrections to Darcy's law at low Reynolds numbers , 1997, Journal of Fluid Mechanics.

[46]  Lianyang Zhang,et al.  Discrete element simulation of mine tailings stabilized with biopolymer , 2017, Environmental Earth Sciences.

[47]  Edward Andò,et al.  Multiscale characterization and modeling of granular materials through a computational mechanics avatar: a case study with experiment , 2016 .

[48]  Eugenio Oñate,et al.  Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems , 2004 .

[49]  Catherine O'Sullivan,et al.  Particle-Based Discrete Element Modeling: Geomechanics Perspective , 2011 .

[50]  D. Wildenschild,et al.  X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems , 2013 .

[51]  Dawei Zhao,et al.  A fast contact detection algorithm for 3-D discrete element method , 2004 .

[52]  Jidong Zhao Hierarchical Multiscale Modeling of Strain Localization in Granular Materials: A Condensed Overview and Perspectives , 2017 .

[53]  Alexander M. Puzrin,et al.  A thermomechanical framework for constitutive models for rate-independent dissipative materials , 2000 .

[54]  T. Sitharam,et al.  Post-liquefaction undrained monotonic behaviour of sands: experiments and DEM simulations , 2009 .

[55]  M. Jiang,et al.  Discrete element analysis of chemical weathering on rock , 2015 .

[56]  Xuxin Tu,et al.  Criteria for static equilibrium in particulate mechanics computations , 2008 .

[57]  Ronaldo I. Borja,et al.  Cam-Clay plasticity. Part V: A mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media , 2004 .

[58]  K. Terzaghi,et al.  Soil mechanics in engineering practice , 1948 .

[59]  Vincent Richefeu,et al.  Contact dynamics as a nonsmooth discrete element method , 2009 .

[60]  Jiansheng Xiang,et al.  A clustered overlapping sphere algorithm to represent real particles in discrete element modelling , 2009 .

[61]  José E. Andrade,et al.  Capturing strain localization in dense sands with random density , 2006 .

[62]  G. Viggiani,et al.  Strain localisation and grain breakage in sand under shearing at high mean stress: insights from in situ X-ray tomography , 2015 .

[63]  Minna Karstunen,et al.  Modelling the variation of degree of saturation in a deformable unsaturated soil , 2003 .

[64]  C. O’Sullivan Particulate Discrete Element Modelling: A Geomechanics Perspective , 2011 .

[65]  Mahesh Prakash,et al.  Discrete–element modelling and smoothed particle hydrodynamics: potential in the environmental sciences , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[66]  Jonny Rutqvist,et al.  Modeling of Coupled Thermo-Hydro-Mechanical Processes with Links to Geochemistry Associated with Bentonite-Backfilled Repository Tunnels in Clay Formations , 2013, Rock Mechanics and Rock Engineering.

[67]  W. Ramberg,et al.  Description of Stress-Strain Curves by Three Parameters , 1943 .

[68]  Richard S. Ladd,et al.  SPECIMEN PREPARATION AND LIQUEFACTION OF SANDS , 1974 .

[69]  P. Delage,et al.  A FORMULATION OF FULLY COUPLED THERMAL–HYDRAULIC–MECHANICAL BEHAVIOUR OF SATURATED POROUS MEDIA—NUMERICAL APPROACH , 1997 .

[70]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[71]  Ting Zhang,et al.  Porous Media Reconstruction Using a Cross-Section Image and Multiple-Point Geostatistics , 2009, 2009 International Conference on Advanced Computer Control.

[72]  Isam Shahrour,et al.  A full 3-D finite element analysis of tunneling–adjacent structures interaction , 2003 .

[73]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[74]  S. Mesarovic,et al.  Micromechanics of dilatancy and critical state in granular matter , 2012 .

[75]  Bernhard A. Schrefler,et al.  Thermo‐hydro‐mechanical analysis of partially saturated porous materials , 1996 .

[76]  Peter M. Byrne,et al.  A Cyclic Shear-Volume Coupling and Pore Pressure Model for Sand , 1991 .

[77]  M. Ortiz,et al.  A finite element method for localized failure analysis , 1987 .

[78]  Glenn R. McDowell,et al.  Discrete element modelling of soil particle fracture , 2002 .

[79]  A. Gens,et al.  Coupled Thermo-Hydro-Mechanical and Chemical Analysis of Expansive Clay Subjected to Heating and Hydration , 2007 .

[80]  Chuanqi Liu,et al.  Multi-scale Modelling of Granular Pile Collapse by Using Material Point Method and Discrete Element Method , 2017 .

[81]  Jean H. Prevost,et al.  A simple plasticity theory for frictional cohesionless soils , 1985 .

[82]  W. Nix,et al.  Instrumented nanoindentation and 3D mechanistic modeling of a shale at multiple scales , 2015 .

[83]  José E. Andrade,et al.  Modeling deformation banding in dense and loose fluid-saturated sands , 2007 .

[84]  C. Callari,et al.  Fem validation of a double porosity elastic model for consolidation of structurally complex clayey soils , 2000 .

[85]  Y. Feng,et al.  Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues , 2007 .

[86]  R. Chambon,et al.  A New Rate Type Constitutive Model for Geomaterials: CloE , 1993 .

[87]  Wenqing Wang,et al.  OpenGeoSys: an open-source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media , 2012, Environmental Earth Sciences.

[88]  D. E. Carlson,et al.  An introduction to thermomechanics , 1983 .

[89]  T. M. Evans,et al.  Discrete element method investigation on thermally-induced shakedown of granular materials , 2017 .

[90]  G. N. Pande,et al.  Analysis of stone-column reinforced foundations , 1998 .

[91]  Antoinette Tordesillas,et al.  Force chain buckling, unjamming transitions and shear banding in dense granular assemblies , 2007 .

[92]  J. Urai,et al.  Meshless numerical modeling of brittle–viscous deformation: first results on boudinage and hydrofracturing using a coupling of discrete element method (DEM) and smoothed particle hydrodynamics (SPH) , 2013, Computational Geosciences.

[93]  J. Bear Dynamics of Fluids in Porous Media , 1975 .

[94]  Jean H. Prevost,et al.  PLASTICITY THEORY FOR SOIL STRESS-STRAIN BEHAVIOR , 1978 .

[95]  Richard S. Ladd,et al.  Specimen Preparation and Cyclic Stability of Sands , 1977 .

[96]  N. Estrada,et al.  Shear strength and force transmission in granular media with rolling resistance. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[97]  Y. Dafalias,et al.  SANISAND-Z: zero elastic range sand plasticity model , 2016 .

[98]  Leonard R. Herrmann,et al.  Bounding surface plasticity. II: application to isotropic cohesive soils , 1986 .

[99]  J. Andrade,et al.  Effects of grain morphology on critical state: a computational analysis , 2016 .

[100]  Mahdi Taiebat,et al.  Study of pore pressure variation during liquefaction using two constitutive models for sand , 2007 .

[101]  J. Andrade,et al.  All you need is shape: Predicting shear banding in sand with LS-DEM , 2018 .

[102]  John R. Williams,et al.  A direct simulation method for particle‐fluid systems , 2003 .

[103]  Kristian Krabbenhoft,et al.  On the contact treatment of non-convex particles in the granular element method , 2014 .

[104]  Di Wu,et al.  A coupled THMC modeling application of cemented coal gangue-fly ash backfill , 2018 .

[105]  Frédéric-Victor Donzé,et al.  YADE‐OPEN DEM: an open‐source software using a discrete element method to simulate granular material , 2009 .

[106]  Dimitrios Kolymbas,et al.  A rate-dependent constitutive equation for soils , 1977 .

[107]  Guy T. Houlsby,et al.  Application of thermomechanical principles to the modelling of geotechnical materials , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[108]  Antonio Gens,et al.  THM-coupled finite element analysis of frozen soil : formulation and application , 2009 .

[109]  M. Manzari,et al.  SIGNIFICANCE OF SOIL DILATANCY IN SLOPE STABILITY ANALYSIS. TECHNICAL NOTE , 2000 .

[110]  A. Carminati,et al.  Investigation of water imbibition in porous stone by thermal neutron radiography , 2006 .

[111]  Christoph Wehrli,et al.  The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function , 1987 .

[112]  D. Penumadu,et al.  High-resolution X-ray and neutron computed tomography of partially saturated granular materials subjected to projectile penetration , 2016 .

[113]  Christoph H. Arns,et al.  Pore Scale Characterization of Carbonates Using X-Ray Microtomography , 2005 .

[114]  Mechanics of origin of flow liquefaction instability under proportional strain triaxial compression , 2016 .

[115]  M. Curtis,et al.  Microstructural investigation of gas shales in two and three dimensions using nanometer-scale resolution imaging , 2012 .

[116]  A. Billi Grain size distribution and thickness of breccia and gouge zones from thin (<1 m) strike-slip fault cores in limestone , 2005 .

[117]  Y. T. Feng,et al.  Discrete thermal element modelling of heat conduction in particle systems: Basic formulations , 2008, J. Comput. Phys..

[118]  Yannis F. Dafalias,et al.  Anisotropic Critical State Theory: Role of Fabric , 2012 .

[119]  José E. Andrade,et al.  Granular element method for three‐dimensional discrete element calculations , 2014 .

[120]  Yu-Shu Wu,et al.  Sequentially coupled THMC model for CO2 geological sequestration into a 2D heterogeneous saline aquifer , 2015, Journal of Natural Gas Science and Engineering.

[121]  J. J. McCarthy,et al.  Thermal expansion effects and heat conduction in granular materials. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[122]  C. O’Sullivan,et al.  Analysis of bender element test interpretation using the discrete element method , 2015 .

[123]  C. O’Sullivan,et al.  Two-dimensional discrete element modelling of bender element tests on an idealised granular material , 2012 .

[124]  Majid T. Manzari,et al.  SIMPLE PLASTICITY SAND MODEL ACCOUNTING FOR FABRIC CHANGE EFFECTS , 2004 .

[125]  O. Zienkiewicz,et al.  An anisotropic hardening model for soils and its application to cyclic loading , 1978 .

[126]  Guy T. Houlsby,et al.  A study of plasticity theories and their applicability to soils , 1981 .

[127]  Alexander M. Puzrin,et al.  Constitutive Modelling in Geomechanics: Introduction , 2012 .

[128]  R. Bathurst,et al.  Analytical study of induced anisotropy in idealized granular materials , 1989 .

[129]  János Urai,et al.  Multi-scale characterization of porosity in Boom Clay (HADES-level, Mol, Belgium) using a combination of X-ray μ-CT, 2D BIB-SEM and FIB-SEM tomography , 2015 .

[130]  Günther Meschke,et al.  A three‐phase thermo‐hydro‐mechanical finite element model for freezing soils , 2013 .

[131]  Jianfeng Wang,et al.  Generation of a realistic 3D sand assembly using X‐ray micro‐computed tomography and spherical harmonic‐based principal component analysis , 2017 .

[132]  Young-Kyo Seo,et al.  Limit state analysis of earthen slopes using dual continuum/FEM approaches , 1999 .

[133]  Ning Guo,et al.  Multiscale insights into classical geomechanics problems , 2016 .

[134]  D. C. Drucker,et al.  Soil mechanics and plastic analysis or limit design , 1952 .

[135]  Gregoire Mariethoz,et al.  The Direct Sampling method to perform multiple‐point geostatistical simulations , 2010 .

[136]  K. Alshibli,et al.  3D finite element modeling of sand particle fracture based on in situ X‐Ray synchrotron imaging , 2016 .

[137]  Gaël Combe,et al.  FEM×DEM multiscale modeling: Model performance enhancement from Newton strategy to element loop parallelization , 2018 .

[138]  Liange Zheng A Coupled THMC model of FEBEX mock-up test , 2010 .

[139]  P. Cundall,et al.  FORMULATION OF A THREE-DIMENSIONAL DISTINCT ELEMENT MODEL - PART II. MECHANICAL CALCULATIONS FOR MOTION AND INTERACTION OF A SYSTEM COMPOSED OF MANY POLYHEDRAL BLOCKS , 1988 .

[140]  Richard J. Bathurst,et al.  Numerical simulation of idealized granular assemblies with plane elliptical particles , 1991 .

[141]  B. C. Martinez,et al.  Biogeochemical processes and geotechnical applications: progress, opportunities and challenges , 2013 .

[142]  A. Skempton THE PORE-PRESSURE COEFFICIENTS A AND B , 1954 .

[143]  K. Roscoe,et al.  ON THE GENERALIZED STRESS-STRAIN BEHAVIOUR OF WET CLAY , 1968 .

[144]  Ning Guo,et al.  Multiscale modeling and analysis of compaction bands in high-porosity sandstones , 2018 .

[145]  Alexander M. Puzrin,et al.  Principles of Hyperplasticity: An Approach to Plasticity Theory Based on Thermodynamic Principles , 2010 .

[146]  J. Moreau Evolution problem associated with a moving convex set in a Hilbert space , 1977 .

[147]  Ronaldo I. Borja,et al.  Cam-Clay plasticity, Part II: implicit integration of constitutive equation based a nonlinear elastic stress predictor , 1991 .

[148]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[149]  Robert Charlier,et al.  A thermo-hydro-mechanical constitutive model and its numerical modelling for unsaturated soils , 2004 .

[150]  Jens Gregor,et al.  High-Resolution Neutron and X-Ray Imaging of Granular Materials , 2013 .

[151]  M. Oda,et al.  Rolling Resistance at Contacts in Simulation of Shear Band Development by DEM , 1998 .

[152]  T. Ng,et al.  A three-dimensional discrete element model using arrays of ellipsoids , 1997 .

[153]  J. Burland,et al.  A logarithmic stress–strain function for rocks and soils , 1996 .

[154]  L. Laloui,et al.  Structural characterization of unsaturated aggregated soil , 2010 .

[155]  Glenn R. McDowell,et al.  On the micro mechanics of one-dimensional normal compression , 2013 .

[156]  Ronaldo I. Borja,et al.  Cam-Clay plasticity, Part VIII: A constitutive framework for porous materials with evolving internal structure , 2016 .

[157]  Malcolm D. Bolton,et al.  Discrete element simulation of crushable soil , 2003 .

[158]  Edmund Perfect,et al.  Neutron imaging of hydrogen-rich fluids in geomaterials and engineered porous media: A review , 2014 .

[159]  François Nicot,et al.  From microscopic to macroscopic second-order work in granular assemblies , 2007 .

[160]  José E. Andrade,et al.  Flow liquefaction instability prediction using finite elements , 2015 .

[161]  D. Dewhurst,et al.  Laboratory characterisation of shale properties , 2012 .

[162]  Ronaldo I. Borja,et al.  Quantifying the heterogeneity of shale through statistical combination of imaging across scales , 2017, Acta Geotechnica.

[163]  Ning Guo,et al.  A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media , 2014 .

[164]  Antoinette Tordesillas,et al.  On the modeling of confined buckling of force chains , 2009 .

[165]  William Pao,et al.  Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir , 2002 .

[166]  A. Whittle,et al.  Mesoscale simulation of clay aggregate formation and mechanical properties , 2016 .

[167]  A. Anandarajah Discrete-element method for simulating behavior of cohesive soil , 1994 .

[168]  A. Yu,et al.  Discrete particle simulation of particulate systems: Theoretical developments , 2007 .

[169]  Jian-Hua Wang,et al.  A multiscale coupling approach between discrete element method and finite difference method for dynamic analysis , 2015 .

[170]  Mei Xu Concurrent Coupling of Atomistic and Continuum Models , 2009 .

[171]  S. Valliappan,et al.  Unified theory of flow and deformantion in double porous media , 1996 .

[172]  Gioacchino Viggiani,et al.  Volumetric Digital Image Correlation Applied to X‐ray Microtomography Images from Triaxial Compression Tests on Argillaceous Rock , 2007 .

[173]  Ken Been,et al.  A STATE PARAMETER FOR SANDS , 1985 .

[174]  Vladimir Mityushev,et al.  Nonlinear correction to Darcy’s law for channels with wavy walls , 2013 .

[175]  R. Borja,et al.  Micropolar hypoplasticity for persistent shear band in heterogeneous granular materials , 2015 .

[176]  Nasser Khalili,et al.  Effective stress in double porous media with two immiscible fluids , 2005 .

[177]  J. Andrade,et al.  Friction in inertial granular flows: competition between dilation and grain-scale dissipation rates , 2015, Granular Matter.

[178]  R. Regueiro,et al.  Concurrent Multiscale Computational Modeling for Dense Dry Granular Materials Interfacing Deformable Solid Bodies , 2011 .

[179]  N. Kalteziotis,et al.  Geotechnical properties of the Corinth Canal marls , 1991 .

[180]  Yi Du,et al.  Reconstruction of porous media using multiple-point statistics with data conditioning , 2015, Stochastic Environmental Research and Risk Assessment.

[181]  José E. Andrade,et al.  Granular Element Method for Computational Particle Mechanics , 2012 .

[182]  Usama El Shamy,et al.  Multiscale Modeling of Flood-Induced Piping in River Levees , 2008 .

[183]  Hai‐Sui Yu,et al.  Plasticity and geotechnics , 2006 .

[184]  C A Coulomb,et al.  ESSAI SUR UNE APPLICATION DES REGLES DE MAXIMIS ET MINIMIS A QUELQUES PROBLEMES DE STATIQUE RELATIFS A L'ARCHITECTURE (ESSAY ON MAXIMUMS AND MINIMUMS OF RULES TO SOME STATIC PROBLEMS RELATING TO ARCHITECTURE) , 1973 .

[185]  A. Schofield,et al.  On The Yielding of Soils , 1958 .

[186]  Matthew R. Kuhn Smooth Convex Three-Dimensional Particle for the Discrete-Element Method , 2003 .

[187]  Dimitrios Kolymbas,et al.  Constitutive Modelling of Granular Materials , 2000 .

[188]  M. Zeghal,et al.  A continuum‐discrete hydromechanical analysis of granular deposit liquefaction , 2004 .