## Abstract

In this paper, we outline a constitutive model capable of describing anisotropy and many other features of the behaviour of multiphase granular soils, together with the computational framework that enables its numerical implementation. The constitutive model is formulated within the framework of bounding surface plasticity. It can simulate monotonic and cyclic loading for a wide range of stress and saturation states, it includes enhanced descriptions of wetting and drying processes, of anisotropy and of changing compressibility. These features are captured using a single set of parameters by using a combination of isotropic and kinematic hardening. The model is formulated based on the concept of effective stress for unsaturated states that guarantees smooth transitions between unsaturated and fully saturated states. Furthermore, we present unified formulations for saturated and unsaturated states in which the isotropic hardening law and the critical state line are described in a bi-logarithmic space defined by the logarithms of the mean effective stress and void ratio. Moreover, the constitutive model is coupled with a soil water characteristic model that allows consideration of the hysteretic nature of the saturation degree changes upon wetting/drying reversals. The paper describes the numerical implementation, which includes several smoothing techniques to enhance the constitutive model’s performance in numerical modelling during transitions between kinematic hardening and isotropic hardening and drying/wetting reversals. The numerical implementation also includes automatic error control and sub-stepping techniques, suitable for explicit integration algorithms, that give users additional control over the accuracy and speed of the analysis. Lastly, several examples are provided to demonstrate the range of application of the computational framework.

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## References

Poorooshasb HB, Pietruszczak S (1985) On yielding and flow of sand; a generalized two-surface model. Comput Geotech 1(1):33–58

Dafalias YF, Manzari MT (2004) Simple plasticity sand model accounting for fabric change effects. J Eng Mech 130(6):622–634

Semnani SJ, White JA, Borja RI (2016) Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity. Int J Numer Anal Methods Geomech 40(18):2423–2449

Zhao Y, Semnani SJ, Yin Q, Borja RI (2018) On the strength of transversely isotropic rocks. Int J Numer Anal Methods Geomech 42(16):1917–1934

Borja RI, Yin Q, Zhao Y (2020) Cam-Clay plasticity. Part IX: on the anisotropy, heterogeneity, and viscoplasticity of shale. Comput Methods Appl Mech Eng 360:112695

Montáns FJ, Borja RI (2002) Implicit J2-bounding surface plasticity using Prager’s translation rule. Int J Numer Meth Eng 55(10):1129–1166

Mroz Z (1967) On the description of anisotropic workhardening. J Mech Phys Solids 15(3):163–175

Oda M, Nakayama H (1989) Yield function for soil with anisotropic fabric. J Eng Mech 115(1):89–104

Lade PV, Inel S (1997) Rotational kinematic hardening model for sand. Part I concept of rotating yield and plastic potential surfaces. Comput Geotechn 21(3):183–216

Oka F et al (1999) A cyclic elasto-plastic constitutive model for sand considering a plastic-strain dependence of the shear modulus. Geotechnique 49(5):661–680

Gao Z, Zhao J (2015) Constitutive modeling of anisotropic sand behavior in monotonic and cyclic loading. J Eng Mech 141(8):04015017

Li X, Dafalias YF (2000) Dilatancy for cohesionless soils. Geotechnique 50(4):449–460

Dafalias YF, Taiebat M (2016) SANISAND-Z: zero elastic range sand plasticity model. Géotechnique 66(12):999–1013

Gens A, Sánchez M, Sheng D (2006) On constitutive modelling of unsaturated soils. Acta Geotech 1(3):137

Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New York

Ghorbani J, El-Zein A, Airey DW (2020) Thermo-elasto-plastic analysis of geosynthetic clay liners exposed to thermal dehydration. Environ Geotech 7(8):566–580

Li P, Schanz M (2011) Wave propagation in a 1-D partially saturated poroelastic column. Geophys J Int 184(3):1341–1353

Ghorbani J, Nazem M, Carter JP (2015) Application of the generalised-α method in dynamic analysis of partially saturated media. In: Computer methods and recent advances in geomechanics: Proceedings of the 14th international conference of international association for computer methods and recent advances in geomechanics, 2014 (IACMAG 2014). Taylor & Francis Books Ltd, pp 129–133

Ghorbani J, Airey D (2019) Mechanism of dissipation of excess flow pressures in unsaturated granular soils subjected to seismic excitations. In: 7th Asia-Pacific conference on unsaturated soils, Nagoya

Ghorbani J, Nazem M, Carter JP (2016) Dynamic analysis of unsaturated soils subjected to large deformations. Appl Mech Mater 846:354–359

Ghorbani J, Nazem M, Carter J, Airey D (2017) Anumerical study of the effect of moisture content on inducedground vibration during dynamic compaction. In: 3rd International conference on performance-based design in earthquake geotechnical engineering (PBD-III Vancouver 2017). International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE-SIMGS), London

Ghorbani J, Nazem M, Carter JP (2016) Numerical study of dynamic soil compaction at different degrees of saturation. In: The twenty-fifth international ocean and polar engineering conference. International Society of Offshore and Polar Engineers

Ghorbani J, Nazem M, Carter JP (2020) Dynamic compaction of clays; a numerical study based on the mechanics of unsaturated soils. Int J Geomech (Forthcoming). https://doi.org/10.1061/(ASCE)GM.1943-5622.0001840

Li X, Zienkiewicz O (1992) Multiphase flow in deforming porous media and finite element solutions. Comput Struct 45(2):211–227

Schrefler BA, Scotta R (2001) A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Comput Methods Appl Mech Eng 190(24):3223–3246

Borja RI (2004) Cam-Clay plasticity. Part V: a mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media. Comput Methods Appl Mech Eng 193(48):5301–5338

Ehlers W, Graf T, Ammann M (2004) Deformation and localization analysis of partially saturated soil. Comput Methods Appl Mech Eng 193(27):2885–2910

Khalili N, Habte MA, Zargarbashi S (2008) A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hystereses. Comput Geotech 35(6):872–889

Ghorbani J, Nazem M, Carter J (2016) Numerical modelling of multiphase flow in unsaturated deforming porous media. Comput Geotech 71:195–206

Sheng D, Sloan S, Gens A (2004) A constitutive model for unsaturated soils: thermomechanical and computational aspects. Comput Mech 33(6):453–465

Bishop AW (1960) The principles of effective stress. Norges Geotekniske Institutt

Schrefler BA (1984) The finite element method in soil consolidation: (with applications to surface subsidence). University College of Swansea

Khalili N, Khabbaz M (1998) A unique relationship of chi for the determination of the shear strength of unsaturated soils. Geotechnique 48(5):681–687

Nuth M, Laloui L (2008) Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int J Numer Anal Meth Geomech 32(7):771–801

Bolzon G, Schrefler B, Zienkiewicz O (1996) Elastoplastic soil constitutive laws generalized to partially saturated states. Géotechnique 46(2):279–289

Santagiuliana R, Schrefler B (2006) Enhancing the Bolzon–Schrefler–Zienkiewicz constitutive model for partially saturated soil. Transp Porous Media 65(1):1–30

Wheeler S, Sharma R, Buisson M (2003) Coupling of hydraulic hysteresis and stress–strain behaviour in unsaturated soils. Géotechnique 53(1):41–54

Mašín D, Khalili N (2008) A hypoplastic model for mechanical response of unsaturated soils. Int J Numer Anal Meth Geomech 32(15):1903–1926

Gallipoli D et al (2003) An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour. Géotechnique 53(1):123–136

Liu C, Muraleetharan KK (2011) Coupled hydro-mechanical elastoplastic constitutive model for unsaturated sands and silts. I: formulation. Int J Geomech 12(3):239–247

Song X, Khalili N (2019) A peridynamics model for strain localization analysis of geomaterials. Int J Numer Anal Methods Geomech 43(1):77–96

Borja RI, Lin C-H, Montáns FJ (2001) Cam-Clay plasticity, part IV: implicit integration of anisotropic bounding surface model with nonlinear hyperelasticity and ellipsoidal loading function. Comput Methods Appl Mech Eng 190(26–27):3293–3323

Dafalias YF (1986) Bounding surface plasticity. I: mathematical foundation and hypoplasticity. J Eng Mech 112(9):966–987

Taylor DW (1948) Fundamentals of soil mechanics, vol 66. LWW

Rowe PW (1962) The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc R Soc Lond Ser A Math Phys Sci 269(1339):500–527

Nova R (1982) A constitutive model for soil under monotonic and cyclic loading. In: Pande GN, Zienkiewicz OC (eds) Soil mechanics: transient and cyclic loads. Wiley, New York

Bolton M (1986) Strength and dilatancy of sands. Geotechnique 36(1):65–78

Manzari MT, Dafalias YF (1997) A critical state two-surface plasticity model for sands. Géotechnique 47(2):255–272

Nova R (1977) On the hardening of soils. Arch Mech 29(3):445–458

Desai C, Somasundaram S, Frantziskonis G (1986) A hierarchical approach for constitutive modelling of geologic materials. Int J Numer Anal Methods Geomech 10(3):225–257

Pestana JM, Whittle AJ (1999) Formulation of a unified constitutive model for clays and sands. Int J Numer Anal Methods Geomech 23(12):1215–1243

Taiebat M, Dafalias YF (2008) SANISAND: simple anisotropic sand plasticity model. Int J Numer Anal Methods Geomech 32(8):915–948

Sloan SW, Abbo AJ, Sheng D (2001) Refined explicit integration of elastoplastic models with automatic error control. Eng Comput 18(1/2):121–194

Ghorbani J et al (2018) A stress integration scheme for elasto-plastic response of unsaturated soils subjected to large deformations. Comput Geotech 94:231–246

Dafalias Y, Popov E (1975) A model of nonlinearly hardening materials for complex loading. Acta Mech 21(3):173–192

Krieg R (1975) A practical two surface plasticity theory. J Appl Mech 42(3):641–646

Bellotti R et al (1989) Interpretation of moduli from self-boring pressuremeter tests in sand. Géotechnique 39(2):269–292

Pestana JM, Whittle AJ (1995) Compression model for cohesionless soils. Géotechnique 45(4):611–631

Crouch RS, Wolf JP (1994) Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part II: calibration and simulations. Int J Numer Anal Methods Geomech 18(11):759–784

Uzuoka R, Borja RI (2012) Dynamics of unsaturated poroelastic solids at finite strain. Int J Numer Anal Methods Geomech 36(13):1535–1573

Richart FE, Hall JR, Woods RD (1970) Vibrations of soils and foundations

Russell AR (2004) Cavity expansion in unsaturated soils. University of New South Wales

Lobo-Guerrero S, Vallejo L (2005) DEM analysis of crushing around driven piles in granular materials. Géotechnique 55(8):617–623

Zhang C, Nguyen G, Einav I (2013) The end-bearing capacity of piles penetrating into crushable soils. Géotechnique 63(5):341–354

Chen L, Qiao L, Li Q (2019) Study on dynamic compaction characteristics of gravelly soils with crushing effect. Soil Dyn Earthq Eng 120:158–169

McDowell G, Nakata Y, Hyodo M (2002) On the plastic hardening of sand. Géotechnique 52(5):349–358

Al-Sharrad M, Gallipoli D, Wheeler S (2017) Experimental investigation of evolving anisotropy in unsaturated soils. Géotechnique 67(12):1033–1049

Sheng D, Fredlund DG, Gens A (2008) A new modelling approach for unsaturated soils using independent stress variables. Can Geotech J 45(4):511–534

Manzanal D, Merodo JAF, Pastor M (2011) Generalized plasticity state parameter-based model for saturated and unsaturated soils. Part 1: saturated state. Int J Numer Anal Methods Geomech 35(12):1347–1362

Russell AR, Khalili N (2004) A bounding surface plasticity model for sands exhibiting particle crushing. Can Geotech J 41(6):1179–1192

Lade PV, Yamamuro JA (1996) Undrained sand behavior in axisymmetric tests at high pressures. J Geotech Eng 122(2):120–129

Miura N, Murata H, Yasufuku N (1984) Stress-strain characteristics of sand in a particle-crushing region. Soils Found 24(1):77–89

Javanmardi Y et al (2018) A reference state curve to define the state of soils over a wide range of pressures and densities. Géotechnique 68(2):95–106

Sheng D, Yao Y, Carter JP (2008) A volume–stress model for sands under isotropic and critical stress states. Can Geotech J 45(11):1639–1645

Been K, Jefferies MG (1985) A state parameter for sands. Géotechnique 35(2):99–112

Wood DM, Belkheir K, Liu DF (1994) Strain softening and state parameter for sand modelling. Geotechnique 44(2):335–339

Nemat-Nasser S, Tobita Y (1982) Influence of fabric on liquefaction and densification potential of cohesionless sand. Mech Mater 1(1):43–62

Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T (1999) Computational geomechanics with special reference to earthquake engineering. Citeseer

Ghorbani J, Airey DW, El-Zein A (2018) Numerical framework for considering the dependency of SWCCs on volume changes and their hysteretic responses in modelling elasto-plastic response of unsaturated soils. Comput Methods Appl Mech Eng 336:80–110

Ghorbani J, Airey DW (2019) Some aspects of numerical modelling of hydraulic hysteresis of unsaturated soils. In: Hertz M (ed) Unsaturated soils: behavior, mechanics and conditions. Nova Science Publishers, New York

Fredlund DG, Xing A (1994) Equations for the soil-water characteristic curve. Can Geotech J 31(4):521–532

Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44(5):892–898

Manzanal D, Pastor M, Merodo JAF (2011) Generalized plasticity state parameter-based model for saturated and unsaturated soils. Part II: unsaturated soil modeling. Int J Numer Anal Methods Geomech 35(18):1899–1917

Sheng D, Sloan S, Yu H (2000) Aspects of finite element implementation of critical state models. Comput Mech 26(2):185–196

Josa A et al (1992) An elastoplastic model for partially saturated soils exhibiting a maximum of collapse. In: 3rd international conference on computational plasticity, Barcelona

Sharma RS (1998) Mechanical behaviour of unsaturated highly expansive clays. University of Oxford

Loret B, Khalili N (2002) An effective stress elastic–plastic model for unsaturated porous media. Mech Mater 34(2):97–116

Russell A, Khalili N (2006) A unified bounding surface plasticity model for unsaturated soils. Int J Numer Anal Methods Geomech 30(3):181–212

Alonso EE, Gens A, Josa A (1990) Constitutive model for partially saturated soils. Géotechnique 40(3):405–430

Pestana JM, Whittle AJ, Gens A (2002) Evaluation of a constitutive model for clays and sands: part II–clay behaviour. Int J Numer Anal Methods Geomech 26(11):1123–1146

Pradhan TB, Tatsuoka F, Sato Y (1989) Experimental stress-dilatancy relations of sand subjected to cyclic loading. Soils Found 29(1):45–64

Verdugo R, Ishihara K (1996) The steady state of sandy soils. Soils Found 36(2):81–91

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## Appendices

### Appendix 1: Additional details on the equations of SWCC

The equation of the SWCC is obtained from [81, 83]

Tables 3 and 4 show the constitutive model and the SWCC data, respectively. Note that a star symbol (*) in front of a number denotes that the value of this parameter is assumed as there was insufficient data to directly obtain the parameter. Also, the SWCC parameters given in Table 4 are obtained by taking Eq. (69), as \( S_{w}^{\alpha } \) in Eq. (39).

All the numerical examples in Sect. 11 have adopted the mechanical parameters of Toyoura sand, the SWCC parameters given in Table 5 and \( b_{1} \) and \( b_{2} \) have been set to 2.0 and 1.5, respectively.

In Table 6, the first label, L(oose) or D(ense), indicates the initial sand states and the second label, D(rained) or U(ndrained), indicates the type of test.

### Appendix 2: Isotropic hardening law

\( f\left( \xi \right) \) is a function of both suction and volume changes (due to the dependency of saturation degree on void ratio). After the expansion of \( f\left( \xi \right) \) in Eq. (9), we obtain

By assuming \( {\mathcal{C}} = - e\frac{\partial f\left( \xi \right)}{\partial e}\frac{1}{f\left( \xi \right)}\left( {1 - \delta_{p}^{\theta } } \right) \) and \( \kappa = \frac{{p^{{\prime }} }}{K}\frac{1 + e}{e} \), and after some manipulations, the following is obtained

By subtracting the elastic strain rate from the total strain rate, the plastic volumetric strain rate is obtained as follows

After some manipulations and the replacement of \( p^{{\prime }} \) by the size of the loading surface, we obtain

### Appendix 3: Transition mechanism based on the gradient of the loading surface

It is also possible to use a general form of the vector of the gradient of the yield surface, \( A^{\sigma } \) as follows

where the first and second components of the vector are the volumetric and deviatoric terms, respectively. It may be noted that \( f_{1} \) and \( f_{2} \) are the loading surfaces constructed based on the radial mapping and deviatoric rule, respectively. Note that when \( \zeta = 1 \) the radial mapping rule will be activated whereas for \( \zeta = 0 \), the deviatoric mapping rule will be the active mechanism.

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Ghorbani, J., Airey, D.W. Modelling stress-induced anisotropy in multi-phase granular soils.
*Comput Mech* **67**, 497–521 (2021). https://doi.org/10.1007/s00466-020-01945-8

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DOI: https://doi.org/10.1007/s00466-020-01945-8