Adaptation of granular solid hydrodynamics for modeling sand behavior

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Adaptation of Granular Solid Hydrodynamics for

Modeling Sand Behavior

by

Andriani-loanna Panagiotidou

MASSACHUSETTS INSTITUTE OF TEGHNOLOGY

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Diploma in Civil Engineering (2009), National Technical University of Athens

Master of Science in Civil and Environmental Engineering (2013), Massachusetts Institute of Technology

Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of

Doctor of Philosophy at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2017

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Adaptation of Granular Solid Hydrodynamics for Modeling

Sand Behavior

by

Andriani-Ioanna Panagiotidou

Submitted to the Department of Civil and Environmental Engineering on June 30, 2017, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

The development of constitutive models that can realistically represent the effective stress-strain-strength of the soil properties is essential for making accurate predictions using finite element analysis. Currently, most of the existing constitutive models are based on the framework of incrementally-linearized elasto-plasticity. However, most of these models do not typically consider energy conservation and are also phenomeno-logical. This means that they can only be used to predict the behavior/ loading conditions for which they have been developed and that they often employ artificial mathematical formulations.

This research proposes an improved constitutive model for sands based on the framework of Granular Solid Hydrodynamics [GSHJ. The GSH framework considers energy and momentum conservation simultaneously and, by combining them with thermodynamic considerations, develops constitutive relations for a given energy ex-pression. This thesis offers a detailed study of the element level behavior of the Tsinghua-Thermosoil model [TTSI (Zhang and Cheng, 2016) based on the GSH. Through this study, we identify and propose a series of modifications to the origi-nal formulation in order to improve predictions of well-established soil behavior.

The proposed formulation, MIT-GH, introduces a new expression of the free en-ergy and modifies the evolution laws and the steady state values for the internal variables. The model can successfully predict phenomena such as a unique compres-sion response at high confining pressures (Limiting Comprescompres-sion Curve [LCC]) and at large shear strain conditions (Critical State Line [CSL]), and a State Boundary Surface [SBS] that limits the peak shear resistance measured in drained shear tests. The LCC and CSL conditions are defined solely from the evolution of elastic strains while the SBS is defined from the free energy expression.

Finally, our work also offers a novel use of the "double" failure mechanism - in-herent in the GSH framework. Using these mechanisms, MIT-GH can model not only Critical State conditions but also localization phenomena. The proposed criterion for the localization is the maximum expected peak friction angle that a specimen can develop at different void ratios and stress levels.

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This study also includes a detailed parametric analysis of the model and a proposal for the calibration of the model. The proposed MIT-GH model should be considered as a first generation formulation based on the principles of granular solid hydrodynamics and how it ties to classic knowledge of soil behavior and prior elasto-plastic models. Further research is now needed to extend the framework to address more complex features of sand behavior including the cyclic response and liquefaction.

Thesis Supervisor: Andrew J. Whittle

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Acknowledgments

First and foremost, I would like to thank my advisor Professor Andrew J. Whittle for his guidance and support throughout these years. He gave me the opportunity to work on very interesting research projects and develop myself in more ways than one.

I would also like to thank my committee members: Professor Germaine,

Profes-sor Kausel and ProfesProfes-sor Kamrin for their help and significant contributions to this research work.

I am also extremely grateful to the Linde Presidential Fellowship at MIT, to the Foundation of Education and European Culture and to the Center for Environmental Sensing and Modeling (CENSAM Phase II) for their generous support of my PhD studies.

Finally, I would also to express my gratitude to my friends and family for their love and support that I have been so lucky to enjoy.

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List of Figures

2-1 Drained triaxial shear tests for dense and loose Sacramento River sand

(Dr = 38% and Dr = 100% respectively) (Lee and Seed, 1967), Dr

-relative density . . . . 45 2-2 Drained triaxial shear tests for dense and loose Sacramento River sand

(Dr = 38% and Dr = 100% respectively) (Lee and Seed, 1967), Dr

-relative density . . . . 46

2-3 Schematic figure of shear strength component for sands (After 1.322 O C W notes). . . . . 47 2-4 Definition of the state parameter, b (after Been and Jefferies (1985)) 48

2-5 Peak friction angle as a function of the state parameter, b (after Been and Jefferies (1985), modified for 1.322 notes) . . . . 49

2-6 Dependence of peak angles of shearing resistance on state parameter at failure for Sacramento River sand, 0: dense sand vo = 1.61 or eo = 0.61

and o: loose sand vo = 1.87 or eo = 0.87 (data from Lee and Seed

(1967), figure after Wood (1990)). . . . . 50 2-7 Empirical Relations at Peak Strength (Bolton, 1986) . . . . 51 2-8 CIUC tests on dense Toyoura sand (after Verdugo and Ishihara (1996)) 53 2-9 CIUC tests on medium-loose Toyoura sand (after Verdugo and Ishihara

(19 9 6 )) . . . . 54

2-10 Critical State of a variety of sands (after Castro et al. (1982)) . . . . 57

2-11 Steady state line for undrained compression tests on contractive sam-ples of Erksak 330/0.7 sand with initial states above the critical state line (after Been et al. (1991)) . . . . 58

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2-12 Curved of the critical state line at lower stress levels for Toyoura sand

in undrained experiments (after Verdugo and Ishihara (1996)) . . . . 58

2-13 Unique critical state line at lower stress levels for Toyoura sand drained

and undrained experiments (after Verdugo and Ishihara (1996)) . . . 59

2-14 Critical state line of Toyoura sand for both undrained and drained ex-periments in the double logarithmic space. Continuous lines represent prediction of MIT-Si for the CSL and the LCC (after Pestana et al. (2002 )) . . . . 60 2-15 Drained and Undrained triaxial tests on (data after Been et al. (1991) 62 2-16 Drained and Undrained triaxial tests on Berlin Sand (data after

Glase-napp (2002) and Becker (2002) and Figure after Nikolinakou et al. (2004) 64

2-17 Global and local evolution of the void ratio in loose and dense Hostun

RF sand specimens submitted to axisymmetric triaxial test under 60 kPa effective confining pressure (global e: open symbols, local e: solid symbols) (after Desrues et al. (1996)) . . . . 65 2-18 Dilatancy in recent and old shear bands: towards a limiting void ratio

(after Desrues (1991)) . . . . 66 2-19 (a) Shear band thickness dependence on particle size. (b) Shear

dis-placement of the shear band dependence on particle size (after Oda et al. (1997)) . . . . 67

2-20 Typical behavior of freshly deposited cohesionless soils in One Dimen-sional Compression: a) Density-Stress Relationship; b) Derived Incre-mental Stiffness Behavior (after Pestana (1994)) . . . . 70

2-21 Compression and shearing behavior of Dog's Bay carbonate sand (Coop,

1990) (a) one dimensional compression and critical states (b) definition

of Relative Breakage (Hardin, 1985) (c) Br measurements for Dog's Bay sand (after Coop and Lee (1993)) . . . . 71

2-22 Cubically packed array of spheres subjected to normal stress, -, and shear stress, T, that produce inter-particle contact forces N and T (after D obry et al. (1982)) . . . . 74

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2-23 Schematic of hardening laws in the HC model . . . . 77

2-24 Schematic of the yield, critical, dilatancy, and bounding lines in q, p space, (after Dafalias and Manzari (2004)) . . . . 79

3-1 Concept of Double Entropy. Dissipative processes produce either gran-ular entropy Sg, or directly thermal entropy S. Eventually, S9 is also converted to S. . . . . 84

3-2 The 'two-block' analog of the degrees of freedom in a general amor-phous system: (after Kamrin and Bouchbinder (2014)) . . . . 85 3-3 Scaling of pressures and stiffness with strain for Hertz contact and TTS

elastic strain energy models . . . . 87

3-4 Effect of hysteretic strain on strain accumulation due to cyclic loading and unloading (after Zhang and Cheng (2013a)) . . . . 93 3-5 Hydrostatic Compression Prediction of TTS model for Toyoura sand

with different initial densities. . . . . 97 3-6 (a) Effective stress paths computed for undrained triaxial compression

shear tests on Toyoura sand using the TTS model (parameters listed in Table 3.2) at selected initial void ratios, CSL-critical state line, SBS-state boundary surface. (b) Shear stress-strain response of the same tests. . . . . 9 9 3-7 Critical state line prediction of TTS for undrained loading for the

se-lected three void ratios. . . . 100

3-8 (a) Relationship between the elastic volumetric and deviatoric strain for undrained triaxial compression shear tests for selected initial void ratios; (b) Relationship between the hysteretic volumetric and devia-toric strain for the same tests. . . . . 101 3-9 (a) Typical TTS predictions of stress-strain response for drained

triax-ial compression shearing of Toyoura sand (b) Typical TTS predictions of void ratio vs. axial strain for the same tests (input parameters Table

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3-10 Typical TTS predictions in void ratio and mean effective stresses for

drained triaxial compression shearing of Toyoura sand in the double logarithmic space (input parameters Table 3.2). The location of the

CSL is also noted. . . . . 107

4-1 Location of State Boundary Surface in the strain invariant space (eqn. (4 .6 )). . . . . 1 18

4-2 Effect of c, and on State Boundary Surface in the strain invariant space. (eqn. (4.6)) . . . . 119

4-3 Predictions of the SBS and Peak Friction Angle and for three different void ratios. . . . . 120 4-4 Predictions of the Peak Friction Angle for void ratio, e = 0.7, for

different sets of c, and b. . . . . 121 4-5 Conceptual Evolution of distance parameter msBs in the elastic strain

invariant space. . . . . 127

4-6 Conceptual Evolution of distance parameter msBs for three drained tests. Two of them (solid and dotted lines) reach the steady state after compression and dilation respectively. Drained Lighted dotted line . . 128

4-7 Typical Results for of MIT-GH predictions during Hydrostatic Com-pression for three void ratios in the double logarithmic void ratio

-mean effective stress space. . . . . 133

4-8 Typical Results of MIT-GH predictions during Hydrostatic Compres-sion for three void ratios for Hysteretic Volumetric Strain Evolution. . 134 4-9 Proposed form for h as a function of void ratio . . . . 138

4-10 Typical Results of MIT-GH predictions for Critical State Elastic and Hysteretic Strain Invariants versus void ratio . . . . 140 4-11 Conceptual Locations of Hydrostatic Loading and Critical State Line 141 4-12 Predictions of MIT-GH for Critical State Friction Angle. Dotted line

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4-13 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at low stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 149 4-14 Typical Results of MIT-GH predictions during Drained Triaxial

Shear-ing for four void ratios at low stresses in the elastic strain invariants space. In this space we also note the locations of the State Boundary Surface and the Critical State. . . . . 150

4-15 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at low stresses presentShear-ing the deviatoric stress and axial strain response. . . . . 151

4-16 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at medium stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 152

4-17 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at medium stresses in the elastic strain invari-ants space... ... 153

4-18 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at medium stresses presentShear-ing the deviatoric stress and axial strain response. . . . . 154 4-19 Typical Results of MIT-GH predictions during Drained Triaxial

Shear-ing for four void ratios at high stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 155

4-20 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at high stresses in the elastic strain invariants space. ... ... 156

4-21 Typical Results of MIT-GH predictions during Drained Triaxial Shear-ing for four void ratios at medium stresses presentShear-ing the deviatoric stress and axial strain response. . . . . 157

4-22 State Boundary Space in Strains Invariant Space and Prediction of Localization . . . . 159

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4-23 Localization Front and Conceptual Space of Expected Behavior During Shearing . . . . 160

4-24 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at low stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 164 4-25 Typical Results of MIT-GH predictions during Undrained Triaxial

Shear-ing for four void ratios at low stresses in the elastic strain invariants space. In this space we also note the locations of the State Boundary Surface and the Critical State- same as the drained case. . . . . 165

4-26 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at low stresses presentShear-ing the deviatoric stress and axial strain response. . . . . 166

4-27 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at medium stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 167

4-28 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at medium stresses in the elastic strain invari-ants space... ... 168

4-30 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at high stresses in the double logarithmic space of void ratio and mean effective stress. . . . . 170

4-31 Typical Results of MIT-GH predictions during Undrained Triaxial Shear-ing for four void ratios at high stresses in the elastic strain invariants space. ... ... 171

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4-33 Locations of State Boundary Surface and Critical State in the strain invariant space along with maximum elastic strain path. . . . . 173

5-1 Parameters controlling Stress and Strain Behavior . . . . 176 5-2 Typical results of CID triaxial tests for quartz sand with shear wave

velocity measurements of small strain shear modulus, Gmax, through-out the test. (a) a/ = 100 kPa, (b) a' = 200 kPa, after Guadalupe et al. (2013) . . . . 181 5-3 Recovered volumetric Strain during hydrostatic unloading from LCC,

after Pestana (1994) ... ... 182

5-4 MIT-GH predictions during Hydrostatic Compression for three values of parameter hmax : (a) Evolution of the elastic volumetric strain versus total volumetric strain. (b) Evolution of the void ratio versus mean effective stress in a double logarithmic space. . . . . 184

5-5 _{MIT-GH predictions during Drained Shearing at low pressures (Po =} 500 kPa) for three values of parameter hmax of the evolutions of (a) the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. . . . . 185 5-6 _{MIT-GH predictions during Drained Shearing at low pressures (Po =}

500 kPa) for three values of parameter hmax of (a) the evolution of the

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 186

5-7 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter hmax. . . . . 188 5-8 MIT-GH predictions for the location of the LCCs and CSLs for three

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5-9 MIT-GH predictions for the Critical State Friction Angle for three values of parameter hmax. . . . . . . . . . . . . . . . . . . ... . . .190

5-10 LCC and steady state predictions for three sets of parameters hmax

and B0 . . . . . . .. . . 193

5-11 Drained Shearing for three sets of parameters hmax and BO at medium

stresses (p' = 500 kPa) . . . . 194

5-12 Drained Shearing for three sets of parameters hmax and BO at high

stresses (p' = 5 M Pa) . . . . 195 5-13 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter e*. . . . . 197

5-14 MIT-GH predictions for the location of the CSLs for three values of param eter e*. . . . . 198 5-15 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter m . . . . 199 5-16 MIT-GH predictions for the location of the CSLs for three values of

param eter m . . . . 200

5-17 MIT-GH predictions during Drained Shearing at low pressures (po =

500 kPa) for three values of parameter m,, of the evolutions of (a)

the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the deviatoric-elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain. and (d) the granular temperature T versus the axial strain in a semi-logarithmic space. 203 5-18 MIT-GH predictions during Drained Shearing at low pressures (pO =

500 kPa) for three values of parameter m, of (a) the evolution of the

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 204

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vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter m eS. . . . . 205

5-20 MIT-GH predictions for the location of the LCCs and CSLs for three

values of parameter mcs. . . . 206

5-21 MIT-GH predictions during Drained Shearing at low pressures (Po =

500 kPa) for three values of parameter mD of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 209

5-22 MIT-GH predictions during Drained Shearing at low pressures (Po = 500 kPa) for three values of parameter mD of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 210

vol-umetric strain and (b) the elastic deviatoric strain for three values of parameter mD . . . . . - -. -. -. . .. . . . . . . . . . . 211 5-24 MIT-GH predictions for the location of the Critical State Lines for

three values of parameter mD. The Limiting Compression Curve is not affected by that parameter. . . . . 212

5-25 MIT-GH predictions for the Critical State Friction Angle for three

values of parameter mD. The higher the value of mD, the lower the critical state friction angle. . . . . 213 5-26 MIT-GH predictions during Hydrostatic Compression for three values

of parameter mg of the evolution of (a) Granular Temperature versus volumetric stains (b) elastic volumetric strain versus volumetric strains. 215

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5-27 MIT-GH predictions during Drained Shearing at high pressures (PO 5000 kPa) for three values of parameter mg of the evolutions of (a) the

elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain, and (d) the granular

temperature T, versus the axial strain in a semi-logarithmic space. . . 216

5-28 MIT-GH predictions during Drained Shearing at high pressures (po =

5000 kPa) for three values of parameter mg of (a) the evolution of the

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 217

values of parameter B1. . . . 220

values of parameter BO. . . . . 222

5-31 MIT-GH predictions during Drained Shearing at low pressures (PO =

500 kPa) for three values of parameter of (a) the evolution of the void

ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 226

5-32 MIT-GH predictions during Drained Shearing at low pressures (Po =

500 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the

elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 227

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5-33 MIT-GH predictions during Undrained Shearing at low pressures (Po = 500 kPa) for three values of parameter of the evolutions of (a) the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . 228 5-34 MIT-GH predictions during Undrained Shearing at low pressures (po =

500 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 229

5-35 MIT-GH predictions for the location of the CSLs for three values of

param eter . . . 230

5-36 Location of State Boundary Surface in the stress invariant space for

three values of parameter , for a constant void ratio, e = 0.7. . ... 231

5-37 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter , for a constant void ratio, e = 0.7. . . . . 232

5-38 MIT-GH predictions for the Critical State Friction Angle for three

values of param eter . . . . 233 5-39 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of parameter . affects only minimally the elastic strains through the distance parameter mSBS . . . . . . . . . . .. . .. . . . . 234 5-40 Location of State Boundary Surface in the stress invariant space for

three values of parameter c,, for a constant void ratio, e = 0.7. . . . . 236 5-41 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter c., for a constant void ratio, e = 0.7. Pa-rameter cs controls the slope of the plotted line in the low stresses. . . 237

5-42 MIT-GH predictions for the Critical State Friction Angle for three

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5-43 Location of State Boundary Surface in the stress invariant space for three values of parameter b, for a constant void ratio, e = 0.7. . . . . 240 5-44 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter b, for a constant void ratio, e = 0.7. . . . . 241 5-45 MIT-GH predictions for the Critical State Friction Angle for three

values of parameter b. The higher the value of b, the more constant the critical state friction angle along the void ratio. . . . . 242 5-46 Major Effect of parameter c,; Comparison of Predicted Friction

An-gle for Shear in Triaxial Compression. eO,3 is the void ratio at o =

29.42 kPa as reported by Fukushima and Tatsuoka (1984) . . . . 244 5-47 Major Effect of parameter ; Comparison of Predicted Friction

An-gle for Shear in Triaxial Compression. eO,3 is the void ratio at a' =

29.42 kPa as reported by Fukushima and Tatsuoka (1984). . . . . 245 5-48 Major Effect of parameter b; Comparison of Predicted Friction

An-gle for Shear in Triaxial Compression. eO,₃is the void ratio at a' = 29.42 kPa as reported by Fukushima and Tatsuoka (1984). . . . . 246 5-49 Effect of m and e* on the CSL at high void ratio . . . . 247

5-50 Effect of mg on predicted stress-strain response for CIDC tests on dense

Toyoura sand . . . . 249

5-51 Effect of mD on predicted stress-strain response for CIDC tests on

dense Toyoura sand . . . 250

5-52 LCC and CSL locations . . . . 251 5-53 Effect of hmax on predicted stress-strain response for CIDC tests on

dense Toyoura sand . . . . 252

5-54 Predictions of MIT-GH for Critical State Friction Angle . . . . 253 5-55 Comparison of predicted response for CIDC tests on dense Toyoura sand255 5-56 Comparison of predicted response for CIUC tests on dense Toyoura sand256

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B-1 Comparison of location of LCC with and without the taking into ac-count the effect of elastic volumetric strains in the evolution of dry den sity . . . 274

E-1 MIT-GH predictions during Hydrostatic Compression for three values

of parameter hmax : (a) Evolution of the Elastic Volumetric Strain versus total volumetric strain. (b) Evolution of the void ratio versus mean effective stress in a double logarithmic space. . . . 314

E-2 MIT-GH predictions during Drained Shearing at low pressures (PO = 500 kPa) for three values of parameter hmax of the evolutions of (a) the

elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 315

E-3 MIT-GH predictions during Drained Shearing at low pressures (Po 500 kPa) for three values of parameter hmax of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 316

E-4 MIT-GH predictions during Drained Shearing at high pressures (PO =

5000 kPa) for three values of parameter hmax of the evolutions of (a)

the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the deviatoric-elastic strain invariants space, and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 317

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E-5 MIT-GH predictions during Drained Shearing at high pressures (po =

5000 kPa) for three values of parameter hmax of (a) the evolution of the

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 318 E-6 MIT-GH predictions during Undrained Shearing at low pressures (PO =

500 kPa) for three values of parameter hmax of the evolutions of (a)

the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the deviatoric-elastic strain invariants space, and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 319 E-7 MIT-GH predictions during Undrained Shearing at low pressures (Po =

500 kPa) for three values of parameter hmax of the evolutions of (a)

the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . . 320 E-8 MIT-GH predictions during Undrained Shearing at high pressures (Po =

the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the deviatoric-elastic strain invariants space, and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 321 E-9 MIT-GH predictions during Undrained Shearing at high pressures (po =

the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . . 322 E-10 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of parameter hmax... . . . ... 323

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E-11 MIT-GH predictions for the location of the LCCs and CSLs for three

values of parameter hmax. . . . . 324

E-12 MIT-GH predictions for the Critical State Friction Angle for three

values of parameter hmax. The higher the value of hmax, the lower the critical state friction angle. . . . 325

E-13 MIT-GH predictions for the critical state values of (a) the elastic

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter e*. . . . 327 E-14 MIT-GH predictions for the location of the CSLs for three values of

param eter e*. . . . 328

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter m . . . 329

E-16 MIT-GH predictions for the location of the CSLs for three values of

param eter m . . . . 330 E-17 MIT-GH predictions during Hydrostatic Compression for three values

of parameter mcs : (a) Evolution of the Elastic Volumetric Strain ver-sus total volumetric strain (b) evolution of the void ratio verver-sus mean effective stress in a double logarithmic space and (c) the granular tem-perature Tg versus the total volumetric strain in a semi-logarithmic space. ... ... 333 E-18 MIT-GH predictions during Drained Shearing at low pressures (PO =

500 kPa) for three values of parameter me, of the evolutions of (a)

the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the deviatoric-elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain. and (d) the granular

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E-19 MIT-GH predictions during Drained Shearing at low pressures (PO = 500 kPa) for three values of parameter m,, of (a) the evolution of the

E-20 MIT-GH predictions during Drained Shearing at high pressures (po = 5000 kPa) for three values of parameter m, of the evolutions of (a) the

elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain, and (d) the granular

temperature T versus the axial strain in a semi-logarithmic space. . . 336

E-21 MIT-GH predictions during Drained Shearing at high pressures (Po =

5000 kPa) for three values of parameter m, of (a) the evolution of the

E-22 MIT-GH predictions during Undrained Shearing at low pressures (Po = 500 kPa) for three values of parameter me, of the evolutions of (a) the

elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. . . . . 338

E-23 MIT-GH predictions during Undrained Shearing at low pressures (PO = 500 kPa) for three values of parameter m,, of the evolutions of (a) the

deviatoric stresses versus axial strain, (b) the stress paths in the stress invariant space and (c) the granular temperature T versus the axial strain in a semi-logarithmic space. . . . 339

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E-24 MIT-GH predictions during Undrained Shearing at high pressures (PO =

5000 kPa) for three values of parameter m, of the evolutions of (a) the

elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. . . . 340

E-25 MIT-GH predictions during Undrained Shearing at high pressures (Po =

5000 kPa) for three values of parameter mc, of the evolutions of (a)

the deviatoric stresses versus axial strain, (b) the stress paths in the stress invariant space and (c) the granular temperature Tg versus the axial strain in a semi-logarithmic space. . . . 341

vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter m es. . . . 343

E-27 MIT-GH predictions for the location of the LCCs and CSLs for three

values of parameter mc,. Parameter m,, plays a very significant role in the determination of the distance between the LCC and the CSL line. It is clear from the figure that mcs affect the CSL more significantly than the LCC... ... 344

values of parameter mc,. The higher the value of m, the lower the critical state friction angle. . . . . 345

E-29 (a) MIT-GH predictions during Hydrostatic Compression for three

val-ues of parameter m, in the double logarithmic void ratio - mean effec-tive stress space. (b) MIT-GH predictions during Hydrostatic Com-pression for three values of parameter m, for Elastic Volumetric Strain E volution . . . . 347

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E-30 MIT-GH predictions during Undrained Shearing at low pressures (PO = 500 kPa) for three values of parameter m, of the evolutions of (a) the

elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 348

E-31 MIT-GH predictions during Undrained Shearing at low pressures (PO = 500 kPa) for three values of parameter m, of the evolutions of (a) the

deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . . 349

E-32 MIT-GH predictions during Drained Shearing at low pressures (po =

500 kPa) for three values of parameter mD of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 351 E-33 MIT-GH predictions during Drained Shearing at low pressures (PO =

500 kPa) for three values of parameter mD of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 352

E-34 MIT-GH predictions during Drained Shearing at high pressures (PO = 5000 kPa) for three values of parameter mD of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 353

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E-35 MIT-GH predictions during Drained Shearing at high pressures (PO = 5000 kPa) for three values of parameter mD of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 354

E-36 MIT-GH predictions during Undrained Shearing at low pressures (Po =

500 kPa) for three values of parameter mD of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 355

E-37 MIT-GH predictions during Undrained Shearing at low pressures (Po 500 kPa) for three values of parameter mD of the evolutions of (a) the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space . . . 356

E-38 MIT-GH predictions during Undrained Shearing at high pressures (PO = 5000 kPa) for three values of parameter mD of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 357 E-39 MIT-GH predictions during Undrained Shearing at high pressures (PO =

5000 kPa) for three values of parameter mD of the evolutions of (a) the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . . 358

E-40 MIT-GH predictions for the critical state values of (a) the elastic vol-umetric strain and (b) the elastic deviatoric strain for three values of parameter mD . . . . . .

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E-41 MIT-GH predictions for the location of the Critical State Lines for three values of parameter mD .. . . . . . . ... 361

E-42 MIT-GH predictions for the Critical State Friction Angle for three

values of parameter mD . . . . . . . ... 362

E-43 MIT-GH predictions during Hydrostatic Compression for three values of parameter mg of the evolution of (a) Granular Temperature versus volumetric stains (b) elastic volumetric strain versus volumetric strains. 364 E-44 MIT-GH predictions during Drained Shearing at low pressures (Po =

500 kPa) for three values of parameter mg of the evolutions of (a) the

elastic deviatoric strain versus axial strain,b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain. and (d) the granular temperature T versus the axial strain in a semi-logarithmic space. . . 365

E-45 MIT-GH predictions during Drained Shearing at low pressures (PO =

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 366

E-46 MIT-GH predictions during Drained Shearing at high pressures (po =

5000 kPa) for three values of parameter mg of the evolutions of (a) the

elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space, (c) the elastic deviatoric strain versus the axial strain, and (d) the granular temperature T9. versus the axial strain in a semi-logarithmic space. . . 367

E-47 MIT-GH predictions during Drained Shearing at high pressures (po =

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . . 368

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E-48 MIT-GH predictions for the location of the LCCs and CSLs for three values of parameter B1. . . 370

E-49 MIT-GH predictions for the location of the LCCs and CSLs for three values of parameter BO. . . . 371

E-50 MIT-GH predictions during Drained Shearing at low pressures (Po = 500 kPa) for three values of parameter of (a) the evolution of the void

ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 374

E-51 MIT-GH predictions during Drained Shearing at low pressures (PO = 500 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, (b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 375

E-52 MIT-GH predictions during Drained Shearing at high pressures (Po = 5000 kPa) for three values of parameter of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 376

E-53 MIT-GH predictions during Drained Shearing at high pressures (pO = 5000 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . 377

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E-54 MIT-GH predictions during Undrained Shearing at low pressures (po =

500 kPa) for three values of parameter of the evolutions of (a) the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space . . . . 378 E-55 MIT-GH predictions during Undrained Shearing at low pressures (PO =

500 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 379 E-56 MIT-GH predictions during Undrained Shearing at high pressures (po

=-5000 kPa) for three values of parameter of the evolutions of (a) the deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space. . . . . 380 E-57 MIT-GH predictions during Undrained Shearing at high pressures (po =

5000 kPa) for three values of parameter of the evolutions of (a) the elastic deviatoric strain versus axial strain, b) elastic deviatoric-elastic volumetric strain path in the elastic strain invariants space and (c) the elastic deviatoric strain versus the axial strain. In this space we also note the locations of the State Boundary Surface and the ending of the test at the Critical State. . . . . 381 E-58 MIT-GH predictions for the location of the CSLs for three values of

param eter .. . . . . 384

E-59 Location of State Boundary Surface in the stress invariant space for

three values of parameter , for a constant void ratio, e = 0.7. . . . . 385

E-60 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter , for a constant void ratio, e = 0.7. . . . . 386 E-61 MIT-GH predictions for the Critical State Friction Angle for three

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vol-umetric strain and (b) the elastic deviatoric strain for three values of param eter . . . . 388 E-63 MIT-GH predictions during Drained Shearing at low pressures (Po =

500 kPa) for three values of parameter hma, of (a) the evolution of the

void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 389 E-64 MIT-GH predictions during Drained Shearing at high pressures (Po =

5000 kPa) for three values of parameter hmax of (a) the evolution of the void ratio versus axial strain and (b) the void ratio and mean effective stress evolution in a double logarithmic space, and (c) the the evolution of the deviatoric stress versus axial strain. . . . 390

E-65 MIT-GH predictions during Undrained Shearing at low pressures (PO = 500 kPa) for three values of parameter c, of the evolutions of (a) the

deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space . . . 391

E-66 MIT-GH predictions during Undrained Shearing at high pressures (Po = 5000 kPa) for three values of parameter c, of the evolutions of (a) the

deviatoric stresses versus axial strain and (b) the stress paths in the stress invariant space . . . 391

three values of parameter c5, for a constant void ratio, e = 0.7.The higher the value of c5, the higher deviatoric stress for a given mean effective stress. _{. . . 392}

E-68 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter cS, for a constant void ratio, e = 0.7.

Param-eter cs controls the slope of the plotted line in the low stresses. The higher the value, -the faster the slope of peak friction angle approaches a vertical line. . . . . 393

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values of parameter c,. The higher the value of c,, the higher the critical state friction angle. . . . . 394

three values of parameter b, for a constant void ratio, e = 0.7. . . .. 396 E-71 Predicted Peak Friction Angle for Shear in Triaxial Compression for

three values of parameter b, for a constant void ratio, e = 0.7. . . . . 397 E-72 MIT-GH predictions for the Critical State Friction Angle for three

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List of Tables

3.1 TTS state parameters . . . . 109

3.2 TTS model parameters . . . 110

4.1 MIT-GH state parameters . . . . 112 4.2 MIT-GH model parameters . . . . 113 5.1 MIT-GH model parameters used as a base case scenario for the

para-m etric investigation . . . . 177 5.2 MIT-GH model parameters for Toyoura sand . . . . 258

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Chapter 1 Introduction

The development of constitutive models that can realistically represent the effective stress-strain-strength of the soil properties is essential for making accurate predictions using finite element analysis. Currently, most of the existing constitutive models are based on the framework of incrementally-linearized, elasto-plasticity such as the Modified Cam-Clay model (MCC, Roscoe and Burland (1968)), (Mroz (1967); Prevost

(1977)), Dafalias and Popov (1977), MIT-E3 (Whittle and Kavvadas, 1994), MIT-Si

model (Pestana, 1994), DM2004 (Dafalias and Manzari, 2004).

Such models generally require four basic components: 1) a description of non-linear, pressure dependent elastic deformations; 2) a yield function that describes the domain of elastic behavior and the loading criterion; 3) hardening laws that describe changes in the size, rotation, shape etc. of the yield locus as the plastic deformations occur; and 4) flow rules that relate increments of plastic strain to the stress state and internal state variables.

Many of these models have very extensive capabilities and have been verified against a large number of experimental data, both monotonic and cyclic. For example MIT-Si (Pestana, 1994) is capable of predicting rate independent effective stress behavior of uncemented sands and clays over a wide range of void ratios and stresses. However most of these models do not typically consider energy conservation (e.g. Zytynski et al. (1978); Collins and Kelly (2002); Houlsby et al. (2005)). The majority of the aforementioned models are also phenomenological, which in principle means

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that they can only be used to predict the behavior/loading conditions for which they have been developed and that they often employ artificial mathematical formula-tions in order to predict more complex phenomena such as liquefaction (Dafalias and Manzari, 2004).

Additionally, it is difficult to expand their predictive capabilities without major revision of the models' framework and introduction of new state variables and material

constants, all of which, in general, require elaborate calibration processes.

It should be pointed out that the mathematical formulation of the more advanced models (Whittle and Kavvadas, 1994; Pestana, 1994; Dafalias and Manzari, 2004)) is in general more complex than the Granular Solid Hydrodynamics (GSH) mathemat-ical formulation, that will be employed in the current research.

This research aims to develop an improved constitutive model for sands based on the framework of GSH. The hydrodynamic theory, pioneered by Landau and Lifshitz

(1987) and Khalatnikov (1965), considers energy and momentum conservation

simul-taneously and combining them with thermodynamic considerations, develops consti-tutive relations for a given energy expression. If the energy expression is known, the inclusion of energy conservation provides the essential constraint that produces unique constitutive relations. The selection of the scalar energy expression is very important and along with the evolution laws of the internal variables can be sufficient to account for material-specific differences.

The GSH formulation is indeed outside of our classical theoretical constitutive modeling training, but it is comprised of two basic building/mathematical blocs, which once mastered can have significant modeling advantages. The author be-lieves it will be possible to achieve comparable predictive capabilities to the current incrementally-linearized, elasto-plastic models using the GSH formulation, achieving closer connection to the particulate (micro-scale) properties of granular soils.

The proposed formulation, MIT-GH, significantly extends the predictive capabil-ities of its predecessor model, TTS (Zhang and Cheng, 2013a), which is based on the theory of GSH, pioneered by Jiang and Liu (2003).This thesis includes also an extensive numerical study at the element-level using the TTS model simulating the

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compression behavior of sands, and drained and undrained conventional triaxial tests. Using the insights gained by this study, this thesis proceeds to modify the free energy expression and the evolution laws of the internal variables. The model introduced here contrary to its predecessor, is able to more accurately predict the monotonic sand behavior.

" It reaches a unique critical state line for both drained and undrained shearing; " It predicts a critical state line that is shallower in the high void ratio range; " It allows better dilation or contraction to reach critical state in drained shearing; " It predicts peak strength conditions at maximum dilation rate;

* It predicts the effect of localization.

Another important advantage of the GSH framework is the ability to derive an-alytical expressions for the "limiting"conditions, namely the Limiting Compression Curve, Critical State Line, State Boundary Surface, critical state friction angle and peak friction angle. This thesis includes systematic derivation of the above expres-sions and suggests an informed calibration procedure. It should be noted, that the above states are not imposed, as in most classical elasto-plastic models, but rather reached organically. They are achieved as a result of the evolution of the internal variables.

Consequently, the analytical expressions are complicated and demonstrate con-voluted dependences on the material parameters. In the current formulation, a pa-rameter does not control a feature of the soil behavior singlehandedly. This is, most probably, the biggest disadvantage of the model in terms of ease of calibration and therefore potential usability. This disadvantage is a direct effect of the framework's main advantage. The constitutive relations include only the free energy and the evo-lution laws of the state variables. MIT-GH comprises of 8 in total equations, that can describe the total monotonic behavior in sands. Inclusion of more complex phenom-ena would not lead to an increased number of governing equations, but rather more