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2014-04-28

An Effective Stress Equation for Unsaturated

Granular Media in Pendular Regime

Khosravani, Sarah

Khosravani, S. (2014). An Effective Stress Equation for Unsaturated Granular Media in Pendular

Regime (Unpublished master's thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/24849

http://hdl.handle.net/11023/1443

master thesis

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UNIVERSITY OF CALGARY

An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime

by

Sarah Khosravani

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF CIVIL ENGINEERING

CALGARY, ALBERTA

April, 2014

c© Sarah Khosravani 2014

Abstract

The mechanical behaviour of a wet granular material is investigated through a microme-

chanical analysis of force transport between interacting particles with a given packing and

distribution of capillary liquid bridges. A single effective stress tensor, characterizing the ten-

sorial contribution of the matric suction and encapsulating evolving liquid bridges, packing,

interfaces, and water saturation, is derived micromechanically.

The physical significance of the effective stress parameter (χ) as originally introduced

in Bishop’s equation is examined and it turns out that Bishop’s equation is incomplete.

More interestingly, an additional parameter that accounts for surface tension forces arising

from the so-called contractile skin emerges in the newly proposed effective stress equation.

Therefore, a so-called capillary stress is introduced which is shown to have two contributions:

one emanating from suction between particles due to air-water pressure difference, and the

second arising from surface tension forces along the contours between particles and water

menisci.

It turns out that the capillary stress is anisotropic in nature as dictated by the spatial

distribution of water menisci, particle packing and degree of saturation, and thus engenders

a meniscus based shear strength that increases with the anisotropy of the particle packing

and the degree of saturation. The newly proposed effective stress equation is analyzed with

respect to packing, liquid bridge distribution and strength issues. Finally, discrete element

modelling is used to verify the micromechanical aspects of the proposed effective stress

equation.

ii

Acknowledgments

First of all, I am deeply indebted to my supervisor, Dr. Richard Wan, for his support,

encouragement and constant guidance during my Master’s degree program. It was an honour

for me to be a member of his research group, and I will be for ever grateful to Dr. Wan

for giving me the opportunity to undertake graduate studies under his supervision and

introducing me to deductive reasoning rather than inductive reasoning.

I also would like to express my deepest gratitude to Dr. Bart Harthong and Mr. Mehdi

Pouragha for their constructive comments and great help during my master’s thesis work.

I am thankful to the Department of Civil Engineering and the Faculty of Graduate

Studies at the University of Calgary for their financial assistance through teaching assis-

tantships. This work was supported by the Natural Science and Engineering Research

Council of Canada throughout my Master’s Program.

Last but not least, I would like to address my sincere gratitude to Dr. Ron Wong, Dr.

Jocelyn Grozic, Dr. Jeffrey Priest and Dr. Marcelo Epstein for accepting the favour of being

in my examination committee.

iii

Dedication

I dedicate this thesis to my parents, for their unconditional love and support!

iv

Table of Contents

1Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Capillary Effect and Matric Suction . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Soil water characteristic curve . . . . . . . . . . . . . . . . . . . . . . 132.3 Experimental Observations on Unsaturated Soil Behaviours . . . . . . . . . . 18

2.3.1 Shear and tensile strengths of unsaturated soils . . . . . . . . . . . . 182.3.2 Collapse behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Studies on Effective Stress of Unsaturated Soils - Existing Frameworks . . . 252.4.1 Phenomenological studies (Macroscale studies) . . . . . . . . . . . . . 26

2.4.1.1 Single effective stress approach . . . . . . . . . . . . . . . . 262.4.1.2 Independent stress state variables approach . . . . . . . . . 30

2.4.2 Micromechanical studies . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 MICROMECHANICS OF EFFECTIVE STRESS INMULTIPHASIC GRAN-

ULAR MEDIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Force Transport in Dry Granular Media . . . . . . . . . . . . . . . . . . . . . 443.3 Force Transport in Saturated Granular Media . . . . . . . . . . . . . . . . . 47

3.3.1 Negligible contact area - rigid particles . . . . . . . . . . . . . . . . . 483.3.2 Finite contact area - compressible particles . . . . . . . . . . . . . . . 493.3.3 Effective stress in a fully saturated idealized compressible particle

packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Force Transport in Unsaturated Granular Media . . . . . . . . . . . . . . . . 56

3.4.1 Effective stress parameters for idealized packing . . . . . . . . . . . . 623.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 COMPUTATION OF CAPILLARY STRESSES IN IDEALIZED GRANU-

LAR PACKINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Idealized Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Simple cubic packing (SCP) . . . . . . . . . . . . . . . . . . . . . . . 70

v

4.2.2 Body-centered cubic packing (BCC) . . . . . . . . . . . . . . . . . . . 714.2.3 Cubic Close Packing or Face Centered Packing (CCP or FCP) . . . . 73

4.3 Theoretical SWCC for Regular Packing in Pendular Regime . . . . . . . . . 754.4 Effective Stress Parameters and Capillary Stress in Regular Packing . . . . . 80

4.4.1 Isotropic packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4.1.1 Effective stress parameters and capillary stresses in SCP and

FCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4.1.2 Isotropic tensile strength in comparison with experimental

results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4.2 Anisotropic packings . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.2.1 Evolution of capillary stress in BCC packing-anisotropy aspects 904.4.2.2 Evolution of degree of anisotropy - link to strength issues . . 93

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 VALIDATION OF THE PROPOSED EQUATION USING DEM SIMULA-

TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Triaxial Tests Simulation at Various Controlled Matric Suctions . . . . . . . 98

5.2.1 Brief review on DEM modelling in unsaturated media . . . . . . . . . 985.2.2 DEM sample description . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.3 DEM triaxial test procedure and results . . . . . . . . . . . . . . . . 1035.2.4 Validation of the proposed effective stress equation with DEM simula-

tion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3 Validation of the proposed effective stress equation using data from literature 1115.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . . 1166.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A Toroidal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

vi

List of Tables

2.1 Review of the conventional modelling approaches in unsaturated soil mechan-ics (Buscarnera, 2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Properties of BCC packings with various l′ . . . . . . . . . . . . . . . . . . . 724.2 SWCC calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 χij calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Bij calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5 Direct tensile test results of clean F-75 sand (Kim, 2001) . . . . . . . . . . . 89

5.1 Simplified steps of DEM modeling of unsaturated granular media . . . . . . 1015.2 DEM sample input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3 Shear strengths of samples with various matric suctions, DEM results . . . . 1055.4 DEM sample properties (Shamy & Groger, 2008) . . . . . . . . . . . . . . . 112

vii

List of Figures and Illustrations

2.1 Illustration of surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Water in capillary tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Free body diagram of forces acting on air-water interface in a capillary tube 92.4 Curved liquid and gas interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Conceptual demonstration of unsaturated sample in different regimes(Lu and

Likos, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Conventional soil water characteristic curve for sand and silt(Lu and Likos,

2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 Demonstration of the ink-bottle effect during:(a)drying process and (b)wetting

process (Marshall et al., 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Theoretical presentation of soil-water characteristic curve of an unsaturated

sample in different regimes (Lu et al., 2007) . . . . . . . . . . . . . . . . . . 162.9 General representation of shear strength in unsaturated samples (Ho and Fred-

lund, 1982) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Yield locus of glass beads R=46 micron (Pierrat et al., 1998) . . . . . . . . . 202.11 Yield locus of glass beads R=90 micron (Pierrat et al., 1998) . . . . . . . . . 202.12 Direct shear test results on cohesionless sands (Donald I., 1956) . . . . . . . 212.13 Tensile strength versus water content (F-75-C),(Kim, 2001) . . . . . . . . . . 232.14 Tensile strength versus water content,(Kim, 2001) . . . . . . . . . . . . . . . 232.15 One-dimensional compression and subsequent soaking tests under constant

void ratio or applied pressure (Jennings and Burland, 1962) . . . . . . . . . 252.16 Effective stress coefficient for unsaturated soil based on experimental results 282.17 Axis translation method in measuring matric suction in laboratory (Hilf, 1956) 312.18 Dimensionless liquid bridge volume versus dimensionless suction (Molenkamp

and Nazemi,2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.19 General scheme of homogenization technique (Oda and Iwashita, 1999) . . . 382.20 Mobilized friction angle in pyramidal packing of various heights and inter-

particle friction angles. Negative inter-particle friction angle represents verti-cal extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.21 Domain of accessible geometrical states based on harmonic representation ofgranular media (Radjai, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Cauchy’s stress in a closed domain . . . . . . . . . . . . . . . . . . . . . . . 443.2 Assembly of dry granular media . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Branch vector between pair of particles . . . . . . . . . . . . . . . . . . . . . 473.4 Free body diagram of inter-particle forces in saturated media . . . . . . . . . 483.5 Free body diagram of saturated media with compressible particles . . . . . . 503.6 Particle in contact with neighboring particles . . . . . . . . . . . . . . . . . . 523.7 Local coordinates on the center of each particle . . . . . . . . . . . . . . . . 523.8 Spherical REV and global coordination system (Quadfel and Rothenburg,2001) 543.9 Schematic anisotropic force distribution in polar system, an = 0.5 & βn = π/6 553.10 Unsaturated media as a three phase system in pendular state . . . . . . . . . 57

viii

3.11 Free body diagram of inter-particle forces . . . . . . . . . . . . . . . . . . . . 583.12 Unequal hydrostatic forces around the particle . . . . . . . . . . . . . . . . . 593.13 Unequal hydrostatic pressure on the air/water interface . . . . . . . . . . . . 603.14 Traction forces between a pair of spherical particles . . . . . . . . . . . . . . 613.15 Concave liquid bridge geometry between a pair of uni-size particles . . . . . 623.16 Dimensionless liquid volume Vrel as a function of the half filling angle α

(Megias-Alguacil and Gaucker, 2009) . . . . . . . . . . . . . . . . . . . . . . 633.17 Positive and negative matric suction zones as a function of the half filling and

wetting angles (Lu and Likos, 2004) . . . . . . . . . . . . . . . . . . . . . . . 643.18 Local coordinates illustration for each liquid bridge . . . . . . . . . . . . . . 65

4.1 Illustration of simple cubic packing (SCP) . . . . . . . . . . . . . . . . . . . 704.2 Illustration of body-Centered Cubic Packing (BCC) . . . . . . . . . . . . . . 714.3 Separation distance between particles , H (Pietsch, 1968) . . . . . . . . . . . 724.4 Arrangement of BCC packing unit cells in 3D space . . . . . . . . . . . . . . 734.5 Illustration of face centered packing (FCP) . . . . . . . . . . . . . . . . . . . 744.6 Porosity of different regular spherical packing . . . . . . . . . . . . . . . . . 744.7 SWCC of SCP and FCP as a function of particle size, H = θ = 0 . . . . . . 754.8 Effect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and

(b) Dense packing (FCP), H = 0 . . . . . . . . . . . . . . . . . . . . . . . . 784.9 Effect of separation distance on SWCC for (a) Loose packing (SCP), and (b)

Dense packing (FCP), θ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.10 The resulting isotropic effective stress coefficient χij while θ = 0(a) Loose

packing (SCP) and (b) Dense packing (FCP) . . . . . . . . . . . . . . . . . . 834.11 Computed relationships between degree of saturation and effective stress pa-

rameter for various packings . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.12 The capillary stress induced by (a) Suction forces, (b) Surface tension forces

in a loose packing (SCP)- R =0.1 mm, H = 0 . . . . . . . . . . . . . . . . . 864.13 The capillary stress induced by (a) Suction forces, (b) Surface tension forces

in a dense packing (FCP)- R =0.1 mm, H = 0 . . . . . . . . . . . . . . . . . 874.14 The total capillary stress in (a) Loose packing (SCP) (b) Dense packing

(FCP)- R =0.1 mm, θ = 30◦ and H = 0 . . . . . . . . . . . . . . . . . . . . 884.15 Compression between measured and predicted tensile strength . . . . . . . . 904.16 Polar plot of anisotropic capillary stresses for various saturation degree, H =

θ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.17 Principal capillary stresses with various contributions in axial and lateral di-

rections, H = θ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.18 Polar plot of anisotropic capillary stresses for various wetting angles, H = 0 . 934.19 Meniscus-based anisotropy as a function of saturation for various anisotropic

BCC packings, θ = H = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 DEM sample consisting of 10,000 mono-sized spherical particles . . . . . . . 1025.2 SWCC of the DEM sample, R=0.024 mm . . . . . . . . . . . . . . . . . . . 1045.3 (a) Deviatoric stress and (b) Volumetric strain versus axial strain for DEM

samples with lateral pressure of 750 Pa . . . . . . . . . . . . . . . . . . . . . 106

ix

5.4 Failure envelope of DEM samples considering the peak shear strength as thefailure point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Anisotropic capillary stress in unsaturated DEM samples,axial strain=20% . 1085.6 Strength of wet granular material based on (a) net stress (q,p) and (b) effective

stress (q′,p′) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.7 Shear strength response based on effective stresses for a confining pressure of

750 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.8 Anisotropy changes for both effective and capillary stresses for a confining

pressure of 750 Pa at matric suction of 30 kPa . . . . . . . . . . . . . . . . 1115.9 Comparisons between SWCC of selected SCP sample and simulated DEM

samples by Shamy and Groger, 2008 . . . . . . . . . . . . . . . . . . . . . . 1135.10 (a) Shear strength response based on net stresses (adopted from Shamy and

Groger, 2008). (b) Shear strength response based on calculated effective stresses114

6.1 Homogenization method in order to develop a constitutive model in unsatu-rated media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2 Micro-CT scan of water menisci of Toyoura sand, courtesy of Profs. Oka andKimoto, Kyoto University, Japan . . . . . . . . . . . . . . . . . . . . . . . . 119

A.1 Concave liquid bridge geometry between a pair of uni-size particles . . . . . 130

x

List of Symbols, Abbreviations and Nomenclature

Roman letter symbols

a anisotropy factor

an anisotropy factor of normal contact forces

aLB anisotropy factor of liquid bridges

aψ anisotropy factor of capillary stress

bi,j ,bi,j body forces, resultant body forces

Bij capillary stress due to surface tensions

c intrinsic cohesion

ca apparent cohesion

cu coefficient of uniformity

e void ratio

ei,j unit vector of surface tension forces

E Young’s modulus of elasticity

f mean value of contact forces

fi,j inter-particle contact forces

f cap inter-particle capillary forces

Fij fabric tensor of contacts

Fx,y,z eigenvalues of fabric tensor

FLBij fabric tensor of liquid bridges

h height parameter of pyramidal packing

hc critical height of water in capillary tube

hw,d capillary rise height during wetting,drying

H separation distance,surface roughness

H dimensionless separation distance

xi

k curve fitting parameter

l, l′ dimensionless parameters of BCC unit cell

li,j branch vectors

L,L′ dimensions of BCC unit cell

m order of approximation

M rotation tensor

n porosity

ni,j unit normal vector

nc number of contacts for each particle

nl number of liquid bridges for each particle

N c number of contacts in REV

Np number of particles in REV

NLB number of liquid bridges in REV

p, p′ mean stress,effective mean stress

p(n) contact normal’s probability density function

q, q′ deviatoric stress,effective deviatoric stress

r radius of capillary tube,radius of inter-particle contact

R radius of particle

R1,2 radius of curvature

Sr degree of saturation

ti,j traction forces

Ts surface tension parameter

Ti,j surface tension forces

ua air pressure

ui,j displacement vector

uw water pressure

xii

uatm atmospheric pressure

V volume of REV

VLB volume of liquid bridge

Vv volume of voids

Vs,w,a volume of solid,water,air

Vrel dimensionless liquid bridge volume

Vcone conical volume

V p volume of particle

w water content

xi,j position vector

xci,j centroid position vector

yLB(x) liquid bridge profile in Cartesian coordinate system

yp(x) particle profile in Cartesian coordinate system

z coordination number

Roman letter symbols

α half filling angle, Biot’s effective stress coefficient

αij tensorial effective stress coefficient

β arbitrary direction in 3D space

βn orientation of major principle direction in 3D space

γw unit weight of water

Γ boundary surface of REV

Γp boundary surface of the particle

Γpw wetted surface of the particle

Γpd total surface of finite contacts on the particle

xiii

Γpc surface of each finite contact on the particle

Γm contractile skin contour

δij Kronecker delta

ǫij strain tensor

θ wetting angle

θr residual volumetric water content

θs saturated volumetric water content

θw volumetric water content

λ number of liquid bridges over the number of contacts

µ conical volume over the total volume

ν Poisson ration

σij , σij, σ′

ij stress, average stress and effective stress tensors

σn, σ′

n normal stress, normal effective stress

σs suction stress

τf shear strength

ϕ friction angle

ϕc inter-particle friction angle

χ Bishop’s effective stress parameter

χij tensorial effective stress parameter due to suction

ψ, ψ matric suction,dimensionless matric suction

ψe air-entry suction

ψij capillary stress tensor

ψx,y,z principle values of capillary stress tensor

xiv

Abbreviations

3D three dimensional

BCC body centered cubic

CCP cubic close packing

DEM discrete element modeling

FCP face centered packing

REV representative elementary volume

SAGD steam assisted gravity drainage

SCP simple cubic packing

SSCC suction stress characteristics curve

SWCC soil water characteristic curve

xv

Chapter 1

INTRODUCTION

1.1 Introduction

The effective stress principle is one of the underpinnings of soil mechanics alongside with the

Mohr-Coulomb failure criterion. In fact, the concept of effective stress makes it possible to

extend conventional theories of dry, deformable, continuous materials to deformable, gran-

ular, multi-phasic materials such as soils. While soils have been conventionally considered

to be either fully saturated or dry in soil mechanics, there has been a need to study the

unsaturated condition as well. For instance, soils are not completely saturated in a variety

of engineering problems such as the construction and operation of earth dams, shallow slope

stability, shallow footing design, and even Steam Assisted Gravity Drainage (SAGD) in pro-

ducing heavy oil or gas bearing sediments. However, most of the constitutive models that

describe the mechanical behaviour of soils have been developed based on the assumption of

fully saturated conditions, even though unsaturated models have been proposed in the past

several decades now.

The unsaturated condition introduces air as an additional phase besides solid and water

phases and, as a result, new internal forces such as capillary forces arise in between the

solid particles. These invariably increase the shear strength of unsaturated soils, such as

frictional sandy soils, through an apparent cohesion or adherence to even give rise to a

tensile strength. Here, cohesion can only be apparent for sand since it does not refer to

the mobilization of physicochemical forces such as van der Waals attractions or double layer

effects among particles, as in clay. The increase in shear strength of unsaturated soils is a

complex function of the degree of saturation and matric suction, the difference between air

and water pressures, which varies during a wetting or drying phase. It is the disappearance

1

of capillary forces during a wetting phase in the absence of any mechanical loads that results

in a collapse type of failure in unsaturated soils, making it highly unstable (materially) in

relation to fluctuations in degree of saturation.

From a mechanics view point, unsaturated soils represent a three-phase system in which

internal forces arise from the interaction of solid (particles), liquid (water) and gas (air)

phases. Therefore, taking partial saturation condition into account has always been com-

plicated, both from the experimental and theoretical point of view. For instance, it is not

quite clear which controlling stress variable under unsaturated conditions substitutes for the

role of effective stress in the saturated case. To the present day, different lines of thoughts

concerning the constitutive modelling of unsaturated soils in a wide range of saturation have

been developed. The diversity of these approaches is based on the choice of an appropri-

ate equation for effective stress, which could play the role of generalized effective stress in

simulating the behaviour of unsaturated soils. Nonetheless, the specific role of the capillary

forces in defining the effective stress and mechanical behaviour of unsaturated soils has re-

mained mostly elusive. Ideally, the stress variable would have to incorporate various internal

processes such as partial pressures of water and air phases in the form of matric suction as

well as evolving volume fractions of the phases such as saturation, and other microstructural

quantities such as fabric.

1.1.1 Objectives

The motivation for this research stems from premises outlined above. The primary ob-

jective is to examine the notion of stress and its definition for a wet three-phase system

composed of idealized soil particles and pore water menisci through a micromechanical anal-

ysis. By considering air and water pressures, surface tension, as well as inter-particle forces

within an assembly of spherical particles with low degree of saturation (pendular regime),

the Cauchy stress tensor can be readily calculated as a volume average of the various con-

stituents (phases) just like in the case of a solid body consisting of interacting point masses

2

in a volume (Love, 1927). The proposed derivation ultimately leads to a tensorial effective

stress equation which can be viewed as a generalized Bishop’s equation, explicitly written as

a function of the spatial distribution of water menisci, matric suction and particle contacts

through an anisotropic tensor, which is a novelty in the unsaturated soil mechanics literature.

Hence, the objectives of this research are to:

1. Develop a micromechanical formulation of force transmission within an unsat-

urated soil as a three-phase system with low degree of water saturation.

2. Derive an analytical expression to define a tensorial effective stress equation,

and discuss the variation of effective stress coefficients as a material parameter

with degree of water saturation and particle packing in light of experimental

data already available for different types of soils.

3. Analyze the effects of anisotropic contribution of pore fluid pressure and con-

tractile skin in unsaturated soil samples and define capillary-based anisotropy

even under isotropic loading, which is fundamental to the understanding of

the strength behaviour of unsaturated soils.

4. Apply DEM (Discrete Element Modeling) as a means to explore the validity

of the proposed analytical equation.

In addition, the effect of a significant contact area between compressible particles on the

effective stress of saturated cohesionless soils is studied from the micromechanical point of

view. The effective stress coefficient (α, so-called Biot’s coefficient) related to soils with

compressible particles is then discussed as a function of micromechanical parameters and

micro-structure anisotropies of the soils fabric.

1.1.2 Organization of Thesis

This thesis is organized into six chapters as outlined below:

3

Chapter 2 presents a literature review on the various issues relevant to this study. The

capillary phenomenon, matric suction and their impact on the mechanical behaviour of

unsaturated soils are discussed using experimental evidence. Different approaches to study

the effective stress in unsaturated soils are presented through both phenomenological and

micromechanical points of view.

Chapter 3 gives a rational approach within which the role of capillary forces and their dis-

tributions is accounted for through the micro-scale physics that governs the state of stress in

an unsaturated soil and its macroscopic engineering properties. Thereafter, a tensorial effec-

tive stress equation for unsaturated soils is proposed by taking into account the anisotropy of

a general particle packing through an appropriate probability density function. Such formu-

lation gives rise to new tensorial effective stress parameters that explain the capillary-based

tensile strength and the shear strength enhancement normally observed in unsaturated soils.

Most importantly, the derived capillary stress is seen to be anisotropic due to inter-particle

capillary forces distribution and particle packing.

Chapter 4 The importance of the capillary forces distribution in the determination of

capillary stresses in the pendular regime is studied in the context of the effective stress equa-

tion derived in previous Chapter. The developed effective stress equation is thus examined

for simple isotropic and anisotropic regular packings comprised of spherical particles, with

results being discussed using experimental data already available in literature.

Chapter 5 examines the shear strength of a wet granular material with random packing

and various matric suctions in a triaxial test specimen using the proposed effective stress

equation. The results are then validated through DEM simulations of the same triaxial

test specimens. It is found that the computation of effective stress based on the proposed

equation for various water contents considering menisci and particle packing effects leads to

a unique Mohr-Coulomb failure envelope with an intrinsic friction angle corresponding to

the dry case, which supports the validity and objectivity of micromechanical derivations and

4

the proposed equation for effective stress. Thus, such proposed generalized effective stress

equation provides a scientifically legitimate substitute for effective stress in the saturated

case, and hence would give the expected irreversible deformations of unsaturated soils when

used in any soil constitutive model without prior modification.

Chapter 6 summarizes the major findings of this thesis and offers recommendations for

future work.

5

Chapter 2

LITERATURE REVIEW

2.1 Introduction

The mechanical behaviour of unsaturated soils is firmly linked to so-called capillary stresses

that arise from interactions between water and air phases which are controlled by the degree

of saturation (water content) as an important state parameter. As such, the resulting capil-

lary forces induced among particles inhibit micro-kinematics such as rolling and slippage so

that unsaturated soils usually possess higher shear strength and display a stiffer behaviour

than saturated soils under the same applied stresses (Fredlund and Rahardjo, 1993). This

Chapter discusses distinctive mechanical behaviours of unsaturated soils over a wide range

of water content based on‘ experimental observations as reported in the literature. These

pertain to gains in both tensile and shear strengths as well as stiffness of unsaturated soils

which, when lost, lead to material collapse behaviours.

2.2 Capillary Effect and Matric Suction

The classic explanation of surface tension is illustrated in Fig. 2.1 where two phases of

water and air interact through an interficial surface over which balance of forces must exist.

Each water molecule on the air-water interface undergoes unequal hydrostatic pressure due

to the pressure deficiency between air and water phases, commonly called matric suction.

As a result, in order to reach mechanical equilibrium in the system, a resultant force called

surface tension (Ts) develops alongside with the interface of water and air phases to give it a

curvature. If there were no pressure difference across the interface (air pressure equals water

pressure), a perfectly flat interficial surface would be expected (Lu and Likos, 2004).

6

wu

au

Figure 2.1: Illustration of surface tension

These induced surface tension forces actually give rise to capillarity or the capillary effect

which is defined as the ability of a liquid such as water to flow in narrow spaces without

the help of external forces. Usually, in order to demonstrate the capillary effect, a small

capillary tube can be used, as is represented in Fig. 2.2. Placing the capillary tube in the

water container, the water would rise inside the capillary tube in order to accomplish the

equilibrium between the adhesive capillary forces and gravity forces.

Writing the equilibrium of forces in the vertical direction, the critical height of the water

column in the tube (hc) can be defined as below (Batchelor, 1967):

γwπr2hc = 2πrTs cos θ ⇒ hc =

2Ts cos θ

γwr(2.1)

where r is the radius of the tube, Ts is the surface tension of the water (force per unit length),

θ is the wetting angle between solid and liquid, and γw is the unit weight of water. As it is

defined in Eq. (2.1), the smaller the tube radius, the greater the rise of the water column.

The pressure deficiency between the air and water phases is called matric suction (ψ)

7

sT

θ θ

2rch

Figure 2.2: Water in capillary tube

and can be calculated as:

ψ = ua − uw = hcγw (2.2)

in which ua and uw represent the hydrostatic pressure in the air and water phases respectively.

The equilibrium of forces acting on both sides of the air-water interface controls the

geometrical shape of the interface between air and water. Considering the free body diagram

of a two-dimensional curved air-water interface in the absence of gravitational forces as

illustrated in Fig. 2.3, the equilibrium of forces in z direction can be written as follows:

2πrTs cos θ − (ua − uw) πr2 = 0 (2.3)

which leads to:

(ua − uw) =2Ts

(r/ cos θ)=

2TsR

(2.4)

in which R is the radius of curvature of the air-water interface surface. As it is defined in Eq.

(2.4), the mean curvature of the air-water interface is a function of the pressure deficiency

between liquid and gas (Lu and Likos, 2004).

8

r

R

θ

au

wu

X

Z

Ts Ts

Figure 2.3: Free body diagram of forces acting on air-water interface in a capillary tube

In the absence of gravitational forces, the pressure deficiency between liquid and gas will

be constant; consequently, as shown in Fig. 2.4a, R will possess a constant value and the

air-water interface will take the form of a spherical arc.

Moreover, in three-dimensional space, two principal radii of curvature (R1 and R2) are

usually introduced to define the geometry of the air-water surface; these curvatures can

have the same concavity or they can possess opposite concavities as shown in Fig. 2.4b and c

respectively. Writing the equilibrium of forces in z direction for a three-dimensional air-water

surface, Eq. (2.4) can be written as:

ua − uw = Ts

(

1

R1

+1

R2

)

(2.5)

Equation (2.5) is conventionally referred to as the Young-Laplace equation (Young, 1805),

which presents a nonlinear, partial differential equation relating the pressure difference of

liquid and gas to the geometry of the interface surface.

In porous materials such as soil, the pore spaces can be seen as capillary tubes. Therefore,

the capillary effect would hold water above the water table at negative hydrostatic pressure

in comparison with the air pressure. The height of the capillary rise is a function of pore size

and its distribution; the smaller the size of the pores in the soil, the greater the capillary rise

9

R

R

R

(a) (b) (c)Z

X

1

2

R 2

R1

Figure 2.4: Curved liquid and gas interfaces

will be. In hydrostatic conditions with no flow, as shown in Fig. 2.5, the soil is completely

saturated below the water table; however, above the water table, the degree of saturation

decreases with height. The unsaturated soil above the water table can be illustrated in three

different states due to the amount of water in the pores and the degree of saturation. These

three states are called pendular, funicular and capillary regimes.

In the residual or pendular state, as shown in Fig. 2.5a, individual liquid bridges are

formed between each pair of particles in contact or in close proximity to each other; therefore

the water phase is assumed to be discontinuous, while the air phase is generally continuous.

The degree of saturation (Sr) in this state is usually smaller than 25%. The matric suction

in this state actually obeys the Young-Laplace equation, and is therefore a function of the

shape of liquid bridges between particles.

In the funicular state, as represented in Fig. 2.5b, liquid clusters comprising more than

a pair of particles are formed in the pore space of the soil and the liquid phase is assumed

to be continuous. In this state, the degree of saturation is within the range of 25% to 90%.

Finally, in the capillary zone with a degree of saturation greater than 90%, all pore space

10

between the particles is filled with liquid, while air bubbles would be entrapped in closed

pore spaces; see Fig. 2.5c.

Ground Surface

a- Pendular State

b- Funicular State

c- Capillary State

Datum

z

Water Table

Figure 2.5: Conceptual demonstration of unsaturated sample in different regimes(Lu andLikos, 2004)

In order to describe the transition between these three states, it is worthwhile to take

a closer look at the wetting and drying processes in a soil sample. During the wetting

process, while the amount of liquid is small, individual liquid bridges are formed between

each pair of particles in the pendular regime. As the amount of liquid gradually increases,

several liquid bridges merge with each other and develop liquid clusters between groups

of soil particles in the funicular regime. Consequently, electrical conduction and chemical

diffusion in the unsaturated soil increase rapidly. The procedure of various liquid bridges

combining with each other in the funicular state is extremely complicated and is controlled

11

by several micromechanical parameters, such as the diversity of shapes and sizes of the soil

particles, the pore size and distribution in between the particles, and the number of contacts

per particle. Furthermore, increasing the degree of saturation results in unsaturated soil

entering the capillary state and all pore spaces between the particles are filled with liquid,

while air bubbles are entrapped in closed pore spaces. Finally, the system becomes saturated

if the amount of liquid is enough to raise the liquid pressure as high as the air pressure and

make all air bubbles dissolve in liquid.

During the drying process in an unsaturated soil sample, as water starts to drain or

evaporate from the saturated soil, the suction pressure increases, and thus the boundary

menisci are pulled inward. This stage is equivalent to the capillary state. While the pressure

difference is enough for the air phase to break into the pores, soil enters the funicular stage

and becomes unsaturated. The pressure at which air bubbles penetrate the pore space of the

soil is called the air-entry value (Aubertin et al., 1998). As the drying process continues, the

liquid bridges begin to form between pairs of particles and the soil enters the pendular state.

The suction pressure increases considerably due to the small curvature of water menisci

between pairs of soil particles.

In fact, in a real unsaturated soil sample, the variety of shapes and sizes in soil particles,

the complicated pore size and its distribution between the particles, and the internal flows

between continuous phases also affect the shape of the water menisci and clusters between

the particles. Therefore, defining the geometry of the air-water interface, and subsequently

determining the pressure differences between the air and water phases (matric suction) and

capillary forces through the micromechanical point of view can be very complicated.

Soil water characteristic curves (SWCC) are usually defined experimentally in order to

identify the relationship between soil suction and water content in unsaturated soils. In the

next subsection, a brief review of SWCC is presented.

12

2.2.1 Soil water characteristic curve

The relationship between soil suction and the amount of water contained in the pores of the

soil is typically illustrated by soil water characteristic curves (SWCC). The amount of water

contained in the soil can be defined using different parameters such as the volumetric water

content (θw) as the ratio of the volume of water over the total volume of the soil sample, or

the degree of saturation (Sr) as the ratio of the volume of water to void volume. These two

parameters can be easily related to each other through the porosity of the soil (n):

θw = Sr n (2.6)

A typical soil-water characteristic curve for a sandy and silty sample as reported by Lu

and Likos (2004) is presented in Fig. 2.6. The amount of zero suction coincides with the

completely saturated state (Sr = 100%). As the matric suction increases, boundary menisci

are pulled inward, but the sample still remains saturated. Eventually, reaching a specific

suction pressure called air entry value (ψe), air starts to enter the largest pores of the soil

and the sample enters the unsaturated state.

Further, as shown in Fig. 2.6, for a specific amount of water (volumetric water content),

soil suction is inversely proportional to the size of the particles; fine-grained soils such as

silts usually possess higher suction over an extensive range of water content as a consequence

of their pore shape, particles size, and pore-size distribution. Throughout the past decades,

several attempts have been made to model the soil-water characteristic curve of unsaturated

soils related to the particle size or pore-size distribution (Arya and Paris, 1981; Fredlund

and Xing, 1994; Leong and Rahardjo, 1997; Assouline et al., 1998). However, it is well-

known that the water content retained in the pore spaces of the soil cannot be uniquely

defined by the value of the matric suction, but it is strongly hysteretic and dependent on

the drying and wetting cyclical processes such as infiltration, capillary rise, evaporation and

gravity drainage. In fact, due to this hysteretic behaviour, no unique relation between the

soil suction and water content can generally be obtained for a real soil. Through the past

13

soil

su

ctio

n

volumetric water content

silt

sand

wθ

ψ

air entry

eψ

Figure 2.6: Conventional soil water characteristic curve for sand and silt(Lu and Likos, 2004)

decade various soil-water hysteresis models have been proposed (Lu and Likos, 2004; Huang

et al., 2005; and Min and Huy, 2010).

Theoretically, several mechanisms can lead to hysteretic behaviour from the micro-scale

or macro-scale point of view. These mechanisms have been classified by Lu and Likos (2004)

as follows:

• Ink-bottle effect: It describes the influence of non-uniformities in the distribu-

tion of pore size and shape. A hypothetical non-uniform pore space described

by two different radii is considered in Fig. 2.7. For a specific matric suction

controlled by a smaller radius, ua − uw = 2Ts/r, the maximum height of the

capillary rise can be different during the drying and wetting processes. As

shown in Fig. 2.7a, the capillary height during the drying process (hd) may

extend beyond the larger part of the pore space (with radius R) while the

14

sample is initially saturated. However, during the wetting process, the cap-

illary rise (hw) will cease before reaching the larger part of the pore space

(Fig. 2.7b). Therefore, the amount of retained liquid in identical pores un-

der the same matric suction is commonly larger during the drying process in

comparison with the wetting process.

dh

wh

r r

R R

( )a ( )b

Figure 2.7: Demonstration of the ink-bottle effect during:(a)drying process and (b)wettingprocess (Marshall et al., 1996)

• Entrapped air effect: It defines the influence of the formation of air bubbles

in pore spaces during the wetting process.

• Deformations: They identify the influence of changes in the pore size, shape

and distribution due to swelling and shrinkage of the soil sample during its

drying and wetting histories.

• Wetting angle hysteresis: It defines the effect of the intrinsic difference in

wetting angles between the soil particles and the pore water during drying

and wetting cycles.

As discussed in the previous subsection, three general regimes of saturation can be defined

15

in an unsaturated domain: the pendular regime, the funicular regime, and the capillary

regime. Within each regime, specific mechanisms play the main role to control the hysteresis

in the suction-water content relationship and affect the shape of the soil-water characteristic

curve. Consequently, as shown in Fig. 2.8, a typical soil-water characteristic curve can be

divided into three sections related to the three different regimes of unsaturated soil (Lu et

al., 2007). Fredlund and Xing (1994) introduced characteristic points on the SWCC such

as θr, the residual water content where a large amount of suction is needed to remove more

water from the soil, and θs, the volumetric water content at the saturated state.

soil

su

ctio

n

volumetric water content wθ

ψ

I II III

I pendular regime

II funicular regime

III capillary regime

drying

wetting

rθ sθ

Figure 2.8: Theoretical presentation of soil-water characteristic curve of an unsaturatedsample in different regimes (Lu et al., 2007)

16

Within the pendular regime, the hysteretic behaviour is mainly affected by wetting angle

hysteresis in the microscopic (particle size) scale. Accordingly, it is possible to theoretically

define the SWCC in the pendular regime. Indeed, Molenkamp and Nazemi (2003) and Lu

and Likos (2004) have defined the SWCC for unsaturated samples consisting of idealized

spherical particles in the pendular state. As the water content increases and approaches the

residual water content θr, wetting angle induced hysteretic behaviour becomes less prominent

in the funicular regime (See Fig. 2.8, region I).

The hysteretic behaviour is most noticeable in the funicular regime, (see Fig. 2.8, re-

gion II) for a different reason. In this region, the actual soil water characteristic curve for

unsaturated soil under arbitrary field conditions will be affected by almost all previously

discussed mechanisms, specifically the Ink-bottle effect. So far, various authors have tried to

develop a hysteresis model to define the shape of the soil-water characteristic curve in this

regime; however, the exact roles and consequence of the various hysteresis mechanisms on

the hysteretic behaviour of unsaturated soils in the funicular regime have remained widely

unclear.

In region III, which indicates the capillary state, the entrapped air bubbles are mostly

in charge of the hysteretic behaviour of unsaturated soil. In fact, as a result of the presence

of entrapped air bubbles, the completely saturated state may not be attained during a re-

wetting phase.

Although the mechanisms controlling the hysteretic behaviour and the shape of the soil-

water characteristic curve in unsaturated samples are different in different regimes, the tran-

sition from one state to another is basically gradual, which leads to capturing a continuous

soil-water characteristic curve for samples experimentally. Based on the above discussions, it

becomes clear that it is possible to analytically describe the SWCC in the pendular regime,

but not in the funicular and capillary states.

17

2.3 Experimental Observations on Unsaturated Soil Behaviours

2.3.1 Shear and tensile strengths of unsaturated soils

The capillary forces in unsaturated soils restrict inter-particle slippage and as such increase

the shear strength of unsaturated soils. As shown in Fig. 2.9, while c represents the classical

cohesion of the soil sample with zero matric suction in the dry or saturated cases, the

shear strength increases due to the additional cohesion resulting from the capillary forces

between soil particles in the unsaturated cases while the matric suction possesses a non-zero

value. Therefore, the cohesion parameter in unsaturated soil mechanics, usually referred

to as apparent cohesion (ca), actually consists of the classical cohesion standing for the

shear resistance due to the physicochemical forces between particles such as van der Waals

attraction and cementation, augmented with the additional cohesion due to capillarity (Lu

and Likos, 2004).

net normal stress

shea

r st

ren

gth

c

1ac

2ac

3ac

{{ {'φ

(

)0

a

w

uu−

=1

(

)0

a

w

uu−

>2

1

(

)(

)

a

w

a

w

uu

uu

−

>−3

2

(

)(

)

a

w

a

w

uu

uu

−

>−

Figure 2.9: General representation of shear strength in unsaturated samples (Ho and Fred-lund, 1982)

Ho and Fredlund (1982) conducted a series of multistage drained triaxial tests on unsat-

18

urated soil samples. Plotting the Mohr-Coulomb failure envelopes for samples with various

suctions and constant confining pressures, they demonstrated an increase in shear strength

with matric suction. They also suggested that the friction angle remains almost the same in

both saturated and unsaturated samples. The strength parameters were determined using

so-called net stress, i.e. total stress minus air pressure.

Performing a series of direct shear box tests on mono-disperse granular glass bead samples

wetted by water and n-hexadecane, Pierrat et al. (1998) examined the effect of the suction

forces on the yielding of wetted granular materials. Yielding was defined as the state at

which the material flows plastically at large deformations and constant stress and as such

the yield locus was found as an envelope of Mohr-circles describing the state of stress of

the material at yield point. Illustrated in Figs. 2.10 and 2.11, the results show the yield

locus of the glass bead samples, with radius of 46µm and 90µm respectively, shifted upward

significantly due to the effect of capillary forces induced by suction. It is worthwhile to note

that even though the moisture content remains the same in both cases, the dissimilarity

between the wetting angles of the water and n-hexadecane changes the amount of capillary

forces induced by them in unsaturated samples, and hence lead to different shear strengths.

Therefore, as shown, the vertical shift distance of the yield locus for samples wetted with

n-hexadecane is smaller than that for samples wetted with water for approximately the same

moisture content.

Donald (1956), Escario and Saez (1986), and Fredlund et al. (1995) studied the non-

linearity between shear strength and matric suction. For instance, Donald (1956) performed

a series of direct shear tests on unsaturated, cohesionless fine sand and coarse silt. The shear

strength rises to a peak value with increasing matric suction, after which there is a decrease

to an almost-steady value (see Figs. 2.12a,b).

Kim (2001) performed a series of direct tension tests on a series of quartz sand samples

with different moisture contents and densities to define the tensile strength of moist sand

19

0

5

10

15

20

25

0 10 20 30 40

shear

stre

ss (

g/c

m2)

normal stress (g/cm2 )

dry

1% n-hexadecane

1.3% water

4% water

Figure 2.10: Yield locus of glass beads R=46 micron (Pierrat et al., 1998)

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40

shear

stre

ss (

g/c

m2)

normal stress (g/cm2)

dry

1% n-hexadecane

1.3% water

4% water

Figure 2.11: Yield locus of glass beads R=90 micron (Pierrat et al., 1998)

20

0

5

10

15

20

25

0 5 10 15 20 25

shea

r st

rength

(kP

a)

matric suction (kPa)(a)

graded frankston sand

brown sand

0

5

10

15

20

25

0 5 10 15 20 25 30

(kP

a)

matric suction (kPa)(b)

fine frankston sand

medium frankston sand

shea

r st

rength

Figure 2.12: Direct shear test results on cohesionless sands (Donald I., 1956)

21

as a function of the water content and relative density. As a result, Kim (2001) observed

that the capillary forces between the soil particles not only lead to an apparent cohesion in

unsaturated samples, but also gave rise to a specific amount of tensile resistance in them. As

demonstrated in Fig. 2.13, the effect of density on the tensile behaviour of the unsaturated

samples becomes negligible at the lower the water content. At higher water contents, the

denser samples experience higher tensile strength induced by capillary forces due to the

presence of more liquid bridges in comparison to the loose samples.

A series of tensile tests on medium-dense sand was also conducted with a wide range

of degree of saturation, see Fig. 2.14. It was found that up to a water content of 15%,

the tensile strength gradually increased with increasing the amount of water content, and

thereafter it started to reduce considerably due to the merging of liquid bridges and loss of

capillary forces.

22

0

200

400

600

800

1000

1200

1400

0 1 2 3 4 5

tensi

le s

tren

gth

(P

a)

water content %

measured data,loose

measured data, medium

measured data, dense

Figure 2.13: Tensile strength versus water content (F-75-C),(Kim, 2001)

0 20 40 60 80

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20

water content %

tensi

le s

tren

gth

(P

a)

degree of saturation (%)

Figure 2.14: Tensile strength versus water content,(Kim, 2001)

23

2.3.2 Collapse behaviour

The collapse behaviour of unsaturated soils conventionally refers to a significantly rapid de-

crease in volume at constant total stresses upon saturation. Barden et al. (1973) indicated

that two factors are necessary to reach a metastable condition and collapse in unsaturated

soils during the wetting process. Firstly, there must be a high enough amount of applied

external stress that develops shear stresses and instability at inter-granular contacts. Sec-

ondly, there must be a high enough amount of suction stress which originally increases the

stability against the applied stresses at inter-granular contacts; however, its reduction during

the wetting process will lead to increased instability between particle contacts, and hence

give rise to a metastable condition.

Jennings and Burland (1962), Lawton et al. (1989), Pereira and Fredlund (2000), and

Sun et al. (2007), conducted a series of oedometer and triaxial laboratory tests to study the

collapse behaviour of partially saturated soils.

Jennings and Burland (1962) compared the results of oedometer and all-round compres-

sion tests on both unsaturated and completely saturated samples. Soaking the unsaturated

samples, under a constant confining pressure or volumetric strain, and plotting the com-

pression curves (void ratio versus applied pressure), they identified that these compression

curves actually crossed the compression curve of the same saturated sample. This behaviour

illustrates that during the wetting process the effective stress of unsaturated soil reduces, due

to the gradual loss of the inter-particle capillary forces; therefore, the unsaturated sample

fails in shear and undergoes additional deformation which can be defined as the collapse

behaviour in unsaturated soils, see Fig. 2.15.

Recalling the definition of effective stress, as a part of the stress on porous media which

controls the mechanical behaviour and deformations, it would be clear that, if one can

define the true effective stress in unsaturated samples, it would be possible to simulate the

mechanical behaviour of unsaturated soils using that effective stress. In the next section,

24

0.65

0.67

0.69

0.71

0.73

0.75

0.77

0.79

0.81

0.83

0.1 1 10

void

rati

o e

applied pressure (t/ft 2 )

compression line of

air-dried samples

soaked at constant

void ratio

soaked at constant

applied pressure

compression line of

saturated samples

Figure 2.15: One-dimensional compression and subsequent soaking tests under constant voidratio or applied pressure (Jennings and Burland, 1962)

a literature review on various efforts to define the effective stress in unsaturated media is

presented.

2.4 Studies on Effective Stress of Unsaturated Soils - Existing Frameworks

Up to now, the concept of effective stress has been very successful in analyzing and predict-

ing the behaviour of saturated or dry porous materials. In fact, it has been acknowledged

as the single most fundamental contribution to the study of granular materials (Khalili et

al., 2004). However, as mentioned previously, natural soils can be unsaturated in various

engineering problems. Yet, as a result of the complexities involved in taking unsaturation

into consideration, most of the theories in conventional soil mechanics have been built based

on two limiting conditions: completely dry or fully saturated. Therefore, appropriate consti-

25

tutive models taking the partial saturation condition into account are required to precisely

deal with a number of engineering problems such as slope instability.

2.4.1 Phenomenological studies (Macroscale studies)

Phenomenological approaches are conventionally used in research on the constitutive be-

haviour of unsaturated soils. Based on mixture theory (Goodman and Cowin, 1972), and

defining a representative elementary volume REV, assumptions regarding material response

are introduced at the macroscopic scale.

Thus far, the conventional phenomenological approach has led to two primary lines of

thought to define a suitable equation for effective stress in unsaturated soils. One is based on

identifying a single suitable stress variable playing the role of the effective stress for unsat-

urated soils; thus, one can easily expand all conventional constitutive models for saturated

case into unsaturated case. By contrast, the other approach usually considers the net stress,

which is defined as the difference between the applied stress and air pressure (σ − ua), and

the matric suction (ψ), as the first and second stress variables respectively.

2.4.1.1 Single effective stress approach

In the late 1950s and 60s, first efforts to define the mechanical behaviour of unsaturated soils

were based on identifying a single suitable stress variable playing the role of the effective

stress for unsaturated soils. As a result, several so-called effective stress equations for unsat-

urated soils were proposed; the most successful equation was proposed by Bishop (1959). He

extended Terzaghi’s effective stress principle to account for the presence of an air phase by

intuitively introducing an average pore fluid pressure weighted over the pore air and water

pressures, i.e.

σ′ = σ − [χuw + (1− χ)ua] (2.7)

where uw and ua are pore water and air pressures respectively, σ′ is the effective stress and

σ is the total Stress due to applied loads. The weighting parameter (χ) is called the Bishops

26

effective stress parameter, which defines the contribution of air and water pressures to the

average pore pressure of unsaturated soil and has been typically related to the degree of

saturation (Sr). The effective stress parameter (χ) is considered to vary gradually from

0 for Sr = 0% to 1 for Sr = 100%, which provides a smooth transition between the dry,

unsaturated and completely saturated states for soil. As such, converting a multiphase

system of unsaturated soil into a mechanically equivalent single phase continuum, Bishop

(1959) proposed a simple single effective stress equation, which encompasses both dry and

fully saturated conditions as special cases.

Several researchers, such as Donald (1961), Blight (1961), Jennings and Burland (1962),

and Escario and Juca (1989), attempted to define χ related to (Sr) based on experiments.

Some of the results of these experimental studies on different soils are shown in Fig. 2.16a

and b.

As one can see, due to practical difficulties in measuring the matric suction and degree

of saturation in the pendular (residual) regime, most of the experimental measurements of

χ have been made for degrees of saturation greater than 25%. Moreover, the relationship

between the effective stress parameter and the degree of saturation is affected by the type

and density of the soil. Matyas and Radhakrishna (1968) noted that the value of parameter

is highly path dependent, and is thus affected by the stress and saturation histories of the

soil. Coleman (1962) also cited that χ is strongly correlated to the micro-structure of the

soil. Hence, defining the relationship between χ and Sr is usually a difficult task and requires

special experimental procedures (Nuth and Laloui, 2008).

Despite these difficulties, a variety of mathematical equations to determine χ have been

proposed so far. For instance, Schrefler (1984) suggested the application of the simple form

of χ = Sr in modelling unsaturated soils. Thereafter, Vanapalli et al. (1996) proposed the

following equation:

27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

coef

fice

nt

of

effe

ctiv

e st

ress

c

degree of saturation Sr %

(a)

breahead silt (Bishop and Donald, 1961)

silt (Jennings and Burland, 1962)

silty clay (Jennings and Burland, 1962)

compacted boulder clay (Bishop et al.,

1960)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

coef

fici

ent o

f ef

fect

ive

stre

ss c

(b)

silt, drained test (Donald, 1961)

silt, constant water content test (Donald, 1961)

Madrid gray clay (Escario and Juca, 1989)

Madrid silty clay (Escario and Juca,1989)

Madrid clay sand (Escario and Juca, 1989)

moraine (Blight, 1961)

boulder clay (Blight, 1961)

clay -shale (Blight, 1961)degree of saturation Sr%

Figure 2.16: Effective stress coefficient for unsaturated soil based on experimental results

28

χ =(

VwVv

)k

= Srk (2.8)

where Vw is the volume of water, Vv is the volume of voids and k is a curve-fitting parameter

employed to attain the best-fit between the experimental and predicted data.

It is worth noting that the above procedures to define χ are fraught with shortcomings,

especially Eq. (2.8) which is purely empirical in nature. For instance, during wetting/drying

cycles hysteritic behaviour cannot be captured. As much, Khalili and Khabbaz (1998) pro-

posed plotting the values of χ versus a more appropriate parameter, such as the suction ratio

defined as the ratio of matric suction (ψ) over the air entry value (ψe), in order to obtain a

unique relationship between χ and the degree of saturation, Sr.

χ =

(

ψψe

)

1

−0.55

ifψ > ψe

ifψ ≤ ψe

(2.9)

Despite all these efforts, the validity of the single effective stress approach has been

questioned by several researchers. For example, Jennings and Burland (1962) and Matyas

and Radhakrishna (1968) examined the application of the single equation for effective stress

in predicting the volume changes and swelling behaviour during collapse in partially satu-

rated soils in the framework of elasticity. Comparing the results of oedometer and all-round

compression tests on partially and fully saturated samples, Jennings and Burland (1962)

demonstrated that the structural changes (void ratio changes) induced by a change in ma-

tric suction in an unsaturated sample during the wetting process are different from structural

changes of a corresponding saturated sample due to an equivalent change in applied stress.

Thus, they concluded that single effective stress cannot provide an adequate explanation

for collapse behaviour. However, Leonards (1962) noted that the collapse behaviour, during

wetting, is actually related to particle sliding with respect to each other, and consequently

is associated to the plasticity framework. Consequently, Jennings’ and Burland’s arguments

are inaccurate as they are founded on the effective stress principle which in no way accounts

29

for plasticity or non-reversibility of deformations.

Subsequently, the application of a single effective stress equation coupled with complete

elasticplastic framework become more appealing. As such, Jommi and di Prisco (1994) and

Sheng et al. (2004) coupled the single effective stress equation with a suitable elasto-plastic

strain-hardening model to capture the stress-strain behaviour of unsaturated soils. Further-

more, Pietruszczak and Pande (1991, 1995) attempted to develop a mathematical framework

(incremental plasticity) based on using a single effective stress equation in order to describe

the mechanical behaviour of unsaturated soils under undrained conditions. In their work,

the effective stress equation was derived based on explicitly calculating the air and water

pressures, including the surface tension forces around particles followed by homogenization

using some simplistic assumptions.

2.4.1.2 Independent stress state variables approach

As a result of arguments on the validity of the single effective stress approach and difficulties

in defining the value of the effective stress coefficient (χ), a rather different hypothesis was

developed considering two independent stress variables.

At the outset, Fredlund and Morgenstem (1977) considered unsaturated soils as a four

phase system. Therefore, writing the equilibrium of forces for each phase, they proposed

that two independent stress variables are necessary to define the soil elements state, which

can be defined according to three proposed stress state variables: (1) (σ−ua) and (ua−uw),

(2) (σ − uw) and (ua − uw), and (3) (σ − ua) and (σ − uw).

In order to investigate the validity of this proposal experimentally, they conducted various

null tests on samples of silt and kaolin. The experimental tests were called null tests since

no overall volume change in the sample was expected as a result of any changes in σ, ua

and uw by the same amount in any pair of the three proposed stress state variables under a

constant degree of saturation condition. Using a new laboratory apparatus, Tarantino et al.

(2000) confirmed the results obtained by Fredlund and Morgenstern (1977) conducting null

30

tests on samples of kaolin.

Usually, the most commonly used variables, which are practically easier to control, are

the net stress (σ − ua) and the matric suction (ua − uw). Considering the air pressure, as

a datum to define other pressures, (ua = uatm = 0), the net stress would be identical to

the total normal stress, and the matric suction would be equal to negative water pressure.

However, from an experimental point of view, as the negative pore water pressure reaches

the absolute zero value (≈ -101.3 kPa), water starts to cavitate, making it almost impossible

to precisely measure the pore water pressure. In order to avoid water cavitation, the pore

air pressure as the datum to define other pressures is increased in laboratory, so that the

pore-water pressure can be referenced to a higher air pressure. This experimental method to

measure the soil suction is called the axis translation method, and was originally proposed by

Hilf (1956). Fig. 2.17 defines the application of the axis translation method in determination

of the matric suction.

-100

-50

0

50

100

150

200

0 50 100 150 200 250 300

po

re w

ater

pre

ssu

re u

w(k

Pa)

air pressure u a (kPa)

sandy clay

weathered state

loess

Figure 2.17: Axis translation method in measuring matric suction in laboratory (Hilf, 1956)

31

Alonso et al. (1987) suggested the combination of an elasto-plastic strain-hardening

model such as Cam Clay with the independent stress state variables in order to define

the stress-strain behaviour of unsaturated soils such as the volumetric changes due to the

wetting. Thereafter, Alonso et al. (1990) proposed the so-called Barcelona Basic Model

(BBM) to describe the stress-strain behaviour of unsaturated soils within the framework of

hardening plasticity using two independent sets of stress variables, i.e. (σ−ua) and (ua−uw).

Subsequently, modifying this developed framework, other researchers such as Wheeler and

Sivakumar (1995), Bolzon et al. (1996) and Sanchez et al. (2005) proposed more extended

models to deal with other complexities in the stress-strain behaviour of unsaturated soils.

Nevertheless, employing the net stress and matric suction as independent effective stress

variables, it is obvious that one cannot express a direct conversion between unsaturated and

saturated cases so as to recover the well-known Terzaghis effective stress. Also, considering

the effects of the net stress and matric suction separately, leads to various complexities in

defining the effects of hydraulic hysteresis on the mechanical stress paths (Nuth and Laloui,

2008).

To summarize, the constitutive models with respect to the two choices of effective stress

can be classified in two most common categories as suggested by Gens et al. (2006):

1. The single effective stress models which are actually based on the use of a

single effective stress equation such as Bishop’s, and

2. the BBM-like models, which are based on the application of two independent

stress variables.

It is worthwhile to emphasize that the main advantage of using the Bishop’s effective

stress in a constitutive model is that it naturally reduces to dry or fully saturated cases.

However, the precise definition of χ has yet to be deciphered. A summary of these two

conventional constitutive models for unsaturated soils, as organized by Buscarnera (2010),

is presented in Table. 2.1.

32

Table 2.1: Review of the conventional modelling approaches in unsaturated soil mechanics(Buscarnera, 2010)

BBM-like models Single effective stress models

- Yield surface is defined in the net stressspace

- Yield surface is defined in the single ef-fective stress space

- Measurement of the net stress is straight-forward

- Measurement of effective stress requiresmore effort

- Transition between saturated and unsat-urated is not clearly defined

- Transition between saturated and unsat-urated is simply defined

- The effect of hydraulic hysteresis is notwell-defined

- the effect of hydraulic hysteresis can becaptured

33

Most of the complications of the phenomenological studies actually arise from the fact

that the analysis is based on the consideration of soil as a continuous medium. However, in

reality, soils are non-homogeneous and discontinuous in nature, so that their global behaviour

is actually governed by their microstructure and the interactions between different phases.

Cundall (2001) referred to the disadvantages of the application of continuum methods in

defining the behaviour of discrete materials such as soils. He stated: ”from continuum point

of view, an appropriate stress-strain law for the material may not exist, or the law may be

excessively complicated with many obscure parameters”. Alternatively, he recommended the

application of the micromechanical approaches, in which the soil is considered as an assembly

of discrete particles, in order to define the complicated behaviour of soils.

2.4.2 Micromechanical studies

As discussed before, the capillary forces due to the liquid bridges in between particles in-

crease the inter-particle forces in unsaturated soils and, thus, alter both soil stiffness and

strength. However, these capillary forces are mostly governed by micro scale properties that

are not systematically taken into account within the framework of conventional continuum

mechanics. By applying micromechanical approaches, the complex overall behaviour of soil

can be automatically recovered from a few simple assumptions and parameters considered

at the micro level (Cundall, 2001).

Therefore, using a micromechanical approach in which micromechanical concepts are

incorporated and macroscopic measures are strictly related to micro-structure can lead to

more precise techniques in order to define the effective stress and the fraction of suction stress

that control the behaviour of unsaturated soils. In order to describe the behaviour of gran-

ular materials, many scalar parameters, both in macro and micro scales, are necessary such

as density, porosity, degree of saturation, particle size and the coordination number, com-

monly characterized as the average number of contacts per particle in the granular domain.

However, these scalar quantities are not usually sufficient to capture all the complexities of

34

granular material microstructure.

Cobbold and Gapais (1979) and Kanatani (1984) were among the first to suggest the ap-

plication of the statistics of directional data in physical and engineering problems. In order

to distinguish the distribution of inter-particle contact directions in granular materials they

introduced the so-called fabric tensor as a directional quantity illustrating the microscopic

texture of the granular material, whether the microstructure presents an isotropic distribu-

tion or some degree of directional preference. Generally, for an approximation of order m,

the fabric tensor of the mth rank is introduced as:

Fij..m =1

2N c

2Nc∑

k=1

niknj

k....nmk (2.10)

whereN c is the number of contact points, and nk represents the normal unit vector associated

with the kth contact. It can be easily proved that if m is odd, all components of the fabric

tensor become zero; however, for even values of m, the fabric tensor contains non-trivial

components defining the statistical details of contact points orientations in the sample. The

extensive physical description and mathematical origin of the fabric tensor can be found in

literature (Oda and Iwashita, 1999).

In the past two decades, the use of micromechanical approaches and computational meth-

ods to capture unsaturated soil behaviour have received a great deal of attention. Several

studies have been carried out to predict the tensile strength and capillary forces between

two idealized spherical particles in contact in a pendular regime. Fisher (1926), Gillespie

and Settineri (1967), and subsequently Megias-Alguacil and Gauckler (2009), defined the

geometrical properties of the concave liquid bridge and capillary forces between two mono-

sized spherical particles using a simple toroidal approximation, in which the shape of the

liquidgas interface is considered as a circular arc. Pietsch (1968) proposed a separation

distance in between two identical spherical particles in order to consider the surface rough-

ness. In addition, Dealy and Cahn (1970), Bisschop and Rigole (1982), and Molenkamp and

35

Nazemi (2003) applied numerical solutions of the Laplace equation to define the shape of

the liquid-gas interface between two idealized spherical particles, and to estimate the inter-

particle capillary forces as a function of wetting angle, volume of liquid, radius of particle,

and surface tensions. However, the difference between the toroidal approximation and exact

numerical solutions for the liquid bridge shape in the pendular state has been proven to be

less than 10%, (see Lian et al., 1993). The comparison between the results of the toroidal

approximation and the analytical solution as reported by Molenkamp and Nazemi (2003)

is presented in Fig. 2.18. In this Figure, the dimensionless liquid bridge volume and the

dimensionless suction between two particles can be defined as:

Vw =VLBR3

(2.11)

ψ =(ua − uw)

TsR (2.12)

where VLB shows the volume of liquid bridge between the particles, and R is the radius of

the particles in contact.

Considering various possible wetted states of three idealized spherical particles in contact

in two dimensional-spaces, Urso et al. (2002) defined the capillary forces and energy of the

system due to the liquid capillary effects. Moreover, Lechman and Lu (2008) analytically

solved the Laplace equation in order to define the shape of the liquid bridge and capillary

forces between two uneven-sized particles in contact. Recently, Nazemi and Majnooni-Heris

(2012) developed a mathematical model to define the geometry of the liquid bridge and

interactions between two rough spherical particles of unequal size and different material.

Chateau et al. (2002) and Molenkamp and Nazemi (2003) used the homogenization

technique to define the strength criterion of soil as a function of its microscopic properties.

As shown in Fig.2.19, the homogenization technique is a double-scale method to define

the global properties of granular materials such as stresses and strains based on their local

particle-size properties such as contact forces (fi) and displacements of particles with respect

to each other at contacts (ui).

36

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.01 0.1 1 10 100 1000

dim

ensi

on

less

liq

uid

vo

lum

e

dimensionless suction

analytical

torodial

ψ

wV

0θ °=

20θ°

=

30θ °=

40θ °=

Figure 2.18: Dimensionless liquid bridge volume versus dimensionless suction (Molenkampand Nazemi,2003)

Figure 2.19 represents a commonly used scheme to arrive at a stress-strain relationship at

the macroscopic level starting from the inter-particle force/displacement at the microscopic

level. Homogenization techniques are used to upscale inter-particle forces and displacements

to stresses and strains respectively. The macro-micro relations can also be formulated by

applying a double-scale perturbation analysis. The extensive mathematical description of

this perturbation analysis can be found in literature (Oda and Iwashita, 1999).

Considering the unsaturated soil as a periodic three phase system, Chateau et al. (2002)

tried to account for the interaction forces between the liquid and gas phases. Furthermore,

applying the homogenization method, they attempted to establish a link between the mi-

croscopic and macroscopic properties of the soil medium. However, their analysis was not

complete due to difficulties in overcoming the solution of Laplace’s equation to determine the

shape of the liquid-gas interface in the funicular saturation state. Molenkamp and Nazemi

37

Macroscopic level

Microscopic level

Local properties

ijσ

Stress Tensor

ijε

Strain Tensor

if

Inter-Particle Forces

iu

Local Displacements

Global constitutive equations

Figure 2.19: General scheme of homogenization technique (Oda and Iwashita, 1999)

(2003) investigated the inter-granular stresses in the solid skeleton due to the pore suction

and surface tension forces in the wetting and drying cycles. The structure of the granular

material was idealized using a pyramidal packing in a periodic cell. Consequently, as shown

in Fig. 2.20, they calculated a mobilized friction sin ϕ induced by the resultant of capillary

forces in pyramidal packings with different inter-particle friction angles (ϕc) and various

anisotropies controlled by the height (h) of the periodic cell. It is worth mentioning that this

mobilized friction angle is in fact induced by capillary forces, and thus should be a function

of the degree of saturation, wetting angle and liquid bridges distribution. However, as shown

in Fig. 2.20, contrary to what one would expect, the mobilized friction angle calculated by

Molenkamp and Nazemi (2003) is only a function of the inter-particle friction angle and the

pyramidal packing anisotropy.

Cho and Santamarina (2001) introduced the equivalent effective stress due to capillary

forces in an unsaturated soil sample as the effective boundary stress that should be applied on

an equivalent saturated sample to generate similar inter-particle contact forces. Therefore,

for a given isotropic packing of mono-sized spherical particles, they defined the equivalent

effective stress as the ratio of the capillary forces induced by liquid bridges in between each

pair of particles over the effective area occupied by each particle. Likos and Lu (2004)

used almost the same approach to define the effective stress parameter in a given packing

of mono-sized spherical particles in the pendular state. Dividing the calculated capillary

38

0cϕ

°

=

10cϕ

°

=

20cϕ

°

=

30cϕ

°

=

10cϕ

°

= −

20cϕ

°

= −

30

cϕ

°

= −

0

-1

0.8

1 1.1 1.2 1.3 1.4

0.2

0.4

0.6

-0.2

-0.4

-0.6

-0.8

1h =

2h =

height parameter h

mob

iliz

ed f

rict

ion

si

nϕ

2h

22(4 )h−

22(4 )h−

Figure 2.20: Mobilized friction angle in pyramidal packing of various heights and inter-par-ticle friction angles. Negative inter-particle friction angle represents vertical extension

forces among two particles on the cross-sectional area between them, Likos and Lu (2004)

distinguished the induced effective stress due to capillary forces. Moreover, dividing the

calculated suction stress over the corresponding matric suction, they recovered Bishop’s

effective stress parameter (χ)theoretically. The shortcoming of this approach is that the

suction stress between two particles is actually a tensorial quantity that depends on the frame

of reference. In an assembly of particles contacts can be oriented following the anisotropy of

the packing and as such the suction stress would result from the integrating over all contact

directions. From this viewpoint, the resulting (χ) between two particles cannot be easily

extended to an assembly of particle with general fabric.

Lu and Likos (2006) proposed the concept of the SSCC (suction stress characteristics

curve) for unsaturated soils using micromechanical inter-particle force considerations to link

matric suction to an apparent cohesion. Then, employing the results of direct and triaxial

39

shear tests on unsaturated samples with different controlled matric suctions, they determined

the apparent cohesion of samples with various degrees of saturation and suctions by plotting

the Mohr-Coulomb failure envelope in shear strength against net total stress space. There-

after, introducing the suction stress (σs) equivalent to the ratio of the apparent cohesion

over the friction angle (ca/ tanϕ) they related suction stress to matric suction which gives

rise to the so-called SSCC, and consequently calculated the true effective stress as:

σ′ = (σ − ua) + σs (2.13)

This true effective stress, calculated from the SSCC, led to a unique Mohr-Coulomb

failure envelope for different samples with different matric suctions. However, this approach

is rather phenomenologically based on experimental observations since there is no explicit

analytical interpretation of inter-particle forces contributing to an apparent cohesion.

Over the course of the past decades, particle-based methods have received a great amount

of attention in the computational modelling of unsaturated soils . Among them, the discrete

element method (DEM) has been used extensively in modelling the behaviour of cohesionless

soils. This method was firstly proposed by Cundall and Strack (1979) in order to predict the

complex macro-scale behaviour of dry granular material using rather simple assumptions and

few number of parameters at the micro-scale. Recently, adding the resultant capillary forces

to the contact forces between particles, due to all bulk and boundary forces acting on the

system, several researchers such as Richefeu et al. (2007), Scholtes et al. (2009) and Radjai

and Richefeu (2009) studied the shear strength and deformation behaviour of unsaturated

granular materials in the pendular regime using DEM.

From the micromechanical point of view, both the coordination number and anisotropy of

the microstructure in a granular material can affect its overall shear strength and mechanical

behaviour. In order to define the relation between the shear strength and anisotropies of

force and micro-fabric in a cohesionless granular material, Radjai (2008) used a probability

density function with a harmonic representation, Pβ(β), so as to define the mean contact

40

force pointing in an arbitrary direction as a function of that direction, i.e.

Pβ (β) =1

4π

{

1 + a[

3cos2 (β − βn)− 1]}

(2.14)

where Pβ(β) represents the probability density function of contact normals, a is a parameter

which defines fabric anisotropy, while β and βn refer to an arbitrary direction in space

and the orientation of the major principal direction respectively. Subsequently, introducing

an average coordination number (particle connectivity z) and a contact anisotropy, Radjai

(2008) proposed a state function, as shown in Eq. (2.15), to model a domain of accessible

geometrical states for granular materials based on a harmonic representation, i.e.

E (β) = z.Pβ (β) =z

4π

{

1 + a[

3cos2 (β − βn)− 1]}

(2.15)

Considering that the geometrical states should stay between two limit isotropic states,

Emin = zmin/4π and Emax = zmax/4π, Radjai (2008) defined a maximum accessible anisotropy,

a, rooted in the value of z, and concluded that the maximum anisotropy and coordination

number cannot be attained simultaneously. Fig. 2.21 represents the main results of this

micromechanical interpretation.

zzzmin max

zC

a m

ax

Figure 2.21: Domain of accessible geometrical states based on harmonic representation of

granular media (Radjai, 2008)

41

2.5 Summary

The inclusion of partial saturation into the analysis of a three phase medium has been

a longstanding issue in unsaturated soils from both experimental and theoretical points of

view. Even though several forms of effective stress equations, inherited either from the single

effective stress or from the independent stress variable approach, are being applied to model

constitutive behaviour of unsaturated soils, it is still not quite clear which controlling stress

variable to choose to substitute for the role of effective stress in the saturated case.

To answer the above question, it is important to understand the interaction between

phases and the resulting suction stress as a function of the difference between air and water

pressures, the degree of saturation and anisotropies of the microstructure. Therefore, mi-

cromechanical approaches using homogenization techniques provide a viable route to defining

a suitable effective stress parameter in unsaturated media.

The main objective of this thesis is to analyze force transport in a three-phase system

composed of idealized soil particles and liquid bridges in the pendular regime, and thereby

introduce the notion of stresses in such a media through micromechanical analysis. Micro-

scale parameters such as geometrical packing, contact distribution, the orientations and

magnitudes of inter-particle forces, including suction forces and surfaces tension forces, are

taken into account in order to describe the macroscopic properties of unsaturated granular

materials.

42

Chapter 3

MICROMECHANICS OF EFFECTIVE STRESS IN

MULTIPHASIC GRANULAR MEDIA

3.1 Introduction

In the field of geomechanics, soil systems are regarded as non-homogeneous, discontinuous

granular media consisting of discrete particles in contact, while their neighbouring void space

is filled by one or more than one liquid or/and gas. Subsequently, their global behaviour

is actually tied to their microstructure and the interactions between the different phases

such as air, water and solid. Basically, using micromechanical concepts and homogenization

techniques allow us to relate micro-structure to well-known macroscopic measures such as

stress and shear strength. As such, a micromechanical approach leads to more precise results

in order to model the mechanical behaviour of soils.

In this chapter, we analyze unsaturated soils as a three-phase medium with micro-

mechanical interpretations to address the longstanding debate on choosing the controlling

stress variable that would substitute for the role of effective stress in such system. First,

the micromechanical derivation of effective stress in dry and saturated media will be exam-

ined as one and two phase systems respectively. Then, the concept of effective stress for

an unsaturated medium as a three-phase system composed of idealized spherical particles

and pore water menisci is examined through a micromechanical point of view. Using the

homogenization technique, stress is linked to the local variables at micro scale based on their

statistical description to arrive at a tensorial effective stress for unsaturated soils.

43

3.2 Force Transport in Dry Granular Media

In the classic definition of stress in a closed continuous medium of volume V , one usually

invokes the notion of force transmission into the interior domain due to body forces (bi) and

external traction forces (ti(x)) acting on its boundary (Γ). Therefore, a stress tensor (σij)

can be assigned to each point of the medium while it should be consistent with the boundary

condition of σijnj = ti where nj is the outward unit normal vector on the surface (Cauchys

stress principle).

Γ

i

i

j

V

b

x

t

Figure 3.1: Cauchy’s stress in a closed domain

The average external stress that arise from such a problem can be computed as the volume

average of all internal stresses acting at every single material point inside the continuous

medium of volume V , i.e.

σij =1

V

∫

Vσij dV (3.1)

The static equilibrium of forces at any material point can be written as:

∂σij∂xj

+ bi = 0 (3.2)

44

where xi and bi indicate the position vector and body forces (force/volume) at each point of

the domain respectively, see Fig. 3.1.

Subsequently, applying the Gauss-Ostrogradski theorem along with the equilibrium con-

dition, Eq. (3.1) can be expressed as an integration over the closed boundary surface of the

domain:

σij =1

V

∫

Vσij dV =

1

V

∫

Γxitj dΓ +

1

V

∫

Vxibj dV (3.3)

in which, xi is the individual position vector of the tractions and body forces.

x

x i

i

R i t j

Figure 3.2: Assembly of dry granular media

When defining the stress tensor for granular system, a transition from a continuum to a

discrete system is required. Thus, a granular system can be described by a representative

elementary volume (REV) composed of an ensemble of distinct particles interacting with each

other and the void space. Then, each distinct particle can be treated as closed continuous

body as introduced in the above with the inter-particle interactions being defined by traction

forces exerted on the boundary surface of each particle, see Fig. 3.2. The void space can

also be treated the same way. This approach can be seen similar to the decomposition of a

granular body into sub-domains (so-called tessellation cells) with traction forces describing

45

their interactions, see Bagi (1996). Thus, the average stress in a dry granular medium in a

REV of volume V can be written as:

σij =1

V

∑

Np

∫

V pσij dV

p =1

V

∑

Np

(∫

Γpxi tjdΓ

p +∫

V pxibj dV

p)

(3.4)

where Np represents the number of particles; V p and Γp indicate the volume and boundary

surface of each particle.

The position vector (xi) can be further expressed as xi = xci +Ri, where Ri indicates the

location vector of the traction forces on the particle with respect to its centroid. Thus, Eq.

(3.4) becomes:

σij =1

V

∑

Np

∫

ΓpRi tjdΓ

p+1

V

∑

Np

(∫

Γpxci tjdΓ

p +∫

V pxibj dV

p)

(3.5)

Introducing the resultant body force acting at the centroid xci of the particle as bj, i.e.

bjxci =

∫

V pxibj dV

p (3.6)

Eq. (3.5) becomes:

σij =1

V

∑

Np

∫

ΓpRi tjdΓ

p+1

V

∑

Np

xci

(∫

Γptj dΓ

p + bj

)

(3.7)

Since each particle is locally in equilibrium, then the last term in Eq. (3.7) vanishes and

thus,

σij =1

V

∑

Np

∫

ΓpRi tjdΓ

p (3.8)

Moreover, the interaction between each pair of particles (α, β) can be described by the

traction forces (fαβj and fβαj ) seen as mutual action and reaction so that fαβj = −fβαj .

Referring to Fig. 3.3 and introducing the branch vector linking the centroids of the same

two particles (lαβi = Rαβi −Rβα

i ), Eq. (3.8) reduces to (Love, 1927):

σij =1

V

∑

Nc

lifj (3.9)

where N c is the total number of contact points in the REV.

46

R

R l

α

α

β

β

βα

Figure 3.3: Branch vector between pair of particles

3.3 Force Transport in Saturated Granular Media

We herein examine a fully saturated cohesionless granular medium in quasi-static state whose

REV is comprised of interacting solid spherical particles in the presence of water in the void

space.

Given that the saturated system is a two-phase (water and solid) system, the total stress

can be written as a volume average of each individual phase stress over the total volume V ,

i.e.

σij =1

V

∫

Vσij dV =

1

V

(∫

V sσij dV

s +∫

V wσij dV

w)

(3.10)

where V α(α = w, s) represents the volume of water and solid phases respectively.

Noting that the water pressure is uwδij, where δij is the Kronecker delta, Eq. (3.10)

becomes:

σij =1

V

∫

V sσij dV

s +V w

Vuwδij (3.11)

Furthermore applying Gauss-Ostrogradski theorem to convert volume into surface inte-

gral just like in Eq. (3.4), Eq. (3.11) becomes:

σij =1

V

∑

Np

(∫

Γpxi tjdΓ

p +∫

V pxibj dV

p)

+V w

Vuwδij (3.12)

Here, the last term in Eq. (3.12) simply refers to the partial pressures due to the water

phase with its respective volume fraction.

47

3.3.1 Negligible contact area - rigid particles

We consider incompressible particles so that the contact area is negligible. Thus, as shown in

Fig. 3.4, in the fully saturated case, the traction forces acting on the surface of each particle

actually consist of the inter-particle forces acting at contact points between the particles and

also the water pressure acting normally toward the surface of the particles. In order to apply

the total boundary condition of the sample, the external forces are considered as tractions

acting on the surface of the particles, which are located on the boundary of the REV.

wu

fβα

Rα

Rβ

βα

cxα

x cxβ

fαβ

wu f αβ

f αβ

fαβ

1

2

3

Figure 3.4: Free body diagram of inter-particle forces in saturated media

As a result, replacing the tractions with the inter-particle forces and water pressure and

applying the local equilibrium condition between surface traction and body forces, Eq. (3.12)

can be written as follows:

σij =1

V

∑

Nc

lifj +uwV

∑

Np

∫

ΓpRinj dΓ

p +V w

Vuwδij (3.13)

where nj is the unit normal vector to the particle surface.

Applying the Gauss-Ostrogradski theorem:

∫

ΓpRinj dΓ

p =∫

V p

∂Ri

∂xjdV p =

∫

V pδij dV

p = V pδij (3.14)

48

where V p indicates the volume of the particle. Thus, noting Eq. (3.14), Eq. (3.13) gives the

total stress as follows:

σij =1

V

∑

Nc

lifj +V s

Vuwδij +

V w

Vuwδij =

1

V

∑

Nc

lifj + uwδij (3.15)

The first term on the right hand side of Eq. (3.15) involves the inter-particle forces, and

hence refers to the effective stress σ′

ij acting in the solid skeleton. Thus, the total stress can

be finally written as:

σij = σ′

ij + uwδij (3.16)

which leads to the well-known Terzaghi’s effective stress equation for saturated media with

negligible inter-particle contact area. Note that in line with soil mechanics convention, we

will consider compressive stresses, including pore water pressure to be positive.

3.3.2 Finite contact area - compressible particles

Consider a fully saturated cohesionless granular medium in quasi-static condition with the

contact area between particles being now finite. As shown in Fig. 3.5, the traction forces (tj)

on a given particle surrounded by several neighbouring particles consist of water pressures

acting over the wetted surface (Γpw), and the inter-particle forces arising from the external

forces acting over finite contact areas (Γpd).

Referring back to Eq. (3.12) which describes the total stress equation in terms of various

force transport components, the consideration of finite contact areas leads to:

σij =1

V

(

∑

Np

∫

Γp

d

Rifj dΓpd +

∑

Np

∫

Γpw

Riuwnj dΓpw

)

+V w

Vuwδij (3.17)

The pore pressure transmission on the wetted parts (Γpw = Γp − Γpd) of the particle can

be expressed as the action of pore pressures on the particle as if it was fully wetted minus

the contributions over contacts of finite area, thus:

∫

Γpw

Riuwnj dΓpw =

∫

ΓpRiuwnj dΓ

p−∫

Γp

d

Riuwnj dΓpd (3.18)

49

jf

αβ

jfαβ

p

wΓ

p

dΓ

wu

Figure 3.5: Free body diagram of saturated media with compressible particles

Noting Eq. (3.14) and substituting Eq. (3.18) into Eq. (3.17), we get:

σij =1

V

(

∑

Np

∫

Γp

d

Rifj dΓpd

)

+ uw

(

δij −1

V

∑

Np

∫

Γp

d

Rinj dΓpd

)

(3.19)

Equation (3.19) essentially describes the force transmission into the fully saturated gran-

ular system with finite inter-particle contact area. External loads applied on the boundaries

of the REV are essentially transmitted to particles such that the inter-particle contact forces

are in equilibrium with pore water pressure forces acting on the wetted parts of the particles.

As such the first term of the right-hand-side of Eq. (3.19) can be identified as the effective

stress tensor (σ′

ij), whereas the second term refers to the pore water pressure contribution

which is anisotropic in general, depending on the spatial distribution of contact areas (αij)

which also encompasses the fabric information. Thus,

σij = σ′

ij + αij uw

σ′

ij =1V

(

∑

Np

∫

Γp

dRifj dΓ

pd

)

; αij =(

δij − 1V

∑

Np

∫

Γp

dRinj dΓ

pd

) (3.20)

The physical interpretation of the integral of the term (Rinj) over the dry contact surface

Γdp is associated with the conical volume formed by the contact surfaces as shown in Fig.

50

3.6. This shows that αij is a function of the fraction of the particle contact surfaces over the

total particle surface areas, as well as micromechanical parameters such as the distribution

of contact normals (fabric) and contact areas. These finite contact areas introduce a length

scale in the definition of stress in Eq. (3.20) through the surface area Γdp normalized to the

particle radius R.

Also, when the contact area tends to zero (Γdp → 0), αij → δij, which leads to Terzaghi’s

effective stress equation. When (Γdp → Γp) as in a continuous medium, αij → 0, giving rise

to Cauchy stress.

In fact, the quantity (αij) relates to α the so-called Skempton’s (Skempton, 1960) or

Biot’s (Biot, 1962) effective stress coefficient in soil and rock mechanics. Typically, different

soil/rock properties such as permeability, compressibility and the area of contact between

particles per unit gross area of the material, have been considered in the literature to deter-

mine this parameter. Skempton (1960) assigns a value of 1 to α for soils of negligible contact

area, which refers to αij = δij . As such, the second order tensor (αij) derived in this thesis

can be seen as a generalized Skepton’s or Biot’s effective stress coefficient.

3.3.3 Effective stress in a fully saturated idealized compressible particle packing

Consider a REV consisting of an assembly of compressible mono-size spherical particles of

radius R interacting through smooth contact areas whose dimension is smaller than the

particle size so that non-conformal contact condition can be assumed. Hence, the radius r

of the contacting area between two spherical particles subjected to a normal contact force f

(Fig. 3.7) can be obtained from Hertz’s law (Hertz, 1882), i.e.

r =3

√

3f R

4E(3.21)

where E is the Young’s modulus and Poisson’s ratio has been assumed to be zero for no

lateral deformations. Therefore, the contact area between two spherical particles can be

51

defined as:

Γpc = πr2 = π

(

3f R

4E

)2/3

(3.22)

A particle within the REV (Fig. 3.6) is in contact with nc local neighboring particles so

that the total contact area per particle is Γpd =nc∑

Γpc . The effective stress in the REV can be

determined from Eq. (3.20) where the contribution of the pore pressures is given by αij, i.e.

p

dΓ

p

1Γ

p

2Γ

p

3Γp

4Γ

p

5Γ

p

cΓ=∑nc

nc, = 5

Figure 3.6: Particle in contact with neighboring particles

αij = δij −1

V

Np∑

nc∑

Bij; Bij =∫

Γpc

Rinj dΓpc (3.23)

where Bij describes the oriented contact area with respect to the local reference frame at

the centroid of a spherical particle (Fig. 3.7).

X

Y

Zf

Hertz law

r

Figure 3.7: Local coordinates on the center of each particle

The number of particles contained in the REV is herein considered large enough so that a

continuous probability distribution function can be used to describe the statistics of normal

52

contacts with associated areas. Thus, the double summation over the particle contacts can

be replaced with an integration over a unit spherical REV in 3D Euclidean space (Fig. 3.8).

Assuming axisymmetry about z axis, we get:

αij = δij =∫ 2π

0

∫ π

0bij(β,φ) p

′(n) sin β dβ dφ (3.24)

where bij(β) is the counterpart of Bij expressed in the global reference for a direction β in

space such that:

bij(β,φ) =Mik(β,φ)BklMjl(β,φ) (3.25)

M(β, φ) =

sin2φ+ (1− sin2φ) cos β − sinφ cosφ(1− cos β) − cosφ sin β

− sinφ cosφ(1− cos β) cos2φ+ (1− cos2φ) cos β − sinφ sin β

cosφ sin β sinφ sin β cos β

(3.26)

and

p′(n) =2N c p(n)

V(3.27)

The contact normal probability density function is given by p(n) which defines the sta-

tistical distribution of unit contact normal vectors over the spherical domain of unit volume,

and thus can be related to the fabric tensor Fij through the following:

Fij =1

2N c

2Nc∑

ninj =∫ 2π

0

∫ π

0(ni nj)p(n) sin β dβ dφ (3.28)

with

p(n) ≥ 0,∫

Vp(n) dV = 1 andp(n) = p(−n) (3.29)

Since p(n) is independent of φ and π−periodic as a function of β, a harmonic approxi-

mation of p(n) in 3D space can be made using a Fourier series as follows (Azema et al.2009):

p(n) =1

4π

[

1 + a(

3cos2β − 1)]

(3.30)

53

x

z

y

n

Γ

φdφ

βdβ

Figure 3.8: Spherical REV and global coordination system (Quadfel and Rothenburg,2001)

where a describes the anisotropy of the fabric tensor.

We next compute the local tensor Bij in Eq. (3.25) with the aid of Eq. (3.23, i.e.

Bij=∫

Γpc

Rninj dΓ

pc = Γpc R

0 0 0

0 0 0

0 0 1

(3.31)

It is worth noting that the isotropic part of Bij, i.e Bii/3 represents the conical volume

formed by the contact surface.

The contact (normal) forces within the REV can be described by a first order harmonic

approximation using Fourier series expansion, over a unit spherical domain as before. Thus,

f(β) = f[

1 + an(

3cos2(β − βn)− 1)]

(3.32)

where an is the anisotropy of contact forces, βn is the orientation of the associated principal

direction, and f represents the mean value of contact forces (see Fig. 3.9). It is clear that

since the contact force distribution is given by an even function, the period of this harmonic

function is equal to π.

54

Figure 3.9: Schematic anisotropic force distribution in polar system, an = 0.5 & βn = π/6

Finally, Bij can be computed by invoking Hertz’s law (Eqs. 3.21 & 3.22) in combination

with the contact force (Eq. 3.32) to get the associated contact area and thus,

Bij = Γpc R

[

1 + an(

3cos2β − 1)]2/3

0 0 0

0 0 0

0 0 1

(3.33)

where Γp

c is the mean contact area over the REV.

Finally, the second order tensor αij required to calculate the effective stress can be com-

puted by inserting Eqs. (3.33, 3.25) into Eq. (3.24), i.e.

αij = δij −(

2N c Γp

c R

3V

)

λx 0 0

0 λy 0

0 0 λz

(3.34)

where

λx = λy = 1− 2

5a− 4

15an ; λz = 1 +

4

5a+

8

15an (3.35)

Furthermore, noting that the term (RΓpc/3) refers to the conical volume formed by the

mean contact surface between the particles as shown in Fig. 3.6, 2N c(RΓpc/3) turns out to

be the total conical volumes (V tcone) formed by the contact surfaces in the REV. Therefore,

55

Eq. (3.34 becomes:

αij = δij − µ

λx 0 0

0 λy 0

0 0 λz

; µ =V tcone

V(3.36)

In conclusion, αij is found to be a function of the ratio of the total conical volumes

formed by the contact surfaces as well as the anisotropies of the contact normal forces and

structural fabric of the granular assembly.

Finally, if we consider an isotropic packing with an isotropic contact force distribution

(a = an = 0), we get

αij = (1− µ)δij (3.37)

In the limiting condition of rigid particles, µ→ 0 and αij → δij .

3.4 Force Transport in Unsaturated Granular Media

The study of force transport in an unsaturated granular system is one of the most interesting

cases to analyze, especially in the range of low water saturation, i.e. the pendular regime.

Building upon the work developed previously for both the dry and saturated cases, we

herein examine the case of three-phase system in the pendular regime where independent

liquid bridges are formed between particles as shown in Fig. 3.10.

Consider an unsaturated granular medium with a REV comprised of interacting solid

particles in the presence of both the water and air phases in the voids. The total stress can

be written as a volume average of each individual phase stress over the total volume V , i.e.

σij =1

V

∫

Vσij dV =

1

V

[∫

V sσij dV

s +∫

V wσij dV

w +∫

V aσij dV

a]

(3.38)

where V α, α = s, w, a represents the volume of solid, water and air phases respectively. Divid-

ing the solid phase into individual solid spherical incompressible particles as sub-domains,

56

x

σ

REV ( V )

air volume, a

V

particle volume, s

V

water volume, wV

a w s

V V V V= ∪ ∪

solid

liquid

gas

Figure 3.10: Unsaturated media as a three phase system in pendular state

and applying the Gauss-Ostrogradski theorem to convert volume averaged stress in these

sub-domains into the surface integrals, we can write:

σij =1

V

[

∑

Np

∫

Γpxitj dΓ

p +∑

Np

∫

V pxibj dV

p

]

+V w

Vuwδij +

V a

Vuaδij (3.39)

where uwδij and uaδij denote the hydrostatic pressures of water and air phases respectively.

As a result, the last two terms of the Eq. (3.39) simply refer to the partial pressures due to

the air and water phases with their respective volume fractions applied to each individual

pressure.

Since we are primarily interested in the transportation of forces in the REV, we will

mainly focus on the first term to the right of the above equation related to particle interac-

tions through tractions tj.

Next, suppose the REV is composed of an ensemble of mono-disperse spherical particles of

radius R joined by independent concave liquid bridges with negligible inter-particle contact

area. Among the various surface tractions exerted on an individual particle, we will find

contributions from inter-particle forces, actions of air and water pressures on dry (Γpd) and

wetted (Γpw) surfaces respectively, and surface tension arising from air/water/solid interfaces

formed by water menisci along contour Γm as illustrated in Fig. 3.11-a and b. It is worth

57

mentioning that such a decomposition of stress the various phases as in Eq. (3.39) naturally

includes various types of interfaces such as air/water and air/water/solid (contractile skin),

including their interactions.

Furthermore, noting that xi = xci+Ri, and considering equilibrium of forces on the closed

surface of each particle, (refer to Eq. (3.7)), we finally get:

σij =1V

∑

Nc

lifj +uaV

∑

Np

∫

Γp

dRninj dΓ

pd +

uwV

∑

Np

∑

nl

∫

ΓpwRninj dΓ

pw

− 1V

∑

Np

∑

nl

∫

ΓmRniTj dΓm + V w

Vuwδij +

V a

Vuaδij

(3.40)

where ni, nj are the unit vectors normal to the particle surface, fj is the inter-particle force, li

represents the so-called branch vector defining the separation distance between two particles,

Tj is the surface tension forces per unit length related to water menisci action on Γm formed

by the intersection of the water meniscus with the particles surface, nlis the number of liquid

bridges on each particle, Γpw is the part of the particle wetted by each liquid bridge, whereas

Γpd is the union of all dry parts of the particles surface (see Fig. 3.11).

au n

mΓp

wΓ

1( )p

dΓ

2( )p

dΓ

3( )p

dΓ

1 2 3( ) ( ) ( )p p p p

d d d dΓ = Γ ∪ Γ ∪ Γ

αβf

T

T

T

(a) center particle with 3 neigbours jointed by menisci (b) free body diagram for center particle

with interacting forces

au

mΓ

mΓ

T

T

Tαβ

f

au

wu

αβf

wu

wu

Figure 3.11: Free body diagram of inter-particle forces

Since the inter-particle forces fj are established based on equilibrium conditions during

the interaction of the various phases with the particle skeleton, including any external loads,

58

the first term of Eq. (3.40) is identified as the effective stress, i.e.

σ′

ij =1

V

∑

Nc

lifj (3.41)

Looking back at the surface traction decomposition illustrated in Fig. 3.11, we observe

that capillary effects induced by a concave liquid bridge between two spherical particles have

two sources.

The first source comes from the unequal hydrostatic pressure of air and water around the

closed boundary of each particle. In other words, as shown in Fig. 3.12, due to the super-

position principle, the unequal hydrostatic forces around the particle induce a component of

capillary forces known as suction force (f cap1 ). This suction force appears due to the second

and third terms on the right hand side of Eq. (3.40).

+ =

au on p

dΓ

wu on

p

wΓ

wu on

p

wΓa

u -

Figure 3.12: Unequal hydrostatic forces around the particle

The second source of capillary forces (f cap2 ) originates from the pressure difference between

air and water acting at the interface of these two phases on the boundary of the liquid bridges

(see Fig. 3.13). As explained in the previous chapter, surface tension forces, induced due to

this pressure deficiency, eventually transfer along to the boundary of the wetted area on the

particle surfaces where solid, air and water meet (Γm), giving rise to the so-called contractile

skin. This component appears as the fourth terms on the right hand side of Eq. (3.40).

In calculating the suction force component, the relationship between integrals over wetted

59

1R

2R

au

wu

sT

Figure 3.13: Unequal hydrostatic pressure on the air/water interface

and dry surfaces must be found, i.e.

∫

Γp

d

Rninj dΓpd = V pδij −

∑

nl

∫

Γpw

Rninj dΓpw (3.42)

Further rearrangement of Eq (3.40) along with Eq. (3.42) leads to the tensorial form of

the effective stress equation for an unsaturated granular medium:

σ′

ij = (σij − uaδij) + χij (ua − uw) + Bij (3.43)

with

χij = (n.Sr)δij +1

V

∑

Np

∑

nl

∫

Γpw

Rninj dΓpw (3.44)

and

Bij =1

V

∑

Np

∑

nl

∫

Γm

RniTj dΓm =RTsV

∑

Np

∑

nl

∫

Γm

niej dΓm (3.45)

where Ts is the surface tension value, n is the porosity, Sr is the degree of saturation, and

ej is the unit vector defining the direction of surface tension forces, whereas χij and Bij are

effective stress parameters, which are actually related to the distributional descriptions of

liquid bridges (menisci) and contractile skin effects respectively through surface integrals of

dyadic products of contact normals and surface tension forces as illustrated in Fig. 3.14.

It is worth noting that χij in Eq. (3.44)is dimensionless tensorial quantity which scales the

matric suction (ua−uw) to account for the spatial distribution (fabric) of liquid bridges and

60

θ T

T

T

T

n

a

b

p

wΓ

p

dΓ

Γ

wu

au

f

au

au

wu

n

countour

α

n

θ T

T

T

T

a

b

c

d

( ) : p

wd c− = Γ ( ) : ma b− = Γ

n

filling angle (here at max); =wetting angleα θ=

m

Figure 3.14: Traction forces between a pair of spherical particles

associated wetted areas. Given that the fabric of the liquid bridges is generally anisotropic,

this makes χij anisotropic, which leads to an anisotropic capillary stress due to matric suction

(χij(ua − uw)) as well in the REV.

On the other hand, Bij in Eq. (3.45) refers to a stress induced by surface tensions acting

along the contractile skins at particle contours Γm over the REV. Here again, this quantity

is seen to involve surface tensions being scaled by the spatial distribution of contractile skins

throughout the REV.

Finally, in line with the discussion of the nature of capillary forces developed among

particles, we define a capillary stress tensor as:

ψij = χij(ua − uw) + Bij (3.46)

In contrast with the current literature, capillary stress always refers to a suction stress

arising from the pressure difference between air and water phases at the particle contacts.

Herein, the capillary stress emerges with two distinct components in the form of suction and

surface tension induced stresses based on a proper decomposition of forces at the particle-

particle contact level in micromechanical derivations.

It is this capillary stress that increases the effective stress in an unsaturated medium,

and thus enhances its shear strength. On the other hand, in the absence of other cohesive

61

forces such as van der Waals or double layer attraction and cementation, this capillary stress

can also give rise to an apparent tensile strength.

More interestingly, the capillary stress as defined in Eq. (3.46) is not isotropic, but

deviatoric in nature by virtue of the matric suction and menisci distribution, the degree

of saturation as well as particle packing. The property of the capillary stress engenders a

meniscus based shear strength that increases with the anisotropy of the particle packing.

3.4.1 Effective stress parameters for idealized packing

The determination of effective stress parameters, χij and Bij, requires defining the wetted

surface (Γpw) and the contour (Γm) at the interface of air-water-solid. These can be obtained

by finding the geometry of the liquid bridge between two spherical particles as illustrated

in Fig. 3.15 for a given volume of liquid, separation distance H and wetting angle θ. The

volume of the liquid V LB is parametrized by the half filling angle α, while its geometry

can be approximated using the so-called toroidal (Megias Alguacil and Gauckler, 2009). An

alternative method would involve solving Young-Laplace equation involving the curvature

of the liquid bridge between two spheres and the pressure difference through a non-linear

differential equation (Molenkamp, 2003).

HN

1R

2RRα

θ 1R

2R

Figure 3.15: Concave liquid bridge geometry between a pair of uni-size particles

62

The toriodal approximation assumes constant pressure difference along the meniscus

which is assumed to have the shape of a surface of revolution with constant curvatures R1

and R2 as function of α, H, and θ, see Fig. 3.15, (Fisher, 1926; Gillespie and Settineri,

1967; Megias-Alguacil and Gauckler, 2009). As such, the volume of the liquid bridge can be

explicitly formulated in terms of parameters such as half filling angle, separation distance,

wetting angle, see Appendix I. Introducing a dimensionless liquid bridge volume (Vrel =

V LB/V p), the corresponding half filling angle α can be readily calculated as shown in Fig.

3.16 for illustration purposes. Once, α is known, the curvaturesR1 and R2 can be determined,

and hence the matric suction corresponding to the specified liquid bridge volume can be found

through Young-Laplace equation:

ua − uw = Ts

(

1

R1

− 1

R2

)

(3.47)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80

Vre

l

H/R=0, θ=40

H/R=0.1, θ=40

H/R=0.5, θ=40

H/R=0, θ=20

H/R=0.1, θ=20

H/R=0.5, θ=20

α

o

o

o

o

o

o

Figure 3.16: Dimensionless liquid volume Vrel as a function of the half filling angle α(Megias-Alguacil and Gaucker, 2009)

It is noted that the solution of Eq. (3.47) can result in R1 > R2 or R1 < R2, which

lead to the development of negative or positive matric suction (ua− uw) in the liquid bridge

63

respectively. Negative matric suction is most likely to occur in real soil-water systems when

the soil is almost saturated (Lu and Likos, 2004). In this thesis, the degree of saturation

is limited to the amount which leads to positive matric suction in water lenses, so that the

water phase is maintained in negative pressure with respect to the air (R1 < R2). The

regions of positive and negative pore water pressure as a function of the half filling angle

and the wetting angle is illustrated in Fig. 3.17.

0

10

20

30

40

50

60

0 10 20 30 40 50 60

hal

f fi

llin

g a

ng

le (

deg

ree)

wetting angle (degree)

Positive matric suction

Negative matric suction

Figure 3.17: Positive and negative matric suction zones as a function of the half filling andwetting angles (Lu and Likos, 2004)

Consequently, estimating the half filling angle for a specific volume of water between two

spherical particles, the integrations in Eqs. (3.44 and 3.45) can be realized. Additionally,

knowing the number of liquid bridges in the REV, the degree of saturation and water content

can be readily calculated.

Here, the unit vectors normal to the surface of the spherical particles can be written

as ni = Ri/|R| where the radius vector Ri varies with the half filling angle α. Also, the

unit vector ei define the direction of the surface tension forces in the contractile skin as

a function the wetting angle and half filling angle as shown in Fig. 3.15. Therefore, the

64

following integral with respect to the local reference (X, Y, Z) at the center of each particle

(as shown in Fig. 3.18) can be calculated as:

Aij =∫

ΓpwRninj dΓ

pw

= πR3

3

(1− cosα)2(2 + cosα) 0 0

0 (1− cosα)2(2 + cosα) 0

0 0 2(1− cos3α)

(3.48)

B′

ij =∫

ΓmRTs niej dΓm

= (πR2Ts)

−sin2αcos(α + θ) 0 0

0 −sin2αcos(α + θ) 0

0 0 sin(2α)sin(α + θ)

(3.49)

X

Y

Z

Figure 3.18: Local coordinates illustration for each liquid bridge

It is interesting to discuss two extreme conditions referring to dry and saturated states.

In dry condition, since (Sr = 0) and α = 0, then χij = Bij = 0 so that Eq. (3.43)

becomes σ′

ij = σij − uaδij. On the other hand, in the saturated case (Sr = 1), α = 180◦,

thus Aij = Vpδij, which leads to χij = δij and Bij = 0. As such Eq. (3.43) becomes

σ′

ij = σij − uwδij .

Next lets assume the REV conatins a large number of particles so that we can consider the

distribution of normal contacts and liquid bridges to be continuous variables. Thus, an ap-

proximated theoretical probability density function can be applied to define the distribution

65

of the contact normal vectors and liquid bridges in the REV.

Considering axi-symmetric condition and the probability density function of the unit

normal vectors p (n) being π-periodic and independent of φ, a harmonic approximation of

p (n) in 3D space can be made using the Fourier series, i.e.

p (n) = 14π

[1 + a (3cos2β − 1)]

p(n) ≥ 0;∫

V p(n) dV = 1; p (n) = p (−n)(3.50)

where a shows the anisotropy of fabric, while the fabric tensor is expressed by Eq. (3.28).

The anisotropy parameter a can be defined as (Radjai, 2008):

a =5

2

(Fz − Fx)

tr(Fij)=

5

2(Fz − Fx) (3.51)

On the other hand, the liquid bridges probability density function, which demonstrates

the liquid bridges distribution in the REV, can be introduced as:

pLB (n) = 14π

[1 + aLB (3cos2β − 1)]

pLB(n) ≥ 0;∫

V pLB(n) dV = 1; pLB (n) = pLB (−n)

(3.52)

where aLB shows the anisotropy of the liquid bridges distribution in the REV, and thus is

only related to the distribution of the unit normal vectors aligned with the liquid bridges.

The liquid bridges fabric tensor can written as:

FLBij =

1

2NLB

∑

2NLB

ninj =∫ 2π

0

∫ π

0(ninj) p

LB (n) sin β dβdφ (3.53)

while NLB is the total number of liquid bridges in the REV and ni is the unit normal vector

associated with these liquid bridges. The anisotropy factor can also be defined as:

aLB =5

2

(

FLBz − FLB

x

)

tr(FLBij )

=5

2

(

FLBz − FLB

x

)

(3.54)

Assuming there is no liquid bridge formed between adjacent particles with no contact,

and the fabric tensor is coaxial with the liquid bridges fabric tensor, while the probability of

finding a liquid bridge on each contact is considered to be directionally independent, pLB(n)

66

can be assumed to be equal to p(n). Therefore, Eqs. (3.44 and 3.45) can be written as

follows:

χij =V w

Vδij +

∫ 2π

0

∫ π

0MikAklMjl p

′(n) sin β dβdφ (3.55)

Bij =∫ 2π

0

∫ π

0MikB

′

klMjl p′(n) sin β dβdφ (3.56)

where

p′ (n) =2NLBpLB (n)

V=

2N cp (n)λ

V(3.57)

with λ is the ratio of number of liquid bridges over the number of contacts in the REV.

Therefore, having the degree of saturation, number and fabric of the contacts, as well as

the probability of finding a liquid bridge on each contact point, χij and Bij can be readily

computed.

3.5 Summary

In this Chapter, the effective stress equation for a three-phase granular medium in the pen-

dular regime was formally derived through a micromechanical analysis. The cases of fully

saturated (two-phase) conditions with and without particle compressibility were also inves-

tigated. As such, the physical significance of the effective stress parameter (χ) as originally

introduced in Bishops equation has been elucidated. More interestingly, an additional pa-

rameter that accounts for surface tension forces arising from the so-called contractile skin

emerges in the newly proposed effective stress equation.

It turns out that χ is generally not a scalar, but is rather a tensorial quantity described

that is generally a function of degree of saturation, particle packing as well as water menisci

distribution. We introduce a so-called capillary stress that is anisotropic in nature as dictated

by the spatial distribution of water menisci and fabric of the solid skeleton evolving during

deformation history. The capillary stress is shown to have two contributions: one emanating

from suction between particles due to air-water pressure difference (related to χij), and the

67

second arising from surface tension forces along the contours between particles and water

menisci (Bij). Issues on the significance of this new formulation in the analysis of capillary

stresses in granular systems in the pendular regime with regular packing will be investigated

in the next chapter.

68

Chapter 4

COMPUTATION OF CAPILLARY STRESSES IN

IDEALIZED GRANULAR PACKINGS

4.1 Introduction

In this chapter, the importance of the fabric of granular media in the determination of

capillary stresses in the pendular regime is studied in the context of the effective stress

equation derived in Chapter 3. To keep calculations tractable, regular packings of mono-

sized granular assemblies are investigated to get valuable insights in the proposed effective

stress formulation. The anisotropy of the capillary stress as a function of packing, liquid

bridge distribution, degree of saturation, wetting angle and particle separation are finally

demonstrated through simple examples.

4.2 Idealized Packing

Various idealized periodic or regular packings of non-overlapping, mono-sized spheres in the

3D Euclidean space are herein introduced. Furthermore, for analysis of fundamentals, a

representative elementary volume (REV) of these packings can be described as a unit cell

which is essentially the simplest repeating unit of the global system. In crystallography,

there are three main varieties of the regular mono-size sphere packings in Euclidean space,

namely: (1) simple cubic (SC), (2) body-centered cubic (BCC), and (3) face-centered (FC),

also known as cubic close-packing (CCP).

69

4.2.1 Simple cubic packing (SCP)

This packing is one of the loosest regular spherical particle assemblies possible as shown

in Fig. 4.1a. A periodic particle arrangement can be extracted from the assembly with a

central particle sharing eight neighbouring basic cubes as depicted in Fig. 4.1b. As such,

this arrangement can be further reduced into a basic unit cell (REV) containing only one

particle which will suffice to represent the whole assembly (Fig. 4.1c).

(a) (b) (c)

Figure 4.1: Illustration of simple cubic packing (SCP)

Working with this basic unit cell, a porosity of 0.476 and a coordination number of 6 can

be calculated (see Fig. 4.1). It is evident that this packing is isotropic because the contact

normals are equally distributed in all three directions of the reference frame. As such recall

the fabric tensor as:

Fij =1

2N c

2Nc∑

ninj (4.1)

where n refer to the unit normal vector defining a contact and N c is the total number

of contacts. Hence, the fabric tensor associated with the simple cubic packing is simply

Fij = δij/3.

70

4.2.2 Body-centered cubic packing (BCC)

Figure 4.2a shows a BCC packing where further examination reveals that the central spherical

particle in each unit cell is in contact with eight more spheres on the corners of the cell. As

seen in Fig. 4.2b), the unit cell contains two particles (one central plus 8 times one-eight of

neighbourbing particle) with its dimensions L and L′ controlled by the radius of the particle

and the separation distance H between two adjacent particles. This separation distance is

conveniently introduced here to mimic the surface roughness of a particle as shown in Fig.

4.3. Therefore, the dimensions of the cubic unit cell can be readily calculated as (Molenkamp

and Nazemi, 2003):

L

L

L'

(a) (b)

z

x

y

Figure 4.2: Illustration of body-Centered Cubic Packing (BCC)

(L′)2 + 2L2 = 16(R +H/2)2, or

(l′)2 + 2l2 = 16(

1 +H/2)2; l = L/R, l′ = L′/R, H = H/R

(4.2)

Due to geometrical compatibility,

1 ≤ l′/2 <√2 and 1 ≤ l/2 <

√

3

2(4.3)

The coordination number associated with a BCC packing is generally 8, but this can

change to 10 depending on the values of l and l′. For instance, in a specific condition where

l′ = 2, the central particle will make contact with two additional particles along the l′

71

H

α

Figure 4.3: Separation distance between particles , H (Pietsch, 1968)

Table 4.1: Properties of BCC packings with various l′

l′ Porosity (n) Fx Fy Fz a

2.1 0.312 0.362 0.362 0.276 -0.222.2 0.318 0.335 0.335 0.330 -0.012

2.3094 0.320 0.333 0.333 0.333 02.4 0.318 0.320 0.320 0.360 0.102.5 0.313 0.305 0.305 0.390 0.212.6 0.303 0.289 0.289 0.422 0.332.7 0.288 0.272 0.272 0.456 0.46

direction to increase the coordination number to 10. Also, the anisotropy, and therefore the

fabric, of the packing changes with the variations in the contact directions controlled by l

and l′. When l = l′ = 4/√3, the packing becomes isotropic. In other cases, the fabric tensor

can be explicitly computed from Eq. (4.1) where the fabric tensor can be written using its

three eigenvalues Fx, Fy, Fz while eigenvectors coincide with the unit vectors of the three

orthogonal axes of the cubic unit cell, i.e.

Fij =

Fx 0 0

0 Fy 0

0 0 Fz

(4.4)

Hence, the porosity and fabric tensor components for a BCC packing as a function of

dimensions l and l′ are given in Table. 4.1. The anisotropy factor a as defined back in Eq.

(3.51) is also included in Table. 4.1.

In anticipation to capillary stresses which will be computed in the next sections, the unit

72

cell for the BCC packing can be equivalently replaced with a polyhedral cell enclosing only

one spherical particle as depicted in Fig. 4.4. This will facilitate the tensorial calculation

of the various contributions of forces acting on a particular central particle whereby no

intersection exists between adjacent unit cells whose boundary conditions are well-defined.

L

L

L'

x

z

y

Figure 4.4: Arrangement of BCC packing unit cells in 3D space

4.2.3 Cubic Close Packing or Face Centered Packing (CCP or FCP)

The cubic close or face-centered packing is actually a special case of a body-centered cubic

packing, in which l′ = 2√2. In this situation, the central spherical particle will come in

contact with four more spherical particles associated to adjacent unit cells in the same

horizontal layer, and thus the coordination number increases to 12 (Fig. 4.5). Calculating

the distribution of the contact normal vectors around the central particle, the packing turns

73

out to be isotropic, i.e. Fij = δij/3. This packing is proven to be the densest possible packing

of the mono-sized spheres, while its porosity is equal to 0.26 (Hales, 2005).

Figure 4.5: Illustration of face centered packing (FCP)

To summarize, simple cubic packing (SCP) and face centred packing (FCP) actually

represent the loosest and densest isotropic packings possible, respectively. Fig. 4.6 illustrates

a comparison between the porosity of the different packings generated for various values of

l′ for BCC as well as SCP and FCP. It is seen that for certain values of l′, there cannot be

any regular packing based on geometrical considerations.

0.25

0.3

0.35

0.4

0.45

0.5

2 2.2 2.4 2.6 2.8 3

poro

sity

(n)

l'

BCCSCPFCP

Figure 4.6: Porosity of different regular spherical packing

74

4.3 Theoretical SWCC for Regular Packing in Pendular Regime

Here, soil-water characteristic curves for SCP and FCP, referring to the loosest and densest

mono-sized spherical packing respectively, are calculated theoretically. Under the absence

of any volumetric strains, the void spaces in the REV are filled with liquid bridges; thereby

increasing the degree of saturation with the resulting matric suction being calculated. Details

of the calculation steps in determining the SWCC can be found in Table. 4.2.

Figure 4.7 shows the theoretical SWCC for SCP and FCP spherical packings as a function

of particle radius, i.e. 0.001 and 0.1 mm. Both the wetting angle and separation distance are

considered to be zero and the surface tension parameter, Ts is assumed to have a value of 74

µN/m. As shown, reducing the size of the grains leads to greater values of matric suction as

expected and in accordance to experimental data on sandy soils (Fredlund and Xing, 1994).

0.1

1

10

100

1000

10000

100000

1000000

0 5 10 15 20 25

mat

ric

suct

ion (

kP

a)

degree of saturation (Sr%)

SCP , R=0.1 mm

SCP, R=0.001 mm

FCP, R=0.1 mm

FCP, R=0.001 mm

Figure 4.7: SWCC of SCP and FCP as a function of particle size, H = θ = 0

75

Table 4.2: SWCC calculation1. k = 1 (k is degree of saturation index)

2. Set the value of θ, R, Ts, V, n,NLB, H

3. Set S0r = Sinitialr ,∆Sr = 0

4. Skr = Sk−1r +∆Sr

5. Compute the volume of liquid bridges:

V kw = V Skrn

V kLB = V k

w/NLB

6. Compute αk, Rk1 , R

k2 , N

kby solving the following system of equations (Toroidal approxi-

mation):

VkLB =

V kLB

2πR3 =[

(Rk1 +R

k2)

2+ (R

k1)

2]

Nk − (N

k)3

3−(

Rk1 +R

k2

)

[

Nk

√

(Rk1)

2− (N

k)2+ (R

k1)

2arcsin

(

Nk

Rk1

)

]

− (Nk−

H2)2

3(3−N

k+ H

2)

Rk1 =

Rk1

R=

(H2+1−cosαk)

cos(αk+θ)

Rk2 =

Rk2

R= sinαk +R

k1(sin

(

αk + θ)

− 1)

Nk= H

2+ 1− cosαk,H = H/R

7. Compute corresponding matric suction (Laplace equation):

(ua − uw)k = Ts

(

1Rk

1

− 1Rk

2

)

8. Calculate dimensionless suction, ψk= (ua − uw)

kR/Ts

9. If αk < αmax (αmax = 45◦for SCP, 30◦ for FCP)

∆Sr = c (c 6= 0 is a constant)

Set k = k + 1 and go to 4

else stop

76

The results are identical to the SWCC calculated by Lu and Likos (2004) for the same

packings. It should be noted that the range of degree of saturation examined in this study

is based on the idealized mono-sized spheres and falls below 25% since the menisci are not

allowed to merge in the pendular regime. As such, the maximum value of the half filling angle

(α) are found to be 45◦ and 30◦ for SCP and FCP respectively. Exceeding this upper limit,

the liquid bridges gradually start to merge with each other, and the geometrical assumptions

in order to solve the Young-Laplace equation are no longer valid.

As discussed in Chapter 2, within the pendular regime, the hysteretic behaviour of SWCC

is mainly affected by wetting angle hysteresis. Commonly, the near zero wetting angles

correspond to the drying process, while larger wetting angles, even as high as 60◦, are

reported to be associated with the wetting process (Bear, 1979). Figure 4.8 demonstrates

the effect of wetting angle on the SWCC shape for both SCP and FCP with θ = 0◦ and

θ = 30◦ referring to a drying and wetting path respectively. Thus, hysteretic behaviour in the

SWCC emerges as a hysteresis in wetting angle. Lu and Likos (2004) reported comparable

results for the same packings.

The effect of various dimensionless inter-particle separation distances on the shape of

SWCC for SCP and FCP is also presented in Fig. 4.9. As a result, when the particles are

in physical contact with each other (H = 0), a near zero saturation can lead to a significant

matric suction, while at higher degrees of saturation, the matric suction decreases. By

contrast, for larger separation distances (H 6= 0), matric suction increases from zero to a

peak value and then decreases as the degree of saturation goes up. Similar results for a

packing of two spherical particles were reported in literature by Molenkamp and Nazemi

(2003).

77

0.1

1

10

100

0 10 20 30 40

dim

ensi

onle

ss m

atri

c su

ctio

n

degree of saturation (%)

(b)

o

0.01

0.1

1

10

100

0 5 10 15 20 25 30

dim

ensi

on

less

m

atri

c su

ctio

n

degree of saturation (%)

(a)

θ = 0

θ = 30o

odrying,

wetting

drying

wetting

drying

wetting,

schematic path

between wetting/drying

cycle

θ = 0

θ = 30

o

o

drying,

wetting,

schematic path

between wetting/drying

cycle

Figure 4.8: Effect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and(b) Dense packing (FCP), H = 0

78

0.1

1

10

100

0 5 10 15 20

dim

ensi

onle

ss

mat

ric

suct

ion

degree of saturation (Sr%)

(a)

0.1

1

10

100

0 5 10 15 20 25 30

dim

ensi

on

less

mat

ric

suct

ion

degree of saturation (%)

(b)

0H =

0.05H =

0.1H =

0H =

0.05H =

0.1H =

Figure 4.9: Effect of separation distance on SWCC for (a) Loose packing (SCP), and (b)Dense packing (FCP), θ = 0

79

4.4 Effective Stress Parameters and Capillary Stress in Regular Packing

In this section, the implications of the newly derived effective stress equation are investigated;

the effective stress parameters and capillary stresses are theoretically evaluated for various

regular spherical packing with one liquid bridge associated to every contact. In this way,

the distribution of liquid bridges is considered to be the same as the distribution of branch

vectors. Recalling Eq. (3.55) and (3.56), the effective stress parameters (χij and Bij), can

be calculated as shown in Table. 4.3 and 4.4 under no deformation as before.

4.4.1 Isotropic packings

4.4.1.1 Effective stress parameters and capillary stresses in SCP and FCP

It is interesting to note that Eq. (3.56) leads to a zero capillary stress due to the surface

tension forces, (Bij) for an isotropic packing with zero wetting angle. From a physical point

of view, the surface tension forces T become orthogonal to a radial outward normal vector n

(refer to Fig. 3.14), and because of symmetry and isotropy reasons there is no contribution

from surface tension forces arising from meniscus/particle interface when summing over all

contacts and liquid bridges within the granular assembly. The vanishing of Bij under such

conditions is to be expected since it represents tensor moment of forces in the granular

assembly.

The isotropy of the packing also leads to an isotropic effective stress parameter χij. The

first invariant of this tensor can be compared with Bishops effective stress parameter (χ)

which can now be analytically calculated for various degrees of saturation for both SCP and

FCP configurations with zero wetting angles as shown in Fig. 4.10. For both SCP and FCP

packing, a larger separation distanceH results in smaller values of χij for the same saturation

degree. As discussed before, the separation distance H can be viewed as an indication of

the particles’ surface roughness (Lian et al, 1993). Therefore, the rougher the particles, the

lesser the induced capillary stress, χij(ua− uw) , due to the same amount of matric suction.

80

Table 4.3: χij calculation1. k = 1 (k is degree of saturation index)

2. Set the value of θ, R, Ts, V, n,NLB, H, aLB

3. Set S0r = Sinitialr ,∆Sr = 0

4. Skr = Sk−1r +∆Sr

5. Compute the volume of liquid bridges:

V kw = V Skrn

V kLB = V k

w/NLB

6. Compute αk, Rk1 , R

k2 , N

kby solving the system of equations, presented in Table. 4.2

7. Compute corresponding matric suction (Laplace’s equation) (ua − uw)k = Ts

(

1Rk

1

− 1Rk

2

)

8. Akij =πR3

3

(

1− cosαk)2(2 + cosαk) 0 0

0(

1− cosαk)2(2 + cosαk) 0

0 0 2(1− cos3αk)

9. For variables β ,φ put

pLB (n) = 14π

{1 + aLB [3cos2 (β)− 1] }

p′(n) = 2NLBpLB (n) /V

M(β, φ) as in Eq. 3.26

10. Compute χkij

χkij =V kw

Vδij +

∫ 2π0

∫ π0 MilA

klmMjmp

′(n) sin βdβdφ

11. If αk < αmax (αmax = 45◦for SCP, 30◦ for FCP)

∆Sr = c (c 6= 0 is a constant)

Set k = k + 1 and go to 4

else stop

81

Table 4.4: Bij calculation1. k = 1 (k is degree of saturation index)

2. Set the value of θ, R, Ts, V, n,NLB, H, aLB

3. Set S0r = Sinitialr ,∆Sr = 0

4. Skr = Sk−1r +∆Sr

5. Compute the volume of liquid bridges:

V kw = V Skrn

V kLB = V k

w/NLB

6. Compute αk, Rk1 , R

k2 , N

kby solving the system of equations, presented in Table. 4.2

7. Compute corresponding matric suction (Laplace’s equation) (ua − uw)k = Ts

(

1Rk

1

− 1Rk

2

)

8. B′kij = πR2Ts

−sin2αk cos(αk + θ) 0 00 −sin2αk cos(αk + θ) 00 0 sin(2αk) sin(αk + θ)

9. For variables β ,φ put

pLB (n) = 14π

{1 + aLB [3cos2 (β)− 1] }

p′(n) = 2NLBpLB (n) /V

M(β, φ) as in Eq. 3.26

10. Compute Bkij

Bkij =

∫ 2π0

∫ π0 MilB

′klmMjmp

′(n) sin βdβdφ

11. If αk < αmax (αmax = 45◦for SCP, 30◦ for FCP)

∆Sr = c (c 6= 0 is a constant)

Set k = k + 1 and go to 4

else stop

82

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20

degree of saturation (%)

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

degree of saturation (%)

(b)

ijχ

ijχ

0H =

0.05H =

0.1H =

0.2H =

0H =

0.05H =

0.1H =

H 0.2=

Figure 4.10: The resulting isotropic effective stress coefficient χij while θ = 0(a) Loosepacking (SCP) and (b) Dense packing (FCP)

83

Numerical results are shown in Fig. 4.11 together with actual experimental data for

various soils in the background. It should be noted that there is herein no attempt to match

the experimental data, given that idealized isotropic packings are considered. The range of

degree of saturation, examined in the numerical computations based on idealized mono-sized

spheres, is well below 30% since the menisci are not allowed to merge to give full saturation.

Also, the experimental data in the range of small degrees of saturation investigated (less

than 30%) is scare and not quite reliable, given known difficulties in measuring low suction in

soils. The observations made in this exercise demonstrate that the effective stress parameter

is surely a function of packing as illustrated, herein, for the isotropic case.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100

Silt, Drained test (Donald, 1961)

Silt, Constant water content test (Donald, 1961)

Madrid gray clay (Escario and Juca, 1989)

Madrid silty clay (Escario and Juca,1989)

Madrid clay sand (Escario and Juca, 1989)

Moraine (Blight, 1961)

Boulder clay (Blight, 1961)

Clay-shale (Blight, 1961)

FCP, n = 0.26

SCP, n = 0.49

0.9

1.0

effe

ctiv

e st

ress

par

amet

er χ

rSχ =

degree of saturation (%)

Figure 4.11: Computed relationships between degree of saturation and effective stress pa-rameter for various packings

Next, the capillary stresses induced by suction and surface tension forces (where θ 6= 0)

are also illustrated separately for SCP and FCP packings with various wetting angles. The

radius of the grains is considered to be 0.1 mm and the surface tension parameter (Ts) is

considered equal to 74 µN/m. Here again, because the liquid bridges distribution is isotropic,

the capillary stress tensor (ψij) is isotropic with the difference that it has both suction

84

(χij(ua − uw)) and surface tension (Bij) contributions. As shown in Fig. 4.12 and 4.13,

while the matric suction increases, the role of suction forces in generating capillary stresses

gradually increases, whereas the effect of surface tension forces progressively disappears for

both packings. Moreover, the wetting angle affects the induced capillary stress in opposing

ways. The largest capillary stress due to suction forces is associated with the lowest wetting

angle; while the larger the wetting angle, the more capillary stress is induced due to surface

tension forces.

It is also interesting to investigate the relative contributions of the surface tension forces

(term Bij) and the suction forces between particles (term χij) to the capillary stress ψij.

We thus re-examine the two packings (SCP and FCP) with now a wetting angle of 30◦ for

illustrative purposes. Fig 4.14, shows the relative contributions of surface tension and suction

cross over at a characteristic matric suction value of about 1 kPa for the loose case.

In Fig. 4.14a for a loose packing, the surface tension effect arising from the parti-

cle/meniscus interface dominates at small matric suctions less than 1 kPa, but is ultimately

overtaken by the suction effect at large matric suctions. For the dense case in Fig. 4.14b, the

contribution of surface tension is also smaller than that of suction above matric suctions of 1

kPa. As seen in Fig. 4.14b, no data can be calculated below a suction of 1 kPa because the

corresponding degree of saturation becomes large so that the assumption of pendular regime

with independent liquid bridges cannot be satisfied. The high degree of saturation requires

the liquid bridges to merge, which invalidates the model. Thus, the results presented in Fig.

4.14 suggests that contractile effect of surface tension is more likely to be important for loose

materials and at low matric suctions, i.e. high water saturations when the menisci are well

developed.

4.4.1.2 Isotropic tensile strength in comparison with experimental results

As discussed, the formation of liquid bridges gives rise to capillary stress that can also be

considered as capillary-induced tensile strength normally observed in unsaturated soils. In

85

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100 1000 10000

matric suction (kPa)(a)

θ= 0

θ=10

θ=20

θ=30

o

o

o

o

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100 1000 10000

matric suction (kPa)

(b)

θ= 0

θ=10

θ=20

θ=30

()

ija

wu

u

χ−

ijB

(kPa)

(kPa)

o

o

o

o

Figure 4.12: The capillary stress induced by (a) Suction forces, (b) Surface tension forces ina loose packing (SCP)- R =0.1 mm, H = 0

86

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000matric suction (kPa)

(a)

θ= 0

θ=10

θ=20

θ=30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100 1000 10000

θ= 0

θ=10

θ=20

θ=30

matric suction (kPa)

(b)

()

ija

wu

uχ

−(k

Pa)

ijB

(kP

a) o

o

o

o

o

o

o

o

Figure 4.13: The capillary stress induced by (a) Suction forces, (b) Surface tension forces ina dense packing (FCP)- R =0.1 mm, H = 0

87

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100 1000 10000

Capil

lary

st

ress

(kP

a)

matric suction (kPa)(a)

ij a w(u u ) χ −

ijB

ijψ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1 10 100 1000 10000

Cap

illa

ry

stre

ss (

kP

a)

ij a w(u u ) χ

ijB

ijψ

−

matric suction (kPa)(b)

Figure 4.14: The total capillary stress in (a) Loose packing (SCP) (b) Dense packing (FCP)-R =0.1 mm, θ = 30◦ and H = 0

88

this section, the validity of this proposed tensile strength (ψij) is examined in comparison

with experimental results of direct tension tests on uniform sandy samples with low degrees

of saturation from literature.

Kim (2001) conducted a series of direct tension tests on samples of washed (free of fines)

and poorly graded Ottawa silica sand (F-75) with various void ratios, and reported the

induced tensile strength due to capillary forces in low degrees of saturation. The coefficient

of uniformity (cu) of the samples was measured and equal to 2, particles specific gravity was

considered equal to 2.65( ASTM standard D854), and the mean particle radius was reported

as 0.11 mm (ASTM standard D422). Table. 4.5 summarizes results from direct tension

testing of F-75 samples with the void ratio of 0.72 and 0.58.

Table 4.5: Direct tensile test results of clean F-75 sand (Kim, 2001)Direct tension test-e =0.72 Direct tension test-e =0.58

w (%) Tensile Strength (Pa) w (%) Tensile Strength (Pa)0.47 412.61 0.43 492.841.04 584.53 1.04 716.332.11 699.14 2.01 962.75

In order to make a comparison between theoretical and experimental data, the tensile

strength (ψij) for FCP and isotropic BCC packings, both consisting of spherical particles

with radius of 0.11mm and wetting angles of θ = 0◦ and 20◦, is calculated. The dimensionless

surface roughness is chosen equal to H = 0.09, whereas the void ratios chosen for FCP (void

ratio, e ≈ 0.56) and isotropic BCC packing (void ratio, e ≈ 0.70) corresponds best to those

of the experimental samples. Because in reality almost no absolutely smooth particle exists

in a soil specimen, Pierrat and Caram (1997) indicated that the most accurate estimation

of dimensionless surface roughness lies between 0.01 to 0.1, which is in agreement with the

assumptions made. Fig. 4.15 illustrates the comparison between measured and predicted

data in the pendular regime. With respect to the variations in shape, size and surface

roughness of the particles in the real sample, all of which affect the experimental results, the

predicted data are in agreement with the measured results.

89

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5 3 3.5

ten

sile

str

eng

th (

Pa)

volumetric water content , (w %)

Predicted tensile strength, FCP, θ =20

Prediced tensile strength, BCP, θ =20

Predicted tensile strength, FCP, θ =0

Predicted tensile strength, BCC, θ =0

Experimental data, F-75 (e=0.72)

Experimental data,F-75 (e=0.58)

o

o

o

o

Figure 4.15: Compression between measured and predicted tensile strength

4.4.2 Anisotropic packings

We next turn to anisotropic packings to demonstrate the anisotropic nature of the capillary

stresses due to suction (χij(ua − uw)) and surface tension (Bij) forces as two controlling

components of the total capillary stress ψij.

4.4.2.1 Evolution of capillary stress in BCC packing-anisotropy aspects

Here, an anisotropic BCC packing with l′ = 2.7 and R = 0.1mm is considered as an example,

see Fig. 4.2 far back at the beginning of this chapter. The distribution of liquid bridges is

considered to be the same as the that of contacts in the domain, thus (λ = 1). Furthermore,

the wetting angle and surface roughness are assumed to be zero.

Figure 4.16 illustrates the evolution of the capillary stress ψij including its individual

components χij(ua − uw) and Bij as function of degree of saturation. It is recalled herein

that the capillary stress has two contributions: one arising from suction (χij(ua − uw)) and

another one from surface tension (Bij). The anisotropic nature of these stresses is clearly

demonstrated with the major and minor principal directions of the capillary stress being

90

aligned with the vertical direction z (axial) and x=y (lateral) respectively.

0.1

0.2

0.3

30

210

60

240

90

270

120

300

150

330

180 0

Sr=14%

Sr=9.8%

Sr=5.6%

Sr=0.5%

Sr=0.02%

ijB

°

1

2

3

30

210

60

240

90

270

120

300

150

330

180 0

°

( )ij u uχ − (kPa)

1

2

3

30

210

60

240

90

270

120

300

150

330

180 0

°

°°

°

°°

° °°°

ijψ

a w (kPa)

(kPa)

Figure 4.16: Polar plot of anisotropic capillary stresses for various saturation degree,H = θ = 0

The capillary stresses component in each one of the principal directions is further shown

in Fig. 4.17 as a function of matric suction. Since the wetting angle is zero, the amount

of capillary stress induced by surface tension forces (Bij) is smaller in comparison with the

capillary stress due to suction forces (χij(ua − uw)).

Moreover, the effect of various wetting angles on capillary stresses in the same packing

while H = 0 and Sr = 14% is also demonstrated in Fig. 4.18. As shown, while the wetting

angle is zero the capillary stress due to surface tension forces (Bij) is minimized. However,

increasing the wetting angle enlarges this capillary stress, and thus significantly increases the

91

0

0.5

1

1.5

2.5

3

3.5

1 10 100 1000 10000

cap

illa

ry s

tres

s (k

Pa)

matric suction (kPa)

(axial)ijB

( )(axial)ij a wu - uχ

(axial)ijψ

2

- 0.5

0

0.5

1

1.5

2

1 10 100 1000 10000

cap

illa

ry s

tres

s (k

Pa)

(lateral)

matric suction (kPa)

ijB

ij a w(u - u )χ

ijψ

(lateral)

(lateral)

Figure 4.17: Principal capillary stresses with various contributions in axial and lateral di-rections, H = θ = 0

92

contribution of surface tension forces to the total capillary stress generated in the packing

(ψij).

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

(kPa)ijB

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

ijψ (kPa)

0θ = o

30θ = o

( )ij u uχ − (kPa)a w

°

°

°

°

°

°°

°

°

°

°

°

Figure 4.18: Polar plot of anisotropic capillary stresses for various wetting angles, H = 0

4.4.2.2 Evolution of degree of anisotropy - link to strength issues

There is a compelling connection between the meniscus-based anisotropic capillary stress,

ψij in unsaturated granular soils and the shear strength contribution that it engenders. As

an extension to the previous discussions we next analyze the evolution of such anisotropy

with the degree of saturation by introducing an anisotropy factor aψ similar to what was

defined for contact fabric, i.e.

93

aψ =5

2

(ψz − ψx)

trace(ψij)=

5

2

(ψz − ψx)

(ψx + ψy + ψz)(4.5)

where ψx, ψy, and ψz are principal values of capillary stress tensor ψij.

The origins of such factor are in the computation of the ratio of deviatoric effective

stress q′ to mean effective stress p′ in a granular assembly (η = q′/p′) as a function of

the anisotropies of microscopic variables such as inter-particle force and contact normal

distribution; see Azema et al. 2009. In this connection, the anisotropy factor aψ represents

an analogous fictitious friction angle that arises due to the presence of water menisci in the

granular assembly. As illustrated in Fig. 4.19, the meniscus-based anisotropy (aψ) for each

packing coincides with the anisotropy of the packing (refer to last column of Table. 4.1).

Upon wetting, the degree of saturation increases such that liquid bridges develop between

particles resulting in an increase in anisotropy of the capillary stresses with the anisotropy

of the packing in the background. This increase in anisotropy is induced by the enlargement

of the wetted contours of the menisci with higher degrees of saturation, while at the same

time, the capillary stresses decrease (Fig.4.16).

The rate of increase in meniscus-induced anisotropy is greater the more prominent the

anisotropy of the packing is, i.e. increasing values of l′ in Fig. 4.19. For the special case of

isotropic packing where l′ = 2.3 , there is obviously no meniscus-based anisotropy induced

upon wetting.

The BCC packing represented by l′ = 2.1 gives a negative anisotropy because of the

rotation of principal axes.

Contrary to the assumption classically made with regard to isotropic pore pressures in

unsaturated soil, the derived tensorial equation for effective stress shows the directional na-

ture of the capillary stresses induced by liquid bridges. One of the consequences of this

finding is that the meniscus-based anisotropy can increase remarkably with water saturation

well below full saturation the more anisotropic the packing is. As such, any perturbation to

94

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20anis

otr

op

y f

acto

r, a

degree of saturation (%)

l '=2.7

l '=2.6

l '=2.5

l '=2.4

l '=2.3

l '=2.1

ψ

Figure 4.19: Meniscus-based anisotropy as a function of saturation for various anisotropicBCC packings, θ = H = 0

such a state of high anisotropy in combination with a decrease in capillary stress will make

the unsaturated sample more prone to material instability. This phenomenon of combined

increase in capillary stress anisotropy and decrease in capillary stress components with sat-

uration can be made the basis of instability failure in unsaturated samples in the absence of

any increase in external mechanical loads.

4.5 Summary

In this Chapter, various regular packings of mono-size spherical particles (SCP,BCC, and

FCP) wetted in the pendular regime are introduced in order to investigate the effect of liquid

bridge existence in the determination of capillary stress.

As such, the soil-water characteristic curves and capillary stresses due to suction and

surface tension forces are examined in these isotropic and anisotropic packings as a function

of liquid bridge distribution, degree of saturation, wetting angle, and separation distance

95

between particles. It is shown that, although the capillary stress induced by surface tensions

may be small at small water saturations, it becomes prominent as water saturation increases.

Apart from the fact that the stress due to contact forces is dependent on fabric, it is

found that the so-called capillary stress arising from liquid bridges is inevitably direction

dependent, i.e. anisotropic. The evolution of such anisotropy with the degree of satura-

tion is also inspected for various BCC packings by introducing a meniscus-based anisotropy

factor aψ. The implication is that granular materials in the pendular regime can engender

an internal capillary-based shear (deviatoric) effect under even isotropic loading, which is

counter intuitive.

96

Chapter 5

VALIDATION OF THE PROPOSED EQUATION

USING DEM SIMULATION

5.1 Introduction

The shear strength and failure envelopes of dry or completely saturated samples are usually

introduced as functions of the effective stress as the controlling parameter for determining

the mechanical behaviour of porous material. For example, the well-known Mohr-Coulomb

failure criterion is introduced as:

τf = σ′

n tanϕ+ c (5.1)

where τf and σ′

n represent the shear strength and the effective normal stress of the sample

at failure, and ϕ and c illustrate the material friction angle and cohesion, respectively.

As discussed in the literature review, the capillary forces in unsaturated granular media

restrict inter-particle slippage and, consequently, increase the shear strength. As such, the

failure envelope of unsaturated granular media is typically determined as a function of the

total net stress and a suction-related parameter, called ”apparent cohesion (ca)”, while the

friction angle is usually considered to be independent of the amount of water saturation, (see

Fig. 2.9).

τf = (σn − ua) tanϕ+ ca (5.2)

From a micro-mechanical point of view, this apparent cohesion characterizes the depen-

dency of the shear strength on the induced inter-particle capillary forces in the unsaturated

medium. Therefore, it is a complex function of the degree of saturation as well as the mi-

cromechanical parameters such as the shape and size of the particles, number and size of the

97

liquid bridges and their spatial distribution, among others.

On the other hand, it should be noted that using an appropriate equation to define the

controlling stress variable, which would play the role of the effective stress in fully saturated

media, will lead to a unique failure envelope whether the soil sample is saturated, dry or

unsaturated. Therefore, if the effective stress σ′

n is well-defined, in order to account for the

effect of inter-particle capillary forces transmitted in unsaturated media, one can still use

Eq. 5.1 to define the shear strength and failure envelope.

Following the above arguments, the validity of the proposed effective stress equation in

this study can be therefore examined. Thus, Discrete Element Method (DEM) calculations

are pursued on granular assemblies (REV) for which the particle size and distribution, wet-

ting angle, degree of saturation, number and distribution of liquid bridges in a unit volume

are all known at any instant during deformation history. As such, the effective stress equation

developed in this work which considers menisci and particle packing effects can be computed

from DEM information using Eqs. (3.43), (3.44) and (3.45). The DEM framework provides

a unique setting in which unsaturated soil behaviour with low degrees of saturation can be

analyzed, given the known difficulties in measuring suctions locally within a sample and

reproducing the same fabric and initial conditions experimentally.

5.2 Triaxial Tests Simulation at Various Controlled Matric Suctions

5.2.1 Brief review on DEM modelling in unsaturated media

During the past decade, the discrete element method, first developed by Cundall and Strack

(1979), has been extensively used to model different geotechnical problems dealing with dry,

cohesionless granular media. The method considers the soil sample as an ideal assembly

of spherical particles, represented by a node located at the center of each sphere. Basic

laws of physics, including Newtons second law, are ruling the interactions between particles

like a mass-spring problem, with an additional algorithm detecting contacts, i.e., updating

98

the existence of a spring between two nodes depending on the distance between them and

a number of other parameters depending on the physics considered: e.g. existence of a

liquid bridge as in this study. As such, interactions between particles are controlled by a

linear force-displacement relationship, which is sufficient for most problems in which small

strains can be assumed at the contact scale (Cundall and Strack, 1979). Thus, micro-scale

deformation and the movements of particles can be calculated for each loading step, and

consequently the overall constitutive behaviour of a sample can be recovered with respect to

the comparatively simple hypothesis at the micro-scale level.

Recently, the method has been expanded to unsaturated soil mechanics while taking into

account the effect of capillary forces in between particles, (Richefeu et al., 2007; Shamy and

Groger, 2008; Scholtes et al.,2009). In this thesis, the open-source DEM code YADE (Kozicki

and Donze, 2008; Smilauer et al., 2010) is used to simulate triaxial tests on unsaturated

samples in the pendular regime.

Scholtes et al. (2009) added the capillary forces induced by independent liquid menisci to

the dry inter-particle forces to enhance the software in order to take into account the effect

of the unsaturated state. To this end, solving the Young-Laplace equation coupled with the

geometry of the liquid bridge and separation distance between a pair of particles, a discrete

set of solutions for the induced capillary forces and liquid bridge volumes associated with

various amounts of matric suction was found.

Thereafter, at each loading step during modelling, considering the amount of matric

suction and specifying the separation distance between each pair of particles in the domain,

associated capillary forces and the volume of the corresponding liquid bridge were defined

using an interpolation technique over the generated set of the solutions of the Young-Laplace

equation. As such, the degree of saturation was calculated as the resultant volume of all

liquid bridges over the volume of pores at each loading step, and the capillary forces related

to each pair of particles were added to their dry inter-particle forces. Subsequently, the

99

related micro-scale deformations and the movements of particles were calculated at each

step of loading, and so the overall constitutive behaviour of a sample was recovered.

A simplified algorithm presenting the basic steps of this modelling of a wet granular

medium is shown in Table. 5.1. More details about this suction controlled DEM simulation

with YADE software can be found in Scholtes et al. (2009) and YADEs website.

100

Table 5.1: Simplified steps of DEM modeling of unsaturated granular media

1. The value of matric suction (ua − uw) is set.

2. External load increment is then applied.

3. Between every pair of particles ,α & β (with or without physical contact):

- The separation distance hαβ is specified.

- The inter-particle force due to external loading is specified:

Is there a physical contact?

No −→ inter-particle contact force due to external loading=0.

Yes −→ inter-particle contact force due to external loading=fαβcon.

- The inter-particle force due to capillarity is specified:

((ua − uw),hαβ) is considered; Is there a solution for Young-Laplace equation?

No −→ inter-particle capillary force & liquid bridge volume=0.

Yes −→ inter-particle capillary force= fαβcap & liquid bridge volume=V αβLB .

- Total inter-particle force is calculated as fαβint=fαβcon+f

αβcap.

4. The corresponding degree of saturation is defined with respect to total volume of liquidbridges.

5. The resulted micro-scale deformations and the movements of particles are defined basedon controlling physics laws. Thus, the constitutive behaviour of the sample is recovered.

6. The external load is increased and steps 3,4 and 5 are repeated till the failure conditionis reached.

101

5.2.2 DEM sample description

The DEM sample is composed of 10,000 mono-dispersed, completely smooth, spherical par-

ticles with a radius of 0.024 mm as shown in Fig. 5.1. For the sake of simplicity and for

comparison purposes, but not by necessity, the assumption is that no liquid bridges are

formed in between particles with no physical contact at the initial state. Here, in order to

maintain the same distribution for contact points and liquid bridges, the liquid bridges are

considered broken once mechanical contact is lost. However, in general, liquid bridges can

still be considered throughout the simulation where mechanical contacts are lost, as long as

they can physically exist according to the Young-Laplace equation.The size of the sample (1

mm3) is assumed to be large enough, in comparison with the size of the particles (0.024mm),

so that the distribution of normal contacts and liquid bridges can be considered continuous

variables.

Figure 5.1: DEM sample consisting of 10,000 mono-sized spherical particles

DEM samples generation is made using the same classical DEM algorithm, as described

by Scholtes et al. (2009). First, a cloud of random spherical particles with no overlapping,

102

and with a given size distribution, is generated in a cubic box with six frictionless fixed

walls. In this first phase, the friction coefficient between the particles is not necessarily the

one that will be used for the simulation. Using a smaller coefficient leads to denser samples.

In the present case, this friction coefficient is set to 0.5◦. Thereafter, the size of spheres is

increased homogeneously, so inter-particle stresses begin to develop between the particles in

contact, and stresses up to 5 kPa appear on the walls of the frictionless box. Due to the

small amount of friction between the particles, the grains are rearranged and the stresses

are stabilized so the sample reaches a quasi-static equilibrium state. The friction angle

between the particles is then increased to 18◦, and the displacements of the boundary walls

of the cube are monitored in order to retain the quasi-static equilibrium condition during

the testing procedure.

Before starting the axial loading simulation, the sample is unloaded so that the confining

pressure is slowly reduced to the amount of desired confining pressure for the test. The

properties of the DEM sample is summarized in Table. 5.2.

Table 5.2: DEM sample input parametersInter-particle friction angle 18◦

Number of particles 10,000Initial volume 1 mm3

Radius of particles 0.024 mmInitial fabric δij/3Wetting angle 0

Initial porosity (n) 0.4Surface tension parameter 0.073 N/mNormal stiffness (Kn) 106Pa

Tangential stiffness (Kt) 0.3 Kn

5.2.3 DEM triaxial test procedure and results

A series of triaxial tests with various matric suctions of 15, 30, 300 kPa and confining

pressures of 250, 500, 750 and 1000 Pa are simulated using the open-source DEM code

YADE. Such low confining pressures are chosen in order to highlight the effects of capillary

103

forces. As discussed in previous section, the initial fabric is made isotropic and the initial

porosity is 0.4 in all simulations. The displacements of the walls are controlled in such a way

that the confining pressure remains constant in the lateral (r = x, y) directions, assuring

an axisymmetric condition during the test. The axial loading is exerted by controlling the

strains in z direction, while the strain rate is restricted so the average resultant force on

the particles is less than 1% of the mean contact force in each loading step, in order to

satisfy the quasi-static condition (Mahboubi et al., 1996). The SWCC of the DEM samples

in comparison with simple cubic packing (SCP) and face centered packing (FCP) is shown

in Fig. 5.2.

0.1

1

10

100

1000

10000

100000

1000000

0 5 10 15 20 25

mat

ric

suct

ion (

kP

a)

degree of saturation (Sr%)

SCP , R=0.1 mm

SCP, R=0.001 mm

FCP, R=0.1 mm

FCP, R=0.001 mm

DEM sample

R=0.024 mm

Figure 5.2: SWCC of the DEM sample, R=0.024 mm

The resulting peak shear strengths of the samples (at failure) with various matric suctions

along with the shear strengths of the dry sample are indicated in Table. 5.3. Moreover, the

deviatoric stress and volumetric strain responses of samples with various matric suctions are

104

shown in Fig. 5.3 at confining pressure of 750 Pa as an example.

Table 5.3: Shear strengths of samples with various matric suctions, DEM resultsDry (matric suction=0 kPa)

Lateral pressure (Pa) 250 500 750 1000Normal stress at failure (Pa) 591 1207 1843 2480

Matric suction=15 kPaLateral pressure (Pa) 250 500 750 1000Normal stress at failure (Pa) 5388 6345 7201 8018

Matric suction=30 kPaLateral pressure (Pa) 250 500 750 1000Normal stress at failure (Pa) 5748 6593 7426 8329

Matric suction=300 kPaLateral pressure (Pa) 250 750Normal stress at failure (Pa) 6380 8289

The Mohr-Coulomb failure envelope for each unsaturated sample with specific matric

suction is then obtained in the mean stress (p = (σz + 2σr)/3) and deviatoric stress (q =

σz − σr) space, by drawing the best line passing through the peak shear strengths of the

sample under various confining pressures, as shown in Figs 5.4. The failure envelope for the

dry case is also plotted as baseline and a friction angle of 25◦ is obtained. As expected, when

using the total stresses p and q, different failure envelopes showing an apparent gain in shear

strength are obtained with increasing matric suction.

As shown in Fig. 5.3 and 5.4 there is a significant jump between the shear strength

envelopes of the dry and unsaturated samples in DEM simulations. Richefeu et al. (2007)

has also pointed to the same significant jump in shear strength while conducting direct

shear tests on assemblies of mono-dispersed smooth glass beads in pendular regime with a

low confinement pressure. The main reason for this behaviour is that large matric suctions

are immediately produced for a very small amount of water, as show in Fig. 5.2, given the

size of the particles and the fact that liquid bridges are considered to exist only between

particles in physical contact (H = 0). Therefore, a large capillary stress (in comparison

with low confinement pressures in these simulations) is induced in unsaturated samples with

very small degrees of saturation, which leads to a significant jump in the shear strengths

105

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20 0.25

q/p

axial strain

(a)

dry sample

suction 15 kPa

suction 30 kPa

suction 300 kPa

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.040.00 0.05 0.10 0.15 0.20 0.25

vo

lum

etri

c st

rain

axial strain

(b)

negative volumetric strain is considered as dilation

dry sample

suction 15 kPa

suction 30 kPa

suction 300 kPa

Figure 5.3: (a) Deviatoric stress and (b) Volumetric strain versus axial strain for DEMsamples with lateral pressure of 750 Pa

106

0

1000

2000

3000

4000

5000

6000

7000

8000

0 1000 2000 3000 4000

q (

Pa)

p (Pa)

dry sample

suction 15 kPa

suction 30 kPa

suction 300 kPa

Figure 5.4: Failure envelope of DEM samples considering the peak shear strength as thefailure point

of the unsaturated samples in comparison with the dry sample. However, as shown in Fig.

5.5, the difference between the capillary stresses induced by various amount of water in

unsaturated samples is less significant once the liquid bridges are formed in the sample,

which causes smaller gaps between the shear strength envelopes of unsaturated samples in

these simulations (see. Fig. 5.3 and 5.4).

It is also worth noting that the slopes of the failure envelopes (corresponding to the peak

friction angle) are increasing slightly with the amount of matric suction (see Fig. 5.4). This

is due to the fact that the peak friction angle is related to the level of particles interlocking

and the sample density; i.e. the denser the sample at peak failure point, the greater the

friction angle. As shown in Fig. 5.3, staring with the same initial density, samples with

a greater amount of suction undergo more compaction at failure point which results in an

increase in their friction angle. This can also partly be attributed to the amount of anisotropy

107

1

2

3

4

5

30

210

60

240

90

270

120

300

150

330

180 0

Suction 300 kPa, Sr=0.05%

Suction 30 kPa, Sr=2.2%

Suction 15 kPa, Sr=5.5%

o

oo

o

kPa

Figure 5.5: Anisotropic capillary stress in unsaturated DEM samples,axial strain=20%

induced by the distribution of liquid bridges in the sample at failure. In other words, the

water presence in the sample leads to a higher friction angle in comparison with the dry

case. Moreover, it is clear that the induced inter-particle forces in unsaturated samples

at a certain axial strain (certain step of the simulation) are greater than those in the dry

sample, which lead to larger displacements between particles in lateral directions. Therefore,

the rate of changes in volumetric strain (compaction and dilation) is greater in unsaturated

samples in comparison with the dry sample. In this thesis, the focus is on the strength of the

unsaturated media, and in order to precisely predict the deformations a constitutive model

needs to be developed in future studies.

5.2.4 Validation of the proposed effective stress equation with DEM simulation results

Here, recalling Eqs. (3.45), (3.55) and (3.56) and considering the micro-scale properties of the

sample such as particles size and roughness, wetting angle, number and distribution of liquid

bridges in a unit volume of the domain, the effective stress can be calculated for each loading

step during the triaxial test simulations with specific matric suctions. According to Fig. 5.6,

108

replacing (a) the net deviatoric and confining stresses by (b) the effective deviatoric and

confining stresses, all failure points for the various matric suctions fall near the failure line for

the dry case, producing a nearly unique Mohr-Coulomb failure line. Furthermore, the failure

envelopes obtained using Bishop’s effective equation with χ = Sr are also plotted to confirm

its shortcomings. This supports the validity of the proposed effective stress equation and its

ability to control the intrinsic behaviour of unsaturated soils since it systematically embeds

meniscus-based information and other particle characteristics at the microscopic level. The

difference between the amount of q and q′, in Fig.5.6, shows the effect of anisotropic nature

of the induced capillary stress in the sample at the failure point.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2000 4000 6000 8000

q-q' (

Pa)

p-p' (Pa)

dry sample

sucion 15 kPa

suction 30 kPa

suction 300 kPa

suction 15 kPa, Bishop

suction 30 kPa, Bishop

suction 300 kPa, Bishop

Figure 5.6: Strength of wet granular material based on (a) net stress (q,p) and (b) effectivestress (q′,p′)

The validity of the effective stress equation is next checked at every stage during defor-

mation history as opposed to limiting the check to solely failure conditions, as was done in

the previous paragraph. Figure. 5.7 shows the plot of stress ratio with axial strain for vari-

109

ous suctions and based on both effective (q′/p′) and net stress (q/p) definitions. All curves

produced using the effective stress definition tend to merge toward the curve representing

the response of the dry material. This indicates that enough micromechanical information

(liquid bridge distribution and fabric) is being accounted for in the effective stress equation

to lead to a unique response which intrinsically belongs to the dry case. However, there is

some discrepancy in the beginning at strain levels less than 2% between the effective stress

and dry curves, probably because of inaccuracies in achieving a stable equilibrium in DEM

calculations within the small strain range. Thus, the statistics of liquid bridge distribu-

tion together with contacts that enter the effective stress equation to calculate the capillary

stress may not have been accurate enough in the early stages of loading history. However,

this matter has to be investigated further.

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20 0.25

q'/p'

axial strain

dry sample

total stress - suction 15 kPa

total stress - suction 30 kPa

total stress - suction 300 kPa

calculated effective stress - suction 15 kPa

calculated effective stress - suction 30 kPa

calculated effective stress - suction 300 kPa

Figure 5.7: Shear strength response based on effective stresses for a confining pressure of750 Pa

Figure 5.8 describes the evolution of the anisotropy factor for both effective and capillary

110

stresses for a confining pressure of 750 Pa and matric suction of 30 kPa as an example.

The anisotropy of effective stress is seen to be much more pronounced than that of capillary

stress since the former is developed mainly by mechanical loading. Although the anisotropy

in capillary stresses is aligned with the anisotropy of contacts (effective stresses), it is limited

by the constraint that the matric suction remains constant during loading history.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.00 0.05 0.10 0.15 0.20 0.25

anis

otr

op

y f

acto

r

axial strain

capillary stress

effective stress

Figure 5.8: Anisotropy changes for both effective and capillary stresses for a confining pres-

sure of 750 Pa at matric suction of 30 kPa

5.3 Validation of the proposed effective stress equation using data from

literature

Further, to support the validity of the proposed effective stress equation, the results from

DEM simulations of shear test on an initially isotropic loose packing consisted of mono-

sized spherical particles in pendular regime under comparatively low net normal stresses are

adopted from literature (Shamy and Groger, 2008). The summary of the sample properties

111

is presented in Table. 5.4.

Table 5.4: DEM sample properties (Shamy & Groger, 2008)Radius of particles 0.5 mmSpecific gravity 2.65

Number of particles 37,867Dimensionless separation distance 0.05

Surface tension of water 0.0727 N/mWetting angle 0Initial Porosity 40%

In contrast with the suction controlled DEM simulations conducted in this thesis, Shamy

and Groger considered constant water content in samples during the loading process. More-

over, they assumed that the liquid bridges were initially generated between particles in phys-

ical contact, while they could exist between each pair of neighboring particles, not necessarily

in physical contact after deformations, as long as a solution to Young-Laplace equation was

possible. The information about the evolution of liquid bridges and contact fabrics dur-

ing this DEM simulations are not available; therefore, an equivalent regular simple cubic

packing with almost the same porosity, particle size, inter-particle distance and SWCC, (see

Fig. 5.9), is considered in order to calculate the effective stress at failure points, using the

proposed equation.

The computation of effective stress based on proposed Eq. (3.45), and assumed packing

(SCP) for same amount of water contents leads to a fairly unique Mohr-Coulomb failure

envelope with almost the same intrinsic friction angle as for the dry case (See Fig. 5.10). It

is evident that estimating the micro-scale properties of the samples with micro-scale charac-

teristics of SCP affects the obtained results. The more precise information available about

the micro-fabric of the liquid bridges and contacts in the sample, the more accurate will be

the calculated effective stress.

112

0

5

10

15

20

25

30

10 100 1000 10000

Sr

%

matric suction (Pa)

SCP

DEM sample

Figure 5.9: Comparisons between SWCC of selected SCP sample and simulated DEM sam-ples by Shamy and Groger, 2008

5.4 Summary

In this Chapter, the validity of the derived generalized effective stress equation (Eq. 3.45)

was investigated using discrete element modelling (DEM) calculations on granular assemblies

(REV) for which the particle size and distribution, wetting angle, degree of saturation,

number and distribution of liquid bridges in a unit volume are all known at any instant

during deformation history. Moreover, DEM simulation results from literature were also

adopted to provide more support to the validity of the derived equation.

Since the proposed equation is derived based on micromechanical interpretations of force

transmission in a discrete granular media, the effect of capillarity interactions are inevitably

taken into account; leading to the true effective stress which controls the behaviour (shear

113

0

100

200

300

400

500

600

0 200 400 600 800 1000

shea

r st

ress

(p

a)

normal net stress (Pa)(a)

water content=0%

water content=0.2%

water content=2.0%

water content=5.2%

0

100

200

300

400

500

600

0 200 400 600 800 1000

shea

r st

ress

(P

a)

normal effective stress (Pa)

water content=0%

water content=0.2%

water content=2%

water content=5%

(b)

Figure 5.10: (a) Shear strength response based on net stresses (adopted from Shamy andGroger, 2008). (b) Shear strength response based on calculated effective stresses

114

strength) of the samples. As such, using the proposed equation, a unique Mohr-Coulomb

failure envelope is obtained for samples with various matric suctions or various amount of

water content.

115

Chapter 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

In emerging geotechnical problems under unsaturated conditions, there is still much debate

on the definition of a proper effective stress and issues surrounding the validity of Bishop’s

effective stress. This thesis examined the force transport in a granular system wetted with

discrete liquid bridges with reference to unsaturated granular materials in the pendular

regime. A micromechanical approach is hereby used to formulate effective stress in such

regime as a function of both liquid bridge and particle contact spatial distributions with

special emphasis on the interactions of the air-water-solid phases, including the interaction

of interfaces. Main findings and conclusions are summarized as follows.

A tensorial equation defining the true effective stress in unsaturated soils in the pendular

regime is proposed:

σ′

ij = (σij − uaδij) + χij (ua − uw) + Bij (6.1)

As such, the effective stress parameter (χ) as initially introduced in Bishop’s equation is

shown to be a tensorial function of degree of saturation, particle packing and liquid bridges

distribution. In the proposed equation, χij is a tensorial quantity which accounts for the

spatial distribution (fabric) of liquid bridges. Given that the fabric of the liquid bridges

is generally anisotropic, this parameter is also anisotropic. Moreover, an accompanying

parameter (Bij), which refers to a stress induced by surface tensions acting along the so-called

contractile skins over the REV, is introduced in the newly proposed equation. This quantity

is also shown to be a function of the spatial distribution of contractile skins throughout the

REV.

116

In contrast to the assumption classically made with regard to isotropic pore pressures

in unsaturated soil, the derived tensorial equation for effective stress shows the directional

nature of the capillary stresses induced by spatial distribution of liquid bridges and fabric of

the solid skeleton evolving during deformation history.

ψij = χij (ua − uw) + Bij (6.2)

This capillary stress is shown to have two components: one originating from suction

between particles induced by air-water pressure difference (related to χij), which leads to an

anisotropic capillary stress due to matric suction (χij(ua−uw)), and the second arising from

surface tension forces along the contours between particles and water menisci (Bij).

This introduced capillary stress is generally anisotropic and therefore generates anisotropic

tensile strength and a meniscus based shear strength in unsaturated sample that varies with

the anisotropy of the packing and the degree of saturation. The implication is that granular

materials in the pendular regime can engender an internal suction based shear (deviatoric)

effect under even isotropic loading. Also, this issue becomes particularly relevant when study-

ing the material instability behaviour of unsaturated media in the pendular regime where

failure is characterized by a sudden collapse. It is shown the meniscus-based anisotropy can

increase remarkably with water saturation well below full saturation the more anisotropic

the packing is. As such, any perturbation to such a state of high anisotropy in combination

with a decrease in capillary stress will make the unsaturated sample more prone to mate-

rial instability. This phenomenon of combined increase in capillary stress anisotropy and

decrease in capillary stress components with saturation can be made the basis of instability

failure in unsaturated samples in the absence of any increase in external mechanical loads.

An example in geotechnical engineering pertains to natural slopes consisting of fine granular

materials as silty sand at low moisture content which are prone to collapse after a rainfall

event, despite their quite shallow angles.

117

6.2 Recommendations for Future Work

The work developed here can be extended to poly-disperse and non-spherical particles pack-

ings. Also, liquid bridges were considered to be distinct and as water saturation increases to

transition into funicular and thereafter capillary states, they are bound to merge. Consid-

ering various scenarios of liquid bridges merging with each other, the same approach can be

used to develop the effective stress equation in variably saturated states.

The proposed effective stress derivation based on micromechanical origins offers a plau-

sible testing ground for the analysis of the constitutive behaviour of unsaturated media. A

constitutive model based on tensorial form of effective stress and distribution of the liquid

bridges can be developed for unsaturated samples, considering the effect of capillary forces

on the micro-scale displacements of the particles (see Fig. 6.1).

Macroscopic level

Microscopic level

ijσStress Tensor

ijεStrain Tensor

if

Inter-particle

contact forces

cap

if

Inter-particle

capillary forces

cap

iuiu

Displacements due to

contact forces

Displacements due to

capillary forces

Figure 6.1: Homogenization method in order to develop a constitutive model in unsaturatedmedia

Finally, the proposed model can be applied to real soil samples using a Micro-CT scan

of water menisci in a localized zone (as shown in Fig. 6.2).

118

Figure 6.2: Micro-CT scan of water menisci of Toyoura sand, courtesy of Profs. Oka andKimoto, Kyoto University, Japan

119

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Appendix A

Toroidal Approximation

M.Alguacil et al. (2009) defined the geometrical properties of a concave liquid bridge between

two mono-sized spherical particles using a simple toroidal approximation as described here.

The liquid bridge profile is considered as a surface of revolution with two constant mean

curvatures as sketched in Fig. A.1.

HN

1R

2RR

α

θ

y

x

Figure A.1: Concave liquid bridge geometry between a pair of uni-size particles

Considering the geometry of the liquid bridge, the mean curvatures are written as:

R1 =R1

R=

(H2+ 1− cosα)

cos(α + θ)(A.1)

R2 =R2

R= sinα +R1(sin (α + θ)− 1) (A.2)

where,

N =H

2+ 1− cosα,H = H/R (A.3)

130

On the other hand, considering a quadrant of the water lens in Cartesian coordinates,

the liquid bridge profile can be described as:

yLB(x) = (R1 +R2)−√

R21 − x2 (A.4)

and the wetted surface of the particle can be written as:

yp(x) =

√

R2 − (x− H

2−R)

2

(A.5)

Therefore, calculating the volumes generated by these two profiles around the basis (x

axes), the volume of liquid bridge can be determined as:

VLB = 2π∫ N

0[yLB(x)]

2dx− 2π∫ N

H/2[yp(x)]

2dx (A.6)

Inserting Eqs. (A.4) and (A.5) in Eq. (A.6), the dimensionless volume of liquid bridge

is defined as:

V LB = VLB

2πR3 =[

(R1 +R2)2+ (R1)

2]

N − (N)3

3−(

R1 +R2

)

[

N√

(R1)2 − (N)

2+ (R1)

2arcsin

(

NR1

)

]

− (N−H2)2

3(3−N + H

2)

(A.7)

As a result, solving Eq. (A.7) with respect to Eqs. (A.1), (A.2) and Eq. (A.3) for a

given liquid bridge volume (VLB), the corresponding half filling angle (α) is calculated and

vice versa.

131