An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime

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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 An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime (Unpublished master's thesis).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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An Eﬀective Stress Equation for Unsaturated Granular Media in Pendular Regime

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 micromechanical analysis of force transport between interacting particles with a given packing and distribution of capillary liquid bridges. A single eﬀective stress tensor, characterizing the tensorial 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

1 Abstract ...... ii Acknowledgments ...... iii Dedication ...... iv Table of Contents ...... v ListofTables ...... vii List of Figures ...... viii ListofSymbols...... xi 1 INTRODUCTION ...... 1 1.1 Introduction...... 1 1.1.1 Objectives ...... 2 1.1.2 Organization of Thesis ...... 3 2 LITERATURE REVIEW ...... 6 2.1 Introduction...... 6 2.2 Capillary Effect and Matric Suction ...... 6 2.2.1 Soil water characteristic curve ...... 13 2.3 Experimental Observations on Unsaturated Soil Behaviours ...... 18 2.3.1 Shear and tensile strengths of unsaturated soils ...... 18 2.3.2 Collapse behaviour ...... 24 2.4 Studies on Effective Stress of Unsaturated Soils - Existing Frameworks . . . 25 2.4.1 Phenomenological studies (Macroscale studies) ...... 26 2.4.1.1 Single effective stress approach ...... 26 2.4.1.2 Independent stress state variables approach ...... 30 2.4.2 Micromechanical studies ...... 34 2.5 Summary ...... 42 3 MICROMECHANICS OF EFFECTIVE STRESS IN MULTIPHASIC GRAN- ULAR MEDIA ...... 43 3.1 Introduction ...... 43 3.2 Force Transport in Dry Granular Media ...... 44 3.3 Force Transport in Saturated Granular Media ...... 47 3.3.1 Negligible contact area - rigid particles ...... 48 3.3.2 Finite contact area - compressible particles ...... 49 3.3.3 Effective stress in a fully saturated idealized compressible particle packing ...... 51 3.4 Force Transport in Unsaturated Granular Media ...... 56 3.4.1 Effective stress parameters for idealized packing ...... 62 3.5 Summary ...... 67 4 COMPUTATION OF CAPILLARY STRESSES IN IDEALIZED GRANU- LAR PACKINGS ...... 69 4.1 Introduction ...... 69 4.2 Idealized Packing ...... 69 4.2.1 Simple cubic packing (SCP) ...... 70

v 4.2.2 Body-centeredcubicpacking(BCC)...... 71 4.2.3 Cubic Close Packing or Face Centered Packing (CCP or FCP).... 73 4.3 Theoretical SWCC for Regular Packing in Pendular Regime ...... 75 4.4 Effective Stress Parameters and Capillary Stress in RegularPacking . . . . . 80 4.4.1 Isotropicpackings ...... 80 4.4.1.1 Effective stress parameters and capillary stresses in SCP and FCP...... 80 4.4.1.2 Isotropic tensile strength in comparison with experimental results...... 85 4.4.2 Anisotropicpackings ...... 90 4.4.2.1 Evolution of capillary stress in BCC packing-anisotropy aspects 90 4.4.2.2 Evolution of degree of anisotropy - link to strength issues. . 93 4.5 Summary ...... 95 5 VALIDATION OF THE PROPOSED EQUATION USING DEM SIMULA- TION ...... 97 5.1 Introduction...... 97 5.2 Triaxial Tests Simulation at Various Controlled Matric Suctions ...... 98 5.2.1 Brief review on DEM modelling in unsaturated media ...... 98 5.2.2 DEMsampledescription ...... 102 5.2.3 DEMtriaxialtestprocedureandresults ...... 103 5.2.4 Validation of the proposed effective stress equation with DEM simula- tionresults ...... 108 5.3 Validation of the proposed effective stress equation using data from literature 111 5.4 Summary ...... 113 6 CONCLUSIONS AND RECOMMENDATIONS ...... 116 6.1 Conclusions ...... 116 6.2 RecommendationsforFutureWork ...... 118 Bibliography ...... 120 A ToroidalApproximation ...... 130

vi List of Tables

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

4.1 Properties of BCC packings with various l′ ...... 72 4.2 SWCCcalculation ...... 76 4.3 χij calculation...... 81 4.4 Bij calculation...... 82 4.5 Direct tensile test results of clean F-75 sand (Kim, 2001)...... 89

5.1 Simpliﬁed steps of DEM modeling of unsaturated granular media ...... 101 5.2 DEMsampleinputparameters...... 103 5.3 Shear strengths of samples with various matric suctions,DEMresults . . . . 105 5.4 DEM sample properties (Shamy & Groger, 2008) ...... 112

vii List of Figures and Illustrations

2.1 Illustrationofsurfacetension ...... 7 2.2 Waterincapillarytube...... 8 2.3 Free body diagram of forces acting on air-water interfaceinacapillarytube 9 2.4 Curvedliquidandgasinterfaces ...... 10 2.5 Conceptual demonstration of unsaturated sample in different regimes(Lu and Likos,2004) ...... 11 2.6 Conventional soil water characteristic curve for sand and silt(Lu and Likos, 2004)...... 14 2.7 Demonstration of the ink-bottle effect during:(a)drying process and (b)wetting process(Marshalletal.,1996) ...... 15 2.8 Theoretical presentation of soil-water characteristic curve of an unsaturated sampleindifferentregimes(Luetal.,2007) ...... 16 2.9 General representation of shear strength in unsaturated samples (Ho and Fred- lund,1982) ...... 18 2.10 Yield locus of glass beads R=46 micron (Pierrat et al., 1998)...... 20 2.11 Yield locus of glass beads R=90 micron (Pierrat et al., 1998)...... 20 2.12 Direct shear test results on cohesionless sands (DonaldI.,1956) ...... 21 2.13 Tensile strength versus water content (F-75-C),(Kim, 2001)...... 23 2.14 Tensile strength versus water content,(Kim, 2001) ...... 23 2.15 One-dimensional compression and subsequent soaking tests under constant void ratio or applied pressure (Jennings and Burland, 1962) ...... 25 2.16 Effective stress coefficient for unsaturated soil based on experimental results 28 2.17 Axis translation method in measuring matric suction in laboratory (Hilf, 1956) 31 2.18 Dimensionless liquid bridge volume versus dimensionless suction (Molenkamp andNazemi,2003)...... 37 2.19 General scheme of homogenization technique (Oda and Iwashita,1999) . . . 38 2.20 Mobilized friction angle in pyramidal packing of various heights and inter- particle friction angles. Negative inter-particle friction angle represents verti- calextension...... 39 2.21 Domain of accessible geometrical states based on harmonic representation of granularmedia(Radjai,2008) ...... 41

3.1 Cauchy’sstressinacloseddomain ...... 44 3.2 Assemblyofdrygranularmedia ...... 45 3.3 Branchvectorbetweenpairofparticles ...... 47 3.4 Free body diagram of inter-particle forces in saturated media...... 48 3.5 Free body diagram of saturated media with compressible particles ...... 50 3.6 Particle in contact with neighboring particles ...... 52 3.7 Local coordinates on the center of each particle ...... 52 3.8 Spherical REV and global coordination system (Quadfel and Rothenburg,2001) 54 3.9 Schematic anisotropic force distribution in polar system, an =0.5 & βn = π/6 55 3.10 Unsaturated media as a three phase system in pendular state...... 57

viii 3.11 Free body diagram of inter-particle forces ...... 58 3.12 Unequal hydrostatic forces around the particle ...... 59 3.13 Unequal hydrostatic pressure on the air/water interface...... 60 3.14 Traction forces between a pair of spherical particles ...... 61 3.15 Concave liquid bridge geometry between a pair of uni-sizeparticles . . . . . 62 3.16 Dimensionless liquid volume Vrel as a function of the half ﬁlling angle α (Megias-AlguacilandGaucker,2009) ...... 63 3.17 Positive and negative matric suction zones as a function of the half ﬁlling and wettingangles(LuandLikos,2004) ...... 64 3.18 Local coordinates illustration for each liquid bridge ...... 65

4.1 Illustration of simple cubic packing (SCP) ...... 70 4.2 Illustration of body-Centered Cubic Packing (BCC) ...... 71 4.3 Separation distance between particles , H (Pietsch,1968) ...... 72 4.4 Arrangement of BCC packing unit cells in 3D space ...... 73 4.5 Illustration of face centered packing (FCP) ...... 74 4.6 Porosity of different regular spherical packing ...... 74 4.7 SWCC of SCP and FCP as a function of particle size, H = θ =0 ...... 75 4.8 Effect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), H =0 ...... 78 4.9 Effect of separation distance on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), θ =0...... 79 4.10 The resulting isotropic effective stress coefficient χij while θ = 0(a) Loose packing(SCP)and(b)Densepacking(FCP)...... 83 4.11 Computed relationships between degree of saturation and effective stress pa- rameterforvariouspackings ...... 84 4.12 The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a loose packing (SCP)- R =0.1 mm, H =0 ...... 86 4.13 The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a dense packing (FCP)- R =0.1 mm, H =0...... 87 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 4.15 Compression between measured and predicted tensile strength ...... 90 4.16 Polar plot of anisotropic capillary stresses for various saturation degree, H = θ =0...... 91 4.17 Principal capillary stresses with various contributions in axial and lateral directions, H = θ =0...... 92 4.18 Polar plot of anisotropic capillary stresses for various wetting angles, H =0. 93 4.19 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 ...... 102 5.2 SWCCoftheDEMsample,R=0.024mm ...... 104 5.3 (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 the failurepoint...... 107 5.5 Anisotropic capillary stress in unsaturated DEM samples,axial strain=20% . 108 5.6 Strength of wet granular material based on (a) net stress (q,p) and (b) effective stress (q′,p′)...... 109 5.7 Shear strength response based on effective stresses for a confining pressure of 750 Pa ...... 110 5.8 Anisotropy changes for both effective and capillary stresses for a confining pressure of 750 Pa at matric suction of 30 kPa ...... 111 5.9 Comparisons between SWCC of selected SCP sample and simulated DEM samplesbyShamyandGroger,2008 ...... 113 5.10 (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- ratedmedia ...... 118 6.2 Micro-CT scan of water menisci of Toyoura sand, courtesy of Profs. Oka and Kimoto,KyotoUniversity,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

LB Fij 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

N p number of particles in REV

N LB number of liquid bridges in REV p,p′ mean stress,eﬀective mean stress p(n) contactnormal’sprobabilitydensityfunction 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

c xi,j centroid position vector yLB(x) liquidbridgeproﬁleinCartesiancoordinatesystem 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

p Γw wetted surface of the particle

p Γd total surface of ﬁnite contacts on the particle

xiii p Γc 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,σij stress, average stress and eﬀective stress tensors

′ σn,σn 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, granular, 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 producing 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 complicated, 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 appropriate 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 remained 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 objective 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 analysis. 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 distributions 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 formulation 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 equation 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 capillary 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 ua

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):

2 2Ts cos θ γwπr hc =2πrTs cos θ hc = (2.1) ⇒ γwr 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 Ts θ θ

2r hc

Figure 2.2: Water in capillary tube

and can be calculated as:

ψ = u u = h γ (2.2) a − w c w

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:

2πrT cos θ (u u ) πr2 =0 (2.3) s − a − w

which leads to: 2T 2T (u u )= s = s (2.4) a − w (r/ cos θ) R 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 T s Ts θ

uw X

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:

1 1 ua uw = Ts + (2.5) − R1 R2 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 R1

R1 R

R 2

R 2 (a) (b) (c) Z

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 Water Table

Datum

Figure 2.5: Conceptual demonstration of unsaturated sample in diﬀerent regimes(Lu and Likos, 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 rmcosaepito iw hs ehnsshv encla follows: been have as mechanisms These view. of point macro-scale or 2010). Huy, and Min and 2005; al., et prop been have models hysteresis soil-water various decade f curve characteristic water soil Conventional 2.6: Figure hoeial,svrlmcaim a edt ytrtcb hysteretic to lead can mechanisms several Theoretically,

soil suction ψ • ytodffrn ai scniee nFg ..Fraspecific a For radius, smaller 2.7. a Fig. by in controlled considered is radii pore different non-uniform two hypothetical by A shape. and size pore of tion Ink-bottle hw nFg .a h ailr egtdrn h rigpr drying the during height capillary wetting the and 2.7a, drying Fig. the in during shown different be can rise capillary xedbyn h agrpr ftepr pc wt radius (with space pore the of part larger the beyond extend ffc:I ecie h nuneo o-nfriisi the in non-uniformities of influence the describes It effect: volumetric water content volumetric content water 14 u a − u w θ w 2 = rsn n itL n io,2004) Likos, and silt(Lu and sand or sand silt T sd(uadLks 04 Huang 2004; Likos, and (Lu osed s /r hvorfo h micro-scale the from ehaviour sfidb uadLks(2004) Likos and Lu by ssified h aiu egto the of height maximum the , airentry ψ e pc described space cs ( ocess arcsuction matric rcse.As processes. R hl the while ) distribu- h d may ) 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.

R R

hd r r

()a ( )b

Figure 2.7: Demonstration of the ink-bottle eﬀect during:(a)drying process and (b)wetting process (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.

I II III

I pendular regime II funicular regime III capillary regime ψ

drying soil suction suction soil

wetting

θ r θ s volumetric water content θ w

Figure 2.8: Theoretical presentation of soil-water characteristic curve of an unsaturated sample 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, region 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 transition 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).

w 1 3 2 − > − shear shear strength ua u w u a u ()()2 − > − 1 ()()ua u w u a− u w > ua u w ) 0 φ ' ( )− 0 = (ua u w

3 ca c 2 a c1 { { a { c net normal stress

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 25

20 ) 2

10 dry shear stress (g/cm stress shear 5 1% n-hexadecane 1.3% water 4% water 0 0 10 20 30 40 normal stress (g/cm2 )

Figure 2.10: Yield locus of glass beads R=46 micron (Pierrat et al., 1998) 18 16 14 ) 2 12 10 8

6 dry

shear stress (g/cm stress shear 4 1% n-hexadecane 2 1.3% water 4% water 0 0 10 20 30 40 normal stress (g/cm2 )

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

20 25

(kPa) 15

graded frankston sand

shear strength shear 5 brown sand 0 0 5 10 15 20 25 matric suction (kPa) (a) 25

20 (kPa) 15

10 ﬁne frankston sand shear strength shear 5 medium frankston sand 0 0 5 10 15 20 25 30 matric suction (kPa) (b)

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 1400

1200

1000

800

600 measured data,loose 400 tensile strength (Pa) strength tensile measured data, medium

200 measured data, dense 0 0 1 2 3 4 5 water content %

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

degree of saturation (%) 0 20 40 60 80 1800 1600 1400 1200 1000 800

tensile strength (Pa) strength tensile 600 400 200 0 0 5 10 15 20 water content %

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 compression tests on both unsaturated and completely saturated samples. Soaking the unsaturated samples, under a constant confining pressure or volumetric strain, and plotting the compression 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 0.83 0.81 compression line of 0.79 air-dried samples 0.77 soaked at constant applied pressure 0.75 compression line of saturated samples 0.73 void ratio e ratio void 0.71 soaked at constant void ratio 0.69 0.67 0.65 0.1 1 10 applied pressure (t/ft 2 )

Figure 2.15: One-dimensional compression and subsequent soaking tests under constant void ratio 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 predicting 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 unsaturated 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 u ), and − a 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 [χu + (1 χ)u ] (2.7) − w − a ′ 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 1

0.9

c 0.8

0.7

0.6 breahead silt (Bishop and Donald, 1961) 0.5 silt (Jennings and 0.4 Burland, 1962)

0.3 silty clay (Jennings coefficent of effective stress effectivestress coefficentof and Burland, 1962) 0.2 compacted boulder 0.1 clay (Bishop et al., 0 1960) 0 20 40 60 80 100 degree of saturation Sr % (a)

1 silt, drained test (Donald, 0.9 1961)

0.8 silt, constant water content test (Donald, 1961) c 0.7 Madrid gray clay (Escario 0.6 and Juca, 1989)

0.5 Madrid silty clay (Escario and Juca,1989) 0.4 Madrid clay sand (Escario and Juca, 1989) 0.3 moraine (Blight, 1961) coefficient of effective stress effectivestress ofcoefficient 0.2

0.1 boulder clay (Blight, 1961) 0 0 20 40 60 80 100 clay -shale (Blight, 1961) degree of saturation Sr% (b)

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

28 k Vw k χ = = Sr (2.8) Vv 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) proposed 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.

−0.55 ψ ψ ifψ>ψe χ =  e (2.9)   1 ifψ ψ ≤ e  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 saturated 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 matric 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 u ) and (u u ), − a a − w (2) (σ u ) and (u u ), and (3) (σ u ) and (σ u ). − w a − w − a − w 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 u ) and the matric suction (u u ). Considering the air pressure, as − a a − w 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.

200

150 (kPa)

w 100

0 sandy clay weathered state -50 pore water pressure wateru pressure pore loess -100 0 50 100 150 200 250 300

air pressure u a (kPa)

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 u ) and (u u ). − a a − w 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 stress - Yield surface is deﬁned in the single ef- space fective stress space

- Measurement of the net stress is straight- - Measurement of eﬀective stress requires forward more eﬀort

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

- The effect of hydraulic hysteresis is not - the effect of hydraulic hysteresis can be well-defined captured

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 increase 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 granular 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, commonly 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:

c 1 2N F = n kn k....n k (2.10) ij..m 2N c i j m kX=1 where N 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 methods 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 roughness. 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:

V V¯ = LB (2.11) w R3 (u u ) ψ¯ = a − w R (2.12) Ts 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 0.45 ° 0.4 θ = 0

V 0.35 θ = 20 ° 0.3

0.25 θ = 30° 0.2

0.15 θ = 40° 0.1 analytical torodial dimensionless liquid volume volume liquid dimensionless 0.05

0 0.01 0.1 1 10 100 1000 dimensionless suction ψ

Figure 2.18: Dimensionless liquid bridge volume versus dimensionless suction (Molenkamp and 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 microscopic 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 σ ε ij ij Macroscopic level Stress Tensor Global constitutive equations Strain Tensor

Microscopic level fi ui Local properties Inter-Particle Forces Local Displacements

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 sinϕ ¯ 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 0.8

0.6 ° = 30 ϕ c ° 0.4 = 20 ϕ c

ϕ ° =10 ϕ c

sin 0.2 h =1 °

= 0

n ϕ c

o °

i 0 t 10

c = −

ϕ c h = 2

ri f -0.2 ° d 20 e = −

z ϕ c

l i

b -0.4

o °

m 30 = − ϕ c -0.6 2h

-0.8

2(4− h2 ) -1 1 1.1 1.2 1.3 1.4 2(4− h2 ) height parameter h

Figure 2.20: Mobilized friction angle in pyramidal packing of various heights and inter-particle 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/ tan ϕ) 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 u )+ σ (2.13) − a s

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. 1 P (β)= 1+ a 3cos2 (β β ) 1 (2.14) β 4π − n − n h io 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. z E (β)= z.P (β)= 1+ a 3cos2 (β β ) 1 (2.15) β 4π − n − n h io 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.

max a a

z C

z z min z max

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, micromechanical 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 micromechanical 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 examined 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).

t i

Γ V

x j

b i

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. 1 σ¯ij = σij dV (3.1) V ZV The static equilibrium of forces at any material point can be written as:

∂σij + bi =0 (3.2) ∂xj

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: 1 1 1 σ¯ij = σij dV = xitj dΓ+ xibj dV (3.3) V ZV V ZΓ V ZV in which, xi is the individual position vector of the tractions and body forces.

R i tj

x i

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:

1 p 1 p p σ¯ij = σij dV = xi tjdΓ + xibj dV (3.4) p p p V p V V p Γ V XN Z XN Z Z where N p represents the number of particles; V p and Γp indicate the volume and boundary

surface of each particle.

c The position vector (xi) can be further expressed as xi = xi + Ri, where Ri indicates the location vector of the traction forces on the particle with respect to its centroid. Thus, Eq.

(3.4) becomes:

1 p 1 c p p σ¯ij = Ri tjdΓ + x tjdΓ + x bj dV (3.5) p p i p i V p Γ V p Γ V XN Z XN Z Z c ¯ Introducing the resultant body force acting at the centroid xi of the particle as bj, i.e.

c p ¯bjx = xibj dV (3.6) i p ZV Eq. (3.5) becomes:

1 p 1 c p σ¯ij = Ri tjdΓ + x tj dΓ + ¯bj (3.7) p i p V p Γ V p Γ XN Z XN Z Since each particle is locally in equilibrium, then the last term in Eq. (3.7) vanishes and

thus,

1 p σ¯ij = Ri tjdΓ (3.8) p V p Γ XN Z Moreover, the interaction between each pair of particles (α, β) can be described by the

traction forces (f αβ and f βα) seen as mutual action and reaction so that f αβ = f βα. j j j − j Referring to Fig. 3.3 and introducing the branch vector linking the centroids of the same

two particles (lαβ = Rαβ Rβα), Eq. (3.8) reduces to (Love, 1927): i i − i 1 σ¯ij = lifj (3.9) V c XN where N c is the total number of contact points in the REV.

46 α

αβ R

αβ l R

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.

1 1 s w σ¯ij = σij dV = σij dV + σij dV (3.10) s w V ZV V ZV ZV 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: w 1 s V σ¯ij = σij dV + uwδij (3.11) s V ZV V Furthermore applying Gauss-Ostrogradski theorem to convert volume into surface integral just like in Eq. (3.4), Eq. (3.11) becomes:

w 1 p p V σ¯ij = xi tjdΓ + xibj dV + uwδij (3.12) p p V p Γ V V XN Z Z 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.

u w f αβ βα 3 f Rβ αβ α β f α 1 c R xα f αβ

x c xβ f αβ 2

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:

w 1 uw p V σ¯ij = lifj + Rinj dΓ + uwδij (3.13) p V c V p Γ V XN XN Z where nj is the unit normal vector to the particle surface. Applying the Gauss-Ostrogradski theorem:

p ∂Ri p p p Rinj dΓ = dV = δij dV = V δij (3.14) p p p ZΓ ZV ∂xj ZV

48 where V p indicates the volume of the particle. Thus, noting Eq. (3.14), Eq. (3.13) gives the total stress as follows:

1 V s V w 1 σ¯ij = lifj + uwδij + uwδij = lifj + uwδij (3.15) V c V V V c XN XN The ﬁrst term on the right hand side of Eq. (3.15) involves the inter-particle forces, and

′ hence refers to the eﬀective stress σij acting in the solid skeleton. Thus, the total stress can be ﬁnally written as:

′ σ¯ij = σij + 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

p acting over the wetted surface (Γw), and the inter-particle forces arising from the external p forces acting over ﬁnite contact areas (Γd). 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:

w 1 p p V σ¯ij = p Rifj dΓd + p Riuwnj dΓw + uwδij (3.17) V p Γ p Γw ! V XN Z d XN Z p p p The pore pressure transmission on the wetted parts (Γ =Γ Γd ) of the particle can w − 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:

p p p Riuwnj dΓw = Riuwnj dΓ Riuwnj dΓd (3.18) Γp Γp Γp Z w Z − Z d

49 uw αβ f j p Γw

p Γd

αβ fj

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:

1 p 1 p σ¯ij = p Rifj dΓd + uw δij p Rinj dΓd (3.19) V p Γ ! − V p Γ ! XN Z d XN Z d Equation (3.19) essentially describes the force transmission into the fully saturated granular 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 (σij), 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,

′ σ¯ij = σij + αij uw (3.20) ′ 1 p 1 p σ = p Rifj dΓ ; αij = δij p Rinj dΓ ij V Γd d V Γd d N p − N p P R P R The physical interpretation of the integral of the term (Rinj) over the dry contact surface

d Γp 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

d scale in the deﬁnition of stress in Eq. (3.20) through the surface area Γp normalized to the particle radius R.

Also, when the contact area tends to zero (Γd 0), α δ , which leads to Terzaghi’s p → ij → ij eﬀective stress equation. When (Γd Γ ) as in a continuous medium, α 0, giving rise p → p ij → 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.

3 3fR r = (3.21) s 4E

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: 2/3 p 2 3fR Γc = πr = π (3.22) 4E ! A particle within the REV (Fig. 3.6) is in contact with nc local neighboring particles so nc p p that the total contact area per particle is Γd = Γc . The eﬀective stress in the REV can be P determined from Eq. (3.20) where the contribution of the pore pressures is given by αij, i.e.

p Γ1 p p Γ5 Γ2

p p Γ3 Γ4

c n p p , n c Γd =∑ Γc = 5

Figure 3.6: Particle in contact with neighboring particles

N p nc 1 p αij = δij Bij; Bij = p Rinj dΓc (3.23) − V Γc X X Z where Bij describes the oriented contact area with respect to the local reference frame at the centroid of a spherical particle (Fig. 3.7).

Z f

Y r

X Hertz law

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:

2π π ′ αij = δij = bij(β,φ) p (n) sin βdβdφ (3.24) Z0 Z0 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) Bkl Mjl(β,φ) (3.25)

sin2φ + (1 sin2φ)cos β sin φ cos φ(1 cos β) cos φ sin β  − − − −  M(β,φ)= 2 2 (3.26)  sin φ cos φ(1 cos β) cos φ + (1 cos φ)cos β sin φ sin β   − − − −       cos φ sin β sin φ sin β cos β      and 2N c p(n) p′(n)= (3.27) V 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:

c 1 2N 2π π Fij = ninj = (ni nj)p(n)sin βdβdφ (3.28) 2N c 0 0 X Z Z with

p(n) 0, p(n) dV = 1 andp(n)= p( n) (3.29) ≥ ZV − 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):

1 p(n)= 1+ a 3cos2β 1 (3.30) 4π − h i 53 z n

β dβ y dφ φ

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.

0 0 0   B = Rn n dΓp =Γp R (3.31) ij p i j c c  0 0 0  ZΓc        0 0 1      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+ a 3cos2(β β ) 1 (3.32) n − n − h i 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,

0 0 0 p 2/3   B = Γ R 1+ a 3cos2β 1 (3.33) ij c n  0 0 0  −   h i      0 0 1      p where Γc is the mean contact area over the REV.

Finally, the second order tensor αij required to calculate the eﬀective stress can be computed by inserting Eqs. (3.33, 3.25) into Eq. (3.24), i.e.

λx 0 0 p 2N c Γ R   α = δ c (3.34) ij ij  0 λy 0  − 3V !        0 0 λz      where 2 4 4 8 λ = λ =1 a a ; λ =1+ a + a (3.35) x y − 5 − 15 n z 5 15 n p Furthermore, noting that the term (R Γc /3) refers to the conical volume formed by the

c p mean contact surface between the particles as shown in Fig. 3.6, 2N (R Γc /3) turns out to

t be the total conical volumes (Vcone) formed by the contact surfaces in the REV. Therefore,

55 Eq. (3.34 becomes:

λx 0 0   V t α = δ µ ; µ = cone (3.36) ij ij  0 λy 0  −   V      0 0 λz      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

α = (1 µ)δ (3.37) ij − ij In the limiting condition of rigid particles, µ 0 and α δ . → ij → ij

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.

1 1 s w a σ¯ij = σij dV = σij dV + σij dV + σij dV (3.38) s w a V ZV V ZV ZV ZV 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 liquid air volume, V a

particle volume, V s gas water volume, V w solid σ VVVV= a ∪ w ∪ s x

REV ( V )

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:

w a 1 p p V V σ¯ij = xitj dΓ + xibj dV + uwδij + uaδij (3.39) p p V " p Γ p V # V V XN Z XN Z

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

p contributions from inter-particle forces, actions of air and water pressures on dry (Γd) and

p wetted (Γw) 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.

c Furthermore, noting that xi = xi +Ri, and considering equilibrium of forces on the closed surface of each particle, (refer to Eq. (3.7)), we ﬁnally get:

1 u p u a p w p p σ¯ij = V lifj + V Γ Rninj dΓd + V Γw Rninj dΓw N c N p d N p nl P P R P P R (3.40) 1 V w V a V Γm RniTj dΓm + V uwδij + V uaδij − N p nl P P R 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

p bridges on each particle, Γw is the part of the particle wetted by each liquid bridge, whereas p Γd is the union of all dry parts of the particles surface (see Fig. 3.11).

u n T a f αβ ()Γ p 3 T d αβ uw f T Γ p Γm w Γm uw T

p 1 ()Γd ua Γ p 2 m ()Γd u T a uw f αβ T

p p1 p 2 p 3 Γd =()()() Γ d ∪ Γ d ∪ Γ d

(a) center particle with 3 neigbours jointed by menisci (b) free body diagram for center particle with interacting forces

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.

′ 1 σij = lifj (3.41) V c XN 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

cap capillary forces known as suction force (f1 ). This suction force appears due to the second and third terms on the right hand side of Eq. (3.40).

+ =

p p p ua on Γ u on Γ u - u on Γ d w w a w w

Figure 3.12: Unequal hydrostatic forces around the particle

cap The second source of capillary forces (f2 ) 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 R1

ua Ts

uw R2

Figure 3.13: Unequal hydrostatic pressure on the air/water interface and dry surfaces must be found, i.e.

p p p p Rninj dΓd = V δij p Rninj dΓw (3.42) Γ − l Γw Z d Xn Z 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′ =(σ u δ )+ χ (u u )+ B (3.43) ij ij − a ij ij a − w ij

with

1 p χij =(n.Sr)δij + p Rninj dΓw (3.44) V p l Γw XN Xn Z and

1 RTs Bij = RniTj dΓm = niej dΓm (3.45) V p l Γm V p l Γm XN Xn Z XN Xn Z 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 (u u ) to account for the spatial distribution (fabric) of liquid bridges and a − w 60 T θ n ua T θ n a c a T T uw n Γ p α w n f p Γd ua T uw b d T countour b Γm T T ua p ():d− c = Γw ():a− b = Γm α= filling angle (here at max); θ =wetting angle

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 (χ (u u )) as well in the REV. ij a − w

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:

ψ = χ (u u )+ B (3.46) ij ij a − w ij

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

p surface (Γw) 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).

θ R1

R R α 2

N H R2

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 curvatures R1 and R2 can be determined, and hence the matric suction corresponding to the speciﬁed liquid bridge volume can be found

through Young-Laplace equation:

1 1 ua uw = Ts (3.47) − R1 − R2

0.8

0.7

0.6 o 0.5 H/R=0, θ=40 o H/R=0.1, θ=40 rel 0.4 o V H/R=0.5, θ=40 o 0.3 H/R=0, θ=20 o H/R=0.1, θ=20 0.2 o H/R=0.5, θ=20 0.1

0 0 20 40 60 80 α

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 (u u ) in the liquid bridge a − w 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.

50 Negative matric suction 40

30 Positive matric suction 20 half filling (degree) angle filling half

0 0 10 20 30 40 50 60 wetting angle (degree)

Figure 3.17: Positive and negative matric suction zones as a function of the half ﬁlling and wetting 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 n = R / R where the radius vector R varies with the half ﬁlling angle α. Also, the i i | | i

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:

p p Aij = Γw Rninj dΓw (1 cosα)2(2 + cosα)R 0 0 − 3   (3.48) = πR 2 3  0 (1 cosα) (2 + cosα) 0   −     3   0 0 2(1 cos α)   −    ′ B ij = Γm RTs niej dΓm sin2αcos(α + θ)0R 0  −  (3.49) =(πR2T ) 2 s  0 sin αcos(α + θ) 0   −       0 0 sin(2α)sin(α + θ)     

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′ = σ u δ . On the other hand, in the saturated case (S = 1), α = 180◦, ij ij − a ij r

thus Aij = Vpδij, which leads to χij = δij and Bij = 0. As such Eq. (3.43) becomes σ′ = σ u δ . ij ij − w ij 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 approximated 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)= 1 [1 + a (3cos2β 1)] 4π − (3.50) p(n) 0; p(n) dV = 1; p (n)= p ( n) ≥ V − R 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):

5 (Fz Fx) 5 a = − = (Fz Fx) (3.51) 2 tr(Fij) 2 − On the other hand, the liquid bridges probability density function, which demonstrates

the liquid bridges distribution in the REV, can be introduced as:

LB 1 2 p (n)= [1 + aLB (3cos β 1)] 4π − (3.52) pLB(n) 0; pLB(n) dV = 1; pLB (n)= pLB ( n) ≥ V − R 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:

1 2π π LB LB n Fij = LB ninj = (ninj) p ( ) sin β dβdφ (3.53) 2N LB 0 0 2NX Z Z LB while N 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:

LB LB 5 Fz Fx 5 a = − = F LB F LB (3.54) LB 2 tr(F LB) 2 z − x ij 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: w V 2π π ′ χij = δij + MikAklMjl p (n)sin β dβdφ (3.55) V Z0 Z0 2π π ′ ′ Bij = MikBklMjl p (n)sin β dβdφ (3.56) Z0 Z0 where 2N LBpLB (n) 2N cp (n) λ p′ (n)= = (3.57) V V 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 pendular regime was formally derived through a micromechanical analysis. The cases of fully saturated (two-phase) conditions with and without particle compressibility were also investigated. 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: c 1 2N F = n n (4.1) ij 2N c i j X 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' z

x y L L (a) (b)

Figure 4.2: Illustration of body-Centered Cubic Packing (BCC)

(L′)2 +2L2 = 16(R + H/2)2, or (4.2) 2 (l′)2 +2l2 = 16 1+ H/2 ; l = L/R, l′ = L′/R, H = H/R Due to geometrical compatibility,

3 1 l′/2 < √2 and 1 l/2 < (4.3) ≤ ≤ s2

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 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 -0.22 2.2 0.318 0.335 0.335 0.330 -0.012 2.3094 0.320 0.333 0.333 0.333 0 2.4 0.318 0.320 0.320 0.360 0.10 2.5 0.313 0.305 0.305 0.390 0.21 2.6 0.303 0.289 0.289 0.422 0.33 2.7 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.

Fx 0 0   F = (4.4) ij  0 Fy 0         0 0 Fz      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' z L x y L

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.

0.5 BCC 0.45 SCP FCP

0.4

0.35 porosity (n) porosity

0.3

0.25 2 2.2 2.4 2.6 2.8 3 l'

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).

1000000 SCP , R=0.1 mm SCP, R=0.001 mm 100000 FCP, R=0.1 mm

10000 FCP, R=0.001 mm

1000

100 matric suction (kPa) suction matric

0.1 0 5 10 15 20 25 degree of saturation (Sr%)

Figure 4.7: SWCC of SCP and FCP as a function of particle size, H = θ =0

75 Table 4.2: SWCC calculation 1. k =1(k is degree of saturation index)

LB 2. Set the value of θ,R,Ts,V,n,N ,H

0 initial 3. Set Sr = Sr , ∆Sr =0

k k−1 4. Sr = Sr +∆Sr

5. Compute the volume of liquid bridges:

k k Vw = VSr n

k k LB VLB = Vw /N

k k k k 6. Compute α ,R1,R2, N by solving the following system of equations (Toroidal approximation):

3 k 2 2 k k VLB k k k k (N ) k k V = 3 = (R + R ) +(R ) N R + R LB 2πR 1 2 1 − 3 − 1 2 2 2 2 2 k k− H k k k k N (N 2 ) k H N (R1) (N ) +(R1) arcsin k (3 N + ) R 1 3 2 " r − # − −

k H − k k R1 ( 2 +1 cos α ) R 1 = R = cos(αk+θ)

k k k R = R2 = sin αk + R (sin αk + θ 1) 2 R 1 − k N = H +1 cos αk,H = H/R 2 − 7. Compute corresponding matric suction (Laplace equation):

k 1 1 (ua uw) = Ts k k R1 R2 − −

k 8. Calculate dimensionless suction, ψ =(u u )kR/T a − w s k ◦ ◦ 9. If α <αmax (αmax = 45 for SCP, 30 for FCP)

∆S = c (c = 0 isaconstant) r 6 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 = 0), matric suction increases from zero to a 6 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).

77 100

o drying, θ = 0o 10 o wetting, θ = 30

drying schematic path between wetting/drying 1 cycle wetting

0.1 dimensionless matric suction dimensionless

0.01 0 5 10 15 20 25 30 degree of saturation (%) (a)

100

o drying, θ = 0 o wetting, θ = 30 10 drying schematic path between wetting/drying cycle wetting

1 dimensionless matric suction dimensionless

0.1 0 10 20 30 40 degree of saturation (%) (b)

Figure 4.8: Eﬀect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), H =0

78 100

H = 0

10 H = 0.05 H = 0.1

1 dimensionless matric suction dimensionless

0.1 0 5 10 15 20 degree of saturation (Sr%) (a) 100

H = 0 H = 0.05 10 H = 0.1

1 dimensionless matric suction dimensionless

0.1 0 5 10 15 20 25 30 degree of saturation (%) (b)

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 distance H 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, χ (u u ) , due to the same amount of matric suction. ij a − w

80 Table 4.3: χij calculation 1. k =1(k is degree of saturation index)

LB 2. Set the value of θ,R,Ts,V,n,N ,H,aLB

0 initial 3. Set Sr = Sr , ∆Sr =0

k k−1 4. Sr = Sr +∆Sr

5. Compute the volume of liquid bridges:

k k Vw = VSr n

k k LB VLB = Vw /N

k k k k 6. Compute α ,R1,R2, N by solving the system of equations, presented in Table. 4.2

k 1 1 7. Compute corresponding matric suction (Laplace’s equation) (ua uw) = Ts k k R1 R2 − − 2 1 cosαk (2 + cosαk) 0 0 πR3 − 2 k  k k  8. Aij = 3 0 1 cosα (2 + cosα ) 0  − 3 k   0 0 2(1 cos α )     −  9. For variables β ,φ put

pLB (n)= 1 1+ a [3cos2 (β) 1] 4π { LB − } p′(n)=2N LBpLB (n) /V

M(β,φ) as in Eq. 3.26

k 10. Compute χij

k k Vw 2π π k ′ χij = V δij + 0 0 MilAlmMjmp (n)sin βdβdφ R R k ◦ ◦ 11. If α <αmax (αmax = 45 for SCP, 30 for FCP)

∆S = c (c = 0 isaconstant) r 6 Set k = k +1 and go to 4 else stop

81 Table 4.4: Bij calculation 1. k =1(k is degree of saturation index)

LB 2. Set the value of θ,R,Ts,V,n,N ,H,aLB

0 initial 3. Set Sr = Sr , ∆Sr =0

k k−1 4. Sr = Sr +∆Sr

5. Compute the volume of liquid bridges:

k k Vw = VSr n

k k LB VLB = Vw /N

k k k k 6. Compute α ,R1,R2, N by solving the system of equations, presented in Table. 4.2

k 1 1 7. Compute corresponding matric suction (Laplace’s equation) (ua uw) = Ts k k R1 R2 − − sin2αk cos(αk + θ)0 0 ′k 2 − 2 k k 8. Bij = πR Ts  0 sin α cos(α + θ) 0  0− 0 sin(2αk)sin(αk + θ)     9. For variables β ,φ put

pLB (n)= 1 1+ a [3cos2 (β) 1] 4π { LB − } p′(n)=2N LBpLB (n) /V

M(β,φ) as in Eq. 3.26

k 10. Compute Bij

k 2π π ′k ′ Bij = 0 0 MilBlmMjmp (n)sin βdβdφ R R k ◦ ◦ 11. If α <αmax (αmax = 45 for SCP, 30 for FCP)

∆S = c (c = 0 isaconstant) r 6 Set k = k +1 and go to 4 else stop

82 0.6

0.5

0.4

ij 0.3 χ H = 0

0.2 H = 0.05

H = 0.1 0.1 H = 0.2 0 0 5 10 15 20 degree of saturation (%) (a)

0.7

0.6

0.5

0.4 ij χ 0.3 H = 0 0.2 H = 0.05 H = 0.1 0.1 H = 0.2 0 0 5 10 15 20 25 30 degree of saturation (%) (b)

Figure 4.10: The resulting isotropic eﬀective stress coeﬃcient χij while θ = 0(a) Loose packing (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.

1.0 Silt, Drained test (Donald, 1961)

0.9 Silt, Constant water content test (Donald, 1961)

χ 0.8 Madrid gray clay (Escario and Juca, 1989)

0.7 Madrid silty clay (Escario and Juca,1989)

0.6 Madrid clay sand (Escario and Juca, 1989) 0.5 Moraine (Blight, 1961) 0.4 Boulder clay (Blight, 1961) 0.3 Clay-shale (Blight, 1961) 0.2 χ =S r effective stress parameter parameter stress effective 0.1 FCP, n = 0.26 0 SCP, n = 0.49 0 20 40 60 80 100 degree of saturation (%)

Figure 4.11: Computed relationships between degree of saturation and eﬀective stress parameter for various packings

Next, the capillary stresses induced by suction and surface tension forces (where θ = 0) 6 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 (χ (u u )) and surface tension (B ) contributions. As shown in Fig. 4.12 and 4.13, ij a − w ij 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 particle/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 suctionof1 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 1.2

1 (kPa) 0.8 u

− 0.6 u () o ij a w θ= 0 χ 0.4 o θ=10 o θ=20 0.2 o θ=30 0 0.01 0.1 1 10 100 1000 10000 matric suction (kPa) (a)

1.2

0.8 (kPa) 0.6 o ij

B θ= 0 o θ=10 0.4 o θ=20 o θ=30 0.2

0 0.01 0.1 1 10 100 1000 10000 matric suction (kPa) (b)

Figure 4.12: The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a loose packing (SCP)- R =0.1 mm, H =0

86 3.5

(kPa) 2.5

u 2 − o u

() θ= 0 1.5 o ij a w θ=10 χ o 1 θ=20 o θ=30 0.5

0 1 10 100 1000 10000 matric suction (kPa) (a)

3.5

3.0

2.5

2.0 o θ= 0

(kPa) o 1.5 θ=10

ij o B θ=20 o 1.0 θ=30 0.5

0.0 1 10 100 1000 10000 matric suction (kPa) (b)

Figure 4.13: The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a dense packing (FCP)- R =0.1 mm, H =0

87 1.2

0.8

0.6 Bij χ (u − u ) 0.4 ij a w

ψ ij Capillary (kPa) stress Capillary 0.2

0 0.01 0.1 1 10 100 1000 10000 matric suction (kPa) (a)

3.0

2.5

2.0

Bij 1.5 χ ij(u a− u w ) 1.0 ψ ij

Capillary (kPa) stress Capillary 0.5

0.0 1 10 100 1000 10000 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 0.43 492.84 1.04 584.53 1.04 716.33 2.11 699.14 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 0.11 mm 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 1200

1000

800

600

tensile strength (Pa) strength tensile Predicted tensile strength, FCP, θ =20 400 o Prediced tensile strength, BCP, θ =20 o Predicted tensile strength, FCP, θ =0 o 200 Predicted tensile strength, BCC, θ =0 Experimental data, F-75 (e=0.72) Experimental data,F-75 (e=0.58) 0 0 0.5 1 1.5 2 2.5 3 3.5 volumetric water content , (w %)

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 (χ (u u )) and surface tension (B ) forces as two controlling ij a − w ij 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 χ (u u ) and B as function of degree of saturation. It is recalled herein ij a − w ij that the capillary stress has two contributions: one arising from suction (χ (u u )) and ij a − w 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.

90 ° 3 90 ° 0.3 120 60 120 60 2 0.2 150 30 150 30 1 0.1

180° 0° 180° 0°

210 330 210 330

240 300 240 300 270° 270° ° χij ()ua− u w (kPa) 90 3 Bij (kPa) 120 60 2 150 30 1

Sr=14% 180° 0° Sr=9.8% Sr=5.6% Sr=0.5% 210 330 Sr=0.02% 240 300 270°

ψ ij (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 (χ (u u )). ij a − w 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 3.5

2.5

2 Bij (axial) 1.5 χij(u a - u w )(axial)

capillary stress (kPa) capillary 1 ψ ij (axial) 0.5

0 1 10 100 1000 10000 matric suction (kPa)

1.5

Bij (lateral) 1 χ ij(u a - u w )(lateral)

0.5 ψ ij (lateral) capillary stress (kPa) capillary 0

- 0.5 1 10 100 1000 10000 matric suction (kPa)

Figure 4.17: Principal capillary stresses with various contributions in axial and lateral directions, H = θ =0

92 contribution of surface tension forces to the total capillary stress generated in the packing

(ψij).

90 ° 2 90 ° 2 120 60 120 60 1.5 1.5

150 1 30 150 1 30 0.5 0.5

180° 0° 180° 0°

210 330 210 330

240 300 240 300 270° 270°

χij ()ua− u w (kPa) Bij (kPa)

90 ° 2 120 60 1.5

150 1 30 0.5

180° 0°

o θ =0 210 330 o θ =30 240 300 270°

ψij (kPa)

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 5 (ψz ψx) 5 (ψz ψx) aψ = − = − (4.5) 2 trace(ψij) 2 (ψx + ψy + ψz) 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 nature 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 0.8

0.6 ψ l '=2.7 0.4 l '=2.6 l '=2.5 0.2 l '=2.4 l '=2.3 0 anisotropy factor, a factor, anisotropy 0 5 10 15 20 l '=2.1 -0.2

-0.4 degree of saturation (%)

Figure 4.19: Meniscus-based anisotropy as a function of saturation for various anisotropic BCC 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 saturation 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 = σn tan ϕ + c (5.1)

′ where τf and σn 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).

τ =(σ u )tan ϕ + c (5.2) f n − a a 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 micromechanical 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 σn 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, wetting 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 (u u ) is set. a − w 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=f αβ . −→ con - The inter-particle force due to capillarity is speciﬁed:

((u u ),hαβ) is considered; Is there a solution for Young-Laplace equation? a − w No inter-particle capillary force & liquid bridge volume=0. −→ Yes inter-particle capillary force= f αβ & liquid bridge volume=V αβ. −→ cap LB

αβ αβ αβ - Total inter-particle force is calculated as fint =fcon+fcap.

4. The corresponding degree of saturation is deﬁned with respect to total volume of liquid bridges.

5. The resulted micro-scale deformations and the movements of particles are deﬁned based on 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 condition is reached.

101 5.2.2 DEM sample description

The DEM sample is composed of 10,000 mono-dispersed, completely smooth, spherical particles 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 (0.024 mm), 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 parameters Inter-particle friction angle 18◦ Number of particles 10,000 Initial volume 1 mm3 Radius of particles 0.024 mm Initial fabric δij/3 Wetting angle 0 Initial porosity (n) 0.4 Surface tension parameter 0.073 N/m 6 Normal stiﬀness (Kn) 10 Pa 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.

1000000 SCP , R=0.1 mm SCP, R=0.001 mm 100000 FCP, R=0.1 mm

10000 FCP, R=0.001 mm DEM sample 1000 R=0.024 mm

100 matric suction (kPa) suction matric

0.1 0 5 10 15 20 25 degree of saturation (Sr%)

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 results Dry (matric suction=0 kPa) Lateral pressure (Pa) 250 500 750 1000 Normal stress at failure (Pa) 591 1207 1843 2480 Matric suction=15 kPa Lateral pressure (Pa) 250 500 750 1000 Normal stress at failure (Pa) 5388 6345 7201 8018 Matric suction=30 kPa Lateral pressure (Pa) 250 500 750 1000 Normal stress at failure (Pa) 5748 6593 7426 8329 Matric suction=300 kPa Lateral pressure (Pa) 250 750 Normal 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 = σ σ ) space, by drawing the best line passing through the peak shear strengths of the z − r 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 2.50

2.00

1.50 q/p 1.00 dry sample 0.50 suction 15 kPa suction 30 kPa suction 300 kPa 0.00 0.00 0.05 0.10 0.15 0.20 0.25 axial strain (a) -0.10

-0.08

-0.06

-0.04

-0.02 volumetric strain volumetric 0.00 dry sample suction 15 kPa 0.02 suction 30 kPa suction 300 kPa 0.04 0.00 0.05 0.10 0.15 0.20 0.25 axial strain (b) negative volumetric strain is considered as dilation

Figure 5.3: (a) Deviatoric stress and (b) Volumetric strain versus axial strain for DEM samples with lateral pressure of 750 Pa

106 8000 7000 6000

5000

4000 (Pa) q 3000 dry sample suction 15 kPa 2000 suction 30 kPa 1000 suction 300 kPa 0 0 1000 2000 3000 4000 p (Pa)

Figure 5.4: Failure envelope of DEM samples considering the peak shear strength as the failure 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 o 90 5 kPa 120 60 4 3 150 30 2 1 o o 180 0

Suction 300 kPa, Sr=0.05% 210 330 Suction 30 kPa, Sr=2.2% Suction 15 kPa, Sr=5.5% 240 o 300 270

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.

9000 8000 7000 6000

(Pa) dry sample

q' 5000 -

q sucion 15 kPa 4000 suction 30 kPa 3000 suction 300 kPa 2000 suction 15 kPa, Bishop 1000 suction 30 kPa, Bishop suction 300 kPa, Bishop 0 0 2000 4000 6000 8000 p-p' (Pa)

Figure 5.6: Strength of wet granular material based on (a) net stress (q,p) and (b) eﬀective stress (q′,p′)

The validity of the eﬀective stress equation is next checked at every stage during deformation 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 distribution 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.

2.50

2.00

1.50 q'/p'

1.00 dry sample total stress - suction 15 kPa total stress - suction 30 kPa 0.50 total stress - suction 300 kPa calculated effective stress - suction 15 kPa calculated effective stress - suction 30 kPa 0.00 calculated effective stress - suction 300 kPa 0.00 0.05 0.10 0.15 0.20 0.25 axial strain

Figure 5.7: Shear strength response based on eﬀective stresses for a conﬁning pressure of 750 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.

0.9 0.8 0.7 0.6 0.5 capillary stress 0.4 effective stress 0.3 anisotropy factoranisotropy 0.2 0.1 0 0.00 0.05 0.10 0.15 0.20 0.25 -0.1 axial strain

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 mm Speciﬁc gravity 2.65 Number of particles 37,867 Dimensionless separation distance 0.05 Surface tension of water 0.0727 N/m Wetting angle 0 Initial 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 physical 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 during 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 characteristics 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 30

SCP 25 DEM sample 20 % r S 15

0 10 100 1000 10000 matric suction (Pa)

Figure 5.9: Comparisons between SWCC of selected SCP sample and simulated DEM samples 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

113 600

500

400

300

shear stress (pa) stress shear 200 water content=0% water content=0.2% 100 water content=2.0% water content=5.2% 0 0 200 400 600 800 1000 normal net stress (Pa) (a)

600

500

400

300 shear stress (Pa) stress shear 200 water content=0% water content=0.2% 100 water content=2% water content=5% 0 0 200 400 600 800 1000 normal effective stress (Pa) (b)

Figure 5.10: (a) Shear strength response based on net stresses (adopted from Shamy and Groger, 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′ =(σ u δ )+ χ (u u )+ B (6.1) ij ij − a ij ij a − w ij 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.

ψ = χ (u u )+ B (6.2) ij ij a − w ij 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 (χ (u u )), and the second arising from ij a − w 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 packings. 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).

σ ε ij Macroscopic level ij Stress Tensor Strain Tensor

cap u cap fi fi i ui Inter-particle Inter-particle Displacements due to Displacements due to contact forces capillary forces contact forces capillary forces

Microscopic level

Figure 6.1: Homogenization method in order to develop a constitutive model in unsaturated media

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 and Kimoto, Kyoto University, Japan

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129 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.

R R α 2 x N H

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:

R ( H +1 cos α) R = 1 = 2 − (A.1) 1 R cos(α + θ)

R R = 2 = sin α + R (sin(α + θ) 1) (A.2) 2 R 1 − where, H N = +1 cos α, H = H/R (A.3) 2 −

130 On the other hand, considering a quadrant of the water lens in Cartesian coordinates, the liquid bridge proﬁle can be described as:

y (x)=(R + R ) R2 x2 (A.4) LB 1 2 − 1 − q and the wetted surface of the particle can be written as:

H 2 y (x)= R2 (x R) (A.5) p s − − 2 −

Therefore, calculating the volumes generated by these two proﬁles around the basis (x axes), the volume of liquid bridge can be determined as:

N N 2 2 VLB =2π [yLB(x)] dx 2π [yp(x)] dx (A.6) Z0 − ZH/2

Inserting Eqs. (A.4) and (A.5) in Eq. (A.6), the dimensionless volume of liquid bridge is deﬁned as:

3 VLB 2 2 (N) V LB = 2πR3 = (R1 + R2) +(R1) N 3 R1 + R2 − −2 h i − H (A.7) 2 2 2 N (N 2 ) H N (R1) (N) +(R1) arcsin (3 N + ) − R1 − 3 − 2 q 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