UNIVERSITY OF CALGARY. An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime. Sarah Khosravani A THESIS

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1 UNIVERSITY OF CALGARY An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime by Sarah Khosravani A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF CIVIL ENGINEERING CALGARY, ALBERTA April, 2014 c Sarah Khosravani 2014

2 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

3 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 assistantships. 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

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

5 Table of Contents 1 Abstract ii Acknowledgments iii Dedication iv Table of Contents v List of Tables vii List of Figures viii List of Symbols xi 1 INTRODUCTION Introduction Objectives Organization of Thesis LITERATURE REVIEW Introduction Capillary Effect and Matric Suction Soil water characteristic curve Experimental Observations on Unsaturated Soil Behaviours Shear and tensile strengths of unsaturated soils Collapse behaviour Studies on Effective Stress of Unsaturated Soils - Existing Frameworks Phenomenological studies (Macroscale studies) Single effective stress approach Independent stress state variables approach Micromechanical studies Summary MICROMECHANICS OF EFFECTIVE STRESS IN MULTIPHASIC GRAN- ULAR MEDIA Introduction Force Transport in Dry Granular Media Force Transport in Saturated Granular Media Negligible contact area - rigid particles Finite contact area - compressible particles Effective stress in a fully saturated idealized compressible particle packing Force Transport in Unsaturated Granular Media Effective stress parameters for idealized packing Summary COMPUTATION OF CAPILLARY STRESSES IN IDEALIZED GRANU- LAR PACKINGS Introduction Idealized Packing Simple cubic packing (SCP) v

6 4.2.2 Body-centered cubic packing (BCC) Cubic Close Packing or Face Centered Packing (CCP or FCP) Theoretical SWCC for Regular Packing in Pendular Regime Effective Stress Parameters and Capillary Stress in Regular Packing Isotropic packings Effective stress parameters and capillary stresses in SCP and FCP Isotropic tensile strength in comparison with experimental results Anisotropic packings Evolution of capillary stress in BCC packing-anisotropy aspects Evolution of degree of anisotropy - link to strength issues Summary VALIDATION OF THE PROPOSED EQUATION USING DEM SIMULA- TION Introduction Triaxial Tests Simulation at Various Controlled Matric Suctions Brief review on DEM modelling in unsaturated media DEM sample description DEM triaxial test procedure and results Validation of the proposed effective stress equation with DEM simulation results Validation of the proposed effective stress equation using data from literature Summary CONCLUSIONS AND RECOMMENDATIONS Conclusions Recommendations for Future Work Bibliography A Toroidal Approximation vi

7 List of Tables 2.1 Review of the conventional modelling approaches in unsaturated soil mechanics (Buscarnera, 2010) Properties of BCC packings with various l SWCC calculation χ ij calculation B ij calculation Direct tensile test results of clean F-75 sand (Kim, 2001) Simplified steps of DEM modeling of unsaturated granular media DEM sample input parameters Shear strengths of samples with various matric suctions, DEM results DEM sample properties (Shamy & Groger, 2008) vii

8 List of Figures and Illustrations 2.1 Illustration of surface tension Water in capillary tube Free body diagram of forces acting on air-water interface in a capillary tube Curved liquid and gas interfaces Conceptual demonstration of unsaturated sample in different regimes(lu and Likos, 2004) Conventional soil water characteristic curve for sand and silt(lu and Likos, 2004) Demonstration of the ink-bottle effect during:(a)drying process and(b)wetting process (Marshall et al., 1996) Theoretical presentation of soil-water characteristic curve of an unsaturated sample in different regimes (Lu et al., 2007) General representation of shear strength in unsaturated samples(ho and Fredlund, 1982) Yield locus of glass beads R=46 micron (Pierrat et al., 1998) Yield locus of glass beads R=90 micron (Pierrat et al., 1998) Direct shear test results on cohesionless sands (Donald I., 1956) Tensile strength versus water content (F-75-C),(Kim, 2001) Tensile strength versus water content,(kim, 2001) One-dimensional compression and subsequent soaking tests under constant void ratio or applied pressure (Jennings and Burland, 1962) Effective stress coefficient for unsaturated soil based on experimental results Axis translation method in measuring matric suction in laboratory (Hilf, 1956) Dimensionless liquid bridge volume versus dimensionless suction (Molenkamp and Nazemi,2003) General scheme of homogenization technique (Oda and Iwashita, 1999) Mobilized friction angle in pyramidal packing of various heights and interparticle friction angles. Negative inter-particle friction angle represents vertical extension Domain of accessible geometrical states based on harmonic representation of granular media (Radjai, 2008) Cauchy s stress in a closed domain Assembly of dry granular media Branch vector between pair of particles Free body diagram of inter-particle forces in saturated media Free body diagram of saturated media with compressible particles Particle in contact with neighboring particles Local coordinates on the center of each particle Spherical REV and global coordination system (Quadfel and Rothenburg,2001) Schematic anisotropic force distribution in polar system, a n = 0.5 & β n = π/ Unsaturated media as a three phase system in pendular state viii

9 3.11 Free body diagram of inter-particle forces Unequal hydrostatic forces around the particle Unequal hydrostatic pressure on the air/water interface Traction forces between a pair of spherical particles Concave liquid bridge geometry between a pair of uni-size particles Dimensionless liquid volume V rel as a function of the half filling angle α (Megias-Alguacil and Gaucker, 2009) Positive and negative matric suction zones as a function of the half filling and wetting angles (Lu and Likos, 2004) Local coordinates illustration for each liquid bridge Illustration of simple cubic packing (SCP) Illustration of body-centered Cubic Packing (BCC) Separation distance between particles, H (Pietsch, 1968) Arrangement of BCC packing unit cells in 3D space Illustration of face centered packing (FCP) Porosity of different regular spherical packing SWCC of SCP and FCP as a function of particle size, H = θ = Effect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), H = Effect of separation distance on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), θ = The resulting isotropic effective stress coefficient χ ij while θ = 0(a) Loose packing (SCP) and (b) Dense packing (FCP) Computed relationships between degree of saturation and effective stress parameter for various packings The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a loose packing (SCP)- R =0.1 mm, H = The capillary stress induced by (a) Suction forces, (b) Surface tension forces in a dense packing (FCP)- R =0.1 mm, H = The total capillary stress in (a) Loose packing (SCP) (b) Dense packing (FCP)- R =0.1 mm, θ = 30 and H = Compression between measured and predicted tensile strength Polar plot of anisotropic capillary stresses for various saturation degree, H = θ = Principal capillary stresses with various contributions in axial and lateral directions, H = θ = Polar plot of anisotropic capillary stresses for various wetting angles, H = Meniscus-based anisotropy as a function of saturation for various anisotropic BCC packings, θ = H = DEM sample consisting of 10,000 mono-sized spherical particles SWCC of the DEM sample, R=0.024 mm (a) Deviatoric stress and (b) Volumetric strain versus axial strain for DEM samples with lateral pressure of 750 Pa ix

10 5.4 Failure envelope of DEM samples considering the peak shear strength as the failure point Anisotropic capillary stress in unsaturated DEM samples,axial strain=20% Strength of wet granular material based on(a) net stress(q,p) and(b) effective stress (q,p ) Shear strength response based on effective stresses for a confining pressure of 750 Pa Anisotropy changes for both effective and capillary stresses for a confining pressure of 750 Pa at matric suction of 30 kpa Comparisons between SWCC of selected SCP sample and simulated DEM samples by Shamy and Groger, (a) Shear strength response based on net stresses (adopted from Shamy and Groger, 2008). (b) Shear strength response based on calculated effective stresses Homogenization method in order to develop a constitutive model in unsaturated media Micro-CT scan of water menisci of Toyoura sand, courtesy of Profs. Oka and Kimoto, Kyoto University, Japan A.1 Concave liquid bridge geometry between a pair of uni-size particles x

11 List of Symbols, Abbreviations and Nomenclature Roman letter symbols a a n a LB a ψ b i,j, b i,j B ij c c a c u e e i,j E f f i,j f cap F ij F x,y,z F LB ij h h c h w,d H H anisotropy factor anisotropy factor of normal contact forces anisotropy factor of liquid bridges anisotropy factor of capillary stress body forces, resultant body forces capillary stress due to surface tensions intrinsic cohesion apparent cohesion coefficient of uniformity void ratio unit vector of surface tension forces Young s modulus of elasticity mean value of contact forces inter-particle contact forces inter-particle capillary forces fabric tensor of contacts eigenvalues of fabric tensor fabric tensor of liquid bridges height parameter of pyramidal packing critical height of water in capillary tube capillary rise height during wetting,drying separation distance,surface roughness dimensionless separation distance xi

12 k l,l l i,j L,L m M n n i,j n c n l N c N p N LB p,p p(n) q,q r R R 1,2 S r t i,j T s T i,j u a u i,j u w curve fitting parameter dimensionless parameters of BCC unit cell branch vectors dimensions of BCC unit cell order of approximation rotation tensor porosity unit normal vector number of contacts for each particle number of liquid bridges for each particle number of contacts in REV number of particles in REV number of liquid bridges in REV mean stress,effective mean stress contact normal s probability density function deviatoric stress,effective deviatoric stress radius of capillary tube,radius of inter-particle contact radius of particle radius of curvature degree of saturation traction forces surface tension parameter surface tension forces air pressure displacement vector water pressure xii

13 u atm V V LB V v V s,w,a V rel V cone V p w x i,j x c i,j y LB (x) y p (x) z atmospheric pressure volume of REV volume of liquid bridge volume of voids volume of solid,water,air dimensionless liquid bridge volume conical volume volume of particle water content position vector centroid position vector liquid bridge profile in Cartesian coordinate system particle profile in Cartesian coordinate system coordination number Roman letter symbols α α ij β β n γ w Γ Γ p Γ p w Γ p d half filling angle, Biot s effective stress coefficient tensorial effective stress coefficient arbitrary direction in 3D space orientation of major principle direction in 3D space unit weight of water boundary surface of REV boundary surface of the particle wetted surface of the particle total surface of finite contacts on the particle xiii

14 Γ p c Γ m δ ij ǫ ij θ θ r θ s θ w λ surface of each finite contact on the particle contractile skin contour Kronecker delta strain tensor wetting angle residual volumetric water content saturated volumetric water content volumetric water content number of liquid bridges over the number of contacts µ conical volume over the total volume ν σ ij,σ ij,σ ij σ n,σ n σ s τ f ϕ ϕ c χ χ ij Poisson ration stress, average stress and effective stress tensors normal stress, normal effective stress suction stress shear strength friction angle inter-particle friction angle Bishop s effective stress parameter tensorial effective stress parameter due to suction ψ, ψ matric suction,dimensionless matric suction ψ e ψ ij ψ x,y,z air-entry suction capillary stress tensor principle values of capillary stress tensor xiv

15 Abbreviations 3D BCC CCP DEM FCP REV SAGD SCP SSCC SWCC three dimensional body centered cubic cubic close packing discrete element modeling face centered packing representative elementary volume steam assisted gravity drainage simple cubic packing suction stress characteristics curve soil water characteristic curve xv

16 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

17 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 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 constituents (phases) just like in the case of a solid body consisting of interacting point masses 2

18 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 unsaturated 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 contractile 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 Organization of Thesis This thesis is organized into six chapters as outlined below: 3

19 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 effective 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

20 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

21 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 (T s ) 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

22 u a u w 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 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 (h c ) can be defined as below (Batchelor, 1967): γ w πr 2 h c = 2πrT s cosθ h c = 2T scosθ γ w r (2.1) where r is the radius of the tube, T s is the surface tension of the water (force per unit length), θ is the wetting angle between solid and liquid, and γ w 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

23 θ θ T s h c 2r Figure 2.2: Water in capillary tube and can be calculated as: ψ = u a u w = h c γ w (2.2) inwhichu a andu w representthehydrostaticpressureintheairandwaterphasesrespectively. 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: which leads to: 2πrT s cosθ (u a u w )πr 2 = 0 (2.3) (u a u w ) = 2T s (r/cosθ) = 2T s R (2.4) in which R is the radius of curvature of the air-water interface surface. As it is defined in Eq. (2.4), the mean curvature of the air-water interface is a function of the pressure deficiency between liquid and gas (Lu and Likos, 2004). 8

24 r Ts θ R Ts u a Z X u w 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 (R 1 and R 2 ) 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: u a u w = T s ( 1 R R 2 ) (2.5) Equation(2.5) is conventionally referred to as the Young-Laplace equation(young, 1805), which presents a nonlinear, partial differential equation relating the pressure difference of liquid and gas to the geometry of the interface surface. In porous materials such as soil, the pore spaces can be seen as capillary tubes. Therefore, the capillary effect would hold water above the water table at negative hydrostatic pressure in comparison with the air pressure. The height of the capillary rise is a function of pore size and its distribution; the smaller the size of the pores in the soil, the greater the capillary rise 9

25 R 1 R R 2 R 1 R 2 Z (a) (b) (c) X 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 (S r ) 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

26 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 Water Table c- Capillary State z 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

27 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

28 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 (θ w ) as the ratio of the volume of water over the total volume of the soil sample, or the degree of saturation (S r ) 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): θ w = S r 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 The amount of zero suction coincides with the completely saturated state (S r = 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 (ψ e ), 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 wellknown 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

29 silt sand ψ soil suction air entry ψ e volumetric water content θ w Figure 2.6: Conventional soil water characteristic curve for sand and silt(lu and Likos, 2004) decade various soil-water hysteresis models have been proposed (Lu and Likos, 2004; Huang et al., 2005; and Min and Huy, 2010). Theoretically, several mechanisms can lead to hysteretic behaviour from the micro-scale or macro-scale point of view. These mechanisms have been classified by Lu and Likos (2004) as follows: Ink-bottle effect: It describes the influence of non-uniformities in the distribution of pore size and shape. A hypothetical non-uniform pore space described by two different radii is considered in Fig For a specific matric suction controlled by a smaller radius, u a u w = 2T s /r, the maximum height of the capillary rise can be different during the drying and wetting processes. As shown in Fig. 2.7a, the capillary height during the drying process (h d ) may extend beyond the larger part of the pore space (with radius R) while the 14

30 sample is initially saturated. However, during the wetting process, the capillary rise (h w ) will cease before reaching the larger part of the pore space (Fig. 2.7b). Therefore, the amount of retained liquid in identical pores under the same matric suction is commonly larger during the drying process in comparison with the wetting process. R R r h d r h w ( 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

31 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 θ r, the residual water content where a large amount of suction is needed to remove more water from the soil, and θ s, the volumetric water content at the saturated state. I II III ψ I pendular regime II funicular regime III capillary regime soil suction drying wetting θ r volumetric water content θ w θ s Figure 2.8: Theoretical presentation of soil-water characteristic curve of an unsaturated sample in different regimes (Lu et al., 2007) 16

32 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 residualwatercontentθ r, wettingangleinducedhystereticbehaviourbecomeslessprominent 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 rewetting phase. Although the mechanisms controlling the hysteretic behaviour and the shape of the soilwater 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

33 shear strength 2.3 Experimental Observations on Unsaturated Soil Behaviours 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 (c a ), 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). 3 c a { { 2 c a { 1 c a c ( u u ) > ( u u ) a w 3 a w 2 ( u u ) > ( u u ) a w 2 a w 1 1 ( u u ) > 0 a w ( u u ) = 0 a w φ ' net normal stress Figure 2.9: General representation of shear strength in unsaturated samples (Ho and Fredlund, 1982) Ho and Fredlund (1982) conducted a series of multistage drained triaxial tests on unsat- 18

34 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 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 nonlinearity 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

35 25 shear stress (g/cm 2 ) dry 1% n-hexadecane 1.3% water 4% water normal stress (g/cm 2 ) Figure 2.10: Yield locus of glass beads R=46 micron (Pierrat et al., 1998) shear stress (g/cm 2 ) dry 1% n-hexadecane 1.3% water 4% water normal stress (g/cm 2 ) Figure 2.11: Yield locus of glass beads R=90 micron (Pierrat et al., 1998) 20

36 25 shear strength (kpa) graded frankston sand brown sand matric suction (kpa) (a) (kpa) shear strength fine frankston sand medium frankston sand matric suction (kpa) (b) Figure 2.12: Direct shear test results on cohesionless sands (Donald I., 1956) 21

37 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 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

38 tensile strength (Pa) measured data,loose measured data, medium measured data, dense water content % Figure 2.13: Tensile strength versus water content (F-75-C),(Kim, 2001) tensile strength (Pa) degree of saturation (%) water content % Figure 2.14: Tensile strength versus water content,(kim, 2001) 23

39 2.3.2 Collapse behaviour The collapse behaviour of unsaturated soils conventionally refers to a significantly rapid decrease 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. Secondly, 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 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

40 void ratio e compression line of saturated samples compression line of air-dried samples soaked at constant void ratio soaked at constant applied pressure 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

41 tutive models taking the partial saturation condition into account are required to precisely deal with a number of engineering problems such as slope instability Phenomenological studies (Macroscale studies) Phenomenological approaches are conventionally used in research on the constitutive behaviour 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 a ), and the matric suction (ψ), as the first and second stress variables respectively 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 unsaturated 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 w +(1 χ)u a ] (2.7) where u w and u a 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

42 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 (S r ). The effective stress parameter (χ) is considered to vary gradually from 0 for S r = 0% to 1 for S r = 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 (S r ) 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 S r 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 = S r in modelling unsaturated soils. Thereafter, Vanapalli et al. (1996) proposed the following equation: 27

43 1 0.9 coefficent of effective stress c degree of saturation S r % (a) breahead silt (Bishop and Donald, 1961) silt (Jennings and Burland, 1962) silty clay (Jennings and Burland, 1962) compacted boulder clay (Bishop et al., 1960) silt, drained test (Donald, 1961) coefficient of effective stress c silt, constant water content test (Donald, 1961) Madrid gray clay (Escario and Juca, 1989) Madrid silty clay (Escario and Juca,1989) Madrid clay sand (Escario and Juca, 1989) moraine (Blight, 1961) degree of saturation S r % (b) boulder clay (Blight, 1961) clay -shale (Blight, 1961) Figure 2.16: Effective stress coefficient for unsaturated soil based on experimental results 28

44 χ = ( Vw V v ) k = S r k (2.8) where V w is the volume of water, V v 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 (ψ e ), in order to obtain a unique relationship between χ and the degree of saturation, S r. ( ψ ψe) χ = ifψ > ψ e ifψ ψ e (2.9) Despite all these efforts, the validity of the single effective stress approach has been questioned by several researchers. For example, Jennings and Burland (1962) and Matyas and Radhakrishna (1968) examined the application of the single equation for effective stress in predicting the volume changes and swelling behaviour during collapse in partially 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

45 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. Furthermore, 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 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 a ) and (u a u w ), (2) (σ u w ) and (u a u w ), and (3) (σ u a ) and (σ u 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, u a and u w 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

46 tests on samples of kaolin. Usually, the most commonly used variables, which are practically easier to control, are the net stress (σ u a ) and the matric suction (u a u w ). Considering the air pressure, as a datum to define other pressures, (u a = u atm = 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 ( kp a), 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 defines the application of the axis translation method in determination of the matric suction. 200 pore water pressure u w (kpa) sandy clay weathered state loess air pressure u a (kpa) Figure 2.17: Axis translation method in measuring matric suction in laboratory (Hilf, 1956) 31

47 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 hardeningplasticityusingtwoindependentsetsofstressvariables, i.e. (σ u a ) and(u a u 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

48 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 space - Measurement of the net stress is straightforward - Transition between saturated and unsaturated is not clearly defined - The effect of hydraulic hysteresis is not well-defined - Yield surface is defined in the single effective stress space - Measurement of effective stress requires more effort - Transition between saturated and unsaturated is simply defined - the effect of hydraulic hysteresis can be captured 33

49 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 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

50 granular material microstructure. Cobbold and Gapais (1979) and Kanatani (1984) were among the first to suggest the application 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 distribution or some degree of directional preference. Generally, for an approximation of order m, the fabric tensor of the m th rank is introduced as: F ij..m = 1 2N c n k 2N c i n k k j...n m k=1 (2.10) wheren c isthenumberofcontactpoints,andn k representsthenormalunitvectorassociated with the k th 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 monosized 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

51 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 interparticle 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 In this Figure, the dimensionless liquid bridge volume and the dimensionless suction between two particles can be defined as: V w = V LB R 3 (2.11) ψ = (u a u w ) T s R (2.12) where V LB 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 (f i ) and displacements of particles with respect to each other at contacts (u i ). 36

52 θ = 0 Vw dimensionless liquid volume θ = 20 θ = 30 θ = 40 analytical torodial 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

53 σ ij Stress Tensor Macroscopic level Global constitutive equations ε ij Strain Tensor f i Inter-Particle Forces Microscopic level Local properties u i 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 sin ϕ induced by the resultant of capillary forces in pyramidal packings with different inter-particle friction angles (ϕ c ) 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

54 0.8 mobilized friction sinϕ h =1 ϕ c = 30 ϕ c = 20 ϕ c ϕ c = 10 ϕ c = 0 = 10 ϕ c = 20 ϕ c = 30 h = 2 2h height parameter h 2 2(4 h ) 2 2(4 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

55 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. Thereafter, introducing the suction stress (σ s ) equivalent to the ratio of the apparent cohesion over the friction angle (c a /tanϕ) they related suction stress to matric suction which gives rise to the so-called SSCC, and consequently calculated the true effective stress as: σ = (σ u a )+σ s (2.13) This true effective stress, calculated from the SSCC, led to a unique Mohr-Coulomb failure envelope for different samples with different matric suctions. However, this approach is rather phenomenologically based on experimental observations since there is no explicit analytical interpretation of inter-particle forces contributing to an apparent cohesion. Over the course of the past decades, particle-based methods have received a great amount of attention in the computational modelling of unsaturated soils. Among them, the discrete element method (DEM) has been used extensively in modelling the behaviour of cohesionless soils. This method was firstly proposed by Cundall and Strack (1979) in order to predict the complex macro-scale behaviour of dry granular material using rather simple assumptions and few number of parameters at the micro-scale. Recently, adding the resultant capillary forces to the contact forces between particles, due to all bulk and boundary forces acting on the system, several researchers such as Richefeu et al. (2007), Scholtes et al. (2009) and Radjai and Richefeu (2009) studied the shear strength and deformation behaviour of unsaturated granular materials in the pendular regime using DEM. From the micromechanical point of view, both the coordination number and anisotropy of the microstructure in a granular material can affect its overall shear strength and mechanical behaviour. In order to define the relation between the shear strength and anisotropies of force and micro-fabric in a cohesionless granular material, Radjai (2008) used a probability density function with a harmonic representation, P β (β), so as to define the mean contact 40

56 force pointing in an arbitrary direction as a function of that direction, i.e. P β (β) = 1 { [ 1+a 3cos 2 (β β n ) 1 ]} (2.14) 4π where P β (β) represents the probability density function of contact normals, a is a parameter which defines fabric anisotropy, while β and β n refer to an arbitrary direction in space and the orientation of the major principal direction respectively. Subsequently, introducing an average coordination number (particle connectivity z) and a contact anisotropy, Radjai (2008) proposed a state function, as shown in Eq. (2.15), to model a domain of accessible geometrical states for granular materials based on a harmonic representation, i.e. E(β) = z.p β (β) = z { [ 1+a 3cos 2 (β β n ) 1 ]} (2.15) 4π Considering that the geometrical states should stay between two limit isotropic states, E min = z min /4π ande max = z max /4π,Radjai(2008)definedamaximumaccessibleanisotropy, a, rooted in the value of z, and concluded that the maximum anisotropy and coordination number cannot be attained simultaneously. Fig represents the main results of this micromechanical interpretation. a max z C z min z z max Figure 2.21: Domain of accessible geometrical states based on harmonic representation of granular media (Radjai, 2008) 41

57 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. Microscale 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

58 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

59 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 (b i ) and external traction forces (t i (x)) acting on its boundary (Γ). Therefore, a stress tensor (σ ij ) can be assigned to each point of the medium while it should be consistent with the boundary condition of σ ij n j = t i where n j 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 Theaverageexternalstressthatarisefromsuchaproblemcanbecomputedasthevolume average of all internal stresses acting at every single material point inside the continuous medium of volume V, i.e. σ ij = 1 V V σ ij dv (3.1) The static equilibrium of forces at any material point can be written as: σ ij x j +b i = 0 (3.2) 44

60 where x i and b i indicate the position vector and body forces (force/volume) at each point of the domain respectively, see Fig Subsequently, applying the Gauss-Ostrogradski theorem along with the equilibrium condition, Eq. (3.1) can be expressed as an integration over the closed boundary surface of the domain: σ ij = 1 σ ij dv = 1 i t j dγ+ V V V Γx 1 x i b j dv (3.3) V V in which, x i is the individual position vector of the tractions and body forces. R i t j x i 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 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

61 their interactions, see Bagi (1996). Thus, the average stress in a dry granular medium in a REV of volume V can be written as: σ ij = 1 V N p V p σ ij dv p = 1 V ( ) x i t j dγ p + x i b j dv p Γ p V p N p (3.4) where N p represents the number of particles; V p and Γ p indicate the volume and boundary surface of each particle. The position vector (x i ) can be further expressed as x i = x c i +R i, where R i indicates the location vector of the traction forces on the particle with respect to its centroid. Thus, Eq. (3.4) becomes: σ ij = 1 R i t j dγ p + 1 ( ) x c V Γ p V Γ p i t j dγ p + x i b j dv p V p N p N p (3.5) Introducing the resultant body force acting at the centroid x c i of the particle as b j, i.e. bj x c i = x i b j dv p (3.6) V p Eq. (3.5) becomes: σ ij = 1 V N p Γ p R i t j dγ p + 1 V ( ) x c i t j dγ p + b j N p Γ p (3.7) thus, Since each particle is locally in equilibrium, then the last term in Eq. (3.7) vanishes and σ ij = 1 V N p Γ p R i t j dγ p (3.8) Moreover, the interaction between each pair of particles (α, β) can be described by the traction forces (f αβ j and f βα j ) seen as mutual action and reaction so that f αβ j = f βα j. Referring to Fig. 3.3 and introducing the branch vector linking the centroids of the same two particles (l αβ i = R αβ i R βα i ), Eq. (3.8) reduces to (Love, 1927): σ ij = 1 V N c l i f j (3.9) where N c is the total number of contact points in the REV. 46

62 α R αβ l R αβ Figure 3.3: Branch vector between pair of particles 3.3 Force Transport in Saturated Granular Media β 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. σ ij = 1 V V σ ij dv = 1 V ( ) σ ij dv s + σ ij dv w V s V w (3.10) where V α (α = w,s) represents the volume of water and solid phases respectively. Noting that the water pressure is u w δ ij, where δ ij is the Kronecker delta, Eq. (3.10) becomes: σ ij = 1 σ ij dv V V s + V w s V u wδ ij (3.11) Furthermore applying Gauss-Ostrogradski theorem to convert volume into surface integral just like in Eq. (3.4), Eq. (3.11) becomes: σ ij = 1 V ( x i t j dγ p + x i b j dv )+ p V w Γ p V p V u wδ ij (3.12) N p 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

63 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 α c xα f βα R α R β β f αβ f αβ 1 u w f αβ 3 x c x β 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: σ ij = 1 V l i f j + u w Γp R i n j dγ p + V w N V c N V u wδ ij (3.13) p where n j is the unit normal vector to the particle surface. Applying the Gauss-Ostrogradski theorem: R i n j dγ p R i = dv p = δ ij dv p = V p δ ij (3.14) Γ p V p x j V p 48

64 where V p indicates the volume of the particle. Thus, noting Eq. (3.14), Eq. (3.13) gives the total stress as follows: σ ij = 1 V l i f j + V s N V u wδ ij + V w V u wδ ij = 1 V c N c l i f j +u w δ ij (3.15) The first term on the right hand side of Eq. (3.15) involves the inter-particle forces, and hence refers to the effective stress σ ij acting in the solid skeleton. Thus, the total stress can be finally written as: σ ij = σ ij +u w δ ij (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 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 (t j ) on a given particle surrounded by several neighbouring particles consist of water pressures acting over the wetted surface (Γ p w), and the inter-particle forces arising from the external forces acting over finite contact areas (Γ p d ). 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: σ ij = 1 V ( N p Γ p d R i f j dγ p d + N p Γ p w R i u w n j dγ p w ) + V w V u wδ ij (3.17) The pore pressure transmission on the wetted parts (Γ p w = Γ p Γ p d) of the particle can be expressed as the action of pore pressures on the particle as if it was fully wetted minus the contributions over contacts of finite area, thus: Γ p w R i u w n j dγ p w = R i u w n j dγ p Γ p Γ p d R i u w n j dγ p d (3.18) 49

65 u w f αβ j p Γ w p Γ d f αβ j 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: σ ij = 1 V ( N p Γ p d ( R i f j dγ p d )+u w δ ij 1 V N p Γ p d ) R i n j dγ p d (3.19) 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 (σ ij), whereas the second term refers to the pore water pressure contribution which is anisotropic in general, depending on the spatial distribution of contact areas (α ij ) which also encompasses the fabric information. Thus, σ ij = σ ij +α ij u w ( ) σ ij = 1 R V i f j dγ p d ; α ij = N p Γ p d ( δ ij 1 V N p Γ p d R i n j dγ p d ) (3.20) The physical interpretation of the integral of the term (R i n j ) over the dry contact surface Γ d p is associated with the conical volume formed by the contact surfaces as shown in Fig. 50

66 3.6. This shows that α ij is a function of the fraction of the particle contact surfaces over the total particle surface areas, as well as micromechanical parameters such as the distribution of contact normals (fabric) and contact areas. These finite contact areas introduce a length scale in the definition of stress in Eq. (3.20) through the surface area Γ d p normalized to the particle radius R. Also, when the contact area tends to zero (Γ d p 0), α ij δ ij, which leads to Terzaghi s effective stress equation. When (Γ d p Γ p ) as in a continuous medium, α ij 0, giving rise to Cauchy stress. In fact, the quantity (α ij ) 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 determine this parameter. Skempton (1960) assigns a value of 1 to α for soils of negligible contact area, which refers to α ij = δ ij. As such, the second order tensor (α ij ) derived in this thesis can be seen as a generalized Skepton s or Biot s effective stress coefficient Effective stress in a fully saturated idealized compressible particle packing Consider a REV consisting of an assembly of compressible mono-size spherical particles of radius R interacting through smooth contact areas whose dimension is smaller than the particle size so that non-conformal contact condition can be assumed. Hence, the radius r of the contacting area between two spherical particles subjected to a normal contact force f (Fig. 3.7) can be obtained from Hertz s law (Hertz, 1882), i.e. r = 3 3f R 4E (3.21) where E is the Young s modulus and Poisson s ratio has been assumed to be zero for no lateral deformations. Therefore, the contact area between two spherical particles can be 51

67 defined as: ( ) 2/3 3f R Γ p c = πr 2 = π (3.22) 4E A particle within the REV (Fig. 3.6) is in contact with n c local neighboring particles so that the total contact area per particle is Γ p d = nc Γ p c. The effective stress in the REV can be determined from Eq. (3.20) where the contribution of the pore pressures is given by α ij, i.e. p Γ 1 p Γ 5 p Γ 2 p Γ 4 p Γ 3 p Γ d = nc, n c = 5 p Γ c Figure 3.6: Particle in contact with neighboring particles α ij = δ ij 1 V N p nc Bij ; B ij = R Γ p i n j dγ p c (3.23) c where B ij 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

68 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: α ij = δ ij = 2π π 0 0 b ij (β,φ)p (n) sinβdβdφ (3.24) where b ij (β) is the counterpart of B ij expressed in the global reference for a direction β in space such that: b ij (β,φ) = M ik (β,φ)b kl M jl (β,φ) (3.25) M(β,φ) = sin 2 φ+(1 sin 2 φ)cosβ sinφcosφ(1 cosβ) cosφsinβ sinφcosφ(1 cosβ) cos 2 φ+(1 cos 2 φ)cosβ sinφsinβ cosφsinβ sinφsinβ cosβ (3.26) and p (n) = 2Nc p(n) V (3.27) The contact normal probability density function is given by p(n) which defines the statistical distribution of unit contact normal vectors over the spherical domain of unit volume, and thus can be related to the fabric tensor F ij through the following: F ij = 1 2N c 2π π ni n 2N c j = (n i n j )p(n)sinβdβdφ (3.28) 0 0 with p(n) 0, V p(n) dv = 1 andp(n) = p( n) (3.29) Since p(n) is independent of φ and π periodic as a function of β, a harmonic approximation of p(n) in 3D space can be made using a Fourier series as follows (Azema et al.2009): p(n) = 1 [ ( 1+a 3cos 2 β 1 )] (3.30) 4π 53

69 z n φ β dβ dφ y Γ x 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 B ij in Eq. (3.25) with the aid of Eq. (3.23, i.e B ij = Rn Γ p i n j dγ p c = Γ p c R (3.31) c It is worth noting that the isotropic part of B ij, i.e B ii /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 n 3cos 2 (β β n ) 1 )] (3.32) where a n is the anisotropy of contact forces, β n 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

70 Figure 3.9: Schematic anisotropic force distribution in polar system, a n = 0.5 & β n = π/6 Finally, B ij can be computed by invoking Hertz s law (Eqs & 3.22) in combination with the contact force (Eq. 3.32) to get the associated contact area and thus, B ij = Γ p c R [ ( 1+a n 3cos 2 β 1 )] / (3.33) where Γ p c is the mean contact area over the REV. Finally, the second order tensor α ij required to calculate the effective stress can be computed by inserting Eqs. (3.33, 3.25) into Eq. (3.24), i.e. ( 2N c Γ p ) c R α ij = δ ij 3V λ x λ y λ z (3.34) where λ x = λ y = a 4 15 a n; λ z = a a n (3.35) Furthermore, noting that the term (RΓ p c/3) refers to the conical volume formed by the mean contact surface between the particles as shown in Fig. 3.6, 2N c (RΓ p c/3) turns out to be the total conical volumes (V t cone) formed by the contact surfaces in the REV. Therefore, 55

71 Eq. (3.34 becomes: α ij = δ ij µ λ x λ y λ z ; µ = V t cone V (3.36) In conclusion, α ij 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 = a n = 0), we get α ij = (1 µ)δ ij (3.37) In the limiting condition of rigid particles, µ 0 and α ij δ 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 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. σ ij = 1 V V σ ij dv = 1 V [ ] σ ij dv s + σ ij dv w + σ ij dv a V s V w V a (3.38) wherev α,α = s,w,arepresentsthevolumeofsolid,waterandairphasesrespectively. Dividing the solid phase into individual solid spherical incompressible particles as sub-domains, 56

72 liquid air volume, a V solid gas σ particle volume, V water volume, V a w s V = V V V 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: σ ij = 1 V [ N p Γ p x i t j dγ p + N p V p x i b j dv p ] + V w V u wδ ij + V a V u aδ ij (3.39) where u w δ ij and u a δ ij 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 interactions through tractions t j. Next, suppose the REV is composed of an ensemble of mono-disperse spherical particles of radius R joined by independent concave liquid bridges with negligible inter-particle contact area. Among the various surface tractions exerted on an individual particle, we will find contributions from inter-particle forces, actions of air and water pressures on dry (Γ p d ) and wetted (Γ p w) surfaces respectively, and surface tension arising from air/water/solid interfaces formed by water menisci along contour Γ m as illustrated in Fig a and b. It is worth 57

73 mentioning that such a decomposition of stress the various phases as in Eq. (3.39) naturally includes various types of interfaces such as air/water and air/water/solid (contractile skin), including their interactions. Furthermore, noting that x i = x c i+r i, and considering equilibrium of forces on the closed surface of each particle, (refer to Eq. (3.7)), we finally get: σ ij = 1 V l i f j + ua N c V Γ p N p d 1 V Rn i n j dγ p d + uw V N p Γ p Rn n l w in j dγ p w (3.40) N p Γ m Rn i T j dγ m + V w u n l V wδ ij + V a u V aδ ij where n i,n j are theunit vectors normal to the particle surface, f j is theinter-particle force, l i represents the so-called branch vector defining the separation distance between two particles, T j is the surface tension forces per unit length related to water menisci action on Γ m formed by the intersection of the water meniscus with the particles surface, n l is the number of liquid bridges on each particle, Γ p w is the part of the particle wetted by each liquid bridge, whereas Γ p d is the union of all dry parts of the particles surface (see Fig. 3.11). f T u a αβ u w T ( ) p 1 Γ d Γ m Γ m ua ( ) p 3 Γ d Γm ( ) p 2 Γ d Γ p w n T u w αβ f T T f u w αβ T u a Γ = ( Γ ) ( Γ ) ( Γ ) p p 1 p 2 p 3 d d d 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 f j are established based on equilibrium conditions during the interaction of the various phases with the particle skeleton, including any external loads, 58

74 the first term of Eq. (3.40) is identified as the effective stress, i.e. σ ij = 1 V N c l i f j (3.41) Looking back at the surface traction decomposition illustrated in Fig. 3.11, we observe that capillary effects induced by a concave liquid bridge between two spherical particles have two sources. The first source comes from the unequal hydrostatic pressure of air and water around the closed boundary of each particle. In other words, as shown in Fig. 3.12, due to the superposition principle, the unequal hydrostatic forces around the particle induce a component of capillary forces known as suction force (f cap 1 ). This suction force appears due to the second and third terms on the right hand side of Eq. (3.40). + = ua on Γ d p uw on Γ w p ua - uw on Γ w p Figure 3.12: Unequal hydrostatic forces around the particle Thesecondsourceofcapillaryforces(f cap 2 ) originates from the pressure difference between airandwateractingattheinterfaceofthesetwophasesontheboundaryoftheliquidbridges (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 (Γ m ), 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

75 R 1 T s u a u w R 2 Figure 3.13: Unequal hydrostatic pressure on the air/water interface and dry surfaces must be found, i.e. Γ p d Rn i n j dγ p d = V p δ ij n l Γ p w Rn i n j dγ p w (3.42) Further rearrangement of Eq (3.40) along with Eq. (3.42) leads to the tensorial form of the effective stress equation for an unsaturated granular medium: σ ij = (σ ij u a δ ij )+χ ij (u a u w )+B ij (3.43) with and B ij = 1 V χ ij = (n.s r )δ ij + 1 V N p n l N p n l Γ p w Γ m Rn i T j dγ m = RT s V Rn i n j dγ p w (3.44) N p n l Γ m n i e j dγ m (3.45) where T s is the surface tension value, n is the porosity, S r is the degree of saturation, and e j is the unit vector defining the direction of surface tension forces, whereas χ ij and B ij 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 Itisworthnotingthatχ ij ineq. (3.44)isdimensionlesstensorialquantitywhichscalesthe matric suction (u a u w ) to account for the spatial distribution (fabric) of liquid bridges and 60

76 θ T a T α n c n u a T θ T a n u w n f p Γ w p Γ d u a T b d T p ( d c) : = Γ ( a b) : = Γm w α = filling angle (here at max); θ =wetting angle u a T T b u w countour Γ m Figure 3.14: Traction forces between a pair of spherical particles associated wetted areas. Given that the fabric of the liquid bridges is generally anisotropic, thismakesχ ij anisotropic, whichleadstoananisotropiccapillarystressduetomatricsuction (χ ij (u a u w )) as well in the REV. On the other hand, B ij in Eq. (3.45) refers to a stress induced by surface tensions acting along the contractile skins at particle contours Γ m 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 ij = χ ij (u a u w )+B ij (3.46) In contrast with the current literature, capillary stress always refers to a suction stress arising from the pressure difference between air and water phases at the particle contacts. Herein, the capillary stress emerges with two distinct components in the form of suction and surface tension induced stresses based on a proper decomposition of forces at the particleparticle 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

77 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 Effective stress parameters for idealized packing The determination of effective stress parameters, χ ij and B ij, requires defining the wetted surface (Γ p w) and the contour (Γ m ) 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 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). R 1 θ R1 R α R 2 N H R 2 Figure 3.15: Concave liquid bridge geometry between a pair of uni-size particles 62

78 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 R 1 and R 2 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 (V rel = V LB /V p ), the corresponding half filling angle α can be readily calculated as shown in Fig. 3.16forillustrationpurposes. Once,αisknown,thecurvaturesR 1 andr 2 canbedetermined, and hence the matric suction corresponding to the specified liquid bridge volume can be found through Young-Laplace equation: u a u w = T s ( 1 R 1 1 R 2 ) (3.47) Vrel H/R=0, θ=40 H/R=0.1, θ=40 H/R=0.5, θ=40 H/R=0, θ=20 H/R=0.1, θ=20 H/R=0.5, θ= α Figure 3.16: Dimensionless liquid volume V rel 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 R 1 > R 2 or R 1 < R 2, which lead to the development of negative or positive matric suction (u a u w ) in the liquid bridge 63

79 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 (R 1 < R 2 ). 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 half filling angle (degree) Positive matric suction Negative matric suction 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 i = R i / R where the radius vector R i varies with the half filling angle α. Also, the unit vector e i 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 Therefore, the 64

80 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: = πr3 3 A ij = Γ p Rn w in j dγ p w (1 cosα) 2 (2+cosα) (1 cosα) 2 (2+cosα) (1 cos 3 α) (3.48) = (πr 2 T s ) B ij = Γ m RT s n i e j dγ m sin 2 αcos(α+θ) sin 2 αcos(α+θ) sin(2α)sin(α + θ) (3.49) Z Y X 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 (S r = 0) and α = 0, then χ ij = B ij = 0 so that Eq. (3.43) becomes σ ij = σ ij u a δ ij. On the other hand, in the saturated case (S r = 1), α = 180, thus A ij = V p δij, which leads to χ ij = δ ij and B ij = 0. As such Eq. (3.43) becomes σ ij = σ ij u w δ ij. NextletsassumetheREVconatinsalargenumberofparticlessothatwecanconsiderthe 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

81 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 4π [1+a(3cos2 β 1)] p(n) 0; V p(n)dv = 1;p(n) = p( n) (3.50) where a shows the anisotropy of fabric, while the fabric tensor is expressed by Eq. (3.28). The anisotropy parameter a can be defined as (Radjai, 2008): a = 5 (F z F x ) 2 tr(f ij ) = 5 2 (F z F x ) (3.51) On the other hand, the liquid bridges probability density function, which demonstrates the liquid bridges distribution in the REV, can be introduced as: p LB (n) = 1 4π [1+a LB(3cos 2 β 1)] p LB (n) 0; V plb (n)dv = 1;p LB (n) = p LB ( n) (3.52) where a LB 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: Fij LB = 1 2π π n 2N LB i n j = (n i n j )p LB (n) sinβdβdφ (3.53) 2N LB 0 0 while N LB is the total number of liquid bridges in the REV and n i is the unit normal vector associated with these liquid bridges. The anisotropy factor can also be defined as: ( ) F LB z Fx LB a LB = 5 2 tr(f LB ij ) = 5 2 ( F LB z ) Fx LB (3.54) Assuming there is no liquid bridge formed between adjacent particles with no contact, and the fabric tensor is coaxial with the liquid bridges fabric tensor, while the probability of finding a liquid bridge on each contact is considered to be directionally independent, p LB (n) 66

82 can be assumed to be equal to p(n). Therefore, Eqs. (3.44 and 3.45) can be written as follows: χ ij = V w V δ ij + B ij = 0 2π π 0 0 2π π 0 M ik A kl M jl p (n)sinβdβdφ (3.55) M ik B klm jl p (n)sinβdβdφ (3.56) where p (n) = 2NLB p LB (n) V = 2Nc p(n)λ V (3.57) with λ is the ratio of number of liquid bridges over the number of contacts in the REV. Therefore, having the degree of saturation, number and fabric of the contacts, as well as the probability of finding a liquid bridge on each contact point, χ ij and B ij 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 parameter 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 χ ij ), and the 67

83 second arising from surface tension forces along the contours between particles and water menisci (B ij ). 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

84 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 monosized 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

85 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 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: F ij = 1 2N c 2N c ni n j (4.1) where n refer to the unit normal vector defining a contact and N c is the total number of contacts. Hence, the fabric tensor associated with the simple cubic packing is simply F ij = δ ij /3. 70

86 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 Therefore, the dimensions of the cubic unit cell can be readily calculated as (Molenkamp and Nazemi, 2003): L' z (a) L L (b) x y Figure 4.2: Illustration of body-centered Cubic Packing (BCC) (L ) 2 +2L 2 = 16(R+H/2) 2,or (l ) 2 +2l 2 = 16 ( 1+H/2 ) 2 ; l = L/R, l = L /R, H = H/R (4.2) Due to geometrical compatibility, 1 l /2 < 2 and 1 l/2 < 3 2 (4.3) The coordination number associated with a BCC packing is generally 8, but this can change to 10 depending on the values of l and l. For instance, in a specific condition where l = 2, the central particle will make contact with two additional particles along the l 71

87 α H Figure 4.3: Separation distance between particles, H (Pietsch, 1968) Table 4.1: Properties of BCC packings with various l l Porosity (n) F x F y F z a 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 F x,f y,f z while eigenvectors coincide with the unit vectors of the three orthogonal axes of the cubic unit cell, i.e. F ij = F x F y F z (4.4) Hence, the porosity and fabric tensor components for a BCC packing as a function of dimensions l and l are given in Table The anisotropy factor a as defined back in Eq. (3.51) is also included in Table In anticipation to capillary stresses which will be computed in the next sections, the unit 72

88 cell for the BCC packing can be equivalently replaced with a polyhedral cell enclosing only one spherical particle as depicted in Fig 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. z y x L L L' Figure 4.4: Arrangement of BCC packing unit cells in 3D space 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

89 out to be isotropic, i.e. F ij = δ ij /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 porosity (n) BCC SCP FCP l' Figure 4.6: Porosity of different regular spherical packing 74

90 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 Figure 4.7 shows the theoretical SWCC for SCP and FCP spherical packings as a function of particle radius, i.e and 0.1 mm. Both the wetting angle and separation distance are considered to be zero and the surface tension parameter, T s 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). matric suction (kpa) SCP, R=0.1 mm SCP, R=0.001 mm FCP, R=0.1 mm FCP, R=0.001 mm degree of saturation (Sr%) Figure 4.7: SWCC of SCP and FCP as a function of particle size, H = θ = 0 75

91 Table 4.2: SWCC calculation 1. k = 1 (k is degree of saturation index) 2. Set the value of θ,r,t s,v,n,n LB,H 3. Set S 0 r = S initial r, S r = 0 4. S k r = S k 1 r + S r 5. Compute the volume of liquid bridges: V k w = VS k rn V k LB = V k w/n LB 6. Compute α k,r k 1,R k 2,N k by solving the following system of equations (Toroidal approximation): V k LB = V LB k = [(R k1 +R k2) 2 +(R k1) 2] N k (Nk ) 3 ( R k 2πR R k ) 2 [N k (R k 1) 2 (N k ) 2 +(R k 1) 2 ( ) ] N arcsin k (Nk H 2 )2 (3 N k + H) 3 2 R k 1 R k 1 = Rk 1 = (H 2 +1 cosαk ) R cos(α k +θ) R k 2 = Rk 2 R = sinαk +R k 1(sin ( α k +θ ) 1) N k = H 2 +1 cosαk,h = H/R 7. Compute corresponding matric suction (Laplace equation): (u a u w ) k = T s ( 1 R1 k ) 1 R2 k 8. Calculate dimensionless suction, ψ k = (u a u w ) k R/T s 9. If α k < α max (α max = 45 forscp,30 forfcp) S r = c (c 0 isaconstant) Set k = k +1 and go to 4 else stop 76

92 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 allowedtomergeinthependularregime. Assuch, themaximumvalueofthehalffillingangle (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 referringtoadryingandwettingpathrespectively. Thus, hystereticbehaviourinthe 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 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 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

93 100 dimensionless matric suction wetting drying drying, wetting, θ = 0 θ = 30 schematic path between wetting/drying cycle degree of saturation (%) (a) dimensionless matric suction 10 1 wetting drying drying, wetting, θ = 0 θ = 30 schematic path between wetting/drying cycle degree of saturation (%) 40 (b) Figure 4.8: Effect of wetting angle hysteresis on SWCC for (a) Loose packing (SCP), and (b) Dense packing (FCP), H = 0 78

94 100 dimensionless matric suction 10 1 H = 0 H = 0.05 H = degree of saturation (Sr%) (a) dimensionless matric suction 10 1 H = 0 H = 0.05 H = 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

95 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 (χ ij and B ij ), can be calculated as shown in Table. 4.3 and 4.4 under no deformation as before Isotropic packings 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, (B ij ) 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 B ij 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 χ ij. 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 For both SCP and FCP packing, alargerseparationdistanceh resultsinsmallervaluesofχ ij forthesamesaturation 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 ij (u a u w ), due to the same amount of matric suction. 80

96 Table 4.3: χ ij calculation 1. k = 1 (k is degree of saturation index) 2. Set the value of θ,r,t s,v,n,n LB,H,a LB 3. Set S 0 r = S initial r, S r = 0 4. S k r = S k 1 r + S r 5. Compute the volume of liquid bridges: V k w = VS k rn V k LB = V k w/n LB 6. Compute α k,r k 1,R k 2,N k by solving the system of equations, presented in Table Compute corresponding matric suction (Laplace s equation) (u a u w ) k = T s ( 1 R1 k ) 1 R2 k 8. A k ij = πr3 3 ( ) 2 1 cosα k (2+cosα k ) ( ) 2 1 cosα k (2+cosα k ) (1 cos 3 α k ) 9. For variables β,φ put p LB (n) = 1 4π {1+a LB[3cos 2 (β) 1]} p (n) = 2N LB p LB (n)/v M(β,φ) as in Eq Compute χ k ij χ k ij = V w k V δ ij + 2π π 0 0 M ila k lmm jm p (n)sinβdβdφ 11. If α k < α max (α max = 45 forscp,30 forfcp) else stop S r = c (c 0 isaconstant) Set k = k +1 and go to 4 81

97 Table 4.4: B ij calculation 1. k = 1 (k is degree of saturation index) 2. Set the value of θ,r,t s,v,n,n LB,H,a LB 3. Set S 0 r = S initial r, S r = 0 4. S k r = S k 1 r + S r 5. Compute the volume of liquid bridges: V k w = VS k rn V k LB = V k w/n LB 6. Compute α k,r k 1,R k 2,N k by solving the system of equations, presented in Table Compute corresponding matric suction (Laplace s equation) (u a u w ) k = T s ( 1 R1 k ) 1 R2 k 8. B k ij = πr 2 T s sin 2 α k cos(α k +θ) sin 2 α k cos(α k +θ) sin(2α k )sin(α k +θ) 9. For variables β,φ put p LB (n) = 1 4π {1+a LB[3cos 2 (β) 1]} p (n) = 2N LB p LB (n)/v M(β,φ) as in Eq Compute B k ij B k ij = 2π 0 π 0 M ilb k lm M jm p (n)sinβdβdφ 11. If α k < α max (α max = 45 forscp,30 forfcp) else stop S r = c (c 0 isaconstant) Set k = k +1 and go to 4 82

98 ij χ H = 0 H = 0.05 H = 0.1 H = degree of saturation (%) (a) ij χ 0.3 H = degree of saturation (%) (b) H = 0.05 H = 0.1 H = 0.2 Figure 4.10: The resulting isotropic effective stress coefficient χ ij while θ = 0(a) Loose packing (SCP) and (b) Dense packing (FCP) 83

99 Numerical results are shown in Fig 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. effective stress parameter χ degree of saturation (%) Silt, Drained test (Donald, 1961) Silt, Constant water content test (Donald, 1961) Madrid gray clay (Escario and Juca, 1989) Madrid silty clay (Escario and Juca,1989) Madrid clay sand (Escario and Juca, 1989) Moraine (Blight, 1961) Boulder clay (Blight, 1961) Clay-shale (Blight, 1961) χ =Sr FCP, n = 0.26 SCP, n = 0.49 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) 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 (T s ) is considered equal to 74 µn/m. Here again, because the liquid bridges distribution is isotropic, the capillary stress tensor (ψ ij ) is isotropic with the difference that it has both suction 84

100 (χ ij (u a u w )) and surface tension (B ij ) contributions. As shown in Fig and 4.13, while the matric suction increases, the role of suction forces in generating capillary stresses gradually increases, whereas the effect of surface tension forces progressively disappears for both packings. Moreover, the wetting angle affects the induced capillary stress in opposing ways. The largest capillary stress due to suction forces is associated with the lowest wetting angle; while the larger the wetting angle, the more capillary stress is induced due to surface tension forces. It is also interesting to investigate the relative contributions of the surface tension forces (term B ij ) and the suction forces between particles (term χ ij ) to the capillary stress ψ ij. 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 kp a, but is ultimately overtaken by the suction effect at large matric suctions. For the dense case in Fig. 4.14b, the contribution of surface tension is also smaller than that of suction above matric suctions of 1 kpa. As seen in Fig. 4.14b, no data can be calculated below a suction of 1 kpa because the corresponding degree of saturation becomes large so that the assumption of pendular regime with independent liquid bridges cannot be satisfied. The high degree of saturation requires the liquid bridges to merge, which invalidates the model. Thus, the results presented in Fig 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 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

101 1.2 (kpa) χ ij ( u a u w) θ= 0 θ=10 θ=20 θ= matric suction (kpa) (a) (kpa) B ij θ= 0 θ=10 θ=20 θ= 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

102 χ ij ( u a u w ) (kpa) θ= 0 θ=10 θ=20 θ= matric suction (kpa) (a) B ij (kpa) θ= 0 θ=10 θ=20 θ= 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

103 1.2 Capillary stress (kpa) χ B ij (u u ) ij a ψ ij w matric suction (kpa) (a) Capillary stress (kpa) B ij χ (u u ) ψ ij ij a w 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

104 this section, the validity of this proposed tensile strength (ψ ij ) 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 (c u ) 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 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 (P a) w (%) Tensile Strength (P a) In order to make a comparison between theoretical and experimental data, the tensile strength (ψ ij ) for FCP and isotropic BCC packings, both consisting of spherical particles withradiusof0.11mmandwettinganglesofθ = 0 and20,iscalculated. Thedimensionless 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 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

105 1200 tensile strength (Pa) Predicted tensile strength, FCP, θ =20 Prediced tensile strength, BCP, θ =20 Predicted tensile strength, FCP, θ =0 Predicted tensile strength, BCC, θ =0 Experimental data, F-75 (e=0.72) Experimental data,f-75 (e=0.58) volumetric water content, (w %) Figure 4.15: Compression between measured and predicted tensile strength Anisotropic packings We next turn to anisotropic packings to demonstrate the anisotropic nature of the capillary stresses due to suction (χ ij (u a u w )) and surface tension (B ij ) forces as two controlling components of the total capillary stress ψ ij 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 ψ ij including its individual components χ ij (u a u w ) and B ij as function of degree of saturation. It is recalled herein that the capillary stress has two contributions: one arising from suction (χ ij (u a u w )) and another one from surface tension (B ij ). The anisotropic nature of these stresses is clearly demonstrated with the major and minor principal directions of the capillary stress being 90

106 aligned with the vertical direction z (axial) and x=y (lateral) respectively χij ( u a uw ) (kpa) 90 B ij (kpa) Sr=14% Sr=9.8% Sr=5.6% Sr=0.5% Sr=0.02% 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 as a function of matric suction. Since the wetting angle is zero, the amount of capillary stress induced by surface tension forces (B ij ) is smaller in comparison with the capillary stress due to suction forces (χ ij (u a u 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 As shown, while the wetting angle is zero the capillary stress due to surface tension forces (B ij ) is minimized. However, increasing the wetting angle enlarges this capillary stress, and thus significantly increases the 91

107 capillary stress (kpa) (axial) matric suction (kpa) B ij χ ( u - u )(axial) ψ ij ij a w (axial) 2 capillary stress (kpa) B ij (lateral) χ (u - u )(lateral) ψ ij ij a w (lateral) matric suction (kpa) Figure 4.17: Principal capillary stresses with various contributions in axial and lateral directions, H = θ = 0 92

108 contribution of surface tension forces to the total capillary stress generated in the packing (ψ ij ) χij ( u a uw) (kpa) B ij (kpa) θ = 0 θ = ψ ij 270 (kpa) Figure 4.18: Polar plot of anisotropic capillary stresses for various wetting angles, H = Evolution of degree of anisotropy - link to strength issues There is a compelling connection between the meniscus-based anisotropic capillary stress, ψ ij 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 a ψ similar to what was defined for contact fabric, i.e. 93

109 a ψ = 5 (ψ z ψ x ) 2trace(ψ ij ) = 5 2 (ψ z ψ x ) (ψ x +ψ y +ψ z ) (4.5) where ψ x, ψ y, and ψ z are principal values of capillary stress tensor ψ ij. 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 In this connection, the anisotropy factor a ψ 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 (a ψ ) 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 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

110 0.8 anisotropy factor, a ψ l '=2.7 l '=2.6 l '=2.5 l '=2.4 l '=2.3 l '= 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

111 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 saturation is also inspected for various BCC packings by introducing a meniscus-based anisotropy factor a ψ. 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

112 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: τ f = σ ntanϕ+c (5.1) where τ f and σ n represent the shear strength and the effective normal stress of the sample at failure, and ϕ and c illustrate the material friction angle and cohesion, respectively. As discussed in the literature review, the capillary forces in unsaturated granular media restrict inter-particle slippage and, consequently, increase the shear strength. As such, the failure envelope of unsaturated granular media is typically determined as a function of the total net stress and a suction-related parameter, called apparent cohesion (c a ), while the friction angle is usually considered to be independent of the amount of water saturation, (see Fig. 2.9). τ f = (σ n u a )tanϕ+c a (5.2) From a micro-mechanical point of view, this apparent cohesion characterizes the dependency 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

113 liquid bridges and their spatial distribution, among others. On the other hand, it should be noted that using an appropriate equation to define the controlling stress variable, which would play the role of the effective stress in fully saturated media, will lead to a unique failure envelope whether the soil sample is saturated, dry or unsaturated. Therefore, if the effective stress σ n is well-defined, in order to account for the effect of inter-particle capillary forces transmitted in unsaturated media, one can still use Eq. 5.1 to define the shear strength and failure envelope. Following the above arguments, the validity of the proposed effective stress equation in this study can be therefore examined. Thus, Discrete Element Method (DEM) calculations are pursued on granular assemblies (REV) for which the particle size and distribution, 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 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

114 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

115 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 More details about this suction controlled DEM simulation with YADE software can be found in Scholtes et al. (2009) and YADEs website. 100

116 Table 5.1: Simplified steps of DEM modeling of unsaturated granular media 1. The value of matric suction (u a u w ) is set. 2. External load increment is then applied. 3. Between every pair of particles,α & β (with or without physical contact): - The separation distance h αβ 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 a u w ),h αβ ) is considered; Is there a solution for Young-Laplace equation? No inter-particle capillary force & liquid bridge volume=0. Yes inter-particle capillary force= f αβ cap & liquid bridge volume=v αβ LB. - Total inter-particle force is calculated as f αβ int=f αβ con+f αβ cap. 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

117 5.2.2 DEM sample description The DEM sample is composed of 10,000 mono-dispersed, completely smooth, spherical particles with a radius of mm as shown in Fig 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 mm 3 )isassumedtobelargeenough, incomparisonwiththesizeoftheparticles(0.024mm), so that the distribution of normal contacts and liquid bridges can be considered continuous variables. Figure 5.1: DEM sample consisting of 10,000 mono-sized spherical particles DEM samples generation is made using the same classical DEM algorithm, as described by Scholtes et al. (2009). First, a cloud of random spherical particles with no overlapping, 102

118 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 Table 5.2: DEM sample input parameters Inter-particle friction angle 18 Number of particles 10,000 Initial volume 1 mm 3 Radius of particles mm Initial fabric δ ij /3 Wetting angle 0 Initial porosity (n) 0.4 Surface tension parameter N/m Normal stiffness (K n ) 10 6 Pa Tangential stiffness (K t ) 0.3 K n DEM triaxial test procedure and results A series of triaxial tests with various matric suctions of 15, 30, 300 kp a and confining pressures of 250, 500, 750 and 1000 P a are simulated using the open-source DEM code YADE. Such low confining pressures are chosen in order to highlight the effects of capillary 103

119 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 matric suction (kpa) SCP, R=0.1 mm SCP, R=0.001 mm FCP, R=0.1 mm FCP, R=0.001 mm DEM sample R=0.024 mm 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 Moreover, the deviatoric stress and volumetric strain responses of samples with various matric suctions are 104

120 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 kp a) Lateral pressure (P a) Normal stress at failure (P a) Matric suction=15 kp a Lateral pressure (P a) Normal stress at failure (P a) Matric suction=30 kp a Lateral pressure (P a) Normal stress at failure (P a) Matric suction=300 kp a Lateral pressure (P a) Normal stress at failure (P a) The Mohr-Coulomb failure envelope for each unsaturated sample with specific matric suction is then obtained in the mean stress (p = (σ z + 2σ r )/3) and deviatoric stress (q = σ z σ r ) space, by drawing the best line passing through the peak shear strengths of the sample under various confining pressures, as shown in Figs 5.4. The failure envelope for the dry case is also plotted as baseline and a friction angle of 25 is obtained. As expected, when using the total stresses p and q, different failure envelopes showing an apparent gain in shear strength are obtained with increasing matric suction. As shown in Fig. 5.3 and 5.4 there is a significant jump between the shear strength envelopes of the dry and unsaturated samples in DEM simulations. Richefeu et al. (2007) has also pointed to the same significant jump in shear strength while conducting direct shear tests on assemblies of mono-dispersed smooth glass beads in pendular regime with a low confinement pressure. The main reason for this behaviour is that large matric suctions are immediately produced for a very small amount of water, as show in Fig. 5.2, given the size of the particles and the fact that liquid bridges are considered to exist only between particles in physical contact (H = 0). Therefore, a large capillary stress (in comparison with low confinement pressures in these simulations) is induced in unsaturated samples with very small degrees of saturation, which leads to a significant jump in the shear strengths 105

121 q/p axial strain (a) dry sample suction 15 kpa suction 30 kpa suction 300 kpa volumetric strain dry sample suction 15 kpa suction 30 kpa suction 300 kpa 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

122 q (Pa) dry sample 2000 suction 15 kpa 1000 suction 30 kpa suction 300 kpa 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

123 kpa Suction 300 kpa, Sr=0.05% Suction 30 kpa, Sr=2.2% Suction 15 kpa, Sr=5.5% 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 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

124 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 = S r 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. q-q' (Pa) p-p' (Pa) dry sample sucion 15 kpa suction 30 kpa suction 300 kpa suction 15 kpa, Bishop suction 30 kpa, Bishop suction 300 kpa, Bishop Figure 5.6: Strength of wet granular material based on (a) net stress (q,p) and (b) 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

125 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 q'/p' dry sample total stress - suction 15 kpa total stress - suction 30 kpa total stress - suction 300 kpa calculated effective stress - suction 15 kpa calculated effective stress - suction 30 kpa calculated effective stress - suction 300 kpa 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

126 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. anisotropy factor capillary stress 0.4 effective stress axial strain Figure 5.8: Anisotropy changes for both effective and capillary stresses for a confining pressure 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 monosized 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

127 is presented in Table 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 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. Moreover, 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

128 SCP DEM sample Sr % matric suction (Pa) Figure 5.9: Comparisons between SWCC of selected SCP sample and simulated DEM samples by Shamy and Groger, 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

129 shear stress (pa) water content=0% water content=0.2% 100 water content=2.0% water content=5.2% normal net stress (Pa) (a) shear stress (Pa) water content=0% water content=0.2% water content=2% water content=5% 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

130 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

131 Chapter 6 CONCLUSIONS AND RECOMMENDATIONS 6.1 Conclusions In emerging geotechnical problems under unsaturated conditions, there is still much debate on the definition of a proper effective stress and issues surrounding the validity of Bishop s effective stress. This thesis examined the force transport in a granular system wetted with discrete liquid bridges with reference to unsaturated granular materials in the pendular regime. A micromechanical approach is hereby used to formulate effective stress in such regime as a function of both liquid bridge and particle contact spatial distributions with special emphasis on the interactions of the air-water-solid phases, including the interaction of interfaces. Main findings and conclusions are summarized as follows. A tensorial equation defining the true effective stress in unsaturated soils in the pendular regime is proposed: σ ij = (σ ij u a δ ij )+χ ij (u a u w )+B ij (6.1) As such, the effective stress parameter (χ) as initially introduced in Bishop s equation is shown to be a tensorial function of degree of saturation, particle packing and liquid bridges distribution. In the proposed equation, χ ij 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(b ij ),whichreferstoastressinducedbysurfacetensionsactingalongtheso-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

132 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 ij = χ ij (u a u w )+B ij (6.2) This capillary stress is shown to have two components: one originating from suction between particles induced by air-water pressure difference (related to χ ij ), which leads to an anisotropic capillary stress due to matric suction (χ ij (u a u w )), and the second arising from surface tension forces along the contours between particles and water menisci (B ij ). 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 studying 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 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. 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

133 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. Considering 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 plausible 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 ij Stress Tensor Macroscopic level ε ij Strain Tensor f i Inter-particle contact forces cap f i Inter-particle capillary forces u i Displacements due to contact forces cap u i Displacements due to 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

134 Figure 6.2: Micro-CT scan of water menisci of Toyoura sand, courtesy of Profs. Oka and Kimoto, Kyoto University, Japan 119