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Stress-strain relations for sand based on particulate considerations Atukorala, Upul Dhananath 1989

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S T R E S S - S T R A I N R E L A T I O N S F O R S A N D B A S E D O N P A R T I C U L A T E C O N S I D E R A T I O N S by U P U L D H A N A N A T H A T U K O R A L A B.Sc. (Engineering), University of Peradeniya, Sri Lanka, 1980 M.A.Sc. University of Brit ish Columbia, Vancouver, Canada, 1983 A THES I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of C i v i l Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A December, 1989 © U p u l Dhananath Atukorala, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C\\r\\ E i A g i n j e x r ' i ^ The University of British Columbia Vancouver, Canada Date l > g ( g > Y i b ^ y 33 , 1959-DE-6 (2/88) With chiselled touch, The stone unhewn and cold Becomes a living mould. The more the marble wastes, The more the statue grows. — Michaelangelo -ii Dedicated to my loving parents. Hi A B S T R A C T Particulate, discrete and frictional systems such as sand constitute a separate class of materials. In order to derive stress-strain relations for these materials, their key features have to be identified and incorporated into the theoretical formulations. The presence of voids, the ability to undergo continuous and systematic spatial rearrangement of particles, the existence of bounds for the developed ratio of tangent and normal contact forces and the systematic variations of the tangent and normal contact force distributions during general loading, are identified as key features of particulate, discrete and frictional systems. The contact normal and the contact branch length distribution functions describe the spatial arrangement of particles mathematically. The distribution of contact normals exhibit mutual ly orthogonal principal directions which coincide with the principal stress directions. Most contacts in frictional systems do not develop l imiting friction during general loading. Sl iding of a few suitably oriented contacts followed by rolling and rigid body rotations and displacements of a large number of particles is the main mechanism causing non-recoverable deformations in frictional systems. As a part of the rearranging process, dominant chains of particles are continuously constructed and destructed, the rates being different at different stages of loading. A change of loading direction is associated with a change of dominant chains of particles resulting in changes in strain magnitudes. Rate insensitive incremental stress-strain relations are derived here using the principle of v i r tual forces. The key features of frictional systems have been incorporated into the stress-strain relations following the theoretical framework proposed by Rothenburg(1980), for analysing bonded systems of uniform spherical particles. For frictional systems, the load-deformation response at particle contacts is assumed to be non-linear. The deforma-tions resulting from all internal activity are quantified denning equivalent incrementally elastic stiffnesses in the tangent and normal directions at contacts and denning loading and unloading criteria. After each increment of loading, the incremental stiffnesses and contact normal distribution are updated to account for the changes resulting from rearrangement of particles. Laws that describe the spatial rearrangement of particles, changes in the ratio between the tangent and normal contact force distributions and the resistance to deforma-tion resulting from changes in dominant chains of particles are established based on the information from laboratory experiments reported in the literature and numerical exper-iments of Bathurst(1985). The stress ratio and the state parameter (defined as the ratio of void ratios at the critical-state to the current state, computed for a given mean-normal stress) are identified as key variables that can be used to quantify the extent of particle rearrangements. The proposed formulations are capable of modelling the non-linear stress-strain response which is dependent on the inherent anisotropy, stress induced anisotropy, density of packing, stress level and stress path. To predict the stress-strain response of sand, a total of 24 model parameters have to be evaluated. A l l the model parameters can be evaluated from five conventional tr iax ia l compression tests. The proposed stress-strain relations have been verified by comparing with laboratory measurements on sand. The data base consists of triaxial tests reported by Negussey(1984), hollow cylinder tests graciously carried out for the author by A. Sayao, and true tr iaxial and hollow cylinder tests made available for the Cleveland Workshop(1987). tv T A B L E O F C O N T E N T S Page No D E D I C A T I O N i i i A B S T R A C T iv T A B L E O F C O N T E N T S v L I ST O F F I G U R E S ix L I ST O F T A B L E S xiv N O M E N C L A T U R E xv A C K N O W L E D G E M E N T S xv i C H A P T E R 1. I N T R O D U C T I O N 1.1 Introduction 1 1.2 Research Objectives 6 1.3 Organization of the Thesis 7 C H A P T E R 2. S T R E S S - S T R A I N R E L A T I O N S F O R S A N D - A L I T E R A T U R E R E V I E W 2.1 Introduction 8 2.2 Observed Behaviour of Sand 9 2.2.1 Density and Stress Level Dependency 11 2.2.2 Existence of a Reference State for Sand 17 2.2.3 Anisotropic Behaviour of Sand 18 2.2.4 Path Dependent Behaviour of Sand 24 2.2.5 The Unload-Reload Response of Sand 27 2.2.6 Discussion 30 2.3 Existing Stress-Strain Models for Sand 31 2.3.1 Elastic Models 32 2.3.1.1 Hooke's Elastic Model 32 2.3.1.2 Cauchy and Green Elastic Models 33 2.3.1.3 Hypoelastic Model 34 2.3.1.4 Discussion 35 2.3.2 Plastic Models 37 2.3.2.1 Constituents of Theory of Plasticity 37 2.3.2.2 Y ie ld and Potential Functions for Sand 38 2.3.2.3 Mult ip le Y ie ld Surface Plasticity 40 2.3.2.4 Discussion 41 2.3.3 Particulate, Discrete and Frictional Models 41 2.3.3.1 Stress Dilatancy Relations 42 2.3.3.2 Microscopic Observations on the Behaviour of C i rcu lar Particles . . . . 44 2.3.3.2.1 Interparticle Contact Forces 45 v 2.3.3.2.2 Normals to Contact Planes 49 2.3.3.3 Discussion 54 2.3.4 Summary 54 C H A P T E R 3. T H E O R E T I C A L F O R M U L A T I O N S F O R B O N D E D S Y S T E M S 3.1 Introduction 55 3.2 Spatial Arrangement of a System of Particles 57 3.3 The Phenomenological Stress Tensor for a Particulate System 64 3.4 Relationship Between Contact Forces and Strains 76 3.5 Stress-Strain Relations for a Bonded System of Particles with an Isotropic Contact Normal Distr ibution 84 3.6 Discussion 86 C H A P T E R 4. T H E O R E T I C A L F O R M U L A T I O N S F O R F R I C T I O N A L S Y S T E M S 4.1 Introduction 88 4.2 Bonded Systems versus Fr ict ional Systems 89 4.3 Stress-Strain Response of Fr ict ional Systems 89 4.3.1 Introduction 91 4.3.1.1 Quantif ication of Developed Contact Forces Resulting from Boundary Loads 93 4.3.1.2 Identifying Geometrically Compatible and Stable Particle Configurations 94 4.3.1.3 Changes in Stable and Compatible Particle Arrangements and Boundary Deformations 96 4.3.2 Stage-1 Relationship Between Boundary Loads and Contact Forces . . . . 98 4.3.3 Stage-2 Stable and Geometrically Compatible Particle Arrangements . . . 98 4.3.4 Stage-3 Quantif ication of Boundary Deformations From Particle Deformations 100 4.3.5 Incremental Stress-Strain Relations 105 4.4 The Contact Normal Distr ibut ion Function S(f i) 112 4.4.1 Two-Dimensional Representation of Contact Normal Distribution . . . .112 4.4.2 Three-Dimensional Representation of Contact Normal Distribution . . . .114 4.4.3 Laws Defining Changes in Contact Normal Distribution 116 4.4.3.1 Influence of Proximity to Crit ical-State Stress Ratio on Aa,j 120 4.4.3.2 Influence of Mean Normal Stress Level and Density of Packing on Aa^ . 121 4.4.3.3 The Changes in Fourth Order Anisotropy Coefficients A6,y« 122 4.5 The Distr ibut ion of /I(fi) 124 4.5.1 Exper imental Data on Two dimensional Distribution of (j,(9) 125 vi 4.5.2 The Theoretical and Experimental Distributions of 126 4.5.3 Three Dimensional Distr ibution of //(•) 129 4.6 The Resistance Function Ht 130 4.7 Numerical Integration of Stress-Strain Equations 140 4.8 Summary 140 C H A P T E R 5. V E R I F I C A T I O N S A N D D I SCUSS IONS 5.1 Introdution 142 5.2 Input Requirements for the Proposed Stress-Strain Relations 145 5.3 The Computer Program S A N D 147 5.4 Model Predictions - I 147 5.4.1 Characteristic Response 147 5.4.1.1 Hydrostatic Loading 150 5.4.1.2 Conventional Triaxial Compression Loading for Different Confining Pressures 150 5.4.1.3 Conventional Tr iaxial Compression Loading and Subsequent Unloading for Different Confining Pressures 152 5.4.2 Sensitivity of Mode l Parameters on the Stress Strain Behaviour 158 5.5 Model Predictions - II 181 5.5.1 Input Parameters from Laboratory Data 181 5.5.2 Predicted Stress-Strain Response of Standard Ottawa Sand From the Conventional Tr iaxia l Test Set Up 182 5.5.3 Predicted Stress-Strain Response of Standard Ottawa Sand From the U B C Hollow Cyl inder Test Set U p 195 5.5.4 Predicted Stress-Strain Response of Hostun and Reid Bedford Sands From Cleveland Da ta Base 209 5.5.4.1 Introduction 209 5.5.4.2 Notation 210 5.5.4.3 Cal ibration Data 210 5.5.4.4 Predictions 226 5.5.4.4.1 Comparisons with the Hollow Cyl inder Test Results 227 5.5.4.4.2 Comparisons with the Cube Test Results 234 5.6 Summary of Predictions and Discussion of Results 240 5.6.1 Summary of Predictions 240 5.6.2 Discussion of Results 241 5.7 Comparisons wi th Other Models 243 5.8 Summary 244 C H A P T E R 6. S U M M A R Y , C O N C L U S I O N S A N D F U R T H E R R E S E A R C H . . . . 6.1 Summary and Conclusions 245 6.2 Further Research 249 vii R E F E R E N C E S 251 A P P E N D I X - A 258 A P P E N D I X - B 261 A P P E N D I X - C 265 A P P E N D I X - D 271 via L I S T O F F I G U R E S Page No Fig.(2.1) Variation of Volumetric Strains Due to Membrane Penetration with Confining Pressure - Ottawa Sand 10 Fig.(2.2) Variation of Volumetric Strains Due to Membrane Penetration with Mean Gra in Size - Japanese Sands 12 Fig.(2.3) Stress-Strain Response Observed for Dense Sacremento River Sand . . . . 14 Fig.(2.4) Stress-Strain Response Observed for Loose Sacremento River Sand . . . . 15 Fig.(2.5) The State Parameter Concept in Vo id Rat io-Mean Normal Stress Space . . 16 Fig.(2.6) Volumetric and Ax i a l Strain Measurements A long Hydrostatic Load and Unload Paths - Ottawa Sand 20 Fig.(2.7) Volumetric and Ax i a l Strain Measurements A long Load-Unload Paths with Varying Stress Rat io - Fine Sand 21 Fig.(2.8) Effect of Strain History on Induced Anisotropy Observed From Directional Shear Cel l - Leighton Buzzard Sand 23 Fig.(2.9) The Variation of Strain Increment Direction and Pr inc ipa l Stress Direction with Appl ied Shear - H a m River Sand 24 Fig.(2.10) Strain Paths for Compression Side Shear Unloading and Extension Side Shear Loading at Constant Mean Normal Stress - Ottawa Sand . . . 26 Fig.(2.11) Incremental Stress and Strain Rat io Variations for Loading Paths -Ottawa Sand 27 Fig.(2.12) The Load-Unload-Reload Response A long Conventional Tr iax ia l Compression Path - Ottawa Sand 29 Fig.(2.13) Frequency Distr ibution of Obl iquity of Contact Forces in Two Dimensional Assemblies of C ircular Discs 46 Fig.(2.14) Variations of Shear Force, Shear Strain and Volume Change Response of Assemblies of Circular Discs 47 Fig.(2.15) Observed Distributions of Normal and Tangent Contact Forces 48 Fig.(2.16) Observed Distributions of Developed Rat io Between Tangent and Normal Contact Forces a) at Peak Shear b) at Residual Shear 50 Fig.(2.17) The Distr ibution of Contact Normals Observed at Various Stages of Loading of Assemblies of C ircular Discs a) Dense Assembly b) Loose Assembly 51 ix Fig.(2.18) Dominant Chains of Particles Observed During Loading of Assemblies of Circular Discs 53 Fig.(3.1) A Three Particle System in Two-Dimensions 58 Fig.(3.2) The Frequency Distribution of Contact Normals 60 Fig.(3.3) The General Shapes of Contact Normal Distribution 63 Fig.(3.4) Definition of Angles in Three Dimensions 65 Fig.(3.5) Load Transfer at Boundary Particles 67 Fig.(3.6) Definition of Various Vector Components Associated with the Theoretical Formulations 69 Fig.(4.1) Representation of a System of Particles as a Network of Contact Branch Vectors 95 Fig.(4.2) Three Dimensional Assembly of Particles 99 Fig.(4.3) Non-linear Load-Deformation Behaviour at Contacts 109 Fig.(4.4) Idealized Particle Arrangements 110 Fig.(4.5) Influence of an on the Developed Friction 127 Fig.(4.6) Incremental Resistance to Deformation at Load-Unload Points 135 Fig.(4.7) The Crit ical-State Surface in a)Principal Stress Space b)Void Rat io - Mean Normal Stress Space 136 Fig.(4.8) The Approximated Shape of Critical-State Surface in Pr inc ipal Stress Space . . . 137 Fig.(5.1) Gra in Size Distr ibution Curves for Ottawa, Hostun and Reid Bedford Sands 144 Fig.(5.2) Flow Chart for the Computer Program S A N D 148 Fig.(5.3) Characteristic Response Along Hydrostatic Load-Unload Path a)Mean Normal Stress vs Volumetric Strain b)Volumetric Strain vs Ax i a l Strain . 151 Fig.(5.4) The Initial States of Sand for Characteristic Predictions 153 Fig.(5.5) Stress-Strain Response A long Conventional Triaxial Path a)Deviator Stress vs A x i a l Strain b)Volumetric Strain vs A x i a l Strain . . 154 Fig.(5.6) Stress-Strain Response Along Conventional Triaxial Path a)Deviator Stress vs A x i a l Strain b)Volumetric Strain vs A x i a l Strain . . 155 Fig.(5.7) Stress-Strain Response A long Conventional Triaxial Path Involving x Loading, Unloading and Reloading a) De viator-Stress vs Ax i a l Strain b)Volumetric Strain vs Ax i a l Strain 156 Fig.(5.8) Stress-Strain Response A long Conventional Tr iax ia l Path Loading, Unloading and Reloading a) De viator-Stress vs Ax i a l Strain b)Volumetric Strain vs Ax i a l Strain 157 Fig.(5.9a) Sensitivity of h^ on the Predicted Response 159 Fig.(5.9b) Sensitivity of hi on the Predicted Response 160 Fig.(5.9c) Sensitivity of h^ on the Predicted Response 161 Fig.(5.9d) Sensitivity of m\ on the Predicted Response 162 Fig.(5.9e) Sensitivity of n on the Predicted Response 163 Fig.(5.9f) Sensitivity of r) on the Predicted Response 164 Fig.(5.9g) Sensitivity of <f>cv on the Predicted Response . 165 Fig.(5.9h) Sensitivity of Xcs on the Predicted Response 166 Fig.(5.9i) Sensitivity of e®s on the Predicted Response 167 Fig.(5.9j) Sensitivity of h,Q on the Predicted Response 168 Fig.(5.9k) Sensitivity of si on the Predicted Response 169 Fig.(5.91) Sensitivity of «2 on the Predicted Response 170 Fig.(5.9m) Sensitivity of ai on the Predicted Response 171 Fig.(5.9n) Sensitivity of c*2 on the Predicted Response 172 Fig.(5.9o) Sensitivity of a$ on the Predicted Response . 173 Fig.(5.9p) Sensitivity of j3\ on the Predicted Response 174 Fig.(5.9q) Sensitivity of /?2 on the Predicted Response 175 Fig.(5.9r) Sensitivity of (i* on the Predicted Response 176 Fig.(5.10) Comparison of Initial Stress and Strain Increment Ratios 186 Fig.(5.11) Comparison of Stress-Strain Response A long Hydrostatic Loading and Unloading Paths a)Mean Normal Stress vs Volumetric Strain b)Volumetric Strain vs A x i a l Strain 188 Fig.(5.12) Comparison of Stress-Strain Response A long A<rr/Ac7a = 0 Paths a)Deviator Stress vs A x i a l Strain b)Volumetric Strain vs A x i a l Strain . . 189 xi Fig.(5.13) Comparison of Stress-Strain Response A long Aar/A<ra = - 1 /2 Paths a) Deviator Stress vs Ax i a l Strain b)Volumetric Strain vs A x i a l Strain . . 191 Fig.(5.14) Comparison of Stress-Strain Response A long Constant Mean Normal Stress Unloading from Compression Side a)Shear-Stress vs Shear-Strain b) Volumetric Strain vs Shear Strain 193 Fig.(5.15) Comparison of Stress-Strain Response Along Constant Mean Normal Stress Unloading from Extension Side a)Shear-Stress vs Shear-Strain b) Volumetric Strain vs Shear Strain 194 Fig.(5.16) Observed Stress-Strain Response for Conventional Tr iaxia l Loading, Unloading and Reloading a) Deviator Stress vs A x i a l Strain b) Volumetric vs A x i a l Strain 196 Fig.(5.17) Predicted Stress-Strain Response for Conventional Tr iaxial Loading, Unloading and Reloading a)Deviator Stress vs A x i a l Strain b)Volumetric vs A x i a l Strain 197 Fig.(5.18) Hollow Cyl inder Test Results a) Appl ied Stress Variation b)Predicted and Observed Strain Variations 203 Fig.(5.19) Comparison of Stress-Strain Response with Constant Mean Normal Stress and Rotat ing Pr incipal Stress Directions a)Appl ied Stress Variat ion b) Predicted and Observed Ax ia l Strain Variations c) Predicted and Observed Cicumferential Strain Variation d) Predicted and Observed Radial Strain Variation e) Predicted and Observed Shear Strain Variation 206 Fig.(5.20) Hydrostatic Load-Unload Data From Cube and Hollow Cyl inder Test Devices - Hostun Sand - a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vertical Strain 211 Fig.(5.21) Hydrostatic Load-Unload Data From Cube and Hollow Cyl inder Test Devices - Reid Bedford Sand - a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vertical Strain 212 Fig.(5.22) Comparison of Predicted and Observed Response for Hydrostatic Load-Unload Path - Hostun Sand - a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vertical Strain 218 Fig.(5.23) Comparison of Predicted and Observed Response for Conventional Tr iaxia l Compression Path - a c^ = 200kPa - Hostun Sand - a) Deviator Stress vs A x i a l Strain b) Volumetric Strain vs A x i a l Strain 219 Fig.(5.24) Comparison of Predicted and Observed Response for Conventional Tr iaxia l Extension Path - ozc = 200A;Pa - Hostun Sand - a) Deviator Stress vs A x i a l Strain b) Volumetric Strain vs Ax i a l Strain 220 xn Fig.(5.25) Comparison of Predicted and Observed Cycl ic Loading Response for Conventional Tr iax ia l Path - o$c = 200kPa - Hostun Sand - a) Deviator Stress vs A x i a l Strain b) Volumetric Strain vs A x i a l Strain 221 Fig.(5.26) Comparison of Predicted and Observed Response for Hydrostatic Load-Unload Path - Reid Bedford Sand - a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vert ical Strain 222 Fig.(5.27) Comparison of Predicted and Observed Response for Conventional Tr iaxial Compression Path - <x3(. = 345kPa - Reid Bedford Sand - a) Deviator Stress - A x i a l Strain b) Volumetric Strain vs A x i a l Strain 223 Fig.(5.28) Comparison of Predicted and Observed Response for Conventional Tr iaxial Extension Path - a^c = 345kP a - Re id Bedford Sand - a) Deviator Stress - A x i a l Strain b) Volumetric Strain vs A x i a l Strain 224 Fig.(5.29) Comparison of Predicted and Observed Cycl ic Loading Response for Conventional Tr iax ia l Pa th - azc — 200kPa - Re id Bedford Sand - a) Deviator Stress - A x i a l Strain b) Volumetric Strain vs A x i a l Strain 225 Fig.(5.30) Comparison of Predicted Response W i t h Hollow Cyl inder Results - HR1 .TST and HH1 .TST 229 Fig.(5.31) Comparison of Predicted Response W i t h Hollow Cyl inder Results - HR2 .TST and HH2 .TST 230 Fig.(5.32) Comparison of Predicted Response W i t h Hollow Cyl inder Results - P H H 3 B . T S T and P H H 3 C . T S T 231 Fig.(5.33) Comparison of Predicted Response W i t h Hollow Cyl inder Results - P H R 3 B . T S T and P H R 3 C . T S T 232 Fig.(5.34) Comparison of Predicted Response W i t h Cube Results - CH1 .TST and C R 1 . T S T 235 Fig.(5.35) Comparison of Predicted Response W i t h Cube Results - CH2 .TST and C R 2 . T S T 236 Fig.(5.36) Comparison of Predicted Response W i t h Cube Results - C H C . T S T . . . 237 Fig.(5.37) Comparison of Predicted Response W i t h Cube Results - C R C . T S T . . . 238 xiii LIST OF TABLES Page No Table 1.1 Constitutive Relations for Materials 3 Table 1.2 Simplifications and Modifications in Theories of Elasticity and Plasticity to Model Sand Behaviour 5 Table 2.1 Comparison of Stress Dilatancy Relations Originating from Particulate Mechanics Considerations 43 Table 4.1 Bonded Systems versus Frictional Systems 90 Table 5.1 Input Parameters for Predicting Characteristic Response of Sand 149 Table 5.2 Input Parameters for Predicting Behaviour of Ottawa Sand wi th D r=50% 184 Table 5.3 U B C Hollow Cylinder Boundary Values - Stage 1 200 Table 5.4 U B C Hollow Cylinder Boundary Values - Stage 2 200 Table 5.5 Input Parameters for Predicting Behaviour of Ottaws Sand wi th D r =42% 202 Table 5.6 Hollow Cylinder Tests for Calibrations 214 Table 5.7 Cube Tests for Calibrations 214 Table 5.8 Input Parameters for Predicting Behaviour of Hostun Sand 215 Table 5.9 Input Parameters for Predicting Behaviour of Re id Bedford Sand . . . . 216 Table 5.10 Hollow Cylinder Tests for Predictions 228 Table 5.11 Cube Tests for Predictions 228 xiv N O M E N C L A T U R E (i* coefficient of l imiting friction a , <7,y stress tensor 6 , strain tensor Ciju tensor of elastic constants Dyu viscosity tensor Vy rate of deformation tensor po hydrostatic stress Ae, Ae,y incremental strain tensor A c , AtTy incremental stress tensor {np} unit normal direction vector - plastic potential surface {riy} unit normal direction vector - yield surface H plastic resistance [D] constitutive matrix C'~u tensor of incremental elastic constants 7J>50 diameter of soil particles for which 5 0% is finer o~3c effective confining stress level evm volumetric strain due to membrane penetration e„ volumetric strain of soil sample a\,°~i,°~z principal stresses e i ; £ 2 ; £ 3 principal strains e void ratio 9 V(l + « ) o~m, Aam mean normal stress, increment ecs void ratio at critical-state 4>p interparticle friction angle (f>cv constant volume friction angle Aip3 change of direction of stress path at deviator stress ={<J\ — 03) es shear strain =(ei — 63) Dr relative density $o , < £ i , < & 2 elastic response functions ro>--;Fii response functions h,h,h strain invariants Wf strain energy density <7ty init ial state of stress e?- init ial state of strain v G shear modulus, number of orientation groups of contact normals xv B bulk modulus E Young's modulus v Poisson's ratio h height M total number of geometric contacts AMg number of geometric contacts with orientation between 9g and 9g + A9 KA, KB unit normals to contact planes at contacts A and B PA,PB contact planes at contacts A and B A 9 , A 9 g increment in orientation in two dimensions A O , AClg increment in orientation in three dimensions 9,9g orientation in two dimensions fl,Qg orientation in three dimensions specified by tp and <f> <f>,ip spherical coordinate angles, <f>:0 to 2n, t/>:0 to 7r S(9), S(9g) contact normal distr ibution in two dimensions S contact normal distr ibution in three dimensions V,v volume #o principal directions of contact normal distribution in two dimensions Co,C2n,D2n Fourier coefficients ai second order anisotropy in two dimensions 04 fourth order anisotropy in two dimensions Of\_,av,otz parameters /?i ;/?2 parameters aB boundary stress tensor R_ position vector of a point on boundary f boundary contact force vector tf position vector of boundary contact point r* position vector of center of particle / * ' m force acting on particle k at contact m nB unit normal vector at boundary contact nc unit normal vector at internal contact f° internal contact force vector /** contact force on particle k from particle 1 P$ distr ibution function of contact forces do, do diameter of particles / average contact force vector on contacts of a given orientation 7 coordination number n„ particle density fn average normal contact force on contacts of agiven orientation xvi ft average tangent contact force on contacts of a given orientation t unit tangent direction vector a,j-,Aa,y second order anisotropy tensor, increment bijn, A 6 , j y fourth order anisotropy tensor, increment f„ internal normal contact force ft internal tangent contact force k'n normal stiffness k[ tangent stiffness l cn, Al°n normal displacement of a physical contact point 1$, Al^ tangent displacement of a physical contact point kn =| tin do kt = ^ k't do uij potential energy stored at a physical contact 7,j engineering strain =2 e,j-wc complementary energy density L, V functional A, j ; AA,j non-symmetric strain tensor A * ; A A * - symmetric strain tensor r t i , / / 2 Lagrangian multipliers 0{j{y) phenomenological stress tensor <;,<;' non-dimensional contact force distr ibution parameters *n N number of particles p distributed boundary traction vector s$ area of boundary contact associated with particle j3 9 ={9 - 90) JI developed ratio between average tangent and normal contact forces in three dimensions kn, equivalent normal stiffness kt,ktt equivalent tangent stiffness 9a principal direction of second order anisotropy in two dimensions 0^ principal direction of fourth order anisotropy in two dimensions fo average normal contact force at tangent contact force anisotropy coefficient an normal contact force anisotropy coefficient A aj increment of tangent contact force anisotropy coefficient A o n increment of normal contact force anisotropy coefficient A d 2 increment of second order anisotropy coefficient XVll increment of second order anisotropy tensor increment of fourth order anisotropy tensor incremental resistance function Toct octahedral shear stress v, volume of solids parameters ho, hi, hi parameters m\, m 2 parameters s i , s 2 , n parameters 0' Lode angle stress ratio = Toctjam 'Hmaz peak stress ratio critical-state stress ratio in conventional triaxial compression critical-state stress ratio in conventional triaxial extension P ratio between r)3Se and r)SSc A C 9 slope of critical-state line in e — logio(am) space Hht incremental resistance function for strain hardening HH incremental resistance function for strain softening hho>h*o parameters Hfluid bulk modulus of fluid Hsolid bulk modulus of soild particles apparent bulk modulus of fluid weighting function P R E prediction O B S observation S I G M mean normal stress S I G C confining stress Subscripts r radial direction a axial direction t tangent direction n normal direction Superscripts e elastic P plastic xviii A C K N O W L E D G E M E N T S Developing a stress-strain model, from particulate considerations, has been a very inter-esting and challenging experience. The author would like to thank the University of Br it ish Columbia for the financial support he received throughout the project without which such an endeavour would have been impossible. There are a large number of individuals who have immensely helped the author in making this research project a success. The author would like to thank each one of them for their helpful suggestions, the optimistic as well as the pessimistic comments and regret his inabil ity to thank them individually. There are three special individuals to whom the author is very much indebted: Prof. Peter Byrne for the guidance, advice and most of al l for the freedom the author enjoyed, Dr. Donald Anderson for his invaluable time spent discussing and reviewing the theoretical formulations and Dr. Dawit Negussey for providing the much needed insight into sand behaviour from an 'experimentalist 's point-of-view'. Prof. Yoginder Vaid 's comments, on the presentation of parts of the thesis are very much appreciated. The author would like to thank Prof. L i a m F inn for the support and encouragement received when it was most needed. The author would like to thank his colleague, A lberto Sayao, for very graciously performing the hollow cylinder tests on Ottawa sand. The author would also like to thank his friends Gerard Canicius, Bla ir Goh l , Fancisco Salgado, L i Yan and Damika Wickremasinghe for the discussions and encouraging comments. Last but not least, the author would like to thank his wife Deepthi, for typing and proof reading parts of the thesis and the support and encouragement received in completing the thesis. xix C H A P T E R 1 INTRODUCT ION 1.1 Introduction Approximately 100 years ago, in 1885, Reynolds experimentally demonstrated that sands undergo significant changes in volume during shearing. It was not unt i l 50 years later that Casagrande established quantitative relationships describing shear induced vol-ume change of sand with density and confining pressure (Lee & Seed(1967)). To date, accomplishments in pursuit of quantifying the shear induced volume change phenomenon in sand are not entirely satisfactory. The ability of sand to change in volume during shearing is considered to be a distinct characteristic of the material. For several decades, only the strength behaviour of sand received attention. Loose as well as dense sands develop a significant portion of strength at small strain levels. Unt i l the recent past, detailed examination of sand behaviour under small strain levels had not been attempted. Dynamic loading such as occurring from machine foundations, earthquakes, ... etc. init ial ly induce small strains in soils. Sands of most densities tend to contract in volume under the application of such dynamic loads. When undrained conditions pre-vail, sands that tend to contract in volume develop significant pore water pressures. The dramatic reduction in strength associated with the high pore water pressures result in the 1 Chapter 1 : 2 phenomenon known as liquefaction. Catastrophic failures of major soil structures such as the Lower San Fernando dam which occurred in 1971, has been attributed to liquefaction. In order to assess the liquefaction potential of a sand deposit it is necessary to quantify the loss of strength as a result of bui ld up of pore water pressure. Development of pore wa-ter pressure under undrained conditions and the volume change characteristics of sand are intimately related to each other. It is important therefore that the volume change charac-teristics of sand be accurately modelled if pore pressure predictions are required. However, the volume change response of sand is extremely complicated being dependent on the stress level, stress path, density of packing and material anisotropy. Sand is a particulate, discrete and frictional material and as such forms a discontinuous medium. Stress at a point denned for a continuous material, or continuum, is no longer valid for a discontinuous material. Instead, stress must be denned as an average force over a finite area. Stresses so denned wi l l fluctuate from area to area. The discrete nature of the material facilitates spatial rearrangement of particles as a result of loading. Also, upon the removal of normal contact forces, sand particles can physically separate from each other. In frictional materials, the maximum ratio of tangent and normal contact forces are related through a coefficient of l imiting friction fj,". A t contacts where there is no relative movement, the developed ratio of tangent and normal contact forces is less than or equal to //*. Being a discrete and frictional material, the state of sand lies between a solid and a l iquid. When loaded hydrostatically, sand behaves more like a solid but when sheared reaches an ultimate or a crit ical state where the behaviour resembles that of a fr ict ional fluid. A comparison of the form of constitutive relations of general solids and liquids is extremely difficult, if not impossible. However, for the simplest Hookean solid and Newtonian l iquid, a comparison of the form of constitutive relations is possible and is shown in Table 1.1. Chapter 1 : 3 Table 1.1 The Constitutive Relations for Materials L IQU IDS (Newtonian) S A N D SOL IDS (Hookean) ? po = static pressure eju - strain tensor Dijki — viscosity tensor (independent of stress and rate of deformation) Cijki = tensor of elastic constants Vu — rate of deformation tensor Cij = stress tensor Oij = stress tensor Chapter 1 : 4 For Hookean solids, the stress tensor depends on the magnitude of strains and elastic modul i. On the other hand, for Newtonian liquids, the stress tensor is dependent on the hydrostatic pressure, rate of deformation and viscosity. Therefore, the stress tensor for sand can be expected to be dependent on the parameters that influence both solids and liquids. Mathematical representation of sand behaviour has been attempted by a great many researchers. Two distinctly different and popular theories are commonly used; one based upon theory of elasticity and the other on theory of plasticity. The final form of the stress-strain relations, common simplifications and the necessary modifications to model sand behaviour, required for each theory, are compared in Table 1.2. It can be seen that there are qualitative similarities between the two theories. In principle, either theory establishes a prescribed set of variations of the terms in the constitutive matrix, [D]. In theory of elasticity, the constitutive matrix [D]e contains elastic constants or modul i. By imposing the manner in which the elastic moduli change during loading and unloading, stress-strain response of non-linear and dissipative materials such as sand can be modelled. In theory of plasticity, the changes in the constitutive matrix [D]p are brought about by allowing the yield and plastic potential surfaces to evolve in stress space so that {ny}, {np} and H change during loading and unloading. Neither theory, even after considerable modifications, has been entirely successful in predicting the shear induced volume change behaviour of sand. Both elasticity and plasticity theories are based on observations on continuum type materials and hence may lack more fundamental variables that describe key features of par-ticulate, discrete and frictional materials. A rational approach requires the identification of key features of these systems from detailed microscopic investigations. Once the key features and the fundamental variables are identified, a theoretical framework can be pos-tulated with in which they can be incorporated into stress-strain relations for particulate, discrete and frictional systems. Chapter 1 : 5 Table 1.2 Simplifications and Modifications in Theories of Elasticity and Plasticity to Model Sand Behaviour T H E O R Y Elasticity Plasticity * {Ae} = \D)e {Ac-} * {Ae} = {np}\{ny}T {Aa}}/H \D\e contains 21 independent material constants that are independent of stress and strain {Ac} = \D]P {Aa} {ny} — unit normal vector to yield surface {np} — unit normal vector to potential surface H = plastic resistance S I M P L I F I C A T I O N S * Orthotropic material - 9 terms Cross anisotropic material - 5 terms Isotropic material - 2 terms Above simplifications come at the expense of shear and normal stress and strain uncoupling. * Assume simplified shapes for yield and potential surfaces. M O D I F I C A T I O N S * Make [D\e dependent on stress, strain and density. Also differentiate between loading and unloading. * Introduce shear-volume coupling v ia an additional parameter. * Introduce laws of evolution of yield and potential surfaces. * Define multiple intersecting yield and potential surfaces. Chapter 1 : 6 1.2 Research Objectives The research work carried out in this dissertation focuses on rate insensitive stress-strain relations for sand derived from particulate considerations. A theoretical framework is described based on a collection of ideas derived from laboratory experiments on real sands and micro-mechanics investigations. The proposed stress-strain relations are expressed in terms of incremental stress and strain quantities and are valid for general loading in three dimensions. Fully drained or undrained loading conditions can be simulated. Theoretical formulations for the stress-strain behaviour of a system of spherical particles that are glued at their contacts are studied first. In glued systems, each glued contact can have a different tangent and normal stiffness that are independent of each other. These glued systems, de-scribed in detail in Chapter-3, wi l l hereafter be referred to as bonded systems of particles. Much insight into the theoretical aspects of bonded systems has been obtained from Rothen-burg(1980). Following the theoretical framework proposed for bonded systems, stress-strain relations for discrete and frictional systems are formulated based on energy considerations in Chapter-4. The frictional aspects of sand behaviour are incorporated through a function that relates the average tangent contact forces to the average normal contact forces. The absolute magnitude of the function relating the average tangent and normal contact forces vary between zero and a l imiting value fj,". In discrete systems, the spatial arrangement of contacts change as a result of loading. Laws that define the extent to which these changes can occur, are established. In deriving the laws describing particle arrangements, labo-ratory information obtained from carefully conducted testing programs together wi th the information obtained from micro-mechanics investigations have been examined in detail. The proposed stress-strain relations are derived for a system of equal size spherical particles and are validated by comparing them with the laboratory response of sand. The data base consists of data from conventional tr iaxial tests, true tr iaxial tests and hollow Chapter 1 : 7 cylinder tests performed under fully drained conditions. 1.3 Organisation of the Thesis This thesis consists of six chapters. In Chapter-2, a literature review on stress-strain relations for sand is presented. First, laboratory observations on stress-strain response of sand are summarized. Thereafter, the material idealizations common in existing stress-strain theories are critically reviewed. The stress-strain relations for a system of bonded particles, as derived by Rothen-burg(1980) are presented in Chapter-3. Some of the formulations have been simplified to suit the objectives of this dissertation. The recapitulated formulations of Rothenburg are contained exclusively in Chapter-3. Chapter-4 describes the stress-strain relations proposed for particulate, discrete and frictional systems. The proposed stress-strain relations are new and differ considerably from those derived for bonded systems of particles. The characteristic stress-strain response of sand as predicted by the proposed model and a study of the effect of the model parameters on the predicted response are presented in the first half of Chapter-5. Thereafter, the proposed stress-strain relations are verified against laboratory stress-strain measurements on sand. The data base consists of stress and strain measurements obtained from conventional tr iaxial tests reported by Negussey(1984) and hollow cylinder tests graciously carried out by A. Sayao for the author together wi th true tr iaxial and hollow cylinder data provided for the Cleveland Workshop(1987). A discussion of the proposed stress-strain relations wi th respect to their ability to capture major aspects of sand behaviour is presented following the comparisons. Chapter-6 of this dissertation summarizes the major findings of the research work, presents the conclusions and make recommendations for further research work. C H A P T E R 2 S T R E S S - S T R A I N R E L A T I O N S FOR S A N D - A L I T E R A T U R E R E V I E W 2.1 Introduction Our current understanding of sand behaviour is based on carefully conducted laboratory experimental programs. W i t h the development of new testing equipment and techniques together with improved accuracy of measurements, our understanding of sand behaviour has improved considerably over the last decade. Aside from physical laboratory experiments, the latest trend of numerically simulating the behaviour of a system of particles has enhanced the understanding of dominant mechanisms causing deformations in particulate systems. Together, both physical and numerical experimental observations form a strong data base for the development of advanced stress-strain relations for sand. There are many different stress-strain relations proposed for sand varying from simple curve fitting type relations to those based on highly complicated multiple yield surface plas-t ic ity theories. A general trend towards idealizing sand as a rate independent elasto-plastic material is seen. Representation of sand as a particulate, discrete and frictional material has not been a popular approach in deriving stress-strain relations. The complicated mechanics of particulate media as well as the evident lack of experimental information are the main causes for this unpopularity. 8 Chapter 2: 9 The first objective of this chapter is to highlight the important features of sand be-haviour as observed from laboratory programmes. This is done in a qualitative manner. The second objective is to examine the material idealizations common in the existing stress-strain relations. The main differences between elastic, plastic and particulate representa-tions of sand are reviewed briefly. 2.2 Observed Behaviour of Sand A summary of the characteristic behaviour of sand as observed from controlled labo-ratory experiments is presented in this section. A t this stage it is important to note that the existing laboratory measurements are subjected to a number of non-homogenities. The laboratory measurements are influenced by; (1) membrane compliance effects, (2) varying rates of loading/deformation, (3) samples of varying size and shape, • (4) varying types of testing equipment and techniques, and (5) varying accuracies of measurements. A n in depth study of the influence of above factors on observed stress-strain behaviour of sand is beyond the scope of this thesis. However, the inaccuracies resulting from membrane compliance effects are worth highlighting since membrane compliance effects can introduce significant errors in volume change measurements for coarse grained materials like sand. Methods of correcting for membrane compliance effects have been suggested by a number of researchers. A n explicit account on these methods has been reported by Vaid &c Negussey (1984). Figure 2.1 shows the comparison between the measured and corrected volumetric strains for standard Ottawa sand with a relative density of 50%, following the method of Chapter 2: 10 Fig. (2.1) Variation of Volumetric Strains Due to Membrane Penetration with Confining Pressure - Ottawa Sand (after Vaid & Negussey(1984)) Chapter 2: 11 correcting for membrane compliance effects outlined by Vaid & Negussey(1984). Note that information in Figure 2.1 has been obtained from Figure 2 of Vaid &: Negussey(1984) but is presented in a different format. It can be seen that the membrane compliance effects account for almost 5 0 % of the measured volumetric strains for standard Ottawa sand. Data shown in Figure 2.1 are for 63.5mm diameter samples 127mm in length surrounded by a 0.3mm thick rubber membrane. W i t h increasing size of sample, decreasing mean grain size and increasing density, membrane compliance effects reduce considerably. The effect of mean grain size on membrane compliance can be seen from Figure 2.2 where other factors have been kept constant(Tokimatsu &; Nakamura(1986)). In Figure 2.2, the volumetric strains due to membrane effects have been computed following the method outlined by Vaid &£ Negussey(1984). The magnitude of correction for membrane compliance effects depends on the assumptions contained in the formulae proposed by different researchers and therefore can introduce a significant degree of inconsistencies among the various stress-strain data bases available for sand. In an attempt to facilitate a simple and clear presentation of experimental observations on sand behaviour, the presentation has been divided into five main sections. Each section highlights an important aspect of sand behaviour. Due to the inconsistencies of experimental observations as outlined earlier, only a qualitative presentation is made herein. 2.2.1 Density and Stress Level Dependency The stress-strain characteristics of sand samples prepared in an identical manner and following a given stress path are dependent on the density of packing and the confining pressure. A long stress paths where the stress ratio increases continuously, sand samples reach a steady-state characterized by deformation at constant volume, stress and velocity (Castro et al (1982)). Upon loading along such stress paths, only loose sands and Chapter 2: 12 CM 0 I 1 1 1 1 1 1 0 0,2 0,4 0,6 0,8 1,0 1,2 VOLUMETRIC STRAIN DUE TO M E M B R A N E P E N E T R A T I O N x 10"2 Fig.(2.2) Variation of Volumetric Strains Due to Membrane Penetration with Mean Grain Size - Japanese Sands (after Tokimatsu & Nakamura(l986)) Chapter 2: 13 sands under high confining pressures exhibit continuous contractive volume change be-haviour al l the way to steady-state. A t most densities and confining pressures, there is an init ial contraction but the steady-state is approached through a dilative response. F ig-ures 2.3 and 2.4 illustrate the influence of confining pressure on the stress-strain response of dense and loose Sacremanto River sand samples, loaded along the conventional triax-ial stress path, respectively(Lee & Seed(1967)). It can be seen that the principal stress ratio-axial strain and volumetric strain-axial strain variations obtained for the dense sand subjected to high confining pressure is similar to that of loose sand under moderate con-fining pressures (compare er 3 c = 120 kg/ cm2 of dense sand with cr 3 c = 12.7 kg/cm2 of loose sand). Stress-strain response of sand shown in Figures 2.3 and 2.4 are for loading under fully drained conditions. If fully saturated sand samples are tested under undrained conditions, then the low compressibility of water imposes a near constant volume condition on the system. Under this constraint, sands that tend to contract in volume during loading develop increasing pore water pressures whereas sands that expand in volume develop decreasing pore water pressures. The volume change characteristics of sand and pore water pressure development, being intimately related to each other, can be understood by imposing the volume constraint resulting from undrained conditions. The influence of both the density and confining pressure can be described using a single parameter termed the state parameter which is defined as the difference in void ratio between the in it ia l state of sand (specified by mean normal effective stress and void ratio) and the steady-state conditions corresponding to the same mean normal effective stress (Been &; Jefferies (1985)). Figure 2.5 illustrates the state parameter for sand of two different init ial states. Sands with similar state paramters have been shown to exhibit similar trends in volume change behaviour (Been & Jefferies(l985)). It can be seen that upon hydrostatic Fig.(2.3) Stress-Strain Response Observed for Dense Sacremento River Sand (after Lee k Seed(1967)) Fig.(2.4) Stress-Strain Response Observed for Loose Sacremento River Sand (after Lee & Seed(1967)) Chapter 2: 16 SSL - s teady state line Fig.(2.5) The State Parameter Concept in Void Rat io -Mean No rma l Stress Space Chapter 2: 17 loading, the state parameter can be made to change from a positive number to a negative number and thus the sand behaviour w i l l change from dilative to contractive. The state parameter concept or the dependency of material response on the relative positions of the current state of a material and a reference state is useful and can be easily incorporated into theoretical formulations describing sand behaviour. 2.2.2 Existence of a Reference State for Sand The concept of a crit ical state for soil, was first proposed by Casagrande (1936). A cr it ical state is postulated as a state where soil and other granular materials undergo un-l imited shear deformations at constant volume and stress. The friction angle developed at the cr it ical state <f>cv, is considered to be a material parameter. It appears that there is no consensus on the relationship between the apparently more fundamental interparticle friction angle </>M and <j>cv. Reported values of </>M are less than the values of <j>cv for the same material (Rowe (1971)). Dur ing undrained shearing, sands reach a steady-state characterized by deformation at constant volume, stress and velocity (Poulos(1981),Castro et a/(1982)). Recent research at the University of Br i t i sh Co lumbia indicates that the friction angle developed at the steady state (interpreted from undrained conventional tr iaxial tests) is identical to the friction angle independently obtained from ring shear tests (Wijewickreme (1986)). The ultimate friction angle obtained from ring shear tests correspond to a state of unlimited shear distortions at constant volume and hence is the best possible estimate of <f>cv. Reported results from ring shear tests indicate that <f>cv is independent of particle size, confining pressure and density. From stress considerations alone, it is postulated that the steady-state and crit ical state appear to be equivalent and unique to a granular material (Wijewickreme (1986), Negussey et al (1986)). Chapter 2: 18 Even though both the steady-state and critical-state are considered to be equivalent and unique in the stress space, laboratory evidence indicate that the two reference states are different in the e — logiofom) space. Chung (1985) has shown that the steady-state line in the e - logio(am) space for extension side undrained loading is dependent on the init ial void ratio of the sample. However, for compression side undrained loading, the steady-state has been shown to be unique (Chem (1985)). No such anomaly exists for the critical-state which is considered to be unique in both extension and compression loading for both drained and undrained conditions. Therefore, the selection of the critical-state as a reference state is more desirable. 2.2.3 Anisotropic Behaviour of Sand When a sand sample is loaded with identical stress probes in different directions, if the strain response is different the material behaviour is considered to be anisotropic. Anisotropy is intimately related to a reference configuration of the material and a coor-dinate system associated with it, wi th respect to which stress and strain are measured (Dafalias (1975)). Based on the postulate that anisotropy may exist prior to straining or it may be in-duced as a result of straining, two different types of anisotropy are defined for sand; inherent anisotropy and induced anisotropy. Inherent anisotropy is defined as a physical character-istic inherent in the material resulting from the process of deposition and influenced by the shape of individual sand grains. Induced anisotropy is defined as a physical characteristic resulting from straining associated with loading after deposition (Arthur & Menzies(1972), Wong & Arthur(1985)). Method of sample preparation significantly influence the degree of inherent anisotropy (Oda(1972)). Also, laboratory experimental results indicate that the inherent anisotropy Chapter 2: 19 is strongly dependent on the density of packing when other factors such as particle size, shape and method of sample preparation are the same and that hydrostatic loading does not change the level of anisotropy ( Negussey (1984)). Figure 2.6 illustrates hydrostatic load-unload strain path data reported by Negussey <k Vaid(1984) for Ottawa sand samples of different relative densities. A l l three samples have been loaded and unloaded between hydrostatic stress states of 50 k P a and 550 kPa. It can be seen that during hydrostatic loading or unloading, a sample of given relative density followed essentially linear strain paths. Note that the slopes of the strain paths for unloading are smaller than for loading. Above data indicate that hydrostatic loading does not change the inherent anisotropy of the samples(i.e. the stain increment ratio is constant). When samples of different densities are considered, the inherent anisotropy increases rapidly with decreasing relative density. Note that from a continuum mechanics point of view, the Mohr circle that correspond to a hydrostatic stress state reduces to a point, implying that the applied shear stresses anywhere in the sample are zero. The hydrostatic loading path is one of many proportional loading paths where the stress ratio is maintained constant. Stress-strain response of sand observed along constant stress ratio loading and subsequent unloading paths have been reported by El-Sohby(1969) for a range of stress ratios. Some of his results are shown in Figure 2.7 and it can be seen that the strain increment ratio remains appreciably constant along constant stress ratio paths. A s the stress ratio increased, the magnitude of strain increment ratio changed in accordance with an expansive volumetric response. Similar conclusions can be made from the stress-strain measurements reported by Pooropshasb et al (1966). Their studies focus only on loading along constant stress ratio paths. Based on expermental observations from directional shear cell, Wong & Arthur(1985) Chapter 2: 20 Fig.(2.6) Volumetric and Ax ia l Strain Measurements A long Hydrostaitc Load and Unload Paths - Ottawa Sand (after Vaid & Negussey(l984)) Chapter 2: 21 ig.(2.7) Volumetric and Ax i a l Strain Measurements A long Load-Unload Paths w i th Vary-ing Stress Rat io - Fine Sand (after El-Sohby(1969)) Chapter 2: 22 conclude that it is the magnitude of cumulative shear strain that control the degree of induced anisotropy. Wong & Ar thur s ' experimental observations consisted of shear loading of Leighton Buzzard sand samples, that were init ial ly isotropic in the horizontal plane, to different levels of shear strains along a stress path where the major principal stress direction was held constant. Thereafter, the samples were unloaded back to the same init ial isotropic stress state, and then reloaded in shear such that the major pr incipal stress direction was different from the init ial direction. Some of their results i l lustrating the influence of shear strain level on the induced anisotropy are shown in Figures 2.8. The first phase of loading with a given direction of major pr inc ipal stress are il lustrated in Figures 2.8a and 2.8b. Three samples were sheared along stress path A to three different major principal strain levels and unloaded along the same stress path. Thereafter, the same three samples were loaded along stress path B where the major pr incipal stress direction was 70° different from stress path A. The observed stress-strain response is shown Figures 2.8c and 2.8d. It can be seen that the sample that was strained most along stress path A shows significant softening and larger changes in volume. Figures 2.8c and 2.8d also contain information on the stress-strain behaviour of samples when they are reloaded without a change of direction of major principal stress (i.e. along stress path A ) . As can be seen from Figures 2.8c and 2.8d, those samples follow the virgin stress-strain curve after loading to the previous unloading state, irrespective of the shear strain level at unloading. The above experimental data clearly show the influence of stress induced anisotropy where it may be seen that the magnitude of straining can lead to considerable softening of the stress-strain behaviour. The directional shear cell and the hollow cylinder test apparatus allow the principal directions of stress to be controlled independently. When the principal stress directions are held fixed at an orientation different from the principal anisotropy directions, the observed principal strain increments are non-coincident wi th the principal stress directions (Symes (1983) Chapter 2: 23 Fig.(2.8) Effect of Strain History on Induced Anisotropy Observed F rom Direct ional Shear Cel l - Leighton Bazzard Sand ( after Wong & Arthur(1985)) Chapter 2: ig.(2.9) The Variation of Strain Increment Direction and Principal Stress Direction with Applied Shear - Ham River Sand (after Symes(1983)) Chapter 2: 25 , Wong &c A r thur (1985)). Typica l variations of principal strain increment directions observed for increasing shear stress level, with principal stress orientations held fixed are shown in Figure 2.9 (Symes (1983)). From Figure 2.9 it can be seen that wi th increasing shear stress level, the deviation between principal strain increment direction and pr incipal stress direction reduce considerably. 2.2.4 Pa th Dependent Behaviour of Sand Materials with a non-unique association of stress and strain, exhibit path dependent behaviour. Experimental test results show that the stress-strain response of sand is path dependent. The strain paths followed by sand samples of different relative densities, over the same range of hydrostatic loading and unloading stress states, were shown in Figure 2.6 earlier. The unloading strain paths were shown to be different from the loading strain paths. For a given hydrostatic stress state, two different strain states exist along loading and unloading paths, exhibiting path dependent behaviour of sand. A long hydrostatic loading and unloading paths, the sense of the stress increment is different. However, even if the sense of the stress increment were maintained, sands sti l l exhibit path dependent behaviour. Experimental data reported by Negussey(1984) on strain paths starting from the compression side and following constant mean normal stress unloading paths, are shown in Figure 2.10. As concluded by Negussey(1984), the change of strain states associated with change of stress states from A; to 5, is very much smaller in magnitude than the changes between stress states B, and C„ and sands therefore exhibit path dependent behaviour even when the sense of stress increment is maintained. The variations of the init ial strain increment ratio, with stress increment ratio for stress paths involving loading in both shear and/or mean normal stress have been explored by Negussey(1984). In his experiments, al l the samples were initial ly loaded to a hydrostatic Fig.(2.10) Strain Paths for Compression Side Shear Unloading and Extension Side Shear Loading at Constant Mean Normal Stress (after Negussey(1984)) Fig.(2.11) Incremental Stress and Strain Ratio Variations for Loading Paths - Ottawa Sand (after Negussey(1984)) Chapter 2: 28 stress state of 50 kPa. Thereafter, stress probes were applied that correspond to different stress paths, as shown in the insert to Figure 2.11. The resulting init ial strain increment ratios were then obtained and when plotted in the space shown in Figure 2.11, fell on two different straight lines. In Figure 2.11, the steeper line corresponds to proportional loading paths where the stress ratio is maintained approximately constant, whereas the natter line corresponds to non-proportional loading paths where the stress ratio continuously changes. A s seen from Figure 2.11, there is an abrupt change in the slopes of the two straight lines. The point at which the bil inearity occurs indicates the stress probe that makes the strain increment (of the samples) in the radial direction change from an inward direction to an outward direction. It can also be seen that the strain increment ratio is quite sensitive to the stress probe along proportional loading paths than for non-proportional loading paths, which results in the different slopes. Note that proportional loading paths are non-failure paths for which case I1 ~ ff£] - 1 w h e r e a s non-proportional paths are failure paths for which case [1 — > 1. Based on the data shown in Figure 2.11, it becomes evident that for a sand sample to fail in shear, the stress changes should be such that the resulting strain increment ratios must be negative. The magnitudes of the slopes of the straight lines shown in Figure 2.11, are dependent on the inherent anisotropy of the samples. Nevertheless, the characteristic response wi l l not be different from what was stated above. 2.2.5 The Unload-Reload Response of Sand The load-unload and reload response of standard Ottawa sand, as observed along con-ventional t r iax ia l load-unload path and reported by Negussey(1984), is shown in Figure 2.12. F rom Figure 2.12, the following can be observed; (a) upon loading, the incremental resistance to deformation decreases continuously. Chapter 2: 29 Fig.(2.12) The Load-Unload-Reload Response A long Conventional Tr iax ia l Pa th - Ottawa Sand (after Negussey(1984)) Chapter 2: 30 (b) upon unloading subsequent to loading, the incremental resistance to deformation abruptly increases and is higher than the in it ia l incremental resistance to deformation during vir-gin loading. W i t h further unloading, the incremental resistance to deformation reduces. (c) upon reloading, the incremental resistance to deformation again increases abruptly. (d) along loading, unloading and reloading paths, the deviator stress, axial strain and volumetric strain variations are non-linear. (e) upon unloading to the in it ia l stress state, non-recoverable strains are produced. This is true particularly for the first cycle of loading, at the end of which non-recoverable axial and volumetric strains of magnitudes 0.0004 and .0001 have been observed. . (f) abrupt changes in the incremental resistance to deformation are associated with abrupt changes in both strain increment magnitude and direction. Dur ing loading and unloading, the hyteresis loops shown in Figure 2.12, indicate energy dissipation or damping with in the material. This is true even for loading at very low stress and strain states. 2.2.6 Discussion The stress-strain response of sand, as observed from laboratory testing programs, is seen to be extremely complicated being dependent on the density, stress level, stress path and material anisotropy. Inherent and induced anisotropy have a strong influence on the stress-strain response of sand. Accurate stress-strain relationships cannot be developed without proper characterization of anisotropy. Laboratory data indicate that the inherent anisotropy of a sample does not change during hydrostatic loading and that the stress ratio can be used as a measure of the stress induced anisotropy. A single parameter known as the state parameter takes account of the influence of both the confining pressure and density of packing. The state parameter establishes the in it ia l Chapter 2: 31 state of the material with respect to a reference state. The reference state could either be the critical-state or the steady-state, the former being preferred because of its uniqueness. Materials of similar state parameters w i l l have similar trends in volume change behaviour. Incorporation of a such a parameter into the theoretical formulations is therefore highly desirable. The constant volume friction angle developed at the critical-state <j>cv, is seen as a fundamental material parameter. (f>cv has been shown to be independent of particle shape, size, effective confining pressure and whether the material is wet or dry. Based upon above observations, <f>cv can be considered to be superior to the interparticle friction angle <f>p, which is difficult to measure. 2.3 Existing Stress-Strain Models for Sand Most stress-strain models proposed for sand make use of the well developed classical theories of elasticity and plasticity, either separately or in a combined form. In theory of elasticity, the strain increment direction is associated with the stress increment, whereas in theory of plasticity the strain increment direction is associated w i th the state of stress. Elasticity and plasticity theories have been developed based on the observations made on materials that can be described in the context of continuum mechanics. To model stress-strain behaviour of sand, the two theories had to be modified. The most cumbersome feature of sand behaviour has been the volume changes resulting from shear loading. Published research attempts to derive stress-strain relations considering sand as a par-ticulate, discrete and frictional material are l imited. The final result of most attempts appear in the form of a stress dilatancy equation (Rowe (1971), Matsuoka (1974b), Oda (1972a), Nemat-Nasser (1980)) which establishes a linear relationship between the strain Chapter 2: 32 increment ratio and stress ratio. The stress dilatancy equation has been helpful in quan-tifying shear induced volume change behaviour of sand in stress-strain formulations based on theories of elasticity and plasticity. 2.3.1 Elastic Models Theory of elasticity deals with methods of calculating stress and strain in elastic solids. In elastic models, the state of stress is uniquely determined by the state of strain so that the stress-strain response is path independent. There are a number of different elastic models. The basic features of a few of these models, that are in common usage, are examined in the following section. 2.3.1.1 Hooke's Elastic Model Hooke's generalized law of elasticity describes an elastic model in its simplest form. In Hooke's model, the stress tensor cr,y and the strain tensor e,y are related through the constitutive tensor (7,y« as shown below. The tensor Cyw, called the tensor of elastic constants, is independent of stress and strain and for the most general case consist of 81 terms. However, on account of symmetry of <7,y and €ki, Cjju can be reduced to 36 terms. When the material behaviour is path independent, dWf there exists a strain energy function Wf with the property = -g^f and the number of terms can be reduced to 21. Often, symmetry of mechanical properties of a material is made use of. When there exist three orthogonal planes about which the mechanical properties of a material are symmetric, the material is described as an orthotropic material. Similarly, when the mechanical properties are symmetrical about one axis, a cross anisotropic material is said aij — Cijki e« (2.1) Chapter 2: 33 to exist. For the special case when the mechanical properties are directionally independent, the simplest isotropic material results. In elastic models proposed for orthotropic, cross anisotropic and isotropic materials, the shear stresses (cr,y : i ^  j) induce shear strains {tij '• i 7^  j) only and vice versa. In effect, in these models the shear and normal stresses and strains are uncoupled from each other. Such models are applicable for materials like steel, where laboratory experiments indicate uncoupled shear and normal components of stresses and strains. The symmetry of mechanical properties, together wi th the uncoupling of shear and normal components of stress and strain, reduce the number of terms describing C,yjy considerably. For orthotropic, cross anisotropic, and isotropic models, C,yju reduce to 9, 5 and 2 terms respectively. The incremental form of equation 2.1 can be written as, ACT,; = C'ijH AeM (2.2) ACT,-,- and A e « now denote incremental stress and strain quantities, and C'-^ now consist of incremental coefficients. The incremental elastic stress-strain relations shown above in equations 2.2, have often been used in modelling sand behaviour assuming isotropic ma-terial behaviour. When C\-u are independent of stress and strain, a key feature of above incremental stress-strain relations is that the strain increment direction is always in the direction of the stress increment. 2.3.1.2 Cauchy and Green Elastic Models Cauchy and Green elastic models are categorized as higher order elastic models. Hooke's isotropic elastic model described earlier is a special case of Cauchy and Green elastic models. The stress-strain relations for a Cauchy elastic model is written as, (Jij = $ 0 $ij + $ 1 £ij + $ 2 e,m emj 1 hj = 1; 2,3 (2.3) Chapter 2: 34 where <&o,<&i,3>2 are elastic response functions dependent on the strain invariants h,h, and Is- • For the Green elastic model, the stress-strain relations are derived from internal energy considerations and for this reason it is called a hyperelastic model. When the strain energy density function Wf, is expressed in terms of strain invariants I\, li and 1%, the stress-strain relations take the form, dWf , , '»=n4 {2A) For the special cases when $ 0 = » i h, $ 1 = a 2 ; $ 2 = 0 and Wf = C\Ii + C1I 2, equations 2.3 and 2.4 reduce to Hooke's isotropic elastic model. In above, C\ and C 2 are constants. Equations 2.3 and 2.4 describe inherently non-linear stress-strain relations. Since the stress-strain response of sand is non-linear, Cauchy or Green elastic models appear to be more appropriate. The number of model constants w i l l depend on the functional form of $ o ; $ i ,®2 and Wf. The incremental form of equations 2.3 and 2.4 can be obtained by considering two different states of stress and strain denoted by e?), (0-?• + A<r,j-; • + Ae,y) and subtracting terms for state 1 from state 2. Associated with such an approach is the necessity to describe a non-zero strain state associated with a non-zero stress state. This is difficult for sand because all laboratory data correspond to an init ial ly zero strain state associated with an init ial ly non-zero stress state. 2.3.1.3 Hypoelastic Mode l In hypoelasticity, the increment of stress is expressed as a function of stress and strain increment. The general form of the equations can be expressed as, Aa{j = Cljkl AeH ; k, I =1,2,3 (2.5) Chapter 2: 35 Where C\-a are functions of stress. For the isotropic case, equations 2.5 can be expanded to take the form(Desai &c Siriwardena(1984)), Ao-y = To.AekkSij + T i . A e y + r 2.Ae f a t(7,y + r3 . < 7 m n A e n m c % + r 4 ( tT , m A€ m ; - + Aeimamj) +r5 -Ae f c k a , m t7 m ; - + r 6 . t 7 m n A e „ m o - , y + r7 . e r ^ t r ^ A e ^ , ; , + Ts-io-imamkAe^ -+- A 6 , - m t T m t a J j ) + r9 .CT m n A€ n m O- ,A (7 i 7 - + T10-0-mnO-nk Ac kmO~ij + ^ 1 . C T m n C T ^ Aejfem<7,r(70 (2.6) where, ITo, ri,...rn are response functions that are dependent on stress invariants. For the special case when r2 = T3 • • • T n = 0 , equations 2.6 reduce to Hooke's isotropic elastic model in its incremental form. In hypoelasticity, the stress-strain behaviour is simulated from increment to increment with C^jy being dependent on the current state of stress, and therefore hypoelastic models exhibit path dependent behaviour. Being able to model path dependent and non-linear be-haviour, the hypoelastic formulation is considered to be more appropriate for mathematical representation of sand behaviour. 2.3.1.4 Discussion The incremental linear elastic stress-strain relations described by either equations 2.2 or 2.5 have been extensively used in modell ing sand behaviour. Because of the convenience resulting from a low number of model parameters, the isotropic model has often been used to predict the complicated stress-strain response of sand. A s outlined earlier, Hooke's isotropic elastic model requires only two model parameters to be specified. Any two of the commonly used elastic parameters i.e. Young's modulus E, Bulk modulus B, Shear modulus G or Poisson's ratio v, can be used in the isotropic elastic model. Since Poisson's ratio is a difficult quantity to measure accurately, combinations of Young's modulus E and bulk modulus B, or shear modulus G , and bulk modulus B, are often used as the two Chapter 2: 36 parameters. After each increment of loading, the two parameters are updated to account for stress level dependency. By doing so, the non-linear and path dependent stress-strain response of sand has been modelled. In essence, the material behaviour is represented by a hypoelastic type model. In the above approach, the bulk modulus is usually assumed to be dependent only on the mean normal effective stress. E and G are assumed to depend on both the mean normal effective stress and shear stress level. The shear volume coupling effects are incorporated via a stress dilatancy equation. The three parameter model proposed by Byrne & Eldridge(l982) is an example of such a stress-strain model. The Cauchy and Green elastic models presented in section 2.3.1.2 lack the ability to model anisotropic and path dependent behaviour of sand. Also the functional forms of $o,<£i> $ 2 and Wf have not been established for sand. A similar situation exist for ro,ri...rn in hypoelastic formulation presented in section 2.3.1.3. When the functional forms of $ 0 ; 3>i, $2 and Wf are assumed, the model parameters can be calculated from a single test. The model parameters can be calculated from selecting a set of stress and strain data points from a single test and solving a system of simultaneous equations treating the model parameters as unknowns. The important question, however, is whether the same set of model parameters can predict the stress-strain response obtained from different tests (i.e. stress paths) . Experience with Cauchy and Green elastic models indicates otherwise (Desai & Siriwardena (1984)). Hooke's elastic representation of a material requires al l 21 model parameters to be defined. W i t h all 21 parameters being fully invoked, observed phenomena such as anisotropy and shear volume coupling effects may be modelled. However, the involvement of a large number of model parameters has discouraged research efforts along these lines. Chapter 2: 37 2.3.2 Plastic Models Plasticity theory has been developed on the basis of observed behaviour of metals. It offers a mathematical framework where the plastically deformed solids can be described. Since sands exhibit plastic or non-recoverable strains, theory of plasticity forms an attractive theoretical framework for the mathematical representation of sand behaviour. However, there are major differences between metals and frictional materials. The presence of voids and tendency to change or maintain volume during shear are two characteristic features that distinguish sand from metals (Lade (1987)). In theory of plasticity, the strain increment direction is uniquely associated with the stress state and the plastic strains. In most elasto-plastic stress-strain relations it is postu-lated that the observed strain increment can be decomposed into an elastic component and a plastic component. The proportion of elastic and plastic strains change during loading and unloading. The plastic strain, increment is calculated from theory of plasticity whereas the elastic strain increment is calculated from theory of elasticity. Making use of above postulates on the observed strain tensor, a family of elasto-plastic stress-strain relations have been proposed for sand. A brief review of the general approach in the development of plastic models is presented in the following sections. 2.3.2.1. Constituents of Theory of Plasticity The existence of yield, potential and hardening functions facilitate the mathematical representation of a material in the context of theory of plasticity. A yield function describes the stress conditions causing plastic strains. Accordingly, a yield surface encloses a volume in the stress space inside of which the strains are fully recoverable. Stress increments directed outward from the yield surface cause plastic strains. A state of stress outside the yield surface is possible through expansion and/or translation Chapter 2: 38 of the yield surface. During yielding, the stress point remains on the yield surface. In plasticity, the condition imposed on the stress point to remain on the yield surface is known as the consistency condition. The direction of plastic strain increment, at the stress point under consideration, is defined by the plastic potential function. When yield and potential functions coincide an associated flow rule exists. Otherwise, the flow rule is described as non-associated. The amplitude of plastic strain increment is specified by the hardening function. In conventional plasticity, the hardening function is derived by imposing the consistency condi-tion. In plasticity, two kinds of hardening have been distinguished; isotropic hardening and kinematic hardening. In a model undergoing isotropic hardening, the yield surface expands radially about its fixed axes. When the yield surface undergoes translation without change of size, the model undergoes kinematic hardening. Once the constituents of the theory of plasticity are defined, the plastic strain increment A e p can be calculated from; {Ae}?= I {np} [{ny}T {ACT}] (2.7) where, {ny} = vector defining the unit normal to yield surface at stress point {fi p} = vector defining the unit normal to potential surface at stress point {ACT} = applied stress increment vector H = plastic resistance 2.3.2.2. Yield and Potential Functions for Sand Yielding of metals is a well defined phenomenon. The stress conditions at which the material cease to exhibit fully recoverable strains, defines the onset of yielding. For sands, Chapter 2: 39 recoverable and non-recoverable strains occur in parallel and cannot be isolated from each other by an experiment (Hardin(1978)). Experimental investigations designed exclusively to establish yield functions for sand have been undertaken (Poorooshasb et o/(1966), Tatsuoka & Ishihara(1974), Tatsuoka &c Molenkap (1983)). In above studies yielding is identified as stress states at which the slope of stress-strain curves change abruptly. In order to model the complicated stress-strain response of sand, a trend of defining separate intersecting yield surfaces for shear induced strains and consolidation strains are common. Following a different approach, the yield surface can be obtained from the failure surface. Here, the yield and the failure conditions are assumed to be described by similar functions so that both surfaces have similar shapes (Ghaboussi &i Momen(1982), Lade & Duncan(1975)). The uniqueness of the yield functions obtained from the above methods, have never been questioned. Often, only isotropic models have been considered. Therefore in the three dimensional pr incipal stress space, yield functions that are symmetric about principal stress axes have often been used in plasticity models. The potential surface can be assumed to be identical to the yield surface or it can be derived separately. Often, potential functions have been derived from stress dilatancy equations (Nova &: Wood(l979), Pastor et a/(l985)). The approach has been straight forward because the stress dilatancy equations proposed in the literature are independent of the stress increment. Laboratory stress-strain measurements of sand indicate that under working stress levels (i.e. prior to failure), the plastic strain increment vector is dependent on the stress increment vector. In theory of plasticity, the strain increment vector is dependent on the stress vector and not on the stress increment vector. To satisfy the dependency of strain increment vector on the stress increment vector, the plastic potential surfaces should contain vertices. Chapter 2: 40 2.3.2.3. Multiple Yield Surface Plasticity The major drawback in the classical theory of plasticity is the presence of a single yield surface. Such a situation imposes a severe constraint in modelling sand behaviour under unloading conditions. Laboratory observations indicate the continuous occurrence of plastic strains during unloading. In the classical plasticity theory, only recoverable strains occur during unloading where the state of stress lies inside of the yield surface. Plastic strains occur only when the state of stress is changed such that yielding occurs in the unloading side. The problem has been solved by replacing the single yield surface with a collection of nested yield surfaces. The theory has been shown to offer great versatality and flexibility in describing the behaviour of real materials. Mu l t ip le yield surface plasticity theory has been successfully implemented to predict soil behaviour (Prevost (1978), F i nn &; Martin(1980)). The theory requires that in field calculations, the positions, sizes and plastic modul i of each of the yield surfaces be stored at each integration point which is very cumbersome. By replacing the field of yield surfaces by an analytical function, the tedious book-keeping procedure can be relieved leading to simple two surface plasticity theories. Adopting a slightly different approach, in 1975, Dafalias introduced the bounding sur-face plasticity theory. The concept and name was motivated by the observation that the stress-strain curves converge with specific bounds at a rate which depends on the distance of the stress point from the bounds. Unlike in conventional plasticity, the bounding surface plasticity theory, explicit control of plastic modulus can be exercised . Plastic deformations can occur on or beneath the bounding surface. A stress point outside the bounding surface can be reached by expanding and/or translating the surface to that point. For each stress point inside the bounding surface, an image point is defined on the bounding surface. The method of defining this image point, however, is arbitrary. The plastic modulus correspond-ing to a given stress state is dependent on the plastic modulus at the corresponding image Chapter 2: 41 point and the distance between the image and stress points. The bounding surface modulus is calculated imposing the consistency condition for the bounding surface. The flow rule can be left open to any definition. The extreme flexibility offerred through direct specification of plastic modulus and flow rule come at the expense of having to prescribe functions describing their variations. These functions form the most important links in the entire formulation. For materials like sand, the currently available laboratory stress-strain data are inadequate to attempt to derive functions describing the variations of the plastic modulus and flow rule for general loading conditions. 2.3.2.4. Discussion Provided that yield and potential functions can be denned for a material, its mathe-matical representation in the context of theory of plasticity is a possibility. Laboratory observations on sand behaviour do not indicate the possibility of straight forward and simple application of theory of plasticity to describe sand behaviour. The uniqueness of the yield and potential functions derived for sand in literature, are not general enough to take account of the anisotropic, stress level and path dependent behaviour of sand. Refinements required to the theory of plasticity to predict observed stress-strain response of sand could be very tedious and involve more extensive laboratory testing programs than currently exist. Nevertheless, theory of plasticity is seen as the most preferred theoretical framework among researchers at present time, for mathematical modelling of stress-strain response of sand. 2.3.3 Particulate, Discrete and Frictional Models Because sands are particulate, it is most appropriate that stress-strain relations be Chapter 2: 42 formulated in a particulate mechanics framework. Mathemat ica l representation of sand be-haviour considering its particulate, discrete and frictional features can be traced back to late fifties (Newland & Al le ly( l959), Rowe(1962) and Horne(1964)). Microscopic investigations on discrete assemblies of particles offer a wealth of qualitative and quantitative information on the possible mechanisms of deformation. Not many detailed microscopic investigations of discrete assemblies of particles have been reported in the literature. A brief summary of some of the important information observed from microscopic investigations on particulate, discrete and frictional systems are presented in the following sections. 2.3.3.1 Stress Dilatancy Relations The stress dilatancy theory proposed by Rowe (1962, 1971) can be viewed as a remark-able effort to explain the shear deformation behaviour of sand. Subsequent to Rowe's work, a number of researchers have published stress dilatancy relations following different ap-proaches including Matsuoka (1974b), Oda(1972a) and Nemat-Nasser(1980). A noticeable difference between Rowe's theory and other theories mentioned above is that Rowe's theory is independent of spatial distribution of interparticle contacts. The basis of Rowe's theory is that sliding occurs on certain favourably oriented contact planes, the orientation being such as to minimize the rate of dissipation of energy in sliding friction between particles with respect to the energy supplied (Home (1964)). Stress dilatancy relations proposed by Rowe(197l), Matsuoka(1974b), Oda(1972a) and Nemat-Nasser(1980) are summarized in Append ix -A. Rowe's and Matsuoka's stress dila-tancy relations have been extensively used in stress-strain modelling of sand. On the other hand, the author is unaware of Oda's and Nemat-Nasser's equations being used in stress-strain models for sand. The capabilities of stress dilatancy relations to model important aspects of sand behaviour can be evaluated as shown in Table 2.1. It can be seen that none Chapter 2: 43 Table 2.1 Comparison of Stress Di latancy Relations Originating from Part iculate Mechanics Considerations. [1] [2] [3] [4] Influence of stress level N ? ? ? Influence of density of packing Y Y Y Y Influence of stress path N N N N Influence of intermediate principal stress N N N N Influence of material anisotropy N Y Y Y Appl icabi l i ty to small strains N N N N Appl icabi l i ty to large strains Y Y Y Y Appl icabi l ity to unloading N N Y Y [1] Rowe(1971) 12] Oda(1972a) [3] Matsuoka( 1974 b) (4! Nemat-Nasser(1980) Y Yes N No Chapter 2: 44 of the stress dilatancy relations that originated from particulate mechanics considerations are capable of capturing al l eight features outlined in Table 2.1. Stress-strain measurements obtained from triaxial testing of sand indicate that the strain increment ratio is dependent on the stress increment ratio, especially at low stress ratios (Negussey (1984)). Such ob-servations l imit the applicability of the stress dilatancy equations, to stress states near failure. 2.3.3.2 Microscopic Observations on the Behaviour of Circular Particles Apart from the stress dilatancy equations, stress-strain relations for sand originating from particulate mechanics considerations have not been developed in proportion to those originating from continuum mechanics considerations. Nevertheless, a few detailed micro-scopic studies on particulate, discrete and frictional systems have been reported in the literature with the aim of understanding the deformation mechanisms dominant in partic-ulate assemblies (Oda &; Konishi(1974), Matsuoka (1974a), Bathurst (1985)). The microscopic investigations of fr ictional assemblies can be divided into two catagories; physical experiments and numerical experiments. Experiments carried out by Oda &; Kon -ishi and Matsuoka fall into the first category whereas Bathurst ' s experiments fall into the second category. The physical experiments have been performed on circular discs comprised of a photo-elastic material. After application of each load increment photographs showing the photoelastic isochromatics have been taken to analyze the contact force development and the changing spatial distr ibution of contacts. The reported numerical experiments are also l imited to simulation of a two dimensional system of circular disc shaped particles. These simulations provide an excellent means of investigating the influence of variables such as interparticle friction, material stiffness, particle shape, size etc. on the behaviour of an assembly of particles. Chapter 2: 45 2.3.3.2.1. Interparticle Contact Forces In frictional assemblies, the ratio of tangent and normal contact forces can vary between 0 and a l imit ing value fi*. The obliquity of the contact force with respect to the direction normal to the contact plane, is a measure of the ratio between developed tangent and normal contact forces. The frequency distribution of the obliquity of the developed contact forces as observed from physical experiments of Oda &: Konishi( l974), are shown in Figure 2.13. These distributions correspond to three different stages of loading; L-2, L-9 and L-12 which are shown in Figure 2.14. The circular discs used in the study had a l imit ing friction angle of 22°. From Figure 2.13 it can be seen that the obliquity of contact forces reached the l imit ing value of 22° only in a few contacts of the assembly. The majority of contacts developed an obliquity that is far less than the l imit ing value, the trend of which did not change with the magnitude of shear load. Yet, the corresponding stress-strain and volume change response, as shown in Figure 2.14, was typical of frictional systems. This is an important observation since it is intutively expected that with increasing shear loads, more number of contacts would develop l imit ing friction. Investigating the behaviour of a system of equal size circular disc shaped particles, Bathurst(1985) arrived at similar conclusions as above on the frequency distr ibution of obliquity of contact forces. His results were obtained from numerical s imulation studies as opposed to the physical experiments of Oda & Konishi, from which the magnitude as well as the spatial distribution of contact forces and contact normals at any given stage of loading could be obtained. The distributions of normal and tangent contact forces that correspond to a stress state along the pure shear loading path, as reported by Bathurst, are shown in Figures 2.15. The block like curves are the actual data and the smooth curves are approximations to the data ( Further reference to this figure wi l l be made in Chapter-4). Chapter 2: L-2 L-9 L-12 150 >-o UJ 110 a 5 o £ 150 110 50 t h i = , 150 contact force u no 50 obliquity particle -20-10 0 10 20 -20-10 0 10 20 -20-10 0 10 20 OBL IQU ITY (DEGREES ) ig.(2.13) Frequency Distribution of Obliquity of Contact Forces in Two Dimensional As-semblies of Circular Discs (after Oda & Konishi(1974)) Chapter 2: Al Fig.(2.14) Variations of Shear Force, Shear Strain and Volume Change Response of Assem-blies of Circular Discs (after Oda & Konishi(1974)) Chapter 2: 48 Fig.(2.15) Observed Distributions of Normal and Tangent Contact Forces (after Bathurst(1985)) Chapter 2: 49 It is important to note that the magnitude of normal contact force is a max imum in the direction of the major principal stress direction which is at 45° to the reference axes. The tangent contact force distribution has its max imum values at ± 4 5 ° to the major pr inc ipal stress direction. The particular shape of the tangent contact force distr ibution shown in Figure 2.15, indicate that most contacts of the system do not develop large tangent forces and that load is carried mainly as normal contact forces. The obliquity of the contact forces or the developed friction at contacts, which is obtained by div iding the tangent force by the normal force, is shown in Figure 2.16. Results shown in Figure 2.16 correspond to two different stages of loading as shown in the insert. It is seen that for both stages of loading, the developed maximum obliquity is contrained to contacts wi th planes inclined at approximately ±45° to the major principal stress direction. For such distributions, the frequency distribution of developed obliquity of contact forces would be very similar to the results shown previously in Figure 2.13. When the major pr inc ipal stress direction rotates, the normal and tangent contact forces distributions have been shown to rotate accordingly (Bathurst(l985)). 2.3.3.2.2. Normals to Contact Planes In discrete assemblies, the spatial distribution of contacts undergo changes as a result of deformation. During deformation, contacts are both created and destroyed, the rates being different in different directions of the system. Figure 2.17 illustrates the spatial distr ibution of normals to the contacts observed at different stages of simple shear loading of dense and loose assemblies of circular disc shaped particles as reported by O d a <Si Konishi(1974). The corresponding states of loading are shown in Figure 2.14. It can be seen from Figure 2.17 that during simple shear loading, as the major pr incipal stress direction is rotated by 45°, the frequency distribution of contact normals also rotated Chapter 2: 50 Fig.(2.16) Observed Distributions of Developed Rat io Between Tangent and Norma l Contact Forces a) at Peak Shear b) at Residual Shear ( after Bathurst(1985)) Fig.(2.17) The Distribution of Contact Normals Observed at Various Stages of Loading of Assemblies of Circular Discs a) Dense Assembly b) Loose Assembly (after Oda & Konishi(1974)) Chapter 2: 52 in the same direction being highest along a direction making 45° with the z axis. The contact normals tend to concentrate in the direction of major principal stress. Upon application of shear loads, interparticle contacts of the assembly are continuously constructed and destructed at a rate depending on the direction and stage of loading. From Figure 2.17, it can be seen that the contact normal distribution indicates principal directions that are orthogonal to each other and the distribution is symmetric about the principal directions. The tendency of the contact normals to concentrate in the direction of major principal stress is considered to be governed by the mobilized stress ratio (Oda &; Konishi(1974)). In discrete assemblies, the interparticle contact planes change so that stable configura-tions are formed to carry the applied contact forces. The inherent and induced anisotropy, described in section 2.2.3, are a direct consequence of the anisotropic distribution of inter-particle contacts in a given assembly. Micro-mechanics investigators view the process of deformation of a discrete system of particles as the outcome of two competing processes one causing order and the other causing disorder. Processes that bring order to a system achieve it through development of chains of particles which carry higher than average contact forces aligning themselves in the direction of major pr incipal stress (see Figure 2.18). Disorder is brought about via expansion of the assembly causing collapse of these chains of particles. The entire process of deformation is viewed as a dynamic one. Tangent contact forces at contacts do not accumulate during deformation to reach l imit ing values except at a few contacts. Tangent contact forces are considered to be essential for all internal activity. However, for the majority of contacts only a nominal ratio between tangent to normal contact forces is sufficient even at failure. Chapter 2: 53 Fig.(2.18) Dominant Chains of Particles Observed During Loading of Assemblies of Circular Discs (after Oda fc Konishi(l974)) Chapter 2: 54 2.3.3.3. Discussion Microscopic investigations of particulate assemblies have been l imited to circular disc shaped particles in two dimensions. Both physical and numerical experimental investiga-tions indicate that loads are primarily carried as normal contact forces. Tangent contact forces, although essential, need not be developed to their l imit ing values at all contacts. The few contacts that develop l imiting friction are aligned in directions that are at ap-proximately ± 4 5 ° to the major principal stress direction. In the absence of crushing of particles and with a relatively few number of particles developing l imit ing tangent forces, rearrangement of particles resulting from sliding of a few suitably oriented contacts followed by roll ing and rigid body rotations and displacements of a large number of particles is seen as the major mechanism causing non-recoverable strains. Exist ing stress-strain models are incapable of accounting for strains resulting from rolling of particles in an explicit manner. 2.3.4 Summary Based on laboratory observations, stress-strain response of sand is seen to be extremely complicated being dependent on the stress level, stress path, density of packing and material anisotropy. A number of different stress-strain models have been proposed in the literature based on theories of elasticity and plasticity and these have not been entirely satisfactory in capturing many aspects of sand behaviour. A n alternate approach is to incorporate the key features of particulate, discrete and frictional materials into the theoretical formulations and derive stress-strain relations that are meaningful and general enough to capature most if not al l observed behaviour. C H A P T E R 3 T H E O R E T I C A L F O R M U L A T I O N S F O R B O N D E D S Y S T E M S 3.1 Introduction Particulate materials form discontinuous media. Stress at a point denned for a con-t inuum is no longer valid for a discontinuous medium. Instead, volume averaged stress quantities must be defined. In particulate assemblies, the applied boundary loads are car-ried by a finite set of interparticle contact forces. Provided the magnitude and obliquity of all contact forces are known, the boundary tractions can be computed. However, because of the statically indeterminate nature of particle assemblies, the contact forces and their obliq-uities cannot easily be calculated from boundary tractions. For theoretical convenience, the stress-strain relations have therefore been derived in terms of average contact forces on con-tacts of similar orientation. Such an averaging process of contact forces is desirable since it provides a method for the spatial variation of contact normals to be introduced into the formulations. This chapter presents the stress-strain formulations derived for a system of equal size spherical particles that are bonded at their contacts. A bonded contact can have a normal stiffness and a tangent stiffness that are independent of each other. In particle assemblies, the spatial arrangement of interparticle contacts through which the load transmission takes 55 Chapter S : 56 place, has a strong influence on the stress-strain response of the material. In bonded assem-blies, the spatial variation of contacts remain fixed during the loading process. For clarity of presentation, the theoretical developments have been divided into four main sections as shown below. a) Spatial arrangement of a system of particles - section 3.2 Here, the mathematical description of spatial arrangement of a system of particles is presented. The concepts of contact normal and contact branch length distribution functions and their mathematical forms are discussed. b) The phenomenological stress tensor - section 3.3 Under this heading, the mathematical formulations involved in deriving the phe-nomenological stress tensor in terms of the interparticle contact forces are presented. The formulations have been divided into two stages; (i) transfer of boundary tractions as boundary contact forces onto particles in the boundary, and (ii) derive relationships between boundary contact forces and internal contact forces considering force and moment equi l ibr ium of the entire assembly. c) Relationship between contact forces and strains - section 3.4 Min imiz ing the complementary strain energy stored at all contacts with respect to the average contact forces, the compatible strain tensor is derived in this section. In doing so, the contact forces are expressed in terms of the strain tensor. d) Assemble stages a),b) and c) to arrive at stress-strain relations for a bonded system of particles - section 3.5 Most of the theoretical formulations and the underlying concepts, presented herein, have been extracted from Rothenburg(l980). They are presented herein for the sake of Chapter 8 : 57 completeness only. Some of the formulations and their interpretations have been simplified to suit the objectives of this dissertation. Expl ic i t mention of such simplifications wi l l appear at relevant locations in the text. The recapitulated formulations of Rothenburg(1980) are contained exclusively in this chapter. 3.2 Spatial Arrangement of a System of Particles Consider the three particle system shown in Figure 3.1. For simplicity only two d i -mensions are considered. In Figure 3.1, A and B represent contact points. Each physical contact A or B is associated with two geometric contacts; one belonging to each particle forming the particular contact. For simplicity, a contact is represented by a point. Asso-ciated with the contacts A and B are the contact planes PA and ps drawn tangential to the two surfaces in contact. The contact normals and %B for particle 2 are drawn at the contact points in an outward direction. The distance between the geometric center of a particle to a contact point of the same particle is called a contact branch length (i.e. G2A, G2B). The orientations of contact branch lengths are different from those of contact normals. In a system of particles, the contact normals and branch lengths are distributed in different directions and can be classified into groups with respect to their orientations. Accordingly, the spatial variations of the frequency of contact normals and average contact branch lengths can be plotted in two dimensions. The resulting contact normal and contact branch length distributions have been extensively referred to in micro-mechanics investiga-tions of particle assemblies. When the assembly consists exclusively of spherical particles of equal diameter, the resulting contact branch length distribution is isotropic at al l times. Yet, the contact normal distributions in these systems can be anisotropic. Consider a two dimensional system of particles consisting of M geometrical contacts. The total number of contact normals is therefore M whereas the total number of physical Chapter S : 58 QjA contact b ranch length A ,B physical contacts n^ contact no rma l vector contact plane Fig.(3.1) A Three Particle System in Two-Dimensions Chapter 3 : 59 contacts is M/2. The frequency distribution of contact normals, for G orientation groups, is shown in Figure 3.2. AMg denotes the total number of contact normals with orientations between 9g and 9g + A9 where A9 = 2n/G. Accordingly a contact normal distr ibution function S(9g), can be defined such that; ^ = S(9g)A9 (3.1) S(0g), in equation 3.1 is generally a discontinuous function. When the number of particles in the system increases, so does the number of geometric contacts. The result is that in such systems, the contact normals are oriented in more number of directions. When the number of particles is high, A9 can be reduced thereby increasing the number of orientation groups G. Ideally, when there are infinite number of particles w i th contact normals oriented in all different directions, AO can approach zero. Systems of particles in which A9 —> 0 as the volume of system V —> oo are hereafter referred to as large systems. For large systems, S(9g) should approach a continuous function. The existence of a continuous function for S(9g) is as abstract as the existence of a continuum. For very large systems, S{9) = l im (3.2) V ' A f l ^ O , K - o o MAO v ; Each physical contact is associated with two contact normals which are in opposite direc-tions. Therefore, S(9) must satisfy the constraint; S{9 + TT) = S{9) ; for all 9 (3.3) In addition to equation 3.3, by definition 5(0) must satisfy the following constraints; / S(8)t Jo »2x )d0 = 1 (3.4) S{9) > 0 ; for all 9 (3.5) Chapter 3 : 60 Chapter 8 : 61 From physical experiments, it was shown in section 2.3.3.2.2 that the contact normal distri-bution has orthogonal directions in which the frequency of contact normals are maximum and minimum. Hereafter, these directions are referred to as pr incipal directions of contact normal distributions or principal directions of anisotropy. The principal directions of the contact normal distribution are approximately the same as the directions of the principal stresses. Based on laboratory experimental evidence on systems of circular disc shaped particles, S(9) is symmetric under reflections about the principal directions of anisotropy. Let #o denote the major principal direction of contact normal distr ibution with respect to a given coordinate system. Therefore, for symmetry of S(6), S{90 + a) = S(60 - a) (3.6) in which a denotes any arbitrary angle measured w i th respect to the principal directions of contact normal distribution. For frame indifference of S(9), S{9) = S(9 - 90) (3.7) 9 and 9Q in equation 3.7 can be measured with respect to any reference frame. Equation 3.3 indicates that S(9) is a periodic function with period n. Therefore, S{9) can be expanded as an infinite Fourier series given by; C °° s0) = -7T + [ C 2 n cos 2n{9 - 9Q) + D2n sin 2n{$ - 90)] (3.8) 2 n=l in which Co, Cm and Dm are Fourier coefficients. For equation 3.8 to satisfy equation 3.6, sin 2n{9 — 9Q) terms must vanish. Therefore, C °° S $) = Y + E C 2 » cos 2n{9 - Oo) (3.9) n=l Any combination of Co, C2„(n = 1,2, ...oo) selected to satisfy equations 3.4 and 3.5 can Chapter S : 62 describe the contact normal distribution function S(9). It has been shown that the first two terms of the Fourier series given by equation 3.9 can be used to approximate the contact normal distributions observed in particulate systems (Rothenburg(l980), Bathurst(1985)). Truncating the Fourier series for re > 2 and normalizing it, S(9) can be represented in a form given by, S(9) = - ! - [ l + a2 cos 2(9 - 90) + a4 cos 4(9 - 60)} (3.10) Equation 3.10 satisfies equations 3.3, 3.4, 3.6 and 3.7. A t this stage, it is important to develop some familiarity with the anisotropy coefficients a 2 and a± and the types of distr i-butions that can be described through various combinations of a 2 and a±. First, consider the case when a\ = 0 wi th a 2 7^  0. For simplicity assume that 9o = 0. In that case, the pr incipal directions of contact normal distribution coincide with the reference coordinate axes as shown in Figure 3.3a. Note that the maximum value of S(9) = 1 + a 2 whereas its m in imum value is 1 — a 2 . Therefore, the quantity a 2 represents the average of the difference between max imum and min imum values of S(9). On the other hand, when o 2 = 0, and #o = 0, the max imum and min imum values of S(6) occur at ± 4 5 ° to the coordinate axes, the general shape of which is shown in Figure 3.3b. When both a 2 ^ 0 and 04 ^ 0, a blend of distributions shown in Figures 3.3a and 3.3b can be obtained which can be used to describe the contact normal distributions shown earlier in Figure 2.17. The two dimensional contact normal distribution function S(9) denned by equation 3.10 can be extended to three dimensions. A given direction in three dimensions fi, can be fully denned by two independent angles (<p,ip) measured from a reference configuration; making use of spherical coordinates. However, the extension of the particular form of S(9) in equation 3.10, to three dimensions describing both 9 and 9Q in terms of two independent angles is cumbersome. It w i l l be shown in Chapter-4 that equation 3.10 can be expressed in a tensorial format which offers greater flexibility in its two and three dimensional Chapter 8 : 64 representations. The kinematic and stress boundary conditions control the development of interparticle contact forces. These contact forces, integrated over the entire volume, must satisfy force and moment equi l ibr ium conditions. The relationship between the boundary tractions and the interparticle contact forces are examined in the next section. A l l formulations presented in the forthcoming section have been derived for three dimensional systems. Analogous to equation 3.2, the contact normal distribution function in three dimensions S(Q), is defined as 5(n) = l im (3.11) In equation 3.11, A n is the elemental solid angle of a sphere with unit radius as shown in Figure 3.4. In spherical coordinates, A Q — sin tp Atp A<f>. 3.3 The Phenomenological Stress Tensor for Particulate Systems In reality, a contact is associated with a finite area. A contact with finite area is capable of transferring moments even when the contacts are not bonded. The stress-strain behaviour of a system of particles consisting of contacts with and without the capability to transfer moments can be very different. For simplicity, the theoretical formulations presented in the following sections assume that the contacts are incapable of transferring moments. In what follows, the phenomenological stress tensor for granular assemblies is derived considering the equi l ibr ium of external loads on the boundary of an assembly of particles which corresponds to a uniform state of stress in continuum mechanics. Such loads corre-spond to tractions on the boundary of the form p,- = n?, where o~f- is a certain second order tensor and n? is the unit normal vector to the boundary. It is argued that with in a given large assembly of particles, a sufficiently small volume enclosing a sufficiently large number of particles can be found where the state of stress is uniform. The volume Chapter S : 65 3 4 ni = cos <f> sin ip ri2 = sin <f> sin tp nz — cos ip. Fig.(3.4) Definition of Angles in Three Dimensions Chapter 8 : 66 under consideration can be selected away from the applied boundary loads so that the gross response of the system can be studied independent of the selected form of boundary trac-tions. The relationship between the interparticle contact forces and the pkenomenological stress tensor can then be derived by surrounding the selected volume with a continuous shell viewed as the boundary of the assembly which is subjected to the selected boundary tractions. Consider the system of spherical particles contained within the boundary B shown in Figure 3.5. Let there be external distributed boundary tractions p(Q) applied at each point R_ along the boundary B. Appl ied to a continuum, boundary tractions p(R_) result in a uniform state of stress of- along the boundary given by, In particulate systems, of- is related to the contact forces within the system. Once the relationship between of- and the contact forces of particles within the whole system is established, the phenomenological stress tensor is introduced by definition for any arbitrary volume of a particulate system. The external distributed tractions p applied over the entire boundary B, are transferred to the boundary particles as boundary forces by subdividing the complete boundary into regions adjacent to each particle that is in contact with the shell boundary. Following this procedure, for a particle /?, the boundary forces can be computed from, Pi{R)=<rfi ; i = 1,2,3 (3.12) (3.13) in which S@ is the area of the boundary B associated with particle /?. The boundary region S@ is selected such that the magnitude of \R — jf\ w i l l not be greater than several particle diameters, r? is the position vector of the point of tangency between the boundary Chapter Fig.(3.5) Load Transfer at Boundary Particles Chapter S : 68 region S@ and the boundary of the particle f3. Note that rf is a constant for the selected boundary region and denotes the point of action of the boundary contact force f . Consider the evaluation of the following integral; I = j RjPi{R)dS ; t,j = 1,2,3 (3.14) In equation 3.14, the variable vector II denotes the position of an arbitrary point on the selected boundary region SP. From the mean-value theorem, equation 3.14 can be rewritten as I=R? J pi{R) dS ; * , j = l ,2,3 (3.15) where, R^ is a selected point on S@ as shown in Figure 3.6. When the origin of the coordinate system is selected far away from the boundary and when \R_ — if\ is not greater than several particle diameters, X-^i (3-16) Combining equations 3.13, 3.14, 3.15 and 3.16, ffr*~J Rj Pi(R) dS ; i,j = 1,2,3 (3.17) It can further be argued that if a judicious selection of the boundary regions S@ is followed, then R? may be selected exactly equal to rather than approximately. Summing for the entire boundary B, equations 3.17 become £ $ 'f = / R i d s ; »'> = l ' 2 ' z (3-18) Combining equations 3.12 and 3.18, E^f »i = / °l nk(B) Rj dS ; i,j = 1,2,3 (3.19) peB J s Chapter 8 : 69 The surface integral in eqation 3.19 can be transformed into a volume integral using the Green-Gauss theorem. Accordingly, J 0% nkB(R) Rj dS = j o% JJ± dV = a% V ; i,j = 1,2,3 (3.20) Combining equations 3.19 and 3.20, the second order tensor aB- can be expressed in terms of boundary contact forces and their positions such that,. 4 = yE>fJf : i,3 = W (3-21) af-, described above, is a volume dependent quantity. The reason for introducing the position vector R in equations 3.17 was to obtain the above volume additive form for aB. In equation 3.21, the position vector r® is defined with respect to a fixed but arbitrary coordinate system. It w i l l be shown later that for a system consisting of equal size particles, aB is independent of the location of the reference coordinate system. For a set of boundary tractions that satisfy moment equilibrium, the resulting boundary contact forces must also satisfy moment equil ibrium. When moment equil ibrium exists, ^ [ / x f ] = 0 (3.22) In tensorial form, equation 3.22 can be written as, E Jf ^ = £ ^ > f ; «^ - = w (3-23) peB p&B Imposing the constraint from moment equi l ibr ium given by equation 3.23, equation 3.21 can be rewritten as, * ? = 2 7 £ [ / f ' f + J ? ' ? ] ; « ' > = W (3-24) peB Imposition of moment equi l ibr ium conditions at this stage marks a deviation from the Chapter 8 : 70 formulations of Rothenburg(l980). The result, as can be seen, leads to a symmetric tensor f o r * J . So far, the forces resulting from boundary tractions on boundary particles have been considered. Next, the process of boundary force transmission to all contacts of the assembly is examined. The process should ultimately lead to an alternate expression for af- in terms of contact forces ft for the entire volume under consideration. The alternate expression is obtained by making use of force equil ibrium conditions at all particle contacts. Let the force acting on particle k at contact m be defined as fk,m. For force equi l ibr ium at all contacts of a particle, £ y f ' m = 0 ; i = 1,2,3 (3.25) m=l where, is the number of contacts of particle k. Equations 3.25 do not contain the position vector r. To obtain a form similar to the right hand side of equations 3.24, the 1TH component of force of particle A; is mult ipl ied by the j " 1 component of the position coordinate of the center of the particle. The above operation transforms equations 3.25 into to a set of equations given by, N ™k Y,H£mrJ=° i «,J' = 1,2,3 (3.26) k=lnv=l in which the outer summation is for the entire set of particles, the number of particles being denoted by N. The position coordinates and r* for those particles adjacent to the boundary are related through the radius of the spheres and the normal vector nP at the boundary contact as shown in Figure 3.6. Accordingly, for a system containing equal size spherical particles, r0 and are related by r* = ^ - ^ n ^ (3.27) Chapter 8 : 71 Fig.(3.6) lations Definition of Various Vector Components Associated w i t h the Theoretical Formu-Chapter S : 72 Equation 3.27 is applicable for a system of equal size spherical particles only. However, if desired, approximate theoretical formulations can be derived for a system of particles containing spherical particles of a l imited range of diameters. The approximate theoretical formulations can be obtained by replacing d$ by an average diameter d®. Mu l t ip ly ing equations 3.27 by ff results in, f f r ^ j f ^ - ^ f f n ^ ; u = 1,2,3 (3.28) which wi l l be substituted later into equations 3.26 for the terms corresponding to boundary particles. In what follows, a differentiation is made between boundary contacts and internal contacts. Boundary contact forces are in equi l ibr ium with boundary tractions whereas internal contact forces are in equil ibrium with internal forces between particles. Each internal contact that is in force equi l ibr ium satisfies, f = - f (3.29) where, / w denotes the contact force acting on particle k from particle 1. Equations 3.26 are for both boundary and internal contacts of the assembly. Each internal contact contributes to two terms that can be related by equation 3.29. Such terms can be grouped as, W + £ $ =-\\# - r§) + £ {i* - ffr] ; t , i = 1,2,3 (3.30) For equal size spherical systems, = * B * i J = 1,2,3 (3.31a) r j - t j = eib n} ; j = l ,2,3 (3.316) in which r* and rf are the position vectors of the centers of the particles k and /, respectively. Chapter 3 : 73 nk and nl are the unit normal vectors drawn at the internal contact. Combining equations 3.31 and 3.30, [f? tf + J? % = - y W fi + ft n\] ; i,j = 1,2,3 (3.32) In effect, equations 3.28 and 3.32 provide alternate expressions for the terms associated with boundary and internal contacts in equations 3.26. Substituting from equations 3.28 and 3.32 into equations 3.26, N mk E E f ^ E t i f f y f ^ Y £ « = o ; i,/ = 1,2,3 (3.33) fc=l m=l 0£B kev-B Combin ing the ff vP- terms from the boundary contacts and jf re* terms from the internal contacts, equation 3.33 reduce to, £ / ? ^ = y £ ^ i U = 1,2,3 (3.34) peB ce v ft in equation 3.34 represents the contact forces of the entire volume including boundary contact forces. Substituting equation 3.34 into equation 3.24, the alternate expression for af- can be obtained. Accordingly, ff.f=^E^rai+^<] i M = 1,2,3 (3.35) cev Equation 3.35 has been derived considering the static equil ibrium of contact forces developed in a system of particles when subjected to boundary tractions specified by pi(R). Considering the static equil ibrium between particles, equation 3.35 has also been reported by Drescher &c de Josselin de Jong (1972), Rothenburg &t Selvadurai (1981) and Cunda l l & Strack (1983). Alternatively, the same relationship has been derived using the principle of v i r tua l work (Christoffersen et al (1981)). Chapter 3 : 74 In equation 3.35, aB is independent of the position vector rf. Moreover, the tensor o~B, which is related to the boundary tractions, is now expressed in terms of contact forces of the entire assembly. As before, o~B is volume dependent and therefore wi l l fluctuate from volume to volume. Consider a large assembly of particles and select an arbitrary region of volume v with in the large assembly. The phenomenological stress tensor Oy(v), for the selected volume v is defined in the following manner; M») = ^ + ! U = 1,2,3 (3.36) In equation 3.36, the summation is carried out only for the contacts with in the selected arbitrary volume. The interparticle contact forces f and the associated contact normals nc which consti-tute the phenomenological stress tensor <7,;(t>), are distributed in many different directions in a given system of particles. In section 3.2, the contact normal distr ibution function was introduced. In what follows, the contact forces on all contacts that have a given orientation of normals are averaged. The resulting equations for <f(/(w) are therefore described in terms of average contact forces and also contain the contact normal distr ibution function 5 ,(n s) (refer to sections 3.2 and 4.3.2 for details) Following a similar approach as in section 3.2, in three dimensions, the contact normals are divided into G groups with respect to their orientation. In the g^ group with orientation fij, let there be AMg contacts. Averaging al l contact forces with respect to AMg contacts, the average contact force vector f{flg) is obtained and given by, 7.W = T I T " £ >? ; t = l , 2 , 8 (3.37) 9 cecig Evaluating equations 3.36 by dividing by AMg and summing over G groups, the average phenomenological stress tensor can be obtained in terms of average contact forces. Accord-Chapter S : 75 ingly, G G s=i 9 ceng g=i  9 ceng (3.38) Substituting from equations 3.1 and 3.37 into equations 3.38, = ^ £ [ n i ( n ^ ) + n,-(n,)] M s(n„) A n ; = 1,2,3 (3.39) V 9=1 For very large systems, v —> oo and A f i —* 0. For such systems, assuming that 7,(H3), n,(n s) and ^ ( f i j ) can be described by continuous functions, equations 3.39 can be written in their integral forms. Accordingly, = i 7 donvJ [/..(n) i»,-(n) +/,(fi) n,(n)] s ( n ) rfn ; »,y = 1,2,3 (3.40) n in which, 7 = l im [^-1 A n — 0 w„ = l im [^1 v—»oo L 1/ J A n - > o In equations 3.40, 7 denotes the average number of contacts per particle of the entire assembly which w i l l be referred to as the average coordination number. nv can be related to the void ratio and particle diameter ao. Both 7 and nv describe scalar quantities that are characteristic to particulate assemblies. The average contact force vector /(O) can be resolved into a normal component / n ( n ) in the direction of the contact normal and a tangent component 7t(H) perpendicular to it. Let the associated unit direction vectors be denoted by n,(f2) and i,(0), respectively. In three dimensional systems, i,(H) can be in any direction perpendicular to n,(fi). Resolving 7(f2) into normal and tangent components, 7,(n) = / n ( n ) n,(n) + 74(n) i,(n) ; . ' = 1 , 2 , 3 (3.4ia) Chapter 3 : 76 7,-(n) =/„(n) n,-(n) +ft(n) t,-(n) ; y = 1,2,3 (3.41&) Combining equations 3.40 and 3.41, an = \invdo J[7„(n) «,-(n) nJ(n)+^ 7t(n)(i1(n) «,-(n)+t,-(n) n,(n))] s(n) dn ; t,y = 1,2,3 n (3.42) Equations 3.42 differ from those of Rothenburg(l980). His formulations do not lead to a symmetrical phenomenological stress tensor. The presence of the terms t,(Q)nj(fi) and £,-(£})n,-(n) which makes in equation 3.42 symmetrical was the result of imposing mo-ment equlibrium of boundary tractions through equation 3.22. Rothenburg(1980) enforces symmetry of the stress tensor after introducing the contact normal distribution function S(f i ) thereby introducing constraints on the anisotropy coefficients that constitute 5(0). Equations 3.42 do not include explicit information on contact properties. The extent to which 7n(^ ) a n d ftity c a n develop, is governed by the mechanical properties of the contacts. Idealizing the load deformation behaviour at each contact to be linear, the relationship between contact deformations and contact forces is established in the next section. 3.4 Relationship Between Contact Forces and Strains A t a microscopic level, the interparticle contact force deformation interactions of real materials are complicated. In general, the contact force deformation relations are non-linear. Non-linear force deformation relations can result from the variation in contact area with the magnitude of contact force. When the contact area does not change with the magnitude of contact forces, the force displacement behaviour of a contact can be approximated by a linear relationship. For the bonded system of particles considered herein, each contact is assumed to have a normal stiffness k!n and a tangent stiffness k\ from which the normal and tangent (i.e. Chapter 8 : 77 shear) deformations at particle contacts are calculated. It is assumed that these stiffnesses remain constant during loading implying a linear contact force-displacement relationship. Moreover, the moment transfer capability of the contacts are assumed to be negligible. In addition to the normal and tangential deformations, bonded systems can undergo only small rotations at particle contacts. In the absence of rigid body rotations, the contribution from small rotations of particles to the overall rotation is insignificant. Even if the particle rotations are significant in comparison to the rigid body rotations, the cumulative effect of particle rotations on the strain tensor can be insignificant. This is because of the random orientations of particles leading to rotations in different directions. Therefore, the rotational deformations of particles are ignored altogether in the formulations. Let A / n and Alt denote the relative displacements in the normal and tangential direc-tions between the centers of two particles forming a given contact point. In effect, the center of each particle undergoes only half the displacements Aln and Alt, since the relative displacements between the two particle centers are due to identical deformations in both particles. Herein, it is assumed that the load-deformation behaviour of a given contact of a particle is independent of the other contacts of the same particle. Invoking the linear contact force-displacement relationship, the developed contact forces can be computed as, (3.43 a) $ = (3-436) where, ft and ft are the normal and tangent contact forces. The potential energy stored in the particles for each physical contact wj, can be expressed as, w}= j ft d(Aln) + j ft d(Alt) (3.44) 0 0 in which the in it ia l displacements of the contact points have been assumed to be zero. Chapter 8 : 78 Combining equations 3.43 and 3.44, •-* = *[tf + C] (345) where, kn = | k*n do and kt — \ k\ do. Note that A n and kt as denned here, no longer have spring stiffness units. The above notation has been adapted from Rothenburg(1980) for compatibil ity of theoretical formulations. For linear systems, the potential energy stored is numerically equal to the complemen-tary energy. Mak ing use of this condition, the complementary energy per unit volume wc, for all contacts of the assembly can be written as, Equation 3.46 contains contact forces of all contacts in the assembly. In section 3.3, an average stress tensor o^-, was derived in terms of contact forces averaged with respect to their contact orientations. Following a similar procedure, the complementary energy of the system per unit volume can be expressed in terms of average contact forces. Accordingly, equation 3.46 w i l l take the form, ^=^E[§-n + ir\ M* S(^) A H (3.47) in which, m ) = T ^ r E w ) 8 (3.48.) /?(n«) = 12 (ft? (3-484) 9 cecig For large systems when AJ1 —> 0 and v —»• oo, the complementary strain energy per unit Chapter S : 79 volume can be written as; w. 'c = r7  nvdo ( 3 4 9 ) where, n„ = l im f—1 A n — o A l l statically admissible contact forces that enter equation 3.49 must satisfy equations 3.42, where the average stress tensor is related to the average contact forces. For three dimensional systems, the orientation of tangent contact forces £,(0), are such that the force distribution gives rise to the most stable system. It can be argued that for a given spatial arrangement of contacts 5 (0 ) , a contact force distribution can always be found such that the complementary energy of the system is a minimum. Min imizat ion of complementary energy is now carried out. The variables involved are the contact forces fi(Q) which can be uniquely defined by 7n(fi), 7t(^ ) a n d U(Q). (Since the complementary energy and the potential energy of the external loads are in terms of 7n, ft and the minimization of complementary energy is carried out w i th respect to these components). Form the functional L such that, The tensor A,y denotes the strains associated with the boundary loads W^. fj,\ and H2 are Lagrangian multipliers required to incorporate the constraints on t and n. For any arbitrary 5 ( 0 ) , <p\ and <f>2 are formed from the constraints as shown below; L — wc - Xij <Jij - Hi (j>i - H2 <t>2 i,j = 1,2,3 (3.50) t t - l = Q (3.51a) t n = 0 (3.516) Chapter S : 80 Therefore, <t>i = ^ 7 nvdo j[tt-1] S(n) dCl = 0 (3.52a) n fa = ~1 nvdo j(t n) S(n) dQ = 0 (3.524) n The complementary energy per unit volume of the system, given by equation 3.49, is in terms of averages of squares of contact forces. To facilitate minimization of the functional L with respect to / n ( f i ) , 7t(^ ) a n d £,{f2), the complementary energy has to be expressed in terms of squares of averages of contact forces. The relationship between f^^O,) and depends on the form of distr ibution of f(U) on contacts of a given orientation. The determination of the exact distr ibution of contact forces on contacts of a given orientation is a problem of enormous complexity. Two approaches can be taken: simply a distribution can be assumed or certain postulates must be introduced. Rothenburg makes use of the postulates from the theory of information and arrives at a distribution function for contact forces. Information theory presents a rational method of assessing the amount of 'missing information ' associated with any distr ibution function. The 'missing information' is related to the possible number of combinations of contact forces that must be tried unti l the right combination is found; the one corresponding to the state of static equilibrium. Selection of the function with the largest amount of missing information assumes the least amount of possibly adverse results for an unsubstantiated guess. Let the density of a spectrum of contact forces for an infinite system be denoted by Pg for which case Pg(fn, ft) dfn dft denotes the probability of a given arbitrary contact of orientation 6 has force components between /„ and /„ + dfn and ft and ft + dft- For il lustration purposes, assume Pg to be of the exponential form given by, Pg = =X, exp [ -h - J ] (3.53) fn ft fn ft For the above distr ibution, it can be readily shown that /„ = 2/^ and ft = 2ff. (Note Chapter S : 81 that to arrive at these relations the variations in /„ and ft have to be assumed to vary between zero and oo, which is more applicable to frictional systems discussed in the next chapter). Alternately, making use of postulates from the theory of information together with the constraints imposed from external load and complementary energy (i.e. equations 3.40 and 3.49), Rothenburg arrives at a Gaussian distr ibution for Pg for both two and three dimensional systems of equal size circular particles. The interested reader should refer to the Chapter-VI of Rothenburg's thesis for more details on the application of the information theory to particulate systems. Mak ing use of the Gaussian distr ibution of contact forces, he demonstrates that, wc = \ i nvdo J [ i + 2-] S{Q) dQ = i 7 nvdo j [ | - - + S(Q) dV (3.54) n n in which, c is a non-dimensional parameter which is related to the dispersion of contact forces on contacts of a given direction. The main steps involved in arriving at equation 3.54, as reported by Rothenburg, are summarized in Append ix -B . Had the distr ibution given in equation 3.53 been assumed, f = 0.5 or alternately if all the contact forces are uniformly distributed c = 1. Substituting from equations 3.42, 3.51 and 3.54 into equation 3.50, the functional L can be rewritten as, a xi:i [7„(n) n,(Q) «i(n)+ § (t,-(fi) n , - ( n ) - M j ( n ) n,(n))] -m (tt-i)-n2 U «)] s(n) dn (3.55) Min imiz ing L with respect to /„, / ( and results in the following relations. d L -=- = o => / B (n) = c kn \{j m{n) ny(n) (3.56) Chapter 3 : 82 T=- = o =• ft(n) = ^ (A t > i,(n) n,-(n) + A,-,- *,-(n) n,-(n)) ; i,j = 1,2,3 (3.57) | £ = 0 =• 2/*i + M2 n,- + ^ - ( A , y ny(Q) + A,-,- ny(Q)) = 0 ; t = 1, 2,3 (3.58) The two Lagrangian multipliers n\ and / i 2 appear in equation 3.58 only. Mak ing use of the identities given in equations 3.51, separate expressions for \i\ and / i 2 can be obtained. Accordingly, tt=i=> 2m = - ^ ^ - ( A y n,-(n) t{(n) + A ;, n,-(n) t,-(n)) (3.59) fn = 0 => /z2 = - ^ ^ - ( A , y n,(Q) ny(n) + A,-,- n,(n) ny(Q)) (3.60) 2 The quantity denned by A,y CT,y, in equation 3.50, is a scalar. Consider all components that enter A,y ff,y. Accordingly, >3 ~ ^ 1 1 ^ 1 1 + A 2 2 <J22 + A33CT33 + ( A 1 2 + A 2 I)CT 12 + (A 13 + A 3 1 ) a 13 + (A 23 + A 3 2)<f 23 (3.61) The symmetry of the stress tensor <F,y has already been imposed in equation 3.60. For the functional L to be unique it is required that (A,y + Ay,) be unique. Symmetry of A,y therefore is not a requirement so long as (A,y + Ay,) is unique. The above interpretations of A,y are different from Rothenburg(1980). In what follows, a new quantity A*- is defined such that, 2A*. = (A,y + Ay,) (3.62) Note that, Kj = A* (3-63) Equations 3.56, 3.57, 3.58, 3.59 and 3.60 can now be expressed in terms of the new tensor A?-. The resulting equations are as follows; 7„(n) = f U ; «,(n) ny(£7) (3.64) Chapter 3 : 83 7t(n) = c h A;- *,(O) n,-(n) (3.65) 2/ii *,(o) + V2 n,-(n) + 7«(n) A;; ra,(o) = o (3.66) 2/*i = -/t(n) A*- t,(0) ray(n) (3.67) = -7«(n) A*- n,-(n) n,-(n) (3.68) Combining equations 3.66, 3.67 and 3.68 the following can be obtained; 7t(n) t,(n) = ^  [AJ n,(n) - ( A ; nr(o) ns{n)) w,(o)] (3.69) In section 3.3, the average stress tensor <f,y was derived in terms of 7„ and ft- From equations 3.42, an = ^ 7 ««do J [7„ n,- n,- + ^ {U n,- + t,- n,)] 5(H) dO (3.70) n Consider the scalar quantity given by | AyO^y. Substituting from equation 3.70, 1 1 / " - 7 -A,y J7,y = - 7 n,«fo j [A,y n,- ny /„ + (A,y «,• + A,y iy n.) ^ ] 5 ( 0 ) dO ; i,j = 1,2,3 (3.71) n Substituting from equations 3.56,3.57 and 3.58 into equation 3.71, \ A,y <f,y = i 7 n„oo 1 + £-] 5 ( 0 ) dO (3.72) n It can be seen that equation 3.72 is identical to the term wc in equation 3.50 which describes the complementary energy per unit volume of the system. For linear systems, the potential energy and complementary energy stored are equal in magnitude. Equations 3.64, 3.69 and 3.70 describe a set of stress-strain equations that are appl-icapable to a system of equal size spherical particles that are bonded at their contacts. The formulations capture the anisotropic distr ibution of contact normals that can exist in Chapter S : 84 particulate assemblies. For a system of particles with an isotropic distribution of contact normals, S(Q) reduces to a constant equal to ^ r . It wi l l now be shown that for a system of particles wi th an isotropic distribution of contact normals, the derived stress-strain relations can be interpreted as those for Hooke's elastic isotropic model described in section 2.3.1.1. 3.5 Stress-Strain Relations for a Bonded System of Particles with an Isotropic Contact Normal Distribution For the special case when the contact normal distribution is isotropic, the stress-strain relations described in equations 3.64,3.69 and 3.70 take a simple and familiar format; that of Hooke's elastic isotropic model. A n isotropic contact normal distribution in three dimen-sions is described by, S(Q) = — (3.73) Combining equations 3.64, 3.69, 3.70 and 3.73 and since A Q = simp Aip A<p, the three dimensional stress-strain relations for a bonded system of particles can be obtained as 2-ir ir Oij = ^7 nvdo J j' [f kn (A*a nr n3) n,- nj+ o o Aj 71; rtj + Ay, n/ n,- , f M— ^ (Ks nr n,) rii n,-}]«n tp d<f> ; i,j = 1,2,3 (3.74) Rearranging terms in equations 3.74 and defining r = equations 3.74 can be rewritten as, Oij = c K 7 nv do j j [(1 - r)(A*, nr n3) n, rij+ 0 0 A!i ni ray + \\ m ra; , r {_« }]sin ip d*P d4> ; i,j = 1,2,3 (3.75) it Stress-strain relations similar to those given by equations 3.75 have been reported by Bathurst &; Rothenburg (1988) for isotropic, spherical systems with linear contact inter-Chapter 8 : 85 actions. Integrating equations 3.75, the following stress-strain relations can be obtained; (3.76) where, in which °~ij — CjjM Ay C 1 1 1 1 fr + ( l _ r ) l 3 5 C l l 2 2 = ^> C l l 3 3 = ^> C 2 2 1 1 [(1~r)u [ 15 M C 2 2 2 2 = [ r + ( 1 _ r )  l 3 5 C2233 [(1~r)u [ 15 M C3311 [ ( 1 _ r ) U 1 15 J A C3322 f ( 1" r )u 1 15 l A ^3333 = f r + ( 1 - f ) [ 3 5 C 1 2 1 2 = C 2 i 2 i r r + ( 1 - r ) [ 3 15 Cl313 = C3131 = [ r + ( 1 ~ r )  [ 3 15 <?2323 = C3232 ,r ( 1 - r ) l 3 15 U A = (3.77) TT 4 ( l + e) and all other C^y i,j,k,l = 1,2,3 are zero. It can be shown that when shear and bulk modul i (i.e. G and B, respectively ) are defined as, and, B ( 7 kn(2 + 3 r ) IOTT dg (1 + e) " 3TT 0^(1 + e) (3.78 a) (3.786) Chapter 3 : 86 the stress-strain equations given by equations 3.75 are identical to those of Hooke's elastic isotropic model. In above equations, the Poisson's ratio v, is given by, -=!^ (3 79) For real materials, the likely range of r would be between 0 and 1. The corresponding range of Poisson's ratio computed from equation 3.79 is between 0.25 and 0.00. In view of the experimentally observed Poisson's ratios for materials such as concrete and cement (0.20 and 0.12 respectively), the values predicted from the theory are seen to be realistic. When the contact normal distribution S(Q) is anisotropic, the components of C,yy are much more complicated containing terms from the anisotropic distribution of 5 (0 ) . 3.6 Discussion In this chapter, stress-strain relations applicable to glued or bonded systems of spheri-cal particles that are of equal diameter, have been derived. W i t h the exception of imposing moment equi l ibr ium of boundary contact forces (i.e. given by equation 3.22) and the inter-pretation of the strain tensor Aty (i.e. given by equations 3.62 and 3.63), all formulations have been extracted from Rothenburg(1980). The theory assumes that the load deformation behaviour of contacts is linear and neglects deformations resulting from particle rotations. The formulations include mathematical functions describing the anisotropic distr ibution of contact normals that is common for particulate systems. When simplified to an isotropic system, the resulting stress-strain equations are identical to those of Hooke's elastic isotropic model. The author is unaware of any published attempts to extend the bonded formulations of Rothenburg to describe discrete and frictional systems of particles. Published work of Rothenburg, subsequent to his Ph.D thesis research work has mainly been focused on investi-Chapter 8 : 87 gating the patterns of contact forces and contact normal distributions that can be developed in systems of frictional particles that are circular in shape. Stress-strain relations derived incorporating conclusions from the above studies have not been reported to date. The main objective of this dissertation is to derive stress-strain relations for a frictional system of particles. The bonded theory presented in this chapter forms the theoretical framework within which the stress-strain relations for frictional systems can be developed. Such an attempt requires an understanding of the differences and similarities between bonded and frictional systems. The forthcoming chapter addresses the question of deriving stress-strain relations for a frictional system of particles. C H A P T E R 4 T H E O R E T I C A L F O R M U L A T I O N S FOR F R I C T I O N A L S Y S T E M S 4.1 Introduction Particulate, discrete and frictional materials like sand constitute a separate class of materials. A sand mass when subjected to boundary tractions, responds with boundary deformations that cannot be fully explained from the stress-strain relations derived for bonded systems. A t a particulate level, these deformations come from, (1) sliding of particles relative to each other, (2) rolling of particles over each other, (3) rigid body rotations and displacements of individual particles, (4) deformation of ind iv idual particles, and/or (5) crushing of particles. Quantitative assessment of the first three activities require information regarding internal stability at each contact between particles and the geometric compatibil ity requirements. Individual particle deformations and the extent of particle crushing depends on the magni-tude of contact forces, stiffness and strength of indiv idual particles. In a system of particles consisting of a significant number of unstable contacts, such as in very loose assemblies, the stress-strain response can be very complicated. In very loose assemblies, violation of 88 Chapter 4 : 89 stability requirements in a localized zone can lead to particle rearrangements of perhaps the entire system under consideration. Mathematical representation of stress-strain relations for sand, based on truly part icu-late considerations, is tedious and impractical. Therefore, a simple mathematical modelling process that is capable of approximating the stress-strain relations for a particulate, discrete and frictional material is proposed herein. The modelling process involves a collection of ideas on sand behaviour observed from laboratory studies and numerical simulation studies of particle assemblies. The stress-strain relations derived for a system of bonded particles, described in Chapter-3, have been used as the main theoretical framework in arriving at stress-strain relations for frictional materials. The theoretical formulations for fr ictional systems, presented in this chapter, are new and differ considerably f rom Rothenburg 's for-mulations for bonded systems. In order to appreciate the proposed stress-strain relations, it is important that the differences between bonded and frictional systems be understood first. 4.2 Bonded Systems versus Frictional Systems The major differences between bonded and frictional systems of particle assemblies are compared in Table 4.1. From the differences outlined in Table 4.1, the stress-strain response of frictional systems can be anticipated to be very different from the bonded systems. In particular, the freedom available for particle rearrangement during loading and the ratio of tangent and normal contact forces to vary between zero and a l imit ing value //* can be seen as two of the most cumbersome features to quantify in frictional systems. 4.3 Stress-Strain Response of Frictional Systems Stress at a point, as defined for a continuum, becomes meaningless for a particulate Chapter 4 : 90 Table 4.1 Bonded Systems versus Frictional Systems B O N D E D S Y S T E M S F R I C T I O N A L S Y S T E M S l ) Particles are bonded at their contacts. 1) Particles are not bonded at their contacts. Interparticle normal contact forces can be compressive or tensile. Zero normal contact forces do not separate particles. Interparticle normal contact forces are always compressive. When the interparticle normal contact force becomes zero, the contact is no longer maintained. 2) Tangent and normal contact forces are independently controlled by tangent and normal stiffness at the contacts. 2) The ratio between tangent and normal contact forces are related through a coefficient that varies between zero and a l imit ing value H'. 3) Spat ia l distr ibution of contacts is fixed. 3) Spatial distribution of contacts change as a result of loading such that most stable configurations are achieved. Therefore, system reacts in direct proportion to external excitation. Therefore, the manner in which the system reacts to external excitation is complicated. Chapter \ : 91 material which consists of both solids and voids. In Chapter-3, it was shown that the stress tensor for a particulate material can be described in terms of volume averaged contact forces on particles. The volume is selected to consist of a sufficiently large number of particles so that the average stress is uniform. Similarly, boundary strains for a particulate material correspond to the average deformations of the selected volume. In this context, the stress-strain relations derived herein for frictional system of particles correspond to the average stresses and strains over a selected volume and do not refer to those at a point as in continuum mechanics. Stress-strain relations proposed for frictional systems are presented herein. The theo-retical treatment presented is based on a collection of ideas derived from laboratory experi-ments on real sand and physical and numerical experiments on circular systems of particles. The formulations involve a series of simplifying assumptions which wi l l be outlined as the theory is presented. 4.3.1 Introduction As noted earlier in section 4.1, the stress-strain response of a fr ictional system of part i -cles is the outcome of a number of different processes occurring among indiv idual particles. The particle movements within a given assembly of particles are governed by, (i) internal stability requirements at particle contacts, and (ii) internal geometric compatibil ity requirements among particles. A l l particles within a given assembly must satisfy force and moment equl ibr ium con-ditions. Furthermore, at each contact the developed ratio of tangent and normal contact forces cannot exceed the coefficient of l imit ing friction of the material fi*. Particles wi th contacts subjected to the l imit ing tangent forces w i l l reach alternate stable configurations by sliding, followed by rolling producing non-recoverable deformations. Dur ing the process Chapter 4 : 92 of deformation of a system of particles, a given particle in the system undergoes varying magnitudes of developed contact forces. The changes in magnitude of contact forces could be rapid since the slightest movement of a particle could relieve or develop normal and tan-gential contact forces. Movement of particles are constrained by the internal stability and geometric compatibi l i ty requirements among particles. The shape and size distributions of particles wi th in a given assembly can influence the geometrically compatible arrangements of particles that can be developed within it. Development of a theoretical framework for describing the stress-strain response of fr ictional systems is carried out in three stages; (i) Developing a method of quantifying the contact forces resulting from boundary loads, (ii) Identifying stable and geometrically compatible particle arrangements that can develop for contact forces obtained in (i) above, and (iii) Relat ing the changes in stable and geometrically compatible particle arrangements, in (ii) above, to boundary deformations. It should be noted that for bonded systems of particles described in Chapter-3, stage (ii) mentioned above is not necessary since particle rearrangement within systems is impossi-ble. Therefore, the theoretical formulations simplify to a considerable extent. In bonded systems, the interparticle contact forces and deformations are directly related through the normal and tangent stiffnesses at the contacts. In frictional systems, the magnitude and direction of deformations are not well defined and because of the discrete and frictional nature, the contact forces and deformations cannot be easily related to each other. The forthcoming sections 4.3.1.1, 4.3.1.2 and 4.3.1.3 discuss some of the features of discrete and frictional systems that lead to above mentioned difficulties in relating the contact forces and deformations. Chapter 4 : 93 4.3.1.1 Quantification of Developed Contact Forces Resulting from Boundary Loads The relationship between internal contact forces and boundary tractions for bonded systems of particles is described by equations 3.42. The derivation of equations 3.42 involves only force and moment equil ibrium considerations of a system of particles and therefore are valid for frictional systems as well. However, for frictional systems, the constraint /t(0) = M ( ^ ) fn{ty has to be enforced at each contact where the magnitude of |/i(^)| varies between 0 and a l imit ing value //*. What proportion of particles would develop /((£}) = fi* /n(n) at any given time during deformation is uncertain. Such an uncertainity makes the derivation of stress-strain relations for fr ictional systems complicated, warranting a considerable amount of understanding on the behaviour of fr ictional systems so that simplifying, yet realistic assumptions could be made regarding the magnitude and direction of contact forces. A t this stage attention is focused once again on the obliquity of contact forces observed in circular disc shaped particle assemblies reported by Oda & Konish i (1974), described previously in section 2.3. As shown in Figure 2.13, the number of contacts that developed the l imiting friction angle of 22° at al l stages of loading was very small. Most contacts developed negligible obliquity of contact forces, the trend of which did not change as the system was loaded to its ful l capacity. A n observation as such suggests that either sliding of particles during deformations is min imal or that sliding is restricted to a few contacts that are aligned in preferred directions. It is diffcult to present experimental evidence in favour or against either hypothesis mentioned above from the data presented by Oda &c Kon i sh i (1974). It should also be mentioned that there are not many comprehensive studies reported in the literature on the contact force development in fr ictional systems. Had it been exclusively sliding of sand particles along preferred orientations that lead to the observed stress-strain response, simple friction block sliding models (Iwan (1967), Zytinsky et al(1978)) should Chapter 4 •' 94 fully explain both stress-strain and volume change response of sand. A lthough friction block sliding models are convenient models to explain the shear stress-shear strain response of sand, they are incapable of quantifying the shear induced volume change response. O n the other hand, rolling of particles can cause changes in the internal arrangement of particles resulting in non-linear and non-recoverable deformations. However, in a system of particles that are in contact w i th each other, it is difficult to perceive rolling without sliding, purely from a kinematic point of view. Sl iding of a few contacts of preferred orientations followed by roll ing of a large number of particles leading to rearrangement of particles is seen as the major mechanism causing most of the observed non-recoverable deformations in frictional systems. For rolling to occur, the tangent contact forces can remain well below their l imit ing values and therefore do not violate the experimental findings of Oda & Konishi (1974). 4.3.1.2 Identifying Geometrically Compatible and Stable Particle Configurations In section 3.2, the concept of a contact branch length distribution was introduced. A system of particles can be represented by a network of contact branch lengths as shown in Figure 4.1. Dur ing deformation, the contact branch vector distribution changes. When the in it ia l contact branch vectors and their subsequent changes are known, the resulting strains can be computed. Laws describing the changes in contact branch vectors can be expected to be very complicated since both stability and geometric considerations must be included in such laws. For the special case when the particles consist of uniform spheres or spheres with a l imited range of diameters, the contact branch length distribution is isotropic and can be represented by the diameter of particles (refer to section 3.2 for details). The spatial arrangement of particles in such systems can be approximated by the contact normal distr ibution function S which is an infinite Fourier series consisting of cosine terms only. It should be noted that the above approximation was based purely on physical experiments Chapter ^ : 95 GjCj contact branch lenyth Cj geometric contacts G; center of particle Fig.(4.1) Representation of a System of Particles as a Network of Contanct Branch Vectors Chapter 4 •' 96 reported by Oda & Konish i (1974) and supported by numerical experiments reported by Bathurst(1985). In section 2.3.3 it was also concluded that the changes in contact normal distr ibution of stable and geometrically compatible arrangements of particles occur in a systematic manner; distribution of contact normals is highest along the major pr incipal stress direction and lowest along the minor principal stress direction. Also, when the pr in-cipal stress directions rotate the high and low concentrations of the distribution of contact normals change accordingly. The new contact normal distributions resulting from rotation of pr incipal stress directions or from shear loading with no change in the principal stress directions can be approximated considering the first two terms of the Fourier series for S. Since these observations have been based on physical experiments where stability and geometric compatibl ity requirements have been satisfied, it can be concluded that the con-tact normal distr ibution function S (outlined in detail in section 3.2), can be used to describe stable and geometically compatible particle arrangements formed in frictional systems con-sisting of uniform spheres or spheres with a l imited range of diameters. 4.3.1.3 Changes in Stable and Compatible Particle Arrangements and Boundary Deformations Dur ing loading, particle arrangements change and deformations occur. Based on pub-lished literature, the microscopic investigators seem to favour the conclusion that deforma-tions are a result of changes in particle arrangements as opposed to changes in particle shape such that most stable and geometrically compatible configurations are acheived. Qual i ta -tively, the stress-strain response of sand is seen as the outcome of two competing processes; one causing order and the other causing disorder of the system. Processes causing order do so by constructing chains of particles in the direction of major principal stress. Processes causing disorder cause collapse of the chains of particles reducing the stability of the entire system of particles. Both processes are present at any given time during deformation, but Chapter 4 •' 97 the rates at which they occur differ. The chains of particles formed during simple shear loading involving pr incipal stress rotations of disc shaped particles, can be clearly seen from Figure 2.16 shown earlier in section 2.3. The fact that the distribution of interparticle con-tact normals are concentrated in the direction of major principal stress and the existence of chains of particles in the same directions are clearly inter-related. In sections 4.3.1.1, 4.3.1.2 and 4.3.1.3 it was pointed out that for frictional systems the contact forces and the resulting changes in deformations cannot be easliy related to each other. Unlike in the bonded theory presented earlier in Chapter-3, simplifications have to be made in order to establish theoretical relations between contact forces and boundary deformations (i.e. stress and strain). Strains are usually measured w i th respect to an in i t ia l strain free reference configura-tion. For frictional materials, such a reference configuration is not well defined because a stress free state associated with zero strains cannot be defined for such systems. A l l labora-tory data on stress and strain measurements correspond to an init ial ly non-zero stress state that correspond to a zero strain state and the reference configuration is defined arbitrarily. Traditionally, in soil mechanics problems the in i t ia l void ratio is used instead of an init ial state of strain and it is common to compute displacements or strains relative to an init ial state of strain and the zero state of strain is not of interest. The difficulties of having to define an in it ia l state of strain in mathematical models have been conveniently overcome by adopting incremental variations in stress and strain quantities such as in hypoelasticity and plasticity theories described earlier in sections 2.2 and 2.3. For most practical problems, in-cremental stress-strain relations are desired. Incremental stress-strain relations derived for frictional systems making simplifying assumptions are presented in the following sections. Chapter 4 : 98 4.3.2 Stage-1 Relationship Between Boundary Loads and Contact Forces Consider a system of particles that is in equil ibrium wi th external stresses CT,J- as shown in Figure 4.2. Then, from equations 3.42 Vij = ^ 1 nv do J j \fn rii rij + i / t (t{ rij + tj n,-)] S sin tp dtp d<j> ; i,j = 1,2,3 (4.1) 0 0 Equations 4.1 are applicable for both bonded as well as frictional systems since they are concerned with equil ibrium only. In frictional systems, let it be assumed that ft = Hfn (4-2) where, JZ defines the spatial variation of the developed ratio of average tangent and normal contact forces with in the assembly. A t this stage, the functional form of JI is not known. By definition, at al l stages of loading JI must satisfy the constraint, 0 < \JI\ < M* (4-3) where, fi" denotes the l imiting coefficient of friction of the material. Substituting from equation 4.2 into 4.1, wv - \ 7 nv do J J fn [«t nj +I (t{ nj + tj n,)] S sin tp dip d<p ; i,j = 1,2,3 (4.4) 0 0 The functional forms of JI and 5 that appear in equations 4.4 w i l l be defined later in sections 4.5 and 4.6, respectively. As outlined earlier in section 4.3.1, equations 4.3 and 4.4 complete the first stage of development of stress-strain relations for frictional systems. 4.3.3 Stage-2 Stable and Geometrically Compatible Particle Arrangements The second stage of development of stress-strain relations involve identification of stable Chapter 4 : 99 Fig.(4.2) Three Dimensional Assembly of Part ic les Chapter 4 : 100 and geometrically compatible particle arrangements that can be formed in fr ictional sys-tems. The contact normal distribution function S, the functional form of which has been proposed based on experimental observations of circular disc systems, establishes stable and geometrically compatible particle arrangement patterns that can be developed in systems consisting of uniform spheres or spherical particles with a l imited range of diameters. The variables that control the changes in S wi l l be addressed later in section 4.4.3 of this chapter. 4.3.4 Stage-3 Quantification of Boundary Deformations From Particle Deformations In the bonded theory presented in Chapter-3, the contact forces and the resulting deformations were related through the contact stiffness parameters, the magnitudes of which were assumed to be constant at all contacts. In section 4.3.1.1 above, based on numerical and laboratory experimental evidence on the behaviour of fr ictional systems of particles, it was postulated that sliding of a few suitably oriented particles followed by rolling and rearranging of a large number of particles is the major mechanism causing most of the deformations. Particle contacts that slide may exhibit reduced stiffness in the tangent direction. Sliding is associated with dissipation of energy which result in non-recoverable deformations. O n the other hand, due to rolling of particles, the spatial arrangement of particles change resulting in a different structure after each increment of loading. The relationship between the applied stress and the developed magnitude of internal contact forces {/„, ft} and their directions {n„ £,•}, for a frictional system of particles consisting of uniform spheres is given by equations 4.3 and 4.4 above. In effect, equations 4.3 and 4.4 are equil ibrium equations for the system under consideration. In what follows, the compatibil ity conditions between the relative contact displace-ments {l°n, lf\ and the average boundary strain tensor Ay of a system of particles that are in equil ibrium are derived using the principle of v irtual forces. The force formulation of Chapter 4 •' 101 virtual work is made use of since the equi l ibr ium conditions are known and it is the geomet-rical relations between contact deformations and boundary strains that are sought. How the system reached the particular equi l ibr ium configuration and the material behaviour is irrevelant. The principle of v i r tual forces states that (Oden (1967)): The strains and displacements in a deformable system are compatible and consistent with the constraints if the total external complementary virtual work is equal to the total internal complementary virtual work for every system of virtual forces and stresses that satisfy the equations of equilibrium. Later on in section 4.3.5, the relationships between the incremental relative contact displacements and incremental average boundary strains wi l l be derived considering the first variation of the compatibi l ity conditions derived above, following the principle of v irtual forces. The a im then would be to arrive at incremental stress-strain relations assuming that the load-deformation behaviour of contacts of a given orientation is incrementally linear. In such incremental relations, the non-linear and dissipative features of frictional systems wi l l be explicitly accounted for by varying the material parameters, spatial arrangement of particles and denning criteria for loading and unloading. Let the load-deformation behaviour of contacts of a given orientation be conceptually represented by deformable springs and that the deformation response of a given contact is independent, of the other contacts of the same particle. Let the system of particles be acted on by an arbitrary v i r tual force system {A/£, A//} at the contacts that are in equil ibrium with the v irtual external loads A<fy. For the system of particles under consideration, from the principle of v i r tua l forces, it follows that where, v is the volume of the system under consideration. A,y are the real strains and lcn (4.5) Chapter 4 : 102 and /j are the relative real displacements at the contacts. Following a similar approach as in equations 3.37, 3.38 and 3.48, the products Afc 1° on contacts of a given orientation are averaged with respect to the number of contacts wi th that orientation. This is necessary because the equlibrium equations 4.4 have been derived in terms of average contact forces on contacts of a given orientation. Accordingly, 2ir x v AWij Xij = M j j c'[Afn I* + Aft k\ S sin xp dyj dxj> (4.6) 0 0 where, M denotes the total number of geometric contacts with in the volume v. In equation 4.6, it has been assumed that Af<l° = c'Afl (4.7) where, f ' is a certain non-dimensional parameter related to the manner in which the applied virtual forces and displacements on contacts of a given orientation are distr ibuted with in the assembly. This parameter, assumed to be independent of 5, is very similar in nature to $ that entered the bonded formulations through the use of information theory when averaging (/ c) 2 terms in equation 3.54. From equation 4.2, it follows that Aft = JI Afn + fn A/7 (4.8) Combining equations 4.6 and 4.8, v Aoij \ij = M j j c'[Afn Qn + Jik) + (fn 1) A/l] S sin t/> dip d<p (4.9) 0 0 Assuming that 7 , nv and S remain constant over the application of the v i rtual forces, variations in are now considered in the form, 2ir TT Ao^ = i 7 n„ do j J [Afn [n,- n,- + |(t,- n,- + tj n,)] + A/l[^(*,- n,- + tj ra,)] 0 0 Chapter 4 : 103 + [~^(Ati rij + At,- n,-)]] S sin ip dip dtp (4.10) In equations 4.10 above, the variations in fn, JI and are considered. Recall that £, and ra,-that appear in equation 4.4 are not independent of each other, but related as 1 = 0 (4.11a) n, 71 , -1 = 0 (4.116) t ,n , = 0 (4.11c) Hence, the variations in £, considered in equations 4.10, are constrained by the variations of given by equations 4.11. It is now possible to substitute for Ai7,j- from equations 4.10 into 4.9 and compare terms corresponding to the arbitrary variations A / n , AJI and At,-. Note that Atj terms do not appear on the right hand side of equation 4.9, and for the variations of tj that appear on the left hand side to be complete, they must include the constraints given in equations 4.11 above. Introducing the symmetric strain tensor A* (= {A,y + Ay,}/2) and substituting from equations 4.10, equation 4.9 can be re-written in the form 2-ir 7r ^ 7 n„ do v J J [Afn [A*- n,- n,- + JI A*- U nj] + Ajl[fn A*- i,- n,] + A i , [ / n JI A*- n3] o o -Hi [2t{ At,]-H2 [At{ n,-] ] Ssin ip dip dtp = M j j c' [A/ n (ln+JI lt)+Ajl(fn lt)] Ssin ipdipcty 0 0 (4.12) where the variations in constraint equations 4.11 have been included into the left hand side of equation 4.12. From equations 4.11, it follows that t , -A i , = 0 (4.13a) n,- At,- = 0 (4.136) Chapter 4 : 104 In equation 4.12, pt\ and are constraint multipliers which are analogous to the Lagrangian multipliers. Equation 4.3 relates the average tangent contact forces to the average normal contact forces through the function JI whose absolute magnitude is constrained between 0 and fi*. Even though the functional form of JI is unknown at this time, it is reasonable to assume that on planes where \jl\ is maximum (i.e. equal to //) the probabil ity of sliding is high and therefore the tangent stiffness kt, is low. Therefore, the influence of on kt is incorporated by assuming that, In equation 4.14, the magnitude of C\ is selected appropriately so that the magnitude of kt of contacts that are in the verge of sliding is low. O n planes where JI = 0 (i.e. no tangent forces), equation 4.14 implies that kt = kn. Note that the resistance to deformation in tangent and normal directions denoted by kt and kn respectively, are dependent on the orientation of the contact groups and do not have the same physical meaning as in the bonded theory. kt and kn defined above are equivalent stiffness parameters and do not represent intrinsic stiffnesses. They are equivalent in the sense that for a given change of loading, the numerical values of the stiffnesses kt and kn are such that the computed deformations include those resulting from sliding as well as rolling. F rom equations 4.2 and 4.14, it follows that-l = [ r J L - n ] ' l n (4-15) where O Q above is equal to c • Combining equations 4.12 and 4.15 and for any arbitrary variations A/„, A/Z and At,-, [ l + , ,_•] f ' In = Y [A*- n,- nj + JI A* U «,-] (4.16) 1 — Oo \fJ,\ I Ck=%[\%U^ (4.17) Chapter 4 : 105 fn A4 A* rij - 2fii U - ^2 rii = 0 (4-18) F rom constraints 4.11 and 4.18, it follows that A • nr — [XI. nr n.) ru , . U= ,r . 1 (4.19) A„ tr ns For most materials the magnitude of //* wi l l be about 0.2 to 0.3. Since JI is alway less than or equal to fx*, the higher order terms of JI that appear in equation 4.16 can be ignored. Accordingly, f ' 'n = j [A*- n,- n,- + /I A*- t{ nj] (4.20) Combining equations 4.2, 4.17 and 4.20 and ignoring higher order terms of JI, 1 A* tr ns , M - ^ T T ^ 4.21) 2 A*.n,- rij Equations 4.15 and 4.20 together with equations 4.19 and 4.21 describe a set of geometrical relations between the contact deformations ln,k and the boundary strain tensor X^. Note that the ratio of tangent to normal contact forces denoted by JI, as defined for a system of equal size spherical particles can be expressed in terms of ratios of compatible strains that can develop with in the system. In above, note that the fact that cannot exceed the coefficient of l imit ing friction of the material has not been incorporated into the equations, but imposed as an external constraint on the compatibility conditions. In this context, the developed relations between the contact deformations and the boundary strains do not explicitly account for sliding in a dissipative sense. However, provision is made for large deformations to occur on planes where JI approaches //*, through relationship 4.14. 4.3.5 Incremental Stress-strain Relations In what follows, incremental stress-strain relations that are applicable for frictional systems are derived. In a strict sense, the derived relations are applicable for infinitesimally Chapter 4 : 106 small increments of loading where the friction distribution fj, and the direction of tangent contact forces i,- remain constant over the loading increment. Let a system of particles that is in equi l ibr ium be subjected to infinitesimally small increment of external loads A<7,y. Let the corresponding changes in the average normal contact forces be A / n and the average contact tangent forces and their directions be Aft and At,-, respectively. From equations 4.19, 4.20 and 4.21, it follows that (AA*„ tr n3)tj + {X*„ Atr n,)ti + (A* s tr ra3)At, = AA*. rar - (AA* S rar ra>, (4.22o) f ' A / n = y [AA*. ra,- ray + /I(AA*. t,- ray + A*- A t , ray) + (A*- t,- ray) Ap] (4.23a) 2(AA,y rii nj)jL.+ 2(A*. ra,- ray)A/J = A A ^ t r ra8 + A ^ A t r n, (4.24a) where, it has been assumed that c' remains constant during the variations A/„, AJI and At,-. Now, if it is assumed that the friction distr ibution and the direction of average tangent contact forces remain constant over the increment of loading (i.e. A/7 ~ 0 and At,- ~ 0), equations 4.22a through 4.24a reduce to (AA* S t r na)tj = AA*. rar - ( A A ^ nr n,)ra,- (4.226) f 'A/n = y [ A A * . ra,- ray + /Z(AA*. t,- ray)] (4.236) 2(AA,y ra,- ray)/I = AA^ S t r n, (4.246) Analogous to equation 4.20, Aln in equation 4.23b describes a set of compatibi l ity condi-tions between the incremental normal contact displacements and the incremental boundary strain tensor. The constitutive relations for the contact model are now introduced to ob-tain the incremental stress-strain relations for the system of particles under consideration. Herein, the incremental normal contact force is related to the incremental normal contact displacement as A / n = k„. Aln, where represents the equivalent incremental normal Chapter 4 : 107 stiffness for the increment of loading. Substituting for A / n in equation 4.10, Aff,) = ^ 7 nv do J j -^-[ni + n,- + tj n,)} X 0 0 [AA* 3 nr n3 + JI AA * 3 tr na] S sin tp dip dxj> (4-25) where, and JI in equations 4.25 are related to the incremental strain tensor through equations 4.26 and 4.27 shown below; = A A - r n r - ( A A ; n r ».) n,-A A ; t, II, 1 AA * tr na , 2 AA^n,- nj Note that for two dimensional conditions, £, given in equation 4.26 is independent of the strain increment ratios. The non-linear and dissipative stress-strain behaviour of frictional systems is modelled herein assuming that the system behaves in an incrementally linear manner. The dissipative effects are accounted for by varying the material parameters k^, S, 7 and nv at the end of each increment and defining a loading or an unloading criterion. The approach followed herein is analogous to that followed in the incremental theory of plasticity. The particular forms of S and JI used in equation 4.25 lead to a constitutive matrix that is symmetric. Symmetric constitutive matrices have been used in the incremental theory of plasticity, when modell ing the behaviour of frictional systems with an associative flow rule. In what follows, a distinction is made between the equivalent incremental normal stiff-ness for loading and unloading. Loading is defined as a change of stress state associated w i th a positive change in stress ratio, AJ7. Unloading is defined as a change of stress state associated w i th a negative change in stress ratio. The magnitude of resistance to defor-mation in loading and unloading are considered separately, the unloading resistance being Chapter 4 : 108 higher. These concepts are il lustrated in Figure 4.3. The detailed descriptions of the man-ner in which the equivalent incremental normal stiffness change during loading, unloading and reloading are presented later in section 4.6. Consider an ideal system of particles as shown in Figure 4.4 where the contact nor-mals are aligned approximately in the vertical direction. It can be seen that the system exhibits stiffer resistance to deformation in the vertical direction. A similar, yet not so obvious influence of the contact normal directions can be expected for the stiffness in the normal direction for more general systems of particles. In other words, the contact normal distribution and the stiffness variations in the normal direction are complementary to each other. The stress-strain relations described by equations 4.27 already contain the contact normal distribution function S. Note that the spatial variation of k^ was introduced in an attempt to incorporate more flexibil ity to the formulations. A s can be seen from the above arguments, it is unnecessary for k^ to be dependent on orientation as the contact normal distribution function S accounts for the spatial variation of normal stiffness. Therefore, the spatial variation of k^ is dropped altogether in the forthcoming sections and the incremen-tal equivalent normal stiffness is assumed to be isotropic. Once an isotropic variation for k^ is assumed, the incremental equivalent normal stiffness parameter can be taken outside the integral equations 4.25 and be lumped together w i th other isotropic parameters such as 7, n„, ao and f'. The functional form of the combination of above isotropic parameters is, Ht= ^ - i k n t l nv do (4.28) where, Ht is termed the resistance function. Substituting from equation 4.28 into equation 4.25 Aa{j - Htj j [tit rij + ^Jx(ti rij + tj n,-)] x 0 0 [AA* 3 nr na + JI AX*rs tr n3)] S sin xp d dcf> ; i,j = 1,2,3 (4.29) Chapter 4 Fig.(4.3) Non-linear Load-Deformation Behaviour at Contacts Chapter 4 \ 110 Fig.(4.4) Idealized Part ic le Arrangements Chapter 4 •' H I Ht which appear in equations 4.29 is analogous to an incremental modulus relating the increment of stress to the increment of strain and consists of particulate parameters, as wi l l be discussed later, that cannot and need not be measured directly. W i t h proper selection of Ht for each increment of loading and constraining the maximum absolute magnitude of JI to fi* in al l directions, equations 4.26, 4.27,4.28 and 4.29 can be used to describe incremental stress-strain relations for frictional systems consisting of uniform spheres. After each increment of loading, Ht, S and JI must be updated to account for the changes occurring during the increment of loading. The contact normal distr ibution function S, that enter the incremental stress-strain equations 4.29 takes account of stable and geometrically compatible particle arrangements that develop in fr ictional systems. It w i l l be shown later that by varying the anisotropy co-efficients that constitute S, changes in the contact normal distribution during shear loading with fixed or rotating pr incipal stress directions can be described. In effect, both inherent and induced structural anisotropy are accounted for through S. The non-linearity of the observed stress-strain response can be modelled through the variations of Ht. As can be seen from equations 4.26 and 4.27, both JI and are dependent on the strain increment ratios ( i.e. the strain path followed). Therefore, the proposed stress-strain relations are capa-ble of describing path dependent behaviour. Parameters describing the effect of void ratio and stress level w i l l be incorporated into the laws defining the changes in the anisotropy coefficients, outl ined later in section 4.4.3. The particulate parameters 7 and (' that enter equations 4.28 cannot be directly mea-sured. Therefore, the variations of 7 and f ' have to be accounted for by means of measure-able parameters. Also, the formulations are incomplete unt i l the variations of S, JI and Ht in equation 4.29 are fully defined. The main objectives of the forthcoming sections is to complete the stress-strain relations by fully describing the variations of S, JI and Ht-Chapter 4 : 112 4.4 The Contact Normal Distribution Function S The basic features of the contact normal distr ibution function S were first introduced in section 3.2. Herein, the tensorial representation of the contact normal distr ibution function S and the laws that define the changes of S associated with loading are addressed in detail. 4.4.1 Two-dimensional Representation of the Contact Normal Distribution Considering a truncated Fourier series, the contact normal distr ibution in two-dimensions (denoted by S(9)) was expressed as; S(9) = ^- [ l + o 2 cos 2(9 - OQ) + o 4 cos 4(9 - Oo)] (4.30) 2n where, the coefficients 02 and 04 have to be selected such that, j S(0) d9 = \ (4.31) 0 S(9) > 0 ;for all 0 (4.32) First consider the special case when 04 = 0. Expanding cos 2(9 — 9Q) terms in equation 4.30, the following expression can be obtained for S(9). S(0) = — [ l + ( o 2 cos 200) cos2 0 + ( — 02 COS 29Q) sin2 9+ (2 02 sin 29Q) sin 0 cos 0\ (4.33) 2ir Equation 4.33 can be identically represented by the following expression; S(0) = [ l + o,y m nj] ; i,j = 1, 2 (4.34) in which a^ is a second order symmetrical tensor and n = (cos 0, sin 9). Expanding equation 4.34 and comparing terms with equation 4.33, the following relations can be obtained; a n =  a2 c o s 20Q (4.35a) Chapter 4 •' 113 = -0,2 cos 2#o (4.356) 0 1 2 = ^21 = 0.2 sin 20 o (4.35c) A s can be seen from equations 4.35, the second order tensor a,y satisfy the following con-straints; akk = 0 ; £ = 1 , 2 (4.36a) S = aii 5 1' ^  J (4.366) The ratios ^ or ^ are a measure of the rotation of the contact normal distribution 0 a , with respect to a given set of coordinate axes. Following a very similar approach, when the coefficient 04 in equation 4.25 is not zero, the two-dimensional form of 5(0) can be written as, 5(0) = [ l + a,y m nj + bijti re,- n, nk nt] ; i,j, k,l=\,2 (4.37) where, n = (cos 9, sin 9). 6,j-y, which is a fourth order tensor, contains 16 terms. However, all 16 terms are not independent of each other since equations 4.31 and 4.32 have to be satisfied as well. A s before, comparing terms of 04 cos 4(9 — 9o) and ( 6 , ^ re,- nj re* re;), the following relations can be obtained; 61111 = 62222 = «4 cos 4#o (4.38a) 6 1 1 2 2 + 61212 + 6 2 1 2 1 + 62112 + 61221 + &2211 = ~ 6 a 4 cos 40o (4.386) 61112 + 61121 + 61211 + 62111 = 4 04 sin 40 o (4.38c) 6 1 2 2 2 + &2122 + 62212 + 62221 = _ 4 04 sin 40o (4.38d) F rom equations 4.38 it can be seen that for a given 04 and 0o, several combinations of 6 , ^ can be found. Any one of these combinations can be used to describe 5(0) in equation 4.37. Chapter 4 •' 114 4.4.2 Three-dimensional Representation of Contact Normal Distribution Following a format similar to equation 4.34, an expression for the three-dimensional contact normal distribution can be obtained. Accordingly, S = [ l + a,y m ray] ; i,j = 1,2,3 (4.39) 47T where, n\ = cos <f> sin xp, n<i = sin <f> sin xp and 713 = cos xp. <p and xp are angles measured in spherical coordinates as shown in Figure 3.4. Analogous to equation 4.36, a,y in equation 4.39 satisfy the following constraints; H = Oji ; i^j (4.40a) akk = 0 ; k= 1,2,3 (4.406) The three-dimensional contact normal distr ibution corresponding to the distribution de-scribed in equation 4.37 can be witten as; S = [ l + a,y ra,- ray + 6,yW w,- ray nk nt] ; A, / = 1,2,3 (4.41) in which ra, are identical to those in equation 4.39. It should be noticed that the tensorial representation of the contact normal distribution in two and three-dimensions differ by an increase in the dimensions of the indices i, j, k and I and a change of the constant 2TT to 4ir so that, 2n t S sin xp dxp dd> = 1 (4.42) 0 0 In equation 4.41, 6,y« contain 81 terms. The combinations of a,y and 6,y« that satisfy the constraint given by equation 4.42, are as follows; «11 + <*22 + <*33 + (3 6 1 m + 3 62222 + 3 63333 + Chapter 4 •' 115 61122 + 62211 + 61212 + 62121 + 61221 + 62112 + 6ll33 + ^3311 + 61313 + 61331 + 63131 + 63113 + 62233 + 63322 + 62332 + 63223 + 62323 + 63232) = 0 (4-43) In addition, a,y and 6,yy must be selected such that S > 0 for all ip,<p. For convenience, equation 4.43 can be separated into two equations as follows; « n + «22 + a 3 3 = 0 (4.44) (3 61111 + 3 62222 + 3 63333)+ 61122 + 62211 + 61212 + 62121 + 61221 + 62112 + 61133 + 63311 + 61313 + 61331 + 63131 + 63113 + 62233 + 63322 + 62332 + 63223 + 62323 + 63232) = 0 (4.45) The reasons behind separating the second order anisotropy coefficients a,) from the fourth order anisotropy coefficients 6 v y, in equations 4.44 and 4.45 are nothing other than simplicity. By doing so, certain combinations of a,y and 6,^ that satisfy equation 4.43 may not satisfy equations 4.44 and 4.45 and are therefore eliminated. However, all combinations of a,y and 6,yy that satisfy equations 4.44 and 4.45 automatically satisfy equation 4.43. It w i l l become evident later that the above separation makes it easier to define the laws that describe the variations of a,y and 6,yy and also reduce the number of coefficients involved in equation 4.43 by one. For the special case when the principal anisotropy directions coincide with the coordi-nate axes, terms 61212, 62121, ^1221, 62112, 61313, 63131, 61331, 63113, 62323, 63232, 62332 and 6 3 2 2 3 are identically zero. If it is assumed that 61122 = 62211,61133 = 63311 and 62233 = 63322, the number of terms in equations 4.44 and 4.45 reduce to 7 terms. Furthermore, if cross anisotropy is imposed, for example about direction-1 as well, 022 = 133, 62222 — 63333 and 61122 = 61133. The result is that the number of terms reduce to a mere 4. Interestingly, when Chapter 4 • H 6 fourth order anisotropy is ignored altogether, for the above case with principal anisotropy directions coinciding with reference axes and with cross anisotropy about direction-1, the number of terms reduce to just 1. 4.4.3 Laws Defining Changes in Contact No rma l D istr ibut ion As outlined in section 4.3.1.2 in assemblies consisting of circular discs, stable and ge-ometrically compatible particle arrangements formed during deformation can be described by the contact normal distribution function S(6) which is given by equation 4.37. The form of equation 4.37 has been proposed based on experimental observations on particle arrangements developed in two dimensional systems that contain particles that are circu-lar in shape. In section 4.4.2, equation 4.37 was extended from two dimensions to three dimensions following conceptual arguments. Implicit in such an extension is that the pro-posed three dimensional distribution of contact normals describes stable and geometrically compatible particle arrangements that can be formed in three dimensional systems. The process of verification of the proposed three dimensional distr ibution involve microscopic examination of three dimensional spherical systems which have not been reported todate in literature. Therefore, equation 4.41 is accepted herein without experimental verification as being capable of describing stable and geometrically compatible particle arrangements in three dimensional systems of spherical particles consisting of uniform spheres or spheres with a l imited range of diameters. The three dimensional distr ibution of contact normals S described by equation 4.41, constitute a large number of anisotropy coefficients a,y and 6,y« ; i,j,k,l = 1,2,3. It was shown earlier that a,y and 6,yy in S, have to satisfy certain constraints (i.e. those given by equations 4.40a, 4.40b and 4.42). Dur ing loading, changes in S occur which can be quantified through the changes in the anisotropy coefficients a,y and 6,yju. Chapter 4 : 117 The variation of S with a,-,- and biju is linear. Since / / S sin xp dtp dtp = 1 at all times, o o it can be shown that the incremental anisotropy coefficients Aa,y and A6,JJU must satisfy the following constraints; A a u + A a 2 2 + A a 3 3 = 0 (4.46) 3 ( A & i n i + A62222 + A63333) +A61122 + A62211 + A&1212 + A62121 + A61221 + A62112 + A 6 1 1 3 3 + A 6 3 3 U + A61313 + A61331 + A63131 + A63113 + A62233 + A63322 +A62323+A62332 +A63232 +A6 3 2 23 = 0 (4-47) Note that the form of equations 4.46 and 4.47 are identical to those given by equations 4.44 and 4.45 except for the incremental quantities of a,y and b^u. Based on the theoretical developments of Rothenburg(1980) and numerical experiments of Bathurst(1985), for circular disc shaped particles the average tangent and normal contact force distributions can be approximated by the following fitting equations; /„(*) = /o [ 1 + o» cos 2(0 - 6j )} (4.48) / t(0) = -atf0 [sin 2(9 -6t)} (4.49) in which an and aj are referred to as force anisotropy coefficients. Some of Bathurst ' s data justifying above forms of average tangent and normal contact forces were shown previously in Figures 2.15a and 2.15b. In two dimensions, the contact forces fn{0) and ft{9) are related to the stress tensor in the following manner (refer to equation 3.39); a* = \ 7 n, do Jo [fn{9) n,- n,- + ^1 (t,. n j + tj n,)] S{B) dd ; i,j = 1, 2 (4.50) Ignoring the fourth order anisotropy coefficients that constitute the contact normal distr i-Chapter 4 • 11.8 bution function 5(0) can be written as, S@) = ^-[1 + "2 cos 2(0 - 0a) ] (4.51) For the special case when 9a = Of = 0* = 0 (i.e. when the principal anisotropy directions coincide with the reference axes ), combining equations 4.48, 4.49, 4.50, and 4.51 and evaluating the integral, Rothenburg(1980) has shown that for W\\ > a"22, The validity of above equation has been independently verified by Bathurst from numerical experimental data. His results indicate that the magnitude of at can be neglected in com-parison to <Z2 and an without loss of accuracy. Note that the left hand side of equation 4.52 denotes the stress ratio in two dimensions. Considering the incremental changes of both left and right hand sides of equation 4.52, From equation 4.51, it can be seen that a change of stress ratio is associated with corre-sponding changes in the anisotropy coefficients a2, an, and at. The manner in which the incremental coefficients A a 2 , Aan and Aat distribute the applied change of stress ratio can-not be obtained from equation 4.53. Such information can only be obtained from physical and numerical experimental studies of particulate systems. Based on numerical experimen-tal results available and the observed behaviour of frictional systems, it is postulated that at any stage of loading, a change of stress ratio is distributed among A a 2 ; A a n and Aat depending on the, a) proximity to the critical-state, and b) mean normal stress level and density of packing. (4.52) (4.53) Chapter 4 : 119 Let the sum of the terms ( A a n + Aa^) be represented as a fraction of A a 2 as follows; ( A a n + Aat) = x A a 2 (4.54) Combining equations 4.53 and 4.54 it can be shown that A a 2 = C 2 A [ (<Tn - (7 2 2)/2 (^11 +^22)/2 (4.55) where, C 2 being a function of x would include the influence of the proximity to critical-state and mean normal stress and density of packing. Since the terms wi th in brackets in the right hand side denotes the stress ratio in two dimensions, equation 4.55 can be written in a more general format given by, in which and am denote deviatoric stress tensor and mean normal stress, respectively. Equations 4.56 establish an important relationship that is capable of explaining the high and low concentrations of contact normals observed in the major and minor pr incipal stress directions, respectively. Without loss of generality, equation 4.56 can be extended to three dimensions as follows; Aatf = C 3 A [^-] ; i,j = 1,2,3 (4.57a) The term C 3 is very similar to C 2 except that C 3 corresponds to three dimensional condi-tions. Expanding the right hand side of equation 4.57a, Aa,y in equation 4.57b above, is dependent on the o\-, am and their changes. Hence, in the case of rotation of principal stress directions for which case the stress components 0"i3 and o"23 change, corresponding changes in the second order anisotropy coeffcients (4.56) (4.576) Chapter 4 : 120 <*12; <*13 and occur according to the variations specified by equation 4.57b. Anisotropy coefficients 012, 013 and 023 are a measure of the rotation of the contact normal distribution S(Cl) in space. Therefore, wi th pr incipal stress rotations, the contact normal distribution rotates in the same direction. Appl icat ion of shear loads without principal stress rotations change the magnitude of anisotropy coefficients such that the contact normal distribution changes maintaining the orientation of the contact normal distribution. In what follows, the term C3 is modified to reflect the influence of proximity to crit ical-state and mean normal stress level and density of packing. 4.4.3.1 Influence of Proximity to Critical-State Stress Ratio on A a,y The proximity of a given sample to the critical-state stress ratio is taken into account by modifying C3 such that, Cs=C3[^L]a* (4.58) in which C'z is dependent on the mean normal stress level and density of packing. a 2 is a positive real constant for the material under consideration. Note that for a given C 3 , when ^ is small the magnitude of C3 is small and therefore the corresponding changes in the incremental anisotropy coefficients Aa,y are small. In other words, at low stress ratios the load is carried by the contact force anisotropy coefficients to a greater extent than by the changes in contact normal distribution. However, when the stress ratio increases, the changes in Aa,j increase, imply ing that at higher ratios of ^ more changes in contact normal distribution occur. So far, only the influence of the ratio -2- has been considered. It wi l l be shown in the ' ICS next section that by modifying C'3 appropriately, the effect of mean normal stress level and density of packing on the variations of Aa,y can be accounted for. Chapter \ : 121 4.4.3.2 Influence of Mean Normal Stress Level and Density of Packing on A ay-Consider the following systems of particles; one that is densely packed and the other loosely packed. Under low to moderate mean normal stress levels, the densely packed sample can undergo large changes in contact normal distribution. In densely packed assemblies the average number of contacts per particle or the coordination number 7 is high. Therefore, changes in contact force distributions may occur at a reduced scale. Even when the mean normal stress level is increased, the contact force distribution may not change very much. However, the ease with which particles can rearrange may reduce thereby reducing the magnitude of Aa,y. On the other hand, loose assemblies have a relatively low average number of contacts per particle. For loose assemblies subjected to moderate mean normal stresses, the changes in contact force distribution may be large when compared to the changes in contact normal distribution. However, the proportion of distr ibution of incremental force and contact normal anisotropy coefficients can be reversed when the mean normal stress level is low where particles have more freedom to rearrange during deformation. In section 2.2, a single parameter called the state parameter was introduced that can account for density and mean normal stress level. A similar concept is proposed herein to describe appropriate variations of A a y - wi th density of packing and mean normal stress level. Accordingly, C 3 , in equation 4.58 is further modified as follows; where, <x\ and 0x2 are positive material constants. ecs is the void ratio at critical-state for a given mean normal stress and e is the void ratio of the sample for the same mean normal stress level. Combining equations 4.57b, 4.58, and 4.59, (4.59) A a. = 1,2,3 (4.60) Chapter 4 : 122 Note that when ^ > 1 such as in loose samples subjected to low mean normal stress levels and for given a\, a.2 and 0:3, the magnitude of C 3 wi l l be larger than for the case when ^ < 1 such as in dense samples subjected to high mean normal stress levels. 4.4.3.3 The Changes in Fourth Order Anisotropy Coefficients A6y« So far, no mention of the fourth order anisotropy coefficients a4 (in two dimensions) or biju (in three dimensions) has been made and both second order and fourth order anisotropy coefficients have been considered independent of each other. However, based on a tedious theoretical treatment of stable particle arrangements that can develop in circular disc sys-tems, Rothenburg (1980) has shown that the fourth order anisotropy is dependent on the second order anisotropy that developes in such systems. His analysis indicates that, in two dimensional systems of circular particles of equal diameter, 04 = C4 a\ (4-61) where C\ is dependent on the coordination number, 7. Based on equation 4.61, the following variations of 6 y y are proposed for three dimensions; bijki = ft I a,>- I =^ ; k, I =1,2,3 (4.62) Note that /3i in equation 4.62 is a material constant. The influence of the coordination number which in turn depends on the mean normal stress level, void ratio, shear stress level and material stiffness has been assumed to be included through a,j-. Expressing equation 4.62 in its incremental form, A6,y« = ft [10*1 - ^ Aam) + ^ |Aa«|] ; k, I =1,2,3 (4.63) 0~ m O m C m The variation of A 6 vH , as defined by equation 4.63 above, is modified further based on Chapter 4 : 123 the sample runs performed by the author to predict stress-strain behaviour of real sands. The results from above sample runs indicated that during loading along constant mean normal stress paths, the developed anisotropy in the material was too large indicating high rates of dilation. Accordingly, the following modifications have been made; (a) introduce dependency on the proximity to critical-state through {^raY2, a n d (b) suppress developed fourth order anisotropy along constant mean normal stress paths introducing the function | [ l - cos (4 f a n - 1 ))]• The new variation of A 6 , ^ takes the form; Abijld = \ ft [1 - cos (4 t a n - \ - ^ - ) ) ] [ | a „ | ( ^ - ^ A ^ M ) + E i | A f l | | |] £ Vcs V 2 Aam o-m o*m <Jm i,j,k,l = 1,2,3 (4.64) In Chapter-5, it wi l l be demonstrated that above variations for A6,yy are capable of predict-ing the correct volumetric strain response along different stress paths. A6,yu, as described by equation 4.64 must satisfy the constraints given by equation 4.47. In fact, one reason for including the absolute magnitude of o,j- in equation 4.62 or in equation 4.64 is to arrive at combinations of A 6 , ^ that satisfy the constraint equation 4.47 relatively easily. In order to satisfy the constraint given by equation 4.47, the following assumptions have to be made; A 61212 = A 6 i 2 2 i = A 6 n 2 2 A61313 = A 61331 = A61133 A 62323 = A62332 = A62233 A62121 = A 62112 = A 6 2 2 i i A 63131 = A 63113 = A 63311 A 63232 = A 63223 = A 63322 (4.65) Equations 4.60, 4.64 and 4.65 contain al l necessary information regarding the possible Chapter 4 : 124 stable and geometrically compatible particle arrangements that can be developed in three dimensional systems of particles consisting of uniform spheres or spheres wi th a l imited range of diameters. In effect, the influence of stress ratio, proximity to critical-state with respect to stress ratio as well as mean normal stress and density of packing on possible changes of particle arrangement patterns have been incorporated into equations 4.60, 4.64 and 4.65. The above equations have been proposed based on a collection of ideas extracted from both experimental and theoretical developments reported in the literature. It should be noted that equation 4.60, 4.64 and 4.65 which quantify the changes in anisotropy coefficients o,y and bijki are entirely new and useful in quantifying the changes in anisotropy that occur in particulate, discrete and frictional systems. Even though both a,y and bijki contain a large number of terms for the most general case, by making simplifying assumptions they can be simplified to a few terms. The adequacy of equations 4.60, 4.64 and 4.65 to model sand behaviour w i l l be assessed in Chapter-5 by comparing model predictions with laboratory stress-strain data on sand. The functional form of JI, and Ht are yet to be fully defined. The next section wi l l address the developed distr ibution of friction JI. 4.5 The Distribution of JI In this section the functional form of JI as derived in section 4.3 from theoretical con-siderations is discussed in detail. In particular, the available experimental information on the spatial distr ibution of JI are compared with the theoretical distributions. There is only a very l imited amount of experimental information available on the distribution of JI in particulate, discrete and frictional systems. Furthermore, this experimental information is l imited to two dimensional systems of circular particles. Therefore, the comparisons are made for two dimensional systems only. Chapter 4 : 125 4.5.1 Experimental Data on Two Dimensional Distribution of fi(9) The distribution of the magnitude of \jl{0)\ computed based on the developed average tangent and normal contact forces in two dimensional systems of circular particles as re-ported by Bathurst(1985), were shown earlier in Figures 2.16a and 2.16b. In above circular systems, particles are either of equal diameter or of a l imited range of diameters. The block like variations shown in Figures 2.16a and 2.16b are a result of the finite number of orien-tation groups considered (A0 = 2ir/G = 10° in this case; refer to section 3.2 for details). The smooth curves are approximations to the block like variations observed from numerical experimental data of Bathurst. Above data correspond to pure shear loading and \jl(0)\ distributions are shown for two different stages of loading; stage-1 correspond to the first time peak shear (Figure 2.16a) and stage-3 correspond to the critical-state conditions (Fig-ure 2.16b). As can be seen from the Figures 2.16a and 2.16b, the developed \ji(6) \ is zero in the directions of major and minor principal stresses which are at 45° to the coordinate axes for pure shear loading path. is max imum along directions which make approximately ± 4 5 ° with the principal stress directions. Note that the maximum |A7(0)| developed at stage-1 is about 0.15 where as at stage-3 it is about 0.10 even though the shear stress level applied at the two stages are not too different (refer to the inset). One physical difference between the two stages is that at stage-1, the dominant chains of particles are about to be disordered wheresas at stage-3 the disordering has been nearly completed. It is not clear whether such disordering would cause a transfer of tangent contact forces to normal contact forces. Such a mechanism of force transfer, however, is not unlikely in systems where particle rolling is a dominant mechanism causing deformations. Further numerical investigations are necessary to examine the above mechanism in detail. The functional form of the approximated distr ibution of \ as proposed by Bathurst Chapter 4 : 126 is, \-ta\\ i at sin 2(9 - 9t) ^ ) l = l [ l + a n cos 2(9-9^ ^ where at, an, Qt and Of are simple curve fitting parameters which vary depending on the observed distr ibution of |/i(0)|. For the distribution shown in Figure 2.16a, at = 0.120, o n = 0.447, Ot = 0.790 and Of = 0.770 (above magnitudes have been obtained from Figures 5.21 and 5.22 of Bathurst(1985)). Note that the magnitude of Of and 0t given above (in radians) are very close to the value | and therefore coincide with the major principal stress direction measured clockwise from the horizontal direction. Had the reference coordinate axes been selected at a direction | to the horizontal direction, equation 4.66 can be approximated as, . at sin 2 0 . , \ji(0)\ = 1 (4.67) I A n 7 1 ' [ l + o„ cos2 0y y ' The max imum value of \jl(0)\ which represents the coefficient of l imit ing friction is given by fi* = . A s can be seen from Figures 4.5a and 4.5b, positive and negative values v i - 4 for an have the effect of producing distributions that are elongated in the direction of the reference axes. 4.5.2 Theoretical and Experimental Distributions for LI(0) in Two Dimensions In section 4.3, the theoretical distribution of JI in three dimensions was derived as, 1 A A?- ti n,-M = ^ A * ' ! «,J = 1,2,3 (4.68) I AA^- n, nj in which, and, 0 < \jl\ < LI* (4.69) A A j l n~ — (AA*„ nr n.) n.- , t<= ' r A , \ " ^ ; t = l , 2 , 3 (4.70) Chapter Chapter 4 : 128 In equations 4.68 and 4.70, n\ = cos <p sin xp, n2 = sin <f> sin ip and 713 = cos tp. The two dimensional counterpart of equation 4.68 is as follows; 1 AA*- U nj in which n = (cos 9, sin 9) and t = ( - s i n 0, cos 0). In addition, /7(0) must satisfy the constraint that, 0 < |/Z(0)| <n* (4.72) Substituting for and n in equation 4.71, /7(0) can be rewritten as, g ,1 - a t s m 2 ( 0 - 0 t ) . P V ' L 2 1 + a t c o s 2 ( 0 - 0 ( ) J v ; / ( A A ^ - A A ^ A A ^ 2 2 A A * 2 in which a t = ( A A * i + A A * 2 ) and tan 291 = ^ J & y . The distr ibution of Jt(9) derived from theoretical considerations (given by equation 4.73) and observed from numerical experiments ( given by equation 4.66) are seen to be very similar in form. The experimental distribution of /7(0) is described by four parameters 0t, 9j, an and at where as the theoretical distribution is described by two parameters at and 6t which are functions of the incremental strain ratios. It was shown earlier that the numerical values of 6f and 0f in the experimental distribution are approximately equal in magnitude. For the same case, the magnitude of at was about 1/4 of an. In the theoretical distr ibution, 9j = $t and at = an/2. The differences in the magnitudes of an and at observed above may be attr ibuted to the relatively small number of contact normal orientation groups considered in the numerical experiments when computing the mobilized friction distribution. In equation 4.73, the term at is a function of the strain increment ratios, the max imum value of which can be shown to be 2 \i* jy/l + 4 /t*2. For most sands, / i * = 0.3 for which <k\max = 0.51. Hence, for /7(0) distributions to be realistic, the value of at computed from the strain increment ratios must be constrained to 2 ji*/y/l + 4 /t*2. Above constraint can Chapter 4 : 129 be imposed numerically without difficulty. The angle 0t measures the rotation of the distr ibution of Jl(0) with respect to the coordinate axes. When shear stresses A c i 2 are applied such as during rotation of principal stress directions, shear strains A A £ 2 are produced and the distribution of Jl($) changes accordingly. 4.5.3 Three Dimensional Distribution of JI The derived distr ibution of JI given by equation 4.68 is in three dimensions. In section 4.5.2, the two dimensional representation of equation 4.68 was shown to agree closely with the observed form of distr ibution of Jt(d). In the absence of observed data on the distribution of JI and based on the comparisons presented in section 4.5.2, it is assumed herein that equation 4.68 can be used to describe the distr ibution of JI in three dimensions without ambiguity. The unit tangent direction vector t, in three dimensions is a function of the strain increment ratios as given by equation 4.70. The determination of the magnitude of t involves solving a system of non-linear simultaneous equations, t enter JI calculations from the term AA?- i , nj(i,j = 1,2,3) in the numerator of equation 4.68. Since the magnitude of t cannot be obtained in a closed form manner, explicit control over the magnitude and distribution of JI cannot be exercised and therefore an alternate function which approximates the distribution of mobil ized friction has been proposed. This is done by comparing the functional forms of the numerator and denominator of equation 4.73. It can be seen that the term at appears in both the denominator and numerator of equation 4.73. For the three dimensional case, the denominator is AA*- re,- rij[i,j = 1,2,3). Rearranging terms associated with AA?- re,- rij(i,j = 1,2,3) and analogous to equation 4.73, the following distribution Chapter 4 : 130 function has been proposed for /x(O); _ _ [o t l sin 2 ((p - <f>t) + sin 2 ( xp - xpt)\ [ 1 + atl cos 2 (<p - <pt ) + at2 cos 2 (xp - xpt )] in which, (4.74) ( ( A A ^ - A A ; 2 ) 2 + 4AA- 2 2) 0- 5 (AA * x + A A ; 2 + 2 A A y ^ ' *> ah = [((2AAg 3 - AX*U - A A 2 2 ) - ( A A i ! - A A 2 2 ) cos2(f> - 2AX*usin2<p)2+ + 1 6 ( A A j 3 c o ^ + AX*23sin4>)2}°-5 [AX*n + A A * 2 + 2AA ; 3 ] [ ' 2AA* t a n 2 % h = 4 ( A A * 3 W + AX*23sin4>) 7 g W t ( ( 2 A A 3 3 - AX*n - A A 2 2 ) - ( A A ^ - AX\2)cos2<p - 2 A A ^ 2 s m 2 (p) K ' ' When simplified to two dimensions (i.e. A A ^ = A A 2 2 , (p = 0, and A A ^ 2 = 0 ), equa-tion 4.74 reduces to equation 4.73. The maximum value of JI w i l l depend on the mag-nitude as well as the signs of and a^. When both a^ and at2 are of the same sign, au\max = : ,—/ , ~ , and at2\max = : ,—/ - • When a 4 and at„ 1 aj 2 / i * iij 2 ( i * are of different signs, au \max = : — / y , and at, mai = - ^ — / y-• 1 1^ 1 " ^ l V 1 + 4 " la<i " V 1 + 4 " 4.6 The Resistance Function # ( The resistance function, H which appears in the incremental stress-strain equations 4.24, controls the magnitude of strain increment for a given stress increment or vice versa and is analogous to a plastic modulus. Laboratory experimental data on real sand indicate that wi th increasing mean normal stress and contractive volumetric strains such as occurring during hydrostatic loading, the resistance to deformation increases continuously. O n the other hand, with increasing stress ratio the resistance to deformation decreases continuously becoming zero at peak stress ratio or at failure. A t unloading and reloading points, abrupt changes occur in the magnitude of resistance to deformation. For the proposed stress-strain Chapter 4 : 131 relations to be meaningful, Ht must therefore consist of parameters that are capable of taking account of the above laboratory observations. Ht is the linear product of the coordination number 7 , particle density nv, particle diameter do, the contact force distr ibution parameter c' and the equivalent incremental normal stiffness parameter k^. For a system consisting of equal size spherical particles, nv can be related to the void ratio e and the particle diameter do. Even for such simple systems parameters such as 7 and c' cannot be measured directly. As noted earlier in section 4.3.2.3.2, k^ is defined differently for loading and unloading stress conditions. Herein, k^ is considered to be a parameter that can be measured or calibrated using laboratory stress-strain data. In what follows, an attempt wi l l be made to study the three particulate parameters 7 , n„ and c' in the light of the numerical experimental results. The main objective of the exercise is to identify their relationships to boundary variables such as stress and strain. (i) average coordination number 7 The average coordination number 7 , is the number of contacts per particle averaged over the entire assembly. For two dimensional systems of equal size particles 7 has been shown to vary between 3 and about 5 (Bathurst (1985)). The lower value is for loosely packed systems and the higher value is for densely packed systems. Irrespective of the init ial coordination number, its magnitude at critical-state is expected to be close to 3. The corresponding numbers of 7 for equal size spherical particles are expected to be between 4 and 6. Results from Bathurst ' s experiments (on two dimensional systems) indicate that 7 is dependent on the mean normal stress am, shear stress roct and the individual particle normal stiffness fc^. A t unloading and reloading points, the magnitude of 7 has not been found to exhibit any abrupt changes. (ii) particle density, n„ Chapter 4 : 132 The particle density nv denotes the ratio of the total number of particles N to the volume of the assembly V. By definition, the void ratio relates the solid volume Vs to the total volume V, in the following manner, For a system consisting of N particles of diameter do, (4.79) 3 1 2 Therefore, nv can be expressed as, V.= ^ l ^ ] ' N (4.80) «. = 77 = — [-F] (4-81) V 7T 0% L V Substituting from equation 4.79 into 4.81, n OQ 1 + e (4.82) Being a function of void ratio, nv does not exhibit abrupt changes in magnitude at the unloading and reloading points. Nevertheless, it is a directly measureable parameter, (iii) contact force distribution parameter, c' The non-dimensional parameter c', is related to the manner in which the normal and tangent contact forces of contacts of a given orientation are distr ibuted. Had the normal and tangential contact forces been distributed uniformly, c' = 1. For a distr ibution such as that given in equation 3.50, c' = 0.5. V i rtual ly no information is available on the behaviour of f ' for changes in stress level, stress path etc. Herein, f ' is treated as a parameter that is likely to vary with stress level and stress path, perhaps wi th the abil ity of exhibiting abrupt changes in magnitude at unloading and reloading points. From the discussions presented above, it is evident that the resistance function Ht con-Chapter 4 : 133 stitute of parameters some of which are not directly measureable. Nevertheless, qualitatively it appears that Ht is capable of modelling the observed variations of the magnitude of strain increments adequately. Un t i l further information becomes available on the variations of the particulate parameters c' and 7 , an alternate form for Ht is proposed that is a function of the stress ratio, mean normal stress, volumetric strain and void ratio. The abrupt changes in Ht at unloading and reloading points are modelled using a concept very similar to that in bounding surface plasticity. Accordingly, the failure surface in the principal stress space is considered to be the bounding surface which can lie outside of the critical-state surface. The resistance function Ht is made to depend on the distance between an image stress point on the bounding surface and the current stress point, as shown in Figure 4.6. The proposed form of Ht is as follows; in which h^, h\ , h2, m, r) and n are model parameters which are constant for a given material, 77,- is a stability parameter which is selected to be around 0.05 to ensure that Ht > 0 prior to reaching peak stress ratio conditions. The manner in which strain softening is handled (for which case Ht < 0) wi l l be outlined later. The parameter TJR denotes the stress ratio at last unloading or reloading point and is considered to be zero for virgin loading. It is postulated herein that the above form of Ht is capable of modelling the laboratory stress-strain response of sand. The influence of the particulate parameters 7 , c', nv and do are impl ic it ly taken account of through the model parameters and the state variables considered in equation 4.83. Laboratory stress-strain data obtained from hydrostatic load-unload tests indicate that Chapter 4 : 134 the unloading modulus is stiffer than the loading modulus. In equation 4.83, when rj = 0; Ht = hho[l + e,h1 + el h2] [ ^ ] m (4.84) O~mo In order to model the hydrostatic load-unload response using eqution 4.84, the exponent m is defined differently for loading and unloading or reloading. Accordingly, m = m\ during virgin loading and m = mi during unloading or reloading upto the max imum past mean normal stress. Typical ly m<i has been found to be about 1.5 to 3.0 times the magnitude of m i and dependent on the type of sand. During unloading or reloading with respect to the stress ratio, r)n is assigned the value of stress ratio at the last unloading point. The magnitude of the denominator [ Vi + + Vcs (1 + f] (1 — i f j ) ) l m equation 4.83, therefore increases abruptly. In effect, the distance between the current stress state and the corresponding failure state stress has been increased. Figure 4.6 shown earlier illustrates the above variations of the distance to failure surface for a general stress path in three dimensions. The critical-state for sand, as defined in the principal stress space and e — logio (crm) space, is shown in Figure 4.7. The particular shape considered for the critical-state surface of sand is based on the assumption that the shapes of the critical-state and failure surfaces are similar. In Figure 4.8, the shape of the failure surface obtained for Cambr ia sand using the true triaxial test device (Ochiai & Lade(1983)), is compared with the proposed shape. In what follows, the critical-state surface in the pr incipal stress space is defined as; 2 ? (4.85, " * [(1 + 3) - ( 1 - ? ) « » 39'] where 0' is the Lode's angle and f3 is a constant that defines the shape of the surface. The critical-state stress ratio along the conventional tr iaxial compression path, r)C3c, is defined Chapter 4 : 135 Fig.(4.6) Incremental Resistance to Deformation at Load-Unload Points Chapter 4 : 136 tfm l o g t 6 ) m Fig.(4.7) The Crit ical-State Surface in a )Pr inc ipa l Stress Space b)Void Rat io - Mean Norma l Stress Space Chapter 4 : 137 Fig.(4.8) The Approximated Shape of Steady-State Surface in P r inc ipa l Stress Space Chapter 4 : 138 as; 2\/2 s i n 4>cv B„x Voc = 75 • T \ ( 4 - 8 6 (3 - stn <f>cv) The different shapes of the critical-state surface that can be obtained for different values of /?, using equation 4.85, are compared with the laboratory data of Ochiai &; Lade(1983) in Figure 4.8. The data correspond to a peak friction angle of 40°. Even though it is understood that the critical-state friction angle of most sands is less than 40°, a comparison is made in Figure 4.8 on the shape of the critical-state surface that can be obtained from the particular functional form adopted. It can be seen that /3 = 0.75 describes the best fit for the data and this value wi l l be used in all model predictions presented in Chapter-5. Ht, described in equation 4.83, does not contain a l l information on the changes in resistance for deformation. In particular, the strain softening behaviour observed in sand, cannot be modelled. A t a particulate level, strain softening is viewed as destruction of particle chains to an extent that the new chains are incapable of carrying the load already in existence. It is first necessary to identify the stress conditions at which strain softening occurs. The criteria for strain softening are described below. when rj > r)cg (1 + r)(l )) —> strain softening (4-87) when r) < rjC3 ( l + r)(1 )) —> strain hardening (4.88) ec3 From equation 4.87, it can be seen that when e < eC3 and r) > 0, the stress ratio r) can exceed the critical-state stress ratio rjC3. Clearly, the peak stress ratio (alternately interpreted as the peak friction angle) is dependent on the ratio ~ which is dependent on the mean normal stress level and density of packing. In what follows, two different forms for Ht applicable for strain hardening and strain Chapter 4 : 139 softening are outlined. Accordingly, for strain hardening •Hht = hhQ [ l + Ai ev + h2e2v] [ ^ } m [ l For strain softening conditions, W ~ VR\ (4.89) HH = [1 + S l sin (,( W _ f / ) » ) ] (4.90) in which 7 7 , ^ denotes the maximum stress ratio which is greater than r)cs. s\ and s2 are model constants and for simplicity h^ can be obtained as a fraction of h^. Note that equations 4.89 and 4.90 are empirical. The above form of HH offers sufficient f lexibi l-ity in representing strain softening behaviour observed in laboratory experiments. When undrained conditions prevail, the fluid bulk modulus is added to the formulations as shown below (Naylor (1978)); {Aa} = [[D\h + [D\s} {AX*} {Ao} = [[D), + [D]f] {AX*} (4.91) (4.92) in which [D\h and [D]s are the constitutive matrices resulting from substituting equations 4.89 and 4.90 to equation 4.24, respectively. The matrix [D\j contains the apparent f luid bulk modulus Hf as shown in equation 4.93 below. (Ao-n\ (Hf. Hf Hf 0 0 o \ fAX*u\ Ac r 2 2 Hf Hf Hf 0 0 0 AA; 2 ACT 33 Hf Hf Hf 0 0 0 AA; 3 A<712 0 0 0 0 0 0 AA* 2 Aff 13 0 0 0 0 0 0 AA* 3 V A<723 y V 0 0 0 0 0 0) V AA; 3 J (4.93) where, -k- = /, , \ v h i,,.\a—- in which En^d is the bulk modulus of f luid and H30ud is "f \l+e)Hfltad Vl+e)Hsolid 1 the bulk modulus of soild particles. Chapter 4 •' 140 4.7 Numerical Integration of the Stress-Strain Equations The stress-strain equations described in equations 4.24 cannot be integrated in a closed form manner. Herein, they have been integrated numerically using the Gaussian quadrature. In the Gaussian quadrature, the fully denned integrand is approximated with a Legendre polynomial of order n. Accordingly, the integration is approximated as, where, (z,-, are sampling points selected with in the prescribed l imits -1 and +1. w, and uij are a set of coefficients selected to minimize the error in approximating the function f(x, y) with the Legendre polynomial, n denotes the number of sampling points as well as the order of the Legendre polynomial. Essentially, the Gaussian quadrature is concerned with selecting the sampling points for evaluating a given function in an optimal manner. The l imits of <j> and i/> that appear in equation 4.29, can be transformed into those shown in equation 4.94 by simple linear transformations. The number of sampling points have to be selected according to the type of function f(x,y). The integrand of equation 4.29 contain sine and cosine functions raised to the powers of 8 or 9. To evaluate a function consisting of cos 8 9 and sine8 9 exactly, a min imum of 24 sampling points must be selected with in the l imits -1 and +1. 4.8 Summary Incremental stress-strain relations that are applicable to frictional systems have been developed. The contact normal distr ibution, developed friction distribution and the laws that define the extent to which changes in contact normal and friction distributions can occur, which form the key features of frictional systems, have been incorporated into the theoretical formulations explicitly. The stress-strain relations so formulated, are applicable (4.94) Chapter 4 • 141 for general loading conditions in three dimensions. The proposed theory assumes the load-deformation behaviour of contacts to be non-linear, both in tangent and normal directions. The non-recoverable deformations resulting from the rearrangement of particles are quantified assuming the system to behave in an incrementally linear manner and updating the material properties at the end of each load-ing increment. Cr i ter ia that distinguish loading from unloading are denned. Loading is defined as a change of stress state associated with a positive change in stress ratio whereas unloading is defined as a change of stress state associated with a negative change in stress ratio. The equivalent tangent stiffness of contacts that are sliding or in the verge of sl iding is reduced to a lower value so that large shear deformations occur on these planes. The compatibil ity conditions between the contact deformations and the boundary strains are obtained using the principle of v irtual forces. Incremental stress strain relations are derived considering the first variation of the compatible contact deformations and boundary strains derived using the principle of v irtual forces. The derived relations are applicable for in -finitesimally small increments of loading where the friction distr ibution and the directions of tangent contact forces are assumed to remain constant. The changes in resistance to de-formation observed in laboratory tests at unloading and reloading points are incorporated by varying the resistance function Ht in a manner similar to that of the plastic modulus in the bounding surface plasticity theory. In the absence of sufficient data to establish the variations of the particulate parameters 7 and c', empirical formulations are proposed for the resistance function Ht which is in principle analogous to a plastic modulus. The proposed incremental stress-strain relations are capable of modell ing the complex response of sand which is dependent on the stress level, stress path, anisotropy of the material and density of packing and are validated in the next chapter. C H A P T E R 5 VERIF ICATIONS A N D DISCUSSIONS 5.1 Introduction In this chapter, the capabilities of the proposed stress-strain model to predict the ob-served behaviour of sand is demonstrated. First, it is shown that the proposed stress-strain model is capable of predicting the characteristic behaviour of sand. The sensitivity of the predicted stress-strain response to a i 10% variation in each of the model parameters is presented thereafter, from which the most sensitive model parameters whose magnitudes need to be evaluated accurately are identified. More refined model capabilities are presented subsequently, comparing model predictions with laboratory observations. The laboratory observations consist of data obtained from three major sources: (a) data reported by Negussey(1984) for standard Ottawa sand using the conventional tr iaxial apparatus, (b) data obtained from the U B C hollow cylinder apparatus for standard Ottawa sand (these tests have been very graciously carried out by A . Sayao for the author. The data wi l l be reported in the forthcoming Ph.D thesis of A . Sayao), and (c) data provided for the International Workshop on constitutive equations for granular non-cohesive soils, Case Western Reserve University at Cleveland, Ohio, during July 142 Chapter 5: 143 1987. The data from this package consist of true triaxial and hollow cylinder test results carried out on Hostun and Reid Bedford sands. The stress-strain data obtained from the conventional triaxial apparatus does not con-tain information on all aspects of sand behaviour. In particular, conventional tr iaxial test data does not contain information on the influence of intermediate principal stress and pr in-cipal stress rotations. However, this information is contained in true triaxial and hollow cylinder test results. Together, the conventional tr iaxial, true triaxial and hollow cylinder data form a complete stress-strain data base that allows the proposed theoretical formula-tions to be verified. The data base for the Cleveland Workshop(1987) consists of two series of true tr iaxial and hollow cylinder data; one set for model calibrations and the other set for verifying model capabilities. The calibration data was made available to the predictors first. Thereafter, the stress-strain response along selected stress paths were to be predicted using the model parameters obtained from the calibration data. A similar procedure was followed by the author when predicting the stress-strain response for the Cleveland data base where only the calibration data was used to derive the model parameters. Researchers engaged in the development of stress-strain models benefit from the Cleve-land Workshop in many ways. The data base provides an easily accessible, accurate, com-plete and homogeneous stress-strain data set. More importantly, the workshop relieves the researcher from the burden of evaluating a newly developed stress-strain model against each of the other popular existing models, since most of the existing models have already been compared w i th the Cleveland data base. The gradation curves for Ottawa, Hostun and Reid Bedford sands are shown in F ig -ure 5.1. A l l three sands that form the data base are fine to medium sands with uniform gradations. The D^Q'S of the three sands varied between 0.25mm to 0.40mm, the Reid M. |.T. GRAIN SrZE SCALE 24 12 Slz« of opening, Incho 6 3 lh I 3/4 W 3/5 4 U. S. S. ii«v« ilzt, mothtt/lnch 10 I , I J 20 40 60 100 200 i, , i , i ,, ,i LEGEND'. — • HOSTUN — s — REID B E D F O R D — « — OTTAWA 100 10 1.0 0.1 0.01 0.001 0.0001 1 1 I 1 1 BOULDER S I ZE COBBLE S IZE c o a r s e med ium f i n e c o a r s e mediuml f i n e f i n e g r a i n e d G R A V E L S I Z E S A N D S I Z E S I L T S I Z E C L A Y S I Z E Fig.(5.1) Grain Size Distributions Curves for Ottawa, Hostun and Reid Bedford Sands Chapter 5: 145 Bedford sand having the lower value and Ottawa sand having the higher value. The model predictions for each of the above sands are presented in detail in the forthcoming sections. Separate sections outl ining the input requirements of the proposed stress-strain relations (i.e. section 5.2) and the method of determining input parameters from laboratory test results (section 5.4) are presented in the forthcoming sections. 5.2 Input Requirements for the Proposed Stress-Strain Relations The input requirements for the proposed stress-strain model can be classified into two groups; (i) parameters specifying the in it ia l state of the material, and (ii) model parameters. Parameters that specify the in i t ia l state of the material are the following; (i) in it ia l void ratio, e° (ii) in i t ia l state of stress, cr~ ; i,j = 1,2,3 (iii) in i t ia l contact normal distribution specified by the anisotropy coefficients a .^ and 6^y. The in i t ia l void ratio and the state of stress, for most controlled laboratory experiments, can be directly measured. However, techniques of directly measuring the init ial contact normal distr ibution have not been developed yet. Therefore, the init ial contact normal distr ibution has to be estimated through indirect measurements. As outlined later, the in i t ia l contact normal distr ibution is estimated indirectly by selecting a°{- and such that correct strain increment directions are predicted for two or more stress increment directions. The model parameters originate from the following formulations; (i) incremental resistance function during hardening, parameters involved are : h^, hi, h2, m\, m2, n, r) Chapter 5: 146 hkQ = hardening tangent modulus parameter for a given mean normal effective stress and relative density (from laboratory data) hi,h2 = hardening parameters (from laboratory data) mi, m2, n = exponents (from laboratory data) r) = parameter that establishes the difference between the critical-state stress ratio and peak stress ratio (from laboratory data). (ii) incremental resistance function during softening, parameters involved are : haQ,si,S2 h3o = softening tangent modulus (selected as a fraction of h^) 81,82= parameters denning variations of tangent modulus (from tr ia l and error matching of predictions with laboratory data). (iii) incremental fluid resistance, Hf parameters included are : Hf (iv) critical-state conditions for the material parameters involved are : <pcv, (\Cs, e°Cs> a<cs) <pcv = constant volume friction angle (from laboratory data) Xcs = the slope of critical-state line in e — logio ( am ) space approximated by a straight line (from laboratory data) (v) parameters defining changes in anisotropy coefficients. parameters involved are : ai, a2, ctz,(3i, f32 a i , 0 : 2 , 0 : 3 = parameters defining changes in second order anisotropy, o,j. 0i,02 = parameters defining changes in fourth order anisotropy, 6,^ Both ol{ and /?,• are obtained from matching one set of deviator stress - axial strain - volumetric strain data obtained from a laboratory test involving change of stress ratio. (vi) coefficient of l imit ing friction Chapter 5: . 147 parameters involved are ; //* (select fi">  a°ij a n d (in (v) above) such that the correct strain increment directions are predicted for varying stress increment directions). 5.3 The Computer Program S A N D The stress-strain behaviour of an element consisting of a particulate, discrete and fric-t ional material can be simulated using the computer program S A N D . Wi th in the element, stress-strain and material properties are assumed to be homogeneously distributed. Bo th stress and strain controlled loading can be imposed. S A N D has been coded in fortran-iv, the flow chart of which is shown in Figure 5.2. 5.4 Model Predictions - I 5.4.1 Characteristic Response The characteristic behaviour of sand, as predicted by the proposed stress-strain model, is presented in this section. The predictions are for conventional tr iaxial loading paths where the intermediate and minor principal stresses are equal and constant in magnitude and correspond to the single set of model parameters shown in Table 5.1. It should be noted that the model parameters used in the characteristic predictions have not been obtained from laboratory experimental data. A l l model predictions on characteristic behaviour of sand correspond to an init ial ly isotropic material (i.e. a?- = 0 = b^-u). The model parameters, summarized in Table 5.1, correspond to an in i t ia l confining pressure of 50 kPa. Initial confining pressures other than 50kPa are attained through hydrostatic loading starting from a hydrostatic stress of 50 kPa , Chapter 5: 148 S T A R T RE- VD I N P U T I N F O R M A T I O N IKI ' r i A L I Z E F O R M U L A T I O N S S T A R T L O A D I N G A N A L Y S I S P R E C E D U R E = D R A I N E D ? IN F L U I D M O D U L U S = 0 F L U I D M O D U L U S = F L U I D M O D U L U S I T F O R M C O M P L I A N C E M A T R I X R E A R R A N G E K N O W N S A N D U N K N O W N S S O L V E F O R U N K N O W N S R E A R R A N G E K N O W N S A N D U N K N O W N S A N D F O R M Aa ii At A N A L Y S I S P R O C E D U R E = D R A I N E D ? N C O M P U T E P O R E P R E S S U R E S ' Y C A L C U L A T E N E W E F F E C T I V E STRESSES C H E C K F O R U N L D N G / R E L D G / S O F T N G 1^ U S E A P P R O P M O D U L I & R E P E A T . L O A D S T E P N, U P D A T E S T R E S S E S , STRA INS , A N I S O T R O P Y , M O B I L I Z E D F R I C T I O N U P D A T E M O D U L I F O R N E X T L O A D S T E P E N D O F L O A D I N G N R E A D N E W L O A D I N G S T O P Fig.(5.2) F low Char t for the Computer Program S A N D Chapter 5: 149 Table 5.1 Input Parameters for Predict ing Characteristic Behaviour of Sand Parameters Specifying Initial State a l l ; a 22 ' a 33 0.0,0.0,0.0 0.0 c° 0.660 Model Parameters N) 34380. kPa 1.30 h2 -0.50 m i , m 2 0.30,0.45 n 1.20 V 1.00 30.0° Xcs 0.15 4 0.705 hsQ 344.0 k P a 2.0 1.5 ai 2.5 <*2 0.50 <*3 1.50 01 2.00 02 1.00 0.25 Chapter 5: 150 using the model parameters shown in Table 5.1. Since the laws that define the changes in anisotropy (described in section 4.4.3) depend on the changes in stress ratio, hydrostatic loading (i.e. a constant stress ratio path) preserve the in it ia l isotropy in the material. The model predictions comprise of the following; (i) hydrostatic loading starting from 50 k P a and loading to 350 kPa followed by unloading to 50 k P a (ii) conventional tr iaxial compression loading starting from confining pressures of 50, 150, 250, and 350 kPa , and (iii) conventional tr iaxial compression loading and subsequent unloading starting from confining pressures of 50, and 250 kPa. 5.4.1.1 Hydrostatic Loading Unlike metals, sands undergo considerable changes in volume during hydrostatic load-ing, of which some are non-recoverable. Dur ing hydrostatic loading, the tangent bulk mod-ulus increases continuously and the unloading response becomes stiffer than for the loading response. Figures 5.3a and 5.3b illustrate the mean normal stress - volumetric strain - axial strain variations for hydrostatic loading and unloading between 50 kPa and 350 kPa. The characteristic stiffening in the incremental bulk modulus, the stiffer unloading response and the occurrence of non-recoverable strains can be predicted from the proposed model. Since the material has been assumed to be isotropic, the volumetric strains predicted are three times the magnitude of axial strains. 5.4.1.2 Conventional Triaxial Compression Loading for Different Confining Pressures The stress-strain response along the conventional tr iax ia l compression loading paths Chapter 5: 151 Q-. o o o - -€> PRE unloading J 1 i i J L (a) L _ 0 0 0.1 0.2 0.3 0.4 VOLUMETRIC STRAIN (X10 " 2 ) 0.5 c\) o I O •—< < ° f— en u cy H ° W o o o Jp 1 oad ing & 0 un load ing S 1 1 l I 1 I I ( b ) i i 0.0 0.1 0.2 0.3 0.4 AXIAL STRAIN (X10 " 2 ) 0.5 Fig.(5.3) Characteristic Response Along Hydrostatic Load-Unload Path a)Mean Normal Stress vs Volumetric Strain b)VoIumetric Strain vs Axial Strain Chapter 5: 152 are predicted for four different confining pressures; 50, 150, 250 and 350 kPa. The init ial conditions for the four cases are shown in Figure 5.4 in the void ratio e - logio (am) space, together with the cr it ical state line assumed for the material. It can be seen that the init ial conditions for the first two cases lie to the left of the critical-state line (i.e. dry of critical state) and the latter to the right of the critical-state line (i.e. wet of critical state). The deviator stress - axial strain - volumetric strain predictions for confining pressures of 50 and 150 kPa are shown in Figure 5.5, and for confining pressures of 250 and 350 kPa are shown in Figure 5.6. The volumetric strain predictions shown in Figure 5.5 are initially contractive tending to be dilative at large axial strains whereas in Figure 5.6, the volumetric strain predictions are contractive, the rate of contraction with axial strain diminishing at large axial strains. The volume change response of sand, as outl ined earlier in section 2.2.1, is dependent on the density of packing as well as the confining pressure. As seen from Figure 5.5, at higher shear loads, the model predicts dilative volumetric strains. It can also be seen that when the confining pressure is increased, the rate of di lation reduces considerably indicating that the model can properly account for shear induced volume changes and its variations associated with density and confining pressure. Note that in model formulations, the effects of both the confining pressure and the density of packing are explicitly accounted for through the ratio ecs/ e which is conceptually similar to the state parameter . 5.4.1.3. Conventional Triaxial Compression Loading and Subsequent Unloading for Different Confining Pressures. The load-unload response predicted for two different confining pressures of 50 k P a and 250 kPa, are shown in Figures 5.7 and 5.8, respectively. The stiffer, deviator stress - axial strain response observed upon unloading subsequent to loading and the associated changes in the incremental ratios of strain, observed in laboratory data (refer to Figure (2.12)), can Chapter 5: 153 Chapter 5: 154 Fig.(5.5) Stress-Strain Response Along Conventional Triaxial Path a)Deviator Stress VB Axial Strain b)Vohunetric Strain vs Axial Strain Chapter 5: 155 0.0 0.02 0.05 0.08 0.10 0.13 0.16 0.18 AXIAL STRAIN ( X 1 0 " 1 ) 0.0 0.02 0.05 0.08 0.10 0.13 0.16 0.18 AXIAL STRAIN (X10"1) Fig.(5.6) Stress-Strain Response Along Conventional Triaxial Path a)Deviator Stress vs Axial Strain b)Volumetric Strain vs Axial Strain Chapter 5: 156 o o — SIGC = 50 KPA o in m o W d _ 04 «° H in 0.0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 AXIAL STRAIN (X10~2) Fig.(5.7) Stress-Strain Response Along Conventional Triaxial Path Involving Loading, Un-loading and Reloading a)Deviator-Stress vs Axial Strain b)Volumetric Strain vs Axial Strain Chapter 5: 157 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10" 1) 0.16 0.18 0.0 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10"1) 0.16 0.18 Fig.(5.8) Stress-Strain Response Along Conventional Triaxial Path Loading, Unloading and Reloading a)Deviator-Stress vs Axial Strain b)Volumetric Strain vs Axial Strain Chapter 5: 158 be predicted adequately. 5.4.2 Sensitivity of Model Parameters on the Stress-Strain Behaviour The sensitivity of the model parameters, listed in section 5.2 earlier, on the predicted stress-strain response of sand, is addressed in this section. Of particular importance is to isolate parameters of high sensitivity so that their magnitudes can be evaluated accurately during the data reduction process or alternately, experiments could be designed so that the parameters under consideration could be evaluated accurately. The determination of the degree of sensitivity of a given model parameter, on the stress-strain response, wi l l depend on the following; (1) the stress path under consideration, and (2) the variables whose variations are evaluated. In what follows, the sensitivity of the deviator stress - axial strain - volumetric strain variables are evaluated for loading along the conventional tr iaxial stress path which has traditionally been referred to in explaining the stress-strain response of sand. The sensitivity analysis has been carried out varying the magnitude of each of the model parameters given in Table 5.1 by ± 10%, keeping the rest of the parameters constant. The predicted deviator stress - axial strain - volumetric strain variations with a +10% change, no change and - 10% change of a given parameter are plotted in the same figure for comparison. The resulting variations predicted are shown in Figures 5.9a through 5.9r. From the results shown in Figures 5.9a through 5.9r, it is evident that the magnitudes of <pcv and eQct have the most significant influence on the deviator stress - axial strain variation as well as the volumetric strain - axial strain variation, particularly at higher shear loads. A ± 10% variation in the magnitudes of a.\, a2, as, f3\, and f32, parameters that explicit ly control the developed anisotropy in the material, influenced the volumetric strain - axial I I Chapter 5: 159 o o — — © N 1 1 1 <J> ( — o PLUS 10 PERCENT ZERO PERCENT MINUS 10 PERCENT © es es o ® o i i i i i i J i i i i 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 Fig.(5.9a) Sensitivity of h^ on the Predicted Response Chapter 5: 160 Fig.(5.9b) Sensitivity of hi on the Predicted Response Chapter 5: 161 X 00 0.02 0.05 0.08 0.10 0.13 0.16 0.18 A X I A L STRA IN ( X 1 0 " 1 ) Fig.(5.9c) Sensitivity of h2 on the Predicted Response Chapter 5: 162 © — 1 — © 1 1 <5> 1 — ? > P L U S 10 P E R C E N T ZERO P E R C E N T M I N U S 10 P E R C E N T o d $ 1 1 1 1 1 1 I I I 0.0 0.02 0.05 0.08 0.10 0.13 0.16 0.18 AX IAL STRA IN ( X 1 0 ' 1 ) o -—- q 0.0 0.02 0.05 0.08 0.10 0.13 0.16 0.18 A X I A L STRA IN ( X 1 0 " 1 ) Fig.(5.9d) Sensitivity of mi on the Predicted Response Chapter 5: 163 Fig.(5.9c) Sensitivity of n on the Predicted Response Chapter 5: 164 •© PLUS 10 PERCENT -+ ZERO PERCENT O MINUS 10 PERCENT - i 1 1 i i i i 0.10 0.13 0.16 0.18 AXIAL STRAIN ( X 1 0 " 1 ) a ,—. o ? 0 , > 9 I I I I l l l l I I I I I I I 0.0 0.02 0.05 0.08 0.10 0.13 0.16 0.18 AXIAL STRAIN ( X 1 0 " 1 ) Fig.(5.9f) Sensitivity of r) on the Predicted Response Chapter 5: 165 o . d (X © | O I 1 O 1 e> ZERO P E R C E N T O O G O G O 00 0.02 0.05 0.08 0.10 0.13 AX IAL STRA IN ( X I O " 1 ) 0.16 0.18 0.05 0.08 0.10 0.13 AX IAL STRA IN ( X 1 0 " 1 ) 0.16 0.18 Fig.(5.9g) Sensitivity of 4>cv on the Predicted Response Chapter 5: 166 d o — — © 1 1 1 0 1 —e> ZERO PERCENT O MINUS 10 PERCENT 0.0 0.02 0.05 0.08 0.10 0.13 A X I A L STRA IN ( X 1 0 " 1 ) 0.16 0.18 0.05 0.08 0.10 0.13 AX IAL STRA IN ( X 1 0 " 1 ) 0.16 0.18 Fig.(5.9h) Sensitivity of A„ on the Predicted Response Chapter 5: d o — © CM 1 1 1 <3> 1 © -o-^-e-o o o © © o i l I l I L 0.0 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN ( X 1 0 " 1 ) 0.16 0.18 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 Fig.(5.9i) Sensitivity of c° on the Predicted Response Chapter 5: 168 p d < -Q _ © 1 — © 1 1 — 1 —<"> J 1 i i i i i I I i i 0.02 0.05 0.08 0.10 0 .13 AXIAL STRAIN ( X 1 0 " 1 ) J L 0.16 0.18 0.0 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 Fig.(5.9j) Sensitivity of h^ on the Predicted Response Chapter 5: O Q P L U S 10 P E R C E N T H 1- Z E R O P E R C E N T <5 0 M I N U S 10 P E R C E N T -J 1 1 1 1 1 I I I I I I J L 00 0.02 0.05 0.08 0.10 0.13 A X I A L S T R A I N ( X I O " 1 ) 0.16 0.18 0 0 0.02 0.05 0.08 0.10 0.13 A X I A L S T R A I N ( X l O " 1 ) 0.16 0.18 Fig.(5.9k) Sensitivity of «i on the Predicted Response Chapter 5: 170 © O P L U S 10 P E R C E N T H »• ZERO P E R C E N T o © M I N U S 10 P E R C E N T J I L I 1 1 I I I I I I I I 0.0 0.02 0.05 0.08 0.10 0.13 A X I A L STRA IN (X10 0.0 0.16 0.18 0.02 0.05 0.08 0.10 0.13 0.16 0.10 AX IAL STRA IN ( X 1 0 " 1 ) Fig.(5.91) Sensitivity of s 2 on the Predicted Response Chapter 5: 171 Chapter 5: 172 d © — © | 1 1 o — I €> o < 2 (X CO W K E— (/J ZERO P E R C E N T 0.0 0.02 0.05 0.08 0.10 0.13 A X I A L STRA IN ( X 1 0 " 1 ) 0.16 0.18 0.0 0.02 0.05 0.08 0.10 A X I A L STRA IN (X10 0.13 0.16 0.18 Fig.(5.9n) Sensitivity of a 2 on the Predicted Response Chapter 5: o — 1 — © I 1 o 1 —e> 0.0 ZERO PERCENT •© MINUS 10 PERCENT j i i i i i i i I i i i 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 0.0 0.02 0.05 0.08 0.10 0.13 AXIAL STRAIN (X10 " 1 ) 0.16 0.18 Fig.(5.9o) Sensitivity of Q3 on the Predicted Response Chapter 5: o © — 1 — © 1 1 o — 1 — © 0.05 0.08 0.10 0.13 AXIAL STRAIN (XIO"1) 0.16 0.18 Fig.(5.9p) Sensitivity of ft on the Predicted Response Chapter 5: o CO CO w H CO K o E-> W Q o 1 — © 1 o 1 —<>> ZERO P E R C E N T J L 00 0.02 0.05 0.08 0.10 0.13 A X I A L STRA IN ( X I O " 1 ) 0.16 0.18 00 0.02 0.05 0.08 0.10 0.13 A X I A L STRA IN ( X 1 0 " 1 ) 0.16 0.18 Fig.(5.9q) Sensitivity of /?2 on the Predicted Response Chapter 5: © — 1 — © 1 1 <3<— 1 — © P L U S 10 P E R C E N T ZERO P E R C E N T M I N U S 10 P E R C E N T £ ) © © © © © © © © 0.02 0.05 0.08 0.10 0.13 AX IAL STRA IN ( X10 " 1 ) 0.16 0.18 0.05 0.08 0.10 0.13 AX IAL STRA IN ( X 1 0 " 1 ) 0.16 0.18 F5g.(5.9r) Sensitivity of on the Predicted Response Chapter 5: 177 strain variation to a degree comparable to that of <pcv but to a lesser degree than of e°cs, although their influence on the deviator stress - axial strain variation was small or insignif-icant. Similar changes in the magnitudes of A„ , (i*, r), n and h^ influenced both the volume change response as well as the deviator stress - axial strain response, to a much lower degree than cpcv. Noticeably, for the stress path under consideration, changes in the considered magnitudes of m i , m2, h\, h2, s\ and s2 had virtually no influence on the stress-strain response. (AA The magnitude of e°C3, together with the corresponding mean normal stress am and Ac,,, specify the location of the critical-state line in the e — logio(o-m) space. A change in the magnitude of eQcs with XC3 and held constant, translates the critical-state line in the e — logio(am) space to parallel positions. Since the anisotropy developed in the sample is governed by the ratio ^ raised to c * 3 , a change in the magnitude of e°C3 introduces significant changes in the developed anisotropy. A s seen from Figure 5.9i, a ± 10% variation in magnitude of e°cs changes the volumetric strain response from a contractive response to a dilative response. The particularly large differences in the volumetric strain response seen in Figure 5.9i is a result of the relatively close in it ia l and critical-state void ratios used as input parameters (that correspond to a mean normal stress of 50 kPa). Note that these numbers for void ratios were selected arbitrarily to illustrate the characteristic response of sand. (ii) 4>cv The magnitude of <pcv enters the theoretical formulations through the critical-state stress ratio r)C3. A higher value of cpcv results in a higher value of r/C3 and vice versa. A s given by equations 4.78, 4.55 and 4.59, the variation of incremental modulus Ht, and the changes in anisotropy coefficients a,y and 6 vy are dependent on the magnitude of r)c3. For Chapter 5: 178 a given state of stress, a higher value of r)cs results in a stiffer value of Ht. Over a given strain increment, a stiffer value of Ht produces larger shear stresses. The changes in the anisotropy coefficients a,j and 6,^ are directly proportional to the changes in shear stresses as well as to the magnitude of ^ raised to an exponent (the exponents are constant, but differ for a,y and 6,JJU). Increased shear stresses therefore increase the changes in o,y and 6,-j-y, whereas an increased value of r)cs would reduce the same. The net effect would be governed by the state of stress and exponents associated with As seen from Figure 5.9g, with a higher value of <pcv, the rate of dilation is seen to increase, indicating the net effect of an increase in <j>cv is to increase the developed anisotropy of the material. Similar arguments can be made regarding the lower rate of dilation observed with the low value of <pcv-(Hi) h^, n h^ and n influence the stress-strain response through the incremental modulus Ht. A higher value of h^ results in an increased value of Ht. On the other hand, a higher value of n would result in lower values of Hf. A s it was outlined above for <pcv, for a given state of stress higher values Ht produces larger shear stresses over a given strain increment producing larger changes in Aa,y and A 6,^ resulting in increased rates of dilation, and therefore the predicted response as seen in Figures 5.9a and 5.9e, respectively. (iv) oty, By Higher values of ay and By, result in larger changes in anisotropy coefficients a,j and bijki, resulting in increased dilation as seen in Figures 5.9m and 5.9p. A s given by equations 4.55 and 4.59, Aa,y oc ( ^ ) a 2 and Abyu oc (^SY2 • When rj < r)cs increased values of a2 and 82 result in reduced values of Aa,j and A6,jju. The corresponding developed anisotropy, therefore should cause reduced rates of dilation of the material. On the other hand when rj > r)cs, such as when close to failure, higher values of Chapter 5: 179 a2 and 02 results in increased values of Aa,y and A6,yju which in turn increase the rate of the material. The results shown in Figure 5.9n and 5.9q correspond to cases when rj < r)C3, and therefore with increased values of a2 and 02 reduced rates of di lation have been predicted. (vi) a3, Xcg The changes in the developed second order anisotropy, Aa,y a (eC3/e)az. Therefore, when e > eC3, a higher value of as results in reduced value of Aa,y resulting in a reduced rate of dilation. On the other hand, when e < eC3, the higher value of 0:3 results in an increased value of Aa,y resulting in an increased rate of di lation. From Table 5.1, e < eC3 and therefore an increase in as should result in an increased rate of di lation. It can be seen from Figure 5.9o that the predicted response is compatible w i th the theoretical formulations. A higher value of XC3 (i.e. steeper slope in e — logio (o~m) space) has the effect of reducing the magnitude of ec$ with increasing mean normal stress. The resulting rates of eC3je and the corresponding changes in Ao,y would therefore be lower. Therefore, as seen from Figure 5.9h , the rate of dilation of the sample would be lower. From equation 4.78, changes in the magnitude of eC3 also affect the denominator [»7» + flcs ( 1 + f] ( 1 _ e/' eC3) ) ]. For r) > 0, when ecs decreases, the magnitude of the denominator decreases thereby decreasing the max imum stress ratio the sample can be loaded to , which has been reflected in the deviator stress-axial strain response in Figure 5.9h. (vii) pL* The contribution of fi* into the constitutive matr ix even though small, is large enough to produce changes in the strain increment direction. A s observed in Figure 5.9r, for frictional systems, the magnitude of (j,* does not influence the shear capacity of the systems. Chapter 5: 180 (viii) r), my, m<i, h\, h<i, s\ ,s2 and h^ For the stress path under consideration, the influence of the considered variations in the magnitudes of parameters r), m\,m2, h\, h2l s\, s2 and A 3 Q were insignificant. For hydrostatic loading, the parameters m\, m2, h\ and h2 dominate the response completely (together wi th h^). The insufficient changes in the mean normal stress with respect to its in i t ia l value of 50 k P a and the low accumulated volumetric strain, for the particular case under consideration, can be cited as possible reasons for the insignificant changes observed in the stress-strain response. O n the surface, the sensitivity analysis indicates that the compounded influence of the two most sensitive model parameters <pcv and e°cs can produce drastic changes in the volumetric strain response when changed by ± 10 % . Such drastic changes in response are highly undesirable, particularly in view of the likely errors in measurements of the parameters. It is generally beleived that the magnitude of <f>cv can be measured to the nearest 0.5 degree which is wi th in a 2 % variation. Therefore, it is the uncertainity associated w i th the measurement of e°C3 that is likely to influence the stress-strain response significantly. Before discussing the matter any further, it should be once again mentioned that the model parameters used in the characteristic predictions have not been obtained from lab-oratory experiments on real sands. Model parameters derived for standard Ottawa sand from laboratory experiments are shown in Table 5.2. A careful examination of the model parameters shown in Tables 5.1 and 5.2 reveal that with the exception of the numerical values of the parameters \CB, cc\, 8\ and a0-, the rest of the parameters in the two tables are identical. Also note that the slope of the critical-state line A C J I, has been magnified seven times for the characteristic predictions. Similarly, the magnitude of a.\ has also been increased from 1.5 to 2.0, whereas the magnitude of 8\ has been decreased from 3.0 to 2.0. The above changes in magnitude of the model parameters were done with the intention of Chapter 5: 181 observing an amplified response for the considered variations in the numerical values of the model parameters during the sensitivity analysis. For real sands, higher values of \ c s are associated with lower values of ot\, a2, 0 : 3 , ft and ft. However, in the analyses carried out herein, no such reductions in the parameters controlling the development of anisotropy were allowed and therefore the response may differ from that of real sands. Therefore, the drastic changes in response predicted during the sensitivity analysis should be viewed with some caution. 5.5 Model Predictions - II So far, only the characteristic stress-strain behaviour of sand has been predicted. As outlined earlier, the characteristic response predictions are based on a single set of compati-ble model parameters that d id not correspond to any specific data base. In this section, the ability of the proposed stress-strain model to predict laboratory observations on Ottawa, Hostun and Reid Bedford sands, obtained from the data bases described in section 5.1, are verified. First, the method of obtaining input parameters is outlined. 5.5.1 Input Parameters from Laboratory Data A l l model parameters that are necessary to predict the stress-strain response of sand can be obtained from the following tr iax ia l compression tests performed for a selected den-sity of packing; (i) hydrostatic loading and in it ia l unloading (A/^, hi, h2, m\, m2) (ii) shearing along conventional tr iaxial path or constant mean normal stress path (prefer-able) a sand sample lying wet of critical-state line all the way to critical-state - starting from hydrostatic conditions =>• (n,<pcv,0:1,0:2,0:3,ft,ft) Chapter 5: 182 (iii) shearing along conventional triaxial path or constant mean normal stress path (prefer-able) of a sand sample with an init ial state dry of critical-state upto the peak stress ratio - starting from hydrostatic conditions and strain controlled loading after peak stress ratio (r), si, s 2) (iv) a sufficient number of conventional triaxial compression tests to establish the cr it ical-state line in the e— log-^ (erm) space =>• ( A M / e ^ u ^ , ) It can be seen that each of the tests described above determines one or more model param-eters (shown within brackets). The in it ia l distribution of contact normals, specified by the coefficients of a?- and and the coefficient of l imit ing friction specified by fi*, remain to be defined. The method of estimating these coefficients for a general case involves a tr ial and error approach. For simpler cases where the principal anisotropy directions (refer to section 2.5 for definition) are coincident with the reference directions and when the material is cross anisotropic, the coefficients a?- reduce to a single term. This term and n* can be estimated as follows; (i) assume 6 ^ = 0 ; k, 1=1,2,3 (ii) select / i * such that LI* ~ tan (\<pCv) (iii) obtain a .^ such that strain increment directions observed from hydrostatic loading and in it ia l unloading are predicted satisfactorily. 5.5.2 Predicted Stress-Strain Response of Standard Ottawa Sand From The Conventional Triaxial Test Set Up Herein, the model predictions are compared with the laboratory stress and strain mea-surements obtained from a series of drained triaxial tests on standard Ottawa sand, as reported by Negussey (1984). For clarity, the comparisons are divided into four parts; (a) comparison of init ia l strain increment ratios for a selected set of stress paths, Chapter 5: 183 (b) comparison of stress-strain response for selected stress paths, (c) comparison of load-unload stress-strain response for constant mean normal stress paths, and (d) comparison of load-unload-reload stress-strain response for the conventional triaxial compression loading path. The bulk of the stress-strain data reported by Negussey (1984) correspond to a relative density of 50%. The model parameters evaluated for standard Ottawa sand with a relative density of 50%, following guidelines outlined in section 5.5.1 above, are shown in Table 5.2. In the absence of critical-state parameters for standard Ottawa sand in the e — log\o (o~m) space, the steady-state parameters reported by Chern(1984) have been assumed represen-tative of critical-state parameters. The above set of model parameters have been used in all predictions presented in section 5.5.2. It should be noted that depending on the stress path followed, the stress-strain variables used in the comparisons, are different. For example, along the hydrostatic stress path, mean normal stress is plotted against the volumetric strain whereas along shear loading paths, the deviator stress is plotted against the axial strain. The stress-strain behaviour has been predicted for two different confining pressures of 50 and 250 kPa, where data is available. Mode l predictions that correspond to proportional loading paths have been obtained feeding the strains as input boundary variables. A long proportional loading paths, the stress-strain variations are strongly dependent on the in i t ia l material anisotropy (as defined by 6°yy) and the developed ratio between the tangent and normal contact forces Note that /J(fi) is a function of strain increment ratios and when stresses are considered as input, an iterative procedure has to be followed in order to obtain compatibil ity between the stress and strain increments. However, when strains are considered an input, there is no necessity for an iterative process. The latter approach was followed herein primarily Chapter 5: 184 Table 5.2 Input Parameters for Predicting Behaviour of Standard Ottawa Sand with Dr = 50% Parameters Specifying Initial State a22> a33 -0.05,-0.05,+0.10 0.0 e° 0.660 Model Parameters 1% 34380.0 kPa hi 1.270 h2 -0.550 mi, m 2 0.30,0.45 n 1.10 n 0.20 <t>cv 30.0° 0.023 e° 0.705 h,0 344.0 kPa Sl 2.0 «2 1.5 ax 1.5 a2 0.50 <*3 1.50 Pi 3.00 82 0.70 0.25 Chapter 5: 185 because strain path data for the samples were available to the author. Along non-proportional loading paths, the strain increment is less sensitive to the applied stress increment and vice versa. Also, fu l l friction is always mobil ized (i.e. that is given by fi*). As a result, either the strains or the stresses can be considered as input variables and no iteration process is necessary. (a) Comparison of Initial Strain Increment Ratios With the Stress Increment Ratios. Figure 5.10 shows the comparison of the predicted init ial strain increment ratio - stress increment ratio variations wi th those observed. In al l cases, loading commenced from a hydrostatic stress of 50 kPa. Therefore, similar in it ia l strain increment ratios should be observed for the same stress paths but subjected to different hydrostatic stresses. As can be seen from Figure 5.10, the model predictions are in good agreement with the laboratory observations, clearly showing the bilinear variation observed in the laboratory. The stress paths that correspond to the data shown in Figure 5.10 are those in which the mean normal stress and shear stress are either maintained constant and/or increased continuously, and therefore do not contain in it ia l strain increment ratios during unloading. The variations in the strain increment ratio during unloading, along selected stress paths, wi l l be compared in the forthcoming sections in (c) and (d). (b) Comparison of Stress-Strain Variations Along Different Stress Paths. The model predictions along the following three stress paths are compared with the laboratory observations; (i) A c r / A o - a = 1.00 - hydrostatic (ii) Ao>/ Ao" a = 0.00 - conventional tr iaxial compression (iii) A<rr/ Ao" a = —0.50 - constant mean normal stress Chapter 5: Chapter 5: 187 For the hydrostatic loading path, the predicted response is compared with the laboratory observations in the mean normal stress - volumetric strain - axial strain frame. Along all other stress paths, the predicted and observed stress-strain variations are compared in the deviator stress - axial strain - volumetric strain frame. (i) When Aar/Aaa = 1.00 The mean normal stress - volumetric strain - axial strain variations predicted are com-pared with the laboratory observations, in Figure 5.11a. The hydrostatic loading curve and the in it ia l part of the unloading curve have been used for model calibration purposes and therefore the results shown are simply back predictions of the calibration data. Loading commenced from an in it ia l mean normal stress of 50 kPa to about 550 kPa, which covers the likely working range of stresses. Higher values of mean normal stress may introduce considerable particle crushing, which is a physically different phenomenon that cannot be captured adequately by the proposed stress-strain formulations. It can be seen from Figure 5.11a that during both hydrostatic loading and unloading, the predicted variations of mean normal stress wi th volumetric strain compare well with the laboratory measurements. The variations of volumetric and axial strains forming the input for the predictions, are shown in Figure 5.11b. (ii) When Aar/Aaa = 0 Mode l predictions along the conventional tr iaxial compression stress path are presented for two different confining pressures of 50 and 250 kPa. The comparison of the deviator stress - axial strain - volumetric strain variations are shown in Figure 5.12a and 5.12b. Data that correspond to a confining pressure of 250 kPa have been used for model calibration purposes. A s it can be seen, the model is capable of predicting the influence of confining pressure, particularly on the volumetric response. When the confining pressure is 250 kPa, the volumetric strain response is mainly contractive, becoming slightly dilative at an Chapter 5: 188 ( b ) °-0 0.06 0.12 0.18 0.24 0.3 AXIAL STRAIN (XlO" 2) Fig.(5.11) Comparison of Stress-Strain Response Along Hydrostatic Loading Unloading Paths a)Mean Normal Stress vs Volumetric Strain b)Volumetric Strain vs Axial Strain Chapter 5: 189 0.0 0.02 0.04 0.06 0.08 0.1 AXIAL STRAIN (XIO"1) 0.12 0.14 O >< f— o < ° f-(/> u g w _J C O O 9 > ? K - X - ~ X  0.0 0.02 0.04 0.06 0.08 0.1 AXIAL STRAIN ( X 1 0 " 1 ) (b) J 1 L 0.12 0.14 Fig.(5.12) Comparison of Stress-Strain Response Along A<rr/Aaa = 0 Paths a)Deviator Stress vs Axial Strain b)Volumetric Strain vs Axial Strain Chapter 5: 190 axial strain of about 1%. However, when the confining pressure is reduced to 50 kPa, the volumetric strain response becomes more dilative subsequent to the in it ia l contractive response. The influence of the confining pressure on the volume change response along conventional tr iaxial loading paths has therefore been modelled satisfactorily. The init ial slopes of the deviator stress - axial strain variations and the l imit ing value of deviator stress predicted are in good agreement with the observations. Note that the max imum deviator stress observed and shown in Figure 5.12a, for a confining pressure of 250 kPa, is below the l imiting value the sample can be loaded. (iii) When AaT/Aaa = -0.50 Volumetric strains resulting from loading along constant mean normal stress paths are exclusively shear induced. As outlined at the begining of the thesis, quantification of shear induced volume change of sand is difficult. Being exclusively shear induced, the volumetric strain data observed along constant mean normal stress paths can be used to verify the ability of a given stress-strain model to predict the shear-volume coupling phenomenon. Deviator stress - axial strain - volumetric strain variations predicted along paths where the mean normal stress was maintained constant at 50 and 250 kPa, are compared with laboratory observations in Figures 5.13a and 5.13b. Both the deviator stress - axial strain variation and the volumetric strain - axial strain variation are in good agreement with the laboratory measurements. Laboratory data showing the variation of volumetric strain with axial strain, for a confining pressure of 50 kPa , are not available for comparison. The predicted volumetric strain response for a confining pressure of 250 kPa, is seen to be slightly dilative when compared w i th the laboratory data. As expected, the predicted volume change response for a confining of 50 kPa , is seen to be more dilative than that for a confining pressure of 250 kPa. Chapter 5: 191 0.16 0.32 0.48 0.64 0.8 A X I A L S T R A I N (X10~ 2 ) 0.06 1.12 0.0 0.16 0.32 0.18 0.64 0.8 A X I A L S T R A I N (X10~ 2 ) 0.96 1.12 Fig.(5.13) Comparison of Stress-Strain Response A long AaT/Aaa = - 1 / 2 Paths a)Deviator Stress vs A x i a l Stra in b)Yolumetric. Strain vs Ax i a l Stra in Chapter 5: 192 (c) Comparison of Load - Unload Response A long Constant Mean No rma l Stress Paths. Herein, the predicted variations of the strain increment directions and the resistance to deformation during loading and unloading along constant mean normal stress paths are compared with laboratory observations. The comparisons have been carried out with the a im of verifying the model capabilities with respect to the following aspects; (i) ability to predict the correct magnitude of strain increment ratios during loading and at unloading points, and (ii) ability to predict the correct modul i of deformation at loading and unloading points. The comparisons presented herein correspond to a constant mean normal stress of 340 kPa. The stress - strain response resulting from loading and subsequent unloading in the compression side and extension sides have been simulated and compared with the available laboratory data. In both compression and extension loading, the deviator stress was increased to a max imum of 171 k P a prior to unloading, maintaining the mean normal stress constant at 340 kPa. Figures 5.14 and 5.15 compare the predicted deviator stress -shear strain - volumetric strain variations with the laboratory data. From Figures, 5.14 and 5.15 it can be seen that the model is capable of predicting the sense as well as the magnitude of volumetric strains satisfactorily. The predicted strain increment ratios as well as the modul i of deformation are seen to be softer than the obser-vations. For unloading from compression, even though the modul i of deformation compare reasonably well with the laboratory data, the predicted strain increment directions are seen to be lower than those observed. A s seen from Figure 5.15b, the predicted strain increment directions for extension side unloading compare well with those observed. It should be noted that during unloading from both compression and extension sides, the predicted as well as the observed changes in volumetric strains are contractive. Chapter 5: o d o C O O W N C4 C O a: ° W 1 co c d -0.3 CM o I O X 2 „ d E-co PRE - SIGM-34 0 KPA OBS - SIGM=34 0 KPA (a) -0.22 -0.14 -0.06 0.02 0 SHEAR STRAIN (XIO"2) 0.18 0.26 (b ) ©--0.3 -0.22 0.18 0.26 -0.1-1 -0.06 0.02 0.1 S H E A R STRA IN ( X I O " 2 ) Fig.(5.14) Comparison of Stress-Strain Response Along Constant Mean Normal Stress Un-loading from Compression Side a)Shear-Stress vs Shear-Strain b) Volumetric Strain vs Shear Strain Chapter 5: 194 Fig.(5.15) Comparison of Stress-Strain Response Along Constant Mean Normal Stress Un-loading from Extension Side a)Shear-Stress vs Shear-Strain b) Volumetric Strain vs Shear Strain Chapter 5: 195 (d) Comparison of Load - Unload - Reload Response Along Conventional Triaxial Compression Path. The load - unload - reload response along the conventional tr iaxial stress path produces volumetric strains due to increasing shear stress, increasing mean normal stress, decreasing shear stress and decreasing mean normal stress, and therefore is more complicated than the load-unload response along a constant mean normal stress path. Figures 5.16a and 5.16b show the observed deviator stress - axial strain - volumetric strain variations during the application of four half cycles of loading and unloading involving no reversal of deviator stress. The test corresponds to a confining pressure of 350 kPa. The deviator stress - axial strain variations are reported in Figure 4.9 of Negussey(l984). The volume change data was very graciously provided to the author by Dr. Negussey so that the complete response can be compared with the model predictions. Discussions with Dr. Neggusey indicated that the particular sand sample has been subjected to a hydrostatic load-unload history between 50 and 200 k P a prior to loading to a confining stress of 350 kPa and this loading history was included in the model predictions. The predicted response is shown in Figure 5.17. The predicted deviator stress-axial strain response is seen to be about 3 0 % softer than the observed response right from the beginning. Nevertheless, the abrupt changes in the incremental resistance to deformation at unload-reload points have been modelled adequately. Noticeably, the ' looping ' effect at the unload-reload points have not been modelled. The volume change response has been predicted satisfactorily wi th correct strain increment directions. 5.5.3 Predicted Stress - Strain Response of Standard Ottawa Sand From the U B C Hollow Cylinder Test Set Up The predicted stress - strain response of standard Ottawa sand for loading with changing intermediate principal stresses and rotating principal stress directions are compared herein Chapter 5: 196 0 0 0.04 0.08 0.12 0.16 0.2 AXIAL STRAIN (X10~ 2 ) 0.24 0.28 0.04 0.08 0.12 0.16 0.2 AX IAL STRA IN ( X 1 0 " 2 ) 0.24 0.28 Fig.(5.16) Observed Stress-Strain Response for Conventional Triaxial Loadmg, Unloading and Reloading a)Deviator Stress vs Axial Strain b)Volumetric vs Axial Strain Chapter 5: 197 Fig.(5.17) Predicted Stress-Strain Response for Conventional Triaxial Loading, Unloading and Reloading a)Deviator Stress vs Ax i a l Strain b)Volumetric vs Ax ia l Stra in Chapter 5: 198 with data obtained from the U B C hollow cylinder apparatus. There were two major reasons that motivated the author for the comparisons; (i) the volume of the sample in the U B C hollow cylinder apparatus is larger than that used in Cleveland Data Base, in which case the measurements are more represen-tative, U B C hollow cylinder apparatus dimensions are: internal diameter = 10.0 cm external diameter = 15.0 cm height = 30.0 cm Dimensions of the hollow cylinder used in generating the Cleveland Data Base are: internal diameter = 5.0 cm external diameter = 7.0 cm height = 12.5 cm (ii) the pr inc ipal stress direction (a) and b(= ^ _ ^ ) variations that can be simulated in the U B C apparatus are more general than the restricted b = sin2 a variations simulated in the hollow cylinder apparatus that was used to produce data for the Cleveland Data Base. The hollow cylinder test wi th which the model predictions are compared , has been performed by A . Sayao and wi l l be reported in his forthcoming Ph.D thesis. The test data has been collected in two stages: Stage - 1 Starting from a hydrostatic stress state of 50 kPa, the sample is loaded to a stress state given by aa = 75 kPa, ar = at = 37.5 kPa for which 6 = 0. Thereafter, o-a,crr and at are changed to vary the magnitude of b from 0.0 to 0.5 such that the mean normal Chapter 5: 199 stress is maintained constant at 50 kPa, and the ratio (J\ja3 constant at 2.0 and the strain components ea, er and et are recorded. Stage - 2 Beginning from a state of stress that correspond to the end state of stage - l ( o " m = 50 kPa, a\ja% = 2.0, b = 0.5), the same sample is now loaded maintaining <J\/as = 2.0 and b = 0.50 to an increased mean normal stress of 300 kPa. The resulting state of stress that correspond to a mean normal stress of 300 k P a is given by ar = 300 kPa, at = 200 k P a and az = 400 kPa. Thereafter, torsion is applied to the sample such that am, aT, a\jaz and 6 are maintained at 300 kPa, 300 kPa, 2.0, and 0.5, respectively and the resulting strain com-ponents ea, er, et and 7 ^ are recorded. The magnitudes of the principal stresses, for the above loading conditions, are constant. It should be noted that in stages - 1 and 2, the loading is applied such that the mean normal stress is maintained constant, although the magnitudes are different for the two stages. Strains recorded in stage - 1 are exclusively shear induced and resulting from changes in the magnitude of the intermediate principal stress. O n the other hand, strains recorded during application of torsion in stage - 2 are also exclusively shear induced, but resulting from the rotation of principal stress directions. In effect, the particular loading conditions under consideration challenge the capabilities of the model to capture the complex shear induced volume change response of sand due to changes in intermediate pr incipal stress and rotation of principal stresses. Tables 5.3 and 5.4 summarize the associated changes in the internal pressure, external pressure, torque and the vertical load for stage - 1 and stage -2 loading conditions. Note that the difference in the internal and external pressures at the end state of stage - 1 is only about 7.0 kPa, the internal pressure being higher. However, the pressure difference across the walls at the beginning of stage - 2 is about 40 kPa, the internal pressure being higher than the external pressure. When a = 60°, the pressure Chapter 5: 200 Table 5.3 U B C Hollow Cyl inder Boundary Values - Stage-1 b Pezt Pint Torque Ax ia l -Load (kPa) (kPa) (Nm) (N) 0.000 37.50 37.50 00.00 333.6 0.200 41.70 44.70 00.00 279.0 0.400 45.40 51.20 00.00 230.5 0.500 47.20 54.30 00.00 207.6 Table 5.4 U B C Hollow Cyl inder Boundary Values - Stage-2 a Pezt Pint Torque Ax ia l -Load (Deg.) (kPa) (kPa) (Nm) (N) 00.00 283.0 325.0 . 00.00 1605.4 10.00 284.0 324.0 23.20 1512.1 20.00 287.0 319.0 43.60 1243.5 30.00 292.0 313.0 58.80 0832.0 40.00 297.0 304.0 66.90 0327.2 50.00 303.0 296.0 66.90 -209.9 60.00 308.5 287.0 58.80 -714.4 Chapter 5: 201 difference is 20 kPa, but this time the external pressure is higher than the internal pres-sure. Such differences in internal and external pressures cause considerable stress gradients across the wall of the sample. The laboratory stress and strain components presented for comparisons later in this section, therefore refer to quantities averaged with respect to the volume of the sample. The loading conditions that correspond to stage - 1 and stage - 2 have been numerically simulated using S A N D and the results are compared with the observed average values of stresses and strains. In the simulation, a cubical element is subjected to uniform stresses as recorded by the averaged sample stresses and the resulting strains are obtained. Ideally, an infinitesimally small element in cyl indrical coordinates is identical to a cube element and the stress difference across the walls of the element are negligible. Since the sample had a relative density of 42% , the in i t ia l anisotropy coefficients <r? , and the volumetric hardening parameters h\ and h2 have been estimated through interpolation of data between relative densities of 30% and 70%. The modified model parameters used in the predictions, are shown in Table 5.5. In what follows, the predicted response of a cubical element subjected to uniform stresses across the boundaries (i.e. walls) are compared with the laboratory measurements. Stage - 1 Comparisons Figure 5.18a illustrates the variation in the stress components o~z, or and at that cor-respond to the loading path followed in stage - 1, as described above. The comparison of predicted and observed variations in strain components ez, er and et w i th b are shown in Figure 5.18b. As seen from Figure 5.18b, first of al l , the sense and the order of magnitude of strains are correctly predicted. The predicted variation of radial strain with b, is lower than those observed whereas the axial strain predicted is higher than that observed. The predicted variation of circumferential strain wi th 6 is seen to be in good agreement with the observed Chapter 5: 202 Table 5.5 Input Parameters for Predicting Behaviour of Standard Ottawa Sand with Dr = 42% Parameters Specifying Initial State o ° a 0 a 0 "11; "22 > "33 -0.10,-0.10,4-0.20 0.0 e° 0.690 Model Parameters 27650.0 k P a hi 1.220 h2 -0.520 mi, m 2 0.30,0.45 n 1.10 n 0.20 <f>cv 30.0° 0.023 0.705 hSQ 277.0 k P a Sl 2.0 1.5 ai 1.5 &2 0.50 <*3 1.50 01 3.00 02 0.70 (** 0.25 Chapter 5: CO ^ © 1 — 0 1 1 0 1 —<?> d-z (5-R 6 - T G ® © © © — -© © © ©--s o—e « $ e e-(a -e> H »" i I i i i i I I I I i i i i 0.0 0.08 0.16 0.24 0.32 b 0.4 0.48 0.56 Fig.(5.18) Hollow Cyl inder Test Results a)Appl ied Stress Var iat ion Chapter 5: co o G © £ - Z O B S ( b ) H — - f - 6 - R O B S o o e-T O B S X - X e - Z P R E • • 6 - R P R E * - -* e-T P R E r ^ ^ ^ f e ^ ^ — e — e — e b 0.0 0.06 0.13 0.2 0.26 0.33 0.4 0.46 b)P red i c tcd and Observed Strain Variations Fig.(5.18) Continued. Chapter 5: 205 values. In Figure 5.18b, note that the predicted variations of ez and et are nearly identical and cannot be visually distinguished. Stage - 2 Comparisons Figure 5.19a shows the variations in the stress components o~z, ar, at and aa associated with the specific torsion loading conditions considered for stage - 2. The predicted variations in strain components ez, er, et and 7 ^ w i th a are compared with observations in Figures 5.19b through 5.19d. As can be seen from Figures 5.19b through 5.19e, over the entire range of variation of a , the sense and the order of magnitude of the predicted strain components are satisfactory only for the axial and circumferential components. The sense and magnitude of radial strain have not been predicted satisfactorily, for values of a greater than about 12°. Even though the sense of shear strain 7 ^ has been predicted satisfactorily, the magnitude of 7 ^ is considerably under predicted for values of a greater than about 30°. Notice, that during unloading (from a = 60° to 0°), the predicted magnitude and shape of 7 ^ - a variation are correct, the mismatch occurring during the loading phase between a = 30° and 60°. The considerable differences in the internal and external pressures as shown earlier in Table 5.4, can be cited as one of the main causes for the discrepancies in the predicted and observed strains, particularly in stage - 2. In addition to the differences in internal and external pressures, the non-homogeneous torque that results across the walls of the sample could cause considerable variations in the state of stress of the actual sample tested from that simulated numerically, particularly at high values of a (i.e. higher values of torque). Chapter 5: 206 Fig.(5.19) Comparison of Stress-Strain Response w i th Constant Mean Normal Stress and Rotat ing P r i nc ipa l Stress Directions a)Appl ied Stress Variat ion b)Predicted and Observed Ax i a l Stra in Var iat ion Chapter 5: o i 6-T OBS H — -+ G -T PRE (c) -J 1 1 1 1 1 i I i i i i i i i 00.0 16.0 32.0 48.0 56.0 ALPHA 40.0 24.0 08.0 £ -R OBS e -R PRE (d) m o b I X o 2 D < OS w g d l o I J I I L J I I ! 1 I I I L 00.0 16.0 32.0 48.0 56.0 40.0 ALPHA c) Pred icted and Observed Circumferential Stra'm Var iat ion d) Pred icted and Observed Radia l Stra in Variat ion 24.0 08.0 Fig.(5.19) Continued. Chapter 5: e) Predicted and Observed Shear Strain Var ia t ion F ig. (5.19) Continued. Chapter 5: 209 5.5.4 Predicted Stress-Strain Response of Hostun and Reid Bedford Sands from Cleveland Data Base 5.5.4.1 Introduction In this section, the model predictions are compared with the stress-strain measurements observed on Hostun and Reid Bedford sands used in generating the Cleveland Data Base. Hostun sand is a uniform medium sand with e m a i = 0.976, e r a n = 0.607 and Gs = 2.667. Reid Bedford sand is a uniform fine to medium sand with enwz — 0.815, emin = 0.523 and Gg = 2.650. A l l of the Hostun sand samples tested had an initial relative density of 9 9 % and were densified by tamping. On the other hand, all of the Reid Bedford sand samples tested had an in it ia l relative density of 50% and were densified by vibration. A l l samples were prepared by pluviation through air. True triaxial tests were carried out on dry samples and hollow cylinder tests were carried out on water saturated samples. In the true tr iaxial apparatus (which wi l l hereafter be referred to as the cube), a cubic sample with sides 10 cm was placed inside a rubber membrane, surrounded by six rigid plates and loaded. Stresses are applied to the sample by symmetrical displacement of opposite faces w i th respect to the center of the sample, that remain fixed. The displacement of each pair of walls was controlled independently. In the hollow cylinder apparatus, samples were subjected to changing axial and torsional loading. The inside and outside cell pressures were kept equal and constant for all tests. During all tests, the radial stress was always the intermediate pr incipal stress which was equal in magnitude to the circumferential stress. A l l of the hollow cylinder samples had an internal diameter of 5 cm, an external diameter of 7 cm and a height of 12.5 cm. Chapter 5: 210 5.5.4.2 Notation Each test carried out, was described by three letters followed by a D A T or a TST . The first letter was either H, representing a hollow cylinder test, or a C, representing a cube test. The second letter identifies the type of sand used in the test, the letter R represents Reid Bedford sand and the letter H represents Hostun sand. The next set of letters specify the stress path. The set of letters D A T identifies data provided for calibration of models, whereas T S T identifies data provided to test the model capabilities. For example, a test descibed by the set of letters C H I . D A T refers to a cube test carried out on Hostun sand following stress path 1 and data is provided for cal ibration of models. 5.5.4.3 Cal ibration Data The calibration data, made available to the predictors consist of: (i) hydrostatic load-unload data for both sands, (ii) conventional triaxial compression loading and unloading data for two different con-fining pressures for both sands, and (iii) conventional triaxial extension loading data for two different confining pressures. Figures 5.20a and 5.21a illustrate the mean normal stress - volumetric strain variation and lateral strains - axial strain variations, observed during hydrostatic loading of Hostun and Reid Bedford sands, respectively. In each figure there are two curves shown, one from the cube test and the other from the hollow cylinder test. Note that both curves are for the same sand with identical densities following the same stress path. However, as can be seen, the two curves are quite different, the hollow cylinder curve being considerably softer than the cube curve. The differences between the two sets of data are a direct consequence of membrane compliance effects prevelant in the hollow cylinder test results, where the sample is surrounded by two rubber membranes on which the internal and external cell pressures Chapter 5: 211 Fig.(5.20) Hydrostatic Load-Unload Data From Cube and Hollow Cylinder Test Devices -Hostun Sand a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain V6 Vertical Strain Chapter 5: 212 a, co co K H co < O 2 : < q W 2 0.0 0.04 0.08 0.12 0.16 V O L U M E T R I C S T R A I N ( X 1 0 " 1 ) 0.2 O C R 6 . D A T - X + H R 6 . D A T - X • C R 6 . D A T - Y X H R 6 . D A T - Y 0.24 0.36 0.48 V E R T I C A L S T R A I N ( X I O " 2 ) ( b ) 0.6 Fig.(5.21) Hydrostatic Load-Unload Data From Cube and Hollow Cylbider Test Devices -Reid Bedford Sand a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vertical Strain Chapter 5: 213 are applied. The horizontal and vertical strain variations shown in the lower Figures of 5.20b and 5.21b are not seen to be linear as observed for Ottawa sand (refer to Figure 2.6). Due to membrane compliance effects, the horizontal strains interpreted from axial and volumetric strains in the hollow cylinder data are significantly in error. In view of the above differences, the hollow cylinder test data for hydrostatic loading and unloading was not considered during model calibrations. As a result, the author was left wi th the hydrostatic stress-strain data obtained from cube tests. In cube tests, being loaded by six rigid plates in a strain controlled manner, the data does not correspond to a truly hydrostatic stress state after each loading increment. Therefore, for model calibration purposes, smooth variations of am — e„ and ehorizontal ~ £axial have been considered. Tables 5.6 and 5.7 summarize the complete set of calibration data made available to the predictors. The extreme right hand column of each table briefly describes the type of tests performed. Of the calibration tests shown in Tables 5.6 and 5.7, the hardening parameters were obtained from the hydrostatic load-unload data from tests C H 6 . D A T and CR6 .DAT . Shear related parameters and the parameters controlling the developed anisotropy were evaluated using tests C R 5 . D A T and CH5.DAT. The likely magnitudes of critical-state void ratios were estimated examining the rest of the calibration data. Thereafter, the stress-strain response for tests C H I . D A T , CR1.DAT, HH1.DAT, HR1 .DAT, CH3 .DAT , C R 3 . D A T , H H 3 . D A T and H R 3 . D A T were predicted to evaluate the adequacy of parameters for loading along other stress paths and confining stress levels. The basic steps followed were as outl ined in section 5.5.1 earlier. Tables 5.8 and 5.9 summarize the model parameters obtained from Hostun and Reid Bedford sands, respectively. In the absence of any information on the critical-state, the model parameters such as <pcv, Xcs, (e° j ; were evaluated with a considerable amount of uncertainity. A c a was taken as the slope of the e — logio (o~m) variation observed at a Chapter 5: 214 Table 5.6 Hollow Cylinder Tests for Calibrations S A N D D E N S I T Y 0~m N A M E T Y P E O F T E S T (gr/cc) (kPa) Hostun 1.65 203 HH1 .DAT Compression 500 HH2 .DAT Compression 203 HH3 .DAT Tension 500 HH4 .DAT Tension 350 HH5 .DAT Compression with unload-reload 69 HH6 .DAT Hydrostatic Reid 1.58 345 HR1 .DAT Compression Bedford 483 HR2 .DAT Compression 345 HR3 .DAT Tension 483 HR4 .DAT Tension 207 HR5 .DAT Compression with unload-reload 103 HR6 .DAT Hydrostatic Table 5.7 Cube Tests for Calibrations S A N D D E N S I T Y N A M E T Y P E O F T E S T (gr/cc) (kPa) Hostun 1.65 200 C H I . D A T Compression 500 C H 2 . D A T Compression 200 C H 3 . D A T Tension 500 C H 4 . D A T Tension 350 CH5 .DAT Compression with unload-reload 100 CH6 .DAT Hydrostatic Reid 1.58 345 C R 1 . D A T Compression Bedford 483 C R 2 . D A T Compression 345 C R 3 . D A T Tension 483 C R 4 . D A T Tension 207 C R 5 . D A T Compression with unload-reload 100 C R 6 . D A T Hydrostatic Chapter 5: 215 Table 5.8 Input Parameters for Predict ing Behaviour of Hostun Sand Parameters Specifying Initial State a l l> a22> a33 -0.11,+0.07,+0.04 0.0 0.616 Mode l Parameters h>K) 27500. k P a hi -0.3420 +0.8810 mli m2 0.30,0.90 n 1.80 r) 7.50 <Pcv 33.0° A M 0.035 el 0.664 h>o 275.0 k P a si 2.0 82 1.5 ai 1.2 Q2 0.50 <*3 1.50 A 2.00 02 0.70 * 0.25 Chapter 5: 216 Table 5.9 Input Parameters for Predict ing Behaviour of Re id Bedford Sand Parameters Specifying Initial State a l l> a22 > a33 -f 0.12,-0.17,4-0.05 0.0 e° 0.677 Model Parameters 38000. k P a h -0.708 hi +1.192 my, mi 0.30,0.75 n 1.80 n 3.40 <Pcv 31.0° 0.030 eg, 0.724 hsQ 380.0 k P a si 2.0 Si 1.5 « l 1.5 OLI 0.50 <*3 1.50 01 2.10 02 0.70 0.25 Chapter 5: 217 mean normal stress level of about 1000 kPa. e°cs and am were evaluated considering the likely void ratios at large strain levels that correspond as closely as possible to critical-state conditions. For each sand, the magnitude of <pcv was estimated considering the residual stress ratios developed. Figures 5.22 through 5.29 illustrate the back predicted calibration data for Hostun and Reid Bedford sands. The model parameters used in above predictions (given in Tables 5.8 and 5.9) were obtained from hydrostatic loading and init ial unloading data together with one set of tr iaxial compression data for the two sands. Tr iaxial extension test data was not used for model calibration purposes. In the cube device, during simulation of conventional tr iaxial compression and extension paths, due to strain controlled loading using three pairs of rigid plates, the two lateral stresses were somewhat different (i.e 5 to 10 kPa). For the data shown in above figures, the deviator stress o\ — 0 3 was calculated as the vertical stress less the average of the two lateral stresses. Some of the differences in the comparisons shown in Figures 5.22 through 5.29 can be attributed to the above assumption. W i t h the exception of the hydrostatic load-unload tests, each figure shows two sets of laboratory data together with the model predictions. The two sets of observations corre-spond to the two test devices; the cube and the hollow cylinder. A s can be seen from the data shown, even if the type of sand, density of sand and the stress path are identical the use of a different test device can significantly change the stress-strain response. In general, the deviator stress-axial strain response observed from the hollow cylinder device is seen to be stiffer than that of the cube (refer to Figures 5.23 through 5.25 and 5.27 through 5.29). The non-uniformities in stress distribution resulting from the rigid end platens of the cube device may have contributed to the differences in the stress-strain response. On the other hand, the relatively small sample used in the hollow cylinder device (i.e. wal l thickness of 10mm), may produce results that are not representative of sand behaviour. In particular, Chapter 5: 218 0.0 0.04 0.08 0.12 0.18 V O L U M E T R I C STRA IN ( X 1 0 " 1 ) 0.2 O X STRA IN + Y STRA IN X STRA IN Y STRA IN OBS OBS P R E P R E b ) o.o 0.12 0.24 0.36 0.48 0.6 VERT ICAL S T R A I N ( X 1 0 ~ 2 ) Fig.(5.22) Comparison of Predicted and Observed Response For Hydrostatic Load-Unload Pa th - Hostun Sand a) Mean Normal Stress vs Volumetric Strain b) Hor izontal Strain vs Vert ica l St ra in Chapter 5: 219 Fig.(5.23) Comparison of Predicted and Observed Response For Conventional Tr iaxia l Com-pression Path - cr 3 c — 200APo - Hostun Sand - a) Deviator Stress vs A x i a l Strain b) Volumetr ic Stra in vs Ax i a l Strain Chapter 5: 220 q b N I Fig.(5.24) Comparison of Predicted and Observed Response For Conventional Triaxial Exten-sion Path - o$c — 200kPa - Hostun Sand a) Deviator Stress vs Axial Strain b) Volumetric Strain vs Axial Strain Chapter 5: 221 o 0.0 0.08 0.16 0.24 0.32 0.4 0.48 AX IAL S T R A I N ( X 1 0 " 1 ) Fig.(5.25) Comparison of Predicted and Observed Cyclic Loading Response For Conven-tional Triaxial Path - <r3c = 200kPa - Hostun Sand a) Deviator Stress vs Axial Strain b) Volumetric Strain vs Axial Strain Chapter 5: 222 (a! 0.0 0.04 0.08 0.12 0.16 VOLUMETRIC STRAIN ( X 1 0 " 1 ) 0.2 O X STRAIN + Y STRAIN X STRAIN Y STRAIN OBS OBS PRE PRE ( b ) 0.0 0.12 0.24 0.36 0.48 VERTICAL STRAIN (X1Q- 2 ) 0.6 Fig.(5.26) Comparison of Predicted and Observed Response For Hydrostatic Load-Unload Path - Reid Bedford Sand a) Mean Normal Stress vs Volumetric Strain b) Horizontal Strain vs Vertical Strain Chapter 5: 223 Fig.(5.27) Comparison of Predicted and Observed Response For Conventional Triaxial Com-pression Path - <73e = Z45kPa - Reid Bedford Sand a) Deviator Stress vs Axial Strain b) Volumetric Strain vs Axial Strain Chapter 5: 224 p b co f-I (a ) -0.6 -0:52 -0.44 -0.36 -0.28 -0.2 AX IAL S T R A I N ( X 1 0 " 1 ) i -0.12 -0.52 -0.44 -0.36 -0.28 -0.2 AX IAL STRA IN ( X I O " 1 ) •0.12 -0.04 -0.04 Fig.(5.28) Comparison of Predicted and Observed Response For Conventional Tiabdal Ex-tension Path - a 3 e = 345*Po - Reid Bedford Sand a) Deviator Stress vs Axial Strain b) Volumetric Strain vs Axial Strain Chapter 5: 225 — CUBE - HOLLOW CYLINDER -O PREDICTION (a) 0.08 0.16 0.24 0.32 0.4 AXIAL STRAIN ( X 1 0 " 1 ) 0.48 0.56 0 16 0.24 0.32 0.4 AX IAL STRA IN ( X 1 0 " 1 ) 0.48 0.56 Fig.(5.29) Comparison of Predicted and Observed Cyclic Loading Response For Conventional Tr iax ia l Pa th - a3c 200kPa - Reid Bedford Sand a) Deviator Stress vs A x i a l Strain b) Volumetr ic Stra in vs A x i a l Strain Chapter 5: 226 the scatter in the strain data, observed for hydrostatic loading and unloading in the cube device, is considerable and the evaluation of init ial anisotropy of the samples may not have been accurate enough. In general, the predicted stress-strain variations are seen to be in agreement w i th the observed laboratory response provided for calibration purposes. The deviations seen be-tween predictions and laboratory data are of similar magnitude as the differences between the two test devices for the same sand, density and stress path. The predicted volume change response along the extension paths are seen to be too dilative when compared with the laboratory data where the volume change reasponse is initially contractive becoming dilative at large strain levels. A s it was concluded from the sensitivity study carried out in section 5.5.1, the location of the critical-state line in the e — /o</io (erm) space, strongly influence the volume change response. The assumed locations of the critical-state lines in the e — logio(o~m) space may have contributed somewhat to the deviations predicted. In what follows, the stress-strain response along selected stress paths are predicted and compared with the observed response. The model parameters used in these predictions have been derived from the calibration data package discussed above. 5.5.4.4 Predict ions The required predictions consist of a set of three different tests for each sand and each testing device. For the hollow cylinder test, the first test was a compression-torsion test wi th a constant inclination of the principal stresses. The second test was a tension-torsion test wi th a constant inclination of principal stresses. The third test was divided into three parts. The first part was a compression test with the vertical stress increased to about one th ird of the failure value. In the second half of the test, 5 cycles of sinusoidally varying shear stress were applied to the sample while keeping the axial stress equal to the value Chapter 5: 227 obtained in the first part. In the th i rd part, the shear stress was monotonically increased to reach failure. Table 5.10 summarizes the required hollow cylinder test predictions. For the cube device, in the first and second tests, the vertical stress was increased and the stress in the x-direction was decreased. The required values of 6 for the tests are summarized in Table 5.11. The third test is a generalized compression up to 560 k P a and a circular trajectory for two cycles. 5.5.4.4.1 Comparisons With the Hollow Cylinder Test Results The stress-strain response predicted along the stress paths summarized in Table 5.10, are compared with the observed response in Figures 5.30 through 5.33. As seen from Figure 5.30, the stress-strain response predicted for tests HH1 .TST and HR1 .TST agree very well w i th the laboratory data. The predicted volume change response for tests HH2 .TST and HR2.TST shown in Figure 5.31, do not compare very well w i th the laboratory observations, being too dilative for both tests. The predicted shear stress-shear strain variations compare well with the observations. A summary of the cal ibration data made available to the predictors was shown earlier in Tables 5.6 and 5.7. Based on the calibration data, it was first mentioned in section 5.5.4.3 that the deviator stress-axial strain variations from hollow cylinder device are consistently stiffer than those from the cube device when other variables are maintained constant. A detailed investigation on the factors causing the differences is beyond the scope of this thesis. However, the differences in stress-strain response between the hollow cylinder and cube devices had to be considered in predicting the cyclic response. This was because cyclic Chapter 5: 228 Table 5.10 Hollow Cylinder Tests for Predictions S A N D D E N S I T Y (gr/cc) 0~m (kPa) N A M E T Y P E O F T E S T Hostun 1.65 500 HH1.TST Comp-Torsion 1.056 a = 32.3° b = 0.286 500 HH2.TST Tension-Torsion -1.414 a = 54.7° b = 0.666 500 H H 3 B . T S T Cyclic-Torsion a=variable invar iab le 500 H H 3 C . T S T Torsion Reid 1.58 345 HR1 .TST Comp-Torsion 1.003 Bedford a = 31.8° 6 = 0.277 345 HR2 .TST Tension-Torsion -1.003 a = 58.2° b = 0.723 345 H R 3 B . T S T Cyclic-Torsion a=variable 6=variable 345 H R 3 C . T S T Torsion Table 5.11 Cube Tests for Predictions S A N D D E N S I T Y (gr/cc) C m (kPa) N A M E T Y P E O F T E S T L\OZ L\OZ Hostun 1.65 500 C H I . T S T b = 0.286 -2.5 500 CH2 .TST b = 0.666 -0.5 500 C H C . T S T Circular-Path Reid 1.58 345 CR1 .TST b = 0.270 -2.7 Bedford 345 CR2 .TST b = 0.720 -0.4 345 C R C . T S T Circular-Path HR1.TST HH1.TST p r e d i c t i o n o b s e r v a t i o n S H E A R S T R A I N S H E A R S T R A I N HR1.TST HH1.TST CO S H E A R S T R A I N S H E A R S T R A I N Fig.(5.30) Comparison of Predicted Response W i t h Hollow Cyl inder Results - HR1 .TST and HH1.TST 1 g. O a. a. pi § -0.15 S T R A I N ( X 1 0 - 1 ) -0.05 0.05 0.15 S3 H CO H o b S H E A R S T R E S S ( K P A ) 0.0 100.0 200.0 300 CO > in pa > o 0  o w o f o *<! 135 Pi to H cc -0.15 S T R A I N ( X I O " 1 ) -0.05 0.05 o b 0.15 CO w > S3 CO H > o 0  0.0 S H E A R S T R E S S ( K P A ) 100.0 200.0 300 o b CO K W > so CO H > o o ro 023 -ff J3?d«»V5 Chapter 5: 231 Fig. (5.32) Comparison of Predicted Response With Hollow Cylinder Results - PHH3B.TST and P H H 3 C . T S T Chapter 5: PHR3B.TST U S CO K ? f— CO u » X * co 1 i o < E-co u 5 12 X D - J o > _L_ -0.03 0.03 0.1 0.18 SHEAR STRAIN (X10~ 2 ) (a) -0.03 0.03 0.1 0.16 SHEAR STRAIN (X10 " 2 ) 0.23 0.3 0.0 PHR3C.TST (d) 0.0 0.02 0.05 0.08 SHEAR STRAIN 0.10 (e) 0.02 0.05 0.08 SHEAR STRAIN o X ° -—' a < a: co g X < -0.1 -0.03 0.03 0.1 0.18 SHEAR STRAIN (X IO" 2 ) 0.23 - p r ed i c t i on ob se rvat i on Fig.(5.33) Comparison of Predicted Response W i t h Hollow Cyl inder Results - P H R 3 B . T S T and P H R 3 C . T S T Chapter 5: 233 loading associated w i th tests P H H 3 . T S T and PHR3 .TST were small in magnitude and any differences in modul i of deformation would magnify the response considerably. Pr ior to predicting the cyclic load response, the model parameters were refined so that the hollow cylinder data from tests HH2 .DAT and HR1 .DAT and the model predictions matched more accurately. Data from above tests correspond to the init ial phase of load-ing for tests P H H 3 B . T S T and P H R 3 B . T S T . For both sands, an accurate match between the predicted and observed deviator stress-axial strain variations was obtained when the magnitude of the exponent m i was increased to a value close to that of mi, when all other model parameters were those given in Tables 5.8 and 5.9. The details of the recalibration operation are given in Append ix -E. Figures 5.32 and 5.33 compare the predicted and observed cyclic load response for tests P H H 3 . T S T and PHR3 .TST , respectively. The variations in shear stress, shear strain, volumetric strain and axial strain are shown in sub-figures (a), (b) and (c) of each figure. The post cyclic monotonic loading response for the two sands are shown in sub-figures (d) and (e) of each figure. In spite of the efforts taken to match the deviator stress-axial strain response for phase-A as accurately as possible (refer to Figures E . la and E. lb, Appendix-E), the predicted in it ia l torsion loading response, as seen from Figures 5.32a and 5.33a, are considerably softer than the observations. The differences are more pronounced for Reid Bedford sand. A s a result, the stress-strain loops for cyclic loading shifted to the right from the origin. Apa r t from the shift, the predicted magnitudes of the unload and reload modul i are in good agreement. The above is true for both sands. The model lacks the necessary parameters to capture the ' looping ' effect during cyclic loading observed in the laboratory data. As summarized in Table E . l of Apend ix -E , the total change of stress ratio r\, involved with cyclic loading in phase-B is not more than ±.06 for both sands. The variations proposed Chapter 5: 234 for the resistance function Ht, are not sensitive enough or do not capture the observed non-linearity in the shear stress-shear strain response for such small changes in stress ratio. The transfer of load in chains of particles during a transition from the axial loading to torsion loading must be examined more carefully wi th further data and is suggested as a topic of further research. As shown in Figures 5.32b and 5.33b, the predicted volumetric strains are contractive for each cycle, diminishing in magnitude with increasing number of cycles. A similar, yet not so apparent response has been predicted for the axial strain variations (see Figures 5.32c and 5.33c). It can be seen that the predicted magnitudes of volumetric and axial strains compare well with the observations. The above statements are true for both sands. The predicted post cyclic torsion loading response, for both sands, are stiffer than the laboratory observations. The differences are more prominent for Hostun sand being about 20% higher than the observation. For Re id Bedford sand, the over prediction is about 10%. The volumetric strain-shear strain variations for the two sands, however, agree well with the laboratory data. 5.5.4.4.2 Comparisons W i t h the Cube Test Results The stress-strain response predicted along the stress paths summarized in Table 5.11, are compared with the observed data in Figures 5.34 through 5.37. The variables SSI, SD2 and ID2 shown in the above Figures are denned in equations 5.1, 5.2 and 5.3 below. 551 = <JZ + (Ty + az (5.1) (5.2) (5.3) V O L U M E T R I C STRA IN ( X 1 0 ~ l ) SD2/SS1 o.o 0.2 0.4 I o I o b co -3 > b 2 1 1 r ~ m m ' V O L U M E T R I C S T R A I N ( X 1 0 " 1 ) O CO SD2/SS1 o.o i o b l o b CO -3 W o > b 0.2 I 0.4 1— m o c r i/i <t> < D O 3 CHC.TST CHC.TST -120.0 0.0 120.0 PHASE ANGLE 240.0 n o < ca E-I (S3 o predict i on observation -120.0 0.0 120.0 PHASE ANGLE 240.0 CHC.TST CHC.TST -120.0 0.0 120.0 PHASE ANGLE 240.0 -120.0 0.0 120.0 PHASE ANGLE 240.0 C H -O I Fig.(5.36) Comparison of Predicted Response W i t h Cube Results - C H C . T S T to CO CRC.TST -120.0 0.0 120.0 PHASE ANGLE 240.0 CRC.TST prediction observation -120.0 0.0 120.0 PHASE ANGLE 240.0 CRC.TST -120.0 0.0 120.0 PHASE ANGLE 240.0 o d < OS H CO I 9 o d CRC.TST -120.0 0.0 120.0 PHASE ANGLE 240.0 Fig.(5.37) Comparison of Predicted Response W i t h Cube Results - C R C . T S T Chapter 5: 239 Stress-strain variations predicted for tests CH1.TST, CR1 .TST , C H 2 . T S T and CR2 .TST , shown in Figures 5.34 and 5.35, compare very well with the observed variations. In these tests, the magnitude of b was maintained constant at the values shown in Table 5.11. A s can be seen from the Figures 5.34 and 5.35, the predicted magnitude and sense of strain components compare well with the observations. A long the circular stress paths C H C . T S T and C R C . T S T , the octahedral shear stress as well as the mean normal stress have been maintained constant. Therefore in the w-plane, the stress paths trace circular paths. The strain components predicted while traversing the circular stress paths are plotted against the phase angle (measured anticlockwise from the x-axis) and are shown in Figures 5.36 and 5.37 for C H C . T S T and C R C . T S T paths, respectively. It can be seen that even though the predicted response compare well w i th the observed response for the test C H C . T S T , the predicted strains for C R C . T S T are larger than those observed. As stated earlier, the tests C H C . T S T and C R C . T S T trace circular stress paths in the 7r-plane. But the shape of the critical-state surface, as shown in Figure 4.9 earlier, is ap-proximately triangular. Therefore, the distance between the critical-state stress conditions and the current stress conditions vary along the stress path becoming smallest at ± 6 0 ° to the x, y and z axes. The resistance function Ht, which controls the magnitude of strain increments varies non-linearly with the distance between the current and image stress point (on the bounding surface), being small in magnitude when the distance between the two stress states is small. Recall that the maximum stress ratio can be higher than the cr it ical -state stress ratio, being controlled by the parameters r)C3, rj and the ratio computed at the point of maximum stress ratio. For Hostun sand, cf>cv = 33° and r) = 7.5 and for Re id Bedford sand <f>cv = 31° and r) — 3.4. Together with the void ratios, the ratio between computed for the two sands are different being larger for Reid Bedford sand. A s a result, the Chapter 5: 240 magnitude of Ht computed for Reid Bedford sand is comparatively lower than that of Hos-tun sand, particulary at stress states corresponding to phase angles of - 6 0 ° , 60° and 180°. As seen from Figure 5.37, the rapid increase in strain components occur at the above phase angles. Note that by selecting a larger value of j3 in equation 4.81, the distance between image point and current stress states can be increased for stress states corresponding to phase angles mentioned above, thereby decreasing the magnitude of strains. Furthermore, the uncertaint ies associated w i th the magnitude of <pcv and critical-state parameters in the void ratio -mean normal stress space were considerable, which may have contributed to the differences in the predicted strains to some extent. 5.6 Summary of Predictions and Discussion of Results 5.6.1 Summary of Predictions In section 5.4 above, the adequacy of the proposed stress-strain formulations to model the characteristic behaviour of sand was verified. It was demonstrated that the proposed formulations are capable of accounting for the shear induced volume change behaviour of sand. Furthermore, the proposed formulations were shown to be capable of modelling the abrupt changes in the magnitude of incremental resistance to deformation and associated changes in strain increment directions as observed in laboratory tests at unloading and reloading points along a given stress path. A sensitivity analysis was carried out with the objective of identifying the sensitve model parameters. The analysis indicated that the constant volume fr ict ion angle cpcv, the location of the critical-state line in the e- logw (<rm) space as defined by e°ca, and the parameters defining the extent of developed anisotropy in the material ot\, 0:2, &3,@i and 02 a r e sensitive parameters for triaxial compression loading. In section 5.5 above, the proposed stress-strain formulations were verified against ob-Chapter 5: 241 served laboratory data obtained from four different types of test devices involving three different sands. Most stress paths considered in sections 5.5.2 and 5.5.3 require the mea-sured volumetric strains to be corrected for membrane compliance effects which could be as high as 5 0 % of the measured volumetric strains. The membrane compliance correction was required since the radial and circumferential strains were not measured, but evaluated from the measured volumetric and axial strains. O n the other hand, the stress paths followed in section 5.5.4, did not require the membrane compliance correction, wi th the exception of the hydrostatic load-unload tests using the hollow cylinder device. In al l other tests, the cell pressures on the membrane was maintained constant and equal. In the tests carried out with the cube device, the three strain components were measured independently. The stresses and strains measured in the laboratory tests were average quantities mea-sured along the boundary of the sample, even though considerable variations in magnitude of stresses could occur for specific loading paths, as outlined earlier in Tables 5.3 and 5.4. The three sands considered were uniform fine to medium sands, wi th their D'h0s varying between 0.25 and 0.40 mm. When the type of sand, stress path and density were identical, the stress-strain response predicted from the cube and hollow cylinder devices were different. The deviator stress-axial strain response observed from the hollow cylinder device was found to be consistently stiffer than from the cube device. The differences in the stress-strain response between different test devices was of the same order as the differences between the observed (i.e. average) and predicted stress-strain response. 5.6.2 Discussion of Results The comparison study carried out in sections 5.5.2, 5.5.3 and 5.5.4 indicate that the deviations observed between the predicted and observed stress-strain variations are mainly for loading paths directed at the extension side. A long such stress paths, the predicted Chapter 5: 242 volumetric strains are found to be too dilative. Such behaviour can be seen from Figures 5.14, 5.24, 5.28, 5.31, 5.35, 5.36 and 5.37. In the proposed model, the volume change response is directly influenced by the magnitude of the developed material anisotropy which in turn depends on the ratio (^*) raised to 0 : 3 which is a positive exponent greater than unity. The magnitude of ecs depends on the position of the critical-state line in the e — logio(am) space. It was assumed in the formulations that the critical-state line can be represented as a straight line in the e — log\o(am) space. Available laboratory data indicate that the critical-state line is actually curved in the e - log\a(am) space. The use of a curved critical-state line in numerical modelling of sand behaviour can be carried out relatively easliy. However, such data is not commonly available for most sands and therefore the influence is difficult to quantify. The critical-state surface in three dimensions, was generated from; r,„ - r,C3c [ ( 1 + j g ) _ ( 1 _ ^ a f B j W I ] i 5 - 4 ) In the model formulations the value of 8 was selected to be 0.75. The above functional form of the critical-state surface has several advantages. It can represent a range of shapes being circular for 8 = 1.0 and gradually becoming triangular in shape when 8 is about 0.70. The value 0.75 was chosen primari ly because it matches the shape of the failure surface observed for Cambr ia sand from laboratory data (Ochiai &; Lade (1983)). Deviations between the predicted and observed stress-strain variations could occur because of the differences in shape of the critical-state surface. For hollow cylinder test results presented in section 5.5.3, the differences in the internal and external cell pressures cannot be modelled accurately using the computer code S A N D . For proper simulation of such stress gradients, the developed stress-strain relations have to be incorporated into a computer program where the geometry effects can be taken account Chapter 5: 243 of. The results, in general, indicate that the resistance function Ht needs refinement. The non-linear stress-strain variations that occur at unload and reload points have not been modelled adequately i.e. stress 'loops'. Furthermore, the response of Ht for small changes in stress ratio involving change of mode of loading have not been captured effectively. The laws denning the extent to which anisotropy can be developed in the material, in a strict sense, have been derived based on observations from systems of circular particles. For sand that do not consist of spherical particles, the variations in the contact branch length distributions have to be incorporated into the theoretical formulations for more accurate comparisons. 5.7 Comparison W i t h Other Models The evaluation of a newly developed stress-strain model wi th respect to the existing models is a difficult but important task. One of the main advantages of a workshop such as the one held in Cleveland during 1987 is that the researchers engaged in development of new stress-strain models are relieved of the burden of having to compare their models wi th the existing popular models. Instead, the predictions from the newly developed model can be compared with a selected data base, which has already been compared w i th most of the existing models, and the new model can be evaluated. A total of 29 predictors took part in the Cleveland Workshop(1987). The stress-strain predictions presented from three of the models that are well established in literature, are shown in Appendix-F. For more details, the reader should refer to the 'Test Results and Predictions ' package of the Cleveland Workshop(1987). From the predictions shown in Appendix -F as well as the other predictions presented for the workshop, the proposed model compares very well w i th most of the existing models. Chapter 5: 244 The large number of model parameters (i.e. a total of 24 ) required to fully activate the proposed formulations, by no means should be considered a drawback. Out of the 24 model parameters, 18 can be evaluated from two simple conventional triaxial tests; a hydrostatic load-unload test and a tr iaxial compression test where the material contracts all the way to failure. W i t h the exception of e°, the remaining parameters can be evaluated relatively easily from three conventional tr iaxial compression tests, of which two tests are to establish the critical-state line in the e — logio (o"m) space. The third test should correspond to a state dry of critical-state and loaded in a strain controlled manner to obtain the parameters r), si and S2-5.8 Summary The theoretical framework presented in chapter-4 to model sand behaviour, has been verified by comparison with laboratory observations. The comparison study indicates that the model is generally capable of predicting the complex stress-strain behaviour of sand satisfactorily. However, along stress paths directed towards the extension side, the model predicts volumetric strains that are too dilative. The proposed model is comparable to the few existing models that predicted the Cleve-land data base satisfactorily. C H A P T E R 6 S U M M A R Y , CONCLUSIONS A N D FURTHER R E S E A R C H 6.1 Summary and Conclusions The observed stress-strain behaviour of sand is extremely complicated, being dependent on the stress level, stress path, density of packing and material anisotropy. Unlike materials such as metals, sands undergo significant changes in volume during hydrostatic loading as well as shear loading. The theoretical frameworks proposed in the literature to model sand behaviour are based on concepts derived from theories of elasticity and plasticity which originated from observations on the behaviour of materials whose shear induced volume change behaviour is insignificant or non-existent. Most models therefore make use of the shear-volume coupling parameters derived from laboratory data, conventionally referred to as stress dilatancy equations, to predict the stress-strain response of sand. The stress dilatancy equations proposed for sand have limited applicablity and do not account for all observed aspects of sand behaviour. Particulate, discrete and frictional materials such as sands, constitute a separate class of materials. Particulate materials form discontinuous media. Stress at a point, as defined for continuous media, is not valid for discontinuous media. Instead, stress has to be defined as the force averaged over a finite volume within the system. Stresses so defined, fluctuate 245 Chapter 6 : 246 from volume to volume within the same system. In discrete materials, the spatial arrange-ment of particles change as a result of shearing. Dur ing rearrangement of particles, the material undergoes deformations resulting in both shear and volumetric strains. As part of the rearranging process, chains of particles are continuously formed and destroyed, and this results in varying magnitude of resistance to deformation. The inherent and induced anisotropy observed in sands, the quantification of which has been difficult, is a direct con-sequence of the anisotropic distributions of contact normals and contact branch lengths. The development of interparticle contact forces, in fr ictional systems, is not well denned. The ratio between the average tangent and normal contact forces JI, can vary between zero and the l imit ing value (i* reaching only at contacts that are sliding. Even at failure, only a very few contacts develop /x*. The stress tensor is influenced by the manner in which JI is distributed within the system. The stress-strain behaviour of a system of particles that are glued or bonded at their contacts have been studied first. Most of the theoretical formulations and the underlying concepts, for bonded systems, have been extracted from Rothenburg(1980) and simplified to suit the objectives of this dissertation. Rate insensitive stress-strain relations applicable for particulate, discrete and frictional systems are derived thereafter following the theo-retical framework proposed for bonded systems and using the principle of virtual forces. The derived stress-strain relations are applicable for a system consisting of unoform spheres when subjected to infinitesimally small increments of loading. For frictional systems, the load-deformation response of contacts of a given orientation, is assummed to be non-linear. The non-linear, continuously rearranging systems are modelled assuming the incremental stiffnesses at the contacts to be linear over a given loading increment, and updating the changes in contact normal distr ibution, resistance to deformation and the ratio of tangent and normal contact force distributions at the end of each increment. The changes in the spatial arrangement of particles, spatial variation in the developed ratio between average Chapter 6 : 247 tangent and normal contact forces and the resistance to deformation are defined. These definitions have been established by examining laboratory data on real sands and numer-ical experiments on circular disc shaped two dimensional systems of particles reported by Bathurst(1985). The stress ratio and the state parameter (defined as the ratio of void ratios between the critical-state of a material to the current state, computed for a given mean-normal stress) have been identified as key parameters that that change the anisotropic response of particulate, discrete and frictional systems. Interparticle contact normals tend to align in the direction of major principal stress, forming chains of particles that carry loads higher than others in the same system. When the loading direction changes, different sets of chains of particles become active. The stiffer incremental resistance observed during unloading followed by loading, is a direct consequence of the above transfer of dominant chains in the system. Mathematical modelling of these dominant chains of particles has been carried out using a concept similar to that in bounding surface plasticity theory. Accordingly, the incremental resistance to deformation varies non-linearly with stress ratio, being dependent on the difference between the current stress ratio and the stress ratio of an image point on the bounding surface. The bounding surface, defined in the stress space, continuosly change with loading coinciding with the critical-state surface at large strains or failure. When the in it ia l state of a given sample is dry of critical-state, the bounding surface init ial ly lies outside of the critical-state surface. Similarly, when the init ial state of a sample is wet of critical-state, the bounding surface initially lies inside of the critical-state surface. Deformations occur, as a result of particle rearrangement. The deformations are the re-sult of a few suitably oriented contacts sliding followed by rearrangement of a large number of particles causing rolling, rigid body rotations and displacements. Separation of deforma-tions resulting from above internal processes is impractical, if not impossible. The strain Chapter 6 : 248 tensor denned in the theoretical formulations, includes deformations resulting from all in-ternal activity, described above. The incremental stress-strain relations, so derived, are applicable for general incremental loading in three dimensions. In order to predict the stress-strain behaviour of sand using the proposed theoretical framework, a total of 24 model parameters have to be evaluated. Of the required 24 pa-rameters, 18 can be evaluated from two simple tests; a hydrostatic load-unload test and a tr iaxial compression test where the material contracts al l the way to failure. W i th the ex-ception of e°, the balance five parameters can be evaluated from three triaxial compression tests of which two are required to establish the critical-state line in the e — logio (crm) space. The th ird test has to correspond to a state dry of critical-state and loaded in a strain controlled manner. The in it ia l anisotropy in the material can also be obtained from the hydrostatic load-unload test data, ignoring the fourth order anisotropy coefficients and assuming that the pr incipal anisotropy directions coincide with the vertical and horizon-tal axes. The constant volume friction angle <pcv, location of the critical-state line in the e — logio (o"m) space and the parameters controlling the development of anisotropy in the material ax, a2, c * 3 , f t and ft have been identified as sensitive parameters. The proposed theoretical formulations are capable of modelling the characteristic be-haviour of sand as observed from laboratory tests. The adequacy of the proposed for-mulations to model real sand behaviour has been verified by comparisons with laboratory measurements on Ottawa, Hostun and Reid Bedford sands. Data for Ottawa sand consist of tr iaxial test results reported by Negussey(1984) and results from hollow cylinder tests carried out using the U B C hollow cylinder device. Data for Hostun and Reid Bedford sands consist of true tr iaxial and hollow cylinder test results presented for the Cleveland Work-shop(l987). The proposed model is capable of modelling the complex stress-strain response of sand which is influenced by the material anisotropy (both inherent and stress induced), Chapter 6 : 249 stress level and path dependency and density of packing. Based on the comparison studies presented in Chapter-5 and subjected to the assump-tions made in the theoretical formulations and the inherent inaccuracies associated with the different test devices, the performance of the model is seen to be satisfactory. 6.2 Further Research The theoretical formulations presented in chapter-4, for frictional systems, have been derived for a system of equal size spherical particles. In such systems, there are two key formulations that govern the stress-strain response; (a) laws that describe the extent of the developed anisotropy Oy, t^y, and (b) variations in the resistance to deformation, Ht. The variations proposed for a,y, 6,j« and Ht, although are general enough, must be refined in the light of accurate stress-strain data from both laboratory and numerical experiments. Being a uniform rounded sand, Ottawa sand is seen as an appropriate material to carry out laboratory experiments with which the theoretical formulations can be verified. Numerical experimental results which undoubtedly wi l l be restricted to two dimensions, must be carried out to examine the changes in the contact normal distribution and dominant chains of particles that are formed along the following stress paths; (i) starting from a hydrostatic state of stress, and loading along different stress paths including unloading and extension side loading, and (ii) starting from an initial stress ratio in both compression and extension sides and following stress paths described in (i) above. Only the changes occurring in the early stages of loading in (i) and (ii) are of main interest. The results, once obtained should be carefully reviewed and the laws defining the changes in developed anisotropy and Ht must be refined. Chapter 6 : 250 Further numerical experimental studies must be undertaken to study the contact force distribution parameter f '. The likely distributions of contact forces on contacts of a given orientation and its changes at unloading and reloading points must be examined. C H A P T E R 7 R E F E R E N C E S Arthur J . R. F. and Menzies B. K. (1972) " Inherent Anisotropy in a Sand " Geotechnique 22, No-1, March 1972, pp 115-118. Bardet J . P. (1983) " App l icat ion of Plasticity Theory to Soil Behaviour; a New Sand Model " Ph.D. Thesis, Cal i fornia Institute of Technology, Pasadena, California, September, 1983. Bathurst R. J . (1985) " A Study of Stress and Anisotropy in Idealized Granular Assemblies " Ph.D Thesis, Dept. of C i v i l Engineering, Queen's University at Kingston, Ontario, Canada, August 1985. Bathurst R. J . , Rothenburg L. (1988) " Micromechanical Aspects of Isotropic Granular Assemblies W i th Linear Contact In-teractions " Journal of App l ied Mechanics, Vol-55, March 1988, pp 17-23. Been K., Jefferies M . G. (1985) " State Parameter for Sand " Geotechnique 35, No-2, 1985, pp 99-112. Byrne P. M . , Eldridge T. L. (1982) " A Three Parameter Di latant Elastic Stress-Strain Model for Sand " Proceedings, International Symposium on Numerical Models in Geotechnics, Zurich, September 1982, pp 73-79. 251 Chapter 7: 252 Casagrande A (1936) " Characteristics of Cohesionless Soils Affecting the Stabil ity of Slopes and Ear th F i l l s " Contributions to Soil Mechanics, 1925-1940, Boston Society of C i v l Engineers, 1936. Castro G., Enos J.L., France J .W. and Poulos S.J . (1982) " Liquefaction Induced by Cycl ic Loading " Report Submitted to Nat ional Science Foundation, Washington, D.C, March, 1982. Chern J . C. (1984) " Undrained Response of Saturated Sands w i th Emphasis on Liquefaction and Cycl ic Mobil ity " Ph.D. Thesis, Dept. of C i v i l Engineering, University of Br i t i sh Columbia, Vancouver, Canada, December 1984. Christoffersen J . , Mehrabadi M . M . , Nemat-Nasser S. (1981) " A Micromechanical Description of Granular Mater ia l Behaviour " , Journal of App l ied Mechanics, Vol-48, June 1981, pp 339-344. Chung E. K. F. (1985) " Effects of Stress Path and Prestrain History on the Undrained Monotonic and Cycl ic Loading Behaviour of Saturated Sand " M.A.Sc. Thesis, Dept. of Civil,Engineering, University of Br i t i sh Columbia, Vancouver, Canada, Ju ly 1985. Cleveland Workshop (1987) " Test Results and Predictions, International Workshop for Granular Non-cohesive Soils " , Department of C i v i l Engineering, Case Western Reserve University, Cleveland, Ohio, July 22-24, 1987. Cundall P. A. , Strack O.D.L. (1983) " Modeling of Microscopic Mechanisms in Granular Mate r i a l " , Mechanics of Granular Materials: New Models and Constitutive Relations, Edited by J.T. Jenkins and M . Satake, pp 137-149, Elsevier Science Publ ishing Company Inc., 1983. Dafalias Y . F. (1975) On Cycl ic and Anisotropic Plasticity; (i) A general model including material behaviour under stress reversals (ii) Anisotropic hardening for init ia l orthotropic materials " Ph.D Thesis, University of California, Berkeley, December 1975. Desai C. S., Siriwardena H. J . (1984) " Constitutive Laws for Engineering Materials w i th Emphasis on Geologic Materials " Chapter 7: 253 Prentice-Hal l Inc., Englewood Clifts, New Jersey, 1984. Drescher A . , de Josselin de Jong G. (1972) " Photoelastic Verification of a Mechanical Model for the Flow of a Granular Mate r i a l " , Journal of the Mechanics and Physics of Solids, Vol-20, pp 337-351, 1972. El-Sohby M.A . ( 1969) " Deformation of Sand Under Constant Stress Ratios " Proceedings of the 7th International Conference on Soil Mechanics and Foundation En -gineering, Vol-1, Mexico, 1969, pp 111-120. Finn W. D. L., Mart in G. R. (1980) " Soil as an Anisotropic K inematic Hardening Soild" A S C E Convention Exposit ion, Florida, October 27-31, 1980. Ghaboussi J . , Momen H. (1982) " Model l ing and Analysis of Cycl ic Behaviour of Sands" Chapter-12, Soil Mechanics-Cycl ic & Transient Loads, Edited by G.N. Pande &; O.C.Zienkiewicz, 1982, John Wi ley & Sons L t d . Hardin B. O. (1978) " The Nature of Stress Strain Behaviour of Soils " Earthquake Engineering and Soil Dynamics, June 1978, Vol-1, Pasadena California, pp 3-90. Home M.R. (1964) " The Behaviour of an Assembly of Rotund, Rigid, Cohesionless Particles " Part I, Proceedings of the Royal Society, A286, pp 62-78. Iwan W. D. ( 1967) " O n a Class of Models for the Yielding Behaviour of Continuous and Composite Sys-tems " Journal for App l i ed Mechanics, Transactions of the A S M E , September 1967, pp 612-617. Katz A . ( 1967) " Principles of Statistical Mechanics" The Information Theory Approach, W.H. Freeman & Company, Chapter-2, pp 14-22, 1967. Lade P. V. , Duncan J . M . (1975) " Elasto-Plastic Stress-Strain Theory for Cohesionless Soil " Chapter 7: 254 Journal of Geotechnical Engineering, A S C E , Vol-101, No-GTIO, October 1975, pp 1037-1051. Lade P. V . (1987) " Behaviour and Plasticity Theory for Metals and Frict ional Mater ia l " Constitutive Laws for Engineering Materials; Theory and Appl ications, Tucson, A r i -zona, 1987. Lee K.L. , Seed H.B. (1967) " Drained Strength Characteristics of Sands" Journal of the Soil Mechanics and Foundations Division, Vol-93, SM6, November 1967, pp 117-141. Matsuoka H. (1974a) " A Microscopic Study on Shear Mechanism of Granular Mater ia l " Soils and Foundations Vol-14, No-1, March 1974, pp 29-43. Matsuoka H. (1974b) " Stress-Strain Relationships of Sands Based on the Mobi l ized Plane " Soils and Foundations, Vol-14, No-2, June 1974, pp 48-61. Naylor D.J. (1978) " Finite Element Methods in Soil Mechanics" Developments in Soil Mechanics-1, 1978. Negussey D. (1984) " A n Experimental Study of the Small Strain Response of Sand " Ph.D. Thesis, Dept. of C i v i l Engineering, University of Br i t i sh Columbia, Vancouver, Canada, December 1984. Negussey D., Wijewickreme W.K .D , Vaid Y . P . (1986) " Constant Volume Friction Angle of Granular Materials " Ist As ian Regional Symposium on Geotechnical Problems and Practices in Foundation Engineering, Colombo, Sri Lanka, January, 1986. Nemat-Nasser S. (1980) " On Behaviour of Granular Mater ia l in Simple Shearing " Soils and Foundations, Vol-20, No-3, September 1980, pp 59-73. Newland P.L., Allely B .H. (1959) " Volume Changes During Undrained Tr iax ia l Tests on Saturated, Di latant, Granular Materials " Chapter 7: 255 Geotechnique vol-9, No-4, pp 174-182. Nova R., Wood D. M . , (1979) " A Constitutive Model for Sand in Triaxial Compression " International Journal for Numerical and Analyt ica l Methods in Geomechanics, Vol-3, 1979, pp 255-278. Ochiai H , Lade P.V (1983) " Three Dimensional Behaviour of Sand with Anisotropic Fabric" Journal of Geotechnical Engineering, A S C E , Vol-109, No-10, October 1983, pp 1313-1328. Oda M . (1972a) " The Mechanism of Fabric Changes During Compressional Deformation of Sand" Soils and Foundations, Vol-12, No-2, June 1972, pp 1-18. Oda M . (1972b) " Init ial Fabric and Their Relations to Mechanical Properties of Granular Mater ia l " Soils and Foundations, Vol-12, No-1, March 1972, pp 18-35. Oda M . , Konishi J . (1974) " Rotat ion of Pr inc ipal Stresses in Granular Mater ia l During Simple Shearing " Soils and Foundations, Vol-14, No-4, December 1974, pp 39-53. Oden T .J . (1967) " Mechanics of Elastic Structures " McGraw-H i l l Book Company, pp 252-262,1967. Pastor M . , Zienkiewicz O. C , Leung E. H.(1985) " Simple Mode l for Transient Soil Loading in Earthquake Analysis - Part II - Non-Associative Models for Sands " International Journal for Numerical and Analyt ica l Methods in Geotechnics, Vol-9, No-5, September-October 1985, pp477-498. Poorooshasb H. B. (1971) " Deformation of Sands in Triaxial Compressions - Part I " 4 th As ian Regional Conference on Soil Mechanics and Foundation Division, 1971, Bangkok, Vol-1, pp 63-66. Poorooshasb H. B., Holubec L, Sherbourne N. (1966) " Y ie ld ing and Flow of Sand in Triaxial Compression - Part I" Chapter 7: 256 Canadian Geotechnical Journal, Vol-3, No-4, November 1966, pp 179-190. Poulos S.J . (1981) " The Steady State of Deformation" Journal of the Geotechnical Engineering Division, A S C E , Vol-107, GT-5 , pp 553-562. Prevost J .H . (1978) " Anisotropic Undrained Stress-Strain Behaviour of C lay " Journal of the Geotechnical Engineering Division, A S C E , Vol-104, pp 1075-1090, 1978. Rothenburg L. (1980) " Micromechanics of Idealized Granular Systems " Ph.D Thesis, Faculty of Engineering, Carleton University, Ottawa, Canada, October 1980. Rothenburg L., Selvadurai A .P .S . (1981) " A Micromechanical Definition of the Cauchy Stress Tensor for Particulate Media " Proceedings of the International Symposium on the Mechanical Behaviour of Structured Media, A.P.S. Selvadurai (Editor), Ottawa, May 1981, pp 469-486. Rowe P.W. (1962) " The Stress Dilatancy Relations for Static Equ i l ib r ium of an Assembly of Particles in Contact " . Proceedings of the Royal Society, A269, 1962, pp 500-527. Rowe P. W. (1971) " Theoretical Meaning and Observed Values of Deformation Parameters for Soils " Stress-Strain Behaviour of Soils, Procedings of the Roscoe Memoria l Symposium, Cam-bridge University, March 1971, pp 143-194. Symes M . J . P. R. (1983) " Rotation of Pr inc ipa l Stresses in Sand " Ph.D. Thesis, Dept. of C i v i l Engineering, Imperial College of Science and Technology, London, January 1983. Tatsuoka F., Ishihara K. (1974) " Y ie ld ing of Sand in Tr iax ia l Compression " Soils and Foundations, Vol-14, No-2, June 1974, pp 63-76. Tatsuoka F., Molenkap F. (1983) " Discussion on Y ie ld Loc i for Sands" Chapter 7: 257 Mechanics of Granular Materials; New Models and Constitutive Relations, Edited by J.T. Jenkins & M . Satake, Elsevier Science Publications, 1983. Tokimatsu K., Nakamura E . (1986) " A Liquefaction Test Without Memebrane Penetration Effects " , Soils and Foundations, Vol-26, No-4, December 1986, pp 127-138. Vaid Y . P., Negussey D. (1984) " A Cr i t i ca l Assessment of Membrane Penetration in the Tr iaxial Tests " A S T M Geotechnical Testing Journal, Vol-7, No-2, June 1984, pp 70-76. Wijewickrema W. (1986) " Constant Volume Friction Angle of Granular Materials " M.A.Sc. Thesis, University of Br it ish Columbia, Vancouver, Canada, February, 1986. Wong R. E . S., Arthur J . R. F. (1985) " Induced and Inherent Anisotropy in Sand " Geotechnique 35, No-4, December 1985, pp 471-481. Zytynski M . , Randolph M . F., Nova R., Wroth C. P. (1978) " O n Model l ing the Unloading Reloading Behaviour of Soils " International Journal for Numerical and Ana lyt ica l Methods in Geotechnics, Vol-2, 1978, pp 87-94. A P P E N D I X - A In Appendix - A , the different stress dilatancy relations proposed by Rowe (1971), Matsuoka (1974a), Oda (1972a) and Nemat-Nasser (1980) are summarized. A . l Rowe's Equation - = [ l - ( ^ r ) J tan2 ( 4 5 + ^ / 2 ) (A. l ) (T3 ae\ where, o~\ and az are the major and minor principal stresses, respectively. ( ^ ) « = the ratio of incremental volumetric strain to major principal strain due to slip 4>f = (pfi = interparticle friction angle for dense sands prior to peak strength <Pf = <Pcv = constant volume friction angle for dense sands at large strains 4>f = <Pcv = constant volume friction angle for loose sands at all strain levels Equation A . l above, is not applicable for unloading involving decreasing stress ratio ^ and reversal of strain directions. 258 A : 259 A.2 Matsuoka's Equation aN d 7 f = - A ® + n ' {A.Z) where, T, <TA' = shear and normal stresses on potential sliding plane <fe/v, <*7 = incremental normal and shear strains on potential sliding plane H = physical constant = fan(0M) A = constant determined by the value of \i ep/, 7 = normal and shear strains on potential sliding plane H 1 = parameter reflecting fabric of material A.3 Oda's Equation , d ev 4 Sz ^ where, 5 I ; 5 2 = summation of the projected area of contact surfaces on the yz and xy planes. x,y,z = reference axes (pfi = angle of interparticle friction o\,o3 = major and minor principal stresses A : 260 A.4 Neinat-Nassers's Equation J_ dV_ V In — / p(y) cos ((pp + u) sin u dv cos 6a J (A.6) tan 4>\i = tan (<f> - v) (A.7) tan (j> — — a (A.8) in which, dV = change of volume of sample V — volume of sample dr\ — change in shear strain of sample <i>M = interparticle friction angle v — dilatancy angle p(v) = distribution of dilatancy angle v within the sample 7,tT = shear and normal stresses at boundary <p = macroscopic friction angle A P P E N D I X - B The main steps involved in arriving at equation 3.51, as reported by Rothenburg(1980), are summarized herein. For simplicity, only the two dimensional formulations are presented. The derivation of the three dimensional equations can be carried out following identical steps. Step-1 Consider an infinite system of particles. Let the spectrum of density of contact forces on contacts of a given orientation be denoted by Pg. Then by definition, OO 00 J J Pgdfndft=l (B.l) —oo —oo 00 oo / / fnPedfndft = Tn (B.2) —oo —oo CO oo 1 j ftPedfndft = Jt (B.Z) -co —co oo oo j j fnPedfndft = pn ( 5 . 4 ) -oo —oo 00 oo j j f?Pedfndft = f? (5.5) 261 B : 262 Step-2 Using postulates from information theory, the amount of missing information associated with Pe is written as (Katz (1967)), oo oo ev = -K I j Pelog Pe dfn dft (J9.6) —oo—oo where i f is a positive arbitrary constant which embodies the choice of units of information. For particulate systems, equation B.6 can be selected in the form, 2ir oo o o 0 —00 —00 Step-3 To arrive at the function Pe, which contains the least amount of possibly adverse information which can result from an unsubstantiated guess, the missing information is maximized (note: this is the non-physical logic underlying the principle of best guess). The missing information is maximized subjected to the constraints imposed by the com-plementary energy of the system and the external loads. Accordingly, form the functional We), L{Pe) = ev do + ^-[wc + A,-, a,-,] (B.8) in which Xw is a Lagrangian multiplier and A,y = c A^ where, c is a non-dimensional param-eter. wc is the complementary energy of the system given by, 2f 00 00 21 / / I Pe[2Tn + 2k) * & S { $ ) " { B 9 ) and <T,y is given by, o-ii=\invdoj j j Pe [fn n, + ft i , n}] dfn dft S{0) d$ (S.10) 2 0 —00 —00 2tr 00 00 0 —00 —00 5 : 263 Maximizing L(Pg) with respect to P#, and normalizing using equation B . l , it can be shown that P = 1 r ( / » ~ fn)2 _ {ft - ft)21 ,g n s 6 2 TT Xw V [ 2 kn Xw 2 kt Xw ' K ' ' where, 7« = f *» A s n, n,- (5.12a) 74 = C A,y i, ny (5.126) From equations B.4, B.5 and B . l l , H = fn + Kkn (5.13a) Jf = Tt+Xwkt (5.13a) Substituting equations B.13a and B.13b into equation B.9, 2*" — 2 —2 Step-4 For dimensional compatibility of equation B.14, A„, must be quadratic in /„ and ft. Accordingly, Au, can be selected in the following form, K = J S(6) [ax/n2 + a2fjt + a3ft2}S (5.15) o Some information on the constants 0 1 , 0 : 2 and 0 3 can be recoverd by minimizing the func-tional, L = wc - XijGij (5.16) with respect to fn and / (. Accordingly, a, = I Z L ( j . , 7 . ) B : 264 For correct directional variations of fn and / t, a2 has no alternative but to satisfy the condition, a 2 = 0 (B.17c) Substituting equations B.17a, B.17b, B.17c and B.15 into equation B.14, 2 —2 which is the two dimensional counterpart of equation 3.51. It can be seen that c above, is related to the dispersion of the contact forces (i.e. equations B.13a and B.13b). A P P E N D I X - C Appendix-C presents information pertaining to recalibration of the model to predict the cyclic loading response of Hostun and Reid Bedford sands. The tests PHH3.TST and PHR3.TST have been carried out in three phases. Phase-A involves loading the respective sand samples monotonically along the triaxial compression path to an axial stress which is one third of the failure value. Thereafter, the samples are subjected to five sinusoidal cycles of torsion involving reversal of shear stresses maintaining the static stress conditions imposed at the end of phase-A which constitutes phase-B. In phase-C, the samples are monotonically loaded in torsion to failure. The associated mag-nitudes of stresses at the different stages of loading for the two sands, are given in Table C . l . Figures C . l a and C . lb illustrate the comparisons of the deviator stress-axial strain variations predicted with the refined model parameters with those observed from hollow cylinder and cube test devices for phase-A loading. These data have been obtained from tests CR1.DAT, HR1.DAT, CH2.DAT and HH2.DAT which correspond to the same con-fining stress levels and stress paths and are only applicable to phase-A loading. The model parameters used for the predictions are summarized in Tables C.2 and C.3. For each sand, all the parameters in Tables C.2 and C.3 are identical to those given in Tables 5.8 and 5.9 265 C : 266 T a b l e d The Boundary Values for PHH3.TST and PHR3.TST Test Phase-A Phase-B Phase-C P H H 3 . T S T crr = at - 500 cr r = at = 500 ar =at = 500 Hostun <7Z=500 to 1020 oz = 1020 c r z = 1020 <73c=500 ;/=.257 ra=± 133 7-^=0 to 500 A r ? = ± .047 P H R 3 . T S T tr r=(7 <=345 ar=o-t—Z4t> o" r =cr<=345 Reid Bedford a 2=345 to 628 cr 2=628 <r,=628 o-3( r=345 r7=.215 TA=± 90 Tzt^O to 300 Ar?= :± .057 Note: Stresses in kPa. C . 267 < Cu CO X - HOLLOW CYL INDER - C U B E O PREDICTION 0.0 0.1 0.2 0.3 0.4 AX IAL STRA IN ( X 1 0 " Z ) 0.5 X - HOLLOW CYL INDER - CUBE •O PREDICTION 0.1 0.2 0.3 0.4 AX IAL STRAIN ( X 1 0 " 2 ) 0.5 Fig. C«1 Comparison of Predicted Deviator Stress vs Axial Strain Variations with a) CR1.DAT and HRl .DAT b) CH2.DAT and HH2.DAT Table C-2 Recalibrated Parameters For Predicting Behaviour of Hostun Sand Parameters Specifying Initial State a l l i a 221 a 33 -0.11,+0.07,+0.04 0.0 e° 0.616 Model Parameters 27500. kPa -0.3420 hi +0.8810 m 1 ; m 2 0.90,0.90 n 1.80 n 7.50 <f>cv 33.0° 0.035 e° 0.664 275.0 kPa Si 2.0 S2 1.5 ax 1.2 a2 0.50 <*3 1.50 Pi 2.00 02 0.70 ¥ 0.25 Table 0 3 Recalibrated Parameters for Predicting Behaviour of Reid Bedford Sand Parameters Specifying Initial State a l l > a 22 > A 3 3 +0.12,-0.17,+0.05 0.0 e° 0.677 Model Parameters 38000. kPa hi -0.708 ho +1.192 mi, m 2 0.75,0.75 n 1.80 n 3.40 <PcV 31.0° Xci 0.030 el 0.724 h*o 380.0 kPa Sl 2.0 S2 1.5 Q l 1.5 " 2 0.50 " 3 1.50 01 2.10 02 0.70 M + 0.25 C : 270 previoulsy, except the magnitude of mi . In the predictions shown in Figures C . l a and C . lb , the magnitude of mi was increased in steps until an accurate match was obtained with the observations. Coincidently, the best fit was observed when mi was replaced by m2 of each sand. A P P E N D L X - D In Appendix-D, the predictions submitted for the Cleveland Workshop by Lade, Mat-suoka and Prevost are presented for comparisons with the predictions of the proposed model. 271 F ^ O L J L . I L _ ^ D E : YEAR OF THE MODEL CLASSIFICATION OF THE MODEL 50IL TYPE APPLICATION LIMITATIONS OF STRESS/STRAIN | D : 272 ] 9 7 7 ' i 9 7 ? C L ^ T O - P L C . ' - T I S A N D S . N . C . C L A Y S L A f - C . E S T R E S S R E V E R S A L S YES NO FAILURE SURFACE * YIELD 5URFACE * FLDW RULE * STRAIN SOFTENING » STRAIN HARDENING * TENSI L E BEHAVIOR * INELASTIC BEHAVIOR * VOLUME DILATANCY ? VOLUME CONTRACTANCY * LOAD RATE * I N I T I A L ANISOTROPY * INDUCED ANI50TROPY * L A D E ' S F A I L U R E S U R F A C E T W O Y I E L D S U R F A C E S N O N A S S O C I A T E D O N C O N I C A L . A S S O C I A T E D O N C O L L A P S E ( C A P ) . ASSUMPTIONS OF THE MODEL I s o t r o p i c h a r d e n i n g and s o f t e n i n g S e p a r a t i o n of s t r a i n s : PARAMETERS OF THE MODEL 14 -for s a n d s , 10 -for n o r m a l l y c o n s o l i d a t e d c l a y . TESTS REQUIRED T h r e e t r i a x i a l . c o m p r e s s i o n t e s t s w i t h u n l o a d i n g r e l o a d i n g b r a n c h . I s o t r o p i c c o m p r e s s i o n . REFERENCES Lade, P.V. " E l a s t o P l a s t i c S t r e s s - S t r a i n T heory f o r C o h e s i o n l e s s S o i l w i t h C u r v e d Y i e l d S u r f a c e s . " I n t e r . J o u r n a l S o l i d s and S t r u c t u r e s . Vol .13, pp. 1019-1035, 1977. Lade, P.V. •'S t r e s s - S t r a l n T h e o r y -for N o r m a l l y C o n s o l i d a t e d CI ay" 3 r d I n t . C o n f e r e n c e on N u m e r i c a l Methods i n Geonifc-c h a r , i c ^ . A a i . M e n . A p r i l . 1979 pp. 1325-1337. D £ . -02 0 .02 .04 .06 .08 .1 .12 •02 r .01 -.01 A l .02 L-.02 0 .02 .04 •06 .08 1 . 1 2 0 .02 .04 .06 .08 .1 £ , -oi r°2 e. A 2 12 HR1 HR2 300 r T £„ -02 0 .02 .04 .06 .08 .1 r -02 £ 250 200 150 100 50 0 12 0 .02 .04 .06 .08 .1 . C I -01 r .02 £ - .01 - .02 L - . 0 2 A 3 0 02 .04 .06 .08 .1 .12 A 4 7 12 7 H H H 3 B T 1 0 0 50 0 - 5 0 - 1 0 0 - 1 5 0 £ .005 .0025 0 .0025 - .001 0 LADE .001 1 1 1 I i J £ 2 .005 - - 0 0 1 0 .001 0 I 1 L_ A 5 - . 0 0 1 7 7 0 .001 - 1 5 0 - . 0 0 1 .001 £ . 0 05 , 0025 0 i ' <-£ z - 0 05 , 0 025 - . 0 0 1 0 ,001 7 . i i i i - i - i A 7 - . 0 0 1 0 .001 7 HR3C 6 0 0 r 7~ 5 0 0 4 0 0 3 0 0 2 0 0 100 IADE 0 . 0 2 . 0 4 . 0 6 . 08 .1 £ v - 0 05 r A 8 - . 0 1 5 L 7 12 HAJJ i ME: MATSUOKA D : 277 Y E A R D F T H E M O D E L C L A S S I F I C A T I O N O F T H E M D D E L 5 0 I L T Y P E A P P L I C A T I O N L I M I T A T I O N S O F S T R E S S / S T R A I N J 9 6 5 E L A S T O P 1 . A 5 T I 1 : 5 A M D S AMD C L A Y S N O N E Y E S N O F A I L U R E S U R F A C E Y I E L D S U R F A C E M O H R - C O U L O M B OR M A T S U O K A - N A f A I 2 Y I E L D 5 U R F A C E 5 A N D O N E DUE TO P R I N C I P A L S T R E S S R O T A T I O N F L O W R U L E S T R A I N S O F T E N I N G S T R A I N H A R D E N I N G T E N S I L E B E H A V I O R I N E L A S T I C B E H A V I O R V O L U M E D I L A T A N C Y V O L U M E C O N T R A C T A N C Y L O A D R A T E I N I T I A L A N I S O T R O P Y I N D U C E D A N I S O T R O P Y A S S U M P T I O N S O F T H E M O D E L 1- H y p e r b o l i c s t r e s s r a t i o v s s h e a r s t r a i n r e l a t i o n e x p r e s s e d in g e n e r a l c o o r d i n a t e s . 2 - S t r e s s r a t i o v s . s t r a i n i n c r e m e n t r a t i o r e l a t i o n e x p r e s s e d i n g e n e r a l c o o r d i n a t e s . 3 - S u p e r p o s i i t i o n o-f t h e " t w o - d i m e n s i o n a l " p r i n c i p a l s t r a i n i n c r e m e n t s . 7 p a r a m e t e r s . 5 m u s t b e d e t e r m i n e d . k : P a r a m e t e r w h i c h d e c i d e s t h e m a g n i t u d e o-f t h e s h e a r s t r a i n d u e t o s h e a r a n d t h e p r i n c i p a l s t r e s s r o t a t i o n . k c : P a r a m e t e r w h i c h d e c i d e s t h e m a g n i t u d e o-f t h e s h e a r s t r a i n d u e t o a n i s o t r o p i c c o n s o l i d a t i o n . K : P a r a m e t e r w h i c h d e c i d e s t h e s t r a i n i n c r e m e n t r a t i o . (S : D e v i a t i o n a n g l e b e t w e e n t h e p r i n c i p a l s t r e s s d i r e c t i o n a n d t h e p r i n c i p a l s t r a i n i n c r e m e n t d i r e c t i o n . <J> : T r i c t i o n a n g l e . C c , C s : C o n s o l i d a t i o n p a r a m e t e r s . 1 t r i a .< i a 1 c o m p r e s s i o n , t e s t d'id 1 f j n i u l u U ; . o r t e i t . M a t s u o k a e t a l l . " A C o n s t i t u t i v e M o d e l o f S a n d s a n d C l a y s 1 c-E v a l u a t i n g t h e l<i< 1 u t - n c o c«f r u l ;;( t h e P r : n t : S t r e s - . : A.- . " 2 i n 1 . I n t e r n a t i o n a l Sy. ip:.>0 : urr. f-.'Lii:(T i >. r. 1 Mc.'Ct-l s i n Gi^ome;. ' <*•••: t L r . G h e r . t . P t - I g i u m . p p . 6"* ''P.. P A R A M E T E R S O F T H E M O D E L T E 5 T 5 R E Q U I R E D R E F E R E N C E S D : 278 HH 1 6 0 0 - T . 0 2 . 0 4 . 0 6 . 0 8 .1 . 1 2 - 0 2 0 2 £ , . 0 2 . 0 4 . 0 6 . 0 8 .1 .12 HH2 3 0 0 - T 2 5 0 2 0 0 1 5 0 1 0 0 5 0 MATSUOKA 01 - . 0 0 5 A _ - . 0 1 L . 0 2 A 2 o . 0 2 . 0 4 . 0 6 . 0 8 .1 7 -i 1 1 i i 0 . 0 2 . 0 4 . 0 6 . 0 8 .1 . 1 2 r 0 2 . 12 0 . 0 2 . 0 4 . 0 6 . 0 8 .1 .12 £ , . 02 r 0 2 £ . .01 - . 0 1 - . 0 2 L - . 0 2 A 3 C 0 2 . 04 . 0 6 . 0 8 .1 . 1 2 A 4 to H R 3 B T 1 0 0 50 0 •50 - 1 0 0 150 £ .005 .0025 0 i £ .005 .0025 A 7 0 i MATSUOKA ,001 0 ,001 7 001 0 .001 7 D : 282 CHC . 0 3 . 0 2 .01 - . 0 1 - 1 2 0 - 6 0 0 6 0 1 2 0 1 8 0 2 4 0 . 0 5 . 0 3 .01 - . 0 1 - . 0 3 - . 0 5 - 1 2 0 - 6 0 0 6 0 1 2 0 1 B 0 2 4 0 -K 1 -r*-A l 3 . 0 3 . 0 2 .01 0 I h - . 0 1 . 0 5 . 0 3 .01 MATSUOKA H 1 1 1 2 4 0 3 0 0 3 6 0 4 2 0 4 8 0 5 4 0 6 0 0 .01 - . 0 3 . 0 5 ^ h 2 4 0 3 0 0 3 6 0 4 2 0 4 8 0 5 4 0 6 0 0 A1 4 - r £ „ . 0 3 .01 .01 - . 0 3 - . 0 5 -H 1 J I . 0 5 . 0 3 .01 •120 - 6 0 0 6 0 1 2 0 1 8 0 2 4 0 .01 . 0 3 - . 0 5 H h - . 0 3 1 2 0 - 6 0 0 6 0 1 2 0 1 8 0 2 4 0 2 4 0 3 0 0 3 6 0 4 2 0 4 8 0 540 600 A l 5 A1 6 CRC .02 .01 o ~: - .01 .03 .02 .01 H 1 1 1 •120 -60 0 60 120 180 240 0 r 1 - .01 .02 -.03 V H 1 b-4- » A ' 7 -120 -60 0 60 120 180 240 .02 .01 • 283 MATSUOKA H 1 1 0 I 1 .01 240 300 360 420 480 540 600 .03 .02 .01 0 I f-. . o , -.02 - .03 H 1 ^ 1 T\ \ A 1 8 240 300 360 420 480 540 600 .02 .01 - .01 \ - .02 .02 .01 '•. 0 I \ I 4- / / .01 \ - .02 -120 -60 0 60 120 180 240 .02 .01 o v*>o -.01 y -\ \ H 1 1 - .02 -120 -60 0 60 120 180 240 A l 9 / M 1 1 1 240 300 360 420 480 540 600 .02 .01 0 .01 .02 / \ \ \ 1 T 240 300 360 420 480 540 600 A 2 0 C H C MATSUOKA A21 C R C MATSUOKA A 2 2 r- R; LT ••V O I"; ~J" Z> : 285 Y E A R D F T H E M O D E L C L A S S I F I C A T I O N O F T H E M O D E L S O I L T Y P E A P P L I C A T I O N L I M I T A T I O N S O F 5 T R E S 5 / S T R A I N Y E S N O F A I L U R E S U R F A C E Y I E L D S U R F A C E S T R A I N S O F T E N I N G S T R A I N H A R D E N I N G T E N S I L E B E H A V I O R I N E L A S T I C B E H A V I D R V O L U M E D I L A T A N C Y V O L U M E C O N T R A C T A N C Y L O A D R A T E I N I T I A L A N I S O T R O P Y I N D U C E D A N I S O T R O P Y A S S U M P T I O N S O F T H E M O D E L P A R A M E T E R S O F T H E M O D E L T E 5 T S R E Q U I R E D R E F E R E N C E S D : 286 .02 .04 .06 .08 .1 .12 £ , .02 r.02 £ .01 -.01 V -.02 L-.02 A 1 0 .02 .04 .06 .08 J .12 0 .02 .04 .06 .08 .1 .12 £, -01 r.02 £ A 2 HR2 300 r T 0 .02 .04 .06 .08 . 1 12 7 1 1 i i .02 .04 .06 .08 .1 .12 £, .02 r.02 £ £ , . 0 1 R . 0 2 £ , H 1 1 1 D 289 CH2 PREVOST T SD2/S1 .2 . 1 0 L_ £ £ P ° x , ° y . c z -.06 -.04 -.02 0 .02 .04 .06 .005 -.005 -.01 -.015 -.02 L A 1 O o H 1 1 1 IID2 \ .02 .04 .06 .06 .1 .12 CR1 PREVOST CR2 PREVOST A l 1 A 1 2 D : 290 CHC .03 . 0 2 .01 .01 .05 .03 .01 .01 .03 •120 -60 0 60 120 180 240 H 1 T+ f \ . -.05 -120 -60 0 60 120 180 240 A l 3 T £ „ .03 . 0 2 .01 0 i h .01 PREVOST H 1 1 1 .05 .03 240 300 360 420 480 540 600 .01 ••' I h -.01 -.03 -.05 H 1 + 1 240 300 360 420 480 540 600 A 1 4 .05 .03 .01 -.01 -.03 -.05 -I h .05 .03 .01 -.01 -.03 -.05 -120 -60 0 60 120 180 240 .05 .03 .01 -.01 -.03 240 300 360 420 460 540 600 .03 H 1 1 . 0 1 . 0 1 I rH-- . 0 3 -.05 -120 -60 0 60 120 180 ?40 . 0 5 H 1 'r-l 1 4, 2 ^ 0 3 0 0 3 6 0 4 2 0 *8C S ^ C 6 C 0 i A l 5 A l 6 

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