study of the undrained static response of sandy soils in the critical state framework

81 Table 4-6– Stresses and strain at phase transformation from undrained triaxial compression tests 5% silty sand.... 88Table 4-15– Stresses and strain at undrained instability state fro

For the degree of

Final examining committee members

, Chair

Approved by Major Professor(s):

Approved by Head of Graduate Program:

Date of Graduate Program Head's Approval:

Tejas Gorur S Murthy

Study of the Undrained Static Response of Sandy Soils in the Critical State

Framework

Doctor of Philosophy

Monica Prezzi Co-Chair

Rodrigo Salgado Co-Chair

CRITICAL STATE FRAMEWORK

A Dissertation Submitted to the Faculty

ofPurdue University

byTejas Gorur S Murthy

In Partial Fulfillment of the

Requirements for the Degree

ofDoctor of Philosophy

August 2006

Purdue University

West Lafayette, Indiana

3239785 2007

UMI Microform Copyright

All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company

300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company

For my parents, Ranganayaki and Sreenivasa Murthy

ACKNOWLEDGMENTS

I am grateful to my advisors Professors Rodrigo Salgado and Monica Prezzi for their tutelage It was my privilege to work on this project I am especially grateful for their guidance, advice and patience throughout graduate school I thank Professors Adolph Altschaeffl and Laura Pyrak-Nolte for serving

on my committee I am indebted to Professor Adolph Altshcaeffl for providing me

with funding all through my stay at Purdue Working with him on the projects from Purdue Physical Plant Services was truly enjoyable I am very obliged to Professor Altschaeffl for providing valuable suggestions during the course of my research His insights into experimentation and soil behavior are truly remarkable and it was my good fortune that I got a chance to interact with him I

will treasure all his kind words, cryptic emails, and advice always

I especially want to thank my colleague Dr D Loukidis for permitting me

to use our joint work in Chapter 4 Dr P Bandini and Dr A Carraro helped me with basic specimen reconstitution techniques I would also like to thank Dr M Santagata for giving me a chance to be her TA so many times It was a remarkable experience I express my deepest gratitude to Ms J Lovell for all her assistance, guidance and discussions during my experimentation and, often

at 4:25 pm I would like to thank Dr C K Chan for providing clarifications on the

design and implementation of the triaxial machine A few triaxial extension tests

were performed by Ms I Z Yildirim, I acknowledge her help

I am very thankful to Drs Srinivasan, and Veda Chandrasekar for their kindness and hospitality I also thank Dr A Shah, Dr P Banada and Euridice Oware for their friendship Finally I am very grateful to my brother Koustuba for his love, support and encouragement throughout my life

TABLE OF CONTENTS

Page

LIST OF TABLES vii

LIST OF FIGURES x

ABSTRACT xiii

LIST OF SYMBOLS xv

CHAPTER 1 INTRODUCTION 1

1.1 Project Rationale 1

1.2 Thesis Scope and Objectives 3

1.3 Organization 4

CHAPTER 2 BACKGROUND 6

2.1 Introduction 6

2.2 Laboratory Element Testing 6

2.3 Stress And Strain Variables 8

2.4 Interpretation of P and Q 13

2.5 Critical-State Soil Mechanics 15

2.6 Mechanical Behavior Of Sands 15

2.6.1 Behavior of Sand Under 1-D and Isotropic Compression 16

2.6.2 Shear Behavior of Sands 17

2.6.3 Stress-Dilatancy Relationships 20

2.6.4 Critical State of Sands 25

2.7 Rationale Behind Laboratory Programs 31

2.8 Summary 32

CHAPTER 3 MATERIALS, EQUIPMENT, AND TESTING PROGRAM 34

3.1 Introduction 34

3.2 Advantages of the Triaxial Test 35

3.3 Equipment Description 37

3.3.1 Application And Measurement of Axial Load 39

3.3.2 Measurement of Axial Displacement 40

3.3.3 Measurement of Pore Pressure 40

3.3.4 Measurement of Volume Change 41

3.3.5 Modifications to the Triaxial Apparatus 41

3.4 Materials Used 44

3.5 Density Characterization of The Soils Studied 48

Page

3.5.1 Emax And Emin of Silty and Clayey Sands 49

3.6 Methods of Sample Preparation 50

3.6.1 Moist Tamping (MT) 51

3.6.2 Water Pluviation Method (WP) 54

3.6.3 Modified Slurry Deposition (SD) 55

3.7 Saturation 58

3.7.1 Vacuum Saturation Method 58

3.7.2 Back-Pressure Saturation 59

3.8 Consolidation 60

3.9 Undrained Triaxial Shear 61

3.10 Disassembly and Water Content Analysis 61

3.11 Corrections to Triaxial Test Data 62

3.12 Summary 64

CHAPTER 4 UNDRAINED RESPONSE OF CLEAN AND SILTY SANDS IN TRIAXIAL COMPRESSION 65

4.1 Introduction 65

4.2 Previous Work 66

4.3 Materials and Equipment 68

4.4 Specimen Preparation 69

4.5 Results 70

4.5.1 Isotropic Compression 70

4.5.2 Undrained Shear in Triaxial Compression 75

4.5.3 Critical State 77

4.5.4 Phase-Transformation State 80

4.5.5 Quasi-Steady State 83

4.5.6 Undrained-Instability State 86

4.6 Discussion of Test Results 89

4.6.1 Critical State 89

4.6.2 Phase Transformation 97

4.6.3 Quasi-Steady State 104

4.6.4 Undrained Instability State 107

4.7 Conclusions 116

CHAPTER 5 EFFECTS OF PLASTICITY OF FINES ON THE UNDRAINED BEHAVIOR OF SANDY SOILS 118

5.1 Introduction 118

5.2 Previous Work 119

5.3 Experimental Program 122

5.3.1 Sample Preparation 122

5.3.2 Test Procedure 123

5.4 Results 123

5.4.1 Critical State 124

5.4.2 Phase-Transformation and Quasi-Steady States 125

5.4.3 Undrained Instability State 127

Page

5.5 Discussion of the Results 128

5.5.1 Critical State 128

5.5.2 Phase-Transformation State 133

5.5.3 Quasi-Steady State 138

5.5.4 Undrained Instability State 139

5.6 Parameters for Comparison of Densities 140

5.6.1 Intergranular Void Ratio 141

5.6.2 Relative Density and Relative State Parameter 145

5.7 Micromechanical Considerations of the Fabric of Sands with Fines 150

5.8 Conclusions 151

CHAPTER 6 UNDRAINED RESPONSE OF SANDY SOILS IN TRIAXIAL EXTENSION 153

6.1 Introduction 153

6.2 Previous Work 154

6.3 Experimental Details 156

6.4 Results 157

6.4.1 Critical State 158

6.4.2 Phase-Transformation State 159

6.4.3 Quasi-Steady State 160

6.4.4 Undrained Instability State 161

6.5 Discussion 162

6.5.1 Critical State 162

6.5.2 Phase-Transformation State 167

6.5.3 Quasi-Steady State 171

6.5.4 Undrained Instability State 172

6.6 Conclusions 175

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 176

7.1 Summary of Results 176

7.1.1 Overview of Research Program 176

7.1.2 Effects of Silt on the Undrained Behavior 177

7.1.3 Effects of Plasticity of Fines 178

7.1.4 Effects of Deformation Mode 178

7.2 Recommendations for Future Research 179

7.2.1 Overview 179

7.2.2 Expansion of the Current Project 180

7.2.3 Evaluation of Microstructure During Shearing 180

7.2.4 Detailed Investigation Deformation Mode 181

7.2.5 Correlation Between Laboratory Behavior and Field Performance 181

LIST OF REFERENCES 183

APPENDIX 193

VITA 227

LIST OF TABLES

Table Page Table 2-1 Summary of Stress-Dilatancy Relationships (Been and Jefferies

2004) 24

Table 3-1–Maximum and minimum void ratios obtained for sands with fines

(from ASTM procedures) 50

Table 3-2– Dry mixture proportion for target fines content (revised from

Carraro 2004) 57

Table 4-1–Results of static undrained triaxial compression tests at critical

state (Clean Sand) 78

state (5% silty sand) 79

state (10% silty sand) 79

state (15% silty sand) 80

triaxial compression tests (clean sand) 81

Table 4-6– Stresses and strain at phase transformation from undrained

triaxial compression tests (5% silty sand) 82

triaxial compression tests (10% silty sand) 82

triaxial compression tests (15% silty sand) 83

Table Page

compression tests (Clean sand) 84

compression tests (5% silty sand) 85

compression tests (10% silty sand) 85

compression tests (15% silty sand) 86

triaxial compression tests (Clean sand) 87

triaxial compression tests (5% silty sand) 88Table 4-15– Stresses and strain at undrained instability state from

triaxial compression tests (10% silty sand) 88

triaxial compression tests (15% silty sand) 89

regression of data 97Table 5-1– Results of static undrained triaxial compression tests at

critical state (5% clayey sand) 125Table 5-2– Stresses and Strains at phase-transformation state

(5% clayey sand) 126Table 5-3– Stresses and strains at quasi-steady state (5% clayey sand) 127Table 5-4–Stresses and strain flow liquefaction state 128Table 5-5–Critical state parameters obtained from least-square

regression of data on clean, silty and clayey sands 132Table 6-1- Stresses reached at the critical state for clean sand tested in

undrained triaxial extension 158

Table Page Table 6-2- Stresses reached at the critical state for 5% silty sand tested in

undrained triaxial extension 159Table 6-3- Stresses and strains at the phase-transformation state in

undrained triaxial extension (clean sand) 159Table 6-4-Stresses and strains at the phase-transformation state in

undrained triaxial extension (5% silty sand) 160Table 6-5- Stresses and strains at the quasi-steady state in undrained

triaxial extension (clean sand) 160Table 6-6- Stresses and strains at the quasi-steady state in undrained

triaxial extension (5% silty sand) 161Table 6-7- Stresses and strains at the undrained instability state in

undrained triaxial extension (clean sand) 161Table 6-8-Stress and strains at the undrained instability state in undrained triaxial extension (5% silty sand) 162Table 6-9- Critical-state parameters for sandy soils tested under undrained

triaxial extension conditions 165Table 6-10- Stress ratios at critical state and at PT state for clean and

5% silty sands in triaxial extension 169

LIST OF FIGURES

Figure Page

Figure 2-1– Soil element subjected to a general state of stress 7

Figure 2-2 – Configuration of the triaxial test 8

Figure 2-3–Schematic view of the S-plane and the space diagonal (Davis and Selvadurai 2002) 14

Figure 2-4– Stiffness vs % strain in log scale 17

Figure 2-5–Behavior of sand under drained and undrained conditions

(Mitchell and Soga 2005) 19

Figure 2-6– Model of Dilatancy (Salgado 2006 and Bolton 1986) 23

Figure 2-7 –The effect of initial state on the stress strain and volume change or pore pressure response to reach the critical-state line (Mitchell and Soga 2005) 30

Figure 3-1– Triaxial Plane (illustration of Baldi et al 1988) 36

Figure 3-2– Schematic diagram of the CKC automatic triaxial system and

details of the triaxial cell 39

Figure 3-3– Granulometric curves of the Ottawa sand, Sil-Co-Sil silt, and

Kaolin clay used in this study 45

Figure 3-4– Scanning electron micrographs of the 1- wet-clay and 2- dry clay (Carraro and Prezzi 2006)(continued) 47

Figure 3-5–Grain size distributions of the clean and silty sands studied 48

Figure 4-1–Isotropic Compression of Clean Sands 71

Figure 4-2–Isotropic Compression of 5% Silty Sands 72

Figure 4-3–Isotropic Compression of 10% silty sands 73

Figure 4-4–Isotropic Compression of 15% silty sands 74

Figure Page

soils in triaxial compression 76

Figure 4-6–Critical state of clean and silty sands 91

Figure 4-7–Critical state lines of clean and silty sands prepared with different specimen preparation methods 93

Figure 4-8–Critical-state locus of clean, 5%, 10% and 15% silty sands in stress space (continued) 96

Figure 4-9–Stress ratio at PT vs State Parameter at PT for 0, 5 and10% silty sands (continued) 99

Figure 4-10 – Flow potential vs void ratio for clean and silty sands (continued) 103

Figure 4-11–Stress ratio at QSS, PTS vs initial state parameter 105

Figure 4-12– Ratio of axial strain at QSS to PT vs state parameter 106

Figure 4-13–Axial strain at QSS vs initial state parameter 107

Figure 4-14–Stress ratio at undrained instability vs initial mean stress (continued) 110

Figure 4-15– Mean stress at undrained instability vs initial mean stress for sandy soils 111

Figure 4-16ʊ Stress ratio vs state parameter at undrained instability state (continued) 115

Figure 5-1– Soil Mixture Classification (Thevanaygam and Martin 2002) 120

Figure 5-2– Critical-state line of clean, silty and clayey sands 129

Figure 5-3– Critical-state loci for clean, silty and clayey sands in (p’-q) space 131 Figure 5-4ʊ State parameter vs stress ratio at phase transformation for clean, silty and clayey sands 134

Figure 5-5ʊ Flow potential vs void ratio for clean, silty and clayey sands 137

Figure 5-6ʊ Stress ratios at the phase-transformation state and at the quasi-steady state vs initial state parameter for clean, silty and clayey sands 139

Figure 5-7ʊ Initial mean stress vs mean stress at UIS for 5% clayey sand 140

Figure Page Figure 5-8ʊ Critical state loci of clean, silty and clayey in eg vs ln p' space 141Figure 5-9ʊ Flow potential vs intergranular void ratio for 0, 5, 10% silty

sand specimens reported in Chapter 4 143Figure 5-10ʊ Flow potential vs intergranular void ratio for clean, 5%silty

and 5%clayey sands 144Figure 5-11ʊ Relative density vs mean stress at critical state for clean, silty and clayey sands 147Figure 5-12ʊ Relative density vs flow potential for clean, silty and clayey

sands 148Figure 5-13ʊ Relative state parameter vs flow potential for clean, silty and clayey sands 149Figure 5-14ʊ Schematic diagram of the placement of fine particles inside a coarse-grained matrix (a) sand-silt mixture (b) sand-clay mixture 150Figure 6-1–Schematic representation of the undrained behavior of sand in

triaxial extension 157Figure 6-2– Critical state of clean and 5% silty sands in undrained triaxial

extension 163Figure 6-3– Critical-state loci of clean and 5% silty sands in stress space 164Figure 6-4– Stress ratio vs state parameter at phase transformation for

clean sands and 5% silty sands 167Figure 6-5– Flow potential vs void ratio at different initial mean stresses for clean and silty sands 170Figure 6-6– Initial state parameter vs stress ratio at phase transformation

and quasi-steady state for clean and silty sands 171Figure 6-7– Void ratio vs stress ratio at the undrained instability state for

undrained triaxial compression and extension (clean and 5% silty sands) (continued) 173Figure 6-8– Initial mean stress vs mean stress at the undrained instability state under triaxial extension conditions 174

ABSTRACT

Murthy, Tejas Gorur S Ph.D., Purdue University, August, 2006 Study of the Undrained Static Response of Sandy Soils in the Critical State Framework

Major Professors: Monica Prezzi and Rodrigo Salgado

The thesis reports on experimental research on the static undrained response of silty and clayey sands The purpose of this testing program was to understand the stress-strain response of sands with small percentages of fines With this objective in sight, the testing program involved understanding and delineating the effect of four variables: density of the soil, percentage of fines in the sand matrix, plasticity of the fines, and the mode of deformation only under isotropically consolidated, and static loading triaxial conditions Four characteristic states were identified in the undrained response of the soils investigated: the undrained instability state, quasi-steady state, phase-transformation state and critical state

The experimental data was analyzed in the context of the critical-state framework The critical state was found to be independent of the initial fabric and

of the pre-shear stress history of the sand The critical-state friction angle increased slightly with the addition of small percentages of silt The effect of silt

on the characteristic states of undrained shear was evaluated and ways to relate the critical state with these states of behavior were proposed The concepts used

in constitutive modeling of sands can be extended to silty sands using the suitable input parameters

The plasticity of the fines had an impact on the response of the sand The variables conventionally used in understanding the behavior of sands with small

amounts of fines were evaluated, and the validity of the different variables was assessed It was found that extension of modeling concepts of clean sands to clayey sands cannot be done directly

Comparison between triaxial extension and triaxial compression tests conducted on 0 and 5% silty sands showed that the behavior under triaxial extension conditions was more contractive than under triaxial compression for both clean and 5% silty sand

This experimental program aims to provide a well designed and complete data set for systematic understanding of the static behavior of sands and sand-fine mixtures This data set can also be used to calibrate constitutive models designed for sandy soils

LIST OF SYMBOLS

p - mean stress

p’ - mean effective stress

pa – normalizing stress =100kPa

q – deviator stress

e – void ratio

esubscript - void ratio type defined by subscript

Dr – relative density

ısubscript – stress in a given direction (direction described in the subscript)

ij’ – friction angle at a given state (state described in the subscript)

QSS – Quasi Steady State

UIS – Undrained Instability State

CS – Critical State

CHAPTER 1 INTRODUCTION

1.1 Project Rationale

Traditionally, research in the field of soil mechanics has focused on understanding the behavior of textbook soils such as clean sands and pure clays The behavior of sands and clays represent the two extremes of the wide spectrum of behavior shown by natural soils The dissimilarity in behavior of sands and clays stems mainly from the striking difference in particle size between these two materials A typical clay particle is much smaller than a sand grain This difference in particle size has important ramifications for the engineering properties of sands and clays The differences in shape of the clay and sand particles, and the physicochemical interactions that affect the clays also manifests in the soil behavior Furthermore, difficulties in replicating field conditions in the laboratory have also led researchers to study either clean sands

or pure clays Different methods of sample reconstitution techniques have been developed to study the behavior of sands in the laboratory, usually under drained conditions, because of the large hydraulic conductivity of sands In the case of clays, undisturbed clay samples are often tested under undrained conditions, as clays have extremely low hydraulic conductivity (testing of clay samples under drained conditions is extremely time consuming)

The pioneers of geotechnical engineering were able to develop design procedures needed to solve geotechnical problems relying heavily on empiricism and accumulated experience Simple experimental results that provided an estimate of the strength of the soil were sufficient for designing geotechnical

structures for simple boundary conditions With the advent of numerical modeling, irregular and complex boundary conditions can be simulated The constitutive relations for the soil are plugged into the numerical models that then provide solutions for the boundary value problems being studied

Parry (1956) carried out a series of well designed experiments on the behavior of reconstituted soft clays This series of experiments led to the development of rigorous mathematical constitutive models based on the principles of plasticity theory This set of models have been referred to as the CAM clay series Similar extensive testing programs by several researchers have led to the development of very advanced constitutive models for both sands and clays

In order to provide a simple framework to describe, interpret and predict soil response to different loading conditions, a unified framework – the critical-state soil mechanics framework – was developed in the 1960’s The critical state framework has been instrumental in providing some good insights into soil behavior and has also been at the core of many constitutive models There is an enormous amount of research that has emerged in this field, but most of it has centered on either clean sands or pure clays

Often, idealizations of transitional materials as either clean sand or pure clay can lead to misleading results Even though enormous developments have occurred in modeling of soils using advanced plasticity theory concepts, more work still needs to be done on transitional soils such as silty and clayey sands

There has been limited research on the behavior of silty and clayey sands, and, in the recent past, it became apparent that there is a strong need to understand the behavior of these transitional soils because of several failures of geotechnical structures involving these soils during recent earthquakes (Yamamuro and Lade 1999) It is known that even though the behavior or silty and clay sands may resemble that of a sand, the presence of even small amounts of fines in the sand brings about several changes in the behavior of the sandy soil Some research has shown that the presence of small amounts of

fines renders the sand more susceptible for liquefaction (Yamamuro and Lade 1999) Even though several researchers have been working on this topic (Carraro et al 2003, Thevanayagam et al 2002), the current state of knowledge

of the behavior of transitional soils is unsatisfactory Difficulties with sample reconstitution methods or undisturbed sampling of these types of soils have contributed to this gap in knowledge With technological advancements in laboratory equipment, extensive testing programs can be performed for different types of soils under non-ideal drainage and stress conditions and hence advance further the state of knowledge in the near future

The main objective of this research program is to understand, compare and contrast the behavior of sands with different percentages of non-plastic and plastic fines The behavior of these transitional soils is also compared with clean sand- considered here as a reference The scope of this thesis is discussed next

1.2 Thesis Scope and Objectives

The main goal of this research was to study the effects of small amounts of fines (both plastic and non-plastic fines) on the behavior of sand The

systematic experimental program to understand the behavior of silty and clayey

state-framework to allow determination of intrinsic parameters for these soils

The behavior of sand with different percentages of non-plastic and plastic fines (called henceforth as silt and clay respectively) was investigated through static triaxial testing performed under undrained conditions For reasons of completeness, the corresponding static drained response of these sands is also presented in this thesis; these tests were performed on the very same materials

by Carraro (2004) In order to understand the behavior of these soils under different stress paths, both triaxial compression and triaxial extension tests were conducted Appropriate laboratory protocols were developed, and the effect of different methods of specimen reconstitution on the load response of these soils was assessed The effect of stress level and the subsequent shear behavior was also gauged The response at medium and large strains was also studied.

The results of this experimental program were analyzed within the state soil mechanics framework The critical-state parameters for the transitional soils studied are reported Correlations between various parameters and the intrinsic critical state parameters are provided Finally, a complete data set is provided that can be used to calibrate and validate constitutive models for the silty and clayey sands investigated

critical-1.3 Organization

The thesis is composed of 7 chapters, including this chapter

Chapter 1 describes the goals and the scope of the work done

Chapter 2 deals with the basics of stress and strain analysis It also discusses the fundamental concepts of the critical-state framework and some important research carried out to date on the critical-state behavior of sands Chapter 3 presents the experimental details of the triaxial tests The different specimen preparation methods and the procedures used for sample saturation, consolidation and shearing are described

Chapter 4 deals with the behavior of sands with small amounts of non-plastic fines The different aspects of the load response of silty sands tested under undrained conditions (the critical state, the phase transformation state, the quasi-steady state and the state of undrained instability) are discussed in detail

Changes in the behavior of the host sand caused by the addition of 0, 5, 10 and 15% silt are highlighted

Chapter 5 deals with the behavior of sands containing different types of fines (silts vs clays) tested under triaxial compression conditions The efficacy of parameters (relative density, state parameter and relative state parameter) traditionally used to study the behavior of sands to describe the behavior of sands containing fines is assessed

Chapter 6 presents a comparison of the behavior of silty sands under undrained triaxial compression and triaxial extension Critical-state parameters obtained under different stress paths are tabulated

Chapter 7 provides some conclusions of the testing program and some recommendations for further research

CHAPTER 2 BACKGROUND

2.1 Introduction

This chapter deals with the use of continuum mechanics to study the behavior of sands The stress and strain variables used to study soil behavior are introduced here The stress and strain components acting on a single soil element and the physical relevance of the choice of stress variables are also presented here A brief description of elemental testing and its implications are presented The critical-state framework used to analyze the experimental data obtained from the triaxial testing conducted as a part of this research program is summarized Finally, some important facets of sand behavior are presented

2.2 Laboratory Element Testing

to six stress components as shown in Figure 2-1 An element of soil is subjected

to changes in stresses in all or any of the stress components depending on the loads the soil element is subjected to Understanding of the corresponding deformations in the soil element upon loading is necessary to formulate constitutive relationships

Figure 2-1– Soil element subjected to a general state of stress

In order to establish the soil stress and strain variables and the resulting constitutive relationships, a backdrop of the triaxial test is provided here Nuances and actual experimental details of the triaxial test are provided in Chapter 3

In a triaxial test, a sample of soil, which is contained in a membrane, is subjected to lateral stresses It is also subjected to axial stresses through rigid end platens as shown in Figure 2-2 Two of the three principal stresses are equal to the cell pressure or lateral stress ır, and one of them is the stress in the

Figure 2-2 – Configuration of the triaxial test

The axial stress is the major principal stress in triaxial compression and is the minor principal stress in triaxial extension Wood (2004) described the triaxial test as a "confined uniaxial test" Although several tests have been developed over the years to better replicate field conditions, the triaxial test is still used extensively in both practice and research

2.3 Stress and Strain Variables

Testing under triaxial conditions allows two degrees of freedom in the control of externally applied stress states The stresses acting on the soil

element constitute the stress tensor Vij (with compressive stresses taken as

the pore pressure

It is clear from the above equation that pore pressure affects the normal stresses alone and the shear stresses are not affected by the pore pressure In soil mechanics, water carries negligible shear stresses consistent with the above equation The stress state (ıca and ıcr) in the triaxial test is axisymmetric, and the corresponding axial and radial strain increments are represented by įİa and

įİr The volumetric strain increment įHp is defined as the sum of the axial strain increment and twice the radial strain increment:

stress q describes the distortional component of the stresses It is beneficial to

consider the deviatoric stress as a stress variable because it can be measured in

a triaxial test

The deviatoric stress is given by:

The ratio of the deviatoric stress to the mean stress is usually denoted by

Ș In frictional materials, such as the ones dealt with in this thesis, the stress ratio

is related to the mobilized friction angle in the soil I'm In this research, as only triaxial compression and extension tests were conducted, detailed discussions are carried out for these types of tests only In the case of triaxial compression, a positive deviatoric stress exists (q>0), and the axial stress on the sample is

greater than the radial stress components The ratio of the major and minor principal stresses is called the flow number (Salgado 2006) and is given by:

The mean stress p can also be defined as:

The values of the principal stresses of a stress tensor do not change with

a change in the coordinate system, and hence they are invariants p is described

in terms of the principal stresses and the stress invariant I1 as follows:

Eq.2-15

When the volumetric stress component is subtracted from the stress tensor, the distortional component of the stress tensor, which is the stress deviator tensor sij, is obtained

The deviatoric stress or the distortional stress is expressed in terms of J2

soil element (Davis and Selvadurai 2002) As depicted in

Figure 2-3, a point on the space diagonal represents an isotropic stress state, and a surface perpendicular to the space diagonal is referred to as the ʌ-plane Upon loading, the stress conditions of the soil element change, and the coordinates of the point in stress space change as well The mean stress acting

on the soil element can be expressed in terms of the distance traversed on the space diagonal (VD) (also shown in

VE and VF using the Pythagorean theorem The stress components VE and VF

are also depicted in the S-plane in Figure 2-3

Figure 2-3–Schematic view of the S-plane and the space diagonal

(Davis and Selvadurai 2002)

2.5 Critical-State Soil Mechanics

Critical state is defined as the state at which the soil continues to shear without further changes in effective stresses or density One important feature of critical-state soil mechanics CSSM is that it incorporates the void ratio e in the failure criterion unlike the traditional Mohr-Coulomb failure criterion which defines only a stress value for failure (Budhu 2000) According to the CSSM conceptual framework, all soils fail on a unique failure space in q, p' and e space When such

a unique failure surface exists, plastic flow due to shearing and due to compression can be unified These concepts enmeshed along with the fundamentals of plasticity theory form the basis of the critical-state framework There has been a surfeit of constitutive models attempting to closely replicate soil behavior incorporating this concept as the essence These models are generally calibrated with high-quality laboratory tests The main objective of this thesis is

to comprehend the behavior of sands with small amounts of fines; hence a brief review of the behavior of clean sands in terms of the critical state framework is provided below

2.6 Mechanical Behavior of Sands

Sand behavior has been extensively studied in soil mechanics The particularities of sand behavior outlined here are described using results from laboratory tests, particularly triaxial tests Sands have relatively large grain sizes and therefore have large hydraulic conductivity (sands are considered free-draining materials)

2.6.1 Behavior of Sand under 1-D and Isotropic Compression

Sands are studied in the laboratory through specimen reconstitution There are different techniques of specimen reconstitution With different methods

of sand specimen reconstitution (e.g., moist-tamping, water-pluviation, deposition, and air-pluviation; these are outlined in Chapter 3), specimens can be reconstituted at any desired density Hence, when sand specimens reconstituted using different techniques are compressed, different compression lines are obtained Only at very large stresses (depending on the mineralogy of the sand),

slurry-a unique normslurry-al consolidslurry-ation line is slurry-approslurry-ached At these lslurry-arge stresses, grslurry-ain crushing becomes prevalent (Jefferies and Been 2000) Pestana and Whittle (1995) referred to this as "high-confining stress compression behavior"

When several such non-unique compression lines exist for sands at stress levels in which grain crushing does not occur, the concept of a unique failure surface in (p’,q and e space) does not exist Hence, the validity and applicability

of critical-state concepts to sands was questioned Jefferies and Been (2000) conducted isotropic-compression tests on Erksak sand and studied the implications of multiple compression lines on the application of the critical-state theories They propose separating the state parameter (state parameter defined

as the difference between the current void ratio and the critical state void ratio at

a given state of stress) and overconsolidation ratio In case of clays these two variables have been treated as alternative forms of the same quantity They proposed isolating the plastic strains occurring in isotropic consolidation and the elastic strains during swelling and recompression The authors concluded that despite the existence of multiple compression lines, critical-state theories could still be extended to sands It is well known that sands with small percentages of fines also can be reconstituted at the same stress level to different densities, similar to that of sands Hence an approach suggested by Jefferies and Been (2000) is considered here and, thus the critical-state framework is also used to understand the behavior of sands with fines in this thesis

2.6.2 Shear Behavior of Sands

The behavior of sands during shear is outlined in the following sections The small-strain behavior of sands is predominantly elastic, with stiffness degrading with increasing strain Figure 2-4 shows a typical stiffness degradation curve of G or E vs % strain in log scale The stiffness degradation curve can be divided into 4 parts: linear elastic, non-linear elastic, pre-yield plastic and fully-plastic zone (Mitchell and Soga 2005) The soil behavior in the linear elastic zone is governed by the packing of the particles (density and the mean effective stress), the contact interactions and the elastic stiffness of the soil grains At such small levels of strains, there is no sliding of the soil grains relative to each other The behavior of sands at these small strain ranges are studied in the laboratory using Bender Element tests Studies on the small strain behavior of sands with silty and clayey fines were done by Carraro (2004)

Figure 2-4– Stiffness vs % strain in log scale

(Mitchell and Soga 2005)

Strain in the non-linear elastic zone ranges between 5×10-4% for sands at low confining stress and about 5×10-2% for sands at high confining stress (Santamarina et al 2001) Irrecoverable plastic strains develop in the zone of pre-yield plastic straining This can be identified through conventional soil testing under both undrained and drained conditions In the case of undrained tests, this state of pre-yield straining can be identified when excess pore pressure develops In drained tests, this stage is determined at the onset of permanent volumetric strains.

A change of slope in a stress-strain curve, defines the point of yielding The strains generated beyond the yielding point are fully plastic The behavior of soil at large strains is discussed in the next section and is the focus of this thesis

Development of irrecoverable plastic strains occurs at strain ranges as small as 7×10-3% to 7×10-2% depending on the type of soil, the density of the soil, and the stress level before shearing (Mitchell and Soga 2005)

Figure 2-5–Behavior of sand under drained and undrained conditions

(Mitchell and Soga 2005)

Figure 2-5 shows the different responses of loose sand during triaxial shearing under both drained and undrained conditions For drained shear, the stress-strain response and the volumetric strain response are outlined For undrained shear, the pore-pressure response and the stress-strain response are provided instead

In general, at the same confining stress, the strength of the soil increases with increasing density In case of drained shear behavior, the peak friction angle increases with increasing density In case of undrained shear, strain softening behavior is observed for loose specimens, whereas strain hardening is often observed for dense specimens The levels of stress reached differ with the different specimen preparation methods used

Sand behavior also depends on the stress state of the soil For soils of a given density or specific volume, the peak strength will increase as the confining stress level falls.

At the particle level, when shear stress is applied to a sand specimen, deformation occurs due to particle rearrangement Depending on the drainage conditions, volume changes also occur upon loading The shear-induced volume change accrues as a result of both rolling and sliding of the particles

Particle sliding tends to reduce the volume of the specimen by packing the particles into a denser state This densification comes about when the initial density of the sand is loose When sliding or slip down occurs, sand particles are filling the voids and they do not move in the direction of shearing Hence, contraction is observed at an early stage in the shearing process for a wide range of packing densities

When sand particles roll over each other, the specimen volume tends to increase Conversely, a large amount of movement is needed for particles to roll over the neighboring particles, and thus dilation occurs when the sand has deformed to large shear strains (Ishihara 1996, Anandarajah 2004) At very large strains, a state of flow commonly referred to as the critical state is reached

The stress-dilatancy relationships discussed in the next section are helpful

in explaining these nuances of the behavior of sands before the sand reaches critical state The critical state of sands is discussed in the last part of this chapter

2.6.3 Stress-Dilatancy Relationships

For any particulate material that is composed of solids and voids, rearrangement of the particles accompanied by changes in volume occurs during shearing The stress-dilatancy relations provide a macroscopic relation between the stress and the volume change during shearing The plastic potential

functions (from fundamental plasticity theory), which describe the development of plastic strains, can be considered as the basis for the stress-dilatancy relations in soils.

The basis of the stress-dilatancy theories is that dilative sands increase in volume during shearing This idea was first proposed by Reynolds in the late

19th century It is also known that the measurement of the mineral to mineral friction angle, yields values lower than that obtained through a continuum approach This led to the understanding that in addition to the frictional strength component, there exists also a dilatational component that adds to the strength

of the sand

A very simple view of the friction-dilatancy relation in plane-strain loading conditions (which does not account for the direction of the principal stress and strain increments) may be expressed as follows:

c

where Ic is the total friction angle, Icc is the critical-state friction angle representing the purely-frictional component of strength and \ the dilatancy angle representing the dilative component of strength

Rowe (1962) proposed the first “stress-dilatancy” theory:

1

1 3

Rowe (1962) assumed that the ratio of the energy supplied by the applied shear stress to the energy dissipated in the soil is a minimum De Josselin de Jong, (1976) proved that the same relation can be derived without recourse to the energy minimization principle by relying on friction laws (Salgado et al 2000).

The Rowe model is also written in the following form:

whereĭc,Ȍ are the critical state friction angle and the dilatancy angle

Figure 2-6 is a popular illustration of the concept of dilatancy, called the saw-blade model When a shear force is applied to the system, slippage takes place when friction between two blocks is overcome, and the two blocks move apart Similarly, when the shear force is increased continuously on this system

of sand particles, the sand particles roll over one another, and volume change is imminent if the shearing is continued in the horizontal direction