investigation of fabric anisotropic effects on granular

INVESTIGATION OF FABRIC ANISOTROPIC EFFECTS ON GRANULAR SOIL BEHAVIOR

by Zhongxuan YANG

A Thesis Submitted to The Hong Kong University of Science and Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Civil Engineering

August 2005, Hong Kong

Authorization

I hereby declare that I am the sole author of the thesis.

I authorize the Hong Kong University of Science and Technology to lend this thesis to other institutions or individuals for the purpose of scholarly research.

I further authorize the Hong Kong University of Science and Technology to reproduce the thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research.

Zhongxuan YANG August 2005

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To my parents

獻給我的父親和母親

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ACKNOWLEDGEMENTS

Grateful acknowledgements are due to the following persons:

— Dr. X. S. Li, my research advisor, for his counsel and guidance over the last four years. The knowledge and unique training acquired will benefit my whole life. — Drs K. M. Lee, and Y. H. Wang, members of my dissertation committee, for their valuable suggestions during the course of this research. — Messrs Kenny Ma, Michael Chung and Jimmy Ko, technicians in the civil lab, for their kind help in the laboratory tests. — Drs. D. Su, Ryan Yan, M. Yoshimine, Z. Y. Cai, and Abraham Chiu, for their valuable discussions and meaningful suggestions. — Colleagues during my study in UST, X. Li, X. J. Zhu, R. Chen, M. Zhang, L. G. Kong, X. G. Shi, X. B. Dong, S. C. Leung, and Simon Choi, for their cooperation and help over the years.

Thanks are also due to Professors D. G. Fredlund, L. Zhang, C. K. Y. Leung, P. Tong, and T. X. Yu, for the knowledge acquired in their classes.

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TABLE OF CONTENTS

Title Page

i

Authorization Page

ii

Signature Page

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Acknowledgement

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Table of Contents

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List of Figures

xi

List of Tables

xvii

Abstract

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Chapter 1 Introduction

1

Chapter 2 Literature Review

5

2.1 Introduction

5

2.2 Rotational Shear

6

2.3 Fabric Anisotropy

11

2.4 Non-Coaxiality

13

2.5 Quantification of Fabric Anisotropy

16

2.5.1 Physical Property Measurement

17

2.5.2 Geometric Quantity Representation

19

Chapter 3 Investigation of Anisotropic Granular Soil Behavior Under Rotational Shear

29

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3.1 Introduction

29

3.2 Hollow Cylinder Apparatus-

32

3.2.1 Loading Application System

33

3.2.2 Pressure Cell

35

3.2.3 Control and Data Collection System

36

3.3 Testing Procedure

37

3.3.1 Sample Preparation

37

3.3.2 Data Interpretation

38

3.3.3 Test Program

41

3.4 Test Results and Discussions

43

3.5 Conclusions

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Chapter 4 Quantifying Fabric Anisotropy ⎯ Electrical Property Measurement

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4.1 Introduction

63

4.2 Theoretical Description of Dielectric Materials

65

4.3 Electrical Measurements for Anisotropic Materials

69

4.3.1 Two-Terminal Electrode Measurement

69

4.3.2 Calibration of Coaxial Cables

70

4.3.3 Measurement

71

4.3.4 Test Results

72

4.4 Discussion

73

4.5 Summary and Conclusion

76

Chapter 5 Quantifying Fabric Anisotropy

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⎯ Using an Image Analysis Approach

83

5.1 Introduction

83

5.2 Test Setup

84

5.3 Materials and Other Equipments Used in the Test

84

5.3.1 Epoxy Resins

84

5.3.2 Sand and Sample Preparation Methods

85

5.3.3 Scanning Electron Microscope (SEM)

88

5.3.4 Tools for Sectioning, Grinding and Polishing

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5.4 Test Procedure

89

5.4.1 Specimen Impregnation

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5.4.2 Epoxy Resin Curing

90

5.4.3 Coupon Surface Preparation

90

5.4.4 Image Capture and Analyze

93

5.5 Conclusions

103

Chapter 6 Investigation on Mechanical Behavior of Granular Material

113

6.1 Introduction

113

6.2 Test Apparatus

114

6.2.1 CKC Triaxial System

114

6.2.2 Lubricated Ends

115

6.3 Test Procedure

116

6.3.1 Sample Preparation

116

6.3.2 Back Pressure Saturation and B-Value Check

118

6.3.3 Isotropic Consolidation

119

6.3.4 Shearing

119

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6.3.5 Evaluation of Initial Void Ratio 6.4 Data Reduction

119 121

6.4.1 Area Correction

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6.4.2 Data Filtering

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6.5 Test Results from Monotonic Loading

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6.5.1 Triaxial Compression Test

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6.5.2 Triaxial Extension Test

123

6.5.3 Torsional Shear Test

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6.6 Discussion on Influence of Fabric Anisotropy

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6.7 Discussion on Critical State Lines

127

6.8 Test Results from Cyclic Loading

132

6.9 Closing Remarks

134

Chapter 7 Constitutive Modeling of Sand Accounting for Different Inherent Fabric Anisotropy Effects

156

7.1 Introduction

156

7.2 Constitutive Modeling of Sand Behavior

157

7.2.1 General Observations of Sand Behavior

157

7.2.2 Description of the State of a Sand

158

7.2.3 State Dependent Dilatancy Theory

159

7.3 An Introduction to a Constitutive Model Accounting for Fabric Anisotropy Effect

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7.3.1 Platform Model with State-Dependent Dilatancy

162

7.3.2 Incorporation of Anisotropy Effects in Response to Proportional Loading

165

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7.4 Model Calibration

169

7.5 Simulation by the Model

172

7.6 Conclusions

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Chapter 8 Conclusions and Recommendations for Future Work

189 189

8.1 Overview 8.1.1 Mechanical Behavior in Laboratory Tests

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8.1.2 Assessment and Quantification of Fabric Anisotropy

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8.1.3 Fabric-Dependent Behavior and Constitutive Modeling

194

8.2 Future Work

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8.2.1 Fundamental Quantification on Fabric Anisotropy

195

8.2.2 Evolving of the Fabric Anisotropy

196

8.2.3 CSLs and Constitutive Modeling

196

Bibliography and References

197

Appendix

213

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LIST OF FIGURES Fig. 2.1

Rotational shearing classification

Fig. 2.2

Patterns of loading paths on the octahedral plane (a) uni-directional 1 (b)

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uni-directional 2 (c) crisscrossing 1 (d) crisscrossing 2 (e) circular (f) elliptic 1 (g) elliptic 2 Fig. 2.3

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Stress ratio versus number of cycles averaged for different loading paths (after Yamada and Ishihara 1983)

Fig. 2.4

Strain increment vector along a circular stress path (after Yamada and Ishihara 1983)

Fig. 2.5

25

25

Effective stress path in p ' − q space for all undrained principal rotation tests (a) Dr=90% (b) Dr=60% (c) Dr=30%

26

Fig. 2.6

Radial shear stress paths (after Yamada and Ishihara 1981)

26

Fig. 2.7

Effect of α on undrained behavior of Toyoura sand (a) stress-strain relationship (b) effective stress path

Fig. 2.8

Fig. 2.9

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Strain increment directions for (a) monotonic loading (b) pure loading (c) combined loading (after Gutierrez et al. 1991)

27

Neutral loading mechanism of the hypoplasticity model (after Li 1997)

27

Fig. 2.10 Formation factor versus density for different sample preparation methods 27 Fig. 2.11

Effective conductivity for mixture of mica flakes with electrolytes at two different ionic concentrations

28

Fig. 2.12 Relation of a particle to neighboring particles

28

Fig. 3.1

Rotational shearing classification

50

Fig. 3.2

Hollow cylindrical torsional shearing apparatus

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Fig. 3.3

Control and data acquisition system (after Cai 2001)

51

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Fig. 3.4

Hollow cylindrical sample after preparation

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Fig. 3.5

Various b values loading conditions in π -plane

52

Fig. 3.6

Stress state in the wall of a hollow cylindrical sample

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Fig. 3.7

Stress path for undrained principal stress rotation tests (a) p ' − q space (b) deviatoric stress space

Fig. 3.8

Fig. 3.9

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Independently controlled force variables against number of cycles (a) outer and inner cell pressure (b) vertical force and toque

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Stress components against number of cycles

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Fig. 3.10 Total principal stresses against number of cycles

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Fig. 3.11

Direction of principal stresses rotate continually in physical space

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Fig. 3.12

Stress trajectory during rotation in the component space

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Fig. 3.13

Stress trajectory during rotation in the component space

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Fig. 3.14

Effective stress path of rotational shearing test

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Fig. 3.15

Number of cycles required approaching stress ratio 1.25

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Fig. 3.16

p ' ~ (σ z − σ θ ) / 2 path of Test RSD70-2

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Fig. 3.17

Strain components against number of cycles in Test RSD70-2

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Fig. 3.18

Deviatoric strains produced in Test RSD70-2

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Fig. 3.19 Enlarged view of dark area of the central part in Fig. 3.18

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Fig. 3.20

Shear stress-strain relation in Test RSD70-2

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Fig. 3.21

Stress path with total strain increment superposed

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Fig. 4.1

Schematic illustration of particle’s orientation in a transversely isotropic sand aggregate (after Dafalias and Arulanandan 1979)

Fig. 4.2

Variation of vertical and horizontal formation factors with porosity for different orientation factors

Fig. 4.3

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78

Variation of the vertical and horizontal formation factor ratios with axial

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ratio R for different orientation factors

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Fig. 4.4

Side view of the container

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Fig. 4.5

Two-terminal electrode system-low frequency measurements

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Fig. 4.6

Calibration of the two-terminal electrode system

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Fig. 4.7

Capacitor-resistor paralleled model

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Fig. 4.8

Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations (a) histogram representation (b) rose diagram representation

Fig. 4.9

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Variation of the vertical and horizontal formation factor ratios with axial Ratio R obtained from the dielectric constants

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Fig. 5.1

Epoxy impregnation system

105

Fig. 5.2

Grain size distribution of Toyoura sand

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Fig. 5.3

Methods of sample preparation (a) dry deposition (b) moist tamping (after Ishihara 1996)

106

Fig. 5.4

Drying system for moist tamped specimen

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Fig. 5.5

A typical SEM image at the magnification of 200

107

Fig. 5.6

Typical image taken by SEM (a) SEM microphotograph (b) B/W mask for image processing

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Fig. 5.7

Measure of particle orientation

108

Fig. 5.8

An example of numbering particles through image processing

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Fig. 5.9

Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations for vertical sections

Fig. 5.10

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Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations for horizontal sections

Fig. 5.11 A set of parallel scanning lines (a) horizontal direction (b) vertical

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109

direction

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Fig. 5.12 Free path resulting from a set of parallel scanning lines (a) horizontal free path (b) vertical free path

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Fig. 5.13 Schematic REAs with scanning lines (a) circular REA (b) annular REA 110 Fig. 5.14 Schematic illumination of the method DDA

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Fig. 5.15 Directional void ratio (a) DD specimen (b) MT specimen

111

Fig. 6.1

CKC triaxial testing system

136

Fig. 6.2

Lubricated-end used in triaxial test

136

Fig. 6.3

Triaxial compression test (DD specimen, Dr=30%)

137

Fig. 6.4

Triaxial compression test (MT specimen, Dr=30%)

138

Fig. 6.5

Triaxial extension test (DD specimen, Dr=30%)

139

Fig. 6.6

Triaxial extension test (MT specimen, Dr=30%)

140

Fig. 6.7

Torsional shear test (DD specimen, Dr=30%)

141

Fig. 6.8

Torsional shear test (MT specimen, Dr=30%)

142

Fig. 6.9

Triaxial compression test (DD specimen, Dr=41%)

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Fig. 6.10 Triaxial compression test (MT specimen, Dr=41%)

144

Fig. 6.11

Triaxial extension test (DD specimen, Dr=41%)

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Fig. 6.12 Triaxial extension test (MT specimen, Dr=41%)

146

Fig. 6.13 Results of special biaxial compression test on bidimensional stacking of cylinders (after Chapuis and Soulié 1981)

147

Fig. 6.14 Ultimate steady-state lines for various shear modes (data replotted after Yoshimine and Ishihara 1998)

147

Fig. 6.15 Pseudo critical state lines for MT and DD specimens under TC and TE conditions

148

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Fig. 6.16 Effects of sample preparation on cyclic strength of sand (after Mulilis et al. 1977)

148

Fig. 6.17

Cyclic loading test (DD specimen, σ d =20kPa)

149

Fig. 6.18


149

Fig. 6.19


150

Fig. 6.20


150

Fig. 6.21

Cyclic loading test (MT specimen, σ d =40kPa)

151

Fig. 6.22


151

Fig. 6.23


152

Fig. 6.24


152

Fig. 6.25


153

Fig. 6.26

Liquefaction resistance under triaxial cyclic loading ( p ' =100kPa, Dr= 30%)

154

Fig. 7.1

Critical state line and definition of state parameter ψ

175

Fig. 7.2

Bounding surface, F1, and bounding cap, F2

175

Fig. 7.3

Mapping rule in deviatoric stress ratio space

175

Fig. 7.4

Anisotropic state variable, A, for MT specimen

176

Fig. 7.5

Anisotropic state variable, A, for DD specimen

176

Fig. 7.6

The final point in experimental tests and CSLs used in the modeling in e − ( p / pa )ξ space

Fig. 7.7

177

Comparison between undrained triaxial compression test results and model response for DD specimen (Dr=30%)

Fig. 7.8

178

Comparison between undrained triaxial compression test results and model response for MT specimen (Dr=30%) xv

179

Fig. 7.9 Comparison between undrained triaxial extension test results and model response for DD specimen (Dr=30%) Fig. 7.10

180

Comparison between undrained triaxial extension test results and model response for MT specimen (Dr=30%)

181

Fig. 7.11 Comparison between undrained torsional shear test results and model response for DD specimen (Dr=30%)

182

Fig. 7.12 Comparison between undrained torsional shear test results and model response for MT specimen (Dr=30%)

183

Fig. 7.13 Comparison between undrained triaxial compression test results and model response for DD specimen (Dr=41%)

184

Fig. 7.14 Comparison between undrained triaxial compression test results and model response for MT specimen (Dr=41%)

185

Fig. 7.15 Comparison between undrained triaxial extension test results and model response for DD specimen (Dr=41%)

186

Fig. 7.16 Comparison between undrained triaxial extension test results and model response for MT specimen (Dr=41%)

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LIST OF TABLES Table 3.1 List of equations used to calculate the stresses and strains

62

Table 3.2 Summary of the rotational shear tests

62

Table 4.1 The techniques for the study of soil fabric

82

Table 5.1 Summary of the results of the magnitude vector ∆

112

Table 5.2 Summary of the results for the mean free path and directional void

112

Table 6.1 Conditions of laboratory tests (cyclic triaxial tests)

155

Table 7.1 Model constants for dry-deposited and moist-tamped Toyoura sand

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Investigation of Fabric Anisotropic Effects On Granular Soil Behavior by Zhongxuan Yang

Department of Civil Engineering The Hong Kong University of Science and Technology

Abstract In the past few decades, within the framework of critical state soil mechanics (CSSM), understanding on soil behavior has become much more developed and extensive. The following are examples of current knowledge: soil behavior is both density and confining pressure dependent; the drainage condition has influence on the soil response; the loading path, characterized not only by the magnitude but also the direction of the load applied, is another influential factor affecting the mechanical behavior of soil. In practical applications, recent geotechnical problems, such as liquefaction analysis and bearing capacity of sandy grounds, have required a comprehensive understanding of the stress-strain-strength relation of sand. Substantial knowledge on the mechanical behavior of granular materials has been gained mostly from the results of well-controlled laboratory testing on homogeneous specimens subjected to uniform stress and strain states. Previous experimental investigations provide a fundamental understanding and give a clear picture of soils, when subjected to various loadings.

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It is known that the importance of the fabric anisotropy has long been recognized. However, previous studies on the effects of fabric anisotropy were mostly carried out in a qualitative manner. No extensive and unified studies have been made to quantify the fabric anisotropy of granular soils and thus its influence on mechanical behavior. In this dissertation, a systematic and comprehensive investigation is presented. Firstly, the anisotropic sand behavior under rotational shear is studied. Secondly, quantification of the fabric anisotropy of granular soils is made using the method of electrical property measurements, as well as the approach using an image analysis on the microstructures of thin sections. The inherent fabric anisotropies for both the dry deposited and moist tamped specimens are identified and further quantified. A series of the conventional laboratory tests are then performed under various loading conditions, using both the dry deposited and moist tamped specimens. The relations between the mechanical behavior and the fabric anisotropy are identified. A constitutive model accounting for the effects of different inherent fabric anisotropies is calibrated based on the experimental test results and observations. The laboratory tests are simulated using this model and a satisfactory performance of this model is obtained.

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CHAPTER 1 INTRODUCTION

When subjected to shear, a loose sand contracts and dense sand dilates. Whether a sand is in a loose or dense state depends not only on the density of the sand but also on the confining pressure applied. Furthermore, for a sand that initially is either in a loose or a dense state, there is an ultimate state of shear failure at which the volumetric strain rate is zero. This ultimate state is the well-known critical state (Rescoe et al. 1958). Laboratory investigations have confirmed that the mechanical behavior of soil is very complex and strongly influenced by density, confining pressure, drainage condition, loading path, and so on (Vaid and Thomas 1995, Verdugo and Ishihara 1996, Yoshimine et al. 1998, Cai 2001). In recent years, efforts have been made to explore the relations between the soil response and the fabric anisotropy. Actually, the importance of the fabric anisotropy has long been recognized. In the past several decades, numerous laboratory investigations have been undertaken to explore the influence of the fabric anisotropy (Oda 1972, Miura and Toki 1982, Nakata et al. 1998, Vaid and Sivathayalan 1996, Zlatović and Ishihara 1997, Yamamuro and Wood 2004). The studies have shown that the soil behavior is fabric-dependent, and the influence of the anisotropy is not negligible, and sometimes very significant. However there is few research linking the fabric anisotropy with the mechanical behavior, in a unified and quantitative way. It is known that a soil mass is essentially composed of discrete soil particles, and its mechanical behavior is influenced, to a great extent, by its discrete nature. 1


Due to the gravitational force existing, either naturally deposited soils or laboratory reconstituted specimens have prominent fabrics, which inevitably affect the mechanical behavior of soils when subjected to loading. To understand the relations between the fabric anisotropy and the mechanical behavior of granular soil, quantification of the fabric anisotropy is required. However, how to quantify the fabric anisotropy is not so easy, and a routine, yet simple, experimental procedure should be developed and followed. In this dissertation, the anisotropic sand behavior under the rotational shear is to be studied firstly. It is known that non-proportional loadings have extensive applications in geotechnical engineering, and the rotational shear is a class of the nonproportional loading with a constant second invariant of the deviatoric stress. The influences of the soil density, the magnitude of the applied shear stress, as well as the intermediate principal stress parameter, on the mechanical behavior of the soils, will be investigated under the rotation of the principal stress axes. Quantification of the fabric anisotropy of granular soils will be made using the method of electrical property measurements, as well as the approach using an image analysis on the microstructures of thin sections. Particularly, a complete experimental procedure, quantifying the fabric anisotropy of granular soils, will be proposed and the inherent fabric anisotropies for both the dry deposited (DD) and the moist tamping (MT) specimens will be quantified using this procedure. In addition, a series of conventional laboratory tests, including triaxial compression, triaxial extension, torsion shear and triaxial cyclic tests, will be performed. The test specimens are prepared by both DD and MT methods, and various loading conditions are imposed. The mechanical behavior of granular soil is investigated, and the influence of the fabric anisotropy is identified, especially the stress strain rela-

2


tion, the dilatancy characteristics, the liquefaction resistance, as well as the critical state. All the test results also provide a batch of direct data for the calibration and verification of the constitutive model proposed. Li and Dafalias (2000) pointed out that the classical stress dilatancy theory (Rowe 1962) in its exact form ignored the extra energy loss due to the static and kinematical constraints at particle contacts which led to a unique relationship between the stress ratio and dilatancy. In this dissertation, the state-dependent dilatancy will be reviewed, and then a platform model developed by Li (2002), with the effect of the fabric anisotropy incorporated, will be introduced. All the model constants are calibrated according to the experimental test results and observations, especially anisotropy property parameters and the critical state lines. The mechanical response of the sand in the triaxial tests and torsional shear tests, for both DD and MT specimens under different initial confining pressures and densities, are simulated using this model. This dissertation is organized as follows: Chapter 1 is the introduction (this chapter). Chapter 2 reviews the literature on the subjects related to the fabric anisotropy as well as the mechanical response of granular soils subjected to various loading conditions. Chapter 3 describes the experimental investigation on the anisotropic granular soil behavior subjected to rotational shear loading. Chapter 4 presents the quantification of the fabric anisotropy of granular soil using the electrical property measurement method. Chapter 5 develops a complete experimental procedure for quantification of fabric anisotropy using an image analysis approach.

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Chapter 6 reports the results and findings of a series of laboratory tests, especially the anisotropic effects on the mechanical behavior of Toyoura sand. Chapter 7 details the concept of the state-dependent dilatancy and a unified modeling framework accounting for different inherent fabric anisotropic effects, and simulates the test results. Chapter 8 concludes this research and comments on future studies.

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CHAPTER 2 LITERATURE REVIEW

2.1

Introduction Extensive laboratory shearing tests have consistently shown that the fabric

plays a vital role on the mechanical behavior of soils under various loading conditions. Quantitative studies on the relations between fabric anisotropy and mechanical behavior of soils have been of particular interest in recent years. However, quantifying the fabric, as well as incorporating it into a constitutive model is of great importance to better understand granular soil behavior. Moreover, laboratory test results also provide direct verifications for checking the performance of any constitutive models proposed. Generally, fabric anisotropy can be classified into two categories, namely inherent anisotropy and load-induced anisotropy. Abundant investigations have shown that fabric anisotropy is an important factor governing the stress-stainstrength relations of granular soil under various loading conditions (Casagrande and Carillo 1944, Vaid and Chern 1985, Oda 1972, Oda et al. 1978, Towhata and Ishihara 1985, Yoshimine et al. 1998). Quantifying fabric behavior was also carried out in the last three decades (Oda 1972, Kuo and Frost 1997, Kannatani 1985, Muhunthan et al. 2000, Masad and Muhunthan 2000, Jang et al. 1999). In this chapter, several topics relevant to this research will be reviewed, which will help improving understanding of soil mechanics fundamentals.

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2.2

Rotational Shear It is well known that soil behavior is strongly influenced by the loading path.

During shearing, the loading path is affected not only by the magnitude of the shear resultant but also by its direction. However, there are several examples of cyclic loading environments in which the mode of load application is associated with the continuous rotation of the principal stress directions. The change in shear stress induced in a seabed deposit by waves passing overhead is a typical example, which was illuminated theoretically by Ishihara and Towhata (1983). In order to identify the influence of the change in shearing direction, a class of loading conditions, called rotational shear, was recognized. Under rotational shearing, the second invariant J 2 D =

1 sij sij , where s ij is the deviatoric stress tensor, 2

is kept constant. According to this definition, rotational shear should include both the change in shearing direction with the π -plane and the change in shearing direction due to a rotation of the π -plane itself, while keeping the shear magnitude constant, as shown in Fig. 2.1 (a) and (b), respectively. Since the early 80s of the last century, a series of experimental studies aimed at understanding the influence of shearing direction on the stress-strain behavior of granular soils have been undertaken. Laboratory cyclic tests, using directional shear, bi-directional simple shear, true triaxial shear and torsional shear devices, on saturated sands, have shown that, compared with the shear reciprocated in one direction only, the multi-directional shear involving shearing direction change may introduce additional plastic volumetric strain, and result in an increase of the rate of pore-pressure built up under undrained conditions. This finding is of great significance because the characteristics of rotational shearing may exist in many practical loading conditions, such as seismic loading, and wave loading, acting on the 6


soil ground. The experimental observation also implies that if the influence of rotational shearing is ignored, the liquefaction potential of a sand deposit could be underestimated. Ishihara and Towhata (1983) conducted a series of cyclic undrained triaxial torsional shear tests on Toyoura sand. Both the stress difference (the difference between the axial stress and horizontal stress) and the torsional shear stress were varied cyclically so that the axes of the principal stress could be rotated continuously, while keeping the amplitude of the combined shear stress (deviator stress) constant. The results of these tests indicated that plastic deformation, as well as pore water pressure, could be produced, even when the deviatoric stress was maintained unchanged. Yamada and Ishihara (1983) studied the deformation and liquefaction characteristics of sand subjected to cyclic stresses, involving changes both in the magnitude and the direction, using a true triaxial apparatus. Various loading paths are included in this study, and are shown in Fig. 2.2. The test results revealed that the cyclic stress ratio causing the onset of the liquefaction of the specimen in a given number of cycles was varied in different loading paths, and the resistance to liquefaction became smaller as the stress paths changed from straight-line to the elliptical and further to the circular, as shown in Fig. 2.3. Detailed analysis on the stress and the strain increment vectors, as given in Fig. 2.4, on the octahedral plane also indicated that the sand exhibited the deformation characteristics of an elastic body at the beginning of cyclic loading where the generated pore water pressure was still very small, but at the end of the cyclic loading, near liquefaction, the sand tended to behave more like a perfectly-plastic body.

7


In the eighth decade of the last century, a series of laboratory tests were performed using a hollow cylinder apparatus (HCA) developed in the Imperial College of Science and Technology, to investigate the effects of principal stress rotation on sandy soil. Hight et al. (1983) introduced the design and demonstrated the capabilities of this hollow cylinder apparatus. The hollow cylindrical specimen may be subjected to axial load, torque and internal and external radial pressures, and thus the magnitude and directions of major and minor principal stresses can be controlled, as well as the magnitude of the intermediate principal stress. Symes et al. (1984, 1988) investigated the effects of principal stress rotation on the saturated sands under both undrained and drained conditions by using the HCA. Under undrained conditions, the rotation of the principal stress direction with the constant shear stress induced positive pore pressures and strains, both of which were cumulative and led to failure when the principal stress directions were rotated cyclically between 0° and 24.5° . As for drained tests, rotation of the principal stress directions resulted in a

contraction of the material volume, irrespective of the sense of the rotation. Furthermore, the magnitude of the contraction was found to depend on the direction of rotation, with the rotation of α (direction of major principal stress inclined to vertical direction) from 0° to 45° producing a greater contraction than that from 45° to 0° . This finding was analogous to the rise in pore pressure during the

undrained rotation of the principal stress directions and to its dependence on the direction of rotation in the corresponding tests. The study by Symes et al. (1982) suggested that it was possible to use an HCA to investigate the effects of the initial anisotropy, the intermediate principal stress parameter b and the principal stress rotation separately. Miura et al. (1986) performed a series of drained tests to investigate the fundamental deformation behavior of sands under more general stress

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conditions, including the rotation of the principal stress axes. Test results showed that in the course of the rotation of the principal stress axes, the direction of the principal strain increment lay between the directions of the principal stress and its increment, and approached the principal stress axes as the shear strain increasing. They also found, despite the values of three principal stresses being kept constant, the specimens contracted accumulatively, caused by the rotation of principal stress axes, although the specimens expanded by the increasing shear stress involving no rotation of principal stress axes within the same stress domain. Towhata and Ishihara (1985) studied the effects of continuous rotation of the principal stress axes on the excess pore water pressure development during the cyclic loading. The device they used was the so-called triaxial torsion shear apparatus, which had only three independent controllable loadings. Some major conclusions could be obtained from those tests, as follows: (1) continuous rotation of principal stress axes led to volume change in drained tests, and to excess pore water pressure development in undrained tests; (2) continuous rotation of principal stress axes reduced the liquefaction resistance of loose Toyoura sand by about 8%; (3) the friction angle was not affected by continuous rotation of the principal stress axes, but was affected by the magnitude of the intermediate principal stress. Yasufuku et al. (1995) presented the results of a series of undrained shear and rotation tests on inherent anisotropic sand in a medium dense state, and the results showed the characteristics of the flow deformation, which caused the specimen to deform rapidly to failure due to the principal stress direction rotation. It was also shown that the limited flow deformation appeared during an undrained principal stress rotation and the occurrence was closely related to the existence of a critical stress ratio.

9


Nakata et al. (1998) carried out a comprehensive investigate on the undrained deformation behavior of sand subject to principal stress rotation. The following conclusions were obtained: (1) excess pore pressures and strains in excess of 5% were accumulated under the cyclic rotation of principal stresses despite the deviatoric stress remaining constant, as shown in Fig. 2.5 (a), (b) and (c); (2) the form of deformation could be classified into non-flow, limited flow and full flow deformation; flow deformation occurred when the effective stress state of the sample reached a critical stress ratio. If the effective stress reached the steady state line after the critical stress ratio was reached, samples demonstrated full flow deformation. On the other hand, if the effective stress reached the phase transformation line after the critical stress ratio was attained, samples demonstrated limited flow deformation; (3) the critical stress ratio was observed to be a function of both relative densities and principal stress directions. Wang et al. (1990) proposed a comprehensive constitutive model for sand, which was formulated within the general framework of bounding surface hypoplasticity. The distinctive feature of this model is the dependence of the loading and plastic strain rate direction on the stress rate direction. The model can simulate different features of sand behavior under various loading conditions, which range from simple monotonic to complex cyclic at different amplitudes and directions. In particular, the successful simulation of the response under rotational shear is one of the distinctive properties of the model, and it is mainly due to its hypoplastic character. Li and Dafalias (2004) presented a constitutive framework for anisotropic sand, including non-proportional loading. This comprehensive hypoplasticity framework arose from a three-stage development. The first stage was the

10


development of a platform model, which operated within the framework of critical state soil mechanics in conjunction with the concept of state-dependent dilatancy, which was introduced by Li and Dafalias (2000) and Li (2002). The platform model allowed description, in a unified manner, of the contractive and dilative responses of a sand over a wide range of densities and pressure levels, but like most existing soil models, it only had a primitive anisotropic mechanism, i.e., conventional kinematic hardening, which is only sufficient for handling simple cyclic responses under proportional loading conditions. The second stage of the development had the objective of being able to handle the passive effects of fabric anisotropy. In particular, it aimed to model the widely observed experimental finding that the residual strength of sand is a function of its fabric orientation, as described by Li and Dafalias (2002). The third stage of development aimed at increasing the model’s capability in simulating non-proportional loading responses, including under the condition of rotation of the principal stress. With the three stages of development, the resulting model was able to describe the response of sand, either isotropic or anisotropic, and either in a loose or in a dense state, to monotonic or cyclic loading, either along proportional or non-proportional paths, using only a set of model constants.

2.3

Fabric Anisotropy

Fabric anisotropy can be classified, in general, as inherent anisotropy and load-induced anisotropy. Casagrande and Carillo (1944) distinguished the inherent anisotropy and induced anisotropy, and suggested that anisotropy might be present before the soil is strained or be induced by straining process. Inherent anisotropy was therefore defined as a physical characteristic inherent in the material and was entirely

11


independent of the applied strains. Induced anisotropy was a physical characteristic due exclusively to the strain associated with an applied stress. Granular materials such as sandy soil, deposited due to gravitational force, possess layered structures and have the inherent anisotropy. A random packing of non-spherical particles, such as sand, possesses statistical characteristics in the spatial arrangement of the particles and associated voids, which is technically termed material fabric (Oda 1999). Oda and Nakayama (1988) listed three sources of inherent anisotropy in a granular material: (1) anisotropic distribution of contact normals, which is indicative of the relationships between particles; (2) the preferred orientation of void spaces; and (3) the preferred orientation of non-spherical particles. In biaxial compression tests on two-dimensional assemblies of rods, Oda et al. (1985) also observed that the inherent anisotropy due to (1) and (2) tended to be completely altered during the early stage of inelastic deformation, while the inherent anisotropy due to (3) still existed at the very late stages of deformation. Since the critical state is always associated with a large shear deformation, the survival of fabric anisotropy at the critical state is expected to mainly be associated with the preferred orientation of the sand particles. This micromechanical finding is in agreement with the hypothesis by Yoshimine et al. (1998). Oda (1981) reported some results of plane strain tests and the results indicated that there were three distinguishable forms of fabric anisotropy, i.e., orientation, packing, and layering. It was found that fabric anisotropy due to packing was crucial in the determination of strength anisotropy for homogeneous sand; fabric anisotropy due to parallel alignment of particles was only one of the influential factors controlling the packing; layering could be another predominant factor in the determination of strength anisotropy.

12


Using a true triaxial shear test apparatus, Yamada and Ishihara (1981) performed a series of static loading tests in the undrained conditions on saturated cubical sand specimens. The three principal stresses were programmed to produce radial stress paths with different directions on the octahedral plane in the threedimensional stress space, as shown in Fig. 2.6. Test results showed that the effects of inherent anisotropy were obvious in the deformation and pore water pressure generation characteristics. Yoshimine et al. (1998) conducted a series of undrained monotonic tests with fixed principal stress direction α , which was the inclination to the vertical direction, and the intermediate principal stress parameter b, on saturated Toyoura sand, using an automated hollow cylindrical torsional shear apparatus. Test results showed that different inclination angles α exhibited different behavior and strength at the steady state, and are shown in Fig. 2.7. Li and Dafalias (2002) clarified that anisotropy effects were affected by the loading direction relative to the fabric orientation, which must be described objectively, i.e., independent of the coordinate system referred to. Therefore, the effects should be described through joint invariants of loading and fabric tensors. A scalar variable, A , was used as an index to characterize the influence of the sand fabric on the stress-strain-strength response. This approach also resulted in a unique critical state surface in e − p − A space.

2.4

Non-Coaxiality

Based on flow rules (either associative or non-associative) of classical plasticity, the plastic strain increments are coaxial with the principal stresses. However, in general, the plastic strain increments are non-coaxial with the principal

13


stresses, as observed during the rotational shear tests (Towhata and Ishihara 1985, Nakata et al. 1997). This phenomenon is believed to be associated with the fabric anisotropy of the material itself. Li and Dafalias (2004) pointed out that the effect of noncoaxial deformation might be observed in the so-called non-proportional loading tests. The definitions of proportional and non-proportional loadings are given as follows. Proportional loading is defined as that during loading where the deviatoric stress components are kept in a constant ratio to each other and the soil element does not rotate with reference to the frame of the principal stresses; conventional triaxial compression and extension are typical cases of proportional loading. Stress paths that produce plastic deformation but do not fit into the above definition are called nonproportional loading paths. This is encountered in almost all practical geotechnical problems. Typical applications having significant non-proportional loading include: wave loading acting on seabed soils, multidirectional earthquake loading acting on level ground, and lateral cyclic loading acting on soils behind retaining structures. In order to examine the non-coaxiality of sand subjected to principal stress rotation, two series of tests by Nakata et al. (1997) were performed using both air pluviated and rodded specimens. It was observed that the rodded specimens were more isotropic compared to the air pluviated ones. The different degrees of noncoaxiality behavior could be identified between these two kinds of specimens. A plasticity formulation within the framework of the bounding surface plasticity theory was presented by Gutierrez et al. (1991). A new stress-dilatancy relation was used to take into account the non-coaxiality, as shown in Fig. 2.8, representing the strain increment directions superposed in the stress plot for three types of loadings in the X-Y plane. Fig. 2.8(a) showed almost coaxiality under the monotonic loading along different constant inclined angles to the vertical direction,

14


called proportional loading. Fig. 2.8(b) and (c) showed non-coaxiality under pure rotation shearing and combined loading, where both the directions and magnitudes of the principal stresses varied respectively, both of which could be termed nonproportional loading. It was seen that non-coaxiality was very much dependent on the stress path and shear stress level. Larger non-coaxiality was observed in the rotational shearing stress path than in the monotonic stress path, and the lower noncoaxiality occurred at higher shear stress levels. In order to investigate the influence of the principal stress rotation on the yielding and plastic flow of sand, Pradel et al. (1990) performed a series of experiments and drew some conclusions: (1) yielding occurred when the stresses reached a particular stress level for a given direction of principal stresses; (2) the yield loci obtained from different stress-strain curves were not exactly identical and seemed to define very smooth curves of elliptical shape that were independent of density; (3) the direction of the plastic strains varied considerably, showing noncoaxiality and the non-existence of a unique plastic potential. Gutierrez et al. (1993) proposed an elastoplastic constitutive model for the deformation of sand during loadings involving rotation of principal stress directions. The unique features of the model were the ability to simulate the dependency of yielding and flow on the stress increment direction, and in the use of a stressdilatancy relation that incorporated the effects of non-coaxiality. Gutierrez and Ishihara (2000) presented a theoretical and experimental study of the effects of non-coaxiality or non-coincidence of the principal stress and plastic principal strain increment directions on the behavior of granular materials. Test results implied that constitutive relations could not be sufficiently formulated in the principal stress space. It was shown that the usual plasticity formulations with plastic

15


potentials that were scalar functions of the stress invariants alone implicitly assumed coaxiality. It was also demonstrated that energy dissipation calculated from the principal components of the stress and plastic strain increment tensors, or from the stress and plastic strain increment invariants would be erroneous and would overestimate the amount of dissipated energy during loading in case of non-coaxial flow. Wang et al. (1990) proposed a comprehensive constitutive model for sand formulated within the general framework of bounding surface hypoplasticity. This model could handle the rotational shearing problem in nature, which was elaborated by Li (1997). As shown in Fig. 2.9, the angle between dr and nN was always less than 90o, even under a neutral loading condition, such that it could predict the nonzero plastic volumetric strain increment. However, as the stress ratio increased and approached failure, the deviation between the stress and strain increments became bigger and bigger until almost reaching 90o, which was in agreement with the experimental observations (Yamada and Ishihara 1983, Gutierrez et al. 1991).

2.5

Quantification on Fabric Anisotropy

In the past several decades, many attempts have been made to measure the fabric anisotropy of soils. Gillott (1970) and Yoshinaka and Kazama (1973) used Xray diffraction measurement and Morgenstern and Tchalenko (1967) adopted an optical method to study the particle orientations in clayey soils. Kanatani (1984a, 1984b, 1985) developed the density function in a tensorial form. Kuo and Frost (1996, 1997), Kuo et al. (1998) and Jang and Frost (1999, 2000) showed that the stereology based technique, together with the image analysis, provided a very powerful tool in studying the fabric of granular soils. However, there are some nondestructive indirect methods for evaluating the soil fabric anisotropy. Arthur and

16


Dunstan (1969) investigated the possibility of using radiography. Dafalias and Arulanandan (1978, 1979), Arulanandan and Dafalias (1979) and Anandarajah and Kuganenthira (1995) used an index, the formation factor, to evaluate sandy soil fabric in terms of electrical conductivity. There are some other methods, such as nuclear magnetic resonance imaging (MRI) and laser-aided tomography, as reported by Oda (1999). Zeng and Ni (1998, 1999) used bender elements measuring Gmax of sands under different directions to study the stress-induced anisotropy. Basically, the techniques for measurement and quantification fabric anisotropy of materials can be classified into two categories, based on different concepts. One is the physical quantities measurement to correlate with the fabric anisotropy of materials. Typical applications include measurements of wave velocity, dielectric dispersion and electrical conductivity, thermal conductivity, and so on. The other kind of approach is collecting and analyzing the information of microstructures of the materials, and using certain quantities to describe the fabric anisotropy.

2.5.1 Physical Property Measurement

Geometrically anisotropic media are generally electrically anisotropic as well. Consider a medium composed of aligned, plate-shaped particles with a lowconductivity, and interparticle pores filled with a conductive fluid. Electrical anisotropy results from the different contributions of surface conduction and the differences in path tortuosity in each direction. The flow of electricity through a soil is a composite of (1) flow through the soil particles alone, which is small owing to the fact that the solid phase is a poor conductor, (2) flow through the pore fluid alone, and (3) flow through both solid and pore fluid. The total electrical flow is influenced by the porosity, the tortuosity of flow paths, and the conditions at the interfaces 17


between the solid and liquid phases. These factors are, in turn, dependent on the particle arrangements and the density. Thus, a simple measurement of electrical conductivity would seem a possible means for evaluation of soil fabric. Mulilis et al. (1977) studied the electrical conductivity of the pore structure of Monterey No. 0 sand for four different sample preparation methods, (1) pluviation of dry sand through air; (2) moist tamping; (3) horizontal high frequency vibrations (dry); and (4) horizontal high frequency vibrations (moist), as shown in Fig. 2.10. Fabric studies and electrical conductivity measurements indicated that differences in the orientation of the contacts between sand grains and in packing were probably the primary reasons for the observed differences in the dynamic strength of the sand. Dafalias and Arulanandan (1979) proposed an analytical expression relating the formation factor F, a certain electrical anisotropy index, to the sand structure. Arulanandan (1991) extended the electromagnetic theory of Maxwell-Fricke for the electrical conduction through heterogeneous media and developed a theoretical relationship for fluid-saturated porous media, accounting for the orientations of the particles, the porosity of the mixture, the dielectric constants of the pore fluid, the solid phase and the mixture. Klein and Santamarina (2003) studied the anisotropy of mica flakes using the effective electrical conductivity σ eff along two orthogonal directions, as shown in Fig. 2.11. Anisotropy in the effective electrical conductivity reflected not only anisotropy in the pore structure (pore fluid conduction) and in particle alignment (surface conduction), but also double layer and spatial polarization losses. In addition, as observed in Fig. 2.11, the effective conductivity was greatest in the mixtures with pores preferentially oriented parallel to the electric field, among all the testing cases.

18


When heat transfers through soils, because the thermal conductivity of soil minerals is much more than that of water and air, the heat transfer is mainly through the soil particles. Therefore, the lower the void ratio, the greater the number and the area of interparticle contacts and the higher the degree of saturation, resulting in higher thermal conductivity. The compression and shear wave velocities, propagating through a soil, depend on the density, confining stress, and fabric of the soil as well. According to elasticity theory, which is applicable for the small deformations associated with wave propagation, the shear wave velocity Vs and the compression wave velocity Vp are related to the shear modulus G and the constrained modulus M by Vs = G ρ and

V p = M ρ . The shear wave velocity is more useful since shear waves can only be transmitted through the soil skeleton of the soil mass. The anisotropic soil structure could be identified by measuring shear wave velocities along different directions (Zeng and Ni 1998, 1999).

2.5.2 Geometric Quantity Representation Orientation of Non-Spherical Particles

Though each particle is irregular in shape, and cannot be idealized as an ellipsoid, the elongation direction can still be specified for each particle, and is denoted by a unit vector m . Orientation of particles in an assembly can therefore be described by introducing a probability density function m . In reality, however, the direct determination of the density function in three-dimensional is very difficult, if not impossible, especially for fine particles. In order to overcome this difficulty, an elongation direction m ' is used on two-dimensional sections so as to give a two-

19


dimensional density function of m ' , instead of m . Using this method, Oda and Koishikawa (1977) showed that particles tended to lie in the horizontal plane with the long axes parallel to it when deposited under gravitational force, and that the orientation fabric of the sand showed axial symmetry, with the symmetry axis parallel to the direction of gravity.

Size and Orientation of Voids (Void Tensorial Form)

Void ratio or its equivalent, has been popularly used as an index for cohesionless soils, but in fact, it only represents the ratio of void volume to solid volume, irrespective of the size and shape. The size distribution of voids together with their orientation is of particular importance for granular soil behavior. Oda (1976) proposed a method to experimentally determine the distribution of the local void ratio on 2-D plane sections. Bhatia and Soliman (1990) used this method and the results showed that the frequency distribution of the local void ratio could be useful in the characterization of sand fabric. However, there is some difficulty existing since voids are irregularly shaped and therefore their sizes can not be uniquely defined. One choice is that each void is isolated from others by connecting centers of adjacent particles on a two-dimensional section, and its size is defined as a diameter of its largest inscribed circle. An optical method, with the help of stereology, is also helpful in quantifying not only the three-dimensional size but also the orientation of voids (Kuo and Frost 1997). This technique consists of measuring void fractions in several scanning directions on three orthogonal sections, and of calculating a second order tensor using relations obtained from the stereological considerations. Pietruszczak and Krucinski (1989a) developed a mathematical framework by employing a continuum measure of microstructural disorder, which

20


was identified with the spatial distribution of the porosity/void ratio, for describing the effects of inherent and induced anisotropy in clays. Based on this framework, Pietruszczak and Krucinski (1989a, 1989b) simulated the anisotropy induced by the

K0-consolidation process, and evaluated the influence of the rotation of principal stress axes, respectively. Muhunthan and Chameau (1997) proposed a method to evaluate the fabric of soils in terms of the directional distribution of the voids in the soils. A fabric index was obtained and employed to quantify the variation of the fabric tensor with deformation and formulate the ultimate state surface within the framework of critical state soil mechanics.

Contact Normal (Interrelation of Particles)

Consider a particle g i , which is in contact with nc neighboring particle g 1 to

g nc at c 1 to c nc , as shown in Fig. 2.12, where n1 to n nc are unit vectors normal to contact surfaces at c 1 to c nc . Since all particles are considered to be convex, the relation of a particle to its neighbors can be fully described not only by counting the numbers of contact points nc , the coordination numbers, but by specifying the directions of all the unit vectors, n , with respect to reference axes. Two density functions, f (nc ) and E (n) , are introduced to deal with the multigrain fabric from a

statistical point of view. The former denotes the density function of the coordination numbers, and the latter density function of unit vectors defined over the entire solid angle Ω(= 4π ) . For an ideal packing of equal spheres, the coordination number nc can be uniquely determined. Graton and Fraser (1935) discussed various systematic assemblies of equal spheres, in which every sphere had the same grain size and the

21


same coordination number. Oda (1977) measured the coordination number in the random assemblies composed of glass balls with different diameters, and the results showed that the coordination numbers were a function of void ratio or porosity. The direction of n could be easily measured as far as a two-dimensional model assembly was concerned (Oda and Konishi 1974). An optical method, using a microscope and thin sections, was introduced by Oda (1972) to determine the threedimensional density function of n for natural sands. The results showed that the unit vectors, n, were oriented anisotropically with a preferred vertical direction, when the particles were deposited under gravity, which could occur even in an assembly of spheres.

Branch Vector

Consider a spherical assembly of volume V consisting of grains having arbitrary shape. The assembly is subjected to external forces fˆ jk acting on boundary points xik with k=1,2,…,Ne. The average stress of an equivalent continuum of volume V is, according to the Gauss-Ostrogradski theorem, given by

σ ij =

1 V

Ne

∑x k =1

k i

fˆ jk

(2.1)

where Ne is the number of external forces. This expression can be applied as a definition of a stress tensor in the granular assemblies. A microstructure definition of the average stress of an assembly of grains with arbitrary shape was given by Christoffersen et al. (1981), and was expressed in terms of the individual contact forces inside the assembly. The analyzed finite-sized domain had to be subjected to a special load t i = σ ijload n j , where σ ijload was a second

22


order tensor and n j was the outward unit normal vector on the boundary of the analyzed representative domain. This load gave rise to contact forces f i k (k = 1,2,…, N c ) between grains. The principle of virtual work led to a constraint

on these forces

σ

load ij

1 = V

Nc

∑l k =1

k i

f jk

(2.2)

where σ ijload must be symmetric tensor to ensure moment equilibrium, and the vector lik , termed the branch vector, connects the centers of the two grains forming the kth

contact. The above constraint could be applied as a definition of the stress tensor

σ ij ≈ σ ijload . It is worth noting that branch vector is clearly a vector, and possesses both direction and magnitude, such that can also be employed to express the fabric tensor, hence describing the anisotropic property of granular materials (Chang and Gao 1996, Chang et al. 2003).

23


(a)

(b) Fig. 2.1 Rotational shearing classification

σz

σz

σz

ZC RS270o

ZE

σx

σy

RS90o

σz

σz

σy

σx

σy

σx

σz

σz

θ

σx

σy

σx

σy

σx

σy

σx

Fig. 2.2 (a)-(f) Patterns of loading paths on the octahedral plane (a)Uni-directional 1 (b) Uni-directional 2 (c) Crisscrossing 1 (d) Crisscrossing 2(e) Circular (f) Elliptic 1 (g) Elliptic 2

24

σy


Fig. 2.3 Stress ratio versus number of cycles averaged for different loading paths (after Yamada and Ishihara 1983)

Fig. 2.4 Strain increment vectors along a circular stress path (after Yamada and Ishihara 1983)

25


(a)

(b)

(c)

Fig. 2.5 Effective stress path in p − q space for all undrained principal rotation tests '

(a) Dr =90%, (b) Dr =60%, (c) Dr =30%

Fig. 2.6 Radial shear stress paths (after Yamada and Ishihara 1981)

Fig. 2.7 Effect of

α

on undrained behavior of Toyoura sand

(a) Stress-strain relationship (b) Effective stress path

26


Fig. 2.8 Strain increment directions for (a) Monotonic Loading

(b) Pure Rotation

(c) Combined Loading

(after Gutierrez et al. 1991)

Fig. 2.9 Neutral loading mechanism of the hypoplasticity model (after Li 1997)

Fig. 2.10 Formation factor versus density for different sample preparation methods

27


Fig. 2.11 Effective conductivity for mixture of mica flakes with electrolytes at two different ionic concentrations

Fig. 2.12 Relation of a particle to neighboring particles

28


CHAPTER 3 INVESTIGATION OF ANISOTROPIC GUANULAR SOIL BEHAVIOR UNDER ROTATIONAL SHEAR

3.1

Introduction It is well known that granular materials, such as sandy soils and natural

sedimentary deposits, due to the action of gravitational force, possess layered structures and inherent anisotropy. Oda (1999) used the technical term fabric to describe the spatial arrangement of particles with the associated voids. Arthur et al. (1972) and Oda (1972) studied the effect of the initial fabric on the mechanical properties of granular material. Several other investigations (Vaid and Chern 1985, Yoshimine et al. 1998) showed that through undrained triaxial tests, the critical state strength of sand measured in extension was much lower than that in triaxial compression under otherwise identical conditions, and that the significant difference was directly associated with the soil dilatancy. Furthermore, through triaxial compression and extension as well as in simple shear tests, Yoshimine and Ishihara (1998) pointed out explicitly that the shearing mode had great influence on the undrained shear behavior of the anisotropic sand. Experimental studies (Vaid et al. 1995, 1996) on flow liquefaction have also revealed that the influence of inherent fabric anisotropy can no longer be ignored in practical applications. In the past few decades, many experimental investigations have shown that the impact of fabric anisotropy on the behavior of granular material is significant.

29


This has been found to be true in both proportional and non-proportional loading cases. During shearing, the loading path is affected not only by the magnitude of the shear resultant but also by its direction. Soil behavior under non-proportional loading is of special interest because the loading encountered in almost all practical geotechnical problems can be categorized as non-proportional loading. Examples are the wave loading acting on seabed soils, multidirectional earthquake loading acting at level ground, and lateral cyclic loading acting on soils behind retaining structures. Rotational shear is a class of non-proportional loading in which the second invariant J2D of the deviatoric stress remains constant (Wang et al. 1990, Li 1997). Among the infinite number of loading conditions belonging to this class, two are distinguishable. The first is the circular loading during which the stress path in the π plane is a circle centered at the origin, whose radius is a measure of J2D, entailing a continuous change of principal stress values but with fixed principal stress directions. It can be realized experimentally in a true triaxial device. The second is the orientational loading that entails a rotation of the principal stress directions but with fixed principal stress values, hence, constant J2D as well as the other isotropic invariants. It can be investigated experimentally in a hollow cylindrical torsional apparatus. Fig. 3.1 (a) and (b) illustrate the rotational shearing classification. Yamada and Ishihara (1983) studied the deformation and liquefaction characteristics of sands subjected to cyclic stresses involving changes both in magnitude and direction, using a true triaxial device. The behavior of sandy soil subjected to rotation of principal stress axes has been investigated in the last few decades, by Arthur et al. 1979, Ishihara and Towhata 1983, Symes et al. 1984, 1988, Miura et al. 1986, Wijewickreme et al. 1993, Nakata et al. 1998, and so on. The common feature associated with rotational shear can be summarized as follows:

30


under undrained conditions, the soil sample is weakened gradually when subjected to cyclic loading, meaning that positive pore water pressure is always generated and the effective stress is lowered; for drained cases, contraction of the material is observed such that the soil is densified. Although some experimental studies have been carried out in the last two decades mainly emphasizing the effects of rotation of the principal stress direction, the influence of the intermediate principal stress is still not fully recognized. As pointed out by Shibuya et al. (1984), in the tests performed by Ishihara and Towhata (1983),

the

intermediate

principal

stress

parameter

b,

defined

as

b = (σ 2 − σ 3 ) /(σ 1 − σ 3 ) , was changing cyclically due to the limitation of the apparatus used. Recent experimental studies using a hollow cylinder torsional apparatus can simulate the condition of pure rotation of the principal stress directions with a constant intermediate principal stress parameter b, but normally with b=0.5. However, it is important to study the effect of the intermediate principal stress σ 2 , which is directly related to the parameter b, on the mechanical behavior of soil to accurately model certain boundary value problems associated with slope stability, foundation design and liquefaction analysis. The influence of parameter b variation on the soil behavior has been studied by many researchers, such as Vaid and Campanella (1974), Lam and Tatsuoka (1988), Kirkgard and Lade (1993), Prashant and Penumadu (2004). These studies included the traditional triaxial compression and extensional tests on solid cylindrical specimens, torsional tests on solid or hollow cylinder specimens, and true triaxial tests using cubic specimens, and all the test results revealed that the intermediate principal stress played a significant role in stress-strain-strength behavior and porepressure response of soil. However, there were very few studies regarding the

31


influence of parameter b on rotational shear behavior. Recalling the influence caused by the shear mode (characterized by parameter b) in the proportional loading on the granular soil behavior, one may speculate that there is also a need to study the influence of b for the tests concerning the rotation of the principal stress directions. The main objective of this chapter is to report the undrained test results of Toyoura sand specimens subjected to the rotation of principal stress axes, under various loading conditions, including different b values. Sand specimens were prepared at relatively dense states by means of the dry deposition method. Rotational shear tests were undertaken using a hollow cylindrical torsional shear apparatus. The influences of the intermediate principal stress parameter b and shear stress magnitude are examined on the excess pore water pressure response of Toyoura sand under undrained conditions. The deformation characteristics are presented and further analyzed in detail

3.2

Hollow Cylinder Apparatus In order to investigate the behavior of soil under different principal stress

directions or magnitudes, or so-called complex stress states, it is necessary to apply shear stress as well as normal stress to the surface of a specimen in an independently controlled manner. In the laboratory, several types of test equipment are capable of delivering both normal and shear stresses to the soil specimen, such as simple shear tests, pure shear tests, ring shear tests, and torsional shear tests, and so on. Difficulties exist in some tests, in terms of interpretation of test results, because of nonuniform stress distributions or lack of means to measure all the stresses applied to the specimens.

32


Among all these tests, the torsional shear test, in which a hollow cylindrical specimen is confined between two membrane sheets, offers the means to individually apply the normal stresses (including vertical normal stress, inner and outer cell pressures), as well as the shear stress, to a specimen. By combining the normal stresses and shear stress, complicated loading conditions can be simulated and realized. A newly developed torsional shear device is used in the current research, which has some special features, such as accommodating large dimension of specimens, a close loop control system, four degrees of freedom in stress oscillation, etc. The torsional shear system is composed of a loading frame, three loading application systems, a pressure cell, a measuring system, and a PC with a controlling and monitoring software installed. The profile of this system is shown in Fig. 3.2.

3.2.1 Loading Application System In order to individually apply the vertical stress, shear stress, inner and outer cell pressures, three loading application systems are employed. A brief description of the system is provided below.

Vertical Load The vertical load is applied by a vibrator, which is mounted on the top of the loading frame. It is mainly composed of a moving coil and a field coil. The moving coil is connected to a shaft that is coaxial with the loading rod in the pressure cell, so that the vertical load can be directly transferred to the top cap of the specimen. By adjusting the amplitude and frequency of the electric current through the moving coil, a monotonic as well as cyclic vertical loading (either compressive or tensive)

33


can be applied. The range of vibrating frequency is around 0-50kHz, and the permissible load is about 2kN. A power amplifier is used for driving the vibrator.

Torque The torque is applied by the DYNASERV DD Servo-Actuator, which is installed beneath the vertical vibrator. The shaft passes through the center of the motor with a special connection so that the shaft can move upward or downward freely and thus can transmit the torque. It is a high torque, high velocity, and high accuracy outer rotor type servo-actuator. The maximum torque is 400 N ⋅ m . There are three control modes: the position control mode, the velocity control mode, and the torque control mode. In the position control mode, motor positioning control is performed according to the command position sent by a higher-level controller. In the velocity control mode, the motor rotating angle is controlled so as to correspond to the velocity command voltage from the higher-level controller. In the torque control mode, the current flows through the motor corresponding to the current command voltage from the higher-level controller. The motor output torque depends on the current. By choosing the position control or velocity control mode, the straincontrolled tests can be performed. On the other hand, by using the torque control mode, the stress-controlled tests can be undertaken. In this investigation, the position control mode is employed. The strain rate can be assigned over a very large range. As the motor can rotate in either clockwise or counter-clockwise direction, both the monotonic loading and cyclic loading can be applied easily.

Inner and Outer Cell Pressures

34


The inner and outer cell pressures are applied by a dual channel pneumatic loading unit, which is originally utilized by the CKC triaxial system (which will be introduced late). The two channels in the loader are directly connected to the inner and outer chambers respectively, but controlled individually. Again the pressures could be controlled either in a monotonic mode or a cyclic mode.

3.2.2 Pressure Cell The pressure cell is similar to that for triaxial tests, but it has its own features. Firstly, it is bigger. The inner diameter and height are 280mm and 474mm, respectively. In order to render the apparatus suitable for testing the saturated specimens, six pressure/drainage lines (two are spare) are drilled through the base pedestal, in which two lines are installed for monitoring the inner and outer cell pressures, and the other two lines, which are connected to the top and bottom of the specimen respectively, are installed for the water circulation through the specimen and for measuring the pore water pressure during each test. There are connectors on the top cap of the cell for the application of the inner and outer cell pressures. The vertical load and shear load are applied through a loading rod that is mounted on the top cap through the linear ball bushings. Two rings connected to the pedestal and the loading cap respectively, are put on the top and bottom of a specimen. In order to ensure the transmission of the torque from the cap and base rings to the specimen, and further to avoid the slippage at their interface, full frictional surfaces are provided on the rings. This is achieved in two ways: firstly, the surfaces facing the soil specimen are sculptured very coarsely; secondly, six tapered stainless steel blades with a base width of 3mm and a height of 4mm are embedded radically in each ring facing the soil specimen.

35


To prepare a hollow cylindrical specimen, two molds are used. The inner specimen mold consists of a split mold with four sections and can be dismantled internally. The outer mold is a common mold composed of two sections.

3.2.3 Control and Data Collection System The control system, together with the data collection system is illustrated in Fig. 3.3. Four subsystems are responsible for the control of the vertical load, torque, and inner and outer cell pressures, respectively. Taking the vertical load control subsystem as an example, the digital control signal is converted into an analog signal by a digital to analog converter (DAC), located in a computer. A power amplifier is used to amplify the signal, which is connected to the coil to drive the vibrator. The axial load generated by the vibrator is monitored by an axial load cell, which is mounted on the loading shaft. The collected signal is further treated by a signal conditioner and converted into a digital signal by an analog to digital converter (ADC), and this digital signal is then sent to the computer. In tests, the torque, the axial load, the rotating angle, the vertical displacement, the volume change of the specimen, the pore pressure, and the cell pressures are all monitored and recorded. Here, only the measurements of the torque, vertical load, and rotating angle are briefly described. The measurements for other quantities are similar to those in the triaxial tests. •

Torque measurement: a TRS-5K torque load cell is mounted on the shaft near the top of the pressure cell. The allowable torque of the cell is 565 N ⋅ m .

36


•

Vertical load measurement: the vertical load is measured by a force load cell mounted on the shaft that is coaxial with the torque cell. The capacity of the load cell is 4.54kN, and the non-linearity is about 0.03% of the full scale.

•

The rotating angle is not directly measured. As in the test, the position control mode is adopted. Therefore, the rotating angle can be assigned precisely.

3.3

Testing Procedure

3.3.1 Sample Preparation Before sample preparation, the inner and outer split molds were installed on the pedestal of the cell, and two membranes with the same thickness of around 0.6mm were stretched tautly on the outer surface of the inner mold and inner surface of the outer mold. The membranes were sealed with o-rings at the base. The dry deposition method was employed in the sample preparation. Oven dried Toyoura sand was weighed and poured into the hollow space between the two molds with a spoon and a funnel. It was attempted to keep zero falling head and then tamp the outer mold gently using a rubber mallet to adjust the density of sample, layer by layer. The sand was deposited slightly above the top of the molds to avoid partial contact between the soil particles and the loading cap, and then the loading cap was put on the top of the specimen with a small pressure. The membranes were fixed with o-rings on the cap. The top drainage line was then connected to the cap, and a vacuum around 15kPa was applied to hold the specimen. The molds were then removed. Carbon dioxide was circulated throughout the specimen from the bottom to the top. The inner and outer diameters and the height of the specimen, at different

37


locations, were then measured at least three times, and the average values were taken as the nominal inner and outer diameters and the height of the specimen. The typical dimensions of a specimen are as follows: height=314mm; inner diameter=200mm; outer diameter=150mm; wall thickness=25mm. The deviations between the average values and the local measurements were expected to be less than 0.2mm. After filling the inner cell and outer chamber with de-aired water, the saturation process was carried out under a differential vacuum pressure 30kPa, and this procedure normally lasted about 6 hours. Fig. 3.4 shows a sample after preparation.

3.3.2 Data Interpretation Based on the equations listed in Table 3.1, one can get the principal stresses and strains, as shown in from Eq. (3.1) to Eq. (3.6).

σ1 =

σ z +σθ

+ (

2

σ z −σθ 2

) 2 + σ zθ

2

(3.1)

σ2 =σr σ3 =

σ z +σθ

ε1 =

− (

2

ε z + εθ 2

+ (

(3.2)

σ z −σθ 2

ε z − εθ 2

) 2 + σ zθ

) 2 + ε zθ

2

(3.3)

2

(3.4)

ε2 = εr ε3 =

ε z + εθ 2

− (

(3.5)

ε z − εθ 2

) 2 + ε zθ

2

(3.6)

The effective stress and deviatoric stress can be calculated through p' =

σ 1' + σ 2 ' + σ 3 ' 3

38

=

σ1 +σ 2 +σ3 3

−u

(3.7)


q=

{

1 ' ' ' ' ' ' (σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2 2

}

(3.8)

The angle of σ 1 inclined to the vertical axis is 2σ zθ 1 tan −1 ( ) σ z −σθ 2

ασ =

(3.9)

The volumetric strain and deviatoric strain can be obtained by

ε v = ε1 + ε 2 + ε 3 εq =

(3.10)

2 (ε1 − ε 2 ) 2 + (ε 2 − ε 3 ) 2 + (ε 3 − ε1 ) 2 9

{

}

(3.11)

For any given α σ , the stress components are given as follows

σz =

σ1 +σ 3

+

2

σ1 −σ 3 2

cos 2α σ

σr =σ2 σθ =

σ1 +σ 3 2

σ zθ =

−

(3.12) (3.13)

σ1 −σ 3 2

σ1 −σ 3 2

cos 2α σ

sin 2α σ

(3.14)

(3.15)

Using the above equations, four individually loads P0, Pi, W and T can be expressed as follows.

P0 =

Pi =

σ 2 (r0 + ri ) + (

σ1 +σ 3 2

−

σ1 −σ 3 2

cos 2α σ )(r0 − ri )

2r0

σ 2 (r0 + ri ) − (

σ1 +σ 3 2

−

σ1 −σ 3

2ri

39

2

cos 2α σ )(r0 − ri )

(3.16)

(3.17)


W =π(

σ1 +σ 3

+

σ1 −σ 3

cos 2α σ )(r0 − ri ) − 2

2

π

2 2 2 σ1 + σ 3 σ1 −σ 3 ⎡ ⎤ 2 2 σ + − ( ) cos 2α σ ⎥ (r0 − ri ) + 2 ⎢ 2 2 ⎣ ⎦ σ + σ 3 σ1 −σ 3 cos 2α σ )(r0 − ri ) σ 2 (r0 + ri ) + ( 1 − 2 2 2 π rd 2r0

(3.18)

(σ 1 − σ 3 ) sin 2α σ

T=

3

3 2π (r0 − ri ) 3

3

+

(3.19)

3

4(r0 − ri ) 3π (r0 − ri )(r0 − ri ) 2

2

4

4

It should be noted that all the stress components listed in Table 3.1, except the vertical stress σ z , which is obtained based on the equilibrium condition, are dependent

on

the

thin-wall

assumption.

Generally

speaking,

the

stress

nonuniformities always exist in a hollow cylinder. Hereafter the derivations for the stress components σ θ , σ r and σ zθ will be briefly introduced. Considering the equilibrium of the wall subjected to both the inner and outer pressures, elementary mechanics shows that both the radial stress σ r and the circumferential stress σ θ vary with the location r, and the expressions can be written as 2 P0 r0 2 ri 2 Pr r0 2 i i σ r = 2 2 (1 − 2 ) − 2 2 (1 − 2 ) r0 − ri r r0 − ri r

(3.20)

2 P0 r0 2 ri 2 Pr r0 2 i i (1 + ) − (1 + ) r0 2 − ri 2 r2 r0 2 − ri 2 r2

(3.21)

σθ =

Averaging the stress components without weighting for r, as

∫

r0

ri

r0

∫

r0

ri

r0

σ r dr / ∫ dr and ri

σ θ dr / ∫ dr , on may obtain the representative stress expressions as shown in ri

Table 3.1.

40


The shear stress σ zθ can be obtained by taking the average value of the elastic and full plastic solutions for the hollow cylinder under torsional loading, as follows ro 3 − ri 3 4 T 3π (ro 2 − ri 2 )(ro 4 − ri 4 )

τ av =

τ av =

3 1 T 3 2π (ro − ri 3 )

(elastic solution)

(3.22)

(full plastic solution)

(3.23)

⎤ ro 3 − ri 3 1⎡ 3 1 4 + T T⎥ 3 3 2 2 4 4 2 ⎣ 2π (ro − ri ) 3π (ro − ri )(ro − ri ) ⎦

σ zθ = τ av = ⎢

(3.24)

Eq. (3.24) is employed to calculate the shear stress, and it has been shown that this equation is accurate enough for most geotechnical applications by Yang (2005), as also shown in the Appendix.

3.3.3 Test Program

The rotational shear test is performed in this study, and the test scheme is summarized as follows. •

The total principal stresses σ 1 , σ 2 and σ 3 are kept constant, while the directions of the principal stresses are continuously rotated in the physical space.

•

b = (σ 2 − σ 3 ) /(σ 1 − σ 3 ) , is held unchanged during the rotation of principal stress directions.

•

The second deviatoric stress invariant

J 2 D (or q) is a constant during the

rotation of principal stress axes. •

In the π − plane, the stress path is concentrated to a fixed point, the location of which is determined by the b value, but the π − plane rotates in the

41


physical space. The loci in the π − plane for different b values are illustrated in Fig. 3.5. There are four independently controlled loads applied to the soil sample, namely torque T, axial load W, inner pressure Pi, and outer pressure P0, as shown in Fig. 3.6(a). Fig. 3.6(b) shows the principal stresses as well as the stress components acting on a hollow cylindrical specimen. The tests are to be performed to investigate the undrained response of granular soil subjected to principal stress rotation, and the stress paths in the p ' − q space, and the deviatoric stress space are illuminated by Fig. 3.7(a) and (b), respectively. When the pressure cell was set up on the pedestal of the loading frame, a computerized initiation process was then carried out. Inner and outer cell pressures of 30kPa were simultaneously applied to the specimen. The initial B-value was checked and an appropriate back pressure (normally 200kPa) was applied to raise the degree of saturation of the soil specimen. The B-values after the application of the back pressure were over 0.97 in all the tests presented. Subsequently, anisotropic consolidation was performed, keeping the constant stress ratio (mean normal stress over deviatoric stress) to a desired stress state. The shearing stage was then commenced. Four independently controlled loads were simultaneously applied to a soil sample, which was enveloped by membranes, and the test was automatically guided and controlled by the software-aided computer. Torque was applied with an angular displacement rate of 0.1o/min, corresponding to .

.

a shear strain rate ( γ = 2 ε zθ ) of 0.05%/min, and the measured torque value was used to control the other three loads/pressures according to the rotational shearing path. The vertical displacement, the volume change, the pore water pressure increment, the 42


vertical load, the rotational angle, and the torque were measured and recorded during each test. A series of undrained principal stress rotation tests for various loading conditions were performed, as tabulated in Table 3.2. In this table, the initial void ratio e0 and relative density Dr refer to the values at the end of the consolidation. All the tests were performed under the same initial mean normal stress of p ' =100kPa.

3.4

Test Results and Discussion

Fig. 3.8(a) and (b) show the applied loads against the number of cycles for one particular test RSD70-2, with a back pressure 200kPa, all of which are sinusoidal functions. The stress components are illustrated in Fig. 3.9(a) to (d), which are also sinusoidal functions against cycle numbers. It can be seen that in Fig. 3.10 the total principal stresses σ 1 , σ 2 and σ 3 are almost constant during the test, while the directions of the principal stresses are cyclically rotated in the physical space, as shown in Fig. 3.11. The major principal stress inclination angle with respect to the vertical

direction,

is

defined

as

ασ = 1/ 2 arctan[2σ zθ /(σ z − σ θ )]

.

In

(σ z − σ θ ) / 2 ~ σ zθ space, the stress trajectory is almost a circle, as shown in Fig. 3.12. There are some unsmoothed parts in Fig. 3.12, located in the unloading phases if the stress direction is moving clockwise, because unloading the shear stress is much easier than loading it. Fig. 3.13(a)-(d) show that, during the rotation of principal stress axes, excess pore pressure is generated gradually as the cycles increase for all the tests listed previously. It can be observed that despite the variation of the density of the specimen, positive excess pore water pressure is always generated under the

43


undrained conditions, which is in agreement with the previous studies on the influence of rotation of principal stress axes (Ishihara and Towhata 1983, Symes et al. 1984, Nakata et al. 1998). It is seen the forms of pore water pressure generation are varied and depend on the initial density of the specimen, the applied deviatoric stress, as well as the intermediate principal stress parameter b, or the Lode angle. It can be further found that for the case b=0.0, the rate of generation of pore water pressure is much slower than those for cases b=0.5 and 1.0. Recall in the triaxial space, when performing the triaxial undrained compression (b=1.0) and extension (b=0.0) tests, the results reveal that a more contractive tendency in the extension side is observed than that in the compression side, either for monotonic or cyclic loadings (Yamada and Ishihara 1983, Yoshimine et al. 1998, Vaid et al 2001). More generally, Yamada and Ishihara (1981) investigated undrained behavior of sands in three-dimensional space using a true triaxial apparatus and the similar phenomenon was observed. The contrasting difference between the compression and extension undrained response of sands has been attributed to the presence of the inherent anisotropy of the specimen, as well as the intermediate principal stress. By analogy, b=0.0 and b=1.0 can be considered as triaxial compression and extension, respectively, without rotation of the π − plane itself. Therefore, it is not surprising that during the rotation of the principal stress axes test, discrepancies in rate of the pore water pressure generation are observed for cases b=0.0 and 1.0. Gutierrez (1989) studied the influence of parameter b on the behavior of sands which were subjected to monotonic loadings, with both fixed and rotating principal stress directions. Test results showed that the influence on the deformation characteristics was apparent, but test conditions were not well designed, which made the results not so convincing, and were limited to the drained condition, and only one

44


cycle loading was undertaken. A series of constant b and ασ (major principal stress direction with respect to vertical axis) undrained shear tests, run by Yoshimine et al. (1998) and Nakata et al. (1998), revealed that the parameter b played a vital role in the stress-strain behavior and the pore water pressure generation. However, rotation tests performed by Ishihara and Towhata (1983) could not simulate the case of pure rotation (no variation in b) because of the limitation of the equipment used (only three independent loads could be applied). Ishihara and Towhata (1984) pointed out that the effects of cyclic variation of the b-parameter in circular loading tests (reported by Ishihara and Towhata 1983) were not so important as compared to that of the rotation of the principal stress axes, in a reply to the discussions by two research groups in England (Shibuya et al. 1984, Arthur et al. 1984). However, the reasons they gave were unconvincing and incomplete. As noted in the current study, the influence caused by variation in the b-parameter is quite significant and may not be overlooked any more. The effect due to the magnitude of deviatoric stress is considerable, and the larger the deviatoric stress applied, the faster the pore water pressure built up. The specimen density also has significant influence on the undrained behavior of sands, and the pore pressure is generated more rapidly in relatively dense samples with otherwise identical conditions, as shown in Fig. 3.13. It seems that the influence due to the magnitude of the deviatoric stress is more pronounced and dominant, compared with that due to the density of the specimen. The effective paths formed by all the undrained tests are shown in Fig. 3.14. It can also be observed that despite the density of the specimen, the effective stress is lessened all the way due to the positive pore water pressure generated, leading the soil to failure, without any dilation being observed. In Fig. 3.14, a straight line is

45


drawn with the slope being the critical stress ratio in triaxial compression for Toyoura sand, Mc=1.25. There is a great difference in the number of cycles required for the stress ratio to reach Mc, if this condition is actually reached. Under the higher deviatoric stress, the sample is more likely to lose effective stress, while the denser soil sample is stronger in resisting the generation of pore water pressure. Therefore, the number of cycles required to failure, for the dense sample is much more than that for the loose one. In most cases, when the stress ratio in terms of q p ' approaches 1.25 (the slopes of the straight lines highlighted in Fig. 3.14), the sample shows the unstable response, and the deformation increases suddenly, which is consistent with the experimental observations by Nakata et al. (1998). Fig. 3.15 illustrates the required number of cycles to approach a stress ratio of 1.25 for some tests. It can be seen from this plot that the case Dr=70%, b=1.0 is most prone to fail, while the case Dr=90%, b=1.0 has the strongest resistance in building up pore pressure, with the case Dr=70%, b=0.5 in the middle. Some other cases not appearing in this plot indicate no failure occurring during the tests. Fig. 3.16 shows the stress path of test RSD70-2 in the p ' ~ (σ z − σ θ ) / 2 space,

where p ' denotes the effective mean normal stress, σ z and σ θ are the axial and circumferential stresses respectively, and Mc=1.25 is the critical stress ratio of Toyoura sand in triaxial compression. This plot provides details of the evolution of the stress state and reduction in the effective mean normal stress during the rotational shearing. The strain components, ε z , ε r , εθ and ε zθ versus the number of cycles of the stress rotation for test RSD70-2 are provided in Fig. 3.17. It is found that all the strain components, as well as the pore water pressure, exhibit significant cyclic oscillations, which is considered to be due to the cyclically changing unequal inner 46


and outer cell pressures (Hight et al. 1983). The strain components are all very small, usually less than 1%, before the stress ratio reaches Mc in all the tests, which is mainly attributed to the medium to dense specimens used and the mild deviatoric stress applied. For test RSD70-2, the shear strain path is illustrated in the component space in Fig. 3.18. An enlarged view of the dark region in Fig. 3.18 is shown in Fig. 3.19. Initially, all the strains are quite small, but after several cycles of shearing, the strains are increased suddenly, indicating failure of the specimen. In Fig. 3.19, although small strains are induced, the hysteretic characteristics of the curve can still be identified. Fig. 3.20 shows the shear stress-strain diagram for test RSD70-2. It is noted that hysteretic loops are generated and the sand is exhibiting not only elastic, but also plastic or permanent deformation. It is observed in this figure that the stiffness of the specimen is decreasing gradually as the number of cycles increasing, especially at the later stage of shearing. As shown previously, continuous rotation of the principal stress directions gives rise to an increase of the pore water pressure and a decrease of the effective stress, and leads the soil to be weakened, or even failed. The total shear strains in the (dε z − dε θ ) ~ 2dε zθ plane are superposed in a circular stress path in the (σ z − σ θ ) ~ σ zθ plane, as shown in Fig. 3.21. In the first cycle, the strain vector is almost moving along the stress path, while in the 10th and 33rd cycles, the strain direction obviously deviates from the stress path, especially in the 33rd cycle. As observed from the strains in Fig. 3.21, the elastic deformation as well as the plastic deformation is induced during the rotation of the principal stress axes. Elastic deformation is dominated at the beginning of the shearing, therefore the direction of the strain increment has a tendency to move along the circular stress path

47


in the (σ z − σ θ ) ~ σ zθ plane. During the subsequent shearing cycles, the plastic deformation becomes more dominant, and the strain increment direction deviates from the stress path gradually and is closer to the stress direction (normal to the circle). Previous investigations (Towhata and Ishihara 1985, Gutierrez et al. 1991, Gutierrez and Ishihara 2000) have shown that in general the plastic strain increments are non-coaxial with the principal stresses, termed noncoaxiality, when the soil is subjected to a variation in the shear direction. During the rotation of the principal stress axes, the direction of the shear resultant is continuously altered without change in the magnitude, so non-coaxial deformation always exists, and is directly associated with the fabric anisotropy of soil (Nakata et al. 1997). For test RSD70-2, in the first cycle, the total strain increment is almost coaxial with the stress increment, and is noncoaxial with the stress direction. In subtracting the elastic part of the strain from the total component, the plastic strain increment is noncoaxial with the stress increment. During the 33rd cycle, since the plastic deformation is much more dominant, the plastic strain increment is almost coaxial with the stress. As noted by Yamada and Ishihara (1983) and Gutierrez et al. (1991), noncoaxiality was gradually eliminated as the stress ratio was increasing, meaning that deviation between the stress and the strain increment was smaller and smaller, which had been demonstrated experimentally. The result of test RSD70-2 also verifies this finding.

3.5

Conclusions

An experimental investigation on anisotropic granular soil, which is subjected to the rotational shear, has been carried out using an automatic hollow cylindrical torsional apparatus. It has been shown that in the undrained rotation of the principal stress axes tests, the generated pore water pressure, the effective stress path, the

48


required cycles to bring the soil to failure (if any), and the deformation characteristics, are generally dependent on the density of the specimen, the applied deviatoric stress level, as well as the intermediate principal stress parameter b. The experimental results indicate that the soil specimen is weakened under the rotation of the principal stress axes in undrained conditions, in spite of the density of soil. The applied deviatoric stress also plays an important role affecting the soil behavior during the shear. It has been observed that the intermediate principal stress parameter b is not a neglectable factor for the soil behavior during the rotation shear, has significant impact. It seems that b=0 (analogous to triaxial compression) has a stronger resistance to the pore pressure build up, while b=0.5 (analogous to torsional shear) and b=1.0 (analogy to triaxial extension) are more prone to generate the pore water pressure and lose the effective stress. The strain components measured in all the tests are very small before failure. The strain path in the component space shows hysteretic characteristics, indicating that permanent deformation has been produced. The strain increment is found to be non-coaxial with the stress direction, which is attributed to the coupling of the elastic and plastic deformation induced. The noncoaxiality, caused by the rotation of the principal stress directions, essentially, is believed to be due to the fabric anisotropy of the soils.

49


(a)

(b)

Fig. 3.1Rotational shearing classification

Fig. 3.2 Hollow cylindrical torsional shearing apparatus

50


Fig. 3.3 Control and data acquisition system (after Cai 2001)

51


Fig. 3.4 Hollow cylindrical sample after preparation

σ1

b=0 b=0.5 b=1.0

σ3

σ2

Fig. 3.5 Various b values loading conditions in

52

π -plane


W

σ1 σ z

T

ασ P0

Pi

σ zθ

σθ σ3

σr =σ2

(a)

(b)

Fig. 3.6 Stress state in the wall of a hollow cylindrical sample

Fig. 3.7 Stress path for undrained principal stress rotation tests (a) p − q space '

(b) deviatoric stress space

53

Chapter 3 Investigation of Anisotropic Granular Soil Behavior Under Rotational Shear 304

Back pressure 200kPa

Po

Pressure (kPa)

302 300 298

Pi

296

(a)

294 0

1

2

3

4

5

Number of cycles

1.2

0.03

1

T

0.02

F

0.6

0

0.4

-0.01 (b)

0.2

-0.02

0

-0.03 0

1

2

3

4

5

Number of cycles Fig. 3.8 Independently controlled force variables against number of cycles (a) Outer and inner cell pressure (b) Vertical force and toque

150

σz (kPa)

Toque (kN.m)

0.01

Foce (kN)

0.8

100

50

(a)

0 0

5

10

15

20

Number of Cycles

54

25

30

35


150

σr(kPa)

100

50

(b)

0 0

5

10

15

20

25

30

35

25

30

35

25

30

35

Number of Cycles 150

σθ(kPa)

100

50

(c)

0 0

5

10

15

20

Number of Cycles

σzθ (kPa)

75

25

-25

(d)

-75 0

5

10

15

20

Number of Cycles Fig. 3.9 Stress components against number of cycles

55


Principle Stresses (kPa)

150

σ1 σ2

100

σ3 50

0 0

5

10

15

20 Number of Cycles

25

30

35

40

Fig. 3.10 Total principal stresses against number of cycles 180

90 45 0 0

5

10

15

20

25

30

Number of Cycles

Fig. 3.11 Direction of principal stresses rotate continually in physical space

30

20

10

σzθ (kPa)

ασ

135

0 -30

-20

-10

0 -10

10

20

30

(σz-σθ)/2 (kPa)

-20

-30

Fig. 3.12 Stress trajectory during rotation in the component space

56

35


PWP (kPa)

Toyoura Sand Rotational Shear p'=100kPa,q=34.65kPa,Dr=70%

(a)

80

b=0.5

b=1.0

60 b=0.0

40 20 0 0

10

20

30

40


PWP (kPa)


(b)

80

b=1.0

60

b=0.0

b=0.5

40 20 0 0

5

10

15

20

Number of Cycles

100 (c)

80 PWP (kPa)

Toyoura Sand Rotational Shear p'=100kPa,q=34.65kPa,Dr=90% b=1.0

60

b=0.5

40 b=0.0

20 0 0

20

40

60

80

100


PWP (kPa)


(d)

80

b=0.5

60

b=0.0 b=1.0

40 20 0 0

10

20

30

Number of Cycles

Fig. 3.13 Excess pore pressures generated during the rotational test

57

40


100

Toyoura Sand Rotational Shear p'=100kPa,b=0.0, Dr=70%

18

10

1(cycle)

q (kPa)

60

80

40 10

40

10

100

60

1(cycle)

q (kPa)

80


q=51.86kPa 1

q=51.86kPa 100 1

40

q=34.65kPa

q=34.65kPa 20

20

(b)

(a) 0

0 0

20

40

60 P' (kPa)

80

100

0

120

100

20

40

60 P' (kPa)

80

100

120

100 Toyoura Sand Rotational Shear p'=100kPa,b=0.5, Dr=70%

32

q (kPa)

60

1(cycle)

33

40

80

q=51.86kPa 1

10

4

60 q (kPa)

80


2

1(cycle) q=51.86kPa 10 1

52

40 q=34.65kPa

q=34.65kPa

20

20 (c)

(d)

0 0

20

40

60 P' (kPa)

80

100

0

120

0

20

40

60

80

100

120

P' (kPa)

100

100


2

q (kPa)

60 27

40

80

1(cycle)

10

4

60 q (kPa)

80


q=51.86kPa 1

10

38

40

2

1(cycle) q=51.86kPa 1 q=34.65kPa

q=34.65kPa 20

20

(f)

(e) 0

0 0

20

40

60 P' (kPa)

80

100

120

0

20

40

60 P' (kPa)

Fig. 3.14 Effective stress path of rotational shearing test

58

80

100

120

Chapter 3 Investigation of Anisotropic Granular Soil Behavior Under Rotational Shear 0.6

Initial stress ratio q/p'

D r =90%, b=1.0 0.4 D r =70%, b=1.0

D r =70%,b=0.5

0.2

0 1

10

100

Number of cycles

Fig. 3.15 Number of cycles required approaching stress ratio 1.25

60 (σz-σθ)/2 (kPa)

40

1

start point of shearing

Mc

20 0 Test ID RSD70-2

-20 -40 0

20

40

60

80

100

120

p' (kPa) Fig. 3.16 p ~ (σ z − σ θ ) / 2 path of Test RSD70-2 '

0.2

εzθ

εθ

Strains (%)

0.1

εr

0

-0.1 Test ID RSD70-2

εz

-0.2 0

5

10

15

20

25

30

Numbers of cycles

Fig. 3.17 Strain components against number of cycles in Test RSD70-2

59

35

Chapter 3 Investigation of Anisotropic Granular Soil Behavior Under Rotational Shear 0.5

Test ID RSD70-2

-2εzθ(%)

0.25

0

0.1

-0.4

-0.9

-1.4

-0.25

-0.5

εz-εθ(%) Fig. 3.18 Deviatoric strains produced in Test RSD70-2

0.2

-2εzθ(%)

0.1

0

-0.1 Test ID RSD70-2 -0.2 0

-0.05

-0.1

-0.15

-0.2

εz-εθ(%) Fig. 3.19 Enlarged view of dark area of the central part in Fig. 3.18

60

-0.25


σzθ(kPa)

15

Test ID RSD70-2

5

-5

-15

-25 -0.3

-0.2

-0.1

0

0.1

εzθ(%) Fig. 3.20 Shear stress-strain relation in Test RSD70-2

dε z − dε θ σ z −σθ

1st cycle

33rd cycle

2dε zθ

σ zθ

10th cycle

Fig. 3.21 Stress path with total strain increment superposed

61

0.2

0.3


Table 3.1 List of the equations used to calculate the stresses and strains Stress Vertical

σz =

W

π (r0 2 − ri 2 )

Strain

2

+

2

P0 (r0 − rd ) − Pi ri 2

r0 − ri

2

z H

εz =

2

P0 r0 + Pi ri r0 + ri

εr = −

u0 − ui r0 − ri

P0 r0 − Pi ri Circumferential σ θ = r0 − ri

εθ = −

u0 + ui r0 + ri

Radial

Shear

σr =

σ zθ

3 3 ⎤ 4(r0 − ri )T 1⎡ 3T = ⎢ + ⎥ 2 ⎢⎣ 2π (r0 3 − ri 3 ) 3π (r0 2 − ri 2 )(r0 4 − ri 4 ) ⎥⎦

ε zθ =

θ (r0 3 − ri 3 ) 2

2

3H (r0 − ri )

Table 3.2 Summary of the rotational shear tests Parameter of Test ID

Initial void ratio

Relative Density

Deviatoric stress

Intermediate

eo

Dr (%)

q (kPa)

principal stress b

RCD70-2

0.707

71.0

34.65

0.0

RSD70-2

0.707

71.2

34.65

0.5

RED70-2

0.711

70.0

34.65

1.0

RCD70-3

0.699

73.1

51.96

0.0

RSD70-3

0.703

72.2

51.96

0.5

RED70-3

0.699

73.3

51.96

1.0

RCD90-2

0.638

89.3

34.65

0.0

RSD90-2

0.633

90.5

34.65

0.5

RED90-2

0.645

87.7

34.65

1.0

RCD90-3

0.633

90.0

51.96

0.0

RSD90-3

0.632

90.9

51.96

0.5

RED90-3

0.633

90.0

51.96

1.0

62


CHAPTER 4 QUANTIFYING FABRIC ANISOTROPY ⎯ ELECTRICAL PROPERTY MEAUSREMENT

4.1

Introduction A soil mass is essentially composed of discrete soil particles, thus its me-

chanical behavior is influenced, to a great extent, by its distinct nature. Granular media, like cohesionless soil, consist of non-spherical particles with random, but still characteristic, and statistical arrangements. The technical term, fabric, has been used to characterize the spatial arrangement of particles and associated voids (Oda 1972). It was not until the mid 1950s that the emergences of suitable techniques made direct measurements of the soil fabric possible. Since the early 1970s, attention has also been focused on the experimental investigations of the fabric of cohesionless soils. Mitchell (1993) summarized the methods which could be used to identify the fabric of soils, as listed in Table 4.1. Among these methods, optical and electron microscopy (SEM or TEM), X-ray diffraction, etc., give a direct measure of specific fabric features, and can be classified destructive methods. On the other hand, some techniques adopting physical indirect measurements to interpret the microstructures of soil attract great interest because of their non-destructive properties. However, the interpretation of data is very much dependent on the model used, and is often not so straightforward.

63


Electrical measurement is considered as a kind of non-destructive or noninvasive technique, and is extensively used for characterization of dielectric materials, such as granular materials and clayey soils. Basically, a flow of electricity through a soil is composed of (1) flow through the soil particles alone, which is small owing to the fact that solid phase soil is a poor conductor, (2) flow through the pore fluid alone, and (3) flow through both solid and pore fluid. The total electrical flow is influenced by the porosity, tortuosity of flow paths, and conditions at the interfaces between the solid and liquid phases as well. Therefore, measurements of electrical conductivity or dielectric constant are deemed to be a possible means for evaluating the soil fabrics. In the theoretical aspect, Dafalias and Arulanandan (1979) proposed an analytical expression, relating the formation factor to the sand structure, which was defined as a ratio of the electrical conductivity of an electrolyte saturating a sand sample to the conductivity of the soil water mixture in a certain direction. Thevanayagam (1993) developed a general frequency-domain solution for electrical response of twophase dilute soils, in which it was found that the formation factor in terms of both resistivity and dielectric constant of soils exhibiting a tensorial character. In experimental studies, Mulilis et al. (1977) investigated the fabric anisotropy of Monterey No. 0 sand for different sample preparation methods, through measuring the electrical conductivity. Arulanandan (1991) measured dielectric constants of saturated soils in horizontal and vertical directions, which could be used to predict the soil porosity when the dielectric constants of the particles and the solution are known. Anandarajah and Kuganenthira (1995) measured the electrical anisotropy in terms of electrical conductivity, the variation of which was correlated with the stress-strain behavior of soils. More recently, Klein and Santamarina (2003) studied the anisotropy of mica

64


flakes using the effective conductivity along two orthogonal directions and the results showed that the directional conductivity could be used to characterize the clayey soil fabric. In this chapter, a theoretical description of the dielectric properties of granular materials will firstly be addressed. Then an experimental study on measuring the electrical properties of granular soils will be undertaken, and the results will be presented. Electrical measurements on anisotropic properties of granular soils will be examined and evaluated, in the light of microscopic observations. Concluding remarks and suggestions will finally be made.

4.2

Theoretical Description of Dielectric Materials Maxwell (1881) derived an expression for the conductivity of a heterogene-

ous media, consisting of spherical particles immersed in a solution. It was assumed that the particles were in dilute suspension (each particle being surrounded by the pore fluid), such that the electric field of one particle did not influence the electric field of another, and he obtained the following expression for the conductivity of the medium, k, as a function of the conductivity of the solution k1, the conductivity of the particle, k2, and the porosity, n:

k 2k1 + k 2 − 2(1 − n )( k1 − k 2 ) = k1 2k1 + k 2 + (1 − n )( k1 − k 2 )

(4.1)

Ficke (1924) extended this expression for ellipsoidal particles in a dilute suspension, for all particles oriented in one direction only. k1 − k 2 1− n fθ = 1+ kθ − k 2 n

where

65

(4.2)


1 3S − 1 fθ = (1 − cos 2 θ ) S 2S S=

S=

(4.3)

⎡ R2 ⎤ 1 1− 1− R2 ln + 1⎥ , 0 ≤ R ≤ 1 2 ⎢ 2(1 − R ) ⎢⎣ 2 1 − R 2 1 + 1 − R 2 ⎥⎦ ⎡ R2 ⎤ 1 tan −1 1 − R 2 − 1⎥ , 2 ⎢ 2 2(1 − R ) ⎣ R − 1 ⎦

1≤ R

(4.4)

(4.5)

The quantity S is called shape factor, and kθ is a complex conductivity of the

medium when all the particles are oriented at an angle θ to the direction of the applied electric field. Dafalias and Arulanandan (1979) proposed an expression for the formation factor, as Fθ =

k1 − k2 kθ − k2

(4.6)

The formation factor can also be written in the form Fθ = 1 +

1− n fθ n

(4.7)

The derivation for Eq. (4.7) is valid for ellipsoidal particles, which are all oriented at an angle θ to the applied electric field. However in most cases, either natural deposits or laboratory prepared specimens, the orientation of the soil particles is rotationally symmetric, with respect to the vertical axis, and is called the axis of transverse isotropy, as shown in Fig. 4.1(a). A probability density function, p(θ ) , characterizing the distribution of particles having major axis orientation with respect to the vertical axis for 0 ≤ θ ≤ π 2 , as shown in Fig. 4.1, is thus introduced to satisfy

∫

π 2

0

p (θ )dθ = 1

Hence, the vertical formation factor, Fv, for such distribution is given by 66

(4.8)

Chapter 4 Quantifying Fabric Anisotropy ⎯ Electrical Property Measurement π 2

Fv = ∫

0

p (θ ) Fθ dθ = 1

(4.9)

Substituting Eq. (4.7) into above equation yields, π 2

Fv = ∫

0

⎡ 1− n ⎤ ⎢⎣1 + n f θ ⎥⎦ p(θ )dθ = 1

(4.10)

Plug Eq. (4.3) into Eq. (4.10), then Eq. (4.10) then can be written in the form Fv = 1 +

1− n fv n

(4.11)

where fv =

2S − Pθ (3S − 1) 2S (1 − S ) π 2

Pθ = ∫

0

p (θ ) cos 2 θdθ

(4.12)

(4.13)

and Pθ is the vertical orientation factor. The major axis OA forms an angle β with OH1 or OH2, as shown in Fig. 4.1(b), which varies from π 2 to π 4 as θ changes from 0 to π 2 , and is related to

θ by the identity cos 2 θ + 2 cos 2 β = 1 . Using the aforementioned method, a horizontal orientation factor Pβ can be defined and related to Pθ by π 2

Pβ = ∫

π 4

p ( β ) cos 2 βdβ =

1 π2 1 p(θ )(1 − cos 2 θ )dθ = (1 − Pθ ) ∫ 2 0 2

(4.14)

Similarly, a horizontal formation factor Fh, is obtained, following a similar procedure Fh = 1 + fh =

1− n fh n

S + 1 + Pθ (3S − 1) 4S (1 − S )

Fv n + (1 − n) f v 4 S (1 − S ) + 2(1 − n)[2 S − Pθ (3S − 1)] = = Fh n + (1 − n) f h 4 S (1 − S ) + (1 − n)[ S + 1 + Pθ (3S − 1)]

67

(4.15)

(4.16)

(4.17)


The foregoing solutions are based on the assumption of very high porosity (the surrounding fields of adjacent particles do not perturb each other appreciably). In geotechnical applications, such assumption is always invalid, therefore, an indirect approximation method, Bruggeman’s integration, was employed by Dafalias and Arulanandan (1979) and Thevanayagam (1993), and the formation factors in the vertical and horizontal directions can be expressed as Fv = n − fv

(4.18)

Fh = n − fh

(4.19)

Thus, formation factor ratio is introduced as Fv n − fv = = n fh − fv Fh n − fh

(4.20)

Fig. 4.2 shows variations of the vertical and horizontal formation factors with porosity for different orientation factors. Recalling the definition of Pθ , increasing Pθ implies an increase in the intensity of the vertical orientation of the particles. This

results in the electric current along the vertical having a smaller non-conductive target area, therefore the electric path, which follows the contours of the particles, becomes less tortuous and consequently the vertical conductivity of the mixture increases, such that there is a decrease of Fv . Similar arguments show that Fh must increase with Pθ . It is noted from Fig. 4.2, the curves for Fv and Fh coincide, provided that the relation 2 Pθv + Pθh = 1 holds ( Pθv and Pθh are the vertical orientation factors calculating Fv and Fh , respectively), which could be directly obtained from Eqs. (4.11), (4.12), (4.15) and (4.16). Fig. 4.3 gives variations of the vertical and horizontal formation factors with axial ratio for different orientation factors (porosity n=0.45), in which the curves for

68


Pθ = 0 and Pθ = 1 are the upper and lower bounds, respectively. As the particle sharp is prolate, the formation factor ratio Fv / Fh decreases from a value greater than one, for Pθ > 1 / 3 , through unit value, as Pθ = 1 / 3 , to a value less than one as Pθ < 1 / 3 . In the case of Pθ > 1 / 3 , it implies a tendency for vertical particles orientation, therefore, Fv > Fh . An opposite result is yielded for the case Pθ < 1 / 3 . As Pθ = 1 / 3 , it means that the material is isotropic, and there are no preferred particle orientation, and Fv = Fh . For Toyoura sand (n=0.45, R=0.61), the upper and lower bounds of the formation factor ratio can be easily obtained as 1.169 and 0.732, respectively, and are highlighted in Fig. 4.3. It is observed that the difference between the upper and lower bounds is not sufficiently large, so one can speculate that in real applications, quantification of fabric anisotropy in terms of electrical quantities may not be an appropriate approach. It is in accord with both experimental results and microstructural observations, which will be elaborated below.

4.3

Electrical Measurements for Anisotropic Materials

4.3.1 Two-Terminal Electrode Measurement

A two-terminal electrode measurement technique is employed in the experimental tests. A plexiglas cubic box is specially manufactured to hold the sand specimen, to which two copper electrodes are attached, forming a parallel-plate capacitor, as shown in Fig. 4.4. The assembly is designed to prevent the leakage of fluids in wet specimens. Electrodes are connected to a bridge-type measuring system. An impedance analyzer is employed to capture impedance or admittance data, and is designed

69


to work over a wide range of frequencies. The data analysis is conducted using equivalent circuits. Coaxial cables are used to connect the cell with the measuring device. Both electrodes act as current and potential terminals: high current H c , high potential H p , low current Lc , and low potential L p connectors. The configuration of the leads and the connection of the shields are shown in Fig. 4.5. The electrical properties of the high potential H p and low potential L p terminals, and the applied current i(t) across the high current H c and low current Lc electrodes are measured. In most cases a sinusoidal excitation is imposed and measurements are repeated at different frequencies. Toyoura sand is chosen as the test material. The sand should be washed using alcohol to remove clay components and impurities, if any. The specimen is prepared by the dry deposition method. Saturation process is then carried out to achieve a high degree of saturation, with deionized water serving as the pore fluid. After saturation, the specimen dimensions are 120×120×120 (mm), and the relative density Dr is around 70%, with void ratio e=0.711, and porosity n=0.42.

4.3.2 Calibration of Coaxial Cables

There are some unwanted series and parallel impedances in the specimens being tested, which are represented by the equivalent circuit shown in Fig. 4.6. Calibration involves the determination of these parameters and facilitates their removal from the measured impedance to obtain the true and desired impedance. Calibration measurements are performed in open-circuit (i.e., air as the dielectric), in short-circuit conditions, and also under a condition with a known impedance (say 5 KΩ resistor)

70


to supersede the specimen. Solving the three linear equations obtained from the aforementioned conditions, one can calibrate the three unknown impedances Z1, Z2, and Z3. The desired impedance can be related to the measured one in the form Z m = Z1 +

(Z 3 + Z u )Z 2 Z2 + Z3 + Zu

(4.21)

in which Z m is the impedance measured by the impedance analyzer and Z u is the unknown impedance of the electrodes in contact with specimen inside.

4.3.3 Measurement

The LF impedance analyzer (HP4192A) should be warm up for about 15 minutes, and the working frequency is set at 10MHz. Connect two pairs of cables to each electrode and the impedance analyzer, and take the corresponding readings. In this study, recording admittances is much easier for the data processing later. A paralleled model, with a resistor and a capacitor, is employed for data interpretation, as shown in Fig. 4.7. The following Eqs. (4.22) and (4.23) are adopted to calculate the dielectric constant k ' and conductivity σ based on the equivalent circuit assumption. k' =

Im(Ysample )

ω

⋅

σ = Re(Ysample ) ⋅

d ε0 A

(4.22)

d A

(4.23)

where k ' ─Dielectric constant

σ ─Conductivity Im(Ysample ) ─Imaginary part of sample admittance Re(Ysample ) ─Real part of sample admittance

71


ω ─Frequency of the impedance analyzer

ε 0 ─Permittivity of free space, 8.85×10-12 F/m

4.3.4 Test Results

Two pairs of electrodes together with coaxial cables are used in this study, which are calibrated at the working frequency of 10MHz. The calibrated impedances are Z1=-0.00393- j0.59833, Z2=0.00193+j0.30283, Z3=-0.00318-j0.57088 in the vertical

direction;

Z1=-0.00390+

j0.00384,

Z2=0.00220-j0.29914,

Z3=0.00345+j0.03840 in the horizontal direction; all the units are in KΩ . As for dry sand, the measured dielectric constants are k v' =3.72 and k h' =3.41, where the subscripts v and h denote vertical and horizontal directions, respectively. The dielectric constants for air k a' = 1 and for the particles k p' = 4.5 are used (after Arulanandan 1991). Based on the linear volume fraction conductivity or dielectric constant relation of the mixture, which is frequently utilized (Santamarina et al. ' = 3.03 (if po2001), the predicted dielectric constant for the dry sand sample is k mix

rosity n=0.42). The deviation between the measurement and prediction is attributed to the test error, such as electrode polarization, contact of electrode with specimen, fringe effect, and so on. The possible sources of test error have been elaborated by Santamarina et al. (2001). These error sources can be minimized, but may not be completely eliminated, therefore, a certain kind of normalization is required. ' Normalizing the measured data with respect to k h' , gives k vnor = 3.31 , and ' k hnor = 3.03 . The corresponding formation factor ratio for the dry case is 0.92.

72


For saturated sand, the measured dielectric constants k v' =56.22, and k h' = 56.14 are obtained. If we use the dielectric constant for deionized water k w' = 80 is used, the predicted dielectric constant for the soil-water mixture is ' ' k mix = 36.21 . Applying the same normalization, we obtain k vnor = 36.26 , and ' k hnor = 36.21 , and the formation factor ratio is 0.999. The measured conductivities

for the mixture are σ v = 8.14 × 10 −3 ( s / m) and σ h = 7.57 × 10 −3 ( s / m) , and the corresponding formation factor ratio is 0.93. The measured formation factor ratio is very close to unity, especially for dielectric constant of saturated soil, which is consistent with the prediction shown previously. It is known that Toyoura sand is a kind of uniform, fine sand consisting of sub-rounded to sub-angular particles and mainly composed of quartz. Referring to Fig. 4.3, since the particle axial ratio R for Toyoura sand is about 0.61 (Oda and Nakayama 1989), the anisotropy of the electrical property is not so remarkable, or visible, due to the fact that particle orientations are randomly distributed in reality. It is believed that in quantification of the fabric anisotropy of granular soils such as sandy soil, the dielectric property measurement may not be a good approach. In contrast, for clayey soil, which is primarily composed of mica, and has a layered fabric, the particle axial ratio R is almost 0, or extremely large, such as in laminae-like particles where R → ∞ , electrical measurement may be appropriate and workable, as reported by Klein and Santamarina (2003).

4.4

Discussion

To further understand the effect on electrical measurements by particle orientations in the practical applications, microstructural information of the sand specimen 73


is required. Preservation of the fabric of a soil specimen can be achieved by impregnating the specimen with epoxy resin and curing. A detailed experimental procedure, as well as a microstructural analysis will be presented in the next chapter, and here only give a brief introduction. Epoxy impregnation is performed by forcing epoxy resin into the soil specimen under a low differential pressure of about 20 to 30kPa. The low pressure and slow epoxy flow are maintained to minimize the disturbance to the soil fabric and prevent air bubbles from being trapped in the soil specimen in order to get a sample with a high degree of saturation. A moist tamped specimen is also impregnated with the epoxy when initial moisture in the specimen is completely removed by drying before the epoxy impregnation. After the epoxy resin is completely cured, coupon surface preparation can be carried out by sectioning, grinding and polishing. Once the coupon surface is suitable as an SEM (Scanning Electronic Microscope) specimen, images then can be taken and microstructure analysis follows. By statistical analysis on quite a few images (REVs), it is found that the dry deposited (DD) specimen has a strong tendency for a preferred orientation in the horizontal direction, while the moist tamped (MT) specimen has no obvious preferential orientation. The angle θ with respect to the horizontal axis in the frequency histogram and rose diagram representation for both DD and MT specimens are shown in Fig. 4.8 (a) and (b), respectively. Based on the frequency histograms angle θ , it is very easy to predict the degree of electrical anisotropy for DD and MT specimens in terms of the formation factor ratio. The vertical orientation factor Pθ can be readily obtained according to Eq. (4.13). The calculated Pθ = 0.395 for the DD specimen, is less than Pθ = 0.467 for the MT one, indicating that the tendency of preferred particle orientation is more horizontal for DD specimen than for MT one. Referring to Fig. 4.3, the upper and

74


lower bounds of the formation factor ratios of Toyoura sand (R=0.61) are 1.169 and 0.732, respectively. By interpolation, the formation factor ratios corresponding to Pθ = 0.395 (DD specimen, n=0.466) and Pθ = 0.467 (MT specimen, n=0.474) are about 0.971 and 0.944, respectively, both of which are very close to unity. Therefore, it is believed that the approach of the electrical conductivity measurements may not discriminate inherent fabric anisotropies, which arise from different specimen preparation methods. In other words, for dielectric materials such as granular soil, the anisotropy may not be reflected in the electrical field, but mechanical anisotropy does exist (Oda 1972, Miura and Toki 1982, Yamamuro and Wood 2004, etc.). The fundamental reason is that the mechanical behavior of sands depends on various factors, for instance, particle orientations, contact normals, and arrangement of particles with associated voids, etc., while the electrical property of granular material is predominantly a flow problem, which is mostly affected by the particle orientation and the tortuosity. Therefore, significant anisotropy in mechanical properties will not result in a pronounced anisotropy in the electrical field. Hence, electrical characterization is not a suitable approach to quantify the fabric anisotropy for sand. All the analysis aforementioned is based on the assumption that soil particles are non-conductive, and the expression for the formation factor is in terms of conductivity. If the frequency applied is sufficiently high (ultra high frequency), the results obtained so far also validate the formation factors in terms of the dielectric constant, provided that the dielectric constant of the solid particle is much less than that of the pore fluid, which holds true for the saturated sand specimen (4.5 for particles and 80 for water). The Eqs. (4.12) and (4.16) are replaced by Eqs. (4.24) and (4.25), respectively, and as an illustration, Fig. 4.3 is re-plotted in terms of the dielectric constant, as shown in Fig. 4.9, which is found to be almost identical to Fig. 4.3.

75


fv =

Pθ 1 − Pθ + 1 − 0.944(1 − 2S ) 1 − 0.944 S

1 − Pθ 1 + Pθ 1 fh = ( + ) 2 1 − 0.944(1 − 2 S ) 1 − 0.944S

4.5

(4.24)

(4.25)

Summary and Conclusion

It is well know that the mechanical properties of any soil, including stressstrain behavior, strength, dilatancy characteristics, etc., depend on the soil fabric. Consequently, quantification of fabric anisotropy is of great importance to better understand the relations between the fabric anisotropy and the mechanical behavior of soil. Quantifying the fabric anisotropy using electrical methods seems an easy option because of its non-invasive nature. In this chapter, theoretical descriptions of the electrical properties of sands, concerning the particle orientations, were presented. Laboratory tests were performed to measure the electrical properties, the conductivity and dielectric constant, for both the dry and saturated specimens, prepared by dry deposition method. The measurements were compared with the prediction, in the light of microstructural observations. Unfortunately, either the theoretical analysis, or test results, shows that it is not an appropriate approach for assessing the fabric anisotropy of granular materials. As noted by Klein and Santamarina (2003), the anisotropy of electrical properties in real soils are much smaller than the extreme anisotropy predicted in their tests, simply because of the multidirectional connectivity of internal porosity of soils, which is the case under most geo-environments. It is inferred that a remarkable anisotropy in mechanical properties might not result in a pronounced anisotropy of physical quantities.

76


To quantification the fabric anisotropy, a direct method must be employed. An approach, using an image analysis on thin sections, will be presented in the next chapter.

77


Fig. 4.1 Schematic illustration of particle’s orientation in a transversely isotropic sand aggregate (after Dafalias and Arulanandan 1979)

5.5 5

1.0 0.8 0.6 0.33 0.2 0

Pθ for Fh

Toyoura sand R =0.61, S =0.395

Formation factor

4.5 4 3.5 3 2.5 2 0.35

0 0.2 0.33 0.6 0.8 1.0

Pθ for Fv 0.375

0.4

0.425

0.45

0.475

0.5

Porosity n Fig. 4.2 Variation of vertical and horizontal formation factors with porosity for different orientation factors

78


2 prolate

oblate

1.6

1.0

Fv /Fh

0.6

F v /F h =1.169

1.2

0.333 0.2

0.8

0.0

F v /F h =0.732

Pθ

0.4 R =0.61 0 0

0.2

0.4

0.6

0.8 1 Axial ratio R

1.2

1.4

1.6

1.8

Fig. 4.3 Variation of the vertical and horizontal formation factor ratios with axial ratio R for different orientation factors

Opening for wire

Plexiglas container

Electrodes

Soil-water mixture Electrodes

Flushing line

Fig. 4.4 Side view of the container

79

2


LF Impedence Analyzer

Lc Lp

Electrode

Specimen

Hp Hc

Fig. 4.5 Two-terminal electrode system-low frequency measurements (Note that connection of shields near the cell denoted by the asterisks)

LF Impedence Analyzer Zm

A single coaxial cable

Z3

Z1 Z2

Sample

Fig. 4.6 Calibration of the two-terminal electrode system

R

C Fig. 4.7 Capacitor-resistor paralleled model

80

Zu


Dry Deposited Specimen Moist Tamped Specimen

10

90 10

120

60

8

Percentage(%)

8

6

150

30

4

6

2 0 180

4

0

2 4

2

6

210

330

8

0 -90

-60

-30

0

30

60

90

240

10

Particle Preferred Orientations

(a) Histogram representation

300 270

(b) Rose diagram representation

Fig. 4.8 Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations

2 prolate

oblate

1.6 1.0 F v /F h =1.151

Fv /Fh

1.2

0.6 0.333 0.2

0.8

0.0

F v /F h =0.755

Pθ

0.4 R =0.61 0 0

0.2

0.4

0.6

0.8 1 1.2 Axial ratio R

1.4

1.6

1.8

Fig. 4.9 Variation of the vertical and horizontal formation factor ratios with axial ratio R obtained from the dielectric constants

81

2


Table 4.1 The techniques for the study of soil fabric Method

Basis

Optical Microscope (Polarizing)

Direct observation of fracture of thin sections

Electron Microscope

Direct observation of particles or fracture surfaces through soil sample (scanning electron microscopeSEM) observation of surface replicas (transmission electron microscopeTEM) Groups of parallel clay plates produce stronger diffraction than randomly oriented plates

X- Ray diffraction

Pore Size Distribution

Acoustical velocity

(1) Forced intrusion of non-wetting fluid (usually mercury) (2) Capillary condensation Particle alignment influences velocity

Dielectric Dispersion and Electrical Conductivity

Variation of dielectric constant and conductivity with frequency

Thermal Conductivity

Particle orientations influence thermal conductivity

Magnetic Susceptibility

Variation in magnetic susceptibility with change of sample orientation relative to magnetic field Properties reflect influences of fabric

Mechanical Properties Strength Modulus Permeability Compressibility shrinkage and Swell

* For a homogeneous sample. Discontinuities, stratification, override effects of microfabric. (after Mitchell 1993)

82

Scale of Observations and Features Discernable Individual particles of silt size and lager, clay particle groups, preferred orientation of clay, homogeneity on a millimeter scale or larger, larger pores, shear zones Useful upper limit of magnification about ×300 Resolution to abut 100A; Large depth of field with SEM; direct observation of particles; particle groups and pore space; details of micro-fabric Orientation in zones several square millimeters in area and several micrometers thick; best in single mineral clays (1) Pores in range from 0.01 to > 10 µ m maximum Anisotropy; measures micro-fabric averaged over a volume equal to sample size* Assessment of anisotropy; flocculation and deflocculation; measure microfabric averaged over a volume equal to sample size* Anisotropy; measures micro-fabric averaged over a volume equal to sample size* Anisotropy; measures micro-fabric averaged over a volume equal to sample size* Micro-fabric averaged over a volume equal to sample size*; anisotropy; macro-fabric features in some cases and so on, on a macroscale can

Chapter 5 Quantifying Fabric Anisotropy — Using an Image Analysis Approach

CHAPTER 5 QUANTIFYING FRBRIC ANISOTRPY ⎯ USING AN IMAGE ANALYSIS APPROACH

5.1

Introduction In the last chapter, the methods for quantifying the soil fabric anisotropy were

reviewed, among which the method through measuring the electrical properties (the dielectric constant and conductivity) and correlating them with the microfeatures of the soils was preferred. However, as shown previously, this kind of method may not be applicable to the granular materials, such like the sandy soils. Therefore, an alternative approach based on the microstructure analysis will be resorted to and elaborated in this chapter. This approach, with the aid of an image analysis, has wide application in recent investigations on the microstructure for the granular soils (Kuo and Frost 1993, Jang et al. 1999, Muhunthan et al. 2000, Masad and Muhunthan 2000). In regarding to this approach, it is required that any pore fluid inside the soil should be removed, and replaced by the epoxy resin to glue the solid particles. Doing this without any disturbance to the original fabric and to keep it intact is very difficult, and in most cases there is no way to quantify how much disturbance there may have. However, the disturbance can be minimized by introducing the epoxy resin to the soil slowly under a low pressure. After the epoxy impregnation, a curing process is required to make certain the sample has enough bonding strength, which is suffi-

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cient to withstand the routine processes such as sectioning, grinding, and polishing. Then, the coupon surface could be treated through sectioning, grinding and polishing, and the image of the coupon surface can thus be captured by using an SEM. Once the high quality image has been obtained, an analysis and assessment of the soil structure could be carried out.

5.2

Test Setup Currently, a traditional triaxial cell is modified for epoxy impregnation, as

shown in Fig. 5.1, and is similar to the apparatus used by Jang et al. (1999). In this modified cell, only some disposable items, such as plastic tubes and filters, and the base and top platens coated with the mold release film can be in contact with the epoxy resin. Openings through the top and bottom platens are lined with the plastic tubing, which will be disposed of once the resin has completely cured. In Fig. 5.1, the left chamber is employed to hold the sample, and right one is used to pressurize the epoxy resin, such that the epoxy can be forced into the sample.

5.3

Materials and Other Equipments Used in the Test

5.3.1 Epoxy Resins The key issue in this study is to select an appropriate epoxy resin, which is applicable to sustain the processes associated with the coupon preparation. Jang et al. (1999) summarized the criterion for choosing the epoxy resin, and the epoxy resin should have the following properties. •

Low viscosity

•

Cure at room temperature

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•

Low shrinkage during curing

•

High hardness value and high bonding strength on curing

•

Good working properties and short curing time

•

Nontoxic

•

Noncreative with soil and test equipment

Epoxy resin is used to preserve the soil structure, and the disturbance should be minimized during the impregnation and later curing process. According to some previous investigations, loose sand specimens are very susceptible to disturbance if the resin has a relatively high viscosity or if significant shrinkage of the resin occurs during curing. In this study, Epo-thin resin and hardener manufactured by Buehler Ltd. Company, Lake Bluff, Illinois, are used to glue the solid particles of the specimen. The ratio, by weight, of resin to hardener is held at 100:40, which is based on the recommendation provided by manufacturer, to allow the mixture to harden at room temperature. In addition, acetone is accessorily added into the mixture to reduce the viscosity and thus to increase the rate of the impregnation and to minimize the disturbance to the soil structure. In the studies of Masad and Muhunthan (2000), Masad et al. (2000), an optimum mix proportion by weight of resin, hardener, and acetone was held to be 100:40:8, and could allow the primary hardening to be completed within 2 hours, which is sufficient to finish the impregnation process and disconnect the saturation lines and valves for cleaning.

5.3.2 Sand and Sample Preparation Methods Toyoura sand is a uniform fine sand consisting of subrounded to subangular particles and composed of 75% quartz, 22% feldspar and 3% magnetite. Some basic 85


data of Toyoura sand are: mean diameter D50=0.23mm, uniformity coefficient: Uc=1.7, maximum void ratio emax=0.977, minimum void ratio emin=0.597, and specific gravity Gs=2.65 (Verdugo and Ishihara 1996). The particle size distribution curve of Toyoura sand is shown in Fig. 5.2. Although the mean particle size of Toyoura is much finer than that of Ottawa sand used by Frost’s group and Muhunthan’s group, the impregnation of Epo-thin epoxy is still working, irrespective of the density of the specimen prepared in this study. Two preparation methods, namely the dry deposition and the moist tamping, are employed to produce specimens with distinctively different fabrics. These two methods are illustrated in Fig. 5.3 and introduced in detail as follows (summarized and modified based on DeGregorio 1990, Ishihara 1996 and Zlatović and Ishihara 1997).

Dry Deposition Oven-dried sand is filled into a forming mold in several layers through a funnel and a tube with a nozzle about 12mm in diameter. In each layer, the sand is poured with almost a zero falling head by keeping the nozzle position slightly above the sand surface, such that the sand is deposited in a loose state. The denser specimen can be prepared by tapping the mold using a rubber mallet. The specimens obtained using this method are more uniform and generally denser as compared to those prepared by the moist tamping method. The air pluviation is another preparation method, and is very similar to the dry deposition method. The dry sand is poured into the mold with a falling height and the density of the specimen can be adjusted by changing this falling height. It is believed that the air pluviation can simulate the

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natural deposition process of wind blown aeolian deposits (Kuerbis and Vaid 1988). Though this method is somewhat different from the dry deposition, the fabric produced by the dry deposition still has certain relation with those in-situ natural deposited soils.

Moist Tamping The moist tamping method basically is placing the moist sand with 5% water content in 10 layers in a mold. First, the weight of the dry sand required to obtain a specimen with the desired void ratio is determined based on the approximate volume of the split mold. The dry sand is weighed and then placed in a container. A certain (5% water content) amount of the de-aired water is added and mixed thoroughly with the sands. The moist sand is then spooned into the split mold and roughly leveled with a spatula. The height of each layer is taken as one-tenth of the specimen height. The caliper is used to position the compacted surface for each layer. The tamper is then lowered into the mold, and the sand is gently compressed to a predetermined height. Since the diameter of the aluminous tamper (65mm) used in this study is slightly smaller than that of the desired specimen (70mm), the entire surface is compacted by moving the tamper around the perimeter of the mold. The top of each tamped layer should be then scarified slightly to promote the bonding between the layers. In addition, an aluminous extension is placed atop the split mold to facilitate the tamping of the last layer. To promote a more uniform specimen, undercompaction idea (Ladd 1978) is utilized during the sample preparation. The lower layers are undercompacted such

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that the compaction from the successive layers will produce a specimen of approximately uniform density throughout when the preparation is finished. The moist tamping method best models the soil fabric of rolled construction fills, for which the method was originally designed (Kuerbis and Vaid 1988). One of the advantages of this method is the versatility in permitting any sample to be prepared within a wide range of void ratios. Therefore the sample could be very loose and highly contractive or dilative in subsequent loading, depending on the initial void ratio. Another major advantage of this method is to prevent segregation of the wellgraded materials.

5.3.3 Scanning Electron Microscope (SEM) A JEOL-6300 scanning electron microscope (SEM) is utilized to take high quality images on the coupon surfaces. The SEM enables the coupon surface to be clearer and more visible. Dried SEM samples must be coated with a conductive film to prevent the surfaces from being charged by the electron beam under the 10kVvoltage. To this end, the coupon surfaces used in this study were sputtered with a thin layer of gold.

5.3.4 Tools for Sectioning, Grinding and Polishing Diamond saw is used to cut the cured specimen into a desired shape. After sectioning, the coupon surface is still very rough and is not qualified as an SEM specimen. Therefore, further grinding and polishing processes should be undertaken by using a Buehler grinder-polisher.

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5.4

Test Procedure

5.4.1 Specimen Impregnation Preservation of the fabric of a soil specimen can be achieved by (1) impregnating the specimen with resin and curing; (2) saturating the specimen with water and freezing. Of these two methods, resin impregnation is of particular interest because it is more practical and flexible. However, a frozen specimen is not strong enough to sustain the extra processes, such as sectioning, grinding, polishing, to obtain a high-quality coupon surface for later image capture and analysis. Epoxy impregnation is performed by forcing epoxy resin into the soil specimen under a low differential pressure of about 20 to 30kPa. The low pressure and slow epoxy flow are maintained to minimize the disturbance to the soil fabric and to prevent air bubbles from being trapped in the soil sample in order to get a sample with a high degree of saturation. It is also found that a very small vacuum, around 5kPa, is helpful in driving away the air bubbles inside the specimen. It is strongly suggested that the sands should be washed using alcohol or methanol, to remove the clay components or any other impurities in the sands. Such treatment will improve the adhesiveness of the sand and epoxy, and maintain the integrity of the coupon surface even after grinding and polishing. The moist tamped specimens are also impregnated with the epoxy for microstructure analysis. Since this kind epoxy will not function well in moist conditions, the initial moisture in the soil specimen should be completely removed by drying before the epoxy impregnation. A system was designed and fabricated to facilitate drying of the moist tamped specimens without causing disturbance, and was shown in Fig. 5.4. With this system, most of water inside the specimens can be removed by circulating the dry air through the specimen. The dry air is obtained by forcing the

89


atmospheric air through a desiccant chamber placed upstream of the system. The desiccant CaSO4 is employed because it functions better in the low moist environment (about 5% moisture in the moist tamped specimen) compared with commonly used silicon gel (physical absorption). In addition, CaSO4 has a moisture indicator and will turn from blue into pink when hydration takes place. In turn, it can be dehydrated in the oven with elevated temperature around 250o, and such that it is reusable. Another desiccant chamber is placed in the downstream, serving as an indicator to check whether the specimen is fully dried or not, according to the change of the color of the desiccant. For the moist tamped specimen, the drying process normally lasts about 24 hours for driving out all the water inside the soil under a vacuum around 5kPa.

5.4.2 Epoxy Resin Curing After the epoxy has been impregnated, the epoxy resin curing process follows. The epoxy used in this study (Epo-thin) should be cured at room temperature for about 18 hours according to the recommendation provided by the manufacturer. Once the specimen is hardened, the mold could be dismantled. Some items that directly contact the epoxy can be removed easily and discarded.

5.4.3 Coupon Surface Preparation Image analysis is performed by capturing and analyzing images of sectioned coupon surfaces taking from the harden specimen. After coupons are sectioned to the desired shape in predetermined locations, the surfaces of the coupons are ground and polished so that the soil structure could be visible by SEM.

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Firstly, the hardened specimen is roughly cut into small patches using a band saw. According to the study by Jang et al. (1999), due to the high cutting speed, sectioning normally inflicts a layer of deformed or cracked material near the surface. To reduce the deformed layer caused by rough sectioning and reduce the effort required to remove this damaged layer during surface preparation, fine sectioning should be conducted using a diamond wheel saw. Here a diamond wafering blade is recommended. The rotational speed of the diamond should be well controlled, since high speeds will damage the integrity of the coupon. Actually in this study the speed is less than 100rpm, to make certain that the cutting surface is smooth and flat. For statistical purposes, the images captured and analyzed should be representative, and lie in different locations. In this study, only the horizontal and vertical sections are chosen for strategic considerations. All the thin layers sectioned are about 10mm-thick, and 20mm×20mm size, which is a workable size for grinding and polishing. The thin layers are then put into the cylindrical molds, and ordinary epoxy resin and hardener mixed in a specific ratio is poured into the molds. Once the epoxy has hardened, the thin layer embedded with epoxy resin can be taken out with one face exposed for grounding and polishing. By such treatment, the thin layers could be installed in a platen, which is specially designed in the Buehler grinder-polisher, for machine grinding and polishing. The platen with the coupons is then mounted in the grinder-polisher. In order to improve the efficiency of grinding and polishing, some key points should be followed, such as abrasive type, abrasive grain size, applied load, rotational speed and lubricant type. In the grinding phase, silicon carbide grinding paper, with different grain sizes, are used. The grain sizes in terms of nominal grain diameter are chosen as 76µm, 35µm, 26µm, and 22µm. The applied load is about 2lbs, and the rotational

91


speed is controlled at 100rpm. Tap water is used as a lubricant. The purpose of the grinding phase is to remove material and produce a planar surface for the later polishing. The principle of the planar grinding is from coarse to fine. It is found that more effort paid to the coarse grinding will effectively reduce the work in the later grinding and polishing phases. During each step of grinding with different particle sizes, frequent checking on the surfaces of the coupons is required, either using naked eyes or an optical microscope. The platen and silicon paper also should be frequently washed using water, because some impurities produced in this process will make the grinding surface uneven and thus affect the quality of the coupon surface. In addition, some silicon carbide powder, with corresponding sizes, can be added as the abrasive, and it will make the coarse planar grinding more effective and save silicon paper. Frankly, the grinding process is very painstaking, and much time is required. Once a planar surface has been obtained after grinding, the polishing phase follows. Polishing is very crucial in the preparation of coupon surfaces for soil structure identification, because quantitative measurement demands high feature contrast and high accuracy of the surface images. In this phase, silicon paper is still used as the abrasive when the polishing grain size of is still 10µm or 5µm. If even finer polishing, such as 3µm, 1µm, 0.3µm and 0.05µm is required, polish cloths will be employed. The polish cloths are made of rayon-fine, embedded with aluminum oxide suspension, which serves as the abrasive, with some water as the lubricant. Now the applied load is adjusted to 1lb, and rotational speed is still kept 100rpm. As the polishing proceeds, the scratches on the sand particles surfaces are gradually erased and the sand particles appear to be brighter. Since the effects of polishing are more apparent on sand particles (comprised mainly of quartz) than on the epoxy matrix, a

92


high contrast image appears, with sand particles much brighter than the surrounding epoxy matrix. The quality of SEM specimens is stricter than that of the optical microscope, so the tiny scratches on the coupon surface must be polished using finer abrasive grain size. In the current study, the specimen is polished up to 0.05µm, and little scratches are observed by SEM. If a higher contrast image is desired, additional surface treatments can be utilized, for instance, etching. It is found for most applications, the aforementioned procedures are sufficient to yield a high contrast image for sandy specimens.

5.4.4 Image Capture and Analysis After sectioning, grinding and polishing, the coupon is qualified as an SEM specimen, but before the image is taken, the coupons must be placed in a desiccant case for complete drying. Since a high degree of vacuum will be applied to specimens in the chamber, any moisture must be removed to avoid contamination of the SEM. Then the dried coupon is sputtered with a thin layer gold to prevent the surface being charged by the electron beam. The image capture will be ready after a 10kVvoltage is applied. Adjusting contrast and focusing at a large magnification, say 200, the desired image will be obtained. For representative purposes, the magnification is fixed at 30, and subsequently yielded image has 1024×819 pixels. Fig. 5.5 shows a typical SEM image at a magnification of 200. This figure clearly shows the sandy particles surrounded by the epoxy matrix. It is also observed that some air bubbles, in dark and circular shapes and lost sandy particles due to the damage from coupon surface preparation. Examples of a typical image captured by SEM and black-and-white mask (binary image) used for image analysis are given in

93


Fig. 5.6. The image contains 300 particles or so, thus it can be treated as a representative elemental volume (REV). The inherent fabric anisotropies for both dry deposited (DD) and moist tamped (MT) specimens will be investigated in this study. It is known that both the DD and MT samples have transverse isotropy, with the vertical axis as the axis of symmetry. Consequently, only the vertical section of the specimen is required to analyze the fabric anisotropy, since the horizontal section is considered to be isotropic.

Particle Orientations and Magnitude Vector ∆ Curray (1956) proposed an index, called the magnitude vector ∆ , to show the intensity of anisotropy of the preferred orientation of the particles. The magnitude vector ∆ is defined as ∆=

2N 2N 1 (∑ cos 2ϕ k ) 2 + (∑ sin 2ϕ k ) 2 2 N k =1 k =1

(5.1)

where ϕk is inclination angle of the kth unit vector n, measured in reference to the H1 axis in a representative V~H1 section, as shown in Fig. 5.7. It is clear that the ∆ value depends on the particle shape and the process of soil deposition. Eq. (5.1) shows that ∆ varies from zero, when the material is isotropic, to unity, when the major axes of all the particles are uniformly distributed in the horizontal plane (plane H1~H2). Through image analysis, each particle is numbered and its orientation can be obtained. An example of the particle numbering is shown in Fig. 5.8. By statistical analysis on quite a few images (REVs) of the vertical sections for both DD and MT coupons, the results in terms of the magnitude vector ∆ , which can be calculated based on Eq. (5.1), are summarized in Table 5.1. It is observed that in the vertical

94


plane, the index ∆ for DD specimens is much greater than zero, indicating that DD specimens possess obvious inherent anisotropy; while the MT samples are slightly anisotropic since ∆ is much closer to zero. The angle ϕ with respect to horizontal axis in the frequency histogram and rose diagram representations for both DD and MT specimens in the vertical sections are shown in Fig. 5.9(a) and (b), respectively. It is clearly shown that for the DD specimen, the preferential particle orientation is the horizontal direction, which is mainly caused by the gravitational force during the deposition process in the sample preparation. While for the MT specimen, the preferred orientation of the particles is more randomly distributed in space, due to the existing initial moisture in the soils. It is believed that the DD specimen is more anisotropic than the MT specimen, and the inherent fabric anisotropies for both DD and MT specimens can be determined quantitatively in terms of magnitude vector ∆ . Fig. 5.10(a) and (b) shows the preferred orientation of the particles for the horizontal sections of both the MT and DD coupons, respectively. It is seen that there is no any preferential particle orientation in the horizontal plane for both DD and MT specimens, and the corresponding values of the magnitude vector ∆ are very close to zero, and are also given in Table 5.1, which in turn provides a justification for the transverse isotropy postulation in the horizontal planes for both DD and MT specimens. A fabric tensor of the second order was defined by Oda (1999) as

Fij =

1 2N

2N

∑n n k =1

k i

k j

(5.2)

where N is the number of particles in a representative volume, and nik and nik are the components of kth vector. The magnitudes of the components in the tensor represent the net portion of the particles that are statistically orientated towards a particular di-

95


rection. It can be seen that Fij is symmetric, and therefore it can be always be represented by three principal values, F1, F2, and F3 (not necessarily implying that

F1 ≥ F2 ≥ F3 ), and three associated principal directions. If the reference frame is coincident with the principal directions, the fabric tensor can be written as ⎡ F1 F = ⎢⎢ 0 ⎢⎣ 0 ' ij

0⎤ 0 ⎥⎥ F3 ⎥⎦

0 F2 0

(5.3)

In most practical cases, soils are transversely isotropic; two of the principal values, say, F2, and F3, are equal to each other, leaving only two independent principal values, F1, and F3, for the tensor. It is further noted that Fij possesses a unit trace, i.e., F1 = 1 − ( F2 + F3 ) = 1 − 2 F3 . Therefore, for a cross-anisotropic soil with a known direction of deposition (usually in the vertical direction), only one scalar quantity is needed to define the fabric tensor. Oda and Nakayama (1988) showed that the fabric tensor for a transversely isotropic material could be written in the following form: 0 0 ⎤ ⎡1 − ∆ 1 ⎢ 0 1+ ∆ 0 ⎥⎥ F = ⎢ 3+ ∆ ⎢⎣ 0 0 1 + ∆ ⎥⎦ ' ij

(5.4)

where ∆ is the magnitude vector. By substituting the ∆ value pertinent to a certain structure (DD or MT specimen) into this expression, the corresponding fabric tensor, characterizing the material anisotropic property, can be obtained.

Vertical Orientation Factor Pθ

Dafalias and Arulanandan (1979) proposed an index, called the vertical orientation factor, for transversely isotropic materials, as π 2

Pθ = ∫

0

96

p (θ ) cos 2 θdθ

(5.5)


in which p(θ ) , a probability density function, was introduced to characterize the dis-

tribution of particles having major axis orientation with respect to the vertical axis for 0 ≤ θ ≤ π 2 , as shown in the last chapter, and satisfied

∫

π 2

0

p (θ )dθ = 1

(5.6)

Based on the particle orientation distribution, as shown in Fig. 5.9, it is also very easy to predict the degree of electrical anisotropy for DD and MT specimens in terms of the formation factor. The vertical orientation factor Pθ can be calculated using Eq. (5.5). The calculated Pθ =0.395 for the DD sample, is less than Pθ =0.467 for the MT sample, indicating that preferred particle orientation tendency is more horizontal for the DD sample than that for the MT one, which is consistent with the description using the magnitude vector ∆ . It is noted that the Pθ values of both DD and MT samples are greater than one third (isotropy), which contradicts our intuition. This apparent difference stems from the two-dimensional interpretation of the obtained histograms of the preferred particle orientation distribution. Here a three-dimensional interpretation is incorporated in the theoretical description as well as the prediction, in terms of formation factors using the histograms of preferred particle orientations. Referring to the last chapter, the upper and lower bounds of the formation factor ratios for Toyoura sand (R=0.61) are 1.169 and 0.732, respectively. By interpolation, the formation factor ratios corresponding to Pθ =0.395 (DD sample, n=0.446) and Pθ =0.467 (MT sample, n=0.474) are about 0.971 and 0.944 respectively, both of which are very close to unity. Therefore, it is believed that electrical conductivity measurements may not discriminate inherent fabric anisotropies, which arise from different sample preparation methods. In other words, for dielectric mate-

97


rials such as granular soil, the anisotropy may not be reflected in the electrical field, but the mechanical anisotropy does exist (Oda 1972, Miura and Toki 1982, Yamamuro and Wood 2004, etc.). The fundamental reason is that the mechanical behavior of sands depends on various factors, for instance, particle orientations, contact normals, arrangement of particles with associated voids, etc., while the electrical property of granular material is predominantly a flow problem, which is mostly affected by the particle orientation and the tortuosity. Therefore, significant anisotropy in mechanical properties will not result in a pronounced anisotropy in the electrical field. Hence, electrical characterization is not a suitable approach to quantify the fabric anisotropy for sand.

Directional Mean Free Path

The free path concept represents the uninterrupted interparticle distance between particles and the mean free path is an average value of the free paths along certain direction. The mean free path is a spatial parameter, of importance in a particulate system, and essentially a mean edge-to-edge distance (Kuo et al. 1998). Frost et al. (1998) investigated the mean free path of a thin section, and obtained a fabric tensor to characterize the anisotropy of the granular assembly in terms of the mean free path. Bowman and Soga (2003) studied the mean free paths along both the horizontal and vertical directions, and the results showed that in general the mean of the vertical distance was lower than that of the horizontal distance during the tests. According to their explanation, the anisotropy of the mean free path was mainly affected by the anisotropic distribution of the particle orientations. In the current study, the mean free paths along the vertical and horizontal directions are studied using the binary images of both the DD and MT coupons. A se-

98


ries of scanning lines are spaced at certain pixels, and are shown in Fig. 5.11. A square of 600×600 pixel is chosen as an REA. The free paths between particles of an example image in the vertical and horizontal directions are shown in Fig. 5.12. A MATLAB code was thus written to calculate the mean free paths along vertical and horizontal directions, and the obtained results were tabulated in Table 5.2. For the DD specimen, the mean free path in the vertical direction is lower than that in the horizontal direction. Conversely, for the MT specimen, the difference of the mean free paths between the vertical and horizontal directions is not so appreciable. The possible explanation is given as follows. For DD specimen, the preferential particle orientation is in the horizontal plane, such that a great number of smaller vertical gaps exist between the more horizontally aligned particles. But for MT specimen, a minor difference is found between the mean horizontal and vertical spacing, suggesting a more isotropic structure has been formed in the MT specimen than in the DD specimen. It has also been confirmed by previous microstructural investigation, that the particle orientation distribution is almost isotropic for the MT specimen. Additionally, in Table 5.2, the horizontal thin sections for both DD and MT coupons are also investigated, and the result shows that the mean free paths in these two horizontal directions are almost identical, which further verifies the postulation of the transverse isotropy for the specimens.

Void Ratio in Tensorial Form

It is recently that some investigators have used the directional void ratio/porosity distribution to characterize the anisotropy of granular materials as well as clayey soils, such as Kuo et al. (1998), Muhunthan et al. (1997), and so on. The void

99


ratio orientation distribution function, N (θ ) , is described by a mean void ratio of the specimen e0 and a void ratio tensor Nij as N (e,θ ) = e0 (1 + N ij ni n j + N ijkl ni n j nk nl + ...)

(5.7)

where Nij , Nijkl are second order and fourth order fabric tensors, respectively, and ni,

nj, nk, nl are unit vectors. It is worth noting that Eq. (5.7) is a general mathematical form expressing the directional distribution of anisotropic parameters, such as contact normal, branch vector and contact vector. The use of high order tensors allows the fluctuations of the density function to be taken into account and yields an accurate representation of the distribution. But for most practical applications, a second-order fabric tensor will be sufficient enough to describe the anisotropy of the soils. Thus Eq. (5.7) can be simplified as

N (e,θ ) = e0 (1 + N ij ni n j )

(5.8)

This expression describes the fabric anisotropy in terms of the void ratio, and suggests that only two measures are sufficient: one is a scalar quantity of the mean value of the void ratio e0, and the other is a second order fabric tensor Nij, giving the directional distribution of the void ratio. Kuo and Frost (1993) and Muhunthan (1993) presented a simplified procedure to experimentally determine the directional void ratio distribution, based on the twodimensional image analysis on the thin sections of the soil samples. As for the twodimensional cases, Eq. (5.8) can be written in terms of Fourier series with a period of 2 π as ∞

N (e,θ ) = e0 [1 + ∑ ( An cos nθ + Bn sin nθ )]

(5.9)

n =0

where

2π

e0 = ∫ N (e, θ )dθ 0

100

(5.10)


⎡ An ⎤ 1 ⎢B ⎥ = ⎣ n ⎦ π e0

∫

2π

0

⎡ cos nθ ⎤ N (e, θ ) ⎢ ⎥ dθ ⎣sin nθ ⎦

(5.11)

where θ is the orientation angle of the vector with respect to the vertical axis. The fabric tensor Nij is of the form ⎛A N ij = ⎜ 2 ⎝ B2

B2 ⎞ ⎟ − A2 ⎠

(5.12)

in which A2 and B2 can be determined through the following steps (Kuo and Frost 1993): Step 1: Select a Cartesian coordinate system in the soil specimen. Step 2:Observe the cross section and choose an REC. Draw a series of lines having orientations θ m = mπ / N , m=0, 1, 2, …, n-1, on a plane that pass through the center of the REV. These lines correspond to the unit vectors ni. Calculate the fraction LL (θ m ) occupied by voids on a particular line. Step 3: Compute A2, B2 and e0 corresponding to Eq. (5.9):

LL (θ m ) N m=0 N −1

e0 = ∑

N −1 ⎡ A2 ⎤ ⎡cos(2π m / N ) ⎤ N −1 = θ L 2 ( ) ∑ L m ⎢ ⎢B ⎥ ⎥ / ∑ LL (θ m ) m=0 ⎣sin(2π m / N ) ⎦ m =0 ⎣ 2⎦

(5.13)

(5.14)

The unit vectors ni and nj can be obtained through ni = cos θ m and n j = sin θ m . Substituting Eq. (5.12) into Eq. (5.8) yields

N (e,θ m ) = e0 (1 + A2 cos 2 θ m + 2 B2 sin θ m cos θ m − A2 sin 2 θ m )

(5.15)

Eq. (5.15) gives an explicit form for the directional distribution of the void ratio. The components of the fabric tensor A2 and B2 represent the deviation from the isotropic distribution of the voids. Therefore, if the voids are isotropically distributed, the

101


components of the void fabric tensor become zero and Eq. (5.15) reduces to the mean value e0. However, it is not easy to choose an appropriate REA. Muhunthan (1993) suggested a circular-type REA could be used, as shown in Fig. 5.13(a). Kuo and Frost (1993) then pointed out that a circular-type REA would lead to a bias fundamentally, while an annular region (as show in Fig. 5.13(b)) between two RECs could efficiently overcome this bias. In this sense, an annular-type REA is chosen in this study, with the radii of the outer and inner circles of 300 and 100 pixels (the image size 1024×819 pixel) respectively. An angle interval of the orientational scanning lines is kept at 10o. The scanning strategy is based on the Digital Differential Analyzer (DDA), a computer graphic terminology. This method is essentially an incremental algorithm, and is schematically illustrated in Fig. 5.14. When x increases one pixel, y will increase one pixel at most. It can be seen that this approach is applicable only when the slope k of the scanning line is not greater than one. If the slope exceeds one, a certain kind of coordinate transformation is required. A MATLAB code was thus written to do the image analysis and compute the directional fractions of the scanning lines occupied by the voids. Fig. 5.15 shows the typical results of the directional distributions of the void ratios for both DD and MT specimens. It is seen that the measured directional void ratios are quite scattered, which is mainly restricted by the specimen size. The directional void ratio can be approximately determined by Eq. (5.15), and the results show that the void ratio distributions are almost isotropic for both DD and MT specimens. Some other investigations are also carried out, and the results sometimes are not very consistent; some results show that the void ratio in horizontal direction is greater than

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that in vertical direction, while some results show oppositely. This is also due to the sample size of the REA. If an REA with the size 600×600 pixel, and a series of scanning lines, the same as those shown in Fig. 5.11, the void ratios in the horizontal and vertical directions can be computed along both the horizontal and vertical directions. Table 5.2 gives the measured void ratios in the horizontal and vertical directions. The result shows that the void ratios in the horizontal and vertical directions are almost the same, implying that the void ratio is completely a scalar quantity, even for an anisotropic packing, just like the DD specimen. The theoretical explanation is that when the scanning line interval becomes very minute, the directional void ratio will approach the globe void ratio e = Av / As . It is thus believed that the void ratio is purely a scalar quantity, and the directional void ratio, which is affected by the sample size in this study and in the literature as well, can never be adopted as a descriptor to quantify the fabric anisotropy of soils.

5.5

Conclusions

This chapter is aiming at quantifying the inherent fabric anisotropy for the laboratory prepared specimens using different sample preparation methods. To this end, a complete experimental procedure was developed to investigate the fabric anisotropy of the granular soils. Some anisotropic indices severing as the descriptors of the fabric anisotropy were introduced and reappraised. The fabric tensors were obtained in terms of the anisotropic indices, collecting the directional data from the observations in a microscopic level and reflecting the anisotropic property of the soils. Some conclusions can be summarized as follows.

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1. Impregnation with the epoxy can preserve the initial fabric of the granular materials efficiently. Using a routine procedure, including sectioning, grinding and polishing, a high quality image at the desired and interested location can be captured by using an SEM, and a binary image can thus be obtained for later image processing and analysis. 2. The particle orientation is of great importance in forming certain structure of the granular soils in a microscopic scale. Statistical analysis showed that the DD specimen had a strong tendency that the particles oriented in the horizontal direction, while the MT specimen had no obviously preferential orientation. It was further believed that the MT specimen had a more isotropic structure than the DD one. The magnitude vector ∆ , representing the intensity of the particle orientations, was introduced and successfully used to characterize the fabric anisotropies for both the DD and MT specimens. 3. The mean free path, a mean edge-to-edge distance for the particles aggregate, is also another good descriptor to characterize the inherent fabric anisotropy. It was found that for the DD specimen, the mean free path in the horizontal direction was greater than that in the vertical direction, but found to be almost the same for the MT specimen. It seems that the mean free path primarily reflects the influence of the particle orientation, but it is expressed in terms of the void phase in the specimen. 4. The directional void ratio concept, which was widely used in the literature in describing the fabric anisotropy for granular materials and also for clayey soils, was found essentially an isotropic quantity and could never be used to quantify and characterize the anisotropy of the soils. This argument was also justified by the experimental results.

104


Fig. 5.1 Epoxy impregnation system

100 90 80 70 % Finer

60 50 D50=0.23mm 40 30 20 10 0 0.01

0.1 Particle size (mm)

Fig. 5.2 Grain size distribution of Toyoura sand

105

1


(a) Dry deposition

(b) Moist tamping

Fig. 5.3 Methods of sample preparation (after Ishihara 1996)

Fig. 5.4 Drying system for moist tamped specimen

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Epoxy matrix

Sandy particle

Lost particle

Air bubble

Fig. 5.5 A typical SEM image at the magnification of 200

(a) SEM microphotograph

(b) B/W mask for image processing

Fig. 5.6 Typical image taken by SEM

107


V Kth particle

V

n

ϕk -n

H1

H1

H2

Fig. 5.7 Measure of particle orientation

Fig. 5.8 An example of numbering particles through image processing

108



10

90 10

120

60

8

8

6

150

30

Percentage(%)

4

6 2 0 180

4

0

2 4

2

210

6

330

8

0 -90

-60

-30

0

30

60

90

240

10


300 270

(a) Histogram


Fig. 5.9 Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations for vertical sections 90


8

8

120

60

6

Percentage(%)

6

4

150

30

2

4 0 180

0

2

2 4

210

330

6

0 -90

-60

-30

0

30

60

90

8

240

300 270


(a) Histogram


Fig. 5.10 Characterization of inherent fabric anisotropy of Toyoura sand with preferred particle orientations for horizontal sections

(a) Horizontal direction

(b) Vertical direction

Fig. 5.11 A set of parallel scanning lines

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(a) Horizontal free path

(b) Vertical free path

Fig. 5.12 Free path resulting from a set of parallel scanning lines

(a) Circular REA

(b) Annular REA

Fig. 5.13 Schematic REAs with scanning lines

(xi+1, Round(yi+k))

(xi, yi)

Scanning Line (xi+1, yi+k)

(xi, Round(yi))

Fig. 5.14 Schematic illumination of the method DDA

110


0 2.5

330

measured data directional approximation 30 isotropic approximation

300

1.5

60

60

300

0.5

0.5 0.0 270

0.0 270

90

90

0.5

0.5

1.0

1.0 240

120

1.5

240

120

2.0

2.0 2.5

30

1.0

1.0

1.5

330

2.0

2.0 1.5

0 2.5

210

150

2.5

210

150 180

180

(a) DD specimen

(b) MT specimen Fig. 5.15 Directional void ratio

111

Chapter 5 Quantifying Fabric Anisotropy — Using an Image Analysis Approach Table 5.1 Summary of the results of the magnitude vector ∆ Sample ID

Plane

Number of particles

Magnitude vector ∆

DD-sample3

Vertical

1754

0.218

DD-sample7

Vertical

1015

0.222

DD-sample2

Vertical

601

0.203

DD-sample6

Horizontal

1193

0.029

MT-sample1

Vertical

1636

0.076

MT-sample2

Vertical

1438

0.113

MT-sample3

Vertical

1100

0.083

MT-sample6

Horizontal

826

0.052

Average value

0.214

0.091

Table 5.2 Summary of the results for the mean free path and directional void ratio Aspect ratio (Horizontal/Vertical) Sample ID

Plane

Number of particles Mean free path

Void ratio

DD-sample3

Vertical

1754

1.096

0.996

DD-sample7

Vertical

1015

1.073

0.990

DD-sample2

Vertical

601

1.135

0.990

DD-sample6

Horizontal

1193

0.974

1.003

MT-sample1

Vertical

1636

1.008

0.998

MT-sample2

Vertical

1438

1.058

0.994

MT-sample3

Vertical

1100

0.994

0.999

MT-sample6

Horizontal

826

1.042

1.046

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CHAPTER 6 INVESTIGATION ON MECHANICAL BEHAVIOR OF GRANULAR MATERIAL

6.1

Introduction Laboratory investigations have shown that the mechanical behavior of soil is

very complex and strongly influenced by its density, confining pressure, drainage condition, loading path, and so on. In recent years, efforts have been made to identify the incidence of fabric anisotropy in soils. In the last chapter, an experimental procedure using an image analysis approach to quantify the fabric anisotropy of granular soils has been presented. Particularly, inherent fabric anisotropies of the dry deposited (DD) and moist tamped (MT) specimens, using Toyoura sand, were successfully studied and quantitatively evaluated. The further effort should also be made to explore the relations between the fabric anisotropy and the mechanical behavior of granular soil. To this end, a series of conventional laboratory experimental tests will be carried out to investigate the mechanical behavior of Toyoura sand and discussed in this chapter. The tests will be performed under various loading conditions, using both DD and MT prepared specimens with different densities and effective mean normal stresses. It will be identified that variations in the stress-strain response among differently prepared specimens, under otherwise identical conditions, should be attributed to the soil fabric anisotropy of. Discussions on the uniqueness of the

113


critical state lines (CSLs) in the e ~ p ' plane will be also made according to experimental findings.

6.2

Test Apparatus All the triaxial tests presented in this study were carried out by an automatic

CKC triaxial system (Li et al. 1988), while the torsional shear tests were conducted using a hollow cylindrical torsional apparatus, the same apparatus for rotational shear tests, which has been introduced in Chapter 3.

6.2.1 CKC Triaxial System The CKC triaxial system was developed for both static and cyclic loading tests. The profile of this system is shown in Fig. 6.1, and is mainly composed of a loading frame (with a loading piston mounted on the top), a triaxial cell, a controllable pneumatic loading unit, a signal conditioner, a process interface unit, as well as a computer, integrated with software for the system control and the data collection. For conventional triaxial tests, five transducers are employed, as follows: 1. Load cell: to measure the axial load 2. LVDT: to monitor the axial displacement 3. Cell pressure transducer 4. Effective pressure transducer 5. Volume change measuring device

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The whole test process is controlled and monitored by a computer software with a user-friendly interface. For the current version of this system, the following functions can be fulfilled automatically: •

Back pressure saturation/B-value check

•

Consolidation

•

Static test

•

Cyclic test

•

K0-line consolidation

•

Dynamic test

•

Constant p test

•

Creep and relaxation

•

Arbitrary stress path test

The pneumatic loading unit is used for regulating the cell pressure and the pressure in the piston for controlling the axial loading. By adjusting the pressure difference between the upper and lower chambers in the piston, either compression or extension loading can be applied to the soil specimen. The system is controlled by a closed-loop feedback scheme, which is capable of performing tests either under strain or stress control mode, depending on the usage in the triaxial stress space. In this study, all triaxial compression and extension tests, as well as triaxial cyclic tests were performed in the CKC system.

6.2.2 Lubricated Ends As discussed by Verdugo and Ishihara (1996), the use of lubricated ends as well as enlarged end plates can not only effectively minimize non-homogeneities in the strain distribution throughout the samples, but also avoid the development of 115


shear bands. In this study, the lubricated ends were used for all the triaxial tests. A lubricated end was formed by two layers of latex membranes, and positioned between the end platen and the specimen. Interfaces between the membranes and the end platen were lubricated with a thin layer of silicone grease (Dow Corning high vacuum grease). The enlarged end plates were used to make certain that the soil specimen was always sitting within the end platen, even when having undergone a large deformation. A detailed view of the lubricated end is shown in Fig. 6.2.

6.3

Test Procedure The following steps were involved in each test, including: •

Sample preparation

•

Back pressure saturation and B-value checking

•

Consolidation

•

Shearing

6.3.1 Sample Preparation All the tests were performed using Japanese standard sand, Toyoura sand. It is generally known that tamping and pluviation are two common sample preparation techniques for granular soils (Ishihara 1996). The basic requirements for either method are (1) the specimen should be as uniform as possible, and can be easily obtained, and (2) a wide range of densities is attainable by the same method. It is known that different specimen preparation methods can create different initial fabrics, which will thus influence the stress-strain responses to the applied shear

116


loadings. In this study, dry deposition and moist tamping techniques were employed to perform the physical tests in the investigation. When the specimen was enclosed by the membrane and O-rings, the top drainage line was then connected to a vacuum source. A very small vacuum pressure, usually around 5kPa, was applied through the top drainage line to hold the specimen, such that the split mould could be removed. After de-airing the specimen, sample preparation was followed by a saturation process. Carbon dioxide (CO2) was allowed to percolate through the specimen from the bottom to the top. Normally this process took about half an hour. After most of the air in the specimen was removed and replaced by the carbon dioxide, a vacuum of 15kPa was applied through the upper drainage line of this setup. De-aired water was then circulated through the specimen from the bottom drainage line to the top. This process should be performed very slowly; otherwise the air bubbles could not be carried out with the water and would be easily trapped inside the specimen. The saturation process was the key step for obtaining a high quality result and required extensive experience and care. The degree of saturation of a specimen not only influenced the effective stress measured, but also affected the void ratio evaluated. The water circulation process normally lasted for 4-6 hours. After the saturation process, the diameter and height of the specimen could be measured. The diameter of the specimen was measured by a perimeter-tape with an accuracy of 0.01mm. The diameters at the top, in the middle and at the bottom were measured respectively and a mean value was taken. The nominal diameter of the specimen could be obtained by subtracting the membrane thickness from the mean diameter, which was measured in advance. Before the sample preparation, a metal

117


dummy specimen with a known height was placed inside the triaxial cell as a reference, and the offset height of the loading rod exposed to the cell was measured by a vernier caliper with an accuracy of 0.01mm. Once the specimen was prepared, the offset height could be measured again. The nominal specimen height was then evaluated by comparing the difference of the offset height measurements. Both the nominal diameter and height of the specimen were taken as the initial dimensions of the specimen. Typically, a solid specimen for the triaxial test had dimensions of 70mm in diameter and 140mm in height. The triaxial cell was then set up on the loading frame. An initial cell and back pressure of 50kPa and 20kPa respectively were applied to the soil specimen, such that an initial effective pressure of 30kPa was maintained.

6.3.2 Back Pressure Saturation and B-Value Check The principle of the back pressure saturation was to dissolve gas (CO2 and air) bubbles inside the voids of a soil specimen by applying back pressure, such that the degree of saturation could be increased. Simultaneously, the cell pressure should also be increased by the same amount to maintain the constant effective stress and deviatoric stress. The Skempton’s B-value was then checked and served as an indicator of the degree of saturation. Usually an increment of 50kPa in cell pressure was used for the checking, which was performed under the undrained condition. The B-value should be equal to or greater than 0.97 before moving to the consolidation stage. Normally a 200kPa back pressure was enough for soil specimen to reach an acceptable B-value, while a larger back pressure, sometimes, was more preferred to prevent cavitation

118


under the undrained condition, when measuring the higher negative pore water pressures, if the soil is highly dilative.

6.3.3 Isotropic Consolidation Once the specimen had been fully saturated, an isotopic consolidation process then followed. The consolidating pressure was chosen as the starting point of the deviatoric shearing. Normally this process was divided into several loading segments, at 20kPa per segment, and each took about 10 to 20 minutes. Meanwhile, the volume change associated with the consolidation process was recorded for evaluating the initial void ratio before the shearing tests.

6.3.4 Shearing Once the specimen was consolidated to the desired stress state, the shearing process was carried out. In this study, soil specimens, prepared by both dry deposition and moist tamping methods, were sheared under various loading paths, including monotonic compression tests, monotonic extension tests, monotonic torsional shear tests, as well as cyclic loading tests.

6.3.5 Evaluation of Initial Void Ratio In general, there are two alternative methods for evaluating the void ratio of a specimen. The first method uses the dimensions of a specimen, and corresponds to a simple procedure based on the direct evaluation of the height and the diameter of a specimen after saturation. The void ratio ei can be calculated from the water density

119


ρ w , the measured initial volume Va, as well as the dry mass and the specific gravity Gs of the soil, based on Eq. (6.1).

ei =

Vs −1 M s /( ρ wGs )

(6.1)

Consequently, the initial void ratio ei before the deviatoric shearing should be updated according to the volume change during the consolidation process. The second method uses the water content of a specimen, and is developed, considering that for a saturated specimen, the void ratio is directly related with the amount of water in the specimen. Based on Eq. (6.2), the final void ratio ef can be evaluated from the water content wf, measured at the end of the test, by assuming that the specimen is fully saturated.

e f = w f Gs

(6.2)

If the final void ratio ef is known, the initial void ratio ei can be thus back-calculated from the volume change (if any) during the deviatoric shearing. The first method is basically the commonly used practice to evaluate the void ratio of a specimen, while the second one is an alternative procedure. According to a study by Verdugo and Ishihara (1996), the scattering in the void ratio evaluated using the second method was much smaller than that obtained by the first one. In addition, the results using these two procedures consistently indicated that the void ratios measured by the second method were smaller than those computed from the first one. The discrepancy between these two methods may arise from the various steps wherein the triaxial cell is assembled, and most likely, some additional volume change takes place, both of which cannot be compensated for by the first method. Hence, the procedure based on the measurement of the water content at the end of

120


the test has been adopted as a more reliable method to evaluate the void ratio. The void ratios estimated by this method were used and reported in this study.

6.4

Data Reduction

6.4.1 Area Correction

During the consolidation process, the volume of a specimen changes. Therefore, both the height and the cross-sectional area of the specimen are also changed. Similar to other conventional triaxial test systems, the axial and volumetric changes of the specimen are recorded during a test. The lateral displacement of the specimen is calculated from these two measurements. Consider a specimen with initial diameter D0, height H0, volume V0, and cross sectional area A0, and ∆H and ∆V are the measured changes in height and volume respectively. The average cross

section area Aavg can be calculated from the following equation.

Aavg ⋅ ( H 0 + ∆H ) = V0 + ∆V

(6.3)

Therefore,

Aavg =

1 + ∆V / V0 1 − ε v = ⋅ A0 1 + ∆H / H 0 1 − ε a

(6.4)

in which ε v and ε a are the volumetric and axial strains respectively, with the sign convention in soil mechanics being followed. Eq. (6.4) is adopted to account for the area correction during the triaxial tests. Usually in the consolidation process, the specimen is uniformly deformed and the area correction adopted here is reliable for evaluating the cross sectional area of the specimen. However in the triaxial compression tests, a bulging type deformation shape occurs at the large strain level, while in the triaxial extension tests, necking deformation can also be observed when

121


the strain exceeds a certain strain level. In this study, the average method is used to represent the stress state during the shearing process, for convenience. It is also noted that for undrained tests, as the volume of the specimen is assumed constant (if the specimen is fully saturated) during the shearing, Eq. (6.4) reduces

Aavg =

1 ⋅ A0 1− εa

(6.5)

The following equations are used to calculate the stresses and strains: effective mean normal stress: p ' = (σ 1' + σ 2 ' + σ 3' )

(6.6)

deviatoric stress: q = 1/ 2{(σ 1' − σ 2 ' ) 2 + (σ 2 ' − σ 3' ) 2 + (σ 3' − σ 3' ) 2 }

(6.7)

pore water pressure u = p − p '

(6.8)

deviatoric strain: ε q = 2 / 9{(ε1 − ε 3 ) 2 + (ε1 − ε 3 ) 2 + (ε1 − ε 3 ) 2 }

(6.9)

6.4.2 Data Filtering

The raw data directly collected during the tests inevitably include some noise. A smoothing process is employed in the data reduction. This method is based on the modern digital signal processing theory, choosing an appropriate window type and a cut-off frequency to filter out the noise.

6.5

Test Results from Monotonic Loading

6.5.1 Triaxial Compression Test

This series of tests were performed with initial effective mean normal stresses (confining pressures) 100kPa, 200kPa, and 400kPa. Both the dry deposited and moist tamped specimens were prepared with almost the same initial void ratio after the

122


isotropic consolidation. There are two different initial densities being utilized in this study for the triaxial tests, Dr=30% (e=0.863) and Dr=41% (e=0.821). Fig. 6.3 and Fig. 6.4 show the triaxial compression test results of the stress-strain curves and the corresponding effective stress paths for both DD and MT specimens, with initial density Dr=30%, respectively, and Fig. 6.9 and Fig. 6.10 show the results with

Dr=41%. It is noted that under the testing conditions, all the specimens exhibit contractive response firstly, and then dilate to large shear strain. All the specimens are not sheared to the critical state, because the bulging (barrel shape) of specimens is observed at large deformations, such that the data interpretation will be very difficult. Nevertheless, a prominent distinction can be observed at the early stage of the shearing. It can be seen that all the specimens behave with a strain hardening response, but for the dry deposited (DD) specimen, the contraction at the initial stage is much more than that for the moist tamped (MT) specimen; in other words, more pore water pressure is generated for the DD specimen than that for the MT specimen. The MT specimen seems to give a more dilative and stiffer response, especially for the initial confining pressures of 100kPa and 200kPa. It is also found that the deviatoric stress is developed much faster for the MT specimen than that for the DD one, at the initial shearing stage, with the same initial confining pressure.

6.5.2 Triaxial Extension Test

Triaxial extension tests were also conducted on both DD and MT specimens, with initial effective mean normal stresses 100kPa, 200kPa, and 400kPa, and almost identical void ratios, referred to the stage after consolidation. Because of the influence caused by the end friction, although much reduced by the lubricated ends, the non-uniform deformation and necking phenomenon of the specimen inevitably

123


occurred around 8 percent of deviatoric shear strain. Fig. 6.5 and Fig. 6.6 show the stress-strain relations and the corresponding effective stress paths in the p ' − q plane for both the DD and MT specimens with initial density Dr=30%, respectively. The responses of initial density Dr=41% are shown in Fig. 6.11 and Fig. 6.12. It is seen that all the DD specimens, despite initial confining pressures, exhibit strain softening behavior, especially when the initial density is Dr=30%, and liquefaction takes place at large shear strains. While the MT specimens show limited unstable response, and the shear stresses approach peak values quickly, then drop slightly, followed by continuous increasing up to the stage when necking occurring. The discrepancies in terms of the strength development between the DD and MT specimens are prominent and remarkable. It is also seen that the generated excess pore water pressures for DD specimens are much larger than those for the MTs. The observed different response is much more significant in the extension mode than in compression, and the response in the extension mode seems to be more affected by the fabric anisotropy (associated with sample preparation method) than the response in compression.

6.5.3 Torsional Shear Test

Torsional shear tests were performed using a hollow cylindrical torsional shear apparatus. The initial confining pressures of 100kPa, 200kPa and 400kpa were chosen in this study, with the specimen density Dr=30%. Due to the large shear strain that the specimens experienced, the shear stress was corrected from the load carried by membranes. Fig. 6.7 and Fig. 6.8 give the deviatoric stress-strain responses and the effective stress paths for the DD and MT specimens respectively. It should be noted that the DD specimens show strain softening responses, while the MT specimens exhibit strain hardening characteristics. It is also observed that the DD 124


specimens show complete contraction with very small shear strength at the large deformations, while the MT specimens under identical conditions exhibit a contractive behavior at first, then the dilative response follows, after passing the phase transformation state. The strength of the MT specimens is much larger than that of the DD specimens, indicating that the DD specimens are weaker than the MTs when subjected to torsional loading, and also means that MT specimens are much easier to generate pore water pressures than DDs.

6.6

Discussion on Influence of Fabric Anisotropy

Monotonic undrained triaxial compression, triaxial extension as well as torsional shear tests were performed using Toyoura sand specimens, which were prepared by both the DD and MT methods, with almost identical initial void ratio, while varying the initial effective confining pressure. As shown in Chapter 5, different sample preparation methods yield various initial fabrics. The MT specimen and DD specimen have different inherent/initial fabrics, which, definitely, lead to significant variations in the mechanical behavior, as well as in the dilatancy characteristics of the sand when subjected to shear loading. It is seen that the influence of the fabric anisotropy is very significant, sometimes even dramatic under certain conditions. For instance, as shown previously, the DD specimen under triaxial compression load contracts to a full liquefaction with almost zero strength at the large deformation, while the MT specimen exhibits a limited liquefaction firstly, and then dilates after passing the phase transformation state. It seems that the MT specimen is much stronger and more stable under the shearing load than the DD specimen. The findings are also consistent with the recent work done by Papadimitriou et al. (2003).

125


Compared with the triaxial compression, the sand behavior under the triaxial extension condition is much more contractive and softer, with the torsional shear in between, which is in agreement with data reported in the literature (Riemer and Seed 1997, Yoshimine et al. 1998, Gennaro et al. 2004). The possible explanation of this prominent difference could be given in terms of the fabric anisotropy and also the loading path. It is also observed that the sand response in both the triaxial extension and the torsional shear modes seem to be much more affected by the specimen preparation method than the response in the triaxial compression mode. The contractive response and softening behavior are observed in all the tests performed using the DD specimens under either the triaxial extension or the torsional shear mode. While for the MT specimens, the dilative behavior always prevails when the shear strain exceeds a certain level. As noted by Chen and Chuang (2001), the MT specimen is more isotropic than the specimen prepared by either the dry tamping or the multiple sieving pluviation technique. The reasoning of this postulation lies in: the steady state line (critical state line) determined by the triaxial extension tests lies to the left of that by the compression tests in the void ratio~effective stress ( e ~ ln p ' ) plane; and the differences between compression and extension lines for the dry tamped and pluviated samples are much larger than for the MT specimens. According to the observations from the microstructure analysis on the thin sections of the DD and MT specimens, it is also confirmed that the MT specimen is much more isotropic than the DD specimen, in terms of either the solid phase—particle preferred orientations, or the void phase—directional mean free path. As compared with the experimental results of the variations in the sand behavior, due to the shear modes, in the mechanical tests, it seems that the MT specimens with various initial effective

126


confining pressures are contractive at first, and then dilate under all the shear modes. Conversely, the DD specimens in the triaxial compression mode are always contractive and then dilative towards the large shear deformation, while in the triaxial extension or the torsional shear mode, the specimens are very much more likely to contract. In the p ' ~ q diagram, in comparing the triaxial compression and extension modes, the MT specimens behave in a more “symmetrical” manner than the DD specimens. If an isotropic specimen is sheared under both the triaxial compression and extension modes, the responses will be somewhat “symmetrical”. Although the effective stress paths for the triaxial compression and extension will not be absolutely identical in the p ' ~ q plane, and are also influenced by the stress state or the intermediate principal stress parameter b [= (σ 2 ' − σ 3' ) /(σ 1' − σ 3' ) ], this observation still can support the idea of the different anisotropy degrees of the MT and DD specimens.

6.7

Discussion on Critical State Lines

The steady state (Poulos 1981) is a state in which the soil mass is continuously deforming at constant volume, constant effective normal stress, constant shear stress, and constant velocity. The steady state was originally used to describe a flow (liquefaction) state under the undrained condition for the contractive soils. Both the steady state and the critical state characterize the continuous deformation at constant volume and stress, and are almost referred to the same state (Been et al.1991, Chu 1995). In this investigation, the critical state is used to depict the ultimate state of sands.

127


The concept of the critical state line as an intrinsic material property has significant applications in engineering practice (e.g. Been and Jefferies 1985, Poulos

et al. 1985). However, the uniqueness of the critical state line for a given soil is still a controversial topic and gives rise to a lot of debate. Some experimental observations show that the uniqueness of the critical state line is questionable. Through a special biaxial compression test on a bidimensional packing of cylinders, Chapuis and Soulié (1981) found that internal structure of granular materials might influence the critical state void ratio, as shown in Fig. 6.13. During the first loading (OA), a dense behavior is observed. After reaching the constant volume stage of deformation, the specimen is unloaded while σ H = σ 2 remains constant. Then, a sharp bend is observed on the curve (BC), when there is an inversion of the major and minor stresses; no peculiar behavior is noticed on the

e~ ε v curve during the stress inversion. After the unloading phase, the specimen is loaded again and a loose behavior is registered (CD). It is surprising that the void ratio during the second stage is quite different from the one obtained from the first loading. This observation invalidates the well-accepted concept that the void ratio at the critical state is solely a function of the mean normal effective stress, and gives evidence that it also depends on the initial state of structural anisotropy (inherent fabric anisotropy). Vaid et al. (1990) studied the influence of the stress path on the critical state. Results from undrained triaxial compression and extension tests on water-deposited sands show that the steady state line of a given sand, though unique in the effective stress plane, is not so in the void ratio-effective stress ( e ~ ln p ' ) plane. Some other similar findings have been reported by Vaid and Tomas 1995, Riemer and Seed 1997, Yoshimine et al. 1999, Chen and Chuang 2001.

128


All these observations and findings implied that the critical state line of the granular soil could be non-unique. This kind of non-uniqueness may be due to the anisotropic fabric, because all the soil structures are formed under a gravitational field and thus possess a certain fabric. It is also due to the loading path (stress anisotropy) imposed on the soil specimen. Particularly, Nakata et al. (1998) and Yoshimine et al. (1998) performed a series of hollow cylinder undrained torsional tests on Toyoura sand. By varying the principal stress directions with respect to the deposition direction of the sand and the ratios among the principal stresses, the test results showed that the soil responses were significantly affected by the loading path, as well as the principal stress directions relative to the fabric coordinates of the soil specimens. Through extensive test data obtained from triaxial compression, triaxial extension, as well as the simple shear in a hollow cylindrical space, Yoshimine and Ishihara (1998) found that the ultimate steady-state line for the simple shear was located between the lines for the triaxial compression and extension, as can be seen in Fig. 6.14, where the ultimate steady-state lines were evaluated from the upper limit of the phase transformation lines. According to the literature, it is well recognized that the critical state lines vary in different shear modes, while how the fabric affects the critical state for a given material is still unclear. In the following part, the critical state from the triaxial tests of the dry deposited (DD) as well as the moist tamped (MT) specimens, including both the compression and extension, will be investigated. It is worth noting that it is very difficult to achieve real critical states under triaxial conditions, especially for triaxial extension, because of the emergence of necking and the formation of the shear band. It is also found that in the triaxial compression tests, compared with the DD specimens, the MT specimens are more

129


likely to reach the critical states. However, for most triaxial compression tests presented previously, the final stages do not meet the requirements for the critical states/steady states, even if the shear strains exceed 20%. The deformation of a specimen beyond this stage will be apparently be non-uniform (bulging shape) due to the boundary conditions (caused by the end frictions) and the specimen itself (nonhomogeneous specimen prepared), thus the data interpretation will become very complex. It can be seen that the mean normal stress and the deviatoric stress still change beyond this stage, but the increments are relatively small and the changing rates also decrease. Therefore, a pseudo critical state for the triaxial compression tests is defined at a shear strain around 20%. However, for the extension tests, because necking and shear band are inevitably formed around 10% of the axial strain, especially for the MT specimens, the real critical state cannot be achieved. Thus, the final state (the stage of the emergence of necking/shear band) will be treated as the pseudo critical state for the triaxial extension tests. Fig. 6.15 shows the pseudo critical states evaluated from those tests presented in the preceding part, together with some extra tests for the sake of integrality. In this figure, the critical state lines are represented in a plane of the void ratio versus the logarithm of the effective confining pressure ( e ~ log p ' ). The critical state lines are evaluated from the least squares fitting. In the triaxial compression tests, the critical state lines for the DD and MT specimens are very close to each other, which is in agreement with the result reported by Ishihara (1993). As observed at the final state of the shearing, the MT specimens are sheared much closer to the critical state, compared with DDs, and it is believed that the real critical state line for the DD specimen will be expected to be located slightly above that for the MT one. While for the triaxial extension tests, the states at 6% of the deviatoric strain and before the

130


necking emergence are plotted respectively for both the DD and MT specimens. The difference between the 6% deviatoric strain state lines of DD and MT specimens is apparent, so is the difference for the necking state lines. Though either the 6% shear strain state or the necking state is not the real ultimate state, the critical state lines for the triaxial extension are still expected to lie beneath those for the triaxial compression. It is also noticed that the difference between the compression and extension lines for the DD specimen is much larger than that for MT. This observation also implies that the specimen prepared by the moist tamping technique is more isotropic than by the dry deposition technique. As discussed before, the fabric can be represented by a spatial arrangement of particles, which can be characterized by the distribution of preferential orientations (orientations of the long axes) of particles. A specimen prepared by pouring particles under gravitational force (the dry deposition method) has contact normals preferred in the vertical direction (the direction of deposition), with elongated particles concentrated in the horizontal plane (the bedding plane). On the contrary, the specimen made by the moist tamping method has more isotropic and uniform distribution of the particle orientations. Thus the mechanical behavior of the MT specimen has more isotropic/symmetrical response in comparing the triaxial compression and extension to that of the DD. The critical compression and extension state lines are also expected to be distributed in a more isotropic manner for MT specimens than for DDs. It is concluded that besides the loading path, the inherent (initial) fabric anisotropy of granular soil, arising from the different sample preparation methods, also has a great influence on the location of critical state lines in the e ~ p ' plane, and thus makes the critical state lines nonunique.

131


6.8

Test Results from Cyclic Loading

It has been shown that the inherent fabric anisotropy, originating from the different sample preparation methods, is of great impact on the mechanical behavior of granular soil, when subjected to the monotonic loadings. Especially, the influence of the inherent anisotropy is found to be significant under the triaxial extension and torsion shear loading conditions. The extent of this influence may lie in the stressstrain relations, the pore water pressure generation forms, the dilatancy characteristics, as well as the uniqueness of the critical state lines. However, in the seismic analysis for the liquefiable soils, attention has been drawn to the identification of the incidence of the fabric anisotropy on the liquefaction susceptibility. Thus, investigations on the liquefaction potential of the granular soils with different inherent fabrics, will be fulfilled by means of the triaxial cyclic loading tests. There are extensive and comprehensive laboratory studies on the subject of the liquefaction potential of saturated sands undergoing cyclic loading. The conclusions of previous studies, as reported by Yoshimi et al. (1977), Seed (1979), and Finn (1981), Ishihara (1996), and so on, have generally confirmed that the resistance to liquefaction of clean sand specimens reconstituted in the laboratory is influenced primarily by factors such as the initial confining stress, the intensity of shaking as represented by the cyclic shear stress, the number of the cyclic stress applications and the void ratio or relative density. In view of the diversity of the mechanical behaviors due to the different sample preparation methods, as reported previously, it should be recognized that granular soil might exhibit varying resistance to the application of cyclic loadings. Mulilis et al. (1977) showed that specimens of Monterey No. 0 sand prepared by air pluviation and moist tamping had the lowest

132


and greatest resistance to the liquefaction among several methods employed in their study, as shown in Fig. 6.16. It can be seen that there is a fairly large range in which the cyclic resistance of sand can vary, depending on the nature of the fabric structures created by different methods of specimen preparation. Thus it is realized that the fabric anisotropy is also another influential factor, governing the liquefaction resistance to cyclic loadings. To explore the inherent anisotropy effects on the liquefaction resistance and cyclic undrained strength, a series of cyclic triaxial tests were performed on both DD and MT specimens with various shear stress amplitudes. The conditions of the cyclic triaxial tests performed in this study are summarized in Table 6.1, in which the stress ratio is defined as σ d / 2 po ' ( σ d is the amplitude of the cyclic deviatoric stress and

po ' is the confining pressure). Fig. 6.17 to Fig. 6.25 show resultant responses under the cyclic loading with almost the same density Dr=30% and with the same initial confining pressure po ' =100kPa, but with different preparation methods, DD and MT. In these figures, the onset of the initial liquefaction is defined when the axial strain exceeds 5% (Ishihara 1996). It is seen that when the specimens are subjected to cyclic loading, excess pore water pressures are gradually built up and the soils are weakened, tending towards liquefaction. However, the forms of the generation of pore water pressure are varied in different sample preparation methods, and sometimes are very dramatic. It is observed that the DD specimen is much weaker to cyclic loading, and the pore water pressure is built up much faster than the MT specimen. Fig. 6.26 shows a comparison of the liquefaction resistance for both DD and MT specimens with Dr=30% and po ' =100kPa. It is shown that the liquefaction resistance of MT specimens is much larger than that of DD specimen. This

133


observation agrees well with those reported in the literature (Mulilis et al. 1977, Yoshimi et al. 1984).

6.9

Closing Remarks

In this chapter, a series of undrained conventional laboratory tests under various loading conditions, including triaxial compression, extension, cyclic and torsional shear tests, using laboratory reconstituted specimens prepared by both DD and MT techniques are reported. Specimens at different densities and consolidating confining pressures were employed to investigate the effects of the initial fabric on the mechanical behavior of granular soil under various shear modes, especially on the critical state lines in the e ~ log p ' plane. Some conclusions can be drawn as follows. 1. The MT specimen seems to give a more dilative and stiffer response than the DD one, irrespective of shear modes, at almost identical void ratio and the same confining pressure. Particularly, comparing the triaxial compression and extension, the response for the MT specimen behaves in a more symmetrical manner than that for the DD specimen, indicating the MT specimen is more isotropic than the DD. 2. It is observed that the response in triaxial extension and torsional shear is much more affected by the sample preparation methods (resulting in different inherent fabric anisotropies) than that in triaxial compression. 3. According to the results from the triaxial cyclic tests, it is found that the MT specimen has stronger liquefaction resistance than the DD one, which is consistent with reports in the literature. Thus it is recognized that the inherent

134


fabric anisotropy is also an influential factor affecting the liquefaction potential of granular soils when subjected to seismic loadings. 4. Both the sample preparation method and the loading path have significant influence on the critical state line. For the same sample preparation, the extension critical state line is always located on the left side of the compression one, with torsional shear in between. The CSL for the DD specimen in compression is very close to that for MT one, while the CSL for the DD specimen in extension is different from that for the MT specimen. The difference between the extension and compression lines is greater for DDs than MTs, implying that the MT specimen possesses more isotropic fabric.

135


Fig. 6.1 CKC triaxial testing system

Fig. 6.2 Lubricated-end used in triaxial test

136


1000 Toyoura sand (Dr=30%) Dry Deposition Triaxial Compression

800

q (kPa)

Test result

600

p'=400kPa

400 p'=200kPa 200

p'=100kPa

e=0.862

e=0.864

e=0.864 0 0

5

10

εq (%)

15

20

600

800


800

q (kPa)

Test result 600

400

200

0 0

200

400 p' (kPa)

Fig. 6.3 Triaxial compression test (DD specimen, Dr=30%)

137


1000

Toyoura sand (Dr=30%) Moist Tamping Triaxial Compression

800

q (kPa)

Test result 600 p'=400kPa e=0.863

400

p'=200kPa e=0.866

200

p'=100kPa e=0.868

0 0

1000

10

15

20

400 p' (kPa)

600

800

εq (%)


800

q (kPa)

5

Test result

600 400 200 0 0

200

Fig. 6.4 Triaxial compression test (MT specimen, Dr=30%)

138


100 Toyoura sand (Dr=30%) Dry Deposition Triaxial Extension

q (kPa)

Test result

50

p'=100kPa

p'=200kPa

e=0.861

e=0.870

p'=400kPa

e=0.869

0 0

2

4

εq (%)

6

8

10

300

400

500

100

q (kPa)

Toyoura sand (Dr=30%) Dry Deposition Triaxial Extension Test result

50

0 0

100

200 p' (kPa)

Fig. 6.5 Triaxial extension test (DD specimen, Dr=30%)

139


300 Toyoura sand (Dr=30%) Moist Tamping Triaxial Extension Test result

200

e=0.867

p'=200kPa p'=100kPa e=0.863

q (kPa)

p'=400kPa e=0.864

100

0 0

2

4

6

εq (%)

8

300 Toyoura sand (Dr=30%) Moist Tamping Triaxial Extension Test result

q (kPa)

200

100

0 0

100

200

300

400

p' (kPa) Fig. 6.6 Triaxial extension test (MT specimen, Dr =30%)

140

500


300

Toyoura sand (Dr=30%) Dry Deposition Torsional Shear Test result

200

p'=400kPa

q (kPa)

p'=200kPa p'=100kPa e=0.863

e=0.857

e=0.863

100

0 0

300

2

4

6

εq (%)

8

Toyoura sand (Dr=30%) Dry Deposition Torsional Shear Test result

q (kPa)

200

100

0 0

100

200

300

400

p' (kPa) Fig. 6.7 Torsional shear test (DD specimen, Dr =30%)

141

500


400 Toyoura sand (Dr=30%) Moist Tamping Torsional Shear

q (kPa)

300

Test result

p'=200kPa e=0.859

p'=100kPa e=0.858

200

p'=400kPa

e=0.863

100

0 0

3

6

εq (%)

9

400 Toyoura sand (Dr=30%) Moist Tamping Torsional Shear

300

q (kPa)

Test result

200

100

0 0

100

200

300 p' (kPa)

Fig. 6.8 Torsional shear test (MT, Dr =30%)

142

400

500



1400 1200

Test result

q (kPa)

1000 800

p'=400kPa e=0.819 p'=200kPa

600 400

p'=100kPa

e=0.819

e=0.825

200 0 0

5

10

15

εq (%)

20


1400 1200

Test result q (kPa)

1000 800 600 400 200 0 0

200

400

600

800

p' (kPa) Fig. 6.9Triaxial compression test (DD, Dr =41%)

143

1000


1600 1400


1200

Test result

q (kPa)

1000 p'=400kPa 800 p'=200kPa 600

e=0.819

e=0.819

400

p'=100kPa e=0.825

200 0 0

5

10 εq (%)

15

20

1600 Toyoura sand (Dr=41%) Moist Tamping Triaxial Compression

1400 1200

Test result

q (kPa)

1000 800 600 400 200 0 0

200

400

600

800

p' (kPa) Fig. 6.10 Triaxial compression test (MT, Dr =41%)

144

1000


300 Toyoura sand (Dr=41%) Dry Deposition Triaxial Extension

200 q (kPa)

Test result

p'=400kPa

e=0.817 p'=200kPa p'=100kPa

100

e=0.819

e=0.823

0 0

3

6

εq (%)

9

12

15

300

400

500

300 Toyoura sand (Dr=41%) Dry Deposition Triaxial Extension Test result

q (kPa)

200

100

0 0

100

200 p' (kPa)

Fig. 6.11 Triaxial extension test (DD, Dr =41%)

145


500 Toyoura san (Dr=41%) Moist Tamping Triaxial Extension

q (kPa)

400

Test result

300

200

100

p'=200kPa e=0.820

p'=400kPa e=0.818

p'=100kPa e=0.818

0 0

2

4

6

εq (%)

8

500 Toyoura sand (Dr=41%) Moist Tamping Triaxial Extension

400

q (kPa)

Test result

300

200

100

0 0

100

200

300 p' (kPa)

400

Fig. 6.12 Triaxial extension test (MT, Dr =41%)

146

500

600


Fig. 6.13 Results of special biaxial compression test on bidimensional stacking of cylinders (after Chapuis and Soulié 1981)

0.95 Toyoura sands Dry deposition TC Void tatio e

0.9 SS

0.85 TE

0.8 1

5

10

50

100

500

Effective mean principal stress p' (kPa) Fig. 6.14 Ultimate steady-state lines for various shear modes (data replotted after Yoshimine and Ishihara 1998)

147


0.95

TE(MT,before necking)

TC(MT) TC(DD)

TE(MT,6% DS)

Void ratio e

0.9

0.85

0.8 TE(DD,6% DS) TE(DD,before necking)

0.75

0.7 0.01

0.1 1 Mean normal stress p' (MPa)

Fig. 6.15 Pseudo critical state lines for MT and DD specimens under TC and TE conditions

Fig. 6.16 Effects of sample preparation on cyclic strength of sand (after Mulilis et al. 1977)

148

10


30

30

15

σd=20kPa

0

Liquefaction

DD Specimen

15 q (kPa)

-15

σd=25kPa

0

Liquefaction

-15

-30

-30 0

20

40

60

80

100

120

0

20

40

p' (kPa)

30

30

σd=20kPa

0

Liquefaction

-15

100

120

σd=25kPa

0

Liquefaction

-15

-30

-30

-10

-5

0

5

10

-10

-5

0

εa (%) 120 DD Specimen

90

σd=20kPa

60

60

30

30 PWP

0 0

5

10

0 15

20

DD Specimen

σd=25kPa

60

60 30

30

PWP

0

0 0

Number of cycles

Fig. 6.17 Cyclic loading test (DD specimen,

90 p'

90 p' (kPa)

90

10

120

PWP (kPa)

p'

5

εa (%)

120

p' (kPa)

80

DD Specimen

15 q (kPa)

q (kPa)

DD Specimen 15

60 p' (kPa)

5

10

15

Number of cycles

σ d =20kPa)


149

σ d =25kPa)

PWP (kPa)

q (kPa)

DD Specimen


40

40

20 0

σd=40kPa

20

σd=30kPa q (kPa)

Liquefaction

-20

0

Liquefaction

-20

-40

-40 0

20

40

60

80

100

120

0

20

40

60

p' (kPa)

40

40

σd=30kPa

0

q (kPa)

q (kPa)

120

σd=40kPa

20 Liquefaction

0

Liquefaction

-20

-20

-40

-40 0

5

εa (%)

120

p'

120

DD Specimen

σd=30kPa

90

-10

10

90

60

60

30

30 0 0

1

90


90 60

60 30

30 PWP

0 1

2

Number of cycles

σ d =30kPa)


150

10

σd=40kPa

0

Number of cycles

5

DD Specimen

0

2

0

p'

PWP

0

-5

εa (%)

120

p' (kPa)

-5

PWP (kPa)

-10

p' (kPa)

100

DD Specimen

DD Specimen

20

80

p' (kPa)

σ d =40kPa)

PWP (kPa)

q (kPa)

DD Specimen

DD Specimen


60

σd=40kPa

MT Specimen

σd=45kPa

30 q (kPa)

30 0 -30

0 -30

-60

-60 0

20

40

60

80

100

120

0

20

40

p' (kPa)

60

60

MT Specimen

σd=40kPa

0 -30

100

120

σd=45kPa

0 -30

-60

-60

-10

-5

0

5

10

-10

-5

εa (%) 120

120

90

90

30

30

p' (kPa)

60

PWP (kPa)

σd=40kPa

60

p'

10

MT Specimen

0

60

60

30

30

20

0

30

5

10

0 15

Number of cycles

Number of cycles

Fig. 6.21 Cyclic loading test (MT specimen,

PWP

0

0 10

90

σd=45kPa

PWP 0

5

120

MT Specimen

p'

90

0

εa (%)

120

p' (kPa)

80

MT Specimen

30 q (kPa)

q (kPa)

30

60 p' (kPa)

σ d =40kPa)


151

σ d =45kPa)

20

PWP (kPa)

q (kPa)

60

MT Specimen


80

σd=50kPa

0 Liquefaction

-40

MT Specimen

σd=55kPa

40 q (kPa)

40

0 -40

-80

Liquefaction

-80 0

20

40

60

80

100

120

0

20

40

p' (kPa)

100

120

MT Specimen

MT Specimen

σd=50kPa

0 Liquefaction

-40

σd=55kPa

40 q (kPa)

40 q (kPa)

80

80

80

0

Liquefaction

-40 -80

-80 -10

-5

0

5

-10

10

-5

0

120

120 90

60

60

30

30 0 0

2

4

6

60

30 0

0 2

4

6

Number of cycles

Number of cycles


30

PWP 0

8

90

60

PWP 0

120

σd=55kPa

90 p' (kPa)

σd=50kPa

90

10

MT Specimen

p'

MT Specimen PWP (kPa)

p'

5

εa (%)

εa (%)

120

p' (kPa)

60 p' (kPa)

σ d =50kPa)


152

σ d =55kPa)

PWP (kPa)

q (kPa)

80

MT Specimen


80

MT Specimen

σd=60kPa

q (kPa)

40 0

Liquefaction -40 -80 0

20

40

60

80

100

120

p' (kPa) 80 MT Specimen

σd=60kPa

q (kPa)

40 0

Liquefaction

-40 -80 -10

-5

0

5

10

εa (%)

120

p' MT Specimen

p' (kPa)

90

σd=60kPa

60 30

PWP

0 0

1

2

3

Number of cycles


153

σ d =60kPa)


0.5 Triaxial Cyclic Test Confining Pressure p0'=100kPa Relative Density Dr=30%

MT Specimens

Cyclic Stress Ratio

0.4

DD Specimens

0.3

0.2

0.1

0 0.1

1 10 No. of Cycles to Cause Initial Liquefaction

Fig. 6.26 Liquefaction resistance under triaxial cyclic loading (p’=100kPa, Dr=30%)

154

100


Table 6.1 Conditions of laboratory tests (cyclic triaxial tests) Test ID

Specimen

e0

Dr

po '

Stress Ratio

TCCD_1

DD

0.865

29.4%

100

0.1

TCCD_2

DD

0.863

30.0%

100

0.125

TCCD_3

DD

0.867

28.9%

100

0.15

TCCD_4

DD

0.856

31.8%

100

0.2

TCCM_1

MT

0.865

29.4%

100

0.2

TCCM_2

MT

0.865

29.4%

100

0.225

TCCM_3

MT

0.861

30.6%

100

0.25

TCCM_4

MT

0.855

32.0%

100

0.275

TCCM_5

MT

0.864

29.7%

100

0.3

155


CHAPTER 7 CONSTITUTIVE MODELING OF SAND ACCOUNTING FOR DIFFERENT INHERENT FABRIC ANISOTROPY EFFECTS

7.1

Introduction The laboratory test results, as shown in the previous chapter, and referring to

those reported in the literature, have confirmed that the inherent fabric anisotropy has a significant influence on the mechanical behavior of granular soils, including the stress-strain relation, the excess pore water pressure generation, the critical state, and so forth. It is generally known that different sample preparation techniques can produce varying initial fabrics, which will in turn govern the granular soil response when subjected to shear loading. In previous chapters, dry deposition (DD) and moist tamping (MT) techniques were employed to obtain the reconstituted specimens in the laboratory, and their initial fabrics were identified and evaluated successfully in a micro-structural scale using an image analysis approach. However, the question of how to model the sand behavior with different fabric effects is still a challenge. In this chapter, general observations of the sand behavior, especially the state dependent dilatancy theory, within the framework of CSSM (critical state soil mechanics), will be reviewed first. A constitutive model accounting for the fabric anisotropy effect, developed by Li and Dafalias (2002), will be introduced. In order to simulate sand behavior with different fabrics, certain modifications are made and model constants are calibrated, especially those critical state parameters, in the light 156


of laboratory experimental results and observations. Finally model simulations, together with their experimental counterparts, are presented. Some limitations in the model simulation are also discussed.

7.2

Constitutive Modeling of Sand Behavior

7.2.1 General Observations of Sand Behavior The mechanical behavior of sand has been experimentally investigated extensively under various loading conditions in the past several decades. A brief summary concerning the dilatancy and critical state, which are the most fundamental and remarkable features of granular soils, is given as follows. •

Dilatancy of sand: subject to shear, a loose sand contracts and a dense sand dilates. Whether a sand dilates or contracts depends not only on the void ratio and the confining pressure, but also the inherent fabric anisotropy associated with the sample preparation methods. Therefore, dilatancy is a function of the void ratio, stress ratio, and other material internal state variables, including the inherent fabric anisotropy of granular soil.

•

Critical state: a sheared dense or loose sand will eventually reach an ultimate state, in which the effective mean normal stress, the deviatoric stress and the volumetric strain are no longer changing, and only the deviatoric strain continues to develop. The ultimate state is the wellknown critical state, sometimes termed steady state. It is found that the CSL in the e − log p plane may not be unique, and depends on the shear mode as well as the material inherent fabric.

157


7.2.2 Description of the State of a Sand

Numerous studies reported in the literature have shown that sand response is state dependent, and the extent of the state dependency includes both the void ratio and the effective mean normal stress. It has also been shown in this study that the mechanical behavior of granular soil is also affected by the loading path and the inherent fabric anisotropy, both of which should also be treated as state descriptors. Critical state is an ultimate state of a material, and is also a good reference state to predict the response of soils to a shear load. Traditionally, the critical state properties are assumed to be independent of the initial state and stress, and the critical state can be defined in p − q − e space and expressed as ∂p ∂q ∂ε v = = =0 ∂ε q ∂ε q ε q

(7.1)

Unlikely clay, the critical state line for sand in the e − log p plane is, in general, not straight. Alternatively, a linear representation of the critical state line (CSL) in the e − ( p / pa )ξ plane was proposed by Li and Wang (1998), where pa was the atmospheric pressure for normalization. Attempts have been made to describe the state with reference to the critical state. Been and Jefferies (1985) defined a state as the difference between the current void ratio and the critical void ratio corresponding to the current effective mean normal stress, i.e., ψ = e − ec , where e and ec are the current void ratio and the critical void ratio corresponding to the current effective mean normal stress respectively, as shown in Fig. 7.1. It is seen that ψ is a measure of how far the material state is from the critical state in terms of density. If ψ is negative, the sand is in a dense state, and it is likely to dilate when subjected to shear. On the contrary,

158


if ψ is positive, the sand can be considered in a loose state, and it is likely to contract during shear. The state parameter can be expressed as

ψ = e − ec = e − ⎡⎣eΓ − λc ( p / pa )ξ ⎤⎦

(7.2)

Besides the state parameter ψ defined in Eq. (7.2), other similar parameters to describe the state were also found in the literature. Bolton (1986) proposed a scalar parameter IR, called the relative dilatancy index, which also combined the influence of density and confining pressure. Ishihara (1993) introduced a scalar quantity IS, called the state index, that bore the information of some characteristic states other than the critical state in the e − p plane.

7.2.3 State Dependent Dilatancy Theory

As discussed previously, the dilatancy for sand must be a function not only of the stress ratio, but also of the material internal state. Li and Dafalias (2000) proposed a general expression for dilatancy, in which the void ratio and other material internal state variables were integrated, as follows: d = f (η , e, Q, C )

(7.3)

where C represented a group of material constants, and Q denoted the internal state variables other than e that may affect d. It can be seen that η, e and Q must join together to describe the dilatancy behavior. Any specific form formulated according to Eq. (7.3) or other similar forms is termed as state-dependent dilatancy. The term state dependent emphasizes the dependence of dilatancy on the stress state, void ratio, and other material internal state variables. With Eq. (7.3), the dilatancy is now uniquely related to an existing state, in terms of the external stress state expressed via

η, and the internal material state expressed via e and Q.

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In the formulation of a specific dilatancy function, some rules must be followed. Firstly, at critical state, η = M , e = ec (void ratio in critical state), and d = 0 . It can be expressed as

d (η = M , e = ec , Q, C ) = 0

(7.4)

The above equation infers that the critical state can be reached only when

η = M , and e = ec , simultaneously. In other words, the condition η = M alone does not guarantee that the critical state has been reached (which is assumed in stress dilatancy theory). It can also be seen that the attainment of critical state is not necessarily associated with particular values of Q and C. Secondly, on examining the dilatancy in a phase transformation state, d ε vp = 0 , which requires d=0, too. The phase transformation condition can be expressed as d (η = M d ≠ M , e ≠ ec , Q, C ) = 0

(7.5)

This indicates that d = 0 is not necessarily associated with a critical state. To obtain a specific function for the dilatancy as expressed by Eq. (7.3), and satisfying the requirement of Eq. (7.4), one should describe adequately the physical conditions of a sand, including both e and p . The state parameter ψ , combining the effects of p and e, describes the physical state of a sand, and can be invoked into the dilatancy function. As such, a dilatancy expression in the following form was proposed (Li et al. 1999, Li and Dafalias 2000) d = d (η ,ψ , C )

(7.6)

Here, for simplicity, the dependence of the dilatancy on Q is suppressed under monotonic loading. Now the requirement at the critical state, as expressed by Eq. (7.4), becomes:

160


d (η = M , ψ = 0, C ) = 0

(7.7)

Based on the observation of the dilative shear behavior of sand, Li and Dafalias (2000) proposed a specific dilatancy function, as follows:

η ⎞ ⎛ d = d1 ⎜ e mψ − ⎟ M⎠ ⎝

(7.8)

in which d 1 and m are two positive model constants. It can be seen that the condition of Eq. (7.7) is automatically satisfied. The condition of Eq. (7.5) can also be satisfied when η = Me mψ = M d , where M d is the stress ratio at the phase transformation. Obviously M d is not a material constant, and depends on the material state, as shown in the literature. Eq. (7.8) can capture most features of sand behavior. Firstly, it is insightful to examine a typical undrained triaxial compression test for a sand initially in a loose state (ψ 0 > 0). At the initial state, η = 0 , so that d = d1 ⋅ e mψ 0 = dinitial . As the shear proceeds, ψ decreases and η increases, therefore e mψ decreases and η M increases, resulting in a decrease in d. As the sand approaches the critical state, ψ approaches 0, and η approaches M , hence d approaches 0. Secondly, consider a sand, initially in a dense state (ψ < 0), sheared under undrained triaxial compression. At the initial state, η = 0 , d = d1 ⋅ e mψ 0 = dinitial >0, so the sand starts with a contractive response. As the shear proceeds and before the stress ratio reaches the phase transformation stress ratio η (η < M d ) and the absolute value of ψ increases, so e mψ decreases, and η M increases, resulting in a decrease in d, but still d > 0 . This indicates that the contraction process for medium to dense sand lasts until reaching the phase transformation state. It has already been shown that at phase transformation

161


state d = 0 . After the phase transformation state, d < 0 holds. Finally the sand reaches the critical state, at which η = M , ψ = 0 , and d = 0 .

7.3

An Introduction to a Constitutive Model Accounting for Fabric Anisotropy Effect

7.3.1 Platform Model with State-Dependent Dilatancy

The platform model (Li 2002) is a double hardening bounding surface sand model with a state-dependent dilatancy (Li and Dafalias 2000). The model follows the basic structure of Wang et al. (1990). A brief description of the model is included here to facilitate a better understanding of the propositions in later sections for the modeling of anisotropic effects. As a notational convention, all the quantities evaluated on the bounding surfaces are distinguished by a superposed bar. There are two bounding surfaces in the model corresponding to two loading mechanisms: a cone with straight surface meridians, F1 , and a flat cap, F2 , as shown in Fig. 7.2(a). The cross-section of the cone in the R − θ plane is shown in Fig. 7.2(b). The cone is associated with the plastic loading induced by a change in the deviatoric stress ratio rij = σ ij p − δ ij = sij p , and the cap is associated with the plastic loading owing to a

change in p = σ kk / 3 under a constant rij . The condition of consistency for the cone, dF1 = 0 , yields the following equation: pnij drij − K p1dL1 = 0

162

(7.9)


where nij is a traceless unit norm tensor (i.e., nij nij = 1 ) normal to F1 at an image stress ratio rij , K p1 and dL1 are the plastic modulus and loading index, respectively, associated with F1 . The mapping rule proposed by Wang et al. (1990) was adopted to map the rij onto the image stress ratio, rij , on F1 , as illustrated in Fig. 7.3. A relocatable projection center, αij , is defined as the stress ratio at which the last reversal of the loading direction from loading to unloading took place ( dL1 changes from positive to negative). The rij is obtained by the projection of rij onto F1 , with αij as the projection center. Correspondingly, the two Euclidian stress-ratio distances,

ρ1 = rij − αij and ρ1 = rij − αij are defined as shown. The condition of consistency for the cap, dF2 = 0 , yields dp − K p 2 dL2 = 0

(7.10)

where K p 2 and dL2 are the plastic modulus and loading index, respectively, associated with the cap F2 . The plastic deviatoric strain increment deijp = dε ijp − (dε vp / 3)δ ij , with ε vp the plastic volumetric strain, is decomposed into two parts as follows deijp = deijp1 + deijp 2 = nij dL1 + mij dL2

(7.11)

where deijp1 and deijp 2 are the plastic deviatoric strain increments associated with the loading indices dL1 and dL2 respectively, and mij = rij / | rij | is a unit norm tensor that defines the direction of the plastic deviatoric strain increment owing to a dp ≠ 0 under a constant rij .

163


The plastic volumetric strain increment is also decomposed into two parts, dε vp1 and dε vp 2 , which are paired with deijp1 and deijp 2 , respectively, as follows dε vp = dε vp1 + dε vp 2 =

)

(

2 2 D1 deijp1deijp1 + D2 deijp 2 deijp 2 = ( D1dL1 + D2dL2 ) 3 3

(7.12) where D1 and D2 are the dilatancy functions for dε vp1 and dε vp 2 , respectively. For the first loading mechanism, the dilatancy function D1 is defined as D1 =

d1 ⎡ R ⎤⎡ mψ ⎢1 + ⎥ ⎢ M c g (θ )e M c g (θ ) ⎣ M c g (θ ) ⎦ ⎣

⎤ ρ1 − R⎥ ρ1 ⎦

(7.13)

where R = 3rij rij 2 and θ = − ⎡⎣sin −1 ( 9rij rjk rki 2 R 3 ) ⎤⎦ 3 (the Lode angle) are two isotropic invariants of rij , g (θ ) is an interpolation function accounting for the influence of θ , M c is the critical stress ratio at triaxial compression, d1 and m are two model constants, and ψ is the state parameter defined in Eq. (7.2). The expression for the plastic modulus K p1 is formulated as follows: ⎡ M c g (θ ) e − nψ ⎛ ρ1 ⎞ ⎤ K p1 = Gh ⎢ ⎜ ⎟ − 1⎥ R ⎝ ρ1 ⎠ ⎦ ⎣

(7.14)

where R and θ are the two invariants of the image stress ratio rij , n is a material constant, and h is a scaling factor, which itself is a function of some state variables, and can be expressed as Eq. (7.15) (Li 2002).

{

}

h = (h1 − h2 e) ( ρ1 / ρ1 ) k + h3 f ( L1 ) ⎡⎣1 − ( ρ1 / ρ1 ) k ⎤⎦

(7.15)

Where h1, h2, and h3 are material constants. The corresponding dilatancy and plastic modulus expressions for the second loading mechanism are given by

164


D2 = d 2

M c g (θ ) dp −1 R dp

(7.16)

a

K p2

⎡ M g (θ ) ⎤ ⎛ ρ 2 ⎞ dp = Gh4 ⎢ c ⎥⎜ ⎟ ⎣ R ⎦ ⎝ ρ 2 ⎠ dp

(7.17)

where d 2 , h4 , and a are positive model constants, and the distances ρ 2 and ρ 2 are measured from the cap. Details on the explanation for the foregoing expressions, as well as the response when R = 0 , can be found in Li (2002). It has been shown (Li and Dafalias 2000, and Li 2002) that the above model is capable of simulating the contractive and dilative responses of a sand over a wide range of densities and stress states, with a single set of model constants. However, the incorporation of a material's inherent anisotropy has not been addressed.

7.3.2 Incorporation of Anisotropy Effects in Response to Proportional Loading

This part of the work began from the experimental observation that a sand having the same density exhibits drastically different undrained critical state strengths under triaxial compression and extension (Vaid and Chern 1985). This finding is of fundamental significance because it suggests that the orthodox view that there is a unique critical state line in the e − p plane could be false. The assumption of a unique critical state line is based on the postulate that the initial soil fabric and the loading history memory are erased after a large shear deformation pertinent to the critical state. However, microstructural studies (e.g. Oda et al. 1985, Oda and Nakayama 1988; and Tobita 1989) have revealed that even after a large shear deformation, the preferred orientation of particles in a granular material may undergo only limited change. In other words, the inherent fabric anisotropy of the sand may

165


well endure after the onset of the critical state, rendering the necessity of treating the critical state line as a function of the fabric anisotropy. It is obvious that a varying critical state line in the e − p plane will also make the state parameter, ψ , a function of soil fabric, which in turn makes the soil dilatancy a function of fabric anisotropy [because D1 is a function of ψ , as shown in Eq. (7.13)]. To this end, a symmetric tensor (Oda and Nakayama 1988) is adopted to describe soil fabric. This fabric tensor is written with respect to its principal direction in the case of transverse isotropy as 0 0 ⎞ ⎛1 − ∆ 1 ⎜ ⎟ 0 1+ ∆ 0 ⎟ Fij = ⎜ 3+ ∆ ⎜ 0 1 + ∆ ⎟⎠ ⎝ 0

(7.18)

where ∆ is called the vector magnitude (Curray 1956), which varies from zero when the material is isotropic to unity when the major axes of all the particles are uniformly distributed in the direction of deposition. The transverse isotropy expressed by Fij in Eq. (7.18) is an accurate enough assumption for most practical applications involving soil deposits under gravity. However, this consideration is not an intrinsic restriction on the proposed approach. An orthotropic tensor can be used to directly replace Eq. (7.18) if it is deemed necessary. While Fij characterizes the inherent fabric anisotropy, its effect must be related to the loading direction. Motivated by the suggestion of Tobita (1988), a modified stress, Tîj , is defined by

1 Tîj = (σ ik Fkj−1 + Fik−1σ kj ) ≡ pˆ (rîj + δ ij ) 6

(7.19)

where Fij−1 is the inverse of the fabric tensor, and the normalized stress σ ij is given by

166


σ ij =

M c g (θ ) rij + δ ij R

(7.20)

This is a stress with a mean normal value of unity, a second stress ratio invariant equal to the critical stress ratio, M c g (θ ) , and deviatoric directions identical to those of the image stress ratio, rij , which is a ready-made candidate for representing a measure of the loading direction in bounding surface models. It can be seen that σ ij , because of its normalization, represents only the loading direction

associated with nij and not the loading magnitude, while rîj in Eq. (7.19) emerges as the modified stress-ratio of importance. As a symmetric second order tensor, Tîj possesses three independent isotropic invariants, among which, the two non-trivial invariants pertinent to rîj are Rˆ = 3rîj rîj / 2 and θˆ = − ⎡⎣sin −1 (9rîj rˆjk rˆki / 2 Rˆ 3 ) ⎤⎦ / 3 . These two invariants are further combined into the following single invariant to characterize the anisotropy, called the anisotropic variable, A=

Rˆ M c g (θˆ)

−1

(7.21)

It follows that if the material is isotropic, then ∆ = 0 and Fij = δ ij / 3 , hence

Tîj = σ ij , and correspondingly, A = 0 . If the material is anisotropic, Tîj deviates from

σ ij , and A can be either positive or negative depending on the orientation of the soil fabric relative to the loading direction and, to a lesser degree, on the fabric intensity. It should be pointed out that A is an objective measure of the fabric anisotropy effect, i.e., it is independent of the reference frame adopted. With this objective measure of the effects of anisotropy, a varying critical state line in the

167


e − p plane can be defined. Considering that the slope of this critical state line is mainly influenced by the shape of the grains, which is an intrinsic property (Poulos et al. 1985), only eΓ [see Eq. (7.2)], the critical state void ratio at intercept p = 0 , is made a function of A and intermediate principal stress parameter b, as follows eΓ = eΓc − kΓ ( Ac − A)(1 − t ⋅ b)

(7.22)

where eΓc (the critical state void ratio for triaxial compression at intercept p = 0 ), k Γ and t are three material constants, and Ac is corresponding to the anisotropic state variable A at triaxial compression. Eq. (7.22) creates a sequence of straight critical state lines running parallel to each other in the e − p ξ plane. It is also noted that Eq. (7.22) is different from the one adopted by Li and Dafalias (2002), because here different inherent fabric anisotropic effects are of concern. However, experimental evidence reported both in the literature and in this study implies that the inherent anisotropy will probably lead to different locations of the critical state line in the e − p space. This argument is implicitly realized via Eq. (7.22), as both A and Ac are functions of the anisotropic parameter ∆ . An examination of test results in the literature has identified that the plastic modulus, K p1 , is also appreciably influenced by the fabric anisotropy. Therefore, the scaling factor h in K p1 in the platform model [see Eq. (7.14)] is modified to be a function of A as follows h = h′

(kh Ac − Ae ) + (1 − kh ) A Ac − Ae

(7.23)

where kh is a new material constant and h′ is in lieu of the original h . It can be seen that h varies linearly with A . When A = Ac (triaxial compression at α = 0D ), h = h′ .

168


When A = Ae (triaxial extension at α = 90D ), h = h′kh , i.e., h is scaled by a factor kh . By including the dependency of the critical state line and K p1 on A , the platform model successfully simulated test results involving various loading directions with respect to the soil fabric coordinate, as reported by Li and Dafalias (2002).

7.4

Model Calibration

The model introduced previously was aimed originally at simulating the effects from various loading paths on anisotropic granular soil, as reported by Li and Dafalias (2002, 2004). In the current study, this model is employed to examine the effects of different inherent/initial fabrics, which stem from the different sample preparation techniques, on sand response, when subjected to monotonic loadings. To better fit the experimental data reported in the last chapter, model constants should be recalibrated. These model constants should be calibrated based on the experimental observations, especially those parameters related to the fabric anisotropy and the critical state, which will be presented in the following part. Other model constants calibration can be referred to Li (2002). As shown in Eq. (7.18), the inherent fabric anisotropy is represented by a second order symmetric fabric tensor. For DD and MT specimens of Toyoura sand, the different initial fabrics have been identified and quantified, using an image analysis approach on thin sections obtained from the resin epoxy impregnated specimens, as reported in Chapter 5. It is preferred to use particle orientations to express the fabric tensor, and thus the vector magnitude ∆ is a measurable quantity. Based on the results presented in Chapter 5, values of ∆ = 0.091 for the MT specimen and ∆ = 0.214 for the DD specimen will be used in the simulation in this

169


study. According to Eq. (7.18), one may easily obtain the corresponding fabric tensor Fij by substituting the ∆ value pertinent to a certain structure (DD or MT specimen). In order to reflect the relative effect of different inherent fabrics on the mechanical response of sand, it is important to take into account the relative orientation of the fabric tensor Fij with reference to the stress state or stress path. To this end, a state variable, A, a scalar quantity, is employed to characterize the influence of the sand fabric on the stress-strain response. It is seen from Eq. (7.21) that besides the stress state (represented by parameters α and b), different initial fabrics also yield different state variables A. Fig. 7.4 and Fig. 7.5 show how the anisotropic state variable A varies with α and b for ∆ = 0.091 (MT specimen) and ∆ = 0.214 (DD specimen) for Toyoura sand, respectively. Note that α = 0 and b = 0

yields a triaxial compression loading condition, and α =90o and b = 1 results in a triaxial extension loading condition. It can be seen that for an MT specimen, the state variable A is less significant that than that for a DD specimen, which is quite understandable because the MT specimen is more isotropic compared with DD one. In the framework of CSSM, the critical state is a good reference state to model the response of soils. The significance of the determination of parameters for CSLs is apparent. Based on the experimental observations, the CSL may not be unique in the e − p plane for different inherent fabrics by comparing the same loading path, such as triaxial compression and extension. It is believed that the difference between triaxial compression and extension CSLs reflects the degree of fabric anisotropy. Laboratory tests show that this difference for an MT specimen is less than that for a DD one. Referring to Eq. (7.22), eΓc , the critical state void ratio ( A = Ac ) intercepting the p = 0 axis, and kΓ and t another two critical state parameters, are treated as material constants. By doing so, the CSL is both stress170


path and fabric dependent, according to Eq. (7.22), which is in agreement with the experimental observations. By inspecting the test results and better fitting the test data, the parameters eΓc , kΓ and t can be calibrated. Li and Wang (1998) proposed the following function for a critical state line in the e − ( p / pa )ξ plane: ec = eΓ − λc ( p / pa )ξ

(7.24)

in which the slope λc is a material constant. The test data at the final state (reported in the last chapter) for the triaxial compression and extension tests can be plotted in the e − ( p / pa )ξ space, as shown in Fig. 7.6. The CSLs employed in the modeling are also showed in this figure. For the triaxial compression, the CSL seems to be independent of the sample preparation methods, which is in accordance with the observation by Ishihara (1993). It is further found that the slopes for MT and DD specimens in triaxial compression are very close to 0.020, which is a material constant in Toyoura sand modeling used by Li and Dafalias (2000, 2002, and 2004), Li (2002), Dafalias and Manzari (2004), Dafalias et al. (2004), etc.. In the triaxial extension, all the final states do not reach the real critical state, and the final points for MT specimen are lying above that for DD specimen. The CSLs in the triaxial extension for both DD and MT specimens used in the modeling are given in Fig. 7.6. According to the investigation by Poulos et al. (1985) that the slope of the critical state line is mainly influenced by the shape of the grains and is an intrinsic property, the slope of the critical state lines in the triaxial extension is tentatively assumed as 0.020, the same as that in the triaxial compression for simplicity in this study. It is noted that most of final points in the triaxial extension are below the CSLs employed in the simulation in this study and the arrows in the graph indicate the probable trends afterwards. 171


According to the experimental results, more dilative and stiffer responses are observed for MT specimens than DD ones, in triaxial compression, extension as well as torsional shear, with otherwise identical conditions. To better fit the experimental data, different hardening parameters for MT and DD specimens should be chosen. Referred Eq. (7.15), only parameter h1 and h2 are altered because only monotonic and proportional loadings are of concern here. There are a total of 18 material constants in the model under consideration. As listed in Table 7.1, these constants can be divided into five groups according to their functions, elastic parameters, critical state parameters, anisotropic parameters and parameters associated with dilatancy D1 and D2.

7.5

Simulation by the Model

Typical triaxial tests, as well as torsional shear tests presented in Chapter 6, are simulated using the anisotropic model with the unified set of model parameters as listed in Table 7.1. All these tests presented show the influence of soil density, confining pressure, soil fabric, as well as principal stress direction, in reference to the orientation of the soil specimens (stress path for simplicity). Fig. 7.7 to Fig. 7.16 show the test results reported in Chapter 6, as well as model simulations under the same conditions. It can be seen that in general, the model can predict both contractive and dilative response, and show the impacts of different inherent fabrics of test specimens, loading path, soil density as well as confining pressure applied. Discrepancies are easily identifiable for the prediction of the response of torsional shear tests, and are believed to be due to the determination of the location of the critical state lines in the e − p plane, which is directly associated with the anisotropic state parameter A. 172


7.6

Conclusions

This study was motivated by the experimental finding that the mechanical response of sand was strongly influenced by the inherent fabric anisotropy as well as the shear mode or loading path. Particularly, the issue of uniqueness of CSLs has emerged, and laboratory tests revealed that CSLs could be non-unique, and the shear mode, as well as the inherent fabric of sandy soils, might result in this kind of nonuniqueness. Though this concept contradicts the traditional treatment of CSL, more and more findings in recent years have implicitly indicated that the uniqueness of CSL is a poor postulate. In this chapter, the state dependent dilatancy theory was reviewed, and a state parameter ψ , defining the difference between the current and the critical state void ratio, was employed to characterize the state of a sand. A platform model proposed by Li (2002) was then introduced, which was a double hardening bounding surface sand model with a state-dependent dilatancy. To incorporate the effect of anisotropy into this model framework, some micromechanical investigation must be carried out. It was realized that the material fabric must be described by tensor quantities, and the influence of material fabric depended on the principal directions of the fabric tensor in reference to the principal directions of the stress tensor. To this end, a joint tensor was proposed, described in terms of the fabric tensor and the stress tensor. The anisotropic state variable A, a scalar, was defined as an invariant of the joint tensor. A model accounting for different inherent fabrics was then constructed, which had some unique features as follows: 1) the critical state line in the e − p plane was a function of the anisotropic state variable, A, such that the dilatancy of the soil became fabric and stress path dependent; 2) the plastic modulus, Kp, was also a

173


function of A, such that the sand stiffness depended on the material fabric and stress path. With a set of 18 unified model constants, the modified model was able to simulate the test results presented in Chapter 6, including both DD and MT specimens, under different confining pressures and densities, with various shear modes. Although no definite data about the critical state lines in the e − p plane was available due to laboratory testing limitations, the model could still capture the important features of the sand response for both DD and MT specimens. The correct trend of the model response indicates that the critical state line, taking as the anisotropic state parameter A dependent, as well as dilatancy, is a good reference to simulate the different characteristics of specimens with different inherent fabrics under shear.

174


Void ratio e

Current State 2 ψ>0 (Contractive))

ψ