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THE UNIVERSITY OF WESTERN ONTARIO
FACULTY OF GRADUATE STUDIES
CERTIFICATE OF EXAMINATION
Supervisor
Dr. Sean Hinchberger
Co- Supervisor
Dr. K.Y. Lo
Examining Board
Dr. Tim Newson
Dr. Ernest Yanful
Dr. John Dryden
Dr. James Blatz
The thesis by
Guangfeng Qu
Entitled
Selected issues on the performance of embankments on clay foundations
is accepted in partial fulfillment of the
Requirement for the degree of
Doctor of Philosophy
Date March, 17,2008 Dr. Jianddong Ren
Chair of Examining Board:
11
ABSTRACT
This thesis examines selected issues related to the performance of earthfill
embankments constructed on soft clay foundations. The primary objectives of the thesis
are: to extend an existing elastic-viscoplastic (EVP) constitutive model to describe the
influence of micro-structure and strength anisotropy on the engineering response of soft
clay, to investigate the impact of clay structure on the performance of a full-scale test
embankment on soft clay, and to evaluate the significance of three-dimensional effects on
the behaviour of three test embankments constructed on soft clay foundations.
Firstly, in this thesis, generalized EVP theory is used to evaluate the viscous
response of 19 clays reported in the literature. It is shown that the viscous response of
clay, including rate-dependent and time-dependent behaviour in different types of
experiments, can be quantitively characterized using a unique set of viscous parameters.
A practical methodology to determine the EVP constitutive parameters is provided.
Next, an existing EVP constitutive model is extended to account for the influence
of micro-structure and anisotropy on the engineering response of rate-sensitive natural
clay. Microstructure and the process of destructuration are mathematically simulated
using a state-dependent fluidity parameter. The EVP model also incorporates a structure
tensor that can be used to describe strength anisotropy of natural clay. The extended
structured and anisotropic models are shown to describe the responses of undisturbed
structured clays, such as Saint-Jean-Vianney clay, Gloucester clay, and St. Vallier clay.
Lastly, four case studies are used to investigate the impact of microstructure and
destructuration on the performance of embankments founded on soft clay and the effects
of 3-dimensional geometry on test embankment behaviour. The Gloucester test
iii
embankment is studied using the structured EVP model. This case is used to examine the
impact of destructuration on strength gain in the Gloucester foundation during staged
construction. In addition, three embankment cases in Vernon British Columbia, St.
Alban Quebec, and Malaysia are studied using 3-dimensional finite element analysis to
examine the impact of 3-dimensional geometry on the performance of test embankments.
This thesis is prepared in accordance with the regulations for Manuscript format
thesis stipulated by the Faculty of Graduate Studies at The University of Western
Ontario.
Chapters 2 and 4 of this thesis are the current versions of manuscripts in
preparation for submission as papers, which will be co-authored by Guangfeng Qu and
S.D. Hinchberger. Chapter 6 is a modified version of a submitted paper co-authored by
G. Qu, S.D. Hinchberger and K.Y. Lo. Chapters 3 and 5 are the manuscripts currently in
review coauthored by S.D. Hinchberger and G. Qu, and S.D. Hinchberger, G. Qu, and
K.Y. Lo, respectively.
Guangfeng Qu conducted numerical analysis and wrote the draft of the chapters.
Dr. Sean Hinchberger assisted in interpretation of the results and the writing of the
chapters. Dr. K.Y. Lo assisted in the interpretation of the results and the writing in
Chapters 2, 5, and 6.
v
ACKNOWLEDGEMENT
The author wishes to express his deepest gratitude and appreciation to his advisor,
Dr. Sean D. Hinchberger for his insightful guidance, friendly encouragement, and
continuous support throughout the research and graduate studies.
The constructive and critical advice given by Dr. K.Y. Lo is greatly appreciated.
The author also thanks Dr. Tim Newson, Dr. Julie Shang, Dr. M. Hesham El Naggar, and
Dr. Ernest Yanful for sharing their knowledge during the general course work.
The author wishes to acknowledge the Geotechnical Research Center, the
Department of Civil and Environmental Engineering at University of Western Ontario for
technical and clerical support.
Many thanks are given to the friends and colleagues for their supports and
interesting discussions during the past four years.
Finally, the author wishes to thank his wife, Yanming, for her love, support, and
patience.
VI
TABLE OF CONTENTS
page
CERTIFICATE OF EXAMINATION ii
ABSTRACT iii
CO-AUTHORSHIP v
ACKNOWLEDGEMENT vi
TABLE OF CONTENTS vii
LIST OF TABLES xi
LIST OF FIGURES xii
NOMENCLATURE xix
CHAPTER 1 INTRODUCTION 1
1.1 Introduction 1
1.2 Definitions 3
1.3 Thesis Objectives and Outline 5
1.4 Original Contributions 7
References 10
CHAPTER 2 EVALUATION OF THE VISCOUS BEHAVIOUR OF NATURAL
CLAY USING GENERALIZED VISCOPLASTIC THEORY 16
2.1 Introduction 16
2.2 Theoretical Background 17
2.2.1 Brief introduction of elastic-viscoplastic theory 17
2.2.2 Strain-rate controlled testing 20
2.2.3 Link with the isotache concept 23
2.2.4 Alternative flow function - the exponential law 24
2.2.5 Stress-controlled testing 25
2.3 Evaluation 27
vn
2.3.1 Rate dependency of preconsolidation pressure 27
2.3.2 Undrained shear strength versus strain-rate 29
2.3.3 Secondary compression 32
2.3.4 Summary 33
2.4 Selection of Parameters 34
2.4.1 The measurement of a 35
2.4.2 The measurement of a™ and yvp 36
2.5 Summary and Conclusion 38
References 41
CHAPTER 3 A VISCOPLASTIC CONSTITUTIVE APPROACH FOR RATE-
SENSITIVE STRUCTURED CLAYS 71
3.1 Introduction 71
3.2 Theoretical Formulation 75
3.2.1 Overstress viscoplasticity 75
3.2.2 Numerical overstress 77
3.2.3 Modification for soil structure 78
3.3 Methodology 81
3.3.1 Laboratory tests 81
3.3.2 Numerical approach 82
3.3.3 Selection of constitutive parameters 83
3.4 Evaluation (Saint-Jean Vianney Clay) 87
3.4.1 Theoretical behaviour of the model for CIU triaxial compression 87
3.4.2 Calculated and measured behaviour for constant rate-of-strain triaxial
compression 88
3.4.3 CIU triaxial creep tests 90
3.4.4 Theoretical response for constant rate-of-strain consolidation 93
3.4.5 Constant rate-of-strain consolidation 94
3.5 Summary and Conclusions 96
References 100
viii
CHAPTER 4 THE STUDY OF STRUCTURE AND ITS DEGRADATION ON
THE BEHAVIOUR OF THE GLOUCESTER TEST
EMBANKMENT 131
4.1 Introduction 131
4.2 Background 132
4.3 Methodology 136
4.3.1 Model 1 -Hinchberger and Rowe Model 136
4.3.2 Model 2 - Structured Elastic-viscoplastic (EVP) Model 139
4.3.3 Finite Element Mesh 142
4.3.4 Constitutive Parameters 142
4.4 Results 144
4.4.1 Analysis using the Unstructured EVP Model (Model 1) 144
4.4.2 Analysis using the Structured EVP Model (Model 2) 147
4.5 Summary and Conclusions 151
References 155
CHAPTER 5 AN ANISOTROPIC EVP MODEL FOR STRUCTURED CLAYS 187
5.1 Introduction 187
5.2 General Approaches to Anisotropic Plasticity 188
5.3 Microstructure Tensor 190
5.4 Application to Tresca's Failure Criterion 193
5.5 Application to an Elastic-Viscoplastic Model 196
5.6 Evaluation 202
5.7 Summary and Conclusions 208
References 210
CHAPTER 6 CASES STUDY OF THREE DIMENSIONAL EFFECTS ON THE
BEHAVIOUR OF TEST EMBANKMENTS 234
6.1 Introduction 234
6.2 Methodology 235
6.3 St. Alban Test Embankment Case 236
ix
6.3.1 Introduction 236
6.3.2 Soil Conditions 237
6.3.3 Geometry 238
6.3.4 Results 238
6.4 Malaysia Trial Embankment Case 239
6.4.1 Introduction 239
6.4.2 Soil Conditions 240
6.4.3 Geometry 241
6.4.4 Results 241
6.5 The Vernon Case 243
6.5.1 Introduction 243
6.5.2 Analysis 244
6.5.2 Results of Vernon Approach Embankment 246
6.5.3 Results of Waterline Test Fill 247
6.6 Discussion 249
6.7 Summary and Conclusion 250
References 252
CHAPTER 7 SUMMARY AND FURTHER WORK 279
7.1 Summary 279
7.2 Suggestions for Future Research 280
References 282
APPENDIXES 283
APPENDIX A 283
APPENDIX B 290
APPENDIX C 296
APPENDIX D 306
APPENDIX E 312
APPENDIX F 324
APPENDIX G 331
CURRICULUM VITAE 337
x
LIST OF TABLES
page
Table 2.1 Geotechnical properties of 19 clays 48
Table 2.2 Summarized a for 19 clays 50
Table 3.1 Properties of Saint-Jean-Vianney clay, (after Vaid et al. 1979) 106
Table 3.2 Constitutive parameters for Saint-Jean-Vianney clay 107
Table 4.1 Material parameters used in both Model 1 and Model 2 for the numerical analysis of the Gloucester test embankment 159
Table 4.2 Viscosity-related parameters for Gloucester clay used by Model
1 and Model 2 160
Table 5.1 Comparison of elastic-viscoplastic models 215
Table 5.2 Constitutive parameters for Gloucester Clay 216
Table 5.3 Constitutive parameters for St.Vallier Clay 217
Table 6.1 Parameters used in the numerical analysis of the three cases 255
XI
LIST OF FIGURES
page
Figure 1.1 Cross-section of embankment and typical undrained strength profile for the underlying foundation clay 13
Figure 1.2 Schematic of an oedometer apparatus and a typical compression
curve. 14
Figure 1.3 Definition of the orientation angle, / 15
Figure 2.1 Illustration of models for elastic viscoplastic materials 52
Figure 2.2 Illustration of relations between strain-rate and yield stress (or
undrained shear strength) in strain-rate controlled tests 53
Figure 2.3 The link between the EVP model and the isotache concept 54
Figure 2.4 The influence of the power law and exponent law flow functions
on the relationship between yield stress and strain-rate 56
Figure 2.5 Typical compression curve for secondary compression. 57
Figure 2.6 Ranges of strain-rates in laboratory tests and in situ (modified from Leroueil and Marques, 1996) 58
Figure 2.7 Relationship between preconsolidation pressure, a'p, and strain-
rate, smial, in log-log scale 59 Figure 2.8 Relationship between undrained strength, Su, and axial strain-
rate> zaxiai > in log-log scale 60
Figure 2.9 Relation between undrained strength and axial strain-rate for Drammen clay and Haney clay 61
Figure 2.10 Comparison of a estimated from rate-controlled oedometer tests and undrained triaxial tests ( See Table 2.2). 62
Figure 2.11 Evaluation on the ability of exponential and power law flow functions to represent the relationship between preconsolidation pressure and strain-rate 63
Figure 2.12 Comparison of a estimated from secondary consolidation tests, rate-controlled oedometer tests, and undrained triaxial tests ( See Table 2.2). 65
Xll
Figure 2.13 Comparisons of a_ac, a_oed, and acreep with aavg 66
Figure 2.14 Typical triaxial compression curves with step-changed strain-rates. 67
Figure 2.15 Illustration of the preferred range of load increment for the measurement of Ca 68
Figure 2.16 Normalized <r'p - s relationship at 10% vertical strain
(sv =10%) for Berthierville clay at a depth of 3.9-4.8m (data
from Leroueil et al. 1988) 69
Figure 2.17 Normalized cr'p - s relationship at 10% vertical strain
(ev =10%) for St. Alban clay from both laboratory tests and in
situ observance (data from Leroueil et al. 1988) 70
Figure 3.1 The influence of structure on the response of Bothkennar clay during oedometer compression (from Burland 1990). 108
Figure 3.2 The influence of structure on the response of London clay during undrained triaxial compression (from Sorensen et al. 2007 and Hinchberger and Qu 2007). 109
Figure 3.3 The state boundary surface, critical state line, and mathematical overstress of the structured soil model. 110
Figure 3.4 Estimation of the aspect ratio, R, for the elliptical cap. I l l
Figure 3.5 Estimation of the yield surface parameter, Moc , in the
overconsolidated stress range. 112
Figure 3.6 Estimation of the intrinsic compressibility, A,, and structure parameter, co0, from oedometer compression for SJV clay. 113
Figure 3.7 Intrinsic compressibility of different clays (adapted from Burland 1990). 114
Figure 3.8 Estimation of n and a ' j^ from undrained triaxial compression
and oedometer compression for SJV clay 115
Figure 3.9 Influence of continued post-peak straining on the power law exponent, n. 118
Figure 3.10 Theoretical behaviour of the structured soil model during CIU triaxial compression. 119
xm
Figure 3.11 Measured and calculated behaviour of SJV clay during CIU triaxial compression. 120
Figure 3.12 Measured and calculated undrained shear strength versus strain-rate for SJV clay. 122
Figure 3.13 Calculated and measured behaviour during CIU triaxial creep
tests on SJV clay 123
Figure 3.14 Calculated and measured creep-rupture life for SJV clay 125
Figure 3.15 Calculated and measured axial strain-rate versus time during CIU triaxial creep on SJV clay. 126
Figure 3.16 Comparison of strain-rate at failure for peak strength and creep rupture - SJV clay. 127
Figure 3.17 Theoretical behaviour of the structured soil model during constant-rate-of-strain K'0 -consolidation. 128
Figure 3.18 Calculated and measured behaviour during oedometer compression. 129
Figure 3.19 Measured and calculated compression curves of SJV clay during constant rate-of-strain consolidation. 130
Figure 4.1 (a) Geometry of the Gloucester test embankment and (b) properties of Gloucester clay 161
Figure 4.2 Influence of clay structure on the behaviour of Gloucester clay in undrained triaxial and oedometer compression tests 163
Figure 4.3 Rate-sensitivity of the undrained shear strength and preconsolidation pressure of Gloucester clay 165
Figure 4.4 Long-term oedometer creep tests on Gloucester clay (data from Lo et al. 1976) 167
Figure 4.5 The state boundary surface and critical state line for Model 1 and Model 2. 168
Figure 4.6 Illustration of the theoretical response of Model 1 (Hinchberger and Rowe Model) 169
Figure 4.7 Illustration of the theoretical response of Model 2 170
xiv
Figure 4.8 Comparison of the measured behaviour in CRS oedometer test on Gloucester clay and the corresponding theoretical response of Model 2 171
Figure 4.9 Comparison of the measured settlement at Gauge SI with the
calculated settlement using Model 1 172
Figure 4.10 Illustration of the linear and bilinear virgin compression curves 173
Figure 4.11 Zones of strength gain due to consolidation, 15 years after the construction of Stage 1- Contours of (Su /Su0 )cons 11A
Figure 4.12 Zones of strength gain due to consolidation, 4 years after the construction of Stage 1. Contours of (Su /Su0 )cons 175
Figure 4.13 Comparison of measured settlement (Gauge SI) with calculated settlement using Model 2 176
Figure 4.14 Comparison of the measured and calculated settlement and excess pore water pressure using Model 1 and Model 2 177
Figure 4.15 Zones of strength loss due to destructuration, 15 years after construction of Stage 1. Contour of [Su /Su0) 179
Figure 4.16 Zones of net strength gain (i.e. consolidation overshadows destructuration), 15 years after construction of Stage 1. Contour ofSJSu0>l 180
Figure 4.17 Zones of net strength loss (i.e. destructuration overshadows consolidation), 15 years after construction of Stage 1. Contour ofSJSu0<l 181
Figure 4.18 Development of zones of net strength gain from the 4th year to the 15th year in Stage 1 182
Figure 4.19 Development of zones of net strength loss from the 4th year to the 15th year in Stage 1 183
Figure 4.20 Zones of net strength increase, 7 years after construction of Stage 2 184
Figure 4.21 Zones of net strength loss 7 years after construction of Stage 2 185
Figure 4.22 Comparison of the compression curve in laboratory test with the measured long-term field compression of Gloucester clay under the Accommodation building (from McRostie and Crawford, 2001) 186
xv
Figure 5.1 Illustration of the microstructure tensor, a]-, and the generalized
stress tensor, a'i}2 for transverse isotropy. 218
Figure 5.2 Sample orientation, i. 219
Figure 5.3 The effect of Aon the anisotropy of cu from Tresca's failure
criterion. 220
Figure 5.4 The effect of stress ratio, a[/a'c, on the anisotropy of cu from
Tresca's failure criterion. 221
Figure 5.5 Conceptual behaviour of the 'structured' soil model. 222
Figure 5.6 The effect of sample orientation, i, on the measured and calculated peak and post-peak undrained strength of Gloucester clay. 223
Figure 5.7 The effect of sample orientation, i, on the measured (Law 1974) and calculated (a) axial stress versus strain and (b) excess pore pressure versus strain for Gloucester clay. 224
Figure 5.8 The comparison for sample orientations, /, of 0° and 90° on the measured (Law 1974) and calculated (a) axial stress versus strain and excess pore pressure versus strain (b) stress paths for Gloucester clay. 225
Figure 5.9 The effect of strain-rate on the peak strength of Gloucester clay (Data from Law 1974). 226
Figure 5.10 The effect of sample orientation, i, on the peak strength of St. Vallier clay during CIU triaxial compression tests. 227
Figure 5.11 The effect of sample orientation, /, on the measured (Lo and Morin 1972) and calculated (a) axial stress versus strain and (b) excess pore pressure versus strain for St. Vallier clay. 228
Figure 5.12 The effect of sample orientation, i, on the measured (Lo and Morin 1972) and calculated stress paths for St. Vallier clay. 229
Figure 5.13 Measured and calculated peak undrained shear strength versus
strain-rate for St. Vallier clay 0=0°). 230
Figure 5.14 Influence of A and co on apparent yield surface 231
Figure 5.15 Influence of destructuration on the apparent yield surface of St. Alban clay 232
xvi
Figure 5.16 Compression curves from oedometer compression tests on intact and destructured specimens of St. Alban clay 233
Figure 6.1 Strength profile assumed and measured using field vane and undrained (UU and CIU) tests (experimental data from La Rochelle et al. 1974) 256
Figure 6.2 Plan view and cross-section of St. Alban test embankment 257
Figure 6.3 Generated Meshes for 3D and 2D FEM model 258
Figure 6.4 Measured and calculated vertical displacement of point 'O' for St. Alban Embankment 259
Figure 6.5 Spatial displacement contour of 3D model for St. Alban embankment (at failure) 260
Figure 6.6 Spatial displacement contour V.S. fissures at failure on the top surface on St. Alban Embankment 261
Figure 6.7 The statistic table for the prediction on the failure thickness of Malaysia test embankment (data from MHA 1989b) 262
Figure 6.8 Strength profiles for the Malaysia case (experimental data from
MHA 1989a) 263
Figure 6.9 Plan view of Malaysia test embankment 264
Figure 6.10 Measured and calculated settlement of Malaysia Trial Embankment 265
Figure 6.11 Velocity field in central cross-section of 2D model for the Malaysia trial embankment (at failure) 266
Figure 6.12 Velocity field in central cross-section of 3D model for the Malaysia trial embankment (at failure) 267
Figure 6.13 Plan view of Vernon embankment (modified after Crawford et al. 1995) 268
Figure 6.14 Longitudinal section through the embankment (after Crawford et
al. 1995) 269
Figure 6.15 Distribution of vane strength with depth 270
Figure 6.16 Vertical displacement of Vernon Approach Embankment in 2D analysis 271
Figure 6.17 Plan view and 3D model of Vernon approach embankment 272
xvii
Figure 6.18 Vertical displacement of Vernon Approach Embankment in 3D
analysis 273
Figure 6.19 Spatial displacement contour of Vernon approach embankment 274
Figure 6.20 Plan view and cross section A-A of Waterline test fill 275
Figure 6.21 Measured and calculated displacement by 2D analysis for the Waterline Test Fill 276
Figure 6.22 Measured and calculated displacement by 3D analysis for the Waterline Test Fill 277
Figure 6.23 Illustration of 3D effect on the bearing capacity and the cases studied. 278
xvin
NOMENCLATURE
st] deviatoric stress tensor
J2 secondary invariant of deviatoric stress tensor
(j'm mean effective stress
p' mean effective stress, /?'= (<J\ +<j'2+cr\ ) /3
q deviatoric stress, q = (cr\ -ar\ )
Sy Kronecker's delta
£tj total strain-rate tensor
seij elastic strain-rate tensor
svpy viscoplastic strain-rate tensor
s^i axial strain-rate
Su undrained compression strength
<j' apparent preconsolidation pressure
<y'^ static yield surface intercept
<j'ny ) dynamic yield surface intercept
a'}P overstress
G stress dependent shear modulus
v Poisson's Ratio
K slope of the e - ln(o^) curve in the overconsolidated stress range
A slope of the e - \n{a'v) curve in the normally consolidated stress range
Cs slope of the e - log(cr^) curve in the overconsolidated stress range
slope of the e-log(cr^) curve in the normally consolidated stress range, C.
Cc = ln(10)A
xix
Ca secondary compression index
e void ratio
n power law exponent
a rate-sensitivity parameter {=11 n)
yvp fluidity parameter, denoting the threshold strain-rate of viscosity
yf, yjp fluidity of the structured state and the destructed state, respectively
4>(F) flow function
Moc
Ccs
slope of the Drucker-Prager envelope in ^2j2 - a'm stress space -
normally consolidated stress range slope of limit state line in yJ2j2 - a'm stress space - over consolidated stress range
</>' effective friction angle
W angle of dilatancy
effective cohesion intercept in ^/2J2 - a'm stress space - normally consolidated stress range
, effective cohesion intercept of the limit state in ^U2 - o'm stress space -
over consolidated stress range
Mv dilation parameter to define plastic potential in O/C zone
R aspect ratio of the elliptical cap
sd damage strain
A weight ing parameter
b destructuration-rate parameter
co coefficient indicating the current degree of structure
co0 parameter indicating the initial degree of structure
i clay orientation respect to vertical direction
xx
A parameter of inherent soil anisotropy
77 coefficient of structure anisotropy at
W crest width of embankment
B base width of embankment
ABBREVIATION
EVP
CRS
3D
2D
elastic viscoplastic
constant rate of strain-rate
3-dimensional
2-dimensional
xxi
1
CHAPTER 1
INTRODUCTION
1.1 Introduction
Recently, some researchers have begun to recognize the important effects of clay
viscosity. The most common effects of viscosity on clay behaviour include: variation of
undrained strength with strain-rate, variation of preconsolidation pressure with strain-
rate, and creep deformation under conditions of constant effective stress (i.e. secondary
compression). These phenomena cannot be accounted for in soil mechanics using
conventional elasto-plasticity theories. For example, in triaxial compression tests, a
reduction of the axial strain-rate by one order-of-magnitude usually results in a decrease
in the undrained strength of about 10% for several clays (e.g. Leroueil et al. 1985;
Graham et al. 1983). This introduces uncertainties for geotechnical designs where
stability is assessed using the undrained strength measured from standard laboratory tests
as a result of the significant difference between strain rates during experiments and those
operating during field performance. In addition, results from long-term consolidation
tests and field observations indicated that secondary compression accounts for a
significant portion of the long-term settlement of embankments founded on some clay
(e.g. Lo et al. 1976; Crawford and Bozozuk 1990; and Hinchberger and Rowe 1998).
Thus, there is a need to consider these viscous effects and their impact on the
performance of structures founded on or in natural clay.
2
Most natural clays are structured to some degree, in addition to being rate-
sensitive. Leroueil and Vaughan (1990) and Burland (1990) have perhaps undertaken the
most comprehensive studies of the general influence of structure on the behaviour of
natural clay. The current and subsequent studies show that structure is as important as
other basic engineering properties, such as void ratio and stress history, governing the
engineering response of natural clay. It has been recognized that the degradation of
structure during loading (destructuration) may lead to a significant reduction in undrained
strength. Burland (1990) has shown that the measured yield stress from oedometer tests
on structured clays is often much higher than that for the corresponding remolded
samples. In most cases, structure permits natural clay to exist at a higher void ratio than
the corresponding destructured or remolded clay leading to high compressibility for
stresses exceeding the in situ yield stress.
Lastly, strength anisotropy is another characteristic of many natural clays, which
has been recognized by many researchers (e.g. Lo 1965; Lo and Morin 1972;
Pietruszczak and Mroz 1983; and Zdravkovic et al. 2002). Lo and Morin (1972)
measured the significant impact of anisotropy on the response of two natural clays from
eastern Canada during undrained triaxial tests on samples trimmed at different angles, / ,
to the vertical axis. As i was increased from 0° to 90°, the undrained strength of both
St. Vallier and Gloucester clay decreased by about 30% ~ 50%. Similar behaviour has
been reported by other researchers (e.g. Jardine et al 1997; Symes et al. 1984).
Hence, extending constitutive models for clay to account for the effects of
viscosity, structure, and anisotropy would be desirable to fully capture the key
engineering characteristics of natural clays. In addition, studying the performance of
3
embankment on natural clays would help to improve modern geotechnical design. This
thesis focuses on these two issues. The following section defines important terms used
throughout the remainder of this thesis.
1.2 Definitions
To facilitate reading of the thesis, this section provides definitions and discussion
of some of the important terms and concepts utilized consistently throughout the thesis.
Embankment
Figure 1.1 shows a typical embankment cross-section and a typical undrained
strength profile for the underlying foundation clay. In Figure 1.1, the symbols, W and B,
represent the crest width and the base width of the embankment, respectively. The side
slope is denoted using H: V. The undrained strength profile of a soft clay deposit
typically comprises three layers: a crust, a transition layer, and a soft clay layer with
depth. The undrained strength is typically constant within the crust layer. It decreases to
a minimum value in the underlying transition layer, and then increases with depth in the
soft clay layer.
Preconsolidation pressure
The preconsolidation pressure (cr'p) of clay represents the maximum effective
vertical stress that the clay has experienced in the past. The preconsolidation pressure
can be estimated using an oedometer compression test. Figure 1.2 shows the schematic
of an oedometer apparatus, together with a typical compression curve of clay plotted as
void ratio versus logarithm of vertical effective pressure. Typically, the compression
curve has two distinct portions. The first portion is relatively flat, representing the elastic
state with low compressibility. The second portion has a greater slope and denotes the
4
plastic state corresponding to high compressibility and irrecoverable strains. The
preconsolidation pressure can be determined from the transition stage between the two
portions using a widely accepted Casagrande procedure (see Holtz and Kovacs 1981 and
Craig 1997).
Structure
The term 'structure' used in this thesis specifically refers to the microstructure of
clay, which arises from fabric effects and inter-particle bonding or cementation. The
effect of structure on the mechanical response of natural clay is significant. Structure
typically imparts additional meta-stable strength to natural clay, which leads to strength
loss with large-strain. In addition, structured clay usually exists at higher void ratio than
the equivalent reconstituted clay. Such a state in clay is meta-stable, and leads to high
compressibility under further loading after yielding. The influence of structure is
illustrated for example by the dashed line in Figure 1.2.
Anisotropy
In this thesis, the term 'anisotropy' specifically refers to the variation of
undrained strength with rotation of principal stresses relative to the axis of natural
deposition of a clay. The undrained strength of clay is usually measured using a triaxial
compression apparatus where the drainage from the specimen is not permitted during
loading. In addition to the vertical specimen, clay can be trimmed at different angles, / ,
to the vertical axis (See Figure 1.3). The angle i denotes the sample orientation. A clay
with anisotropy typically yields different undrained strengths depending on the sample
orientation. In this case, to accurately access the stability of embankments and slopes,
the strength anisotropy has to be taken into account in accordance with the different
5
orientation of the major principal stress along the potential failure surface (see Lo and
Milligan 1967).
Yield surface
Clays have a yield surface in generalized stress state. The yield surface is defined
as a surface in stress space, which denotes stress states at which yielding begins. Inside
of the yield surface, stress states are elastic. The classic yield surfaces include: Cam-clay
yield surface, Modified Cam-clay yield surface, and elliptical yield surface (see Roscoe
and Schofield 1963; Roscoe and Burland 1968; Chen and Mizuno 1990; Atkinson 1993).
For inviscid soil, the yield surface is mainly governed by stress history. For viscous clay,
the location of yield surface in stress space is also dependent on the loading strain-rate.
1.3 Thesis Objectives and Outline
This thesis has two aims: (i) to develop a general and efficient constitutive
framework, which can take into account the viscosity, structure, and anisotropy of natural
clays, and (ii) to study selected issues affecting the performance of earth-fill
embankments built on deposits of natural clay.
In Chapter 2, a general elastic-viscoplastic (EVP) theory is described and used to
derive the relationships between undrained strength and strain-rate, preconsolidation
pressure and strain-rate and the coefficient of secondary compression in terms of two
EVP viscosity parameters. Nineteen clays from the literature are used to show that a
unique set of viscous parameters can be used to describe the rate-sensitivity and time-
dependency of many natural clays.
Chapter 3 extends the EVP model to account for clay structure by introducing a
state-dependent fluidity parameter, and a damage law to describe the destructuration
6
process. Calculated and measured behavior of Saint-Jean-Vianney clay is compared for
constant-rate-of-strain /^-consolidation, and both isotropically consolidated undrained
triaxial compression (CIU) tests and constant stress creep tests. The ability of the
extended constitutive framework is evaluated by comparing measured and calculated
creep rupture response, and the measured and calculated influence of strain-rate on the
peak undrained shear strength, post-peak undrained shear strength, and apparent
preconsolidation pressure of Saint-Jean-Vianney clay.
Chapter 4 further investigates the influence of structure degradation on the field
behaviour of a test embankment constructed at Canadian Forces Base (CFB) in
Gloucester, Ontario. The calculated long-term settlement obtained using both structured
and non-structured EVP models are compared with the measured response. This
comparison suggests that the extended EVP model gives improved predictions of
embankment behaviour. Then, the spatial distribution of 'destructuration' in the
Gloucester foundation is examined numerically with time after construction. The
locations of possible weakened zones (destructured) in soil foundation are identified and
the mechanism governing the formation of these zones is investigated. The results may
have implications for the design and analysis of stage constructed embankments
Chapter 5 introduces a tensor approach, which enables the EVP model to account
for the strength anisotropy of natural clays. The advantages and limitations of this
approach are discussed with reference to other constitutive alternatives. Then the new
model is evaluated by comparing the calculated behaviour in triaxial compression tests
with the measured behaviour of two anisotropic natural clays. The comparison shows
that the extended EVP model is able to simulate the anisotropic undrained strength and
7
pore water pressure response of Gloucester clay and St. Vallier clay for various sample
orientations.
Chapter 6 examines three cases involving full-scale test embankments built on
soft clay deposits. The cases are examined using both two-dimensional (2-D) plane strain
finite element and three-dimensional (3D ) finite element analysis taking account of the
true 3D geometry of each case. By comparing the calculated collapse fill thickness from
2D and 3D analyses, it is shown that 3D effects are quite significant for all of the test
embankments examined. Finally, by comparing Finite Element (F.E.) results with a well
known bearing capacity factor, it is shown that the use of bearing capacity factors
commonly used for shallow foundations can be used to approximately assess 3D test
embankments with an aspect ratio of base length to base width less than 2. The analysis
and results presented provide practical insight into some of the key factors that should be
taken into account for the design and construction of embankments and test fills on soft
clay deposits.
Chapter 7 presents a summary of this study and suggestions for further work.
1.4 Original Contributions
The original contributions of this thesis are summarized as following:
Chapter 2 shows that the viscosity of clays can be mathematically quantified
using a unique set of constitutive parameters. In addition, practical guidance is given to
select and measure the viscous parameters directly from experiments without trial and
error. The research in this chapter has been presented in the following manuscripts:
8
Qu, G. and Hinchberger S.D. (2007) Evaluation of the viscous behaviour of natural clay
using a generalized viscoplastic theory. Geotechnique, Submitted October 2007. Prepared
from the research presented in Chapter 2.
Hinchberger, S.D. and Qu, G. (2007) Discussion: the Influence of structure on the time-
dependent behaviour of a stiff sedimentary clay. Geotechnique. Accepted.
The structure and strength anisotropy effects of natural clays are accounted for
within a generalized elastic viscoplastic model described in Chapters 3 and 5. The
research in this chapter has been presented in the following manuscripts:
Hinchberger, S.D. and Qu, G.(2006) A viscoplastic constitutive approach for structured
rate-sensitive natural clays. Canadian Geotechnical Journal, Re-Submitted November
2007. Prepared from the research presented in Chapter 3.
Hinchberger, S.D., Qu, G. and Lo, K.Y.(2007) A simplified constitutive approach for
anisotropic rate-sensitive natural clay. International Journal of Numerical and Analytical
Methods in Geotechnical Engineering. Submitted January 2007, resubmitted October
2007. Prepared from the research presented in Chapter 5
The case studies in Chapter 4 and 6 highlight the significant influence of clay
structure and 3D geometry on the performance of test embankments founded on soft clay.
The research in these chapters has been presented in the following manuscripts:
9
Qu, G. and Hinchberger, S.D. (2007) Clay microstructure and its effect on the
performance of the Gloucester test embankment. Geotechnical Research Centre Report
No. GEOT2007-15, the University of Western Ontario, London, Ontario, CAN. Prepared
from Chapter 4.
Qu, G. Hinchberger, S.D., and Lo, K.Y. (2007) Case studies of three dimensional effects
on the behaviour of test embankments. Canadian Geotechnical Journal. Submitted
August 2007. Prepared from Chapter 5.
10
References
Atkinson, J.H. 1993. An introduction to the mechanics of soils and foundations : through
critical state soil mechanics. McGraw-Hill Book Co., New York.
Burland, J.B. 1990. On the compressibility and shear strength of natural clays.
Geotechnique, 40(3): 329-378.
Chen, W.F., and Mizuno, E. 1990. Nonlinear analysis in soil mechanics : theory and
implementation. Elsevier Science Publishing Company Inc., New York, NY,
U.S.A.
Craig, R.F. 1997. Soil Mechanics. E & FN Spon, New York.
Crawford, C.B., and Bozozuk, M. 1990. Thirty years of secondary consolidation in
sensitive marine clay. Canadian Geotechnical Journal, 27(3): 315-319.
Graham, J., Noona, M.L., and Lew, K.V. 1983. yield states and stress-strain relationships
in a natural plastic clay. Canadian Geotechnical Journal, 20(3): 502-516.
Hinchberger, S.D., and Rowe, R.K. 1998. Modelling the rate-sensitive characteristics of
the Gloucester foundation soil. Canadian Geotechnical Journal, 35(5): 769-789.
Holtz, R.D., and Kovacs, W.D. 1981. An introduction to geotechnical engineering.
Prentice Hall, Inc., Toronto.
Jardine, R.J., Zdravkovic, L., and Porovic, E. 1997. Anisotropic consolidation, including
principal stress axis rotation: Experiments, results and practical implications. In
Proc. 14th Int. Conf. Soil Mech. Found. Engng. Hamburg, Vol.4, pp. 2165-2168.
11
Leroueil, S., Kabbaj, M., Tavenas, R, and Bouchard, R. 1985. Stress-strain-strain rate
relation for the compressibility of sensitive natural clays. Geotechnique, 35(2):
159-180.
Leroueil, S., Bouclin, G., Tavenas, F., Bergeron, L., and La Rochelle, P. 1990.
Permeability anisotropy of natural clays as a function of strain. Canadian
Geotechnical Journal, 27(5): 568-579.
Lo, K.Y. 1965. Stability of slopes in anisotropic soils. Journal of the Soil Mechanics and
Foundations Division American Society of Civil Engineers, 91(SM4): 85-106.
Lo, K.Y., and Milligan, V. 1967. Shear strength properties of two stratified clays.
American Society of Civil Engineers Proceedings, Journal of the Soil Mechanics
and Foundations Division American Society of Civil Engineers, 93(SM1): 1-15.
Lo, K.Y., and Morin, J.P. 1972. Strength anisotropy and time effects of two sensitive
clays. Canadian Geotechnical Journal, 9(3): 261-277.
Lo, K.Y., Bozozuk, M., and Law, K.T. 1976. Settlement analysis of the gloucester test
fill. Canadian Geotechnical Journal, 13(4): 339-354.
Pietruszczak, S., and Mroz, Z. 1983. On hardening anisotropy of ko-consolidated clays.
International Journal for Numerical and Analytical Methods in Geomechanics,,
7(1): 19-38.
Roscoe, K.H., and Burland, J.B. 1968. On the generalised stress-strain behaviour of wet
clay. Cambridge Univesrity Press, Cambridge.
Roscoe, K.H., Schofield, A.N., and Thurairajah, A. 1963. Yielding of clays in states
wetter than critical. Geotechnique, 13(3): 211-240.
12
Symes, M.J.P.R., Gens, A., and Hight, D.W. 1984. Undrained anisotropy and principal
stress rotation in saturated sand. Geotechnique, 34(1): 11-27.
Zdravkovic, L., Potts, D.M., and Hight, D.W. 2002. The effect of strength anisotropy on
the behaviour of embankments on soft ground. Geotechnique, 52(6): 447-457.
13
Figure 1.1 Cross-section of embankment and typical undrained strength profile for
the underlying foundation clay
Clay foundation
Depth Depth
Typical Strength Profile Su (kPa)
Transition Layer
Soft Clay Layer
14
Figure 1.2 Schematic of an oedometer apparatus and a typical compression curve.
Elastic stage Elasto-plastic stage
Ae Schematic of an oedometer apparatus (after Holtz and Kovacs 1981)
*»K)
\ Meta-stable i
15
Figure 1.3 Definition of the orientation angle, i
t Vertical Direction (Direction of deposition/gravity)
A j = 0°
"S. 1/ * J /
\ l - Sample !•----•
Ground level
l-45° See details. ' l . y \ Clay layer * ^_._.=^=^.-_._._ | p,- . t k i
"J Horizontal Direction " 7 7 ^ Z ^ 7 7 ^ s7^T
16
CHAPTER 2 EVALUATION OF THE VISCOUS BEHAVIOUR OF NATURAL CLAY
USING GENERALIZED VISCOPLASTIC THEORY
2.1 Introduction
In 1957, Suklje (1957) proposed the isotache concept to describe the time-
dependent behaviour of clay in one-dimensional compression. The isotaches were
defined as a series of t-a'v compression curves constructed from tests performed at
various constant strain-rates. Since then, the concept of isotaches has been extended
gradually over time to general stress space. For example, Tavenas et al. (1978) estimated
isotaches in p'-q stress space for St. Alban clay using a series of drained and undrained
triaxial compression tests and creep tests. Graham et al. (1983) studied the influence of
strain-rate on Belfast clay using undrained triaxial compression, undrained triaxial
extension and one-dimensional oedometer compression tests. The results of Graham et
al. (1983) were expressed in terms of isotaches also plotted in p'-q stress space.
In addition to rate-sensitivity, many natural clays exhibit significant creep or
secondary compression at constant effective stress during incremental oedometer tests.
Such behaviour is indicative of the viscous response of clay. Although it is generally
recognized that there are similarities between the time-dependent response of clay during
undrained and drained compression, so far there has not been a comprehensive study to
generalize the viscous characteristics of clay for these different stress paths (e.g. triaxial
compression and oedometer compression).
This chapter uses a generalized viscoplastic theory to examine the viscous
A version of this chapter has been submitted to Geotechnique 2007
17
response of 19 clays reported in the literature. The main objectives of this study are: (i)
to investigate if a unique set of viscous parameters can be used to describe the rate-
sensitivity of clay during drained and undrained triaxial compression tests and the
secondary compression (or creep) response exhibited in incremental oedometer tests, (ii)
to link the isotache concept (Suklje 1957; Tavenas et al. 1977; Graham et al. 1983; and
Leroueil et al. 1985) with generalized elastic viscoplastic constitutive theory, and (iii) to
provide guidance for the selection of viscosity parameters for viscous clays. To achieve
these objectives, theoretical relationships are derived from viscoplastic theory for
undrained strength and preconsolidation pressure versus strain-rate expressed in terms of
two viscosity parameters called the fluidity parameter and rate-sensitivity parameter. In
addition, a theoretical relationship is derived relating the fluidity and rate-sensitivity
parameters to the secondary compression index. The measured behaviour of 19 clays is
evaluated using the derived relationships to show that a unique set of viscoplastic
parameters exist for the viscous response of clays during loading along stress paths
involving drained compression or undrained shear. Such a study should be of interest to
engineers and researchers in the field of soil mechanics.
2.2 Theoretical Background
2.2.1 Brief introduction of elastic-viscoplastic theory
Figure 2.1 illustrates the main characteristics of elastic-viscoplastic theory.
Figure 2.1a shows a general 1-D rheologic model for elastic-viscoplastic theory, which
comprises a linear elastic spring in series with a plastic slider and viscous dashpot in
parallel. For this type of model, the strain-rate, s , can be expressed in terms of elastic
18
(se) and viscoplastic (svp) components as follows:
s =ee+svp
[2.1]
Perzyna (1963) proposed an overstress viscoplastic theory to describe the rate-
sensitivity of materials at yield during uniaxial tension. For a steel bar in tension (see
Figure 2.1b), the viscoplastic strain-rate proposed by Perzyna (1963) is:
axial
(. V
r vp axial
(1)
0
for
for O'axial- Vy > 0
° axial ~<Ty ^ 0 [2.2]
where y^is the fluidity parameter with unit of inverse time, n is a power law exponent,
<raxial is the rate-dependent axial yield stress, ay is the yield stress mobilized at very low
strain-rate. The plastic potential from von Mises failure envelop is unity (1).
Perzyna's theory has been extended to geologic materials (e.g. Desai and Zhang
1987, Katona and Mullert 1984, Adachi and Oka 1982), which typically possess a state
boundary surface denoted by A-C-E-0 in Figure 2.1(c). Stress states inside the state
boundary surface are elastic while stress states lying outside the surface are considered to
be viscoplastic. The yield surfaces A-C-E and B-D-F are typically referred to as static
and dynamic yield surfaces, respectively. The static yield surface (A-C-E) defines the
initial onset of time-dependent viscoplastic behaviour. The dynamic yield surface (B-D-
F) passes through the current stress state and is used to define the plastic potential for
associated flow and the degree of overstress. The term 'dynamic' yield surface is used
since viscoplasticity has typically been viewed as a dynamic process (see Sheahan 1996
and Adachi and Oka 1982). In accordance with Adachi and Oka (1982) and Hinchberger
and Rowe (1998), the EVP constitutive equation for normally-consolidated soil is:
19
*, = * ; + # = < V T ; + ( * F ) > OF
da' [2.3]
where Cjjkl is the elastic compliance tensor and a- is the effective stress tensor. The
scalar function, ^ (F) , is the flow function governing the magnitude of the viscoplastic
strain-rate and F can be any valid yield surface function from plasticity theory. The
associated plastic potential, 8F
do' is a unit vector normal to the dynamic yield surface
in <y'm - yJ2J2 space (see Appendix A for details). The theoretical relationships derived
in the following sections also apply to the state boundary surface (ACEO) depicted in
Figure 1(d), which is commonly found for soils.
Two different flow functions, 0(F) , are evaluated. The first is based on the
power law (Norton 1929) extended to general stress states as follows:
<W = H<}/<S))" t2-4]
and
m)-{«? o-'W-o-'W >o my my
a>W _ ,(s) < Q my my
where yvp is the fluidity parameter and n is the power law exponent. As shown in Figure
2.1c, a'^ is the static yield surface intercept (Point A in Figure 2.1c) and a'm(^ is the
intercept of dynamic yield surface with the mean stress axis (Point B). The stress state
denoted by Point D in Figure 2.1c is a state of overstress (o'^-o'^ >0). This type of
flow function has been adopted by Adachi and Oka (1982) and Hinchberger and Rowe
(1998).
20
The second form of <f>(F) considered is an exponential flow function:
^F) = f-cxp[n(-^-l)] [2.5] my
where again yvph the fluidity parameter and n governs the rate-sensitivity.
Finally, although there are many different hardening laws, in the following
discussion, kinematic strain hardening is assumed. Expansion or contraction of the static
yield surface ( c r ^ ) is governed by the viscoplastic volumetric strain (s^) viz.:
d < s ) = | ^ < s ) d ^ [2.6]
where e is void ratio, and A and K are the compression index and recompression index,
respectively.
2.2.2 Strain-rate controlled testing
In this section, the influence of strain-rate in rate controlled laboratory tests is
evaluated by deriving relationships between strain-rate and yield stress and strain-rate
and undrained shear strength for these tests.
The relationship between yield stress and strain-rate
First, considering CRS (constant rate of strain) isotropic compression, a
relationship between loga ' ^ and log(^ a , ) can be derived explicitly from elastic-
viscoplastic theory using Equations [2.3] and [2.4] as follows:
where all of the above parameters have been defined above, and 1/3 is the plastic
potential for axial strain in isotropic compression (see Appendix A). This plastic
£VP
axial fi^s/^s)" dF
da' axial .
21
potential would apply to yield surfaces such as the modified Cam-clay model or the
elliptical cap model (See Figure 2.1c). Taking the logarithm of [2.7] gives:
l°&Zai) = nto& my
V ">y J
+ log(rv;') + log(l/3) [2.8a]
and re-arranging yields:
l o g ( < } ) = a l o g ( ^ ) + [ log(< s ) ) - a logfrvp) - a log(l/3) ] [2.8b]
where a( =11 n) is the rate-sensitivity parameter, and o'^ is the strain-rate dependent
isotropic yield stress corresponding to the axial strain-rate, sv£ial.
At yield and failure, the elastic component of strain, e°., can be neglected without
significant influence on the rate sensitivity relationship (see Appendix B). Hence, the
viscoplastic strain-rate in Equation [2.8b] can be expressed in terms of the total strain-
rate, viz.
log^i - 0 )= a \og(eaxial) + log(aJ'>)- a l og (^ ) - a log(i) [2.9]
Equation [2.9] shows that the power law flow function in Equation [2.4] implies a
linear relationship between log^cr^j and log(£Ta/), which is plotted as a straight line
A-B in Figure 2.2a.
Using a similar approach to that described above, relationships between log(S*d))
and l og fo^ ) and log\(T'p(d)) and l o g ^ ^ ) can be derived, where S(d) is the rate-
dependent undrained shear strength and <j'pd) is the rate-dependent preconsolidation
pressure. For most commonly used yield surfaces (e.g. Cam-clay, Modified Cam-clay,
and the elliptical cap), there is a fixed relationship between the top of the yield surface
22
(see Point F in Figure 2.1c) and the yield surface intercept with the mean stress axis (see
Point B in Figure 2.1c) e.g.:
g(d) g(s)
A r ' ( d )
my my
[2.10]
Substituting Equation [2.10] into [2.9] and modifying the plastic potential for axial strain
Backebol clay Belfast clay Winnipeg clay Ottawa Leda clay Remolded Boston blue clay Gloucester Clay St.Jean Vianney clay Sackfill clay Hong Kong Marine clay San Francisco Bay Mud London clay
1 1
10-' 10"! 10"' 10-: 10": 10-' 10°
Strain rate, /min
61
Figure 2.9 Relation between undrained strength and axial strain-rate for Drammen
clay and Haney clay
1 -
0.9
!E o.8 E
T—
II
ate
^ 0.7 CO —
3
CO II 2
W 0.6
0.5
0 Drammen clay o Haney clay
s 1 1
•
1 1 I
i
I o y
a=0.046| . r £ r
i<5/^ i f9dX s ^T JZf
•° T i X 1 !
1 y | i i
— c o - \ w \ ! | 0 1 ! ! 1
\ I 1 ^ I 1 4 E - d / m i n 1 • ' ' i
' 5
2El-5/min |
10-7 1Q-6 10-5 10-4 10-3 1Q-2 10-1 10°
Strain rate, /min
62
Figure 2.10 Comparison of a estimated from rate-controlled oedometer tests and
undrained triaxial tests ( See Table 2.2).
CO CO CD
CD
c CO k-
T3 C
T3 CD
C o o cb *-* CO k_
E p
a o e d V S - a c r e e p
0.00 0.00
a from rate-controlled oedometer tests, a
63
Figure 2.11 Evaluation on the ability of exponential and power law flow functions to
represent the relationship between preconsolidation pressure and strain-
rate
(a) Experiment results from Batiscan clay, Backebol clay, Gloucester clay, and Drammen
clay
a. 400
300
E 200 c o
' • * - •
CO ;u "o » c o o £ 100
c 2 cc Q. Q. <
A •
Batiscan clay Backebol clay Gloucester Clay Drammen clay Regression line using exponential flow function Regression line using power law flow function
-10-3 10-2 -10-1
Strain rate, /min
Figure 2.11 Evaluation on the ability of exponential and power flow functions to
represent the relationship between preconsolidation pressure and strain-
rate (Cont.)
(b) Results measured from lab and in situ for Gloucester clay (data from Leroueil et al.
1983)
200 CO
a.
CO
3 (A V)
C
,o CO
TO o CO c o o CD
c 5 CO O. CL <
10-2
Strain rate, /min
65
Figure 2.12 Comparison of a estimated from secondary consolidation tests, rate-
controlled oedometer tests, and undrained triaxial tests (See Table 2.2).
O aoedv.s.au
a> o
(A • * - » (0
E o
T3 a; o
•o
c o o a>
£ o
of v s a
o o o
o a. CD
•4-*
CO
0
C o o </i tn a) CO
E p
0.00
a from rate-controlled undrained tests, a uc
66
2.13 Comparisons of a.uc , a_ot&, and a with a avg
.10
.08 lower bound
005
06 -Hong Kong njailrrerclay^J Raticr»an H a w 1
.04
.02 4
0.00 0.00 .02 .04 .06 .08 .10
avg
67
Figure 2.14 Typical triaxial compression curves with step-changed strain-rates.
o
i
0.6 -
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
0.0 -
°1c
r
Confining pressure.kPa Axial strain rate
16%/h
0.255/h ~""~-'
i i
= 5%/h 0.5%/h
0.05%/h
i
f— Belfast clay (Graham, et al. 1983)
Winnipeg clay (Graham, et al. 1983)
i i
10 15 20 25 30
Axial Strain, %
2.15 Illustration of the preferred range of load increment for the measurement
of C„
Compression curve on intact specimen
Structure effect
Compression curve on remolded specimen
Estimate Ca
for load increments exceeding Point A
5*
Vertical Effective Stress, a'v, in log scale (kPa)
Figure 2.16 Normalized a' -e relationship at 10% vertical strain (sv -10%) for
Berthierville clay at a depth of 3.9-4.8m (data from Leroueil et al. 1988)
i
o
¥ • C O
Atev=10%
lab: A
in situ:—
Berthierville at a depth of 3.9-4.8;m
0.9
0.8 1 0 - 9 -I o-4 -to-3
Strain rate, /min
70
re 2.17 Normalized a'p-s relationship at 10% vertical strain (ev = 10%) for St.
Alban clay from both laboratory tests and in situ observance (data from
Leroueil et al. 1988)
Ate =10%
lab: • ^aint Alban clay jat a depth of 3.1 -4.9m
in situ:—
10-4 10-3
Strain rate, /min
71
CHAPTER 3
A VISCOPLASTIC CONSTITUTIVE APPROACH FOR RATE-
SENSITIVE STRUCTURED CLAYS
3.1 Introduction
It is generally recognized that most geologic materials are structured to some
degree (e.g. Leroueil and Vaughan 1990; and Burland 1990; Malandraki and Toll 2000).
For natural clay, there are two general forms of structure: (i) macrostructure which refers
to visible features such as fissures, joints, stratification and other discontinuities in an
otherwise intact soil mass (Lo and Milligan 1967; Lo 1970; Bishop and Little 1967; and
Lo and Hinchberger 2006) and (ii) microstructure which arises from fabric effects and
inter-particle bonding or cementation (Mitchell 1970). Although both types of structure
can strongly influence the engineering response of natural clay, macrostructure such as
fissures and joints can be seen with the naked eye and treated in engineering mechanics
either by introducing joints and/or contacts between discrete elements of intact material
(Cho and Lee 1993; Chen et al. 2000; Li et al. 2007) or by adopting a mass strength for
the clay (Lo 1970 and Lo and Hinchberger 2006). In contrast, the influence of
microstructure is comparatively more difficult to assess in part due to its microscopic
nature. Consequently, the majority of studies reported in the literature over the past 20 to
30 years have focused on either characterizing the influence of microstructure on the
strength and stiffness of natural clay (Leroueil and Vaughan 1990; Burland 1990;
Gasparre et al. 2007; Sorensen et al. 2007; etc.) or on constitutive proposals that include
A version of this chapter has been submitted to Canadian Geotechnical Journal 2007
72
the effects of microstructure (Baudet and Stallebrass 2004; Callisto and Rampello 2004;
Karstunen et al. 2005).
Typically, clay microstructure, hereafter referred to as structure, is mechanically
characterized by comparing the response of natural intact clay to that of the
corresponding reconstituted material. Examples of the influence of structure on the
mechanical response of natural clay are given in Figures 3.1 and 3.2. Figure 3.1
compares the response of undisturbed and reconstituted Bothkennar clay during
oedometer compression (Burland 1990) and Figure 3.2 compares similar behaviour for
London clay during triaxial compression (see Sorensen et al. 2007 and Hinchberger and
Qu 2007). Additional examples of the behaviour of structured clay during oedometer and
triaxial compression tests can be found in Mesri et al. (1975), Philibert (1976), and Locat
and Lefebvre (1985).
As shown in Figure 3.1, structure permits natural clays to exist at higher void
ratios than the equivalent reconstituted materials. Such a state in clay is typically
metastable leading to high compressibility when loaded past its preconsolidation pressure
(Vaid et al. 1979; Leroueil et al. 1985). In addition, structure imparts additional strength
to the soil skeleton above that which can be typically accounted for by state-parameters
such as void ratio and stress history (see Figure 3.2). Again, this additional strength is
typically metastable leading to significant post-peak strength loss with large-strain (Lo
and Morin 1972) and creep-rupture at deviator stresses exceeding the large-strain post-
peak strength (Philibert 1976). Behaviour such as that depicted in Figures 3.1 and 3.2 has
lead various researchers to conclude: (i) that the effect of structure on the mechanical
response of natural clay is as significant as state parameters such as void ratio and stress
73
history, which are commonly used in traditional soil mechanics models (Leroueil and
Vaughan 1990) and (ii) it is critical to include structure and loss of structure during
straining in constitutive models for natural clays (Baudet and Stallebrass 2004).
Recently, both rate-independent (Liu and Carter 1999; Baudet and Stallebrass
2004; Callisto and Rampello 2004; and Karstunen et al. 2005) and rate-dependent (e.g.
Kim and Leroueil 2001 and Rocchi et al. 2003) constitutive models have been proposed
to model the mechanical response of structured clay. However, since most structured
clays exhibit significant strain-rate sensitivity, creep and stress relaxation (Vaid et al.
1979; Leroueil et al. 1983; Silvestri et al. 1984; Leroueil et al. 1985), constitutive models
that account for the viscous behaviour of clay are desirable. In terms of time-dependent
constitutive proposals, the 1-dimensional elastic-viscoplastic model described by Kim
and Leroueil (2001) has been shown to provide an encouraging description of the
Berthieville test embankment (Kim and Leroueil 2001). Although 1-dimensional models
can be sufficient for practical problems involving 1-dimensional settlement, they are not
suited for the study of problems involving 2- or 3-dimensional behaviour. Rocchi et al.
(2003) proposed an elastic-viscoplastic constitutive model for 2-dimensional analysis of
structured clay. This model (Rocchi et al. 2003) was a useful step forward, however, it
has been shown to only roughly describe the engineering behaviour of structured clay
during K'0 compression. Currently, a time-dependent constitutive model capable of
describing the mechanical behaviour of structured clay for generalized 2-dimensional
loading and stress-paths other than K'a -compression does not exist.
The primary objective of this chapter is to describe the extension of an existing
elastic-viscoplastic constitutive model (Hinchberger 1996; Rowe and Hinchberger 1998;
74
Hinchberger and Rowe 1998) to describe the influence of structure on the engineering
behaviour of rate-sensitive structured natural clay. In the extended model, soil structure
is accounted for mathematically using a state-dependent viscosity parameter, and a
damage law that describes 'destructuration' of the clay. The model is tested by
comparing calculated and measured behavior of Saint-Jean-Vianney (SJV) clay for
constant-rate-of-strain K'a -consolidation, and both isotropically consolidated undrained
triaxial compression (CIU) tests and constant load CIU triaxial creep tests. Though these
comparisons, it is shown that a single elastic-viscoplastic constitutive model can describe
behaviour such as accelerated creep rupture, the influence of strain-rate on the peak
undrained shear strength, large-strain post-peak undrained shear strength, and the
apparent preconsolidation pressure of a structured natural clay. In addition, the
constitutive model does not rely on multiple or nested yield surfaces, which simplifies the
formulation. The research presented in this chapter suggests a potential mathematical
link between the time-dependent response of natural clay during tests involving either
constant volume shear or volumetric compression. The model and its extensions should
be of interest to researchers and practitioners in the field of soil mechanics or
geomechanics.
75
3.2 Theoretical Formulation
3.2.1 Overstress viscoplasticity
In the following sections, the elastic-viscoplastic model proposed by Hinchberger
and Rowe (1998) is extended to account for the effect of 'structure' on the engineering
behaviour of natural rate-sensitive clay. The Hinchberger and Rowe (1998) model is a
three-parameter elastic-viscoplastic formulation based on the elliptical cap yield surface
(Chen and Muzino 1990), Drucker-Prager failure envelope, Perzyna's theory of
overstress viscoplasticity (Perzyna 1963) and concepts from the critical state framework
(Roscoe et al. 1963). Full details of the constitutive model are presented by Rowe and
Hinchberger (1998) and Hinchberger (1996). The following is a brief summary of the
model.
In the normally-consolidated stress range, the constitutive relationship is:
" y y 2G 3(1 + e)(j'm ll y 2G 3(1 + e)a'm
lJ r ^ V " da'
W))- (<V<Ni M < ^ < ] [31]
0 M<><< s ) j
where stj is the deviatoric stress tensor, a'm is the mean effective stress, Stj is
Kronecker's delta, G is the stress dependent shear modulus, K is the slope of the
e - ln(ciy) curve in the over-consolidated stress range, e is the void ratio, yvp is the
fluidity parameter, &^ I o'^ is mathematically the overstress ratio (described below)
anddF/dcrJj is the plastic potential, which is derived as a unit norm vector. The flow
function, <|>(F), in Equation [3.1] is a power law (Norton 1929) similar to functions used
76
by Adachi and Oka (1982) and Kantona (1984).
For normally consolidated clay, associated flow has been adopted (see Figure
3.3). Accordingly, the plastic potential, dFjda'y , is derived using the elliptical cap
equation:
F=(o'm - l ) 2 +2J2R2 -(og> -if =0 [3.2]
where cyjj^is the dynamic yield surface intercept, 1 and R are parameters defining the
aspect ratio of the elliptical cap, and J2 is the second invariant of the deviatoric stress
tensor, stj. In the constitutive formulation, the Drucker-Prager failure envelope is used to
define the critical state viz.:
F=Mcsa^+c;s-V2l7=0 [3.3]
where Mcs is the slope of the Drucker-Prager envelope and c'cs is the effective cohesion
intercept in *J2J2 - o'm stress space. The cap parameters 1 and R are determined so that
the top of the cap (point B in Figure 3.3) is coincident with the critical state line or
Drucker-Prager envelope. Lastly, strain hardening of the static yield surface, da'$, is
proportional to incremental plastic volumetric strain, <9e , viz.:
( X - K )
Thus, for normally consolidated clay, there are eight constitutive parameters that
must be determined for this model: the compression and recompression indices, X
and K , the critical state parameters, Mcs and c'cs, the static yield surface intercept, cr' -1,
the aspect ratio of the elliptical cap, R, the power law exponent, n , and the fluidity
77
parameter, yvp . The distinction between static and dynamic yield surface will be
addressed below.
In the overconsolidated stress range, the constitutive equation (Equation [3.1])
incorporates the Drucker-Prager envelope:
F=M 0 C a ' m + c ; c -V2J^ [3.5]
where M o c and c'oc define the slope and cohesion intercept of the yield envelop in
•yJ2i2 - <*'m stress space. As a result, the state boundary surface or yield surface of the
soil is denoted by A-B-C in Figure 3.3 and defined by Equations [3.2] and [3.5]. As
noted above, the Drucker-Prager equation is also used to define the critical state line. In
this study, a non-associated flow rule was required to describe the volumetric response of
Saint-Jean-Vianney Clay in the overconsolidated state. Accordingly, the parameter M
has been utilized with Equation [3.5] to define the plastic potential, dg/da'^ (see point
D in Figure 3.3) and the resultant dilatant behaviour of Saint-Jean-Vianney clay for
plastic states of stress approaching the critical state from the dry side.
In total, eleven constitutive parameters must be measured to fully define the
elastic-viscoplastic material behaviour. Although the number of constitutive parameters
is significant, the parameters can be estimated from standard incremental oedometer
consolidation, and undrained triaxial compression tests undertaken at different strain-
rates.
3.2.2 Numerical overstress
Figure 3.3 illustrates the static (or reference) yield surface, the dynamic yield
surface and the definition of overstress adopted in the elastic-viscoplastic formulation. In
78
viscoplastic theory, the static yield surface defines the yield loci mobilized at very low
strain-rates. Stress states that lie inside the static yield surface are elastic. The intercept
of the static yield surface with the mean stress axis is a ' ^ . The dynamic yield surface is
used to define the level of overstress and the plastic potential, dF/dcr'y, for time-
dependent plastic flow. The intercept of the dynamic yield surface with the mean stress
axis is c'^y*. In accordance with overstress viscoplasticity (Perzyna 1963), stress states
are permitted to exceed the yield surface of the material (in this case the static yield
surface). Points D and E in Figure 3.3 illustrate two states of overstress. Referring to
Figure 3.3, a dynamic yield surface is defined passing through states of overstress (e.g.
Points D and E) and, a'jff/a'®, defines the overstress ratio (&lff/a'£j= 1.1 implies
10% overstress). The resultant rate-of-plastic flow, sjf , is governed by the flow
function, <|>(F), in Equation [3.1]. As a result, a series of isotaches exists in yJ2J2 -a'm
stress space (see Figure 3.3), which defines states of equivalent overstress or flow
potential, (|)(F). Suklje (1957) proposed similar isotaches for equal volumetric strain-
rates in the e-a'v plane. In this chapter, the concept is applied to the magnitude of the
viscoplastic strain-rate tensor in ^2J2 -<y'm stress space.
3.2.3 Modification for soil structure
Most structured soils exhibit characteristics such as creep rupture during both
drained and undrained triaxial creep tests (Vaid et al. 1979; Lefebvre et al. 1982).
Previously, this type of behaviour has been modeled using overstress viscoplasticity
theory (Perzyna 1963). For example, Adachi et al. (1987) introduced a state-dependent
79
fluidity parameter in the Adachi and Oka model (1982) to account for accelerated creep
rupture of Umeda clay (Sekiguchi, 1984). Aubrey et al. (1985) describe a similar
modification of the Adachi and Oka (1982) model utilizing a damage law. Recently,
Kimoto et al. (2004) incorporated state-dependent viscosity parameters in the Adachi and
Oka (1982, 1995) model to describe strain softening of structured clay. However, in spite
of the potential of this approach, relatively little attention has been given to the use of
state-dependent viscosity parameters to describe the behaviour of time-dependent
structured clay for generalized stress states.
Here, Equation [3.1] is extended using a state-dependent fluidity parameter to
describe the engineering behaviour of structured rate-sensitive clay. In the new
formulation, the parameter, oa0, is introduced to mathematically define the structure viz.:
coo={rTlfsPYn [3.6]
and
n=lla [3.7]
where yjp is the fluidity of the structured or undisturbed clay fabric, y^p is the fluidity of
the destructed or intrinsic fabric, and n is the power law exponent from the power law in
Equation [3.1], and a is the rate-sensitivity parameter (see Chapter 2). The structure
parameter, oo0, is related to common engineering parameters as shown below.
Next, the concept of damage strain, ed , is used to define the transition from an
initially viscous state (the structured state) to a more fluid destructed state viz.:
ded = A/(l-A)(de:P1)2+A(desvp)2 [3.8]
In Equation [3.8], originally proposed by Rouainia and Wood (2000), ded is the
80
incremental damage strain, d e ^ and de^p are incremental plastic volumetric and plastic
octahedral shear strains, respectively. A is a weighting parameter, which is assumed to
be 0.5 similar to Baudet and Stellebrass (2004). Lastly, an exponential damage law is
introduced to describe the process of structure degradation:
aX8d)=[l + (co0n-l .o)e-b e d]1 / n [3.9]
where b is a material parameter governing the rate of destructuration, ed is the damage
strain, 6)0 defines the initial structure and co(ed) defines the state-dependent structure
level. On inspection of Equation [3.9], it can be seen that co(ed)=a)0 for undisturbed
clay (e.g. ed = 0 ) and that co(ed) decreases to 1 as the plastic strain and consequent
damage strain becomes large. Accordingly, the fluidity parameter is a function of
damage strain:
fp(ed)=yjp lcon{ed) [3.10]
and the viscoplastic strain-rate tensor is:
^p=Yvp(ed)(<t)(F))^ [3.H]
In the extended elastic-viscoplastic constitutive model, the fluidity of the
structured clay fabric, yJp, is assumed to be significantly lower than the fluidity of the
destructured fabric, y^p. For the undisturbed structured state, plastic deformation of the
clay fabric is initially restrained by the low structured fluidity, yjp , permitting overstress
to build up relative to the static state boundary surface. However, with continued plastic
straining, damage causes the soil viscosity to break down and the clay fabric to become
more fluid. This process is commonly referred to as destructuration (Baudet and
81
Stallebrass 2004). Thus, it is hypothesized that structure is caused by viscous bonding
between particles and that destructuration is a stress relaxation phenomenon whereby the
viscosity of the structured soil is gradually reduced due to plastic strain until eventually
the destructured or intrinsic state is reached (governed by the Hinchberger and Rowe
(1998) model). This hypothesis is tested using Saint-Jean-Vianney clay. The following
sections describe the methodology of this study and selection of material parameters for
the structured soil model.
3.3 Methodology
3.3.1 Laboratory tests
Vaid et al. (1979) reported the results of constant rate-of-strain K„ -consolidation,
and CIU undrained triaxial compression and triaxial creep tests performed to study the
time-dependent behaviour of SJV clay. The experimental program and methodology is
described in detail by Vaid et al. (1979) and Campanella and Vaid (1972). Only those
details required for the present study are repeated here. The specimens utilized in the
laboratory program (Vaid et al., 1979) were trimmed from block samples retrieved from
the site of the SJV slide. Consequently, the samples are considered to be of a high
quality and the measured laboratory behaviour is considered representative of the
undisturbed natural clay; notwithstanding that limitations of the apparatus used by Vaid
et al. (1979) may have affected the measured response. In addition, Vaid et al. (1979)
observed that the block sample used in their testing program had distinct upper and lower
layers and that the behaviour of these layers was significantly different during the
laboratory testing. Thus, laboratory results are separated into upper and lower layers.
82
Properties of Saint-Jean-Vianney clay are summarized in Table 3.1 and the experimental
results are reproduced and compared with the constitutive model in Figures 3.6, 3.8 and
3.11 to 3.19, inclusive.
3.3.2 Numerical approach
Calculated behaviour has been obtained by modeling laboratory test conditions
using the finite element program AFENA (Carter and Balaam 1995), which has been
modified by the authors to include a rate-sensitive 'structured' clay model. In all cases,
8-noded rectangular isoparametric elements were used assuming axisymmetric geometry.
For each test, the number of elements and time-steps were varied to ensure convergence
of the calculations. An undrained finite element formulation was used to obtain the
calculated behaviour of SJV clay for isotropically consolidated undrained (CIU) triaxial
compression tests. A fully drained formulation (e.g. uncoupled) was adopted for constant
rate-of-strain K'0 -consolidation tests. Chen and Muzino (1990) describe similar
formulations for elastoplastic analysis. A fully coupled formulation was used to model
undrained creep and creep-rupture (Hinchberger 1996). In all cases, an incremental
solution approach was adopted.
For all compression tests on SJV clay, compression was simulated by applying
boundary displacements at a rate that matched the displacement-rate in the corresponding
laboratory test. Smooth rigid end conditions were assumed. For triaxial compression of
overconsolidated clay, there is a limitation on the stress state that can be applied to the
specimen. It is normally assumed that the excess pore pressure in a triaxial specimen
cannot exceed the cell pressure. Thus, for calculated behaviour during CIU triaxial
compression and creep tests, stress states exceeding the triaxial limit were corrected back
83
to the triaxial limit and the nodal forces required to make this correction were applied in
the force vector for the subsequent increment to maintain equilibrium. This type of
approach is commonly used in incremental elastoplastic analysis to correct stress states
that exceed the failure criterion.
As noted above, CIU triaxial creep tests were modeled using a fully coupled
formulation. To simulate the behaviour of SJV clay during CIU creep, uniform deviator
stresses were specified at the top mesh boundary. This approach ignores stress
concentrations within the sample due to the relatively stiff end-caps. Axial loads were
applied incrementally over a period of 30 seconds and maintained at a constant level for
the duration of the test. This was done to avoid numerical instability. A smooth rigid
boundary was adopted at the bottom of the finite element mesh and pore pressures were
constrained by the triaxial limit as described above (in this case ue < G'celi = 40 kPa).
3.3.3 Selection of constitutive parameters
The material parameters utilized in this chapter can be divided into three groups
as summarized in Table 3.2. The three groups include: (i) conventional elastoplastic
constitutive parameters, which define the variation of soil stiffness and strength versus
the state variables void ratio and stress history, (ii) the intrinsic viscosity parameters
(y^p and n), which govern the fluidity and rate sensitivity of the soil skeleton, and (iii)
structure and destructuration parameters ( co0andb ), which govern the structure
component of the constitutive behaviour. The following is a brief description of how the
parameters were derived from the experimental tests. Additional details can be found in
Qu and Hinchberger (2007).
A single set of elastic-plastic parameters were used for the analyses reported
84
below. First, the critical state parameter, Mcs = 1.34, was determined from the measured
structured friction angle (()>' = 40°) of SJV clay reported by Vaid et al. (1979). Figures
3.4 and 3.5 illustrate selection of the state boundary surface parameters from laboratory
results in Saihi et al. (2002). In the N/C stress range, the aspect ratio of the elliptical cap
yield surface, R = 0.7 , was estimated from the stress path response of normally
consolidated SJV clay during CIU triaxial compression. Figure 3.4 illustrates this
parameter selection. Similarly, in the O/C stress range, the yield surface parameter,
Moc = 0.48, was estimated from CIU triaxial tests on overconsolidated specimens of SJV
clay as shown in Figure 3.5. The intercept of the Drucker-Prager envelop in the O/C
stress range, c^., is a dependent parameter determined by the yield surface intercept
(either a ^ or a ^ ) . The initial void ratio, e0 =1.15, was estimated from the natural
moisture content and specific gravity reported in Table 3.1 and Poisson's Ratio, v=0.33,
was assumed. Only one of the plastic constitutive parameters, M ¥ =0.01, was obtained
by some trial and error. Initially, calculations were performed assuming associated flow
in the O/C stress range; however, such calculations tended to overestimate the post-peak
pore pressures at large-strain by about 20%. Consequently, reduced dilatancy was
assumed. Further details are provided in Chapter 3.
In the constitutive model, compressibility of the intrinsic soil skeleton is defined
by the recompression and compression indexes, K and X, respectively. To estimate A,,
normalized one-dimensional compression curves for SJV clay where plotted as shown in
Figure 3.6 (e.g. data from each test was normalized by the mobilized preconsolidation
pressure c'p). Figure 3.6 also shows the assumed intrinsic compression line (ICL). For
85
SJV clay, the intrinsic compression index, A,, was taken to be 0.26 giving an ICL parallel
too but below that of clays from Drammen, Tilbury, St. Andrews, Tilbury, Alvangen and
several ocean cores over the stress range lOOkPa to l,000kPa (see Figure 3.7). From
Figure 3.7, it can be seen that the assumed ICL for SJV clay is parallel to that measured
for other natural clays up to a vertical stress of about 2000kPa. For stresses exceeding
2000kPa, it is anticipated that the intrinsic compression index, X, of SJV clay would
reduce since ICL's are typically concave upward (Burland 1990). The recompression
index, K , was taken to be 0.02, which can be easily deduced from the measured
compression behaviour shown in Figure 3.6.
The power law exponent, n, can be estimated from data presented in Figure 3.8.
For elastic-viscoplastic models based on a power law flow function, (|)(F), n can be
obtained by plotting either undrained shear strength, S u , or apparent preconsolidation
pressure, a ' , versus strain-rate on a log-log scale. The power law exponent (n = 22) is
inversely proportional to the slope of this plot (see Figures 3.8a, b and c). In addition, it
should be noted that for many natural clays, including SJV clay, the power law exponent,
n , remains constant during 'destructuration' or straining as shown in Figure 3.9 for
Winnipeg clay, Belfast clay and London clay (see Figure 3.2 for the stress-strain response
of London clay). Exceptions to this have been noted by Sorensten et al. (2007).
The fluidity parameters (yjp and y,vp) and the static yield surface intercept ( a ^ )
are inter-related parameters and the most difficult parameters to assess for overstress
viscoplastic formulations. Detailed guidance on the selection of these parameters can be
found in Chapter 3. In the absence of specific testing to determine the fluidity of SJV
clay, a structured fluidity of lxlO"10 min"1 was assumed from long-term observations at
86
the Berthierville, Gloucester and St. Alban test sites (see Leroueil 2006). These are also
Champlain clay deposits. The structured fluidity (lxlO10 min"1) defines the transition
from inviscous behaviour for strain-rates less than or equal to yjp to viscous behaviour
for strain-rates greater than yj p . Referring to Figure 3.8(a) and (b), the static yield
surface intercept for both the upper (a£y} =405kPa) and lower (a^y* =518kPa) layers
of SJV clay can be estimated from Point 'A' using Equations [3.2] and [3.5], which
define the state boundary surface. Alternatively, the static yield surface intercept can be
deduced from the apparent preconsolidation pressure versus strain-rate assuming
ysvp =lxl0"10min_1 , K'o=0.5 and using Equation [3.2]. Referring to Figure 3.8(c), the
static yield surface intercept estimated from the consolidation response varies from
370kPa to 410kPa, which is comparable to that deduced in Figure 3.8(a) for the upper
clay layer. Thus, it has been inferred that the constant rate of strain consolidation tests
were performed on clay from the upper layer although this is not explicitly stated by Vaid
et al. (1979).
The structure parameter, co0, can be obtained from either: (i) undrained shear
strength versus strain-rate (see Figures 3.8a and 3.8b) or (ii) from the intrinsic, o'p_{, and
structured, a ' s , preconsolidation pressures in oedometer compression (see Figure 3.6).
In this chapter, the latter approach is used. From Figure 3.6 and Equation [3.2], it can be
shown that the structure parameter is a>0 = a'p_s IG'^ =1.68. Lastly, from Equation [3.9],
the constitutive parameter b governs the rate-of-destructuration and the magnitude of
strain at which the intrinsic state is reached. Referring again to Figure 3.6, it can be seen
that SJV clay reaches the intrinsic state during 1-D compression at an axial strain of
87
approximately 13% (e.g. 0.28/(1 + 1.15)=0.13), which can be achieved using Equation
[3.9] for b = 120. Similar analysis of the behaviour during CIU triaxial compression can
be used to deduce b = 4000. The rate of damage is significantly higher during CIU
triaxial compression, which can be attributed to strain localization.
3.4 Evaluation (Saint-Jean Vianney Clay)
3.4.1 Theoretical behaviour of the model for CIU triaxial compression
Figure 3.10 illustrates the basic features of the extended constitutive model during
CIU triaxial compression test on heavily over-consolidated structured clay. Initially, in
accordance with the test conditions, the clay is isotropically consolidated to a stress state
significantly lower than the static yield surface (see Point 1 in Figure 3.10). During
undrained triaxial compression, the effective stress-path moves from point 1 to 2 where
the triaxial limit is reached and then from point 2 to 3 on the Drucker-Prager envelope
where first yield occurs. Continued strain-rate controlled compression causes the stress-
state to exceed the static yield surface moving from point 2 to point 4 along the triaxial
limit. During this stage of the test, significant overstress builds in the model due to the
low fluidity of the structured soil skeleton, yjp . The initial low fluidity, yjp restrains
plastic deformation of the soil skeleton and the resultant load-displacement response is
essentially elastic from point 1 to 4. From point 1 to 4, components of the plastic strain-
rate tensor are finite but very small: non-associated flow is assumed (see Point 3).
At point 4, the overstress becomes large enough to cause significant plastic
straining and consequent destructuration, ed , which begins to dominate the constitutive
behaviour. During further compression, the soil fluidity increases from yjp to y^ in
88
accordance with Equations [3.9] and [3.10] and the overstress built up during
compression from point 1 to point 4 dissipates from point 4 to 6. The strain-softening is
treated as a stress-relaxation problem and the rate of softening is governed by the
parameter b in Equation [3.9].
3.4.2 Calculated and measured behaviour for constant rate-of-strain triaxial
compression
As discussed above, Vaid et al. (1979) carried out constant rate-of-strain CIU
triaxial compression tests on specimens of SJV clay consolidated to an isotropic effective
stress of 40kPa. A low consolidation pressure was chosen to study the engineering
behaviour of SJV clay in a highly overconsolidated state. Figures 3.11(a) and (b) show
calculated and measured behaviour for the upper and lower layers during CIU triaxial
compression. As noted above, Vaid et al. (1979) reported the existence of both upper
and lower clay layers in the block sample used for their investigation. Figure[G36] 3.12
compares the measured and calculated peak and post-peak (large-strain) deviator
stress, a d , versus strain-rate and Table 3.2 summarizes the constitutive parameters used
in the computations.
Focusing on Figure 3.11(a) and (b), it can be seen that there is reasonable
agreement between the calculated and measured stress-strain response of SJV clay.
Differences between measured and calculated behaviour prior to reaching the peak
strength can be attributed to the elastic properties used in the analyses. The finite element
calculations were undertaken using an average elastic modulus deduced from all of the
undrained compression tests whereas the lower clay appears to be stiffer and the upper
clay is less stiff than the average value selected. Other than some variations, which can
89
be attributed to natural variation of SJV clay, the agreement between measured and
calculated behaviour is reasonable.
Referring to the excess pore pressure response plotted in Figures 3.11(a) and (b),
there is generally good agreement between the measured and calculated excess pore
pressures at the peak strength and at the large-strain post-peak state (e.g. for axial strains
exceeding 2.5%). After reaching the peak strength, however, there is considerable
divergence of the calculated and measured behaviour particularly for the upper layer as
shown in Figure 3.11(a). Variations between the measured and calculated excess pore
pressure are less for the lower layer shown in Figure 3.11(b). Overall, the calculated and
measured response is for the most part similar and variances can be attributed to the use
of average constitutive parameters (E and b [G37]) for the computations and the natural
variation of the SJV clay.
Figure 3.12 compares calculated and measured peak and post-peak shear strength
for both the upper and lower clay layers. From Figure 3.12, it can be seen that there is
very good agreement between measured and calculated peak undrained shear strength
versus strain-rate for both upper and lower clay layers. The constitutive model
overestimates the large-strain post-peak strength ( e^^ > 3%) versus strain-rate for the
lower clay layer but gives good predictions of the post-peak strength for the upper layer.
In general, the data presented in Figure 3.12 suggests that the lower clay is more
structured than the upper clay and over prediction of the post-peak strength of the lower
layer is a consequence of using the same structure parameter for both layers (see Table
3.2). A higher structure parameter is[G38] also indicated by Figure 3.8(b) for the lower
layer. Considering that the structure parameter used in the computations was estimated
90
from the e- loga^ response of SJV clay, the general behaviour of the model is quite
satisfactory.
3.4.3 CIU triaxial creep tests
In addition to triaxial compression, CIU triaxial creep tests were performed at
constant deviator stresses ranging from 430 kPa to 630 kPa. Figures 3.13(a) and (b)
compare measured and calculated axial strain versus log-time for the upper and lower
layers, respectively. Figure 3.14 shows measured and calculated creep rupture time. The
creep tests were simulated using the same constitutive parameters used in the previous
section (see Table 3.2).
Referring to Figures 3.13(a) and (b), all of the laboratory test specimens were able
to initially support the applied deviator stress, od, for some time prior to failure. Failure
was manifest by accelerated creep rates (creep rupture) accompanied by a rapid reduction
in pore pressure. The constitutive formulation described in this chapter is able to
simulate such behaviour. The theoretical response shown in Figures 3.13(a) and (b) is
very similar to the measured response. In the theoretical calculations, the applied
deviator stresses exceed the long-term static state boundary surface of the material. The
material begins to creep at a rate governed by the structured fluidity, yjp . However, with
time, damage or destructuration occurs (governed by Equations [3.8] and [3.9]) causing
the fluidity of the soil skeleton to increase from yjp to eventually y,vp. This process of
destructuration leads to accelerating axial creep rates and creep rupture. For the structure
parameter assumed in the analysis, co0 =1.68, the clay fluidity increases by almost five
orders of magnitude during the calculations due to damage strain.
91
Detailed inspection of Figure 3.13(a) for the upper clay shows that the calculated
initial strain during each test exceeds the measured strain. This can be attributed to the
use of an average modulus of elasticity for the computations. For the upper layer, the
measured and calculated times to failure are close. For the lower clay layer shown in
Figure 3.13(b), the calculated and measured initial strains are in good agreement except
for the test undertaken at a deviator stress, ad , of 630kPa. The high initial strain
measured during this test is not consistent with the other experimental observations
suggesting possible sample disturbance. Again from Figure 3.13(b), there is good
agreement between calculated and measured time to creep rupture (rupture life) at a
deviator stress of 575 kPa. For od =630kPa and 440 kPa, however, the calculated creep
rupture times become approximate as discussed below. For both upper and lower layers
(see the lower parts of Figs. 13(a) and (b)), the rapid generation of excess pore pressure
that occurs at failure (rupture) is predicted by the model.
Figure 3.14 compares calculated and measured creep rupture life during CIU
triaxial creep tests. As observed above for the upper clay, there is good agreement
between the theoretical and measured rupture life. For the lower clay, the agreement
between measured and calculated behaviour is less accurate and can be characterized as
approximate. The difference between calculated and measured rupture life at
Gd = 630kPa can be attributed at least in part to the time required to apply loads to the
specimen during the computations (30 seconds). Considering, however, that natural
variation of Champlain clays can be quite significant (Robertson 1975), the constitutive
model appears to give useful predictions of creep rupture life even for the lower clay
layer. From an engineering point of view, predicting instability or meta-stability is an
92
important characteristic whereas estimating the exact time of the instability is of lesser
importance.
Figure 3.15 shows the measured and calculated creep rates prior to creep rupture
illustrating the main limitation of the constitutive model. The constitutive model predicts
constant creep rates prior to creep rupture whereas the measured creep rates reduce with
time. Similar measured behaviour has been reported for other clays (Bishop and
Lovenby 1973 and Tavenas et al. 1978). However, on reinspection of Figures 3.13(a)
and (b), the axial strain that accumulates prior to creep rupture is generally less than
0.1%, which would be difficult to detect or measure in situ. Thus, from a practical point
of view, the inability to predict diminishing creep rates with time is not a significant
limitation. The phenomenon, however, can be predicted by allowing some strain-
hardening of the Drucker-Prager envelop similar to that done by Lade and Duncan[G39]
(1973) or by assuming some rotational hardening of the state boundary surface
introducing additional constitutive parameters (see Appendix F).
Figure 3.7 Intrinsic compressibility of different clays (adapted from Burland
4H
3H
•g
2 2H
LL: Liquit Limit
LL=109
LL=80
LL=63
LL=46
Assumed A,=0.26 for St. Jean Viartney cla' with LL=35
-Q-
-*-
• St. Jean Vianney clay St. Andrew clay Oslofjord Ocean cores Tilbury Alvangen G. of mexico Gosport Pisa clay Avonmouth Drammen Grangemou Drammen Detroit Milazzo S. Joaqui Milazzo Po Valley
\ A L L = 6 4
= - - 1 1 = 6 2
LL=4oS5*fe - 1 -
101 10"1 10° 102 103 104 105
Vertical Effective Stress, a' , kPa
115
Figure 3.8 Estimation of n anda'my from undrained triaxial compression and
CD test on undisturbed London sample CU test on undisturbed London sample CU test on reconsituted London sample CU test on undisturbed Belfast clay CU test on undisturbed Winnipeg clay
— Data from Sorensen et al. (2007) —i and Hinchberger and Qu (2007) J Data from Figure 3.2
Data from Figure 3.9(a)
Best fit n=43 for London clay
Best fit n=29 for Winnipeg clay
Axial Strain,%
119
Figure 3.10 Theoretical behaviour of the structured soil model during CIU triaxial
compression[G49].
(c)
lg(<0)
•••-
1 (b)
• 2 /
IY
,**
Slight *Iegati
. Triaxial Limit 1 14* Critical State Line
• 5 /
Jtry ^ ^ 6 / > »« . / X-<—Dynamic Yield
^vj(— Initial Static Yield Surface
Contraction Due to —uft 1
11 1 — • Vertical Strata €v 1
Slope is governed by the parameter b
J my6 ^ my u my6 0
6 f Destructured State
Vertical Straia e»
120
Figure 3.11 Measured and calculated behaviour of SJV clay during CIU triaxial
compression,
(a) Upper layer
700 I p r
Axial Strain (%)
Axial Strain (%)
Figure 3.11 Measured and calculated behaviour of SJV clay during CIU triaxial
compression. (Cont.)
(b) Lower layer[G50]
(0 0.
<D
55 Q 2 • >
<D Q
800
700
600
500
400
300
200
100 1,
Measured - 2.8x10"1%/mirt
Measured - 2.0x10"2%/min
Measured - 7.2x10"4%/min
Calculated -2.8x10"1%/mir
Calculated - 2.0x10_2%/mii
Calculated - 7.2x10"4%/mii
1.0 1.5 2.0 Axial Strain (%)
2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
Axial Strain (%)
Figure 3.12 Measured and calculated undrained shear strength versus strain-rate for
SJVclay[G51].
2000 CO Q_ -*
' CO
~b "c -t—•
CD *—» CO i —
CO CD
CO T3 CD
'c0 -T3
C Z)
1000 900 800 700 600
500
400
300
200
Calculated Peak and Residual Strength (Lower: Layer) Measured Peak and Residual Strength (Lower: Layer) Calculated Peak and Residual Strength (Upper; Layer) Measured Peak and Residual Strength (Upper; Layer)
100 0.0001
iJ-Largeistrain Strength
(3%)
0.001 0.01 0.1
Strain Rate (%/min)
10
Figure 3.13 Calculated and measured behaviour during CIU triaxial creep tests on
SJV clay
(a) Upper layer[G52]
35
Stra
in,
xial
<
1.5-
1.0-
.5-
on-
—
i
I
•—[
l
1
._
!
ji i
ii [i ii ii
fr > i j
S
i I ii ;
ii I 1 ! 1 1
i
1 • 1
Stress Level=430KPa n M e a s u r e d
Rtrpss I evel=47nKPa J , , " = a - : > u l = u * . — _ Stress Level=470KPa
Stress Level=430KPa - i ~ „ , „ , , i „ f „ j stress Levei=470KPa J Calculated
100 Time (min)
1000 10000
•
20-
0 -
20-
40-
_ * A .
I t •*•* - <
! I |
n j
1 1
N i !|
! 1
1 1 ! ! ! i
1 i i 1 1 1
100 Time (min)
10000
Figure 3.13 Calculated and measured behaviour during CIU triaxial creep
tests on SJV clay
(b) Lower layer[G53]
» Stress Level=630KPa » Stress Level=575KPa n Stress Level=550KPa
Table 4.2 Viscosity[i69]-related parameters for Gloucester clay used by Model 1 and
Model 2
Model 1 (Hinchberger and Rowe Model)
a = l/n YVP
0.033 ixlO'8 /min
Model 2 Structured EVP model
a = lln 77 <°o
0.033 lxl0"8 /min 1-18
b
50
161
Figure 4.1 (a) Geometry[i70] of the Gloucester test embankment and (b) properties of
Gloucester clay
(a) Geometry[i71] of the Gloucester test embankment and the according finite element
mesh (modified from Hinchberger 1996)
Distance from the centre (m) 5 10 15 20
Fissures
Preconsoiidation pressure
(kPa)
I 50 100 150
Boundaries: A-B: Smooth, rigid, no drainage; B-C: Smooth, rigid, drained; C D : Smooth, rigid, no drainage
10 15
1 r
1 * Soft clay • # layer
clay layer
• Oedometer(NRC) I—Assumed a' (sii
20 25
ure 4.1 (b) Comparison[i72] of the adopted parameters with laboratory results of
Gloucester clay (from Hinchberger 1996)
4h
6b
8h
£ 10 Q.
Q
12
14
16
18
20
Compression
Index, X
1 2 — I —
8
T~
NUMERICAL
m
4QIJ&. o
*p o
a o
<3D <JE>
o (TO
C)
O
O
...o..
o o
LAB
Hydraulic
Conductivity (m/min)
5 10 15 50
Preconsolidation
Pressure (kPa)
100 150
FIELD
-t-O
— I — I — I —
NUMERICAL
o o
o
o
O"
Oi
o
LAB
D
-|—l—l—l—l—|—l—l—l—i—|—r-GROUNDWATER : —
i O LEVEL!
s Jo-
£ To D"
"b"
<\c o
\<8>.
O \Q O
YIELD SURFACE INTERCEPT -ELLIPTICAL CAP MODEL • « . ' M •:
°o\. & o
\ 0
my0
O LABORATORY
O oedometer (NRC) preconsolidation; pressure
Q o
h-
_ J ; i i L
\ o \
. . . . . . . . . \
_ i i iL_i_
10
12
14
16
18
20
Figure 4.2 Influence of clay structure on the behaviour of Gloucester clay in
undrained triaxial and oedometer compression tests
a) Typical undrained[i73] triaxial compression[i74] [G75]response
2.0
1.5 H
(0
1.0 H
(0
0.5 4
0.0
Structured Specimen (a'c=40kPa)
Destructured Specimen (a'c=83kPa)
Data from Law 1974
8 10
Axial Strain,%
Figure 4.2 Influence of clay structure on the behaviour of Gloucester clay in
undrained triaxial and oedometer compression tests (Cont.)
b) Typical oedometer[i76][G77] compression curve
a p - i a p - s
0.0
-0.1
-0.2 -
O
(0
•5 "0.3 'o > H—
o w g -0.4 4 to
JO
O
-0.5 -
-0.6 4
-0.7
10
0'p, =80.5kPa
f p-s ==95KPa
<*>o = G , p-s / t ' p - i
i =1.18
ICU: Intrinsic compression line
Measured Data from Leroueil et al. (1983) on Gloucester clay obtainted from 4.05-4.18m
10 100
Vertical effective stress (a'u), kPa, in log scale
165
Figure 4.3 Rate-sensitivity of the undrained shear strength and pre[G78]consolidation
pressure of Gloucester clay
(a) Undrained strength
Data from Law 1974 • CAU peak strength o CAU large strain strength
10-3 10-2 10-1
Strain Rate (/min)
166
Figure 4.3 Rate-sensitivity of the undrained shear strength and pre[G79]consolidation
pressure of Gloucester clay (Cont.)
(b) Preconsolidation pressure
200 CO
tn tn a)
c o CD
"5 CO c o o CD
c CD Q.
<
100
90
80 -
70 -
60 -
50
40
Measured data by Leroueil et al. (1983) at a depth of 4.08-4.15m Measured data by Leroueil et al.(1983) at a depth of 3.45-3.9m Usipg Constant Rate of Strain (CRS) oedometer tests
Estimated in-situ preconsolidation pressure from depths between 2.4 and 4.9m
(Leroueil et al 1983)
• Estimated ranges of preconsolidation pressure from conventional oedometer tests
(Leroueil et al 1983)
-10-8 10-' 1 0 -e -i 0-5
Strain rate, /min
10" 10": 10":
167
Figure 4.4 Long-term oedometer creep[G80] tests on Gloucester clay (data from Lo
Figure 4.5 The state boundary[i81] surface and critical state line for Model 1 and
Model 2.
pj2
B3
A3
o
State boundary surfac Dynamic yield surfa
^ ^ " - " A 2 /
" ^ — 1Y1oc /
/ / ^—-,
• /
/ f
;e:Al-A2-A3 , :e: B1-B2-B3 /c lass ical Critical State Line
AM«
/ Elliptical Caps B 2 / ^ /
^ \ G
Elas t ic^/ \ domain State boundary/ \
surface A l \
dF
P \~~~ State of overstress at 'G'
\ / Dynamic yield \ * surface \ Bl
•
a, /(s)
a '(d) a
169
Figure 4.6 Illustration of the theoretical response[i82] of Model 1 (Hinchberger and
Rowe Model)
(a)
£A < eB < ec
(c)
log(eO
(b)
log(e)
(d)
I s r
a = \ln
log(e)
170
Figure 4.7 Illustration of the theoretical[i83] response of Model 2
(a) (c)
Metastable Structure
&A < ^B < £c
CJ » C
1
*j£ 1 _~B'
C
B
A
S„c
S„B
$uA
ec
m
U £A
log(crp)
(b) (d)
= 77 = 7) log(£)
I ~— "™ ~~ Y ™" " T B ' P o s t P ^ strength
= r7 srf log(e)
gure 4.8 Comparison of the measured behaviour in CRS oedometer test on
Gloucester clay and the corresponding theoretical response of Model 2
-0.7
-0.8
200
Measured Data from Leroueil et al. (1983) on Gloucester clay obtained from 3.45-3.90m Theoretical response (Model 2)
50 100 150
Vertical effective stress (o'v), kPa 200
Figure 4.9 Comparison of the measured settlement at Gauge SI with the calculated
settlement using Model 1
1.5 years 5 years
1 I 15 years 20 years
1000 2000 3000 4000 5000 6000 7000 8000
Time (days)
173
Figure 4.10 Illustration of the linear and bilinear virgin compression curves
(a) linear approach
Ae
Long Term Compression Curve
ln(<7„) •
>e v0
(b) Bilinear approach
Transition Residual phase
Figure 4.11 Zones of strength gain due to consolidation, 15 years after the
construction of Stage 1- Contours[i84] of {Su/Su0)cons
25
Distance from the centre (m) 5 10 15 20
Zone A
Stage 1
T
Contours of ( 4 /5 „ 0 X, ) m
T
10 15 20
25
Preconsolidation pressure
(kPa)
50 100 150
- ! — r ~
tj> Soft clay layer
a . Medium to stiff clay layer
• •
• Oedometer(NRC) - Assumed a' J*\
25
Figure 4.12 Zones of strength gain due to consolidation, 4 years after the construction
of Stage 1. Contours[i85] of (Su /Su0)cgns
Distance from the centre (m)
10 15
4 years after the construction of Stage 1 15 years after the constructio of Stage 1
Soft clay
K layer
, Medium to stiff clay layer
• Oedometer(NRC) -Assumed a'my
(s^
10 15 20 25
Figure 4.13 Comparison of measured settlement (Gauge SI) with calculated
settlement using Model 2
? o
4-1
c
E <u C/)
0 i
-10 -
-20 -
-30 -
-40 -
-50 -
-60 -
-70 -
-80 -
1.5
• o
""^^^~
years 5 years
1 10 years
A
Stage 1
Measured Data in Stage 1 Measured Data in Stage 2 Caculated by Model 2
• • i • - — i — " i
15 years
J
i
°\
- — I — i-
20 years
\
Stage 2
— i ^ ^
1000 2000 3000 4000 5000 6000 7000 8000
Time (days)
Figure 4.14 Comparison of the measured and calculated settlement and excess pore
water pressure using Model 1 and Model 2
(a) Comparison of the measured settlement (Gauges SI and S3) with the calculated
settlement using Model 1 and Model 2
c
<D C/D
-40 4
-50
-60
-70
-80
1.5 years 5 years 10 years 13 years
i. i
Measured Settlment (S1) in Stage 1 Calculated by Model 2 Calculated by Model 1 Measured Settlement (S3) in Stage 1 Calcualted by Model 2 Calculated by Model 1
1000 2000 3000 4000 5000 6000 7000 8000
Time (days)
Figure 4.14 Comparison of the measured and calculated settlement and excess pore
water pressure using Model 1 and Model 2 (Cont.)
(b) Comparison of the measured extra pore water pressure[i86] with the calculated
settlement using Model 1 and Model 2
c g
'•4-»
CO >
LU
1 Year after the constructure of Stage 2
1 2
O
o Geonor Hydraulic Standpipe IRAD Vibrating Wire Calculated using Model 1 Calculated using Model 2
3 4 5
Excess Pressure Head (m)
Figure 4.15 Zones of strength loss due to destructuration, 15 years[i87] after
construction of Stage 1. Contour of [Su I Su0) 'Str
25
Distance from the centre (m)
5 10 15 20 T T T
25
Contours of (sjSua\
10 15 20
Preconsolidation pressure
(kPa)
50 100 150
" i — r
i f Soft clay f . layer
, .Medium to stiff
' clay layer
>Oedometer(NF\C) I— Assumed &
25
Figure 4.16 Zones of net strength gain (i.e. consolidation overshadows
destructuration), 15 years[i88] after construction of Stage 1. Contour of
SJSU0>1
25
Distance from the centre (m) 5 10 15 20
T T 25
20 I Stage 1
Zone A
c g 5 10 i
LU
Zone C
Contours of Su I Su(j > 1
10 15
Preconsolidation pressure
' (kPa)
50 100 150
i — r
^ Soft clay ' layer
a # Medium to stiff clay layer
• Oedometer(NPtC) — Assumed o'my
<s)>
20 25
Figure 4.17 Zones of net strength loss (i.e. destructuration overshadows
consolidation), 15 years[i89] after construction of Stage 1. Contour of
su/su0<i
25
Distance from the centre (m) 10 15 20 25
20 4
Zone A
E, 15 C O
"«^ ro >
LU
K
Stage 1 .NylZone B
Contours of ; S„ / Sll0 < 1
10
Preconsolidation pressure
(kPa)
50 lio 150 T
f Soft clay £• layer
# ^Medium to stiff clay layer
• Oedometer(NRC) I— Assumed a' (s!l
15 20 25
Figure 4.18 Development of zones of net strength gain from the 4th year to the 15th
year in Stage 1
Distance from the centre (m)
5 10 15 20
15 years after the construction of Stage 1 4 years after the construction of Stage 1
Preconsolidation pressure
(kPa)
5D 100 150
i f Soft clay *9 layer
Medium to stiff clay layer
• Oedometer(NRC) — Assumed a' Js\
10 15 20 25
Figure 4.19 Development of zones of net strength loss from the 4th year to the 15th
year in Stage 1
25
Distance from the centre (m)
5 10 15 20 T T T
25
20
Zone A
15
c q > m 10
5 |
Contours of S„ / S,„, < 1
15 years after the construction of Stage 1 4 years after the construction of Stage 1
10 15 20
Preconsolidation pressure
(kPa)
50 100 150
i r
£ Soft clay ' layer
a , Medium to stiff clay layer
• •
• Oedometer (NFJC) I— Assumed o',
25
Figure 4.20 Zones of net strength increase, 7 years [i90] after construction of Stage 2
Distance from the centre (m) 5 10 15 20 25
54 Contours of Sh/Sll0 > 1
Preconsolidation pressure
(kPa)
I 50 100 150
At the end of Stage1(i.e. just before the construction of Stage 2) 7 years after the construction of Stage 2
10 15 20
I S Soft clay * layer
a . Medium to stiff clay layer
• Oedometer (NF\C) I— Assumed 0'
25
Figure 4.21 Zones of net strength loss 7 years after construction of Stage 2
25
Distance from the centre (m) 10 15 20 25
20
Zone A
15
Stage 2
Stage 1
y ; Lateral shift due to \ 1.0 the additional loads in Stage 2
Contours of S^ I Sll0 < 1
At the end of Stage 1 (i.e. just before the construction^ Stage 2) 7 years after the construction of Stage 2
10 15
Preconsolidation pressure
(kPa)
50 100 150
i — r
£ Soft clay ' layer
a s Medium : to stiff
* clay layer
• Oedometer(NRC) — Assumed a'my
(s))
20 25
186
Figure 4.22 Comparison of the compression curve in laboratory test with the
measured long-term field compression of Gloucester clay under the
Accommodation building (from McRostie and Crawford, 2001)
en a> a. E o o
4H
10
Average of laboratory tests
80 100
In situ observation
0.5 year 1 year
2 years
— i —
20 — i —
40 — i —
60 80 100
Vertical effective stress (cr'v), kPa
187
CHAPTER 5
AN ANISOTROPIC EVP MODEL FOR STRUCTURED CLAYS
5.1 Introduction
Structured[G91] clay deposits are widely distributed throughout the world. As a
result, many countries build significant infrastructure on or in these difficult soils.
During loading, these clays can exhibit engineering characteristics such as rate-
sensitivity, drained and undrained creep, accelerated creep rupture and significant
anisotropy (Lo et al. 1965; Lo et al. 1972; Tavenas et al. 1978; and Vaid et al. 1979).
Some of these characteristics, in particular anisotropy, have been attributed to the
microscopic structure of clay.
For many structured clays, both anisotropy and viscosity appear to be significant.
Lo and Morin (1972) found that anisotropy, strain-rate and time effects were pronounced
for St. Louis and St. Vallier clay from Eastern Canadian,. Tavenas et al. (1978) observed
similar behaviour for other clays from eastern Canada. The engineering significance of
both anisotropy and strain-rate effects has been well established. Recently, Hinchberger
and Rowe (1998) and Kim and Leroueil (2001) demonstrated the importance of viscous
effects for embankments founded on soft clay deposits. Similarly, Zdravkovic et al.
(2002) demonstrated the effect of anisotropy on embankment behaviour. Thus, a
constitutive model that can describe both anisotropy and viscous effects in 'structured'
clays would be useful in geomechanics.
This chapter describes a constitutive approach to model the time-dependent
188
plastic behaviour of rate-sensitive anisotropic structured clay. The main objective of the
chapter is to demonstrate a novel approach to the anisotropic behaviour of viscous
'structured' clay at yield and failure. As a consequence of this study, some observations
are also made regarding the anisotropic elastic behaviour of 'structured' clay. The
constitutive approach described in the following sections utilizes non-linear elasticity
theory, overstress viscoplasticity (Perzyna 1963), a Drucker-Prager failure envelope, and
an elliptical cap yield surface (Chen and Mizuno 1990). Structure is accounted for by
adopting a viscosity parameter that is initially high (the structured viscosity) and that
decreases to the residual or intrinsic viscosity due to plastic strain or damage strain (see
chapter 3). The structured viscosity is made anisotropic using a tensor approach similar
to that described by Boehler (1987), Pietruszczak and Mroz (2001) and Cudny and
Vermeer (2004). The intrinsic viscosity is assumed to be isotropic. Theoretical
behaviour is compared with the measured response of Gloucester clay and St. Vallier clay
(Lo and Morin 1972) during undrained triaxial compression tests on samples trimmed in
different orientations, / , to the vertical axis. The comparisons show that the constitutive
model is capable of accounting for both anisotropy and strain-rate effects on the
engineering behaviour of these clays.
5.2 General Approaches to Anisotropic Plasticity
In general, four main approaches have been developed to describe the anisotropic
behaviour of clayey soils at yield and failure excluding those based on nested yield
surfaces. The approaches are:
(i) Rotational Kinematic Hardening Laws: The yield surface is assumed to
rotate under the influence of an anisotropic stress field (Davies and Newson
189
1992; Whittle and Kavvadas 1994; Wheeler et al. 2003). Rotational
hardening models have been used to describe the response of embankments
built on natural clay soils (Zdravkovic et al. 2002; Oztoprak and Cinicioglu
2005).
(ii) Transformed[G92] Stress Tensor: A fabric tensor is used to modify the stress
tensor, <r',,, obtaining the transformed stress tensor, T' Yield and failure y *^ y
criterion are subsequently developed using T~ instead of &tj (Miura et al.
1986; Tobita 1988; Tobita and Yanagisawa 1992; Sun et al. 2004;).
(iii) The Fabric Tensor[G93] Approach: A fabric tensor is used to modify the
plastic energy dissipation formulation to develop new state boundary surfaces
(Muhunthan et al. 1996).
(iv) The Structure Tensor Approach: Boehler (1987), Pietruszczak and Mroz
(2001), and Cudny and Vermeer (2004) used the stress tensor, c'iy, and a
microstructure tensor, atj, to obtain an anisotropic scalar coefficient, n , that
can be used to give anisotropic characteristics to scalar parameters such as the
cohesion intercept, c', and effective friction angle, <f>'. Pietruszczak and
Mroz (2001) demonstrated the use of this approach to obtain an anisotropic
Mohr-Coulomb failure criterion.
In summary, all of these approaches are useful, however, the common limitations
of the first three are generally: (i) complex formulations, (ii) numerous material
parameters required and (iii) parameters that generally cannot be determined using
conventional laboratory tests. However, the fourth approach described by Pietruszczak
and Mroz (2001) is relatively straightforward and it can be implemented into viscoplastic
190
formulations (Pietruszczak et al. 2004) with the introduction of only one additional
constitutive parameter for the case of transverse isotropy.
5.3 Microstructure Tensor
Transverse Isotropy
In accordance with Pietruszczak and Mroz (2001), material anisotropy can be
described using a microstructure tensor, atj, which describes the spatial distribution of
microstructure. In its general form, the microstructure tensor is:
a = •J
a a. xx xy xz
a a yx yy yz avc azy az
[5.1]
For clays deposited under the influence of gravity with horizontal bedding or laminations,
the principal directions of anisotropy are vertical and horizontal. In this case, the
microstructure tensor, ay , is coaxial with the axes of orthotropy of the material and it can
be simplified
a- = y
to:
axx
0
0
0 T
0
0 "
0 T
[5.2]
where the superscript, T, denotes transverse isotropy. For a transverse isotropic material,
the microstructure tensor can also be written in terms of the mean and deviatoric
components viz.:
191
a,-, =
[5.3]
r
0
0
0
0
o" 0 T
a3 _
=
am -arA/2
0
0
0
a,
0
a -a A/2 m m
0
0
a + a A m m
fit[ + flj + a3 a3 — am 2a3 — 1ax
a — - , A — V a„ 2ax +a3
where a is the mean structure and A describes deviations from the mean. When aT is
normalized by am, the microstructure tensor becomes:
T
11 a m m a
a —a A/2 m m
0 a n
0
0 0
-amA/2 0
0 am +amA m m
l - A / 2 0 0
0 l - A / 2 0
0 0 1 + A
[5.4]
where the normalized microstructure tensor, a j , quantifies the spatial distribution of
structure with respect to the mean structure and the parameter, A, defines the degree of
inherent material anisotropy. The absolute magnitude of A is zero in the case of isotropy
and it increases as the degree of anisotropy increases. Researchers such as Oda and
Nakayama (1989) have shown that it may be possible to relate A to measurements of soil
fabric.
For most naturally deposited clays, the major principal anisotropic direction is
vertical (e.g. azz>axx=a ) . Correspondingly A is positive (see Figure 5.1a).
However, for heavily overconsolidated clays such as London clay (Ward et al. 1959) high
192
horizontal stresses may lead to higher undrained strength in specimens of horizontal
orientation compared to those of vertical orientation. Accordingly, A could be negative
if the major principal direction of anisotropy is horizontal (see Figure 5.1b).
For clays with sub-horizontal bedding or laminations, a transform tensor, Q, can
be applied. For example, in the case of plane strain:
fl = = gx =
cos" i sin" i sin2/
sin2/ cos2/ -sin 2/
- 0.5 sin 2i 0.5 sin 2/ cos 2i
[5.5]
where i is the angle of the bedding or laminations relative to the horizontal axis. Thus,
the transform tensor can be applied to cases where the major principal directions of the
microstructure tensor are not oriented along the vertical direction.
From Pietruszczak and Mroz (2001), a scalar parameter, rj, can be derived to
define the anisotropy of a material using the generalized effective stress state, otj , and
_ r • 2
microstructure tensor, atj or atj . The diagonal components of atj represent the
resultant stresses on each of the principal planes of orthotropy (see Figure 5.1c):
( .2 \ ' 2 '2 ' 2 ' 2
9 U=(yx =°xx +(Jxy +°xz \(J'2)yy=C7'y2=axy2+°yy
2+°yz ( .l\ .2 . 2 . 2 . 2
F )«=<** = < T ^ +°zy +°zz
yi
2
[5.6]
The anisotropic scalar parameter, r\, can be obtained by taking the normalized
projection of the microstructure tensor on the generalized stress state viz.:
193
2 _ . 2 . fr>.r - • • - _ - - r U i 0"« )
tr\fT9 j
[5.7]
which in the case of a vertically orientated (e.g. i=0°) specimen subject to a triaxial stress
state simplifies to:
( 1 - A / 2 W 2 +(1-A/2)cr '2 +(1 + A)(T'2
77 = •2 . 2 . 2
^ +<7y +(Jz
[5.8]
Equation [5.7] conforms to the Representation Theorem of Isotropic Functions (Wang et
al. 1970) and as such, r\ is independent on the choice of orthogonal coordinate system
viz.
[5.9]
where Q is the transform tensor. The scalar parameter, r\, accounts for the influence of
stress orientation and material orientation as illustrated in the following section.
5.4 Application to Tresca's Failure Criterion
To illustrate the use of the microstructure tensor, consider Tresca's failure criteria,
which is often used in soil mechanics:
f(a\,r])=a\ -a'3 -r]cu0 (<J\ >cr'3)
[5.10]
194
where cu0 is the isotropic undrained shear strength and 77 represents the influence of
anisotropy on the undrained strength of clay. As shown above, the scalar coefficient, 77,
is derived from the microstructure tensor, aj,, and the stress tensor, o'„. The magnitude
of 77 depends on the relative orientation and magnitude[G94] of both a I and <7y.
Consider a series of undrained triaxial compression tests on clay specimens
trimmed at different orientations, i, to the vertical. In accordance with Equation [5.9],
the effect of sample rotation can be taken into account by transforming the structure
tensor using Equation [5.5] taking i equal to the angle formed by the specimen axis and
the vertical (see Figure 5.2). Now, given the following arbitrary triaxial stress state:
au
a\ 0 0 0 a\ 0 0 0 a\
=
"l 0 0"
0 1 0
0 0 4
[5.11]
where a\ is the cell pressure in a triaxial test, and &'a is the axial stress, the influence of
sample orientation, i, on 77 is shown in Figure 5.3 for A equal to 0, ±0.1 and ±0.2,
respectively.
Referring to Figure 5.3, when the anisotropic parameter A equals zero, 77 is
constant and equal to one. For this case, the resultant undrained shear strength is
isotropic. As A is increased from 0.0 to 0.1 and 0.2, respectively, the undrained shear
strength becomes increasingly more anisotropic and the strength of vertical samples
( i = 0°) exceeds that of horizontal samples ( i = 90°). Conversely, the strength of
horizontal samples ( i = 90°) exceeds that of vertical samples ( i = 0°) when A is
195
negative. Thus, the parameter r\ can be used to modify Tresca's failure criteria obtaining
anisotropic undrained shear strength similar to that observed by Lo and Milligan (1967).
It can be shown that, for soils that reach a unique effective stress ratio at failure:
Tli = T W + (TW _ 1 W )C0S2i t5-12]
which is identical to the relationship used by Lo (1965) to describe the anisotropic
undrained shear strength of Welland clay in Canada.
Figure 5.4 illustrates the influence of the stress ratio, o\jo\ , on r]cu0. Consider
the following triaxial stress state:
[5.13]
which permits investigation of the influence of o\ \&'c , on the anisotropic parameter T|.
Referring to Figure 5.4, for stress ratios less than one, the undrained shear strength is
higher for horizontal specimens than for vertical specimens since the major principal
stress is acting in the radial direction. For stress ratios that exceed one, r\ increases to a
maximum of almost 1.2 for vertical specimens and stress ratios in the order of 6. Similar
trends can be observed for specimens trimmed at i = 45° and i = 90°, respectively. Thus,
the anisotropy is not only dependent on the orientation of the stress field relative to the
microstructure of the clay, i, but also on the stress ratio, o\ /CJ'C . It should be noted that
the parameter 77 approaches its upper limit as &a > 5<r'c. Although this complicates the
°ij =
°\ 0
0
0
<y'c
0
0
0
<*\.
= 0'c
1 0 0
0 1 0
_0 0 a\l&
determination of A somewhat, it has benefits that will be explored later in this chapter.
196
5.5 Application to an Elastic-Viscoplastic Model
Overstress Viscoplasticity
The formulation presented in the following sections is based on the Hinchberger
and Rowe Model (Hinchberger 1996; Hinchberger and Rowe 1998). This model has a
state boundary surface defined by an elliptical cap yield function (Chen and Mizuno
1990) and Drucker-Prager envelope (see Figure 5.5a); it has provision for either isotropic
and anisotropic non-linear behaviour in the elastic stress range; and the plastic response is
defined within the framework of Perzyna's theory of overstress viscoplasticity (Perzyna
1963) utilizing concepts from critical state soil mechanics (Roscoe et al. 1963). A
summary of the Hinchberger and Rowe (1998) model can be found in Table 5.1;
however, in principal the following constitutive formulation could be adapted to any
overstress viscoplastic model.
The basic constitutive equation (from Hinchberger and Rowe, 1998) is:
£ =£e+evp
2G Xl + e)a'm y ^ >' -+-
dF
dot
[5.14a]
where the flow function, <|)(F) , is a power law viz.
<KF)= my os
my
- 1
[5.14b]
In Equations [5.13] and [5.14], e~ is the strain-rate tensor, stj is the deviatoric stress
tensor, a ' is the mean effective stress, 8U is Kronecker's delta, G is the stress
197
dependent shear modulus, K is the slope of the e - l n ( a ^ ) curve in the over-
consolidated stress range, and e is the void ratio. The scalar function <j)(F) is called the
flow function, o,{^ is the overstress (see Hinchberger and Rowe 2005), c ' j^ is the static
yield surface intercept and dF/dcr'j is the normalized plastic potential for associated
plastic flow. An associated flow rule has been adopted in this chapter. It should be noted
that although the associated flow rule and isotropic plastic potential simplify the
formulation, such an assumption introduces a limitation in the model since the plastic
potential of most clays is anisotropic and in some cases non-associated (Graham et al.
1983 and Newson 1998).
The time-dependent plastic behaviour of clay is thus governed by the viscosity
parameter, u,, and the strain-rate exponent n . Viscosity, u., is the inverse of fluidity
(y'1' = l /( i) and as u, increases the soil becomes less fluid and viscous effects increase.
The rate-sensitivity is governed by n . As n increases the rate-sensitivity decreases.
Consequently, through varying n and \x, viscous rate-sensitive, viscous rate-insensitive
and inviscous plasticity can be modeled. The latter can be obtained by using an iterative
solution scheme to keep the stress-state on either the static yield surface or the Drucker-
Prager envelope (Zienkiewicz and Cormeau 1974).
Modification for Structure
Burland (1990) suggested that the engineering behavior of natural clays can be
described with reference to the remolded or intrinsic state. In accordance with this
concept, it has been hypothesized (See Chapter 3) that the viscous component of clay
structure can be defined in terms of the intrinsic and structured viscosities viz.:
0)0 =
1
198
[5.15]
V r -mt J
where a}, is the initial structure parameter, fistr is the initial viscosity of the undisturbed
'structured' clay and jj,^ is the remolded or intrinsic viscosity (Hichberger and Qu 2007).
As a result, 'structured' clay is considered to have a high initial viscosity relative to the
residual or intrinsic viscosity[G95].
During loading, it is assumed that the initial viscosity, (0,str, is gradually damaged
by plastic strain until eventually the clay is completely destructured and the viscosity has
degraded to the intrinsic viscosity, jx^ . This process is commonly referred to as
'destructuration' (Rouainia and Wood 2000). Degradation of the clay viscosity is
assumed to occur as a function of damage strain viz.:
tfed) = /"int + <A,r - # * )«"*'" [5-16]
where b is a parameter that controls the rate-of-destructuration of clay and the damage
strain, s d , is:
ded=V(l-A)(de70l)2+A(de;
p)2 [5.17]
In Equation [5.17], (see Rouainia and Wood 2000), A is a weighting parameter and e vp vol
and e^ are plastic volumetric and octahedral shear strains (V3yoct), respectively. In this
chapter, the weighting parameter, A, has been assumed to be 0.5. It is also recognized
that the current model does not account for shear banding or strain localization and that
the parameter & in Equation [5.16] includes these effects for shearing modes of failure.
In summary, the Hinchberger and Rowe model (Hinchberger and Rowe 1998) has
199
been modified by adopting a state-dependent viscosity parameter and the resultant plastic
strain-rate tensor is:
The conceptual behaviour of the 'structured' clay model is described below.
Conceptual Behaviour of the 'Structured' Model
The conceptual behaviour of the structured model has been described extensively
by Hinchberger and Qu (2007) for over consolidated materials such as St. Vallier (Lo and
Morin 1972) and Saint-Jean Vianney clays (Vaid and Campanella 1977; Vaid et al.
1979). Figure 5.5 illustrates the model behaviour for lightly over consolidated materials
during CIU triaxial compression tests.
Referring to Figures 5.5b and 5.5c, after initial isotopic consolidation to point 1 in
Figure 5.5b, triaxial compression of the soil specimen at a constant rate of strain will
cause the effective stress path to move on the elastic wall from point 1 to 2 where
yielding occurs. During continued compression, the 'structured' soil skeleton will
undergo plastic straining as the stress path moves from 2 to 3; however, the plastic strain-
rate during this phase of compression is very low due to the high viscosity of the
'structured' soil skeleton. Thus, the material behaviour is still predominantly elastic from
2 to 3 as shown in Figure 5.5b as overstress builds up relative to the long-term or static
yield surface (Hinchberger and Rowe 2005).
At point 3, the overstress and resultant plastic strain-rate becomes high enough to
begin destructuration of the clay and consequent increased fluidity of the clay skeleton.
From point 3 to 4, there is stabilization of the overstress during which the peak strength is
BF
da'. HM (<KF))
dF [5.18]
200
reached. From point 4 to 5, however, the damage rate is high and there is a significant
reduction of overstress (stress relaxation) caused by the shear thinning or degrading soil
viscosity. Thus, strain softening is modeled as a stress-relaxation phenomenon. As
compression continues, it is assumed that eventually the plastic strain causes the viscosity
of the soil skeleton to decrease to the intrinsic viscosity; although this state may not be
reached during triaxial compression.
The conceptual behaviour described in Figure 5.5 applies to lightly over
consolidated materials such as Gloucester clay. In addition, during undrained triaxial
creep tests, application of a constant deviator stress exceeding that denoted by point A in
Figure 5.5a and 5.5b will cause time-dependent plastic creep followed by eventual creep
rupture of the material. However, applied deviator stresses below that denoted by point
A will cause time-dependent plastic creep that will eventually stabilize when the stress
state reaches the static yield surface. Sheahan (1995) summarizes such behaviour for
natural clays. In general, the model adopted in this chapter is identical that described in
Chapter 3 except that, in this study, an associated flow rule has been assumed in
conjunction with separate 'structured' and 'destructured' bounding surfaces. Appendix G
describes an alternative approach utilizing non-associated plasticity.
Modification for Anisotropy
To account for both time-dependency and anisotropy of natural clay at yield and
failure, it is hypothesized that the 'structured' viscosity of clay is anisotropic whereas the
intrinsic viscosity is isotropic. Studies by Law (1974) and Lo and Morin (1972) contain
experimental observations supporting this assumption for some clays from Eastern
Canada.
201
Using the microstructure tensor, the structured viscosity can be modified as
follows:
H{ed,r\) = (jimt + {wstr ~/"int yhSd) [5.19]
where T| is the anisotropic scalar parameter defined by Equations [5.7] and [5.8].
Equation [5.19] can also be expressed in terms of the initial structure, CO;, viz.:
co(sd ,7])= (l + (ryo/ - l)e-*« ) ' " [5.20]
where oo(ed,T|) defines the remaining structure at any point after some destructuration,
ed , has occurred. The resultant anisotropic viscoplastic strain-rate tensor is:
Thus, a structure parameter and microstructure tensor have been used to extend
the Hinchberger and Rowe (1998) model to obtain an anisotropic rate-sensitive
constitutive model for structured clays. Clay structure is treated as a viscous bonding
phenomenon and the source of anisotropy is assumed to be the anisotropic distribution of
viscous bonds. The main characteristics of the constitutive model are summarized in
Table 5.1.
It is noted that for a tensor approach, adopting an associated plastic potential law
would lead to underestimation of the deviatoric plastic strain for natural clay subject to at
isotropic stress path. A non-associated flow rule can be utilized to overcome this
limitation and improve the prediction of the tensor approach on plastic strain under
loading. However, additional parameters have to be introduced for the non-associated
plastic potential law and these parameters must be deduced from non-standard laboratory
dF
a<rf, ^(ed,r?) (<KF))
dF [5.21]
202
tests. Thus, for the sake of simplicity, the tensor approach in this chapter adopts an
associated plastic potential law and consequently only one parameter, A, is required to
describe the anisotropy characteristic of natural clay.
5.6 Evaluation
Methodology
This section compares calculated and measured behaviour of both Gloucester clay
and St. Vallier clay during undrained triaxial compression tests on specimens trimmed at
various orientations, i, to the vertical. Only tests performed at consolidation pressures
less than the in situ overburden stress were considered in the analysis. In addition, the
test results used below were obtained using high quality triaxial specimens trimmed from
block samples.
For Gloucester clay and St. Vallier clay, a series of isotropically consolidated
undrained (CIU) triaxial compression tests were evaluated. The measured behaviour has
been reported by Law (1974) for Gloucester clay and Lo and Morin (1972) for St. Vallier
clay. The calculated behaviour presented in Figure 5.6 through 5.13 was obtained using
the finite element (FE) program AFENA (Carter and Balaam 1990), which has been
modified by the authors to account for time-dependent plasticity and structure. A FE
analysis was undertaken for each test starting from the initial stress state reported during
the test. The sample was loaded by prescribing displacements to the top of the mesh at a
rate corresponding to the compression rate reported for each test. The FE calculations
were performed using 6-noded linear strain triangles in conjunction with axi-symmetric
conditions. The top and bottom mesh boundaries were assumed to be smooth (e.g.
203
friction was neglected) and rigid. The FE calculations are summarized in Figure 5.6
through 5.9 for Gloucester clay and Figure 5.10 through 5.13 for St. Vallier clay. The
constitutive parameters used in the analysis are listed in Tables 5.2 and 5.3 for Gloucester
and St. Vallier clay, respectively.
Gloucester clay
Law (1974) conducted a series of CIU triaxial compression tests on specimens of
Gloucester clay trimmed at 0°, 30°, 45°, 60° and 90° to the vertical[i96] (see Figure 5.2).
The test results are summarized in Figure 5.6, which shows the measured and calculated
peak undrained shear strength of Gloucester clay versus sample orientation, i. Figure 5.6
also shows the measured and calculated post-peak strength at 8% axial strain and the
calculated intrinsic or residual strength of Gloucester clay at large-strain. The intrinsic or
residual state was assumed in the FE interpretation even though it is difficult to reach the
residual state in a triaxial apparatus.
From Figure 5.6, it is evident that the measured and calculated peak undrained
shear strength of Gloucester clay are strongly anisotropic. The peak strength of vertical
specimens (e.g. /=0°) is typically 40% higher than for horizontal specimens (e.g. i = 90°).
In general, there is overall good agreement between the calculated and measured peak
strength for all sample orientations, i.
At an axial strain of 8%, there is also good agreement between the calculated and
measured post-peak shear strength. The measured strength of Gloucester clay at 8% axial
strain is only slightly lower than the calculated strength for the values of / considered. At
the intrinsic state, which is reached at 12% axial strain (assumed), the theoretical strength
of Gloucester clay is isotropic (see Figure 5.6). Overall, it is concluded that the trends of
204
calculated undrained shear strength are comparable to the measured trends of undrained
strength versus sample orientation, i.
Calculated and measured deviator stress and excess pore pressure versus axial
strain are compared in Figure 5.7 up to 12% axial strain. From Figure 5.7, it can be seen
that there is adequate agreement between the measured and calculated behaviour
notwithstanding the notable differences in the elastic range as discussed below. In the
post-peak stress range, the measured rate of strength reduction is somewhat higher than
the calculated rate for specimens trimmed at / of 0° and 30°. However, the theoretical
response is considered to be a reasonable idealization considering the probable impact of
such factors as natural variability on the laboratory measurements. In addition, the
calculated excess pore pressures are generally within 15% of the measured excess pore
pressures for axial strains up to 10%. The difference between calculated and measured
pore pressures can be attributed to the isotropic elastic theory used to obtain the
calculated behaviour as discussed in the following paragraph.
Figure 5.8 shows the calculated and measured stress path during triaxial
compression tests on specimens at i=0° and 90°, respectively. In accordance with
Graham and Houlsby (1983), an anisotropic elastic parameter, (3 = E V / E h , can be
derived from the deviation of the measured stress path from the theoretical isotropic
stress path for / = 0° (see Figure 5.8). For Gloucester clay, the anisotropic parameter, (3,
is approximately 1.6 assuming a Poisson's ratio of 0.3, where Ev and Eh are the vertical
and horizontal elastic modulus, respectively. Reanalysis of the CIU triaxial test using
cross-anisotropic elastic theory in the elastic-viscoplastic constitutive model produced
Curve '2' in Figure 5.8, which is in close agreement with the measured stress path.
205
To conclude, Figure 5.9 summarizes the effect of strain-rate on the undrained
strength of Gloucester clay. Again, the constitutive model is capable of describing the
overall variation of undrained shear strength versus strain-rate, and as such, the
constitutive results are considered to be encouraging. For Gloucester clay, the
constitutive model is capable of describing both the variation of peak and post-peak
strength versus sample orientation and the effects of strain-rate on the mobilized strength.
The peak strength of Gloucester clay varies by about 10% per order of magnitude change
in the strain-rate. This is quite significant and in many cases it should not be ignored by
engineers (Marques et al. 2004).
St. Vallier clay
To complete the evaluation, the anisotropic behaviour of St. Vallier clay during
CIU triaxial compression tests was also considered. St. Vallier clay is considered
because it exhibits different anisotropy behaviour from Gloucester clay, which may be of
interest. The behaviour of St Vallier clay during CIU triaxial compression was reported
by Lo and Morin (1972). Figure 5.10 through 5.13, inclusive, compare the calculated and
measured behaviour and the constitutive parameters for this case are summarized in
Table 5.3.
Overall, there is also good agreement between the calculated and measured peak
undrained shear strength versus sample orientation of St. Vallier clay (see Figure 5.10).
From Figure 5.10, it can be seen that the undrained shear strength of St. Vallier clay is
highly anisotropic. The peak undrained shear strength of vertical specimens, /=0°, is 1.8
times that of the horizontal specimens (i=90°), which is a significant difference. Figure
5.11 compares the calculated and measured deviator stress and excess pore pressure
206
versus axial strain for vertical and horizontal specimens. The overall trends in the
measured and calculated data are considered to be consistent. Similar to Gloucester clay
(see Figure 5.9), slight differences between the measured and calculated data may also be
attributed to the anisotropic elastic response of St. Vallier clay.
Figure 5.12 shows the calculated and measured stress paths for triaxial
compression tests on specimens at i = 0° and 90°. For St. Vallier clay, the elastic
anisotropic parameter, (3, is 1.14 . Similarly, Curve 2 in Figure 5.12 shows the stress
path calculated using cross-anisotropic elasticity in conjunction with the structured
elastic-viscoplastic model for i = 0°. Again, the calculated and measured behaviour
agree. Thus, it appears that the constitutive framework is able to also account for the
variation of peak undrained shear strength of St. Vallier clay versus sample orientation.
The effect of strain-rate on the measured and calculated undrained peak shear
strength of St. Vallier clay is summarized in Figure 5.13. Referring to Figure 5.13, an
order of magnitude increase in the applied strain-rate causes a 15% increase in the peak
undrained shear strength of St. Vallier clay. In comparison, the peak strength of
Gloucester clay increased by only 10% for an order of magnitude increase in the applied
strain-rate. The increased rate-sensitivity is accounted for by decreasing the exponent, n,
in the constitutive model for St. Vallier clay (see Table 5.3). The results in Figure 5.13
further highlight the significant influence of strain-rate on the engineering behaviour of
clays from Eastern Canada.
The influence of destructuration on anisotropy
Figure 5.14 illustrates apparent yield states derived from the anisotropic
structurered constitutive model assuming a>0 = 1.52 and A = 0.45. Since the yield stress
207
in an EVP model is governed by strain-rate, the term ' apparent' is used to denote
isotaches in general stress space and the apparent yield surfaces depicted in Figures 5.14a
and 5.14b correspond to normalized isotaches. From Figure 5.14a, it can be seen that as
A increases, the apparent yield surface predicted by the proposed constitutive model
becomes increasingly anisotropic. Similarly, in Figure 4.15b, as the structure
parameter, a> , decreases, the apparent yield surface becomes more isotropic. For
comparison purposes, Figure 5.15 shows the effect of destructuration on St. Alban clay.
This figure summarizes the influence of anisotropic consolidation (K'0 ranges from 0.5
to 0.6) on the apparent yield surface of St. Alban clay measured using drained triaxial
probing tests (Leroueil et al. 1979). From Figure 5.15, it can be seen that as the
volumetric strain increases from 8% to 20% during K'0 consolidation, the apparent yield
surface of St. Alban clay becomes more isotropic. The behaviour depicted in Figure 5.15
is consistent with that shown in Figure 5.14b for CD0 =1.52 and A = 0.45. The parameter
co0 =1.52 can be estimated from the structured and intrinsic compression curve for St.
Alban clay shown in Figure 5.16[197]. Figure 5.16 also demonstrates the destructuration
of St. Alban clay with increase of volumetric strain. Based on Figure 5.16, the structured
and intrinsic compression curves for St. Alban clay are almost equal for volumetric
strains exceeding 20% (e.g. the intact clay is destructured).
Figuress 5.15 and 5.16 suggest that destructuration of a natural anisotropic clay
can lead to a reduction of its anisotropy even for K\ -consolidation where K'0 =0.5-0.6.
Similar behaviour has been reported for Winnipeg clay and Onsoy Clay, by Graham et al.
(1983) and Lune et al. (2006), respectively. Furthermore, as shown in Figure 5.6, the
Figure 5.9 The effect of strain-rate on the peak strength[ilOO] of Gloucester clay
(Data from Law 1974).
10-5 io-<
ratio
-c
reng
t
To
shea
r U
nd ra
ined
1 -i
.9
.8
.7
.6
Calculated -^
o
i
o
i
'
Lab CAU — * £ s ^
•^\a=Un=0.033
1
i
- LabUU
^—p^ 1
i
103 10"2
Strain rate, %/min
10-1 10°
227
Figure 5.10 The effect of sample orientation, i, on the peak strength of St. Vallier clay during CIU triaxial compression tests.
140
S. 120
P 100 CD
co CD
. C (/) co CD
Q_ " O CD C
'CO
•o c
80
60
40 4
20 4
Calculated
Measured J r o m i o and MorirL(1912X
45 90
Orientation angle, i
228
Figure 5.11 The effect of sample orientation, /, on the measured (Lo and Morin 1972) and calculated (a) axial stress versus strain and (b) excess[G101] pore pressure versus strain for St. Vallier clay.
180
CO
a. en
D i
Calculated,i=0
Measured, r=Q
1 1 1 1 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Vertical Strain (%)
o Calculated
MoaGuk)d,i=0°
i=90°
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Vertical Strain (%)
229
Figure 5.12 The effect of sample orientation, i, on the measured (Lo and Morin 1972) and calculated stress paths for St. Vallier[G102] clay.
Cr and Cc are the recompression index and compression index respectively.
Figure CI shows a typical response of clay in an oedometer compression test, in terms of
the void ratio versus the effective vertical stress in a semi-log scale. Cr and Cc can be
determined by the following equations:
Cr=Ae/A\og(a'v) for<7'v<<7'p [CI]
Cc = Ae/Alog(cr'v) for <r'v > &p [C2]
where e is the void ratio, a\ is the effective vertical compression pressure, and o' is
the preconsolidation pressure.
The determination of Cr and Cc is graphically shown in Figure CI, where Cr
characterizes the pseudo - elastic segment of the compression curve and Cc describes the
plastic segment.
Ca is the secondary compression index. Figure C2 shows a typical compression
curve from a drained constant stress creep test. It can be seen that Ca is measured from
the segment of compression curve after the dissipation of excess pore water pressure (see
EOP in Figure C2). Raymond and Wahls (1976) and Mesri and Godlewski (1979)
defined Ca viz:
C a =Ae/Al0g(O [C3]
where e is the void ratio and t is the time elapse after the beginning of creep tests. In
297
addition, Figure C2 graphically shows the measurement of Ca . Alternatively,
C^ = A£/Alog(0 is also used to describe the secondary compression. The relationship
between Ca and C^ is:
Ca£=Caex{l + e0) [C4]
where e0 is the initial void ratio. It is noted that Cr, Cc, Ca , and C^ are all
dimensionless parameters[Gl 16][G117]
298
Figure CI Measurement[G118] of Cr and Cc
CD o"
' •*•-•
TO Cd
-a o >
Vertical Effective Stress, a'v, in log scale (kPa)
299
Figure C2 Measurement[G119] of Ca
Ae
EOP: End of pore pressure dissipation
log(time)
300
Figure C3 Measurement of Ca from secondary compression tests on London[il20]
clay (data from Lo 1961)
• & X «
sJTO w**0 o^50 1^w\J
** 330 3X> 350 IB 15
41
to 340380 360 1825 t-
345 385 365 1S30
2 3 4 5 104
Time (min) 2 3
301
Figure C4 Measurement of Ca from secondary compression tests on[G121]
Gloucester clay (data from Lo et al. 1976)
"5 o o o
12 h
°Ooo o
Au=0
^
LABORATORY DATA
O Loetal. 1976
Depth Stress Increment 4.3m 43.2 - 82.7 kPa
CCCE=0.022, e0=1.8
Ca=0.061
^ 'Q.
« S ^
S.
I I I I I I I I I \ I | | I 1 | J I I I I I I M 1 I I I t I M I I I I I I I I I | I | | | I i I I I I
10 100 1000 10000 100000 100000C
Elapsed Time (min)
Figure C5 Measurement of Ca from secondary compression tests on Drammen
(data from Bjerrum 1967; Berre and Bjerrum 1973)
TIME IN YEARS 0.1 ! 10 100 1000 3000
Figure Co Measurement of Ca from secondary compression tests on Sackville clay
(data from Hinchberger 1996)
-1 -
-2
^ -3r-c CO
55 1? -4 x ^ <
- °---o.... " ' 0 . .
o. o.
o.
-
— _ LABORATORY DATA
O Data from Hinchberger (1996)
Depth Stress Increment • 3.8m 50-100kPa
Cocs=0.0115 e0=1.7
- Ca=0.0311
i i i i i i 111 i l l
Q
O
Q * -
'—.
1 1 1 1
^o. \
M i l
""-'Q.. ^
•
^ Cas r^-.
1 ^O
i i i i i i 11
10 100 1000 Elapsed Time (min)
304
Figure C7 Measurement of Ca from secondary compression tests on
Berthierville[il22] clay (data from Leroueil et al. 1988)
0.00 LABORATORY DATA
- Data from Leroueil et al. 1988 Depth Stress Increment
I I
-.05 -
.10 h
.15 h
-.20
-.25
2.23-3.48m 135kPa
Cae=0.01 e0=1.7 Ca=0.027
_i i i i 1111
Note:
The test at the highest increment stress (135kPa) is chosen to obtain Ca, because the influence of clay destructuration on the secondary compression is assumed to be less significant for the clay sample at high increment stress than the clay samples at low increment stress.
Cas T — — —-1 135kPa
J i i i 1111
10 100 1000 _l_uj
10000
_ 1 I I I I 1 1 1
100000
Elapsed Time (min)
305
Figure C8 Measurement of Ca from secondary compression tests on St. Alban[il23]
clay (data from Tavenas et al. 1988)
St Alban Clay
LABORATORY DATA (Tavenas et al. 1978)
Long-term Oedometer creep test Depth Stress Increment 3m 28.0 kPa
Ccce=0.015 e0=2.43
Ccc=0.05
j i i i 11 in i i i i 1 1 1 1 1 _i i ' i i ' ' i ^uL
10 100 1000 10000 100000 100000C
Elapsed Time (min)
306
APPENDIX D
FACTORS AFFECTING a
This appendix investigates several factors that may have an impact on a, such as
temperature, plasticity index, sensitivity, liquidity index, and destructuration.
The influence of temperature has been investigated by several researchers (e.g.
Boudali et al.1994; Graham et al. 2001;Marques et al. 2004). Marques et al. (2004)
presented a detailed study on the temperature effect on the behaviour of St-Roch-de-
F Achigan clay. In Figure Dl, it can be seen that the slope for the log( <r' ) and log(eaiaal)
relationships appears to be independent on the change of temperature from 10°C to 30°C
and 50°C. Similar observations were reported by Boudali et al. (2004). Therefore, the
parameter a appears not to be sensitive to temperature.
The parameter, a , seems independent on the plastic index (PI). Table 2.1
summarizes the soil properties (e.g. water content and plasticity index) for the clays. The
values of a are plotted against the plasticity index (PI) for 18 clays in Figure D2. There
is no clear evidence for the correlation between a and PI. Thus, it seems that the rate-
sensitivity, represented by a, is independent on PI. This finding is consistent with the
study by Graham et al. (1983).
The correlations of a with St (Sensitivity), and LI (Liquidity index) are presented
in Figures D3, and D4 respectively. As shown in Figures D3, the correlation of a with
St can be approximately represented by a linear line, which shows the trend for most
clays presented except St. Alban clay and Batiscan clay. In Figure D4, the correlation
between a and LI can be represented by the following equations:
307
Best fit line: a = 0.05xLI [Dl]
Upper bound: a = 0.08xL7 [D2]
Lower bound: a = 0.03xZi [D3]
As shown in Figure D4, most clays fall in the range defined by the two bound
lines. However, it is noticed that the three Leda clays (St. Alban clay, Batiscan clay, and
Ottawa Leda clay) are located outside of the range defined by Equations [D2] and [D3].
These three Leda clays are the Champlain Sea Clay from eastern Canada, which is
characterized by the extraordinarily high water content and liquidity index. Therefore,
the proposed relationship between a and LI may be not applicable for some Leda clay.
Hinchberger and Qu (2007) discussed the influence of destructuration on a. The
comparison of a measured at different strains for London clay, Belfast clay and
Winnipeg clay respectively shows that the a measured at various strains appears to be
consistent. Thus, a is considered independent on the structure damage during
loading[il24]. (more details is referred to Appendix E).
308
Figure Dl Influence of temperature on the rate-sensitivity parameter, a forSt-
Roch-de-F Achigan clay (modified from Marques et al. 2004)
200
13 o ieo.
.5 140<
Temperature • 10 °c
309
Figure D2 Variation of viscosity exponent, a, with Plasticity index[il25] ( for clays listed in Tables 2.1 and 2.2)
.10
.08
8 o
.06 4
>
.04
O
O 0 0
O O
o
o
.02 4 o
o o
0.00 10
—r-
20 30 40 50 60 70
Plasticity lndex.%
310
Figure D3 The correlation between[i 126] a and St (Sensitivity)
.12
.10
.08 4
.06
.04
.02
0.00
a /
...v/.n... /
/
a =0.025 + 0.0016*St
: /
/
/
a Batiscan clay
""SrAlbah clay
20 40 60 80 100 120 140
St
D4 The correlation between a and[il27] LI (Liquidity index)
31
.12
.10 4
.08
.06 4
Upper bound
a=g.08*LI Best fit line oc=0.05*LI
/
; /
/ .
Ottawa Leda clay
.04
.02
0.00
Lower bound a=0.03*LI
_. Batiscan clay " 0 ~ St. Alban clay
Leda ciay
LI
312
APPENDIX[G128] E
INFLUENCE OF STRUCTURE ON THE TIME-DEPENDENT BEHAVIOUR OF
A STIFF SEDIMENTARY CLAY
Sorensen et al. (2007) have decided to study the influence of microstructure on
the time-dependent response of undisturbed and reconstituted London clay using drained
and undrained triaxial compression tests (CIU and CID) with step changes in the applied
strain-rate. The paper presents interesting behaviour and Sorensen et al. (2007) should be
commended for demonstrating the viscous response of London clay.
The primary influence of microstructure on the engineering response of London
clay can be seen in Figure El a, which compares the stress-strain response in the
undisturbed and reconstituted states. Figure Elb shows similar behaviour from triaxial
compression tests on Rosemere clay from Eastern Canada (Philibert 1976). From Figure
El, it can be seen that there are similarities in the relative stress-strain response of both
materials in spite of their vastly different index properties (e.g. IL = 0 versus I I ~ 1.2).
The stress-strain response of both clays during triaxial compression is characterized by:
(i) reaching a peak shear strength followed by post-peak strength reduction with large-
strain, (ii) predominantly strain hardening response of the reconstituted or disturbed
materials, and (iii) at large-strain, the post-peak strength of the undisturbed clay
approaches that of the reconstituted and 'cut' materials, respectively. The difference in
behaviour (the shaded areas in Figures El a and Elb) is typically attributed to the effects
of microstructure or weak bonding between the clay particles and aggregates of clay
particles. Such behaviour is analogous to that typically observed in oedometer
consolation tests on undisturbed and reconstituted materials (Burland 1990).
A version of this appendix has been accepted in Geotechnique 2007
313
Regarding the time-dependency or rate-sensitivity of London clay, Sorensen et al.
(2007) quantify viscous effects using the jump in deviatoric stress induced immediately
after changing the axial strain-rate. Although such an approach has merit, the following
presents an alternative interpretation of the rate-sensitive response of London clay using
the theory of overstress viscoplasticity (Perzyna 1963). The current authors hope that this
alternative interpretation will provide additional insight into the viscous response of
undisturbed and reconstituted London clay.
Theoretical Background
Perzyna (1963) originally proposed the theory of overstress viscoplastic for the
yielding of steel at high temperature. This theory has been subsequently adapted to
geologic materials by researchers such as Adachi and Oka (1982), Katona and Mulert
(1984), Desai and Zhang (1987) and Hinchberger and Rowe (1998) to name a few. For
an elastic-viscoplastic material, the strain-rate tensor can be decomposed into elastic and
viscoplastic components as follows:
e„ = 85+63" [El]
At yield or failure, the viscoplastic strain-rate typically dominates (Chapter 2). A
form of the viscoplastic strain-rate tensor is (e.g. Katona and Mulert 1984 and Desai and
Zhang 1987):
^=^(f))y/^ijhj{^/^y-^k^^j] [E2]
where ^ is a viscosity constant with units of inverse time (typically s_I), f is the yield
function from classical plasticity theory, §(f) is called the flow function and it is derived
from f , and [3f /da^ J is the plastic potential, which is derived as a vector of unit length.
314
The Macauley brackets ( ) in Equation [E2] imply (j)(f) = 0 for f < 0 and
HfHq/qoY-liorf>0.
The flow function, (|>(f), in Equation [E2] is a power law (Norton 1929) where q0
represents the long-term strength (reached at very low strain-rates), q is the strain-rate
dependent deviator stress at yield and the term q/q0 is the overstress (e.g. q/q0 =1.1
implies 10% overstress). An upper bound estimate of q0 , q 0 =125kPa±, can be
obtained for London clay from the deviator stress reached after 4 days of stress relaxation
(see Figure 3 in Sorensen et al. 2007).
Considering axial strain-rate only, the viscoplastic strain-rate at yield is
approximately:
e;W((4/<7j"-l)(V273) [E3]
where V2/3 is an estimate of the plastic potential, 3f / 3 o u , derived assuming constant
volume deformation. Although London clay exhibits dilatant behaviour during the
triaxial tests (see the pore pressure response in Figure 8, Sorensen et al. 2007), the plastic
potential has a negligible impact on the following discussion and derivation. Taking the
logarithm of Equation [E3] and rearranging, it can be shown that (Qu and Hinchberger
2007):
log(q) = alog(eaxial)+As [E4]
for q/q0 > 1.1. In Equation [E4], As =log(q0u.a) and a = \ln . Leroueil and Marques
(1996) and Soga and Mitchell (1996) have used a similar relationship to evaluate the rate-
sensitivity of various clays.
Thus, elastic-viscoplastic constitutive models based on a power law flow function
315
(e.g. Adachi and Oka 1982, Katona and Mulert 1984, Hinchberger 1996, Hinchberger
and Rowe 1998, and Desai and Zhang 1987) imply a linear relationship between log(q)
and log(e) for stress states at yield or failure. In such a theory, the rate-sensitivity
(variation of q versus £) at yield or failure is governed by a , which is the inverse of the
power law exponent, n . The following is a re-evaluation of the strain-rate effects
measured by Sorensten et al. (2007) for London clay using the above theoretical
framework.
Interpretation of Rate-Effects
Figure El a shows the deviator stress, q, versus axial strain response reported by
Sorensen et al. (2007). The data is re-plotted in Figure E2 using a semi-log scale. From
Figure E2, it can be seen that there is relatively uniform variation of log(q) versus axial
strain, notwithstanding that rate-effects appear to be less pronounced for the reconstituted
material at axial strains in excess of about 5%.
Extracting deviator stress versus axial strain-rate from Figures El (a) and E2, a
E3 summarizes the log(q) versus log(e) data extracted from undrained triaxial
compression tests on undisturbed London clay at axial strains of 1, 1.5, 2, 2.5, 3, 4, and
4.5%. Figure E4 shows similar data for the reconstituted material at axial strains of 1, 2,
3, 4, and 5%. The slope, a , of the lines in Figures E3 and E4 represents the rate-
sensitivity of London clay. When compared in Figure E5, the data suggests that the mean
value of a is about 0.023 (n=44) and that both the undisturbed and reconstituted
materials have essentially the same rate-sensitivity. Furthermore, the rate-sensitivity
316
parameter, a , estimated from drained triaxial tests on intact material (see Figure 9 in
Sorensen et al. 2007) is also plotted in Figure E5. It can be seen that the rate sensitivity
parameter estimated from CID triaxial tests is the same as that deduced from the CIU
tests. Thus, the rate-sensitivity is identical for both drained and undrained triaxial
compression and for the intact and remolded materials.
For comparative purposes, Figure E6 shows the results of step tests on Belfast and
Winnipeg clay (Graham et al. 1983). The strain-rate parameter, a , is plotted in Figure
E7 for both clays. From Figure E7, it can be seen that a varies from 0.035 to 0.041 (24<
n < 29) for Belfast clay, and from 0.033 to 0.036 (28 < n < 30) for Winnipeg clay. Both
clays are more rate-sensitive than London clay. In addition, Belfast, and Winnipeg clay
do not show reduced rate-sensitivity with continued straining (or destructuration) after
reaching the peak strength; even for axial strains in excess of 15%. In contrast, the rate-
sensitivity of London clay diminishes with large axial strains in excess of about 5%;
however, additional testing is required to confirm this behaviour.
Summary
From the above discussion and interpretation, it can be concluded that the rate-
sensitivity of undisturbed London clay is the same as that of the reconstituted material.
Thus, the structure of London clay appears to have a negligible impact on its rate
sensitivity, whereas, the primary influence of structure appears to be exhibited by the
shaded areas in Figures la and 2. The above interpretation, has utilized a power law in
conjunction with Perzyna's theory of overstress viscoplastic (Perzyna 1963) and clearly
other interpretations are possible. However, Sorensen et al. (2007) hope that this
discussion provides an alternative perspective to that of Sorensen et al. (2007) for
317
consideration.
318
Figure El Stress-strain behaviour during triaxial compression tests on London clay (Reconstituted and Undisturbed) and Rosemere clay (Undisturbed and 'Precut').
a=0.022 at e*4%; 0=0.022 at p=3% I \ ^ "\ i ! i ix=0.023 at e=2%
!|a=0.023:ate=1<^i
10-7 10-6 10-5 10"4 103 102
Strain rate, /min, in log scale
Figure E5 Summary of a obtained from undrained triaxial compression tests on reconstituted London clay, and drained and undrained tests on undisturbed London clay
.10
.08
O .06 •o
c 03
8 .04
.02
0.00
-Jt— CD test on undisturbed sample -•— CU test on undisturbed sample -O— CU test on reconsituted sample
0.00
Axial Strain
323
Figure E6 Stress-strain relations in CAU tests on Belfast clay and Winnipeg clay
.1 H
0.0
<J1C: Confining pressure.kPa
0.00
Axial strain rate = 5%/h 0,.5°/<^
Belfast clay (Graham, et al. 1983)
Winnipeg clay (Graham, et al. 1983)
.05 .10 .15 .20 .25 .30
Axial Strain, %
324
Figure E7 Parameter a measured for Belfast clay and Winnipeg clay
Belfast clay (drahaml Hlj al. 1983)
V-oc=0.035 M Reak a=0.040 M *f W/o post-peak o oc=6.041 i H e(=N5% post-peak
2i
Winnipeg clay (JGrahanfi,! M bil. 1983)
a=0.033 at Peak ] I
a=0.036 at e=10% post-piak]
a=0.033 at e=15% post-pieak
10" 7 10£ 10E 10J 10-3 10-2
Strain rate, /min
325
APPENDIX F
ON THE DECREASE OF STRAIN-RATE IN THE O/C CREEP TESTS
During the undrained creep test on Saint-Jean-Vianney clay at dry side in stress
space, the axial strain-rate was found to decrease with time prior to creep rupture.
Considering the incremental strain from the completion of loading to the creep rupture
was less than 0.2% for each creep test, the decrease of strain-rate is negligible from an
engineering point of view. However, theoretically, the overstress[G129] concept alone
can not explain this phenomenon.
It is noted that the influence of this decrease of strain rate in. creep is minor
considering-4he~4elal4ftefe^ ereep-was-ftet-e*eeed 0.2%, which is out of
engineering irrtefestrAlse-the in-situ creeps are often' in drained'Conditions, wMeh-fetiew
has been successfully used to simulate "the •s#-ain-rate -decrease- during -the drained creep
test and the undrained creep test with stress state in the "wet: side (e.g. Kutter et al. 1.992
and-Hinehberger 4 996).
To investigate the possible reasons for this phenomenon, this appendix re
evaluates this creep tests on SJV clay using modified approaches with various
assumptions to simulate the decrease of strain-rate. The hypotheses adopted in the
modified approaches are described below, together with the comparison of the calculated
and measured response of SJV clay.
In the first approach, it is assumed that the Drucker-Prager envelop would be
hardened due to plastic work, as suggested by Lade and Duncan (1973). The slope of the
326
Drucker-Prager envelop in the -J2J2 -<Jm stress space can be represented using the
effective friction angle viz:
M = 2 S w * ' [Fl] CS -> • , i L J
3-srn0
where Mcs is the slope of the Drucker-Prager envelop in Figure Fl, and 0' is the
effective friction angle.
The hardening law can be expressed using an exponential equation:
Mcs=M1-(Mz-Ml)xe'CWp [F2]
wp=jcr'yde? [F3]
where Mt and M2 represent the initial and final slopes, respectively, c is the hardening
parameter, and wp is the plastic work. The magnitude of the final slope, M2=1.34, was
obtained according to <p'= 40° reported by Vaid et al.(1979). The other two parameters,
M[ =1 and c =20 were obtained using a trial and error approach.
As shown in Figure Fl, the increase of the slope of the Drucker-Prager envelope
due to hardening leads to a contraction of dynamic yield surface and consequently a
decrease in the overstress. As a result, the calculated strain-rate during the creep tests
would reduce with time.
Figure F2 shows the comparison of the measured and calculated strain-rate versus
time during the undrained creep tests. It appears that the decrease of strain-rate can be
simulated by accounting for the hardening of the Drucker-Prager envelope.
In the second approach, it is assumed that the stress path in the central part of the
triaxial specimen was permitted to follow the elastic stress path during initial loading, not
327
the triaxial limit (see the dash line in Figure F3). In this approach[G130], more overstress
develops relative to the static yield surface in the central part of the specimen: a
consequence of the assumed stress state. Compared with analyses where the triaxial limit
was enforced throughout the specimen (see the solid stress-path line in Figure F3), the
higher level of overstress in the modified analysis causes significantly higher strain-rates
and more dilatancy early on in the simulation. Thus the overstress and consequent creep
rates reduce with time as the stress state moves right toward the static yield surface,
producing calculated creep rates similar to measured creep rates, as shown in Figure F4.
In summary, both of the two approaches used in this appendix are capable of
simulating the decrease of strain-rate and subsequent creep rupture during the undrained
triaxial creep tests on SJV clay. Another alternative is to assume rotational hardening of
the state boundary surface, which would give similar results with those two approaches.
In addition, the decrease of strain-rate can also be attributed to external factors, for
example, the sample bulging under constant loads and consequent stress decrease on the
specimen top. However, given the lack of experimental evidence to support these
hypotheses, a definitive conclusion can not be drawn as to the reason for the decrease of
strain-rate during the undrained creep tests at the dry side in stress space for SJV clay.
Further experiments on the overconsolidated natural clay are desired to testify these
hypotheses or investigate the external factors.
328
Fl Illustration of the hardening[il31] of the Drucker-Prager envelope.
Inereaseof M due lo .Hardening effect
Dynamic yield surfaces corresponding MI and M2
, N \ Contraction of the N \ dviuunlc vield
N x surfaces during creep \J*\ tests
\ 'X
\ \ \ \
m
329
Figure F2 The measured and calc[G132]ulated strain-rate variation during the creep
tests accounting for the hardening of the Drucker-Prager envelope
430
j—Calculated
10-H
Measured
10-5
10 100 Time (min)
1000 10000
330
Figure F3 Comparison of stress paths in CIU undrained creep on Saint-Jean-Vianney clay
Critical State Line
y'U) my
331
Figure F4 The measured and calculated stress-rate versus time using the second approach
10° •
10-' •
Rat
e, %
/MIN
C
reep
Stra
in
5
Axi
al
1 0 4 '
10* •
i
ad=470-
\
rr— 1 1 1
7 I / I / I / 1
/ 1 r /
-o
(»
X^F i\ ^^=\v^ I
; ^* • , A (Measured i
Calculated |
ad=430
1000 10000
Time (min)
332
APPENDIX G
A NON-ASSOCIATED VISCOPLASTIC APPROACH
The main body of the research in Chapter 5 has focused on the use of associated
viscoplasticity to describe the engineering behaviour of 'structured' anisotropic time-
dependent clay. This appendix describes an alternative approach based on a non-
associated flow rule in the over consolidated stress range (i.e. the dry side) and an
associated flow rule in the normally consolidated stress range (see Figure Gl). Based on
the results presented below, it can be seen that the engineering behaviour of Gloucester
clay can be described using either the approach presented in the main body or using the
approach summarized in Figure Gl.
Figures G2 to G4 compare the calculated and measured behaviours of Gloucester
clay during undrained triaxial compression tests. Figure G2 shows the measured and
calculated peak and post-peak strengths for specimens with the orientations of
i=0°,30o,45°,60o,and90°. The corresponding curves of deviator stress versus axial
strain and excess pore pressure versus axial strain are presented in Figure G3. Deviator
stress versus axial strain curves for i = 0° and i = 90° are compared in Figure G4.
As shown in Figure G2, a non-associated approach is also able to reproduce the
measured peak and post-peak strengths of Gloucester clay. The calculated deviator
stresses increase up to the peak strength after which there is a reduction of strength with
axial strain after mobilization of the peak strength (see Figures G3 and G4). The
calculated behavior agrees well with the measured behaviour. From Figures G3 and G4,
the general trends of the calculated and measured pore water pressure with strain are
333
comparable, although the excess pore pressures are underestimated by the non-associated
approach.
Overall, the non-associated approach can reproduce the major characteristics of
the stress- strain behavior and strength for Gloucester clay in undrained triaxial tests.
Figure Gl Conceptual behaviour of the non-associated soil model
A / 2 ^
M , = -0.03
V
Unassociated Plastic Potential Law Destructured Critical State Line
I I
M,j=0.0 Af.=0.03
M=0.9
Associated Plastic Potential Law
Typical Stress Path - —> M Static yield surface
335
Figure G2 The effect of sample orientation, i, on the measured and calculated peak and post-peak undrained strength of Gloucester clay, (using a non-associated approach)
30 Using a non-associated approach
Peak
O Measured strength (Law 1975) • Calculated strength (This paper)
10
5 -4
Measured post-peak strength at 8% strain Calculated post-peak strength at 8% strain Calculated post-peak strength at 20% strain
-20 -10 10 20 30 40 50 60 — i —
70
— i —
80 90
Orientation angle, i
336
Figure G3 The effect of sample orientation, i, on the measured and calculated (a) axial stress versus strain and (b) excess pore pressure versus strain for Gloucester clay, (using a non-associated approach)
Figure G4 The comparison for sample orientations, /, of 0° and 90° on the measured and calculated axial stress versus strain and excess pore pressure versus strain
a. .*_
CO
D
(0 Q.
l l £
8 10 12 Vertical Strain (%)
8 10 12
Vertical Strain (%)
338
CURRICULUM VITAE
Name : Guangfeng Qu
PLACE OF BIRTH: Hehei, China
POST-SECONDARY EDUCATION AND DEGREES:
Ph.D University of Western Ontario 2003-2008
Master of Science University of Tianjin 2000-.2003
Bachelor University of Tianjin 1996-.2000
HONORS & SCHOLARSHIPS
2006 Novak Award
2005 John Booker Award
2003-2006 IGSS (International graduate student scholarship)
2003-2007 Graduate Special Scholarship
2003 Outstanding Graduation Thesis of Master of Science
1999 Tianjin University Academically Outstanding Student Honor with privilege of
being directly admitted into the graduate school without Mandatory Admission
Examinations
1998 and 2000 Tianjin University People's Scholarship (The First Class)
RELATED WORK EXPERIENCE:
2000 Engineer in Jun Hua Foundation Engineering Technology Group
2003-2007 Teaching and Research Assistant, University of Western Ontario
Publications
Qu, G. and Hinchberger S.D. (2007) Evaluation of the viscous behaviour of natural clay using a generalized viscoplastic theory. Geotechnique, In review
Hinchberger, S.D. and Qu, G. (2007) Discussion: the Influence of structure on the time-dependent behaviour of a stiff sedimentary clay. Geotechnique. In press
Qu, G. Hinchberger, S.D., and Lo, K.Y. (2007) Case studies of three dimensional effects on the behaviour of test embankments. Canadian Geotechnical Journal. In review.
Hinchberger, S.D. and Qu, G.(2006) A viscoplastic constitutive approach for structured rate-sensitive natural clays. Canadian Geotechnical Journal, Re-Submitted November 2007
Hinchberger, S.D., Qu, G. and Lo, K.Y.(2007) A simplified constitutive approach for anisotropic rate-sensitive natural clay. International Journal of Numerical and Analytical Methods in Geotechnical Engineering. In review
Qu, G. and Hinchberger, S.D. (2007) Clay microstructure and its effect on the performance of the Gloucester test embankment. Geotechnical Research Centre Report No. GEOT2007-15, the University of Western Ontario, London, Ontario.
