Guangfeng Qu _PhD Thesis
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Transcript of Guangfeng Qu _PhD Thesis
SELECTED ISSUES ON THE PERFORMANCE OF EMBANKMENTS ON
CLAY FOUNDATIONS
(Spine title: Selected issues on the performance of embankments)
(Thesis format: Integrated-article)
By
Guangfeng Qn
Graduate Program
in
Engineering Science
Department of Civil and Enviromental Engineering
A thesis submitted in partial fulfillment
of the requirement for the degree of
Doctor of Philosophy
Faculty of Graduate Studies
The University of Western Ontario
London, Ontario,Canada
©Guangfeng Qu 2008
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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.
Key words: elastic-viscoplastic, viscosity, rate-sensitivity, natural clay, microstructure,
anisotropy, case study, three-dimension.
iv
CO-AUTHORSHIP
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
during undrained tests gives:
log{s^)=alog(saxial) + log(5H( s>)-alog( r-)-«log(J |) [2.11]
and by similar argument, since S(ud) I cr'^A) is also constant:
logfo™ ) = a \og(saxial) + log(cx;(s>)- a logfr *) - a l o g ^ ) [2.12]
Equations [2.11] and [2.12] are also straight lines in log-log scale as shown in
Figures 2.2(c) and 2.2(b), respectively. Equation [2.12] is essentially consistent with the
following relationship used by Leroueil and Marques (1996) for the variation of
preconsolidation pressure versus strain-rate in oedometer tests:
log{a'p)=alog(eaxial) + A [2.13]
where a and A are constants.
Referring to Figure 2.2a, there are 3 characteristics of Equation [2.9] that should
be discussed. First, the power law flow function (Equation[2.4]) implies a linear log-log
relation between isotropic yield stress and axial strain-rate, represented by a straight A-B
line in Figure 2.2a. Second, the slope of line A-B, a = IIn , represents the rate-sensitivity
of the isotropic yield stress: as a increases, the yield stress becomes more rate-sensitive.
Third, the linear A-B line terminates at Point A, whose coordinates are <j'^} (the static
23
yield surface) and yvp73 . Point A denotes the static yield surface intercept in Figure 2.1c
(i.e. (y'^y) = cr'^)). The value of yvp 13 can be considered as a threshold strain-rate above
which strain-rate effects are mobilized. For axial strain-rates less than yvp 13, the
isotropic yield stress is rate-insensitive (e.g. the EVP model retrogresses an elastic plastic
model). Similar principles apply to the preconsolidation pressure and undrained strength,
as shown in Figure 2.2(b) and 2.2(c) where again the slope, a, increases with the rate
sensitivity.
In summary, if a power law flow function (see Equation. [2.4]) is used in
conjunction with elastic-viscoplastic theory, then there will be linear log-log relationships
between S„d) - eaxial and cr'£A) - £axial, with the same magnitude of slope, a , as shown in
Equations [2.11] and [2.12]. In the following sections, these derived relationships will be
tested by evaluating the behaviour of 19 clays in rate-controlled undrained triaxial and
oedometer compression tests reported in the literature.
2.2.3 Link with the isotache concept
Figure 2.3a demonstrates the main characteristics of a power law EVP model, in a
generalized stress space with an additional axis of strain-rate. In contrast with
conventional critical state theory, the EVP model implies a family of dynamic yield
surfaces, which, as shown by the dashed lines (e.g. 1-2-3, 4-5-6, and 7-8-9) in Figure
2.3a, expand with increasing strain-rate. The static yield surface (A-C-E) in Figure 2.3a
defines the onset of time-dependent behaviour. Viscous behaviour is mobilized only
when the stress state exceeds the static yield surface. In addition, lines A-B, C-D, and E-
F shown in Figure 2.3a correspond to those shown in Figures 2.2(a), (b), and (c), and the
24
lines A-B, C-D, and E-F can be linked to each other by the yield surface function.
As shown in Figure 2.3b, a series of isotaches can be constructed by projecting
the dynamic yield surfaces in Figure 2.3a onto the yJ2J2 -a'm plane. The projected
dynamic yield surfaces define the rate-dependent yielding in generalized stress space and
are essentially consistent with the isotache concept proposed by Suklje (1957), and
extended by Tavenas et al. (1978) and Leroueil et al. (1985). This correlation can be
attributed to the common assumption of the existence of a unique cr'-e -e*9 relationship
shared by both the EVP model and the isotache concept. It can be further deduced from
the EVP model that the spacing among isotaches is governed by the parameter, a . For
example, the spacing between isotaches 4-5-6 and 7-8-9 in Figure 2.3b is controlled by
the vertical distance between the points 4 and 7 in Figure 2.2a, which is governed by the
magnitude of a . Thus, a higher a leads to a series of isotaches with larger spacing, as
shown in Figure 2.3c.
There is one key difference between the isotache concept and EVP theory. In the
EVP model, the distribution of dynamic yield surfaces has a lower limit, the static yield
surface (ACE), below which the behaviour of clay is elastic and rate-insensitive. In
contrast, the isotache concept assumes that isotaches contract infinitely in stress space
with the reduction of strain-rate.
2.2.4 Alternative flow function - the exponential law
Several viscoplastic flow functions have been proposed for use in overstress
models (e.g. Adachi and Okano, 1974; Perzyna, 1963; Desai and Zhang, 1987; Fodil et
al., 1997). Recently, an exponent flow function was used by Rocchi et al. (2003) who
reported good agreement with the measured viscous behaviour of Batiscan clay.
25
The basic exponential function is presented in Equation [2.5]. From Equations
[2.3] and [2.5], the following relationship between strain-rate and yield stress during
isotropic compression can be derived:
,'(*) acr'^ log(saxial) + l - a logO v / , ) - a log( - ) a my [2.14a]
where a-lln. As shown in Equation [2.14a], an exponential-law flow function implies
a linear relationship between \og(saxial) and <J'^ . Similarly, the following equations can
be obtained for undrained strength 5„ ' and preconsolidation pressure, a Ad) .
:(d)
,'(d)
«Sriog(*w) + l-a\og(yvp)-a\of>Q-) '(*)
a;*' =acr'p(s)log(eaxial) + l-alog(rvp)-a\ogQ-) a ,(.„)
[2.14b]
[2.14c]
Figure 2.4 illustrates the difference between a power law flow function and an
exponential flow function (see Equation [2.9] and [2.14a], respectively). Using a log-log
scale, the solid line representing a power law flow function is linear with a constant
slope; whereas the dash line representing an exponential flow function is convex and its
tangent slope gradually decreases with increased strain-rate.
2.2.5 Stress-controlled testing
Bjerrum (1967) defined secondary compression as delayed compression
(reduction of void ratio) at constant effective stress. Conventional plasticity theories can
not account for secondary deformation. Raymond and Wahls (1968) utilized Ca to
characterize creep deformation under constant effective stress as follows:
Ae = Ca-Alog(t) [2.15]
26
In this section, a relationship between Ca and a = 1/n is explicitly derived from
elastic viscoplastic theory.
From Figure 2.5, considering secondary compression over the time interval
At = t2-t1, the volumetric strain is:
Aevol =Ca /(l + e0)-log(t2 ltx) = Ca /(l + c j - l og te I s2) [2.16]
where e0 is the initial void ratio and£x and e2 are the volumetric strain-rate at tx and
12 (Note: s1/s2 =t2lt\ fr°m Equation [2.15]). From EVP theory, the strain-rate at tx
and t2 are:
t =Yvp{a'w /(j'(s) )"
b \ I \vmy l u my-tl) da' [2.17a]
and
s =YVP((T'W/a'(s) )" b2 / \umy ' umy-tl)
JL dcrl
[2.17b]
where <r'^y)_n and cr'j£lt2 are the static yield surface intercepts at tt and t2, respectively.
Substituting [2.17a] and [2.17b] into [2.16] gives:
A*vo, = Ca 1(1 + e0) • n • l o g « > ( 1 / <r'£2)
From the hardening law, Equation [2.6]:
X-K
[2.18a]
A £ vol = l + en
l n \ u my-tl I u my-tl) [2.18b]
Combining equations [2.18a] and [2.18b] gives:
- = Cj[\n(10)-(A-K)] [2.18c]
Simplifying Equation [2.18c] yields:
27
a=- = CJ(Cc-Cr) [2.19] n
where Cc and Cr are the compression index and recompression index, respectively
(Ce = ln(10)A and Cr = ln(10)/c).
Equation [2.19] defines the theoretical connection between the rate-sensitivity
parameter, a( = l/n), and the secondary compression index, Ca. The parameter, a,
plays an important role in this interrelationship. The above discussions in this section
describe the context in which the reported viscous behaviour of 19 clays will be evaluated
to assess if there is evidence supporting a unique set of viscosity parameters for clays.
It should be noted that a relationship similar to Equation [2.19] has been used by
other researchers (e.g. Mesri and Choi 1979; Kim and Leroueil 2001) viz.:
a=CJCc
Considering Cr is typically 10% of Cc for clays (Holtz and Kovacs 1981), the above
relationship is practically consistent with Equation [2.19] derived in this chapter.
2.3 Evaluation
This section first examines the influence of strain-rate on the preconsolidation
pressure of some natural and remolded clays, followed by undrained strength versus
strain-rate, and then the secondary compression behaviour. Table 2.1 summarizes the
conventional geotechnical properties of the clay, which originated from Hong Kong,
Norway, Northern Ireland, Britain, Sweden, the United States, and Canada.
2.3.1 Rate dependency of preconsolidation pressure
The preconsolidation pressure (cr'p) is very important in settlement calculations.
28
For many clays, the preconsolidation pressure and the e-log(<r'v) curve are rate-
dependent ( e.g. Leonardo and Ramiah 1959; Crawford 1965; Bjerrum 1967; Vaid et al.
1979; Graham et al.1983; Leroueil et al. 1983; and Leroueil et al. 1985). Figure 2.6
summarizes the usual range of strain-rates used in laboratory tests and the range
mobilized in situ. It appears that in situ strain-rates are much lower than those used in
laboratory. In addition, long-term field observations by Crawford and Bozozuk (1990)
and Kabbaj et al. (1988) have shown that the use of laboratory measured a'p and
compression curve without accounting for rate effects may lead to significant
underestimation of in situ long-term settlement for viscous clay deposits.
The preconsolidation pressure, er'p , versus strain-rate can be obtained from
oedometer tests using any of the following test procedures: (a) Constant rate of strain
(CRS) oedometer test, where cr'p and s^^ can be measured directly, (b) Conventional
incremental oedometer tests (e.g. Drammen and Winnipeg clay) undertaken using
different load increment duration. For this case, the strain-rate corresponding to a'p is
estimated from the average strain-rates in the loading increment straddling <j'p (Graham
et al. 1983). (c) Incremental creep tests (e.g. Batiscan clay). Leroueil et al.(1985)
introduced a procedure using a series of incremental oedometer creep tests to construct
the CRS compression curves by connecting stress points associated with the same strain-
rate in Ae - ln(cr'p ) space.
Figure 2.7 summarizes the measured <j' plotted against saxial for 12 clays. From
Figure 2.7, it appears that the relationship between log(cr'p) and log(f(a..a/) is essentially
29
linear regardless of the dramatically different magnitudes of cr'p ( from 40 to lOOOkPa)
and eaxial ( from 10E-8/min to 10E-2/min). As noted above, the effect of strain-rate on
(j'p can be represented by a, defined in Equation [2.12]. The parameter, a, can be
measured from the best fit line through the test data in log-log scale. It can be clearly
seen in Figure 2.7 that Ottawa Leda clay with a =0.104 exhibits greater rate-sensitivity
(<y'p) than Winnipeg clay with a =0.03. Table 2.2 summaries the measured values of a
from rate-controlled oedometer tests for 12 of the 19 clays studied. The magnitudes of a
fall in a range from 0.02 to 0.1. It should be noted that a has been measured over 2 to 3
orders of magnitude strain-rate for most of the clays reported in Figure 2.7. Only 2 clays
have been studied over 4 orders of magnitude strain-rate (See Berthierville and Batiscan
clays).
2.3.2 Undrained shear strength versus strain-rate
As a key parameter in stability analysis, the undrained shear strength (Su) of
many clays is also a function of strain-rate during loading. This observation was
confirmed by numerous studies both in the laboratory and using field vane tests (e.g.
Taylor, 1943; Bjerrum 1973; Torstensson 1977; Graham et al. 1983; Kulhawy and Mayne
1990; and Hinchberger 1996). Kulhawy and Mayne (1990) concluded that the average
change of Su is about 10% per log cycle change in strain-rate. In this section, the rate-
dependency of Su for each clay is studied and compared with the rate-dependence of the
preconsolidation pressure for the corresponding clay. This is done to investigate the
possible correlation between a from log(Su)-log(s ) relationship and a from
30
log(cr'p ) - log(s ) relationship.
For 13 clays in Table 2.2, the Su and associated saxial were obtained either from
undrained triaxial compression tests at different constant strain-rates, undrained tests with
step-changed strain-rates (e.g. Winnipeg clay, London clay, and Belfast clay), or field
vane tests with different rotation-rates (e.g. Backebol clay).
Figure 2.8 plots the normalized undrained shear strength, SuN, (normalized by the
strength, Su, at eaxial = 1.0 min-1) versus strain-rate in a log-log scale for 11 clays. For
most clays in Table 2.2, the relationship between l o g ^ ^ ) and l o g ^ ^ , ) is apparently
linear (See Figure 2.8). Two exceptions are Haney clay and Drammen clay, as shown in
Figure 2.9. For Haney clay, when the strain-rate reduces to a threshold value of
2xlO~5mnT1 , the undrained strength becomes constant and unaffected by further
reduction in the strain-rate. Thus, the undrained strength of Haney clay becomes rate-
independent at strain-rates less than 2 xlO-5 min"1. Similar phenomenon can be seen for
Drammen clay, which has a threshold strain-rate of 5xl0~6 min"1 (Berre and Bjerrum,
1973). Even so, it is noted that when s > 2x10 5 min"1 for Haney clay and
E >5xl0~6 min-1 for Drammen clay, the relation of l o g ^ ^ ) -\og(s) is essentially
linear as seen for the other clays.
Therefore, a linear l o g ^ ^ ) - log(^) relationship can be obtained for each of 11
clays shown in Figure 2.8 for the range of strain-rates investigated (up to 4 orders of
magnitude). For Haney and Drammen clays, the linear relationship has a termination
point, corresponding to a threshold strain-rate below which Su is rate-insensitive. This
phenomenon may also occur in other natural clays at very low strain-rate.
31
Interrelationship between a.uc and a_Qti
As summarized in Table 2.2 and Figure 2.10, the rate-sensitivity parameter, a,
measured from rate-controlled undrained tests (a . u c ) and oedometer tests (a.oed) appears
to be unique even though the stress paths in these two tests are different. The best
agreement has been found for Winnipeg clay where the difference between a_uc and a_oed
is only 0.001; whereas St. Jean Vianney clay has the worst agreement between a_uc and
«.oed, where the discrepancy is 0.007: This may be attributed to the natural variability of
clay samples and experimental error due to the triaxial equipment used ( See Vaid et al.
1979 and Roberson 1975). As also shown in Figure 2.10, the hollow circles representing
the interrelationship between a.uc and a.oed fall within the range of the bounded lines
that deviate +0.005 from the 1:1 line for the clays except Belfast clay (-0.006) and SJV
clay (-0.007). In general, the rate-sensitivity parameter, a , measured from rate-
controlled undrained and rate-controlled oedometer tests appear to be consistent.
However, more data over a wider range of the a values would be required for a more
definite conclusion to be drawn.
Exponential Flow Function
Figure 2.11(a) shows a comparison of the relationship between <j'p and saxial
derived from the power law (Equation [2.13]) and exponential law (Equation [2.14]) flow
functions, together with data from select clays (e.g. Batiscan, Winnipeg, Gloucester, and
Drammen clay). For Batiscan clay, Winnipeg clay, Drammen, and Gloucester clay, both
power law and exponential laws seem to fit well with the laboratory data over the range
of strain-rate measured. The difference between these two flow functions is negligible
32
compared with the data. However, Figure 2.11(b) shows the results from laboratory tests
and field observations by Leroueil et al. (1983) for the Gloucester case. The data in
Figure 2.11(b) does not agree well with the exponential flow rule and the data suggest a
linear to slightly concave up relationship. From this comparison, it appears that a power
law is more representative of data over a large range of strain-rates than the exponential
law.
2.3.3 Secondary compression
As previously shown, the rate-sensitivity in oedometer compression tests and in
undrained triaxial compression tests can be quantified by the magnitude of a .
Furthermore, it is not clear if the parameter a can be used to characterize other viscous
behaviour, such as the time-dependent creep deformation that occurs in secondary
compression.
Table 2.2 summarizes a obtained using Equation [2.19], and Cr, Cc, and Ca for
11 clays reported in the literature. In some cases, Cr was not reported (e.g. for Winnipeg
clay, Belfast clay and Ska Edeby clay). For these cases Cr was deduced from the typical
relationship of Cr = Cc/10 from Holtz and Kovacs (1981). For all other clays, however,
Cr, Cc, and Ca were obtained directly from published experimental data, using higher
incremental loading stresses to minimize the possible influence of clay structure and
destructuration during secondary compression (Leroueil and Vaughan 1990; Burland
1990). Appendix C summarized the determination of these parameters.
Referring to Table 2.2 and Figure 2.12, it can be seen that acreep obtained from
Cr, Cc, Ca, and Equation [2.19] is in a good agreement with a_otAand a_uc . The best
33
match was found for Sackville clay, where the difference is only 0.0001; whereas the
worst match was for San Francisco clay with a discrepancy of 0.012. It should be noted
that the samples used to measure a and a.uc were obtained respectively by
independent researchers (Arulanandan et al. 1971; Lacerda 1976) from different locations
and different depths. Again, considering the natural variability of clay and the potential
influence of clay structure during compression, the general match among acreep, a_otd and
or.uc is very encouraging.
2.3.4 Summary
In the preceding sections, the viscous behaviour in constant-stress tests and rate-
controlled tests was evaluated, using experimental data for 19 clays from Europe to North
America (see Table 2.1 for the references). The key observations are summarized below:
All of the clays summarized in this study exhibit an essentially linear relationship
between sa and Su , or between sa and ay p for strain-rates over 2 to 4 orders of
magnitude. This observation is consistent with Equations [2.11] and [2.12], which were
derived from EVP theory. For Haney clay and Drammen clay, the linear relationship has
a termination point at a threshold strain-rate, below which the undrained strength is rate-
independent. This is consistent with the concept of a static yield surface.
The strain-rate parameter, a, is unique for each of the 14 clays in Figure 2.13
regardless the type of experiment used to measure it. As shown in Table 2.2, the value of
acreep obtained from secondary compression using Equation [2.19] agrees well with a.oed
and aac measured from rate-controlled tests (oedometer and undrained tests,
respectively). Figure 2.13 compares auc, a and a.oed with the corresponding «avg
34
for 14 clays. From this figure, it can be seen that most symbols fall in a narrow range of
± 0.005 from the 1:1 line. This is encouraging from an engineering point of view and it
suggests the viscous behaviour in rate-controlled tests (rate-dependency of yield stress)
and stress-controlled test (time-dependency of secondary deformation) have an inherent
correlation through the parameter, a .
Appendixes D and E discusse other factors that may have an impact on a, such
as temperature, plasticity index, liquidity index, and destructuration.
2.4 Selection of Parameters
An EVP model can be developed by coupling viscosity with conventional
elastoplastic theory. Thus, the parameters required in such a model can be divided into
two groups: elastoplastic and viscosity-related. If critical state concepts are adopted, the
elastoplastic parameters include e , v , A (= Cc /ln(10)), K (= Cr /ln(10)), M , and a
suitable yield surface function Ffo'^ ,e^ol), where e is void ratio, v is Poisson's ratio,
A and K can be estimated from oedometer compression tests, M can be obtained from
the constant volume effective friction angle, and svvpol is the plastic volumetric strain. The
parameters, A and K , can be determined from oedometer compression test or isotropic
compression test, and M is the slope of the critical state line. Methods to determine these
elastoplastic parameters can be found in the literature (e.g. Roscoe and Burland 1968;
Chen and Mizuno 1990; and Atkinson 1993). In addition, the yield surface can be
estimated from the stress path in undrained triaxial compression tests or determined more
precisely from a series of stress-path probing tests (e.g. Tavenas et al. 1979, Leroueil et
al. 1979; DeNatale 1983). To avoid repetition, only measurement of the three viscosity-
35
related parameters, a, o*^, and yvp, are discussed below.
2.4.1 The measurement of a
The parameter, a , can be determined from either multiple CRS drained isotropic
compression tests, multiple CRS drained oedometer compression tests, or multiple CRS
undrained triaxial compression tests undertaken using various strain-rates. To minimize
the influence of natural variation, compression tests can be performed on one specimen
with step-changed strain-rates, from which the stress-strain curves at various constant
strain-rates can be interpolated (e.g. Richard and Whiteman 1963; Graham et al. 1983).
Examples of the step-changed strain-rate approach are shown in Figure 2.14. The
measured yield stresses can be plotted against the corresponding strain-rates in a log-log
scale, as illustrated previously in Figure 2.2abc. The slope of the regression line passing
through experimental data gives the magnitude of a.
If CRS drained oedometer or isotropic compression tests are adopted, the strain-
rate should be low enough to avoid generating significant excess pore water pressure,
since the pore water pressures vary within a specimen causing difficulties in
determination of the applied effective stress. The study on Batiscan clay (Leroueil et al.
1985) indicated that excess pore pressure was undetectable for strain-rates lower than
3xl0~6/min in CRS oedometer tests on specimens 19mm high and 50.8mm in diameter.
In addition, a can be obtained using Equation [2.19], where Ca , Cc, and Cr can
be measured from an oedometer consolidation test or K'0 triaxial test. This approach
provides an additional way to check a measured from CRS compression tests and it
permits evaluation of the consistency of different experiments. It is noted that this
36
approach should be carefully used for structured natural clays, considering the influence
of destructuration on the measured values of Ca and Cc during the compression loading.
Ideally, Ca should be measured for load increments in the intrinsic state as shown in
Figure 2.15.
2.4.2 The measurement of <r'g a n d x p
cr'j^ and yvp can be determined from the linear log-log relationship obtained
during the measurement of a (see Figure 2.2). The first step is to examine the log-log
plot, and if the clay has a threshold strain-rate below which rate effects stop, Option A
can be used. If such a termination point does not exist, Option B can be adopted. Option
A and B are discussed below.
Option A
Referring to Figure 2.9, the undrained shear strength of Haney clay is rate-
sensitive for strain-rates in excess of 2xl0~5 /min . This signifies that Point E on the
static yield surface has been reached at eaxial = 2xl0"5 /min (see Figures 2.2c and 2.3a).
Correspondingly, as shown in Figure 2.9, the normalized static strength S^ is 0.65,
which gives S(u
s) = 268kPa (see Vaid and Campanella,1979). As a result, the static yield
surface intercept, <r'j^, is:
^ A S W [2.20]
where A is a constant which can be derived from the yield surface function (see Point E
and A in Figure 2.3b). For Haney clay, cr'^ =515kPa since A s 0.5 (see Vaid and
Campanella, 1977), and the fluidity parameter is yvp = V3?2x 2xl0"5 /min
37
«2xlO~5min_1 (see Figure 2.2c). This approach can be applied to obtain the
parameters, cr'j^ and yvp, for Drammen clay (see Figure 2.9) and any other clay which
exhibit a threshold strain-rate in their log(iS'K ) versus log(£aj.ia/) relationships.
Alternatively, cr'^ a n d ^ can be obtained from the log(<r'p) versus log(f raa /)
relationship. For example, Figure 2.16 shows the variation of <r' with strain-rates for
Berthierville clay at a depth of 3.9-4.8m (Leroueil et al. 1988). The termination point
corresponding to static state can be reached at eCDdal = 9xl0~7 /min. As a result, the
fluidity parameter is yvp = V5/"3x 9xl(T7 /min «lxl0~6 /min (see Figure 2.2b).
Correspondingly, in Figure 2.16, the normalized static preconsolidation pressure is 0.92,
which gives cr'^)=79.8kPa (see Leroueil et al. 1988). The resultant static yield surface
intercept, cr'JJ, is:
< ) = A p X < [2-21]
where Ap is a constant which can be derived from the yield function and K'0 (see Point
A and C in Figure 2.3b).
OptionB
For cases where laboratory testing has not been done at strain-rates slow enough
to identify a termination point (e.g. Points C or E in Figures 2.2b and 2.2c), the fluidity
parameter, yvp, can be estimated from engineering cases.
Figure 2.17 shows the log(cr'p) versus l o g ( ^ a , ) relationship obtained from
laboratory and in situ observations for St. Alban clay (3.1-4.9m). An additional example
is presented in Figure 2.11b for Gloucester clay. For these two cases (Leroueil et al.
38
1983 and 1988), the termination point has not been reached for strain-rates as low as
1(T8 min -1. Consequently, it appears that the strain-rate, 10~8 min-1, can be taken as the
upper bound of the fluidity parameter, yvp. Figure 2.7 shows that for strain-rates lower
than 10~10 muT1, the completion of 1% strain requires more than 190 years, which is out
of practical interest for engineers. Thus, for St. Alban clay and Gloucester clay, the
fluidity parameter, yvp , can be taken from the range between the upper bound
(10~8 min"1) and lower bound (1(T10 min-1).
Next, according to the assumed yvp, the parameter a'^ or S^s) can be easily
estimated from the log( a'^) versus log( saxial) relationship or the log( S^s)) versus
l°g(£axiai) relationship (see Figures 2.2b and 2.2c). Consequently, the resultant <r'^ can
be obtained through Equations [2.21] or [2.20]. However, the value of cr'^ deduced
from these upper and lower bounds yvp should not result in yield stresses below the
initial in situ stress state of the clay.
Using Option B, the EVP constitutive model would be conservative and capable
of accounting for rate-effects for the service life of most engineering projects (e.g. for 50
years if the strains do not exceed 5% ± ). In addition, if the actual yvp is higher than the
assumed value, the prediction on long-term settlement and stability by the EVP model
will be on the conservative side.
2.5 Summary and Conclusion
This chapter has evaluated the viscous response of the clays reported in the
39
literature with the intent to examine if the viscous response of clay during drained and
undrained CRS laboratory tests and during drained constant stress tests can be described
within a general elastic viscoplastic theory. Both power law and exponential flow laws
have been used in conjunction with EVP theory to explicitly derive equations relating Su
and a'p versus strain-rate. The folio wings summarize the main findings:
1) A linear log -log relationship between strain-rate and preconsolidation pressure
(cr'p) or undrained shear strength (Su) can be obtained from the rate-sensitivity response
of all 19 clays for the ranges of strain-rate studied. This is consistent with the theoretical
equations derived from the power law EVP model.
2) This study shows strong evidence suggesting that the rate-sensitivity
parameter, a , measured from three different types of experiments (CRS undrained
triaxial compression, CRS oedometer compression, and oedometer creep tests) is
consistent for the clays studied. This consistency indicates that the viscous responses of
clay are inherently related and consequently it is possible to account for these apparently
different viscous behaviours using a single phenomenological constitutive theory.
3) In the EVP framework, a power law flow function appears more appropriate
than an exponential law flow function, especially for the cases involving a large range of
strain-rates and it is consistent with Ca from secondary compression theory.
4) The viscous response of the clays presented in this chapter can be fully
interpreted using the EVP framework with a power law flow function, provided a suitable
yield function, ^(cr^,^.) with an appropriate aspect ratio, A, is chosen. The rate-
sensitivity of & and 5„ and the time-dependency during secondary compression can be
40
described using a single set of viscosity parameters, a, cr'^, and^vp. Particularly, the
minimum undrained strengths observed from the behaviour of Haney, Drammen, and
Berthierville (3.8-4.8m) clays appears to confirm the concept of the static yield surface in
the EVP model.
5) A link between the isotache concept and the EVP framework has been
demonstrated. Both of these two theories have been developed based on a unique
<j' -svp -e relation.
6) Novel and straightforward guidance has been provided to select and measure
all three viscosity-related parameters.
41
References
Adachi, T., and Okano, M. 1974. A constitutive equation for normally consolidated clay.
In Soils and Foundations, pp. 55-73.
Adachi, T., and Oka, F. 1982. Constitutive equations for normally consolidated clay
based on elasto-viscoplasticity. Soils and Foundations, 22(4): 57-70.
Arulanandan, K., Shen, C.K., and Young, R.B. 1971. Undrained creep behaviour of a
coastal organic silty clay. Geotechnique, 21(4): 359-375.
Atkinson, J.H. 1993. An introduction to the mechanics of soils and foundations : through
critical state soil mechanics. McGraw-Hill Book Co., New York.
Bjerrum, L. 1967. Engineering geology of Norwegian normally-consolidated marine
clays as related to settlements of buildings. Geotechnique, 17(2): 81-118.
Bjerrum, L. 1973. Problems of soil mechanics and construction on soft clays. State of the
art report Session IV.. In Pore. 8th Int. Conf. Soil Mech. Mescow, Vol.3, pp. 111-
159.
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.
Crawford, C.B. 1965. Resistance of soil structure to consolidation. Canadian
Geotechnical Journal, 2(2): 90-115.
Crawford, C.B., and Bozozuk, M. 1990. Thirty years of secondary consolidation in
sensitive marine clay. Canadian Geotechnical Journal, 27(3): 315-319.
42
DeNatale, J.S. 1983. On the calibration of constitutive models by multivariate
optimization.. Ph.D Thesis, University of California.
Desai, C.S., and Zhang, D. 1987. Viscoplastic model for geologic materials with
generalized flow rule. In International Journal for Numerical and Analytical
Methods in Geomechanics, pp. 603-620.
Fodil, A., Aloulou, W., and Hicher, P.Y. 1997. Viscoplastic behaviour of soft clay.
Geotechnique, 47(3): 581-591.
Gasparre, A., Nishimura, S., Coop, M.R., and Jardine, R.J. 2007. The influence of
structure on the behaviour of London Clay. Geotechnique, 57(1): 19-31.
Graham, J., Crooks, J.H.A., and Bell, A.L. 1983. Time effects on the stress-strain
behaviour of natural soft clays. Geotechnique, 33(3): 327-340.
Hinchberger, S.D. 1996. The behaviour of reinforced and unreinforced embankments on
rate senstive clayey foundations. Ph.D Thesis, University of Western Ontario,
London.
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 engineeering.
Prentice Hall, Inc., Toronto.
Kabbaj, M., Tavenas, F., and Leroueil, S. 1988. In situ and laboratory stress-strain
relationships. Geotechnique, 38(1): 83-100.
Kaliakin, V.N., and Dafalias, Y.F. 1990. Verification of the elastoplastic-viscoplastic
bounding surface model for cohesive soils. Soils and Foundations, 30(3): 25-36.
43
Katona, M.G. 1984. Evaluation of Viscoplastic Cap Model. Journal of Geotechnical
Engineering, 110(8): 1106-1125.
Kavazanjian, E., and Mitchell, J. 1980. Time-dependent deformation behaviour of clays.
Journal of Geotechnical Engineering, 106(6): 611-630.
Kim, Y.T., and Leroueil 2001. Modeling the viscoplastic behaviour of clays during
consolidation: Application to Berthierville clay in both laboratory and field
conditions. Canadian Geotechnical Journal, 38(3): 484-497.
Kulhawy, F.H., and Mayne, P.W. 1990. Manual on estimating soil properties for
foundation design. Electric Power Research Institute, Palo Alto, Calif.
Lacerda, W.A. 1976. Stress-relaxation and creep effects on soil deformation. Ph.D.,
University of California, Berkeley, United States — California.
Law, K.T. 1974. Analysis of Embankments on Sensitive Clays, University of Western
Ontario, London, Ontario.
Lehane, B.M., Jardine, R.J., Bond, A.J., and Frank, R. 1993. Mechanisms of shaft friction
in sand from instrumented pile tests. Journal of Geotechnical Engineering, 119(1):
19-35.
Leonardo and Ramiah 1959. Time effects in consolidation of clays ASTM special
technical publication No.254
Leroueil, S., and Vaughan, P.R. 1990. The general and congruent effects of structure in
natural soils and weak rocks. Geotechnique, 40(3): 467-488.
Leroueil, S., Samson, L., and Bozozuk, M. 1983. Laboratory and field determination of
preconsolidation pressures at Gloucester. Canadian Geotechnical Journal, 20(3):
477-490.
Leroueil, S., Kabbaj, M., and Tavenas, F. 1988. Study of the validity of a a - sv - £v rate
model in in istu conditions. Soils and Foundations, 28(3): 13-25.
Leroueil, S., Kabbaj, M., Tavenas, F., and Bouchard, R. 1985. Stress-strain-strain-rate
relation for the compressibility of sensitive natural clays. Geotechnique, 35(2):
159-180.
Lo, K.Y. 1961. Secondary compression of clays. Journal of Geotechnical and
Geoenvironmental Engineering, 87(4): 61-87.
Lo, K.Y., Bozozuk, M., and Law, K.T. 1976. Settlement analysis of the gloucester test
fill. Canadian Geotechnical Journal, 13(4): 339-354.
Mesri, G., and Choi, Y.K. 1979. Discussion on 'Strain-rate behaviour of St. Jean Vianney
clay'. Canadian Geotechnical Journal, 4: 831-834.
Mesri, G., Feng, T.W., and Shahien, M. 1995. Compressibility parameters during primary
consolidation. In Proc. Int. Symp. on Compression and Consolidation of Cayey
Soils. Hiroshima, Jpn, Vol.2, pp. 1021-1037.
Mesri, G., and Castro, A. 1987. Ca / Cc concept and k'o during secondary compression.
Journal of geotechnical engineering, 113(3): 230-247.
Oldecop, L.A., and Alonso, E.E. 2001. A model for rockfill compressibility.
Geotechnique, 51(2): 127-139.
Oldecop, L.A., and Alonso, E.E. 2007. Theoretical investigation of the time-dependent
behaviour of rockfill. Geotechnique, 57(3): 289-301.
Perzyna, P. 1963. Constitutive equations for rate sensitive plastic materials. Quarterly of
Applied Mathematics, 20(4): 321-332.
Philibert, A. (1976). "Etude de la resistance au cisaillement d'une argile Champlain. ,
M.Sc. Thesis, Universite de Sherbrooke, Quebec.
Raymond, G.P., and Wahls, H.E. 1976. Special Report: Estimating 1-dimensional
consolidation, including secondary compression, of clay loaded from
overconsolidated to normally consolidated state, National Research Council,
Transportation Research Board.
Richardson, A.M., and Whitman, R.V. 1963. Effect of strain-rate upon undrained shear
resistance of saturated remoulded fat clay. Geotechnique, 13(4): 310-324.
Robertson, P.K. 1975. Strain-rate behaviour of Saint-Jean-Vianney clay. Ph.D Thesis,
University of British Columbia, British Columbia,Canada.
Rocchi, G., Fontana, M., and Da Prat, M. 2003. Modelling of natural soft clay destruction
processes using viscoplasticity theory. Geotechnique, 53(8): 729-745.
Roscoe, K.H., and Burland, J.B. 1968. On the generalized stress-strain behaviour of 'wet'
clay. In Engineering Plasticity, pp. 535-609.
Rowe, R.K., and Hinchberger, S.D. 1998. The significance of rate effects in modelling
the Sackville test embankment. Canadian Geotechnical Journal, 35(3): 500-516.
Sallfors, G. 1975. Preconsolidation pressure of soft high plastic clays, Chalmers
University of Technology, Gothenburg, Sweden.
Sheahan, T.C. 1995. Interpretation of undrained creep tests in terms of effective stresses.
Canadian Geotechnical Journal, 32(2): 373-379.
Sheahan, T.C, Ladd, C.C., and Germaine, J.T. 1996. Rate-dependent undrained shear
behavior of saturated clay. Journal of Geotechnical Engineering, 122(2): 99-108.
Sorensen, K.K., Baudet, B.A., and Simpson, B. 2007. Influence of structure on the time-
dependent behaviour of a stiff sedimentary clay. Geotechnique, 57(1): 113-124.
Soga, K., and Mitchell, J. K. (1996). "Rate dependent deformation of structured natural
clays." Measuring and modeling time dependent soil behaviour, Geotechnical
special publication No. 61, ASCE, Washington D.C., 243-257.
Suklje, L. 1957. The analysis of the consolidation process by the isotache method. In
Proc. 4th Int. Conf. on Soil Mech. and Foun. Engen. London, Vol.1.
Tavenas, F., and Leroueil, S. 1978. Effects of stresses and time on yielding of clays. In
Proc of the Int Conf on Soil Mech and Found Eng, 9th, Jul 11-15 1977. Edited by
P.C.O.X. ICSMFE. Tokyo, Jpn. Jpn Soc of Soil Mech and Found Eng, Tokyo, pp.
319-326.
Tavenas, F., Leroueil, S., La Rochelle, P., and Roy, M. 1978. Creep behaviour of an
undisturbed lightly overconsolidated clay. Canadian Geotechnical Journal, 15(3):
402-423.
Taylor, D.W. 1948. Fundamentals of soil mechanics. John Wiley, New York.
Torstensson, B.A. 1977. Time-dependent effects in the field vane test. In Int. Symp. on
Soft Clays. Bangkok, pp. 387-397.
Vaid, Y.P., and Campanella, R.G. 1977. Time-dependent behavior of undisturbed clay.
Journal of the Geotechnical Engineering Division, 103(7): 693-709.
Vaid, Y.P., Robertson, P.K., and Campanella, R.G. 1979. Strain-rate behaviour of Saint-
Jean-Vianney clay. Canadian Geotechnical Journal, 16(1): 35-42.
Wiesel, C.E. 1973. Some factors influencing in situ vane test results. In Proc. 8th
ICSMFE. Moscow, Vol. 1.2, pp. 475-479.
47
Yin, J.-H., Zhu, J.-G., and Graham, J. 2002. A new elastic viscoplastic model for time-
dependent behaviour of normally and overconsolidated clays: Theory and
verification. Canadian Geotechnical Journal, 39(1): 157-173.
Zhu, G., and Yin, J.-H. 2000. Elastic visco-plastic consolidation modelling of clay
foundation at Berthierville test embankment. International Journal for Numerical
and Analytical Methods in Geomechanics, 24(5): 491-508.
Tab
le 2
.1
Geo
tech
nica
l pro
perti
es o
f 19
cla
ys
1 2 3 4 5 6 7 8 9 10
Cla
y N
ame
Rec
onsi
tute
d L
ondo
n cl
ay,
Eng
land
Rem
olde
d B
osto
n bl
ue
clay
, Uni
ted
Stat
es
Win
nipe
g, C
entr
al C
anad
a
Glo
uces
ter
clay
, C
anad
a
Bat
isca
n cl
ay,
Can
ada
St. A
lban
cla
y, C
anad
a
Han
ey c
lay,
Can
ada
Hon
g K
ong
mar
ine
clay
, H
ong
Kon
g
Dra
mm
en c
lay,
Nor
way
St. J
ean
Via
nney
cla
y,
Can
ada
Ref
eren
ces
Lo
1961
;Sor
ense
n et
al.2
007;
G
aspa
rre
et a
l.200
7
Tay
lor
1948
Gra
ham
et
al.1
983
Law
, 197
4; L
o et
al,1
976;
L
erou
eil
et.
al,1
983
Ler
ouei
l et
al.
1985
;
Tav
enas
and
Ler
ouei
l,197
8;
Gra
ham
et a
l,198
3;
Vai
d an
d C
ampa
nella
,197
7
Yin
et
al. 2
002
Bje
rrum
196
7;
Ber
re a
nd B
jerr
um,
1973
;
Vai
d et
al,1
979
Wat
er
Con
tent
(%
)
23
39.9
60
60
79.6
90
57.4
51
42
Liqu
id
Lim
it (%
)
60
45.4
77
52
43
50
44
60
62
36
LI
(%)
0.08
0.77
0.62
1.27
2.74
2.74
0.92
0.65
1.38
PI
(%)
40
23.7
45
30
21
23
18
31.5
31
16
St
N/A
N/A
3
N/A
125
14
8
N/A
N/A
100
c c
c
Cc =
0.38
6, C
r =0.
184,
Ca
=0.0
36
CJC
C
=0.
018,
Cr =
CJ1
0
Cc
=1.4
95,
Cr =
0.05
8,
Ca
=0.
061
Cc=
0.24
, C
r = C
c/10
CJC
C
=0.
03
Cc
=0.
0793
, C
r =0.
018,
Ca
=0.0
025
Cc=
0.45
, C
r =0.
055,
Ca
=0.0
16
-1^
00
Tab
le 2
.1
Geo
tech
nica
l pro
perti
es o
f 19
cla
ys (C
ont.)
11
12
13
14
15
16
17
18
19
Cla
y N
ame
Bel
fast
cla
y, N
.Ire
land
Sack
ville
cla
y,
Can
ada
Ska
Ede
by c
lay,
Sw
eden
Bac
kebo
l cl
ay (
7m),
Sw
eden
Ber
thie
rvill
e cl
ay,
Can
ada
St C
esai
re c
lay,
C
anad
a
Lou
isev
ille
clay
, C
anad
a
San
Fran
cisc
o B
ay M
ud.
Uni
ted
Stat
es
Led
a cl
ay,
Can
ada
Ref
eren
ces
Gra
ham
et
al,1
983
Hin
chbe
rger
199
6;R
ow a
nd
Hin
chbe
rger
199
8.
Wie
sel,1
973;
Mes
ri e
t al
. 199
5
Sallf
ors
1975
; Tor
sten
sson
19
77;L
erou
eil
et a
l 19
85.
Ler
ouei
l et
al.
1988
; Z
hu a
nd Y
in 2
000;
K
im a
nd L
erou
eil,2
001;
Ler
ouei
l et
al.
1985
Ler
ouei
l et
al.
1985
Aru
lana
ndan
et a
l. 19
71;
Lac
erda
197
6;
Kav
azan
jian
and
Mit
chel
l 19
80
Cra
wfo
rd,
1965
Wat
er
cont
ent
(%)
70
61
102
62
84.8
76.5
100
55
Liq
uid
Lim
it (%
)
90
47
99
46
70
70
93
31
LI
(%)
0.67
1.74
1.05
1.67
1.34
1.15
1.15
4
PI
(%)
60
19
65
24
43
43
48
8
St
7.5
N/A
25
22
28
~7
>50
c c
c
CJC
C =
0.05
,
Cr=
CJ1
0
Cr
=0.0
7;, C
c =
0.64
6,
Ca
=0.
0312
CJC
C =
0.05
,
Cr=
CJW
Cr
=0.
027,
Cc =
0.49
7,
Ca
=0.0
27
Cr=
0.1,
Cc
=0.7
5, C
a =0
.05
-1^
Tab
le 2
.2
Sum
mar
ized
a
for
19 c
lays
1 2 3 4 5 6 7 8 9 10
11
Cla
y N
ame
Rec
onsi
tute
d L
ondo
n cl
ay,
Eng
land
R
emol
ded
Bos
ton
blue
cla
y ,
Uni
ted
Stat
es
Win
nipe
g, C
entr
al C
anad
a
Glo
uces
ter
clay
, C
anad
a B
atis
can
clay
, C
anad
a St
. Alb
an c
lay,
C
anad
a
Han
ey c
lay,
Can
ada
Hon
g K
ong
mar
ine
clay
Dra
mm
en c
lay,
Nor
way
St. J
ean
Via
nney
cla
y,
Can
ada
Bel
fast
cla
y, N
.Ire
land
Ref
eren
ces
Lo
1961
;Sor
ense
n et
al.2
007;
G
aspa
rre
et a
l.200
7
Tay
lor
1948
;
Gra
ham
et
al.1
983
Law
,197
4; L
o et
al,1
976;
L
erou
eil
et.
al,1
983
Ler
ouei
l et
al.
1985
; M
esri
et a
l. 19
95
Tav
enas
and
Ler
ouei
l, 19
78;
Gra
ham
et
al,1
983
Vai
d an
d C
ampa
nella
,197
7
Yin
et a
l. 20
02
Bje
rrum
196
7;
Ber
re a
nd B
jerr
um,
1973
Vai
d et
al.1
979
Gra
ham
et
al,1
983
"-«
fr
om
tria
xial
co
mpr
essi
on
(13
clay
s)
0.02
3
0.02
4
0.03
1
0.03
7
0.04
1
0.04
4
0.05
1
0.04
5
0.04
6
«-oe
d
from
oe
dom
eter
co
mpr
essi
on
(12
clay
s)
0.03
0
0.03
5
0.04
7
0.04
1
0.04
8
0.05
2
0.05
2
CI cr
eep
From
E
q.[2
.19]
(1
1 cl
ays)
0.01
8
0.02
0
0.04
3
0.03
3
0.04
1
0.04
1
0.05
6
(the
av
erag
e va
lue)
0.02
1
0.02
4
0.02
7
0.03
8
0.04
0
0.04
1
0.04
1
0.04
3
0.04
7
0.04
9
0.05
1
o
Tab
le 2
.2
Sum
mar
ized
a
for
19 c
lays
(Con
t.)
12
13
14
15
16
17
18
19
Cla
y N
ame
Sack
ville
cla
y,
Can
ada
Ska
Ede
by c
lay,
Sw
eden
Bac
kebo
l cl
ay (
7m),
Sw
eden
Ber
thie
rvill
e cl
ay,
Can
ada
St C
esai
re c
lay,
C
anad
a
Lou
isev
ille
clay
, C
anad
a
San
Fran
cisc
o B
ay
Mud
. U
nite
d St
ates
Led
a cl
ay, C
anad
a
Ref
eren
ces
Hin
chbe
rger
199
6;
Row
and
Hin
chbe
rger
199
8.
Wie
sel,1
973;
Mes
ri e
t al.
1995
Sallf
ors
1975
; Tor
sten
sson
19
77;L
erou
eil
et a
l 198
5.
Ler
ouei
l et
al.
1988
; Z
hu a
nd Y
in 2
000;
K
im a
nd L
erou
eil,2
001;
Ler
ouei
l et
al.
1985
Ler
ouei
l et
al.
1985
Aru
lana
ndan
et
al. 1
971;
L
acer
da 1
976;
K
avaz
anjia
n an
d M
itche
ll 19
80
Cra
wfo
rd,
1965
«-u
c
from
tr
iaxi
al
com
pres
sion
(1
3 cl
ays)
0.05
3
0.05
4
0.05
3
0.06
5
«-oe
d
from
oe
dom
eter
co
mpr
essi
on
(12
clay
s )
0.05
8
0.05
6
0.06
7
0.06
9
0.10
4
cree
p
From
E
q.[2
.19]
(1
1 cl
ays)
0.05
3
0.05
6
0.05
7
0.07
7
(the
av
erag
e va
lue)
0.05
3
0.05
5
0.05
6
0.05
7
0.06
7
0.06
9
0.07
1
0.10
4
52
Figure 2.1 Illustration of [G24]models for elastic viscoplastic materials
b)
c'P -V«P ^axial I
a , —a axial v
V °> J
c) d)
Critical state line
Figure 2.2 Illustration of relations between strain-rate and yield stress (or
shear strength) in strain-rate controlled tests
(a)
- Derived from the EVP mbdel with a power law flow function
-
-.
-
'.. 0, ( »L- ' - A
— _ . rpy _ -r
1 1
* 1 1-
:
; ...^ —
1 i —
4 y
, j .
t
j : ~
• < 1 i
'
*
Ta
i \ •
i
-, . -
' !
Axial Strain Rate, min"1, in log scale
(b)
I
• Derived from the EVP model with a power taw flow function
-
, ,
'7\JfvZ-'Qyt _~"~~•*—'•— ~-y
1 ; ' ~ \ — i —
- - 'eyS
1 / * • / - X * "
t
1 1 *1 1 — -
Sf \
\
i
> D
or
•
(c) Axial Strain Rate, min '\ in log scale
I xittfi
u
....
._
I l ' l .
mvi
i . i
'• - I ' l l
.,i..L
J_L,
diffpn
5<
-:-f
S~7 "TIT
;v
j lheif
.......... 1)
(7 ill Lrv*f 3 !:
U =V
i m
: i
%
I
utfel with'ra p
•J f-i4-
-2-1 -r I r !
<
• • < •
^
H--
Swef:
*>
j —
I -
IS iv tlov
a
ur
MtinSJj
r-t-f
4-4 H
i | fill
5h:
"*p
J_j
• : i i i . i
.iillJRLI!
^ ,,L|J
i iii i ill : ..1
i ;H
i : l j
nt i..f...
r~i|:|--
-•
i i
-1:1
11
J Axial Strain Rate, min1 , in log scale
Figure 2.3 The link between the EVP model and the isotache concept
(a) Yield surface family for the EVP model
(b) The correlation between dynamic yield surfaces and isotaches on stress space.
Critical state line
Isotach contour
a(s) 4 7 B a my m
Figure 2.3 The link between the EVP model and the isotache concept (Cont.)
(c) The relationship between the magnitude of a and the spacing for isotaches
y[2J2
0 I -104 v«Wi
Isotaches
high
Figure 2.4 The influence of the power law and exponent law flow functions on the
relationship between yield stress and strain-rate
CO Q. j £
CO o CO
o
b CD i _ 3 CO CO
Q . ^ CO > O Q. 2 o CO
Strain Rate, mirf1, in log scale
57
Figure 2.5 Typical compression curve for secondary compression.
Ae
EOP: End of excess pore pressure dissipation
h log(time)
Figure 2.6 Ranges of strain-rates in laboratory tests and in situ (modified from
Leroueil and Marques, 1996)
' ^taiirrate range" SniFifeldtohaVieur
l1
strain rate range ,1 In abpratory test
\ f -j- i ' ' "S . I. ... . . . . j ^ . -1 f i 1- -t Berttatefvite-Ol -
-J|..M t
|M&?4 'jpRS)oeBomet«r
strain rate range in Varje shear test
-A,
5b3 pictocpster
* !r - 1 j i " " " — • |T « H ^ Jriftrsined shear test
Time required for"!*?* strpfn Years years
1 i . . . , i MsJwtris i,daysi!.;
foMfl^strair. ^ r $ y ^ year l.'tto^h
• <
$e$ofids i
1 Mins >.
10-13 -(o-12 1011 10-'° 10"9 10"8 10"7 10"6 10"s 1CH 10"3 10"2 10"1 10° Strain rate, /m in
58
10-9 10"8 10"7 10"6 10s 10"4 10"3 10-2 10-1 10° 101 102 103
Equivalent strain rate with unit of %/hr
Figure 2.7 Relationship between preconsolidation pressure, a'p, and strain-rate,
£ m i a i , i n log-log scale
CD Q .
CO CO
0
c o TO
"o (/) C
8 CD
c CO
co Q. Q. <
102 10"1
Strain rate, /min
A • O
• « n T O + V X
»
Batiscan clay Backebol clay Louiseville clay St Cesaire clay Gloucester Clay Belfast clay Winnipeg clay Drammen clay St.Jean Vianney clay Ottawa Leda clay Berthierville clay St. Alban clay
60
Figure 2.8 Relationship between undrained strength, Su, and axial strain-rate, saxial,
in log-log scale
1
.9
.8
.7
I 6
n • w
CO
CO II z
co3
.5
.3
I
a=0.023 I
- a=0.0244\^= = 5 j f =«==3S==
o|=0.031'Jf ^
a=0.0441 -
a=0.046 a
a=0.065\
0
•
a=0.053/
|
|
- ^3^ <*^^"^
• D • V A
e A
• ©
'
a=0. 0 3 7 . ^ - *
a=0.045
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).
Lastly, Figure 3.16 compares calculated and measured strength (or creep stress)
versus strain-rate in both creep tests and triaxial compression tests. Figure 3.16(a) shows
that the relationship between undrained strength and strain-rate in the constant rate-of-
strain tests is consistent with the relationship between creep stress and the minimum
strain-rate in the constant stress tests (Vaid et al. 1979). This measured behaviour agrees
well with the theoretical relations. As shown in Figure 3.16(b), the measured creep rate
versus Gd exhibits a similar pattern, but with significant natural variation for the lower
clay layer whereas the data for the upper layer is more uniform. In general, from Figure
3.16, it can be seen that the constitutive model provides a good estimate. In addition,
93
from a theoretical point of view, this figure shows that for a given clay the power law
exponent, n, in the constitutive model can be deduced from the minimum strain-rate
measured immediately prior to creep rupture.
3.4.4 Theoretical response for constant rate-of-strain consolidation
Figures 3.17(a) though (e) illustrate the theoretical behaviour of the constitutive
formulation for constant rate-of-strain K^ -consolidation. Figures 3.17(a) and (b) show a
typical stress path during strain-rate controlled K^,-compression. Figure 3.17(c) shows
the theoretical void ratio versus mean effective stress during K^ -consolidation. The
structure parameter is shown versus mean effective stress, a'm, and damage strain, ed , in
Figures 3.17(d) and (e), respectively.
Referring to Figure 3.17(a), the structured soil is assumed to initially behave as an
elastic material during K^ -compression as the stress state moves from point 1 to 2. At
point 2, the stress state reaches the static yield surface where there is a transition from
elastic to plastic behaviour. During continued compression, the material response from
point 2 to 4 remains predominantly elastic even though the stresses exceed the static yield
surface. This is evident in Figures 3.17(a), (b) and (c), inclusive. From point 2 to 4, the
structured fluidity and level of overstress are too low to cause significant plastic flow.
Thus, although the magnitude of the viscoplastic strain-rate tensor is finite, it is very
small and the material response is predominantly elastic (Figure 3.17c). As a result, the
ratio of vertical to horizontal stress from 1 to 4 is governed by Poisson's ratio.
At point 4, the combined overstress and structured fluidity are such that the
viscoplastic strain-rate becomes sufficiently large to dominate the constitutive behaviour
94
causing destructuration. There is a change in the stress path from 4 to 5 as the specimen
adjusts to the plastic strain-rate tensor (or plastic potential). During the transition from 4
to 5, the static yield surface begins to expand due to strain hardening. Continued
compression beyond point 5 causes further destructuration and consequently increased
fluidity. Eventually, at point 6, the initial structure is destroyed and the compression
curves of the structured and intrinsic soil skeleton merge (see Figure 3.17c).
Although it is not entirely evident in Figures 3.17(a) through (e), the theoretical
behaviour during K^ -compression is analogous to that described for CIU triaxial
compression. Initially, significant overstress builds in the model from point 2 to 4 due to
the low fluidity of the structured soil fabric. The overstress reaches a maximum at point
4 and begins to dissipate as the magnitude of the plastic strain-rate tensor becomes large
enough to dominate the material behaviour. From point 4 to 6 in Figure 3.17, there is
stress-relaxation similar to that described in Figure 3.2 for CIU triaxial compression. As
shown in Figure 3.17(c), the overstress ratio, G'^ lo'^, is greatest at the mobilized
apparent preconsolidation pressure (point 4) and lowest at point 6 where destructuration
is complete. In accordance with the theory of overstress viscoplasticity, reducing the
rate-of-compression relative to that illustrated in Figure 3.17(c) will cause points 3, 4, 5,
6 and 7 to shift to the left toward 1, 2 and 7: the long-term compression curve.
Conversely, increasing the strain-rate will cause points 3, 4, 5, 6 and 7 to shift to the
right. The strain-rate parameter, n, controls the degree of rate-sensitivity (or shift).
3.4.5 Constant rate-of-strain consolidation
Vaid et al. (1979) conducted a series of constant rate-of-strain consolidation tests
on 6.1 cm diameter and 2.5cm high specimens using a K^-triaxial cell (Campanella and
95
Vaid, 1972). Drainage was permitted from the top of each specimen and excess-pore
pressures measured at the bottom. The compressibility of SJV clay was studied using
strain-rates slow enough to allow almost complete dissipation of excess pore pressures.
However, Vaid et al. (1979) reported that the maximum excess pore pressure in the
fastest test was 7% of the applied vertical stress for stresses above 600kPa. Leakage was
also observed during the slowest tests as discussed below (Robertson 1975).
Figure 3.18(a) shows calculated and measured normalized compression curves
during K^ -compression. Figure 3.18(b) shows variation of the tangent virgin
compression index (Cc) versus vertical effective stress. Although Vaid et al. (1979) did
not specify the origin of the test specimens, it has been inferred that all tests were
performed on the upper layer (see previous discussion of Figure 3.8c). In addition, the
constitutive parameters used to obtain the calculated behaviour are the same as those
listed in Table 3.2 and used in prior computations.
From Figure 3.18(a), there is good agreement between the measured and
calculated normalized compression curves. The measured response is affected by natural
variation. Referring to Figure 3.18(b), the calculated and measured variation of Cc versus
vertical effective stress agrees well. In the proposed constitutive model, the initial high
compressibility mobilized after reaching the apparent yield stress is due to the high
structured fluidity, yjp , which permits the material to exist in a metastable state at a
higher void ratio than would be expected for the equivalent destructured or reconstituted
material. The post yield compressibility is governed by Equations [3.8] and [3.9].
To conclude, Figure 3.19 shows measured and calculated variation of the
e-loga'v response versus strain-rate. The calculated behaviour was obtained using a
96
static yield surface intercept of o Sy = 405 kPa for tests undertaken at rates of 1.68xl0~2
and 6.78xl0"2%/min and o'^^TOkPa for the tests undertaken at 1.17xl0"3 and 3.48x10"
3%/min. Both static yield surface intercepts were deduced from the data plotted in Figure
3.8(c). Overall, there is good agreement between the calculated and measured response
of SJV clay during strain-rate controlled K^ -compression. The data presented in this
figure further illustrates the challenge of working with natural clays, which often exhibit
significant natural variation. However, in spite of the need for some interpretation of the
results, the proposed constitutive model is able to describe the general trends in behaviour
of SJV clay during drained K^ -compression using parameters that are readily deduced
from standard rate controlled laboratory tests.
3.5 Summary and Conclusions
In this chapter, an existing elastic-viscoplastic constitutive model has been
extended using a state-dependent viscosity parameter to describe the response of rate-
sensitive structured clay. The formulation has been tested by comparing calculated and
measured behaviour of Saint-Jean-Vianney clay during CJU triaxial compression, CIU
triaxial creep and constant rate-of-strain K„ -consolidation tests. Considering the
challenges introduced by natural variation, the agreement between calculated and
measured response was found to be reasonable notwithstanding some differences. The
proposed formulation has been shown to describe the predominant effects of stain-rate on
the engineering behaviour of Saint-Jean-Vianney clay. With the exception of the damage
parameter, b , in Equation [3.9], the behaviour of Saint-Jean-Vianney clay could be
97
adequately described during both drained and undrained tests using a single set of
constitutive parameters.
Many natural clays develop strain-localization or shear banding at failure.
Accordingly, the higher damage parameter, b , required to describe shear failure
(b =4000 versus b =120) may be due to strain localization. In addition, it has been
recognized that the stiffness of triaxial apparatus can influence the rate of strain softening
(Lo 1972). Consequently, it must be recognized that the apparatus used by Vaid et al.
(1979) may have had an impact on the post-peak response of Saint-Jean-Vianney clay.
Most likely, both strain-localization and apparatus effects are incorporated in the damage
parameter, b , deduced above.
In terms of the rate-sensitivity of SJV clay, the research reported in this chapter
suggests that rate sensitivity and structure during drained and undrained laboratory
compression can be linked mathematically using state-dependent viscosity terms and
overstress viscoplasticity theory (Perzyna 1963). As shown above, the structure
exhibited in undrained compression and creep tests performed in a triaxial cell could be
described using a structure parameter deduced from oedometer compression tests. This is
considered to be new and of interest to researchers in the field of geomechanics.
Lastly, many natural structured clays exhibit significant anisotropy (Lo and Morin
1972). Both anisotropy and rate-sensitivity seem to be inherently linked for structured
clays such as St. Louis clay and St. Vallier clay (Lo and Morin 1972). In this study
anisotropy has been neglected. This is considered to be satisfactory for the present
investigation given the stress paths investigated (triaxial and K^ -consolidation) and the
absence of principal stress rotations. Regardless, there are normally significant principal
98
stress rotations in situ and the impact of anisotropy on the engineering response of
structured clays could be quite significant.
Overall, based on the analyses presented, the following conclusions are drawn:
The proposed formulation is able to generally describe much of the engineering
behaviour of SJV clay using constitutive parameters that are determined from standard
rate-controlled laboratory tests. The formulation can describe metastable phenomena
such as accelerated creep rupture and high compressibility after reaching the apparent
preconsolidation pressure of structured clay in addition to the influence of strain-rate on
the peak shear strength, post-peak shear strength and the apparent preconsolidation
pressure of SJV clay.
It appears that the rate-dependent behaviour during undrained and drained
laboratory tests can be simulated with a power law and unique single power law exponent
for SJV clay and for the range of strain-rate studied.
The structure parameter deduced from drained oedometer compression tests can
also simulate the influence of structure during undrained triaxial creep and compression
tests.
At times, the proposed constitutive model becomes approximate particularly
where natural variations may have affected the measured response and average soil
parameters have been used in the computations. Specifically, there are differences
between calculated and measured post-peak excess pore pressures during CIU
compression (see Figure 3.10a - upper layer) and creep rupture life (see Figure 3.14 -
lower layer).
The main limitation of the proposed constitutive model is that it cannot describe
99
the reduction of creep rates with time that are normally measured prior to creep rupture.
This is a minor limitation, however, since the strains that accumulate prior to creep
rupture are small and probably immeasurable in situ.
With the exception of the damage parameter b , a single set of constitutive
parameters can describe both drained and undrained behaviour of Saint-Jean-Vianney
clay over the range of strain-rates considered. Accordingly, it appears that it may be
possible to mathematically link the time-dependent behaviour of structured clay for stress
paths causing either shearing or compression. Traditionally, there has been an artificial
distinction between such behaviours (Tavenas et al. 1978).
The damage parameter, b , appears to be affected by strain localization. This
should be investigated further.
The constitutive model is proposed as an alternative to structured models that use
multiple or nested yield surfaces (Rocchi et al. 2003). Although the model can describe
most of the rate-sensitive and structured behaviour of SJV clay, the model should be
extended to include the effects of anisotropy, which can be pronounced in natural clays.
References
Adachi, T., and Oka, F. 1982. Constitutive equations for normally consolidated clay
based on elasto-viscoplasticity. Soils and Foundations, 22(4): 57-70.
Adachi, T., and Oka, F. 1995. An Elastoplastic Constitutive Model for Soft Rock with
Strain-Softening. International Journal for Numerical and Analytical Methods in
Geomechanics, 19(4): 233-247.
Adachi, T., Oka, F., and Mimura, M. 1987. Mathematical structure of an overstress
elasto-viscoplastic model for clay. Soils and Foundations, 27(3): 31-42.
Aubry, D., Kodaissi, E., and Meimon, Y. 1985. A viscoplastic constitutive equation for
clays including a damage law. In Fifth International Conference on Numerical
Methods in Geomechanics. Nagoya, pp. 421-428.
Baudet, B., and Stallebrass, S. 2004. A constitutive model for structured clays.
Geotechnique, 54(4): 269-278.
Bishop, A.W., and Little, A.L. 1967. Influence of size and orientation of sample on
apparent strength of London clay at Maldon, Essex. Oslo, Norway, Vol.1, pp. 89-
96.
Bishop, A.W., and Lovenbury, H.T. 1969. Creep characteristics of two undisturbed clays.
In Proc. 7th ICSMFE. Mexico, Vol.1, pp. 29-37.
Burland, J.B. 1990. On the compressibility and shear strength of natural clays.
Geotechnique, 40(3): 329-378.
Callisto, L., and Rampello, S. 2004. An interpretation of structural degradation for three
natural clays. Canadian Geotechnical Journal, 41(3): 392-407.
Campanella, R.G., and Vaid, Y.P. 1972. A simple Ko triaxial cell. Canadian Geotechnical
Journal, 9(3): 249-260.
Carter, J.P., and Balaam, N.P. 1990. AFENA-A general finite element algorithm: users
manual, School of Civeil Engineering and Mining Engineering,University of
Sydney, Australia.
Chen, S.G., Cai, J.G., Zhao, J., and Zhou, Y.X. 2000. Discrete element modelling of an
underground explosion in a jointed rock mass. Geotechnical and Geological
Engineering, 18(2): 59-78.
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.
Cho, T.F., and Lee, C. 1993. New discrete rockbolt element for finite element analysis.
International Journal of Rock Mechanics and Mining Sciences & Geomechanics
Abstracts, 30(7): 1307-1310.
Gasparre, A., Nishimura, S., Coop, M.R., and Jardine, R.J. 2007. The influence of
structure on the behaviour of London Clay. Geotechnique, 57(1): 19-31.
Graham, J., Crooks, J.H.A., and Bell, A.L. 1983. Time effects on the stress-strain
behaviour of natural soft clays. Geotechnique, 33(3): 327-340.
Hinchberger, S.D. 1996. The behaviour of reinforced and unreinforced embankments on
rate senstive clayey foundations. Ph.D Thesis, University of Western Ontario,
London.
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.
Hinchberger, S.D., and Qu, G. 2007. Discussion: the Influence of structure on the time-
dependent behaviour of a stiff sedimentary clay. Geotechnique, Accepted.
Karstunen, M., Krenn, H., Wheeler, S.J., Koskinen, M, and Zentar, R. 2005. Effect of
anisotropy and destructuration on the behavior of Murro test embankment.
International Journal of Geomechanics, 5(2): 87-97.
Katona, M.G. 1984. Evaluation of Viscoplastic Cap Model. Journal of Geotechnical
Engineering, 110(8): 1106-1125.
Kim, Y.T., and Leroueil 2001. Modeling the viscoplastic behaviour of clays during
consolidation: Application to Berthierville clay in both laboratory and field
conditions. Canadian Geotechnical Journal, 38(3): 484-497.
Kimoto, S., Oka, F., and Higo, Y. 2004. Strain localization analysis of elasto-viscoplastic
soil considering structural degradation. Computer Methods in Applied Mechanics
and Engineering, 193(27-29): 2845-2866.
Lade, P.V., and Duncan, J.M. 1973. Ccubical triaxial tests on cohesionless soil. Journal
of Soil Mechanics and Foundations Division ASCE(99(SM10)): 793-812.
Lefebvre, G., Langlois, P., Lupien, C , and Lavallee, J.-G. 1982. Laboratory testing and
in situ behaviour of peat as embankment foundation. In Canadian Geotechnical
35th Conference: Water Retaining Structures. Montreal, Quebec, Canada.
Canadian Geotechnical Soc, Montreal, Que, Can, pp. 113-142.
Leroueil, S., and Vaughan, P.R. 1990. The general and congruent effects of structure in
natural soils and weak rocks. Geotechnique, 40(3): 467-488.
Leroueil, S., Samson, L., and Bozozuk, M. 1983. Laboratory and field determination of
preconsolidation pressures at Gloucester. Canadian Geotechnical Journal, 20(3):
477-490.
Leroueil, S., Kabbaj, M., Tavenas, F., and Bouchard, R. 1985. Stress-strain-strain rate
relation for the compressibility of sensitive natural clays. Geotechnique, 35(2):
159-180.
Li, S.H., Wang, J.G., Liu, B.S., and Dong, D.P. 2007. Analysis of critical excavation
depth for a jointed rock slope using a face-to-face discrete element method. Rock
Mechanics and Rock Engineering, 40(4): 331-348.
Liu, M.D., and Carter, J.P. 1999. Virgin compression of structured soils. Geotechnique,
49(1): 43-57.
Lo, K.Y. 1970. The operational strength of fissured clays. Geotechnique, 20(1): 57-74.
Lo, K.Y. 1972. An approach to the problem of progressive failure. Canadian
Geotechnical Journal, 9: 407-429.
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., and Hinchberger, S.D. 2006. Stability analysis accounting for macroscopic and
microscopic structures in clays. In Proc. 4th International Conference on Soft Soil
Engineering. Vancouver, Canada, pp. pp. 3-34.
Locat, J., and Lefebvre, G. 1985. Laboratory investigations on the lime stabilisation of
sensitive clays. In Proc. 40th Can. Geotech. Conf. Regina, pp. 121-130.
Malandraki, V., and Toll, D. 2000. Drained probing triaxial tests on a weakly bonded
artificial soil. Geotechnique, 50(2): 141-151.
Mesri, G., Rokhsar, A., and Bohor, B.F. 1975. Composition and compressibility of
typical samples of Mexico city clay. Geotechnique, 25(3): 527-554.
Mitchell, J.K. 1976. Foundamental of soil behavioiur. Wiley, New York.
Norton, F.H. 1929. The Creep of Steel at High Temperature. McGraw-Hill Book Co.,
New York.
Perzyna, P. 1963. Constitutive equations for rate sensitive plastic materials. Quarterly of
Applied Mathematics, 20(4): 321-332.
Philibert, A. 1976. Etude de la resistance au cisaillement d'une argile Champlain. M.Sc.
Thesis, Universite de Sherbrooke, Quebec.
Qu, G., and Hinchberger, S.D. 2007. Evaluation of the viscous behaviour of natural clay
using a generalized viscoplastic theory. Geotechnique, submitted.
Robertson, P.K. 1975. Strain rate behaviour of Saint-Jean-Vianney clay. Ph.D Thesis,
University of British Columbia, British Columbia,Canada.
Rocchi, G., Fontana, M., and Da Prat, M. 2003. Modelling of natural soft clay destruction
processes using viscoplasticity theory. Geotechnique, 53(8): 729-745.
Roscoe, K.H., Schofield, A.N., and Thurairajah, A. 1963. Yielding of clays in states
wetter than critical. Geotechnique, 13(3): 211-240.
Rouainia, M., and Wood, D.M. 2000. Kinematic hardening constitutive model for natural
clays with loss of structure. Geotechnique, 50(2): 153-164.
Rowe, R.K., and Hinchberger, S.D. 1998. The significance of rate effects in modelling
the Sackville test embankment. Canadian Geotechnical Journal, 35(3): 500-516.
Saihi, F., Leroueil, S., La Rochelle, P., and French, I. 2002. Behaviour of the stiff and
sensitive Saint-Jean-Vianney clay in intact, destructed, and remoulded conditions.
Canadian Geotechnical Journal, 39(5): 1075-1087.
Sekiguchi, H. 1984. Theory of undrained creep rupture of normally consolidated clay
based on elasto-viscoplasticity. Soils and Foundations, 24(1): 129-147.
Silvestri, V. 1984. The preconsolidation pressure of Champlain clay, Part II. Canadian
Geotechnical Journal, 21(3): 600-602.
Sorensen, K.K., Baudet, B.A., and Simpson, B. 2007. Influence of structure on the time-
dependent behaviour of a stiff sedimentary clay. Geotechnique, 57(1): 113-124.
Suklje, L. 1957. The analysis of the consolidation process by the isotache method. In
Proc. 4th Int. Conf. on Soil Mech. and Foun. Engen. London, Vol.1.
Tavenas, F., Leroueil, S., La Rochelle, P., and Roy, M. 1978. Creep behaviour of an
undisturbed lightly overconsolidated clay. Canadian Geotechnical Journal, 15(3):
402-423.
Vaid, Y.P., Robertson, P.K., and Campanella, R.G. 1979. Strain rate behaviour of Saint-
Jean-Vianney clay. Canadian Geotechnical Journal, 16(1): 35-42.
106
Table 3.1 Properties of Saint-Jean-Vianney clay, (after Vaid et al. 1979)
Liquid Limit 36%
Plastic Limit 20%
Plasticity index 16%
Natural water content 42%
Degree of saturation 100%
Specific gravity of solid 2.75
Percent finer than 2um 50% Unconfined compressive strength
(test duration approx. 3min) 640kPa
Sensitivity Approx. 100
Activity P.I.(%<2um) 32%
Table 3.2 Constitutive parameters for Saint-Jean-Vianney clay
Parameters
Initial Structure, Q)Q
Destructuration Parameter, b
Weighting Parameter, A
Power Law Exponent, n Structured Fluidity, yv
sp, (min"1)
Intrinsic Fluidity, yjp, (min-1)
Aspect Ratio of Elliptical Cap, R
Moc
Mv
Ccs
Poisson's Ratio, v Initial Void Ratio, e0
Recompression Index, K Compression Index, X
Clay
1.68 4000 (120)
0.5
22
l.OxlO"10
9-lxlO"6
0.7 1.34
0.48
0.011
0 0.33 1.15 0.02 0.26
* The damage exponent, b , was 120 for K'0 compression and 4000 for triaxial
compression
108
Figure 3.1 The influence of structure on the response of Bothkennar clay during
oedometer compression (from Burland 1990[G40]).
2.2
2.0
1.8
1.6
O
'•a 1-t en •o O 1-2 >
1.0
- Attributed to structure \ f ^ ) / A \
" BothKennar clay from 6.5m depth \ N .
o In situ state
ICL (Intrinsic compression line)
, ,
\
1 10 100 1000 10000
Vertical Effective Stress, o ' v , kPa
109
ure 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[G41]).
£a,a = 0.05% lh = 8.3£ - 6/ min
— i 1 1 1 1 1 1 1 1 1 1 1 1 1 i
0.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
Axial Strain, %
110
Figure 3.3 The state boundary surface, critical state line, and mathematical overstress
of the structured soil model [G42].
Triaxial limit Critical state line
Resultant isotaches of constant (/>{F)
Associated flow rule \
\ \
\ \ \ \ \ V • -*-4— Dynamic yield surface
/(d) \ l (typical)
_ ' ( J ) '(d) my my
m
I l l
Figure 3.4 Estimation of the aspect ratio, R , for the elliptical cap[G43].
nrj- 1UUU
(kPa) 900 -
800 -
700 -
600 -
500 -
400 -
300 -
200 -
100 -
0 -
Crjti
y/ Elliptical cap with
/ ~-^S3'^ ^ ^ Q
r i i - T — T - i r i
cal state line M=1.34 :
an aspect ratio of 0.7
\ - r " i
100 200 300 400 500 600 700 800 900 1000
<ym , kPa
112
3.5 Estimation of the yield surface parameter, Moc, in the overconsolidated
stress range[G44].
uuu -
900 -
800 -
700 J
600 -
500 -
400 -
300 -
200 -
100 -
0 -
/ I W
/ u^i
I/'X r > X
Triaxial lirnit
/ The slope of the Lin
i : C
— 1 - • — — i — (
)
)
3
V-
yCr\t\ca\ state line
/ ' ' \ Mcs=1.34
)K/state line
"-- - -^--Dynamic yield surface
i - O -
\-w-i
Undrained triaxial tests on intact SJV clay \ (Saihi et al. 2002) \
i i
100 200 300 400 500 600 700 800 900 1000
cxm , kPa
Figure 3.6 Estimation of the intrinsic compressibility, X, and structure parameter,
co0, from oedometer compression for SJV clay[G45].
0.0
to
CO
a: •g o > a> o> c CO x: O
0.1 4
0.2 4
0.3 4
0.4 4
0.5 4
0.6
=*?=
Estimated
IntFinsic Gdmpression
line k=0.26
Data from Vaidetal. 1979
Deviation due to natural variation
1.68x10"^%/min
3.48x10"°%/min
1.17x10"3%/min
! , —
e"2 e^1 e° e1
Normalized Vertical Effective Stress, c'v /a'p (kPa), in natural log scale
114
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
oedometer compression for SJV clay
(a) Peak Undrained Shear Strength - Upper layer[G47].
05 Q.
1000
900
800 |-
700
600
500
400 h
300
200 -
100
i i i i i i r ...... Measured Peak Strength {Upper Layer)
Data from Vaid et al. 1979
n r
'2 J , = 224kPa
Note: From Figure 3.3, at Point A:
r ; ' ' = lxlO"")min"'
At I I
1Q-12 1Q-11 1Q-10 10-9 1Q-8 10-7 1Q-6 1Q-5 1Q-4 -| Q 3 10" 2 10"1 1 0 °
Strain Rate (/min)
116
Figure 3.8 Estimation of n ando'^ from undrained triaxial compression and
oedometer compression for SJV clay (cont.)
(b) Peak Undrained Shear Strength - Lower layer.
(kP
a)
CM
CM
1000 900 800 700
600
500
400
300
200
100 -10-12 -| Q-11 1 0-io <| Q-9 10"8 10"7 10"6 10"5 10"4 10"3 10"2 10"1 10°
Strain Rate (/min)
i i i i I i i r
Measured Peak Strength (Lower Layer)
Note: From Figure 3.3, at Point A:
\rlp =1x10-'° din"
117
Figure 3.8 Estimation of n anda'^y from undrained triaxial compression and
oedometer compression for SJV clay(Cont)
(c) Preconsolidation pressure versus strain-rate.
o-'1,:' = 52UPa a"''. = 0.796CT'''' = 417 kPa
<T"„" = 45lkPa o-f,! = 0.796er"„" = 358/tPu
a=0.045 n=1/a=22
Upper Layer]
B Measured apparent preconsolidation pressures (Vaidetal. 1979)
X;"=lxlO""min
J I I L .
-10-12 10-n -\Q-W 10'9 10"8 10"7 10"6 10"5 10"4 10-3 10"2 10"1 10°
Strain Rate (/min)
118
Figure 3.9 Influence of continued post-peak straining on the power law exponent, n.
(a) Stress-strain behaviour for Belfast clay and Winnipeg clay during CU tests
0.7
o.i H
0.0
CT1C: Confining pressure.kPa Axial strain rate = 5%/h
0.5%/h
Belfast clay (Graham, et al. 1983)
Winnipeg clay (Graham, et al. 1983)
10 15 20 25 30
Axial Strain, %
(b) Relationshi[G48]p between axial strain and n
c CD c o a. x 0 £ i—
c i—
100
80
60
40
20
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
Stress Level=630KPa Stress Level=575KPa ^ — — ^ — Stress Level=550KPa
Measured
Calculated
10000
125
Figure 3.14 Calculated and measured creep-rupture life for SJV clay[G54]
650
600 H
550 H
500
450 -\
400
f Upper Layer A Lower Layer
-V— Upper Layer -A— Lower Layer Df
• Measured'
Calculated
10 100 1000 10000
Rupture Life, tf, (min)
126
Figure 3.15 Calculated and measured axial strain-rate versus time during CIU triaxial
creep on SJV clay.
(a) Measured (b) Calculated
I i :070.1 550-
10000
Figure 3.16 Comparison of strain-rate at failure for peak strength and creep rupture
SJV clay,
(a) Upper layer
900
800
700
jk. Measured Critical Creep Rate In Creep Tests # Measured Strain Rate at Failure for Triaxial Shear Tests
Theoretical
1e-7 1e-6 1e-5 1e-4
Strain Rate /min
1e-3 1e-2
(b) Lower layer
900
800
£ 700
600
500
A Measured Critical Creep Rate In Creep Tests # Measured Strain Rate at Failure for Triaxial Shear Tests
Theoretical
1e-7 1e-6
For Lower Layer Clay
1e-5 1e-4
Strain Rate /min
1e-3 1e-2
Figure 3.17 Theoretical behaviour of the structured soil model during constant-rate-
of-strain K^ -consolidation.
Elasti
BasticZone
Ae.
1
(c)
2
Vlscoplastic Zone
C 3 ^ 5
\ v\
\ v \ \
Long Term p Compression Curve
am
Overstress Created by Structure, <a
\ 6
\ \ \ 7 V
Points:
1. Starting Stress State
2. Static Yield Point
3. Apparent Yield (No Structure)
4. Apparent Yield (Structured Soil)
5. Stress State Adjusted To The Plastic Potential
6. End of Destructuratton
CD ^ Elastic Zone
(Or
1.0L (d)
Viscoplastic Zone
4,5 L1,2,3,4,5
Structured Zone I Non-structured Zone •
129
Figure 3.18 Calculated and measured behaviour during oedometer compression[G55].
(a) Normalized compression curves.
Normalized Vertical Effective Stress, <f /o' kPa, in log scale
0.1 1 5
0.0
0.1
* 0.2
S 0.3
o 0.4 -
0.5
0.6
1 1 1 i • . i i
v% Estimated; ^ X y r
••Intrinsic compression line * J&
_ .
Measrued data
from Vaid et al. 1979!
. .
, o
I 1 1
— Structure effect
Theoretical
(Structured mode \
6.78x10"2%/min
1.68x10"2%/min
3.48x10"3%/min
1.17x10"3%/min
(b) Variation of compressibility (Cc) during loading
2.0
1.5
c o CO
£ 1.0 o O
0.5 A
0.0
Calculated Cc variation * Measured Cc variation
under the loading rate of 1.68 x 10"2%/min
500 1000 1500 2000 2500
Vertical Effective Stress, c'v, kPa
gure 3.19 Measured and calculated compression curves of SJV clay during constant
rate-of-strain consolidation[G56].
100 1000
Vertical Effective Stress, a'v, kPa
131
CHAPTER 4 THE STUDY OF STRUCTURE AND ITS DEGRADATION ON THE
BEHAVIOUR OF THE GLOUCESTER TEST EMBANKMENT
4.1 Introduction
Staged construction is often used to build embankments founded on soft clay
deposits. This construction method can be used to improve the stability of embankments
during construction by permitting time for excess pore pressure dissipation and
consolidation to occur, leading to increased strength in most cases. However, for some
embankments built on clay, staged construction has failed to produce significant strength
gain even after large vertical settlements and several years of consolidation (Holtz and
Broms 1972 and Stermac et al. 1967). Such behaviour has been attributed to strength
loss due to 'disturbance' or 'yielding' of the clay, which exceeds that normally expected
due to consolidation and reduction of void ratio. This phenomenon is thought to be due
to the breakdown of microstructure, which is a combination of fabric effects and
cementation bonding between clay particles (Mitchell 1976).
Many natural clays have microstructure to some extent (here after referred to as
structure). Structure can have a pronounced impact on the stress-strain-time response of
clay and the performance of embankments built on those deposits of clay. As discussed
in Chapter 3, structure has two important effects: (i) it imparts additional strength above
and beyond that normally anticipated from conventional soil mechanics models and (ii) it
permits clay to exist at higher void ratios than expected for the corresponding
132
'unstructured' or reconstituted material. With respect to structured clay, only Karstunen
et al. (2005) has studied the influence of clay structure on the performance of
embankments (the Muro test embankment). However, most structured clays are very rate
sensitive and Karstunen et al. (2005) undertook their study using a rate-independent
elastic plastic constitutive model. As such, an analysis that takes into account both the
time dependent response of structured clay and the influence of structure on the
performance of a test embankment should be of interest to geotechnical engineers.
In this Chapter, the Gloucester test embankment (Bozozuk and Leonards 1972) is
examined using the finite element method with a constitutive model that accounts for the
viscous behaviour of Gloucester clay and its structure (see Chapter 3). The primary
objective of this chapter is to evaluate the impact of structure on the long-term settlement
of the Gloucester test embankment. The second objective is to numerically study the
distribution of 'destructuration' in the Gloucester foundation clay with time and to
compare the strength loss due to 'destructuration' to the strength increase that would
normally be expected due to compression and decreasing void ratio. The results of this
study should provide insight into the response of natural clay deposits to embankment
construction and/or staged loading.
4.2 Background
Gloucester test embankment
The Gloucester test embankment is located on a site near the southern outskirts of
Ottawa, Canada. The site is commonly referred to as Canadian Geotechnical Research
Site No. 1 at Gloucester. McRostie and Crawford (2001) recently summarized the
history of this site and the historic research activities undertaken there, including the
133
Gloucester test embankment and performance monitoring of an Accommodation
building. Stage 1 of the Gloucester test embankment was constructed in 1967. After 15
years, a second stage was constructed beginning in June, 1982. Prior to construction of
the Stage 1 embankment, the upper 1.2m of the foundation deposit was excavated to
minimize the influence of the upper fissured clay. Then both stages 1 and 2 were built
within the excavation. Figure 4.1(a) shows the geometry of Gloucester test embankment.
The Gloucester test embankment was heavily instrumented and well documented
(Bozozuk and Leonards 1972; Lo et al. 1976; Fisher et al. 1982), as shown in Figure
4.1(a). However, only the centre settlement gauges (SI and S3) shown in Figure 4.1(a)
are considered in the following discussion.
The Gloucester foundation comprises an upper soft clay layer and a lower
medium to stiff clay layer (see Figure 4.1b). The upper clay layer extends from a depth
of 0 to 4m and comprises grey-brown, oxidized clay that is occasionally stratified with
silt. The average preconsolidation pressure in this layer is about 60kPa. Below that, the
lower layer extends from 4m to 19m and possesses relatively high preconsolidation
pressures, ranging from 80 kPa to 170 kPa. Finally, this layer is underlain with varved
clay and glacial till (Bozozuk and Leonards 1972; Lo et al. 1976; Leroueil et al. 1983).
The performance of the Gloucester test embankment has been studied by several
researchers. Fisher[i57] et al. (1982) predicted the field behaviour prior the construction
of the Stage 2 fill using an elastoplastic finite element model and the Gibson and Lo
(1967) model, and incorporating some engineering judgment. Lo et al. (1976) simulated
the performance of the Stage 1 fill, also using the Gibson and Lo (1967) model. Finally,
Hinchberger and Rowe (1998) utilized an elasto-viscoplastic constitutive model to
134
simulate both Stages 1 and 2 of the Gloucester case. In spite of past efforts to model the
Gloucester test embankment, so far the case has not been studied using a constitutive
model that accounts for the impact of structure and destructuration on the response of the
Gloucester clay.
Clay structure
Figures 4.2a and 4.2b illustrate the influence of clay structure on the behaviour of
the Gloucester clay during undrained triaxial compression and oedometer compression
tests (Law 1974 and Leroueil et al. 1983). Figure 4.2a shows the stress-strain curves for
two specimens consolidated isotropically using different confining stresses (<T'C = 83kPa
and a'c = AOkPa). The apparent preconsolidation pressure of these two specimens[i58]
obtained from a depth of 2.4m is about 60 kPa (Lo et al. 1976). In Figure 4.2a, the
specimen consolidated to stresses less than & (<J\ = 40kPa <60kPa) exhibits a peak
strength and subsequent reduction of strength with continued compression. When the
consolidation pressure exceeds the preconsolidation pressure of the clay
(cr'c = 83kPa >60kPa), the clay structure is degraded during the consolidation process and
the specimen exhibits a strain-hardening response. The shaded area in Figure 4.2a can be
attributed to bonding and fabric or more generally structure, assuming the intrinsic or
fully destructurated state is reached during CRS triaxial compression (Such may not
always be the case). Figure 4.2b shows a typical compression curve from an oedometer
compression test[i59] on Gloucester clay (Leroueil et al. 1983). After yielding, high
compressibility is observed and with further straining, the compression index Cc reduces
135
and becomes approximately constant for stresses exceeding Point A in Figure 4.2b. The
compression curve at large-strain in Figure 4.2b is essentially the intrinsic compression
line (Burland 1990). These characteristics of Gloucester clay are consistent with the
studies performed on other natural clays by Kabbaj et al. (1988), Burland (1990), and
Leroueil and Vaughan[i60] (1990).
Rate-sensitivity
The undrained shear strength and preconsolidation pressure of Gloucester clay are
both rate-sensitive. Laboratory data from both Leroueil et al. (1983) and Law (1974)
effectively characterize the rate-sensitivity of Gloucester clay. Figures 4.3a and 4.3b
show that the undrained shear strength and preconsolidation pressure versus strain rate is
essentially linear in a log-log plot. For an order change in strain rate, both the strength
and preconsolidation pressure vary by about 8%. A straight line can be fit through the
data with a slope a = 0.033. It should also be noted that the slope of the log-log plots is
consistent for both cr'p and Su.
Secondary compression
Lo et al.(1976) presented the results of long-term oedometer creep tests on
undisturbed Gloucester clay using a Rowe Cell. The data is reproduced in Figure 4.4,
which shows that more than half of the total compression during these tests occurred after
complete dissipation (see Aw=0) of the excess pore pressures. Such behaviour is
commonly referred to as secondary compression or delayed compression (Bjerrum 1967)
and is clearly an important characteristic of Gloucester clay.
Summary
136
As discussed above, Gloucester clay is a structured, rate-sensitive, and time-
dependent natural clay. From the data presented above, the structure and rate-sensitivity
are significant characteristics of Gloucester clay and ideally both characteristics should
be accounted for in the analysis of this case. Below, a structured EVP model is used to
investigate the influence of structure and rate-sensitivity on the performance of the
Gloucester test embankment.
4.3 Methodology
This chapter uses an unstructured elastic-viscoplastic constitutive model
(Hinchberger and Rowe 1998) and a structured elastic-viscoplastic constitutive model
(see Chapter 3) to examine the performance of the Gloucester test embankment. Both
models are described in detail elsewhere. The following is a brief summary of both
models and their constitutive parameters.
4.3.1 Model 1 -Hinchberger and Rowe Model
Hinchberger and Rowe (1998) developed an unstructured EVP model, which
hereafter is referred to as Model 1 in this chapter. Model 1 is an overstress elastic-
viscoplastic (EVP) model based on Perzyna's theory of overstress viscoplasticity
(Perzyna 1963 and 1971). The constitutive equation is:
r dF e„ = e* +£? =^-+—^—^j-8u +ejf =—+-J^~Sii+yvp(<l)(F)) ,J ,J ,J 2G 3(l + e)(r'm y " 2G 3(1+ e)tr' J X ' W
[4.1]
137
* ( / ) =
my
^ my
- 1 a' ( J >>(7' ( l )
^ my ^ my
o,(d)<o'{s)
my my
[4.2]
where stj is the deviatoric stress tensor, a'm is the mean effective stress, dtj is
Kronecker's delta, G is the stress dependent shear modulus, K is the slope of the
e- ln(o^) curve in the over-consolidated stress range, yvp is the fluidity parameter,
a( = l/n) is rate-sensitivity parameter, n = \la is the power law exponent, t r ^ i s the
intercept of static yield surface with mean stress axis, o'^ is the intercept of dynamic
yield surface, and a ^ / a ^ is the overstress ratio. dF
is the plastic potential,
which is derived as a unit norm vector. The flow function, <j>{F), in Equation [4.2] is a
power law function.
Figure 4.5 shows the state boundary surface for the Hinchberger and Rowe (1998)
model. In the normally consolidated stress range, the state boundary surface is defined
using an Elliptical cap equation:
/ = K - / ) 2+ 2 J 2 / ? 2 - ( < > - / ) 2 = 0 [4.3]
where / and R are parameters defining the aspect ratio of the elliptical cap, and J2 is
the second invariant of the deviatoric stress tensor, s;i. In the overconsolidated stress
range, the Drucker-Prager equation is used as:
f=MocG'm+c'-j2T2=0 [4.4]
where Moc is the slope of the Drucker-Prager envelope and c is the effective cohesion
138
intercept in -yj2J2 - o'm stress space. The resultant state boundary surface, which
delineates elastic and viscoplastic stress states, is illustrated by A1-A2-A3 in Figure 4.5.
The kinematic strain-hardening of the state boundary surface occurs according to:
^=^-<^eZ [4.5] A — K
where do'^ is the incremental expansion of the state-boundary surface, e is void ratio,
A and K are the compression and recompression indices, and devvp
ol is the incremental
plastic volumetric strain.
In accordance with overstress viscoplastic theory, stress states are permitted to exceed
the state boundary surface of the soil. Point G in Figure 4.5 shows a typical state of
overstress. In accordance with Equation [4.1], the overstress ratio at Point G is
amv} I amv} a n ^ m e resultant plastic strain-rate tensor is: my w m y
ejf = fp mv
a>U) . w my ,
•1 dF
do': [4-6] V
Lastly, the Drucker-Prager equation is also used to define the critical state for the
Hinchberger and Rowe model (1998) viz.:
f=Mcsa'm -J2T2=0 [4.7]
where Mcs is the slope of the classic critical state line (see Figure 4.5). The elliptical cap
is defined such that the top of the cap coincides with the critical state line.
In summary, there are nine constitutive parameters required for this model. Each of
the constitutive parameters, e0, A , K , Moc, Mcs, a'^ , R, yvp, and a , can be derived
from standard laboratory tests as described in Chapter 2.
139
Figure 4.6 illustrates the constitutive response of Model 1 during CRS (constant rate-
of-strain) oedometer and CRS undrained triaxial compression tests. In accordance with
EVP theory, Model 1 gives classic idealized compression behaviour during oedometer
compression. The compressibility is governed by the recompression K and compression
X indices in the overconsolidated (elastic) and normally consolidated (viscoplastic) stress
ranges, respectively. In addition, the preconsolidation pressure, o'p , is rate-sensitive.
Since a power law flow function is used in the Hinchberger and Rowe (1998) model (see
(j)(F)in Equation [4.2] ), there is a linear relationship between log(a')-log(e) with a
slope of a. The static yield surface intercept, a'^J, and fluidity parameter, yvp, define
the viscous range of the material. For strain-rates less than yvp (e.g. eMial < yvp), the
model becomes a rate-independent elastic-plastic model and the preconsolidation
pressure is rate-insensitive. The model becomes viscous and rate-sensitive for strain-
rates greater than yvp (see also Chapter 2).
Similar behaviour occurs during CRS triaxial compression. For strain-rates less
thanyvp, the material response is rate-independent and there is a classic elastic-plastic
strain-hardening stress-strain response. For strain-rates exceeding yvp, the undrained
shear strength is rate-sensitive and there is a linear relationship between log(Su) - log(e)
with a slope of a.
4.3.2 Model 2 - Structured Elastic-viscoplastic (EVP) Model
The Gloucester test embankment was also analyzed in this study using the
structured EVP model described in Chapter 3 (Model 2). The following is a brief
140
summary of the model and the additional parameters required.
Model 2 is an extended Hinchberger and Rowe (1998) model. In this model,
structure is defined by:
where oo0 is the initial structure, cc = l / n ( n is the power law exponent in Equation
[4.2]), yjp is the fluidity of the undisturbed structured clay, Y^p is the fluidity of the
corresponding remoulded or destructured clay, and c ' and G' { are the structured and
intrinsic preconsolidation pressures as defined in Figure 4.2. The structured fluidity, yjp ,
is much lower than the destructured fluidity, yfp, and consequently co0 is greater than
one.
To simulate destructuration, the structure parameter is assumed to be a function of
plastic strain viz.:
aK£d)=[l + ((O0n-1.0)e-b£^ [4.9]
and
ed=\'o ^\-A){deZl)2+A(de:P)2dt [4.10]
where ed is the plastic damage strain, e ^ and ejp are plastic volumetric and octahedral
shear strains, A is a weighting parameter (assumed to be 0.5), and b is a parameter that
controls the rate-of-destructuration and the magnitude of plastic damage strain required to
reach the intrinsic state. This damage law was first proposed by Rouainia and Wood
(2000) for use with a rate-insensitive elastoplastic model.
The resultant viscoplastic strain-rate tensor of Model 2 is:
141
vvp ?F
where (f)(F) = \<j'^) lo'^J - 1 is a power law flow function and the term a = \ln
defines the slope of the Su and o'p versus strain-rate in a log-log scale. On inspection of
Equations [4.8] through [4.11], the clay fluidity term in Equation [4.11], yl" l[co{ed)],
increases from the initial value, yvsp, and it approaches yjp with large plastic damage
strain. As noted above, the magnitude of plastic damage strain required to reach the
destructured state is governed by b in Equation [4.9].
Figure 4.7 shows the response of Model 2 during CRS oedometer and CRS
triaxial compression tests. Similar to Model 1, Model 2 implies linear log(Su)-log(e)
behaviour and linear log(a' ) - log(e) behaviour. These relationships apply to both the
undisturbed structured state and the intrinsic or destructured state. The initially low
structured fluidity, yjp , imparts additional strength to the clay above the post-peak
strength; whereas the post-peak strength is mobilized at large-strain or in the intrinsic
state. In Model 2, the additional strength is attributed to viscous effect[i61]s and is
denoted by the shaded area in Figure 4.7(c), which represents additional overs tress
caused by the initially low structured fluidity, yjp . Similarly, during the CRS oedometer
compression, the initially low structured fluidity, yjp» permits the clay to exist at a higher
void ratio after yielding than would otherwise be expected for the corresponding
destructured or remoulded clay. The shaded area in Figure 4.7(a) is additional overstress
due to yjp (structure). As a consequence of using EVP theory, the structured state in both
142
CRS oedometer and CRS triaxial compression tests is metastable, since the fluidity will
increase with plastic damage strain and eventually reach the intrinsic fluidity as a
consequence of the soil being in a state of overstress.
The above discussions have described the constitutive models used to examine the
performance of the Gloucester test fill. Model 1 (Hinchberger and Rowe 1998) has 11
constitutive parameters; whereas, Model 2 (the Structured EVP model) has 13
constitutive parameters. The following sections describe the finite element mesh and
constitutive parameters used to re-examine this case.
4.3.3 Finite Element Mesh
The Gloucester test embankment is examined using finite element analysis. The
corresponding finite element mesh, shown in Figure 4.1a, consisted of 800 six-noded
plane-strain triangles and 1702 nodes. A rigid boundary condition was set at a depth of
20.2m and smooth boundaries are assumed at the embankment centerline and 75m
beyond the centerline. The construction process for Stages 1 and 2 was simulated by
adding the corresponding elements representing the embankment fill, layer by layer, into
the finite element mesh and incrementally increasing the body force due to gravity within
the added elements.
4.3.4 Constitutive Parameters
The constitutive parameters for Gloucester clay have been summarized in Table
4.1 and 4.2. The elastoplastic parameters, e0, v , A , K, Mcs, Moc , and R , were
estimated by Hinchberger (1996), from standard laboratory tests provided by the National
Research Council of Canada (NRC). The hydraulic conductivity, k, of the Gloucester
foundation soil was assumed to be dependent on the void ratio as follows:
143
k = k0cxp(^-fi-) [4.12]
where k0 is the hydraulic conductivity at the reference void ratio, and Ck is the slope of
the e - log(fc) plot. The elastoplastic constitutive parameters are summarized in Table 4.1
and compared with laboratory results in Figure 4.1b. The same elastoplastic parameters
were used with Models 1 and 2.
Table 4.2 summarizes the viscosity-related parameters. The parameters, yvp and
n = \la , required in Model 1 for Gloucester clay were directly adopted from the study
by Hinchberger and Rowe (1998). The structure-dependent parameters (y]p, yjp, co0 and
b) required in Model 2 were estimated using a similar approach described in Chapter 3.
As shown in Figure 4.2, the structure parameter is coo =<7'p-s / °V, =1.18, which was
estimated from the intrinsic, a* {, and structured, <y'p_s, preconsolidation pressures in
oedometer compression on intact Gloucester clay specimens (Leroueil et al. 1983). For
analysis purpose, the structured fluidity, yvsp , was taken to be lxlO^min"1 (from
Hinchberger and Rowe 1998). The intrinsic fluidity yvp is an inter-dependent parameter,
which can be calculated using Equation [4.8]. The destructuration-rate parameter, b [i62],
was estimated using Equation [4.9] and to the magnitude of plastic strain required to
reach the intrinsic state during oedometer compression (see Point A in Figure 4.2).
Figure 4.8 compares the theoretical response of Model 2 during CRS oedometer
compression test with the actual response of Gloucester clay obtained at depth between
3.45m and 3.90m (Leroueil et al. 1983).
The embankment fill was modeled as a Mohr-Coulomb material with an effective
144
friction angle </>' = 35°, a dilation angle y/ = 0°, c'=0kPa, and a unit weight of 18.4
kN/m3.
4.4 Results
This section presents the results of finite element analysis of the Gloucester test
embankment. Calculated long-term settlements using Model 1 and Model 2 are
compared with the measured data to assess the relative importance of structure in the
Gloucester case. In addition, the calculated change of strength in the Gloucester
foundation is studied. In most cases, strength gain due to consolidation is relied on
during the design of staged embankment. However, for natural clay, destructuration due
to plastic strain in the clay foundation may result in strength loss. In situ tests such as
field vane and cone penetration tests are typically used to investigate changes in the
undrained strength of foundation deposits to verify the design assumption (Stermac et al.
1967, Holtz and Broms 1972, and Koskinen et al. 2002). In the following sections, the
distribution of strength gain due to consolidation and strength loss due to destructuration
are studied with the intent to gain insight into the effect of structure and destructuration
on the staged construction.
4.4.1 Analysis using the Unstructured EVP Model (Model 1)
Settlement versus time response
Figure 4.9 shows the measured long-term settlement at Settlement Gauge SI
beneath the center of the Gloucester test embankment (see Figure 4.1 for the location). In
addition, the estimated time required for the primary consolidation (1.5[G63][G64] years)
is also shown in Figure 4.9 as reported by Lo (1976). From Figure 4.9, it can be seen that
145
a significant amount of the total settlement of Stage 1 (34cm) can be attributed to
secondary compression (20cm +/-). With further load of the Stage 2 fill, the settlement
increased from 34cm up to about 70cm.
The calculated settlements versus time using Model 1 are presented in Figure 4.9.
Two types of theoretical curves have been generated using: (i) a linear virgin
compression curve, and (ii) bilinear virgin compression curves, as shown in Figures
4.10(a) and 4.10(b). For the bilinear calculations, the clay was assumed to be more
compressible immediately after reaching the preconsolidation pressure and less
compressible at stresses of either 20 or 40 kPa higher than the preconsolidation pressure.
Such an approach can approximately account for destructuration in compression.
From Figure 4.9, it can be seen that Model 1 gives calculated behaviour that is in
reasonable agreement with measured behaviour for Stage 1. For Stage 2, however, the
use of a linear virgin compression curve (see Figure 4.10a) leads to calculated settlements
significantly higher than those measured in Stage 2. Good agreement between calculated
and measured settlements can be obtained using a bilinear virgin compression curve with
a transition stress of 20kPa (see Figure 4.10b). Although a bilinear virgin compression
curve is common for structured clay (see Figure 4.2), the transition of compression index
from X to X12 occurs at stresses about 60kPa higher than the preconsolidation pressure
(see Point A in Figure 4.2). Consequently, the bi-linear curve required to fit the field
response is not entirely consistent with the actual compression response of Gloucester
clay.
Strength Increase Due To Consolidation
One of the important objectives of staged construction is to improve the strength
146
of the clay deposit by allowing time for consolidation and consequent decrease in void
ratio. Here, the term 'consolidation' includes both primary and secondary consolidation.
Studies on remoulded clay have shown that the decrease in void ratio during
consolidation leads to expanding of yield surface and a corresponding increase in
undrained shear strength. In addition, Bjerrum and Lo (1963) have shown that the
undrained strength increased with the period of aging as well. Thus, it was considered to
be insightful to examine the relative strength increase in the Gloucester foundation
caused by consolidation.
Model 1 uses a classical critical state concept in which consolidation and the
resultant expansion of the yield surface (see Equation [4.9]) causes strength increase.
Thus, the undrained shear strength, S u , is a function of void ratio and stress history.
From the elliptical cap equation, it can be shown that the initial undrained strength prior
to Stage 1 constructing, Su0, is related to (j'^l viz.:
S B 0 = A O i 6 5 ] [4.13]
where A is a constant and depends on the aspect ratio of the yield surface, and a ^ is the
initial yield surface intercept prior to the construction in Stage 1 (see Figure 4.1b). In
addition, the relative increase in undrained strength due to hardening during consolidation
is:
rS >i o'^+do'}* mv„
o-'(s) [4.14]
V u° JCon my„
where d o ^ is the incremental expansion of the yield surface due to strain-hardening
(from Equation [4.9]).
147
Figure 4.11 shows the contours of (Su I Su0 )cms 15 years after construction of
Stage 1. The contours represent zones of relative strength increase due to consolidation.
From Figure 4.11, it can be seen that the magnitude of (Su ISM)cgm is 1.4 in Zone A
below the embankment centerline. This indicates a 40% increase in undrained strength.
The zones influenced by consolidation extend 13m from the centre line and over 10m
deep into the foundation. Thus, from Model 1, significant increase in the undrained
strength of the Gloucester clay would be anticipated after 15 years of consolidation.
Figure 4.12 shows the contours of (Su I Su0)cgm 4 years after construction of Stage
1. The expansion of the (Su I'Su0)cons =1.2 contour from the 4th year to the 15th year of
Stage 1 suggests that the magnitude of strength gain due to secondary consolidation
increases considerably with time; whereas the extent of strength gain does not change
significantly. From Model 1, up to 20% strength gain would be anticipated 4 years after
construction of Stage 1.
4.4.2 Analysis using the Structured EVP Model (Model 2)
Long-term settlement
Figure 4.13 shows the calculated settlement versus time at Gauge SI, using the
structured EVP model (Model 2). From Figure 4.13, it can be seen that Model 2 gives
settlement predictions that are very close to those measured for both Stages 1 and 2. This
is an improvement over the unstructured EVP model, since referring back to Figure 4.8,
Model 2 gives a stress-strain response during CRS oedometer compression that is
consistent with that measured for Gloucester clay (see Figure 4.8) and using structure
parameters that are easily deduced from laboratory tests. Figure 4.14(a) presents the
148
measured and calculated settlement during Stage 1 at Gauge SI and S3, using Model 1
and Model 2 respectively. This figure shows that the results from Model 2 agree better
with the measured data, compared with that from Model [i66] 1. Furthermore, both
models were found to give settlement with depth comparable to that measured in the
case. Figure 4.14(b) shows the measured and calculated excess pore water pressure 1
year after the construction of Stage 2. Both Model 1 and Model 2 slightly overestimate
the excess pore water pressure at all depths. Model 1 gives slightly higher excess pore
pressures within the upper soft clay layer and lower excess pore pressure within the
underlying medium to stiff clay layer, compared with Model 2.
Strength Loss Due To Destructuration
For Model 2, there are two competing effects on the strength change during
consolidation. The first is strength increase due to consolidation and consequent
hardening of the yield surface. This strength increase with time during the Gloucester
case can be examined by contouring (Su ISM)com using Equation [4.14]. However, for
the structured EVP model, there is also loss of strength caused by destructuration. From
Equations [4.9] and [4.10], the relative strength loss due to destructuration can be
represented viz:
V""0 )Str wo
where co0 is the intact structure, co{ed) represents the state-dependent structure.
Referring to Equation [4.15], destructuration causes a reduction of peak undrained
strength. For the Gloucester clay with &)0=1.18, the maximum strength loss is 15%,
representing complete transformation to the intrinsic state.
149
The distribution of destructuration 15 years after the construction of Stage 1 is
shown in Figure 4.15. The degree of destructuration is denoted by contours of
[Su/Su0 ) . From Figure 4.15, it can be seen that the strength loss due to
destructuration is concentrated in two zones (Zone A and Zone B). Zone A is located in
the upper 5m-thick soft clay layer close to the centerline, where most of the vertical
compression in the foundation occurs ( Leroueil et al. 1983 and Figure 4.14(b). The large
plastic strain in this layer causes considerable destructuration. The magnitude of
[Su/Su0 ) in the center of Zone A ranges from 0.9 to 0.85 suggesting the undrained
strength would decrease by 10% to 15%. In Zone B under the embankment toe, the
magnitude of \SU I Su0 J varies from 0.95 to slightly higher than 0.9, indicating a
relative decrease in undrained strength by 5% tol0%.
Combined influence of Consolidation and destructuration
In reality, the variation of undrained strength is affected by both destructuration
and consolidation. The combined effects can be deduced from Model 2 by plotting the
ratio,5„/5M0:
( c \ ( ^ \ ( <i \ [4.16]
v "V V " ° J Com V "° JStr su
where \SU I Su0 J is the ratio representing the relative strength loss due to
destructuration, and the ratio, [Su/Su0)Cgnt, denotes the relative strength gain due to
consolidation.
Figures 4.15 and 4.16 present the calculated strength variation in the Gloucester
foundation under the combined influence of destructuration and consolidation at the end
150
of Stage 1. Figure 4.16 shows zones of net strength gain (i.e. Su /Su0 >1) and Figure
4.17 shows zones of net strength loss (i.e. Su /Su0 <1). Referring Figure 4.16, it appears
that the strengthened zones are located mainly within the upper soft clay layer in Zone A.
However, the actual strength increase is only 30% in Zone A, compared to 40%, which
was deduced from conventional unstructured soil mechanics (Model 1). Figure 4.17
presents the contour of Su/Su0 <1, denoting the net strength loss in the foundation.
From Figure 4.17, it can be seen that there is an extensive weakened zone (Zone B) under
the toe of embankment, which reaches a depth of 3m. A second weakened zone (Zone C)
is located between depths of 8m and 13m and it extends laterally extending about 7m
from the centre line of the embankment.
In summary, 15 years after the construction of Stage 1, the strengthening effect
due to consolidation is significant; however, destructuration reduces the magnitude and
extent of the strengthened zone that would be expected from conventional soil
mechanics. In addition, there is some weakening in Zone B, which is likely on the
potential failure surface and may be of practical interest for stability analysis and design
of staged construction on structured natural clay.
Time effect on the strength variation
Figures 4.18 and 4.19 show the development of strengthened and weakened zones
in the Gloucester foundation from the 4th year to the 15th year after the construction of
Stage 1. As shown in Figure 4.18, the extent of strengthened zones does not change
significantly during this period. However, the magnitude of strengthening increases
considerably, as suggested by the expanded Su / Su0 =1.2 contour. The weakened zone in
Zone B under the embankment toe maintains the same size while the intensity of
151
weakening increases, as suggested by the expanded SM/Su0=0.95 contour in Figure
4.19,. The other weakened zone located in Zone C expands in size slightly (about lm
downward). Simultaneously the degree of strength loss in this zone increases, but it does
not exceed 5%.
Influence of additional load (Stage 2)
Figure 4.20 shows the geometry after construction of the Stage 2 fill.
Correspondingly, the construction of Stage 2 increases the stresses imposed by the
embankment on the clay foundation. As a result, additional consolidation and
destructuration is induced by Stage 2, as discussed below.
Figures 4.20 and 4.21 show the distribution of net strength gain (Su /Su0 >1) and
strength loss( Su /SuQ <1) .respectively, at the end of Stage 1 and 7 years after the
construction of Stage 2. The net strength gain has been estimated using Equation [4.16].
Figure 4.20 shows that the zone of strength increase expands laterally by about 2m due to
the increased width of the fill base. In addition, the magnitude of strengthening also
increases, as shown by the expanded contour line for Su /Su0=1.2. In Figure 4.21, the
weakened zone in Zone B shifts laterally by about 2m outward, as a result of the new
location of the embankment toe. The weakened zone in Zone C also expands downward
and further outward from the centerline[G67].
4.5 Summary and Conclusions
This chapter has examined the influence of clay structure and its degradation on
the field performance of the Gloucester test embankment. Two different constitutive
152
models, an unstructured EVP model and structured EVP model, have been used to
evaluate the long-term settlement of the Gloucester test embankment and to assess the
change in undrained strength with time. The following is a summary of the conclusions
drawn from this study:
1. More than 50% of Stage 1 settlements occurred at constant effective stress, due
to secondary compression. A similar observation on the long-term (33 years) settlement
of an Accommodation building founded on the same site has been reported by McRostie
and Crawford (2001), as shown in Figure 4.22. As a result, it can be concluded that a
constitutive model that can account for the viscous behaviour of Gloucester clay is
required to predict the long-term performance of infrastructure founded on Gloucester
clay.
2. The structured EVP model (Model 2) was capable of describing the long-term
performance of the Gloucester case and gave improved results compared to those
obtained using the unstructured EVP model, and using soil parameters consistent with
those measured in the Gloucester case.
3. During Stage 1 of the Gloucester case (the first 15 years), the undrained
strength of the foundation deposit is subject to the combined influence of destructuration
and consolidation. In Zone A (see Figure 4.16) directly below the embankment, the
strengthening effect due to consolidation overshadows the strength loss due to
destructuration. In Zone B near the toe and Zone C at depth below centerline, however,
the opposite was observed. In these zones, destructuration overshadows consolidation
causing a net decrease in undrained strength of between 5% and 10% in Zone B and 5%
in Zone C.
153
4. The distribution of net strengthening and weakening in the Gloucester
foundation can be interpreted as follows. In Zone A below the embankment centerline,
the clay is confined and the predominantly volumetric compression results in hardening
of the yield surface. This hardening effect leads to strength gain that overshadows
strength loss caused by destructuration. In Zones B and C, however, there is more
deviatoric or shear strain and comparatively less volumetric strain. The net effect is a
reduction of strength due to the limited volumetric strains.
5. There is additional evidence reported in the literature of other cases that show
the existence of weakened and strengthened zones in clay foundations below
embankments. For example, eight years after construction of the 2-meter-high
Murro[G68] Embankment, an in situ investigation (Koskinen et al. 2002) was carried out
to access the increase of the undrained shear strength. As pointed out by Karstunen et al.
(2005) within the top 7 meters in the foundation, the undrained shear strength increased;
Below that, however, the undrained shear strength decreased. Another example is the
1.5-meter-high the Ska-Edeby embankment (Holtz and Lindskog 1972). For this case,
vane shear tests in 1961 and 1970 suggested that the vane shear strength increased about
5kPa within the upper 3 meter in the clay deposit, whereas there was a reduction in shear
strength for the clay at the depth between 3m and 5m. These trends of strength variation
under the Murro Embankment and the Ska-Edeby embankment agree well with that
calculated for the Gloucester embankment, where the undrained strength is strengthened
in the top 10m, but weakened between the depths of 10m and 15m.
6. Analysis of the Gloucester case has shown the possible existence of a
weakened or disturbed zone of clay near the toe of the Gloucester test embankment. It is
154
conceivable that such a zone may influence the stability of embankments founded on
highly structured clay deposits and reduce the effectiveness of staged construction. As a
result, it is concluded that the importance of stress concentrations near the toe of
embankments should be explored further.
7. The structure of natural clay plays an important role in the distribution of
strengthened and weakened (disturbed) zones in embankment foundations. For a clay
deposit with more structure than the Gloucester case, the structure degradation would
lead to a greater reduction in undrained strength. As a result, the weakened zone would
expand; whereas the strengthen zone would contract relative to those deduced for the
Gloucester case. For Gloucester clay with &)0=1.18, the maximum reduction of
undrained strength is 15% ; While for St. Jean Vianney clay with coQ =1.1, the maximum
reduction ratio would be 42%. It can be inferred that the strength loss due to
destructuration would be more serious for the same embankment founded on St. Jean
Vianney clay than that on Gloucester clay.
155
References
Adachi, T., and Oka, F. 1982. Constitutive equations for normally consolidated clay
based on elasto-viscoplasticity. Soils and Foundations, 22(4): 57-70.
Baudet, B., and Stallebrass, S. 2004. A constitutive model for structured clays.
Geotechnique, 54(4): 269-278.
Bjerrum, L. 1967. Engineering geology of Norwegian normally-consolidated marine
clays as related to settlements of buildings. Geotechnique, 17(2): 81-118.
Bjerrum, L., and Lo, K.Y. 1963. Effect of aging on shear-strength properties of normally
consolidated clay. Geotechnique, 13(2): 147-157.
Bozozuk, M., and Leonards, G.A. 1972. The Gloucester test fill. In Proceedings of the
ASCE Specialty Conference on Performance of Earth and Earth-Supported
Structures, pp. 299-317.
Burland, J.B. 1990. On the compressibility and shear strength of natural clays.
Geotechnique, 40(3): 329-378.
Callisto, L., and Rampello, S. 2004. An interpretation of structural degradation for three
natural clays. Canadian Geotechnical Journal, 41(3): 392-407.
Carter, J.P., and Balaam, N.P. 1990. AFENA-A general finite element algorithm: users
manual, School of Civeil Engineering and Mining Engineering,University of
Sydney, Australia.
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.
156
Fisher, D.G., Rowe, R.K., and Lo, K.Y. 1982. Prediction of the second stage behaviour of
Gloucester Test Fill. In Geotechnical Research Report.
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.
Hinchberger, S.D., and Rowe, R.K. 2005. Evaluation of the predictive ability of two
elastic-viscoplastic constitutive models. Canadian Geotechnical Journal, 42(6):
1675-1694.
Holtz, R.D., and Lindskog, G. 1972. Soil movement below a test embankment. In
Proceedings of the ASCE Specialty Conference on the Performance of Earth-
Supported Structures. Lafayette, Indiana. Purdue University, pp. 273-284.
Holtz, R.D., and Broms, B.B. 1972. Long-term loading tests at Ska - Edeby. In
Proceedings of the ASCE Specialty Conference on Performance of Earth and
Earth-Supported Structures. Sweden. Purdue University, Vol.1, pp. 435- 464.
Kabbaj, M., Tavenas, F., and Leroueil, S. 1988. In situ and laboratory stress-strain
relationships. Geotechnique, 38(1): 83-100.
Karstunen, M., Krenn, H., Wheeler, S.J., Koskinen, M., and Zentar, R. 2005. Effect of
anisotropy and destructuration on the behavior of Murro test embankment.
International Journal of Geomechanics, 5(2): 87-97.
Katona, M.G. 1984. Evaluation of Viscoplastic Cap Model. Journal of Geotechnical
Engineering, 110(8): 1106-1125.
Kim, Y.T., and Leroueil 2001. Modeling the viscoplastic behaviour of clays during
consolidation: Application to Berthierville clay in both laboratory and field
conditions. Canadian Geotechnical Journal, 38(3): 484-497.
Law, K.T. 1974. Analysis of Embankments on Sensitive Clays. Ph.D Thesis, University
of Western Ontario, London, Ontario.
Leroueil, S., and Vaughan, P.R. 1990. The general and congruent effects of structure in
natural soils and weak rocks. Geotechnique, 40(3): 467-488.
Leroueil, S., Samson, L., and Bozozuk, M. 1983. Laboratory and field determination of
preconsolidation pressures at Gloucester. Canadian Geotechnical Journal, 20(3):
477-490.
Liu, M.D., and Carter, J.P. 2002. A structured Cam Clay model. Canadian Geotechnical
Journal, 39(6): 1313-1332.
Lo, K.Y., Bozozuk, M., and Law, K.T. 1976. Settlement analysis of the gloucester test
fill. Canadian Geotechnical Journal, 13(4): 339-354.
McRostie, G.C., and Crawford, C.B. 2001. Canadian Geotechnical Research Site No. 1 at
Gloucester. Canadian Geotechnical Journal, 38(5): 1134-1141.
Perzyna, P. 1963. Constitutive equations for rate sensitive plastic materials. Quarterly of
Applied Mathematics, 20(4): 321-332.
Perzyna, P. 1971. Thermodynamics of rheological materials with internal changes, 10(3):
391-408.
Rouainia, M., and Wood, D.M. 2000. Kinematic hardening constitutive model for natural
clays with loss of structure. Geotechnique, 50(2): 153-164.
Rowe, R.K., and Hinchberger, S.D. 1998. The significance of rate effects in modelling
the Sackville test embankment. Canadian Geotechnical Journal, 35(3): 500-516.
Silvestri, V. 1984. The preconsolidation pressure of Champlain clay, Part II. Canadian
Geotechnical Journal, 21(3): 600-602.
Stermac, A.G., Lo, K.Y., and Barsvary, A.K. 1967. The performance of an embankment
on a deep deposit of varved clay. Canadian Geotechnical Journal, 2(3): 234-253.
Yin, J.-H., and Graham, J. 1996. Elastic visco-plastic modelling of one-dimensional
consolidation. Geotechnique, 46(3): 515-527.
159
Table 4.1 Material parameters used in both Model 1 and Model 2 for the numerical
analysis of the Gloucester test embankment
Depth (m) K X e0 v Mcs R ° Ck M oc
(xlO m/min)
0.0-2.0 0.025 0.65 1.8 0.3 0.9 1.65 10.0 0.25 0.9
2.0-5.2 0.025 0.65 1.8 0.3 0.9 1.65 7.2 0.25 0.9
5.2-7.2 0.025 0.32 1.8 0.3 0.9 1.65 6.0 0.5 0.9
7.2-13.4 0.025 1.35 2.4 0.3 0.9 1.65 6.0 0.25 0.9
13.4-20.2 0.025 0.75 1.8 0.3 0.9 1.65 7.2 0.25 0.9
160
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
et al. 1976)
c o "co CO
O
o
2 4
4 4
6 4
8 4
Long-term consolidation tests (Loetal. JI976) AU=P
Sarjnple at -2.4 rpetre Sarinple at - 4.2 pnetre
10 10° 101 102 103 104 105 106
Time, mins
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
undrained peak strength of Gloucester clay exhibits significant anisotropy (e.g. 40%
208
difference for undrained strengths for / = 0° and i = 90°); whereas the post-peak strength
of Gloucester clay is nearly isotropic (e.g. 7% different for i = 0° and / = 90°). This
suggests that destructuration in the undrained triaxial compression tests[G98] reduces the
strength anisotropy of Gloucester clay.
5.7 Summary and Conclusions
This chapter presented a constitutive approach for modeling the rate-dependent,
anisotropic behaviour of structured clay. The foundation of the constitutive framework is
an existing overstress elastic viscoplastic model (Hinchberger 1996; Hinchberger and
Rowe 1998), which has been extended using a state-dependent viscosity parameter to
account for the effects of clay 'structure' (Hinchberger and Rowe 2005). A tensor
approach similar to that described by Boehler (1987), Pietruszczak and Mroz (2001) and
Cudny and Vermeer (2004) has been used to incorporated anisotropic viscoplasticity into
the model, which has been shown to describe some of the key engineering characteristics
of two clays from Eastern Canada.
For St. Vallier and Gloucester clay, the effects of strain-rate (or time) and sample
orientation are clearly significant and a constitutive framework that can account for these
two effects is considered to be desirable. Other factors that may affect the response of
natural clay during triaxial compression tests include but are not limited to end effects
and strain localization both of which have been ignored. In accordance with Equations
[5.16] and [5.17], the damage parameter, b , governs the rate of structural degradation of
the clay skeleton. Presently, it is not clear how this parameter may be affected by strain
localization and future development should focus on this important issue. However, the
model captures most of the anisotropic and time-dependent characteristics of these clays,
209
which are clearly very significant.
Based on the analyses and discussions presented above, the following
observations and conclusions may be made:
(i). The extended Hinchberger and Rowe (1998) model can describe the effect of strain-
rate and sample orientation on the peak undrained shear strength of Gloucester clay
and St. Vallier clay.
(ii). The constitutive approach described above can also approximate the nearly
isotropic post-peak strength of Gloucester clay during CIU triaxial compression.
(iii). The plastic response of St. Vallier clay is more anisotropic ( A = 0.3 and
cui=0/cui=90=1.8 ) than that of Gloucester clay (A = 0.15 and cui=0/cui=90=1.4 ).
However, the opposite can be observed for the elastic anisotropy where (3 = 1.6 for
Gloucester clay compared to (5 = 1.15 for St. Vallier clay. As a result, it is
concluded that the degree of elastic and viscoplastic anisotropy may not necessarily
be interrelated for structured clay.
(iv). For Gloucester clay, it is concluded that both cross-anisotropic elasticity (e.g.
Graham and Houlsby 1983; Love 1927) and anisotropic viscoplasticity should
ideally be accounted for in a constitutive model for this material. The need to
account for the anisotropic elasticity of St. Vallier clay is less evident.
210
References
Boehler, J.P. 1987. Applications of tensor functions in solid mechanics. Springer,
Wien.
Burland, J.B. 1990. On the compressibility and shear strength of natural clays.
Geotechnique, 40(3): 329-378.
Carter, J.P., and Balaam, N.P. 1990. AFENA-A general finite element algorithm:
users manual, School of Civeil Engineering and Mining
Engineering,University of Sydney, Australia.
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.
Cudny, M., and Vermeer, P.A. 2004. On the modelling of anisotropy and
destructuration of soft clays within the multi-laminate framework. Computers
and Geotechnics, 31(1): 1-22.
Davies, M.C.R., and Newson, T.A. 1992. Critical state constitutive model for
anisotropic soil. In Proceedings of the Wroth Memorial Symposium, 07/27-
07/29/92. Oxford, UK. Publ by Thomas Telford Services Ltd, London, Engl, p.
219.
Graham, J., and Houlsby, G.T. 1983. Anisotropic elasticity of a natural clay.
Geotechnique, 33(2): 165-180.
211
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. 1996. The behaviour of reinforced and unreinforced embankments
on rate senstive clayey foundations. Ph.D Thesis, University of Western
Ontario, London.
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.
Hinchberger, S.D., and Rowe, R.K. 2005. Evaluation of the predictive ability of two
elastic-viscoplastic constitutive models. Canadian Geotechnical Journal,
42(6): 1675-1694.
Hinchberger, S.D., and Qu, G. 2007. A viscoplastic constitutive approach for
structured rate-sensitive natural clay. Canadian Geotechnical Journal, Re-
Submitted November 2007.
Kim, Y.T., and Leroueil 2001. Modeling the viscoplastic behaviour of clays during
consolidation: Application to Berthierville clay in both laboratory and field
conditions. Canadian Geotechnical Journal, 38(3): 484-497.
Law, K.T. 1974. Analysis of Embankments on Sensitive Clays. Ph.D Thesis,
University of Western Ontario, London, Ontario.
Lo, K.Y. 1972. An approach to the problem of progressive failure. Canadian
Geotechnical Journal, 9: 407-429.
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.
Love, A.E.H. 1927. A treatise on the mathematical theory of elasticity. Cambridge
University Press, Cambridge,England.
Marques, M.E.S., Leroueil, S., and de Almeida, M.d.S.S. 2004. Viscous behaviour of
St-Roch-de-1'Achigan clay, Quebec. Canadian Geotechnical Journal, 41(1):
25-38.
Miura, K., Miura, S., and Toki, S. 1986. Deformation behavior of anisotropic dense
sand under principal stress axes rotation. Soils and Foundations, 26(1): 36-52.
Muhunthan, B., Cudny, M., and Masad, E. 1996. Fabric effects on the yield behavior
of soils. Soils and Foundations, 36(3): 85-97.
Newson, T.A. 1998. Validation of a non-associated critical state model. Computers
and Geotechnics, 23(4): 277-287.
Oda, M., and Nakayama, H. 1989. Yield function for soil with anisotropic fabric.
Journal of Engineering Mechanics, 115(1): 89-104.
Oztoprak, S., and Cinicioglu, S.F. 2005. Soil behaviour through field instrumentation.
Canadian Geotechnical Journal, 42(2): 475-490.
Perzyna, P. 1963. Constitutive equations for rate sensitive plastic materials. Quarterly
of Applied Mathematics, 20(4): 321-332.
Pietruszczak, S., and Mroz, Z. 2001. On failure criteria for anisotropic cohesive-
frictional materials. International Journal for Numerical and Analytical
Methods in Geomechanics, 25(5): 509-524.
Pietruszczak, S., Lydzba, D., and Shao, J.F. 2004. Description of creep in inherently
anisotropic frictional materials. Journal of Engineering Mechanics, 130(6):
681-690.
Roscoe, K.H., Schofield, A.N., and Thurairajah, A. 1963. Yielding of clays in states
wetter than critical. Geotechnique, 13(3): 211-240.
Rouainia, M., and Wood, D.M. 2000. Kinematic hardening constitutive model for
natural clays with loss of structure. Geotechnique, 50(2): 153-164.
Sun, D.A., Matsuoka, H., Yao, Y.P., and Ishii, H. 2004. An anisotropic hardening
elastoplastic model for clays and sands and its application to FE analysis.
Computers and Geotechnics, 31(1): 37-46.
Tavenas, F., and Leroueil, S. 1978. Effects of stresses and time on yielding of clays.
In Proc of the hit Conf on Soil Mech and Found Eng, 9th, Jul 11-15 1977.
Edited by P.C.O.X. ICSMFE. Tokyo, Jpn. Jpn Soc of Soil Mech and Found
Eng, Tokyo, pp. 319-326.
Tavenas, F., Leroueil, S., La Rochelle, P., and Roy, M. 1978. Creep behaviour of an
undisturbed lightly overconsolidated clay. Canadian Geotechnical Journal,
15(3): 402-423.
Tobita, Y. 1988. Yield condition of anisotropic granular materials. Soils and
Foundations, 28(2): 113-126.
214
Tobita, Y., and Yanagisawa, E. 1992. Modified stress tensors for anisotropic behavior
of granular materials. Soils and Foundations, 32(1): 85-99.
Vaid, Y.P., and Campanella, R.G. 1977. Time-dependent behavior of undisturbed
clay. Journal of the Geotechnical Engineering Division, 103(7): 693-709.
Vaid, Y.P., Robertson, P.K., and Campanella, R.G. 1979. Strain rate behaviour of
Saint-Jean-Vianney clay. Canadian Geotechnical Journal, 16(1): 35-42.
Wang, C.C. 1970. A new representation theorem for isotropic functions. Archive For
Rational Mechanics And Analysis, 36: 166-223.
Ward, W.H., Samuels, S.G., and Butler, M.E. 1959. Further studies of the properties
of London Clay. Geotechnique, 9(2): 33-59.
Wheeler, S.J., Naatanen, A., Karstunen, M., and Lojander, M. 2003. An anisotropic
elastoplastic model for soft clays. Canadian Geotechnical Journal, 40(2): 403-
418.
Whittle, A.J., and Kavvadas, M.J. 1994. Formulation of MIT-E3 constitutive model
for overconsolidated clays. Journal of Geotechnical Engineering, 120(1): 173-
198.
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.
Zienkiewicz, O.C., and Cormeau, I.C. 1974. Viscoplasticity - plasticity and creep in
elastic solids - a unified numerical solution approach. International Journal for
Numerical Methods in Engineering, 8(4): 821-845.
Tab
le 5
.1
Com
pari
son
of e
last
ic-v
isco
plas
tic m
odel
s
Hin
chbe
rger
and
R
owe
(199
8)
Hin
chbe
rger
and
Q
u (2
007)
P
rese
nt C
hapt
er
Ela
stic
Mod
el
Yie
ld F
unct
ion
Lim
it Su
rfac
e
Flow
Fun
ctio
n,
4>(F)
K=
(l
+ e
)ar'
IK
a
nd
G,V
m
1 m
y os
my
-1
f=(<
-lf-
2J
2R
2+(<
;»-l
f=0
f =
CJ^
M+
C^C
-
V^
T =
°
" N/c
Cri
tica
l S
tate
f=C
T^M
+<
c-7
^7 =
0 -
O/C
Yie
ld
my
os
MM
m
y M
^»7)
m
y os
<j;
my
-1
Har
deni
ng L
aw
rK.)
_(
! +
£)
„-C
(X
-K)
to
Table 5.2 Constitutive parameters for Gloucester Clay
Parameter Initial void ratio, e
tic Yield Surface Intercept o'^ , (kPa)
Visoplastic Strain-rate Exponent, n Mstruc
V
Recompression Index, K Compression Index, X
Aspect Ratio of Elliptical Cap, R A (weighting Parameter) Anisotropy Parameter A
Structured viscosity, |j,str, (min) Damage Exponent, b
Structure parameter, co0
Value 1.8
48.5
30 1.20 0.3
0.02 0.63 2.5 0.5
0.15 1E8 50
1.18
Table 5.3 Constitutive parameters for St.Vallier Clay
Parameter Initial void ratio, e
tic Yield Surface Intercept o'^, (kPa)
Visoplastic Strain-rate Exponent, n Mstruc
V Recompression Index, K Compression Index, A
Aspect Ratio of Elliptical Cap, R A (weighting Parameter) Anisotropy Parameter, A
Structured viscosity, /ustr, (min) Damage Exponent, b
Structure parameter, co0
Value 1.6
70
25 1.85 0.3
0.01 0.65 1.8 0.5 0.3
2.5E9 200
1.37
218
Figure 5.1 Illustration of the microstructure tensor, al, and the generalized stress
tensor, ov for transverse isotropy.
(a) Structure Tensor with positive A (b) Structure Tensor with negative A (c) Generalized stress state
X X
t
(a) (b)
( C )
219
Figure 5.2 Sample orientation, /.
a.
4 &
i = 45°
"7 /
/ 1/ Y
/ = 60°
1 / \
H* LW d£ / = 90°
. . . - • .1 X
.3 The effect of A on the anisotropy of cu from Tresca's failure criterion.
2 h
/: Orientation angle
7]: Anisotropy scalar
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
T) i=90°
221
Figure 5.4 The effect of stress ratio, (5'al<5'c, on the anisotropy of cu from Tresca's
failure criterion.
o o 3
o II
_cc CO o W
>> Q. O
1.4
1.2
o <n
' c
<
1.0 - —
0.8
F 0 . 6 -
0.4
0.2
0.0
- < ^ -~ / F ^'~-—.
-
n i=0 , A - 0.2 i=45°, A = 0.2 i=90°,A = 0.2
i i i I I i i i i
6 7 8 9 10
Vertical Stress Ratio, a'fa'
222
Figure 5.5 Conceptual behaviour of the 'structured' soil model.
fij2
Structured Overconsolidated Yield envelope
/ /
/ Af =1.2
Destructured Critical State Line
(Classical)
M = 0.9
Associated Plastic Flow
Dynamic yield surface for point '3' (Expands to states of overstress e.g.
'3'defining dFfdcj'y )
my my
(a) m
/277A
(b) (c)
ure 5.6 The effect of sample orientation, i, on the measured and calculated peak
and post-peak undrained strength of Gloucester clay.
30
25
20
15
10
_o-~- , Measured (Lavy 1974)
Calculated
Intrinsic strength at ea=20%(assumed)
—i 1 1 1 1 r -
0 10 20 30 40 50 60 70 80 90 I I
-20 -10
Orientation angle, i
224
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.
(a) cfl 60 Q.
50
40
30
20
10
- i=0 Measured - i=30 Measured - i=45 Measured
—o— i=90 Measured
8 10 12
Vertical Strain (%)
30
20
10
60
50
40
30
20
10
(b)
- i=0 Calculated - i=30 Calculated - i=45 Calculated
T i=ou uaicuiaiea —•— i=90 Calculated
8 10 12
Vertical Strain (%)
8 10 12
Vertical Strain (%)
8 10 12
Vertical Strain (%)
225
Figure 5.8 The comparison for sample orientations, /, of 0° and 90° on the measured
(Law 1974) and calculated[i99] (a) axial stress versus strain and excess
pore pressure versus strain (b) stress paths for Gloucester clay.
(a)
CO Q . -*_
<r> D
60
50
40
30
20
10
irf\
I?
r
^vj
O
o —
"°~"-~ ft==
• ^ *
\-0 Measured i=90 Measured i=0 Calculated i=90 Calculated
"•^ir 3
"~*
Q.
</> CO
2 a. So
o a. <n o> CD
a HI
8 10 12
Vertical Strain (%)
8 10 12
Vertical Strain (%)
(b)
226
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.
i i u -
120 -
100 -
80 -
60 -
40 -
20 -
0 -
O i
Calculated i=0° J W
K./L* ; < ~ _jfjpf i i
/ t Calculated i=90° .
••••[••
I
i
1 (
<?l
I 1— O /
1/
9=2.55° in (ar a3) - am' space
p = 1.14 = Ev /Eh ( See Graham and Houlsby, 1983)
Curve '2' (Cross-anisotropic elasticity)
— 0 — Measured Stress Path i=0 --*- Measured Stress Path i=90
i i
50 100 150 200
a ', kPa
230
Figure 5.13 Measured and calculated peak undrained shear strength versus strain-rate for St. Vallier clay (/=0°).
85
60
55
•! CO D.
4 - »
treng
CO
lear
CO T3 CD C CO
ou
75 -
70 -
65 -
i
I j
j »
GaJeiilated-
Measured by Lo aniMoriri (1972)
10": 10' 10-3 10-2
Strain rate, %/min
Figure 5.14 Influence of A and co on apparent yield surface
(a[G103]) Variation of A
V 2-V<" A For a given structure parameter
1.32Su
Su
6) 0 =1.52
(b[G104]) Variation of m
my Julio*. T For a given anisotropic parameter
A = 0.45
1.32Su
Su
M=0.91
C00 =1 .52 (Intact state)
CO = 1.3—1 Partly G) = 1.1 -T destructured
CO = 1 (Intrinsic state)
<7' lo'w
m my
1.0
Figure 5.15 Influence of destructuration on the apparent yield surface of St. Alban
clay
^ 2 / a ' ( d )
my
1.4
1.2 4
a 1c K' Sv/V
Surfaced 53kPa 0.5 Surfaced 58kPa 0.6 Surface ;3 80kPa 0,5
8% 14% 20%
Surface! 1(53KPa)
^yv o\c : Vertical consolidation stress
K'0 : Consolidation stress ratio, cr'1c/a'3c.
5vA/: Volumetric compression strain in consolidation
Compression curves from oedometer compression tests on intact and
destructured specimens of St. Alban clay
(J „ i <j _ _ p-i p-s
10
-10
-15
-20 4
-25
100
= ^ C 3 T T -a'p.s=35kPa \ ^
\
: : _ , ' , _ ! \ "
°>o _ a P-S' GP-\ -1-52 \ \ Intact sample
a j=23kPa
Remolded sample
10 100 Mean effective stress (o'm), kPa, in log scale
234
CHAPTER 6 CASES STUDY OF THREE DIMENSIONAL EFFECTS ON THE
BEHAVIOUR OF TEST EMBANKMENTS
6.1 Introduction
Trial embankments or test fills are usually constructed to assist in the design of
important embankments on difficult foundations (Dascal et al. 1972; Tavenas et al. 1974;
La Rochelle et al. 1974). In some cases, these structures are built to investigate
geotechnical technology such as geosynthetic reinforcement (Rowe and Soderman 1984;
Alfaro and Hayashi 1997; Rowe et al. 1995) or prefabricated wick drains (Crawford et
al.1992; Burgado et al. 2002). Since trial embankments are often heavily instrumented
and the foundation soils extensively investigated, these structures make interesting cases
for researchers and engineers to study. For most studies involving trial embankments, the
geometry is usually simplified to the two-dimensional (2D) plane strain case neglecting
the three-dimensional (3D) geometry and its effect (Tavenas et al.1974; Indraratna et
al.1992; Crawford et al. 1995; Zdravkovic et al. 2002). Thus, it is important to evaluate
to what extend 3D effects may influence the behaviour of test fills so as to improve the
interpretation of their behaviour and performance.
This chapter uses the finite element software ABAQUS to investigate 3D effects
on the behavior of tiiree full-scale test fills: the St. Alban test embankment (La Rochelle
et al. 1974), the Malaysia trial embankment(MHA 1989b; Indraratna et al. 1992), and the
Vernon test fill (Crawford et al. 1995). Back analysis of these cases highlights some
important considerations in the design and interpretation of test embankments. From this
A version of this chapter has been submitted to Canadian Geotechnical Jouranl 2007
235
study, a shape factor commonly used in bearing capacity calculations is evaluated for use
with 2D embankment collapse calculations. The analysis and evaluation described in the
following sections should be of interest to geotechnical engineers and researchers
involved in the study of embankments on soft soils.
6.2 Methodology
In this chapter, the 2D and 3D behaviour of embankments built on soft cohesive
soil was studied using the elasto-plastic finite element software ABAQUS. In each case,
the embankment fill and foundation soils were discretized using linear eight-node brick
elements and four-node plane strain elements for 3D and 2D analysis, respectively. The
typical finite element mesh comprised a rough rigid boundary which was extended at the
bottom of the soft foundation soil and smooth rigid boundaries on its lateral sides. Some
typical meshes are illustrated throughout the paper.
The foundation soil was modeled as an elastic-perfectly plastic material. The
constitutive parameters include: undrained strength (cu), undrained elastic modulus( Eu),
Poisson ratio( v), bulk unit weight( 7), and coefficient of earth pressure( K0) defined in
terms of total stresses. A typical undrained strength profile comprised a crust underlain
by soft soil layers. In all cases, the foundation clay was assumed to have a constant Eu
for each layer and failure was assumed to be governed by the Mohr-Coulomb failure
criterion with (j)u = 0°, cu varying with depth, and a dilation angle, y/, of 0°. Typical
foundation layers were idealized as having an undrained shear strength, cu0, at the top of
the layer and a gradient of cu with depth, pc . The fill was also modeled as an
elastoplastic material with a constant Young's modulus(£'), Poisson ratio(v = 0.3), bulk
236
unit weight( y), effective friction angle, ((/)') , and cohesion intercept, (c'), and dilation
angle(i^). Soil properties for each case are summarized in Table 6.1 and discussed
below.
The construction process was numerically simulated by activating both the weight
(body force) and stiffness of fill elements layer by layer using an incremental, iterative,
and load-adjusting solution scheme to ensure convergence of elastoplastic solutions.
6.3 St. Alban Test Embankment Case
6.3.1 Introduction
In 1972, the geotechnical research group of Laval University built four test
embankments at Saint Alban to investigate the behavior of embankments on sensitive
Champlain clay. Deposits of Champlain clay are widespread in eastern Canada. In this
case, three of the four test embankments were built with different side slopes to study the
influence of slope inclination on the deformation behavior (La Rochelle et al. 1974). One
test embankment was constructed to failure to study the collapse of embankments on
sensitive soft clays. This case was used by Zdravkovic et al.(2002) to demonstrate the
effect of strength anisotropy on the embankment behaviour using the 2D MIT-E3
constitutive model (Whittle and Kavvadas 1994) which requires 15 material parameters.
In contrast, this chapter investigates 3D effects using a simple undrained strength profile
and elastoplastic analysis.
The St. Alban case has also been investigated by Trak et al.(1980), who used 2D
limit equilibrium analysis to show that the factor of safety of test embankment 'A' at
failure(La Rochelle et al. 1974) was 1.20 and 0.93 using the vane strength profile and the
237
relationship cu - 0.22cr' , respectively. From this assessment, Trak et al. (1980)
concluded that using cu = 0.22c'' for the strength profile was slightly conservative in
this case. It is noted that 3D geometric effect was neglected in the limit equilibrium
analysis; however, in this chapter, 2D and 3D FEM analyses are performed to investigate
the St. Alban test embankment.
6.3.2 Soil Conditions
The foundation at St. Alban comprised a 1.5m-thick weathered crust and a 8m
thick layer of soft silty marine clay, which was underlain by a 4m thick soft clayey silt.
Below the clayey layers was a deposit of fine to medium sand extending to 24.4m depth.
For subsequent analysis of this case, undrained conditions were assumed due to
the relatively short period of construction (10 days from Oct. 4 to Oct. 13, 1972) and the
low permeability of Champlain clays ranging from 10~10 m/s to 10~9 m/s (Tavenas et al.
1983). As shown by Tavenas et al. (1983), such an assumption is not strictly correct near
the drainage boundaries of the clay but should be adequate to assess the relative
behaviour of 2D and 3D embankments.
The Champlain clay deposit has been studied utilizing in situ field vane tests (17
in situ vane tests), cone tests ( 8 static cone tests), and laboratory tests on tube and block
samples (La Rochelle et al. 1974; Tavenas et al. 1974; Leroueil et al. 1979). As pointed
out by La Rochelle et al. (1974), the vane and cone tests across the site at St. Alban
indicated that there was no significant variation of undrained shear strength horizontally
across the site but a relatively typical variation of undrained shear strength with depth.
The clay sensitivity varied between 14 and 22. The measured undrained strength from
field vane and triaxial tests are shown in Figure 6.1, together with the undrained strength
238
profile used in subsequent FE analysis in this case (the solid line in Figure 6.1). The
undrained strength profile for the 1.6m-thick weathered crust layer was corrected to
15kPa due to the probable existence of fissures. The undrained shear strength was then
assumed to increase at a rate of 2.1kPa per meter from 7kPa at a depth of 1.5m. The
assumed undrained elastic modulus and bulk unit weight are summarized in Table 6.1.
The fill materials consisted of uniform medium to coarse sand with an effective
friction angle, <p', of 44° based on drained triaxial tests (La Rochelle et al. 1974). For
analysis, the dilation angle, y/, was assumed to be half of the effective friction angle.
6.3.3 Geometry
A plan view and cross section of the test embankment are shown in Figure 6.2.
The ratio of crest length to width was 4:1 at the designed height of 4.6m and the
corresponding ratio of length to width at the base was about 2:1. The right (front) side
slope in Figure 6.2a was 1.5H:1V and the other slopes were 2H:1V. A 1.5m high berm
was placed on the left side and at the ends to ensure the failure occurred on the right side.
The test embankment failed at a fill thickness of 4.0m before reaching its design height.
For analysis, the 2D analysis considered the central cross section A-A as shown in
Figure 6.2b;whereas, the 3D analysis took into account symmetry and thus only half of
the trial embankment was modeled. The 2D and 3D finite element meshes are shown in
Figure 6.3a and 6.3b.
6.3.4 Results
As shown in Figure 6.4, the calculated failure fill thickness for 2D and 3D FE
analysis were 3.6m and 4.0m respectively, based on the vertical displacement curves of
the embankment at a central point "O" (See Figure 6.2). Despite the 10% difference
239
between predicted failure thicknesses, the magnitude of settlement from both 2D and 3D
analysis at the same fill thickness are very similar. The contour of spatial displacement at
failure is shown on the 3D model in Figure 6.5, which indicates that the development of
failure in the longitudinal direction is restricted by the length of the fill (due to the
stabilization effect of both end- slopes). In Figure 6.6, the observed and computed
extents (plan view) of failure are compared, and the agreement is reasonable. It can also
be seen that the mobilized failure mass for the St. Alban embankment has strong 3D
characteristics. As such, failure of the St. Alban test fill is 3D in nature and use of a 2D
model results in underestimation of the failure thickness by about 10% compared with
that obtained by 3D analysis.
It is interesting to note that 3D analysis yields a higher factor of safety than 2D
analysis for the St. Alban case. As a result, cu = 0.22<r'p investigated by Trak et al.
(1980) is actually a reasonable estimate of strength profiles, considering that 3D analysis
would give a factor of safety 10% higher than that obtained by Trak et al. (1980) using
2D analysis (0.93).
6.4 Malaysia Trial Embankment Case
6.4.1 Introduction
The Malaysia trial embankment was built to failure at Muar flat in the valley of
the Muar River in Malaysia. Muar clay is a very soft clay which caused frequent
instability problems during construction of the Malaysian North-South Expressway. The
Malaysia trial embankment was built between 27th Oct. 1988 and 4th Feb. 1989. The fill
was placed at a rate of about 0.4m/week until it failed at a fill thickness of 5.4m.
The Malaysia case was fully instrumented and well documented (MHA 1989a).
240
A series of comprehensive field and laboratory tests were carried out before the
embankment construction, which provided parameters for researchers and engineers to
predict the embankment behavior. An International Symposium entitled: "Trial
Embankment on Malaysian Marine Clays" was held in November 1989 and 31 class A'
predictions of the embankment performance were received from experienced researchers
and engineers (MHA 1989b); each employing different methods of analysis ranging from
stability charts and limit equilibrium analysis, to undrained and drained finite element
analysis (e.g. Brand and Premchitt 1989). In one case, a centrifuge model test was used
(Nakase and Takemura 1989). Subsequent to the symposium, Indraratna et al. (1992)
reported a calculated failure thickness of 5.0m using a modified Cam-clay model and 2D
FEM analysis. All predictions of the failure thickness are summarized in Figure 6.7,
where it can be seen that there was a wide variation of predicted failure thickness ranging
from 2.8m to 9.5m. The majority of predictors underestimated the failure thickness: the
average predicted failure thickness was 4.7m, whereas the actual failure thickness was
5.4m. This discrepancy reflects the difficulty of geotechnical prediction and also
suggests there may be some characteristics that have not been fully explored by the
predictors.
6.4.2 Soil Conditions
Figure 6.8 summarizes the Malaysian soil profile. As reported by MHA( 1989a),
the subsoil consists of a 2m-thick weathered crust underlain by a 6m-thick deposit of very
soft silty clay and a 10m thick layer of silty clay. The upper clay deposits overly a 0.5m
peat layer, 3.5m sandy clay, and then dense sand. According to field and laboratory tests,
the undrained strength increases linearly with depth below the weathered crust.
241
For the analysis of this case, the undrained strength of the crust was corrected to
one third of the field vane strength to account for the likely presence of fissures (Lo and
Hinchberger 2006). The engineering parameters of the fill and foundation subsoil used in
the analysis are summarized in Table 6.1.
6.4.3 Geometry
A plan view and cross-section of the Malaysia trial embankment are presented in
Figure 6.9a and 6.9b. With respect to the designed thickness of 6m, the ratio of crest
length to width was 2, while the aspect ratio at the base of the main fill was 1.4. Similar
to the St. Alban case, a 2.5m high berm was placed around three sides of the fill to force
the failure toward one side.
6.4.4 Results
Figure 6.10a shows the calculated fill thickness versus vertical displacement at
point "O" (See Figure 6.9) beneath the center of the embankment. From this Figure, it
can be seen that the displacement increases linearly as the fill thickness increases during
the initial stage of construction. When approaching to the critical height, however, a
small increase in fill thickness results in large displacement. As shown in Figure 6.10b,
initially the net fill height increases with fill thickness until reaching a maximum value at
a critical point, where upon it decreases with the addition of fill indicating that the
incremental vertical displacement exceeds the corresponding increment in fill thickness.
When further loads are applied, the net fill height decreases and full collapse occurs.
Thus, the state with the maximum net height is considered as a critical state and the
corresponding thickness is taken as the calculated failure thickness. According to Figure
6.10, the calculated failure thickness for 2D and 3D analysis are 4.2m and 5.2m
242
respectively. Therefore, in the case of Malaysia test fill, consideration of 3D geometric
effects results in a 20% increase in the calculated failure thickness relative to the 2D
analysis.
In Figure 6.10a, the deformation curve of the 2D model follows closely with that
of the 3D model until failure occurs. It appears that in this case the 3D geometry does
not significantly influence the deformation prior to imminent collapse. This is consistent
with the fact that the calculated settlement by predictors using 2D analysis agree well
with the measured data of the Malaysia trial embankment in spite of the relatively large
discrepancy of predicted failure thickness (Brand and Premchitt 1989).
Figure 6.11 and 6.12 show the velocity fields at failure for 2D and 3D models,
respectively. The trend of movement of the failure mass is shown by the direction of the
velocity vectors, and the length of velocity vectors represents the relative magnitude of
movement. Both 2D and 3D failure surfaces were estimated based on the velocity fields
at the respective failure thickness ( 4.2m for 2D analysis and 5.2m for 3D analysis). As
shown in Figures 6.11 and 6.12, the estimated failure surfaces for 2D and 3D analysis are
generally comparable though 2D and 3D models predicted different failure thickness.
As shown above, the failure thicknesses predicted using 2D and 3D analysis of
the Malaysia trial fill differ by about 20%. Thus neglecting 3D geometric effects will
lead to underestimation of the failure thickness. From an engineering point of view, it
seems to be conservative to adopt the plane strain assumption. However, to evaluate the
strength of subsoil based on the behaviour of a trial embankment, the 2D model will, in
return, lead to an overestimation of the available foundation strength profile, which could
lead to inadequate designs for long embankment on such soft soils. This will be
243
discussed and highlighted further during evaluation of the Vernon case below.
6.5 The Vernon Case
6.5.1 Introduction
The final case considered is the Vernon embankment presented by Crawford et al.
(1992, 1995). In this case, two consecutive failures occurred during construction of an
approach embankment on soft clay in British Columbia. The embankment failures
occurred in spite of the fact that two test fills were built successfully on either side of the
failures and that the test fills were higher than the approach embankment that failed.
Figure 6.13 shows a site plan of the Vernon approach embankment and the
location of the two test fills. The Vernon case comprised the West Abutment Test fill
which was constructed to a maximum fill thickness of 11.5m, with wick drains in the
foundation; and the east test fill or Waterline Test Fill which was built to a maximum fill
thickness of 12m, without wick drains. Both test fills were constructed in 1986 and
remained stable for approximately 3 years before being incorporated into the Vernon
approach embankment. Figure 6.14 shows a cross-section of both the Waterline and
West Abutment Test fills. Since the wick drains may influence the test fill behaviour,
only the Waterline test fill is selected for comparative analysis with the Vernon approach
embankment.
Construction of the Vernon approach embankment commenced in early
December 1988 and progressed slowly to a fill thickness of between 7m and 9.5m by
June 30th, 1989. At this time, the embankment failed (first failure) on the north side
encompassing a portion of the West Abutment Test Fill. The extent of the first failure is
shown in Figures 6.13 and 6.14. At the time of the first failure, the West Abutment Test
244
Fill had been in place for approximately 3 years, and according to the results of
monitoring, the excess pore pressures generated during construction of this test fill had
dissipated (see Crawford et al. 1992).
The failed approach embankment was redesigned with 5m thick and 30m wide
berms on both sides of the original embankment and reconstruction commenced in
August 1989 at a very slow rate. In March 1990, a second failure occurred that was
much larger in extent and included most of the first failure. The second failure occurred
at a fill thickness of about 11.2m and it involved both sides of the approach fill. The
extent of the second failure is also shown in Figures 6.13 and 6.14. The approach
embankment was eventually completed using berms and lightweight fill; however, the
case raises an obvious but perplexing issue: In what way were the results of the two test
fills misleading?
6.5.2 Analysis
The subsurface conditions in the Vernon Case are summarized in Figure 6.15. In
the Vernon case, the foundation soils comprised about 4m of interlayered sand, silt and
clay underlain by a 5m thick crust comprising stiff to very stiff clay then a deep deposit
of soft to firm silty clay. Figure 6.15 summarizes the results of field vane tests done in
1960 and 1985 in addition to the undrained strength profiles investigated in this study.
For the purpose of analyzing the Vernon case, the undrained strength of the crust was
reduced to 40kPa in accordance with Lo (1970) and Lo and Hinchberger (2006) to
account for the probable effect of fissures on the mass strength of the crust. The
undrained strength was assumed to be constant at 40kPa from the ground surface to 6m
deep then it was assumed to decrease linearly from 6m to a depth of 9m below which the
245
strength increased linearly with depth. Three different strength profiles were
investigated: Profiles L(0.84M), M and H(1.08M) which denote lower, middle and upper
strength profiles (see Figure 6.15). The L and H profiles are 84% and 108% of the M
profile as a whole.
Lo and Hinchberger (2006) studied the 3D effect in this case using 2D
axisymmetric FE analysis, where the 3D geometry of test embankment was simplified as
axisymmetric. In addition, the three profiles used by Lo and Hinchberger (2006) have the
same crust strength but slightly different strength for the soft clay underlying the crust.
This chapter utilizes a real 3D model to account for the geometry characteristic of the test
embankments.
Table 6.1 summarizes the material parameters used in the analysis. The
foundation clay was modeled as an undrained material with a unit weight of 16 kN/m3,
friction angle (pu = 0°, and undrained shear strength, cu, that varied with depth (see
Figure 6.15). The fill was considered to be a drained material with a unit weight of 20.4
kN /m3 , and effective friction angle, <f>', of 30°, in accordance with that reported in the
case (Crawford et al. 1992, 1995)
The Vernon case was studied using both 2D and 3D finite element analysis as
described in the following: (i) The first failure was evaluated using 2D and 3D finite
element analyses to assess the mobilized undrained strength of the clayey foundation and
to investigate the role of 3D effects on the approach embankment performance, (ii) Next,
the Waterline Test Fill was analyzed using 2D and 3D finite element analyses. The back-
calcualted strength profiles by 2D (H) and 3D (L) analysis of the Waterline Fill are
compared with the mobilized strength obtained from the first failure of the Vernon
246
approach embankment. The purpose of these analyses was to investigate the degree to
which 3D effects may have affected the performance and consequent lessons learned
from the failure of the Vernon approach embankment.
6.5.2 Results of Vernon Approach Embankment
As shown in Figure 6.13 and 6.14, the Vernon approach embankment was
constructed between the Waterline test fill and West Abutment test fill. In order to
evaluate the first failure, Station 27+80 was considered for 2D analysis since it is situated
at the midpoint of the first failure.
Figure 6.16 shows the calculated failure thickness (8.2m, 9.8m, and 10.8m) for
the L-profile, M- profile, and H-profile respectively. Compared with the actual failure
thickness of 9.9m, the M-profile provides the best fit and it is thus considered to be the
approximate mobilized strength profile for the first failure from 2D analysis,
notwithstanding that there could be other interpretations.
However, the assumed plane strain condition of the 1st failure may not strictly
satisfy the actual condition for the Vernon approach embankment. As shown in Figure
6.13 and 6.17, the height and width of the approach embankment increase toward the
bridge site. The longitudinal slope of the embankment crest was also about 3.2% (See
Figure 6.14). Thus, each cross section in the approach embankment varied geometrically
and consequently the degree of divergence from a plane-strain condition is unknown.
In light of this, a 3D analysis was done to compare with the 2D analysis discussed
above and to explore to what extent the first failure may have been affected by 3D
effects. Accordingly, the true 3D geometry of the approach embankment was modelled
as shown in Figure 6.17. The crest width was constant at 22m with 1.5H:1V side slopes.
247
The crest aspect ratio, length/width, was approximately 9.8 and the average base aspect
ratio was 4.2. The plan view and cross sections of both ends are shown in Figure 6.17.
Figure 6.18 compares the results of 3D analysis and 2D analysis using the M-
profile. The predicted failure thickness from 3D analysis was 10.3m, which was 0.4m
higher than the 2D prediction. Since the calculated failure thickness of the Vernon
approach embankment is only 4% higher for the 3D case compared to that calculated for
2D analysis, it is concluded that the choice of Station 27+80 for 2D analysis of the first
failure was acceptable.
Figure 6.19 shows the displacement contours of the 3D model at the failure
thickness, together with the vectors indicating the direction and relative magnitude of
movement of the ground surface. The extent of the calculated failure mass is between
station 27+35 and station 28+20 and spreading about 50m outward from central line.
Referring to Figure 6.13 and 6.14, the observed limit of the first failure agrees well with
that calculated by 3D analysis.
6.5.3 Results of Waterline Test Fill
Since the Waterline fill was used to conclude the final approach embankment
would be stable, the performance of this test fill was analyzed in detail.
For 2D analysis, the central cross section of the Waterline test fill was considered
because of its symmetrical geometry. Figure 6.20 shows the geometry considered. The
measured and calculated displacement of centre point 'O' below the Waterline fill are
presented in Figure 6.21. From Crawford et al (1995), the measured vertical
displacement curve was essentially linear, which indicates that the behaviour of the
Waterline fill was predominantly elastic. The predicted failure thickness from 2D
248
analysis were 8.3m for the L-profile, 10.8m for the M-profile, and 11.8m for the H-
profile, respectively. Considering that the Waterline test fill was stable at a thickness of
11.8m, the 2D analysis suggests that the H-profile is the lower bound strength available
in situ. The M-profile, however, was back calculated from the first failure of the Vernon
approach embankment and this discrepancy warrants further investigation.
Accordingly, a 3D model was undertaken to account for the geometry of the
Waterline test fill. The results of the 3D analysis are presented in Figure 6.22. From
Figure 6.22a, it can be seen that the predicted failure thickness is 11.8m using the L-
profile and that the embankment is stable at 11.8m for both the M- and H-strength
profiles. Thus, it can be deduced from the 3D analysis that the L-profile is a lower bound
for the available in situ foundation strength.
Based on the analysis and discussion above, there is a consistent interpretation of
the Vernon case. If the M-strength profile shown in Figure 6.15 is adotped, then a 3D
analysis indicates that the Waterline test fill is stable at a fill thickness of 11.8m whereas
the approach embankment fails at a fill thickness of 9.8m. This is what was observed in
this case. A 2D-analysis on the Waterline test fill, however, yielded the H-profile as the
lower bound strength profile, which may lead to an inadequate design for the approach
embankment. The FE analyses suggest that 3D effects may have contributed to the
failure of the approach embankment before reaching the height of the adjacent Waterline
Fill, notwithstanding that natural soil variability may have also played a role. From the
FE analysis, the ratio of 3D collapse thickness of the Waterline test fill to the 2D collapse
thickness is 1.4, which is significant.
249
6.6 Discussion
From the detailed analysis of the above cases, the base aspect ratio (L/B) of length
over width can be utilized to represent the 3D geometry of test embankments. In
addition, the ratio of the calculated failure thickness by 3D and 2D FE analysis
(Hf3D I Hf2D) can be used to quantify 3D effects. Hf3D I Hf 2D is plotted against the
base aspect ratio (L/B) in Figure 6.23. As shown in Figure 6.23, the 3D effect
represented by Hf3D/Hf2D are inversely proportional to the base aspect ratio. It is
qualitatively consistent with the shape factor equation utilized by Skempton (1951) to
account for geometric effects on the bearing capacity of spread foundations, e.g.:
^ ^ = 1 + — [6.1] Quit,2D L I B
where L and B are the length and width of the foundation; qultiD and qult 2D are the
ultimate bearing capacity of a rectangular foundation and the bearing capacity of a
infinitely long foundation, respectively.
Equation [6.1] is plotted in Figure 6.23 for comparison with the St. Alban,
Malaysia, and Vernon cases. For the Vernon approach embankment and the St. Alban
test embankment with aspect ratios (L/B) equal to or larger than 2.0, the corresponding
case points in Figure 6.23 plot very close to Equation [6.1]. This suggests that 3D
geometry effects on test embankments with a aspect ratio greater than 2 are reasonably
close to those deduced from Equation [6.1]. For the cases of Malaysia trial fill and the
Waterline test fill, the difference between the predicted failure thicknesses by 2D and 3D
analysis increases from 20% up to 40% as the aspect ratio decreasing from 1.4 to 1.2.
The corresponding points representing these two cases in Figure 6.23 (from 2D and 3D
250
FE analyses) lie above Equation [6.1], indicating that 3D effects on test embankments
with an aspect ratio less than 2 is greater than that expected for foundation bearing
capacity.
It is noted that other factors such as side slopes and berms of the test fills may
also influence the 3D effect to some degree, and that these factors may account for some
of the difference noted in Figure 6.23.
6.7 Summary and Conclusion
Three full-scale test fills have been evaluated utilizing finite element analysis
(ABAQUS) accounting for 2D and 3D geometries, respectively. The key findings
resulting from these case studies are summarized as follows:
Considerable difference (10% to 40%) of die predicted failure thicknesses
obtained by 2D and 3D analysis was found for all three cases. Considering 3D geometry
results in an increase in the predicted failure thickness of test fills. This finding is
qualitatively consistent with the shape factor equation in bearing capacity theory
(Skempton, 1951) for base aspect ratios (L/B) greater than 2. However, when the base
aspect ratio is less than 2, the 3D effect on test embankments becomes considerably
greater than that suggested by bearing capacity theory.
3D analysis agrees better with the field behaviour. Beside the influence of 3D
geometry, the calculated extent of failure by 3D analysis agrees fairly well with the field
observations in the St. Alban and Vernon cases.
Assuming plane strain conditions in the analysis of test fills may potentially lead
to the overestimation of the available soil strength and consequently inadequate design of
long embankments on the same site. As shown in Vernon case, neglecting the 3D
251
geometry and its impact on the behaviour of the Waterline test fill could have been
misleading for the design of the long approach embankment.
It is recommended that 3D geometry should be considered in the design and
interpretation of test fills whose base ratios of length to width are less than 2. In this
situation, the influence of 3D geometry should be taken into account to reasonably
evaluate the behaviour of the trial fills, including failure thickness, strength profiles, and
stress in the reinforcement in the test embankment.
252
References
Alfaro, M.C., and Hayashi, S. 1997. Deformation of reinforced soil wall-embankment
system on soft clay foundation. Soils and Foundations, 37(4): 33-46.
Bergado, D.T., Fannin, R.J., Holtz, R.D., and Balasubramaniam, A.S. 2002. Prefabricated
vertical drains (PVDs) in soft Bangkok clay: A case study of the new Bangkok
International Airport project. Canadian Geotechnical Journal, 39(2): 304-315.
Brand, E.W., and Premchitt, J. 1989. Comparison of the predicted and observed
performance of the Muar test embankment. In Proceeding of the international
symposium on trial embankments Malaysia marine. Edited by R.R. Hudson, C.T.
Toh, and S.F. Chan. Kuala Lumpur. The Malaysian Highway Authority, Vol.2, pp.
10-18.
Crawford, C.B., Jitno, H., and Byrne, P.M. 1994. Influence of lateral spreading on
settlements beneath a fill. Canadian Geotechnical Journal, 31(2): 145-150.
Crawford, C.B., Fannin, R.J., and Kern, C.B. 1995. Embankment failures at Vernon,
British Columbia. Canadian Geotechnical Journal, 32(2): 271-284.
Dascal, O., Tournier, J.P., Tavenas, F., and La Rochelle, P. 1972. Failure of test
embankment on sensitive clay. In Proceeding of ASCE Specialty Conference on
Performance of Earth and Earth-Supported Structures. Purdue University,
Lafayette, Vol.1, pp. 129-158.
Indraratna, B., Balasubramaniam, A.S., and Balachandran, S. 1992. Performance of test
embankment constructed to failure on soft marine clay. Journal of Geotechnical
Engineering, 118(1): 12-33.
La Rochelle, P., Trak, B., Tavenas, F., and Roy, M. 1974. Failure of a test embankment
on a sensitive Champlain clay deposit. Canadian Geotechnical Journal, 11(1):
142-164.
Leroueil, S., Tavenas, F., Brucy, F., La Rochelle, P., and Roy, M. 1979. Behavior of
destructured natural clays. Journal of the Geotechnical Engineering Division,
105(6): 759-778.
Lo, K.Y. 1970. The operational strength of fissured clays. Geotechnique, 20(1): 57-74.
Lo, K.Y., and Hinchberger, S.D. 2006. Stability analysis accounting for macroscopic and
microscopic structures in clays. In Proc. 4th International Conference on Soft Soil
Engineering. Vancouver, Canada, pp. pp. 3-34.
MHA 1989a. Factual report on performance of the 13 trial embankments. In Proceeding
of the international symposium on trial embankments Malaysia marine. Edited by
R.R. Hudson, C.T. Toh, and S.F. Chan. Kuala Lumpur. The Malaysian Highway
Authority, Vol.1.
MHA 1989b. The Embankment built to failure. In Proceeding of the international
symposium on trial embankments Malaysia marine. Edited by R.R. Hudson, C.T.
Toh, and S.F. Chan. Kuala Lumpur. The Malaysian Highway Authority, Vol.2.
Nakase, A., and Takemura, J. 1989. Prediction of behaviour of trial embankment built to
failure. In International Symposium On Trial Embankments On Malaysia Marine
Clays. Kuala Lumpur. November 6-8, Vol.2, pp. 3-1,3-13.
Rowe, R.K., and Soderman, K.L. 1985. Approximate method for estimating the stability
of geotextile-reinforced embankments. Canadian Geotechnical Journal, 22(3):
392-398.
254
Rowe, R.K., Gnanendran, C.T., Landva, A.O., and Valsangkar, A.J. 1995. Construction
and performance of a full-scale geotextile reinforced test embankment, Sackville,
New Brunswick. Canadian Geotechnical Journal, 32: 512-534.
Skempton, A.W. 1951. The bearing capacity of clay. In Building Research Congress.
London.
Tavenas, F., Leblond, P., Jean, P., and Leroueil, S. 1983. Permeability of natural soft
clays, part I: methods of laboratory measurement. Canadian Geotechnical Journal,
20(4): 629-644.
Tavenas, F.A., Chapeau, C , La Rochelle, P., and Roy, M. 1974. Immediate settlements
of three test embankments on champlain clay. Canadian Geotechnical Journal,
11(1): 109-141.
Trak, B., La Rochelle, P., Tavenas, F., Leroueil, S., and Roy, M. 1980. New approach to
the stability analysis of embankments on sensitive clays. Canadian Geotechnical
Journal, 17(4): 526-544.
Whittle, A.J., and Kavvadas, M.J. 1994. Formulation of MIT-E3 constitutive model for
overconsolidated clays. Journal of Geotechnical Engineering, 120(1): 173-198.
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.
Tab
le 6
.1
Para
met
ers
used
in
the
num
eric
al a
naly
sis
of t
he th
ree
case
s
Cla
y D
epos
it
Dep
th, m
U
ndra
ined
Sh
ear
Stre
ngth
, kPa
y s
at(K
N/m
3)
K0
EK(K
Pa)
St. A
lban
test
em
bank
men
t
Cru
st
0-1.
6
c u0
= 1
5kP
a
A,
=0
19
1.0
1.5E
7
Soft
cla
y
1.6-
40
c„0
= I
kPa
p c>
=2.
\kP
alm
17
0.9
2.5E
7
Mal
aysi
a
Cru
st
0-2
c„0
= 2
5kP
a
A.=
0
19.8
1.0
2.5E
7
tria
l em
bank
men
t U
nder
lyin
g so
ft c
lay
2-40
c„0
= 8
kPa
p c_
=lA
8kP
a/m
16.0
0.9
8E6
Ver
non
case
Cru
st
0-6
c u0
= A
QkP
a
20
1.04
1.8E
7
(M-P
rofi
le)
Tra
nsiti
on
laye
r 6-
9
c u0
= 4
0kP
a
p Cj
=-2
.67k
Pa/
m
17
0.85
1.5E
7
Soft
cla
y
9-80
c u0
= 3
2kP
a
p c_
=l.0
3kP
a/m
17
0.85
2.5E
7
0'=
44°
yr
= 2
2°
Fill
Mat
eria
ls
E'=
2E
8kP
a c'
= 5
kPa
v =
0.3
y s
al=
19kN
/m3
^'=3
1°
^=
15
° E
'=5.
1E6k
Pa
c'=
5kP
a v
= 0
.3
y .
=20
.5kN
/m3
^'=
33
° ^
=1
6°
E'=
\5E
lhP
a c'
=5k
Pa
v =
0.3
y s
al=
20.5
kN/m
3
256
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)
0 10 2m i i i i i r i
Om
Undrained Strength, (kPa)
20 30 40 50 60 70
-2m
o . -4m <D Q
-6m 4
-8m 4
-10m
i i I i ; i i i i i i i I i i i i I I I I — I I I
Assumed Strength Profile
7kPa
In-situ Average Vane Strength
In-situ Minimum Vane Strength;
In-situ Maximum Vane Strength
ncreasing rate = 2.1
257
Figure 6.2 Plan view and cross-section of St. Alban test embankment
(a) Plan view
- 4 . 6 —i 6.1 1 7.6 , 6 . 9 -
A — t — ^ * " \^^^2^
X ' X 6 ?
Point 'O'
(b) A-A Section
258
Figure 6.3 Generated Meshes for 3D and 2D FEM model
(a) Plane strain analysis
(b) 3D analysis
259
Figure 6.4 Measured and calculated vertical displacement of point 'O' for St. Alban
Embankment
o.oo
-.02
- . 0 4 - • • • •
-.06 -
-.08
« -.10
-.12 4-
-.14 H
0.0
2D analysis . — 3D analysis I R23 Centre Point
1.0
Hf2D=3.6m B(3D=4.0m
1.5 2.0 2.5 3.0 3.5 4.0
Embankment thickness,m
4.5 5.0
260
Figure 6.5 Spatial displacement contour of 3D model for St. Alban embankment (at
failure)
U, Magnitude + 9.842e-01 +9.022e-01 + 8.202e-01 +7.382e-01 + 6.SSle-01 + S.741e-01 + 4.921e-01 +4.101e-01 +3.281e-01
t +2.4Sle-01 +1.640e-01 + 8.202e-02 + 0.000e+00
261
Figure 6.6 Spatial displacement contour V.S. fissures at failure on the top surface on
St. Alban Embankment
Projected contours from Figure 6.5
Observed extent of fissures
Measured (La Rochelle et al. 1974)
Calculated extent of failure
3DFEM Analysis
......
sm
•—7*—~t—
X.
262
Figure 6.7 The statistic table for the[il05] prediction on the failure thickness of
Malaysia test embankment (data from MHA 1989b)
C/2
a o o
• a
pre
o ^ 1) ran
Z
11 10 9
8 7 6 5
4 3 2 1 0
A.ctual failure thickness = 5.4 m
4 5 6 7
Predicted thickness, m
263
Figure 6.8 Strength profiles[i 106] for the Malaysia case (experimental data from
MHA 1989a)
Dense Sand i i i i | i i i i [ i i i i I i f i i I i i i i I i i i i | i i
0 10 20 30 40 50 60 70
Undrained Strength, (kPa)
6.9 Plan view of Malaysia test embankment
h -
! _^^ Berm
i
_ / ^ Main Fill
H _ ^ ^ j
^ ^ s .
A-A Section
s 71
+
Berm
Berm
IZ
Main Fill
Point '0 '
Plan View
Figure 6.10 Measured and calculated settlement[il07] of Malaysia Trial Embankment
b)
Failure Thickness : Hf 3D = 5.2m
2D analysis 3D analysis
8
Fill Thickness,m
266
Figure 6.11 Velocity field in central cross-section of 2D model for the Malaysia trial
embankment (at failure)
Scale: [ 5m
267
Figure 6.12 Velocity field in central cross-section of 3D model for the Malaysia trial
embankment (at failure)
Scale: p ^
Velocity Field on A-A section
268
Figure 6.13 Plan view of Vernon embankment (modified after Crawford et al. 1995)
LIMIT OF SECOND FAILURE
LIMIT FIRST
STRUCTURE
KEY : 0 1960 BORINGS & 1985 BORINGS • 1990 BORINGS
269
Figure 6.14 Longitudinal section through the embankment (after Crawford et al. 1995)
— i r FAILURE • 30 JUNE '89
FAILURE • 10 MAR. '90
SURFACE, 10 MAR. M90
FINAL ffAVEMENT SFC.
WATERLINE FILL
: X * « /
WEST ABUTMENT FILL
SETTLEMENT PLATE
j . 1 . . .
MAXIMUM SETTLEMENT
- A * "
| - WICKS 1986 -l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
8 +
8 +
WEST SURVEY STATION EAST
ure 6.15 Distribution of vane strength [i 108]with depth
Crust
Transition Layer
-10
-20 -4-Clay Layer
-30 +
Measured Vane Strength 1985 (Crawford 1992) Measured Vane Strength 1960 (Crawford 1992)
-40
-10
4 -20 H Profile .
Strength Profile(M)
L Profile
4 -30
_L J_ 0 20 40 60 80 100 120 140
Undrained Strength, (kPa)
-40
Figure 6.16 Vertical displacement of Vernon Approach Embankment in 2D
analysis [G109]
10
E «-- 8
2D ^na fyg jg^^g^^^ iQf i i g j r
2D analysis(M Strength Profile)
2D analysis(L Strength Profile)
• * f
4f
Fill Thickness,m
E c CD
E CD O co Q. CO
b "CO _o V-4 L . CD >
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
"^^r~Z: Z * * * J M *
' A ^ * ^ 2D analysis(H Strength Profile)
2D analysis(M Strengthi Woffle!)''
- 2D analysis(L StrengthProfile)
* measured data (Crawford et al.,1995) - i
12 8 10
Fill Thickness,m
272
Figure 6.17 Plan view and 3D model of Vernon approach embankment[G 110]
Z 2^ Station 26+20
Station 27+8C
Figure 6.18 Vertical displacement of Vernon Approach Embankment in 3D
analysis [ G i l l ]
1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 Fill Thickness,m
o.o
-.5
-1.0 -
-1.5 -
-2.0
-2.5
^"W^'^^^gi^gmci^
3D analysis(M Strength Profile)
J2D analysis(M sirength Profile)
— i 1 1 1 1 1 1 1 1 1 ^i 1
0 1 2 3 4 5 6 7 8 9 10 11 12
Fill Thickness,m
274
Figure 6.19 Spatial displacement contour of Vernon approach embankment[Gl 12]
Station 26+20
Station 27+35
Station 27+80
Station 28+20
275
Figure 6.20 Plan view and cross section A-A of Waterline test [Gl 13]fill
54.2
16.; 21 , 7,6 32
1 I
t r
A
11,40 Poorer l-»
4
17,1 - 1 0 •>
x
17,1
Point O
A-A Section
Figure 6.21 Measured and calculated displacement by 2D analysis for the Waterline
Test Fill
12
11
10
9
8
7
6
5
4
3
2
1
0
J2D analysis(H Strength Profile) = . - . . = , , = . ^ = 4 * ^ ;
|2D analysis(M Strength Profile)
2D ahalysjs(L Strength Profile)
A
y
. /
^
The fill thickness of the Waterline test embankment
H=11.8ni
* measured data (Crawford et al. 1a92)
6 10 11 7 8 9 10 11 12 13
Fill Thickness,m
10 11 12 13
Fill Thickness,m
Figure 6.22 Measured and calculated displacement by 3D analysis for the Waterlme
Test Fill
(a)
e
Heig
ht,
Fill
N
et
12
11
10
9
8
7
6
5
4
3
2
1
0
(b)
3D knalysis(M Strength'Profile) -—^ £ 3D jtna|ysis(L Strength Profile)
; The fill thickness of the Waterlinq test embankment: j 4 _ _ 4 ..._, 4 ; ~H=Tx.8m.|.i.
8 9 10 11 12 13 Fill Thickness,m
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Fill Thickness,m
278
Figure 6.23 Illustration of 3D effect on the bearing capacity and the cases studied.
1.5
1.4 4
Q 1.3 CM
X
1.2 4
1.1
1.0
Waterline fill . • • - •
Shape factor from Equation [6.1]
Malaysia
St. Alban fill
Vernon approach embankment
T i : r T 1 1 r
0 1 2 3 4 5 6 7 8 9 10 11
Base aspect ratio, L/B
279
CHAPTER 7
SUMMARY AND FURTHER WORK
7.1 Summary
In this thesis, a general constitutive framework has been developed to account for
viscosity, structure, and strength anisotropy of natural clay. In addition, selected issues
affecting the performance of embankments on clay foundations were investigated.
This thesis first introduces a simple elastic viscoplastic model and describes an
approach to determine the viscosity-related parameters required in this model. Some of
the fundamental principles in EVP theory are validated based on the viscous responses of
19 clays reported in the literature by different researchers.
By introducing a state-dependent fluidity parameter, an existing EVP model is
extended to account for structure and its degradation on the engineering behaviour of
natural clay. The extended constitutive model was successfully used to simulate the
rupture response and rate sensitivity of Saint-Jean-Vianney clay.
Then a tensor approach was coupled with the EVP model to simulate the strength
anisotropy of natural clays. This model was shown to be able to simulate the measured
orientation-dependent strengths and pore water pressure responses in undrained triaxial
tests on two natural clays.
The Gloucester test embankment was examined to investigate the influence of
structure and destructuration on its field performance. This study shows that the use of a
structured EVP model improves the numerical simulation over long-term settlement of
the Gloucester test embankment, compared with the use of an unstructured EVP model.
280
In addition, contours of strength change in the Gloucester foundation highlight the
influence of destructuration, which reduces the strength gain due to consolidation and
even leads to net strength loss in some local zones in the clay deposit. Thus, considering
destructuration is important to evaluate the in situ settlement and the stability of
infrastructures on or in structured natural clays.
Lastly, three full-scale test embankments built on soft clay deposits were studied
to investigate the influence of three-dimensional (3-D) geometry on their in situ
performance. Both two-dimensional (2-D) plane strain finite element analysis and three-
dimensional (3-D) finite element analysis are performed for 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 test embankments, which have dramatically different fill
thicknesses and underlying clay deposits. Finally, a suggestion is provided to estimate the
3D effect based on the aspect ratio of the fill base length to the base width.
7.2 Suggestions for Future Research
Although the developed constitutive model is able to simulate the main
characteristics of the natural clays studied, it is acknowledged that the behaviour of
natural clay is complicated and the numerical simulation for some clays is very
challenging. The following summarizes several interesting issues deserving further
investigation.
There are currently few studies on the static yield state and the threshold strain-
rate require to reach it except those reported by Sheahan et al (1995) and Hinchberger
(1996). The possible reason is that the magnitude of the threshold strain rate according to
the static yield state is far lower than the strain-rates commonly used in laboratory tests.
281
As an alternative, the static yield state may be evaluated using long-term consolidation
tests, where a very low strain-rate would be reached after a long period of creep.
This thesis studied the viscosity of clay macroscopically. It would be helpful to
explore the microscopic mechanism of the viscosity behaviour for clays. The rate-
process theory, proposed for the atom and molecule level by Glasstone et al. (1941),
assumes a balance of input energy and the energy barrier among the equilibrium position
for particles. This theory has been introduced into soil mechanics to interpret the
viscosity mechanism (Mitchell et al. 1968; Feda 1989). Obviously, as a mixture of water,
particles, and possibly air, clay is far more complicated than metal and consequently the
application of Gloasstone's rate-process in soil mechanic faces considerable challenges.
However, it is worth further study to gain insight into the viscosity mechanism at a
microscopic level for clay viscosity.
Clay structure has been modeled by adopting a state-dependent fluidity parameter
in this thesis. However, as mentioned previously, the rates of structure damage with
plastic strain during undrained triaxial compression and oedometer compression tests
appear different. This discrepancy may be attributed to the strain localization during
undrained triaxial compression. To investigate this issue, a numerical simulation
including strain localization in triaxial tests may be helpful to address this discrepancy
and improve the understanding in the development of shear bands in specimens during
undrained compression tests.
282
References
Feda, J. 1989. Interpretation of creep of soils by rate process theory. Geotechnique, 39(4):
667-677.
Glasstone, S., Laidler, K.J., and Eyring, H. 1941. The theory of rate process. McGraw
Hill, New York.
Hinchberger, S.D. 1996. The behaviour of reinforced and unreinforced embankments on
rate senstive clayey foundations. Ph.D Thesis, University of Western Ontario,
London.
Mitchell, J.K., Campanella, R.G., and Singh, A. 1968. Soil creep as rate process.
American Society of Civil Engineers Proceedings, Journal of the Soil Mechanics
and Foundations Division, 94(SM1): 231-253.
Sheahan, T.C. 1995. Erratum: interpretation of undrained creep tests in terms of effective
stresses. Canadian Geotechnical Journal, 32(3): 557.
Sheahan, T.C. 1995. Interpretation of undrained creep tests in terms of effective stresses.
Canadian Geotechnical Journal, 32(2): 373-379.
283
APPENDIXES
APPENDIX A
ON THE PLASTIC POTENTIAL IN EVP MODEL
This appendix introduces the definition of plastic potential, and addresses the
necessity of the plastic potential normalization in EVP theory. Lastly, the typical plastic
potentials for undrained triaxial compression tests, oedometer compression tests, and
isotropic compression tests are summarized.
Definition of plastic potential
The plastic potential defines the direction of plastic strain increments. In stress
space, the plastic potential governs the relative magnitudes of volumetric and deviatoric
incremental plastic strains. For example, in Figure Al, the plastic potential for Point B is
vertical, suggesting no volumetric incremental plastic strain at the current stress state;
whereas the plastic potential for Points Tl or T2 suggests no deviatoric incremental
plastic strain at an isotropic stress state. Point A depicts a stress state with both
volumetric and deviatoric components of strain.
Necessity of normalization on plastic potential
This section illustrates the need of the normalization on plastic potential in EVP
theory through a simple example.
As shown in Figure Al, an elliptical yield surface is assumed in this appendix
with an associated flow rule.
284
/ ' < T : - 2 C T 1 / 3 ^ 2
2 m my
v + {fir2)
2-(2<7'my/3)2 [Al]
where / is the yield surface function, which can be used to derive the plastic potential if
an associated flow rule is assumed.
Considering two isotropic stress states (see Points Tl and T2 in Figure Al), their
volumetric plastic potentials are shown as following:
df _v'm-2<j'my/3 a'm
do'm 2 6
Consequently,
[A2]
a/ TI = 1 0 JL °>6° do' ao-: IU =15 ^
Equation [A3] shows that the magnitude of plastic potential, — — , increases with
the magnitude of o'm , which is inconsistent with EVP theory in which [G114]—-— is a
a°"m
unit vector. As previously mentioned, the plastic potential defines the direction of plastic
strain according to the stress state. In this case, the stress states for both Tl and T2 are
isotropic, and consequently the plastic potentials for these two points should be same.
Thus, there is a need to normalize plastic potential in the EVP theory to address this
discrepancy.
In this appendix, the plastic potential function, —— , is normalized viz.:
285
d<rf.
do'.
a<x'
\2
+ 3/
\2 [A4]
In Equation [A4], the plastic potential is normalized as a unit vector in the o'm -
yj2J2 stress space. As a result, the normalized plastic potentials for Points Tl and T2 can
be expressed as:
do'
71
a' =60
Tl [A5]
3<r' #
3^2^
a<xl
n
a' =90
3a' T2
+ 3/
\ 2 ' < = 9 0
= 1 [A6]
dJlT2
It can be clearly seen that the normalized plastic potentials for the stress points Tl
and T2 are same and consistent with the definition of plastic potential in EVP theory.
Typical plastic potentials for standard experiments
For oedometer compression tests, the corresponding boundary condition can be
represented by: £horizontal =0 , ewrf = eatfa, , and £dev= yf2/3(eaxua-ehorizimlal )=4lT$ e ^ .
Assuming that the elastic strain at yielding is negligible compared to the plastic strain, the
axial plastic potential in oedometer test can be expressed viz.:
286
3f df
^O'axial
dv'axial de axial
oedometer
da' v
+ j
3/ 3^2^
V(j£v0/)2+feJ2 [A7]
y
For undrained triaxial compression tests, the boundary condition includes
mial . For isotropic compression, the ^horizontal = " £ axial ^ £vol = 0 > a n d £dev = ^ ^ £ ,
boundary condition includes: e .fl/ =e axial horizontal ' vol axial ' evoi - 3 £fflrifl/' a n d £*„ - 0 .
The normalized axial plastic potentials for undrained triaxial tests and isotropic
compression tests are derived respectively:
3/ . axial
de axial
undrained AdeJ^deJ j ( 0 ) ^ M ^3
d£axial
4faJ+faJ W^f
[A8]
[A9]
Tables Al and A2 summarize the boundary conditions and the normalized axial
plastic potentials for undrained triaxial compression test, oedometer compression test,
and isotropic compression test, respectively. These values are utilized in Equations from
[2.7] to [2.12] in Chapter 2.
Table Al Summarized boundary conditions in standard experiments
Isotropic Oedometer Undrained triaxial
compression compression compression
test test test
'axial axial axial 'axial
'horizontal ' axial 0 ' £ axial ' ^
^vol ~ V^ £ horizontal "*" ^ axial > 3e axial 'axial 0
^dev V r, V & axial ' ^horizontal * axial 'axial
288
Table A2 Summary of normalized plastic potentials for standard experiments
Normalized Isotropic Oedometer
plastic potential compression compression
Undrained triaxial
compression
1 V375 0
#
l/3(=0.33)
V275
(=0.82) (=0.77)
289
Figure Al Illustration of plastic potential in stress space
(V^-K/3)2
Om ,kPa
290
APPENDIX B
THE RELATIVE MAGNITUDE OF e' AND ej IN EVP MODEL y y
Typically, in an EVP model, the total strain-rate comprises two components,
elastic strain-rate and viscoplastic strain-rate. The objective of this appendix is to
evaluate the assumption that the elastic strain-rate at or after yielding is negligible
compared with viscoplastic strain-rate. A numerical simulation of compression tests was
performed to investigate the relative magnitudes of the elastic, et, and viscoplastic, ejf,
strain-rates.
A schematic diagram (Figure Bl) shows the stress path of a constant rate of strain
(CRS) isotropic compression test. The isotropic compression test is chosen because the
plastic potential along this stress path can be conveniently assumed to be unity, although
this is approximate. The assumed constitutive parameters for the numerical analysis are
listed as following: o™ =50kPa, a =0.033( n = 30 ), f =1.0 xl0~8 /min, X =0.65,
K 7 A = 0 . 1 , and the applied volumetric strain-rate,eml =1x10"^/min.
Figure B2a compares the relative magnitudes of the elastic and viscoplastic strain-
rates during the isotropic compression. At beginning of loading, the ratio of elastic strain-
rate to total strain-rate is close to unity (see the dashed line in Figure B2a), suggesting
elastic strain-rate dominates during this period. Then, elastic strain-rate abruptly
decreases as the stress approaches the apparent yield stress or isotache corresponding to
the applied strain-rate (see Figure B2b for the determination of the apparent yield stress).
Figure B2a shows that at or after yielding, the elastic strain-rate is about 4% of the total
strain-rate. Accordingly the viscoplastic strain-rate increases up to 96% of the total strain-
291
rate at or after yielding (see the solid line in Figure B2a).
A sensitivity study was done to investigate the influence of K IX. This study
found that the ratio, Kl X, has some influence on the ratio of the elastic strain-rate to the
total strain-rate, eeml to evol, at yielding. As shown in Figure B3, the ratio of £e
vollsvol at
yielding increases with the ratio of Kl X. For KIX values ranging from 5% to 30%, the
ratio of £evolleml at yielding varies from 2% to 12%. Thus, the maximum difference
between eeml andevo/ at yielding is 12%. For the typical KlX ratio of 0.1 for most soils
(Holtz and Kovacs 1981), the elastic strain-rate is within 5% of the magnitude of
viscoplastic strain-rate. Therefore, from an engineering point of view, the strain-rate at or
after yielding is approximately equal to the viscoplastic strain-rate.
The following further evaluates the assumption of neglecting the elastic strain-rate
in the context of the rate-sensitivity analysis. Figure B4 shows the log( a'p ) and log( e )
relations in terms of the strain-rate accounting for elastic strain-rate and the strain-rate
neglecting elastic strain-rate, respectively. The measured data from Batiscan clay is also
shown in Figure B4. It can be seen that the effect of neglecting ee is minor, even in the
case of KlX =0.3. It is then concluded that elastic strain-rate component can be neglected
in the rate-sensitivity analysis of clays.
Although this conclusion is derived from the isotropic compression tests, this
statement is considered applicable for oedometer compression tests and undrained triaxial
compression tests.
292
Figure B1 Stress path in CRS isotropic compression test
Dynamic yield surface „ A with increased £ "p
Static yielding surface Stress path in CRS isotropic compression
Figure B2 Comparison of the elastic strain-rate with the viscoplastic strain-rate during
CRS isotropic compression^ 115].
(a) Variation of elastic and viscoplastic strain-rates with loading stress
o '•«->
CO
CD
CD i— •
'co -I—»
CO
1.4 35 40 45 50
Effective vertical stress, kPa
55 60 65
1.2 -
1.0
.8
.6
•4 -|
.2
0.0
Visocplastic strain-rate / Total strain-rate Elastic strain-rate / Total strain-rate
Apparent yield stress
(b) Determining the apparent yield stress
0.000
,- -.005
it)
"eg -.010 o •c CD
> -.015
.020 40 50 60
Effective vertical stress, kPa, in log scale
294
Figure B3 Relationship between eevolleml and KIX in isotropic compression tests
0.00
0.04
KA,
Figure B4 Influence of neglecting elastic strain-rate in the rate-sensitivity analysis.
Neglecting elastic strain-rate (assuming KJ'X = 0.3) Neglecting elastic strain-rate (assuming K/X = 0.1) Total strain rate Measured rate-senstivity for Batiscan Clay
-10-3 1Q-2 -10-1
Strain rate, /min
296
APPENDIX C
DETERMINATION OF THE PARAMETERS, Cr,Cc, Ca
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
series of essentially parallel linear lines can be obtained in log(q) - log© space. Figure
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) London Clay (Sorensen et al. 2007)
600
Axial Strain, % (b) Rosemere Clay (Philibert 1976)
Axial Strain (%)
319
Figure E2 Stress-strain response of London clay in semi-log scale (Re-plotting Figure la in a semi-log scale)
Figure E3 The rate sensitivity parameter, a, measured from undrained triaxial compression tests on undisturbed London clay
Undr^inkttriaxia! triaxial corripression test)onjurjdisthibed Lohdon clay:
oc=0.021 at e=2.5% prerp#akj-4
apO.Q21 at e=2% pre-pedk!—
apO.026 at e=^1.5% ;pre-peak-
a^0.b2!4ate=1% pre-peak—
(k=0.023 at Peak
a=0.021 af!e=4.0% post-peaks a=0.017ate=4.5% post-peak!
10-7 10e 10-5 10"4 10- 3 10-2
Strain rate, /min, in log scale
Figure E4 The rate sensitivity parameter, a , from undrained triaxial compression tests on OC reconstituted London clay
UndraineditrWxjial compressiqn!te?t qn OC! reconstituted Londonjclay
a=0.016ate=?5%
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)
(a)
8 10 12
Vertical Strain (%)
30
20
10
i > 8 10 12
Vertical Strain (%)
(b)
CO 0.
o Q_ <n to
g 111
60 CO Q .
e o- 50
40
30
20
10
- i=0 Calculated - i=30 Calculated - i=45 Calculated
—•— i=90 Calculated
i-^
8 10 12 Vertical Strain (%)
ol 8 10 12
Vertical Strain (%)
337
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.