modelling of desiccation crack depths in clay soils€¦ · speedy pathways for water ingress,...

Modelling of Desiccation Crack Depths in Clay Soils

A thesis submitted in fulfilment of the requirement for the

Degree of Doctor of Philosophy

By

R M SASIKA D WIJESOORIYA

BSc. (Honours)

Department of Civil Engineering

Monash University

Australia

December 2012

To

Kularatne and Leela, my beloved parents

Dileepa, my loving husband

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COPYRIGHT NOTICES

1. Under the Copyright Act 1968, this thesis must be used only under the normal

conditions of scholarly fair dealing. In particular no results or conclusions should be extracted from it, nor should it be copied or closely paraphrased in whole or in part without the written consent of the author. Proper written acknowledgement should be made for any assistance obtained from this thesis.

2. I certify that I have made all reasonable efforts to secure copyright permissions

for third-party content included in this thesis and have not knowingly added copyright content to my work without the owner's permission.

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DECLARATION

I hereby declare that this thesis contains no material which has been accepted for the

award of any other degree or diploma at any university or equivalent institution and

that, to the best of my knowledge and belief, this thesis contains no material previously

published or written by another person, except where due reference is made in the text

of the thesis. Where sections of this thesis include the results of joint research or

scholarly publication clear acknowledgement of the relative contributions of the

respective authors is made.

Sasika Dilrukshi Wijesooriya

Department of Civil Engineering

Monash University

Clayton, Australia

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ABSTRACT

Desiccation cracking is a major problem in many fields. In addition to introducing

speedy pathways for water ingress, cracks can also compromise the structural integrity

of the geo structures. In this regard, prediction of the depth of cracking is an important

aspect in evaluating system performance. Modelling of desiccation cracks is a major

concern for the past few decades. Despite the numerous attempts to model the crack

depths, no comprehensive modelling method is available. In this study, an attempt has

been made to model the desiccation crack depths using analytical and numerical

approaches.

The thesis presents a review of the literature identifying the gaps of the knowledge in

this field, numerical modelling of desiccation crack depths under various conditions for

different soils and laboratory experimentation to support the numerical model. The

existing theoretical methods used to analyse desiccation crack depths and new methods

have been developed to describe the predictions from the numerical program. Further

the cyclic change of climate conditions are considered for developing the numerical

model. Finally a new approach is used to predict the crack depths in which moisture

content change was used instead of suction. Finally a more rigorous approach of

predicting crack depths incorporating cohesive properties at the crack is presented using

the computer program UDEC.

The results from the numerical approach are presented and discussed in the thesis. The

crack depths are compared with either theoretical results or a bench-mark model,

highlighting pros and cons of current approaches. More accurate crack depths agreeing

with the published field observation data can be predicted when using the moisture

content change instead of suction as a model parameter. Furthermore, cohesive

properties of the crack should be considered for fracture modelling provided that soils

are not subjected to extreme dry conditions.

However, it is recommended to conduct comprehensive field experimentation to

measure the desiccation crack depths and compare the results with the numerical

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modelling predictions of the same soil in same field conditions to draw fully validated

conclusions.

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ACKNOWLEDGEMENT

The author would like to express her sincere appreciation to Professor Jayantha

Kodikara, under whose supervision this work was done. His patience, guidance,

suggestions and freely available time were invaluable. Financial support for this study,

which was provided in form of a scholarship by the Monash University, is gratefully

acknowledged.

Special gratitude is extended to Dr. Nathan Rajeev for his help as associate supervisor,

especially in experimental work. Thanks are also extended to Dr. Aruna Amarasiri for

the help given in numerical modelling work. I greatly appreciate the assistance from

civil engineering academic staff including Prof. Abdelmalek Bouazza, A/Prof. Ranjith

Gamage, Dr. Asadul Haque, Dr. Ha Bui, Dr. Dilan Robert and Dr. Susanga Costa at

Monash University.

I would like to thank all the administrative and technical staff members at the

Department of Civil Engineering, Monash University especially to Chris Powell, Jenny

Manson, Long Goh, Mike Leach, and Alan Taylor for their precious support in my

experiments and other administrative aspects.

My deepest gratitude is given to my husband, Dileepa not only for his support,

understanding and encouragement. I offer my sincere gratitude and deep respect to my

loving parents and sisters for their caring advises encouragement and love.

I would greatly appreciate the valuable discussions, assistance and above all the

friendship from my room-mates Ben Shannon and Senthil Kumar. I would also like to

extend my sincere thanks to my fellow postgraduates of the Department of Civil

Engineering for their friendship and the wonderful time I shared with them at Monash

University.

Sasika Wijesooriya

June 2012

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TABLE OF CONTENT

DECLARATION ............................................................................................................. iv

ABSTRACT ..................................................................................................................... vi

ACKNOWLEDGEMENT ............................................................................................. viii

TABLE OF CONTENT .................................................................................................... x

LIST OF PUBLICATIONS ........................................................................................... xvi

NOMENCLATURE ..................................................................................................... xviii

ABBREVIATION ........................................................................................................... xx

LIST OF TABLES ........................................................................................................ xxii

LIST OF FIGURES....................................................................................................... xxv

Chapter 1 ........................................................................................................................... 1

INTRODUCTION............................................................................................................. 1

1.1 General ............................................................................................................... 1

1.2 The Scope of this Study ...................................................................................... 3

1.3 The Structure of the Thesis................................................................................. 4

Chapter 2 ........................................................................................................................... 8

LITERATURE REVIEW.................................................................................................. 8

2.1 Introduction ........................................................................................................ 8

2.2 Clay Liners of Landfills...................................................................................... 9

2.2.1 Evaluation of clay liner designs ................................................................ 11

2.2.2 Types of clay liners ................................................................................... 13

2.2.3 Major issues related to clay liners ............................................................. 19

2.2.4 Field Performances of clay liners .............................................................. 22

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2.3 Desiccation Cracking........................................................................................ 43

2.3.1 Initiation and evaluation ............................................................................ 48

2.3.2 Factors affecting desiccation cracking ...................................................... 51

2.3.3 Effects of cracking .................................................................................... 52

2.3.4 Theoretical developments for desiccation cracking .................................. 53

2.4 Numerical Models ............................................................................................ 71

2.4.1 Water balance modelling software ............................................................ 72

2.4.2 Fracture modelling software ..................................................................... 77

2.5 Concluding Remarks ........................................................................................ 80

Chapter 3 ......................................................................................................................... 83

COMPARISON OF CRACK MODELLING APPROACHES ...................................... 83

3.1 Introduction ...................................................................................................... 83

3.2 Existing Analytical Approaches for Predicting Crack Depth........................... 84

3.2.1 Constant suction profile ............................................................................ 84

3.2.2 Linearly decreasing suction profile ........................................................... 88

3.2.3 Parabolic suction variation ........................................................................ 89

3.3 Numerical Modelling Approach for Crack Depth Prediction .......................... 92

3.3.1 Overview of UDEC program .................................................................... 93

3.3.2 Numerical model implementation ............................................................. 94

3.4 Comparison of Numerical and Theoretical Results........................................ 102

3.4.1 Constant suction profile (suction profile 1) ............................................ 104

3.4.2 Linearly decreasing suction profile (suction profile 2) ........................... 109

3.4.3 Parabolic suction variation (suction profile 3) ........................................ 114

3.5 Comparison of Results from Different Suction Profiles ................................ 118

3.6 Comparison of Different Theoretical Approaches ......................................... 121

3.7 Conclusion ...................................................................................................... 128

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Chapter 4 ....................................................................................................................... 130

EXPERIMENTAL INVESTIGATION OF SHRINKAGE AND SWELLING BEHAVIOUR ............................................................................................................... 130

4.1 Introduction .................................................................................................... 130

4.2 Hydric constant (𝛼) in Stress Analysis ........................................................... 131

4.3 Experimental Procedure ................................................................................. 136

4.3.1 Materials .................................................................................................. 136

4.3.2 Sample preparation and set-up ................................................................ 144

4.4 Results ............................................................................................................ 156

4.4.1 Swell-Shrink cycles ................................................................................. 156

4.4.2 Swelling or shrinking paths ..................................................................... 157

4.4.3 Variation of hydric coefficient (𝛼 ∗) ....................................................... 160

4.5 Other Research on Wet-Dry Cycles ............................................................... 162

4.5.1 Experimental data analysis of Sharma (1998) ........................................ 162

4.5.2 Experimental data analysis of Romero (1999) ........................................ 170

4.5.3 Experimental data analysis of Tripathy (2000) ....................................... 176

4.5.4 Experimental data analysis of Montanez (2002) ..................................... 182

4.5.5 Experimental data analysis of Monroy (2006) ........................................ 186

4.6 Summary and Discussion ............................................................................... 194

4.7 Conclusion ...................................................................................................... 199

Chapter 5 ....................................................................................................................... 201

MODELLING OF STABLE DESICCATION CRACK DEPTHS DURING CYCLIC WETTING AND DRYING .......................................................................................... 201

5.1 Introduction .................................................................................................... 201

5.2 Suction Profiles .............................................................................................. 202

5.3 The Suction in Different Climate Conditions ................................................. 206

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5.4 Development of Numerical Model for Compacted Clay Layers under Cyclic Atmospheric Conditions ............................................................................................ 208

5.4.1 Stress change by the direct addition of suction change........................... 209

5.4.2 Fraction of suction on change of the stress ............................................. 210

5.4.3 Stress change on the basis of moisture content change........................... 212

5.5 Soils Used for the Analysis ............................................................................ 213

5.5.1 Regina clay .............................................................................................. 213

5.5.2 Horsham clay .......................................................................................... 216

5.5.3 Altona clay .............................................................................................. 218

5.6 Results ............................................................................................................ 221

5.6.1 Crack depth prediction under different climatic condition ..................... 221

5.6.2 Crack opening and closing with time ...................................................... 223

5.6.3 Effect of placement conditions of the clay liner on initial desiccation ... 226

5.6.4 Effect of the Poisson’s ratio .................................................................... 229

5.6.5 Effect of equilibrium suction .................................................................. 231

5.7 Conclusions .................................................................................................... 233

Chapter 6 ....................................................................................................................... 235

INHERENT PROPERTIES OF UDEC ........................................................................ 235

6.1 Introduction .................................................................................................... 235

6.2 The Numerical Program UDEC ..................................................................... 236

6.2.1 UDEC operation ...................................................................................... 236

6.2.2 Theoretical background of UDEC .......................................................... 237

6.3 Problem Analysing using UDEC.................................................................... 241

6.4 Behaviour of the UDEC Model ...................................................................... 244

6.4.1 Effect of mesh size (l) ............................................................................. 244

6.4.2 Effect of number of increments............................................................... 248

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6.4.3 Effect of crack length .............................................................................. 252

6.4.4 Effect of damping value .......................................................................... 254

6.4.5 Effect of block size.................................................................................. 256

6.4.6 Scaling up the model geometry ............................................................... 257

6.5 Summary and Conclusion............................................................................... 264

Chapter 7 ....................................................................................................................... 265

MODELLING OF DESICCATION CRACK DEPTHS INCORPORATING SOIL FRACTURE ENERGY ................................................................................................. 265

7.1 Introduction .................................................................................................... 265

7.2 Basics of Linear Elastic Fracture Mechanics (LEFM) ................................... 266

7.2.1 Griffith’s criterion ................................................................................... 267

7.2.2 Irwin's modification ................................................................................ 269

7.3 Past Approaches for Numerical Modelling of Fracture ................................. 273

7.3.1 Numerical modelling attempts using LEFM ........................................... 273

7.3.2 Modelling attempts with cohesive crack ................................................. 274

7.4 Modelling Crack Depths with Cohesive Properties ....................................... 276

7.4.1 Cohesive crack implementation .............................................................. 276

7.4.2 Compacted clay soils............................................................................... 279

7.4.3 Modelling Crack Depths in Soft Soils .................................................... 280

7.5 Predicting Crack Depths in Compacted Clay Soils ........................................ 285

7.5.1 Results obtained using the linearly decreasing suction profile ............... 285

7.5.2 Effects of method of implementing cohesive law ................................... 287

7.5.3 Effect of friction angle with respect to suction ....................................... 289

7.5.4 Depth of desiccation cracks with the cohesive properties ...................... 291

7.6 Predicting Crack Depths in Soft soils ............................................................. 297

7.6.1 Results for crack depth prediction using single crack ............................. 298

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7.6.2 Results for crack depth prediction using multiple cracks ....................... 299

7.7 Conclusion ...................................................................................................... 302

Chapter 8 ....................................................................................................................... 304

CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH...... 304

8.1 Conclusions .................................................................................................... 304

8.2 Future Research Recommendations ............................................................... 307

REFERENCES .............................................................................................................. 309

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LIST OF PUBLICATIONS

• Wijesooriya, S., Amarasiri, A. and Kodikara, J. A Numerical Modelling Approach

for Predicting Crack Depths in Drying Clay Soils, International Journal of

Geomechanics, (in preparation).

• Wijesooriya, S. D. and J. K. Kodikara (2012). Experimental study of shrinkage and

swelling behaviour of a compacted expansive clay soil. 11th Australia New Zealand

Conference on Geomechanics (ANZ 2012). Melbourne, Australia.

• Wijesooriya, S. and Kodikara, J. (2011) Prediction of desiccation crack depths

allowing for shear failure, Asia-Pacific conference on unsaturated soils, Thailand.

• Wijesooriya, S., Amarasiri, A. and Kodikara, J. (2011) Modelling of desiccation

crack depths using UDEC in drying clay soils, International Conference of the

International Association for Computer Methods and Advances in Geomechanics,

Melbourne.

• El Maarry, Kodikara, J., Wijesooriya, S. D. and Markiewicz, W. (2011). Numerical

modelling of a desiccation mechanism for formation of Crater Floor Polygons on

Mars and Giant Polygons on Earth: Results from a Pre-Fracture Model, Earth &

Planetary Science Letters.

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NOMENCLATURE

𝐴 Ratio of effective and total stress differences (𝛥𝜎3′/𝛥𝜎3)

𝐴𝑐 Cross sectional area 𝐴𝑠 Surface area 𝑎 Crack length 𝑎𝑣 Matric suction compression

index 𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝐶5 Constants Cc Compression index Cs Swelling/recompression index 𝑐 Apparent cohesion, bc φtan= 𝑐′ Effective cohesion intercept 𝑐𝑢 Undrained cohesion 𝐷𝑖𝑗𝑘𝑙 Tangent stiffness tensor 𝑑 Depth of the layer 𝐸 Young’s modulus with respect

to total stress relative to pore air pressure

𝐸𝑠 Young’s modulus at saturated level

𝐸𝑇 Evapotranspiration 𝑒 Void ratio 𝑒0 Initial void ratio 𝑒𝑎 Vapour pressure of the air in

the atmosphere above the water surface

𝑒𝑠 Saturation vapour pressure of water at the temperature of the surface

𝑒𝑤 Moisture ratio 𝐹𝑖 Resultant external force 𝑓(𝑢) Turbulent exchange function 𝐺 Shear modulus 𝐺𝑓 Fracture energy

𝐺𝐼𝐶 Rate of release of strain energy with critical crack extension

𝐺𝑠 Specific gravity 𝑔𝑖 Gravitational acceleration 𝐻 Modulus with respect to matric

suction 𝐻𝑑 Depth to permanent water table

from surface Hi Initial height Hs Depth of seasonal changes in

water content from surface 𝐾 Bulk modulus 𝐾𝑎𝑝𝑝 Apparent fracture toughness 𝐾𝐼 Stress intensity factor 𝐾𝐼𝐶 Critical stress intensity factor /

Fracture toughness k Interface shear stiffness 𝑘𝑜 Earth pressure coefficient at

rest 𝑘𝑛 Normal stiffness 𝑘𝑠 Shear stiffness 𝐿 Length of the layer l Size of the mesh M Slope of failure envelope in p, q

space 𝑛 Frequency of seasonal variations in cycles/year 𝑛𝑗 Unit normal to 𝐴𝑠 𝑃 Horizontal stresses at the tip of

the crack

𝑝 Mean stress, (𝜎1 + 2𝜎3)3�

𝑝𝑛𝑒𝑡 Mean stress corrected for air pressure, (𝑝 − 𝑢𝑎)

𝑄 Horizontal stresses at the top surface of the crack

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q Deviator stress, (σ1 - σ3) 𝑅 Rainfall 𝑅𝐸 Net recharge from the

surrounding soil 𝑅𝐸𝑇 Rate of evaporation 𝑅𝑂 Runoff 𝑟 Distance from the crack tip 𝑆 Suction, (𝑢𝑎 − 𝑢𝑤) 𝑆0 Suction at the surface 𝑆𝑒 Equilibrium suction 𝑆𝑠 Gradient of the SWCC 𝑠 Spacing between cracks 𝑡 Time 𝑈 Stored energy 𝑢 Displacement 𝑢𝑎 Pore air pressure 𝑢𝑤 Pore water pressure 𝑢𝚤̈ Acceleration �̇� Velocity 𝑊 Depth to water table / Depth of

the soil layer 𝑤 Moisture content 𝑧 Depth measured from surface 𝑧𝑐 Depth of cracking 𝑧𝑡 Depth of tension crack 𝑧𝑖 Initial depth of cracking 𝛼 Hydric constant 𝛼∗ Hydric coefficient 𝛼𝑇 Coefficient relating cohesion to

tensile strength, 𝛼𝑇 = 0.5 tan ϕ ′ 𝛼𝑑𝑐 Diffusion coefficient 𝛽 Given by �𝑘 𝐸𝐴𝑐⁄ 𝛾 Unit weight

dγ Dry unit weight ∆ Change of the parameter 𝛿𝑘𝑙 Kronecker delta 𝜀 Strain 𝜀𝑘𝑙 Observed strain 𝜀𝑠ℎ Shrinkage strain

shvε Volumetric shrinkage strain 𝜁 Specific surface energy

𝜃 Angle to the failure surface κ Compressibility parameter for

unload / reload condition λ compressibility parameter 𝜈 Poisson’s ratio 𝜇 Lamé constant 𝜉 Lamé constant 𝜙 Angle of shearing resistance 𝜙𝑏 Angle of shearing resistance

with respect to total stress relative to pore air pressure

𝜙′ Effective friction angle 𝜎 Normal stress 𝜎′ Effective stress 𝜎1, 𝜎3 Major principal stress, Minor

principal stress 𝜎𝑐 Stress required to opening the

crack 𝜎𝑓 Failure stress 𝜎𝑖𝑗 Stress tensor 𝜎𝑡 Tensile strength 𝜏 Shear stress 𝜏𝑓 Shear strength 𝜒 Fraction to modify the suction

contribution towards the shear stress

𝜓 Parameter given by 𝑎 W� Subscripts: x, y, z horizontal, vertical, horizontal

coordinate directions 1, 3 major, minor a, w air, water 0 surface max maximum min minimum

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ABBREVIATION

AE Actual rate of evaporation

CCL Compacted clay layer

EPFM Elasto plastic fracture mechanics

FEM Finite element modelling

FPZ Fracture process zone

GCL Geo-synthetic clay liner

GM Geo-membrane

LCRS Leachate collection and removal system

LE Linear Elastic

LEFM Linear Elastic Fracture Mechanics

LL Liquid Limit

PE Potential rate of evaporation

PI Plasticity Index

PL Plastic Limit

SF Cracking with shear failure method

SWCC Soil-Water characteristic curve

TMI Thornthwaite Moisture Index

UDEC Universal Distinct Element Code

USCS Unified soil classification system

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LIST OF TABLES

Table 2-1 Summary of Desiccation crack depths recorded ............................................ 46

Table 2-2 Variation of tensile strength with test method ................................................ 54

Table 2-3 Summary of research used numerical models ................................................ 73

Table 2-4 Comparison of several programs .................................................................... 76

Table 4-1 Mineralogy content of Altona clay ............................................................... 143

Table 4-2 Summary of the soil classification test results .............................................. 144

Table 4-3 Properties of the soil ..................................................................................... 162

Table 4-4 Summary of tests by Sharma ........................................................................ 164

Table 4-5 Summary of soil parameters ......................................................................... 170

Table 4-6 Summary of tests by Romero ....................................................................... 173

Table 4-7 Soil Properties ............................................................................................... 176

Table 4-8 Summary hydric coefficients of tests by Tripathy ........................................ 178

Table 4-9 Soil Parameters for sand bentonite mixes ..................................................... 182

Table 4-10 Summary of the tests by Montanez ............................................................ 184

Table 4-11 Properties of London Clay .......................................................................... 187

Table 4-12 Summary of tests by Monroy ..................................................................... 189

Table 4-13 Summary of hydric coefficient values ........................................................ 197

Table 5-1 Soil parameters for Regina clay .................................................................... 214

Table 5-2 Soil properties of Horsham clay ................................................................... 217

Table 5-3 Soil properties of Altona clay ....................................................................... 219

Table 6-1 Typical input parameters of the model ......................................................... 243

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LIST OF FIGURES

Figure 2-1 Cross section of a landfill (http://www.groundwateruk.org/Image-Gallery.aspx) .............................................................................................. 10

Figure 2-2 Layers of base (bottom) lining systems ......................................................... 14

Figure 2-3 Acceptable zone based on design objectives for hydraulic conductivity, volumetric shrinkage, and unconfined compressive strength (Daniel and Wu 1993) ............................................................................................. 15

Figure 2-4 Different types of cover layers (Dwyer, 1998; Albright et al., 2004) ........... 16

Figure 2-5 Variation of moisture content with depth, Curve 1, equilibrium water content under a surface covering; curve 2, dry season; curve 3, wet season (after Kraynski, (Chen 1988)) ........................................................ 24

Figure 2-6 Effect of drying and rewetting on the ultimate moisture content of Leda clay (after warkentin (1961)) ..................................................................... 25

Figure 2-7 Typical seasonal soil suction variations with depth (Morris et al., 1992) ..... 26

Figure 2-8 Soil water characteristic curves (Fredlund and Anqing 1994) ...................... 27

Figure 2-9 Schematic Profile View of a Typical Hazardous Waste Landfill ................. 29

Figure 2-10 The relationship between the rate of actual evaporation and potential evaporation (i.e., AE/PE) and water availability (Wilson, Fredlund et al. 1994) ..................................................................................................... 34

Figure 2-11 Leachate generation rates at a modern domestic landfill in Pennsylvania (USA), Average annual precipitation at the landfill site is 1.0 m/year (after Bonaparte 1995, (Bouazza and Van Impe 1998)) ...... 36

Figure 2-12 Daily precipitation and resulting drainage rate for periods (a) before and after (b) the fall 2000 drought at Albany, GA. Daily precipitation is shown as vertical bars, drainage rate as a continuous line (Albright, Benson et al. 2006). ................................................................................... 39

Figure 2-13 Time-movement plots of several points inside and outside the National Art Gallery building, (Richards, Peter et al. 1983) .................................... 40

Figure 2-14 Variation of vertical deformation with several wet-dry cycles, (Tripathy, Subba Rao et al. 2002) .............................................................. 41

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Figure 2-15 Shrinkage phases of clay upon drying (Kodikara et al., 1999) ................... 42

Figure 2-16 Theoretical patterns of desiccation cracks (a) parallel (b) square (c) hexagonal (after bezant and Cedolin, 1991, (Kodikara, Barbour et al. 2000)) ......................................................................................................... 43

Figure 2-17 Parallel cracks observed in long thin moulds.(Costa, Kodikara et al. 2008) .......................................................................................................... 43

Figure 2-18 Irregular shaped crack pattern observed on a playa surface, southern Nevada, Hammer and handle measured 330mm (Longwell 1928). Playa surface means a flood plain clay surface or dried-out lake surface. ....................................................................................................... 44

Figure 2-19 Orthogonal Cracking pattern observed in coal mine tailings, Queensland, Australia (Morris, Graham et al. 1992) ................................. 44

Figure 2-20 Hexagonal mud crack pattern observed on playa surface, Las Vegas quadrangle, Nevada. (Longwell 1928) ...................................................... 45

Figure 2-21 Vertical crack pattern of a compacted clay layer with 3 lifts (Yesiller et al. 2000) ..................................................................................................... 45

Figure 2-22 Schematic illustration of cracking (Konrad and Ayad 1997a) .................... 49

Figure 2-23 Influence of dry densities on σt of the clay with different water contents (Wang, Zhu et al. 2007) ............................................................... 55

Figure 2-24 Influence of water contents on σt of the clay with different dry densities (Wang, Zhu et al. 2007) .............................................................. 56

Figure 2-25 Details of tensile strength with constant natural density, (c) tensile strength with moisture content with polynomial fit showing the trend (d) tensile strength with degree of saturation (Lakshmikantha, 2009) ...... 56

Figure 2-26 Mohr-Coulomb failure criterion .................................................................. 60

Figure 2-27 Strength envelopes indicating tensile strength after Lee and Ingles in 1968 (Morris, Graham et al. 1992) ............................................................ 60

Figure 2-28 Variation of stress intensity factor with crack depth (Morris, Graham et al. 1994) ..................................................................................................... 65

Figure 2-29 Stress analysis for desiccation cracking (Harison and Hardin, 1994) ......... 67

Figure 2-30 Schematic representation of the proposed model by Kodikara and Choi (2006 (a)) ................................................................................................... 69

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Figure 2-31 Schematic diagrams for the (a) stress relief approach (b) energy balance approach (Costa 2009) .................................................................. 70

Figure 2-32 Flow chart of the proposed model by Konrad and Ayad (1997(a)) ............ 78

Figure 3-1 Suction profile (a) and tensile strength profile (b) with depth ...................... 85

Figure 3-2 Depths of cracking with different Poisson’s ratios tensile strength values when the constant suction profile is assumed throughout the depth .......... 87

Figure 3-3 Suction and tensile strength profiles when linearly decreasing with depth .. 88

Figure 3-4 Predicted crack depth values for linearly decreasing suction profile through linear elastic (LE) approach and allowing for shear failure (SF) approach with different surface suction values.................................. 89

Figure 3-5 Parabolic suction and tensile strength profiles .............................................. 90

Figure 3-6 Predicted Crack depths when the water table is 4m below the surface (LE-linear elastic approach, SF-elastic allowing for shear failure and LEFM-linear elastic fracture mechanics approach) ................................... 91

Figure 3-7 Predicted Crack depths using parabolic suction profile when the surface suction is 50kPa ......................................................................................... 92

Figure 3-8 Cross section of the clay layer used for the model ........................................ 95

Figure 3-9 Variation of depths of cracking with layer width when E =5MPa, ν =0.3, σt=0kPa and S0=50kPa ............................................................................. 96

Figure 3-10 Zones and blocks of the model .................................................................... 97

Figure 3-11 Variation of crack depths with mesh size when E =5MPa, ν =0.3, σt=0kPa and S0=50kPa ............................................................................. 98

Figure 3-12 Stress conditions in a soil particle at each increment ................................ 100

Figure 3-13 Effect of the Poisson’s ratio on crack depth when E =5MPa, σt =0.5 S tanϕb, W =4m and S0=50kPa ........................................................ 103

Figure 3-14 Crack depths changing with the depth to water table when Es=5MPa, E = Es + 10S , ν =0.3, σt = −αTS tanϕbcotϕ′ , W =4m, S0 =50kPa and ϕ =300 ............................................................................................... 104

Figure 3-15 Appling suction with time ......................................................................... 105

Figure 3-16 Crack of a model after opening ................................................................. 106

xxvii

Figure 3-17 Variation of crack depth with surface suction when E =5MPa, ν =0.3, W =10m, σt=constant and S0=50kPa ...................................................... 107

Figure 3-18 Variation of crack depth with surface suction when E =5MPa, ν =0.3, W =4m σt=constant and S0=50kPa ......................................................... 108

Figure 3-19 Changing the suction in UDEC ................................................................. 109

Figure 3-20 Variation of crack depth with depth to water table when E =5MPa, ν =0.3, σt = 0.5 S tanϕb ............................................................................ 110

Figure 3-21 Variation of crack depth with the surface suction when when E =5MPa, ν =0.3, σt = 0.5 S tanϕb and W =4m ..................................................... 111

Figure 3-22 (a) Strength envelope indicating tensile failure showing the effect of tensile strength, (b) Schematic of failure envelope in tension ................. 112

Figure 3-23 Variation of depth of cracking with depth to water table when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 ........... 113

Figure 3-24 Variation of crack depth with surface suction when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, W =4m and ϕ =300 ......................................... 114

Figure 3-25 Applying suction in increments for parabolic variation ............................ 114

Figure 3-26 Effect of surface suction on depth of cracking predictions for parabolic suction variation; when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′ , W =4m and ϕ =300 .................................................................................. 115

Figure 3-27 Effect of depth to water table on depth of cracking predictions for parabolic suction variation; when E =5MPa, ν =0.3, σt =−αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 ............................................. 116

Figure 3-28 Depth of cracking variation with surface suction for parabolic suction profile when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, W=4m and ϕ =300 ...................................................................................................... 116

Figure 3-29 Depth of cracking variation with water table depth for parabolic suction profile when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′ , S0=50kPa and ϕ =300 ............................................................................................... 117

Figure 3-30 Behaviour of crack depth predictions with surface suction based on the suction profile; when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′ , W=4m and ϕ =300 .................................................................................. 118

xxviii

Figure 3-31 Behaviour of crack depth predictions with depth to water table based on the suction profile; when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 .................................... 119

Figure 3-32 Behaviour of crack depth predictions with surface suction based on suction profile allowing for shear failure; when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, S0=50kPa and ϕ=300 ...................................... 120

Figure 3-33 Behaviour of crack depth predictions with surface suction based on the suction profile; when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′ , W=4m and ϕ =300 .................................................................................. 120

Figure 3-34 Comparison of results obtained through different theoretical assumptions with depth to water table variation when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 (SS – allowing for shear failure, LE – Linear elastic) ...................................................... 121

Figure 3-35 Comparison of results obtained through different theoretical assumptions with surface suction variation, when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 (SS – allowing for shear failure, LE – Linear elastic) ............................................................ 122

Figure 3-36 Comparison of results obtained through theoretical assumptions with depth to water table variation when E =5MPa, ν =0.3, σt =−αTS tanϕbcotϕ′, S0=50kPa and ϕ =300 (SS – allowing for shear failure, LE – Linear elastic) ..................................................................... 123

Figure 3-37 Comparison of results obtained through different theoretical assumptions with surface suction variation when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′ , S0=50kPa and ϕ =300 (SS – allowing for shear failure, LE – Linear elastic) ............................................................ 124

Figure 3-38 Comparison of values of crack depth with surface suction variation when E =5MPa, ν =0.3, σt = −αTS tanϕbcotϕ′, W=4m and ϕ =300 . 126

Figure 3-39 Design curve; zcW vs. S0γW curve when ϕ =300, ν =0.3 for suction profile given in Figure 3-19 ..................................................................... 128

Figure 4-1 Particle size distribution of Altona clay ...................................................... 138

Figure 4-2 Automatic soil compactor designed for proctor and CBR Compaction and removing sample after modified compaction test ............................. 140

Figure 4-3 Compaction curves for Altona clay ............................................................. 141

Figure 4-4 Mechanical soil mixer ................................................................................. 142

xxix

Figure 4-5 LoadTrac III consolidation machine ........................................................... 142

Figure 4-6 Compression curve for Altona clay ............................................................. 143

Figure 4-7 Sample preparation using method one ........................................................ 147

Figure 4-8 Sample preparation using method two ........................................................ 148

Figure 4-9 Values for unit weight and moisture content obtained as initial conditions. * - UW stands for Unit Weight, ** - MC stands for Moisture Content ..................................................................................... 149

Figure 4-10 Monash designed Oedometer .................................................................... 150

Figure 4-11 Schematic diagram of test set-up and placement of soil sample ............... 151

Figure 4-12 (a) Saturating the sample for swelling, (b) Wetting the sample to get the swell before saturation ....................................................................... 152

Figure 4-13 Vertical displacement of three samples compacted to similar initial conditions when subjected to wetting. ..................................................... 153

Figure 4-14 Vertical displacement of three samples compacted to similar initial conditions when subjected to drying after full swelling .......................... 154

Figure 4-15 Photos of the sample at different stages .................................................... 155

Figure 4-16 Marginal vertical displacements during wet-dry cycles for several samples ..................................................................................................... 156

Figure 4-17 Typical average vertical displacements for Altona clay ........................... 157

Figure 4-18 Swelling and shrinking paths for the first cycle ........................................ 158

Figure 4-19 Swelling and shrinking paths for the second cycle ................................... 159

Figure 4-20 Swelling and shrinking paths for the third cycle ....................................... 159

Figure 4-21 Swelling and shrinking paths for the fourth cycle ..................................... 160

Figure 4-22 Variation of α ∗ with number of cycles ..................................................... 161

Figure 4-23 Variation of α ∗ with moisture ratio .......................................................... 161

Figure 4-24 Compaction curves and the initial positions of samples ........................... 163

Figure 4-25 Swelling and shrinking curves for samples under 10kPa vertical stress, Test 4 – initially compacted under 800kPa. Test19 & 20 – initially

xxx

compacted under 3200kPa. Other tests – initially compacted under 400kPa ..................................................................................................... 167

Figure 4-26 Swelling and shrinking curves for samples under 20kPa (Test 5) and 50kPa (Test 2) vertical stress, Test 3 – initially compacted under 800kPa. Test 2 & 5 – initially compacted under 400kPa ......................... 167

Figure 4-27 Variation of α ∗ with wet-dry cycles ......................................................... 168

Figure 4-28 Variation of α ∗ with degree of saturation in each wetting or drying process ..................................................................................................... 169

Figure 4-29 Compaction curves for Boom clay ............................................................ 171

Figure 4-30 Swelling shrinking curves tested under different pressures ...................... 172

Figure 4-31 Variation of α ∗ with number of cycles ..................................................... 175

Figure 4-32 Standard Proctor curves for Soil A and Soil B .......................................... 177

Figure 4-33 Swelling and shrinking curves for several wet-dry cycles ........................ 179

Figure 4-34 Hydric coefficient change with wetting and drying for the Soil A with 6.25kPa .................................................................................................... 180

Figure 4-35 Hydric coefficient variation during several wet-dry cycles ...................... 180

Figure 4-36 Summary of variation of hydric coefficient in each wetting or drying process ..................................................................................................... 181

Figure 4-37 Compaction curves for samples with different compositions; WG – Well Graded U – Uniform and B – Bentonite ......................................... 183

Figure 4-38 Behaviour of sand samples with different initial conditions under wetting and drying ................................................................................... 185

Figure 4-39 Hydric coefficient values for wet-dry processes ....................................... 186

Figure 4-40 Compaction characteristics of London clay from standard compaction test ............................................................................................................ 188

Figure 4-41 Variation of void ratio with moisture ratio during wetting and drying ..... 193

Figure 4-42 Variation of α ∗ values with different wetting and drying paths ............... 194

Figure 4-43 Moisture content variation in a suction controlled test ............................. 195

Figure 4-44 Typical paths of expansive soils subjecting to partial wet dry cycles ....... 198

xxxi

Figure 4-45 Typical variations of α ∗ with full wet dry cycles ..................................... 200

Figure 5-1 Effect of various environmental conditions on the matrix suction profile (Peter 1979) .............................................................................................. 203

Figure 5-2 Values of suction from road site installations and postulated design curves after Richards, B.G. (1985) .......................................................... 204

Figure 5-3 Typical Suction Profile for arid and semi-arid conditions used in the present study ............................................................................................ 208

Figure 5-4 Predicted crack depths using different stress changing approaches for Regina clay soil ........................................................................................ 210

Figure 5-5 SWCC for Regina clay after (Vu, Hu et al. 2007) ...................................... 215

Figure 5-6 Void ratio vs. water content graph for Regina clay ..................................... 216

Figure 5-7 SWCC for Horsham clay (after Richards, 1985) ........................................ 218

Figure 5-8 Void ratio vs. water content curve for Horsham clay (after Richards, 1985) ........................................................................................................ 218

Figure 5-9 SWCC for Altona clay (after Chan, 2012) .................................................. 220

Figure 5-10 Void ratio vs. water content curve for Altona clay ................................... 220

Figure 5-11 Suction profiles under different climatic conditions ................................. 221

Figure 5-12 Predicted depth of cracking change with climatic conditions ................... 222

Figure 5-13 Suction profile variation due to seasonal climate change in (a) Arid area (b) Semi-arid area ............................................................................. 223

Figure 5-14 Predicted depth of cracking with seasonal change in an arid area ............ 224

Figure 5-15 Predicted depth of cracking with seasonal change in an semi-arid area ... 225

Figure 5-16 Predicted crack depth variation with initial density of the layer in an arid climate .............................................................................................. 227

Figure 5-17 Predicted crack depth variation with initial density of the layer in an semi-arid climate ...................................................................................... 228

Figure 5-18 Predicted depth of cracking variation with the Poisson's ratio in an arid area ........................................................................................................... 229

Figure 5-19 Predicted depth of cracking variation with the Poisson's ratio in an semi-arid area ........................................................................................... 230

xxxii

Figure 5-20 Suction profiles below the ground surface with different equilibrium suction values in (a) an arid climate (b) a semi-arid climate ................... 231

Figure 5-21 Predicted depth of cracking with the change of equilibrium suctions in an arid climate .......................................................................................... 232

Figure 5-22 Predicted depth of cracking with the change of equilibrium suctions in an semi-arid climate ................................................................................. 233

Figure 6-1 Sign convention for positive shear stress components ................................ 237

Figure 6-2 Actions performed during one computation cycle ...................................... 238

Figure 6-3 Area associated to grid point P .................................................................... 239

Figure 6-4 (a) Problem geometry and modelled problem (b) analysed problem due to symmetry ............................................................................................. 242

Figure 6-5 Change of mesh size in a constant size block ............................................. 245

Figure 6-6 Selection of Failure stress............................................................................ 246

Figure 6-7 Effect of mesh size; K = 5e9 Pa, G = 2e9 Pa, kn = 5e9 Pam, Damp = 0.2, N = 10000, W = 0.6m a = 0.2 m and σt = 1000Pa. .......................... 247

Figure 6-8 Stress vs. displacement curve using to get Overall E .................................. 250

Figure 6-9 Effect of number of stress increments; K = 5e9Pa, G = 2e9Pa, kn=5e9Pa, Damp=0.2, W =0.6m, σt=1000Pa, a =0.02m and l =0.2m ...................... 251

Figure 6-10 Change of load vs. displacement plot with the change of number of cycles ....................................................................................................... 252

Figure 6-11 Model geometry change with changing crack length ................................ 253

Figure 6-12 Effect of crack length, K = 5e9Pa G = 2e9Pa kn=5e9Pam, Damp=0.2, N=10000, W =1.2m, σt=1000Pa, l =0.04m and KIC=530Pam0.5 ............ 254

Figure 6-13 Effect of damping value, K = 5e9Pa, G = 2e9Pa, kn =5e9Pam, N=10000, W =0.6m, σt=1000Pa and al =10 (a =0.02m, l =0.2m) .......... 255

Figure 6-14 Stress vs. displacement curves when changing the damping value .......... 255

Figure 6-15 Change of Size of block when the crack length kept constant .................. 256

Figure 6-16 Normal Stresses from UDEC and LEFM ahead of the crack.................... 257

Figure 6-18 Scale effect on failure stress, K = 5e9Pa, G = 2e9Pa, Damp=0.2, N=10000 and σt=1000Pa ......................................................................... 258

xxxiii

Figure 6-17 Plot of models changing the size of geometry .......................................... 258

Figure 6-19 Inherent fracture energy present in current UDEC formulation................ 259

Figure 6-20 Scale effect on Fracture energy, K = 5e9Pa, G = 2e9Pa, Damp=0.2, N=10000 and σt=1000Pa ......................................................................... 260

Figure 6-21 Effect of normal stiffness on failure stress, K = 5e9Pa, G = 2e9Pa, Damp=0.2, N=10000 and σt=1000Pa ...................................................... 261

Figure 6-22 Effect of normal stiffness on fracture toughness, K = 5e9 Pa, G = 2e9 Pa, Damp=0.2, N=10000 and σt=1000Pa ................................................ 261

Figure 6-23 Effect of modulus on failure stress kn=5e9Pam, Damp=0.2, N=10000 and σt=1000Pa ......................................................................................... 263

Figure 6-24 Effect of modulus on failure stress, kn=5e9 Pam, Damp=0.2, N=10000 and σt=1000Pa ......................................................................................... 263

Figure 7-1 Semi infinite plate with the central crack of the length 2a and the directions for near tip stress field ............................................................. 269

Figure 7-2 Actual incremental fracture process in Load-Displacement space ............. 270

Figure 7-3 Distribution of the stress normal to the crack plane (Wang, 1996) ............. 272

Figure 7-4 Bridging stresses at the crack tip while crack opening considered in the cohesive crack models ............................................................................. 275

Figure 7-5 Softening curves .......................................................................................... 277

Figure 7-6 Linear softening law used in the numerical model ...................................... 278

Figure 7-7 Moisture content profiles of Saint-Alban clay test (Konrad and Ayad, 1997a) ...................................................................................................... 281

Figure 7-8 SWCC for Saint-Alban clay and the empirical equations for different moisture content ranges ........................................................................... 282

Figure 7-9 Depth of cracking variation for cohesive model with surface suction when E =5MPa, ν =0.3, σt = −αTStanϕbcotϕ′, W =4m and ϕ =300 ... 286

Figure 7-10 Depth of cracking variation for cohesive model with depth to water table when E =5MPa, ν =0.3, σt = −αTStanϕbcotϕ′, S0=50kPa and ϕ =300 ...................................................................................................... 286

Figure 7-11 Different methods of implementing cohesive law (a) Softening Law change-1 (b) Softening Law change-2 (c) Softening Law change-3 ....... 287

xxxiv

Figure 7-12 Depth of cracking for different implementations of softening law with the change of surface suction ................................................................... 288

Figure 7-13 Effect of friction angle with respect to suction on desiccation cracks ...... 290

Figure 7-14 Depth of cracking for Regina clay using different softening laws with changing surface suction .......................................................................... 292

Figure 7-15 Depth of cracking for Horsham clay using different softening laws with changing surface suction .......................................................................... 293

Figure 7-16 Depth of cracking for Altona clay using different softening laws with changing surface suction .......................................................................... 294

Figure 7-17 Fracture energies with different softening laws ........................................ 295

Figure 7-18 Effect of fracture energy on desiccation cracking in terms of residual limit .......................................................................................................... 296

Figure 7-19 Effect of fracture energy on desiccation cracking ..................................... 297

Figure 7-20 Crack depth with time ............................................................................... 298

Figure 7-21 Block with displacement and joints........................................................... 299

Figure 7-22 Block with opened cracks and displacement vectors ................................ 300

Figure 7-23 Progression of multiple cracks with time .................................................. 301

xxxv

Chapter 1

INTRODUCTION

1.1 General

Desiccation takes place from a soil surface when it is exposed to the atmosphere. The

degree of desiccation depends on soil properties, the climate condition and the depth to

the water table mainly. When desiccation progresses, especially during the dry seasons,

water escapes from the surface and soil water suctions develop making soils shrink.

While the soil is free to shrink vertically, it is normally laterally constrained against

shrinkage. This leads to formation of desiccation cracks at the upper portion of the soil

surface.

Desiccation cracking can induce severe problems in agricultural, geotechnical, and

environmental applications. In agricultural engineering, cracks can influence the

undesirable water and solute flow through soil in irrigated lands. In geotechnical

engineering, the presence of tension cracks due to desiccation may also badly influence

the stability of natural slopes (Baker and Leshchinsky, 2003; Zhan et al., 2007) and

1

2

vertical cuttings, as well as the bearing capacity of foundations (Silvestri et al., 1990). In

addition to that, they may lead to pipe settlements and breakages and dam failures

(Vasil'ev et al., 1988). In the field of environmental engineering, desiccation cracking

has the potential to render a low conductivity barrier constructed of a clay soil layer

ineffective within a short period of time (Melchior, 1997; Khire et al., 1999; Albright et

al., 2006). Deep desiccation cracks have been observed in arid areas especially in

expansive clay soils and in coal mine tailings deposits in Queensland, Australia (Morris

et al., 1992) with typical depths of about 1 to 1.5 m and, rarely, with a maximum depth

of about 4 m.

Due to these adverse effects, desiccation cracking has become a concern in geotechnical

and geo-environmental engineering, especially in design and construction of landfill

covers in arid regions. Significant amount of research (e.g., Konrad and Ayad, 1997b;

Philip et al., 2002; Kodikara and Choi, 2006; Tang et al., 2008) has been conducted for

a long period extending over decades on various aspects of this problem. However,

there are still further advancements to be made, in particular in the area of prediction of

depths of desiccation cracks in a given climate.

Soils which normally contain significant amount of clay with minerals of high

shrink/swell potential, can undergo large volume changes due to moisture content

variations. These types of soils are often called ‘expansive or reactive soils’. Expansive

soils cause distress to light-weight (e.g., foundations of light buildings) or surficial

structures (e.g., road pavements, pipelines) making considerable damage annually. As a

damage mitigative measure, vertical moisture barriers are used commonly, particularly

in the case of road pavements (Picornell and Lytton, 1987; Jayatilaka and Lytton, 1997;

Introduction 3

Aubeny and Long, 2007). However, knowledge of the maximum crack depth is valuable

to design damage mitigative measures such as moisture barriers. Furthermore, the depth

of desiccation cracks that are likely at the crest and down the slope are important

considerations in assessing slope stability (e.g., Dyer et al., 2009).

In compacted soil barriers, the infiltration rate depends on the configuration and the

depth of the fracture. On the basis of a numerical study on the infiltration of water

through a fractured clay soil, Moore and Ali (1982) remarked that the depth of cracking

played a significant role, whereas the cracking frequency was less important for overall

infiltration. However, the depth and the frequency of cracking should be considered

together to determine the overall infiltration. The worst condition would occur when

the depth of cracking and frequency are both very high.

Many analytical and numerical models have been used to model the desiccation crack

depths (Morris et al., 1992; Konrad and Ayad, 1997b; Amarasiri and Kodikara, 2011c).

In crack development in field conditions, mostly linear elastic stress analysis or linear

elastic fracture mechanics is used. Only recently, Kodikara and co-workers have

introduced methods to incorporate significant non-linearity present in fracture process.

Therefore, there is significant scope for undertaking research on crack depth prediction

under field conditions.

1.2 The Scope of this Study

The primary purpose of this study is to develop a reliable numerical modelling

technique to simulate the development of desiccation cracking in clay soils with

4

particular emphasis on compacted clay liners. Hence, this research focuses on the

phenomenon of desiccation cracking. Numerical model predictions are compared with

empirical solutions and experimental observations and the critical parameters involved

in the prediction of depth of desiccation cracking. The basic research aims can be listed

as:

i. To predict the level of desiccation cracking in a clay layer subject to certain

climate conditions;

ii. To find the range of moisture and suction change causing the cracks and their

effect on depth of cracking in compacted clay layers;

iii. To identify and obtain the value ranges of critical parameters that affect the depth

of cracking using laboratory experiments, available literature and numerical

analysis;

iv. To propose an analytical model to predict the depth of cracking; and

v. To evaluate the effects of different soil parameters on the prediction of crack

depths.

To achieve these aims, the study presented in this thesis was conducted comprising a

literature review, numerical analysis, analytical development and laboratory

experiments to support the numerical approach proposed.

1.3 The Structure of the Thesis

This thesis is divided into eight chapters, a bibliography and a list of publications

produced from the work described in this thesis.

Introduction 5

Chapter 1: Introduction

This chapter introduces the problem; “Desiccation cracking in clay liners” generally,

and explains the basic points of the thesis and points to the directions of the research. It

will also give a guide for the thesis structure.

Chapter 2: Literature Review

This chapter presents a wide review of the past research which is relevant to the

desiccation cracking in clay liners to identify the seminal work done and the gaps in the

literature.

The existing knowledge on desiccation problem of landfill liners and modelling

techniques developed so far is summarised and discussed. Fundamentals of the soil

behaviour are reviewed with emphasis on mechanics of unsaturated soils. Finally, the

gaps in the state of the art are presented.

Chapter 3: Comparison of crack modelling approaches

The crack depths are calculated analytically and numerically through different

theoretical assumptions such as:

• Linear elastic approach;

• Shear failure approach;

• Linear elasticity fracture mechanics approach; and

• Incorporation of fracture energy using cohesive crack modelling.

For numerical modelling, the UDEC computer program is selected. The results are

obtained by changing the depth to water table from the surface and the suction within

the block. These results are compared and discussed with analytical results.

6

Chapter 4: Experimental Investigation of shrinkage and swelling behaviour

The behaviour of the soil layer with the moisture content is necessary to be observed to

predict the accurate suction profile. Hence, the laboratory tests are mainly focused on

the shrinking and swelling behaviour to represent the field soil behaviour when

subjected to climatic wet dry cycles.

Through these tests the input parameters required for the progress of the research are

obtained. The main difficulty of these tests is measuring the soil volumes accurately

with the presence of cracks.

Chapter 5: Modelling of stable desiccation crack depths during cyclic wetting and drying

In this chapter, the stable crack depths are modelled for compacted clay liners by

considering the cyclic effect of wet and dry cycles. Several clay soils, namely Regina,

Horsham and Altona, from different parts of Australia and from some other countries,

are used in the analysis. The evaluation of crack depth in different climate conditions

and initial placement conditions of the clay layer is considered.

Chapter 6: Inherent properties of UDEC for fracture modelling

This chapter is used to understand the inherent properties of UDEC for fracture

modelling. First, the fracture modelling approach of UDEC analysis is explained. The

effect of different input parameters is then analysed and discussed on the basis of

generated results.

Introduction 7

Chapter7: Modelling of desiccation crack depths incorporating soil fracture energy

In this chapter, the stress reduction while opening a crack is included in the code using

FISH, the inbuilt programming language of the numerical program UDEC. The values

of the softening curve are calculated on the basis of the fracture energy of the problem

geometry. Then the crack depths are predicted for the same soils used in Chapter 5. The

effect of cohesive properties of the fracture during its initiation and development is

studied in this chapter.

Chapter 9: Conclusions and Future recommendations

Conclusions derived from this study and recommendations for future research are

presented in this chapter.

Chapter 2

LITERATURE REVIEW

2.1 Introduction

Issues related to desiccation have been studied first qualitatively and then quantitatively

since the 1950s or so. The use of landfill liners received increasing attention with the

advent of geo-environmental engineering in the 1970s, where the importance of

addressing desiccation was highlighted. In general, the improved knowledge developed

on landfill liner design and performance has led to more efficient designs, precautions

and solutions for a wider range of existing problems of the landfill liners.

This chapter presents the existing knowledge on landfill liners, and their desiccation

problem and modelling techniques developed. Fundamentals of the soil behaviour will

be reviewed with emphasis on mechanics of unsaturated soils, as applicable to

compacted soils.

8

Literature Review 9

2.2 Clay Liners in Landfills

A landfill or a contaminated site should always be isolated from the natural environment

to minimize or eliminate the potential pollution of groundwater resources. To achieve

this, low permeable layers are placed around the waste mass. Clay layers are often used

as these low permeable layers due to the availability of clay soils, cost effectiveness of

construction and general good performance of the clay layers. The other materials used

for liners include concrete, asphalt and geo-composites such as geosynthetic clay liners.

The low conductivity layers, placed at the sides, bottom and top of the waste body are

referred to as side, base and cover liner respectively. The term ‘liners’ refers to both

side and base liners and a liner can basically can be identified as a compacted clay layer

that:

• Slows and retards the leachate migration into the ground;

• Prevents bio-gas escape to the environment; and

• Provides mechanical support to the waste mass.

However the role of a properly designed and constructed landfill cover liner is more

complex than that of bottom and side liners. The role of cover liner can be identified as

(Holzlohner et al., 1995) as to:

• Limit the percolation of surface waters through to the waste below and down to

the bottom liner;

• Prevent the direct uptake of contaminants by organisms;

• Control gas fluxes which create a hazard in the vicinity;

10

• Reduce dissolving contaminants and formation of leachate which can pollute

both soil and groundwater;

• Enhance aesthetic appearance; and

• Provide support for the aftercare options.

Gas extraction pipe

Leachate drain pipesLeachate collection and

drainage system

Impermeable clay cap

Synthetic membrane liner

Waste

Compacted low-permeability clayLeachate

collection sumpNeutral rock foundation

Figure 2-1 Cross section of a landfill (http://www.groundwateruk.org/Image-Gallery.aspx)

The knowledge of designing landfill cover liners have developed considerably over the

last two to three decades. Most of countries have their own regulatory requirements for

the construction of landfill liners. For example in Victoria, Australia, information

bulletins are published by environment agency (EPA, 2008). Similarly, other countries

have also developed various guidelines for landfill design. Holzlohner et al. (1995) and

Manassero et al. (2000) summarized major typical requirements adapted by different

countries for the design of landfill liners for hazardous and municipal waste disposal.

Accordingly, the design of covers mainly depends on several factors such as:

• climatic conditions of the site area;

• geo-mechanical properties;

http://www.groundwateruk.org/Image-Gallery.aspx

Literature Review 11

• environmental risks of the contaminated area;

• waste type (i.e. hazardous, municipal or organic waste);

• intended time period; and

• cost.

In addition to government publications, design techniques and guidelines have also been

published by many researchers in technical journals and conference proceedings

(Bouazza and Van Impe, 1998; Kodikara, 2001).

2.2.1 Evaluation of clay liner designs

Prior to 1975, the emphasis and hence the regulatory restrictions placed on environment

pollution or protection was significantly less. Therefore, clay liners to isolate waste

were not designed on a proper scientific basis. The recognition of the importance of

construction procedures on the field scale performances of compacted clay liners

(CCLs) and the consequent development of guidelines for improved installation were

undertaken in late 1980s. Notable contributions towards proper engineering of

composite barriers consisting of mineral liners (CCLs) were made by Daniel (1989),

Jessberger (1994), and Benson (1994).

Since 1990, the main steps in the progress of modern solid waste containment systems

can be presented approximately in chronological order as follows (Manassero et al.,

2000):

12

• Recognition of the importance of construction procedures on the field scale

performances of compacted clay liners (CCLs) and the consequent set up of

guidelines;

• Current and correct employment of composite barriers consisting of mineral

liners (CCLs) or geo-synthetic clay liners (GCLs), placed in close contact with a

geo-membrane (GM);

• Introduction and in some cases standardization of laboratory and field tests for

the evaluation of barrier components and monitoring systems for assessing full

scale liner performances;

• Recognition of potential stability problems of different types of waste deposits,

looking in particular at sliding surfaces involving interfaces of composite barrier

systems;

• Recognition of the importance of compatibility, diffusive transport and sorption

phenomena on the overall performance of barrier systems;

• Recognition of the importance of biogas migration from landfills and the

consequent research work on gas-barrier interaction;

• Recognition of the importance of natural and manmade attenuation layers below

waste deposits in order to reduce impact of pollutants on groundwater;

• Recognition of the role played by the deformations and settlements of subgrade

layers on the performance of mineral barriers;


• Introduction of geo-synthetic clay liners (GCLs) as pollutant containment

barriers;

• Use (in some cases after appropriate modifications) of traditional geotechnical

construction techniques to install barriers around polluted subsoils (e.g. simple

and composite slurry walls, grouted and jet grouted bottom barriers, reactive

diaphragm walls, etc.);

• Introduction of new barrier types for cover systems (e.g. capillary barrier and

natural buffer barriers); and

• Introduction of performance design and related risk analyses overcoming the

prescriptive design procedures. Performance design has become feasible and

reasonably reliable owing to the goals of the aforementioned theoretical and

experimental work and practical experiences. These have led to substantial

improvements in terms of both modelling techniques and knowledge about the

different input parameters that are necessary to simulate the actual behaviour of

waste containment systems.

2.2.2 Types of clay liners

As described earlier, many different types of liners are used in landfills. Bottom layers

can be basically divided into two types, namely, single bottom liner systems and double

or multiple bottom liner systems (Bagchi, 2004). However, cover layers can be

considered in many different types such as compacted clay covers (prescriptive covers),

evapotranspiration covers (Zornberg et al., 2003), capillary barrier covers (Dwyer,

14

1998) (alternative covers) and so on. These alternative covers are designed to overcome

the problems in prescriptive cover designs. Unfortunately, however, alternative covers

do not completely solve those problems because of the complexity of the nature.

Waste

Synthetic impermeable layer

Waste

Compacted base

Compacted clay layerLeachate

collection pipes

Drainage layer

Drainage layer

Compacted clay layer

Leachate collection

pipes

Compacted subbase

Figure 2-2 Layers of base (bottom) lining systems

The main components of the bottom lining systems are the drainage layers or leachate

collection and removal system (LCRS), the compacted clay layer (CCL) and the

attenuation layer or geological barrier. Apart from the geological barrier, which of

course can only be of the natural type, the other layers can be natural or constructed of

amended soils. Generally, regulations require a (saturated) hydraulic conductivity value

of the CCL less than 10-9m/s to be achieved. These low values of hydraulic conductivity

could only be obtained if clods and inter-clod pores are eliminated, which can most

likely be achieved if the soil is compacted wet of optimum, with high compaction effort

and methods producing large shear strains, such as the use of sheep foot rollers. The

permeability of a clay soil compacted at few percentage points wet of optimum may be

10 to 1000 times less than the permeability of the same soil compacted to the same dry

density at a water content dry of optimum. The thickness of a CCL is usually between

600 and 900mm, but occasionally the thickness may reach 1.2 to 3.0m. The most

common type of compacted clay liner is one that is constructed from naturally occurring


soils that contain a significant quantity of clay (i.e., soils that are classified as CL, CH,

or SC in the Unified Soil Classification System (USCS) outlined in ASTM D2487).

Some typically acceptable zones (as shown in Figure 2-3) for selecting initial dry

density and moisture content were defined by Daniel and Wu (1993) based on low

hydraulic conductivity, low desiccation induced shrinkage and high unconfined

compressive strength.

20

Shrinkage

Strength

Hydraulic conductivity

Overall

Moisture content (%)

Dry

uni

t wei

ght (

kN/m

3 )

221816141210815

16

17

18

19

20

Figure 2-3 Acceptable zone based on design objectives for hydraulic conductivity, volumetric shrinkage, and unconfined compressive strength (Daniel and Wu 1993)

The cover layer of a landfill can be easily affected by many different other conditions

than the bottom layer. Therefore extra research attention has been placed on the landfill

cover systems, e.g., Benson (2001), Dwyer (1998). This has led to development of

various cover systems including simple soil cover, compacted clay covers, geosynthetic

16

clay covers, monolithic covers and capillary barriers (Dwyer, 1998; Albright et al.,

2004).

Compacted Clay Cover

Geosynthetic Clay Cover

Monolithic Cover

Capillary Barrier

Finetextured

soil

Finetextured

soil

Finetextured

soilFine

textured soil

Finetextured

soil

WasteWaste

Waste

Waste

WasteCompacted

Clay

GCL

CoarseSoil

Simple soil Cover

Figure 2-4 Different types of cover layers (Dwyer, 1998; Albright et al., 2004)

The prescriptive or traditional cover designs are usually based on the existence of layers

of natural fine grained soils depending on the most type of waste (i.e. municipal or

hazardous). The percolation control in these barriers is typically achieved by

constructing a compacted clay liner with low saturated hydraulic conductivity, thereby

minimizing the infiltration through the layer and also by maximizing the overland flow.

As the knowledge of cover layers grew, a strong emphasis has been placed on the

possibilities and advantages of using alternative sealing materials and cover designs.

The main objective here was to minimise the costs for cover layer systems while


increasing the effectiveness. Thereby, the requirements on the soils used and the design

of the cover layer become more and more complex and the layer thickness can increase

as much as 3m. In addition, synthetic materials such as geo-membranes and geo-

synthetic clay liners have also been incorporated into the design process.

Alternative designs mostly rely on water storage principles (i.e., controlling percolation

by water storage during wet periods and evapotranspiration during dry periods) and are

often referred to as "evapotranspirative covers". A capillary barrier layer (i.e., a layer of

fine-grained soil over a layer of coarse soil) is sometimes added to increase the water

storage capacity of the cover. However when considering the different types of designs,

some major and most popular alternative covers can be identified as designs composed

of capillary barriers, evapotranspiration layers and composite layers (Melchior, 1997;

Dwyer, 1998; Albright et al., 2004). A capillary barrier is a cover employing a finer

grained layer overlying a coarser-grained layer. This contrast in particle size limits

downward migration of water by exploiting the contrasting unsaturated hydraulic

properties of soils with different gradation. An evapotranspiration layer acts not as a

barrier, but as a sponge that stores moisture during precipitation events, and then

releases it back to the atmosphere as evapotranspiration. Composite barriers consist of

mineral layers (CCLs) or geo-synthetic clay liners (GCLs), placed in close contact with

a geo-membrane (GM).

Capillary barriers are practical for semiarid and arid regions because there is no need for

moisture conditioning, which reduces construction costs. In addition, because capillary

barriers do not have a moist, compacted clay layer, they are less susceptible to

degradation caused by desiccation cracking. Capillary barriers are constructed in

18

various forms, ranging from a simple design consisting of two layers to more complex

designs that include multiple layers of finer-grained and coarser-grained soils (Morris

and Stormont, 1997).

The hydraulic conductivity of soils typically used in evapotranspiration (ET) covers is

higher under saturated conditions than the hydraulic conductivity of typical clay barrier

materials. However, under unsaturated conditions, the hydraulic conductivity of these

soils is typically less than that of clays. ET covers have also been referred in the

technical literature as monocovers, monolithic, store-and-release and soil-plant covers.

They are usually vegetated with native plants that survive on the natural precipitation. In

addition, ET covers are less vulnerable than clay barriers to desiccation and cracking

during and after installation, are relatively simple to construct, require low maintenance,

and can be constructed with a reasonably broad range of soils (Zornberg et al., 2003).

Cote et al. in 1996 suggested a ‘self-sealing, self-healing liner’, capable of self-repairing

in situ when damaged, surrounding the waste with two layers of porous material (Kwon

and Cho, 2011). The two layers, each containing a sufficient amount of interactive

components, act as a waste-encompassing interface. Upon placement, the components

in the two layers form precipitates that fill the pores at the interface of the two layers,

and finally form a waste-encompassing layer with reduced permeability. The special

benefit of this method is that the interactive components form a new seal if seal rupture

occurs, thus preventing loss of noxious substances through the rupture. However, the

precipitates from the self-recovery reaction of the layer might be dissolved in the acidic

rains with long term conditions.


To meet the economic constraints and to fulfil the technical requirements, it is important

to know the pros and cons of the technical performance of the components and capping

systems and the uncertainties in its assessment, especially in view of the importance of

their long-term behaviour.

2.2.3 Major issues related to clay liners

The basic purpose of a well designed and constructed clay liner is to prevent or

minimize the infiltration. It is doubtful whether this expectation is satisfied in long term

conditions, where the clay liner is in operation. The deficiencies of a clay liner can

occur due to inappropriate designs or construction faults. In addition:

• Cracking due to desiccation;

• Distortion due to differential settlement in the waste body; and

• Difficult placement over compressible waste

can also be identified as factors that can compromise their long term performance.

When a compacted clay soil is exposed to the atmosphere water evaporates, potentially

inducing shrinkage in all directions. However, only the vertical direction is free to

shrink and the lateral direction is normally restrained. As a result, the soil layer can

crack, which could lead to substantial increase in seepage rates. In this context, the

relative effect of the drying/shrinkage increases with the as compacted water content.

Consequently, shrinkage strains and therefore desiccation cracks increase with

compaction water content (Istok, 1989).

20

Clay liners are normally compacted with high water content (on the wet side of the

optimum moisture content) to achieve low hydraulic conductivity. However, a small

decrease in the water content can cause a large increase in soil water tension or soil

suction. When laterally restrained, water tension induces tensile stresses on the soil

matric, which eventually leads to cracking of the clay and lateral shrinkage opening the

cracks further. However, it is found that two processes counteract desiccation in a

bottom layer (Holzlohner et al., 1997). Firstly, capillary rise of liquid water from the

subgrade compensates for the water loss by water vapour transport and the water

content can equilibrate at a reasonably high value under appropriate site conditions (i.e.,

depending on the capillarity of the soils and conductivity of the subgrade and depth to

the water table). Secondly, the compressive stress caused by the high load of the waste

body can suppress the tensile stress generation and prevent the formation of cracks.

In a cover system the desiccation processes are driven by water evaporation into the

atmosphere and by the transpiration of the plants, when present. Under this condition

the CCL becomes attractive as a water reservoir for the plants thereby accelerating the

desiccation process. In cover systems, the compressive vertical stress caused by the

overburden on the clay liner is limited. It is, therefore, questionable whether stable

conditions can be achieved in a cover system in equilibrium with its environment

without significant changes in water content from the as compacted state occuring. At

least very thick restoration layers with high storage capacity for plant-available water

are necessary to achieve such conditions (Simon and Müller, 2004).

Cracking reduces the effective thickness of the low permeability earthworks layer,

thereby compromising the overall integrity of the barrier, and conditions that would


enable self-sealing of cracks cannot be entirely relied upon (Savadis & Mallwitz 1997).

During its lifetime a landfill clay liner may be subjected to different periods of wetting

and drying. The moisture loss can be more complicated due to heat generation within

landfills. As the landfill acts as a bio-reactor, heat is generated and moisture loss can be

due to a combination of moisture migration away from the heat source and, if linked air

voids are available, evaporative losses at the interface between the geo-membrane and

the mineral barrier can take place. Doll (1997) has modelled moisture migration away

from such a heat-source in landfill liners and found that significant drying can occur

unless downward vapour diffusion due to temperature gradients can be balanced by

capillary moisture rise from underlying strata.

In addition, clay barriers may not maintain their initial as-compacted moisture content

but instead the moisture will redistribute over time until constant soil water potential

throughout the layered soil profile is attained and maintained. Compacted clay liners

placed on coarse grained soils (common practice due to installation of under drains and

leachate detection layers) will equilibrate matric potential with the soil beneath and

decrease in moisture content. Through monitoring a test pad, some researchers found

that incorporation of a drainage barrier or capillary layer beneath a compacted clay liner

causes significantly more moisture loss and associated increases in suctions in the

compacted clay liner, when compared with a clay liner underlain by a fine soil (Philip et

al., 2002).

When a landfill matures, the waste decays reducing its volume. This can cause

differential settlements of the landfill cover layer leading to differential settlements,

under which a liner can crack and fail. Similarly, when placing a layer on top of

22

compressible wastes, it may not be possible to compact to the required level and

consequently the required permeability may not be achieved. This may also lead to

undesired performance.

To overcome the above discussed issues and limitations of compacted clay layers and

understand how those problems could occur, there should be a good understanding of

field performances of compacted clay layers. The effects of these problems could be

understood by observing the field performance of clay liners. The next section describes

the literature which explains the behaviour of clay layers in field conditions.

2.2.4 Field performances of clay liners

The long-term performance of clay barrier covers is considered on two main factors:

(i) Proper construction of the barrier layer (the hydraulic conductivity requirement is

met at field scale); and

(ii) Long-term maintenance of the barrier layer (to maintain the low hydraulic

conductivity)

Factors contributing to proper construction of CCLs have been studied in detail and are

well understood (Benson et al., 1999; Kodikara, 2001). Far less attention has been given

to evaluating whether the integrity of clay barriers can be maintained under field

conditions. However, when the field performance of compacted clay liner is considered,

the moisture content variation within the layer and hydraulic conductivity and water

balance of the layer are recognized as the most important factors. Hence the

performance of the landfill liner after installation such as, evaporation, moisture content


variation, infiltration, drainage, suction and effect of wet/dry cycles has been widely

monitored (Singh and J., 2003). These field observations provide some evidence to

indicate that clay barrier covers may not be performing as intended (Corser et al., 1992).

2.2.4.1 Moisture content and suction variation within the soil layer

The variation in water content due to the evaporation mainly affects the soils close to

the ground surface. This depth changes with the soil type, location of water table and

the climate conditions. In a clay soil, plants cause significant drying only within the root

zone (Smethurst et al., 2006).

Figure 2-5 (after Kraynski, in (Chen, 1988)) shows the moisture content variation with

the depth of the soil layer. Hd and Hs represent the depth to permanent water table and

depth of seasonal changes in water content respectively. The curve number 1 is for

equilibrium water content profile expected below an ideal surface covering in which the

moisture content decreases smoothly from top on the basis of an equilibrium suction

profile. There is no gain or loss of moisture to the atmosphere. Curves number 2 and 3

show the moisture content variation with uncovered soil natural conditions. The typical

variation of water content with depth corresponding to dry-season and wet-season

conditions are shown as curve number 2 and 3 respectively. In the wet season the water

content increases rapidly near the surface of the soil. Similarly in the dry season, water

content near the surface decreases. Seasonal fluctuations are restricted to a depth Hd,

which has a potential to expand if water deficiency increases. However, this depth is

typically 1 to 2m beneath normal ground cover in Eastern Australia, about 0.5 to 1.0m

below the non-vegetated surface of Australian coal tailings, and about 3m in Manitoba,

24

Canada (Morris et al., 1992). Further, AS2870-2011 presented the depth of design soil

suction change for different locations around Australia which changes from 1.5m to

4.0m. However, the depths provided in the standard could be subject to seasonal

variations. The change in suction at the surface is considered to be the same value

throughout Australia with a value of 1.2pF. This value is used later in this study to

predict the suction profiles.

Hs

Hd

Water content

Dep

th

Curve 3Curve 2

Curve 1

Figure 2-5 Variation of moisture content with depth, Curve 1, equilibrium water content under a surface covering; curve 2, dry season; curve 3, wet season (after Kraynski, (Chen 1988))

Moisture content of the soil layer responds reasonably fast to the climatic conditions.

However, after several climatic condition cycles, the soil reaches an equilibrium

condition in laboratory conditions (Warkentin and Bozozuk, 1961). Warkentin and


Bozozuk have dried cylindrical samples of undisturbed Leda clay to three different

water contents and allowed it to swell and shrink in sub subsequent cycles. They stated

that if the drying and wetting cycles are repeated, the increment of water content

decreases in further cycles. The net loss after each cycle is small and could be due to

further orientation of particles with each drying cycle. After many cycles the volume

change becomes reversible. A typical plot showing the effect of drying and rewetting of

Leda clay by Warkentin and Bozozuk (1961) is presented in Figure 2-6. The figure

illustrates the gradual decrease of the saturated moisture content with the number of wet

dry cycles.

Initial point (water content 76.6%)

Dry

ing

curv

e

Wet

ting

curv

e

Number of wetting and drying cycles

Moi

stur

e co

nten

t (%

)

Figure 2-6 Effect of drying and rewetting on the ultimate moisture content of Leda clay (after warkentin (1961))

26

Theoretically, if climatic conditions remain unchanged long enough for the complete

equilibrium conditions and a constant water table to be reached in the soil profile, the

matric suction would decrease linearly with depth from maximum suction at the ground

surface to zero suction (atmospheric pressure) at the water table. However, this

condition is hardly ever reached in practice and generally a transient nonlinear decay of

matric suction from a value at the surface depending on the transient event (drying or

wetting) to zero at the water table is observed. The transient osmotic suctions that

depend on the salt concentration in the pore water are also considered in total suctions.

Within the depth of cracking, changes in matric suction occur much more rapidly than

changes in soil chemistry and solute suction. Hence, it is considered that matric suction

has a greater effect on cracking than solute suction within the usual timeframe of

engineering interest.

3

2 2 2

3 3 3

5 55 33

111

Suction, pF

Dep

th, m

W

W

WD

DD

Melbourne AdelaideSydney

W = Wet season Total suction D = Dry season Matrix suction

Figure 2-7 Typical seasonal soil suction variations with depth (Morris et al., 1992)


Figure 2-7 shows wet and dry season suction profiles with depth measured in southern

Australia (Morris et al., 1992). The unit of suction was given in pF which is defined as

log, (suction head in centimetres of water). It appears that the depth of shrinkage cracks

typically coincides approximately with the depth of seasonal changes in water content

and suction. The ground water table is at or is more likely to be below this depth. The

profiles of total suction (the sum of matric suction and osmotic suction) in Figure 2-7

have an approximately constant suction of about 4pF (about 1000kPa) at higher depths

mainly due to solute suction. The solute suction reaches a peak below the water table

due to the greater salt concentration produced there by downward leaching of salt from

above. It can reach 4.2pF or 4.6pF near trees (Morris et al., 1992). Suggested matric

suction profiles are shown as broken lines in Figure 2-7. They tend to be negative at the

water table, corresponding to zero suction on the logarithmic pF scale, and join the total

suction profiles towards the surface, where matric suction dominates over solute

suction.

Figure 2-8 Soil water characteristic curves (Fredlund and Anqing 1994)

(a) (b)

28

Under unsaturated conditions, the interactions between suction, water content and

hydraulic conductivity (Singh and Gupta, 2000) control the rate of water loss in the soil.

Typical plots of the relation between the soil suction and water content, known as soil-

water characteristic curves, for different soil types are shown in Figure 2-8 (a) and (b)

(Fredlund and Anqing, 1994). Figure 2-8 (a) shows the soil-water characteristic curve

for a silty soil, along with some of its key characteristics. The air-entry value of the soil

is the matric suction where air starts to enter the largest pores in the soil. The residual

water content is the water content where a large suction change is required to remove

additional water from the soil. The adsorption curve differs from the desorption curve as

a result of hysteresis of water flow. The saturated water content and the air-entry value,

generally increase with the plasticity of the soil. Other factors such as stress history also

affect the shape of the soil-water characteristic curves. The total suction corresponding

to zero water content appears to be essentially the same for all types of soils (Figure 2-8

(b)).

2.2.4.2 Water balance of the soil layer with the surrounding environment

There are three distinct moisture migration mechanisms operating over the lifetime of a

clayey barrier (Philip et al., 2002):

(1) Evaporative losses from the surface of clay layers;

(2) Equilibration of soil moisture potential between clay barrier and surrounding

soil; and

(3) Moisture migration down the thermal gradient produced by high temperatures

generated during decay of waste in the landfill.


For basal liners mechanism (1) can generally also be categorized as moisture losses that

occur during construction and pre-waste disposal (temporary works) whilst mechanism

(3) generally occurs post waste disposal and during bio-reaction (permanent works). For

basal liners mechanism (2) will take place over the entire lifetime of a landfill from

initial construction until after bio-reaction. For compacted clay final covers,

mechanisms (1) and (2) will be the dominant processes. Figure 2-9 shows a schematic

profile view of a landfill liner indicating water balance.

Top soil – percolation layer

Sand – lateral drainage layer

Clay – Barrier layer

Waste

Sand – Drainage layer

Clay – Barrier layerGeomembrane layer

Geomembrane layer

Precipitation Evapotranspiration

Runoff

Infiltration

Lateral drainage

Leachate collection

Cov

er li

ner s

yste

mB

otto

m li

ner

syst

em

Figure 2-9 Schematic Profile View of a Typical Hazardous Waste Landfill

30

According to the Blight (2003), for the zone of major seasonal wetting and drying at the

slope surface, the full water balance may be written as

�(𝑅 − 𝑅𝑂) − � 𝐸𝑇 + ∆𝑤 − 𝑅𝐸 [2-1]

where

𝑅 = rainfall,

𝑅𝑂 = runoff,

𝐸𝑇 = actual evapotranspiration,

∆𝑤 = change in stored water within the soil, and

𝑅𝐸 = net recharge from the surrounding soil.

The rainfall is relatively simple to measure, although it is very site specific, and long

records are available for many countries. Runoff is also site specific but measurable.

The upward flow of water resulting from the suction gradient into the drying (root) zone

at the slope surface is negligible in comparison with rainfall and evapotranspiration, and

may be ignored in water balance calculations (Smethurst et al., 2006).

2.2.4.3 Evaporative losses from the surface of clay layers

In the absence of artificial recharge or irrigation, water may enter the soil through the

ground surface from rainfall and leave again through the surface due to evaporation and

as a result of evapotranspiration by plants. Predicting this exchange of moisture between

the soil surface and the atmosphere is a critical issue in the design of soil covers for

acid-generating mine tailings and waste rock, and other land-based disposal systems,


since the rate of evaporation depends on both soil layer properties and environmental

conditions. Three key soil layer properties influence this exchange:

1. The supply of water and evaporative demand;

2. The ability of soil to store and transmit water; and

3. The influence of vegetation absorption through roots.

The main environmental conditions govern the evaporation (Yanful et al., 2003)

namely:

1. Air temperature;

2. Relative humidity;

3. Net radiation;

4. Wind speed; and

5. Surface cover, such as turf.

Engineers have traditionally used a term defined as potential evaporation, PE, to

estimate evaporation or evapotranspiration rates. Potential evaporation may be defined

as the upper limit or maximum rate of evaporation from a pure water surface under

given climatic conditions. The potential rate of evaporation may be computed using the

Dalton type equation (Gray, 1970);

𝑅𝐸𝑇 = 𝑓(𝑢)(𝑒𝑠 − 𝑒𝑎) [2-2]

where:

𝑅𝐸𝑇 = rate of evaporation (mm/day),

𝑒𝑠 = saturation vapour pressure of water at the temperature of the surface (mm Hg or

kPa),

32

𝑒𝑎 = vapour pressure of the air in the atmosphere above the water surface (mm Hg or

kPa),

𝑓(𝑢) = turbulent exchange function that depends on the mixing characteristics of the air

above the evaporating surface.

The use of the apparently simple expression given in equation [2-2] is considered a

direct approach (Granger, 1989). However, the application of equation [2-2] to field

problems can be difficult. Accurate evaluation of the turbulent exchange function

requires either an empirical approach or the application of rigorous aerodynamic profile

methods. Furthermore, equation [2-2] is often indeterminate when applied to field

studies because of difficulties associated with the evaluation of surface temperatures

and vapour pressure (Granger, 1989). The Dalton equation is generally not applied in

the elementary form stated in equation [2-2]. However, equation [2-2] forms the basis

for the widely used Penman method. Penman (1948) resolved the difficulty associated

with surface temperature in equation [2-2] by combining it with a second simultaneous

equation for the sensible heat flux at the surface. Penman (1948) also provides a

relatively simple method for determining the turbulent exchange function on the basis of

mean wind speed. The Penman method assumes the surface to be saturated at all times

and therefore provides an estimate of the potential rate of evaporation.

Numerous other methods are also available for calculating the rate of potential

evaporation. These include the temperature-based method proposed by Thornthwaite

(1948) and the energy-based method developed by Priestley and Taylor (1972).

Although all of the various methods for potential evaporation may predict different rates

of evaporation when applied to a specific site (Granger, 1989) the fundamental


assumption used by the methods outlined above is that water is freely available at the

surface for evaporation. In other words, the surface is an open water surface or a

saturated soil surface. However, when we consider the actual condition it is not freely

available on the surface.

The actual rate of evaporation, AE, begins to decline as the surface becomes unsaturated

and the supply of water to the surface becomes limited (Gray, 1970). Figure 2-10 shows

a typical relationship for the ratio of AE and PE, AE/PE, with water availability for a

sand surface. Gray (1970) presents similar curves for sand and clay surfaces. The rate of

actual evaporation is approximately equal to the potential rate (i.e., AE/PE equal to

100%) when the sand is saturated or nearly saturated (i.e., water content at or above the

field capacity). The rate of AE/PE decreases as the sand surface becomes drier and

eventually falls to a low, relatively constant value as the sand surface desiccates to the

permanent wilting point for plants. Similar variation can be assumed for clay soils as

well.

The shape of the drying curve shown in Figure 2-10 is well known and has been

described by others including Hillel (1980). In general, the curve is described as having

three stages of drying. Stage I drying is the maximum or potential rate of drying that

occurs when the soil surface is at or near saturation and is determined by climatic

conditions. Stage II drying begins when the conductive properties of the soil no longer

permit a sufficient flow of water to the surface to maintain the maximum potential rate

of evaporation. The rate of evaporation continues to decline during Stage II drying as

the surface continues to desiccate and reaches a slow residual value defined as Stage III

drying. Hillel (1980) states that the slow rate of evaporation during Stage III drying

34

occurs after the soil surface become sufficiently desiccated to cause the liquid-water

phase to become discontinuous. The flow of liquid water to the surface ceases and water

molecules may only migrate to the surface through the process of vapour diffusion. In

summary, it can be seen that the rate of actual evaporation from a soil surface is

controlled by both climatic conditions, which defines the potential rate of evaporation,

and soil properties such as hydraulic conductivity and vapour diffusivity.

AE

/PE

(%)

Permanent wilting point

Field capacity

Water or moisture availability

Stage IIIStage II

Stage I

Typical evaporation curve for sand

0

50

100

Figure 2-10 The relationship between the rate of actual evaporation and potential evaporation (i.e., AE/PE) and water availability (Wilson, Fredlund et al. 1994)

Since the actual rate of evaporation is controlled by both climatic conditions and soil

properties, accurate prediction of the actual rate of evaporation from soil surfaces

requires a method of analysis that includes both factors. The methods previously

outlined (i.e., the use of equation [2-2] or the Penman method) are based on climatic

conditions such as temperature, relative humidity, wind speed, and net radiation. These

methods are reliable only for special conditions where the rate of evaporation is

controlled solely by climatic conditions (i.e., Stage I drying) in Figure 2-10. The


climate-based methods of analysis for potential evaporation often over-estimate actual

evaporation rates, since the actual rate of evaporation is soil limited. This is frequently

the case for unsaturated soil surfaces in arid and semi-arid environments (Wilson,

1990).

Geotechnical engineers are often required to predict evaporative fluxes under both

environmental and soil conditions. Wilson (1990) and Wilson et al. (1994; 1997) have

conducted research on coupling soil and atmosphere for soil evaporation. They

proposed a model that includes both atmospheric conditions and soil properties and

proved its applicability for sandy, silty and clayey soils. This model can be used to

modify the potential rate of evaporation obtained from Penman method to obtain the

actual evaporation for unsaturated conditions, known as modified Penman method.

2.2.4.4 Moisture migration downward the soil layer

Moisture migration downward or percolation through the soil layer worsens the

problems created by waste in a landfill. Therefore, as noted earlier, all the precautions

are to be taken for minimizing downward leakage while designing and construction of

landfill liners. However, preventing percolation is impossible using compacted soil

layers and geo-synthetics in the field. Hence percolation rates of landfill liners have

been monitored and measured in the field widely (Benson et al., 2001; Albright et al.,

2006; Adu-Wusu et al., 2007; Henken-Mellies and Schweizer, 2011).

36

Figure 2-11 Leachate generation rates at a modern domestic landfill in Pennsylvania (USA), Average annual precipitation at the landfill site is 1.0 m/year (after Bonaparte 1995, (Bouazza

and Van Impe 1998))

The leachate amount collected at the bottom of the waste container with the installation

of cover layers was recorded by Bonaparte (1995), who also plotted the LCRS flow rate

with time (Figure 2-11). In the absence of supplemental moisture addition, the leachate

generation rate will be highest early in the facility active life, with the rate decreasing as

the landfill is filled and progressively closed. These covers can virtually eliminate

infiltration and thus the long-term leachate generation provided that the cover layer

performs as expected in long term conditions similar to at the newly built conditions.

Several laboratory studies have shown that environmental conditions, especially those

that result in desiccation and freeze–thaw cycling, cause cracking of the soil and

increases in saturated hydraulic conductivity of two or more orders of magnitude

(Drumm et al., 1997; Albrecht and Benson, 2001; Albright et al., 2006; Henken-Mellies

and Schweizer, 2011). However in general, the amount of drainage discharge per year is


independent of the geometry of the landfill such as inclination of the cover (Melchior,

1997).

Henken-Mellies (2011) presented a field testing programme under site-specific

conditions for mineral landfill surface cover systems, of which the performance has

been monitored in a long-term study in Aurach, Bavaria (Germany). In that study they

have studied three cover systems; (1) Simple soil barrier cover, (2) Geo-synthetic clay

liner, and (3) Compacted clay liner. They have monitored and plotted the precipitation,

surface runoff drainage flow, the percolation through the layer and the soil water

content in large scale lysimeter test fields from 2002 to 2008 for a CCL. After 7 years of

observation there has been no deterioration of the mineral barrier recorded. They

suggested that the proper design of a clay liner can mitigate the high amounts of

percolation.

Albright et al. (2006) monitored the field scale hydrology of clay barrier covers at three

sites located in warm–humid, cool–humid, and warm–arid climates. In that research

they monitored precipitation, drainage, volumetric soil water content, and the surface

flow over two to four years. In their records, it is noticeable that for some time after the

layer installation the percolation rate was relatively constant (precipitation also

negligible) and showed little temporal response to precipitation events. In contrast,

when precipitation resumed after the low precipitation period, percolation was

transmitted within hours of precipitation events (Figure 2-12) and there was no good

relationship between the soil moisture content and the drainage rate. So, it was

concluded that cracks had developed and those cracks were providing preferential flow

paths causing corresponding increase of hydraulic conductivity.

38

It is unknown whether preferential flow paths in the soil barriers will close over the long

term. However, other studies suggest that these flow paths are likely to persist. Melchior

(1997) reports that preferential flow regularly occurred through desiccation cracks in the

clay barrier of a conventional final cover for 4 years, when the cracks formed during a

dry period. Hydraulic conductivity tests on desiccated clays conducted by Albrecht and

Benson (2001) over a period of approximately 1 year have also shown that preferential

flow paths in clays persist even with continuous access to water. Albrecht and Benson

(2001) describe observations of soil barriers in landfill covers in Minnesota and

Wisconsin, USA, where extensive cracking of the barrier occurred. Roots were present

in many of the cracks and moisture was present on the crack surfaces, suggesting that

preferential flow through the cracks had occurred for an extended period. No evidence

was found of infilling or other mechanisms that would impede or eliminate preferential

flow through the barrier over time.

Nevertheless, the drainage rate of a clay layer is not only dependent on the cracking and

hence preferential flow paths. According to the Hillel (1980) redistribution or internal

drainage in soils can occur due to gravitational and suction gradients within the deep

part of the soil profile and leads to downward water movement. Hence, proper design

and close monitoring on the performance after the cover installation is vital.


Figure 2-12 Daily precipitation and resulting drainage rate for periods (a) before and after (b) the fall 2000 drought at Albany, GA. Daily precipitation is shown as vertical bars, drainage

rate as a continuous line (Albright, Benson et al. 2006).

2.2.4.5 Swell-shrink behaviour

The seasonal changes make the soils wet and dry cyclically. These wet-dry cycles do

not only cause desiccation cracking but also significant damage to the engineered

40

structures due to shrinking and swelling. Furthermore, it may affect not only the

geometry of the layer but also the soil-water characteristic curves (SWCC) (Jayanth et

al., 2012) and hydraulic conductivity of the clay (Thakur et al., 2005). Several field and

laboratory research works have reported this behaviour of soil (Azam, 2007; Ito and

Azam, 2010; Yu and Wei, 2011).

Figure 2-13 Time-movement plots of several points inside and outside the National Art Gallery building, (Richards, Peter et al. 1983)

The effect of swell-shrink behaviour was monitored at Adelaide, Australia for several

years (Richards et al., 1983). The ground levels both near and away from the trees had


been monitored at regular intervals on a building and the adjacent roadway, as shown in

the time-movement plots in Figure 2-13. At the end of 1971, the trees were removed

and the effect of this removal is clearly shown in Figure 2-13.

Tripathy et al. (2002) examined the swelling and shrinking effects of compacted soil

through laboratory experiments. They found that the swelling and shrinkage path

becomes reversible in terms of water content and void ratio once the specimen reaches a

stable condition after a number of wet/dry cycles. Under this stable condition, the

vertical deformations during swelling and shrinkage are the same. This stabilized

condition generally occurred after about four swell–shrink cycles. The stable swell–

shrink path changes with the changes in the surcharge pressure and swell–shrink

pattern.

Figure 2-14 Variation of vertical deformation with several wet-dry cycles, (Tripathy, Subba Rao et al. 2002)

42

Hanafy (1991) proposed a characteristic S-shaped curve to describe the potential

volume change of an expansive clayey soil for the change in void ratio relative to the

changes in water content resulting from desiccation and water absorption (Figure 2-15)

during one complete swelling–shrinkage cycle. The main phases of soil shrinkage that

accompanied water withdrawal included normal, residual and structural shrinkage

phases (Haines, 1923; Kodikara et al., 1999). During structural shrinkage a few large,

stable pores are emptied and the decrease in volume of the soil is less than the volume

of water lost. Normal shrinkage occurs when the change in soil volume equals the water

lost. Residual shrinkage occurs when air enters the soil and the reduction in soil volume

is less than volume of water lost. At the no shrinkage phase, soil does not shrink upon

further drying. Similarly the swelling curve is also divided into three phases known as

primary swell, secondary swell and no swell (Day, 1999).

Figure 2-15 Shrinkage phases of clay upon drying (Kodikara et al., 1999)


2.3 Desiccation Cracking

Desiccation cracking due to matric suction has been widely observed and reported in the

literature. Kodikara et al.(2000) has presented theoretically plausible aerial patterns of

desiccation cracks in idealised media (Figure 2-16). These aerial patterns observed in

the laboratory and field are shown in the following figures. The parallel cracks can be

observed (Figure 2-17) when thin long layers desiccate (Nahlawi and Kodikara, 2006;

Costa et al., 2008). The cracks observed in real situations are showing mostly random

(Figure 2-18), orthogonal (Figure 2-19) and hexagonal (Figure 2-20) variation. The

cross sectional view of cracks was observed (Figure 2-21) by Yesiller et al. (2000) in a

laboratory experiment for a compacted clay layer which compacted in three lifts.

Figure 2-16 Theoretical patterns of desiccation cracks (a) parallel (b) square (c) hexagonal (after bezant and Cedolin, 1991, (Kodikara, Barbour et al. 2000))

Figure 2-17 Parallel cracks observed in long thin moulds.(Costa, Kodikara et al. 2008)

44

Figure 2-18 Irregular shaped crack pattern observed on a playa surface, southern Nevada, Hammer and handle measured 330mm (Longwell 1928). Playa surface means a flood plain clay

surface or dried-out lake surface.

Figure 2-19 Orthogonal Cracking pattern observed in coal mine tailings, Queensland, Australia (Morris, Graham et al. 1992)


Figure 2-20 Hexagonal mud crack pattern observed on playa surface, Las Vegas quadrangle, Nevada. (Longwell 1928)

Figure 2-21 Vertical crack pattern of a compacted clay layer with 3 lifts (Yesiller et al. 2000)

Measuring crack depth is an approximate method due to the irregular shape and

complex geometry (Yesiller et al., 2000). The depth, width and spacing of cracks are not

46

uniform throughout the area. Generally the highest depths and widths are recorded.

However the crack depths and spacing have been reported by many researchers. Table

2-1 summarizes some of the results available in literature.

Table 2-1 Summary of Desiccation crack depths recorded

Reference Soil Type Description of area

Depth of Cracking

Width of Crack

Spacing of Cracking

(Longwell, 1928)

Playa sediments

Desert of Southern Nevada

75-150mm

After (Simpson, 1936; Lau, 1987)

Clay In a dry season at Western Texas

6.1m

After (Jahn, 1950; Lau, 1987)

Silt Layer deposited under water

1.5-3.0m

(Horberg, 1951) Clay At Lake Agassiz in North Dakota

>3.0m 150mm

(Knechtel, 1952)

Playa sediments

Playa de los pinos, New Mexico

24.5-27.5m

(Willden and Mabey, 1961)

Playa sediments

On Black Rock and Smoke Creek deserts of Nevada

>1.22m 30.5-76.3m

After (Blight and Williams, 1971; Lau, 1987)

In South Africa 0.65-1.45m

(Zein el Abedine and Robinson, 1971)

Vertisol with more than 30% clay content

In Sudan 0.65-1.35m 0.28-0.51m

After (Dasog, 1986; Lau, 1987)

Highly plastic clay

In Saskatchewan 0.28-0.6m 9-22mm 0.7-3.45m

After (Mitchell, 1986; Lau,

Intermixed and inter

Dried out surface of a hydraulic fill

>1.5m 0.6-1.0m


1987) layered silty fine sand and moderately to highly plastic clayey silt

site in Los Angeles

(Miller and Mishra, 1989)

clay Landfill liner 0.3m (up to entire depth)

>10mm n/a

(Morris et al., 1992)

Adelaide (From Peter P.) 1.8-2.0m - -

New South Wales, West Moreton, Bowen basin coalfields in Queensland

Typically 0.5m

- -

Melbourne 2.0-3.0m - -

clay Western Australia, Mine tailings

4m 1m

Lake Agassiz clay

Winnipeg 6.0-8.0m - -

After (Montgomery and Parsons, 1989; Yesiller et al., 2000)

n/a Cover layer of a landfill in Wisconsin

1.0m 13mm n/a

After (Basnett and Brungard, 1992; Yesiller et al., 2000)

clay At slopes of a liner during construction

0.3m 13-25mm n/a

After (Corser et al., 1992; Yesiller et al., 2000)

n/a Compacted cover sections in an arid part of California

>0.1m n/a n/a

(Dyer et al., 2009)

clay flood embankments in the UK

0.6-1.0m 5-25mm 0.1m

48

Lachenbruch (1961) presented a theoretical analysis for tension crack formation

considering the boundary effects through a Linear Elastic Fracture Mechanics (LEFM)

approach. Corte & Higashi (1964) conducted notable field desiccation experiments in

glass and wooden boxes. They presented some important characteristics and

relationships between shrinkage cracking related parameters. Following that research,

many workers have analysed and investigated crack formation, development and mostly

by modelling.

The state of the art work on shrinkage cracking was reviewed by Kodikara et al. (2002)

and has identified most of the studies as qualitative and behavioural. However, a

significant amount of research effort has been dedicated to predict and relate the crack

depth and spacing and to provide a theoretical base for the mechanics of crack

formation.

2.3.1 Initiation and evaluation

As compacted clays lose water through evaporation, volume change/shrinkage occurs as

the gravimetric water content decreases from the initial value (Figure 2-22(a)). When

certain restraints are applied to the material during shrinkage, full potential free

shrinkage may not develop. Then the difference between the free shrinkage and the

actual shrinkage will contribute to the development of tensile stresses within the

material (Figure 2-22(b)).


Figure 2-22 Schematic illustration of cracking (Konrad and Ayad 1997a)

When the tensile stresses exceed the tensile strength of the material, shrinkage cracks

may initiate as shown in Figure 2-22(c) in a direction perpendicular to that of the

maximum tensile stress (Mitchell and Soga, 2005, Kodikara, 2006). In the early stages

of drying from a horizontal surface of initially saturated soil, decreases in water content

are largely accommodated by reorganization of the soil particles into successively closer

arrangements. This involves one-dimensional volumetric straining under k0 conditions,

since the loss of water is largely one dimensional towards the drying surface, where it is

dominantly removed by evaporation.

50

As the developed horizontal (or lateral) stress reaches the tensile strength of the material

cracks can initiate as shown in Figure 2-22c. As drying proceeds shrinkage continues

and dominant cracks grow deeper (Figure 2-22(d)). Cracks were found continuing to

propagate rapidly once they have initiated (Morris et al., 1992). The cracks spread to

join together at the surface as roughly a polygonal crack network, which then gets

stabilized with further desiccation (Philip et al., 2002). Once the crack network had

formed, cracks continued to propagate downwards and widen, although the rate of

widening reduced and this is thought to be related in part to residual shrinkage (i.e.,

water loss greater than volume change) (Philip et al., 2002). Although the orientation of

cracks is random, they all intersect at approach angles closely approximating 90°.(Philip

et al., 2002). Depth is ultimately constrained by increasing stresses due to self- weight

of soil or overburden and the crack length is limited by intersection with other cracks.

Soil will eventually stop changing volume with moisture content decreases when it

reaches a constant void ratio with reduction in moisture content (i.e., shrinkage limit)

during the zero shrinkage phase of the shrinkage characteristic curve.

After initial crack development, soil blocks can further subdivide by repeated

occurrences of the same process within already divided blocks. Initially, the tensile

stresses are highest at the middle of the longest side of an existing block of soil formed

by earlier cracking (Morris et al., 1992).

The tensile stress at the tip of any given crack is reduced by the presence of adjacent

cracks. The reduction in tensile stress at the tips of shorter cracks is also greater than

that at the tips of longer cracks. Consequently, the growth of smaller cracks is retarded

and eventually suppressed by the growth of adjacent larger cracks. Cracking thus


becomes concentrated at a series of relatively large cracks. In a roughly uniform

horizontal stress field beneath level ground, the large cracks are roughly uniformly

spaced in any given direction (Morris et al., 1992).

This subdivision stops when the decreasing size of the block in plan (and hence the

increasing relative depth of cracking) overcomes the restraining stresses at the base of

the block and matric suctions attain equilibrium with the atmosphere; that is, when the

maximum suction consistent with the ambient conditions is reached (Morris et al.,

1992).

2.3.2 Factors affecting desiccation cracking

The onset of cracking depends on the mineralogy of the soil, climatic conditions such as

temperature and rainfall and surface vegetation cover. At a selected strength level,

plastic clays contain more water than lean clays. They therefore experience larger

volumetric contractions on drying (Philip et al., 2002). They may also have relatively

large effective cohesions and tensile strengths (Baker, 1981). These properties can lead

to development of wider, deeper cracks in plastic clays than in lean clays.

Also, high temperatures alone do not produce wide, deep cracks if high temperatures

occur during wet seasons. Wide, deep cracking is associated with plastic soils and high

temperatures during dry seasons when the water table drops to considerable depth in the

soil profile. Furthermore, small changes in moisture content during post-compaction can

lead to large changes in volume and increased crack potential (Philip et al., 2002).

52

Landfill covers are relatively thin systems, placed over a relatively unstable and

consolidating mass. Their placement (at an angle) and the likely differential movement

at their base (Harison and Hardin, 1994), will also subject them to stresses, which

cannot be supported without cracking as the soil loses its plasticity once dried to below

the plastic limit.

2.3.3 Effects of cracking

A cracked soil has much higher hydraulic conductivity than the same soil at the same

water content in a no crack state. Thus any consolidation can be expected to proceed

much more quickly. However, this effect can be masked (or indeed reversed) by low

hydraulic conductivity in regions of desiccated soil between cracks (Blake et al., 1973)

and in the zone of shrinkage ahead of the crack tips. Hydraulic conductivity tests

performed by Albrecht and Benson (2001) indicated that shrinkage cracking of clays

increases their hydraulic conductivities by one to three orders of magnitude. In addition,

permanent changes in SWCC occur after an initial drying.(Sadek et al., 2007).

Other than that, cracks affect a soil's compressibility, its time rate of consolidation, its

strength, and the rate at which water can re-enter. Thus, much geotechnical construction

is affected directly or indirectly by the presence of cracks in a soil mass. A soil with

cracks is more compressible than an intact version of the same soil at the same water

content.


2.3.4 Theoretical developments for desiccation cracking

With the increased concern about desiccation cracking researchers tried to understand

and explain the phenomena using theoretical approaches. The crack depth, spacing,

initiation and evaluation were analysed in detail through several different theoretical

approaches. Basically it was believed that cracks occur when the tensile stress within

the block exceeds the tensile strength of the soil. Hence the behaviour of tensile strength

of a soil is also important to understand when dealing with desiccation cracking.

2.3.4.1 Tensile strength of soil

Tensile strength of a soil is a critical parameter when observing the stress related issues

in soil. It varies with several different parameters. However, it is not commonly

examined by researchers, perhaps due to the experimental difficulty in measuring it. The

problem with tensile strength is that values measured for a given material vary with the

type of test used and with the specimen size (Harison et al., 1994). Furthermore,

determination of tensile strength should be changed according to the condition of the

sample such as saturated and unsaturated or consolidated, unconsolidated and over

consolidated samples (Lakshmikantha, 2009).

The most common experimental methods of measuring tensile strength and the values

they obtained have been presented in Table 2-2 (Harison et al., 1994). V95 refers to the

volume of that part of the specimen subjected to at least 95% of maximum tensile stress.

The ratios of tensile strength from the ring test to the tensile strength from each of the

54

other tests are listed in the last column of Table 2-2. This analysis shows that the tensile

strength should not be treated as a material property.

Table 2-2 Variation of tensile strength with test method

Test method Specimen dimensions (mm) V95(mm3) Strength ratio

Ring test Ro/Ri = 8; R = 50 and t = 25 8.29 1

Three-point bending 200 × 50 × 25 ; 100 × 50 × 25 156 ; 78.1 1.78 ; 1.56

Four-point bending 200 × 50 × 25 2187 3.00

Brazilian test R = 50 and t = 25 10596 4.10

Direct-pull dog bone d = 25 and L = 35 17181 4.50

Note: Ro = Outside radius; Ri = Inside radius; t = Thickness; d = Diameter; and L = Length

Wang et al. (2007) has investigated the tensile strength of the clay by a uniaxial tension

loading assembly on cylindrical compacted specimens. The results have been presented

in graphs showing the influence of the dry density (Figure 2-23) and the influence of the

water content (Figure 2-24) on the tensile strength (𝜎𝑡) of clay. The results showed that

the values of 𝜎𝑡 increase with dry density which can be regarded as the result from the

change of compaction effort because a greater compaction effort is required to compact

a denser specimen. The tensile strength (𝜎𝑡) of the tested clay was decreasing with

increasing water contents. This behaviour can be regarded as the macroscopic exhibition

of the change of interaction force among soil particles and suction potential in the soil

columns induced by the change of water contents.

Figure 2-25 shows the experimental data for tensile strength variation with moisture

content and degree of saturation presented by Lakshmikantha (2009). Tensile strength

was determined using equipment designed by Rodriquez in 2002. The reduction in


tensile strength at lower degrees of saturation was explained as it may be attributed to

weak or broken capillary bonds. Finally he concluded that, the effect of density is more

pronounced for specimens with lower moisture content than those with higher moisture

content and the tensile strength does not increase continually with increasing suction,

but reaches a peak and then reduces with further increase in suction.

Figure 2-23 Influence of dry densities on 𝜎𝑡 of the clay with different water contents (Wang, Zhu et al. 2007)

56

Figure 2-24 Influence of water contents on 𝜎𝑡 of the clay with different dry densities (Wang, Zhu et al. 2007)

Figure 2-25 Details of tensile strength with constant natural density, (c) tensile strength with moisture content with polynomial fit showing the trend (d) tensile strength with degree of

saturation (Lakshmikantha, 2009)


2.3.4.2 Theoretical approaches describing crack depth and spacing

In soil mechanics, cracking can be described as soil failure. For that reason, different

theoretical assumptions can be used through different approaches. Elastic theory,

elastic-plastic behaviour and LEFM theory have widely been used in past research.

Apart from that, the energy balance approach and stress relief approach also have been

used. In early days, the crack depth was predicted using the Rankine theory of earth

pressure which was proposed by Taylor in 1948 (Bagge, 1985; Terzaghi et al., 1996).

For the drained condition,

𝑧𝑐 =2𝑐′𝛾

𝑡𝑎𝑛 (45 +𝜙′2

) [2-3]

and for the undrained condition,

𝑧𝑐 =2𝑐𝑢

𝛾 [2-4]

where,

𝑧𝑐 = depth of tension crack.

𝑐′ =effective cohesion intercept.

𝑐𝑢= undrained shear strength

𝜙′ = effective friction angle.

𝛾 = the unit weight of soils.

58

2.3.4.3 Linear elastic approach

Morris et al. (1992) have reviewed the behaviour of unsaturated soils and the mechanics

of cracking based on three different approaches. The linear elastic approach will be

discussed here. They suggested that the immediate surface of a soil deposit is dominated

by the matric suction (𝑢𝑎 − 𝑢𝑤) during desiccation and assumed that it drives the

shrinkage and cracking process during drying. The strains were assumed as one

dimensional just before the cracking and then the suction stresses decrease linearly from

the maximum at the surface to zero at the ground water table. The relation for the

suction profile was given by,

𝑆 = 𝑆0(1 −𝑧𝑐

𝑊) [2-5]

The tensile strength of the soil was selected as function of suction. Hence the relation to

obtain the tensile strength was considered as,

𝜎𝑡 = 𝛼𝑇𝑆𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′ [2-6]

Then they have presented a relationship to predict crack depths which is given by,

𝑧𝑐 =𝑆0

𝑆0𝑊 + 𝜈𝛾

1 − 2𝜈 − (1 − 𝜈)𝛼𝑇 𝑡𝑎𝑛 𝜙𝑏 𝑐𝑜𝑡 𝜙

[2-7]

Although, the relationship presented was simple and easy to understand, the stress

conditions are not one dimensional after a crack starts to open, especially close to the

tips of the opened crack. Hence the authors recommend further refinements of the

presented solution.


2.3.4.4 Elastic-Plastic approach

When considering the elastic-plastic behaviour, the most widely used failure criterion is

the Mohr-Coulomb theory as illustrated in Figure 2-26. The shear strength of a soil at a

point on a particular plane in terms of effective stresses can be expressed as a linear

function given by equation [2-8].

𝜏𝑓 = 𝑐′ + 𝜎′𝑡𝑎𝑛𝜙′ [2-8]

where,

𝜏𝑓 = shear strength.

𝑐′ = effective cohesion intercept.

𝜎′ = effective normal stress.


The relationship for principle stresses and shear strength parameters can be obtained

from Figure 2-26 as,

(𝜎1′ − 𝜎3′) = 2𝑐′ 𝑐𝑜𝑠𝜙′ + (𝜎1′ + 𝜎3′)𝑠𝑖𝑛𝜙′ [2-9]

where,

𝜎1′ , 𝜎3′ = effective principal stresses

There are other failure criteria, such as Mohr-Paul, Griffith, Griffith-Brace and

Modified Mohr-Coulomb theories (Figure 2-27) which have been proposed to describe

the soil failure under tensile forces.

60

τ

τtFailure envelope

τf

σt

σ1'

σ3'

σ1'

σ1' σ'c'

ϕ′

σf'2θ

σ3' σ3'

σf'

θ

Figure 2-26 Mohr-Coulomb failure criterion

τ

σn

Mohr-Coulomb

Mohr-Paul

Griffith-Brace

Griffith

Figure 2-27 Strength envelopes indicating tensile strength after Lee and Ingles in 1968 (Morris, Graham et al. 1992)


A method to determine the maximum depth of tension cracks for both drained and

undrained conditions has been presented by Bagge (1985) . The analysis was based on

effective stresses and includes the effect of negative pore pressures (Bagge, 1985). The

relationship for the crack depth under undrained conditions is given by,

𝑧𝑐 =2𝐴𝑐𝑢 − 𝛾𝑤𝑊�𝑘𝑜 + 𝐴(1 − 𝑘𝑜)� − 𝛼𝑇𝑐′

�𝑘𝑜 + 𝐴(1 − 𝑘𝑜)�(𝛾 − 𝛾𝑤)

[2-10]

And for the drained conditions,

𝑧𝑐 =2𝑐′(𝑐𝑜𝑠𝜙′ − (1/2)𝛼𝑇(1 + 𝑠𝑖𝑛𝜙′)

1 − 𝑠𝑖𝑛𝜙′ ) − 𝛾𝑤𝑊

𝛾 − 𝛾𝑤

[2-11]

𝑧𝑐 = depth of tension crack.

𝑐′ =effective cohesion intercept.

𝑐𝑢= undrained shear strength


𝛾, 𝛾𝑤 = the unit weight of soils and water

𝐴 = ratio of effective and total horizontal stress differences, (i.e.; Δσ3′/Δσ3)

𝑊 = depth to ground water table

𝑘𝑜 = earth pressure coefficient at rest

𝛼𝑇 = coefficient relating cohesion with shear strength at tensile failure

In 1992, Morris et al. has presented a similar solution for crack depth prediction

obtained directly from the Mohr-Coulomb diagram (Figure 2-26). Here failure was

assumed to take place in shear when the minor principal stress attempts to exceed the

62

tensile strength (Morris et al., 1992). The suction profile was assumed to be linearly

decreasing as given in equation [2-5]. The tensile strength was also considered to be

varying according to equation [2-6]. The derived relationship for crack depth was given

by,

𝑧𝑐 =𝑆0

𝑆0𝑊 + (1 − 𝑠𝑖𝑛𝜙′)𝛾

𝑡𝑎𝑛𝜙𝑏[2𝑐𝑜𝑠𝜙′ − 𝛼𝑇(1 + 𝑠𝑖𝑛𝜙′]𝑐𝑜𝑡𝜙′

[2-12]

Furthermore, they have derived another relation to obtain the crack depth by relating

cracking with a transition between tensile and shear failure. This is an extension to the

work (Bagge, 1985) into a more general stress system and for unsaturated soils more

generally. The relation was then presented as,

'cottan)23('sin3

tan'cos18)3(0

0

φφαφφφ

γ

bT

b

c

M

MWS

Sz

+−−

−+

=

[2-13]

where,

φφ

sin3cos6−

=M

Suction variation with depth and the tensile strength is given by equations [2-5] and

[2-6] respectively. Both equations [2-12] and [2-13] predict similar results for crack

depth as claimed by the authors.


2.3.4.5 Linear Elasticity Fracture Mechanics (LEFM) approach

Although fracture mechanics developed mainly in the 1960's, the basic concept of crack

propagation was established by Griffith in the 1920s. This theoretical approach has been

followed by many researchers to predict the crack depth.

In 1961, Lachenbruch analysed the stresses near the crack using the LEFM approach to

predict the tension crack depths. He presented several relationships to predict the stable

and unstable crack depths which are induced by different stress conditions such as

diurnal thermal stresses, surficial tensile stresses due to cooling, and deep tensile

stresses due to tectonic forces. Equation [2-14] represents the stable cracking depth due

to diurnal thermal activities of permafrost (Lachenbruch, 1961).

𝑧𝑐 ≈1

1.21𝜎2𝐸𝐺𝑐

𝜋(1 − 𝜈2) [2-14]

Where,

𝑧𝑐 = critical crack depth

𝐸 = Young’s modulus

𝐺𝑐 = the rate of release of strain energy with critical crack extension

𝜎 = constant tensile stress

𝜈 = Poisson’s ratio

After one crack has formed, the stresses around the opened crack area would relieve.

Hence no parallel crack can be initiated close to it in which the stress relief exceeded

10%, for the stress would be below the strength there (Lachenbruch, 1961). Also, the

64

second crack should initiate within the area where about 5% stress relief occurred.

Hence the author suggests calculating the spacing of the crack based on the above

predicted crack depths.

A LEFM solution for predicting crack depths has been presented in Morris et al. (1992).

That research presented the stress intensity factors corresponding to three modes of

stress distribution. The summation of three stress intensity factors equal to the critical

stress intensity factor, which can be obtained in a separate calculation,

𝐾𝐼𝐶2 =

−2𝜁𝐸1 − 𝜈2 [2-15]

This work have been extended and presented in more detail in a later publication by

Morris et al. (1994). They developed a graph showing the relation between the mode 1

stress intensity factor and the crack depth as shown in Figure 2-28. The curve intersects

the crack depth axis at 𝑧𝑐 = 0 and at,

𝑧𝑚𝑎𝑥 =1.6420𝑆0

𝑆0𝑊 + 𝜈𝛾

(1 − 2𝜈)

[2-16]

where,

𝑧𝑚𝑎𝑥 is the maximum possible crack depth for the steady state suction profile. The

critical crack depth (𝑧𝑐) values can be obtained by the intersections of the critical stress

intensity factor line and the mode 1 stress intensity factor curve. The smaller depth is

the pre-existing crack depth and the higher depth is the propagated final crack depth

(Morris et al., 1994). The authors concluded that LEFM over-predicts crack depths as

clay cracking requires significant dissipation of plastic energy.


Harison and Hardin (1994) have presented methods for analysing earth structures

subjected to tensile stresses that may produce cracking based on LEFM. Methods of

solution are proposed for desiccation cracking of a clay layer above the water table;

cracking of layered earth structures subjected to external loads; cracking of clay caps for

landfills due to differential settlement; and radial cracking in pressurized boreholes.

Figure 2-28 Variation of stress intensity factor with crack depth (Morris, Graham et al. 1994)

The crack depths were derived using a stress intensity factor which was determined

using stress analysis for a cracked half-space (Figure 2-29) by applying tractions to the

crack surface, where the magnitude of applied traction is determined by the stresses in

the uncracked body (Harison and Hardin, 1994). They presented several equations to

66

obtain the stress intensity factors. From those equations, the variation of stress intensity

factor with crack depth have been calculated and were illustrated in a figure similar to

the graph of Morris et al. shown in Figure 2-28. Here the variation of different surface

suction values was also shown. Although, the trend of the variation of results is similar

to Morris et al. (1994), the values obtained for crack depth are much less.

Konrad and Shen (1997) presented a simplified theoretical approach for the prediction

of the spacing between thermal cracks in asphalt pavements based on a finite element

formulation of linear elastic fracture mechanics applied to an idealized layered system

in which the properties of the frozen layers are temperature dependent. They have

presented equations by which one can determine the extent of stress relief caused by the

fictitious normal stress distributed on the crack walls over a depth equal to the thickness

of the asphalt layer. The size of the stress relief zone depends upon the crack depth and

the neighbouring crack will form at the edge of this stress relief zone. Spacing between

cracks then readily follows (Konrad and Shen, 1997). The calculated crack spacing was

found to be between 25 and 90m for asphalt base/subbases while the observed crack

spacings are 8 to 9m.

In another study, cracks were analysed by postulating that they occur successively (Sun

et al., 2009). Formulae for the secondary crack spacing were then derived after stress

analysis. The effect of the Poisson ratio on the crack spacing was analysed and a linear

relation between the Poisson’s ratio and crack spacing was obtained. The authors

concluded that the Poisson’s ratio of a soil is important in soil cracking studies.


Figure 2-29 Stress analysis for desiccation cracking (Harison and Hardin, 1994)

2.3.4.6 Other approaches

Two other approaches; namely, stress-relief and energy balance have been used to

predict the crack spacing and depth in past research. This section will summarize the

key findings from that research briefly.

68

(i) Stress relief approach

A simplified analytical model to explain desiccation cracking of clay layers in

laboratory cracking tests was presented by Kodikara and Choi (2006) based on a stress

relief approach. This model predicts the consecutively propagating cracks on the basis

of the maximum tensile stress that develops at the mid sections of the clay layer due to

the restraints provided at the basal interface. The highly non-linear process was

simulated by recursively applying the analytical model incorporating the variation of

material properties through correlations with the moisture content and following the

moisture reduction during desiccation experiments. The schematic diagram of the

analytical model presented here is shown in Figure 2-30.

The free shrinkage strain was observed correlating linearly to moisture content

reduction which can be identified as a new approach in this field. The relation to obtain

the maximum tensile stress at the middle of the layer for a fully elastic interface

condition was then given by,

𝜎𝑥,𝑚𝑎𝑥 = 𝐸𝛼𝛥𝑤 �1 −1

𝑐𝑜𝑠ℎ(𝛽𝐿 2⁄ )� [2-17]

where, 𝛼 is the Hydric constant and 𝛥𝑤 is the moisture reduction, giving 𝜀𝑠ℎ = 𝛼𝛥𝑤,

𝜎𝑥,𝑚𝑎𝑥 is the maximum tensile stress, 𝐸 is the Young’s modulus of the clay, 𝛽 =

�𝑘 𝐸𝐴𝑐⁄ , 𝐴𝑐 is the cross sectional area, 𝑘 (in units of pressure) is the interface shear

stiffness and 𝐿 is the length of the clay layer. They used empirical correlations to

represent non-linear material properties of clay and the basal interface.


Figure 2-30 Schematic representation of the proposed model by Kodikara and Choi (2006)

The approximate spacing to depth ratio of parallel cracks that form in long desiccating

soil layers subjected to uniform tensile stress was examined by Kodikara et al.(2011).

The formation of sequential crack patterns was examined theoretically on the basis of an

energy balanced and stress relief approach.

Kodikara and Choi (2006) have derived a relation for non-dimensional crack spacing as

shown in equation [2-18] to capture the stress release by a crack penetrating to the full

depth of an elastic clay layer. It is assumed that the crack occurs reasonably fast and

there is no change in tensile stress and tensile strength due to further drying, and that a

second crack would initiate when the horizontal tensile stress reaches 𝛼 (≤1) times the

tensile strength.

�𝑠𝑑

� = −�1

(1 − 𝜈)+

𝐸𝑘𝑑(1 − 𝜈2)

𝑙𝑛(1 − 𝛼) [2-18]

where, s is the spacing between cracks, 𝐸 is the elastic modulus, 𝜈 is the Poisson’s ratio,

𝑘 is the interface shear stiffness and d is the depth of the layer.

c

L/2 L/2

evaporation

tensile stressdistribution

basex

Clay layer

Interface

τf

uup

τ

stiffness, k

characteristics

70

Figure 2-31 Schematic diagrams for the (a) stress relief approach (b) energy balance approach (Costa 2009)

(ii) Energy balanced approach

The formation of crack simultaneously and sequentially was analysed under this energy

balanced approach and relations derived to predict the non dimensional strain energy

loss (Kodikara et al., 2011). In this method, it was postulated that the strain energy

released during crack formation was balanced by the energy consumed during the

formation of cracks. By using the non-dimensional energy balance at crack formation

the authors have derived a relation given by,

�𝑠𝑑

� = −1.82𝑙𝑛 (1 − (𝐾𝐼𝐶

𝜎𝑡)2 (1 − 𝜈2)

𝑑

[2-19]

where, 𝜎𝑡 is the tensile strength of the material, 𝐾𝐼𝐶 is the fracture toughness and the

other parameters are same as in equation [2-18].

Kodikara et al. (2011) have then compared both the stress-relief and energy balanced

approaches and presented solutions to predict non-dimensional energy loss and crack


spacing for the general case. The relation to predict non-dimensional crack spacing was

given by,

�𝑠𝑑

�𝑚𝑖𝑛

≈ −1.5�1

(1 − 𝜈)+

𝐸𝑘𝑑(1 − 𝜈2)

𝑙𝑛 (1 − (𝐾𝐼𝐶

𝜎𝑡)2 (1 − 𝜈2)

𝑑

[2-20]

The authors recommend further research in this area due to the large number of

simplifying assumptions made in the analysis.

2.4 Numerical Models

Demonstrating and predicting the effectiveness of a cover over the long term presents a

challenge, as there is usually not enough data available from long-term field monitoring

of soil covers. The laboratory-measured values are often not quite representative of field

conditions due the challenges associated with obtaining representative samples for

laboratory tests and the differences that exist in field and laboratory conditions. Hence

as a reasonable substitute, numerical models have been used widely in predicting long

term conditions. However, their ability to accurately simulate conditions existing in the

field must be verified before the predictions can be trusted.

Generally, the modelling approaches can be considered as numerical modelling,

analytical modelling and empirical modelling. Empirically based models may not be

applicable to general conditions outside where they were developed whereas theoretical

methods can be too restrictive to cover a wide range of conditions that exist in field.

Research undertaken using numerical modelling is presented in this section.

72

To improve the accuracy of computer predictions, it is important that the input data, for

example, soil properties and the data that describes the physical processes in the field,

be representative of field conditions. Soil properties may be derived by comparing

computer model simulations to experimental field results, and modifying uncertain soil

properties and physical conditions until the model reasonably simulates the field results.

This process, referred to as model calibration, allows for the determination of soil

properties that are more representative of field conditions (Adu-Wusu et al., 2007).

Model calibration also helps to understand the processes and identify important factors

that influence the moisture movement in soils. The calibration process helps to gain

insight into the internal workings and sensitivities of a particular model. The calibrated

model can then be used as a tool to aid in the design process, for example, to predict

long-term behaviour of a soil cover or to compare different design scenarios. The

numerical models discussed here are based on two main categories; (i) Water balance

modelling, (ii) Fracture modelling.

2.4.1 Water balance modelling software

Existing models used for water balance modelling include: Groundwater Loading

Effects from Agricultural Management Systems: GLEAMS (Leonard et al., 1987),

Hydrologic Evaluation of Landfill Performance: HELP,(Schroeder et al., 1994),

UNSAT-H (Fayer, 2000), HYDRUS-1D (Simunek et al., 1998), Soil Water Infiltration

and Movement: SWIM (Krysanova et al., 1997), Soil Cover (Wilson G. W., 1994) and

Vadose/W (Krahn, 2004). Generally, these models simulate subsurface flow based on

either a simple water balance approach (GLEAMS and HELP) or more complex


numerical solutions of Richard’s equation (UNSAT-H, HYDRUS-1D, VS2DTI, SWIM,

SoilCover and Vadose/W) (Adu-Wusu et al., 2007).

The usefulness of most of these as design tools has been assessed by researchers. The

following is a summary of some of their research findings.

Table 2-3 Summary of research used numerical models

Author and Paper Title

Aim of the research

Liner/ Cover Details

Observations Model

1 Milind V. Khire, Craig H. Benson, and Peter J. Bosscher, 1999

Field data from a capillary barrier and Model predictions with UNSAT-H

To compare the field measured water balance data and estimated water balance data using UNSAT-H

Capillary barrier;

• 75 cm of clean, uniformly graded medium sand (SP) • 15 cm uncompacted, sparsely vegetated sandy silt(SM-ML) • Slope 37%

• Precipitation

• Climatic data • Run off • Soil water

content • Percolation

UNSAT-H and HELP

• Volumetric water content

• Soil water storage

• Evapotranspiration

2 Smethurst, J. A., Clarke, D. & Powrie, W. (2006)

Seasonal changes in pore water pressure in a grass-covered cut slope in London Clay

To quantify the hydrological environment and pore water pressures in a cut slope in London Clay

Cut Slope

• 0.4m Top soil

• 2.5m weathered London clay

• 20m London clay

• Lambeth Group Deposit

• Slope 160

• Water content Variation with Soil Suction along the depth when drying & wetting (1 Yr)

• Evapotranspiration

• Daily Rainfall

• Soil moisture deficit

• Volumetric Soil water content

• Volumetric water

CROPWAT

• Soil moisture deficit

74

content along the depth (lab & Field)

• Suction

• Pore water Pressure

3 Sadek, S.,Ghanimeh, S.,El-Fadel, M., 2007

Predicted performance of clay-barrier landfill covers in arid and semi-arid environments

To predict the impact of cracks in the clay barrier, and estimate their potential effect on the percolation rates through the cover system

• 15cm vegetative Soil

• 45cm Barrier Soil (Ks =10-

5 cm/s) • Prepared Sub

Grade

• Daily Precipitation (1 Yr)

• Relative Humidity

• Maximum and minimum Temperature

• Potential Evapotranspiration

• Volumetric Water Content

Hydrus-2D numerical model

• Volumetric Water content along the depth

• Suction along the depth

• Cumulative Precipitation

• Cumulative Evapotranspiration (3 yrs)

• Cumulative Flux

• Daily Flux

• Monthly Flux

4 Yanful, E. K., Mousavi, S. M.

Yang, M., 2003

Modelling and measurement of evaporation in moisture-retaining soil covers

To model water flow through a single clayey till cover and a layered soil cover using a coupled liquid flow, vapour diffusion and heat transfer finite-element model (SoilCover)

Clayey till cover (25cm)

• Halton clayey till

• Evaporation

• Drainage • Total water flux • Volumetric water

content along the depth

SoilCover

• Volumetric water content along the depth

Layered soil covers

• 47mm Coarse Sand • 125mm Clayey Till • 78mm Fine Sand

• Relative humidity

• Temperature • Evaporation • Drainage • Total water flux • Volumetric water

content along the

SoilCover

• Evaporation

• Drainage

• Total water flux


depth • Volumetric water content along the depth

5 Adu-Wusu, C.,Yanful, E. K., Lanteigne, L., O'Kane, M., 2007

Prediction of the water balance of two soil cover systems

To Confirm the in situ soil properties of the various cover materials used for the test covers through model calibration using Vadose/W and to Determine the suitability of the model in describing existing field conditions and hence its effectiveness as a design tool

TP # 1

• 90cm Non compacted Layer

• GCL Barrier • Waste rock

platform • Slope 20%

• Daily minimum and maximum temperature (1 Yrs)

• Daily minimum and maximum relative humidity

• Daily wind speed

• Daily precipitation

• Soil porosity

• Volumetric specific heat capacity

• Thermal conductivity

• Coefficient of compressibility

• Saturated hydraulic conductivity

• Unsaturated hydraulic conductivity function

• Soil water characteristic curves

Vadose/W 2D model

• Potential evaporation

• Temperature • Suction • Soil water

storage • Percolation • Runoff

TP # 3

• 90cm Non compacted Layer

• 60cm Sandy silt Barrier

• Waste rock platform

• Slope 20%

76

Dwyer (2003) compared water balance simulations of six test covers applying the

computer models HELP and UNSAT-H to field data to verify the accuracy of the

models in describing field conditions. He reported that neither model predicted runoff

and percolation with reasonable accuracy. Furthermore, HELP does not contain

algorithms to simulate unsaturated flow rigorously, which can result in gross errors

when simulating capillary breaks (Khire et al., 1999). Roesler and Benson in 2002

(Adu-Wusu et al., 2007) reported that UNSAT-H could not simulate field data with

reasonable accuracy even after input parameters were modified similarly with HELP.

Table 2-4 Comparison of several programs

Software Water flow theory Evaporation modeling Lateral drainage

Vadose W Caters for unsaturated water flow by SWCC and hydraulic conductivity function

Flux boundary by coupling ground heat mass and vapor flow

Run off allowed

Unsat H liquid water flow using the Richards equation, water vapor diffusion using Fick’s law, and sensible heat flow using the Fourier equation using SWCC and hydraulic conductivity function

two methods are used, isothermal mode and thermal mode. In isothermal mode the potential evapotranspiration concept uses by solving Penman equation. In thermal mode evaporation calculates as a function of the vapour density difference between the soil and the reference height

No lateral drainage

Help assumes Darcian flow by gravity influences through homogeneous layers

Penman method, incorporating wind and humidity effects as well as long wave radiation losses

lateral drainage allowed

SoilCover is a one-dimensional finite-element package that models transient liquid and

water vapour flow, based on a theoretical model for predicting the rate of evaporation


from soil surfaces presented by Wilson et al. (1994) on the basis of a system of

equations for coupled heat and mass transfer in soil (Yanful et al., 2003). Simulations of

water balance using SoilCover were compared favourably with measured data from soil

column experiments in the laboratory (Yanful et al., 2003). The model’s ability to

predict field response data was also validated by Shuniark in 2003 (Adu-Wusu et al.,

2007), who reported that the model reasonably simulated measured data from four test

covers in the field. The program SoilCover has been subsequently replaced by

Vadose/w program.

2.4.2 Fracture modelling software

A rational highly idealized framework for the prediction of the spacing between primary

shrinkage cracks in cohesive soils undergoing desiccation was presented by Konrad and

Ayad (Konrad and Ayad, 1997b). The proposed framework was based on the theory of

LEFM, which is used to describe, in a simple manner, the phenomenon of crack

propagation. The principle of effective stresses was used to describe stress partitioning

in soils, and a fictitious stress superposition concept is used to predict the average

spacing between primary cracks. Crack propagation was analysed with a trapezoidal

distribution of total horizontal tensile stress as derived from the material constitutive

equations. A model named CRACK (Figure 2-32) was proposed that can be applied to

soft soils, consolidated natural soils, and compacted clays to predict crack spacing.

The model CRACK appears to possess most of the essential features of the analysis of

shrinkage crack development in clayey soils. It requires the knowledge of several soil

properties, which are relatively difficult to determine in the laboratory. The computed

78

results, however, are sensitive to input parameters. Nevertheless, the model CRACK

was shown to produce reasonable values at crack initiation and time of crack initiation

when compared with the field observations (Ayad et al., 1997).

Figure 2-32 Flow chart of the proposed model by Konrad and Ayad (1997(a))


Kodikara et al. (2004) modelled the curling deformation of desiccating clay using the

FLAC computer program. They observed that, the computer program simulated the

observed curling behaviour reasonably well despite the simplified assumptions made

during the analysis.

Inci (2008) used Finite Element Modelling (FEM) to simulate the strain response of soil

due to the loss of moisture through thermal shrinkage. He has observed that FEM

simulates the strains and the displacements correctly. However, the author recommends

including crack potential determination for alternative water content profiles and

various crack depths in future work. Experimental validation was also expected to be

done.

Amarasiri et al. (2010b) analysed long thin layers of slurry clay contained in moulds of

different dimensions using the computer program UDEC duplicating laboratory

experiments carried out on similar specimens. The computer program was successful in

modelling fracture which can be identified in the form of number of cracks, residual soil

height, moisture content at crack initiation and time progression of cracks etc. The

authors recommend the programme for desiccation crack modelling stating that

extension of the methodology could potentially be of value in modelling desiccation

cracking of field geo-engineering applications and other instances where time varying

material properties are prevalent.

Further studies using UDEC model have been undertaken as described in a number of

publications (Amarasiri et al., 2010a; Amarasiri and Kodikara, 2011b; Amarasiri et al.,

2011) The modelling approach proposed by them appeared to capture the essential

80

behaviour of the desiccation cracking with varying behaviour of soil properties and

deformability of soil.

The studies mentioned above emphasise the importance of validating computer codes

and suggest the need for improvement to existing codes or development of new ones.

Inability to reproduce field data may not be due only to suspect or inaccurate computer

programs but also to the accuracy of the parameters used as input. In some of the studies

mentioned above, better results were obtained when the input parameters were

modified. However, modelling of unsaturated soils involves many parameters and

factors that could easily influence the outcome. Given the influence of soil properties

such as the SWCC and unsaturated hydraulic conductivity-suction functions on model

output, it is necessary that variability in soil properties measured in the laboratory and

field be accounted for during modelling.

2.5 Conclusions

From the review of the literature related to desiccation cracking, several conclusions

were drawn as given below.

Waste containment systems have used different types of surrounding layers to minimize

pollution. The selection of layer design mainly depends on the regulations of the area,

waste type, geology, climatic conditions and the budget allocated. Since these

conditions are highly variable from one place to another there is no any particular type

of liner design recommended for waste containment systems. Furthermore, there are


still further developments to be made on more effective designs such as liners with self-

healing ability.

Almost all liner designs consist of a compacted clay layer. The main reason for using a

compacted clay layer is to minimize infiltration pollutants and water. However, given

the long term conditions it is impossible to minimise the infiltration due to preferential

flow paths which are developed with desiccation. Long term monitoring of the

performance of liners is an expensive and difficult task. However, careful monitoring

for long term stability after a reasonable period following liner installation is yet to be

observed in order to understand the liner behaviour with the maturity of landfill.

Desiccation is one of the severe problems a clay liner can face. Cracks develop in clay

liners when they desiccate. Much research needs to carried out in the form laboratory

and field experiments, theoretical models and numerical models in order to understand

the mechanism of cracking. Factors that affect desiccation cracking have been

understood qualitatively. However, the precise mechanism of cracking is imperfectly

understood and the depth and spacing of the cracks are not predicted accurately by any

of the proposed models.

When observing the cracking behaviour in the field, it has been observed that the crack

initiation and propagation highly depend on the cyclic change of the climate.

Desiccation crack modelling related to the moisture content variation during wet-dry

cycles seems to be important, but has not been undertaken in the past. Therefore, this is

considered as a subject of the current study.

Of the various computer programs used to predict the crack depths in the literature,

most are not sufficiently capable to capture the actual behaviour taking into account the

82

likely variation in moisture contents, material properties and boundary conditions.

However, the modelling approach used in UDEC appears to be able to reproduce the

desiccation fracturing process. This program will be further considered in the research

undertaken in the current study.

Ground moisture modelling has been also undertaken by several approaches using a

number of computer programs. Among them, Vadose/W appears to be able to

accurately simulate trends in the field response and more representative values of soil

properties.

Chapter 3

COMPARISON OF CRACK MODELLING APPROACHES

3.1 Introduction

The previous chapter presented a review of desiccation cracking and associated

mechanisms. This chapter presents results from some analytical models and numerical

models capable of producing crack depths in clay soils. The numerical modelling

approach for predicting desiccation crack depths of clay layers is presented using the

computer program, UDEC. Using this model, non-dimensional results are produced to

estimate crack depths. Numerical results are compared with the results obtained from

the analytical model of Morris et al. (1992), which has been referred to extensively by

researchers in the geo-environmental field.

83

84

3.2 Existing Analytical Approaches for Predicting Crack Depth

In the previous chapter, it was highlighted that the suction profile that develops in the

soil is very important in crack development. In this regard, three suction profiles as

given below are considered for the analysis as suggested by Morris et al. (1992).

i. Constant suction

ii. Linearly decreasing suction

iii. Parabolic suction variation

Three methods are used to estimate the crack depths, namely linear elastic, elastic with

shear failure and LEFM approaches.

3.2.1 Constant suction profile

Constant suction profile is an idealised condition which can happen in clay layers when

the soil dries uniformly down the depth. Usually, slow drying conditions can lead to this

development. The schematic view of the profile is shown in Figure 3-1 (a). Using

Equation [2-6], the tensile strength of the soil can be expected to be uniform in the

ground as shown in Figure 3-1(b).

Under elastic conditions, the relation between the components of stress and the

components of the strain can be described according to the Hooke’s law. In the

following derivations, stresses were analysed to obtain crack depths from the basic

theoretical assumptions. The x-direction is considered as horizontal and perpendicular

to the crack, the y-direction is horizontal and parallel to the crack, and the z-direction is

Comparison of Crack Modelling Approaches 85

vertical (down towards the earth). Tensile stresses and strains are considered to be

negative.

Dep

th

Suction

W

S0

z

Dep

th

Tensile strength

W

σt

z z

Figure 3-1 Suction profile (a) and tensile strength profile (b) with depth

The plane strain condition is also assumed meaning the crack is infinitely long and the

problem could be analysed in 2 dimensions. Hence assuming plane strain conditions,

strains in the x and y directions can be considered as zero (z direction = vertically

down). When the strain in the y direction is 𝜀𝑦 and the stresses in the x, y and z

directions are 𝜎𝑥, 𝜎𝑦 and 𝜎𝑧 respectively, the relationships for strain using applied stress

parameters in x, y and z directions can be written as,

𝜀𝑦 = 0 =1𝐸

�𝜎𝑦 − 𝜈(𝜎𝑥 + 𝜎𝑧)�

∴ 𝜎𝑦 = 𝜈(𝜎𝑥 + 𝜎𝑧) [3-1]

and,

𝜀𝑥 = 0 =1𝐸

�𝜎𝑥 − 𝜈�𝜎𝑦 + 𝜎𝑧��

86

∴ 𝜎𝑥 = 𝜈�𝜎𝑦 + 𝜎𝑧� [3-2]

where 𝐸 is a compression modulus for changes in total stress and 𝜈 is Poisson's ratio.

By substituting the value of 𝜎𝑦 from equation [3-1] in eq. [3-2], it is possible to obtain a

relationship between stresses in x and y directions.

𝜎𝑥 = 𝜈2(𝜎𝑥 + 𝜎𝑧) + 𝜈𝜎𝑧

Hence,

𝜎𝑥 =𝜈

1 − 𝜈𝜎𝑧 [3-3]

In natural ground conditions, however, matric suction also takes a part of the stress and

strain values. When considering the effect of matric suctions as the potential strain, the

effective strain in y direction can be written as

𝜀𝑦 =1𝐸

�𝜎𝑦 − 𝜈(𝜎𝑥 + 𝜎𝑧)� − 𝛼𝛥𝑤

where, 𝛼Δ𝑤 = 𝑆 𝐻⁄ . 𝑆 is the matric suction and 𝐻 is the modulus with respect to matric

suction. It should be noted that Morris et al. (1992) used directly 𝑆 𝐻⁄ as the strain

component without using the change in moisture content. As noted previously, for plane

strain condition, strains in the x and y directions are equal to zero,

𝜀𝑦 = 0 =1𝐸

�𝜎𝑦 − 𝜈(𝜎𝑥 + 𝜎𝑧)� − 𝛼𝛥𝑤

𝜎𝑦 = 𝜈(𝜎𝑥 + 𝜎𝑧) + 𝐸(𝛼𝛥𝑤) [3-4]

and

𝜀𝑥 = 0 =1𝐸

�𝜎𝑥 − 𝜈�𝜎𝑦 + 𝜎𝑧�� − 𝛼Δ𝑤


𝜎𝑥 = 𝜈�𝜎𝑦 + 𝜎𝑧� + 𝐸(𝛼Δ𝑤) [3-5]

Using equation [3-4] and [3-5]

𝜎𝑥 = �𝜈

1 − 𝜈� 𝜎𝑧 + 𝐸

(1 + 𝜈)(1 − 𝜈2) 𝛼Δ𝑤

or in terms of suction, S,

𝜎𝑥 = �𝜈

1 − 𝜈� 𝜎𝑧 +

𝐸𝐻

1(1 − 𝜈) 𝑆 [3-6]

where 𝜎𝑧 = 𝛾𝑧, 𝛾 is the unit weight of the soil, 𝛼 is the hydric constant and 𝛥𝑤 is the

moisture content change. A similar relationship was derived by Morris et al. (1992)

using elastic theory.

𝑧𝑐 =(1 − 2𝜈)

𝜈𝛾𝑆0 +

(1 − 𝜈)𝜈𝛾

𝜎𝑡 [3-7]

The results obtained from equation [3-7] are presented in Figure 3-2.

Figure 3-2 Depths of cracking with different Poisson’s ratios tensile strength values when the constant suction profile is assumed throughout the depth

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100

Dep

th o

f Cra

ckin

g (m

)

Suction (kpa)

v=0.3

v=0.35

v=0.4

t = -10kPa

t = -20kPa σt

σt

88

3.2.2 Linearly decreasing suction profile

In this section, the boundary conditions of the clay layer are assumed to be a constant

atmosphere flux boundary at the top and steady state conditions of moisture equilibrium

with the bottom boundary as the water table. The resulting suction profile would be a

linearly decreasing profile with depth as shown in Figure 3-3 (Morris et al., 1992).

Dep

th

Suction

W

S0S

zD

epth

Tensile strength

W

σt(0)σt

Figure 3-3 Suction and tensile strength profiles when linearly decreasing with depth

The values of suction in various depths were calculated using equation [2-5] and the

tensile strength was calculated similarly to the previous section using equation [2-6].

The crack depth corresponding to this type of suction profile through linear elastic

analysis is given by equation [2-7]. When allowed for shear failure in elastic approach,

the relationship becomes the form in equation [2-13]. The results from those equations

are presented in Figure 3-4. It can be seen that the shear failure criterion for fracture


gives higher crack depths than those obtained using the linear elastic method with

tensile strength failure.

Figure 3-4 Predicted crack depth values for linearly decreasing suction profile through linear elastic (LE) approach and allowing for shear failure (SF) approach with different surface

suction values

3.2.3 Parabolic suction variation

The real field suction profiles are neither linearly decreasing nor parabolic exactly. The

actual variations were given in Figure 2-5 for moisture content and in Figure 2-7 for

suctions. However, in many cases, a parabolic variation can be considered to be a better

approximation for the field suction profile. The variation of suction with depth for this

2 4 6 8 10 0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

Depth to water table (m)

Dep

th o

f cra

ckin

g (m

) LE-Suction=100kPa LE-suction=50kPa LE-Suction=25kPa SF-Suction=100kPa SF-Suction=50kPa SF-Suction=25kPa

90

variation is given by equation [3-8]. The tensile strength has the same relation with

suction as used in previous sections.

𝑆 = 𝑆0 �1 −𝑧2

𝑊2� [3-8]

D

epth

Suction

W

S0S

z

Dep

th

Tensile strength

W

σt(0)σt

z

Figure 3-5 Parabolic suction and tensile strength profiles

Following the same analysis approach as used for constant and linear suction profiles,

the crack depth is calculated for a parabolic suction profile. The corresponding crack

depth for linear elastic approach is given in equation [3-9] while the shear failure

criterion gives equation [3-10].

𝑧𝑐 =𝑊2

2�−

𝐶1𝛾𝑆0𝐵

± ��𝐶1𝛾𝑆0𝐶2

�2

+4

𝑊2� [3-9]

where,

𝐶1 = 𝜈1−𝜈

and 𝐶2 = 1−2𝜈1−𝜈

− 0.5𝑡𝑎𝑛𝜙𝑏


𝑧𝑐 =𝑊2

2�−

𝛾𝑆0𝐶3

± ��𝛾

𝑆0𝐶3�

2+

4𝑊2�

[3-10]

where,

𝐶3 =1 − 𝑠𝑖𝑛𝜙′

𝑡𝑎𝑛𝜙𝑏(2𝑐𝑜𝑠𝜙′ − 0.5𝑠𝑖𝑛𝜙′ − 0.5)

Figure 3-6 Predicted Crack depths when the water table is 4m below the surface (LE-linear elastic approach, SF-elastic allowing for shear failure and LEFM-linear elastic fracture

mechanics approach)

The crack depths obtained from the above equations are presented in Figure 3-6 and

Figure 3-7 below with different surface suction values and water table depths. Those

figures show the crack depths for parabolic suction variation within the soil mass in the

LE and SF approaches. However, the LEFM approach uses its own stress distribution

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 20 40 60 80 100

Crac

k D

epth

(m)

Surface suction (kPa)

LE SF LEFM

92

profile. The values of the LEFM approach have been calculated using equation [2.16]

which is discussed in section 2.3.4.5.

Figure 3-7 Predicted Crack depths using parabolic suction profile when the surface suction is 50kPa

These theoretical results will be compared and discussed with the numerically obtained

values later in this chapter.

3.3 Numerical Modelling Approach for Crack Depth Prediction

Numerical modelling is extensively used in understanding and predicting complex

natural phenomena due to its effectiveness. Hence, it is being used herein for modelling

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10

Dep

th o

f Cra

ckin

g (m

)


LE SF LEFM


desiccation cracking. As presented in section 2.4. The Universal Distinct Element Code

(UDEC) computer program is chosen for the analysis of desiccation cracks in a

hypothetical compacted clay liner based on the linear elasticity method and allowing for

shear failure in bulk media.

3.3.1 Overview of UDEC program

UDEC is a widely used numerical program in rock modelling. It simulates two

dimensional discontinuous media based on the distinct element method (Itasca, 2004).

The discontinuous media are represented as assemblages of discrete blocks. The

discontinuities or joints are treated as boundary conditions between blocks. Individual

blocks behave as either rigid or deformable material. Deformable blocks are subdivided

into a mesh of finite-difference elements, and each element responds according to a

prescribed linear or nonlinear stress-strain law. UDEC has several built-in material

behaviour models, for both the intact blocks and the discontinuities, which permit the

simulation of the response of discontinuous geologic or other similar materials.

The method of analysis in UDEC is based on the Newton’s equation. This equation of

motion is damped to reach a force equilibrium state under the applied initial and

boundary conditions. Two forms of velocity proportional damping are provided in

UDEC; first, adaptive global damping (referred as auto), second, local damping. Auto

damping is used to adjust the damping constant automatically using the viscous

damping forces. In local damping the damping force on a node is proportional to the

magnitude of the unbalanced force. UDEC also contains a built-in programming

language FISH. User defined functions can be written with FISH, to extend UDEC’s

usefulness. FISH offers a unique capability to UDEC users who wish to tailor analyses

94

to suit specific needs. Therefore this software has been identified as an attractive

program to model cracking.

The desiccation cracking is a three dimensional problem in the field, where cracks

would propagate down the depth and generally across the depth. However when one

crack is considered in a finite soil block, the 2D conditions are acceptable since any soil

block is reasonably uniform along the crack length which is the normal plane of the

plane of analysis. The size of the model and the modeling time can be greatly reduced

using the plane strain approximation. It should be remembered, however, that the 2D

approximation has limitations, especially in relation to prediction of cracking in the x-y

plane.

3.3.2 Numerical model implementation

This section is to reproduce the theoretical results shown in the section 3.2 numerically.

Hence, a compacted clay layer was analysed as shown in Figure 3-8. Since the intention

is to study the propagation of an isolated crack in plane strain, an interface (or joint) is

placed where the crack is intended to propagate. The boundary conditions are such that

the top surface is free of stresses, the edge boundaries are restrained against lateral

deformation and the bottom boundary is restrained against vertical deformations. The

groundwater level is placed at the bottom of the layer.


Depth of clay liner

Water table

Possible Crack

Length of the clay liner

ZcZ

3.3.2.1 Problem geometry

The clay block was created in the UDEC program using co-ordinates. For crack

simulation in UDEC the blocks should be split into several subsidiary blocks, using pre-

decided joints which open up with the change of stresses representing crack formation.

The water table was considered to be at the bottom boundary of the soil block. This

condition was accomplished by using the suction stresses within the block. Hence, the

depth to water table was changed by changing the depth of the block. Since a finite

block is considered, the distance to the boundaries of the block from the crack

influences in the results considerably. The size of the zones within the deformable

blocks is also a considerable factor in obtaining precise results. Both sides and bottom

of the soil layer were supported by roller supports.

Figure 3-8 Cross section of the clay layer used for the model

96

Several test models were conducted with the same input parameters but changing the

size of the block to find the distance to the boundary from the crack to reduce boundary

effects. Some typical properties of compacted clay were used in the model setup. The

width of the clay layer was increased from 5m to 50m until the model gave consistent

results. This was done with two different block heights of 4 m and 10 m. Figure 3-9

gives the results obtained.

Figure 3-9 Variation of depths of cracking with layer width when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡=0kPa and 𝑆0=50kPa

Initially, with the increase of the length of the clay layer the resulting crack depth also

increased. Finally the crack depths became stabilized after a certain layer length. For

both layers with 4m and 10m heights, the consistent crack depth is almost 3 m and the

2

2.2

2.4

2.6

2.8

3

3.2

0 10 20 30 40 50

Dep

th o

f Cra

ckin

g (m

)

Length of the layer (m)

W = 4m

W = 10m


layer length needs to be at least 30 m. According to this observation, a relationship was

deduced for the crack depth and layer length so that the layer length should be more

than 10 times the crack depth (i.e. Length ≥ 10 × 𝑧𝑐, where 𝑧𝑐 is the crack depth). In

other words, the boundary of the layer should be more than 5 times the crack depth

distant from the crack. Initially the crack depth can be estimated from a theoretical

method and according to that the length of the clay layer can be decided. The advantage

of selecting the correct size of the block without selecting larger sizes is minimizing the

run time of models.

Figure 3-10 Zones and blocks of the model

The effect of mesh size on the results was examined as the next step. A block of 4 m

height× 5 m (𝑊 =4 m) width and 10 m× 10 m (𝑊 =10 m) layers were selected to get

the optimum mesh size. The mesh consists of triangular zones with the maximum edge

length of 0.5m as shown in Figure 3-10. The models were run until the results did not

change as the mesh size was reduced further. Figure 3-11 shows the variation of crack

Join

Block Zone

Zone edge length

98

depth with mesh size. In all models in this series, the soil was assigned zero tensile

strength, 5MPa compression modulus (𝐸), 1000 suction increments and 50kPa surface

suction. The tensile strength and the suction were kept constant throughout the depth.

Here also two model series were analysed with 10m and 4m deep water tables.

When decreasing the mesh size in both models with different block heights, (𝑊) the

crack depths reduce and finally achieve a constant value. In both cases, the mesh

dependency ceases when the mesh size is less than 0.1m. Hence, a mesh size of 0.1m

was selected as the optimum mesh size for all the models used in this analysis.

Figure 3-11 Variation of crack depths with mesh size when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡=0kPa and 𝑆0=50kPa

Other than the mesh size and size of the block, the damping method and method of

changing stresses with time also generally affects the results in UDEC models. Hence,

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 0.1 0.2 0.3 0.4 0.5

Dep

th o

f Cra

ckin

g (m

)

Mesh Size (m)


those factors also tested for the above model. As discussed earlier, damping auto and

local was used for the testing. Although, no significant change was observed, damping

local was selected for the modelling due to the fact that local damping is more suitable

to minimize oscillations that may arise when abrupt failure occurs in the model (Itasca,

2004). Local damping is also recommended for static analysis in most instances.

UDEC needs many time steps to reach the equilibrium state since it was explicit method

of analysis. Hence, the effect of number of time steps between increments on results

was also tested. However, the number of time steps used was not too low starting from

500 steps. The results obtained showed only a negligible difference and, therefore, 1000

steps were selected as a reasonable number considering the unbalanced force and the

model run time. However, this approach was reviewed depending on the reduction of

unbalanced forces, which was the ultimate criterion for achieving equilibrium.

The suction while desiccating the soil layer was changed by increasing the tensile

stresses in cumulative increments. The following constitutive relation was used for

modelling suction stress (Rajeev and Kodikara, 2011).

𝛥𝜎𝑖𝑗 = 𝐷𝑖𝑗𝑘𝑙(𝛥𝜀𝑘𝑙 − 𝛥𝜀𝑠ℎ𝛿𝑘𝑙) [3-11]

where 𝛥𝜎𝑖𝑗 represents the incremental stresses induced by expansion of the soil

(compressive stress/strain are negative), 𝐷𝑖𝑗𝑘𝑙 is the tangent stiffness tensor and 𝛥𝜀𝑘𝑙 is

the observed (or mechanical) strain and 𝛿𝑘𝑙 is the Kronecker delta. The strain increment

𝛥𝜀𝑠ℎ is the isotropic free shrinkage increment experienced by the soil due to a suction

increment or a moisture decrease. This approach is acceptable for continuous drying

events as simulated in this chapter.

100

Z

ΔS = ∆σx

ΔS = ∆σy

ΔS = ∆σz

x

y

Three suction profiles were utilized similar to Section 3.2: a constant suction (Figure

3-1), a linearly decreasing suction along the depth (Figure 3-3) and a parabolic suction

variation (Figure 3-5). The linearly decreasing suction profile can be given by the

equation,

𝑆 = 𝑆0(1 − 𝑧 𝑊⁄ ) [3-12]

and the parabolic variation follows equation,

𝑆 = 𝑆0(1 − 𝑧2 𝑊2⁄ ) [3-13]

According to those, the suction is decreasing from the top surface (maximum suction) to

the ground water table (zero suction). The tensile strength of the soil and the joint was

also changed with the suction according to the relation,

𝜎𝑡 = 𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′ [3-14]

in each case. The value of 𝜙𝑏 is selected as (𝜙 − 50) following Morris et al. (1992)

according to their interpretation of data presented by Fredlund et al. (1978).

When plane strain conditions are assumed for the numerical analysis, the strains in x

and y directions and stress in z direction are, 𝜀𝑥 = 0, 𝜀𝑦 = 0 and 𝜎𝑧 = 𝛾𝑧.

Figure 3-12 Stress conditions in a soil particle at each increment


The following description shows how the suction increments are applied in the UDEC

model. If we consider an element within the desiccating soil, the stress increments can

be shown as in Figure 3-12. The suction increment (Δ𝑆) is applied in all three directions

which will increase the stress in each direction by Δ𝜎. This consideration is approximate

only for soil closest to saturation. Since clay liners are generally compacted wet of

optimum, it is considered this is reasonable.

At the beginning before applying suction, the strains in each direction can be written as,

𝜀𝑥 =1𝐸

�𝜎𝑥 − 𝜈�𝜎𝑦 + 𝜎𝑧�� [3-15]

𝜀𝑦 =1𝐸

�𝜎𝑦 − 𝜈(𝜎𝑥 + 𝜎𝑧)� [3-16]

𝜀𝑧 =1𝐸

�𝜎𝑧 − 𝜈�𝜎𝑥 + 𝜎𝑦�� [3-17]

After the first cycle 𝜎𝑥 , 𝜎𝑦 and 𝜎𝑧 become 𝜎𝑥 + Δ𝑆, 𝜎𝑦 + Δ𝑆 and 𝜎𝑧 + Δ𝑆 respectively

where Δ𝑆 equals, Δ𝜎𝑥, Δ𝜎𝑦 and Δ𝜎𝑧 respectively. Then the new strains become,

𝜀𝑥 =1𝐸

�(𝜎𝑥 + Δ𝑆) − 𝜈�𝜎𝑦 + 𝜎𝑧 + 2Δ𝑆��

Since 𝜀𝑥 = 0,

𝜎𝑥 = 𝜈�𝜎𝑦 + 𝜎𝑧� − (1 − 2𝜈)Δ𝑆 [3-18]

Similarly, in the y direction,

𝜀𝑦 =1𝐸

��𝜎𝑦 + Δ𝑆� − 𝜈(𝜎𝑥 + 𝜎𝑧 + 2Δ𝑆)�

Since 𝜀𝑦 = 0,

𝜎𝑦 = 𝜈(𝜎𝑥 + 𝜎𝑧) − (1 − 2𝜈)Δ𝑆 [3-19]

102

But for the z direction where 𝜀𝑧 is not equal to zero,

𝜀𝑧 =1𝐸

�𝜎𝑧 − 𝜈�𝜎𝑥 + 𝜎𝑦�� +(1 − 2𝜈)

𝐸Δ𝑆

When substituting the value of yσ (equation [3-19]) in equation [3-18],

𝜎𝑥 = 𝜈[𝜎𝑧 + {𝜈(𝜎𝑥 + 𝜎𝑧) − (1 − 2𝜈)Δ𝑆}] − (1 − 2𝜈)Δ𝑆

𝜎𝑥 =

𝜈(1 − 𝜈) 𝜎𝑧 −

(1 − 2𝜈)(1 − 𝜈) Δ𝑆

[3-20]

The stress relationship for stress in the x direction (equation [3-20]) is similar to the

relationship derived by Morris et al. (1992).

3.4 Comparison of Numerical and Theoretical Results

Numerical models were run to predict the analytical results discussed in section 3.2. The

same three suction profiles were followed with two different theoretical assumptions,

namely the linear elastic method and the cracking with shear failure method. For these

analyses, the same input parameters as used by Morris et al. (1992) were selected; in

order to compare both results easily.

Numerical models were run with two layer thicknesses, 4m and 10m at which depths

the water table was placed in each case. For all models, the Poisson’s ratio (ν) was

chosen as 0.3. The results significantly change with the Poisson’s ratio as shown in

Figure 3-13. Hence the Poisson’s ratio should be carefully selected when modelling

crack depth numerically. The effective cohesion was assumed to be zero for all the

models in the linear elasticity method.


Figure 3-13 Effect of the Poisson’s ratio on crack depth when 𝐸 =5MPa, 𝜎𝑡 = 0.5 𝑆 𝑡𝑎𝑛𝜙𝑏, 𝑊 =4m and 𝑆0=50kPa

The modulus of elasticity was selected as a constant value of 5MPa although it is a

function of suction. However, the effects of varying modulus of elasticity were also

examined. Young’s modulus was changed with suction based on 𝐸 = 𝐸𝑠 + 10 × 𝑆 where

𝐸 is the Young’s modulus, 𝐸𝑠 is the modulus at saturation, and 𝑆 is suction at any depth

based on relations between modulus and suction provided by Kodikara et al. (2004) and

Sawangsuriya et al. (2009). 𝐸𝑠 was taken to be 5MPa in a study on the effects of varying

modulus.

Figure 3-14 shows the results obtained numerically with and without changing Young’s

modulus as a function of suction. There is no significant difference between the two

results especially when the lower water table depths are used. Hence, it was decided that

changing Young’s modulus has no discernible effect on the results obtained

numerically.

0

0.5

1

1.5

2

2.5

3

3.5

0.25 0.3 0.35 0.4

Dep

th o

f Cra

ckin

g (m

)

Poisson's Ratio

104

Figure 3-14 Crack depths changing with the depth to water table when 𝐸𝑠=5MPa, 𝐸 = 𝐸𝑠 +10𝑆, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊 =4m, 𝑆0=50kPa and 𝜙 =300

3.4.1 Constant suction profile (suction profile 1)

In this section, the desiccation cracks were produced using the numerical model UDEC

similar to the theoretical assumptions in Section 3.2.1. The constant suction profile was

applied incrementally as shown in Figure 3-15. Suction increment Δ𝑆 was calculated by

dividing the final suction by the number of suction increments. For the linear elastic

analysis the in-built elastic isotropic constitutive model was selected for the block

material and for the joint materials area contact-coulomb slip with residual strength was

selected.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2 4 6 8 10

Dep

th o

f Cra

ckin

g (m

)

Depth to Water Table (m)

Analytical, Morris et al. (1992)

Numerical with constant E

Numerical with changing E


Dep

th

Suction

Wf

S0,fS0,4S0,3S0,2S0,1

ΔS

S0,f-1

Figure 3-15 Appling suction with time

When the suction increases in the zones within blocks with time, the joint opens

creating a gap inbetween the blocks. Figure 3-16 shows the joint of the model and

velocity vectors of zones after opening the crack. The length of that opening from top to

the last displaced point was recorded as the crack depth. The crack depths obtained are

shown in Figure 3-17 and Figure 3-18 in comparison to the analytical results.

The crack depths obtained from the numerical model were larger than the calculated

results when the tensile strength of the soil was zero in Figure 3-17 and Figure 3-18.

This is possible since Morris et al. (1992) do not allow for stress redistribution after

crack opening and in contrast the numerical model does open the crack, which

redistributes the developed tensile stresses. In order to verify this explanation, a

numerical model was also designed with a very high tensile strength at the joint to

106

preclude cracking and obtain the stress distribution expected numerically. When the

tensile strength was set to a high value, there was no crack propagation in the soil block,

but the stress distribution profile assumed in the numerical analyses by Morris et al.

(1992) could be replicated. The depth from the surface to the point where stresses

change from tension to compression was considered as the crack depth.

Figure 3-16 Crack of a model after opening

With high tensile strength (in this case 100kPa was selected) the predicted results were

similar to the analytical results using Morris et al. (1992) (Figure 3-17) with a 10 m

deep clay layer. The same kind of variation was observed when the depth to water table

was decreased to 4 m (i.e., decrease the depth of suction profile) while keeping other

parameters constant as shown in Figure 3-18. This variation would be due to the

differences in implementation of the results in two models, where the UDEC model

Depth of the crack


used the incremental implementation in contrast to the Morris et al analytical model.

This checking was not undertaken for 10m depth water table since results would be

expected to be similar and it would require substantial computer time to complete.

Figure 3-17 Variation of crack depth with surface suction when 𝐸 =5MPa, 𝜈 =0.3, 𝑊 =10m, 𝜎𝑡=constant and 𝑆0=50kPa

The comparison shows that the numerical model produces deeper desiccation cracks

than the Morris et al. (1992) analytical model would produce. This analytical model

gives the value for the crack depth at a depth where the horizontal tensile force equals or

exceeds the given tensile strength. However, in reality the crack develops progressively.

When the crack opens by a small amount, the stresses applied at the face of the crack

then becomes zero and stresses concentrate at the tip of the crack. This causes the crack

0

1

2

3

4

5

6

10 20 30 40 50

Dep

th o

f Cra

ckin

g (m

)

Surface Suction (kPa)

Linear Elasticity Solutions (W = 10m)

Analytical (Morris et al; 1992)

Numerical, σt=100kPa

Numerical , σt=0kPa

σt

σt

108

to extend further. This phenomenon happens until the stresses at the tip of the crack are

getting less than the tensile strength of the soil. UDEC model allows for progressive

crack opening and stress redistribution along the crack which is unaccounted for in the

analytical model. This reasoning was verified by the results obtained for the numerical

models with high tensile strength where no physical crack propagation took place

(Figure 3-17 and Figure 3-18). In those models, the stresses were not redistributed since

no cracks opened up. Hence those crack depths obtained are similar to the

corresponding analytical results.

Figure 3-18 Variation of crack depth with surface suction when 𝐸 =5MPa, 𝜈 =0.3, 𝑊 =4m 𝜎𝑡=constant and 𝑆0=50kPa

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

10 20 30 40 50

Dep

th o

f Cra

ckin

g (m

)


Linear Elasticity Solutions (W = 4m)

Analytical Numerical, σt=100kPa Numerical, σt=0kPa σt

σt

(Morris et al; 1992)


3.4.2 Linearly decreasing suction profile (suction profile 2)

When the suction profile decrease linearly with the depth as discussed in Section 3.2.2,

the crack depths were computed numerically in order to compare with the analytical

results. Both linear elastic and elastic allowing for shear failure was used for

comparison. In the linear elastic approach, shear failure in the bulk medium is not

accommodated for, similarly to Morris et al.’s analytical formulation. Therefore, the

models were rerun with shear strength of the bulk medium specified as Mohr-Coulomb

model in UDEC.

Depth

Suction

Wf

S0,fS0,4S0,3S0,2S0,1

W1

W2

W3

W4

ΔS

ΔW

Figure 3-19 Changing the suction in UDEC

As shown in Figure 3-19, suction at the surface increases by an amount of Δ𝑆 and the

depth of suction profile goes down by an amount of Δ𝑊 in each step. This

representation is similar to progression of drying due to diffusion of soil moisture by

evaporation. Δ𝑆 was calculated by dividing the final surface suction by a number of

suction increments and Δ𝑊 was calculated by dividing the final depth to water table by

110

the same number of suction increments. The suction values inbetween the top surface

and the bottom water table were calculated using the corresponding suction profile

given in equation [3-12].

With this suction profile, the variation of crack depth was obtained while changing the

depth of suction profile and surface suction. Results obtained through the approaches of

linear elastic and allowing for shear failure are presented in separate figures. The

variation of crack depth was obtained with the depth to water table and surface suctions

in both approaches. In all figures the numerical results are compared with the analytical

results.

Figure 3-20 Variation of crack depth with depth to water table when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = 0.5 𝑆 𝑡𝑎𝑛𝜙𝑏

0.0

1.0

2.0

3.0

4.0

5.0

1 2 3 4 5 6 7 8 9 10

Dep

th o

f Cra

ckin

g (m

)


Analytical Results, S0=25kPa Numerical Results, S0=25kPa Analytical Results, S0=50kPa Numerical results, S0=50kPa

Linear Elastic Approach


Figure 3-21 Variation of crack depth with the surface suction when when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = 0.5 𝑆 𝑡𝑎𝑛𝜙𝑏 and 𝑊 =4m

Figure 3-20 shows the variation of the depth of cracking with the depth to water table

when subjected to 50kPa and 25kPa surface suction. Figure 3-21 shows the variation of

depth of cracking with different surface suctions comparing with the corresponding

analytical results. Similarly to the results previously presented, the numerical model

predicts higher crack depths than the analytical model. In fact, the numerically predicted

crack depths are almost double the analytical predictions. When surface suction

increases beyond 80kPa in Figure 3-21, the clay layer cracks in its entire depth.

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1.50

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Linear Elastic Approach, W = 4m

112

Capp = C' + S tanϕb

τ

ϕ

σt σn

q

S

M

pnet

q

pnet

σt = 0.5 S tan ϕb

σt = σt (0)

(b)(a)

In the modelling allowing for shear failure, the shear failure envelope and the tensile

strength limit were imposed as shown in Figure 3-22. The shear failure envelope

modified for tensile failure is shown in Figure 3-22(a) (after Bagee (1985)), which is

known as Mohr-Paul failure diagram. Figure 3-22(b) shows the effect of suction on

failure criteria developed to include tension (after Morris (1992)). In this method, the

soil blocks were discretized as continuum elements with Mohr-Coulomb plasticity

behaviour and the joint in the middle with coulomb slip with residual strength behaviour

in shear.

The results are shown in Figure 3-23 and Figure 3-24. First one compares the change of

crack depth with the change of depth to water table with analytical results. Then the

variation of crack depth with surface suction is given in the next figure. The Young’s

modulus and the Poisson’s ratio were selected as 5MPa and 0.3 respectively. Cohesion

and tensile strength properties were set to change as functions of changing suction

within the soil block as the crack progressively opened. Following the functional

Figure 3-22 (a) Strength envelope indicating tensile failure showing the effect of tensile strength, (b) Schematic of failure envelope in tension


relationships used previously, the cohesion was given by 𝑐 = 𝑆0(1 − 𝑧 𝑊⁄ )𝑡𝑎𝑛𝜙𝑏 and the

tensile strength was given by 𝜎𝑡 = 0.5𝑐.

Figure 3-23 Variation of depth of cracking with depth to water table when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300

From the results shown in Figure 3-23 and Figure 3-24 it is clear again that the

numerical model predicts higher results. However, the difference predicted by the

curves is getting less compared to the results obtained from the linear elastic method.

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Numerical, UDEC

shear failure approach

114

Figure 3-24 Variation of crack depth with surface suction when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 =−𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊 =4m and 𝜙 =300

3.4.3 Parabolic suction variation (suction profile 3)

Dep

th

Suction

Wf

S0,fS0,4S0,3S0,2S0,1

W4

W3

W2

W1

Figure 3-25 Applying suction in increments for parabolic variation

0

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Shear Failure method solutions, W=4m


As the final suction profile, a parabolic variation was followed as discussed in section

3.2.3 following equation [3-8] which is a better representation of the real field

condition. In this series of models similar input parameters were also used as the soil

properties and the suction applied in cumulative increments as shown in Figure 3-25.

Both linear elastic approach and shear failure approach were used in this analysis. The

results were then compared with the analytical results. Figure 3-26 and Figure 3-27

illustrate the results obtained through the linear elastic approach with surface suction

variation and depth to water table respectively. Also Figure 3-28 and Figure 3-29 show

the comparison of crack depth predictions through shear failure approach as before,

with surface suction and depth to water table.

Figure 3-26 Effect of surface suction on depth of cracking predictions for parabolic suction variation; when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊 =4m and 𝜙 =300

0

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)


Analytical Numerical

Linear elastic approach

116

Figure 3-27 Effect of depth to water table on depth of cracking predictions for parabolic suction variation; when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300

Figure 3-28 Depth of cracking variation with surface suction for parabolic suction profile when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊=4m and 𝜙 =300

0

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Analytical

Numerical

Linear elastic approach

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Analytical Numerical

Shear failure approach


Figure 3-29 Depth of cracking variation with water table depth for parabolic suction profile when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300

The parabolic suction profile in clay layers is more realistic and predicts more

reasonable values for crack depths in comparison to the observed crack depths in the

field given in Table 2-1. When the linear elastic theoretical assumptions are used, the

numerical model predicts larger crack depths in comparison to those of the analytical

model ((Morris et al., 1992). In the shear failure approach, the results from both

methods appear to produce closer crack depth values.

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Shear failure approach

118

3.5 Comparison of Results from Different Suction Profiles

The suction profile is the most important input information in desiccation crack

modelling. This is evident from the fact that the crack depths change quite significantly

as the suction profile changes. It can be expected that a more realistic suction profile

leads to more realistic crack depth values. However, due to the high complexity of the

suction profile below the ground surface it is reasonable to assume a representative

simple profile for analysis.

Figure 3-30 Behaviour of crack depth predictions with surface suction based on the suction profile; when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊=4m and 𝜙 =300

The suction profiles 1, 2 and 3 were compared in Figure 3-30 and Figure 3-31 to

examine the effects of suction profiles. The results shown in both figures are for the

linear elastic approach with the same input parameters. Both figures show that the least

crack depths are predicted by the suction profile 3. In addition, the minimum difference

0

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6

7

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suc prof 1-Analytical Suc prof 1-Numerical suc prof 2-Analytical suc prof 2-Numerical suc prof 3-Analytical suc prof 3-Numerical


between the numerical and analytical results is also observed when the suction profile 3

is used for the stress variation.

Figure 3-31 Behaviour of crack depth predictions with depth to water table based on the suction profile; when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300

Similar results can be observed when shear failure is allowed for in the model (Figure

3-32 and Figure 3-33). The results given in Figure 3-32 and Figure 3-33 illustrate that

both analytical and numerical results produce similar results when the shear failure was

allowed. As shown in Figure 3-33, the crack depths predicted by the numerical method

are less than those predicted by the analytical method for the suction profile 3. A

possible reason for this occurrence is that the numerical method redistributes stresses

during crack opening and therefore, shear failure may occur to a larger extent thereby

reducing the crack depths. However, it can be expected that this result will depend on

the parameters used to define the shear failure of the soil.

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Suc prof 2-Analytical Suc prof 3-Analytical Suc prof 2-Numerical Suc prof 3-Numerical

Linear Elastic Approach

120

Figure 3-32 Behaviour of crack depth predictions with surface suction based on suction profile allowing for shear failure; when 𝐸=5MPa, 𝜈=0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙=300

Figure 3-33 Behaviour of crack depth predictions with surface suction based on the suction profile; when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊=4m and 𝜙 =300

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suc prof 2-Analytical suc prof 2-Numerical suc prof 3-Analytical suc prof 3-Numerical

Shear failure Approach

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suc prof 2-Analytical suc prof 2-Numerical suc prof 3-Analytical suc prof 3-Numerical

Shear failure Approach


3.6 Comparison of Different Theoretical Approaches

The values obtained through the different theoretical approaches are compared in this

section for the suction profiles 2 and 3. The suction profile 1 is highly idealistic in the

case of deep clay layers and predicts larger crack depths than expected. The suction

profile 1 is more applicable to a case where the ground is covered by an impermeable

barrier and the suction has equilibrated with the water table.

Figure 3-34 Comparison of results obtained through different theoretical assumptions with depth to water table variation when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa

and 𝜙 =300 (SS – allowing for shear failure, LE – Linear elastic)

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SS-Analytical SS-Numerical LE-Numerical LE-Analytical

122

Figure 3-35 Comparison of results obtained through different theoretical assumptions with surface suction variation, when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300 (SS –

allowing for shear failure, LE – Linear elastic)

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5

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Dep

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)

Surface Suction (m)

SS- Analytical

SS- Numerical

LE- Numerical

LE- Analytical


Figure 3-36 Comparison of results obtained through theoretical assumptions with depth to water table variation when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙 =300

(SS – allowing for shear failure, LE – Linear elastic)

Figure 3-34 to Figure 3-37 show the comparisons of results from linear elastic and shear

failure approaches with depth to water table and surface suction. Generally, the crack

depths from the numerical results are higher than those from the analytical results, for

the reasons explained earlier. Similarly, when shear failure is allowed for, the

discrepancy between the crack depths is reduced more than in the linear elasticity

approach.

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LE-Numerical

LE-Analytical

124

Figure 3-37 Comparison of results obtained through different theoretical assumptions with surface suction variation when 𝐸=5MPa, 𝜈=0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑆0=50kPa and 𝜙=300 (SS –

allowing for shear failure, LE – Linear elastic)

In order to improve the analytical model with linear elasticity to incorporate stress

redistribution, it was attempted to redistribute the stresses due to opening of the crack in

the following section. With redistribution of stresses, the crack tip attracts higher

stresses, which may be able to be captured using LEFM theoretical approach. As

appears in LEFM theory, the stress intensity factor was selected as proposed by

Bentham et al. (1973) based on asymptotic approximation, given in equation [3-21].

𝐾𝐼 = (0.6825𝑃 + 0.439𝑄)�𝜋𝑧𝑐 [3-21]

0.0

0.5

1.0

1.5

2.0

2.5

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4.0

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)

Surface Suction (m)

SS- Analytical

SS- Numerical

LE- Numerical

LE- Analytical


where 𝐾𝐼 is the stress intensity factor, 𝑃 and 𝑄 are the horizontal normal stresses at the

tip of the crack and top surface of the crack respectively. The stresses based on stress

intensity factor were calculated using equation given by,

𝜎𝑥 =𝐾𝐼

√2𝜋𝑟 [3-22]

where 𝑟 is the distance from the crack tip. The stress from equation [3-22] is added to

that in equation [3.20] to get the updated tensile stress, which is then used to calculate

the depth of cracking compared with the tensile stress value as undertaken by Morris et

al. (1992). The new equation to calculate the depth of cracking comes out as a cubic

equation as given below.

𝐶42𝑧𝑐

2 − (2𝐶4𝐶5 − 𝐶4𝑧𝑖)𝑧𝑐2 + �𝐶5

2 + 2𝐶4𝐶5𝑧𝑖�𝑧𝑐 − �𝐶52𝑧𝑖 +

𝐾𝐼2

2𝜋� = 0

[3-23]

𝐶4 =𝜈

(1 − 𝜈) 𝛾 +(1 − 2𝜈)(1 − 𝜈)

𝑆0

𝑊−

0.5𝑆0𝑡𝑎𝑛𝜙𝑏

𝑊

𝐶5 =(1 − 2𝜈)(1 − 𝜈)

𝑆0

𝑊− 0.5𝑆0𝑡𝑎𝑛𝜙𝑏

where 𝑧𝑖 is the depth of cracking from equation [3-20]. Equation [3-23] was solved and

the results are shown in Figure 3.38 referring to it as ‘modified analytical’.

The depth of cracking also can be produced using LEFM approach and considering the

fracture energy as presented by Morris et al. (1994). In this instance the crack depth is

determined by the condition that the stress intensity factor falls below the critical stress

intensity factor. The method proposed by this approach is discussed in the Section

2.3.4.2 under the LEFM approach. By using the equation proposed for crack depths

(equation [2-16]) the crack depths were calculated and are shown in Figure [3-38]

126

referred to as ‘LEFM-Morris et al. (1994)’. The results from this method depend on the

used critical stress intensity factor used.

Figure 3-38 Comparison of values of crack depth with surface suction variation when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛𝜙𝑏𝑐𝑜𝑡𝜙′, 𝑊=4m and 𝜙 =300

The results obtained through the LEFM approach are similar to the results produced by

UDEC model through the linear elastic approach. Similarly, the linear elastic analytical

approach with stress redistribution (‘modified analytical’) produces quite similar results

to both above methods. Therefore, it is possible to say that an improved analytical

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LEFM- Morris et al. (1994) Modified analytical LE- Analytical SS- Numerical SS- Analytical LE- Numerical


solution has been developed to predict the crack depths when shear failure is not taking

place. As explained earlier, the shear strength approach produces smaller crack depths.

It is useful to present the results obtained so far in non-dimensionless form to generalise

the results for other geometries. The properties that govern the desiccation cracking

process were identified as surface suction, tensile strength, suction profile, depth to

water table, soil unit weight, friction angle and the Poisson’s ratio. The crack depth (𝑧𝑐)

can then be written as,

𝑧𝑐 = 𝑓(𝜈, 𝑊, 𝜎𝑡, 𝑆0, 𝛾, 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑓𝑖𝑙𝑒 𝑎𝑛𝑑 𝜙) [3-24]

By considering Poisson’s ratio, suction profile and friction angle constant (and therefore

tensile strength variation known), only 𝜈, 𝑊, 𝜎𝑡 , 𝑆0 and 𝛾 need to be considered.

Therefore, from the Buckingham 𝜋 theorem (Buckingham, 1914; Buckingham, 1915),

two non-dimensional quantities, namely 𝑧𝑐 𝑊⁄ and 𝑆0 𝛾𝑊⁄ are considered. A parametric

study was undertaken to produce sufficient results to develop a relationship between

these two non-dimensional quantities. The results are given in Figure 3-39, for 𝜈 = 0.3,

𝜙 = 300 for the linearly decreasing suction profile (suction profile 2). The curves have

been produced for both linear elasticity and allowing shear failure. These curves can be

used to estimate crack depths approximately for the given set of other parametric values.

It is possible to develop similar crack depth curves to cover other ranges of parameters

that were held constant in this instance.

128

Figure 3-39 Design curve; 𝑧𝑐 𝑊⁄ vs. 𝑆0 𝛾𝑊⁄ curve when 𝜙 =300, 𝜈 =0.3 for suction profile given in Figure 3-19

3.7 Conclusions

This chapter undertook a study of crack depth evaluation using the UDEC program

utilising its existing capabilities. Three suction profiles were considered, namely

constant, linearly varying and parabolically varying with depth. Perhaps the last suction

profile is the most realistic for a deep clay layer. The results obtained from the

numerical analyses were compared with the analytical model developed by Morris et al.

(1992) to predict crack depths. Two methods of analysis were considered, namely

treating the soil as linear elastic allowing tensile failure and allowing shear failure to

occur within the soil. The results indicated that in the case of the linear elastic approach

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

z c/W

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LE, UDEC Modified analytical SS, UDEC


with tensile failure, the UDEC results predicted much deeper cracks than the analytic

model did. This result was attributed to the stress redistribution that takes place when a

crack opens up, which was not considered in the Morris et al’s analytical model. For

shear failure however, the numerical and analytical models produce closer results.

Morris et al’s analytical model was then extended to allow for stress distribution, which

eventually gave results closer to those predicted by UDEC. As a typical example, the

results were presented in non-dimensional form for general use.

A third approach where the LEFM theory was also proposed by Morris et al. The results

from this analysis was also compared with the UDEC results and it was found that

Morris et al results produced slightly higher crack depths than the UDEC results.

However, these results are highly dependent on the critical crack intensity factor or

fracture toughness chosen for the soils. In the current form of UDEC, fracture energy

(or facture toughness) associated with a joint or crack opening is not considered

directly. In Chapters 6 and 7, incorporation of fracture energy into UDEC model will be

considered in more detail.

The main issue of using UDEC is the run time in incremental nonlinear analysis. The

model run time changes significantly with the size of the block, number of grid points

and the method of analysis used. When running 10m deep 40m wide clay layer with

50cm mesh for the shear failure method, one model took about a month to complete

with a run in a core 2 duo, 4GB ram computer.

Chapter 4

EXPERIMENTAL INVESTIGATION OF SHRINKAGE AND SWELLING BEHAVIOUR

4.1 Introduction

Through various field and laboratory studies, it is well established that expansive clay

soils experience seasonal movement in the ground surface in wet and dry periods. Some

of these studies focused on the swelling and the shrinkage behaviour of expansive soils

(Richards et al., 1983; Sharma, 1998; Tripathy et al., 2002). However, very few studies

have been conducted to observe the cyclic swell-shrink behaviour of expansive soils for

several numbers of cycles.

The degree of swelling and shrinking depends on the amount of clay minerals within the

soil and the type of the clay minerals. However, the behaviour can be assumed similar

for any expansive soil and varies only in degree (Sharma, 1998; Kodikara, 2012). When

a soil layer is subjected to a sufficient number of wet-dry cycles, it can act

predominantly in an elastic or reversible manner, which is generally referred to as

130

Experimental Investigation of Shrinkage and Swelling Behaviour 131

environmental stabilization (Gould et al., 2011) or aging. Aging of newly compacted

clay soil and its effect on soils behaviour has been studied in the past by several workers

including Tripathy et al. (2002). How the aging of expansive soils will affect their

swell-shrink behaviour is not fully explained and more experimental work needs to be

undertaken in this area.

In Chapter 2, it was highlighted that stress induced by desiccation as needed for fracture

modelling can be represented either by suction increment or moisture increment. The

stress induced by the moisture content change of soil can be analysed using a hydric

constant (𝛼) with the change of moisture content (Kodikara and Choi, 2006; Peron et

al., 2007). However the behaviour of this α value due to the cyclic swell-shrink

behaviour is unknown. Hence this chapter is dedicated to present the laboratory

experiments carried out to examine the variation of 𝛼 value by analysing the data

published in the literature and undertaking new experiments. During experiments, the

behaviour of the soil subjected to wet-dry cycles will be analysed for a number of

different expansive soils with different initial and test conditions. A more general

variation of 𝛼 will be presented on the basis of the results obtained.

4.2 Hydric constant (𝜶) in Stress Analysis

In order to analyse the behaviour of unsaturated soils accurately, it is important to

compute the stress-strain behaviour of the soil subject to boundary and initial

conditions. Unfortunately, there is still not a universally accepted model to capture the

complex behaviour observed by compacted soils. For instance, D’onza et al. (2011)

132

presented a blind test on different constitutive models independently calibrated by

different teams of researchers on the same set of suction-controlled triaxial and

oedometer tests performed on compacted silty soil samples. These calibrated

constitutive models were then used to predict the behaviour of some other soil tests and

to compare the results blindly. None of the models agreed fully with the experimental

data showing the need for improvements of constitutive models that are currently

available to predict the unsaturated soil behaviour. None of these test results included

desiccation cracking of soils, and hence, such behaviour was not modelled.

Generally, the stresses induced due to moisture content change were analysed by most

researchers using the suction variation. Hence the swell-shrink behaviour was also

described as a function of suction; e.g., Barcelona Basic Model, BBM, (Alonso et al.,

1987; Kodikara, 2012). However, it is evident that the measurements of suction in

expansive soils are not straightforward especially under field conditions. Currently, a

number of techniques such as axis translation (increasing air pressure to generate a

curved meniscus while keeping water pressure atmospheric) tensiometer, thermal

conductivity sensor and osmotic methods need to be employed to get the full suction

range covered by laboratory methods. Some of these techniques are not able to be

employed in the field or may not work well. Therefore, although suction is a well-

defined potential of the soil for water flow, its measurement and its use as a stress state

variable is still posing problems to advancement of unsaturated soil mechanics.

In 2009, Baker and Frydman questioned the use of suction in geotechnical constitutive

theory (Baker and Frydman, 2009). In their review, they pointed out that all devices for

measuring soil suction actually measure soil-water potential (internal or matric) rather


than the state of pressure (tension) in the soil water, except for the tensiometer. The

actual water tension in unsaturated soil was recognized as difficult to be measured or

controlled. Furthermore they explained that the matric potential consists of two major

components namely, adsorption and capillary, and only the capillary component may be

explained in terms of mechanical pressure (tension). They argued that the geotechnical

literature generally ignores the adsorption potential when interpreting “suction”

measurements although under a significant range of conditions adsorption may be the

main component of the matric potential. This omission results in prediction of

unrealistically large water tensions which cannot be realized (due to cavitation) under

field conditions, when the air pressure is normally atmospheric.

In addition, suction variation does not give the full range of moisture content variation

in repeating wet-dry cycles (to be discussed later in this chapter) and displays

significant hysteresis during drying and wetting, increasing the complexity of the

problem. Due to these drawbacks, in this research moisture content variation is used

over suction as advocated by some other researchers (Costa, 2009; Gould et al., 2011;

Kodikara, 2012).

When the stresses are analysed in terms of moisture content variation, the potential

strain can be obtained using a constant of proportionality known as the hydric constant

(𝛼) and the difference of moisture content similar to heat flow analysis (i.e. ∆𝜀𝑠ℎ =

𝑘∆𝑇, where 𝑘 is the thermal conductivity, ∆𝑇 is the thermal gradient and ∆𝜀𝑠ℎ is the

shrinkage strain change).

∆εvsh = α∆w [4-1]

134

However, unlike for thermal expansion coefficient, α is not a constant and its behaviour

has not been studied in detail with the cyclic wetting and drying for compacted soils. A

theoretical explanation of the hydric coefficient is presented as follows. According to

the MPK framework presented by Kodikara (2012), the net stress in unsaturated soil can

be presented as a function of void ratio, e and the moisture ratio, 𝑒𝑤, where:

𝑒𝑤 = 𝑤𝐺𝑠 [4-2]

Therefore, this functional relationship can be presented as:

𝜎 = 𝑓(𝑒, 𝑒𝑤) [4-3]

By partially differentiating [4-3] gives:

𝑑𝜎 = �𝜕𝜎𝜕𝑒

�𝑒𝑤

𝑑𝑒 + �𝜕𝜎

𝜕𝑒𝑤�

𝑒𝑑𝑒𝑤 [4-4]

From the cyclic formula,

�𝜕𝜎

𝜕𝑒𝑤�

𝑒= −

� 𝜕𝑒𝜕𝑒𝑤

�𝜎

�𝜕𝑒𝜕𝜎�

𝑒𝑤

[4-5]

Substituting equation [4-3] in equation [4-4] gives:


�𝑒𝑤

𝑑𝑒 −� 𝜕𝑒

𝜕𝑒𝑤�

𝜎

�𝜕𝑒𝜕𝜎�

𝑒𝑤

𝑑𝑒𝑤


�𝑒𝑤

𝑑𝑒 − �𝜕𝑒

𝜕𝑒𝑤�

𝜎�

𝜕𝜎𝜕𝑒

�𝑒𝑤

𝑑𝑒𝑤


�𝑒𝑤

(𝑑𝑒 − �𝜕𝑒

𝜕𝑒𝑤�

𝜎𝑑𝑒𝑤) [4-6]


when,

�𝜕𝑒

𝜕𝑒𝑤�

𝜎= 𝛼∗ [4-7]

and,

�𝜕𝜎𝜕𝑒

�𝑒𝑤

= 𝐾

where 𝐾 is the bulk modulus at the particle moisture ratio and 𝛼∗ is the hydric

coefficient given by the partial differential of void and moisture ratios at a particular net

stress (equation [4-7]). Then equation [4-6] becomes,

𝑑𝜎 = 𝐾(𝑑𝑒 − 𝛼∗𝑑𝑒𝑤) [4-8]

If this relation is for only one directional stress then the 𝐾 should be replaced by 3𝐾.

𝛼∗𝑑𝑒𝑤 represents the free shrinkage. The relation to obtain the potential free shrinkage

can be written as,

𝛼∗ 𝑑𝑒𝑤

(1 + 𝑒0)= ∆𝜀𝑣𝑠ℎ [4-9]

Hence, from equations [4-1] and [4-9],

𝛼∗ 𝑑𝑒𝑤

(1 + 𝑒0)= 𝛼∆𝑤 [4-10]

Using the relation presented in equation [4-2] and substituting it in equation [4-10] leads

to the relation,

𝛼 = 𝛼∗ 𝐺𝑠

(1 + 𝑒0) [4-11]

136

The relation given by equation [4-11] can be used to calculate the hydric constant. It is

clear that the hydric coefficient depends on the stress level. This coefficient (𝛼∗) can be

measured experimentally by following a relatively easy test. The following sections

explain the swelling and shrinking experiments conducted to develop the relationships

of void and moisture ratios and thereby the variation of the hydric coefficient.

4.3 Experimental Procedure

Some swell-shrink tests were conducted to calculate the hydric coefficient. The soils

used in the investigations, their basic characteristics, the method of sample preparation

and experimentation adopted, observations made and methods of calculations are

presented in this section.

4.3.1 Materials

The expansive soil material was collected from the clay deposits in North Altona in

Melbourne, Australia at the depths of 0.4m to 2.0m. It is light brown in colour when

dry, becoming dark when wet. This clay is referred to as Altona clay in this thesis.

Several basic geotechnical tests were carried out according to the Australian Standards

in order to characterise the properties of the clay used here and these are listed below:

• Particle size distribution analysis;

• Atterberg limits;

• Specific gravity;


• Compaction tests (Standard and Modified);

• One dimensional consolidation test.

4.3.1.1 Particle size distribution analysis

Soil particles down to 75μm sieve (clay soil), were prepared by the wet sieving method

in accordance with the Australian Standard (AS1289.3.6.1, 2009), and were analysed

using an optical particle size analyser instead of the hydrometer as noted in the

Standard, due to its higher level of accuracy. The soil samples were soaked in water for

one hour before being agitated and then washed on a 425μm sieve until the water was

clear and all of the material had been washed through the sieve. The washed material

with water was collected in a tray placed under the sieve and was oven dried at 105°C to

collect the washed soil particles in powder form.

A mass of 5g of the soil powder was suspended in distilled water and poured into the

sample dispersion unit of the particle size analyser for the analysis. In this device, the

soil particles within the solution were passed through a focused laser beam. The laser

light is scattered at an angle that is inversely proportional to the soil particle size, and

the angular intensity of the scattered light was then measured by a series of

photosensitive detectors. The map of scattering intensity versus angle was used to

calculate the particle size using a model in the analyser software. The optical particle

size analysis was repeated three times and the average results are plotted in Figure 4-1.

The results show that Altona clay is relatively well graded with 63 % of clay, 30 % of

silt and less than 7% of sand.

138

Figure 4-1 Particle size distribution of Altona clay

4.3.1.2 Atterberg limits and specific gravity

The Australian Standard was followed in determining the liquid limit (AS1289.3.1.1,

2009) and the plastic limit (AS1289.3.2.1, 2009). The moisture content was also

measured in accordance with the instructions in Australian Standards (AS1289.2.1.1,

2005). The value of the liquid limit for Altona clay was found to be 70.2% and the

plastic limit was 21.8%. The plasticity index was obtained as 48.4%. Under the unified

soil classification system (USCS), this soil is classified as clay of high plasticity (CH)

on the basis of Atterberg limits.

The specific gravity of Altona clay was determined as 2.614. The test was repeated

twice to get the value more accurately. The linear shrinkage value was also measured

according to the standard (AS1289.3.4.1, 2008 ) giving a value of 16%. This means that

0

10

20

30

40

50

60

70

80

90

100

0.0001 0.001 0.01 0.1 1 10

Perc

enta

ge P

assi

ng (%

)

Practicle Size (mm)

Clay Silt Sand

Gravel


the soil has the potential to shrink 16% when dried from the liquid limit to oven dry

condition.

4.3.1.3 Compaction tests

Standard and modified compaction tests were conducted according to the Australian

Standards, (AS1289.5.1.1, 2003 ) and (AS1289.5.2.1, 2003 ) respectively. Soil was

prepared and the test was conducted following the standard. An automatic soil

compactor (shown in Figure 4-2) was used to eliminate the inconsistencies in

compaction measured by an operator.

The automatic blow pattern of the compactor ensures the optimum compaction for each

layer of soil. The rammer automatically travels across the mould and the table rotates

the mould in equal steps on a base that is extremely stable. The number of blows,

weight of the rammer and height of the rammer drop can be set at the beginning.

140

Figure 4-2 Automatic soil compactor designed for proctor and CBR Compaction and removing sample after modified compaction test

The results of the compaction tests are shown in Figure 4-3. The optimum moisture

content for standard proctor compaction was approximately 23.5% whereas for

Automatic compaction machine

Controls

Compaction hammer

Compaction mould

Equipments for Compaction test

Removing jack


modified compaction the optimum moisture content was around 16%. The dry densities

for both standard and modified compaction tests at the optimum points are 1.5g/cm3

and 1.8g/cm3 respectively.

Figure 4-3 Compaction curves for Altona clay

4.3.1.4 One dimensional consolidation test

One-dimensional consolidation tests were also carried out in accordance with Australian

standards (AS1289.6.6.1, 1998) using an oedometer. Soil was prepared to a slurry of

152% moisture content which is approximately twice the liquid limit to ensure that air

bubbles were minimised and workability was increased using a mechanical soil mixer

(Figure 4-4). The slurry was tested after leaving for at least 24 hours for equalization.

The slurry was placed in a 63 mm diameter (18.2 mm height) cylinder and LoadTrac III

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

4 8 12 16 20 24 28 32 36

Dry D

ensi

ty (g

/cm

3 )

Gravimetric Water Content (%)

Modified ProctorStandard Proctor100% Saturation80% Saturation60% Saturation40% Saturation

142

(Geomech Corp.) which is an automated consolidation oedometer rig. The experimental

set up is shown in Figure 4-5.

Using the software provided with the LoadTrac, the user inputs the stress step values (in

kPa) and minimum and maximum times (in hours). Various pressures were preset to

allow measurements of the soil strain under predetermined pressures. The machine

places the predetermined stress onto the sample based on the area given and the load is

applied for the inputted amount of time. The pressure is built up slowly to allow the

pore water pressure to dissipate. The sample was allowed to swell at each unloading

step, which was also performed at predetermined steps. The linear variable differential

transformer (LDVT) and load cells were calibrated prior to use.

Figure 4-4 Mechanical soil mixer Figure 4-5 LoadTrac III consolidation machine


Figure 4-6 represents the variation of void ratio with the pressure on a log scale. From

this figure, the soil compressibility parameters for loading (λ0) and unloading (κ0) under

saturated conditions were obtained as λ0 of 0.391 and κ0 of 0.038 respectively.

Figure 4-6 Compression curve for Altona clay

4.3.1.5 Mineralogy

Table 4-1 Mineralogy content of Altona clay

Quartz Albite Orthoclase Kaolin Smectite Calcite Halite Anatase

59% 2% 3% 2% 31% 3% <1% <1%

The study of mineralogy of the soil was not within the scope of this study. Hence,

mineralogy analysis was not conducted in this research. However, the similar soil

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100 1000 10000

Void

Rat

io

Pressure (kPa)

144

obtained from the same area has been analysed in the past (Gallage et al., 2008) from

which it is possible get the results for the composition of this soil , as given in Table 4-

1.

4.3.1.6 Summary of results

The properties of soil obtained from the tests described above are summarised in Table

4-1.

Table 4-2 Summary of the soil classification test results

Colour Light brown / beige

Linear shrinkage 16%

Liquid limit 70.2%

Plastic limit 21.8%

Plasticity index 48.4%

Soil class Inorganic clays of high plasticity (CH)

Standard optimum moisture content 23.5%

Compressibility parameters for loading 0.391

Compressibility parameters for unloading 0.038

Specific gravity 2.614

4.3.2 Sample preparation and set-up

A targeted experimental procedure needed to prepare a large number of samples with

the same density, moisture content and soil homogeneity was essential. Hence, selection


of an appropriate method for sample preparation and initial conditions required careful

consideration and balance.

4.3.2.1 Sample size

The samples were prepared in a basic size oedometer ring of 76mm in diameter. When

deciding the height of the sample, three conditions were taken into consideration:

1. Loss of energy input due to boundary friction;

2. Maintaining homogeneity throughout the sample; and

3. Producing sufficient swelling and shrinking effect while testing.

The height of the sample needed to be small enough to avoid a significant degree of

energy loss and to avoid heterogeneity of the sample (Romero, 1999). However, the

height should be sufficient to produce considerable volume change with the moisture

change. Considering these facts, the sample height was selected as 12mm.

4.3.2.2 Sample preparation

As stated before, samples were prepared carefully and identically in order to maintain

homogeneity and repeatability. The initial conditions of the samples were selected as

17±1% moisture content and 14.6±0.2kN/m3 dry unit weight. The starting point is

shown in the compaction curve in Figure 4-3, where it lies in the dry side of the

optimum. The behaviour of the soil compacted at wet side of optimum is reasonably

well known and some results can be found already in the literature (Romero, 1999;

Tripathy et al., 2002; Monroy, 2006). The dry side of the optimum has been neglected

146

although in actual field conditions there is a possibility of ending up on the dry side of

compaction under different compaction energies. Hence, the dry side of optimum was

considered in this research.

The soil was first oven dried and sieved to remove the sand size particles. The clay

powder was then mixed with the required amount of water manually for around 15

minutes by spraying until moisture was distributed evenly. The wet soil was placed in a

resealable polythene bag and sealed. This soil bag was sealed again in another bag to

make sure, the soil sample was completely air tight and left for at least 24 hours prior to

use.

Stainless steel oedometer rings were used in the test. The diameters of the rings were

76.5±0.1mm and heights were 19.1±0.1mm. The rings were cleaned, greased and

weighed at the beginning of the test.

Static compaction was selected for compacting samples. Static compaction shows much

higher potential in repeatability and uniformity over dynamic compaction (Sivakumar,

1993). Two methods were tested for ultimate sample preparation. Method one involves

compacting soil in one large ring and then cutting samples out of the compacted soil

using the cutting ring. The bottom of the sample was trimmed off and was then inserted

into the oedometer ring. Figure 4-7 shows the method followed in obtaining samples

using this method (method one). The second method involves compacting the soil

directly into the oedometer ring. The required quantity of soil was calculated on the

basis of the dry density to be achieved. Figure 4-8 illustrates method two.


Figure 4-7 Sample preparation using method one

Load

Cutting ring

Soil block

Loading plate

Cutting a sample

Cutting a sample

Compacting large volume

Before Compaction

Pushing the sample out

of the cutting ring

Oedometer cell

Prepared sample

148

Figure 4-8 Sample preparation using method two

Several samples were compacted using both preparation methods. The initial unit

weights and moisture contents of samples after compacting were compared. Figure 4-9

shows the results obtained from the sample preparation methods one and two. The

results for unit weight from method one show some scatter although the moisture

content is almost the same in all samples. Unit weight and moisture content results

obtained from method two show a small variation. The variation of unit weight in

different samples for method one may be due to the uneven distribution of soil in the

large container. The sample disturbance during cutting and moving the sample from one

ring to another may also have influenced conditions to some extent. Other than the un-

Prepared sample

Before Compaction

Oedometer cell

Compacting sample


acceptable initial condition variation, the first method is also difficult to execute

precisely in comparison to the second method. Hence, it was decided to compact the

samples directly into the oedometer ring using static compaction for all tests performed

in this research.

Figure 4-9 Values for unit weight and moisture content obtained as initial conditions. * - UW stands for Unit Weight, ** - MC stands for Moisture Content

4.3.2.3 Set-up

The intension of the test programme was to observe the change of void ratio over the

time when the sample was subjected to controlled wet and dry conditions. Several types

of apparatus can be used for measuring swell-shrink properties, and the conventional

oedometer is one of them. Monash University Civil Engineering laboratory has

oedometers designed and manufactured at Monash as shown in Figure 4-10. This

oedometer is relatively simple, easy to use and takes less space than the conventional

oedometer. Furthermore, wetting and drying tests take long periods of time and it was

10

12

14

16

18

20

22

24

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6 7 8 9 10

Moi

stur

e Co

nten

t (%

)

unit

wei

ght (

kN/m

3)

Sample Number

Method 1 - UW* Method 2 - UW

Method 1 - MC** Method 2 - MC

150

not possible to utilise a conventional oedometer for a long period of time due to the

requirements of other students. Hence it was decided to use the Monash designed

oedometer for this study.

As shown in Figure 4-11, the Monash oedometer is loaded from above and the load is

transferred through a loading rod. An electronic dial gauge can be attached to it for the

measurement of displacements. The equipment can accommodate different sizes of

rings, but a conventional sized cell and ring were used. However, the main limitation of

the test set-up is the pressure that can be applied by the loads that can be placed above.

Typically about 100kPa pressure is easily achievable.

Figure 4-10 Monash designed Oedometer

Loading plate Load

Dial-gauge

Oedometer Cell


Figure 4-11 Schematic diagram of test set-up and placement of soil sample

The tests conducted had two main parts, wetting and drying, and a total of 50 samples

was tested. The samples were subjected to several wetting and drying cycles until the

volume change followed the same path for both wetting and drying. This condition was

considered as the environmentally stabilized condition for the environment imposed.

The tests were conducted with 5kPa constant pressure applied on the sample in a

temperature controlled room. The temperature was maintained at 25°C during wetting

and increased up to 40±5°C while drying using halogen working lamps. The pressure

used was typically a nominal pressure and is representative of an unloaded clay layer.

When obtaining the swelling path, the moisture content was increased by adding the

required amount of water through the holes on the top cap (Figure 4-12(a)) and by

inundating the sample in water when the full potential of swell was assumed to have

been obtained (Figure 4-12(b)). To trace one wetting or drying path 6 to 10 samples

were tested, and each sample provided only one point in the 𝑒 − 𝑒𝑤 space. The quantity

Oedometer Cell

Dial-gauge

Load Loading plate

Porous disks

Lid with holes

152

of water to be added was decided allowing 3 to 5mm water evaporation from the sample

during the swelling period. During other periods the cell was covered with plastic to

prevent evaporation.

Figure 4-12 (a) Saturating the sample for swelling, (b) Wetting the sample to get the swell before saturation

Before selecting the test duration for swelling and shrinking, several tests were

conducted letting the sample swell until it did not show a significant increase in height.

For drying similar tests were performed. Figure 4-13 and Figure 4-14 show the vertical

displacement behaviour with time for the first swelling and shrinking after first swelling

respectively. Figure 4-13 illustrates that, almost 90% of the swelling occurred during the

very first day after wetting. Hence it was decided to allow the samples to swell for three

days while carrying out the wet dry cycles. However, the drying process usually took a

longer period than swelling took and it was necessary to allow for about 8 days for

drying, as shown in Figure 4-14. In wet dry cycles, the samples were allowed to dry for

10 days ensuring more accurate results.

(a) (b)

Syringe with water

Water surrounding the sample


Figure 4-13 Vertical displacement of three samples compacted to similar initial conditions when subjected to wetting.

While running a test, the vertical displacement of the sample was recorded every 24hrs.

Then at the end of the test, the moisture content and unit weight were measured

immediately after dismantling the test set-up. Even before the full shrinkage, cracks and

considerable lateral shrinkage were observed. Figure 4-15 shows the photos of the

sample at different stages of the test. The moisture content of the sample was measured

by oven drying the soil obtained from different places of the sample.

To measure the unit weight the following procedure was used in a water displacement

technique. Immediately after dismantling the sample, a small portion of the sample was

removed without applying too much force to avoid any change in properties. The

specimen was then weighed and coated with paraffin wax (density = 0.96g/cm3) and

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12

Vert

ical

dis

plac

emen

t (m

m)

No. of days

Sample 1 Sample 2 Sample 3

154

again weighed with the paraffin wax coat by hanging it underneath an electronic

balance. After that, the specimen was submerged in a beaker of distilled water and the

submerged mass was measured. Using equation [4-12] the unit weight was calculated.

𝑈𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 =𝑤(𝑖𝑛 𝑎𝑖𝑟)

𝑤 (𝑖𝑛 𝑎𝑖𝑟) − 𝑤 (𝑖𝑛 𝑤𝑎𝑡𝑒𝑟) × 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 [4-12]

where, w is the mass of the sample. The calculated soil unit weight was corrected for the

wax volume. The measured vertical displacement of the soil was not used in the void

ratio calculation due to the cracks and lateral strain.

Figure 4-14 Vertical displacement of three samples compacted to similar initial conditions when subjected to drying after full swelling

-2.5

-2

-1.5

-1

-0.5

0 0 2 4 6 8 10 12

Vert

ival

dis

plac

emen

t (m

m)

No. of days

Sample 1 Sample 2 Sample 3


Figure 4-15 Photos of the sample at different stages

After 1st drying

After 2nd wetting After 2nd drying

After 1st wetting

After 3rd wetting

After 4th wetting

After 3rd drying

After 4th drying

156

4.4 Results

Trends and variations of different parameters captured during the tests were presented in

this section. The vertical displacements and void ratios were taken into consideration.

4.4.1 Swell-Shrink cycles

The maximum and minimum vertical displacements observed during wetting and drying

cycles were plotted in Figure 4-16 and Figure 4-17. The samples were compacted at the

same initial conditions under the same vertical pressure as discussed previously. The

tests were conducted continually for several wet dry cycles.

Figure 4-16 Marginal vertical displacements during wet-dry cycles for several samples

The results indicate that the initial wetting and initial drying (first cycle) have different

vertical displacements variations. The next cycles show a repetitive variation. From

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1st Wetting

1st Drying

2nd wetting

2nd Drying

3rd wetting

3rd Drying

4th wetting

4th Drying

Swel

ling

heig

ht (m

m)

Sample 45

Sample 46

Sample 47

Sample 48

Sample 49

Sample 50


Figure 4-17, it can be seen that Altona clay shows more than 30% net expansion from

the initial height. The change in vertical displacement during wet dry cycles is expected

to be around 20% after aging. This results show a good agreement with the results of

Tripathy et al. (2002).

Figure 4-17 Typical average vertical displacements for Altona clay

4.4.2 Swelling or shrinking paths

The variation of void ratio in each cycle is examined and the void ratio is plotted against

the moisture ratio. Figure 4-18, Figure 4-19, Figure 4-20 and Figure 4-21 show the first,

second, third and fourth cycles respectively. In each figure wetting is followed by

drying to obtain the full cycle.

In Figure 4-18, the initial point indicates the point where the sample was compacted.

The lower part of the swelling curve from the initial point was obtained by drying the

0

5

10

15

20

25

30

35

40

45

1st Wetting

1st Drying 2nd wetting

2nd Drying

3rd wetting

3rd Drying

4th wetting

4th Drying

Swel

ling

Perc

enta

ge (s

wel

ling

heig

ht/I

nitia

l hei

ght,

%)

158

sample. It can be observed that the curves behave wildly at the initial wetting and drying

cycles before converging to a stabilized curve achieved in the fourth cycle.

The stabilized curve is shown in Figure 4-21 where drying and wetting both are lying on

top of each other. In the beginning, the void ratio does not change with the moisture

content very much. However, after approximately 0.2 moisture ratio the void ratio

shows a significant change until close to the saturation followed by a flatter part again

to reach the saturation line. This behaviour agrees with results of Tripathy et al. (2002)

and the explanation given by Gould et al.(2011).

Figure 4-18 Swelling and shrinking paths for the first cycle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Void

Rat

io, e

Moisture Ratio, wGs

1st cycle

Initial point

Wetting curve

Drying curve

Starting point of 1st

cycle

End point of 1st

cycle


Figure 4-19 Swelling and shrinking paths for the second cycle

Figure 4-20 Swelling and shrinking paths for the third cycle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Void

Rat

io, e

Moisture Ratio, wGs

2nd cycle

Drying curve

Wetting curve

Starting point of 2nd cycle

End point of 2nd

cycle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Void

Ratio

, e

Moisture Ratio, wGs

3rd cycle

Drying curve

Wetting curve

Starting point of 3rd cycle

End point of 3rd

cycle

160

Figure 4-21 Swelling and shrinking paths for the fourth cycle

4.4.3 Variation of hydric coefficient (𝜶∗)

The variation of hydric coefficient was examined in relation to number of cycles and

moisture content in Figure 4-22 and Figure 4-23.

It can be seen that 𝛼∗ shows similar behaviour in all swell shrink cycles, although the

first cycle shows little variation in comparison to the other cycles, as shown in Figure

4-22. However, for both shrinking and swelling curves similar variation of 𝛼∗ values

can be observed. A careful examination of the figure shows slightly higher values for 𝛼∗

while shrinking than while swelling.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Void

Rat

io, e

Moisture Ratio, wGs

4th cycle

Starting point of 4th cycle

End point of 4th

cycle


Figure 4-22 Variation of 𝛼∗ with number of cycles

At low moisture contents, 𝛼∗ is low and increases with the moisture content

approximately up to 1 and remains relatively constant for a long range of moisture

content. Finally, the value of 𝛼∗ drops drastically as shown in Figure 4-23.

Figure 4-23 Variation of 𝛼∗ with moisture ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Hyd

ric C

oeff

icie

nt (α

*)

Wetting

Drying

1st cycle 2nd cycle 3rd cycle 4th cycle

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2

Hyd

ric C

oeff

icie

nt (α

*)

Moisture Ratio, wGs

1st wetting 1st Drying 2nd wetting 2nd Drying 3rd Wetting 3rd Drying 4th Wetting 4th Drying

162

4.5 Other Research on Wet-Dry Cycles

It was decided to analyse different types of soils in order to draw a general conclusion.

Hence, existing data obtained by digitising the results in several published research

works were analysed.

4.5.1 Experimental data analysis of Sharma (1998)

In this study, 10% Wyoming sodium bentonite with speswhite kaolin was used as

expansive soil represented by ‘BK’. Speswhite kaolin alone was denoted by “K’. The

basic properties of the soil are given in Table 4-3. The compaction curve and the initial

position of the soil are shown in Figure 4-24. Tri-axial tests were used to measure the

swell-shrink behaviour and the axis translation technique was the basic approach in

measuring suction. The characteristics of the tests used for this analysis are shown in

Table 4-4.

Table 4-3 Properties of the soil

Parameter Value

BK K

Specific Gravity (𝐺𝑠) 2.65

Liquid limit 93% 65%

Plasticity Index 60% 33%

Optimum Moisture content 29%

Optimum dry density 1.44Mg/m3


Figure 4-24 Compaction curves and the initial positions of samples

1

1.1

1.2

1.3

1.4

1.5

15 20 25 30 35 40

Dry

Den

sity

(Mg/

m3)

Moisture Content (%)

164

Table 4-4 Summary of tests by Sharma

Test No.

Material Static compaction Pressure (kPa)

Drainage

No. of days

Test method Mean net stress (kPa)

Suction changing rate (kPa/hr)/Value of k for non-linear variation of suction

No of cycles

Suction Hydric coefficient (𝜶∗)

1 2 3

wet dry wet dry wet

1 BK 400 Single 68 wetting/drying 10 2.5 2 400-50-380-0-370

0.298 0.721 0.357 0.869

2 BK 400 Single 28 wetting/drying 50 1.6 1 400-100-400 -0.172

0.596

3 BK 800 Single 67 wetting/drying 50 0.8 1.5 400-100-400-10 0.295 0.563 0.338

4 BK 800 Double 37 wetting/drying 10 2.0 1.5 400-100-400-10 0.374 0.726 0.569

5 BK 400 Double 56 wetting/drying 20 1.6 1.5 400-100-400-10 0.321 0.630 0.523

8 BK 400 Double 42 1. wetting/drying 10 ±0.00016 1 300-20-300 0.187 0.884

2. loading 10-175 300

10 BK 400 Double 66 1.loading/unloading

10-100-10

200

2. wetting/drying 10 ±0.00016 1 200-20-200 0.504 0.851

3.loading/unloadi 10-250- 200


ng 10

12 K 400 Double 13 wetting/drying 10 Wetting- ±0.00048

Drying- ±0.00024

1 400-100-400 0.468 0.497

14 K 400 Double 76 wetting/drying 10 ±0.00016 2.5 400-50-400-20-400-5

0.467 0.661 0.453 0.636 0.480

16 BK 400 Double 71 1.wetting/drying 10 ±0.00016 1 400-20-400 0.195 0.905

2. loading 10-40 400

3.wetting/drying, P’ varied to v kept constant

40-70-1 ±0.00016 1 400-20-105

18 K 400 Double 39 1. wetting/drying 10 ±0.00016 1 200-20-200 0.495 0.684

2. loading 10-300 200

19 BK 3200 Double 36 wetting/drying 10 ±0.00016 1 400-50-175 0.785 1.044

20 BK 3200 Double 48 wetting/drying 10 ±0.00016 1 100-1-100 0.905 1.007


The wetting and drying paths obtained from the tests are shown in Table 4-4, and are

plotted in the void ratio-moisture ratio space as shown in Figure 4-25 and Figure 4-26.

The results of the samples tested under 10kPa net vertical stress are shown in Figure

4-25 while the results at 20kPa and 50kPa pressures were plotted in Figure 4-26. The

curves generally show a flatter line while swelling and a steeper line while shrinking. It

can be observed that some samples (Tests 2, 5, 8 and 16) collapsed while wetting when

they reached the pressure line corresponding to the net vertical stress in comparison

with the MPK framework. Also shown in these figures are the likely compaction curves

interpolated from the two results given.

Figure 4-27 shows the variation of the hydric coefficient with wet dry cycles. The value

of 𝛼∗ shows a rough variation due to the small intervals selected in determining the

gradients. However, a general trend can be observed. While the initial swelling 𝛼∗ is

about 0.5, it drops at the end of the swell and again increases up to approximately 0.8

during the shrinking period. A similar trend follows for the next cycle as well with 𝛼∗

values of 0.65 and 0.9 for wetting and drying respectively.

To observe the variation of 𝛼∗ closely, that variation was plotted against the degree of

saturation for each cycle as shown in Figure 4-28. Figure 4-28(a) shows the variation of

𝛼∗ during the first wetting period. 𝛼∗ for different samples between 0.55 and 0.1 while

slightly decreasing with the increased degree of saturation. Figure 4-25 shows the first

wetting, where 𝛼∗ ranges between 0.4 and 1. During the first drying 𝛼∗ increases rapidly

and then tends to decrease at higher degrees of saturation. The second wetting shows the

typical increasing and deceasing behaviour of 𝛼∗ for full swell or full shrink. The last

figure (Figure 4-28(d)) shows that the hydric coefficient increased up to the value 1.


Figure 4-25 Swelling and shrinking curves for samples under 10kPa vertical stress, Test 4 – initially compacted under 800kPa. Test19 & 20 – initially compacted under 3200kPa. Other

tests – initially compacted under 400kPa

Figure 4-26 Swelling and shrinking curves for samples under 20kPa (Test 5) and 50kPa (Test 2) vertical stress, Test 3 – initially compacted under 800kPa. Test 2 & 5 – initially compacted

under 400kPa

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Void

Rat

io

Moisture Ratio

Test 1 Test 4 Test 8 Test 14 Test 16 Test 19 Test 20

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Void

Rat

io

Moisture Ratio

Test 2 Test 3 Test 5

168

Figure 4-27 Variation of 𝛼∗ with wet-dry cycles

Sharma’s results were not carried out for full swelling and full shrinking as it was

undertaken in this thesis and by Tripathy et al. (2002). Hence, the typical behaviour of

the hydric coefficient is hard to observe. However, as for previous tests described, the

general trend of increasing close 1.0 during wetting and dropping to a small value under

drying passing through 1.0 or more is clear.

Another important observation one can make from Sharma’s data is that wetting always

causes a lesser change in void ratio leading to low hydric coefficient values. On the

other hand, drying causes a higher amount of void ratio change for a small moisture

reduction leading to higher hydric coefficients generally close 1.0. From this

observation, it may be concluded that a hydric coefficient close to 1.0 may be relevant

for shrinking from the compacted state.

0

0.2

0.4

0.6

0.8

1

1.2

Hyd

ric co

effic

ient

test 1

test 3

test 4

test 5

test 10

test 12

test 14

test 18

Wetting Wetting Wetting Drying Drying


Figure 4-28 Variation of 𝛼∗ with degree of saturation in each wetting or drying process

170

4.5.2 Experimental data analysis of Romero (1999)

The material tested is known as Boom Clay. Boom clay powder at 𝛾𝑑 = 10.8kN/m3 and

starting at an initial water content of 27.4% was controlled air dried to a final target of

15.4%. Then the samples were compacted statically to achieve the desired dry unit

weight (i.e. 16.7 or 13.7kN/m3). Other material properties of the soil are listed in Table

4-5. The static compaction curve for Boom clay with similar compaction energy to

Proctor compaction curve and the initial state of the soil specimens are shown in Figure

4-29.

Table 4-5 Summary of soil parameters

Parameter Value


Liquid limit 55.7±0.9%

Plasticity Index 26.9±1.0%

Soil Class (USCS, ASTM D2487) CH

De-structured dry unit weight 10.8 kN/m3

De-structured void ratio 1.46

Compressibility parameter for saturated conditions [λ(0)]

0.150 (𝛾𝑑 = 13.7 kN/m3) 0.136 (𝛾𝑑 = 16.7 kN/m3)

Compressibility parameter for unload/reload conditions [κ]

0.01

This research was mainly focused on the volumetric behaviour of unsaturated clays

(swelling, collapse, and shrinkage) under suction, stress and temperature changes. The

tests were performed in two types; oedometer and isotropic tests. In both methods,

isothermal wetting and drying cycles were carried out at a constant net vertical stress at

two different temperatures.


Figure 4-29 Compaction curves for Boom clay

For oedometer tests, specimens with a diameter of 50mm and a height of 10mm were

used for temperature controlled oedometer testing and these were compacted in a single

layer. The maximum pre-consolidation net vertical stress is 4.5±0.16MPa at 22°C and

4.07±0.12MPa at 80°C for the heavily over-consolidated sample. The maximum net

horizontal stress measured at ambient temperature is around 1.74±0.07MPa. For

isotropic tests, specimens 38mm in diameter and 76mm in height were statically

compacted in a stainless steel rigid mould in three layers using static one dimensional

compression.

In oedometer tests, the samples at two different temperatures (22°C and 80°C) were

loaded at approximately constant water content (around 15%) and at constant air

pressure (𝑢𝑎=0.5MPa) until the desired net vertical stresses were reached. The matric

13

14

15

16

17

18

19

4 9 14 19 24

Dry

Uni

t Wei

ght,

γ d (k

N/m

3 )

Water content (%) Standard Proctor energy: 0.59 compaction MJ/m3 - 22 C Standard Proctor energy: 0.59 compaction MJ/m3 - 80 C Wan (1996) Modified dynamic compaction test:2.69 MJ/m3 Sr = 100%

Initial Conditions

172

suction paths were applied systematically by maintaining a constant air pressure of

𝑢𝑎 =0.5MPa and controlling the water pressure. The following suction paths were

followed in the first wetting path: 0.45MPa, 0.2MPa, 0.06MPa and 0.01MPa.

Subsequently, the process was reversed following the same suction steps for drying. The

next cycles also followed the same suction path until the desired matric suction was

achieved.

Figure 4-30 Swelling shrinking curves tested under different pressures

In isotropic tests, the wet-dry cycles, under constant isotropic stress and temperature,

were applied by varying the matric suction. The samples were isotropically loaded at

constant water content until the desired net mean stresses were reached. The suction

paths were then imposed in steps similarly to the oedometer tests. All the tests analysed

in this section are summarised in Table 4-6.

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Void

ratio

moisture ratio

0.026 MPa 0.085 MPa 0.3 MPa 0.55 MPa 0.6 MPa 1.2 MPa I 0.085 MPa I 0.6 MPa


Table 4-6 Summary of tests by Romero

Test No.

Initial

Unit

Weigh

t (𝜸𝒅),

kN/m3

Tempe

rature

(°C)

Height/

Radius

Ratio

No.

of

days Test

Mean

net

stress

(MPa)

No. Of

Suction

changin

g steps

No of

cycles Suction (MPa)

Gradient of 𝒆 − 𝒆𝒘 curve

(𝜶∗)

1 2

wet dry wet dry

C17-

0.026C 16.7 22 0.4 74 Oedometer 0.026 11 1.5 2-0.01-0.45-0.01-0.3

0.547 0.473

H17-

0.026C 16.7 80 0.4 103 Oedometer 0.026 11 1.5 2-0.01-0.45-0.01-0.3

0.596 0.465

C17-

0.085A 16.7 22 0.4 160 Oedometer 0.085 16 2

2-0.01-0.45-0.01-0.45-

0.01 0.412 0.452 0.514

C17-

0.085B 16.7 22 0.57 105 Oedometer 0.085 12 2 2-0.01-0.45-0.01-0.2

0.408 0.373 0.345

H17-0.085 16.7 80 0.4 124 Oedometer 0.085 10 1.5 1.6-0.01-0.45-0.01 0.51 0.449 0.578

C17-0.300 16.7 22 0.4 89 Oedometer 0.3 10 1.5 2-0.01-0.45-0.01 0.254 0.314 0.358

H17-0.300 16.7 80 0.4 92 Oedometer 0.3 10 1.5 1.6-0.01-0.45-0.02 0.315 0.222 0.149

C17-0.550 16.7 22 0.4 36.5 Oedometer 0.55 12 2 2-0.01-0.45-0.01-0.45 0.094 0.033 0.089 0.12

174

1

H17-0.550 16.7 80 0.4 42 Oedometer 0.55 12 2 1.6-0.01-0.45-0.01-0.45 0.441 0.225 0.223 0.68

C14-

0.600B 13.7 22 0.4 66 Oedometer 0.6 11 2 2-0.01-0.45-0.01-0.2

0.548 0.108

C14-

0.600C 13.7 22 0.57 70 Oedometer 0.6 11 2 2-0.01-0.45-0.01-0.2

0.456 0.117

H14-

0.600D 13.7 80 0.4 73 Oedometer 0.6 11 2 1.6-0.01-0.45-0.01-0.2

0.628 0.104

C14-1.200 13.7 22 0.4 92.5 Oedometer 1.2 10 2 1.8-0.01-0.45-0.01 0.461 0.197

H14-

1.200B 13.7 80 0.4 68 Oedometer 1.2 10 1.5 1.6-0.01-0.45-0.01

0.308 0.051

I17-0.085A 16.7 99 Isotropic 0.085 12 2 2-0.01-0.45-0.01 0.606 0.653 0.599

I14-0.600 13.7 111 Isotropic 2-0.01-0.45-0.01 0.883 0.78


Figure 4-30 shows the void ratio variation with moisture content in different tests under

different vertical pressures. The constant pressure lines shown in the figure are

estimated based on two points calculated using a compressibility parameter. I 0.085MPa

and I 0.6MPa denote isotropic tests while the rest are oedometer tests. It can be

observed that three samples have undergone collapse at the pressure line corresponding

to the vertical stress (C14-0.600B, C14-0.600 and I14-0.600) in comparison with the

MPK framework. Others show a similar trend to Sharma’s data. The swelling curve

shows a flatter variation following a steeper shrinking curve. The average gradients of

each curve are shown in Figure 4-31.

Figure 4-31 Variation of 𝛼∗ with number of cycles

From Figure 4-31, it can be seen that, generally, the hydric coefficient is greater in

shrinking periods than in swelling periods, although there are variations from this trend

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1st Swelling 1st Shrinking 2nd Swelling 2nd Shrinking

Hyd

ric C

oeff

icie

nt

C17-0.085A

C17-0.085B

C17-0.300

H17-0.550

C14-0.600B

C14-0.600C

H14-0.600D

C14-1.200

H14-1.200B

I17-0.085A

I14-0.600

176

in some instances. In this data set the hydric coefficient is changing from 0.05 to 0.7 in

swelling periods and from 0.25 to 0.9 in shrinking periods.

4.5.3 Experimental data analysis of Tripathy (2000)

Two types of black cotton soils were used in these experiments, referred to as Soil A

and Soil B hereafter. The basic properties of the soil are shown in Table 4-7. The

samples were tested in an oedometer cell under controlled environment. The initial

positions of the samples used in this study are shown in Figure 4-32 with proctor

compaction curves for each soil.

Table 4-7 Soil Properties

Soil Property Soil A Soil B

Liquid limit (%) 100 74

Plasticity index (%) 58 42

Shrinkage limit (%) 10.6 13.5

Specific gravity 2.68 2.73

% passing sieve No. 200 98 80

Clay content (<0.002 mm; %) 62 52

Silt content (%) 36 28

Fine sand content (%) 2 20

Free swell (%) 340 225

During the tests, full swelling - full shrinking and full swelling – partial shrinking were

tested for several wetting drying cycles. The paths during swelling and shrinking were

plotted in void ratio – moisture content space. The tests were conducted under different


vertical net stresses, 6.25kPa and 50kPa. The swelling and shrinking paths for different

tests are shown in Figure 4-33.

It can be that when the vertical pressure is less the soil shows substantial or even wild

expansive/shrinking behaviour during first 2 cycles and then gradually comes to an

equilibrate state. All 4 tests, given in Figure 4-33 show similar behaviour after several

wet dry cycles.

Figure 4-32 Standard Proctor curves for Soil A and Soil B

The variation of hydric coefficient with moisture content in each wet or dry process is

shown in Figure 4-34 for soil A tested under 6.25kPa vertical net stress. This shows that

the variation of hydric coefficient is almost similar in every process except in 1st wetting

process and 2nd wetting process.

11

12

13

14

15

16

17

18

10 20 30 40 50

Dry

Den

sity

(kN

/m3)


Soil A

11

12

13

14

15

16

17

18

10 20 30 40 50

Dry

Den

sity

(kN

/m3)


Soil B

178

Table 4-8 Summary hydric coefficients of tests by Tripathy

Test Soil A with 6.25kPa

Soil A with 6.25kPa – Partial Shrinkage

Soil A with 50kPa

Soil A with 100kPa

Soil B with 6.25kPa

Soil B with 50kPa

Cycle 1 wetting 0.946 0.943 0.913 0.858

drying 1.002 1.011 0.956 0.879

Cycle 2 wetting 2.429 0.954 1.091 1.696

drying 0.997 0.994 0.877 0.840

Cycle 3 wetting 1.037 0.949 0.951 0.927

drying 0.908 0.973 0.927 0.876

Cycle 4 wetting 0.957 0.952 0.953 0.989

drying 0.957 0.952 0.953 0.989

Cycle 5 wetting 0.979 0.938 0.958 0.936 0.859

drying 0.979 0.938 0.958 0.936 0.859

The variation of hydric coefficient in tests conducted with soil A under 6.25kPa, soil A

under 50kPa and Soil B under 6.25kPa for several cycles are shown in Figure 4-35. The

first wetting and second wetting shows somewhat abnormal behaviour compared to the

rest of the tests. Furthermore, it can be observed that for low vertical net stress,

abnormality of the expansive behaviour increases in the initial wetting cycles. Another

observation is that when the vertical stress is low, the number of cycles required for soil

to stabilize increases. However after the 2nd cycle soil becomes mostly stable.


Figure 4-33 Swelling and shrinking curves for several wet-dry cycles

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.0 0.5 1.0 1.5 2.0

Void

Rat

io

Moisture Ratio

(a) Soil A under 6.25kPa

Cycle 1 Cycle 2 Cycle 3 Cycle 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Void

Rat

io

Moisture Ratio

(b)Soil A under 50kPa


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0.8 1 1.2 1.4 1.6

Void

Rat

io

Moisture Ratio

(c) Soil A under 6.25kPa - Partial Shrinkage


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Void

Rat

io

Moisture Ratio

(d) Soil B under 6.25kPa


180

Figure 4-34 Hydric coefficient change with wetting and drying for the Soil A with 6.25kPa

Figure 4-35 Hydric coefficient variation during several wet-dry cycles

The hydric coefficient was closely examined in each process for all the tests as shown in

Figure 4-36. All tests show a similar variation for the hydric coefficient α∗, except for

the partial drying test. This varies from zero (at lower and higher degrees of saturation)

to one at the intermediate saturation levels, which is significant, after stabilization.

However the degree of saturation corresponding to the same value of α∗ is considerably

different for different soil types and different vertical stresses (Figure 4-36).

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80

Hyd

ric C

oeff

icie

nt

Moisture Content

Cycle 1-wetting Cycle 2-wetting Cycle 1-drying Cycle 2-drying Cycle 3-wetting Cycle 3-drying Cycle 4-wetting Cycle 4-drying

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Hyd

ric C

oeff

icie

nt

No of Cycles

A 6.25 A 50 B 6.25


Figure 4-36 Summary of variation of hydric coefficient in each wetting or drying process

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt

Degree of saturation

First Wetting A 6.25 A 6.25 half A 50 B 6.25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt


First Drying A 6.25 A 6.25 half A 50 B 6.25

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt


Second Wetting A 6.25 A 6.25 half A 50 B 6.25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt


Second Drying A 6.25 A 6.25 half A 50 B 6.25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt


Third Wetting A 6.25 A 6.25 half A 50 B 6.25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5

Hyd

ric C

oeff

icie

nt


Third Wetting A 6.25 A 6.25 half A 50 B 6.25

0 0.2 0.4 0.6 0.8

1 1.2 1.4

0 0.2 0.4 0.6 0.8 1 1.2

Hyd

ric C

oeff

icie

nt


Forth Wetting & Drying A 6.25 A 6.25 half A 50 B 6.25

182

4.5.4 Experimental data analysis of Montanez (2002)

In this study, sand bentonite mixes were used to analyse the expansive behaviour of soil.

Well and uniformly graded sand was mixed with different proportions of bentonite. The

soil properties are presented in Table 4-9.

Table 4-9 Soil Parameters for sand bentonite mixes

Well-graded sand Uniform sand Bentonite

Bentonite (%) 0 5 10 15 0 10 100

Specific gravity 2.655 2.657 2.658 2.660 2.657 2.660 2.688

Liquid Limit (%) 69 110 150 71 337

Plasticity Index (%)

46 88 122 45 286

Classification (BS) SPg SW SW SPC CE

Samples were prepared by the static compaction technique. Soil was compacted in a

larger mould and the samples were then cut out in required the size using a cutting ring

which was finally transferred into the oedometer ring. The initial conditions of the

samples are shown in Figure 4-37.

A sample was then subjected to several wet dry cycles in order to evaluate the

behaviour of several properties including void ratio and water content with suction. The

tests were conducted with different initial moisture contents, dry densities, bentonite

percentages and sand types. The summary of the tests is presented in Table 4-10.


Figure 4-37 Compaction curves for samples with different compositions; WG – Well Graded U – Uniform and B – Bentonite

The results obtained from these experiments are shown in Figure 4-38. Figure 4-38(a)

shows the behaviour with different initial moisture contents. The trends are not very

clear although, when the moisture content is less, it is apparent that soil reaches its

stable position more quickly than the soils compacted to higher moisture contents.

Considering the tests analysed previously, it may be that the soil shows erratic

1

2

3

4

5

6

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2 7 12 17 22

Dry

Den

sity

(Mg/

m3 )


WG sand with 10% B

Standard

7

1.6

1.7

1.8

1.9

2

2.1

2.2

2 7 12 17

Dry

Den

sity

(Mg/

m3 )


WG sand with 5% B

Standard Heavy

8

1.6

1.7

1.8

1.9

2

2.1

2.2

2 7 12 17

Dry

Den

sity

(Mg/

m3 )


WG sand with 15% B

Standard Heavy

9

1.4

1.5

1.6

1.7

1.8

1.9

2 12 22

Dry

Den

sity

(Mg/

m3 )


U sand with 10% B

Standard Heavy

184

behaviour during the first cycle when the initial moisture content is low. However, it is

hard to explain the behaviour using one cycle only.

Table 4-10 Summary of the tests by Montanez

Sample No.

initial 𝒘

Initial dry

density (Mg/m3)

Sand type

% of Bentonite

Sample type

𝜶∗ value

1st drying

1st wetting

2nd drying

1 5.94 1.82 WG 10 c 0 0.441

2 11.95 1.85 WG 10 c 0.032 0.469 0.413

3 12.11 1.76 WG 10 c 0.032 0.416 0.446

4 12.14 1.66 WG 10 c 0.028 0.328 0.312

5 13.97 1.88 WG 10 c 0.022 0.37

6 22 1.66 WG 10 r 0.074 0.187 0.146

7 11.74 1.78 WG 5 c 0.083 0.149 0.133

8 11.41 1.79 WG 15 c 0.013 0.506 0.441

9 12.15 1.49 U 10 c 0.008 0.331 0.328

c = compacted r = reconstituted WG = well-graded sand U = Uniform sand

Figure 4-38(b) shows the behaviour of samples with different initial dry densities. It can

be clearly seen that the soil is moving towards a stable condition during cycling. The

curves for each wetting and drying do not show much difference when the dry densities

are lower. This may be due to the fact that soil structure does not need to rearrange

significantly if the soil has not been compressed heavily at the beginning. Figures 4-38

(c) and (d) show the effect of composition, indicating the commonly known fact that the

more reactive fines are in the soil, the higher the swelling and shrinking effect are.


Some soils show that the shrinking curve is above the swelling curve, which is different

to other results examined so far. This behaviour can be expected when the initial

condition of the soil is below its equilibrium position.

Figure 4-38 Behaviour of sand samples with different initial conditions under wetting and drying

According to the results presented in Figure 4-38, the gradients obtained during the first

drying (the tests were started by drying the sample) are much lower than seen in the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Void

Rat

io

Moisture Ratio

(a) Different initial moisture contents

6%, Sample 1 11.50%, Sample 2 14%, Sample 5 22 %, Sample 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8

Void

Rat

io

Moisture ratio

(b) Different initial dry densities

1.65 g/cm3, Sample 4 1.75 g/cm3, Sample 3 1.85 g/cm3, Sample 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8

Void

ratio

Moisture Ratio

(c) Different Bentonite contents

B 5%, Sample 7 B 10%, Sample 2 B 15%, Sample 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Void

Rat

io

Moisture Ratio

(d) Different sand Types with 10% B

WG Sand, Sample 4

186

previous results. Also, the wetting stage shows higher 𝛼∗ values than the drying stage.

The values of the hydric coefficient were computed and then plotted in Figure 4-39. It

can be seen that the hydric coefficient is generally lower than in previous tests,

presumably due to sand providing a different form of resistance to swelling and

shrinking.

Figure 4-39 Hydric coefficient values for wet-dry processes

4.5.5 Experimental data analysis of Monroy (2006)

The material chosen for this study consisted of brown, weathered London Clay,

obtained from Harlesden, in north London. The basic properties of the soil are listed in

Table 4-11. The compaction characteristics of the soil are shown in Figure 4-40.

0

0.1

0.2

0.3

0.4

0.5

0.6

1st drying 1st wetting 2nd drying

Hyd

ric C

oeff

icie

nt

1

2

3

4

5

6

7

8

9


Table 4-11 Properties of London Clay

Parameter Value


Liquid limit 85%

Plasticity Index 60%

Soil Class (BS5930: 1981) CV (very high plasticity clay)

Clay fraction 58%

Silt fraction 40% Sand fraction 2%

Compressibility parameter for saturated conditions [Cc(0)]

0.349

Compressibility parameter for unload/reload conditions [Cs(0)]

0.081

The soil was compacted statically into the oedometer ring under increasing load. The

maximum load applied to compact the sample was approximately 720kPa. The initial

condition of the sample is marked in Figure 4-40. The suction was changed in order to

examine the behaviour during wetting and drying. Loading and unloading stages were

also carried out in the oedometer, and the vertical pressure was kept at a constant value

during these tests. However, in some tests the vertical stress has been changed. A

complete summary of the tests used in this study is given in Table 4-12.

188

Figure 4-40 Compaction characteristics of London clay from standard compaction test

1.36

1.4

1.44

1.48

1.52

1.56

1.6

1.64

14 16 18 20 22 24 26 28 30 32 34

Dry

Den

sity

(g/c

m3)


Initial Condition of the sample


Table 4-12 Summary of tests by Monroy

Test Initial 𝒆

Initial dry

density (Mg/m3)

Initial 𝑺𝒓 (%)

No of

days

Test path Suction (kPa) Vertical net stress

(kPa)

Hydric Coefficient 𝜶∗

Cycle 1 Cycle 2 Cycle 3

wet dry wet dry wet dry

o13 0.942 1.39 67.6 93 wetting 840 - 405 7 - 100 -0.096

Loading

Unloading

405

405 - 430

100 – 630

630 - 100

Wetting (V Constant) 430 - 115 100 - 270 0.076

Loading

Unloading

115

115 - 130

270 – 600

600 - 415

wetting 130 - 0 415 0.106

Loading 0 415 - 700

o14 0.947 1.387 67.4 51 wetting 940 - 430 7 -

Loading 430 7 - 600

Unloading 430 - 445 600 - 220

190

wetting 445 - 0 220 0.058

Drying 0 - 340 220 0.349

Wetting 340 - 0 220 0.184

o15 0.941 1.391 66.7 146 Loading 720 - 610 105 - 615

Unloading 610 - 630 615 - 105

Wetting 630 - 0 105 0.376

Drying 0 - 400 105 0.332

Loading 400 - 340 105 - 585

o16 0.945 1.388 67.7 43 Wetting 870 - 415 7 -

Loading 415 7 - 220

Wetting 415 - 0 220 -0.371

Drying 0 - 300 220 0.472

Wetting 300 - 0 220 0.211

o21 0.947 1.387 68.2 73 Wetting 1010 - 620 7 - 65 0.095

Wetting 620 - 0 65 0.164


Drying 0 - 120 65 0.216

Wetting 120 - 0 65 0.115

Drying 0 - 120 65 0.105

Loading 120 - 85 65 - 590

Unloading 85 - 120 590 - 180

o25 0.962 1.376 69.8 80 Wetting 820 - 0 7 0.415

Loading 0 7-30

Drying 0 - 100 30 0.385

Wetting 100 - 0 30 0.068

Drying 0 - 100 30 0.164

Wetting 100 - 0 30 0.107

Drying 0 - 100 30 0.127

o26 0.953 1.382 68.7 102 wetting 810 - 0 7 -

Loading 0 7 - 30

Drying 0 - 50 30 -

192

Loading 50 30 - 65

Drying 50 - 90 65 -

Loading 90 65 - 205

wetting 90 - 0 205 0.217

Drying 0 - 100 205 0.143

wetting 100 - 0 205 0.597

Drying 0 - 100 205 0.639

o27 0.956 1.38 68.4 89 wetting 870 - 0 7 0.428

Loading 0 7 - 30

Drying 0 - 80 30 0.249

Loading 80 30 - 425


The behaviour of soil during the tests is plotted in the void ratio and moisture ratio

space as shown in Figure 4-41. The figure shows a complete path in each test including

the loading and unloading cycles. Constant pressure lines shown in the figure were

estimated using the standard proctor curve and compressibility parameter for the

saturated condition. The gradient of the wetting portion gives lower values than the

drying portion of the curves. This observation is in agreement with the observations

from most of the other results presented earlier. Some samples (o16, o26) showed

collapse when they reached the corresponding pressure line during wetting, which is in

line with the MPK framework suggested by Kodikara (2012).

Figure 4-41 Variation of void ratio with moisture ratio during wetting and drying

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Void

Rat

io

Moisture Ratio

o13 o14 o15 o16 o21 o25 o26 o27

194

The gradients of wetting and drying curves (𝛼∗) in Figure 4-41 are plotted in Figure

4-42. The values corresponding to wetting are lower than the values corresponding to

the drying that followed.

Figure 4-42 Variation of 𝛼∗ values with different wetting and drying paths

4.6 Summary and Discussion

The present study and Tripathy’s study considered the full swell shrink cycles for

several cycles. The other studies involved conducting one cycle of wetting and drying

and in most cases, the cycles are not full drying or full wetting. Most of the experiments

were carried out in isotropic conditions and suction was controlled to change the

moisture content. The current research, Tripathy’s research and Montanez’s research

have been selected initial conditions of the sample below its equilibrium curve (at a

certain moisture ratio) while other research selected a position above the likely

equilibrium position.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1st wetting 1st drying 2nd wetting 2nd drying 3rd wetting 3rd drying

Hyd

ric C

oeff

icie

nt

o13

o14

o15

o16

o21

o25

o26


The suction (𝑆 = 𝑢𝑎 − 𝑢𝑤) of the sample is controlled by changing the pore water

pressure using the axis translation technique. The basic principle of the axis translation

technique is to elevate the air pressure, while keeping the water pressure zero or

positive. By changing the pore water pressure from a negative to a positive value, the

possibility of cavitation is prevented, not only in the measuring system, which was the

intention of changing pore water pressure, but also within the soil pores. This implies

that any influence of cavitation of the pore water which may be an important

phenomenon under in-situ stress conditions, is not accounted for in the laboratory tests

using the axis translation technique (Sharma, 1998).

Controlling suction by changing (increasing and decreasing) using same value for

several cycles may not represent the moisture content also change (decrease and

increase) in constant value for every cycle as expected. Figure 4-43 shows the moisture

content variation in a suction controlled test given by Romero (1999). The moisture

content changes considerably for the same suction in various cycles due to hysteresis.

Figure 4-43 Moisture content variation in a suction controlled test

0

0.1

0.2

0.3

0.4

0.5

0.51 0.53 0.55 0.57 0.59 0.61 0.63

Suct

ion

(Mpa

)

Moisture Ratio

196

The summary of the hydric coefficient values is presented in Table 4-13. From these

values it is obvious that 𝛼∗ values very much depend on the initial conditions of the soil.

Further results of 𝛼∗ variation during partial wet-dry cycles are given in Figure 4-44.

Full wet-dry cycles show similar 𝛼∗ values close to 1.0 during both wetting and drying

when the moisture ratio range is approximately 0.3 to 1.0. At lower moisture ratio

values, 𝛼∗ value decreases gradually and becomes zero. Similar variation can be seen at

higher moisture ratio values more than 1.0.


Table 4-13 Summary of hydric coefficient values

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5

wetting drying wetting drying wetting drying wetting /drying wetting/drying

𝛼∗ range Current 0 - 1.15 0 - 1.7 0.15 - 1.04 0.45 - 0.9 0.6 - 0.84 0.55 - 0.84 0.6 - 1.05

Sharma 0 – 0.91 0.5 – 1.04 0.34 – 0.57 0.64 – 0.87 0.48

Romero 0.09 – 0.6 0.22 – 0.88 0.05 – 0.78 0.12 – 0.68

Tripathy 0.85 – 0.95 0.88 – 1.01 0.95 - 2.43 0.84 – 1.00 0.93 - 1.04 0.88 – 0.97 0.95 – 0.99 0.86 – 0.98

Montanez 0.01 -0.08 0.149 – 0.51 0.15 – 0.45

Monroy 0.06 – 0.43 0.14 – 0.47 0.07 – 0.6 0.11 – 0.64 0.11 – 0.11 0.13

Overall 0 – 1.15 0 - 1.7 0.05 – 2.43 0.11 – 1.00 0.11 – 1.04 0.13 – 0.97 0.6 – 1.05 0.86 – 0.98

Average 𝛼∗

Current 1.00 1.30 0.95 0.90 0.83 0.83 0.97 0.97

Sharma 0.39 0.75 0.45 0.75 0.48

Romero 0.42 0.43 0.30 0.40

Tripathy 0.92 0.96 1.54 0.93 0.97 0.92 0.96 0.93

Montanez 0.03 0.36 0.32

Monroy 0.21 0.31 0.21 0.30 0.11 0.13

Overall 0.59 0.63 0.63 0.60 0.60 0.63 0.96 0.95


As discussed before, the initial conditions highly relate to the values of 𝛼∗. Figure 6.45

shows a conceptual understanding of how the soils approach the stable condition when

subjected to several partial wetting drying cycles. It is also apparent 𝛼∗ depends on the

net stress level.

Figure 4-44 Typical paths of expansive soils subjecting to partial wet dry cycles

The wetting path is generally curved, which starts flatter and gets steeper in the middle

then can get flatter towards the end especially if the saturations are very high. When the

initial conditions are above the equilibrium line, 𝛼∗ values in the drying path are almost

Moisture Ratio

Void Ratio

Typical path a soil can follow, when the initial conditions of the soil lie below the stable curve and are subjected to partial wetting and drying (e.g. Figure 4-33, Figure 4-38)

Stable curve corresponding to the stress level at which drying and wetting are

Typical path a soil can follow, when the initial conditions of the soil lie above the stable curve and are subjected to partial wetting and drying (e.g. Figure 4-25, Figure 4-26, Figure 4-30, Figure 4-41)


1.0 and 𝛼∗ values at wetting paths generally less. This makes the soil approach the

stable curve as cycles progress. However, when the initial conditions are selected below

the equilibrium line, the 𝛼∗ values during wetting and drying approach the stable curve

from behind as illustrated in Figure 4-44.

4.7 Conclusions

Modelling the behaviour of partly saturated soils during cycles of wetting and drying,

especially those containing a significant amount of reactive clay minerals, seems to pose

a high level of difficulty. Behaviour of soil depends on many factors, such as initial

moisture content, initial dry density, net vertical stress, composition of the soil and

degree of wetting and drying processes.

Despite all these restrictions, any expansive soil appears to come to an equilibrium

condition where the soil behaves reversibly while wetting and drying. The path

followed by the soil to reach the equilibrium curve depends on the factors listed in the

previous paragraph. Typical paths followed by expansive soils are shown in Figure 4-44

when they are subjected to partial wetting and drying.

The first two cycles of the wetting and drying process can be highly unpredictable. Most

soils show an erratic behaviour when adjusting the soil structure generated by

compaction to initial wet/dry cycles, particularly if the soil is heavily compacted

initially and the vertical stress on the soil during cycling is small.

200

The hydric coefficient is less when the soil swells and comparatively high when the soil

shrinks, if the soil is above the equilibrium line. If it is below, the opposite can happen

where the hydric coefficient is less during drying and high during wetting.

For full wetting drying cycles, typical variations of 𝛼∗ for several cycles are shown in

Figure 4-45. After the first few cycles, α∗ starts from zero and then increases up to one

or closer to one, and remains at this value for a significant amount of wetting or drying.

At the extreme of wetting or drying, the value will drop close to zero and the cycle

continues as shown in Figure 4-45. The first few cycles can show a deviation from this

behaviour when the soil structure is significantly adjusting to the drying wetting

environment.

Figure 4-45 Typical variations of 𝛼∗ with full wet dry cycles

𝛼∗ Value

1st Wetting

2nd Wetting

3rd Wetting

1st Drying

2nd Drying

3rd Drying

1

Chapter 5

MODELLING OF STABLE DESICCATION CRACK DEPTHS DURING CYCLIC WETTING AND DRYING

5.1 Introduction

As discussed in Chapter 2, seasonal movements of soil can be observed in response to

the change of soil moisture content resulting in a change of suction stresses. It was also

highlighted in Chapter 4 that this soil shrinking and swelling becomes stable with the

aging of soil. After the soil becomes stabilised, it behaves in reversibly during wetting

and drying (Tripathy et al., 2002; Wijesooriya and Kodikara, 2012). Mitchell (2012)

provided models to predict the suction profiles below the ground surface after this stable

condition had been reached. The standard values of suctions to be used under such

conditions generally were also given in the Australian Standards (Residential slabs and

footings, (Shannon, 2012)) for design purposes.

Significant differences in suction profiles can be observed depending on the water table

depth. With the use of the suction profile proposed by Mitchell (2012), the stable crack

201

202

depths can be calculated using elastic theory. In this chapter crack depths were

calculated using the above method for several clay soils from different parts of Australia

and from some other countries. The evaluation of crack depth in different climate

conditions and initial placement conditions of the clay layer was considered.

5.2 Suction Profiles

The suction variation below the ground has been measured by several researchers and

presented as the observed suction profiles during different climate conditions (Corte and

Higashi, 1964; Richards, 1985; Morris et al., 1992; Wijesooriya and Kodikara, 2012).

The position of the water table was considered as a key factor in each of these works

and the depth of soil from the surface was also identified as critical when the suction

profile changes significantly according to seasonal effects. The water table was

considered as shallow when the water table is less than 6m deep in clay soil, 3m in

sandy clay and silts and 1m in sand (Corte and Higashi, 1964). Otherwise it was

considered as a deep water table.

When shallow water table depths were presented, several appropriate suction profiles

were presented by Morris et al. (1992) as discussed in Chapter 2 (section 2.3.4.2) and

Chapter 3 (section 3.2). However, the depth to water table could increase when the soil

layer is subjected to arid climatic conditions.

Modelling of Stable Desiccation Crack Depths during Cyclic Wetting and Drying 203

Figure 5-1 Effect of various environmental conditions on the matrix suction profile (Peter 1979)

Peter (1964) suggested that the equilibrium suction profile should be given by the

extension of the hydrostatic water table, regardless of the climate, where shallow water

tables exist. However, it is well known that the suction profile fluctuates seasonally and

during wet and dry conditions. The effect of the position of the water table and various

environmental factors on the matric suction profile is presented in Figure 5-1. The

broken line in the figure shows the equilibrium suction profile and the solid line gives

the observed actual suction profile.

When deep water tables are present, the moisture conditions within the soil are

controlled by the moisture balance between rainfall and evapotranspiration (Russam and

Coleman, 1961; Richards, 1985). Richards (1985) predicted equilibrium suction profiles

using empirical relations between soil suction and climatic indices such as

Thornthwaite’s Moisture Index (TMI) in terms of total suction. These suction profiles

204

are based on the curve proposed by Russam and Coleman (1961) as shown in Figure 5-2

for optimum drainage conditions considering the practical purposes.

Perera et al. (2004) later modified the equilibrium suction prediction model for different

soils based on the particle size and plasticity index, especially with reference to

pavement layers. Although these models indicated great predictive capability based on

the error analysis, the Russam and Coleman (1961) model was used in this analysis

since it was more applicable in Australian conditions and has direct reference to clay

layers.

Figure 5-2 Values of suction from road site installations and postulated design curves after Richards, B.G. (1985)


A more detailed theoretical model for predicting the suction profile below the ground

has been developed by Mitchell (2012). In his model, the suction variation with depth

was represented according to the soil characteristics and the time. The model given in

equation [5-1] was derived by solving the moisture diffusion equation (equation [5-2])

as the solution of a linear homogeneous equation of the fourth order (equation [5-3]).

𝑆(𝑧, 𝑡) = 𝑆𝑒 + ∆𝑆𝑒−�

𝑛𝜋𝛼𝑑𝑐

𝑧𝑐𝑜𝑠 �2𝑛𝜋𝑡 − �

𝑛𝜋𝛼𝑑𝑐

𝑧� [5-1]

where 𝑆(𝑧, 𝑡) is the suction at any depth 𝑧 in metres at the time of 𝑡 in years, 𝑆𝑒 is the

equilibrium suction below the depth of seasonal suction change (or reactive zone depth),

∆𝑆 is the amplitude of suction variation at the surface. All the suction values have the

unit pF, which is defined as pF=log10 |suction in cm of water|. 𝑛 is the frequency of

seasonal variations given by cycles per year, 𝛼𝑑𝑐 is the diffusion co-efficient of soil with

the units of m2 per year.

𝜕𝑆𝜕𝑡

= 𝛼𝑑𝑐𝜕2𝑆𝜕𝑧2

[5-2]

The empirical relationship to calculate the diffusion co-efficient was proposed by

Lytton (1994) for expansive soils. This empirical equation is given by,

𝛼𝑑𝑐 = 0.0029 − 0.000162𝑆𝑠 − 0.0122(𝑎𝑣) [5-3]

where 𝛼𝑑𝑐 is given in cm2/s, 𝑆𝑠 is the slope of the suction water content curve and 𝑎𝑣 is

the matric suction compression index (slope of the void ratio-suction curve). The value

of 𝑆𝑠 can be obtained from equation [5-4] (Lytton, 1994) using the Atterberg limits of

the soil.

206

𝑆𝑠 = −20.29 + 0.1555(𝐿𝐿%) − 0.117(𝑃𝐼%) + 0.0684(% 𝑁𝑜. 200) [5-4]

where 𝐿𝐿% is the percentage of the liquid limit, 𝑃𝐼% is the percentage of the plasticity

index and % 𝑁𝑜. 200 is the percentage of soil passing through the US No. 200 sieve

equivalent to 75μm mesh size.

Equation [5-1] defines the suction decay along the depth and its oscillation with time

which is symmetric about the equilibrium suction 𝑆𝑒. Although this model covers the

seasonal effects on the suction profile and the variation of suction with the depth it is

based on some unrealistic assumptions. Basically, it assumes that the suction varies

purely due to the climatic conditions and the moisture with in the soil is neglected,

which is not accurate if the water table is not deeper than 10m (Aubeny and Long,

2007). Furthermore, at or near the surface the model (equation [5-1]) is valid only if the

soil layer has a very shallow root depth or bare surface which is highly unlikely.

However due to the usefulness in modelling the seasonal variation of suction profile

with depth as represented by equation [5-1], this model was selected as the basis for the

numerical model.

5.3 The Suction in Different Climate Conditions

Despite all the different empirical, theoretical and numerical models developed to

predict the suction below ground, it is essential to measure the actual suction profile in

the field under natural conditions. These real observations should be used as a

benchmark to validate the other developed models. Hence the literature provides some

useful data on observed suction profiles in different areas of the world.


Peter (1964) has observed the matric and total suction profiles throughout the different

parts of Australia giving especial consideration to Adelaide. He reported that in semi-

arid climates, the matric suction varies from 50kPa to 200kPa in different locations of

Adelaide. In desert environments such as in Woomera, he reported matric suctions of

1.5MPa to 2.0MPa. However, under extreme conditions, the total suction was measured

as high as 15.0MPa.

According to the chart provided by Richards (1985) as shown in Figure 5-2, suctions

can be estimated on the basis of a climatic index. Considering that Adelaide, Horsham

and Bordertown are in semi arid climates, the matric suctions at approximately 0.5m

below the ground can be obtained from the Russam and Coleman curve in Figure 5-2.

The values are between 100kPa (3pF) and 2000kPa (4.3pF) when the climatic index is

considered to change from 20 to -20 for semi arid conditions. Furthermore, Richards

observed more than 10MPa at the surface at a road site in Horsham, Victoria although

the area was considered a semi-arid area. Assuming arid conditions occur when the TMI

is less than -20, then matric suctions below 0.5m are observed greater than 2MPa. In

deserts during the dry season the exposed surface soil displayed suctions from 9.8MPa

to 98MPa and during wet season suctions from 4MPa to 12.4MPa (Russam and

Coleman, 1961).

Mitchell (1979) used surface suction value of 30MPa during the drying season and

1MPa during the wetting season in semi-arid climates. The difference between the

equilibrium suction and the surface value was selected as 1.5pF. However, Australian

Standards for slabs and footings (AS2870, 2011) suggest a value of 1.2pF for the

change of suction with depth for every part of Australia for design purposes.

208

Considering the observed suctions in the past, it was decided to select a matric suction

variation from 10kPa to 2.5MPa (2.0pF to 4.4pF, with 1.2pF difference to equilibrium

suction) from wet season to dry season in semi-arid climate conditions. Also, in arid

climate conditions the suction was changed from 20kPa to 5MPa (2.3pF to 4.7pF, with

1.2pF difference to equilibrium suction) from the wet season to dry season. The suction

profiles are illustrated in Figure 5-3 for arid and semi-arid conditions.

Figure 5-3 Typical Suction Profile for arid and semi-arid conditions used in the present study

5.4 Development of Numerical Model for Compacted Clay Layers under Cyclic Atmospheric Conditions

A numerical model was developed similar to that in Chapter 3 except that the suction

profile is different. Furthermore, the stress change was calculated on the basis of water

0

1

2

3

4

5

6

7

8

1 2 3 4 5

Dep

th (m

)

Suction (pF)

Semi-arid - wettest profile Semi-arid - driest profile Arid - wettest profile Arid - driest profile


content change instead of the suction change. Initially three methods were considered to

achieve the applicable stress change with the change in moisture content:

• Stress change by the direct addition of suction difference (same as Chapter 3);

• Fraction of suction added to change the stress to allow for desaturation; and

• Stress change on the basis of moisture content change.

The procedure of implementing stress change in each method is discussed in the

following section.

5.4.1 Stress change by the direct addition of suction change

The suction values were calculated using equation [5-1] corresponding to the midpoint

depth in each zone in the problem model geometry then the value obtained was applied

as the tensile stress of that zone. This application of stress was based on the assumption

that the degree of saturation will remain high close to the tip of the propagating crack,

and that the soil will behave similarly whether it is subjected to an externally applied

stress change or the internally developed suctions. This was discussed in more detail in

the Chapter 3.

However, this condition is not applicable when the soil is subjected to higher suctions as

soil gets drier, although heavy clays can remain saturated to suctions of 1MPa or so.

Therefore, the applied suctions will not be correct especially under arid conditions. The

results from this section are shown in Figure 5-4 in comparison to the other methods of

suction application. When this approach is followed, 8m crack depth was observed for

2MPa surface suction value, which represented semi arid conditions. Morris et al.

210

(1992) and some other researchers (Table 2-1, Chapter 2) have indicated that typically

observed crack depths in semi arid condition do not exceed 4m. Therefore, deeper crack

depths of 8m are unlikely in semi arid climates. Accordingly, the use of this assumption

is less appropriate to predict the crack depths for arid conditions.

Figure 5-4 Predicted crack depths using different stress changing approaches for Regina clay soil

5.4.2 Fraction of suction on change of the stress

As the suction increases beyond the air entry value the soil may significantly desaturate

and the 𝐸 𝐻⁄ ratio will fall below the value (1 − 2𝜈). Lau (1987) has investigated the

𝐸 𝐻⁄ ratio experimentally for Regina clay soil and he found that at low suction 𝐸 𝐻⁄ is

not a constant but at higher applied stresses, it is almost constant. He suggested that the

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Dep

th o

f cra

ckin

g (m

)

Maximum surface suction (MPa)

Theoretical value Directly added suction Fraction of suction Stress due to W change


normal range of the 𝐸 𝐻⁄ should vary between 0 and 0.2, with the latter corresponding

to near saturated condition.

The relation between the water content change and the suction change can be written as,

1(1 − 𝜐) 𝐸𝛼∆𝑊 =

𝐸𝐻

1(1 − 𝜐) Δ𝑆 [5-5]

Δ𝜎 =𝐸𝛼∆𝑊

(1 − 2𝜈) =𝐸𝐻

Δ𝑆(1 − 2ν)

Hence, the stress change in one direction in UDEC, Δ𝜎, can be written as:

Δ𝜎 =𝐸

3𝐻(1 − 2𝜈) Δ𝑆 [5-6]

The values for 𝐸 𝐻⁄ ratio were used as suggested by Lau in his experimental

investigation using the data from Fredlund (1964). The typical 𝐸 𝐻⁄ value for Regina

clay with 0.4 Poisson’s ratio was taken as 0.11. Then the stresses for the UDEC model

(Δ𝜎) were calculated using equation [5-7].

The results obtained through the above method were shown in Figure 5-4. As expected,

the compacted crack depths are smaller than those obtained by using direct suction

change as the stress change and show a better correlation with the theoretical crack

depth values shown in Figure 5-4. However, the 𝐸 𝐻⁄ ratio can vary with the soil type

and desaturation. Hence, it was decided to select another method that can predict crack

depths more rationally.

212

5.4.3 Stress change on the basis of moisture content change

Both methods described earlier used stress change through the suction change.

However, Chapter 4 described the possibility of considering the moisture content

change instead of suction change. In the third approach adopted, the moisture content

change was used to incorporate the stress change within the soil.

The suction profiles assumed for arid climates and semi-arid climates are the same as

shown in Figure 5-3. The suction corresponding to each zone was calculated using

equation [5-1]. However to relate suction to water content, the soil water characteristics

curve (SWCC) is used to obtain the water content change corresponding to the suction

change applied.

The applied stress is given by equation [5-7]. The horizontal strain due to matric suction

can be written as,

𝜀𝑥 =∆𝑆𝐻

= 𝛼∆𝑊 [5-7]

By substituting Equation [5-8] in equation [5-7]

Δ𝜎 =𝐸𝛼∆𝑊

(1 − 2𝜈) [5-8]

By substituting the value of α in Equation [4-11] in Chapter 4 for Equation [5-9] the

stress change becomes,

Δ𝜎 =𝐸𝐺𝑠

(1 − 2𝜈) ∆𝑊𝛼∗

(1 + 𝑒0) [5-9]

Equation [5-10] was then used to calculate the stress change in the continuum. The void

ratio was obtained from either void ratio vs. suction or void ratio vs. water content


curve. Then, from the void ratio vs. moisture ratio curve the hydric coefficient (𝛼∗) was

obtained.

The results obtained using this method, are presented in Figure 5-4. From these results it

can be seen that the predicted crack depths are less than those compacted by the other

two methods and the theoretical results. However, the actual observed crack depths for

typical clay soils are well matched with these results as shown in Table 2-1 in Chapter

2. Hence it was decided continue use of this method for further analysis.

5.5 Soils Used for the Analysis

In order to examine typical crack depths, it was decided to use several clay soils from

different areas. Two Australian soils and a Canadian soil were selected. Three soils

were named on the basis of their original location as Regina clay, Horsham Clay and

Altona clay.

5.5.1 Regina clay

Regina Saskatchewan, Canada, has a semi-arid continental climate with warm summers

and cold, dry winters. The layers of clay and silt in the area are generally between 3.5m

and 6.5m (Strunk et al., 2009). Measured suctions in these soils are about 3MPa for the

upper 4 m of clay (Vu et al., 2007).

Regina clay is widely used in unsaturated soils research and hence the soil properties are

widely available in the literature (Azam et al., 2012). For this study the soil parameters

214

for Regina clay were selected from the papers published by Fredlund and his team

(Fredlund and Rahardjo, 1993; Fredlund, 2002; Vu et al., 2007). The parameters used

for this study are shown in Table 5-1.

Table 5-1 Soil parameters for Regina clay

Parameter value

Density 15.4 kN/m3

Young’s modulus 10 MPa

Poisson’s Ratio 0.4

Liquid limit 70

Plasticity Index 38

% Passing No. 200 97.8

av2 (gradient of 𝑒 vs. 𝑆 curve) 0.09


Friction angle 20˚

For the numerical model the relation between the water content and suction (i.e. SWCC)

is required. According to the SWCC presented in the literature (Vu et al., 2007), the

curve was represented in two equations as shown in Figure 5-5 to simplify the

calculation in the numerical model. If the calculated suction is greater than 387kPa, the

logarithmic equation was used otherwise the 2nd order polynomial equation was used.


Figure 5-5 SWCC for Regina clay after (Vu, Hu et al. 2007)

Then to calculate the void ratio change, the void ratio vs. water content graph (after

Fredlund, 2002) was used as shown in Figure 5-6 at low stress levels. The hydric

coefficient was obtained from the same figure. However, for simplicity, only three

values were used for hydric coefficient (𝛼∗) as shown in Figure 5-6. The gradient of the

curve changes with the water content. However, only three gradient values (0, 0.5 and

1.0) were used in the numerical model for specific water content ranges. These

gradients in each water content range are shown in Figure 5-6.

216

Figure 5-6 Void ratio vs. water content graph for Regina clay

5.5.2 Horsham clay

Horsham is an area with a semi-arid climate with an annual average of 102 wet days and

mean annual maximum and minimum temperatures respectively of 21.7˚C and 7.7˚C.

The TMI of the area is -23, which is on the arid side. The months May to October fall

below the annual means. Differences in elevation between the brown soils on the higher

parts and the lower-lying grey clay soils are small (Skene, 1959).

Richards (1985) used Horsham clay for his research and published soil properties as

shown in Table 5-2.

y = -7E-06x3 + 0.001x2 - 0.0105x + 0.4876 R² = 0.9988

0

0.5

1

1.5

2

2.5

0 20 40 60 80

Void

ratio

, e

Water content (%)

α*=0.5 α*= 1

α*= 0


Table 5-2 Soil properties of Horsham clay

Parameter value

Density 16.81 kN/m3

Young’s modulus 12 MPa


Liquid limit 65

Plastic limit 22

Plasticity Index 43

% Passing No. 200 75

av2 (gradient of 𝒆 vs 𝑺) 0.084


Friction angle 20˚

𝒔𝒔 (gradient of SWCC) -10.0835

Diffusion coefficient, 𝜶𝒅𝒄 (m2/yr) 11.06

The curves used to obtain the relationship of water content with suction and void ratio

are shown in Figure 5-7 and Figure 5-8 respectively (Richards, 1985). Equations shown

in these figures were used in the numerical model to calculate water content and void

ratio. The hydric coefficient values used for different water content ranges are shown in

Figure 5-8.

218

Figure 5-7 SWCC for Horsham clay (after Richards, 1985)

Figure 5-8 Void ratio vs. water content curve for Horsham clay (after Richards, 1985)

5.5.3 Altona clay

Altona is a temperate to semi-arid area in Western Melbourne. This area has a residual

clay layer 1.5 m below the ground surface with basalt rock found at varying depth

limiting the thickness of the layer (Gallage et al., 2008). The Monash Geomechanics

y = -3.448ln(x) + 43.582 R² = 0.9679

0

10

20

30

1 10 100 1000 10000

Wat

er c

onte

nt, %

Suction (kPa)

y = -7E-05x3 + 0.0047x2 - 0.0777x + 0.7607 R² = 0.9988

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20 25 30 35

Void

ratio

, e

Water content, %

α*=0 α*=0.5 α*= 1


Research Group often uses soils from this area and the experimental investigations in

the previous chapter were also carried out using the same soil. The parameters used for

the numerical modelling in this section are listed in Table 5-3.

Table 5-3 Soil properties of Altona clay

Parameter Value

Density 15.3kN/m3

Young’s modulus 6.34MPa (Shannon, 2012)


Liquid limit 70.2

Plastic limit 21.8

Plasticity Index 48.4

% Passing No. 200 93

av2 (gradient of 𝒆 vs 𝑺) 0.033 (Chan, 2012)


Friction angle (at joint) 20

𝒔𝒔 (gradient of SWCC) -8.6755

Diffusion coefficient, 𝜶𝒅𝒄 (m2/yr) 12.31

The SWCC and void ratio vs. water content curve for Altona clay are shown in Figure

5-9 (after Chan, 2012) and Figure 5-10 respectively. The relationships used to calculate

220

the water content and void ratio are shown in the figures. The hydric coefficient values

are also shown in Figure 5-10.

Figure 5-9 SWCC for Altona clay (after Chan, 2012)

Figure 5-10 Void ratio vs. water content curve for Altona clay

y = 2E-08x2 - 0.0002x + 0.4501 R² = 1

y = -0.071ln(x) + 0.8159 R² = 0.9863

0

0.1

0.2

0.3

0.4

0.5

0.01 0.1 1 10 100 1000 10000 100000 1000000

Volu

met

ric m

oist

ure

cont

ent

Suction (kPa)

y = 1.68x2 + 1.4941x + 0.3128 R² = 0.9945

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5

Void

ratio

Water Content

α*=0.5 α*= 1

α*=0


5.6 Results

The variation of predicted crack depth values with different influential parameters for

different clay soils are presented here.

5.6.1 Crack depth prediction under different climatic condition

The position of the suction profile beneath the ground surface changes significantly with

the climate condition of the area. Figure 5-11 shows the different suction profiles under

different climatic conditions which were used to obtain the depths of cracking as shown

in Figure 5-12. All three clay soils used in the study showed similar crack depths under

the same climatic condition.

Figure 5-11 Suction profiles under different climatic conditions

0

1

2

3

4

5

6

7

8

1.5 2 2.5 3 3.5 4 4.5 5 5.5

Dep

th (m

)

Suction (pF)

Surface suction=10MPa Surface suction=8MPa Surface suction=6MPa Surface suction=5MPa Surface suction=4MPa Surface suction=2.5MPa Surface suction=1.5MPa Surface suction=0.5MPa Surface suction=0.1MPa

222

The wettest profile used was 100kPa at the surface and the equilibrium suction at depth

was 6.2kPa, which showed no cracks at all in all three soils. The depth of cracking value

of 4.5m was observed for Regina and Altona clay soils when 10MPa surface suction

was applied although for Horsham clay crack depth was about 0.5m less.

Figure 5-12 Predicted depth of cracking change with climatic conditions

In the field generally, 2 to 4m depths of cracks were observed when the matric suction

at shallow depths was recorded around 3MPa values (Russam and Coleman, 1961;

Corte and Higashi, 1964; Richards, 1985; Wijesooriya and Kodikara, 2012). The

predicted crack depths appear to match those observed crack depths since the crack

0

1

2

3

4

5

6

7

8

0.0 2.0 4.0 6.0 8.0 10.0

Dep

th o

f cra

ckin

g (m

)


Regina clay Horsham clay Altona clay


depths for 3MPa matric suction are around 2m for all three soils. It should be noted

however, in these predictions, that the matric suction values given for the surface and

the in the field suctions are measured at shallow depths. Hence, considering the fact that

the surface suction is always higher than the suction values at shallow depths during the

drying periods, the observed crack depths should be a little higher than the predicted

values confirming the accuracy of the predicted results.

5.6.2 Crack opening and closing with time

Figure 5-13 Suction profile variation due to seasonal climate change in (a) Arid area (b) Semi-arid area

0

1

2

3

4

5

6

7

8

1.5 2.5 3.5 4.5 5.5

Dep

th (m

)

Suction (pF)

t=0.1 t=0.2 t=0.3 t=0.4 t=0.5 t=0.6 t=0.7 t=0.8 t=0.9 t=1

(a) Arid 0

1

2

3

4

5

6

7

8

1.5 2.5 3.5 4.5 5.5

Dep

th (m

)

Suction (pF)

t=0.1 t=0.2 t=0.3 t=0.4 t=0.5 t=0.6 t=0.7 t=0.8 t=0.9 t=1

(b) Semi-arid

224

During the year with the change of seasons, the suction profile also changes

significantly. Assuming that a year has only one cycle of climatic change Figure 5-13

was produced. The driest time of the year was obtained when t=0.5yr which represents

around January in Australia. The wettest profile was obtained when t=1yr representing

around July. The broken lines in Figure 5-13 show the change of suction profile from

the wettest condition to driest condition and the solid lines show the change from driest

to wettest condition in each time step.

Figure 5-14 Predicted depth of cracking with seasonal change in an arid area

Depending on the suction or water content profile in the soil, the depth of the crack may

vary. The cracks can be opened further if drying continues or the opened cracks can

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

Dep

th o

f cra

ckin

g (m

)

Time (years)

Regina clay

Horsham clay

Altona clay


close due to soil wetting, along with erosion from the sides of the crack or plastic flow.

Hence it is important to observe the behaviour of the cracks during the seasonal

movements. However, the results shown in the following figures do not show the

continuation of suction profiles through the seasonal variation. Instead they show the

crack depths observed when the suction profile changes from the initial conditions to

the relevant suction profile at a particular time. Hence the results may deviate from the

actual results, when actual progression of suction and associated changes in crack depth

and width are considered with the moisture dynamics associated with actual wetting and

drying.

Figure 5-15 Predicted depth of cracking with seasonal change in an semi-arid area

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

Dep

th o

f cra

ckin

g (m

)

Time (years)

Regina clay

Horsham clay

Altona clay

226

Figure 5-14 and Figure 5-15 show the crack depths corresponding to the suction profiles

at times of 0.1yr steps in arid climate and semi-arid climate respectively. For Australian

conditions the cycle will start from the wettest month July (0yr or 1yr) and reach the

driest condition in January at 0.6yrs. The results show that the crack depths increase

quickly after a certain suction level and then reach a peak when the suction becomes a

maximum. During wetting the crack depths decrease sharply as suction decreases and

then remain unchanged. These results suggest that the crack depth can show significant

change during the year with its depth peaking rapidly during the dry period. These

results appear to be consistent with the field observations that cracks close during the

winter and again open up during summer. However, their actual dynamics may be

affected by debris flowing into the cracks and causing changes in the normal pattern of

behaviour.

5.6.3 Effect of placement conditions of the clay liner on initial desiccation

As shown in Chapter 4, after a clay liner is placed it undergoes several cycles before it

comes to a stable condition. However, the first drying period a clay layer is subjected to

after the initial placement is critical in developing desiccation cracks (Kodikara, 2006).

In this context, studying the effect of initial placement conditions is important.

Generally, the maximum density based on the standard proctor compaction curve was

selected as the initial placement density in the numerical model. However, in order to

capture the effect of placement conditions on cracking the density of the soil was

changed.


In the models for both arid and semi-arid climates, the results obtained by changing the

density only are shown in Figure 5-16 and Figure 5-17. The results in both figures do

not show a significant effect on the depth of cracking. All three soils have produced

similar results although Horsham clay is a little different from the other two in showing

little decrease in crack depth with density.

Figure 5-16 Predicted crack depth variation with initial density of the layer in an arid climate

2

2.2

2.4

2.6

2.8

3

3.2

1400 1500 1600 1700 1800 1900

Dep

th o

f cra

ckin

g (m

)

Initial Placement density (kN/m3)


228

Figure 5-17 Predicted crack depth variation with initial density of the layer in an semi-arid climate

The results suggest that the initial placement density used in the UDEC does not

influence on crack depth prediction significantly.

0

0.5

1

1.5

2

2.5

3

1400 1500 1600 1700 1800 1900

Dep

th o

f cra

ckin

g (m

)

Initial Placement density (kN/m3)



5.6.4 Effect of the Poisson’s ratio

The effect of Poisson’s ratio on the depth of cracking was considered for both arid and

semi-arid conditions. The values of Poisson’s ratio were changed from 0.36 to 0.46

where the general values of the Poisson’s ratio for clay soils can be considered to lie

within this range. The results are shown in Figure 5-18 and Figure 5-19 for the arid

climates and semi-arid climates respectively, which show that the depth of cracking

reduces with the increase of Poisson’s ratio.

Figure 5-18 Predicted depth of cracking variation with the Poisson's ratio in an arid area

All soils show the same trend and similar values of crack depth at any value of

Poisson’s ratio. In an arid climate (Figure 5-18) the depth of cracking has decreased

0

0.5

1

1.5

2

2.5

3

3.5

0.3 0.35 0.4 0.45 0.5

Dep

th o

f cra

ckin

g (m

)

Poisson's ratio


230

from 5m to 3m for 0.1 changes in the Poisson’s ratio value. Also in semi-arid climates

(Figure 5-19) the change of depth of cracking is 4m to 1.5m given the same change in

Poisson’s ratio.

Figure 5-19 Predicted depth of cracking variation with the Poisson's ratio in an semi-arid area

These results indicate that at lower suctions the effect of the Poisson’s ratio is higher

than at higher suctions. However, soils with higher values of the Poisson’s ratios are

less problematic in any climate condition against the desiccation cracking.

0

0.5

1

1.5

2

2.5

3

0.3 0.35 0.4 0.45 0.5

Dep

th o

f cra

ckin

g (m

)

Poisson's ratio



5.6.5 Effect of equilibrium suction

As the next parameter the effect of change of equilibrium was selected. Several values

of amplitude of suction at the surface variation (∆𝑆) have been used in the past research.

Mitchell (1979) has used 1.5pF and the Australian footing standard suggests the use of

1.2pF suction for design purposes. However, the equilibrium value can be changed

according to variation in other conditions such as the depth to water table and soil

properties.

Figure 5-20 Suction profiles below the ground surface with different equilibrium suction values in (a) an arid climate (b) a semi-arid climate

To capture the change in the effect of equilibrium suction due to different reasons, the

models were run with suction profiles as shown in Figure 5-20. The suction profiles

0

1

2

3

4

5

6

7

8

3 3.5 4 4.5 5

Dep

th (m

)

Suction (pF)

Se=0.80MPa

Se=0.63MPa

Se=0.50MPa

Se=0.40MPa

Se=0.32MPa

Se=0.25MPa

Se=0.20MPa

Se=0.16MPa

(a) Arid

0

1

2

3

4

5

6

7

8

2.5 3 3.5 4 4.5 D

epth

(m)

Suction (pF)

Se=0.39MPa

Se=0.31MPa

Se=0.25MPa

Se=0.20MPa

Se=0.16MPa

Se=0.12MPa

Se=0.10MPa

Se=0.08MPa

(b) Semi-arid

232

representing arid and semi-arid climates are shown in Figure 5-20 (a) and (b)

respectively. The amplitude of suction at the surface was changed from 0.8pF to 1.5pF

while keeping the surface suction constant in both climate conditions. The equilibrium

suctions were changed from 0.16MPa to 0.8MPa in arid climate conditions and from

0.08MPa to 0.39MPa in semi-arid climates.

Figure 5-21 Predicted depth of cracking with the change of equilibrium suctions in an arid climate

The results are shown in Figure 5-21 and Figure 5-22 for arid and semi-arid climates

respectively. As expected, the crack depths increase similarly with the increase of

equilibrium suction in all soils both in arid and semi-arid climate conditions.

0

1

2

3

4

5

6

0.00 0.20 0.40 0.60 0.80 1.00

Dep

th o

f cra

ckin

g (m

)

Equlibrium suction (MPa)

Regina clay

Horsham clay

Altona clay


Figure 5-22 Predicted depth of cracking with the change of equilibrium suctions in an semi-arid climate

5.7 Conclusions

The UDEC program was used in this chapter to predict the crack depths for several soil

types. The predicted crack depths using the numerical model represent the generally

observed values of depths of cracking in the field reasonably well.

Three soils were selected to represent the clay soils in different climatic locations and

soil types. Generally all three clay soils behave similarly with the change of influencing

parameters. Especially for Regina and Altona clay, the predicted crack depths are

almost the same while the Horsham clay shows around 0.5m less value. However, it is

0

0.5

1

1.5

2

2.5

3

3.5

0.00 0.10 0.20 0.30 0.40 0.50

Dep

th o

f cra

ckin

g (m

)

Equlibrium suction (MPa)


234

reasonable to conclude that all three soils produce similar depths of cracking under the

same conditions.

The effects of several parameters on depth of cracking were analysed and discussed.

The climate condition of the area has a great influence on the depth of cracking and the

predicted crack depths range between zero and 8m when the aridity increases. The

higher compaction during the placement of a layer tends to decrease the depth of

cracking while a lower Poisson’s ratio increases the crack depth. In a situation where

the water table is significantly below the ground level, increasing equilibrium suction

while keeping the surface suction constant tends to increase the crack depth. This may

represent a scenario where the atmospheric conditions remain the same (therefore the

surface suction remains the same) but the water table is dropping thereby increasing the

equilibrium suction level.

While the analysis presented in this chapter encompasses the major stresses that control

desiccation cracking in a certain climatic location, the actual crack development and

dynamics may be influenced by other factors that were not considered in this analysis.

These factors include the effects of continuous drying and wetting dynamics of the soil

and associated influence on the crack dynamics, and the effects of other intervening

events such as debris flow into the cracks and associated changes in stress and moisture

development. Nevertheless, the computed crack depths appear to be reasonable on the

basis of the field observations.

Chapter 6

INHERENT PROPERTIES OF UDEC FOR FRACTURE MODELLING

6.1 Introduction

In previous chapters, the UDEC program was used for fracture modelling utilising the

capabilities available in the program. In other words, the fracture was considered to

initiate when the tensile strength of the soil was reached. However, the fracture energy

associated with opening a new surface was not controlled. It was not clear whether the

UDEC inherently has a kind of fracture energy in its current formulation. Therefore, this

chapter investigates the inherent properties of UDEC formulation with respect to

fracture formation.

235

236

6.2 The Numerical Program UDEC

6.2.1 UDEC operation

The newest UDEC version 5.0 (released in 2011) and the previous version (released in

2004) were used for the modelling in this research. UDEC in its general form:

• allows simulation of motion of blocks (including slip and opening) in a

discontinuous medium;

• treats the discontinuous medium as an assembly of discrete polygonal blocks

with round corners. The motion and interaction of blocks are computed on the

basis of an explicit solution scheme which allows tracing of the mechanical

evaluation of the system even if the process becomes mechanically unstable;

• uses the linear and non-linear force displacement laws to govern the relative

motion along the discontinuities for the normal and shear directions;

• assumes blocks to be rigid or deformable. In the deformable configuration (as

used in this research) it discretises a block into zones and assigns continuum

constitutive models to each of them; and

• provides an in built library of materials models for deformable blocks and

discontinuities to suit the problem.

In UDEC, the geometry of the model is defined in terms of a fixed orthogonal

horizontal-vertical (x-y) system where x is positive in the rightwards direction and y is

positive in the upward direction. Displacements and force refer to the fixed x-y system

and counter clockwise moments and rotations are positive. The tensile stresses and

corresponding material elongations are positive and compressive stresses and

Inherent Properties of UDEC 237

corresponding contractions are negative for the block material. Joint normal stress is

positive in compression while joint normal displacement is positive in opening. Positive

shear stresses are shown in Figure 6-1.

Figure 6-1 Sign convention for positive shear stress components

UDEC only handles problems in two-dimensions either in plane stress or plane strain.

Real problems are generally three-dimensional. However, useful predictions of the

behaviour of systems can often be obtained from two-dimensional analyses depending

on the problem.

6.2.2 Theoretical background of UDEC

UDEC computation is based on a time marching algorithm in which governing

equations are integrated explicitly in time in a series of successive cycles or steps.

Actions performed in one cycle are shown in Figure 6-2. A central finite difference

scheme is used to integrate equations of motion. Any grid point in a deformable block is

238

associated with the surrounding zones. The area influencing one grid point is shown in

Figure 6-3.

The quation of motion for velocities at grid point P is calculated as,

𝑢𝚤̈ = ∫ 𝜎𝑖𝑗 𝑛𝑗 𝑑𝐴𝑠+𝐹𝑖

𝑚+ 𝑔𝑖 [6-1]

𝜎𝑖𝑗 is the zone stress tensor. 𝐴𝑠 is the surface enclosing the area lumped at the grid point

P and 𝑛𝑗 is the unit normal to 𝐴𝑠 . 𝑢𝚤̈ is the acceleration. 𝑔𝑖 is the gravitational

acceleration. 𝐹𝑖 is the resultant external forces applied to the grid point P, can be

obtained as a sum of three terms,

𝐹𝑖 = 𝐹𝑖𝑙 + 𝐹𝑖

𝑐 + 𝐹𝑖𝑧 [6-2]

Apply Equations of motion to all

gridpoints in blocks

Update block grid point co-ordinates and

contacts between blocks (delete or create

contacts as needed)

Compute unbalanced force and exit loop if force if force is below limit and

solve command has been issued

Apply material constitutive equations to all zones in blocks and contacts between

blocks

Derive stresses and nodal forces

Derive velocities and displacements

Figure 6-2 Actions performed during one computation cycle


Forces 𝐹𝑖𝑙 are the external applied loads. Forces 𝐹𝑖

𝑐 result from the contact forces and

exist only for grid points along the block boundary. The force contributions from the

internal stresses in the zones adjacent to the grid point are calculated as,

𝐹𝑖𝑧 = ∫ 𝜎𝑖𝑗 𝑛𝑗 𝑑𝐴𝑠 [6-3]

By finite difference integration of equation [6-1],

𝑢(𝑡+∆𝑡) = 𝑢(𝑡) + �̇�(𝑡+∆𝑡/2)∆𝑡 [6-4]

With velocities stored at the half time step point, it is possible to express displacement

as in equation [6-4] where the superscripts denote the time at which the corresponding

variable is evaluated. Because the force depends on displacement, the

force/displacement calculation is done at one time instant.

Zone 1

P

Zone 2

Zone 3

Zone 4Zone 5

Zone 6

Fi

Figure 6-3 Area associated to grid point P

During each time step, strains are related to nodal displacements in the usual fashion,

240

𝜖�̇�𝑗 = 12

��̇�𝑖,𝑗 + �̇�𝑗,𝑖� [6-5]

This equation (equation [6-5]) does not imply a restriction to small strains due to the

incremental treatment. The constitutive relations for deformable blocks are used in an

incremental form so that implementation on nonlinear problems can be accomplished

easily. The actual form of the equation is,

∆𝜎𝑖𝑗𝑒 = 𝜉Δ𝜖𝜐𝛿𝑖𝑗 + 2𝜇Δ𝜖𝑖𝑗 [6-6]

where, ξ and µ are the Lamé constants, ∆𝜎𝑖𝑗𝑒 is the elastic increment of the stress tensor,

Δ𝜖𝑖𝑗 is the incremental strain, Δ𝜖𝜐 is the increment of volumetric strain and 𝛿𝑖𝑗 is the

Kronecker delta function.

In equations of motion, damping is used to simulate dissipation of kinetic energy in geo-

materials as deformation takes place. Several damping methods can be used in UDEC.

Local damping and auto damping are generally used for quasi-static problems (i.e. when

loading or unloading rates are low enough that inertia effects can be disregarded). The

damping scheme is designed to converge to the static solution (if it exists) as fast as

possible.

For local damping, the direction of the damping force is such that energy is always

dissipated. For deformable blocks, the equation of motion given by equation [6-7] is

replaced by

𝑢𝚤̇ (𝑡+∆𝑡/2) = �̇�(𝑡−∆𝑡/2) + �∑ 𝐹𝑖(𝑡) − 𝛼�∑ 𝐹𝑖

(𝑡)�𝑠𝑔𝑛 ��̇�(𝑡−∆𝑡/2)�� ∆𝑡𝑚𝑛

[6-7]

to incorporate local damping. Viscous damping is used in auto damping. However, the

viscosity constant is continuously adjusted in such a way that the power absorbed by


damping is a constant proportional to the rate of change of kinetic energy in the system.

The adjustment to the viscosity constant is made by a numerical servo-mechanism that

seeks to keep the following ratio, 𝑅, equal to a given ratio.

𝑅 =∑ 𝑃∑ �̇�𝑘

[6-8]

where, 𝑃 is the damping power for a node, �̇�𝑘 is the rate of change of nodal kinetic

energy.

A limiting time step for integration of the dynamic equations of motion is chosen to lead

to a stable computation of internal block deformation and stable computation of inter-

block relative displacements. Generally higher numbers of time steps are preferred with

local or auto damping conditions.

6.3 Description of a problem analysed using UDEC

In order to analyse the scale effects, the problem analysed in this chapter was a centre

crack in a finite square body subjected to far field constant stress (𝜎𝑓) normal to the axis

of the crack, as shown in Figure 6-4(a). The problem was simplified due to the

symmetry of the problem, as shown in Figure 6-4(b).

In the numerical model, two blocks were connected to each other through a joint to

represent the geometry shown in Figure 6-4. The block can be defined as the

fundamental geometric entity for the distinct element calculation in the program UDEC.

The deformable blocks were to be composed of elastic material following an isotropic

elastic constitutive model and the joint was behaving according to the Mohr-Coulomb

242

area of contact with tensile strength. The existing crack differentiates from the joint by

applying no tensile strength for the length of crack. 2𝑊 is the width of the block and 2 𝑎

is the length of the crack.

The bulk modulus (𝐾 ) and shear modulus (𝐺 ) were calculated using the Young’s

modulus (𝐸) and the Poisson’s ratio (𝜈) as:

𝐾 = 𝐸3(1−2𝜈)

[6-9]

𝐺 = 𝐸2(1+𝜈)

[6-10]

and the joint normal stiffness (𝑘𝑛) and shear stiffness (𝑘𝑠) were calculated using the

equation,

𝑘𝑛 = 𝑘𝑠 = 10 �𝑚𝑎𝑥 �𝐾+4/3𝐺∆𝑧𝑚𝑖𝑛

�� [6-11]

𝜎𝑓

𝜎𝑓

2a 2W

2W

𝜎𝑓

2W a

W

𝜎𝑓

Figure 6-4 (a) Problem geometry and modelled problem (b) analysed problem due to symmetry


where ∆𝑧𝑚𝑖𝑛 is the minimum size of the zones. The input parameters given in Table 6-1

are not general but specifically related to a soil since the intention was to obtain a clear

understanding of the program behaviour with the change of different parameters.

Table 6-1 Typical input parameters of the model

Parameter Block Joint Crack

Size 2 𝑊 × 𝑊 (𝑊 − 𝑎) 𝑎

Density 1835 kg/m3 n/a n/a

Bulk modulus 5e8 Pa n/a n/a

Shear modulus 2e8 Pa n/a n/a

Joint normal stiffness n/a 5e8Pa.m 5e8Pa.m

Joint shear stiffness n/a 5e8Pa.m 5e8Pa.m

Tensile strength n/a 1000Pa 0

Cohesion n/a 0 0

Friction angle n/a 25˚ 25˚

Dilation angle n/a 0 0

Residual cohesion n/a 0 0

Residual friction n/a 25˚ 25˚

Residual tensile strength n/a 1000Pa 0

In the model far field stress was applied at the boundary of the block. To avoid unstable

conditions in the model the stress was applied consecutively from zero to the final stress

limit in small increments and then the failure stress was obtained. The procedure

followed in applying the stress is discussed in a later section.

244

6.4 Behaviour of the UDEC Model

The behaviour of UDEC with the change of each parameter was analysed and is

discussed in this section. Mesh size, sizing effect, damping, modulus etc were selected

specifically for the analysis.

6.4.1 Effect of mesh size (l)

Each material block is divided into finite difference zones as discussed before. The

maximum edge length of those triangular zones was given as the input which is referred

to mesh size.

The effect of mesh size change was observed while keeping the other inputs constant,

such as block size, properties of the bulk media and joint, size of the crack and so on.

The change of the size of the mesh in the block is shown in Figure 6-5. From Figure

6-5(a) to (f) mesh size decreases from 0.2 m to 0.005 m. The block size was selected as

1.2 × 0.6 m and the length of the crack was 0.2m.

The failure stress for the above shown different models was obtained from stress vs.

(load step) time curve, where the time at the failure was found from the unbalanced

force vs. time graph. The method followed for obtaining failure stress is shown below in

Figure 6-6.


Figure 6-5 Change of mesh size in a constant size block

Since there was no clear point for the failure in the stress vs. displacements graph, an

alternative method to find the failure stress was needed. The unbalanced force (Figure

6-6(a)) at the beginning shows almost zero value before the joint contacts start to open.

However, as the failure stress was reached and the blocks started to separate from each

other, a sudden change in the unbalanced force curve can be observed. The time when

(a)

(b)

(d)

(c)

(e)

(f)

Mesh size = 0.2m

Mesh size = 0.005m

Mesh size = 0.05m

Mesh size = 0.01m

Mesh size = 0.1m

Mesh size = 0.02m

246

this happened was identified as the failure point and the corresponding time and the

failure stress were determined from the stress vs. time graph as shown in Figure 6-6 (b).

Figure 6-6 Selection of Failure stress

(a)

(b)

σ’


A typical failure stress obtained using this method is shown in Figure 6-6. All results

are shown in Figure 6-7 where log (failure stress) vs. log (mesh size) is shown. These

results show the effect of mesh size indicated as l. 𝑘𝑛 is the joint normal stiffness and N

denotes the number of stress increments used to achieve the final stress. N is described

in more detail in Section 6.4.2. As can be seen from Figure 6-7, there is slight increase

in failure stress with increasing mesh size. However, it can be ignored for the cases

considered.

Figure 6-7 Effect of mesh size; 𝐾 = 5e9 Pa, 𝐺 = 2e9 Pa, 𝑘𝑛 = 5e9 Pam, Damp = 0.2, N = 10000, 𝑊 = 0.6m 𝑎 = 0.2 m and 𝜎𝑡 = 1000Pa.

𝝈 is the failure stress and 𝝈𝒕 is the tensile strength of the joint.

Potyondy and Cundall (2004), (as proposed by (Anderson, 1991)) has presented a

relationship for failure stress as,

𝜎΄ = 𝐾𝐼𝐶𝐶(𝜓) √𝜋

𝑎−12� [6-12]

0.1

1.0

0.004 0.04

log

(σ/σ

t)

log L

kn=5e9Pa kn=5e8Pa

248

where, 𝐶(𝜓) = [𝑠𝑒𝑐 �𝜋𝜓2

�]12� [1 − 0.025𝜓2 + 0.06𝜓4] and 𝜓 = 𝑎

W� . 𝐾𝐼𝐶 is the mode 1

fracture toughness, σ΄ is the failure tensile stress, 𝑎 is the crack length, and 2𝑊 is the

height of the soil block.

According to equation [6-12] there is no direct relationship between the failure stress

and mesh size, similar to the results predicted from UDEC in Figure 6-7. However, the

maximum stress which is governed by tensile strength shows proportionality with the

length of joint. So, when the tensile stresses are applied on a length of 𝑊, failure occurs

in a length of (𝑊 − 𝑎), and the ratio between the applied tensile stress vs. failure stress

can be written as,

𝜎′𝜎𝑡

= 𝑊−𝑎𝑎

[6-13]

When, 𝑊 =0.6m, applied tensile stress=1000Pa and 𝑎 =0.2m (as the values used to

obtain results in Figure 6-7), the failure stress is 666Pa. The results produced by the

numerical program are also closer to this value.

6.4.2 Effect of number of increments

The effect of change of the number of stress increments (N) was examined in this

section. In UDEC, the stresses should be applied consecutively in small steps to avoid

instability of the model. The following part of the code was used to apply stresses at the

boundaries.

def bstress

str1=1.0e3 ; Failure stress


sdif=str1/10000 ; Stress increment, N = 10000

nsb1=0 ; Initial normal stress at the boundary

nsblt=1*str1 ; Normal stress limit at the boundary

loop while nsb1 < nsblt

nsb1=nsb1 + sdif

command

bound st 0 0 sdif range yr -0.001,0.001 ; Applying stresses at bottom boundary

bound st 0 0 sdif range yr 9.599,9.601 ; Applying stresses at top boundary

solve force 0.1

endcommand

endloop

end

As it can be seen in the FISH code, the final stress was divided into 10000 divisions and

each division was added cumulatively to the stress. This number of divisions was

referred to as the number of stress increments (N).

Here, the effect of the number of suction increments was captured, while keeping all

other parameters constant. The variation of failure stress and overall modulus of the

model was observed. The failure stress was obtained as described in Section 6.4.1.

Overall modulus was obtained from the stress vs. displacement curve as shown in

Figure 6-8.

250

Figure 6-8 Stress vs. displacement curve using to get Overall 𝐸

The failure stress and overall modulus values are shown in Figure 6-9. Both failure

stress values and overall modulus values do not show significant change when changing

the number of increments, indicating no major effect from the number of increments.

∆𝜎

∆𝑢

E* = ∆𝜎∆𝑢


Figure 6-9 Effect of number of stress increments; 𝐾 = 5e9Pa, 𝐺 = 2e9Pa, 𝑘𝑛=5e9Pa, Damp=0.2, 𝑊 =0.6m, 𝜎𝑡=1000Pa, 𝑎 =0.02m and 𝑙 =0.2m

However, when the load vs. displacement curves are examined it can be observed that,

these were changing from a rough variation to a smooth linear variation when

increasing the number of stress increments at the initial part of the curve (Figure 6-10).

Hence it can be recommended to use small increments when changing the conditions of

a model such as stress, velocity etc.

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0

100

200

300

400

500

600

0 2000 4000 6000 8000 10000 12000 14000

Mod

ulus

*109 ,

E (P

a)

Stre

ss, σ

(Pa)

Number of increments

Stress

Overall E

252

Figure 6-10 Change of load vs. displacement plot with the change of number of cycles

6.4.3 Effect of crack length

In this section, models were used to check the effect on failure stress. The crack lengths

of the models were changed from 0.1m to 1.0m as shown in Figure 6-11 and the results

were plotted in Figure 6-12. By manipulating equation [6-12], the failure stress over

tensile strength can be given by the relationship,

(a) (d)

(e) (b)

(c) (f)

N=2000

N=4000

N=8000

N=6000

N=10000

N=120000


𝜎΄𝜎𝑡

=𝐾𝐼𝐶

𝜎𝑡 √𝜋 (

1𝐶(𝜓) √𝑎

) [6-12a]

Hence in order to obtain a straight line response log (failure stress/tensile strength) vs.

log (1/ C(ψ) √𝑎 ) was plotted. The results show good agreement to the theoretical

gradient as shown in Figure 6-12.

It can be seen that when the crack length increases the failure stress reduces. When the

crack length increases the uncracked length reduces, which reduces the energy required

to break the material and the joint can fail with lesser stress. This highlights that the

model behaves according to the traditional fracture mechanics theoretical concepts.

a2.4m

1.2m

aaa

1.2m1.2m1.2m

Figure 6-11 Model geometry change with changing crack length

254

Figure 6-12 Effect of crack length, 𝐾 = 5e9Pa 𝐺 = 2e9Pa 𝑘𝑛=5e9Pam, Damp=0.2, N=10000, 𝑊 =1.2m, 𝜎𝑡=1000Pa, 𝑙 =0.04m and 𝐾𝐼𝐶=530Pam0.5

6.4.4 Effect of damping value

In UDEC, a solution is reached when the rate of change of kinetic energy in the model

approaches a negligible value (Itasca, 2004). This is accomplished by damping the

equations of motion, to reach a force equilibrium state as quickly as possible under the

applied initial and boundary conditions. Hence, the effect of damping was also

examined as another parameter, since it may affect the crack propagation specifically.

Here the damping value was changed from 6% to 25%, and other parameters kept

constant. The failure stresses and fracture toughness values were plotted in Figure 6-13.

As the graph shows, the effect of damping within this range is negligible. However, to

see the minor effects of damping on the stability of the model, the stress vs.

displacement curves of each model were examined (Figure 6-14). The slight increase in

the failure stress with increasing damping value can be noted. Nevertheless, these

0.1

1

0.3 3

Log

(σ′/

σ t)

log [1/a0.5*C(ψ)]

Theoretical Gradient Line


effects can be considered negligible and, therefore, small. A 6% damping value was

selected as a reasonable value.

Figure 6-13 Effect of damping value, 𝐾 = 5e9Pa, 𝐺 = 2e9Pa, 𝑘𝑛=5e9Pam, N=10000, 𝑊 =0.6m, 𝜎𝑡=1000Pa and 𝑎 𝑙� =10 (𝑎 =0.02m, 𝑙 =0.2m)

Figure 6-14 Stress vs. displacement curves when changing the damping value

0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Damping value

Stress

KIC

D=6% D=25%

256

6.4.5 Effect of block size

In this section, the size of the block has been changed while keeping the crack length

and other soil and joint properties constant. The change of the geometries of the models

is shown in Figure 6-15.

To find the applicability of the equivalence between the theoretical assumptions in

Linear Elastic Fracture Mechanics (LEFM) and the theoretical formulation in the UDEC

analysis, the stress along the axis of the crack was plotted in Figure 6-16, in which the

LEFM results were derived considering the centre cracked rectangular plate under

uniform tension.

Figure 6-15 Change of Size of block when the crack length kept constant

For this configuration, the stress intensity factor, 𝐾𝐼 is given by,

𝐾𝐼 = 𝜎𝑓√𝜋𝑎 . 𝑓(𝛼, 𝛽) [6-14]

nW

nW

nW

nW

2n

2n

2n

2n a

a

a

a

n = 1

n = 2

n = 3

n = 4


where, 𝛼 = 2𝑎𝑊

, 𝛽 = 2𝐻𝑑𝑊

, where 𝐻𝑑 is the height of the block and 𝑓(𝛼, 𝛽)= 1.0415 in this

particular case (Murakami, 1986) and 𝜎𝑓is the far field stress applied on the finite plate.

The normal stress near the crack tip was given by:

𝜎𝑛 = 𝐾𝐼√2𝜋𝑟

[6-15]

where r is the distance from the crack tip.

Figure 6-16 Normal Stresses from UDEC and LEFM ahead of the crack

6.4.6 Scaling up the model geometry

The failure stress is affected by scale of the problem in fracture mechanics. Hence, the

geometry of the model was scaled up by a factor ‘n’ to examine the behaviour of the

model response, as shown in Figure 6-17. The effect of problem scale on failure stresses

and the fracture toughness was observed in this series of models. The soil and joint

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Nor

mal

Str

ess (

Pa)

Distance from the crack tip (m)

Numerical

Theoretical

258

properties were selected as constant values. Furthermore, the mesh size was kept

constant in all the models.

6.4.6.1 Effect of joint normal stiffness while scaling up

Figure 6-18 Scale effect on failure stress, 𝐾 = 5e9Pa, 𝐺 = 2e9Pa, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

0.100

1.000

0.02 0.2 2

log

(σ'/σ t

)

log na

kn=5e8Pa kn=5e9Pa kn=5e10Pa kn=5e11Pa

n = 1

n = 2

n = 3

n = 4

Figure 6-17 Plot of models changing the size of geometry

nW

nW

n

nW

2n

2n

2n

2nW na

na

na

na


The log of failure stress was plotted against log of crack length with different joint

stiffness values as shown in Figure 6-18.

When lower fracture energies were used associated with higher normal stiffness values,

the curves tend to show close to a gradient of -1/2 representing LEFM behaviour. When

the fracture energy is increased by lowering the normal stiffness values, the normalised

failure stress becomes flatter for small crack lengths prior to changing to approximately

-1/2 gradient. The flatter part of the curve indicates the influence of fracture process

zone for smaller crack lengths, showing quasi-brittle or more ductile behaviour.

However, as the crack length increases, the influence of the fracture process zone

diminishes, eventually approaching the LEFM response. From this graph we can

suggest that the relevance of cohesive law is crucial when higher fracture energies are

used with higher crack lengths and when the lower fracture energies are considered.

The fracture energy related to the curve representing 𝑘𝑛 = 5𝑒9𝑃𝑎 is around 0.3Pam,

which is more relevant to very soft clay soils and this value is lower than that of natural

compacted clay.

Displacement

Tens

ile st

ress

Fracture energy

Figure 6-19 Inherent fracture energy present in current UDEC formulation

260

Therefore, in the current implementation of UDEC, there is an inherent fracture energy

associated with the initial elastic response of the joint. This fracture energy component

is shown in Figure 6-19. However, normal stiffness 𝑘𝑛 should be not be used to control

the fracture energy after failure because it is related to the physical behaviour prior to

cracking. A more rational way is to control the post failure stiffness to represent the

actual fracture energy of crack formation. This aspect will be considered in the next

chapter.

Figure 6-20 Scale effect on Fracture energy, 𝐾 = 5e9Pa, 𝐺 = 2e9Pa, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

Figure 6-20 presents theoretical and UDEC results for fracture energy with different

normal stiffness values and block size. When the normal stiffness value is constant, the

size of the block does not have an effect on fracture energy. This can be expected since

the fracture energy is related to the tensile strength and 𝑘𝑛 as noted earlier (both of them

are constant).

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

0 0.5 1 1.5

log

Gf

a^0.5

kn=5e8 UDEC

kn=5e9 UDEC

kn=5e10 UDEC

kn=5e11 UDEC

kn=5e8 Theoretical

kn=5e9 Theoretical

kn=5e10 Theoretical

kn=5e11 Theoretical


Figure 6-21 Effect of normal stiffness on failure stress, 𝐾 = 5e9Pa, 𝐺 = 2e9Pa, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

Figure 6-22 Effect of normal stiffness on fracture toughness, 𝐾 = 5e9 Pa, 𝐺 = 2e9 Pa, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

0.100

1.000

1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

log

(σ′/

σ t)

log Kn

1.0

10.0

100.0

1000.0

1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

K IC

log kn

UDEC changing kn

Theoretical

262

Figure 6-23 shows the effect of normal stiffness on the normalised strength while all

other aspects of the model were kept constant. When the normal stiffness is low, the

joint imparts a significant resistance to failure due to associated high fracture energy.

However, when the joint stiffness becomes very large, the strain energy coming from

the continuum section (its modulus was kept constant in this instance) can easily break

the joint overcoming the fracture energy resulting from the low joint stiffness.

Therefore, the curve tends to flatten out as the joint stiffness increases.

When increasing the normal stiffness it can be seen that the numerical results deviate

from the theoretical value, although for smaller normal stiffness values both values

show similar results. That happens when smaller fracture energies using unloading

modulus also act same but in numerical model which does not allow the fracture

toughness to fall in high numbers.

6.4.6.2 Effect of bulk and shear modulus

After observing the variation of joint stiffness, the effect of bulk and shear modulus was

tested in this section.

Figure 6-23 shows the effect of modulus on failure stress. When the stiffness of the

block material increases to higher values then the LEFM is no longer applicable since

the failure stress does not show considerable variation with the size of the geometry.

This shows that the inherent fracture behaviour in UDEC comes into the picture only for

deformable blocks. That indicates that when the blocks have a smaller modulus,

deformable material can store and release energy for fracture propagation.


Figure 6-23 Effect of modulus on failure stress 𝑘𝑛=5e9Pam, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

Figure 6-24 Effect of modulus on failure stress, 𝑘𝑛=5e9 Pam, Damp=0.2, N=10000 and 𝜎𝑡=1000Pa

0.100

1.000

0.02 0.2 2

log

(σ′/

σ t)

log a

K=5e8 K=5e9 K=5e10 K=5e11

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.2 0.4 0.6 0.8 1 1.2 1.4

K IC

a0.5

K=5e8 UDEC K=5e9 UDEC K=5e10 UDEC K=5e11 UDEC K=5e8 Theoretical K=5e9 Theoretical K=5e10 Theoretical K=5e11 Theoretical

264

6.5 Summary and Conclusions

The behaviour of the output results from the UDEC numerical model was observed with

the change of different input parameters focused on energy associated with the

numerical modelling.

The failure stress increases with the increase of mesh size. This result shows good

agreement with the results presented in Chapter 3. This shows the need of selecting the

precise size of the mesh before using the program for the actual problem.

The increments used to implement the stress change need to be selected carefully so that

no high unbalanced forces are generated in the model. Similarly, the studies should

include a selection of an appropriate damping value to be used in the model. This can

become important when cracks are close to each other where opening of one crack may

affect the response of the other crack.

The hidden fracture energy associated with current implementation of UDEC was found

to be due to the initial normal stiffness of the joint. When the normal stiffness is low,

the associated fracture energy can lead to high fracture process zones and therefore,

plastic-like behaviour in fracture development. Under these circumstances, the model

results will deviate from classical LEFM behaviour. However, the fracture energy

cannot be properly controlled with the current implementation. This aspect will be

considered in more detail in the next chapter.

Chapter 7

MODELLING OF DESICCATION CRACK DEPTHS INCORPORATING SOIL FRACTURE ENERGY

The numerical modelling work undertaken up to now did not consider soil fracture

energy directly. This chapter will detail how it can be incorporated in the numerical

modelling. The cohesive fracture approach is presented as a way of consistently and

conveniently incorporating soil fracture energy in desiccation crack formation. The

cohesive fracture method is considered to be particularly suited to soil since soil can

exist in a wide range of consistencies during desiccation.

7.1 Introduction

Fracture mechanics is a failure theory that determines the material failure by fracture

using an energy criterion possibly in conjunction with yielding. It considers that in order

to generate a surface within a material, energy is required and this energy is referred to

as the fracture energy. Basically fracture mechanics can be considered in two main

265

266

forms: Linear Elastic Fracture Mechanics (LEFM) and Elasto-Plastic Fracture

Mechanics (EPFM). LEFM is more applicable for brittle materials such as glass and

some metals while EPFM gives excellent results for ductile materials such as certain

metals, alloys and polymers.

LEFM considers that there is a stress singularity at the crack tip and any plastic zone

that develops due to this stress singularity is relatively small compared to the size of the

structure subject to loading. For soil, however, it is not very clear whether LEFM will

be applicable for all soil consistency states from the liquid limit to below the plastic

limit. Arguments for the use of LEFM (Lakshmikantha et al., 2012) and for the use of

EPFM (Hallett and Newson, 2001) have been raised by various researchers. Therefore,

a common approach that caters for both conditions is needed for soil cracking.

In this chapter, the use of a cohesive crack approach that uses the stress softening while

opening the crack is followed. This approach is applicable to both elastic and plastic

conditions and hence can cater for all soil consistency states. The values of the softening

curve are calculated on the basis of the fracture energy. The approach is implemented

in the numerical code of UDEC and is applied to predict the crack depths for the same

soils used in the Chapter 6. The effect of cohesive properties of the fracture during the

initiation and progression of fracture is studied.

7.2 Basics of Linear Elastic Fracture Mechanics (LEFM)

LEFM was originated by Griffith (1921; 1924) formulating an energy based criterion

for propagation of cracks and then was extended by Irwin (1957) developing the stress

Modelling of desiccation crack depths incorporating soil fracture energy 267

intensity factor and fracture process zone. Later Rice (1968b (a); 1968a (b)) introduced

the J-integral as another way of computing the critical energy release rate applicable to

fracture. However, LEFM is valid as long as the material behaves elastically and the

fracture process zone is small compared to the loaded area of the structure. In other

words, the main assumption is that the whole area of the material is elastic except the

vanishingly small area at the tip of the crack.

7.2.1 Griffith’s criterion

It is well known that the growth of a crack requires the creation of two new surfaces and

hence consumes more energy because the surfaces carry more energy than material

bodies. Based on this concept, Griffith found an expression for the stored elastic energy

U of the crack by solving the elasticity problem (Figure 7-1) of a finite elliptical crack

with a length 2𝑎 at the centre of a semi finite, homogeneous, isotropic plate with

Young’s modulus 𝐸 which is subjected to applied tensile stress σ given by,

𝑈 =𝜋𝜎2𝑎2

𝐸

[7-1]

assuming that the thickness of the plate is of unit length. When the surface energy per

unit area of the material is 𝜁, the total required energy to open the crack by creating two

new surfaces of a crack of length 2a is,

𝑊 = 4𝑎𝜁 [7-2]

268

When the crack extends further, the required energy is provided by the stored energy,

which can be expressed mathematically as in equation [7-3] based on the assumption

that the energy consumed in crack propagation consisted of the surface energy only.

𝑑𝑈𝑑𝑎

=𝑑𝑊𝑑𝑎

[7-3]

By substituting equations [7-1] and [7-2] in equation [7-3] and rearranging it for the

stress as in equation [7-4] this gives the stress required for crack opening according to

Griffith as:

𝜎𝑐 = �2𝐸𝜁𝜋𝑎

[7-4]

Two important conclusions can be drawn from equation [7-4]. First, the critical stress

level for a given crack length varies with the material due to the variation of the surface

energy. Second, the critical stress level decreases with increase of the crack length and

the critical stress level is inversely proportional to the square root of the prevailing crack

length. Another important observation is that the crack would not propagate if the initial

crack length is zero since the stress required becomes infinite. This means that LEFM

cannot predict the crack initiation in its pure from.


2a

σ

σ

x

y r

θ

Figure 7-1 Semi infinite plate with the central crack of the length 2a and the directions for near tip stress field

7.2.2 Irwin's modification

Griffith’s findings are based on very brittle materials such as glass. Hence, for ductile

materials the sole consideration of surface energy for cracking comprises the results. As

a remedy for this drawback, other parameters were introduced such as critical energy

release rate 𝐺𝐼𝐶 and stress intensity factor 𝐾𝐼𝐶.

7.2.2.1 Critical energy release rate

The critical energy release rate is a material property and within the elastic range it is

constant and can be graphically shown in a diagram as shown in Figure 7-2. The

shaded area illustrates the potential energy change due to the crack opening for ∆𝒂

length. The potential energy change is the difference between the external work done

and the stored but recoverable elastic strain energy. The energy release rate (ERR) can

be mathematically expressed as:

270

𝐺𝐼𝐶 =𝑑𝑈𝑑𝑎

=𝜋𝜎𝑐

2𝑎𝐸

[7-5]

By rearranging the terms the relation for the critical stress can be given as:

𝜎𝑐 = �𝐸𝐺𝐼𝐶

𝜋𝑎

[7-6]

This equation describes the inter-relation between the material property, stress level and

the crack size by means of critical ERR, critical stress and crack length. Furthermore, it

gives the advantage of omitting the surface energy parameter and hence omitting the

error of not considering the plastic flow near the crack and associated non linearity.

Load

DisplacementO

a

a + Δa

A

Area representing

energy supply for fracture

B

Area representing elastic strain

energy

Load-displacement

curve

Figure 7-2 Actual incremental fracture process in Load-Displacement space


7.2.2.2 Stress Intensity Factor (SIF)

When a body containing a crack is subjected to tensile stresses, a strong stress

concentration develops around the crack tip. This stress concentration varies in a

manner inversely proportional to the square root of the length from the crack tip

regardless of the shape and boundary conditions, on which, however, the intensity of the

stress concentration will depend. For the same intensity, the stresses around and close to

the crack tip are identical (Bažant and Planas, 1998). The intensity of stress is greatly

affected by the mode of the fracture also. However, in the present research, only mode I

fracture associated with pure tensile loading is considered as applicable to desiccation

cracking predominantly.

The in plane stress acting at a point ahead of the crack tip was described by Irwin using

the stress intensity factor as,

𝜎𝑖𝑗 =𝐾𝐼

√2𝜋𝑟𝑓𝑖𝑗(𝜃) [7-7]

The expression 𝑓𝑖𝑗(𝜃) is a known function of 𝜃 and 𝑟 and 𝜃 are the cylindrical polar

coordinates as shown in Figure 7-1. 𝐾𝐼 is the stress intensity factor for mode I crack

opening. This factor 𝑓𝑖𝑗(𝜃) is dimensionless and hence is independent of stress and the

size of the structure except the shape factors of the geometry.

Clearly the actual stress acting on the body is greater than the stress given by equation

[7-7]. Hence the solution is valid only near the crack tip, which is referred as the

fracture process zone as shown in Figure 7-3.

272

The expression to obtain the SIF changes with the geometry of the structure. The

expressions for most possible problems have been summarised by Sih (1973). The SIF

in an infinite body as shown in Figure 7-1, can be expressed as:

𝐾𝐼 = 𝜎√𝜋𝑎 [7-8] St

ress

Fracture Singularity dominated zone

Distance

KI/√2πr

σ (theoretical with externally applied load)

Stress limit due to plastic zone

Figure 7-3 Distribution of the stress normal to the crack plane (Wang, 1996)

7.2.2.3 The relationship between the SIF and ERR

Since the near tip stress field is unique to the material of the structure and the energy

flow rate into the crack tip should depend on this stress field, a unique relationship

between SIF and ERR can be derived.

Simply by equating equation [7-6] and [7-8], the expression for critical conditions can

be derived as,


𝜎𝑐 = �𝐸𝐺𝐼𝐶

𝜋𝑎 =𝐾𝐼𝐶

√𝜋𝑎

𝐾𝐼𝐶2 = 𝐸𝐺𝐼𝐶 [7-9]

This expression applies in the plane stress condition and if the plane strain conditions

are considered equation [7-9] should be adjusted to,

𝐾𝐼𝐶2 =

𝐸𝐺𝐼𝐶(1 − 𝜈2) [7-10]

These equations (equation [7-9] and [7-10]) can also be used for the other conditions

less than the critical condition. Hence they can be identified as the general relationship

connecting the SIF and ERR.

7.3 Past Approaches for Numerical Modelling of Fracture

As discussed in the Chapter 2, several numerical models have been used by researchers

to model the desiccation process. However, LEFM and the cohesive crack method

appear to be leading in the state of the art fracture analysis.

7.3.1 Numerical modelling attempts using LEFM

Many computer programs used for fracture modelling (Haberfield, 1987; Haberfield,

1990; Lim et al., 1994) have considered the fracture toughness and the SIF at the crack

tip. On most occasions, the fracture was allowed to open when the measured SIF at the

crack tip stress field has reached the critical value. When the critical SIF (𝐾𝐼𝐶 ) is

involved in the numerical analysis, a finer mesh is often required near the crack tip and

274

re-meshing near the crack tip is required with the advancement of the fracture. This can

be very time-consuming and computationally expensive. Furthermore, it assumes that

the applicability of material linear elasticity in the presence of an infinite tensile stress at

the crack tip. Hence it ignores the presence of a finite tensile strength for the material.

LEFM is more valid for brittle soils (Prat et al., 2008). It assumes infinite tensile

stresses near the crack tip, which is not valid when the material has limited tensile

strength and displays significant plastic behaviour around the crack process zone. The

significant plastic behaviour in clay beams as experienced in tests by (Hallett and

Newson, 2005) shows the requirement of Elasto-Plastic Fracture Mechanics (EPFM) for

some soil consistencies.

7.3.2 Modelling attempts with cohesive crack

As the drawbacks and difficulties identified in fracture mechanics approach, in

geotechnical engineering a cohesive crack approach was introduced as a powerful

alternative for fracture modelling.

The cohesive crack method was introduced in the early 1960s by Dugdale (Dugdale,

1960) and Barenblatt (Barenblatt, 1962). It was introduced to represent different

nonlinear processes occurring at the front of a pre-existing crack of quasi-brittle

materials by which the peak and post-peak fracture behaviour can be determined. In the

late 1970s, this method was extended by proposing that the cohesive crack may be

assumed to develop anywhere, even if no pre-existing macro crack is actually present as

considered before (Hillerborg et al., 1976; Modeer, 1979).


w

FPZ

0

Stress

Distance from the crack tip

Tensile strength

Figure 7-4 Bridging stresses at the crack tip while crack opening considered in the cohesive crack models

In the cohesive crack model, the entire fracture process zone is considered to be lumped

into the crack line and is characterised in the form of a stress-displacement law which

exhibits softening. In other words, the cohesive crack law lets the bridging stresses

across a crack that is opening up drop progressively from the tensile strength to zero as

the two crack faces move apart, as shown in Figure 7-4. Hence it allows for the

recognition of the finite tensile strength as well as the plastic energy dissipation in the

crack-processing zone.

Recently Amarasiri et al. used this cohesive modelling approach for soft rock

modelling. They used a cohesive model featuring a bilinear softening curve to model the

mode I fracture in a three point bending test in a single edge notched beam (Amarasiri

276

and Kodikara, 2011b). The change of displacement with the applied load at the crack

mouth which was obtained from the UDEC modelling was compared with the

laboratory results and found good agreement. They have concluded that the cohesive

law approach is a convenient and rational method for fracture modelling in soft rock

geo-mechanics.

Later Amarasiri has successfully reproduced another three point bending test laboratory

results for clay beam (Amarasiri and Kodikara, 2011a) confirming the applicability of

the cohesive modelling approach for fracture modelling in geo materials.

7.4 Modelling Crack Depths with Cohesive Properties

7.4.1 Cohesive crack implementation

Cohesive crack models are used in fracture modelling since they describe in full the

progressive cracking process. Different analytical softening curves can be used

depending on the different softening behaviours in various materials used for the

analysis, which can be referred as rectangular, linear and bilinear (Bažant and Planas,

1998). These typical curves are shown in Figure 7-5. The fractures in compacted clay

layers follow the linear softening curve according to the findings of Amarasiri et al.

(2010a). Therefore, a linear softening law was described in the following models.


Dugdale

Crack opening (w)

Linear

Bilinear

Nor

mal

Stre

ss (σ

)

Rectangular

Figure 7-5 Softening curves

The constitutive model for the block material was selected as linear elastic with the

same soil parameters used in the previous sections. Then the joint model was selected as

the area contact-Coulomb slip with residual strength. Linearly decreasing suction profile

(suction profile 2) was used in cumulative incremental fashion as discussed in previous

sections.

The tensile strength of the joint is considered as a function of suction similarly to

previous sections. The cohesive law used in the numerical model is shown in Figure

7-6. The cohesive law is applied to the model in several steps in the FISH code.

According to the tensile stress at the point the fracture toughness has been calculated

using the power law equation

𝐾𝑎𝑝𝑝 = 0.154𝜎𝑡1.022

[7-11]

as given in Amarasiri et al.(2010a). Then the fracture energy is calculated from Irwin’s

equation,

278

𝐺𝑓 =𝐾𝑎𝑝𝑝

2

𝐸

[7-12]

Crack opening (w)

Nor

mal

Stre

ss (σ

)

b

yield limit

ca residual

limit

Figure 7-6 Linear softening law used in the numerical model

The gradient of line ab in Figure 7-6 is the joint normal stiffness from which the yield

limit can be calculated. Fracture energy (𝐺𝑓) is the area under the abc curve in Figure

7-6. Hence from the geometry of the curve the residual limit can be calculated. Since

the yield limit and residual limit is known, the tensile stress can be changed according to

the softening curve. As the normal displacement exceeds the yield limit the tensile stress

will be changed according to the softening tensile value from the bc arm of Figure 7-6.

Also when the normal displacement of the joint exceeds the residual limit, the tensile

stress can be assigned as zero.

The use of cohesive cracks to model desiccation cracking is more complex than in

cracking where material properties are held constant because the cohesive softening


laws themselves may change while the crack is opening up. This changing process has

been described in detail by Kodikara and Amarasiri (1961).

7.4.2 Compacted clay soils

7.4.2.1 Using the linearly decreasing suction profile

According to the results of Chapter 3, the UDEC program generally produces higher

crack depths than the analytical solutions which are similar to the predictions from

LEFM approach. By this method it was decided to apply fracture energy externally to

the crack tip. Hence several models were run with cohesive law implemented to the

joint.

To implement the cohesive law, the linear softening law was used. And the suction

profile was applied as in Chapter 3. The material properties and boundary conditions

were the same as in the models in Chapter 3 excepting the cohesive properties at the

crack.

7.4.2.2 Using observed suction profiles

Following from Chapter 6, development of cracks under different climate conditions

was analysed incorporating soil fracture energy. The analysis method was improved by

incorporating cohesive properties for Regina, Altona and Horsham clay soils. The

procedure used for applying cohesive crack was similar to the method described in the

Section 3.7 in the Chapter 3. However, two softening laws were followed in this section,

280

namely, the Dugdale softening law and linear softening law. The overall method used in

the numerical model is described briefly in the following paragraph.

The fracture energy was calculated on the basis of the tensile strength (Equation 7-11

and 7-12). Then the yield limit as shown in Figure 7-6 was calculated by dividing the

tensile strength by the normal stiffness and then the residual limit was calculated using

the fracture energy. Using these values the residual tensile stress is calculated and

assigned to the crack as a compression force. The stress change due to the suction

change was calculated as in equation [5-10] (in Chapter 5) and applied progressively to

the block of length 40m and height 8m.

7.4.3 Modelling crack depths in soft soils

In order to demonstrate the modelling of soft natural soils, the field test undertaken by

Konrad and Ayad (1997a) was used. However, only one crack was allowed to propagate

in order to compare the effect of cohesive properties on the joint.

7.4.3.1 Test results and soil parameters

Konrad and Ayad (1997a) used the natural clay at Saint-Alban in Saint Lawrence

Valley for the desiccation test. The test was carried out for the clay soil at 2m below the

ground level since the clay layer was almost homogeneous between 2m and 7m depths

from the ground according to the vane and piezocone penetration tests. The moisture

contents at various depths were obtained using time domain reflectometry (TDR) probes


up to 2.7m depth. Then the moisture content profiles were recorded with time as shown

in Figure 7-7.

Figure 7-7 Moisture content profiles of Saint-Alban clay test (Konrad and Ayad, 1997a)

The clay between 1.9m and 2.7m depth showed 25% and 50% plastic and liquid limits

respectively with gravimetric moisture content greater than 100%, hence could be

considered as soft clay. The Poisson’s ratio for Saint Alban clay was considered as 0.3

and the Young’s modulus at 2m depth was selected as 5MPa according to the recorded

range from 4MPa to 6MPa (Ayad et al., 1997). The specific gravity of the soil was 2.8

(Konrad and Seto, 1994).

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100 120 D

epth

from

2m

bel

ow th

e su

rfac

e (c

m)

Moisture content (%)

t=0h

t=18h

t=24h

t=42h

t=65h

t=73h

t=97h

t=145h

t=193h

t=241h

282

7.4.3.2 Material properties in the numerical analysis

Similarly to the previous chapter (Chapter 6) the suction was obtained from the SWCC

as given by Konrad and Ayad (1997a) for Saint-Alban clay. The empirical equations

used for inputting the soil characteristics in the numerical model for different moisture

content ranges are shown in Figure 7-8. Here the gravimetric moisture content was used

instead of volumetric moisture content given in the above reference.

The Young’s modulus of the Saint-Alban clay was selected as 5MPa. However, due to

the fact that the soil properties changed with the moisture content, it was decided to

incorporate this change in the numerical model, although it had not indicated a

considerable effect in previous analyses undertaken on other soils.

Figure 7-8 SWCC for Saint-Alban clay and the empirical equations for different moisture content ranges

0

20

40

60

80

100

120

1 10 100 1000

Gra

vim

etric

moi

stur

e co

nten

t, w

(%)

Suction, S (kPa)

S = 3843.4e-0.109w R² = 0.9975

S = 520.78e-0.03w R² = 0.9993

S = 19727e-0.08w R² = 0.9746

25.5

74.5


The Young’s modulus increase with the increase of the suction (Kodikara et al., 2004)

and the following relation was proposed for Werribee clay,

𝐸 = 24(1 − 2𝜈)(𝑆)0.92 [7-13]

The Werribee clay testing was undertaken from a slurry state, whereas the Saint-Alban

clay had some strength and stiffness at the beginning. Hence, equation [7-13] was

modified to incorporate the initial conditions and the new equation is given as:

𝐸 = 𝐸0 + 24(1 − 2𝜈)(𝑆 − 𝑆0) [7-14]

where, 𝐸 is the Young’s modulus, 𝐸0 is the initial Young’s modulus which was selected

to be 5MPa, 𝜈 is the Poisson’s ratio and 𝑆0 is the initial suction which is taken to be

5.6kPa calculated from the SWCC.

The variation of tensile strength with drying was also considered. It is reasonable to

assume that tensile strength increases linearly with the suction (Morris et al., 1992;

Nahlawi, 2004). According to Ayad (1997), the tensile strength of the material at the in

situ moisture content was about 9kPa, as obtained by pressure testing on cylindrical

specimens. Taking the soil suction at this point to be 5.6kPa as consistent with the

SWCC, the following relation was derived:

𝜎𝑡 = 1.74𝑆 [7-15]

The fracture energy of the Saint-Alban clay soil was calculated from the experimental

data presented in Ayad et al. (1997). This was calculated by dividing the work done on

the specimen by the surface area fractured. The work done on the specimen was

obtained from the area under the load displacement curve. Although the fracture energy

obtained through this method is a rough estimation, the calculated average value from

284

three samples was considered as reasonable. The fracture energy value of 0.74N/m was

obtained for the clay at the in situ moisture content to be used in the numerical analysis.

7.4.3.3 Model implementation and applying stresses

After deciding the initial soil parameters and their variations with others, it was required

to implement those in the numerical program. In UDEC the implementation of the stress

and other soil parameter variation can be done using the in-built language FISH. The

method of applying stress change is described below.

The successive moisture content profiles shown in Figure 7-7 as drying progressed were

implemented progressively in the numerical model. The soil properties such as Young’s

modulus and tensile strength were modified according to the moisture content change or

the corresponding suction change, as described above. The suction due to the moisture

content change was applied as a stress change to the soil layer.

The stress increment in the model was applied similarly to the method used in Chapter

5. However, equation [5-10] that was used to calculate the stress increment in Chapter 5

was modified to incorporate moisture content instead of void ratio in this series of

models. The modified equation is given as:

∆𝜎 =𝐸

3(1 − 2𝜈)∆𝑤𝐺𝑠

1 + 𝑤𝐺𝑠 [7-16]

This stress increment was implemented in the FISH code according to the moisture

content profiles. A soil block of 3m length and 1.2m height was selected for the


numerical model with a very fine mesh. A linear softening cohesive law was used to

manage residual tensile stresses while opening the crack.

7.5 Predicting Crack Depths in Compacted Clay Soils

After implementing the model as described in the section 7.4.1, the results for the crack

depths were obtained. The effect of the different implementations of cohesive law and

the effect of the shear angle with respect to the suction were checked and the results are

discussed below.

7.5.1 Results obtained using the linearly decreasing suction profile

The results obtained from the cohesive model are compared with the previously

obtained similar linear elastic model results as shown in Figure 7-9 and Figure 7-10

(LE-UDEC curve and cohesive-UDEC (normal) curve). The implementation of

cohesive crack does not seem to change the results significantly especially when the

watertable is smaller. Hence, to make sure that the cohesive part of the code is working

properly, a series of models was analysed by making artificially high fracture energy

values. The results shown in Figure 7-9 for cohesive-UDEC (high) indicate the fracture

energy to be 100 times higher than it should be according to equation [7-11].

When the fracture energy is extremely high, as expected, the crack length is low

compared with the previous values. The higher fracture energy increases the residual

286

limit, that is the gradient of bc line in Figure 7-6 also increases. Hence the opening of

the crack makes much more difficult resulting in lesser crack depths.

Figure 7-9 Depth of cracking variation for cohesive model with surface suction when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛 𝜙𝑏 𝑐𝑜𝑡 𝜙′, 𝑊 =4m and 𝜙 =300

Figure 7-10 Depth of cracking variation for cohesive model with depth to water table when 𝐸 =5MPa, 𝜈 =0.3, 𝜎𝑡 = −𝛼𝑇𝑆 𝑡𝑎𝑛 𝜙𝑏 𝑐𝑜𝑡 𝜙′, 𝑆0=50kPa and 𝜙 =300

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

20 40 60 80 100

Dep

th o

f Cra

ckin

g (m

)


LE-UDEC Cohesive-UDEC (Normal) Cohesive-UDEC (high)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

2 4 6 8 10

Dep

th o

f Cra

ckin

g (m

)


LE - UDEC Cohesive-UDEC (normal)


7.5.2 Effects of method of implementing cohesive law

The cohesive law was implemented in the numerical code using a large number of steps.

This final cohesive profile was achieved through three different ways as shown in

Figure 7-11.

A AA C

B

B1

B1

B B

B2

B2

B3B3

C3 C3C2 C2C1C1 CCDisplacement

Tens

ile S

tress

(a) (c)(b)

Figure 7-11 Different methods of implementing cohesive law (a) Softening Law change-1 (b) Softening Law change-2 (c) Softening Law change-3

Figure 7-11 illustrates the progressions of cohesive laws to the final stage shown as

ABC curve. B1, B2, B3, C1, C2 and C3 are intermediate positions during the stepping

process in the numerical model. In the final curve given as ABC, the point B represents

the yield limit on the x axis and the tensile strength on the y axis. The point C is the

residual limit on the displacement axis. In Figure 7-11(a), both the yield limit and

residual limit were changed in the numerical process to reach the final ABC while

maintaining a constant stiffness (i.e. the gradient of the AB line). This approach was

named Softening Law change (SL change)-1. In Figure 7-11(b) both the yield limit and

the residual limit were kept constant. Only the tensile strength and the stiffness were

changed. This approach was referred to as Softening Law change (SL change)-2.

Finally, as shown in Figure 7-11(c), the stiffness, tensile strength and the yield limit

288

were maintained unchanged during stepping and only the residual limit was increased

until the final curve was reached. This approach was called Softening Law change (SL

change)-3 for reference. The depths of cracking obtained through these approaches were

shown in Figure 7-12.

Figure 7-12 Depth of cracking for different implementations of softening law with the change of surface suction

In Figure 7-12, SL change 1, 2 and 3 refers to the above mentioned three softening law

implementing approaches and no SL refers to the crack model without any user defined

cohesive properties at the crack opening. As Figure 7-12 illustrates all models show

similar results for crack depths through each approach despite the without cohesion

approach. However, with further drying, the curves tend to separate from each other

0

1

2

3

4

5

6

7

8

0.0 5.0 10.0 15.0 20.0

Dep

th o

f cra

ckin

g (m

)


H-no SL H-SL change-1 H-SL change-2 H-SL change-3


although that difference is not more than 100mm. From this result, it was decided that

the method of softening law implementation is not significant and softening law change-

1 approach was used in all the later modelling, since it can be considered as the most

realistic approach among the three approaches as the tensile strength increases with the

suction.

7.5.3 Effect of friction angle with respect to suction

In previous sections of this thesis, the friction angle with respect to suction, 𝜙𝑏, was

considered as a constant based on Morris et al. (1992). However, it was based on a

broad interpretation of data published in literature and may not be relevant for the entire

range of suction.

Bishop (1959) proposed an equation for shearing resistance using the effective stress

based Mohr–Coulomb failure criterion given in equation [7-17]. A 𝜒 parameter was

used to modify the suction value that contributes to the effective stress.

𝜏 = 𝑐΄ + [(𝜎 − 𝑢𝑎) + 𝜒(𝑢𝑎 − 𝑢𝑤)𝑡𝑎𝑛𝜙΄] [7-17]

Fredlund et al. (1978) used an approach that proposed another relation for shear strength

with normal stress and matric suction. This relation is given as:

𝜏 = 𝑐΄ + (𝜎 − 𝑢𝑎) tan 𝜙΄ + (𝑢𝑎 − 𝑢𝑤)𝑡𝑎𝑛𝜙𝑏 [7-18]

By rearranging equation [7-18],

𝜏 = 𝑐΄ + �(𝜎 − 𝑢𝑎) + (𝑢𝑎 − 𝑢𝑤) �𝑡𝑎𝑛𝜙𝑏

tan 𝜙΄�� tan 𝜙΄

[7-19]

290

By comparing equation [7-17] with equation [7-19],

𝜒 =𝑡𝑎𝑛𝜙𝑏

tan 𝜙΄

[7-20]

Khalili and Khabbaz (1998) proposed a relationship for χ parameter to determine the

shear strength in unsaturated soils as given below.

𝜒 = �(𝑢𝑎 − 𝑢𝑤)

(𝑢𝑎 − 𝑢𝑤)𝑏�−0.55

[7-21]

where, (𝑢𝑎 − 𝑢𝑤)𝑏 is the suction at the air entry. On the basis of experimental data, they

indicated that 𝜒 is not a constant (so that the internal angle of friction with respect to

matric suction) and decreases with increasing matric suction.

Figure 7-13 Effect of friction angle with respect to suction on desiccation cracks

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10

Dep

th o

f cra

ckin

g (m

)


A-no SL-Constant ϕ^b A-no SL-Varing ϕ^b A-with SL-Constant ϕ^b A-with SL-Varing ϕ^b


By equating equation [7-20] and [7-21],

𝑡𝑎𝑛𝜙𝑏 = tan 𝜙΄ �(𝑢𝑎 − 𝑢𝑤)

(𝑢𝑎 − 𝑢𝑤)𝑏�−0.55

[7-22]

According to equation [7-22], the internal friction angle with respect to matric suction

was changed in the FISH code to obtain the depths of cracking and the results are shown

in Figure 7-13.

Figure 7-13 has shown the development of depth of desiccation cracking with different

modelling conditions. In the legend the first capital A refers to the soil which was

Altona clay in this scenario, SL is Softening Law, showing the application of cohesive

properties to the crack or not and 𝜙𝑏 is the friction angle with respect to matric suction.

The above figure shows that in both cases, i.e. with and without cohesive crack

properties, changing 𝜙𝑏 does not show any effect at wet stages (lower suctions).

However when the desiccation progresses with higher suction stresses, changing 𝜙𝑏

shows a slight increase in crack depth. With higher suction values, the 𝜒 parameter

decreases and controls the suction effect on the shear strength and the tensile strength

that leads to increase the depth of cracking. However, since the difference between the

two curves is not significant and at low suctions no difference is detected, the effect of

𝜙𝑏 was considered as not significant.

7.5.4 Depth of desiccation cracks with the cohesive properties

The crack depths while desiccating a soil layer were obtained for Regina, Horsham and

Altona clay soils. For each soil the effectiveness of using cohesive properties at the

292

crack was analysed using two different softening laws and then compared with the crack

depths without any cohesive properties. Figure 7-14, Figure 7-15 and Figure 7-16 show

the results for each soil.

Figure 7-14 Depth of cracking for Regina clay using different softening laws with changing surface suction

0

1

2

3

4

5

6

7

8

0.0 2.0 4.0 6.0 8.0 10.0

Dep

th o

f cra

ckin

g (m

)


R-no SL R-SL-1 R-SL-2


Figure 7-15 Depth of cracking for Horsham clay using different softening laws with changing surface suction

In all three figures, the highest crack depths were observed when no cohesive properties

were assigned to the crack or, in other words, when there is no actual fracture energy

specified to the crack propagation. On the other hand, with the Dugdale softening law

the lowest crack depths were recorded. These results are acceptable since the fracture

energy tends to reduce the crack depth since some of the strain energy is consumed by

opening the fracture. It is obvious that as shown in Figure 7-16 with the increase of

fracture energy, the depth of cracking reduces. With the use of cohesive properties,

there is a fictitious stress between the cracked surfaces that holds the two surfaces

0

1

2

3

4

5

6

7

8

0.0 2.0 4.0 6.0 8.0 10.0

Dep

th o

f cra

ckin

g (m

)


H-no SL H-SL-1 H-SL-2

294

together, which is also referred to as residual tensile strength. Therefore, this residual

strength limits the development of deep cracks.

Figure 7-16 Depth of cracking for Altona clay using different softening laws with changing surface suction

(In the legends of Figure 7-14, Figure 7-15 and Figure 7-16, first capital letter refers to the name of the soil such as A for Altona clay, H for Horsham clay and R for Regina clay. Then the SL-1 and 2 are the two softening laws where SL-1 refers to the Dugdale softening law and SL-2 refers to the linear softening law. Finally no SL is the control model where no softening law was used, in other words no cohesive properties were given by the user to the numerical program.)

However, it is noticeable for the above figures that the cohesive properties at the crack

show a considerable effect only at low suctions. When the soil is subjected to higher

0

1

2

3

4

5

6

7

8

0.0 2.0 4.0 6.0 8.0 10.0

Dep

th o

f cra

ckin

g (m

)


A-no SL A-SL-1 A-SL-2


suctions the effect of cohesive properties appears to become negligible. It may be that

with high suctions the soil will develop high strain energy to propagate the crack and

the fracture energy may not have a big influence on the final crack depth.

Displacement

Ten

sile

str

ess

Without cohesive properties Linear softening law(SL-2)

Dugdale softening law (SL-1)

Fracture energy Fracture energy

Fracture energy

Yield limit

Yield limit

Yield limit

Residual limit

Figure 7-17 Fracture energies with different softening laws

With the fracture energies used, the predicted crack depths do not show considerable

reduction compared to the crack depths without cohesive properties especially at higher

suctions. Hence to confirm the effect of fracture energy in crack development, several

series of models were run by increasing fracture energy artificially.

Generally fracture energy in soils up to about 173N/m (0.173kPa.m) have been reported

according to the literature (Costa, 2009). However for illustration of its effect, from

small to very high fracture energies were used here. Figure 7-18 and Figure 7-19

illustrate the effect of fracture energy in terms of the residual limit and fracture energy

itself. The fracture energy was increased from zero up to 3.5kPa.m in predicting the

crack depths while keeping the other conditions the same.

296

From Figure 7-18 and Figure 7-19, it can be clearly seen that the development of crack

depth is highly dependent on the increase of fracture energy, as expected. The softening

law used in this series of models was the linear softening law as shown in Figure 7-17.

When the fracture is increased, the residual limit increases but not the tensile strength or

the yield limit. Hence, with the increase of fracture energy, the residual strength

between the cracking surfaces increases and the tensile strength does not decrease to

zero until the crack opens wider. For these reasons, the crack depth drops considerably

with the increase of fracture energy. However, although this demonstrates the influence

of fracture energy and residual limit, it should be noted that these results are artificial

since such large residual limits are not practical for clay soils.

Figure 7-18 Effect of fracture energy on desiccation cracking in terms of residual limit

0

1

2

3

4

5

6

7

8

0 50 100 150 200 250

Dep

th o

f cra

ckin

g (m

)

Residual limit (mm)

20MPa 14MPa 10MPa 8MPa


Figure 7-19 Effect of fracture energy on desiccation cracking

(20MPa, 14MPa, 10MPa and 8MPa in above figures, indicate the surface suction value of the soil block.)

7.6 Predicting Crack Depths in Soft soils

Cohesive properties at the crack mostly have an influence when low suctions are

incorporated. Hence, the soft soils response to the cohesive properties would be an

interesting observation.

As discussed in the section 7.4.2, the model was developed to reproduce the test

conducted by Konrad and Ayad (1997a) using slurry clay. However, other than the

0

1

2

3

4

5

6

7

8

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Dep

th o

f cra

ckin

g (m

)

Fracture Energy (kPa.m)

20MPa 14MPa 10MPa 8MPa

298

multiple cracks as observed during the field test, single crack was also used in this

study.

7.6.1 Results for crack depth prediction using single crack

Figure 3-25 shows the depth of the crack predicted with the same moisture content

profiles with time. But a single joint was allowed to open. The progression of crack

depth was modelled according to the increment of suction as given in Figure 7-7

profiles with time. The crack depths were modelled with and without cohesive

properties. As with the previously experience, the crack depths are similar in both cases

especially when the suctions reach higher values.

Figure 7-20 Crack depth with time

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20 40 60 80 100

Dep

th o

f cra

ckin

g (m

)

Time (hr)

With cohesion

Without cohesion


7.6.2 Results for crack depth prediction using multiple cracks

The block of soil was modelled allowing for multiple cracks to occur. This was

undertaken by providing joints at equal spacing of 50 mm. The distribution of cracks

and the displacements of zones within the block after 18 hours are shown in Figure

7-21. The figure shows only the top left section of the soil block analysed for clarity. As

the cracks progress, the displacement vectors become more lateral from the initial

vertical downward direction with the increasing suction.

Figure 7-21 Block with displacement and joints

300

Figure 7-22 Block with opened cracks and displacement vectors

The condition of soil block after 65 hours is shown in Figure 7-22. It is clear that the

opening of the joints provided is selective, where several cracks at some spacing open


first and these cracks extend deeper before other joints start to open in between. Several

minor cracks also initiated at the surface, but they did not extend further. Some of these

cracks closed as the dominant cracks propagate deeper. This behaviour may be partially

attributed to the current modelling approach, where cracks can open anywhere when the

tensile stress exceeds the tensile strength. However, a similar response is also seen in

field situations, where bifurcation of crack patterns occurs with minor cracks at small

spacing which stop growing and even close giving way to the progression of major

cracks to greater depths.

Figure 7-23 Progression of multiple cracks with time

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Dep

th o

f cra

ckin

g (m

)

Time (hr)

maximum crack depth

most obseverd crack depth

302

The depth of the largest crack was recorded with time as the information on primary

crack growth. The depths of the minor cracks were monitored and the average depth

was recorded with time. These results are shown in Figure 7-23.

The primary cracks develop at considerable speed for almost three days and then limit

the progression. The secondary cracks show fast development within the first 24 hours

and subsequently slow down their progression.

7.7 Conclusions

Compacted clay and soft soils from different geographical regions were modelled

incorporating the fracture energy as cohesive properties. The results of these analyses

were compared with the results from numerical models that did not incorporate fracture

energy (i.e., without using cohesive models). In order to model compacted clays with

cohesive cracks, three types of softening laws were used and results were obtained for a

wide range of suctions. Soft soils were analysed with actual ground and suction

conditions observed in the field. A single crack and multiple cracks were used in the

analysis for soft soils.

The softening cohesive laws used included linearly decreasing and constant as described

by Dugdale model. However, experimental evidence indicates that linear softening law

is more appropriate for soil modelling.

When the suctions in the soil become very large, the influence of the soil fracture

energy diminishes and therefore the use of the cohesive model becomes less important.

Hence, the results obtained are similar to models that do not use fracture energy.


Therefore, in modelling crack depths in extremely dry conditions such as in deserts or

during droughts, the use of cohesive properties does not have any considerable effect on

the crack depth predictions. Hence, it is fair to say that the normal fracture analysis with

a tensile criterion would be sufficient to model extremely dry conditions.

However, for low to moderate dry conditions observed generally in temperate or semi-

arid areas, cohesive properties could have a significant influence on the crack depth

predictions. Without cohesive properties, the numerical analysis over-predicts the crack

depth and when the cohesive properties are introduced, the desiccation crack depths are

reduced. It can be noted, that the crack depths predicted using the cohesive cracks are

within the range typically observed in the field, although the field measurements of

depth of cracking are not accurate on most occasions, due to the extreme difficulty in

measuring the actual depths.

Analyses also indicated that the use of single isolated crack in crack depth prediction

could lead to deeper cracks in comparison to crack depths obtained from multiple crack

analysis. In some field conditions, however, it is possible to generate isolated cracks. In

addition, the advantages of using single crack analysis are the ease of modelling and

reduced model run times. It can also be argued that the use of isolated crack depth could

be conservative in design as it provides the maximum crack depth possible. Multiple

cracking can be used as a solution to replicate actual field conditions.

Chapter 8

CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH

The conclusions drawn from the present study are summarised and presented in this

chapter. Following the conclusions, improvements for the research and new pathways

are suggested based on the knowledge gathered from this study.

8.1 Conclusions

1. Desiccation is one of the major problems a clay liner can face due to the fractures.

Due to its complexity, the precise mechanism of cracking has remained

undiscovered for a long time.

2. Desiccation crack modelling related to the moisture content variation during wet-

dry cycles seems to be more accurate and relatively new in this field. No research

has been conducted to predict the desiccation crack depths using numerical model

considering the moisture content variations.

304

Conclusions and Recommendations for Further Research 305

3. Most of the various computer programs used to predict the crack depths in the

literature are not capable enough to use for fracture modelling in geo-materials.

However the computer programme UDEC was found to be suitable software to

reproduce the actual behaviour of soil in the field when fracturing.

4. In UDEC program, the mesh size does not show much effect on the failure stress.

The number of steps used to change the initial conditions does not affect the final

results. A similar result was obtained when changing the damping value.

However, it is recommended to use a higher number of steps while changing the

conditions of the model and more careful consideration for damping should be

given for dynamic model problems.

5. All other parameters such as, crack length, size of the block and scale of the

model geometry have an influence in failure stress, indicating that clay type soils

with higher fracture energies behaves more plastically than following LEFM

theories unless large problem geometries are used. The soils with lower fracture

energies (i.e. higher stiffness) clearly follow the LEFM theories. Therefore,

applying cohesive properties to the fracture opening may increase the accuracy of

the model in desiccation fracture modelling.

6. The numerical program generally predicts higher crack depths than the existing

analytical predictions due to the relaxation of stresses that are concentrated at the

crack tip just after opening a crack.

7. The moisture content can successfully be used to overcome the over-prediction of

crack depths in the numerical program. The hydric co-efficient is required to

apply moisture content changes to the numerical program.

306

8. Expansive soils come to an equilibrate condition where the soil behaves elastically

while wetting and drying. The path followed by the soil to reach the equilibrium

curve depends on initial moisture content, initial dry density, net vertical stress,

composition of the soil and level of wetting and drying processes.

9. The value of hydric coefficient is dependent on the initial condition of the soil

before equilibrium and the applied stress level. After reaching the equilibrium

stage, in each full wetting or drying process, the α* remains at a value of one for

more than 80% except during the beginning and the end of the process.

10. The cracks seem to grow deeper very quickly once they open up at the surface

within a small amount of time. The closure is also fast. The climate condition of

the area has a great influence on the depth of cracking. The lower Poisson’s ratios

increase the crack depth. However, the deviation of equilibrium suction from the

surface suction towards the wet side decreases the crack depth.

11. To apply cohesive properties to the fractures in soil, linear softening law is more

appropriate according to the experimental and numerical evidences. The fractures

in soils that are subjected to extremely dry condition are not influenced by

cohesive properties. Hence, normal fracture analysis without cohesive properties

would be sufficient to model extremely dry conditions. However, for typical dry

conditions observed generally in arid or semi-arid areas, cohesive properties can

have a significant influence on the crack depth prediction.

12. Both single and multiple conditions can be used to model crack depths effectively,

however, using multiple cracks is recommended only if the spacing of the cracks

is known. Modelling using single crack can be recommended as a safer method to

Conclusions and Recommendations for Further Research 307

be used in design calculations, since this method leads to the maximum crack

depths possible.

13. The predicted crack depths may vary from the actual observed values. Becuase, in

actual conditions continuous desiccation or continuous wetting is highly unlikely

and other disturbing activities for cracking can also happen, such as animals

movements, plastic flow within the soil and erosion by rain or wind. The lack of

accurate field observation data for depth of desiccation cracks to compare with the

numerical predictions is a serious limitation.

14. Finally, all clay soils produce similar depths of cracking under the same

conditions. Hence it is reasonable to predict the crack depths for any clay soil

from the results presented here, provided that particle size and other properties of

the soil are similar to the tested soils.

8.2 Future Research Recommendations

Based on the knowledge gathered from this study, the following further steps research

are recommended.

The most important recommendation would be developing a comprehensive data base

of field desiccation tests. The observations of crack initiation, growth measuring the

depth, width and spacing of crack under different climate conditions and soil types

should be recorded.

Considering the difficulty of the above research, laboratory experiments can be planned

using transparent boxes in order to measure the correct crack depth. However, the boxes

308

must be large enough to observe the natural crack depths. Further, this laboratory test

should be conducted with different soil types replicating different climate conditions.

The effect of cyclic climate variation on depth of cracking need to be monitored either

using a laboratory test or using a field test. Development of the crack and the healing

behaviour should be modelled physically for several wet and dry cycles.

The numerical model proposed in this study should be developed further to model the

spacing of the surficial cracks and the precise sequence of the crack development.

Further, the behaviour of the cracks under several wet dry cycles should be modelled.

Three dimensional numerical modelling also should be developed for a better

representation of the actual field situation.

A 3-D numerical model should be developed by coupling the atmospheric and climatic

conditions and the fracture properties giving more realistic results.

References 309

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