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2018-09-12

Determination of Mode I Fracture Toughness, Tensile

Strength and Adhesion of Compacted Clays

Cao, Keda

Cao, K. (2018). Determination of Mode I Fracture Toughness, Tensile Strength and Adhesion of

Compacted Clays (Unpublished master's thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/32930

http://hdl.handle.net/1880/107754

master thesis

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UNIVERSITY OF CALGARY

Determination of Mode I Fracture Toughness, Tensile Strength and Adhesion of Compacted

Clays

by

Keda Cao

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN CIVIL ENGINEERING

CALGARY, ALBERTA

SEPTEMBER, 2018

© Keda Cao 2018

ii

Abstract

Tensile strength and fracture toughness are two parameters used to define the ability of resisting

fracture under tensile stresses. Adhesion of clays to metals is the strength of soil-metal interface.

These properties of cohesive soils are fundamental and essential for geotechnical problems. Two

types of compacted clays were used in this research — Calgary till and Regina clay.

The fracture toughness KIc and tensile strength σt of two clays were measured by straight notched

disk bending (SNDB) method and uniaxial direct tensile test, respectively. Results showed that the

variations of σt with respect to moisture content and dry density were very similar with that of KIc.

A strong positive correlation between these two properties was found.

The tensile strength of compacted clays was curve fitted based on a double-porosity concept which

considers the water among micropores and macropores in clay structure and their contribution to

the strength of soils.

The adhesion of compacted clays to stainless steel was measured using a modified direct shear box

test. To obtain the adhesion factors, the adhesion was correlated with unconfined compressive

strength of compacted clays. The effects of moisture content and pre-consolidation pressure on

adhesion were investigated.

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Acknowledgements

First of all, I would offer my deepest thanks to my supervisor, Dr. Ron Chik-Kwong Wong, for

guiding me through a fantastic world of science, exploring interesting problems and understanding

the principles behind. This thesis is the picture of our journey across geotechnics.

Drs. Richard Wan and Mingzhe Dong, are thanked for providing constructive comments for my

work as my thesis committee.

Meanwhile, the technical staff are highly appreciated for their important support. Special thanks

to Mirsad Berbic for helping me deal with experimental design and setup.

In addition, great fellow graduate students and friends whom I shared happy time with are sincerely

acknowledged: Qing Jia, Qiang Chen, Yangdong Zhang, Junwei Guo, Jiechun Wu, Xv Gong,

Longyang Shi, Kevin Yuen, and Chee Wong.

Also, I want to show gratitude to my big family for their unconditional love to me, in particular to

my grandparents who provided additional financial support for my Master’s studies.

Last but not least, I’m grateful to my girlfriend Jieyu Zhang for helping me through all the difficult

times.

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Table of Contents

ABSTRACT .................................................................................................................................. II

ACKNOWLEDGEMENTS ....................................................................................................... III

TABLE OF CONTENTS ........................................................................................................... IV

LIST OF TABLES .................................................................................................................... VII

LIST OF FIGURES AND ILLUSTRATIONS ..................................................................... VIII

LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURE .............................. XIII

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

1.1. BACKGROUND ....................................................................................................................... 1 1.2. RESEARCH SCOPES ................................................................................................................ 2

1.2.1. Mode I Fracture Toughness .......................................................................................... 2 1.2.2. Tensile Strength ............................................................................................................ 3 1.2.3. Role of Matric Suction .................................................................................................. 3 1.2.4. Adhesion to Metal ......................................................................................................... 4

1.3. OUTLINE OF THESIS............................................................................................................... 4

BASIC PROPERTIES AND SWCC OF THE SOILS ....................................... 7

2.1. BASIC MATERIAL PROPERTIES .............................................................................................. 7 2.1.1. Atterberg Limits and Soil Classifications ..................................................................... 7 2.1.2. Compaction Curves ....................................................................................................... 7

2.2. SOIL-WATER CHARACTERISTIC CURVES (SWCC) ................................................................ 8 2.2.1. Introduction ................................................................................................................... 8 2.2.2. Methods....................................................................................................................... 11 2.2.3. Modified Filter Paper Method .................................................................................... 12 2.2.4. Results ......................................................................................................................... 14

MODE I FRACTURE TOUGHNESS ............................................................... 21

3.1. INTRODUCTION.................................................................................................................... 21 3.2. REVIEW OF LEFM THEORY ................................................................................................ 21

3.2.1. Griffith Criterion (Energy Release Rate) .................................................................... 22 3.2.2. Stress Intensity Factor ................................................................................................. 25 3.2.3. Applicability of LEFM ............................................................................................... 28 3.2.4. Summary ..................................................................................................................... 29

3.3. REVIEW OF TESTING METHODS FOR KIC .............................................................................. 29 3.3.1. Methods Already Used for Soils ................................................................................. 30 3.3.2. ISRM Suggested Methods for Rocks .......................................................................... 35

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3.3.3. Straight Notched Disk Bending (SNDB) .................................................................... 37 3.3.4. Summary ..................................................................................................................... 38

3.4. REVIEW OF PREVIOUS KIC TESTS ON COMPACTED CLAYS ................................................... 39 3.5. SNDB TESTS FOR FRACTURE TOUGHNESS KIC .................................................................... 41

3.5.1. Test Setup.................................................................................................................... 41 3.5.2. Test Program ............................................................................................................... 42 3.5.3. Test Procedure ............................................................................................................ 42 3.5.4. Test Results ................................................................................................................. 45 3.5.5. Summary ..................................................................................................................... 49

UNIAXIAL TENSILE STRENGTH ................................................................. 70

4.1. INTRODUCTION.................................................................................................................... 70 4.2. REVIEW OF PREVIOUS TESTING METHODS .......................................................................... 70

4.2.1. Direct Methods............................................................................................................ 70 4.2.2. Indirect Methods ......................................................................................................... 71

4.3. UNIAXIAL TENSILE TEST ..................................................................................................... 72 4.3.1. Test Apparatus ............................................................................................................ 73 4.3.2. Test Procedure ............................................................................................................ 73 4.3.3. Test Results ................................................................................................................. 75

4.4. SUMMARY ........................................................................................................................... 78

CHARACTERISTICS OF FRACTURE TOUGHNESS AND TENSILE STRENGTH ................................................................................................................................ 87

5.1. INTRODUCTION.................................................................................................................... 87 5.2. CORRELATION BETWEEN KIC AND ΣT ................................................................................... 87

5.2.1. Rocks........................................................................................................................... 87 5.2.2. Soils............................................................................................................................. 89 5.2.3. Calgary Till and Regina Clay in This Study ............................................................... 89 5.2.4. Descriptions for the Size of Fracture Process Zone (FPZ) ......................................... 92

5.3. CURVE-FITTING FOR VARIATION TRENDS OF TENSILE STRENGTH ...................................... 93 5.3.1. Theories for Strength of Unsaturated Soils ................................................................. 95 5.3.2. Curve-fitting for Tensile Strength ............................................................................... 98

5.4. SUMMARY ......................................................................................................................... 102

ADHESIVE PROPERTY OF COMPACTED CLAYS ................................. 113

6.1. INTRODUCTION.................................................................................................................. 113 6.2. UNCONFINED COMPRESSION TEST .................................................................................... 118 6.3. MODIFIED DIRECT SHEAR TEST ........................................................................................ 119

6.3.1. Test Apparatus, Program, and Procedure ................................................................. 119 6.3.2. Test results ................................................................................................................ 121

6.4. SUMMARY ......................................................................................................................... 124

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CONCLUSION .................................................................................................. 136

7.1. OVERVIEW ........................................................................................................................ 136 7.2. CONCLUSIONS ................................................................................................................... 137 7.3. RECOMMENDATIONS ......................................................................................................... 138

APPENDIX A : LOAD-DISPLACEMENT CURVES OF KIC TEST ................................. 140

APPENDIX B : LOAD-DISPLACEMENT CURVES OF UNIAXIAL TENSILE TESTS 143

APPENDIX C : THE SETUP OF SPRING GAUGE AND ITS CALIBRATION ............. 146

APPENDIX D : KIC TESTS ON CLAY SHALE ................................................................... 149

APPENDIX E : UNSUCCESSFUL KIIC TESTS ON COMPACTED CLAYS ................... 153

REFERENCE ............................................................................................................................ 155

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List of Tables

Table 2.1 Physical properties of the soils ................................................................................16

Table 2.2 Suction measurement for Regina clay .....................................................................16

Table 2.3 Suction measurement for Calgary till ......................................................................17

Table 2.4 Curve-fitting parameters for SWCC of Calgary till and Regina clay ......................17

Table 3.1 Values of m and n for stress intensity factor estimation (after Tutluoglu & Keles 2011) ................................................................................................................................50

Table 3.2 Testing scheme of KIc for two clays ........................................................................50

Table 3.3 Details of mode I fracture toughness test on Calgary till ........................................51

Table 3.4 Details of mode I fracture toughness test on Regina clay........................................52

Table 4.1 Detailed results of uniaxial direct tensile test for Calgary till .................................79

Table 4.2 Detailed results of uniaxial direct tensile test for Regina clay ................................80

Table 5.1 Expressions for shear strength of unsaturated soils .................................................96

Table 6.1 Unconfined compression and direct soil-steel shear tests results and adhesion factors under different moisture contents for Calgary till and Regina clay at pre-consolidation pressure of 30 kPa ...................................................................................125

Table 6.2 Direct soil-steel shear test results of Regina clay under different pre-consolidation pressures .........................................................................................................................126

Table D.1 Matric suction of four clay shale samples .............................................................150

Table D.2 Summary of geometries and test tesults................................................................150

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List of Figures and Illustrations

Figure 1.1 Three modes of cracking (after Broek, 1982) ..........................................................6

Figure 2.1 Compaction curve of Calgary till ...........................................................................18

Figure 2.2 Compaction curve of Regina clay ..........................................................................18

Figure 2.3 Illustration for modified suction measurement method..........................................19

Figure 2.4 Vacuumized specimen for matric suction measurement ........................................19

Figure 2.5 SWCC of Calgary till .............................................................................................20

Figure 2.6 SWCC of Regina clay ............................................................................................20

Figure 3.1 Stress state near crack tip of 2a crack length in an infinite plate under tension.....53

Figure 3.2 Energy change with respect to crack length ...........................................................53

Figure 3.3 (a) Theoretical infinite stress, (b) theoretical plastic zone and the corresponding stress, and (c) true plastic zone and stress distribution (corrected by Irwin) in the plane θ = 0 ....................................................................................................................................54

Figure 3.4 Configuration and geometries of SENB .................................................................55

Figure 3.5 Horizontal testing system for SENB (after Wang et al., 2007) ..............................55

Figure 3.6 Typical test setup with counter-balance loads (after Hallett & Newson, 2001) .....56

Figure 3.7 (a) Configuration and (b) experimental setup of CT Test (after Lakshmikantha, 2009) ................................................................................................................................56

Figure 3.8 Flattened Brazilian Disk test ..................................................................................57

Figure 3.9 Typical load-displacement curve for FBD Test .....................................................57

Figure 3.10 Diagonal and secondary cracks in FBD specimen of oven-dried soil (after Agaiby, 2013) ..................................................................................................................58

Figure 3.11 Configuration of Ring Test ...................................................................................58

Figure 3.12 Specimens with different geometries obtained from core ....................................59

Figure 3.13 CB test configuration and the cross section of the specimen at the notch ...........59

Figure 3.14 SR test configuration and the cross section of the specimen at the notch ............60

Figure 3.15 CCNBD test configuration and the cross section of the specimen at the notch ...60

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Figure 3.16 SCB specimen geometry ......................................................................................61

Figure 3.17 Small crack generated in SCB soil specimen due to sawing ................................61

Figure 3.18 SNDB geometry and fixture .................................................................................62

Figure 3.19 Layout of the instruments .....................................................................................62

Figure 3.20 (a) Sketch and (b) experimental setup for SNDB .................................................63

Figure 3.21 Mold for compaction (collar of standard compaction mold)................................64

Figure 3.22 Number of blows for points around the compaction curve of Calgary till ...........64

Figure 3.23 Sawing sample with a hand saw guided by a miter box .......................................65

Figure 3.24 Film-wrapped specimen .......................................................................................65

Figure 3.25 Sealed specimens in zip-locked bag .....................................................................65

Figure 3.26 Typical load-displacement curves of SNDB tests for (a) Calgary till, ρd = 1.85 g/cm3, (b) Calgary till, ρd = 1.75 g/cm3, and (c) Regina clay ..........................................66

Figure 3.27 Variation of KIc with respect to moisture content and degree of saturation for Calgary till (a) & (b) and Regina clay (c) & (d) ..............................................................67

Figure 3.28 Variation of moduli with respect to moisture content and degree of saturation for Calgary till (a) & (b) and Regina clay (c) & (d) ..............................................................68

Figure 3.29 Ratios of KIc/Kaverage for Calgary till with different a/t .........................................69

Figure 4.1 Direct methods of tensile tests for compacted clays (after Zhang et al. 2015) ......81

Figure 4.2 Indirect methods of tensile tests for compacted clays: a) four-point beam bending, b) Brazilian disk test, and c) unconfined penetration test ................................................82

Figure 4.3 Configuration of direct tensile test .........................................................................83

Figure 4.4 Dog-bone specimen (a) compacted in the mold, and (b) sealed in bags ................83

Figure 4.5 Typical load-displacement curves of uniaxial tensile tests of (a) and (b) Calgary till, and (c) Regina clay for varying dry densities and moisture contents ........................84

Figure 4.6 Variations of tensile strength with respect to moisture content and the degree of saturation for (a) & (b) Calgary till and (c) & (d) Regina clay ........................................85

Figure 4.7 Tensile strength versus matric suction for (a) Calgary till and (b) Regina clay .....86

Figure 5.1 KIc versus σt relationship of dolostone, limestone and sandstone.........................103

x

Figure 5.2 KIc versus σt relationship of rocks (after Zhang, 2002) ........................................103

Figure 5.3 KIc versus σt relationship of marble, gabbro and granite ......................................104

Figure 5.4 KIc and σt correlation for rocks (after Haberfield and Johnston, 1989) ................104

Figure 5.5 Linear correlations between KIc and σt for Calgary till and Regina clay ..............105

Figure 5.6 Linear regression of KIc and σt for compacted clays (including data from Wang et al., 2007 and Lakshmikantha, 2009) ..............................................................................105

Figure 5.7 Power-law correlation between KIc and σt for different soils prepared from different methods ...........................................................................................................106

Figure 5.8 SEM images of compacted clays for different soil packings. Pores are black and particles are white for lower graphs. (after Romero, 1999) ...........................................107

Figure 5.9 Adsorbed water and capillary free water in micropores and macropores ............108

Figure 5.10 Effective degree of saturation expressed by (a) piecewise function, (b) continuous function, and (c) piecewise and continuous functions for comparison .......109

Figure 5.11 Curve-fitted shear cohesions measured by Wong et al. (2017) using piecewise function ..........................................................................................................................110

Figure 5.12 Tensile-shear failure envelope for compacted soil (after Wong et al., 2017) ....110

Figure 5.13 Curve-fitted σt of Regina clay using piecewise function of Sre (Equation 5.19)

with Srm = 0.56 ...............................................................................................................111

Figure 5.14 Curve-fitted σt of Regina clay using continuous function of Sre (Equation 5.20)

with ν = 0.5 ....................................................................................................................111

Figure 5.15 Curve-fitted tensile strength of Calgary till using piecewise function of Equation 5.19 .................................................................................................................................112

Figure 6.1 Axial stress along the pipeline resulted from longitudinal displacement .............127

Figure 6.2 (a) Schematic and (b) fixture of modified direct shear box test for determining adhesion of compacted clay ...........................................................................................127

Figure 6.3 Setup of unconfined compression test ..................................................................128

Figure 6.4 Specimen wrapped with plastic sheet to prevent moisture loss during testing ....128

Figure 6.5 Axial load-displacement curves of unconfined compression test at varying moisture content for (a) Regina clay and (b) Calgary till ..............................................129

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Figure 6.6 Unconfined compressive strength versus moisture content relationship for (a) Regina clay and (b) Calgary till .....................................................................................129

Figure 6.7 Shear stress-displacement curves of modified direct shear (soil-steel) tests with pre-consolidation pressure of 30 kPa and total stress of σn ≈ 0 kPa for compacted Regina clay with moisture contents of (a) 19.5%, (b) 22%, (c) 25%, (d) 28.5%, and (e) 31% ................................................................................................................................130

Figure 6.8 Illustration for the movement of the modified shear box .....................................131

Figure 6.9 Shear stress-displacement curves of modified direct shear (soil-steel) tests with pre-consolidation pressure of 30 kPa and total stress of σn ≈ 0 kPa for compacted Calgary till with moisture contents of (a) 11%, (b) 13%, (c) 15%, (d) 17%, and (e) 19.5% (ρd = 17.5 g/cm3) .................................................................................................132

Figure 6.10 Adhesion versus moisture content relationship for compacted Calgary till and Regina clay .....................................................................................................................133

Figure 6.11 Adhesion factor versus half unconfined compressive strength relationship for compacted Calgary till and Regina clay ........................................................................133

Figure 6.12 Adhesion factors for pipelines (after ASCE 1984; Rizkalla et al., 1996; Paulin et al., 1998) ........................................................................................................................134

Figure 6.13 Load-displacement curves of tests for Regina clay under different pre-consolidation pressures ..................................................................................................135

Figure 6.14 Adhesions versus different pre-consolidation pressures for compacted Regina clay at optimum moisture content ..................................................................................135

Figure A.1 Load-displacement curves of Calgary till with ρd = 1.75g/cm3 in KIc tests .........140

Figure A.2 Load-displacement curves of Calgary till with ρd = 1.85g/cm3 in KIc tests .........141

Figure A.3 Load-displacement curves of Regina clay in KIc test ..........................................142

Figure B.1 Load-displacement curves of uniaxial tensile tests of Calgary till specimens with ρd = 1.75 g/cm3 ...............................................................................................................143

Figure B.2 Load-displacement curves of uniaxial tensile tests of Calgary till specimens with ρd = 1.85 g/cm3 ...............................................................................................................144

Figure B.3 Load-deformation curves of uniaxial tensile tests of Regina clay specimens .....145

Figure C.1 Setup of spring gauge on clamps .........................................................................147

Figure C.2 Change of secant length versus strain with different initial secant lengths .........147

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Figure C.3 Deformation versus strain ....................................................................................148

Figure C.4 Slope versus initial secant length .........................................................................148

Figure D.1 Core samples of clay shale preserved in tubes ....................................................151

Figure D.2 Core 1 from 320.6 m to 327.8 m .........................................................................151

Figure D.3 Core 2 from 144.2 m to 148.6 m .........................................................................151

Figure D.4 Unsuccessfully sawed specimen PM-1 ................................................................152

Figure D.5 Load- displacement curves of specimens ............................................................152

Figure D.6 Fractured specimens of clay shale .......................................................................152

Figure E.1 Punch Through Shear (PTS) test ..........................................................................154

Figure E.2 Wooden compaction mold for PTS specimen ......................................................154

Figure E.3 PTS setup .............................................................................................................154

Figure E.4 Unwanted crack in PTS sample ...........................................................................154

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List of Symbols, Abbreviations and Nomenclature

A Cross-section area of the specimen neck

a Crack length

a' A curve fitting parameter for SWCC

Ao Initial cross-section area of the UCS specimen

B Thickness of specimens

BD Brazilian disk

c' Effective stress cohesion

ca Adhesion of clay to metal or pipe

CB Chevron bend

CCNDB Cracked chevron notched Brazilian disc

cs Shear cohesion

CT Compact tension

ct Tensile cohesion

cUCS Corrected unconfined compressive strength

D Diameter

Df Deformation of the specimen neck

E Elastic modulus

e Void ratio

fbase Friction stress between steel base and carriage

FBD Flattened Brazilian disk

Fmax Maximum load

FPZ Fracture process zone

xiv

fshear box Friction stress between upper shear box and steel base

fsoil Friction stress between soil and steel base

G Energy release rate

GIc Critical energy release rate

Gs Specific gravity

KI Stress intensity factor under mode I opening

KIc Mode I fracture toughness

KIc(average) Average mode I fracture toughness of a group of tests

KII Stress intensity factor under mode I opening

KIIc Mode II fracture toughness

Knc The lateral stress ratio under normal consolidation

L Height of unconfined compression test specimen

LEFM Linear Elastic Fracture Mechanics

Li Initial secant length

LL Liquid limit

m A parameter in stress intensity factor function

m' A curve fitting parameter for SWCC

m'' A curve fitting parameter for the exponential change of lateral

stress ratio with OCR

M Mass of specimen

M1 Mass of wet filter paper and cold tare mass

M2 Mass of dry filter paper and hot tare mass

Mf Mass of dry filter paper

xv

n A parameter in stress intensity factor function

n' A curve fitting parameter for SWCC

n'' A curve fitting parameter for the correlation between undrained

shear stress and vertical effective stress

OCR Overconsolidation ratio

P Load

PI Plastic Index

PL Plastic Limit

Pmax Maximum load or critical load

Pmin Minimum load at the valley of load-displacement curve

PTS Punch through shear

qs Shaft resistance

R Radius of disk sample

Radhesion Adhesion resistance stress

Rc Crack resistance

Ri Inner radius of ring test sample

Ro Outer radius of ring test sample

r Radial coordinate in polar coordinates

rp* Theoretical radius of plastic zone at crack tip

rp Corrected radius of plastic zone at crack tip

s A curve fitting parameter for the correlation between undrained

shear stress and vertical effective stress

S Length of loads span

xvi

S’ Length of half loads span

SCB Semi-circular bend

SENB Single edge notched bending

SIF Stress intensity function

SL Secant Length

SNDB Single Notched Disk Bend

SR Short Rod

Sr Degree of saturation

Sre Effective degree of saturation

SrM Macroscopic degree of saturation

Srm Microscopic degree of saturation

Sresidual Residual degree of saturation

su Undrained shear strength

SWCC Soil-water characteristic curves

T Tensile force in total

t Thickness of disk specimens

Tc Cold tare mass

Th Hot tare mass

tu Axial soil resistance

Ua Stored strain energy before cracking

ua Pore air pressure

UCS Unconfined compressive strength

Ur Required energy for crack propagation

xvii

uw Pore water pressure

Vmold Volume of dog-bone compaction mold

W Width of specimens

W1 Weight of upper half of the specimen

W2 Weight of upper clamp

w Moisture content

wafter Moisture content after tests

wf Water content of filter paper

wpre Moisture content at sample preparation

w0 Optimum moisture content

Y’ Stress intensity factor function for SCB

YI Mode I stress intensity factor function

α Adhesion factor

α' A curve sitting parameter for effective degree of saturation

αc Modified adhesion factor

β Shaft resistance factor

β' Ratio of notch length to specimen radius

ΔL Deformation of unconfined compression test specimen

ε Strain

γd Dry unit weight of soil

γw Unit weight of water

γp Plastic energy

γs Specific surface energy

xviii

δ Friction angle of the soil-pile or soil-pipe interface

η Dimensionless factor for the calculation of stress intensity

factor

Θ Normalized volumetric moisture content

θ Angular coordinate in polar coordinates

θ' Load angle of FBD specimen

θr Residual volumetric moisture content

θs Saturated volumetric water content

θv Volumetric moisture content

κ Installation factor

λ A curve fitting parameter for volumetric moisture content

ν A curve fitting parameter for tensile strength

ρd Dry density

ρd, max Maximum dry density under standard proctor

σ Tensile stress in a semi-finite plate for Griffith theory

σc Critical stress that initiates crack propagation

σh’ Horizontal effective stress

σn Total stress in soil

σt Tensile strength

σv’ Vertical effective stress

σx Stress in x direction

σy Stress in y direction

σys Yield strength

xix

τ Shear stress

τmax Maximum shear stress

τd Shear strength of unsaturated soil

τresidual Residual shear stress

υ Poisson's ratio

Φ Stress intensity function

φ' The angle of shearing resistance of soil

φb Friction angle as a function of saturation

χ A parameter that defines the contribution of matric suction to

the strength of soil

Ψ Matric suction

Ψa Air entry value of suction

Ψaverage The average matric suction of the upper and lower surfaces

Ψr Suction corresponding to the residual moisture content

1

Introduction

1.1. Background

The cohesive soils naturally exhibit cohesion within the materials themselves and adhesion at the

interfaces with other materials. Loss of cohesion usually leads to formation of cracks in the body

of soils, while disappearance of adhesion is possibly associated with substantial soil-structure

interactions. Understanding the cohesive strength of soils and their adhesive characters with other

materials is helpful to learning the mechanical behaviours of soils in various geotechnical issues.

On one hand, attention has been increasingly paid to cohesive strength of soils because cracks in

geotechnical structures bring about negative impacts. For instance, cracks appearing prior to slope

failure could potentially deteriorate slope instability and lower the factor of safety (e.g. Li & Yang,

2016); desiccation-induced cracks in clay liners were likely to change the hydraulic conductivity,

causing leakage (e.g. Miller et al., 1998); cracks at the upstream face of earth-rock-fill dam core

might propagate quickly under rapid impounding in the way of hydraulic fracturing (Wang et al.,

2016); tensile cracks around pipelines were observed from soil-pipe interactions (Wong et al.,

2016). However, the cohesive strength of soils that ought to be used to describe the formation and

extension of cracks are still uncertain. Based on classical strength-of-material theory, crack occurs

once the stress exceeds the tensile strength of material; but recent considerations about the issue

have extended to fracture mechanics which provides the pre-conditions for crack propagation

when cracks have been already generated in soils due to temperature variation, moisture loss or

stress state change. With the development of cracks, structures may fail ultimately. To understand

2

the rupturing behaviour of soils, the cohesive strength is the fundamental that needs to be

investigated.

On the other hand, adhesion of cohesive soils to structures has been studied and applied to

engineering practice, such as pile design and pipeline analysis. The adhesion of soil partially

contributes to the skin friction at the structure surface. For piles, the adhesion along the axial

direction is a part of bearing capacity; for pipelines, the adhesion transfers loads to the structures

when subjected to external ground movement. Adhesion is a basic factor that influences the soil-

structure interactions.

In this research, the cohesive strength and adhesive property of unsaturated compacted clays have

been studied. Due to the complex nature of unsaturated clays, these properties usually do not stay

constant with changing phases of water, air, and solid in soils. Normally, more water content means

less stiffness, lower strength and higher ductility, and vice versa. Determining how these properties

will change under moisture conditions is the main purpose of this thesis.

1.2. Research Scopes

1.2.1. Mode I Fracture Toughness

In fracture mechanics, fracture toughness of brittle materials reflects the capability to resist crack

propagation under the circumstances that a crack has been pre-existed somehow. According to the

stress conditions upon cracking, the modes of fracture are classified into: opening (mode I), in-

plane shearing (mode II), and out-of-plane tearing (mode III), as illustrated in Figure 1.1. Tensile

cracking under mode I stress condition is the most common scenario in geotechnical structures.

3

The first objective of this thesis is to determine the mode I fracture toughness of compacted clays

with different moisture contents and dry densities. Fracture toughness will be measured through

laboratory tests, for which Linear Elastic Fracture Mechanics (LEFM) will be applied to estimate

the fracture toughness of compacted clays.

1.2.2. Tensile Strength

On the perspective of classical strength-of-material concept, tensile strength defines the maximum

stress that the material can withstand under tensile loads. Although its suitability of being applied

to cracking is questionable, it is still a material property that could be easily defined and

determined. Throughout previous studies on geomaterials, it was found that the tensile strength

and mode I fracture toughness were strongly positively correlated, either in a linear or power-law

manner (e.g. Wang et al., 2007).

The second objective is to measure the tensile strength of compacted clays with the same moisture

contents and dry densities as fracture toughness tests. Tensile strength will be determined by means

of laboratory tests as well. Linear and power-law correlations will be conducted to evaluate the

relationship between tensile strength and fracture toughness of the materials.

1.2.3. Role of Matric Suction

From the researches in the past decades, it has been generally accepted that the mechanical

properties of unsaturated clays are affected by matric suction, the surface tension in soil pores due

to capillary effect. Nonetheless, as to compacted clays, there is a problem remaining unexplained

4

that under the relatively dry condition, the measured matric suction is much higher than the

strength of material, up to several magnitudes. The principle of how matric suction influences the

mechanical behaviour of compacted clays is still a matter of debate.

Another objective of this research is to measure the matric suction of compacted clays and to curve

fit the tensile strength with a lately proposed double-porosity concept, attempting to explain the

variation trends of properties with respect to moisture content.

1.2.4. Adhesion to Metal

In pile and pipeline design manuals (e.g. CGS, 2006; ASCE, 1984), the adhesive strength to metal

is empirically related to the undrained shear strength of soils. Practical expression of adhesion for

piles has been widely applied in real engineering settings with large referable data bases; at the

same time, available data are limited for pipelines.

The final objective of this thesis is to examine the adhesion of unsaturated clays to stainless steel,

a common anti-corrosion material for pipelines. The adhesions will be obtained by shearing the

soil-metal interface using a direct shear box. The effects of moisture content as well as the pre-

load pressure on adhesion will be studied in this research.

1.3. Outline of Thesis

Chapter 1 provides the general background of the issues and states the objectives of this research.

5

Chapter 2 gives the basic properties and soil-water characteristic curves of two types of clays that

are studied in this research.

Chapter 3 reviews the Linear Elastic Fracture Mechanics (LEFM) Theory succinctly and presents

the testing programs for mode I fracture toughness. A most viable method named Single-Notched

Disk Bend (SNDB) was selected from the most frequently used testing methods for determining

mode I fracture toughness of geomaterials. The test setup and results are included in this chapter

as well.

Chapter 4 concentrates on the testing programs and results of uniaxial direct tensile tests.

Chapter 5 shows the correlations of mode I fracture toughness and tensile strength of geomaterials

and a tentative curve fit for the changing tendencies of tensile strength using a microstructure-

based concept.

Chapter 6 presents the adhesions of clays to stainless steels under different moisture contents and

different pre-consolidation pressures.

Chapter 7 summarizes the conclusions of this research and perspective studies on this topic.

6

Mode I Mode Ⅱ Mode Ⅲ

Opening Shearing Tearing

Figure 1.1 Three modes of cracking (after Broek, 1982)

7

Basic Properties and SWCC of the Soils

Two types of soils used in the investigation were collected from the construction sites. Calgary till

was obtained from Sage Hill in Calgary, Alberta, whilst Regina clay was excavated near Mosaic

Stadium in Regina, Saskatchewan, Canada. Prior to the tests, all of impurities, like leaves and

branches, were removed, and then the soils were air-dried, pulverized and sieved through a 2-mm

sieve.

In this Chapter, basic properties of the soils as well as their soil-water characteristic curves

(SWCC) will be presented.

2.1. Basic Material Properties

2.1.1. Atterberg Limits and Soil Classifications

Atterberg limits are basic properties that provide an idea for distinguishing the state of soil under

the effect of moisture content. Liquid Limit (LL), Plastic Limit (PL), and Plasticity Index (PI) are

determined from the methods of ASTM D4318. The values are listed in Table 2.1. Following the

description in ASTM D2487, Calgary till is classified as CL and Regina clay is CH.

2.1.2. Compaction Curves

Since the specimens used in this study are prepared through compaction, it is necessary to

determine the compaction curves of the soils and to determine the optimum dry densities and the

corresponding moisture contents. Compaction was done using standard effort following ASTM

D698. Peaks of dry densities (ρd, max) are obvious for both soils as seen from Figures 2.1 and 2.2,

when moisture content is optimal (w0). Full saturation lines were drawn to assist with selecting

8

density-moisture points for the tests later. In addition, the specific gravities and clay contents are

listed in Table 2.1 as well.

2.2. Soil-Water Characteristic Curves (SWCC)

2.2.1. Introduction

Because of the coexistence of air, water, and solid phases in unsaturated soil, the classical effective

stress (σ - uw) for saturated soil cannot describe the stress state that involves pore-air pressure

(Fredlund and Rhardjo, 1993). Amid the pores of soil structures, air and water are trapped, between

which thin water films form with the aid of surface tensions. The water film separates the air phase

and water phase in different spaces where the air possesses a pressure of ua and the water possesses

uw, producing a pressure difference (ua - uw) that is borne by the water film. This pressure

difference is defined as matric suction. Matric suction can be deemed as one of the two stress state

variables for partially saturated soil when ua is the reference pressure and the other one is (σ - ua).

The combination of (ua - uw) and (σ - ua) is the most satisfactory for use in engineering practice as

the effects of total stress change can be separated from the pore-water pressure (Fredlund, 1979).

As a stress state variable, matric suction plays a governing role in the mechanical properties of

unsaturated soils. At the same time, matric suction changes with moisture content, and the soil-

water characteristic curve (SWCC) depicts the relationship between them (Tinjum et al., 1997).

Normally, low moisture content contributes to high suction pressure and vice versa. Besides water,

other factors such as soil structure, mineralogy, grain size distribution, etc., are also relevant to the

suction (Lakshmikantha 2009). Therefore, obtaining SWCC is crucial to reflecting the mechanical

features of unsaturated soils.

9

Apart from matric suction, osmotic suction is another pressure component, which is induced by

differential chemical concentrations when water is prone to be in equilibrium through a

semipermeable membrane from low to high concentrations in soil (Krahn and Fredlund, 1972). As

the water used for sample preparation is potable, which is approximately as fresh as pure water,

only matric suction is considered in this study.

The ‘J’ or ‘S’ shape of SWCC is controlled by three parameters: the air entry value (Ψa) that is the

smallest suction pressure that allows water start to move out of the pores and is also the inflection

point of SWCC, saturated volumetric water content (θs) when soil is saturated, and residual

volumetric water content (θr) where the suction approaches an asymptote at high suction level and

lower moisture content (Babu et al., 2005). Based on the parameters above, several equations were

proposed to characterize the shape of SWCC. Three frequently used equations are briefly

introduced herein:

1) Brooks and Corey (1964) suggested Equation 2.1. This equation is valid for the suction greater

than Ψa. The normalized volumetric moisture content Θ can be replaced by the degree of saturation

Sr (Fredlund and Xing, 1994).

a

λa

rs

rv ΨΨΨΨ

θθθθΘ ≥

=

−−

= Equation 2.1a

as ΨΨθθΘ <== and1 Equation 2.1b

where θv is volumetric moisture content; Θ is the normalized volumetric moisture content; Ψ is

suction; λ is a curve fitting parameter.

10

2) van Genuchten (1980) developed Equation 2.2 in an attempt to obtain a closed-form expression

for hydraulic conductivity. This form overcomes the sharp discontinuity of Equation 2.1 for

suction near saturation.

( )[ ] ''1mn

r a'ΨS−

+= Equation 2.2

where the degree of saturation Sr is expressed as a function of suction Ψ; a’, n’ and m’ are curve

fitting parameters. The condition of n’ = 1/ (1-m’) is often assumed to simplify the curve fitting.

3) Fredlund and Xing (1994) proposed Equation 2.3 that can describe the full range of suction

accurately. However, this expression requires calibrations of four parameters a’, n’, m’, and Ψr. In

addition, θr corresponding to Ψr has been out of the moisture range concerned in this study.

[ ] ( ){ } '' ]'/ln[)/1000000(1ln)1ln(

mns

r

rv

aΨe

θΨ

Ψ/Ψθ+

⋅+

+= Equation 2.3

where θv is the volumetric moisture content; a’, n’, and m’ are curve fitting parameters; Ψr is the

suction corresponding to the residual moisture content θr; 106 is a maximum value of suction at

zero water content estimated from their experimental results.

In this study, SWCCs of Calgary till and Regina clay will be determined through experiments.

Details are going to be described in the following sections. The methods for measuring matric

suction will be reviewed first; then the matric suction of compacted specimens will be determined

through a modified method using filter papers; finally the data will be curve fitted by van

Genuchten’s equation (due to its accuracy and simplicity compared with the other two) to reveal

the variation trends of matric suction of compacted clays.

11

2.2.2. Methods

Several devices can be used for measuring suction, such as psychrometer, pressure plate extractor,

tensionmeter, and filter paper. Psychrometer is an instrument that measures relative humidity in

the air phase of a soil pores or the region near the soil, from which the suction of the soil sample

can be obtained from a calibration curve. Nonetheless the technique requires constant temperature

environment and is not recommended if temperature fluctuation could not be controlled

restrictively (Fredlund and Rhardjo, 1993). Pressure plate extractor is a device that is able to

maintain pore water pressure constant and increase the pore air pressure simultaneously so as to

obtain desired matric suction (Tinjum et al., 1997); but the measuring capability of this method is

limited to a range of 0-1500 kPa (Fredlund and Rhardjo, 1993). Tensionmeter basically measures

matric suction directly when pore-air pressure is atmospheric; however its measuring range is

confined in 0-90 kPa (Fredlund and Rhardjo, 1993). Filter paper method is a cost-effective way

that is capable of measuring both total suction and matric suction, based on the assumption that

the dry filter paper absorbs vapor or liquid from a wet specimen until the suction value of the filter

paper is equal to the soil (Bulut et al., 2001); after the equilibrium, the moisture content of filter

paper can be used to determine the suction of soil, for which ASTM D5298 provides detailed

description for the measurement.

From the review above, filter papers stand out among these methods with its low costs, simple test

equipment, and established measurement techniques. In this study, considering the merits of filter

paper method, Whatman No.42 filter papers were used, for which the calibration curves of matric

suction were provided by Leong et al. (2002):

12

)47(0229.0909.2log)47(0673.0945.4log

≥−=

<−=

ff

ff

wwΨwwΨ

Equation 2.4

where Ψ is the matric suction in kPa, and wf is the moisture content of filter paper after

measurement.

2.2.3. Modified Filter Paper Method

Though the means of using filter papers have been well described in ASTM D5298, the way of

measuring matric suction is not perfect to conduct because it requires two separate specimens with

similar sizes and smooth surfaces so that the filter papers and the specimens can closely fit each

other. In addition, the specimens must be taped tightly by electrical tape and sealed in an air-tight

jar. However, making smooth surfaces is difficult, particularly under low moisture condition, and

sealing specimens is inconvenient if the specimens are large. So an enhanced measurement with

two filter papers attached on the upper and lower surfaces of one single specimen was proposed

and used in this study, as illustrated in Figure 2.3.

The specimens used for matric suction measurement were compacted in the cylindrical collar of

standard compaction mold, with a dimension of diameter D = 101.6 mm (4 inches) and thickness

t = 50.5 mm. Similar to the standard method, two 75 mm protective papers are set in between the

55 mm filter papers and the specimen, preventing the filter papers from contacting the soil directly;

then the whole assembly of protective papers, filter papers, and specimen will be wrapped by an

aluminum foil, placed in a sealing bag and vaccumized exerting pressure to provide contact for the

filter papers and the specimen.

13

Since the aim of this research is to examine the effects of moisture content and dry density on the

cohesive and adhesion of soils, a series of moisture contents and different dry densities are included

in the testing program of matric suction. Calgary till was compacted to two different dry densities

ρd = 1.75 and 1.85 g/cm3, over a moisture content range of 11% - 19.5%; while Regina clay was

compacted using consistent standard compaction energy from the dry of optimum 19.5% to the

wet of optimum 31.5%. This testing scheme will be carried out in the next chapters for fracture

toughness, tensile strength, and adhesion tests.

The tests for determination of matric suction were conducted according to following steps:

1) Prepare the soils to the designated moisture contents and store them for at least 3 days for

moisture equilibrium.

2) Weigh the required mass of soil and compact soil into the mold with a standard compaction

hammer; make sure the surfaces of compacted specimens are smooth, using a small knife

to help flatten the surfaces if necessary.

3) Pre-treat the filter papers and protective papers with 2% formaldehyde solution to prevent

mold growth, oven dry them for 24 h, and store them in an air-tight bag prior to use.

4) Cut a piece of aluminum foil that is sufficiently large to cover the entire specimen; place

the lower filter paper on the foil and then put the lower protective paper on the filter paper,

using tweezers; next, put the specimen on top of them, and then place the upper protective

paper and the upper filter paper.

5) Wrap the assembly with the aluminum foil and insert it into a sealing bag immediately.

6) Vacuumize and seal the bag tightly (Figure 2.4).

14

7) Label the specimen and with the date of preparation, and store it in an insulation box for at

least 10 days for moisture equilibrium of the specimen-filter papers system.

8) Finally, after equilibrium, the filter papers are moved quickly and carefully into metal tins

using tweezers; the weights of empty tins, of tins containing wet filter papers, and of tins

with dry filter papers after oven drying must be recorded.

2.2.4. Results

The results are listed in Tables 2.2 and 2.3. The specimens were labeled by ρd / w, where the suffix

a and b represent the upper and lower filter papers, respectively. It was observed that the difference

of matric suction between the two filter papers for one single specimen was noticeable. For

example, as to 1.48/28.5 specimen, the matric suction measured by the upper filter paper was

190.511 kPa, while the lower was 401.791 kPa which is twice as much as the upper one. In the

author’s opinion, this difference was a result of two issues: 1) compaction is not ideally uniform

throughout the entire specimen, thus there must be a density difference between the two faces of

specimen, which inevitably caused a difference of measurement; 2) the calibration curve of filter

paper for matric suction is expressed in terms of semi-log equation, which means even a very small

measurement error of the moisture content of filter paper may be magnified to a significant error

in matric suction — for instance, still as to the 1.48/28.5 specimen, the moisture content of filter

papers were 39.60% and 34.78%, of which the difference was as little as 14%, however the

corresponding matric suction values were different to a large extent, 191 kPa and 402 kPa

respectively. To reduce the measurement errors from above-mentioned factors, the average matric

suction of two filter papers will be used as the representative value of each specimen.

15

Data curve-fitted by van Genucheten’s Equation 2.2 using least square method are plotted in

Figures 2.5 and 2.6, and the parameters for curve fitting are listed in Table 2.4. The variation trends

of matric suction with respect to the degree of saturation are clear for both soils, where the matric

suction increases exponentially as the degree of saturation drops; moreover, for Calgary till, the

trend line of 1.85 g/cm3 is located above the 1.75 g/cm3, which is in line with the observation of

Lakshmikantha (2009) where specimens with higher dry densities possess higher matric suction

under the same degree of saturation; for Regina clay, the magnitudes of matric suction are fairly

close to that measured by Chowdhury (2013) and Wong et al. (2017).

16

Table 2.1 Physical properties of the soils

Parameter Calgary till Regina clay LL 37% 69% PL 19% 29% PI 18% 40%

Classification CL CH Clay content 15% 50%

ρd, max 1.85 g/cm3 1.50 g/cm3 w0 15% 25% Gs 2.75 2.83

Table 2.2 Suction measurement for Regina clay

# (ρd/w) Tc M1 M2 Th Mf Mw wf Ψ (kPa) Ψ average 1.45/31a 28.3371 28.6464 28.5498 28.3338 0.2160 0.0933 43.19 109

114 1.45/31b 28.2809 28.6008 28.5020 28.2777 0.2243 0.0956 42.62 119 1.48/28.5a 38.0880 38.4021 38.3060 38.081 0.2250 0.0891 39.60 191

296 1.48/28.5b 37.1742 37.4776 37.3914 37.1663 0.2251 0.0783 34.78 402 1.50/25a 37.7238 38.0204 37.9422 37.7184 0.2238 0.0728 32.53 570

768 1.50/25b 26.1122 26.3955 26.3269 26.1075 0.2194 0.0639 29.12 966 1.48/22a 28.0586 28.3514 28.2843 28.0571 0.2272 0.0656 28.87 1004

1396 1.48/22b 28.4933 28.7625 28.7064 28.4913 0.2151 0.0541 25.15 1788 1.45/19.5a 27.9235 28.1975 28.1442 27.9207 0.2235 0.0505 22.60 2657

2811 1.45/19.5b 28.5130 28.7792 28.7287 28.5103 0.2184 0.0478 21.89 2965

where Tc = cold tare mass (g);

M1 = mass of wet filter paper + cold tare mass (g);

M2 = mass of dry filter paper + hot tare mass (g);

Th = hot tare mass (g);

Mf = mass of dry filter paper = M2 - Th (g);

Mw = mass of water in filter paper = M1 - M2 – Tc + Th (g).

wf = water content of filter paper = Mw / Mf (%);

Ψ = matric suction (kPa);

Ψaverage = the average matric suction of the upper and lower surfaces (kPa).

17

Table 2.3 Suction measurement for Calgary till

# (ρd/w) Tc M1 M2 Th Mf Mw wf Ψ (kPa) Ψ avearge 1.85/17a 27.9226 28.3087 28.1494 27.9229 0.2265 0.1596 70.46 20

21 1.85/17b 28.4932 28.8630 28.7130 28.4924 0.2206 0.1492 67.63 23 1.85/15a 28.2803 28.5947 28.5054 28.2792 0.2262 0.0882 38.99 209

295 1.85/15b 28.3360 28.6330 28.5547 28.3349 0.2198 0.0772 35.12 381 1.85/13a 37.7239 38.0101 37.9381 37.7190 0.2191 0.0671 30.63 765

932 1.85/13b 26.1129 26.4040 26.3356 26.1087 0.2269 0.0642 28.29 1098 1.85/11a 28.5137 28.8007 28.7419 28.5100 0.2319 0.0551 23.76 2218

1832 1.85/11b 28.0593 28.3427 28.2809 28.0569 0.2240 0.0594 26.52 1447

1.75/19.5a 28.3371 28.7539 28.5563 28.3337 0.2226 0.1942 87.24 8 8

1.75/19.5b 28.2810 28.6916 28.4946 28.2780 0.2166 0.1940 89.57 7 1.75/17a 28.5126 28.8390 28.7363 28.5117 0.2246 0.1018 45.33 78

90 1.75/17b 28.0588 28.3757 28.2780 28.0574 0.2206 0.0963 43.65 102 1.75/15a 38.0868 38.3947 38.2949 38.0749 0.2200 0.0879 39.95 180

259 1.75/15b 37.7236 38.0233 37.9365 37.716 0.2205 0.0792 35.92 337 1.75/13a 27.9237 28.1982 28.1288 27.9201 0.2087 0.0658 31.53 665

598 1.75/13b 28.4946 28.7723 28.7002 28.4914 0.2088 0.0689 33.00 530 1.75/11a 26.1125 26.3946 26.3300 26.1078 0.2222 0.0599 26.96 1351

1368 1.75/11b 37.1737 37.4434 37.3795 37.1668 0.2127 0.0570 26.80 1385

Table 2.4 Curve-fitting parameters for SWCC of Calgary till and Regina clay

Parameter Calgary till Regina clay 1.85 g/cm3 1.75 g/cm3

a' 0.031 0.005 0.003 n' 1.161 1.200 1.214 m’ 0.139 0.167 0.194

18

Figure 2.1 Compaction curve of Calgary till

Figure 2.2 Compaction curve of Regina clay

19

Figure 2.3 Illustration for modified suction measurement method

Figure 2.4 Vacuumized specimen for matric suction measurement

Specimen

ProtectivePapers

FilterPapers

20

Figure 2.5 SWCC of Calgary till

Figure 2.6 SWCC of Regina clay

21

Mode I Fracture Toughness

3.1. Introduction

To define the critical state that a crack would extend under tensile stresses, a parameter, called

fracture toughness, from Linear Elastic Fracture Mechanics (LEFM) ought to be used to be on

behalf of the ability of resisting fracturing for brittle or quasi-brittle materials. Fracture toughness

is a material property that can be obtained from experiments.

In this chapter, the theory of LEFM is introduced briefly at first, and then the existing methods for

determining mode I fracture toughness of geomaterials, both rocks and soils, are reviewed.

Subsequently, a most viable method named Single Notched Disk Bend (SNDB) is chosen for test

implementation. Finally the tests results are shown and analyzed.

3.2. Review of LEFM Theory

Evaluating the state of a pre-existed crack in a material that whether it will propagate under certain

stress conditions is the key principle of fracture mechanics. The evaluation criterion was firstly

raised by Griffith (1921) based on the energy change in fracturing process, while an equivalent

stress-based criterion was proposed by Irwin (1957) later. These two criteria are the bases of LEFM

for determining the state of cracks. The theories behind two approaches are presented below

briefly.

22

3.2.1. Griffith Criterion (Energy Release Rate)

Griffith criterion considers the condition of crack elongation on account of energy change. The

reason why he proposed this energy approach was that he found due to the existence of micro-

cracks in a brittle material the actual strength of the material would be much less than the

theoretical solution, which might be attributed to the high stress concentrations around crack tips

in the body of the material, even if the average stresses were at low magnitudes. This high level of

stress concentration could not be described by linear elasticity since its solution tends to be infinite

at a crack tip. Alternatively, the energy approach avoids the use of elasticity theory.

To go over his theory, an infinite, isotropic, and unit-thick plate is hypothesized. The plate is under

the conditions of plane stress in tension and ideal elastic deformation. Now a 2a long elliptical

flaw is created within the tensioned plate, perpendicular to the direction of tensile stresses (Figure

3.1). With the creation of crack, stored energy from the tensioned plate shall release and transfer

to the surface energy of crack because of the law of energy conservation. From the theory of

elasticity, Griffith found the stored strain energy (Ua) of the cracked area:

EaπσUa

22

−= Equation 3.1

where σ is the tensile stress in the infinite plate; a is the crack length; E is the elastic modulus of

the material.

And the surface energy contained in the newly created area from cracking, which is also the

required energy (Ur) for crack creation, is:

23

sr γaU 4= Equation 3.2

where γs is the specific surface energy, and 4a = 2a × 2 since there are two crack surfaces.

So the total energy change of the system because of the formation of the new crack surfaces equals

to:

sra γaE

aπσUU 422

+−=+ Equation 3.3

Both the stored elastic energy and surface energy change when the crack length changes, and so

does the summation of them, as seen in Figure 3.2. For the system energy, a peak point appears as

the crack increases, which is the critical point of a crack state. At the left side of it, the crack keeps

stable as the released energy is insufficient to create new surfaces, whilst at the right side the crack

transforms to unstable state and propagates spontaneously as the released energy from storage is

higher than the required energy for creating new surfaces. Since the threshold is an extremum, it

can be equivalently expressed as:

0/)4()( 22

=+−=+ daγa

Eaπσd

daUUd

sra Equation 3.4

or:

sra γ

Eaπσ

dadU

dadU 42 2

===− Equation 3.5

From the equations above, the critical stress that initiates the crack self-propagating is:

24

aπEγ

σ sc

2= Equation 3.6

In other words, to obtain the critical stress extending a crack, the release of stored energy should

be not less than the surface energy of the crack.

However, this theory is only applicable to extremely brittle materials, such as glass, inorganic

crystal material and ultra-high-strength steel. As to other materials in engineering practice, such

as plain steel, aluminum, polymer or other materials that exhibit more ductility, the theory

underestimates the real energy required for crack extension because the energy absorbed by plastic

deformations around the crack tip is not taken into account. Owing to stress concentration around

a crack tip, plastic behaviour must exist within a small area, unavoidably consuming energy, which

should also be a part of required energy for crack propagation. So the Griffith criterion was

modified by Orowan (1949) with plastic energy γp included:

aπEγγ

σ psc

)(2 += Equation 3.7

Irwin (1957) moved one more step forward to generalize Griffith criterion. He suggested that the

energy stored in the material releases at a certain rate as crack grows, which was denominated by

him as G in memory of Griffith. G is defined as “energy release rate” per crack tip, per unit

thickness and per unit crack extension, equal to dUa/da:

GE

aπσda

dUa ==2

Equation 3.8

25

To extend a crack, the energy release rate must exceed the needed energy for per unit crack

extension Rc = dUa/da, which is defined as crack resistance. The energy release rate that just

sufficiently satisfies the required energy, a combination of surface energy and plastic energy, for

fracturing is called critical energy release rate GIc:

EaπσGR

dadU c

Iccr

2

=== Equation 3.9

aπEG

σ Icc

2= Equation 3.10

3.2.2. Stress Intensity Factor

As aforementioned, the elastic solutions of stresses near a crack tip prone to go to infinity under

any external loads, so an alternative energy approach was proposed by Griffith. Besides that way,

Irwin (1957) developed a stress state approach which avoided to use stress as a criterion but was

still capable of describing the state of stress concentration around a crack tip on the basis of

elasticity.

From the theory of elasticity, given all the boundary conditions, there must be a stress function

called Airy Stress Function that can be used to find the stresses at any point within a body. For the

boundary conditions presented in Figure 3.1 under uniaxial tension (mode I), the corresponding

Airy Stress Function had been solved by Westergaard (1939) in polar coordinates:

−=

23sin

2sin1

2cos

2θθθ

raσσ x Equation 3.11a

26

+=

23sin

2sin1

2cos

2θθθ

raσσ y Equation 3.11b

23cos

2cos

2sin

2θθθ

raστ xy = Equation 3.11c

These equations had been rewritten by Irwin to:

)(2

θfrπ

Kσ ijI

ij = Equation 3.12

where fij (θ) is a function of θ; KI is defined as the “stress intensity factor”, which is a description

of the stress singularity.

In order to both satisfy the dimension of Equation 3.11and the proportional relation between the

stresses and external loads, KI is supposed to take the form of:

aπησKI = Equation 3.13

where η is a dimensionless factor that equals to 1 for an infinite plate and is dependent on crack

length and geometry for a finite body.

The proposal of stress intensity factor is of significance because it gives a measure for the whole

stress field around a crack tip. The stress fields of different cracks under different loads can be

compared using the stress intensity factor to evaluate the singularity at crack tips. For example, if

two elastic cracks have same KI, they may have similar stress fields in the vicinity of tips.

27

Undoubtedly, as there is a direct link between KI and σ as Equation 3.13, if a failure stress σc can

be determined from the fracture load at the moment of failure, a critical KIc exists. For an infinite

plate:

aπσK cIc = Equation 3.14

According to Equation 3.14, it could be concluded that KIc could be a parameter that evaluates the

state of crack. As long as the stress intensity factor KI < KIc, failure will not occur and the crack

stays stable; at the point of KI = KIc, the crack may grow further with increasing load or may arrest

without addition input of energy; once KI > KIc, crack extension is unavoidable. Hence, KIc is

thought as a material parameter representing the critical stress field around a crack tip prior to

fracturing or the ability of resisting crack propagation, called “Fracture Toughness”.

Interestingly, the critical state in terms of energy (Equation 3.9) and stress state (Equation 3.14)

can be converted in plane stress condition:

EKG Ic

Ic

2

= Equation 3.15

and in plane strain condition:

EKυG Ic

Ic

22 )1( −= Equation 3.16

Both GIc and KIc are material properties and they are equivalent in defining cracking-resistance

capability of materials in LEFM theory.

28

3.2.3. Applicability of LEFM

Now it is clear that the stresses are unrealistically infinite at the crack tip when linear elasticity

theory is applied. The true stresses around a tip must be finite otherwise the crack shall definitely

open. So a question that why a crack stays stable prior to fracturing was raised. The answer was

provided by Irwin. He claims a fracture process zone (FPZ) will take place at the crack tip where

plasticity dominates the behaviour and the stresses will not go to infinity because the yield strength

impedes the growth of stresses, and the actual stresses distribute as shown in Figure 3.3. The size

of plastic area is assumed to be an rp*- radius circle expressed in terms of the stress intensity factor

KI and yield strength σys:

2

2

2

2*

22 ysys

Ip σ

aσπσKr == Equation 3.17

But the assumption does not reflect the actual plastic zone, as shown in Figure 3.3(c), which should

be larger. Irwin afterwards modified the radius to:

*2 pp rr = Equation 3.18

Usually, the applicability of LEFM can be justified as long as the fracture process zone is

reasonably small compared with the size of specimen because the stress distribution beyond the

zone still follows the solution of linear elasticity, for which KI remains to be able to determine the

stress field. On this basis, a sufficiently large specimen size is required for LEFM so that the size

of FPZ can be neglected; additionally, the specimen is supposed to deform ideally elastically so as

a check on the linearity of the load-displacement curve prior to the peak is necessary (ASTM

E399).

29

3.2.4. Summary

To overcome the unrealistic description of elasticity theory on the stresses around crack tips, two

inter-convertible criteria, energy approach and stress state approach, were proposed to define the

critical point when a crack is about to extend. On the perspective of energy change, once the energy

release from storage is larger than the required energy for propagation, fracturing initiates. From

the point of stress intensity factor, it can determine the singularity of stress fields and the size of

plastic zones at crack tips, where same stress intensity factor gives similar stress distribution and

plastic zone. The critical stress intensity factor that initiates fracturing is defined as fracture

toughness. KIc shall be applied as a material property to help understand the cracking-resistance

ability of soils in this study.

3.3. Review of Testing Methods for KIc

Though KIc tests of several materials, such as metals, rocks, concretes, soils, woods and ceramics,

have been conducted, standard testing methods have not been well-established except for metallic

materials, for which experimental setups are introduced by ASTM E399 in detail. As to other

materials, only suggested methods or tentative methods are available. Especially for soils, few

references could be found. In this section, methods that have been used for rocks and soils, are

going to be reviewed, among which the most suitable method, Single Notched Disk Bend (SNDB),

is chosen for this study considering the convenience of sample preparation, experimental setup,

and the low-strength nature of soils.

30

3.3.1. Methods Already Used for Soils

3.3.1.1. Single Edge Notched Bending (SENB) Test

SENB is one of the test methods mentioned in ASTM E399 that has been used for compacted clays

(Nichols & Grismer, 1997; Wang et al., 2007; Amarasiri et al., 2011), saline soils (Lima &

Grismer, 1994) and frozen soil (Li & Yang, 2000). The specimen of this test is rectangular with a

single notch formed at one edge, and the specimen is loaded by three points as shown in Figure

3.4, with two supporting rollers at the bottom and one rod loading on the top. This configuration

is also called three-point bending.

There are some requirements for the geometries of SENB specimen according to ASTM E399.

The span S should be four times of the width W; the acceptable notch length a is supposed to be a

ratio to width W, over the range of 0.2 ≤ a/W < 1; the thinckness B is set to W/B = 2. From LEFM,

the mode I fracture toughness is calculated from the critical load Pmax through Equation 3.19.

=Waf

BWSPKIc 2/3

max Equation 3.19

where the shape function f(a/W) is given by Srawley (1976):

2

3/2

1.99 ( )(1 ) 2.15 3.93 2.7( )3

2(1 2 )(1 )

a a a aa a W W W Wf a aW W

W W

− − − + = ⋅ + −

Equation 3.20

With established testing standard and simple test configuration, SENB method has been vastly

adopted to different materials. But most of these materials are strong, and few materials as weak

as soils have been tested in this way before. For weak materials, the primary concern is the self-

31

weight effect that the maximum load exerting on the specimen at failure may be close to, even

smaller than, the self-weight of specimen. Consequently, the self-weight of specimen cannot be

excluded from the failure load, and meanwhile the contribution of self-weight can hardly be

explained.

To eliminate the self-weight effect on the tests for soils, two solutions are available.

The first one, raised by Wang et al. (2007), is to set up the experiment horizontally as shown in

Figure 3.5. The specimen is put on several steel balls or Omni-directional wheels so that the

specimen does not bend vertically and is able to move freely on the horizontal plane. The positions

of two rollers and loading rod present in Figure 3.4 are moved to ⑥ and ④ in Figure 3.5 so that

the specimen is loaded horizontally where gravity does not point to the direction of failure load.

The second one is to balance the self-weight by changing the dimensions of specimen and putting

counter-balance loads (e.g. Amarasiri et al., 2011). The typical setup is illustrated in Figure 3.6. If

the dimensions of specimen are altered, the shape function should be changed as well. For counter-

balance weights and slides involved in the system, the balancing loads ought to be calculated

precisely.

3.3.1.2. Compact Tension (CT) Test

CT is also well described in ASTM E399. This type of specimen has been attempted on testing

soil cracking as well (Lee et al., 1988; Ayad et al., 1997; Konrad & Cummings, 2001;

32

Lakshmikantha, 2009; Lakshmikantha et al., 2012). As illustrated in Figure 3.7, the specimen,

supported by a metal bar, is pre-cracked at the top edge, and two horizontal tensile loads exerting

on two sides of the specimen are pulling in opposite directions. Fracturing is expected to develop

vertically along the notch.

The most up-to-date ASTM E399 (2013) provides calculation of the stress intensity factor for CT

from LEFM:

1/2IP aK f

BW W =

Equation 3.21

2 3 4

3/2

(2 ) 0.886 4.64 13.32( ) 14.72( ) 5.6( )

(1 )

a a a a aa W W W W Wf aW

W

+ + − + − = −

Equation 3.22

However, it seems to be not a suitable method for testing soils. Though the loading directions are

perpendicular to the gravity which avoids the self-weight impacts, the complexity of CT test brings

about the difficulties of experimental setup, for which special molds and customized parts shall be

designed and machined. In addition, it is uneasy to compact the specimen uniformly within the

mold where two large holes take up spaces. In addition, the influence of loading pins on stress

distribution in such a compact specimen is unknown for soils.

3.3.1.3. Flattened Brazilian Disk (FBD) Test

Flattened Brazilian Disk test is an enhanced method of Brazilian Disk (BD) Test. As known, BD

is a well-described in ASTM D3967 for tensile strength of rocks. During BD test, a tensile crack

33

initiates and then develops along the diameter of specimen in the direction of compressive loads.

This offers an idea to measure mode I fracture toughness through tensile splitting methods. Based

on the observation after crack initiation, the crack extends unstably first but will not keep growing

if the load P is not increased, therefore there is a valley on the load-displacement curve where the

crack turns to arrest under the minimum load Pmin. But the conventional BD test proposed by Guo

et al. (1993) has inherent drawbacks, as pointed out by Zhao et al. (1994), that 1) crack initiation

cannot be guaranteed at the disk centre and 2) the load distribution at the arc cannot be ensured to

be uniform. These two concerns made this method for KIc determination unconvincing

To solve the problems, Flattened Brazilian Disk Test was proposed by Wang and Xing (1999).

The geometry of FBD specimen is present in Figure 3.8. Upper and lower arcs of the disk are

flattened and these two ends will be loaded by two plates crushing inwards. It was thought that the

uniformity of tractions on two flattened sides would be better than the original BD. The load angle

should meet the condition 2θ’ > 19.5° if a center cracking is anticipated. KIc can be calculated from

the minimum load Pmin at the valley of the curve in Figure 3.9 through Equations 3.23 and 24 when

2θ’ = 20°.

minmax1/2 ( / )Ic

PK a RR t

= Φ Equation 3.23

7 6 5 4

3 2

( / ) 4.2892 26.6765 84.9054 93.0870

50.7763 14.3776 2.7408

a a a aa RR R R R

a a aR R R

Φ = − − + −

+ − +

Equation 3.24

34

where t is the thickness of disk specimen; R is the diameter of specimen; a is the crack length;

Φ(a/R) is the stress intensity function (SIF) dependent on crack position and crack length.

However, this modification does not provide substantial solutions to the problems of original

design. Agaiby (2013) tested a series of oven-dried clay-like soils by FDB. Although most of his

specimens cracked as expected, a few of them still failed along the diagonal lines instead of the

central diameter, and secondary cracks were generated within the specimen as shown in Figure

3.10. Unpredictability of fracturing path makes this method not practically applicable.

3.3.1.4. Ring Test

As seen in Figure 3.11, ring test specimen looks like a centre-holed Brazilian disk with outer radius

R0 and inner radius Ri forming a ring-shape sample, which is loaded by two rigid platens to induce

the cracks to propagate upwards and downwards along the vertical diameter. Though the specimen

of Ring Test is slightly different from Brazilian Test, the principle behind is basically the same

that trying to find out the smallest load required for crack propagation and obtain fracture

toughness through the corresponding largest stress intensity factor.

The earliest investigation on fracture toughness of compacted clay through Ring Test was

conducted by Harison et al. (1994), and it was for the first time that the tensile strength and fracture

toughness of soil had been gained simultaneously so as to examine the relationship between the

two properties.

35

The main limitation of the method is the drilling works required for sample preparation. Soils are

fragile materials, so drilling across the specimen is highly likely to cause sample disturbance.

3.3.2. ISRM Suggested Methods for Rocks

The suggested methods are designed for rock cores since rock sample are typically obtained

through coring. The cored samples can be split easily in the way shown in Figure 3.12, by which

cylindrical, circular and semi-circular specimens are readily to be prepared and small specimens

might be made out of tested big ones so that precious samples will be used as sufficiently as

possible. Four methods in Figure 3.12 are going to be reviewed herein.

3.3.2.1. Chevron Bend (CB), Short Rod (SR) and Cracked Chevron Notched Brazilian Disc

(CCNBD)

CB and SR were the earliest suggested methods published by ISRM (1988). At a later time,

CCNDB was released in 1995 (Fowell et al., 1995), which was a modification from generic

Brazilian Disk test. For all of these three suggested methods, chevron or V shape notches need to

be cut for specimens, which helps generate stable cracking pattern under increasing loading.

CB uses the same experimental setup as SENB three-point bending (Figure 3.13) with the

cylindrical specimens; from the critical load at specimen failure, fracture toughness can be

determined. SR fractures in another manner by simply pulling the notch apart; same as CB, a

chevron notch is cut as shown in Figure 3.14. CCNBD (Figure 3.15) is loaded by two plates to

36

induce radial crack extension from the tips of chevron notches. Calculations on KIc, which can be

found in the papers, are not presented herein.

Even these three methods are reliably conductible for rocks, their applicability to soils is uncertain.

Manufacturing cylindrical samples for soils is easy, yet making chevron grooves requires special

tool, and it is difficult to control the accurate dimensions of chevron notches when cutting soils.

3.3.2.2. Semi-Circular Bend (SCB)

SCB is the most recently suggested method introduced in 2013 by ISRM (Kuruppu et al., 2013).

The simplicity of this test is the greatest merit. The specimen size is small compared with others

and machining a straight notch, rather than a chevron, out of a cylindrical disk is easy.

Experimental fixture is quite similar to three-point bending as well (Figure 3.16).

The fracture toughness is calculated through Equations 3.25 and 3.26 when a/R ≥ 0.2. The span

ratio S/2R is recommended as 0.5 ≤ S/2R ≤ 0.8. A high ratio approaching to 0.8 is preferred, unless

the material is too weak to be tested, then a lower ratio range could be used (Kuruppu et al., 2013).

'max

2IcP aK Y

RBπ

= Equation 3.25

[ ] [ ]' 2' '1.297 9.516( / 2 ) 0.47 16.457( / 2 ) 1.071 34.401( / 2 )Y s R s R s R= − + − + + +β β Equation 3.26

where β’ = a/R, notch length ratio; Y’ is the stress intensity factor function.

37

Due to its simplicity, the author has attempted to make some dummy samples for trial. However,

soil is too fragile to be sawed for this specimen because small cracks are easily to occur at the tip

of notch (Figure 3.17) during and after sawing. Once there are secondary cracks around the notch,

the actual notch depth is unknown and the test results are consequently questionable.

3.3.3. Straight Notched Disk Bending (SNDB)

SNDB is a newly developed method by Alkilicgil (2006) for rocks. The specimen of this method

is quite convenient to prepare, for which only a straight notch is machined at one face of the disk

sample. The test fixture and specimen geometry are shown in Figure 3.18, where P is the

compressive load exerting on the loading rod, S’ is the length of half span, D is the diameter of the

disk, R is the radius, t is the thickness of disk, and a is the notch depth. No necessity of special-

designed tools for a chevron cut and direct machining on a cylindrical sample makes the

preparation process easy and time-saving.

For the calculation of KIc, Tutluoglu & Keles (2011) offered Equations 3.27 and 3.28 derived from

finite element analysis with ABAQUS program. The parameters in equations change with

specimen geometries a/t, t/R and S’/R. But not all of geometries give effective solutions for fracture

toughness computation. When t/R > 2.0 and S’/R < 0.4, or a/t < 0.5 and S’/R < 0.4, the stress

intensity factor appears to be negative, which means it is compressive stresses that distribute

around the crack tip rather than tensile stresses. It is preferable that S’/R ≥ 0.5 and the stress

38

intensity factor function YI could be expressed as a function of m, n and S’/R, where m and n could

be found in Table 3.1 according to the geometry.

2Ic IPK Y aDt

π= Equation 3.27

nRSmYI +

=

'

Equation 3.28

Due to the simplicity of the specimen and test setup, SNDB is a suitable choice for testing soils,

although it is originally designed for rocks. Cylindrical specimen can be easily obtained from

compaction molds or coring tubes, and meanwhile cutting straight notches does not need special

tools but just an affordable band saw that can cut soils.

3.3.4. Summary

In this section, the methods for soils and rocks have been reviewed with their advantages and

disadvantages. It could be concluded that the existing methods for soils and ISRM-suggested

methods are not easy to conduct. SENB has such a large span that either horizontal loading systems

or counter balance loads are needed so as to negate the self-weight effect; CT has a complicated

geometry that requires customized compaction molds as well as unique loading system; tests

through FBD may not stably yield central cracking, thus consequently less convincing; Ring Test

causes sample disturbance quite easily because of drilling works at the specimen centre; special

tools are necessary to machining chevron notches for CB, SR, and CCNBD specimens; SCB is

simple but trial specimen shows it is not applicable to soils as its fragility would produce unwanted

small cracks around the sawed notch.

39

Fortunately, SNDB, originally for rocks, was examined and considered to be a viable method for

soils, in which cylindrical samples with straight notches that can be readily prepared with common

tools available in any geotechnical labs.

In summary, SNDB is selected as the testing method of this study for compacted clays. Details

about sample preparation and experimental conduction are provided in Section 3.5 at a later time.

3.4. Review of Previous KIc Tests on Compacted Clays

As stated before, the investigations on the characteristics of KIc of soils are still on an early stage,

and only a limited number of reference for KIc determination of compacted clays are available. The

testing methods for compacted clays have been reviewed in section 3.3.1 already, so only the

sample preparation strategies and the results are briefly introduced below.

Harison et al. (1994) and Amarasiri et al. (2011) tested air-dried compacted clays by compacting

clays to nearly saturated state first, and then air drying them to designated moisture contents by

controlling the weight losses from evaporation. They gained a regularity that low moisture contents

contribute to high KIc of air-dried clays and yielded the ranges of fracture toughness: 3.0 ~ 152.2

kPa·m0.5 for Werribee clays studied by Amarasiri et al. (2011), and 0.01 ~ 0.18 MPa·m0.5 for

Kentucky soils (classified as CL and ML) tested by Harison et al. (1994). However, under this way

of preparation, the specimens are very easy to crack during desiccation, in particular for high-

40

clayey soils, based on the author’s experience. Besides, whether the moisture content can distribute

uniformly in the sample is questionable.

Wang et al. (2007) and Lakshmikantha (2009) investigated directly compacted clays by

compacting calculated amount soils prepared with desired moisture contents into the molds to

reach the required degrees of saturation and dry densities. This preparation methodology is much

easier to adopt and it is consistent with the land fill procedures in real engineering settings.

Nonetheless, only a very small range over 16.3% ~ 19.3% of moisture contents were studied by

Wang et al. (2007) because of the difficulties of sample preparation on drier and wetter sides.

Besides, stress-controlled loading was applied in Lakshmikantha’s (2009) study by adding small

weights gradually, which was not as controllable as strain-controlled method and could not record

post-peak behaviours of load-displacement curves. The range of fracture toughness of Nuozhadu

clay investigated by Wang et al. (2007) was 7 ~ 32 kPa·m0.5, while for Barcelona clay in

Lakshmikantha’s (2009) research the range was 0.50 ~ 2.47 kPa·m0.5 which are relatively small

compared with the other soils.

Considering the above-mentioned issues, in this study SNDB specimens will be compacted

directly to the designated dry densities over a wide and conductible moisture content range, then

loaded under strain control, recording both pre-peak and post-peak stages. The values of fracture

toughness of Calgary till and Regina clay will be compared with the soils above.

41

3.5. SNDB Tests for Fracture Toughness KIc

3.5.1. Test Setup

The concept of SNDB has been previously introduced in section 3.3.3. In this section, details of

the experimental setup for compacted soils will be described.

The experiments are conducted on a load frame where the instruments are mounted on. As seen

from Figure 3.19, the load cell (SENSOTEC, Model 41/572-01-01, capacity range 0 - 2000 LBS)

is connected to a triaxial frame; the base seats on a piston which moves upwards or downwards

vertically at controlled constant rates (minimum rate: 0.00001 mm/min). By zooming in to Figure

3.20, the arrangement of loading parts could be seen clearly. An adaptor that holds the loading rod

is installed on the load cell, and in between the adaptor and the load cell is a balance strip that

helps LVDTs reduce measurement error induced from rotation. Two same LVDT (resolution:

0.001mm), also called displacement gauges, are placed symmetrically on both sides, for which the

symmetry must be guaranteed since measurement will be corrected by taking the average of the

two readings. The specimen is loaded by the upper loading rod and two lower loading rollers

supported by bearings. The size of bearings just fits the loading rollers, and that setup prevents the

rollers from waggling and ensures their smooth rotations along with bearings. Bearings are fixed

tightly on supporters with fixtures, while supporters are immobilized tightly on the base with

screws. After assembling, the integrity of the whole parts fixed on the piston and the mobility of

the load cell are checked so that a desired three-point bend can be realized.

42

3.5.2. Test Program

The test program was designed on the purposes of examining 1) the effect of dry densities over a

certain moisture range, 2) the influence of moisture content for a given dry density, and 3) the

impact of moisture content for a given compaction energy. The first two can be realised by varying

compaction energy to different samples, through which the specimens of Calgary till were

compacted to two different dry densities 1.75 g/cm3 and 1.85 g/cm3, with moisture contents from

11% to 19.5%. Specimens of Regina clay were compacted by using same compaction energy so

that the points selected were located on the standard proctor compaction curve over the range of w

= 19.5% - 31%. Specimens with moisture contents in low and high ranges were too fragile to saw

and too soft to prepare, respectively. The testing scheme was presented in Table 3.2. Each group

of tests was repeated for 4 times to check the repeatability, so 56 specimens were prepared in total

for investigating the influence of dry density and moisture content on KIc.

The matric suction measurements mentioned in Chapter 2 were made at the same dry densities and

moisture contents included in Table 3.2.

3.5.3. Test Procedure

Sample preparation and testing are the main stages of procedures. The tests were conducted

according to the following steps:

1) Oven-dried and pulverized soils are homogeneously mixed with water to the required water

contents, and then sealed in air-tight zip-lock bags for at least three days for moisture equilibrium.

43

Before compaction, the mass of specimen is calculated through the volume of compaction mold

and the desired density so that the needed amount of soil could be weighted.

2) Compaction is done by a standard-effort compaction rammer. Soils are compacted into a

cylindrical container, which is actually the collar of a standard compaction mold (Figure 3.21).

The specimen is compacted in three layers with equal mass of soil. The way of compaction is of

great consideration in order to ensure reasonably uniform and repeatable compactions. The

proposed compaction schemes in this study are similar to the standard compaction in ASTM D698.

In the standard, the rammer should blow 25 times for each layer when compacting soils in the

standard compaction mold. But in this study, because the collar is used instead, which is shorter

than the standard mold, the number of blows should be reduced at the ratio of 50.5 mm/ 116.3 mm

(ratio of the heights) to 50.5 mm/ 116.3 mm × 25 ≈ 10.86. Based on the author’s experience, the

number of blows had better been rounded down to 10 so as to avoid over-compacting the

specimens, and the energy input could be compensated later by a small hammer to make sure that

weighed soils will fully fill the compaction mold. This number of blows can be used for

compacting the specimens of which the dry density and moisture content are located at the

compaction curve. However, for those specimens not on the compaction curve, the energy input

must be modified empirically. The number of blows for Calgary till are labeled in Figure 3.22,

which were earned from the author’s experience when practicing sample preparations. As long as

following the number of blows in Figure 3.22, the collar can be filled with pre-weighed soils very

easily. Only flattening works with a small hammer is needed in the end.

44

3) After compaction, careful sawing is followed immediately. A commercial 4-inch miter box is

used to help guide the hand saw (Figure 3.23), making sure the notches are straight and vertical

and the disturbance to samples is avoided. The width of saw is about 0.5 mm, which will make a

1.0 mm notch for the specimen after sawing. Care and patience are essential during sawing to

avoid any possible disturbance to fragile specimens. Although the notch length ratio from a/t = 0.1

to 0.9 were provided from numerical modelling of KIc calculation, it is only practical when a/t is

not higher than 0.4 because sawing further causes cracks easily for soil specimens. Notched sample

must be sealed by plastic film (Figure 3.24) right after sawing to prevent moisture loss, and then

stored in a sealed bag (Figure 3.25), standing for at least 24 hours to eliminate excess pore pressure

induced by compaction and sawing.

4) Before conducting experiment, firstly, the specimen should be inspected carefully to identify

any additional cracks that possibly occur during standing; if none, the specimen is then placed on

the apparatus as shown in Figure 3.20. Due to the weak strength of material, to minimize the self-

weight effect, S’/R = 0.5 for Calgary till and S’/R = 0.6 for Regina clay should be used as suggested

by Kuruppu et al. (2013). Loading rate is set to 0.2 mm/min, also recommended by Kuruppu et al.

(2013) on the purpose of avoiding any dynamic effects. Readings of LVTDs and load cell are

recorded by a computer simultaneously at the frequency of 2 points per second. At this loading

rate, each test will finish in 15 minutes so that moisture loss can be reasonably controlled. Vertical

cracking propagation is expected to be observed when the load achieves the maximum.

45

5) Moisture content should be checked after the test and compared with the pre-test value. The

periphery of the sample, where moisture loss could be significant during the test, is cut, and only

the core of the sample is used for moisture measurement. At least 200 g soil shall be collected and

oven-dried for 24 hours. As long as the moisture loss is within 0.5%, the test is deemed valid,

otherwise abandoned. The remaining part of soils will be oven-dried as well as pulverized later by

machines for reuse.

3.5.4. Test Results

The load P and load-point displacement were recorded over the range of both pre-peak and post-

peak stages. The fracture toughness KIc was calculated from the peak load Pmax of the load-

displacement curve.

Typical load-displacement curves of Calgary till and Regina clay specimens tested at the condition

of a/t = 0.2 are presented in Figure 3.26, from which obvious peaks could be observed. Each curve

could be divided into three stages: Run-in stage, loading stage, and post-peak stage. Run-in stage

is commonly seen from all kinds of displacement curves, at which the loading parts in the system

are still loosely assembled. After running in, the loading parts start to closely work with each other,

and the curve enters a loading range where the load hikes steadily until it approaches the peak.

Subsequently, the critical load initiates crack extension rapidly where remarkable weakening

behaviour can be seen after the peak. These curves reveal that the compacted clays are brittle or

quasi-brittle materials within the testing range of moisture content.

46

From typical curves, a tendency can be seen that specimens of lower moisture content fail more

quickly after the critical loads, because the points recorded at the post-peak weakening section of

the load-displacement curves are less. Thus specimens of low moisture content behave more brittle

than these with the higher moisture content. In addition, samples with higher failure loads possess

steeper slopes, or higher moduli, at the pre-peak stages of the load-displacement curves, which

reflects a fact that the stronger the strength is, the higher the sample stiffness is. These load-

displacement curves can be used as references for numerically modelling the fracturing behaviours

of soils, when the parameters in numerical models need to be calibrated to make sure that the

numerical curves will closely follow the experimental curves.

Pmax is assumed as the critical force that initiates crack propagation and used for calculating KIc in

this study. The results of the two clays are summarized in Tables 3.3 and 3.4. The load-

displacement curves of all the 56 specimens are attached in Appendix A.

3.5.4.1. Effects of Moisture Content and Dry Density

As stated before, from the peak loads values of fracture toughness could be determined from

Equations 3.27 and 3.28. Obtained KIc of the two clays following the testing program will be shown

in Figure 3.27.

For Calgary till, Figure 3.27 (a) plots the KIc values changing with respect to moisture content

under two dry densities and exhibited two distinct data trends. In general, KIc increases with a

decrease of moisture content. Meanwhile, the dry density also has an impact on the value of KIc,

47

but it indicates that the influences on specimens with different moisture contents are different. At

low moisture contents of 11% - 13%, KIc hikes noticeably when the dry density increases from

1.75 g/cm3 to 1.85 g/cm3. At the same time, at high moisture range of 15% - 17%, the values of

KIc of two dry densities overlap, which implies the contribution of dry density to the fracture

toughness is unobvious within the range, even though more compaction energy has been input to

obtain higher dry density. Interestingly, if the moisture content is switched to the degree of

saturation by Equation 3.29, two apparently distinct bands are observed in Figure 3.27 (b). At the

same degree of saturation, the fracture toughness of the samples with higher dry densities is always

larger.

ewGS s

r = Equation 3.29

For Regina clay, specimens compacted under a constant standard compaction energy were tested.

Figure 3.27 (c) plots the variation trend of KIc with respect to moisture content. The variation curve

shows a peak of KIc at w = 22%. For specimens with lower or higher moisture contents, the fracture

toughness are significantly smaller than the peak value. This increase-and-decrease trend of KIc

indicates that the contribution of compaction energy to the strength of soil may vary with moisture

content.

48

The moduli of load-displacement curves of KIc tests were calculated from the loading-stage

segments. Similar effects of moisture content and dry density on the moduli of both soils were

found as Figure 3.28.

3.5.4.2. Effect of Notch Length

The effect of notch length was examined by the specimens of Calgary till. The ratio of a/t = 0.2 -

0.4 were used for most of specimens, except for w = 11% scenario in which only a/t = 0.2 or 0.3

were adopted because the specimens were too brittle to saw if the notch was deeper than 2 cm. The

length ratio of each notch is listed in Table 3.3. To examine the influence of notch length, the

fracture toughness of each test has been normalized by the average value of each group and then

plotted in Figure 3.29.

Figure 3.29 shows that all of the normalized fracture toughness lie in a narrow band with only ±

20% variation. For geomaterials, this level of variation is perceived reasonably acceptable. Similar

fluctuation was observed by Haberfield et al. (1990) when soft rocks were tested. Since the data

are confined in a small horizontal band, it could be concluded that the fracture toughness of

compacted specimen is reasonably constant within the length range of a/t = 0.2 - 0.4.

Owing to unnoticeable effect of notch length on fracture toughness, for specimens of Regina clay,

only a/t = 0.2 and 0.3 were adopted, considering that Regina clay contains more clayey contents

and hence more easily adheres to the saw. Sawing shorter notch helps save time and is less likely

to create unwanted cracks in specimens.

49

3.5.5. Summary

The fracture toughness of Calgary till with dry densities of 1.75 g/cm3 and 1.85 g/cm3 over the

range w = 11% - 19.5% and of Regina clay compacted by a given compaction energy with the

moisture content 19.5% - 31% have been determined using SNDB method. For Calgary till, KIc

increases when moisture content drops, and meanwhile two distinct trends of KIc occur for the

specimens of two different dry densities. For Regina clay, a peak value of KIc appears when the

moisture content decreases from the wet side of optimum to the dry side. The fracture toughness

of the two compacted clays is in the magnitudes of 3.0 to 30.0 kPa·m0.5. This range is consistent

with the results of Wang et al. (2007)’s tests on Nuozhadu clay, a soil similar with Calgary till that

is also classified as CL with LL = 29.1% and PL = 20.2%, using compacted specimens with ρd =

1.60 - 1.76 g/cm3 and w = 16.9% - 19.3%. The change of moduli of load-displacement curves has

the same tendencies as the fracture toughness. In addition, the notch length has an unnoticeable

influence on fracture toughness within the ratio of a/t = 0.2 - 0.4.

50

Table 3.1 Values of m and n for stress intensity factor estimation (after Tutluoglu & Keles 2011)

a/t

t/R

0.5 1.0 1.5

m n m n m n

0.1 11.8930 -0.5768 5.2801 0.1421 4.8564 -0.4053

0.2 12.0560 -0.7241 5.8903 -0.4229 5.6252 -1.2537

0.3 12.9210 -0.8442 6.5884 -0.7633 5.8341 -1.4529

0.4 14.5360 -0.9225 7.4728 -0.9269 5.9777 -1.3607

0.5 17.2960 -0.9789 8.8301 -0.9966 6.4837 -1.2188

0.6 22.0450 -1.0140 11.1730 -1.0349 7.7662 -1.1242

0.7 31.2320 -1.0969 15.8170 -1.0957 10.6776 -1.1150

0.8 52.5460 -1.3484 26.8770 -1.2154 18.0195 -1.2300

0.9 128.5000 -1.4060 68.6170 -1.6095 46.7509 -1.6090

Table 3.2 Testing scheme of KIc for two clays

Parameter Calgary till Regina clay

ρd (g/cm3) 1.75 & 1.85 1.75 1.45 1.48 1.50 1.48 1.45

w (%) 11 13 15 17 19.5 19.5 22 25 28.5 31

51

Table 3.3 Details of mode I fracture toughness test on Calgary till

ρd (g/cm3)

wpre (%) Sr test # a (mm) a/t wafter

(%) Pmax (N)

KⅠc

(kPa·m0.5) KIc(average) (kPa·m0.5)

1.75

19.5 0.938

1 0.010 0.2 19.17 82.08 3.62

3.75 2 0.015 0.3 19.11 83.16 4.50 3 0.020 0.4 19.33 55.08 3.82 4 0.010 0.2 19.38 69.40 3.06

17 0.818

5 0.010 0.2 16.87 122.06 5.38

6.35 6 0.015 0.3 16.66 110.17 5.96 7 0.020 0.4 16.62 91.81 6.37 8 0.010 0.2 16.51 174.98 7.71

15 0.722

9 0.015 0.3 14.55 237.63 12.86

11.34 10 0.020 0.4 15.02 146.90 10.19 11 0.010 0.2 14.68 204.14 8.99 12 0.015 0.3 15.04 246.28 13.33

13 0.626

13 0.020 0.4 12.60 151.22 10.49

12.08 14 0.010 0.2 12.81 267.88 11.80 15 0.015 0.3 12.79 240.87 13.04 16 0.010 0.2 12.74 294.88 12.99

11 0.529

17 0.010 0.2 10.76 371.57 16.37

14.76 18 0.010 0.2 10.59 349.97 15.41 19 0.010 0.2 11.07 279.75 12.32 20 0.010 0.2 11.09 339.17 14.94

1.85

17 0.961

21 0.010 0.2 16.56 119.89 5.28

6.46 22 0.020 0.4 16.49 109.09 7.57 23 0.020 0.4 16.89 93.96 6.52 24 0.015 0.3 16.95 112.23 6.07

15 0.848

25 0.010 0.2 14.52 284.07 12.51

11.04 26 0.015 0.3 14.82 217.10 11.75 27 0.015 0.3 14.60 169.57 9.18 28 0.020 0.4 14.69 154.46 10.72

13 0.735

29 0.010 0.2 12.70 441.78 19.46

20.79 30 0.010 0.2 12.93 550.24 24.23 31 0.015 0.3 12.58 384.54 20.81 32 0.020 0.4 12.70 268.95 18.66

11 0.622

33 0.010 0.2 10.85 614.62 27.07

26.75 34 0.015 0.3 10.55 460.47 24.92 35 0.010 0.2 10.64 575.72 25.36 36 0.010 0.2 10.78 672.94 29.64

52

Table 3.4 Details of mode I fracture toughness test on Regina clay

ρd (g/cm3)

wpre (%) Sr test # a (mm) a/t wafter

(%) Pmax (N)

KⅠc

(kPa·m0.5) KIc(average) (kPa·m0.5)

1.45 31 0.922

1 0.010 0.2 30.65 112.33 6.10

8.59 2 0.015 0.3 30.55 138.26 9.43 3 0.010 0.2 30.53 179.30 9.74 4 0.010 0.2 30.72 167.02 9.07

1.48 28.5 0.884

5 0.010 0.2 28.01 203.06 11.03

12.25 6 0.010 0.2 28.50 204.14 11.09 7 0.010 0.2 28.32 245.19 13.32 8 0.010 0.2 28.10 249.51 13.56

1.50 25 0.798

9 0.010 0.2 24.91 329.45 17.90

16.26 10 0.010 0.2 24.77 260.32 14.14 11 0.010 0.2 24.81 326.28 17.73 12 0.010 0.2 24.90 280.84 15.26

1.48 22 0.683

13 0.010 0.2 21.60 287.32 15.61

18.82 14 0.010 0.2 21.57 394.34 21.42 15 0.010 0.2 22.02 399.66 21.71 16 0.010 0.2 21.81 304.61 16.55

1.45 19.5 0.580

17 0.010 0.2 19.58 209.55 11.38

11.94 18 0.010 0.2 19.32 258.15 14.03 19 0.010 0.2 19.49 237.63 12.91 20 0.010 0.2 19.23 173.90 9.45

53

Figure 3.1 Stress state near crack tip of 2a crack length in an infinite plate under tension

Figure 3.2 Energy change with respect to crack length

2a

σ

σ

x

yσy

σx

τxy

rθ

54

(a)

(b)

(c)

Figure 3.3 (a) Theoretical infinite stress, (b) theoretical plastic zone and the corresponding stress, and (c) true plastic zone and stress distribution (corrected by Irwin) in the plane θ = 0

r

σy

Crack Tip

r

σy

σys

rp*

r

σy

σys

rp

55

Figure 3.4 Configuration and geometries of SENB

Figure 3.5 Horizontal testing system for SENB (after Wang et al., 2007)

56

Figure 3.6 Typical test setup with counter-balance loads (after Hallett & Newson, 2001)

(a)

(b)

Figure 3.7 (a) Configuration and (b) experimental setup of CT Test (after Lakshmikantha, 2009)

57

Figure 3.8 Flattened Brazilian Disk test

Figure 3.9 Typical load-displacement curve for FBD Test

58

Figure 3.10 Diagonal and secondary cracks in FBD specimen of oven-dried soil (after Agaiby, 2013)

Figure 3.11 Configuration of Ring Test

59

Figure 3.12 Specimens with different geometries obtained from core

Figure 3.13 CB test configuration and the cross section of the specimen at the notch

60

Figure 3.14 SR test configuration and the cross section of the specimen at the notch

Figure 3.15 CCNBD test configuration and the cross section of the specimen at the notch

61

Figure 3.16 SCB specimen geometry

Figure 3.17 Small crack generated in SCB soil specimen due to sawing

62

Figure 3.18 SNDB geometry and fixture

Figure 3.19 Layout of the instruments

63

(a)

(b)

Figure 3.20 (a) Sketch and (b) experimental setup for SNDB

64

Figure 3.21 Mold for compaction (collar of standard compaction mold)

Figure 3.22 Number of blows for points around the compaction curve of Calgary till

65

Figure 3.23 Sawing sample with a hand saw guided by a miter box

Figure 3.24 Film-wrapped specimen

Figure 3.25 Sealed specimens in zip-locked bag

66

(a)

(b)

(c)

Figure 3.26 Typical load-displacement curves of SNDB tests for (a) Calgary till, ρd = 1.85 g/cm3, (b) Calgary till, ρd = 1.75 g/cm3, and (c) Regina clay

67

(a) (c)

(b) (d)

Calgary till Regina clay

Figure 3.27 Variation of KIc with respect to moisture content and degree of saturation for

Calgary till (a) & (b) and Regina clay (c) & (d)

68

(a) (c)

(b) (d)

Calgary till Regina clay

Figure 3.28 Variation of moduli with respect to moisture content and degree of saturation for

Calgary till (a) & (b) and Regina clay (c) & (d)

69

Figure 3.29 Ratios of KIc/Kaverage for Calgary till with different a/t

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

10 11 12 13 14 15 16 17 18 19 20

Nor

mal

ized

Fra

ctur

e To

ughn

ess

KIc

/Kav

erag

e

w (%)

a/R=0.2 a/R=0.3 a/R=0.4a/R=0.2 a/R=0.3 a/R=0.4

ρd=1.75 g/cm3

ρd=1.85 g/cm3

70

Uniaxial Tensile Strength

4.1. Introduction

In Chapter 3, mode I fracture toughness, the capability of resisting crack propagation under tension,

has been investigated. As another property of resisting tensile rupture from the perspective of

classical strength-of-material theory, tensile strength was considered as the primary parameter that

governs the tensile behaviour of soils. Since no standard for testing tensile strength of soils has

been established, existing methods are reviewed first, from which the uniaxial direct tensile test

will be selected for determining the tensile strength of compacted clays in this study. Test setup,

procedure, calculation, and the results are included in this chapter.

4.2. Review of Previous Testing Methods

4.2.1. Direct Methods

Tensile strength can be obtained easily through direct tensile test. It equals to the ultimate failure

load divided by the area of failure cross-section. Figure 4.1 outlines the existing direct tensile test

methods for measuring the tensile strength of clayey soils. In Figure 4.1, the vertical arrows

represent that the specimen is pulled vertically, while the horizontal arrows are the loads exerting

on the specimen in a horizontal plane. These tests are performed under load or displacement control.

For compacted clayey soils, dog-bone shaped specimen with wide ends and a narrow neck is

commonly adopted. This geometry imposes the failure location around the specimen neck. The

earliest dog-bone direct tensile tests were carried out horizontally under displacement control by

Tschebotarioff et al. (1953) for bentonite, illite, and kaolinite clays (Figure 4.1(a)). In the decades

71

later, similar configurations were applied in other researches. For example, dumb-bell specimens

and clamps in shown in Figure 4.1(b) were used by Towner (1987); rectangular molds shown in

Figure 4.1(c) were created by Nahlawi et al. (2004); circular molds as Figure 4.1(d) were employed

by Tamrakar et al. (2005); trapezoidal equipment in Figure 4.1(e) was adopted by Lakshmikantha

(2009), Tang et al. (2015), and Stirling et al. (2015).

In addition, specimens with constant cross-section were utilized for clays. To capture complete

post-peak behaviour of ductile clays, Zhang et al. (2015) performed tests using long cuboid

specimens as shown in Figure 4.1(f). To fix the specimens, two ends need to be glued to the loading

parts. Similarly, Wang et al. (2007) used cylindrical specimens that were held by special grips in

the tests as shown in Figure 4.1(g). In addition, as seen in Figure 4.1(h) a horizontally positioned

cylindrical specimen splitting at the diameter plane was proposed by Tang and Graham (2000).

By comparing the test apparatus of the dog-bone and constant cross-section specimens, it is much

simpler to conduct the test in the former way which does not require special gripping parts, time-

consuming gluing works, or adhesives that could possibly change the properties of soils.

4.2.2. Indirect Methods

Indirect method derives tensile strength indirectly from compressive test by generating tensile-

splitting crack in the specimen and calculating the stresses at failure. For instance, for four-point

bending tests, tensile stress can be computed from the beam theory of solid mechanics which

assumes the deflection is small and the cross sections remain planar during bending (e.g. Narvaez

72

et al. 2015). For Brazilian test, a cylindrical or disk specimen is crushed by compressive loads and

the splitting tensile strength is obtained from the elasticity theory (e.g. Stirling et al. 2015). For

unconfined penetration test developed by Fang and Fernandez (1981), a cylindrical specimen is

penetrated by two disks to induce tensile fracture at the centre of the specimen, for which plasticity

theory is used to calculate the tensile strength at failure. Illustrations for these three indirect

methods are shown in Figure 4.2. Cracks drawn in the figure are the position where tensile ruptures

are anticipated to occur upon specimen failure.

However, whether the tensile strength obtained from these indirect methods is reliable is still

debatable. In bending beam test, it is impossible to ignore the self-weight effect on the results since

soil is such a weak material. For Brazilian test, Stirling et al. (2015) indicated that the results at

high moisture contents might be potentially overestimated when compared with that of uniaxial

tensile tests. Kim et al. (2012) found that the tensile strength determined from the penetration test

is dependent on the size of loading disks.

Considering the limitations of indirect methods above, uniaxial tensile tests which are simple and

easy to conduct are generally preferable for soils.

4.3. Uniaxial Tensile Test

In this study, the direct method with ‘dog-bone’ specimens was adopted. Calgary till and Regina

clay were tested using the same testing program as the fracture toughness tests in Chapter 3. In

this section, experimental setup, apparatus, test procedures, and results will be presented.

73

4.3.1. Test Apparatus

Figure 4.3 shows the configuration of uniaxial tensile test conducted in this study. The specimen

is held by two clamps tightly, for which only a small section around the neck is left for breakup.

The ends of clamps are fixed by steel wire ropes to make sure that the tensile forces will exert

along the centre line of clamps vertically. The upper wire rope is linked to the load cell, while the

lower one encircled to a steel rod is fixed on the base. The base on the bottom is much heavier

than the force needed to break the specimen so that the lower clamp goes down steadily at a

designated rate.

A spring gauge is installed on one side of the clamps, recording the displacement near the neck.

The spring gauge consists of two parts: a metal strip that is able to deform largely within its elastic

range, and a strain gauge that can read the deformation. The strain gauge is mounted at the centre

of the metal strip. When the metal strip deforms, it forms a curvature which corresponds to a strain

at the position of the strain gauge. Thus, it is possible to figure out a relation between the

deformation of metal strip and the reading of strain gauge. As long as the strains are recorded, the

deformations shall be known using calibration curves. Details about the spring gauge and its

calibration are described in Appendix C.

4.3.2. Test Procedure

The procedure is similar to the SNDB tests. The testing program for two clays is the same as that

of fracture toughness test as listed in Table 3.2.

74

1) Specimen preparation: Calculated amount of soils, prepared to the required moisture content in

advance, is compacted in a metal mold by a small hammer. The interior side of mold must be

greased with vacuum lubricant before compaction, which will allow specimen removal easily from

the mold after compaction. The specimen is compacted in two layers with equal mass of soils. A

typical compacted specimen is shown in Figure 4.4(a). Compacted specimens are wrapped by

plastic sheet and then sealed in zip-lock bags (Figure 4.4(b)) for at least one day waiting for the

formation of cohesion and dissipation of excess pore water pressure.

2) Specimen installation: The specimen is carefully mounted on two clamps with the installation

of spring gauge. The initial length of the curvature secant of the spring gauge is recorded for

calibrations. Vertical alignment among the load cell, the specimen, the loading wire ropes, and the

clamps needs to be checked prior to loading.

3) Loading and data acquisition: The loading rate is set to 0.2 mm/min so as to complete each test

in 15 minutes without significant moisture loss. The readings of load cell and spring gauge are

recorded by a data acquisition system at a frequency of 10 HZ.

4) Moisture content check: After the test, the entire tested specimen is oven-dried to check the

post-test moisture content.

75

5) Data processing: The reading of spring gauge is converted to displacements based on calibration

curves. The tensile strength of material is calculated through the equation:

AWWPσt

21max −−= Equation 4.1

where W1-(the weight of upper half of the specimen) = 0.5·ρd × (1 + w) × Vmold, Vmold = 61.1 cm3;

W2-(the weight of upper clamp) = 108.55/1000 kg × 9.8 N/kg = 1.064 N;

Pmax-(the peak force of the load-displacement curve);

A-(cross-section area of the neck of specimens) = 25.16 mm × 25.55 mm.

4.3.3. Test Results

For Calgary till, compacted specimens were prepared at two dry densities of 1.75 g/cm3 and 1.85

g/cm3, with varying moisture contents of 11%, 13%, 15%, 17%, and 19.5%. For Regina clay

specimens were compacted at a given compaction energy, with moisture contents of w = 19.5%,

22%, 25%, 28.5%, and 31%. The test of each moisture content was repeated 4 times as a group.

The group of 4 specimens with same moisture content was used to check the moisture loss after

the tests. As long as the average moisture loss of a group is less than 0.5%, the test group is thought

valid. Because the dog-bone specimen is small, it is hard to precisely control the moisture content

of an individual specimen at compaction and the moisture loss during the test.

Experimental results are summarized in Tables 4.1 and 4.2 for compacted Calgary till and Regina

clay, respectively. The data of each test group with same moisture content fluctuated more

significantly than the test of fracture toughness. This is possibly because it is difficult to ensure

76

the uniformity of compaction for the dog-bone specimens so that the soil density around a short

section of the specimen neck is not consistent for each test. The average tensile strength of each

test group will be used as the representative value when considering the variation trend of tensile

strength.

The load-displacement curves labeled by the numbers of tests are attached in Appendix B. Typical

load-displacement curves of different moisture contents were selected and drawn in Figure 4.5 for

comparison. (Due to the extreme weakness of w = 19.5% specimens of Calgary till, it is hard to

install the spring gauge without breaking the specimens, so the load-displacement curves were not

recorded). From the curves of Calgary till specimens in Figures 4.5(a) and 4.5(b), it could be seen

that low moisture content resulted in high failure load and large modulus. From Figure 4.5(c), the

curves of Regina clay with moisture content higher than 22% behaved similarly with those of

Calgary till, but both the modulus and failure load of the curve with w = 19.5% are significantly

smaller than the case of w = 22%. These characteristics of load-displacement curves are in line

with those of KIc tests. For all of the curves for two compacted clays, drops of loads, or strength,

after the peaks with strain elongation were observed, which is defined as post-peak softening. This

post-peaking softening behaviour is noticeable in the curves of tensile tests, and higher moisture

contents result in more significant post-peak softening behaviour. For instance, Calgary till

specimen with w = 17% has up to 0.36 mm post-peak elongation at the specimen neck but the

specimen with w = 11% only has less than 0.04 mm. This is due to the increase of ductility with

increasing moisture content.

77

The variations of tensile strength for two clays are shown in Figure 4.6. For Calgary till, Figure

4.6 (a) shows that the tensile strength increases with decreasing moisture content. Meanwhile, two

trends for the specimens with two different dry densities appear when plotted against moisture

content. Once the moisture content is converted to the degree of saturation, two clear trends

without overlapping area are illustrated in Figure 4.6 (b). For Regina clay, peak values appear at

w = 22% in Figure 4.6(c) when plotted against moisture content and at Sr = 0.683 in Figure 4.6(d)

when plotted against the degree of saturation, as the moisture content increases from 19.5% to

31%. Interestingly, the variation tendencies of tensile strength of two compacted clays are quite

similar with those of KIc. It is reasonable to quantify that there should be a positive correlation

between these two parameters.

In Figure 4.7, the tensile strength of two compacted clays was plotted against the average measured

matric suction of specimens presented in Chapter 2. For partially saturated soils, it was commonly

thought that matric suction would govern the mechanical properties. The tensile strength of

Calgary till did increase with increasing matric suction; however, the tensile strength of Regina

clay behaved differently—an increase of matric suction from 1396 kPa to 2811 kPa did not lead

to an increase of tensile strength but a dramatic decrease. Besides, the magnitudes of matric suction

are much higher than the tensile strength of compacted clays. Thus, the variations and the

magnitudes of tensile strength can be hardly explained by matric suction. From the author’s point

of view, the matric suction is probably not the only factor that affects the strength of compacted

clays, and the complexity of the structure of clays should also be considered as a contributory

factor.

78

4.4. Summary

Uniaxial tensile tests were carried out on dog-bone shaped specimens to determine the tensile

strength of two clays. Specimens were compacted to the designated densities over a certain

moisture range following the same testing program of fracture toughness test. The results showed

that the variation trends with respect to dry densities and moisture contents of tensile strength are

very similar to those of mode I fracture toughness. Positive correlations between these two

properties could possibly be found. Matric suction is probably not the only factor that governs the

strength of clay. The structure of clay is supposed to be taken into consideration for the strength.

79

Table 4.1 Detailed results of uniaxial direct tensile test for Calgary till

ρd (g/cm3)

wpre (%) Sr Test # SL

(mm) Pmax (N)

T (N)

wafter (%)

σt

(kPa)

σt

(average) (kPa)

wafter

(average) (%)

1.75

19.5 0.938

1 -- 8.03 6.33 18.73 9.85

9.84 19.32 2 -- 7.55 5.85 19.79 9.10 3 -- 5.71 4.01 19.60 6.24 4 -- 10.80 9.10 19.16 14.16

17 0.818

5 94.97 9.71 8.02 16.74 12.48

13.32 16.72 6 95.16 18.36 16.67 16.50 25.94 7 94.89 7.55 5.86 16.77 9.12 8 94.88 5.39 3.70 16.86 5.76

15 0.722

9 95.00 24.85 23.17 14.20 36.05

27.63 14.53 10 95.13 15.12 13.44 14.59 20.91 11 94.37 17.28 15.60 14.53 24.27 12 93.94 20.51 18.83 14.80 29.30

13 0.626

13 94.33 22.68 21.01 12.50 32.69

33.53 12.62 14 94.47 15.12 13.45 11.98 20.93 15 94.10 36.72 35.05 13.08 54.53 16 94.05 18.36 16.69 12.92 25.97

11 0.529

17 94.63 22.68 21.03 9.91 32.71

42.79 10.54 18 94.80 33.48 31.83 10.54 49.51 19 93.84 25.92 24.27 10.54 37.75 20 94.05 34.57 32.92 11.16 51.20

1.85

17 0.961

21 95.95 15.12 13.40 16.24 20.84

15.90 16.57 22 95.34 8.64 6.92 16.61 10.76 23 94.44 17.28 15.56 16.35 24.20 24 94.04 6.74 5.02 17.08 7.81

15 0.848

25 95.09 22.68 20.97 15.06 32.62

47.32 14.51 26 94.80 39.96 38.25 13.86 59.50 27 95.60 41.04 39.33 14.65 61.18 28 94.48 24.83 23.12 14.45 35.96

13 0.735

29 94.47 35.64 33.94 12.37 52.80

57.42 12.52 30 94.64 51.85 50.15 11.74 78.01 31 94.06 27.00 25.30 13.19 39.36 32 93.96 39.96 38.26 12.76 59.52

11 0.622

33 94.24 54.00 52.31 10.07 81.38

91.81 10.60 34 94.67 61.56 59.87 10.78 93.14 35 94.71 66.79 65.10 10.56 101.27 36 94.69 60.48 58.79 11.01 91.46

80

Table 4.2 Detailed results of uniaxial direct tensile test for Regina clay

ρd (g/cm3)

wpre (%) Sr Test # SL

(mm) Pmax (N)

T (N)

wafter (%)

σt

(kPa)

σt

(average) (kPa)

wafter

(average) (%)

1.45

31

0.922

1 94.69 20.81 19.18 29.65 29.83

23.19 30.86 2 94.80 20.51 18.88 29.87 29.37 3 95.24 10.80 9.17 32.32 14.26 4 95.38 14.03 12.40 31.59 19.29

1.48

28.5

0.884

5 95.49 27.00 25.37 28.65 39.46

31.10 28.51 6 94.71 20.62 18.99 28.03 29.54 7 94.70 22.68 21.05 28.00 32.74 8 95.74 16.19 14.56 29.35 22.64

1.50

25

0.798

9 95.11 23.75 22.12 24.47 34.42

41.20 24.52 10 94.63 21.58 19.95 25.89 31.04 11 95.10 26.07 24.44 23.30 38.03 12 94.92 41.04 39.41 24.40 61.31

1.48

22

0.683

13 95.04 60.48 58.88 21.20 91.59

71.36 21.50 14 94.78 44.12 42.52 22.01 66.14 15 94.27 37.80 36.20 21.71 56.31 16 93.64 47.52 45.92 21.07 71.43

1.45 19.5 0.580

17 94.72 27.03 25.45 19.41 39.59

27.83 19.07 18 95.69 21.60 20.02 18.76 31.14 19 94.31 14.13 12.55 19.43 19.52 20 94.51 15.12 13.54 19.01 21.06

81

Figure 4.1 Direct methods of tensile tests for compacted clays (after Zhang et al. 2015)

82

a)

b)

c)

Figure 4.2 Indirect methods of tensile tests for compacted clays: a) four-point beam bending, b)

Brazilian disk test, and c) unconfined penetration test

83

Figure 4.3 Configuration of direct tensile test

(a) (b)

Figure 4.4 Dog-bone specimen (a) compacted in the mold, and (b) sealed in bags

84

(a) Calgary till, ρd = 1.85 g/cm3

(b) Calgary till, ρd = 1.75 g/cm3

(c) Regina clay

Figure 4.5 Typical load-displacement curves of uniaxial tensile tests of (a) and (b) Calgary till,

and (c) Regina clay for varying dry densities and moisture contents

85

(a) (c)

(b) (d)

Calgary till Regina clay

Figure 4.6 Variations of tensile strength with respect to moisture content and the degree of

saturation for (a) & (b) Calgary till and (c) & (d) Regina clay

86

(a)

(b)

Figure 4.7 Tensile strength versus matric suction for (a) Calgary till and (b) Regina clay

87

Characteristics of Fracture Toughness and Tensile Strength

5.1. Introduction

In Chapters 3 and 4, mode I fracture toughness and tensile strength have been determined for two

types of compacted clays. The results show that the variations of these two properties with respect

to dry density and moisture content are very similar, so there is possibly a positive correlation

between them. In this Chapter, considering the similar variations of two properties, a correlation

between them will be attempted. Previous similar correlations carried out by other researchers will

be presented as well.

In addition, for Calgary till both fracture toughness and tensile strength increase with decreasing

moisture content for a given dry density, and higher dry density contributes to stronger fracture

resistance and tensile strength. At the same time, for Regina clay, peak values of fracture toughness

and tensile strength occur when moisture content declines from the wet side of optimum to the dry

side of optimum, even if the matric suction keeps increasing with decreasing moisture content.

The influence of matric suction on the strength of clay is still uncertain. Therefore, the variation

characteristics of tensile strength with respect to moisture content are going to be tentatively curve

fitted by a double-porosity concept equation for unsaturated clays, taking the effect of matric

suction into account.

5.2. Correlation between KIc and σt

5.2.1. Rocks

Gunsallus and Kulhawy (1984) developed the correlations between fracture toughness and tensile

strength to curve fit the data (drawn in Figure 5.1) of three types of rocks using linear equation:

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KIc = 0.0736 σt + 0.76; R2 = 0.5223 Equation 5.1

However, the coefficient of determination R2 of this linear regression equation is not satisfactorily

high, and the correlation is unrealistic since for materials of σt = 0, the KIc should be zero as well

(Zhang 2002). In other words, the true correlation function is supposed to have a zero intercept.

Zhang (1998, 2002) conducted a regression between the KIc and σt of several types of materials

including basalt, coal, dolostone, granite, limestone, marble, sandstone, oil shale, siltstone, syenite,

and tuff, using his own test results and the data collected from others. A linear relation was found:

94.0;1453.0 2 == RσK tIc Equation 5.2

From above, linear equations can reasonably describe the relationship between the two parameters.

However, this linear relationship is not perfectly suitable for all types of materials. Some others

found the performance of power-law correlation was better than the linear under certain

circumstances. From the data of Zhang (1998) of marble, gabbro and granite plotted in Figure 5.3,

both linear (Equation 5.3) and power-law (Equation 5.4) equations are sufficiently reasonable for

the curve fitting, but the power-law equation does have a higher coefficient of determination R2 =

0.9257 than the linear one. Haberfield and Johnston (1989) also found the power law is ideally

suitable for a variety of rocks (Figure 5.4 and Equation 5.5).

1590.0;1278.0 2 == RσK tIc Equation 5.3

9257.0;0391.0 24667.1 == RσK tIc Equation 5.4

962.0;208.3 2909.0 == RσK tIc Equation 5.5

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Thus, it could be seen that for rocks both linear and power-law equations could fit the data very

well, but the power-law correlation may perform better for certain materials from the perspective

of coefficient of determination.

5.2.2. Soils

Similar correlations were also applicable for soils. Harison et al. (1994) derived a linear correlation

for air-dried compacted clays based on ring test as defined by Equation 5.6. Wang et al. (2007)

provided their results from SNDB and uniaxial tensile tests of compacted clay using Equation 5.7.

Amarasiri et al. (2011) found a perfect linear relationship (Equation 5.8) for compacted clay

specimens that were air-dried to the designated moisture contents. Agaiby (2013) tested strong

oven-dried clays by FBD, of which the data were fitted by Equation 5.9.

93.0;0706.0 2 == RσK tIc Equation 5.6

88.0;3546.0 2 == RσK tIc Equation 5.7

990.0;1700.0 2 == RσK tIc Equation 5.8

9227.0;1434.0 2 == RσK tIc

Equation 5.9

5.2.3. Calgary Till and Regina Clay in This Study

To examine the relationship between the fracture toughness and tensile strength of the compacted

Calgary till and Regina clay in this study, the average values of each group of KIc and σt are used

for the correlation. When the data points of two types of clays are curve fitted separately, the linear

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slopes are 0.3155 and 0.3226 for Calgary till and Regina clay respectively, whilst the coefficient

of determination R2 are 0.8957 and 0.3542 (Figure 5.5). R2 of Regina clay is small which is

considered as a result of limited data sample. The slopes of two clays are so close to each other

that they can be deemed equal. As the data points of two clays are located nearly within a narrow

band, it is reasonable to curve fit all of these points together. The result of correlation is:

8262.0;3179.0 2 == RσK tIc Equation 5.10

For the correlation of compacted clays, only Wang et al. (2007) and Lakshmikantha (2009)

attempted before. Their data are plotted in Figure 5.6 along with this study. Interestingly, all the

data of compacted clays lie on a narrow band, so a trial linear regression for these data could be:

9038.0;3504.0 2 == RσK tIc Equation 5.11

From above, the linear correlation shows a reasonably high coefficient of determination for

compacted clays. At the same time, power-law regression shows even higher R2 as Equations 5.12

and 5.13 for the data points in Figures 5.5 and 5.6, respectively.

9239.0;7400.0 27919.0 == RσK tIc Equation 5.12

967.0;5831.0 28777.0 == RσK tIc Equation 5.13

It seems the power-law correlation performs better for compacted clays than the linear regression.

This is due to the number of curve fitting parameters—the linear equation only has one parameter,

while the power-law function has two.

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The correlations between KIc and σt of all categories of soils that have been tested so far are

presented in log-log scale in Figure 5.7, where several data trends appear. Harison et al. (1994)

compacted ring test specimens first and then air dried them to designated moisture content by

controlling the weight loss from evaporation, and Amarasiri et al. (2011) used the same method to

determine KIc from compacted SENB specimens and then obtained σt from cohesive crack models

that were calibrated by KIc experiment. Their data bands almost lie in parallel as seen from Figure

5.7. Agaiby (2013) used FBD to test oven-dried specimens of four types of clay, of which the data

are located in between the air-dried compacted clays. Surprisingly, the extension line of the fitted

function for Agaiby’s data goes through the results of frozen loess SENB test carried out by Li &

Zhu (2002). As seen, these data are reasonably confined in several bands, for which the regression

lines are straight. From above, it can be seen that the power-law correlation is appropriate to all

types of soils.

Different data trends indicate that the relationship between KIc and σt possibly depends on the

method of sample preparation. As seen in Figure 5.7, samples compacted directly without air-

drying lie around the black-dashed line; samples compacted and then air dried to the designated

moisture contents are located in two parallel belts (green and brown lines); strong oven-dried and

frozen soils are within another band. In author’s opinion, the method of sample preparation may

change the mechanical properties of soils. For clays, directly compacted specimen and air-dried

specimen with same moisture content may have different soils structure because the initial

moulding moisture content has considerable influence on the resulting structure of clays, which

results in different matric suction (Vanapalli et al., 1999). For oven-dried clays, the material

becomes extremely strong due to the loss of water. For frozen soils, the strength will be

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significantly impacted by temperature and moisture content because the strength and portion of

ice in the frozen soil are the dominant factors to the strength of specimen (Li & Yang, 2000).

5.2.4. Descriptions for the Size of Fracture Process Zone (FPZ)

In Chapter 3, the applicability of LEFM was discussed considering the size of Fracture Process

Zone (FPZ) or plastic zone around the crack tip. Equation 3.17 indicates that the radius of the

assumed circular FPZ is proportional to KIc2/σys

2. Only when the size of FPZ is sufficiently small

compared with the size of specimen, LEFM is deemed applicable, and the estimated fracture

toughness is valid.

Nonetheless, the criterion of the applicability has not been established for any types of soil. In this

case, the suggested criterion for metals and rocks will be referred to. According to ASTM E399

for metals, the thickness, the crack length, and the ligament length of SENB specimens must be

larger than 2.5 (KIc/σys) 2. But, for geomaterials, the yield strength could be hardly defined on load-

displacement curves. Thus, the tensile strength, which is supposed to be equal or larger than the

yield strength, is used instead. The ISRM-suggested method of SCB recommends a size

requirement of D ≥ 2.0 (KIc/σt) 2 for rocks.

From Equation 5.10, (KIc/σt) 2 of the two clays in this study could be calculated using the linear

relationship of the two parameters. The value is approximately equal to 101.06 mm, which is much

larger than the thickness of the SNDB specimen already. Similar magnitudes of (KIc/σt) 2 for clays

were also found by others—around 125 mm from SENB specimens of Wang et al. (2007), 20-500

mm from CT samples of Lakshmikanatha et al. (2008), and 26.8-45 mm from SENB clay beams

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of Amarasiri et al. (2011). Nevertheless, the above-mentioned values of (KIc/σt) 2 are unreasonably

large compared with the size of specimens. Therefore, a use of (KIc/σt) 2 to represent the size of

FPZ is questionable.

Considering uncertain expression of FPZ size with (KIc/σt) 2 for soils, Amarasiri et al. (2011)

attempted to figure out the length of FPZ with cohesive crack model using distinct element

program UDEC. They obtained a range of FPZ length from 4 ~ 7.5 mm for SENB clay samples in

dimensions of W = B = 30 mm and a = 10 mm. The sizes of FPZ in their numerical model are

reasonably smaller than the size of specimen and the values of (KIc/σt) 2. Amarasiri et al. (2011)

claimed that the standards for other materials may be too restrictive for soil testing.

Because the standard for testing LEFM KIc of soils has not been established, the size requirement

of specimen is uncertain. To make sure the FPZ is much smaller than the size of specimen, a size

requirement must be examined by employing a series of specimens with different sizes, from small

to large, among which the smallest specimen that generates KIc values consistent with larger

specimens will be the minimum size required for a valid test (Kuruppu et al., 2013).

5.3. Curve-fitting for Variation Trends of Tensile Strength

For unsaturated soils, matric suction is believed to be dependent on the moisture content or degree

of saturation, which undoubtedly plays an important role on the mechanical properties of soils.

But, how it influences the strength of soils is still uncertain. That could be noticed from the results

of this study.

94

In Chapter 2, the matric suction of the two compacted clays presented in Figures 2.4 and 2.5

clearly show that the matric suction increases exponentially when the moisture content drops. This

changing behaviour is also observed from the fracture toughness and tensile strength of Calgary

till, as seen from Figures 3.27(a) and 4.7(a) where the variation tendencies of KIc and σt with respect

to dry density and moisture content are almost identical. However, that was not observed from the

results of Regina clay in Figures 3.27(c) and 4.7(c). Instead, well-defined peaks were observed

over the testing moisture range of Regina clay. In this case, although suction keeps increasing with

decreasing moisture content, the values of fracture toughness and tensile strength still descend

dramatically after they reach the peaks. Same observations were also found in other researches

(e.g. Tamrakar et al, 2005; Lakshmikantha, 2009; Tang et al, 2015).

Moreover, the magnitudes of matric suction are much larger than the tensile strength of the

materials, as seen in Figure 4.7. The tensile strength of compacted soils in this study is roughly

within a range of 5 kPa ~ 100 kPa. By contrast, the matric suction roughly ranges from 8 kPa to

3000 kPa. For high moisture contents, the magnitudes of matric suction are close to the tensile

strength, but for low moisture content, the difference could be as large as 30 times.

If matric suction is the only factor that governs the mechanical properties of compacted clays, the

difference between the matric suction and the strength of material above can hardly be explained.

Thus, to quantify the effect of matric suction on the strength of soils, several postulates were

proposed in the past decades, which will be presented in the next section.

95

5.3.1. Theories for Strength of Unsaturated Soils

Bishop (1959) was the first one who proposed an expression for the shear strength τd of unsaturated

soil, which takes matric suction into account:

[ ] 'tan)()(' φuuχuσcτ waand −+−+= Equation 5.14

where c’ is the effective stress cohesion; σn is the total stress in soil; φ’ is the angle of shearing

resistance; (ua - uw) is the matric suction; χ is a parameter between 0 and 1 that defines the

contribution of matric suction to the strength.

Later on, some other researchers proposed similar expressions, where the parameter χ is expressed

in comparable ways. Fredlund and Morgenstren (1977) defined an additional frictional angle φb

that is related to matric suction (Equation 5.15, Table 5.1). Vanapalli et al. (1996) introduced

normalized volumetric water content Θ (equivalent to the degree of saturation) and a fitting

parameter κ (Equation 5.16, Table 5.1). Lu and Likos (2010) defined an effective saturation Sre,

which is the ratio of free water volume to the total available voids (Equation 5.17, Table 5.1). All

the works above were intended to figure out the contribution of matric suction to the shear strength.

However, from their studies, χ and φb are non-unique parameters that vary with the degree of

saturation. Although Θ and Sre take the degree of saturation as a variable, the physical concepts

behind them are still unclear. Hence the application of these expressions is quite limited.

96

Table 5.1 Expressions for shear strength of unsaturated soils

Expression Comparable χ

bwaand φuuφuσcτ tan)('tan)(' −+−+= 'tan/tan φφb Equation 5.15

[ ] 'tan)()(' φuuΘuσcτ waκ

and −+−+= κΘ Equation 5.16

[ ] 'tan)()(' φuuSuσcτ waerand −+−+=

residual

residualrer S

SSS−−

=1

Equation 5.17

Recently, Alonso et al. (2010) attempted to suggest a double-porosity concept that includes

macropores formed by aggregates and micropores inside the aggregates, based on the observation

of Scanning Electron Microscope (SEM) technique on the structure of compacted clays from

Romero (1999). From Figure 5.8 of SEM pictures, aggregates and inter-aggregate pores are clearly

seen, which indicates that the compacted clay is not formed directly by clay particles but by

aggregates that consist of micro clay particles. Within aggregates, spaces between clay particles

are defined as intra-aggregate pores. Meanwhile, the space between aggregates are defined as inter-

aggregate pores. Since the inter-aggregate pores are larger than intra-aggregate pores, they are also

called macropores and micropores, respectively. Both macropores and micropores are supposed to

contain water.

For describing the occupation of water in the micropores and macropores, a macroscopic degree

of saturation SrM and a microscopic degree of saturation Sr

m were proposed. The combination of

the two is the total degree of saturation of soil:

mr

Mrr SSS += Equation 5.18

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Water held in micropores is predominantly controlled by particle surface forces and attached to

the solid phase by physico-chemical bonds (Alonso et al. 2010). Water trapped in macropores is a

combination of adsorbed water and capillary water (illustrated as Figure 5.9, Baker & Sam, 2009).

The adsorbed water in macropores are usually immobile and does not contribute to the capillary

actions, which thus should be included in the microscopic degree of saturation Srm. Only the free

water causing capillary effect is included in SrM. They also indicated that it is the capillary water

in the macropores that governs the macro mechanical behaviour of soils because the water

adsorbed in micropores makes the aggregates themselves too strong to break prior to the failure of

macro connection between aggregates. Hence, an effective degree of saturation Sre is defined by

Equation 5.19 (Piecewise function), of which the definition is a measure of free water filling the

macroporosity. Sre extends from 0, when all the water is stored and adsorbed in aggregates, to 1,

when fully saturated. Srm is approximated to be constant by Alonso et al. (2010). The variation of

Sre with respect to Sr and Sr

m is shown in Figure 5.10 (a).

Piecewise: mr

mr

rmr

mr

mrre

r SSS

SSSSS

−−

−

=−−

=11

11

Equation 5.19

Continuous: ')( αr

er SS = Equation 5.20

But as seen in Figure 5.10 (a), the effective degree of saturation reaches zero when the degree of

saturation is still non-zero, and the function is piecewise. Alonso et al. (2010) approximated the

Equation 5.19 by a continuous expression of Equation 5.20 where α’ is a fitting parameter not less

than 1.0.

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Interestingly, the expressions given by Alonso et al. (2010) are quite similar to Equations 5.16

and 5.17. The curve-fitting results in Alonso et al. (2010)’s study for shear strength of compacted

clays revealed that the predictive capabilities of both piecewise and continuous functions did not

show superiority over each other. Next, curve fittings for the tensile strength of compacted Calgary

till and Regina clay will be presented to examine the predictabilities of this double-porosity

concept proposed by Alonso et al. (2010).

5.3.2. Curve-fitting for Tensile Strength

The tensile strength data of Regina clay will be curve fitted first using double porosity concept.

Firstly, the matric suction of Regina clay has already been curve fitted by van Genuchten (1980)’s

equation in Chapter 2 as:

[ ] [ ]{ } 194.0241.1194.0241.1 )(003.01)003.0(1−−

−+=+= war uuΨS Equation 5.21

where Ψ = (ua - uw)

For Regina clay, when it is saturated, the effective stress cohesion c’ does not exist (evidence was

provided by Wong et al., 2017: specimen soaked in water eventually broke at the location where

the surface was not covered by waterproof material). So the apparent cohesion of shear failure cs

could be expressed as Equation 5.22 in terms of piecewise effective degree of saturation:

'tan)(1

'tan)( φuuSSSφuuSc wamr

mrr

waers −

−−

=−= Equation 5.22

99

Given the matric suction expression of van Genuchten’s Equation 5.21, the shear cohesion is

expressed in terms of matric suction and Srm:

{ } 'tan)(1

)](003.0[1 194.0241.1

φuuS

Suuc wamr

mrwa

s −−

−−+=

−

Equation 5.23

Apparent shear cohesions and frictional angles measured by Wong et al. (2017) along with matric

suction will be used for calibrating the parameter Srm in Equation 5.23. φ’ = 23 º was observed in

their study which is believed to be an intrinsic property of compacted Regina clay. The apparent

cohesions were extracted from the results of their direct shear tests and curve-fitted using Srm =

0.56 as shown in Figure 5.11. With the calibrated expression of shear cohesion, tensile strength

can be fitted following a transitional behaviour from tensile to shear mode postulated by Wong et

al. (2017).

Figure 5.12 illustrates the transitional behaviour: two distinctive Mohr-Column envelopes with

different frictional angles φ’ and φt can be seen, where the intercepts of shear failure envelope and

tensile failure envelope represent the shear cohesion cs and tensile cohesion ct, respectively.

Apparently, ct is smaller than cs, so it is reasonable to assume that there is a link between the two

cohesions as represented by Equation 5.24 (the meaning of the factor ν will be discussed later):

)0.1( ≤= νcνc st Equation 5.24

)sin1(tansin2

)sin1(tansin2

tt

ts

tt

tttt φφ

φcνφφ

φccσ+

=+

= Equation 5.25

100

)sin1(tansin2'tan003.0/)1(

1241.11

194.01

tt

trm

r

mrr

t φφφφS

SSSνσ

+⋅

−

−−

⋅=−

Equation 5.26

With Equation 5.24, the direct tensile strength can be expressed in terms of shear cohesion and

tensile frictional angle by Equation 5.25 according to the geometric relations of the lines in Figure

5.12. After substituting the matric suction (ua - uw) with the degree of saturation (Sr) using Equation

5.21, Equation 5.25 can be transformed to Equation 5.26 where the direct tensile strength is simply

expressed by the degree of saturation Sr, curve-fitting parameters ν and Srm, and material properties

φt and φ’.

Now, φ’ = 23º, and Srm = 0.56 used for the curve fitting of the shear cohesion data of Wong et al.

(2017) will be applied to tensile strength. According to Wong et al. (2017), φt lies in a range within

42 - 46º. An average of φt = 44º is selected, because the variation of tensile frictional angle does

not make much difference within such a small range, based on the author’s experience. Fitted

curves are presented in Figure 5.13. If ν = 1.0, namely ct = cs, the purple double dot dash line shows

that the variation of data is fitted very well, though the predicted results are significantly larger

than the data. However, if ν = 0.5, the green dot dash line shows a better match with the tensile

strength obtained from the uniaxial tensile tests. In the author’s opinion, the factor ν is considered

to be affected by the mode of failure and loading rate, since tensile failure only occurs in a very

small zone, where strains develop quickly, whilst shear failure develops across the entire shear

plane of which the strains grow relatively slowly.

101

Curve fitting by the continuous function of Equation 5.20 was made as well, as shown in Figure

5.14, with α’ = 3, 4, 5, 6, and 7, and with ν = 0.5, φ’ = 23º and φt = 44º. Nonetheless, none of them

can fit the trend of data well.

( ))sin1(tan

sin2'tan003.0/)1(

241.11

194.01'

tt

tr

αrt φφ

φφSSνσ

+⋅

−⋅=

−

Equation 5.26

Curve fitting for Calgary till was also conducted in this study. Since basic properties of shear and

tensile frictional angles were not measured, the curves were fitted by assuming φ’ = 29º, φt = 50º,

Srm = 0.45, and ν = 0.4 using piecewise function. The trends of variation are reasonably fitted as

seen in Figure 5.15.

Therefore, from the curve fitting results of tensile strength for Regina clay, it could be concluded

that the piecewise function of effective degree of saturation Sre would perform better than the

continuous function. The data of Calgary till was reasonably fitted by the function as well. The

results imply that the microscopic water adsorbed by aggregates could probably be constant. Only

the free water in macro pores that induces capillary effect varies with a change degree of saturation

and eventually makes an impact on the mechanical properties of compacted clays.

In addition, due to the positive correlation between the tensile strength and fracture toughness, it

is possible to use this curve-fitting strategy to predict the variation of fracture toughness using the

expression of tensile strength.

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5.4. Summary

In this Chapter, fracture toughness and tensile strength have been correlated. The relationship

between the fracture toughness and tensile strength could be linear or power law. The data of

compacted Calgary till and Regina clay, as well as other soils investigated in other studies, are

well described by the power law equation. The methodology of soil specimen preparation is

probably a factor that influences the relation between KIc and σt.

The ratio of (KIc/σt)2, indicating the size of FPZ, was obtained from linear correlation, but it is even

larger than the size of specimen and it does not satisfy the existing size requirement for metals and

rocks. Whether the ratio of (KIc/σt)2 can be used to describe the size of FPZ for soils is still

uncertain, because Amarasiri et al. (2011) showed the sizes of FPZ in their numerical model are

much smaller than the values of (KIc/σt)2 . To investigate the size effect of FPZ for soils, further

research is required.

A double-porosity concept proposed by Alonso et al. (2010) successfully curve fitted the tensile

strength of two clays. The piecewise function of the effective degree of saturation Sre yields a better

prediction than the continuous one in predicting the variation of tensile strength, with which the

variation trends of fracture toughness could also be predicted using its relationship with tensile

strength.

103

Figure 5.1 KIc versus σt relationship of dolostone, limestone and sandstone

(after Gunsallus and Kulhawy, 1984)

Figure 5.2 KIc versus σt relationship of rocks (after Zhang, 2002)

104

Figure 5.3 KIc versus σt relationship of marble, gabbro and granite

(after Zhang et al., 1998)

Figure 5.4 KIc and σt correlation for rocks (after Haberfield and Johnston, 1989)

105

Figure 5.5 Linear correlations between KIc and σt for Calgary till and Regina clay

Figure 5.6 Linear regression of KIc and σt for compacted clays (including data from Wang et al.,

2007 and Lakshmikantha, 2009)

106

Figure 5.7 Power-law correlation between KIc and σt for different soils prepared from different

methods

107

w = 15%, γd = 16.7 kN/m3

(a)

w = 15%, γd = 13.7 kN/m3

(b)

Figure 5.8 SEM images of compacted clays for different soil packings. Pores are black and

particles are white for lower graphs. (after Romero, 1999)

108

Figure 5.9 Adsorbed water and capillary free water in micropores and macropores

(after Hueckel et al., 2001; Baker and Frydman, 2009)

109

(a)

(b)

(c)

Figure 5.10 Effective degree of saturation expressed by (a) piecewise function, (b) continuous

function, and (c) piecewise and continuous functions for comparison

110

Figure 5.11 Curve-fitted shear cohesions measured by Wong et al. (2017) using piecewise

function

Figure 5.12 Tensile-shear failure envelope for compacted soil (after Wong et al., 2017)

111

Figure 5.13 Curve-fitted σt of Regina clay using piecewise function of Sre (Equation 5.19) with

Srm = 0.56

Figure 5.14 Curve-fitted σt of Regina clay using continuous function of Sre (Equation 5.20) with

ν = 0.5

112

Figure 5.15 Curve-fitted tensile strength of Calgary till using piecewise function of Equation

5.19

113

Adhesion of Compacted Clays

6.1. Introduction

Attention has been paid to the interaction between soils and structures in various engineering

practice. One of the most important properties of the interaction is the ability of soil to adhere to a

structure. When soil adheres to a structure, the relative displacement between the structure and soil

may result in non-negligible stresses. This adhesive property is defined as adhesion.

Adhesion is a critical element in pipeline design, because the interaction between the pipeline and

the surrounding soil that induces loads in the pipeline is often of concern. When a longitudinal

displacement is imposed upon a pipeline, possibly caused by surface faulting or lateral spreading,

axial soil resistance is generated (as illustrated in Figure 6.1). For this axial load transfer between

pipeline and clay, the resulting maximum axial soil resistance per unit length of a fully buried

pipeline is expressed by Equation 6.1 according to the design manual of ASCE (1984):

uau αsπDπDct ⋅== Equation 6.1

where D is the diameter of pipeline; ca is the adhesion of clay to pipeline; su is the average

undrained shear strength of soil; α is the adhesion factor, an empirical coefficient varying with su.

Equation 6.1 was inferred from the alpha α method, a total stress approach, for calculating shaft

resistance qs in pile design, which has been widely and successfully used in engineering practice.

The parameter α in Equation 6.2 for piles buried in clays ranges from 0.5 to 1.0 as recommended

by Canadian Foundation Engineering Manual (CGS, 2006).

us sαq = Equation 6.2

114

However, the original assumption for Equation 6.2 was incorrect. It implied that the mobilized

interface strength of soil is a portion of the undrained shear strength, but the reality is that the

distortion of soil is confined to a thin layer around the pile where drainage occurs quickly under

static loading (Cooke & Price, 1978). Instead, the interface resistance was suggested to be

expressed in terms of effective stress as Equation 6.3 (CGS, 2006), which is conventionally called

beta β method.

's vq βσ= Equation 6.3

where qs is the shaft resistance of pile; σv’ is the vertical effective stress at the depth; β is the shaft

resistance factor, an empirical correlation parameter for qs and σv’.

As a matter of fact, α and β approaches for pile design are equivalent, as suggested by Sladen

(1992). The transformation of equations is shown herein.

The shaft resistance qs can be expressed in terms of the horizontal effective stress of soil σh’ and

the friction angle of the soil-pile interface δ in Equation 6.4.

' tans hq σ δ= Equation 6.4

The horizontal effective stress σh’ is supposed to be related to the vertical effective stress σv

’ as

Equation 6.5.

( ) ''' 'OCR mh nc vσ κK σ= Equation 6.5

115

where Knc is the lateral stress ratio under normal consolidation; σv’ is the vertical effective stress at

the depth; OCR is the overconsolidation ratio; m'' is a curve fitting parameter for exponential

change of lateral stress ratio with respect to OCR; κ is an installation factor.

By substituting Equation 6.5 into Equation 6.4, the shaft resistance qs is now expressed in terms

of the vertical effective stress σv’ according to Equation 6.6. The expression in square brackets is

the beta β.

( ) '' ' 'tan OCR ms nc v vq δ κK σ βσ = ⋅ = Equation 6.6

The vertical effective stress σv’ can be linked with undrained shear strength su by Equation 6.7,

which is called SHANSEP Equation based on Cam Clay model, proposed by Ladd & Foott (1974).

Substituting σv’ from Equation 6.7 into Equation 6.6 yield Equation 6.8. Equation 6.8 has been

adopted to back calculate qs with existing su data by Saye et al. (2012), and the expression in the

bracket can be deemed as the adhesion factor of α approach in Equation 6.2. So α is expressed by

Equation 6.9.

'' '(OCR)nu vs s σ= ⋅ Equation 6.7

''- '' tanOCRm n ncs u u

κK δq s αss

= ⋅ =

Equation 6.8

''- ' ' tanOCRm n ncκK δαs

= ⋅ Equation 6.9

where s and n’’ are curve fitting parameters for the correlation between undrained shear strength su

and vertical effective stress σv’.

116

Substitution of OCR from Equation 6.7 into Equation 6.9 gives Equation 10. From Equation 10, α

is a function of su.

''''''

'

tan nnm

v

unc

σss

sδKκα

−

= Equation 6.10

Therefore, expressing the shaft resistance (or adhesion) of pile in clay with the corresponding

undrained shear strength is reasonable even though the original physical hypothesis behind is

incorrect. There is indeed a reasonable correlation between the shaft resistance (or adhesion) and

the undrained shear strength. The expression of α method has been vastly used for piles due to its

simplicity and large existing empirical data base.

For a similar cylindrical buried structure, the axial resistance of pipeline was reasonably assumed

according to Equation 6.1. However, so far, only a limited number of tests have been conducted

for determining the empirical adhesion factors for pipelines. ASCE (1984) recommended a range

from 0.5 to 1.5 for the imposed displacement problems related to pipelines, where the adhesion

factor α decreases with increasing undrained shear strength su. Later, Rizkalla et al. (1996) obtained

a few field data which are much smaller than the ASCE-recommended values. From the laboratory

investigation conducted by Paulin et al. (1998), even lower adhesion factors were found (data

present in Figure 6.12). Since the adhesion factor is an empirical parameter, it would not be

applicable unless a sufficiently large empirical data base is available. For the clays used in this

study, the factors of neither of them have been measured before. The measurement of the empirical

factors will provide references for the pipeline design around local area. However, due to the

compacted samples used in this study were unsaturated, for which the OCR is infinite, the

117

derivation for the correlation between adhesion and undrained shear strength is not reasonable. On

the purpose of establishing a practical way of expressing the adhesion of unsaturated soils, the

adhesion factor in this study will be the ratio of adhesion and half of the unconfined compressive

strength.

For unsaturated soils, adhesion could be expressed by Equation 6.11 where the stress variable (σn

- ua) is associated with external loads and another stress variable (ua - uw) is a direct contribution

to soil strength. In this study, only the influence of (ua - uw) that is independent of external load

condition is of interest.

[ ]( ) ( ) tana n a a wc σ u χ u u δ= − + − ⋅ Equation 6.11

where ca is the adhesion of soil to pipeline; σn is the total stress in soils; δ is the frictional angle of

the soil-pipeline interface.

In the following tests, compacted clays will be used as the in-situ backfill for buried pipes. The

variation of adhesion factor with respect to the half unconfined compressive strength (0.5cUCS)

under different moisture contents will be examined. The unconfined compressive strength (cUCS)

will be determined from unconfined compression (UC) tests. The adhesion is obtained from a

modified direct shear box test (illustration provided in Figure 6.2), where the soil specimen is

firstly compacted in a mold, which is installed on a stainless steel plate, and then is loaded by a

vertical pre-consolidation pressure that helps create a direct contact between the soil and metal.

The vertical pre-consolidation pressure is removed before shearing the soil-metal interface so that

the total stress σn is approximately equal to 0 kPa.

118

Regina clay and Calgary till will be tested with the moisture contents from the dry side of optimum

to the wet side of optimum to obtain the variations of adhesion. Besides, the effect of vertical pre-

consolidation pressure, exerted to the specimen prior to test, on the adhesion will be investigated

using Regina clay specimens at the optimum moisture content.

6.2. Unconfined Compression Test

Soils, with desired moisture contents and dry densities, were compacted in three layers in a

cylindrical standard compaction mold using a standard compaction rammer. The specimen is 101.6

mm in diameter and 116.03 mm in height. For Regina clay, standard compaction energy of 25

blows for each layer was used to prepare specimens with moisture contents of 19.5%, 22%, 25%,

28.5%, and 31%. For Calgary till different energy was applied: 42, 37, 25, and 37 blows were used

respectively for specimens with moisture contents of 11%, 13%, 15%, and 17% to yield a constant

dry density of 1.85 g/cm3. In addition, a specimen of w = 19.5% and ρd = 1.75 g/cm3 compacted

by standard proctor was tested for Calgary till.

The setup of unconfined compression test is shown in Figure 6.3, where the specimen is loaded by

two steel disks. Axial displacements are recorded by LVDTs. The loading rate is set to 0.2

mm/min. Since each specimen fails approximately in 40 - 70 minutes, it is covered by a plastic

sheet during the test to prevent moisture loss (Figure 6.4). Once a decrease of load is observed

from the load-displacement curve, the test is stopped immediately. Moisture contents must be

checked after the tests. The load-displacement curves of unconfined compressive tests are

presented in Figure 6.5. The unconfined compressive strength is calculated from Equation 6.12, as

the cross section of specimen changes under compression. The average cross-section area equals

119

to A0/(1 - ΔL/L) assuming the total volume of specimen does not change during the test (ASTM

D2166). The variation trends of cUCS with respect to moisture content are plotted in Figure 6.6.

)/Δ1/(0

max

LLAFcUCS −

= Equation 6.12

where cUCS = unconfined compressive strength; Fmax = maximum load of the load-displacement

curve; A0 = initial cross-section area of the specimen; ΔL = axial deformation; L = specimen height.

6.3. Modified Direct Shear Test

6.3.1. Test Apparatus, Program, and Procedure

The apparatus for direct shear test for adhesion are modified from conventional direct shear box

test for soils as illustrated in Figure 6.2. The lower half of the direct shear box is replaced by a

stainless steel base, which is machined to the size of the original lower shear box. Threaded holes

were also machined for the bolts that hold the upper box and the steel base together for the

compaction. The upper shear box is a 100 mm-diameter cylindrical chamber with a height 47.5

mm. Required amount of soil is calculated according to the volume of chamber subtracting the

volume of loading cap above the specimen. The specimen was compacted by a small hammer in

one layer.

The test program for determining the effect of moisture content on adhesion under a given pre-

consolidation vertical pressure of 30 kPa is same as that of the UCS test in section 6.2. For

examining the influence of pre-consolidation pressure on adhesion, a series of pre-consolidation

120

vertical pressures 30 kPa, 50 kPa, 100 kPa, and 300 kPa will be tested with Regina clay specimens

compacted at the optimum moisture content w = 25%.

Tests were conducted according to the following steps:

1) The steel base is cleaned with soap and water before the test. The bottom of the upper box is

greased with vacuum grease, which can help seal the gap and prevent moisture loss of the

specimen during the test. Then the steel base and the upper box are tightly held with bolts.

Reconstituted soil now could be compacted in the chamber of the upper shear box using a small

hammer.

2) After compaction, a loading cap is placed on top of the specimen. The assembly of direct shear

box, specimen, and loading cap is moved to the testing machine. The pre-consolidation

pressure is applied by the loading unit on the specimen first. A vertical pressure of 30 kPa is

exerted on the specimen when examining the effects of moisture content, whilst a series of

pressures from 30 kPa to 300 kPa are applied when investigating the influence of pre-

consolidation pressure. The pressure should be kept for at least 24 hours before removal to

ensure a close contact between the soil and steel, and soil consolidation. Then the bolts can be

removed from the assembly after the specimen is loaded by the loading unit.

3) Before starting to shear, the loading unit is dismounted. The shearing rate is 0.02 mm/min to

allow dissipation of pore water pressure, during which the data are recorded every 0.2 minutes

by the computer system. The system can automatically convert the shear load to shear stress

by dividing the area of the specimen cross section.

4) The moisture content of specimen is checked after each test.

121

6.3.2. Test results

A typical stress-displacement curve of compacted Regina clay is shown in Figure 6.7(a). As seen,

the curve goes up first, level off, and increases again until the peak after which it goes down to a

constant value. The curve could be generally divided into three stages: the first stage is called run-

in stage, where the stress gradually goes up and then enters a level section at which the steel base

slips towards left because of an imperfect gap between the shear box and the steel base as

illustrated by Figure 6.8(i). At this stage τ = fbase < (Radhesion + fsoil + fshear box) so there is no movement

between the upper shear box and the steel base. The second stage is the loading stage when the

carriage makes contact with the steel base as shown in Figure 6.8(ii). Now it is the adhesion of soil

that takes the shear stress, where the peak τmax = (Radhesion + fsoil + fshear box)max. At the last stage, after

reaching the peak, the adhesion starts to diminish until a residual stress occurs.

Similar three-stage curves are also seen from the curves of Calgary till at w = 17% & 19.5%.

However, at low moisture contents w = 11%, 13%, and 15%, the level sections of run-in stage

disappear. From Figure 6.9(c), the curve rises to the peak and then declines to the residual value.

This is a result of small adhesion strength. Because the adhesion is so small that the combination

of (Radhesion + fsoil + fshear box) is lower than the friction fbase between the steel base and the carriage,

the steel base will not move during shearing. Under this situation, the test enters the loading stage

where τ = (Radhesion + fsoil + fshear box) at the beginning.

122

The residual stress is a result of adhesion, where τresidual = Radhesion, because the specimen still

contacts the steel base at the stage and the sum of fsoil and fshear box are approximated to zero. It could

be also concluded that τmax is the peak adhesion of the specimen.

The effects of moisture content on the adhesion of compacted clays, under a given pre-

consolidation pressure of 30 kPa, were examined first. Results of peak adhesions are plotted in

Figure 6.10 against moisture content. According to Equations 6.2 and 6.10, the adhesion factor α

is a function of undrained shear strength su of normally or over consolidated soil. For compacted

soil, su is different to define and determine due to the uncertain effect of (ua - uw) and stress history.

The measured peak adhesion in this study was empirically correlated to half unconfined

compressive strength (0.5cUCS) assuming τmax = αc ∙ 0.5cUCS, and plotted in Figure 6.11. From the

graphs, obvious ascending trends of adhesion with increasing moisture content could be seen for

both clays, and meanwhile the adhesion factor decreases with an increase of unconfined

compressive strength. In both graphs, the values of Calgary till are smaller than those of Regina

clay, which reflects a fact that Calgary till is a less adhesive material containing less clayey content

and suction effect. The χ parameters of the specimens with different moisture contents were

calculated according to Equation 6.11. The friction angles at soil-pipe interface used for the

calculation were 2/3 of the soils, namely δ = 15º for Regina clay and δ = 19º for Calgary till. The

matric suctions of specimens were assumed to be equal to the measured suctions in Chapter 2. The

results are listed in Table 6.1.

The load-displacement curves of different pre-consolidation pressures for Regina clay specimens

are drawn in Figure 6.13. Adhesions obtained from the curves are presented in Figure 6.14. The

123

triangular points in Figure 6.14 were obtained from the tests following the procedure specified in

the former section. A dramatic increase of adhesion can be seen from the increasing pre-

consolidation pressures 30 kPa to 100 kPa. However, the adhesion tested at 300 kPa does not

exhibit much difference than that at 100 kPa. The author repeated the test at 300 kPa for the second

time, but the result still did not show substantial discrepancy. In this case, the pre-consolidation

pressure may only have influences on the adhesion of compacted Regina clay with optimum

moisture content within the range of 30 kPa – 100 kPa, whereas for 100 kPa – 300 kPa the increase

of pre-consolidation pressure does not have significant impact on the adhesion. This is probably

due to the contact between the soil and steel base. With an increment of pre-consolidation pressure

from 30 kPa to 100 kPa, the clay aggregates at the soil-metal interface might slightly deform and

create more capillary effects, which consequently increased the χ parameter in Equation 6.11.

Nonetheless, a further increase of pre-consolidation pressure from 100 kPa to 300 kPa was

ineffective to the further deformation of aggregates and to the creation of more capillary effects,

and hence the χ parameter would not increases remarkably.

The red star in Figure 6.14 is the adhesion of a specimen tested under 100 kPa pre-consolidation

pressure, but this specimen was tested at 50 kPa before. As seen, the adhesion of this test is less

than one half of the regular compacted and pre-consolidationed specimen under 100 kPa. The

reason was possibly that after shearing the soil at the soil-metal interface largely deformed where

the aggregates significantly rotated and rearranged. The reloading pressure of 100 kPa may not

produce a close contact between the soil and metal as the directly compacted specimen. Therefore,

the χ parameter of this reloaded specimen could be smaller than the directly compacted one, and

so as the value of adhesion.

124

6.4. Summary

The adhesion of clays to metal can be correlated to the undrained shear strength or expressed in

terms of effective stress. Even though the physical meaning behind the correlation between the

adhesion and undrained shear strength is questionable, it is still a practical method that have been

vastly applied in engineering practice with large data bases for piles. The correlation between

adhesion and undrained shear strength is called α method, which has been justified as an equivalent

method as the effective stress method (β method). For pipelines, the lack of the adhesion factor α

data limits the application of this method.

In this study, because the compacted samples are unsaturated soils, it is unreasonable to correlate

the adhesion with the undrained shear strength. However, in order to establish a practical way of

expressing the adhesion of unsaturated soils, in this study, the adhesion is correlated with half

unconfined compressive strength (0.5cUCS) to obtain the adhesion factors. The variation trends of

adhesions and adhesion factors, excluding the effects of external loads, were investigated using

compacted Calgary till and Regina clay first. It could be seen that the adhesion and adhesion factor

decreased with decreasing moisture content and increasing cUCS respectively under a constant pre-

consolidation pressure of 30 kPa. In addition, the impacts of pre-consolidation pressure were

studied from 30 kPa to 300 kPa with Regina clay with optimum moisture content; the result showed

that an increase of pre-consolidation pressure from 30 kPa to 100 kPa would increase the

magnitudes of adhesion, but would not be effective with a further increment from 100 kPa to 300

kPa.

125

Table 6.1 Unconfined compression and direct soil-steel shear tests results and modified adhesion factors under different moisture

contents for Calgary till and Regina clay at pre-consolidation pressure of 30 kPa

Parameter wpre

(%)

Fmax

(N)

ΔL

(mm)

cUCS

(kPa)

wafter,

UCS

(%)

wafter, shear

(%)

τmax

(kPa)

τresidual

(kPa)

Modified

adhesion factor

αc

χ

Regina

clay

19.5 2273.77 5.57 269.31 19.12 19.22 8.44 3.31 0.0627 0.0112

22 2765.25 9.55 315.74 21.55 21.70 9.85 3.93 0.0624 0.0263

25 1897.87 8.62 218.60 24.49 24.62 9.91 3.85 0.0907 0.0482

28.5 1498.20 7.99 173.57 28.07 28.40 10.54 3.47 0.1214 0.1328

31 764.76 8.72 88.00 30.69 30.55 13.28 5.23 0.3018 0.4340

Calgary

till

19.5 556.29 13.28 61.29 19.44 19.17 8.95 3.34 0.2920 3.3845

17 1432.31 11.83 160.04 16.78 16.53 7.00 3.21 0.0875 0.9531

15 2434.72 10.13 276.49 14.83 14.81 4.80 3.67 0.0347 0.0472

13 3096.57 8.89 355.76 12.51 12.93 3.82 2.62 0.0215 0.0119

11 3430.64 7.93 397.64 10.76 11.01 3.44 2.97 0.0173 0.0055

wpre = the moisture content at preparation; Fmax = the maximum load of UCS test; τresidual = the residual stress of direct shear test

wafter, UCS = the moisture content after UCS test; wafter, shear= moisture content after direct shear test

126

Table 6.2 Direct soil-steel shear test results of Regina clay under different pre-consolidation

pressures

Parameter Pre-consolidation

pressure (kPa) wafter (%) τmax (kPa) τresidual (kPa)

Regina

clay

30 24.87 9.91 3.85

50 24.56 30.51 3.13

100 24.50 38.55 3.32

300 24.66 39.65 3.36

300 24.90 41.97 3.31

100* 25.01 18.26 2.89

wafter = the moisture content after the test

* represents that the test was tested before under 50 kPa pre-consolidation

pressure prior to this test

127

Figure 6.1 Axial stress along the pipeline resulted from longitudinal displacement

(a)

(b)

Figure 6.2 (a) Schematic and (b) fixture of modified direct shear box test for determining

adhesion of compacted clay

128

Figure 6.3 Setup of unconfined compression test

Figure 6.4 Specimen wrapped with plastic sheet to prevent moisture loss during testing

129

(a)

Figure 6.5 Axial load-displacement curves of unconfined compression test at varying moisture content

for (a) Regina clay and (b) Calgary till

(a)

(b)

Figure 6.6 Unconfined compressive strength versus moisture content relationship for (a) Regina clay and

(b) Calgary till

130

(a) 19.5%

(b) 22%

(c) 25%

(d) 28.5%

(e) 31%

Figure 6.7 Shear stress-displacement curves of modified direct shear (soil-steel) tests with pre-

consolidation pressure of 30 kPa and total stress of σn ≈ 0 kPa for compacted Regina clay with

moisture contents of (a) 19.5%, (b) 22%, (c) 25%, (d) 28.5%, and (e) 31%

131

Figure 6.8 Illustration for the movement of the modified shear box

132

(a) w = 11%

(b) w = 13%

(c) w = 15%

(d) w = 17%

(e) w = 19.5%, ρd = 17.5 g/cm3

Figure 6.9 Shear stress-displacement curves of modified direct shear (soil-steel) tests with pre-

consolidation pressure of 30 kPa and total stress of σn ≈ 0 kPa for compacted Calgary till with

moisture contents of (a) 11%, (b) 13%, (c) 15%, (d) 17%, and (e) 19.5% (ρd = 17.5 g/cm3)

133

Figure 6.10 Adhesion versus moisture content relationship for compacted Calgary till and Regina

clay

Figure 6.11 Modified adhesion factor versus half unconfined compressive strength relationship

for compacted Calgary till and Regina clay

134

Figure 6.12 Adhesion factors for pipelines (after ASCE 1984; Rizkalla et al., 1996; Paulin et al.,

1998)

135

Figure 6.13 Load-displacement curves of tests for Regina clay under different pre-consolidation

pressures

Figure 6.14 Adhesions versus different pre-consolidation pressures for compacted Regina clay at

optimum moisture content

The red star represents that the specimen was tested under a pre-consolidation pressure of 50 kPa prior to the test at 100 kPa

136

Conclusion

7.1. Overview

This thesis presented the studies on fracture toughness, tensile strength, and adhesion of two types

of compacted clays (Calgary till and Regina clay) along with their soil-water characteristic curves.

Mode I fracture toughness and tensile strength, which define the critical state of propagation for

an existing crack by linear elastic fracture mechanics and the condition of rupture on the

perspective of classical strength-of-material theory, were investigated for characterizing the

mechanical properties of compacted clays under tensile loading. First of all, mode I fracture

toughness was obtained through SNDB test, and then tensile strength was measured through

uniaxial direct tensile test. Interestingly, the variations of two properties were almost identical with

respect to the change of moisture content and dry density. As a further step, the relationship

between fracture toughness and tensile strength was positively curve fitted by linear and power-

law equations. Moreover, a double-porosity concept was adopted to curve fit the data of tensile

strength, trying to explain the contribution of matric suction to the variation trends of tensile

strength and fracture toughness.

In addition, the adhesions of two compacted clays to stainless steel were determined from modified

direct shear box tests. Adhesions of clays with different moisture contents and pre-load pressures

were studied to shed light on the adhesive interaction between compacted clays and stainless steel.

137

This chapter summarizes the main conclusions of the thesis and proposes further investigations in

the future.

7.2. Conclusions

(1) The mode I fracture toughness of compacted clays was determined using SNDB specimens

with different moisture contents and dry densities. Within the measurable range of moisture

content, two compacted clays exhibited different variations. For Calgary till, fracture toughness

increased with decreasing moisture content under a given dry density, and two distinct trends of

fracture toughness appeared for the specimens with two different dry densities. For Regina clay,

of which the specimens were compacted by a given compaction energy, the fracture toughness

increased first and then decreased when the moisture content declined from the wet side of

optimum to the dry side of optimum. The magnitudes of compacted clays were within a range of

3.0 to 30.0 kPa·m0.5 in this study.

(2) Tensile strength of compacted clays were obtained using uniaxial direct tensile tests with dog-

bone shaped specimens. The testing program of tensile test was identical to the fracture toughness

test. The variations of tensile strength were similar with fracture toughness.

(3) Due to similar variation trends of fracture toughness and tensile strength for the two compacted

clays, it is reasonable to establish a positive relationship between the two properties. Linear and

power-law equations were applied for curve fitting their relationship. It was found that both linear

and power law equations could reasonably curve fit the data.

138

(4) A double-porosity concept proposed by Alonso et al. (2010) was adopted to curve fit the tensile

strength of two compacted clays with their matric suction. The data of Regina clay was curve fitted

first, for which a transition from tensile mode to shear mode postulated by Wang et al (2017) was

employed. The data showed that the piecewise function for the effective degree of saturation Sre

yielded a better prediction than the continuous one in predicting the variation of tensile strength.

The curve fitting for tensile strength could possibly be used to predict the variation of fracture

toughness considering their positively correlated relationship.

(5) The adhesions of compacted Calgary till and Regina clay to stainless steel were obtained from

a series of modified direct shear test. For practical pipeline design, the modified adhesion factor,

which is a ratio of adhesion and 0.5cUCS, was determined as well. For both compacted clays, the

adhesion decreased with decreasing moisture content, and meanwhile the adhesion factor

decreased with increasing unconfined compressive strength under a given pre-load pressure of 30

kPa. In addition, for different pre-load pressures from 30 kPa to 300 kPa, the adhesion of Regina

clay with optimum moisture content increased with an increment of pre-load pressure from 30 kPa

to 100 kPa, but did not increase further when the pre-load pressure rise from 100 kPa to 300 kPa.

7.3. Recommendations

(1) For mode I fracture toughness, further investigations on the size effect of fracture process zone

are needed to establish a criterion for evaluating the KIc test of soils. Special instruments may be

139

designed to measure the crack opening displacement during fracturing, because currently gauges

available for metals and rocks can be hardly applied to the specimens of soils.

(2) For uniaxial tensile test, measurement for the development of tensile strains will be useful for

studying the fracture process behaviour during the test. More reasonable compaction method

should be developed to ensure the uniformity of compaction for dog-bone shaped specimens, and

hence to reduce the fluctuation of the data of tensile strength.

(3) For the correlations between mode I fracture toughness and tensile strength, more types of soils

need to be tested to evaluate the relationships for different materials. The impact of specimen

preparation method on the correlation is also of interest for future research.

(4) For the curve fitting based on the double-porosity concept, more attempts for different

compacted clays are supposed to be made to verify the applicability of the concept, for which the

parameters used to predict the variations of data should be measured or calibrated accordingly.

The development of new techniques on detecting the volume of water in micropores and

macropores among clay structures will be a foundation for practical application of the double-

porosity concept.

(5) The adhesion and adhesion factor of compacted clays to stainless steel for pipelines may be

measured under in-situ condition using full-scale pipes in the future, which will be valuable to the

design of pipeline in engineering practice.

140

Appendix A : Load-displacement curves of KIc test

(a) w = 19.5%

(b) w = 17%

(c) w = 15%

(d) w = 13%

(e) w = 11%

Figure A.1 Load-displacement curves of Calgary till with ρd = 1.75g/cm3 in KIc tests

(Note: Legend number represents test number in the corresponding table)

0

10

20

30

40

50

60

70

80

90

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

1 2 3 4

0

20

40

60

80

100

120

140

160

180

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

5 6 7 8

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

9 10 11 12

0

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

13 14 15 16

0

50

100

150

200

250

300

350

400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

17 18 19 20

141

(a) w = 17%

(b) w = 15%

(c) w = 13%

(d) w = 11%

Figure A.2 Load-displacement curves of Calgary till with ρd = 1.85g/cm3 in KIc tests

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

21 22 23 24

0

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

25 26 27 28

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

29 30 31 32

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Load

(N)

Displacement (mm)

33 34 35 36

142

(a) w = 31%

(b) w = 28.5%

(c) w = 25%

(d) w = 22%

(e) w = 19.5%

Figure A.3 Load-displacement curves of Regina clay in KIc test

143

Appendix B : Load-displacement curves of uniaxial tensile tests

(a) w = 17%

(b) w = 15%

(c) w = 13%

(d) w = 11%

Figure B.1 Load-displacement curves of uniaxial tensile tests of Calgary till specimens with ρd = 1.75

g/cm3

(Note: Legend number represents test number in the corresponding table)

144

(a) w = 17%

(b) w = 15%

(c) w = 13%

(d) w = 11%

Figure B.2 Load-displacement curves of uniaxial tensile tests of Calgary till specimens with ρd = 1.85

g/cm3

Note: Curves 35 and 36 in Figure B.2 (d) exhibit much larger displacements than the curves 33

and 34. This is because the clamps were not installed properly so that some slips occurred during

pulling. So these two curves cannot reflect the true tensile deformation around the specimen neck.

Similar curves are also found in Figure B.3 (d). But since the slip between the clamps and the

specimen was insignificant (< 2.5 mm), the specimen still broke at the neck. Therefore, the clamp

slip did not influence the calculation of tensile strength.

145

(a) w = 31%

(b) w = 28.5%

(c) w = 25%

(d) w = 22%

(e) w = 19.5%

Figure B.3 Load-deformation curves of uniaxial tensile tests of Regina clay specimens

146

Appendix C : The setup of spring gauge and its calibration

Figure C.1 shows an installed spring gauge, which consists of a metal strip and a strain gauge. The

spring gauge is mounted on the clamps after the installation of specimen. The metal strip, which

is made of brass with dimensions of 100 mm in length and 7 mm in width, is initially straight, so

the two ends need to be gently squeezed and inserted into the holders glued on the clamps. When

the test starts, two ends move outwards with the holders. During loading, the secant length (SL) is

becoming larger while the strain at the vertex of curvature is changing. The change of secant length

can be calculated through the reading of strain gauge. The reading of strain gauge is set to zero at

the beginning, so the relation between the strain and the change of secant length is different

depending on the initial secant length prior to the test, (e.g. with a range of 94 ~ 97 mm) as seen

in Figure C.2. Thus, the initial secant length must be recorded for calculating the change of secant

length before each test. Given the initial secant length, the deformation at the specimen neck could

be obtained from the change of secant length converted from the reading of strain gauge.

Figure C.3 provides the deformation with the corresponding reading of strain gauge and initial

secant length. When the deformation is smaller than 2.5 mm, the relation could be approximated

to be linear. The slopes of linear equations with larger initial secant length are steeper, and hence

the slope and the initial secant length can be correlated as a linear expression as well, for which

the equations are given in Figure C.4.

There is an evitable variation of initial secant length because the installation position of specimen

is impossible to be exactly the same for every time. Instead it is within a range from 93 mm to 96

147

mm. Given the initial secant length Li and the calibration curve in Figure C.4, the deformation Df

could be calculated through:

5 3( 1.8323 10 1.9513 10 )f iD L µε− −= − × × + × × Equation C.1

Figure C.1 Setup of spring gauge on clamps

Figure C.2 Change of secant length versus strain with different initial secant lengths

148

Figure C.3 Deformation versus strain

Figure C.4 Slope versus initial secant length

149

Appendix D : KIc Tests on Clay Shale

SNDB tests are also applicable for testing clay shale. In this appendix, the procedure of testing and

the results are briefly introduced.

Clay shale was initially preserved in plastic tubes. Two tubes were opened to obtain the cylindrical

samples for the tests. After opening, long cylindrical cores were cut into desired segments using

an electrical saw. To ensure smoothness and flatness, both top and bottom surfaces of the

specimens were sanded by a machine quickly. Four specimens, AM-1and AM-2 from core 1, PM-

1and PM-2 from core 2, were obtained for trial tests.

Cylindrical specimens were used to measure matric suction immediately after being machined to

small segments, using filter paper method as described in Chapter 2. Results are listed in Table

D.1. Afterwards, SNDB tests were conducted to gain KIc.

For all of the four specimens, the notches were sawed by hand to the ratio of a/t = 0.2. The

thicknesses were approximately equal to the radius. The geometries of specimens measured by a

caliper are presented in Table D.2. Each measured length is an average of three measurements.

The span of SNDB tests is fixed to the ratio of S’/R = 0.7, where m = 5.8903, n = -0.4229, and YI

= 3.70 for KIc calculation. Specimens were weighted before the tests and moisture contents were

checked after the tests.

150

Only three specimens were ultimately tested successfully because PM-1 was broken during sawing

as shown in Figure D.4. The loading rate was set to 0.18 mm/min. The load-displacement curves

of AM-1, AM-1, and PM-2 are drawn in Figure D.5.

Figure D.6 presents the cracking patterns of failed specimens. The cracked plane is not flat but

layered. The nature of parallel layering is considered as the cause of zigzag crack propagation

paths. It is believed that the strength of clay shale along the deposition direction is non-uniform,

where the interfaces between layers are weaker than the material itself. This non-uniformity might

contribute to the zigzag patterns and unsuccessful preparation of PM-1.

Table D.1 Matric suction of four clay shale samples

# Tc M1 M2 Th Mf Mw Wf Ψ (kPa) Ψ average am-1u 27.9235 28.1155 28.0853 27.9211 0.1642 0.0278 16.93% 6391

6983 am-1l 28.4934 28.6858 28.6580 28.4919 0.1661 0.0263 15.83% 7575 am-2u 28.5142 28.6881 28.6630 28.5107 0.1523 0.0216 14.18% 9784

9839 am-2l 28.3373 28.4950 28.4720 28.3338 0.1382 0.0195 14.11% 9895 pm-1u 28.0591 28.2595 28.2105 28.0573 0.1532 0.0472 30.81% 744

1080 pm-1l 28.2806 28.4645 28.4233 28.2781 0.1452 0.0387 26.65% 1417 pm-2u 37.7255 37.9094 37.8640 37.7198 0.1442 0.0397 27.53% 1236

1961 pm-2l 26.1134 26.2853 26.2497 26.1094 0.1403 0.0316 22.52% 2687

Table D.2 Summary of geometries and test tesults

Sample #ρ d

(g/cm3)w (%) M(g) R (mm) t (mm) t/R

S' (mm)

S'/Ra

(mm)a/t P max (N)

K Ⅰc

(kPa·m0.5)

AM-1 2.09 13.25 647.65 43.22 46.66 1.08 30.48 0.705 9.33 0.2 1497.12 117.59AM-2 1.99 12.84 593.62 43.19 45.15 1.05 30.48 0.706 9.03 0.2 2648.59 211.63PM-2 1.71 19.77 570.53 43.90 45.89 1.05 30.48 0.694 9.18 0.2 784.20 61.15PM-1 Failed to prepare the notch

151

Figure D.1 Core samples of clay shale preserved in tubes

Figure D.2 Core 1 from 320.6 m to 327.8 m Figure D.3 Core 2 from 144.2 m to 148.6 m

152

Figure D.4 Unsuccessfully sawed

specimen PM-1

Figure D.5 Load- displacement curves of specimens

AM-1

AM-2

PM-2

Figure D.6 Fractured specimens of clay shale

153

Appendix E : Unsuccessful KIIc Tests on Compacted Clays

After the investigation of KIc of compacted clays, KIIc tests were designed and prepared. However,

due to technical problems involved in sample preparation, the tests were unsuccessful because of

the fragility of specimens. Though this series of test failed, the author still included the information

about the tests in this Appendix to review the lessons learned from sample preparation.

Punch Through Shear (PTS) is an ISRM-suggested method that has been used for strong

geomaterials such as rock (Backers & Stephansson, 2012) and mortar (e.g. Davies and So, 1986)

for KIIc. To obtain nearly pure shear stress state (ratio of KII/KI = 29.9) around the notch tips, the

notch lengths should be made according to the dimensions illustrated in Figure E.1. The PTS

specimen was tentatively applied for compacted clays in this study.

The specimen was compacted in designed wooden mold, for which the interior sides were fully

greased with vacuum grease. The two long notches were made by inserting two steel plates into

the mold and then taking them out after compaction. The two short notches were sawed by a

handsaw.

The setup of the PTS is shown in Figure E.3. However, the specimen breaks easily at the middle

as shown in Figure E.4, which is probably a consequence of specimen disturbance when two steel

plates are pulled out after compaction. This difficulty makes the result of test questionable.

Therefore, this test was not successfully conducted with PTS method.

154

Figure E.1 Punch Through Shear (PTS) test

Figure E.2 Wooden compaction mold for PTS

specimen

Figure E.3 PTS setup

Figure E.4 Unwanted crack in PTS sample

155

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1. (Li and Yang 2016)(Miller et al. 1998)(Wang et al. 2016)(Wong et al. 2016)(ASTM 2007) (Broek 1982) 2. ASTM D4318(ASTM 2017), ASTM 2487(ASTM 2006), ASTM D5298(ASTM 2018) (Leong et al. 2002) (van Genuchten 1980) . (Fredlund and Rahardjo 1993) (Fredlund and Xing 1994)(Krahn and Fredlund 1972)(Bulut et al. 2001)(Babu et al. 2005)(Brooks and Corey 1966) (Chowdhury 2013) (Tinjum et al. 1997)(Fredlund 1979) 3. (Griffith and Eng 1921)(Westergaard 1939, Orowan 1948, Irwin 1957)(Nichols and Grismer 1997)(Wang et al. 2007)(Amarasiri et al. 2011)(Lima and Grismer 1994)(Li and Yang 2000)(Srawley 1976) (Ayad et al. 1997)(Konrad and Cummings 2001)(Lakshmikantha 2009)(Lakshmikantha et al. 2012)(Lee et al. 1988)(ASTM 2008) (ASTM 2013a)(Guo et al. 1993)(Zhao et al. 1994)(Wang and Xing 1999)(Agaiby 2013)(Harison et al. 1994)(Kuruppu et al. 2013)(ISRM 1988)(Fowell et al. 1995) (Alkilicgil 2006)(Tutluoglu and Keles 2011). 4. (Tschebotarioff et al. 1953)(Towner 1987)(Tamrakar et al. 2005)(Tang and Graham 2000)(Tang et al. 2015)(Nahlawi et al. 2004)(Stirling et al. 2015)(Zhang et al. 2015)(Narvaez et al. 2015)(Fang and Fernandez 1981)(Kim et al. 2012) 5.(Bishop 1959) (Gunsallus and Kulhawy 1984)(Zhang et al. 1998)(Zhang 2002)(Haberfield and Johnston 1989)(Alonso et al. 2010)(Romero et al. 1999)(Romero Morales 1999)(Wong et al. 2017)(Vanapalli et al. 1996) (Hueckel et al. 2002, Baker and Frydman 2009)(Fredlund and Morgenstern 1977)(Likos et al. 2010)(Vanapalli et al. 1999)(Li and Yang 2000) 6. (ASCE 1984)(Saye et al. 2012)(Sladen 1992)(Rizkalla et al. 1996)(Paulin et al. 1998)(Cooke and Price 1978)(CGS 2006) (ASTM 2013b) 7. Appendix: (Davies and So 1986)(Backers and Stephansson 2012)