Dynamic Tensile, Flexural and Fracture Tests of ...tspace.library.utoronto.ca/bitstream/1807/...Granite Feng Dai Doctor of Philosophy Department of Civil Engineering University of

Dynamic Tensile, Flexural and Fracture Tests of Anisotropic Barre Granite

by

Feng Dai

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Civil Engineering University of Toronto

© Copyright by Feng Dai 2010

ii

Dynamic Tensile, Flexural and Fracture Tests of Anisotropic Barre

Granite

Feng Dai

Doctor of Philosophy

Department of Civil Engineering University of Toronto

2010

ABSTRACT

Granitic rocks usually exhibit strongly anisotropy due to pre-existing microcracks induced by

long-term geological loadings. The understanding of anisotropy in mechanical properties of

rocks is critical to a variety of rock engineering applications. In this thesis, the anisotropy of

tension-related failure parameters involving tensile strength, flexural strength and Mode-I

fracture toughness/fracture energy of Barre granite is investigated under a wide range of loading

rates.

Three sets of dynamic experimental methodologies have been developed using the modified split

Hopkinson pressure bar system; Brazilian test to determine the tensile strength; semi-circular

bend method to determine the flexural strength; and notched semi-circular bend method to

determine the Mode-I fracture toughness and fracture energy. For all three tests, a simple quasi-

static data analysis is employed to deduce the mechanical properties; the methodology is

assessed critically against the isotropic Laurentian granite. It is shown that if dynamic force

balance is achieved in SHPB, it is reasonable to use quasi-static formulas. The dynamic force

balance is obtained by the pulse shaper technique.

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To study the anisotropy of these properties, rock blocks are cored and labeled using the three

principal directions of Barre granite to form six sample groups. For samples in the same

orientation group, the measured strengths/toughness shows clear loading rate dependence. More

importantly, a loading rate dependence of the strengths/toughness anisotropy of Barre granite has

been first observed: the anisotropy diminishes with the increase of loading rate.

The reason for the strengths/toughness anisotropy can be understood with reference to the

preferentially oriented microcracks sets; and the rate dependence of this anisotropy is

qualitatively explained with the microcracks interaction. Two models abstracted from

microscopic photographs are constructed to interpret the rate dependence of the fracture

toughness anisotropy in terms of the crack/microcracks interaction. The experimentally observed

rate dependence of the anisotropy is successfully reproduced.

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To my family

v

ACKNOWLEDGEMENTS

The completion of this thesis also completes my career as a student. In retrospect, the joyful days

in the University of Toronto not only opened my sights for advanced science and techniques, but

also brought numbers of friends into my life.

Foremost, I would like to express my sincerest thanks to my advisor, Professor Kaiwen Xia, for

his support, guidance and tolerance during my study at the University of Toronto. We first met in

China, while I was still lecturing courses in China and getting confused about the future. When

he decided to offer me a position to work with him as a Ph.D student in the University of

Toronto, I knew this would be a great opportunity for me to make a difference; and I cherished it

very much. So far, we have coauthored eight papers in world leading peer-review journals; and

two more in preparation. I am happy that I did not disappoint him.

Many thanks to Professor Bibu Mohanty, who had been my mentor for the passing four years.

The warmhearted advices for my future career as well as the financial supports from him are

greatly appreciated. I would also like to thank the other members in my defense committee,

Professor Evan Bentz and Professor Giovanni Grasselli and Professor Ming Cai from Laurentian

univerdity for providing valuable advices and constructive comments in improving the draft of

this thesis. Professor Murray Grabinsky in Geotechnical laboratory is appreciated for always

being nice to me.

I am grateful to Professor Qingyuan Wang and Professor Zheming Zhu in Sichuan University,

P.R.China for taking care of me before I leave for University of Toronto; Professor Lizhong

Tang from Central South University for sharing personal experience with me of working in

academia. Special thanks to Mr. Javid Iqbal, who helped me like my old brother, especially in

the first year of my doctoral program. I thank Mr. Rong Chen, Mr. Sheng Huang and Mr. Tubing

Yin for pleasant cooperation and insightful discussion during the course of this study. I am also

lucky to have been in the company of my friends and fellows in Geotechnical laboratory in the

department of Civil Engineering, Dr. M. H.B. Nasseri, Dr. Dragana Simon, Dr. Abdolreza Saebi

Moghaddam, Mr. Leonardo Trivino, Mr. Abdullah Galaa Abdelaal, Mr. Bryan Tatone, and Mr.

Omid Khajeh Mahabadi. To the technical staffs of the structure laboratory, Renzo Basset,

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Giovanni Buzzeo, John MacDonald, Joel Babbin, I thank you all for helping me in running my

experiments smoothly and efficiently.

I am indebted to my wife, Xiaoli Jia, for her love, encouragement, support and tolerance in the

passing four years. I would also like to thank my parents and parents-in-law who have been

always supporting me. Years ago, my father failed to enroll in the best university of China due to

the Culture Revolution; my doctoral degree awarded from a world-class university is the best

consolation to him.

The eternal love from the family fosters my strength to conquer the difficulties in rainy days,

past, present and future.

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TABLE OF CONTENTS

ABSTRACT................................................................................................................................... ii

ACKNOWLEDGEMENTS ..........................................................................................................v

TABLE OF CONTENTS ........................................................................................................... vii

LIST OF TABLES ....................................................................................................................... xi

LIST OF FIGURES .................................................................................................................... xii

LIST OF ACRONYMS AND ABBREVIATIONS ............................................................... xxiii

LIST OF SYMBOLS .................................................................................................................xxv

CHAPTER 1 INTRODUCTION ..................................................................................................1

1.1 Background..........................................................................................................................1

1.2 Problem Statement ...............................................................................................................5

1.3 Research Objectives.............................................................................................................6

1.4 Research Contribution .........................................................................................................7

1.5 Thesis Organization .............................................................................................................9

CHAPTER 2 LITERATURE REVIEW ...................................................................................11

2.1 Barre Granite and Its Anisotropy.......................................................................................11

2.1.1 Microstructural Investigation.................................................................................11

2.1.2 Mechanical Properties............................................................................................14

2.2 Tension Tests .....................................................................................................................20

2.2.1 Static Tension Tests ...............................................................................................20

2.2.2 Dynamic Tension Tests..........................................................................................21

2.3 Fracture Tests.....................................................................................................................23

viii

2.3.1 Static Fracture Tests...............................................................................................23

2.3.2 Dynamic Fracture Tests .........................................................................................29

CHAPTER 3 EXPERIMENTAL SETUP AND TECHNIQUES ...........................................35

3.1 Samples Preparations .........................................................................................................35

3.1.1 Laurentian Granite .................................................................................................35

3.1.2 Barre Granite..........................................................................................................37

3.2 MTS Hydraulic Servo-control System...............................................................................40

3.3 Split Hopkinson Pressure Bar ............................................................................................41

3.3.1 Working Principle..................................................................................................41

3.3.2 Pulse Shaping.........................................................................................................45

3.3.3 Momentum Trap ....................................................................................................48

3.4 Laser Gap Gauge System...................................................................................................51

3.4.1 Principles and Setup...............................................................................................52

3.4.2 Calibration of the System.......................................................................................53

CHAPTER 4 DYNAMIC TENSION TESTS...........................................................................57

4.1 Background Studies ...........................................................................................................57

4.2 Dynamic Brazilian Test .....................................................................................................59

4.3 Validation of Dynamic Brazilian Test ...............................................................................61

4.3.1 Dynamic Brazilian Test without Pulse Shaping ....................................................61

4.3.2 Dynamic Brazilian Test with Careful Pulse Shaping ............................................67

4.4 Tensile Strength of Barre Granite ......................................................................................74

4.4.1 Determination of Anisotropic Tensile Strength.....................................................74

4.4.2 Tensile Strength Anisotropy ..................................................................................83

4.4.3 Interpretation of the Results...................................................................................91

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4.5 Summary ............................................................................................................................93

CHAPTER 5 DYNAMIC FLEXUAL TESTS..........................................................................95

5.1 Background studies............................................................................................................95

5.2 Dynamic Semi-circular Bend Flexural Test ......................................................................99

5.2.1 The Semi-circular Bend Testing in a SHPB System .............................................99

5.2.2 Determination of Flexural Strength .....................................................................100

5.3 Validation of Semi-Circular Bend Tests..........................................................................103

5.3.1 Failure Sequences of the Specimen in the Dynamic SCB Test ...........................103

5.3.2 Dynamic SCB Test without Pulse Shaping .........................................................104

5.3.3 Dynamic SCB Test with Careful Pulse Shaping..................................................107

5.4 Flexural Strength of Barre Granite ..................................................................................111

5.4.1 Determination of Anisotropic Flexural Strength .................................................111

5.4.2 Flexural Strength Anisotropy...............................................................................120

5.4.3 Interpretation of the Results.................................................................................127

5.5 Summary ..........................................................................................................................133

CHAPTER 6 DYNAMIC FRACTURE TESTS.....................................................................135

6.1 Background Studies .........................................................................................................135

6.2 Dynamic Notched Semi-circular Bend Fracture Test ......................................................139

6.2.1 The Notched Semi-circular Bend Testing in an SHPB System...........................139

6.2.2 Determination of Mode-I Fracture Toughness ....................................................140

6.2.3 Determination of Dynamic Fracture Energy........................................................142

6.3 Validation of Dynamic Notched Semi-Circular Bend Test .............................................146

6.3.1 Dynamic Analysis and Fracture Time .................................................................146

6.3.2 Dynamic NSCB Test without Pulse Shaping.......................................................147

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6.3.3 Dynamic NSCB Test with Careful Pulse Shaping...............................................150

6.4 Fracture Toughness Anisotropy of Barre Granite............................................................155

6.4.1 Determination of Anisotropic Stress Intensity Factor .........................................155

6.4.2 Determination of Fracture Toughness of Barre Granite ......................................160

6.4.3 Fracture Toughness Anisotropy...........................................................................165

6.5 Crack-Microcrack Interaction..........................................................................................173

6.5.1 Background..........................................................................................................173

6.5.2 Microstructural Investigation and Featuring Models...........................................175

6.5.3 The Crack-Microcrack Interaction.......................................................................178

6.5.4 Finite Element Analysis of Two Models .............................................................183

6.5.5 Simulated Fracture Toughness Anisotropy..........................................................195

6.5.6 Concluding Remarks............................................................................................199

6.6 Summary ..........................................................................................................................201

CHAPTER 7 SUMMARY AND FUTURE WORK ..............................................................202

7.1 Summary of the Thesis Work ..........................................................................................202

7.2 Future Work .....................................................................................................................206

7.2.1 Confining Effects .................................................................................................207

7.2.2 Thermal Effects....................................................................................................208

BIBLIOGRAPHY ......................................................................................................................211

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LIST OF TABLES

Table 4.1 The material properties used in the finite element model of BD samples of Barre

granite along six directions. .......................................................................................................... 82

Table 4.2 Tensile strengths of Barre granite along six directions from both static and

dynamic Brazilian tests. ................................................................................................................ 88

Table 5.1 The material properties used in the finite element model of SCB samples of Barre

granite along six directions. ........................................................................................................ 119

Table 5.2 Flexural strengths of Barre granite with corresponding loading rates as well as the

non-local reconciliation for both static and dynamic SCB tests. ................................................ 126

Table 5.3 Summary of the parameters deduced using non-local failure model for all six

sample groups of Barre granite. .................................................................................................. 130

Table 6.1 The normalized stress intensity factor aKK II πσ/* = , for an edge crack in an

infinite orthotropic strip with remote uniform traction σ............................................................ 163

Table 6.2 The material properties used in the finite element model of NSCB samples of

Barre granite along six directions. .............................................................................................. 164

Table 6.3 Fracture toughness and fracture energy of Barre granite with corresponding

loading rates from both static and dynamic NSCB fracture tests. .............................................. 172

Table 6.4 Stress intensity factor of the main crack with one collinear microcrack at different

distances to the main crack tip. ................................................................................................... 185

Table 6.5 The fracture toughness and corresponding loading rates for three models (Intact,

Model 1 and Model 2)................................................................................................................. 197

Table 6.6 The simulated Mode-I fracture toughness anisotropic index (αk) of Barre granite

with loading rates........................................................................................................................ 198

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LIST OF FIGURES

Figure 2.1 Mineral and microcracks traced from three orthogonal planes for Barre granite;

after (Nasseri and Mohanty, 2008). .............................................................................................. 12

Figure 2.2 3D block diagram showing microcracks orientations in Barre granite; rose

diagrams show the alignment of microcracks and mineral fabric orientation for each plane;

reproduced after (Nasseri and Mohanty, 2008); the letters in the braskets are the directions used

in this thesis. ............................................................................................................................... 13

Figure 2.3 3D block diagram showing location of CCNBD specimens prepared along each

plane with respect to microcracks orientations in Barre granite (dominant fracture planes shown

in heavy exaggerated lines); reproduced after (Nasseri and Mohanty, 2008); the letters in the

braskets are the directions used in this thesis................................................................................ 16

Figure 2.4 Variation of fracture toughness measured along six directions with the number of

tests along each direction in Barre granite; after (Nasseri and Mohanty, 2008)........................... 17

Figure 2.5 Strain rate effects of the maximum compressive stress for X-, Y- and Z- samples

of Barre granite; reproduced after (Xia et al., 2008); the letters in the braskets are the directions

used in this thesis. ......................................................................................................................... 19

Figure 2.6 The three basic modes of crack propagation: (a) Mode I, opening mode; (b) Mode

II, in-plane shearing; (c) Mode III, tearing mode. ........................................................................ 24

Figure 2.7 Definition of the local coordinate axis ahead of a crack tip. Z direction is normal

to the plane. ............................................................................................................................... 25

Figure 2.8 Comparison of the fracture mechanics approach to design with the traditional

strength of material approach: (a) strength approach (b) fracture toughness approach................ 27

Figure 3.1 Procedures for preparing three types of samples: Brazilian disc (BD), semi-

circular bend (SCB) and notched semi-circular bend (NSCB) samples. ...................................... 36

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Figure 3.2 3D block diagram showing longitudinal wave velocities and the sampling location

of Brazilian discs prepared along each plane with respect to microcrack orientations in Barre

granite; the first index for sample numbering represents the direction normal to the splitting

plane, and the second index indicates the propagation direction of the crack, e.g. Sample YX of

(a) BD sample; (b) SCB sample; (c) NSCB sample; the dashed lines depict the failure plane.... 39

Figure 3.3 Photoes of (a) semi-circular bend and (b) Brazilian test of rock samples in the

MTS hydraulic servo-control testing system. ............................................................................... 40

Figure 3.4 Photo of a split Hopkinson pressure bar (SHPB) system in the Department of

Civil Engineering, University of Toronto..................................................................................... 42

Figure 3.5 Schematics of a split Hopkinson pressure bar (SHPB) system and the x-t diagram

of stress waves propagation in SHPB. .......................................................................................... 43

Figure 3.6 Strain-gauge data, after signal conditioning and amplification, from a SHPB

compression test of a Barre granite sample showing the three stress waves measured as a

function of time............................................................................................................................. 44

Figure 3.7 Pulse shapers in SHPB (a) schematic of the assembly (b) unshaped and shaped

incident stress pulses..................................................................................................................... 48

Figure 3.8 The momentum-trap system: (a) the actual image and (b) the x–t diagram showing

its working principle. .................................................................................................................... 49

Figure 3.9 Comparison of stress waves from the incident bar, with and without momentum

trap; the legends refer to the stress wave with trap. ...................................................................... 51

Figure 3.10 Photo and schematics of the laser gap gauge (LGG) system set up perpendicular

to the bar axis of SHPB................................................................................................................. 53

Figure 3.11 Static calibration of the LGG system using a gap gauge blocking the collimated

beam: schematic setup and the calibration result.......................................................................... 54

Figure 3.12 Dynamic calibration of the LGG system: schematic setup and a typical dynamic

testing result compared to the predictions by Equation (3.6). ...................................................... 56

xiv

Figure 4.1 Schematic of the Brazilian test in a SHPB system. The Brazilian disc, with a

thickness B = 16 mm and diameter D= 40 mm, is sandwiched between the incident and

transmitted bars. A strain gauge is mounted on the specimen near the disc centre. ..................... 60

Figure 4.2 Dynamic forces on both ends of the Laurentian granite disc specimen tested using

a traditional SHPB without pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted. ........... 61

Figure 4.3 High-speed video images of a typical dynamic Brazilian test on Laurentian

granite without pulse shaping. ...................................................................................................... 62

Figure 4.4 Mesh of the Brazilian disc for the finite element analysis with ANSYS; P1 and P2

are the diametrical forces on both loading ends............................................................................ 64

Figure 4.5 (a) Tensile stress σx (b) compressive stress σy histories at the center of a Brazilian

disc from dynamic finite element analysis and quasi-static equation in a typical SHPB Brazilian

test on Laurentian granite without pulse shaping. ........................................................................ 65

Figure 4.6 Comparison of strain gage signal with the dynamic forces on both loading ends of

the disc in a dynamic Brazilian test on Laurentian granite using a traditional SHPB without pulse

shaping. ............................................................................................................................... 66

Figure 4.7 Dynamic forces on both ends of a Laurentian granite disc specimen tested using a

modified SHPB with careful pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted. ......... 67

Figure 4.8 High-speed video images of two typical dynamic Brazilian tests on Laurentian

granite with careful pulse shaping. ............................................................................................... 68

Figure 4.9 (a) Tensile stress σx (b) compressive stress σy histories at the center of a Brazilian

disc on Laurentian granite from both dynamic and quasi-static finite element analyses in a typical

SHPB Brazilian test with pulse shaping. ...................................................................................... 70

Figure 4.10 Comparison of the strain gage signal with the transmitted force for a dynamic

Brazilian test on Laurentian granite using a modified SHPB with careful pulse shaping............ 71

Figure 4.11 The measured tensile strength of Laurentian granite from dynamic Brazilian tests

with and without employing jaws. ................................................................................................ 73

xv

Figure 4.12 Schematics of a Brazilian test in (a) the material testing machine and (b) the

SHPB system. ............................................................................................................................... 74

Figure 4.13 Stress trajectories of a Brazilian disc under quasi-static deformation. (a) fxx, (b) fyy

and (c) fxy with isotropic model, and (d) fxx (e) fyy and (f) fxy for sample YX using anisotropic

model (positive for compression, negative for tension)................................................................ 76

Figure 4.14 Stress trajectories of a Brazilian disc of Barre granite under quasi-static

deformation. (a) fxx, (b) fyy and (c) fxy with isotropic model. ........................................................ 78


deformation. (a) fxx, (b) fyy and (c) fxy for sample XY, and (d) fxx (e) fyy and (f) fxy for sample XZ

(positive for compression, negative for tension)........................................................................... 79


deformation. (a) fxx, (b) fyy and (c) fxy for sample YX, and (d) fxx (e) fyy and (f) fxy for sample YZ



deformation. (a) fxx, (b) fyy and (c) fxy for sample ZX, and (d) fxx (e) fyy and (f) fxy for sample ZY


Figure 4.18 Dynamic force balance check for a typical dynamic Brazilian test of Barre granite

with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted................................................. 84

Figure 4.19 Virgin Brazilian discs of Barre granite prepared for the test; each division in the

scale denotes 1 mm. ...................................................................................................................... 85

Figure 4.20 Recovered Brazilian discs of Barre granite after tests; each division in the scale

denotes 1 mm. ............................................................................................................................... 85

Figure 4.21 The variation of static tensile strength of Barre granite along six directions, i.e.

XY, XZ, YX, YZ, ZX and ZY, using (a) orthotropic model (b) isotropic model. ....................... 87

Figure 4.22 The variation of tensile strength with loading rates for six sample groups of Barre

granite. ............................................................................................................................... 89

xvi

Figure 4.23 The tensile strength with loading rates for samples splitting in the plane normal to

(a) X axis (b) Y axis (c) Z axis; and (d) the tensile strength anisotropic index (αt) of Barre granite

with loading rates.......................................................................................................................... 90

Figure 5.1 Schematics of the determination of the flexural strength of concrete by ASTM

standards: a) ASTM C293, i.e. center point loading; the entire load is applied at the center of the

span. The maximum tensile stress only occurs at the center of the span; b) ASTM C78, i.e. four

points loading; half of the load is applied upon each third of the span length. Maximum tensile

stress is present over the center 1/3 portion of the span. .............................................................. 97

Figure 5.2 Schematic of the semi-circular bending (SCB) testing in a SHPB system. The

semi-circular specimen, with a thickness B = 16 mm and radius R = 20 mm, is sandwiched

between the incident and transmitted bars. A strain gauge is mounted on the specimen near the

point O. ............................................................................................................................. 100

Figure 5.3 Meshing scheme of the SCB specimen for finite element analysis. F1 and F2

denote forces applied on the contact points. ............................................................................... 101

Figure 5.4 Y as a function of the dimensionless geometry parameter S/2R from the quasi-

static finite element analysis; the coefficient of determination of the fitting curve R2 is 0.9999.....

............................................................................................................................. 102

Figure 5.5 High-speed video images of a dynamic semi-circular bend test on Laurentian

granite. ............................................................................................................................. 103

Figure 5.6 Samples recovered from the SCB testing on Laurentian granite in a SHPB system

(a) without pulse shaping, and (b) with pulse shaping; each division in the scale denotes 1 mm....

............................................................................................................................. 104

Figure 5.7 Force histories on both ends of the specimen in the SCB-SHPB test on Laurentian

granite without pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted. ............................ 105

Figure 5.8 Tensile stress histories at the failure spot O of the Laurentian granite specimen

from the dynamic finite element and quasi-static analyses for the SCB-SHPB test without pulse

shaping. ............................................................................................................................. 106

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Figure 5.9 Strain gauge signal and the transmitted force P2 in the SCB-SHPB test on

Laurentian granite without pulse shaping. .................................................................................. 106

Figure 5.10 Demonstration of dynamic force equilibration on both ends of the specimen in the

SCB-SHPB test on Laurentian granite with appropriate pulse shaping. In.: incident; Re.:

reflected; Tr.: transmitted............................................................................................................ 108

Figure 5.11 Tensile stress histories at the specimen failure spot from dynamic and quasi-static

finite element analyses for the SCB-SHPB test on Laurentian granite with appropriate pulse

shaping. ............................................................................................................................. 109

Figure 5.12 Strain gauge signal and the transmitted force P2 in the SCB-SHPB test on

Laurentian granite with pulse shaping. ....................................................................................... 109

Figure 5.13 Schematics of the semi-circular bend test in (a) the material testing machine and

(b) the SHPB system................................................................................................................... 112

Figure 5.14 Stress trajectories of a semi-circular bend sample under quasi-static deformation.

(a) qxx, (b) qyy and (c) qxy with isotropic model, and (d) qxx (e) qyy and (f) qxy for ZX sample

using anisotropic model (positive for compression, negative for tension). ................................ 113


(a) qxx, (b) qyy and (c) qxy with isotropic model. .......................................................................... 115


(a) qxx, (b) qyy and (c) qxy for XY sample and (d) qxx (e) qyy and (f) qxy for XZ sample (positive for

compression, negative for tension). ............................................................................................ 116


(a) qxx, (b) qyy and (c) qxy for YX sample, and (d) qxx (e) qyy and (f) qxy for YZ sample (positive

for compression, negative for tension)........................................................................................ 117


(a) qxx, (b) qyy and (c) qxy for ZX sample, and (d) qxx (e) qyy and (f) qxy for ZY sample using

anisotropic model (positive for compression, negative for tension)........................................... 118

xviii

Figure 5.19 Dynamic force balance check for a typical dynamic semi-circular bend test on

sample XZ of Barre granite with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.. 121

Figure 5.20 (a) Virgin semi-circular bend samples of Barre granite; (b) Recovered semi-

circular bend samples of Barre granite after tests. ...................................................................... 121

Figure 5.21 The variation of static flexural strength of Barre granite along six directions, i.e.

XY, XZ, YX, YZ, ZX and ZY.................................................................................................... 122

Figure 5.22 The variation of flexural strength with loading rates along six directions of Barre

granite. ............................................................................................................................. 123

Figure 5.23 The flexural strength with loading rates for samples splitting in the plane normal

to (a) X axis (b) Y axis (c) Z axis; and (d) The flexural strength anisotropic index (αf) of Barre

granite with loading rates............................................................................................................ 125

Figure 5.24 Normalized tensile stress along the prospective fracture path in a SCB XY sample;

x is the distance of a point along the prospective fracture path to the failure spot of the SCB

sample (see the insert); the fitting curve has a coefficient of determination R2 of 0.9999. ........ 128

Figure 5.25 Comparison of strengths of sample group XY of Barre granite from dynamic SCB

test and BD test as well as the reconciliation by non-local failure model. ................................. 130

Figure 5.26 Comparison of strengths of sample group XZ of Barre granite from dynamic SCB


Figure 5.27 Comparison of strengths of sample group YX of Barre granite from dynamic SCB


Figure 5.28 Comparison of strengths of sample group YZ of Barre granite from dynamic SCB


Figure 5.29 Comparison of strengths of sample group ZX of Barre granite from dynamic SCB


Figure 5.30 Comparison of strengths of sample group ZY of Barre granite from dynamic SCB


xix

Figure 6.1 Schematics of the notched semi-circular bend (NSCB) specimen in the spit

Hopkinson pressure bar (SHPB) system with laser gap gauge (LGG) system. A strain gauge is

mounted on the specimen surface near the crack tip. ................................................................. 140

Figure 6.2 Finite element model of the NSCB specimen system (a) the half model of NSCB

sample (b) close view of the crack tip mesh (c) crack tip coordinate system............................. 141

Figure 6.3 Typical loading history and CSOD history of the NSCB specimen tested in SHPB

on Laurentian granite. ................................................................................................................. 143

Figure 6.4 Selected high speed camera images showing the fracture and fragmentation of a

NSCB Laurentian granite specimen............................................................................................ 144

Figure 6.5 Dynamic forces on both ends of the NSCB specimen tested using a conventional

SHPB on Laurentian granite. In.: incident; Re.: reflected; Tr.: transmitted. .............................. 148

Figure 6.6 Comparison of CSOD and strain gage signal with the transmitted force of the

NSCB specimen tested using a conventional SHPB on Laurentian granite (the unit for CSOD is

0.05 mm). ............................................................................................................................. 149

Figure 6.7 The evolution of SIF of the NSCB specimen tested using a conventional SHPB on

Laurentian granite with both quasi-static analysis and dynamic analysis. ................................. 150

Figure 6.8 Dynamic forces on both ends of the NSCB specimen tested using a modified

SHPB on Laurentian granite. In.: incident; Re.: reflected; Tr.: transmitted. .............................. 151

Figure 6.9 Comparison of CSOD and strain gage signal with the transmitted force of the

NSCB specimen tested using a modified SHPB test on Laurentian granite (the unit for CSOD is

0.05 mm). ............................................................................................................................. 152

Figure 6.10 The evolution of SIF of the NSCB specimen tested using a modified SHPB on

Laurentian granite with both quasi-static analysis and dynamic analysis. ................................. 153

Figure 6.11 The effect of loading rate on the fracture toughness and fracture energy of

Laurentian granite. ...................................................................................................................... 154

xx

Figure 6.12 Local coordinate system for the stress and displacement fields near the crack tip

of an orthotropic solid................................................................................................................. 156

Figure 6.13 Schematics of the straight through notched semi-circular bend fracture test in (a)

the material testing machine and (b) the SHPB system.............................................................. 161

Figure 6.14 An infinite orthotropic strip with an edge crack under remote uniform tractions

normal to the edge crack. ............................................................................................................ 161

Figure 6.15 The overall mesh of the strip and a close-view of the mesh in the vicinity of the

crack tip; the length of the trip is modeled as ten times of the width W. .................................... 162

Figure 6.16 Mesh for the NSCB specimen and crack tip local coordinate system (a) mesh of

the half model (b) close view of the crack tip mesh (c) crack tip coordinate system................. 164

Figure 6.17 Dynamic force balance check for a typical NSCB fracture test of Barre granite

with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted............................................... 165

Figure 6.18 (a) Virgin NSCB samples and (b) recovered NSCB samples of Barre granite. . 166

Figure 6.19 The variation of static fracture toughness of Barre granite on six sample groups,

i.e. XY, XZ, YX, YZ, ZX and ZY. ............................................................................................. 167

Figure 6.20 The variation of fracture toughness with loading rates on six directions of Barre

granite. ............................................................................................................................. 168

Figure 6.21 The variation of fracture energy with loading rates on six directions of Barre

granite. ............................................................................................................................. 169

Figure 6.22 The fracture toughness with loading rates for sample group of (a) XY, splitting in

the plane normal to X axis; (b) ZX, splitting in the plane normal to Z axis; and (c) the fracture

toughness anisotropic index αk of Barre granite. ........................................................................ 171

Figure 6.23 (a) Photo of microscopic thin section showing microcracks in a tested Barre

granite sample; Case 1: the main crack inclines at an angle of o45 to microcracks; (b) Model 1:

the crack-microcracks configuration for Case 1. ........................................................................ 176

xxi

Figure 6.24 (a) Photo of microscopic thin section showing microcracks in a tested Barre

granite sample; Case 2: The main crack is collinear to microcracks; (b) Model 2: the crack-

microcracks configuration for Case 2. ........................................................................................ 177

Figure 6.25 One arbitrarily located microcrack near the crack tip of a semi-infinite crack. . 178

Figure 6.26 The original problem and the three sub-problems decomposed from the original

one based on the superposition method. ..................................................................................... 179

Figure 6.27 The phase diagram of amplification and shielding effects of main crack due to the

presence of a unique microcrack using 0th-order and 1st-order approximate solution.............. 183

Figure 6.28 Finite element mesh (a) global mesh of Model 1; Case 1 (b) close-view of the

mesh at the vicinity of the main crack of the Intact Model. The main crack and its tip are

indicated with arrows.................................................................................................................. 186


mesh at the vicinity of the main crack and the inclined microcrack of Model 1; Case 1. The main

crack and its tip are indicated with arrows.................................................................................. 187


mesh at the vicinity of the main crack and the collinear microcrack of Model 2; Case 2. The main

crack and its tip are indicated with arrows and the collinear microcrack is also marked........... 188

Figure 6.31 The deformation and stress intensity trajectories at the vicinity of the main crack

for the semi-circular band specimen in the absence of microcracks. ......................................... 189

Figure 6.32 The deformation and stress intensity trajectories of the main crack and the

inclined microcrack of the semi-circular bend specimen in Model 1. ........................................ 190

Figure 6.33 The deformation and stress intensity trajectories of the main crack and the

collinear microcrack of the semi-circular bend specimen in Model 2........................................ 190

Figure 6.34 The dynamic load exerted on both ends of the NSCB specimen for three

configurations (Intact, Model 1 and Model 2), the load comes from an actual measurement in a

modified SHPB tests with force balance achieved on both ends of the sample. ........................ 192

xxii

Figure 6.35 The evolution of SIF of the NSCB specimen for three configurations (Intact,

Model 1 and Model 2) from both quasi-static analysis and dynamic analysis; force balance has

been guaranteed using a modified SHPB tests with careful pulse shaping. ............................... 193

Figure 6.36 The linear load exerted on both ends of the NSCB specimen for three

configurations (Intact, Model 1 and Model 2), assuming force balance on both loading ends of

the sample. ............................................................................................................................. 194

Figure 6.37 The evolution of the dynamic dimensionless SIFs and corresponding loading rates

of the NSCB specimen for three configurations (Intact, Model 1 and Model 2) with linear

dynamic loading, assuming force balance on both loading ends of the sample. ........................ 194

Figure 6.38 The simulated dynamic fracture toughness of Barre granite with loading rates for

three configurations (Intact, Model 1 and Model 2). .................................................................. 197

Figure 6.39 The simulated Mode-I fracture toughness anisotropic index (αk) of Barre granite

with loading rates based on crack-microcracks interaction model. ............................................ 199

Figure 7.1 Schematic of the Brazilian test under hydrostatic confining pressure on SHPB

system. ............................................................................................................................. 208

Figure 7.2 Schematic of the Brazilian test on SHPB system under in-situ thermal heating.209

xxiii

LIST OF ACRONYMS AND ABBREVIATIONS

1PB One Point Bend

2D Two Dimensional

3D Three Dimensional

3PB Three Point Bend

ASTM American Society for Testing and Materials

BD Brazilian Disc

CB Chevron Bend

CCNBD Cracked Chevron Notch Brazilian Disc

CCS Compact Compressive Specimens

COD Crack Opening Displacement

CSOD Crack Surface Opening Displacement

CSOV Crack Surface Opening Velocity

CSTBD Cracked Straight Through Brazilian Disc

FEA Finite Element Analysis

LORD Laser Occlusive Radius Detector

ISRM International Society of Rock Mechanics

LEFM Linear Elastic Fracture Mechanics

LVDT Linear Variable Displacement Transducers

LGG Laser Gap Gauge

SCB Semi-Circular Band

SHPB Split Hopkinson Pressure Bar

SIF Stress Intensity Factor

SR Short Rod

xxiv

TEM Transmission Electron Microscope

NSCB Notched Semi-Circular Bend

WLCT Wedge-Loaded Compact Tension

xxv

LIST OF SYMBOLS

A Cross-sectional area of the bar

Ac Area of the new generated crack surfaces

a Depth of the notch

ija Elastic compliance constants of the material

ija′ Compliance constants in the local x-y coordinate system

B The thickness of the sample

C Elastic wave velocity

c Half length of the microcrack

Cijkl Stiffness constants of Barre granite

D The diameter of the sample

d Distance of the main crack tip to the center of the microcrack

Dij Factors to determine the anisotropic fracture toughness

E Young’s Modulus

Ei Young’s Modulus in the i principle direction

fxx, fyy, fxy Dimensionless stress components for BD

F Dimensionless tensile stress component at the disc center

fc The cutoff frequency of a bar

G Shear Modulus

GC Fracture energy

Gij Shear Modulus in the i-j plane

sl Length of the striker

K Kinetic energy of the cracked fragments

k Calibrated parameter of the LGG system

xxvi

KI Mode I SIF

KIC Mode I fracture toughness

PIK Mode I propagation fracture toughness

KII Mode II SIF

KIIC Mode II fracture toughness

KIII Mode III SIF

KIIIC Mode III fracture toughness

0IK Prescripted far field loading in terms of Mode I SIF

0IIK Mode II SIF of the main crack for Intact Model

•0IK Prescribed far field loading rate

0ICK Nominal fracture toughness from measurements

1MIK Local SIF of the main crack for Model 1

•1IK Loading rate of SIF of the main crack for Model 1

2MIK Local SIF of the main crack for Model 2

•2IK Loading rate of SIF of the main crack for Model 2

LIK Local SIF of the main crack

•LIK Local loading rate of the main crack

P1 Dynamic forces on the incident end

P2 Dynamic forces on the transmitted end

Pf Failure load

qxx, qyy, qxy Dimensionless stress components for SCB

Q Dimensionless tensile stress at the failure spot

R Radius of the sample

xxvii

R0 Radius of the bar

S Distance of the two supporting pins

u Displacement vector

u&& Second time derivative of the displacement vector

u1 Displacement of the incident bar end

u2 Displacement of the transmitted bar end

X Principal axis of Barre granite with the slowest P-wave velocity

Y Principal axis of Barre granite with intermediate P-wave velocity

Z Principal axis of Barre granite with the fast P-wave velocity

V Half crack opending displacement

V0 Veloctity of the striker

W Energy carried by the stress wave

Wi Energy carried by the incident stress wave

Wr Energy carried by the reflected stress wave

Wt Energy carried by the transmitted stress wave

WG Energy consumed to create new crack surfaces

ρ Density

v Poisson’s ratio

vij Poisson’s ratio for strain in j direction by strain in i direction

εi, εr, εt Incident, reflected and transmitted strain pulse

)(tε& Strain rate

σ Tensile stress

σm Maximum tensile stress

σ0(x), τ0(x) Undisturbed normal/shear stress along the location of the microcrack

σp(x), τp(x) A pair of pseudo-tractions

σt Tensile strength

xxviii

σf Flexural strength

σt,N Tensile strength by non-local reconciliation

κ Ratio of the flexural strength to tensile strength

σ& Loading rate of the tensile stress/flexural stress

σij Stress tensor

ijε Strain tensor

σx,σy, τxy Components of the stress tensor

δ Characteristic material length

ΔU Amount of voltage reading of LGG output

ω Angular velocity

ξ Ratio of the local SIF at the main crack tip to the loading

tα Anisotropic index of tensile strength

fα Anisotropic index of flexural strength

kα Anisotropic index of Mode I fracture toughness

θ Angle of x -axis to the line linking main crack tip to the microcrack center

φ Microcrack orientation as the angle from x -axis to the 'x -axis

CHAPTER 1: INTRODUCTION 1

CHAPTER 1

INTRODUCTION

This chapter presents the background statements, thesis objectives and the organization of the

entire thesis.

1.1 Background

Under tectonic loading, rocks may naturally exhibit anisotropy with two mechanisms: 1)

anisotropic elasticity of rock forming minerals and the alignment of the grains in preferred

directions; 2) oriented pores and/or microcracks (Phillips and Phillips, 1980). It has been

reported that the alignment of microcrack in granitic rocks correlates well with the anisotropy of

physical properties, such as uniaxial compressive strength (Douglass and Voight, 1969) and

tensile strength (Peng and Johnson, 1972). Using optical techniques, Schedl et al. concluded that

the splitting planes and anisotropy in Barre granite are mainly caused by microcracks (Schedl et

al., 1986). A good correlation between microcrack density, microcrack length, microcrack sets

orientation and fracture toughness has been demonstrated recently (Nasseri and Mohanty, 2008;

Nasseri et al., 2005).

Rocks are much weaker in tension than in compression. The mechanical properties resisting the

tension type failure (i.e. tensile strength, flexural strength and Mode-I or tension mode fracture

toughness/fracture energy) are critical to rock engineering practice such as the stability of mine

roofs, galleries, tunnel boring, cutting, crushing, drilling and blasting. By definition, tensile


strength is the rupture stress in a pure tensile uniaxial stress state. The tensile strength measured

from a bending configuration is termed flexural strength. Mode-I fracture toughness is the

critical stress intensity factor of a Mode-I (i.e. tension mode) crack. It is thus important to

characterize these tension-related properties of anisotropic rocks in general and to understand the

correlation between properties and the microcrack-induced anisotropy in particular. Barre granite

is chosen in this study because it is a well-known anisotropic granite and its microcracks

embedded structure has been well characterized (Nasseri and Mohanty, 2008). In addition, it was

designated as part of a standard rock suite by the U.S. Bureau of Mines (Goldsmith et al., 1976).

Various methods have been proposed for measuring the static tensile strength and Mode-I

fracture toughness of rocks. For tensile strength measurement, direct pull test appear to the most

straightforward choice. However, given the difficulties associated with experimentation in direct

tensile tests, indirect methods serve as convenient alternatives to measure the tensile strength of

rocks; some examples are the Brazilian disc test (Bieniawski and Hawkes, 1978; Coviello et al.,

2005; Hudson et al., 1972; Mellor and Hawkes, 1971), the ring test (Coviello et al., 2005;

Hudson, 1969; Hudson et al., 1972; Mellor and Hawkes, 1971), and the bending test (Coviello et

al., 2005). These indirect methods aim at generating tensile stress in the sample by far-field

compression, which are much easier and cheaper in both sample preparation and experimental

instrumentation than the direct pull test.

Among these indirect methods, the Brazilian test is probably the most popular one due to its

superior features like convenient specimen preparation and easy experimentation. It has been

suggested by the International Society for Rock Mechanics (ISRM) as a recommended method

for measuring the tensile strength of rocks (Bieniawski and Hawkes, 1978). For anisotropic rocks,

many researchers have investigated the tensile strength mostly using Brazilian tests, such as

Berenbaum and Brodie on coals (Berenbaum and Brodie, 1959), Evans on coals (Evans, 1961),

Hobbs on siltstones, sandstones and mudstones (Hobbs, 1964), Mclamore and Gray on shales

(Mclamore and Gray, 1967) and Barla on gneisses and schists (Barla, 1974), Chen et al. on four

types of bedded sandstones (Chen et al., 1998a) and Dai et al. on Barre granite (Dai and Xia,

2010).

Another indirect method is the bending test. Bending of one dimensional specimens (i.e. beams

with circular or rectangular cross section) is very popular in many branches of civil engineering


(Coviello et al., 2005). Three points bending (3PB) and four points bending (4PB) tests are even

adopted as a standard for determining the flexural strength of materials such as natural and

artificial building stones, rocks, cement and concrete (ASTM C99 / C99M-09, 2009; ASTM

C880 / C880M-09, 2009; ASTM Standard C78-09, 2009; ASTM Standard C293-07, 2007; BS

EN 12372, 1999; BS EN 13161, 2008). The measured tensile strength from bending tests, or

flexural strength is generally higher than the tensile strength measured from direct pull or

Brazilian tests (Coviello et al., 2005). Since rocks are usually obtained in the form of rocks cores,

it is thus convenient to use core-based specimens. A semi-circular bend technique is thus

developed in this work to measure the flexural strength of rocks, featuring core-based sample

geometry and bending loading configuration.

To measure Mode-I fracture toughness of rocks, myriads of techniques have also been

documented in the literature, methods including radial cracked ring (Shiryaev and Kotkis, 1982),

notched semi-circular bend (NSCB) (Chong and Kuruppu, 1984; Lim et al., 1994a; Lim et al.,

1994b; Lim et al., 1994c), chevron-notched SCB (Kuruppu, 1997), Brazilian disc (Guo et al.,

1993), and cracked straight through Brazilian disk (CSTBD) (Atkinson et al., 1982; Chen et al.,

1998b; Fowell and Xu, 1994). International Society of Rock Mechanics (ISRM) also proposed

short rod (SR) and chevron bending (CB) tests in 1988 (Ouchterlony, 1988) and cracked chevron

notched Brazilian disc (CCNBD) in 1995 (Fowell et al., 1995). All of those specimens are core-

based, which facilitate sample preparation obtained directly from cores of natural rock masses.

For anisotropic rocks, Kirby and Mazur investigated the fracture toughness on coal and studied

the effects of anisotropic nature of coal to the fracture toughness both experimentally and

analytically (Kirby and Mazur, 1985). Chen and his coworkers determined the mixed-mode (I–II)

fracture toughness of a shale with CSTBD tests (Chen et al., 1998b) and an anisotropic Hualien

marble using the cracked ring test (Chen et al., 2008) and CSTBD tests (Ke et al., 2008). Nasseri

and Mohanty measured fracture toughness of four types of granite with CCNBD (Nasseri and

Mohanty, 2008; Nasseri et al., 2005; Nasseri et al., 2006).

In many mining and civil engineering applications, such as quarrying, rock cutting, drilling,

tunneling, rock blasts, and rock bursts, rocks are stressed dynamically. Accurate

characterizations of rock mechanical properties over a wide range of loading rates are thus

crucial. Researchers also have extended the static method to the regime of dynamic testing. For

the dynamic tensile strength measurement, Zhao and Li (2000) measured the dynamic tensile


properties of granite with the Brazilian tests, with the loading driven by air and oil. To attain

tensile strength of rocks under higher loading rates, a Brazilian test is adopted in the standard

dynamic testing device, the split Hopkinson pressure bar (SHPB). For examples, conventional

SHPB tests were conducted on Brazilian discs of marble (Wang et al., 2006; Wang et al., 2009)

and argillite (Cai et al., 2007) to measure the dynamic tensile strengths. Quasi-static analysis had

been used in these works to relate far-field peak load to the tensile strength of the sample but

without sufficient justification. For Mode-I fracture toughness measurement, Tang tried to

measure dynamic fracture toughness of rocks by three point impact using a single Hopkinson bar

(Tang and Xu, 1990), and Zhang employed the SHPB technique to measure the rock dynamic

fracture toughness with short rod (SR) specimen (Zhang et al., 2000; Zhang et al., 1999). In these

attempts with Hopkinson bar, the evolution of the stress intensity factor (SIF) and the fracture

toughness were calculated using quasi-static analysis without careful consideration of the loading

inertial effects; this will lead to significant errors of the measurements.

The SHPB technique is increasingly becoming the standard method of measuring material

dynamic mechanical properties in the strain rate range 102~104 s-1 for a variety of engineering

materials, such as metals (Gray, 2000), composites (Ninan et al., 2001), concrete (Ross et al.,

1996; Ross et al., 1995), ceramics (Chen and Ravichandran, 2000; Chen and Ravichandran, 1996;

Chen and Ravichandran, 1997), and rocks (Dai et al., 2010c; Dai and Xia, 2010; Shan et al.,

2000; Xia et al., 2008; Zhang et al., 2000; Zhang et al., 1999). Recently, novel techniques in

SHPB tests have emerged. The pulse-shaper technique eliminates the high frequency oscillations

of the stress waves in the dynamic tests, resulting in a smooth loading pulse and a significant

improvement in the interpretation of the dynamic response (Frew et al., 2001; Frew et al., 2002).

It is especially useful for investigating dynamic response of brittle materials such as rocks (Frew

et al., 2001; Frew et al., 2002). The momentum-trap technique in SHPB (Song and Chen, 2004)

can prohibit multiple loading due to the reflection of the stress waves, thus is best suited for

quantitatively assessing the loading wave induced damage to the sample. With these newly

developed techniques in SHPB, the tensile, flexural and fracture tests can be accommodated on

the SHPB system to characterize the corresponding mechanical properties of brittle rocks.


1.2 Problem Statement

Preferentially oriented microcracks in Barre granite are thought to be responsible for the

anisotropic behavior of physical/mechanical properties. It is of interest to characterize the

mechanical properties of Barre granite in general and to understand the correlation between

mechanical properties and the microcrack-induced anisotropy in particular. Researches on some

static physical/mechanical properties of anisotropic Barre granite have been reported. However,

dynamic tests on the Barre granite are rarely investigated in the literature.

Early dynamic compression and tension tests were conducted on Barre granite to investigate the

loading rate effect and the correlation between the micro-structure induced anisotropy and

material mechanical properties (Goldsmith et al., 1976). However, as pointed out by Xia et al.

(2008), the effect of micro-structures on the dynamic behavior of Barre granite was inconclusive

due to lack of control of the loading rate and other deficiencies in the experimental design. For

instance, the pulse-shaper technique (Frew et al., 2001; Frew et al., 2002) is especially useful to

modify the loading pulse and thus facilitate dynamic stress equilibrium for quasi-static stress-

strain analysis in compression tests. In addition, through a careful design of the geometry of the

pulse-shaper, the resulting loading rate or strain rate can be a constant. It is thus necessary to

revisit the tension tests on Barre granite in a systematic manner with newly developed techniques

in SHPB tests. For the dynamic flexural tests and the Mode-I fracture tests of Barre granite, these

has never been attempted in the literature.

Dynamic tension, flexural and fracture tests of rocks are much harder to carry out than their

static counterparts. In contrast to the static tests, there are no ISRM suggested methods for the

dynamic testing of rocks. On the other hand, the extent of anisotropy of some mechanical

properties of Barre granite could be subtle. A delicate and systematic dynamic testing method is

thus urgent. In this thesis, a set of dynamic rock tension, flexural and fracture testing methods are

proposed based on core-based rock samples using SHPB. These methods are then critically

assessed before they are applied to characterizing the anisotropy of Barre in tension, bending and

fracture.

This thesis builds on previous investigation on the effect of microcrack-induced anisotropy of

dynamic compressive strength (Xia et al., 2008) to further investigate the anisotropy of tension,

flexural and fracture properties: tensile strength, flexural strength and Mode-I fracture


toughness/fracture energy of Barre granite under a wide range of loading rates as well as their

relationship to the embedded microcracks preferentially oriented in the granite. The validity of

proposed testing methods in SHPB is carefully checked with the aid of high speed photography.

The tensile strength, flexural strength and Mode-I fracture toughness anisotropy and their micro-

structural correlations are investigated.

1.3 Research Objectives

The ultimate research objectives of this thesis are 1) to quantify the anisotropy of tension-related

failure parameters, including the tensile strength, the flexural strength and the Mode-I fracture

toughness/fracture energy of anisotropic Barre granite over a wide range of loading rates and 2)

to establish the relationship between the preferentially oriented microcrack sets in granitic rocks

and the anisotropy of these properties.

To achieve so, four sub-objectives have to be addressed in turn.

• First, to accurately characterize the tensile strength, the flexural strength and the Mode-I

fracture toughness/fracture energy of rocks; systematic testing methods in conjunction with

data reduction have to be developed. Three sets of dynamic testing methodologies

involving experimentation and calculation equations using the standard dynamic testing

machine (i.e. SHPB) will be proposed to measure these properties.

• Second, the reliability and robustness of the proposed dynamic testing methodologies for

measuring dynamic tensile strength, flexural strength and Mode-I fracture toughness

should be rigorously validated.

• Third, the dynamic tensile strength, flexural strength and Mode-I fracture toughness of

Barre granite are to be investigated with respect to six directions under a wide range of

loading rates. The correlation between the preferred oriented microcrack sets in the Barre

granite and the apparent anisotropy of these mechanical properties will be established.


• Last, the degree of anisotropy for all three parameters appears to be rate dependent: it

diminishes with the dynamic loading rate. Qualitative interpretations are to be given on the

loading rate dependence of the apparent mechanical properties anisotropy. Specifically,

two representative models from two microscopic photos of recovered samples are used to

explain the observed rate dependence of the anisotropy of fracture toughness in the

theoretical framework of crack-microcrack interaction.

1.4 Research Contribution

1．Examined the dynamic Brazilian tests using split Hopkinson pressure bar for measuring the

dynamic tensile strength of rocks. It has been proved that the dynamic Brazilian test is valid,

provided dynamic force balance has been achieved on both ends of the Brailian disc. The

discussion has been published in the journal of Rock Mechanics and Rock Engineerings:

• Dai, F., Huang, S., Xia, K. and Tan, Z., 2010. Some fundamental issues in dynamic

compression and tension tests of rocks using split Hopkinson pressure bar. Rock

Mechanics and Rock Engineering, doi: 10.1007/s00603-010-0091-8.

2．The evaluated dynamic Brazilian testing methods are then used to investigate the tensile

strength anisotropy of Barre granite under a wide range of loading rates. This work has been

summerized in the journal of Pure and Applied Geophysics:

• Dai, F. and Xia, K., 2010. Loading Rate Dependence of Tensile strength anisotropy of

Barre granite. Pure and Applied Geophysics, doi: 10.1007/s00024-010-0103-3.

3．Proposed and evaluated the dynamic semi-circular Bend method using split Hopkinson

pressure bar for measuring the dynamic flexural strength of rocks. The method evaluation has

been detailed in the journal of Review of Scientific Instruments; the rate dependence of the

flexural strength of a granite has been reported in the jouranl of International Journal of Rock

Mechanics and Mining Sciences, as shown below.


• Dai, F., Xia, K. and Luo, S.N., 2008. Semicircular bend testing with split Hopkinson

pressure bar for measuring dynamic tensile strength of brittle solids. Review of Scientific

Instruments, 79(12).

• Dai, F., Xia, K.W. and Tang, L.Z., 2010. Rate dependence of the flexural tensile strength

of Laurentian granite. International Journal of Rock Mechanics and Mining Sciences, 47(3):

469-475.

• The invesigation of the flexural strength anisotropy of the anisotropic Barre granite has

also been reported in this thesis and a draft on this topic will soon be submitted to a jouranl.

4 ． Proposed and evaluated the dynamic notched semi-circular Bend method using split

Hopkinson pressure bar for measuring the dynamic Mode-I fracture toughness of rocks. The

method evaluation has been published in the journal of Experimental Mechanics; Using a laser

gap gauge (LGG) developed by Chen, R., the author explored the method of using the same

notched semi-circular bend to measure the fracture energy of rocks. The collaboration on this

work ends up with a co-authored paper published in the journal of Engineering Fracture

Mechanics, as listed below.

• Dai, F., Chen, R. and Xia, K., 2010. A semi-circular bend technique for determining

dynamic fracture toughness. Experimental Mechanics, doi:10.1007/s11340-009-9273-2.

• Chen, R., Xia, K., Dai, F., Lu, F. and Luo, S.N., 2009. Determination of dynamic fracture

parameters using a semi-circular bend technique in split Hopkinson pressure bar testing.

Engineering Fracture Mechanics, 76(9): 1268-1276.

• The invesigation of the Mode-I fracture toughness and fracture energy of the anisotropic

Barre granite has also been investigated in this thesis and a draft on this topic is under

preparation, to be submitted to a journal.


1.5 Thesis Organization

This thesis comprises seven chapters. The key contents for each chapter are outlined below.

Chapter 1: This chapter presents the background statements, thesis objectives and the

organization of the entire thesis.

Chapter 2: A review of the existing research on the microscopic characterization of the

microstructure of Barre granite, as well as investigation of its mechanical properties is covered.

The methodology for rock tension and fracture tests under both static and dynamic loadings are

reviewed in details. Attention is paid on the dynamic tension and fracture tests of rocks using the

split Hopkinson pressure bar.

Chapter 3: The experimental setup and the working principles are presented, along with

procedures of sample preparations for tension, flexural and fracture tests. Novel techniques in the

split Hopkinson pressure bar, including pulse shaping technique, momentum trap technique and

laser gap gauge system are discussed.

Chapter 4: In this chapter, a Brazilian disc testing method is proposed to measure the

dynamic tensile strength of rocks. Both traditional and pulse shaped split Hopkinson pressure bar

tests are conducted to validate the dynamic Brazilian tests method with isotropic granite

Laurentian granite for demonstration. This method is then applied to investigate tensile strength

of anisotropic Barre granite along six directions. The rate dependence of the tensile strength

anisotropy has been observed and the correlation to the microstructure of Barre granite has been

stated.

Chapter 5: In this chapter, a semi-circular bend flexural testing method is proposed to

measure the dynamic flexural strength of rocks with split Hopkinson pressure bar system. To

validate the dynamic flexural testing method, both traditional and pulse shaped split Hopkinson

pressure bar tests are conducted on isotropic Laurentian granite; and the data reduction method is

critically assessed. This method is then adopted to investigate the loading rate dependence of

flexural strength anisotropy of Barre granite. The result is then interpreted. The flexural strength

is consistently higher than the tensile strength by Brazilian test for all directions; and this has

been interpreted with a non-local failure approach.


Chapter 6: In this chapter, a notched semi-circular bend testing method is proposed to

measure the dynamic Mode-I fracture toughness and fracture energy of rocks; and this novel

method is critically assessed using isotropic Laurentian granite. This method is then applied to

investigating the loading rate dependence of Mode-I fracture properties of anisotropic Barre

granite. The rate dependence of the fracture toughness anisotropy is observed and two conceptual

models abstracted from microscopic thin section photos are constructed to qualitatively

reproduce the rate dependence of the fracture toughness anisotropy in terms of the interaction of

the main crack with pre-existing microcracks preferred oriented along different directions of

Barre granite.

Chapter 7: This chapter summarizes the overall conclusions of the thesis from the preceding

chapters. Future work has also been outlined.

CHAPTER 2: LITERATURE REVIEW 11

CHAPTER 2

LITERATURE REVIEW

A review of the existing research on microscopic characterization of microstructure of Barre

granite, as well as investigation of its mechanical properties is covered. The methodology for

rock tension and fracture tests under both static and dynamic loadings are reviewed in details.

Attention is paid to the dynamic tension and fracture tests of rocks using the split Hopkinson

pressure bar.

2.1 Barre Granite and Its Anisotropy

2.1.1 Microstructural Investigation

Barre granite, the rock chosen for current study in this thesis is obtained from the same source as

that reported by Nasseri and Mohanty (2008). By virtue of recent development of computer-

aided image analysis programs, it is feasible to characterize the microstructure of rocks through

analysis of digital images obtained from thin sections (Launeau and Robin, 1996; Nasseri et al.,

2005).

As shown in 4Figure 2.1, three thin sections are sliced along three orthogonal planes normal to the

three axes along which P-wave velocities were measured. Intermediate, fast and slow directions

were assigned X, Y, and Z axes, respectively (see 4Figure 2.2 also). The mineral and microcracks


can thus be optically traced. The microcracks are of either the intragranular or intergranular type

and are found in quartz and feldspar grains, and along cleavage planes of biotite grains (Nasseri

and Mohanty, 2008).

Figure 2.1 Mineral and microcracks traced from three orthogonal planes for Barre granite;

after (Nasseri and Mohanty, 2008).


Figure 2.2 3D block diagram showing microcracks orientations in Barre granite; rose diagrams show the alignment of microcracks and mineral fabric orientation for each plane; reproduced after (Nasseri and Mohanty, 2008); the letters in the braskets are the directions used in this thesis.

In the YZ plane, microcracks are preferably oriented with an average length of 1.07 mm and

maximum length of 3.5 mm cutting through the larger quartz grains. The 3D block diagram of

microcracks orientation in 4Figure 2.2 reveals the larger microcracks (2~3 mm long) are evident

to run parallel to the Y-axis while the shorter ones (~1 mm) run parallel to the Z-axis. Mineral

fabric orientation is aligned with the direction of longer microcrack orientation with a minimal

angular disagreement of 3˚ with respect to the Y-axis in that plane, and the former yields a shape

ratio of 1.25.

In the XY plane, the intermediate size microcracks (~2 mm) are oriented parallel to the Y-axis

and the shorter ones are again parallel to the X-axis in this plane. The mineral fabric orientation

follows the longer microcrack preferred orientation direction and the former reveal a shape ratio

of 1.25 in that plane.


In the XZ plane, the longer microcracks are aligned sub-parallel to the X-axis, whereas the

smaller microcracks are found to be nearly parallel to the Z-axis. The mineral fabric direction

follows a similar trend as that of preferred microcrack, and shows a mineral shape ratio of 1.07

along the XZ plane. The rose diagram representing the microcrack orientations and length along

specific direction for each plane is shown in 4Figure 2.2.

Douglass and Voight (1969) had pointed out that the microcracks in Barre granite are

preferentially oriented and there is a strong concentration of microcracks within the rift plane

(plane of easiest splitting) and the secondary concentration was found within the grain plane.

According to Freleigh Fitz Osborne (1935), the rift, grain, and hardway are planes approximately

at right angles to one another along which granites fail most easily under tension. The rift is due

to the peculiar properties of quartz and is approximately horizontal in granites. The grain is in the

direction of foliation. The hardway may be a direction at right angles to the other two or may be

determined by tectonic cracks or other features. With reference to the dominant three sets of

microcracks in 4Figure 2.3, it can be concluded that 1) XY plane (normal to the Z axis with the

slowest P-wave velocity) is recognized to be parallel to the rift plane with the dominant

microcracks; 2) YZ plane (normal to the X axis with the intermedial P-wave velocity) is parallel

with the sub-dominant microcrack second set (grain plane) and 3) the least dominant third set

(hard way or most resistive plane) runs parallel with the XZ plane (normal to the Y axis with the

fast P-wave velocity) in Barre granite (Nasseri and Mohanty, 2008). The XY plane, YZ plane

and XZ plane correspond to the quarryman’s description of “rift plane”, “grain plane” and “hard-

way plane” respectively.

2.1.2 Mechanical Properties

The mechanical properties of Barre granite had been investigated by many scholars, mostly on

static behaviors. For examples, Riley and Brace considered the influence of the confining

pressure on the static compressive strength of Barre granite; no consideration has been given to

its anisotropy (Riley and Brace, 1971). Hardy and Jayaraman (1970) measured the static tensile

strength along three orthogonal directions. The strength yields negligible difference in two


orthogonal directions, but drastically different from the third axis, indicating a transversely

isotropic of Barre granite regarding its tensile strength. For fracture toughness measurement,

Iqbal and Mohanty compared the measured toughness of Barre granite with three methods: the

chevron bend (CB) test, the short rod (SR) test and the cracked chevron notch Brazilian disc

(CCNBD) test (Iqbal and Mohanty, 2007). The main purpose of their research is to assess the

ISRM proposed three standard methods, the microcracks induced anisotropy has not been taken

into account (Iqbal and Mohanty, 2007).

After the “rift plane” and the “hard plane” are identified with respect to the three orthogonal axis

sorted with P-wave velocities, Nasseri and Mohanty (2008) measured fracture toughness of

Barre granite with cracked chevron notched Brazilian disc (CCNBD) (Fowell et al., 1995)

prepared along six different directions. 4Figure 2.3 illustrates the 3D block diagram showing

location of CCNBD specimens prepared along each plane with respect to microcracks

orientations in Barre granite. The dominant fracture planes are shown in heavy exaggerated lines

so that the “rift plane” is explicitly visualized. The sample is named XY if it fractures in the

plane normal to X axis and the fracture propagates along Y axis.


Figure 2.3 3D block diagram showing location of CCNBD specimens prepared along each plane with respect to microcracks orientations in Barre granite (dominant fracture planes shown in heavy exaggerated lines); reproduced after (Nasseri and Mohanty, 2008); the letters in the braskets are the directions used in this thesis.

4Figure 2.4 shows the variation of the measured fracture toughness along six directions with the

number of tests along each direction in Barre granite. It is evident that sample ZY yields the least

fracture toughness while sample YX yields the highest. Referring to the identification of the “rift

plane” and the “hard plane”, it is easy to interpret the result. Sample ZY fractures within the

plane normal to Z axis, i.e. the rift plane, and it is thus easier to be broken, leading to the lowest

fracture toughness. In contrast, the plane normal to Y axis is the hardest to split, thus sample YX

split in plane normal to Y axis yields the highest fracture toughness.


Figure 2.4 Variation of fracture toughness measured along six directions with the number of tests along each direction in Barre granite; after (Nasseri and Mohanty, 2008).

Goldsmith et al. (1976) conducted quasi-static tension and compression experiments on an

Instron testing machine and dynamic direct tension and compression tests on a split Hopkinson

bar. In the notation of this work, they used orientation 2 (maximum static Young’s modulus),

orientation 3 (minimum static Young’s modulus), and orientation 1 (intermediate Young’s

modulus) to denote the three orthogonal planes in Barre granite. With respect to the P-wave

velocity, orientation 1 corresponds to the direction with the intermediate P-wave velocity;

orientation 2 corresponds to the direction with the maximum P-wave velocity; orientation 3

corresponds to the direction with the minimum P-wave velocity. Thus, it is easier to identify that

the “rift plane” corresponds to the plane normal to the orientation 3; the “grain plane”

corresponds to the plane normal to the orientation 1; and “hard-way plane” corresponds to the

plane normal to the orientation 2.

Goldsmith et al. (1976) showed that (a) direction 2 has the highest tensile strength and (b) the

tensile strength for each direction has clear rate dependence. With reference to 4Figure 2.3,

because most of the microcracks are parallel with orientation 2, the plane normal to orientation 2

is the “hard-way plane”. It is thus expected that the tensile strength is the highest in direction 2 as

shown by Goldsmith et al. (1976). On the other hand, microcracks are mostly perpendicular to


the orientation 3 (with the minimum P-wave velocity), and thus facilitate opening and linking of

themselves in the direct pull test. The measured tensile strength in direction 3 thus should be the

lowest.

Xia et al. (2008) revisited the dynamic compression tests of Barre granite with SHPB, the effects

of microcrack induced anisotropy on the dynamic response of Barre granite are investigated. The

recently developed techniques in the last a couple of years have been employed in the SHPB

tests. They are pulse shaper technique (Frew et al., 2001; Frew et al., 2002) for achieving stress

equilibrium and momentum trap technique (Nemat-Nasser et al., 1991) for ensuring single

loading pulse for soft recovery of samples. The axial directions of the samples are chosen to be

parallel to the preferred direction of microcracks and the samples are grouped and denoted by Y

(lowest P wave velocity), Z (highest P wave velocity), and X (intermediate P wave velocity).

The results are re-plotted in 4Figure 2.5. It is noted that σm in 4Figure 2.5 designates the maximum

stress achieved during the tests; it is not equivalent to the compressive strength because some

samples tested at low strain rate (~70/s) remained intact after the test. For samples cracked and

fragmented, σm can be taken as dynamic compressive strength. It is shown from 4Figure 2.5 that

for all three sample groups, they have obtained the compressive strength under strain rate ~100/s.

Since the strain rate is constant, they can thus examine the strengths with respect to principal

directions, excluding the rate effects of the strength. Y samples have the highest measured

compressive strength; while the X and Z samples are much less.


Figure 2.5 Strain rate effects of the maximum compressive stress for X-, Y- and Z- samples of Barre granite; reproduced after (Xia et al., 2008); the letters in the braskets are the directions used in this thesis.

The Y axis of the Barre granite block by Xia et al.(2008) is the direction with the lowest P wave

velocity, corresponding to the Z axis in the notation by Nasseri and Mohanty (2008), which

certainly has the lowest P-wave velocity; and the rift plane is normal to this axis. Previous

investigations on the compressive tests of brittle rocks have generally agreed that all the rock

samples failed along macroscopic fractures, which is formed by the growth and coalescence of

microcracks oriented parallel to the maximum compressive stress (Ashby and Sammis, 1990;

Horii and Nemat-Nasser, 1986; Rawling et al., 2002). The rift plane essentially has the most

microcracks, which are oriented perpendicular to the loading direction of Y sample. The

compression in Y direction of Barre granite causes the majority of the microcracks to close

rather than open. It is thus much harder to fail the sample in this direction, resulting in the

highest measured compressive strength. In the other two directions, microcracks in the rift plane

(together with other microcracks sets) contribute to failing of the sample, yielding a lower


compressive strength (4Figure 2.5). This mechanism explains why Y samples have higher strength

than X and Z samples.

2.2 Tension Tests

2.2.1 Static Tension Tests

Like all other brittle solids, rocks are considerably weaker in tension than in compression.

Understanding of tensile strength of rocks and other brittle materials thus bears important

engineering and geophysical applications, such as quarrying, rock drilling, cutting, rock blasting

and rockbursts.

Various methods have been proposed for measuring the tensile strength of rocks. Direct tensile

or pull test is a natural approach to measuring the tensile strength of brittle solids and

International Society for Rock Mechanics (ISRM) has suggested a method for determining direct

tensile strength of rocks (Bieniawski and Hawkes, 1978). However, the stress concentration due

to the sample gripping often induces damage near sample ends, causing premature failure and

deviation from the desired uniaxial stress state. In addition, bending in direct tensile tests due to

imperfections in the sample preparation and misalignment makes it difficult to interpret the

testing results (Coviello et al., 2005).

Given the difficulties associated with experimentation with direct tensile tests, indirect methods

serve as convenient alternatives to measure the tensile strength of rocks; some examples are the

Brazilian disc test (Bieniawski and Hawkes, 1978; Coviello et al., 2005; Hudson et al., 1972;

Mellor and Hawkes, 1971), the ring test (Coviello et al., 2005; Hudson, 1969; Hudson et al.,

1972; Mellor and Hawkes, 1971), and the bending test (Coviello et al., 2005). These indirect

methods aim at generating tensile stress in the sample by far-field compression, which are much

easier and cheaper in both sample preparation and experimental instrumentation than the direct

pull test. Among these indirect methods, the diametrical compression of thin disc specimen is

probably the most popular one due to advantages of convenient specimen preparation and

experimental implementation. This method is termed the Brazilian test, because a Brazilian


engineer Fernando L.L.B. Carneriro first developed and presented this method of measuring the

tensile strength of concrete under quasi-static loading in 1943 at the fifth meeting of the

Brazilian association for technical rules. It has also been suggested by the International Society

for Rock Mechanics (ISRM) as a recommended method for tensile strength measurement of

rocks (Bieniawski and Hawkes, 1978). The Brazilian test has been chosen by many researchers

to measure the indirect tensile strength of rocks and investigate the effect of anisotropy on the

tensile strength. Examples are Berenbaum and Brodie (1959) on coals, Evans (1961) on coals,

Hobbs (1964) on siltstones, sandstones and mudstones, Mclamore and Gray (1967) on shales and

Barla (1974) on gneisses and schists, and Chen et al. (1998a) on four types of bedded sandstones.

2.2.2 Dynamic Tension Tests

Existing attempts to measure rock tensile strength are mostly limited to quasi-static loading,

primarily due to the difficulties in experimentation and subsequent data interpretation for

dynamic tests. However, in many mining and civil engineering applications, such as quarrying,

rock cutting, tunneling, rock blasts, and rockbursts, rocks are stressed and failed dynamically.

Accurate measurement of dynamic tensile strength is thus critical. Direct dynamic tensile testing

is rare (Goldsmith et al., 1976), and existing research efforts have concentrated on extending the

indirect methods from quasi-static to dynamic loading.

Researchers first tried to modify the material testing machine to achieve fast loading. For

example, Zhao and Li (2000) measured the dynamic tensile properties of Bukit Timah granite

from Singapore with the Brazilian disk and three point bending flexural methods using a self-

designed fast-loading material testing machine. The tensile strengths obtained by both methods

increase as the loading rate increases. It was also observed that the flexural tensile strength

determined by the 3-point flexural method is about 2.5 times of the tensile strength determined

by the Brazilian method at the same loading rate, but the reason for it was not given.

It has been generally recognized that the tensile strength of rocks is loading rate dependent. To

characterize the tensile strength under higher loading rates, most researchers tried to use the


popular dynamic testing facility, split Hopkinson pressure bar (SHPB) to achieve wider range of

dynamic loading. The SHPB is increasingly becoming a standard dynamic testing machine for

measuring dynamic mechanical properties in the high loading range for a variety of engineering

materials, such as metals (Gray, 2000), composites (Ninan et al., 2001), concretes (Ross et al.,


Chen and Ravichandran, 1997) and rocks (Xia et al., 2008; Zhang et al., 2000; Zhang et al.,

1999). SHPB has also been adopted to conduct indirect tension tests for measuring the tensile

strength of brittle solids like rocks. Core-based samples have also received wide popularity in

dynamic tests. The Brazilian disk test technique was recently implemented in a two-bar

Hopkinson type loading technique. Tedesco et al. perhaps were the first to perform a Brazilian

test in a Hopkinson bar setup (Hughes et al., 1993; Tedesco et al., 1994; Tedesco and Ross, 1998;

Tedesco et al., 1989; Tedesco et al., 1991). They used this loading technique to measure the

dynamic tensile strength of concrete at the strain rate range of 1 s-1~8 s-1 (Tedesco et al., 1989).

So far an increasing number of researchers have employed a Brazilian disk in Hopkinson bar

compression tests for measuring the dynamic tensile strength of ceramic materials (Johnston and

Ruiz, 1995; Rodriguez et al., 1994), concretes (Ross et al., 1995; Ross et al., 1989) and mortars

(Ross et al., 1989), and an explosive stimulant (Grantham et al., 2004).

For rocks, spalling tests were performed on rocks from three different Canadian mines (Mohanty,

1987) and granites and tuffs (Cho et al., 2003) to study the strain rate dependency of the dynamic

tensile strength. Conventional SHPB tests were conducted using Brazilian disc method on

marbles (Wang et al., 2006; Wang et al., 2009) and argillites (Cai et al., 2007). In these dynamic

Brazilian tests on SHPB, the dynamic loads to the sample are taken as the average of the two

interfacial loads on both ends of the sample; and a standard quasi-static equation for determining

the tensile strength are utilized. For quasi-static and low speed Brazilian tests, it is reasonable to

use the standard static equation to calculate the tensile strength. However, for dynamic Brazilian

test conducted with SHPB featuring stress wave loading, the application of the quasi-static

equation to the data reduction has not yet been rigorously checked. The pulse-shaper technique

(Frew et al., 2002; Frew et al., 2005; Song and Chen, 2004) can facilitate dynamic force balance

and thus reduce the inertial effect, but the extent of such reduction is not adequately examined.

Recently, two types of dynamic indirect tension tests were proposed and examined, i.e. Brazilian

tests and semi-circular bend flexural tests performed on SHPB to measure the tensile strength


(Dai et al., 2010c) and flexural strength (Dai et al., 2008) of Laurentian granite. The differences

between the two strength values measured from two methods are discussed (Dai et al., 2010d).

The methods are then employed to research on the anisotropic Barre granite (Dai and Xia, 2010).

The details are presented in later chapters.

2.3 Fracture Tests

2.3.1 Static Fracture Tests

2.3.1.1 Fracture toughness

In recent years, rock fracture mechanics has been applied as a possible tool for solving a variety

of rock engineering problems, including rock cutting, hydro-fracturing, explosive fracturing,

underground excavation, and rock mass stability (Chen et al., 2008). Rock fracture mechanics is

established within the framework of linear elastic fracture mechanics, which assumes the

material of interest is linear elastic. Rock fracture mechanics is essentially extended from the

Griffith theory (1920) and Irwin’s modification (1957) which recognizes the importance of stress

intensity near a crack tip (Chen et al., 2008).

Irwin introduced the concept of stress intensity factor (SIF) to describe the stress and

displacement field near a crack tip. Depending on the applied stress experienced by the crack, a

crack propagates with the superposition of three basic failure modes, as shown in 4Figure 2.6:

Mode I is the tension/opening mode, where the principal load is applied in a direction normal to

the crack plane, tending to open the cracks (4Figure 2.6a); Mode II is the in-plane shear mode, in

which the load tend to slide one crack face with respect to the other (4Figure 2.6b); Mode III is the

tearing mode or out of plane mode, in which the crack faces are sheared parallel to the crack

front (4Figure 2.6c) (Anderson, 2005). Thus, corresponding to the three fracture modes, there are

three types of SIFs: Mode I (KI), Mode II (KII) and Mode III (KIII).


Figure 2.6 The three basic modes of crack propagation: (a) Mode I, opening mode; (b) Mode II, in-plane shearing; (c) Mode III, tearing mode.

Mode-I fracture is the most encountered fracture mode in nature as well as engineering practice.

Take Mode-I fracture as an example, to express the stress field and displacement field ahead of

the crack tip. 4Figure 2.7 illustrates an element near the tip of a crack in an elastic material; the

stress components of the element are also denoted. The stress fields ahead of the Mode-I crack

tip in an isotropic linear elastic material can be written as the following:

⎥⎦⎤

⎢⎣⎡ −= )

2sin()

2sin(1)

2cos(

2θθθ

πσ

rK I

xx (2.1a)

⎥⎦⎤

⎢⎣⎡ += )

23sin()

2sin(1)

2cos(

2θθθ

πσ

rK I

yy (2.1b)


)2

3cos()2

sin()2

cos(2

θθθπ

σr

K Ixy = (2.1c)

StrainPlaneStressPlane

v yyxxzz

⎩⎨⎧

+=

)(0σσ

σ (2.1d)

0== yzxz ττ (2.1e)

Figure 2.7 Definition of the local coordinate axis ahead of a crack tip. Z direction is normal to the plane.

The displacement relationship for Mode-I are:

⎥⎦⎤

⎢⎣⎡ +−= )

2(sin21)

2cos(

222 θκθ

πr

GK

u Ix (2.2a)

⎥⎦⎤

⎢⎣⎡ −+= )

2(cos21)

2sin(

222 θκθ

πr

GK

u Iy (2.2a)


where G is the shear Modulus. v43−=κ for plane strain and )1/()3( vv +−=κ for plane stress.

The critical value of the SIFs when crack propagation initiates is defined as the fracture

toughness (i.e. for Mode-I, KIC; Mode-II, KIIC; and Mode-III, KIIIC). Fracture toughness thus

serves as a measure of the ability of a material to resist the growth of a preexisting crack under

loading. Fracture toughness of rocks is the most important material property in rock fracture

mechanics. For quasi-brittle geological materials, crack propagation is the major cause of

material failure in many cases. Thus, assessment of fracture toughness is important to the

understanding of failure behavior of structures involving geological materials (Chang et al.,

2002). As stressed by Sun and Ouchterlony (1986), some applications of the fracture toughness

of rocks are listed as below:

(i) A parameter for classifying rock materials;

(ii) An index for fragmentation processes such as tunnel boring and rock blasting;

(iii) A material property in the modeling of rock fragmentation like hydraulic fracturing,

explosive simulation of gas wells, radial explosive fracturing, and crater blasting as well as in

stability analysis.

4Figure 2.8 contrasts the fracture mechanics approach with the traditional tensile strength

approach for structural design and material selection. In the traditional strength approach, the

material is believed to be adequate or the structure is believed to be safe if the induced tensile

stress by the applied stress is lower than the tensile strength. In the fracture mechanics approach,

the fracture toughness is an analogy of the tensile strength; but instead of only one variable, i.e.

applied stress, additional variable is the flaw size, and both the flaw size and the applied stress

contribute to the stress intensity factor (SIF).


Figure 2.8 Comparison of the fracture mechanics approach to design with the traditional strength of material approach: (a) strength approach (b) fracture toughness approach.

2.3.1.2 Rock fracture tests

As a material parameter of rocks, the fracture toughness of rocks can be obtained by designed

experiments. Thus, laboratory testing of fracture toughness of rocks aim at developing

convenient rock samples to determine the critical SIFs using analytical or numerical methods

with experimental recorded loadings as input. In this thesis, the Mode-I fracture will be dealt

with because the opening mode is the most often encountered failure mode; and even for

macroscopic shear or mixed mode failure, opening mode fracture has been observed

microscopically (Ashby and Sammis, 1990; Horii and Nematnasser, 1986; Rawling et al., 2002).


While in earlier attempts of measuring fracture toughness of rocks, ASTM-E399 standard

(ASTM Standard E399-09, 2009) has been followed, which is developed for measuring plane

strain fracture toughness of metallic materials. For brittle geo-materials like rocks, the nature of

fracture process in rocks (brittle) is different from that in most metals (plastic yielding); the

fracture specimens are generally sampled directly from rock cores to avoid the pre-damage

during sample fabrication. Thus, direct application of such standards to rocks remains

inconvenient. Special sample geometries should be developed for fracture toughness

measurements of rocks. Various methods have been proposed in the literature to measure

fracture toughness of rocks, including radial cracked ring (Chen et al., 2008; Shiryaev and Kotkis,

1982), semi-circular bend (SCB) (Chong and Kuruppu, 1984; Lim et al., 1994a; Lim et al.,

1994b; Lim et al., 1994c), chevron-notched SCB (Kuruppu, 1997), Brazilian disc (Guo et al.,

1993), and cracked straight through Brazilian disk (CSTBD) (Atkinson et al., 1982; Chen et al.,

1998b; Fowell and Xu, 1994).

The International Society of Rock Mechanics (ISRM) also proposed short rod (SR) and chevron

bending (CB) tests in 1988 (Ouchterlony, 1988) and cracked chevron notched Brazilian disc

(CCNBD) in 1995 (Fowell et al., 1995). Three types of specimens have been widely used for

determining the pure Mode I fracture toughness of rocks by Ingraffea et al. (1984) on limestone

and granite, Swan and Alm (1982) and Sun and Ouchterlony (1986) on Stripa granite, Gunsallus

and Kulhawy (1984) on sandstone, Ouchterlony (1987) on granite and marble, Shetty et al. (1985)

and Fowell and Xu (1994) on ceramics and rocks, and Backers et al. (2003) on sandstone. All of

those specimens are core-based, which facilitate sample preparation from cores obtained from

natural rock masses.

To calculate the pure Mode I SIF, myriads of analytical and numerical methods have been used

in the literature. Some examples summarized by Ke et al. (2008) are as follows: an approximate

integral solution (Libatskii and Kovchik, 1967); the Fredholm equation (Rooke and Tweed,

1973); boundary collocation method (Isida, 1975); the modified mapping-collocation method for

orthotropic rectangular plates (Gandhi, 1972); an analytical expression to solve an infinite

cracked plate; the dislocation and boundary collocation methods with the superposition

procedure for a Brazilian disc with a central crack (Guo et al., 1993), the dislocation and

boundary collocation methods with the superposition procedure for a Brazilian disc with a

central crack (Awaji and Sato, 1978), a distributed dislocation method for cracked Brazilian disc


(Atkinson et al., 1982); the dislocation method combined with the superposition technique for

cracked straight through Brazilian disc (CSTBD) (Fowell and Xu, 1994); the weight function

method for Brazilian disc with a central crack (Dong et al., 2004). Finite element method has

also been widely used to calculate the Mode I SIF. For example, Murakami (1976) proposed a

simple procedure for the accurate determination of stress intensity factors by the conventional

finite element method; Fischer et al. (1996) used the finite element method combined with the

modified ring test; Lim et al. (1993) used the finite element method to calibrate the SIF for

semicircular specimen under three-point bending loading; Wang et al. (2003) used the ANSYS

sub-model technique in finite element analysis for cracked chevron notched Brazilian disc

(CCNBD).

2.3.2 Dynamic Fracture Tests

2.3.2.1 Early fracture tests

Dynamic fracture plays a vital role in geophysical processes and engineering applications (e.g.,

earthquakes, airplane crashes, projectile penetrations, rock bursts and blasts). Dynamic fracture

problems are branched into two categories: one is the fracture initiation of a static crack under

dynamic loading; the other is a dynamic fracture propagation or arrest of a propagating crack

(Freund, 1990). The dynamic fracture toughness is the key parameter determining the dynamic

fracture initiation of a static crack; and the dynamic fracture energy is associated with the

process of a propagating crack. Accurate measurements of these dynamic fracture parameters

involving fracture toughness and fracture energy are prerequisites for understanding mechanisms

of dynamic fracture and also useful for engineering applications.

Dynamic fracture tests under higher loading rates are more complicated than static ones. This is

due to the inertia effects associated with stress wave propagation in the samples. In early

dynamic fracture experiments, Charpy pendulum impact was widely used to investigate dynamic

fracture behavior of materials. There is a recommended ASTM standard (i.e. ASTM E24.03.03,

1980, proposed standard method for instrumented impact testing of pre-cracked Charpy


specimens of metallic materials) for dynamic fracture toughness tests using Charpy impact

testing with loading rate less than 105 MPa.m1/2s-1. As pointed out by Jiang and Vecchio, many

studies have shown that the conventional Charpy impact test has significant drawbacks as

follows (Jiang and Vecchio, 2009):

1) The dynamic load is realized by physic motion of the hammer. Lack of understanding of the

inertial forces associated with stress wave propagation in the Charpy specimen, the experimental

results are hard to interpret. The stress intensity factor is greatly influenced by the complex

waves (Böhme, 1988).

2) The load obtained from strain gauges mounted on the hammer is different from that applied on

the specimen due to the strong inertial effect. The load may be underestimated due to inertial

effects. In addition, the significant frequency oscillations in the recorded loading evolution make

it difficult to accurately determine the critical load for a fracture (Lorriot et al., 1998).

3) The conventional quasi-static data analysis is no longer applicable for dynamic fracture

toughness reduction since the bending sample is in a non-equilibrium stress condition, due to the

huge inertial effects through the course of the impacting (Böhme and Kalthoff, 1982).

4) The rampant impacting of the pendulum on the sample can cause an unexpected “lost-of-

contact” between the sample and the supporting bases. In that case, the so-called three point

impacting test is actually degenerated into one point impacting (Böhme and Kalthoff, 1982;

Kalthoff, 1985; Marur, 2000).

2.3.2.2 Fracture tests with Hopkinson bar

Hopkinson pressure bar testing, originally developed for dynamic compression tests of

engineering materials, has been modified for conducting dynamic fracture tests of materials

based on one dimensional stress-wave propagation theory.

The loading stress pulse includes both compressive and tensile pulses, and the loading methods

adopted for Hopkinson bar experimental technique include one-bar, two-bar, and three-bar


setups (Jiang and Vecchio, 2009). For example, compressive stress pulse loaded bending fracture

tests may involve one-bar/one-point bend (1PB) impact (unsupported) (Homma et al., 1991;

Rizal and Homma, 2000; Wada, 1992; Weisbrod and Rittel, 2000), one-bar/three-point bend

(3PB) impact (Bacon, 1993; Bacon et al., 1994; Irfan and Prakash, 2000; Mines and Ruiz, 1985),

two-bar (incident and transmitted bars)/3PB (Jiang et al., 2004b; Tanaka and Kagatsume, 1980),

two bar/4PB (Weerasooriya et al., 2006), and three bar (one incident bar and two transmitted

bars, either of transmitted bars as a support)/3PB impact (Yokoyama and Kishida, 1989). Several

specimen geometry configurations corresponding to different Hopkinson bar testing systems

have been proposed. Under tensile stress pulse loading, configurations include edge notched

tensile samples (Owen et al., 1998), double-edge notched tensile sheet samples (Xia et al., 1994),

and center notched tensile samples (Lambert and Ross, 2000). Under compressive stress pulse,

configurations include notched bending samples (Jiang et al., 2004b; Tanaka and Kagatsume,

1980; Weerasooriya et al., 2006), wedge-loaded compact tension (WLCT) specimens loaded by

a compressive pulse (Klepaczko, 1979), compact compressive specimens (CCSs) (Rittel et al.,

1992), and Brazilian disk specimens (Dai et al., 2010a; Lambert and Ross, 2000; Zhou et al.,

2006).

For determining dynamic fracture parameters such as load, displacement, and fracture time,

strain gauge techniques and optical techniques have been used. It can be concluded from the

literature that: 1) Hopkinson pressure bar compressive stress pulse is more popular than the

tensile stress pulse loading technique; 2) that the strain gauge method is widely utilized for

measuring crack initiation time; 3) that quasi-static fracture mechanics theory is applicable for

dynamic fracture toughness measurement under the condition of stress-equilibrium, and 4) that

finite element analysis (FEA) is a fundamental and frequently used method for computing the

dynamic stress-intensity factor. Major developments of the Hopkinson bar based fracture tests in

the literature are highlighted here below 2chronologically according to a recent critical review

(Jiang and Vecchio, 2009).

1977 Hopkinson tensile pulse is employed by Costin et al. (1976) for loading a pre-fatigued

cylinder sample (long bar sample) for fracture toughness measurement, initiating the application

of Hopkinson bar technique in material fracture toughness testing.


1978 Reflected tensile pulse is introduced for loading single-edge cracked samples in a

pressure bar setup by Stroppe et al. (1978).

1979 WLCT sample experimental method is presented by Klepaczko (1979) for determining

dynamic fracture toughness, KId, using Hopkinson pressure bar.

1980 Two-bar (transmitted tube)/3PB loading testing system is proposed for measuring

dynamic load and deflection responses by Tanaka and Kagatsume (1980).

1983 Stress-state equilibrium issue is first considered in Hopkinson compressive bar loaded

CT sample fracture tests by Corran et al. (1983).

1985 One-bar (incident bar)/3PB fracture test method is established by Mines and Ruiz (1985)

as an improvement to classical Charpy impact testing.

1989 Three-bar/3PB fracture test is proposed for dynamic fracture toughness measurement,

and loss-of-contact under stress-wave loading is identified by a simple transverse wave

propagation analysis by Yokoyama and Kishida (1989).

1990 Pulse shaping is employed for reducing the dynamic effect in two-bar/3PB fracture

testing by Ogawa and Higashida (1990).

1991 One-bar bending fracture test is proposed by Homma et al. (1991) for measuring fracture

toughness of polymethyl mechacrylate.

1992 Two-bar/CCS fracture test is presented by Rittel et al. (1992) for determining fracture

toughness of steel.

1992 Brazilian disk samples are used by Nakano et al. (1992) in measuring Mode I and Mixed

Mode I/II fracture toughnesses for brittle materials.

1993 Two-point strain gauge measurement is employed for determining load and load-point

displacement in one-bar/3PB testing by Bacon (1993).

2003 Two-bar/3PB fracture toughness test is proposed by Nwosu et al. (2003) for measuring

Mode II delamination fracture toughness of composite material.


2004 Momentum-tripping technique is adopted in an improved Hopkinson pressure bar loaded

fracture test by Jiang et al. (2004b).

2006 Two-bar/four-point bend (4PB) is proposed by Weerasooriya et al. (2006) for

determining dynamic fracture toughness of ceramic materials.

2007 “Loss-of-contact” is investigated experimentally in a two-bar/3PB testing system by

Jiang and Vecchio (2007a; 2007b) using a novel contact voltage measurement method, and

pulse-shaping effect is reexamined in a two-bar/4PB setup.

2.3.2.3 Dynamic rock fracture tests

Limited attempts have been made to measure the dynamic initiation fracture toughness of rocks.

Using specially designed fast loading material testing machines, Costin (1981), Wu (1986) and

Bazant et al. (1993) measured the fracture toughness of oil-shale, marble, granite and limestone

using three-point bending specimens.

Tang and Xu (1990) tried to measure dynamic fracture toughness of rocks by three point impact

using a single Hopkinson bar. The dynamic load was measured by a load sensor attached to the

impacting bar, and the groove-opening displacement of the sample was also measured optically.

To do this, a synchronous light-chink device was mounted on the sample. By recording the

luminous flux passing through the narrow chink during the test, the groove-opening

displacement was simultaneously monitored. The load is found to increase linearly until a point

Cp, after which the displacement rate D(t) increases sharply. This turning point Cp, in their view,

is the critical point of cracking. The dynamic fracture toughness KIc is then determined from the

stress intensity factor at the critical point using a standard quasi-static equation.

Zhang et al. (1999, 2000) conducted first SHPB measurements on dynamic fracture toughness of

rocks using short-rod specimens through wedge loading. The time-resolved dynamic loadings on

both ends of the specimen were deduced using the standard SHPB data process with waves

recorded by pairs of strain gauges mounted on the incident bar and transmitted bar respectively.


A dynamic Moiré method was used to determine the fracture initiation time. Two optical

gratings were glued to each side of the specimen separated by the main crack plane. The centers

of both gratings are placed on the same section of the tip of the pre-machined crack. During

dynamic fracture, the crack-open-displacement (COD) increases with time. As soon as the crack

approaches the critical state, the rate of the COD reaches an extreme value. This moment was

considered as the critical time tc. The critical compressive force acting on the wedge was thus

determined, and so did the critical stress intensity factor (i.e. fracture toughness) through a quasi-

static equation.

Wang et al. (2010) used two types of holed cracked flattened Brazilian disc samples

diametrically impacted by SHPB to measure the dynamic fracture toughness of marbles. In their

methods, the dynamic loading P(t) on the sample was taken as the average of loads on both

loading interfaces of the sample and the bar; and this loading P(t) was used as input in the

dynamic finite element analysis to deduce the dynamic stress intensity factor KI(t). The fracture

initiation time tf of the disc specimen was resolved from the signal of the strain gauge cemented

on the sample surface near the crack tip. The dynamic fracture initiation toughness was then

taken as the stress intensity factor at the fracture time tf from the evolution curve of dynamic

stress intensity factor.

Compared to the dynamic fracture toughness tests of rocks, even fewer researches have been

tried to measure dynamic fracture energy of rocks directly. There is only one report on the direct

measurement of the energy consumption during the dynamic fracture of rocks. This work done in

2000 by Zhang et al. (2000) was a continuing research of their previous paper published in 1999

(Zhang et al., 1999) to further investigate the energy dissipation during the dynamic rock fracture.

In their work, a high-speed camera was used to estimate the fragment velocities, from which, the

residual fragment kinetic energy was calculated. The total energy consumption can be deduced

from the strain gauge signals considering both the kinetic energy of bar material particles and the

elastic strain energy. The fracture energy and the damage energy thus can be obtained based on

the first law of thermodynamics.

CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 35

CHAPTER 3

EXPERIMENTAL SETUP AND TECHNIQUES

The experimental setup and the working principles are presented, along with procedures of

sample preparations for tension, flexural and fracture tests. Novel techniques in SHPB,

including pulse shaping technique, momentum trap technique and laser gap gauge system are

discussed.

3.1 Samples Preparations

3.1.1 Laurentian Granite

An isotropic fine-grained granitic rock, Laurentian granite is one of the two rocks used in this

research. The mineralogical and mechanical characteristics of Laurentian granite are well

documented (Nasseri et al., 2005). This granite is used to demonstrate and validate the proposed

methods for dynamic tensile, flexural and fracture tests conducted in SHPB. Laurentian granite is

taken from the Laurentian region of Grenville province of the Precambrian Canadian Shield,

north of St. Lawrence and north-west of Quebec City, Canada. The mineral grain size of

Laurentian granite varies from 0.2 to 2 mm with the average quartz grain size of 0.5 mm and the

average feldspar grain size of 0.4 mm, with feldspar being the dominant mineral (60%) followed

by quartz (33%). Biotite grain size is of the order of 0.3mm and constitutes 3–5% of this rock

(Nasseri et al., 2005).


4Figure 3.1 illustrates the procedures for preparing testing samples of Brazilian disc (BD) samples

for tension tests, semi-circular bend (SCB) samples for flexural tests and notched semi-circular

bend (NSCB) samples for fracture tests. Rock cores with a nominal diameter of 40 mm are first

drilled from a rock block and then sliced to obtain discs with an average thickness of 16 mm. All

the disc samples are polished afterwards resulting in a surface roughness variation of less than

0.5% of the sample thickness. These discs are the samples for Brazilian disc (BD) tests. The

semi-circular bend (SCB) samples are subsequently made from the discs by diametrical cutting.

These SCB samples are prepared for flexural tests. A notch with approximately 1 mm thickness

is then machined using a rotary diamond-impregnated saw from the center of the disc

perpendicular to the diametrical cut. These are the notched semi-circular bend (NSCB) samples

for fracture tests.

Figure 3.1 Procedures for preparing three types of samples: Brazilian disc (BD), semi-circular bend (SCB) and notched semi-circular bend (NSCB) samples.

It is noted that sufficient crack tip sharpness is necessary for accurately measuring fracture

initiation toughness (Bergmann and Vehoff, 1994; Suresh et al., 1987). For an ideal crystal, the

naturally formed crack has a finite thickness of the order of atomic spacing; for a polycrystalline

solid, the thickness is comparable to its grain size. The thickness of an intergranular crack is on

the order of the characteristic material length (e.g., the average grain size in a polycrystalline


solid). In the experiments, a 1 mm wide notch was first made in the semi-circular rock disc and

then sharpen the tip with a diamond wire saw to achieve a tip diameter of 0.5 mm. The average

grain size of Laurentian granite is about 0.59 mm (Iqbal and Mohanty, 2007; Nasseri and

Mohanty, 2008), so the diameter of the crack tip is similar to the thickness of naturally formed

cracks. This will ensure accurate measurements of fracture toughness. Indeed, as discussed by

Lim et al. (1994b) and references therein, if the notch radius is smaller than 0.8 mm, there is no

change of measured fracture toughness for rocks they used.

3.1.2 Barre Granite

Barre granite is the anisotropic rock chosen for this study because it exhibits a high degree of

anisotropy (Nasseri and Mohanty, 2008). In addition, it was designated as part of a standard rock

suite by the U.S. Bureau of Mines (Goldsmith et al., 1976). It is aimed to investigating the rate

dependency of anisotropy of mechanical properties tested from three tests, as well as their

relationship with pre-existing microcracks aligned in preferred directions. Barre granite is

obtained from the southwest region of Burlington in Vermont, USA. It is an intrusive deposit of

Devonian age, concordant on a regional scale but discordant at local contacts. It is a fine to

medium grained rock with mineral grain sizes ranging from 0.25 to 3 mm. Quartz makes up 25%

(by volume) of this rock and has an average grain size of 0.9 mm. Feldspar is the dominant

mineral (65%) and has an average grain size of 0.83 mm. The average grain size for biotite (6%)

is 0.43 mm. The microcracks are of either the intragranular or intergranular type and are found in

quartz and feldspar grains, and along cleavage planes of biotite grains (Xia et al., 2008).

Microcracks orientation in Barre granite has been investigated and it has been reported that there

is a strong concentration of microcracks within the rift plane (plane of easiest splitting) and the

hard way (plane of hardest splitting) (Nasseri and Mohanty, 2008; Nasseri et al., 2005). The

Barre granite block used in this research is directly taken from quarried stones with clear

identification of three principal planes. P-wave velocities are then measured along three

orthogonal axes of the block, which is labeled as X, Y and Z axes with respect to slow (3.57

km/s), intermediate (4.00 km/s) and high P-wave velocity (4.75 km/s) respectively (4Figure 3.2).

Sano et al. (1992) determined the principal axes of Barre granite by measuring the P-wave and S-


wave velocities in various directions of propagation and polarization. In their paper, the P-wave

velocities along three principal axes are 3.540 km/s, 3.985 km/s and 4.655 km/s respectively,

very similar to current measurements. Micro-structural examination of thin sections is then

conducted to further confirm the three orthogonal principal directions of the chosen block. 4Figure

3.2 illustrates the 3D relationships between the three sets of microcracks inferred from the

petrographical studies along the three orthogonal axes marked with P-wave velocities. The first

microcrack set runs parallel with the YZ plane (rift plane), the second microcrack set is found to

be parallel to the XZ plane (grain plane), and the third set runs parallel with the XY plane (hard

way or most resistive plane) in Barre granite using the convention of directions in the paper.

When preparing the three types of samples, a similar procedure of fabrication is followed as that

for Laurentian granite. The samples of Barre granite are cored and labeled using the three

principal anisotropic directions shown in 4Figure 3.2. Rock cores with a nominal diameter of 40

mm are first drilled along X- Y- and Z- directions from the same rock block. For each core, the

other two principal directions are also marked. The rock cores are then sliced to obtain disc

samples with an average thickness of 16 mm. All the disc samples are polished afterwards; and

two in-plane principal directions are labeled accordingly. The BD samples were prepared in this

way. The diametrical loading directions are chosen along the two in-plane principal directions.

The rule of nomenclature for the Brazilian disc groups is also shown schematically in 4Figure 3.2

(e.g. 4Figure 3.2a for sample YX), with the first index representing the normal of the disc fracture

plane and the second index indicating the loading direction. Therefore six groups

(directions/configurations), namely XY, XZ, YX, YZ, ZX, and ZY are prepared in this research.

When preparing SCB samples and NSCB samples, the BD samples are first cut diametrically

along the other two in-plane principal material axes to obtain SCB samples. The NSCB samples

are fabricated from SCB samples following the sample fabrication procedures as that for

Laurentian granite samples. The rule of nomenclature is the same as that for the BD samples of

Barre granite, with the first index representing the normal of the potential fracture plane and the

second index indicating the loading direction or the splitting direction. An example of the YX

sample of SCB and NSCB are schematically shown in 4Figure 3.2b and c. In this way, six sample

groups, namely XY, XZ, YX, YZ, ZX, and ZY are prepared for all three types of samples (i.e.

BD, SCB, NSCB) in this research.


Figure 3.2 3D block diagram showing longitudinal wave velocities and the sampling location of Brazilian discs prepared along each plane with respect to microcrack orientations in Barre granite; the first index for sample numbering represents the direction normal to the splitting plane, and the second index indicates the propagation direction of the crack, e.g. Sample YX of (a) BD sample; (b) SCB sample; (c) NSCB sample; the dashed lines depict the failure plane.


3.2 MTS Hydraulic Servo-control System

The static tests are conducted on an MTS hydraulic servo-control testing system (4Figure 3.3).

Testing Star-II (digital controller) is used to control the testing process and MTS Testing Ware-

SX software is used to set the testing parameters. The loading rate applied is based on the

standard testing of rocks in tension (Bieniawski and Hawkes, 1978) and in fracture (Fowell et al.,

1995). For example, for Brazilian tests, a constant loading rate of 200 N/s is applied on all the

tests. The entire load and displacement histories are measured with linear variable displacement

transducers (LVDT) and a 50 kN load cell respectively.

Figure 3.3 Photoes of (a) semi-circular bend and (b) Brazilian test of rock samples in the MTS hydraulic servo-control testing system.


3.3 Split Hopkinson Pressure Bar

The Hopkinson bar experimental technique rooted in the stress wave experiments of iron wires

performed by John Hopkinson (Hopkinson, 1872; Hopkinson, 1901), and later by his son

Bertram Hopkinson (Hopkinson, 1905). A decade later, B. Hopkinson (Hopkinson, 1914)

developed the pressure bar technique to experimentally determine the pressure produced by an

explosive. In 1948, Davies used electrical condenser units in conjunction with oscilloscopes to

record the wave propagation in the pressure bar for the first time (Davies, 1948). The following

year, H. Kolsky proposed and used the split Hopkinson pressure bar (SHPB) to determine the

dynamic compression stress-strain behavior of different materials (Kolsky, 1949). In this

modification, he divided the pressure bar into two parts, which are later called incident/input bar

and transmitted/output bar respectively. These classic studies have established the foundation for

the experimental methods and data analysis strategy of the state-of-the-art SHPB.

The SHPB technique is increasingly becoming the standard method of measuring material

dynamic mechanical properties in the strain rate range 102~104 s-1 for a variety of engineering

materials, such as metals (Gray, 2000), composites (Ninan et al., 2001), concrete (Ross et al.,


Chen and Ravichandran, 1997), and rocks (Dai et al., 2010c; Xia et al., 2008; Zhang et al., 2000;

Zhang et al., 1999).

3.3.1 Working Principle

A 25 mm in diameter SHPB system is used in the study. 4Figure 3.4 shows the photo of the SHPB

setup at the department of Civil Engineering and Lassonde Institute of the University of Toronto.


Figure 3.4 Photo of a split Hopkinson pressure bar (SHPB) system in the Department of Civil Engineering, University of Toronto.

SHPB is composed of three bars: striker bar, incident bar, and transmitted bar (Gray, 2000). The

specimen is sandwiched between the incident bar and the transmitted bar. For the system used in

this research, the length of the striker bar is 200 mm. The incident bar is 1500 mm long and the

strain gauge location is 733 mm from the impact end of the bar. The transmission bar is 1200

mm long and the stain gauge station is 655 mm away from the sample. An infrared detector

system is used together with a two-channel TDS1021 digital oscilloscope to measure the velocity

of the striker bar. An eight-channel Sigma digital oscilloscope by Nicolet is used to record and

store the strain signals collected from the Wheatstone bridge circuits after amplification.

As shown in 4Figure 3.5, during the test, a striker bar is launched by the gas gun; and the impact

of the striker bar on the free end of the incident bar induces a longitudinal compressive wave

propagating in both directions. The left-propagating wave is fully released at the free end of the

striker bar and forms the trailing end of the incident compressive pulse (4Figure 3.5). Upon

reaching the bar-specimen interface, part of the incident wave is reflected (reflected wave) and

the remainder passes through the specimen to the transmitted bar (transmitted wave). The time of


passage and magnitude of these three elastic pulses through the incident and transmitted bars are

recorded by strain gauges.

Figure 3.5 Schematics of a split Hopkinson pressure bar (SHPB) system and the x-t diagram of stress waves propagation in SHPB.

4Figure 3.6 shows the strain gauge data measured as a function of time for the three wave signals

during the dynamic compression testing of a Barre granite sample. The incident and transmitted

wave signals represent compressive loading pulses, while the reflected wave is a tensile wave.

Using the wave signals from the strain gauges on the incident and transmitted bars as a function

of time, the forces and velocities at the two interfaces between the pressure bars and the

specimen can be determined. The input strain pulse, reflected strain pulse and transmitted strain

pulse are denoted as εi(t), εr(t) and εt(t), respectively (4Figure 3.5).


0 200 400 600 800-0.12

-0.09

-0.06

-0.03

0.00

0.03

0.06Reflected wave

Transimitted wave

Out

put (

V)

Time (μs)

Incident wave

Figure 3.6 Strain-gauge data, after signal conditioning and amplification, from a SHPB compression test of a Barre granite sample showing the three stress waves measured as a function of time.

Based on the one dimensional stress wave theory, the dynamic forces on the incident end (P1)

and the transmitted end (P2) of the specimen are (Kolsky, 1949; Kolsky, 1953):

)(1 riAEP εε += , tAEP ε=2 (3.1)

The displacement of the incident bar end (u1) and the transmitted bar end (u2) integrated from the

respective velocities (v1 and v2):

dtCut

ri∫ −=01 )( εε , dtCu

t

t∫=02 ε (3.2)

In the above equations, E is the Young’s Modulus of the bar material, A is the cross-sectional

area of the bar, and C is the one dimensional longitudinal stress wave velocity of the bar.

The histories of strain rate )(tε& , strain )(tε and stress )(tσ within the sample in the dynamic

compression tests (4Figure 3.5) can be calculated as:


⎪⎩

⎪⎨

⎧

++=−−=

−−=

∫)()(

)()()()(

02

0

triAA

t

triLC

triLC

Etdtt

t

εεεσεεεε

εεεε&

(3.3)

where L is the length of the sample and A0 is the initial area of the sample. Assuming the stress

equilibrium prevails during dynamic loading (i.e., tri εεε =+ ), the commonly used formulas are

obtained:

⎪⎩

⎪⎨

⎧

=−=

−=

∫tA

A

t

rLC

rLC

Etdtt

t

εσεε

εε

0)()()(

02

2&

(3.4)

When the specimen deforms uniformly, the strain rate within the specimen is directly

proportional to the amplitude of the reflected wave. Similarly, the stress within the sample is

directly proportional to the amplitude of the transmitted wave. The reflected wave is also

integrated to obtain strain and is plotted against stress to give the dynamic stress-strain curve for

the specimen.

3.3.2 Pulse Shaping

The loading pulse of the conventional SHPB system for materials testing at high strain rates have

an approximately trapezoidal shape acompanied with high level of oscillations. The oscillations

induced by the sharp rising portion of the incident wave causes difficulty in achieving dynamic

stress equilibrium state in the sample. However as discussed before, all the calculation equations

deduced in the SHPB tests requires stress equilibrium in the sample. The results determined from

these calculation equations will induce sizeable errors if a stress non-equilibrium dominates the

sample.


In a review paper by Franz et al. discussing the incident pulse shaping for SHPB experiments

with metal samples (Frantz et al., 1984), the authors emphasized that a slowly rising incident

pulse is a preferred loading pulse in order to minimize the effects of dispersion and inertia; and

thus facilitate dynamic stress equilibrium of the sample. Franz presented experimental results to

show that a properly chosen tip material or pulse shaper can not only provide stress equilibrium

in the sample but also generate a nearly constant strain rate in the sample (Frantz et al., 1984).

Gray and Blumenthal also discussed these issues in their recent review paper (Gray, 2000). To

shape the incident pulse, one way is to modify the shape of the striker. For example, Christensen

et al. used striker bars with a truncated-cone on the impact end in an attempt to produce ramp

pulses (Christensen et al., 1972); Franz used a machined striker bar with a large radius on the

impact face to generate a slowly rising incident pulse for the tests (Frantz et al., 1984); Li et al.

used tapered striker to generate an approximate half-sine loading waveform (Li et al., 2000).

Another way, and a more convenient way is to place a small, thin disc made of soft materials

between the striker and the incident bar. The disc is called the pulse shaper and can be made of

paper, aluminum, copper or stainless steel, with 0.1–2.0 mm in thickness. For examples, Wu and

Gorham used paper shapers on the impact surface of the incident bar to eliminate high frequency

oscillations in the incident pulse for SHPB tests (Wu and Gorham, 1997). Togami et al. used a

thin plexiglass disk to produce non-dispersive compression pulses in an incident bar (Togami et

al., 1996). Chen used a polymer disk to spread the incident compressive pulses for experiments

on silicone rubber (Chen et al., 1999). Song and Chen employed a C11000 half-hardened copper

disk as the front pulse-shaper, and two C11000 annealed copper disks as the rear pulse shapers to

control the profiles of the loading and unloading portions of the incident pulse so that dynamic

stress–strain loops of the subject material can be accurately determined (Song and Chen, 2004).

Given the wide application of the pulse shaper techniques in the SHPB tests, models have been

developed by researchers to guide the design parameters of the shaper. Nemat-Nasser et al.

modeled the plastic deformation of an OFHC copper pulse shaper, predicted the incident strain

pulse, and showed good agreement with some measured incident strain pulses (Nemat-Nasser et

al., 1991). Ravichandran and Subhash presented a method of characteristics analysis for wave

motions in a ceramic sample and provided a criterion for dynamic stress equilibrium

(Ravichandran and Subhash, 1994). Frew et al. extended the work by Ravichandran and Subhash


(1994) to obtain high rate stress–strain data for limestone samples (Frew et al., 2001). They also

presented data showing that the ramp pulse durations could be controlled such that samples could

be unloaded just prior to failure (Frew et al., 2001).

The pulse shaping technique virtually eliminates high frequency oscillations in stress waves,

resulting in a smooth loading pulse and a significant improvement in the interpretation of the

dynamic response. It is especially useful for investigating dynamic response of brittle materials

such as rocks (Frew et al., 2001; Frew et al., 2002). Without proper pulse shaping, it is difficult

to achieve dynamic stress equilibrium in such materials because the sample may fail immediately

from its end in contact with the incident bar when it is impacted by the incident wave. In

hlaboratory, the pulse shaper technique has been used to achieve the dynamic force equilibrium,

during dynamic rock tension and fracture testing. To transform the incident wave from a

rectangular shape to a ramped shape, the main pulse shaper is made up of a thin C11000 copper

disc (with 0.64 mm in diameter and 0.7 mm in thickness). In addition, a small rubber disc (0.64

mm in diameter and 0.3 mm in thickness) is placed in front of the copper shaper to further reduce

the slope of the pulse to a desired value, as schematically shown in 4Figure 3.7a.

During tests, the striker impacts the pulse shapers before the incident bar, thus generating a non-

dispersive ramp pulse propagating into the incident bar and thus facilitating the dynamic force

balance in the specimen (Frew et al., 2001; Frew et al., 2002). The function of the pulse shaper is

to 1) fill out the high frequence noice generated during the impacting and 2) maintain force

equilibrium across the sample. A wide variety of incident pulses can be produced by varying the

geometry of the copper disks as shown in 4Figure 3.7b. Curve A is obtained using a C11000

copper disc as a pulse shaper with 0.64 mm in diameter and 0.2 mm in thickness; Curve B, a

C11000 copper disc shaper with 0.64 mm in diameter and 0.35 mm in thickness; Curve C, a

C11000 copper disc shaper with 0.64 mm in diameter and 0.7 mm in thickness and a small

rubber disc with 0.64 mm in diameter and 0.3 mm in thickness; Curve D, a C11000 copper disc

shaper with 0.64 mm in diameter and 0.8 mm in thickness and a small rubber disc with 0.64 mm

in diameter and 0.3 mm in thickness. Depending on the materials of testing as well as the loading

rate of interest, different loading pulse is needed and can be achieved with proper shaper design.


Striker Bar Incident Bar

CopperRubber

0 50 100 150 200 2500.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

D

CB

Vol

tage

(v)

Time (μs)

without shaper

A

(a) (b)

Figure 3.7 Pulse shapers in SHPB (a) schematic of the assembly (b) unshaped and shaped incident stress pulses.

3.3.3 Momentum Trap

In the traditional SHPB system, the stress pulse travels along the incident bar and load the

sample at its ends; part of the wave is transmitted to the transmitted bar through the specimen

between the incident bar and the transmitted bar. The remaining part of the stress pulse is

reflected back, propagating along the incident bar as tension wave, termed as reflection wave. At

the free end of the incident bar, this tension wave is reflected one more time as compression

wave and reloads the sample. By the same token, the sample may be subjected to multiple

loading due to the stress waves traveling back and forth along the bars. This will make it

impossible to correlate the loading history to the deformation or fracture profiles from the

recovered samples.

To ensure a single loading to be applied on the sample, many applications have been attempted

to prohibit additional loadings on the sample. For example, in compression tests, special fixtures,

such as “stopper rings,” can be used to limit the total axial strain of the sample; as long as the


sample length equals that of the ring, the stopper ring “stops” the loading on the sample by

sustaining the remaining compression pulse. However, for hard and brittle materials with very

small failure strain, such as ceramics and rocks, the “stopper-ring approach” is difficult to

implement. This is because for this class of materials, once microcracks have been generated in

these brittle solids by the initial pulses, the subsequent reflected compression pulses will

inevitably shatter the specimen, making recovery of the sample essentially impossible (Nemat-

Nasser et al., 1991).

To remedy this problem, a momentum trap system similar to that proposed by Song and Chen

(2004) has been developed in this research. A photo of the momentum trap system of SHPB

setup is shown in 4Figure 3.8a, which is composed of a momentum transfer flange that is attached

to the impact end of the input bar and a rigid mass that is attached to the supporting I beam for

the whole bar system.

Figure 3.8 The momentum-trap system: (a) the actual image and (b) the x–t diagram showing its working principle.

As showed in 4Figure 3.8b inset, there is a gap between the flange and the rigid mass. The

distance of the gap d is determined by the velocity of the striker 0v , the length of the input bar l


and the shape of the input pulse. It takes Clt /20 = for the reflected wave to arrive at the impact

end of the incident bar. The reflection wave is then reflected and changed from the tensile wave

to compression wave at the input end. As a result, it will exert dynamic compression on the

sample for a second time. In a similar manner, the sample in a conventional SHPB set up

experiences multiple compressive loading. The main idea of the momentum-trap method is to

absorb the first reflection by a big mass that can be considered as rigid because of its large

impedance (which is equal to CAρ , where ρ is density) compared to the bar. It is required that

when the reflection wave arrives at the front end of the input bar, the flange is in contact with the

big mass. As a result, the reflection wave is stopped by the big mass. This requirement is

expressed as:

∫= 0

0)(

t

i dttCd ε (3.5)

If there is no pulse-shaper between the striker and the input bar, the particle velocity of the input

bar after impact is 1/2 0V , for the case where the striker and input bar are made of the same

material. Denote the length of the striker by sl , the total duration of the loading pulse is Cls /2 ,

which is usually smaller than Clt /20 = . The total displacement of the end of the incident bar

(flange), which is equal to the gap between the flange and the rigid mass that need to be set in

advance is then ∫= 0

0)(

t

i dttCd ε = ∫1

0 02/1t

dtV = ClV s /0 . If there is a pulse-shaper between the

striker and the incident bar, the measured incidence pulse should be used to determine the size of

the gap using Equation (3.5).

As an example shown in 4Figure 3.9, the second compression is indeed reduced a lot so that the

sample will experience essentially single pulse loading (positive: compression; negative: tension).

It is noted that after the 1st incident and reflection, the 2nd incident wave is tensile, which will

separate the incident bar from the sample. The single loading pulse is thus ensured in the

dynamic SHPB tests with momentum-trap. The 2nd reflected wave in 4Figure 3.9 is compressive,

induced by the total reflection of the 2nd incident wave from the free surface of the incident bar

end that is separated from the specimen.


Figure 3.9 Comparison of stress waves from the incident bar, with and without momentum trap; the legends refer to the stress wave with trap.

3.4 Laser Gap Gauge System

The application of the photoelectronic method on the SHPB testing was first suggested by

Wright and Lyon (1959) and was later utilized by Griffith and Martin (1974) to monitor the

displacements of the end faces of cylindrical carbon-fibre composite specimen in the dynamic

SHPB tests. By recording the luminous flux of light passing through the fabricated notch of the

sample, Tang and Xu (1990) measured the groove-opening displacement of the three point

bending fracture sample during dynamic one bar impact tests. The source light used in their tests

was regular white light. With the development of modern photoelectronic techniques, white light

was replaced by a better and stable source of light, the laser. Using self-developed laser line

velocity sensor system, Ramesh and Kelkar (1995) invented a technique for continuously

measuring the projectile velocities in the plate impact experiments. A similar laser system, Laser

Occlusive Radius Detector (LORD), was applied on the Kolsky bar (split Hopkinson pressure


bar) tests by Ramesh and Narasimhan (1996) to measure the radial strain of the deformed

samples of plastic materials. Li and Ramesh (2007) extended the optical technique, LORD, to the

tension Kolsky bar testing for measuring dynamic tension properties of four viscoplastic

materials.

3.4.1 Principles and Setup

In the dynamic fracture tests reported in Chapter 6, a laser gap gauge (LGG) system was

developed by Chen et al., (2009) to monitor the opening of the notch of the NSCB sample and

thus reduce the opening velocity of the cracked fragments. As shown in 4Figure 3.10, the system

consists of two major components: the collimated line laser source and the sensing system.

The laser operates at 670 nm with a 5 mW output power. It has a large field depth and minimal

variations in thickness across the line length. The line is 30 μm thick at 185 mm away, and the

divergence angle is 5°. A cylindrical lens is used to achieve a parallel laser sheet. The plano-

convex cylindrical lens is made from coated BK7 glass. The high performance multilayer anti-

reflection coatings have an average reflectance of less than 0.5 % (per surface) in the wavelength

range of 650-1050 nm. The light detection part consists of a collecting lens and a photodiode

light detector. The collecting lens focuses the incoming laser light into the photodiode detector,

which is placed near its focal point. A narrow-band-pass filter centered at 670 nm is placed in

front of the detector window. The photodiode detector output is pre-amplified, and the

optoelectronics and the preamplifier together have a bandwidth of 1.5 MHz. The output voltage

of the detector is proportional to the total amount of laser light collected. The whole system has a

noise level less than 0.4 mV.

The LGG is mounted perpendicular to the bar axis and the laser passes through the notch in the

center of the specimen. During the test, as the notch opens up, the amount of light passing

through increases, leading to higher voltage output from the detector. The voltage is linearly

proportional to the gap width and thus the crack surface displacement distance can be reduced.


Figure 3.10 Photo and schematics of the laser gap gauge (LGG) system set up perpendicular to the bar axis of SHPB.

3.4.2 Calibration of the System

3.4.2.1 Static calibration

Calibration of LGG is conducted under both static and dynamic conditions. For static calibration,

a set of high precision gauges is used to partly block the probe laser (4Figure 3.11). The blocking

width ranges from 0 to 10 mm at a step of 0.1 mm. A specific blocking width (d) corresponds to

a light-passing width Δd and a certain amount of voltage reading (ΔU) in the detector output

( 4Figure 3.11).


0.0 0.2 0.4 0.6 0.8 .00

1

2

3

4

Δd (m

m)

ΔU (V)1

X

d

Collimated beam Gap gauge

Measurement Linear fitting

Figure 3.11 Static calibration of the LGG system using a gap gauge blocking the collimated beam: schematic setup and the calibration result.

The Δd -ΔU curve exhibits a good linearity, indicating the high uniformity of the laser sheet:

Ukd Δ=Δ (3.6)


where k=4.08 mm/V is the calibration parameter of the LGG system. Denote the standard

deviation as σ, and the error propagation follows from Equation (3.6) as:

22 )/()/(/ Ukd Ukd Δ+≈Δ ΔΔ σσσ (3.7)

Here σk=0.03 mm/V, and σΔU=0.4 mV. As an example, for ΔU = 0.25 V, Δd = 1.02 mm and the

relative error of the displacement measurement is around 0.8 %.

3.4.2.2 Dynamic calibration

The dynamic calibration is carried out with a single Hopkinson bar. In order to get higher cutoff

frequency, a miniaturized 6.35 mm diameter Hopkinson bar is utilized to calibrate the LGG

system. The cutoff frequency fc of a long bar is (Graff, 1975):

vRC

fc02

23.0

π= (3.8)

where R0 is the bar radius, ν and C are Poisson’s Ratio and elastic wave velocity, respectively. fc

is 400 kHz for the 6.35 mm diameter bar, much higher than 100 kHz for the 25 mm diameter bar.

One end of the incident bar is impacted with the striker bar and the LGG is used to monitor the

motion of the other end. When the compressive wave arrives at the free end of the incident bar, it

is reflected as a tensile wave. The incident and reflected waves measured from the strain gauge

glued on the incident bar. The strain signals are corrected for the travel time from the gauge to

the free end of bar. Then the displacement of the free end follows as (Kolsky, 1953):

∫ +=Δt

ri Cdd0

)( τεε (3.9)

where t denotes time. With the LGG output, the displacement of the free end of the incident bar

can be calculated using Equation (3.6) for comparing the values obtained with Equation (3.9).


The comparison shows excellent agreement (4Figure 3.12), demonstrating that the LGG system

has a wide bandwidth sufficient for valid measurements in SHPB.

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

D

ispl

acem

ent (

mm

)

Laser Gap Gauge Strain Gauge

t (μs)

Laser beamX

Incident bar

Figure 3.12 Dynamic calibration of the LGG system: schematic setup and a typical dynamic testing result compared to the predictions by Equation (3.6).

CHAPTER 4: DYNAMIC TENSION TESTS 57

CHAPTER 4

DYNAMIC TENSION TESTS

In this chapter, a dynamic Brazilian disc testing method is proposed to measure the dynamic

tensile strength of rocks using split Hopkinson pressure bar (SHPB). Both traditional and pulse

shaped SHPB tests are conducted to validate the dynamic Brazilian tests method on SHPB with

isotropic Laurentian granite for demonstration. This method is then applied to investigate

tensile strength of anisotropic Barre granite along six directions. The rate dependence of the

tensile strength anisotropy has been observed and the correlation to the microstructure of Barre

granite has been stated.

4.1 Background Studies

Granites may naturally exhibit anisotropy due to pre-existing microcracks that are preferentially

oriented (Sano et al., 1992; Takemura et al., 2003). The granite chosen in this research is Barre

granite, a widely investigated anisotropic granite. It has been confirmed that the splitting planes

and anisotropy of Barre granite are mainly caused by microcracks with scanning electron

microscope (SEM) and transmission electron microscope (TEM) techniques (Schedl et al., 1986).

Tension-type failure is encountered in a wide range of rock engineering applications. It is thus

important to characterize the tensile strength of anisotropic rocks (i.e. Barre granite) in general

and to understand the correlation between strength and the microcracks in specific. In many


mining and civil engineering applications, such as quarrying, rock cutting, drilling, tunnelling,

rock blasts, and rock bursts, rocks are stressed dynamically. Accurate characterizations of rock

tensile strength over a wide range of loading rates are thus crucial.

For static tension tests, there are various methods that have been proposed for measuring the

tensile strength of rocks. Due to the difficulties associated with experimentation in direct tensile

tests, indirect methods were proposed to serve as convenient alternatives to measure the tensile

strength of rocks; some examples are the Brazilian disc test (Bieniawski and Hawkes, 1978;

Coviello et al., 2005; Hudson et al., 1972; Mellor and Hawkes, 1971), the ring test (Coviello et

al., 2005; Hudson, 1969; Hudson et al., 1972; Mellor and Hawkes, 1971), and the bending test

(Coviello et al., 2005). The sample preparation and experimental instrumentation for these

indirect tests are much easier than the direct pull test. Among these indirect methods, the

diametrical compression of thin disc specimen, generally referred to as the Brazilian test, is

probably the most popular one. It has been suggested by the International Society for Rock

Mechanics (ISRM) as a recommended method for tensile strength measurement of rocks

(Bieniawski and Hawkes, 1978). The disc sample used is thus termed Brazilian disc (BD).

Brazilian tests have also been chosen by many researchers to measure the indirect tensile

strength of anisotropic rocks and investigate the effect of anisotropy on the tensile strength.

Examples are Berenbaum and Brodie on coals (Berenbaum and Brodie, 1959), Evans on coals

(Evans, 1961), Hobbs on siltstones, sandstones and mudstones (Hobbs, 1964), Mclamore and

Gray on shales (Mclamore and Gray, 1967), Barla on gneisses and schists (Barla, 1974), and

Chen et al. on four types of bedded sandstones (Chen et al., 1998a).

Brazilian test has also been extended to the dynamic tests for measuring the dynamic tensile

strength of brittle solids like rocks. Using Brazilian test, Zhao and Li (2000) measured the

dynamic tensile properties of granite with a fast hydraulic loading system. For achieving even

higher loading rates, researchers resort to the split Hopkinson pressure bar (SHPB), which is

widely considered as a standard dynamic testing machine. For examples, dynamic Brazilian tests

were conducted in conventional SHPB system on marbles (Wang et al., 2006; Wang et al., 2009)

and argillites (Cai et al., 2007). These attempts followed the pioneer work on dynamic Brazilian

tests of concretes using SHPB (Ross et al., 1995; Ross et al., 1989). For quasi-static and low

speed Brazilian tests, it is reasonable to use the standard static equation to calculate the tensile

strength. However, for dynamic Brazilian tests conducted with SHPB featuring stress wave


loading, the application of the quasi-static equation to the data reduction has not been rigorously

checked yet.

In this chapter, both static and dynamic Brazilian tests are conducted to investigate the

anisotropic Barre granite. For dynamic Brazilian test conducted with SHPB, a rigorous

assessment will be carried out on the validation of the tests using both traditional and pulse

shaped tests with the aid of high speed photograph. This chapter is arranged as follows. Section

4.2 will present the schematics of the dynamic Brazilian test in SHPB. Section 4.3 validates the

proposed dynamic Brazilian test. In section 4.4, the tensile strength anisotropy of Barre granite is

characterized under a wide range of loading rates with evaluated methods. The result is

interpreted in terms of the microstructure of Barre granite. Section 4.4 summarizes the chapter.

4.2 Dynamic Brazilian Test

A close-view of the dynamic Brazilian test in the SHPB system is schematically shown in 4Figure

4.1, where the disc sample is sandwiched between the incident bar and the transmitted bar. The

principle of Brazilian test comes from the fact that rocks are much weaker in tension than in

compression. The diametrically loaded rock disc sample fails first due to the tension along the

loading diameter near the centre. The calculation equation of tensile strength is based on the 2D

elastic analysis as (Michell, 1900):

DBPf

t πσ

2= (4.1)

where Pf is the load when the failure occurs, σt is the tensile strength, D and B are the diameter

and the thickness of the disc, respectively.


B

D

Incident Bar Transmitted Bar

strain gauge

P1 P2

Figure 4.1 Schematic of the Brazilian test in a SHPB system. The Brazilian disc, with a thickness B = 16 mm and diameter D= 40 mm, is sandwiched between the incident and transmitted bars. A strain gauge is mounted on the specimen near the disc centre.

A strain gauge is glued on the disc surface with 5 mm away from the centre of the disc (4Figure

4.1) to detect the rupture onset. This is only for evaluation purpose in Section 4.3. The center

point of the disc emits elastic release waves upon cracking, and this wave causes sudden strain

drop in the recorded strain gauge signal (Jiang et al., 2004a). The peak point of the strain gauge

signal right before the sudden drop corresponds to the arrival of the release wave due to fracture

initiation. It is noted that the original strain gauge signal should be shifted considering the time

the elastic wave propagates from disc centre to the strain gauge.


4.3 Validation of Dynamic Brazilian Test

4.3.1 Dynamic Brazilian Test without Pulse Shaping

4.3.2.1 Dynamic forces and failure sequences with high speed camera

Traditionally, by the direct impact of the striker on the free end of the incident bar in a SHPB test,

the generated incident wave is a square compressive stress wave with a very sharp arising

portion, which is accompanied by high frequency oscillations. As a result, the dynamic forces on

both ends of the sample vary significantly. 4Figure 4.2 depicts a large oscillation of dynamic force

occurring on the incident side and a sizeable distinction between P1 and P2.

0 30 60 90 120

-40

-20

0

20

40

60

Forc

e (k

N)

Time (μs)

In. Tr.(P2) Re. In.+Re.(P1)

Figure 4.2 Dynamic forces on both ends of the Laurentian granite disc specimen tested using a traditional SHPB without pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.

For a valid Brazilian test, the disc sample should break first along the loading direction

somewhere near the centre of the disc. To verify this, a Photron Fastcam SA1 high speed camera


is used to monitor the fracture processes of the Brazilian disc (BD) without pulse shaping. The

high speed camera is placed perpendicular to the sample surface with images taken at an inter

frame interval of 3.8 μs. The failure process of this test without shaping the loading incident

wave is shown as 4Figure 4.3.

Figure 4.3 High-speed video images of a typical dynamic Brazilian test on Laurentian granite without pulse shaping.

The time zero corresponds to the moment when the incident pulse arrives at incident bar-sample

interface. It can be observed that the first breakage emanates from the incident side of the sample

at around 36 μs after the incident wave arrives at the bar/ sample interface. Soon after that,

damages also appear from the transmitted side of the sample (see the image at 55 μs). Thus, the

splitting of the disc (see, the image at 93 μs) is triggered by the damages at the loading points

through a “wedging” process to the centre of the disc. It thus can be concluded that in this case,

the working principle of a Brazilian test is violated. The rectangular incident loading wave with a


sharp rising edge (4Figure 4.2) seems to affect the failure mode of the testing sample significantly.

Since the cracking of the BD initiates from the loading ends, not from somewhere near the centre

of the disc, the standard equation [i.e. Equation (4.1)] is invalid for reducing the tensile strength

from the tensile stress history at the disc centre.

4.3.2.2 Evaluation of the quasi-static BD equation

The dynamic finite element analysis represents the accurate stress history at any point inside the

disc. A commercial finite element software ANSYS is employed in the calculation. The finite

element model is meshed with quadrilateral eight-node element PLANE82, with a total of 4,800

elements and 14,561 nodes (4Figure 4.4).

Assuming linear elasticity, this analysis solves the following equation of motion with the

Newmark time integration technique:

u&&ρ=⋅∇ σ (4.2)

where σ is the stress tensor, ρ denotes density, and u&& is the second time derivative of the

displacement vector u. The input loads in the finite element model are taken as the dynamic

loading forces exerted on the incident and transmitted side of the specimen, which are calculated

using Equation (4.1) with measured waves.


P1 P2

Figure 4.4 Mesh of the Brazilian disc for the finite element analysis with ANSYS; P1 and P2 are the diametrical forces on both loading ends.

4Figure 4.5 shows the evolutions of tensile stress and compressive stress at the disc centre

calculated from both static analysis (i.e. standard Brazilian equation) and dynamic finite element

analysis. The static analysis is carried out with Equation (4.1) using the transmitted force on the

sample ( 4Figure 4.1). The overall trends of the two curves match with each other but the dynamic

ones feature fluctuations. Furthermore, the dynamic tensile stress is far from linear and therefore

it is difficult to achieve a constant tensile loading rate. Consequently, the tensile stress from the

quasi-static data reduction with the far-field load recorded from the transmitted bar cannot reflect

the transient tensile stress history in the Brazilian disc. The usage of the far-field loads such as

the transmitted force to obtain the tensile stress with standard Brazilian test will lead to very

large errors in the result.


0 30 60 90 120-100

-80

-60

-40

-20

0

20

40

Com

pres

sive

Stre

ss (M

Pa)

Time (μs)

Quasi-static Dynamic

0 30 60 90 120

-10

0

10

20

30

40

Tens

ile S

tress

(MP

a)

Time (μs)


(b)

(a)

Figure 4.5 (a) Tensile stress σx (b) compressive stress σy histories at the center of a Brazilian disc from dynamic finite element analysis and quasi-static equation in a typical SHPB Brazilian test on Laurentian granite without pulse shaping.


The peak value of the transmitted force P2(t) is normally taken as Pf for standard Brazilian

equation [Equation (4.1)] to calculate the material tensile strength (Cai et al., 2007). Therefore,

the onset instant of fracture as identified from the strain gauge history recorded on the specimen

should coincide (approximately) with the peak instant of P2(t) after appropriate time corrections.

4Figure 4.6 compares the strain gage signal with the dynamic forces P1(t) and P2(t). The strain

gage signal features significant fluctuation with three peaks. This scenario cannot be traced from

the transmitted wave which shows only signal peak. In contract, the strain gauge signal is

markedly affected by the force on the incident side of the sample [P1(t)], which also exhibits

large fluctuation. The inertial effect dominates in the dynamic test. The transmitted force in this

case cannot be regarded as the bearing load to the sample, and its peak does not coincide with the

rupture time. Therefore, the tensile strength in the sample cannot be deduced from far-field

measurement via quasi-static analysis.

0 30 60 90 120-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Strain gauge Tr. In.+Re.

Time (μs)

Vol

tage

(V)

-10

0

10

20

30

40

50

Force (kN)

Figure 4.6 Comparison of strain gage signal with the dynamic forces on both loading ends of the disc in a dynamic Brazilian test on Laurentian granite using a traditional SHPB without pulse shaping.


4.3.2 Dynamic Brazilian Test with Careful Pulse Shaping

4.3.2.1 Dynamic forces and failure sequences with high speed camera

4Figure 4.7 illustrates the time-varying forces in a typical test with careful pulse shaping. The

incident wave is shaped to a ramp pulse with a rising time of 180 µs, and a total pulse width of

300 µs. It is evident that the time-varying forces on both sides of the samples are almost identical

before the peak point is reached during the dynamic loading. The resulting forces on both side of

the sample also feature a linear portion before the peak, thus facilitating a constant loading rate

via σ& =2k2/(πDB), where the parameter k2 is illustrated in 4Figure 4.7.

0 50 100 150 200 250 300-30

-20

-10

0

10

20

30

40

Forc

e (k

N)

Time (μs)

In. Tr.(P2) Re. In.+Re.(P1)k2

Figure 4.7 Dynamic forces on both ends of a Laurentian granite disc specimen tested using a modified SHPB with careful pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.

High speed camera is also used to capture the failure sequences of the BD sample with force

balance achieved in the test. 4Figure 4.8 presents the key frames with representative features. In

sharp contrast to the images from the non-pulse-shaped Brazilian tests, this disc cracks near the


centre and the primary crack occurs at around 160 μs. The crack then propagates bilaterally to

the loading ends. The next two frames illustrate the splitting trajectory of the sample before it is

completely split into two fragments approximately along the centre line of the sample in the last

frame (4Figure 4.8). It is also noted that after the initiation of the primary crack, one secondary

crack is visible near the loading ends at time instant 236 μs. Since the splitting of the disc

initiates near the centre, the tensile strength can be determined as long as the tensile stress of the

disc at failure can be accurately determined. Next, we need to evaluate whether the standard BD

equation [i.e. Equation (4.1)] can be used to deduce the tensile strength.

Figure 4.8 High-speed video images of two typical dynamic Brazilian tests on Laurentian granite with careful pulse shaping.


4.3.2.2 Validation of quasi-static BD equation

For a conventional dynamic compression test with SHPB or direct tension test with split

Hopkinson tension bar (SHTB), the sample is cylindrical and thus the force balance on the ends

ensures the stress equilibrium throughout the sample. However, the disc is two dimensional (2D);

force balance on the boundaries (4Figure 4.7) does not necessarily ensure dynamic equilibrium

within the entire sample. A further comparison of the stress history at a point of interest from full

dynamic analysis with that from quasi-static analysis is necessary.

The transient dynamic stress history at the disc centre (potential failure spot) is calculated and

compared with that from a quasi-static analysis using Equation (4.1). The histories of the stress

components σx (in tension) and σy (in compression) for dynamic and quasi-static finite element

analyses are compared in 4Figure 4.9(a) and (b) respectively. The stress states at the disc centre

from both quasi-static and dynamic data reductions match with each other. Thus, provided force

balance on the sample ends, the quasi-static analysis with the far-field loading measured as input

can accurately represent the stress history in the sample.

4Figure 4.10 shows the signal of the strain gauge mounted on the sample, compared with the

transmitted force. Only one peak (A) of the signal is registered by the stain gauge, occurring at

time 149 µs. Thus, the breakage initiation time is designated by the peak A at the time of 149 µs.

Because the peak transmitted force occurs at time 152.5 µs, it is thus delayed only by 3.5 µs after

the measured onset of breakage. It is concluded that in this case, the peak far-field load matches

with the breakage onset with negligibly small time difference. The small time difference of 3.5

µs can be interpreted as follows. The release waves travel at the sound speed of the rock material

(around 5 km/s) and the distance between the fracture location and the supporting pin is 20 mm.

It thus takes around 4 µs for the first release wave to reach the supporting pins, where the

transmitted wave is recorded and also illustrated in 4Figure 4.10. Due to the interaction between

the release wave and the pins, the load on the transmitted side decreases (4Figure 4.10). Thus, the

peak of the transmitted force can be regarded as synchronous with the single peak of the strain

gauge signal (the rupture onset).


0 50 100 150 200 250 300

-60

-45

-30

-15

0

15

Com

pres

sive

Stre

ss (M

Pa)

Time (μs)


0 50 100 150 200 250 300-5

0

5

10

15

20

Tens

ile S

tress

(MP

a)

Time (μs)


(b)

(a)

Figure 4.9 (a) Tensile stress σx (b) compressive stress σy histories at the center of a Brazilian disc on Laurentian granite from both dynamic and quasi-static finite element analyses in a typical SHPB Brazilian test with pulse shaping.


0 50 100 150 200 250 300

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

3.5 μs

Strain gauge Tr.

Time (μs)

Vol

tage

(V)

0

5

10

15

20

Force (kN)

A

B

Figure 4.10 Comparison of the strain gage signal with the transmitted force for a dynamic Brazilian test on Laurentian granite using a modified SHPB with careful pulse shaping.

From 4Figure 4.9 and 4Figure 4.10, it thus can be concluded that, provided force balance has been

achieved on both ends of the Brazilian disc, the dynamic tensile strength can be calculated from

the quasi-static equation. For the particular test shown above, the measured tensile strength in the

SHPB Brazilian test with proper pulse shaping is calculated to be 18.9 MPa at a loading rate of

233 GPa/s.

The above validation demonstrates that in a modified SHPB test with proper pulse shaping, the

dynamic force balance within the Brazilian disc can be achieved. Thus, the tensile stress state at

the disc centre can be calculated with simple quasi-static analysis. Moreover, high-speed

photography visualizes that the disc sample failures near the center of the disc rather than the

loading ends. The rupture time synchronizes with the peak of the transmitted pulse recorded in

the SHPB system after corrections for travel time. Therefore, the dynamic tensile strength can be


calculated from the peak of the transmitted wave measured in the SHPB system with quasi-static

analysis.

Thus, despite the 2D configuration of the Brazilian disc in the SHPB testing, as long as the force

balance on both ends of the sample can be guaranteed, it is highly feasible to achieve stress

equilibrium in the sample and also the synchronization of the rupture onset with the peak of the

transmitted pulse in brittle rocks as shown above, and the simple quasi-static analysis is valid for

data reduction. This method thus provides an efficient way of determining the dynamic tensile

strength of rocks.

4.3.2.3 Necessity of using the loading jaws in dynamic BD tests

It is noted that in the ISRM suggested Brazilian test method, two steel loading jaws are used to

transfer the load to the disc shaped rock samples diametrically over an arc angle of

approximately 10° at failure (Bieniawski and Bernede, 1979). The jaws are designed to reduce

the localized stress concentration at the loading ends and thus to prevent the failure at the loading

ends. This technique works well for static loading while for SHPB tests, the extra interfaces

between the bar and the jaw will complicate the wave stress propagation and increase the

difficulties of experimentation. Furthermore, in the foregoing high speed camera snapshots of the

dynamic Brazilian test without jaws, no obvious pre-mature breakages are observed. It is thus

concluded that for SHPB tests, the loading jaws might not be necessary. To further assess this

postulation, two sets of dynamic BD tests, one with jaws and one without jaws, were conducted.

These tests were conducted with careful pulse shaping and thus the dynamic force balance was

achieved in all tests.

The radius of the jaws is chosen as 30 mm, 1.5 times of the radius of the disc sample as

suggested by the ISRM standard (Bieniawski and Bernede, 1979). 4Figure 4.11 illustrates the

measured tensile strength of Laurentian granite from dynamic Brazilian tests with and without

employing curved jaws at the loading ends; the insert in 4Figure 4.11 shows the sample assembly

with the loading jaws. The consistency of the strength values from two sets of tests clearly


confirmed previous assumption. The simplicity of the experimentation will facilitate the

standardization of the dynamic BD method using SHPB.

0 1000 2000 3000

20

30

40

50

Tens

ile S

treng

th (M

Pa)

Loading Rate (GPa/s)

With jaws Without jaws

Front jaw Rear jaw

Figure 4.11 The measured tensile strength of Laurentian granite from dynamic Brazilian tests with and without employing jaws.


4.4 Tensile Strength of Barre Granite

4.4.1 Determination of Anisotropic Tensile Strength

4.4.1.1 Stress distribution in the disc sample

For the static test, the disc samples are compressed diametrically with loading platens in the

MTS hydraulic servo-control testing system. 4Figure 4.12(a) schematically shows the loading

scheme of a Brazilian disc, where D and B are the diameter and the thickness of the disc,

respectively and P is the diametrical load. For the dynamic test, the disc specimen in the SHPB

system is shown schematically in 4Figure 4.12(b), where the sample disc is sandwiched between

the incident bar and the transmitted bar.

P1

B

D

P2

(a)

B

D

P

P

(b)


x

y

x

y

y

z

z

y

Figure 4.12 Schematics of a Brazilian test in (a) the material testing machine and (b) the SHPB system.

Let x, y, z be a global Cartesian coordinate system shown in 4Figure 4.12, the y-axis defines the

loading direction and the z-axis denotes the axial direction of the disc. In the quasi-static elastic


equilibrium, the components of the stress tensor for any point (x, y) in the disc can be expressed

as follows:

xxx fDBP

⋅=π

σ 2, yyy f

DBP

⋅=π

σ 2, xyxy f

DBP

⋅=π

τ 2 (4.3)

where σx, σy and τxy are three components of the stress tensor and fxx, fyy and fxy are the

corresponding components of the dimensionless stress tensor and can be calculated using

numerical tools according to Equation (4.4). Compression is positive in the following

calculations.

DBP

f xxx

π

σ2

= ,

DBP

f yyy

π

σ2

= ,

DBP

f xyxy

π

τ2

= (4.4)

To measure the indirect tensile strength of anisotropic rocks by Brazilian test, a thorough

analysis of the stress state in the anisotropic disc is required. In this work, finite element analysis

using ANSYS is conducted to analyze the stress state in the anisotropic rock disc for all the six

sample configurations. Quadrilateral eight-node element PLANE82 is used in the analysis, and

the finite element model consists of 4,800 elements and 14,561 nodes in total. Barre granite is

considered orthotropic and the nine stiffness constants Cijkl has also been documented (Sano et al.,

1992) as C1111= 32.70 GPa, C2222= 41.69 GPa, C3333= 56.17 GPa, C2323= 20.59 GPa, C3131= 17.67

GPa, C1212= 15.78 GPa, C2233= 6.43 GPa, C3311= 3.93 GPa, C1122= 3.45 GPa. For comparison

purpose, the stress state of the Brazilian disc is also analyzed assuming that the rock is isotropic.


(a)

(b)

(c)

(d)

(e)

(f)

-0.8

-0.6

-0.4

-0.2

0.0

0.8

0.2

0.4

0.20.

4

0.8

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.8 0.8

0.20.4

0.00.0

0.0

5.0

5.0

3.3

4.2

3.3

4.2

2.5

1.7

0.8 2.5

1.7

0.8

-1.0

-0.75

-0.25

-0.5

-1.0

-0.75

-0.5

-0.25

1.0

0.75

0.250.5

0.750.50.25

1.0

0.0 0.0

0.0

0.0

0.20.

4

0.80.8

0.2

0.4

-0.8

-0.6

-0.4

-0.2

-0.8

-0.6

-0.4

-0.2

0.0

0.00.0

0.0

0.2

0.4

0.8 0.8

0.20.4

2.5

1.7

0.8 2.5

1.7

0.8

5.0

5.0

3.3

4.2

3.3

4.2

-1.0

-0.75

-0.5

-0.25

0.0

0.750.50.25

1.0

0.0

0.0

0.0

1.0

0.75

0.250.5

-1.0

-0.75

-0.25

-0.5

Figure 4.13 Stress trajectories of a Brazilian disc under quasi-static deformation. (a) fxx, (b) fyy and (c) fxy with isotropic model, and (d) fxx (e) fyy and (f) fxy for sample YX using anisotropic model (positive for compression, negative for tension).


For the isotropic case, 4Figure 4.13a, b and c show the distribution of the dimensionless stress

components fxx, fyy and fxy respectively. The calculated values of fxx, fyy and fxy at the centre of the

disc (potential failure spot) are fxx ~ -1, fyy ~ 3 and fxy = 0, respectively. For anisotropic case, eight

sample configurations XY, XZ, YX, YZ, ZX, and ZY are analyzed; and similar symmetrical

stress contours as the isotropic case are observed (see 4Figure 4.15, 4Figure 4.16 and 4Figure 4.17).

The stress trajectories of the fxx, fyy and fxy for sample YX are illustrated and compared with that

for isotropic case in the 4Figure 4.13d to f. The tensile stress distribution near the centre of the

disc is quite uniform for the anisotropic YX sample (4Figure 4.13d and e), very similar to the

isotropic case ( 5Figure 4.13a and b). The shear stress components (5Figure 4.13f) along the loading

diameter and the horizontal diameter are zero due to the intentional coring and loading along

three predetermined material symmetrical plane X, Y and Z. Therefore, the fxx and fyy along the

loading direction in the 5Figure 4.13d and 5Figure 4.13e accurately represent the dimensionless in-

plane principal stress σ1 and σ2.

A complete suite of stress trajectories of the fxx, fyy and fxy for Brazilian disc under quasi-static

deformation for the isotropic case and six anisotropic sample configurations are illustrated in

5Figure 4.14, 5Figure 4.15, 5Figure 4.16 and 5Figure 4.17.


(a)

(b)

(c)

Figure 4.14 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) fxx, (b) fyy and (c) fxy with isotropic model.


(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.15 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) fxx, (b) fyy and (c) fxy for sample XY, and (d) fxx (e) fyy and (f) fxy for sample XZ (positive for compression, negative for tension).


(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.16 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) fxx, (b) fyy and (c) fxy for sample YX, and (d) fxx (e) fyy and (f) fxy for sample YZ (positive for compression, negative for tension).


(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.17 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) fxx, (b) fyy and (c) fxy for sample ZX, and (d) fxx (e) fyy and (f) fxy for sample ZY (positive for compression, negative for tension).


4.4.1.2 Tensile strength

Using Equation (4.3), the stress state at any point within the disc can be fully determined by the

three dimensionless stress components fxx, fyy and fxy. 5Figure 4.13 shows that for points along the

loading diameter of the anisotropic Brazilian disc, the shear stress is zero and the tensile stress is

almost constant near the centre of the disc. In addition, the corresponding compressive stress is

very similar to the isotropic case with a value around three times of the tensile stress. The

vanishing of the shear stress components along the loading diameter implies the coincidence of

the in-plane principal stress σ1 and σ2 with σx and σy. For the tensile strength determination for

anisotropic Barre granite in this study, we use the same assumption as Chen et al. (1998a) that

the indirect tensile strength is given by the maximum absolute value of the tensile stress σx

perpendicular to the loading diameter at the disc centre:

FDBPf

t ⋅=π

σ2

(4.5)

where σt is the tensile strength and Pf is the load when the failure occurs. F is fxx at the centre of

the disc with coordinates (0, 0). The calculated dimensionless factors F for all sample

configurations as well as the material properties used in the finite element analysis are tabulated

in 5Table 4.1.

Table 4.1 The material properties used in the finite element model of BD samples of Barre

granite along six directions.

Sample Suites

F

xE (GPa)

yE (GPa)

xyG (GPa)

xyv

XY 0.9215 40.7 32.2 15.8 0.093 XZ 0.8334 54.8 32.2 17.7 0.105 YX 1.0452 32.2 40.7 15.8 0.073 YZ 0.9112 54.8 40.7 20.6 0.146 ZX 1.1058 32.2 54.8 17.7 0.062 ZY 1.0769 40.7 54.8 20.6 0.108


For Brazilian tests conducted in the MTS system, the quasi-static equation, Equation (4.4) is

accurate while for dynamic Brazilian tests in the SHPB system, a quasi-static stress state in the

sample disc during the test has to be checked before Equation (4.4) is used. This is because in

dynamic tests, there exists the so-called inertial effect associated with stress wave loading as

shown by Böhme and Kalthoff (1982). This inertial effect will lead to errors in data reduction if a

quasi-static analysis is used. Using the pulse-shaper technique in SHPB tests (Frew et al., 2002),

it has been demonstrated in the previous section that the dynamic forces on both ends of the

specimen can effectively minimize the inertial effect even for complicated sample geometry as

Brazilian disc specimen (Dai et al., 2010c).

4.4.2 Tensile Strength Anisotropy

4.4.2.1 Dynamic equilibrium

In order to guarantee a quasi-static state in the dynamic Brazilian test, pulse shaping technique is

deployed for all the dynamic tests. The dynamic force balance on the two loading ends of the

sample is critically assessed. To compare the force histories of these two, the time zeros of the

incident and reflection stress waves are shifted to the sample-incident bar interface and the time

zero of the transmitted stress wave is shifted to the sample-transmitted bar interface invoking 1D

stress wave theory. 5Figure 4.18 compares the time-varying forces on both ends of the sample in

the typical test with pulse shaping. It is evident from 5Figure 4.18 that with pulse shaping, the

dynamic forces on both sides of the samples are almost identical before the critical failure point

is reached during the dynamic loading.


0 40 80 120 160-40

-20

0

20

40

In. Re. In.+Re. (P1) Tr. (P2)

Forc

e (k

N)

Time (μs)

Figure 4.18 Dynamic force balance check for a typical dynamic Brazilian test of Barre granite with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.

5Figure 4.19 and 5Figure 4.20 shows the exemplars of the virgin Brazilian discs and recovered disc

samples after the tests. It is noted that for all the dynamic Brazilian disc tests, the momentum

trap technique in the SHPB system is also used. Thus, all BD samples are subjected to single-

pulse loading with the momentum trap technique, which prevents further damage to the sample

due to the multiple loading pulses. The disc samples are split diametrically along the loading

directions, as shown in 5Figure 4.20. Several secondary cracks can be seen in some recovered

samples. As proven previously by high-speed camera images, this scenario will not affect the

tensile strength determination with dynamic Brazilian tests via SHPB.


Figure 4.19 Virgin Brazilian discs of Barre granite prepared for the test; each division in the scale denotes 1 mm.

Figure 4.20 Recovered Brazilian discs of Barre granite after tests; each division in the scale denotes 1 mm.


4.4.2.2 Static tensile strength anisotropy

The static tensile strength values are taken as the average of three individual tests for each

sample group. 5Figure 4.21a depicts the variation of static tensile strength measured along six

different directions of Barre granite. The measured static tensile strength exhibits very strong

anisotropy. The average tensile strength for the two sample configurations with the same

splitting plane X (i.e. XY, XZ) yields the lowest tensile strength of 9.5±0.14 MPa and 8.8±0.11

MPa respectively; configurations with the splitting plane Y (i.e. YX, YZ) owns intermediate

tensile strength of 13.0±0.17 MPa and 11.8±0.10 MPa; whereas the configurations with the

splitting plane Z (i.e. ZX, ZY) exhibit the highest strength values of 17.1±0.15 MPa and

16.5±0.17 MPa, respectively. The highest strength value (17.1 MPa from sample ZX) is almost

twice of the lowest one (8.8 MPa from sample XZ).

5Figure 4.21b also shows the apparent tensile strength if an isotropic rock is assumed for all the

sample groups. The X plane remains the weakest plane to split and Z plane stays the strongest.

However, with the isotropic model, sample XY has the lowest tensile strength of 10.3 MPa,

rather than sample XZ (8.8 MPa) from the orthotropic model; sample ZX remains to be the

toughest to split but the strength value drops from 17.1 MPa (5Figure 4.21a) to 15.5 MPa ( 5Figure

4.21b). In addition, the ratio of the highest tensile strength to the lowest one decreases to 1.50

with isotropic model. The ineligible discrepancy between those two treatments reveals that the

consideration of the material anisotropic elasticity is necessary for tensile strength determination

of anisotropic granite. Therefore, the material anisotropic elasticity of Barre granite is considered

for both static and dynamic data analyses.


(a)

(b)

XY XZ YX YZ ZX ZY0

4

8

12

16

Anisotropic Model

16.517.1

11.813.0

8.8

Tens

ile s

treng

th (M

Pa)

9.5

XY XZ YX YZ ZX ZY0

4

8

12

16 15.315.5

12.912.5

10.610.3

Tens

ile s

treng

th (M

Pa)

Isotropic Model

Figure 4.21 The variation of static tensile strength of Barre granite along six directions, i.e. XY, XZ, YX, YZ, ZX and ZY, using (a) orthotropic model (b) isotropic model.


4.4.2.3 Dynamic tensile strength anisotropy

All the tensile strength values with corresponding loading rates are tabulated in 5Table 4.2.

Table 4.2 Tensile strengths of Barre granite along six directions from both static and

dynamic Brazilian tests.

XY XZ

No. σ& (GPa/s) tσ (MPa) No. σ& (GPa/s) tσ (MPa)

1 1.8E-04 9.4 1 1.7E-04 8.9 2 1.8E-04 9.4 2 1.7E-04 8.7 3 1.8E-04 9.7 3 1.7E-04 8.8 4 239.6 19.2 4 175.0 15.1 5 458.5 24.9 5 355.4 21.3 6 745.5 29.2 6 746.7 25.0 7 876.6 32.3 7 845.0 28.6 8 1261.8 34.0 8 1197.4 31.3 9 1480.1 37.7 9 1339.8 34.7

10 1550.0 38.2 10 1558.9 36.6

YX YZ

No. σ& (GPa/s) tσ (MPa) No. σ& (GPa/s) tσ (MPa)

1 2.1E-04 13.2 1 1.8E-04 11.8 2 2.1E-04 12.9 2 1.8E-04 12.0 3 2.1E-04 12.9 3 1.8E-04 11.6 4 303.1 24.4 4 200.4 19.3 5 371.2 27.3 5 433.7 26.2 6 591.0 31.2 6 735.3 30.5 7 867.5 33.3 7 896.0 34.6 8 1226.4 37.6 8 1222.1 33.8 9 1400.2 38.9 9 1498.0 38.7

10 1617.6 40.9 10 1801.1 41.7

ZX ZY

No. σ& (GPa/s) tσ (MPa) No. σ& (GPa/s) tσ (MPa) 1 2.2E-04 17.2 1 2.1E-04 16.4 2 2.2E-04 16.9 2 2.1E-04 16.7 3 2.2E-04 17.1 3 2.1E-04 16.4 4 243.3 28.4 4 236.9 24.9 5 332.1 32.1 5 540.2 33.1 6 656.4 35.6 6 630.8 34.8 7 988.6 38.0 7 892.8 36.4 8 1698.5 44.9 8 1171.5 41.8 9 1281.1 45.1 9 1816.2 44.4


5Figure 4.22 illustrates the variation of strength values with loading rates. Within the range of

loading rates available, the tensile strength increases with the loading rate for each of the six

sample groups in a non-linear manner. There seems to be a transition of loading rate sensitivity

at the loading rate of 500 GPa/s. The rock tensile strength is more rate-sensitive when it is loaded

below this transition loading rate. The reason for this transition is not clear, however, similar

trend was found in the literature (Cai et al., 2007).

It is also observed from 5Figure 4.22 that the splitting plane of the disc (the first index in the

sample terminology) has the dominant influence on the tensile strength while the fracture

propagation direction (the second index in the sample terminology) only has a slight influence.

In view of this, all the results are sorted into three groups according to three different splitting

planes normal to X axis (sample XY and XZ), Y axis (sample YX and YZ) and Z axis (sample

ZX and ZY).

0 500 1000 1500

10

15

20

25

30

35

40

45

ZX ZY YX YZ XY XZ

Tens

ile s

treng

th (M

Pa)

Loading rate (GPa/s)

Figure 4.22 The variation of tensile strength with loading rates for six sample groups of Barre granite.


Within these three groups, the ratio of the maximum tensile strength to the minimum tensile

strength is defined as the anisotropic index of tensile strength, denoted as tα . The strength values

for all three splitting planes are shown in 5Figure 4.23a, b and c.

400 800 1200 1600

15

20

25

30

35

40

45

X Plane (XY & XZ)

Tens

ile s

treng

th (M

Pa)


400 800 1200 1600

15

20

25

30

35

40

45

Z Plane (ZX & ZY)

Tens

ile s

treng

th (M

Pa)


400 800 1200 1600

15

20

25

30

35

40

45

Y Plane (YX & YZ)

Tens

ile s

treng

th (M

Pa)


(a)

(c)

(b)

(d)

0 400 800 1200 1600

1.2

1.4

1.6

1.8

α t


Figure 4.23 The tensile strength with loading rates for samples splitting in the plane normal to (a) X axis (b) Y axis (c) Z axis; and (d) the tensile strength anisotropic index (αt) of Barre granite with loading rates.


The samples with splitting plane normal to Z axis owns the highest values of tensile strength

while samples with splitting plane normal to X axis yields the lowest. The variation of the

anisotropic index of tensile strength tα with loading rates is shown in 5Figure 4.23d. For the

static case, αt equals to 1.83, with the highest strength of 16.8 MPa for samples splitting in the

plane normal to Z axis; and the lowest of 9.2 MPa with splitting plane normal to X axis.

Compared to the static one, the dynamic tensile strength anisotropy is much lower. For example,

under the loading rate around 200 GPa/s, sample with splitting plane normal to Z axis owes the

highest tensile strength of 28.9 MPa and splitting plane normal to X axis shows the lowest value

of 18.9 MPa, and αt is 1.53. As the loading rate is around 1800 GPa /s, αt is about 1.13 and the

maximum tensile strength still occurs in samples with splitting plane normal to Z with a value of

47.3 MPa and the lowest one is fixed in samples split in plane normal to X axis as 41.7 MPa.

Barre granite obviously exhibits stronger anisotropy under static loading than the counterpart

under dynamic loading. In addition, the αt curve in 5Figure 4.23d drops quickly approaching the

isotropic value of 1. This implies that under very high loading rates (e. g. shock wave loading),

the tensile strength anisotropy may disappear.

4.4.3 Interpretation of the Results

The main purpose of the study is to characterize the microcrack induced tensile strength

anisotropy of Barre granite under both static and dynamic loading conditions. As shown in

5Figure 4.23d, Barre granite exhibits strong anisotropy under static loading. This tensile strength

anisotropy is mainly attributed to the preferred distribution and orientation of microcracks sets.

Douglass and Voight (1969) studied the microcrack orientation in Barre granite and

demonstrated that a strong concentration of microcracks lies within the rift plane and the

secondary concentration was found within the grain plane. In this study, with reference to the

dominant three sets of microcracks in Figure 3.2 in Chapter 3, YZ plane is recognized to be

parallel to the rift plane with the dominant microcracks, and XZ plane is the secondary

concentration of microcracks for Barre granite. The YZ plane, XZ plane and XY plane

correspond to the quarryman’s description of “rift plane”, “grain plane” and “hard-way plane”


respectively. This explains that in the static tensile strength measurements, the minimum tensile

strength is obtained from sample XY and XZ, both split in the rift plane YZ (normal to X axis);

while the maximum are obtained from sample ZX and ZY with a hard-way splitting plane XY

(normal to Z axis). The relationship of the microcracks induced tensile strength anisotropy with

the principal directions is also consistent with those reported by Goldsmith et al. (1976), who

used orientation 2 (maximum static Young’s modulus), orientation 3 (minimum static Young’s

modulus) and orientation 1 (intermediate static Young’s modulus) to denote the three orthogonal

planes in Barre granite. In my notation, direction 1 is Y, direction 2 is Z, and direction 3 is X.

Under dynamic loading, the anisotropy of tensile strength is much lower than that under static

loading. The anisotropic index of tensile strength drops drastically from the static value of 1.83

to the dynamic value of 1.13 with a loading rate of 1800 GPa/s. The tensile strength anisotropy

of Barre granite appears to be sensitive under quasi-static loading while rather insensitive under

dynamic loading rates. Similar phenomenon has also been observed by Kipp et al. (1980) on a

fine-grained sedimentary rock, oil shale, which also has a pre-existing flaw structure. They found

that in oil shale, the static fracture stress is on the order of 5~20 MPa, quite sensitive to the

loading orientation relative to the bedding planes (Schmidt, 1977); in contrast, the fracture stress

at strain rates from 104 s-1 to 105 s-1 (on the order of 100 MPa, obtained by spalling tests) is

insensitive to orientations (Grady and Hollenbach, 1979).

When a rock sample with an array of cracks is loaded statically, the critical flaw or crack will

dominate the response of the rock, yielding the maximum bearing load. If a preferred orientation

of the largest flaws exists, the material will also show a dependence on the orientation for the

fracture stress (Kipp et al., 1980). For tension tests on six groups of Barre granite samples, the

splitting direction (normal to the loading direction) with the most microcracks (i.e., X direction)

thus has the smallest strength while the direction with the least microcracks (i.e., Z direction) has

the largest strength. In sharp contrast, under dynamic loading, however, the critical flaw no

longer dominates; rather, myriads of pre-existed cracks with a wide range of sizes are activated

nearly simultaneously. Thus, the material is fractured into more pieces through multiple crack

growth. Even with some preferred flaws/microcracks orientation, the dynamic fracture stress

tends to be independent of orientation (Kipp et al., 1980). Hence, the anisotropic property of

Barre granite due to the presence of preferred microcracks has less influence on the dynamic

catastrophic failure. In addition, based on their study, Kipp et al. (1980) found the insensitivity of


the fracture stress over a large range of crack sizes, which suggests that the inherent flaws in the

rock are the basis for the rate dependence of fracture stress, i.e., it is a geometric but not a

material effect (Kipp et al., 1980, Grady and Kipp, 1980). Thus, the effects of anisotropy on the

tensile strength of Barre granite are overshadowed by this dynamic effect.

The dynamic load is qualitatively very different from static load. Although in the case the sample

is essentially loaded under a quasi-static condition, it takes time for the load to reach a certain

level and this time is shorter for a faster loading case. As a result, only a small volume V of the

sample is indeed stressed to a high value during such a short time and this volume is not affected

by its neighboring small volumes. Since crack densities are quite different for various

orientations, and because cracks interact more when aligned in the same plane, for a given small

volume V, the number of “strongly interactive” cracks will be different for the three orientations.

When V decreases, the number of “strongly interactive” cracks may decrease more for a low

crack density orientation than for a high crack density orientation (clustering effect). This will

lead to less anisotropy for dynamic rock tensile strength.

4.5 Summary

In this Chapter, a dynamic Brazilian test with SHPB system is proposed to measure the dynamic

tensile strength of rocks. A simple quasi-static data reduction method similar to the static

standard Brazilian method is used to calculate the strength which assumes a quasi-static stress

state dominates the dynamic test.

To validate this method, two types of dynamic tests were conducted: 1) non-pulse shaped

incident loading wave featuring a rectangular shape; 2) carefully pulse shaped incident wave

with a ramped shape. It was observed with the aid of a high speed camera that in a pulse shaped

SHPB test, the splitting of the disc starts approximately from the centre. This is not the case for

tests without pulse shaping. The usage of static analysis of dynamic BD tests given dynamic

force balance is then further examined. It is demonstrated that with dynamic force balance

achieved by the pulse shaping technique, the peak of the far-field load synchronizes with the


fracture time of the crack gauge at the disc centre and the time-varying dynamic forces on both

ends of the sample are almost identical. Furthermore, the evolutions of dynamic compressive

stress and tensile stress at the centre of the disc obtained from the dynamic finite element

analysis agree with those from quasi-static analysis.

These results fully verified that with dynamic force balance in SHPB, the inertial effect is

minimized for samples with complex geometries like Brazilian disc. The dynamic force balance

thus enables the regression of tensile strength from dynamic Brazilian test using quasi-static

approach. To conclude, the dynamic tensile strength of rocks measured using SHPB are reliable

with careful experimental implementations.

The dynamic Brazilian test is then applied to investigate the tensile strength of anisotropic Barre

granite. Rate dependence of the tensile strength of Barre granite has been observed along all six

directions. The Barre granite exhibits strong tensile strength anisotropy under static loading

while diminishing anisotropy in dynamic loading. Under high loading rates, it is anticipated that

the tensile strength anisotropy can be ignored and the dynamic tensile strength appear to be

isotropic. The reason for the tensile strength anisotropy may be understood using the microcracks

orientations and the rate dependence of the anisotropy is explained with the microcracks

interaction.

CHAPTER 5: DYNAMIC FLEXUAL TESTS 95

CHAPTER 5

DYNAMIC FLEXUAL TESTS

In this chapter, a dynamic semi-circular bend (SCB) flexural testing method is proposed to

measure the flexural strength of rocks with split Hopkinson pressure bar (SHPB) system. To

validate the dynamic flexural testing method, both traditional SHPB and pulse shaped SHPB

tests are conducted using isotropic Laurentian granite; and the data reduction method is

critically assessed. This method is then adopted to investigate the loading rate dependence of

flexural strength anisotropy of Barre granite. The result is then interpreted. The flexural strength

is consistently higher than the tensile strength by Brazilian test for all directions; and this has

been interpreted with a non-local failure approach.

5.1 Background studies

As stated in the previous chapter, although direct tensile or pull test has been a natural approach

for measuring the tensile strength of brittle solids like rocks, the stress concentration due to the

sample gripping often induces damage near sample ends, causing its pre-mature failure and

deviation from the desired uniaxial stress state. In addition, bending in direct tensile tests due to

imperfections in the sample preparation and misalignment makes it difficult to interpret the

testing results (Coviello et al., 2005). Consequently, indirect methods have been developed to

determine the tensile strength of rocks. Examples are Brazilian disc (BD) test (Bieniawski and

Hawkes, 1978; Coviello et al., 2005; Hudson et al., 1972; Mellor and Hawkes, 1971), ring test


(Coviello et al., 2005; Hudson et al., 1972; Mellor and Hawkes, 1971), and bending test

(Coviello et al., 2005).

Apart from the Brazilian tests, the tensile strength can also be measured from bending tests

(Coviello et al., 2005). Generally, the tensile strength measured from a bending configuration is

termed flexural strength. This test aims at generating tensile stress at a critical point in the

sample with bending configuration by far-field compression, which is also much easier in

instrumentation than direct tensile tests. The apparent merit of the bending tests over the other

indirect tension methods is that the tensile stress at the failure point of the bending tests is pure

uni-axial, while all other indirect tests, the stress state at the failure spot is bi-axial. Bending of

one dimensional specimens (i.e. beams with circular or rectangular cross section) is very popular

in many branches of civil engineering (Coviello et al., 2005). Three points bending (3PB) and

four points bending (4PB) tests are adopted as a standard for determining the flexural strength of

materials such as natural and artificial building stones, rocks, cement and concrete (ASTM C99 /

C99M-09, 2009; ASTM C880 / C880M-09, 2009; ASTM Standard C78-09, 2009; ASTM

Standard C293-07, 2007; BS EN 12372, 1999; BS EN 13161, 2008). For example, there are two

ASTM standards to guide the testing of flexural strength of concrete. One is ASTM standard

C293 using central point loading. As schematically shown in 5Figure 5.1a, the entire load is

applied at the center of the span; and the maximum tensile stress only occurs at the center of the

span (ASTM Standard C293-07, 2007). The other standard is ASTM Standard C78 with four

points loading, as 5Figure 5.1b depicts. In this method, half of the load is applied upon each third

of the span length and the maximum tensile stress is present over the center one third portion of

the span. For both methods, the critical tensile strength causing the failure of the beam is the

flexural strength (ASTM Standard C78-09, 2009).

The 3PB and 4PB tests have been used by researchers to measure the nominal tensile strength or

flexural strength of rocks; and it has been found that the measured tensile strength from bending

tests, or flexural strength is generally higher than the tensile strength measured from direct pull

or Brazilian tests under quasi-static loading cases (Coviello et al., 2005), even under fast

loadings cases with a modified material testing machine (Zhao and Li, 2000).


Figure 5.1 Schematics of the determination of the flexural strength of concrete by ASTM standards: a) ASTM C293, i.e. center point loading; the entire load is applied at the center of the span. The maximum tensile stress only occurs at the center of the span; b) ASTM C78, i.e. four points loading; half of the load is applied upon each third of the span length. Maximum tensile stress is present over the center 1/3 portion of the span.

The only paper available in the literature on the dynamic bending tests of rocks is that by Zhao

and Li (2000), who tested the dynamic flexural properties of a granite with three point bending

techniques. The loading was driven by air and oil, and thus the highest loading rate they reached

is rather limited. To characterize dynamic flexural strength under higher loading rates, the

bending tests are adopted on the split Hopkinson pressure bar (SHPB).


It is well known that for dynamic tests, there is an inertial effect caused by dynamic stress wave

loading. The pulse shaping technique in SHPB (Frew et al., 2002; Frew et al., 2005; Song and

Chen, 2004) can facilitate dynamic force balance and thus reduce the inertial effect, but the

extent of such reduction is not adequately examined. To justify the quasi-static assumption for

reducing the data from indirect dynamic tensile tests with SHPB, two conditions remain to be

verified rigorously: the dynamic stress equilibrium in the sample and the synchronization of the

peak loading with the rupture onset. For dynamic compressive or direct tension testing, the

samples are cylindrical and thus the force balance on the ends ensures the stress equilibrium

throughout the sample (Frew et al., 2001). However, the samples used for indirect dynamic

tensile testing are two dimensional (2D); force balance on the boundaries does not necessarily

ensure stress equilibrium within the entire sample. One needs to compare the stress history at a

chosen point obtained from full dynamic analysis with that from quasi-static analysis. In the

quasi-static analysis, the peak load is used to calculate the flexural strength. Examining the

match in time between the loading peak and the rupture onset is thus also necessary.

It is intented to develop and validate an applicable method for characterizing the dynamic

flexural strength of rocks and potentially for other brittle solids: semi-circular bending (SCB)

testing with a modified SHPB system. The SCB method has been developed for tensile strength

measurements under quasi-static conditions (Aravani and Ferdowsi, 2006), and this concept is

adopted in our dynamic SHPB testing. It is a convenient alternative to the Brazilian test method,

and it has certain advantages including convenience in sample preparation and less stress

concentration at the contact points. For brittle solids such as rocks, the sample is susceptible to

damage induced by sample preparation. For example, the rocks are normally sampled as

cylindrical cores, thus favoring current SCB method and the Brazilian disk tests. However,

traditional three-point bend tests use rectangular samples. Furthermore, the failure force required

in a SCB test is much less than the Brazilian disk tests for a given material. Consequently, the

stress concentration at the contacts, which may lead to inaccuracy in the measurements, is less

for SCB. In addition, for dynamic tests, it is commonly believed that it takes several round trips

of a wave in the sample before the stress reaches the equilibrium state (Song and Chen, 2004).

Shorter samples are used in the SCB tests than in the Brazilian disk tests. Hence for a given

sample diameter, it is easier to achieve the desired equilibrium state in SCB tests.


This chapter is organized as follows. Section 5.2 presents the methodology including the

modified SHPB system, SCB testing and finite element analysis, followed by the rigorous

evaluation of the proposed dynamic SCB flexural testing method in Section 5.3. Section 5.4

presents the flexural strength results of anisotropic Barre granite along six different directions;

and the consistent higher measures of the flexural strength to the tensile strength characterized in

the preceding chapter are interpreted using a non-local failure approach. Section 5.5 summarizes

the whole work in this chapter.

5.2 Dynamic Semi-circular Bend Flexural Test

5.2.1 The Semi-circular Bend Testing in a SHPB System

The SCB method has been developed for flexural strength measurements under quasi-static

conditions (Aravani and Ferdowsi, 2006), and this concept is adopted in our dynamic SHPB

testing. The SCB testing in the SHPB system and the sample geometry are schematically shown

in 5Figure 5.2. The curved end of the specimen is in tangential contact with the incident bar, and

the flat end is in contact with the transmitted bar through two supporting pins separated by a

distance of S = 21.8 mm. Upon impact, bending and fracture are induced in the specimen. As

introduced in Chapter 3, the SCB samples for both Laurentian granite and Barre granite are

prepared accordingly, with a nominal radius of R = 20 mm and average thickness of B = 16 mm.

In order to determine the rupture initiation instant, tm, a strain gauge is mounted on the surface of

the specimen L = 4 mm away from the center O where the maximum tensile stress occurs and

thus the rupture initiates (5Figure 5.2). tm is signaled by a rapid drop in strain (Weisbrod and Rittel,

2000). The travel time of the unloading wave from the failure spot O to the strain gauge is ∆t.

Then the rupture instant follows as tr = tm − ∆t, where ∆t = L /c and c is the material wave speed.

The determination of rupture instant via strain gauge is only for evaluation purpose in Section

5.3.


P1P2/2

B

R

S

Strain gauge

P2/2Failure spot

Figure 5.2 Schematic of the semi-circular bending (SCB) testing in a SHPB system. The semi-circular specimen, with a thickness B = 16 mm and radius R = 20 mm, is sandwiched between the incident and transmitted bars. A strain gauge is mounted on the specimen near the point O.

5.2.2 Determination of Flexural Strength

It still remains a challenge to measure in situ the full field stress history in the specimen. One

practical way yet with reasonable accuracy is to measure the far-field loading and then input it

into a finite element analysis to deduce the stress in the tested sample. Both quasi-static and

dynamic finite element analyses are feasible. The finite element analysis is performed with

ANSYS. Taking advantage of the specimen symmetry in our SCB tests, only half of the

specimen is necessary for constructing the finite element model. Quadrilateral eight-node

element PLANE82 is used in the analysis, and the finite element model consists of 2357

elements and 7252 nodes in total (5Figure 5.3).


Figure 5.3 Meshing scheme of the SCB specimen for finite element analysis. F1 and F2 denote forces applied on the contact points.

For quasi-static analysis, the forces are equal, i.e., F1=F2=P(t)/2. The tensile stress σs(t) history

near the failure spot O can be determined as:

)2

()()(RSY

BRtPts ⋅=

πσ . (5.1)

where P(t) is assumed to be the transmitted force (P2) deduced from the SCB-SHPB tests. Y is a

function of the dimensionless geometry parameter S/2R, which needs to be calibrated with static

finite element analysis. 5Figure 5.4 shows Y as a function of S/2R calculated from the finite

element analysis, and a polynomial fitting yield:

2)2

(54.4)2

(87.222.2RS

RSY ++= . (5.2)

The coefficient of determination R2 is 0.9999 for the fitting curve in Equation 5.2 (Figure 5.4).

For our configuration, S/2R = 0.5461 and Y = 5.132.


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.03

4

5

6

7

8

9 Data Fitting curve

Y

S/2R

R = 0.99992

Figure 5.4 Y as a function of the dimensionless geometry parameter S/2R from the quasi-static finite element analysis; the coefficient of determination of the fitting curve R2 is 0.9999.

The dynamic flexural strength (σf) within the quasi-static analysis can then be calculated with the

peak value in the measured loading history (Pmax) as:

)2

(max

RSY

BRP

f ⋅=π

σ (5.3)

The dynamic finite element analysis is conducted to obtain the elastodynamic response of the

SCB specimen. Assuming linear elasticity, this analysis solves the following equation of motion

with the Newmark time integration technique (Weisbrod and Rittel, 2000):

u&&ρ=⋅∇ σ (5.4)

where σ is the stress tensor, ρ denotes density, and u&& is the second time derivative of

displacement vector u. The input loads F1 and F2 are taken as half of the dynamic loading forces

exerted on the incident side and transmitted side of the specimen, respectively, i.e., F1 =P1/2 and

F2 =P2/2.


5.3 Validation of Semi-Circular Bend Tests

5.3.1 Failure Sequences of the Specimen in the Dynamic SCB Test

To visualize the dynamic fracture process of SCB rock specimen, a Photron Fastcam SA1 high

speed camera is utilized to monitor the dynamic SCB test on Laurentian granite. The high speed

camera is placed perpendicular to the sample surface with images taken at an inter frame interval

of 8 μs. Frames with representative features are illustrated in 5Figure 5.5.

Figure 5.5 High-speed video images of a dynamic semi-circular bend test on Laurentian

granite.

After around 170 μs, the newly generated cracks of the SCB sample become visible. This

macroscopic crack initiates from the failure spot O (5Figure 5.2), where the tensile stress is the


maximum. After that, the macroscopic crack propagates to the incident bar end of the sample

along the loading axis, resulting in the final catastrophic failure of the SCB sample. The fracture

pattern of the recovered SCB sample for this test is shown in the last frame of 5Figure 5.5. This

shows that the primary failure of the SCB test is tensile and the failure indeed starts from the

failure spot O, where the tensile stress is the largest.

5.3.2 Dynamic SCB Test without Pulse Shaping

The results and discussions of post-mortem examination and strain (stress) histories deduced

from the strain gauge measurements on the incident and transmitted bars and on the specimen,

and from the finite element analysis are presented below.

5Figure 5.6 shows two recovered samples from the SCB tests in the SHPB system without pulse

shaping [5Figure 5.6a] and with proper pulse shaping [5Figure 5.6b].

Figure 5.6 Samples recovered from the SCB testing on Laurentian granite in a SHPB system (a) without pulse shaping, and (b) with pulse shaping; each division in the scale denotes 1 mm.


Both samples are split cleanly into two halves without noticeable damage at the three loading

points. The fracture pattern indicates that the principal crack initiates from the failure spot O of

the sample as expected. Note that both samples are subjected to single-pulse loading with the

momentum trap technique, which prevents further damage to the sample due to multiple loading

pulses (Xia et al., 2008).

5Figure 5.7, 5Figure 5.8 and 5Figure 5.9 show the results for the SCB test without pulse shaping.

The direct SHPB measurements include the force histories for the incident, reflected and

transmitted waves ( 5Figure 5.7). The rising time is about 20µs, and the pulse width is about 140

µs for the incident wave. In order to check whether the force balance is achieved between both

ends of the specimen, the force on the transmitted side (P2) is compared to the force on the

incident side (P1); the latter shows pronounced fluctuations, and the force balance is not achieved

during the whole loading duration.

Figure 5.7 Force histories on both ends of the specimen in the SCB-SHPB test on Laurentian granite without pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.


Figure 5.8 Tensile stress histories at the failure spot O of the Laurentian granite specimen from the dynamic finite element and quasi-static analyses for the SCB-SHPB test without pulse shaping.

Figure 5.9 Strain gauge signal and the transmitted force P2 in the SCB-SHPB test on Laurentian granite without pulse shaping.


Using the forces (stress) history on both ends of the sample acquired from the test without pulse

shaping as the inputs to the dynamic finite element analysis, the stress histories at the failure spot

O are obtained. This is then compared with that from quasi-static analyses where the transmitted

force is used as loading input (5Figure 5.8). The former shows more pronounced fluctuations than

the latter, and the agreement between them is poor. Because the dynamic finite element analysis

represents the real stress history, the quasi-static analysis with the far-field loading as input can

not adequately represent the stress history in the sample without force balance.

The peak value of P2(t) is normally taken as Pmax for static analysis to calculate the material

tensile strength [Equation. (5.3)] (Wang et al., 2006). Therefore, the onset instant of fracture as

identified from the strain gauge history recorded on the specimen should coincide

(approximately) with the peak instant of P2(t) after appropriate time corrections as discussed

above. 5Figure 5.9 shows the comparison between these two histories. Two troughs are visible

from the stain gauge signal. The first trough occurs at about 39 µs and the second at 68 µs. The

second (lower) trough is believed to be indicative of the rupture onset. The peak of P2 occurs at

96 µs, delayed by 28 µs with respect to the rupture onset. This is apparently caused by the

inertial effect in the specimen due to the stress wave loading. As a result of the inertial effect, the

far-field load P2 does not synchronize with the local load at point O (5Figure 5.8). The failure

occurs once the local stress reaches the material strength and thus its onset can be earlier than the

peak of the far-field load P2. Therefore, the flexural stress in the sample cannot be deduced from

far-field measurement via quasi-static analysis, i.e., the quasi-static analysis is not valid for

deducing the flexural strength in the SCB-SHPB experiments if the far-field dynamic force

balance is not achieved.

5.3.3 Dynamic SCB Test with Careful Pulse Shaping

For the SHPB test with proper pulse shaping ( 5Figure 5.10, 5Figure 5.11 and 5Figure 5.12), the

results are in sharp contrast to those from the test without pulse shaping. The incident wave is

shaped to a ramp pulse with a rising time of 150 µs, and a total pulse width of 300 µs. The force


balance is fully achieved on both ends of the sample before the peak is reached in the incident

pulse, since the pre-peak forces on both sides of the samples are almost identical (5Figure 5.10).

Figure 5.10 Demonstration of dynamic force equilibration on both ends of the specimen in the SCB-SHPB test on Laurentian granite with appropriate pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.


Figure 5.11 Tensile stress histories at the specimen failure spot from dynamic and quasi-static finite element analyses for the SCB-SHPB test on Laurentian granite with appropriate pulse shaping.

Figure 5.12 Strain gauge signal and the transmitted force P2 in the SCB-SHPB test on Laurentian granite with pulse shaping.


With the measured forces as inputs, the tensile stress histories for dynamic and quasi-static finite

element analyses (5Figure 5.11) are found to agree with each other. 5Figure 5.12 compares the

strain signal from the gauge mounted on the specimen and the transmitted force P2 measured by

the SHPB system. The single trough registered by the strain gauge on the specimen occurs at

174.3 µs, indicating the onset of the rupture. The peak of P2 occurs at 176.1 µs, with a delay of

only 1.8 µs relative to the rupture initiation. This small difference can be understood as follows.

The load on the SCB specimen increases with the incident pulse before it reaches the peak. At

the onset of rupture, release waves are emitted from point O. These waves travel at the sound

speed of the rock material. The distance between point O and the supporting pin is 10.9 mm and

it thus takes around 2.2 µs for the first release wave to reach the supporting pins. Due to the

interaction between the release wave and the pins, the load on the transmitted side decreases

despite that the load in the incidence pulse is still rising (5Figure 5.10). Furthermore, the measured

peak transmitted force is 6.628 kN, only 0.2% higher than the force of 6.615 kN at the rupture

initiation time. Thus, the single trough in the strain gauge signal (the rupture onset) can be

regarded as synchronous with the peak of the transmitted force.

The measured flexural strength in the dynamic SCB test with proper pulse shaping is calculated

to be 34.1 MPa at a loading rate of 373 GPa/s. The loading rate is calculated by fitting the linear

portion of the tensile stress evolution. The dynamic flexural strength of Laurentian granite has

not been reported before. The dynamic flexural strength of another type of granite measured with

the three point bending technique ranges from 20 MPa to 30 MPa (Zhao and Li, 2000). Our

result is higher than those by Zhao and Li (2000). This is because the highest loading rate they

achieved is only 10 GPa/s, an order of magnitude lower than ours. The higher value of the

flexural strength in our experiments is expectedly due to the loading rate effect on flexural

strength.

The above results demonstrate that in a modified SHPB test with proper pulse shaping, the

dynamic force balance within the sample is achieved. Thus, the tensile stress state at the failure

spot O in the sample can be calculated with either quasi-static analysis or dynamic finite element

analysis using the far-field measurements as inputs. Moreover, the rupture time synchronizes

with the peak of the transmitted pulse recorded in the SHPB system after corrections for travel

time. Therefore, the dynamic flexural strength can be calculated from the peak of the transmitted

wave measured in the SHPB system with quasi-static analysis.


The dynamic SCB technique allows indirect tensile testing with a well established dynamic

compression setup. Despite the 2D configuration in the SCB testing, it is highly feasible to

achieve stress equilibrium and the synchronization of the rupture onset with the peak of the

transmitted pulse in rocks as shown above, and the simple quasi-static analysis is valid for data

reduction. This method is thus an efficient way of determining the dynamic flexural strength in

brittle solids like rocks.

5.4 Flexural Strength of Barre Granite

5.4.1 Determination of Anisotropic Flexural Strength

5.4.1.1 Stress distribution in the disc sample

Static measurement is conducted with an MTS hydraulic servo-control testing system ( 5Figure

5.13a). Dynamic test is conducted using a 25 mm SHPB system ( 5Figure 5.13b). The specimen is

sandwiched between the incident and transmitted bars. The dynamic forces on both ends of the

sample P1 and P2 are recorded by the two strain gauges mounted on the incident bar and

transmission bar, respectively.

Let x, y, z be a global Cartesian coordinate system shown in 5Figure 5.13, the y-axis defines the

loading direction and the z-axis denotes the axial direction of the disc. In the quasi-static elastic

equilibrium, the components of the stress tensor for any point (x, y) in the disc can be expressed

as follows:

xxx qRBP

⋅=π

σ , yyy qRBP

⋅=π

σ , xyxy qRBP

⋅=π

τ (5.5)

where σx, σy and τxy are three components of the stress tensor and qxx, qyy and qxy are the

corresponding components of the dimensionless stress tensor and can be calculated using


numerical tools according to Equation (5.6). Compression is positive in the following

calculations.

RBP

q xxx

π

σ= ,

RBP

q yyy

π

σ= ,

RBP

q xyxy

π

τ= (5.6)

(a)

P1P2/2

B

R

S

P2/2

P

P/2

B

R

SP/2

o


(b)

x

yx

y

o

Figure 5.13 Schematics of the semi-circular bend test in (a) the material testing machine and (b) the SHPB system.

To measure the flexural strength of anisotropic rocks by SCB test, a thorough analysis of the

stress state in the anisotropic disc is required to deduce the calculating equation. In this work,

finite element analysis using ANSYS is conducted to analyze the stress state in the anisotropic

Barre granite half disc for all our six types of SCB sample configurations. Quadrilateral eight-

node element PLANE82 is used in the analysis, and the finite element model consists of 11,397

elements and 34,592 nodes in total. Same as adopted in the previous Chapter 4, Barre granite is

considered orthotropic and the nine stiffness constants Cijkl has also been documented (Sano et al.,


1992). For comparing purpose, the stress state of the SCB half disc is also analyzed assuming

that the rock is isotropic.

(a)

(b)

(c)

(d)

(e)

(f)

3

2

1

0 0

-1-2

-3

-1-2

-3

0 0

1 1

2 2

0 0

0

00

0 0

-2

-1

-2 -2

-4-3

-1 -1

-3-3

-4-4

0

0 0

0 0

00

0

2

4

-2

-4

2-2

4-4 -224 -4

0

0

0

0

0 0

11 1

2

3

-3-3-2

-1-2

-1

-1-1

0 0

00

00 0

0 0

-1-1

-2 -2-2-3

-4

-1-3-3

-4 -4

0 0

0

0

00

00

2

2-2

-2

4-4

2 -2-4 4

Figure 5.14 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) qxx, (b) qyy and (c) qxy with isotropic model, and (d) qxx (e) qyy and (f) qxy for ZX sample using anisotropic model (positive for compression, negative for tension).


For the isotropic case, 5Figure 5.14a, b and c show the distribution of the dimensionless stress

components qxx, qyy and qxy respectively. The calculated values of qxx, qyy and qxy at the disc centre

of the SCB half disc (potential failure spot) are qxx ~ -5.132, qyy ~ 0 and qxy = 0, respectively. For

anisotropic case, eight sample configurations XY, XZ, YX, YZ, ZX, and ZY are analyzed; and

similar symmetrical stress contours as the isotropic case have been observed (see, 5Figure 5.15,

5Figure 5.16, 5Figure 5.17 and 5Figure 5.18). As a demonstration, the stress trajectories of the qxx,

qyy and qxy for sample ZX are illustrated in the 5Figure 5.14d to f. The tensile stress distribution

near the centre of the disc is quite uniform for the anisotropic ZX sample (5Figure 5.14d and 4e),

very similar to the isotropic case (5Figure 5.14a and b). The shear stress components (5Figure 5.14f)

along the loading diameter and the horizontal diameter are zero due to the intentional coring and

loading along three predetermined material symmetrical plane X, Y and Z. Therefore, the qxx and

qyy along the loading direction in the 5Figure 5.14d and 5Figure 5.14e acurrately represent the

dimensionless in-plane principal stress σ1 and σ2. 5Figure 5.14e also depicts that the compressive

stress of a point on the diameter of the half disc is zero, which suggests that the stress state at the

failure spot O, i.e. the disc centre, is pure uniaxial tension.

A complete suite of stress trajectories of the qxx, qyy and qxy for semi-circular bend sample under

quasi-static deformation for the isotropic case and six anisotropic sample configurations are

illustrated in 5Figure 5.15, 5Figure 5.16, 5Figure 5.17 and 5Figure 5.18.


(a)

(b)

(c)

Figure 5.15 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) qxx, (b) qyy and (c) qxy with isotropic model.


(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.16 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) qxx, (b) qyy and (c) qxy for XY sample and (d) qxx (e) qyy and (f) qxy for XZ sample (positive for compression, negative for tension).


(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.17 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) qxx, (b) qyy and (c) qxy for YX sample, and (d) qxx (e) qyy and (f) qxy for YZ sample (positive for compression, negative for tension).


(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.18 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) qxx, (b) qyy and (c) qxy for ZX sample, and (d) qxx (e) qyy and (f) qxy for ZY sample using anisotropic model (positive for compression, negative for tension).


5.4.1.2 Flexural strength

Using Equation (5.6), the stress state at any point within the disc can be fully determined by the

three dimensionless stress components qxx, qyy and qxy. 5Figure 5.14 shows that for points along

the loading diameter of our anisotropic SCB half disc, the shear stress is zero and the tensile

stress is the highest at the centre of the disc. In addition, the corresponding compressive stress is

also very similar to the isotropic case. The vanishing of the shear stress components along the

loading diameter implies the coincidence of the in-plane principal stress σ1 and σ2 with σx and σy.

For the flexural strength determination for anisotropic Barre granite in this study, a similar

assumption as Chen et al. (1998a) is used that the indirect flexural strength is given by the

maximum absolute value of the tensile stress σx perpendicular to the loading diameter at the disc

centre (maximum local tensile stress):

QBRPf

f ⋅=π

σ (5.7)

where σf is the flexural strength and Pf is the load when the failure occurs. Q is qxx at the centre

of the disc with coordinates (0, 0). Note that at the potential failure spot O, the centre of the disc,

in-plane principal stress σ2 is zero; thus the stress state at O is pure uni-axial tension. For our

configuration, S/2R = 0.5461 and Q = 5.132, for isotropic model, as reported before; and the

calculated dimensionless factors Q for all six sample configurations as well as the material

properties used in the finite element analysis are tabulated in 5Table 5.1.

Table 5.1 The material properties used in the finite element model of SCB samples of Barre

granite along six directions.

Sample Suites

Q

xE (GPa)

yE (GPa)

xyG (GPa)

xyv

XY 5.138 40.7 32.2 15.8 0.093 XZ 5.115 54.8 32.2 17.7 0.105 YX 5.257 32.2 40.7 15.8 0.073 YZ 5.094 54.8 40.7 20.6 0.146 ZX 5.395 32.2 54.8 17.7 0.062 ZY 5.218 40.7 54.8 20.6 0.108


5.4.2 Flexural Strength Anisotropy


For SCB tests conducted in the MTS system, the quasi-static Equation (5.3) is accurate while for

dynamic Brazilian tests in the SHPB system, a quasi-static stress state in the sample half disc

during the test has to be checked before Equation (5.3) can be used. This is because the inertial

effect induced in the dynamic tests will lead to errors in data reduction if a quasi-static analysis is

used without justification. In the evaluation of the dynamic SCB method in Section 5.3, it has

been demonstrated that the dynamic forces on both ends of the SCB specimen can effectively

minimize the inertial effect even for complicated sample geometry as SCB (Dai et al., 2008) by

employing pulse shaping technique in SHPB tests (Frew et al., 2002). The quasi-static equation

thus can be used to determine the dynamic flexural strength of rocks (Dai et al., 2010d).

Under these circumstances, in order to guarantee a quasi-static state in the dynamic SCB test and

thus employ the quasi-static equation for data reduction, the time-resolved dynamic forces on

both loading ends of the SCB samples should match and this should be critically assessed for

each test. To do so, pulse shaping technique is employed for all our dynamic tests and the

dynamic force balance on the two loading ends of the sample is compared before data processing.

5Figure 5.19 compares the time-varying forces on both ends of the sample in a typical test on

sample XZ with pulse shaping. It is evident that with pulse shaping (5Figure 5.19), the dynamic

forces on both sides of the samples are almost identical before the maximum loading (i.e. critical

failure point) is reached during the dynamic loading.

Following this strategy, the force balance on both ends of the sample can be guaranteed for all

the dynamic SCB tests. In addition, the momentum trap technique was also applied to all

dynamic tests to ensure single pulse loading pulse to the samples. 5Figure 5.20a and b show the

examples of the virgin SCB samples and recovered samples after tests, respectively. Along the

loading directions, the half disc samples are split into two approximate quarter disc, as shown in

5Figure 5.20b.


Figure 5.19 Dynamic force balance check for a typical dynamic semi-circular bend test on sample XZ of Barre granite with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.

(a) (b)

Figure 5.20 (a) Virgin semi-circular bend samples of Barre granite; (b) Recovered semi-circular bend samples of Barre granite after tests.


5.4.2.2 Static flexural strength anisotropy

5Figure 5.21 depicts the variation of static flexural strength measured along six different

directions of Barre granite. For each group, three independent tests have been conducted and the

average strength over the three is taken as the flexural strength for the sample group. The

measured static flexural strength exhibits very strong anisotropy. The average flexural strength

for the two sample configurations with the same splitting plane X (i.e. XY, XZ) yields the lowest

flexural strength of 13.5±0.17 MPa and 12.9±0.25 MPa respectively; configurations with the

splitting plane Y (i.e. YX, YZ) owns intermediate flexural strengths of 17.9±0.17 MPa and 16.4±

0.15 MPa; whereas the configurations with the splitting plane Z (i.e. ZX, ZY) exhibit the highest

strength values of 25.1±0.32 MPa and 24.1± 0.24 MPa, respectively. The highest strength value

(25.1 MPa from sample ZX) is nearly twice of the lowest one (12.9 MPa from sample XZ).

XY XZ YX YZ ZX ZY0

5

10

15

20

25 24.125.1

17.916.4

12.913.5

Flex

ural

Stre

ngth

(MP

a)

Figure 5.21 The variation of static flexural strength of Barre granite along six directions, i.e. XY, XZ, YX, YZ, ZX and ZY.


5.4.2.3 Dynamic flexural strength anisotropy

All the flexural strength values with corresponding loading rates are tabulated in 5Table 5.2.

5Figure 5.22 illustrates the variation of strength values with loading rates. Within the range of

loading rates available, the flexural strength increases with the loading rate for each of the six

sample groups in a non-linear manner. Similar to the dynamic tensile strength determined from

Brazilian tests in the Chapter 4, there seems to be a transition of loading rate sensitivity at the

loading rate of 500 GPa/s. The rock flexural strength is more rate-sensitive when it is loaded

below this transition loading rate.

0 500 1000 1500 2000

10

20

30

40

50

60

70

ZX SCB ZY SCB YX SCB YZ SCB XY SCB XZ SCB

Flex

ural

Stre

ngth

(MP

a)


Figure 5.22 The variation of flexural strength with loading rates along six directions of Barre granite.


5Figure 5.22 further evaluate the same splitting scenario that what is observed in Chapter 4 that

the fracture plane of the disc (the first index in the sample terminology) has a dominant influence

on the flexural strength while the fracture propagation direction (the second index in the sample

terminology) only has a relative slight influence. It is thus logic to divide all the flexural

strengths data points into three groups according to three different splitting planes normal to X

axis (sample XY and XZ), Y axis (sample YX and YZ) and Z axis (sample ZX and ZY),

respectively.

Within these three groups, the ratio of the maximum flexural strength to the minimum flexural

strength can be defined as the anisotropic index of flexural strength, denoted as αf. The flexural

strength values for all three splitting planes are shown in 5Figure 5.23a, b and c. The samples with

splitting plane normal to Z axis own the highest values of flexural strength while samples with

splitting plane normal to X axis yields the lowest. The variation of the anisotropic index of

flexural strength αf with loading rates is shown in 5Figure 5.23d. For the static case, αf equals to

1.86, with the highest strength of 24.6 MPa for samples splitting in the plane normal to Z axis;

and the lowest of 13.2 MPa with splitting plane normal to X axis. Compared to the static one, the

dynamic flexural strength anisotropy is much lower. For example, under the loading rate around

200 GPa/s, sample with splitting plane normal to Z axis owes the highest flexural strength of

41.8 MPa and splitting plane normal to X axis shows the lowest value of 27.0 MPa, and αf is

1.55. As the loading rate is up to 2000 GPa /s, αf is about 1.24 and the maximum flexural

strength still remains in samples with splitting plane normal to Z with a value of 68.9 MPa and

the lowest one is fixed in samples split in the plane normal to X axis as 55.6 MPa.

Thus, Barre granite obviously exhibits stronger anisotropy under static loading, while relatively

lower anisotropy during dynamic loading. In addition, the αf curve in 5Figure 5.23d drops quickly

approaching the isotropic value of 1. This implies that the flexural strength anisotropy may

disappear under very high loading rates (e. g. shock wave loading).


(a)

(c)

(b)

(d)

0 500 1000 1500 2000

1.2

1.4

1.6

1.8

α f


0 500 1000 1500 2000

10

20

30

40

50

60

X Plane (XY & XZ)

Flex

ural

Stre

ngth

(MP

a)


0 500 1000 1500 200010

20

30

40

50

60

Y Plane (YX & YZ)

Flex

ural

Stre

ngth

(MP

a)


0 500 1000 1500 200020

30

40

50

60

70

Z Plane (ZX & ZY)

Flex

ural

Stre

ngth

(MP

a)


Figure 5.23 The flexural strength with loading rates for samples splitting in the plane normal to (a) X axis (b) Y axis (c) Z axis; and (d) The flexural strength anisotropic index (αf) of Barre granite with loading rates.


Table 5.2 Flexural strengths of Barre granite with corresponding loading rates as well as the

non-local reconciliation for both static and dynamic SCB tests.

XY

XZ

No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) 1 ~8E-04 13.3 10.2 1 ~8E-04 13.1 10.1 2 ~8E-04 13.6 10.4 2 ~8E-04 12.6 9.7 3 ~8E-04 13.6 10.4 3 ~8E-04 13.0 10.0 4 418.2 32.2 24.8 4 330.3 25.3 19.5 5 707.6 36.9 28.4 5 450.3 29.7 22.9 6 637.9 38.3 29.4 6 760.6 30.9 23.8 7 727.6 39.9 30.7 7 900.1 35.1 27.0 8 1093.4 43.9 33.7 8 980.7 40.5 31.2 9 1400.7 45.0 34.6 9 1338.2 43.8 33.7

10 1599.3 50.4 38.8 10 1516.2 47.4 36.5 11 1796.9 54.8 42.2 11 1702.2 48.0 36.9

YX YZ

No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) 1 ~8E-04 18.1 13.9 1 ~8E-04 16.5 12.7 2 ~8E-04 17.8 13.7 2 ~8E-04 16.2 12.5 3 ~8E-04 17.8 13.7 3 ~8E-04 16.5 12.7 4 470.1 37 28.5 4 372.2 33.4 25.7 5 676.1 38.1 29.3 5 471.5 33.5 25.8 6 535.7 39.7 30.6 6 491.3 36.2 27.9 7 860.5 47.2 36.3 7 694.8 38.8 29.9 8 1078.6 44.4 34.2 8 992.6 48.8 37.6 9 1382.9 49.7 38.2 9 1470 48.6 37.4

10 1702.1 54.5 41.9 10 1585.2 47.8 36.8 11 2001.3 57.0 43.8 11 2052.5 53.0 40.8

ZX

ZY

No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) No. σ&

(GPa/s) σf

(MPa) σt,N

(MPa) 1 ~8E-04 24.8 17.7 1 ~8E-04 24.4 17.4 2 ~8E-04 25.4 18.1 2 ~8E-04 23.9 17.0 3 ~8E-04 25.1 17.9 3 ~8E-04 24.0 17.1 4 473.1 45.0 32.1 4 386.4 42.0 30.0 5 662.3 45.2 32.3 5 691.4 47.6 34.0 6 620.2 49.4 35.3 6 1016.8 53.5 38.2 7 760.0 53.5 38.2 7 908.0 55.1 39.4 8 998.7 54.5 38.9 8 1300.0 57.0 40.7 9 1061.8 56.4 40.3 9 1118.4 59.0 42.1

10 1314.1 62.1 44.4 10 1525.1 59.9 42.8 11 1926.9 65.9 47.1 11 1700.0 63.2 45.1


5.4.3 Interpretation of the Results

5.4.3.1 Non-local failure theory

Comparing the flexural strength with SCB method and the tensile strength with BD method, in

several aspects similar scenarios have been evidenced. They are: 1) rate dependence of the

calculated strengths along six sample groups; 2) strong strength anisotropy in the static case; 3)

weak anisotropy in the dynamic case. The interpretation for the rate dependence of flexural

strength anisotropy is the same as that for the tensile strength anisotropy in Chapter 4 in terms of

the microstrucural specifications and thus will not be repeated here.

However, it is noted that the measured flexural strength of Barre granite with the SCB method is

consistently higher than the tensile strength determined with BD method for a given loading rate.

This phenomenon has been observed in the static tensile measurements with a stress gradient

around the potential failure spot (Coviello et al., 2005; Hudson et al., 1972; Lajtai, 1972; Mellor

and Hawkes, 1971). In the tensile property measurement of a granite under intermediate loading

rates, Zhao and Li (2000) also reported the higher value of dynamic tensile strength (i.e. flexural

strength) from their 3-point flexural test as compared to that obtained by BD test. No quantitative

interpretation has been made for this discrepancy.

A non-local approach (Carter, 1992; Lajtai, 1972; Van de Steen and Vervoort, 2001) is utilized

here to reconcile the discrepancy of measured dynamic strengths from SCB and BD tests. Since

the dynamic equilibrium is ensured for all SCB tests, the non-local approach should work for our

dynamic tests. This theory states that the material fails when the local stress averaged over a

distance δ along the prospective fracture path reaches the tensile strength σt (Van de Steen and

Vervoort, 2001):

dll

lt ∫+

= 0

0

δσδσ (5.8)

where δ is designated as an characteristic material length scale and σ is the distribution of the

tensile stress over δ. Numerical method is used here to determine σt for a given sample geometry.

The tensile stress gradient along the prospective fracture path of our SCB sample can be


calculated numerically with finite element analysis. The polynomial fit of the normalized stress

gradient results in the following equation (see, 5Figure 5.24):

cbxaxm

++= 2

σσ (5.9)

where σ is the tensile stress along the prospective fracture path, σm is the tensile stress at the

failure spot (also the maximum tensile stress in the sample), x is the distance of a point along the

fracture path to the centre of the SCB sample (see the insert in 5Figure 5.24), a, b and c are fitting

factors. If δ is known, substituting Equation (5.9) to Equation (5.8), the ratio κ (i.e. σf /σt)

between flexural strength σf and the resulting tensile strength σt can be determined. On the other

hand, if the ratio κ can be estimated by comparing flexural strength σf to the resulting tensile

strength σt, the characteristic material length δ can be obtained for the material.

Figure 5.24 Normalized tensile stress along the prospective fracture path in a SCB XY sample; x is the distance of a point along the prospective fracture path to the failure spot of the SCB sample (see the insert); the fitting curve has a coefficient of determination R2 of 0.9999.


5.4.3.2 Characteristic material length by matching measures

Take sample group XY for an example to show how to determine the characteristic material

length δ employing non-local failure theory. The tensile stress along the prospective fracture path

in a XY SCB sample can be calculated with finite element analysis and the stress is normalized

with the maximum tensile stress at the potential failure spot O (see, 5Figure 5.24). A polynomial

fit of the normalized stress gradient along the failure path of XY sample yield Equation (5.10):

997.01965.00121.0 2 +−= xxXYm

XY

σσ (5.10)

For sample XY, the ratio κ equal to 1.3, determined by matching the tensile strength with the

reduced strength coming from the flexural strength divided by κ. 5Figure 5.25 shows comparable

strengths of sample XY from dynamic SCB test and BD test as well as the reconciliation by non-

local failure model employing a ratio of κ=1.3.

3.1/ == κσσ XYt

XYf (5.11)

Substituting Equation (5.10) into (5.8),

dxxxdx XYXY XYmXYXY

XYt ∫∫ +−==

δδσσδσ

0

2

0)997.01965.00121.0( (5.12)

From Equation (5.11), 3.1/XYf

XYt σσ = , rearranging Equation (5.12),

XYmXYXYXYXY

XYf σδδδδ

σ]997.0)(

21965.0)(

30121.0[

3.123 +−= (5.13)

At failure, XYm

XYf σσ = , thus Equation (5.13) can be further simplified as:

0)3.1

1997.0()(2

1965.0)(3

0121.0 2 =−+− XYXY δδ (5.14)

Solving Equation (5.14), δXY can be calculated as δXY= 2.6 mm.


Following the same strategy, the flexural strengths for all six sample groups of Barre granite are

reconciled. A summary of the results involving the fitting factors a, b and c, the ratio κ and the

determined characteristic length δ are tabulated in 5Table 5.3. 5Figure 5.25~5Figure 5.30 illustrate

both flexural and tensile strengths of sample group XY, XZ, YX, YZ, ZX and ZY of Barre

granite as well as the reconciliation by non-local failure model, respectively.

Table 5.3 Summary of the parameters deduced using non-local failure model for all six

sample groups of Barre granite.

Groups a b c κ δ (mm) XY 0.0121 -0.1965 0.9970 1.30 2.6 XZ 0.0110 -0.1875 0.9961 1.30 2.7 YX 0.0149 -0.2188 0.9980 1.30 2.3 YZ 0.0115 -0.1918 0.9973 1.30 2.7 ZX 0.0176 -0.2388 0.9978 1.35 2.5 ZY 0.0148 -0.2185 0.9990 1.35 2.7

0 500 1000 1500 2000

10

20

30

40

50

60

XY SCB XY SCB (Non-local) XY BD

Stre

ngth

(MP

a)


Figure 5.25 Comparison of strengths of sample group XY of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.


0 500 1000 1500

10

20

30

40

50

XZ SCB XZ SCB (Non-local) XZ BD

Stre

ngth

(MP

a)


Figure 5.26 Comparison of strengths of sample group XZ of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.

0 500 1000 1500 200010

20

30

40

50

60

YX SCB YX SCB (Non-local) YX BD

Stre

ngth

(MP

a)


Figure 5.27 Comparison of strengths of sample group YX of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.


0 500 1000 1500 2000

10

20

30

40

50

YZ SCB YZ SCB (Non-local) YZ BDS

treng

th (M

Pa)


Figure 5.28 Comparison of strengths of sample group YZ of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.

0 500 1000 1500 2000

20

30

40

50

60

70

ZX SCB ZX SCB(Non-local) ZX BD

Stre

ngth

(MP

a)


Figure 5.29 Comparison of strengths of sample group ZX of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.


0 500 1000 1500 2000

20

30

40

50

60

ZY SCB ZY SCB(Non-local) ZY BD

Stre

ngth

(MP

a)


Figure 5.30 Comparison of strengths of sample group ZY of Barre granite from dynamic SCB test and BD test as well as the reconciliation by non-local failure model.

5.5 Summary

In this chapter, a dynamic flexural strength testing method, the dynamic SCB method is proposed

for measuring the dynamic flexural strength of such brittle solids as rocks. The SCB technique

allows indirect tensile testing with a well established dynamic compression setup. Using a very

simple quasi-static data analysis, the flexural strength of the sample can be deduced.

To evaluate this method, the dynamic SCB tests are performed on Laurentian granite with and

without pulse shaping, and conduct quasi-static and dynamic finite element analyses. A strain

gauge is mounted near the failure spot on the specimen to determine the onset instant of fracture.

It is demonstrated that in a modified SHPB test with proper pulse shaping, the dynamic force

balance within the sample can be achieved. Thus, the tensile stress state at the failure spot O in

the sample can be calculated with either quasi-static analysis or dynamic finite element analysis


using the far-field measurements as inputs. Moreover, the rupture time synchronizes with the

peak of the transmitted pulse recorded in the SHPB system after corrections for travel time.

Therefore, the dynamic flexural strength can be calculated from the peak of the transmitted wave

measured in the SHPB system with quasi-static analysis. This method is thus an efficient way of

determining the dynamic flexural strength in brittle solids.

The dynamic SCB test is then applied to investigate the flexural strength of anisotropic Barre

granite. Rate dependence of the flexural strength of Barre granite has been observed. Similar to

the tensile strength measured from Brazilian tests in the previous chapter, the Barre granite

exhibits strong flexural strength anisotropy under static loading while diminishing anisotropy in

dynamic loading. Under very high loading rates, it is anticipated that the tensile strength

anisotropy disappears.

The reason for the flexural strength anisotropy may be understood using the microcrack

orientations and the rate dependence of the anisotropy is explained with the microcrack

interaction, the same reason as the rate dependence of the tensile strength anisotropy of Barre

granite discussed in Chapter 4. The flexural strengths of Barre granite on all six directions are

consistently higher than the tensile strength measured from Brazilian tests in both static test and

dynamic tests. The tensile stress gradient along the potential failure path is believed to be the

main reason for this distinction, since the tensile strength is defined under a homogeneous tensile

stress state. A non-local failure theory is adopted to qualitatively explain the differences of the

measured strengths; and the gap between the two is bridged as well.

CHAPTER 6: DYNAMIC FRACTURE TESTS 135

CHAPTER 6

DYNAMIC FRACTURE TESTS

In this chapter, a dynamic notched semi-circular Bend (NSCB) testing method is proposed to

measure the Mode-I fracture toughness and fracture energy of rocks; and this novel method is

critically assessed using isotropic Laurentian granite. This method is then applied to

investigating the loading rate dependence of Mode-I fracture properties of anisotropic Barre

granite. The rate dependence of the fracture toughness anisotropy is observed and two

conceptual models abstracted from microscopic thin section photos are constructed to

qualitatively reproduce the rate dependence of the fracture toughness anisotropy in terms of the

interaction of the main crack with pre-existing microcracks oriented along preferred different

directions of Barre granite.

6.1 Background Studies

Dynamic fracture plays a vital role in geophysical processes and engineering applications (e.g.,

earthquakes, airplane crashes, projectile penetrations, rock bursts and blasts). Accurate

measurements of dynamic fracture parameters are prerequisite for understanding mechanisms of

fracture and are also useful for engineering applications. For brittle materials such as rocks, one

can not simply use the standard methods of fracture measurement developed for metals. Special

sample geometries have been adopted for fracture toughness measurements of ceramics, rocks

and concretes (Fowell and Xu, 1993; Hanson and Ingraffea, 1997; Ouchterlony, 1989). Various


methods were proposed in the literature to measure fracture toughness of rocks, including radial

cracked ring (Shiryaev and Kotkis, 1982), notched semi-circular bend (NSCB) (Chong and

Kuruppu, 1984; Lim et al., 1994a; Lim et al., 1994b; Lim et al., 1994c), chevron-notched SCB

(Kuruppu, 1997), Brazilian disc (Guo et al., 1993), and cracked straight through Brazilian disk

(CSTBD) (Atkinson et al., 1982; Chen et al., 1998b; Fowell and Xu, 1994). International Society

of Rock Mechanics (ISRM) proposed short rod (SR) and chevron bending (CB) tests in 1988

(Ouchterlony, 1988) and cracked chevron notched Brazilian disc (CCNBD) in 1995 (Fowell et

al., 1995). All of those specimens are core-based, which facilitate sample preparation from cores

obtained from natural rock masses.

In many mining and civil engineering applications, rocks may be loaded dynamically. By

definition, the dynamic SIF at the fracture onset is the dynamic fracture toughness. Thus, a good

dynamic fracture toughness measurement method should be able to accurately 1) determine the

transient evolution of SIF and 2) detect the fracture initiation time. Unfortunately, the inertial

effect associated with dynamic loading in a dynamic test leads to unreliable data reduction.

Böhme and Kalthoff first showed the inertia effects in dynamic fracture tests (Bohme and

Kalthoff, 1982), where a three point bending configuration was used with the load exerted by a

drop weight. They demonstrated that the measured load histories at the loading point and the two

supporting points did not synchronize with the SIF history of the crack, which was independently

measured with optical method. Tang tried to measure dynamic fracture toughness of rocks by

three point impact using a single Hopkinson bar (Tang and Xu, 1990), and Zhang deployed the

split Hopkinson pressure bar (SHPB) technique to measure the rock dynamic fracture toughness

with SR specimen (Zhang et al., 2000; Zhang et al., 1999). In these two attempts with Hopkinson

bar, the evolution of SIF and the fracture toughness were calculated using a quasi-static analysis

without careful consideration of the loading inertial effect.

One way to minimize the measurement error induced by the inertial effect is to combine

experiments with full dynamic numerical simulations (Bui et al., 1992; Rittel and Maigre, 1996;

Weisbrod and Rittel, 2000). From the experiments, the loading histories on the sample

boundaries are measured. These data are then used as inputs to a full dynamic numerical code to

determine the local SIF history at the crack tip. A strain gauge or a crack gauge is glued on the

sample close to the initial crack tip to measure the fracture initiation time. The dynamic fracture

toughness is then determined as the SIF at the crack initiation time. This method is rather tedious


and complicated to apply. An obviously better way to handle the inertial effect is to find a

method to minimize or eliminate it. Without the inertial effect, one can employ quasi-static

analysis to relate the measured far-field loads to the local SIF at the crack tip. Because the static

numerical analysis can be performed beforehand, the material fracture toughness can be then

readily calculated using the peak far-field load.

Recently, a pulse shaping technique in conventional SHPB tests was proposed to facilitate

dynamic force equilibrium and thus eliminate inertial effect for dynamic compressive tests (Frew

et al., 2002; Frew et al., 2005; Song and Chen, 2004). The pulse shaper reduces the slope of the

loading pulse and thus allows more time for a compression sample (cylindrical) to achieve an

almost stress equilibrium state during the loading. Using this technique, Frew et al. obtained the

compressive stress-strain data for a few rocks (Frew et al., 2001). Recently, Owen et al. observed

that the stress intensity factors obtained by directly measuring the crack tip opening are

consistent with those calculated with the quasi-static equation when the dynamic force balance of

the specimen is roughly achieved in the split Hopkinson tension bar testing (Owen et al., 1998).

This concept was further applied to measure the fracture toughness of ceramics using a four-

point bend in SHPB with pulse-shaping (Weerasooriya et al., 2006). However, methods like this

were not fully verified. For cylindrical samples in dynamic compressive tests, the sample stress

state is 1D and thus the force balance at the ends guarantees the equilibrium stress state

throughout the sample. However, samples used for dynamic fracture test have much more

complicated geometry; force balance on the boundary does not necessarily ensure the stress

equilibrium of the entire sample. To fully justify quasi-static analysis for data regression in

dynamic fracture tests using SHPB, one must show the dynamic stress equilibrium in the sample

and the matching of the peak far-field load with the sample fracture onset.

Another important parameter in dynamic fracture is the fracture energy. The fracture energy is

directly related to the energy consumption during dynamic failures. To our best knowledge, there

is only one attempt to measure the dynamic propagation toughness of rocks (Bertram and

Kalthoff, 2003), where an array of strain gauges was used to measure the strain field associated

with fracture propagation. For dynamic fracture, the shrinkage of the domain of small scale

yielding may lead to significant error for methods based only on the singular term of a stress

field (e.g., the strain gauge and caustics methods) (Freund, 1990). Indeed, six terms of expansion

of the stress field is required to fit the photoelastic fringe patterns (Xia et al., 2006).


Fracture energy can be easily measured with optical methods for transparent polymers or

polished metals (Owen et al., 1998; Xia et al., 2006). The concept of optical techniques in SHPB

testing was initiated by Griffith and Martin (1974), who used white light to monitor the

displacements at the end faces of a cylindrical specimen. Tang and Xu measured the crack

surface opening displacement (CSOD) using a line source of white light (Tang and Xu, 1990),

and took the turning point of the CSOD history as the fracture initiation time. Zhang et al. used

the Moiré method to monitor the CSOD of short-rod specimens, and assumed that the peak point

of the opening velocity curve obtained from CSOD corresponds to the onset of fracture (Zhang

et al., 1999). Furthermore, Ramesh and Kelkar adopted a line laser source to measure the

velocity history of flyer in planer impact (Ramesh and Kelkar, 1995). Later, Ramesh and

Narasimhan used this technique to measure the radial expansion of specimens in SHPB tests

(Ramesh and Narasimhan, 1996).

In this work, a new method is proposed to measure the dynamic fracture toughness and fracture

energy of anisotropic Barre granite using the NSCB specimen, loaded dynamically with a

modified SHPB system. Using an isotropic fine-grained granitic rock, Laurentian granite, it is

demonstrated that the dynamic fracture toughness can be calculated by substituting the

experimental measured peak load to the quasi-static equation if the dynamic force balance is

achieved. This method thus provides a useful and cost-effective way to quantify dynamic

fracture toughness of rocks involving Barre granite in particular. For fracture energy

determination, a laser gap gauge (LGG) is used to monitor CSOD of a straight through notched

semi-circular bend (NSCB) specimen during SHPB testing. The residue kinetic energy in the

fragments can be obtained from the fragment velocity, which is the temporal derivative of the

CSOD history. Given the residual fragment kinetic energy and total energy consumption

(deduced from the strain gauge signals), the dynamic fracture energy are determined. A similar

method was attempted by Zhang et al., who used a high-speed camera to estimate the fragment

velocities (Zhang et al., 2000).

This chapter is organized in the following sequence. The overall methodology, including the

experimental setup, the fracture toughness and fracture energy determination are proposed in

Section 6.2. The validation of the dynamic fracture method using a quasi-static data reduction

method is detailed in Section 6.3 using both traditional and pulse shaped SHPB tests. Section 6.4

presents the methodology for calculating anisotropic stress intensity factor and the equation for


fracture toughness determination; the fracture toughness and fracture energy of anisotropic Barre

granite are measured for all six directions. The rate dependence of fracture toughness anisotropy

is carefully simulated using crack-microcrack models abstracted from microscopic thin section

photography is covered in Section 6.5. Main conclusions are summarized in Section 6.6.

6.2 Dynamic Notched Semi-circular Bend Fracture Test

6.2.1 The Notched Semi-circular Bend Testing in an SHPB System

Dynamic fracture tests on rock materials usually resort to compression-induced tension in order

to avoid failure due to gripping in purely tensile testing. Chong and Kuruppu (1984) adopted a

notched semi-circular bend (NSCB) rock specimen to measure fracture toughness in a

compression setting. This static fracture testing method can be extended to the dynamic tests

with SHPB.

A schematic of the sandwiched NSCB specimen in the SHPB system is shown in 5Figure 6.1, its

radius is R and thickness is B, the depth of the notch is a, and the span of the supporting pins is S.

The force applied on the side without any supporting pins is P1; the force is P2 on the other side

with two pins, with P2/2 is exerted on each pin. 5Figure 6.1 also illustrates the setup of the laser

gap gauge (LGG) system, which is adjusted orthogonal to the sample surface as well as the bar

axis. The LGG is used in the dynamic fracture tests to quantify the flying velocity of the two

cracked half fragments of the NSCB sample, from which the kinetic energy of the fragments can

be calculated.


DetectorGap

Cylindrical lens Sample

Laser

Incident bar Transmitted bar

Collecting lens

Striker

P1 P2a

B

SR

Pulse Shaper Strain Gauge

Figure 6.1 Schematics of the notched semi-circular bend (NSCB) specimen in the spit Hopkinson pressure bar (SHPB) system with laser gap gauge (LGG) system. A strain gauge is mounted on the specimen surface near the crack tip.

6.2.2 Determination of Mode-I Fracture Toughness

A quasi-static data reduction method has been borrowed to determine the Mode-I fracture

toughness of the sample. A complete evaluation of the method is detailed in the Section 6.3.

Referring to ASTM standard E399-06e2 (E399-06e2, 2006), the quasi-static stress intensity

factor (SIF) of the notched SCB specimen is calculated according to the following static equation:

)/,/(2/3 RSRaYBR

SPK I ⋅= (6.1)

where IK is the quasi-static Mode-I SIF, P is the time-varying load, and Y (a/R, S/R) is a

function of the dimensionless crack length a/R and the dimensionless geometrical parameter S/R.

For a given configuration, the numerical value of Y(a/R, S/R) is a constant and is calculated using

the finite element software ANSYS according to Equation (6.2).


2/3

)/,/(

BRSP

KRSRaY I= (6.2)

Due to symmetry, a half-crack model is employed to construct the finite element model.

Quadrilateral eight-node element PLANE82 is used in the analysis. To better simulate the stress

singularity of the crack tip, quarter nodal elements (singular elements) (Barsoum, 1977) are

applied to the vicinity of the crack tip in the mesh of the finite element model (5Figure 6.2b). The

entire model has 2357 elements and 7252 nodes (5Figure 6.2a). For static analysis, the input loads

F1 and F2 equal to half of the transmitted forces recorded in the experiment (i.e., F1= F2 = P2/2).

Symmetry Crack tip

14

34

ω

LGG

Δlv/2

A

F2

F1

(c)

y

x

r

θv

symmetry

(a) (b)

Figure 6.2 Finite element model of the NSCB specimen system (a) the half model of NSCB sample (b) close view of the crack tip mesh (c) crack tip coordinate system.

In the local crack tip coordinate system (5Figure 6.2c), assuming plain strain, the near-tip crack

opening displacement (COD) for a stationary crack under static loading is related to the SIF as:


ππθ 2)1(8 2

,r

EKvCOD I

r−

== (6.3)

For the half-crack model, the SIF is then determined as:

rvEVK I

π2)1(4 2−

⋅= (6.4)

where V is the half COD determined using Equation (6.3).

Using Equation (6.1), the evolution of the SIF (i.e. KI) in the dynamic tests can be determined.

The peak of KI is the Mode-I fracture toughness KIC and the slope of the pre-peak linear portion

of the KI (i.e.•

IK ) is the loading rate for the dynamic test.

6.2.3 Determination of Dynamic Fracture Energy

5Figure 6.3 shows a typical loading history (P2) and the corresponding crack surface opening

displacement (CSOD) history during a NSCB test with dynamic force balance achieved

throughout the test (i.e. P1=P2). The loading history is recorded by the strain gauges on the bars

and the CSOD is monitored by the LGG system. It will be proved in the next section that the

sample is in a quasi-static state as long as the dynamic force balance is achieved. The peak point

of the loading (A) thus corresponds to the fracture initiation in the specimen, as in a quasi-static

experiment. The temporal derivative of the CSOD history is the crack surface opening velocity

(CSOV) history. CSOV increases with time and then approaches a terminal velocity of v=13.9

m/s at the turning point B.

The two vertical lines passing through points A and B divide the whole deformation period into

three stages I-III. It is believed that in stage I the crack opens up elastically, in stage II the crack

propagates dynamically, and in stage III the fracture separates the sample into two pieces and the

two fragments rotate away from each other. The separation velocity of the two fragments

(normal to the bar axis) is approximately the terminal velocity of CSOV (for small angle of


rotation in stage III), and doubles the fragment velocity. The crack propagation process lasts

about ΔtAB = 53 μs as seen from CSOD and CSOV. Given the crack distance Ls=R-a=16 mm for

this test, the average crack growth velocity vf can be estimated to be about 300 m/s.

Figure 6.3 Typical loading history and CSOD history of the NSCB specimen tested in SHPB on Laurentian granite.

A high speed camera (Photron Fastcam SA1) is used to monitor the fracture initiation and

propagation process of the test as well as the trajectories of the fragments. The high speed

camera is placed perpendicular to the SHPB and specimen. Images are recorded at an interframe

interval of 8 μs; the sequence shown in 5Figure 6.4 represents only the frames of representative

features. The first two images show the pre-fabricated notch and the crack opening can be barely

seen. The opening of the NSCB crack becomes visible at t > 40 μs. At 80 μs, the NSCB

specimen is split completely into two fragments. The fragments then rotate about the contact


point between the specimen and the incident bar (A point in 5Figure 6.2). The rotation angle of

the fragment is measured to be 9° at 160 μs, 21° at 480 μs, and 32° at 800 μs. This indicates that

the angular velocity of the fragments is almost constant during the period (about 314 rad/s), and

the motion of the fragments is rotational.

Figure 6.4 Selected high speed camera images showing the fracture and fragmentation of a NSCB Laurentian granite specimen.

The high speed camera imaging indicates that the fragments rotate around the axis along the

loading point at the incident bar side of the sample. The LGG system measures CSOD and the

fragment angular velocity can be deduced. The linear velocity of the two rotating fragments at

the LGG point is approximately the terminal velocity in the CSOV curve (5Figure 6.3). The

distance between the LGG and the rotating axis Δl = 18 mm, so the angular velocity ω= v/2/Δl =

313 rad/s for the snapshot shown in 6Figure 6.4, in excellent agreement with the result obtained


from high speed imaging. A similar method was used by Zhang et al., who used a high-speed

camera to estimate the fragment residual velocities (Zhang et al., 2000).

The energy conservation principle is then used to calculate the propagation fracture energy.

During the dynamic test, the energy dissipation (ΔW) pertaining to the fracture specimen is the

energy difference between the input energy (Wi) and the summation of the energy reflected (Wr)

and transmitted (Wt):

tri WWWW −−=Δ (6.5)

where W is the energy carried by the stress wave, which can be calculated as follows (Song and

Chen, 2006):

∫=t

ACdEW0

2 τε (6.6)

where E and C are the Young’s modulus and wave speed of the bar material respectively. A is

the cross-sectional area of the bar andε denotes the time-resolved strain induced by the stress

wave.

This energy dissipation ΔW has two parts: the energy consumed to create new crack surfaces WG

and the residue kinetic energy in the two cracked fragments K. For the rotating fragments (6Figure

6.4), the moment of inertia is I, and the total rotational kinetic energy is K = Iω2/2, where the

fragment angular velocity ω is estimated from the CSOD data with our optical device. The

energy consumed in generating new cracks thus can be reduced as WG = ΔW - K. Consequently,

the average propagation fracture energy is determined with Equation (6.6) below:

cGc AWG /= (6.7)

where Ac is the area of the new generated crack surfaces. For isotropic material, assuming plane

strain, the average dynamic propagation fracture toughness can be attained:

)1/( 2ν−= EGK cPI (6.8)

where E and ν are the Young’s modulus and Poisson’s ratio of the sample material respectively.


6.3 Validation of Dynamic Notched Semi-Circular Bend Test

6.3.1 Dynamic Analysis and Fracture Time

6.3.1.1 Dynamic analysis

Dynamic finite element analysis is carried out to determine the SIF evolution by solving the

equation of motion with Newmark time integration method in ANSYS (Weisbrod and Rittel,

2000):

u&&ρ=⋅∇ σ (6.9)

where σ is the stress tensor, ρ denotes density, and u&& is the second time derivative of

displacement vector u . Assuming linear elastic material model, the elastodynamic response of

the NSCB specimen is solved. The finite element meshing is the same as 6Figure 6.2 while taking

the input loading F1 and F2 as half of the dynamic loading forces exerted on the incident side and

transmitted side of the sample, respectively (i.e., F1=P1/2, F2=P2/2 ). For a sample with a

stationary crack subjected to dynamic loading, the near-tip crack opening displacement (COD)

are similar to the static case as the following (Freund, 1990):

ππθ 2)()1(8)(

2

,r

EtKvtCOD I

r−

== (6.10)

For the half-crack model used in ANSYS, dynamic SIF is determined as:

rvEtVtK I

π2)1(4

)()( 2−= (6.11)

6.3.1.2 Fracture initiation detection

The fracture initiation time is detected by the strain gauge mounted on the sample surface near

the crack tip. The crack emits elastic release waves upon fracture initiation, and this wave causes

sudden drop in the recorded strain gauge signal (Jiang et al., 2004a). The lowest point of the drop


corresponds to the arrival of the release wave due to complete fracture initiation. It is noted that

this fracture initiation time should be corrected considering the time for the elastic wave to

propagate from the crack tip to the strain gauge. Thus, the adjusted strain gauge signal ε1(t)

follows as:

)()(1 ttt Δ+= εε (6.12)

where ε(t) is the original strain gauge signal. The travel time of the unloading wave from the

crack front to the strain gauge is determined as Δt=L/c, where L is the distance from the strain

gauge to the crack tip and c is the material elastic wave speed. The fracture initiation time is then

detected by the significant drop of the corrected strain signal ε1(t).

6.3.2 Dynamic NSCB Test without Pulse Shaping

In a conventional SHPB tests, impact of the striker on the incident bar generates a square shaped

incident stress wave. The rising portion of the incident wave is so sharp that high frequency

oscillations are inevitably introduced. 6Figure 6.5 shows the forces on both ends of the sample.

From Equation (6.1), the dynamic force on one side of the specimen P1 is the sum of the incident

(In.) and reflected (Re.) waves, and the dynamic force on the other side of the specimen P2 is the

transmitted wave (Tr.). It is evident from 6Figure 6.5 that a large fluctuation of dynamic force

occurs on the incident side and a sizeable distinction exists between forces on the two ends of the

specimen.


0 50 100 150

-30

-20

-10

0

10

20

30

Forc

e (k

N)

Time (μs)

In. Tr. Re. In.+Re.

Figure 6.5 Dynamic forces on both ends of the NSCB specimen tested using a conventional SHPB on Laurentian granite. In.: incident; Re.: reflected; Tr.: transmitted.

The measured CSOD of the NSCB specimen by LGG and the transmitted force in a conventional

SHPB test are illustrated in 6Figure 6.6. Two force peaks A and B are identified in the transmitted

force signal, occurring at time 62 μs and 94 μs respectively. More interestingly, over a rather

long time period, around 85 μs after the incident stress wave arrives at the sample, the measured

CSOD is negative. This means that the crack surfaces at the measuring site close rather than

open, a manifestation of the loading inertia effect. The closing of the crack surface may lead to

“loss of contact” between the transmitted side of the sample and the two pins (Bohme and

Kalthoff, 1982). This explains why after the first peak A of the transmitted force, an obvious

unloading is observed ( 6Figure 6.6). This unloading ends at trough C and then the load

continuously rises until the second peak B. From the CSOD signal, it can be seen that the trough

D almost synchronizes with C, indicating the completion of the unloading and the restart of the

loading phase.


0 50 100 150-2

0

2

4

6

8

10

C

D

BA

20 μ s

Tr. Strain gauge CSOD

Time (μs)

Tran

smitt

ed fo

rce

(kN

)C

SO

D (0

.05

mm

)

-25

-20

-15

-10

-5

0

5

E

F

Strain gauge signal (m

V)

Figure 6.6 Comparison of CSOD and strain gage signal with the transmitted force of the NSCB specimen tested using a conventional SHPB on Laurentian granite (the unit for CSOD is 0.05 mm).

The signal from the strain gauge mounted on the sample surface is also depicted in 6Figure 6.6.

Two troughs E and F are visible from the strain gauge signal. The first trough E occurs at time 39

µs and the second trough F occurs at time 76 µs. The second trough F is lower and believed to

coincide with the fracture initiation time at 76 µs. Because the peak transmitted force occurs at

time 96 µs, the fracture initiation of the NSCB sample is thus 20 µs ahead of the peak

transmitted load. These observations show that due to the inertial effect, the far-field loads on the

sample boundary do not synchronize with the local load at the sample crack-tip. This kind of

loading inertial effect is similar to what was observed by Böhme and Kalthoff in a different

testing configuration (Bohme and Kalthoff, 1982).

6Figure 6.7 shows the evolution of SIF from both quasi-static and dynamic data reductions. The

static analysis is carried out using the transmitted force on both end of the sample ( 6Figure 6.2).

The overall trends of the two curves match with each other but the dynamic SIF features huge

fluctuation. Furthermore, the dynamic SIF is far from linear and therefore it is difficult to


achieve a constant loading rate. Consequently, the SIF from the quasi-static data reduction with

the far-field load recorded from the transmitted bar cannot reflect the transient SIF history in the

NSCB sample. The usage of the far-field loads such as the transmitted force to obtain the

fracture toughness with a quasi-static analysis will lead to tremendous error in the results. The

quasi-static equation is not valid for determining fracture toughness in a conventional SHPB test.

0 50 100 150

-3

-2

-1

0

1

2

3

4

5

SIF

(MP

a m

1/2 )

Time (μs)

Dynamic Quasi-static

Figure 6.7 The evolution of SIF of the NSCB specimen tested using a conventional SHPB on Laurentian granite with both quasi-static analysis and dynamic analysis.

6.3.3 Dynamic NSCB Test with Careful Pulse Shaping

A composite pulse shaper, a combination of a C11000 copper (with 0.64 mm in diameter and 0.7

mm in thickness) and a thin rubber shim (with 0.64 mm in diameter and 0.3 mm in thickness) is

utilized to shape the loading pulse. In a test with the same speed of striker as previous case, the

incident wave is shaped to a ramp pulse with a rising time of 150 µs, and a total pulse width of

300 µs ( 6Figure 6.8). Also shown in 6Figure 6.8 are the forces on both ends of the specimen. In


contrast to 6Figure 6.5, the forces on the two ends of the specimen exhibit no fluctuation and they

are almost identical before the maximum value is reached. The balance of dynamic forces on

both end of the sample is clearly achieved.

0 50 100 150 200 250 300-10

-5

0

5

10

15

Fo

rce

(kN

)

Time (μs)

In. Re. In.+Re. Tr.

Figure 6.8 Dynamic forces on both ends of the NSCB specimen tested using a modified SHPB on Laurentian granite. In.: incident; Re.: reflected; Tr.: transmitted.

6Figure 6.9 illustrates the measured CSOD of the NSCB specimen by LGG and the transmitted

force in a modified SHPB test. The measured CSOD is always positive and there is a single peak

A in the transmitted force (6Figure 6.9), occurring at time 164 μs. The phenomenon of crack

closing due to inertia effects vanishes completely in this case.


0 50 100 150 200 250 300-1

0

1

2

3

4A

B

Tr. Strain gauge CSOD

Time (μs)

-30

-20

-10

0

10

4 μ s

Strain gauge signal (m

V)Tr

ansm

itted

forc

e (k

N)

CS

OD

(0.0

5 m

m)

Figure 6.9 Comparison of CSOD and strain gage signal with the transmitted force of the NSCB specimen tested using a modified SHPB test on Laurentian granite (the unit for CSOD is 0.05 mm).

6Figure 6.9 also shows the signal from the strain gauge mounted on the sample surface near the

crack tip. Only one trough B is registered by the stain gauge, occurring at time 160 µs. Thus, the

fracture initiation time is designated by the unique trough B at time 160 µs. Because the peak

transmitted force occurs at time 164 µs, it is thus only 4 µs after the measured fracture onset. It

can be concluded that in this case, the peak far-field load matches with the fracture onset with

negligibly small time difference. The small time difference between them can be partially

interpreted as follows. The load on the specimen increases with the incident pulse before it

reaches the peak. At the fracture onset, release waves are emitted from the crack tip at the sound

speed of the rock material. The distance between the crack tip and the supporting pin is 12 mm

and it thus takes around 2.4 µs for the first release wave to reach the supporting pins. Due to the

interaction between the release wave and the pins, the load on the transmitted side decreases

( 6Figure 6.9). In addition, between 160 µs and 164 µs, the curve of transmitted force is almost flat

( 6Figure 6.9). The 4 µs time difference will thus lead to negligibly small error in the fracture

toughness.


By carefully shaping the loading wave, the dynamic force balance on the boundary of the sample

is achieved ( 6Figure 6.8). However, with a 2D geometric configuration, the force balance on the

boundary does not necessarily guarantee the dynamic stress equilibrium in the entire specimen.

To address this issue, the SIF evolution is evaluated by dynamic finite element analysis, and the

result is compared with that from a quasi-static analysis (6Figure 6.10). The dynamic SIF exhibits

no fluctuation at all in contrast to that shown in 6Figure 6.7. The evolutions of SIF from both

static and dynamic methods match reasonably well.

100 150 200 250

0

1

2

3

4

SIF

(MP

a m

1/2 )

Time (μs)

Dynamic Quasi-static

Figure 6.10 The evolution of SIF of the NSCB specimen tested using a modified SHPB on Laurentian granite with both quasi-static analysis and dynamic analysis.

From the above discussion, it is verified that with dynamic force balance in SHPB, the peak far-

field load coincides with the fracture onset. The fracture toughness can thus be confidently

deduced from the peak far-field load by virtue of quasi-static equations. For the case examined,

the dynamic fracture toughness is 3.47 MPa.m1/2, with the loading rate of 79.7 GPa.m1/2/s. It is

also noted that when there is no pulse shaper, the failure time is at 76 µs. The corresponding


dynamic stress intensity factor is 1.5 MPa.m1/2 ( 6Figure 6.6). This value can not be used as the

dynamic fracture toughness because it carries significant errors. First, the loading condition is

not well defined due to the oscillation of the load. Secondly, the oscillation is due to the

dispersion of stress waves in the bar system, and thus it only represents the trend of the dynamic

load but not the accurate force at individual measurement points. As a matter of fact, the static

fracture toughness of this rock is about 1.5 MPa.m1/2 (Nasseri and Mohanty, 2008). The dynamic

fracture toughness should be much higher. Hence, again, the test without pulse shaper is not

reliable.

Following proposed methods, the fracture initiation toughness and fracture energy of Laurentian

granite are measured. As illustrated in 6Figure 6.11, the dynamic fracture initiation toughness and

fracture energy increase almost linearly with increasing loading rates.

20 40 60 80 1000

1

2

3

4

5

6

Gc (kJ/m

2)

KIC

Gc

KIC

(MP

a m1/

2 )

KI (GPa m1/2 s-1)

0

1

2

3

4

5

6

Figure 6.11 The effect of loading rate on the fracture toughness and fracture energy of Laurentian granite.


6.4 Fracture Toughness Anisotropy of Barre Granite

6.4.1 Determination of Anisotropic Stress Intensity Factor

Suppose ija are the elastic compliance which defines the relationship between the stress ijσ and

strain ijε . For orthotropic material in plane stress, one has

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

12

22

11

662616

262212

161211

12

22

11

σσσ

γεε

aaaaaaaaa

(6.13)

111

1E

a = (6.14a)

222

1E

a = (6.14b)

2

21

1

1212 E

vEv

a−

=−

= (6.14c)

1266

1G

a = (6.14d)

where iE is the Young’s modulus in the i principle direction, 12G is the shear modulus in the

21− plane, ijv is the Poisson’s ratio define the extensional strain in the j direction produced by

a unit compressive strain in the i direction.

For plane strain, ija is replaced by ijb according to the following equations:

33

33

aaa

ab jiijij −= ; 2,1, =ji (6.15)

where


3

323 E

vEva i

i

ii

−=

−= (6.16)

and

333

1E

a = (6.17)

6Figure 6.12 defines a local coordinate system in relation with the orthotropic material directions.

The stress and displacement of fields in the vicinity of the crack tip can be expressed analytically

below in turn (Tan and Gao, 1992).

y

x

r

θ

ϕE1E2

crack

Figure 6.12 Local coordinate system for the stress and displacement fields near the crack tip of an orthotropic solid.

Stress fields are:


⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−=

2/11

21

2/12

22

21

2/11

12/1

2

2

21

21

)sin(cos)sin(cos1Re

2

)sin(cos)sin(cosRe

2

θθθθπ

θθθθπσ

ss

ss

ssrK

ss

ss

ssss

rK

II

Ixx

(6.18a)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−=

2/12

2/1221

2/11

12/1

2

2

21

)sin(cos1

)sin(cos11Re

2

)sin(cos)sin(cos1Re

2

θθθθπ

θθθθπσ

ssssrK

ss

ss

ssrK

II

Iyy

(6.18b)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−

+−=

2/12

22/1

2

1

21

2/11

2/1221

21

)sin(cos)sin(cos1Re

2

)sin(cos1

)sin(cos1Re

2

θθθθπ

θθθθπσ

ss

ss

ssrK

ssssss

rK

II

Ixy

(6.18c)

Displacements are:

{ }

{ }⎥⎦

⎤⎢⎣

⎡+−+

−+

⎥⎦

⎤⎢⎣

⎡+−+

−=

2/111

2/122

21

2/1112

2/1221

21

)sin(cos)sin(cos1Re2


θθθθπ

θθθθπ

spspss

rK

spsspsss

rKu

II

Ix

(6.19a)

{ }

{ }⎥⎦

⎤⎢⎣

⎡+−+

−+

⎥⎦

⎤⎢⎣

⎡+−+

−=

2/111

2/122

21

2/1112

2/1221

21



θθθθπ

θθθθπ

sqsqss

rK

sqssqsss

rKu

II

Iy

(6.19b)

where,

iii saasap 16122

11 ′−′+′= (6.20a)

i

iii s

saasaq 26222

12 ′−′+′= ; 2,1=i (6.20b)


ija′ are the compliance constants in the local x-y coordinate system. For orthotropic material, it is

related to ija according to the following equations,

ϕϕϕϕ 422

226612

41111 sincossin)2(cos aaaaa +++=′ (6.21a)

ϕϕϕϕ 422

226612

41122 coscossin)2(sin aaaaa +++=′ (6.21b)

ϕϕ 22661222111212 cossin)2( aaaaaa −−++=′ (6.21c)

ϕϕ 22661222116666 cossin)2(4 aaaaaa −−++=′ (6.21d)

ϕϕϕϕ 2sin]2/2cos)2(cossin[ 66122

112

2216 aaaaa ++−−=′ (6.21e)

ϕϕϕϕ 2sin]2/2cos)2(cossin[ 66122

112

2226 aaaaa ++−−=′ (6.21f)

1s and 2s are related to 1T and 2T , which are the roots of the following characteristic equation for

an orthotropic material, according to Lekhnitskii (1963).

0)2( 222

66124

11 =+++ aaaa μμ (6.22)

The roots of this equation are either complex or purely imaginary and cannot be real. They are:

1111 ηζμ iT +== (6.23a)

2222 ηζμ iT +== (6.23b)

13 μμ = (6.23c)

24 μμ = (6.23d)

where, 1−=i , the overbar denotes the complex conjugate here. iζ and iη are real constants. It

is always true that 0,0 21 >> ηη , 21 ηη ≠ .

The relationship between ks and kT is shown (Lekhnitskii, 1963) to be:


;sincossincos

ϕϕϕϕ

k

kk T

Ts

−+

= .2,1=k (6.24)

The stresses yyσ and xxσ ahead of the crack tip and in the plane of the crack could be obtained

by substituting 0=θ into Equation (6.18),

rKI

yy πσ

2= ;

rKII

xy πσ

2= (6.25)

By substituting o180=θ into Equation (6.19), the near-tip displacement u and v on the crack

face may be given in matrix form as:

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

II

I

y

x

KK

DDDDr

uu

2221

12112π

(6.26)

where

⎥⎦

⎤⎢⎣

⎡−−

=21

211211 Im

sspspsD (6.27a)

⎥⎦

⎤⎢⎣

⎡−−

=21

2112 Im

ssppD (6.27b)

⎥⎦

⎤⎢⎣

⎡−−

=21

211221 Im

ssqsqsD (6.27c)

⎥⎦

⎤⎢⎣

⎡−−

=21

2122 Im

ssqqD (6.27d)

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡∗∗

∗∗

y

x

II

I

uu

DDDD

rKK

2221

1211

2π (6.28)

and

DDD /2211 =∗ DDD /2121 −=∗ (6.29a)


DDD /1212 −=∗ DDD /1122 −=∗ (6.29b)

211222112221

1211 DDDDDDDD

D −== (6.30)

If the crack lies parallel to the global x-axis, and also parallel to one of the material symmetry

plane, in the pure Mode-I case,

21

12 Dr

uK y

I ⋅=π (6.31)

Let BrAr

u y += , two nearest nodes at the crack tip are used to determine the factor A. IK could

then be calculated using the following equation:

212 DAK I ⋅=

π (6.32)

6.4.2 Determination of Fracture Toughness of Barre Granite

6.4.2.1 Calculating equation

Static measurement is conducted with an MTS hydraulic servo-control testing system ( 6Figure

6.13a). Dynamic test is conducted using a 25 mm SHPB system (6Figure 6.13b) and the specimen

is sandwiched between the incident and transmitted bars. The specimens used in this study have

a nominal thickness B= 16 mm, radius R= 20 mm and a crack length a= 5 mm. The calculating

equation of the fracture toughness of anisotropic Barre granite follows the same equation as

before [Equation (6.1)], except that the stress intensity factor for anisotropic fracture problems

has to be taken into account via Equation (6.32).


Figure 6.13 Schematics of the straight through notched semi-circular bend fracture test in (a) the material testing machine and (b) the SHPB system.

6.4.2.2 Testing example

First, a crack problem is presented below to demonstrate the validation of the approach for the

determination of stress intensity factor in a plane of orthotropic elasticity. An infinite strip with a

finite width W, an edge crack with length of a, subjected to a remote uniform traction σ

perpendicular to the plane of the crack is illustrated in 6Figure 6.14.

E1

E2 σσa

W

Figure 6.14 An infinite orthotropic strip with an edge crack under remote uniform tractions normal to the edge crack.


Using an integral equation approach, Kaya and Erdogan solved this problem for a special case of

orthotropy (Kaya and Erdogan, 1980). Tan and Gao also researched on this problem to calibrate

their integral equation with boundary element method (Tan and Gao, 1992). The material chosen

in their investigation was a boron-epoxy composite with material properties as

follows: 1E =170.65 GPa, 2E =55.16 GPa, 12G =4.83 GPa and 12v =0.1114.

Six cases are calculated with dimensionless crack length vary from 0.1 to 0.6. By virtue of the

symmetry, half of the strip is modeled. The length of the strip is modeled ten times of the width

W. 6Figure 6.15 illustrates a mesh of 530 PLANE82 elements and 1683 nodes for a typical edge

crack problem with a dimensionless crack length a/W=0.6.

Close view of crack tip elements

1434

Figure 6.15 The overall mesh of the strip and a close-view of the mesh in the vicinity of the crack tip; the length of the trip is modeled as ten times of the width W.

The normalized stress intensity factor aKK II πσ/* = for all six cases are calculated and

tabulated in 6Table 6.1, along with the results reported by Kaya and Erdogan (1980) and Tan and

Gao (1992) for comparison. The results are highly satisfactory, with the maximum error less than

1 % for most of the cases.


Table 6.1 The normalized stress intensity factor aKK II πσ/* = , for an edge crack in an

infinite orthotropic strip with remote uniform traction σ.

a/W

KEIK )( *

TIK )( *

ANIK )( *

KEANIKΔ

(%)

TANIKΔ

(%) 0.1 1.128 1.114 1.134 0.5 1.8 0.2 1.320 1.334 1.320 0.0 -1.0 0.3 1.607 1.624 1.612 0.3 -0.7 0.4 2.042 2.057 2.050 0.4 -0.3 0.5 2.720 2.715 2.734 0.5 0.7 0.6 3.860 3.832 3.878 0.5 1.2

KEIK )( * : Results by Kaya and Erdogan (1980)

TIK )( * : Traction formula result by Tan and Gao (1992)

ANIK )( * : Results by ANSYS in this study

KEANIKΔ = 100

)()()(

*

**

×−

KEI

KEIANI

KKK

TANIKΔ = 100

)()()(

*

**

×−

TI

TIANI

KKK

6.4.2.3 Anisotropic fracture toughness of Barre granite

With confidence on the numerical method to determine the anisotropic stress intensity factor, the

critical stress intensity factor can then be calculated with the critical load recorded from

experiments. For the six samples suites, XZ, XZ, YX, YZ, ZX and ZY, the factor 21D are

determined and tabulated in 6Table 6.2.


Table 6.2 The material properties used in the finite element model of NSCB samples of

Barre granite along six directions.

Sample Suites

12D (Pa-1)

1E (GPa)

2E (GPa)

12G (GPa)

12v

XY 5.95E-11 40.7 32.2 15.8 0.093 XZ 5.58E-11 54.8 32.2 17.7 0.105 YX 5.30E-11 32.2 40.7 15.8 0.073 YZ 4.59E-11 54.8 40.7 20.6 0.146 ZX 4.28E-11 32.2 54.8 17.7 0.062 ZY 3.95E-11 40.7 54.8 20.6 0.108

Taking advantage of the symmetry of the NSCB specimen, half model is constructed for the

finite element analysis. PLANE82 element (2D eight-node structural solid) is also used in the

analysis. Quarter-nodal element (singular element) (Barsoum, 1977) is applied to the vicinity of

the crack tip in meshing the finite element model, to better simulate the stress singularity near the

crack tip (r is the radius to the crack tip). The total model is meshed with 2357 elements and

7252 nodes as shown in 6Figure 6.16. The orthotropic material constants in the finite element

model for six sample suites are also listed in 6Table 6.2, with reference to 6Figure 3.2 in Chapter 3.

Figure 6.16 Mesh for the NSCB specimen and crack tip local coordinate system (a) mesh of the half model (b) close view of the crack tip mesh (c) crack tip coordinate system.


6.4.3 Fracture Toughness Anisotropy


As demonstrated in the previous subsection, as long as the dynamic forces on both ends of the

NSCB sample have been achieved during the test, the inertial effect in the sample can be

effectively minimized, and the quasi-static data reduction scheme can be utilized to determine

the fracture toughness of rocks. We thus only need guarantee the balance of the time-resolved

dynamic force on both ends of the NSCB sample. To do so, pulse shaping technique is employed

for all the dynamic tests and the dynamic force balance on the two loading ends of the sample

has been compared before data processing. 6Figure 6.17 compares the time-varying forces on both

ends of the sample in the typical test on sample YZ with pulse shaping.

Figure 6.17 Dynamic force balance check for a typical NSCB fracture test of Barre granite with pulse shaping. In.: incident; Re.: reflected; Tr.: transmitted.


It is evident that with pulse shaping, the dynamic forces on both sides of the samples match with

each other up to the maximum loading (i.e. critical failure point), from which the dynamic

fracture toughness can be calculated via Equation (6.1).

The dynamic NSCB method is then applied to all six sample groups of Barre granite. 6Figure

6.18a and b show the examples of the virgin NSCB samples and recovered ones after tests,

respectively. Along the loading directions, the NSCB samples are split into two quarter disc

approximately (see 6Figure 6.18b). It is noted that for all dynamic fracture tests conducted on the

SHPB system, the force balance on both ends of the sample has been checked; and all samples

are subject to single-pulse loading with the momentum trap technique, which prevents further

damage to the sample due to the multiple loading pulses.

(a) (b)

Figure 6.18 (a) Virgin NSCB samples and (b) recovered NSCB samples of Barre granite.


6.4.3.2 Static fracture toughness anisotropy

Three independent static tests have been conducted on each of the six groups of Barre granite

samples (i.e. XY, XZ, YX, YZ, ZX and ZY) and the fracture toughness for each group is taken

as the average value. 6Figure 6.19 shows the variation of fracture toughness measured along six

different directions for Barre granite. The highest fracture toughness of Barre granite is

1.74±0.06 MPa·m1/2, from sample ZX split within Z plane. The other sample group split within Z

plane (i.e. ZY) owns the second highest fracture toughness of 1.57±0.04 MPa·m1/2. The average

fracture toughness measured from samples split in Y planes (i.e. YZ and YX) yields intermediate

fracture toughness of 1.42±0.04 and 1.20±0.06 MPa·m1/2, respectively. The lowest and the

second lowest fracture toughness are measured along two fracture directions split within X plane

(i.e. XY and XZ), exhibiting average KIC values of 1.02±0.04 MPa·m1/2 and 1.11±0.06 MPa·m1/2,

respectively. The measured static fracture toughness exhibits very strong anisotropy, with the

highest measured fracture toughness (1.74 MPa·m1/2, for sample ZX) 1.7 times of the lowest

(1.02 MPa·m1/2, for sample XY).

XY XZ YX YZ ZX ZY0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.81.57

1.74

1.421.20

1.111.02

KIC

(MPa

m1/

2 )

Figure 6.19 The variation of static fracture toughness of Barre granite on six sample groups, i.e. XY, XZ, YX, YZ, ZX and ZY.


6.4.3.3 Dynamic fracture toughness anisotropy

All the dynamic fracture toughness and fracture energy values with corresponding loading rates

are tabulated in 6Table 6.3. 6Figure 6.20 illustrates the variation of fracture toughness values with

loading rates along six directions. Within the range of loading rates available, the fracture

toughness increases with the loading rate for each direction in approximately the same rate. This

reveals that: 1) the rate of the increment of the dynamic fracture toughness with loading rate

appear to be the same for all directions and 2) the order of the highest fracture toughness to the

lowest among six directions remains approximately the same as that in the static case.

0 40 80 120 160 2000

2

4

6

8

10

ZX ZY YZ YX XZ XY

KIC

(MP

a m1/

2 )

KI (GPa m1/2 s-1)

Figure 6.20 The variation of fracture toughness with loading rates on six directions of Barre granite.

As before, it is also apparent from Figure 3.2 in Chapter 3 that the fracture plane of the disc

(noted as the first index in the sample terminology) has a sizeable influence on the fracture


toughness; while the fracture propagation direction (the second index in the sample terminology)

also has some influence on the measured fracture toughness. The samples in Z plane own the

highest fracture toughness; while samples in X plane, the least.

The variation of dynamic fracture energy with loading rates along six directions is shown in

6Figure 6.21. Generally, within the range of loading rates available, the fracture energy increases

with the loading rate for each direction, but in different rate. The data are much more scattered

than the fracture toughness measurements; and samples split in Z plane appear to have higher

fracture energy than the other two planes, showing the influence of the pre-existing microcracks

to the measured fracture energy to some extent.

40 80 120 160 2000

1

2

3

4

5

6

7

8 ZX ZY YZ YX XZ XY

GC (k

J/m

2 )

KI (GPa m1/2 s-1)

Figure 6.21 The variation of fracture energy with loading rates on six directions of Barre granite.


Similar to previous two anisotropic indexes introduced in Chapter 4 and Chapter 5, the

anisotropic index of fracture toughness, αk is defined as the ratio of the maximum fracture

toughness to the minimum fracture toughness within six sample groups. 6Figure 6.22a and b show

the two extremes of the variation of fracture toughness. The sample group ZX with splitting

plane normal to Z axis owns the highest fracture toughness while sample XY with splitting plane

normal to X axis yields the lowest.

The variation of the anisotropic index of fracture toughness αk with loading rates is shown in

6Figure 6.22c. Under static loading, αk equals to 1.70; under dynamic loading, it decreases rapidly.

For example, under the loading rate around 20 MPa·m1/2s-1, sample ZX axis owes the highest

fracture toughness of 3.27 MPa·m1/2 and XY axis shows the lowest value of 2.34 MPa·m1/2, and

αk is 1.40. As the loading rate is 220 MPa·m1/2s-1, αk is about 1.20 and the maximum fracture

toughness remains still in samples ZX with a value of 11.0 MPa·m1/2 and the lowest one is fixed

in samples XY as 9.1 MPa·m1/2. Thus, Barre granite obviously exhibits stronger anisotropy under

static loading, while relatively lower anisotropy during dynamic loading. In addition, as shown in

6Figure 6.22c, αk drops quickly towards the isotropic value of 1. This suggests that under very

high loading rates (e. g. shock wave loading) the fracture toughness anisotropy is negligible and

the fracture toughness under such circumstances appear to be isotropic.

In the next section, a crack-microcrak interaction model is constructed to interpret the apparent

loading rate dependence of the fracture toughness anisotropy of Barre granite. Both static and

dynamic analyses have been conducted. The pronounced feature of the same rate of increase of

the loading rate dependence of the Mode-I fracture toughness is also mimicked.


(a)

(c)

(b)

0 50 100 150 200 250

1.2

1.3

1.4

1.5

1.6

1.7

α

KI (GPa m1/2 s-1)

0 50 100 150 2000

2

4

6

8

XY (X Plane)KIC

(MPa

m1/

2 )

KI (GPa m1/2 s-1)0 50 100 150 200

2

4

6

8

10

ZX (Z Plane)KIC

(MPa

m1/

2 )

KI (GPa m1/2 s-1)

k

Figure 6.22 The fracture toughness with loading rates for sample group of (a) XY, splitting in the plane normal to X axis; (b) ZX, splitting in the plane normal to Z axis; and (c) the fracture toughness anisotropic index αk of Barre granite.


Table 6.3 Fracture toughness and fracture energy of Barre granite with corresponding

loading rates from both static and dynamic NSCB fracture tests.

XY

XZ

No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 1 8E-5 1.06 N/A 1 1E-4 1.10 N/A 2 8E-5 1.02 N/A 2 1E-4 1.18 N/A 3 8E-5 0.98 N/A 3 1E-4 1.05 N/A 4 43.8 2.88 0.51 4 28.2 2.51 0.43 5 62.0 4.02 0.68 5 37.3 3.25 0.55 6 73.4 3.89 1.01 6 73.6 4.41 0.78 7 100.0 4.94 1.25 7 87.0 5.31 2.02 8 106.6 5.69 1.90 8 124.9 6.42 2.15 9 123.3 6.23 1.91 9 135 6.60 2.59

10 134.5 6.05 2.45 10 158.5 7.22 2.28 11 175.7 7.31 3.20 11 159.0 7.58 3.53 12 180.0 7.80 3.61 12 198.4 8.72 3.83

YX YZ

No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 1 1E-4 1.19 N/A 1 9E-5 1.40 N/A 2 1E-4 1.26 N/A 2 9E-5 1.39 N/A 3 1E-4 1.15 N/A 3 9E-5 1.47 N/A 4 28.4 2.22 0.46 4 40.4 3.54 0.54 5 81.1 4.35 1.28 5 47.5 3.43 0.72 6 86.9 4.57 1.45 6 48.2 4.01 0.93 7 101.0 5.50 1.95 7 79.4 4.45 1.33 8 114.2 6.31 2.08 8 82.2 4.94 1.57 9 165.0 7.96 3.38 9 94.4 5.53 2.28

10 179.0 8.56 4.05 10 135.8 6.77 2.35 11 187.4 8.38 4.55 11 140.0 7.59 3.87 12 191.2 8.93 4.80 12 171.8 8.30 4.14

ZX ZY

No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 No. ICK&

GPa m1/2s-1 ICK

MPa m1/2 cG

kJ/m2 1 9E-5 1.79 N/A 1 1E-4 1.55 N/A 2 9E-5 1.67 N/A 2 1E-4 1.54 N/A 3 9E-5 1.76 N/A 3 1E-4 1.62 N/A 4 29.0 3.46 0.78 4 24.5 2.96 0.58 5 45.4 4.27 1.17 5 60.5 4.62 1.16 6 68.8 5.27 1.51 6 81.1 5.30 1.73 7 76.5 5.36 2.55 7 100.0 5.60 2.05 8 85.4 5.67 2.57 8 103.5 6.16 3.00 9 113.1 7.20 4.10 9 140.0 7.70 3.40

10 156.1 8.75 4.61 10 166.8 8.77 4.23 11 179.9 9.60 6.27 11 190.2 9.24 4.19 12 201.1 9.90 7.26 12 199.0 9.36 5.62


6.5 Crack-Microcrack Interaction

6.5.1 Background

The phenomenon of a microcracking zone near the main crack tip and its effects on the

propagation of main crack in brittle materials such as ceramics, rocks and concretes (Claussen et

al., 1977; Evans and Faber, 1981; Hoagland et al., 1973) has been discussed by many

researchers. Existing theoretical analysis has focused on the effect of microcracking on the stress

field near the main crack tip. Due to the complexity of the problem, except for a few simple

cases, closed-form solutions are not available. As a result, most previous studies are confined to

semi-analytical solutions. Generally speaking, researchers attempted to model this problem in

two perspectives. One is the continuum mechanics model, which aims at building a constitutive

framework in the continuously damaged material mimicking the overall effect of microcracking

(Hutchinson, 1987; Ortiz, 1987). However, due to the complexity of the interacting problem, no

agreement has been reached among researchers on what is the best material model to simulate

the configuration of crack-microcrack interaction. The second approach considers the multiple

microcracks in the microcrack zone near the tip of a main crack as discrete entities; in which

case, interaction is assessed using the stress function or assumed stress state with superposition

principle. Examples of this type of modeling include the work of Chudnovsky and Kachanov

(1983), Rose (1986) and Chudnovsky et al.(1987). However, most of these models adopted some

inconsistent and/or unrealistic assumptions and consequently their accuracy is rather limited

(Gong and Horii, 1989).

The method of pseudo-tractions, proposed by Horii and Nemat-Nasser (1985), further improved

by Gong and Horii (1989), has been proved to be an effective approach to analyze the crack

microcrack interaction problem. Based on the complex potentials of Muskhelishvili (1953) and

the principle of superposition, this method can treat general problems with any number of

interacting cracks or other inhomogeneities (Gong and Horii, 1989). Herein, the concept of

crack-microcrack interaction is demonstrated using the 0-order and 1st-order solution of the

pseudo-traction method by Gong and Horri (1989). The stress shielding and amplification region

due to the location and orientation of microcracks with respect to the main crack are also

discussed.


Although there are many interesting discussions on the effects of shielding and amplification of

the main crack due to the presence of microcracks, few of them have a strong experimental basis.

Mode I fracture toughness KIC measurements on four types of granites (Nasseri and Mohanty,

2008; Nasseri et al., 2006) were carried out under the standard procedure by the ISRM

recommended method in 1995 (Fowell et al., 1995), showed varying degree of anisotropy. In that

measurement of fracture toughness anisotropy of Barre granite, an isotropic material model was

assumed to determine the fracture toughness values (Nasseri and Mohanty, 2008; Nasseri et al.,

2006).

In this research with semi-circular bend fracture tests, orthotropic material model with material

constants measured from literature (Sano et al., 1992) is used to estimate the true stress intensity

near the crack tip. As discussed before, fracture toughness anisotropy has also been observed

from testing results, featuring different degree of anisotropy regarding to the loading rates. For

the first time, the loading rate dependence of fracture toughness anisotropy of Barre granite is

reported in the rock community. The variation of fracture toughness with respect to the loading

rates will be interpreted on the basis of crack-microcrack interaction in the microscopic scale.

Microstructural observation with a newly developed technique of computer-aided image analysis

program clearly showed the microcrack density and orientation around the propagation path of

the main crack. From two images of thin sections that correspond to the two confronting cases of

fracture toughness measurements, two physical crack microcrack models are constructed. The

existing theoretical formulas are not applicable to the proposed models due to the assumptions

made in the theoretical derivation. Thus, finite element analysis with the commercial software

package ANSYS is used to determine the effects of embedded microcracks on the disturbance of

the stress field at the vicinity of the main crack. The finite element method has been proved to be

an effective method to handle this type of problem (Meguid et al., 1991), especially when the

microcrack is very close to the main crack. The numerical results can simulate the experimental

measurements very well, which implies that the preferred distribution and orientation of pre-

existing mircocracks in the nominally homogeneous rocks is responsible for the anisotropy of

fracture toughness.


6.5.2 Microstructural Investigation and Featuring Models

In order to understand the physical mechanism of fracture toughness anisotropy, two thin

sections of rock samples corresponding to the two confronting measurements of fracture

toughness of Barre granite are captured and illustrated in 6Figure 6.23a for Case 1 and 6Figure

6.24a for Case 2, (Courtesy of Dr. M. H.B. Nasseri). 6Figure 6.23a corresponds to the crack-

microcracks orientation in Barre granite for the situation in which the fracture toughness is the

highest. In Case 1, the main crack propagates from the tip of the notch with an acute angle to

most microcracks. 6Figure 6.24a shows the microcracks density and orientation in Barre granite

for the case in which the fracture toughness is the lowest. In Case 2, the main crack is collinear

to most microcracks.

6Figure 6.23b (Model 1) and 6Figure 6.24b (Model 2) depict two conceptual models based on

6Figure 6.23a and 6Figure 6.24a respectively, taking only into account the nearest few microcracks

near the tip of the main crack. Only the nearest microcracks are considered because it has been

shown from both continuum and discrete models that the interaction is dominated by the nearest

microcracks (Hutchinson, 1990). Based on the statistics of the configuration of microcracks in

6Figure 6.23a, Model 1 is constructed ( 6Figure 6.23b), where two symmetric microcracks near the

notch tip are oriented at an angle of o45 to the horizontal main crack surface. Denote the length

of the microcrack by 2c, the distance from the right tip of the microcrack to the tip of the main

crack is 0.2c. The conceptual model shown in 6Figure 6.24b is based on the thin section photo of

6Figure 6.24a. In this case, existing microcracks are mostly parallel to the propagation path of the

main fracture. For comparison purposes, the same length of the microcrack of 2c and the same

distance from the main crack tip to the closest microcrack tip of 0.2c are used. Particular

attention is directed to the crack-microcracks interaction of the two conceptual models for an

insight on the observed anisotropy of KIC.


Figure 6.23 (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample; Case 1: the main crack inclines at an angle of o45 to microcracks; (b) Model 1: the crack-microcracks configuration for Case 1.


Figure 6.24 (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample; Case 2: The main crack is collinear to microcracks; (b) Model 2: the crack-microcracks configuration for Case 2.


6.5.3 The Crack-Microcrack Interaction

Consider a general problem of a semi-infinite main crack and an arbitrarily located and oriented

microcrack, as shown in 6Figure 6.25. Denote the distance between the main crack tip and the

center of the microcrack by d and the length of the microcrack by 2c. The angle measured from

the x -axis to the line connecting the tip of the main crack and the center of the microcrack is θ

and the microcrack orientation is defined by the angle φ from x -axis to the 'x -axis.

Figure 6.25 One arbitrarily located microcrack near the crack tip of a semi-infinite crack.

In the following, 0IK denotes the stress intensity factor of the main crack without microcracks;

MAIK denotes the local stress intensity factor of the main crack. As a demonstration, the method

proposed by Gong and Horii (1989) are used to calculate the ratio between MAIK and 0

IK with

both 0th order approximation and 1st order approximation for the case where 2/ =cd .

The original problem was decomposed into three sub-problems: I, II and III (6Figure 6.26).


=

+

+

Original Problem Sub-Problem I

Sub-Problem II Sub-Problem III

K0I ,K

0II

σ0(x),τ0(x)

−σ0(x0),−τ0(x0)

δKI,δKII

σp(x),τp(x)

σ(x0),τ(x0)

−[τ0(x)+τp(x)]

−[σ0(x)+σp(x)]

Figure 6.26 The original problem and the three sub-problems decomposed from the original one based on the superposition method.

Sub-problem I contains the main crack subjected to applied Mode-I stress intensity factor 0IK and

Mode-II stress intensity factor 0IIK . The undisturbed stresses )(0 xσ and )(0 xτ along the location

of the microcrack can be determined by virtue of the stress function of Muskhelishvili (1953).

In sub-problem II, in order to satisfy the boundary conditions of the original problem, a pair of

pseudo-tractions σp(x) and τp(x) is applied to the microcrack; and the microcrack is stressed under

the traction pair of –[σ0(x)+σp(x)] and –[τ0(x)+τp(x)]. In this case, the induced stresses along the

main crack can be expressed and denoted as σ(x0) and τ(x0).


In the sub-problem III, the main crack is subjected to tractions equivalent to the negative values

of the induced stresses at the same location in sub-problem II. Assessing the whole processes, it

is obvious that traction-free condition along the main crack surfaces is satisfied automatically.

The traction-free condition along the microcrack requires the induced stresses at the position of

microcrack equal to the pseudo-tractions σp(x) and τp(x) and this will lead to a couple of

consistency equations to finally complete the series of equations (Gong and Horii, 1989).

For fracture tests, the resulting fracture mode is pure Mode-I, thus in the current analysis, the far-

field loading of 0IIK is zero. Consequently, the 0th-order approximate solution, which gives the

first two terms of the stress intensity factors of the main crack, is as follows (Gong and Horii,

1989):

0)21(0)12(2)11(0)11(2

20 ][)(41

IIMAI KABAB

dcKK ⋅⋅+⋅⋅+= (6.33)

The 1st-order approximate solution, which gives the first three terms of the stress intensity factors

for the main crack, is as follows (Gong and Horii, 1989):

{

} 0)22(02)12(2)12(02)11(2)21(0)21(02)12(2)11(02)11(2)11(0

)21(1)12(3)11(1)11(3)21(2)12(2)11(2)11(2)21(0)12(4

)11(0)11(440

)21(0)12(2)11(0)11(220

)(8)(8

)(4)(3)

(24)(128

1][)(41

I

IIMAI

KDBDBADBDBA

ABABABABAB

ABdcKABAB

dcKK

⋅⋅+⋅+⋅+⋅+

⋅+⋅+⋅+⋅+⋅+

⋅+⋅⋅+⋅⋅+=

(6.34)

where

])25()2cos[()

21(

21

])21()2cos[(

41])

21(cos[)2(

21

)11(

θφ

θφθφ

+−++−

+−+++−+=

nnn

nnnnnAn

(6.35a)

])25()2sin[()

21(

21

])21()2sin[(

43])

21(sin[)2(

21

)12(

θφ

θφθφ

+−++−

+−+−+−+=

nnn

nnnnnAn

(6.35b)


])25()2sin[()

21(

21

])21()2sin[(

41])

21(sin[

2)21(

θφ

θφθφ

+−++−

+−+++−=

nnn

nnnnnAn

(6.35c)

])25()2cos[()

21(

21

])21()2cos[(

43])

21(cos[

2)22(

θφ

θφθφ

+−+++

+−+++−−=

nnn

nnnnnAn

(6.35d)

])21sin[()sin(2])2()

21cos[(

])21cos[()2( 11)11(

φθθφθ

φθ

kkkpkk

kpkkpkB

k

kkk

−++−−−×

+−−−−= −−

(6.35e)

])21cos[()sin(2])2()

21sin[(

)2(])21sin[()2( 11)12(

φθθφθ

φθ

kkkpkk

pkkkpkB

k

kkk

−++−−−×

−+−−−−= −−

(6.35f)

])21cos[()sin(2])2()

21sin[(

])21sin[( 11)21(

φθθφθ

φθ

kkkpkk

kpkkkpB

k

kkk

−++−−−×

−−−= −−

(6.35g)

])21sin[()sin(2])2()

21cos[(

)2(])21cos[( 11)22(

φθθφθ

φθ

kkkpkk

pkkkkpB

k

kkk

−+−−−−×

−+−−= −−

(6.35h)

)]45cos()44cos(2)43cos()23cos(2

)2cos(6)2cos()cos(2)2cos(415[)cos1(16

1)11(02

φθφθφθφθ

φθθθφθ

−−−−−−−+

−−−−++

=D (6.35i)

)]45sin()44sin(2)43sin(

)23sin()2sin(3)2sin(2[)cos1(16

1)21(02)12(02

φθφθφθ

φθφθφθ

−+−+−+

−−−++

== DD (6.35j)


)]45cos()44cos(2

)43cos()2cos()cos(27[)cos1(16

1)22(02

φθφθ

φθθθθ

−+−+

−+−−+

=D (6.35k)

In the above expressions, 22 )!(2)!2(

kkp kk = . The factorial !k is defined as a positive integer k as:

12)1(! ⋅⋅⋅⋅−= kkk .

6Figure 6.27 shows the different regions in which the stress intensity factor at the tip of the main

crack may be either increased (amplification, 1/ 0 >IMAI KK ) or decreased (shielding,

1/ 0 <IMAI KK ), depending on the location and orientation of the microcracks relative to the main

crack. This effect of amplification or shielding of the stress intensity factor of the main crack due

to the presence of the microcracks can be of vital significance to the anisotropy of rocks and

hence can result in sizeable variations of the measured fracture toughness in the experiments.

It has been recognized from both continuum and discrete perspectives that the nearest

microcracks have the dominant effects on the stress redistribution at the tip of the main crack

(Hutchinson, 1990). The influence of microcracks far from the immediate tip region of the main

crack appears to be less important. Therefore, it is accurate enough to consider only one or two

microcracks closest to the tip of the main crack, just as the crack/microcracks configurations in

6Figure 6.23b (Case 1) and 6Figure 6.24b (Case 2). In the following, the two proposed conceptual

models in 6Figure 6.23b and 6Figure 6.24b are analyzed with finite element method to quantify the

shielding or amplification effects of main crack due to the presence of the nearest microcracks.


0 20 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180

KMAI

/K0I<1

(shielding) KMA

I/K0

I>1

(amplification)

0 order 1st order

θ (d

egre

e)

φ (degree)

KMAI

/K0I<1

(shielding)

Figure 6.27 The phase diagram of amplification and shielding effects of main crack due to the presence of a unique microcrack using 0th-order and 1st-order approximate solution.

6.5.4 Finite Element Analysis of Two Models

The approximate solutions proposed in the literature (Gong and Horii, 1989; Rose, 1986) were

developed based on the assumption of 1/ >cd (Figure 6.25), when 1/ ≤cd , the results are

invalid. If cd / is close to 1, these methods become impractical because many higher orders of

expansion are necessary to achieve reasonable accuracy. In addition, these analytic solutions are

derived in the situation that the elastic solid containing the main crack/microcracks is infinite.

However, the two models deal with an elastic solid with a specific geometry of a half disc. In

these cases, finite element analysis can be a good alternative and this fact has been confirmed by

Meguid et al. (1991).


Herein, finite element analysis is carried out using ANSYS software. Taking advantage of the

symmetry of both models, half-crack model is used to build the finite element model. PLANE82

(eight-node) element is used in the analysis. To better simulate the stress singularity of r -1/2 near

the crack tip (r is the radius to the crack tip), 1/4 nodal element (singular element) (Barsoum,

1977) is applied to mesh the vicinity of the crack tip. Theoretically, the stress intensity factor can

be calculated from either the stress field or displacement field near the tip of the crack. Since the

stress field in ANSYS is deduced from the displacement field, the stress intensity factor is

calculated based on the displacement field in the vicinity of the crack tip. A loading force of 200

N is applied to the finite element model; and the Young’s modulus is taken as 80 GPa and the

Possion’s ratio is 0.21.

To first verify the accuracy of our numerical calculation, a semi-infinite main crack with one

collinear microcrack under Mode I loading (correspond to the case of 0==θφ in 6Figure 6.27) is

considered as a testing problem. The exact solution (Rose, 1986) of this problem is given below,

where )(kK and )(kE denote the complete elliptic integrals of the first kind and the second kind

respectively (Abramowitz and Stegun, 1972).

)'()(

0 kkk

KK

I

MAI

Κ⋅Ε

= (6.36)

where, cdcdk

+−

= 21' kk −=

The calculated stress intensity factor via ANSYS are normalized by 0IK and compared with the

exact results (Gong and Horii, 1989) for different cases of d/c in 6Table 6.4. The meshed elements

are between 1663 and 2139; the nodes are between 5032 and 6486. The maximum error as

tabulated in 6Table 6.4 is less than 1.03 %, which occurs when d/c=1.1


Table 6.4 Stress intensity factor of the main crack with one collinear microcrack at different

distances to the main crack tip.

d/c 1.1 1.2 1.3 1.4 1.5 2 3 Exact value 1.652 1.387 1.274 1.209 1.167 1.076 1.030

ANSYS value 1.635 1.398 1.287 1.221 1.176 1.086 1.024 Error (%) -1.029 0.721 1.020 0.992 0.771 0.999 -0.582

Error (%) =100× (ANSYS value- Exact value)/ Exact value

In the following numerical analysis, the phenomena of stress shielding and stress amplification

due to crack/microcrack interaction are quantified. Three numerical models are built to achieve

this goal: 1) Intact Model, corresponding to the microcrack-free case ( 6Figure 6.28), 2) Model 1,

corresponding to Case 1 (6Figure 6.29), and 3) Model 2, corresponding to Case 2 ( 6Figure 6.30).

With the same boundary conditions, the stress intensity factor 0IK for Intact Model, 1M

IK for

Model 1 and 2MIK for Model 2 can be calculated. 0

IK can be recognized as the far field loading.

For the microcrack-free case (Intact Model), a similar meshing scheme is used as reported before

( 6Figure 6.28). 6Figure 6.29 and 6Figure 6.30 show the meshing of our finite element Model 1 and

Model 2, respectively, in which 6Figure 6.29a and 6Figure 6.30a are the global meshes, featuring

increasing grids density towards the main crack tip; and 6Figure 6.29b and 6Figure 6.30b show the

mesh at the vicinity of the main crack and the microcracks for Model 1 and Model 2 respectively.


Figure 6.28 Finite element mesh (a) global mesh of Model 1; Case 1 (b) close-view of the mesh at the vicinity of the main crack of the Intact Model. The main crack and its tip are indicated with arrows.


Figure 6.29 Finite element mesh (a) global mesh of Model 1; Case 1 (b) close-view of the mesh at the vicinity of the main crack and the inclined microcrack of Model 1; Case 1. The main crack and its tip are indicated with arrows.


Figure 6.30 Finite element mesh (a) global mesh of Model 2; Case 2 (b) close-view of the mesh at the vicinity of the main crack and the collinear microcrack of Model 2; Case 2. The main crack and its tip are indicated with arrows and the collinear microcrack is also marked.


6Figure 6.31, 6Figure 6.32 and 6Figure 6.33 compare the stress intensity (i.e. the stress difference

between the maximum and minimum principal stress) contours around the main crack tip. 6Figure

6.31 shows the stress intensity contours near the tip of the main crack in the absence of

microcracks, in which case 0IK is evaluated. 6Figure 6.32 shows the stress intensity contours due

to the presence of two symmetric microcracks in Case 1. 6Figure 6.33 shows the magnified view

of the stress intensity contours due to the presence of a collinear microcrack in Case 2. Nine

contour lines denoted by A to I, represent the stress intensity values from 40,000 Pa to 200,000

Pa with a stress increment of 20,000 Pa. These contour lines indicate the concentrated stress

distribution of stress field near the tip of the crack: the closer to the tip of the crack, the higher

the stress intensity.

Figure 6.31 The deformation and stress intensity trajectories at the vicinity of the main crack for the semi-circular band specimen in the absence of microcracks.


Figure 6.32 The deformation and stress intensity trajectories of the main crack and the inclined microcrack of the semi-circular bend specimen in Model 1.

Figure 6.33 The deformation and stress intensity trajectories of the main crack and the collinear microcrack of the semi-circular bend specimen in Model 2.


Examining these contours, it can be seen that the microcracks in Model 1 tend to decrease the

intensity of the stress field of the main crack, thus yielding stress shielding effect of the main

crack; whereas for Model 2, the microcrack tends to increase the intensity of the stress field of

the main crack, thus yielding stress amplification of the main crack. This can be judged by

comparing the distance of the crack tip to the contour with the same stress level.

The finite element calculations can quantify the shielding or amplification effects for both cases.

Within the framework of linear elastic fracture mechanics (LEFM), only the ratio between stress

intensity factors is required in this study: 01 / IMI KK is 0.880 (shielding) for Model 1 and

02 / IMI KK is 1.233 (amplification) for Model 2. The quantification of shielding or amplification

effects using finite element analysis can help interpret the anisotropy of measured fracture

toughness in the experiments.

For the dynamic fracture tests conducted in the modified SHPB system with careful pulse

shaping, it has been proved in the previous section that for the Intact Model, the microcrack-free

case, as long as the dynamic force balance has been achieved on both ends of the NSCB sample,

the time-varying SIF deduced from the quasi-static data analysis is the same as that from a full

dynamic analysis. Following the same strategy, Model 1 and Model 2 will be examined as well.

Dynamic finite element analysis are carried out on Model 1 and Model 2 to determine the SIF

evolution by solving the equation of motion with Newmark time integration method in ANSYS

(Weisbrod and Rittel, 2000). The finite element meshing for Model 1 and Model 2 is the same as

that for 6Figure 6.29 and 6Figure 6.30, while taking the input loading F1 and F2 as half of the

dynamic loading forces exerted on the incident and transmitted side of the sample, respectively.

For pulse shaped SHPB tests, dynamic force balance can also be achieved (i.e. F1= F2), thus in

the finite element analysis, the bearing load (e.g. F2) is applied on both loading ends of the

sample. As demonstration, a typical dynamic load as shown in 6Figure 6.34 has been applied to

all three finite element models (i.e., Intact, Model 1 and Model 2). This time-varying load comes

from an actual measurement in a typical dynamic NSCB test with force balance achieved on both

ends of the sample.


0 50 100 150 200

0

1

2

3

4

Forc

e (k

N)

Time (μs)

Figure 6.34 The dynamic load exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and Model 2), the load comes from an actual measurement in a modified SHPB tests with force balance achieved on both ends of the sample.

For the three finite element models (Intact, Model 1 and Model 2), the SIF evolutions calculated

by dynamic finite element analysis are compared with that from a quasi-static analysis in 6Figure

6.35. For all three models (Intact, Model 1 and Model 2), the evolutions of SIFs from both static

and dynamic methods match reasonably well. Therefore, with remote force balance, the static

analysis can fully reproduce the transient SIF evolution of the main crack tip for all three models.

Recall that the static SIF evolution is proportional to the bearing load (e.g. F2) according to the

static equation which can be calibrated with static finite element analysis (6Figure 6.29 and 6Figure

6.30). Thus, at any instant throughout the loading, 01 / IMI KK is 0.880 (shielding) for Model 1,

and 02 / IMI KK is 1.233 (amplification) for Model 2.


0 50 100 150 200

0

1

2

3

4

5

SIF

(MP

a m

1/2 )

Time (μs)

Intact_Dynamic Intact_Static Model 1_Dynamic Model 1_Static Model 2_Dynamic Model 2_Static

Figure 6.35 The evolution of SIF of the NSCB specimen for three configurations (Intact, Model 1 and Model 2) from both quasi-static analysis and dynamic analysis; force balance has been guaranteed using a modified SHPB tests with careful pulse shaping.

It is also noted that for each test, the loading rate is determined as the slope of the pre-peak linear

portion of the SIF evolution. Thus, the ratio of the loading rate for Model 1 and Model 2 to the

loading rate for the Intact Model should be ••

01/ II KK is 0.880 (shielding) for Model 1 and ••

02 / II KK =1.233 (amplification) for Model 2.

This can be proven as follows. Assume there is a linearly increased force as shown in 6Figure

6.36 exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and

Model 2) under force balance. Using dynamic finite element analysis aforementioned, the

evolution of the dynamic dimensionless SIF Y (normalized by 2/3/ BRPS , see Figure 6.13) and

corresponding loading rates of the NSCB specimen for three configurations (Intact, Model 1 and

Model 2) can be calculated, as illustrated in 6Figure 6.37. The loading rate for each case, as

denoted by the slope of the dashed line, exhibits exactly the same ratio of 01 / IMI KK equals to

0.880 for Model 1 (shielding), and 02 / IMI KK equals to 1.233 for Model 2 (amplification).


0 20 40 60 80 1000

50

100

150

200

Forc

e (N

)

Time (μs)

Figure 6.36 The linear load exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and Model 2), assuming force balance on both loading ends of the sample.

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4Model 2

Model 1

Y

Time (μs)

Intact_Dynamic Intact_Slope Model 1_Dynamic Model 1_Slope Model 2_Dynamic Model 2_Slope Intact

Figure 6.37 The evolution of the dynamic dimensionless SIFs and corresponding loading rates of the NSCB specimen for three configurations (Intact, Model 1 and Model 2) with linear dynamic loading, assuming force balance on both loading ends of the sample.


6.5.5 Simulated Fracture Toughness Anisotropy

All of the rock fracture toughness measurement methods including those proposed by ISRM are

based on the linear elastic fracture mechanics (LEFM) theory, in which the fracture toughness is

considered to be unique and the crack of the rock sample initiates when the stress intensity factor

at the main crack reaches the fracture toughness. By recording the loading exerted on the

samples in the experiments, fracture toughness can be calculated based on proposed formulas.

For convinience, in the crack/microcracks configurations, the ratio of the local stress intensity

factor at the main crack tip and the loading is denoted byξ (i.e. ξ=0/ ILI KK ). Thus, 1=ξ

corresponds to the microcrack-free case. 1>ξ corresponds to the stress amplification where the

stress intensity factor of the main crack is increased due to the presence of microcracks. 1<ξ ,

corresponds to the stress shielding where the stress intensity factor of the main crack decreased

due to the presence of microcracks

For the SHPB tests for dynamic Mode-I fracture toughness characterization, the observed rate

dependence of fracture toughness anisotropy can also be interpreted using the proposed models,

but rather more complicated than the static case. This is because the dynamic fracture toughness

is no longer considered to be a unique value; in contrast, it is recognized to be dependent on the

loading rates. As proven before with dynamic finite element analysis, the ratio of the SIF loading

rate at the main crack tip and the loading is the same as the ratio of the local SIF at the main

crack tip to the loading at the same loading rate, i.e. ξ=0/ ILI KK and ξ=

••0/ I

LI KK .

Let 0ICK be the measured Mode-I fracture toughness (the global fracture toughness) of rocks from

the material testing device (either the material testing machine or SHPB system) based on the

maximum value of load, if a quasi-static state has been guaranteed. 0ICK is the maximum

apparent value of 0IK measured during material testing, which is usually considered to be ICK .

Experimental methods for measuring fracture toughness of rocks never “see” the effect of

microcracks on the disturbance of the local stress intensity field of the main crack. If 1≠ξ ,

during the experiment, the local stress intensity factor LIK ( 0

IK⋅= ξ ) increases with the loading 0IK until the L

IK is equal to the fracture toughness ICK . Under this circumstance, if the


corresponding loading rate at far field is apparently•

0IK , then the local stress intensity factor L

IK

( 0IK⋅= ξ ) actually increases with a loading rate of

•

⋅ 0IKξ . Thus,

ξξ /)/max()max( 00IC

LIIIC KKKK === (6.37a)

ξ/0••

= LII KK (6.37b)

This suggests that the intrinsic material toughness (mircrocrack-free cases) and the measured

fracture toughness (microcracks embedded) are related:

0ICIC KK ⋅= ξ (6.38a)

••

⋅= 0IIC KK ξ (6.38a)

Given the relationship of the fracture toughness for mircrocracks-free case and the measured

fracture toughness for microcracks embedded cases, the apparent measured fracture toughness

for the adopted Model 1 and Model 2 can be calculated. Based on the experimental results, it is

assumed that the intrinsic dynamic Mode-I fracture toughness can be described with the data

points tabulated in 6Table 6.5, where fracture toughness ranges from 3.2 MPa.m1/2 to 8.5

MPa.m1/2 and corresponding loading rate ranges from 40 GPa.m1/2/s to 180 GPa.m1/2/s with an

increment of loading rate of 20 GPa.m1/2/s. This variation of dynamic fracture toughness

corresponds to the microcrack-free case with 1=ξ , as denoted before.

The measured toughness values and corresponding loading rates for Model 1 and Model 2 are

calculated and tabulated in 6Table 6.5; and also illustrated in 6Figure 6.38. It is obvious that Model

1 own the highest fracture toughness while Model 2, the least, at any given loading rates.

Physically, it is quite reasonable because, with collinear microcracks right in front of the

potential fracture path, Model 2 facilitates fracturing of rock pieces than the Model 1 case, with

microcracks inclined to the fracture path. In addition, the numerical simulation on the fracture

toughness of these two models in 6Figure 6.38 reflects that: 1) the rate of increment of the

dynamic fracture toughness with respect to the loading rate appear to be the same for Barre

granite samples along different groups; 2) the ranking of the six sample groups with respect to


the magnitude of the fracture toughness at any given loading rate remains the same for always.

These two scenarios reproduced what were observed in the experiments.

Table 6.5 The fracture toughness and corresponding loading rates for three models (Intact,

Model 1 and Model 2). Intact Model Model 1 Model 2

Loading Rates

(GPa.m1/2/s) Fracture Toughness

(MPa.m1/2) Loading Rates(GPa.m1/2/s)

Fracture Toughness(MPa.m1/2)

Loading Rates (GPa.m1/2/s)

Fracture Toughness(MPa.m1/2)

~0 1.2 ~0 1.4 ~0 1.0 40 3.2 45.5 3.6 32.4 2.6 60 4.0 68.2 4.5 48.7 3.2 80 4.7 90.9 5.3 64.9 3.8 100 5.5 113.6 6.3 81.1 4.5 120 6.2 136.4 7.0 97.3 5.0 140 7.0 159.1 8.0 113.5 5.7 160 7.7 181.8 8.8 129.8 6.2 180 8.5 204.5 9.7 146.0 6.9

0 40 80 120 160 200

2

4

6

8

10 Intact Model 1 Model 2

KIC

(MP

a m1/

2 )

KI (GPa m1/2 s-1)

Figure 6.38 The simulated dynamic fracture toughness of Barre granite with loading rates for three configurations (Intact, Model 1 and Model 2).


In the preceding chapter, the anisotropic index of Mode-I fracture toughness (αk) has been

defined as the ratio of the maximum fracture toughness to the minimum of fracture toughness.

Specifically in current numerical simulations, for static case, 01 / IMI KK =0.880 for Model 1 and

02 / IMI KK =1.233 for Model 2. Thus, during the experiments for fracture toughness

measurements, the measured fracture toughness for Model 1 should be

ICICMIC KKK 136.1880.0/1 == and that for Model 2 should be ICIC

MIC KKK 811.0233.1/2 == .

Thus, during static loading, the anisotropic index of Mode-I fracture toughness, αk

is 21 / MIC

MIC KK =1.136KIC/0.811KIC =1.40. For dynamic cases, the featuring loading rates are first

picked up and then the corresponding fracture toughness values at given selected loading rates

are deduced with interpolation within data points of Model 1 and Model 2, respectively.

Afterwards, αk can be calculated as the ratio of the toughness from Model 1 to that from Model 2

at giving loading rates. All related results are tabulated in 6Table 6.6.

Table 6.6 The simulated Mode-I fracture toughness anisotropic index (αk) of Barre granite

with loading rates.

Fracture Toughness (MPa.m1/2) Loading Rates (GPa.m1/2/s) Intact Model 1 Model 2

Anisotropy Index αk

~0 1.2 1.400 1.000 1.400 40 3.2 3.449 2.892 1.193 60 4.0 4.201 3.644 1.153 80 4.7 4.953 4.396 1.127

100 5.5 5.705 5.148 1.108 120 6.2 6.457 5.900 1.094 140 7.0 7.209 6.652 1.084 160 7.7 7.961 7.404 1.075 180 8.5 8.713 8.156 1.068 200 9.2 9.465 8.908 1.063

The variation of the anisotropic index of the simulated Mode-I fracture toughness, αk with

loading rates is also plotted in 6Figure 6.39. The data points of αk in 6Figure 6.39 drops quickly

approaching the isotropic value of 1. Thus, with proposed Models, the same trend of the loading

rate dependence of Mode-I fracture toughness has been reproduced as that reported previously in

the experimental characterizations, shown in 6Figure 6.22c.


In real experimental tests for the two extreme cases corresponding to Model 1 and Model 2

shown in 6Figure 6.23 and 6Figure 6.24 respectively, the measured fracture toughness leads to an

anisotropic ratio of ~1.65. Although it is not intended to reproduce the exact ratio from

experiments by adjusting the geometrical parameters in these models, the numerical result yields

a very good agreement with experimental measurements.

Figure 6.39 The simulated Mode-I fracture toughness anisotropic index (αk) of Barre granite with loading rates based on crack-microcracks interaction model.

6.5.6 Concluding Remarks

Laboratory measurements of ICK of Barre granite under a wide range of loading rates were

carried out statically with MTS machine and dynamically with SHPB system. The fracture

toughness of the Barre granite investigated exhibited a decreasing anisotropy with the increase of


loading rates. Microstructural investigation of thin sections showed that there are three dominant

embedded microcracks orientated in preferred directions. The crack-microcrack interaction

model and its effect on the stress intensity factor of the main crack are used to explain this

loading rate dependence of Mode-I fracture toughness anisotropy.

Two models are constructed to investigate the influence of the stress intensity factor of the main

crack due to the presence of microcracks based on microscopic thin section pictures. Our finite

element analysis indicate that for the case of two symmetric microcracks near the vicinity of the

main crack (Model 1), the stress intensity factor is lower than the stress intensity factor of the

same main crack in the absence of microcracks (shielding); for the case of a collinear microcrack

near the main crack (Model 2), the stress intensity factor is higher than the stress intensity factor

of the same main crack in the absence of microcracks (amplification). Under our ideal

crack/microcrack models, the measured static fracture toughness could be different by a factor of

1.40; while in the dynamic case, for example, under a loading rate of 200 GPa.m1/2/s, the factor

decreases to 1.063.

In engineering design against fracture failure, the intrinsic fracture toughness is one of the key

parameters. For rocks with pre-existing microcracks, the measured fracture toughness is not

necessarily the intrinsic fracture material toughness ICK as discussed before. If the measured

fracture toughness value corresponding to stress shielding case is used, the design tends to be

overly aggressive; if the measured fracture toughness value corresponding to stress amplification

is used, the design tends to be overly conservative. However, it is able to deduce the intrinsic

fracture toughness ICK combining with the microstructural analysis of the rock sample and

numerical simulation. It is then feasible to use ICK and in-situ microstructural analysis to

determine the global fracture toughness LoadIK and make our design accordingly. As a result,

different toughness values will be used in different directions and these values can be evaluated

easily with ξ evaluated from numerical analysis and the relation of ξ/ICLoadI KK = . On the other

hand, under very higher loading rates, the apparent measured fracture toughness only has

negligible difference with the intrinsic dynamic fracture toughness. The material appears to be

isotropic in the perspective of practical significance.


6.6 Summary

In this Chapter, a dynamic NSCB method is proposed in conjunction with the LGG system

(Chen et al., 2009) to measure the dynamic fracture toughness and fracture energy of rocks.

Because the quasi-static analysis can be carried out independently of the detailed forms of the

loading history, the dynamic NSCB method thus provides a much more convenient way to

quantify the dynamic fracture toughness of brittle materials such as rocks.

To validate the new method, the inertial effect in the dynamic NSCB test of rocks using SHPB is

systematically examined. It was found that without pulse shaping, the dynamic forces on both

ends of the specimen are very different. The resulting inertial effect causes two peaks in the

transmitted force pulse, and a huge delay of the peak transmitted force with respect to the crack

onset. The SIF history obtained from a full dynamic finite element analysis is very different from

that obtained from a quasi-static analysis using the transmitted dynamic force as loads. With

careful pulse shaping, the dynamic force balance for the entire dynamic loading period can be

achieved. In this case, the peak far-field load matches with the fracture onset reasonably well. In

addition, the SIF history obtained from full dynamic finite element analysis agrees well with that

from quasi-static analysis. It is thus verified that with far-field force balance, the inertial effect is

minimized and quasi-static analysis is thus valid to deduce the fracture toughness.

The new method is then applied to research on the fracture properties of anisotropic Barre

granite under both static and dynamic loadings. Rate dependence of the fracture toughness and

fracture energy of Barre granite is observed. The Barre granite exhibits strong fracture toughness

anisotropy under static loading and diminishing anisotropy in dynamic loading. Under high

loading rates, it is anticipated that the fracture toughness anisotropy can be ignored. The rate

dependence of the anisotropy is explained with proposed microcrack interaction models, built

from microstructural investigation of two thin sections showing the pre-existing microcracks

orientated in preferred directions. These two thin sections are taken from recovered Barre granite

samples in two different groups with distinct measured fracture toughness. The crack microcrack

interaction models reproduced the apparent rate dependence of fracture toughness anisotropy.

CHAPTER 7: SUMMARY AND FUTURE WORK 202

CHAPTER 7

SUMMARY AND FUTURE WORK

This chapter summarizes the overall conclusions of the thesis from the preceding chapters.

Future work is also outlined.

7.1 Summary of the Thesis Work

This thesis investigated the anisotropy of tension-related failure parameters, i.e. tensile strength,

flexural strength and Mode-I fracture toughness/fracture energy of anisotropic Barre granite

under a wide range of loading rates, and explored the relationship between the fabric of

preferentially embedded microcracks in the Barre granite and the measured anisotropy of these

physical properties.

The following lines summarize the main items completed in this thesis and the major

concussions:

• Three sets of dynamic experimental methodologies involving experimentation and

calculation equations using the modified dynamic testing machine, i.e. SHPB system, are

developed to measure the dynamic tension-related mechanical properties of rocks. These


methods are: the dynamic BD method to determine the dynamic tensile strength of rocks,

the dynamic SCB method to determine the dynamic flexural strength of rocks, and the

dynamic NSCB method to determine the dynamic Mode-I fracture toughness of rocks. The

samples chosen are all core-based, facilitating sample preparation from rock blocks. The

data reduction is readily applicable for employing quasi-static equations to the dynamic

tests, provided that the time-resolved dynamic forces are balanced at both loading ends of

the BD, SCB and NSCB samples.

• The reliability and robustness of the proposed dynamic testing methodologies, for dynamic

tensile strength, flexural strength and Mode-I fracture toughness characterization are

rigorously validated. To do so, both pulse shaped and non-pulse shaped tests were

conducted to assess under which circumstance the proposed dynamic testing methods are

valid. A strain gauge was mounted near the failure spot or crack tip on the specimen to

determine the onset instant of fracture. It was demonstrated that in a modified SHPB test

with proper pulse shaping, the dynamic force balance within the sample is achieved. Thus,

the tensile stress at the failure spot or the Mode-I SIF at the crack tip in the sample can be

calculated with either quasi-static analysis or dynamic finite element analysis using the far-

field measurements as inputs. Moreover, the rupture time synchronizes with the peak of the

transmitted pulse recorded in the SHPB system after corrections for travel time. Therefore,

the dynamic tensile strength, flexural strength and fracture toughness can be calculated

from the peak of the transmitted wave measured in the SHPB system with quasi-static

analysis.

• The loading rate dependence of the tensile strength, flexural strength and Mode-I fracture

toughness and fracture energy of Barre granite has been observed along all six directions.

Take sample XY for example, the tensile strength under a quasi-static loading rate of 1.8E-

4 GPa/s is 9.5 MPa, while under dynamic loading rate up to 1500 GPa/s, the tensile

strength is 38.2 MPa, four times of the static strength; the static flexural strength of XY is

13.5 MPa under a loading rate of 8E-4 GPa/s, while the dynamic flexural strength is 54.8

MPa, also four times of the static tensile strength, with a dynamic loading rate ~1800

GPa/s; the static Mode-I fracture toughness of XY is 1.03 MPa·m1/2 under loading rate of

~8E-5 MPa·m1/2s-1; and the dynamic counterpart is more than seven times of the static one

(i.e. 7.8 MPa·m1/2), under a loading rate of 180 MPa·m1/2s-1.


• The flexural strengths of Barre granite along all six directions are consistently higher than

the tensile strength measured from Brazilian tests in both static test and dynamic tests. The

tensile stress gradient along the potential failure path is believed to be the main reason of

this distinction, since the tensile strength is defined under a homogeneous tensile stress

state. A non-local failure theory is adopted to qualitatively explain the differences of the

measured strengths; and the gap between these two is bridged as well. This can be done by

determining the ratio κ (i.e. σf /σt) first by comparing the static flexural strength to the static

tensile strength. Then this ratio κ can be utilized to reconcile the dynamic tensile strength

from the measured dynamic flexural strength, compared with the direct measures of

dynamic tensile strengths from Brazilian tests for all six sample directions. The reconciled

tensile strength from flexural strength matches very well with the tensile strength from

direct measures via Brazilian tests.

• Under static loading, Barre granite exhibits strong anisotropy for the three tension-related

parameters, i.e. tensile strength, flexural strength and Mode-I fracture toughness. For the

static case, the tensile strength anisotropic index equals to 1.83, with the highest strength of

16.8 MPa for samples splitting in the plane normal to Z axis; and the lowest of 9.2 MPa

with splitting plane normal to X axis. The flexural strength anisotropic index equals to 1.86,

with the highest strength of 24.6 MPa for samples splitting in the plane normal to Z axis;

and the lowest of 13.2 MPa with splitting plane normal to X axis. Under static loading, the

fracture toughness index equals to 1.70, as the loading rate is up to 220 MPa·m1/2s-1, the

index drops to 1.20; the maximum fracture toughness remains still in samples ZX and the

lowest one is fixed in samples XY for both cases.

• Under dynamic loading, in sharp contrast to the static loading, Barre granite exhibits much

weaker anisotropy for the three tension-related parameters, i.e. tensile strength, flexural

strength and Mode-I fracture toughness. The anisotropic index of 1) tensile strength drops

drastically to the dynamic value of 1.13 with a loading rate of 1800 GPa/s; 2) flexural

strength drops to 1.24 under a loading rate of 2000 GPa/s; 3) Mode-I fracture toughness

drops to 1.20, as the loading rate is up to 220 MPa·m1/2s-1. The tensile strength, flexural

strength and Mode-I fracture toughness anisotropy of Barre granite appears to be strong

under quasi-static loading while rather weak under dynamic loading rates.


• It is identified that anisotropy of the mechanical properties explored in this research is

correlated closely to the preferentially oriented microcracks sets. With reference to the

dominant three sets of microcracks in Chapter 3, YZ plane is recognized to be parallel to

the rift plane with the dominant microcracks, and XZ is the secondary concentration of

microcracks for Barre granite. The YZ plane, XZ plane and XY plane correspond to the

quarryman’s description of “rift plane”, “grain plane” and “hard-way plane” respectively.

This explains that in our static tests of tensile strength flexural strength and Mode-I

fracture toughness measurements, the minimum strength or toughness is obtained from

sample XY and XZ, both split in the rift plane YZ (normal to X axis), while the maximum

are obtained from sample ZX and ZY with a hard-way splitting plane XY (normal to Z

axis).

• Qualitative interpretation on the anisotropy of tensile strength and flexural strength has

been given. When a rock sample with an array of cracks is loaded statically, the critical

flaw or crack will dominate the response of the rock, yielding the maximum bearing load.

If a preferred orientation of the largest flaws exists, the material will also show a

dependence on the orientation for the fracture stress. In contrast, the dynamic load is

qualitatively very different from static load. Given the rapid, short time loading, only a

small volume V of the sample is indeed stressed to a high value and this volume is not

tremendously affected by its neighboring volumes. This will lead to a less anisotropy of the

dynamic rock tensile/flexural strength.

• The crack-microcrack interaction model is utilized to reproduce the apparent rate

dependence of Mode-I fracture toughness anisotropy. Two models were built from

microstructural investigation of thin sections showing the pre-existing microcracks

orientated in preferred directions. The two thin sections were taken from recovered Barre

granite fracture samples along two directions with distinct measured fracture toughness.

Both analytical and numerical analysis reveled that the stress intensity of the main crack

tends to be shielded due to the microcracks at an angle of o45 to main crack, while the

stress intensity of main crack is amplified as it is collinear to microcracks. The measured

fracture toughness is reversely proportional to the shield/amplification effects due to the

microcracks, and this yields the apparent fracture toughness anisotropy for the two models.

By the same token, a dynamic analysis was conducted employing the same models. Using


the crack-microcrack interacting models, the trend of the rate dependence of fracture

toughness for the two cases is reconstructed, from which, descending fracture toughness

anisotropy with ascending loading rates was also explicitly reproduced. In addition, the

models explained why the rate of increment of the dynamic fracture toughness with respect

to the loading rate appears to be the same for Barre granite samples along different groups.

7.2 Future Work

A straightforward extension of current research is the measurement of the fracture surface

topology, e.g. surface roughness and fractal dimensions, of the newly generated fracture surfaces

of Barre granite samples recovered from the dynamic tensile test, dynamic flexural test and the

dynamic Mode-I fracture tests. The employed novel technique on SHPB, i.e. Momentum Trap

technique, prohibits multiple loading to the sample, and thus makes it possible to quantitatively

relate the surface topology to the loading. The objective of this investigation is twofold: first, to

examine the relation between these mechanical properties (i.e. tensile strength, flexural strength

and Mode-I fracture toughness) and fracture surface topology as a function of loading rates; and

second, to look into the loading rate dependence of fracture surface topology anisotropy of the

Barre granite using the tested samples on six directions preserved after the dynamic tensile,

flexural and Mode-I fracture toughness tests.

Apart from this, two other motivating branches of interest can be directed to systematically and

thoroughly investigate the environmental influences involving confining effects and thermal

effects on the mechanical properties of rocks in general and anisotropic Barre granite in specific.

Both confining effects and thermal effects may influence the mechanical properties of rocks

through its microcracks.


7.2.1 Confining Effects

It has been well-known that the rock strength properties are markedly influenced by the

confining pressure. The rock strength characterization under lateral confinement is thus

important to actually understand the mechanism of rock behaviors in engineering applications.

The quasi-static triaxial test on rocks is mature and has already been standardized by ISRM

(Vogler and Kovari, 1978).

Experiments have also been carried out to study the strain rate effects on rock material properties

in the triaxial compression (Li et al., 1999; Masuda et al., 1987; Sangha and Dhir, 1975; Yang

and Li, 1994). While most believe that as the confining pressure increases, the strength of rocks

increases more when the strain rates increases up to the same order of magnitude (Masuda et al.,

1987; Sangha and Dhir, 1975). Others reported that the strength increment is less under a higher

confining pressure as the strain rate increases under the same range, such as Sangha and Dhir

(1975) on a sandstone and Yang and Li (1994) on a granite. The opposing results reveal that the

rate of increase of the mechanical properties of rocks could be different for different rocks and

different confining pressure; and exactly these contrasting observations inspire the curiosity to

look into the anisotropic Barre granite.

The strain rates achieved in these tests are between 10-7 to 100/s, limited by the dynamic testing

machine used. To achieve higher strain rates, triaxial Hopkinson method has been adopted to

simultaneously subject the samples to lateral confinement and axial loading (Nemat-Nasser et al.,

2000). Christensen (1972) pioneered the usage of triaxial Hopkinson method on rock testing. The

lateral compression was mostly applied through a pneumatic pressure vessel; and a similar vessel

is to be employed in our SHPB design to accommodate the dynamic tensile tests via BD sample,

flexural tests via SCB samples and fracture tests via NSCB samples. As an example, 6Figure 7.1

illustrates the design of the dynamic Brazilian test under hydrostatic confining pressure on SHPB

system; the pneumatic pressure vessel is filled with high pressured oil. The BD sample can be

replaced with the other two types of samples.


Figure 7.1 Schematic of the Brazilian test under hydrostatic confining pressure on SHPB system.

It is aimed at in the near future to further investigate the effects of confining pressure on the

tensile strength, flexural strength and Mode-I fracture toughness of anisotropic Barre granite

under dynamic loading cases. The objective of this research is threefold: 1) to develop a set of

reliable triaxial Hopkinson bar methods to conduct strength and toughness measurements of

rocks under high loading rates; 2) to study the effects of the confining pressure and the coupled

effects of the confining pressure and loading rates on the anisotropy of material properties of

Barre granite; 3) to correlate the properties to the microstructure of Barre granite and look into

the micro-mechanism of the observations.

7.2.2 Thermal Effects

Temperature variation occurs as a company of a variety of rock engineering practices, such as

rock cutting, drilling, blasting and fragmentation, ore crushing, tunneling boring, etc. It has been

recognized that temperature markedly influences the mechanical properties of rocks (Duclos and

Paquet, 1991; Homandetienne and Houpert, 1989; Inada and Yokota, 1984; Nasseri et al., 2007;

Wai et al., 1982), such as coefficient of thermal expansion, Young’s modulus, Poisson’s ratio,

the compressive/tensile strength and fracture toughness; in addition, different heating/cooling

rates yields different changes to the mechanical properties.


The research on the coupled effects of temperature and loading rates on the mechanical

properties of rocks has been rarely reported in the literature. Several pioneering attempts are as

follows. Lindholm (1974) constructed the relationship between rock strength, temperature and

the strain rate. Zhang et al. (2001) extended their early works (Zhang et al., 2001) to investigate

the dynamic fracture toughness of a gabbro and a marble subjected to two cases of thermal

circumstances: case 1, samples are tested at high temperature; case 2, samples are pre-heat

treated at varying high temperatures.

Given available investigations in the literature, the effect of temperature on the dynamic tensile

strength and flexural strength of rocks are missing. As well, there is no research on the

mechanical properties, e.g. tensile strength, flexural strength and fracture toughness of

anisotropic rock like Barre granite under a wide range of loading rates. With current available

novel techniques on SHPB, it is time to initiate the study in these areas.

It is of interest to investigate thermal effects on the dynamic tensile strength, flexural strength

and fracture toughness of anisotropic Barre granite. Two types of thermal effects will be

considered: 1) pre-heat treated 2) in-situ heating. For case 1, three types of Barre granite samples

(BD, SCB, and NSCB) along six directions are first pre-heat treated at varying high temperature,

say 200 ˚C, 400 ˚C, 600 ˚C and 800 ˚C. After all cooled down to the room temperature, these

samples are used to conduct the SHPB tests as discussed before. For case 2, a schematic of the

modified SHPB design for conducting dynamic Brazilian tests with in-situ heating is illustrated

in 7Figure 7.2. This modified SHPB system can also host the flexural test using SCB samples and

Mode-I fracture tests using NSCB samples.

Figure 7.2 Schematic of the Brazilian test on SHPB system under in-situ thermal heating.


For both thermal conditions proposed herein, different heating/cooling rates may be tried as well

to look into the dependence of this on the material properties. This research is expected to

understand the relationship between the thermal cracking on the host rock physics and the

mechanical properties under a wide range of loading rates. The first exploration of the anisotropy

of Barre granite on these mechanical properties due to thermal cracking under a wide range of

loading rates is also of primary interest in the near future.

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