Dynamic Tensile, Flexural and Fracture Tests of Anisotropic Barre Granite

Transcription

1 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 010

2 Dynamic Tensile, Flexural and Fracture Tests of Anisotropic Barre Granite Feng Dai Doctor of Philosophy Department of Civil Engineering University of Toronto 010 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 quasistatic 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. ii

3 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. iii

4 To my family iv

5 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, v

6 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. vi

7 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 Background Problem Statement Research Objectives Research Contribution Thesis Organization...9 CHAPTER LITERATURE REVIEW Barre Granite and Its Anisotropy Microstructural Investigation Mechanical Properties Tension Tests Static Tension Tests Dynamic Tension Tests Fracture Tests...3 vii

8 .3.1 Static Fracture Tests Dynamic Fracture Tests...9 CHAPTER 3 EXPERIMENTAL SETUP AND TECHNIQUES Samples Preparations Laurentian Granite Barre Granite MTS Hydraulic Servo-control System Split Hopkinson Pressure Bar Working Principle Pulse Shaping Momentum Trap Laser Gap Gauge System Principles and Setup Calibration of the System...53 CHAPTER 4 DYNAMIC TENSION TESTS Background Studies Dynamic Brazilian Test Validation of Dynamic Brazilian Test Dynamic Brazilian Test without Pulse Shaping Dynamic Brazilian Test with Careful Pulse Shaping Tensile Strength of Barre Granite Determination of Anisotropic Tensile Strength Tensile Strength Anisotropy Interpretation of the Results...91 viii

9 4.5 Summary...93 CHAPTER 5 DYNAMIC FLEXUAL TESTS Background studies Dynamic Semi-circular Bend Flexural Test The Semi-circular Bend Testing in a SHPB System Determination of Flexural Strength Validation of Semi-Circular Bend Tests Failure Sequences of the Specimen in the Dynamic SCB Test Dynamic SCB Test without Pulse Shaping Dynamic SCB Test with Careful Pulse Shaping Flexural Strength of Barre Granite Determination of Anisotropic Flexural Strength Flexural Strength Anisotropy Interpretation of the Results Summary CHAPTER 6 DYNAMIC FRACTURE TESTS Background Studies Dynamic Notched Semi-circular Bend Fracture Test The Notched Semi-circular Bend Testing in an SHPB System Determination of Mode-I Fracture Toughness Determination of Dynamic Fracture Energy Validation of Dynamic Notched Semi-Circular Bend Test Dynamic Analysis and Fracture Time Dynamic NSCB Test without Pulse Shaping ix

10 6.3.3 Dynamic NSCB Test with Careful Pulse Shaping Fracture Toughness Anisotropy of Barre Granite Determination of Anisotropic Stress Intensity Factor Determination of Fracture Toughness of Barre Granite Fracture Toughness Anisotropy Crack-Microcrack Interaction Background Microstructural Investigation and Featuring Models The Crack-Microcrack Interaction Finite Element Analysis of Two Models Simulated Fracture Toughness Anisotropy Concluding Remarks Summary...01 CHAPTER 7 SUMMARY AND FUTURE WORK Summary of the Thesis Work Future Work Confining Effects Thermal Effects...08 BIBLIOGRAPHY...11 x

11 LIST OF TABLES Table 4.1 The material properties used in the finite element model of BD samples of Barre granite along six directions Table 4. Tensile strengths of Barre granite along six directions from both static and dynamic Brazilian tests Table 5.1 The material properties used in the finite element model of SCB samples of Barre granite along six directions Table 5. Flexural strengths of Barre granite with corresponding loading rates as well as the non-local reconciliation for both static and dynamic SCB tests Table 5.3 Summary of the parameters deduced using non-local failure model for all six sample groups of Barre granite Table 6.1 The normalized stress intensity factor K * = K /σ πa, for an edge crack in an infinite orthotropic strip with remote uniform traction σ Table 6. The material properties used in the finite element model of NSCB samples of Barre granite along six directions Table 6.3 Fracture toughness and fracture energy of Barre granite with corresponding loading rates from both static and dynamic NSCB fracture tests Table 6.4 Stress intensity factor of the main crack with one collinear microcrack at different distances to the main crack tip Table 6.5 The fracture toughness and corresponding loading rates for three models (Intact, Model 1 and Model ) Table 6.6 The simulated Mode-I fracture toughness anisotropic index (α k ) of Barre granite with loading rates I I xi

12 LIST OF FIGURES Figure.1 Mineral and microcracks traced from three orthogonal planes for Barre granite; after (Nasseri and Mohanty, 008) Figure. 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, 008); the letters in the braskets are the directions used in this thesis Figure.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, 008); the letters in the braskets are the directions used in this thesis Figure.4 Variation of fracture toughness measured along six directions with the number of tests along each direction in Barre granite; after (Nasseri and Mohanty, 008) Figure.5 Strain rate effects of the maximum compressive stress for X-, Y- and Z- samples of Barre granite; reproduced after (Xia et al., 008); the letters in the braskets are the directions used in this thesis Figure.6 The three basic modes of crack propagation: (a) Mode I, opening mode; (b) Mode II, in-plane shearing; (c) Mode III, tearing mode Figure.7 Definition of the local coordinate axis ahead of a crack tip. Z direction is normal to the plane Figure.8 Comparison of the fracture mechanics approach to design with the traditional strength of material approach: (a) strength approach (b) fracture toughness approach... 7 Figure 3.1 Procedures for preparing three types of samples: Brazilian disc (BD), semicircular bend (SCB) and notched semi-circular bend (NSCB) samples xii

13 Figure 3. 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 Figure 3.3 Photoes of (a) semi-circular bend and (b) Brazilian test of rock samples in the MTS hydraulic servo-control testing system Figure 3.4 Photo of a split Hopkinson pressure bar (SHPB) system in the Department of Civil Engineering, University of Toronto... 4 Figure 3.5 Schematics of a split Hopkinson pressure bar (SHPB) system and the x-t diagram of stress waves propagation in SHPB 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 Figure 3.7 Pulse shapers in SHPB (a) schematic of the assembly (b) unshaped and shaped incident stress pulses Figure 3.8 The momentum-trap system: (a) the actual image and (b) the x t diagram showing its working principle 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 Figure 3.10 Photo and schematics of the laser gap gauge (LGG) system set up perpendicular to the bar axis of SHPB Figure 3.11 Static calibration of the LGG system using a gap gauge blocking the collimated beam: schematic setup and the calibration result Figure 3.1 Dynamic calibration of the LGG system: schematic setup and a typical dynamic testing result compared to the predictions by Equation (3.6) xiii

14 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 Figure 4. 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 Figure 4.3 High-speed video images of a typical dynamic Brazilian test on Laurentian granite without pulse shaping Figure 4.4 Mesh of the Brazilian disc for the finite element analysis with ANSYS; P 1 and P are the diametrical forces on both loading ends Figure 4.5 (a) Tensile stress σ x (b) compressive stress σ y 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 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 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 Figure 4.8 High-speed video images of two typical dynamic Brazilian tests on Laurentian granite with careful pulse shaping Figure 4.9 (a) Tensile stress σ x (b) compressive stress σ y 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 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 Figure 4.11 The measured tensile strength of Laurentian granite from dynamic Brazilian tests with and without employing jaws xiv

15 Figure 4.1 Schematics of a Brazilian test in (a) the material testing machine and (b) the SHPB system Figure 4.13 Stress trajectories of a Brazilian disc under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy with isotropic model, and (d) f xx (e) f yy and (f) f xy for sample YX using anisotropic model (positive for compression, negative for tension) Figure 4.14 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy with isotropic model Figure 4.15 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample XY, and (d) f xx (e) f yy and (f) f xy for sample XZ (positive for compression, negative for tension) Figure 4.16 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample YX, and (d) f xx (e) f yy and (f) f xy for sample YZ (positive for compression, negative for tension) Figure 4.17 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample ZX, and (d) f xx (e) f yy and (f) f xy for sample ZY (positive for compression, negative for tension) 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 Figure 4.19 Virgin Brazilian discs of Barre granite prepared for the test; each division in the scale denotes 1 mm Figure 4.0 Recovered Brazilian discs of Barre granite after tests; each division in the scale denotes 1 mm Figure 4.1 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 Figure 4. The variation of tensile strength with loading rates for six sample groups of Barre granite xv

16 Figure 4.3 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 (α t ) of Barre granite with loading rates Figure 5.1 Schematics of the determination of the flexural strength of concrete by ASTM standards: a) ASTM C93, 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 Figure 5. 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 = 0 mm, is sandwiched between the incident and transmitted bars. A strain gauge is mounted on the specimen near the point O Figure 5.3 Meshing scheme of the SCB specimen for finite element analysis. F 1 and F denote forces applied on the contact points Figure 5.4 Y as a function of the dimensionless geometry parameter S/R from the quasistatic finite element analysis; the coefficient of determination of the fitting curve R is Figure 5.5 High-speed video images of a dynamic semi-circular bend test on Laurentian granite 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 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 xvi

17 Figure 5.9 Strain gauge signal and the transmitted force P in the SCB-SHPB test on Laurentian granite without pulse shaping 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 5.1 Strain gauge signal and the transmitted force P in the SCB-SHPB test on Laurentian granite with pulse shaping Figure 5.13 Schematics of the semi-circular bend test in (a) the material testing machine and (b) the SHPB system Figure 5.14 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy with isotropic model, and (d) q xx (e) q yy and (f) q xy for ZX sample using anisotropic model (positive for compression, negative for tension) Figure 5.15 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy with isotropic model Figure 5.16 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for XY sample and (d) q xx (e) q yy and (f) q xy for XZ sample (positive for compression, negative for tension) Figure 5.17 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for YX sample, and (d) q xx (e) q yy and (f) q xy for YZ sample (positive for compression, negative for tension) Figure 5.18 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for ZX sample, and (d) q xx (e) q yy and (f) q xy for ZY sample using anisotropic model (positive for compression, negative for tension) xvii

18 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.. 11 Figure 5.0 (a) Virgin semi-circular bend samples of Barre granite; (b) Recovered semicircular bend samples of Barre granite after tests Figure 5.1 The variation of static flexural strength of Barre granite along six directions, i.e. XY, XZ, YX, YZ, ZX and ZY... 1 Figure 5. The variation of flexural strength with loading rates along six directions of Barre granite Figure 5.3 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 (α f ) of Barre granite with loading rates Figure 5.4 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 R of Figure 5.5 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 Figure 5.6 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 Figure 5.7 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 Figure 5.8 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 Figure 5.9 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 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 xviii

19 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 Figure 6. 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 Figure 6.3 Typical loading history and CSOD history of the NSCB specimen tested in SHPB on Laurentian granite Figure 6.4 Selected high speed camera images showing the fracture and fragmentation of a NSCB Laurentian granite specimen 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 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) 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 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 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) 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 Figure 6.11 The effect of loading rate on the fracture toughness and fracture energy of Laurentian granite xix

20 Figure 6.1 Local coordinate system for the stress and displacement fields near the crack tip of an orthotropic solid 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 Figure 6.14 An infinite orthotropic strip with an edge crack under remote uniform tractions normal to the edge crack 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 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 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 Figure 6.18 (a) Virgin NSCB samples and (b) recovered NSCB samples of Barre granite 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 Figure 6.0 The variation of fracture toughness with loading rates on six directions of Barre granite Figure 6.1 The variation of fracture energy with loading rates on six directions of Barre granite Figure 6. 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 α k of Barre granite Figure 6.3 (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample; Case 1: the main crack inclines at an angle of o 45 to microcracks; (b) Model 1: the crack-microcracks configuration for Case xx

21 Figure 6.4 (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample; Case : The main crack is collinear to microcracks; (b) Model : the crackmicrocracks configuration for Case Figure 6.5 One arbitrarily located microcrack near the crack tip of a semi-infinite crack Figure 6.6 The original problem and the three sub-problems decomposed from the original one based on the superposition method Figure 6.7 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 Figure 6.8 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 6.9 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 ; Case (b) close-view of the mesh at the vicinity of the main crack and the collinear microcrack of Model ; Case. The main crack and its tip are indicated with arrows and the collinear microcrack is also marked 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 6.3 The deformation and stress intensity trajectories of the main crack and the inclined microcrack of the semi-circular bend specimen in Model 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 Figure 6.34 The dynamic load exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and Model ), the load comes from an actual measurement in a modified SHPB tests with force balance achieved on both ends of the sample xxi

22 Figure 6.35 The evolution of SIF of the NSCB specimen for three configurations (Intact, Model 1 and Model ) from both quasi-static analysis and dynamic analysis; force balance has been guaranteed using a modified SHPB tests with careful pulse shaping Figure 6.36 The linear load exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and Model ), assuming force balance on both loading ends of the sample 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 ) with linear dynamic loading, assuming force balance on both loading ends of the sample Figure 6.38 The simulated dynamic fracture toughness of Barre granite with loading rates for three configurations (Intact, Model 1 and Model ) Figure 6.39 The simulated Mode-I fracture toughness anisotropic index (α k ) of Barre granite with loading rates based on crack-microcracks interaction model Figure 7.1 Schematic of the Brazilian test under hydrostatic confining pressure on SHPB system Figure 7. Schematic of the Brazilian test on SHPB system under in-situ thermal heating.09 xxii

23 LIST OF ACRONYMS AND ABBREVIATIONS 1PB D 3D 3PB ASTM BD CB CCNBD CCS COD CSOD CSOV CSTBD FEA LORD ISRM LEFM LVDT LGG SCB SHPB SIF One Point Bend Two Dimensional Three Dimensional Three Point Bend American Society for Testing and Materials Brazilian Disc Chevron Bend Cracked Chevron Notch Brazilian Disc Compact Compressive Specimens Crack Opening Displacement Crack Surface Opening Displacement Crack Surface Opening Velocity Cracked Straight Through Brazilian Disc Finite Element Analysis Laser Occlusive Radius Detector International Society of Rock Mechanics Linear Elastic Fracture Mechanics Linear Variable Displacement Transducers Laser Gap Gauge Semi-Circular Band Split Hopkinson Pressure Bar Stress Intensity Factor SR Short Rod xxiii

24 TEM NSCB WLCT Transmission Electron Microscope Notched Semi-Circular Bend Wedge-Loaded Compact Tension xxiv

25 LIST OF SYMBOLS A A c a a ij Cross-sectional area of the bar Area of the new generated crack surfaces Depth of the notch Elastic compliance constants of the material a ij Compliance constants in the local x-y coordinate system B C c C ijkl D d D ij E E i f xx, f yy, f xy F f c G G C G ij l s K k The thickness of the sample Elastic wave velocity Half length of the microcrack Stiffness constants of Barre granite The diameter of the sample Distance of the main crack tip to the center of the microcrack Factors to determine the anisotropic fracture toughness Young s Modulus Young s Modulus in the i principle direction Dimensionless stress components for BD Dimensionless tensile stress component at the disc center The cutoff frequency of a bar Shear Modulus Fracture energy Shear Modulus in the i-j plane Length of the striker Kinetic energy of the cracked fragments Calibrated parameter of the LGG system xxv

26 K I K IC P K I K II K IIC K III K IIIC 0 K I Mode I SIF Mode I fracture toughness Mode I propagation fracture toughness Mode II SIF Mode II fracture toughness Mode III SIF Mode III fracture toughness Prescripted far field loading in terms of Mode I SIF 0 K II Mode II SIF of the main crack for Intact Model 0 I K Prescribed far field loading rate 0 K IC Nominal fracture toughness from measurements M 1 K I Local SIF of the main crack for Model 1 1 I K Loading rate of SIF of the main crack for Model 1 M K I Local SIF of the main crack for Model I K Loading rate of SIF of the main crack for Model L K I Local SIF of the main crack L I K Local loading rate of the main crack P 1 P P f q xx, q yy, q xy Q R Dynamic forces on the incident end Dynamic forces on the transmitted end Failure load Dimensionless stress components for SCB Dimensionless tensile stress at the failure spot Radius of the sample xxvi

27 R 0 S u u& & u 1 u X Y Z V V 0 W W i W r W t W G ρ v v ij ε i, ε r, ε t Radius of the bar Distance of the two supporting pins Displacement vector Second time derivative of the displacement vector Displacement of the incident bar end Displacement of the transmitted bar end Principal axis of Barre granite with the slowest P-wave velocity Principal axis of Barre granite with intermediate P-wave velocity Principal axis of Barre granite with the fast P-wave velocity Half crack opending displacement Veloctity of the striker Energy carried by the stress wave Energy carried by the incident stress wave Energy carried by the reflected stress wave Energy carried by the transmitted stress wave Energy consumed to create new crack surfaces Density Poisson s ratio Poisson s ratio for strain in j direction by strain in i direction Incident, reflected and transmitted strain pulse ε& (t) Strain rate σ σ m σ 0 (x), τ 0 (x) σ p (x), τ p (x) Tensile stress Maximum tensile stress Undisturbed normal/shear stress along the location of the microcrack A pair of pseudo-tractions σ t Tensile strength xxvii

28 σ f σ t,n Flexural strength Tensile strength by non-local reconciliation κ σ& σ ij ε ij σ x,σ y, τ xy δ ΔU ω ξ α t α f Ratio of the flexural strength to tensile strength Loading rate of the tensile stress/flexural stress Stress tensor Strain tensor Components of the stress tensor Characteristic material length Amount of voltage reading of LGG output Angular velocity Ratio of the local SIF at the main crack tip to the loading Anisotropic index of tensile strength Anisotropic index of flexural strength α k θ φ Anisotropic index of Mode I fracture toughness Angle of x -axis to the line linking main crack tip to the microcrack center Microcrack orientation as the angle from x -axis to the x' -axis xxviii

29 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; ) 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, 197). 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, 008; Nasseri et al., 005). 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

30 CHAPTER 1: INTRODUCTION 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, 008). 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., 005; Hudson et al., 197; Mellor and Hawkes, 1971), the ring test (Coviello et al., 005; Hudson, 1969; Hudson et al., 197; Mellor and Hawkes, 1971), and the bending test (Coviello et al., 005). 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, 010). 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

31 CHAPTER 1: INTRODUCTION 3 (Coviello et al., 005). 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, 009; ASTM C880 / C880M-09, 009; ASTM Standard C78-09, 009; ASTM Standard C93-07, 007; BS EN 137, 1999; BS EN 13161, 008). 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., 005). 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, 198), 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., 198; 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 corebased, 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., 008) and CSTBD tests (Ke et al., 008). Nasseri and Mohanty measured fracture toughness of four types of granite with CCNBD (Nasseri and Mohanty, 008; Nasseri et al., 005; Nasseri et al., 006). 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 (000) measured the dynamic tensile

32 CHAPTER 1: INTRODUCTION 4 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., 006; Wang et al., 009) and argillite (Cai et al., 007) 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., 000; 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 10 ~10 4 s -1 for a variety of engineering materials, such as metals (Gray, 000), composites (Ninan et al., 001), concrete (Ross et al., 1996; Ross et al., 1995), ceramics (Chen and Ravichandran, 000; Chen and Ravichandran, 1996; Chen and Ravichandran, 1997), and rocks (Dai et al., 010c; Dai and Xia, 010; Shan et al., 000; Xia et al., 008; Zhang et al., 000; 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., 001; Frew et al., 00). It is especially useful for investigating dynamic response of brittle materials such as rocks (Frew et al., 001; Frew et al., 00). The momentum-trap technique in SHPB (Song and Chen, 004) 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.

33 CHAPTER 1: INTRODUCTION 5 1. 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. (008), 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., 001; Frew et al., 00) is especially useful to modify the loading pulse and thus facilitate dynamic stress equilibrium for quasi-static stressstrain 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., 008) to further investigate the anisotropy of tension, flexural and fracture properties: tensile strength, flexural strength and Mode-I fracture

34 CHAPTER 1: INTRODUCTION 6 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 microstructural 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 ) 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.

35 CHAPTER 1: INTRODUCTION 7 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., 010. Some fundamental issues in dynamic compression and tension tests of rocks using split Hopkinson pressure bar. Rock Mechanics and Rock Engineering, doi: /s 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., 010. Loading Rate Dependence of Tensile strength anisotropy of Barre granite. Pure and Applied Geophysics, doi: /s 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.

36 CHAPTER 1: INTRODUCTION 8 Dai, F., Xia, K. and Luo, S.N., 008. Semicircular bend testing with split Hopkinson pressure bar for measuring dynamic tensile strength of brittle solids. Review of Scientific Instruments, 79(1). Dai, F., Xia, K.W. and Tang, L.Z., 010. Rate dependence of the flexural tensile strength of Laurentian granite. International Journal of Rock Mechanics and Mining Sciences, 47(3): 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., 010. A semi-circular bend technique for determining dynamic fracture toughness. Experimental Mechanics, doi: /s Chen, R., Xia, K., Dai, F., Lu, F. and Luo, S.N., 009. Determination of dynamic fracture parameters using a semi-circular bend technique in split Hopkinson pressure bar testing. Engineering Fracture Mechanics, 76(9): 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.

37 CHAPTER 1: INTRODUCTION 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 : 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.

38 CHAPTER 1: INTRODUCTION 10 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.

39 CHAPTER : LITERATURE REVIEW 11 CHAPTER 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..1 Barre Granite and Its Anisotropy.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 (008). By virtue of recent development of computeraided 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., 005). As shown in 4Figure.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. also). The mineral and microcracks

40 CHAPTER : LITERATURE REVIEW 1 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, 008). Figure.1 Mineral and microcracks traced from three orthogonal planes for Barre granite; after (Nasseri and Mohanty, 008).

41 CHAPTER : LITERATURE REVIEW 13 Figure. 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, 008); 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. reveals the larger microcracks (~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 1.5. In the XY plane, the intermediate size microcracks (~ 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 1.5 in that plane.

42 CHAPTER : LITERATURE REVIEW 14 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.. 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.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; ) 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, 008). The XY plane, YZ plane and XZ plane correspond to the quarryman s description of rift plane, grain plane and hardway plane respectively..1. 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

43 CHAPTER : LITERATURE REVIEW 15 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, 007). 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, 007). 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 (008) measured fracture toughness of Barre granite with cracked chevron notched Brazilian disc (CCNBD) (Fowell et al., 1995) prepared along six different directions. 4Figure.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.

44 CHAPTER : LITERATURE REVIEW 16 Figure.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, 008); the letters in the braskets are the directions used in this thesis. 4Figure.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.

45 CHAPTER : LITERATURE REVIEW 17 Figure.4 Variation of fracture toughness measured along six directions with the number of tests along each direction in Barre granite; after (Nasseri and Mohanty, 008). 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 (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 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. Goldsmith et al. (1976) showed that (a) direction has the highest tensile strength and (b) the tensile strength for each direction has clear rate dependence. With reference to 4Figure.3, because most of the microcracks are parallel with orientation, the plane normal to orientation is the hard-way plane. It is thus expected that the tensile strength is the highest in direction as shown by Goldsmith et al. (1976). On the other hand, microcracks are mostly perpendicular to

46 CHAPTER : LITERATURE REVIEW 18 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. (008) 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., 001; Frew et al., 00) 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.5. It is noted that σ m in 4Figure.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, σ m can be taken as dynamic compressive strength. It is shown from 4Figure.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.

47 CHAPTER : LITERATURE REVIEW 19 Figure.5 Strain rate effects of the maximum compressive stress for X-, Y- and Z- samples of Barre granite; reproduced after (Xia et al., 008); the letters in the braskets are the directions used in this thesis. The Y axis of the Barre granite block by Xia et al.(008) is the direction with the lowest P wave velocity, corresponding to the Z axis in the notation by Nasseri and Mohanty (008), 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., 00). 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

48 CHAPTER : LITERATURE REVIEW 0 compressive strength (4Figure.5). This mechanism explains why Y samples have higher strength than X and Z samples.. Tension Tests..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., 005). 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., 005; Hudson et al., 197; Mellor and Hawkes, 1971), the ring test (Coviello et al., 005; Hudson, 1969; Hudson et al., 197; Mellor and Hawkes, 1971), and the bending test (Coviello et al., 005). 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

49 CHAPTER : LITERATURE REVIEW 1 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... 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 (000) measured the dynamic tensile properties of Bukit Timah granite from Singapore with the Brazilian disk and three point bending flexural methods using a selfdesigned 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.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

50 CHAPTER : LITERATURE REVIEW 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, 000), composites (Ninan et al., 001), concretes (Ross et al., 1996; Ross et al., 1995), ceramics (Chen and Ravichandran, 000; Chen and Ravichandran, 1996; Chen and Ravichandran, 1997) and rocks (Xia et al., 008; Zhang et al., 000; 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., 004). For rocks, spalling tests were performed on rocks from three different Canadian mines (Mohanty, 1987) and granites and tuffs (Cho et al., 003) 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., 006; Wang et al., 009) and argillites (Cai et al., 007). 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., 00; Frew et al., 005; Song and Chen, 004) 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

51 CHAPTER : LITERATURE REVIEW 3 (Dai et al., 010c) and flexural strength (Dai et al., 008) of Laurentian granite. The differences between the two strength values measured from two methods are discussed (Dai et al., 010d). The methods are then employed to research on the anisotropic Barre granite (Dai and Xia, 010). The details are presented in later chapters..3 Fracture Tests.3.1 Static Fracture Tests 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., 008). 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 (190) and Irwin s modification (1957) which recognizes the importance of stress intensity near a crack tip (Chen et al., 008). 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.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.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.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.6c) (Anderson, 005). Thus, corresponding to the three fracture modes, there are three types of SIFs: Mode I (K I ), Mode II (K II ) and Mode III (K III ).

52 CHAPTER : LITERATURE REVIEW 4 Figure.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.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: K I θ θ θ σ xx = cos( ) 1 sin( )sin( ) πr (.1a) K I θ θ 3θ σ yy = cos( ) 1 + sin( )sin( ) πr (.1b)

53 CHAPTER : LITERATURE REVIEW 5 K θ θ 3θ σ I xy = cos( )sin( )cos( ) (.1c) πr σ 0 Plane Stress zz = v( σ xx + σ yy ) Plane Strain (.1d) τ τ = 0 (.1e) xz = yz Figure.7 to the plane. Definition of the local coordinate axis ahead of a crack tip. Z direction is normal The displacement relationship for Mode-I are: K I r θ θ u x = cos( ) κ 1+ sin ( ) G π (.a) K I r θ θ u y = sin( ) κ + 1 cos ( ) G π (.a)

54 CHAPTER : LITERATURE REVIEW 6 where G is the shear Modulus. κ = 3 4v for plane strain and κ = ( 3 v ) /(1 + v) for plane stress. The critical value of the SIFs when crack propagation initiates is defined as the fracture toughness (i.e. for Mode-I, K IC; Mode-II, K IIC ; and Mode-III, K IIIC ). 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., 00). As stressed by Sun and Ouchterlony (1986), some applications of the fracture toughness of rocks are listed as below: (i) (ii) A parameter for classifying rock materials; 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.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).

55 CHAPTER : LITERATURE REVIEW 7 Figure.8 Comparison of the fracture mechanics approach to design with the traditional strength of material approach: (a) strength approach (b) fracture toughness approach 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., 00).

56 CHAPTER : LITERATURE REVIEW 8 While in earlier attempts of measuring fracture toughness of rocks, ASTM-E399 standard (ASTM Standard E399-09, 009) 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., 008; Shiryaev and Kotkis, 198), 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., 198; 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 (198) 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. (003) 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. (008) 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, 197); 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

57 CHAPTER : LITERATURE REVIEW 9 (Atkinson et al., 198); 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., 004). 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. (003) used the ANSYS sub-model technique in finite element analysis for cracked chevron notched Brazilian disc (CCNBD)..3. Dynamic Fracture Tests.3..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 E , 1980, proposed standard method for instrumented impact testing of pre-cracked Charpy

58 CHAPTER : LITERATURE REVIEW 30 specimens of metallic materials) for dynamic fracture toughness tests using Charpy impact testing with loading rate less than 10 5 MPa.m 1/ s -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, 009): 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). ) 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, 198). 4) The rampant impacting of the pendulum on the sample can cause an unexpected lost-ofcontact 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, 198; Kalthoff, 1985; Marur, 000)..3.. 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

59 CHAPTER : LITERATURE REVIEW 31 setups (Jiang and Vecchio, 009). 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, 000; Wada, 199; Weisbrod and Rittel, 000), one-bar/three-point bend (3PB) impact (Bacon, 1993; Bacon et al., 1994; Irfan and Prakash, 000; Mines and Ruiz, 1985), two-bar (incident and transmitted bars)/3pb (Jiang et al., 004b; Tanaka and Kagatsume, 1980), two bar/4pb (Weerasooriya et al., 006), 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, 000). Under compressive stress pulse, configurations include notched bending samples (Jiang et al., 004b; Tanaka and Kagatsume, 1980; Weerasooriya et al., 006), wedge-loaded compact tension (WLCT) specimens loaded by a compressive pulse (Klepaczko, 1979), compact compressive specimens (CCSs) (Rittel et al., 199), and Brazilian disk specimens (Dai et al., 010a; Lambert and Ross, 000; Zhou et al., 006). 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; ) 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 chronologically according to a recent critical review (Jiang and Vecchio, 009) 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.

60 CHAPTER : LITERATURE REVIEW Reflected tensile pulse is introduced for loading single-edge cracked samples in a pressure bar setup by Stroppe et al. (1978) WLCT sample experimental method is presented by Klepaczko (1979) for determining dynamic fracture toughness, K Id, using Hopkinson pressure bar Two-bar (transmitted tube)/3pb loading testing system is proposed for measuring dynamic load and deflection responses by Tanaka and Kagatsume (1980) Stress-state equilibrium issue is first considered in Hopkinson compressive bar loaded CT sample fracture tests by Corran et al. (1983) One-bar (incident bar)/3pb fracture test method is established by Mines and Ruiz (1985) as an improvement to classical Charpy impact testing 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) Pulse shaping is employed for reducing the dynamic effect in two-bar/3pb fracture testing by Ogawa and Higashida (1990) One-bar bending fracture test is proposed by Homma et al. (1991) for measuring fracture toughness of polymethyl mechacrylate. 199 Two-bar/CCS fracture test is presented by Rittel et al. (199) for determining fracture toughness of steel. 199 Brazilian disk samples are used by Nakano et al. (199) in measuring Mode I and Mixed Mode I/II fracture toughnesses for brittle materials Two-point strain gauge measurement is employed for determining load and load-point displacement in one-bar/3pb testing by Bacon (1993). 003 Two-bar/3PB fracture toughness test is proposed by Nwosu et al. (003) for measuring Mode II delamination fracture toughness of composite material.

61 CHAPTER : LITERATURE REVIEW Momentum-tripping technique is adopted in an improved Hopkinson pressure bar loaded fracture test by Jiang et al. (004b). 006 Two-bar/four-point bend (4PB) is proposed by Weerasooriya et al. (006) for determining dynamic fracture toughness of ceramic materials. 007 Loss-of-contact is investigated experimentally in a two-bar/3pb testing system by Jiang and Vecchio (007a; 007b) using a novel contact voltage measurement method, and pulse-shaping effect is reexamined in a two-bar/4pb setup 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 C p, after which the displacement rate D(t) increases sharply. This turning point C p, in their view, is the critical point of cracking. The dynamic fracture toughness K Ic is then determined from the stress intensity factor at the critical point using a standard quasi-static equation. Zhang et al. (1999, 000) 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.

62 CHAPTER : LITERATURE REVIEW 34 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 t c. 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 quasistatic equation. Wang et al. (010) 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 K I (t). The fracture initiation time t f 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 t f 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 000 by Zhang et al. (000) 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.

63 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 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., 005). 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. to 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., 005).

64 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 36 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), semicircular 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

65 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 37 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, 007; Nasseri and Mohanty, 008), 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 Barre Granite Barre granite is the anisotropic rock chosen for this study because it exhibits a high degree of anisotropy (Nasseri and Mohanty, 008). 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 0.5 to 3 mm. Quartz makes up 5% (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., 008). 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, 008; Nasseri et al., 005). 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.). Sano et al. (199) determined the principal axes of Barre granite by measuring the P-wave and S-

66 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 38 wave velocities in various directions of propagation and polarization. In their paper, the P-wave velocities along three principal axes are km/s, km/s and 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. 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.. 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. (e.g. 4Figure 3.a 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.b 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.

67 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 39 Figure 3. 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.

68 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 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 00 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.

69 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES Split Hopkinson Pressure Bar The Hopkinson bar experimental technique rooted in the stress wave experiments of iron wires performed by John Hopkinson (Hopkinson, 187; 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 10 ~10 4 s -1 for a variety of engineering materials, such as metals (Gray, 000), composites (Ninan et al., 001), concrete (Ross et al., 1996; Ross et al., 1995), ceramics (Chen and Ravichandran, 000; Chen and Ravichandran, 1996; Chen and Ravichandran, 1997), and rocks (Dai et al., 010c; Xia et al., 008; Zhang et al., 000; Zhang et al., 1999) Working Principle A 5 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.

70 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 4 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, 000). 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 00 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 100 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 TDS101 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

71 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 43 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 ε i (t), ε r (t) and ε t (t), respectively (4Figure 3.5).

72 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES Reflected wave Output (V) Transimitted wave Incident wave Time (μs) 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 (P 1 ) and the transmitted end (P ) of the specimen are (Kolsky, 1949; Kolsky, 1953): P1 = AE( ε i + ε r ), P = AEε t (3.1) The displacement of the incident bar end (u 1 ) and the transmitted bar end (u ) integrated from the respective velocities (v 1 and v ): t 0 u1 = C ( ε ε ) dt, u = C ε dt (3.) i r 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 ε& (t), strain ε (t) and stress σ (t) within the sample in the dynamic compression tests (4Figure 3.5) can be calculated as: t t 0

73 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES = = = ) ( ) ( ) ( ) ( ) ( ) ( 0 0 t r i A A t t r i L C t r i L C E t dt t t ε ε ε σ ε ε ε ε ε ε ε ε& (3.3) where L is the length of the sample and A 0 is the initial area of the sample. Assuming the stress equilibrium prevails during dynamic loading (i.e., t r i ε ε ε = + ), the commonly used formulas are obtained: = = = t A A t r L C r L C E t dt t t ε σ ε ε ε ε 0 ) ( ) ( ) ( 0 & (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 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.

74 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 46 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, 000). 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., 197); 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., 000). 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 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, 004). 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

75 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 47 (1994) to obtain high rate stress strain data for limestone samples (Frew et al., 001). 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., 001). 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., 001; Frew et al., 00). 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 nondispersive ramp pulse propagating into the incident bar and thus facilitating the dynamic force balance in the specimen (Frew et al., 001; Frew et al., 00). The function of the pulse shaper is to 1) fill out the high frequence noice generated during the impacting and ) 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. 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.

76 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES without shaper Rubber Copper Voltage (v) A B C D Striker Bar Incident Bar (a) Time (μs) (b) Figure 3.7 Pulse shapers in SHPB (a) schematic of the assembly (b) unshaped and shaped incident stress pulses 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

77 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 49 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 (004) 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 v 0, the length of the input bar l

78 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 50 and the shape of the input pulse. It takes t 0 = l / C 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 ρ CA, 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: t 0 d = C ε ( t) dt (3.5) 0 i If there is no pulse-shaper between the striker and the input bar, the particle velocity of the input bar after impact is 1/V 0, for the case where the striker and input bar are made of the same material. Denote the length of the striker by l s, the total duration of the loading pulse is l s / C, which is usually smaller than t 0 = l / C. 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 t 0 d = C ε i( t) dt = 1 1/ V dt = V0 l s / C. If there is a pulse-shaper between the 0 t 0 0 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 1 st incident and reflection, the nd 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 nd reflected wave in 4Figure 3.9 is compressive, induced by the total reflection of the nd incident wave from the free surface of the incident bar end that is separated from the specimen.

79 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 51 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

80 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 5 bar) tests by Ramesh and Narasimhan (1996) to measure the radial strain of the deformed samples of plastic materials. Li and Ramesh (007) extended the optical technique, LORD, to the tension Kolsky bar testing for measuring dynamic tension properties of four viscoplastic materials Principles and Setup In the dynamic fracture tests reported in Chapter 6, a laser gap gauge (LGG) system was developed by Chen et al., (009) 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 planoconvex cylindrical lens is made from coated BK7 glass. The high performance multilayer antireflection coatings have an average reflectance of less than 0.5 % (per surface) in the wavelength range of 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.

81 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 53 Figure 3.10 Photo and schematics of the laser gap gauge (LGG) system set up perpendicular to the bar axis of SHPB Calibration of the System 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).

82 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 54 X d Collimated beam Gap gauge 4 3 Measurement Linear fitting Δd (mm) ΔU (V) 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: Δ d = kδu (3.6)

83 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 55 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: σ Δd / Δd ( σ k / k) + ( σ ΔU / ΔU ) (3.7) Here σ k =0.03 mm/v, and σ ΔU =0.4 mv. As an example, for ΔU = 0.5 V, Δd = 1.0 mm and the relative error of the displacement measurement is around 0.8 % 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 f c of a long bar is (Graff, 1975): C = 0.3 (3.8) πr v f c 0 where R 0 is the bar radius, ν and C are Poisson s Ratio and elastic wave velocity, respectively. f c is 400 khz for the 6.35 mm diameter bar, much higher than 100 khz for the 5 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): Δd t = ( ε + ε ) Cdτ (3.9) 0 i r 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).

84 CHAPTER 3: EXPERIMENTAL SETUP AND TECHNIQUES 56 The comparison shows excellent agreement (4Figure 3.1), demonstrating that the LGG system has a wide bandwidth sufficient for valid measurements in SHPB. Incident bar Laser beam X 1.0 Displacement (mm) Laser Gap Gauge Strain Gauge t (μs) Figure 3.1 Dynamic calibration of the LGG system: schematic setup and a typical dynamic testing result compared to the predictions by Equation (3.6).

85 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., 199; Takemura et al., 003). 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

86 CHAPTER 4: DYNAMIC TENSION TESTS 58 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., 005; Hudson et al., 197; Mellor and Hawkes, 1971), the ring test (Coviello et al., 005; Hudson, 1969; Hudson et al., 197; Mellor and Hawkes, 1971), and the bending test (Coviello et al., 005). 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 (000) 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., 006; Wang et al., 009) and argillites (Cai et al., 007). 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

87 CHAPTER 4: DYNAMIC TENSION TESTS 59 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. 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. 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 D elastic analysis as (Michell, 1900): P f σ t = (4.1) πdb where P f is the load when the failure occurs, σ t is the tensile strength, D and B are the diameter and the thickness of the disc, respectively.

88 CHAPTER 4: DYNAMIC TENSION TESTS 60 D B Incident Bar Transmitted Bar P 1 P strain gauge 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., 004a). 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.

89 CHAPTER 4: DYNAMIC TENSION TESTS Validation of Dynamic Brazilian Test Dynamic Brazilian Test without Pulse Shaping 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. depicts a large oscillation of dynamic force occurring on the incident side and a sizeable distinction between P 1 and P In. Tr.(P ) Re. In.+Re.(P 1 ) Force (kn) Time (μs) Figure 4. 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

90 CHAPTER 4: DYNAMIC TENSION TESTS 6 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

91 CHAPTER 4: DYNAMIC TENSION TESTS 63 sharp rising edge (4Figure 4.) 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 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 PLANE8, 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: σ = ρ& u& (4.) 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.

92 CHAPTER 4: DYNAMIC TENSION TESTS 64 P1 P Figure 4.4 Mesh of the Brazilian disc for the finite element analysis with ANSYS; P 1 and P 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.

93 CHAPTER 4: DYNAMIC TENSION TESTS 65 (a) Tensile Stress (MPa) Quasi-static Dynamic (b) Compressive Stress (MPa) Time (μs) Time (μs) Quasi-static Dynamic Figure 4.5 (a) Tensile stress σ x (b) compressive stress σ y 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.

94 CHAPTER 4: DYNAMIC TENSION TESTS 66 The peak value of the transmitted force P (t) is normally taken as P f for standard Brazilian equation [Equation (4.1)] to calculate the material tensile strength (Cai et al., 007). 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 P (t) after appropriate time corrections. 4Figure 4.6 compares the strain gage signal with the dynamic forces P 1 (t) and P (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 [P 1 (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 Voltage (V) Strain gauge -1.5 Tr. In.+Re Force (kn) Time (μs) 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.

95 CHAPTER 4: DYNAMIC TENSION TESTS Dynamic Brazilian Test with Careful Pulse Shaping 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& =k /(πdb), where the parameter k is illustrated in 4Figure k In. Tr.(P ) Re. In.+Re.(P 1 ) Force (kn) Time (μs) 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

96 CHAPTER 4: DYNAMIC TENSION TESTS 68 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 36 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.

97 CHAPTER 4: DYNAMIC TENSION TESTS 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 (D); 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 σ x (in tension) and σ y (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 15.5 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 0 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 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).

98 CHAPTER 4: DYNAMIC TENSION TESTS 70 (a) 0 Quasi-static Dynamic (b) Tensile Stress (MPa) Compressive Stress (MPa) Time (μs) Quasi-static Dynamic Time (μs) Figure 4.9 (a) Tensile stress σ x (b) compressive stress σ y 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.

99 CHAPTER 4: DYNAMIC TENSION TESTS Strain gauge Tr. A B Voltage (V) Force (kn) μs Time (μs) 0 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 33 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

100 CHAPTER 4: DYNAMIC TENSION TESTS 7 calculated from the peak of the transmitted wave measured in the SHPB system with quasi-static analysis. Thus, despite the D 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 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

101 CHAPTER 4: DYNAMIC TENSION TESTS 73 confirmed previous assumption. The simplicity of the experimentation will facilitate the standardization of the dynamic BD method using SHPB. 50 With jaws Without jaws Tensile Strength (MPa) Front jaw Rear jaw Loading Rate (GPa/s) Figure 4.11 The measured tensile strength of Laurentian granite from dynamic Brazilian tests with and without employing jaws.

102 CHAPTER 4: DYNAMIC TENSION TESTS Tensile Strength of Barre Granite Determination of Anisotropic Tensile Strength 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 4.1(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 4.1(b), where the sample disc is sandwiched between the incident bar and the transmitted bar. P y D z B y y Incident Bar Transmitted Bar x z D P 1 x P y P B (a) (b) Figure 4.1 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 4.1, the y-axis defines the loading direction and the z-axis denotes the axial direction of the disc. In the quasi-static elastic

103 CHAPTER 4: DYNAMIC TENSION TESTS 75 equilibrium, the components of the stress tensor for any point (x, y) in the disc can be expressed as follows: P σ = π DB P σ = f π DB P = f π DB x f xx, y yy, xy xy τ (4.3) where σ x, σ y and τ xy are three components of the stress tensor and f xx, f yy and f xy 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. f xx σ x P πdb =, f yy σ y =, P πdb f xy = τ xy P (4.4) πdb 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 PLANE8 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 C ijkl has also been documented (Sano et al., 199) as C 1111 = 3.70 GPa, C = GPa, C 3333 = GPa, C 33 = 0.59 GPa, C 3131 = GPa, C 11 = GPa, C 33 = 6.43 GPa, C 3311 = 3.93 GPa, C 11 = 3.45 GPa. For comparison purpose, the stress state of the Brazilian disc is also analyzed assuming that the rock is isotropic.

104 CHAPTER 4: DYNAMIC TENSION TESTS 76 (a) (d) (b) (e) (c) (f) Figure 4.13 Stress trajectories of a Brazilian disc under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy with isotropic model, and (d) f xx (e) f yy and (f) f xy for sample YX using anisotropic model (positive for compression, negative for tension).

105 CHAPTER 4: DYNAMIC TENSION TESTS 77 For the isotropic case, 4Figure 4.13a, b and c show the distribution of the dimensionless stress components f xx, f yy and f xy respectively. The calculated values of fxx, f yy and f xy at the centre of the disc (potential failure spot) are f xx ~ -1, f yy ~ 3 and f xy = 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 f xx, f yy and f xy 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 f xx and f yy along the loading direction in the 5Figure 4.13d and 5Figure 4.13e accurately represent the dimensionless inplane principal stress σ 1 and σ. A complete suite of stress trajectories of the f xx, f yy and f xy 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.

106 CHAPTER 4: DYNAMIC TENSION TESTS 78 (a) (b) (c) Figure 4.14 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy with isotropic model.

107 CHAPTER 4: DYNAMIC TENSION TESTS 79 (a) (d) (b) (e) (c) (f) Figure 4.15 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample XY, and (d) f xx (e) f yy and (f) f xy for sample XZ (positive for compression, negative for tension).

108 CHAPTER 4: DYNAMIC TENSION TESTS 80 (a) (d) (b) (e) (c) (f) Figure 4.16 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample YX, and (d) f xx (e) f yy and (f) f xy for sample YZ (positive for compression, negative for tension).

109 CHAPTER 4: DYNAMIC TENSION TESTS 81 (a) (d) (b) (e) (c) (f) Figure 4.17 Stress trajectories of a Brazilian disc of Barre granite under quasi-static deformation. (a) f xx, (b) f yy and (c) f xy for sample ZX, and (d) f xx (e) f yy and (f) f xy for sample ZY (positive for compression, negative for tension).

110 CHAPTER 4: DYNAMIC TENSION TESTS 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 f xx, f yy and f xy. 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 σ 1 and σ with σ x and σ y. 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 σ x perpendicular to the loading diameter at the disc centre: P f σ t = F π DB (4.5) where σ t is the tensile strength and P f is the load when the failure occurs. F is f xx 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 granite along six directions. Sample Suites The material properties used in the finite element model of BD samples of Barre F E x (GPa) E y (GPa) G xy (GPa) XY XZ YX YZ ZX ZY v xy

111 CHAPTER 4: DYNAMIC TENSION TESTS 83 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 (198). 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., 00), 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., 010c) Tensile Strength Anisotropy 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.

112 CHAPTER 4: DYNAMIC TENSION TESTS 84 Force (kn) In. Re. In.+Re. (P 1 ) Tr. (P ) 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 4.0 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 singlepulse 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 4.0. 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.

113 CHAPTER 4: DYNAMIC TENSION TESTS 85 Figure 4.19 Virgin Brazilian discs of Barre granite prepared for the test; each division in the scale denotes 1 mm. Figure 4.0 Recovered Brazilian discs of Barre granite after tests; each division in the scale denotes 1 mm.

114 CHAPTER 4: DYNAMIC TENSION TESTS Static tensile strength anisotropy The static tensile strength values are taken as the average of three individual tests for each sample group. 5Figure 4.1a 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 4.1b 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 4.1a) to 15.5 MPa (5Figure 4.1b). 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.

115 CHAPTER 4: DYNAMIC TENSION TESTS 87 (a) Tensile strength (MPa) Anisotropic Model XY XZ YX YZ ZX ZY (b) Isotropic Model Tensile strength (MPa) XY XZ YX YZ ZX ZY Figure 4.1 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.

116 CHAPTER 4: DYNAMIC TENSION TESTS Dynamic tensile strength anisotropy All the tensile strength values with corresponding loading rates are tabulated in 5Table 4.. Table 4. Tensile strengths of Barre granite along six directions from both static and dynamic Brazilian tests. XY XZ No. σ& (GPa/s) σ t (MPa) No. σ& (GPa/s) σ t (MPa) 1 1.8E E E E E E YX YZ No. σ& (GPa/s) σ t (MPa) No. σ& (GPa/s) σ t (MPa) 1.1E E E E E E ZX ZY No. σ& (GPa/s) σ t (MPa) No. σ& (GPa/s) σ t (MPa) 1.E E E E E E

117 CHAPTER 4: DYNAMIC TENSION TESTS 89 5Figure 4. 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., 007). It is also observed from 5Figure 4. 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). 45 Tensile strength (MPa) ZX ZY YX YZ XY XZ Loading rate (GPa/s) Figure 4. granite. The variation of tensile strength with loading rates for six sample groups of Barre

118 CHAPTER 4: DYNAMIC TENSION TESTS 90 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 asα. The strength values for all three splitting planes are shown in 5Figure 4.3a, b and c. t (a) (b) Tensile strength (MPa) X Plane (XY & XZ) Tensile strength (MPa) Y Plane (YX & YZ) Loading rate (GPa/s) Loading rate (GPa/s) (c) (d) Tensile strength (MPa) Z Plane (ZX & ZY) α t Loading rate (GPa/s) Loading rate (GPa/s) Figure 4.3 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 (α t ) of Barre granite with loading rates.

119 CHAPTER 4: DYNAMIC TENSION TESTS 91 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 α t with loading rates is shown in 5Figure 4.3d. For the static case, α t 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. 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 00 GPa/s, sample with splitting plane normal to Z axis owes the highest tensile strength of 8.9 MPa and splitting plane normal to X axis shows the lowest value of 18.9 MPa, and α t is As the loading rate is around 1800 GPa /s, α t 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 α t curve in 5Figure 4.3d 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 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 4.3d, 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. 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

120 CHAPTER 4: DYNAMIC TENSION TESTS 9 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 (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 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 5~0 MPa, quite sensitive to the loading orientation relative to the bedding planes (Schmidt, 1977); in contrast, the fracture stress at strain rates from 10 4 s -1 to 10 5 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

121 CHAPTER 4: DYNAMIC TENSION TESTS 93 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; ) 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

122 CHAPTER 4: DYNAMIC TENSION TESTS 94 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.

123 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., 005). 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., 005; Hudson et al., 197; Mellor and Hawkes, 1971), ring test

124 CHAPTER 5: DYNAMIC FLEXUAL TESTS 96 (Coviello et al., 005; Hudson et al., 197; Mellor and Hawkes, 1971), and bending test (Coviello et al., 005). Apart from the Brazilian tests, the tensile strength can also be measured from bending tests (Coviello et al., 005). 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., 005). 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, 009; ASTM C880 / C880M-09, 009; ASTM Standard C78-09, 009; ASTM Standard C93-07, 007; BS EN 137, 1999; BS EN 13161, 008). For example, there are two ASTM standards to guide the testing of flexural strength of concrete. One is ASTM standard C93 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 C93-07, 007). 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, 009). 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., 005), even under fast loadings cases with a modified material testing machine (Zhao and Li, 000).

125 CHAPTER 5: DYNAMIC FLEXUAL TESTS 97 Figure 5.1 Schematics of the determination of the flexural strength of concrete by ASTM standards: a) ASTM C93, 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 (000), 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).

126 CHAPTER 5: DYNAMIC FLEXUAL TESTS 98 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., 00; Frew et al., 005; Song and Chen, 004) 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., 001). However, the samples used for indirect dynamic tensile testing are two dimensional (D); 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, 006), 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, 004). 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.

127 CHAPTER 5: DYNAMIC FLEXUAL TESTS 99 This chapter is organized as follows. Section 5. 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. Dynamic Semi-circular Bend Flexural Test 5..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, 006), 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.. 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 = 1.8 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 = 0 mm and average thickness of B = 16 mm. In order to determine the rupture initiation instant, t m, 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.). t m is signaled by a rapid drop in strain (Weisbrod and Rittel, 000). The travel time of the unloading wave from the failure spot O to the strain gauge is t. Then the rupture instant follows as t r = t m 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.

128 CHAPTER 5: DYNAMIC FLEXUAL TESTS 100 R B P Strain gauge P / 1 S Failure spot P / Figure 5. 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 = 0 mm, is sandwiched between the incident and transmitted bars. A strain gauge is mounted on the specimen near the point O. 5.. 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 PLANE8 is used in the analysis, and the finite element model consists of 357 elements and 75 nodes in total (5Figure 5.3).

129 CHAPTER 5: DYNAMIC FLEXUAL TESTS 101 Figure 5.3 Meshing scheme of the SCB specimen for finite element analysis. F 1 and F denote forces applied on the contact points. For quasi-static analysis, the forces are equal, i.e., F 1 =F =P(t)/. The tensile stress σ s (t) history near the failure spot O can be determined as: s P( t) S σ ( t) = Y ( ). (5.1) π BR R where P(t) is assumed to be the transmitted force (P ) deduced from the SCB-SHPB tests. Y is a function of the dimensionless geometry parameter S/R, which needs to be calibrated with static finite element analysis. 5Figure 5.4 shows Y as a function of S/R calculated from the finite element analysis, and a polynomial fitting yield: S Y + R S R =. +.87( ) 4.54( ). (5.) The coefficient of determination R is for the fitting curve in Equation 5. (Figure 5.4). For our configuration, S/R = and Y = 5.13.

130 CHAPTER 5: DYNAMIC FLEXUAL TESTS 10 9 Data Fitting curve 8 7 Y 6 5 R = S/R Figure 5.4 Y as a function of the dimensionless geometry parameter S/R from the quasistatic finite element analysis; the coefficient of determination of the fitting curve R is The dynamic flexural strength (σ f ) within the quasi-static analysis can then be calculated with the peak value in the measured loading history (P max ) as: Pmax S σ f = Y ( ) (5.3) π BR R 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, 000): σ = ρ& u& (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 F 1 and F are taken as half of the dynamic loading forces exerted on the incident side and transmitted side of the specimen, respectively, i.e., F 1 =P 1 / and F =P /.

131 CHAPTER 5: DYNAMIC FLEXUAL TESTS Validation of Semi-Circular Bend Tests 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 granite. High-speed video images of a dynamic semi-circular bend test on Laurentian 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.), where the tensile stress is the

132 CHAPTER 5: DYNAMIC FLEXUAL TESTS 104 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 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.

133 CHAPTER 5: DYNAMIC FLEXUAL TESTS 105 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., 008). 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 0µs, 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 (P ) is compared to the force on the incident side (P 1 ); 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.

134 CHAPTER 5: DYNAMIC FLEXUAL TESTS 106 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 P in the SCB-SHPB test on Laurentian granite without pulse shaping.

135 CHAPTER 5: DYNAMIC FLEXUAL TESTS 107 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 P (t) is normally taken as P max for static analysis to calculate the material tensile strength [Equation. (5.3)] (Wang et al., 006). 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 P (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 P occurs at 96 µs, delayed by 8 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 P 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 P. 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 Dynamic SCB Test with Careful Pulse Shaping For the SHPB test with proper pulse shaping (5Figure 5.10, 5Figure 5.11 and 5Figure 5.1), 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

136 CHAPTER 5: DYNAMIC FLEXUAL TESTS 108 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.

137 CHAPTER 5: DYNAMIC FLEXUAL TESTS 109 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 5.1 Strain gauge signal and the transmitted force P in the SCB-SHPB test on Laurentian granite with pulse shaping.

138 CHAPTER 5: DYNAMIC FLEXUAL TESTS 110 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 5.1 compares the strain signal from the gauge mounted on the specimen and the transmitted force P measured by the SHPB system. The single trough registered by the strain gauge on the specimen occurs at µs, indicating the onset of the rupture. The peak of P occurs at 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. 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 6.68 kn, only 0.% higher than the force of 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 0 MPa to 30 MPa (Zhao and Li, 000). Our result is higher than those by Zhao and Li (000). 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.

139 CHAPTER 5: DYNAMIC FLEXUAL TESTS 111 The dynamic SCB technique allows indirect tensile testing with a well established dynamic compression setup. Despite the D 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 Determination of Anisotropic Flexural Strength 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 5 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 P 1 and P 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: P = π RB P = q π RB σ x q xx, y yy σ, xy xy P τ = q (5.5) π RB where σ x, σ y and τ xy are three components of the stress tensor and q xx, q yy and q xy are the corresponding components of the dimensionless stress tensor and can be calculated using

140 CHAPTER 5: DYNAMIC FLEXUAL TESTS 11 numerical tools according to Equation (5.6). Compression is positive in the following calculations. q xx σ x P πrb =, q yy σ y =, P πrb q xy = τ xy P (5.6) πrb R P B B Incident Bar Transmitted Bar P/ y o S x P/ R P P / 1 x S y o P / (a) (b) 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 eightnode element PLANE8 is used in the analysis, and the finite element model consists of 11,397 elements and 34,59 nodes in total. Same as adopted in the previous Chapter 4, Barre granite is considered orthotropic and the nine stiffness constants C ijkl has also been documented (Sano et al.,

141 CHAPTER 5: DYNAMIC FLEXUAL TESTS ). For comparing purpose, the stress state of the SCB half disc is also analyzed assuming that the rock is isotropic. (a) (d) (b) (e) (c) -4 4 (f) Figure 5.14 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy with isotropic model, and (d) q xx (e) q yy and (f) q xy for ZX sample using anisotropic model (positive for compression, negative for tension).

142 CHAPTER 5: DYNAMIC FLEXUAL TESTS 114 For the isotropic case, 5Figure 5.14a, b and c show the distribution of the dimensionless stress components q xx, q yy and q xy respectively. The calculated values of q xx, q yy and q xy at the disc centre of the SCB half disc (potential failure spot) are q xx ~ -5.13, q yy ~ 0 and q xy = 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 q xx, q yy and q xy 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 q xx and q yy along the loading direction in the 5Figure 5.14d and 5Figure 5.14e acurrately represent the dimensionless in-plane principal stress σ 1 and σ. 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 q xx, q yy and q xy 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.

143 CHAPTER 5: DYNAMIC FLEXUAL TESTS 115 (a) (b) (c) Figure 5.15 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy with isotropic model.

144 CHAPTER 5: DYNAMIC FLEXUAL TESTS 116 (a) (d) (b) (e) (c) (f) Figure 5.16 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for XY sample and (d) q xx (e) q yy and (f) q xy for XZ sample (positive for compression, negative for tension).

145 CHAPTER 5: DYNAMIC FLEXUAL TESTS 117 (a) (d) (b) (e) (c) (f) Figure 5.17 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for YX sample, and (d) q xx (e) q yy and (f) q xy for YZ sample (positive for compression, negative for tension).

146 CHAPTER 5: DYNAMIC FLEXUAL TESTS 118 (a) (d) (b) (e) (c) (f) Figure 5.18 Stress trajectories of a semi-circular bend sample under quasi-static deformation. (a) q xx, (b) q yy and (c) q xy for ZX sample, and (d) q xx (e) q yy and (f) q xy for ZY sample using anisotropic model (positive for compression, negative for tension).

147 CHAPTER 5: DYNAMIC FLEXUAL TESTS 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 q xx, q yy and q xy. 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 σ 1 and σ with σ x and σ y. 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 σ x perpendicular to the loading diameter at the disc centre (maximum local tensile stress): P f σ f = Q (5.7) π BR where σ f is the flexural strength and P f is the load when the failure occurs. Q is q xx 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 σ is zero; thus the stress state at O is pure uni-axial tension. For our configuration, S/R = and Q = 5.13, 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 granite along six directions. Sample Suites The material properties used in the finite element model of SCB samples of Barre Q E x (GPa) E y (GPa) G xy (GPa) XY XZ YX YZ ZX ZY v xy

148 CHAPTER 5: DYNAMIC FLEXUAL TESTS Flexural Strength Anisotropy Dynamic equilibrium 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., 008) by employing pulse shaping technique in SHPB tests (Frew et al., 00). The quasi-static equation thus can be used to determine the dynamic flexural strength of rocks (Dai et al., 010d). 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 5.0a 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 5.0b.

149 CHAPTER 5: DYNAMIC FLEXUAL TESTS 11 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 5.0 (a) Virgin semi-circular bend samples of Barre granite; (b) Recovered semicircular bend samples of Barre granite after tests.

150 CHAPTER 5: DYNAMIC FLEXUAL TESTS Static flexural strength anisotropy 5Figure 5.1 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 1.9±0.5 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 5.1±0.3 MPa and 4.1± 0.4 MPa, respectively. The highest strength value (5.1 MPa from sample ZX) is nearly twice of the lowest one (1.9 MPa from sample XZ) Flexural Strength (MPa) XY XZ YX YZ ZX ZY Figure 5.1 The variation of static flexural strength of Barre granite along six directions, i.e. XY, XZ, YX, YZ, ZX and ZY.

151 CHAPTER 5: DYNAMIC FLEXUAL TESTS Dynamic flexural strength anisotropy All the flexural strength values with corresponding loading rates are tabulated in 5Table 5.. 5Figure 5. 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. 70 Flexural Strength (MPa) ZX SCB ZY SCB YX SCB YZ SCB XY SCB XZ SCB Loading rate (GPa/s) Figure 5. granite. The variation of flexural strength with loading rates along six directions of Barre

152 CHAPTER 5: DYNAMIC FLEXUAL TESTS 14 5Figure 5. 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 α f. The flexural strength values for all three splitting planes are shown in 5Figure 5.3a, 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 α f with loading rates is shown in 5Figure 5.3d. For the static case, α f equals to 1.86, with the highest strength of 4.6 MPa for samples splitting in the plane normal to Z axis; and the lowest of 13. 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 00 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 7.0 MPa, and α f is As the loading rate is up to 000 GPa /s, α f is about 1.4 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 α f curve in 5Figure 5.3d 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).

153 CHAPTER 5: DYNAMIC FLEXUAL TESTS 15 (a) (b) Flexural Strength (MPa) X Plane (XY & XZ) Loading rate (GPa/s) Flexural Strength (MPa) Y Plane (YX & YZ) Loading rate (GPa/s) (c) (d) 70 Flexural Strength (MPa) Loading rate (GPa/s) Z Plane (ZX & ZY) α f Loading rate (GPa/s) Figure 5.3 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 (α f ) of Barre granite with loading rates.

154 CHAPTER 5: DYNAMIC FLEXUAL TESTS 16 Table 5. 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) σ f (MPa) σ t,n (MPa) No. σ& (GPa/s) σ f (MPa) σ t,n (MPa) 1 ~8E ~8E ~8E ~8E ~8E ~8E YX YZ No. σ& (GPa/s) σ f (MPa) σ t,n (MPa) No. σ& (GPa/s) σ f (MPa) σ t,n (MPa) 1 ~8E ~8E ~8E ~8E ~8E ~8E ZX ZY No. σ& (GPa/s) σ f (MPa) σ t,n (MPa) No. σ& (GPa/s) σ f (MPa) σ t,n (MPa) 1 ~8E ~8E ~8E ~8E ~8E ~8E

155 CHAPTER 5: DYNAMIC FLEXUAL TESTS Interpretation of the Results 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; ) 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., 005; Hudson et al., 197; Lajtai, 197; Mellor and Hawkes, 1971). In the tensile property measurement of a granite under intermediate loading rates, Zhao and Li (000) 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, 199; Lajtai, 197; Van de Steen and Vervoort, 001) 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 σ t (Van de Steen and Vervoort, 001): δ l t + 0 σ δ = σ dl (5.8) l 0 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 σ t for a given sample geometry. The tensile stress gradient along the prospective fracture path of our SCB sample can be

156 CHAPTER 5: DYNAMIC FLEXUAL TESTS 18 calculated numerically with finite element analysis. The polynomial fit of the normalized stress gradient results in the following equation (see, 5Figure 5.4): σ σ m = ax + bx + c (5.9) where σ is the tensile stress along the prospective fracture path, σ m 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 5.4), a, b and c are fitting factors. If δ is known, substituting Equation (5.9) to Equation (5.8), the ratio κ (i.e. σ f /σ t ) between flexural strength σ f and the resulting tensile strength σ t can be determined. On the other hand, if the ratio κ can be estimated by comparing flexural strength σ f to the resulting tensile strength σ t, the characteristic material length δ can be obtained for the material. Figure 5.4 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 R of

157 CHAPTER 5: DYNAMIC FLEXUAL TESTS 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 5.4). A polynomial fit of the normalized stress gradient along the failure path of XY sample yield Equation (5.10): σ σ XY XY m = 0.011x x (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 5.5 shows comparable strengths of sample XY from dynamic SCB test and BD test as well as the reconciliation by nonlocal failure model employing a ratio of κ=1.3. XY XY σ f / σ t = κ = 1.3 (5.11) Substituting Equation (5.10) into (5.8), σ XY t δ XY δ δ XY XY XY = σ XYdx = ( 0.011x x ) σm dx (5.1) 0 0 XY XY From Equation (5.11), σ = σ /1. 3, rearranging Equation (5.1), t f XY σ f δ 1.3 XY XY = [ ( δ XY ) ( δ XY ) δ XY ] σ m (5.13) 3 XY XY At failure, σ = σ, thus Equation (5.13) can be further simplified as: f m ( δ XY ) ( δ XY ) + (0.997 ) = 0 (5.14) Solving Equation (5.14), δ XY can be calculated as δ XY =.6 mm.

158 CHAPTER 5: DYNAMIC FLEXUAL TESTS 130 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 Figure 5.5~5Figure 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 XZ YX YZ ZX ZY Strength (MPa) XY SCB XY SCB (Non-local) XY BD Loading rate (GPa/s) Figure 5.5 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.

159 CHAPTER 5: DYNAMIC FLEXUAL TESTS Strength (MPa) XZ SCB XZ SCB (Non-local) XZ BD Loading rate (GPa/s) Figure 5.6 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 Strength (MPa) YX SCB YX SCB (Non-local) YX BD Loading rate (GPa/s) Figure 5.7 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.

160 CHAPTER 5: DYNAMIC FLEXUAL TESTS Strength (MPa) YZ SCB YZ SCB (Non-local) YZ BD Loading rate (GPa/s) Figure 5.8 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 Strength (MPa) ZX SCB ZX SCB(Non-local) ZX BD Loading rate (GPa/s) Figure 5.9 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.

161 CHAPTER 5: DYNAMIC FLEXUAL TESTS Strength (MPa) ZY SCB ZY SCB(Non-local) ZY BD Loading rate (GPa/s) 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

162 CHAPTER 5: DYNAMIC FLEXUAL TESTS 134 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.

163 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

164 CHAPTER 6: DYNAMIC FRACTURE TESTS 136 methods were proposed in the literature to measure fracture toughness of rocks, including radial cracked ring (Shiryaev and Kotkis, 198), 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., 198; 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 ) 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, 198), 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., 000; 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., 199; Rittel and Maigre, 1996; Weisbrod and Rittel, 000). 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

165 CHAPTER 6: DYNAMIC FRACTURE TESTS 137 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., 00; Frew et al., 005; Song and Chen, 004). 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., 001). 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 fourpoint bend in SHPB with pulse-shaping (Weerasooriya et al., 006). 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, 003), 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., 006).

166 CHAPTER 6: DYNAMIC FRACTURE TESTS 138 Fracture energy can be easily measured with optical methods for transparent polymers or polished metals (Owen et al., 1998; Xia et al., 006). 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., 000). 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.. 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

167 CHAPTER 6: DYNAMIC FRACTURE TESTS 139 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 Dynamic Notched Semi-circular Bend Fracture Test 6..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 P 1 ; the force is P on the other side with two pins, with P / 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.

168 CHAPTER 6: DYNAMIC FRACTURE TESTS 140 Striker Incident bar P a 1 P R Transmitted bar S Pulse Shaper Strain Gauge Cylindrical lens Sample Collecting lens Laser B Gap Detector 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.. 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 E399-06e (E399-06e, 006), the quasi-static stress intensity factor (SIF) of the notched SCB specimen is calculated according to the following static equation: K I P S = Y ( a / R, S / R) (6.1) 3 / BR where K I 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.).

169 CHAPTER 6: DYNAMIC FRACTURE TESTS 141 K I Y ( a / R, S / R) = (6.) P S BR 3 / Due to symmetry, a half-crack model is employed to construct the finite element model. Quadrilateral eight-node element PLANE8 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.b). The entire model has 357 elements and 75 nodes (5Figure 6.a). For static analysis, the input loads F 1 and F equal to half of the transmitted forces recorded in the experiment (i.e., F 1 = F = P /). y r v θ x ω F (c) symmetry Δl v/ F1 A Symmetry LGG Crack tip (a) (b) Figure 6. 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.c), assuming plain strain, the near-tip crack opening displacement (COD) for a stationary crack under static loading is related to the SIF as:

170 CHAPTER 6: DYNAMIC FRACTURE TESTS 14 COD For the half-crack model, the SIF is then determined as: r, θ 8(1 v ) K I r = π = (6.3) E π K I V E π = (6.4) 4(1 v ) r where V is the half COD determined using Equation (6.3). Using Equation (6.1), the evolution of the SIF (i.e. K I ) in the dynamic tests can be determined. The peak of K I is the Mode-I fracture toughness K IC and the slope of the pre-peak linear portion of the K I (i.e. K ) is the loading rate for the dynamic test. I 6..3 Determination of Dynamic Fracture Energy 5Figure 6.3 shows a typical loading history (P ) and the corresponding crack surface opening displacement (CSOD) history during a NSCB test with dynamic force balance achieved throughout the test (i.e. P 1 =P ). 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

171 CHAPTER 6: DYNAMIC FRACTURE TESTS 143 rotation in stage III), and doubles the fragment velocity. The crack propagation process lasts about Δt AB = 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 v f 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

172 CHAPTER 6: DYNAMIC FRACTURE TESTS 144 point between the specimen and the incident bar (A point in 5Figure 6.). The rotation angle of the fragment is measured to be 9 at 160 μs, 1 at 480 μs, and 3 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//δl = 313 rad/s for the snapshot shown in 6Figure 6.4, in excellent agreement with the result obtained

173 CHAPTER 6: DYNAMIC FRACTURE TESTS 145 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., 000). 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 (W i ) and the summation of the energy reflected (W r ) and transmitted (W t ): Δ W = W W W (6.5) i r t where W is the energy carried by the stress wave, which can be calculated as follows (Song and Chen, 006): = t Eε ACdτ (6.6) W 0 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 W G 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 = Iω /, 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 W G = ΔW - K. Consequently, the average propagation fracture energy is determined with Equation (6.6) below: G = W / A (6.7) c G c where A c is the area of the new generated crack surfaces. For isotropic material, assuming plane strain, the average dynamic propagation fracture toughness can be attained: P K I = Gc E /(1 ν ) (6.8) where E and ν are the Young s modulus and Poisson s ratio of the sample material respectively.

174 CHAPTER 6: DYNAMIC FRACTURE TESTS Validation of Dynamic Notched Semi-Circular Bend Test Dynamic Analysis and Fracture Time 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, 000): σ = ρ& u& (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. while taking the input loading F 1 and F as half of the dynamic loading forces exerted on the incident side and transmitted side of the sample, respectively (i.e., F 1 =P 1 /, F =P / ). 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): COD( t) r, θ 8(1 v ) K I ( t) r = π = (6.10) E π For the half-crack model used in ANSYS, dynamic SIF is determined as: V ( t) E π t) = (6.11) 4(1 v ) r K I ( 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., 004a). The lowest point of the drop

175 CHAPTER 6: DYNAMIC FRACTURE TESTS 147 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 ε 1 (t) follows as: ε ( t) = ε ( t + Δ ) (6.1) 1 t 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 ε 1 (t) 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 P 1 is the sum of the incident (In.) and reflected (Re.) waves, and the dynamic force on the other side of the specimen P 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.

176 CHAPTER 6: DYNAMIC FRACTURE TESTS 148 Force (kn) 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 6 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, 198). 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.

177 CHAPTER 6: DYNAMIC FRACTURE TESTS 149 Transmitted force (kn) CSOD (0.05 mm) Tr. Strain gauge CSOD A E C 0 F -0 D - 0 μs B Time (μs) Strain gauge signal (mv) 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 0 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, 198). 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.). 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

178 CHAPTER 6: DYNAMIC FRACTURE TESTS 150 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. 5 4 SIF (MPa m 1/ ) 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 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

179 CHAPTER 6: DYNAMIC FRACTURE TESTS 151 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. Force (kn) In. Re. In.+Re. Tr Time (μs) 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.

180 CHAPTER 6: DYNAMIC FRACTURE TESTS 15 Transmitted force (kn) CSOD (0.05 mm) Tr. Strain gauge CSOD A B 4 μ s Time (μs) Strain gauge signal (mv) 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 1 mm and it thus takes around.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.

181 CHAPTER 6: DYNAMIC FRACTURE TESTS 153 By carefully shaping the loading wave, the dynamic force balance on the boundary of the sample is achieved (6Figure 6.8). However, with a D 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. SIF (MPa m 1/ ) Dynamic Quasi-static Time (μs) 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 farfield 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.m 1/, with the loading rate of 79.7 GPa.m 1/ /s. It is also noted that when there is no pulse shaper, the failure time is at 76 µs. The corresponding

182 CHAPTER 6: DYNAMIC FRACTURE TESTS 154 dynamic stress intensity factor is 1.5 MPa.m 1/ (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.m 1/ (Nasseri and Mohanty, 008). 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. 6 5 K IC G c 6 5 K IC (MPa m 1/ ) Gc (kj/m ) K I (GPa m 1/ s -1 ) Figure 6.11 The effect of loading rate on the fracture toughness and fracture energy of Laurentian granite.

183 CHAPTER 6: DYNAMIC FRACTURE TESTS Fracture Toughness Anisotropy of Barre Granite Determination of Anisotropic Stress Intensity Factor Suppose a ij are the elastic compliance which defines the relationship between the stress σ ij and strainε ij. For orthotropic material in plane stress, one has ε11 a = ε a γ 1 a a a a 1 6 a a a σ 11 σ σ 1 (6.13) 1 a 11 = (6.14a) E 1 1 a = (6.14b) E a 1 v E 1 1 = = (6.14c) 1 v E 1 a 66 = (6.14d) G 1 where E i is the Young s modulus in the i principle direction, G1 is the shear modulus in the 1 plane, v ij 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, a ij is replaced by b ij according to the following equations: b ij ai3a j3 = aij ; i, j = 1, (6.15) a 33 where

184 CHAPTER 6: DYNAMIC FRACTURE TESTS 156 a v v i 3i i3 = = (6.16) Ei E3 and 1 a 33 = (6.17) E 3 6Figure 6.1 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, 199). y E E1 ϕ r crack θ x Figure 6.1 Local coordinate system for the stress and displacement fields near the crack tip of an orthotropic solid. Stress fields are:

185 CHAPTER 6: DYNAMIC FRACTURE TESTS = 1/ 1 1 1/ 1 1/ 1 1 1/ 1 1 ) sin (cos ) sin (cos 1 Re ) sin (cos ) sin (cos Re θ θ θ θ π θ θ θ θ π σ s s s s s s r K s s s s s s s s r K II I xx (6.18a) = 1/ 1/ 1 1/ 1 1 1/ 1 ) sin (cos 1 ) sin (cos 1 1 Re ) sin (cos ) sin (cos 1 Re θ θ θ θ π θ θ θ θ π σ s s s s r K s s s s s s r K II I yy (6.18b) = 1/ 1/ 1 1 1/ 1 1/ 1 1 ) sin (cos ) sin (cos 1 Re ) sin (cos 1 ) sin (cos 1 Re θ θ θ θ π θ θ θ θ π σ s s s s s s r K s s s s s s r K II I xy (6.18c) Displacements are: { } { } = 1/ 1 1 1/ 1 1/ 1 1 1/ 1 1 ) sin (cos ) sin (cos 1 Re ) sin (cos ) sin (cos 1 Re θ θ θ θ π θ θ θ θ π s p s p s s r K s p s s p s s s r K u II I x (6.19a) { } { } = 1/ 1 1 1/ 1 1/ 1 1 1/ 1 1 ) sin (cos ) sin (cos 1 Re ) sin (cos ) sin (cos 1 Re θ θ θ θ π θ θ θ θ π s q s q s s r K s q s s q s s s r K u II I y (6.19b) where, i i i s a a s a p = (6.0a) i i i i s s a a s a q = ; = 1, i (6.0b)

186 CHAPTER 6: DYNAMIC FRACTURE TESTS 158 a ij are the compliance constants in the local x-y coordinate system. For orthotropic material, it is related to a ij according to the following equations, 4 4 a 11 = a11 cos ϕ + (a1 + a66)sin ϕ cos ϕ + a sin ϕ (6.1a) 4 4 a = a11 sin ϕ + (a1 + a66)sin ϕ cos ϕ + a cos ϕ (6.1b) a 1 = a1 + ( a11 + a a1 a66 )sin ϕ cos ϕ (6.1c) a 66 = a66 + 4( a11 + a a1 a66)sin ϕ cos ϕ (6.1d) a = a sin ϕ a cos ϕ + (a + )cosϕ / ]sin ϕ (6.1e) 16 [ 11 1 a66 a = a sin ϕ a cos ϕ + (a + )cos ϕ / ]sin ϕ (6.1f) 6 [ 11 1 a66 s1 and s are related to T 1 and T, which are the roots of the following characteristic equation for an orthotropic material, according to Lekhnitskii (1963). a 11 ( 1 66 = 4 μ + a + a ) μ + a 0 (6.) The roots of this equation are either complex or purely imaginary and cannot be real. They are: T = + (6.3a) 1 μ1 = ζ 1 iη1 T = + (6.3b) μ = ζ iη μ 3 = μ 1 (6.3c) μ 4 = μ (6.3d) where, i = 1, the overbar denotes the complex conjugate here. ζ i and ηi are real constants. It is always true that η 1 > 0, η > 0, η1 η. The relationship between sk and Tk is shown (Lekhnitskii, 1963) to be:

187 CHAPTER 6: DYNAMIC FRACTURE TESTS 159 s k Tk cosϕ + sinϕ = ; k = 1,. (6.4) cosϕ T sinϕ k The stresses σ yy and σ xx ahead of the crack tip and in the plane of the crack could be obtained by substituting θ = 0 into Equation (6.18), σ yy K I = ; πr σ xy K II = (6.5) πr By substituting o θ =180 into Equation (6.19), the near-tip displacement u and v on the crack face may be given in matrix form as: u u x y = r D π D 11 1 D D 1 K K I II (6.6) where s p1 s1 p D 11 = Im (6.7a) s1 s p 1 p D 1 = Im (6.7b) s1 s s q1 s1q D 1 = Im (6.7c) s1 s q 1 q D = Im (6.7d) s1 s K K I II = π D r D 11 1 D D 1 u u x y (6.8) and D D / D 11 = D = D / D 1 1 (6.9a)

188 CHAPTER 6: DYNAMIC FRACTURE TESTS 160 D = D / D 1 1 D = D / D 11 (6.9b) D D D 11 1 = = D11D D1D1 (6.30) D1 D 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, K I = u y r π 1 D 1 (6.31) u y Let = A + Br, two nearest nodes at the crack tip are used to determine the factor A. K I could r then be calculated using the following equation: K I A = π (6.3) D Determination of Fracture Toughness of Barre Granite Calculating equation Static measurement is conducted with an MTS hydraulic servo-control testing system (6Figure 6.13a). Dynamic test is conducted using a 5 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= 0 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 (6.3).

189 CHAPTER 6: DYNAMIC FRACTURE TESTS 161 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 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 σ W a E E1 σ Figure 6.14 An infinite orthotropic strip with an edge crack under remote uniform tractions normal to the edge crack.

190 CHAPTER 6: DYNAMIC FRACTURE TESTS 16 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, 199). The material chosen in their investigation was a boron-epoxy composite with material properties as follows: E 1= GPa, E =55.16 GPa, G 1 =4.83 GPa and v 1 = 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 PLANE8 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 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 K * = K /σ πa for all six cases are calculated and I tabulated in 6Table 6.1, along with the results reported by Kaya and Erdogan (1980) and Tan and Gao (199) for comparison. The results are highly satisfactory, with the maximum error less than 1 % for most of the cases. I

191 CHAPTER 6: DYNAMIC FRACTURE TESTS 163 Table 6.1 The normalized stress intensity factor K * = K /σ πa, for an edge crack in an infinite orthotropic strip with remote uniform traction σ. a/w (K ) * I KE (K ) * I T I (K ) * I AN I Δ K I (%) (K ) * I (K ) * I (K ) * I KE T AN : Results by Kaya and Erdogan (1980) : Traction formula result by Tan and Gao (199) : Results by ANSYS in this study KE AN Δ K I (%) T AN * * KE ( K I ) AN ( K I ) KE Δ K I AN = 100 * ( K ) I KE * * T ( K I ) AN ( K I ) T Δ K I AN = 100 * ( K ) I T 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 D 1 determined and tabulated in 6Table 6.. are

192 CHAPTER 6: DYNAMIC FRACTURE TESTS 164 Table 6. The material properties used in the finite element model of NSCB samples of Barre granite along six directions. Sample Suites D 1 (Pa -1 ) E 1 (GPa) E (GPa) G 1 (GPa) XY 5.95E XZ 5.58E YX 5.30E YZ 4.59E ZX 4.8E ZY 3.95E v 1 Taking advantage of the symmetry of the NSCB specimen, half model is constructed for the finite element analysis. PLANE8 element (D 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 357 elements and 75 nodes as shown in 6Figure The orthotropic material constants in the finite element model for six sample suites are also listed in 6Table 6., with reference to 6Figure 3. 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.

193 CHAPTER 6: DYNAMIC FRACTURE TESTS Fracture Toughness Anisotropy Dynamic equilibrium 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.

194 CHAPTER 6: DYNAMIC FRACTURE TESTS 166 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.

195 CHAPTER 6: DYNAMIC FRACTURE TESTS 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 m 1/, 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 m 1/. The average fracture toughness measured from samples split in Y planes (i.e. YZ and YX) yields intermediate fracture toughness of 1.4±0.04 and 1.0±0.06 MPa m 1/, 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 K IC values of 1.0±0.04 MPa m 1/ and 1.11±0.06 MPa m 1/, respectively. The measured static fracture toughness exhibits very strong anisotropy, with the highest measured fracture toughness (1.74 MPa m 1/, for sample ZX) 1.7 times of the lowest (1.0 MPa m 1/, for sample XY). K IC (MPa m 1/ ) XY XZ YX YZ ZX ZY 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.

196 CHAPTER 6: DYNAMIC FRACTURE TESTS Dynamic fracture toughness anisotropy All the dynamic fracture toughness and fracture energy values with corresponding loading rates are tabulated in 6Table Figure 6.0 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 ) the order of the highest fracture toughness to the lowest among six directions remains approximately the same as that in the static case K IC (MPa m 1/ ) 6 4 ZX ZY YZ YX XZ XY K I (GPa m 1/ s -1 ) Figure 6.0 granite. The variation of fracture toughness with loading rates on six directions of Barre As before, it is also apparent from Figure 3. 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

197 CHAPTER 6: DYNAMIC FRACTURE TESTS 169 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 6.1. 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. G C (kj/m ) ZX ZY YZ YX XZ XY K I (GPa m 1/ s -1 ) Figure 6.1 granite. The variation of fracture energy with loading rates on six directions of Barre

198 CHAPTER 6: DYNAMIC FRACTURE TESTS 170 Similar to previous two anisotropic indexes introduced in Chapter 4 and Chapter 5, the anisotropic index of fracture toughness, α k is defined as the ratio of the maximum fracture toughness to the minimum fracture toughness within six sample groups. 6Figure 6.a 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 α k with loading rates is shown in 6Figure 6.c. Under static loading, α k equals to 1.70; under dynamic loading, it decreases rapidly. For example, under the loading rate around 0 MPa m 1/ s -1, sample ZX axis owes the highest fracture toughness of 3.7 MPa m 1/ and XY axis shows the lowest value of.34 MPa m 1/, and α k is As the loading rate is 0 MPa m 1/ s -1, α k is about 1.0 and the maximum fracture toughness remains still in samples ZX with a value of 11.0 MPa m 1/ and the lowest one is fixed in samples XY as 9.1 MPa m 1/. Thus, Barre granite obviously exhibits stronger anisotropy under static loading, while relatively lower anisotropy during dynamic loading. In addition, as shown in 6Figure 6.c, α k 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.

199 CHAPTER 6: DYNAMIC FRACTURE TESTS 171 (a) (b) 8 10 K IC (MPa m 1/ ) 6 4 XY (X Plane) K IC (MPa m 1/ ) ZX (Z Plane) K I (GPa m 1/ s -1 ) K I (GPa m 1/ s -1 ) (c) k α K I (GPa m 1/ s -1 ) Figure 6. 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 α k of Barre granite.

200 CHAPTER 6: DYNAMIC FRACTURE TESTS 17 Table 6.3 Fracture toughness and fracture energy of Barre granite with corresponding loading rates from both static and dynamic NSCB fracture tests. K & IC GPa m 1/ s -1 XY K IC G c K & IC GPa m 1/ s -1 XZ K IC G c kj/m No. MPa m 1/ kj/m No. MPa m 1/ 1 8E N/A 1 1E N/A 8E N/A 1E N/A 3 8E N/A 3 1E N/A K & IC GPa m 1/ s -1 YX K IC G c K & IC GPa m 1/ s -1 YZ K IC G c kj/m No. MPa m 1/ kj/m No. MPa m 1/ 1 1E N/A 1 9E N/A 1E N/A 9E N/A 3 1E N/A 3 9E N/A K & IC GPa m 1/ s -1 ZX K IC G c K & IC GPa m 1/ s -1 ZY K IC G c kj/m No. MPa m 1/ kj/m No. MPa m 1/ 1 9E N/A 1 1E N/A 9E N/A 1E N/A 3 9E N/A 3 1E N/A

201 CHAPTER 6: DYNAMIC FRACTURE TESTS Crack-Microcrack Interaction 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.

202 CHAPTER 6: DYNAMIC FRACTURE TESTS 174 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 K IC measurements on four types of granites (Nasseri and Mohanty, 008; Nasseri et al., 006) 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, 008; Nasseri et al., 006). In this research with semi-circular bend fracture tests, orthotropic material model with material constants measured from literature (Sano et al., 199) 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 preexisting mircocracks in the nominally homogeneous rocks is responsible for the anisotropy of fracture toughness.

203 CHAPTER 6: DYNAMIC FRACTURE TESTS 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 6.3a for Case 1 and 6Figure 6.4a for Case, (Courtesy of Dr. M. H.B. Nasseri). 6Figure 6.3a corresponds to the crackmicrocracks 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 6.4a shows the microcracks density and orientation in Barre granite for the case in which the fracture toughness is the lowest. In Case, the main crack is collinear to most microcracks. 6Figure 6.3b (Model 1) and 6Figure 6.4b (Model ) depict two conceptual models based on 6Figure 6.3a and 6Figure 6.4a 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 6.3a, Model 1 is constructed (6Figure 6.3b), where two symmetric microcracks near the notch tip are oriented at an angle of o 45 to the horizontal main crack surface. Denote the length of the microcrack by c, the distance from the right tip of the microcrack to the tip of the main crack is 0.c. The conceptual model shown in 6Figure 6.4b is based on the thin section photo of 6Figure 6.4a. 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 c and the same distance from the main crack tip to the closest microcrack tip of 0.c are used. Particular attention is directed to the crack-microcracks interaction of the two conceptual models for an insight on the observed anisotropy of K IC.

204 CHAPTER 6: DYNAMIC FRACTURE TESTS 176 Figure 6.3 (a) Photo of microscopic thin section showing microcracks in a tested Barre o granite sample; Case 1: the main crack inclines at an angle of 45 to microcracks; (b) Model 1: the crack-microcracks configuration for Case 1.

205 CHAPTER 6: DYNAMIC FRACTURE TESTS 177 Figure 6.4 (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample; Case : The main crack is collinear to microcracks; (b) Model : the crackmicrocracks configuration for Case.

206 CHAPTER 6: DYNAMIC FRACTURE TESTS 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 6.5. Denote the distance between the main crack tip and the center of the microcrack by d and the length of the microcrack by c. 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 6.5 One arbitrarily located microcrack near the crack tip of a semi-infinite crack. In the following, 0 K I denotes the stress intensity factor of the main crack without microcracks; MA K I 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 K both 0 th order approximation and 1 st order approximation for the case where d / c =. MA I and 0 K I with The original problem was decomposed into three sub-problems: I, II and III (6Figure 6.6).

207 CHAPTER 6: DYNAMIC FRACTURE TESTS 179 σ 0 (x),τ 0 (x) = K 0 I,K0 II + Original Problem Sub-Problem I [σ 0 (x)+σ p (x)] [τ 0 (x)+τ p (x)] σ p (x),τ p (x) σ(x 0 ),τ(x 0 ) + σ 0 (x 0 ), τ 0 (x 0 ) δk I,δK II Sub-Problem II Sub-Problem III Figure 6.6 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 0 K I and Mode-II stress intensity factor K. The undisturbed stresses σ ( ) and τ ( ) along the location 0 II 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 σ p (x) and τ p (x) is applied to the microcrack; and the microcrack is stressed under the traction pair of [σ 0 (x)+σ p (x)] and [τ 0 (x)+τ p (x)]. In this case, the induced stresses along the main crack can be expressed and denoted as σ(x 0 ) and τ(x 0 ). 0 x 0 x

208 CHAPTER 6: DYNAMIC FRACTURE TESTS 180 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 σ p (x) and τ p (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 farfield loading of K is zero. Consequently, the 0 th -order approximate solution, which gives the 0 II first two terms of the stress intensity factors of the main crack, is as follows (Gong and Horii, 1989): K MA I 1 4 c d 0 0 = K I + ( ) [ B(11) A0(11) + B(1) A0(1) ] K I (6.33) The 1 st -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): K MA I 0 1 c = K I + ( ) 4 d + B A + 8A 4(1) 0(11) ( B 0(1) (11) [ B (11) ) + 3( B D 0(11) A (11) 0(11) A + B + B (11) (1) (1) + B D A (1) 0(1) 0 1 c 4 ] K I + ( ) 4( B 18 d A ) + 4( B A + B 0(1) (1) ) + 8A 0(1) ( B (11) 3(11) D { 0(1) 1(11) + B 4(11) 3(1) (1) A D 0(11) A ) 1(1) 0 0() )} K I (6.34) where A n (11) = ( n + )cos[ nφ ( n + ) θ ] + cos[( n + ) φ ( n + ) θ ] ( n + )cos[( n + ) φ ( n + ) θ ] (6.35a) A n (1) = ( n + ) sin[ nφ ( n + ) θ ] sin[( n + ) φ ( n + ) θ ] ( n + ) sin[( n + ) φ ( n + ) θ ] (6.35b)

209 CHAPTER 6: DYNAMIC FRACTURE TESTS 181 ] ) 5 ( ) )sin[( 1 ( 1 ] ) 1 ( ) sin[( 4 1 ] ) 1 ( sin[ (1) θ φ θ φ θ φ = n n n n n n n n A n (6.35c) ] ) 5 ( ) ) cos[( 1 ( 1 ] ) 1 ( ) cos[( 4 3 ] ) 1 ( cos[ () θ φ θ φ θ φ = n n n n n n n n A n (6.35d) ] ) 1 ) sin[( sin( ] ) ( ) 1 cos[( ] ) 1 cos[( ) ( 1 1 (11) φ θ θ φ θ φ θ k k kp k k kp k k p k B k k k k = (6.35e) ] ) 1 )cos[( sin( ] ) ( ) 1 sin[( ) ( ] ) 1 sin[( ) ( 1 1 (1) φ θ θ φ θ φ θ k k kp k k p k k k p k B k k k k = (6.35f) ] ) 1 )cos[( sin( ] ) ( ) 1 sin[( ] ) 1 sin[( 1 1 (1) φ θ θ φ θ φ θ k k kp k k kp k k kp B k k k k + + = (6.35g) ] ) 1 )sin[( sin( ] ) ( ) 1 cos[( ) ( ] ) 1 cos[( 1 1 () φ θ θ φ θ φ θ k k kp k k p k k k kp B k k k k + + = (6.35h) )] 4 cos(5 ) 4 cos(4 ) 4 cos(3 ) cos(3 ) 6cos( ) cos( ) cos( ) 4cos( [15 ) cos 16(1 1 0(11) φ θ φ θ φ θ φ θ φ θ θ θ φ θ = D (6.35i) )] 4 sin(5 ) 4 sin(4 ) 4 sin(3 ) sin(3 ) 3sin( ) [sin( ) cos 16(1 1 0(1) 0(1) φ θ φ θ φ θ φ θ φ θ φ θ = = D D (6.35j)

210 CHAPTER 6: DYNAMIC FRACTURE TESTS 18 D 0() 1 = [7 cos( θ ) cos(θ ) + cos(3θ 4φ ) 16(1 + cosθ ) + cos(4θ 4φ ) + cos(5θ 4φ )] (6.35k) (k)! In the above expressions, pk =. The factorial k! is defined as a positive integer k as: k ( k!) k! = k( k 1) 1. 6Figure 6.7 shows the different regions in which the stress intensity factor at the tip of the main MA crack may be either increased (amplification, K / K 0 > 1 I I ) or decreased (shielding, K MA I / K 0 I < 1), 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 6.3b (Case 1) and 6Figure 6.4b (Case ). In the following, the two proposed conceptual models in 6Figure 6.3b and 6Figure 6.4b are analyzed with finite element method to quantify the shielding or amplification effects of main crack due to the presence of the nearest microcracks.

211 CHAPTER 6: DYNAMIC FRACTURE TESTS θ (degree) K MA I /K 0 I <1 (shielding) K MA I /K 0 I >1 (amplification) 0 order 1 st order K MA I /K 0 I <1 (shielding) φ (degree) Figure 6.7 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 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 d / c > 1 (Figure 6.5), when d / c 1, the results are invalid. If d / c 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).

212 CHAPTER 6: DYNAMIC FRACTURE TESTS 184 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. PLANE8 (eight-node) element is used in the analysis. To better simulate the stress singularity of r -1/ 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 00 N is applied to the finite element model; and the Young s modulus is taken as 80 GPa and the Possion s ratio is 0.1. 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 φ = θ = 0 in 6Figure 6.7) is considered as a testing problem. The exact solution (Rose, 1986) of this problem is given below, where K (k) and E (k) denote the complete elliptic integrals of the first kind and the second kind respectively (Abramowitz and Stegun, 197). K K MA I 0 I Ε( k) = (6.36) k Κ( k') where, k = d c d + c k' = 1 k The calculated stress intensity factor via ANSYS are normalized by K 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 139; the nodes are between 503 and The maximum error as tabulated in 6Table 6.4 is less than 1.03 %, which occurs when d/c=1.1 0 I

213 CHAPTER 6: DYNAMIC FRACTURE TESTS 185 Table 6.4 Stress intensity factor of the main crack with one collinear microcrack at different distances to the main crack tip. d/c Exact value ANSYS value Error (%) 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 6.8), ) Model 1, corresponding to Case 1 (6Figure 6.9), and 3) Model, corresponding to Case (6Figure 6.30). With the same boundary conditions, the stress intensity factor Model 1 and M K I for Model can be calculated. K 0 I for Intact Model, M 1 K I for 0 K I 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 6.8). 6Figure 6.9 and 6Figure 6.30 show the meshing of our finite element Model 1 and Model, respectively, in which 6Figure 6.9a and 6Figure 6.30a are the global meshes, featuring increasing grids density towards the main crack tip; and 6Figure 6.9b and 6Figure 6.30b show the mesh at the vicinity of the main crack and the microcracks for Model 1 and Model respectively.

214 CHAPTER 6: DYNAMIC FRACTURE TESTS 186 Figure 6.8 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.

215 CHAPTER 6: DYNAMIC FRACTURE TESTS 187 Figure 6.9 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.

216 CHAPTER 6: DYNAMIC FRACTURE TESTS 188 Figure 6.30 Finite element mesh (a) global mesh of Model ; Case (b) close-view of the mesh at the vicinity of the main crack and the collinear microcrack of Model ; Case. The main crack and its tip are indicated with arrows and the collinear microcrack is also marked.

217 CHAPTER 6: DYNAMIC FRACTURE TESTS 189 6Figure 6.31, 6Figure 6.3 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 K is evaluated. 6Figure 6.3 shows the stress intensity contours due 0 I 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. Nine contour lines denoted by A to I, represent the stress intensity values from 40,000 Pa to 00,000 Pa with a stress increment of 0,000 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.

218 CHAPTER 6: DYNAMIC FRACTURE TESTS 190 Figure 6.3 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.

219 CHAPTER 6: DYNAMIC FRACTURE TESTS 191 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, 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: M 0 I / I K K is (shielding) for Model 1 and M 1 0 I / I K K is 1.33 (amplification) for Model. 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 will be examined as well. Dynamic finite element analysis are carried out on Model 1 and Model to determine the SIF evolution by solving the equation of motion with Newmark time integration method in ANSYS (Weisbrod and Rittel, 000). The finite element meshing for Model 1 and Model is the same as that for 6Figure 6.9 and 6Figure 6.30, while taking the input loading F 1 and F 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. F 1 = F ), thus in the finite element analysis, the bearing load (e.g. F ) 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 ). 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.

220 CHAPTER 6: DYNAMIC FRACTURE TESTS Force (kn) Time (μs) Figure 6.34 The dynamic load exerted on both ends of the NSCB specimen for three configurations (Intact, Model 1 and Model ), 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 ), the SIF evolutions calculated by dynamic finite element analysis are compared with that from a quasi-static analysis in 6Figure For all three models (Intact, Model 1 and Model ), 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. F ) according to the static equation which can be calibrated with static finite element analysis (6Figure 6.9 and 6Figure 6.30). Thus, at any instant throughout the loading, 0 and K K is 1.33 (amplification) for Model. M I / I K K is (shielding) for Model 1, M 1 0 I / I

221 CHAPTER 6: DYNAMIC FRACTURE TESTS 193 SIF (MPa m 1/ ) Intact_Dynamic Intact_Static Model 1_Dynamic Model 1_Static Model _Dynamic Model _Static Time (μs) Figure 6.35 The evolution of SIF of the NSCB specimen for three configurations (Intact, Model 1 and Model ) 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 to the loading rate for the Intact Model should be 1 0 I / K I K is (shielding) for Model 1 and 0 I / K I K =1.33 (amplification) for Model. 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 ) under force balance. Using dynamic finite element analysis aforementioned, the 3 / evolution of the dynamic dimensionless SIF Y (normalized by PS / BR, see Figure 6.13) and corresponding loading rates of the NSCB specimen for three configurations (Intact, Model 1 and Model ) can be calculated, as illustrated in 6Figure The loading rate for each case, as denoted by the slope of the dashed line, exhibits exactly the same ratio of K K M 1 0 I / I for Model 1 (shielding), and K K equals to 1.33 for Model (amplification). M I / I equals to

222 CHAPTER 6: DYNAMIC FRACTURE TESTS Force (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 ), assuming force balance on both loading ends of the sample. Y Intact_Dynamic Intact_Slope Model 1_Dynamic Model 1_Slope Model _Dynamic Model _Slope Model Intact Model Time (μs) 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 ) with linear dynamic loading, assuming force balance on both loading ends of the sample.

223 CHAPTER 6: DYNAMIC FRACTURE TESTS 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 by ξ (i.e. L 0 K K = ξ ). Thus, ξ = 1 corresponds to the microcrack-free case. ξ > 1 corresponds to the stress amplification where the stress intensity factor of the main crack is increased due to the presence of microcracks. ξ < 1, 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 I / I crack tip to the loading at the same loading rate, i.e. L 0 L 0 K I / K I = ξ and K I / K I = ξ. Let K 0 IC 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. K 0 IC is the maximum apparent value of 0 K I measured during material testing, which is usually considered to be K IC. 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 ξ 1, during the experiment, the local stress intensity factor 0 K ( = ξ ) increases with the loading L I K I 0 K I until the L K I is equal to the fracture toughness K IC. Under this circumstance, if the

224 CHAPTER 6: DYNAMIC FRACTURE TESTS L corresponding loading rate at far field is apparently K I, then the local stress intensity factor K I 0 ( = ξ ) actually increases with a loading rate of ξ K I 0 K I. Thus, 0 0 L K = max( K ) = max( K / ξ ) = K /ξ (6.37a) IC I I IC 0 L K I = K I /ξ (6.37b) This suggests that the intrinsic material toughness (mircrocrack-free cases) and the measured fracture toughness (microcracks embedded) are related: K = ξ (6.38a) 0 IC K IC 0 IC = K I K 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 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. MPa.m 1/ to 8.5 MPa.m 1/ and corresponding loading rate ranges from 40 GPa.m 1/ /s to 180 GPa.m 1/ /s with an increment of loading rate of 0 GPa.m 1/ /s. This variation of dynamic fracture toughness corresponds to the microcrack-free case withξ = 1, as denoted before. The measured toughness values and corresponding loading rates for Model 1 and Model are calculated and tabulated in 6Table 6.5; and also illustrated in 6Figure It is obvious that Model 1 own the highest fracture toughness while Model, 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 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; ) the ranking of the six sample groups with respect to

225 CHAPTER 6: DYNAMIC FRACTURE TESTS 197 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 Model 1 and Model ). Loading Rates (GPa.m 1/ /s) The fracture toughness and corresponding loading rates for three models (Intact, Intact Model Model 1 Model Fracture Toughness (MPa.m 1/ ) Loading Rates (GPa.m 1/ /s) Fracture Toughness (MPa.m 1/ ) Loading Rates (GPa.m 1/ /s) ~0 1. ~0 1.4 ~ Fracture Toughness (MPa.m 1/ ) 10 8 Intact Model 1 Model K IC (MPa m 1/ ) K I (GPa m 1/ s -1 ) Figure 6.38 The simulated dynamic fracture toughness of Barre granite with loading rates for three configurations (Intact, Model 1 and Model ).

226 CHAPTER 6: DYNAMIC FRACTURE TESTS 198 In the preceding chapter, the anisotropic index of Mode-I fracture toughness (α k ) 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, K K M 0 I / I K K M 1 0 I / I =0.880 for Model 1 and =1.33 for Model. Thus, during the experiments for fracture toughness measurements, the measured fracture toughness for Model 1 should be K 136 M 1 M IC = K IC / = 1. K IC and that for Model should be K IC K IC /1.33 = K IC =. Thus, during static loading, the anisotropic index of Mode-I fracture toughness, α k is K M 1 IC / K M IC =1.136K IC /0.811K IC =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, respectively. Afterwards, α k can be calculated as the ratio of the toughness from Model 1 to that from Model at giving loading rates. All related results are tabulated in 6Table 6.6. Table 6.6 with loading rates. The simulated Mode-I fracture toughness anisotropic index (α k ) of Barre granite Fracture Toughness (MPa.m 1/ ) Loading Rates (GPa.m 1/ /s) Intact Model 1 Model Anisotropy Index α k ~ The variation of the anisotropic index of the simulated Mode-I fracture toughness, α k with loading rates is also plotted in 6Figure The data points of α k 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.c.

227 CHAPTER 6: DYNAMIC FRACTURE TESTS 199 In real experimental tests for the two extreme cases corresponding to Model 1 and Model shown in 6Figure 6.3 and 6Figure 6.4 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 (α k ) of Barre granite with loading rates based on crack-microcracks interaction model Concluding Remarks Laboratory measurements of K IC 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

228 CHAPTER 6: DYNAMIC FRACTURE TESTS 00 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 ), 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 00 GPa.m 1/ /s, the factor decreases to 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 K IC 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 K IC combining with the microstructural analysis of the rock sample and numerical simulation. It is then feasible to use K IC and in-situ microstructural analysis to determine the global fracture toughness K Load I and make our design accordingly. As a result, different toughness values will be used in different directions and these values can be evaluated Load easily with ξ evaluated from numerical analysis and the relation of K = K / ξ. 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. I IC

229 CHAPTER 6: DYNAMIC FRACTURE TESTS Summary In this Chapter, a dynamic NSCB method is proposed in conjunction with the LGG system (Chen et al., 009) 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.

230 CHAPTER 7: SUMMARY AND FUTURE WORK 0 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

231 CHAPTER 7: SUMMARY AND FUTURE WORK 03 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 farfield 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. 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 m 1/ under loading rate of ~8E-5 MPa m 1/ s -1 ; and the dynamic counterpart is more than seven times of the static one (i.e. 7.8 MPa m 1/ ), under a loading rate of 180 MPa m 1/ s -1.

232 CHAPTER 7: SUMMARY AND FUTURE WORK 04 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. σ f /σ t ) 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. MPa with splitting plane normal to X axis. The flexural strength anisotropic index equals to 1.86, with the highest strength of 4.6 MPa for samples splitting in the plane normal to Z axis; and the lowest of 13. 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 0 MPa m 1/ s -1, the index drops to 1.0; 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; ) flexural strength drops to 1.4 under a loading rate of 000 GPa/s; 3) Mode-I fracture toughness drops to 1.0, as the loading rate is up to 0 MPa m 1/ s -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.

233 CHAPTER 7: SUMMARY AND FUTURE WORK 05 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 o 45 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

234 CHAPTER 7: SUMMARY AND FUTURE WORK 06 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. 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.

235 CHAPTER 7: SUMMARY AND FUTURE WORK 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 10 0 /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., 000). Christensen (197) 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.

236 CHAPTER 7: SUMMARY AND FUTURE WORK 08 Figure 7.1 system. Schematic of the Brazilian test under hydrostatic confining pressure on SHPB 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; ) 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.. 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., 007; Wai et al., 198), 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.

237 CHAPTER 7: SUMMARY AND FUTURE WORK 09 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. (001) extended their early works (Zhang et al., 001) 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, 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 ) 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 00 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, a schematic of the modified SHPB design for conducting dynamic Brazilian tests with in-situ heating is illustrated in 7Figure 7.. This modified SHPB system can also host the flexural test using SCB samples and Mode-I fracture tests using NSCB samples. Figure 7. Schematic of the Brazilian test on SHPB system under in-situ thermal heating.