The present work describes investigations on the anisotropic strength behavior of rocks in the splitting tensile test (Brazilian test). Three transversely isotropic rocks (gneiss, slate and sandstone) were studied in the Lab. A total of more than 550 indirect tensile strength tests were conducted, with emphasis was placed on the investigation of the influence of the spatial position of anisotropic weakness plane to the direction of the load on the fracture strength and fracture or fracture mode. In parallel, analytical solutions were evaluated for stress distribution and developed 3D numerical models to study the stress distribution and the fracture mode at the transversely isotropic disc. There were new findings on the fracture mode of crack propagation, the influence of the disc thickness, the influence of the applying loading angle and angle of the loading-foliation for transversely isotropic material. / Inhalt der Arbeit sind Untersuchungen zum anisotropen Festigkeitsverhalten von Gesteinen beim Spaltzugversuch (Brazilian Test). Laborativ wurden drei transversalisotrope Gesteine (Granit, Schiefer und Sandstein) untersucht. Insgesamt wurden mehr als 550 Spaltzugversuche durchgeführt, wobei der Schwerpunkt auf die Untersuchung des Einflusses der räumlichen Lage der Anisotropieebene zur Richtung des Lasteintrages auf die Bruchfestigkeit und das Bruchbild bzw. den Bruchmodus gelegt wurde. Parallel dazu wurden analytische Lösungen zur Spannungsverteilung ausgewertet sowie numerische 3D-Modelle entwickelt, um die Spannungsverteilung sowie den Bruchmodus bei einer transversalisotropen Scheibe zu untersuchen. Es wurden neue Erkenntnisse zum Bruchmodus, der Rissausbreitung, des Einflusses der Scheibendicke, dem Einfluss des Lasteinleitungswinkel sowie des Winkels Lasteintrag - Anisotropieebene für transversalisotropes Material gewonnen.

The present work describes investigations on the anisotropic strength behavior of rocks in the splitting tensile test (Brazilian test). Three transversely isotropic rocks (gneiss, slate and sandstone) were studied in the Lab. A total of more than 550 indirect tensile strength tests were conducted, with emphasis was placed on the investigation of the influence of the spatial position of anisotropic weakness plane to the direction of the load on the fracture strength and fracture or fracture mode. In parallel, analytical solutions were evaluated for stress distribution and developed 3D numerical models to study the stress distribution and the fracture mode at the transversely isotropic disc. There were new findings on the fracture mode of crack propagation, the influence of the disc thickness, the influence of the applying loading angle and angle of the loading-foliation for transversely isotropic material.:ACKNOWLEDGMENTS 5 ABSTRACT 7 TABLE OF CONTENTS 9 LIST OF FIGURES 13 LIST OF TABLES 19 I. INTRODUCTION 21 Objective of this work 22 Scope of work 23 Research procedure 23 Significance of the work 24 Layout 24 1 STATE OF THE ART 27 1.1 Review of the Brazilian tensile strength test 27 1.1.1 General overview 27 1.1.2 Development of the Brazilian tensile strength test 29 1.1.3 The Brazilian tensile strength test on anisotropic rocks 31 1.1.4 Summary 32 1.2 Analytical aspects 33 1.2.1 Hypotheses for the conventional Brazilian test 34 1.2.2 Failure criteria 36 1.2.3 Crack initiation and propagation 39 1.2.4 Summary 41 1.3 Numerical considerations 41 1.3.1 Numerical methods 42 1.3.2 Summary 42 1.4 Conclusion 43 2 DIAMETRAL COMPRESSION IN A SOLID DISC – COMPILATION OF ANALYTICAL AND SEMI-ANALYTICAL SOLUTIONS 45 2.1 Introduction 45 2.2 Diametral compressive stress distribution in an isotropic elastic disc 45 2.2.1 Elastic theory of line load 46 2.2.2 2D analytical solutions 47 2.2.3 3D disc under line and diametral compressive distributed loads 55 2.2.4 3D solution under diametral compressive distributed load 56 2.3 Stress and strain in an isotropic solid disc 59 2.4 Stress and strain in anisotropic rocks 61 2.5 Conclusion 65 3 LABORATORY TESTS 69 3.1 Introduction 69 3.2 Laboratory test program 70 3.3 Sample preparation 71 3.4 Ultrasonic measurements 72 3.5 Uniaxial and triaxial compression tests 73 3.5.1 Uniaxial compression test 73 3.5.2 Triaxial compression tests 74 3.6 Brazilian tensile strength tests 76 3.6.1 Test apparatus 76 3.6.2 Laboratory test results 77 3.6.3 Interpretation of the test results 89 3.7 Conclusion 96 4 NUMERICAL SIMULATION OF ISOTROPIC MATERIALS - COMPARISON WITH ANALYTICAL SOLUTIONS 97 4.1 Introduction 97 4.2 Numerical simulation of isotropic materials 97 4.2.1 FLAC3D simulation program 97 4.2.2 Simulation procedure 98 4.2.3 Numerical model setup 98 4.2.4 Influence of mesh type 99 4.2.5 Influence of specimen thickness 100 4.2.6 Influence of Poisson’s ratio 102 4.2.7 Influence of loading angle (2) 106 4.2.8 Comparison of 3D analytical and numerical results 110 4.2.9 Influence of stress concentration at the loading jaws 112 4.3 Comparison with experimental results of Postaer Sandstone (FG.Ss) 112 4.4 Conclusion 114 5 NUMERICAL SIMULATION OF ANISOTROPIC MATERIALS - COMPARISON WITH LABORATORY TESTS 117 5.1 Introduction 117 5.2 General procedure for simulating the Brazilian test using FLAC3D 117 5.2.1 Conceptual model 119 5.2.2 Boundary Conditions 119 5.2.3 Numerical model set-up 120 5.3 Constitutive model 121 5.3.1 Choice of constitutive model 121 5.3.2 Bilinear Strain-Hardening/Softening Ubiquitous-Joint Model [98] 121 5.4 Parameter calibration 124 5.4.1 Material parameters used 124 5.4.2 Contact between disc and loading jaws 126 5.4.3 Post-failure deformation properties 128 5.4.4 Tension cut-off 129 5.5 Numerical simulation results 131 5.5.1 Introduction 131 5.5.2 Stress distribution and failure state 133 5.5.3 Stress state in an isotropic elastic medium with arbitrary orientation planes 136 5.5.4 Plasticity states 139 5.5.5 Damage and fracture process 141 5.5.6 Fracture patterns – Comparison of lab results and numerical simulations 148 5.6 Tensile strength – Comparison of lab results and numerical simulations 149 5.6.1 Tensile strength of Le.Gs Gneiss 150 5.6.2 Tensile strength of My.Sc Slate 155 5.7 Summary and Review 159 5.7.1 Potential failure state deduced from pure elastic considerations 159 5.7.2 Tensile strength distribution 160 5.7.3 Tensile strength – determining the anisotropy factor 161 5.7.4 Tensile strength – different procedures - different results 163 6 CONCLUSION AND RECOMMENDATIONS 165 APPENDICES 171 Appendix 3.1 - Fracture patterns in FG.Ss samples 171 Appendix 3.2 - Fracture patterns in FG.Gs samples 177 Appendix 3.3 - Fracture patterns in Le.Gs samples 183 Appendix 3.4 - Fracture patterns in My.Sc samples 190 Appendix 4.1 - Influence of loading angle 197 Appendix 4.2 - Influence of material properties 203 Appendix 5.1 - Failure zone state in Le.Gs Gneiss 209 Appendix 5.2: Failure zone state in My.Sc Slate 216 REFERENCES 223 / Inhalt der Arbeit sind Untersuchungen zum anisotropen Festigkeitsverhalten von Gesteinen beim Spaltzugversuch (Brazilian Test). Laborativ wurden drei transversalisotrope Gesteine (Granit, Schiefer und Sandstein) untersucht. Insgesamt wurden mehr als 550 Spaltzugversuche durchgeführt, wobei der Schwerpunkt auf die Untersuchung des Einflusses der räumlichen Lage der Anisotropieebene zur Richtung des Lasteintrages auf die Bruchfestigkeit und das Bruchbild bzw. den Bruchmodus gelegt wurde. Parallel dazu wurden analytische Lösungen zur Spannungsverteilung ausgewertet sowie numerische 3D-Modelle entwickelt, um die Spannungsverteilung sowie den Bruchmodus bei einer transversalisotropen Scheibe zu untersuchen. Es wurden neue Erkenntnisse zum Bruchmodus, der Rissausbreitung, des Einflusses der Scheibendicke, dem Einfluss des Lasteinleitungswinkel sowie des Winkels Lasteintrag - Anisotropieebene für transversalisotropes Material gewonnen.:ACKNOWLEDGMENTS 5 ABSTRACT 7 TABLE OF CONTENTS 9 LIST OF FIGURES 13 LIST OF TABLES 19 I. INTRODUCTION 21 Objective of this work 22 Scope of work 23 Research procedure 23 Significance of the work 24 Layout 24 1 STATE OF THE ART 27 1.1 Review of the Brazilian tensile strength test 27 1.1.1 General overview 27 1.1.2 Development of the Brazilian tensile strength test 29 1.1.3 The Brazilian tensile strength test on anisotropic rocks 31 1.1.4 Summary 32 1.2 Analytical aspects 33 1.2.1 Hypotheses for the conventional Brazilian test 34 1.2.2 Failure criteria 36 1.2.3 Crack initiation and propagation 39 1.2.4 Summary 41 1.3 Numerical considerations 41 1.3.1 Numerical methods 42 1.3.2 Summary 42 1.4 Conclusion 43 2 DIAMETRAL COMPRESSION IN A SOLID DISC – COMPILATION OF ANALYTICAL AND SEMI-ANALYTICAL SOLUTIONS 45 2.1 Introduction 45 2.2 Diametral compressive stress distribution in an isotropic elastic disc 45 2.2.1 Elastic theory of line load 46 2.2.2 2D analytical solutions 47 2.2.3 3D disc under line and diametral compressive distributed loads 55 2.2.4 3D solution under diametral compressive distributed load 56 2.3 Stress and strain in an isotropic solid disc 59 2.4 Stress and strain in anisotropic rocks 61 2.5 Conclusion 65 3 LABORATORY TESTS 69 3.1 Introduction 69 3.2 Laboratory test program 70 3.3 Sample preparation 71 3.4 Ultrasonic measurements 72 3.5 Uniaxial and triaxial compression tests 73 3.5.1 Uniaxial compression test 73 3.5.2 Triaxial compression tests 74 3.6 Brazilian tensile strength tests 76 3.6.1 Test apparatus 76 3.6.2 Laboratory test results 77 3.6.3 Interpretation of the test results 89 3.7 Conclusion 96 4 NUMERICAL SIMULATION OF ISOTROPIC MATERIALS - COMPARISON WITH ANALYTICAL SOLUTIONS 97 4.1 Introduction 97 4.2 Numerical simulation of isotropic materials 97 4.2.1 FLAC3D simulation program 97 4.2.2 Simulation procedure 98 4.2.3 Numerical model setup 98 4.2.4 Influence of mesh type 99 4.2.5 Influence of specimen thickness 100 4.2.6 Influence of Poisson’s ratio 102 4.2.7 Influence of loading angle (2) 106 4.2.8 Comparison of 3D analytical and numerical results 110 4.2.9 Influence of stress concentration at the loading jaws 112 4.3 Comparison with experimental results of Postaer Sandstone (FG.Ss) 112 4.4 Conclusion 114 5 NUMERICAL SIMULATION OF ANISOTROPIC MATERIALS - COMPARISON WITH LABORATORY TESTS 117 5.1 Introduction 117 5.2 General procedure for simulating the Brazilian test using FLAC3D 117 5.2.1 Conceptual model 119 5.2.2 Boundary Conditions 119 5.2.3 Numerical model set-up 120 5.3 Constitutive model 121 5.3.1 Choice of constitutive model 121 5.3.2 Bilinear Strain-Hardening/Softening Ubiquitous-Joint Model [98] 121 5.4 Parameter calibration 124 5.4.1 Material parameters used 124 5.4.2 Contact between disc and loading jaws 126 5.4.3 Post-failure deformation properties 128 5.4.4 Tension cut-off 129 5.5 Numerical simulation results 131 5.5.1 Introduction 131 5.5.2 Stress distribution and failure state 133 5.5.3 Stress state in an isotropic elastic medium with arbitrary orientation planes 136 5.5.4 Plasticity states 139 5.5.5 Damage and fracture process 141 5.5.6 Fracture patterns – Comparison of lab results and numerical simulations 148 5.6 Tensile strength – Comparison of lab results and numerical simulations 149 5.6.1 Tensile strength of Le.Gs Gneiss 150 5.6.2 Tensile strength of My.Sc Slate 155 5.7 Summary and Review 159 5.7.1 Potential failure state deduced from pure elastic considerations 159 5.7.2 Tensile strength distribution 160 5.7.3 Tensile strength – determining the anisotropy factor 161 5.7.4 Tensile strength – different procedures - different results 163 6 CONCLUSION AND RECOMMENDATIONS 165 APPENDICES 171 Appendix 3.1 - Fracture patterns in FG.Ss samples 171 Appendix 3.2 - Fracture patterns in FG.Gs samples 177 Appendix 3.3 - Fracture patterns in Le.Gs samples 183 Appendix 3.4 - Fracture patterns in My.Sc samples 190 Appendix 4.1 - Influence of loading angle 197 Appendix 4.2 - Influence of material properties 203 Appendix 5.1 - Failure zone state in Le.Gs Gneiss 209 Appendix 5.2: Failure zone state in My.Sc Slate 216 REFERENCES 223