1 Introduction ................................................. 1
2 Experimental Procedures ...................................... 5
2.1 The Triaxial Test ....................................... 5
2.1.1 Principle and Terminology ........................ 5
2.1.2 Triaxial Testing Apparatus ....................... 7
2.1.3 Preparation and Jacketing of Specimens .......... 12
2.1.4 Measurements, Control, and Data Reduction ....... 13
2.2 Other Types of Tests ................................... 15
3 Experimental Studies on the Brittle Fracture Stress ......... 17
3.1 Introduction ........................................... 17
3.2 Types of Fractures ..................................... 18
3.3 Observed Stresses at Brittle Failure ................... 21
3.4 Influence of Strain Rate, Temperature, and Time ........ 27
3.5 Size and Shape Effects ................................. 30
3.6 Influence of Intermediate Principal Stress ............. 35
3.7 Anisotropy ............................................. 38
3.7.1 Observations .................................... 38
3.7.2 Theoretical Analysis ............................ 41
4 Griffith Theory of Brittle Failure .......................... 45
4.1 Introduction ........................................... 45
4.2 Uniaxial Tension ....................................... 46
4.3 Uniaxial Compression and Biaxial Stresses .............. 49
4.4 Triaxial Stresses ...................................... 51
4.5 Modified Griffith Theory (Closed Cracks) ............... 53
4.6 Extension of the Griffith Approach to Other Modes
of Localized Inelastic Failure ......................... 55
4.6.1 Shear Failure ................................... 55
4.6.2 Compressive Failure ............................. 55
4.7 General Comment on Griffith-Type Theories .............. 56
5 Evolution of Physical Properties during Brittle Failure ..... 59
5.1 Introduction ........................................... 59
5.2 Pre-peak and Post-peak Behaviour ....................... 60
5.2.1 Role of the Testing Machine ..................... 60
5.2.2 Complete Force-Displacement Curves .............. 62
5.3 Volume Changes ......................................... 68
5.4 Elastic Wave Speeds and Attenuation .................... 76
5.4.1 General ......................................... 76
5.4.2 Background Theory and Observation ............... 77
5.4.3 Stress-Induced Changes in Elastic Wave Speeds ... 81
5.4.4 Attenuation ..................................... 87
5.5 Transport Properties ................................... 88
5.6 Acoustic Emission ...................................... 93
5.7 Microscopical Observations ............................. 99
5.7.1 General ......................................... 99
5.7.2 Observational Methods ........................... 99
5.7.3 Initial Structure .............................. 101
5.7.4 Effects of Stressing ........................... 105
5.8 A Synoptic View of Brittle Failure .................... 1ll
6 Micromechanics of Brittle Fracture ......................... 115
6.1 Introduction .......................................... 115
6.2 Micromechanical Theory for Compact Rock ............... 116
6.2.1 Initiation of Stress-Induced Microcracking ..... 116
6.2.2 Microcrack Propagation and Stability ........... 119
6.2.3 Microcrack Interaction and Damage Mechanics
of Brittle Failure ............................. 120
6.2.4 General Comments on Micromechanics-Based
Damage Mechanics Models for Compact Rock ....... 127
6.2.5 Stochastic Model of Brittle Failure in
Compact Rock ................................... 130
6.2.6 Time-Dependent Failure of Compact Rock ......... 133
6.3 Micromechanical Theory for Porous Rock ................ 134
6.3.1 General ........................................ 134
6.3.2 Strongly Cohesive Porous Rock .................. 134
6.3.3 Weakly Cohesive Porous Rock .................... 135
6.4 Localization Theory ................................... 137
6.4.1 Application of Plasticity Theory and
Bifurcation Analysis ........................... 137
6.4.2 Comments on Localization Theory ................ 143
7 The Role of Pore Fluids .................................... 147
7.1 Introduction .......................................... 147
7.2 The Notion of Effective Stress ........................ 148
7.3 Elastic Properties of Porous Media .................... 149
7.3.1 Linear Poroelasticity .......................... 149
7.3.2 Experimental Measurement of Poroelastic
Properties ..................................... 152
7.4 Brittle Failure - Experimental ........................ 155
7.4.1 Drained Tests .................................. 155
7.4.2 Undrained Tests ................................ 158
7.4.3 Chemical Effects ............................... 160
7.5 Brittle Failure - Theory of Pore Pressure Effects ... 161
8 Friction and Sliding Phenomena ............................. 165
8.1 Introduction .......................................... 165
8.2 Coefficient of Friction ............................... 166
8.2.1 General ........................................ 166
8.2.2 Influence of Variables ......................... 168
8.3 Phenomenology of Frictional Behaviour ................. 173
8.3.1 General ........................................ 173
8.3.2 Time Dependence of Static Friction ............. 173
8.3.3 Velocity Dependence of Kinetic Friction ........ 176
8.3.4 Rate- and State-Dependent Friction ............. 183
8.3.5 Experimental Observations on the Occurrence
of Stick-Slip .................................. 188
8.3.6 Influence of Friction Constitutive Behaviour
and Machine Stiffness on Stick-Slip ............ 194
8.3.7 Nucleation and Propagation of Dynamic Shear
Rupture ........................................ 196
8.4 Physical Aspects of Rock Friction ..................... 198
8.4.1 Frictional Slip, Wear and the Formation of
Gouge .......................................... 198
8.4.2 Gouge Deformation .............................. 200
8.4.3 Physical Factors Determining the Frictional
Strength ....................................... 203
8.4.4 Physical Basis for the Direct Effect
(Velocity Parameter a) ......................... 205
8.4.5 Physical Basis for the Ageing and Evolution
Effects (Parameters β and b) ................... 207
9 Brittle-Ductile Transition ................................. 211
9.1 Introduction .......................................... 211
9.2 Experimental Observations ............................. 212
9.2.1 Role of Confining Pressure ..................... 212
9.2.2 Role of Temperature and Strain Rate ............ 217
9.2.3 Physical Properties and Pressure Sensitivity ... 219
9.2.4 Microstructural Observations ................... 221
9.3 Physical Basis of the Brittle-Ductile Transition ...... 222
9.3.1 Preliminaries .................................. 222
9.3.2 Brittle-Ductile Transition with Increasing
Pressure ....................................... 225
9.3.3 Continuum Models and Localization Analysis ..... 228
9.3.4 Influence of Temperature, Strain Rate and
Water .......................................... 232
9.4 Ductile to Brittle Transitions ........................ 233
9.4.1 High Pressure Embrittlement .................... 233
9.4.2 High Temperature Embrittlement ................. 234
A1 Appendix 1 Fracture Mechanics ............................. 239
A1.1 Stress Intensity Analysis ............................. 239
A1.2 Critical Parameters for Failure ....................... 242
A1.3 Dynamic and Kinetic Effects ........................... 244
A2 Appendix 2 Plasticity Theory and Localization Analysis .... 247
A2.1 Introduction .......................................... 247
A2.2 Constitutive Relations for a Material without
Pressure Dependence or Dilatancy ...................... 248
A2.3 Constitutive Model for Dilatant, Pressure-Sensitive
Materials ............................................. 250
A2.4 Bifurcation, Instability and Shear Localization ....... 254
A3 Appendix 3 Constitutive Relations for Friction ............ 257
A3.1 Introduction .......................................... 257
A3.2 Friction Law .......................................... 258
A3.3 Evolution Law ......................................... 259
A3.4 Time-Dependent Evolution during Stationery Contact .... 261
A3.5 Analysis of Frictional Stability - Stick-Slip ......... 262
A3.6 Influence of Normal Stress and Temperature ............ 263
References .................................................... 267
Index ......................................................... 335
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