intro struct geo allmendinger_part 1&2 جيولوجيا تركيبية
TRANSCRIPT
Contents i
GEOL 326
Cornell University
Introduction to Structural Geology
Spring 1999
by
Richard W. AllmendingerDepartment of Geological Sciences
Snee HallCornell University, Ithaca, NY 14853-1504 USA
R. W. Allmendinger © 1999
Contents i i
Contents
Lecture 1—Introduction, Scale, & Basic Terminology ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Introduction .................................................................................................................................. 1
1.2 Levels of Structural Study............................................................................................................ 2
1.3 Types of Structural Study............................................................................................................. 2
1.4 Importance of Scale..................................................................................................................... 3
1.4.1 Scale Terms.................................................................................................................... 3
1.4.2 Scale Invariance, Fractals............................................................................................... 4
Lecture 2 —Coordinate Systems, etc. ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Introduction .................................................................................................................................. 8
2.2 Three types of physical entities.................................................................................................... 8
2.3 Coordinate Systems..................................................................................................................... 9
2.3.1 Spherical versus Cartesian Coordinate Systems..........................................................10
2.3.2 Right-handed and Left-handed Coordinate Systems.................................................... 10
2.3.3 Cartesian Coordinate Systems in Geology................................................................... 11
2.4 Vectors.......................................................................................................................................12
2.4.1 Vectors vs. Axes ........................................................................................................... 12
2.4.2 Basic Properties of Vectors...........................................................................................12
2.4.3 Geologic Features as Vectors.......................................................................................14
2.4.4 Simple Vector Operations ............................................................................................. 17
2.4.5 Dot Product and Cross Product .................................................................................... 18
Lecture 3 — Descriptive Geometry: Seismic Reflection ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1
3.1 Echo Sounding...........................................................................................................................21
3.2 Common Depth Point (CDP) Method......................................................................................... 23
3.3 Migration .................................................................................................................................... 25
3.4 Resolution of Seismic Reflection Data.......................................................................................26
3.4.1 Vertical Resolution ........................................................................................................26
3.4.2 Horizontal Resolution ....................................................................................................27
3.5 Diffractions.................................................................................................................................28
3.6 Artifacts...................................................................................................................................... 29
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3.6.1 Velocity Pullup/pulldown ...............................................................................................29
3.6.2 Multiples........................................................................................................................ 29
3.6.3 Sideswipe......................................................................................................................30
3.6.4 Buried Focus .................................................................................................................31
3.6.5 Others ...........................................................................................................................32
Lecture 4 — Introduction to Deformation & Strain .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3
4.1 Introduction ................................................................................................................................33
4.2 Kinematics .................................................................................................................................33
4.2.1 Rigid Body Deformations .............................................................................................. 33
4.2.2 Strain (Non-rigid Body Deformation) .............................................................................34
4.2.3 Continuum Mechanics...................................................................................................36
4.2.4 Four Aspects of a Deforming Rock System: ................................................................. 37
4.3 Measurement of Strain............................................................................................................... 38
4.3.1 Change in Line Length:................................................................................................. 39
4.3.2 Changes in Angles:....................................................................................................... 40
4.3.3 Changes in Volume (Dilation): ......................................................................................41
Lecture 5 — Strain II: The Strain Ellipsoid ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 2
5.1 Motivation for General 3-D Strain Relations ..............................................................................42
5.2 Equations for Finite Strain.......................................................................................................... 43
5.3 Extension of a Line .................................................................................................................... 43
5.4 Shear Strain...............................................................................................................................45
Lecture 6 — Strain III: Mohr on the Strain Ellipsoid ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 7
6.1 Introduction ................................................................................................................................47
6.2 Mohr’s Circle For Finite Strain ...................................................................................................47
6.3 Principal Axes of Strain..............................................................................................................48
6.4 Maximum Angular Shear ........................................................................................................... 49
6.5 Ellipticity.....................................................................................................................................50
6.6 Rotation of Any Line During Deformation ..................................................................................50
6.7 Lines of No Finite Elongation.....................................................................................................51
Lecture 7 — Strain IV: Finite vs. Infinitesimal Strain .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3
7.1 Coaxial and Non-coaxial Deformation .......................................................................................54
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7.2 Two Types of Rotation............................................................................................................... 55
7.3 Deformation Paths .....................................................................................................................55
7.4 Superposed Strains & Non-commutability .................................................................................58
7.5 Plane Strain & 3-D Strain........................................................................................................... 58
Lecture 8—Stress I: Introduction ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 0
8.1 Force and Stress........................................................................................................................ 60
8.2 Units Of Stress...........................................................................................................................61
8.3 Sign Conventions:......................................................................................................................61
8.4 Stress on a Plane; Stress at a Point ..........................................................................................62
8.5 Principal Stresses ......................................................................................................................63
8.6 The Stress Tensor .....................................................................................................................64
8.7 Mean Stress...............................................................................................................................64
8.8 Deviatoric Stress........................................................................................................................ 64
8.9 Special States of Stress.............................................................................................................65
Lecture 9—Vectors & Tensors ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 6
9.1 Scalars & Vectors ......................................................................................................................67
9.2 Tensors...................................................................................................................................... 68
9.3 Einstein Summation Convention................................................................................................69
9.4 Coordinate Systems and Tensor Transformations ....................................................................70
9.5 Symmetric, Asymmetric, & Antisymmetric Tensors................................................................... 71
9.6 Finding the Principal Axes of a Symmetric Tensor ....................................................................73
Lecture 10—Stress II: Mohr’s Circle ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 4
10.1 Stresses on a Plane of Any Orientation from Cauchy’s law..................................................... 74
10.2 A more “Traditional” Way to Derive the above Equations........................................................ 75
10.2.1 Balance of Forces....................................................................................................... 76
10.2.2 Normal and Shear Stresses on Any Plane.................................................................. 77
10.3 Mohr’s Circle for Stress............................................................................................................78
10.4 Alternative Way of Plotting Mohr’s Circle.................................................................................80
10.5 Another Way to Derive Mohr’s Circle Using Tensor Transformations .....................................81
10.5.1 Transformation of Axes ...............................................................................................81
10.5.2 Tensor Transformations.............................................................................................. 82
10.5.3 Mohr Circle Construction............................................................................................. 82
Contents v
Lecture 11—Stress III: Stress-Strain Relations
11.1 More on the Mohr’s Circle........................................................................................................85
11.1.1 Mohr’s Circle in Three Dimensions .............................................................................86
11.2 Stress Fields and Stress Trajectories ......................................................................................86
11.3 Stress-strain Relations.............................................................................................................87
11.4 Elasticity...................................................................................................................................88
11.4.1 The Elasticity Tensor...................................................................................................88
11.4.2 The Common Material Parameters of Elasticity..........................................................89
11.5 Deformation Beyond the Elastic Limit ......................................................................................90
Lecture 12—Plastic & Viscous Deformation ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2
12.1 Strain Rate...............................................................................................................................92
12.2 Viscosity...................................................................................................................................93
12.3 Creep .......................................................................................................................................94
12.4 Environmental Factors Affecting Material Response to Stress................................................ 95
12.4.1 Variation in Stress....................................................................................................... 95
12.4.2 Effect of Confining Pressure (Mean Stress)................................................................95
12.4.3 Effect of Temperature ................................................................................................. 96
12.4.4 Effect of Fluids ............................................................................................................96
12.4.5 The Effect of Strain Rate............................................................................................. 97
12.5 Brittle, Ductile, Cataclastic, Crystal Plastic ..............................................................................97
Lecture 13—Deformation Mechanisms I: Elasticity, Compaction .... . . . . . . . . . . . . . . . . . . . . . .100
13.1 Elastic Deformation................................................................................................................ 100
13.2 Thermal Effects and Elasticity................................................................................................ 102
13.3 Compaction............................................................................................................................ 103
13.4 Role of Fluid Pressure ........................................................................................................... 104
13.4.1 Effective Stress ......................................................................................................... 105
Lecture 14—Deformation Mechanisms II: Fracture ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
14.1 Effect of Pore Pressure.......................................................................................................... 111
14.2 Effect of Pre-existing Fractures.............................................................................................. 112
14.3 Friction ................................................................................................................................... 113
Lecture 15—Deformation Mechanisms III: Pressure Solution & Crystal Plasticity 114
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15.1 Pressure Solution................................................................................................................... 114
15.1.1 Observational Aspects .............................................................................................. 114
15.1.2 Environmental constrains on Pressure Solution ....................................................... 117
15.2 Mechanisms of Crystal Plasticity ........................................................................................... 117
15.2.1 Point Defects............................................................................................................. 118
15.2.2 Diffusion .................................................................................................................... 118
15.2.3 Planar Defects........................................................................................................... 119
Lecture 16—Deformation Mechanisms IV: Dislocations ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
16.1 Basic Concepts and Terms.................................................................................................... 121
16.2 Dislocation (“Translation”) Glide ............................................................................................ 123
16.3 Dislocations and Strain Hardening......................................................................................... 123
16.4 Dislocation Glide and Climb................................................................................................... 125
16.5 Review of Deformation Mechanisms ..................................................................................... 126
Lecture 17—Flow Laws & State of Stress in the Lithosphere .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
17.1 Power Law Creep .................................................................................................................. 127
17.2 Diffusion Creep...................................................................................................................... 129
17.3 Deformation Maps.................................................................................................................. 129
17.4 State of Stress in the Lithosphere.......................................................................................... 130
Lecture 18—Joints & Veins ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133
18.1 Faults and Joints as Cracks................................................................................................... 133
18.2 Joints .....................................................................................................................................133
18.2.1 Terminology .............................................................................................................. 134
18.2.2 Surface morphology of the joint face:........................................................................ 135
18.2.3 Special Types of Joints and Joint-related Features .................................................. 136
18.2.4 Maximum Depth of True Tensile Joints..................................................................... 136
18.3 Veins .....................................................................................................................................137
18.3.1 Fibrous Veins in Structural Analysis.......................................................................... 138
18.3.2 En Echelon Sigmoidal Veins..................................................................................... 139
18.4 Relationship of Joints and Veins to other Structures ............................................................. 140
Lecture 19—Faults I: Basic Terminology ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
19.1 Descriptive Fault Geometry ................................................................................................... 141
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19.2 Apparent and Real Displacement .......................................................................................... 142
19.3 Basic Fault Types .................................................................................................................. 143
19.3.1 Dip Slip...................................................................................................................... 143
19.3.2 Strike-Slip.................................................................................................................. 143
19.3.3 Rotational fault .......................................................................................................... 144
19.4 Fault Rocks ............................................................................................................................ 144
19.4.1 Sibson’s Classification .............................................................................................. 144
19.4.2 The Mylonite Controversy ......................................................................................... 146
Lecture 20—Faults II: Slip Sense & Surface Effects .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
20.1 Surface Effects of Faulting..................................................................................................... 147
20.1.1 Emergent Faults........................................................................................................ 147
20.1.2 Blind Faults ............................................................................................................... 149
20.2 How a Fault Starts: Riedel Shears........................................................................................ 149
20.2.1 Pre-rupture Structures............................................................................................... 150
20.2.2 Rupture & Post-Rupture Structures .......................................................................... 151
20.3 Determination of Sense of Slip .............................................................................................. 151
Lecture 21—Faults III: Dynamics & Kinematics ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
21.1 Introduction ............................................................................................................................ 157
21.2 Anderson’s Theory of Faulting............................................................................................... 158
21.3 Strain from Fault Populations................................................................................................. 161
21.3.1 Sense of Shear ......................................................................................................... 161
21.3.2 Kinematic Analysis of Fault Populations ................................................................... 161
21.3.3 The P & T Dihedra .................................................................................................... 162
21.4 Stress From Fault Populations1............................................................................................. 164
21.4.1 Assumptions.............................................................................................................. 164
21.4.2 Coordinate Systems & Geometric Basis................................................................... 165
21.4.3 Inversion Of Fault Data For Stress............................................................................ 167
21.5 Scaling Laws for Fault Populations........................................................................................ 169
Lecture 22—Faults IV: Mechanics of Thrust Faults .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
22.1 The Paradox of Low-angle Thrust Faults.............................................................................. 170
22.2 Hubbert & Rubey Analysis..................................................................................................... 170
22.3 Alternative Solutions.............................................................................................................. 174
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Lecture 23—Folds I: Geometry ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
23.1 Two-dimensional Fold Terminology....................................................................................... 178
23.2 Geometric Description of Folds.............................................................................................. 180
23.2.1 Two-dimensional (Profile) View:................................................................................ 180
23.2.2 Three-dimensional View:........................................................................................... 181
23.3 Fold Names Based on Orientation......................................................................................... 182
23.4 Fold Tightness ....................................................................................................................... 183
Lecture 24 — Folds II: Geometry & Kinematics ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184
24.1 Fold Shapes........................................................................................................................... 184
24.2 Classification Based on Shapes of Folded Layers................................................................. 185
24.3 Geometric-kinematic Classification:....................................................................................... 186
24.3.1 Cylindrical Folds........................................................................................................ 186
24.3.2 Non-Cylindrical Folds................................................................................................ 188
24.4 Summary Outline ................................................................................................................... 189
24.5 Superposed Folds.................................................................................................................. 189
Lecture 25—Folds III: Kinematics ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191
25.1 Overview................................................................................................................................ 191
25.2 Gaussian Curvature............................................................................................................... 191
25.3 Buckling ................................................................................................................................. 192
25.4 Shear Parallel to Layers......................................................................................................... 193
25.4.1 Kink folds................................................................................................................... 195
25.4.2 Simple Shear during flexural slip............................................................................... 196
25.5 Shear Oblique To Layers....................................................................................................... 196
25.6 Pure Shear Passive Flow....................................................................................................... 197
Lecture 26—Folds IV: Dynamics ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198
26.1 Basic Aspects ........................................................................................................................ 199
26.2 Common Rock Types Ranked According to “Competence” .................................................. 199
26.3 Theoretical Analyses of Folding............................................................................................. 199
26.3.1 Nucleation of Folds ................................................................................................... 200
26.3.2 Growth of Folds......................................................................................................... 201
26.3.3 Results for Kink Folds ............................................................................................... 202
Contents i x
Lecture 27—Linear Minor Structures ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
27.1 Introduction to Minor Structures............................................................................................. 203
27.2 Lineations............................................................................................................................... 203
27.2.1 Mineral Lineations..................................................................................................... 203
27.2.2 Deformed Detrital Grains (and related features)....................................................... 204
27.2.3 Rods and Mullions..................................................................................................... 205
27.3 Boudins.................................................................................................................................. 205
27.4 Lineations Due to Intersecting Foliations............................................................................... 206
Lecture 28—Planar Minor Structures I .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
28.1 Introduction to Foliations........................................................................................................ 207
28.2 Cleavage................................................................................................................................ 207
28.2.1 Cleavage and Folds .................................................................................................. 208
28.3 Cleavage Terminology........................................................................................................... 209
28.3.1 Problems with Cleavage Terminology....................................................................... 210
28.3.2 Descriptive Terms..................................................................................................... 210
28.4 Domainal Nature of Cleavage................................................................................................ 211
28.4.1 Scale of Typical Cleavage Domains ......................................................................... 212
Lecture 29—Planar Minor Structures II: Cleavage & Strain .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213
29.1 Processes of Foliation Development ..................................................................................... 213
29.2 Rotation of Grains.................................................................................................................. 213
29.2.1 March model ............................................................................................................. 214
29.2.2 Jeffery Model............................................................................................................. 214
29.2.3 A Special Case of Mechanical Grain Rotation .......................................................... 214
29.3 Pressure Solution and Cleavage ........................................................................................... 215
29.4 Crenulation Cleavage ............................................................................................................ 216
29.5 Cleavage and Strain .............................................................................................................. 217
Lecture 30—Shear Zones & Transposition ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219
30.1 Shear Zone Foliations and Sense of Shear........................................................................... 219
30.1.1 S-C Fabrics ............................................................................................................... 219
30.1.2 Mica “Fish” in Type II S-C Fabrics............................................................................. 219
30.1.3 Fractured and Rotated Mineral Grains...................................................................... 220
Contents x
30.1.4 Asymmetric Porphyroclasts....................................................................................... 220
30.2 Use of Foliation to Determine Displacement in a Shear Zone............................................... 221
30.3 Transposition of Foliations..................................................................................................... 222
Lecture 31—Thrust Systems I: Overview & Tectonic Setting .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
31.1 Basic Thrust System Terminology ......................................................................................... 225
31.2 Tectonic Setting of Thin-skinned Fold & Thrust Belts............................................................ 226
31.2.1 Andean Type:............................................................................................................ 227
31.2.2 Himalayan Type: ....................................................................................................... 227
31.3 Basic Characteristics of Fold-thrust Belts.............................................................................. 228
31.4 Relative and Absolute Timing in Fold-thrust Belts................................................................. 229
31.5 Foreland Basins..................................................................................................................... 229
Lecture 32—Thrust Systems II: Basic Geometries ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231
32.1 Dahlstrom’s Rules and the Ramp-flat (Rich Model) Geometry.............................................. 231
32.2 Assumptions of the Basic Rules ............................................................................................ 232
32.3 Types of Folds in Thrust Belts ............................................................................................... 233
32.4 Geometries with Multiple Thrusts........................................................................................... 234
32.4.1 Folded thrusts ........................................................................................................... 234
32.4.2 Duplexes ................................................................................................................... 235
32.4.3 Imbrication................................................................................................................. 237
32.4.4 Triangle Zones .......................................................................................................... 237
Lecture 33—Thrust Systems III: Thick-Skinned Faulting ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .239
33.1 Plate-tectonic Setting............................................................................................................. 239
33.2 Basic Characteristics ............................................................................................................. 240
33.3 Cross-sectional Geometry ..................................................................................................... 240
“Upthrust” Hypothesis ........................................................................................................... 240
33.3.1 Overthrust Hypothesis............................................................................................... 240
33.3.2 Deep Crustal Geometry ............................................................................................ 241
33.4 Folding in Thick-skinned Provinces ....................................................................................... 242
33.4.1 Subsidiary Structures................................................................................................ 242
33.5 Late Stage Collapse of Uplifts................................................................................................ 243
33.6 Regional Mechanics............................................................................................................... 244
Contents x i
Lecture 34—Extensional Systems I ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245
34.1 Basic Categories of Extensional Structures........................................................................... 245
34.2 Gravity Slides......................................................................................................................... 245
34.2.1 The Heart Mountain Fault ......................................................................................... 246
34.2.2 Subaqueous Slides ................................................................................................... 246
34.3 Growth Faulting on a Subsiding Passive Margin ................................................................... 247
34.4 Tectonic Rift Provinces .......................................................................................................... 248
34.4.1 Oceanic Spreading Centers...................................................................................... 248
34.4.2 Introduction to Intracontinental Rift Provinces........................................................... 249
Lecture 35—Extensional Systems I I .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .250
35.1 Basic Categories of Extensional Structures........................................................................... 250
35.2 Rotated Planar Faults ............................................................................................................ 250
35.3 Listric Normal Faults.............................................................................................................. 252
35.4 Low-angle Normal Faults....................................................................................................... 253
35.5 Review of Structural Geometries........................................................................................... 254
35.6 Thrust Belt Concepts Applied to Extensional Terranes ......................................................... 254
35.6.1 Ramps, Flats, & Hanging Wall Anticlines:................................................................. 254
35.6.2 Extensional Duplexes:............................................................................................... 254
35.7 Models of Intracontinental Extension..................................................................................... 255
35.7.1 Horst & Graben: ........................................................................................................ 255
35.7.2 “Brittle-ductile” Transition & Sub-horizontal Decoupling:........................................... 255
35.7.3 Lenses or Anastomosing Shear Zones:.................................................................... 255
35.7.4 Crustal-Penetrating Low-Angle Normal Fault:........................................................... 256
35.7.5 Hybrid Model of Intracontinental Extension............................................................... 256
Lecture 36—Strike-slip Fault System s .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
36.1 Tectonic setting of Strike-slip Faults ...................................................................................... 257
36.1.1 Transform faults ........................................................................................................ 257
36.2 Transcurrent Faults and Tear Faults...................................................................................... 258
36.3 Features Associated with Major Strike-slip Faults................................................................. 259
36.3.1 Parallel Strike-slip ..................................................................................................... 259
36.3.2 Convergent-Type ...................................................................................................... 262
36.3.3 Divergent Type.......................................................................................................... 262
36.4 Restraining and Releasing bends, duplexes.......................................................................... 263
Contents x i i
36.5 Terminations of Strike-slip Faults........................................................................................... 264
Lecture 37—Deformation of the Lithosphere ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
37.1 Mechanisms of Uplift.............................................................................................................. 265
37.1.1 Isostasy & Crust-lithosphere thickening.................................................................... 265
37.1.2 Differential Isostasy................................................................................................... 266
37.1.3 Flexural Isostasy ....................................................................................................... 267
37.2 Geological Processes of Lithospheric Thickening ................................................................. 269
37.2.1 Distributed Shortening:.............................................................................................. 269
37.2.2 “Underthrusting”: ....................................................................................................... 269
37.2.3 Magmatic Intrusion:................................................................................................... 269
37.3 Thermal Uplift......................................................................................................................... 270
37.4 Evolution of Uplifted Continental Crust .................................................................................. 270
Lecture 1 1Terminology, Scale
LECTURE 1—INTRODUCTION, SCALE, & BASIC TERMINOLOGY
1.1 Introduction
Structural Geology is the study of deformed rocks. To do so, we define the geometries of rock
bodies in three dimensions. Then, we measure or infer the translations, rotations, and strains experienced
by rocks both during, and particularly since, their formation based on indicators of what they looked like
prior to their deformation. Finally, we try to infer the stresses that produced the deformation based on
our knowledge of material properties. Structure is closely related to various fields of engineering mechanics,
structural engineering, and material science.
But, there is a big difference: In structural geology, we deal almost exclusively with the end
product of deformation in extremely heterogeneous materials. Given this end product, we try to infer the
processes by which the deformation occurred. In engineering, one is generally more interested in the
effect that various, known or predicted, stress systems will produce on undeformed, relatively homogeneous
materials.
?
Engineering:
Structural Geology:
Key Point: What we study in structural geology is strain and its related translations and rotations;
this is the end product of deformation. We never observe stress directly or the forces responsible for the
deformation. A famous structural geologist, John Ramsay, once said that "as a geologist, I don't believe in
stress". This view is perhaps too extreme -- stress certainly does exist, but we cannot measure it directly.
Stress is an instantaneous entity; it exists only in the moment that it is applied. In Structural Geology we
study geological materials that were deformed in the past, whether it be a landslide that formed two
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hours ago or a fold that formed 500 Ma ago. The stresses that were responsible for that deformation are no
Lecture 1 2Terminology, Scale
longer present. Even when the stresses of interest are still present, such as in the test of the strength of a
concrete block in an engineering experiment, you cannot measure stress directly. What you do is measure
the strain of some material whose material response to stress, or rheology, is very well known.
If you learn nothing else in this course, it should be the distinction between stress and strain, and
what terms are appropriate to each:
Stress Strain
note that terms in the same row are not equal but have somewhat parallel meanings. As we will see later in the course, the relations among these terms is quite
compression shortening (contraction)
tension lengthening (extension)
1.2 Levels of Structural Study
There are three basic level at which one can pursue structural geology and these are reflected in
the organization of this course:
• Geometry basically means how big or extensive something is (size or magni-
tude) and/or how its dimensions are aligned in space (orientation). We will
spend only a little time during lecture on the geometric description of structures
because most of the lab part of this course is devoted to this topic.
• Kinematics is the description of movements that particles of material have
experienced during their history. Thus we are comparing two different states
of the material, whether they be the starting point and ending point or just
two intermediate points along the way.
• Mechanics implies an understanding of how forces applied to a material
have produced the movements of the particles that make up the material.
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1.3 Types of Structural Study
Lecture 1 3Terminology, Scale
• Observation of natural structures, or deformed features in rock. This observa-
tion can take place at many different scales, from the submicroscopic to the
global. Observation usually involves the description of the geometry and
orientations of individual structures and their relations to other structures.
Also generally involves establishing of the timing relations of structures (i.e.
their order of formation, or the time it took for one feature to form).
• Experimental -- an attempt to reproduce under controlled laboratory conditions
various features similar to those in naturally deformed rocks. The aim of
experimental work is to gain insight into the stress systems and processes that
produced the deformation. Two major drawbacks: (1) in the real earth, we
seldom know all of the possible factors effecting the deformation (P, T, t,
fluids, etc.); (2) More important, real earth processes occur at rates which are
far slower than one can possibly reproduce in the laboratory (Natural rates:
10-12 to 10–18 sec-1; in lab, the slowest rates: 10-6 - 10-8 sec-1)
• Theoretical -- application of various physical laws of mechanics and thermo-
dynamics, through analytical or numerical methods, to relatively simple struc-
tural models. The objective of this modeling is to duplicate, theoretically, the
geometries or strain distributions of various natural features. Main problem is
the complexity of natural systems.
1.4 Importance of Scale
1.4.1 Scale Terms
Structural geologists view the deformed earth at a variety of different scales. Thus a number of
general terms are used to refer to the different scales. All are vague in detail. Importantly, all depend on
the vantage point of the viewer:
• Global -- scale of the entire world. ~104-105 km (circumference = 4 x 104 km)
• Regional or Provincial -- poorly defined; generally corresponds to a physio-
graphic province (e.g. the Basin and Range) or a mountain belt 103-104 km (e.g.
the Appalachians).
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• Macroscopic or Map Scale -- Bigger than an area you can see standing in one
Lecture 1 4Terminology, Scale
place on the ground. 100-102 km (e.g. the scale of a 7.5' quadrangle map)
• Mesoscopic -- features observable in one place on the ground. An outcrop of
hand sample scale. 10-5-10-1 km (1 cm - 100s m) (e.g. scale of a hand sample)
• Microscopic -- visible with an optical microscope. 10-8-10-6 km
• Submicroscopic -- not resolvable with a microscope but with TEM, SEM etc.
< 10-8 km.
Two additional terms describe how pervasive a feature or structure is at the scale of observation:
• Penetrative -- characterizes the entire body of rock at the scale of observation
• Non-penetrative -- Does not characterize the entire body of rock
These terms are totally scale dependent. A cleavage can be penetrative at one scale (i.e. the rock
appears to be composed of nothing but cleavage planes), but non-penetrative at another (e.g. at a higher
magnification where one sees coherent rock between the cleavage planes):
The importance of scale applies not only to description, but also to our mechanical analysis of
structures. For example, it may not be appropriate to model a rock with fractures and irregularities at the
mesoscopic scale as an elastic plate, whereas it may be totally appropriate at a regional scale. There are
no firm rules about what scale is appropriate for which analysis.
1.4.2 Scale Invariance, Fractals
Many structures occur over a wide range of scales. Faults, for example, can be millimeters long
or they can be 1000s of kilometers long (and all scales in between). Likewise, folds can be seen in thin
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sections under the microscope or they can be observed at map scale, covering 100s of square kilometers.
penetrative
non-penetrative
Lecture 1 5Terminology, Scale
Geologists commonly put a recognizable feature such as a rock hammer or pencil in a photograph (“rock
hammer for scale”) because otherwise, the viewer might not know if s/he was looking at a 10 cm high
outcrop or a 2000 m high cliff. Geologic maps commonly show about the same density of faults, regardless
of whether the map has a scale of 1:5,000,000 or 1:5,000.
These are all examples of the scale invariance of certain structures. Commonly, there is a consistent
relationship between the size of something and the frequency with which it occurs or the size of the
measuring stick that you use to measure it with. The exponent in this relationship is called the fractal
dimension.
The term, fractal, was first proposed by B. Mandelbrot (1967). He posed a very simple question:
“How long is the coast of Britain?” Surprisingly, at first, there is no answer to this question; the coast of
Britain has an undefinable length. The length of the coast of Britain depends on the scale at which you measure
it. The longer the measuring stick, the shorter the length as illustrated by the picture below. On a globe
with a scale of 1:25,000,000, the shortest distance you can effectively measure (i.e. the measuring stick) is
10s of kilometers long. Therefore at that scale you cannot measure all of the little bays and promontories.
But with accurate topographic maps at a scale of 1:25,000, your measuring stick can be as small as a few
tens of meters and you can include much more detail than previously. Thus, your measurement of the
coast will be longer. You can easily imagine extending this concept down to the scale of a single grain of
sand, in which case your measured length would be immense!
Length of coastline as determined with:
ruler "a"
ruler "b"
ruler "c"
ruler "a"
ruler "b"ruler "c"
landocean
ocean
true coastline
"True" Geography Measurements with Successively Smaller Rulers
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Mandelbrot defined the fractal dimension, D, according to the following equation:
Lecture 1 6Terminology, Scale
L(G) ~ G1 - D
where G is the length of the measuring stick and L(G) is the length of the coastline that you get using that
measuring stick.
The plot below, from Mandelbrot’s original article, shows this scale dependence for a number of
different coasts in log-log form.
4.0
3.5
3.0
1.0 1.5 2.0 2.5 3.0 3.5
west coast of Britain (D = 1.25)
coast of South Africa (D = 1.02)land frontier of Germany, 1900 (D = 1.15)
coast of Australia (D = 1.13)
circle
land frontier of Portugal (D = 1.13)
log (length of "measuring stick" in km)10
log
(
tota
l len
gth
in k
ilom
eter
s)10
Fractals have a broad range of applications in structural geology and geophysics. The relation
between earthquake frequency and magnitude, m, is a log linear relation:
log N = -b m + a
where N is the number of earthquakes in a given time interval with a magnitude m or larger. Empirically,
the value of b (or “b-value”) is about 1, which means that, for every magnitude 8 earthquakes, there are
10 magnitude 7 earthquakes; for every magnitude 7 there are 10 magnitude 6; etc. The strain released
during an earthquake is directly related to the moment of the earthquake, and moment, M, and magnitude
are related by the following equation:
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Lecture 1 7Terminology, Scale
log M = c m + d
where c and d are constants. Thus, the relation between strain release and number is log-log or fractal:
log N =−b
clogM + a +
bd
c
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Lecture 2 8Vectors, Coordinate Systems
LECTURE 2 —COORDINATE SYSTEMS, ETC.
2.1 Introduction
As you will see in lab, structural geologists spend a lot of time describing the orientation and
direction of structural features. For example, we will see how to describe the strike and dip of bedding,
the orientation of a fold axis, or how one side of a fault block is displaced with respect to the other. As
you might guess, there are several different ways to do this:
• plane trigonometry.
• spherical trigonometry
• vector algebra
All three implicitly require a coordinate system. Plane trigonometry works very well for simple problems
but is more cumbersome, or more likely impossible, for more complex problems. Spherical trigonometry
is much more flexible and is the basis for a wonderful graphical device which all structural geologists
come to love, the stereonet. In lab, we will concentrate on both of these methods of solving structural
problems.
The third method, vector algebra, is less familiar to many geologist and is seldom taught in
introductory courses. But it is so useful, and mathematically simple, that I wanted to give you an
introduction to it. Before that we have to put the term, vector, in some physical context, and talk about
coordinate systems.
2.2 Three types of physical entities
Let’s say we measure a physical property of something: for example, the density of a rock.
Mathematically, what is the number that results? Just a single number. It doesn’t matter where the
sample is located or how it is oriented, it is still just a single number. Quantities like these are called
scalars.
Some physical entities are more complex because they do depend on their position in space or
their orientation with respect to some coordinate system. For example, it doesn’t make much sense to talk
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about displacement if your don’t know where something was originally and where it ended up after the
Lecture 2 9Vectors, Coordinate Systems
displacement. Quantities like these, where the direction is important, are called vectors.
Finally, there are much more complex entities, still, which also must be related to a coordinate
system. These are “fields” of vectors, or things which vary in all different directions. These are called
tensors.
Scalars
Vectors
Tensors
Examples
mass, volume, density, temperature
velocity, displacement, force, acceleration, poles to planes, azimuths
stress, strain, thermal conductivity, magnetic susceptibility
Most of the things we are interested in Structural Geology are vectors or tensors. And that means
that we have to be concerned with coordinate systems and how they work.
2.3 Coordinate Systems
Virtually everything we do in structural geology explicitly or implicitly involves a coordinate
system.
• When we plot data on a map each point has a latitude, longitude, and elevation.
Strike and dip of bedding are given in azimuth or quadrant with respect to
north, south, east, and west and with respect to the horizontal surface of the
Earth approximated by sea level.
• In the western United States, samples may be located with respect to township
and range.
• More informal coordinate systems are used as well, particularly in the field.
The location of an observation or a sample may be described as “1.2 km from
the northwest corner fence post and 3.5 km from the peak with an elevation of
6780 m at an elevation of 4890 m.”
A key aspect, but one which is commonly taken for granted, of all of these ways of reporting a
location is that they are interchangeable. The sample that comes from near the fence post and the peak
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could just as easily be described by its latitude, longitude, and elevation or by its township, range and
Lecture 2 1 0Vectors, Coordinate Systems
elevation. Just because I change the way of reporting my coordinates (i.e. change my coordinate system)
does not mean that the physical location of the point in space has changed.
2.3.1 Spherical versus Cartesian Coordinate Systems
Because the Earth is nearly spherical, it is most convenient for structural geologists to record their
observations in terms of spherical coordinates. Spherical coordinates are those which are referenced to
a sphere (i.e. the Earth) and are fixed by two angles and a distance, or radius (Fig. 2.1). In this case the
two angles are latitude, φ, and longitude, θ, and the radius is the distance, r, from the center of the Earth
(or in elevation which is a function of the distance from the center). The rotation axis is taken as one axis
(from which the angle φ or its complement is measured) with the other axis at the equator and arbitrarily
coinciding with the line of longitude which passes through Greenwich, England. The angle θ is measured
from this second axis.
We report the azimuth as a function of angle from north and the inclination as the angle between
a tangent to the surface and the feature of interest in a vertical plane. A geologist can make these
orientation measurements with nothing more than a simple compass and clinometer because the Earth’s
magnetic poles are close to its rotation axis and therefore close to one of the principal axes of our
spherical coordinate system.
Although a spherical coordinate system is the easiest to use for collecting data in the field, it is
not the simplest for accomplishing a variety of calculations that we need to perform. Far simpler, both
conceptually and computationally, are rectangular Cartesian coordinates. This coordinate system is
composed of three mutually perpendicular axes. Normally, one thinks of plotting a point by its distance
from the three axes of the Cartesian coordinate system. As we shall see below, a feature can equally well
be plotted by the angles that a vector, connecting it to the origin, makes with the axes. If we can assume
that the portion of the Earth we are studying is sufficiently small so that our horizontal reference surface
is essentially perpendicular to the radius of the Earth, then we can solve many different problems in
structural geology simply and easily by expressing them in terms of Cartesian, rather than spherical,
coordinates. Before we can do this however, there is an additional aspect of coordinate systems which we
must examine.
2.3.2 Right-handed and Left-handed Coordinate Systems
The way that the axes of coordinate systems are labelled is not arbitrary. In the case of the Earth, it
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matters whether we consider a point which is below sea level to be positive or negative. That’s crazy,
Lecture 2 1 1Vectors, Coordinate Systems
you say, everybody knows that elevations above sea level are positive! If that were the case, then why do
structural geologists commonly measure positive angles downward from the horizontal? Why is it that
mineralogists use an upper hemisphere stereographic projection whereas structural geologists use the
lower hemisphere? The point is that it does not matter which is chosen so long as one is clear and
consistent. There are some simple conventions in the labeling of coordinate axes which insure that
consistency.
Basically, coordinate systems can be of two types. Right-handed coordinates are those in which,
if you hold your hand with the thumb pointed from the origin in the positive direction of the first axis,
your fingers will curl from the positive direction of the second axis towards the positive direction of the
third axis (Fig. 2.2). A left-handed coordinate system would function the same except that the left hand
is used. To make the coordinate system in Fig. 2.2 left handed, simply reverse the positions of the X2 and
X3 axes. By convention, the preferred coordinate system is a right-handed one and that is the one we
shall use.
2.3.3 Cartesian Coordinate Systems in Geology
What Cartesian coordinate systems are appropriate to geology? Sticking with the right-handed
convention, there are two obvious choices, the primary difference being whether one regards up or down
as positive:
X1 = East
X2 = North
X3 = Up
X2 = East
X1 = North
X3 = Down
East, North, Up North, East,Down
Cartesian coordinates commonly used in geology and geophysics
In general, the north-east-down convention is more common in structural geology where positive
angles are measured downwards from the horizontal. In geophysics, the east-north-up convention is
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more customary. Note that these are not the only possible right-handed coordinate systems. For example,
Lecture 2 1 2Vectors, Coordinate Systems
west-south-up is also a perfectly good right-handed system although it, and all the other possible combi-
nations are seldom used.
2.4 Vectors
Vectors form the basis for virtually all structural calculations so it’s important to develop a very
clear, intuitive feel for them. Vectors are a physical quantity that have a magnitude and a direction; they
can be defined only with respect to a given coordinate system.
2.4.1 Vectors vs. Axes
At this point, we have to make a distinction between vectors, which are lines with a direction (i.e.
an arrow at one end of the line) and axes, which are lines with no directional significance. For example,
think about the lineation that is made by the intersection between cleavage and bedding. That line, or
axis, certainly has a specific orientation in space and is described with respect to a coordinate system, but
there is no difference between one end of the line and the other.1 The hinge — or axis — of a cylindrical
fold is another example of a line which has no directional significance. Some common geological examples
of vectors which cannot be treated as axes, are the slip on a fault (i.e. displacement of piercing points),
paleocurrent indicators (flute cast, etc.), and paleomagnetic poles.
2.4.2 Basic Properties of Vectors
Notation. Clearly, with two different types of quantities — scalars and vectors — around, we
need a shorthand way to distinguish between them in equations. Vectors are generally indicated by a
letter with a bar, or in these notes, in bold face print (which is sometimes known as symbolic or Gibbs
notation):
V = V = [V1, V2, V3] (eqn. 1)
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1[It should be noted that, when structural geologists use a lower hemisphere stereographic projectionexclusively we are automatically treating all lines as axes. To plot lines on the lower hemisphere, wearbitrarily assume that all lines point downwards. Generally this is not an issue, but consider theproblem of a series of complex rotations involving paleocurrent directions. At some point during thisprocess, the current direction may point into the air (i.e. the upper hemisphere). If we force that line topoint into the lower hemisphere, we have just reversed the direction in which the current flowed! Generallypoles to bedding are treated as axes as, for example, when we make a π-diagram. This, however, is notstrictly correct. There are really two bedding poles, the vector which points in the direction of stratalyounging and the vector which points towards older rocks.]
Lecture 2 1 3Vectors, Coordinate Systems
Vectors in three dimensional space have three components, indicated above as V1, V2, and V3. These
components are scalars and, in a Cartesian Coordinate system, they give the magnitude of the vector in
the direction of, or projected onto each of the three axes (b). Because it is tedious to write out the three
components all the time a shorthand notation, known as indicial notation, is commonly used:
Vi , where [i = 1, 2, 3]
X1
V
β
γ
X2
X3
V1
V3
V2
V
|V|V2
V1X1
X2
αβ
(a) (b)
α
(V12 + V2
2 )1
2
|V|
(V12 + V22 )
12
Components of a vector in Cartesian coordinates (a) in two dimensions and (b) inthree dimensions
Magnitude of a Vector . The magnitude of a vector is, graphically, just the length of the arrow. It is
a scalar quantity. In two dimensions it is quite easy to see that the magnitude of vector V can be
calculated from the Pythagorean Theorem (the square of the hypotenuse is equal to the sum of the
squares of the other two sides). This is easily generalized to three dimensions, yielding the general
equation for the magnitude of a vector:
V = |V| = (V12 + V2
2 + V32) 1/2 (eqn. 2)
Unit Vector. A unit vector is just a vector with a magnitude of one and is indicated by a
triangular hat: V . Any vector can be converted into a unit vector parallel to itself by dividing the vector
(and its components) by its own magnitude.
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ˆ V =V1
V,
V2
V,
V3
V
(eqn. 3)
Lecture 2 1 4Vectors, Coordinate Systems
Direction Cosines. The cosine of the angle that a vector makes with a particular axis is just equal to
the component of the vector along that axis divided by the magnitude of the vector. Thus we get
cosα =V1
V, cosβ =
V2
V, cosγ =
V3
V . (eqn. 4)
Substituting equation eqn. 4 into equation eqn. 3 we see that a unit vector can be expressed in terms of the
cosines of the angles that it makes with the axes. These cosines are known as direction cosines:
ˆ V = cosα , cosβ , cosγ[ ] . (eqn. 5)
Direction Cosines and Structural Geology. The concept of a unit vector is particularly important in
structural geology where we so often deal with orientations, but not sizes, of planes and lines. Any
orientation can be expressed as a unit vector, whose components are the direction cosines. For example,
in a north-east-down coordinate system, a line which has a 30° plunge due east (090°, 30°) would have the
following components:
cos α = cos 90° = 0.0 [α is the angle with respect to north]
cos β = cos 30° = 0.866 [β is the angle with respect to east]
cos γ = (cos 90° - 30°) = 0.5 [γ is the angle with respect to down]
or simply [ cos α, cos β, cos γ ] = [ 0.0 , 0.866 , 0.5 ] .
For the third direction cosine, recall that the angle is measured with respect to the vertical, whereas
plunge is given with respect to the horizontal.
2.4.3 Geologic Features as Vectors
Virtually all structural features can be reduced to two simple geometric objects: lines and planes.
Lines can be treated as vectors. Likewise, because there is only one line which is perpendicular to a
plane, planes — or more strictly, poles to planes — can also be treated as vectors. The question now is,
how do we convert from orientations measured in spherical coordinates to Cartesian coordinates?
Data Formats in Spherical Coordinates. Before that question can be answered, however, we have to
examine for a minute how orientations are generally specified in spherical coordinates (Fig. 2.6). In
North America, planes are commonly recorded according to their strike and dip. But, the strike can
correspond to either of two directions 180° apart, and dip direction must be fixed by specifying a geographic
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quadrant. This can lead to ambiguity which, if we are trying to be quantitative, is dangerous. There are
Lecture 2 1 5Vectors, Coordinate Systems
two methods of recording the orientation of a plane that avoids this ambiguity. First, one can record the
strike azimuth such that the dip direction is always clockwise from it, a convention known as the right-hand
rule. This tends to be the convention of choice in North America because it is easy to determine using a
Brunton compass. A second method is to record the dip and dip direction, which is more common in
Europe where compasses make this measurement directly. Of course, the pole also uniquely defines the
plane, but it cannot be measured directly off of either type of compass.
N30°
40°
strike
dip direction
dip
Quadrant:
Azimuth & dip quadrant:
Azimuth, right-hand rule:
Dip azimuth & dip:
Pole trend & plunge:
N 30 W, 40 SW
330, 40 SW
150, 40
240, 40
060, 50
Alternative ways of recording the strike and dip of a plane. The methods whichare not subject to potential ambiguity are shown in bold face type.
Lines are generally recorded in one of two ways. Those associated with planes are commonly
recorded by their orientation with respect to the strike of the plane, that is, their pitch or rake. Although
this way is commonly the most convenient in the field, it can lead to considerable ambiguity if one is not
careful because of the ambiguity in strike, mentioned above, and the fact that pitch can be either of two
complementary angles. The second method — recording the trend and plunge directly — is completely
unambiguous as long as the lower hemisphere is always treated as positive. Vectors which point into the
upper hemisphere (e.g. paleomagnetic poles) can simply be given a negative plunge.
Conversion from Spherical to Cartesian Coordinates. The relations between spherical and Cartesian
coordinates are shown in Fig. 2.7. Notice that the three angles α , β, and γ are measured along great circles
between the point (which represents the vector) and the positive direction of the axis of the Cartesian
coordinate system. Clearly, the angle γ is just equal to 90° minus the plunge of the line. Therefore ,
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cos γ = cos (90 - plunge) = sin (plunge) (eqn. 6a)
Lecture 2 1 6Vectors, Coordinate Systems
N
E
D
cos γ
cos β
cos αtrend
90 - trend
plunge
90 - plunge
Perspective diagram showing therelations between the trend andplunge angles and the directioncosines of the vector in theCartesian coordinate system. Grayplane is the vertical plane in whichthe plunge is measured.
cos (plunge)
unit vector
The relations between the trend and plunge and the other two angles are slightly more difficult to
calculate. Recall that we are dealing just with orientations and therefor the vector of interest (the bold
arrowhead in Fig. 2.8) is a unit vector. Therefore, from simple trigonometry the horizontal line which
corresponds to the trend azimuth is equal to the cosine of the plunge. From here, it is just a matter of
solving for the horizontal triangles in Fig. 2.8:
cos α = cos (trend) cos (plunge), (eqn. 6b)
cos β = cos (90 - trend) cos (plunge) = sin (trend) cos (plunge). (eqn. 6c)
These relations, along with those for poles to planes, are summarized in Table 1:
North
East
Down
Table 1: Conversion from Spherical to Cartesian Coordinates
Direction Cosine Lines Poles to Planes(right-hand rule )
Axis
cos α cos(trend)*cos(plunge) sin(strike)*sin(dip)
cos β sin(trend)*cos(plunge) –cos(strike)*sin(dip)
cos γ sin(plunge) cos(dip)
The signs of the direction cosines vary with the quadrant. Although it is not easy to see an
orientation expressed in direction cosines and immediately have an intuitive feel how it is oriented in
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space, one can quickly tell what quadrant the line dips in by the signs of the components of the vector.
Lecture 2 1 7Vectors, Coordinate Systems
For example, the vector, [–0.4619, –0.7112, 0.5299], represents a line which plunges into the southwest
quadrant (237°, 32°) because both cos α and cos β are negative.
Understanding how the signs work is very important for another reason. Because it is difficult to
get an intuitive feel for orientations in direction cosine form, after we do our calculations we will want to
convert from Cartesian back to spherical coordinates. This can be tricky because, for each direction cosine,
there will be two possible angles (due to the azimuthal range of 0 - 360°). For example, if cos α = –0.5736,
then α = 125° or α = 235°. In order to tell which of the two is correct, one must look at the value of cos β;
if it is negative then α = 235°, if positive then α = 125°. When you use a calculator or a computer to
calculate the inverse cosine, it will only give you one of the two possible angles (generally the smaller of
the two). You must determine what the other one is knowing the cyclicity of the sine and cosine
functions.
-1
-0.5
0
0.5
1
0 30 60 90 120 150 180 210 240 270 300 330 360
Sin
e or
Cos
ine
Angle (degrees)
cosinesine
Graph of sine and cosine functions for 0 - 360°. The plot emphasizes that forevery positive (or negative) cosine, there are two possible angles.
2.4.4 Simple Vector Operations
Scalar Multiplication. To multiply a scalar times a vector, just multiply each component of the
vector times the scalar.
xV = [ xV1, xV2, xV3 ] (eqn. 7)
The most obvious application of scalar multiplication in structural geology is when you want to reverse
the direction of the vector. For example, to change the vector from upper hemisphere to lower (or vice
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versa) just multiply the vector (i.e. its components) by –1. The resulting vector will be parallel to the
Lecture 2 1 8Vectors, Coordinate Systems
original and will have the same length, but will point in the opposite direction.
Vector Addition. To add two vectors together, you sum their components:
U + V = V + U = [ V1 + U1 , V2 + U2 , V3 + U3 ] . (eqn. 8)
Graphically, vector addition obeys the parallelogram law whereby the resulting vector can be constructed
by placing the two vectors to be added end-to-end:
U
VU + V U – V
U
–V
(a) (b)
(a) Vector addition and (b) subtraction using the parallelogram law.
Notice that the order in which you add the two vectors together makes no difference. Vector
subtraction is the same as adding the negative of one vector to the positive of the other.
2.4.5 Dot Product and Cross Product
Vector algebra is remarkably simple, in part by virtue of the ease with which one can visualize
various operations. There are two operations which are unique to vectors and which are of great importance
in structural geology. If one understands these two, one has mastered the concept of vectors. They are
the dot product and the cross product.
Dot Product. The dot product is also called the “scalar product” because this operation produces
a scalar quantity. When we calculate the dot product of two vectors the result is the magnitude of the
first vector times the magnitude of the second vector times the cosine of the angle between the two:
U • V = V • U = U V cos θ = U1V1 + U2V2 + U3V3 , (eqn. 9)
draft date: 1/20/99
The physical meaning of the dot product is the length of V times the length of U as projected onto V (that
Lecture 2 1 9Vectors, Coordinate Systems
is, the length of U in the direction of V). Note that the dot product is zero when U and V are perpendicular
(because in that case the length of U projected onto V is zero). The dot product of a vector with itself is
just equal to the length of the vector:
V • V = V = |V|. (eqn. 10)
Equation (eqn. 9) can be rearranged to solve for the angle between two vectors:
θ = cos−1 U • VUV
. (eqn. 11)
This last equation is particularly useful in structural geology. As stated previously, all orientations are
treated as unit vectors. Thus when we want to find the angle between any two lines, the product of the
two magnitudes, UV, in equations (eqn. 9) and (eqn. 11) is equal to one. Upon rearranging equations
(eqn. 11), this provides a simple and extremely useful equation for calculating the angle between two
lines:
θ = cos-1 ( cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 ). (eqn. 12)
Cross Product. The result of the cross product of two vectors is another vector. For that reason,
you will often see the cross product called the “vector product”. The cross product is conceptually a
little more difficult than the dot product, but is equally useful in structural geology. It’s primary use is
when you want to calculate the orientation of a vector that is perpendicular to two other vectors. The
resulting perpendicular vector is parallel to the unit vector, ˆ l , and has a magnitude equal to the product
of the magnitude of each vector times the sine of the angle between them. The new vector obeys a
right-hand rule with respect to the other two.
V × U = V ∧ U = ( V U sin θ ) ˆ l (eqn. 13)
and
V × U = [ V2U3 - V3U2 , V3U1 - V1U3 , V1U2 - V2U1] (eqn. 14)
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The cross product is best illustrated with a diagram, which relates to the above equations:
Lecture 2 2 0Vectors, Coordinate Systems
lU
V
V × U
θ(unit
vector)
U × V
Diagram illustrating the meaning of the cross-product. The hand indicates theright-hand rule convention; for V × U, the finger curl from V towards U and thethumb points in the direction of the resulting vector, which is parallel to the unit
vector ˆ l . Note that V × U = - U × V
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Lecture 3 2 1Seismic Reflection Data
LECTURE 3 — DESCRIPTIVE GEOMETRY: SEISMIC REFLECTION
3.1 Echo Sounding
Geology presents us with a basic problem. Because rocks are opaque, it is very difficult to see
through them and thus it is difficult to know what is the three-dimensional geometry of structures.
This problem can be overcome by using a remote sensing technique known as seismic reflection.
This is a geophysical method which is exactly analogous to echo sounding and it is widely used in the
petroleum industry. Also several major advances in tectonics have come from recent application of the
seismic reflection in academic studies. I’m not going to teach you geophysics, but every modern structural
geologist needs to know something about seismic reflection profiling.
Lets examine the simple case of making an echo first to see what the important parameters are.
rock wall
ρ air v air v rock ρ rock
a very smallamount of soundcontinues into therock
most sound is reflectedback to the listener
Why do you get a reflection or an echo? You get an echo because the densities and sound velocities of air
and rock are very different. If they had the same density and velocity, there would be no echo. More
specifically
velocity= V =E
ρ(E = Young’s modulus)
and
reflection coefficient = R=amplitude of reflected wave
amplitude of incident wave=
ρ2V2 − ρ1V1
ρ2V2 + ρ1V1
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Lecture 3 2 2Seismic Reflection Data
In seismic reflection profiling, what do you actually measure?
groundsurface
1st subsurface layer
2nd subsurface layer
time sound was made
time to go down to the 1st layer and return
time to go down to the 2nd layer and return
depth
time
The above illustration highlights three important things about seismic reflection profiling:
1. Measure time, not depth,
2. The time recorded is round trip or two-way time, and
3. To get the depth, we must know the velocity of the rocks.
Velocities of rocks in the crust range between about 2.5 km/s and 6.8 km/s. Most sedimentary
rocks have velocities of less than 6 km/s. These are velocities of P-waves or compressional waves, not
shear waves. Most seismic reflection surveys measure P- not S-waves.
Seismic reflection profiles resemble geologic cross-sections, but they are not. They are distorted
because rocks have different velocities. The following diagram illustrates this point.
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3 km
6 km
6 km/s
3 km/s
time
dept
h
1 s
2 s
6 km horizontal reflector
3 s
Lecture 3 2 3Seismic Reflection Data
3.2 Common Depth Point (CDP) Method
In the real earth, the reflectivity at most interfaces is very small, R ≈ 0.01, and the reflected energy
is proportional to R2. Thus, at most interfaces ~99.99% of the energy is transmitted and 0.01% is reflected.
This means that your recording system has to be able to detect very faint signals coming back from the
subsurface.
source receivers (geophones)
one ray through point
two rays through point
three rays through point
The black dot, and each point on the reflector with a ray going through it, is a common depth point.
Notice that there are twice as many CDPs as there are stations on the ground (where the geophones are).
That is, there is a CDP directly underneath each station and a CDP half way between each station (hence
the name “common midpoint”)
Also, in a complete survey, the number of traces through each midpoint will be equal to one half
the total number of active stations at any one time. [This does not include the ends of the lines where
there are fewer traces, and it also assumes that the source moves up only one station at a time.] The
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number of active stations is determined by the number of channels in the recording system. Most
Lecture 3 2 4Seismic Reflection Data
modern seismic reflection surveys use at least 96 (and sometimes -- but not often -- as many as 1024
channels), so that the number of traces through any one CDP will be 48.
This number is the data redundancy, of the fold of the data. For example, 24 fold or 2400%
means that each depth point was sampled 24 times. Sampling fold in a seismic line is the same thing as
the “over-sampling” which you see advertised in compact disk players.
Before the seismic reflection profile can be displayed, there are several intermediate steps. First,
all of the traced through the same CDP have to be gathered together. Then you have to determine a set of
velocities, known as stacking or NMO velocities, which will correct for the fact that each ray through a
CDP has a path of a different length. These velocities should line up all of the individual “blips”
corresponding to a single reflector on adjacent traces
far offset
near offset
sour
ce
distance from source, x
time
[in practice, there is no geophone at the source because it is too noisey]
CDP Gather CDP Gather with NMO
∆ t = normal moveout (NMO)
the NMO velocity is whatever velocitythat lines up all the traces in a CDP gather. It is not the same as the rock velocity
t o
t x
The relation between the horizontal offset, x, and the time at which a reflector appears at that
offset, tx, is:
tx2 = t0
2 +x2
Vstacking2
or
∆t t t tx
Vtx
stacking
= − = +
−0 0
22
2
1
2
0
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If you have a very simple situation in which all of your reflections are flat and there are only
Lecture 3 2 5Seismic Reflection Data
vertical velocity variations (i.e. velocities do not change laterally), then you can calculate the rock interval
velocities from the stacking velocities using the Dix equation:
Vi 1 2=
Vst2
2 t2 −Vst1
2 t1
t2 − t1
1
2
where Vi12 is the interval velocity of the layer between reflections 1 and 2, Vst1 is the stacking velocity of
reflection 1, t1 is the two way time of reflection 1, etc. The interval velocity is important because, to
convert from two-way time to depth, we must know the interval, not the stacking, velocity.
Once the correction for normal moveout is made, we can add all of the traces together, or stack
them. This is what produces the familiar seismic reflection profiles.
Processing seismic data like this is simple enough, but there are huge amounts of data involved.
For example a typical COCORP profile is 20 s long, has a 4 ms digital sampling rate (the time interval
between numbers recorded), and is 48 fold. In a hundred station long line, then, we have
200 CDPs( ) 48 sums( ) 20 s( )0.004 s
data sample
= 48×106 data samples.
For this reason, the seismic reflection processing industry is one of the largest users of computers in the
world!
3.3 Migrat ion
The effect of this type of processing is to make it look like the source and receiver coincide (e.g.
having 48 vertical traces directly beneath the station). Thus, all reflections are plotted as if they were
vertically beneath the surface. This assumption is fine for flat layers, but produces an additional distortion
for dipping layers, as illustrated below.
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Lecture 3 2 6Seismic Reflection Data
actual raypaths
actual position of reflector in space
position of reflection assuming reflecting point is vertically beneath the station
surface
Note that the affect of this distortion is that all dipping reflections are displaced down-dip and have a
shallower dip than the reflector that produced them. The magnitude of this distortion is a function of the
dip of the reflector and the velocity of the rocks.
The process of migration corrects this distortion, but it depends on well-determined velocities
and on the assumption that all reflections are in the plane of the section (see “sideswipe”, below). A
migrated section can commonly be identified because it has broad “migration smiles” at the bottom and
edges. Smiles within the main body of the section probably mean that it has been “over-migrated.”
3.4 Resolution of Seismic Reflection Data
The ability of a seismic reflection survey to resolve features in both horizontal and vertical
directions is a function of wavelength:
λ = velocity / frequency.
Wavelength increases with depth in the Earth because velocity increases and frequency decreases. Thus,
seismic reflection surveys lose resolution with increasing depth in the Earth.
3.4.1 Vertical Resolution
Generally, the smallest (thinnest) resolvable features are 1/4 to 1/8 the dominant wavelength:
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Lecture 3 2 7Seismic Reflection Data
layered sequence in the Earth
At low frequencies (long wavelengths) these three beds will be "smeared out" into one long wave form
At higher frequencies (shorter wavelengths) the three beds will be distinguishable on the seismic section
3.4.2 Horizontal Resolution
The horizontal resolution of seismic reflection data depends on the Fresnel Zone, a concept which
should be familiar to those who have taken optics. The minimum resolvable horizontal dimensions are
equal to the first Fresnel zone, which is defined below.
λ4 λ
4
first Fresnel Zonefirst Fresnel Zone
higher frequency lower frequency
Because frequency decreases with depth in the crust, seismic reflection profiles will have greater horizontal
resolution at shallower levels.
At 1.5 km depth with typical frequencies, the first Fresnel Zone is ~300 m. At 30 km depth, it is
about 3 km in width.
Consider a discontinuous sandstone body. The segments which are longer than the first Fresnel
Zone will appear as reflections, whereas those which are shorter will act like point sources. Point sources
and breaks in the sandstone will generate diffractions, which have hyperbolic curvature:
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Lecture 3 2 8Seismic Reflection Data
Fresnel zonereflections diffractions
3.5 Diffractions
Diffractions may look superficially like an anticline but they are not. They are extremely useful,
especially because seismic reflection techniques are biased toward gently dipping layers and do not image directly
steeply dipping or vertical features. Diffractions help you to identify such features. For example, a vertical
dike would not show up directly as a reflection but you could determine its presence by correctly
identifying and interpreting the diffractions from it:
dike diffraction from dike
geologic section seismic section
raypaths
High-angle faults are seldom imaged directly on seismic reflection profiles, but they, too, can be located
by finding the diffractions from the truncated beds:
geologic section seismic section
The shape and curvature of a diffraction is dependent on the velocity. At faster velocities, diffractions
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become broader and more open. Thus at great depths in the crust, diffractions may be very hard to
Lecture 3 2 9Seismic Reflection Data
distinguish from gently dipping reflections.
3.6 Artifacts
The seismic reflection technique produces a number of artifacts -- misleading features which are
easily misinterpreted as real geology -- which can fool a novice interpreted. A few of the more common
“pitfalls” are briefly listed below.
3.6.1 Velocity Pullup/pulldown
We have already talked about this artifact when we discussed the distortion due to the fact that
seismic profiles are plotted with the vertical dimension in time, not depth. When you have laterally
varying velocities, deep horizontal reflectors will be pulled up where they are overlain locally by a high
velocity body and will be pushed down by a low velocity body (as in the example on page 2).
3.6.2 Multiples
Where there are very reflective interfaces, you can get multiple reflections, or multiples, from
those interfaces. The effective reflectivity of multiples is the product of the reflectivity of each reflecting
interface. For simple multiples (see below) then,
Rmultiple = R2primary.
If the primary reflector has a reflection coefficient of 0.01 then the first multiple will have an effective
reflection coefficient of 0.0001. In other words, multiples are generally only a problem for highly reflective
interfaces, such as the water bottom in the case of a marine survey or particularly prominent reflectors in
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sedimentary basins (e.g. the sediment-basement interface).
Lecture 3 3 0Seismic Reflection Data
Multiple from a flat layer:
geologic section seismic section
simple raypath
multiple raypath
multiple at twice the travel time of the primary
primary reflection
dept
h
time
Multiple from a dipping layer (note that the multiple has twice the dip of the primary):
geologic section seismic section
simple raypath multiple
raypath
multiple at twice the
travel time of the primary
primary reflection
dept
h
time
Pegleg multiples:
geologic section seismic section
simple raypaths
pegleg raypath
pegleg multiple
primary reflections
dept
h
time
3.6.3 Sideswipe
In seismic reflection profiling, we assume that all the energy that returns to the geophones comes
from within the vertical plane directly beneath the line of the profile. Geology is inherently three-dimensional
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so this need not be true. Even though geophones record only vertical motions, a strong reflecting
Lecture 3 3 1Seismic Reflection Data
interface which is out-of-the-plane can produce a reflection on a profile, as in the case illustrated below.
seismic reflection survey along this line
in plane ray path from sandstone
out of plane ray path ("sideswipe") from dike
Reflections from out of the plane is called sideswipe. Such reflections will cross other reflections and will
not migrate out of the way. (Furthermore they will migrate incorrectly because in migration, we assume
that there has been no sideswipe!) The main way of detecting sideswipe is by running a sufficient
number of cross-lines and tying reflections from line to line. Sideswipe is particularly severe where
seismic lines run parallel to the structural grain.
3.6.4 Buried Focus
geologic section seismic section
dept
h
time
a
b
c a
b
c
d
e
f
f
e
d
Tight synclines at depth can act like concave mirrors to produce an inverted image quite unlike the actual
structure. Although the geological structure is a syncline, on the seismic profile it looks like an anticline.
Many an unhappy petroleum geologist has drilled a buried focus hoping to find an anticlinal trap! The
likelihood of observing a buried focus increases with depth because more and more open structures will
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produce the focus. A good migration will correct for buried focus.
Lecture 3 3 2Seismic Reflection Data
3.6.5 Others
• reflected refractions
• reflected surface waves
• spatial aliasing
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Lecture 4 3 3Introduction to Deformation
LECTURE 4 — INTRODUCTION TO DEFORMATION
4.1 Introduction
In this part of the course, we will first lay out the mechanical background of structural geology
before going on to explain the structures, themselves. As stated in the first lecture, what we, as geologists,
see in the field are deformed rocks. We do not see the forces acting on the rocks today, and we certainly
do not see the forces which produced the deformation in which we are interested. Thus, deformation
would seem to be an obvious starting point in our exploration of structural geology.
There is a natural hierarchy to understanding how the Earth works from a structural view point:
• geometry
• kinematics
• mechanics (“dynamics”)
We have briefly addressed some topics related to geometry and how we describe it; the lab part of this
course deals almost exclusively with geometric methods.
4.2 Kinematics
“Kinematic analysis” means reconstructing the movements and distortions that occur during
rock deformation. Deformation is the process by which the particles in the rock rearrange themselves
from some initial position to the final position that we see today. The components of deformation are:
Rigid body deformation
Translation
Rotation
Non-rigid Body deformation (STRAIN)
Distortion
Dilation
4.2.1 Rigid Body Deformations
Translation = movement of a body without rotation or distortion:
draft date: 20 Jan 1999
Lecture 4 3 4Introduction to Deformation
particle paths
in translation, all of the particle paths are straight, constant length, and parallel to each other.
Rotation = rotation of the body about a common axis. In rotation, the particle paths are curved
and concentric.
curved particle paths
The sense of rotation depends on the position of the viewer. The rotation axis is defined as a vector
pointing in the direction that the viewer is looking:
Right-handedclockwise
dextral
Left-handedcounter-clockwise
sinestral
Translation and rotation commonly occur at the same time, but mathematically we can treat them completely
separately
4.2.2 Strain (Non-rigid Body Deformation)
Four very important terms:
draft date: 20 Jan 1999
Continuous -- strain properties vary smoothly throughout the body with no abrupt changes.
Lecture 4 3 5Introduction to Deformation
Discontinuous -- abrupt changes at surfaces, or breaks in the rock
fold is continuous fault is discontinuous
Homogeneous -- the properties of strain are identical throughout the material. Each particle of
material is distorted in the same way. There is a simple test if the deformation is homogeneous:
1. Straight lines remain straight
2. Parallel lines remain parallel
Heterogeneous --the type and amount of strain vary throughout the material, so that one part is
more deformed than another part.
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Lecture 4 3 6Introduction to Deformation
This diagram does not fit the above test so it is heterogeneous. You can see that a fold would be a
heterogeneous deformation.
4.2.3 Continuum Mechanics
Mathematically, we really only have the tools to deal with continuous deformation. Thus, the
study of strain is a branch of continuum mechanics. This fancy term just means “the mechanics of
materials with smoothly varying properties.” Such materials are called “continua.”
Right away, you can see a paradox: Geological materials are full of discontinuous features:
faults, cracks, bedding surfaces, etc. So, why use continuum mechanics?
1. The mathematics of discontinuous deformation is far more difficult.
2. At the appropriate scale of observation, continuum mechanics is an adequate approximation.
We also analyze homogeneous strain because it is easier to deal with. To get around the problem
of heterogeneous deformation, we apply the concept of structural domains. These are regions of more-or-less
homogeneous deformation within rocks which, at a broader scale, are heterogeneous. Take the example
of a fold:
The approximations that we make in order to analyze rocks as homogeneous and continuous again
depend on the scale of observation and the vantage point of the viewer.
Let’s take a more complex, but common example of a thrust-and-fold belt:
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Lecture 4 3 7Introduction to Deformation
4.2.4 Four Aspects of a Deforming Rock System:
1. Position today 2. Displacement
initial(beginning)
final(present)
Position today is easy to get. It’s just the latitude and longitude, or whatever convenient measure you
want to use (e.g. “25 km SW of Mt. Marcy” etc.).
The displacement is harder to get because we need to know both the initial and the final positions
of the particle. The line which connects the initial and final positions is the displacement vector, or what
draft date: 20 Jan 1999
we called earlier, the particle path.
Lecture 4 3 8Introduction to Deformation
3. Dated Path3. Path
initial(beginning)
final(present)
35 Ma31
2613
11
10
8.5
60 Ma
Ideally, of course, we would like to be able to determine the dated path in all cases, but this is usually just
not possible because we can’t often get that kind of information out of the earth. There are some cases,
though:
Hawaii(0 Ma)
Midway(40 Ma)
80 Ma
EmperorSeamounts
Hawaii Ridge
Pacific Ocean
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Lecture 4 3 9Strain, the basics
4 .3 Measurement of Strain
There are three types of things we can measure:
1. Changes in the lengths of lines,
2. Changes in angles
3. Changes in volume
In all cases, we are comparing a final state with an initial state. What happens between those two
states is not accounted for (i.e. the displacement path, #3 above, is not accounted for).
l i l f
4.3.1 Change in Line Length:
Extension:∆ l = ( li – lf )
li
lf
we define extension (elongation) e≡∆l
li=
l f − li( )li
=l f
li−1 (4.1)
shortening is negative
Stretch: S≡l f
li=1+ e (4.2)
Quadratic elongation: λ = S2 = 1+ e( )2(4.3)
if λ = 1 then no change
if λ < 1 then shortening
draft date: 20 Jan 1999
if λ > 1 then extension
Lecture 4 4 0Strain, the basics
λ ≥ 0 because it is a function of S2. It will only be 0 if volume change reduces lf to zero.
4.3.2 Changes in Angles:
ψ
α
y
x
There are two ways to look at this deformation:
1. Measure the change in angle between two originally perpendicular lines:
change in angle = 90 - α = ψ ≡ angular shear
2. Look at the displacement, x, of a particle at any distance, y, from the origin (a
particle which does not move):
x
y= γ ≡ shear strain (4.4)
The relationship between these two measures is a simple trig function:
γ = tan ψ (4.5)
γ and ψ are very useful geologically because there are numerous features which we know were originally
perpendicular (e.g. worm tubes, bilaterally symmetric fossils, etc.):
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Lecture 4 4 1Strain, the basics
4.3.3 Changes in Volume (Dilation):
Dilation = ∆ ≡−( )V V
Vf i
i
(4.6)
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Lecture 5 4 2The Strain Ellipsoid
LECTURE 5 — STRAIN II: THE STRAIN ELLIPSOID
5 .1 Motivation for General 3-D Strain Relations
Last class, we considered how to measure the strain of individual lines and angles that had been
deformed. Consider a block with a bunch of randomly oriented lines:
Point out how each line and angle change and why.
Well, we now have equations to describe what happens to each individual line and angle, but
how do we describe how the body as a whole changes?
We could mark the body with lines of all different orientations and measure each one -- not very
practical in geology. There is, however, a simple geometric object which describes lines of all different
orientations but with equal length, a circle:
Any circle that is subjected to homogeneous strain turns into an ellipse. In three dimensions, a sphere turns
draft date: 20 Jan, 1999
into an ellipsoid. You’ll have to take this on faith right now but we’ll show it to be true later on.
Lecture 5 4 3The Strain Ellipsoid
5 .2 Equations for Finite Strain
Coming back to our circle and family of lines concept, let’s derive some equations that describe
how any line in the body changes length and orientation.
3
1
3
1
1
1
unit radius
[sometimes you'll see the 1 and 3 axes referred to as the "X" and "Z" axes, respectively]
l =f 3S =3 λ 3 l =f 1
S =1 λ 1
l fl i
S = λ = = lf
The general equation for a circle is: x2 + z2 = 1,
and for an ellipse:x
a
z
b
2
2
2
2 1+ = (5.1)
where a & b are the major and minor axes.
So, the equation of the strain ellipse is:
x z2
1
2
3
1λ λ
+ = (5.2)
5.3 Extension of a Line
Now, let’s determine the strain of any line in the deformed state:
draft date: 20 Jan, 1999
Lecture 5 4 4The Strain Ellipsoid
3
1
3
1
1
(x, z)
θ
ψ
θ '
(x', z')
λ 3
λ 1
λ
From the above, you can see that:
′ = ′z λ θsin and ′ = ′x λ θcos (5.3)
Substituting into the strain ellipse equation (5.2), we get
λ θλ
λ θλ
sin cos2
3
2
1
1′ + ′ = . (5.4)
Dividing both sides by λ, yields:
sin cos2
3
2
1
1′ + ′ =θλ
θλ λ
. (5.5)
We can manipulate this equation to get a more usable form by using some standard trigonometric
double angle formulas:
cos 2α = cos2 α - sin2 α = 2cos2 α - 1 = 1 - 2sin2 α . (5.6)
Cranking through the substitutions, and rearranging:
λ λ λ λ θλ λ λ
3 1 3 1
1 3
2
21+ + −( ) ′
=cos
. (5.7)
draft date: 20 Jan, 1999
If we let
Lecture 5 4 5The Strain Ellipsoid
λ ' = 1λ
, λ1' = 1
λ1 , and λ3
' = 1
λ3 ,
then
′ + ′( ) −′ − ′( ) ′ = ′
λ λ λ λθ λ3 1 3 1
2 22cos . (5.8)
5.4 Shear Strain
To get the shear strain, you need to know the equation for the tangent to an ellipse:
x z′ + ′ =x z
1 3λ λ1 . (5.9)
Substituting equations 5.3 (page 44) into 5.9:
x zλ θλ
λ θλ
cos sin′ + ′ =1 3
1 , (5.10)
we can solve for the intercepts of the tangent:
(x', y')λ 3
λ sin θ'
θ '
ψψ + θ'
90 − θ'
90 − ψ
λ 1
λ cos θ'From equation 5.10 and settingfirst x = 0 and then z = 0, andsolving for the other variable
From the trigonometry of the above triangle (from here, it can be solved in a lot of different ways):
tantan tan
tan tantanψ θ ψ θ
ψ θλλ
θ+ ′( ) = + ′− ′
= ′1
1
3
.
draft date: 20 Jan, 1999
Lecture 5 4 6The Strain Ellipsoid
Recall that: tan ψ = γ .
Lots of substitutions later:
γλ λ θ θ
λ θ λ θ=
−( ) ′ ′′ + ′
1 3
32
12
sin cos
cos sin .
The denominator is just λ1λ3
λ, which you get by multiplying eqn. 5.4 by λ1λ3 and dividing by λ.
Eventually, you get
γλ λ λ
θ= −
′12
1 12
3 1
sin .
and with the same reciprocals as we used before (top of page 45):
′ = =′ − ′( ) ′γ γ
λλ λ
θ3 1
22sin (5.11)
draft date: 20 Jan, 1999
Next time, we’ll see what all this effort is useful for…
Lecture 5 4 7Mohrs Circle for Finite Strain
LECTURE 6 — STRAIN III: MOHR ON THE STRAIN ELLIPSOID
6 .1 Introduction
Last time, we derived the fundamental equations for the strain ellipse:
′ =′ + ′( ) −
′ − ′( ) ′λλ λ λ λ
θ3 1 3 1
2 22cos (6.1)
and
′ = =′ − ′( ) ′γ γ
λλ λ
θ3 1
22sin (6.2)
These equations are of the same form as the parametric equations for a circle:
x = c - r cos α
y = r sin a ,
where the center of the circle is located at (c, 0) on the X-axis and the circle has a radius of “r”. Thus, the
above equations define a circle with a center at
c, , 02
03 1( ) = ′ + ′
λ λ
and radius
r = ′ − ′
λ λ3 1
2 .
These equations define the Mohr’s Circle for finite strain.
6.2 Mohr’s Circle For Finite Strain
The Mohr’s Circle is a graphical construction devised by a German engineer, Otto Mohr, around
the turn of the century. It actually is a graphical solution to a two dimensional tensor transformation,
draft date: 20 Jan, 1999
which we mentioned last time, and can be applied to any symmetric tensor. We will see the construction
Lecture 6 4 8Mohrs Circle for Finite Strain
again when we talk about stress. But, for finite strain, it looks like:
λ '
γ '
λ '1 λ '3
2θ'
λ '1
λ '3
2
−
λ '1λ '32
+
ψ
λ '
γ
λ
You can prove to yourself with some simple trigonometry that the angle between the λ '-axis and a line
from the origin to the point on the circle that represents the strain of the line really is ψ:
tanψ γλ
γλ
λ
γ= ′′
= =1
6.3 Principal Axes of Strain
λ1 and λ3, the long and short axes of the finite strain ellipse, are known as the principal axes of
strain because they are the lines which undergo the maximum and minimum amounts of extension.
From the Mohr’s Circle, we can see a very important property of the principal axes. They are the only
two points on the circle that intersect the horizontal axis.
Thus, lines parallel to the principal axes suffer no shear strain or angular shear. All other lines in the
body do undergo angular shear.
draft date: 20 Jan, 1999
Lines are perpendicular before and after the deformation because they are parallel to the principal axes
Lecture 6 4 9Mohrs Circle for Finite Strain
6 .4 Maximum Angular Shear
You can also use the Mohr’s Circle to calculate the orientation and extension of the line which
undergoes the maximum angular shear, ψmax, and shear strain, γmax:
γ '
2θ'
λ '1
λ '3
2
−
λ '1λ '32
+
ψmax
tangent line
λ '
From the geometry above,
sin maxψλ λ
λ λλ λλ λ
=
′ − ′
′ + ′ = ′ − ′′ + ′
3 1
3 1
3 1
3 1
2
2
,
or
ψ λ λλ λmax sin= ′ − ′
′ + ′
−1 3 1
3 1
. (6.3)
To get the orientation of the line with maximum angular shear, θ'ψmax:
cosmax
2 2
2
3 1
3 1
3 1
3 1
′ =
′ − ′
′ + ′ = ′ − ′′ + ′
θλ λ
λ λλ λλ λψ ,
or
′ = ′ − ′′ + ′
−θ λ λλ λψ max
cos12
1 3 1
3 1
. (6.4)
draft date: 20 Jan, 1999
You could also easily solve this problem by differentiating with respect to θ, and setting it equal
Lecture 6 5 0Mohrs Circle for Finite Strain
to zero:
d
d
γθ′
= 0 .
6.5 El l ipt ic i ty
This is a commonly used parameter which describes the aspect ratio (i.e. the ratio of the large and
small axes) of the strain ellipse. Basically, it tells you something about the two-dimensional shape of the
strain ellipse.
Re
e
S
S=
+( )+( ) =
1
11
3
1
3
. (6.5)
Note that, because S1 is always greater than S3 (by definition), R is always greater than 1. A circle has an
R of 1.
6.6 Rotation of Any Line During Deformation
It is a simple, yet important, calculation to determine the amount that any line has rotated during
the deformation:
(x, z)
(x', z')
θ θ'
tan θ = zx
tan θ' = z'x'
The stretches along the principal axes, 1 and 3, are:
Sx
xx x1 1 1= = ′ ⇒ ′ =λ λ
and
draft date: 20 Jan, 1999
Sz
zz z3 3 3= = ′ ⇒ ′ =λ λ .
Lecture 6 5 1Mohrs Circle for Finite Strain
Substituting into the above equations, we get a relation between θ and θ':
tan tan tantan′ = = = =θ
λλ
θλλ
θ θz
x
S
S R3
1
3
1
3
1
. (6.6)
The amount of rotation that any line undergoes then is just (θ - θ').
6.7 Lines of No Finite Elongation
In any homogeneous deformation without a volume change, there are two lines which have the
same length both before and after the deformation. These are called “lines of no finite elongation”
(LNFE):
lines of "no finite elongation"
l = l = 1i f
λ ' = S = 12
We can solve for the orientations of these two lines by setting the Mohr Circle equation for elongation to
1,
′ =′ + ′( ) −
′ − ′( ) ′ =λλ λ λ λ
θ3 1 3 1
2 22 1cos ,
and solving for θ':
cos cos22
2 13 1
3 1
2′ =′ + ′ −( )
′ − ′( ) = ′ −θλ λ
λ λθ ,
and
cos2 3
3 1
1′ =
′ −( )′ − ′( )θ
λλ λ
. (6.7)
draft date: 20 Jan, 1999
Lecture 6 5 2Mohrs Circle for Finite Strain
There are alternative forms which use θ instead of θ' and λ instead of λ':
tan2 1
3
1
1θ
λλ
=−( )
−( )and
draft date: 20 Jan, 1999
tan2 3
1
1
3
1
1′ =
−( )−( )θ λ
λλ
λ .
Lecture 6 5 3Infinitesimal & Finite Strain
LECTURE 7 — STRAIN IV: FINITE VS. INFINITESIMAL STRAIN
Up until now, we’ve mostly been concerned with describing just the initial and final states of
deformed objects. We’ve only barely mentioned the progression of steps by which things got to their
present condition. What we’ve been studying is finite strain -- the total difference between initial and
final states. Finite strain can be thought of as the sum of a great number of very small strains. Each small
increment of strain is known as Infinitesimal Strain. A convenient number to remember is that an
infinitesimal strain is any strain up to about 2%; that is:
el l
lf i
i
=−
≤ 0 02.
With this concept of strain, at any stage of the deformation, there are two strain ellipsoids that represent
the strain of the rock:
Finite Strain Ellipse Infinitesimal Strain Ellipse
This represents the total deformation from the beginning up until the present.
This is the strain that the particles will feel in the next instant of deformation
You can look at it this way:
Start with a box
strain it a finite amount carve a new box out of it
and deform that new box by a very small amount
Finite Strain Infinitesimal Strain
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Lecture 7 5 4Infinitesimal & Finite Strain
Key aspect of infinitesimal strain:
• The maximum angular shear is always at 45° to the principal axes
7.1 Coaxial and Non-coaxial Deformation
Notice that, in the above drawing, I purposely made the axes of the infinitesimal strain ellipse
have a different orientation than those of the finite strain ellipse. Obviously, this is one of two cases -- in
the other, the axes would be parallel. This is a very important distinction for understanding deformation:
• Coaxial -- if the axes of the finite and infinitesimal strain ellipses are parallel
• Non-coaxial -- when the axes of finite and infinitesimal are not parallel
These two terms should not be confused (as they, unfortunately, usually are in geology) with the following
two terms, which refer just to finite strain.
• Rotational -- when the axes of the finite strain ellipse are not parallel to their
restored configuration in the undeformed, initial state
• Non-rotational -- the axes in restored and final states are parallel
In general in the geological literature, rotational/non-coaxial deformation is referred to as simple shear
and non-rotational/coaxial deformation is referred to as pure shear. The following table may help
organize, if not clarify, this concept:
Finite Strain Infinitesimal Strain
Non-rotational ⇒ pure shear
Rotational ⇒ Simple shear
Coaxial ⇒ progressive pure shear
Non-coaxial ⇒ progressive simple shear
In practice, it is difficult to apply these distinctions, which is why most geologists just loosely refer to
pure shear and simple shear. Even so, it is important to understand the distinction, as the following
draft date: 20 Jan, 1999
diagram illustrates:
Lecture 7 5 5Infinitesimal & Finite Strain
A non-coaxial, non-rotational deformation
7.2 Two Types of Rotation
Be very careful to remember that there are two different types of rotations that we can talk about
in deformation:
1. The rotation of the principal axes during the deformation. This occurs only
in non-coaxial deformation.
2. The rotation of all other lines in the body besides the principal axes. You can
easily calculate this from the equations that we derived in the last two classes
(e.g., eqn. 6.6, p. 51). This rotation affects all lines in the body except the
principal axes. This rotation has nothing to do with whether or not the
deformation is by pure or simple shear.
If we know the magnitudes of the principal axes and the initial or final position of the line, it is
always possible to calculate the second type of rotation. Without some external frame of reference, it is
impossible to calculate the first type of rotation. In other words, if I have a deformed fossil and can
calculate the strain, I still do not know if it got to it’s present condition via a coaxial or non-coaxial strain
path.
Many a geologist has confused these two types of rotation!!
7.3 Deformation Paths
Most geologic deformations involve a non-coaxial strain path. Thus, in general, the axes of the
infinitesimal and finite strain ellipsoids will not coincide. In the diagram below, all the lines which are
within the shaded area of the infinitesimal strain ellipse [“i(+)”] will become infinitesimally longer in the
next tiny increment of deformation; they may still be shorter than they were originally. In the shaded
draft date: 20 Jan, 1999
area of the finite strain ellipse [“f(+)”], all of the lines are longer than they started out.
Lecture 7 5 6Infinitesimal & Finite Strain
i (-)
i (+)
f (-)f (+)
note: LNIE at 45° to principal axes
note: LNFE at < 45°to principal axes
Infinitesimal strain Finite Strain
Thus, the history of deformation that any line undergoes can be very complex. If the infinitesimal
strain ellipse is superposed on the finite ellipse in the most general possible configuration, there are four
general fields that result.
f (+), i (-)
f (-), i (-)
f (-), i (+)
f (+), i (+)
I
II
III
IV
Most general case:
An arbitrary superposition of the infinitesimal ellipse on the finite ellipse. Not very likely in asingle progressive deformation
• Field I: lines are shorter than they started, and they will continue to shorten in
the next increment;
• Field II: lines are shorter than they started, but will begin to lengthen in the
next increment;
• Field III: lines are longer than they started, and will continue to lengthen in
the next increment; and
• Field IV: lines are longer than they started, but will shorten in the next
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increment.
Lecture 7 5 7Infinitesimal & Finite Strain
The case for a progressive simple shear is simpler, because one of the lines of no finite extension
coincides with one of the lines of no infinitesimal extension. To understand this, think of a card deck
experiment.
Note that the individual cards never change length or orientation. Thus, they are always parallel to one of the lines of no infinitesimal and no finite extension
cards
ψ
f (-), i (+)
f (+), i (+)f (-), i (-)
Simple Shear
III
III
Thus, lines will rotate only in the direction of the shear, and lines that begin to lengthen will
never get shorter again during a single, progressive simple shear.
In progressive pure shear, below, you only see the same three fields that exist for simple shear,
so, again, lines that begin to lengthen will never get shorter. The difference between pure and simple
shear is that, in pure shear, lines within the body will rotate in both directions (clockwise and
counterclockwise).
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f (+), i (+)
f (-), i (-)f (-), i (+)
III
III
Pure Shear
Lecture 7 5 8Infinitesimal & Finite Strain
7 .4 Superposed Strains & Non-commutability
In general, the order in which strains and rotations of different types are superimposed makes a
difference in terms of the final product. This property is called “non-commutability”.
Two strains:
Area = 4.13 sq. cm
1. Simple shear, ψ = 45°
2. Pure shear, e = 1x
1. Pure shear, e = 1
2. Simple shear, ψ = 45°
x
A strain & a rotation:
1. Stretch = 2 2. Rotation = 45°
2. Stretch = 21. Rotation = 45°
7.5 Plane Strain & 3-D Strain
So far, we’ve been talking about strain in just two dimensions, and implicitly assuming that
there’s no change in the third dimension. Strain like this is known as “plane strain”. In the most general
case, though, strain is three dimensional:
draft date: 20 Jan, 1999
Lecture 7 5 9Infinitesimal & Finite Strain
Z = λ3Y = λ2
X = λ1
Note that, in three dimensional strain, the lines of no extension become cones of no extension. That is
because an ellipsoid intersects a sphere in two cones.
Three-dimensional strains are most conveniently displayed on what is called a Flinn diagram.
This diagram basically shows the ratio of the largest and intermediate strain axes, X & Y, plotted against
the ratio of the intermediate and the smallest, Y & Z. A line with a slope of 45° separates a field of
“cigar”-shaped strain ellipsoids from “pancake”-shaped ellipsoids. All plane strain deformations plot on
this line, including, for example, all simple shears.
k = 1k = ∞
k = 0
Y
Z S 3
S 2
1 + e3
1 + e2= =
X
Y S 2
S 1
1 + e2
1 + e1= =
prolate speroids"cigars"
oblate speroids"pancakes"pla
ne st
rain
most geological deformations
draft date: 20 Jan, 1999
Lecture 8 6 0Introduction to Stress
LECTURE 8—STRESS I: INTRODUCTION
8.1 Force and Stress
I told you in one of the first lectures that we seldom see the forces that are responsible for the
deformation that we study in the earth because they are instantaneous, and we generally study old
deformations. Furthermore, we cannot measure stress directly. Nonetheless, one of the major goals of
structural geology is to understand the distribution of forces in the earth and how those forces act to
produce the structures that we see.
There are lots of practical reasons for wanting to do this:
• earthquakes
• oil well blowouts
• what makes the plates move
• why landslides occur, etc.
Consider two blocks of rock. I’m going to apply the same forces to each one:
FF
draft date: 20 Jan, 1999
Lecture 8 6 1Introduction to Stress
Your intuition tells you that the smaller block is going to “feel” the force a lot more than the
larger block. That’s because there are fewer particles in it to distribute the force. Thus, although the two
blocks are under the same force, it is more “concentrated” in the little block. To express this, we need to
define a new term:
Stress = Force / Area
or as an equation:
rr
σ = F
A(8.1)
Note that, because force is a vector and area is a scalar, stress defined in this way must also be a vector. For
that reason, we call it the stress vector or more correctly, a traction vector. When we talk about tractions,
it is always with reference to a particular plane.
8.2 Units Of Stress
Stress has units of force divided by area. Force is equal to mass times acceleration. The “official”
unit is the Pascal (Pa):
ForceArea
mass accelerationArea
kgms
mNm
Pa2
2 2= × =
= =
In the above equation, N is the abbreviation for “Newton” the unit of force. In the earth, most stresses are
substantially bigger than a Pascal, so we more commonly use the unit “megapascal” (Mpa):
1 MPa = 106 Pa = 10 bars = 9.8692 atm.
8.3 Sign Conventions:
Engineering: compression (-), tension (+)
Geology: compression (+), tension (-)
In geology, compression is more common in the earth (because of the high confining pressure).
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Engineers are much more worried about tensions.
Lecture 8 6 2Introduction to Stress
8 .4 Stress on a Plane; Stress at a Point
An arbitrary stress on a plane can be resolved into three components:
X1 X2
X3
random stresson the plane
normal stress
shear stress // 2 axis
shear stress // 1 axis
We can extend this idea to three dimensions to look at stress at a single point, which we’ll represent as a
very small cube:
X1 X2
X3
σ23
σ11σ12
σ13
σ31σ32
σ33
σ22
σ21
In three dimensions, there are nine tractions which define the state of stress at a point. There is a
convention for what the subscripts mean:
the first subscript identifies the plane by indicating the axis which is
perpendicular to it
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the second subscript shows which axis the traction vector is parallel to
Lecture 8 6 3Introduction to Stress
These nine vectors can be written in matrix form:
σσ σ σσ σ σσ σ σ
ij =
11 12 13
21 22 23
31 32 33
(8.2)
As you may have guessed, σij is the stress tensor. If my cube in the figure, above, is in equilibrium so that
it is not rotating, then you can see that
σ12 = σ21 , σ13 = σ31 , and σ32 = σ23
Otherwise, the cube would rotate about one of the axes. Thus, there are only six independent components
to the stress tensor. This means that the stress tensor is a symmetric tensor.
8.5 Principal Stresses
Notice in the “stress on a plane” figure (page 62) that the gray arrow labeled “random stress on a
plane” is larger than any of the normal or shear stresses. If we change the orientation of the plane so that
it is perpendicular to this arrow then all the shear stresses on the plane go to zero and we are left with
only with the gray arrow which is now equal to the normal stresses on the plane. Now let’s extend this
idea to the block. It turns out that there is one orientation of the block where all the shear stresses on all
of the face go to zero and each of the three faces has only a normal stress on it. Then, the matrix which
represents the stress tensor reduces to:
σσ
σσ
ij =
1
2
3
0 0
0 0
0 0
(8.3)
In this case the remaining components -- σ1, σ2, and σ3 -- are known as the principal stresses. By
convention, σ1 is the largest and σ3 is the smallest. People sometimes refer to these as “compression” and
“tension”, respectively, but this is wrong. All three may be tensions or compressions.
You can think of the three principal axes of stress as the major, minor, and intermediate axes of
an ellipsoid; this ellipsoid is known as the stress ellipsoid.
draft date: 20 Jan, 1999
Lecture 8 6 4Introduction to Stress
σ3
σ1 σ2
8.6 The Stress Tensor
As you may have guessed from the lecture on tensors last time, σij is the stress tensor. The stress
tensor simply relates the traction vector on a plane to the vector which defines the orientation of the plane
[remember, a tensor relates two fields of vectors]. The mathematical relation which describes this relation
in general is known as Cauchy’s Law:
p li ij j= σ (8.4)
I can use this equation to calculate the stress on any plane in the body if I know the value of the stress
tensor in my chosen coordinate system.
8.7 Mean Stress
This is just the average of the three principal stresses. Because the sum of the principal diagonal
is just the first invariant of the stress tensor (i.e. it does not depend on the specific coordinate system), you
do not have to know what the principal stresses are to calculate the mean stress; it is just the first
invariant divided by three:
σ σ σ σ σ σ σm = + + = + +1 2 3 11 22 33
3 3 . (8.5)
8.8 Deviatoric Stress
With this concept of mean stress, we can break the stress tensor down into two components:
draft date: 20 Jan, 1999
σ σ σσ σ σσ σ σ
σσ
σ
σ σ σ σσ σ σ σσ σ σ σ
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
0 0
0 0
0 0
=
+−
−−
m
m
m
m
m
m
. (8.6)
Lecture 8 6 5Introduction to Stress
The first component is the isotropic part or the mean stress; it is responsible for the type of deformation
mechanism as well as dilation. The second component is the deviatoric stress; it is what actually
produces the distortion of a body. Note that when you talk about deviatoric stress, the maximum stress
is always positive (compressional) and the minimum is always negative (tensional).
8.9 Special States of Stress
• Uniaxial Stress: only one non-zero principal stress, i.e. σ1 or σ3 ≠ 0
• Biaxial Stress: one principal stress equals zero, the other two do not
• Triaxial Stress: three non-zero principal stresses, i.e. σ1, σ2, and σ3 ≠ 0
• Axial Stress: two of the three principal stresses are equal, i.e. σ1 > σ2 = σ3
• Lithostatic Pressure: The weight of the overlying column of rock:
P gdz gzlithostatic
z
ave= ≈∫ ρ ρ0
• Hydrostatic Pressure: (1) the weight of a column of fluid in the interconnected
pore spaces in a rock (Suppe, 1986):
P gzfluid ave f= ρ
(2) The mean stress (Hobbs, Means, & Williams, 1976):
σ σ σ σ σ σ σm = + + = + +1 2 3 11 22 33
3 3
(3) When all of the principal stresses are equal (Jaeger & Cook, 1976):
P = σ1 = σ2 = σ3
Although these definitions appear different, they are really all the same. Fluids at rest can
support no shear stress (i.e. they offer no resistance to shearing). That is why, by the way, we know that
draft date: 20 Jan, 1999
the outer core of the earth is a fluid -- it does not transmit shear waves from earthquakes.
Lecture 8 6 6Introduction to Stress
Thus the state of stress is the same throughout the body. This type of stress is also known as
Spherical Stress. It is called the spherical stress because it represents a special case in which the stress
ellipsoid is a sphere. Thus, every plane in a fluid is perpendicular to a principal stress (because all axes of
a circle are the same length) and there is no shear on any plane.
draft date: 20 Jan, 1999
Lecture 9 6 7Vectors & Tensors
LECTURE 9—VECTORS & TENSORS
Last time, I called stress a tensor; today, I want to give you a glimpse of what that statement
actually means. At the same time, we will see a different way of looking at stress (and other tensor
properties such as strain) which is very efficient, mathematically. It is much more important that you try
to understand the concepts, rather than the specific equations. The math itself, is a part of linear algebra.
We “derived” the stress tensor by considering a small cube whose faces were perpendicular to
the axes of an arbitrary coordinate system (arbitrary with respect to the stress on the cube). In other
words, we are trying to find something which relates the tractions themselves to the orientations of the
planes on which they occur.
9.1 Scalars & Vectors
In your math courses, you have no doubt heard about two different types of quantities:
1. Scalar -- represented by one number. Just a point in space. Some examples:
• temperature
• density
• mass
2. Vector -- represented by three numbers. A line showing direction and
magnitude. It only makes sense to talk about a vector with respect to a
coordinate system, because of the direction component. Some examples:
• velocity
• force
• displacement
Remember that a vector relates two scalars. For example, the relation between temperature A
draft date: 20 Jan, 1999
and B is the temperature gradient which is a vector.
Lecture 9 6 8Vectors & Tensors
9 .2 Tensors
Now we come back to our original question: what type of physical property relates two vectors,
or two fields of vectors to each other?
That type of property is called a Tensor:
3. Tensor -- represented by nine numbers. Relates a field of vectors to each other. Generally
can be represented as an ellipsoid. Some examples:
• electrical conductivity
• thermal conductivity
• stress
• strain
The stress tensor relates the orientation of a plane—expressed as the direction cosines of the pole
to the plane—to the tractions on that plane. In the diagram, below, if we know the stress tensor, σij, then
we can calculate the tractions p1 and p2 for a plane of any orientation given by α and β:
X1
X2
α
β
p1
p2
edge-on view of a plane (i.e.the plane contains the X3 axis)
We can express this relationship by the simple mathematical expression, which is known as Cauchy’s
Law:
draft date: 20 Jan, 1999
p li ij j= σ . (9.1)
Lecture 9 6 9Vectors & Tensors
9 .3 Einstein Summation Convention
The above equation is written in a form that may not be familiar to you because it uses a simple
mathematical shorthand notation. We need the shorthand because that equation actually represents a set
of three linear equations which are somewhat cumbersome to deal with and write down all the time.
There are nine coefficients, εij , which correspond to the values of the strain tensor with respect to whatever
coordinate system you happen to be using. Those three equations are:
p l l l1 11 1 12 2 13 3= + +σ σ σ ,
p l l l2 21 1 22 2 23 3= + +σ σ σ , (9.2)
p l l l3 31 1 32 2 33 3= + +σ σ σ .
We could write the same in matrix notation:
p
p
p
l
l
l
1
2
3
11 12 13
21 22 23
31 32 33
1
2
3
=
σ σ σσ σ σσ σ σ
, (9.3)
but this is still awkward, so we use the notation above, known as dummy suffix notation, or Einstein
Summation Convention. Equations 8-2 can be written more efficiently:
p l l l lj jj
1 11 1 12 2 13 3 11
3
= + + ==∑σ σ σ σ ,
p l l l lj jj
2 21 1 22 2 23 3 21
3
= + + ==∑σ σ σ σ ,
p l l l lj jj
3 31 1 32 2 33 3 31
3
= + + ==∑σ σ σ σ .
From here, it is just a short step to equation 9.1:
p li ij j= σ , where i and j both can have values of 1, 2, or 3.
p1, p2, and p3 are the tractions on the plane parallel to the three axes of the coordinate system, X1, X2, and
draft date: 20 Jan, 1999
X3, and l1, l2, and l3 are equal to cosα, cosβ, and cosγ, respectively.
Lecture 9 7 0Vectors & Tensors
In equation 9.1, because the “j” suffix occurs twice on the right hand side, it is the dummy suffix,
and the summation occurs with respect to that suffix. The suffix, “i”, on the other hand is the free suffix;
it must occur once on each side of each equation.
You can think of the Einstein summation convention in terms of a nested do-loop in any
programming language. In a FORTRAN type language, one would write the above equations as follows:
Do i = 1 to 3p(i) = 0Do j = 1 to 3
p(i) = sigma(i,j)*l(j) + p(i)repeat
repeat
9.4 Coordinate Systems and Tensor Transformations
The specific values attached to both vectors and tensors -- that is the three numbers that represent
a vector or the nine numbers that represent a tensor -- depend on the coordinate system that you choose.
The physical property that is represented by the tensor (or vector) is independent of the coordinate
system. In other words, I can describe it with any coordinate system I want and the fundamental nature
of the thing does not change. As you can see in the diagram, below, for vectors:
X1
X2
X3
V
V3
V1V2
X3'
X1'
X2'
V'
V2'V3'
V1'
(note that the length and relative orientationof V on the page has not changed; only theaxes have changed)
draft date: 20 Jan, 1999
Lecture 9 7 1Vectors & Tensors
The same is true of tensors; a strain ellipse has the same dimensions regardless of whether I take
a coordinate system parallel to geographic axes or a different one. In the earth, we can use a variety of
different coordinate systems; the one most commonly used when we’re talking about vectors and tensors
is the Cartesian system with direction cosines described earlier:
• north, east, down .
There are times when we want to look at a problem a different way: For example, we are
studying a fault and we want to make the axes of the coordinate system parallel to the pole to the fault
and the slip direction;
There is a simple way to switch between geographic and fault coordinates: Coordinate
transformation, and the related transformations of vectors and tensors.
We’re not going to go into the mathematics of transformations (although they are reasonably
simple). Just remember that the difference between a tensor and any old random matrix of nine numbers
is that you can transform the tensor without changing its fundamental nature.
The nine numbers that represent an infinitesimal strain tensor, or any other tensor, can be
represented as a matrix, but not all matrices are tensors. The specific values of the components change
when you change the coordinate system, the fundamental nature does not. If I happen to choose my
coordinates so that they are parallel to the principal axes of stress, then the form of the tensor looks like:
σσ
σσ
ij =
1
2
3
0 0
0 0
0 0
9.5 Symmetric, Asymmetric, & Antisymmetric Tensors
Coming back to our original problem of describing the changes of vectors during deformation,
the tensor that relates all those vectors in a circle to their position is known at the displacement gradient
draft date: 20 Jan, 1999
tensor.
Lecture 9 7 2Vectors & Tensors
The displacement gradient tensor, in general, is an asymmetric tensor. What that means is that it
has nine independent components, or, if you look at it in matrix form:
e
e e e
e e e
e e eij =
11 12 13
21 22 23
31 32 33
, where e12 ≠ e21, e13 ≠ e31, and e32 ≠ e23.
If eij were a symmetric tensor, then e12 = e21, e13 = e31, and e32 = e23, and it would have only 6
independent components.
It turns out that any asymmetric tensor can be broken down into a symmetric tensor and an
antisymmetric tensor. So, for the displacement gradient tensor, we can break it down like:
ee e e e
ij
ij ji ij ji=+( )
+−( )
2 2
=
+( ) +( )
+( ) +( )
+( ) +( )
+
−( ) −( )
−( ) −( )
−( )
ee e e e
e ee
e e
e e e ee
e e e e
e e e e
e e e
1112 21 13 31
21 1222
23 32
31 13 32 2333
12 21 13 31
21 12 23 32
31 13 32
2 2
2 2
2 2
02 2
20
2
2
−−( )
e23
20
Writing the same equation in a more compact form:
eij ij ij= +ε ω ,
where
εij
ij jie e=
+( )2
and ωij
ij jie e=
−( )2
.
The symmetric part is the infinitesimal strain tensor and the antisymmetric part is the rotation tensor.
Written in words, this equation says:
“the displacement gradient tensor = strain tensor + rotation tensor”.
draft date: 20 Jan, 1999
Note that the infinitesimal strain tensor is always symmetric. Thus, you can think of pure shear as ωi j = 0
Lecture 9 7 3Mohrs Circle for Stress
and simple shear as ωi j ≠ 0.
9.6 Finding the Principal Axes of a Symmetric Tensor
The principal axes of a second order tensor can be found by solving an equation known as the
“Characteristic” or “secular” equation. This equation is a cubic, with the following general form:
λ λ λ3 2 0− − − =Ι ΙΙ ΙΙΙ
The three solutions for λ are called the eigenvalues; they are the magnitudes of the three principal axes.
Knowing those, you can calculate the eigenvectors, which give the orientations of the principal axes. The
calculation is generally done numerically using a procedure known as a Jacobi transformation. The
coefficients, Ι, ΙΙ and ΙΙΙ are known as the invariants of the tensor because they have the same values
regardless of the orientation of the coordinate system. The first invariant, Ι, is particularly useful because
it is just the sum of the principal diagonal of the tensor. Thus, for the infinitesimal strain tensor, it is
always true that:
σ σ σ σ σ σ1 2 3 11 22 33+ + = + + .
This is particularly useful when we get to stress and something known as hydrostatic pressure.
draft date: 20 Jan, 1999
Lecture 1 0 7 4Mohrs Circle for Stress
LECTURE 10—STRESS II: MOHR’S CIRCLE
10.1 Stresses on a Plane of Any Orientation from Cauchy’s law
We would like to be able to calculate the stress on any plane in a body. To do this, we will use
Cauchy’s Law, which we derived last time.
X1
X3
α
γ
p1
p3
α
αγ
γ
p1Np1S
p3N
p3S
We will assume that we know the orientations of the principal stresses and that we have chosen our
coordinate system so that the axes are parallel to those stresses. This gives us the following matrix for the
stress tensor:
σσ
σσ
ij =
1
2
3
0 0
0 0
0 0
(10.1)
The general form of Cauchy’s Law is:
p li ij j= σ (10.2)
which, if we expand it out for the case shown above will be:
draft date: 20 Jan, 1999
p l
p l1 1 1 1
3 3 3 3 3 390
= == = = −( ) =
σ σ ασ σ γ σ α σ α
cos
cos cos sin
Lecture 1 0 7 5Mohrs Circle for Stress
If we want to find the normal and shear stresses on the plane, σn and σs respectively, then we have to
decompose the tractions, p1 and p3, into their components perpendicular and parallel to the plane. First
for p1:
p p
p p
N
S
1 1 1 12
1 1 1
= = ( ) =
= = ( )cos cos cos cos
sin cos sin
α σ α α σ α
α σ α α
and then for p3:
p p
p p
N
S
3 3 3 32
3 3 3
= = ( ) =
= = ( )sin sin sin sin
cos sin cos
α σ α α σ α
α σ α α
Now, the normal stress arrows point in the same direction, so we add them together:
σ σ α σ αn N Np p= +( ) = +1 3 12
32cos sin (10.3)
The shear stress arrows point in opposite directions so we must subtract them:
σ σ α α σ α α σ σ α αs S Sp p= −( ) = − = −( )1 3 1 3 1 3cos sin cos sin cos sin (10.4)
10.2 A more “Traditional” Way to Derive the above Equations
In this section, I will show you a derivation of the same equations which is found in more
traditional structural geology text books. The diagram, below, was set up so that there is no shear on the
faces of the block. Thus, the principal stresses will be perpendicular to those faces. Also, a very important
point to remember in these types of diagrams: You must always balance forces, not stresses. So, the basic
idea is to balance the forces, find out what the stresses are in terms of the forces, and then write the
expressions in terms of the stresses. From the following diagram, you can see that:
draft date: 20 Jan, 1999
Lecture 1 0 7 6Mohrs Circle for Stress
θθ
F3
F1
Area = A
Area = A sin θ
Area = A cos θ
10.2.1 Balance of Forces
θ
θF3
F1
F1NF1S
F3S
F3N
Force normal to the plane:
F1N F3NFN = +
F1S F3SFS = −
Force parallel to the plane:
Now, we want to write the normal forces and the parallel (or shear) forces in terms of F1 and F3. From
simple trigonometry in the above diagram, you can see that:
F1N = F1 cos θ , F1S = F1 sin θand
F3N = F3 sin θ , F3S = F3 cos θ.
So, substituting these into the force balance equations, we get:
draft date: 20 Jan, 1999
FN = F1N + F3N = F1 cos θ + F3 sin θ (10.5)
Lecture 1 0 7 7Mohrs Circle for Stress
and
FS = F1S − F3S = F1 sin θ − F3 cos θ. (10.6)
10.2.2 Normal and Shear Stresses on Any Plane
Now that we have the force balance equations written, we just need to calculate what the forces
are in terms of the stresses and substitute into the above equations.
FN and FS act on the inclined plane, which has an area = A. The normal and shear stresses then,
are just those forces divided by A:
σ σnn
ss
A A= =F F
and . (10.7)
F1 and F3 act on the horizontal and vertical planes, which have different areas as you can see from the
first diagram. The principal stresses then, are just those forces divided by the areas of those two sides of
the block:
σθ
σθ1
13
3= =F FA Acos sin
and . (10.8)
Equations 10.7 and 10.8 can be rewritten to give the forces in terms of stresses (a step we skip here) and
then we can substitute into the force balance equations, 10.5 and 10.6. For the normal stresses:
F F F A A AN n= + = = +1 3 1 3cos sin cos cos sin sinθ θ σ σ θ θ σ θ θ
The A’s cancel out and we are left with an expression just in terms of the stresses:
σ σ θ σ θn = +12
32cos sin (10.9)
For the shear stresses:
F F F A A AS s= − = = −1 3 1 3sin cos cos sin sin cosθ θ σ σ θ θ σ θ θ .
As before, the A’s cancel out and we are left with an expression just in terms of the stresses:
draft date: 20 Jan, 1999
σ τ σ σ θ θs = = −( )sin cos1 3 (10.10)
Lecture 1 0 7 8Mohrs Circle for Stress
Note that the shear stress is commonly designated by the Greek letter tau, “τ”. Also note that we have
made an implicit sign convention that clockwise (right-handed) shear is positive. Equations 16.9 and
16.10 are identical to 10.3 and 10.4.
10.3 Mohr’s Circle for Stress
Like we did with strain, we can write these equations in a somewhat different form by using the
double angle formulas:
cos 2α = cos2 α - sin2 α = 2cos2 α - 1 = 1 - 2sin2 α .
Using these identities, equations 10.9 and 10.10 (or 10.3 and 10.4) become:
σ σ σ σ σ θn = +
+ −
1 3 1 3
2 22cos (10.11)
σ τ σ σ θs = = −
1 3
22sin (10.12)
The graphs below show how the normal and shear stresses vary as a function of the orientation of the
plane, θ:
θ
2
90° 180°
σ1 + σ3σn
σ1
σ3
The above curve shows that:
• maximum normal stress = σ1 at θ = 0°
• minimum normal stress = σ3 at θ = 90°
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Lecture 1 0 7 9Mohrs Circle for Stress
2θ
90° 180°
σ s
σ1 – σ3
This curve shows that:
• shear stress = 0 at θ = 0° or 90°
In other words, there is no shear stress on planes perpendicular to the principal stresses.
• maximum shear stress = 0.5 (σ1 - σ3) at θ = 45°
Thus, the maximum shear stress is one half the differential stress.
The parametric equations for a circle are:
x = c - r cos α and y = r sin α ,
so the above equations define a circle with a center on the x-axis and radius:
c r, , 02
02
1 3 1 3( ) = +
= −σ σ σ σ
and
The Mohr’s Circle for stress looks like:
draft date: 20 Jan, 1999
Lecture 1 0 8 0Mohrs Circle for Stress
σn
2
2
2θ
σs
σ3 σ1
σ1 – σ3
σ1 + σ3
10.4 Alternative Way of Plotting Mohr’s Circle
Sometimes you’ll see Mohr’s Circle plotted with the 2θ angle drawn from σ3 side of the circle:
2θ
σ s
σ n
θ
θ
σ1
σ 3
In this case, θ is the angle between the pole to the plane and σ3, or between the plane itself and σ1. It is not
draft date: 20 Jan, 1999
the angle between the pole and σ1.
Lecture 1 0 8 1Mohrs Circle for Stress
10.5 Another Way to Derive Mohr’s Circle Using Tensor
Transformations
The derivation of Mohr’s Circle, above, is what you’ll find in most introductory structure textbooks.
There is a far more elegant way to derive it using a transformation of coordinate axes and the corresponding
tensor transformation. In the discussion that follows, it is much more important to get an intuitive feeling
for what’s going on than to try and remember or understand the specific equations. This derivation
illustrates the general nature of all Mohr’s Circle constructions.
10.5.1 Transformation of Axes
This refers to the mathematical relations that relate to orthogonal sets of axes that have the same
origin, as shown in the figure, below.
X1
X2
X3
X3'
X2'
X1'
cos a-121
cos a-123
cos a-122
In the diagram, a21 is the cosine of the angle between the new axis, X2’, and the old axis, X1, etc. It is
important to remember that, conventionally, the first suffix always refers to the new axis and the second
suffix to the old axis. Obviously, there will be three angles for each pair of axes so that there will be nine
in all. They are most conveniently remember with a table:
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Lecture 1 0 8 2Mohrs Circle for Stress
X1 X2 X3
X3'
X2'
X1'
a21 a22 a23
a13a12a11
a33a31 a32
Old Axes
New Axes
or, in matrix form:
a
a a a
a a a
a a aij =
11 12 13
21 22 23
31 32 33
.
Although there are nine direction cosines, they are not all independent. In fact, in the above
diagram you can see that, because the third angle is a function of the other two, only two angles are
needed to fix one axis and only one other angle -- a total of three -- is needed to completely define the
transformation. The specific equations which define the relations between all of the direction cosines are
known as the “orthogonality relations.”
10.5.2 Tensor Transformations
If you know the transformation matrix, you can transform any tensor according to the following
equations:
′ =σ σij ik jl kla a (new in terms of old)
or
σ σij ki lj kla a= ′ (old in terms of new).
[These transformations are the key to understanding tensors. The definition of a tensor is a physical
quantity that describes the relation between two linked vectors. The test of a tensor is if it transforms
according to the above equations, then it is a tensor.]
10.5.3 Mohr Circle Construction
draft date: 20 Jan, 1999
Any second order tensor can be represented by a Mohr’s Circle construction, which is derived
Lecture 1 0 8 3Mohrs Circle for Stress
using the above equations simply by making a rotation about one of the principal axes. In the diagram,
below, the old axes are parallel to the principal axes of the tensor, σi j, and the rotation is around the σ1
axis.
X1
X1'
X3
X3'
θ
σ i j = σ1 0 00 σ2 00 0 σ3
With a rotation of θ about the X2 axis, the transformation matrix is:
aij =−
cos sin
sin cos
θ θ
θ θ
0
0 1 0
0
After a tensor transformation according to the above equations and using the identities cos(90 - θ) = sin θ
and cos(90 + θ) = - sin θ, the new form of the tensor is
′ =+( ) −( )( )
−( )( ) +( )
σσ θ σ θ σ σ θ θ
σσ σ θ θ σ θ σ θ
ij
12
32
3 1
2
1 3 12
32
0
0 0
0
cos sin sin cos
sin cos sin cos
.
Rearranging using the double angle formulas, we get the familiar equations for Mohrs Circle
′ = +
+ −
σ σ σ σ σ θ11
1 3 1 3
2 22cos
and
′ = − −
σ σ σ θ13
1 3
22sin
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Lecture 1 0 8 4Mohrs Circle for Stress
σ1
σ3 σ'33σ'13
σ'31σ'11
σ'ii
σ'ij( i ≠ j)
2θ
draft date: 20 Jan, 1999
Lecture 1 1 8 5Stress-Strain Relations
LECTURE 11—STRESS III: STRESS-STRAIN RELATIONS
11.1 More on the Mohr’s Circle
Last time, we derived the fundamental equations for Mohr’s Circle for stress. We will use Mohr’s
Circle extensively in this class so it’s a good idea to get used to it. The sign conventions we’ll use are as
follows:
Tensile stresses
σ n negative σ n
Compressive stresses
positive
clockwise (right lateral)negative
counterclockwise (left lateral)positive
Mohr’s circle quickly allows you to see some of the relationships that we graphed out last time:
2θ = 90°
σS max
=σ
1σ
3−2
θ = 45°
σ1
σ3
plane with maximum shear stress
You can see that planes which are oriented at θ = 45° to the principal stresses (2θ = 90°) experience the
maximum shear stress, and that that shear stress is equal to one half the difference of the largest and
smallest stress.
draft date: 20 Jan, 1999
The general classes of stress expressed with Mohr’s circle are:
Lecture 1 1 8 6Stress-Strain Relations
general tension uniaxial tension general tension & compression
pure shear stress uniaxial compression general compression
11.1.1 Mohr’s Circle in Three Dimensions
The concepts that we’ve been talking about so far are inherently two dimensional [because it is a
tensor transformation by rotation about the σ2 axis]. Even so, the concept of Mohr’s Circle can be
extended to three dimensions if we consider three separate circles, each parallel to a principal plane of
stress (i.e. the plane containing σ1-σ2, σ1-σ3, or σ2-σ3):
σn
σ1σ2σ3
σs
stresses on planes perpendicular to σ1-σ3 plane (i.e. what we plotted in two dimensions)
stresses on planes perpendicular to σ1-σ2 plane
stresses on planes perpendicular to σ3-σ2 plane
All other possible stresses plot within the shaded area
11.2 Stress Fields and Stress Trajectories
Generally within a relatively large geologic body, stress orientation will vary from place to place.
draft date: 20 Jan, 1999
This variation constitutes what is known as a stress field.
Lecture 1 1 8 7Stress-Strain Relations
Stress fields can be portrayed and analyzed using stress trajectory diagrams. In these diagrams,
the lines show the continuous variation in orientation of principal stresses. For example, in map view
around a circular pluton, one might see the following:
σ3
σ1
Note that the σ1 trajectories are always locally perpendicular to the σ3 trajectories. A more complicated
example would be:
σ 1
σ 3
this might be an example of a block being pushed over a surface
11.3 Stress-strain Relations
So far, we’ve treated stress and strain completely separately. But, now we must ask the question
of how materials respond to stress, or, what is the relation between stress and strain. The material
response to stress is known as Rheology.
Natural earth materials are extremely complex in their behavior, but there are some general
classes, or models, of material response that we can use. In the most general sense, there are two ways
that a material can respond to stress:
draft date: 20 Jan, 1999
1. If the material returns to its initial shape when the stress is removed, then the
Lecture 1 1 8 8Stress-Strain Relations
deformation is recoverable.
2. If the material remains deformed after the stresses are removed, then the
strain is permanent.
11.4 Elasticity
Imagine a body of rock; each time I apply a little more stress, it deforms a bit more:
Stress Strain2.5 0.5%5.0 1.0%7.5 1.5%10.0 2.0%0.0 0.0%
Notice that when I removed the stress in the last increment, the material popped back to its
original shape and the strain returned to zero. You can plot data like this on what is known as a
stress-strain curve:
Str
ess
Strain
The straight line means that there is a constant ratio between stress and strain.
This type of material behavior is known as elastic .
Note that part of the definition of elastic behavior is that the material response is instantaneous. As soon
as the stress is applied, the material strains by an appropriate amount. When the stress is removed, the
material instantly returns to its undeformed state.
11.4.1 The Elasticity Tensor
The equation that expresses this linear relation between stress and strain in its most general form
is:
draft date: 20 Jan, 1999
σi j = Ci j k l εk l .
Lecture 1 1 8 9Stress-Strain Relations
Ci j k l is the elasticity tensor. It is a fourth order tensor which relates two second order tensors. Because
all of the subscripts can have values of 1, 2, and 3, the tensor Ci j k l has 81 separate components! However,
because both the stress and strain tensors are symmetric, the elasticity tensor can have, at most, 36
independent components.
Fortunately, most of the time we make a number of simplifying assumptions and thus end up
worrying about four material parameters.
11.4.2 The Common Material Parameters of Elasticity
l i l f
wf
wi
ell f l i
l i
−=
et w i
wf w i−=
σ
With the above measurements, there are several parameters we can derive Young’s Modulus:
Ee
Cl
= =σ1111 .
This is for simple shortening or extensions. For the the ratio of the transverse to longitudinal strain we
use Poissons Ratio:
υ = = −e
e
E
Ct
l 1122
For volume constant deformation (i.e., an incompressible material), υ = 0.5, but most rocks vary between
0.25 and 0.33. For simple shear deformations, Modulus of Rigidity:
draft date: 20 Jan, 1999
Lecture 1 1 9 0Stress-Strain Relations
σ = G e
For for uniform dilations or contractions, Bulk Modulus or Incompressibility:
σ = K e
All of these parameters are related to each other by some simple equations:
GE K=+( )
= −( )+( )2 1
3 1 22 1υ
υυ
11.5 Deformation Beyond the Elastic Limit
What happens if we keep applying more and more stress to the rock? Intuitively, you know that
it can’t keep on straining indefinitely. Two things can happen
• the sample will break or rupture, or
• the sample will cease deforming elastically and will start to strain faster than
the proportional increase in stress.
These two possibilities look like this on stress strain curves:
draft date: 20 Jan, 1999
Lecture 1 1 9 1Stress-Strain Relations
σ
e
σ
e
yield strength
non-recoverable strain"anelastic" or "plastic"
rupture strength
permanent strain if stress removed befor rupture
ultimate strength
max elastic strain
plastic strain
hypothetical paths when stress removed
rupture in elastic realm plastic deformation
Note that the maximum elastic strains are generally <<5%. There are two forms of plastic deformation:
σ
e
σ
e
yield strength
yield strength
perfect plastic strain hardening
strain hardening part of curve
draft date: 20 Jan, 1999
Lecture 1 2 9 2Plastic & Viscous Deformation
LECTURE 12—PLASTIC & VISCOUS DEFORMATION
12.1 Strain Rate
So far, we haven’t really said anything about time except to say the elasticity is instantaneous.
You can think of two different graphs:
strain, e time, t
stre
ss, σ
stra
in, e
Time-dependent deformation would have a different response. Suppose I took the same material and did
three different experiments on it, each at a different constant stress level:
σ c
time, t
stra
in, e
time, t
stra
in, e
time, t
stra
in, e
σ a σ b
In other words, for different constant stresses, the material deforms at different strain rates. In the above
graphs, the strain rate is just the slope of the line. Strain rate is the strain divided by time. Because strain
has no units, the units of strain rate are inverse time. It is commonly denoted by an “e” with a dot over it:
e . Geological strain rates are generally given in terms of seconds:
draft date: 1/20/99
10 1016 1 12 1− − − −≤ ≤s e sgeol˙ .
Lecture 1 2 9 3Plastic & Viscous Deformation
Note that strain rate is not a velocity. Velocity has no reference to an initial shape or dimension and has
units of distance divided by time.
12.2 Viscosity
With this idea of strain rate in mind, we can define a new type of material response:
stre
ss, σ
•strain rate, e
•σ = η e
The slope of the curve, η, is the viscosity.
It is a measure of the resistance of the
material to flow
slope
= η
A material with a high viscosity flows very slowly. Low viscosity materials flow rapidly. Relative to
water, molasses has a high viscosity. When the above curve is straight (i.e. the slope is constant) then we
say that it is a Newtonian fluid. The important difference between viscous and elastic:
• Viscous -- time dependent
• Elastic -- time independent
Real rocks commonly have a combination of these:
draft date: 1/20/99
time
stra
in
time
stress removed
delayed recovery
elastic
viscous
Viscoelastic Elasticoviscous
stra
in
Lecture 1 2 9 4Plastic & Viscous Deformation
The difference between perfect viscous and perfect plastic:
Perfect viscous -- the material flows under any applied stress
Perfect plastic -- material flows only after a certain threshold stress (i.e. the
yield stress) has been reached
12.3 Creep
The viscous material curve on page 93 is idealized. Geological materials deformed under constant
stress over long time spans experience several types of rheological behaviors and several strain rates.
This type of deformation at constant stress for long times is called creep. In general, in long term creep
rocks have only 20 - 60% of their total short term strength. As shown in the following diagram, there are
three fields:
stra
in, e
time, t
I II III
t 2t 1
stress removed at times 1 & 2
elastic
visco- elastic
rupture
delayed recovery permanent
(plastic) strain
0 -- Instantaneous elastic strain
I -- Primary or transient creep; strain rate decreases
II -- Secondary or steady state creep; strain rate constant
III -- Tertiary or accelerated creep; strain rate goes up
draft date: 1/20/99
This curve is constructed for constant stress; i.e. stress does not change during the entire length
Lecture 1 2 9 5Plastic & Viscous Deformation
of time. The creep curve has considerable importance for the possibility of predicting earthquakes.
Consider some part of the earth’s crust under a constant stress for a long period of time. At first the strain
is fast (in fact instantaneous) and then begins to slow down until it reaches a steady state. Then, after a
long time at steady state, the strain begins to accelerate, just before rupture, that is the earthquake, occurs.
12.4 Environmental Factors Affecting Material Response to Stress
There are several factors which change how a material will respond to stress. Virtually all of
what we know along these lines comes from experimental work. Usually, when you see stress strain
curves for experimental data, the stress plotted is differential stress, σ1 - σ3.
12.4.1 Variation in Stress
Failure field
Elastic field
stra
in, e
time, t
σyield
σrupture
time, t
stra
in, e
stress, σincreasing differential
stress
As you can see in the above graph, increasing the differential stress drives the style of deformation from
elastic to viscous to failure. At low differential stresses, the deformation is entirely elastic or viscoelastic
and recoverable. At higher differential stress, the deformation becomes viscous, and finally, at high
differential stresses, rupture occurs.
12.4.2 Effect of Confining Pressure (Mean Stress)
An increase in confining pressure results in an increase in both the yield stress, σy, and the
rupture stress, σr. The overall effect is to give the rock a greater effective strength. Experimental data
draft date: 1/20/99
shows that:
Lecture 1 2 9 6Plastic & Viscous Deformation
strain, e
100 Mpa
30 Mpa
3.5 Mpa
1 Mpa
confining (mean) stressesσy
σy
σy
σy
diffe
rent
ial s
tres
s
σy σr
[the confining pressure at the base of the continental crust is on the order of 1000 Mpa]
12.4.3 Effect of Temperature
An increase in temperature results in a decrease in the yield stress, σy, and an increase in the
rupture stress, σr. The overall effect is to enlarge the plastic field.
strain, e
diffe
rent
ial s
tres
s
σy σr
25°C 100°C
300°C
500°C
800°C
these may never rupture
12.4.4 Effect of Fluids
Fluids can have two different effects on the strength of rocks, one at a crystal scale, and one at the
scale of the pore space in rocks.
1. Fluids weaken molecular bonds within the crystals, producing an effect similar to temperature;
at laboratory strain rates, the addition of water can make a rock 5 to 10 times weaker. With the addition
of fluids, the yield stress, σy, goes down and the rupture stress,σr, goes up:
draft date: 1/20/99
Lecture 1 2 9 7Plastic & Viscous Deformation
strain, e
diffe
rent
ial s
tres
s
σy σr
400°C, dry
900°C, dry
1000°C, dry
900°C, wet
2. If fluid in the pores of the rock is confined and becomes overpressured, it can reduce the
confining pressure.
Peffective = Pconfining - Pfluid
As we saw above, a reduced confining pressure tends to reduce the overall strength of the rock.
12.4.5 The Effect of Strain Rate
Decreasing the strain rate results in a reduction of the yield stress, σy. In the laboratory, the
slowest strain rates are generally in the range of 10-6s-1 to 10-8s-1. An “average” geological strain rate of
10-14s-1 is equivalent to about 10% strain in one million years.
strain, e
diffe
rent
ial s
tres
s
σy
10 sec-1-4
10 sec-1-7
10 sec-1-6
10 sec-1-5
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12.5 Brittle, Ductile, Cataclastic, Crystal Plastic
Lecture 1 2 9 8Plastic & Viscous Deformation
There are several terms which describe how a rock fails under stress. These terms are widely
misused in geology. Your will see them again when we talk about fault zones.
Brittle -- if failure occurs during elastic deformation (i.e. the straight line part of the stress-strain
curve) and is localized along a single plane, it is called brittle. This is non-continuous deformation, and
the piece of rock which is affected by brittle deformations will fall apart into many pieces.
Ductile -- This is used for any rock or material that can undergo large changes in shape (especially
stretching) without breaking. Ductile deformation can occur either by cracking and fracturing at the scale
of individual grains or flow of individual minerals. In lab experiments, you would see:
Brittle Ductile
[internal deformation could be by grain-scale fracturing or by plastic flow of minerals; i.e. thedeformation mechanism is not specified]
When people talk about the “brittle-ductile” transition, it should be with reference to the above
two styles of deformation. Brittle is localized and ductile is distributed. Unfortunately, people usually
have a specific deformation mechanism in mind.
Cataclasis (cataclastic deformation) -- Rock deformation produced by fracturing and rotating
of individual grains or grain aggregates. This term implies a specific mechanism; both brittle and ductile
deformation can be accomplished by cataclastic mechanisms.
Crystal Plastic -- Flow of individual mineral grains without fracturing or breaking. We will talk
about the specific types of mechanisms later; for those with some background in material science,
however, we are talking in general about dislocation glide and climb and diffusion.
It may help to remember all of these terms with a table (after Rutter, 1986):
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Lecture 1 2 9 9Plastic & Viscous Deformation
Distribution of Deformation
Localized Distributed
Cataclastic
Crystal Plastic
Brittle faulting
Cataclastic Flow
Plastic shear zone
Homogeneous plastic FlowM
echa
nism
of
Def
orm
atio
n
incr Temp, Conf. Press.
incr strain rate
Brittle Ductile
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Lecture 1 3 100Elasticity, Compaction
LECTURE 13—DEFORMATION MECHANISMS I: ELASTICITY,COMPACTION
So far, we’ve been talking just about empirical relations between stress and strain. To further
understand the processes we’re interested in, we now have to look in more detail to see what happens to
a rock on a granular, molecular, and atomic levels.
13.1 Elastic Deformation
If a deformation is recoverable, what does that mean as far as what happens to the rock at an
atomic level? It means that no bonds are broken.
r
r = bond length
In elastic strain, we increase or decrease the bond length, r, but we don’t actually break the bond. For
example, an elastic simple shear of a crystal might look like:
original state stress applied stress removed
When the stress is removed, the molecule “snaps” back to its original shape because each bond has a
preferred length. What determines the preferred length? It’s the length at which the bond has the
minimum potential energy. There are two different controls on that potential energy (U):
Potential energy due to attraction between oppositely charge ions
Urattraction ∝ − 1
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PE due to repulsion from electron cloud overlap
Lecture 1 3 101Elasticity, Compaction
Urrepulsion ∝ − 1
12
The total potential energy, then, can be written as:
UA
r
B
rtotal = − + 12
where C1 and C2 are constants. A graph of this function highlights its important features:
Pot
entia
l Ene
rgy,
U
Bond length, r
minimumpotentialenergy
[the solid curve is the sumof the other two]
bond lengthwith Umin, ro
repulsion term =Br12
attraction term = –A
r
To get the bond force, you have to differentiate the above equation with respect to r:
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FdU
dr
A
r
B
r= = − +2 13
12
Lecture 1 3 102Elasticity, Compaction
Bon
d F
orce
, F
= d
U/d
r
Bond Length, r
Repulsion
Attraction
Note that repulsion due to electron cloud overlap acts only over very small distances, but it is very
strong. The attraction is weaker, but acts over greater distances. These curves show that it is much
harder to push the ions together than it is to pull them apart (i.e. the repulsion is stronger than the
attraction). At the most basic level, this is the reason for a virtually universal observation:
• rocks are stronger under compression than they are under tension
13.2 Thermal Effects and Elasticity
A rise in temperature produces an increase in mean bond length and decrease in potential energy
of the bond. This is why rocks have a lower yield stress, σy, at higher temperature. The strain due to a
temperature change is given by:
e Tij ij= α ∆
α ≡ coefficient of thermal expansion
The temperature change, ∆T, is a scalar so the coefficient of thermal expansion, α i j, is a symmetric, second
order tensor. It can have, at most, six independent components. The actual number of components
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depends on crystal symmetry and thus varies between 1 and 6.
Lecture 1 3 103Elasticity, Compaction
A good example of the result of thermal strain are cooling joints in volcanic rocks (e.g. columnar
joints in basalts).
Flow erupted at a temperature of 1020°C
Flow cooled to a surface temperature of 20°C
∆T = (Tf - Ti) = -1000°C
1000 m
1000 m
wnw1
If α = 2.5 x 10-6 °C-1 and ∆T = -1000°C, then the strain on cooling to surface temperature will be
e = α ∆T = 2.5 x 10-6 °C-1 x -1000°C = -2.5 x 10-3 .
If the initial length of the flow is 1000 m, then the change in length will be:
ew w
w
w
wf i
i i
=−
= ∆ ⇒ ∆w = e wi = –2.5 x 10-3 (1000 m) = –2.5 m.
The joints form because the flow shrinks by 2.5 m. Because the flow is welded to its base, it cannot shrink
uniformly but must pull itself apart into columns. If you added up all the space between the columns (i.e.
the space occupied by the joints) in a 1000 m long basalt flow, it would total 2.5 m:
1000 m - Σ wn = 2.5 m .
13.3 Compaction
Compaction is a process that produces a permanent, volumetric strain. It involves no strain of
individual grains or molecules within the grains; it is the result of the reduction of pore space between
the grains.
Porosity is defined as:
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Lecture 1 3 104Elasticity, Compaction
φ = Vp
Vp + Vs = volume of the pores
total volume ,
and the void ratio as:
θv = Vp
Vs = volume of the pores
volume of the solid
Much compaction occurs in a sedimentary basin during diagenesis and is not tectonic in origin.
There is an empirical relationship between compaction and depth in a sedimentary basin known
as Athy’s Law:
φ = φo e- az
where z = the depth, a = some constant, and φo is the initial porosity [“e” means exponential not strain].
13.4 Role of Fluid Pressure
Compaction is usually considered hand in hand with fluid pressure. This is just the pressure of
the fluids which fill the pores of the rock. Usually, the fluid is water but it can also be oil, gas, or a brine.
We shall see in the coming days that fluid pressure is very important for the overall strength of the rock.
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fluid presses out equally in all directions[every plane is ⊥ to a principal stress so no
shear stress]
Lecture 1 3 105Fracture
13.4.1 Effective Stress
The role of fluids in a rock is to reduce the normal stress across the grain to grain contacts in the
rock without changing the shear stresses. We can now define a new concept, the effective stress which
originally comes from Terzaghi in soil mechanics, but appears equally applicable to rocks.
σ i j* =
σ 11 - Pf σ 12 σ 13σ 21 σ 22 - Pf σ 23σ 31 σ 32 σ 33 - Pf
Note that only the principal diagonal (i.e. the normal stresses) of the matrix is affected by the pore
pressure.
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Lecture 1 4 106Fracture
LECTURE 14—DEFORMATION MECHANISMS II: FRACTURE
A very important deformation mechanism in the upper part of the Earth’s crust is known as
fracture. Fracture just means the breaking up into pieces. There are two basic types as shown in our
now familiar stress-strain curves:
σ
e
σ
e
yield strength
rupture strength
brittle fracture ductile fracturerupture strength
In brittle fracture, there is no permanent deformation before the rock breaks; in ductile fracture, some
permanent deformation does occur before it breaks. Fracture is strongly dependent on confining pressure
and the presence of fluids, but is not as strongly dependent on temperature.
14.1 The Failure Envelop
The Mohr’s circle for stress is a particularly convenient way to look at fracture. Suppose we do
an experiment on a rock. We will start out with an isotropic stress state (i.e. σ1 = σ2 = σ3) and then
gradually increase the axial stress, σ1, while holding the other two constant:
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σ1
σ2
σ3
σs
σn
initial isotropic
stress
Stress state at time of fracture
2θ
Lecture 1 4 107Fracture
If we look in detail at the configuration of the Mohr’s Circle when fracture occurs, there is something very
curious:
2θ
φ φ = angle of internal friction
2θ = 90° + φ
[detail of previous figure]
The fracture does not occur on the plane with the maximum shear stress (i.e. 2θ = 90°). Instead, the angle, 2θ, is
greater than 90°. The difference between 2θ at which the fracture forms and 90° is known as the angle of
internal friction and is usually given by the Greek letter, φ.
Now lets do the experiment again at a higher confining pressure:
σs
σnnew initial isotropic
stress
New stress state at time of fracture
2θ 2θ
same diferential stress as before (circle is the same size) but it doesn't break this time
In fact, we can do this sort of experiment at a whole range of different confining pressures and
each time there would be a point at which the sample failed. We can construct an “envelop” which links
the stress conditions on each plane at failure. Stress states in the rock with Mohr’s circles smaller than
this envelop would not result in failure; any stress state in which the Mohr’s Circle touch or exceeded the
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envelop would produce a fracture of the rock:
Lecture 1 4 108Fracture
σs
σn
2θ 2θ
In general, we see a failure envelop which has four recognizable parts to it:
φφ
2θσ n
So
To
IV
III
II
I
failure envelopeσs
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Field I -- Tensile fracture: You can see that the Mohr’s circle touches the failure envelop in only one
Lecture 1 4 109Fracture
place. The 2θ angle is 180°; thus, the fractures form parallel to σ1 and perpendicular to σ3. The point To is
known as the Tensile strength. Note that, because the Mohr’s circle intersects the failure envelope at a
principal stress, there is no shear stress on the planes in this case. The result is that you make joints
instead of faults.
σ1
σ3
-30 ≤ To ≤ -4 Mpa
Field II -- Transitional tensile behavior: this occurs at σ1 ≈ |3To|. The circle touches the
envelop in two places, and, 120° ≤ 2θ ≤ 180°:
σ1
σ3
< 30°
30° ≤ φ ≤ 90°
The shape of the trans-tensile part of the failure envelop is determined by cracks in the material. These
cracks are known as Griffith Cracks after the person who hypothesized their existence in 1920. Cracks
are extremely effective at concentrating and magnifying stresses:
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Lecture 1 4 110Fracture
d
l
lines of equal stress
in plan view:
2l
The tensile stress at the tip of a crack is given by:
σ ≈ 23
σ3 (2l)2
d
The sizes of cracks in rocks are proportional to the grain size. Thus, fine-grained rocks will have shorter
cracks and be stronger under tension than coarse-grained rocks. The equations for the trans-tensile part
of the failure envelope, predicted by the Griffith theory of failure are:
σs2 - 4 To σn - 4 To
2 = 0
or
σs = 2 To σn + To
Field III -- Coulomb behavior: This portion of the failure envelop is linear, which means that
there is a linear increase in strength with confining pressure. This is very important because it is
characteristic of the behavior of the majority of rocks in the upper crust of the earth. The equation for this
part of the failure envelop is:
σs = so + σn tan φ = so + σn µ
In the above equation:
µ = coefficient of internal friction
and
so = the cohesion
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Lecture 1 4 111Fracture
σ1
σ3
~ 30°
φ ≈ 30°
2θ ≈ 120°
θ ≈ 60°
Field IV -- Ductile failure (Von Mises criterion): This occurs at high confining pressure and
increasing temperature. Here the fracture planes become nearer and nearer to the planes of maximum
shear stress, which are located at 45°. There is a constant differential stress at yield.
σ1
σ3
30° - 45°
0° ≤ φ ≤ 30°
90° ≤ 2θ ≤ 120°
45° ≤ θ ≤ 60°
14.2 Effect of Pore Pressure
Last time, we saw that the pore fluid pressure counteracts, or reduces, the normal stress but not
the shear stress:
Effective stress = σ i j* =
σ 11 - Pf σ 12 σ 13σ 21 σ 22 - Pf σ 23σ 31 σ 32 σ 33 - Pf
Taking this into account, the equation for Coulomb fracture then becomes:
σs = so + (σn - Pf) tan φ = so + σn* µ
The result is particularly striking on a Mohr’s Circle:
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Lecture 1 4 112Fracture
σ1 - P σ1σ3σ3 - P
σn
σs
Because the pore fluid pressure changes the effective normal stress but does not affect the shear
stress, the radius of the Mohr’s Circle stays the same but the circle shifts to the left. A high enough pore
fluid pressure may drive the circle to the left until it hits the failure envelop and the rock breaks. Thus,
pore pressure weakens rocks.
This effect is used in a practical situation when one wants to increase the permeability and
porosity of rocks (e.g. in oil wells to help petroleum move through the rocks more easily, etc.). The
process is known as hydrofracturing or hydraulic fracturing. Fluids are pumped down the well and
into the surrounding rock until the pore pressure causes the rocks to break up.
14.3 Effect of Pre-existing Fractures
Rock in the field or virtually anywhere in the upper part of the Earth’s crust have numerous
preexisting fractures (e.g. look at the rocks in the gorges around Ithaca). These fractures will affect how
the rock subsequently fails when subjected to stress. Two things occur:
• So, the cohesion, goes virtually to zero
• µ, the coefficient of friction changes to a coefficient of sliding friction
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Lecture 1 4 113Fracture
σ1σ3
σn
σs
2θ2
2θ1
oS
pristine rock would only fail on this plane
failure envelop
envelop for
pre-existing
fractures
any pre-exisitng fracture with an angle between 2θ1 and 2θ2 will slip in this stress state
The equation for the failure envelop for preexisting fractures is
σs = σn* µf
This control by preexisting features can be extended to metamorphic foliations.
θc
θf 90°
60°
30°
0°0° 30° 60° 90°
θf
θc
fault
par
allel
to cl
eava
ge
14.4 Fr ict ion
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The importance of friction was first recognized by Amontons, a French physicist, in 1699. Amontons
Lecture 1 4 114Pressure Solution & Crystal Plasticity
presented to the French Royal Academy of Science two laws, the second of which was very controversial:
• Amontons First Law -- Frictional resisting force is proportional to the normal
force
• Amontons Second Law -- Frictional resisting force is independent of the area
of surface contact
The second law says in effect that you can change the surface area however you want but, if the
normal force remains the same, the friction will be the same. You have to be intellectually careful here.
The temptation is to think about increasing the surface area with the implicit assumption that the mass of
the object will change also. But if that happens, then the normal force will change, violating the first law.
So, when you change the surface area, you must also change the mass/area.
m m
Fn = mg Fn = mg
Much latter, Bowden provided an explanation for Amontons’ second law. He recognized that the
microscopic surfaces are very much rougher than it appears from our perspective. [Example: if you
shrunk the Earth down to the size of a billiard ball, it would be smoother than the ball.] Thus its surface
area is very different than the macroscopic surface area:
Fnasperities
voids
At the points of contact, or asperities, there is a high stress concentration due to the normal stress.
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Lecture 1 5 115Pressure Solution & Crystal Plasticity
LECTURE 15—DEFORMATION MECHANISMS III:PRESSURE SOLUTION & CRYSTAL PLASTICITY
15.1 Pressure Solution
15.1.1 Observational Aspects
One of the very common deformation mechanisms in the upper crust involves the solution and
re-precipitation of various mineral phases. This process is generally, and loosely, called pressure solution.
Evidence that pressure solution has occurred in rocks:
crinoid stem or other fossil
material removed by pressure solution
Stylolites
Classic morphology: jagged teeth with concen- trations of insoluable residue. This is common in marbles (e.g. particularly well seen in polished marble walls). Many stylolites don't have this form.
σ1
Although we commonly think of stylolites as forming in limestones and marbles, they are also very
common in siliceous rocks such as shale and sandstone.
Sometimes, we see veins and stylolites nearby, indicating that volume is preserved on the scale of
the hand sample or outcrop. In this case, the veins are observed to be approximately perpendicular to the
stylolites:
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Lecture 1 5 116Pressure Solution & Crystal Plasticity
More commonly, there is much more evidence for removal of material than for the local re-
precipitation. Then, there is a net volume decrease; you see shortening but no extension. The rocks in
the Delaware Water Gap area, for example, have experienced more than 50% volume loss due to pressure
solution.
What actually happens to produce pressure solution? No one really knows, but the favored
model is that, because of the high stress concentration at grain contacts, material there is more soluble.
Material dissolved from there migrates along the grain boundary to places on the sides of the grains,
where the stress concentration is lower, and is deposited there. This model is sometimes called by the
name “fluid assisted grain boundary diffusion” because the material diffuses along a thin fluid film at the
boundary of the grain:
σ 1
solution of material at grain-to-grain contact
there may be a thin fluid film between
the grains
redeposition at the grain margins
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This process is probably relatively common during diagenesis.
Lecture 1 5 117Pressure Solution & Crystal Plasticity
Not all pressure solution can be called a diffusional process because, as we will see later, diffusion
acts slowly and over short distances. In the case where there is a net volume reduction at hand sample or
outcrop scale, there has to be has to be large scale flushing of the material in solution out of the system by
long distance migration of the pore fluids.
15.1.2 Environmental constrains on Pressure Solution
Temperature -- most common between ~50° and 400°C. Thus, you will see it best developed in
rocks that are between diagenesis and low grade metamorphism (i.e., greenschist facies).
Grain Size -- at constant stress, pressure solution occurs faster at smaller grain sizes. This is
because grain surface area increases with decreasing grain size.
Impurities/clay -- the presence of impurities such as clay, etc., enhances pressure solution. It
may be that the impurities provide fluid pathways.
The switch from pressure solution to mechanisms dominated by crystal plasticity is controlled by
all of these factors. For two common minerals, the switch occurs as follows:
Upper Temperature Limit for Pressure Solution
Grain Size Quartz Calcite
100 µm 450°C 300°C
1000 µm 300°C 200°C
These temperatures are somewhere in lower greenschist facies of metamorphism.
15.2 Mechanisms of Crystal Plasticity
Many years ago, after scientists had learned a fair amount about atom structure and bonding
forces, they calculated the theoretical strengths of various materials. However, the strengths that they
predicted turned out to be up to five orders of magnitude higher than what they actually observed in
laboratory experiments. Thus, they hypothesized that crystals couldn’t be perfect, but must have defects
in them. We now know that there are three important types of crystal defects:
• Point
• Linear
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• Planar
Lecture 1 5 118Pressure Solution & Crystal Plasticity
15.2.1 Point Defects
To general types of point defects are possible:
• Impurities
Substitution
Interstitial
• Vacancies
Impurities occur when a “foreign” atom is found in the crystal structure, either in place of an
atom that is supposed to be there (substitution) or in the spaces between the existing atoms. Vacancies
occur when an atom is missing from its normal spot in the crystal lattice, leaving a “hole”. These are
illustrated below:
Substitution Impurity -- Atom of a similar atomic radius is substituted for a regular one
Interstitial Impurity -- Atom of a much smaller atomic radius "squeezes" into a space in the crystal lattice
Vacancy -- Atom missingfrom crystal lattice
Because the crystal does not have its ideal configuration, it has a higher internal energy and is therefore
weaker than the equivalent ideal crystal.
15.2.2 Diffusion
In general, crystals contain more vacancies at higher temperature. The vacancies facilitate the
movement of atoms through the crystal structure because atoms adjacent to a vacancy can “jump” into it.
This general process is known as diffusion. This is illustrated in the following figure:
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Lecture 1 5 119Pressure Solution & Crystal Plasticity
1. 2. 3.
4. 5. 6.
[the darker gray atoms have all moved from their original position by jumping into the adjacent vacancy. Atoms and vacancies diffuse in opposite directions]
There are two types of diffusion:
• Crystal lattice diffusion (Herring Nabarro creep) -- This type is important
only at high temperatures (T ≈ 0.85 Tmelting) such as one finds in the mantle of
the earth because it occurs far too slowly at crustal temperatures. [shown
above]
• Grain boundary diffusion (Coble creep) -- This type occurs at lower
temperatures such as those found in the Earth’s crust.
15.2.3 Planar Defects
There are several types of planar defects. Most are a product of the movement of dislocations.
Several are of relatively limited importance and some are still poorly understood. These include:
• Deformation bands -- planar zones of deformation within a crystal
• Deformation lamellae -- similar to deformation bands; poorly understood
• Subgrain boundaries
• Grain boundaries
• Twin lamellae
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Lecture 1 5 120Pressure Solution & Crystal Plasticity
The last three are illustrated, below:
Grain boundaries -- ("high-angle tilt boundaries") there is a large angle mismatch of the crystal latices. This would be seen under the microscope as a large difference in extinction angles of the crystals
Subgrain boundaries -- ("low-angle tilt boundaries") there is a small angle mismatch of the crystal laticesThis would be seen under the microscope as a small difference in extinction angles of the crystals
> 5°1° - 5°
38.2°
e - lamellae in calcite[Ca-ions at the corners of the rhombs]
Twin Lamellae
Narrow band in which there has been a symmetric rotation of the crystal lattice, producing a "mirror image". The twin band will have a different extinction angle than the main part of the crystal
c - axis (optic axis)
The formation of twin lamellae is called “Twin gliding”. This is particularly common in calcite, dolomite,
and plagioclase (in which twin glide produces “albite twins”). In plagioclase, twin lamellae commonly
form during crystal growth; in the carbonates, it is usually a product of deformation. Because of its
consistent relationship to the crystal structure, twins in calcite and dolomite can be used as a strain gauge.
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Lecture 1 6 121Dislocations
LECTURE 16—DEFORMATION MECHANISMS IV: DISLOCATIONS
16.1 Basic Concepts and Terms
Linear defects in crystals are known as dislocations. These are the most important defects for
understanding deformation of rocks under crustal conditions. The basic concept is that it is much easier
to move just part of something, a little at a time, than to move something all at once. I’m sure that that is
a little obscure, but perhaps a couple of non-geological examples will help.
The first example is well known: How do you move a carpet across the floor with the least
amount of work? If you just grab onto one side of it and try and pull the whole thing at once, it is very
difficult, especially if the carpet has furniture on it, because you are trying to simultaneously overcome
the resistance to sliding over the entire rug at the same time. It is much easier to make a “rumple” or a
wave at one side of the rug and then “roll” that wave to the other side of the rug:
1. 2.
3. 4.
rug has now moved one full "unit" to the right
b
Freight trains also provide a lesser known example. A long train actually starts by backing up.
There is a small amount of play in the connections between each car. After backing up, when the train
moves forward for a small instant it is just moving itself, then just itself and the car behind it, etc. This
way, it does not have to start all of the cars moving at one time.
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Crystals deform in the same way. It is much easier for the crystal to just break one bond at a time
Lecture 1 6 122Dislocations
than to try and break all of them simultaneously.
b
1. 2. 3.
4.The the line of atoms in gray in each step represents the extra half plane for that step. the atoms that comprised the intial extra half plane are indicated by black dots.
The dislocation line is the bottom edge of the extra half plane. In this diagram, it is perpendicular to the page. In each step, only a singe bond is broken, so that the dislocation moves in increments of one lattice spacing each time. This distance that the dislocation moves is known as the Burgers Vector , and is indicated by b in the diagram on the left.
Note that there is no record in the crystal of the passage of a dislocation; the dislocation leaves a perfect crystal in its "wake". Thus, a dislocation is not a fault in the crystal.
As you can see in the above figure, we describe the orientation of the dislocation and its direction of
movement with two quantities:
• Tangent vector -- the vector parallel to the local orientation of the dislocation
line
• Burgers vector -- slip vector parallel to the direction of movement. It is
directly related to the crystal lattice spacing
These two quantities allow us to define two end member types of dislocations:
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Lecture 1 6 123Dislocations
b
t
t
extra half "plane"
dislocation line
crystalographic glide plane
"Cut-away" view of part of a dislocation loop
[the previous figure could have been of
this face of the block]
edge segmentscrew
segment
Edge dislocation : b ⊥ t
Screw dislocation: b // t
Most dislocations are closed loops which have both edge and screw components locally.
16.2 Dislocation (“Translation”) Glide
When the movement of a dislocation is confined to a single, crystallographically determined
plane, it is known as dislocation glide (or translation glide by some). A particular crystallographic plane
combined with a preferred slip direction is called a slip system.
The number of slip systems in a crystal depends on the symmetry class of the crystal. Crystals
with high symmetry will have many slip systems; those with low symmetry will have fewer. Slip will
start on planes with the lowest critical resolved shear stress. That is, slip will start on planes where the
bonds are easiest to break.
16.3 Dislocations and Strain Hardening
After dislocations begin to move or glide in their appropriate slip planes, there are three things
that happen almost immediately which make it more difficult for them to continue moving:
1. Self stress field: there is a stress field around each dislocation line which is
related to the elastic distortion of the crystal around the extra half plane. In
this case, the dislocations repell each other so that it takes more stress to get
them to move:
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Lecture 1 6 124Dislocations
self-stress field [schematic]
2. Dislocation Pinning (pile-up): This occurs when an impurity point defect
lies in the glide plane of a dislocation. If the impurity atom is tightly bound in
the crystal lattice, the dislocation, which is everywhere else in its glide plane
slipping freely, will become pinned by the atom. Other dislocations in the
same glide plane will also encounter the same impurity, and will tend to pile
up at that point.
b
impurity atomglide plane
Dislocation lines
3. Jogs: When dislocations of different slip systems pass through each other,
one produces a jog or step in the other. This jog makes it much more difficult
to move because the “jogged” segment quite probably requires a different
critical resolved shear stress to move. In the diagram, below, the extra half
planes are shown in shade of gray:
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Lecture 1 6 125Dislocations
t 1
b 1
b 2
t 2
b 2
t 2
t 1
b 1
Jog
Before the two dislocations run into each other
After they pass, a jog has been produced in dislocation 1
16.4 Dislocation Glide and Climb
If there are a sufficient number of vacancies in a crystal, when a dislocation encounters an
impurity atom in its glide plane the dislocation can avoid being pinned by jumping to a parallel crystal-
lographic plane. This jump is referred to as dislocation climb.
The process of dislocation climb is markedly facilitated by the diffusion of vacancies through the
crystal. Thus, climb occurs at higher temperatures because there are more vacancies at higher temperatures.
It is important to understand that diffusion has two roles in deformation: It can be the primary deformation
mechanism (but probably only in the mantle for crystal lattice diffusion), or it can aid the process of dislocation glide
and climb.
When dislocation glide and climb occurs, strain hardening no longer takes place. The material
either acts as a perfect plastic, or it strain softens.
There are several new terms that can be introduced at this point:
Cold Working -- Plastic deformation with strain hardening. The main process is dislocation
glide.
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Hot working -- Permanent deformation with little or no strain hardening or with strain softening.
Lecture 1 6 126Dislocations
The main process is dislocation glide and climb.
Annealing -- Heating up a cold worked, strain hardened crystal to the point where diffusion
becomes rapid enough to permit the glide and climb of dislocations. Then the dislocations either climb
out of the crystal, into sub-grain walls, or they cancel each other out, producing a strain free grain from
one that was obviously deformed and strained.
16.5 Review of Deformation Mechanisms
• Elastic deformation -- Very low temperature, small strains
• Fracture -- Very low temperature, high differential stress
• Pressure Solution -- Low temperature, fluids necessary
• Dislocation glide -- Low temperature, high differential stress
• Dislocation glide and climb -- Higher temperature, high differential stress
• Grain boundary diffusion -- Low temperature, low differential stress
• Crystal lattice diffusion -- High temperature, low differential stress
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Lecture 1 7 127Flow Laws & Stress in Lithosphere
LECTURE 17—FLOW LAWS & STATE OF STRESS IN THE
LITHOSPHERE
Experimental work over the last several years has provided data which enable us to determine
how stress and strain -- or more specifically stress and strain rate -- are related for crystal plastic mechanisms.
The relationship for dislocation glide and climb is known as power law creep, for diffusion, diffusion
creep.
17.1 Power Law Creep
The basic equation which governs dislocation glide and climb is:
˙ expe CQ
RTo
n= −( ) −
σ σ1 3 . (17.1)
The variables are:
e = strain rate [s-1]
Co = a constant [GPa-ns-1; experimentally determined]
σ1 - σ3 = the differential stress [GPa]
n = a power [experimentally determined]
Q = the activation energy [kJ/mol; experimentally determined]
R = the universal gas constant = 8.3144 × 10-3 kJ/mol °K
T = temperature, °K [°K = °C + 273.16°]
It is called “power law” because the strain rate is proportional to a power of the differential
stress. Because temperature occurs in the exponential function, you can see that this sort of rheology is
going to be extremely sensitive to temperature. To think of it another way, over a very small range of
temperatures, rocks change from being very strong to very weak. The exact temperature at which this
occurs depends on the lithology.
Using this equation and the data from Appendix B in Suppe (1985) you can easily calculate the
differential stress that aplite can support at 300°C assuming that power law creep is the deformation
mechanism. First of all, rearranging the above equation:
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Lecture 1 7 128Flow Laws & Stress in Lithosphere
σ σ1 3
1
− = −
˙
exp
e
CQ
RTo
n
Substituting in the actual values:
σ σ1 3
14
2 8
1
3 1
10
10163
273 16 300
− =( ) −
× +( )
− −
− −−
s
GPa s kJ mol
8.3144 10 kJ mol K K
1
3.1 11
-3 -1 -1.
.
exp.
After working through the math, you get:
σ1 - σ3 = 0.236 GPa = 236 MPa .
These curves can be constructed for a variety of rock types and temperature (just by iteratively
carrying out the same calculations we did above), and we get the following graph of curves:
Tem
pera
ture
(°C
)
Max Shear Stress (Mpa)
200 400 600 800 1000
400
800
1200
olivine (dry)
olivine (wet)
clinopyroxenite
diabasefeldspar-bearing rocks
quartzite (dry)
quartzite (wet)
limestone
granite (dry)
Power Law Creep Curves
[strain rate = 10 s ]-14 -1
gray show range for-15 ≤ log[strain rates] ≤ -13
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Lecture 1 7 129Flow Laws & Stress in Lithosphere
Note that, for a geothermal gradient of 20°C/km and a 35 km thick continental crust, the temperature at
the base of the crust would be 700°C; there, only olivine would have significant strength.
17.2 Diffusion Creep
This mechanism is a linear function of the differential stress and is more sensitive to grain size
than temperature:
˙ ( )e C TD
do n=−( )σ σ1 3 . (17.2)
Again, the variables are:
e = strain rate [s-1]
Co(T) = a temperature dependent constant [experimentally determined]
σ1 - σ3 = the differential stress
n = a power [experimentally determined]
D = the diffusion coefficient [experimentally determined]
d = the grain size
In diffusion, the strain rate is inversely proportional to the grain size. Thus, the higher the grain size, the
slower the strain rate due to diffusional processes. Although crystal lattice diffusion requires high overall
temperatures, it is not nearly so sensitive to changes in temperature.
17.3 Deformation Maps
With these flow laws, we can construct a diagram known as a deformation map, which shows
what deformation mechanism will be dominant for any combination of strain rate, differential stress,
temperature, and grain size. Generally there are two types:
• differential stress is plotted against temperature for a constant grain size;
different curves on the diagram represent different strain rates.
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• differential stress is plotted against grain size for a constant temperature;
Lecture 1 7 130Flow Laws & Stress in Lithosphere
different curves on the diagram represent different strain rates. This one is
generally easier to construct.
The diagram below shows an example of the first type for the mineral olivine.
Grain boundary diffusion[Coble Creep]
Dislocation glide & climb
Dislocation glide
Lattice diffusion[Nabarro Herring Creep]
σ −
σ
(MP
a)1
3
1
10
100
1000
0.1
0.01
0 800 1600
10 s-15 -1
10 s-14 -1
10 s-13 -1
T (°C)
17.4 State of Stress in the Lithosphere
By making a number of assumptions, we can use our understanding of the various deformation
mechanisms and their related empirically derived stress-strain relations (or flow laws) to predict how
stresses vary in the earth’s crust. Four basic assumptions are made; two relate to the deformation
mechanisms and two relate to the lithologies:
• The upper crust is dominated by slip on pre-existing faults. Thus we will use
the Coulomb relation for the case of zero cohesion:
σs = σn* µs . (17.3)
• The lower crust is dominated by the mechanism of power law creep as described
by the equation developed above (eqn 17.1).
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Lecture 1 7 131Flow Laws & Stress in Lithosphere
• The crust is dominantly composed of quartz and feldspar bearing rocks.
• The mantle is composed of olivine.
The basic idea is that the crust will fail by whatever mechanism requires less differential shear
stress. [Remember that the maximum shear stress is just equal to one-half the differential stress.] The
resulting curve has the following form:
5
10
15
20
25
30
35
slip on pre-existing faults
power law creep for quartz-dominated lithologies
power law creep for olivine
Moho
σ3σ1 − 1000
Differential Stress (MPa)
Dep
th (
km)
Maximum stress in crust
strong
weak
strong
weak
Lithospheric Column
CR
US
TM
AN
TLE
[the only possible stresses in the lithosphere are in the shaded fields]
This model is sometimes humorously referred to as the “jelly sandwich” model of the crust. It predicts
that the lower crust will be very weak (supporting differential stresses of < 20 Mpa) relative to the upper
crust and upper mantle; it will behave like jelly between two slices of (stiff) bread. In general, the most
numerous and the largest earthquakes tend to occur in the region of the stress maximum in the middle
crust, providing at least circumstantial support to the model.
These curves are sometimes incorrectly referred to as “brittle-ductile transition” curves. Because
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we have used very specific rheologies to construct them, they should be called “frictional crystal-plastic
Lecture 1 7 132Flow Laws & Stress in Lithosphere
transition” curves.
Now, we should review some of the important “hidden” assumptions and limitations of these
curves, which have been very popular during the last decade:
• Lithostatic load and confining pressure control the deformation of the upper crust --
notice that there is no depth term in eqn. 17.3, even though the vertical axis of
the graph is plotted in depth. The depth is calculated by assuming that the
vertical stress is either σ1 or σ3 and that it is equal to the lithostatic load:
σ1 or σ3 = Pl i t h = ρgz
• Temperature is the fundamental control on deformation in the lower crust -- Again,
there is no depth term in the Power Law Creep equation (17.1). Depth is
calculated by assuming a geothermal gradient and calculating the temperature
at that depth based on the gradient. So really, two completely different things are
being plotted on the vertical axis and neither one is depth!
• Friction is assumed to be the main constraint on deformation in the upper crust -- The
value of friction is assumed to be constant for all rock types. [This follows
from “Byerlee’s Law” which we will discuss in a few days.]
• Laboratory strain rates are extrapolated over eight to ten orders of magnitude to get
the power law creep curves -- the validity of this extrapolation is not known.
• Other deformation mechanisms are not considered to be important -- The most
important of these would include pressure solution, the unknown role of
fluids in the lower crust, and diffusion.
• There is a wide variation in laboratory determined constants for all of the flow laws --
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Basically, do not take the specific numbers too seriously.
Lecture 1 8Joints & Veins 133
LECTURE 18—JOINTS & VEINS
18.1 Faults and Joints as Cracks
We’ll start our exploration of structures with discontinuous structures and later move on to
continuous structures. There are two basic types of discontinuous structures:
• Faults -- discontinuities in which one block has slipped past another, and
• Joints -- where block move apart, but do not slip past each other.
Most modern views of these structures are based on crack theory, which we had some exposure
to previously when we talked about the failure envelop. There are three basic “modes” of cracks:
Mode I : opening Mode II : sliding Mode III : tearing
Looked at this way, faults are mode II or mode III cracks, while joints are mode I cracks. Notice the gross
similarity between mode II cracks and edge dislocations and mode III cracks and screw dislocations.
Although they are similar, bear in mind that there are major differences between the two.
Definition of a joint: a break in the rock across which there has been no shearing, only extension.
Basically, they are mode I cracks. If it is not filled with anything, then it is called a joint; if material has
been precipitated in the break, then it is called a vein.
18.2 Joints
Joints are characteristic features of all rocks relatively near the Earth’s surface. They are of great
practical importance because they are pre-fractured surfaces. They have immediate significance for:
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• mining and quarrying
Lecture 1 8Joints & Veins 134
• civil engineering
• ground water circulation
• hydrothermal solutions and mineral deposits
Despite their ubiquitous nature and their practical importance, there are several reasons why
analyzing joints is not easy and is subject to considerable uncertainty:
• age usually unknown
• they are easily reactivated
• they represent virtually no measurable strain
• there are many possible mechanisms of origin
18.2.1 Terminology
Systematic joints commonly are remarkably smooth and planar with regular spacing. They
nearly always occur in sets of parallel joints. Joint sets are systematic over large regions. Joint systems
are composed of two or more joint sets. Joints which regularly occur between (i.e. they do not cross) two
member of a joint set are called cross joints.
Most joints are actually a joint zone made of “en echelon” sets of fractures:
A right-stepping, en echelon joint
detail shows how the end of the en echelon segments curve towards each other
Joint systems are consistent over large regions indicating that the scale of processes that control
jointing is also regional in nature. For example, in the Appalachians, the joints are roughly perpendicular
to the fold axes over broad regions:
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a joint
Lecture 1 8Joints & Veins 135
New York
Pennsylvania
New Jersey
Ohio
0 100 km
Lake Erie
folds
Joints are not always perpendicular to fold axes or even related to regional folds in any systematic
way. On the Colorado Plateau, for example, joints in sedimentary rocks are parallel to the metamorphic
foliation in the basement.
18.2.2 Surface morphology of the joint face:
twist hackles
plumose markings
direction of propagation
"Butting relation" (map view)
younger joint
older joint
This kind of morphology indicates that the fracture propagates very rapidly. Younger joints nearly
always terminate against older joints at right angles. This is called a butting relation. As we will see
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later in the course, this occurs because the older joint acts like as free surface with no shear stress
Lecture 1 8Joints & Veins 136
18.2.3 Special Types of Joints and Joint-related Features
Although many joints are tectonic in origin (e.g. the joints in sedimentary rocks of the Ithaca
area), others are totally unrelated to tectonics. Some special types:
Sheet structure or exfoliation -- This is very common in the granitoid rocks and other rocks are
were originally free from other types of joints. Sheet joints form thin, curved, generally convex-upward
shells which parallel the local topography. The sheets get thicker and less numerous with depth in the
earth and die out completely at about 40 m depth. The sheets are generally under compression parallel to
their length; the source of this compression is not well understood. In general, they are related to
gravitational unloading of the granitoid terrain. In New England, they have been used to construct the
pre-glacial topography because they formed before the last glaciation:
present land surface
pre-glacial land surface
Spalling and rock bursts in mines and quarries -- In man made excavations, the weight of the
overburden is released very suddenly. This creates a dangerous situation in which pieces of rock may
literally “explode” off of the newly exposed wall or tunnel (it is released by the formation of a joint at
acoustic velocities). For this reason quarries, especially deep ones, after miners make a new excavation,
no one is allowed to work near the new face of rock for a period of hours or days until the danger of rock
bursts has passed.
Cooling joints in volcanic rocks—The process involved is thermal contraction; as the rock
cools it shrinks, pulling itself apart. This is the source of the well known columnar joints in basaltic rocks,
etc.
18.2.4 Maximum Depth of True Tensile Joints
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True tensile joints, with no shear on their surfaces, occur only in the very shallow part of the
Lecture 1 8Joints & Veins 137
Earth’s crust. The shape of the Mohr failure envelop gives us some insight into the maximum depth of
formation of true joints:
σn
σs
σ∗ 1 = | 3To |To
If we assume that, near the surface of the earth, σ1 is vertical, then we can write the stress as a function of
depth, the density of rocks, and the pore fluid pressure:
σ1 = ρgz (1 - λ)
where λ is the fluid pressure ratio: λ = Pf / ρgz.
The maximum depth of formation of tensile joints, then, is:
ZT
go
max ( )=
−31ρ λ
Thus, except at very high pore fluid pressures, the maximum depth of formation of joints is about 6 km,
given that the tensile strength of rocks, To, is usually less than 40 MPa.
18.3 Veins
Veins form when joints or other fractures in a rock with a small amount of shear are filled with
draft date: 9 March, 1999
material precipitated from a fluid. For many reasons, veins are extremely useful for studying local and
Lecture 1 8Joints & Veins 138
regional deformations:
• record incremental strains
• many contain dateable material
• fluid inclusions in the vein record the temperature and pressure conditions at the time the vein formed
In addition, veins have substantial economic importance because many ore deposits are found in veins.
The Mother Lode which caused the California gold rush in 1849 is just a large gold-bearing quartz vein.
18.3.1 Fibrous Veins in Structural Analysis
An extremely useful aspect of many veins is that the minerals grow in a fibrous form as the walls
of the vein open up, with the long axes of the fibers parallel to the incremental extension direction.
ε1
ε1
Step 1 Step 2
There are two types of fibrous veins, and it is important to distinguish between them in order to use them
in structural analysis:
Syntaxial veins form when the vein has the same composition as the host rock (e.g. calcite veins
in limestone). The first material nucleates on crystals in the wall of the vein and grows in optical
continuity with those. New material is added at the center of the vein (as in the example, above).
Antitaxial veins form when the vein material is a different composition than the host rock (e.g.
calcite vein in a quartzite). New material is always added at the margins of the vein.
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Lecture 1 8Joints & Veins 139
step
1
step
2
step
3
step
3
step
2
new material added at margins
new material added at center
step
3
step
2step
2
step
1
step
1
Antitaxial Vein
vein material a different composition than wall rocks
Syntaxial Vein
vein material the same composition than wall rocks
These are among the very few natural features which show the rotational history of a deformation and
thus are particularly useful for studying simple shear deformations.
It is important to remember that the fibers are not deformed. They are simply growing during
the deformation.
18.3.2 En Echelon Sigmoidal Veins
Veins in which the tip grows during deformation (so that the entire vein gets larger) also provide
information on the incremental history of the deformation. The tip always grows perpendicular to the
incremental (or infinitesimal) principal extension), even though the main part of the vein may have
rotated during the simple shear. These veins are called sigmoidal veins or sometimes “tension gashes.”
They can also be syntaxial or antitaxial, thus providing even more information.
The formation of all of these types of veins in a simple shear zone is illustrated below:
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Lecture 1 8Joints & Veins 140
ε 3 ε 1
45°45°
infinitesimal strain axes
ε 3 ε 1
45°45°
finite strain axes
Recall that, in a shear zone, the axes of the infinitesimal strain ellipse are always oriented at 45° to the
shear plane. Because the tips of the sigmoidal veins always propagate perpendicular to the infinitesimal
extension direction, the tips will also be at 45° to the shear zone boundary. If the veins grow in a
syntaxial style, as in the above diagram, the fibers at the tips and in the center of the vein will also be at
45°.
18.4 Relationship of Joints and Veins to other Structures
Faults& Shear Zones
Folds
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Lecture 1 9Faults I: Basic Terminology 141
LECTURE 19—FAULTS I: B ASIC TERMINOLOGY
19.1 Descriptive Fault Geometry
For faults that are not vertical, there are two very useful terms for describing the blocks on either
side of the fault. These terms can be used either for normal or reverse faults:
• Hanging Wall, so called because it “hangs” over the head of a miner, and
• Footwall, because that’s the block on which the miners feet were located.
Hanging Wall
Footwall
The three dimensional geometry of a fault surface can be quite variable, and there are several
terms to describe it:
• Planar -- a flat, planar surface
• Listric (from the Greek word “listron” meaning shovel shaped) -- fault dip
becomes shallower with depth, i.e. concave-upward
• Steepening downward or convex up
• Anastomosing -- numerous branching irregular traces
In three dimensions, faults are irregular surfaces. All faults either have a point at which (a) their displacement
goes to zero, (b) they reach a point where the intersect another fault, or (c) they intersect the surface of the
Earth. There are three terms to describe these three possibilities:
• Tip Line -- Where fault displacement goes to zero; it is the line which separates
slipped from unslipped rock, or in the above crack diagrams, it is the edge of
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the crack. Unless it intersects the surface of the Earth or a branch line, the tip
Lecture 1 9Faults I: Basic Terminology 142
line is a closed loop
• Branch Line -- the line along which one fault intersects with or branches off
of another fault
• Surface trace -- the line of intersection between the fault surface and the land
surface
19.2 Apparent and Real Displacement
The displacement of one block relative to another is known as the slip vector. This vector
connects two points which were originally adjacent on either side of the fault. It is extremely unusual to
find a geological object which approximates a point that was “sliced in half” by a fault.
Fortunately, we can get the same information from a linear feature which intersects and was offset
across the fault surface. Such lines are known as piercing points. Most such linear features in geology
are formed by the intersection of two planes:
• intersection between a dike and a bed
• intersection of specific beds above and below an angular unconformity
• fold axis
It is however, much more common to see a planar feature offset by a fault. In this case, we can only talk
about separation, not slip:
strike separation
dip separation
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Lecture 1 9Faults I: Basic Terminology 143
There are an infinite number of possible slips that could produce an observed separation of a planar
feature. If you just saw the top of the above block, you might assume that the fault is a strike slip fault. If
you just saw the front, you might assume a normal fault. However, it could be either one, or a combination
of the two.
19.3 Basic Fault Types
With this basic terminology in mind, we can define some basic fault types:
19.3.1 Dip Slip
Normal -- The hanging wall moves down with respect to the footwall. This movement results in
horizontal extension. In a previously undeformed stratigraphic section, this would juxtapose younger
rocks against older.
High-angle -- dip > 45°
Low-angle -- dip < 45°
Reverse -- the hanging wall moves up with respect to the footwall. This movement results in
horizontal shortening. In a previously undeformed stratigraphic section, this would juxtapose older
rocks against younger.
High-angle -- dip > 45°
Thrust -- dip < 45°
19.3.2 Strike-Slip
Right lateral (dextral)-- the other fault block (i.e. the one that the viewer is not standing on)
appears to move to the viewers right.
Left lateral (sinistral)-- the other fault block appears to move to the viewers left.
A wrench fault is a vertical strike-slip fault.
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Oblique Slip -- a combination of strike and dip slip
Lecture 1 9Faults I: Basic Terminology 144
19.3.3 Rotational fault
In this case one block rotates with respect to the other. This can be due to a curved fault surface
[rotation axis is parallel to the fault surface], or where the rotation axis is perpendicular to the fault
surface. The latter case produces what is commonly known as a scissors or a hinge fault:
Scissors Fault:
19.4 Fault Rocks
The process of faulting produces distinctive textures in rocks, and those textures can be classified
according to the deformation mechanism that produced it. Again, the two general classes of mechanisms
that we discussed in class are: Frictional-Cataclastic (“Brittle mechanisms”), and crystal-plastic mechanisms.
19.4.1 Sibson’s Classification
Presently, the most popular classification method of fault rocks comes from the work by Sibson.
He has two general categories, based on whether the texture of the rock is foliated or random:
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Lecture 1 9Faults I: Basic Terminology 145
Random Fabric Foliated FabricIn
cohe
sive
Coh
esiv
e
Fault breccia(visable fragments > 30%)
Fault gouge(visable fragments < 30%)
crush breccia(fragments > 0.5 cm)
fine crush breccia(fragments 0.1 - 0.5 cm)
crush micro-breccia(fragments < 0.1 cm)
Protocataclasite
Cataclasite
Ultracataclasite
Protomylonite
Mylonite
Ultramylonite
0 - 10 %
10 - 50 %
50 - 90 %
90 - 100 %
Proportion of M
atrix
These rock types tend to form at different depths in the earth:
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cohesive cataclasite series(non-foliated)
non-cohesive gouge &breccia
cohesive mylonite series(foliated)
1 - 4 km
10 - 15 km250-350°C
Lecture 1 9Faults I: Basic Terminology 146
19.4.2 The Mylonite Controversy
There exists to this day no generally accepted definition f the term “mylonite” despite the fact
that it is one of the most commonly used fault rock names. There are two or three current definitions:
• A fine grained, laminated rock produced by extreme microbrecciation and
milling of rocks during movement on fault surfaces. This definition is closest
to the original definition of Lapworth for the mylonites along the Moine thrust
in Scotland
• Any laminated rock in which the grain size has been reduced by any mechanism
during the process of faulting. This is an “intermediate” definition.
• A fault rock in which the matrix has deformed by dominantly crystal-plastic
mechanisms, even though more resistant grains may deform by cracking and
breaking. This definition tends to be that most used today.
The problem with these definitions is that they tend to be genetic rather than descriptive, and they don’t
take into account the fact that, under the same temperature and pressure conditions, different minerals
will deform by different mechanisms.
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Lecture 2 0 147Faults II: Slip Sense & Surface Effects
LECTURE 20—FAULTS II: SLIP SENSE & SURFACE
EFFECTS
20.1 Surface Effects of Faulting
Faults that cut the surface of the Earth (i.e. the tip line intersects the surface) are known as
emergent faults. They produce a topographic step known as a scarp:
fault scarp fault-line scarp
The scarp can either be the surface exposure of the fault plane, in which case it is a fault scarp or it can
simply be a topographic bump aligned with, but with a different dip than, the fault (a fault-line scarp).
Where scarps of normal faults occur in mountainous terrain, one common geomprohic indicator of the
fault line are flat irons along the moutain front:
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flat irons
Lecture 2 0 148Faults II: Slip Sense & Surface Effects
These are particularly common in the Basin and Range of the western United States. In areas of strike-slip
faulting, features such as off-set stream valleys, and sag ponds — wet swampy areas along the fault trace
— are common (sag ponds can also be seen along normal and thrust fault traces).
sag pond
off-set stream
Faults which do not cut the surface of the Earth (i.e. their tip lines do not intersect the surface) are
called blind faults. They can still produce topographic uplift, particularly if the tip line is close to the
surface, but the uplift is broader and more poorly defined than with emergent faults. Blind faults have
stirred quite a bit of interest in recent years because of their role in seismic hazard. The recent Northridge
Earthquake occurred along a blind thrust fault.
20.2 How a Fault Starts: Riedel Shears
clay cake
Much of our basic understanding of the array of structures that develop during faulting comes from
experiments with clay cakes deformed in shear, as in the picture, above. These experiments show that
strike-slip is a two stage process involving
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Lecture 2 0 149Faults II: Slip Sense & Surface Effects
• pre-rupture structures, and
• post-rupture structures.
20.2.1 Pre-rupture Structures
Riedel Shears :
R (synthetic)
R' (antithetic)
φ2
90 -
φ2
90 - φ
The initial angles that the synthetic and antithetic shears form at is controlled by their coefficient of
internal friction. Those angles and the above geometry mean that the maximum compression and the
principal shortening axis of infinitesimal strain are both oriented at 45° to the shear zone boundary.
With continued shearing they will rotate (clockwise in the above diagram) to steeper angles.
Because the R' shears are originally at a high angle to the shear zone they will rotate more quickly and
become inactive more quickly than the R shears. In general, the R shears are more commonly observed,
probably because they have more displacement on them.
Riedel shears can be very useful for determining the sense of shear in brittle fault zones.
Extension Cracks: In some cases, extension cracks will form, initially at 45° to the shear zone:
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Lecture 2 0 150Faults II: Slip Sense & Surface Effects
45°
These cracks can serve to break out blocks which subsequently rotate in the shear zone, domino-style:
Note that the faults between the blocks have the opposite sense of shear than the shear zone itself.
20.2.2 Rupture & Post-Rupture Structures
A rupture, a new set of shears, called “P-shears”, for symmetric to the R-shears. These tend to
link up the R-shears, forming a through-going fault zone:
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R (synthetic)
R' (antithetic)
φ2
P-shears
Lecture 2 0 151Faults II: Slip Sense & Surface Effects
20.3 Determination of Sense of Slip
To understand the kinematics of fault deformation, we must determine their slip. The slip vector
is composed of two things: (1) the orientation of a line along which the blocks have moved, and (2) the
sense of slip (i.e. the movement of one block with respect to the other).
Geological features usually give us one or the other of these. Below, I’ll give you a list of features,
many of which may not mean much to you right now. Later in the course, we will describe several in
detail. I give you their names now just so that you’ll associate them with the determination of how a fault
moves.
Orientation
Frictional-Cataclastic faults
grooves, striae, slickensides, slickenlines
Crystal plastic
mineral lineations
Sense
Frictional-cataclastic
Riedel shears, steps, tool marks, sigmoidal gash fractures, drag folds, curved mineral
fibers
Crystal plastic mechanisms
Sheath folds, S-C fabrics, asymmetric c-axis fabrics, mica fish, asymmetric augen,
fractured and rotated mineral grains
draft date: 9 March, 1999
Lecture 2 0 152Faults II: Slip Sense & Surface Effects
draft date: 9 March, 1999
"RO"-Type (top): The fault surface is totallycomposed of R and R' surfaces. There are no Psurfaces or an average surface of the fault plane.Fault surface has a serrated profile. Not verycommon.
Riedel Shears
These features are well described in the classic papers by Tchalenko (1970), Wilcox et al. (1973),etc. The discussion below follows Petit (1987). It is uncommon to find unambiguous indicators ofmovement on the R or R' surfaces and one commonly interprets them based on striation and anglealone In my experience, R shears can be misleading and one should take particular care in usingthem without redundant indicators or collaborative indicators of a different type.
diagrams modified after Petit (1987)
"RM"-Type (middle): The main fault surface iscompletely striated. R shears dip gently (5-15°)into the wall rock; R' shears are much lesscommon. The tip at the intersection of R and themain fault plane commonly breaks off, leaving anunstriated step.
Lunate fractures (bottom): R shears commonlyhave concave curvature toward the fault plane,resulting in "half moon" shaped cavities ordepressions in the fault surface.
Orientations of Common Fault-Related Features
90° − φ/2
φ/2
45°45°
R
R'P
~10°
Shear Fractures Veins
R = synthetic Riedel shearR' = antithetic Riedel shearP = P-shear; φ = angle of internal friction Same sense of shear applies to all following diagrams
RR'
[sense of shear is top (missing) block to the right in all the diagrams on this page]
Lecture 2 0 153Faults II: Slip Sense & Surface Effects
draft date: 9 March, 1999
"PO"-Type (bot tom) : T surfaces are missingentirely. Striated P surfaces face in direction ofmovement of the block in which they occur. Leeside of asperities are unstriated.
T
P
Striated P-Surfaces
These features were first described by Petit (1987). The fault plane is only partially striated, andthe striations only appear on the up-flow sides of asperities.
"PT"-Type (top & middl e): ~ planar, non-striatedsurfaces dip gently into the wall rock. Petit (1987)calls these "T" surfaces because of lack ofevidence for shear, but they commonly form atangles more appropriate for R shears. Striated Psurfaces face the direction in which that blockmoved. Steep steps developed locally atintersection between P and T. P surfaces may berelatively closely spaced (top) or much farther apart(middle).
diagrams modified after Petit (1987)
diagrams modified after Petit (1987)
Unstriated Fractures ("T fractures")Although "T" refers to "tension" it is a mistake to consider these as tensile fractures. Theycommonly dip in the direction of movement of the upper (missing) block and may be filled withveins or unfilled.
Crescent Marks (bottom) Commonly concave inthe direction of movement of the upper (missing)block. They virtually always occur in sets and areusually oriented at a high angle to the fault surface.They are equivalent the "crescentic fractures"formed at the base of glaciers.
"Tensile Fractures" (top): If truely tensile in originand formed during the faulting event, these shouldinitiate at 45° to the fault plane and then rotate tohigher angles with wall rock deformation. Manynaturally occuring examples are found with anglesbetween 30° and 90°. They are referred to as"comb fractures" by Hancock and Barka (1987).
T
veins or empty fractures
[sense of shear is top (missing) block to the right in all the diagrams on this page]
Lecture 2 0 154Faults II: Slip Sense & Surface Effects
draft date: 9 March, 1999
" S-C" Fabri cs
Although commonly associated with ductile shear zones, features kinematically identical to S-Cfabrics also occur in brittle fault zones. There are two types: (1) those that form in clayey gouge inclastic rocks and (2) those that form in carbonates. They have not been described extensively inthe literature. This is somewhat odd because I have found them one of the most useful, reliable,and prevalent indicators.
Clayey Gouge fabric (top ): Documented byChester and Logan (1987) and mentioned by Petit(1987). Fabric in the gouge has a sigmoidal shapevery similar to S-surfaces in type-1 mylonites. Thisimplies that the maximum strain in the gouge anddisplacement in the shear zone is along the walls.Abberations along faults may commonly be relatedto local steps in the walls.
Carbonate fabric (top ): This feature isparticularly common in limestones. A pressuresolution cleavage is localized in the walls of a faultzone. Because maximum strain and displacementis in the center of the zone rather than the edges,the curvature has a different aspect than the clayeygouge case. The fault surface, itself, commonlyhas slip-parallel calcite fibers.
gouge
pressure solution cleavage
Mineral Fibers & Tool Marks
Tool Marks (bottom): This feature is most com-mon in rocks which have clasts much harder thatthe matrix. During faulting, these clasts gougethe surface ("asperity ploughing" of Means[1987]), producinig trough shaped grooves.Although some attempt to interpret the groovesalone, to make a reliable interpretation, one mustsee the clast which produced the groove as well.Other- wise, it is impossible to tell if the deepestpart of the groove is where the clast ended up orwhere it was plucked from.
Mineral Fibers and Steps (top): When faultingoccurs with fluids present along an undulatoryfault surface or one with discrete steps, fiberousminerals grow from the lee side of the asperitieswhere stress is lower and/or gaps open up.These are very common in carbonate rocks andless so in siliceous clastic rocks.
[sense of shear is top (missing) block to the right in all the diagrams on this page]
Lecture 2 0 155Faults II: Slip Sense & Surface Effects
draft date: 9 March, 1999
Lecture 2 1 157Faults III: Dynamics & Kinematics
LECTURE 21—FAULTS III: DYNAMICS & K INEMATICS
21.1 Introduction
Remember that the process of making a fault in unfractured, homogeneous rock mass could be
described by the Mohr’s circle for stress intersecting the failure envelope.
2θ
σ *1σ *3
σ n
σ sfailure envelop
Under upper crustal conditions, the failure envelope has a constant slope and is referred to as the
Coulomb failure criteria:
σs = So + σn* µ, where µ = tan φ.
What this says is that, under these conditions, faults should form at an angle of 45° - φ/2 with respect to
σ1. Because for many rocks, φ ≈ 30°, fault should form at about 30° to the maximum principal stress, σ1:
σ1
45 + φ/2
45 - φ/2
Lecture 2 1 158Faults III: Dynamics & Kinematics
Furthermore the Mohr’s Circle shows that, in two dimensions, there will be two possible fault
orientations which are symmetric about σ1.
σ1
45 + φ/2
45 - φ/2
Such faults are called conjugate fault sets and are relatively common in the field. The standard
interpretation is that σ1 bisects the acute angle and σ3 bisects the obtuse angle between the faults.
21.2 Anderson’s Theory of Faulting
Around the turn of the century, Anderson realized the significance of Coulomb failure, and
further realized that, because the earth’s surface is a “free surface” there is essentially no shear stress parallel to
the surface of the Earth. [The only trivial exception to this is when the wind blows hard.]
Therefore, one of the three principal stresses must be perpendicular to the Earth’s surface, because
a principal stress is always perpendicular to a plane with no shear stress on it. The other two principal
stresses must be parallel to the surface:
σ vertical,1
σ vertical, or2
σ vertical3
This constraint means that there are very few possible fault geometries for near surface deformation.
They are shown below:
Lecture 2 1 159Faults III: Dynamics & Kinematics
σ1
σ3
45 - φ/2
Thrust faults
dip < 45°
45 + φ/2
σ1
σ3 Normal faults
dip > 45°
Strike-slipσ1
σ3σ2
Anderson’s theory has proved to be very useful but it is not a universal rule. For example, the
theory predicts that we should never see low-angle normal faults near the Earth’s surface but, as we shall
see later in the course, we clearly do see them. Likewise, high-angle reverse faults exist, even if they are
not predicted by the theory.
There are two basic problems with Anderson’s Theory:
• Rocks are not homogeneous as implied by Coulomb failure but commonly
have planar anisotropies. These include bedding, metamorphic foliations,
and pre-existing fractures. If σ1 is greater than about 60° to the planar anisotropy
then it doesn’t matter; otherwise the slip will probably occur parallel to the
Lecture 2 1 160Faults III: Dynamics & Kinematics
anisotropy.
• There is an implicit assumption of plane strain in Anderson’s theory -- no
strain is assumed to occur in the σ2 direction. Thus, only two fault directions
are predicted. In three-dimensional strain, there will be two pairs of conjugate
faults as shown by the work of Z. Reches.
σ1σ3
σ2four possible fault sets in 3D strain
Listric faults and steepening downward faults would appear to present a problem for Anderson’s
theory, but this is not really the case. They are just the result of curving stress trajectories beneath the
Earth’s surface:
Because the stress trajectories curve, the faults must curve. The only requirement is that they intersect the surface at the specified angles
Lecture 2 1 161Faults III: Dynamics & Kinematics
21.3 Strain from Fault Populations
Anderson’s law is commonly too restrictive for real cases where the Earth contains large numbers
of pre-existing fractures of various orientations in a variety of rock materials. Thus, structural geologists
have developed a number of new techniques to analyze fault populations. There are two basic ways to
study populations of faults: (1) to look at them in terms of the strain that they produce — e.g. kinematic
analysis, or (2) to interpret the faults in terms of the stress which produced them, or dynamic analysis.
Both of these methods have their advantages and disadvantages and all require knowledge of the sense
of shear of all of the faults included in the analysis.
21.3.1 Sense of Shear
Brittle shear zones have been the focus of increasing interest during the last decade. Their
analysis, either in terms of kinematics or dynamics, require that we determine the sense of shear. Because
piercing points are rare, we commonly need to resort to an interpretation of minor structural features
along, or within the shear zone itself. In general, these features include such things as (listed roughly in
order of decreasing reliability):
• sigmoidal extension fractures
• steps with mineral fibers
• shear zone foliations (“brittle S–C fabrics”)
• drag folds
• Riedel shears (with sense-of shear indicators)
• tool marks
21.3.2 Kinematic Analysis of Fault Populations
The simplest kinematic analysis, which takes it’s cue from the study of earthquake fault plane
solutions is the graphical P & T axis analysis. Despite their use in seismology as “pressure” and
“tension”, respectively, P and T axes are the infinitesimal strain axes for a fault. Perhaps the greatest
advantage of P and T axes are that, independent of their kinematic or dynamic significance, they are a
simple, direct representation of fault geometry and the sense of slip. That is, one can view them as simply
a compact alternative way of displaying the original data on which any further analysis is based. The
results of most of the more sophisticated analyses commonly are difficult to relate to the original data;
such is not the problem for P and T axes. For any fault zone, you can identify a movement plane, which
Lecture 2 1 162Faults III: Dynamics & Kinematics
is the plane that contains the vector of the fault and the pole to the fault. The P & T axes are located in the
movement plane at 45° to the pole:
movem
ent plane
striae & slip sense (arrowshows movement of
hanging wall)
P-axis
pole to fault plane
b-axis
T-axisfa
ult p
lane
21.3.3 The P & T Dihedra
MacKenzie (1969) has pointed out, however, that particularly in areas with pre-existing fractures
(which is virtually everywhere in the continents) there may be important differences between the principal
stresses and P & T. In fact, the greatest principal stress may occur virtually anywhere within the
P-quadrant and the least principal stress likewise anywhere within the T-quadrant. The P & T dihedra
method proposed by Angelier and Mechler (1977) takes advantage of this by assuming that, in a population
of faults, the geographic orientation that falls in the greatest number of P-quadrants is most likely to
coincide with the orientation of σ1. The diagram, below, shows the P & T dihedra analysis for three
faults:
Lecture 2 1 163Faults III: Dynamics & Kinematics
3
3 3 3 3 3 2 2
3 3 3 3 3 3 2 1 1 1 0
3 3 3 3 3 3 3 1 1 1 1 1 0
3 3 3 3 3 3 1 1 1 1 1 1 0
3 3 3 3 3 3 2 1 1 1 0 0 0 1 2
3 3 3 3 3 3 1 0 0 0 0 0 0 0 2
3 3 3 3 3 1 0 0 0 0 0 0 0 0 2
3 3 3 3 3 1 1 0 0 0 0 0 0 0 0 2 3
3 3 3 2 1 0 0 0 0 0 0 0 0 0 3
3 2 2 2 1 0 0 0 0 0 0 0 0 2 3
3 2 2 1 1 0 0 0 0 0 0 0 1 3 3
1 2 1 1 0 0 0 0 0 0 1 2 3
1 1 2 1 0 0 0 0 1 1 2 3 3
1 2 2 1 1 1 1 1 3 3 3
1 2 2 2 3 3 3
3
In the diagram, the faults are the great circles with the arrow-dot indicating the striae. The conjugate for
each fault plane is also shown. The number at each grid point shows the number of individual P-quadrants
that coincide with the node. The region which is within the T-quadrants of all three faults has been
shaded in gray. The bold face zeros and threes indicate the best solutions obtained using Lisle’s (1987)
AB-dihedra constraint. Lisle showed that the resolution of the P & T dihedra method can be improved by
considering how the stress ratio, R, affects the analysis. The movement plane and the conjugate plane
divide the sphere up into quadrants which Lisle labeled “A” and “B” (see figure below). If one principal
stress lies in the region of intersection of the appropriate kinematic quadrant (i.e. either the P or the T
quadrant) and the A quadrant then the other principal stress must lie in the B quadrant. In qualitative
terms, this means that the σ3 axis must lie on the same side of the movement plane as the σ1 axis.
Lecture 2 1 164Faults III: Dynamics & Kinematics
S
O
A
BA
Bpole to fault
fault plane
conjugate plane
movement plane
σ1
σ3
σ3
possible positions of σσσσ given σσσσas shown
3 1
21.4 Stress From Fault Populations 1
Since the pioneering work of Bott (1959), many different methods for inferring certain elements of
the stress tensor from populations of faults have been proposed. These can be grouped in two broad
categories: graphical methods (Compton, 1966; Arthaud, 1969; Angelier and Mechler, 1977; Aleksandrowski,
1985; and Lisle, 1987) and numerical techniques (Carey and Brunier, 1974; Etchecopar et al., 1981; Armijo
et al., 1982; Angelier, 1984, 1989; Gephart and Forsyth, 1984; Michael, 1984; Reches, 1987; Gephart, 1988;
Huang, 1988).
21.4.1 Assumptions
Virtually all numerical stress inversion procedures have the same basic assumptions:
1. Slip on a fault plane occurs in the direction of resolved shear stress (implying
that local heterogeneities that might inhibit the free slip of each fault plane --
including interactions with other fault planes -- are relatively insignificant).
1This supplemental section was co-written John Gephart and Rick Allmendinger and is adapted from the1989 Geological Society of America shortcourse on fault analysis.
2. The data reflect a uniform stress field (both spatially and temporally)—this
Lecture 2 1 165Faults III: Dynamics & Kinematics
requires that there has been no post-slip deformation of the region which
would alter the fault orientations.
While the inverse techniques may be applied to either fault/slickenside or earthquake focal
mechanism data, these assumptions may apply more accurately to the latter than the former. Earthquakes
may be grouped in geologically short time windows, and represent sufficiently small strains that rotations
may be neglected. Faults observed in outcrop, on the other hand, almost certainly record a range of
stresses which evolved through time, possibly indicating multiple deformations. If heterogeneous stresses
are suspected, a fault data set can easily be segregated into subsets, each to be tested independently. In
any case, to date there have been many applications of stress inversion methods from a wide variety of
tectonic settings which have produced consistent and interpretable results.
21.4.2 Coordinate Systems & Geometric Basis
Several different coordinate systems are use by different workers. The ones used here are those
of Gephart and Forsyth (1984), with an unprimed coordinate system which is parallel to the principal
stress directions, and a primed coordinate system fixed to each fault, with axes parallel to the pole, the
striae, and the B-axis (a line in the plane of the fault which is perpendicular to the striae) of the fault, as
shown below:
X3
X1'
X3'
X2'
X1
X2
cos -113
faultstriae
3
1
2
[note -- for the convenience ofdrawing, both sets of axes areshown as left handed]
Lecture 2 1 166Faults III: Dynamics & Kinematics
X1'
X2'
X3'
X1
X2
X3
σσσσ1
σσσσ3
σσσσ2
faultpole
faul
tpl
ane
striae
cos β-113
The relationship between the principal stress and the stress on the one fault plane shown is given
by a standard tensor transformation:
σij′ = βik βjl σkl .
In the above equation, bik is the transformation matrix reviewed earlier, skl are the regional stress
magnitudes, and sij' are the stresses on the plane. Expanding the above equation to get the components of
stress on the plane in terms of the principal stresses, we get:
σ11′ = β11β11σ1 + β12β12σ2 + β13β13σ3 [normal traction],
σ12′ = β11β21σ1 + β12β22σ2 + β13β23σ3 [shear traction ⊥ striae],
and σ13′ = β11β31σ1 + β12β32σ2 + β13β33σ3 [shear traction // striae].
From assumption #1 above we require that σ12' vanishes, such that:
0 = β11β21σ1 + β12β22σ2 + β13β23σ3 .
Combining this expression with the condition of orthogonality of the fault pole and B axis:
Lecture 2 1 167Faults III: Dynamics & Kinematics
0 = β11β21 + β12β22 + β13β23 .
yieldsσ2 − σ1σ3 − σ1
≡ R = − β1 3 β2 3
β1 2 β2 2 . (21.1)
where the left-hand side defines the parameter, R, which varies between 0 and 1 (assuming that σ1 ≥ σ2 ≥
σ3) and provides a measure of the magnitude of σ2 relative to σ1 and σ3. A value of R near 0 indicates that
σ2 is nearly equal to σ1; a value near 1 means σ2 is nearly equal to σ32. Any combination of principal stress
and fault orientations which produces R > 1 or R < 0 from the right-hand side of (21.1) is incompatible
(Gephart, 1985). A further constraint is provided by the fact that the shear traction vector, σ13′ , must have
the same direction as the slip vector (sense of slip) for the fault; this is ensured by requiring that σ13′ > 0.
Equation (21.1) shows that, of the 6 independent components of the stress tensor, only four can be
determined from this analysis. These are the stress magnitude parameter, R, and three stress orientations
indicated by the four βij terms (of which only three are independent because of the orthogonality relations).
21.4.3 Inversion Of Fault Data For Stress
Several workers have independently developed schemes for inverting fault slip data to obtain
stresses, based on the above conditions but following somewhat different formulations. In all cases, the
goal is to find the stress model (three stress directions and a value of R) which minimizes the differences
between the observed and predicted slip directions on a set of fault planes.
The first task is to decide: What parameter is the appropriate one to minimize in finding the
optimum model? The magnitude of misfit between a model and fault slip datum reflects either: (1) the
minimum observational error, or (2) the minimum degree of heterogeneity in stress orientations, in order
to attain perfect consistency between model and observation. Two simple choices may be considered:
Many workers (e.g. Carey and Brunier, 1974; Angelier, 1979, 1984) define the misfit as the angular
2An similar parameter was devised independently by Angelier and coworkers (Angelier et al., 1982;Angelier, 1984, 1989):
Φ = −−
σ σσ σ
2 3
1 3
.
In this case, if Φ = 0, then σ2 = σ3, and if Φ = 1, then σ2 = σ1. Thus, Φ = 1 – R.
difference between the observed and predicted slip vector measured in the fault plane (referred to as a
Lecture 2 1 168Faults III: Dynamics & Kinematics
“pole rotation” because the angle is a rotation angle about the pole to the fault plane). This implicitly
assumes that the fault plane is perfectly known, such that the only ambiguity is in the orientation of the
striae (right side of figure below). Such an assumption may be acceptable for fault data from outcrop for
which it is commonly easier to measure the fault surface orientation than the orientation of the striae on
the fault surface. Alternatively, one can find the smallest rotation of coupled fault plane and striae about
any axis that results in a perfect fit between data and model (Gephart and Forsyth, 1984)—this represents
the smallest possible deviation between an observed and predicted fault slip datum, and can be much
smaller than the pole rotation, as shown in the left-hand figure below (from Gephart, in review). This
“minimum rotation” is particularly useful for analyzing earthquake focal mechanism data for which
there is generally similar uncertainties in fault plane and slip vector orientations.
σ1
σ2
σ3
4.8°
σ3
σ1
σ215.3°
faultplane
striae
calc. striae
minimum rotation pole rotation
conjugate plane
faultplane
Because of the extreme non-linearity of this problem, the most reliable (but computationally
demanding) procedure for finding the best stress model relative to a set of fault slip data involves the
application of an exhaustive search of the four model parameters (three stress directions and a value of R)
by exploring sequentially on a grid (Angelier, 1984; Gephart and Forsyth, 1984). For each stress model
examined the rotation misfits for all faults are calculated and summed; this yields a measure of the
acceptability of the model relative to the whole data set—the best model is the one with the smallest sum
of misfits. Following Gephart and Forsyth (1984), confidence limits on the range of acceptable models
Lecture 2 1 169Faults III: Dynamics & Kinematics
can then be calculated using statistics for the one norm misfit, after Parker and McNutt (1980). In order to
increase the computational efficiency of the inverse procedure, a few workers have applied some approx-
imations which enable them to linearize the non-linear conditions in this analysis (Angelier, 1984; Michael,
1984); naturally, these lead to approximate solutions which in some cases vary significantly from those of
more careful analyses. The inversion methods of Angelier et al. (1982, eq. 9 p. 611) and Michael (1984)
make the arbitrary assumption that the first invariant of stress is zero (σ11 + σ22 + σ33 = 0). Gephart (in
review) has noted that this implicitly prescribes a fifth stress parameter, relating the magnitudes of
normal and shear stresses (which should be mutually independent), the effect of which is seldom evaluated.
Following popular convention in inverse techniques, many workers (e.g. Michael, 1984; Angelier
et al., 1982) have adopted least squares statistics in the stress inversion problem (e.g. minimizing the sum
of the squares of the rotations). A least squares analysis, which is appropriate if the misfits are normally
distributed, places a relatively large weight on extreme (poorly-fitting) data. If there are erratic data (with
very large misfits), as empirically is often the case in fault slip analyses, then too much constraint is
placed on these and they tend to dominate a least squares inversion. One can deal with this by rejecting
anomalous data (Angelier, 1984, suggests truncating the data at a pole rotation of 45°), or by using a
one-norm misfit, which minimizes the sum of the absolute values of misfits (rather than the squares of
these), thus placing less emphasis on such erratic data, and achieving a more robust estimate of stresses
(Gephart and Forsyth, 1984).
21.5 Scaling Laws for Fault Populations
Much work over the last decade has shown that fault populations display power law scaling
characteristics (i.e., “fractal”). In particular, the following features have been shown to be scale invariant:
• trace length vs. cummulative number
• displacement vs. cummulative number
• trace length vs. displacement
• geometric moment vs. cummulative number
If the power law coefficients were known with certainty, then these relationships would have important
predictive power. Unfortunately, there are very few data sets which have been sample with sufficient
completeness to enable unambiguous determination of the coefficients.
Lecture 2 2 170Faults IV: Mechanics of Thrust Faults
LECTURE 22—FAULTS IV: MECHANICS OF THRUST FAULTS
22.1 The Paradox of Low-angle Thrust Faults
In many parts of the world, geologists have recognized very low angle thrust faults in which
older rocks are placed over younger. Very often, the dip of the fault surface is only a few degrees. Such
structures were first discovered in the Alps around 1840 and have intrigued geologists ever since. The
basic observations are:
1. Faults are very low angle, commonly < 10°;
2. Overthrust blocks of rock are relatively thin, ~ 5 - 10 km;
3. The map trace of individual faults is very long, 100 - 300 km; and
4. The blocks have been displaced large distances, 10s to 100s of km.
What we have is a very thin sheet of rock that has been pushed over other rocks for 100s of kilometers.
This process has been likened to trying to push a wet napkin across a table top: There’s no way that the
napkin will move as a single rigid unit.
The basic problem, and thus the “paradox” of large overthrusts, is that rocks are apparently too
weak to be pushed from behind over long distances without deforming internally. That rocks are so
weak has been noted by a number of geologists, and was well illustrated in a clever thought experiment
by M. King Hubbert in the early 1950's. He posed the simple question, “if we could build a crane as big
as we wanted, could we pick up the state of Texas with it?” He showed quite convincingly that the
answer is no because the rocks that comprise the state (any rock in the continental crust) are too weak to
support their own weight.
22.2 Hubbert & Rubey Analysis
In 1959, Hubbert along with W. Rubey wrote a classic set of papers which clearly laid out the
mechanical analysis of the paradox of large overthrusts. I want to go through their analysis because it is a
superb illustration of the simple mechanical analysis of a structural problem.
Lecture 2 2 171Faults IV: Mechanics of Thrust Faults
The simplest expression of the problem is to imagine a rectangular block sitting on a flat surface.
When we push this block on the left side, the friction along the base, which is a function of the weight of
the block times the coefficient of friction, will resist the tendency of the block to slide to the right. The
basic boundary conditions are:
X
Z
x
z
zzσ
xxσσ zx
Note that indices used in the diagram above are the standard conventions that were used when we
discussed stress.
When the block is just ready to move, the applied stress, σxx, must just balance the shear stress at
the base of the block, σzx . We can express this mathematically as:
σ σxx
z
zx
xdz dx
0 0∫ ∫= (22.1)
We can get an expression for σzx easily enough because it’s just the frictional resistance to sliding,
which from last time is
σs = µ σn ,
or, in our notation, above
σzx = µ σzz . (22.2)
The normal stress, σzz, is just equal to the lithostatic load:
σzz = ρ g z .
Lecture 2 2 172Faults IV: Mechanics of Thrust Faults
So,
σzx = µ ρ g z .
We can now solve the right hand side of equation 22.1 (p. 171):
σ µρxx
z xdz gz dx
0 0∫ ∫=and
σ µρxx
zdz gzx
0∫ = .
Now we need to evaluate the left side of the equation. Remember that we want to find the
largest stress that the block can support without breaking internally as illustrated in the diagram below.
σ1 σ
xx
The limiting case then, is where the block does fracture internally, in which case there is no shear on the
base. So, in this limiting case
σ1 = σxx and σ3 = σzz .
Now to solve this problem, we need to derive a relationship between σ1 and σ3 at failure, which we can
get from Mohr’s circle for stress. From the geometry of the Mohr’s Circle, below, we see that:
σ σ σ σφ
φ1 3 1 3
2 2− = + +
So
tansin
Lecture 2 2 173Faults IV: Mechanics of Thrust Faults
σ 3σ 1 -
2
σ 3 σ 1φ
S o
σ 3σ 1 +
2
S otan φ
Solving for σ1 in terms of σ3 we get:
σ1 = Co + K σ3 , (22.3)
where
C S Ko o= 2 and K = +−
11
sinsin
φφ
. (22.4)
So,
σxx = σ1 = Co + K σ3 = Co + K σzz .
But σzz = ρgz, so
σxx = Co + K ρgz .
Now, we can evaluate the left side of equation 22.1 (p. 171):
σ µρxx
zdz gzx
0∫ =
C K gz dz gzxo
z+( ) =∫ ρ µρ
0
Lecture 2 2 174Faults IV: Mechanics of Thrust Faults
C zK gz
gzxo + =ρ µρ2
2 .
Dividing through by z and solving for x, we see that the maximum length of the block is a linear function
of its thickness:
xC
g
Kzo= +µρ µ2
. (22.5)
Now, let’s plug in some realistic numbers. Given
φ = 30°
µ = 0.58
So = 20 Mpa
ρ = 2.3 gm/cm3,
we can calculate that
Co = 69.4 Mpa
K = 3.
With these values, equation 20-5 becomes:
xmax = 5.4 km + 2.6 z .
Thus,
Thickness Maximum Length
5 km 18.4 km
10 km 31.4 km
22.3 Alternative Solutions
These numbers are clearly too small, bearing out the paradox of large thrust faults which we
stated at the beginning of this lecture. Because large thrust faults obviously do exist, there must be
something wrong with the model. Over the years, people have suggested several ways to change it.
Lecture 2 2 175Faults IV: Mechanics of Thrust Faults
1. Rheology of the basal zone is incorrect—In our analysis, above, we assumed that friction
governed the sliding of the rock over its base. However, it is likely that in some rocks, especially shales or
evaporites, or where higher temperatures are involved, plastic or viscous rheologies are more appropriate.
This would change the problem significantly because the yield strengths in those cases is independent of
the normal stress.
2. Pore Pressure—Pore pressure could reduce the effective normal stress on the fault plane [σzz*
= σzz - Pf] and therefore it would also reduce the frictional resistance due to sliding, σzx (from equation
20-2). There is, however, a trade off because, unless you somehow restrict the pore pressure to just the
fault zone, excess fluid pressure will make the block weaker as well (and we want the block as strong as
possible).
Hubbert and Rubey proposed that pore pressure was an important part of the answer to this
problem and they introduced the concept of the fluid pressure ratio:
λρf
fP
gz= = pore fluid pressure
lithostatic stress
0 50 100 150 200
Maximum length, x (km)max
fF
luid
Pre
ssur
e in
faul
t zon
e,
λλλλ 0.9
1.0
0.8
0.7
0.6
0.5
z = 5 km
z = 10 km
[after Suppe, 1985]
λ = 0.435b
λ = λ
bf
λ = λb
fλ =
0.435
b
The graph above show how pore pressure in the block (plotted as λ b) and pore pressure along the fault
(λf) affect the maximum length of the block. For blocks 5 and 10 km thick, two cases are shown, one
where there is no difference in pore pressure between block and fault, and the other where the pore
pressure is hydrostatic (assuming a density, ρ = 2.4 gm/cm3). The diagram was constructed assuming Co
Lecture 2 2 176Faults IV: Mechanics of Thrust Faults
50 MPa and K = 3.
3. Thrust Plates Slide Downhill—This was the solution that Hubbert and Rubey favored
(aided by pore pressure), but the vast amount of seismic reflection data in thrust belts which has been
collected since they wrote their article shows that very few thrust faults move that way. Most major
thrust faults moved up a gentle slope of 2 - 10°.
There are major low-angle fault bounded blocks that slide down hill. The Heart Mountain
detachment in NW Wyoming is a good example.
4. Thrust Belts Analogous to Glaciers—Several geologists, including R. Price (1973) and D.
Elliott (1976) have proposed that thrust belts basically deform like glaciers. Like gravity sliding, the
spreading of a glacier is driven by its own weight, rather than being pushed from behind by some
tectonic interaction. Glaciers, however, can flow uphill as long as the topographic slope is inclined in the
direction of flow.
horizontal extension ("spreading")
thrust faulting at toe
This model was very popular in the 1970’s, but the lack of evidence for large magnitude horizontal
extension in the rear of the thrust belt, or “hinterland” has made it decline in popularity.
5. Rectangular Shape Is Not Correct—This is clearly an important point. Thrust belts and
individual thrust plates within them are wedge-shaped rather than rectangular as originally proposed by
Hubbert and Rubey. Many recent workers, including Chapple (1978) and Davis, Suppe, & Dahlen (1983,
and subsequent papers) have emphasized the importance of the wedge.
α
β
Lecture 2 2 177Faults IV: Mechanics of Thrust Faults
The wedge taper is defined the sum of two angles, the topographic slope, α , and the slope of the
basal décollement, β, as shown above. Davis et al. (1983) proposed that the wedge grows “self-similarly”,
maintaining a constant taper.
basal décollement slope, β
topographic slope, α
In their wedge mechanics, they propose the following relation between α and β when the wedge is a
critical taper:
α β λ µ βλ
+ =−( ) +−( ) +
1
1 1k
where µ is the coefficient of friction, λ is the Hubbert-Rubey pore pressure ratio, and k is closely related to
the “earth pressure coefficient” which was derived above in equations 20-3 and 20-4.
If the basal friction increases, either by changing the frictional coefficient, µ, or by increasing the
normal stress across the fault plane (which is the same as decreasing λ ), the taper of the wedge will
increase. Note that, as λ → 1, α → 0. In other words, when there is no normal stress across the fault
because the lithostatic load is entirely supported by the pore pressure, there should be no topographic
slope.
If the wedge has a taper less than the critical taper, then it will deform internally by thrust faulting
in order to build up the taper. If its taper is greater than the critical taper, then it will deform by normal
faulting to reduce the taper.
Lecture 2 3 178Fold geometries
LECTURE 23—FOLDS I: GEOMETRY
Folding is the bending or flexing of layers in a rock to produce frozen waves. The layers may be
any planar feature, including sedimentary bedding, metamorphic foliation, planar intrusions, etc. Folds
occur at all scales from microscopic to regional. This first lecture will probably be mostly review for you,
but it’s important that we all recognize the same terminology.
23.1 Two-dimensional Fold Terminology
Antiform
Folds that are convex upward:
Synform
Folds that are concave upward:
To use the more common terms, anticline and syncline, we need to know which layers are older
and which layers are younger. Many folds of metamorphic and igneous rocks should only be described
using the terms antiform and synform.
Anticline
oldest rocks in the center of the fold
Syncline
youngest rocks in the center of the fold
older
younger
It may, at first, appear that there is no significant difference between antiforms and anticlines and synforms
and synclines, but this is not the case. You can easily get antiformal synclines and synformal anticlines,
Lecture 2 3 179Fold geometries
for example:
Synformal anticline
Folds that are concave upward, but the oldest beds are in the middle:
younger
older
Folds can also be symmetric or asymmetric. The former occurs when the limbs of the folds are
the same length and have the same dip relative to their enveloping surface. In asymmetric folds, the
limbs are of unequal length and dip:
enveloping surface
limb
Symmetric folds:
enveloping surface
limb
Asymmetric folds:
Overturned folds:
the tops of the more steeply dipping beds are facing or verging to the eastin this picture
EW
In asymmetric and overturned folds the concept of vergence or facing is quite important. This is the
direction that the shorter, more steeply dipping asymmetric limb of the fold faces, or the arrows in the
above pictures.
Lecture 2 3 180Fold geometries
Numerous different scales folds can be superimposed on each other producing what are known
as anticlinoria and synclinoria:
anticlinoria synclinoria
Two final terms represent special cases of tilted or folded beds:
Monocline Homocline
23.2 Geometric Description of Folds
23.2.1 Two-dimensional (Profile) View:
The most important concept is that of the hinge, which is the point or zone of maximum curvature
in the layer. Other terms are self-explanatory:
Lecture 2 3 181Fold geometries
wavelength
amplitude
hinge pointhinge zone
limb
inflection point
Note that the amplitude is the distance from the top (or bottom) of the folds to the inflection point.
23.2.2 Three-dimensional View:
In three dimensions, we can talk about the hinge line, which may be straight or curved, depending
on the three-dimensional fold geometry. The axial surface contains all of the hinge lines. It is more
commonly referred to as the “axial plane” but this is a special case where all of the hinge lines lie in a
single plane.
hinge line -- the line connecting allthe points of maximum curvaturein a single layer
crest line -- the line which liesalong the highest points in afolded layer
trough line -- the line which liesalong the lowest points in afolded layeraxial surface -- the surface containing
all of the hinge lines of all of the layers
In practice, you specify the orientation of the hinge line by measuring its trend and plunge. This information,
alone, however, is insufficient to totally define the orientation of the fold. For example, all of the folds
below have identical hinge lines, but are clearly quite different:
Lecture 2 3 182Fold geometries
To completely define the orientation of a fold you need to specify both the trend and plunge of the hinge
line and the strike and dip of the axial surface. The orientation of the axial surface alone is not sufficient
either.
Most of the time, you will be representing the fold in two-dimensional projections: cross-sections,
structural profiles or map views. In these cases what you show is the trace of the axial surface, or the
axial trace. This is just the intersection between the axial surface and the plane of your projection.
23.3 Fold Names Based on Orientation
The hinge line lies within the axial plane, but the trend of the hinge line is only parallel to the
strike of the axial surface when the hinge line is horizontal. If the hinge line is not horizontal, then we say
that the fold is a plunging fold. The following table give the complete names for fold orientations:
090
0
90
Dip of the Axial Surface
Plu
nge
of th
e H
inge
Lin
e
inclined
plunging
sub-vertical
sub-horizontal
upright recumbent
Reclin
ed
Lecture 2 3 183Fold geometries
23.4 Fold Tightness
Another measure of fold geometry is the interlimb angle, shown in the diagram below.
interlimb angle
With this concept, there are yet more descriptive terms for folded rocks:
Name Interlimb Angle
Flat lying, Homocline 180°
Gentle 170 - 180°
Open 90 -170°
Tight 10 - 90°
Isoclinal 0 - 10°
Lecture 2 3 184Fold geometry & kinematics
LECTURE 24 — FOLDS II: GEOMETRY & K INEMATICS
24.1 Fold Shapes
We have been drawing folds only one way, with nice smooth hinges, etc. But, there are many
different shapes that folds can take:
Chevron folds Kink Bands
Cuspate folds Box folds
Disharmonic folds
Lecture 2 4 185Fold geometry & kinematics
24.2 Classification Based on Shapes of Folded Layers
One way of quantifying fold shape is by construction dip isogon diagrams. Dip isogons are lines
which connect points of the same dip on different limbs of folds:
dip isogon
lines of constant dip
Construction of dip isogons:
By plotting dip isogons, you can identify three basic types of folds:
Class 1: Inner beds more curved than outer beds. Dip Isogons fan outward
1A -- isogons on limbs make an obtuse angle with respect to the axial surface
1B -- isogons are everywhere perpendicular to the beds, on both innner and outer surfaces. These are Parallel folds
1C -- isogons on limbs make anacute angle with respect to the axial surface
Class 2 : Inner and outer surfaceshave the same curvature. Dip isogonsare parallel to each other and to theaxial surface. These are Similar folds
Class 3 : Inner surface is less curvedthan the outer surface. Dip isogonsfan inwards
Lecture 2 4 186Fold geometry & kinematics
24.3 Geometric-kinematic Classification:
24.3.1 Cylindrical Folds
Cylindrical folds are those in which the surface can be generated or traced by moving a line
parallel to itself through space. This line is parallel to the hinge line and is called the fold axis. Only
cylindrical folds have a fold axis. Thus, the term fold axis is properly applied only to this type of fold.
If you make several measurements of bedding on a perfectly cylindrical fold and plot them as
great circles on a stereonet, all of the great circles will intersect at a single point. That point is the fold
axis. The poles to bedding will all lie on a single great circle. This is the practical test of whether or not a
fold is cylindrical:
Fold axis
ß diagram π diagram
Fold axis
There are two basic types of cylindrical folds:
Parallel Folds -- In parallel folds, the layer thickness, measured perpendicular to bedding
remains constant. Therefore, parallel folds are equivalent to class 1B folds described above. Some special
types of parallel folds:
Concentric folds are those in which all folded layers have the same center of curvature and the
radius of curvature decreases towards the cores of the folds. Therefore, concentric folds get tighter
towards the cores and more open towards the anticlinal crests and synclinal troughs. The Busk method
Lecture 2 4 187Fold geometry & kinematics
of cross-section construction is based on the concept of concentric folds. These types of folds eventually
get so tight in the cores that the layers are “lifted-off” an underlying layer. The French word for this is
“décollement” which means literally, “unsticking”.
flow of weak rocks
Kink Folds have angular axes and straight limbs. The layers do not have a single center of
curvature. As we will see later in the course, these are among the easiest to analyze quantitatively
γγ
The axis of the kink has to bisect the angles between the two dip panels or the layer thickness will not be preserved
Similar Folds -- The other major class of cylindrical folds is similar folds. These are folds in
which the layer thickness parallel to the axial surface remains constant but thickness perpendicular to the
layers does not. They are called similar because each layer is “similar” (ideally, identical) in curvature to
the next. Thus, they comprise class 2 folds. In similar folds, there is never a need for a décollement
because you can keep repeating the same shapes forever without pinching out the cores:
Lecture 2 4 188Fold geometry & kinematics
Similar Folds
24.3.2 Non-Cylindrical Folds
These folded surfaces cannot be traced by a line moving parallel to itself. In practice what this
means is that the fold shape changes geometry as you move parallel to the hinge line. Thus, they are
complex, three dimensional features. Some special types:
Conical folds -- the folded surfaces in these folds are in the shape of a cone. In other words, the
folded layers converge to a point, beyond which the fold does not exist at all.
There is a very distinct difference between plunging cylindrical fold and conical folds. The conical fold
simply does not exist beyond the tip of the cone. Thus, the shortening due to fold of the layers changes
along strike of the hinge. Conical folds are commonly found at the tip lines of faults.
Sheath folds -- These are a special type of fold that forms in environments of high shear strain,
such as in shear, or mylonite, zones. They are called “sheath” because they are shaped like the sheath of a
knife.
Lecture 2 4 189Fold geometry & kinematics
Sheath folds are particularly useful for determining the sense of shear in mylonite zones. The upper plate
moved in the direction of closure of the sheath. They probably start out as relatively cylindrical folds and
then get distorted in the shear zone.
24.4 Summary Outline
• Cylindrical
Parallel
Concentric
Kink
Similar
• Non-cylindrical
Conical
Sheath
24.5 Superposed Folds
Multiple deformations may each produce their own fold sets, which we label F1, F2, etc., in the
order of formation. This superposition of folds can produce some very complex geometries, which can be
very difficult to distinguish on two dimensional exposures. Ramsay (1967; Ramsay & Huber, 1985) have
come up with a classification scheme based on the orientations of the fold axis (labeled F1, below) and
axial surface (the black plane, below) of the first set of folds with respect to the fold axis (labeled b2,
Lecture 2 4 190Fold geometry & kinematics
below) and the sense of displacement of the layer during the second folding (labeled a2, below). With this
approach, there are four types of superposed fold geometries:
Type 0:
+ =axial plane of 1
fold axis of 1
a 2
b 2
Type 1:
+ =
Type 2:
+ =
Type 3:
+ =
Type 0 results in folds which are indistinguishable from single phase folds. Type 1 produces the
classic “dome and basin” or “egg-carton” pattern. Type 2 folds in cross-section look like boomerangs.
Type 3 folds are among the easiest to recognize in cross-section.
Lecture 2 5 191Fold Kinematics
LECTURE 25—FOLDS III: K INEMATICS
25.1 Overview
Kinematic models of fold development can be divided into five types:
1. Gaussian Curvature,
2. Buckling,
3. Layer parallel shear,
4. Shear oblique to layers, and
5. Pure shear passive flow.
The first two treat only single layers while the third and fourth address multilayers. The final one treats
layers as passive markers, only. All are appropriate only to cylindrical folds. Thus, you should not think
of these as mutually exclusive models. For example, you can have buckling of a single layer with shear
between layers.
25.2 Gaussian Curvature
The curvature of a line, C, is just the inverse of the radius of curvature:
Crcruvature
= 1 .
In any surface, you can identify a line (really a family of parallel lines) with maximum curvature and a
line of minimum curvature. These two are called the “principal curvatures.” The product of the
maximum and minimum curvatures is known as the Gaussian curvature, a single number which describes
the overall curvature of a surface:
C C CGauss = max min .
There is a universal aspect to this: the Gaussian curvature of a surface before and after a deformation remains
constant unless the surface is stretched or compressed (and thereby distorted internally). Although few people
realize it, we deal with this fact virtually daily: corrugated cardboard boxes get their strength from the
fact that the middle layer started out flat before it folded and sandwiched between the two flat outer
Lecture 2 5 192Fold Kinematics
layers. Because its Gaussian curvature started out at zero, it must be zero after folding, meaning that
bending it perpendicular to the folds is not possible without internally deforming the surface. Corrugated
tin roofs are the same. In general, by folding a flat layer in one direction, you give the layer great
resistance to bending in any other direction.
Because bedding starts out flat or nearly so, its minimum curvature after folding must be zero if
the layer is not to have significant internal deformation. In other words, the fold axis must be a straight line.
The folds which meet this criteria are cylindrical folds; non-cylindrical folds do not because their hinge
lines (the line of minimum curvature) are not straight. [Now you see why we distinguish between axes
and hinges!]
line of maximum curvature
line of minimum curvatureA non-cylindrical folded layer in which Gaussian curvature is not equal to zero after folding
25.3 Buckling
Buckling applies to a single folded layer of finite thickness, or to multiple layers with high
cohesive strength between layers:
A B
C D
A' B'
C' neutral surface
perpendicular before and after deformation, so no shear parallel to the folded layer
D'
Lecture 2 5 193Fold Kinematics
Note how in the above picture the outer arc gets longer (i.e. A'B' > AB) and the inner arc gets shorter
(C'D' < CD). In the middle, there must be a line that is the same length before and after the folding. In
three dimensions, this is called the neutral surface.
Bedding thickness remains constant; thus, the type of fold produced is a parallel or class 1b fold.
Because a line perpendicular to the layer remains perpendicular, there can be no shear strain parallel to
the layer. In an anticline-syncline pair, the maximum strains would be in the cores of the folds, with zero
strain at the inflection point on the limbs:
You can commonly find geological evidence of buckling of individual beds during folding:
thrust faults, stylolites, etc.
veins, boudings, normal faults, etc.
25.4 Shear Parallel to Layers
There are two end member components to this kinematic model. The only difference between
them is the layer thickness:
Lecture 2 5 194Fold Kinematics
• Flexural Slip -- multiple strong stiff layers of finite thickness with low
cohesive strength between the layers
• Flexural Flow -- The layer thickness is taken to be infinitesimally thin.
Because they’re basically the same, we’ll mostly concentrate on flexural slip.
opposite sense of shear on the limbs
no shear in the hinge
ψψψψ
Because shear is parallel to the layers, it means that one of the two lines of no finite and no infinitesimal
elongation will be parallel to the layers. Thus, the layers do not change length during the deformation.
The slip between the layers is perpendicular to the fold axis. You can think of this type of deformation as
“telephone book” deformation. When you bend a phone book parallel to its binding, the pages slide past
one another but the individual pages don’t change dimensions; they are just as wide (measured in the
deformed plane) as they started out.
Note that the sense of shear changes only across the hinge zones but is consistent between
anticlinal and synclinal limbs:
Lecture 2 5 195Fold Kinematics
When you have an incompetent layer, such as a shale, between two more competent layers which are
deforming by this mechanism, the shear between the layers can produce drag folds, or parasitic folds, on
the limbs of the larger structure:
Because the layers of flexural slip (as opposed to flexural flow) folds have finite thickness, you
can see that they must deform internally by some other mechanism, such as buckling. Thus, buckling and
flexural slip are not by any means mutually exclusive.
25.4.1 Kink folds
Kink folds are a special type of flexural slip fold in which the fold hinges have infinite curvature
(because the radius of curvature is equal to zero).
γi
γe
no shear in horizontal layers, only in dipping layers
if layer thickness is constant, then
=γiγe
If the internal kink angle γi < γe then you will have thinning of the beds in the kink band; if γi > γe then
the beds in the kink band will thicken.
Lecture 2 5 196Fold Kinematics
25.4.2 Simple Shear during flexural slip
For kink bands: tan tanψ δ=
2
2[δ = dip of bedding]
average slip = =
s h2
2tan
δ
For curved hinges: tan . ψ π δ δ=°
=180
0 0175
average slip = =°
s hπ δ
180
The following graph show the relationship between bedding dip and shear on the limbs for kink and
curved hinge folds:
1
2
3
60° 120° 180°
tan ψψψψ
dip, δδδδ
kink folds
curved hinges
∞ at δ = 180°
25.5 Shear Oblique To Layers
This type of mechanism will produce similar folds. In this case, the shear surfaces, which are
commonly parallel to the axial surfaces of the folds, are parallel to the lines of no finite and infinitesimal
elongation.
Lecture 2 5 197Fold Kinematics
To make folds by simple shear without reversing the shear sense, you have to have heterogeneous
simple shear zone with the layer dipping in the same direction as the sense of shear in the zone.
25.6 Pure Shear Passive Flow
In this type of mechanism, the layers, which have already begun to fold by some other mechanism
behave as passive markers during a pure shear shortening and elongation. The folds produced can be
geometrically identical to the previous kinematic model:
Lecture 2 5 198Fold Kinematics
volume constant, pure shear
volume reduction, no extension pure shear(e.g. pressure solution)
Lecture 2 6 199Fold Dynamics
LECTURE 26—FOLDS IV: DYNAMICS
26.1 Basic Aspects
There are two basic factors to be dealt with when one attempts to make a theoretical analysis of
folding:
1. Folded layers do not maintain original thickness during folding, and
2. Folded rocks consist of multiple layers or “multilayers” in which different
layers have different mechanical properties.
These two basic facts about folding have the following impact:
1. There is layer-parallel shortening before folding and homogeneous shortening
during folding. The latter will tend to thin the limbs of a fold and thicken the
hinges.
2. In multilayers, the first layers that begin to fold will control the wavelength of
the subsequent deformation. Incompetent layers will conform to the shape, or
the distribution and wavelength, of the more competent layers.
26.2 Common Rock Types Ranked According to “Competence”
The following list shows rock types from most competent (or stiffest) at the top to least at the
bottom:
Sedimentary Rocks Metamorphic Rocksdolomite meta-basaltarkose granitequartz sandstone qtz-fspar-mica gneissgreywacke quartzitelimestone marblesiltstone mica schistmarlshaleanhydrite, halite
26.3 Theoretical Analyses of Folding
In general, theoretical analyses of folding involve three assumptions:
Lecture 2 6 200Fold Dynamics
1. Folds are small, so gravity is not important
2. Compression is parallel to the layer to start
3. Plane strain deformation
26.3.1 Nucleation of Folds
If layers of rock were perfect materials and they were compressed exactly parallel to their layering,
then folds would never form. The layers would just shorten and thicken uniformly. Fortunately (at least
for those of us who like folds) layers of rock are seldom perfect, but have irregularities in them. Folds
nucleate, or begin to form, at these irregularities.
Bailey Willis, a famous structural geologist earlier in this century performed a simple experiment
while studying Appalachian folds. He showed that changes in initial dip of just 1 - 2° were sufficient to
nucleate folds.
As folds begin to form at irregularities, a single wave length will become dominant. Simple
theory shows that the dominant wavelength is a linear function of layer thickness:
for elastic deformation: L tE
Edo
= 26
3π
for viscous deformation: L tdo
= 26
3π ηη
where Ld = dominant wavelength
t = thickness of the stiff layer
E = Young’s modulus of the stiff layer
Eo = Young’s modulus of the confining medium
η = viscosity modulus of the stiff layer
ηo = viscosity modulus of the confining medium
Viscous deformation will also depend on the layer parallel shortening:
Lecture 2 6 201Fold Dynamics
L tS
Sdo
= −( )( )2
1
6 2 23π ηη
S = λλ
1
3
where λ is the quadratic elongation. Thus, the thicker the layer, the longer the wavelength of the fold:
For a single layer,
4 6≤ ≤L
td ,
and for multilayers:L
td ≈ 27 .
26.3.2 Growth of Folds
At what stage does this theory begin to break down? Generally around limb dips of ~15° [small
angle assumptions were used to derive the above equations]. For more advanced stages of folding, it is
common to use a numerical rather than analytical approach.
A general result of numerical folding theory: As the viscosity contrast between the layers decreases,
layer parallel shortening increases and folding becomes less important:
Lecture 2 6 202Fold Dynamics
ηοη
ηοη
ηοη
= 42 : 1
= 5 : 1
26.3.3 Results for Kink Folds
Experimental work on kink folds indicates that kinks form in multilayers with high viscosity
contrast and bonded contacts (i.e. high frictional resistance to sliding along the contacts). Compression
parallel to the layers produces conjugate kink bands at 55 - 60° to the compression. Loading oblique to
the layering (up to 30°) produces asymmetric kinks.
Lecture 2 7Linear Minor Structures 203
LECTURE 27—LINEAR MINOR STRUCTURES
27.1 Introduction to Minor Structures
Minor structures are those that we can see and study at the outcrop or hand sample scale. We
use these features because they contain the most kinematic information. In other words, the strain and
strain history of the rock is most commonly recorded in the minor structures.
There are several types of minor structures, but they fall into two general classes: linear and
planar, which we refer to as lineations and foliations, respectively.
Lineations Foliations
mineral fibers veins
minor fold axes stylolites
boudins joints
intersection lineations cleavage
rods & mullions S-C fabrics
The lineations and foliations in a rock comprise what is known as the rock fabric. This term is analogous
to cloth fabric. Rocks have a texture, an ordering of elements repeated over and over again, just like cloth
is composed of an orderly arrangement of threads.
27.2 Lineations
Any linear structure that occurs repeatedly in a rock is called a lineation; it is a penetrative linear
fabric. Lineations are very common in igneous and sedimentary rocks, where alignment of mineral
grains and other linear features results from flow during emplacement of the rock. However, we’re most
interested in those lineations which arise from, and reflect, deformation. Of primary importance is to
remember that there is no one explanation for the origin of lineations.
27.2.1 Mineral Lineations
These are defined by elongations of inequant mineral grains or aggregates of grains.
Lecture 2 7Linear Minor Structures 204
common minerals:
hornblendesillimanitefeldsparquartzbiotite
Mineral lineations can form in
Folds
parallel to the hinge
perpendicular to the hinge
anywhere in between
Fault zones -- parallel to the slip direction
Regional metamorphism
The preferred orientation of elongate mineral grains can form by three different mechanisms:
1. Deformation of grains -- straining the grains into ellipsoidal shapes
2. Preferential growth -- no strain of the mineral crystal but may, nonetheless, reflect the
regional deformation
3. Rigid body rotation -- the mineral grains themselves are not strained but they rotate as
the matrix which encloses them is strained.
It is, occasionally, difficult to tell these mechanisms apart.
27.2.2 Deformed Detrital Grains (and related features)
This category differs from the previous only in that pre-existing sedimentary features, or features
formed in sedimentary rocks are deformed. The basic problem with their interpretation is that such
features commonly have very different mechanical properties than the matrix of the rock. thus the strain
of the deformed object which you measure may not reflect the strain of the rock as a whole.
Lecture 2 7Linear Minor Structures 205
common features:
ooidspebblesreduction spots
27.2.3 Rods and Mullions
Rods are any elongate, essentially monomineralic aggregate not formed by the disruption of the
original rock layering. They are generally cylindrical shaped and striated parallel to their length. They
are almost always oriented parallel to fold hinge lines and occur in the hinge zones of minor folds. Rods
are thought to form by metamorphic or fluid flow processes during tectonic deformation.
Mullions are elongate bodies of rock, partly bounded by bedding planes and partly by newer
structures. They generally have a cylindrical, ribbed appearance and are oriented parallel to the fold
hinges. They form at the interface between soft and stiff layers.
soft (e.g. argillite)
stiff (e.g. quartzite)
27.3 Boudins
Boudin is the French word for sausage. They are formed by the segmentation of pre-existing
layers and appear similar to links of sausages. The segmented layers certainly can be, but need not be,
sedimentary layering. The segmentation can occur in two or three dimensions.
Lecture 2 7Linear Minor Structures 206
Chocolate tablet boudinage
For simple boudinage (upper right), the long axis of the boudin is perpendicular to the extension
direction. Chocolate tablet boudinage forms when you have extensions in two directions.
The shapes of boudins in cross section are a function of the viscosity contrast between the layers:
low viscosity contrast
high viscosity contrast
"pinch & swell"
"fish mouths"
27.4 Lineations Due to Intersecting Foliations
A type of lineation can form when two foliations, usually bedding and cleavage, intersect. When
this occurs in fine-grained, finely bedded rocks, the effect is to produce a multitude of splinters. The
resulting structure is called pencil structure. There are good examples at Portland Pt. quarry. Pencils
are usually oriented parallel to fold hinges.
Lecture 2 8Planar Minor Structures I: Cleavage 207
LECTURE 28—PLANAR MINOR STRUCTURES I
28.1 Introduction to Foliations
The word foliation comes from the Latin word folium which means “leaf” (folia = leaves). In
structural geology, we use foliation to describe any planar structure in the rocks. Under the general term
foliation there are several more specific terms:
• bedding
• cleavage
• schistosity
• gneissic layering
These collective foliations were sometimes referred to in older literature as “S-surfaces”. Geologists
would determine the apparent relative age relations between foliations and then assign them numbers
from oldest to youngest (with bedding, presumably being the oldest, labeled S0). In the last decade, this
approach has fallen out of favor because, among other things, we know that foliations can form
simultaneously (as well will see with “S-C fabrics” in a subsequent lecture). Furthermore, structural
geologists used to correlate deformational events based on their relative age (e.g. correlating S3 in one are
with S3 in another are 10s or 100s of kilometers away). With the advent of more accurate geochronologic
techniques, we now know that such correlation is virtually worthless in many cases.
28.2 Cleavage
Many rocks have the tendency to split along certain regular planes that are not necessarily
parallel to bedding. Such planes are called cleavage . Roofing slates are an excellent example. Cleavage
is a type of foliation that can be penetrative or non-penetrative. An important point to remember is that:
rock cleavage ≠ mineral cleavage
The two are generally unrelated.
Lecture 2 8Planar Minor Structures I: Cleavage 208
28.2.1 Cleavage and Folds
Cleavage is commonly seen to be related in a systematic way to folds. When this occurs, the
cleavage planes are nearly always parallel or sub-parallel to the axial surfaces of the folds. This is known
as axial planar cleavage.
If examined in detail, the cleavage usually is not exactly parallel to the axial surface every where but
changes its orientation as it crosses beds with different mechanical properties. This produces a fanning of
the cleavage across the fold. In a layered sandstone and shale sequence, the cleavage is more nearly
perpendicular to bedding in the sandstone and bends to be at a more acute angle in the shale. This is
known as cleavage refraction.
sandstone
shale
Cleavage Fanning & Refration
As we will see next time, cleavage refraction is related to the relative magnitudes of strain in the different
layers and the orientation of the lines of maximum shear strain. As a side light, cleavage refection can be
used to tell tops in graded beds. This property can be very useful in metamorphic terranes where the
grading includes only medium sand and finer.
Lecture 2 8Planar Minor Structures I: Cleavage 209
finin
g up
war
dfin
ing
upw
ard
cleavage is steep in the coarse beds but shallows upward as the grain size gets smaller
Sharp "kink" in the cleavage at the boundaries between the graded sequences
Cleavage can also be very useful when doing field work in a poorly exposed region with overturned
folds. If the cleavage is axial planar, then the cleavage with dip more steeply than bedding on the upright
limbs of the folds but will dip more gently than bedding on the overturned limbs:
dips more gently than bedding
dips more steeply than bedding
cleavage dips more steeply than bedding
inferred position and form of overturned anticline
28.3 Cleavage Terminology
Lecture 2 8Planar Minor Structures I: Cleavage 210
Cleavage can take on a considerable variety of appearances, but at its most basic level, there are
two types of cleavage:
• Continuous cleavage occurs in rocks which have an equal tendency to cleave (or
split) throughout, at the scale of observation. In other words, the cleavage is penetrative.
• Spaced or Discontinuous cleavage it not penetrative at the scale of observation.
28.3.1 Problems with Cleavage Terminology
Because of its economic importance (i.e. in quarries, etc.) some of the names for various types of
cleavage are very old and specific to a particular rock type. Furthermore, cleavage terminology has been
overrun with genetic terms, which are still used by some, long after the particular processes implied by the
name have been shown to not be important. The following is an incomplete list of existing terms which
should not be used when describing cleavage because they are all genetic:
• fracture cleavage
• stylolitic or pressure solution cleavage
• Shear foliation
• strain-slip cleavage
These terms have their place in the literature, but only after you have proven that a particular process is
important.
28.3.2 Descriptive Terms
Anastomosing
Conjugate
Lecture 2 8Planar Minor Structures I: Cleavage 211
Crenulation
symmetric asymmetric
S 2
S 1
S 2
Crenulation is particularly interesting. In it, a pre-existing alignment of mineral grains is deformed
into microfolds. This is accompanied by mineral differentiation such that the mineral composition in the
zones of second foliation (or crenulation cleavage, labeled “S2” above) is different than that part of the
rock between the cleavage planes. Crenulation cleavage has been called “strain slip cleavage” but that
term has now thankfully fallen into disuse.
28.4 Domainal Nature of Cleavage
Most cleaved rocks have a domainal structure at one scale or another which reflects the mechanical
and chemical processes responsible for their formation.
cleavage domains
microlithons
The rocks tend to split along the cleavage domains, which have also been called “folia”, “films”, or
“seams”.
In fine-grained rocks, cleavage domains are sometimes called “M-domains” because mica and
other phyllosilicates are concentrated there, whereas the lenticular microlithons are the “QF-domains”
because of the concentration of quartz and feldspar. As in the discussion of crenulation cleavage, above,
we see that mineralogical and chemical differentiation is a common aspect of cleavage.
Lecture 2 8Planar Minor Structures I: Cleavage 212
28.4.1 Scale of Typical Cleavage Domains
10 cm 1 cm 1 mm 0.1 mm 0.01 mm
slaty
crenulation
spaced
anastomosing
limit of resolution of the optical microscope
Lecture 2 9Cleavage & Strain 213
LECTURE 29—PLANAR MINOR STRUCTURES II:CLEAVAGE & STRAIN
29.1 Processes of Foliation Development
There are four basic processes involved in the development of a structural foliation:
1. Rotation of non-equant grains,
2. Change in grain shape through pressure solution,
3. Plastic deformation via dislocation mechanisms, and
4. Recrystallization.
The first two are the most important in the development of cleavage at low to moderate metamorphic
grades and will be the focus of this lecture.
29.2 Rotation of Grains
This process in very important in compaction of sediments and during early cleavage development.
The basic idea is:
After strain, particles are the same length but have rotated to closer to perpendicular with the maximum shortening direction
There are two similar models which have been devised to describe this process. Both attempt to predict
the degree of preferred orientation of the platy minerals (how similarly oriented they are) as a function of
strain. The preferred orientation is usually displayed as poles to the platy particles; the more oriented
they are, the higher the concentration of poles at a single space on the stereonet.
Lecture 2 9Cleavage & Strain 214
29.2.1 March model
rotation of purely passive markers that have no mechanical contrast with the confining medium.
We solved this problem already for two-dimensional deformation when we talked about strain.
tan tantan′ = =θ θ θ
xz xzz
x
xz
xz
S
S R
In three dimensions it is a little more complex but still comprehensible:
tan tan sin sin′ = +( )δ δ φ φR Rxy yz xz yz2 2 2 2
where φyz is the azimuth with respect to the y axis, δ and δ' are the dips of the markers before and after the
strain, and R is the ellipticity measured in a principal plane of the strain ellipse (i.e. a plane that contains
two of the three principal axes, as indicated by the subscripts).
29.2.2 Jeffery Model
Rotation of rigid bodies in a viscous fluid (the former modeled as rigid ellipsoidal particles). For
elongate particles, there is little difference between the Jeffery and March models. For example, detrital
micas in nature have aspect ratios between 4 and 10. For this range of dimensions, the Jeffery model
predicts 12 to 2 % lower concentrations than a March model.
Both of these models work only for loosely compacted material (i.e. with high porosity). At
lower porosities, the grains interfere with each other, resulting in lots of kinking, bending and breaking of
grains.
29.2.3 A Special Case of Mechanical Grain Rotation
In 1962, John Maxwell of Princeton proposed that the cleavage in the Martinsburg Formation at
the Delaware Water Gap was formed during dewatering of the sediments and thus this theory of cleavage
formation has come to be known as the dewatering hypothesis. He noted that the cleavage was parallel
to the sandstone dikes in the rocks:
Lecture 2 9Cleavage & Strain 215
Maxwell suggested that expulsion of water from the over-pressured sandstone during dewatering resulting
in alignment of the grains by mechanical rotation. We now know that this is incorrect for the Martinsburg
because
1. Cleavage in the rocks there is really due to pressure solution, and
2. Internal rotations during strain naturally results in sub-parallelism of cleavage and the
dikes.
Mechanical rotation does occur during higher grade metamorphism as well. The classic example
is the rolled garnet:
29.3 Pressure Solution and Cleavage
We’ve already talked some about the mechanical basis for pressure solution. The basic observation
in the rocks which leads to an interpretation of pressure solution is grain truncation in the microlithons:
Lecture 2 9Cleavage & Strain 216
Most people associate pressure solution with carbonate rocks, but it is very common in siliceous rocks as
well.
There are two general aspects that pressure solution and related features that you can observe in
the rocks:
local overgrowths and vein formation means limited fluid circulation. Volume is more-or-less conserved
more commonly, you see no evidence for redeposition, which means bulk circulation and volume reduction were important
In the Martinsburg Formation that Maxwell studied, a volume reduction of greater than 50% has been
documented by Wright and Platt.
29.4 Crenulation Cleavage
Crenulation cleavage is probably a product of both pressure solution and mechanical rotation. It
has two end member morphologies:
Discrete -- truncation of grains against the cleavage domains. Very strong alignment of grains within cleavage domains
Zonal -- initial fabric is continuous across the cleavage domains. Clearly a case of microfolding
Both types of the same characteristics:
1. No cataclastic textures in cleavage domains (i.e. they are not faults),
2. There is mineralogical and chemical differentiation. Quartz is lacking from the cleavage
Lecture 2 9Cleavage & Strain 217
domains and there is enrichment of Al2O3 and K2O in the cleavage domains relative to
the microlithons,
3. Thinning and truncation are common features, and
4. No intracrystalline plastic deformation.
Probably what occurs is rotation of phyllosilicates by microfolding accompanied by pressure solution of
quartz and/or carbonate.
29.5 Cleavage and Strain
There are two opposing views of how cleavage relates to strain:
1. J. Ramsay, D. Wood, S. Treagus -- Cleavage is always parallel to the XY plane of the finite
strain ellipsoid (i.e. it is perpendicular to the Z-axis). Thus, there can be no shear
parallel to the planes.
Z = principal axis of shortening
cleavage
The basis for this assertion is mostly observational. These workers have noted in
many hundreds of instances that the cleavage is essentially perpendicular to the strain
axes as determined by other features in the rock.
2. P. Williams, T. Wright, etc. -- cleavage is commonly close to the XY-plane but can deviate
significantly and, at least at some point during its history, may be parallel to a plane of
shear.
There are two issues here which are responsible for this debate:
First, at high strains the planes of maximum shear are very close to the planes of maximum
elongation (the X-axis). Thus it is very difficult in the field to measure angles precisely enough that you
can resolve the difference between a plane of maximum shear and a principal plane.
Lecture 2 9Cleavage & Strain 218
Second, cleavage becomes a material line. If the deformation is by pure shear then it could be
that cleavage remains perpendicular to the Z-axis. However, in a progressive simple shear, it cannot
remain perpendicular to the Z-axis all the time (because it is a material plane) and thus must experience
shear along it at some point.
Lecture 3 0Shear Zones, Transposition 219
LECTURE 30—SHEAR ZONES & TRANSPOSITION
30.1 Shear Zone Foliations and Sense of Shear
Within ductile shear zones, a whole array of special structures develop. Because of the progressive
simple shear, the structures that develop are inherently asymmetric. it is this asymmetry that allows us to
determine the sense of shear in many shear zones.
30.1.1 S-C Fabrics
S = schistosité
C = cisaillement (shear)
S-C fabrics are an example of two planar foliations which formed at the same time (although
there are many examples of the S-foliation forming slightly or considerably earlier than the C-foliation).
The S planes are interpreted to lie in the XY plane of the finite strain ellipsoid and contain the maximum
extension direction (as seen in the above figure). The C-planes are planes of shear. As the S-planes
approach the C-planes they curve into and become sub-parallel (but technically never completely parallel)
to the C-planes.
Two types of S-C fabrics have been identified:
• Type I -- found in granitoid rocks rich in quartz, feldspar, and biotite. Both the S- and
C-planes are well developed.
• Type II -- form in quartzites. The foliation is predominantly comprised of C-planes,
with S-planes recorded by sparse mica grains (see below)
30.1.2 Mica “Fish” in Type II S-C Fabrics
The S-planes are recorded by mica grains in rock. In general, the cleavage planes of all the mica
Lecture 3 0Shear Zones, Transposition 220
grains are similarly oriented so that when you shine light on them (or in sunlight) they all reflect at the
same time. This effect is referred to somewhat humorously as “fish flash”.
(001)
(001)
C-planes
S-planes
fine-grained "tails" of recrystallized mica
30.1.3 Fractured and Rotated Mineral Grains
Minerals such as feldspar commonly deform by fracture rather than by crystal plastic mechanisms.
One common mode of this deformation is the formation of domino blocks. The fractured pieces of the
mineral shear just like a collapsing stack of dominos:
note that the sense of shear on the microfaults is opposite to that of the shear zone
30.1.4 Asymmetric Porphyroclasts
There are two basic types of asymmetric porphyroclasts:
reference plane
σσσσ - type δδδδ - type
Lecture 3 0Shear Zones, Transposition 221
In the σ-type, the median line of the recrystallizing tails does not cross the reference plane, whereas in the
δ-type, the median line of the recrystallizing tails does cross the reference plane. The ideal conditions for
the development of asymmetric porphyroclasts are:
1. Matrix grain size is small compared to the porphyroclasts,
2. Matrix fabric is homogeneous,
3. Only one phase of deformation,
4. Tails are long enough so that the reference plane can be constructed, and
5. Observations are made on sections perpendicular to the foliation and parallel to the
lineation.
30.2 Use of Foliation to Determine Displacement in a Shear Zone
Consider a homogeneous simple shear zone:
d
ψψψψ
θθθθ'd = γ x
γ = tan ψ
x
In the field, we can’t measure ψ directly, but we can measure θ', which is just the angle between the
foliation (assumed to be kinematically similar to S-planes) and the shear zone boundary. If the foliation is
parallel to the XY plane of the strain ellipsoid then there is a simple relationship between θ' and γ:
tan22′ =θγ
Although it is trivial in the case of a homogeneous shear zone, we could compute the displacement
graphically by plotting γ as a function of the distance across the shear zone x and calculating the area
Lecture 3 0Shear Zones, Transposition 222
under the curve (i.e. the integral shown):
d
ψψψψy
shear strain, γγγγ
y d
d dyy
= ∫ γ0
For a heterogeneous shear zone -- the usual case in geology -- the situation is more complex, but
you can still come up with a graphical solution as above. The basic approach is to (1) measure the angle
between the foliation and the shear zone boundary, θ', at a number of places, (2) convert those measurements
to the shear strain, γ, (3) plot γ as a function of perpendicular distance across the shear zone, and (4)
calculate the displacement from the area under the resulting curve:
y
shear strain, γγγγ
y d
θ'3
θ'1
θ'2
foliation
30.3 Transposition of Foliations
In many rocks, you see a compositional layering that looks like bedding, but in fact has no
stratigraphic significance. The process of changing one foliation into another -- thereby removing the
frame of reference provided by the first foliation -- is known as transposition. There are two basic
Lecture 3 0Shear Zones, Transposition 223
processes involved:
1. Isoclinal folding of the initial foliation (i.e. bedding) into approximate parallelism with
the axial surfaces, and
2. attenuation and cutting out of the limbs by simple shear.
macroscopically, the bedding trends E-W, with the younger and older relations as indicated
younger
older
On the outcrop, the bedding trends N-S. If the fold hinges are very obscure, then you may interpret the layering as a normal stratigraphicsequence
Obscuring of the fold hinges is an important part of the process of transposition:
This sequence of deformation would produce transposed layering in which all of the beds (really just a single bed) were right side up
Transposition is most common in metamorphic rocks, but can also occur in mélanges. It is
difficult to recognize where extreme deformation is involved. In general one should look for the
following:
• look for the fold hinges
Lecture 3 0Shear Zones, Transposition 224
• look for cleavage parallel to compositional layering
• Walk the rocks out to a less deformed area.
Lecture 3 1 225Thrust Systems: Tectonics
LECTURE 31—THRUST SYSTEMS I: OVERVIEW & TECTONIC
SETTING
31.1 Basic Thrust System Terminology
Before starting on the details of thrust faults we need to introduce some general terms. Although
these terms are extensively used with respect to thrust faults, they can, in fact, be applied to any low
angle fault, whether thrust or normal.
Décollement -- a French word for “unsticking”, “ungluing”, or “detaching”. Basically, it is a
relatively flat, sub-horizontal fault which separates deformed rocks above from undeformed rocks, below.
Thin-skinned -- Classically, this term has been applied to deformation of sedimentary strata
above undeformed basement rocks. A décollement separates the two. My own personal use applies the
term to any deformation with a décollement level in the upper crust. This definition includes décollement
within shallow basement. In general, the term comes from Chamberlain in 1910 and 1919; he termed the
Appalachians a “thin-shelled” mountain range. John Rodgers, a well known Yale structural geologist
gave the term its present form in the 1940’s.
Thick-skinned -- Again, the classic definition involves deformation of basement on steep reverse
faults. My own definition involves décollement at middle or deep crustal levels, if within the crust at all.
Allochthon -- A package of rocks which has been moved a long way from their original place of
deposition. The word is commonly used as an adjective as in: “these rocks are allochthonous with respect
to those…”
Autochthon -- Rocks that have moved little from their place of formation. These two terms are
commonly used in a relative sense, as you might expect given that the plates have moved around the
globe! You will also see the term “parautochthon” used for rocks that probably have moved, but not as
much as some other rocks in the area you are studying.
Klippe -- An isolated block of rocks, once part of a large allochthon, which has become separated
from the main mass, usually by erosion but sometimes by subsequent faulting.
Fenster -- This is the German word for “window”, and it means literally that: a window or a
hole through an allochthon, in which the underlying autochthon is exposed. A picture best illustrates
these last two terms:
Lecture 3 1 226Thrust Systems: Tectonics
window (fenster)
klippe
Map view
cross- section
31.2 Tectonic Setting of Thin-skinned Fold & Thrust Belts
Long linear belts of folds and thrusts, known as foreland thrust belts, occur in virtually all major
mountain belts of the world. Characteristically, they lie between the undeformed craton and the main
part of the mountain belt itself. Some well-known examples include:
• Valley & Ridge Province (Appalachians)
• Jura Mountains (Alps)
• Canadian Rockies (Foothills, Front & Main Ranges)
• Sub-Himalayan Belt
• Subandean belt
Foreland thrust belts occur in two basic types of plate settings:
Lecture 3 1 227Thrust Systems: Tectonics
31.2.1 Andean Type:
AccretionaryWedge
HinterlandForeland
CratonForearc Back Arc (retroarc)
subduction zone
ForelandBasin
This type of foreland thrust belt is sometimes called an antithetic belt because the sense of shear is
opposite to that of the coeval plate margin subduction zone.
31.2.2 Himalayan Type:
Foreland(peripheral)
Basin
sutureForeland
Hinterland
Indian continental crust
Tibet
The Himalayan type is sometimes called a synthetic thrust belt because the sense of shear is the same as
the plate margin that preceded it. At this point, we need to introduce two additional terms:
Foreland is a stable area marginal to an orogenic belt toward which rocks of the belt were folded
and thrusted. It includes thin-skinned thrusting which does not involve basement. In active mountain
belts, such as the Andes or the Himalaya, the foreland is a region of low topography.
Hinterland refers to the interior of the mountain belt. There, the deformation involves deeper
Lecture 3 1 228Thrust Systems: Tectonics
structural levels. In active mountain belts, the hinterland is a region of high topography which includes
everything between the thrust belt and the magmatic arc (where there is one). “Hinterland” in particular
is a poorly defined term about which there is no general agreement. You should always state what you
mean by it.
31.3 Basic Characteristics of Fold-thrust Belts
1. Linear or arcuate belts of folds and low-angle thrust faults
2. Form in subhorizontal or wedge-shaped sedimentary prisms
3. Vergence (or facing) generally toward the continent
4. Décollement zone dips gently (1 - 6°) toward the interior of the mountain belt
5. They are the result of horizontal shortening and thickening.
2 - 15 km
100 - 600 km
1000's km
shelfmiogeocline hinge
The typical fold-thrust belt in North America and many other parts of the world is formed in a passive
margin sequence (or “miogeocline”) deposited on a rifted margin.
This geometry is responsible for numbers one through four in the list above because:
• miogeocline is laterally continuous
•wedge-shape responsible for the vergence
• planar anisotropy of layers produces décollement
Lecture 3 1 229Thrust Systems: Tectonics
31.4 Relative and Absolute Timing in Fold-thrust Belts
A general pattern in mountain belts is that deformation proceeds from the interior to the exterior
(or from hinterland to foreland):
oldest faults
youngest fault
Interior Exterior
This progression has been demonstrated both directly and indirectly. The more interior faults are seen to
be folded and deformed by the more exterior ones and the erosion of the individual thrust plates produces
an inverted stratigraphy in the foreland basin in which deposits derived from the oldest thrust plate are
found at the bottom of the sedimentary section.
The duration of thrust belts is quite variable. In the western North America, the thrust belt
spanned nearly 100 my; in the Andes it has been active for only the last 10 - 15 my, and in Taiwan it is
only 4 my old.
Rates of shortening in foreland thrust belts is similarly variable. In general, they range from
mm/yr to cm/yr. Antithetic thrust belts are 1 to 2 orders of magnitude slower than plate convergence
rates whereas synthetic thrust belts are 30 - 70% of the total convergence rate.
31.5 Foreland Basins
The horizontal shortening of the rocks in a thrust belt is accompanied by vertical thickening. This
thickening means that there is more weight resting on the upper part of the continental lithosphere than
there was before. Thus, the lithosphere bends or flexes under this load, just like a diving board does
when you stand on the end of it. As we will see in a few lectures (last week of classes), this large scale,
broad wavelength deformation of the lithosphere is known as flexural isostasy.
The loading by the thrust belt produces an asymmetric depression, with its deepest point right
next to the belt. Material eroded from the uplifted thrust belt is deposited in the depression, forming a
Lecture 3 1 230Thrust Systems: Tectonics
type of sedimentary basin known as a foreland basin.
load
depression of basement under the load (exaggerated)
asymmetric foreland basin
asymmetric foreland basin
forebulge
load
The Cretaceous deposits of western Wyoming and eastern Idaho are perhaps some of the best known
foreland basin deposits.
Lecture 3 2 231Thrust Systems: Basic Geometries
LECTURE 32—THRUST SYSTEMS II: B ASIC GEOMETRIES
32.1 Dahlstrom’s Rules and the Ramp-flat (Rich Model) Geometry
The basic geometries of fold and thrust belts are summarized in three “rules” proposed by
Dahlstrom (1969, 1970), based on his work in the Canadian Rockies:
1. Thrusts tend to cut up-section in the direction of transport
2. Thrusts parallel bedding in incompetent horizons and cut across bedding in
competent rocks
3. Thrusts young in the direction of transport
Deformation following these rules produces a stair step or “ramp and flat” geometry. This geometry was
first recognized by J. L. Rich (a former Cornellian) in 1934:
trace of future thrust fault
footwall flat
hanging wall ramp
hanging wall flat
footwall flatfootwall ramp
normal thickness structurally thickened
normal thickness
hanging wall anticline
Lecture 3 2 232Thrust Systems: Basic Geometries
The important points to remember about this “ramp-flat” model are:
• structural thickening occurs only between the footwall and the hanging wall
ramps
• thrusts cut up-section in both the footwall and the hanging wall ramps
• Stratigraphic throw is not a good indication of the amount of thrust displacement
• Anticline is only in the hanging wall, not the footwall
• Thrust puts older rocks on younger rocks
Suppe calls this process “Fault bend folding”. He has made it more quantitative by assuming a strict kink
geometry. In his terminology, the dipping beds located over the footwall ramp are referred to as the
“back-limb” and those over the hanging wall ramp the “forelimb”. These limbs define kink bands which
help you find where the ramps are located in the subsurface. Suppe has derived equations to show that
the forelimb dips (or “fore-dips”) should be steeper than the back limb dips (or “back-dips”).
It is important to remember that the conclusions we have listed above do not depend on having a
kink geometry. You get the same results with curved folds and listric faults.
32.2 Assumptions of the Basic Rules
Before we get too carried away with this elegantly simple geometry, lets explore an important
underlying assumptions of Dahlstrom’s rules:
• Thrusts cut through a previously undeformed, flat-lying sequence of layered sedimentary
rock. As long as this is true, a thrust fault will place older rocks over younger
rocks. However, you can easily conceive of geometries where this will not be
true:
Lecture 3 2 233Thrust Systems: Basic Geometries
Prior folding
younger-over-older
older-over-younger
older-over-younger
Prior thrusting
younger-over-older
Thrusting along an unconformity
32.3 Types of Folds in Thrust Belts
The hanging-wall anticline shown above is not the only type of fold which can form in thrust
belts. In general, there are four types which are commonly found:
Lecture 3 2 234Thrust Systems: Basic Geometries
I. Fault Bend Folds
mode I
mode II
(also “hanging wall”or “ramp” anticlines)
II. Fault Propagation Folds(also “tip-line folds”)
tip line
IV. Detachment FoldsIII. Wedge Fault-folds
(also “Lift-off” or“pop-up” folds)
32.4 Geometries with Multiple Thrusts
32.4.1 Folded thrusts
In general, younger faults will form at lower levels and cut into undeformed layering. When
they move over ramps, they will deform any older thrusts higher in the section as illustrated in the
diagram below. This provides one of the best ways to determine relative ages of faults.
younger thrust fault with primary ramp
older thrust fault folded by movement over the ramp in the younger thrust fault
Lecture 3 2 235Thrust Systems: Basic Geometries
32.4.2 Duplexes
Commonly, a fault will splay off of an older thrust fault but then will rejoin the older fault again.
This produces a block of rock complete surrounded by faults, which is known as a horse. Several horses
together make a duplex.
floor thrust
roof thrust
trajectory of next fault
horse
direction of transport
1234
numbers indicate sequence of formation
Notice that the sequence of formation of the horses is in the direction of transport (i.e. from the hinterland
to the foreland). This is mostly observational. If the horses formed in the other direction, then you would
see “beheaded” anticlines:
trajectory of next fault
direction of transport
"beheaded anticlines"
Lecture 3 2 236Thrust Systems: Basic Geometries
The exact shape of a duplex depends upon the height of the ramps, the spacing of the ramps, and
the displacement of the individual horses. For example, as shown on the next page, if the displacement is
equivalent to the initial spacing of the ramps you get a compound antiformal structure known as an
antiformal stack:
For
mat
ion
of a
n an
tifor
mal
sta
ck b
y m
ovem
ent o
n a
serie
s of
hor
ses,
eac
h w
ith d
ispl
acem
ent e
quiv
alen
t to
the
initi
al s
paci
ng
betw
een
the
ram
ps.
The
top
sect
ion
form
ed fi
rst.
Lecture 3 2 237Thrust Systems: Basic Geometries
32.4.3 Imbrication
Imbrication means the en echelon tiling or stacking of thin slices of rocks. Imbricate zones are
similar to duplexes except that they do not all join up in a roof thrust. There are two basic types of
imbrications, illustrated below:
Hanging Wall Imbrication:
Footwall Imbrication:
32.4.4 Triangle Zones
At the leading edge of a thrust belt, one commonly sees a curious syncline (or monocline). The
best documented example is in the southern Canadian Rockies, where the Alberta syncline forms the
eastern edge of the orogen:
?
?Extra spaceThe problem with frontal synclines:
Lecture 3 2 238Thrust Systems: Basic Geometries
The problematical space is triangular in shape so it is known as a triangle zone. The solution to this
dilemma of frontal synclines is to fill the space with a type of duplex:
displacement goes to zero
shaded area is the triangle zone
This duplex differs from the ones that we discussed above, in that the roof thrust has the opposite sense
of shear than the floor thrust, where as in “normal” duplexes they have the same sense of shear. For this
reason, triangle zones have sometime been referred to as passive roof duplexes. You can best visualize
the kinematics of this structure by imagining driving a wedge into a pack of cards:
There is more than academic reasons to be interested in triangle zones. They can be prolific hydrocarbon
traps, and to date have been among the most productive parts of fold-thrust belts.
Lecture 3 3 239Thrust Systems: Thick-skinned
LECTURE 33—THRUST SYSTEMS III: THICK-SKINNED
FAULTING
33.1 Plate-tectonic Setting
The two classic areas displaying thick-skinned structures are the Rocky Mountain Foreland (“Lara-
mide Province”) of Wyoming, Colorado, and surrounding states, and the Sierras Pampeanas of western
Argentina. Both of these areas are associated with flat subduction beneath the continent and a gap in arc
magmatism:
no vo
lcanic
arc
thin-
skinn
ed
thru
st be
lt
little or no asthenospheric wedge between the two plates
Thick-skinned Province(deforms most of crust)
Note that coeval thin and thick-skinned deformation can be found in both the Argentine and western US
examples. Some workers have proposed that the flat subduction is related to, or caused by, subduction of
buoyant pieces of oceanic crust such as ridges and oceanic plateaus; this relationship has not been
definitely proven.
There are parts of many other mountain belts in the world which have thick-skinned style geome-
tries. It is not clear that flat subduction plays a role in many of these cases. These include:
• Mackenzie Mountains, Canada
• Wichita-Arbuckle Mountains, Oklahoma, Texas
• Foreland of the Atlas Mountains, Morocco
• Iberian-Catalán Ranges, Spain
• Cape Ranges, South Africa
• Tien Shan, China
Lecture 3 3 240Thrust Systems: Thick-skinned
33.2 Basic Characteristics
1. Involve crystalline basement;
2. Commonly occur in regions of thin sedimentary cover;
3. Structural blocks commonly only two or three times longer than they are
wide;
4. Blocks exhibit a variety of structural orientations;
5. Bounding structures commonly reverse faults with a wide variety of dips (<5°
to 80°);
6. Broad flat basins separate the mountain blocks.
33.3 Cross-sectional Geometry
In the western United States, there has, for many years, been a debate about the structural
geometry of the uplifts in vertical sections. Several hypotheses have been proposed, but the can be
grouped in two basic categories:
“Upthrust” Hypothesis
33.3.1 Overthrust Hypothesis
Lecture 3 3 241Thrust Systems: Thick-skinned
A large amount of seismic reflection and borehole data basically confirm that the overthrust model is
more correct. In the Rocky Mountain foreland, the deepest overhang of basement over Paleozoic strata
that has been drilled is ~14,000 ft (the total depth of the hole was 19,270 ft).
33.3.2 Deep Crustal Geometry
Insight into the deep crustal geometry of thick-skinned uplifts has come from three basic sources
of information:
• seismic reflection profiling
• earthquake hypocenters and focal mechanisms
• inferences from the dip slope of the blocks.
The COCORP deep seismic reflection profile across the Wind River Mountains of western Wyoming
provided the most complete look at the deep structure of the uplift. That profile showed a 36°-dipping
thrust fault which could be traced on the seismic section to times of 8 - 12 s (24 - 36 km). More recent
processing and a reinterpretation of that seismic line indicates that the fault may have a listric geometry
and flatten at between 20 and 30 km depth. This listric geometry would help explain the dip slope of the
range.
Earthquake focal mechanisms from the still-active Sierras Pampeanas of western Argentina uni-
formly show thrust solutions with dips between 30 and 60°. There is virtually no evidence for seismic
faulting on near vertical planes or with normal fault geometries. The earthquakes also provide important
insight into the crustal rheology during deformation. They occur as deep as 35 - 40 km in the crust,
indicating that virtually the entire crust is deforming by brittle mechanisms, at least at short time scales.
These depths are deeper than would be predicted from power law creep equations, unless the strain rate
was unusually fast, the heat flow were abnormally low, or the lithology were unusually mafic. All three
of these are reasonable possibilities for this part of the Andean foreland.
Finally the dip slope observed on many thick-skinned blocks is useful because it suggests that the
blocks have been rotated. This rotation can be accomplished by listric faults or faults with bends in them.
The scale of the ramp part of the fault, or the depth at which the fault flattens, can be deduced from the
scale of the dip slope.
Lecture 3 3 242Thrust Systems: Thick-skinned
33.4 Folding in Thick-skinned Provinces
Older views of folds in thick-skinned regions suggested that the folds were formed by “draping”
of the sedimentary section over faulted basement, hence the term “drape folds”. This interpretation,
however, runs into problems, particularly if the fault beneath the sedimentary section is thought to be
steep. It would require one of the following geometries:
décollement at sediment-basement
interface
Ductile or brittle thinning of the steep limb of the
structure
or
The most successful modern view is that the folds are fault-propagation folds, formed at the tip of a
propagating thrust fault. In this scenario, overturned beds beneath basement overhangs can be interpreted
to have formed when the fault propagated up the anticlinal axis, leaving an overturned syncline in the
footwall.
propagation path for next increment of thrust movement
This syncline will be left in the footwall
present tipline
33.4.1 Subsidiary Structures
A very important family of structures are formed because the synclines underlying many of the
uplifts are very tight and their deformation can no longer be accommodated by strictly layer-parallel slip.
These structures are known as out-of-the-syncline or “crowd” structures. Basically, in the core of a
syncline, there is not enough room so some of the layers get “shoved out”.
Lecture 3 3 243Thrust Systems: Thick-skinned
out-of-the-syncline thrusts
"rabbit ear"structure
Similar structures can occur on a larger scale, where they are called out-of-the-basin faults. And example
of this latter type of structure would be Sheep Mountain on the east side of the Bighorn Basin in northwestern
Wyoming:
out-of-the-basin structure
major uplift
major basin
33.5 Late Stage Collapse of Uplifts
In the Rocky Mountain foreland, at least, and perhaps in other thick-skinned provinces which are
no longer active, it is common to see the uplifts “collapse” by normal faulting. Thus, certain major
structural blocks such as the Granite Mountains of central Wyoming have relatively little morphologic
expression because most the structural relief has been destroyed by normal faulting.
In map and cross-section, this looks like:
Lecture 3 3 244Thrust Systems: Thick-skinned
Map Cross-section
A
A'A' A
33.6 Regional Mechanics
In the Rocky Mountain foreland, basement surfaces define regional “folds” at 100 - 200 km length
scales. A model by Ray Fletcher suggests that the wavelength of the first order flexures should be four to
six times the thickness of the highly viscous upper layer (i.e. the upper crust). In a rough sense, this
model fits the basic observations from Wyoming if one uses a reasonable depth to the frictional crystal
plastic transition zone. It is not highly successful everywhere.
Just like thrust belts, thick-skinned uplifts load the crust, producing subsidence and creating a
sedimentary basin. The mechanics of these basins, known as broken foreland basins, is somewhat
different, however, because one must model a broken beam, rather than an unbroken elastic beam.
Lecture 3 4 245Extensional Systems I
LECTURE 34—EXTENSIONAL SYSTEMS I
34.1 Basic Categories of Extensional Structures
There are three basic categories of extensional structures. They differ primarily in how deep they
affect the lithosphere:
1. Gravity slides (i.e. landslides, etc.)
2. Subsiding passive margins (Gulf coast growth structures)
3. Tectonic rift provinces
• Oceanic spreading centers (e.g. Mid-Atlantic Ridge)
• Intracontinental rifts (e.g. Basin and Range)
All are produced by essentially vertical σ1 and horizontal σ3.
34.2 Gravity Slides
Subaerial gravity slides include landslides, slumps, etc., as well as much larger scale regional
denudation features. Only the last one is commonly preserved in the geologic record.
normal faults
thrust faults
fault comes back to ground surface in down-dip direction
commonly intensely brecciated internally
break away scarp rotated surface
These can occur at all different scales. The underlying similarity is that the fault cuts the ground surface
at both its up-dip and its down-dip termination so that only very shallow levels of the crust are involved.
Although commonly caused by tectonic deformation, these are not, themselves considered to be “tectonic
Lecture 3 4 246Extensional Systems I
structures”. At the very largest scales, gravity slides are difficult to distinguish from thrust plates in
mountain belts.
34.2.1 The Heart Mountain Fault
One of the largest known detachment structures is located in northwestern Wyoming and is
called the Heart Mountain fault.
region of the Heart Mountain fault
approximate orientation of cross-section, below
Wyoming
Yellowstone
110 km
Ordovician Bighorn dolomite
bedding plane detachment overrode former land surface
at Heart Mtn., there is an apparent thrust relation
(Ordovician/Cretaceous)
K
O
regional slope < 2°
The mechanism of emplacement of the detachment is still much debated. It is possible that it was
emplaced very rapidly.
34.2.2 Subaqueous Slides
Gravity slides of unlithified or semi-lithified sediments on submarine slopes produces a very
intensely deformed rock which has been termed an olistostrome. These are also known as “sedimentary
mélanges”, the term mélange being French for mixture. Mélanges can also be tectonic in origin, forming
at the toe of an accretionary prism in a subduction zone.
Lecture 3 4 247Extensional Systems I
34.3 Growth Faulting on a Subsiding Passive Margin
Passive continental margins with high sedimentation rates commonly experience normal faulting
related primarily to the local loading by the additional sediments. The Gulf Coast is an excellent example.
Such structures are commonly called “down-to-the-basin” faults. You should be careful to distinguish
them from rift-stage structures describe in detail in the next lecture.
rift phase extensional basins
growth faults detached within drift phase sediments
subsidence
sedimentation
In detail, an individual growth fault looks like:
antithetic faultssynthetic faults
roll-over anticline showing "reverse drag"
fault parallels bedding
(Jurassic salt in the Gulf Coast)
true listric fault geometry
syn-fault deposits much thicker in hanging wall
The key to recognizing growth structures is that sediments of the same age are much thicker on the
hanging wall than they are on the footwall. This means that the fault was moving while the sediments
accumulated preferentially in the depression made by the fault.
Lecture 3 4 248Extensional Systems I
34.4 Tectonic Rift Provinces
34.4.1 Oceanic Spreading Centers
The largest tectonic rift provinces in the world are represented by the earth’s linked oceanic
spreading centers. These are sometimes inaccurately referred to as “Mid-ocean ridges” because the
spreading center in the Atlantic happens to be in the middle of the ocean. We know about the structure
of the oceanic spreading centers primarily from studies of their topography (or really their bathymetry).
That topography represents an important interplay between structure, magmatism, and thermal subsid-
ence.
isolated volcanoes
rough topo due mostly to normal faulting
Slow spreading (e.g. Mid-Atlantic Ridge)
Intermediate spreading (e.g. Galapagos Rise)continuous volcanic axis
smoother topography
axial high
smooth topography
Fast spreading (e.g. East Pacific Rise)
At slow spreading rates (~2.5 cm/yr), normal faulting dominates the topography. There is a
distinct rift valley. Even though there are greater local reliefs, overall the ridge is lower because there is a
smaller thermal component to the topography.
At intermediate spreading rates (7 cm/yr half rates), volcanic processes become more important as
magma can reach the surface every where along the axis. There is still a subdued rift valley due to
normal faulting but the topography is smoother and higher.
Lecture 3 4 249Extensional Systems I
At high spreading rates (~15 cm/yr) the regional topography is dominated by thermal effects and
abundant volcanism, with little or no axial rift valley.
34.4.2 Introduction to Intracontinental Rift Provinces
Intracontinental rift provinces form within continental crust (hence the prefix “intra”). They may
lead to the formation of an ocean basin, but there are many examples which never made it to that stage.
Such rifts are call failed rifts or aulacogens. Many such features are found at hot spot triple junctions
formed during the breakup of the continents:
faile
d ar
m
Hot spot
Pre-Breakup Post-Breakup
Most intracontinental rifts have a gross morphology similar to that of their oceanic counterparts. This
reflects the importance of lithospheric scale thermal processes in extensional deformation. Generally, the
regional thermal upwarp is much larger than the zone of rifting.
marginal highs
extended region with "basin & range" style morphology
thinned crust and lithosphere
50 - 800 km
Lecture 3 4 250Extensional Systems II
LECTURE 35—EXTENSIONAL SYSTEMS II
35.1 Basic Categories of Extensional Structures
Until about 15 years ago, our understanding of extensional deformation was dominated by Ander-
son’s theory of faulting. The resulting geometric model is known as the horst and graben model:
horstgraben graben
Faults in this model are planar and dip at 60° (assuming an angle of internal friction of 30°). Superficially,
this model appeared to fit the observations from many rifted areas (e.g. the Basin and Range, Rhine
Graben, etc.).
The basic problems with it are:
• non-rotational, even though tilted beds are common in rift provinces
• only small extensions are possible, and we now know of extensions >100%
These problems forced people to seek alternative geometries
35.2 Rotated Planar Faults
In this geometry, the faults are planar but they rotate as they move, much as a stack of dominoes
collapses. For that reason it is commonly called the domino model. The resulting basins which form at
the top of the dominoes are call asymmetric half graben because they are bounded by a fault only on
one side. This model produces the commonly observed rotations in rift provinces:
Lecture 3 5 251Extensional Systems II
xφθ
w
If you know the dip of the rotated bedding and the dip of the fault, you can calculate the horizontal
extension assuming a domino model from the following equation (from Thompson, 1960):
% extension = x - w
w 100 =
sin (φ + θ)sin φ
- 1 100 .
When the faults rotate to a low angle, they are no longer suitably oriented for slip. Then, a new
set of faults may form at a high angle. Several episodes of rotated normal faults can result in very large
extensions.
Lecture 3 5 252Extensional Systems II
35.3 Listric Normal Faults
In listric normal faults, only the bedding in the hanging wall rotates. This is in contrast with the
domino model in which the faults and bedding in both hanging wall and footwall rotate.
problem: how does block deform to fill space?
The shape of a listric block poses interesting space problems. How does the hanging wall deform to fill
the space. The solutions to this problem are illustrated below.
solution 1: simple shear of hanging wall solution 2: decreasing slip down dip
solution 3: oblique of vertical simple shear of hanging wall
In both the listric and the rotated planar faults cases, the dip of bedding is directly related to the percent
horizontal extension. For the same bedding dip, the amount of extension predicted by the rotated planar
faults is much greater than that predicted by the listric faults as shown schematically by the graph below
[the graph is not accurate, but is for general illustration purposes only].
Lecture 3 5 253Extensional Systems II
90°
45°
Dip
of B
eddi
ng,
θ
100 200
% Extension
listric fault
rotated planar fault
35.4 Low-angle Normal Faults
Planar, or very gently listric, normal faults which formed initially at a low angle (in contrast to
faults rotated to a low angle) and move at a low angle are called low angle normal faults. These faults
are very controversial because they are markedly at odds with Anderson’s Law of faulting. Given the
weakness of rocks under tension, it seems likely that they move under their own weight and over
virtually friction-free surfaces (which could be simulated by pore pressure close to lithostatic, i.e. λ ≈ 1).
Their mechanics are still poorly known and much debated. These faults accommodate more extension
than high-angle normal faults, but less than either of the geometries discussed above.
All of the above structural styles can be combined in a single extensional system. The picture, below, is
similar to cross-sections drawn across many of the metamorphic core complexes in the western U.S.
upwarp due to unloading of the footwall
Lecture 3 5 254Extensional Systems II
35.5 Review of Structural Geometries
The following table, after Wernicke and Burchfiel, summarizes the structural styles discussed
above:
Planarfault
Curvedfault
Rotational Non-rotational
Faults (domino style)& strata both rotated
Faults (listric-style) HW strata only rotated
High-angle & low-angle normal faults
compaction after faulting
35.6 Thrust Belt Concepts Applied to Extensional Terranes
35.6.1 Ramps, Flats, & Hanging Wall Anticlines:
35.6.2 Extensional Duplexes:
Lecture 3 5 255Extensional Systems II
35.7 Models of Intracontinental Extension
A major question is, “what happens in the middle and lower crust in extensional terranes?”
Because extensional provinces are generally characterized by high heat flow and therefore probably a
weak plastic rheology at relatively shallow depths, it is not at all clear that the faults that we see at the
surface should continue deep into the crust. There are now four basic models:
35.7.1 Horst & Graben:
35.7.2 “Brittle-ductile” Transition & Sub-horizontal Decoupling:
35.7.3 Lenses or Anastomosing Shear Zones:
Lecture 3 5 256Extensional Systems II
35.7.4 Crustal-Penetrating Low-Angle Normal Fault:
35.7.5 Hybrid Model of Intracontinental Extension
ductile stretchingof the lower crust
mechanical (cold) rifting of theupper crust, syn-rift strata
volcanoes, high topography, post-rift thermal subsidence strata
Thermal lithosphericthinning
crust
mantle
Lecture 3 6 257Strike-slip Provinces
LECTURE 36—STRIKE-SLIP FAUL T SYSTEMS
36.1 Tectonic setting of Strike-slip Faults
There are three general scales of occurrence of strike-slip faults:
1. Transform faults
1a. Oceanic transforms
1b. Intracontinental transforms
2. Transcurrent faults
3. Tear faults
36.1.1 Transform faults
Oceanic transforms occur at offsets of oceanic spreading centers. Paradoxically, the sense of
shear on an oceanic transform is just the opposite of that implied by the offset of the ridge. this arises
because the ridge offset is probably inherited form the initial continental break-up and is not produced by
displacement on the transform.
Oceanic transform
Lecture 3 6 258Strike-slip Provinces
36.2 Transcurrent Faults and Tear Faults
Large strike-slip faults within continents which are parts of plate boundaries are call intraconti-
nental transforms. Examples include:
• San Andreas fault (California)
• Alpine fault (New Zealand)
• North Anatolian fault (Turkey)
Other large intra-continental strike-slip faults —called transcurrent faults by Twiss and Moores—
are not clearly the plate boundaries include
• Altyn Tahg fault (China)
• Atacama fault (Chile)
• Garlock fault (California)
• Denali fault (Alaska)
All of these structures have a characteristic suite of structures associated with them.
A tear fault is a relatively minor strike-slip fault, which usually occurs in other types of structural
provinces (e.g. thrust or extensional systems) and accomodates differential movements of individual
allochthons. when a tear fault occurs within a thrust plate, it usually is confined to the hanging wall and
does not cut the footwall:
Lecture 3 6 259Strike-slip Provinces
fault does not continue into underlying plate
A wrench fault is basically a vertical strike slip fault whereas a strike slip fault can have any
orientation but must have slipped parallel to its strike.
36.3 Features Associated with Major Strike-slip Faults
In general there are three types of structures, all of which can occur along a single major strike
slip fault:
1. Convergent -- the blocks move closer or converge as they slide past each
other
2. Divergent -- the blocks move apart as they move past one another
3. Parallel -- they neither converge nor diverge.
36.3.1 Parallel Strike-slip
clay cake
Much of our basic understanding of the array of structures that develop during parallel strike-slip faulting
comes from experiments with clay cakes deformed in shear, as in the picture, above. These experiments
show that strike-slip is a two stage process involving
• pre-rupture structures, and
• post-rupture structures.
Lecture 3 6 260Strike-slip Provinces
1 Pre-rupture Structures
1. En echelon folds:
45°
The folds in the shear zone form initially at 45° to the shear zone walls, but then rotate to smaller angles.
2. Riedel Shears (conjugate strike-slip faults):
R (synthetic)
R' (antithetic)
φ2
90 -
φ2
90 - φ
The initial angles that the synthetic and antithetic shears form at is controlled by their coefficient of
internal friction. Those angles and the above geometry mean that the maximum compression and the
principal shortening axis of infinitesimal strain are both oriented at 45° to the shear zone boundary.
With continued shearing they will rotate (clockwise in the above diagram) to steeper angles.
Because the R' shears are originally at a high angle to the shear zone they will rotate more quickly and
become inactive more quickly than the R shears. In general, the R shears are more commonly observed,
probably because they have more displacement on them.
Riedel shears can be very useful for determining the sense of shear in brittle fault zones.
Lecture 3 6 261Strike-slip Provinces
3. Extension Cracks: In some cases, extension cracks will form, initially at 45° to the shear zone:
45°
These cracks can serve to break out blocks which subsequently rotate in the shear zone, domino-style:
Note that the faults between the blocks have the opposite sense of shear than the shear zone itself.
2 Rupture & Post-Rupture Structures
A rupture, a new set of shears, called “P-shears”, for symmetric to the R-shears. These tend to
link up the R-shears, forming a through-going fault zone:
Lecture 3 6 262Strike-slip Provinces
R (synthetic)
R' (antithetic)
φ2
P-shears
36.3.2 Convergent-Type
Convergent type structures have sometimes been referred to as transpressional structures, a
horrible term which is both genetic and confuses stress and strain.
In convergent structures, you see
• enhanced development of the en echelon folds
• development of thrust faults sub-parallel to folds axes
• formation of “flower structures”
In map view: In cross-section:
cross-section
A T
36.3.3 Divergent Type
In the divergent type, extensional structures dominate over compressional. It has the following
characteristics:
• folds are absent
• development of normal faults
Lecture 3 6 263Strike-slip Provinces
• formation of “inverted flower structures”
In cross-section:
A T
Extensional basins formed along strike-slip faults are called “pull-apart” basins.
36.4 Restraining and Releasing bends, duplexes
You can have both convergent and divergent structures formed along a single strike-slip fault
system. They usually form along bends in the fault:
rhombochasm orpull-apart basin"extensional
(releasing)bend"
"contractional(restraining)
bend"(e.g. Transverse Ranges
in S. California)
right step in a right-lateral fault system
left step in a right-lateral fault system
Restraining or releasing bends can be the site of formation of strike-slip duplexes, in which the faults can
either be contractional or extensional, repsectively. Extensional or contractional structures can also be
concentrated at the overlaps in en echelon strike-slip fault segments:
Lecture 3 6 264Strike-slip Provinces
thrust faults in overlap region normal faults in overlap region
36.5 Terminations of Strike-slip Faults
Transform faults, either oceanic or intracontinental, can only terminate at a triple-junction. Tran-
scurrent faults may terminate in a splay of strike-slip faults sometimes referred to as a horsetail structure:
In this way, the deformation is dtributed throughout the crust. Alternatively, they may terminate in an
imbricate fan of normal faults (for a releasing bend) or thrust faults (for a restraining bend).
Lecture 3 6 265Vertical Motions: Isostasy
LECTURE 37—DEFORMATION OF THE LITHOSPHERE
So far, we’ve mostly talked about “horizontal tectonics”, that is horizontal extension or horizontal
shortening. Yet the most obvious manifestation of deformation is the mountains! That is the vertical
displacements of the lithosphere.
There are two parts to the topographic development question:
1. What are the mechanisms by which mountains are uplifted? and
2. Once they are uplifted, how do they evolve?
37.1 Mechanisms of Uplift
37.1.1 Isostasy & Crust-lithosphere thickening
Imagine that you have an object (an iceberg, piece of wood, etc.) floating in water:
topography
ρwater
ρiceρ
ice
The way to get more topography is to make the ice (or wood) thicker. The topography itself and the ratio
of the part of the iceberg above and below water is a direct function of the ratio of the densities of ice and
water. This basic principle is known as isostasy.
There are two basic models for isostasy. The Pratt model assumes laterally varying densities; the
Airy model assumes constant lateral densities:
Lecture 3 7 266Vertical Motions: Isostasy
2.67 2.59 2.52 2.57 2.62 2.76
3.3 gm/cm3
compensation level
2.75
3.3 gm/cm3
2.75 2.75 2.75 2.75 2.75
Pratt Model Airy Model
[identical topo]
We now know that, in general, Airy Isostasy applies to the majority of the world’s mountain belts. Thus
most mountain belts have roots, just like icebergs have roots.
37.1.2 Differential Isostasy
Two relations make it simple to calculate the isostatic difference between two columns of rock:
1. The sum of the changes in mass in a column above the compensation level is
zero:
∆ ∆ ∆ ∆ρ ρ ρ ρw w s s c c m mh h h h( ) + ( ) + ( ) + ( ) = 0
where “w” refers to water, “s” to sediments, “c” to crust, and “m” to mantle.
2. The changes in elevation of the surface of the earth, ∆E, equals the sums of the
changes in the thickness of the layers:
∆ ∆ ∆ ∆ ∆E h h h hw s c m= + + +
This gives us two equations and two unknowns. Thus, if we know the densities and the changes in
elevations, we can predict the changes in crustal thicknesses.
Take as an example the Tibetan Plateau, which is 5 km high. If we assume a crustal density of
2.75 gm/cm3 and a mantle density of 3.3 gm/cm3 then:
∆ ∆ ∆E h hc m= + = 5 km
Lecture 3 7 267Vertical Motions: Isostasy
and
∆ ∆ ∆ ∆ρ ρc c m m c mh h h h( ) + ( ) = + =2 75 3 3 0. . .
Solving for ∆hc:
∆hc 2 75 3 3 5 3 3. . .−( ) = ∗and
∆hc = 30 km .
What this means is that the crust beneath the Tibetan Plateau should be 30 km thicker than a crust of
equivalent density, whose surface is a sea level. The base of the crust beneath Tibet should be 25 km
deeper than the base of the crust at sea level (because of the 5 km elevation). Note that the root is about five
times the size of the topographic high.
37.1.3 Flexural Isostasy
So far in our discussion of isostasy we’ve made the implicit assumption that the crust has no
lateral strength. Thus, when we increase the thickness by adding a load, you get vertical faults:
lithosphere
load
The Earth usually doesn’t work that way. More commonly, you see:
lithosphere
load
In other words, the lithosphere has finite strength and thus can distribute the support of the load over a
much broader area. The bending of the lithosphere is call flexure and the process of distributing the load
Lecture 3 7 268Vertical Motions: Isostasy
is called flexural isostasy. The equations which, to a first order, describe flexure are:
z zx x x
o=
+
−
cos sin exp
α α α
Where x is the distance from the center of the load, z is the vertical deflection at x, and zo is the maximum
deflection at x = 0. zo, α , and related constants are given by the following equations:
zV
Doo= α 3
8
αρ ρ
=−( )
41
4D
gm w
and
DEhe=
−( )
3
212 1 υ
This last equation is what really determines the amplitude and wavelength of the deflection. D is known
as the flexural rigidity, a measure of a plate’s resistance to bending. The flexural rigidity is in fact the
plate’s bending moment divided by its curvature. A high flexural rigidity will result in only very gentle
flexure.
As you can see from the above equation, D depends very strongly on he, the thickness of the plate
being bent, or in the case of the earth, the effective thickness of the elastic lithosphere; it varies as the cube
of the thickness. In simple terms, thin plates will flex much more than thick plates will.
If a mountain range sits on a very strong or thick plate, the load is distributed over a very broad
area and the mountains do not have a very big root.
In the Himalayan-Tibetan system we see both types of isostasy:
Lecture 3 7 269Vertical Motions: Isostasy
Himalayas flexural isostasy
India55 km
70 km
Tibetlocal isostasy
In general, the degree to which flexural vs. local isostasy dominate depend on a number of factors,
including heat flow, the age of the continental crust being subducted and the width of the mountain belt.
37.2 Geological Processes of Lithospheric Thickening
37.2.1 Distributed Shortening:
37.2.2 “Underthrusting”:
37.2.3 Magmatic Intrusion:
Lecture 3 7 270Vertical Motions: Isostasy
37.3 Thermal Uplift
Because things expand when they are heated, their density is reduced. This has a profound effect
on parts of the Earth’s lithosphere which are unusually hot, thin, or both. Thermal uplift is most
noticeable in rift provinces such as the oceanic spreading centers or intracontinental rifts where the
lithosphere is being actively thinned and the asthenosphere is unusually close to the surface. It can also,
however, be an important effect in compressional orogens with continental plateaus such as the Andes or
the Himalaya.
For the oceanic spreading centers, the change in elevation with time can be computed from:
∆E T Tkta
a ww a=
−−( )
ρρ ρ
απ
2
1
2 ,
where, α is the coefficient of thermal expansion, k is the thermal diffusivity (8 x 10-7 m2 s-1), Tw is the
temperature of seawater, Ta is the temperature of the asthenosphere (~1350°C), and t is time. In continental
areas the maximum regional elevation which you commonly can get by thermal uplift alone is between
1.5 and 2.0 km.
37.4 Evolution of Uplifted Continental Crust
Once uplifted what happens to all that mass of rock in mountain belts? There are some simple
physical reasons why mountain belts don’t grow continuously in elevation. At some point the gravitational
potential of the uplifted rocks counteracts and cancels the far field tectonic stresses and then the mountain
belt grows laterally rather than vertically.
Generally, the higher parts of mountain ranges, especially in the Himalaya and the Andes are in a
delicate balance between horizontal extension and horizontal compression. Small changes in plate interac-
tions, rheology of the crust, or erosion rates can cause the high topography to change from one state to
another.
We used to think of orogenies as being all “compressional” or all extensional. However, with this
understanding of the simple physics of mountain belts, it is clear the you can easily find normal faults
forming in the interior of the range at the same time as thrust faults are active along the exterior margins.
Peter Molnar makes an excellent analogy between mountain belts and medieval churches. Both
are built up high enough so that they would collapse under their own weight if it weren’t for their
Lecture 3 7 271Vertical Motions: Isostasy
external lateral supports. In the case of the churches, flying buttresses keep them from collapsing. In the
case of mountain belts, plate convergence and the horizontal tectonic stresses that it generates, keeps the
mountains from collapsing.
Many people now think that a very common sequence of events is for large scale intracontinental
rifting to follow a major mountain building episode. When the horizontal compression that built the
mountains is removed, the uplifted mass of rocks collapses under its own weight, initiating the rifting.
This sequence of events is observed, for example in the Mesozoic compressional deformation and the
Cenozoic Basin and Range formation in the western United States.
It is important to realize that there can be two type of extension in over-thickened crust: (1) a
superficial effect just due to the topography, and (2) a crustal-scale effect in which the positive buoyancy
of the root contributes significantly to the overall extension.
Index 272
Aactivation energy...................................127Alberta syncline.....................................237albite twins.............................................120allochthon...............................................225Alpine fault ............................................258Amontons........................................113-114analytical methods ....................................3Anderson’s theory..........................159-160Anderson’s theory of faulting .............250Andes ..............................................241, 270angle of internal friction........ 107, See also
fractureangular shear ...........................................40annealing ................................................126anticline...................................................178antiform ..................................................178Appalachians .........................................134Appalachians .........................................225Argentina........................................239, 241asperities.................................................114asthenosphere........................................270asymmetric half graben......... 250, See also
extensional structuresasymmetric porphyroclasts ..........220-221Atacama fault.........................................258Athy’s Law.........104, See also compactionAtlas Mountains ....................................239augen.......................................................152aulacogens..............................................249autochthon..............................................225axes............................................................12
Bb-value ........................................................6basalt .......................................................103Basin and Range............................250, 271Basin and Range........................................3bedding...................................................207bending moment ...................................268Bighorn Basin.........................................243bond
attraction...................................100, 102force...................................................101length ................................................102potential energy........................100-102repulsion...................................100, 102
boudinage...............................................206
boudinage, chocolate tablet .................206boudins ...........................................203, 205Bowden...................................................114Britain, coast of ..........................................5brittle.........................................................98“brittle-ductile transition” ...................131Brunton compass.....................................15buckling ...........................................192-193Burgers vector........................................122Busk method ..........................................186
Ccalcite.......................................................120Canada....................................................239Canadian Rockies..................226, 231, 237Cape Ranges...........................................239Cartesian coordinates .................10, 14, 17cataclasis ...................................................98Cauchy’s Law.....................................64, 68CDP ......................................................23-24Chamberlain, T. C. ................................225characteristic equation............................73China.......................................................239cleavage ...................................203, 207-208
anastomosing...................................210conjugate...........................................210crenulation................................211, 216
climb..........................................................98Coble creep.............................................119COCORP.................................................241COCORP...................................................25coefficient of friction .............................112coefficient of internal friction110, See also
fracturecoefficient of thermal expansion.........102cohesion..........................................110, 112cold working..........................................125Colorado.................................................239Colorado Plateau...................................135columnar joints ......................................103compaction......................................103-104confining pressure.....................95, 97, 107continuum mechanics.............................36coordinate transformation .....................71cosine.........................................................17cracks, modes I, II, III............................133creep..........................................................94creep curve ...............................................95
Index 273
Crenulation cleavage ............................216cross product.......................................18-19crust.........................................................131crystal plastic ...................................98, 146curvature ................................................191cylindrical fold.........................................12
DDahlstrom...............................................231décollement....................................177, 187décollement....................................225, 228defects .....................................................117
impurities .........................................118interstitial..........................................118linear..................................................121planar ................................................119point ..................................................118substitution ......................................118vacancies...........................................118
deformation bands................................119deformation lamellae............................119deformation map...................................129deformation paths...................................55deformation, crystal plastic .127, 146, 152Delaware Water Gap ....................116, 214Denali fault.............................................258denudation .............................................245deviatoric stress.......................................65diagenesis ................................104, 116-117diffractions ..........................................27-28diffusion.....................98, 118, 125-126, 129diffusion
diffusion coefficient ........................148erosional ...........................................147
diffusion creep...............................127, 129crystal lattice ....................119, 126, 129grain boundary................116, 119, 126
Dilation ...............................................33, 41dip and dip direction..............................15dip isogon...............................................185direction cosines..........................14, 16, 82dislocation glide ..............................98, 123dislocation
jogs.....................................................124climb..................................................125edge ...................................................123glide............................................125-126glide and climb.........................125-127
pinning..............................................124screw .................................................123self stress field..................................123strain hardening ..............................123
dislocations..............................119, 121-122displacement gradient tensor...........71-72displacement vector................................37Distortion..................................................33Dix equation.............................................25dolomite..................................................120dome and basin .....................................190dominos ..................................................220dot product...............................................18drag folds................................................195ductile .......................................................98dummy suffix notation...........................69duplex ...................235, See also thrust belt
extensional........................................254passive roof238, See also triangle zone
dynamic analysis...................................161
Eearthquake..................................................6earthquake................................................95east-north-up convention.......................11eigenvalues...............................................73eigenvectors .............................................73Einstein summation convention ......69-70elastic.........................................................93elastic deformation................................100electrical conductivity.............................68ellipticity...................................................50en echelon...............................................134engineering mechanics.............................1exfoliation...............................................136Experimental..............................................3extension...................................................39
domino model..................................250down-to-the-basin...........................247
Ffabric........................................................203failed rifts................................................249failure envelop........ 108, 110, 113, See also
fracturefault bend folding232, See also thrust beltfault plane solutions..............................161fault rock.................................................146
Index 274
fault rocks...............................................144fault scarp...............................................147fault-line scarp.......................................147fault-propagation folds.........................242faults........................................................133
anastomosing...................................141blind ..................................................149branch line........................................142conjugate sets...................................158dip slip ..............................................143emergent...........................................147footwall.............................................141hanging wall ....................................141hinge fault ........................................144left lateral (sinistral) ........................143listric..................................................141listric..................................................160normal...............................................143oblique slip.......................................143piercing points.................................142planar ................................................141reverse...............................................143right lateral (dextral).......................143rotational ..........................................144scissors ..............................................144sense-of-slip......................................151separation.........................................142slip vector .........................................142surface trace .....................................142tip line ...............................................141wrench ..............................................143
fenster......................................................225flat irons..................................................148flexural rigidity......................................268flexure .....................................................267Flinn diagram...........................................59fluid inclusions ......................................138fluid pressure.104, See also pore pressurefluid pressure ratio................................175fluids .........................................................96fold axis...................................................186folding
“competence”...................................199buckling ............................................193dominant wavelength.....................200elastic.................................................200flexural flow.....................................194flexural slip.......................................194
multilayers........................................199neutral surface .................................193nucleation.........................................200passive flow......................................197theoretical analysis..........................199viscous ..............................................200
foldsanticlinoria........................................180asymmetric.......................................179axial surface...............................181-182axial trace..........................................182axis.....................................................192class 1 ................................................185concentric..........................................186conical ...............................................188cylindrical.........................................186dip isogon.........................................185drag ...................................................195enveloping surface..........................179facing.................................................179hinge..................................................180interlimb angle.................................183isoclinal.............................................183kink............................................187, 195non-cylindrical.................................188parallel ..............................................186parasitic.............................................195plunging ...........................................182reclined .............................................182recumbent.........................................182sheath................................................188similar ...............................................187superposed.......................................189symmetric.........................................179synclinoria........................................180vergence............................................179
foliation...........................................203, 207folium......................................................207footwall...................................................232force.............................................................2force...........................................................61foreland...........................................227, 235foreland basin ........................................230
broken ...............................................244foreland basins.......................................229foreland thrust belts....226, See also thrust
beltFORTRAN................................................70
Index 275
fractal dimension.......................................5fractals......... 4, 6, See also scale invariancefracture............................................106, 126
brittle.................................................106Coulomb....................................110-111ductile ...............................................106ductile failure...................................111tensile ................................................108transitional tensile...........................109
fractures, pre-existing...........................112Fresnel Zone.............................................27Fresnel zone .............................................27friction..............................................113-114
GGarlock fault ..........................................258Gaussian curvature...............................191geometry..................................................2-3geothermal gradient..............................129Gibbs notation..........................................12gneissic layering....................................207graded beds............................................208grain boundaries ...................................119grain size.........................................110, 129Granite Mountains ................................243gravity slides..........................................245Griffith Cracks .......................................109Griffith cracks ........................................110growth faulting.. 247, See also extensional
structuresGulf Coast...............................................247
Hhand sample...............................................4hanging wall ..........................................232Heart Mountain detachment ...............176Heart Mountain fault............................246Herring Nabarro creep.........................119Himalaya ................................................270hinterland.......................................227, 235homocline.......................................180, 183horse......................235, See also thrust belthorsetail structure .................................264horst and graben....................................250hot working............................................125Hubbert & Rubey ..................................170hydraulic fracturing..............................112hydrostatic pressure .............................175
hydrostatic pressure ...............................65
IIberian-Catalán Ranges ........................239Idaho .......................................................230indicial notation.......................................13interval velocity.......................................25isostasy....................................................265
Airy.............................................265-266differential........................................266flexural...............................229, 267-269local ...................................................269Pratt ...................................................265
JJeffery Model..........................................214jelly sandwich ........................................131joint sets ..................................................134joint systems...........................................134joints........................................................133
butting relation................................135cooling...............................................136cross joints ........................................134sheet structure..................................136systematic joints ..............................134twist hackles.....................................135
joints........................................................203joint zone ..........................................134
Jura Mountains ......................................226
Kkinematic analysis.................................161kinematic analysis...................................33kinematics...................................................2klippe ......................................................225
Llaboratory .................................................97landslides................................................245Laramide Province................................239latitude......................................................10left-handed coordinates 11, See also right-
handed coordinateslinear algebra ...........................................67lineation..................................................203lines ...........................................................14listric normal faults ...............................252lithosphere..............................................130
Index 276
lithosphere......................................229, 270lithostatic pressure..................................65longitude...................................................10low angle normal faults........................253
MMackenzie Mountains ..........................239magnitude ..................................................2magnitude, earthquake ............................6Mandelbrot, B. ...........................................5mantle .....................................................131March model..........................................214Martinsburg Formation................214, 216material properties....................................1mean stress...............................................65mechanics ................................................2-3mélange...................................................246mélanges.................................................223metamorphic core complexes..............253metamorphic foliation..........................113metamorphism ......................................117mica “fish” .............................................219mica fish..................................................152microlithon.....................................215, 217microlithons ...........................................211mid-ocean ridges...................................248migration .............................................25-26mineral fibers .................................152, 203mineral lineations..................................152minor structures ....................................203miogeocline ............................................228Modulus of Rigidity................................89Mohr’s Circle............................................82Mohr’s Circle, 3-D...................................86Mohr’s Circle, finite strain .....................47Mohr’s circle, for stress.........................106Mohr’s Circle, for stress................107, 111Mohr’s Circle, stress......78-79, 85, See also
stressmoment, earthquake.................................6monocline...............................................180Morocco ..................................................239Mother Lode...........................................138movement plane....................................161mullions..................................................203multiple.....................................................30multiples...................................................29mylonite..................................................146
Nneutral surface .......................................193New England .........................................136Newton .....................................................61Newtonian fluid ......................................93non-penetrative .........................................4North Anatolian fault ...........................258north-east-down convention ...........11, 14numerical methods ...................................3
Ooceanic spreading centers ....................248Oklahoma...............................................239olistostrome............................................246olivine .....................................................130optical microscope ....................................4orientation ..................................................2orientations...............................................14orthogonality relations ...........................82
PP and T axes ...........................................161P-shears...................................................151P-shears...................................................261P-waves.....................................................22paleocurrent indicators ..........................12paleomagnetic poles ...............................12parallelogram law ...................................18parasitic folds.........................................195parautochthon........................................225particle path .............................................37particle paths............................................34passive margin...............................228, 247pencil structure......................................206penetrative..................................................4permeability ...........................................112piercing points.......................................142pitch...........................................................15plagioclase..............................................120plane strain...............................................58plane trigonometry ...................................8planes ........................................................14plastic, perfect..........................................94plate convergence rates ........................229point source..............................................27Poissons Ratio..........................................89poles ....................................................12, 14
Index 277
pore fluid..................................................97pore fluid pressure................................175pore pressure ..........................105, 111-112pore space.................................................96porosity....................................103-104, 112power law creep ....................................127pressure solution....................115-117, 126pressure solution....................213, 215-217
grain size...........................................117impurities .........................................117temperature......................................117
principal stresses .....................................63pure shear.................................................54
Qquadrangle map ........................................4quadratic elongation..... 39, See also strain
RR-shears ..................................................151R-shears ..................................................261rake............................................................15reflection coefficient................................21reflectivity.................................................23rheology....................................................87Rhine Graben .........................................250Rich, J. L..................................................231Riedel Shears..........................150, 152, 260rift provinces..........................................249right-hand rule..............................15, 19-20Right-handed coordinates11, See also left-
handed coordinatesrigid body deformation..........................33rock bursts..............................................136Rocky Mountain Foreland ...................239Rocky Mountain foreland....................243Rodgers, J................................................225rods..........................................................203rotation........................................................1Rotation.....................................................33rotation......................................................34
left-handed .........................................34right-handed ......................................34
rupture stress ...........................................96
SS-C fabrics...............................................152S-C fabrics...............................203, 207, 219
S-surfaces................................................207S-waves.....................................................22sag ponds................................................148San Andreas fault..................................258sandstone dikes .....................................214scalar ..............................................17-18, 67scalar product ..... 18, See also dot productscale .......................................................3, 36scale invariance..........4-5, See also fractalsscale
global.....................................................3macroscopic..........................................3map.....................................................3-4mesoscopic ...........................................4microscopic ..........................................4provincial..............................................3regional .................................................3submicroscopic ....................................4
schistosity ...............................................207secular equation.......................................73sedimentary basin .................................104seismic reflection
artifacts................................................29fold.......................................................24
shear strain...............................................40shear stress .............................................158shear zone...............................................140
displacement....................................221sense-of-shear ..................................219
sheath folds ....................................152, 188Sheep Mountain ....................................243Sierras Pampeanas ........................239, 241sign conventions
engineering.........................................61geology................................................61
simple shear ...........................................139simple shear .............................................54sine.............................................................17slickenlines .............................................152slickensides ............................................152slip system..............................................123slip vector ...............................................142soil mechanics........................................105Spain........................................................239spalling....................................................136spherical coordinates..................10, 14, 17stacking.....................................................24stereographic projection.........................12
Index 278
strain.........................................................1-2strain........................................................100strain..........................................................68strain ellipse .............................................42strain ellipsoid .......................................217strain hardening ....................................123strain rate....................................................3strain rate................. 92-93, 96-97, 127, 129strain softening......................................125strain
angles .............................................39-40coaxial .................................................54continuous..........................................34discontinuous.....................................35finite ....................................................53heterogeneous....................................35homogeneous.....................................35infinitesimal .......................................53lines .....................................................39lines of no finite elongation .............51maximum angular shear ..................49non-coaxial .........................................54non-commutability............................58non-rotational ....................................54principal axes.....................................48pure shear...........................................54rotational ............................................54simple shear .......................................54superposition .....................................58volume ................................................39volumetric ........................................103
stress.........................................................1-2stress................................100, 107, 110, 114stress...............................................61, 67-68stress field.................................................86stress tensor.........................................63-64stress trajectory........................................87stress vector..............................................61stress
axial .....................................................65biaxial..................................................65deviatoric............................................64differential..........................................95effective.............................................105isotropic ............................................106mean...............................................64-65normal.................................................77principal..............................................63
principal plane...................................86shear ....................................................77spherical..............................................66triaxial .................................................65uniaxial ...............................................65units.....................................................61
stretch........................................................39striae........................................................152strike and dip...........................................14strike-slip, convergent ..........................259strike-slip, convergent type .................262strike-slip, divergent.............................259strike-slip, divergent type....................262strike-slip, en echelon folds .................260strike-slip, parallel.................................259strike-slip, transpression ......................262structural domains..................................36stylolites..................................................115stylolites..................................................203sub-grain walls ......................................126Sub-Himalayan Belt ..............................226Subandean belt ......................................226subduction, flat......................................239subgrain boundaries .............................119syncline...................................................178synform...................................................178
Ttangent vector ........................................122tear fault....... 258, See also strike-slip faulttemperature............................................117temperature......................96, 102, 127, 129tension gashes........................................139tensor transformation .......................70, 81tensor
antisymmetric ....................................72asymmetric.........................................72infinitesimal strain ............................72invariants............................................73principal axes...............................73, 83symmetric...........................................72
tensors.......................................................68Terzaghi..................................................105Texas........................................................170Texas........................................................239Theoretical..................................................3thermal conductivity...............................68thermal diffusivity ................................270
Index 279
thermal expansion.................................270thermal subsidence ...............................248thermodynamics........................................3thrust belt
Andean-type ....................................227antiformal stack...............................236antithetic ...................................227, 229basic characteristics.........................228Dahlstrom’s rules .....................231-232duration ............................................229folded thrusts...................................234Himalayan-type...............................227imbrication .......................................237ramp and flat geometry..................231rates ...................................................229synthetic....................................227, 229timing................................................229triangle zones...................................237types of folds in ...............................233
thrust faults ............................................170gravity gliding .................................176gravity sliding..................................176paradox of ........................................170wedge shape ....................................176
thrustsout-of-the-syncline ..........................242thick-skinned ...........................225, 239thin-skinned.....................................225
Tibetan Plateau...............................266-267Tien Shan................................................239tool marks...............................................152topography, pre-glacial ........................136traction vector..........................................61traction vectors ........................................62transcurrent fault...................................258transform, intracontinental .. 258, See also
strike-slip faulttransformation matrix........................82-83transformation of axes............................81translation.............................................1, 33transposition ...................................222-223trend and plunge.....................................15triangle zone...........................................238triple-junction ........................................264twin glide................................................120twin lamellae...................................119-120
U
unit vector ...............................13-14, 16, 19universal gas constant ..........................127
Vvacancies.............118, 125, See also defectsValley & Ridge Province ......................226vector.......................................13, 14, 17, 67vector product.. 19, See also cross product
addition...............................................18cross product.................................18-19dot product.........................................18magnitude ..........................................13scalar multiplication .........................17subtraction..........................................18
veins ........................................115, 133, 137antitaxial ...........................................138sigmoidal ..........................................139syntaxial............................................138"tension" gashes...............................139
veins ........................................................203velocity......................................................26
pullup/pushdown............................29rock......................................................22
vergence..................................................179viscosity ..................................................200viscosity ....................................................93viscous, perfect ........................................94void ratio ................................................104Von Mises...............................................111
Wwavelength...............................................26wedge taper............................................177wedge, critical taper..............................177Wichita-Arbuckle Mountains..............239Wind River Mountains.........................241window...................................................225wrench fault ...........................................259Wyoming.........230, 239, 241, 243-244, 246
Yyield stress...........................................96-97Young’s modulus ..................................200Young’s Modulus....................................89