Introduction to structural geology by Maria Villarreal

Contents

GEOL 326 Cornell University

Introduction to Structural Geology Spring 1999

Richard W. Allmendinger Department of Geological Sciences Snee Hall Cornell University, Ithaca, NY 14853-1504 USA rwa1@cornell.edu

R. W. Allmendinger ÂŠ 1999

Contents

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

Contents

iii

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

Contents

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

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.......................... 1 0 0

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............................................... 1 0 6

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 1 1 4

Contents

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 ....................................... 1 2 1

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 ................................ 1 2 7 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 ..................................................................................1 3 3 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 ............................................................. 1 4 1

19.1 Descriptive Fault Geometry ................................................................................................... 141

Contents

vii

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............................................. 1 4 7

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.................................................... 1 5 7

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..............................................1 7 0

22.1 The Paradox of Low-angle Thrust Faults.............................................................................. 170 22.2 Hubbert & Rubey Analysis..................................................................................................... 170 22.3 Alternative Solutions.............................................................................................................. 174

Contents

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Lecture 23—Folds I:

Geometry ............................................................................ 1 7 8

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.................................................... 1 8 4

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........................................................................1 9 1

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 ..........................................................................1 9 8

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

Lecture 27—Linear Minor Structures.................................................................... 2 0 3 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.................................................................. 2 0 7 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 .................................. 2 1 3

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............................................................ 2 1 9 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

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................................ 2 2 5

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 ............................................... 2 3 1

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......................................2 3 9

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

Lecture 34—Extensional Systems I ...................................................................... 2 4 5 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 II .....................................................................2 5 0 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 Systems ............................................................... 2 5 7 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

xii

36.5 Terminations of Strike-slip Faults........................................................................................... 264

Lecture 37—Deformation of the Lithosphere ........................................................2 6 5 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 Terminology, Scale

LECTURE 1â&#x20AC;&#x201D;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 hours ago or a fold that formed 500 Ma ago. The stresses that were responsible for that deformation are no

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Lecture 1 Terminology, 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

compression

shortening (contraction)

tension

lengthening (extension)

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

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: â&#x20AC;˘ Geometry basically means how big or extensive something is (size or magnitude) 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. â&#x20AC;˘ 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. â&#x20AC;˘ Mechanics implies an understanding of how forces applied to a material have produced the movements of the particles that make up the material.

1.3

Types of Structural Study

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Lecture 1 Terminology, Scale

• Observation of natural structures, or deformed features in rock. This observation 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 thermodynamics, through analytical or numerical methods, to relatively simple structural 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 S c a l e

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. ~104-10 5 km (circumference = 4 x 104 km) • Regional or Provincial -- poorly defined; generally corresponds to a physiographic province (e.g. the Basin and Range) or a mountain belt 103-10 4 km (e.g. the Appalachians). • Macroscopic or Map Scale -- Bigger than an area you can see standing in one

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Lecture 1 Terminology, 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):

non-penetrative penetrative

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 sections under the microscope or they can be observed at map scale, covering 100s of square kilometers.

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Lecture 1 Terminology, 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! "True" Geography

Measurements with Successively Smaller Rulers true coastline

ocean

land

ruler "b" ruler "c"

ocean

ruler "a"

Length of coastline as determined with: ruler "a" ruler "b" ruler "c"

Mandelbrot defined the fractal dimension, D, according to the following equation:

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Lecture 1 Terminology, Scale

L(G) ~ G 1 - 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 Mandelbrotâ&#x20AC;&#x2122;s original article, shows this scale dependence for a number of different coasts in log-log form.

log10 (total length in kilometers)

coast of A

ustralia (D

4.0

= 1.13)

circle coast of South Africa (D = 1.02) 3.5

land fron

tier of Ge

rmany, 1

west

3.0

1.0

coas

900 (D =

land fron

t of B

ritain

(D =

tier of Po

1.5

rtugal (D

1.15)

1.25)

= 1.13)

2.0 2.5 3.0 log 10 (length of "measuring stick" in km)

3.5

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 â&#x20AC;&#x153;b-valueâ&#x20AC;?) 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 Terminology, 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 =

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−b  bd   log M +  a +  c c

Lecture 2 Vectors, Coordinate Systems

LECTURE 2 —COORDINATE SYSTEMS, 2.1

ETC .

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 about displacement if your don’t know where something was originally and where it ended up after the

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Lecture 2 Vectors, 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.

Examples Scalars

mass, volume, density, temperature

Vectors

velocity, displacement, force, acceleration, poles to planes, azimuths

Tensors

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 could just as easily be described by its latitude, longitude, and elevation or by its township, range and

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Lecture 2 Vectors, 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

matters whether we consider a point which is below sea level to be positive or negative. That’s crazy,

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Lecture 2 Vectors, 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:

North, East, Down

East, North, Up

X1 = North

X 3 = Up X2 = East

X 2 = North

X 1 = East

X 3 = 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 more customary. Note that these are not the only possible right-handed coordinate systems. For example,

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Lecture 2 Vectors, Coordinate Systems

west-south-up is also a perfectly good right-handed system although it, and all the other possible combinations 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 = [V 1, V 2, V 3]

(eqn. 1)

[It should be noted that, when structural geologists use a lower hemisphere stereographic projection exclusively we are automatically treating all lines as axes. To plot lines on the lower hemisphere, we arbitrarily assume that all lines point downwards. Generally this is not an issue, but consider the problem of a series of complex rotations involving paleocurrent directions. At some point during this process, the current direction may point into the air (i.e. the upper hemisphere). If we force that line to point into the lower hemisphere, we have just reversed the direction in which the current flowed! Generally poles to bedding are treated as axes as, for example, when we make a π-diagram. This, however, is not strictly correct. There are really two bedding poles, the vector which points in the direction of stratal younging and the vector which points towards older rocks.]

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Lecture 2 Vectors, 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]

X3 X2

(V12 + V22 )

( V 12 + V 22 )

V3 V

|V| β

|V|

X1 (a)

(b)

Components of a vector in Cartesian coordinates (a) in two dimensions and (b) in three 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| = (V 12 + V 22 + V3 2) Unit Vector.

1/2

(eqn. 2)

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.

ˆ = V1 , V2 , V3  V V V V 

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(eqn. 3)

Lecture 2 Vectors, 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 V , cos β = 2 , cosγ = 3 . V V 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 = [cosα , cos β , cosγ ] . V

(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:

or simply

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]

[ 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 quadrant. This can lead to ambiguity which, if we are trying to be quantitative, is dangerous. There are

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Lecture 2 Vectors, 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.

N 30°

rik

dip direction 40°

dip

Quadrant:

N 30 W, 40 SW

Azimuth & dip quadrant:

330, 40 SW

Azimuth, right-hand rule:

150, 40

Dip azimuth & dip:

240, 40

Pole trend & plunge:

060, 50

Alternative ways of recording the strike and dip of a plane. The methods which are 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 , cos γ = cos (90 - plunge) = sin (plunge)

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(eqn. 6a)

Lecture 2 Vectors, Coordinate Systems

N cos α

cos (plunge)

trend 90 - trend plunge

Perspective diagram showing the relations between the trend and plunge angles and the direction cosines of the vector in the Cartesian coordinate system. Gray plane is the vertical plane in which the plunge is measured.

cos β

90 - plunge cos γ unit vector D

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:

Table 1: Conversion from Spherical to Cartesian Coordinates Poles to Planes (right-hand rule)

Axis

Direction Cosine

Lines

North

cos α

cos(trend)*cos(plunge)

sin(strike)*sin(dip)

East

cos β

sin(trend)*cos(plunge)

–cos(strike)*sin(dip)

Down

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 space, one can quickly tell what quadrant the line dips in by the signs of the components of the vector.

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Lecture 2 Vectors, 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

Sine or Cosine

sine

cosine

0.5 0 -0.5 -1 0

120

150 180 210 Angle (degrees)

240

270

300

330

360

Graph of sine and cosine functions for 0 - 360°. The plot emphasizes that for every 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, xV 2, 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 versa) just multiply the vector (i.e. its components) by –1. The resulting vector will be parallel to the

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Lecture 2 Vectors, 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 V

U+

U (a)

–V

U–

(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)

The physical meaning of the dot product is the length of V times the length of U as projected onto V (that

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Lecture 2 Vectors, 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:

U•V . θ = cos−1  UV 

(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 β 1 cos β2 + cos γ 1 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, ˆl , 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 )

ˆl

(eqn. 13)

and V × U = [ V2U3 - V3U2 , V3U1 - V1U3 , V1U2 - V2U1]

(eqn. 14)

The cross product is best illustrated with a diagram, which relates to the above equations:

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Lecture 2 Vectors, Coordinate Systems

V× U U

l (unit vector)

θ V U×V

Diagram illustrating the meaning of the cross-product. The hand indicates the right-hand rule convention; for V × U, the finger curl from V towards U and the thumb points in the direction of the resulting vector, which is parallel to the unit vector ˆl . Note that V × U = - U × V

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Lecture 3 Seismic Reflection Data

LECTURE 3 — D ESCRIPTIVE GEOMETRY: S EISMIC 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. ρ air

v air

v rock

ρ rock

a very small amount of sound continues into the rock most sound is reflected back to the listener

rock wall

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 =

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amplitude of reflected wave ρ 2V2 − ρ 1V1 = amplitude of incident wave ρ2 V2 + ρ 1V1

Lecture 3 Seismic Reflection Data

In seismic reflection profiling, what do you actually measure? ground surface

time sound was made

time to go down to the 1st layer and return

time

depth

1st subsurface layer

time to go down to the 2nd layer and return 2nd subsurface layer

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

depth

because rocks have different velocities. The following diagram illustrates this point.

3 km/s 3 km

6 km/s

6 km

time

1s 2s

6 km horizontal reflector

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Lecture 3 Seismic 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 number of active stations is determined by the number of channels in the recording system. Most

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Lecture 3 Seismic 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 CDP Gather distance from source, x

CDP Gather with NMO

to time

∆t = normal moveout (NMO)

tx near offset

source

far offset the NMO velocity is whatever velocity that lines up all the traces in a CDP gather. It is not the same as the rock velocity

[in practice, there is no geophone at the source because it is too noisey]

The relation between the horizontal offset, x, and the time at which a reflector appears at that offset, tx, is:

tx = t0 + 2

x2 2

Vstacking

or 1

 x2  2 ∆t = t x − t0 =  t02 + 2 − t0 Vstacking  

If you have a very simple situation in which all of your reflections are flat and there are only

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Lecture 3 Seismic 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: 1

Vi 1 2

 Vst2 t 2 − Vst2 t1  2 1 = 2  t − t   2 1

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

Migration 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 Seismic Reflection Data

surface

actual raypaths actual position of reflector in space position of reflection assuming reflecting point is vertically beneath the station

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 Seismic Reflection Data

At low frequencies (long wavelengths) these three beds will be "smeared out" into one long wave form

layered sequence in the Earth

3.4.2

At higher frequencies (shorter wavelengths) the three beds will be distinguishable on the seismic section

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.

higher frequency

lower frequency

Îť 4

first Fresnel Zone

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 Seismic Reflection Data

Fresnel zone reflections

3.5

diffractions

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: geologic section

seismic section

raypaths

dike

diffraction from dike

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 become broader and more open. Thus at great depths in the crust, diffractions may be very hard to

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Lecture 3 Seismic 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 â&#x20AC;&#x153;pitfallsâ&#x20AC;? 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 sedimentary basins (e.g. the sediment-basement interface).

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Lecture 3 Seismic Reflection Data

Multiple from a flat layer: geologic section

seismic section

simple raypath

time

depth

primary reflection multiple raypath multiple at twice the travel time of the primary

Multiple from a dipping layer (note that the multiple has twice the dip of the primary): geologic section

seismic section

primary re

simple raypath

multiple raypath

mu trav ltiple at el tim tw e of ice the the prim ary

time

depth

flection

Pegleg multiples: geologic section

seismic section

3.6.3

time

depth

primary reflections simple raypaths

pegleg raypath

pegleg multiple

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 so this need not be true. Even though geophones record only vertical motions, a strong reflecting

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Lecture 3 Seismic Reflection Data

interface which is out-of-the-plane can produce a reflection on a profile, as in the case illustrated below. out of plane ray path ("sideswipe") from dike

seismic reflection survey along this line

in plane ray path from sandstone

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.

Buried Focus

geologic section

depth

seismic section

d c

f e

a c

time

3.6.4

f b

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 produce the focus. A good migration will correct for buried focus.

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Lecture 3 Seismic Reflection Data

3.6.5

Others • reflected refractions • reflected surface waves • spatial aliasing

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Lecture 4 Introduction to Deformation

LECTURE 4 — INTRODUCTION

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:

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Lecture 4 Introduction 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-handed clockwise dextral

Left-handed counter-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: Continuous -- strain properties vary smoothly throughout the body with no abrupt changes.

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Lecture 4 Introduction 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 Introduction 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 Introduction to Deformation

4.2.4 Four Aspects of a Deforming Rock System:

final (present)

initial (beginning)

1. Position today

2. Displacement

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 we called earlier, the particle path.

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Lecture 4 Introduction to Deformation

6 final (present) 26

0 Ma 8.5

10 11

31 35 Ma

initial (beginning)

3. Path

3. Dated Path

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 canâ&#x20AC;&#x2122;t often get that kind of information out of the earth. There are some cases, though:

80 Ma Emperor Seamounts Midway (40 Ma) Hawaii Ridge

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Pacific Ocean

Hawaii (0 Ma)

Lecture 4 Strain, 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).

4.3.1 Change in Line Length: Extension: ∆ l = ( li – lf )

we define extension (elongation)

e≡

(

)

lf ∆l l f − li = = −1 li li li

(4.1)

shortening is negative Stretch: Quadratic elongation:

S≡

lf li

=1 + e

(4.2)

λ = S 2 = (1 + e )

if λ = 1 then no change if λ < 1 then shortening if λ > 1 then extension

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(4.3)

Lecture 4 Strain, the basics

λ ≥ 0 because it is a function of S 2. It will only be 0 if volume change reduces lf to zero.

4.3.2 Changes in Angles:

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 = γ ≡ shear strain y

(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 Strain, the basics

4.3.3 Changes in Volume (Dilation):

Dilation =

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∆≡

− Vi Vi

)

(4.6)

Lecture 5 The 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 into an ellipsoid. You’ll have to take this on faith right now but we’ll show it to be true later on.

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Lecture 5 The 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 unit radius lf

=S3 = 3

λ3

=S1 =

λ1

1 1 1

S =

λ =

lf li

= lf

[sometimes you'll see the 1 and 3 axes referred to as the "X" and "Z" axes, respectively]

The general equation for a circle is:

x2 + z2 = 1,

and for an ellipse:

x2 z2 + =1 a2 b2

(5.1)

x2 z2 + =1 λ1 λ3

(5.2)

where a & b are the major and minor axes. So, the equation of the strain ellipse is:

5.3

Extension of a Line Now, let’s determine the strain of any line in the deformed state:

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Lecture 5 The Strain Ellipsoid

3 3

(x, z)

(x', z')

1 λ3

θ' λ1

From the above, you can see that:

z ′ = λ sin θ ′

and

x ′ = λ cosθ ′

(5.3)

Substituting into the strain ellipse equation (5.2), we get

λ sin 2 θ ′ λ cos2 θ ′ + = 1. λ3 λ1

(5.4)

sin 2 θ ′ cos2 θ ′ 1 + = . λ3 λ1 λ

(5.5)

Dividing both sides by λ, yields:

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:

λ3 + λ1 + (λ3 − λ1 ) cos 2θ ′ 1 = . 2λ1λ3 λ If we let

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(5.7)

Lecture 5 The Strain Ellipsoid

' 1 , and λ ' = 1 , λ' = 1 , λ 1 = 3 λ3 λ1 λ

then

(λ3′ + λ1′) − (λ3′ − λ1′) cos2θ ′ = λ ′ . 2

5.4

(5.8)

Shear Strain To get the shear strain, you need to know the equation for the tangent to an ellipse:

xx ′ zz′ + =1. λ1 λ3

(5.9)

x λ cosθ ′ z λ sin θ ′ + =1 , λ1 λ3

(5.10)

Substituting equations 5.3 (page 44) into 5.9:

we can solve for the intercepts of the tangent:

ψ + θ' λ3 λ sin θ'

(x', y') 90 − ψ

90 − θ'

θ' λ1

From equation 5.10 and setting first x = 0 and then z = 0, and solving for the other variable

λ cos θ'

From the trigonometry of the above triangle (from here, it can be solved in a lot of different ways):

tan(ψ + θ ′) =

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tan ψ + tan θ ′ λ1 = tan θ ′ . 1 − tan ψ tan θ ′ λ3

Lecture 5 The Strain Ellipsoid

tan ψ = γ .

Recall that:

Lots of substitutions later:

γ =

The denominator is just

λ 1λ 3 λ

(λ1 − λ3 ) sin θ ′ cosθ ′

λ3 cos2 θ ′ + λ1 sin 2 θ ′

, which you get by multiplying eqn. 5.4 by λ 1λ3 and dividing by λ.

Eventually, you get

γ 1 1 1  =  −  sin 2θ ′ . λ 2  λ3 λ1  and with the same reciprocals as we used before (top of page 45):

γ′=

γ (λ3′ − λ1′) sin 2θ ′ = 2 λ

Next time, we’ll see what all this effort is useful for…

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(5.11)

Lecture 5 Mohrs Circle for Finite Strain

LECTURE 6 — STRAIN III: MOHR 6.1

ON THE

STRAIN ELLIPSOID

Introduction Last time, we derived the fundamental equations for the strain ellipse:

λ′ =

(λ3′ + λ1′) − (λ3′ − λ1′) cos2θ ′ 2

(6.1)

and

γ′=

γ (λ3′ − λ1′) sin 2θ ′ = 2 λ

(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, 0) =  and radius

λ3′ + λ1′  , 0  2

 λ3′ − λ1′  .  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, which we mentioned last time, and can be applied to any symmetric tensor. We will see the construction

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Lecture 6 Mohrs Circle for Finite Strain

again when we talk about stress. But, for finite strain, it looks like:

γ'

λ' − λ' 3 1 2

λ ψ

2θ'

λ'1

λ'3

λ'

λ'3 + λ'1 2

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:

γ′ tan ψ = = λ′ 6.3

γ λ

1 λ

=γ

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.

Lines are perpendicular before and after the deformation because they are parallel to the principal axes

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Lecture 6 Mohrs 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'

li nt

λ' − λ' 3 1 2

tan

2θ'

max

λ'3 + λ'1

λ'

From the geometry above,

sin ψ max

λ3′ − λ1′ λ ′ − λ1′ = 2 λ′ + λ′ = 3 , 3 1 λ3′ + λ1′ 2

 λ ′ − λ1′  ψ max = sin −1  3  .  λ3′ + λ1′ 

(6.3)

To get the orientation of the line with maximum angular shear, θ'ψ max:

cos 2θψ′ max

λ3′ − λ1′ λ ′ − λ1′ , = 2 λ′ + λ′ = 3 3 1 λ3′ + λ1′ 2

θψ′ max =

 λ ′ − λ1′  1 cos −1  3  . 2  λ3′ + λ1′ 

(6.4)

You could also easily solve this problem by differentiating with respect to θ, and setting it equal

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Lecture 6 Mohrs Circle for Finite Strain

to zero:

dγ =0. dθ ′ 6 . 5 Ellipticity 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.

(1 + e1 ) = S1 (1 + e3 ) S3

(6.5)

Note that, because S 1 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 = z x

tan θ' = z' x'

The stretches along the principal axes, 1 and 3, are:

S1 = λ1 =

x′ ⇒ x ′ = x λ1 x

S3 = λ3 =

z′ ⇒ z ′ = z λ3 . z

and

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Lecture 6 Mohrs Circle for Finite Strain

Substituting into the above equations, we get a relation between θ and θ':

tan θ ′ =

z λ3 λ3 S tan θ = tan θ = tan θ 3 = . S1 R x λ1 λ1

(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 i = lf = 1 λ' = S2 = 1

We can solve for the orientations of these two lines by setting the Mohr Circle equation for elongation to 1,

λ′ =

(λ3′ + λ1′) − (λ3′ − λ1′) cos2θ ′ = 1, 2

and solving for θ':

cos 2θ ′ =

(λ3′ + λ1′ − 2) = 2 cos2 θ ′ − 1 , (λ3′ − λ1′)

and

cos2 θ ′ =

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(λ3′ − 1) (λ3′ − λ1′)

(6.7)

Lecture 6 Mohrs Circle for Finite Strain

There are alternative forms which use θ instead of θ' and λ instead of λ':

tan 2 θ =

(λ1 − 1) (1 − λ3 )

and

tan 2 θ ′ =

draft date: 20 Jan, 1999

λ3 (λ1 − 1) . λ1 (1 − λ3 )

Lecture 6 Infinitesimal & 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:

l f − li ≤ 0.02 li

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 This represents the total deformation from the beginning up until the present.

Infinitesimal Strain Ellipse This is the strain that the particles will feel in the next instant of deformation

You can look at it this way: Finite Strain

strain it a finite amount Start with a box

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Infinitesimal Strain

carve a new box out of it

and deform that new box by a very small amount

Lecture 7 Infinitesimal & 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

Coaxial ⇒ progressive pure shear

Rotational ⇒ Simple 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 diagram illustrates:

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Lecture 7 Infinitesimal & 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 area of the finite strain ellipse [“f(+)”], all of the lines are longer than they started out.

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Lecture 7 Infinitesimal & Finite Strain

note: LNIE at 45° to principal axes

i (-)

f (+) f (-) i (+) 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 (-) I Most general case:

f (+), i (+)

III

f (-), i (+)

An arbitrary superposition of the infinitesimal ellipse on the finite ellipse. Not very likely in a single progressive deformation

f (-), i (-) • 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 increment.

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Lecture 7 Infinitesimal & 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.

f (-), i (+) Ď&#x2C6;

f (+), i (+)

f (-), i (-) cards

III I II

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

Simple Shear

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).

f (-), i (+) f (-), i (-) f (+), i (+)

III

Pure Shear

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Lecture 7 Infinitesimal & 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°

1. Pure shear, ex = 1

2. Pure shear, ex = 1

2. Simple shear, ψ = 45°

A strain & a rotation: 1. Rotation = 45°

1. Stretch = 2

7.5

2. Stretch = 2

2. Rotation = 45°

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:

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Lecture 7 Infinitesimal & Finite Strain

Y =λ2

Z = λ3

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=∞

most geological deformations

1 + e1

k=1

1 + e2

oblate speroids "pancakes"

prolate speroids "cigars"

k=0 Y Z

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S2 S3

1 + e2 1 + e3

Lecture 8 Introduction 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:

F F

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Lecture 8 Introduction 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:

r r F σ= 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):

Force mass × acceleration = = Area Area

m kg 2  s  N = 2 = Pa 2 m m

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).

Engineers are much more worried about tensions.

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Lecture 8 Introduction to Stress

8.4

Stress on a Plane; Stress at a Point An arbitrary stress on a plane can be resolved into three components:

normal stress random stress on the plane

shear stress // 2 axis

X2 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:

X3 σ33

σ31

σ32 σ23

σ13

σ11

σ12 σ 21

σ22

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 the second subscript shows which axis the traction vector is parallel to

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Lecture 8 Introduction to Stress

These nine vectors can be written in matrix form:

σ 11 σ 12 σ 13  σ ij = σ 21 σ 22 σ 23  σ 31 σ 32 σ 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:

σ 1 0 0  σ ij =  0 σ 2 0   0 0 σ 3 

(8.3)

In this case the remaining components -- σ1, σ 2, 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.

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Lecture 8 Introduction to Stress

σ3

σ1

8.6

σ2

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:

pi = σ ij l j

(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 M e a n 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:

σm =

σ 1 + σ 2 + σ 3 σ 11 + σ 22 + σ 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:

0  σ 11 − σ m σ 12 σ 13  σ 11 σ 12 σ 13  σ m 0 σ     σ 22 − σ m σ 23  .  21 σ 22 σ 23  =  0 σ m 0  +  σ 21 σ 31 σ 32 σ 33   0 0 σ m   σ 31 σ 32 σ 33 − σ m 

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(8.6)

Lecture 8 Introduction 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. σ 1 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:

Plithostatic = ∫ ρgdz ≈ ρave gz 0

• Hydrostatic Pressure: (1) the weight of a column of fluid in the interconnected pore spaces in a rock (Suppe, 1986):

Pfluid = ρave gz f (2) The mean stress (Hobbs, Means, & Williams, 1976):

σm =

σ 1 + σ 2 + σ 3 σ 11 + σ 22 + σ 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 the outer core of the earth is a fluid -- it does not transmit shear waves from earthquakes.

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Lecture 8 Introduction 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.

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Lecture 9 Vectors & 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

and B is the temperature gradient which is a vector.

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Lecture 9 Vectors & 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:

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:

p1 β α X1 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:

pi = σ ij l j .

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(9.1)

Lecture 9 Vectors & 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:

p1 = σ 11l1 + σ 12l2 + σ 13l3 , p2 = σ 21l1 + σ 22l2 + σ 23l3 ,

(9.2)

p3 = σ 31l1 + σ 32l2 + σ 33l3 . We could write the same in matrix notation:

 p1  σ 11 σ 12 σ 13   l1   p  = σ     2   21 σ 22 σ 23  l2  ,  p3  σ 31 σ 32 σ 33  l3 

(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: 3

p1 = σ 11l1 + σ 12l2 + σ 13l3 = ∑ σ 1 j l j j =1

p2 = σ 21l1 + σ 22l2 + σ 23l3 = ∑ σ 2 j l j , j =1 3

p3 = σ 31l1 + σ 32l2 + σ 33l3 = ∑ σ 3 j l j . j =1

From here, it is just a short step to equation 9.1:

pi = σ ij l j , 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, X 2, and X3, and l1, l2, and l3 are equal to cosα, cosβ, and cosγ, respectively.

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Lecture 9 Vectors & 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 3 p(i) = 0 Do 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:

X 3'

V V3 V 3' V2

V 1'

(note that the length and relative orientation of V on the page has not changed; only the X 1' axes have changed)

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V 2'

X 2'

Lecture 9 Vectors & 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:

σ 1 0 0  σ ij =  0 σ 2 0   0 0 σ 3 

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 tensor.

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Lecture 9 Vectors & 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:

e11 e12 eij = e21 e22 e31 e32

e13  e23  , e33 

e12 ≠ e21 ,

where

If eij were a symmetric tensor, then e12 = e21 ,

e13 = e31,

e13 ≠ e31,

and

e32 ≠ e23 .

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:

eij =

  e11  (e + e ) =  21 12  2  (e31 + e13 )  2 

+ e ji 2

) + (e

− e ji

)

(e12 + e21 ) (e13 + e31 )  2 e22

(e32 + e23 ) 2

(e12 − e21 ) (e13 − e31 ) 

 0   2   (e23 + e32 )  +  (e21 − e12 )   2 2   (e31 − e13 ) e33   2  

2 0

(e32 − e23 ) 2

 2 (e23 − e32 )   2  0  

Writing the same equation in a more compact form:

eij = ε ij + ω ij , where

ε ij =

+ e ji 2

)

and

ω ij =

− e ji 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”. Note that the infinitesimal strain tensor is always symmetric. Thus, you can think of pure shear as ωi j = 0

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Lecture 9 Mohrs 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:

λ3 − Ιλ2 − ΙΙλ − ΙΙΙ = 0 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 1 + σ 2 + σ 3 = σ 11 + σ 22 + σ 33 . This is particularly useful when we get to stress and something known as hydrostatic pressure.

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Lecture 1 0 Mohrs 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.

p3N γ

p3S γ γ

p1S

p1N

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:

σ 1 0 0  σ ij =  0 σ 2 0   0 0 σ 3 

(10.1)

pi = σ ij l j

(10.2)

The general form of Cauchy’s Law is:

which, if we expand it out for the case shown above will be:

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Lecture 1 0 Mohrs 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:

p1 N = p1 cos α = (σ 1 cos α ) cos α = σ 1 cos2 α p1S = p1 sin α = (σ 1 cos α ) sin α and then for p3:

p3 N = p3 sin α = (σ 3 sin α ) sin α = σ 3 sin 2 α p3 S = p3 cos α = (σ 3 sin α ) cos α

Now, the normal stress arrows point in the same direction, so we add them together:

σ n = ( p1 N + p3 N ) = σ 1 cos2 α + σ 3 sin 2 α

(10.3)

The shear stress arrows point in opposite directions so we must subtract them:

σ s = ( p1S − p3 S ) = σ 1 cos α sin α − σ 3 cos α sin α = (σ 1 − σ 3 ) cos α sin α

10.2

(10.4)

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:

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Lecture 1 0 Mohrs Circle for Stress

Area = A sin θ Area = A cos θ F1 θ

Area = A

10.2.1 Balance of Forces F3N F3

Force normal to the plane:

F1 θ F1N

FN = F1N + F3N

F3S

Force parallel to the plane:

FS = F1S − F3S

F1S

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 θ

F3N = F3 sin θ ,

F3S = F3 cos θ.

and

So, substituting these into the force balance equations, we get:

FN = F1N + F3N = F1 cos θ + F3 sin θ

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(10.5)

Lecture 1 0 Mohrs 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 F S act on the inclined plane, which has an area = A. The normal and shear stresses then, are just those forces divided by A:

σn =

Fn A

and

σs =

Fs . A

(10.7)

F1 and F 3 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:

σ1 =

F1 A cosθ

and

σ3 =

F3 . A sin θ

(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:

FN = F1 cosθ + F3 sin θ = σ n A = σ 1 A cosθ cosθ + σ 3 A sin θ sin θ The A’s cancel out and we are left with an expression just in terms of the stresses:

σ n = σ 1 cos2 θ + σ 3 sin 2 θ

(10.9)

For the shear stresses:

FS = F1 sin θ − F3 cosθ = σ s A = σ 1 A cosθ sin θ − σ 3 A sin θ cosθ . As before, the A’s cancel out and we are left with an expression just in terms of the stresses:

σ s = τ = (σ 1 − σ 3 )sin θ cosθ

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(10.10)

Lecture 1 0 Mohrs 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.

1 0 . 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 - 2sin 2 α .

Using these identities, equations 10.9 and 10.10 (or 10.3 and 10.4) become:

σn =  

σ1 + σ 3   σ1 − σ 3  + cos 2θ 2   2 

σs = τ =  

(10.11)

σ1 − σ 3  sin 2θ 2 

(10.12)

The graphs below show how the normal and shear stresses vary as a function of the orientation of the plane, θ:

σ1 σ1 + σ3

σn

σ3 90°

The above curve shows that: • maximum normal stress = σ1 at θ = 0° • minimum normal stress = σ3 at θ = 90°

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180°

Lecture 1 0 Mohrs Circle for Stress

σs σ1 – σ3 2 θ 90°

180°

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

y = r sin α ,

and

so the above equations define a circle with a center on the x-axis and radius:

(c, 0) =  The Mohr’s Circle for stress looks like:

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σ1 + σ 3  , 0  2

and

σ1 − σ 3 2

Lecture 1 0 Mohrs Circle for Stress

σs σ1 – σ3 2 2θ σ3

σn

σ1

σ1 + σ3 2

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:

σ3

σs

2θ

σn

σ1

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 the angle between the pole and σ1.

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Lecture 1 0 Mohrs Circle for Stress

1 0 . 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.

X3 X3'

cos-1a 23

X2'

cos-1a 22 cos-1a 21

X1 X1'

In the diagram, a21 is the cosine of the angle between the new axis, X2’, and the old axis, X 1, 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 Mohrs Circle for Stress

Old Axes

New Axes

X1'

a 11

a 12

a 13

X2'

a 21

a 22

a 23

X3'

a 31

a 32

a 33

or, in matrix form:

 a11 aij =  a21   a31

a12 a22 a32

a13  a23  .  a33 

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 ij′ = aik a jlσ kl

(new in terms of old)

σ ij = aki aljσ kl′

(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 Any second order tensor can be represented by a Mohr’s Circle construction, which is derived

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Lecture 1 0 Mohrs 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.

X3 X3'

σij

X1'

σ1 0 0 = 0 σ2 0 0 0 σ3

X1 With a rotation of θ about the X2 axis, the transformation matrix is:

 cosθ aij =  0   − sin θ

0 sin θ  1 0   0 cosθ 

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

(σ 1 cos2 θ + σ 3 sin 2 θ ) 0  σ ij′ =  0 σ2  ((σ 1 − σ 3 ) sin θ cosθ ) 0 

((σ

− σ 1 ) sin θ cosθ )   0  . 2 2 (σ 1 sin θ + σ 3 cos θ ) 3

Rearranging using the double angle formulas, we get the familiar equations for Mohrs Circle

σ 11′ =  

σ1 + σ 3   σ1 − σ 3  + cos 2θ 2   2 

and

σ 13′ = − 

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σ1 − σ 3  sin 2θ 2 

Lecture 1 0 Mohrs Circle for Stress

σ'ij ( i ≠ j) σ3

σ'33 σ'31

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σ' 2θ 13 σ1 σ'11

σ'ii

Lecture 1 1 Stress-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

Compressive stresses σ n positive counterclockwise (left lateral) positive

clockwise (right a l teral) negative

Mohr’s circle quickly allows you to see some of the relationships that we graphed out last time:

σ3

σ S max =

σ1 − σ 3 2

2θ = 90°

σ1

θ = 45° 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. The general classes of stress expressed with Mohr’s circle are:

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Lecture 1 1 Stress-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): stresses on planes perpendicular to σ1-σ2 plane

σs

σn σ3

σ2

stresses on planes perpendicular to σ3-σ2 plane

σ1

stresses on planes perpendicular to σ1-σ3 plane (i.e. what we plotted in two dimensions)

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.

This variation constitutes what is known as a stress field.

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Lecture 1 1 Stress-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:

σ3 σ1

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: 1. If the material returns to its initial shape when the stress is removed, then the

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Lecture 1 1 Stress-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 2.5 5.0 7.5 10.0 0.0

Strain 0.5% 1.0% 1.5% 2.0% 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:

Stress

The straight line means that there is a constant ratio between stress and strain. This type of material behavior is known aselastic .

Strain

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:

Ď&#x192;i j = Ci j k l Îľk l .

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Lecture 1 1 Stress-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

wf li

el =

l f− l i li

et =

wf − w i wi

With the above measurements, there are several parameters we can derive Young’s Modulus:

σ = C1111 . el

This is for simple shortening or extensions. For the the ratio of the transverse to longitudinal strain we use Poissons Ratio:

υ=

et −E = el C1122

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:

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Lecture 1 1 Stress-Strain Relations

= Ge

For for uniform dilations or contractions, Bulk Modulus or Incompressibility: σ

= Ke

All of these parameters are related to each other by some simple equations:

11.5

E 3K (1 − 2υ ) = 2(1 + υ ) 2(1 + υ )

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:

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Lecture 1 1 Stress-Strain Relations

rupture in elastic realm

plastic deformation non-recoverable strain "anelastic" or "plastic"

rupture strength

ultimate strength yield strength

σ hypothetical paths when stress removed

max elastic strain permanent strain if stress removed befor rupture

plastic strain

Note that the maximum elastic strains are generally <<5%. There are two forms of plastic deformation:

perfect plastic yield strength

yield strength

strain hardening part of curve

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strain hardening

Lecture 1 2 Plastic & Viscous Deformation

LECTURE 12—PLASTIC & VISCOUS DEFORMATION 1 2 . 1 Strain Rate So far, we haven’t really said anything about time except to say the elasticity is instantaneous.

strain, e

stress, σ

You can think of two different graphs:

strain, e

time, t

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:

σc

strain, e

time, t

strain, e

σb

σa

time, t

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: 10 −16 s −1 ≤ e˙geol ≤ 10 −12 s −1 .

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Lecture 1 2 Plastic & 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.

1 2 . 2 Viscosity With this idea of strain rate in mind, we can define a new type of material response: •

stress, σ

σ = η e e op

The slope of the curve, η, is the viscosity.

It is a measure of the resistance of the material to flow •

strain rate, e

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: Viscoelastic

Elasticoviscous

stress removed

ous

strain

visc

delayed recovery

time

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elastic

time

Lecture 1 2 Plastic & 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

strain, e

three fields:

viscoelastic

III

rupture

stress removed at times 1 & 2 delayed recovery

elastic

time, t

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 This curve is constructed for constant stress; i.e. stress does not change during the entire length

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Lecture 1 2 Plastic & 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

strain, e

increasing differential stress

strain, e

σ rupture

Failure field

s, σ

es str

yield

Elastic field

time

time, t

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 shows that:

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Lecture 1 2 Plastic & Viscous Deformation

differential stress

100 Mpa

σy

30 Mpa

σy σy

3.5 Mpa

σr

confining (mean) stresses

σy 1 Mpa

[the confining pressure at the base of the continental crust is on the order of 1000 Mpa]

strain, e

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. 100°C 300°C

differential stress

25°C

500°C

these may never rupture

σy

σr

800°C

strain, e

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:

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Lecture 1 2 Plastic & Viscous Deformation

differential stress

400°C, dry

σy

900°C, dry

σr

1000°C, dry

900°C, wet

strain, e

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.

differential stress

10 10 10 10

-4 -5

-6 -7

-1

sec

-1

sec

-1

sec

-1

sec

strain, e 12.5

Brittle, Ductile, Cataclastic, Crystal Plastic

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σy

Lecture 1 2 Plastic & 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:

[internal deformation could be by grain-scale fracturing or by plastic flow of minerals; i.e. the deformation mechanism is not specified]

Brittle

Ductile

When people talk about the â&#x20AC;&#x153;brittle-ductileâ&#x20AC;? 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 Plastic & Viscous Deformation

Distribution of Deformation Localized

Distributed incr strain rate

Mechanism of Deformation

Cataclastic

Brittle faulting

Cataclastic Flow incr Temp, Conf. Press.

Crystal Plastic

Plastic shear zone

Brittle

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Homogeneous plastic Flow

Ductile

Lecture 1 3 Elasticity, Compaction

100

LECTURE 13—DEFORMATION MECHANISMS I: ELASTICITY, C OMPACTION 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.

1 3 . 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

Uattraction ∝ − PE due to repulsion from electron cloud overlap

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1 r

Lecture 1 3 Elasticity, Compaction

101

Urepulsion ∝ −

1 r12

The total potential energy, then, can be written as:

Utotal =

−A B + 12 r r

where C1 and C2 are constants. A graph of this function highlights its important features:

Potential Energy, U

repulsion term =

B r12

Bond length, r bond length with Umin, ro

[the solid curve is the sum of the other two]

minimum potential energy

attraction term = –

A r

To get the bond force, you have to differentiate the above equation with respect to r:

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dU − A 12 B = 2 + 13 dr r r

Lecture 1 3 Elasticity, Compaction

102

Bond Force, F = dU/dr

Repulsion

Bond Length, r

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

1 3 . 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:

eij = α ij ∆T 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 depends on crystal symmetry and thus varies between 1 and 6.

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Lecture 1 3 Elasticity, Compaction

103

A good example of the result of thermal strain are cooling joints in volcanic rocks (e.g. columnar joints in basalts).

1000 m Flow erupted at a temperature of 1020°C

1000 m Flow cooled to a surface temperature of 20°C w1

∆T = (Tf - Ti) = -1000°C

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:

w f − wi ∆w = wi wi

⇒

∆w = e w i = –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 .

1 3 . 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 Elasticity, Compaction

104

φ =

Vp = volume of the pores , total volume Vp + Vs

and the void ratio as:

θv =

Vp = volume of the pores volume of the solid Vs

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 = φ o 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.

fluid presses out equally in all directions [every plane is ⊥ to a principal stress so no shear stress]

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Lecture 1 3 Fracture

105

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 ij =

σ 1 1 - Pf σ 21 σ 31

σ 12 σ 2 2 - Pf σ 32

σ 13 σ 23 σ 3 3 - 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 Fracture

106

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: brittle fracture

ductile fracture rupture strength

rupture strength

yield 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:

σ1

σs

2θ

σ3

σ2 initial isotropic stress

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σn

Stress state at time of fracture

Lecture 1 4 Fracture

107

If we look in detail at the configuration of the Mohr’s Circle when fracture occurs, there is something very curious: [detail of previous figure]

φ = angle of internal friction

θ 2= 90° +φ

2θ

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

2θ

σn new initial isotropic stress

New stress state at time of fracture

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 envelop would produce a fracture of the rock:

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Lecture 1 4 Fracture

108

σs

2θ

σn

In general, we see a failure envelop which has four recognizable parts to it:

failure envelope

σs So

2θ σn

I II

III

Field I -- Tensile fracture: You can see that the Mohr’s circle touches the failure envelop in only one

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Lecture 1 4 Fracture

109

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

-30 ≤ To ≤ -4 Mpa

σ3

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° ≤ φ ≤ 90°

< 30° 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 Fracture

110

lines of equal stress

in plan view:

d l

The tensile stress at the tip of a crack is given by:

σ ≈

2 (2l)2 σ3 3 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:

σ 2s - 4 To σn - 4 T2o = 0 or

σ s = 2 To σ n + 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 Fracture

111

σ1 φ ≈ 30° 2θ ≈ 120°

σ3

θ ≈ 60° ~ 30°

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 0° ≤ φ ≤ 30° 90° ≤ 2θ ≤ 120°

σ3

45° ≤ θ ≤ 60° 30° - 45° 1 4 . 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 ij =

σ 1 1 - Pf σ 21 σ 31

σ 12 σ 2 2 - Pf σ 32

σ 13 σ 23 σ 3 3 - 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 Fracture

112

σs

σn σ3 - P

σ3

σ1 - P

σ1

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.

1 4 . 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 Fracture

113

pristine rock would only fail on this plane

σs

for lop ting e v en -exis pre tures c fra

elo

nv ee

fail

2θ 2 σn 2θ 1

σ3

σ1

any pre-exisitng fracture with an angle between 2θ 1 and 2θ 2 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.

θf

90°

60°

θf

30° fa

θc 0° 0°

30°

θc

60°

90°

1 4 . 4 Friction The importance of friction was first recognized by Amontons, a French physicist, in 1699. Amontons

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Lecture 1 4 Pressure Solution & Crystal Plasticity

114

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.

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:

asperities

voids

At the points of contact, or asperities, there is a high stress concentration due to the normal stress.

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LECTURE 15â&#x20AC;&#x201D;D EFORMATION 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

Ď&#x192;1

Stylolites Classic morphology: jagged teeth with concentrations of insoluable residue. This is common in marbles (e.g. particularly well seen in polished marble walls). Many stylolites don't have this form.

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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More commonly, there is much more evidence for removal of material than for the local reprecipitation. 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 â&#x20AC;&#x153;fluid assisted grain boundary diffusionâ&#x20AC;? because the material diffuses along a thin fluid film at the boundary of the grain:

Ď&#x192;1

solution of material at grain-to-grain contact

redeposition at the grain margins

This process is probably relatively common during diagenesis.

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there may be a thin fluid film between the grains

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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 • Planar

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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

Vacancy -- Atom missing from crystal lattice

Interstitial Impurity -- Atom of a much smaller atomic radius "squeezes" into a space in the 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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[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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The last three are illustrated, below:

1° - 5°

> 5°

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 latices This would be seen under the microscope as a small difference in extinction angles of the crystals

38.2° 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) e - lamellae in calcite [Ca-ions at the corners of the rhombs]

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 Dislocations

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LECTURE 16—DEFORMATION MECHANISMS IV: DISLOCATIONS

1 6 . 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:

b rug has now moved one full "unit" to the right 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. Crystals deform in the same way. It is much easier for the crystal to just break one bond at a time

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122

than to try and break all of them simultaneously.

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: â&#x20AC;˘ Tangent vector -- the vector parallel to the local orientation of the dislocation line â&#x20AC;˘ 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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"Cut-away" view of part of a dislocation loop extra half "plane"

dislocation line

t b

screw t segment crystalographic glide plane

edge segment

Edge dislocation :

b ⊥ t

Screw dislocation:

b // t

[the previous figure could have been of this face of the block]

Most dislocations are closed loops which have both edge and screw components locally.

1 6 . 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.

1 6 . 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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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. Dislocation lines

glide plane

impurity atom

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 â&#x20AC;&#x153;joggedâ&#x20AC;? 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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125

t1 t1 b1 b1 b2 t2 Jo g

b2 t2

Before the two dislocations run into each other

After they pass, a jog has been produced in dislocation 1

1 6 . 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 crystallographic 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. Hot working -- Permanent deformation with little or no strain hardening or with strain softening.

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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.

1 6 . 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 Flow Laws & Stress in Lithosphere

LECTURE 17—FLOW LAWS & STATE LITHOSPHERE

127

STRESS

I N THE

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.

1 7 . 1 Power Law Creep The basic equation which governs dislocation glide and climb is:

−Q  n . e˙ = Co (σ 1 − σ 3 ) exp  RT 

(17.1)

The variables are:

= strain rate [s-1]

= a constant [GPa-ns -1; experimentally determined]

σ1 - σ3 = the differential stress [GPa] n

= a power [experimentally determined]

= the activation energy [kJ/mol; experimentally determined]

= the universal gas constant = 8.3144 × 10-3 kJ/mol °K

= 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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n  e˙   σ1 − σ 3 =   −Q     Co exp  RT    Substituting in the actual values:

  −14 −1   10 s  σ1 − σ 3 =    −163 kJ mol −1  2.8 −3.1 −1  (10 GPa s ) exp 8.3144 × 10 -3 kJ mol -1 K -1 (273.16 + 300) K    

1 3.1

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: Max Shear Stress (Mpa) 200

400

600

800

1000

quartzite (wet) limestone

quartzite (dry) granite (dry)

Temperature (°C)

400

feldspar-bearing rocks diabase clinopyroxenite olivine (wet) olivine (dry)

800

gray show range for -15 ≤ log[strain rates] ≤ -13 Power Law Creep Curves -14 -1

[strain rate = 10 1200

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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.

1 7 . 2 Diffusion Creep This mechanism is a linear function of the differential stress and is more sensitive to grain size than temperature:

e˙ = Co (T )

D(σ 1 − σ 3 ) . dn

(17.2)

Again, the variables are:

= strain rate [s-1]

Co(T) = a temperature dependent constant [experimentally determined] σ1 - σ3 = the differential stress n

= a power [experimentally determined]

= the diffusion coefficient [experimentally determined]

= 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.

1 7 . 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. • differential stress is plotted against grain size for a constant temperature;

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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.

Dislocation glide

(MPa)

1000 Dislocation glide & climb

100

-13 -1

σ1 − σ 3

10 -14 -1

Lattice diffusion [Nabarro Herring Creep]

0.1 0.01

Grain boundary diffusion [Coble Creep]

-15 -1

800

1600

T (°C)

1 7 . 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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• 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: Differential Stress (MPa)

σ1 − σ3

Lithospheric Column 1000

5 power law creep for quartz-dominated lithologies

strong

-e

xis

weak

tin

Depth (km)

sli

CRUST

Maximum stress in crust

lts

30 35

Moho

for olivine

[the only possible stresses in the lithosphere are in the shaded fields]

strong

weak

MANTLE

p power law cree

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 we have used very specific rheologies to construct them, they should be called “frictional crystal-plastic

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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 -Basically, do not take the specific numbers too seriously.

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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:

• mining and quarrying

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• 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 joint

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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New York

100 km

Lake Erie folds

Ohio New Jersey

Pennsylvania

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:

ol oint

direction of propagation

n oi

plumose markings

ger j

youn

"Butting relation" (map view) twist hackles

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 later in the course, this occurs because the older joint acts like as free surface with no shear stress

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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:

pre-glacial land surface

present 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 True tensile joints, with no shear on their surfaces, occur only in the very shallow part of the

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Earth’s crust. The shape of the Mohr failure envelop gives us some insight into the maximum depth of formation of true joints:

σs

σ∗1 = | 3To |

σn

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:

Zmax =

3To ρg(1 − 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 material precipitated from a fluid. For many reasons, veins are extremely useful for studying local and

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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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Antitaxial Vein vein material a different composition than wall rocks

step 1

step 2

step 3

step 2

step 1

new material added at center

step 3

step 2

step 1

step 2

step 3

new material added at margins

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 â&#x20AC;&#x153;tension gashes.â&#x20AC;? 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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infinitesimal strain axes

ε3

ε1

45°

ε3

ε1

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 19—FAULTS I: BASIC 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 the crack. Unless it intersects the surface of the Earth or a branch line, the tip

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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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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 > 45째

Low-angle -- dip < 45째 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 > 45째

Thrust -- dip < 45째

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. Oblique Slip -- a combination of strike and dip slip

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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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Incohesive

Random Fabric

145

Foliated Fabric

Fault breccia (visable fragments > 30%) Fault gouge (visable fragments < 30%)

crush breccia (fragments > 0.5 cm)

0 - 10 %

crush micro-breccia (fragments < 0.1 cm)

Protocataclasite

Protomylonite

10 - 50 %

Cataclasite

Mylonite

50 - 90 %

Ultracataclasite

Ultramylonite

90 - 100 %

These rock types tend to form at different depths in the earth:

non-cohesive gouge & breccia

1 - 4 km

cohesive cataclasite series (non-foliated) 250-350Â°C cohesive mylonite series (foliated)

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10 - 15 km

Proportion of Matrix

Cohesive

fine crush breccia (fragments 0.1 - 0.5 cm)

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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 20â&#x20AC;&#x201D;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:

flat irons

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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 â&#x20AC;&#x201D; wet swampy areas along the fault trace â&#x20AC;&#x201D; are common (sag ponds can also be seen along normal and thrust fault traces).

off-set stream sag pond

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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• pre-rupture structures, and • post-rupture structures.

20.2.1 Pre-rupture Structures Riedel Shears :

90 -

φ 2

90 - φ

R (synthetic) R' (antithetic)

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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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:

P-shears

φ 2

R (synthetic) R' (antithetic)

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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, Iâ&#x20AC;&#x2122;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 youâ&#x20AC;&#x2122;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

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Orientations of Common Fault-Related Features Shear Fractures

Veins 45°

~10° φ/2 R

90° − φ/2

45°

R' R = synthetic Riedel shear R' = antithetic Riedel shear P = P-shear; φ = angle of internal friction

Same sense of shear applies to all following diagrams

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 si uncommon to find unambiguous indicators of movement on the R or R' surfaces and one commonly interprets them based on striation and angle alone In my experience, R shears can be misleading and one should take particular care in using them without redundant indicators or collaborative indicators of a different type.

"RO"-Type (top): The fault surface is totally composed of R and R' surfaces. There are no P surfaces or an average surface of the fault plane. Fault surface has a serrated profile. Not very common.

"RM"-Type (middle): The main fault surface is completely striated. R shears dip gently (5-15°) into the wall rock; R' shears are much less common. The tip at the intersection of R and the main fault plane commonly breaks off, leaving an unstriated step.

Lunate fractures (bottom): R shears commonly have concave curvature toward the fault plane, resulting in "half moon" shaped cavities or depressions in the fault surface.

diagrams modified after Petit (1987) [sense of shear is top (missing) block to the right in all the diagrams on this page]

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Striated P-Surfaces These features were first described by Petit (1987). The fault plane is only partially striated, and the striations only appear on the up-flow sides of asperities.

P "PT"-Type (top & middle): ~ planar, non-striated surfaces dip gently into the wall rock. Petit (1987) calls these "T" surfaces because of lack of evidence for shear, but they commonly form at angles more appropriate for R shears. Striated P surfaces face the direction in which that block moved. Steep steps developed locally at intersection between P and T. P surfaces may be relatively closely spaced (top) or much farther apart (middle).

"PO"-Type (bottom) : T surfaces are missing entirely. Striated P surfaces face in direction of movement of the block in which they occur. Lee side of asperities are unstriated.

diagrams modified after Petit (1987)

Unstriated Fractures ("T fractures") Although "T" refers to "tension" it is a mistake to consider these as tensile fractures. They commonly dip in the direction of movement of the upper (missing) block and may be filled with veins or unfilled. "Tensile Fractures" (top): If truely tensile in origin and formed during the faulting event, these should initiate at 45Â° to the fault plane and then rotate to higher angles with wall rock deformation. Many naturally occuring examples are found with angles between 30Â° and 90Â°. They are referred to as "comb fractures" by Hancock and Barka (1987). veins or empty fractures Crescent Marks (bottom) Commonly concave in the direction of movement of the upper (missing) block. They virtually always occur in sets and are usually oriented at a high angle to the fault surface. They are equivalent the "crescentic fractures" formed at the base of glaciers.

diagrams modified after Petit (1987) [sense of shear is top (missing) block to the right in all the diagrams on this page]

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"S-C" Fabrics Although commonly associated with ductile shear zones, features kinematically identical to S-C fabrics also occur in brittle fault zones. There are two types: (1) those that form in clayey gouge in clastic rocks and (2) those that form in carbonates. They have not been described extensively in the literature. This is somewhat odd because I have found them one of the most useful, reliable, and prevalent indicators. Clayey Gouge fabric (top ): Documented by Chester and Logan (1987) and mentioned by Petit (1987). Fabric in the gouge has a sigmoidal shape very similar to S-surfaces in type-1 mylonites. This implies that the maximum strain in the gouge and displacement in the shear zone is along the walls. Abberations along faults may commonly be related to local steps in the walls. gouge Carbonate fabric (top ): This feature is particularly common in limestones. A pressure solution cleavage is localized in the walls of a fault zone. Because maximum strain and displacement is in the center of the zone rather than the edges, the curvature has a different aspect than the clayey gouge case. The fault surface, itself, commonly has slip-parallel calcite fibers. pressure solution cleavage

Mineral Fibers & Tool Marks Mineral Fibers and Steps (top): When faulting occurs with fluids present along an undulatory fault surface or one with discrete steps, fiberous minerals grow from the lee side of the asperities where stress is lower and/or gaps open up. These are very common in carbonate rocks and less so in siliceous clastic rocks.

Tool Marks (bottom): This feature is most common in rocks which have clasts much harder that the matrix. During faulting, these clasts gouge the surface ("asperity ploughing" of Means [1987]), producinig trough shaped grooves. Although some attempt to interpret the grooves alone, to make a reliable interpretation, one must see the clast which produced the groove as well. Other- wise, it is impossible to tell if the deepest part of the groove is where the clast ended up or where it was plucked from.

[sense of shear is top (missing) block to the right in all the diagrams on this page]

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LECTURE 21—FAULTS III: DYNAMICS & KINEMATICS 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. op

vel

n ee

fail

σs

2θ σ3 *

σ1 *

σn

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

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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:

σ1

vertical,

σ2

vertical, or

σ3

vertical

This constraint means that there are very few possible fault geometries for near surface deformation. They are shown below:

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45 - φ/2 Thrust faults dip < 45°

σ1 σ3 45 + φ/2

σ3

Normal faults dip > 45°

σ1

σ2

σ3

Strike-slip

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

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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.

σ2

four possible fault sets in 3D strain

σ3

σ1

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

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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

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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 45Â° to the pole:

P-axis

b-axis

lt p

T-axis

fau

pla

lan

ent

vem

pole to fault plane

striae & slip sense (arrow shows movement of hanging wall)

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 Ď&#x192;1. The diagram, below, shows the P & T dihedra analysis for three faults:

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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.

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movement plane

σ3 O

B A pole to fault

σ1

A B

fault plane S

conjugate plane

σ3

possible positions of σ 3 given σ 1 as shown 21.4 Stress From Fault Populations1 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). 2. The data reflect a uniform stress field (both spatially and temporally)—this

This supplemental section was co-written John Gephart and Rick Allmendinger and is adapted from the 1989 Geological Society of America shortcourse on fault analysis.

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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:

X1' cos-1

X3'

striae

fault

X2'

[note -- for the convenience of drawing, both sets of axes are shown as left handed]

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cos -1 β 13

σ3

X1 ' X1

σ1

fault pole

striae

X2 X2'

ne pla

fau

X3 '

σ2

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 σ kl . In the above equation, b ik 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:

and

σ11′ = β11β11σ1 + β12β 12σ2 + β13β13σ3

[normal traction],

σ12′ = β11β21σ1 + β12β 22σ2 + β13β23σ3

[shear traction ⊥ striae],

σ13′ = β11β31σ1 + β12β 32σ2 + β13β33σ3

[shear traction // striae].

From assumption #1 above we require that σ12' vanishes, such that: 0 = β11β21σ1 + β12β 22σ2 + β13β23σ3 . Combining this expression with the condition of orthogonality of the fault pole and B axis:

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0 = β11β 21 + β12β22 + β13β 23 . yields

β13 β23 σ2 − σ1 ≡ R=− σ3 − σ1 β12 β22 .

(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 difference between the observed and predicted slip vector measured in the fault plane (referred to as a

An similar parameter was devised independently by Angelier and coworkers (Angelier et al., 1982; Angelier, 1984, 1989):

Φ=

σ2 − σ3 . σ1 − σ 3

In this case, if Φ = 0, then σ 2 = σ3, and if Φ = 1, then σ2 = σ1. Thus, Φ = 1 – R.

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“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.

conjugate plane

σ2

σ2 σ3

fault plane

15.3°

σ3 fault plane

4.8° calc. striae striae

σ1 minimum rotation

σ1 pole rotation

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

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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 approximations 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 Faults IV: Mechanics of Thrust Faults

LECTURE 22—FAULTS IV: MECHANICS

170

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.

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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 σ xx

σ zx

z X

σ zz

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: z

∫

σ xx dz = ∫ σ zx dx

(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 . The normal stress, σzz, is just equal to the lithostatic load: σzz = ρ g z .

(22.2)

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So, σzx = µ ρ g z . We can now solve the right hand side of equation 22.1 (p. 171): z

∫

σ xx dz = ∫ µρgz dx

and z

∫σ 0

dz = µρgzx .

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:

σ1 − σ 3  σ1 + σ 3 S  = + o  sin φ 2 tan φ   2

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σ1 - σ3 2

So σ3

σ1

σ1+ σ3 2

So tan φ

Solving for σ1 in terms of σ3 we get: σ1

= Co + K σ3 ,

where

Co = 2 So K

and

(22.3)

1 + sin φ . 1 − sin φ

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): z

∫σ 0

∫ (C z

dz = µρgzx

+ Kρgz ) dz = µρgzx

(22.4)

Lecture 2 2 Faults IV: Mechanics of Thrust Faults

Co z +

174

Kρgz 2 = µρgzx . 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:

Co Kz + . µρg 2 µ

(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.

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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:

λf =

Pf pore fluid pressure = lithostatic stress ρgz

Fluid Pressure in fault zone,

λ f

1.0

0.9

z = 5 km

f = λ 435 λ b = 0. = λ f λ b λ b 0.435 λ b=

0.8

0.7 z = 10 km 0.6

0.5

[after Suppe, 1985] 0

100 Maximum length, x

150

200

(km)

max

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

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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

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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.

topographic slop

e, α

e, β

llement slop

basal déco

In their wedge mechanics, they propose the following relation between α and β when the wedge is a critical taper:

α +β =

(1 − λ )µ + β (1 − λ )k + 1

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.

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LECTURE 23â&#x20AC;&#x201D;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 itâ&#x20AC;&#x2122;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

Syncline

oldest rocks in the center of the fold

youngest rocks in the center of the fold younger

older

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,

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for example:

old

Synformal anticline Folds that are concave upward, but the oldest beds are in the middle:

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: Symmetric folds:

limb enveloping surface

Asymmetric folds:

limb enveloping surface

Overturned folds: W

the tops of the more steeply dipping beds are facing or verging to the east in this picture

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.

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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:

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hinge zone

hinge point

amplitude b lim

inflection point

wavelength

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 â&#x20AC;&#x153;axial planeâ&#x20AC;? but this is a special case where all of the hinge lines lie in a single plane. hinge line -- the line connecting all the points of maximum curvature in a single layer

crest line -- the line which lies along the highest points in a folded layer

axial surface -- the surface containing all of the hinge lines of all of the layers

trough line -- the line which lies along the lowest points in a folded layer

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:

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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: 90

upright

Dip of the Axial Surface

inclined

ed ec

sub-vertical 90

lin

plunging R

Plunge of the Hinge Line

sub-horizontal

recumbent

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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°

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LECTURE 24 â&#x20AC;&#x201D; FOLDS II: GEOMETRY & KINEMATICS 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

Cuspate folds

Kink Bands

Box folds

Disharmonic folds

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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: Construction of dip isogons:

lines of constant dip dip isogon

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

Class 2: Inner and outer surfaces have the same curvature. Dip isogons are parallel to each other and to the axial surface. These are Similar folds

1C -- isogons on limbs make an acute angle with respect to the axial surface

Class 3: Inner surface is less curved than the outer surface. Dip isogons fan inwards

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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

Ă&#x; diagram

Fold axis

Ď&#x20AC; diagram

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

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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

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:

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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 â&#x20AC;&#x153;sheathâ&#x20AC;? because they are shaped like the sheath of a knife.

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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

â&#x20AC;˘ Cylindrical Parallel Concentric Kink Similar

â&#x20AC;˘ 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 F 1, below) and axial surface (the black plane, below) of the first set of folds with respect to the fold axis (labeled b2,

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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: fold axis of 1

= a2

axial plane of 1

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.

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LECTURE 25—FOLDS III: KINEMATICS 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:

1 rcruvature

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:

CGauss = Cmax Cmin . 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

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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 minimum curvature A non-cylindrical folded layer in which Gaussian curvature is not equal to zero after folding

line of maximum curvature

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 perpendicular before and after deformation, so no shear parallel to the folded layer

neutral surface

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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: veins, boudings, normal faults, etc.

thrust faults, stylolites, 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:

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• 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.

no shear in the hinge

opposite sense of shear on the limbs 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:

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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

if layer thickness is constant, then

γe

γi

γe

no shear in horizontal layers, only in dipping layers

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.

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25.4.2 Simple Shear during flexural slip tan ψ = 2 tan

For kink bands:

δ   2

[δ = dip of bedding]

average slip = s = 2 h tan

δ   2

π δ = 0.0175 δ 180° π average slip = s = hδ 180° tan ψ =

For curved hinges:

The following graph show the relationship between bedding dip and shear on the limbs for kink and curved hinge folds: ∞ at δ = 180° 3 kink folds 2 tan ψ

curved hinges

60°

120°

180°

dip, δ

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.

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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:

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volume constant, pure shear

volume reduction, no extension pure shear (e.g. pressure solution)

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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 dolomite arkose quartz sandstone greywacke limestone siltstone marl shale anhydrite, halite

Metamorphic Rocks meta-basalt granite qtz-fspar-mica gneiss quartzite marble mica schist

26.3 Theoretical Analyses of Folding In general, theoretical analyses of folding involve three assumptions:

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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:

where

for elastic deformation:

Ld = 2πt

for viscous deformation:

Ld = 2πt

E 6 Eo

η 6ηo

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:

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Ld = 2πt

η( S − 1) 6ηo (2 S 2 ) λ1 λ3

where λ is the quadratic elongation. Thus, the thicker the layer, the longer the wavelength of the fold:

For a single layer,

4≤

Ld ≤6, t

and for multilayers:

Ld ≈ 27 . t

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:

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ηο = 42 : 1 η ηο η

ηο = 5:1 η

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.

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LECTURE 27â&#x20AC;&#x201D;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, weâ&#x20AC;&#x2122;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.

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common minerals: hornblende sillimanite feldspar quartz biotite

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.

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common features: ooids pebbles reduction 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.

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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

"pinch & swell"

"fish mouths"

high viscosity contrast

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.

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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.

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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.

Cleavage

fining upward

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fining upward

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 steeply than bedding

dips more gently than bedding

inferred position and form of overturned anticline

cleavage dips more steeply than bedding

28.3 Cleavage Terminology

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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

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Crenulation

symmetric

asymmetric

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.

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28.4.1 Scale of Typical Cleavage Domains

anastomosing spaced

slaty crenulation

10 cm

1 cm

1 mm

0.1 mm limit of resolution of the optical microscope

0.01 mm

Lecture 2 9 Cleavage & Strain

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LECTURE 29â&#x20AC;&#x201D;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.

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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 θ xz′ = tan θ xz

Sz tan θ xz = Sx Rxz

In three dimensions it is a little more complex but still comprehensible:

(

tan δ ′ = tan δ Rxy2 sin 2 φ yz + Rxz2 sin 2 φ yz

)

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:

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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 Weâ&#x20AC;&#x2122;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:

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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

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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.

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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.

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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

C = cisaillement (shear)

S = schistosité

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

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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”. S-planes

(001) C-planes

(001)

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

σ - type

δ - type

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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 x

γ = tan ψ

d = γ x

θ'

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:

tan 2θ ′ =

2 γ

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

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under the curve (i.e. the integral shown):

d y

shear strain, γ y

d = ∫ γ dy 0

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:

θ'3

foliation

θ'2

θ'1

shear strain, γ

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

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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.

younger macroscopically, the bedding trends E-W, with the younger and older relations as indicated 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 mĂŠlanges. It is difficult to recognize where extreme deformation is involved. In general one should look for the following:

â&#x20AC;˘ look for the fold hinges

Lecture 3 0 Shear Zones, Transposition

â&#x20AC;˘ look for cleavage parallel to compositional layering â&#x20AC;˘ Walk the rocks out to a less deformed area.

224

Lecture 3 1 Thrust Systems:

225 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:

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Map view

window (fenster)

klippe crosssection

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:

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31.2.1 Andean Type:

Forearc

Back Arc (retroarc)

Craton

Hinterland Foreland

Accretionary Wedge

sub

Foreland Basin

duc

tion

zon

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:

Hinterland suture Foreland Foreland (peripheral) Basin

Tibet

Indian con

tinental cru

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

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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.

1000's km miogeocline

hinge

shelf

100 - 600 km

2 - 15 km

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

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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): Interior

Exterior oldest faults youngest fault

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

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type of sedimentary basin known as a foreland basin.

load

asymmetric foreland basin

load

asymmetric foreland basin forebulge

depression of basement under the load (exaggerated)

The Cretaceous deposits of western Wyoming and eastern Idaho are perhaps some of the best known foreland basin deposits.

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LECTURE 32—THRUST SYSTEMS II: BASIC 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

hanging wall anticline

hanging wall ramp hanging wall flat

amp

ll r twa

footwall flat

foo

footwall flat normal thickness

structurally thickened normal thickness

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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:

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Prior folding

older-over-younger Prior thrusting

younger-over-older

older-over-younger

Thrusting along an unconformity

younger-over-older

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:

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234 Basic Geometries

I. Fault Bend Folds

(also “hanging wall” or “ramp” anticlines)

II. Fault Propagation Folds (also “tip-line folds”)

mode I

tip line mode II

IV. Detachment Folds III. 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. older thrust fault folded by movement over the ramp in the younger thrust fault

younger thrust fault with primary ramp

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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.

trajectory of next fault

horse

roof thrust 4

direction of transport 2

floor thrust 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 â&#x20AC;&#x153;beheadedâ&#x20AC;? anticlines: trajectory of next fault

direction of transport

"beheaded anticlines"

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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

Formation of an antiformal stack by movement on a series of horses, each with displacement equivalent to the initial spacing between the ramps. The top section formed first.

antiformal stack:

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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:

? The problem with frontal synclines:

Extra space

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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 â&#x20AC;&#x153;normalâ&#x20AC;? 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.

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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 (“Laramide 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

th in th -sk ru in st n be ed lt

vo ar lcan c ic

magmatism:

Thick-skinned Province (deforms most of crust)

little or no asthenospheric wedge between the two plates

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 geometries. 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

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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

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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 uniformly 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.

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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

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”.

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243 Thick-skinned

"rabbit ear" structure out-of-the-syncline thrusts

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 â&#x20AC;&#x153;collapseâ&#x20AC;? 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:

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244 Thick-skinned

Map

Cross-section A'

33.6 Regional Mechanics In the Rocky Mountain foreland, basement surfaces define regional â&#x20AC;&#x153;foldsâ&#x20AC;? 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 Extensional Systems I

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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 σ 3.

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.

break away scarp

rotated surface normal faults thrust faults

commonly intensely brecciated internally

fault comes back to ground surface in down-dip direction

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 Extensional Systems I

structures”.

246

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. Yellowstone

region of the Heart Mountain fault approximate orientation of cross-section, below

Wyoming

bedding plane detachment

overrode former land surface O regional slope < 2°

Ordovician Bighorn dolomite

at Heart Mtn., there is an apparent thrust relation (Ordovician/Cretaceous)

110 km

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.

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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 â&#x20AC;&#x153;down-to-the-basinâ&#x20AC;? faults. You should be careful to distinguish them from rift-stage structures describe in detail in the next lecture. sedimentation growth faults detached within drift phase sediments

rift phase extensional basins

subsidence

In detail, an individual growth fault looks like:

synthetic faults

antithetic faults

syn-fault deposits much thicker in hanging wall

true listric fault geometry

roll-over anticline showing "reverse drag"

fault pa

rallels b

edding

(Jurassic salt in the Gulf Coast)

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.

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34.4 Tectonic Rift Provinces

34.4.1 Oceanic Spreading Centers The largest tectonic rift provinces in the world are represented by the earthâ&#x20AC;&#x2122;s linked oceanic spreading centers. These are sometimes inaccurately referred to as â&#x20AC;&#x153;Mid-ocean ridgesâ&#x20AC;? 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 subsidence. Slow spreading (e.g. Mid-Atlantic Ridge) isolated volcanoes rough topo due mostly to normal faulting

Intermediate spreading (e.g. Galapagos Rise) continuous volcanic axis smoother topography

Fast spreading (e.g. East Pacific Rise) axial high smooth topography

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.

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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 â&#x20AC;&#x153;intraâ&#x20AC;?). 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

fai

led

arm

formed during the breakup of the continents:

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

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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 Anderson’s theory of faulting. The resulting geometric model is known as the horst and graben model: graben

horst

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:

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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 sin (φ + θ) - 1 100 . 100 = w sin φ

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.

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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 Extensional Systems II

Dip of Bedding, θ

90°

253

listric fault

45°

rotated planar fault

100

200

% Extension

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

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35.5 Review of Structural Geometries The following table, after Wernicke and Burchfiel, summarizes the structural styles discussed above: Rotational

Non-rotational

Planar fault

Faults (domino style) & strata both rotated

High-angle & low-angle normal faults

Curved fault

Faults (listric-style) HW strata only rotated

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:

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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:

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35.7.4 Crustal-Penetrating Low-Angle Normal Fault:

35.7.5 Hybrid Model of Intracontinental Extension

volcanoes, high topography, postrift thermal subsidence strata

mechanic al (cold) rifting of the upper crust, syn-rift strata

ductile stretching of the lower crust crust mantle

Thermal lithospheric thinning

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257

LECTURE 36â&#x20AC;&#x201D;STRIKE -SLIP FAULT 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

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36.2 Transcurrent Faults and Tear Faults Large strike-slip faults within continents which are parts of plate boundaries are call intracontinental 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:

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259

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

â&#x20AC;˘ pre-rupture structures, and â&#x20AC;˘ post-rupture structures.

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260

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):

90 -

φ 2

90 - φ

R (synthetic) R' (antithetic)

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261

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:

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262

P-shears

φ 2

R (synthetic) R' (antithetic)

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

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

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263

• formation of “inverted flower structures”

In cross-section:

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:

right step in a rightlateral fault system

rhombochasm or pull-apart basin

"extensional (releasing) bend"

left step in a rightlateral fault system

"contractional (restraining) bend"

(e.g. Transverse Ranges in S. California)

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 Strike-slip Provinces

thrust faults in overlap region

264

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. Transcurrent 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 Vertical Motions: Isostasy

265

LECTURE 37—DEFORMATION

LITHOSPHERE

OF THE

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

ice

water

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 Vertical Motions: Isostasy

266

Pratt Model

Airy Model [identical topo]

2.67 2.59 2.52 2.57 2.62 2.76

2.75 2.75 2.75 2.75 2.75 2.75

3.3 gm/cm 3 3.3 gm/cm 3 compensation level

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:

∆( ρw hw ) + ∆( ρs hs ) + ∆( ρc hc ) + ∆( ρm hm ) = 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 = ∆hw + ∆hs + ∆hc + ∆hm

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 = ∆hc + ∆hm = 5 km

Lecture 3 7 Vertical Motions: Isostasy

and

267

∆( ρc hc ) + ∆( ρm hm ) = 2.75∆hc + 3.3∆hm = 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:

load

lithosphere

The Earth usually doesn’t work that way. More commonly, you see:

load

lithosphere

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

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268

is called flexural isostasy. The equations which, to a first order, describe flexure are:

x x  −x  z = zo 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:

Voα 3 zo = 8D 1

 4 4D α=   ( ρ m − ρ w )g  and

 Ehe3  D= 2  12(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 Vertical Motions: Isostasy

269

Himalayas flexural isostasy

Tibet local isostasy

55 km India 70 km

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 â&#x20AC;&#x153;Underthrustingâ&#x20AC;?:

37.2.3 Magmatic Intrusion:

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270

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: 1  2 kt ρa  2α (Tw − Ta )   , ∆E = π  ρa − ρw   

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 interactions, 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

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271

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 A activation energy...................................127 Alberta syncline.....................................237 albite twins .............................................120 allochthon...............................................225 Alpine fault ............................................258 Amontons........................................113-114 analytical methods ....................................3 Anderson’s theory..........................159-160 Anderson’s theory of faulting .............250 Andes ..............................................241, 270 angle of internal friction........ 107, See also fracture angular shear ...........................................40 annealing ................................................126 anticline...................................................178 antiform ..................................................178 Appalachians .........................................134 Appalachians .........................................225 Argentina........................................239, 241 asperities.................................................114 asthenosphere ........................................270 asymmetric half graben......... 250, See also extensional structures asymmetric porphyroclasts ..........220-221 Atacama fault.........................................258 Athy’s Law .........104, See also compaction Atlas Mountains ....................................239 augen.......................................................152 aulacogens..............................................249 autochthon..............................................225 axes............................................................12 B b-value ........................................................6 basalt .......................................................103 Basin and Range ............................250, 271 Basin and Range ........................................3 bedding...................................................207 bending moment ...................................268 Bighorn Basin.........................................243 bond attraction...................................100, 102 force...................................................101 length ................................................102 potential energy........................100-102 repulsion...................................100, 102 boudinage...............................................206

272

boudinage, chocolate tablet .................206 boudins ...........................................203, 205 Bowden...................................................114 Britain, coast of ..........................................5 brittle.........................................................98 “brittle-ductile transition” ...................131 Brunton compass.....................................15 buckling ...........................................192-193 Burgers vector........................................122 Busk method ..........................................186 C calcite.......................................................120 Canada....................................................239 Canadian Rockies..................226, 231, 237 Cape Ranges...........................................239 Cartesian coordinates .................10, 14, 17 cataclasis ...................................................98 Cauchy’s Law.....................................64, 68 CDP ......................................................23-24 Chamberlain, T. C. ................................225 characteristic equation............................73 China.......................................................239 cleavage ...................................203, 207-208 anastomosing...................................210 conjugate...........................................210 crenulation................................211, 216 climb..........................................................98 Coble creep.............................................119 COCORP.................................................241 COCORP...................................................25 coefficient of friction .............................112 coefficient of internal friction110, See also fracture coefficient of thermal expansion.........102 cohesion ..........................................110, 112 cold working..........................................125 Colorado .................................................239 Colorado Plateau...................................135 columnar joints ......................................103 compaction......................................103-104 confining pressure.....................95, 97, 107 continuum mechanics.............................36 coordinate transformation .....................71 cosine.........................................................17 cracks, modes I, II, III............................133 creep ..........................................................94 creep curve ...............................................95

Index Crenulation cleavage ............................216 cross product.......................................18-19 crust.........................................................131 crystal plastic ...................................98, 146 curvature ................................................191 cylindrical fold.........................................12 D Dahlstrom...............................................231 dĂŠcollement....................................177, 187 dĂŠcollement....................................225, 228 defects .....................................................117 impurities .........................................118 interstitial..........................................118 linear..................................................121 planar ................................................119 point ..................................................118 substitution ......................................118 vacancies...........................................118 deformation bands................................119 deformation lamellae............................119 deformation map...................................129 deformation paths...................................55 deformation, crystal plastic .127, 146, 152 Delaware Water Gap ....................116, 214 Denali fault.............................................258 denudation .............................................245 deviatoric stress.......................................65 diagenesis ................................104, 116-117 diffractions ..........................................27-28 diffusion.....................98, 118, 125-126, 129 diffusion diffusion coefficient ........................148 erosional ...........................................147 diffusion creep...............................127, 129 crystal lattice ....................119, 126, 129 grain boundary................116, 119, 126 Dilation ...............................................33, 41 dip and dip direction..............................15 dip isogon...............................................185 direction cosines..........................14, 16, 82 dislocation glide ..............................98, 123 dislocation jogs.....................................................124 climb..................................................125 edge ...................................................123 glide............................................125-126 glide and climb .........................125-127

273

pinning..............................................124 screw .................................................123 self stress field..................................123 strain hardening ..............................123 dislocations..............................119, 121-122 displacement gradient tensor...........71-72 displacement vector................................37 Distortion..................................................33 Dix equation.............................................25 dolomite..................................................120 dome and basin .....................................190 dominos ..................................................220 dot product...............................................18 drag folds................................................195 ductile .......................................................98 dummy suffix notation...........................69 duplex ...................235, See also thrust belt extensional........................................254 passive roof238, See also triangle zone dynamic analysis...................................161 E earthquake..................................................6 earthquake................................................95 east-north-up convention.......................11 eigenvalues...............................................73 eigenvectors .............................................73 Einstein summation convention ......69-70 elastic.........................................................93 elastic deformation................................100 electrical conductivity.............................68 ellipticity...................................................50 en echelon...............................................134 engineering mechanics.............................1 exfoliation...............................................136 Experimental..............................................3 extension...................................................39 domino model..................................250 down-to-the-basin...........................247 F fabric........................................................203 failed rifts................................................249 failure envelop........ 108, 110, 113, See also fracture fault bend folding232, See also thrust belt fault plane solutions..............................161 fault rock.................................................146

Index fault rocks...............................................144 fault scarp...............................................147 fault-line scarp .......................................147 fault-propagation folds.........................242 faults........................................................133 anastomosing...................................141 blind ..................................................149 branch line........................................142 conjugate sets...................................158 dip slip ..............................................143 emergent...........................................147 footwall.............................................141 hanging wall ....................................141 hinge fault ........................................144 left lateral (sinistral)........................143 listric..................................................141 listric..................................................160 normal...............................................143 oblique slip.......................................143 piercing points.................................142 planar ................................................141 reverse...............................................143 right lateral (dextral).......................143 rotational ..........................................144 scissors ..............................................144 sense-of-slip......................................151 separation .........................................142 slip vector .........................................142 surface trace .....................................142 tip line ...............................................141 wrench ..............................................143 fenster......................................................225 flat irons..................................................148 flexural rigidity......................................268 flexure .....................................................267 Flinn diagram...........................................59 fluid inclusions ......................................138 fluid pressure.104, See also pore pressure fluid pressure ratio................................175 fluids .........................................................96 fold axis...................................................186 folding â&#x20AC;&#x153;competenceâ&#x20AC;?...................................199 buckling ............................................193 dominant wavelength.....................200 elastic.................................................200 flexural flow.....................................194 flexural slip.......................................194

274

multilayers........................................199 neutral surface .................................193 nucleation .........................................200 passive flow......................................197 theoretical analysis..........................199 viscous ..............................................200 folds anticlinoria........................................180 asymmetric.......................................179 axial surface...............................181-182 axial trace..........................................182 axis.....................................................192 class 1 ................................................185 concentric..........................................186 conical ...............................................188 cylindrical.........................................186 dip isogon.........................................185 drag ...................................................195 enveloping surface..........................179 facing.................................................179 hinge..................................................180 interlimb angle.................................183 isoclinal.............................................183 kink............................................187, 195 non-cylindrical.................................188 parallel ..............................................186 parasitic.............................................195 plunging ...........................................182 reclined .............................................182 recumbent.........................................182 sheath ................................................188 similar ...............................................187 superposed.......................................189 symmetric.........................................179 synclinoria........................................180 vergence............................................179 foliation...........................................203, 207 folium......................................................207 footwall...................................................232 force.............................................................2 force...........................................................61 foreland...........................................227, 235 foreland basin ........................................230 broken ...............................................244 foreland basins.......................................229 foreland thrust belts....226, See also thrust belt FORTRAN................................................70

Index

275

fractal dimension.......................................5 fractals......... 4, 6, See also scale invariance fracture............................................106, 126 brittle.................................................106 Coulomb ....................................110-111 ductile ...............................................106 ductile failure...................................111 tensile ................................................108 transitional tensile...........................109 fractures, pre-existing...........................112 Fresnel Zone.............................................27 Fresnel zone .............................................27 friction..............................................113-114

hydrostatic pressure ...............................65

G Garlock fault ..........................................258 Gaussian curvature ...............................191 geometry..................................................2-3 geothermal gradient..............................129 Gibbs notation..........................................12 gneissic layering ....................................207 graded beds............................................208 grain boundaries ...................................119 grain size.........................................110, 129 Granite Mountains ................................243 gravity slides..........................................245 Griffith Cracks .......................................109 Griffith cracks ........................................110 growth faulting.. 247, See also extensional structures Gulf Coast...............................................247

J Jeffery Model..........................................214 jelly sandwich ........................................131 joint sets ..................................................134 joint systems...........................................134 joints........................................................133 butting relation ................................135 cooling...............................................136 cross joints ........................................134 sheet structure..................................136 systematic joints ..............................134 twist hackles.....................................135 joints........................................................203 joint zone ..........................................134 Jura Mountains ......................................226

H hand sample...............................................4 hanging wall ..........................................232 Heart Mountain detachment ...............176 Heart Mountain fault............................246 Herring Nabarro creep .........................119 Himalaya ................................................270 hinterland .......................................227, 235 homocline .......................................180, 183 horse......................235, See also thrust belt horsetail structure .................................264 horst and graben....................................250 hot working............................................125 Hubbert & Rubey ..................................170 hydraulic fracturing..............................112 hydrostatic pressure .............................175

I Iberian-Catalรกn Ranges ........................239 Idaho .......................................................230 indicial notation.......................................13 interval velocity.......................................25 isostasy....................................................265 Airy.............................................265-266 differential........................................266 flexural...............................229, 267-269 local ...................................................269 Pratt...................................................265

K kinematic analysis.................................161 kinematic analysis...................................33 kinematics...................................................2 klippe ......................................................225 L laboratory .................................................97 landslides................................................245 Laramide Province................................239 latitude......................................................10 left-handed coordinates 11, See also righthanded coordinates linear algebra ...........................................67 lineation ..................................................203 lines ...........................................................14 listric normal faults ...............................252 lithosphere..............................................130

Index lithosphere......................................229, 270 lithostatic pressure ..................................65 longitude...................................................10 low angle normal faults........................253 M Mackenzie Mountains ..........................239 magnitude ..................................................2 magnitude, earthquake ............................6 Mandelbrot, B. ...........................................5 mantle .....................................................131 March model..........................................214 Martinsburg Formation................214, 216 material properties....................................1 mean stress...............................................65 mechanics ................................................2-3 mélange...................................................246 mélanges.................................................223 metamorphic core complexes..............253 metamorphic foliation ..........................113 metamorphism ......................................117 mica “fish” .............................................219 mica fish..................................................152 microlithon.....................................215, 217 microlithons ...........................................211 mid-ocean ridges...................................248 migration .............................................25-26 mineral fibers .................................152, 203 mineral lineations..................................152 minor structures ....................................203 miogeocline ............................................228 Modulus of Rigidity................................89 Mohr’s Circle............................................82 Mohr’s Circle, 3-D ...................................86 Mohr’s Circle, finite strain .....................47 Mohr’s circle, for stress.........................106 Mohr’s Circle, for stress................107, 111 Mohr’s Circle, stress......78-79, 85, See also stress moment, earthquake.................................6 monocline ...............................................180 Morocco ..................................................239 Mother Lode...........................................138 movement plane....................................161 mullions..................................................203 multiple.....................................................30 multiples...................................................29 mylonite..................................................146

276

N neutral surface .......................................193 New England .........................................136 Newton .....................................................61 Newtonian fluid ......................................93 non-penetrative .........................................4 North Anatolian fault ...........................258 north-east-down convention ...........11, 14 numerical methods ...................................3 O oceanic spreading centers ....................248 Oklahoma ...............................................239 olistostrome............................................246 olivine .....................................................130 optical microscope ....................................4 orientation ..................................................2 orientations...............................................14 orthogonality relations ...........................82 P P and T axes ...........................................161 P-shears...................................................151 P-shears...................................................261 P-waves.....................................................22 paleocurrent indicators ..........................12 paleomagnetic poles ...............................12 parallelogram law ...................................18 parasitic folds.........................................195 parautochthon........................................225 particle path .............................................37 particle paths............................................34 passive margin...............................228, 247 pencil structure......................................206 penetrative..................................................4 permeability ...........................................112 piercing points.......................................142 pitch...........................................................15 plagioclase..............................................120 plane strain...............................................58 plane trigonometry ...................................8 planes ........................................................14 plastic, perfect..........................................94 plate convergence rates ........................229 point source..............................................27 Poissons Ratio..........................................89 poles ....................................................12, 14

Index pore fluid ..................................................97 pore fluid pressure................................175 pore pressure ..........................105, 111-112 pore space.................................................96 porosity....................................103-104, 112 power law creep ....................................127 pressure solution....................115-117, 126 pressure solution....................213, 215-217 grain size...........................................117 impurities .........................................117 temperature......................................117 principal stresses .....................................63 pure shear.................................................54 Q quadrangle map ........................................4 quadratic elongation..... 39, See also strain R R-shears ..................................................151 R-shears ..................................................261 rake............................................................15 reflection coefficient................................21 reflectivity.................................................23 rheology....................................................87 Rhine Graben .........................................250 Rich, J. L..................................................231 Riedel Shears..........................150, 152, 260 rift provinces..........................................249 right-hand rule..............................15, 19-20 Right-handed coordinates11, See also lefthanded coordinates rigid body deformation..........................33 rock bursts..............................................136 Rocky Mountain Foreland ...................239 Rocky Mountain foreland ....................243 Rodgers, J................................................225 rods..........................................................203 rotation........................................................1 Rotation.....................................................33 rotation......................................................34 left-handed .........................................34 right-handed ......................................34 rupture stress ...........................................96 S S-C fabrics...............................................152 S-C fabrics...............................203, 207, 219

277

S-surfaces................................................207 S-waves.....................................................22 sag ponds................................................148 San Andreas fault..................................258 sandstone dikes .....................................214 scalar ..............................................17-18, 67 scalar product ..... 18, See also dot product scale .......................................................3, 36 scale invariance..........4-5, See also fractals scale global.....................................................3 macroscopic..........................................3 map.....................................................3-4 mesoscopic ...........................................4 microscopic ..........................................4 provincial..............................................3 regional .................................................3 submicroscopic ....................................4 schistosity ...............................................207 secular equation.......................................73 sedimentary basin .................................104 seismic reflection artifacts................................................29 fold.......................................................24 shear strain ...............................................40 shear stress .............................................158 shear zone...............................................140 displacement....................................221 sense-of-shear ..................................219 sheath folds ....................................152, 188 Sheep Mountain ....................................243 Sierras Pampeanas ........................239, 241 sign conventions engineering.........................................61 geology................................................61 simple shear ...........................................139 simple shear .............................................54 sine.............................................................17 slickenlines .............................................152 slickensides ............................................152 slip system..............................................123 slip vector ...............................................142 soil mechanics........................................105 Spain........................................................239 spalling....................................................136 spherical coordinates..................10, 14, 17 stacking.....................................................24 stereographic projection.........................12

Index strain.........................................................1-2 strain........................................................100 strain..........................................................68 strain ellipse .............................................42 strain ellipsoid .......................................217 strain hardening ....................................123 strain rate....................................................3 strain rate................. 92-93, 96-97, 127, 129 strain softening ......................................125 strain angles .............................................39-40 coaxial .................................................54 continuous..........................................34 discontinuous.....................................35 finite ....................................................53 heterogeneous....................................35 homogeneous.....................................35 infinitesimal .......................................53 lines .....................................................39 lines of no finite elongation .............51 maximum angular shear ..................49 non-coaxial .........................................54 non-commutability............................58 non-rotational ....................................54 principal axes.....................................48 pure shear...........................................54 rotational ............................................54 simple shear .......................................54 superposition .....................................58 volume ................................................39 volumetric ........................................103 stress.........................................................1-2 stress................................100, 107, 110, 114 stress...............................................61, 67-68 stress field.................................................86 stress tensor.........................................63-64 stress trajectory........................................87 stress vector..............................................61 stress axial .....................................................65 biaxial..................................................65 deviatoric............................................64 differential..........................................95 effective.............................................105 isotropic ............................................106 mean...............................................64-65 normal.................................................77 principal..............................................63

278

principal plane...................................86 shear ....................................................77 spherical..............................................66 triaxial .................................................65 uniaxial ...............................................65 units.....................................................61 stretch........................................................39 striae........................................................152 strike and dip...........................................14 strike-slip, convergent ..........................259 strike-slip, convergent type .................262 strike-slip, divergent.............................259 strike-slip, divergent type....................262 strike-slip, en echelon folds .................260 strike-slip, parallel.................................259 strike-slip, transpression ......................262 structural domains ..................................36 stylolites..................................................115 stylolites..................................................203 sub-grain walls ......................................126 Sub-Himalayan Belt..............................226 Subandean belt ......................................226 subduction, flat......................................239 subgrain boundaries .............................119 syncline ...................................................178 synform...................................................178 T tangent vector ........................................122 tear fault....... 258, See also strike-slip fault temperature............................................117 temperature......................96, 102, 127, 129 tension gashes........................................139 tensor transformation .......................70, 81 tensor antisymmetric ....................................72 asymmetric.........................................72 infinitesimal strain ............................72 invariants............................................73 principal axes...............................73, 83 symmetric...........................................72 tensors.......................................................68 Terzaghi..................................................105 Texas........................................................170 Texas........................................................239 Theoretical..................................................3 thermal conductivity...............................68 thermal diffusivity ................................270

Index thermal expansion.................................270 thermal subsidence ...............................248 thermodynamics........................................3 thrust belt Andean-type ....................................227 antiformal stack ...............................236 antithetic ...................................227, 229 basic characteristics.........................228 Dahlstrom’s rules .....................231-232 duration ............................................229 folded thrusts...................................234 Himalayan-type...............................227 imbrication .......................................237 ramp and flat geometry..................231 rates ...................................................229 synthetic....................................227, 229 timing................................................229 triangle zones...................................237 types of folds in ...............................233 thrust faults ............................................170 gravity gliding .................................176 gravity sliding..................................176 paradox of ........................................170 wedge shape ....................................176 thrusts out-of-the-syncline ..........................242 thick-skinned ...........................225, 239 thin-skinned .....................................225 Tibetan Plateau ...............................266-267 Tien Shan................................................239 tool marks...............................................152 topography, pre-glacial ........................136 traction vector..........................................61 traction vectors ........................................62 transcurrent fault...................................258 transform, intracontinental .. 258, See also strike-slip fault transformation matrix........................82-83 transformation of axes............................81 translation.............................................1, 33 transposition ...................................222-223 trend and plunge.....................................15 triangle zone...........................................238 triple-junction ........................................264 twin glide................................................120 twin lamellae...................................119-120 U

279

unit vector ...............................13-14, 16, 19 universal gas constant ..........................127 V vacancies.............118, 125, See also defects Valley & Ridge Province ......................226 vector.......................................13, 14, 17, 67 vector product.. 19, See also cross product addition...............................................18 cross product.................................18-19 dot product.........................................18 magnitude ..........................................13 scalar multiplication .........................17 subtraction..........................................18 veins ........................................115, 133, 137 antitaxial ...........................................138 sigmoidal ..........................................139 syntaxial............................................138 "tension" gashes...............................139 veins ........................................................203 velocity......................................................26 pullup/pushdown ............................29 rock......................................................22 vergence..................................................179 viscosity ..................................................200 viscosity ....................................................93 viscous, perfect ........................................94 void ratio ................................................104 Von Mises...............................................111 W wavelength...............................................26 wedge taper............................................177 wedge, critical taper..............................177 Wichita-Arbuckle Mountains..............239 Wind River Mountains.........................241 window...................................................225 wrench fault ...........................................259 Wyoming.........230, 239, 241, 243-244, 246 Y yield stress...........................................96-97 Young’s modulus ..................................200 Young’s Modulus....................................89