000 0

Exercise (ii)

0

20->009

6->092

50->255

0

Bedding145/60

Problem 1. The plane containing the elongation direction and the hinge must be the axial plane. Note that this is a strange orientation for the hinge line (more often than not, fold hinges are par­allel to the stretching lineation.

0

F1

F2

Trueattitude

of B 151/31

Problem 2

0

15->249

0

Drill: 60 -> 134

Acute bisector

P1 P2

P3

P4 P5

0 0

0

Bedding

0

Slicks

In order for there to be no apparent offset, the intersection of the fault and the bed must be parallel to the slip dir­ection, as in ferred from the orientation of the slicks.

Fold hinge: 34 -> 350

The orientation of two beds does not provide enough information to determine whether the fold in question is an anticline or a syncline. Consider the cartoon cross-sections below. Plotting the limbs on a stereonet amounts only to plotting the orientation of the dashed lines.

Original bedding 140/46

( )

( )

(

( )Original bedding 140/46

( )

( )

Pole to unconformity (unrotated)

Pole to unconformity rotated

Pole to bedding unrotated

Pole to bedding rotated)

Pole to unconformity unrotated

Pole to unconformity rotated

Pole to bedding (unrotated)

Pole to bedding rotated

Lab 3: Stereonets

Fall 2005

1 Introduction

In structural geology it is important to determine the orientations of planes and lines and their intersections. Working out these relationships as we have in Cartesian x-y-z coordinates, however, is a cumbersome and tedious task. The easiest way to handle orientation problems of lines and planes is through the use of stereographic projections. The use of stereographic projection � or stereonets � is the bread and butter of structural analysis. They are used to work out many tricky three dimensional relationships; they are used to plot and represent all kinds of geometric data that you collect in the �eld; they are used in the analysis of that data. From now, until the end of the semestre, hardly a lab will go by that won't use these. The purpose of this lab is to make you all masters of the stereographic projection. We will develop these techniques using paper, pencil and a stereonet, but will introduce software programmes that plot data stereographically.

In stereographic projection, planes and lines are drawn as they would appear if they intersected the bottom of a transparent sphere viewed from above1 . To do this on a �at sheet of paper we use a two dimensional projection of the sphere called a stereonet. The stereonet shows the projection of a set of great circles and a set of small circles that are perpendicular to one another (just like longitude and latitude lines, respectively, on the globe). These form a grid that we can use to locate the position of variously oriented planes and lines.

Great circle: A circle on the surface of a sphere made by the intersection with the sphere of a plane that passes through the center of the sphere. The longitude lines on a globe are great circles.

Small circle: A circle on the surface of a sphere made by the intersections of a plane that does not pass through the center of the sphere. The latitude lines on a globe are small circles. Note also that the latitude and longitude lines on a globe are perpendicular to each other.

A stereonet should be visualized as the bottom half of a sphere. Planes intersect the sphere as great circles and lines intersect the sphere as points. Its helpful when starting out with stereonets to visualize the plane or line as it cuts through a 3-dimensional bowl (props may be helpful).

1.1 Basic techniques

Plotting a plane: Example: plot a plane with attitude 060/20. 1. On tracing paper mark a north arrow through the north pole of the net. 2. To locate the line of strike, count 60° east of north on the outer circle. Mark this point on the outside circle of the net, and on the opposite side (180° away). 3. Rotate the tracing paper until the strike line intersects the north pole of the net. This positions the tracing paper so that dip may be plotted using the great circle grid as a reference. 4. To plot dip, count o� 20° inward from the right hand side of the outer circle along the EW diameter of the net (always the right hand side if using the right hand rule, otherwise decide which direction to count in from based on the direction of dip). Trace, from pole to pole, the great circle arc that intersects this point. 5. Rotate back to the starting position and check that your plotted plane makes sense. Visualize!

Plotting a line: Example: plot a line with attitude 40/0252 . 1. On tracing paper mark the north arrow. 2. Locate the direction of bearing by counting o� 25° west of north on the outer circle. Mark this point. 3. Rotate the bearing mark to coincide with the nearest great circle diameter of the net (the N, S, E or W poles) and count inward 40° from the outer circle. 4. Rotate back to check if your plotted line makes sense.

1 Note that using the bottom hemisphere of this sphere is just a convention. Mineralogists use the top hemisphere, but they are a weird lot, to be sure

2 Note the convention we use: trend and plunge data are plotted as plunge and trend. The azimuthal measurement � strike or trend � is always written down as three digits, and the inclination measurement is always two measurements. This prevents a great deal of confusion

1

Figure 1:

Pole to a plane: Planes are awkward to deal with, but any plane can be represented more simply as a line that intersects it at a right angle. Example: plot the pole to a plane with attitude N74E, 80N 1. On tracing paper, mark the north arrow. 2. Mark the strike N74E on the stereonet and rotate it to north as if plotting the plane. 3. Count 80 in from the edge as you would for �nding the dip of the plane. Now count an additional 90°. Alternatively, count 80 from the center of the stereonet rather than the outer edge. Mark this point, its is the pole to the plane. 4. Check to make sure your pole makes sense.

Line of intersection of two planes: 1. Draw the great circle for each plane. 2. Rotate the tracing paper so that the point of intersection lies on the N-S or E-W line of the net. Mark the outer circle at the closest end of the N-S or E-W line. 3. Before rotating the paper back, count the number of degrees on the N-S or E-W line from the outer edge to the point of intersection. This is the plunge. 4. Rotate back. Find the bearing of the mark made on the outer edge of the circle. This is the trend.

Angles within planes: Angles within planes are measured along the great circle of the plane. The most common need is to plot the pitch or rake of a line within a plane. Example: a fault surface of N52W/20NE contains a slickenside lineation with a pitch of 43° to the east (Figure 1a). Figure 1b shows the lineation plotted on the stereonet.

True dip from strike and apparent dip: 1. Draw a line representing the strike line of the plane. This will be a straight line across the center of the stereonet intersecting the outer circle at the strike bearing. 2. Plot the apparent dip as a pole. 3. We now have two points on the outer circle (the two ends of the strike line) and one point within (the apparent dip point), all three of which must lie on the same plane. Turn the strike line to lie on the NS line of the net and draw the great circle that passes through these points. 4. Measure true dip of plane along EW line of net.

Strike and dip from two apparent dips: 1. Plot both points representing the apparent dips lines. 2. Rotate the tracing paper until both points lie on the same great circle. This plane is the true strike and dip of the bed.

Vertical axis rotations: 1. Rotations about a vertical axis a�ect only the strike of the plane, while dip remains unchanged. Rotations are measured along the outer circle.

Example: What is the new attitude of a plane oriented N60W/45NE after a rotation of 30° clockwise about R (vertical axis)?

Answer: N30W/45NE Rotations about a horizontal axis parallel to the strike Rotation about the strike line a�ecst only the dip of

the plane while the strike remains unchanged. In this case the overlay is rotated such that the strike of the plane (which is the rotation axis) coincides with the NS line, and the rotation is measured along the great circle grid.

Example: What is the attitude of a plane oriented N20E/80SE after a rotation of 50° counterclockwise about R? Answer: N20E/30SE

Note that during rotation all points on the original great circle projection of the plane move along small circles to points on the rotated arc of the plane. In this way, arc lengths of the initial projection of the plane are preserved during rotation. The rake of a linear element in the plane is thus constant regardless of the orientation of the plane.

General rotations Rotations about any other axis (the usual case in geology) are trickier and are most easily done with poles to the planes rather than the planes themselves. We simplify the problem by �rst rotating the rotation axis to be horizontal, performing the required rotation, and �nally returning the rotation axis to its original orientation.

Example: Find the new attitude of a plane oriented N30E/30SE after a rotation of 60° counterclockwise about a rotation axis (R) EW/30E 1. Plot R, plane and pole to plane (Figure 4a) 2. Rotate R to horizontal position (R'). To maintain constant angular relationship between P and R, P must be rotated by the same amount along small circle. (Figure 4b) 3. Position R' parallel to the NS axis of the net and perform 60° counterclockwise rotation by moving P' to P� (60° measured on small circle). (Figure 4c) 4. Restore R' to true orientation. Simultaneous rotation of P� gives true attitude of pole to rotated plane (P� '). Plane itself is then reconstructed from the pole. (Figure 4d)

Cones: Because a drill core rotates as it is extracted, the orientation of a bedding plane cannot be determined, but a range of possible orientations can be de�ned. Lines perpendicular to the sides of the cone, representing poles to bedding, pass through the center of the sphere and intersect the lower hemisphere as two half-circles or one circle. 1. Plot the borehole. 2. Rotate the tracing paper so that the borehole lies along a great circle (I usually start with the straight line through the center and work out along each great circle in 10° increments). Count out

2

the number of degrees between the angle of the borehole and the angle of the bedding in both directions along the great circle line and make a mark. Do the same for each great circle line that you can rotate the borehole point to. 3. These marks should de�ne a circle or two curved lines. You get a circle when the entire cone intersects the lower hemisphere (usually for a small angle between the borehole and the bedding or for a steeply dipping borehole). Two lines result when portions of the lower and upper cone both intersect the lower hemisphere of the stereonet (for a large angle between the borehole and bedding or a shallowly dipping borehole). A horizontal borehole always results in two symmetric lines.

More on Rotations When �nding the original orientation of features that have been tilted, rotate the tilted plane back to horizontal using its strike line as the rotation axis. Remember to rotate everything else on the plot by the same number of degrees along the small circles perpendicular to the rotation axis.

-To rotate the limbs of plunging folds back to horizontal, �rst rotate the fold axis back to horizontal (rotate this about a horizontal axis perpendicular to the trend of the fold axis). Now rotate the fold limbs back to horizontal using the (now horizontal) fold axis as the rotation axis.

-Always keep in mind that the true tilting could be much more complex that what we assume in our simple stereonet manipulations.

-Keep the following in mind when rotating objects in a stereonet. Planes, lines and cones all pass through the origin and have a corresponding upper portion that we do not see. They sometimes, however, come into play when we do rotations or plot cones. For example, imagine a line trending N and plunging 45 degrees. It would plot as a point halfway between the edge and the center of the circle on the north south line. If we rotate the line around a horizontal, north striking axis, the pole traces a path towards the edge of the stereonet along a small circle line. When rotated 90° the pole plots directly on the edge of the stereonet. The other end of the line also plots as a pole 180° around the edge of the stereonet, and these two points are essentially the same thing and de�ne the same line. Continuing the rotation, the pole in the northern half of the stereonet disappears (intersects the upper rather than lower hemisphere) and the pole now follows a small circle line in the southern half of the stereonet.

2 Resources

Stereographic plotting software is easily found on the Internets. Check out Rick Allmendinger's Stereonet (Mac and Windows) programme at:

http://www.geo.cornell.edu/geology/faculty/RWA/maintext.html

Rod Holcombe's GEOrient is promising, as well (Windows):

http://www.holcombe.net.au/software/index.html

3 Exercises

For each problem you should turn in a separate piece of tracing paper on which you should label everything that you plot. Make sure you draw the primitive (the outline of the stereonet), and label the cardinal directions. Make sure you indicate your answer clearly and include any comments you feel are necessary for someone to �gure out what you've done.

3

3.1 Basic operations

1. Plot and label the following planes as both traces (great circles) and poles (points). For quadrant or dip/dipdirection convention, convert to azimuth and right hand rule �rst.

(a) N20W/40W

(b) 065/90

(c) N5E/10E

(d) Dip 70 towards 030

(e) Horizontal plane

2. Plot and label the following lines (trend and plunge):

(a) 20/S45E

(b) 00/322

(c) 60/S85W

(d) Vertical line

3. Plot the following planes and the lines within them. Figure out the trend and plunge of each line:

(a) Plane strikes N30E, dips 45W; line pitches/rakes 30N (30° down from N end of plane)

(b) Plane is 075/20; line pitches/rakes 18◦from E

(c) Plane strikes N15W, dips 50W; line pitches/rakes 90W

4. Rotate the following lines 30◦counterclockwise around a pole plunging 15◦, trending 334. Figure out trend and plunge of each line after rotation:

(a) 42 towards 312

(b) Plunge 23, trend N20E

(c) 42→210

5. Using either pencil and paper, or a downloaded stereonet programme, plot the measurements taken at Beaver-tail point. First, you will have to compile all the measurements from everyone. Carefully distinguish between bedding, cleavage and fold axes. Are there any relationships between the di�erent kinds of measurements? I hope so.

4

3.2 Problem solving

1. a. Bedding in a sedimentary rock strikes N35W, dips 60SW. Plot this surface as a trace and a pole. b. plot the direction of grain elongation pitching 32NW in the bedding of (a). c. The hinge of a small fold plunges 46° in the direction of N68E; plot this hinge line. d. Determine the attitude (strike and dip) of the plane containing the hinge and the grain elongation direction. Why is this plane interesting?

2. On a vertical exposure striking N75W (F1) the apparent dip of bedding (B) is 23W.On a second vertical exposure (F2) striking due north the apparent dip of B is 16S. What is the true strike and dip of B?

3. A vein (V) striking N18W and dipping 16SW intersects a fault (F) striking N70E and dipping 87S. What is the trend and plunge of the line of intersection of F and V.

4. The attitude of a bore-hole is trend S22E, plunge 40. On a core sample, the angle between the core axis and the pole of bedding is 60°. Show all possible positions of the pole of bedding on the stereonet.

5. Two veins (V1 and V2) are found in a region where ore bodies lie along the intersection of the two vein systems. Past experience shows that the most information is obtained if a drill hole cuts the line of intersection of the two veins at 90° and lies in the plane bisecting the veins acutely. What should the trend and plunge of the drill hole be if V1 strikes N62W and dips 64NE, and V2 strikes N34W and dips 70SW.

6. In a series of uniformly dipping sediments in which the bedding is generally obscure, a fault striking N15W and dipping 60SW produces no o�set where it cuts a layer of conglomerate. Direction of movement on the fault is given by slickensides (elongated mineral �bers on the fault plane) trending N87W and plunging 59. From an aerial photograph, the general strike of the bedding is determined to be N45W. What is the dip of the bedding?

7. On one planar limb of a plunging fold the strike of bedding is N25E and the dip is 50NW; on the other limb the strike is N24W and the dip is 70NE. What is the trend and plunge of the fold axis? Is it an anticline or a syncline? Explain your reasoning.

8. The following data are from 3 non-parallel drill holes: Hole A: trend N42E, plunge 65; angle between core axis and bedding pole is 45° Hole B: trend S60E, plunge 44; angle between core axis and bedding pole is 70° Hole C: trend S5W, plunge 59; angle between core axis and bedding pole is 40° Assuming that the bedding is uniform, �nd the attitude of the bedding.

9. Below an unconformity striking N37E and dipping 32NW a series of planar beds strikes N18W and dips 60W. What was the strike and dip of these beds at the time the unconformity was formed? What assumption do you have to make to arrive at an answer?

10. On one side of a fault, upright beds strike N43E and dip 75NW. On the other side of the fault, the beds are again upright and strike N74E and dip 54NW. From an air photo, the fault is seen to strike N20E. Assuming rotational movement on the fault, �nd: a. the dip of the fault b. the amount of rotational movement on the fault in terms of an angle between lines on opposite sides of the fault that were parallel before rotation. (Note that these lines must be perpendicular to the axis of rotation.)

5

Lab 4: Cross­sections

Solutions

Fall 2005

1 Two beds

A

B

2 Groovy section

Note that the location of the cross section line may not be coincident with where you placed it. Note the consis­

tent dips and thicknesses of the beds, the use of kink­band geometries and the deflection of the anticline trace

by the topography.

1

3 Map from section

Here is the map. Believe!

2

4 Bad section I

The cross­section in question is not a balanced section because it fails to meet the criterion of admissibility.

That is, it fails to satisfy the basic requirement of geological plausibility. In particular: at the point B, the section

shows an unlabeled fault switching its sense of slip from thrust motion to normal motion, with no other splays or

transfer structures that might account for this. Quite apart from it being unlikely that normal faults are present

in this area, this is geometrically impossible. At point C, there is a contact that is clearly a fault contact, since

there is considerable stratigraphic separation and structural discordance across the contact, but it is not labeled

as such. Even if this were fixed, the Lazeart syncline itself presents an apparent problem. The syncline has to

form somewhere off the section and then be transported into its present location (because the bottom of the

syncline is faulted off ). This is not impossible, but it is unclear how this is accomplished.

The Absaroka thrust presents some problems: the bottom splay appears fine, but should be straightened out.

The top splay has no stratigraphic separation across it, and appears to have been drawn only to somehow ac­

count for the overturned (?) section of Triassic rocks in the hangingwall of the lower splay. There is also a cryptic

contact at point D, that apparently "separates" Triassic rocks from Triassic rocks. This seems unparsimonious,

at the very least. The Commissary fault, too, is problematic since stratigraphic separation is inconsistent along

the trace of the fault.

Finally, minor infelicities mar the section: an unlabeled region at A is sloppy (or reflects a major lack of imag­

ination) and the contact between Cretaceous and Jurassic rocks at point E, where the contact is conformable in

one place, but cuts – as a fault – down (!!) section.

The cross­section probably fails the criterion of retrodeformability, but given the problems outlined above,

it hardly seems worth the bother to even try.

AB

C

DE

F

<z

3

5 Retrodeforming a simple section

Upon retrodeforming the section, it is clear that the section is not balanced. A loose line placed on the left side

of the section becomes incredibly distorted as unequal amounts of shortening affect the three contacts. Another

Loose lineLoose line (restored)

Loose line

Pin line

Pin line

way to look at it is to track the positions of footwall and hangingwall cut­offs before and after deformation.

4

Lab 7: Fold and thrust belts

Solutions

Fall 2005

1 Definitions, et. al.

See K. R. McClay, Glossary of thrust tectonics terms, scanned and posted on the website.

Backthrust . In many thin­skinned fold and thrust belts, most of the fold and thrust structures have a definite,

consistent vergence to them. That is, the sense of overturn on the folds and the dip and transport direction on

the faults suggest consistent transport of material towards the foreland. A backthrust is a thrust fault that dips in

a direction opposite to that of most of the structures in the belts.

Foreland . Thin skinned fold and thrust belts are often found on the flanks of mountain belts. The area out­

board of the mountain belt consisting of undeformed sediments is known as the foreland. Since thin­skinned

fold and thrust belts typically detach on previously flat lying sediments and propagate deformation towards the

foreland, they are often referred to as foreland fold and thrust belts.

Hinterland . The core of a mountain belt, often characterized by rocks of high metamorphic grade and ductile

deformation histories is the hinterland of the range. Thin­skinned fold and thrust belts are generally found

between the undeformed foreland and the strongly deformed core, or hinterland of the range. Tectonic transport

in foreland thrust belts is generally directed from the hinterland to the foreland.

Thrust nappe . A nappe is a recumbent, often isoclinal fold with definite asymmetry (vergence). Nappes are

commonly observed with sheared out lower limbs, or thrust faults. Both the direction of shear or thrust faulting

on the lower limb of the fold and the asymmetry of the fold have consistent vergence or direction of tectonic

transport. Such a structure is a thrust nappe.

Duplex Low angle faults are often characterized by alternating ramps and flats. Ramps are often associated

with a series of imbricate (parallel, or "shingled") faults joined by faults above and below them. In a thrust envi­

ronment, the structural association is: two flat segments (called the floor and roof thrusts), connected by several

parallel ramp segments. The rock masses bounded by these faults are called horses, and the entire structural

association is a duplex.

1

Duplex structure

Roof thrust

Floor thrust

HorseHorse

Out­of­sequence thrust . In a fold and thrust belt, deformation commonly propagates towards the foreland.

That is, thrusts become progressively younger as you go towards the foreland (in the direction of transport). This

is in sequence thrusting. A fault that is younger, but is located more hinterland ward than some other fault is,

in contrast, out of sequence. Note that this has nothing to do with the transport direction or dip of the out of

sequence thrust: it need not be a backthrust.

Blind thrust A blind thrust is a thrust fault that does not break the surface. Instead, the tip of the fault is buried

in a fold. This fold – a fault­propagation fold – is geometrically required to accommodate slip on a fault past the

fault tip.

2 Thrust related folds – cross­sections

2.1

This is a series of fault­bend folds: a syncline above the transition from flat to ramp, and an anticline above

the transition from the ramp back to the upper flat. The calculation for shortening is shown schematically as an

inset. Note that any measure of shortening will somewhat depend on your choice for the “deformed” line length,

since this will scale the results. I made these calculations based on the using the distance from one side of the

section to the other as the original length.

2

2.2

Lo - Lf / Lo ~ 1/4 (~25% shortening)

KK

J

Tr

P

pC

P

Tr

J

This is a fault propagation fold. To calculate shortening, I measured along the dashed line.

3

Ldeformed

Loriginal

Loriginal

Ldeformed= 1/5 (20 % shortening)Loriginal -

3 Sandboxes and critical taper theory

3.1

The sandbox experiment is an attempt to make a scale analog model of thin­skinned fold and thrust belts. The

box was constructed of plexiglass, which is rigid and transparent. In this box, layers of sand and coffee were laid

down on a sheet of mylar paper resting on an inclined ramp. The starting thickness of the sand was around 4cm,

and the ramp was inclined at 4◦. To simulate the transport of material in a thrust belt towards the foreland, the

mylar sheet was pulled underneath the sediment, translating the "foreland" towards the back wall ("backstop")

of the box. Very quickly, a stable wedge of sand was formed. This wedge was two­sided: towards the foreland,

the top of the wedge formed a 6◦angle. Towards the backstop, an early formed backthrust and backfold made a

steeper ( 25 ◦) angle. This geometry was basically stable: even as more material was incorporated into the wedge

by continued pulling on the mylar sheet, the wedge grew, but maintained a reasonably constant angle.

Deviations from a perfect wedge resulted from the top surface being deformed about folds verging towards

the foreland. Viewed from the top, at the end of deformation, five or six major structures dominated the top

surface. Most of the shortening structures were folds, although these were presumably cored by faults. In one

instance, material from the middle layer broke the surface.

4

The presence of a plexiglass sidewall created some edge effects, in that frictional drag along the wall resulted

in less shortening. Another edge effect was the abrupt thinning of the original package of sediment. Numerous

tear faults formed were the sediment package thinned laterally.

Most of the deformation was localized in the toe of the wedge: once folds and faults had formed in the back

of the wedge, there was little or no continued deformation. Thus, most of the structures were developed "in

sequence", with the youngest structures closest to the foreland and vice versa.

We tested the idea that significant erosion can affect the wedge deformation dynamics by removing a large

portion of the wedge top material. Upon continued shortening, the original back thrust and back fold was reac­

tivated, presumably in an attempt to restore the original stable wedge geometry.

3.2

The geometry of a wedge is set by the strength of the material deforming within it, and the frictional resistance

of the decollement upon which the wedge forms. In particular, the weaker the decollement, the lower the wedge

angle; strong wedge material has the same effect. In a material like sand, these parameters can be captured

by the internal friction angles of loose sand and sand on mylar. In thin­skinned fold and thrust belts, rocks

presumably deform according to the Mohr­Coulomb criterion so sand is not a horrible choice as an analog

material.

3.3

This section asks you to take the concept of self­similar wedge growth to an absurd level. If the wedge angle

remains at 5◦, and the wedge tip remains at sea level, self­similar growth to a 180km long wedge suggests that

the top of the wedge be at elevations in excess of 15.5 kilometers. This is three times higher than the highest

regions of the Earth today (individual peaks in the Himalaya reach 8km, but the average elevation at the crest of

the range is a bit over 5km).

This analysis neglects several important parameters. First, isostatic compensation is neglected. We know

that for every 1km of topography, there is a corresponding 6 or 7 kilometers of crust present as a "root", much

like most of the volume of an iceberg is below the ocean. So we might expect that isostatic subsidence would

take care of most of our 15km high wedge. Second, critical wedge theory assumes constant strength, but we

know that the strength of rocks varies considerably with depth. While the increase in strength with depth due to

increasing pressure is accounted for by appealing to Mohr­Coulomb rheology, above certain temperatures, rocks

deform ductilely and according to viscous or viscous­plastic flow laws. Finally, since we expect that erosion to

scale – at least to a first order – with average slope and therefore elevation, the higher we make mountains, we

expect erosion rates to increase as well. It could be that geomorphology, and not crustal strength is the real limit

for the height of mountains on Earth.

3.4

The backstop in the sandbox experiment is probably the most unsatisfying part of the whole set­up. What,

in nature, corresponds to a vertical, unyielding wall? Early papers on critically tapered wedges had cartoons

showing bulldozers pushing wedges in front of them, but this is surely just trading one suspect metaphor for

another.

One thing to realize is that the critical taper models and sandbox experiments are meant to simulate or

describe fold and thrust belts or accretionary prisms. That is, they are models of a small part of the anatomy of

an entire mountain range, in particular, the exterior parts. The backstop then, is just the interior (hinterland) of

the mountain range, and all the model requires is that this part of the mountain range consists of thicker crust

5

and higher elevations. How that part of the range became thickened and whether sandbox experiments shed

any light into this is beside the point.

Alternatively, smaller ranges might be described as two Coulomb wedges back to back. Along these lines,

our experiment yielded a clue as to what the backstop was all about. Recall that the crest of the wedge did

not occur at the backstop. Instead, one of the earliest structures was a back thrust / back fold. The wedge we

created was a two­sided wedge, one with a gentle foreland dipping angle of about 5◦, the other with a hinterland

/ backstop dipping angle of about 20◦. In essence, there were two wedges, backing up against one another. Each

wedge forms the backstop to the other. In some experiments, researchers have pulled the underlying mylar sheet

through a slit in the middle of the original pile of sediment. What happens is very similar to what happened in

our experiment: two wedges form, each making the backstop to the other. A often­cited example of a double­

sided mountain belt is the island of Taiwan, which has been described as two thin­skinned wedges verging in

opposite directions on either flank of the mountain range.

4 The Mechanical Paradox of large overthrusts

FT

h

FR

L

FTσxx = ⇒ FT = σxx h

h

FR = σy x L

σy x = µσy y = µρg h (= tan φρg h)

if FR = FL

σy x L σxx = = µρg h

h

Supposing a horizontal tectonic stress of 100MPa, µ = 0.038. In terms of the angle of internal friction, φ ∼

2◦ . Price (1988) cites a value for µ of 0.577 and φ = 30◦for typical values of rock strength known from rock

deformation experiments. Twiss and Moores (page 171, eg.) describe results from the deformation of sandstone

samples that yield φ = 28.7 ± 7.4. In other words, our analysis seems to predict much, much weaker faults than

we expect from experimental results.

Supposing we assume a far more reasonable value for µ = 0.6. Then, to initiate sliding along the base of the

rigid block, we require σxx =∼ 1.6 GPa. Twiss and Moores (p. 207) cite 250 MPa as being a maximum value of

stress based on the stress required to fracture rock. The actual value will depend on the confining pressure (and

hence the height of the block), but 250 MPa is a very permissive number. (TM discuss this problem in terms

of the maximum length of block that you can push from behind, using 250MPa as a maximum stress. They get

17km.)

Hubbert and Rubey get around the apparent paradox by appealing to a mechanism that will greatly reduce

the effective frictional resistance at the base. In particular, the expression for frictional resistance, modified for

pore fluid pressure, becomes: ∗ σy x = µσy y = µ(1 − λ)ρg h

6

where λ is the pore fluid factor, the ratio between the pore fluid pressure p and the lithostatic pressure ρg h. Even

hydrostatic pore fluid pressure (i.e. p = ρw g h, where ρw is the density of water) greatly reduces the frictional

resistance along the base of the fault (λ ∼ 0.4). If pore fluid pressures approach lithostatic pressures, then λ ∼ 1

and the frictional resistance approaches zero.

The question then becomes: do we have evidence of such high pore fluid pressures in nature. Certainly, in

some environments, very high pore fluid pressures exist. On the other hand, field observations of many faults

suggest that this cannot be a general mechanism. In particular, Clark showed a few slides of the Keystone Thrust

in Nevada where field evidence clearly indicated that the thrust sheet was emplaced over a subaerially exposed

erosion surface. The Keystone thrust sheet rode over deposits of gravel streams and unconsolidated alluvial

deposits, which are not the sorts of rocks that could sustain near­lithostatic fluid pressures.

Price (1988) suggests that the main problem to the so­called "mechanical paradox of large overthrusts" is that

the model description is at fault. That is, its only a paradox to the extent that we buy into a specific mechanical

description (a model) of how large thrust sheets are emplaced. Price argues that if we go out and look at real

thrust faults, both ancient (such as faults in the Canadian Rockis) and active (such as the great Alaska earthquake

of 1964), we would realize that this mechanical description was entirely inappropriate. Toss out the model and

you also get rid of the paradox. (At some level, the existence of the mechanical paradox of thrust faults should

have alerted us to the possibility that the model was deeply flawed).

In particular, the mechanical model assumes that thrust sheets move (1) entirely rigidly; (2) are pushed from

behind; (3) slip along the base of the thrust sheet occurs simultaneously over the entire fault surface. Price points

out that all three assumptions are ruled out by observations of real faults in nature. Thrust sheets are not rigid:

deformation – folding and fracturing – occurs throughout the entire thrust sheet and the amount of slip along

the fault is variable both along strike and in the direction of motion. More to the point, slip along thrust faults

takes place by the addition of many small slip events that affect only a small amount of the fault at any one time.

Even in one slip event, rupture does not take place simultaneously, but instead propagates at rates that scale

with shear wave velocity. He quotes Oldow: "thrusts did not move simultaneously over the whole of their extent,

but partially, first in one part then in another ... the movement would not be like that of a sledge, pushed bodily

forward over the ground, but more akin to the crawl of a caterpillar which advances one part of its body at a time,

and all parts in succession".

Washington’s reply is actually fairly subtle. He doesn’t want to rescue the Hubbert and Rubey model, but

doesn’t like Price’s explanation either. In particular, he dismisses Price’s explanation that the fact that fault mo­

tion occurs non­simultaneously over the whole surface resolves the paradox. This is a subtle point: he doesn’t

dispute – for example – the observations that Price summarizes from the 1964 Alaska earthquake. He just argues

that the fact that slip occurs non­simultaneously makes no difference to the paradox. His claim is that fault slip

and earthquakes are simply the release of elastic strain built up along a fault; that at any given time, the built­up

elastic strains are such that the pre­failure shear stresses along an active fault are generally at or near the stresses

required for failure. He argues, therefore, that the need to explain how the entire fault surface comes to this point

of critical balance is essentially the same thing as the Hubbert and Rubey problem of balancing the basal resis­

tance with the tectonic driving stress at the back of the thrust sheet. His solution to the paradox also involves

tossing out a basic part of the model, but what he tosses out is the conceptualization that thrust sheets move as

tabular bodies being pushed from behind.

Washington appeals to the general wedge geometry of thrust belts. Thrust belts can be translated along the

basal decollement because the area of surface across which the driving stresses are applied increase towards the

back of the wedge. Individual thrust sheets move along with the entire wedge, so a large part of the motion of

any given thrust sheet might be due to drag along the upper surface of the thrust sheet. What Washington seems

to be saying, in effect, is that part of the problem is considering a thrust sheet in isolation. Thrust belts consist

of series of faults, stacked shingle­like. Thrust sheets move along a fault at their base, but typically also have

7

another thrust bounding the top of the sheet, whose motion may contribute importantly to transmitting the

appropriate stresses down to the base of the sheet. (Note: when I first read this paper, I thought that Washington

was simply off­base. Upon re­reading it a few times, I now think that there is a lot more to his argument than I first

gave him credit for. I do think that his argument could be re­stated much more clearly.

Price’s response is two­fold. First, he disputes Washington’s assertion that active thrust faults are everywhere

near failure (a claim that Washington provided without much in the way of evidence). The point stands: if thrusts

do not slip simultaneously along their entire surface, then there is no need to balance a resisting force that is in

large part a function of the surface area. It is true, however, that having demolished this model of a thrust sheet,

Price fails to explain how stresses are transmitted across thrust sheets, or what the origin of those stresses are.

Price resolves the paradox by eliminating the model, but provides no alternative model.

Second, Price takes Washington to task for his appeal to critical wedges as a model that can explain fault

motion. Critical wedge models (sandbox models) are idealized as a penetratively deforming mass of material

that are slip along their base. Price is correct that, apart from the basal decollement, there are no faults in these

models. Washington’s figure 1 certainly appears a little ad hoc, and its easy to see why Price, a geologist who had

spent over 20 years looking at thrust faults in the field, would have nothing but disdain for this totally unrealistic

cartoon of a thrust sheet. But what Washington is actually trying to do is show that there is another source of

stress driving individual thrust sheets that has to do with their being located in a larger deforming mass (some­

thing like a critically tapered wedge). At least he provides some hand­waving in the direction of a model (whose

details are, at a minimum, a bit unclear).

8

12.113 Structural Geology

Part 1: Continuum mechanics andrheology

Fall 2005

2

Contents

1 Stress and strain 51.1 Stress (T&M pp.128 ­ 164) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Strain: part one 72.1 Reading assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Strain I: displacement, strain and terminology . . . . . . . . . . . . . . . 7

2.2.1 Measurement of strain . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 The strain ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Displacement vector fields and strain . . . . . . . . . . . . . . . . . . . . 92.5 Mohr circles for strain I : Infinitesimal strain . . . . . . . . . . . . . . . . 102.6 Mohr circles II: Finite strain . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6.1 Unstrained state reference frame . . . . . . . . . . . . . . . . . . . 122.6.2 Strained reference frame . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Week 3 notes: Progressive deformation 133.1 Reading assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Progressive strain in pure shear . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Rheology part 1: Ideal material behaviours 174.1 Reading assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Ideal behaviours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Elastic behaviour – Hooke’s law . . . . . . . . . . . . . . . . . . . . 184.2.2 Viscous behaviour – Newtonian fluids . . . . . . . . . . . . . . . . 184.2.3 Visco­elasticity I: Maxwell bodies . . . . . . . . . . . . . . . . . . 194.2.4 Visco­elasticity II: Kelvin bodies . . . . . . . . . . . . . . . . . . . 204.2.5 Other ideal rheologies . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Study and review questions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Rheology part 2: Experimental rock deformation, stress­strain curves 255.1 Reading assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Experimental rock deformation . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Experimental rock deformation: phenomenology . . . . . . . . . . . . . 255.4 Ductile deformation: the effects of strain rate and temperature . . . . . 275.5 Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3

4

Chapter 1

Stress and strain

1.1 Stress (T&M pp.128 ­ 164)

1.1.1 Introduction

In geology, we seldom observe the forces responsible for the deformations that we are interested in. In fact, it turns out that you can’t actually measure stress directly (stress measurements are made by observing the deformation of reference materials whose response to stress is known). Nevertheless, one of the main objectives of earth science is to try and understand the mechanisms whereby observed deformations (faults, folds, mountain ranges and so on) are produced.

The key concept in this regard is stress, which is related to the familiar concept of a force in a reasonably straightforward way:

Stress = Force / Area By this definition, it is clear that a given force acting on a larger area results in a

smaller stress than the same force acting on a smaller area. Since Force is a vector quantity (i.e. its direction and magnitude are defined), we denote the above relation as

−→ F = −→σ (1.1) A

−→σ is termed a stress vector, or a traction, and is defined with reference to a par­ticular plane. Stress has units of force (Newtons) divided by area (square meters). A

N is a Pascal (Pa). In the Earth, most stress of interest are of the order of 106 ­ 109 m2

Pa, and stresses are commonly reported in megapascals (MPa). In engineering and material science, the convention is that positive stresses are

tensional, and compressional stresses are negative. In the Earth, true tensional stresses are exceedingly rare, however, so we use the opposite sign convention.

1.1.2 The stress tensor

An arbitrary stress acting on a plane can be resolved into three components: one normal to the plane (normal stress) and two mutually orthogonal components tan­gential to the surface of the plane (shear stresses). Since we are interested in contin­uous volumes (the interior of the earth), this is further generalized:

Consider a small volume whose faces are oriented with the coordinate axes. We keep track of the faces by identifying each face x, y, z with the coordinate axis normal to that face. The reason we deal with this volume is that the surface stresses on each

5

pair of faces is independent of the surface stresses on the other two pairs of faces. As it turns out, if we know the surface stresses on any three orthogonal faces defining an infinitesimal volume, we have completely characterised the stress and we can calculate the surface stress on any plane crossing this region.

6

Chapter 2

Strain: part one

2.1 Reading assignment

Twiss and Moores: chapter 15, in particular, pages 292 – 302. Discussion of specific special types of strain (pure shear, simple shear) begin on page 303. Pages 304 – 310 deal with progressive strain, and are useful background material for the lab.

W. Means (1976) Stress and Strain is a great text, very clear, well written and reads easily. J. Ramsay and M. Huber (1983) The Techniques of Modern Structural Geology, Volume 1: Strain Analysis is amazingly detailed, with many, many examples of de­tailed strain analysis. It can, however, be “a bit much”.

2.2 Strain I: displacement, strain and terminology

Given enough differential stress, a material responds by deforming. We distinguish: between rigid body deformations and non­rigid body deformations. The first in­cludes translation and rotations of a body. The second includes distortion and di­lation. Other important distinctions are: continuous vs. discontinuous strain and homogenous vs. heterogeneous strain. Whether strain is homogenous or heteroge­nous is often a function of the scale of observation. Also, when strain in natural sys­tems is analyzed, a common approach is to identify structural domains wherein the strain is continuous and approximates homogeneity. The point of doing so is that we can use the tools of continuum mechanics – the physics of continuous deformation.

2.2.1 Measurement of strain

1. Changes in the lengths of lines

2. Changes in angles

3. Changes in areas or volumes

Changes in line length:

7

3 important measures: Elongation

e ≡ Δl =

l f − li = l f − 1

li li li

Stretch

S ≡ l f

li = 1 + e

Quadratic elongation

λ ≡ S2 = (1 + e)2

So λ = 1 means no change in length; λ < 1 reflects shortening and λ > 1 is extension.

Changes in angles

Figure 2.1: Tracking strain of two initially perpendicular lines

1. Consider 2 originally perpendicular lines. The change in angle between those lines is

90 − α = ψ ≡ angular shear

2. Consider a particle on the y­axis. Measure displacement at some distance y from the origin, in the x direction:

x = γ ≡ shear strain y

Note that:

γ = tan ψ

Change in volume (or area) The dilation is similar to the definition of elongation:

Vf − Vi ΔV Δ ≡ =

Vi Vi

Note that all these measurements take the undeformed state as the point of ref­erence. It is equally feasible to take the length or angles in the deformed state as the reference state. There are no particularly good reasons for doing one or the other, neither is good for large strains. Alternatively, the infinitesimal strain is often a very useful concept, corresponding to the strain accrued in a vanishingly small instant of deformation.

8

� �

�

Figure 2.2: Distortion of material lines and strain markers

2.3 The strain ellipsoid

In a deforming body, material lines will rotate and change shape. We want to be able to characterize the rotation and elongation of any arbitrary line. A circle (sphere in 3D) allows us to keep track of all possible orientations of lines. As it turns out, any homogenous strain turns a circle into an ellipse and a sphere into an ellipsoid.

Consider a circle of unit radius, deformed into an ellipse oriented such that the major and minor semi­axes are parallel with the coordinate axes. The elongation, stretch and quadratic elongation of the major semi­axis are given by:

l f x − 1 ex =

1 Sx = 1 + ex = l f x

λx = l 2 f x

Alternatively, l f x = λx . Similarly, l f z = λz . Since the equation of an ellipse is 2 2x

b2 = 1, where a and b are the lengths of the semi­axes, the equation of the strain ellipse is just

2 2

a2 + z

x z+ = 1 λx λz

In three dimensions: 2 2 2x y z+ + = 1

λx λz λz

More generally, the semi­axes of the strain ellipse (ellipsoid in 3D) are the prin­cipal strains, analogous to the principal stresses we saw earlier. The length of the semi­axes (S1, S2, S3) are the magnitudes of the principal strains. The strain ellip­soid can have various shapes, corresponding to uniaxial, biaxial or triaxial strain. Circular sections through biaxial strain ellipses will be undistorted; in triaxial strain they will be distorted, though equally shortened or lengthened in all directions.

If you’re lucky, the rock body you are looking at will contain initially circular or spherical markers. Deformation will turn these into ellipses, whose long and short axes are the principal strain axes. Examples of reasonably nice spherical strain mark­ers are ooids or (perhaps) pebbles in a conglomerate. The spherical cross sections of worm tubes can also be used in this way. Initially elliptical or ellipsoidal markers can also be used to characterize strain, though this is more complicated.

2.4 Displacement vector fields and strain

One approach to analyzing strain is to keep track of particle displacements. The po­sition (x, y) of a particle before deformation to its position after deformation (x , y �)

9

� � � �

can be related by a set of coordinate transformation equations of the form:

x � = ax + by

y � = cx + d y

Figure 2.3: Displacement field for general, homogenous strain

If a, b, c, d are constants, then the strain is homogenous (fig 2.3). Two partic­ularly important strain regimes are simple shear and pure shear (fig. 2.4). Their coordinate transformation equations, expressed in matrix notation are:

1 γ k 0simple shear: pure shear:

0 1 0 1/k

Deformation is usually not instantaneous: strains accumulate over time. One kind of strain can follow one another. Mathematically, this is equivalent to multiply­ing the strain matrices. Note that matrix multiplication is not commutative, so, for example, simple shear followed by pure shear does not yield the same final result as pure shear followed by simple shear.

If the orientations of the principal strain axes do not rotate during deformation, then the strain is said to be irrotational or coaxial.

2.5 Mohr circles for strain I : Infinitesimal strain

The infinitesimal strain is a useful concept that can be thought of as representing the instantaneous material response to stress. The accumulation of infinitesimal strain increments over geological time results in the deformation that the geologist observes and tries to understand in the outcrop – known as the finite strain. For our purposes we can consider "pretty small" strains – say, less than 1% – to be infinites­imal. For the purposes of deriving the Mohr circle equations, this case also allows the use of small angle approximations, in particular, γ = ψ.

As with stress, we want a Mohr circle construction that allows us to read off (1) the elongation (for infinitesimal strain, call it �) and (2) the shear strain (γ) that af­fects a line of some given orientation. The principal strains are denoted �1, �2, �3. A

10

�

Figure 2.4: Displacement fields for pure and simple shear

fairly tedious derivation (for details, you might check out Hobbes (1976)) yields: � � �1 − �2 �� �1 + �2

� = + cos 2α 2 2

γ � �1 − �2 = sin 2α 2 2

From which a Mohr circle construction can be made (see figure 3.1). α is the an­gle between a material line and the principal strain axes. Note that this Mohr circle is drawn in � vs. γ/2 coordinates. Examination of the Mohr circle for infinitesimal strain yields the following important relations: 1. There are two lines that experience the maximum shear strain, and they are lo­cated at 45◦to the principal strain axes. 2. The maximum shear strain is given by γ/2 ± (�1 − �2)/2, i.e. γ ± (�1 − �2). 3. Any 2 lines perpendicular to one another are 180◦apart on the Mohr circle, so they suffer shear strains equal in magnitude but opposite in sign.

2.6 Mohr circles II: Finite strain

As the result of finite strain, lines are lengthened or shortened and the angles be­tween intersecting lines are usually changed. Considering a unit circle deformed into an ellipse whose axes are parallel with the coordinate frame, we can derive rela­tionships that track the elongations and rotations (shear strains) for any line. As you might expect, a Mohr circle construction is the ticket to the big time. The derivations of these are so tedious that even Ramsay and Huber relegate them to an appendix (cf. Ramsay and Huber, appendix D if you can’t help yourself ). Two separate con­structions are available: one identifies lines according to the angles they make with the principal strain directions in the "unstrained state" – this is of somewhat limited

11

�

use since we rarely know what the orientations of lines used to be. The other deals with the orientations in the strained state. Again, as with stress and infinitesimal strain, the orientation of a line is defined by the angle it makes with the principal strain axes; the Mohr circle constructions are created by deriving equations that re­late elongations, shear strains and line orientation that also form parametric equa­tions for a circle.

2.6.1 Unstrained state reference frame

The unstrained reference frame refers back to the pre­deformation orientation of a line P . The line makes an angle θ with the principal strain axes. Upon deformation, it suffers an elongation and rotates into a new position P � (in general, we will use primes to distinguish the reference frames) making an angle θ� with the principal strain directions. The elongation and shear strain of an arbitrary line (in 2D) are given by equations that should provoke a sense of deja vue:

(λ1 +λ2) (λ1 −λ2)λ = + cos 2θ

2 2 λ1 −λ2

γ = � sin 2θ 2 λ1λ2

Note: these fail to produce a circle (instead, you get a Mohr ellipse) unless λ1λ2 = 1, i.e. there is no dilation.

2.6.2 Strained reference frame

Usually, we are confronted with good rocks gone bad, and so the angles we measure are those of lines in the strained state, i.e. we measure θ�, not θ. Conversion between the two reference frames can be done using:

�/λ1/2sin θ =λ1/2 sin θ 2

cos θ =λ1/2 cos θ�/λ1/2 1

and pulling a little definitional slight of hand. In particular, we define: 1. A new strain parameter γ� = γ/λ.; and 2. The reciprocal quadratic extensions γ� = 1/γ1 and γ� = 1/γ2.1 2

The Mohr equations become:

1 +λ� λ�λ� 1 −λ�

= 2 − 2λ� cos 2θ� 2 2 λ�

2 −λ� � 1γ = sin 2θ�

2

12

Chapter 3

Week 3 notes: Progressive deformation

3.1 Reading assignment

Progressive strain histories are covered in section 15.5 (pages 308ff.) of Twiss and Moores. Also read pages 352 to 357 ("Strain in shear zones"). Nice treatment of this sort of thing is also found in the introductory chapters of Passchier et. al. (2005) Microtectonics.

3.2 Progressive strain in pure shear

In lecture, we discuss possible strain histories for lines of various orientations that undergo progressive strain. (In lab, we deal with the case of pure shear).

By progressive strain, we simply indicate that in our analysis we will break down the deformation history into many small steps. At each step we will consider both the infinitesimal strain that accrues at that step, and the finite strain that has ac­crued up to that point. Even in very simple cases, some rather complex behavior can result.

The analysis considers one of the "simplest" kinds of strain: pure shear in two­dimensions, i.e.:plane strain, �2 = 0 and λ2 = 1.

The following four figures accompany the in­class discussion. These figures are meant to be incomplete. Use these to help you follow the class discussion; you should annotate and complete these.

13

Figure 3.1: Mohr circle for Infinitesimal strain

Figure 3.2: Mohr circle for finite strain, unstrained state reference frame

Figure 3.3: Mohr circle for finite strain, strained state reference frame

14

Line of no fin

ite elongatio

n

45 deg.

Shorteningonly

Lengthening only

Figure 3.4: "Real world" reference frame. Shaded area shows region where material lines have suffered shortening followed by lengthening. Line of no finite elongation separates sub­regions defined by whether extension exceeds shortening.

15

16

Chapter 4

Rheology part 1: Ideal material behaviours

4.1 Reading assignment

Chapter 18 in Twiss and Moores, pages 361 – 385, deals with both ideal models for rock deformation as well as experimental investigation of ductile flow. Chapter 9, pages 165 – 190, deals with the mechanics of brittle fracture. This is good stuff, be­lieve.

4.2 Ideal behaviours

The next section of the course will review what we know about the relationship be­tween stress and strain; that is, the laws that govern deformation. We start with "ideal" behaviours, and then compare these to experimental results. A material’s re­sponse to an imposed stress is called the rheology. A mathematical representation of the rheology is called a constitutive law.

stress

}

dx

Figure 4.1: Simple, 1­D Hooke body (linear elasticity)

17

rate

stress

Figure 4.2: In a Newtonian fluid, stress and strain­rate are linearly proportional to each other. The slope of the stress­strain rate line is the viscosity.

4.2.1 Elastic behaviour – Hooke’s law

The characteristics of elasticity are: 1. strain is instantaneous upon application of stress; 2. stress and strain are linearly related; 3. strain is perfectly recoverable.

σ ∝ �

The constant of proportionality is called the elastic modulus; the exact modulus you use depends on whether the strain is volumetric, uniaxial or in shear. Respectively, these are:

σh = K �v

σn = E �n

σs = 2µ�s

Where K , E , µ are the bulk modulus, Young’s modulus and the shear modulus. Another important elastic constant is ν, Poisson’s ratio. Poisson’s ratio relates the elastic strain for orthogonal directions.

4.2.2 Viscous behaviour – Newtonian fluids

Strain in a viscous fluid is time dependant and non­recoverable. Time­dependance is the fundamental difference with elastic behaviour: instead of a linear relationship between stress and strain, viscous materials exhibit a linear relationship between stress and strain­rate.

τ = η�̇

18

strain hardening:(”shear rate thickening”)

strain softening: (”shear rate thinning”)

Figure 4.3: Strain­rate softening or hardening

The constant of proportionality, η, is the viscosity, and has units of Pascal­seconds. A plot of strain against time is a line, provided that the viscosity remains constant. Viscosity is a measure of the strength of a material: higher viscosity materials are stronger. Typical geological viscosities are:

MaterialWaterLavaGlacier iceSaltWindow glassAthenospheric mantle

Viscosity (Pa­s) 10−3

0.1 ­ 10 1013

1014 ­ 1020

1021

1021

The effective viscosity can change during the course of deformation. If the vis­cosity increases, we call the process strain hardening or shear rate thickening; con­versely, if the material becomes weaker and the viscosity decreases, the process is strain softening or shear rate thinning.

4.2.3 Visco­elasticity I: Maxwell bodies

If we represent elastic behaviour with a spring, and viscous behaviour with a dash­pot1, we can also combine dashpots and springs in a number of different ways to simulate various possible ideal material responses or rheologies.

1What the heck is a dashpot? They show up a lot in material science, geodynamics and differential equations, and are invariably introduced as though they are a familiar, intuitive part of one’s day to day experience. Except that they aren’t, really. Ask the random person on the street what a dashpot is and you’ll get a pretty blank look. Dashpots are simple pistons combined – usually – with a hydraulic fluid. Examples of objects that include dashpots are car shocks, bike shocks, and are found on some doors.

19

� �

Figure 4.4: Maxwell visco­elastic body

The most basic combination is to put a spring and a dashpot in series. This combination is known as Maxwell viscoelasticity. The constitutive relationship for Maxwell viscoelasticity comes from the linear addition of the relationships for elas­ticity and viscous fluids2:

σ̇ σ �̇= +

2µM 2ηM

The subscript M just indicates that we’re dealing with Maxwell­rigidity and Maxwell­viscosity. This equation can be manipulated by stipulating either constant stress (σ = σ0) or constant strain ( �̇= 0). Under constant stress:

σ0 σ0 � = + t

2µM 2ηM

On a strain vs. time plot (fig. 4.5), this shows instantaneous elastic strain, fol­lowed by steady state linear viscous strain. The case of constant strain is a little more difficult to intuit, but the equation, and plot of stress against time are clear enough.

µMσ = σ0 exp − t

ηM

A plot of stress against time demonstrates the viscous stress relaxation of a Maxwell body (fig. 4.5). That is, a Maxwell body should relax to an isotropic (i.e. hydrostatic) stress state on a time scale that is captured by the ratio of ηM : µM . Expressed as a fraction, this ratio has units of time, and is known as the Maxwell time. The Maxwell time of the asthenospheric mantle is on the order of a thousand years and is what sets the time scale of phenomena like post glacial rebound.

4.2.4 Visco­elasticity II: Kelvin bodies

Another way to combine a spring and a dashpot (that is, an elastic response and a viscous response) are to put them in parallel, rather than in series. This material is known as a Kelvin body, and sometimes called fermoviscous behaviour. This is the idealization of a phenomenon (that is, something that actually happens in the real world) called elastic afterworking, which is just that, in the real world, most springs

2The notation I use differs somewhat from what Clark used in class. Mine follows standard texts such as Ranalli, G. (1995) Rheology of the Earth, and Turcotte, D. and Schubert (2002) Geodynamics.

20

� � ��

� �

viscous relaxation

viscous

elastic}

} recovery

Constant StressConstant Strain

timetime

stress strain

Figure 4.5: Stress and strain versus time for Maxwell viscolelastic bodies

and supposedly elastic materials don’t always respond instantaneously to the im­posed stress. A Kelvin body shows time­dependant, recoverable strain. Kelvin be­haviour is given by

σ = 2µK � + 2ηK �̇

Upon loading, the elastic response of the spring is damped by the dashpot, and goes asymptotically to σ0/2µK :

� = σ0

1 − exp − µK

t 2µK ηK

When the load is removed, the strain is recovered, but not instantaneously. Sup­pose, a strain �0 had been accummulated, then �(t ) is

µK� = �0 exp − t

ηK

4.2.5 Other ideal rheologies

Four other ideal behaviours are worth mentioning. The first are two end­members of Newtonian viscosity (fig. 4.7): the Pascal liquid and the Euclid solid, which have viscosities 0 and ∞, respectively. The last two are types of plastic behaviour. A plas­tic material is one that has a yield stress, but otherwise behaves as a Newtonian fluid (or a non­Newtonian, power­law fluid). The St. Venant body (also called an elastic­plastic body) is idealized as a friction block being pulled by an elastic spring (fig. 4.8). Such a body is a material that exhibits linear elasticity up to a certain point, the yield point. Then, the material fails abruptly. This sort of rheology is relevant to understanding elastic strain accumulations that drive the earthquake cycle.

A Bingham body, also called viscoplastic, is idealized as a spring in series with a friction block itself in parallel with a dashpot 3. Its behaviour is governed by the equations:

σ = 2µ� σ < σY

σ = σY + 2ηB �̇ σ ≥ σY

3Note Twiss and Moores leave out the leading spring, which is a special case.

21

constant strain

viscous recovery

remove stress

time

Figure 4.6: Kelvin visco­elasticity, strain versus time.

infinite strain, for no stress

no strain, regardless of stress

Figure 4.7: Idealized end­members of Newtonian viscosity

22

static friction threshold“Yield Point”

friction

Friction Block Analog - St.Venant body (stick-slip)

Figure 4.8: Model and stress­strain plot for St. Venant (or "stick­slip") rheology

The Bingham body behaves elastically at stresses lower than the yield stress, and flows as a linear fluid above the yield strength with a strain rate proportional to σ − σY . Examples of Bingham materials are certain clays, some kinds of submarine debris flows, oil paintings, drilling muds, toothpaste and bread dough.

4.3 Study and review questions

You should be aware of the differences between: linear elastic, linear viscous, Maxwell viscoelastic, and Kelvin behaviour. St. Venant and Bingham bodies are also relevant since they introduce the concept of a yield stress.

For each of the above, draw a spring and dashpot idealization, and draw curves that relate stress and strain (or strain­rate). For Maxwell and Kelvin bodies, also plot stress against time for constant strain, and strain (or strain rate) against time for constant stress. Think of a real world example or two for each behaviour.

23

yield point

Figure 4.9: Bingham viscoplastic rheology

24

Chapter 5

Rheology part 2: Experimental rock deformation, stress­strain curves

5.1 Reading assignment

The latter part of Chapter 18 in TM is essential reading for this section (pp. 369 – 385). Other useful sources are chapter 5 in Ranalli, G. (1995) Rheology of the Earth and Nicholas, A. and Poirier, J.P. (1976) Crystalline Plasticity and Solid State Flow in Metamorphic Rocks. Hobbes, et. al. (1982) An Outline of Structural Geology is the closest textbook to the order and logic of the lectures as presented in class.

5.2 Experimental rock deformation

Most of the results of experimental rock deformation come from a fairly standard experimental set­up: a triaxial deformation apparatus.

The triaxial deformation apparatus is basically a pressure vessel surrounding a piston. This kind of apparatus allows careful control of the principal stresses, tem­perature and strain­rates. The effect of pore fluids or chemical solutions can also be controlled. Major limitations are: (1) the amount of finite strain that the apparatus can accommodate (on the order of 10%); (2) the slowest strain­rates possible (about 10−7sec−1. Geological strain rates, by comparison, range from 10−12 to 10−20.

5.3 Experimental rock deformation: phenomenology

Figure 5.2 shows the relationship between stress (this is shorthand for "differential stress", i.e. σ1 − σ3) and (total, finite) strain. The first part of the curve shows a linear relationship between stress and strain – characteristic of elasticity. This strain is recoverable, meaning that the strain goes back to zero once the stress is removed. The second part of the curve is known as primary creep. Stress and strain are no longer linearly related; strain is recoverable, but the recovery turns out to be time­dependant. That is, this part of the curve shows behaviour characteristic of Kelvin visco­elasticity. The third part of the curve is referred to as secondary creep. When

25

Figure 5.1: Triaxial deformation rig: the so­called Patterson vessel. Pore pressure, confining pressure

Furnace

Internal Load Cell

Press. Compensator

P Inlet c Atmospheric Pressure

Sample

26

� �

Figure 5.2: Stress­strain curves

stre

ss

stre

ss

strain strainrecoverable

SECONDARYELASTIC PRIMARY

stress is removed, you recover the elastic part instantaneously, the secondary part in a time dependant part, but you accumulate permanent strain.

5.4 Ductile deformation: the effects of strain rate and temperature

Consider a series of experiments (these results were first described by Heard and his colleagues from experiments on marble, see references in Hobbes, et. al. 1982). Fig­ure 5.3 shows three plots. The first shows a family of curves corresponding to differ­ent temperatures at a constant strain­rate. The second shows a family of curves cor­responding to different strain­rates at a constant (high) temperature. The first­order observation is that both increasing temperature and decreasing strain­rate have the effect of weakening the rock (less stress for the same strain). The third shows a plot of stress against strain rates for experiments on marbles. The thin black lines shows how the lines might be extrapolated to geological strain rates, but, more importantly since many of the lines are parallel, it suggests that for certain deformation mech­anisms, temperature can substitute for strain­rate. Since we can heat up a sample much easier than we can control the passage of time, this represents a strategy for simulating natural deformation in the laboratory.

Experimental data can be fit by

�̇= A exp − E

σN

RT

where T is the absolute temperature, R is the gas constant, E is the activation energy, and N is a constant ranging from 1 to 8. For materials deforming such that N > 1, this is called "power­law creep", since stress and strain rate are related by a power N . If N = 1, then strain rate is linearly proportional to stress, which is the definition of a Newtonian viscous body. For constant strain­rates or constant stresses, the equa­tion can be manipulated to yield an "effective viscosity", which at the very least is a useful shorthand for the strength of materials deforming by ductile flow. The effec­tive viscosity of the upper mantle, as inferred from studies of post­glacial rebound are in the 1021−22 range.

5.5 Review questions

What are typical strain rates for tectonic systems? What are the slowest strain­rates typically achievable in laboratory experiments? How can we safely extrapolate ex­

27

Figure 5.3: Experimental stress­strain curves. These plots are taken from Hobbs, H., Means, W., and Williams, P. (1982) An Outline of Structural Geology, New York: Wiley and Sons, inc., pp. 64 – 66.

28

perimental results to natural systems? In the plot with a family of temperature curves in stress vs. strain­rate space, not

all the lines are, in fact parallel. This and similar plots are often subdivided into high, moderate and low stress regions.

Outline how the environmental variables of confining pressure, temperature and strain­rate affect the behaviour of rock deformation.

29

12.113 Structural Geology

Part 3: Ductile deformation, folds andfabrics

Fall 2005

2

Contents

1 Notes XIFolds 51.1 Acknowledgements for figures . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Jargon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Terminology and classification . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Geometric considerations for folds . . . . . . . . . . . . . . . . . . . . . . 71.6 Models of fold formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.1 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6.2 Flexural slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6.3 Passive flow folding . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.4 Combination of fold mechanisms . . . . . . . . . . . . . . . . . . 101.6.5 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Other kinds of folds: kink folds, chevron folds, diapirs . . . . . . . . . . 131.8 Review questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Fabrics and deformation mechanisms 172.1 Fabrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Lineations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3

4

Chapter 1

Folds

1.1 Acknowledgements for figures

With the exception of the Busk construction figure, all figures for this section are drawn from Hobbs, H., Means, W., and Williams, P. (1982) An Outline of Structural Geology, New York: Wiley and Sons, inc., pp. 64 – 66.

1.2 Reading

Begin by re­examining the lab on folds, especially the handout. Two chapters are devoted to folds in TM. The first (ch. 11) is rather dreary stuff about the description and classification of folds. The second is a far better investment of your time and energy.

1.3 Jargon

As usual, a laundry list of key terms.

Fold hinge, fold axis; hinge surface, axial surface; cylindrical / non­cylindrical fold. Types of folds: concentric, parallel, similar. Detachment folds, chevron folds, fault propagation folds, diapir or deroulement folds. Harmonic / disharmonic folds, ptygmatic folds. Neutral surface. Axial planar foliation.

1.4 Terminology and classification

Field measurements of folds: standard list of measurements and observations in­clude: cylindricity of fold, style of folding of multilayers (harmonic or disharmonic), thickness of folded layers, strike/dip of beds around fold; orientation (trend/plunge) of hinge line; orientation of axial surface (hinge surface); asymmetry of folds (S or Z?); presence of axial planar foliation, intersection lineations, stretching lineations;

5

Figure 1.1: Ramsay’s classification of folds.

secondary structures on limbs or in hinge (eg. evidence of flexural slip between lay­ers; evidence of dilatancy – like fractures – in hinge).

Careful measurement and documentation of even small minor folds in the field is crucial because (1) minor fold geometries are often related to geometry of large map­scale folds that control the distribution of rock units; (2) deformation style of folds reflects conditions of deformation (i.e. ductile vs. brittle); (3) folding can occur as a result of shortening or tectonic transport so constraining the geometry and style of folding is crucial for unraveling tectonic history of an area.

1.4.1 Classification

Field measurement and description of folds forms the basis for a classification of folds. Note that fold classification schemes can be based on geometry (purely de­scriptive classification) vs. mechanics (a classification that infers a model of forma­tion). Geometric classification schemes can get pretty abstruse but a good classifi­cation scheme reflects differences in process and conditions of formation. Detailed fold classification schemes are described in pp. 220–235 of TM. Broadly speaking though, two (or three) common fold styles are especially worthy of note: parallel and similar folds.

Parallel folds

Parallel folds (Class IB in Ramsay’s scheme) are folds where the orthogonal thick­ness of the layer is constant about the fold. Concentric folds are a special case of parallel folds where the outer and inner layers define arcs that have a common cen­ter of curvature. These kinds of folds are common in upper crustal tectonic settings, where most deformation occurs by proceses that only permit limited ductile flow of rock. Instead, most of the deformation is accommodated by slip on bedding or layer boundaries (so­called flexural slip folding, see below).

6

Figure 1.2: Busk construction method for flexural slip detachment folds. Note the geometric necessity for a detachment surface and ductile deformation in the very core of the folds

Similar folds

Similar folds (Ramsay’s class II) are characterized by parallel dip isogons, but more to the point, by relative thinning of fold limbs and thickening of hinge zones. These folds are common in metamorphic terranes, i.e. where most deformation occurs by mechanisms permitting extensive ductile flow of rock.

1.5 Geometric considerations for folds

Especially for concentric folds, geometry creates constraints for the continuation of the folds at depth as well as local space problems.

The Busk construction method is a way to extapolate fold geometries (fig. 1.2). Busk folds are arcs of circles centered on common points. Therefore, this construc­tion method produces concentric fold geometries. However, Busk construction re­quires that folds eventually “die out” on an essentially unfolded surface. Comparing the length of folded layers at the top of the construction (l3) to the bottom surface, it is clear that the lengths are rather unequal, and that the bottom of the folds (surface l0) is a detachment surface. These kinds of folds are therefore called detachment folds (figure 11.2 in TM shows a real world example of detachment folds – note that these folds depart from ideal concentric geometry).

Another requirement is that some amount of material has to flow ductilely into the bottom of the anticlines in order to solve a "space problem" : if the folds are truly concentric this geometry has to break down as the radii of curvature get smaller and smaller. If the folds form above weak rocks (which might explain why the folds detach where they do), this material has to flow into the core of the fold. In some cases, this material flows according to pressure gradients and the pressurization of the ductile layer can produce piercement diapirs in the core of the structure (fig. 6.9 page 102, for a general idea of what this might look like).

7

Figure 1.3: Strain for buckle folds. Figure from Hobbes, Means and Williams (1982)

1.6 Models of fold formation

The common models of fold formation present a series of end­member possibili­ties. The models are not mutually exclusive, and the combination of various fold mechanisms may explain observed geometries better than any single mechanisms. Despite much work on these problems, you might feel that some models fail to be completely compelling. You would not be alone.

1.6.1 Buckling

Buckling is what happens when you push on the ends of fairly rigid layer (put a piece of paper on a flat surface and push the edges towards one another: this is buckling). Buckling of a layer will produce parallel fold geometries, since the thickness of the layer is unaffected. The important thing to realize is that the model predicts a char­acteristic pattern of strain, and so is testable by going out into the field and checking to see whether that pattern actually obtains.

Characteristic features of buckle folds: (1) the upper part of the layer folded anticlinally will be in extension, the lower half in compression. You can define a neutral surface that separates areas of compres­sion and extension. On this surface, material points experience no strain. (2) Deformation occurs only by bending about the fold axis. Ideally, there is no ex­tension parallel to the fold axis. That is, this is an example of plane strain. (3) Compressive and extensional strain increase with distance from the neutral sur­face.

1.6.2 Flexural slip

This is “phonebook folding” or “deck of cards” folding. The idea is that folds are pro­duced by shear on surfaces parallel to the layer being folded. This model producesparallel folds.Important features:1) Deformation occurs by bending about the fold axis and shear on the slip surfacesin directions normal to the fold axis. It is also plane strain. Fold axis is parallel to the

8

Figure 1.4: Strain in flexural slip folds. Figure from HMW(1982)

intermediate principal strain axis.2) Layer maintains its thickness, but there is no neutral surface.3) Folded layer is a circular section through strain ellipses.4) Profile of fold shows a divergent fan of strain ellipses.5) A passive linear marker on the fold surface will maintain its orientation wrt to thefold axis. After folding, a stereonet plot of the lineation will have it define a smallcircle about the fold axis.

1.6.3 Passive flow folding

The layer being folded is assumed to exert no mechanical effect on the folding pro­cess. It is just a passive marker. Folds form by differential flow or slip along closely spaced surfaces oblique to the layer being folded. This will produce similar folds.

Characteristic features: 1) Also plane strain. Deformation by simple shear on shear planes. The shear planes are circular sections through the strain ellipsoid. 2) The direction of shearing can be quite variable. The only requirement is that shear plane is not parallel to the layer. However, the maximum fold amplitude obtains when shear direction is normal to the fold axis. 3) In the plane parallel to the axial plane of the fold, no changes in layer thickness will be observed. However, viewing the fold normal to the shear plane and parallel to the presumed shear direction, you will see great variation between the thickness of hinges and limbs. This variation requires no flow of material from limbs to hinges. 4) Shear sense on the shear planes changes from limb to limb on the same fold. Strain ellipses, and principal planes of strain make a divergent fan. 5) There is no neutral layer, and strains are constant at all points within a layer. 6) These folds can be harmonic over large lengths, as opposed to concentric folds, where geometry requires them to be detached. 7) A passive linear marker is distorted and rotates towards the slip direction. Since the slip direction is never parallel to the hinge of the fold produced, initially linear markers will never be oriented parallel to the fold hinge. 8) There is no relationship between layer thickness and wavelength of folds.

9

Figure 1.5: Passive flow folding, from HMW(1982)

Why this model is inadequate.

OK. You can make folds by passive slip or passive flow. But is it really what hap­pens? Recall that what we need is a model to explain the kinds of folds commonly observed in metamorphic terranes. In these areas, folds are typically characterized by 1) similar fold geometry, in particular thinning at limbs and (relative) thickening at hinges; 2) stretching lineations parallel to fold hinges; 3) commonly disharmonic folding; 4) common apparent relationship between layer thickness and competency and wavelength and style of folding. Passive flow folding can do (1) but utterly fails at (2), (3) and (4).

More importantly, is there any evidence for the existence of slip surfaces? Sim­ilar folds are commonly associated with axial planar foliation, so on the face of it, a potential slip surface exists. But offsets (i.e. evidence for slip) across the cleavage or foliation planes are not common, and where present are more likely the effects of pressure solution. If related to shear on these surfaces, we might expect to see a stretching lineation oriented at high angle to the fold axis. More commonly, stretch­ing lineations are parallel to the fold axis. Finally, axial planar surfaces are principal planes of strain. There should be no shear on them at all.

1.6.4 Combination of fold mechanisms

Our 3 simple models suffer from some problems.

1. They produce either ideal parallel or ideal similar folds. In Ramsay classifica­tion, that means class IB and class II folds. But in nature, class IC and class III

10

folds are common

2. They predict folds forming under conditions of plane strain and simple shear. That is, they assume what is pretty much the simplest strain regime. This is unlikely to be correct.

3. All the above mechanisms predict that the intermediate strain axis λ2 lies par­allel to the fold axis. That is, everywhere we look, we should see stretching lin­eations perpendicular to the fold axis. Far more common, especially in high grade zones, is for the stretching lineation to be parallel to the fold hinge.

4. One of the most common secondary structures associated with folds is the presence of an axial planar foliation or axial planar cleavage. This is devel­oped in rocks of a wide variety of rock types, fold geometries and metamor­phic grades: from disjunctive fracture cleavages at low grade to high grade fabrics formed from crystal­plastic deformation. Only passive flow folds (in­directly) accounts for the role of foliation development during deformation – but in this case, the interpretation that the axial planar foliation is a slip sur­face is questionable.

Ideal, theoretical models also ignore what seems to be a fundamental property of folds: folds form in layered sequences, with competency contrasts between the layers. In fact, competency contrasts appear to be necessary for folds to form (this is certainly true for metamorphic rocks.

Wavelength of folds: a function of layer thickness

Theoretical and experimental work on folding of viscous materials produces results that accord with that last observation: that fold wavelength is a function of layer thickness, as well as competency contrast. For a relatively strong layer bounded above and below by weak material, the dominant wavelength of folds is predicted to be �

η1λi = 2πh 3 1/6

η2

where h is the thickness of the layer, and η is the viscosity of each layer. Note how the wavelength is a directly related to both layer thickness and viscosity contrast.

Wavelength of folds: multiple layers

For a stack of multiple layers – say alternating layers of viscosities η1 and η2, the number of layers changes the details of the relationship described in the equation above: �

λi = 2πh 3 1 η1

6n2 η2

where n is the number of layers. That is to say, there is a dependence on the number of layers. But the relative viscosities and layer thickness still exert a first order control.

Complications arise when you have a number of competent layers, with different characteristic wavelengths (that is, the wavelength that they would fold at if a layer of such a thickness and viscosity were to fold alone), being folded together. It turns

11

Figure 1.6: Combining a buckle fold with homogeneous shortening to generate more realistic geometries and strain distributions

out that if the layers are sufficiently far apart, each will behave independently of the others, each folding at its own characteristic wavelength, and so this would produce disharmonic folding. “Sufficiently far” in this case means something like 1/4 to 1/2 the characteristic wavelength of that layer.

If the layers are closer together, though, there is interference between them. The thinner layer will often show two wavelengths of folding: the larger wavelength con­trolled by an adjacent, thicker layer and its own characteristic wavelength superim­posed upon it.

Polygenetic models: the quest for more realistic fold geometries

More realistic fold geometries can be produced by combining various mechanisms with each other, or with an additional amount of homogeneous shortening. Figure 1.6 shows the distribution of strain in a slab folded so that it has a neutral surface (a); then subject to additional homogeneous shortening of 20% (b) and 50 % (c). (See discussion and figures, pp. 243–245, figs. 12.12 and 12.13).

1.6.5 Numerical models

Figure 1.7 shows some results from numerical simulation of fold formation. This is an early simulation of the deformation of a single, viscous layer enclosed in a ho­mogenous, viscous medium. Thin lines are drawn perpendicular to the principal

12

axis of shortening at each point. Three cases are shown, for different viscosity con­trasts. Note that as viscosity contrast decreases, the importance of folding (versus homogeneous pure shear shortening) decreases. Note also the deflection of princi­pal axes of strain near the boundary between the two layers.

1.7 Other kinds of folds: kink folds, chevron folds, di­apirs

Kink folds are asymmetric folds with straight limbs and sharp hinges. They occur as a short limb connecting two long limbs. Kink geometries are often adopted for idealizing fold geometries when constructing cross sections of fold and thrust belts. But beyond just an idealization, they actually exist! Various models of kink band formation include:

1. Migration of the kink band boundary into undeformed material

2. Kink band boundaries don’t migrate, but instead mark the boundaries of a small shear zone within which layers rotate and shear past one another.

Chevron folds are symmetric folds with completely straight limbs and very tight fold hinges. One model for chevron fold formation has them being the result of interference of growing kink bands. Alternatively, chevron folds can form by flexural slip folding with strain concentrated in the hinge. If the layers are of some finite thickness, voids will form in the hinge zones. These voids are places that you might expect to be a locus of veining and fluid migration.

Diapirs are antiformal domes that appear to have been intruded vertically into their host rock. Salt and pressurized shales are common materials involved in di­apiric emplacement. On larger scales, domal culminations of high grade gneisses are common features of the internal parts of mountain belts. Commonly, diapir em­placement is thought to require density inversion, a view motivated by the obser­vation that salt is typically less dense than compacted, lithified sediments1 and by early modeling of diapiric and domal culminations using materials with viscous rhe­ology. Leaving aside considerations of buoyancy for the time being, material within a diapir is necessarily weak, and therefore flows according to pressure gradients. For various reasons, pressures on the flanks of diapirs are higher than in the center of the diapir, so material moves laterally into the diapir, and then up into the diapir. Diapir­like flow of weak material is also noted filling the cores of anticlines in de­tachment fold settings.

1.8 Review questions

1. Draw a flexural slip and a passive flow fold. Draw the expected sense of shear along the folded layers

1In fact, even though many people still labour under this assumption, modern modelling and inves­tigation of salt tectonics environments has conclusively shown that density inversion is not needed for diapirism, nor is plausibly inferrable in many cases

13

Figure 1.7: Numerical models of fold simulation. HMW(1982)

14

2. Consider a layer with a pre­existing lineation that acts as a passive marker. What happens to the lineation when the layer is folded according to the three main models of fold formation. The key will be to determine whether the fold­ing will cause rotation of the passive marker. That is: is the layer with the lin­eation a circular section of the strain ellipsoid (no rotation), or not?

3. In an perfect, upright flexural slip fold, what happens to the fold with depth and why?

15

16

Chapter 2

Fabrics and deformation mechanisms

2.1 Fabrics

A fabric is a particular arrangement of component features (called fabric elements) in a rock. This arrangement is usually regular: for instance the horizontal layers of sedimentary rocks are a kind of fabric, but we also refer to random fabric, which is by definition not regular. We distinguish between primary and secondary (tec­tonic) fabrics: the first formed with the rock and includes, igneous flow lineation, sedimentary lamination and bedding and flow foliation. Secondary fabrics are ones that have formed as a result of deformation. Another key distinction is between penetrative or continuous fabric – which is observable at all scales, as opposed to non­penetrative or spaced fabric, where there is obvious spacing between the fabric elements (eg. solution cleavage seams or fractures). Finally we distinguish between planar fabrics – called foliation – and linear fabrics (lineations).

2.1.1 Foliations

A foliation is the general term for any kind of planar fabric, including: bedding, lamination, flow foliation, cleavage, schistosity and gneissosity. Many foliations are products of specific tectonic environments, including a particular orientation of principal strain axes. Depending on the rock, tectonic foliations may include spaced fracture cleavage, slaty cleavage, crenulation cleavage, schistosity, gneissosity.

Rocks can record many generations of fabrics, from the primary (eg. bedding) to fabrics associated with many generations of tectonic episodes. The relative timing of these is worked out using cross­cutting relations. The physical appearance of the fabrics ultimately reflects the materials, metamorphic grade and strain­rates and so careful observation and analysis of fabrics is crucial to teasing out the tectonic history of a deformed rock.

Cleavage

Cleavage is a secondary (tectonic) fabric which imparts on the rock a tendency to part or split along it. The classic example is slate, where the cleavage is so perfect that the rock is easily quarried into thin, flat slabs for pool tables and roofing mate­rials.

17

Disjunctive cleavage A type of spaced cleavage defined by an array of more or less parallel fabric domains (called cleavage domains). Within each fabric domain, there is typically evidence of pressure solution (hence this is often called solution cleavage or stylolitic cleavage). These domains are separated by intervals called microlithons. The spacing of the cleavage domains (and hence the width of the mi­crolithons) is quite variable, but commonly observed on a cm­scale.

Pencil cleavage This is characterized by the rock breaking into elongate, pencil­like shards. Pencil cleavage appears to result from the intersection of two spaced cleavages, one of which forms due to the primary preferred orientation of clays (im­parted during sedimentation and compaction), the second may form axial planar to folds or perpendicular to layer­parallel shortening. This causes detrital clays to fold and rotate, fine grained soluble materials to undergo pressure solution and new clay minerals to crystallize. When the original sedimentary parting and the new (spaced, and probably pretty weakly developed) cleavage intersect at fairly high angles, pen­cil cleavage results. With more strain, it is possible to get the complete erasure of the original parting and production of slaty cleavage.

Slaty cleavage Slaty cleavage forms by the same processes as outlined above: ro­tation and folding of detrital clays, pressure solution and recrystallization of clay minerals. The difference is that the cleavage domains are so tightly spaced that the fabric is completely penetrative and continuous at all but high power magnification. The formation of slaty cleavage is coincident with the onset of low grade metamor­phism, particularly the transition from smectite (a poorly ordered clay) to illite (a much more ordered clay). This transition is temperature sensitive and measurable by XRD.

Phyllitic cleavage and schistosity Phyllitic cleavage is characteristic of low grade (low greenschis). It is produced by the preferred alignment of clay and mica miner­als. At the lowest metamorphic grade these will be illite, transitioning to white mica (sericite / muscovite) and chlorite at higher grades. If the cleavage is formed synde­formationally, the phyllosilicate minerals grow with a strong preferred orientation. At the hand sample scale, the preferred orientation of the phyllosilicates imparts a very strong parting or cleavage, although the phyllosilicates are very fine grained and individual minerals are usually not visible without magnification. At higher meta­morphic grades, sericite­chlorite assemblages are replaced with coarser muscovite and / or biotite, and the individual mica grains are visible under close inspection. At this point, the rock is referred to as a schist. Schistose rocks are usually named ac­cording to the assemblage of metamorphic minerals contained (biotite, muscovite, garnet, staurolite, kyanite, etc.).

Crenulation cleavage If rocks containing an early, closely spaced cleavage (this can be slaty cleavage, phyllitic cleavage or schistocity) are shortened in a direction at a low angle to the original fabric, the older fabric folds at a very small scale. This produces a characteristic "wrinkled" appearance, somewhat like the baffles in an accordion. When a very regularly closely spaced cleavage is crenulated, the spac­ing of the small folds is very uniform, and fold hinges and fold limbs line up. This, and possibly pressure solution or shear thinning in the limbs defines a new folia­tion, which is called a crenulation cleavage. Crenulation cleavage can be symmetric or asymmetric. Eventually, pressure solution (which will tend to transport material from the limbs to the hinges) and shear of the limbs can lead to an almost complete

18

obliteration of the early fabric. This process is called transposition, and the new fabric is called a transposition foliation.

gneissic layering . Gneissic layering is a coarse (mm to tens of cm scale) composi­tional layering in metamorphic rocks. This compositional banding can be produced a variety of different ways. Primary layering is inherited from the original sedimen­tary layering and compositional differences of a sedimentary protolith. Transposi­tion layering is produced by isoclinal folding and shear of fold limbs of the original layering. Compositional layering can also be produced from metamorphic differen­tiation, or the injection of melts either along the layering or rotated through shear into the foliation.

Foliations and folds

Planar fabrics can form in a shear zone, but often form in a specific geometrical relationship to a series of folds. Specifically, in folded rocks, planar fabrics are com­monly parallel to the axial plane of the fold. You might find it useful to think of exam­ples where this might not be the case (consider an interbedded quartzite and shale being folded). Foliations in folded terrains can be extremely useful: if they are truly parallel (or close to parallel) to the axial plane, the intersection of the layer being folded and the foliation will result in a line that is parallel to the fold axis. Moreover, there is a specific geometrical relationship between bedding and axial planar folia­tion that allows you to infer the position of fold closures. These are key relationships that are the bread and butter of field geology in folded terrains. See TM, section 13.5. In areas of multiple deformations, you can have synformal anticlines and antiformal synclines and so on. The key to working this out is to be able to establish “younging” (or “way up”), and the geometrical relationship between the limb of a structure and an axial planar fabric.

Once a foliation is formed, it can be modified by further deformation. Some processes: rotation during shear (see fig 14.4 for different mechanisms of rotation), small scale folding (crenulation), recrystallisation. See TM chapters 13 and 14 for material on the formation and modification of planar and linear fabrics (they are short chapters). The preservation of earlier generations of fabrics within porphy­roblasts (eg. garnet) and within low strain zones (the hinges of later folds) is ex­ploited by geologists who then label them S1, S2, S3 etc... Of particular importance is the concept of transposition foliation. Such a fabric is a composite fabric: mul­tiple generations of planar features are found to be parallel. The common case is to find compositional layering (e.g. bedding, called S0) parallel to a tectonic fabric. How does this happen? (Chapter 13).

2.1.2 Lineations

A lineation is the generic term for a linear fabric in a rock. An intersection lineation is formed by the intersection of two planes. Typically, an intersection lineation is formed by the intersection of a layer being folded and a cleavage parallel to the axial plane of that fold. In this case, the intersection lineation will form parallel to the fold hinge, so observation of the orientation of the intersection lineation can yield very useful information about the geometry of the folds in a region. A mineral lineation is defined by the preferred orientation of minerals, especially minerals that have a well­defined long axis, such as amphiboles. Since the orientation of metamorphic minerals often reflects the state of strain, it is very common to find minerals ori­ented parallel to a principal strain axis especially parallel to fold hinges. Stretching

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lineations are defined by the orientation of parts of the rock that have been strained such that they have a well­defined long axis. For example, cobbles in a conglomerate that have suffered strain will often be observed to a well­defined preferred orienta­tion of their long axis. At a smaller scale, individual minerals – such as quartz grains – can also be stretched, and form a lineation in the rock. Stretching lineations are usually interpretable as being parallel to the maximum principal strain axis.

While many lineations are parallel to either regional fold axis orientations or the maximum extension direction, this is not always the case. A boudin line lineation is defined by the stretching of a relatively competent layer in a ductile rock. Instead of homogeneously stretching, this layer is likely to break into swells and necks. With further stretching the pinch and swell structure gives way to a broken line of typically elliptical fragments, called boudins due to their supposed resemblance to sausages. The alignment of the boudins defines a linear fabric element, but this lineation is, in fact, perpendicular to the stretching direction.

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Fabrics, folds and deformation mechanisms

Or:

“What went down in the last month?”

Fall 2005

The purpose of these notes is to give you an overview of the material that will be the focus of the final exam.

This, and the review questions, should help you focus your attention. Also, I intend to have the topic by topic

course notes out, but this should help you for the time being.

1 Folds

1.1 Descriptive terminology

There’s a considerable amount of descriptive terminology for folds. While I doubt that any question on the exam

will directly ask you to, for instance, define what it means for a fold to be non­cylindrical, it would be useful for

you to make sure that you are familiar with the descriptive terminology for folds. The introductory material on

the lab on folds is a good starting point; chapter 11 in Twiss and Moores covers this in greater depth.

1.2 Kinematic models of folding

Be familiar with the general ideas behind flexural slip, buckling and passive flow models and the effects of homo­

geneous shortening superimposed on these models. Make sure you can draw the strain ellipses in the different

regions of a folded layer for each of these models. Consider the behaviour of a linear marker thatŠs folded in

these ways. Pages 315 – 320 in the text are useful in this regard. Also to review is the control on wavelength of

folding by thickness and viscosity contrasts and the behaviour of folded multi­layers. Don’t worry about memo­

rizing the exact equations, but know qualitatively the first­order controls. These relationships, of course, assume

a specific kinematic model.

1.3 Super­posed folding

The patterns of super­posed folding are typically categorised into Types I, II, and III. These different patterns

reflect the relative orientations of the fold hinges and axial surfaces of the first and second folds. Make sure you

understand how you can generate these patterns.

2 Fabrics

2.1 Foliations

Planar fabrics can form in a shear zone, or may form in a specific geometrical relationship to a series of folds.

Depending on the rock, foliations may include spaced fracture cleavage, slaty cleavage, crenulation cleavage,

1

schistosity, gneissosity. (Other planar fabrics include flow foliation, which may not be related to tectonic defor­

mation per se). In folded rocks, planar fabrics are commonly parallel to the axial plane of the fold. You might

find it useful to think of examples where this might not be the case (consider an interbedded quartzite and shale

being folded). Foliations in folded terrains can be extremely useful: if they are truly parallel (or close to parallel)

to the axial plane, the intersection of the layer being folded and the foliation will result in a line that is parallel to

the fold axis. Moreover, there is a specific geometrical relationship between bedding and axial planar foliation

that allows you to infer the position of fold closures. These are key relationships that are the bread and butter

of field geology in folded terrains. See section 13.5. Once a foliation is formed, it can be modified by further de­

formation. Some processes: rotation during shear (see fig 14.4 for different mechanisms of rotation), small scale

folding (crenulation), recrystallisation. See chapters 13 and 14 for material on the formation and modification

of planar and linear fabrics (they are short chapters). The preservation of earlier generations of fabrics within

porphyroblasts (eg. garnet) and within low strain zones (the hinges of later folds) is exploited by geologists who

then label them S1, S2, S3 etc... Of particular importance is the concept of transposition foliation. Such a fabric is

a composite fabric: multiple generations of planar features are found to be parallel. The common case is to find

compositional layering (e.g. bedding, called S0) parallel to a tectonic fabric. How does this happen? (Chapter

13). In areas of multiple deformations, you can have synformal anticlines and antiformal synclines and so on.

The key to working this out is to be able to establish “younging” (or “way up”), and the geometrical relationship

between the limb of a structure and an axial planar fabric. Study and be able to reproduce figure 13.15 – it is a

favorite of Clark’s.

2.2 Lineations

Distinguish between intersection lineations, mineral lineations and stretching lineations. Be able to describe

a few examples of each. Again, chapters 13 is the key one for that. Note that some mineral lineations will be

parallel to a principal strain axis, but some will not. Stretching lineations are usually indicative of the maximum

stretching orientation. Intersection lineations often reflect a fold geometry. Explain the relationship between a

boudin line lineation and the maximum extension direction.

3 Deformation mechanisms

What’s the difference between brittle and plastic deformation? Is plastic the same as ductile deformation? Plastic

deformation mechanisms include twin glide, dislocation glide, dislocation creep, recovery, static recrystallisa­

tion. Dislocation creep, in common usage, refers to a set of processes: dislocation glide, recovery via subgrain

rotation and grain boundary migration. You should review chapter 19 on this, or Passchier and Trouw. Section

19.7 and onwards deals with issues surrounding the formation of crystallographic preferred orientation. Don’t

worry too much about this, but do know the difference between fabric as defined by the preferred orientation of

minerals (mica flakes making a foliation, or stretched quartz grains defining a stretching lineation) and a crys­

tallographic preferred orientation. An important thing to note about plastic deformation mechanisms: they are

sensitive to temperature, strain­rate, grain size, the presence of fluids but more than that, the specific behaviour

is material dependant. That is, twin gliding occurs in calcite but not quartz, dislocation glide occurs in marbles

and limestones at low temperatures, quartz at moderate temperatures and so on. Some minerals, such as garnet,

never deform plastically. This means that for any given polymineralic rock, the overall bulk strain is going to be

accommodated in a heterogeneous fashion in a rock.

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12.113 Structural Geology

Part 3: Normal faults and extensionaltectonics

Fall 2005

Contents

1 Reading assignment 1

2 Growth strata 1

3 Models of extensional faults 2 3.1 Listric faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Planar, rotating fault arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.3 Stratigraphic signature of normal faults and extension . . . . . . . . . . 2 3.4 Core complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Slides 7

1 Reading assignment

Read Chapter 5.

2 Growth strata

Although not particular to normal faults, relative uplift and subsidence on either side of a surface breaking fault leads to predictable patterns of erosion and sedi­mentation. Sediments will fill the available space created by slip on a fault. Not only do the characteristic patterns of stratal thickening or thinning tell you about the

1

Figure 1: Model for a simple, planar fault

style of faulting, but by dating the sediments, you can tell the age of the fault (since sediments were deposited during faulting) as well as the slip rates on the fault.

3 Models of extensional faults

The simplest model of a normal fault is a planar fault that does not change its dip with depth. Such a fault does not accommodate much extension. (Figure 1)

3.1 Listric faults

A listric fault is a fault which shallows with depth. Compared to a simple planar model, such a fault accommodates a considerably greater amount of extension for the same amount of slip. Characteristics of listric faults are that, in order to maintain geometric compatibility, beds in the hanging wall have to rotate and dip towards the fault. Commonly, listric faults involve a number of en echelon faults that sole into a low­angle master detachment. (Figure 2)

3.2 Planar, rotating fault arrays

An array of parallel, normal faults slipping on a sub­horizontal detachment will ro­tate with fault slip, much like books on a book shelf. As they rotate, the dip of the faults will get shallower, and the beds between the faults will also rotate. Again, the amount of extension can be calculated for a given angle between beds and the fault, and the total amount of slip on the fault.

Sometimes, normal fault systems are, in fact, best described as a combination of listric faults with rotating, planar faults bounding rotating blocks in the hanging wall. The "master fault" – called a detachment fault is a listric fault, with a set of rotating planar faults in its hanging­wall or upper plate. The steep part of the de­tachment is sometimes called the break­away fault. Figure 4

3.3 Stratigraphic signature of normal faults and extension

If, prior to extensional faulting, stratigraphy was undeformed, then the characteris­tic of normal faulting is that younger rocks will be juxtaposed over older rocks, but

2

Figure 2: Geometry of listric normal faults. The bottom figure shows the geometric relations for calculating the amount of extension, given the angle between beds and

3

the fault, and the total fault slip

Figure 3: A system of rotating, planar faults. The bottom figure shows how this ge­ometry compares to simple listric normal faults in terms of the amount of extension that can be produced

4

Figure 4: Listric fault, with rotating upper plate blocks and breakaway fault

Figure 5: Stratigraphic signature of extensional tectonics where low angle faults are present

5

Figure 6: Model for core complex formation along a low angle detachment

with "missing section" at the fault.Particularly where the normal faults are low­angle, this relationship is often mis­interpreted. Prior to the recognition that low angle normal faults are widespreadfeatures of the extended crust, they were often mapped as thrusts (which makes nosense) or unconformities. Where low angle faults are common, a "stratigraphic sec­tion" will show many apparent gaps. Figure 5

3.4 Core complexes

A common feature of highly extended terrains is that large amounts of extension on relatively low angle normal faults juxtaposes rocks of high metamorphic grade against unmetamorphosed rocks and even surficial deposits. The exposure of high grade rocks typically domal, or antiformal geometry, with normal faults completely bounding the high­grade rocks. Rocks in the immediate footwall of the bounding normal faults show a characteristic series of fabrics: since rocks are brought up from depth, fabrics reflecting progressively colder and more brittle deformational envi­ronments are superimposed upon one another. So, walking from the interior of the core complex out towards the low­grade upper plate rocks ductile fabrics and my­lonites will be overprinted by brittle­ductile transitional fabrics, which themselves are often overprinted by thick gouge and breccia and then a discrete, brittle fault.

The conceptual model of core complexes – high grade rocks brought to the sur­face along low angle detachments has some important implications for how exten­sion is accommodated in the lower crust or mantle. In figure 6, large amounts of slip on a detachment obviously creates a great deal of "space" in the lower crust. The question then is: does the detachment offset the Moho, and accommodation is then

6

Figure 7: Two models for accommodating large amounts of extension in the upper crust. From Block and Royden 1984

taken up by flow of mantle rocks. Or does the detachment sole into the middle or lower crust, with lower crustal flow solving the space problem. See figure 7.

4 Slides

7

Figure 8: Rocks above a low angle detachment are broken up by normal faults that sole into the main detachment. The yellow unit is the same in all the outcrops

8

Figure 9: Range scale tilted normal fault block

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Figure 10: Series of planar rotated normal faults. Scale is about 2 meters from top to bottom of the slide. This faulting has accommodated about 60% extension.

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Figure 11: Listric fault

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Figure 12: Listric fault merging into a detachment surface at the feet of geologist.

Figure 13: Low angle normal fault. Some people still don’t believe these are real.

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Figure 14: Low angle normal fault; rocks in the hanging wall of the fault are late Quaternary. There are folks who believe that low angle faults do exist, but believe they were rotated from an initially steep attitude. This a more reasonable attitude than total denial, to be sure.

Figure 15: The Whipple mountains are one of the archetypical core complexes of the western U.S. extensional provinces. This is a cross­section from the work of Lister and Davis. Note the multiple rotations.

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Figure 16: A view looking east at the northern Panamint Rnage: lower plate high grade rocks to the left, upper plate rocks to the right. The surface sloping to the left in the slide is the detachment fault surface.

14

Figure 17: View to the east along the Whipple detachment. Dark rocks are Tertiary aged sediments and volcanic rocks in tilted and rotated blocks. Pale rocks are the lower plate mylonites

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Figure 18: Spot the low angle detachment

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Figure 19: Detachment surface in the Clark mountains, California

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Figure 20: High grade fabrics representative of extensional strain at deep crustal lev­els. These rocks are lower plate rocks below a detachment in Death Valley.

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Figure 21: High grade fabrics with lower grade (i.e. progressively more brittle) defor­mation overprinted on them.

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Figure 22: High grade extensional shear fabrics

Figure 23: C/S fabric, ductile deformation

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Figure 24: Low grade (brittle) gouge and breccia; lower plate of a detachment fault.

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