3 Strain With Figures

7/28/2019 3 Strain With Figures

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Fossen Chapter 3

Strain in Rocks


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Deformed Bygdin Conglomerate, with quartzite pebbles

and quartzite matrix, Norway. Similar pebble and

matrix compositions minimize strain partitioning and

enhance strain estimates


3/41

Block diagrams showing sections through the

strain ellipsoid, with Flinn diagram

Direction of instantaneous stretching axes and fields of instantaneouscontraction (black) and extension (white) for dextral simple shear


4/41

Map of the

conglomerate

layer


5/41

Conglomerate

in a

constrictionfield


6/41

Part of a stretched belemnite boudins with quartz and calcite

infill. The space between the broken pieces of the belemnite are

filled with pricipitated material. The more translucent materialin the middle of the gaps is quartz, the material closer to the

pieces is calcite. Photo from the root zone of the Morcles nappe

in the Rhone valley, Switzerland by Martin Casey

http://www.see.leeds.ac.uk/structure/strain/gallery/belpart.html
http://www.see.leeds.ac.uk/structure/strain/gallery/belpart.htmlhttp://www.see.leeds.ac.uk/structure/strain/gallery/belpart.htmlhttp://www.see.leeds.ac.uk/structure/strain/gallery/belpart.htmlhttp://www.see.leeds.ac.uk/structure/strain/gallery/belpart.html


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Elongated belemnites in Jurassic limestone in the

Swiss Alps. The upper one has enjoyed sinistral

shear compared to the lower one which has

stretched


8/41

Stretched belemnite. Stretching in the upper right, lower left

direction has broken and extended the fossil. The gaps between

the pieces are filled with a precipitate. Photo from the root zone

of the Morcles nappe, Rhone valley, Switzerland by Martin Casey



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Elliptical reduction spots in a slate from North Wales. The spots

were originally round in section and are deformed to ellipses.

(photo: Rob Knipe)



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Reduction spots in Welsh slate. The green spots

are reduced, and used to be spherical before

deformation. Now they are pancakes.


11/41

Deformed Ordovician Pahoe-hoe lava (sketched in

1880s). The ellipses used to be more circular

originally. Can use Rf/, center-to-center, or Fry

method techniques.


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Measurement of Strain

The simplest case: Originally circular objects

When markers are available that areassumed to have been perfectly circular and

to have deformed homogeneously, the

measurement of a single marker defines thestrain ellipse


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Direct Measurement of Stretches

Sometimes objects give us the opportunity to

directly measure extension

Examples:

Boudinaged burrow

Boudinaged tourmaline

Boudinaged belemnites

Under these circumstances, we can fit an ellipse

graphically through lines, or we can analytically

find the strain tensor from three stretches


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Direct Measurement of Shear Strain

Bilaterally symmetrical fossils are anexample of a marker that readily gives shear

strain

Since shear strain is zero along the principal

strain axes, inspection of enough distorted

fossils (e.g. brachiopods, trilobites) can

allow us to find the directions!


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Wellman's Method Relies on a theorem in geometry that says that if

two chords together cover 180 of a circle, the anglebetween them is 90

In Wellmans method, we draw an arbitrarydiameter of the strain ellipse

Then we take pairs of lines that were originally at90 and draw them through the two ends of thediameter

The pairs of lines intersect on the edge of the strainellipse


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Wellmans Method Uses deformed variably oriented lines which were originally

perpendicular (e.g., hinge and median lines of brachiopods, trilobites)

Measurement: Trace the deformed lines on a the image with a pencil

Draw a reference line between two arbitrary points (A and B)

Put A at the intersection of the two originally perpendicular lines

on a fossil, and draw the two lines (e.g., hinge and median lines) While line AB is un-rotated, bring B where A was, and repeat

Place dots where the pairs of deformed lines cross

Do this for all fossils, while AB is in the same constant orientation

For each fossil, the pairs of lines intersect on the edge of the strainellipse

Draw a smooth ellipse through the dots. This is the strain ellipse;

measure its long and short semi-axis.

Determine the strain ratio, Rs and orientation of S1 relative to AB


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Wellman method

used for deformed

trilobites and

brachiopods with

two originally

perpendicular lines


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Breddin Method Requires presence of many fossils

Draw a reference line on the image of fossils

Measure the angle () between the hingeline of the fossil w.r.tthe reference line (e.g., trace of foliation)

Do this for all fossils (see the angle on next slide)

Measure the angular shear () for all fossils (e.g., the anglebetween deformed hinge and median lines)

Measure the shear strain () Plot vs. Compare the plot (by transferring to a an overlay) with a

standard Breddin Graph centered at =0 and shows the Rscontours

The fossils with the =0 give the orientation of the S1 axis See next slide


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Data from two

slides before,

plotted onBreddin graph.

Date plot on the

curve for Rs=2.5


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Straight lines are

drawn betweenneighboring grain

centers.

The line lengths (d)

are plotted vs. the

angle () that thelines make with the

reference line.

The max (X) and min

(Y), give the Rs = X/Y

The center-to-center method


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Center to Center Method

Ramsay, J. G., and Huber, M. I., 1983

Modern Structural Geology. Volume 1: Strain Analysis


22/41

Frys Method

Depends on objects that originally were

clustered with a relatively uniform inter-object distance.

After deformation the distribution is non-uniform

Extension increases the distance betweenobjects; shortening reduces the distance

Maximum and minimum distances will be alongS1 and S2, respectively


23/41

From:

http://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdf
http://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdfhttp://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdfhttp://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdfhttp://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdf


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Undeformed and deformed oolitic

limestone

h d


25/41

Fry Method Is a variant of the center-to-center method

Could be used for ooids that may dissolve, and phenocrysts in

igneous and metamorphic rocks. Measures the closeness of grains

Measurement:

On a transparent overlay make a dot at the center of each grain;

number the grains (1, 2, 3, ., ., n)

Draw an arbitrary reference line or draw a box around the image

Have another overlay, and mark a dot at its center

Put the dot on grain 1, trace the reference line, and mark all the

other points with dots (label them with numbers)

While the top overlay is kept in the same orientation, put the dot

on grain number 2, and mark other grains with dots

Repeat for all grains

An empty ellipse, or an elliptical area full of points appears; this is

the strain ellipse

Determine the strain ratio (Rs) and the orientations of S1 and S3


26/41

a. Grain centers are transferred to an overlay

b. A central point () is defined and moved on

grain 1, while copying the other points while

overlays orientation is kept constant

c. An empty ellipse develops with gives the strain

ellipse.

Fry Method


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Pros:

Frys Method is fast and easy, and can be used onrocks that have pressure solution along grain

boundaries, with some original material lost

Rocks can be sandstone, oolitic limestone, and

conglomerate

Cons:

The method requires marking many points (>25)

The estimation of the strain ellipses eccentricity issubjective and inaccurate

If grains had an original preferred orientation, this

method cannot be used

Fry Method


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Rf/ Method In many cases originally, roughly circular markers

have variations in shape that are random, e.g., grains in sandstone or conglomerate

In this case the final ratio Rfof any one grain is a

function of the original ratio Ri and the strain ratio Rs

Rf max = Rs.RiRf min = Ri/Rs


29/41

Rf/ Method Could be used for grains with initial spherical or non-

spherical shapes (i.e., initial grain ratio ofRi=1 or Ri >1) Measurement:

Measure the long and short axes of each grain on the

deformed rock, or its image

Find its final ratio (Rf)

Find the angle () between the long axis of each grainand a reference line

The reference line could be the trace of the foliationor bedding

Plot the log ofRfvs. Note the pattern (e.g., drop- or onion-shaped)


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http://a1-structural-geology-software.com/The_rf_phi__prog_page.html
http://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.html


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Rf/ contd Rf max = Rs.Ri

Rf min = Ri/Rs

If Rs < Ri (strain ellipticity is < the initial grain ellipticity)

Rs = (Rf max/Rf min)Rimax = (Rfmax Rf min)

If Rs > Ri (strain ellipticity is > the initial grain ellipticity)

Rs = (Rfmax Rf min)Ri max = (Rf max/Rf min)

The direction of the maximum is the orientation of S1


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http://a1-structural-geology-software.com/The_rf_phi__prog_page.html
http://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.htmlhttp://a1-structural-geology-software.com/The_rf_phi__prog_page.html


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Mohr Circle two deformed brachiopods

This method is good when there are only few fossils available

Step 1. Measure the angle between the hinge lines of the twobrachiopods ()

Measure the angular shear (A and B) for each fossil

Step 2. Plot a circle on tracing paper of any size. Draw two

radii (A and B), with an angle of 2 Draw (on graph paper) the Coordinates of the Mohr Circle

( vs. )

Step 3. Draw (on graph paper) two lines from the origin

inclined at angles to the horizontal axis. Step 4. Overlay the tracing paper on the graph paper, and put

the center of the circle on the x-axis. Rotate the tracing circle

until each of the radii (on graph paper) intersects its

corresponding line (on tracing) that emanates from the origin


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Note that the sense (ccw or cw) of the angles are not correctly

plotted. The senses of must be the same in the real world and the

Mohr circle world!

Tracing paper

Graph paper

Tracing paper overlaid

on graph paper

photograph


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Deformed Trilobite

http://courses.eas.ualberta.ca/eas421/lecturepages/strain.html

h d f d b h d
http://courses.eas.ualberta.ca/eas421/lecturepages/strain.htmlhttp://courses.eas.ualberta.ca/eas421/lecturepages/strain.html


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Three deformed brachiopods Measure the angle between fossils A and B (), and B and C ()

Measure the angular shear for each fossil (A, B, C)

Set up the coordinate system ( vs. ) with arbitrary scale

Draw three lines of any length at A, B, C from the origin

Draw a circle of any size on a tracing paper

Draw angles 2 (between A & B) and 2 (between B & C) from the

center of the circle. Mark points A, B, & C on the circle

Move the center of the circle (tracing paper) along the x-axis, and

rotate it until lines A, B, C intersect their corresponding points A, B,

and C on the circle. Fix the tracing paper with tape.

Read the values for and 1 and 3, and S1 and S3(scale does not mattersince we want to get Rs = S1/S3

Read the amount and sense of the angles 2A, 2B,or 2C

Draw 1 from say fossil A on the rock, in the same sense (e.g., cw or

ccw) as it is for the 2 in the Mohr circle


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A B

C

A B

C

22

31

A

B

C

cw

cw

cw


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Three section provide data for 3D strain


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Strain obtained from deformed conglomerate

plotted on Flinn diagram (Norway)


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Moderatelydeformed

Neoproterozoic

quartzconglomerate.

Strain exposed in

sections parallel tothe principal planes

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3 Strain With Figures