3 strain with figures
TRANSCRIPT
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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
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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
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Map of the
conglomerate
layer
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Conglomerate
in a
constrictionfield
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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
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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
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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Elliptical reduction spots in a slate from North Wales. The spots
were originally round in section and are deformed to ellipses.
(photo: Rob Knipe)
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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Reduction spots in Welsh slate. The green spots
are reduced, and used to be spherical before
deformation. Now they are pancakes.
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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
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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
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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
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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
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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
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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