Fossen – Chapter 3

1
Fossen Chapter 3

• Strain in Rocks

2
Deformed Bygdin Conglomerate, with quartzite
pebbles and quartzite matrix, Norway. Similar
pebble and matrix compositions minimize strain
partitioning and enhance strain estimates
3
Block diagrams showing sections through the
strain ellipsoid, with Flinn diagram
Direction of instantaneous stretching axes and
fields of instantaneous contraction (black) and
extension (white) for dextral simple shear
4
Map of the conglomerate layer
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Conglomerate in a constriction field
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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
material in 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/galler
y/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
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/galler
y/belpart.html
9
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/galler
y/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 are assumed to
  have been perfectly circular and to have deformed
  homogeneously, the measurement of a single marker
  defines the strain 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 an example 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!

15
Wellman's Method

• Relies on a theorem in geometry that says that if
  two chords together cover 180 of a circle, the
  angle between them is 90
• In Wellmans method, we draw an arbitrary diameter
  of the strain ellipse
• Then we take pairs of lines that were originally
  at 90 and draw them through the two ends of the
  diameter
• The pairs of lines intersect on the edge of the
  strain ellipse

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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 strain ellipse
• 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.t the 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 angle between 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 Rs contours
• The fossils with the ?0 give the orientation of
  the S1 axis
• See next slide

19
Data from two slides before, plotted on Breddin
graph. Date plot on the curve for Rs2.5
20
Straight lines are drawn between neighboring
grain centers. The line lengths (d) are
plotted vs. the angle (?) that the lines 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 between objects
  shortening reduces the distance
• Maximum and minimum distances will be along S1
  and S2, respectively

23
From http//seismo.berkeley.edu/burgmann/EPS116/
labs/lab8_strain/lab8_2009.pdf
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Undeformed and deformed oolitic limestone
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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 constantc. An
empty ellipse develops with gives the strain
ellipse.
Fry Method
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Fry Method

• Pros
• Frys Method is fast and easy, and can be used on
  rocks 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 (gt25)
• The estimation of the strain ellipses
  eccentricity is subjective and inaccurate
• If grains had an original preferred orientation,
  this method cannot be used

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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 Rf of any one grain
  is a function of the original ratio Ri and the
  strain ratio Rs
• Rf max Rs.Ri
• Rf 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 of Ri 1 or Ri gt1)
• 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
  grain and a reference line
• The reference line could be the trace of the
  foliation or bedding
• Plot the log of Rf vs. ?
• Note the pattern (e.g., drop- or onion-shaped)

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http//a1-structural-geology-software.com/The_rf_p
hi__prog_page.html
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Rf/? contd

• Rf max Rs.Ri
• Rf min Ri/Rs
• If Rs lt Ri (strain ellipticity is lt the initial
  grain ellipticity)
• Rs ?(Rf max/Rf min)
• Ri max ?(Rf max Rf min)
• If Rs gt Ri (strain ellipticity is gt the initial
  grain ellipticity)
• Rs ?(Rf max Rf min)
• Ri max ?(Rf max/Rf min)
• The direction of the maximum is the orientation
  of S1

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A pure shear with Rs ??1/??3 1.5 is applied,
(i.e., ??1 1.5 and ??3 1) ??1 0
?1.5 0 0 ??3 or 0
1/?1.5where ??3 1/??1followed by pure
shear of Rs ??1/??3 3(i.e., ??1 3 and ??3
1) Notice the coaxial strain (see strain
ellipses ? is around 0).
Rf/? Method Grains had constant Ri La/Sa The
plot on the right shows Ri2.
Rs ??1/??3 S1/S3
Undeformed
Ri gt Rs
Rs gt Ri
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http//a1-structural-geology-software.com/The_rf_p
hi__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 two brachiopods (?)
• 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!
photograph
Tracing paper
Tracing paper overlaid on graph paper
Graph paper
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Deformed Trilobite
http//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 matter since we want to get Rs
  S1/S3
• Read the amount and sense of the angles 2?A,
  2?B,or 2?C
• 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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cw
?A
cw
?
?B
?
?C
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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Moderately deformed Neoproterozoic quartz
conglomerate. Strain exposed in sections
parallel to the principal planes