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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 instantaneous contraction (black) and extension (white) for dextral simple shear

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**Map of the conglomerate layer**

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**Conglomerate in a constriction field**

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1 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 (fibers grown parallel to 1). 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

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**Elongated belemnites in Jurassic limestone in the Swiss Alps**

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

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**Elliptical reduction spots in a slate from North Wales**

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**

Reduction spots in Welsh slate. The green spots are reduced (Fe++), and used to be spherical before deformation. Now they are pancakes. The top plane is the XY plane of the strain ellipsoid!

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**Deformed Ordovician Pahoe-hoe lava (sketched in 1880s)**

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 Ooids, reduction spots 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, tourmaline, 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 principal 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 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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Wellman’s Method Uses deformed variably oriented lines which were originally perpendicular (e.g., hinge and median lines of brachiopods, trilobites) Procedure: Trace the deformed lines on the image (photo) with a pencil Draw a box around the objects Draw a reference line between two arbitrary points (A and B), preferably parallel to the long edge of the box 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 (kept parallel to the box), bring B where A was, and repeat the drawing Place a dot () 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. Find the strain ratio, Rs = (long semi-axis)/(short semi-axis), and the 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 (photo) of the fossils Measure the angle (’) between the hingeline of the fossil w.r.t the reference line (e.g., trace of foliation) Measure the angular shear (’) for all fossils (e.g., the angle between deformed hinge and median lines) Repeat these for all fossils (see next slide) Plot ’ against ’ Compare the plot to an overlay of a transparent standard Breddin Graph centered at ’=0 that shows the Rs contours 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 on the Breddin graph**

Data from two slides before, plotted on the Breddin graph. Date plot on the curve for Rs=2.5

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**The center-to-center method**

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 ratio of the max (X) and min (Y), give the Rs = X/Y

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Fry’s 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 The maximum and minimum distances will be along S1 and S2, respectively

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**From: http://seismo. berkeley**

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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 put a dot () at the center of each grain; number the grains (1, 2, 3, ., ., through n, whatever number is) Draw an arbitrary reference line and/or a box around the image Have a transparent overlay, and mark a plus sign (+) at its center Put the overly on the image and trace the reference line on the overlay Put the + sign on 1 (center of grain 1), keep reference lines parallel, and mark all the other points on the overly with dots Put the + sign on 2 (center of grain 2), keep reference lines parallel, and mark all the other points on the overly with dots Repeat for all grains The final product is an empty ellipse, or an elliptical area full of points, which approximates the strain ellipse. Measure the major semi-axes: S1 and S3 Determine the strain ratio Rs= S1/S3 and the orientations of S1 and S3

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Fry Method Grain centers are transferred to an overlay. A central point () on the overlay is defined and moved on the center of grain 1, while copying the other points and overlay’s orientation is kept constant (sides of the boxes remain parallel) An empty ellipse develops which gives the strain ellipse.

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**Center to Center Method**

Undeformed Deformed Ramsay, J. G., and Huber, M. I., 1983 Modern Structural Geology. Volume 1: Strain Analysis

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Fry Method Pros: Fry’s 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 (>25) The estimation of the strain ellipse’s 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 initial grain ratio Ri and the strain ratio Rs The final ratio depends on the relative orientation of the long axis of the strain ellipse and that of the grain’s long axis

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**Rf/’ cont’d Rf max = Rs.Ri Rf min = Ri/Rs**

If Rs < Ri (strain ellipticity is < the initial grain ellipticity) Rs = (Rf max/Rf min) Ri max = (Rf max Rf min) If Rs > Ri (strain ellipticity is > 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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Rf/’ Method Could be used for grains with initial spherical or non-spherical shapes (i.e., initial grain ratio of Ri =1 or Ri >1) Procedure: Measure the long and short axes of each grain on the deformed rock, or on its image Find its final ratio (Rf) Find the angle (’) between the long axis of each grain and a reference line (e.g., trace of foliation or bedding) Plot the log of Rf against ’ Note the pattern (e.g., drop- or onion-shaped) Fit a theoretical curve on a transparent overlay to the distribution. Read the RS and Ri.

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**Rf/’ Method Grains had constant Ri = La/Sa The plot on the right shows Ri=2.**

Rs = 1/3 = S1/S3 A pure shear with Rs= 1/3 = 1.5 is applied, (i.e., 1 = 1.5 and 3 = 1/1.5) | S | | | | S3| or | /1.5| where 3 = 1/1 for pure shear Or a pure shear of Rs= 1/3 = 3 (i.e., 1 = 3 and 3 = 1/3) Notice the coaxial strain (see strain ellipses ’ is around 0). Undeformed Ri > Rs Rs > Ri

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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 (’). Note: this angle is doubled in the Mohr circle! Measure the angular shear (A and B) for each fossil Step 2. Plot a circle on a tracing paper of any size. Draw two radii (A and B) from the center of the circle, with an angle of 2’ Draw (on a graph paper) the coordinates of the Mohr Circle ( ’ vs. ’) with an arbitrary scale Step 3. Draw (on the same graph paper) two lines from the origin inclined at the 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, keeping the center on the x-axis, until each of the lines on the graph paper intersects its corresponding radius that emanates from the origin.

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Note: A is CCW (+) and B is CW (-) (see next slide) The senses of are the same in the real world and the Mohr circle space! photograph B Tracing paper A Tracing paper overlaid on graph paper Graph paper

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**Real world r Mohr world c**

CW + Real world r Mohr world CW + + O c

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Deformed Trilobite

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**Example: 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 The angle between the hinges of neighboring fossils are indicated by ’ and ’, and the angular shears are given by A, B, and A (all are CW) A cw ’ B A B C A B C 2’ 2’ ’ ’ 3’ 1’ The dashed circle and the rosette are on the tracing paper ’ C c cw The hinge and median lines of three brachiopods are traced on a photo The angles ’ and ’are doubled (2’ between A and B, and 2’ between B and C), and plotted as two radii, while c is kept on the x-axis. The three angles are all CW, and plotted on the graph paper.

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**Three 2D section provide data for the 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**

Moderately deformed Neoproterozoic quartz conglomerate. Strain exposed in sections parallel to the principal planes

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