1. How strain is measured and quantified in ductile regime
Fossen – Chapter 3
How strain is measured and quantified in ductile regime

2. Strain analysis Allows exploring the state of strain in rocks
Find the variation of strain at different scales (microscopic, outcrop, and region)
Is used to map shear zones, and estimate the displacement
Helps us to find the folding mechanisms in folded rocks
The shape of the strain ellipse (e.g., oblate -> flattening; prolate -> constriction) provides information on how deformation occurred.
In slate with reduction spots, for example, it tells us about shortening
The orientation of the strain ellipse can tell us if strain was simple shear or pure shear; used for kinematics
Strain markers in sedimentary rocks can let us reconstruct original sedimentary thickness.

3. Strain markers
Strain markers reveal the state of strain in a deformed rock.
Markers are used to find 1D and 2D strain (later extended into 3D by combining three 2D sections)
In 1D, strain analysis measures change in length
This leads to finding the amount and direction of shortening or stretching. For example, from:
Boudinaged dike, elongate minerals, linear fossils

4. Deformed pebbles in a conglomerate (constriction field), seen in plane in an outcrop. The length of these pebbles my parallel the maximum principal stretch axis (X)

5. Deformed Bygdin Conglomerate, with quartzite pebbles and quartzite matrix, Norway. Similar pebble and matrix compositions minimize strain partitioning and enhance strain estimates

6. calcite
quartz 1
Part of a stretched belemnite boudins with quartz and calcite infill. The space between the broken pieces of the belemnite are filled with precipitated 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

7. 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 l’
lo4 l’
lo3

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

9. Strain in 2D
We use strain markers of known initial shape or linear markers oriented variably in a rock
2D strain is done, for example, in sections through conglomerate, oolites, vesicles, pillow lava.
These sections can be oriented thin sectionss, or surfaces of a joint plane in the field.

10. Block diagrams showing 2D sections through the strain ellipsoid
Block diagrams showing 2D sections through the strain ellipsoid. Flinn diagram represent the shape of the strain ellipse
Direction of instantaneous stretching axes (ISA) and fields of instantaneous contraction (black) and extension (white) for dextral (right-lateral) simple shear

11. Map of the conglomerate layer and variation of strain over a large area.

12. Change in angle
Can be done if the original angular relation between two lines is known.
The change in the angles can be used to find shear strain () from, for example:
dikes, foliation, and bedding, which may be seen in neighboring undeformed and deformed zones (e.g., around and in a shear zone).
two originally orthogonal lines of symmetry in fossils, such as trilobite, brachiopods, worm burrows (relative to the bedding).
The deviation from orthogonal relation () gives the shear strain.

13. Deformed Precambrian pillow lava, Superior Province, Ontario, Canada
Deformed Precambrian pillow lava, Superior Province, Ontario, Canada. Can use Rf/, center-to-center, or Fry method techniques.

14. Deformed pillow lava.Swamp R., Ontario, Canada.

15. 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

16. 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)

17. Reduction spots in slate
Reduction spots in slate. The green spots are reduced from Ferric iron (Fe3+) (Fe2+) by fluids (turned from red to green) and then Fe2+ was leached out. Spots are mostly spherical before deformation. After deformation they are elliptical, and give the strain ellipse.

18. 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

19. 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!
The Wellman and Breddin methods are commonly used!

20. Undeformed brachiopods originally oriented randomly

21. Wellman Method
Uses deformed, variably oriented pair of 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 (image is real world)
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 of the two lines
Place a dot () where the pairs of deformed lines, going through A and B, 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 line AB

22. Wellman method used for deformed trilobites and brachiopods with two originally perpendicular lines

23. 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 the 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

24. Data from two slides before (traced below), plotted on the Breddin graph. Data plot on the curve for Rs=2.5, where Rs is the strain ellipse

25. 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. Note: reference line may be the trace of foliation! The ratio of the max (X) and min (Y), give the Rs = X/Y

26. Fry 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

27. Undeformed medium grained oolitic grainstone dominated by ooliths (70 vol% of allochems), bioclasts (30%) and minor peloids and intraclasts. 

28. From: http://seismo. berkeley

29. Undeformed and deformed oolitic limestone

30. 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

31. 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.

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

33. 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

34. Moderately deformed Neoproterozoic quartz conglomerate
Moderately deformed Neoproterozoic quartz conglomerate. Strain exposed in sections parallel to the principal planes
Newport, Rhode Island, Purgatory conglomerate

35. 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
Rfmax = Rs.Ri
Rfmin = Ri/Rs

36. 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 the final ratio (Rf) for each grain
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.

37. Rf/’ Method Case: Grains had constant Ri = La/Sa The plot on the right shows Ri=2.
Rs = 1/3 = S1/S3
For pure shear: S3 = 1/S1 or 3 = 1/1 Apply a pure shear with Rs= 1/3 = 1.5 (i.e., 1 = 1.5 and 3 = 1/1.5) | S | |  | | S3| or | /1.5| Note: Grain #7 has the max Rf; grain #1 has the min Rf Or apply a pure shear with 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

40. Digital restoration of single deformed trilobite
Application of digital method on a single fossil. A: Image of deformed trilobite Angelina showing non-orthogonal relationship between hinge line h and median line m. Image has been rotated to make stretching lineation L (long arrow) vertical. White circle is reference circle. Solid square dots 1–8 are dragging handles. B: Shape restoration by pulling handle 4 to right until h and m become orthogonal. R is strain ratio. C: Size restoration

41. Digital restoration of single deformed brachiopod
A: Initial stage. Undeformed brachiopod fossil and reference circle enclosed in square of side l0. X and Z are directions of maximum and minimum stretching, h is hinge line, m is median line. B: Homogeneous deformation of A results in change in perpendicularity between h and m and transformation of reference circle into strain ellipse (gray) of axial ratio l2/l1. Another reference circle (white) is drawn at this stage.
C: Retrodeformation: rectangle in B transforms into square of side l2. Reference circle and strain ellipse in B transform into reciprocal strain ellipse (white) and circle (gray), respectively.

42. 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 (2’) in the Mohr circle!
(Please read the powerpoint slide on strain Mohr circle!)
Measure the angular shear (A and B) for each fossil (see next slide!)
Step 2. Plot a circle on a tracing paper of any size with center c.
Draw two radii (A and B) from the center of the circle, with an angle of 2’
Draw (on a graph paper) the Cartesian 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. Watch for cw and ccw senses!
Step 4. Overlay the tracing paper on the graph paper, and put the center (c) 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.

43. 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
ccw B 2
Tracing paper cw c A 
Tracing paper overlaid on graph paper B 2 c A
Graph paper

44. Real world r Mohr world c
CW +
Real world r
Mohr world CW +  + O c

45. Deformed Trilobite

46. Example: Three deformed brachiopods see next slide for pictures!
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

47. 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 (rosette), while c is kept on the x-axis. The three  angles are all cw, and plotted on the graph paper.

48. Three 2D section provide data for the 3D strain

49. Strain obtained from deformed conglomerate plotted on Flinn diagram (Norway)

50. Rf/’ cont’d
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