Fossen – Chapter 3
How strain is measured and
quantified in ductile regime
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.
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
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)
Deformed Bygdin Conglomerate, with quartzite pebbles and
quartzite matrix, Norway.
Similar pebble and matrix compositions minimize strain
partitioning and enhance strain estimates
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
http://www.see.leeds.ac.uk/structure/strain/gallery/belpart.html
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’
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
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.
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
Map of the
conglomerate
layer and
variation of
strain over a
large area.
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.
Deformed Precambrian pillow lava, Superior Province,
Ontario, Canada.
Can use Rf/, center-to-center, or Fry method techniques.
http://myweb.facstaff.wwu.edu/talbot/cdgeol/Structure/Strain/Strain_in_igrox.html
Deformed pillow lava.Swamp R.,
Ontario, Canada.
http://myweb.facstaff.wwu.edu/ta
lbot/cdgeol/Structure/Strain/Strai
n_in_igrox.html
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
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
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.
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
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!
Undeformed brachiopods
originally oriented randomly
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
Wellman method
used for deformed
trilobites and
brachiopods with
two originally
perpendicular lines
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
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
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
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
Undeformed medium grained oolitic grainstone dominated by ooliths (70 vol% of
allochems), bioclasts (30%) and minor peloids and intraclasts.
https://wwwf.imperial.ac.uk/earthscienceandengineering/rocklibrary/viewrecord.php?SampleNo=sp5
From: http://seismo.berkeley.edu/~burgmann/EPS116/labs/lab8_strain/lab8_2009.pdf
Undeformed and deformed oolitic
limestone
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
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.
Center to Center Method
Undeformed
Deformed
Ramsay, J. G., and Huber, M. I., 1983
Modern Structural Geology. Volume 1: Strain Analysis
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
Moderately deformed
Neoproterozoic quartz
conglomerate.
Strain exposed in sections
parallel to the principal
planes
Newport, Rhode Island,
Purgatory conglomerate
http://blogs.agu.org/mountainbeltway/2010/
08/20/purgatory-conglomerate/
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
Rf/’ Method
• Could be used for grains with initial spherical or nonspherical 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.
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
Undeformed
Apply a pure shear with
Rs= 1/3 = 1.5
(i.e., 1 = 1.5 and 3 = 1/1.5)
Ri > Rs
Rs > Ri
| S1 0 |
| 1.5
0|
| 0 S3| or |0 1/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).
http://a1-structural-geology-software.com/The_rf_phi__prog_page.html
http://a1-structural-geology-software.com/The_rf_phi__prog_page.html
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
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
http://geology.gsapubs.org/content/34/7/593.figures-only
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. http://geology.gsapubs.org/content/34/7/593.figures-only
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 xaxis, until each of the lines on the graph paper intersects its corresponding
radius that emanates from the origin.
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
cw
A
Tracing paper
c

Tracing paper overlaid
on graph paper
B
2
c
Graph paper
A
CW +
Real world
r
Mohr world
CW
O
+
+
c
Deformed Trilobite
http://courses.eas.ualberta.ca/eas421/lecturepages/strain.html
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
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
’
The dashed circle
and the rosette are
on the tracing paper
 1’
c
’
 3’
2’
2’
C
C
A A
B
’
cw
C
The hinge and median
lines of three brachiopods
are traced on a photo
B
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.
Three 2D section provide data for the 3D strain
Strain obtained from deformed conglomerate
plotted on Flinn diagram (Norway)