Geology

you have to finish the Excel sheet and I will provide you with the two java programs for the homework.

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Homework 4: Plane Strain
GEOL 314: Structural Geology Lab
Fall 2013 NAME:_________________
Due Thursday October 31th
______________________________________________________________________________

Strain: Deformation resulting from stress. Strain is what we observe in a rock. Stress is
transferred to strain via the rheology of a rock.

In this homework you will examine 2-D or “plane” strain. Plane strain is the special case
of 3-D strain in which material points remain in a single plane throughout deformation. It is often
useful to think of strain in two dimensions, as normal, reverse, and transverse faulting produce a
2-D strain (oblique faulting is an exception).

The two end members of plane strain deformation are pure shear and simple shear. The
major difference between pure and simple shear is the rotation of the major strain axes (in 2-D,
e1 and e3, or X and Z). Pure shear is also known as coaxial deformation, because the principal
axes of the strain ellipse maintain the same orientation throughout the deformation history (long
axis parallel to σ3, short axis parallel to σ1). However, in simple shear, the principal strain axes
rotate during deformation, an example of non-coaxial deformation. “General shear” is a
combination of these two end members. In all cases, the ellipticity of the strain ellipse increases
as strain progresses.

In this homework, you will be using the computer programs SHEAR BOX and FLOW
LINES to simulate pure shear, simple shear, and general shear. The aim here is to more fully
understand the rotation and behavior of material lines, points, and strain axes during these types
of shear. The programs are written in Java and will run on both Mac and PC computers and a
copy of them is located on Moodle. The programs SHEAR BOX and FLOW LINES allow you to
input the amount of pure and/or simple shear and the orientation of a material line. The program
gives you that axial ratio and orientation of the strain ellipse for each increment, and the length
and orientation of the material line you specified.

Turn in excel spreadsheets with recorded data and graphs (or email them to me at
rxg0121@louisiana.edu). Answer questions related to strain programs on this handout.

These programs are fairly self-explanatory, but you will understand the input and results
MUCH more if you fully read the “BACKGROUND” button for each of the programs.
That said, a few points can be summarized for all of the programs:

kx = stretch along the x-axis
ky = stretch along the y-axis, where k = final length / original length
kx and ky = 1 means no stretch in the material in the directions of x and y.
Gamma = shear strain
θ Angle = angle of a material line from the horizontal plane (+ counterclockwise)

NOTE: These, and other, programs are available for you to download for your own use at:
http://www.geology.wisc.edu/~struct/software.html

GEOL 314 Fall 201

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mailto:rxg0121@louisiana.edu

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Part I

PURE SHEAR

For pure shear, there is a stretch along the x-axis (kx) so that kx≠1. In order to preserve
area, the stretch along the y-axis (ky) must be the reciprocal of the stretch along the x-axis (kx).
So, if kx = 2, then ky = 0.5. However, area does not always have to be preserved in this program
or in reality.

Run the program SHEAR BOX using a pure shear component (ratio of
the final long axis to the initial) of 2 and 3. Consider material lines oriented
0°, 30°, 65°, and 90° from horizontal. In this example of pure shear, the
material line at 0° will always be the long axis of the ellipse, which will show
up as a RED line. Any other material lines specified (i.e., 30° and 90°) will
show up as BLUE lines.

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For each of the angles listed in the table below, record the length and orientation of these

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lines during each of the increments. Because we are dealing with the case of pure shear, γ = 0.
Use 10 increments for each line and conserve area such that ky is the reciprocal of kx.

Trial Pure Shear Simple Shear (γ) Orientation of Line
1 2 0 0° (blue line)
2 2 0 30° (blue line)
3 2 0 65° (blue line)
4 2 0 90° (blue line)
5 3 0 0° (blue line)
6 3 0 30° (blue line)
7 3 0 65° (blue line)
8 3 0 90° (blue line)

Record your results in a labeled table in Excel. Graph the length vs. orientation for each
increment of the four material lines when pure shear = 3. Mark the direction of increasing
increments along each line on the graph. (2 pts)

1) For each of the 4 material lines, describe in words what happened to the lines during
deformation (i.e. changes in length, angle) for the case of pure shear = 3. (3 pts)
That is:

a) What can you say about the deformation history of each line?
(hints: were any of them elongated after being shortened or visa versa?)

b) Which lines got both extended and shortened if this is the case?

c) Which ones didn’t change at all?

GEOL 314 Fall 2013

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Now, open the program FLOW LINES in order to visualize the movement of material points
during pure shear. Use the same pure shear component of 2 and 3 (as above) with a gamma of 0
and 10 steps.

2) a) What type of pattern do the flow lines make (sketches can help explain what you see)?
(1 pt)

b) Is this what you expect based on what you know about the orientation of the finite
strain axes in pure shear? Why? (1 pt)

GEOL 314 Fall 2013

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SIMPLE SHEAR
During simple shear, there is no stretch in the x or y axis, and thus in the programs the

original length = 1, final length =1, and thus kx=ky=1. Rather, for simple shear, we include a
component gamma (γ), which characterizes the movement of the top of the box past the bottom.
If the thickness (y-direction) of the box is 1 unit, then γ=1 is when the box has moved 1 unit to
the right.

Run the SHEAR BOX program using a γ = 1 and then a γ = 3 with pure shear = 1
(because this is simple shear) in both cases. Use the orientations of three material lines: 30°, 90°,
and 120° (BLUE lines). The angle θ is the orientation of the long axis of the strain ellipse (the
RED line) relative to the top and bottom of the box. Record axial ratios and θ angles of the
ellipse, and length and orientations of the material lines through all of the increments. Use at
least 10 increments for each line.

Trial Pure Shear Simple Shear (γ) Orientation of Line
1 1 1 30°
2 1 1 90°
3 1 1 120°
4 1 3 30°
5 1 3 90°
6 1 3 120°

GRAPH axial ratio (R) vs. θ for the strain ellipse when γ = 1 and γ = 3. (2 pts)

3) a) Can you make any generalizations about how the axial ratio and θ angle changes in
these two cases? (1 pt)

b) On a second graph, plot the length vs. orientation for the three material lines for γ = 3.
Mark the direction of increasing increments along each line on the graph. (2 pts)

4) Which of the 3 lines showed both shortening and elongation? Why? (1 pt)

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5) How did the deformation change for lines with the same initial orientation when you
changed γ? (1 pt)

Now, open the program FLOW LINES in order to visualize the movement of material points
during simple shear. Use the same parameters as above for the γ = 1 and γ = 3 case.

6) a) What type of pattern do the flow lines make (sketches can help explain what you see)?

(1 pt)

b) How do these flow lines contrast to those in pure shear? (1 pt)

c) Is this what you expect based on what you know about the orientation of the finite
strain axes in simple shear? I suspect it wasn’t intuitively obvious but why? (1 pt)

GEOL 314 Fall 2013

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GENERAL SHEAR
In the SHEAR BOX program, input pure shear = 2 and γ = 3. Put in three material lines

with initial orientations of 30°, 45°and 60°. Observe the axial ratios and θ of the ellipse, and
length and orientations of the material lines. Use at least 10 increments for each line. Repeat this
exercise with pure shear = 2 and γ = 1.

Trial Pure Shear Simple Shear (γ) Orientation of Line
1 2 3 30°
2 2 3 45°
3 2 3 60°
4 2 1 30°
5 2 1 45°
6 2 1 60°

Notice that the behavior of the three material lines is not the same for the two cases.

7) When pure shear = 2 and γ = 3 the line at 45° shows an interesting behavior relative to the
finite strain axis. What is it? Do you know why (or can you suppose why) the line does this?
(2 pts)

Now, open the program FLOW LINES in order to visualize the movement of material points
during general shear. Use the same parameters above for the two cases above.

8) a) In the two cases, which component of general shear (pure or simple) contributes to the

asymmetry the most? (1 pt)

b) Based on this is the flow behavior in simple shear or pure shear more complex? (1 pt)

GEOL 314 Fall 2013

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DISCUSSION QUESTIONS

9) According to the data graphed and what you observed in the programs for the behavior of
material lines and the flow lines of points, summarize briefly the major important differences
between pure, simple, and general shear. (5 pts)

GEOL 314 Fall 2013

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Part II

In the next photographs, you will find examples of structures formed by pure shear, simple shear
or general shear. Please indicate what type of shear is responsible for each case and explain
why. (Hint: pay attention to the symmetry of structures!!)

1- Conjugate faults (2 pts)

GEOL 314 Fall 2013

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2- Asymmetric boudins (2 pts)

GEOL 314 Fall 2013