Download Problem Set #1

Geol 542: Advanced Structural Geology
Fall 2011
Problem Set #4: Principal Stresses and Mohr Circles
1. At some point in the Earth's crust, the three principal stresses are: 1 = 100 MPa; 2 = 80 MPa; 3 = 65 MPa
(compression is positive).
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a)
What term can we use to describe this state of stress?
(2)
b) What is the mean normal stress?
(2)
c)
(2)
What is the maximum shear stress?
d) What is the differential stress?
(2)
e)
What are the three deviatoric stresses?
(3)
f)
Use the calculated deviatoric stress to determine in what directions the rock will contract, and in what
directions the rock will extend.
(3)
g)
Given the result above, comment on whether or not tensile stresses are needed in the Earth in order to
allow extension (e.g., through normal faulting), or if extension is possible with all-round compression. (2)
2. Use graph paper to construct a Mohr circle, as directed below, then answer the related questions. Remember, a
Mohr diagram can only work if the n and s axes have an identical scale. Use an entire graphing page for your
Mohr diagram (i.e., in landscape format).
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a.
Construct a Mohr diagram to represent a state of stress in which the principal stresses are
measured to be: 1 = 40 MPa, 2 = 22 MPa, and 3 = 18 MPa (compression is positive).
(6)
b.
If the minimum compressive stress, 3, is the vertical (lithostatic) component of stress, use your
Mohr diagram to read off the approximate normal (n) and shear (s) stresses acting on a fault
plane dipping at 40 (remember,  is defined in physical space as the angle between the 1
direction and the normal vector n to the plane). Assume that only 1 and 3 resolve onto the
fault plane (i.e., the strike of the fault is parallel to 2). Include a sketch showing your fault
configuration.
(5)
c.
What would the normal and shear stress be on a fault plane having the same strike and dip as
above but dipping in the opposite direction?
(2)
d.
In geologic problems, why would we even care about the sign of s?
(2)
e.
Corroborate your result in (b) by using the Mohr equations (of the form shown below) to
calculate the precise magnitudes of the shear and normal stresses.
(6)
n = 1 cos2  + 3 sin2 
s = (1 – 3) sin  cos 
f.
In this state of stress, in which directions would the rock contract and in which directions would
the rock extend (show your calculations).
(6)
Geol 542: Advanced Structural Geology
Fall 2011
g.
What would we call this type of tectonic environment based on the relative orientations of the
principal stresses in 3D space?
(2)
h.
Now assume the fault exists within saturated rocks having a fluid pressure of 15 MPa. Draw a
new Mohr circle representing this effective stress condition on the SAME graph as before.
(4)
i.
How has n and s changed for this fault in response to the addition of fluid? Intuitively, what
would this mean for the likelihood of sliding along the fault?
(2)
j.
How did the differential stress and deviatoric stresses change in response to the addition of a
fluid pressure? Show your calculations.
(7)
k.
Given your answers to i and j above, how does the addition of fluid pressure affect the way rocks
in the Earth’s crust deform (in terms of the distortion they experience)?
(2)
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