The Mohr Stress Diagram

The Mohr Stress Diagram
Edvard Munch as a young geologist!

The Mohr Stress Diagram
A means by which two stresses acting on a plane of known
orientation can be plotted as the components of normal and
shear stresses (derived separately from each of the two stresses).
The Mohr circle is thus an elegant way to determine the shear and
normal stresses for a pair of stresses oriented obliquely to the plane
in question. The Mohr circle allows you to quickly read this for
planes of any orientation.
It also makes it easy to visualize mean stress and differences in
stress, or deviatoric stress and relate these to deformation.

Setting up the Problem (we did this previously for one stress)

Add another stress & calculate their normal and shear components
The angle theta is the angle between the maximum !
principle stress and the plane it is acting on…

Decompose two original stresses into their x and y components!
After several trigonometric and algebraic manipulations,
the two equations left are ……

Stress Equations
Two perpendicular stresses oriented at any
angle to a plane!
Normal Stress
σ n = (σ 1 + σ 3 ) - (σ 1 - σ 3 ) cos 2Θ!
2!
2!
Shear Stress
σ s = (σ 1 - σ 3 ) sin 2Θ!
2!

The Mohr Stress Diagram
σ n = (σ 1 + σ 3 ) - (σ 1 - σ 3 ) cos 2Θ!
2!
2!
Theta = angle between !
the maximum stress and
the plane it is acting on.!
σ s = (σ 1 - σ 3 ) sin 2Θ!
2!

The Mohr Circle - (mean or average stress)
Mean Stress!

The Mohr Circle – radius or deviatoric stress
Deviatoric Stress!

The Mohr Circle – diameter or differential stress
Differential Stress!

Laboratory Experiments in Rock Deformation
Deformed marble rock cylinders

Laboratory Experiments in Rock Deformation
Stress a rock sample until it fractures (or flows)

In-class exercise, work in groups of 4, turn in write up of answers.
Given these samples, discuss how the magnitude of stresses likely
varied relative to one another in these four experiments. Each
cylinder was deformed in a different experiment, each with its
own axial and radial load (which varied relative to one another).

In-class problem!
1. For maximum and minimum stresses of 600 and 200
mega-pascals (MPa) oriented as a vertical vector and a
horizontal, E-W striking vector (respectively), determine
the normal and shear stresses on a plane oriented North-
South, 45 degrees East. It helps to first draw a block
diagram.
2. So max stress is oriented vertically and equal to 600 MPA
3. Min stress is horizontal, oriented east-west and = 200 MPa

In-class problem!
There are two ways to solve these problems.
Use the Mohr Stress Diagram or
Use the equations
(extra credit if you do both…)

Determine the normal and shear stresses on a plane oriented N-S, 45 o E
Maximum stress is oriented vertically and equal to 600 MPA
Minimum stress is horizontal, oriented east-west and = 200 MPa
Use the Mohr Stress Diagram!

For maximum and minimum stresses of 600 and 200 (MPa) oriented as
a vertical vector and a horizontal, E-W striking vector (respectively),
determine the normal and shear stresses on a plane oriented North-
South, 45 degrees East. It helps to first draw a block diagram.
Use the Equations!
σ n = (σ 1 + σ 3 ) - (σ 1 - σ 3 ) cos 2Θ!
2!
2!
σ s = (σ 1 - σ 3 ) sin 2Θ!
2!
For the minimum and maximum principle stresses of 600 and 200
megapascals (MPa) oriented as a vertical vector and a horizontal,
E-W striking vector (respectively), determine the normal and shear
stresses on a plane oriented North-South, 45 degrees East

2 For the stress state in the previous problem, determine the differential
stress and mean stress. Start by plotting the solution for normal and
shear stresses on the Mohr Stress Diagram.
Problem 2!
3 Determine whether decreasing the dip of the fault will decrease or
increase the shear stress acting on it.
4 Discuss how a change in differential stress might affect whether a
rock might be more or less likely to break. It may help by arbitrarily
varying the stresses and looking at how they plot on the circle, or by
imagining stress on a cube.
5 Now discuss whether increasing the mean stress would cause a rock
to break more readily. Would this be more or less likely with
increasing depth in the crust
6 Draw the stress state where the minimum and maximum stresses are
both equal to 600 Mpa.

Following set of slides should be copied and handed out to students for
exercises. Work in groups of 4, so for a class of 80 students, print out 20
sets.

In-class exercise, work in groups of 4 & turn in a write up of your answers.
Problem 1.1 Given these samples, discuss how the magnitude and
mean stresses likely varied relative to one another in these four samples.
Each cylinder was deformed in a different experiment, each with its
own axial and radial load (which varied relative to one another).

In-class problem 1.2
For maximum and minimum stresses of 600 and 200 megapascals
(MPa) oriented as a vertical vector and a horizontal, E-
W striking vector (respectively), determine the normal and shear
stresses on a plane oriented North-South, 45 degrees East. It
helps to first draw a block diagram.
So max stress is oriented vertically and equal to 600 MPA
Min stress is horizontal, oriented east-west and = 200 MPa

For maximum and minimum stresses of 600 and 200 megapascals
(MPa) oriented as a vertical vector and a horizontal, E-W striking
vector (respectively), determine the normal and shear stresses on a
plane oriented North-South, 45 degrees East. It helps to first draw a
block diagram.
Use the Equations!
σ n = (σ 1 + σ 3 ) - (σ 1 - σ 3 ) cos 2Θ!
2!
2!
σ s = (σ 1 - σ 3 ) sin 2Θ!
2!

Or use the Mohr Circle!

For the stress state in the previous problem, determine the differential
stress and mean stress. Start by plotting the solution for normal and shear
stresses on the Mohr Stress Diagram.
Determine whether decreasing the dip of the fault will decrease or
increase the shear stress acting on it.
Discuss how a change in differential stress might make the sample more
or less likely to break. It may help by arbitrarily varying the stresses and
looking at how they plot on the circle, or by imagining stress on a cube.
Now discuss whether increasing just the mean stress would cause a rock
to break more readily. Would this be more or less likely with increasing
depth in the crust
Draw the stress state where the minimum and maximum stresses are both
equal to 600 MPa. What is the differential stress. Would you expect the
rock to deform under these conditions

In-class problem 2.2
1. For the stress state in the previous problem, determine the
differential stress and mean stress. Start by plotting the
solution for normal and shear stresses on the Mohr Stress
Diagram.

In-class problem 2.3
1. Determine whether decreasing the dip of the fault will
decrease or increase the shear stress acting on it.

In-class problem 2.4
Discuss how a change in differential stress might make the
sample more or less likely to break. It may help by
arbitrarily varying the stresses and looking at how they
plot on the circle, or by imagining stress on a cube.

In-class problem 2.5
1. Now discuss whether increasing the mean stress would cause
a rock to break more readily. Would this be more or less likely
with increasing depth in the crust

In-class problem 2.6
1. Draw the stress state where the minimum and maximum
stresses are both equal to 600 Mpa. What is the differential
stress Would you expect the rock to deform under these
conditions

In-class problem 2
For the stress state in the previous problem, determine the differential stress
and mean stress. Start by plotting the solution for normal and shear stresses
on the Mohr Stress Diagram.
Determine whether decreasing the dip of the fault will decrease or increase
the shear stress acting on it.
Discuss how a change in differential stress might make the sample more or
less likely to break. It may help by arbitrarily varying the stresses and looking
at how they plot on the circle, or by imagining stress on a cube.
Now discuss whether increasing just the mean stress would cause a rock to
break more readily. Would this be more or less likely with increasing depth in
the crust
Draw the stress state where the minimum and maximum stresses are both
equal to 600 Mpa. What is the differential stress. Would you expect the rock
to deform under these conditions