The Mohr Stress Diagram

Edvard Munch as a young geologist!

Material in the chapter is covered

in Chapter 7 in Fossen’s text

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.

σn = (σ1 + σ3) + (σ1 - σ3) cos 2Θ!2!2!

σs = (σ1 - σ3) sin 2Θ!2!

Stress Equations

Normal Stress

Shear Stress

Two perpendicular stresses oriented at any angle to a plane!

Theta is the angle between the maximum stress and the pole

to the plane the stresses are acting upon.!

The Mohr Stress Diagram σn = (σ1 + σ3) + (σ1 - σ3) cos 2Θ!

2!2!

σs = (σ1 - σ3) sin 2Θ!2!

Theta = angle between !The normal to the plane and the maximum principle stress (see Fig 4.1 in Fossen)!

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, 45o E

Use the Mohr Stress Diagram!

Maximum stress is oriented vertically and equal to 600 MPA

Minimum stress is horizontal, oriented east-west and = 200 MPa

Use the Equations!

σn = (σ1 + σ3) + (σ1 - σ3) cos 2Θ!2!2!

σs = (σ1 - σ3) 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

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.

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.

3  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.

4  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?

5  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 mean stress 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 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 cross section.

So max stress is oriented vertically and equal to 600 MPA

Min stress is horizontal, oriented east-west and = 200 MPa

Use the Equations!

σn = (σ1 + σ3) + (σ1 - σ3) cos 2Θ!2!2!

σs = (σ1 - σ3) sin 2Θ!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 cross section.

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 for planes of all orientations on the Mohr Stress Diagram.

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 (i.e. not deviatoric or differential stress).

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 for planes of all orientations on the Mohr Stress Diagram.

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 (i.e. not deviatoric or differential stress).

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?