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diagram mohr

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    The Mohr Stress Diagram

    Edvard Munch as a young geologist

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    Material in the chapter is covered

    in Chapter 7 in Fossens text

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    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.

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    !n= (!1+ !3) + (!1- !3) cos 2"

    2

    2

    !s= (

    !1-!

    3) sin 2"

    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.

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    The Mohr Stress Diagram!n= (!1+ !3) + (!1- !3) cos 2"

    22

    !s= (!1- !3) sin 2"

    2

    Theta = angle between

    The normal to the planeand the maximum

    principle stress (see Fig

    4.1 in Fossen)

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    The Mohr Circle - (mean or average stress)

    Mean Stress

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    The Mohr Circle radius or deviatoric stress

    Deviatoric Stress

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    The Mohr Circle diameter or differential stress

    Differential Stress

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    Laboratory Experiments in Rock Deformation

    Deformed marble rock cylinders

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    Laboratory Experiments in Rock Deformation

    Stress a rock sample until it fractures (or flows)

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    In-class exercise, work in groups of 4, turn in write up of answers.

    Given these samples, discuss how the magnitude of stresses likelyvaried relative to one another in these four experiments. Each

    cylinder was deformed in a different experiment, each with itsown axial and radial load (which varied relative to one another).

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    In-class problem

    1. For maximum and minimum stresses of 600 and 200

    mega-pascals (MPa) oriented as a vertical vector and ahorizontal, E-W striking vector (respectively), determinethe 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

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    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)

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    Determine the normal and shear stresses on a plane oriented N-S, 45oE

    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

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    Use the Equations

    !n= (!1+ !3) + (!1- !3) cos 2"

    22

    !s= (!1- !3) sin 2"

    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 asa 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.

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    2 For the stress state in the previous problem, determine the differentialstress 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 arock 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 byimagining 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 withincreasing depth in the crust?

    5 Draw the stress state where the minimum and maximum stresses are

    both equal to 600 Mpa.

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    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.

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    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 likelyvaried 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).

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    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 shearstresses on a plane oriented North-South, 45 degrees East. Ithelps 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

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    Use the Equations

    !n= (!1+ !3) + (!1- !3) cos 2"

    22

    !s= (!1- !3) sin 2"

    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 across section.

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    Or use the Mohr Circle

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    For the stress state in the previous problem, determine the differentialstress 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 moreor 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 rockto 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?

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    In-class problem 2.2

    1. For the stress state in the previous problem, determine thedifferential stress and mean stress. Start by plotting thesolution for normal and shear stresses for planes of allorientations on the Mohr Stress Diagram.

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    In-class problem 2.4

    Discuss how a change in differential stress might make thesample more or less likely to break. It may help byarbitrarily varying the stresses and looking at how theyplot on the circle, or by imagining stress on a cube.

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    In-class problem 2.5

    1. Now discuss whether increasing the mean stress would causea rock to break more readily (i.e. not deviatoric or differentialstress).

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    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 differentialstress? Would you expect the rock to deform under theseconditions?


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