|
Normal Stress (sn)
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Shear Stress (t)
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| (20) |
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(22) |
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| (21) |
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(23) |
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Adding equations (21) and (23) gives:
The fundamental stress equations derived in Chapter 5.2 can be rearranged as follows:
|
Normal Stress (sn)
|
Shear Stress (t)
|
||
| (20) |
|
(22) |
|
| (21) |
|
(23) |
|
Adding equations (21) and (23) gives:
| (24) |
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| (25) |
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Now recall that:
 |
substituting this relationship into the right hand side of equation (25) results in:
| (26) |
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Recall from your college algebra class that the equation of a circle drawn in a x-y coordinate system is:
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(27)
|
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where r is the radius of the circle and (h,k) are the coordinates of the center of the circle. Comparing equations (26) and (27) reveals that the fundamental stress equations define a circle in sn-t space centered on the point:
|
Center
|
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with a radius of:
|
Radius
|
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Figure 15 illustrates a circle constructed from equation (26).
This circle represents the locus of all possible normal and shear stresses for a given state of stress acting on planes whose normals make an angle of q degrees to s1 (Figure 16).
Structural geologists refer to Figures 15 and 16 as "Mohr diagrams" after the German engineer Otto Mohr (1835-1918) who first introduced them over a century ago. Note further that the average stress is simply:
|
Average Stress
|
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or, in three dimensions:
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Average Stress in Three Dimensions
|
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In a Mohr diagram, average stress is the center of the Mohr Circle. Differential stress is the difference between the maximum and minimum principal stresses. In a 2-dimensional Mohr diagram it is the diameter of the Mohr circle.