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Mohr Circles & States of Stress
Ch. 4, p. 75-‐
1. Representing Stress on a Mohr Diagram: The infinite number of normal and shear stresses acting on all possible planes passing through a point can be represented simply and graphically using a Mohr diagram. This σn vs. σs (normal stress vs. shear stress) graph shows all (σn , σs ) combinations for planes of any orientation relative to the principal stresses.
[Fig. 4.6. Mohr diagram representation of stress] [Figure. Stresses represented in physical space (Twiss & Moores, 2007)]
2. Representing Stress on a Mohr Diagram: For the 2D case represented by Mohr diagrams, the orientation of any plane relative to σ1 and σ3 is given by θ: the angle measured from σ1 toward the normal vector to the plane.
All possible orientations of planes passing through a point are represented by θ in the range 0° to 180°.
[Fig. 4.6. Mohr diagram representation of stress]
3. Representing Stress on a Mohr Diagram: When all possible combinations of σn , σs are plotted in Mohr space for all possible plane orientations, they trace out a circle called a Mohr circle, centered on the σn axis. The circle has both +ive and –ive values of shear stress σs to represent the sign convention for dextral and sinistral shearing. For σn , +ive values are compressive whereas –ive values are tensile.
[Fig. 4.6. Mohr diagram representation of stress]
4. Representing Stress on a Mohr Diagram: Note that angle θ in physical space is always doubled to 2θ in Mohr space. So the full range of plane orientations is 360° in Mohr space (hence, the circle).
The angle 2θ is measured CCW away from the right side of the circle, along the σn axis, for a CCW angle from σ 1 to the normal to the plane in physical space (i.e., +ive shear stress). Else, the angle is measured CW (–ive shear stress).
[Fig. 4.6. Mohr diagram representation of stress]
5. Representing Stress on a Mohr Diagram: The Mohr circle shows us that any plane (red star in figure) will have two complementary planes that respectively contain the same amount of shear stress but a different normal stress (orange star), or the same normal stress but an opposite sign of shear stress (yellow star). 6. Constructing a Mohr Circle: By definition, any surface containing zero shear stress (no sliding possible) has a normal stress that is also a principal stress (i.e., a principal plane). Therefore, principal stresses must plot along the σn axis, where σs = 0. For the 2D case, we only plot σ 1 and σ 3.
Because σ 1 > σ 3 in magnitude, it plots further to the right on the σn axis.
[Figure. Mohr diagram axes]
7. Constructing a Mohr Circle: We typically only represent +ive shear and normal stresses, so a semi-‐circle is drawn that connects σ 1 and σ 3. Given the definition of θ, if σ 1 is perpendicular to a plane, θ = 0° (i.e., 2θ = 0°). This is why 2 θ is measured from the right side of the circle, where 2θ = 0°.
In physical space, the range of θ between σ 1 and σ 3 is 90°, so in Mohr space this angle must be 180°. This allows both σ 1 and σ 3 to plot on the σn axis, which is necessary as they are normal stresses to principal planes.
[Figure. Constructing a Mohr diagram] [Figure. Representation of stresses and theta relative to a plane]
8. Using a Mohr Circle: Read off the coordinates of (σn , σs ) along the circle for any value of 2θ to determine the normal and shear stress along any plane at angle θ to σ 3.
[Figure. Using a Mohr circle diagram] [Figure. Representation of stresses and theta relative to a plane]
9. Representing Stress on a Mohr Diagram: We can see from the Mohr circle that the maximum shear stress possible occurs where 2θ = ±90°. These represent planes at 45° to both σ 1 and σ 3 in physical space (two conjugate planes of maximum shear stress).
[Fig. 4.6. Mohr diagram representation of stress]
10. Representing Stress on a Mohr Diagram: The center of the Mohr circle is simply the average of σ 1 and σ 3 and so is called the mean stress (
" n ) or the mean normal stress, (σ 1 +σ 3 )/2. The radius of the Mohr circle is half the difference between σ 1 and σ 3 and is equal to the maximum shear stress, (σ 1 – σ 3 )/2.
[Fig. 4.6. Mohr diagram representation of stress]
11. Mohr Equations: These two quantities can be used to find the coordinates of any point along the Mohr circle, and thus σn and σs for any plane. Using trigonometry:
σn =
" n + r cos 2θ and σs = r sin 2θ.
Using the equations for mean stress and maximum shear stress:
σn = [(σ 1 + σ 3 )/2] + [(σ 1 – σ 3 )/2] cos 2θ and σs = [(σ 1 – σ 3 )/2] sin 2θ.
These are the Mohr equations and can always be used instead of constructing a Mohr circle.
[Fig. 4.6. Mohr diagram representation of stress]
If a rock contains pressurized fluid in pores or cracks, this fluid pushes out against the confining stress, resulting in a stress state that is effectively less than it otherwise would be.
All principal stresses are reduced equally by p (^) f , causing the Mohr circle to shift to the left by p (^) f without changing in size. Thus, fluid pressure reduces the normal stress but leaves the shear stress unaffected.
[Fig. 4.6. Mohr diagram representation of stress]
20. Mechanics of Brittle Failure: We can think of a material like rock undergoing some sort of structural failure once the stresses reach some critical value for brittle fracturing. Rocks thus have a limit to their strength.
But what is strength and how do we measure it?
This has typically been done in the laboratory using a pressurized vessel containing a rock cylinder placed between hydraulic pistons (called a rock testing machine).
[Figure. Rock testing machine (Pollard & Fletcher, 2005)]
21. Mechanics of Brittle Failure: A range of rock failure styles may occur depending on the loading conditions. For uniaxial loading, opening fractures form parallel to σ 1 (the long axis of the cylinder). This type of failure is called axial splitting or longitudinal splitting.
[Figure. Brittle failure of rock specimens (Twiss & Moores, 2007; Pollard & Fletcher, 2005)]
22. Mechanics of Brittle Failure: For axial compression (cylinder inside a pressurized fluid), shear failure occurs through the development of a diagonal break through the sample. In some cases, conjugate fractures may form. They form at <45° to σ 1 (usually ~30°) so are not planes of maximum shear stress.
These two styles of failure warrant the development of explicit failure criteria to explain brittle fracture of rock.
[Figure. Brittle failure of rock specimens (Twiss & Moores, 2007; Pollard & Fletcher, 2005)]
