We’re taking a short break from optical mineralogy this week–a bit ironic considering the fact that optical is a large discussion topic at the moment–and going to talk instead about a structural geology tool: Mohr circles.

What are Mohr circles supposed to bring to the table:

- the relationship between the principle stresses acting on a region, the shear & normal stresses for a specific plane, and the orientation of that plane
- no geographical frame of reference, but will reflect whether dextral or sinistral motion is occurring along a plane
- the relationship between rock strength and under what conditions the rock will fracture

In order to construct Mohr circle, we’re basically going to ignore the fact the world is three-dimensional and simply work in 2D. On any object, there are a number of forces at work:

- the force pulling us towards the center of the planet = gravity
- the force pulling us towards every other object on Earth (so small an effect, that the force of gravity wins out by far)
- the force of the air in the atmosphere around us

Stress is simply the amount of force divided by the area its applied to. If you take the same force and apply it to the area the size of a piece of paper vs. the size of a postage stamp, the latter will have the higher amount of stress applied to it. You can try this at home: make two blocks of play-doh, one 10 x 10 x 10 cm and one 10 x 2 x 2 cm; take a heavy book and place it on block 1; how long did it take for the block to squish? how much did it actually squish?; repeat it with block 2.

In our world, one of the principle stresses is almost always perpendicular to the Earth’s surface and the other is parallel to the Earth’s surface. The maximum stress is “sigma 1″ and the minimum is “sigma 3″ (in a 3D world, “sigma 2″ is an intermediate value & is oriented perpendicular to both sigma 1 & 3).

However, usually the plane we’re interest in is at an angle to the Earth’s surface and therefore not parallel / perpendicular to sigma 1 & 3. In order to better deal with what’s going on with the stress on our plane, we resolve sigma 1 into the normal stress (perpendicular to the oriented plane) and the shear stress (parallel to the oriented plane). To do the calculation, we need to know what the angle between either sigma 1 or 3 and our plane plus some trigonometry. (Yes, we use trig in structure!) For a Mohr Circle we need the angle between sigma 3 & the plane in question OR sigma 1 & the line normal to the plane in question. Both of these angles will be the same (go ahead–take a moment a prove that to yourself) and we call it “theta.”

On a Mohr diagram, we’ll use the normal stress (“sigma N”) as the x-axis and the shear stress (“sigma S” or “tau”) as the y-axis. To draw the circle, we plot sigma 1 & 3 on the sigma N axis; use the formula (sigma 1 + sigma 3) / 2 to find the center of the circle (O or C); and use a compass to draw the circle. [In geology, we plot compressional stresses as positive (to the right on the diagram) & tensional stresses as negative (to the left)--that is not true in engineering or materials studies.]

If we know what theta is, we can plot the oriented plane on the Mohr diagram. Since a Mohr Circle does not have a geographical frame of reference, instead of having to deal with 360 degrees of possible plane orientations, we only have to worry about 180 degrees. (N45E dip 60 SW will be oriented the same as S45W dip 60 SW and plot the same on a Mohr diagram.) Instead of plotting theta, we plot 2*theta on the Mohr Circle starting from sigma 1 and then measure either CW or CCW depending on whether the stress field is dextral or sinistral. Dextral shear will plot on the lower portion of the Mohr Circle (negative values) and sinistral shear on the upper portion (positive values). We can then either read the normal stress & shear stress off of the diagram (less precise) or use trig (more precise).

Up to this point, you could do this all with some trig. The real power is when we start considering under what conditions a rock will fracture. If the rock hasn’t previously been fractured, we can plot a Coulomb fracture envelope on the Mohr diagram. When the circle intersects the envelope, the rock will break. By drawing the line perpendicular to the intersection of the circle & envelope, we can discover the orientation of the plane along which the rock broke.

In the diagram above, C is the cohesion of the rock, which is the amount of shear stress the rock can accommodate without breaking when the normal stress is zero. Different types of rock with have a different C value (*e.g. *basalt higher than limestone), which will impact where the fracture envelope intersects the y-axis. Phi on the diagram is angle of internal friction within the rock and is directly related to mu, the Coulomb coefficient or coefficient of internal friction (mu = tan phi). Internal friction is the resistance of the interior of a substance (in this case a rock) to deformation. At low internal friction values, the rock will be more likely to break and the Coulomb fracture envelope will have a shallow slope. At high values, the rock will be hard to break and the slope of the fracture envelope will be very steep.

On the following diagram, the original cataclastite will break earlier if the normal stress was zero, but is stronger internally than the original reservoir rocks. If the difference between the principle stresses is low -> produce a smaller circle (blue) -> will intersect with the cataclasite first -> cataclastite breaks. If the difference is large -> a large circle results (green-yellow) -> the circle will intersect with the reservoir rocks first -> reservoir breaks.

The other two lines on the above diagram represent what occurs if the rock is already fractured. The rocks lose their cohesion (C) and so the fracture envelope intersects at the origin of the diagram. The slope of the lines may be the same as pre-fracture, but usually frictional sliding dictates how easy / hard it is to re-activate along a fracture plane and have a different slope. On the diagram above, the circles will intersect the re-sheared cataclasite or reservoir rocks before hitting the original fracture envelope. If the previous fractures intersect at that time, the fractures will be reactivated. However, imagine that we originally fractured a rock. At a later point in time, we re-oriented the plane vs. the prevailing stress field and now theta for the fractured plane is less than 45 degrees. In this case, the Mohr Circle might intersect the unfractured envelope before the pre-fractured plane hit resheared envelope and a new set of fractures may develop.

The final case I want to talk about has some real-world implications that have come up a few times recently. Though the differential stress (the difference between sigma 1 & sigma 3) may stay the same, if we change the pore fluid pressure in a system, we can move the circle on the Mohr diagram. We calculate the effective stress based on the differential stress minus the pore fluid pressure. The more fluid in the system, the further to the left the circle will move and the closer to the fracture envelope will be.

How does this have practical implications? There have been some recent earthquakes in Arkansas, which is not along a plate boundary. One of the suggestions for why is that fluid (due to fracking to get hydrocarbons out) is being pumped into the system, reducing the effective stress and causing slip along pre-existing faults. Earthquakes were also reported in Basel, Switzerland in 2006-2007 due to water being pumped into the ground for a geothermal power plant.

There are some other uses for Mohr circles, but this is what I normally cover & use myself. If you want to add other applications, please feel free to leave a comment!

Next week: back to optical & interference figures.

on 8. March, 2011 at 7:48 |Callan BentleyIt’s also worth noting that in addition to depicting the fundamental stress equations and the numerous interesting situations and utilities you note, Mohr Circles can be employed in graphing the fundamental strain equations, and can tell us exactly the amount of stretching and angular shear on a line of any orientation inside a deformed object. I start with Mohr strain circles even though they are more abstract (with reciprocal quadratic stretch and gamma/lambda ratios, students think “what the heck?”) and then move on to Mohr stress circles, which I feel are more inherently intuitive. Maybe I’ll do a post on the strain side of Mohr circle utility…

on 8. March, 2011 at 11:16 |Elli GoekeI don’t tend to use them for strain in my classes–probably because I wasn’t taught that way. I’ll have to look into it & see if it would work well into the way I teach about strain. Thanks for the suggestion! And please, post about what you do with strain Mohr circles!

on 3. February, 2013 at 22:41 |Dan WalshHi,

I recently realized there weren’t too many easily usable digital representations of the Mohr circle for stress online. Today I created an applet to simulate the Mohr Circle using a really interesting (and free) tool called geogebra. If you want to try it out it I uploaded it to:

danielrwalsh.com/mohrs_circle.html

It isn’t the fastest loading app, but once it gets going it’s really useful. I’ve been using it for in class demonstrations. You can also use geogebra for a lot of other cool things.

on 12. March, 2013 at 10:39 |Tijani Galican the cohesion be negative??