Posts Tagged ‘structure’

Structure: Mohr Circles

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).

relationship between principle stresses & the normal and shear stress on a given plane; Fig 3.7 from van der Pluijm & Marshak 2004: http://www.ic.ucsc.edu/~casey/eart150/Lectures/Stress/stress.htm

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.

Fig. 12 from Dewhurst & Jones, 2002 (AAPG)

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.

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In addition to mineralogy, structure is also reading a paper each week for discussion.   Since structure is a more advanced class and the students have experience with reading research papers, instead of assigning papers from Geology, I’m looking for papers that are a bit longer, require more structural-based knowledge, and relate to the topic covered in lecture that week.   The first paper (which I didn’t blog about) was from GSA Bulletin, which is where I’m going to start my weekly search.   This week I branched out to Lithosphere because I couldn’t find a recent rheology-themed paper.   Next week for microstructures, we’ll be back in GSA Bulletin.

Post-lunch, structure discussed a relatively short paper on using Field-based constraints on finite strain and rheology of the lithospheric mantle, Twin Sister, Washington by Tikoff et al.   In lab last week, the students did an analogue materials lab by Dyanna Czech and we’ve been talking about rheology in lecture, so the paper fit well into where the class currently is.

The questions the students considered when reading the paper:

  • what was the motivation for this study?
  • what kinds of data were used?
  • how was the data analyzed?
  • what kind of assumptions were made by the authors?
  • terms you didn’t understand?
  • concepts that were difficult to comprehend?

Unfortunately, there are a few issues with my structure paper discussions that don’t appear in mineralogy:

  1. it’s post-lunch and the students are ready for their naps
  2. there are only three students, so no one can have an “off” day
  3. so far I’ve chosen papers that require the students to actually remember material learned during previous semesters in other classes (e.g. how thermobarometry works; sedimentary basin formation), which I’ve had some backlash to

I feel like I have to lead the discussion more strongly with structure, which is something I want to move away from.   I’m considering assigning the student’s in rotating fashion ownership of the day’s discussion, but with three they’ll have to take a turn fairly frequently.   I’d welcome suggestions.

Full citation (paper is behind a paywall):

Tikoff, B., Larson, C.E., Newman, J., and Little, T., 2011, Field-based constraints on finite strain and rheology of the lithospheric mantle, Twin Sisters, Washington: Lithosphere, v.

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One of the great issues that arises in structural geology is trying record three-dimensional data on paper, which is inherently two-dimensional.   Usually, students struggle with trying to visualize things in three-dimensions and have a hard time being able to see how two different planes or lines will intersect each other.   But why do we care about planes or lines in space?

Quite a few rocks in the world have some form of parallel alignment due a variety of factors.   Sedimentary rocks are usually deposited in horizontal layers that are then compacted & cemented together (lithification) to form the layer-cake rocks that we frequently think of, especially when envisioning the US southwest.

Metamorphic rocks that have been squished & heated frequently have minerals that either re-align or grow in parallel planes called foliations.

mineralogy students examine metamorphic rocks in Dutchess County, NY

Though the planar features aren’t always exactly flat, we approximate their orientations by describing the strike (general NSEW orientation) and dip (angle from horizontal) of a similar plane.

Lines form in rocks due to several different factors and are recorded as trends (NSEW orientation) and plunges (angle off of horizontal):

  • during metamorphism, elongate minerals may re-align or grow in specific directions

lineated gneiss (Callan's foot for scale); http://farm4.static.flickr.com/3075/3109216762_1b7572f138_o.jpg

  • a line will form at the intersection of two planes (e.g. a set of sedimentary beds is cut by a fault)
  • a group of beds is folded around a hinge axis, that is also a line

Ok, so now at least we have an idea about what kinds of lines & planes might be found in rocks.   So, how do we represent them in 2D?   We could simply leave them as a list of measurements, but most people when examining:

  • N50W, 25 NE
  • N74W, 27 NE
  • N89W, 32 NE
  • N27W, 27 NE
  • N9W, 32 NE

wouldn’t be immediately able to tell you whether or not these planes were related to each other by some geologic process or not.   In order to plot 3D data in 2D, we use a spherical projection:

Fig. A1.3, Twiss & Moores, 2007

Image a sphere with either a plane or a line going exactly through the center of the sphere.   How the plane or line intersects with the exterior of the sphere will depend on its NSEW orientation as well as the angle it is relative to horizontal.   When planes intersect the exterior of the sphere, they’re going to form a line, which we’ll call a “great circle.”   When a line intersects the exterior of the sphere, its going to be a point called a “pole.”   We’re still in 3D, so in order to get to 2D we’re going to image cutting off the top half of the sphere and just using the lower hemisphere.   To flatten down to a 2D representation, we’re going to project down from a point (called the zenith) through the equatorial (horizontal) plane to where our plane or line is intersecting the exterior of the sphere.

Fig. H.2, Davis & Reynolds, 1997

If the plane has a steep dip (closer to vertical than horizontal), it will be close to the center of the sphere.   For shallow dips, the great circle will plot closer to the edge.

The graph we plot the data on using this method is called a stereographic net or just simply a stereonet.

Let’s go back to the data I gave you earlier.   If we were to plot that data as great circles on a stereonet, it would look like:

All the great circles intersect at a pole, which would indicate to a structural geologist that the various planes that were measured may be related to each other.   Specifically, the intersection at a point is a strong sign that these planes were folded during the same deformation event.

Stereonets are useful to determine a variety of different things about rocks such as the original orientation of the rocks prior to folding, where an ore-bearing dike might intersect the surface, and whether or not a paleomagnetic signature occurred pre- or post-folding.   However, they are not usually intuitive and students frequently struggle a bit until they reach an “ah ha!” moment.   I actually always have to do a few every year to get me back into practice, since I don’t use them on a daily basis in my own research.   But its like riding a bike: once you’ve learned how to manipulate the data, the methods always come back.

If you want a look at the lab my students will be tackling this week on stereonets, its here.

Anyone have a favorite stereonet story to share?

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One of the problems that I’ve had with structure in the past is that working on labs, the students can manage to get off track & continue down that road for quite awhile.   I try hard to give the students worked examples of problems before they have to tackle them on their own, but sometimes it just isn’t enough.   My last round of structure, it wasn’t until a week later when the labs were handed in that suddenly I realized that some concepts were still shaky and then I had to quickly come up with ways to get the several students off-track back in the fold.   Part of this is due to the students who just don’t want to speak up even if I’m in the room to answer questions, but a larger part is due to students who get the assignment, work for a few minutes, and then “disappear” out of lab several hours early.   There are legitimate reasons to leave lab early, but most of the time the students don’t seem to understand that this is time that has been set aside for them to work on the lab & have me right there to answer questions.   I’ve tried talking to my students.   I’ve tried insisting that they stay.   I’ve pointed out that their grades are better when they don’t disappear and then try to work on the lab at 2 am the night before its due.   But the students still disappear.

This upcoming semester, my structure class is tiny–3 students.   Its great in that my grading turn around is going to be swift, I know all three students well already from petrology in the fall, and I can easily focus on things the students are struggling with.   But I also know the students and that if I don’t do something, they’re going to be working on their stereonet labs at 2 am rather during the three hours on Tuesday that are set aside for the class.   Its been something that’s rattled around in the back of my mind for a few weeks now and today, I suddenly had an epiphany: I’m going to ask the students to turn in whatever they have done before they leave lab on Tuesdays and then finish the rest up on their own time.   Why do I think this will work?   I know that two of the students are competitive and will want egg each other on to get as much as possible done before handing things in (the third will simply go along with it and stick it out), which means they’ll tend to stay for the full 3 hours.   I’m going to be able to go through and grade at least part of the lab early, which will give me a chance to catch detours by the students much earlier.   It will also give the students the chance to go back and fix whatever they’ve missed, which hopefully will increase how much they learn through the lab work.   Now, I’m not sure this would work with a larger class due to the turn-around on grading I’m giving myself (my aim will be to get things Tuesday and have them back Wednesday by 1.30 when class meets again).   But, I’m going to give it shot.

Anyone tried other ways to keep the students on the right path during lab?   Things that have and haven’t worked?

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