Salamander shear

Whilst discussing how to quantify strain with my GMU structural geology students recently, I hit upon a cool analogy. In order for you to understand the analogy (assuming you’re not a structural geologist), I’ll have to review some background information first. Stick with it, and I promise you a salamander at the end.

Structural geologists are interested in how rocks deform. If we have some idea of the original shape of a rock element (say, an oncolite, or a cooling column, or a fossil, or more prosaically, a sedimentary layer), and we find a deformed version of it, then we can use that object as a strain marker. By measuring its features, we can determine how parts of the strain marker have elongated, shortened, or rotated.

Elongation is pretty easy to calculate. You compare the final dimensions to the original dimensions via the equation e = (l_f-l_o)/l_o where “l_o” is the original length and “l_f” is the final length.

If the strain marker has been elongated, then the value of e will be positive. If it has been shortened, then the value of e will be negative. Another value, S (for ‘stretch’) is also a useful parameter to calculate. S = l_f/l_o, which is the same as saying: S = e + 1

As a quick example, a line which has been deformed from an original length of 20 cm to a final length of 30 cm has an elongation of 0.5. It has a stretch of 1.5. Put another way, the line has been increased in length by 50% (of its original length), and it is now 150% as long as it originally was.

Here’s a little diagram to summarize:

Okay — so far, so good. We can now calculate length changes. But many of the structural elements in a strain marker not only change their length, but also their orientation. That is to say, they rotate. In order to quantify that, we need to thinking about shear strain. Shear strain (γ), is calculated from angular shear (ψ), a quantity that is measurable in a rock element (again, provided you have a clear idea of what it looked like before deformation).

For any given line in a strain marker, angular shear is defined as the deviation (in degrees) from perpendicular for a line which was originally perpendicular to the line we care about. That is a ridiculous definition at first glance, and it seems to many students quite non-intuitive why one would bother looking for a line originally perpendicular to the line on which we are actually trying to measure the angular shear! I mean, is that abstractified or what?

Before I reveal my stunning new analogy for angular shear, let me diagram the definition for you, and relate it to shear strain.

The line we care about (blue) looks quite the same after deformation as it looked before deformation. So we can’t actually measure anything about it directly that has changed. But when we look at a line which we know was originally perpendicular to it (gray/red), we can measure how it has changed.

The angle between the old perpendicular-line and the new ‘perpendicular’- line is ψ, the angular shear. By convention, clockwise rotations of the perpendicular-line are given positive values, while counter-clockwise rotations are assigned negative values. In the above example, the rotation is clockwise by 35°, so we note it as “ψ = +35°”.

Another way to think about this situation is to consider any point on our perpendicular-line. Here, let’s consider point A:

A was originally at the end of the old perpendicular-line (gray), in position A_o. After deformation, it is now at the end of the new ‘perpendicular-line’ (red), in position A_f. You can see that this outlines a right triangle (yellow):

Right triangles are lots of fun, because they allow us to practice trigonometry. You may remember the mnemonic phrase “SOHCAHTOA” from your high-school trigonometry class. Basically, this relates the angle ψ to the lengths of the sides of the right triangle, where S refers to sin(ψ), C refers to cos(ψ), and T refers to tan(ψ). (That’s sine, cosine, and tangent, respectively.) “O” is the length of the side opposite the corner of the triangle with the ψ angle. “A” is the length of the non-hypotenuse side adjacent to the ψ-angled corner. “H” is the length of the hypotenuse itself. So the change in the position of point A, which could be expressed as Δ(A), is equal to the length of the ‘opposite’ side relative to the length of the ‘adjacent’ side of the triangle. The length of the adjacent side isn’t changing through this deformation, but the length of the opposite side is changing. With a little deformation, it changes a little. With a lot of deformation, the length of that O side increases dramatically.

The formula for calculating shear strain from angular shear is: γ = tan(ψ)

So, with a large value of ψ, you get a large value for tan(ψ). The shear strain (γ) increases along the line we care about (blue) with increasing rotation of the ‘perpendicular’-line (red). The maximum value of ψ would be approaching 90°, which means that your γ would be close to infinity. The minimum angular shear, 0°, would yield a shear strain of 0. Here’s a table of other possible values.

Okay, got it? …(Whew!)

Now for the salamander!

In order to express to my structure students why we bother to pay attention to that ‘perpendicular’-line, I wanted to get them into the perspective of the line we care about: I wanted to get them into its mindset. But of course, lines are geometric entities, and don’t have minds. They are hard to identify with. So I opted for a salamander, because they are cute. Furthermore, salamanders are roughly linear organisms, and have legs poking out of them at roughly right-angles to their spinal column. Here’s my blackboard sketch:

So if you think about what the salamander experiences as it gets sheared (right-lateral), you can imagine what shear strain feels like. The legs go from an original orientation (purple chalk) to a deformed orientation (blue chalk), rotating by about 45°. Along the length of the salamander’s body (orange chalk), the angular shear is 45°, and the tangent of 45° is 1. So the shear strain along the length of this poor salamander is 1. What this means is that the tips of the salamander’s toes (the equivalent of point A in the earlier diagrams) have moved one whole leg-length to the right of their original positions. You can put some numbers to this if you want: Say the leg length is 1.5 cm, and the toes are displaced 1.5 cm. Remember SOHCAHTOA… γ = tan(ψ) = opp/adj = 1.5/1.5 = 1.

There’s something else on the diagram, too: a yellow line. This makes the point that if the leg-line of the salamander is the “line you care about,” and not the spine-line, then the spine-line can be “the line originally perpendicular to the line you care about.” If you’re measuring the angular shear along the leg lines, you would want to measure how the spine-line has rotated relative to its original position. As a result of the shearing deformation in this example, the spine-line has rotated by 45° counter-clockwise. So we’d say: “ψ = -45°”. The tangent of -45° is -1. The shear strain on the leg-line is therefore exactly the opposite of the shear strain on the spine-line.

But the key point of this analogy is to put your mind in the salamander’s perspective, and “feel” the shear strain along the length of your body as deformation proceeds. I hope that intuiting shear strain from that perspective will be useful to structure students learning these concepts.

Focused photo of turbidite

Since yesterday’s live-blogging the rock sample prep routine turned out blurry, I figured I owed it to this sample, and to you, to give everyone a better look. So I scanned it this morning. Penny for scale.

Assuming that your computer screen is vertical, this is in the same orientation as when it was deposited: coarse at the bottom, fine at the top.

Dalmatian pluton

Continuing with some photos from eastern California…

After checking out the faulted moraine, but before heading up the hill to check out the indurated shear zone (which you can just see in the background of this photo), we stopped to check out this visually-striking outcrop:

Look at the glee on the faces of Kurt (green shirt) and Marcos (running; he’s so excited!). We were all pretty jazzed by this polka-dotted outcrop. But in spite of being titillated, we weren’t quite able to figure this thing out.

Here’s Jeff expressing his understanding of just what we’re looking at:

Wes Hildreth, our Bishop Tuff man from the USGS, takes a closer look:

So here’s an even closer look, with my pen for scale:

We’re looking at two main rock types: a dark colored one which is present in grapefruit- to basketball-sized chunks, most with high sphericity and round cross-sections, and a light-colored granite which was present between the dark orbs. There were some more angular dark chunks, too.

Our best bet was that these were xenoliths, stoped off the edges of the granitic magma chamber as it intruded, then piled up on the floor of the magma chamber. If you went “in” to this rock face by a few feet, the ‘xenoliths’ disappeared and you would be swimming in pure granite. The problem with the chamber-floor-debris-layer interpretation is that this “layer” of putative xenoliths was oriented subvertically. So if it represents the floor of an ancient magma chamber, then the whole pluton would have to be rotated by ~90° in order to produce this particular outcrop.

An alternative explanation is that the subvertical orientation of this layer of dark blobs was the wall of a magma chamber, and we’re looking at an outcrop face that was ~parallel to that wall. Probing fingers of granitic magma were working their way into the surrounding rock, perhaps thermally eroding it as they went, producing rounded shapes. Later, once everything had solidified, a fault developed through the transitional zone which divided pure pluton from pure wall rock, and gave us this dual-composition outcrop.

I think I prefer the xenolith interpretation, but to be honest, none of us were really sure what was going on here. We were all struck by the beautiful polka-dot pattern, but among this crew of 30 professional geologists, not one could say for sure just what the hell was going on with this outcrop.

…And I think that’s pretty cool.

Graded bed sample

Today during Physical Geology lab, I used our grinding wheel to plane down a turbidite sample than I collected this past December down in Chilean Patagonia. Thanks to the technological miracle of blogging via iPhone, I can send it to you in a mere 45 seconds. Enjoy the graded bed: let it transport you back to the Cretaceous, in a deep marine basin with periodic influxes of mud and sand. Ahhh, turbidite; take me away!

Mini debris flow

Small-scale debris flows as a muddy slurry flowed over weathered-out chunks of the Bishop Tuff. Probably about two days old. Southern edge of the Volcanic Tableland, last September.

Shear bands in amphibolite

Check out these cool structures in one of the amphibolite bodies exposed along the Billy Goat Trail (C&O Canal NHP, near Potomac, Maryland):

Those are shear bands — basically small shear zones that are discretely localized within a larger body of less-deformed rock. Note the grain-size reduction visible in the shear bands, their dextral sense of offset, and their induration (making them more resistant to the forces of weathering and erosion: they stand up at least a centimeter higher than the rest of the amphibolite outcrop). We have seen a larger indurated shear zone before.

Note that the upper photo is truncated by the format of this blog template — click on it to go to the original image on Flickr, which allows you to see the sense of scale, and a wider view.

Here’s a cool YouTube video showing the process by which these things form (in a nice conjugate set given a homogenous material and plane strain):

Lola “helps” with grading

My cat loves to sit on, or lie on, paper. Maps are here favorite, but she will take a pile of structural geology labs instead, if that’s all that’s available.

Faulted moraine

Continuing with the recounting of geological sights in the Owens Valley, California, area… This one is in the Pine Creek area. Take a look at this photo:

No, that’s not just a portrait of Jeff Lee and his awesome handlebar mustache. Look behind Jeff, on the hillside above.

See the little step down that the hill takes? Let’s zoom in:

Still can’t see it? Here, allow me to annotate it for you:

That’s a fault! A normal fault, with the Jeff side of the landscape dropping down relative to the mountain side (in the distance). Great, you might think. A subtle fault scarp. Big deal.

Oh, but you should not be so quick to dismiss it! After all, the material that the fault cuts across turns out to be a significant clue to the timing of when this fault happened.

This Google Map shows these two very well-developed lateral moraines extending out of Pine Creek Canyon:

In the Pleistocene, a valley glacier glided down out of the Sierran highlands into the Owens Valley to the east. As it flowed, it brought ground-up Sierran rocks down with it, depositing the sedimentary debris as glacial till. The fault above cuts through the northern lateral moraine. The moraines (made of till) are therefore Pleistocene in age, and since the fault cuts across the moraines, it must be more recent than the Pleistocene.

This is not a shocker: the boundary between the Sierra Nevada and the Owens Valley is well known to be a normal fault, as are most of the recent faults in the Basin and Range province. But being able to say “ten feet of offset have occurred on this fault since the Pleistocene” is a significant piece of data.

Cool, huh?

A small shear zone

Back at NOVA Geoblog, I spent a portion of September and October 2009 reviewing the geological wonders I witnessed as part of a GSA field forum in the Owens Valley of California. However, I got distracted by other things, and never finished the series.

I’d like to pick up on that today, looking at a feature which is a typical part of mountain belts like the Mesozoic-aged Sierra Nevada magmatic arc. Here, we will look at a small shear zone exposed in the edge of the Sierra Nevada batholith, in the eastern Sierras, where they meet the Owens Valley, west of Bishop and north of the Tungsten Hills.

Let’s start off with a photo of the area: this is looking to the southeast, with the Sierras to the right, and the Tungsten Hills in the middle distance, and the White Mountain range beyond that. Simon Kattenhorn (blue t-shirt) provides a sense of scale, as do the vehicles parked at the more distant hairpin turn.

You can see those two hairpin turns in the road in this Google Map:

So you can see both in the Google Map and in the first photo that there is a prominent ridge poking up from the hillside there. (It’s the dark green stripe trending ~095° on the map.) From where the cars were parked, this looked like a dike, perhaps of granite, that was weathering out in positive relief. Several of us decided to climb up there and check it out. I’m glad we did, for it turns out to be a positively-weathered shear zone.

Here’s a decent shot that shows well the undeformed granite (bottom third) and the highly-foliated shear zone which cuts across it and deforms it to various degrees (upper two-thirds):

Shear zones are the deep, hot, ductile equivalent of faults. They were first described in the Scottish Hebrides in 1970 by John Ramsay and R.H. Graham¹. The idea is that two big blocks of rock move relative to one another, and if conditions are sufficiently high-temperature and high-pressure, in between will develop a zone of smooshed and squished rocks. The textural patterns that result are called a deformational “fabric,” and it is that fabric that calls our attention to the shear zone. You can see some of the more-deformed areas and the less-deformed areas in this photo:

The photo above also shows a top-to-the-left sense of shear, with the bands of dark minerals “tipping over” to the left.

Why this particular shear zone was weathered out in positive relief (standing up above the surrounding hillside) is unknown to me, but I guess that it may have to do with induration: the phenomenon that sometimes faulting or shearing makes rocks harder than they were pre-deformation.

Anyhow, let’s take a look at the various varieties of fabrics that this rock demonstrates. The shear zone cuts through granite, and here’s a relatively undeformed (~equigranular) exposure of the granite:

Small shear band running through the granite, again exhibiting the asymmetric fabric that suggests top-to-the-left kinematics.

A more pervasively-deformed sample, showcasing several decent augen (hard chunks, in this case of feldspar, that the foliation wraps around):

Nicely-developed S-C fabric, again with top-to-the-left shear sense:

Annotated version of the same photo, highlighting the orientation of the S- and C-foliation surfaces. S-C fabrics are typical of ductile deformation in transpressional shear zones…

Another sample, more pervasively deformed, showing smaller grain size (mylonitization):

Ditto, and with a more-fully-developed transposition foliation:

Lastly, here’s the same face that I annotated above, rotated 90°, which to me brings a different sense of perspective to the outcrop:

What really jumps out at my eye about this outcrop is the more-deformed (highly-foliated) and less-deformed (more-equigranular) domains. This is typical of my experience with shear zones: strain tends to be localized in certain bands, with other areas in the same shear zone being markedly less deformed (an old example from the old blog). This makes it tricky to accurately measure the amount of strain the rock has experienced, because no single square inch of this outcrop surface is “typical” of the overall strain in the shear zone.

To learn more about rock fabrics in shear zones, check out this site.

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1: Ramsay, J.G., and Graham, R.H., 1970. Strain variation in shear belts. Canadian Journal of Earth Sciences 7, 786-813.

Snoverkill, 4pm

A look out the back window as DC breaks its annual snowfall record…