Of Mammoths and Mechanics

2013 May 8
by Daniel Lakeland

I was at the La Brea Tar Pits museum with my kids monday. It's nice the first time, ok the second time, but a little bit too small to go very often. They could really use some updates to the exhibits with some scientific information about what they've discovered.

One of the exhibits, right at the entrance, is where they have buckets of asphalt each bucket containing two cylinders. One about the size of a deer's foot (about 4 cm in diam?) and one about the size of a baby mammoth, or a bear (15 cm or so?). Attached to these is a handle and you can try to "pull  your foot out" of the asphalt.

There are two things to be noticed here, the first is that if you pull lightly it will come out slowly but steadily. I doubt asphalt has a single viscosity, more likely it's a shear-rate dependent fluid. At very high shear rates, like impacts, it's elastic. It's also highly temperature dependent which doesn't enter into this issue because the temperature is controlled in the museum.

The other thing you obviously notice is that if you try your hardest to pull out the cylinders, the narrow one comes out much easier. This isn't too surprising, the thicker cylinder mobilizes much more of the asphalt. But one thing they fail at in this exhibit is to make the exhibit dimensionless. The relevant question is how much force does it take to pull your foot out, as a fraction of the force that an animal that size could generate? Since this is a rate dependent issue, you'd also like to relate it to how quickly the animal normally moves its leg. I imagine it could easily happen that an animal like a deer gets trapped, and it has a fast-twitch muscle and it tries to just pluck its foot out. This rapid application of force causes the asphalt to lock up like a solid and trips the animal leading to it falling down and becoming stuck. A large mammoth lumbers along normally, when it gets its foot stuck it pulls steadily with a mighty muscle and slowly frees its foot. It's also much more stable, so less likely to fall over. So even though we perceive that the deer, with its 4cm diam leg might have had an easier time based on our experiment, it may have been the other way around.

One way to address this would be to plot force necessary to remove the leg at a rate appropriate to the animal's normal leg movement rate and as a fraction of the animals weight vs diameter of the leg as a fraction of the diameter of a large mammoth leg. The x axis would go from 0 to 1 and the y axis would show you who had the easiest time removing their leg. To do this you could start by solving equations of motion for the asphalt flow axisymmetrically around the cylinder. This is a 1D model in the radial direction and should be not too hard. As a first approximation a Newtonian tar could be used. That's relevant to high temperature asphalt but as I mentioned above at cold temperatures you'd need to take rate dependence into account.

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