Reading 2.8-2.9

2.8: I'd like to try and address Max's question about the quasistatic approximation. While reading this section, I too was frustrated by this blatant "physics move" where you assume something is so that doesn't really make sense and then things work out the way you want it to.

For the short story, skip this paragraph: However, just like most blatant physics things, this one ended up having a purpose that I realize if you bash your head against your book enough. This section is all about understanding the relationship between (the) stress (tensor) and (the) strain (tensor), i.e. the c_ijkl. If we think about the simpler Hooke's Law that we already know, F=-kx, we are making the same quasistatic approximation. We are saying, "irrespective of the way in which we got to displacing a spring by x, what is the force exerted between either ends of the spring?" If we think about "what actually happens" as we push a spring to a new displacement if the acceleration is fast, we see a wave propagate down the axis of the spring. However, in the quasistatic approximation of pushing a spring really slowly (i.e. not very much acceleration) toward the new displacement, we see that Hooke's Law still does exactly what we want it to do (even though it would do exactly what we want it to do once the wave died out, and the displacement of the spring became fixed). Even though the law works all the time (hence "law") we needed to derive it from the quasistatic situation, since it's waaaaay easier to see it in action that doing all of the simulations of a wave along a spring in your head to come up with the F=-kx relationship.

Long story short, in this scenario we want to know, like the spring, what happens once the deformation has occured and not what is happening during it, just to understand the Hookean relationship at work. Once we understand this, I think we can extrapolate to the moving case, saying that at every instantaneous moment the stress is related to strain as such, because every moment is quasistatic...right?

Wow, I went on a lot longer than I had expected to. Does this help, Max? Is it right, professors?

Also, nice work, Higgy, on your 6-choose-2 method of finding the 21 independent constants. That's slick.

2.9: This is a pretty section, except for the means-to-an-end of 2.100. I am a little put-off by the "Now consider the specific case i=k=1, j=l=2." I mean, they're all δ's, so we really only need i=k, j=l,i≠k...right? However, I know really see even where this is something that's recommendable. It is late, and I'm not thinking very hard, so that could also be the problem. Also, I bashed my head against the book for the last section. It hurts now. Someone else can pay it forward and bash their head on section 2.9 for me. Otherwise, Lamé constants are sweet!

First Post

Well, I got to this a little later than everyone else, but hey, I'm not enrolled in the course! I'm just going to learn everything you're going to learn and not get credit for it. Tah Dah!

My name is David Gross. I am a senior Mathematics Major at HMC and I have a Humanities concentration in Literature. I am taking this class because I would really like to learn some higher mathematics my last semester at Mudd, even if I don't have room in my schedule for it. I took Fields and Waves from Sahakian and learned a lot at the time, but can't quite use tensor notation fluidly (a.k.a. I may have done poorly or less than my best during that course). As such I feel this would be another way to robustify my skills.

My favorite equation...that's tricky. Here's one of my favorites though. Check out this dimensionless ODE:

dx/d&tau = rx(1- x/k) - x^2/(1+x^2)

You can think about x as the percentage amount that a population of bugs is over a critical amount at which birds start preying on them. You can think of r as the growth rate of the bugs and you can think of k as the carrying capacity of the bugs in whatever environment then live in.

What's interesting about it? In the two dimensional parameter space, (r,k)-space, there is a region where the bugs go into refuge, and there is a region where there is a bug outbreak. However, there is also a bistable region where both stable states can exist. This is because the parameter space is the projection of a cusp of a catastrophe surface, and seasonal changes in the predation parameters, carrying capacity, and growth rate (which happen) can lead to a hysteresis loop (like it does). Hooray!

Here's a fact: My favorite cheeses are Bleu Cheeses, even though my mom hates them!