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!

Reading 2.4-2.6

So I'm trying to get ahead a bit since I have my first papers and presentations due next week. Luckily I can choose to get ahead in a course that doesn't give me any credit!

Section 2.4 really was great. Even though I've seen the continuity equation derived before, this was done in such a blatantly analogous way to the derivation of any kind mass conservation equation...I wonder what I must have been paying attention to instead of this the last time I saw it.

I find the argument used to enforce the symmetry of &Pi_ij in section 2.5 very interesting, and it's motivation comes from making sure that the thing that we derived from physical examples makes physical sense. Additionally, the 3D conic section representation of the information in a symmetric rank-2 tensor is awesome, and completely new to me. Sweet!

Strain in 2.6 is great, but it is nothing needed without the equation of state in 2.7. That's some good stuff.

Reading 1.6-2.3

This is some good stuff in this reading. So I've got a question but it's mainly about how the stress tensor has to do with thinking about in regard to fluids:

Is the stress tensor the location in your e.o.m. where you would effectively specify whether a blob of fluid or gunk is suspended in air as opposed to sitting in a tank? I mean, gravity has something to do with the difference between those situations. Ok--lets say I actually meant the difference between fluid sitting in a tank and fluid in a blob doing infinite free-fall into the fiery pits of Pandemonium. What about then?