Reading 2/10/2008

Section 2.8: This section was very cool. I'm guessing that the energy function defined herein is going to be the key to many later results. That said, I'm still trying to get used to the physicist differential notation. Usually I need to convert this in my mind to something that is actually mathematically well-defined, but the only way I see to do that here is to change d[whatever] to Delta[whatever] and then take limits.

I'm a little confused on the implication of (2.70) leading to (2.71). The notion of exact differential is not quite rigorously defined; is this perhaps equivalent to the existence of a gradient function in a conservative field? That's the explanation that makes sense. Then (2.72) would become a dot product of the gradient with dr, which would make considerably more sense to my mathematical sensibilities.

Equation (2.75) is very cool and intuitive. (Although I feel like there's got to be a more elegant, if less physical, way to get here.) We've got a function which is literally the gradient of the stress tensor, when considered as a vector in R⁹.

I don't necessarily see why (2.78) needs explanation - the derivative of component with respect to another component is very clearly a delta function under any circumstances.

We get a beautifully simple equation in the end, but my only question is how are we assured that the components of the tensor c are space invariant? Is this an assumption or is this universally true?

Section 2.9: This is very intuitive stuff. In essence, if we can find rotations that leave the stress tensor (and therefore the potential energy) unchanged, then these rotations are symmetries, and through equation (2.90), force symmetries into the elastic constants.

In the text, we've essentially noted that without knowing the tensor itself, we can argue physically that a certain structure must have certain elastic symmetries. But it seems we could actually reverse this argument. If we construct the elastic tensor of a structure, we can find all of its symmetries, even if they are not obvious due to coordinate choice.

So let's denote the elastic 4-tensor by T. If I remember correctly, we denote the action of a rotation A on T (which is really a pullback, but...) by writing

A^*T(u,v,w,z) = T(A^Tu,A^Tv,A^Tw,A^Tz)

where u,v,w,z are vectors. This agrees with (2.87). Note that we are using here that A^T = A^-1. Writing it this way, we're actually looking for rotations A such that

A^*T = T

In essence, we can just find all rotations that pull back under * to a transformation fixing T. I still feel like this could be formulated as an eigenvalue problem, which was my initial motivation for this tangent, making such a task even simpler, but I can't think of how at the moment.

On another note, I'd prefer a slightly better justification for why (2.97) is a sufficiently general isotropic tensor.

The derivation at the end is slick, though.