Confusing

Okay everyone, this posting will be a bit heavier than some; sorry it it's over the top.

I'm supposed to be doing a project wherein I measure the elastic moduli of simple fluids by simulation--we are particularly interested in the shear modulus. In essence, it is simple. If I put a shearing force on an object, it will push back. For small perturbances, the restoring force (the pushing back) is proportional to how much I deform the solid.

dF/A = -G du/l,

where dF is a differential amount of force, G is called the shear modulus, and du is a differential amount of "strain" (deformation). A is the area over which the force is exerted (ie, we normalize the force with the area) and l is the length between two points in the object, relative to which I measure the distortion (the strain). If you're already lost and you actually care about this, just comment and I'll offer you the explanation specially designed for people with no physical intuition (please realize when I say things like that I don't mean them). For those of you who know some physics, this is precisely analagous to the harmonic oscillator: F = -k x. We just normalize it to make it a property of the substance, rather than the system and/or situation.

So far so good, but I'm reading this paper, and they define the strain tensor as:

u,ij = (1/2)[(du,i/dx,j) + (du,j/dx,i)],

where u,i is the ith component of the displacement and x,i is the ith coordinate of the displacement. It seems like this derivative would always be either 1 or 0.

This had me confused for quite some time, untill I figured out what they meant by those derivatives. Well here's what it is. As I move along coordinate x,j, AFTER the deformation, I will observe that the amount the object is displaced from its former conformation changes as I move through the object. If that last sentence didn't make sense, don't feel bad. It's very hard to explain in words. I finally understood when my friend Dan found a nice article at Wikipedia that actually had pictures to explain this stuff. If you want to understand, go there. It turns out that this derivative is equal to the strain for small perturbations (it's a first order expansion).

I just wanted to mention this, since this work is actually a major part of my life.