I understand the stress tensor at a specific point $P$ as follows. You draw a cube with its center at $P$ and look at its faces. The face with its normal vector in the positive $x$ direction has a certain force acting on it. $\sigma_{xx}$ is the limit of the $x$-component of that force over the area of the face as the cube shrinks to size $0$. Similarly, $\sigma_{xy}$ is the limit of the $y$-component and so on.
Specifically, the force on a face is essentially the sum of the forces on all the particles on the face. If the force inside the material is roughly continuous (is that true for the insides of materials?) then as you take the limit as the cube shrinks to size $0$, shouldn't the force on all the particles on each face of the cube be essentially the same, thus leading to the forces on each face being the same and thus making, for example, $\sigma_{xx}=\sigma_{yx}=\sigma_{zx}$?
It seems to me somewhat obvious that all stress tensor components whose second element is $x$ should be the same (along with $y$ and $z$) but given how that is not stated in physics books that is definitely not the case.
Where does my logic go wrong? Also, could people give me some examples of where the forces on the different faces of the cube are clearly different even as the cube size gets extremely small? I would like to gain a better understanding of stresses.
Best Answer
The stress tensor is a continuum mechanics concept, and you should expect it to break down (or become far more complicated) at the particle scale anyway.
However, a problem with your logic is that stress isn't really concerned with the net force on particles. Assume that your material is stationary: the net force on every single particle is zero, regardless of whether there is stress present or not. (I'm also not sure you're aware that $\sigma_{xy}=\sigma_{yx}$, nor that you appreciate $\vec\sigma[-\hat n]=-\vec\sigma[\hat n]$.)
Instead, stress is all about the relationship between adjacent particles (and how they respond to this deviating from their relaxed or "preferred" configuration). Think of a (stationary) material body that is subjected to large (static) external forces, and imagine surgically removing a cube (or other test volume) from the interior of this body. When you remove this cube, what forces would you have to apply to the cube's sides to keep it a cube (that is, to maintain it in exactly the same configuration as before you removed it from the bulk, and prevent it relaxing or "springing" back to some other shape)? Alternatively, what forces would you have to apply to the internal surfaces of the void (the hole left in the middle of the original body) to prevent it from deforming in response to removal of that piece of material?
A simple example (described in Landau & Lifshitz, and probably any other elasticity textbook) is a block of material in a vice, such that the stress tensor is homogeneous (uniform and constant, and the same for any sized cube) and $\sigma_{xx}$ is its only non-zero component (i.e. $\sigma_{xx}\neq\sigma_{yx}$).