Why is pressure always non directional?

Question

While hammering a nail (before it is in the wall) it is pretty evident that the tip of the nail is going to be applying a force directed along th axis of the nail, then why is it said that pressure is always non directional ?

Answer 1

I think it's a bit of a wording issue. The Cauchy stress tensor can be decomposed as the sum of a hydrostatic and a deviatoric component as $\sigma=\mathbf{s}+\frac{1}{3}\mbox{Tr}(\sigma)\mathbf{I}=\mathbf{s}+p\mathbf{I}$ where $p$ is the pressure, and since $F=\sigma\cdot\hat{\mathbf{n}}$, the force related to the hydrostatic pressure component simply reduces to $F_{hyd}=p\hat{\mathbf{n}}$, whose magnitude is evidently independent of direction $\hat{\mathbf{n}}$.

More intuitively, if you imagine a bubble floating in the air, the interior walls of the bubble are being bombarded by air molecules flying in all directions, but the air molecules have no preferred direction, and they exert force in all directions. So we say that pressure is independent of direction in a region where the pressure is equal everywhere.

This does not necessarily mean that pressure can't be manipulated to exert a force in a certain direction to do work, though. For example, inside a gun as it fires, you essentially have a cylinder (with one end blocked, and the other end fitted with a bullet) with a high pressure gas inside. The gas exerts force isotropically (in all directions equally), so the back face of the bullet is feeling the same force per area as the walls of the gun are. But, the bullet is free to move (while the walls aren't, unless the pressure so high that the gun explodes), and so it is forced out of the gun. But throughout the whole process, the gas was in fact exerting force isotropically in all directions.

Similarly, in the case of a nail being driven into a piece of wood, at the nail-wood interface there is an extremely high pressure, and it exerts force in all directions. However, just like the bullet (and not the walls) was able to yield in the case of the gun, the wood (and not the nail, nor the person driving it in) is able to physically yield under the pressure. Thus the wood is moved out of the way under the pressure, even though pressure at the interface applies in all directions.