# What is the meaning of pressure of a solid?

Currently I'm taking an introductory course on thermodynamics. I've got a problem with understanding what is the meaning of pressure of a solid body. The question arose when I looked at phase diagram in P, T coordinates.

When we're talking about pressure of gas or liquid, it can be defined thus: it is the force per unit area with which the molecules of the substance hit the surface. This force is due to the molecular motion.

But in the case of solids, molecules don't "move", they just oscillate about their fixed positions. Therefore, they don't hit (in usual meaning) a testing surface and we have problems with pressure's definition. How then the pressure can be defined?

The operational macrosopic definition of pressure in a solid is the same as that for a gas : when the solid's wall is in mechanical equilibrium with its surroundings, that means the force exerted on the wall by the surroundings is exactly compensated by the pressure within the solid times the area of the wall.

First, contrary to the OP's implication, average force acting on a certain test area in a solid need not be necessarily caused by a sequence of impacts only, each of which would result in a certain momentum transfer per time (aka force), as we know it from a gas.

It may also be the result of static forces from the neighbouring atoms or molecules in the solid. Since these forces can be attractive or repulsive in the current deformation state, the pressure in a solid may also become negative (tensile), without necessarily tearing a vacuum bubble into the solid (as opposed to a liquid or gas). And since this causation of pressure does not rely on molecular motion, it can even be present in the limit $$T\to 0$$.

Second, the solid is locally described not only by a scalar pressure, but actually by an axis system and a set of three eigen-stresses along these axes. In other words, stress (the generalization of isotropic pressure for a solid) depends on the direction of the test surface.

Instead of referring to an explicit axis system, one usually uses the stress tensor $$\sigma_{ij}$$ which describes this directionality of loading more easily. We have for the force in a solid

$$d F_i=\sum_{j}\sigma_{ij}d A_j$$

In such a multiaxial stress state, an equivalent of hydrostatic pressure can nevertheless be defined. It is mainly given by the trace of the stress tensor, which is a scalar invariant (i.e. does not depend on direction). We have

$$p=-\frac{1}{3}\sum_j \sigma_{jj}$$

The minus sign is a result of stress being usually defined as positive for tension (which tries to extend the material locally), while pressure is usually defined as positive for compression.