How to see intuitively that $\kappa_T$ is constant for solids? In thermodynamics we define
$$\kappa_T = -\dfrac{1}{V}\left(\dfrac{\partial V}{\partial p}\right)_T.$$
By this definition, if we are treating $V$ as a function of $p$ and $T$, then $\kappa_T$ is also a function of $p$ (since we are keeping $T$ fixed), thus it can depend on pressure and temperature.
I've read, though, that $\kappa_T$ is nearly constant for a solid which allows us to treat it as a constant in whatever we have to do.
My question here is: how can we see conceptually that $\kappa_T$ is constant and doesn't depend on the pressure?
Obviously there are other ways to see this: I believe (although I haven't tried yet) that using Statistical Mechanics we could deduce $\kappa_T$ in the same way as we get the specific heat of solids considering the solid as a lattice of harmonic oscillators. This would be a mathematical deduction which could show why $\kappa_T$ is nearly constant.
Another way would be experimentally. We could perform experiments and looking at the data conclude that $\kappa_T$ is nearly constant.
But is there any other, conceptual way, to infer that $\kappa_T$ is nearly constant? Is there any intuitive point of view here?
 A: The force between atoms in a solid is going to look something like:

This is obviously not linear, but for small displacements $\delta r$ around the equilbrium position we can expand the force as a polynomial:
$$ F(r_0-\delta r) = A\delta r + \mathcal O (\delta r^2) $$
then drop the terms in $\delta r^2$ on the grounds that they are small. So for sufficiently small displacements we can assume the displacement is linear in the applied force, which will give us a constant compressibility.
And most solids and liquids have a very low compressibility so under most circumstance the displacements of the atoms $\delta r$ are indeed very small. That's why the compressibility is roughly constant.
Once you apply enough pressure to move outside the linear area the compressibility cannot be assumed constant. For example a quick Google found this data for the variation of the compressibility of water with pressure:

A: We know intuitively from everyday experience that solids are nearly incompressible.  No matter how hard we press on them from all sides, the volume of the solid hardly changes.  It takes a huge increase in pressure to cause a decrease in the volume of a solid (or liquid).  In fluid mechanics, we often treat liquids as being perfectly incompressible.  A liquid or solid being nearly incompressible means that $\kappa$ is very close to zero.
