Compressibility of Solids

Question

This is an extension of the question asked here, with the same setup.

Question 1

In 3D compressibility is: $$\beta = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_T$$

In 1D compressibility is: $$\beta = -\frac{1}{L}\frac{\partial L}{\partial F}=\frac{1}{\kappa x_{eq}}$$

My question is the units seem to be different in the 3D case vs the 1D case. In 3D, we have: $$\frac{1}{V} \Longrightarrow \frac{1}{m^3}$$ $$\frac{\Delta V}{\Delta P} = \frac{\Delta V}{\frac{\Delta F}{A}} \Longrightarrow \frac{m^3}{\frac{N}{m^2}}$$ So in total we get: $$\frac{1}{m^3} \times \frac{m^3}{\frac{N}{m^2}} = \frac{m^2}{N}$$

In 1D we have: $$\frac{1}{L} \Longrightarrow \frac{1}{m}$$ $$\frac{\partial L}{\partial F} \Longrightarrow \frac{m}{N}$$ So in total we get: $$\frac{1}{m} \times \frac{m}{N} = \frac{1}{N}$$

Which seems to suggest the units of compressibility are different, which confuses me because shouldn't they have the same units?

Question 2

Why does $P \to F$ from 3D to 1D? This was asked in the link attached but there was no answer given there.

Question 3

Why is $$F = - \frac{d V}{d x}?$$ I know that force is the derivative of potential energy, but here $V$ is potential, not potential energy. And from here, potential energy and potential have different units, so if force is the derivative of potential energy, it seems to me that it cannot be the derivative of potential too.

What am I missing here?

Answer 1

It's unremarkable for properties of systems of different dimensionality to have different units. For 1D, 2D, and 3D idealized systems exposed to equiaxial, equibiaxial, and equitriaxial stress* (meaning the same normal stress applied in every available dimension), the compressibility has units of $\frac{1}{\mathrm{N}}$, $\frac{\mathrm{m}}{\mathrm{N}}$, and $\frac{\mathrm{m}^2}{\mathrm{N}}$, respectively. No conflict arises because these units are never used simultaneously; a system can't be simultaneously modeled as 2D and 3D, for example.

*$\sigma_{xx}$, $\sigma_{xx}=\sigma_{yy}$, and $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=-P$, respectively, where the pressure $P$ is taken as negative (tensile stresses as positive). In 1D, if a certain cross-sectional area $A$ is assumed, then $F=-\sigma_{xx}A$ might be used instead for an axial compressive force.

Does that answer your first two questions? I don't understand your third question, which reuses $V$ (volume?) to probably mean a potential energy, sometimes called a potential.