assuming you cannot compress more than the true density, is there a formula to determine how much force corresponds to the density achieved?
I scanned through some literature and found this book to lead to useful information. Chapter 2 "Bulk Solid Characterization", section 4 "Compressibility", references several sets of equations, each set from a different paper about compressibility of solids:
Heckel Equation
See: Heckel, R.W., 1961. An analysis of powder compaction phenomena. Transactions of the Metallurgical Society of AIME 221, 1001e1008
$$ln \left( \frac{1}{1-D} \right)=kP+A$$
where $D$ is the relative density of the bulk solid at the applied pressure $P$ and is equal to $1-\varepsilon$, and $k$ is a material-related parameter and is inversely related to the mean yield pressure of the bulk solid. A larger value of $k$ indicates a smaller yield pressure and hence a greater degree of plasticity. $A$ is a parameter related to the densification due to die filling (e.g. initial powder packing) and rearrangement of particles.
Note: $\varepsilon$ is void fraction, or "proportion of the total volume not occupied by particles".
<PERSON> states in Powder Technology 127 (2002) ("Compaction equations: a comparison of the <PERSON> and <PERSON> equations") that:
The powder metallurgy area tends to use the <PERSON> equation because metal all compact by the same mechanism (plastic deformation), and <PERSON> did his work using metal powders. The ceramics area tends to use that first used by <PERSON>, which plots the relative density ($D$) against the logarithm of applied pressure as shown in the following:
$$D=\frac{1}{V}=a_2+K_2\ln{P_a}$$
Also, regarding the Heckel Equation:
$$k=\frac{1}{3\sigma_0}$$
$$A=\ln \left(\frac{1}{\varepsilon} \right)+B$$
where $\sigma_0$ is the yield strength of the material being studied. 453
$K$ is thus inversely related to the ability of the material to deform plastically. Heckel studied mainly powders and the equation was only meant to apply to materials that compact by plastic deformation. The term $3\sigma_0 \space (=1/K)$ is often called the yield pressure. Heckel found that, for metal powders, the value of the yield pressure was increased considerably by the presence of oxide surface layers (which are especially likely to be significant with ultrafine metal powders).
$B$ appears to be another material-specific empirical constant.
<PERSON> also indicates the significant variance in Heckel plots can appear for low pressures especially when the powder consists of an aggregate of a various particle sizes.
Kawakita Equation
See: <PERSON>, K., Lüdde, K.H., 1971. Some considerations on powder compression equations. Powder Technology 4, 61-68. https://doi.org/10.1016/0032-5910(71)80001-3
$$C=\frac{V_0 - V}{V_0}=\frac{abP}{1+bP}$$
where $C$ is the degree of volume reduction, $V_0$ is the initial volume of the bulk solid, $V$ is the volume of the bulk solid at pressure $P$, and $a$ and $b$ are two material constants with $a$ indicating the initial powder porosity before compression (i.e., the total proportion of reducible volume at maximum pressure), and $b$ being a constant related to the yield stress of particles.
[The equation] can be expressed as
$$\frac{P}{C}=\frac{P}{a}+\frac{1}{ab}$$
For given compression data, one can plot $P/C$ as a function of the compression pressure, which is often referred to as the Kawakita plot.
Adams Equation
See: <PERSON>, M.J., Mullier, M.A., Seville, J.P.K., 1994. Agglomerate strength measurement using a uniaxial confined compression test. Powder Technology 78, 5e13. 193