What are equations related to the compaction of composite metal powders?

Say I am trying to create a metal composite and have a mixture of 3 different metal powders each with varying weight %'s. I'm going to compact them into the shape of a ring by applying a force P. The inner and outer radius of the ring are given and I am asked to determine the thickness of this ring after compaction.

My first question is, each of these metals have a true density (ie. 8900 kg/m3 for Al). Looking at this powder alone, if density is reached, can it be further compressed or does adding any more force P do nothing because it physically cannot obtain a greater density (due to atoms themselves etc.).

Second question, assuming you cannot compress more than the true density, is there a formula to determine how much force corresponds to the density achieved?

• You can't reach full density without heat ( forging). I don't know about equations , but powdered metal part manufacture has been used for over 80 years ; There must be much published information. May 27 '19 at 15:50

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".

P.J. Denny states in Powder Technology 127 (2002) ("Compaction equations: a comparison of the Heckel and Kawakita equations") that:

The powder metallurgy area tends to use the Heckel equation because metal all compact by the same mechanism (plastic deformation), and Heckel did his work using metal powders. The ceramics area tends to use that first used by Brusch, 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. $$K$$ 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.

Denny 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: Kawakita, 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.

See: Adams, M.J., Mullier, M.A., Seville, J.P.K., 1994. Agglomerate strength measurement using a uniaxial confined compression test. Powder Technology 78, 5e13.

$$\ln \left( \frac{\tau_0}{c_0} \right) + c_0 \epsilon + \ln (1 - e^{-c_0 \epsilon}) \space \space \space \space \text{(for large strains)}$$

where $$\tau_0$$ is the apparent single particle strength and $$c_0$$ is a constant related to the friction between particles.

... the natural strain $$\epsilon$$ is defined as

$$\epsilon=\ln \left( \frac{h_0}{h} \right)$$

where $$h$$ is bed height of a load-bearing column.

Summary

The increase in pressure required to compress metal powders is roughly proportional to the $$\ln\left(\frac{1}{\varepsilon}\right)$$ where $$\varepsilon$$ is the void coefficient (percentage of space in the mixture that isn't occupied by metal). Different empirical correlations attempt to model nonlinear deviations for low compaction pressures but most Heckel plots ($$\ln\frac{1}{1-D}$$ versus $$P$$) for metal powders eventually become linear for sufficiently high $$P$$. The papers I skimmed show that metals powders do not reach what you call the "true density" ($$\varepsilon=1-D=0$$) for pressures tested. The Denny paper has several plots for different groups of metals for compaction pressure up to $$700 \space\text{MPa}$$. I would recommend reading it.

Many factors come into play, such as the size of the grains, their roundness, and most important the granular aggregate distribution, so that in microscopic scale they are of not similar size but of a range of sizes which leave the least amount of cavity.

As to how much force needed to achieve maximum compactness again depends on the shape of the grains. If the grains have sharp corners, they require less forces to flatten and compact. One would estimate that after compression beyond just shaking and initial compression, the force required to bring the grains past the yield point would increase fast.