Densification of Powders under Pressure

Densification of powders under pressure was treated in the past by a number of authors as related to the compaction process, which is an important step in conventional powder metallurgy. Expressions used to describe density-pressure relationship were empirical in nature. For example, Heckel (Ref 21) used the following expression for compaction of powders in a rigid die:

where /Jis relative density, P is applied pressure, and k and A are constants. The expression is in good agreement with the experimental results obtained for a number of powders at conventional pressures (<0.9 GPa) (Ref 6), as well as for the consolidation of powders under high pressures (Ref 22). The deviations from straight lines observed in ln[l/(l - P)] versus P plots for stainless steel powders (see Fig. 3) were attributed to a martensite transformation at higher pressures. The empirical relationship developed by Balshin (Ref 23):

where K and B are constants, is also in good agreement with experimental results. Figure 4 shows the experimental results of Fig. 3 in terms of Balshin's expression.

Fig. 3 Compaction curves (P = f(P)) of a stainless steel powder: ln[l(l - P)] versus P
Powder Metallurgy Process
Fig. 4 Experimental data of Fig. 3 as /-Versus InP

In the frames of the theory of plasticity, Torre (Ref 24) obtained an expression for the final density P, as a function of the applied external pressure, P, and the yield stress of the material, Oy, for spherical particles with spherical voids:

According to this expression, Pf = 87.7% at P = <Ty, Pt = 95.3% at P = 2<?y, f-\ = 98.9% at P = 3rry, and Pf = 99.75% at P = 4(Ty. Experimental results obtained during high pressure consolidation of powders are in good agreement with this expression. For example, the data for pure iron and pure nickel powders in Fig. 5 fit the expression, if for iron iry = 0.55 and 0.7 GPa for iron and nickel, respectively. Experimental values for yield stress in compression of samples prepared from these powders at P = 2 GPa are rry = 0.45 and 0.5 GPa for iron and nickel, respectively.

Fig. 5 Compaction curves for fine iron and nickel powders

Attempts were made to introduce yield stress and strain hardening of the material as densification parameters of powders (Ref 21). Such an approach usually fails because during powder densification, and especially at its final stage, the flow stress and strain hardening in the region of still open pores can be quite high and not typical for plastic deformation of a bulk sample in compression.

The final stage of densification of powders under pressure is worth additional discussion and requires a critical analysis of up-to-date results. In a number of materials densified by plastic flow, cusp-shaped pores <0.1 /'m in size have been observed. The radius of material on the pore surface is much smaller than the particle radius. The material surrounding the pore has a hemispherical shape resembling the edge of a wire produced by extrusion or deep drawing (see Fig. 6). The pore radius decreases with deformation. Such cusp-shaped pores are less stable under applied pressure than the spherical pores considered by Torre (Ref 24) formed by diffusional flow. What in principle opposes the closure of cusp-shaped pores is the increasing surface tension force on the concave surfaces, resulting in LaPlace compressive stresses on these surfaces, <7= 7s/rcurv (7S is surface energy and rcurv is the radius of curvature). These stresses oppose tensile stresses caused by stress/pressure gradient. Enhanced surface diffusion from the concave surfaces to the convex parts of the cusp-shaped pores can be important for the final stage of densification even at a relatively low temperature.

Fig. 6 Schematic of a cusp-shaped pore (triple point) in compacted spherical powders; r1 is the radius of curvature of the pore after densification by plastic deformation.

What can be a more important problem for the final stage of densification by plastic flow is that generation of mobile dislocations is required in order to provide necessary plastic strain. Indeed, if we consider a powder particle under external pressure during the final stage of densification, the gradient of stress/pressure will result in shear stresses only in the regions adjacent to the surface of the pore. The inner part of the particle is under hydrostatic pressure. Regions adjacent to the pore become very small as the pore becomes smaller. For a pore ~0.1 /''m in size, the particle region that can supply dislocations for plastic flow should be about the same size ("-'0.1 /''in). Even for a microcrystalline material, this will most probably be a single grain. Most fine micron/submicron size metal particles are single crystals. For nanoscale particles <0.1 /'m in size, the pore size will be <0.01 /'m. The probability of the presence of an operative dislocation source in such a small region is very low. Thus, for the pore closure by plastic flow at the final stage of densification, shear stresses exceeding the theoretical shear strength are required that are able to nucleate dislocations in a perfect lattice. This requirement can indicate a real change of the densification mechanism at the final stage.

Stresses necessary for dislocation nucleation in a perfect lattice are usually estimated to be in the range of G/10 to G/30, where G is shear modulus. If the shear stresses, cr,h. acting on the material adjacent to the pore surface are assumed equal to the applied external pressure, the pressure P = 3 GPa yields <Tsh = G/10 for aluminum (G = 28.6 GPa) and i7sh = G/30 for iron (G = 86.9 GPa). Higher shear stresses are expected around cusp-shaped pores. In the solution of the Hertzian problem for two spherical bodies of equal radius in contact under pressure (Ref 25), the maximum pressure is given by:

Riax = 0.388-3V2MP

and the maximum shear stress for U= 0.3 is:

= 0.31 Pm where P is applied pressure, E is Young's modulus, and Vis Poisson's ratio.

Maximum shear stresses for a number of metals calculated using a simplified approach that is based on the solution of the Hertzian problem and considers regions adjacent to pores as spheres being in contact under pressure are summarized in Table 1. The results indicate that increasing shear stress can result in nucleation of dislocations even in materials with high shear modulus, e.g. tungsten.

Table 1 Calculated shear stresses "sh in regions adjacent to a cusp-shaped pore

Material

Young's modulus (E), GPa

Shear modulus (G), GPa

^IilîlW

GPa, at applied P, GPa

in G, at applied P, GPa

1

2

3

1

2

3

Aluminum

77.4

28.6

2.7

3.4

4.0

G/11

G/8

G/7

Iron

223

87

5.6

7.0

8.0

G/15

G/12

G/11

Copper

127

47

3.8

4.8

5.5

G/12

G/10

G/8

It should be noted that stresses required to move dislocations in high performance materials with high concentrations of very fine precipitates or in supersaturated solid solutions can exceed the stresses needed for dislocation nucleation in a perfect lattice of the pure metal, especially if superposition of different strengthening mechanisms takes place. According to the estimations made for iron, nickel, and cobalt base alloys (Ref 14), grain boundary strengthening for a nanocrystalline material with —10 nm grain size is —3 GPa, whereas dispersion strengthening due to the presence of fine particles with —10 nm interparticle spacing is —5 GPa. Because the high stresses required for the consolidation of prealloyed powders with very fine microstructures are rarely attainable in practice, another approach has been suggested, namely consolidation of very fine elemental powder blends. The flow stress of such blends will be affected by strain hardening only and will be controlled, at the final stage of consolidation, by stresses required to nucleate dislocation in a perfect lattice.

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