Density Distributions in Powder Metal

Compacts: Prediction and Control

Report No. 01-2 Research Team: Jayant Khambekar (508) 831-5981 jayant@wpi.edu Mark Richman (508) 831-5556 mrichman@wpi.edu Focus Group: Bill Jandeska (GM) Michael Krehl (Sinterstahl GMBH) Owe Mars (Hoganas N.A.) Sim Narasimhan (Hoeganaes) Renato Panelli (Mahle Metal Leve S.A.) Daniel Pfister (Osterwalder AG) Prasan Samal (OMG Americas) Achievements this Work Period (April 2001 to November 2001):

In this work period, we extended our previous work on single punch compaction of hollow, cylindrical bushing-like parts to cases in which the coefficients of friction between the die-wall and the powder, and between the core rod and the powder depend on the density of the evolving compact. It is critical that the friction between the compact and the containing surfaces be properly modeled because the friction forces that develop at the core rod and die wall induce the density variations that we wish to predict.

During compaction, the compact evolves from its initially loose powdered state to

its final green solid state. As a result, the nature of the interactions between the compact and the containing surfaces change dramatically from the beginning to the end of compaction. Until now, our analysis was restricted to cases in which the coefficients of friction were taken to be constant throughout the compaction process. The problem with such an analysis is that it ignores the roughly two-fold increase in density that occurs in the course of a typical compaction process. Moreover, it ignores the spatial variations in density that occur at any instant of the compaction. Clearly the first of these variations is more substantial than the second because the changes in average density from the beginning to the end of compaction are greater than the spatial variations that occur at

1

any instant. However, the percent top-to-bottom density variations at intermediate stages of compaction can be considerably larger than the final top-to-bottom density variations in the green compact. Consequently, even these variations exert a significant effect on the manner in which the densities evolve to their final green states.

In our work concerning density-dependent coefficients of friction, we focus on

bushing-like parts that have rotational symmetry about their centerlines. Such parts require two-dimensional models to predict both the axial and radial variations of the density. However, the radial variations in the densities of these parts is small relative to the axial variations that occur due to corresponding pressure differences.. For this reason, we employ a one-dimensional model that is capable of predicting the axial variations in density and pressure. The model has the virtues of both mathematical and numerical simplicity.

We have surveyed the literature and have identified several workers who have

measured either directly or indirectly the dependence of the coefficient of friction on density. In this report, we propose a mathematical dependence that is qualitatively similar to that measured, and employ it to predict how the density distributions evolve during compaction. We compare these predictions to those that would be obtained using constant values for the coefficients of friction. Typically, the constant values are taken to be either the value corresponding to the average density at the final stage of compaction, or the value corresponding to the average density throughout compaction.

Current Project Goals: To develop models that predict density variations in simple but realistic green P/M compacts. The models should account for the effects of:

• density dependent powder properties, such as compressibility and fluidity;

• density dependent coefficients of friction between the evolving compact and the die wall, and between the evolving compact and the core rod;

• non-uniform powder fill before compaction begins;

• friction between the punch and the powder;

• single- versus double-punch compaction.

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Report No. 01-2

DENSITY DISRIBUTIONS IN POWDER METAL COMPACTS:

PREDICTION AND CONTROL

Jayant Khambekar and Mark Richman

Powder Metal Research Center Metal Processing Institute

Worcester Polytechnic Institute Worcester, MA 01609

1. INTRODUCTION: In this report, we modify our previous model to improve predictions of density and pressure distributions during compaction of hollow, cylindrical, bushing-like parts. The work presented here advances our previous work by accounting for the fact that the coefficients of friction between the die wall and the powder, and between the core rod and the powder vary with density. In our previous work, these friction coefficients had been taken to be constant throughout the compaction process and throughout the compact at any instant of compaction. Our calculations include the dependence of fluidity on density that we described in our last report. As in our previous work, we assume that the frictional interactions between the evolving compact and the surfaces that contain it are governed by Coulomb’s law. Accordingly, the shear stress that develops at any such surface is proportional to the normal pressure that develops there. The factor of proportionality is the coefficient of friction appropriate to the surfaces in contact. Until now, we have carried out compaction analyses using the assumption that these coefficients are independent of the density as compaction evolves. Ignoring the variations with density, therefore, accounts for neither the significant increases in average density that occur as the compaction proceeds from beginning to end, nor the smaller variations in density that exist throughout the compact at any single stage of compaction. In order to model the dependence of friction coefficient on density during compaction, we adapt the experimental results of Sinka et.al. [2001] and Solimanjad et.al. [2001]. In the first case, Sinka et.al. [2001] measure the measure the density dependence indirectly, while in the second case, Solimanjad [2001] do so directly. In both cases, the results are qualitatively similar. In this report, we propose a mathematical dependence that is qualitatively similar to that measured, and employ it to predict how the density

3

distributions evolve during compaction. We compare these predictions to those that would be obtained using constant values for the coefficients of friction. Because our previous work has demonstrated that the radial variation of density in bushing-like compacts is relatively small, we focus only on the variations of the density and pressure in the axial direction of the compact. Because of the rotational symmetry of the parts, we are able carry out a simple one-dimensional analysis of the compaction. The virtue of such an analysis is that it is possible to account for the dependence of fluidity on density without introducing significant computational complications to the calculations. 2. COMPACTION ANALYSIS WITH PRESSURE- AND DENSITY- DEPENDENT COEFFICIENTS OF FRICTION: In this section, we present the governing equations of force and mass balance in a simple one-dimensional model that accounts for frictional effects. The force balance derived here is similar that employed in elementary compaction analysis (see German [1994], for example) of solid cylindrical parts. However, it is a generalization of that equation because it incorporates the effects of friction at both the core rod and the die wall. A more complete description of the derivation is given in PMRC Report 01-1. The abbreviated derivation given here is included so that this report will be self contained. We also present the appropriate form of the balance of mass and a required equation of state that relates the pressure at every point in the compact to the corresponding density. We then describe experiments conducted to measure the dependence of friction coefficient on compact pressure, and propose a mathematical model that gives a pressure dependence that is consistent with the experimental results. By combining this equation with the equation of state, we predict the explicit dependence of friction coefficient on density. Finally, we numerically calculate the axial variations of pressure and density as well as the required compaction load, and compare the results to those that would be obtained if the coefficients of friction were taken to be constants. 2.1 Force Balance, Mass Balance, and the Equation of State:

We are concerned with the single punch compaction of a powder in a hollow cylindrical die of inside radius Ri and outside radius Ro. The height of the powder before compaction is L, and the height after compaction is H. Of special interest are the ratio a=Ri/Ro, and the aspect ration h=H/Ro of the final compact. The post-compaction geometry is shown in Figure 1. The average pressure applied over the top surface of the compact is po, so that the total compaction load F is π(Ro

2- Ri2) po. We establish a

cylindrical coordinate system in which the axial z−coordinate measures distance from the lower face of the compact, and the r-radial coordinate measures radial distance from the

4

centerline. Because of symmetry, there are no variations with angle of rotation about the z-axis.

In principle, the axial pressure p, the radial pressure σ, and the shear stress τ vary with r and z throughout the compact. In terms of the axial pressure p and the shear stress τ, the axial equilibrium equation is given by,

rr

rzp

∂∂

=∂∂ )(1 τ . (1)

Figure 1: Post-compaction geometry as seen in avertical plane through the center of the compact.In the pre-compaction state, the height H isreplaced by L.

H r

z

Ro Ri

F=π(Ro2-Ri

2)po F

Our earlier work has demonstrated that the radial variations of the pressures p, σ, and τ do not produce significant radial variations in density ρ. For this reason, we take p,σ, and τ to be functions of axial z-coordinate only.

Multiplying equation (1) by r and integrating with respect to r from Ri to Ro

yields,

),(),(2

)( 22

zRrRzRrRdzdpRR

iiooio =−==

−ττ , (2)

Boundary conditions at the core rod (r=Ri) and outer die wall (r=Ro) relate the shear stress to the radial pressure through Coulomb friction. If µi is the coefficient of friction between the core rod and the powder compact, and µo is the coefficient of friction between outer die wall and the powder compact, then the boundary conditions are,

),(),( zRrzRr iii =−== σµτ and ),(),( zRrzRr ooo === σµτ . (3)

In our previous work, the dependence of the coefficients of friction on either pressure or density has been ignored. Much of our effort this work period has been devoted to incorporating the dependence of µi and µo on axial pressure p and therefore density ρ.

In order to write the shear stresses at the die wall and core rod and then the force balance (2) explicitly in terms of pressure p, we recognize that the local radial pressure σ is proportional to the local axial pressure p. That is,

5

pασ = , (4)

in which the coefficient α is referred to as the fluidity and depends on the local value ρ of the density. The dependence of the fluidity on pressure or density has been described in detail in PMRC Report 01-1.

Employing boundary conditions (3) and pressure relation (4) in force balance (2) yields,

pRR

RRdzdp

io

iioo

)()(2

22 −+

=µµα

. (5)

Integration of equation (5) introduces a constant that is determined by the requirement that pressure p(z=H) at the top of the compact must be equal to the pressure po applied by the punch,

opHzp == )( . (6)

If the coefficients α, µi and µo were constants, then equation (5) could be integrated to show that the pressure distribution p(z) scales with the top pressure po. Under these circumstances, the ratio p/po can be determined without determining the applied pressure po. If, on the other hand, the coefficients α, µi and µo depend on p (or density ρ), then p does not scale with po, and the ratio p/po itself depends on po.

The relation between the applied pressure po and the height H of the compact is determined by the balance of mass. If Z is the axial coordinate that measures distance from the bottom of the powder in the pre-compaction state, and if η(Z) is the pre-compaction density distribution in the die, then the balance of mass of a thin disk of pre-compaction thickness dZ and corresponding post compaction thickness dz requires that,

)()(

Zz

dzdZ

ηρ

= , (7)

where ρ(z) is the final density distribution in the compact. The boundary conditions that ensure that the top and bottom surfaces of the powder before compaction correspond to the top and bottom surface of the compact after compaction are simply,

and LHzZ == )( 0)0( ==zZ . (8)

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On of these boundary conditions is used to initiate the integration of equation (7) while the other will not be satisfied unless po is chosen correctly. An alternative form of the balance of mass is given in integral form by,

. (9) ∫∫ =HL

dzzdZZ00

)()( ρη

In order to relate the pressure p(z) to the density ρ(z), we employ the equation of state,

[ ][ ]pM

MpMMM ηαβη

αβηηρ)1()()1()(

−+−−+−

= , (10)

where M is the maximum theoretical density of the powder and β is the compressibility of the powder in its loose state. If the initial pre-compaction density η depends on Z, then equation (10) determines the dependence of the green density ρ(z) only after equations (5) and (7) are simultaneously integrated to determine Z(z). 2.2 Dependence of Friction Coefficents on Pressure: In the past, several workers have investigated the relation between sliding friction and such parameters as temperature, contact resistance, and surface area that have relevance to powder metallurgy. Earlier work of this type include those conducted by Bockstiegel et.al [1971], Mallender et.al. [1972, 1974], and Ernst et.al. [1990]. However, only recently have measurements been made that characterize the changes in the coefficient of friction during compaction as the compact evolves from loose powder to green solid. Of course, without such a characterization, the variation of friction coefficient with density is unknown. Consequently, it is not possible to predict the manner in which the density distributions distort during compaction. Here, we focus on the more recent work of Sinka et.al. [2001], who indirectly measured the friction coefficient as it changed during the compaction of pharmaceutical powders into solid cylindrical tablets. Direct measurements maded by Solimanjad et.al. [2001] during the compaction of a .5% lubricated iron powder yielded variations of the friction coefficient that were qualitatively similar to those obtained by Sinka et.al. [2001]. The measurements of Sinka et.al. [2001] are based on the following simple analysis that applies to a solid cylindrical compact (i.e. Ri=0). If the coefficients α and µo were constants, then equation (5) could be integrated to yield,

)]()/2exp[()( HzRpzp ooo −= αµ , (11)

7

where po is the compaction pressure applied at the top (z=H) of the compact. Equation (11) can be re-written as,

Hz

o

zpp

zpzp

/

)0()0()(

=

==

, (12)

where p(z=0) is the pressure at the bottom of the compact. Solving equation (11) for the coefficient of friction, employing equation (4) to eliminate α, and using equation (12) to eliminate p(z) from the final result, we obtain

=

=

==

)0(ln

)0()()0(

2

/

zpp

zpp

zzp

HR o

Hzoo

o σµ . (13)

Equation (13) forms the basis for the indirect measurement of µo. Sinka et.al. [2001] employ a computer-controlled press that measures H using linear variable differential transformers (LVDTs), load cells that measure the top and bottom pressures po and p(z=0), and piezo-electric sensors to measure the radial pressure σ(z). Reliable results were obtained provided that sensors were at least 20 percent of H away from the bottom of the compact and at least 30 percent of H away from the top. With measurements of friction coefficient carried out in this manner, the experimental results of Sinka et.al. [2001] indicate that the coefficient of friction decreases monotonically with increasing pressure, so that it is greatest when the specimen is in its loose powdered state. As the pressure increases from zero, the coefficient of friction deceases rapidly at first, more slowly for larger pressures, and hardly at all as the pressures become extremely large. Their results fit well to the mathematical variation,

kp

pko +

+=

112 µµ

µ , (14)

where µ2 is the coefficient of friction between the die wall and the loose powder (when p=0), and µ1 is the corresponding coefficient of friction when the compact is at maximum theoretical density (when p becomes unbounded). Notice that when µ2 is greater than µ1, the rate of change of µo with p,

221

)1()(

kpk

dpd o

+−

=µµµ

, (15)

8

is always negative and that it decreases to zero with increasing p. Although there was a dependence found on punch velocity, typical values found for these parameters were: µ2=.85, µ1=.10, and k=.14 (MPa)-1. Because the density increases with pressure, these results may also be used to infer that the coefficient of friction decreases monotonically with increasing density, and that it approaches a minimum value as the density reaches its theoretical maximum value. These trends are in qualitative agreement with the direct measurements of Solimanjad et.al. [2001], who employed a powder friction measuring device (PFMD) to measure the torque exerted on a flat punch in contact with rotating cylindrical specimens of controlled average densities. 2.3 Isolating the Effects of Pressure-Dependent Coefficients of Friction: The effects caused by the dependence of the friction coefficients on the pressure (and density) are not easy to isolate because the ratio α of the induced normal stress to the applied axial stress also depends on pressure. In general, then, changes in friction forces at the die wall and core rod are due to changes in both normal stress and coefficient of friction. In this section, in order to avoid this complication, we present an illustrative example that isolates the effects of pressure-dependent friction coefficients. We do so by taking α to be a constant throughout the compaction. In the sections that follow, we show numerical results in which both the stress ratio α and the friction coefficients µI and µo depend on density. For simplicity, we consider the case in which the friction coefficients µi at the core rod, and µo at the die wall are equal. Provided that they vary according to equation (14), the pressure variations may be obtained by integrating equation (5) in closed form for constant values of α. In this manner, we obtain,

, (16) )](exp[)()( 21212 HzQpkppkp noo

n −+=+ µµµµµ

where the constants Q and n are defined by Q≡2α/(Ro-Ri) and n≡(µ2-µ1)/µ1, respectively. Equation (16) determines p(z) in terms of the unknown constant po, which is determined by satisfying the balance of mass. If the coefficients of friction were independent of pressure, then µ2=µ1 and n=0, expression (16) for p(z) would reduce to the elementary solution given by equation (11), and clearly the pressure p would scale with the applied pressure po. For non-zero values of the friction parameter n≡(µ2-µ1)/µ1, equation (16) demonstrates that p does not scale simply with po. In general, equation (16) can not be solved explicitly for p(z). However, if the initial value µ2 of the friction coefficient is twice the final value µ1, then n=1 and equation (16) becomes a quadratic equation for p. Under these circumstances, the explicit expression for the axial pressure profile is,

9

[ ])](exp[)2(1112 HzQkpkp

kppp

oooo

−+++−= µ , (17)

which shows clearly that the pressure profile does not scale with the applied pressure po; rather, it depends on the parameter kpo. In order to determine the applied pressure, it is necessary to employ equation (17) in equation of state (10), and to use the intermediate result in balance of mass (7). For fixed heights H/L and parameter ratio β/k, both boundary conditions (8) are satisfied only by the correct choice for kpo. By contrast, we consider the pressure profile that would develop if the coefficients of friction were constants throughout compaction. To be consistent, we again consider the case in which the friction coefficients µi at the core rod, and µo at the die wall are equal. In this simple case, force balance (5) yields,

)](exp[ HzQpp

oo

−= µ , (18)

where, as in equation (17), the constant Q is defined by Q≡2α/(Ro-Ri). For the purpose of comparison, we take the friction coefficients µi=µo to be equal to the average of the maximum value µ2 and minimum value µ1. When µ2=2µ1, this means that µi=µo=3µ2/4. In Figure 2, we compare the scaled pressure profile given by equation (18) with those given by equation (17) when µ1=.3, µ2=.6, and µi=µo=.45. The thick solid curve corresponds to the single profile that occurs when the coefficients of friction are constant throughout compaction. As compaction proceeds, the applied pressure po increases, the profile p(z) increases proportionally, but the shape of the profile remains unchanged. The precent top-to-bottom pressure decrease, for example, remains unchanged during compaction. The thin solid curves for kpo=0, 1, 10, and 100, on the other hand, correspond to the pressure profiles that occur when the coefficients of friction evolve during compaction. For relatively low values of kpo, the compaction is in its early stages, the pressures are relatively small, the coefficients of friction are relatively high (i.e. near µ2=.6), and the percent top-to-bottom pressure decreases are relatively high. For relatively high values of kpo, the compaction is in its later stages, the pressures are relatively high, the coefficients of friction are relatively low (i.e. near µ1=.3), and the percent top-to-bottom pressure decreases are relatively low. In this case, then, not only do the overall pressures increase as compaction proceeds, but the shapes of the pressure profiles change as well. 2.4 Combining the Effects of Pressure–Dependent Coefficients of Friction and

Density-Dependent Radial-to-Axial Pressure Ratio: In this section, we study the compaction process when the coefficients of friction µI and µo depend on pressure, and the radial-to-axial pressure ratio α (also referred to as

10

Figure 2: Axial profiles of p/p0 for µi=µo=µ(p) given by equation (19) with ϕ=1 when kpo = 0,1,10,100 (thin solid curves) and when µi=µo=.45 (dark solid curve, independent of kpo). Here, α=.5, Ri/Ro=.5, and H/Ro=4.0.

Normalized Pressure (p/p0)0.0 0.2 0.4 0.6 0.8 1.0

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

µi= µo=.45µi= µo=µ(p)

kp0= 100

10

10

11

fluidity) depends on density. First we introduce a pressure dependence into the coefficients of friction that is qualitatively similar to that observed experimentally by Sinka et.al. [2001] and Solimanjad et.al. [2001]. To this end, we introduce the following empirical pressure dependence:

)exp()( 121 poi βφµµµµµ −−+== , (19)

where µ2 is the coefficient of friction of the loose powder, µ1 is the coefficient of friction of the compact in its solid green state, and ϕ is a parameter analogous to k in equation (14) that governs how rapidly (with respect to increasing pressure) the friction coefficient approaches its minimum value µ1. In Figure 3, we show the variation of µi=µo with βp for values of ϕ that range from .01 to 10. Based on the curves in Figure 3, it is difficult to determine the values of ϕ that are appropriate for our purposes. This is because the data collected by Sinka et.al. [2001] applied to pharmaceutical powders, and were therefore collected at pressures that were much lower than those required to compact than the metal powders of interest here. Solimanjad et.al. [2001], on the other hand, worked with iron powder and measured the explicit dependence of the friction coefficient of friction on density between 3.7 g/cm3 to over 7 g/cm3. It is therefore more convenient to examine the explicit dependence of friction coefficients on density, from which we can choose values of ϕ that give reasonable quantitative agreement with the data collected by Solimanjad et.al. [2001]. If we first invert the equation of state (10) to find the pressure as a function of density, then we can use the intermediate result in equation (19) to obtain the explicit dependence of µi and µo on density ρ. However, inversion of the equation of state equation of state yields,

))(1(

)]/(1)[(M

Mp−−

−−=

ραηρηβ , (20)

where β is the compressibility of the powder in its loose state. The complication here is that the pressure βp depends on the radial-to-axial pressure ratio α, which itself depends on density. To address this complication, we consider a powder blend which is 99.5% by weight of Distalloy AE, .5% by weight of graphite, and 1% wax Hoechst micropulver admixed as internal lubricant. Distalloy AE is a diffusion alloyed iron powder with composition 4 wt% Ni, 1.5 wt% Cu, and .5 wt% Mo. Particle sizes range from 20 to 180 µm. The apparent density η of the powder is 3.04 g/cm3, and the theoretical maximum density M is 7.33g/cm3 (so that the relative apparent density η/M is equal to .41). The plasticity theory for powder compaction described by Trasorras and Parameswaran [1998] indicates that the radial-to-axial pressure ratio is given by,

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Figure 3: Variations of µi and µo with βp for parameter ϕ = .01,.05,.1,.5,1,5, and 10 when µ2=.9 and µ1=.1

Axial Pressure (βp)0.001 0.01 0.1 1 10 100 1000

Coe

ffici

ents

of F

rictio

n (µ

i=µ o)

0.0

0.2

0.4

0.6

0.8

1.0

φ= 10 5 1 .05 .01.5 .1

13

b

b23

3+−

=α , (21)

where b is a function of relative density ρ/M. By curve fitting to data obtained from triaxial consolidation experiments, Trasorras and Parameswaran [1998] have found that,

−

=2516.

)/(33.76.7ln8287.21)/( MMb ρρ . (22)

Equations (21) and (22) determine the explicit dependence of α on ρ/M, which for completeness is shown in Figure 4. As the density of the compact reaches its theoretical maximum value, the deformation becomes incompressible, the induced radial pressure equals the applied axial pressure, and according to equation (4) the parameter α must approach unity. For prescribed values of the parameters ϕ, equations (19) through (22) fix the dependence of µi=µo on ρ for the Distalloy AE powder described above.. In Figure 5, we plot the variations of µi and µo with ρ for the same range of ϕ that appeared in Figure 3. Because of the fairly rapid decrease in friction coefficient with increase in density reported by Solimanjad et.al. [2001], we expect that values of ϕ between .5 and 5 are probably reasonable for the metal powders of interest here. Finally, it remains to calculate the manner in which the axial pressure and density profiles evolve during compaction when both the friction coefficients (µi=µo)and the radial-to-axial pressure ratio (α) vary with density. Force balance (5) for the pressure is complicated by the fact that α depends explicitly on ρ. In fact, ρ can be determined directly as follows. Equation (20) gives p=p[ρ,α(ρ),η(Z)], where we have accounted for the possibility that the initial fill density η could vary with initial axial location Z. The axial gradient of the pressure can be expressed in terms of the density and its axial gradient as follows:

dzdZ

dZdp

dzd

ddpp

dzdp η

ηρ

ρα

αρ ∂∂

+

∂∂

+∂∂

= , (24)

where equation (20) can be used to calculate the derivatives of p with respect to ρ, α, and η. Note that by including the term involving dη/dZ, we have accounted for the possibility that the initial fill density η may be non-uniform. When the coefficients of friction at the die wall and core rod are equal, force balance (6) becomes,

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Figure 4: Variation of α with ρ/M given by equations (21) and (22) for lubricated Distalloy AE.

Relative Density (ρ/M)0.4 0.5 0.6 0.7 0.8 0.9 1.0

Rad

ial-t

o-Ax

ial P

ress

ure

Rat

io (α

)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

15

Figure 5: Variation of coefficients of friction (µi=µo) with ρ/M for parameter φ =.01,.05,.1, .5,1,5, and 10 when η/M = .41 and α is given by equations (21) and (22) for lubricated Distalloy AE.

Relative Density (ρ/Μ)

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Coe

ffici

ents

of F

rictio

n (µ

i=µ o)

0.0

0.2

0.4

0.6

0.8

1.0

φ= 0.01

0.05

0.1

0.5

1

5

10

16

∂∂

−

∂∂

+∂∂

=−

dZdppq

ddpp

dzd

oη

ηηραµ

ρα

αρρ

1

, (25)

where q≡2/(Ro-Ri), )(ρα is given by equations (21) and (22), p[ρ,α(ρ),η(Z)] is given by equation (20), and µo{p[ρ,α(ρ),η(Z)]} is given by equation (19). Note that we have used equation (7) to eliminate dZ/dz from equation (25). Numerical integration of equations (25) and (7) may be carried out simultaneously to determine ρ(z) and Z(z). If, for example, integrations are initiated at the top of the compact, then we employ the first of boundary conditions (8), guess at the value ρ(z=H), and iterate on our guess until the second of boundary conditions (8) is satisfied. With ρ(z) and Z(z)completely determined in this manner, the pressure variation p(z) is fixed by the inverted equation of state (20). If the initial fill density is uniform, then η does not depend on Z, and integration of equations (25) and (7) need not be done simultaneously. In this case, we guess at the value ρ(z=H), integrate equation (25) alone to get ρ(z), use this result to integrate equation (7), and iterate on ρ(z=H) until both boundary conditions (8) are satisfied. 3. RESULTS AND DISCUSION: In this section, we present a selection of results that best demonstrate the underlying physics associated with the dependence of friction coefficients on pressure and density. The results are obtained by employing the solution procedure outlined in the previous section for uniform initial fills, and apply to the Distalloy AE powder blend for which equations (21) and (22) are valid. For this reason, in all cases we take η/M=.41. In addition, the geometric parameters are a=Ri/Ro=.5, and L/Ro=8, the coefficients of friction at the core rod µi and die wall µo are equal, the maximum value of the friction coefficients (i.e. in the loose powdered state) is µ2=.9, the minimum value the friction coefficients (i.e. near maximum theoretical density) is µ1=.1, and the parameter that governs the rate at which the friction decays with pressure and density is ϕ=1. The pre-compaction normalized height L/Ro=8 is chosen so that when the relative compaction height H/L is equal to .5, the aspect ratio H/2Ro of the green compact is equal to 2. In Figure 6, we show how the percent top-to-bottom density variations evolve during compaction as H/L decreases from 1 at the start of compaction to .5 at the end, when µi=µo is the function µ(p) of pressure given by equation (19). Interestingly, the nonuniformity induced in the density distribution increases to a maximum (42 percent) at an intermediate height of H/L=.72 that is approximately four times the nonuniformity (10.3 percent) that is present in the final green compact. The increase in nonuniformity that occurs in the early stages of compaction is due to the fact that (according to the equation of state) even small absolute top-to-bottom pressure differences give rise to fairly large density differences when the magnitudes of the pressures are low. The decrease in nonuniformity that occurs in the later stages of compaction is due to two

17

Figure 6: Variations of percent top-to-bottom density decrease with H/L for µi=µo=µ(p) given by equation (19) with ϕ=1, and for µi=µo=µavg= .302, and for µi=µo=µ1 = 0.1 when Ri/Ro= .5, L/Ro=8.0, η/M=.41 and α(ρ) is given by equations (21) and (22).

Compaction Height ( H/L )0.50.60.70.80.91.0

Perc

ent T

op-to

-Bot

tom

Den

sity

Dec

reas

e

0

10

20

30

40

50

µ (p)

µavg= .302

µ1 =.1

18

effects. First, even if the coefficients of friction were constant throughout the compaction, the equation of state demonstrates that even large absolute top-to-bottom pressure differences give rise to only small density differences when the magnitudes of the pressures are high. Second, the coefficient of friction, and therefore the percent top-to-bottom pressure variations diminish as compaction proceeds. For the purpose of comparison, we have also shown on Figure 6 the corresponding variations of top-to-bottom density decrease when the coefficients of friction µi=µo are taken to be constants. In one comparison, we take µi=µo equal to the final minimum value µ1=.1, and in the other we take µi=µo equal to an average value µavg=.302, which (from Figure 5) is appropriate to the average relative density (ρ/M=.615) between .41 at the start of compaction and .82 at the end. As expected, the nonuniformities predicted for minimum coefficients of friction are generally smaller than those predicted as the friction coefficients decrease to their minimum values, and the predictions approach one another as compaction concludes. The nonuniformities predicted by the µi=µo=µavg=.302 curve, on the other hand, are smaller than those predicted by the µi=µo=µ(p) curve whenever µavg is less than µ(p), are larger than those on the µi=µo=µ(p) curve whenever µavg is greater than µ(p), and agree very near the instant (H/L=2/3) at which µ(p) is equal to µavg. In Figure 7, we show detailed profiles of axial pressure βp and relative density ρ/M at three instants H/L=.9, .667, and .5 for the three cases of friction shown in Figure 6. At the earliest stage of compaction, the coefficient of friction µ(p) is still near µ2=.9, so that the pressure and corresponding density variations are much greater than those for µi=µo=µavg=.302 and µi=µo=.1. As H/L and µ(p) decrease, the differences between the details of the profiles for the variable coefficient of friction and the minimum coefficient of friction diminish until they are virtually indistinguishable at H/L=.5. Interestingly, at no point in the compaction do the variable friction coefficient µ(p) and average friction coefficient µavg profiles become indistinguishable. Even at H/L=2/3=.667, when the average relative density in the compact is equal to .615, the large variations in pressure and density at that instant yield differences between profiles generated based on uniform coefficients of friction and those based on friction coefficients that depend on pressure. In Figure 8, we show the variation of the applied pressure βpo during compaction for the three cases discussed above. In the earliest stages of compaction, the predictions based on pressure-dependent friction coefficients yield the largest required compaction pressures because the friction coefficients are highest in this case. There is a cross-over between the compaction loads predicted by the variable friction model and the µavg-model, because at the final stages of compaction the µavg-model employs the largest friction coefficients and therefore predicts the largest required pressures. The gradual approach and eventual coincidence of the compaction loads predicted by the variable friction model and those predicted by the minimum friction model is in every way consistent with the observations made concerning the profiles of pressure shown in Figure 7.

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Axial Pressure (βp)

1 10 100 1000

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.5

Relative Density (ρ/Μ)

0.6 0.7 0.8 0.9 1.0

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.5

Axial Pressure (βp)0.1 1 10

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.667

Relative Density (ρ/Μ)

0.4 0.5 0.6 0.7 0.8

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.667

Figure 7: Axial variations of βp and ρ/M for H/L= .9, .667, .5 when µi=µo=µ(p) given by equation (19) with ϕ=1, and for µi=µo=µavg=.302, and for µi=µo=µ1=0.1, when Ri/Ro= .5, L/Ro=8.0, η/M =.41 and α(ρ) is given by equations (21) and (22).

Axial Pressure (βp)0.1 1 10

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.9

Relative Density (ρ/Μ)

0.4 0.5 0.6 0.7 0.8

Axia

l Loc

atio

n (z

/H)

0.0

0.2

0.4

0.6

0.8

1.0

H/L=.9

µ(p)µ1= .1µavg= .302

µ(p)µ1= .1µavg= .302

µ(p)µ1= .1µavg= .302

µ(p)µ1= .1µavg= .302

µ(p)µ1= .1µavg= .302

µ(p)µ1= .1µavg= .302

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Figure 8: Variation of βp0 with H/L for µi=µo=µ(p) given by equation (19) with ϕ=1, and for µi=µo=µavg=.302, and for µi=µo=µ1=.1, when Ri/Ro= .5, L/Ro=8.0, η/M = .41 and α(ρ) is given by equations (21) and (22).

Compaction Height ( H/L)0.50.60.70.80.91.0

Appl

ied

Pres

sure

(βp 0

)

0

5

10

15

20

25

30

35

40

µ(p)µ1= .1µavg=.302

21

The differences in predicted compaction loads for differing friction models suggests that a significant fraction of the total load must be devoted to overcoming the frictional forces exerted at the die wall and core rod. To isolate this phenomenon, in Figure 9 we calculate the fraction of the total load that is exerted to overcome friction, and show how it varies during compaction in the three cases (µi=µo=µ(p), µi=µo=µavg, and µi=µo=µ1) of interest here. The portion of the load that is devoted to friction is essentially the difference between the pressure at the top of the compact and the pressure at the bottom. Figure 9 demonstrates that in all cases the fraction of the load devoted to overcoming friction decreases as the compaction proceeds. In the case of constant friction coefficients (µavg and µ1), this is because the specimen becomes less compressible and requires a larger fraction of the load to overcome the specimen’s inherent resistance to axial contraction. In the case of variable friction (µ(p)) this effect is present, but it is overwhelmed by a larger effect due to the drastic reduction in friction coefficient from µ2=.9 to µ1=.1.

Based on our discussion to this point, it is not surprising that, according to Figure 9, the variable friction model requires the largest friction force early in the compaction, that there is a crossover between the variable friction and average friction models, and that the variable friction and minimum friction models nearly coincide by the final stage of compaction. What is striking, however is that Figure 9 demonstrates that in all cases a significant fraction of the compaction load is used to overcome friction at all stages of compaction, even when the friction coefficient is as low as .1. 4. REFERENCES: Bockstiegel, G., Svenson, O., Modern Developments in Powder Metallurgy, (ed. Hausner, H.H.), Plenum Press, New York, Vol. 4, p. 87-, 1971. Ernst, E., Schroder, C., Arnhold, V., Wahling, R., Beiss, P., Friction Measurements During Powder Compaction, MPIF/AMPI, Princeton, NJ, Vol 1, p. 307-321, 1990. German, R.M., Powder Metallurgy Science, MPIF, Princeton, NJ, 1994. Mallender, R.F., Dangerfield, C.J., Coleman, D.S., Powder Met., Vol. 15, No. 30, p. 130, 1972. Mallender, R.F., Dangerfield, C.J., Coleman, D.S., The Variation of the Coefficient of Friction with Temperature and Compaction Variables for Iron Powder Stearate-Lubricated Systems,” Powder Metallurgy, Vol. 17, No. 34, pp. 288-301, 1974. Sinka, I.C., Cunningham, J.C., Zavaliangos, Experimental Characterization and Numerical Simulation of Die Wall Friction in Pharmceutical Powder Compaction, Proc. 2001 Int’l. Conf. On Powd. Met. and Part. Mat’ls., PM2TEC, New Orleans, LA., May 2001.

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Figure 9: Variations of the ratio of the friction force to the compaction load for µi=µo=µ(p) given by equation (19) with ϕ=1, and for µi=µo=µavg=.30 , and for µi=µo=µ1=.1, when Ri/Ro= .5, L/Ro=8.0, η/M = .41 and α(ρ)is given by equations (21) and (22).

Compaction Height (H/L)0.50.60.70.80.91.0

Fric

tion

Load

/ App

lied

Load

0.6

0.8

1.0

µ(p)µ1= .1µavg= .302

23

Solimanjad, N., Wikstrom, H., Larsson, R., A new Device for Measurement of Friction During Metal Powder Compaction, Proc. 2001 Int’l. Conf. On Powd. Met. and Part. Mat’ls., PM2TEC, New Orleans, LA., May 2001. Trasorras, J.R.L., Parameswaran, R., Cocks, A.C.F., Mechanical Behavior of Metal Powders and Powder Compaction Modeling, ASM Handbook, Vol. 7, ASM International, Materials Park, OH, pp. 326-342, 1998.

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