13ernst/pubarchive/kobayashietal/chapter… · plasticity equations. on the basis of this theory,...

13

C O M P A C T I O N A N D F O R G I N G

O F P O R O U S M E T A L S

13.1 Introduction

Powder forming, once considered a laboratory curiosity, has evolved into a manufacturing technique for producing high-performance components economically in the metal-working industry because of its low manufacturing cost compared with conventional metal-forming processes [1,2]. Generally, the powder-forming process consists of three steps: (1) com- pacting a precise weight of metal powder into a "green" preform with 10-30% porosity (defined by the ratio of void volume to total volume of the preform); (2) sintering the preform to reduce the metal oxides and form strong metallurgical structures; (3) forming the preform by repressing or upsetting in a closed die to less than 1% residual porosity.

Powder forming has disadvantages in that the preform exhibits porosity. Because of this porosity, the ductility of the sintered preform is low in comparison with wrought materials [3]. In forging compacted and sintered powdered-metal (P/M) preforms, where large amount of deformation and shear is involved, pores collapse and align in the direction perpendicular to that of forging and result in anisotropy. However, repressing-type deformation, where very little deformation and shear are present, does not lead to marked anisotropy [4]. A low-density preform will result in more local flow and a higher degree of anisotropy than will a preform of high initial density [5]. These anisotropic structures can lead to nonuniform impact resistances of the forged P/M parts. Also, in forming of sintered preforms, materials are more susceptible to fracture than in forming of solid materials, and the analysis is of particular importance in producing defect-free components by determining the effect of various parameters (preform and die geometries, sintering conditions, and the friction conditions) on the detailed metal flow. In this chapter, the plasticity theory for solid materials is extended to porous materials, applicable to the deformation analysis of sintered powdered-metal preforms.

In characterizing the mechanical response of porous materials, a phenomenological approach (introducing a homogeneous continuum model) is employed. For the finite-element formulations of the equilibrium and energy equations based on the infinitesimal theory, the following assumptions are made: the elastic portion of deformation is neglected

244

Compaction and Forging of Porous Metals 245

because the practical forming process involves very large amounts of plastic deformation; the normality of the plastic strain-rates to the yield surface holds; anisotropy that occurs during deformation is negligible; and thermal properties of the porous materials are independent of the temperatures.

13 .2 Y i e l d C r i t e r i o n a n d F l o w R u l e s

For porous metals, a number of plasticity theories have been proposed with a yield function f ( e o ) of the following form [6, 7]:

f (oi j ) = A J2 + BI 2 (13.1)

where It is the linear invariant of stress tensor, J2 is the quadratic invariant of deviatoric stress tensor, and A and B are functions of void ratio or relative density. The invariants/1 and J2 are defined as

I1= ox + oy + o~

and

J5 = 6[(ox - 0 , ) 5 + ( o , - ° , ) 5 + ( o z - o )51 + + +

Starting with eq. (13.1), Oyane and his colleagues [8, 9, 10] derived the plasticity equations. On the basis of this theory, they derived the slip-line field equations and the upper-bound theorem applicable to porous metals.

The yield surface can be defined by

f ( o o) = A J2 + BI 2 = YzR

where YR is the apparent yield stress of porous materials determined by uniaxial tension or compression. It can then be shown that B = 1 - (A/3) . The yield function f ( o q ) is now expressed by

f ( o o ) = A[~{(ox - or)2+ (ey - oz)2+ (o, - Ox) 5} + (r~zv+ fez,+ zz)]

A + ( 1 - ~-)(ox + Oy+ Oz) 2 (13.2)

With this yield function, flow rules are expressed by

e,J = Of i (13.3) 0oij

where oq and k 0 are apparent stresses and strain-rates, respectively, considering a porous metal to be a continuum. The proportionality factor )~ in the flow rule, eq. (13.3), is given by

2YR (13.4)

where the apparent strain-rate kn is defined according to the equivalence

246 Metal Forming and the Finite-Element Method

of the work-rate, namely, a0k~ j = YR~R [11, 12] and expressed by

=~- [ ] { (~x - ,~)2 + (~ , _ ~z)2 + (~ , _ ~ ) 2 ) + (~,~y + .2 (~ . )2 ryz + ~,L)]

1 _ _ .2 (13.5)

+ 3(3 - A) ev

where ~'0 is the engineering shear strain rate and ko is the volumetric strain rate.

The apparent yield stress YR depends on the property of the base metal and the relative density R (ratio of the volume of base metal to the total volume of porous metal) [13] according to

f (oi i) = y 2 = fly2 (13.6)

where Yb is the yield stress of the base metal and ~/is a function of relative density. The effects of strain, strain-rate, and temperature on the yield stress are included in Yb = Yb(~b, ~b, Tb), where ~b, ~b, and Tb are strain, strain-rate, and temperature of the base metal. The relationship between the apparent strain and strain-rate and those of the base metal are given by

YReR = RYb~b or ~b = ~ eR

and (13.7)

~b=f-~d~ In order to complete the constitutive equations, A in eq. (13.2) and T/in eq. (13.6) must be determined as functions of relative density by experiment. Among the proposed constitutive equations, those proposed by Doraivelu et al. [13] and by Shima et al. [8], appear to agree with experimental measurements quite well. The expressions for A and ~ by Doraivelu et al. are

A = 2 + R 2 and r /= 2 R 2 - 1 (13.8)

It is to be noted that for R = 0.707, r/-- 0, which implies that the apparent yield stress is zero according to eq. (13.6). For the analysis in this chapter, the constitutive equation (13.8) is used. Further discussion on the validity of the constitutive equations for porous materials can be found in References [14] and [15].

13.3 Finite-Element Modeling and Numerical Procedures

The variational form of the equilibrium equation, as a basis for discretization, is derived as

fvyRr~RdV-fs F~ 6,dS=O (13.9)

where 6 denotes variations.


Discretization The element used for discretization is an isoparametric quadrilateral element with bilinear shape function (see Fig. 8.1 and Fig. 9.1 for the two-dimensional and the axisymmetric deformations, respectively). The elemental velocity field is approximated by

U = N T v (13.10)

where v is the velocity vector of nodal-point value and N is the shape function matrix. Applying differentiation to eq. (13.10),

= By (13.11)

The matrices N and B have been defined for a quadrilateral element in the previous chapters. Substitution of eq. (13.11) into eq. (13.5) leads to

(~R) 2 = ~TD~ = vTBTDBv = vTpv (13.12)

and the variation tSaR becomes, because of symmetry of the matrix P,

6~R = 1 6vTp v (13.13) ER

Substituting eqs. (13.10) and (13.13) into eq. (13.9) at the elemental level and assembling the element equations with global constraints, we obtain

(/) = 0 (13.14)

where (j) indicates the jth element. Because the variation 6v is arbitrary, eq. (13.14) results in the stiffness equations. The matrix P in eq. (13.12) is obtained as follows. The inversion of the flow rules (13.3) can be expressed by

Y~ a = -z-- D/~

~R

where, for the axisymmetric case,

4 - A A - 2

A(3 - A) A(3 - A)

4 - A

A(3 - A ) D =

Sym.

A - 2 0

A(3 - A )

A - 2 0

A(3 - A )

4 - A 0

A(3 - A )

1

A

with the stress and strain-rate vectors are defined according to

o T= {tr,, oz, Oo, r,z} and kv= (~r, ~z, Go, ~rz)

(13.15)


respectively. Then, from the requirement that o"r~ = Yn~n, (~n) 2 = i~TDi~ and P = BTDB

Updating of Relative Density The volumetric strain-rate is related to the density-rate according to

~o = CTV = -- -- (13.16) R

where the relative density R is defined by

Vb R ~ - -

vb +vv

with Vb being volume of the base metal and V~ as the volume of void. Integrating eq. (13.16), we have

R = Roexp(- f ~vdt)= Ro(i - Aev) (13.17)

In eq. (13.17), R0 is the current relative density and Aeo is the change of volumetric strain in one deformation step. The average relative density Ra is defined by

E R,V/ (13.18)

R " = ~ ' V '

where R i and V~ are the relative density and the volume of an element, respectively.

Fully Dense Materials For fully dense materials, R = 1.0 and A = 3. Then, the matrix D of eq. (13.15) becomes infinity. Consequently, convergence behavior for the solution becomes erratic when the relative density in the element approaches unity.

A constraint was incorporated in the numerical procedures such that an element with R = 0.9990 was considered as a fully dense element. This constraint was helpful in obtaining well-behaved convergence for the solution. Osakada et al. [16] and Mori et al. [17], also using this theory, developed the finite-element method and applied it to the rigid-plastic deformation analysis of fully dense materials.

Volume Integration The program has been tested by analyzing compression of cylinders and rings of porous materials [18]. During the test it was found that the reduced integration must be applied to the terms involving the volumetric strain-rate. Such a reduced integration strategy is straightforward in the formulation of fully dense material, since the term of the volumetric strain


energy-rate is fully uncoupled from that of the distortional energy-rate. In the present formulation, the terms with volumetric strain-rate cannot be decoupled from those with distortional strain-rate, as seen in eq. (13.14). Thus, the matrix P is decomposed into two components, P~ and P2- Here, P~ contains the terms with distortional strain-rate and P2 involves the terms with volumetric strain-rate. The matrices P~ and P2 are obtained by decomposing the matrix D into the two components D~ and D2 as

D = D1 + D2 where

and

D 1 =

4 2 2 0

3A 3A 3A

4 2 0

3A 3A

4 Sym. ~-~ 0

1

A

(13.19a)

I1 lil 1 1 (13.19b) Dz= 3(3 - a--------~ 1

/Sym.

Then, P = P1 + Pz = BTDIB + BTD2 B. For evaluating the matrix P, the matrix P2 is evaluated at the reduced integration point, while P1 is evaluated at the regular integration points. The evaluation of the matrix P is then obtained at the regular integration points. It should be noted again that for solid materials A = 3 in eq. (13.19), which gives

~TD~ = ~TD1/~

and 1 / [3(3- A)] is replaced by K, the penalty constant.

13.4 Simple Compression [14]

In simple compression, a cylindrical sintered P/M preform is compressed between two fiat dies. The preform was 50.4 mm (2 in.) in diameter and 50.4 mm (2 in.) in height. It was assumed that the preform has uniform relative density of 0.8. The simulations were carried out with two different friction factors, m = 0.2 and 0.5. The flow stress of the matrix material was assumed to be expressed by 0 = 1.0 + 0.01~ and the constitutive eq. (13.8) was used.

Figure 13.1 shows the predicted grid distortions at 20, 50 and 70% reduction in height for the friction factor m --- 0.5. It can be seen from the


I . l O 0

O. IlO0

~_ O. IN )O

X

LLI "T 0 .300

0 . 0 0 0

O. I100

I I

1 1 I

) , 0 0 0

I I

I I

I I I I

RAD I US

I

I • SO0

2 0 %

I

| . 0 0 0

X ta [ I ! I I

~ o . i e e l I I I ;3E: "il

: : ,, t 0.000 i

0 .000 O . S O 0

X

OimO ° .

O.ImO

I I

! !

I • 0 0 0 I • SO0

R A D I U S

I

~.000

5 0 %

7 0 %

1t.1420 I .OO0 I . S O 0 I . O 0 0

R A D I U S

FIG. 13.1 Simple compression of a cylindrical sintered P/M preform. The predicted grid distortions during simple compression at 20, 50, and 70% reductions in height. The dashed lines are the predicted boundaries with fully dense initial preform. Initial relative density = 0.8; friction factor = 0.5.

figure that the predicted grid distortions resemble those expected in the compression of a fully dense preform. The predicted boundaries of the deformed workpiece for a fully dense initial preform are shown in dashed lines for comparison. It is seen that the porous preform changes its volume during the compression process. The average density change as a function of height strain is shown in Fig. 13.2. The height strain is defined by ln(H/Ho) where H0 is the height of the undeformed preform and H is the deformed height. It is interesting to note that the average density varies very little with differerent friction conditions.


1.0-

0.9-

o

Friction Factor

O.e, ~ m=O.O o m = 0 2 A m=0.5

0.7 o.o 0:2 0:4 o:~' o:a ;.o ,:z

Height Strain,*,~n (H/Ho)

FIG.. 13.2 T h e predicted average relative densi ty changes as a function of height strain in simple compression.

Although the average density changes are not affected by different frictions, the differences in the local density distributions are considerable. Figures 13.3a and b show the predicted relative density distributions at 20, 50, and 70% reductions in height for the friction factors m = 0.2 and 0.5, respectively. It is seen from Fig. 13.3a that at 20% reduction the density is lowest near the center of the die contact surface where the deformation is restricted by friction. The highest density is observed at the center of the workpiece and also near the outside radius of the die contact surface. At 50% reduction in height, the density is lowest at the equator of the side surface, and at this stage, the side surface barrels considerably. The higher mean stress at the equator results in the low density at this point. It is also noted that at 50% reduction in height, densification near the center of the die contact surface has been accelerated. This can be explained by the fact that the pressure near the z axis increases as radius-to-thickness ratio increases. At 70% reduction in height, a large portion of the workpiece near the z axis becomes fully dense while the material near the side surface remains porous. In fact, the density of the workpiece near the free surface decreases as the deformation progresses from 50% to 70% in height.

Comparing Figs. 13.3a and b, it is seen that the overall densifications of the workpiece, with the two different interface frictions, are almost the same. However, it is noted that at 20% reduction, the gradient of the density distribution is larger with the higher friction. This fact supports the above argument that the density near the center of the die contact surface increases because of the increase in mean pressure near the axis. It is well known that higher pressure near the axis can be achieved with higher friction when a thin disk is compressed between two flat dies. It is also

~.85 ~ ~ .87 "-- ' - -~ '86"~ f

Radius

20%

/. 1 A ''° Radius

50%

I .99 .95

. .9'r / i -I i Radius

.8 ~ . ~ -.89

z~ ~ . 8 7 ~ .87

Radius

l .93 7 .99

er I 9r-------~ 9593" -

Radius .gr, 9s ~3

z I "~- Ro(lius

20%

50%

70%

FIG. 13.3 The predicted relative density distributions in simple compression at 20, 50, and 70% reductions in height. Initial density=0.8; (a) friction factor=0.2 and (b) friction factor = 0.5.

252


observed that the density near the free surface is lower with higher friction, owing to the larger barrelling. At 70% reduction in height, it is noted that the size of the fully dense region near the axis is larger with higher friction, while the density near the free surface decreases considerably as the workpiece undergoes deformation from 50 to 70% reduction in height. In practice, fracturing is often observed at the equatorial surface because of this density reduction.

Figure 13.4 shows the predicted load-displacement curves for two different friction conditions. For comparison, the predicted load- displacement curve for the fully dense preform with friction factor m = 0.5 is also shown. The load is lower with the porous preform than that for the fully dense preform. It can be seen from the figure that the difference between the forming loads with different frictions is very small during the early stage of deformation. However, during the later stage, the load begins to increase faster in the case of higher friction.

13.5 Axisymmetric Forging of Flange-Hub Shapes [18] In a closed-die forging, the stress state favors complete densification, because of compressive mean stress. Experimental studies on closed-die

r~

0 _J

1.400 i O|

1-200

1-000

O . l O 0

0-600

0.400

0.200

/ - m = 0 . 5 , R o = 0 . 8

= 0 , 2 , R o =0 .8

0 • 0 0 0 i ! ! 1 I o.ooo 0.200 o. 4oo o.soo o. loo ~.ooo

DIE STROKE

FIG. 13.4 The predicted load versus die stroke curves in simple compression. Ro indicates the initial relative density and m indicates the friction factor.


forging of metal powders were presented by Downey and Kuhn [19]. These studies allowed qualitative determination of the preform shapes for forging to full density with a sound metallurgical structure and without fracture. One of the forgings investigated had a flange-hub shape. The forging dies and two preform specimens, with the initial mesh system used for the analysis, are shown in Fig. 13.5. Since the specimens are axisymmetric, only a quarter of the workpiece is used for calculations.

Equation (13.8) was used with the stress-strain relationship given by

Yb = Y0[1 + ~ b

-10 ,28

0.3518J

for the base metal (OFHC copper). The initial relative density was Ro = 0.800. Yo is the initial yield stress of the base metal and was given as 42061 psi (290MN/m2). The frictional condition is given by inky (ky represents the apparent yield stress in shear) and the friction factor m was assumed to be 0.1.

The results, illustrating the deformation zone and the extent of densification, are given for two preforms in Fig. 13.6, where the distributions of apparent effective strain and relative density are shown at 24% reduction in height. In this figure, the pattern of equistrain and density contours are remarkably similar. This is because the effective strain contains not only the term of distortion but also the volume change that is associated with the relative density. The effective strains are largest at the center of the forging and at the edge of the die-specimen interface, as are the relative densities.

For preform I, the central region under the die shows no deformation and no change in relative density from the initial value (Ro -- 0. 800) for

t i

~ E]~ORM

11 | ~ ' 1 1 I I

FIG. 13.5 Preform shapes and mesh systems in flange-hub forging.

O. . f


FIG. 13.6 Effective strain (right) and relative density (left) distributions at 24% reduction in height in forging (a) preform I and (b) preform II.

small reduction. Another observation of importance is that the relative density near the bulged free surface increases at first, then begins to decrease as reduction increases. Thus, the equatorial free surface is a possible fracture site and this was indeed observed in an experiment [19].

For preform II, the workpiece is deforming radially as well as axially relative to the die motion to fill the cavity. Characteristics of strong singularity, where the gradients of strain and density are large, are seen along the die-workpiece contact surface. This results in a distinctly different deformation pattern from that obtained for preform I. It was noted for preform II that friction at the die-workpiece interface is sensitive to the metal flow. It was also found that at a certain stage of deformation the severe distortions are localized around the die corners and remeshing was needed for continuing forging. A need for remeshing occurred at 37% reduction in height for the case of preform II.

The technique for remeshing has been discussed in detail in Chap. 7. The area-weighted average method is adopted for the problem because of its simplicity and computational efficiency. In this method, the nodal point value of relative density or apparent strain is determined from the average of the adjacent element values surrounding that nodal point weighted by the associated element areas. Further details of the method can be found in Reference [20].

Figure 13.7a shows the mesh system at 37% reduction in height before and after remeshing. The comparisons of relative density and effective strain fields before and after remeshing are shown in Fig. 13.7b. For remeshing, no change was made in the number of elements and nodal points, and therefore the element connectivity was not changed.

In Fig. 13.7b close similarity between the relative density contours before and after remeshing is found. Other comparisons of the apparent average relative densities before and after remeshing confirm the credibi- lity of the program as follows: At 37% reduction in height, Ra = 0.9378

r

Metal Forming and the Finite-Element Method

Z

256

(a)

0.94 ~ . ~ / ' / , ~ / 0 . 9 0 ~

(b) FIG. 13.7 (a) Mesh systems and (b) relative density contours before (left) and after (right) remeshing in flange-hub forging.

and 0.9373 before and after remeshing, respectively; and at complete filling, at 46% reduction in height, Ra =0.9933 in both cases. Relative density and effective strain contours at complete filling are shown in Fig. 13.8, which shows that the relative density reaches full density over the entire forging for preform II.

13.6 Axisymmetric Forging of Pulley Blank

The finite-element program has been applied to the simulation of closed-die forging of a pulley blank for the two preforms shown in Fig. 13.9 [20]. The sintered preforms used in the simulations were made of

z

r 0"995~0.8 1 0 . 4

FIG. 13.8 Distributions of relative density (left) and apparent strain (right) at the completion of forging the flange-hub shape.

E ;co.

PPJCIPORIdA PRZPOP.JdB

I~1111 l i l l l I I I I I I I I l i l i l l l i l l l l I I I I I l l l l l

i R'OI~_ : l l t l l I

I I I I I : l l l l l t I I I I I , , ;

+ i l l l l ! : I I I I I I

i t i i l l l l l I i i l l ] l l

, ! l l J l l l , • : ! i l l i l l ! i

I

"]° -4o -]o -zo -~o ~o 1'o zo ~o 40 RADL~ (mm)

FIG. 13.9 Preform shapes and mesh systems in axisymmetric closed-die forging of a pulley blank.

257


aluminum powders. The yield stress of the preform was determined from that of the base metal of commercially pure aluminum and the constitutive eq. (13.8). The rate-sensitivity of the material property and the temperature effect on deformation were not included in this simulation. The initial relative density was assumed to be uniform and to be 0.780 for both preforms. Two values of m were assumed for friction, m = 0 . 1 and m =0.5.

The nonsteady-state forging process was simulated by a step-by-step method with increments of 2% of the initial height for preform A and 5% of the initial height for preform B. In order to complete the calculation successfully, four and ten remeshings were required for preforms A and B, respectively. Preform A. The calculation was stopped at 38% and 43% reductions in height for m =0.1 and m =0.5, respectively, in forging preform A. The program is designed to automatically discontinue the calculation if the relative density of any element falls below the limiting value of the relative density for which the apparent yield stress becomes zero, because this indicates the initiation of fracture.

In Figs. 13.10a and b, the distributions of relative density and hydrostatic stress at the final stage of forging are shown for two friction conditions. The patterns of densification and hydrostatic stress are similar to each other. For the two friction conditions, the relative density distributions differ somewhat in the rim area, as do the hydrostatic stress distributions. It was found that the effect of friction on the densification of the materials was not evident at an early stage of forging. As deformation continues, however, the densification and the compressive hydrostatic stress are greater with higher friction. It can also be seen that the locations of the large and small values of relative density are the same for both friction conditions and lie at the flange part around the die comers and near the free surfaces, respectively. Preform B. Contrary to preform A, complete forging was possible with preform B. This was due to the differences in material flow in the two preforms.

The distributions of relative density and hydrostatic stress with low friction were compared with those with high friction at the final stage of forging in Fig. 13.11. The distribution patterns are similar to each other for the two friction conditions at the same stage of forging except that for higher friction, the densification and the compressive hydrostatic stress were greater. Because of this effect of friction on densification, the final forged part in preform B with high friction did not completely fill the cavity at 60% reduction in height due to the volume change, while complete filling was achieved for low friction at the same reduction in height (see Figs. 13.11a and b).

As in forging of preform A, densification and the compressive hydrostatic stress are concentrated at the flange part near the die comers. The figure also shows that the weakest mechanical properties will be near the


(o)

0.84-

).98 L~ I P -94°.88

0.84"

~ -50

-I0 -20

(b)

0.82-

0.90-

0.94 0.92

09:~0.9/8 ~

I

i I I i

0.88 i

0.82 i

I

-50 -60 J ~

-40

~ - ao

-20

-30

FIG. 13.10 Relative density (left) and hydrostatic stress (3o,,, units of ksi) (right) distributions at the final stage for the two friction conditions in forging preform A (darkened areas indicate the possible fracture sites): (a) m = 0.1; (b) m = 0.5.

tips of the rim section. However, the relative density in most of the forged hub section reached full density, R = 0.999, in the final stage.

The experiments [19] also showed that preform B produced defect-free pulley blanks. This agreement allows the prediction of the probable locations of fracture during forging using the existing finite-element program.

13.7 Heat Transfer in Porous Materials

Heat transfer analysis for solid metals was presented in Chap. 12. For porous materials the energy balance equation, corresponding to eq. (12.1)


(a)

0.82 0.94- 0.98-

0.84

0.90

0.999

- 2 4 0 x ~ J / ~ -60

: -90

-,2o

(b)

0 . 8 8 -

0.92- 0.98-

0 . 8 6

0.999

I I

I , i

~ J ~ " 2 - 2 1 0 -270

~,~/- 240 / - 1 8 0

-30 - 6 0

-120

FIG. 13.11 Relative density (left) and hydrostatic stress, (30,. units of ksi) (right) distributions at the final stage for the two friction conditions in forging preform B: (a) m =0.1, (b) m =0.5.

in Chap. 12, is expressed by

kRTR.i i -- DRCRT" R + KYR~ R = 0 (13.20)

where the subscript R denotes apparent (or equivalent) quantities, representing a porous material as an equivalent continuum. Thus, using the apparent thermal properties, kR and CR derived from the base-metal properties, the heat transfer analysis follows exactly the same procedure described in Chap. 12.

In porous materials, heat transfer takes place in three ways: conduction through base metal, convection, and radiation through the pores. When the size of the pores is sufficiently small, convection is negligible.


Radiation through pores can be neglected when the temperature is low. Therefore, conduction plays a crucial role in heat transfer in porous materials.

Change in thermal conductivity, as determined by porosity, has been studied extensively and numerous attempts have been made to correlate the complex effects of the pores with experimental results in a simplified form [21]. Among published proposals, the most generally employed are the expressions by Russel [22] and by Eucken [23]. Both expressions were derived from Maxwell's relation for conductors and resistors. Although the expressions differ from each other in form, they lead to similar results. They assume that the pores are discontinuous, spherical, and evenly distributed throughout a continuous base-metal matrix.

Im and Kobayashi [24] derived a simple linear relationship between the thermal conductivities of the powdered metal and the base metal. Under the assumptions that the heat flow is unidirectional and that the volume pore fraction is equal to the linear pore fraction and to the cross-sectional pore fraction, as well as that the thermal properties are homogeneous, the heat balance equation yields

k_~n = 1 - (1 - R)(1 - k,,/kb) (13.21) kb kv R

14 kbl - R

where kR is the apparent thermal conductivity, kb is the thermal conductivity of the base metal, and ko is the thermal conductivity of the voids.

According to eq. (13.21), kR depends on both the volume pore fraction and the ratio between the thermal conductivity of the base metal and air. In addition to the conduction of air, at high temperatures the radiation across the pores contributes to heat transfer. In this case, the radiation portion of heat transfer can be added to the thermal conductivity of air and lowers the ratio between the thermal conductivity of the base metal and air in eq. (13.21). By neglecting k,,/kb terms compared to 1, eq. (13.21) becomes

kR = kbR (13.22)

Since the effect of temperature on the yield stress is determined by the temperature of the base metal, the relationship between the temperatures of the base metal and the porous material should be determined. By introducing the apparent density PR and the specific heat, CR of porous materials, the total change in internal energy can be expressed by

dTR dTb dT~ pRCR VR "--~- = pbCb Vb --~- + p,,c,, V,, -dt (13.23)

where the subscripts R, b, and v denote the quantity related to the total porous materials, base metals, and voids, respectively, t is the time, and p, c, and V are the density, the specific heat, and the volume, respectively.

The temperature-rate of voids is at most of the same order as the


temperature-rate of the base metal, since the heat generation due to plastic deformation is limited to that of the base metal. The change in the internal energy of voids can also be neglected, since the thermal capacity of the voids is much smaller than that of the base metal. Therefore, eq. (13.23) can be reduced to

dTR dTb pRCR VR - -~ ~--" pbCb Vb -~ (13.24)

where the apparent specific heat of the porous material, cR, can be determined by

cR = cbn + coo - R) (13.25)

Combining this equation with eq. (13.24), noting that pRVR = pbVb, gives

dTR dTb (13.26) [cbR + c o o - R) I ~ = Cb dt

Integration of eq. (13.26) leads to the expression of the temperature of the base metal as,

rb = f [R + (CJCb)(1 - R)] dTR (13.27) 3

13.8 Hot Pressing Under the Plane-Strain Condition

Much experimental work has been performed on powdered-metal hot forging. Preform ductility and transient cracking in forging were analyzed by Fischmeister et ai. [25], who showed that large amounts of plastic deformation are beneficial in obtaining good impact and fatigue properties. Malik [26] has published work on the producibility of titanium powdered- metal shapes for aerospace structural applications. Ferguson et al. [27] have analyzed the hot isostatic process, which involves a consolidation of loose powders or sintered preforms.

Im and Kobayashi [24] applied the finite-element technique to the analysis of plane-strain compression at elevated temperatures. The workpieces, made of sintered iron powders, were assumed to have uniform initial relative densities of 0.802 and 0.743. Computed results were compared in terms of macroscopic densification and forging pressure with experimental values published by Fischmeister et al. [28]. In upsetting long bars between flat dies, (see Fig. 13.12), the plane-strain condition (kx = 0, Ly = t~z =0) was assumed at the central cross-section, owing to the restraint of longitudinal material flow.

The conditions used in the computation were as follows. The dimensions of the workpieces were 10 x 10 × 100 mm. (0.39 × 0.39 x 3.94 in.) Two initial relative densities, Ro=0.743 and R0=0.802, were selected from four experimental cases reported in Reference [28]. Since no lubricant was used between the workpiece and dies in the experiments, two friction factors, 1.0 and 0.5 were assumed for both cases. Initially the temperatures of the dies and workpieces were assumed to be 293 K (68°F), and 1433 K

(o)

Compaction and Forging of Porous Metals

Z

263

(b) Z

aT= 0

I0-

5 Fz=O Uy =0

aT/an =0

0 uz=O OT/On =0 Fy=O

T=T

Fy= mk q= q°n+ qnf UZ= -I

q=qn c Fi=O

_ C r

q-qn+qn I

5 I0

T=T

r5

FIG. 13.12 Schematic of plane-strain compression: (a) geometry of the die and workpiece; (b) boundary conditions. (Superscripts c, r, f on the heat flow q , refer to convection, radiation, and friction, respectively.)

(120°F), respectively. The material proper ty of the base metal was assumed to be rigid-plastic, and the yield stress was taken as 28717 psi (198MN/m 2) from the handbook. Again, the consitutive eq. (13.8) was used.

The thermal properties necessary for computat ion were taken from the handbook as follows:

kb = 55 N/(s K), ko = 0.045 N/(s K), ka = 19 N/(s K),

Cbp b = 3.6 N/ (mm 2 K), cop,, = 0.4 N / (mm 2 K), ¢dPd = 3.77 N / ( m m 2 K),

oe = 3.6 × 10 N/ (mm z s K4),

hc = 5.5 N/ (mm s K), h® = 0.01 N / (mm s K),

r = 0.85


Because of geometrical symmetry, only a quarter-section of the central cross-section of the bar was analyzed for deformation and temperature. The boundary conditions are also shown in Fig. 13.12. Deformation was analyzed by the step-by-step method with increments of 0.4% of the initial height of the workpiece. Total reduction in height was 70% for Ro = 0.743 with low friction. For Ro = 0.802 with low friction, the calculation stopped at 67% owing to the severe distortion in the corner element of the deformed free surface in contact with the die. For both Ro = 0.743 and R0 = 0.802 with high friction, the calculations stopped at 58% and 64%, respectively, because the relative density of an element fell below the limiting value.

The computed and the experimental results are compared in terms of the macroscopic deformation behavior. The changes of lateral flow during the plane-strain compression are given in Fig. 13.13. The average width Wo of the deformed free surface is used to calculate the true strain in the y direction, ey = ln(Wo/Wo), where Wo is the original width. According to the figure, the results show good agreement between computation and experiment. The straight line in the figure is for the fully dense material ( R = I . 0 ) , where ez=-ey because of incompressibility. In a porous material, volume change occurs during deformation and the increase in width becomes less than the decrease in height.

The macroscopic level of densification as function of axial compression is given in Fig. 13.14. As seen in Fig. 13.14 the predictions are excellent for all cases. It is also seen that the densification occurs mainly at an early stage of upsetting. As deformation proceeds, the densification becomes saturated and lateral flow is enhanced.

Figure 13.15a reproduces the local hardness distribution in the homoge-

EE.M. 1.2

Friclion Foctor, m ....... .~ .a Ro LO 0.5 ....... ~ . . i" . .

0.802 .....

o8 _o. ?43 .............. ~..," ....

¢ 0.~ l~e

(/1 ==

0 O2 0.4 0.6 0 8 1.0 True Stroin in y-direct ion

F[G. 13.13 Comparison of variations in lateral flow between theory and experiment [28].

0 m

I00

I / , , : . . . . r

I l l ,ok'e! • Focior, m

' % I.o o5 I

80.2 74.3 . . . . . . . . . .

E I p e r i m e n t

• 80 .2 i 74.3

6 C I I , I I I

0 0.4 0 .8 1.2

True S i r o i n in z - c l i r e c t i o n

FIG. 13.14 Variations of average relative density during compression, and comparison with experiment [28].

(Q)

0.9~ I 0 98

l"~ ,~:, ~ ,t~, ~';.~0 9~ "7"%,~ , ' , . / / i I i I

=i o ojo! { "---I J " • ~ ~,,,iII ,, 1o

tl, ~,'4, I 0.98 i

(b) FIG. 13.15 (a) Experimental local hardness distribution in the center cross-section of a bar [28]. (b) Relative density distributions at 40% reduction in height for R o = 0.743 with two friction conditions: left (m = 0.5); right (m = 1.0).

265


i 0.98 I 0 84 090 \0"940 90 / / / o.98 ~ ~ = ~ 0.99 ~~/~ ; y

! I; I FIG. 13.16 Relative density distributions at the final stages for Ro = 0.743 with two friction conditions: (left) 70% reduction and m = 0.5; (right) 58% reduction and m = 1.0.

neous center region, obtained from the experimental results given in Reference [28]. This information can be correlated with the computed relative density distributions, such as shown in Fig. 13.15b. These figures show that densification is effective near the edge (owing to folding during compression) and in the central region of the deformed workpieces. Excellent agreement of relative density and hardness distributions between experiment and computation can be seen from these figures.

Figure 13.16 shows the relative density distributions at the final stages of compression for Ro = 0.743 with two friction conditions. It can be seen that most of the central region is fully dense and that less densification occurs near the free surface. In some elements at the free surface, the relative density reaches the limiting value and then tends to decrease. This is interpreted as possible fracturing and the critical elements are shown as darkened areas in the figures.

The temperature profiles in the workpiece and dies are presented in Figs. 13.17a and b for Ro = 0.743. Figure 13.17a shows that the effect of friction on the temperature distribution is negligible. It is also seen that the conductive heat flow from the workpiece to the die is dominant when compared to heat generation due to plastic deformation at an early stage. Consequently, the isothermal lines in the workpiece are almost linear. As deformation increases, the isothermal lines encircle the center, because heat generation due to plastic deformation increases and is maximal at the center, as seen in the left-hand side of Fig. 13.17b. A transition between these two types of temperature distributions can be observed at 58% reduction in height, shown in the right-hand side of Fig. 13.17b.

13.9 Compaction In the process of compaction, a P/M preform is placed in a cylindrical container and pressed in a double-action press, as shown in Fig. 13.18. For the simulation [14], the container was 50.8 mm (2.0 in.) in diameter and 152.4 mm (6.0 in.) in height. Because of symmetry, the analysis was performed for the half-height. The flow stress behavior of the matrix material was assumed to be Yb -0 .1 . The constant-friction factor law was

1 6 0

I z /

#

J

\ /

,/ 1 6 ( . . . .

• 16.0 12.0 6 0 4 0 0 0 4 0 6.0 12,0 16.0

W I D [ H qMM)

(e)

1 6 0 1 2 0 6.0 4 0 0 .0 4.0 6 0 1 2 0 16.0

Wl[) ~ H [MM)

(b) FIO. 13.17 (a) Temperature distributions for R o = 0."/43 at 40% reduction in height with two friction conditions: (left) m = 0 . 5 ; (right) m = l . 0 . (b) Temperature distributions for Ro = 0.'/43 at the final stages with two friction conditions: (left) m = 0.5; (right) m -- 1.0.

267


/ / / /

/

i / / / CONTAINER / / /

FIG, 13.18 Schematic diagram of simple die compaction.

assumed, and the friction factors used in simulation were m = 0.1, 0.2, and 0.5. The punch velocities were 76.2 mm/s (3.0 in./s) for both the upper and lower punches. The initial relative density in the simulation was taken to be 0.8. It was also assumed that the air and the lubricant trapped in the pores did not affect the compaction process.

Figures 13.19a and b show the predicted relative density distributions for the friction facors m = 0.2 and 0.5, respectively. It is seen that the density is greatest near the outside radius of the moving punch and decreases in regions remote from the punch, particularly near the container die wall. As the deformation progresses towards full compaction, the density distribution becomes more uniform; that is, the difference between the highest and the lowest density becomes smaller. It can be seen that the trend of density distributions for both friction conditions are similar, but the density is more uniform with lower friction.

From the predicted density distribution (which is a function of radial as well as height location), the average density was calculated as a function of height. The average density was obtained by integrating the relative density distribution over the cross-sectional area and dividing by that cross-sectional area at a given height.

Figures 13.20a and b show the predicted density variation as a function of height for different friction factors, m -- 0.2 and m = 0.5, respectively. It can be seen from the figures that the density is lowest at the mid-height, and that it increases toward the punch. It is also seen that the distribution becomes more uniform with increasing deformation and lower friction.


i

I 4: I

I

2%

.825

- -82- - - - - ' -

._ 815.--- - - ~

81 ---- ' - -

Radius

- 8 3

8%

_865

.855 _ / 8!

Radius

--89

14%

.~.. ,935.~____

_ .]2

Rodius

/ . !

.r

/.?1'

18%

.97~.

i .975

f .973

I Radius

.97

(a)

m

2%

__88 -- .87 --.86 .65~______

.82 ~

° ._ . . . .~ 81 ~

8%

. '__.._-.89 /

__._..87 /

, ~ . 8 3 f

.81 j

Radius Radius

--95

--93

(b)

14%

/ .9.5

.95

~ . 9 4 ~

--.91 ~

rTf Rodi~

- . 9 8

- .97

-- . !

18%

.99~

.98

i , 9 7 ~

_ I . , ~ 1 / . 9 4

Radius

FIG. 13.19 (a) Predicted relative density distribution during die pressing; initial relative density=0.8; friction factor=0.2. (b) Predicted relative density distribution during die pressing; initial relative density = 0.8; friction factor = 0.5.

In order to validate the prediction of the current model, experimental data for pressing of a powder preform were sought from the literature. None that were directly comparable could be found. However, density measurements for powder compaction in a closed die were found in a German doctoral dissertation [29]. Figure 13.21 shows measured density distributions of P/M preforms that were prepared by die compaction and isostatic compaction. From this figure, it is apparent that the density

270

(a)

Metal Forming and the ~ n i t e - E l e m e n t Method

DENSITY DIS . H=0 .2 1.000

O. 950

I O.9OO

0.850 o ,o .,.°3 °"' rl 14 PERCENT x 1S PERCENT

°'9°°o.ooo o'.soo ,'.ooo ,'.5oo ;.ooo ;.500 ~'.ooo HEIGHT

(b) 1.OOO

0.950

I O.gO0 Y

0.950

0,800 O.OOO

DENSITY DIS M=0.5

~ N HEIGHT

!i o'.soo ,'.0oo ,'.soo ;.ooo ;.~oo ~'.ooo

HEIGHT

FIG. 13.20 (a) Predicted average density as function of height; initial relative density = 0.8; friction factor =0.2. (b) Predicted average density as function of height; initial relative density = 0.8; friction factor =0.5.

distributions of the preform obtained by die compaction are similar to those predicted.

Figure 13.22 shows the predicted punch load vs. punch stroke curves for friction factors m = 0.1, 0.2 and 0.5. The figure illustrates that the pressing load is higher for higher friction. The load is important in a die pressing operation because the maximum allowable load determines the level of achievable average density.

8intered RZ 150

d o - 20.15 mm ho/d o " 3.0

In Closed Die

R e " 0.7

0.8

0.9

Hydrostatic Pressing

m / . e / o 8 / "

0.92

0.00

0.88

. ~ R e " 0.9

A ~ A B. A

° 8 ~ - A-'~,~=_.~..~A."

.m

E

| o.6o

~ 0 . 7 8

o.~

0.70

0.68

R e " 0.7

8 d

'Jam J

lp,

f e

O

J 0 10 20 3'0 40 mm 60

Billet Height

FIG. 13.21 Measured average relative densities obtained in compaction of powder in closed dies and by hydrostatic pressing [29].

271

272

12.0

Metal Forming and the Finite-Element Method

I0.0

6.0

o 6.0 0

..J

4.0 ~ / y ~ / ~ x

2 .0-

I Complete Fill = I o.o 1

o.o o~2 o~4 o.e Punch Stroke (in.)

FIG. 13.22 The predicted punch load during die pressing.

ole

References

1. Wisker, J. W., and Jones, P. K., (1974), The Economics of Powder Forging Relative to Competing Processes---Present and Future, in "Modem Develop- ments in Powder Metallurgy," (Edited by H. H. Hausner, and W. E. Smith), Vol. 7, American Powder Metallurgy Institute, Princeton, NJ, p. 33.

2. Jones, P. K., (1973), "New Perspectives in Powder Metallurgy," Vol. 6, Plenum Press, New York.

3. Kaufman, S. M., and Moearski, S., (1971), "Effect of Small Amount of Residual Porosity on the Mechanical Properties of P/M Forgings," Int. J. Powder Metal., Vol. 7, p. 19.

4. Dieter, G. E., (1976), "Mechanical Metallurgy," 2d Ed., McGraw Hill, New York.

5. Moyer, K. H., (1974), A Comparison of Deformed Iron-Carbon Alloy Powder Preforms with Commercial Iron-Carbon Alloys, in "Modern Developments in


Powder Metallurgy," (Edited by H. H. Hausner and W. E. Smith), Vol. 7, American Powder Metallurgy Institute, Princeton, NJ, p. 235.

6. Kuhn, H. A., and Downey, C. L. Jr., (1971), "Deformation Characteristics and Plasticity Theory of Sintered Powdered Materials," Int. J. Powder Metal., Vol. 7, p. 15.

7. Green, R. J., (1972), "A Plasticity Theory for Porous Metals," Int..I. Mech. Sci., Vol. 14, p. 215.

8. Shima, S., and Oyane, M., (1976), "Plasticity Theory for Porous Metals," Int. J. Mech. Sci., Vol. 18, p. 285.

9. Tabata, T., and Masaki, S., (1975), "Plane-Strain Extrusion of Porous Materials," Memoirs of the Osaka Institute of Technology, Series B., Vol. 19, No. 2.

10. Tabata, T., and Oyane, M., (1975), "The Slip-Line Field Theory for a Porous Material," Memoirs of the Osaka Institute of Technology, Series B., Vol. 18, No. 3.

11. Hill, R., (1950), "The Mathematical Theory of Plasticity." Oxford University Press, London.

12. Johnson, W., and Miller, P. B., (1980), "Engineering Plasticity." Van Nostrand Reinhold, London.

13. Doraivelu, S. M., Gegel, H. L., Gunasekera, J. S., Malas, J. C., and Morgan, J. T., (1984), "A New Yield Function for Compressible P/M Materials," Int. J. Mech. Sci., Vol. 26, p. 527.

14. Oh, S. I., and Gegel, H. L., (1986), "ALPIDP--Modeling of P/M Forming by the Finite Element Method," Proc. NAMRC XIV, Minneapolis, MN, p. 284.

15. Oh, S. I., Wu, W. T., and Park, J. J., (1987), "Application of the Finite Element Method to P/M Forming Processes," Proc. 2nd ICPT, Stuttgart, West Germany, p. 961.

16. Osakada, K., Nakano, J., and Mori, K., (1982), "Finite Element Method for Rigid-Plastic Analysis of Metal Forming--Formulation for Finite Deforma- tion," Int. J. Mech. Sci., Vol. 24, p. 459.

17. Moil, K., Shima, S., and Osakada, K., (1980), "Finite Element Method for the Analysis of Plastic Deformation of Porous Metals," Bull. JSME, Vol. 23, No. 178.

18. Im, Y. T., and Kobayashi, S., (1985), Finite Element Analysis of Plastic Deformation of Porous Materials, in "Metal Forming and Impact Mechanics," (Edited by S. R. Reid), Pergamon Press, Oxford, p. 103.

19. Downey, C. L. Jr., and Kuhn, H. A., (1975), "Application of a Forming Limit Concept to the Design of Powder Preforms for Forging," J. Engr. Mat. Tech, Vol. 97, p. 121.

20. Im, Y. T., and Kobayashi, S., (1986), "Analysis of Axisymmetric Forging of Porous Materials by the Finite Element Method," Advanced Manufacturing Processes, Vol. 1, p. 473.

21. Austin, J. B., (1939), Factors Influencing Thermal Conductivity of Non- metallic Materials, in "Symposium on Thermal Insulating Materials," Ameri- can Society for Testing Materials, Philadelphia, p. 3.

22. Russel, H. W., (1935), "Principles of Heat Flow in Porous Insulators," J. Am. Ceram. Soc., Vol. 18, p. 1.

23. Eucken, A., (1932), VDI, Forschungsheft, No. 353 (in German), Forschung auf dem Gebiete des Ingenieurwesens Ausgabe B, Band 3, March-April.

24. Ira, Y. T., and Kobayashi, S. (1986), "Coupled Thermo-Viscoplastic Analysis in Plane-Strain Compression of Porous Materials," Advanced Manufacturing Processes, Vol. 1, p. 269.

25. Fischmeister, H., Sjoberg, G., Elfstrom, B. O., Hamberg, K., and Mironov, V., (1977), Preforms Ductility and Transient Cracking in Powder Forging, in "Modem Developments in Powder Metallurgy," (Edited by H. H. Hausner


and P. V. Taubenblat), Vol. 9, American Powder Metallurgy Institute, Princeton, NJ. p. 437.

26. Malik, R. K., (1974), "Hot Pressing of Titanium Aerospace Components," Int. J. Powder Metal. and Powder Tech., Vol. 10, No. 2.

27. Ferguson, B., Kuhn, A., Smith, O. D., and Hofstatter, F., (1984), "Hot Consolidation of Porous Preforms Using Soft Tooling," Int. J. Powder Metal. and Powder Tech., Vol. 20, No. 2, p. 131.

28. Fischmeister, H. F., Aren, B., and Eastering, K. E., (1971), "Deformation and Densification of Porous Preforms in Hot Forging," Powder Metal., Vol. 14, No. 27, p. 144.

29. Schacher, H. D., (1978), Kaltmassivumformen von Sintermetall, Ph.D. Dissertation, Institute fur Umformtechnik, Universitat Stuttgart.

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