Int. J. Mechanical Science, 1976, Vol. 18, pp. 285-291.
Pergamon Press. Printed in Great Britain
PLASTICITY THEORY FOR POROUS METALS
S. SHIMA a n d M. OYANE
Department of Mechanical Engineering, Faculty of Engineering, Kyoto University, Sakyo-ku, Kyoto,
Japan
(Received 1 N o v e m b e r 1975)
Summary--A plasticity theory for porous metals is proposed. From the stress-strain curves for
sintered copper with various apparent densities, the stress-strain curves for pore-free copper is
calculated by utilizing the basic equations.
The equations are applied to frictionless closed-die compression and the stress in the direction of
compression is evaluated in relation to the relative density and is compared with experimental results.
1. INTRODUCTION
IN RECENT years p o w d e r metallurgy techniques
h a v e been c o m b i n e d with conventional d e f o r m a tion p r o c e s s e s and h a v e been successful in
fabricating various engineering products. 1-5 In this
n e w c o m b i n e d technique, sintered porous metals
are e m p l o y e d as starting materials in ordinary
working processes, such as forging and extrusion.
U p to now, this has b e e n d e v e l o p e d and e m p l o y e d
without any theoretical background. Theoretical
approaches are, h o w e v e r , of importance for analysing and predicting, for example, the working load
required to cause plastic deformation and the
density of the products.
Although, for analysing problems in ordinary
metal working processes, various theories and
methods of analysis h a v e b e e n d e v e l o p e d , they are
not capable of being applied to the d e f o r m a t i o n of
porous materials. In conventional plasticity theory,
on which these theories and methods are based,
v o l u m e c o n s t a n c y is a s s u m e d for the material
undergoing deformation, and this assumption applies in effect to pore-free metals v e r y well. In the
d e f o r m a t i o n of porous metals, h o w e v e r , the volu m e does not remain constant.
It was not until v e r y recently that a plasticity
theory for porous metals was p r o p o s e d for the first
time. K u h n et al. 6 and G r e e n 7 h a v e independently
p r o p o s e d yield criteria and stress-strain relations
for these materials. B o t h of the theories suggest
that the yield criterion is a function of the first
invariant of stress, J1, and the second invariant of
deviatoric stress, J~, that is,
F = {c~J,2 + ~j~}m
form,
de, = d A ( t r i - q ~ r s )
(i = 1,2,3)
w h e r e a, /3 a n d ~ b are functions of the relative
density, dA is a non-negative constant and ~r~ the
hydrostatic stress. Since in both cases dA has not
been determined these approaches h a v e not been
utilized satisfactorily for the analysis of practical
deformation p r o c e s s e s ; they h a v e been used only
for simple stress states such as uni-axial compression and tension or plane-strain compression.
This paper is c o n c e r n e d with the d e v e l o p m e n t of
basic equations of plasticity theory for porous
materials which are similar to those described
a b o v e and s o m e e x a m p l e s of the application of this
theory are discussed.
2. BASIC EQUATIONS
2.1 Yield criterion
For a yield criterion for sintered metals we may begin
with equation (1), which may be rewritten as
F = [{(o.1 - 02)2+ (0"2- 03)2+ (0-3- ~r,)2}/2 + (0-~//)211/z
(2)
where f represents the degree of influence of the
hydrostatic stress component, 0-,, on the onset of yielding
of porous bodies and may be a function of the relative
density. F in the above equation may be related to the
yield stress of the matrix metal, 0"0.Thus equation (2) may
be rewritten as
f'0"o = [{(o"1- o'2)2+ (o'2 - o"3)2+ (o'3- o',)2}/2
+ (0-,/f)2],,2
(I)
and that the stress-strain tr-e, relations are of the
285
(3)
where f' represents the ratio of the apparent stress
applied to the porous bodies and the effective stress
applied to the matrix, and may be again a function of the
relative density. The functions / and f' can be determined
286
S. SHIMA and M. OYANE
theoretically from a simple modelfl but they do not
necessarily provide good a g r e e m e n t between equation (3)
and experimental results. Therefore, they will be determined experimentally in a later section.
The yield surface of a porous body with a relative
density p expressed by equation (3) is an ellipsoid whose
major axis coincides with the ~ axis (Fig. 1).
If o-0 is replaced by o-eq, that is the equivalent stress
applied to the matrix, then equation (3) m a y be applicable
to porous materials with work-hardening matrices,
(3)'
2.2 Stress-strain relations
F r o m equation (3) we m a y write
dA
g = ~ [{(o'1 - o'2)2 + (o'~ - o-3)2 + (o% - o',)2}/2 + (o-~ [f)2],/2
O" o .
If g is a s s u m e d to be a plastic potential, then the principal
strain increments dE~, de2 and de~ are derived by partially
differentiating g with respect to tr,, ~ and try:
Og_
- dA {o-~ - (1 - 2/9f2)o-m }
(7)
Eliminating o-m from equations (4) and (5) and rearranging, ~ , ~r2 and o-~ can be expressed in terms of dE~, de2, de3
and de,,. Substituting these in equation (6), coupled with
equation (3)', we have
(8)
2.3 Determination of f and f'
To determine the f o r m s of f u n c t i o n s / and [ ' we m a y
substitute into equations (4) and (5) o'2 = tr3 = 0 and
o% = ~ , / 3 , these being the conditions for uni-axial
compression. T h u s we have
de,-de:
dE,
9 2
2[
Thus
i=
de2 = dA' 0g_g= dA{o-2- (1 - 2/9/2)o'm }
3or2
Og_
de3 = dA' ~
3p de,.__.
2(f') 2 o-e.
d~¢--~= f ' [(2/9){(de, - dez) ~ + (d~2 - de3) ~+ (de3 - de,)2}
p
+ (f deo )2],,L
1
de, = dA' ~
relative density p consists of the matrix material of
volume p, then d W is not equal to o-e, de,,.
Substituting equation (4) into equation (6) and rearranging, we have
- dA {o% - (1 - 2/9f:)o-~ }
(4)
(de,d,2/"2
\ de~-~/
or
1_ 3 /
dEo ] ''2
f ~/2 \ d e , - d E 2 /
"
and further
deo = de, + de2 + de3 = - d o / P = dA (2/3F)o-~
(9)
(9)'
(5)
F r o m equation (3)
where dA is a proportionality constant which will be
determined later.
Now, if an equivalent strain increment referred to the
matrix is denoted by de.~, then the plastic work done per
unit volume of the porous body, dW, is expressed by
or0 = (1 + 1/9[:)":ltr,l/f'
or
Io-,I =
d W = o-,de~ + or2dE2 + O-3 de3 = pO'e,~de,,.
(6)
Note that since a unit volume of a porous body with a
t5
•
O'm
/-~o2,Mcr3
FIG. 1. Schematic illustration of yield surfaces for porous
materials.
f'o-0
(1 + 1/9/2) ' n
(10)
If changes in the relative density in simple c o m p r e s s i o n or
tension are obtained experimentally, f m a y be determined
from equation (9). Then if the yield stress is plotted
against relative density, f ' m a y be determined. T h u s to
find f ' simple tension and c o m p r e s s i o n tests were carried
out on copper c o m p a c t s of various densities.
T h e experimental procedure was as follows:
(1) C o m p a c t s in the shape of cylinders (20 m m dia. and
25 m m in height) and rectangular blocks (10 x 10 x 80 m m )
were made with various densities from copper powders,
the details of which are listed in Table 1. So that the
density should be uniform throughout the compacts, zinc
stearate was employed as a lubricant on the die surfaces
of the compaction apparatus.
(2) T h e c o m p a c t s were sintered for 2 hr at 900°C in
v a c u u m , and the density was m e a s u r e d after sintering.
(3) Tensile s p e c i m e n s were machined from the rectangular blocks and these were further heated at 600°C for
1~ hr in v a c u u m . The diameter of the parallel portion of
the test-pieces was 6 m m and the gauge length was 20 mm.
F r o m the tensile test results the yield strength was noted.
Plasticity theory for porous metals
287
1.0
T A B L E I . CHARACTERISTICS OF COPPER POWDER USED IN
T HE PRESENT EXPERIMENT
Apparent density
(untapped)
Purity of copper
Oxygen (loss in
weight in hydrogen)
g/cm3
%
2.0 ~ 2.3
min. 99.6
%
max. 0.2
Initial
~-o--e-
o
._~ 0.7
Mesh
+80
particle size distribution
100
145
200
0.74 5
250
350 -
t
'~ 0.6
I
%
Relative Density
0.869
0-830
max.
max.
10
10
max'l
3 5 ~ 15 3 0 - 4 5 25 ~ 35 5 - 15
0-5
I
(4) The cylindrical compression specimens were loaded
incrementally; after each incremental loading the height
and diameter of the specimens were measured. From
these tests the yield strengths of the specimens were
derived. In the compression tests p.t.f.e, films (thickness
0.02 mm) were used as a lubricant.
Fig. 2 shows the relationship between relative density
and strain in compression for five different initial relative
densities of the sintered copper. From these curves it is
possible to derive the relationships between volumetric
strain, ev, and compressive strain, et, and between lateral
strain, ¢2, and e,; these are shown in Figs. 3(a) and (b),
respectively.
To obtain values of the function 11[, the right-hand side
of equation (9)' is evaluated in the following manner: after
an increment of compressive strain Ae~ of 0.1, A~o and Ae2
are found from Figs. 3(a) and (b) for various values of e,.
Substituting these values in
\tie, - AEU
-0-
0-680
--a-
0-620
,
,
0
0.5
1[0
1iS
Compressive Strain (log)
FIG. 2. Change in relative density with compressive strain
for uni-axial compression of sintered copper.
values of the parameter 1/f are obtained. These values
are plotted against relative density in Fig. 4(a). If we
assume that f is of the form
[ = a(1 _p)m
then from Fig. 4(a) [ may be determined by the
least-squares method to be given by
.f = 1/2.49(1 - p)°-~".
Initil
Relative Density
0.869
-4~
0.830
"¢-
0.7~;5
0.4
0.680
9-0.3
d
•...0-
0.620
I
I~1'
o
-o-
0.869
-~-
0.e20
-~-
07~
+
0..80
- . - o =
0.3
t~-~
._c-0.2
/'/
/ 1 /¢
1//
c
~ o.2
//,d
..I-,
~ -0.1
,
Ag
0.1
V
-0.5
Compressive Strain
(a)
-1.0
¢, (log)
(1 lb)
Now we should determine the function f'. In Fig. 4(b)
the yield stress of sintered copper is plotted against
relative density. Fig. 4(b) shows that the yield stress is
independent of the type of loading and is only dependent
on the relative density of the material (assuming the
matrix has the same strength throughout). We will assume
Initial
Relative Density
--o-
(lla)
P
0
- 0.5
-I.0
Compressive Strain el (log)
(b)
FIG. 3. Relationships between (a) volumetric strain and compressive strain and (b) radial strain and
compressive strain in simple compression.
288
S. SHIMA and M. OYANE
o
°B[Eq
(llb)]
1.5
¢
[Eq ( l i b ) ]
o~
•
Compression
O
Tension
~
\
1.0
3-
ii
0.5
-
~---_
i.O
_
0.9
o
0869
•
0 830
¢
0 745
~>
0680
0
0620
--lo i
c~
2--
t
o
r
0.8
0.7
0.6
Relative Density
O
1.0
015
i
0.8
Relative
0.4
0.6
Density
FIG. 4. Examination of (a) function [ and (b) parameter n.
Fig. 4 (a); for comparison as simple form of [ suggested
by Oyane et al. '° and given by
that
f ' = p".
(12a)
f = I / 2 . 5 ~ / ( I - p)
W h e n n = 1, the ratio of the apparent stress applied to a
porous body to the effective stress applied to the matrix is
equal to the ratio of the effective or real area of the matrix
to its apparent cross-sectional area, that is p.
Substituting equations ( l i b ) and (12a) into equation
(10), the yield stress of the sintered copper, i.e. ~ , can be
calculated for given values of n and cro providing it is
possible to draw the solid curves in Fig. 4(b) for n = 1, 2
and 2.5. The values of ~o is determined in such a way that
the yield stress of solid copper (p = 1) from Fig. 4(b) is
5 - 6 kg/mmL F r o m Fig. 4(b) the value of n is taken to be
n = 2.5
(I Ib)'
is plotted as curve B. Substitution of equation (1 lb)' into
equation (10) provides an almost identical curve of yield
stress against relative density as that of equation (1 lb) for
the same value of ~ro. T h u s equation (1 lb)' is also a usable
expression for the function f.
It will be of some use to see if equations (11) and (12) m a y
also describe the behaviour of other sintered
materials such as iron and aluminium. K u h n e t al. 6
measured, for sintered iron and aluminium, the P o i s s o n ' s
ratio, v, in the plastic range in simple compression. From
equation (4), u is expressed in terms of the relative density,
p, by
(12b)
and thus ~ro is 6 k g / m m 2.
The functions f and [ ' have now been determined by
experiment. Equation (1 lb) is represented by curve A in
u = 0.5(1 - 2/9f~)/(1 + 1/9f2).
This equation is plotted in Fig. 5(b) as curves A and B
1 2 - O
E x p e r i m e n t a l no]
10
Theoretical
E
~8==
0.5
: Eq.
(lib)
B : Eq.
(llb~
A
.9
0.4
"o 4
g
0.3
o
0..
2 -0
5.5
6.0
65
70
75
Density (glcm 3)
1,0
o
SintereO
•
5entered Aluminiurn 1"6]
0.95
Relative
Iron[63
G90
0.85
0.80
Density
FIG, 5. Variation of (a) yield stress and (b) P o i s s o n ' s ratio with density for sintered iron and sintered
aluminium.
Plasticity theory for porous metals
w h i c h h a v e been calculated by substituting equations
( l i b ) and ( l i b ) ' , respectively.
Fig. 5(a) s h o w s the variation of the yield stress of
sintered iron with relative d e n s i t y ? A s s u m i n g that the
density of pure iron is 7 . 8 7 g / c m 3 and that ~ro =
1 0 . 0 k g / m m ~, the yield stress m a y be calculated from
equation (10) and equation (1 lb) a s s u m i n g n = 2.5. T h u s
we m a y say that the f u n c t i o n s f and f', determined by
e x p e r i m e n t s on sintered copper, are also applicable to
sintered iron and possibly to sintered aluminium.
S u m m a r i z i n g the above equations we have
__
289
material by utilizing the basic equations. This section
deals with an examination of the s t r e s s - s t r a i n curve
converted from those for sintered copper in uni-axial
c o m p r e s s i o n and tension.
Substituting tr2 = ~r3 = 0 in equation (13) and (15), with
n = 2.5, we have
--
1
cr.q = - ~ ( 1 + 1/9f~)1'21~,1
/9
and
1
de2=de2=
~.q = p-: [{(~1 _ ~)2 + (~2 - ~3) 2
+ (or3 - tr1)2}/2 + (tr,/f)2],,2
(13)
1 + 9f2/2
l + 9 f 2 de1.
Substituting the above equations into equation (14), we
have
da,q = p" '[(2/9){(de,-de2) 2
+ (de2 - de3) 2 + (de3 - de1) 2} + (f de~)2] 1/2
(13)'
de,~ = {p 15/(1 + l/9f2)ll2}ldell.
(14)
(14)'
Combining the first equation in equation (15) and equation
(16), dp is e x p r e s s e d in terms of de1 as
de,
3 1 de,_._~{crt-(1-2/9f2)cr~}
2 p2.-, 0%
d~2
3 1
2 p2.-1 o',~
de3
3 1 (le,q{o.3_(l_2/9f2)o. }
2 p2,-1 o'~q
dp = -{3p [(i + 9/z)} d6,.
__d~"{o'2-(l-2/9fZ)o"
}
d~o = de, + de2 + d~3 = - d p / p = I
(15)
de,,__tr,.
p2.-I
(16)
O.ea f 2
W h e n p = 1, then f = o% [' = 1 and dp = de~ = 0, t h e s e
equations reduce to the well k n o w n equations for
conventional plasticity theory.
2.4 Calculation of stress-strain curve
In the basic equations of plasticity theory for porous
materials, established above, tre~ and e,~ refer to the
equivalent stress and the equivalent strain respectively to
which the matrix material is subjected. Therefore, if a
s t r e s s - s t r a i n c u r v e for a sintered material is obtained, this
can be converted to s t r e s s - s t r a i n curve for the matrix
Fig. 6(a) s h o w s s t r e s s - s t r a i n curves for sintered copper
in compression, where the stress has been obtained f r o m
the load divided by the i n s t a n t a n e o u s cross-sectional area,
that is the true stress.
Fig. 6(b) s h o w s l o a d - e x t e n s i o n curves for sintered
copper. T h e s p e c i m e n s are same as those described in 2.3.
Since the cross-sectional area could not be m e a s u r e d in
tensile testing, the s t r e s s - s t r a i n curves could not be
obtained directly as in the case of the c o m p r e s s i o n test.
F r o m the curves in Fig. 6(a) and equations (13)', (14)'
and (16)', the s t r e s s - s t r a i n curve for the matrix is obtained
by a step-wise calculation, the results being s h o w n by the
thick solid lines in Fig. 7.
F r o m the l o a d - e x t e n s i o n curves of Fig. 6(b), the tensile
stress, or,, m u s t first be evaluated. Since the m a s s (not the
volume) of the e x t e n d e d portion of the test-piece is
c o n s t a n t during testing, we have
Alp =- A~l~p~
where A is the cross-sectional area, l the gauge length and
fL-
600
40
/
Relative Density
A
--O--
0-869
E 30 - - I ~
E
0.795
0.624
~
500
Initial'
~3o0/
_
20
o
a~1../.
/
/
P~ =0-96
I
f
Pi =0.79
200/1/,.-
~D "~"
10
0
(I6)'
100
0-1
0.2
0-3
Compressive
0.4 0.5 0-6
Strain (log)
~7
0
1
2
3
Extension
5
6
7
(mm)
(a)
(b)
FIG. 6. (a) S t r e s s - s t r a i n curves in simple c o m p r e s s i o n and (b) l o a d - e x t e n s i o n curves in simple tension
for sintered copper.
290
S. SHIMA and M. OYANE
Since de2 = de3 = 0, tr: and tr~ are therefore, from
equation (15), e x p r e s s e d in terms of o"l as
~0
tr2 = ~r3 = {(/2 _ 2/9)/(/2 + 4/9)}o',
E
3o -
(17)
./
p,=o.96j
~..]
°,.°'-
Substituting this equation into equation (13) and rearranging, we have
20
~/~iore-frle Copper
ffl
10
Io-,1/o-.,
r
0
0.2
0.4
0.6
Strain (log)
FIG. 7. Calculated s t r e s s - s t r a i n curves for the matrix
copper (pore-free).
suffix " i " refers to the original state of the test-piece. If an
extension is given, p can be calculated from equation (16)'
and thus A is determined from the above equation, trl is
then evaluated. Thus, by the same procedure as for
compression, the stress-strain curve for the matrix is
calculated as s h o w n by the broken lines in Fig. 7. For
comparison, a true s t r e s s - s t r a i n curve obtained by tensile
testing a fully annealed commercially pure copper is
s h o w n by the thin solid line in the figure. T h e calculated
curves agree well with each other regardless of the type of
loading and of the initial density of sintered copper. T h e y
are also in substantial a g r e e m e n t with the curve for the
fully d e n s e copper. T h e s e results confirm that the basic
plasticity equations are appropriate.
3. A P P L I C A T I O N O F T H E BASIC T H E O R Y
This section deals with an application of the basic
equations to closed-die compression, or re-compression
of sintered copper without friction at the tool-work-piece
interfaces. In this case the c o m p r e s s i n g direction is a
principal direction. Let or, be the stress in this direction
and the related expressions be derived.
= p~'(/~ + 419)"L
Equation (18) provides a relationship between applied
pressure and relative density w h e n the yield stress or flow
stress of the matrix material is known.
Further, substituting dE2= d ~ 3 = 0 into equation (14)
and considering that d~, = deo = - d p / p , we have
d~.q = {p(4/9 + f2)},,2 do.
2-0
test, the flow stress tr,q can be calculated from equation
(13)'. T h u s I@,l/~.qm a y be plotted against relative density
as in Fig. 8(a). In the figure, the solid line s h o w s the
1.5
1-0
t5
., •
E~perimenlal
¢
60
5(
o
0
Pi
0-689
0631
•
0-583
Theoretical
A
8
Experimental
C
Theoretical
Pi
0.689
0.631
-/
O
C//~
0583
0.5
aZ~rl
0
1.0
0-9
0-8
0.7
Relative Density
~.
0.6
(19)
T h u s , from equation (18) and equation (19), the theoretical
relationship between stress and relative density can be
obtained if the stress-strain curve for the matrix is
known.
Closed-die c o m p r e s s i o n was p e r f o r m e d employing
sintered copper in a similar w a y to that already described
in 2.3. Two types of testing were carried out:
(1) Specimens, fully annealed or work-hardened, were
c o m p r e s s e d in a die and the load was m e a s u r e d to
evaluate the stress or,. The specimens were taken out
f r o m the die, and the relative density p was measured.
Then, they were immediately c o m p r e s s e d uni-axially and
the current yield stress o'ic was measured.
(2) Fully annealed s p e c i m e n s were c o m p r e s s e d in a die.
T h e load and the density were m e a s u r e d and thus a
c o m p r e s s i v e stress-relative density plot was obtained.
In all of these test, zinc stearate or p.t.f.e, sheets were
used as lubricants.
Utilizing the m e a s u r e d values of p and tr,c in the first
70
2-5
(18)
0.6
0.7
0.8
0.9
Relative Density
1.0
FIG. 8. Variation of (a) non-dimensional tool pressure, pressure exerted on the die wall and (b) tool
pressure with relative density.
291
Plasticity theory for porous metals
theoretical curve due to equation (18). The figure shows
that there is good agreement between the two results.
On the basis of the above information coupled with a
given stress-strain curve for the matrix, let the tool
pressure vs relative density be evaluated and
compared with experimental results: Integration of
equation (19) would provide a relationship between e,~
and p, but the integration is impossible, and therefore an
approximate equation is given as follows:
/
0.068\
d ~ = ~0.633 + -1- - ~ ) do.
and are given b y
f = 1/2.49(1 - p)o.5~4
and
n = 2.5.
A s s u m i n g that the yield criterion s e r v e s as a
plastic potential, the s t r e s s - s t r a i n relations h a v e
b e e n d e r i v e d as
dEi = d),(o-~ -- (1-- 2/9f)o-m }
This equation gives a very good approximation for
p > 0.60. Integration of the above equation yields
e,~ = 0.633p - 0.068 In (1 - p) + e,
(i = 1 , 2 , 3 )
w h e r e dA is a p r o p o r t i o n a l i t y c o n s t a n t given by
(19)'
where E~is an integration constant which is determined by
substituting ~ = 0 for p = p~ (initial relative density).
The material used in the experiments was copper, the
stress-strain curve of which has already been given in Fig.
7. From the figure it may be reasonable to employ the
curve obtained from the specimen whose initial relative
density is 0.869. The curve is expressed by the following
two equations:
2 p2n-I o-eq
w h e r e dE,~ is the e q u i v a l e n t strain i n c r e m e n t f o r the
matrix e x p r e s s e d by
dE,q = p"-'[(2/9){(dE, -- de2)2 + (de2 - de3) 2
+ (dE3 - dEl) 2} + (.1t dEo):] m.
tr,q = 96.56(tr,, + 0.003)°58~3 for
0 -< e,, < 0.034
(20a)
m
__
Cr,q = 49.26(E,q- 0.0105) 0.3333 for
e , q - 0.034
(20b)
which were determined by the least squares method.
From equations (18), (19)' and (20) the tool pressure (O-l)
required to achieve a given density can be evaluated if the
initial density is known. Fig. 8(b) shows the plot of
compressive stress, that is the tool pressure, vs relative
density in closed-die compression of sintered copper with
various initial relative densities due to the second test.
The solid lines in the figure show the theoretically derived
curves. The figure shows that there is good agreement
between the theoretical and experimental results.
In Fig. 8(a), the broken line shows o'dcr, vs relative
density, from which the stress exerted on the die wall in
re-compression of sintered metals can be estimated.
If the frictional coefficient at the tool-work-piece
interfaces is not equal to zero, which is the case in
practice, the stress distribution is not uniform throughout
the body. Therefore the solution of the problem would not
be as simple as in the above method. This will be
presented in the future.
4. CONCLUSIONS
A yield criterion f o r p o r o u s materials has b e e n
p r o p o s e d w h i c h is o f the f o r m
- -
1
o'~. = ~
[{(o'1 - o-z)2 + (o-z - o'3)2 + (o-3 - o-,)2}/2
-
(o-~, If)q
':~.
F u n c t i o n s f a n d n w e r e d e t e r m i n e d f r o m simple
c o m p r e s s i o n and t e n s i o n t e s t s on s i n t e r e d c o p p e r
T h e s e basic e q u a t i o n s h a v e b e e n applied to the
analysis of c l o s e d - d i e c o m p r e s s i o n w i t h o u t friction
at the t o o l - w o r k - p i e c e i n t e r f a c e s . It was s h o w n that
the calculated tool p r e s s u r e - r e l a t i v e d e n s i t y relat i o n s h i p for s i n t e r e d c o p p e r agreed well with the
e x p e r i m e n t a l results.
A l t h o u g h the f u n c t i o n f and the p a r a m e t e r n in
the e q u a t i o n s h a v e b e e n d e t e r m i n e d f r o m experimental results on s i n t e r e d c o p p e r , it w a s c o n f i r m e d
that t h e y are also applicable to s i n t e r e d iron and
p o s s i b l y s i n t e r e d aluminium. This suggests that the
basic e q u a t i o n s w o u l d be a p p r o p r i a t e for o t h e r
materials as well as the c o p p e r .
Acknowledgements--The authors wish to thank Dr. B.
Dodd of the Department of Engineering Science, University of Oxford for correcting the English.
REFERENCES
1. K. FARREL, Int. J. Powder Met. 2, 3 (1966).
2. R. A. HUSEaY and M. A. SCHEIL, Proc. Int. Powder
Metallurgy Con[., Vol. 4, p. 395, New York (1970).
3. H. W. ANTES, Ibid., p. 415.
4. G. LUSA, Ibid., p. 425.
5. Y. ISnIMARU,Y. SArro and Y. NISHINO, Ibid. p. 441.
6. H. A. KUHN and C. L. DOWNEY, Int. J. Powder Met. 7,
5 (1971).
7. R. J. GREEN, lnt. J. mech. Sci. 14, 215 (1972).
8. M. OYANE, S. SHIMA and Y. KONO, Bull. Japan Soc.
Mech. Engrs. 16, 1254 (1973).
9. K. KUROKI, T. IDE a n d Y. TOKUNAGA, J. Japan Soc.
Powder and Powder Met., 21, 43 (1974).
10. M. OYANE, T. KAWAKAMIand G. SHIMA, Ibid., 20, 143
(1973).