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Modelling of hot compaction of metal powder by homogenization
Mechanics of Materials 13 (1992) 247-255 Elsevier
247
Modelling of hot compaction of metal powder by homogenization J.L. Auriault, D. Bouvard, C. Dellis and M. Lafer Institut de M~canique de Grenoble, BP 53 X, 38041 Grenoble Cedex, France Received 5 March 1991; revised version received 31 January 1992
An homogenization technique is presented and applied to the modelling of the compaction of metal powders at high temperatures. Experiments performed on stainless steel powder suggest that the densification rate during hot isostatic pressing varies as the external pressure to the power of n, where n is the exponent of the creep law of the constituent metal. The homogenization process applied to a porous body made up of non-linearly creeping particles permits rigorous proof of this result. More generally it shows that the power-law structure of the local description is preserved at the macroscopic level.
I. Introduction
Hot compaction of metal powder - in particular hot isostatic pressing - is increasingly used to make fully dense parts of complex shape and fine microstructure. Advanced materials such as superalloys, titanium alloys, heavy metals, composites, etc., are considered. Modelling of the mechanics of consolidation has developed through the necessity of controlling densification kinetics and shape change to optimize industrial processes. The difficulty lies mainly in the significant changes of porosity -- in both proportion and morphology - induced by the consolidation and in the non-linear visco-plastic constitutive behaviour of the powder particles. Hence the models developed up to now consist in either phenomenological constitutive laws with parameters fitted from experiments, which are particularly heavy and sometimes inaccurate (Abouaf and Chenot, 1986; Nohara et al., 1986; Bouvard and Lafer, 1989b), or micromechanical equations, more physically-based but still limited to simple Correspondence to: J.L. Auriault, Institut de M6canique de Grenoble, BP 53 X, 38041 Grenoble Cedex, France.
geometry (Sofronis and McMeeking, 1991) or isotropic loading (Matthews, 1980; Arzt et al., 1983; Bouvard and Ouedraogo, 1987; Bouvard and Lafer, 1990). In this paper an homogenization technique is proposed to find out the constitutive law of metal powders at high temperatures from the equations governing the deformation at the particle scale. The macroscopic modelling is obtained without any prerequisite at the macroscale. The only assumption to be made is a good separation of the scales, i.e., l and L being the characteristic lengths of the particles and the bulk medium, respectively, IlL << I. The reader will refer to Auriault (1991) for the methodology to be applied. Hot isostatic pressing tests performed on stainless steel powder are first presented with emphasis on the influence of the external pressure on the densification kinetics. Next the homogenization process is explained and applied to the problem of a porous body made up of non-linearly creeping particles. The resulting equations are developed particularly in the case of isotropic macroscopic deformation, leading to a densification relation of the same form as that derived from previous experiments.
0167-6636/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
248
J.L Auriault et al. / Modelling hot compaction
pressure. The densification rate f~/p was calculated by derivation of the analytical functions fitted to the experimental points. Observation of the compacted samples by scanning electronic microscopy showed a fairly homogeneous microstructure during consolidation and suggested that the densification mainly results from deformation of the particles by power-law creep (Bouvard and Lafer, 1989a). A compression test was next performed at 1200°C on a fully dense sample. Various strain rates were successively applied (Bouvard, 1989). The resulting strain-stress curve appears in Fig. 2. A jump in the strain rate results in a steep rise in the stress, followed by a quasi-stationary state. Repetitions of the first loading sequence show a slight hardening. The relation between strain rate and stress at 12% of axial strain is presented in Fig. 3. A power function perfectly fits the experimental points:
2. Experiments
Direct measurement of volume changes during hot isostatic pressing (HIP) is still impossible since the pressure is applied to the sealed can, containing the powder, by a gas. Various dilatometers were recently designed to measure a characteristic dimension - usually the median diameter - of the cylindrical sample, from which density changes are calculated (McCoy et al., 1985; Wadley et al., 1987). Development of such advanced devices is tricky and derivation of density often imprecise; their use is therefore still limited. An alternative way consists of a succession of HIP tests interrupted after different holding times. Measurement of the density of the compacted samples provides discrete points of the density versus time curve, from which densification kinetics can be estimated. This method was used in particular on 316L stainless steel powder. Detailed presentation of the procedure can be found in Bouvard and Lafer (1989a). Densification curves obtained at 1200°C with two different pressures - 10 and 15 MPa - are presented in Fig. 1. As expected, for the same holding time, the relative density p increases with
d: = A o r n,
with n = 4.07 and A = 1.7 × 10 -9, the stress tr being expressed in MPa and the strain rate ~ in s-1. As the tested material presented the same
O.95 0.9 ~ 0.85'
l 0.8!
P J I
1o ] ~ A
1200°C
15 MI~
0.75 0,7-
°"6 o
2b
4b
6b
+b
16o 6o Tree (m)
t4o
2oo
Fig. 1. Results of HIP on stainless steel powder at 1200°C; densificationcurves fitted from experimental points.
J.L. Auriault et al. / Modelling hot compaction
249
25" 30~ma/um 300nm~nm
20"
"~ 15o~ o~
-~ < i0x <
5 " /
0
o
,amom
o:o2
o;o4
o.'o6
0.08
0~1
0.'12
0.'14
0.16
AXIAL STRAIN Fig. 2. Result of a simple compression test on fully dense steel at 1200"C.
microstructure as powder particles during consolidation, the above creep law is supposed to apply also to particle deformation. Most models assume - or result in - a variation of the densification rate with the external
pressure P to the power of n, n being the exponent of the creep law. In order to test this relation, the densification rate calculated from experiments, divided by A P " is plotted in Fig. 4 as a function of the density. The curves corresponding
-3.0'
-", - 3 . 4 v
<
'~ -3.8 <
.~ - 4 . 2 x
<
~
-4.6
.O7
-5.0
0.
1~0
'
i~1
'
1~2
1~3
log [ AXIAL STRESS (MPa) ] Fig. 3. Creep law of dense steel at 1200°C fitted from experimental results.
1.4
J.L. Auriault et aL/ Modellinghot compaction
250 -2
c
-3
\
-7
o.65
0?z
o. 5
0?s
0:s
llm,k11VlZl~m4m'l'/ Fig. 4. Normalized densification rate as a function of relative density.
to both pressures are very close, and this supports the assumption.
Odqvist's constitutive law, which is classical at high temperatures:
Orij plij + Sij,
(1)
3. T h e h o m o g e n i z a t i o n process
Vo- = O,
(2)
We aim at discovering the bulk (macroscopic) description of metal powders under compaction from the description at the particle level. This passage from a microscopic to a macroscopic level is named an homogenization process. We use here the most powerful one which introduces the smallest possible number of assumptions. In the first part the local description of the material is given. The homogenization of periodic structures is presented in the second part. Finally, the process is applied to our problem in the third part. The main results are that the structure of the description is obtained for a quite general anisotropic behaviour and this structure shows a similar power law to the local one.
Oij = A S n - l s i j ,
3.1. Description of the metal powder
cNIr=O.
The material is composed of metal particles and pores. We assume the metal to follow
Discontinuity lines can exist in the bulk material. In this case we assume the normal stress and
=
Vv=O,
S = tr(1/~-SS) ,
A > 0,
l(Ovi
Ovj.]
Dii=O, Dij='~ OXj + ~ ]
(3)
(4)
Here p is the pressure, S the stress deviator, and D the rate of the deformation deviator. Equation (2) expresses the momentum balance, where inertial effects are neglected. Equation (3) is Odqvist's law for an incompressible material following (4); S appears as the second invariant of the stress deviator and n is the scalar already introduced in Section 2. On the surface of the particles, the normal stress is zero:
J.L. Auriault et al. / Modellinghot compaction
251
the displacement to be continuous along these lines. It will be convenient to rewrite the constitutive law (3) in the reverse form
Sij =A-l/nDl-"/"Oi;,
D
=
~/tr(D.D) .
(5)
In all practical cases the bulk volume of the considered material is very large compared to the particle size. Therefore a very good separation of scales exists which enables us to predict an equivalent continuous macroscopic description.
3. 2. Homogenization principle We use here a double-scale asymptotic development based on the existence of two well-separated characteristic lengths (see Bensoussan et al., 1978 and Sanchez-Palencia, 1980, for the method). If l is a characteristic length of the particles or pores and L a characteristic length of the material sample: l
Z- = n <
X 1 Fig. 5. Schematicview of the material at the microscopiclevel: unit cell. between them. The geometry of the pores inside the unit cell is arbitrary. In particular, the pores can be either interconnected or separated. A change of the internal geometry of the unit cell does not modify the structure of the macroscopic description but only the effective coefficients appearing in it. The two characteristic lengths 1 and L introduce two dimensionless space variables, and each physical quantity q~ is a function of these two variables. We will rather use the physical space variables x, and y = x/~7 =
y).
The variable x is the slow space variable or macroscopic space variable well adapted to macroscopic variation. On the contrary, y is the fast space variable well suited for local description. Due to the small parameter r/, it is realistic to look for the physical quantities in the form of an asymptotic development. For example the velocity v will be looked for in the form:
v(x, y) = v(°)(x, y) +TqvO)(x, y) + "'" On the other hand, the 12 periodicity of the structure induces the same periodicity for v with respect to y. The v (i) are then 12 periodic in y. Let us investigate the influence of such a form of v on the other physical quantities. Noticing that d ~ +r/-1 dx
ax
~y'
252
J.L. Auriault et aL / Modelling hot compaction
we obtain for the rate of deformation deviator D
Oij = Dxiy( v ) + "rl - l D r i j ( v ) , where the subscripts x and y show the variable for the derivation. Therefore we have D = ~/- ~D(- 1) + D(0) + ~/D(1) + . . . ,
where the unknown v (°) and p(-1) are O periodic. Let us multiply the two members of Eq. (6) by v (°) and integrate on O s. Then we have
fa v(O)aO.i~-1)d O Oyj
with D (- 1)
=
Dr( V(0)),
D(°) ----'~x(V(0)) +~r(V(1)) ' where ~ is the rate of deformation. It is now easy to deduce the corresponding development for S:
S -~-vn-1/n(V
-ls(-1)
And, by using the divergence theorem, the condition (8), and the periodicity on 5 0 s N 5 0 , we write
f, o( v,,o,o-,?1,) d O = [g~oy~°)o'i~,- ~ 3yy
"4- S (0) + '178 (1) "4- *" • ).
d S = O.
We assume p and S to be of the same order with respect to the powers of ~/. Therefore we have
Then, the remark that the symmetry of 0 "(-1) enables us to put
p = ~q,-W,(n-lp(-O
__0r(
av~°)
+p(0) + ~TpO)+ . . . ).
0yj
Remark: As V -- 10 -3, a pressure which is equal to 3 × 10 -2 times the value of the deviator can be said to be of the same order of magnitude as the deviator itself! This fact justifies our assumption. 3. 3. Macroscopic description
3.3.1. Lower orders For the lower orders we obtain the following boundary-value problem on /2: o'(-1)=p(-1)l
The equation simplifies to: fo~riy (v`0)) q/~.-1, d O = 0. NOW, taking into account the incompressibility condition (7), we are left with
Let us introduce the above developments in the local description. Looking for like powers in each equation gives successive problems to be solved for a unit cell. We begin by the lower orders.
Vror(-1)=0,
.- 1) = ~ r r i j ( / 5 ( 0 ) ) O.i(-1).
+ S (-1),
fo Drij(
v O)) S~j- 1) d O = 0.
s
The integrand corresponds to a dissipation density and represents a positive quantity. Therefore we deduce
or(v(°) ) = 0
(6)
and, with the periodicity,
(7)
At the local level the first order velocity is a translational velocity. The consequences are: S (-1)= 0 and Eq. (6) reduces to V r p ( - l ) = 0, so that p ( - 1 ) = p ( - l ) ( x ) and Eq. (8) implies p ( - O = 0.
S ( - 1) = A - l i n D ( - 1X1 - n / n ) D ( - 1),
~(-1) = I t r ( D r (v(0>) . Dr(v(0>)), Vyv(°) = 0
or Dyii(v (0)) = 0,
( r ( - ° N Ir = 0,
(8)
J.L. Auriault et al. / Modelling hot compaction
3.3.2. The fundamental cell problem
At the following orders, taking into account the preceding results gives V @CO) = O3 Y
&?(O) = A
D’O’
&a) = p’o’Z + $0) 7
- l/npol
-n/n@)
9
p
=
@
\/e(D’o’
tr(
=
(gyst(X.lk)
@”
+
‘D’o’)
=
a’OQvlr=O.
This is a boundary-value problem for the L4 periodic unknown parameters Y(‘) and p(O),where O,(vCo’) is considered as being given. Let us admit the existence of a solution u(l) and p(O). The velocity u(l) appears as a homogeneous vectorial function of degree 1 in the rate of deformation O,(uCo’) defined to an arbitrary local translation i?(‘)(x). For the sake of simplicity, we are using the same notations as for a linear relation: U!” 1 = Xijkaxjk( u’“‘) + i$“( X), where the third-order / 0,
We now express (S(O)) and ( p(O)) as functions of the macroscopic rate of deformation 0,(~‘~‘). For (S(O)) this is done easily by using the expression (10) for u(l). We obtain successively:
(9)
=a, ( u’“‘) +a; ( Y(l)) )
vv”‘+vu’o’=o, x Y
253
(10)
tensor x is of zero average]
zs,z,k)gx,k~u’o’~~
( gyst(x.lk)
’
x
(~yst(x.mn>
’
Oxlk(
u(o))
bztk)
+
Oxmn(
LLi)
u(o))
and $0
=A-l/n(trD(0)D(O))l-n’2n ’ (Oyij(X.pq>
+ zipzjq)gxpq(u(o))*
Then taking the average on fl gives (SC’)>. It appears to be related to 0Ju(‘)) by a homogenous anisotropic constitutive relation of the same degree n as the original local constitutive relation. But here, since shrinkage occurs, the kinematics is described by 0 in place of the deviator D.
xdR=O.
The structure of the solution p(O) is more involved. Since it is not needed in the following, we disregard it. 3.3.3. The macroscopic description At the following order, Eq. (2) gives v Y CT(l)+ vx (T(O)= 0.
(11)
The stress a(‘) being a periodic quantity, the balance (11) implies that the source term VXaCo) be of zero volume average. Noticing
For (p(O)) the way to be followed is somewhat different. We first consider a vector field (Yon 0, (Y being 0 periodic with a given divergence a V,a=a. This field is not unique, but the result is independent of this uncertainty as well as of the value of a. Secondly we multiply the momentum balance (9) by (Y and integrate on R,
( p"'Zij + Sl’i”‘)dR
j~~i ~
I = /, -& ( ai( p’“‘Zij+ S$‘,>) dR 5 I
vx (a(O)>= 0, -
where (a’O’> = (p’O’)Z+
2
I, I
(S(O)).
( p”‘Zij + S$‘)) dR = 0. J
The first integral on the right hand cancels out
J.L. Auriault et al. / Modelling hot compaction
254
by using the divergence theorem, the periodicity, and the condition on F. We are left with
fo O°~in(°)l.. dO = a f. p(O) dO Oyj ~
-u
S2s
= -
with
Bi: = (b)
= (½A-I/nl~(.~yst(X.l,)+ ]st) X(-,~yst(X.l,)+ 2.)I'-°/=°
(aa---Z/S~9)dO
Jl2s Oyj u
= - f
''~yi'/(e) S:°)dO.
O~
And finally the average of p(0) is given by l (p(O)> = _ - -
J.[.~yij(a)S: 7) dO.
a l O I as Here again the relation shows an anisotropic power-like relation with exponent n. Remark 1. In practice, p(0) will be given by solving the fundamental cell problem of Section 3.3.2, together with v (1). Remark 2. It is easy to show that the result is thermodynamically consistent: the average of the local dissipation density equals the macroscopic dissipation.
3.3.4. Isotropic shrinkage We restrict ourselves here to isotropic shrinkage. Therefore the rate of the deformation tensor reduces to:
The velocity v (~) is now a linear vectorial function of the trace VxV(°) of the rate of deformation, and we have successively,
O~°) = ( ~yst( X.lk ) "{- Islltk ) 3 VxU(O)llk, Dj°t ) = ( ~ , , t f x . , )
+ I,,)½ Vxv(°),
tr( D (°)" D (°)) = (~yst(X.tz) + Ist ) (-~y,t( X.t, ) + I~t )
and ( S~°')= n,jlVxv(°)ll-n/nvN,°),
X (~Iyij(X.pp)-l-~ij)). For (pO> a similar relation holds:
< p(0)> __ cl v::)l ]-n/n VxV °,, l ( p(O)) = _ __( ~ y i j ( ot)b )"
a
In the case of an isotropic macroscopic material, (S ~°)) = 0 and we recover an isotropic relation as was assumed in Section 2 for the experimental investigation. In this simple case the power-law structure of the macroscopic description is evident.
4. Conclusions The homogenization process applied to the investigation of the modelling of hot compaction of metal powders enables us to rigorously demonstrate that the power-law structure of the local description is preserved at the macroscopic level. This is in agreement with experimental investigations concerning isotropic hot isostatic pressing tests. But it gives the more general description at the macroscopic level, for an anisotropic material under any excitation. These are precisely the cases where experimental data are difficult to obtain. Working out explicit constitutive relations requires solving the fundamental problem (obtaining the third-order tensor X) for the unit cell, which can be achieved through numerical calculations. Such work is in progress.
Acknowledgements The authors wish to thank the Vallourec Company for providing the powder and for supporting
J.L. Auriault et al. / Modelling hot compaction
part of the experiments, and the Tecphy Company for funding C. Dellis jointly with CNRS. References Abouaf, M. and J.L. Chenot (1986), Simulation num6rique de la ddformation h chaud de poudres mdtalliques, J. Mec. Theor. Appl. (France) 5, 121. Arzt, E., M.G. Ashby and K.E. Easterling (1983), Practical applications of hot isostatic pressing diagrams: Four case studies, Metall. Trans. 14A, 211. Auriault, J.L. (1991), Heterogeneous medium. Is an equivalent macroscopic description possible?, Int. J. Eng. Sci. 29 (7), 785. Bensoussan, A., J.L. Lions and G.C. Papanicolaou (1978), Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam. Bouvard, D. and E. Ouedraogo (1987), Modelling of hot isostatic pressing: A new formulation using random variables, Acta Metall. 35, 2323. Bouvard, D. (1989), Rh6oiogie des poudres mdtalliques au cours de la mise en forme ~ haute tempdrature, Th~se d'Etat, Grenoble. Bouvard, D. and M. Lafer (1989a), Determination of the densification kinetics of metal powders by interrupted hot isostatic pressing tests, Powder metall. Int. 21, 11. Bouvard, D. and M. Lafer (1989b), Rheological characteriza-
255
tion of metal powder at high temperature, in: T.G. Gasbarre, ed., Advances in Powder Metallurgy, MPIF, Princeton, NJ, Vol. 1. Bouvard, D. and M. Lafer (1990), Recent developments in modelling of hot isostatic pressing: Taking account of particle size distribution and contacts interaction, in: R.J. Schaefer and M. Linzer, eds., Proc. 2nd Int. Conf. on Hot lsostatic Pressing, Gaithersburg, Maryland, 1989, American Society of Metals, Cleveland, OH. Matthews, J.R. (1980), Indentation hardness and hot pressing, Acta Metall. 28, 31. McCoy, J.K., L.E. Muttart and R. Wills (1985), Continuous monitoring of volumetric changes in ceramic powder compacts during hot isostatic pressing, Am. Ceram. Soc. Bull. 64, 1240. Nohara, A., T. Soh and T. Nakagawa (1986), Numerical solution of the hot isostatic pressing, in: Proc. 3rd Int. Conf. on lsostatic Pressing, London, 1986, MPR Publishing Services, Stratford. Sanchez-Palencia, E. (1980), Non-homogeneous media and vibration theory, Lecture Notes in Physics, Springer, Berlin-New York. Sofronis, P. and R.M. McMeeking (1991), Creep of a power law material containing spherical voids, 3. Appl. Mech., to be published. Wadley, H.N.G., A.H. Kahn, Y. Gefen and M. Mester (1987), Eddy current measurement of density during hot isostatic pressing, in: Proc. Conf. on Progress in Quantitative Nondestructive Evaluation, Williamsburg, 1987.
247
Modelling of hot compaction of metal powder by homogenization J.L. Auriault, D. Bouvard, C. Dellis and M. Lafer Institut de M~canique de Grenoble, BP 53 X, 38041 Grenoble Cedex, France Received 5 March 1991; revised version received 31 January 1992
An homogenization technique is presented and applied to the modelling of the compaction of metal powders at high temperatures. Experiments performed on stainless steel powder suggest that the densification rate during hot isostatic pressing varies as the external pressure to the power of n, where n is the exponent of the creep law of the constituent metal. The homogenization process applied to a porous body made up of non-linearly creeping particles permits rigorous proof of this result. More generally it shows that the power-law structure of the local description is preserved at the macroscopic level.
I. Introduction
Hot compaction of metal powder - in particular hot isostatic pressing - is increasingly used to make fully dense parts of complex shape and fine microstructure. Advanced materials such as superalloys, titanium alloys, heavy metals, composites, etc., are considered. Modelling of the mechanics of consolidation has developed through the necessity of controlling densification kinetics and shape change to optimize industrial processes. The difficulty lies mainly in the significant changes of porosity -- in both proportion and morphology - induced by the consolidation and in the non-linear visco-plastic constitutive behaviour of the powder particles. Hence the models developed up to now consist in either phenomenological constitutive laws with parameters fitted from experiments, which are particularly heavy and sometimes inaccurate (Abouaf and Chenot, 1986; Nohara et al., 1986; Bouvard and Lafer, 1989b), or micromechanical equations, more physically-based but still limited to simple Correspondence to: J.L. Auriault, Institut de M6canique de Grenoble, BP 53 X, 38041 Grenoble Cedex, France.
geometry (Sofronis and McMeeking, 1991) or isotropic loading (Matthews, 1980; Arzt et al., 1983; Bouvard and Ouedraogo, 1987; Bouvard and Lafer, 1990). In this paper an homogenization technique is proposed to find out the constitutive law of metal powders at high temperatures from the equations governing the deformation at the particle scale. The macroscopic modelling is obtained without any prerequisite at the macroscale. The only assumption to be made is a good separation of the scales, i.e., l and L being the characteristic lengths of the particles and the bulk medium, respectively, IlL << I. The reader will refer to Auriault (1991) for the methodology to be applied. Hot isostatic pressing tests performed on stainless steel powder are first presented with emphasis on the influence of the external pressure on the densification kinetics. Next the homogenization process is explained and applied to the problem of a porous body made up of non-linearly creeping particles. The resulting equations are developed particularly in the case of isotropic macroscopic deformation, leading to a densification relation of the same form as that derived from previous experiments.
0167-6636/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
248
J.L Auriault et al. / Modelling hot compaction
pressure. The densification rate f~/p was calculated by derivation of the analytical functions fitted to the experimental points. Observation of the compacted samples by scanning electronic microscopy showed a fairly homogeneous microstructure during consolidation and suggested that the densification mainly results from deformation of the particles by power-law creep (Bouvard and Lafer, 1989a). A compression test was next performed at 1200°C on a fully dense sample. Various strain rates were successively applied (Bouvard, 1989). The resulting strain-stress curve appears in Fig. 2. A jump in the strain rate results in a steep rise in the stress, followed by a quasi-stationary state. Repetitions of the first loading sequence show a slight hardening. The relation between strain rate and stress at 12% of axial strain is presented in Fig. 3. A power function perfectly fits the experimental points:
2. Experiments
Direct measurement of volume changes during hot isostatic pressing (HIP) is still impossible since the pressure is applied to the sealed can, containing the powder, by a gas. Various dilatometers were recently designed to measure a characteristic dimension - usually the median diameter - of the cylindrical sample, from which density changes are calculated (McCoy et al., 1985; Wadley et al., 1987). Development of such advanced devices is tricky and derivation of density often imprecise; their use is therefore still limited. An alternative way consists of a succession of HIP tests interrupted after different holding times. Measurement of the density of the compacted samples provides discrete points of the density versus time curve, from which densification kinetics can be estimated. This method was used in particular on 316L stainless steel powder. Detailed presentation of the procedure can be found in Bouvard and Lafer (1989a). Densification curves obtained at 1200°C with two different pressures - 10 and 15 MPa - are presented in Fig. 1. As expected, for the same holding time, the relative density p increases with
d: = A o r n,
with n = 4.07 and A = 1.7 × 10 -9, the stress tr being expressed in MPa and the strain rate ~ in s-1. As the tested material presented the same
O.95 0.9 ~ 0.85'
l 0.8!
P J I
1o ] ~ A
1200°C
15 MI~
0.75 0,7-
°"6 o
2b
4b
6b
+b
16o 6o Tree (m)
t4o
2oo
Fig. 1. Results of HIP on stainless steel powder at 1200°C; densificationcurves fitted from experimental points.
J.L. Auriault et al. / Modelling hot compaction
249
25" 30~ma/um 300nm~nm
20"
"~ 15o~ o~
-~ < i0x <
5 " /
0
o
,amom
o:o2
o;o4
o.'o6
0.08
0~1
0.'12
0.'14
0.16
AXIAL STRAIN Fig. 2. Result of a simple compression test on fully dense steel at 1200"C.
microstructure as powder particles during consolidation, the above creep law is supposed to apply also to particle deformation. Most models assume - or result in - a variation of the densification rate with the external
pressure P to the power of n, n being the exponent of the creep law. In order to test this relation, the densification rate calculated from experiments, divided by A P " is plotted in Fig. 4 as a function of the density. The curves corresponding
-3.0'
-", - 3 . 4 v
<
'~ -3.8 <
.~ - 4 . 2 x
<
~
-4.6
.O7
-5.0
0.
1~0
'
i~1
'
1~2
1~3
log [ AXIAL STRESS (MPa) ] Fig. 3. Creep law of dense steel at 1200°C fitted from experimental results.
1.4
J.L. Auriault et aL/ Modellinghot compaction
250 -2
c
-3
\
-7
o.65
0?z
o. 5
0?s
0:s
llm,k11VlZl~m4m'l'/ Fig. 4. Normalized densification rate as a function of relative density.
to both pressures are very close, and this supports the assumption.
Odqvist's constitutive law, which is classical at high temperatures:
Orij plij + Sij,
(1)
3. T h e h o m o g e n i z a t i o n process
Vo- = O,
(2)
We aim at discovering the bulk (macroscopic) description of metal powders under compaction from the description at the particle level. This passage from a microscopic to a macroscopic level is named an homogenization process. We use here the most powerful one which introduces the smallest possible number of assumptions. In the first part the local description of the material is given. The homogenization of periodic structures is presented in the second part. Finally, the process is applied to our problem in the third part. The main results are that the structure of the description is obtained for a quite general anisotropic behaviour and this structure shows a similar power law to the local one.
Oij = A S n - l s i j ,
3.1. Description of the metal powder
cNIr=O.
The material is composed of metal particles and pores. We assume the metal to follow
Discontinuity lines can exist in the bulk material. In this case we assume the normal stress and
=
Vv=O,
S = tr(1/~-SS) ,
A > 0,
l(Ovi
Ovj.]
Dii=O, Dij='~ OXj + ~ ]
(3)
(4)
Here p is the pressure, S the stress deviator, and D the rate of the deformation deviator. Equation (2) expresses the momentum balance, where inertial effects are neglected. Equation (3) is Odqvist's law for an incompressible material following (4); S appears as the second invariant of the stress deviator and n is the scalar already introduced in Section 2. On the surface of the particles, the normal stress is zero:
J.L. Auriault et al. / Modellinghot compaction
251
the displacement to be continuous along these lines. It will be convenient to rewrite the constitutive law (3) in the reverse form
Sij =A-l/nDl-"/"Oi;,
D
=
~/tr(D.D) .
(5)
In all practical cases the bulk volume of the considered material is very large compared to the particle size. Therefore a very good separation of scales exists which enables us to predict an equivalent continuous macroscopic description.
3. 2. Homogenization principle We use here a double-scale asymptotic development based on the existence of two well-separated characteristic lengths (see Bensoussan et al., 1978 and Sanchez-Palencia, 1980, for the method). If l is a characteristic length of the particles or pores and L a characteristic length of the material sample: l
Z- = n <
X 1 Fig. 5. Schematicview of the material at the microscopiclevel: unit cell. between them. The geometry of the pores inside the unit cell is arbitrary. In particular, the pores can be either interconnected or separated. A change of the internal geometry of the unit cell does not modify the structure of the macroscopic description but only the effective coefficients appearing in it. The two characteristic lengths 1 and L introduce two dimensionless space variables, and each physical quantity q~ is a function of these two variables. We will rather use the physical space variables x, and y = x/~7 =
y).
The variable x is the slow space variable or macroscopic space variable well adapted to macroscopic variation. On the contrary, y is the fast space variable well suited for local description. Due to the small parameter r/, it is realistic to look for the physical quantities in the form of an asymptotic development. For example the velocity v will be looked for in the form:
v(x, y) = v(°)(x, y) +TqvO)(x, y) + "'" On the other hand, the 12 periodicity of the structure induces the same periodicity for v with respect to y. The v (i) are then 12 periodic in y. Let us investigate the influence of such a form of v on the other physical quantities. Noticing that d ~ +r/-1 dx
ax
~y'
252
J.L. Auriault et aL / Modelling hot compaction
we obtain for the rate of deformation deviator D
Oij = Dxiy( v ) + "rl - l D r i j ( v ) , where the subscripts x and y show the variable for the derivation. Therefore we have D = ~/- ~D(- 1) + D(0) + ~/D(1) + . . . ,
where the unknown v (°) and p(-1) are O periodic. Let us multiply the two members of Eq. (6) by v (°) and integrate on O s. Then we have
fa v(O)aO.i~-1)d O Oyj
with D (- 1)
=
Dr( V(0)),
D(°) ----'~x(V(0)) +~r(V(1)) ' where ~ is the rate of deformation. It is now easy to deduce the corresponding development for S:
S -~-vn-1/n(V
-ls(-1)
And, by using the divergence theorem, the condition (8), and the periodicity on 5 0 s N 5 0 , we write
f, o( v,,o,o-,?1,) d O = [g~oy~°)o'i~,- ~ 3yy
"4- S (0) + '178 (1) "4- *" • ).
d S = O.
We assume p and S to be of the same order with respect to the powers of ~/. Therefore we have
Then, the remark that the symmetry of 0 "(-1) enables us to put
p = ~q,-W,(n-lp(-O
__0r(
av~°)
+p(0) + ~TpO)+ . . . ).
0yj
Remark: As V -- 10 -3, a pressure which is equal to 3 × 10 -2 times the value of the deviator can be said to be of the same order of magnitude as the deviator itself! This fact justifies our assumption. 3. 3. Macroscopic description
3.3.1. Lower orders For the lower orders we obtain the following boundary-value problem on /2: o'(-1)=p(-1)l
The equation simplifies to: fo~riy (v`0)) q/~.-1, d O = 0. NOW, taking into account the incompressibility condition (7), we are left with
Let us introduce the above developments in the local description. Looking for like powers in each equation gives successive problems to be solved for a unit cell. We begin by the lower orders.
Vror(-1)=0,
.- 1) = ~ r r i j ( / 5 ( 0 ) ) O.i(-1).
+ S (-1),
fo Drij(
v O)) S~j- 1) d O = 0.
s
The integrand corresponds to a dissipation density and represents a positive quantity. Therefore we deduce
or(v(°) ) = 0
(6)
and, with the periodicity,
(7)
At the local level the first order velocity is a translational velocity. The consequences are: S (-1)= 0 and Eq. (6) reduces to V r p ( - l ) = 0, so that p ( - 1 ) = p ( - l ) ( x ) and Eq. (8) implies p ( - O = 0.
S ( - 1) = A - l i n D ( - 1X1 - n / n ) D ( - 1),
~(-1) = I t r ( D r (v(0>) . Dr(v(0>)), Vyv(°) = 0
or Dyii(v (0)) = 0,
( r ( - ° N Ir = 0,
(8)
J.L. Auriault et al. / Modelling hot compaction
3.3.2. The fundamental cell problem
At the following orders, taking into account the preceding results gives V @CO) = O3 Y
&?(O) = A
D’O’
&a) = p’o’Z + $0) 7
- l/npol
-n/n@)
9
p
=
@
\/e(D’o’
tr(
=
(gyst(X.lk)
@”
+
‘D’o’)
=
a’OQvlr=O.
This is a boundary-value problem for the L4 periodic unknown parameters Y(‘) and p(O),where O,(vCo’) is considered as being given. Let us admit the existence of a solution u(l) and p(O). The velocity u(l) appears as a homogeneous vectorial function of degree 1 in the rate of deformation O,(uCo’) defined to an arbitrary local translation i?(‘)(x). For the sake of simplicity, we are using the same notations as for a linear relation: U!” 1 = Xijkaxjk( u’“‘) + i$“( X), where the third-order / 0,
We now express (S(O)) and ( p(O)) as functions of the macroscopic rate of deformation 0,(~‘~‘). For (S(O)) this is done easily by using the expression (10) for u(l). We obtain successively:
(9)
=a, ( u’“‘) +a; ( Y(l)) )
vv”‘+vu’o’=o, x Y
253
(10)
tensor x is of zero average]
zs,z,k)gx,k~u’o’~~
( gyst(x.lk)
’
x
(~yst(x.mn>
’
Oxlk(
u(o))
bztk)
+
Oxmn(
LLi)
u(o))
and $0
=A-l/n(trD(0)D(O))l-n’2n ’ (Oyij(X.pq>
+ zipzjq)gxpq(u(o))*
Then taking the average on fl gives (SC’)>. It appears to be related to 0Ju(‘)) by a homogenous anisotropic constitutive relation of the same degree n as the original local constitutive relation. But here, since shrinkage occurs, the kinematics is described by 0 in place of the deviator D.
xdR=O.
The structure of the solution p(O) is more involved. Since it is not needed in the following, we disregard it. 3.3.3. The macroscopic description At the following order, Eq. (2) gives v Y CT(l)+ vx (T(O)= 0.
(11)
The stress a(‘) being a periodic quantity, the balance (11) implies that the source term VXaCo) be of zero volume average. Noticing
For (p(O)) the way to be followed is somewhat different. We first consider a vector field (Yon 0, (Y being 0 periodic with a given divergence a V,a=a. This field is not unique, but the result is independent of this uncertainty as well as of the value of a. Secondly we multiply the momentum balance (9) by (Y and integrate on R,
( p"'Zij + Sl’i”‘)dR
j~~i ~
I = /, -& ( ai( p’“‘Zij+ S$‘,>) dR 5 I
vx (a(O)>= 0, -
where (a’O’> = (p’O’)Z+
2
I, I
(S(O)).
( p”‘Zij + S$‘)) dR = 0. J
The first integral on the right hand cancels out
J.L. Auriault et al. / Modelling hot compaction
254
by using the divergence theorem, the periodicity, and the condition on F. We are left with
fo O°~in(°)l.. dO = a f. p(O) dO Oyj ~
-u
S2s
= -
with
Bi: = (b)
= (½A-I/nl~(.~yst(X.l,)+ ]st) X(-,~yst(X.l,)+ 2.)I'-°/=°
(aa---Z/S~9)dO
Jl2s Oyj u
= - f
''~yi'/(e) S:°)dO.
O~
And finally the average of p(0) is given by l (p(O)> = _ - -
J.[.~yij(a)S: 7) dO.
a l O I as Here again the relation shows an anisotropic power-like relation with exponent n. Remark 1. In practice, p(0) will be given by solving the fundamental cell problem of Section 3.3.2, together with v (1). Remark 2. It is easy to show that the result is thermodynamically consistent: the average of the local dissipation density equals the macroscopic dissipation.
3.3.4. Isotropic shrinkage We restrict ourselves here to isotropic shrinkage. Therefore the rate of the deformation tensor reduces to:
The velocity v (~) is now a linear vectorial function of the trace VxV(°) of the rate of deformation, and we have successively,
O~°) = ( ~yst( X.lk ) "{- Islltk ) 3 VxU(O)llk, Dj°t ) = ( ~ , , t f x . , )
+ I,,)½ Vxv(°),
tr( D (°)" D (°)) = (~yst(X.tz) + Ist ) (-~y,t( X.t, ) + I~t )
and ( S~°')= n,jlVxv(°)ll-n/nvN,°),
X (~Iyij(X.pp)-l-~ij)). For (pO> a similar relation holds:
< p(0)> __ cl v::)l ]-n/n VxV °,, l ( p(O)) = _ __( ~ y i j ( ot)b )"
a
In the case of an isotropic macroscopic material, (S ~°)) = 0 and we recover an isotropic relation as was assumed in Section 2 for the experimental investigation. In this simple case the power-law structure of the macroscopic description is evident.
4. Conclusions The homogenization process applied to the investigation of the modelling of hot compaction of metal powders enables us to rigorously demonstrate that the power-law structure of the local description is preserved at the macroscopic level. This is in agreement with experimental investigations concerning isotropic hot isostatic pressing tests. But it gives the more general description at the macroscopic level, for an anisotropic material under any excitation. These are precisely the cases where experimental data are difficult to obtain. Working out explicit constitutive relations requires solving the fundamental problem (obtaining the third-order tensor X) for the unit cell, which can be achieved through numerical calculations. Such work is in progress.
Acknowledgements The authors wish to thank the Vallourec Company for providing the powder and for supporting
J.L. Auriault et al. / Modelling hot compaction
part of the experiments, and the Tecphy Company for funding C. Dellis jointly with CNRS. References Abouaf, M. and J.L. Chenot (1986), Simulation num6rique de la ddformation h chaud de poudres mdtalliques, J. Mec. Theor. Appl. (France) 5, 121. Arzt, E., M.G. Ashby and K.E. Easterling (1983), Practical applications of hot isostatic pressing diagrams: Four case studies, Metall. Trans. 14A, 211. Auriault, J.L. (1991), Heterogeneous medium. Is an equivalent macroscopic description possible?, Int. J. Eng. Sci. 29 (7), 785. Bensoussan, A., J.L. Lions and G.C. Papanicolaou (1978), Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam. Bouvard, D. and E. Ouedraogo (1987), Modelling of hot isostatic pressing: A new formulation using random variables, Acta Metall. 35, 2323. Bouvard, D. (1989), Rh6oiogie des poudres mdtalliques au cours de la mise en forme ~ haute tempdrature, Th~se d'Etat, Grenoble. Bouvard, D. and M. Lafer (1989a), Determination of the densification kinetics of metal powders by interrupted hot isostatic pressing tests, Powder metall. Int. 21, 11. Bouvard, D. and M. Lafer (1989b), Rheological characteriza-
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