Unloading of powder compacts and their resulting tensile strength

Acta Materialia 51 (2003) 4589–4602 www.actamat-journals.com

Unloading of powder compacts and their resulting tensile strength C.L. Martin ∗ Laboratoire GPM2, Institut National Polytechnique de Grenoble, UMR CNRS 5010, ENSPG, BP46, 38402 Saint Martin d’He`res Cedex, France Received 2 December 2002; received in revised form 15 May 2003; accepted 26 May 2003

Abstract We examine the unloading of an agglomerate of perfectly plastic spherical particles that have been compacted beyond their elastic limit and that exhibit some adhesion. The behaviour at the contact scale is derived from the model of Mesarovic and Johnson for the decohesion of two spherical particles [1]. The resulting springback that the powder compact experiences is calculated using the Discrete Element Method. An analytical equation for the extent of springback is proposed and coincides well with simulation results. The alteration of the compact during unloading is investigated through contact loss and the decrease of the contact area. The tensile strength of the resulting compact is calculated in isostatic conditions. While isostatic conditions represent the basis of this work, the effect of closed die conditions on the anisotropy of springback is also investigated. Finally, the consequences of adding hard elastic particles to form a composite are analysed in terms of the springback extent and the resulting loss of tensile strength of the agglomerate.  2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Powder consolidation; Elastic behaviour; Springback; Discrete Element Method; Plastic deformation

1. Introduction During the plastic compaction of metallic powders, the elastic behaviour of the particles does not influence significantly the macroscopic response of the compact. This is because the elastic mutual indentation of particles is superseded by plasticity very early on during compaction. However, elasticity is of great importance during the unloading of the aggregate that follows its compaction. The

∗

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problem is of practical interest since it controls the degree of springback and the strength of the resulting green compact. Excessive springback may produce catastrophic flaws within the compact and voids that have been created by expansion act as incipient cracks. Hence modelling of the strength of green compacts should start with a good understanding of the unloading phase and of the microstructural alterations that come with it. The degree of springback is important by itself for the numerous powder applications for which control of the part dimensions is critical. Springback may also have some influence on the operations that follow compaction. For example, it has been observed

1359-6454/03/$30.00  2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00296-9

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that anisotropy during sintering partly results from the anisotropic springback experienced by the agglomerate during closed die compaction [2]. The modelling of the unloading stage has been mainly approached by considering the compact as a porous medium with appropriate effective elastic properties [3,4]. However the particulate nature of the agglomerate has not been addressed. In particular the possibility of decohesion of previously contacting particles cannot be modelled from macroscopic approaches. Wu et al. [5] have used finite element simulations to study the compaction and the unloading of powder composites. Their work is limited to the compaction and the unloading of 2D hexagonal and square packings. The unloading of a plastically compacted aggregate involves several interacting phenomena at the contact length-scale: plasticity at the contact between two particles before unloading, elastic recovery and potential decohesion of the two particles during unloading, and possible relative rearrangement between particles. In order to build a discrete description of the unloading process, we restrict the following discussion to aggregates of spherical particles. The first step consists of looking for analytical models that describe the behaviour of the contact during loading and unloading. The problem of the contact between two adhering elastic spheres is now well understood [6–8] and the analysis of the plastic indentation of spheres (with the possibility of strain-hardening and/or rate effects) is also well documented [9,10]. Only recently, the full process of unloading and decohesion of two spheres that have been indented plastically has been examined with some detail by Mesarovic and Johnson [1]. Their analysis is limited to perfectly plastic materials and to normal indentation followed by normal unloading. However, it provides the necessary analytical tools to undertake the modelling of the unloading of an aggregate of plastically deformed spheres. In the present report, we introduce the formulation of Mesarovic and Johnson into a discrete element model to analyse the unloading behaviour of powder compacts with a random initial arrangement of particles. Using this model, numerical simulations of the unloading process allow the quantification of the role of the different material

parameters. We will show that springback can not be solely considered as an elastic problem but that it involves inelastic features such as decohesion between particles. The information that we seek at the compact level is the degree of springback once external stress is removed and the alteration of the microstructure during unloading. In particular, we investigate the loss of contacts that occurs during unloading, and the evolution of the contact area. By continuing to impose tensile conditions, we determine the strength of the compact. We study both homogeneous aggregates that are formed of a single population of particles and composite aggregates that are a mixture of soft plastic particles and hard elastic particles. Closed die compaction followed by unloading is also investigated in order to get some information on anisotropic springback.

2. Model The compact is modelled as a particulate medium with spherical particles that indent each other. We consider both the case of a homogeneous compact, and that of a composite made of two populations of spherical particles with different material properties. The particles are modelled as elasto-plastic. Elastic constants are noted Ep and np (p = 1,2). The materials are considered as perfectly plastic with yield stress ⌺p, and no strain-hardening is considered. For two contacting particles p and q with radius rp and rq, we define the following parameters: rpq ⫽ Epq ⫽

rprq , rp ⫹ rq

冉

p,q ⫽ 1,2

1⫺n2p 1⫺n2q ⫹ Ep Eq

⌺pq ⫽ min(⌺p,⌺q),

冊

⫺1

,

(1) p,q ⫽ 1,2

p,q ⫽ 1,2

(2) (3)

2.1. Constitutive equations for the elasto-plastic loading of the contact The compaction of homogeneous and composite compacts has already been treated in earlier works [11,12], and is not detailed here. The elastic load-

C.L. Martin / Acta Materialia 51 (2003) 4589–4602

ing of the contact follows a Hertzian law while in the plastic regime the normal indentation force, Np, for a perfectly plastic material is: Np(a) ⫽ 3p⌺pqa2,

p,q ⫽ 1,2

(4)

with a, the contact radius: pa2 ⫽ 2pc2rpqh,

(5)

where h is the indentation between the two particles and c2 is a constant equal to 1.43 in the case of a perfectly plastic material [10]. The use of Eqs. (4) and (5) with c 2 = 1.43 assumes that the contacts undergo only two regimes upon loading, namely the elastic Hertzian regime, and the fully plastic similarity regime derived by Stora˚ kers et al. [10]. Hence we neglect the intermediate elasto-plastic and finite deformation regimes as discussed by Mesarovic and Fleck [13]. Note also that the writing of Eq. (4) neglects any adhesion effects during loading. We note a0 and N0 the contact radius and the normal load at the end of the plastic compaction process, before the load is removed. The pressure distribution under the contact depends on the strain hardening law [1,10,14]. For strain hardening material, the pressure increases towards the edges of the contact. For perfectly plastic materials, however, the pressure distribution is almost uniform throughout the contact area even for large plastic contacts [1]. We denote p0 this uniform pressure which is simply 3 times ⌺pq. The uniformity of the pressure under the contact allows a simple treatment of the unloading stage. 2.2. Constitutive equations for the unloading of the contact Mesarovic and Johnson [1] have examined the evolution of a contact that is unloading in the case of a uniform pressure distribution throughout the contact area. Under this assumption, it is possible to derive in close form the profile of the elastically recovered crown. It is given by the normal displacement of an elastic half-space under uniform pressure acting on a circular area. For the general case of a contact between two elastic-perfectly plastic spheres with different material properties, we expect that the uniformity of pressure on the contact area should remain valid as discussed by

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Mesarovic and Johnson [1]. In the following, we describe the results of the study of Mesarovic and Johnson that are pertinent for our simulations. It should be noted that we use only the singular model described in their analysis that relates to a fully elastic unloading with no plastic flow involved at the contact level. Typically the separation of two plastically deformed spheres involves a significant amount of plasticity in the case of very soft metals and/or very small contacts. The limit between a fully elastic unloading and a partially plastic unloading may be captured by the ratio of the adhesive energy to the elastic energy stored in the crown [1]: c⫽

p wEpq , 2p⫺4 p20a0

(6)

with w the work of adhesion. The correct value of w is difficult to ascertain for a given material system since it depends on the deformation mode and on the degree of contamination of the surfaces. When no significant plastic flow is involved, as assumed here, w is equal to the effective surface energy and ranges typically between 0.01 and 10 J/m2. Mesarovic and Johnson [1] have shown that for c⬍0.0085, the singular model used in the following provides an asymptotic limit of more complex models that include the effect of a constant adhesive stress acting between separated surfaces at the contact. In any case, when the unloading is predominantly elastic (c⬍0.1) the singular model provides a good approximation for the behaviour of the contact [1]. When c is larger than 2, the unloading is fully plastic across the contact zone and the separation occurs by bulk plastic flow with the normal tensile force given by the same force necessary for the plastic indentation. The domain c⬎0.1 is thus beyond the scope of this study but should be of interest only for a limited amount of practical powder applications. In the context of a fully elastic unloading, for a constant pressure, p0, applied on a circular area of radius a, the normal displacement on the surface is [15]: u(r,a) ⫽

冢冉 冊

p r 4p0a0 E ⫺ pEpq a0 2

冪

冉 冊冣

1⫺

a a0

2

r ⱕa,

(7)

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冢冉

冉 冊冊

4p0a0 r a u(r,a) ⫽ E ,arcsin pEpq a0 r

冉 冊冪 冉 冊 冣

⫺arcsin

a r

1⫺

a a0

2

(8)

r ⬎ a,

(9)

where h0 is the plastic indentation before unloading and hu(a) is linked to the normal displacements at the surface by hu(a) = u(0,a0)⫺u(0,a). Hence using Eq. (7), hu(a) can be written (E(0) = p / 2): hu(a) ⫽

冪

2p0a0 Epq

冉冊

1⫺

a 2 . a0

冢 冉冊

Nu(a) ⫽ 2p0a20 arcsin

where E(.) and E(.,.) are the complete and incomplete elliptic integrals of the second kind, and r is the radial coordinate. We define hu(a) as the indentation recovered through elastic unloading between two spheres with the same yield stress and having a contact area a (Fig. 1). The distance between the centres of the two indented spheres that have been pressed together is given by: lpq ⫽ rp ⫹ rq⫺h0 ⫹ hu(a),

unloading process, to the maximum separation due 2p0a0 . to full unloading hu(0) = Epq During unloading, the normal load is given by [1]:

(10)

hu(a) varies between 0 (a = a0) at the onset of the

冉 冊冣

冪

1⫺

a a0

2

(11)

⫺2冑2p w Epq a2, 3

where the first term on the right hand side gives the load in the absence of adhesion while the second term relates to the adhesion traction. Note that size effects are included in the model because of the second term. This can be seen by rewriting Eq. (11) in terms of c which is inversely proportional to the contact radius a0:

冢 冉 冊 冪 冉 冊冣 冉 冉 冊 冊冉 冊

a Nu 2 a ⫽ arcsin ⫺ N0 p a0 a0 2 1 ⫺4 1⫺ c p p

1 2

1⫺

a a0

2

(12)

3

a 2 . a0

Fig. 2 shows the typical evolution of the normalised load Nu/N0 as a function of the normalized value h˜ u = hu(a) / hu(0) at the end of unloading for typical values of c. Towards the very end of the unloading process, the maximum tensile load is reached when dNu / da = 0, and the two surfaces snap apart at this point. The value of (a/a0) at this point is well approximated by:

冉冊 冉 a a0

Fig. 1. Schematic of a contact undergoing unloading. The initial contact at the end of plastic loading is characterised by its radius a0, while the current contact radius during the unloading process is a. The indentation recovered from unloading is hu for two contacting particles.

a a ⫺ a0 a0

snap

冊

1

9(p⫺2) 3 1 ⫽ c 3, 4

(13)

for small values of c [1]. The complete loading and unloading process of an individual contact is shown in Fig. 3 as a function of the indentation between two perfectly plastic spheres. Note the two non-linear portions of the curve that corresponds to elastic loading at the very early stage of indentation, and to unloading. Contacts that have not been loaded beyond elasticity during compaction are unloaded using a Hertzian law instead of Eq. (11). For a monosize

C.L. Martin / Acta Materialia 51 (2003) 4589–4602

Fig. 2. Evolution of Nu/N0 with h˜ u at the end of the unloading process for two values of c. h˜ u never reaches unity since the contact snaps when Nu/N0 attains an extremum in tensile conditions.

agglomerate made of perfectly plastic particles, the relative density at which the first plastic contact occurs during compaction was determined to be 0.637 in our prior simulations on plastic compaction of powder agglomerates [11]. The fraction of elastic contacts decreases rapidly with compaction. For material parameters typical of metallic powders, it is 20% at 0.65 relative density and falls to 2% at 0.70 relative density. Hence contacts having a Hertzian behaviour will not play a significant role for relative densities higher than 0.70. 2.3. Discrete Element Method (DEM) description The details concerning the methodology adopted for the Discrete Element Method can be found in earlier works [11,12]. We use an explicit scheme for deriving the displacements of the particles knowing the resolved force on each of them. Mass upscaling permits the use of reasonable values for the time step as proposed in [16]. Forces and strains are unaffected by mass upscaling provided that the particles are at equilibrium. Hence the equilibrium positions of the particles are not alt-

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Fig. 3. Evolution of Nu/N0 with h/h0 during a typical loadingunloading process at a contact. Material parameters are E = 120 GPa, n = 0.34, ⌺ = 400 MPa, m = 0.1, R = 150 µm, w = 0.5 J/ m2 (c⬇5.10⫺4) and h 0 / R = 0.15 that would be typical of a metallic powder compacted to 0.80 relative density. The tensile part of the unloading is barely detectable (Nu/N0 at failure is approximately –3.10⫺4 for these material parameters).

ered by the value of the upscaling factor if an adequately small strain rate is imposed (e˙ 0 = 10⫺5s⫺1) and checks are included to ensure that each particle is indeed close to equilibrium. Boundary conditions are taken as periodic and we have verified that 4000 particles are more than sufficient to represent the bulk properties of the compact by checking that simulations using only 400 particles give essentially the same results in terms of average properties. Concerning the tangential force at the contact, we simply write that the contact is either in a sticking state or in a state of gross sliding. The sticking case is ensured by giving a sufficiently large tangential stiffness to the contact. When the contact is sliding, a simple Coulomb law of friction is used: T ⫽ ⫺mNt,

(14)

where t is the unit vector parallel to the contact plane in the direction of the relative velocity, and m is a friction coefficient. This friction law is applied

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whatever the status of the contact (elastic loading, fully plastic loading, elastic unloading). At this point, we should emphasize that we use normal indentation force expressions (Eqs. (4) and (5)) and a contact pressure expression (p 0 = 3⌺pq) that have been derived for frictionless contacts with normal indentation. It has been shown by the similarity solution that the contact stiffness for the frictionless problem is only about 1% less than the fully sticking contact, so the contact status (sticking or frictionless) should not be a problem [17]. Of more concern is the effect of the obliquity of the indentation and of friction on the contact pressure and on its spatial distribution under the contact. Plane strain FEM calculations of a rigid indenter and a perfectly plastic half-space indenting each other obliquely show that obliquity of the contact indentation does not have a large effect on the contact pressure nor on its uniformity [18]. Within the similarity regime, sticking friction leads to 7% higher contact pressure (and 8% smaller contact area) compared to the frictionless solution [17]. Hence the frictionless solution might underestimate the pressure under the contact. Concerning the effect of friction on the uniformity of the contact pressure, Carlsson et al. [19] have investigated in great detail the effect of Coulomb friction on the indentation behaviour of viscoplastic materials. Their conclusions are essentially in accord with [17] concerning the negligible effect of friction on the contact stiffness. Also, they show that in the case of a perfectly plastic material, the spatial homogeneity of the contact pressure is not affected by friction. The large finite contact areas formed during the plastic indentation should inhibit any rotation of the particles during the unloading of the compact. Hence rotations are not accounted for in these simulations and moment equilibrium is not enforced as discussed in [11,12]. The macroscopic stress tensor is calculated by using the following expression [20]: sij ⫽

冉冘

1 V

c

lpqNni.nj ⫹

冘 c

冊

lpq Tni.tj ,

tact plane and V is the volume of the sample. The summation is carried out on all contacts. 3. Unloading of homogeneous compacts 3.1. Unloading in isostatic conditions, analytical solution In order to investigate more clearly the role of the different material parameters during the unloading process, it is useful to propose an analytical solution of the accumulated bulk strain once the compact has been fully unloaded. We consider the case of a set of monosize spherical particles (radius R) indenting each other up to a given relative density. At the onset of unloading the average indentation, h0, is related to the relative density, D0 [21,22]:

冉 冉 冊冊

h0 ⫽ 2R 1⫺

1 3

,

(16)

where Di is the relative density of the compact before plastic compaction. For monosize packings, Di is set to the random close packed density (0.637) that coincides with the value obtained with our simulations [11]. During unloading, the initial indentation, h0, is decreased by hu(a), and Eq. (16) can be rewritten:

冉 冉 冊冊

Di h0⫺hu(a) ⫽ 2R 1⫺ D

1 3

,

(17)

where D is the current relative density. We consider first the case of cohesionless particles (c = 0). When the load is fully recovered at the contact (a = 0), Eqs. (16) and (17) together with Eq. (10) give the accumulated bulk strain eˆ 0u due to unloading:

冉冊 冉

冉 冊冊

p0a0 D0 D0 ⫽ 3ln 1 ⫹ eˆ ⫽ ln D EpqR Di 0 u

1 3

.

(18)

The contact radius a0 can be evaluated using Eqs. (5) and (16) and together with the small strains involved in unloading, eˆ 0u can be approximated by:

(15)

冉 冊冪 冉 冉 冊冊

3p0 D0 eˆ ⫽ Epq Di 0 u

where n denotes the unit normal vector to the con-

Di D0

1 3

Di 2c 1⫺ D0 2

1 3

.

(19)

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Eq. (19) yields an expression for the springback of the compact that neglects some important features of the unloading process. In particular, it does not take into account adhesion between particles and their potential decohesion. Hence particles size effects are not included. Also, this expression does not take into account the possible rearrangement that takes place between particles during unloading. However it already includes some important features of the unloading of a particulate compact. In particular it shows that springback is proportional to the ratio p0/Epq (where p0 is 3 times the yield stress ⌺ for perfectly plastic materials), and increases with increasing plastic strain during the compaction stage from Di to D0. We can further refine the model by including adhesion effects. This is done by considering that the unloading is complete when on average Nu(a) = 0. In that case, Eq. (11) leads to the following equation for a/a0:

冉冊

a a ⫺ a0 a0

冪 冉 冊 冉冊 a a0

2

(20)

Fig. 4. Typical macroscopic pressure-density curves during loading and unloading from various initial relative densities obtained after isostatic compaction. Material parameters are those used in Fig. 3.

that can be solved numerically to yield the approximate value of a/a0 at zero load, which we denote as (a/a0)0. The bulk strain due to unloading for partially cohesive particles (c ⫽ 0), eˆ u, is then simply related to the bulk strain due to unloading for cohesionless particles, eˆ 0u:

sequence is stiff, while the end of the unloading is clearly non-linear. Various combinations of material parameter values that appear in the expression of c (Eq. (6)) have been used in order to test the influence of the work of adhesion, elastic and plastic properties and particle size (Table 1). Note that we only simulate compacts of monosize spheres. We use an average value of c for the agglomerate, denoted as c¯ . This is necessary because there exists a distribution of contact sizes in the compact for a given relative density [11]. We have verified that this distribution is narrow enough to allow the use of the average value of a0 in the calculation of c¯ that characterises the compact. It should be pointed out that contacts that may involve substantial plastic flow during unloading (c0.1) are those that are very small compared to the average value. We have verified that these small contacts do not contribute significantly to the macroscopic response of the compact both because they are not numerous, and because they do not carry significant loads. In any case, we have restricted the

arcsin

1⫺

3

a 2 ⫽ 冑2c(2p⫺4) , a0

eˆ u ⫽ eˆ 0u

冪1⫺冉冉a 冊 冊 , a

2

(21)

0 0

where the second term on the right hand side accounts for adhesion between particles. 3.2. Unloading in isostatic conditions, DEM simulations Typical unloading curves from various initial relative densities are shown in Fig. 4. These curves have been obtained with the same material parameter values as those used in Fig. 3. The macroscopic unloading of the agglomerate exhibits the same qualitative features as the unloading of each contact (Fig. 3). The beginning of the unloading

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Table 1 Material parameters used for the DEM simulations w (J/m2)

E (GPa)

⌺ (p0/3) (MPa)

R (µm)

m (friction)

0.01–0.1–0.5–1–5–10–50

120–1200

300–400

15–150

0.1

simulations to values of c¯ smaller than 0.1, for which plastic flow at the contact should be limited. We define a normalised strain, e˜ u, using the above-calculated strain, eˆ 0u, for cohesionless packings as e˜ u = eu / eˆ 0u, where eu is the bulk strain from the DEM calculation when the macroscopic pressure attains zero. Results of numerical simulations are shown as a function of c¯ in Fig. 5 for various initial relative densities (0.70–0.95). The different points plotted on the figure correspond to different sets of material parameters, leading to different values of the parameter c¯ as given in Table 1. Fig. 5 shows that all simulation points follow approximately a master curve that tends towards unity for small values of c¯ . Fig. 5 confirms that the numerical simulations are in good agreement with Eq.

(21) except for low values of c¯ , for which rearrangement between particles starts to play a significant role. Fig. 5 indicates together with Eq. (19) that springback increases with the relative density attained during the plastic compaction. For material parameters that are given in Fig. 3, the bulk strain, eu, increases from 2% at D 0 = 0.70, to 4% at D 0 = 0.95. Mesarovic and Johnson [1] have shown that for c⬇2, contact separation is fully plastic. We have represented this point in Fig. 5 to indicate the extent of the elasto-plastic domain of the unloading process in between the fully elastic unloading and the fully plastic domain. We have run a few simulations with a larger friction coefficient between the particles (Eq. (14), m = 0.2 instead of 0.1). These simulations show that springback increases with the friction coefficient but not to a large extent. It is likely that because tangential rearrangement is further hindered by higher friction, kinematics constraints increase, hence forcing particles to separate normally. In any case tangential rearrangement of particles during isostatic unloading is of little extent, thus explaining the small effect of the friction coefficient in isostatic conditions. 3.3. Loss of contacts and evolution of the contact area in isostatic conditions

Fig. 5. Evolution of the normalized bulk strain, e˜ u, with c¯ for different sets of material parameters (Table 1) and various initial relative densities obtained after isostatic compaction. The no adhesion limit (Eq. (19)) is attained for very small values of c¯ , while the fully adhesive limit (c¯ ⬇2) leads to e˜ u = 0. Eq. (21) gives the effect of adhesive contacts on the bulk strain.

At a given stage during the unloading process, contacts may be in different states depending on their loading history. Some of them may be unloading with a compressive load, some of them may be tensile because of adhesion, while others may have snapped. Fig. 6 shows the loss of contacts during unloading, ⌬Z, as a function of c¯ for various initial relative densities. This figure indicates that compacts that lose most contacts during unloading are those that have been densified most. However, the final coordination number is still a monotonically increasing function of relative den-

C.L. Martin / Acta Materialia 51 (2003) 4589–4602

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Fig. 6. Contact loss at the completion of unloading versus c¯ for different sets of material parameters (Table 1) and various initial relative densities obtained after isostatic compaction. The initial (before unloading) coordination numbers are denoted Z0 and listed in front of each curves.

Fig. 7. Evolution of the contact size versus c¯ at the end of unloading for different sets of material parameters (Table 1) and various initial relative densities obtained after isostatic compaction. (a/a0)0 refers to the solution of Eq. (20) that gives the value of a/a0 at zero load while (a/a0)snap refers to the value of a/a0 when the contact snaps (Eq. (13)).

sity. The loss of contacts predicted by the simulations is rather large for values of c less than 0.01. These contacts that have snapped during unloading will not participate to the strength of the agglomerate if further isostatic tensile strain is imposed on the agglomerate. However, the contacts that have snapped during unloading may be reloaded if some compressive component is imposed. This explains the S shape curve that is exhibited upon the beginning of the reloading stress-strain curve in simple compression of green samples [23,24]. The ratio (a/a0) decreases as unloading proceeds. Fig. 7 shows the area of contact at the end of unloading versus c¯ . The curve (a/a0)0 given by the numerical solution of Eq. (20) is in good agreement with the values calculated by the Discrete Element Method. Again, for low values of c¯ , the analytical solution that neglects rearrangement between particles underestimates the value of (a/a0) given by the DEM. Also shown in Fig. 7 is the value of (a/a0)snap at which a contact snaps. The chronology of contact loss is symmetric to the chronology of their creation, i.e., the last contacts that have been formed (and thus smallest) are the

first to snap upon unloading. This is because as shown by Eq. (10), the contacts with the smallest value of a0 are those that have also the smallest value of (a/a0) for a given elastically recovered indentation hu. 3.4. Strength of the agglomerate under isostatic tensile conditions The continuation of isostatic tensile conditions on the agglomerate leads to its fracture. The study of the complete mechanism of fracture of the agglomerate with the possibility of strain localisation is beyond the scope of this report. Our simulations show that the macroscopic tensile pressure (Eq. (15)) goes through an extremum during tensile loading before the total ruin of the compact (indicated also by a catastrophic loss of contacts). We define this extremum as the tensile pressure of failure of the agglomerate, pf. Fig. 8 shows the evolution of the ratio pf /p0 as a function of c¯ on a log–log plot. pf /p0 increases with c¯ , and also, for a given value of c¯, with the relative density at the end of the compaction process. It should be

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the particles over the whole range of c¯ studied here. Green strength is typically measured experimentally with diametral compression tests for which the stress state is quite far from the isostatic stress state used here. The closest stress state available experimentally is simple tension. Keeping in mind this restriction, we present the experimental data obtained under these conditions by Pavier and Doremus [25] on an iron base powder tested between 0.73 and 0.92 relative density. The value of c¯ is estimated from their data and by assuming that w ranges in between 0.5 and 1 J/m2. These results seem in broad agreement with the simulated values of the compact strength. 3.5. Unloading after closed die compaction

Fig. 8. Evolution of the tensile strength in isostatic conditions normalised by p0 versus c¯ for different sets of material parameters (Table 1) and various initial relative densities before unloading. The lines indicate the experimental data from Pavier and Doremus on an iron base powder [25] for which c¯ is estimated: particle diameter 80 µm, E = 210 GPa, ⌺ = 370 MPa, D 0 = 0.73–0.92, a0 is calculated from Eqs. (5) and (16) and w is estimated to range in between 0.5 J/m2 (lhs line) and 1 J/m2 (rhs line) [1].

recalled that strain hardening effects are not taken into account here so that p0 is simply 3 times the yield stress ⌺ of the material. Because adhesion is not included during the compaction stage, the size of the particles does not play any role in our simulations of compaction. For the unloading stage, size effects can be evaluated by looking at Fig. 5 and Fig. 8 and by recalling that c is inversely proportional to the particle size (for a given relative density). For powder compacts characterised by a low value of c¯ (c¯ ⬍ 10⫺3), Fig. 5 shows that keeping the same powder material but decreasing the particle size does not bring any significant decrease in springback. Significant gain in the decrease of the springback is attained only for systems with high values of c¯ (c¯ ⬎ 10⫺2). Conversely, Fig. 8 shows that significant gain in strength can be attained by decreasing the size of

Closed die compaction is the most common route for producing parts from powders. The anisotropy resulting from closed die compaction has already been described both through analytical models [26], and through numerical simulations [11,27]. The anisotropic structure of the agglomerate may be due both to anisotropy in the distribution of contact orientations, and to anisotropy in the distribution of contact sizes. We have shown that the anisotropy in contact orientations is not significant, provided that some rearrangement between particles is allowed. The strongest source of anisotropy comes from the fact that the area of those contacts whose normals are close to the compacting direction is larger on average than the area of contacts whose normals are close to the transverse directions [11]. We can expect that the anisotropic structure resulting from uniaxial compaction also exhibits some anisotropic behaviour upon unloading. We denote x3 as the pressing direction, and x1 and x2 as the two other directions. While the kinematics of unloading for isostatically pressed sample is rather simple (isostatic compaction followed by isostatic unloading), the unloading kinematics after closed die compaction may follow different paths. We study two types of unloading paths to evaluate the effect of unloading kinematics. The first type consists in unloading all three directions with the following rule:

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e˙ ii ⫽ e˙ 0abs(sii / max(sjj)),

(22)

with sii the diagonal term of the macroscopic stress tensor given by Eq. (15). At the onset of unloading, for perfectly plastic material, the ratio sii / s33 for i ⫽ 3 is less than unity (typically in between 0.55– 0.7 depending on the relative density [11]). Hence Eq. (22) ensures that unloading proceeds on all three directions from the very beginning but not at the same strain rate. At some point during unloading, the three macroscopic stresses become identical and the end of unloading is nearly isostatic. We refer to this type of kinematics as proportional. The second type of unloading consists of unloading only the x3 direction until the following condition is met: abs(s33) ⬍ 0.7min(abs(s11),abs(s22)).

(23)

At this point, a very low strain-rate is imposed to the x3 direction. We denote this type of kinematics as non-proportional. The differences between the two kinematics are illustrated in Fig. 9 showing stresses versus axial and transverse strains. We believe that the non-proportional loading is qualitatively closer to the practical situation of the

Fig. 9. Illustration of the two types of kinematics imposed during unloading after closed die compaction. The simulation is for a sample unloaded from 0.75 relative density with c¯ ⬇ 10⫺3.

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unloading of a die since; in that case, the pressing direction is unloaded first. However, we do not attempt to reproduce the precise conditions of unloading. Those conditions depend strongly on the rigidity of the die and on the friction with the walls. We have observed that the bulk strains are approximately the same in closed die and in isostatic conditions for a given set of material parameters. Furthermore the bulk strain in the closed die case is unaffected by the kinematics chosen for unloading. However, the strains in the axial and transverse directions are quite different due to the anisotropy created during compaction. This is illustrated in Fig. 10 where we have plotted the ratio of the axial to the transverse strain as a function of the initial relative density. Note that this ratio is insensitive to the value of c¯ . Fig. 10 confirms that the kinematics of unloading has an important influence on this ratio. This figure shows that if the pressing direction is unloaded first as often done in the actual process, the ratio of the axial to the transverse strain is increased. This is in accord with experimental data collected by Kuroki [2] on iron

Fig. 10. Ratio of the axial to transverse springback as a function of the initial relative density, D0, for the two types of kinematics illustrated in Fig. 9. The ratio is essentially independent of c¯ .

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powder compacts who has observed ratios of the axial to the transverse strain of the order of six.

4. Unloading of composite compacts We now consider a mixture of perfectly plastic particles with elastic particles. The elastic particles are chosen so that their elastic modulus is ten times larger than the elastic modulus of the soft particles, and are hereafter referred to as hard. This approximates, for example, the common case of the compaction of a mixture of metallic and ceramic particles, for which springback effects and green strength might depart from the homogeneous case. We have already treated the cold compaction of such mixtures [12]. We have shown in particular that the additional macroscopic pressure necessary to compact a mixture of soft plastic particles and hard elastic particles to a given relative density depends on the rearrangement possibility of the hard particles and on the strain-hardening behaviour of the soft particles. Here we limit our study to perfectly plastic particles in order to treat the springback problem. Hard–hard contacts load and unload following a Hertzian law. No cohesion is considered for these contacts whereas soft–hard contacts have the same value for the work of adhesion w as soft–soft contacts. The friction coefficient is set to 0.1 for all contacts. We denote f2 as the fraction of hard elastic particles relative to the total volume of solid. The volume fraction of hard elastic particles relative to the total volume of the compact is: ⌽2 = f2D. Mixtures have been compacted following the method described in [12] with four different amounts of hard particles: f2 = 10, 20, 30 and 40%. The mean value of c¯ for the resulting agglomerate has been calculated by considering only soft–soft contacts for which Eq. (6) is used. This allows for a direct comparison with homogeneous agglomerates made only of soft particles. The introduction of hard elastic particles brings some specific features concerning the behaviour of plastic particles under unloading. We have observed that during macroscopic unloading some soft–hard contacts reload beyond the plastic indentation due to compaction alone. The fraction num-

ber of such contacts is small in any case (less than 5% of the total number of soft–hard contacts for f 2 = 40%). It shows however that hard elastic particles may constrain neighbouring soft particles while the compact is macroscopically unloading. We look first at the extent of springback experienced by mixtures of soft and hard particles as compared to homogeneous agglomerates. Fig. 11 shows that for large amounts of hard particles (⌽2 ⬎ 0.30), springback is larger than for the homogeneous agglomerate while it is smaller for small amounts of hard particles. The quantitative values in Fig. 11 are obviously linked to our choice that the elastic modulus of the hard phase is ten times larger than the elastic modulus of the soft phase. However it still bears some qualitative significance. Each type of contact unloads differently. We have measured that, on average and for the material parameters used for these simulations, the indentation retrieved by unloading of soft–hard contacts is roughly half the indentation retrieved

Fig. 11. Springback of a mixture of soft and hard particles normalised by the springback of a homogeneous agglomerate with soft particles only. ⌽2 represents the volume fraction of hard particles relative to the total volume of the compact prior to unloading. e0u is the springback for the homogeneous compact made of soft particles only.

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by soft–soft contacts. However, the full Hertzian unloading of a hard–hard contact yields larger springback than soft–soft contacts for the material parameters used here. Typically the indentation retrieved from hard–hard contacts is 2–4 times larger than the indentation retrieved from the unloading of a soft–soft contact. Of course these ratios depend on the value of c¯ and on the elastic properties of the hard phase and on the distribution of forces in the compact. For small amounts of hard particles, hard–hard contacts are very few (they increase with the square of f2) and the unloading behaviour of the mixture is dominated by soft–soft contacts and soft–hard contacts that yield lower springback. As the amount of hard particles increases, hard–hard contacts start to play a significant role and the springback of the composite increases. Additionally, Fig. 11 shows that compacts with soft particles that are cohesive (or with large values of c) are more affected than those that exhibit small cohesion between soft particles. This is linked to our assumption that hard–hard contacts have no cohesion. Next we turn our attention to the loss of strength (under isostatic tensile conditions) that the addition of hard particles may induce, bearing in mind that hard–hard contacts are considered cohesionless. Fig. 12 shows that small amounts of hard phase (⬍10%) decrease the strength by a factor between 1.5–2, while larger amounts of hard phase dramatically decrease the strength of the composite. Again, compacts with large c¯ are more affected by the introduction of hard non-cohesive particles. Note that we do not show results for c¯ ⬍ 10⫺3 because the strength of such composites is too small to yield significant perspective on the effect of the presence of hard particles.

5. Concluding remarks The unloading of an assembly of plastically deformed particles has been treated with special attention to the discrete nature of the problem. This allows for a better understanding of the contact alteration that the compact sustains during unloading. We have shown that springback depends on the material parameters of the powder

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Fig. 12. Loss of tensile strength under isostatic conditions due to the presence of hard elastic particles. p0f is the failure pressure for the homogeneous compact made of soft particles only.

itself (elastic and plastic properties, adhesion), on the compaction history (relative density attained during compaction) and on the process route (isostatic versus closed die compaction). Springback is often referred solely as an elastic problem. We have shown that in fact, although elasticity is the primary mechanism involved, decohesion at the contact level has also an important role during unloading. Nevertheless, a natural extension of the present work would consist in determining effective elastic coefficients of the unloading compact. Periodic boundary conditions permit the description of the bulk of the material, but these conditions are obviously not able to capture the effect of walls. In particular, the important effect of walls friction is omitted. Particles that are compacted and unloaded close to the walls are submitted to a loading history that is quite different from particles in the bulk of the compact. This layer of particles may exhibit specific features that may affect springback and that may generate internal stresses. Another restriction to the present work is the limited domain of relative density for which the assumption of non-interaction between contacts is valid. Contact impingement may become significant at

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relative densities as low as 0.85–0.90, and even at lower densities for composite systems as demonstrated recently [28]. For sake of completeness, we have shown results up to a relative density of 0.95. However, it should be clear that the present simulations should be taken with some care for relative densities larger than 0.85–0.90. Related to this issue is the fact that the similarity solution that is used for deriving the contact pressure (p 0 = 3⌺pq) and the contact stiffness might not hold for large indentation values that are expected at large relative densities. It has been shown for example that contact pressure for frictionless indentation decreases significantly when the finite deformation regime is attained (p 0⬇2.4⌺pq for indentation values typical of a 0.90 relative density compact) [17]. Note however that friction has the beneficial effect (from a modelling point of view) of retarding greatly the departure from the similarity solution [17]. Green strength is clearly related to the unloading problem. Yield surfaces that include tensile conditions have been derived analytically [26] or numerically [27]. However they have been developed under the simplifying assumptions of a fully cohesive or of a cohesionless aggregate. It seems natural to treat the fracture of an assembly of plastically deformed particles as a discrete problem with constitutive equations that incorporate the cohesive nature of contacts. This has been done here for the special case of isostatic loading. This is because the constitutive equations used for the contact are derived for the normal decohesion of two particles [1] and should remain approximately valid for small departure from normal unloading. For more deviatoric straining, it remains to be seen if a simple friction law associated with the constitutive equations derived by Mesarovic and Johnson can be used.

Acknowledgements The author thanks his colleagues Didier Bouvard and Sean Sweeney for their helpful suggestions concerning the manuscript.

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