Nuerical siulations of isotropic and die copaction of powder by the discrete eleent ethod J-F. Jerier, B. Harthong, B. Chareyre, D. Ibault, F-V. Donzé & P. Doréus Laboratoire Sols, Solides, Structures, Grenoble, France Jerier@geo.hg.inpg.fr, Harthong@geo.hg.inpg.fr, Bruno.Chareyre@hg.inpg.fr, Didier.Ibault@hg.inpg.fr, Frederic.Donze@hg.inpg.fr, Pierre.Doreus@hg.inpg.fr ABSTACT New constitutive equations ust be forulated relating the icroechanical properties to the acroscopic ones to describe accurately the behaviour of powders during a cold copaction process. To contribute to this new forulation, a Discrete Eleent Method (DEM) has been used to siulate the densification process of the powder icrostructure undergoing copaction. However, the contact laws used in classical DEM forulation do not reproduce correctly the evolution of the stress evolution during the copaction process. To overcoe this liitation at a high density value (over 0.85), the change in local density is taken into account. The first nuerical siulations using the open-source YADE software show a fairly good INTODUCTION Powder etallurgy (PM) has long been an attractive process technology for both advanced and conventional aterials. In a foring operation, powder is consolidated into a desired shape, norally by applying pressure. After foring, the green body (i.e., the copacted body) is sintering so that the echanical resistance of the final coponent is effective. Using powder copaction can save both tie and oney in the anufacture of echanical parts in serial production. One of the ajor advantages is that near net shape parts produced by this ethod need little or no achining. Another significant advantage is the possibility to design the aterial properties by ixing different powder aterials so that the final product achieves a specifically desired echanical behaviour. This results in a very high density (i.e., copact density superior to 0.9) conducive to foring a ore hoogeneous aterial suitable for high strength applications. But in the powder copaction process, plasticity and elasticity phenoena, internal friction of porous ediu and frictional effects between the die walls ay induce inhoogeneous distributions of density and residual stress. As a consequence, cracks appear into the copacted zone during the pressing process and the echanical behaviour of that part is difficult to predict [1]. In order to understand the behaviour of - 1
powder etallurgy under copaction, the industrials have perfored expensive trials and error procedures. On the other hand, nuerical ethods can help to predict the product properties, but the two nuerical approaches known based on the finite eleent ethod (FEM) and discrete eleent ethod (DEM) [2] reproduce these phenoena with soe difficulties. These obstacles are ainly due to the nonlinear aterial behaviour during pressing step [3]. Nevertheless, understanding and predicting the echanical behaviour of final coponent can be done by the icro-echanic siulation with adequate phenoenological odels. These odels ay iic the copaction process [4], because the DEM has the advantage to give a good insight on the different physical phenoena present during the early stage of powder copaction [5] [6]. In this paper, we use a new icroechanic odel to perfor discrete nuerical siulations of powder copaction up to a density equal to 0.95. For now, the phenoenological equations used in the DEM siulations to reproduce the powder copaction are only valid for a density lower than 0.85. In section 2, we present the existing and new discrete odels which are developed to siulate the powder process. Then in section 3, the siulations of isotropic and die copaction are perfored in a cubic cell with both the existing and the new discrete odels. The results are then copared to the ultiparticle finite eleent ethod (MPFEM) siulations, because it was shown to give an accurate description of the particle s deforation even for high densities [7][8]. THE DISCETE ELEMENT MODELS In DEM, each particle of an assebly is odelled separately and its otion is deterined fro the interactions with neighbouring particles. Dynaic schees follow the otion of each particle using a siple explicit finite difference schee that resolves Newton s second law [9]. The ain advantage of these schees is that they can deal with the non-linearity in contact laws in a rather siple way. Note that nuerous siulations of powder copaction are perfored via this ethod in integrating nonlinear contact laws [10]. Then, we present in this section these phenoenological odels used to odel the powder copaction and we also introduce the new contact law developed to siulate the copaction up to a high density value. Existing odels In the siulations of copaction, the powder is often represented by onosize spheres ( i ) with an elastic or elasto-plastic behaviour. In the case of elastic behaviour (E,ν), the siulations integrate an elastic contact law such as the Hertzian contact [11] between a particle I and a particle j. Hence, we note the elastic noral force F elas equal to: F elas = 4 1-ν ² ( 3 E i j 1 + ) i 1-ν ² E j * 1.5 h (1) * = i i j + j (2) With this contact law, a force is created between two spheres i and j when an overlap (hi j) appears. In general, the elastic contact law is coupled with a plastic contact law because the copaction process is done with soft particles which follow a hardening relation in the plastic regie equal to: 1 σ = σ 0 ε (3) Where σ 0 is a aterial constant and is the hardening coefficient. - 2
If two spherical particles i and j undergo plastic deforation, a constitutive equation controls the plastic contact force. The plasticity appears at contact when the elastic contact force (F elas) is higher than the plastic contact (F plas), then the noral contact force during the siulation is F = in(f elas; F plas). Following the work of Storåkers [5] the noral plastic contact force is defined by: 2 3 1 2+ 1 0 ) 2 1 2+ 1 2 * F plas = πσ 2 3 c( 2 h 2 (4) Where c( = 1.43e 0.97 is a function that is related to the size of the contact area. Then, for a plastic Von Mises type aterial with strain hardening, soe authors have used odel of Storåkers Eq (4) coupled with the Hertz contact Eq (1) to perfor the siulations of powder copaction DEM. Nevertheless, this approach is liited in density, because it considers that the aterial transported away by plastic flow fro the contact does not interact with neighbouring particles. Hence particles are considered as spheres that are siply truncated at the contact as copaction proceeds. This geoetrical assuption liits the doain of validity of the siulations to the doain of density for which contact ipingeent is negligible (i.e., density inferior or equal to 0.85). In order to overcoe this liitation and to take into account the influence of the news contacts at high density, a new odel has been developed. New odel The copaction analysis ust absolutely account for the new contacts between the particles created during the deforation. Then to know the nuber of contacts per particle for a given situation, the concept of an average Voronoï cell has been suggested by Arzt [12]. A cell is ade up of a polyhedron containing one powder particle where the nuber of faces of adjacent cells is deterined by the nuber of nearest neighbours for the particle in question. With this concept various inforation on an individual particle such as its nuber of nearest Vparticle neighbours and the local density (i.e., local density equal to ρ local = ) can be obtained. Vcell In the new odel developed by Harthong [13], the notion of average local density (ρ local) is created and integrated in the DEM siulations. This additional paraeter is used in the Harthong s law in order to accoodate the siulations of copaction at high densities. Thus, the force between two particles i and j can be expressed with an increental forulation such as: With the average local density equal to: F ( t + Δt) = F ( t) + S ( h, ρ local) δh ( t + Δt) (5) ρilocal + ρ jlocal ρlocal = (6) 2 And the stiffness S depends on aterial and geoetrical paraeters: - 3
S h β ( β ( ρlocal + ρinitial * * 1 * = σ 0 [ α( e + α1( e + α 2( 1 ρ initial h (7) Where α, α 1, α 2, β and β 1 are constants or functions of and ρ initial is the average local density when the contact is identified for the first tie. In the following, the coefficients used will be: 0.97 0.58 α ( = 15 α1( = ( ) 4 3 1 + 1 α 2 ( = 15( 2 + 1 β ( = 1.75( 2 β1 = 8 In the next section, we use this new contact law for nuerical siulations of isotropic and die copaction. Then, we copare the results obtained with reference odels (Hertzian contact and Storåkers s odel) and a reference ethod (MPFEM). SIMULATIONS AND ESULTS To assess the efficiency of the new contact law, we perfor a siulation of isotropic and die copaction with the sae rando packing (32 onosize spheres with a radius = 0.15 without friction. These 32 spheres of lead (σ 0 = 20.5MPa, = 4.16) are copressed by rigid planes in a cubic box (1x1x1 3 ) via the DEM with different odels and via the MPFEM (see Figure 1). This nuber of spheres is liited to keep a reasonable MPFEM calculation tie. For the DEM siulations, we perfor the with both the open-source DEM code YADE [14]. The calculation of both the local relative density and the contact law derived fro the new contact law could be ipleented. For coparison needs, Hertzian contact and Storåkers s odel were also ipleented. (A) (B) (C) FIGUE 1. The sae assebly of 32 onosize spheres used for the MPFEM (A) and DEM (B) siulations and the associated voronoï cells (C). - 4
Isotropic copaction In the case of isotropic copaction, all six walls are translated inward in the box, the aplitude of the displaceent being proportional to the initial distance between the planes and the sphere s centre. This copaction is close to an isotropic copaction of a rando packing, without any rearrangeent (see Figure 2). For these siulations, the results obtained by the MPFEM are close to experiental results [15] and they represent the reference results for our analysis. The curves resulting fro the siulations of isotropic presented in Figure 2 show a good correlation between MPFEM results and DEM results with the new contact law until a density value equal to 0.97. These curves also show that the Hertzian contact is not appropriate to odel the copaction of soft powder particles. In the case of Storåkers s odel, it overestiates a little bit the response during the early stage and it cannot reproduce the copaction when powder particles are incopressible. Figure 2. Coparison DEM curves coputed with YADE and MPFEM curve coputed with ABAQUS for isotropic copaction of 32 spheres in rando packing, without friction. Die copaction In the case of die copaction, the horizontal walls are translated towards one another at a constant speed, whereas the vertical walls are fixed. The curves resulting fro the siulations of die presented in Figure 3 show a good accordance in the three directions between MPFEM results and DEM results with the new contact law until a density value equal to 0.95. In Storåkers s odel for this siulation with very little spheres, inforation is issing for the response to copaction. For the high density value, this odel cannot reproduce the fact that the aterial cannot flow. - 5
Figure 3. Coparison DEM curves coputed with YADE and MPFEM curve coputed with ABAQUS for die copaction of 32 spheres in rando packing, without friction. CONCLUDING EMAKS The study of the contact forces in a rando packing of spheres subitted to copaction is essential for nuerical siulation. The MPFEM ethod used is based on precise and quantifiable data. But for the oent, it still requires a really long calculation tie copared to the DEM. With the introduction of a new paraeter which represents the average local density, the effects of plastic incopressibility can be odelled in the DEM fraework, allowing the description of high-density effects. The results presented above are encouraging, and show that the approach proposed here can potentially lead to an interesting accuracy. To perfor realistic DEM siulations, this approach will be kept and the odel will also need to account for the elasticity of the sphere s constitutive aterial, polydisperse sphere packings, different aterial types, and for the cohesive behaviour of the contacts. EFEENCES 1. C. Martin, Journal of the Mechanics and Physics of Solids 52, 1691 1717 (2004). 2. P. Cundall, and O. Strack, Geotechnique 29, 47 65 (1979). 3. A. Cocks, Progress in Materials Science 46, 201 229 (2001). 4. N. Fleck, Journal of the Mechanics and Physics of Solids 43, 1409 1431 (1995). 5. B. Storåkers, S. Biwa, and P.-L. Larsson, International Journal of Solids and Structures 34, 3061 3083 (1997). 6. S. Mesarovic, and K. Johnson, Journal of the Mechanics and Physics of Solids 48, 2009 2033 (2000). 7. A. Procopio, and A. Zavaliangos, Journal of the Mechanics and Physics of Solids 53, 1523 1551 (2005). 8. Y. Chen, D. Ibault, and P. Doréus, Materials Science Foru 534-536, 301 304 (2006). - 6
9. C. Thornton, and S. Antony, Philosophical transactions oyal Society Matheatical, physical and engineering sciences 356, 2763 2782 (1998). 10. O. Skrinjar, and P.-L. Larsson, Coputational Materials Science 31, 131 146 (2004). 11. H. Hertz, J. eine und Angewandte Matheatik 92, 156 171 (1882). 12. E. Artz, Acta Materiall 30, 1883 1890 (1982). 13. B. Harthong, J.-F. Jerier, P. Doréus, D. Ibault, and F.V. Donze, International Journal of Solids and Structures subitted (2009). 14. J. Kozicki, and F.-V. Donzé, Coputer Methods in Applied Mechanics and Engineering 197, 4429 4443 (2008). 15. A. Procopio, and A. Zavaliangos, Journal of the Mechanics and Physics of Solids 53, 1523 1551 (2005). - 7