## Numerical Modeling of Powder Compaction in Dies

In this section, practical illustrations of finite element analysis of powder compaction are presented. The first one is the compaction of a two-level component with emphasis on the effect of compaction kinematics on punch loads. The second example, the compaction of a long bushing, illustrates density predictions.

### Compaction of a Two-Level Part

The model presented in the section "A Constitutive Model for Metallic Powders with Ductile Particles" is used to examine the compaction of an axisymmetric two-level component. This part was studied experimentally by Kergadallan et al. (Ref 4). The part was pressed with five different tooling motions; two of those cases, labeled as part 30 and part 34, are presented here. The compaction experiments and the finite element model are described, and numerical predictions are compared to experimental results.

Compaction Experiments. Following are the conditions of the compaction tests used to study an axisymmetric two-level component.

Part Geometry. The part is axisymmetric with a thin outer rim, a hub, and a bore. This geometry is representative of many common powder metal parts (e.g., engine camshaft timing pulleys, one-way mechanical diode clutch plates). Geometry and dimensions are shown in Fig. 20. The outer diameter is 78 mm and the overall height is 26 mm.

7S mm

7S mm

68 mm

Fig. 20 Nominal dimensions of axisymmetric part used in compaction experiments

The powder blend is based on the diffusion alloyed iron powder, Hoeganaes Distalloy AE, as described in section "Calibration of Material Parameters for an Iron Powder Blend."

Press and Tooling. Figure 21 shows a schematic of the press and tooling. The part was pressed in a hydraulic COSMO press. The tooling consists of four moving components: an outer die and an inner core that move together, a top punch, and a lower inner punch. The tooling also includes a stationary lower outer punch. The press was instrumented with strain gauges to measure loads on tooling members (see Fig. 21) and potentiometer displacement transducers to measure tooling displacements.

Initial Conditions. The fill positions of the punches are given by Ri and R2 as shown in Fig. 22 and Table 2. The initial density in the die cavity was estimated by assuming uniform density within the rim section and the hub section of the part, as shown schematically in Fig. 23. The initial density values (Table 2) were estimated by weighing the different sections of the part and assuming no mass transfer between the top and bottom sections.

Conditions |
Part 30 |
Part 34 |

Rim fill (Rj), mm |
54.71 |
55.82 |

Hub fill (R2), mm |
27.23 |
25.18 |

PHub, g/cm3 |
3.51 |
3.63 |

pRim, g/cm3 |
3.21 |
2.97 |

Mass, g |
498.09 |
482.42 |

Fig. 22 Actual tooling motions for axisymmetric component with piecewise linear approximations. Source: Ref 4

Fig. 23 Estimated initial fill density distribution for parts used in compaction experiments. Source: Ref 4

Compaction Kinematics. The displacement histories for each of the moving tooling members are shown in Fig. 22. Tabulated information on punch displacements for part 30 and part 34 can be found in Ref 4. As explained later in this section, compaction of part 30 results in low density in the rim section. Also, cracks appear on the inner and outer surface of the rim. The aim of the compaction kinematics used with part 34 was to reduce the density imbalance between the hub and rim sections and to eliminate the cracks.

The Finite Element Model. The powder material model described in the section "A Constitutive Model for Metallic Powders with Ductile Particles" was implemented in the finite element code Abaqus/Standard (Ref 2) through a user-defined material subroutine. Details of the numerical implementation are beyond the scope of this article. Aravas (Ref 3) has described the numerical integration of constitutive models of this class.

Finite Element Model. An axisymmetric model of the compact was set up using four noded axisymmetric elements (Ref 2). Figure 24 shows a three-dimensional representation of the finite element mesh obtained by revolving the part about the axis of symmetry.

Fig. 24 Three-dimensional representation of the axisymmetric finite element discretization of the powder compact

Tooling. For simplicity, the tooling components were modeled as rigid surfaces. In a more detailed model, the tooling can also be modeled assuming elastic material behavior. Tooling stresses can be computed this way. However, modeling the interfaces between the powder and the elastic tooling is numerically very intensive.

Material Model. Constant values were used for the elastic properties. A fixed value of Young's modulus (141 GPa), corresponding to the final average density, was calculated using the expression for elastic modulus determined by Pavier et al. (Ref 14). It was mentioned earlier that the elastic properties are strong functions of density. However, in an analysis such as this, severe numerical instabilities can result as a consequence of the soft elastic properties at low density. Further, the effect of elastic behavior is dominant only when the relative density approaches its pore-free value.

The friction coefficient between iron powder compacts and steel/carbide tooling varies with the normal stress (Ref 44, 45). For lubricated iron powders, the coefficient of friction varies in the range 0.1 to 0.2 for normal stresses in the range 100 to 700 MPa. For the present model, a Coulomb friction model was used and a coefficient of friction of 0.15 was assumed.

Compaction Kinematics. A piecewise linear approximation to the actual tooling motions was used as depicted in Fig. 22.

Part Fill Density. The initial fill density was set as represented schematically in Fig. 23.

Experimental and Modeling Results. Figure 25 shows the compaction forces measured for part 30. The forces exerted on the bottom and upper punches increase from the initial instant corresponding to the moment of contact between the powder and the upper punch. At instant tL1 (Fig. 22), the speed of the lower inner punch (LIP) is modified such that it descends at practically constant force over the rest of its travel through time tL2. While the LIP is in motion, the upper punch (UP) and the lower outer punch (LOP) withstand forces that increase progressively up to their peak values. The rest of the cycle, for time >tL2, continues while the LIP remains at a fixed position. The forces on the LOP and the UP increase rapidly to their maximum values. Compression finishes at time tF, corresponding to the maximum displacement of the UP. The maximum loads recorded are shown in Table 3. Densities were measured in five sections of the part as shown schematically in Fig. 26. Measurements are shown in Table 4. For part 30, the hub and rim densities are approximately 7.06 g/cm3 and 6.90 g/cm3.

Punch |
Part 30 loads, MN |
Part 34 loads, MN |

Measured Predicted |
Measured Predicted | |

LIP |
2.10 2.12 |
1.61 1.75 |

LOP |
0.59 0.36 |
0.61 0.5 |

Section of Fig. 26 |
Density, g/cm3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Part 30 |
Part 34 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

1 |
7.01 |
6.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

2 |
7.07 |
6.93 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

3 |
7.09 |
7.03 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

4 |
6.90 |
ra o
Fig. 25 Comparison of compaction tooling loads for part 30: predicted versus experimental Fig. 25 Comparison of compaction tooling loads for part 30: predicted versus experimental Fig. 26 Comparison of compaction tooling loads for part 34: predicted versus experimental After ejection, the parts were visually inspected for cracks. Part 30 presented a distinct crack around the rim ID, 2 mm below the hub. Very fine cracks were also present on the outer surface of the rim, close to the bottom end. Although it is not possible to state exactly when the cracks were formed, it is very likely that they appeared during load removal/ejection. For part 30, the maximum loads on the inner and outer lower punches are 2.1 MN and 0.59 MN, respectively. The high load on the lower inner punch results in high punch deflection. During ejection, the elastic recovery of the lower inner punch results in tension on the rim causing cracking. To eliminate the cracks that appeared in part 30, part 34 was pressed with higher fill in the rim section and reduced fill in the hub section (Table 2). The tool motions applied were modified to accommodate the different fill positions, otherwise they were very similar to the motions used with part 30. The forces measured are shown in Fig. 27. The peak load values are given in Table 3. The load patterns for parts 30 and 34 are similar; however, for part 34, the lower inner punch load is reduced to 1.61 MN, while the outer lower punch force increases slightly to 0.61 MN. The hub and rim densities are approximately 6.97 g/cm3 and 6.94 g/cm3 (Table 3). The reduced load in the lower inner punch results in lower deflection and eliminates tension during load removal/part ejection. Part 34 was defect free. Figure 28 illustrates the compaction process as predicted by the numerical model. The deformed mesh is plotted for part 34. The model shows that there is very limited transfer of powder between the top and bottom (rim) sections of the part. This behavior is expected because the tooling motions were designed to minimize powder transfer to avoid the formation of a defect at the corner between the two sections. Figure 29 plots the evolution of density in the compact during pressing of part 34. The loads computed for parts 30 and 34 are plotted in Fig. 26 and 27 and compared to the measured loads. There is good overall agreement between the experiments (Ref 4) and the calculations. The computed maximum load values for each punch are shown in Table 3. Again the agreement with the experiments is very good, showing that the model is capable of predicting the effects of punch kinematics and fill position on punch loads. One observation on the nature of the load computations is pertinent. As the compact approaches its pore-free density, it becomes incompressible. Consequently, towards the end of the compaction stroke, minor variations in tool motions result in sharp changes in load. Thus, it is very important to ascertain the exact displacements of tooling members from their reference positions if accurate load predictions are sought. ## Compaction of a Long BushingThis example illustrates how the numerical model can be applied to predict density distributions. This part was studied experimentally by Trasorras et al. (Ref 40). A steel bushing was pressed in a 150 ton mechanical press. The green bushing dimensions are outside diameter (OD) = 19.05 mm, inner diameter (ID) = 12.7 mm, and height = 25.4 mm. The powder used was a blend of atomized steel powder (Ancorsteel 1000) with 0.75 wt% zinc stearate as lubricant. The punch motions comprise the following sequence. After powder filling, the top punch moves down to compact the powder, then rises and exits the die cavity. At the end of top punch motion, the die is stripped to eject the compact. Finally, the core rod is stripped. The lower punch and the die remain stationary throughout the compaction. As compaction starts, a density gradient develops in the bushing due to the friction between the compact and the tooling members. With continued top punch motion, the bushing densifies with the top always being at a higher density than the bottom. During ejection, there is some additional densification of the bottom end of the compact. Finally, the compact expands as it exits the die cavity thereby reducing the overall density. The axial density distribution in the bushing was determined by successively sectioning and weighing the compact. Compaction of the bushing was modelled in PCS (Ref 42), a powder compaction modeling system based on the finite element code NIKE2D (Ref 46). Figure 30 shows the finite element discretization of the tooling and powder, with the punches shown at their fill position. The material model described earlier was used with the constitutive functions b(D), c(D), and elastic properties calibrated for the atomized iron powder. The initial apparent density of the powder was 3.2 g/cm3. A complete model of the tooling was used (Fig. 30) and elastic isotropic behavior was assumed. The friction between the compact and the tooling members was assumed to follow Coulomb's model with a friction coefficient of 0.2. Figure 31 compares the axial density distribution predicted by the numerical model with the experimental results. The model properly represents the densification that takes place during both compaction and ejection and the predicted final density distribution is in good agreement with the experiments. The experiments show a sharp increase of density at the powder layer in contact with the top and bottom punches. The numerical model, with the discretization level used, was not able to predict this effect. This article has examined the general structure of constitutive laws for the compaction of powder compacts and demonstrated how these material models can be used to model the response of real world components to a series of complex die operations. It identified the general structure of the constitutive law and described a number of models that have been proposed in the literature. This field is still evolving, and it is evident that there will be significant developments in this area over the next few years as a wider range of experimental studies are conducted, providing greater insights into the compaction process. At the current time, there is no universally accepted model. Therefore, a pragmatic approach and a relatively simple form of empirical model were adopted requiring, for the determination of the unknown functions, a limited range of experiments. This selection allowed an examination of the compaction of axisymmetric components in detail and a comparison of general features of the component response with practical measurements. Similar procedures could have been adopted for any of the methods described, although in general, more sophisticated experiments are required in order to determine any unknown function or coefficients, particularly if the shape of the yield function is not known, or assumed, a priori. References cited in this section 2. ABAQUS/Standard User's Manual, Version 5.7, Vol 1-3, Hibbitt, Karlsson, & Sorensen, Inc., Providence, RI, 1997 3. N. Aravas, On the Numerical Integration of a Class of Pressure-Dependent Plasticity Models, Int. J. Numer. Meth. Eng., Vol 24, 1987, p 1395-1416 4. Y. Kergadallan, G. Puente, P. Doremus, and E. Pavier, Compression of an Axisymmetric Part, Proc. of the Int. Workshop on Modelling of Metal Powder Forming Processes (Grenoble, France), 1997, p 277-285 14. E. Pavier and P. Doremus, Mechanical Behavior of a Lubricated Powder, Advances in Powder Metallurgy & Particulate Materials-1996, Vol 2 (Part 6), Metal Powder Industries Federation, 1996, p 27-40 40. J.R.L. Trasorras, S. Krishnaswami, L.V. Godby, and S. Armstrong, Finite Element Modeling for the Design of Steel Powder Compaction, Advances in Powder Metallurgy & Particulate Materials-1995, Vol 1 (Part 3), Metal Powder Industries Federation, 1995, p 31-44 42. Powder Compaction Simulation Software (PCS Elite) User's Manual, Concurrent Technologies Corp., Johnstown, PA 44. B. Wikman, H.A. Haggblad, and M. Oldenburg, Modelling of Powder-Wall Friction for Simulation of Iron Powder Pressing, Proc. of the Int. Workshop on Modelling of Metal Powder Forming Processes (Grenoble, France), July 1997, p 149-158 45. E. Pavier and P. Doremus, Friction Behavior of an Iron Powder Investigated with Two Different Apparatus, Proc. of the Int. Workshop on Modelling of Metal Powder Forming Processes (Grenoble, France), July 1997, p 335-344 46. J. Hallquist, "NIKE2D-A Vectorized, Implicit, Finite Deformation, Finite-Element Code for Analyzing the Static and Dynamic Response of 2-D Solids," Technical report UCRL-19677, Lawrence Livermore National Laboratory, Livermore, California, 1993 Mechanical Behavior of Metal Powders and Powder Compaction Modeling J.R.L. Trasorras and R. Parameswaran, Federal-Mogul, Dayton, Ohio; A.C.F. Cocks, Leicester University, Leicester, England |

## Post a comment