- Email: [email protected]
Experimental and numerical investigation of idealized consolidation
Acta mater. 48 (2000) 4323–4330 www.elsevier.com/locate/actamat
EXPERIMENTAL AND NUMERICAL INVESTIGATION OF IDEALIZED CONSOLIDATION PART 1: STATIC COMPACTION W. WU, G. JIANG, R. H. WAGONER and G. S. DAEHN* Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA ( Received 24 April 2000; received in revised form 19 June 2000; accepted 19 June 2000 )
Abstract—In order to investigate the compaction of homogeneous and composite powders under carefully controlled conditions, idealized two-dimensional compacts were constructed using rods of aluminum and hardened steel. Special attention was paid to the material behavior of the rods, the die boundary conditions, the periodic packing arrangement, and the shape change during ejection from the dies (springback). Corresponding finite element simulations of uniaxial compaction were carried out. Pressure–density relationships, over a wide range of pressures and densities, were achieved by matching initial packing density and accounting for springback. Springback consists of two significant contributions: particle compressibility and void reopening. Deformed particle shapes were predicted well and the role of inter-particle friction was found to be minor for the idealized arrangements studied. The ratio of contact pressure over current yield stress is close to 3 for homogeneous compaction, and much higher values are seen for heterogeneous compaction. As expected, the presence of hard particles greatly inhibits the densification process. 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Powder consolidation; Composites; Finite element method
1. INTRODUCTION/BACKGROUND
The uniaxial compaction of metal powders or mixed metal/ceramic powders is widely used to produce netor near-net-shaped components and to consolidate materials that are subjected to further mechanical working. The process typically consists of die filling with homogeneous or heterogeneous (composite) powders, pressing to a desired pressure (and possibly holding the sample under pressure), then releasing the pressure and ejecting the compact from the die. Compaction can be done at room temperature (cold pressing) or elevated temperature (hot pressing). At room temperature the effects of creep and diffusion are negligible for most engineering materials. The important processes in cold compaction include powder rearrangement and reorientation, elastic deformation, plastic deformation, and fragmentation (for brittle materials). Relocation takes place as soon as pressure is applied. At the same time, particle surface flattening appears. This regime is often referred to as stage I [1]. At higher pressures, particle relocation becomes insignificant (because particles conform more closely
* To whom all correspondence should be addressed. E-mail address: [email protected] (G.S. Daehn)
by deformation), and plastic deformation becomes the predominant densification mechanism [2]. This is referred to as stage II. Transition from stages I to II typically occurs when the pressure exceeds the bulk yield stress of the material and plastic flow becomes relatively homogeneous [3]. Friction plays an important role during compaction. Inter-particle and particle/die wall friction hinders pressure transmission and therefore produces density gradients inside the compact [4–7]. Inter-particle friction also hinders the relative movement of particles during compaction. Springback, the change of shape and density upon removal from a die, is of practical concern. The size of a green compact is greater than the die size. Excessive springback, as well as density gradients and die wall friction, can cause cracking in compacts and may be responsible for warping during sintering [8, 9]. Two basic approaches have been used to simulate homogeneous powder compaction: continuum methods and discrete-particle methods. In the continuum models, powder compacts are considered to consist of continuum porous bulk material with a yield criterion that permits flow in a hydrostatic stress state. From the first widely used models of Shima and Oyane [10] and Gurson [11, 12], various models have
1359-6454/00/$20.00 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 2 0 6 - 8
4324
WU et al.: IDEALIZED CONSOLIDATION—PART 1
been proposed based on theoretical considerations and experimental measurements [13–18]. The plastic deformation behavior of individual particles during compaction is not explicitly considered, thus linkages to microstructure, particle properties and friction are generally not available. These methods are well suited to approximate solutions of large-scale boundaryvalue problems, such as die compaction. Discrete-particle approaches focus on the deformation of individual particles and often use simplified particle interactions to obtain a density–pressure relationship. Implementation of discrete-particle modeling can take many forms, including finite element analysis and slip-line field methods. Arzt et al. [19, 20] proposed a particularly simple and useful approach. For stage I (density ⬍90%), they calculated the average contact area and coordination number based on the particle cumulative radial distribution function. Material at contact interfaces was assumed to be yielded at an average contact pressure of three times the yield stress (based on the indentation solution for rigid plastic material [21]). Punch pressure was calculated based on the average contact pressure, particle size and statistical coordination number. The approach has been successful in calculating pressure–density diagrams due to plastic yielding for a number of single-phase powders [1, 22]. The model is appropriate for isolated contact at low density; at high density, contact zones interact and deformation becomes severe, violating the isolated contact assumptions. Also at high density, the unavoidable spatial heterogeneity of the compact becomes problematic. Higher densities can be treated numerically. Akisanya et al. [23, 24] used the slip-line field method to study the consolidation of simple systems, and the finite element method (FEM) for analysis of materials with non-hydrostatic loading or strain-hardening behavior up to 96% density. Only single particle-toparticle interactions were simulated locally, and frictionless conditions and idealized packing were required for numerical tractability. The FEM simulation was in agreement with the experimental measurement of the compaction of Plasticine cylinders within a scatter band. FEM was also applied by other researchers on the particle level for local contact [25, 26] and bulk yield [27] analysis. Relatively few studies have been carried out on the densification of composite powders [28–30]. The addition of rigid inclusions greatly hinders densification. At higher volume fractions, the inclusions can form networks to support some of the applied pressure elastically, and thereby reduce the pressure on the plastic phase [28]. Quantitative comparison of compaction simulations and experiments has been difficult to achieve because of (1) difficulties in characterizing the material and surface properties of fine powders, (2) difficulties in describing a random initial packing in a manner suitable for simulation, and (3) the limitations of current
computation power and numerical methods for simulation of random assemblies of a large number of realistic particles. In order to provide a well-characterized starting point for simulation and measurement of an idealized compaction, a regular two-dimensional array of rods was constructed of materials with measured properties. Homogeneous compaction was studied with aluminum rods and heterogeneous cases were carried out with interspersed hardened steel rods. All rods were constrained along the length by the die cavity. Square and hexagonal packing were used, and friction conditions were modified by adding lubricant. Special care was taken in the experiments to account for surface packing accuracy, overall initial density, and the change of dimensions after removal from the dies. Simulation of these idealized compacts was carried out using two-dimensional (2D) solid elements in ABAQUS [31]. 2. EXPERIMENTAL PROCEDURE
Pure aluminum (99.999%) and oil-hardened highcarbon steel (0.9% C, 1.2% Mn) rods 2 mm in diameter were used as the respective deformable and nondeformable phases. The steel rods remained elastic throughout our experiments (sy ⫽ 1050 MPa) while the aluminum rods deformed plastically. The elastic– plastic behavior of the aluminum was measured by uniaxial tensile testing, and verified at higher strains by rolling of rods followed by tensile testing, as shown in Fig. 1. The tensile curve was used in simulations along with the standard Von Mises yield criterion and associated plastic flow theory. The packed rods were compacted uniaxially at room temperature with a standard servohydraulic testing machine as illustrated in Fig. 2. The die walls and punches were lubricated with a light coating of white petroleum jelly before each test to reduce friction. Hexagonal and square packing arrangements of aluminum and steel rods were compacted in a 20 mm⫻20 mm square die (Fig. 2). The 20 mm long
Fig. 1. Measured stress–strain curve for aluminum rods and assumed numerical constitutive relationship.
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 2. Compaction method used in the experiments.
rods were fit precisely into the die: five layers of rods were packed horizontally in both hexagonal and square arrangements, as shown in Fig. 3. In each packing arrangement, four layers of round rods and two layers of half-round rods were used. In the experiments the rods were constrained so they could not expand lengthwise during compaction. Correspondingly, in simulations, plane strain conditions were used along the axial direction. In experiments, to ensure there were no gaps between the rods and the die wall, each rod was ground from two ends until it could be fit precisely into the die. Also, some half and quarter-cut rods were used (as shown in Fig. 3) to satisfy the boundary conditions and periodicity in the FEM simulation. To obtain fairly precise half
4325
rods, the whole rods were mounted in lucite and ground until the surface width of the rods was not less than 1.96 mm. Similarly, quarter-cut rods were obtained by grinding off 1 mm from both the top and lateral directions. To free the rods, the mounting lucite was dissolved in acetone after the target sizes were achieved. For both arrangements two kinds of experiments were carried out: homogeneous compaction of aluminum rods only, and composite compaction of aluminum and steel rods. The theoretical relative densities of hexagonal and square packs, before any consolidation, are 0.9068 and 0.7854, respectively. The real initial densities in experiments were 0.8991 and 0.7784 for homogeneous compacts, and 0.8978 and 0.7721 for the composite compacts. The measured initial densities were slightly lower than the theoretical densities because of imperfections in the packing arrangements. To study the effects of inter-rod friction on consolidation of rods, zinc stearate was applied to the rod surfaces as a lubricant. A mixture of zinc stearate powder and acetone was poured over the rods. When the acetone evaporated, about 0.5 wt% zinc stearate remained on the rod surface. Using a pin-and-disk method, Mallender et al. found that the friction coefficients of unlubricated pins of iron powder compacts were 0.35–0.50, while friction coefficients were 0.07– 0.10 when the pins were lubricated by zinc stearate [32]. During compaction, the change in density of the compact was determined by the change in displacement of the upper punch. An extensometer was attached between the upper punch and the die, as shown in Fig. 2, to record the displacement of the punch as a function of pressure p, denoted Upun(p).
Fig. 3. Packing arrangements used in this study. The compacted volume is a rectangular prism.
4326
WU et al.: IDEALIZED CONSOLIDATION—PART 1
The density change during compaction was calculated from the upper punch displacement, with knowledge of the final geometry and mass of the materials in the die. The elastic deformation of the system, Uelsys, was measured while releasing the pressure to zero. The measurement of elastic deformation was subtracted from the upper punch displacement data, proportionally to the pressure, to obtain Upl comp(p), the net displacement of the pressing punch caused by plastic deformation of the compact during compaction. This calculation eliminates the error introduced by the elastic deformation of the punch and compact. Upl comp(p) was used to calculate the density change during compaction, assuming the cross-section area and mass of the compact are constant. The elastic deformation of the punch, Uelpun, was measured by applying and releasing pressure while the die was empty. The springback of the compact was calculated as the difference between the elastic deformation of the compaction system, Uelsys, and the deformation of the pressing punch, Uelpun. 3. ANALYTICAL PROCEDURE
Finite element simulations of the unit cells shown in Fig. 3 were carried out using ABAQUS. Mesh arrangements for the simulations of heterogeneous compaction are shown in Fig. 4. The aluminum side of the interface was finely meshed to increase the accuracy of contact simulation. Element types CPE3 and CPE4R were employed. These are respectively three- and four-noded, elastic–plastic, plane strain solid elements, with hourglass control and reduced integration for CPE4R (one integration point for each element). A quarter of an aluminum rod is meshed with 885 elements and 943 nodes. The steel rods are represented by rigid bounding surfaces. Different meshes were used to study sensitivity.
Fig. 4. Unit cells for FEM simulation. Aluminum rods are meshed with plane strain elements and steel rods are represented by rigid circles. The cell boundaries are rigid.
The Von Mises yield criterion and associated flow theory were used in the simulations, with an isotropic hardening law following the experimental data shown in Fig. 1. The elastic properties of the aluminum were taken from handbook values: Young’s modulus as 62 GPa and Poisson’s ratio as 0.345. An inter-particle friction coefficient of 0.4 was adopted for heterogeneous compaction simulation, and 0.5 for homogeneous compaction. Sensitivity simulations were carried out to investigate the dependence of simulation results on the friction coefficient. The unit cell boundaries are represented by rigid surfaces, with symmetry conditions imposed. Because of the symmetry, cell boundaries are modeled as frictionless. To simulate the punch motion, the top cell surface is moved rigidly downward until a target load is attained, while maintaining the wall positions. During unloading, the top wall is lifted until the contact force drops to zero. For comparison, unloading was also performed by removing the top and lateral walls simultaneously until there were no contact forces from any direction. Relative density equals the ratio of the area occupied by materials to the area of the unit cell. The relative density so defined plus porosity equals unity. The area occupied by aluminum is the sum of all element areas, which can be calculated from the nodal coordinates recorded in the ABAQUS output. The area occupied by steel is always πr2/4, where r is the radius of the steel rods. The unit cell geometry can also be monitored via the ABAQUS output. 4. RESULTS/DISCUSSION
The experimental and simulated pressure–density curves for pure aluminum and aluminum/steel rods are shown in Figs 5 and 6, respectively. Initially the density increases rapidly and approximately linearly with the pressure, then approaches the theoretical limit asymptotically. Comparison of the deformed rod shapes is presented in Fig. 7 for aluminum rods at the punch pressure of 172 MPa. Simulated results are
Fig. 5. Pressure–density relation for the homogeneous compaction of aluminum rods.
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 6. Pressure–density relation for the heterogeneous compaction of aluminum and steel rods.
Fig. 7. Deformed shapes of aluminum rods at 172 MPa. Simulated results are superimposed on the experimental photos.
overlapped on photos from experiments. Triangular voids appear at the particle junctions in each case. The experimental and simulated pressure–density curves and deformed aluminum shapes show very good correspondence. It is essential to use half and quarter-cut rods in experiments to match the periodicity and symmetrical conditions in simulations. Figure 8 compares the pressure–density curves of simulation and experiments in which only six layers of whole rods (no half and quarter-cut rods) were used. The poor agreement is the result of decreased initial density caused by the rods in the top and bottom layers. During compaction,
4327
Fig. 8. Poor agreement of using whole rods only (no half and quarter-cut rods) in aluminum–steel compaction with hexagonal packing.
the contact between the whole steel rod layer and the die punch shielded the pores between them. The choice of the simulation unit cell [Fig. 4(a)] is only effective when the half and quarter-cut rods are placed, as in Fig. 3(a). Friction is difficult to characterize in particle systems. Friction coefficients may not really be constant during a process because of changes in contact geometry, relative sliding and interface temperatures. In order to assess the importance of friction coefficients in these simulations, a sensitivity study was performed. In the case of pure aluminum rods with square packing, no inter-rod sliding takes place during compaction. Hence, friction has no effect. For this reason, only hexagonal packing was used in the friction sensitivity study. Figure 9 shows that the value of the friction coefficient has almost no effect in the homogeneous case, and only a small effect for very small values (⬍0.2) in the composite case. In experiments, inter-particle friction was reduced by adding zinc stearate on the rod surfaces. Figure 10 shows the pressure–density curves of the hexagonal compactions of both dry and lubricated aluminum/steel rods. The same trend was followed both in experiments and simulations. Inter-particle friction only modestly inhibits the densification of the aluminum and steel composite compacts. The pressure distribution at the particle interfaces was carefully examined in simulation. Figure 11(a) shows the ratio of the contact pressure (Pcont) to the current yield stress (sy), for nodes along the contact surface at different punch displacements and pressures (P), for pure aluminum rods with hexagonal packing. The corresponding relative densities (RD) are also shown in the figures. Figure 11(b) shows the same simulation results, but the ratio of contact pressure to initial yield stress (s0y) is shown. Figure 11(c) also shows Pcont/sy for homogeneous compaction, but in this case elastic–ideally plastic behavior with a fixed yield stress of s0y for the deformable phase, and frictionless contact, were assumed. Figure 12 shows
4328
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 9. Simulated friction effect on the rod interface with hexagonal packing: (a) aluminum rods, (b) aluminum and steel rods.
Fig. 11. Ratio of contact pressure to yield stress for nodes along the contact surface for homogeneous compaction with hexagonal packing at different punch pressures (relative densities): (a) current yield stress, (b) initial yield stress, (c) ideal elastic– plastic and frictionless. Fig. 10. Measured friction effect on the rod interface of aluminum and steel rods with hexagonal packing.
results for heterogeneous composite compaction. In Fig. 12(a), the contact friction coefficient is 0.4 and the material strain-hardens normally. Figure 12(b) shows the behavior with strain-hardening material in frictionless conditions. In Fig. 12(c) an elastic–ideally plastic deformable phase is studied with frictionless contact. The ratios (Pcont/sy and Pcont/s0y) continually
increase as contact area increases (pressure and density increase also). For homogeneous compaction, the mean values of Pcont/sy (具Pcont/sy典), for both strainhardening [Fig. 11(a)] and elastic–ideally plastic [Fig. 11(c)] materials, are very close to 3, the value used by Arzt et al. [19, 20]. The ratio is higher in heterogeneous compacting with frictional contact [Fig. 12(a)], but frictionless contact lowers the normalized contact pressure [Fig. 12(b)]. These results show that the approach of Arzt et al. is inherently approximate,
WU et al.: IDEALIZED CONSOLIDATION—PART 1
4329
Fig. 13. Springback upon unloading for heterogeneous compaction with hexagonal packing: (a) under full load and after removal of punch load (1D) and all contact loads (2D), (b) change in volume due to void re-opening and material compressibility.
Fig. 12. Ratio of contact pressure to current yield stress for nodes along the aluminum side of the contact surface for heterogeneous compaction with hexagonal packing at different punch pressures (relative densities): (a) friction coefficient of 0.4, (b) frictionless, (c) ideal elastic–plastic and frictionless.
as a ratio of interface pressure to yield stress is implicitly assumed. For homogeneous compaction, if current yield stress is used, this may be a reasonable approximation [see Fig. 11(a) and (b)]. Arzt et al.’s approach will considerably underestimate compaction pressure for composite consolidation. Figure 13(a) shows the porosity change under full load and after unloading. The “load” curve shows the amount of porosity predicted under load. Upon unloading, springback will re-open some voids. The
1D-unload curve shows the total porosity after removing only the punch load. The 2D-unload curve shows the porosity after removing both the punch and the lateral-wall loads. The maximum porosity change found in simulation upon unloading is 0.03% in 1Dunload, and 0.17% in 2D-unload. The latter result has more practical interest. Figure 13(b) shows the total volume change upon unloading, and the volume change due to material compressibility and void reopening upon unloading. The result shows that void re-opening contributes more at low pressure and reaches a plateau at high pressure, while the contribution from compressibility increases linearly. This characteristic of springback would not be observed if elasticity was not included in the model used. 5. CONCLUSIONS
The compaction of idealized 2D powders (rods) has been simulated using finite element modeling, and measured using aluminum and steel rods to eliminate uncertainties of particle rearrangement and unknown material properties. The following principal conclusions were reached.
4330
WU et al.: IDEALIZED CONSOLIDATION—PART 1
1. When initial packing density is carefully matched and springback upon die removal is accounted for, finite element simulation and experimental pressure–density curves are in very close agreement over a wide range of pressures and densities (77– 100%). Homogeneous and composite compaction results are equally satisfactory. 2. Measured and simulated particle shapes after compaction are in excellent agreement. 3. The ratio of contact pressure to current yield stress is near 3 for homogeneous compaction. The ratio is considerably higher for heterogeneous compaction. Friction does not affect the ratio in homogeneous compaction, but does in heterogeneous compaction (the ratio increases by a factor of two from frictionless to a friction coefficient of 0.4). 4. The effect of inter-particle friction on simulated pressure–density behavior is negligible for homogeneous powders in a cellular array. The effect is small for composite powders with friction coefficients greater than 0.2, but densification occurs more readily for friction coefficients less than 0.2. 5. Springback upon die removal must be considered for careful comparison of compact densities and volumes. Springback stemming from the re-opening of voids was observed to produce up to 0.5% volume change. 6. Simulation allows the separation of springback effects from those arising from particle compressibility, and from those arising from void re-opening upon load removal. At low pressures (less than 100 MPa), most of the springback is related to void re-opening. As pressure increases, the component resulting from void re-opening reaches a plateau, and the component from compressibility continues to increase. Thus the compressibility effect dominates at high pressure. 7. As expected, the presence of hard particles inhibits the densification of composite compacts as compared with a homogeneous compact of deformable particles. This is because much higher pressure is needed to make the deformed phase conform to the hard particles.
Acknowledgements—This material is based upon work supported by the National Science Foundation under Grant No. DMR-9705558, Dr B. A. MacDonald contract monitor, and the Ohio Supercomputer Center, The Ohio State University (Grant No. PAS 080). The authors would like to thank Christine D. Putnam for proofreading and editing assistance.
REFERENCES 1. Arzt, E., Ashby, M. F. and Easterling, K. E., Metall. Trans. A. 1983, 14A(February), 211. 2. Fischmeister, H. F., Artz, E. and Olsson, L. R., Powder Metall., 1978, 4, 179. 3. Hewitt, R. L., Wallace, W. and Malherbe, M. C., Powder Metall., 1974, 17(33), 1. 4. Thompson, R. A., Ceram. Bull., 1981, 60(2), 237. 5. Briscoe, B. J. and Rough, S. L., Colloids Surf. A: Physicochem. Eng. Aspects. 1998, 137, 103. 6. Armstrong, S., Aesoph, M. D. and Gurson, A. L., paper presented at International Conference and Exhibition on Powder Metallurgy and Particulate Materials, Seattle, WA. 1995, 3, 31. 7. Bocchini, G. F., paper presented at International Conference and Exhibition on Powder Metallurgy and Particulte Materials, Seattle, WA. 1995, 2, 115. 8. Blumenthal, W. R., Sheinberg, H. and Bingert, S. A., MRS Bull., 1997, 22, 29. 9. Glass, S. J. and Ewsuk, K. G., MRS Bull., 1997, 22, 24. 10. Shima, S. and Oyane, M., Int. J. Mech. Sci., 1976, 18, 285. 11. Gurson, A. L., Trans. ASME. J. Eng. Mater. Technol., 1977, 99, 2. 12. Tvergaard, V., Int. J. Fract., 1981, 17(4), 389. 13. Fleck, N. A., Kuhn, L. T. and McMeeking, R. M., J. Mech. Phys. Solids. 1992, 40(5), 1139. 14. Lee, D. N. and Kim, H. S., Powder Metall., 1992, 35(4), 275. 15. Kwon, Y. S., Lee, H. T. and Kim, K. T., Trans. ASME, J. Eng. Mater. Technol., 1997, 119(October), 366. 16. Park, S., Han, H. N., Oh, K. H. and Lee, D. N., Int. J. Mech. Sci., 1999, 41, 121. 17. Tszeng, T. C., in Simulation of Materials Processing: Theory, Methods and Applications. ed. J. Huetink and F. P. T. Baaijens, Balkema, Rotterdam, 1998, p. 391. 18. Williamson, R. L., Knibloe, J. R. and Wright, R. N., Trans. ASME, J. Eng. Mater. Technol., 1992, 114(January), 105. 19. Arzt, E., Acta metall., 1982, 30, 1883. 20. Fischmeister, H. F. and Arzt, E., Powder Metall., 1983, 26(2), 82. 21. Hill, R., in The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1950, p. 254. 22. Helle, A. S., Easterling, K. E. and Ashby, M. F., Acta metall., 1985, 33(12), 2163. 23. Akisanya, A. R., Cocks, A. C. F. and Fleck, N. A., J. Mech. Phys. Solids. 1994, 42(7), 1067. 24. Akisanya, A. R. and Cocks, A. C. F., J. Mech. Phys. Solids. 1995, 43(4), 605. 25. Li, H., Saigal, S. and Wang, P. T., Acta mater., 1996, 44(7), 2591. 26. Delo, D. P. and Piehler, H. R., Acta mater., 1999, 47(9), 2841. 27. Pavanachand, C. and Krishnakumar, R., Acta mater., 1997, 45(4), 1425. 28. Lange, F. F., Atteraas, L. and Zok, F., Acta metall. mater., 1991, 39(2), 209. 29. Turner, C. D. and Ashby, M. F., Acta mater., 1996, 44(11), 4521. 30. Singh, B. N., Powder Metall., 1971, 14(28), 277. 31. ABAQUS/Standard User’s Manual. Vol. 5.8. Hibbitt, Karlsson and Sorensen, Inc, Pawtucket, RI, 1998. 32. Mallender, R. F., Dangerfield, C. J. and Coleman, D. S., Powder Metall., 1974, 17(34), 288.
EXPERIMENTAL AND NUMERICAL INVESTIGATION OF IDEALIZED CONSOLIDATION PART 1: STATIC COMPACTION W. WU, G. JIANG, R. H. WAGONER and G. S. DAEHN* Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA ( Received 24 April 2000; received in revised form 19 June 2000; accepted 19 June 2000 )
Abstract—In order to investigate the compaction of homogeneous and composite powders under carefully controlled conditions, idealized two-dimensional compacts were constructed using rods of aluminum and hardened steel. Special attention was paid to the material behavior of the rods, the die boundary conditions, the periodic packing arrangement, and the shape change during ejection from the dies (springback). Corresponding finite element simulations of uniaxial compaction were carried out. Pressure–density relationships, over a wide range of pressures and densities, were achieved by matching initial packing density and accounting for springback. Springback consists of two significant contributions: particle compressibility and void reopening. Deformed particle shapes were predicted well and the role of inter-particle friction was found to be minor for the idealized arrangements studied. The ratio of contact pressure over current yield stress is close to 3 for homogeneous compaction, and much higher values are seen for heterogeneous compaction. As expected, the presence of hard particles greatly inhibits the densification process. 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Powder consolidation; Composites; Finite element method
1. INTRODUCTION/BACKGROUND
The uniaxial compaction of metal powders or mixed metal/ceramic powders is widely used to produce netor near-net-shaped components and to consolidate materials that are subjected to further mechanical working. The process typically consists of die filling with homogeneous or heterogeneous (composite) powders, pressing to a desired pressure (and possibly holding the sample under pressure), then releasing the pressure and ejecting the compact from the die. Compaction can be done at room temperature (cold pressing) or elevated temperature (hot pressing). At room temperature the effects of creep and diffusion are negligible for most engineering materials. The important processes in cold compaction include powder rearrangement and reorientation, elastic deformation, plastic deformation, and fragmentation (for brittle materials). Relocation takes place as soon as pressure is applied. At the same time, particle surface flattening appears. This regime is often referred to as stage I [1]. At higher pressures, particle relocation becomes insignificant (because particles conform more closely
* To whom all correspondence should be addressed. E-mail address: [email protected] (G.S. Daehn)
by deformation), and plastic deformation becomes the predominant densification mechanism [2]. This is referred to as stage II. Transition from stages I to II typically occurs when the pressure exceeds the bulk yield stress of the material and plastic flow becomes relatively homogeneous [3]. Friction plays an important role during compaction. Inter-particle and particle/die wall friction hinders pressure transmission and therefore produces density gradients inside the compact [4–7]. Inter-particle friction also hinders the relative movement of particles during compaction. Springback, the change of shape and density upon removal from a die, is of practical concern. The size of a green compact is greater than the die size. Excessive springback, as well as density gradients and die wall friction, can cause cracking in compacts and may be responsible for warping during sintering [8, 9]. Two basic approaches have been used to simulate homogeneous powder compaction: continuum methods and discrete-particle methods. In the continuum models, powder compacts are considered to consist of continuum porous bulk material with a yield criterion that permits flow in a hydrostatic stress state. From the first widely used models of Shima and Oyane [10] and Gurson [11, 12], various models have
1359-6454/00/$20.00 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 2 0 6 - 8
4324
WU et al.: IDEALIZED CONSOLIDATION—PART 1
been proposed based on theoretical considerations and experimental measurements [13–18]. The plastic deformation behavior of individual particles during compaction is not explicitly considered, thus linkages to microstructure, particle properties and friction are generally not available. These methods are well suited to approximate solutions of large-scale boundaryvalue problems, such as die compaction. Discrete-particle approaches focus on the deformation of individual particles and often use simplified particle interactions to obtain a density–pressure relationship. Implementation of discrete-particle modeling can take many forms, including finite element analysis and slip-line field methods. Arzt et al. [19, 20] proposed a particularly simple and useful approach. For stage I (density ⬍90%), they calculated the average contact area and coordination number based on the particle cumulative radial distribution function. Material at contact interfaces was assumed to be yielded at an average contact pressure of three times the yield stress (based on the indentation solution for rigid plastic material [21]). Punch pressure was calculated based on the average contact pressure, particle size and statistical coordination number. The approach has been successful in calculating pressure–density diagrams due to plastic yielding for a number of single-phase powders [1, 22]. The model is appropriate for isolated contact at low density; at high density, contact zones interact and deformation becomes severe, violating the isolated contact assumptions. Also at high density, the unavoidable spatial heterogeneity of the compact becomes problematic. Higher densities can be treated numerically. Akisanya et al. [23, 24] used the slip-line field method to study the consolidation of simple systems, and the finite element method (FEM) for analysis of materials with non-hydrostatic loading or strain-hardening behavior up to 96% density. Only single particle-toparticle interactions were simulated locally, and frictionless conditions and idealized packing were required for numerical tractability. The FEM simulation was in agreement with the experimental measurement of the compaction of Plasticine cylinders within a scatter band. FEM was also applied by other researchers on the particle level for local contact [25, 26] and bulk yield [27] analysis. Relatively few studies have been carried out on the densification of composite powders [28–30]. The addition of rigid inclusions greatly hinders densification. At higher volume fractions, the inclusions can form networks to support some of the applied pressure elastically, and thereby reduce the pressure on the plastic phase [28]. Quantitative comparison of compaction simulations and experiments has been difficult to achieve because of (1) difficulties in characterizing the material and surface properties of fine powders, (2) difficulties in describing a random initial packing in a manner suitable for simulation, and (3) the limitations of current
computation power and numerical methods for simulation of random assemblies of a large number of realistic particles. In order to provide a well-characterized starting point for simulation and measurement of an idealized compaction, a regular two-dimensional array of rods was constructed of materials with measured properties. Homogeneous compaction was studied with aluminum rods and heterogeneous cases were carried out with interspersed hardened steel rods. All rods were constrained along the length by the die cavity. Square and hexagonal packing were used, and friction conditions were modified by adding lubricant. Special care was taken in the experiments to account for surface packing accuracy, overall initial density, and the change of dimensions after removal from the dies. Simulation of these idealized compacts was carried out using two-dimensional (2D) solid elements in ABAQUS [31]. 2. EXPERIMENTAL PROCEDURE
Pure aluminum (99.999%) and oil-hardened highcarbon steel (0.9% C, 1.2% Mn) rods 2 mm in diameter were used as the respective deformable and nondeformable phases. The steel rods remained elastic throughout our experiments (sy ⫽ 1050 MPa) while the aluminum rods deformed plastically. The elastic– plastic behavior of the aluminum was measured by uniaxial tensile testing, and verified at higher strains by rolling of rods followed by tensile testing, as shown in Fig. 1. The tensile curve was used in simulations along with the standard Von Mises yield criterion and associated plastic flow theory. The packed rods were compacted uniaxially at room temperature with a standard servohydraulic testing machine as illustrated in Fig. 2. The die walls and punches were lubricated with a light coating of white petroleum jelly before each test to reduce friction. Hexagonal and square packing arrangements of aluminum and steel rods were compacted in a 20 mm⫻20 mm square die (Fig. 2). The 20 mm long
Fig. 1. Measured stress–strain curve for aluminum rods and assumed numerical constitutive relationship.
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 2. Compaction method used in the experiments.
rods were fit precisely into the die: five layers of rods were packed horizontally in both hexagonal and square arrangements, as shown in Fig. 3. In each packing arrangement, four layers of round rods and two layers of half-round rods were used. In the experiments the rods were constrained so they could not expand lengthwise during compaction. Correspondingly, in simulations, plane strain conditions were used along the axial direction. In experiments, to ensure there were no gaps between the rods and the die wall, each rod was ground from two ends until it could be fit precisely into the die. Also, some half and quarter-cut rods were used (as shown in Fig. 3) to satisfy the boundary conditions and periodicity in the FEM simulation. To obtain fairly precise half
4325
rods, the whole rods were mounted in lucite and ground until the surface width of the rods was not less than 1.96 mm. Similarly, quarter-cut rods were obtained by grinding off 1 mm from both the top and lateral directions. To free the rods, the mounting lucite was dissolved in acetone after the target sizes were achieved. For both arrangements two kinds of experiments were carried out: homogeneous compaction of aluminum rods only, and composite compaction of aluminum and steel rods. The theoretical relative densities of hexagonal and square packs, before any consolidation, are 0.9068 and 0.7854, respectively. The real initial densities in experiments were 0.8991 and 0.7784 for homogeneous compacts, and 0.8978 and 0.7721 for the composite compacts. The measured initial densities were slightly lower than the theoretical densities because of imperfections in the packing arrangements. To study the effects of inter-rod friction on consolidation of rods, zinc stearate was applied to the rod surfaces as a lubricant. A mixture of zinc stearate powder and acetone was poured over the rods. When the acetone evaporated, about 0.5 wt% zinc stearate remained on the rod surface. Using a pin-and-disk method, Mallender et al. found that the friction coefficients of unlubricated pins of iron powder compacts were 0.35–0.50, while friction coefficients were 0.07– 0.10 when the pins were lubricated by zinc stearate [32]. During compaction, the change in density of the compact was determined by the change in displacement of the upper punch. An extensometer was attached between the upper punch and the die, as shown in Fig. 2, to record the displacement of the punch as a function of pressure p, denoted Upun(p).
Fig. 3. Packing arrangements used in this study. The compacted volume is a rectangular prism.
4326
WU et al.: IDEALIZED CONSOLIDATION—PART 1
The density change during compaction was calculated from the upper punch displacement, with knowledge of the final geometry and mass of the materials in the die. The elastic deformation of the system, Uelsys, was measured while releasing the pressure to zero. The measurement of elastic deformation was subtracted from the upper punch displacement data, proportionally to the pressure, to obtain Upl comp(p), the net displacement of the pressing punch caused by plastic deformation of the compact during compaction. This calculation eliminates the error introduced by the elastic deformation of the punch and compact. Upl comp(p) was used to calculate the density change during compaction, assuming the cross-section area and mass of the compact are constant. The elastic deformation of the punch, Uelpun, was measured by applying and releasing pressure while the die was empty. The springback of the compact was calculated as the difference between the elastic deformation of the compaction system, Uelsys, and the deformation of the pressing punch, Uelpun. 3. ANALYTICAL PROCEDURE
Finite element simulations of the unit cells shown in Fig. 3 were carried out using ABAQUS. Mesh arrangements for the simulations of heterogeneous compaction are shown in Fig. 4. The aluminum side of the interface was finely meshed to increase the accuracy of contact simulation. Element types CPE3 and CPE4R were employed. These are respectively three- and four-noded, elastic–plastic, plane strain solid elements, with hourglass control and reduced integration for CPE4R (one integration point for each element). A quarter of an aluminum rod is meshed with 885 elements and 943 nodes. The steel rods are represented by rigid bounding surfaces. Different meshes were used to study sensitivity.
Fig. 4. Unit cells for FEM simulation. Aluminum rods are meshed with plane strain elements and steel rods are represented by rigid circles. The cell boundaries are rigid.
The Von Mises yield criterion and associated flow theory were used in the simulations, with an isotropic hardening law following the experimental data shown in Fig. 1. The elastic properties of the aluminum were taken from handbook values: Young’s modulus as 62 GPa and Poisson’s ratio as 0.345. An inter-particle friction coefficient of 0.4 was adopted for heterogeneous compaction simulation, and 0.5 for homogeneous compaction. Sensitivity simulations were carried out to investigate the dependence of simulation results on the friction coefficient. The unit cell boundaries are represented by rigid surfaces, with symmetry conditions imposed. Because of the symmetry, cell boundaries are modeled as frictionless. To simulate the punch motion, the top cell surface is moved rigidly downward until a target load is attained, while maintaining the wall positions. During unloading, the top wall is lifted until the contact force drops to zero. For comparison, unloading was also performed by removing the top and lateral walls simultaneously until there were no contact forces from any direction. Relative density equals the ratio of the area occupied by materials to the area of the unit cell. The relative density so defined plus porosity equals unity. The area occupied by aluminum is the sum of all element areas, which can be calculated from the nodal coordinates recorded in the ABAQUS output. The area occupied by steel is always πr2/4, where r is the radius of the steel rods. The unit cell geometry can also be monitored via the ABAQUS output. 4. RESULTS/DISCUSSION
The experimental and simulated pressure–density curves for pure aluminum and aluminum/steel rods are shown in Figs 5 and 6, respectively. Initially the density increases rapidly and approximately linearly with the pressure, then approaches the theoretical limit asymptotically. Comparison of the deformed rod shapes is presented in Fig. 7 for aluminum rods at the punch pressure of 172 MPa. Simulated results are
Fig. 5. Pressure–density relation for the homogeneous compaction of aluminum rods.
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 6. Pressure–density relation for the heterogeneous compaction of aluminum and steel rods.
Fig. 7. Deformed shapes of aluminum rods at 172 MPa. Simulated results are superimposed on the experimental photos.
overlapped on photos from experiments. Triangular voids appear at the particle junctions in each case. The experimental and simulated pressure–density curves and deformed aluminum shapes show very good correspondence. It is essential to use half and quarter-cut rods in experiments to match the periodicity and symmetrical conditions in simulations. Figure 8 compares the pressure–density curves of simulation and experiments in which only six layers of whole rods (no half and quarter-cut rods) were used. The poor agreement is the result of decreased initial density caused by the rods in the top and bottom layers. During compaction,
4327
Fig. 8. Poor agreement of using whole rods only (no half and quarter-cut rods) in aluminum–steel compaction with hexagonal packing.
the contact between the whole steel rod layer and the die punch shielded the pores between them. The choice of the simulation unit cell [Fig. 4(a)] is only effective when the half and quarter-cut rods are placed, as in Fig. 3(a). Friction is difficult to characterize in particle systems. Friction coefficients may not really be constant during a process because of changes in contact geometry, relative sliding and interface temperatures. In order to assess the importance of friction coefficients in these simulations, a sensitivity study was performed. In the case of pure aluminum rods with square packing, no inter-rod sliding takes place during compaction. Hence, friction has no effect. For this reason, only hexagonal packing was used in the friction sensitivity study. Figure 9 shows that the value of the friction coefficient has almost no effect in the homogeneous case, and only a small effect for very small values (⬍0.2) in the composite case. In experiments, inter-particle friction was reduced by adding zinc stearate on the rod surfaces. Figure 10 shows the pressure–density curves of the hexagonal compactions of both dry and lubricated aluminum/steel rods. The same trend was followed both in experiments and simulations. Inter-particle friction only modestly inhibits the densification of the aluminum and steel composite compacts. The pressure distribution at the particle interfaces was carefully examined in simulation. Figure 11(a) shows the ratio of the contact pressure (Pcont) to the current yield stress (sy), for nodes along the contact surface at different punch displacements and pressures (P), for pure aluminum rods with hexagonal packing. The corresponding relative densities (RD) are also shown in the figures. Figure 11(b) shows the same simulation results, but the ratio of contact pressure to initial yield stress (s0y) is shown. Figure 11(c) also shows Pcont/sy for homogeneous compaction, but in this case elastic–ideally plastic behavior with a fixed yield stress of s0y for the deformable phase, and frictionless contact, were assumed. Figure 12 shows
4328
WU et al.: IDEALIZED CONSOLIDATION—PART 1
Fig. 9. Simulated friction effect on the rod interface with hexagonal packing: (a) aluminum rods, (b) aluminum and steel rods.
Fig. 11. Ratio of contact pressure to yield stress for nodes along the contact surface for homogeneous compaction with hexagonal packing at different punch pressures (relative densities): (a) current yield stress, (b) initial yield stress, (c) ideal elastic– plastic and frictionless. Fig. 10. Measured friction effect on the rod interface of aluminum and steel rods with hexagonal packing.
results for heterogeneous composite compaction. In Fig. 12(a), the contact friction coefficient is 0.4 and the material strain-hardens normally. Figure 12(b) shows the behavior with strain-hardening material in frictionless conditions. In Fig. 12(c) an elastic–ideally plastic deformable phase is studied with frictionless contact. The ratios (Pcont/sy and Pcont/s0y) continually
increase as contact area increases (pressure and density increase also). For homogeneous compaction, the mean values of Pcont/sy (具Pcont/sy典), for both strainhardening [Fig. 11(a)] and elastic–ideally plastic [Fig. 11(c)] materials, are very close to 3, the value used by Arzt et al. [19, 20]. The ratio is higher in heterogeneous compacting with frictional contact [Fig. 12(a)], but frictionless contact lowers the normalized contact pressure [Fig. 12(b)]. These results show that the approach of Arzt et al. is inherently approximate,
WU et al.: IDEALIZED CONSOLIDATION—PART 1
4329
Fig. 13. Springback upon unloading for heterogeneous compaction with hexagonal packing: (a) under full load and after removal of punch load (1D) and all contact loads (2D), (b) change in volume due to void re-opening and material compressibility.
Fig. 12. Ratio of contact pressure to current yield stress for nodes along the aluminum side of the contact surface for heterogeneous compaction with hexagonal packing at different punch pressures (relative densities): (a) friction coefficient of 0.4, (b) frictionless, (c) ideal elastic–plastic and frictionless.
as a ratio of interface pressure to yield stress is implicitly assumed. For homogeneous compaction, if current yield stress is used, this may be a reasonable approximation [see Fig. 11(a) and (b)]. Arzt et al.’s approach will considerably underestimate compaction pressure for composite consolidation. Figure 13(a) shows the porosity change under full load and after unloading. The “load” curve shows the amount of porosity predicted under load. Upon unloading, springback will re-open some voids. The
1D-unload curve shows the total porosity after removing only the punch load. The 2D-unload curve shows the porosity after removing both the punch and the lateral-wall loads. The maximum porosity change found in simulation upon unloading is 0.03% in 1Dunload, and 0.17% in 2D-unload. The latter result has more practical interest. Figure 13(b) shows the total volume change upon unloading, and the volume change due to material compressibility and void reopening upon unloading. The result shows that void re-opening contributes more at low pressure and reaches a plateau at high pressure, while the contribution from compressibility increases linearly. This characteristic of springback would not be observed if elasticity was not included in the model used. 5. CONCLUSIONS
The compaction of idealized 2D powders (rods) has been simulated using finite element modeling, and measured using aluminum and steel rods to eliminate uncertainties of particle rearrangement and unknown material properties. The following principal conclusions were reached.
4330
WU et al.: IDEALIZED CONSOLIDATION—PART 1
1. When initial packing density is carefully matched and springback upon die removal is accounted for, finite element simulation and experimental pressure–density curves are in very close agreement over a wide range of pressures and densities (77– 100%). Homogeneous and composite compaction results are equally satisfactory. 2. Measured and simulated particle shapes after compaction are in excellent agreement. 3. The ratio of contact pressure to current yield stress is near 3 for homogeneous compaction. The ratio is considerably higher for heterogeneous compaction. Friction does not affect the ratio in homogeneous compaction, but does in heterogeneous compaction (the ratio increases by a factor of two from frictionless to a friction coefficient of 0.4). 4. The effect of inter-particle friction on simulated pressure–density behavior is negligible for homogeneous powders in a cellular array. The effect is small for composite powders with friction coefficients greater than 0.2, but densification occurs more readily for friction coefficients less than 0.2. 5. Springback upon die removal must be considered for careful comparison of compact densities and volumes. Springback stemming from the re-opening of voids was observed to produce up to 0.5% volume change. 6. Simulation allows the separation of springback effects from those arising from particle compressibility, and from those arising from void re-opening upon load removal. At low pressures (less than 100 MPa), most of the springback is related to void re-opening. As pressure increases, the component resulting from void re-opening reaches a plateau, and the component from compressibility continues to increase. Thus the compressibility effect dominates at high pressure. 7. As expected, the presence of hard particles inhibits the densification of composite compacts as compared with a homogeneous compact of deformable particles. This is because much higher pressure is needed to make the deformed phase conform to the hard particles.
Acknowledgements—This material is based upon work supported by the National Science Foundation under Grant No. DMR-9705558, Dr B. A. MacDonald contract monitor, and the Ohio Supercomputer Center, The Ohio State University (Grant No. PAS 080). The authors would like to thank Christine D. Putnam for proofreading and editing assistance.
REFERENCES 1. Arzt, E., Ashby, M. F. and Easterling, K. E., Metall. Trans. A. 1983, 14A(February), 211. 2. Fischmeister, H. F., Artz, E. and Olsson, L. R., Powder Metall., 1978, 4, 179. 3. Hewitt, R. L., Wallace, W. and Malherbe, M. C., Powder Metall., 1974, 17(33), 1. 4. Thompson, R. A., Ceram. Bull., 1981, 60(2), 237. 5. Briscoe, B. J. and Rough, S. L., Colloids Surf. A: Physicochem. Eng. Aspects. 1998, 137, 103. 6. Armstrong, S., Aesoph, M. D. and Gurson, A. L., paper presented at International Conference and Exhibition on Powder Metallurgy and Particulate Materials, Seattle, WA. 1995, 3, 31. 7. Bocchini, G. F., paper presented at International Conference and Exhibition on Powder Metallurgy and Particulte Materials, Seattle, WA. 1995, 2, 115. 8. Blumenthal, W. R., Sheinberg, H. and Bingert, S. A., MRS Bull., 1997, 22, 29. 9. Glass, S. J. and Ewsuk, K. G., MRS Bull., 1997, 22, 24. 10. Shima, S. and Oyane, M., Int. J. Mech. Sci., 1976, 18, 285. 11. Gurson, A. L., Trans. ASME. J. Eng. Mater. Technol., 1977, 99, 2. 12. Tvergaard, V., Int. J. Fract., 1981, 17(4), 389. 13. Fleck, N. A., Kuhn, L. T. and McMeeking, R. M., J. Mech. Phys. Solids. 1992, 40(5), 1139. 14. Lee, D. N. and Kim, H. S., Powder Metall., 1992, 35(4), 275. 15. Kwon, Y. S., Lee, H. T. and Kim, K. T., Trans. ASME, J. Eng. Mater. Technol., 1997, 119(October), 366. 16. Park, S., Han, H. N., Oh, K. H. and Lee, D. N., Int. J. Mech. Sci., 1999, 41, 121. 17. Tszeng, T. C., in Simulation of Materials Processing: Theory, Methods and Applications. ed. J. Huetink and F. P. T. Baaijens, Balkema, Rotterdam, 1998, p. 391. 18. Williamson, R. L., Knibloe, J. R. and Wright, R. N., Trans. ASME, J. Eng. Mater. Technol., 1992, 114(January), 105. 19. Arzt, E., Acta metall., 1982, 30, 1883. 20. Fischmeister, H. F. and Arzt, E., Powder Metall., 1983, 26(2), 82. 21. Hill, R., in The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1950, p. 254. 22. Helle, A. S., Easterling, K. E. and Ashby, M. F., Acta metall., 1985, 33(12), 2163. 23. Akisanya, A. R., Cocks, A. C. F. and Fleck, N. A., J. Mech. Phys. Solids. 1994, 42(7), 1067. 24. Akisanya, A. R. and Cocks, A. C. F., J. Mech. Phys. Solids. 1995, 43(4), 605. 25. Li, H., Saigal, S. and Wang, P. T., Acta mater., 1996, 44(7), 2591. 26. Delo, D. P. and Piehler, H. R., Acta mater., 1999, 47(9), 2841. 27. Pavanachand, C. and Krishnakumar, R., Acta mater., 1997, 45(4), 1425. 28. Lange, F. F., Atteraas, L. and Zok, F., Acta metall. mater., 1991, 39(2), 209. 29. Turner, C. D. and Ashby, M. F., Acta mater., 1996, 44(11), 4521. 30. Singh, B. N., Powder Metall., 1971, 14(28), 277. 31. ABAQUS/Standard User’s Manual. Vol. 5.8. Hibbitt, Karlsson and Sorensen, Inc, Pawtucket, RI, 1998. 32. Mallender, R. F., Dangerfield, C. J. and Coleman, D. S., Powder Metall., 1974, 17(34), 288.