Ev

given orientation to the plastic strain lJ. The surface for frictionless closed-die compaction, normalized by the hydrostatic stress, P, required to compact the material to the same density is shown in Fig. 9 for '/= 0, where it is compared with the comparable surface for hydrostatic compaction. There are a number of important points to note:

• The surface for closed-die compaction is extended in the direction of loading and contracted in the transverse direction compared to the surface for hydrostatic compaction. Thus, the yield behavior cannot be described in terms of the relative density alone.

• The vertex on the yield surface is at the loading point. Similar results are obtained for crystal plasticity theory applied to fully dense materials.

• The yield surfaces expand in a self-similar manner. Thus, the different components of stress increase in proportion to each other when the straining is proportional.

• It is not possible to obtain a closed form solution for arbitrary loading paths. Thus, the full micromechanical model is required in order to compute the constitutive response.

The predictions of this model are compared with the experimental results of Akisanya et al. (Ref 10), shown in Fig. 10 for hydrostatic and closed die compaction. It is evident that the general form of these surfaces is well represented by this model.

- Closed die

Hydrostatic

11 / A // / A

/ V

Fig. 9 Yield surfaces for isostatic and closed-die compaction predicted by the anisotropic model of Fleck (Ref 17)

Fig. 10 Comparison of Fleck's (Ref 17) model with the experimental results of Akisanya et al. (Ref 10)

So far only the early stages of compaction, when distinct necks exist between the particles, have been discussed. In the latter stages of compaction (D > 0.9), the pores are isolated. Constitutive models employed in this regime have been taken from studies originally concerned with ductile failure. In these models, the pores are assumed to remain spherical. The most widely used model is developed by Gurson (Ref 20), which predicts a yield surface given by:

This expression suffers from the same problem as Eq 31, in that the effects of material and geometric hardening are coupled. Cocks (Ref 21) proposed that a more appropriate form for the constitutive model, which does not differ significantly from that proposed by Gurson (Ref 20), is given by Eq 34, with where

Equations 36, 37, 38, and 39 for the constitutive response again apply, where, now, G(D) = D.

In practice, the pores do not achieve a spherical shape during the latter stages of compaction. Instead, they maintain a cusp shaped profile. Liu et al. (Ref 22) and Akisanya et al. (Ref 23) have demonstrated that bodies with cusp-shaped pores have greater compressibility than bodies with spherical pores at the same density. Qian et al. (Ref 24) have suggested how the influence of the cusp-shaped pores can be incorporated into the constitutive models for creeping materials. They still assume that the body is isotropic macroscopically. Extending their analysis to a perfectly plastic material would simply result in different definitions of Gi(D) and G2(D) in Eq 43.

Under general loading situations, the expectation might be that the pores would become squashed in the main direction of compaction. Ponte Castenada (Ref 25) has examined the effect of oriented ellipsoidal-shaped pores on the constitutive response, but there have not been any studies of the influence of distorted cusp-shaped pores on the material behavior. In all the stage 2 models, it has always been assumed that the contact patches between the distorted particles are able to support tensile tractions and the contacting particles are unable to slide over each other. There has been no attempt to analyze the effect of the strength of these contacts on the constitutive response.

The yield surfaces for stage 1 and stage 2 compaction can have different shapes. In order to ensure a smooth transition, a gradual change from one shape to another is assumed to occur over a given range of relative density. Assuming a linear variation from the yield expression for stage 1 to that for stage 2 over the range D1 to D2 gives:

where Si is the effective stress for stage 1 compaction and £2 is the effective stress during stage 2. In practice, the transition is generally assumed to occur over the range D\ = 0.77 to D2 = 0.9.

Empirical Yield Functions. In the empirical models of powder compaction, the different form of porosity in stage 1 and stage 2 is not considered. Simple continuous expressions for the yield function are developed that satisfy the condition that the material response is incompressible when D = 1. The most widely used expressions are those due to Kuhn and Downey (Ref 6) and Shima and Oyane (Ref 7). These models have the same general structure as those described in the section "Yield Functions from Micromechanical Models." The yield function can again be represented in the form of Eq 34. For simplicity, a quadratic function of the von Mises effective stress, £e, and mean stress, £m, is assumed for £. It is_further assumed that the material response can be described in terms of a single state variable, the relative density, D. Scan then be expressed in the form of Eq 43, where GA{D) and G2(D) are determined experimentally by assuming that the material yield strength <Ty remains constant. Shima and Oyane (Ref 7) determined these quantities by compacting samples in a closed die to a given density, sintering the samples and then performing uniaxial tensile and compressive tests to determine the yield properties. By measuring the uniaxial yield strength and Poisson's ratio, Gj(D) and G2(D) could be determined at the chosen density; and by repeating these tests at different densities, the complete functions Gi(D) and G2(D) could be determined. Their results for copper and iron powders are well approximated by:

The above form has been employed by a number of authors in experimental and computational studies of powder compaction. Kim and Suh (Ref 26) recognized that a material could harden as a result of shear deformation as well as a result of densification. They proposed a modified form of expression for the yield function:

with

Here ftis the effective yield strength of the compact. The evolution law for is assumed to be of the form:

which takes into account geometric and material hardening. Now, from Eq 28 and 51, find:

and the individual plastic strain increments are given by Eq 27. The yield function of Eq 49 was found to give a slightly better fit of the tension/torsion experiments of Kim et al. (Ref 5) on sintered iron compacts discussed earlier than the model of Shima and Oyane (Ref 7).

These models were validated by performing experiments on sintered compacts. The effect of the sintering process is to anneal the particles and to bond them together. As a result, the yield strength is expected to be the same in tension and compression, as observed by Shima and Oyane (Ref 7), and it is not unreasonable for a quadratic function, such as those presented in Eq 43 and 50, to adequately describe the shape of the yield surface. A consequence of this choice of yield function is that it will always predict a negative volumetric strain-rate when the mean stress is negative (i.e., compaction will always occur under net compressive stress states). In experiments on compacted, but not sintered, powders,Brown and Abou-Chedid (Ref 27) and Watson and Wert (Ref 9) observed that the volumetric strains were positive in uniaxial compression.

Based on their experimental observations, Watson and Wert (Ref 9) proposed the adoption of Drucker and Prager's (Ref 28) two part yield surface, which has been employed to model the response of soils. This surface is shown schematically in Fig. 11 and consists of a shear surface and a spherical cap. Watson and Wert (Ref 9) assumed associated flow for both surfaces (i.e., the strain-increment vector is normal to the surface). Thus, dilation occurs for stress states on the shear surface and compaction occurs if the loading point lies on the spherical cap.

Fig. 11 Schematic of Watson and Wert's (Ref 9) two surface Drucker-Prager model

Following Gurson and McCabe (Ref 29), Brown and Abou-Chedid (Ref 27) proposed a modified form of the Cam-Clay model (Ref 30, 31) to model their experiments. In this model the yield surface is represented by Eq 34, with

Brown and Abou-Chedid (Ref 27) did not propose forms for the functions A(D) and B(D). More detailed experiments are required to determine the exact form of these expressions.

The macroscopic effective stress of Eq 53 is similar to the quadratic forms of Eq 43 and 50, but now the elliptic surface is centered on the point (0, Q in £e - £ m space. The quantity C represents the degree of cohesion between the particles. If they are perfectly bonded, C = 0, and the surface reduces to the symmetric expressions of Eq 43 and 50, which is appropriate for sintered materials. If the particles are smooth and the contacts are unable to support a normal tensile stress, then the value of C can be chosen so that the yield surface passes through the origin, as in the micromechanical model of Eq 41 for ri= 0. Then, and Eq 53 becomes

Equation 55 is equivalent to the most common form of Cam-Clay model used to evaluate the response of soils (Ref 31).

These last two models were derived from constitutive laws that have been developed for soils. A wide range of models have been proposed in the literature for granular materials, in which the yield surfaces are allowed to adopt different shapes, as well as being able to translate in stress space, in a similar manner to the kinematic models (Ref 31).

References cited in this section

5. K.T. Kim, J. Suh, and Y.S. Kwon, Plastic Yield of Cold Isostatically Pressed and Sintered Porous Iron under Tension and Torsion, PowderMetall., Vol 33, 1990, p 321-326

6. H.A. Kuhn and C.L. Downey, Material Behavior in Powder Preform Forging, J. Eng. Mater. Technol., 1990, p 41-46

7. S. Shima and M. Oyane, Plasticity Theory for Porous Metals, Int. J. Mech. Sci., Vol 18, 1976, p 285-291

8. S.B. Brown and G.A. Weber, A Constitutive Model for the Compaction of Metal Powders, Modern Developments in Powder Metallurgy, Vol 18-21, 1988, MPIF, p 465-476

9. T.J. Watson and J.A. Wert, On the Development of Constitutive Relations for Metallic Powders, Metall. Trans. A, Vol 24, 1993, p 2071-2081

10. A.R. Akisanya, A.C.F. Cocks, and N.A. Fleck, The Yield Behaviour of Metal Powders (1996), Int. J. Mech. Sci., Vol 39 (No. 12), 1997, p 1315-1324

11. S. Brown and G. Abou-Chedid, Yield Behaviour of Metal Powder Assemblages, J. Mech. Phys. Solids, Vol 42 (No. 3), 1994, p 383-399

12. W. Prager, Proc. Inst. Mech. Eng., Vol 169, 1955, p 41

13. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, 1950

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

15. C.J. Yu, R.J. Henry, T. Prucher, S. Parthasarathi, and J. Jo, Advances in Powder Metallurgy & Particulate Materials, Vol 6, Metal Powder Industries Federation, 1992, p 319-332

16. N.A. Fleck, L.T. Kuhn, and R.M. McMeeking, Yielding of Metal Powder Bonded by Isolated Contacts, J. Mech. Phys. Solids, Vol 40, 1992, p 1139-1162

17. N.A. Fleck, On the Cold Compaction of Powders, J. Mech Phys. Solids, Vol 43 (No. 9), 1995, p 1409-1431

19. R.M. Govindarajan and N. Aravas, Deformation Processing of Metal Powders, Part 1: Cold Isostatic Pressing, Int. J. Mech. Sci., Vol 36, 1994, p 343-357

20. A.L. Gurson, Continuum Theory of Ductile Rupture by Void Nucleation and Growth, Part 1: Yield Criteria and Flow Rules for Porous Ductile Media, J. Eng. Mater. Technol., Vol 99, 1977, p 2-15

21. A.C.F. Cocks, The Inelastic Deformation of Porous Materials, J. Mech. Phys. Solids, Vol 37 (No. 6), 1989, p 693-715

22. Y-M. Liu, H.N.G. Wadley, and J. Duva, Densification of Porous Materials by Power-Law Creep, Acta Metall. Mater., Vol 42, 1994, p 2247-2260

23. A.R. Akisanya, A.C.F. Cocks, and N.A. Fleck, Hydrostatic Compaction of Cylindrical Particles, J. Mech. Phys. Solids, Vol 42 (No. 7), 1994, p 1067-1085

24. Z. Qian, J.M. Duva, and H.N.G. Wadley, Pore Shape Effects during Consolidation Processing, Acta Metall. Mater., Vol 44, 1996, p 4815

25. P. Ponté Castañeda and M. Zaidman, Constitutive Models for Porous Materials with Evolving Microstructure, J. Mech. Phys. Solids, Vol 42, 1994, p 1459-1497

26. K.T. Kim and J. Suh, Elastic-Plastic Strain Hardening Response of Porous Metals, Int. J. Eng. Sci., Vol 27, 1989, p 767-778

27. S. Brown and G. Abou-Chedid, Appropriate Yield Functions for Powder Compacts (1992), Scr. Metall. Mater., Vol 28, 1993, p 11-16

28. D C. Drucker and W. Prager, Q. Appl. Math., Vol 10, 1952, p 157-165

29. A.L. Gurson and T.J. McCabe, Experimental Determination of Yield Functions for Compaction of Blended Powders, Proc. MPIF/APMI World Cong., on Powder Metallurgy and Particulate Materials (San Francisco), Metal Powder Industries Federation, 1992

30. A. Schofield and CP. Wroth, Critical State Soil Mechanics, McGraw-Hill, 1968

31. D.M. Wood, Soil Behavior and Critical State Soil Mechanics, Cambridge University Press, 1990

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

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