## Constitutive Model for Metallic Powders with Ductile Particles

Several constitutive models have been applied to the numerical simulation of powder compaction in dies. The most widely used are variations of two types of models: models with an empirical quadratic yield function (Ref 32, 33, 34, 40, and 41) and Cap models (Ref 35, 36, 38, and 39).

The main practical advantages of a model of the first type as developed in Ref 34, 36, 40, and 41 include:

• A reasonable representation of the behavior of metal powders with ductile particles under monotonic loading

• A continuous yield function that facilitates its numerical implementation

• Experimental calibration with a relatively small number of tests

Furthermore, this model has been implemented in the compaction modeling finite element code PCS (Ref 42) that is in use by several parts manufacturers in the United States. The overall strategy presented here can be employed for any of the models described in the section "Constitutive Models for Metal Powder Compaction." The only difference lies in the range of experiments that need to be performed in order to determine any unknown functions in the models.

Model Formulation

The model considered in detail here uses an empirical yield function and fits the general framework presented earlier with assumptions:

• As the powder aggregate is compacted, the particles deform plastically according to the behavior described by classical plasticity with isotropic hardening

• The powder compact displays macroscopic elastic-plastic behavior that is isotropic and independent of strain rate

• The total strain can be decomposed into elastic and plastic components

• The contributions of particle sliding to the overall deformation are negligible

• The state of the powder aggregate is represented by two state variables, the relative density, D, and the yield strength of the powder particle cry

Rigorously, for elastic-plastic models, a slightly different definition of the relative density D is necessary when D is used as a state variable. Earlier D was defined as the ratio of the density of powder to the density of the fully dense material. However, the plastic state of the material should not be affected by the changes in D that are due to elastic deformation. A more appropriate definition of D is the ratio of the density of unloaded powder to the density of the fully dense material, the relaxed relative density of the powder aggregate.

Following the general approach presented earlier, consider an element of a powder compact, which is subjected to a macroscopic stress state and experiences strains En. The strain can be decomposed into macroscopic elastic and plastic components as per Eq 21:

Here assume that the elastic response is isotropic. The elastic constitutive law can then be expressed by Eq 22:

where G and K are functions of D.

An empirical yield function F of the general form of Eq 26, which is a function of the stress state and two state variables, is given by:

where P = - 3 = -£m, is the pressure, a form of the first invariant of the macroscopic stress. The specific form of the yield function is as defined in Eq 34 and 43:

As the powder aggregate densifies and reaches high relative density, its response will approach the incompressible plastic behavior of fully dense metals. Therefore, the function G1(D) is expected to increase monotonically with an increase in relative density and the function G2(D) is expected to decrease monotonically with an increase in relative density.

The yield functions developed by Trasorras et al. (Ref 34) involved functions b(D) and c(D) that are expressible in terms of the functions Gi(D) and G2(D) in the following manner: Gi(D) = c(D) and G2{D) = 3c(D)/2b(D).

In the following development, the yield function is cast in the form used in Ref 34.

The functions b(D) and c(D) are to be determined through experiments. The associated flow at the macroscopic level (from Eq 27) yields:

From Eq 61, the volumetric and deviatoric components of the plastic strain are given by:

Eliminating clA allows

The conservation of mass is expressed in the form of the continuity equation (from Eq 33):

On integration, the continuity equation yields the following result for the evolution of relaxed relative density,

The isotropic hardening law for the powder particle (from Eq 18) is:

The particle hardening h may be constant (isotropic linear hardening), or may be a function of equivalent plastic strain in the particle.

If the energy dissipated by particle sliding is negligible, as assumed earlier, the external and internal plastic work of deformation can be equated (from Eq 37 with G(D) = D):

### Application to Powder Blends

In the previous sections, the constitutive models have assumed that the powder consists of an aggregate of metal particles of a single kind. In powder compaction in dies, the powder blend will contain lubricant, and in the case of ferrous alloys, often graphite and other alloying elements as well. The theories presented here are expected to be valid for powder blends, provided that the additions to the base metal powder occupy a small volume of the total aggregate (typically a few volume percent). That being the case, the models can be used with the constraints that:

• The relative density D be defined as the ratio of the total density of the powder to the pore-free density of the powder to enforce a plastically incompressible behavior when the material reaches the pore-free density, as opposed to the density of the fully dense metal particle.

• The elastic properties, which are strongly dependent on the presence of nonmetallic constituents, be determined experimentally for the specific blend.

• The constitutive functions b(D) and c(D) be experimentally determined for the specific blend.

References cited in this section

32. S. Shima, "A Study of Forming of Metal Powders and Porous Metals," Ph.D. thesis, Kyoto University, 1975

33. Y. Morimoto, T. Hayashi, and T. Takei, Mechanical Behavior of Powders in a Mold with Variable Cross Sections, Int. J. Powder Metall. Powder Technol., Vol 18 (No. 1), 1982, p 129-145

34. J.R.L. Trasorras, S. Armstrong, and T.J. McCabe, Modeling the Compaction of Steel Powder Parts, Advances in Powder Metallurgy & Particulate Materials-1994, Vol 7, American Powder Metallurgy Institute, 1994, p 33-50

35. J. Crawford and P. Lindskog, Constitutive Equations and Their Role in the Modeling of the Cold Pressing Process, Scand. J. Metall., Vol 12, 1983, p 271-281

36. J.R.L. Trasorras, T.M. Krauss, and B.L. Ferguson, Modeling of Powder Compaction Using the Finite

Element Method, Advances in Powder Metallurgy, Vol 1, T. Gasbarre and W.F. Jandeska, Ed., American Powder Metallurgy Institute, 1989, p 85-104

38. H. Chtourou, A. Gakwaya, and M. Guillot, Assessment of the Predictive Capabilities of the Cap Material Model for Simulating Powder Compaction Problems, Advances in Powder Metallurgy & Particulate Materials-1996, Vol 2 (Part 7), Metal Powder Industries Federation, 1996, p 245-255

39. D.T. Gethin, R.W. Lewis, and A.K. Ariffin, Modeling Compaction and Ejection Processes in the Generation of Green Powder Compacts, Net Shape Processing of Powder Materials, 1995 ASME Int. Mechanical Engineering Congress and Exposition, AMD-Vol 216, S. Krishnaswami, R.M. McMeeking, and J.R.L. Trasorras, Ed., The American Society of Mechanical Engineers, 1995, p 27-45

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

41. S. Krishnaswami and J.R.L. Trasorras, Modeling the Compaction of Metallic Powders with Ductile Particles,Simulation of Materials Processing: Theory, Methods and Application, Shen and Dawson, Ed., Balkema, Rotterdam, 1995, p 863-858

42. Powder Compaction Simulation Software (PCS Elite) User's Manual, Concurrent Technologies Corp., Johnstown, PA

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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