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 fry

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 j and experiences strains Ej. 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 kk=- 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 G^D) and G2(D) in the following manner: G^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):

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