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If appropriate forms for the yield function in terms of the stress and accumulated plastic strain can be found, the above equations can be employed to determine the full constitutive response. In the following sections, different possible forms of the yield function are examined.

Yield Functions from Micromechanical Models. The section "Deformation of Powder Compacts: Experimental Observations" states the nature of the porosity changes as a material densifies. Initially, the porosity is open and distinct necks exist between the contacting particles (stage 1). At high relative densities (D > 0.9), the porosity is closed, which is referred to as stage 2. Different forms of micromechanical models have been developed for these two stages. In the stage 1 models, it is assumed that there is no interaction between the deformation zones that form in the vicinity of the different contacts. Analyses of the contact of two isolated bodies can then be used to obtain the appropriate contact law for a given microplasticity model. The macroscopic response is then determined by combining the contributions from each contact. Fleck et al. (Ref 16) employed a perfectly plastic microplasticity model and assumed that the material was isotropic and could be described using a single state variable, which they took to be the relative density, D. This assumption is equivalent to assuming that the material had previously been compacted isostatically to a given density. From their analysis, Fleck et al. (Ref 16) propose an approximate yield function:

where Py is the yield strength in hydrostatic compression;

where D0 is the initial dense random packing density, which is generally taken to be 0.64.

The full response can be obtained from Eq 26 to Eq 30 by noting that the densification rate is given by (neglecting elastic volume changes):

In practice, the matrix material can strain harden as the contact zones deform plastically. Fleck et al. (Ref 16) demonstrate how the effects of strain hardening can be taken into account. A more compact form of model can be obtained by rewriting Eq 31 in the form of Eq 7, that is,

where Sis an effective stress for the porous material, which is a function of the macroscopic stress and internal geometry described by D. Rearranging Eq 31 then gives:

For loading conditions on the yield surface, Eq 29 becomes:

Fleck et al. (Ref 16) take &y to be the average yield strength in the plastically deforming material, and they define an fJcP

average internal effective strain increment, e. by equating the internal and external work. Using the current terminology,

where G(D) is the effective volume fraction of plastically deforming material, given by Fleck (Ref 17) as:

Combining Eq 36, 37, 33, and 18 gives:

where dE is a macroscopic effective plastic strain increment not to be confused with the von Mises effective strain increment, dEe. The advantage of this form is that it separates out the effects of geometric and material hardening and it leads to a more compact expression than the relationships proposed by Fleck et al. (Ref 16).

In the development of the above model, two major assumptions were made: the strengths of the contact zones were assumed to be the same in tension and compression, and the material was assumed to be isotropic. It was demonstrated in the section "Deformation of Powder Compacts: Experimental Observations" that the strengths of the contacts are likely to be different in compression and tension, particularly for smooth particles. Also, when a compact is loaded along stress paths other than hydrostatic, the microstructure (the distribution of necks) becomes anisotropic.

Fleck (Ref 17) has proposed a model in which the tensile strength of a contact patch is less than the compressive strength by a factor '/. If Py (Eq 33) is the magnitude of the pressure required to plastically deform a compact, then the magnitude of the hydrostatic tensile stress required to initiate plastic flow is '¡Py. Fitting a quadratic to Fleck's results and adopting the form of expression presented in Eq 34, the following equation is obtained:

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