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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 cl I7!- 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:

This expression reduces to Eq 35 when '/= 1. When '/= 0, it becomes:

The yield surfaces are plotted in Fig. 8 for '/= 0 and 1, where they are compared with the surfaces obtained by Fleck (Ref 17). It should be noted that these surfaces have vertices where they meet the Em axis. The direction of the strain increment vector is nonunique for pure hydrostatic stress states, which can lead to computational difficulties when implementing this model. In practice, this nonuniqueness has been bypassed by inserting a circular arc at the vertex to ensure a smooth continuous yield surface (Ref 19).

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