Considering particle morphology in a constitutive model for metal powders compaction
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F 5 j FI 1 (1 2 j )FS
[1]
where FI and FS are the functions associated with the plastic deformation of powder systems of highly irregular shape and spherical shape, respectively. The term j is a state variable to be discussed later. The plastic potential of Eq. [1] is motivated by the postulation that the energy of powder compaction can be mainly dissipated through particle deformation and particle (relative) displacement. The division among these two mechanisms depends on the stress state and particle morphology. In deriving the plastic potential for a power system consisting of highly irregular particles such as the iron powder used by Brown and Abou-Chedid,[1] we found that the calculations based upon the following plastic potential agree very well with the experimental test:[7] FI 5 AJ82 1 B(J1 1 k)2 2 Y 2I 5 0
[2]
T. CALVIN TSZENG, Assistant Professor, is with the Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL 60616. Manuscript submitted February 13, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
where A, B, and k are functions of the relative density (D); sij is the macroscopic stress; and YI is the apparent flow stress of the irregular powder, which is at least a function of the relative density. Further, J1 5 skk 5 3sm, and J 82 5
1 s8 s8 , s8 5 sij 2 smdij 2 ij ij ij
[3]
where dij is the Kronecker delta function. We also observed[7] that the deformation of an assemblage of cohensionless spherical particles, as represented by the gas atomized copper powder, may be described by the Coulomb rule in the cases when the hydrostatic pressure is relatively low. In this respect, the plastic potential is taken to be the approximate of Drucker and Prager:[11] FS 5 (aJ1 1 !J 82 2 z )2 5 0
[4]
where z and a are given in Eq. [6]. Note that a quadratic form is used in Eq. [4] to maintain the correct dimension. In Eq. [1], j is a state variable, which, in general, is a function of the particle morphology and stress state. Similar to the previous study,[7] we assume that the state variable is j 5 j (M, r), where a morphology index, M P (0, 1), is used to prescribe the effect of morphology. We assume that the particle morphology can be quantified in the way that M 5 0 for spherical particles and M 5 1 for highly irregular particles. On the other hand, the effect of stress is represented by the deviatoric ratio, r, which is defined by[7]
r5
! J2 1 p 1 z /3a 3a
[5]
where
z 5 c!1 2 12a2,
3a 5 sin w !1 2 3a2
[6]
The cohesion in shear, c, is related to the cohesion in tension, t, by c5
1 1 sin w t 2 cos w
[7]
where w is the angle of friction. A schematic showing the relation between these variables is given in Figure 1. The specific form of the state variable, j, will be discussed in a later part of this article. It is desirable to derive the relations between the coefficients A, B, and k and the measurable quantities so that the yield loci of a specific powder system can be calculated. In the sequel, we assume that two powder systems of M 5 0 and
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