Eq 2

More general definitions of these stresses are provided in the following section.

Alternatively, instead of using £m, the material response can be presented in terms of the hydrostatic component of stress (or pressure) P = -£m. These definitions are used interchangeably throughout the article.

Many of the experimental methods used today to evaluate the behavior of metal powders have been inspired by techniques used to test soils. Most published data has been generated using such prodedures. Before examining the material data in detail, it proves instructive to first examine the general features of the most common testing methods.

It is convenient to represent the yield function, as well as the stress paths during testing, in the P - plane. In this representation, the closer a point is to the ordinate (£e axis), the more shear experienced in the material; the closer it is to the abscissa (P axis), the more hydrostatic pressure experienced. The typical powder compaction process involves stress states that have a high-pressure component, a natural consequence of the confinement of the powder in a die cavity.

Figure 1 shows a schematic representation of the load paths corresponding to the different test procedures:

Simple shear test: This is marked by a complete absence of any hydrostatic load. This test has little value in the context of loose powders and can only be applied to sintered powders or porous materials. Simple compression test: This is once again a shear-dominated test, with friction at the interface of the powder and dies assumed absent. As with the previous case, this test can realistically only be used for sintered powders or porous materials.

• Tri axial test: The primary advantage of this test is that a variety of stress paths can be examined through a combination of compressive stresses in axial S7 and radial £r directions. A detailed description of this test is provided in the section "Experimental Determination of Powder Material Constitutive Properties and Functions" in this article. The effective and hydrostatic components of stress for this loading situation are given by Ee = |SZ - Sr| andi5 = 3(2Sy + Ez).

• Closed die compaction: The state of stress in the powder in this test is possibly the best description of the actual stress state in a typical compaction. This test does not lend itself to parameter extraction procedures because friction is present at the die walls. It is possible, however, to simulate frictionless closed die compaction in a triaxial cell.

• Hydrostatic compaction: The state of stress during this test is one of pressure alone with shear components completely absent. This test is easy to set up and provides the compressibility of the material.

Fig. 1 Schematic representation of the load paths corresponding to the different test procedures. Source: Ref 31

All the cases illustrated in Fig. 1 involve the testing of cylindrical specimens. The inherent symmetry in the shape and loading can be exploited to simplify the analytical treatment required for the extraction of material parameters.

It should be noted that results from a particular testing procedure might be used to extract material parameters for different constitutive models. Simplifying assumptions are an essential ingredient in such determinations. For instance, friction is inevitably present in most testing procedures. However, it often becomes necessary to ignore the effects of friction in the extraction procedure, because not doing so may seriously impair the ability to obtain reasonable values for material parameters.

Kim et al. (Ref 5) cold isostatically pressed tubes of iron powder (Hoeganaes ASC 100-29) to relative densities in the range D = 0.8 to 0.85. These tubes were then sintered at 1150 °C for 1 h in a hydrogen atmosphere. The tubes were then tested in combined tension, S. and torsion, T. Typical yield surfaces obtained from these experiments are shown in Fig. 2, where the results are normalized with respect to the yield strength of the fully dense material, (Ty. The sintering process has bonded the particles, imparting significant tensile strength to the compact. A number of studies for determining the mechanical properties of powder compacts employ the same type of procedure. For example, the experimental studies of Kuhn and Downey (Ref 6) and Shima and Oyane (Ref 7), which form the basis for a widely used constitutive law for powder compaction, were performed on sintered compacts. Brown and Weber (Ref 8) demonstrated that the mechanical response of compacted and then sintered powders is very different from that of identical powders that have only been compacted. Watson and Wert (Ref 9) compacted aluminum powders hydrostatically and by using closed-die compaction. They performed uniaxial tensile and compressive tests on these samples to construct the yield surface.

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