Deformation of Powder Compacts Experimental Observations

Before examining the structure of constitutive laws for powder compaction, it is instructive to identify the major physical processes that occur within the compact as it is deformed and to determine how these are likely to influence the macroscopic behavior. When the powder is initially poured into a die or mold, the particles are arranged randomly with particles only in point contact with each other.

Consider the situation where the compact is densified hydrostatically. As the pressure is increased, plastic flow occurs in the vicinity of the contacts. As a result, the contact zones spread, the centers of the particles move closer together, and the material densifies. During the initial stages of this process, the porosity remains connected, consisting of a network of interconnected channels threading through the material. In the micromechanical models, this is referred to as stage 1 compaction. As the material is densified further, the channels pinch off, leaving a distribution of isolated pores (stage 2). Throughout this process, because of the initial random structure, the porosity is randomly distributed throughout the material, particularly during stage 1 compaction. If, however, the sample is much larger than the mean particle size, no preferential orientations or distributions of pores develop, and the structure remains macroscopically isotropic. A single state variable can then be used to describe the structure and macroscopic response. A convenient state variable is the relative density D, which is defined as the ratio of the density of the compact to the density of the material, so that at full density D = 1.

In practical compaction processes, the densifying material can see complex stress states and histories. As a result, the structure that develops is no longer isotropic, and it may not be appropriate to describe this structure and the macroscopic response using a single state variable. For example, consider the case of frictionless closed-die compaction. As a cylinder is compacted along its axis, there is no straining in the transverse direction. The contact patches that develop between the deforming particles are therefore larger normal to the direction of loading than along the axis. Also, the size of these patches is different from those in a compact that has been densified hydrostatically to the same relative density. Thus, the relative density does not uniquely describe either the microstructure or the macroscopic response.

Any constitutive law that is developed needs to be validated experimentally. An examination of the types of experiments that have been performed and how the results from these tests are commonly presented follows. The analysis is based on the identification of a yield surface for the material (which is a surface in stress space within which the material responds elastically; plastic deformation can only occur for stress states on the yield surface). The size and shape of the yield surface is a function of the history of loading. The yield surface can be determined using a number of different methods. A number of samples can be prepared in the same way (for example, hydrostatically compacted to the same relative density). Each sample is then loaded along a different path in stress space until it yields. The yield surface is then formed by connecting the different yield points. Alternatively, a single specimen can be used. After compacting to the desired state, a series of probing experiments can be performed by unloading and reloading along a range of different stress paths until yield occurs. The yield surface can be mapped out in exactly the same way as before. When using this method, it is important to ensure that only a small amount of plastic straining occurs during each probe so that there is no significant change of microstructure (i.e., state) over the series of probing tests.

When examining a range of different stress states, it proves convenient to present the results in terms of global measures of stress. Two convenient quantities are the von Mises effective stress, Se, and the mean (or hydrostatic) stress, Em. If £ i, £2, and £3 are the principal stresses,

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.

• Triaxial 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 Br 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 Se = |EZ - and P = + £z).

• 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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Fig. 2 Yield surfaces determined by Kim et al. (Ref 5) from tension/torsion tests on sintered iron powders

Akisanya et al. (Ref 10) used the probing method to determine yield surfaces for samples of spherical powders, which were either hydrostatically compacted or compacted by simulating closed-die compaction. These tests were performed using a triaxial cell in which hydrostatic pressure and uniaxial compression could be applied independently. Yield surfaces obtained for a relative density of 0.8 are shown in Fig. 3 using the axes of Ss = Sz - and Sm. Note that in these experiments S, > Sz and thus £s is negative with £s = -Ee. It is important to note that the shape of the yield surfaces obtained by compacting the samples using these two different methods are significantly different from each other. Thus, the states of the materials are different even though the densities are the same.

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Fig. 3 Yield surface for spherical copper powders determined by Akisanya et al. (Ref 10) for isostatically pressed and close-die compacted samples

Another convenient way of presenting experimental data is in the form of isodensity plots. These plots are constructed by loading samples proportionally (either by maintaining the stresses or the strains proportional to each other) and then determining the stress states that correspond to a given density. If the material response can be described using the relative density as the only state variable, then these surfaces would be equivalent to the yield surfaces of the material. It is evident from the results of Fig. 3 that more than one state variable is required to describe the response. However, in many practical situations, an element of material experiences near proportional loading and isodensity or similar plots provide valuable information about the constitutive behavior. Brown and Abou-Chedid (Ref 11) constructed a series of isodensity plots in - ¿Lm space for spherical copper and sponge iron (Hoeganaes MH-100) powders (Fig. 4, 5). The solid symbols in these figures represent the yield strength in uniaxial compression following biaxial compression to the desired density. The major observations here are the different forms of the surface for the different powders, and the fact that the spherical powder has limited strength in uniaxial compression. These conclusions are consistent with the experiments of Akisanya et al. (Ref 10) presented in Fig. 3. The irregular powders, however, exhibit significant strength in uniaxial compression. Also, the irregular powders tested by Watson and Wert (Ref 9) had a significant yield strength when loaded in tension, while samples of spherical powder compacts can readily be broken in tension, often by hand, due to the lack of interlocking of the particles. These plots will be discussed and evaluated in the following sections. There are however, three important points to note at this stage:

• It is possible to identify a yield surface for a powder compact.

• The state of the material is a function of the prior history of loading. This influences the size and shape of the yield surface.

• The yield behavior depends on the morphology of the powder particles. Irregular particles interlock, giving tensile strength, while compacts of smooth spherical particles can be readily pulled apart.

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Fig. 4 Isodensity surfaces for spherical copper powders tested by Brown and Abou-Chedid (Ref 11)

Fig. 4 Isodensity surfaces for spherical copper powders tested by Brown and Abou-Chedid (Ref 11)

Fig. 5 Isodensity surfaces for irregular iron powders tested by Brown and Abou-Chedid (Ref 11)

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