The Structure of Constitutive Laws for Powder Material

Consider an element of a powder compact (Fig. 7), subjected to a macroscopic stress state and experiencing strains Eij. The strain can be decomposed into macroscopic elastic and plastic components, as before:

Assume that the elastic response is isotropic. The elastic constitutive law can then be expressed in the form of Eq 5:

where G and K are functions of the state of the material. Several techniques for determining the elastic moduli G and K are summarized in Table 1. It is sometimes more practical to measure Young's modulus E instead of shear modulus G. Shear modulus can be determined using G = 3KE/9(K - E).

Table 1 Experimental techniques for the determination of elastic properties of powder compacts







Triaxial test unloading


Can determine E at low density (D is "^0.65 for atomized steel powders). Experimental apparatus is involved.

Ref 14

Resonant frequency


Very accurate and repeatable. Simple experimental setup. Simple sample preparation. Commercial laboratories available that use this technique. Minimum density D is "~0.75 for atomized steel powders.

Ref 15



Higher scatter than resonant frequency. Able to determine anisotropic properties in a transverse rupture bar.

Ref 15

Hydrostatic test unloading


Can determine A'at low density (D is ^0.65 for atomized steel powders). Experimentally difficult at very high density.

Ref 14

Fig. 7 Macroscopic element of powder compact subjected to stresses £(J

The variables E and K can be measured during the unloading of a triaxial compaction test and a hydrostatic compaction test, respectively. Another technique used to determine E is resonant frequency. The powder is compacted into a beam, typically a transverse rupture bar. The beam is excited through an impact load, and the natural vibration frequency is measured. This frequency can be used to compute E. Ultrasound has also been applied to the determination of E in compacted specimens.

Watson and Wert (Ref 9) measured the variation of G and K as a function of D for samples which had been compacted in closed die. They found that:


and Gm and Vm representing the shear modulus and Lamé constant for the fully dense material, respectively.

If a microscopic element deforms plastically according to any of the models described, and there are no other dissipative mechanisms operating, then it is possible to identify a yield surface in stress space:

which is a function of stress and the state of the material, described in terms of a number of state variables, <Sds. Plastic flow can only occur if the stress state is on the yield surface. It can further be shown that associated flow at the microlevel guarantees associated flow at the macroscopic level, thus:

where A is the macroscopic plastic multiplier.

If frictional sliding contributed to the internal dissipative processes, then it is still possible to identify a yield surface, but associated plastic flow cannot be guaranteed. Fleck (Ref 16) has examined the two extremes of frictional behavior: free sliding; and sticking, whereby sliding can only occur by shear yielding in the vicinity of a contact. In each case, there is no frictional dissipation and associated flow can be guaranteed. The yield surface for intermediate frictional conditions should lie between the yield surfaces for these two situations. Fleck (Ref 16) found that there is only a small difference between the surfaces for a given assumed state, indicating that only a small amount of energy is dissipated by shearing between the particles. Thus, even if frictional sliding occurs it will only have a small influence on the macroscopic response, and it is appropriate to assume that an associated flow rule is valid.

During compaction the stress point remains on the yield surface, therefore, following Eq 19:

In order to complete the model, evolution laws for the state variables in terms of the stress, strain, and the state of the material are needed. In the models, the state could be described in terms of suitable measures of strain, or alternatively, plastic strains can be considered the state variables. Then Eq 28 becomes:

3F dF

3Ltl dSu

Combining Eq 29 with Eq 27 gives


with the individual strain components given by (Eq 27).

To simplify constitutive models, isotropic behavior is often assumed. This consideration dictates the use of mechanical and material descriptions independent of a particular choice of coordinate system. This freedom allows one to use rotation invariant or coordinate system independent quantities in the constitutive model. For the mechanical stress or strain, which are mathematically represented as tensors of the second order, these quantities are the first and second tensor invariants.

The yield function can then be expressed as a function of

= yS/2SijSij the macroscopic von Mises effective stress in the powder aggregate, a form of the second invariant of the macroscopic stress and S m = 3 the mean stress in the powder aggregate, a form of the first invariant of the macroscopic stress. Equation 26 then assumes the form: F =

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 (Ty 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,

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