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Thus, where ¡' is an effective strain increment. The volumetric component of the strain increment at yield is:

which is consistent with the initial assumption of incompressibility.

This manipulation of equations is standard, but there are some important points to note:

• The effective stress, which drives the deformation, is only a function of the deviatoric stresses (i.e., it is independent of the magnitude of the mean stress).

• The strain increments at yield are proportional to the deviatoric components of stress.

• The deviatoric component of stress, and therefore the effective stress, drives changes of shape of an element.

• Volume changes are driven by the mean stress.

In the above description the magnitude of the strain increment at yield has not been determined; only the relative magnitudes of the strain components have been determined. To determine the magnitude, the evolution of the state of the material with time must be understood. In Eq 7, the state is represented in terms of the yield strength of the material. Use of a single state variable of this type implies isotropic behavior. Below, two particular forms of isotropic models are examined.

Perfect Plasticity. The simplest assumption is that the yield strength remains constant, in other words, the yield dEv surface retains the same size and shape in stress space. The plastic strain increment lJcan then have any value. In a structural problem, the exact value of the strain increment is determined either from the full elastoplastic response of the body or from any resulting geometry changes.

Isotropic Hardening. As the material is deformed plastically, the yield stress can increase. Consider the situation

where yield strength is a function of the accumulated plastic strain c e :

o that is,

where h(/ry) is the slope of a plot of yield strength against plastic strain in a standard uniaxial material test.

Plastic flow can only occur if the stress state is on the yield surface, in other words, plastic flow only occurs as the stress is increased from (7,; to <Ju + defy if/= 0 throughout the increment and (dCTv)(()f()£v) > 0. Then, differentiating (Eq 7) gives:

Combining Eq 18 and 19 gives

The individual strain increments can be obtained by combining Eq 20 with Eq 12 and 14.

In the above model, it is assumed that the material remains isotropic as it deforms plastically, so that the yield surface maintains the same shape and simply changes in size as plastic strain is accumulated. In practice, the material response can become anisotropic. One means of representing anisotropy is through the use of a kinematic hardening model, wherein the center of the yield surface in stress space is allowed to translate according to some rule. See Ref 12 for further information. As a different approach towards anisotropy, Hill (Ref 13) presents a generalized theory of plasticity that allows for a change in the shape of the yield surface, while the basic requirements of convexity and incompressibility are maintained.

There are many other types of constitutive laws that have been developed to model the response of plastically deforming materials. As in the models described here, these alternatives mainly consist of identifying a yield surface and developing appropriate laws for the expansion, translation, and change in shape of the surface as the material deforms plastically. These laws can contain many state variables, together with complex evolution laws. The choice of constitutive law depends on the type of loading histories being evaluated. If the stress history experienced by the material point is simple, then relatively simple constitutive laws adequately describe the material response. For example, for monotonic proportional loading, the kinematic and isotropic hardening models predict the same response provided they are fit to the same base data. A detailed knowledge of the shape and orientation of the entire yield surface and how these evolve with time is only required if complex loading histories are being considered. It should also be remembered that the more complex the constitutive law, the wider the range of tests that need to be devised to determine all the material parameters and the evolution equations for each of the state variables. When selecting the structure of a constitutive law, it is therefore important to first determine the type of stress history that each material point is likely to experience and to select the equations capable of capturing the major feature of the response.

In the following section, assume that the individual particles of a compact deform according to the constitutive models described in this section. This understanding provides a structure for the development of the constitutive laws for the compact.

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).

 Experimental Elastic Comments Reference technique property Triaxial test unloading E(D) Can determine E at low density (D is —0.65 for atomized steel powders). Experimental apparatus is involved. Ref 14 Resonant frequency E(D) 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 Ultrasound E(D) Higher scatter than resonant frequency. Able to determine anisotropic properties in a transverse rupture bar. Ref 15 Hydrostatic test unloading K(D) 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:

with 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, S(*. 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

0FidEËKdFidLtù

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 macroscopic von Mises effective stress in the powder aggregate, a form of the second invariant of the macroscopic stress and = 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 =

0 0