## Elastoplastic Constitutive Behavior

The constitutive behavior is discussed within the framework of small deformation theory of plasticity, which allows the decomposition of the total strain into elastic and plastic components and development of relatively simple expressions for the constitutive response. In practice, large plastic straining of a body can occur during the compaction process. The general forms of expression are still applicable, provided appropriate measures of stress, strain, stress-rate, and deformation-rate for these problems are defined. At this stage, it is not necessary to precisely determine what these definitions are. All necessary information can be determined by making the simple assumption of small strains. Start by considering the response of a microscopic continuum element of material (i.e., the response of the individual particles that form the compact).

Assume that the total strain, £u. a body experiences can be decomposed into elastic and plastic components, such that:

Where appropriate, the index notation for tensorial quantities is used to represent the constitutive response. The indices /' and j can have a value in the range of 1 to 3. A repeating suffix implies summation over this range (thus, £,y = £n + £22 + £33).

Elastic Behavior. The elastic strains depend on the current stress. For a linear elastic material,

Sh=CimaM (Eq4)

where Cijkl is the elastic compliance matrix. If the material response is isotropic, it is more convenient to decompose the strain into deviatoric, v3 , and volumetric, c v, components, such that:

where G is the shear modulus, K is the bulk modulus, .v,; = au - 3 O//J,( = u,} - <J,Jhj are the deviatoric components of stress, £Tm = 5 cf 1,1, is the mean stress, and Su is the Kronecker delta function (J,,. = 0 for i 5u = 1 for i = /). For a fully dense material, G and K are material constants. If the material is porous, both these moduli can be a function of the relative density of the material. This point is revisited later in the article in an examination of the constitutive response of powder compacts.

Classical Plasticity for Incompressible Materials. Central to the development of constitutive laws for plastically deforming materials is the concept of a yield surface. Here a more precise definition of ways in which the changing state of the material can be modeled is presented. The discussion is limited to classical plasticity concepts for fully dense materials, where plastic flow occurs at constant volume. Porous bodies are presented in subsequent sections.

A yield surface is shown schematically in Fig. 6 in principal plane stress space, in which there are two measures of stress <71 and cr2. Mathematically, this surface can be defined by the equation:

where /(C,;) is a function of stress. For example, for an isotropic incompressible material, a suitable form of this expression is:

where G is the yield strength of the material in uniaxial tension and fre is the von Mises effective stress, given by:

In principal plane-stress space, cr, - Vcr^H- C; - (T|a:

Fig. 6 Schematic of yield surface in plane-stress space, showing strain increment vector normal to the yield surface

Equation 9 then represents the von Mises ellipse shown in Fig. 6. Iff < 0, the material response is elastic, and iff = 0 (i.e., if the stress state is on the yield surface), the material is able to deform plastically. The increments of plastic strain at yield are related through the associated flow rule:

where Ais a plastic multiplier. This equation simply states that the strain increment vector is normal to the yield surface in stress space. Additional information is needed to determine the exact value of A.

For the yield condition of Eq 7,

Now,

Also,

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