## Cry try te

where h{rry) 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 (d<rv)(()fl()Ev) > 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.

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