4 of

Thus,

tip P

where iL 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 :

that is,

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