The description about the different work hardening rules: isotropic, kinematics and combined, are given below.

 

 

Isotropic Hardening

 

The isotropic hardening assumes that the center of the yield surface remains stationary in the stress space, but that the size of the wield surface expands, due to work hardening. The change of the von Mises yield surface is plotted ahead:

 

 

 

A review of the load path of a uniaxial test that involves both the loading and unloading of a specimen will assist in describing the isotropic work hardening rule. The specimen is first loaded from stress free (point 0) to initial yield at point 1, as shown in the next scheme:

 

 

 

 

It is then continuously loaded to point 2. Then, unloading from 2 to 3 following the elastic slope E (Young’s modulus) and then elastic reloading from 3 to 2 takes place. Finally, the specimen is plastically loaded again from 2 to 4 and elastically unloaded from 4 to 5. Reverse plastic loading occurs between 5 and 6.

 

It is obvious that the stress at 1 is equal to the initial yield stress and stresses at points 2 and 4 are larger than , due to work hardening. During unloading, the stress state can remain elastic (for example, point 3), or it can reach a subsequent (reversed) yield point (for example, point 5). The isotropic work hardening rule states that the reverse yield occurs at current stress level in the reversed direction. Let  be the stress level at point 4. Then, the reverse yield can only take place at a stress level of   (point 5).

 

The isotropic work hardening model (with a work slope of 0) is the default option in CivilFEM. For many materials, the isotropic workhardening model is inaccurate if unloading occurs (as in cyclic loading problems). For these problems, the kinematic hardening model or the combined hardening model represents the material better.

 

 

Kinematics Hardening

 

Under the kinematic hardening rule, the von Mises yield surface does not change in size or shape, but the center of the yield surface can move in stress space, as shown in the next figure:

 

 

 

Prager’s law is used to define the translation of the yield surface in the stress space. On the other hand, the loading path of a uniaxial test is illustrated ahead:

 

 

The specimen is loaded in the following order: from stress free (point 0) to initial yield (point 1), 2 (loading), 3 (unloading), 2 (reloading), 4 (loading), 5 and 6 (unloading).  As in isotropic hardening, stress at 1 is equal to the initial yield stress , and stresses at 2 and 4 are higher than , due to work hardening. Point 3 is elastic, and reverse yield takes place at point 5. Under the kinematic hardening rule, the reverse yield occurs at the level of , rather than at the stress level of .

 

Similarly, if the specimen is loaded to a higher stress level (point 7), and then unloaded to the subsequent yield point 8, the stress at point 8 is . If the specimen is unloaded from a (tensile) stress state (such as point 4 and 7), the reverse yield can occur at a stress state in either the reverse (point 5) or the same (point 8) direction.

 

For many materials, the kinematics hardening model gives a better representation of loading/unloading behavior than the isotropic hardening model. For cyclic loading, however, the kinematic hardening model can represent neither cyclic hardening nor cyclic softening.

 

 

Combined Hardening

 

The figure ahead shows a material with highly nonlinear hardening. the initial hardening is assumed to be almost entirely isotropic, but after some plastic straining, the elastic range attains an essentially constant value (that is, pure kinematics hardening).

 

 

The basic assumption of the combined hardening model is that such behavior is reasonably approximated by a classical constant kinematics hardening constraint, with the superposition of initial isotropic hardening. The isotropic hardening rate eventually decays to zero as a function of the equivalent plastic strain measured by:

 

 

This implies a constant shift of the center of the elastic domain, with a growth of elastic domain around this center until pure kinematics hardening is attained. In this model, there is a variable proportion between the isotropic and kinematics contributions that depends on the extent of plastic deformation (as measured by ).

 

Exits two type of formulation. One of them occurs whenever plasticity is not included as a parameter. The work hardening data at small strains governs the isotropic behavior, and the data at large strains governs the kinematics hardening behavior. If the last work hardening slope is zero, the behavior is the same as the isotropic hardening model. The second type of formulation, which includes the plasticity parameter, allows greater generality by introducing the fractional contribution to kinematics hardening as a user input.

 

As the next three figures show, f is the value between 0 and 1:

 

 

 

 

Combined hardening model utilizing kinematics fraction factor is available for von Mises, Hill and Barlat models in isotropic, orthotropic, and anisotropic options.