<CRYSTAL_KINEMATIC>#
Description#
Because the crystal potential models takes into account the strains on the individual slip planes, kinematic hardening represents a linear measure of a slip offset distance \(X_i\), or backstress, on the slip line. As described in the <POTENTIAL> crystal section, the yield criterion on each slip plane \(i\) is of the form \(\vert \tau_i - \sum_{j} X_i^j\vert - r_i\), where \(\tau_i\) is the resolved shear stress on slip plane \(i\) and \(r_i\) is the sum of all isotropic hardenings on that slip plane.
The internal variables \(\alpha_i\), to which the backstresses are
associated, are thus scalar and distinct from the tensorial
kinematic hardening laws in the macroscopic potentials. They are related
to the backstresses \(X_i\) through
\(X_i = {\tt x0} + {\tt C} \alpha_i\) with x0 and C
parameters having the dimensions of a stress. Trick: a large negative
offset x0 might be used to model unidirectional slip, provided
that the same positive offset is added to the corresponding isotropic
hardening.
These hardening laws will apply uniquely for the single crystal potentials. Permissible crystalline kinematic types are with their additional parameters are
linear\(\dot{\alpha}_i = \dot{\gamma}\)
nonlinearNonlinear kinematic hardening with dynamic and static recoveries:
\(\dot{\alpha}_i= \dot{\gamma} - {\tt D}X_i\left|\dot{\gamma}\right|/{\tt C}- \left(\dfrac{|X_i|}{\tt M}\right)^{\tt m} {\rm sign}(X_i)\)
where \(\tt D\), \(\tt m\) and \(\tt M\) are material parameters.
nonlinear_phi- \[\dot{\alpha}_i= \dot{\gamma}\phi (v_i) - {\tt D}\left(X_i-{\tt Xbar}\right)\left|\dot{\gamma}\right|/{\tt C}\]
with
\[\phi(v_i) = (1-{\tt phi}) + {\tt phi}~e^{-{\tt delta}~v_i}\]