Like the flow-stress hardening method, the decoupled flow-stress hardening implementation is a generalization of the hardening behaviors to allow greater flexibility. In differentiating the two, for the decoupled model the rate dependence may be separately specified for the yield and hardening portions of the flow stress. As such, the generic flow-stress definition of

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p\right)=\sigma_y\hat{\sigma}_{\text{y}}\left(\dot{\bar{\varepsilon}}^p\right)+K\left(\bar{\varepsilon}^p\right)\hat{\sigma}_{\text{h}}\left(\dot{\bar{\varepsilon}}^p\right),

is used in which :math:`\hat{\sigma}` are rate multipliers that by default are unity (such that the response is rate independent) with subscripts y and h denoting functions associated with yield and hardening. The isotropic hardening is described by :math:`K\left(\bar{\varepsilon}^p\right)` and :math:`\sigma_y` is the constant initial yield stress. It may also be seen that if the yield and hardening dependencies are the same (:math:`\hat{\sigma}_{\text{y}}=\hat{\sigma}_{\text{h}}`) the decoupled flow stress model reduces to that of the flow stress case and mirrors the general structure of the Johnson-Cook model [:footcite:`mat:ref:johncook`, :footcite:`mat:ref:johncook2`]. 

Given the aforementioned default to rate dependence, the corresponding multiplier need not be specified. A representation for the isotropic hardening, however, *must* be specified and can be defined via linear, power-law, Voce, or user-defined representations. For the user-defined case, an *isotropic* hardening function should be used and it must be highlighted that the interpretation differs from the general user-defined hardening model. *In this case, as the specified function represents the isotropic hardening, it should start from zero -- not yield.* 
 
Although the decoupled flow-stress hardening model defaults to rate independent, a multiplier may be defined. For rate-dependence, either the previously discussed Johnson-Cook or power-law breakdown models or a user-defined multiplier may be used. For the user-defined capability, the multiplier should be input as a strictly positive function of the equivalent plastic strain rate with a value of one in the rate-independent limit.  
