Unlike the previously described models, the flow-stress hardening method is less a specific physical representation and more a generalization of the hardening behaviors to allow greater flexibility in separately describing isotropic hardening and rate-dependence. As such, the generic flow-stress definition of 

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p\right)=\tilde{\sigma}_y\left(\bar{\varepsilon}^p\right) \hat{\sigma}\left(\dot{\bar{\varepsilon}}^p\right),

is used in which :math:`\hat{\sigma}` is the rate multiplier that by default is unity (such that the response is rate independent) and :math:`\tilde{\sigma}_y` is the isotropic hardening component that may also be specified as,

.. math::

   \tilde{\sigma}_y=\sigma_y+K\left(\bar{\varepsilon}^p\right),

with :math:`\sigma_y` being the constant yield stress and :math:`K` is the isotropic hardening that is initially zero and a function of the equivalent plastic strain. A multiplicative decomposition such as this mirrors the general structure used by Johnson and Cook [:footcite:`mat:ref:johncook`, :footcite:`mat:ref:johncook2`] although greater flexibility is allowed in terms of the specific form of the rate multiplier. 

Given the aforementioned default for 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 is required 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 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.
