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

.. math::

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

is used in which :math:`\hat{\sigma}` and :math:`\breve{\sigma}` are rate and temperature multipliers, respectively, that by default are unity (such that the response is rate and temperature independent).  The isotropic hardening component, :math:`\tilde{\sigma}_y`, is 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 and temperature multipliers. 

Given the aforementioned defaults for rate and temperature dependence, the corresponding multipliers 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 and temperature independent, a multiplier may be defined for either (or both) of the terms. 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.

In terms of temperature dependence, the multiplier may be specified given a Johnson-Cook dependency [:footcite:`mat:ref:johncook`, :footcite:`mat:ref:johncook2`],

.. math::

   \breve{\sigma}\left(\theta\right)=1-\left(\frac{\theta-\theta_{\text{ref}}}{\theta_{\text{melt}}-\theta_{\text{ref}}}\right)^M,

with :math:`\theta_{\text{ref}},~\theta_{\text{melt}}` and :math:`M` being the reference temperature, melting temperature, and temperature exponent.  The temperature multiplier may also be specified via a user defined function. 
