.. _failure-jc:

Johnson-Cook Failure
====================

The Johnson-Cook model [:footcite:`mat:ref:johncook2`] is implemented with the form,

.. math::
   :label: mat:eq:modularfailure_johnsoncook

   d = \int_0^{\bar{\varepsilon}^p} \frac{ d\bar{\varepsilon}^{p} }{ \left(D_1 + D_2 \exp(D_3 \eta)\right) \left(1 + D_4 \langle \ln \frac{\dot{\bar{\varepsilon}}^p }{\dot{\varepsilon}_0 } \rangle \right) \left( 1 + D_5 T^* \right) },

where :math:`\{D_1, D_2, D_3, D_4, D_5\}` are fitting constants and :math:`\dot{\varepsilon}_0` is a reference strain rate. The term :math:`\eta` represents stress triaxiality, the ratio of mean hydrostatic stress to von-Mises stress: :math:`\eta = \frac{p}{\sigma_{vm}}`. The term :math:`T^*` represents the homologous temperature, given as a function of the temperature :math:`T` by, 

.. math::

   T^* = \frac{T - T_{\text{ref}} }{T_{\text{melt}} - T_{\text{ref}}},

where :math:`T_{\text{ref}}` is a reference temperature and :math:`T_{\text{melt}}` is the melting temperature.

The Johnson-Cook failure model form :eq:`mat:eq:modularfailure_johnsoncook` is formulated as a multiplicative combination of triaxiality, strain-rate, and temperature effects, and the denominator may be interpreted as the critical failure strain. The failure process initiates once the total quantity reaches :math:`d = 1`. 

User Guide
----------

.. code-block:: sierrainput

   #
   # JOHNSON_COOK_FAILURE Failure model definitions
   #
   JOHNSON COOK D1       = <real>
   JOHNSON COOK D2       = <real>
   JOHNSON COOK D3       = <real>
   JOHNSON COOK D4       = <real>
   JOHNSON COOK D5       = <real>
   #
   #Following Johnson-Cook parameters can only be defined once.  As such, only 
   #  needed if not previously defined via Johnson-Cook multipliers
   #  w/ flow-stress hardening.  Does need to be defined 
   #  w/ Decoupled Flow Stress
   #
   REFERENCE RATE        = <real>
   REFERENCE TEMPERATURE = <real>
   MELTING TEMPERATURE   = <real>

.. raw::
   html

   <hr>

.. footbibliography::
