Model::Solid::Affine::J2
========================

:bdg-link-secondary-line:`src/Model/Solid/Affine/J2.H <../../../../doxygen/J2_8H_source.html>`

This models an isotropic elastic-perfectly-plastic, non-time-dependent solid model.

The energy and derivatives are:

.. math::
   :nowrap:

   \begin{gather}
   W = \frac{1}{2}(\varepsilon - \varepsilon_p):\mathbb{C}(\varepsilon-\varepsilon_p) \\
   DW = \mathbb{C}(\varepsilon-\varepsilon_p) \\
   DDW = \mathbb{C}
   \end{gather}

where :math:`\mathbb{C}` is an isotropic :code:`Set::Matrix4` and :math:`\varepsilon_p` is 
is stored in the :code:`F0` eigenstrain.

The plastic strain is evolved according to the following:

#. Calculate the deviatoric stress :math:`\sigma_v=\sigma - \frac{1}{3}tr(\sigma)\mathbf{I}`
#. Calculate :math:`J_2=\sqrt{\frac{3}{2}\sigma_v:\sigma_v}`
#. If :math:`J_2<\sigma_0` then quit - no plasticity occurs
#. Calculate :math:`\Delta\sigma = (1-\frac{\sigma_0}{J_2})`, which projects the stress 
   back on the yield surface.
#. Convert to change in plastic strain, :math:`\Delta\varepsilon=\mathbb{C}^{-1}\Delta\sigma`
#. Update plastic strain: :math:`\varepsilon_p += \Delta\varepsilon`

Notes:

* This does not implement any kind of hardening model. Rate hardening, isotropic hardening,
  and kinematic hardening have yet to be implemneted.

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