Perfect plasticity
==================

Overview
--------

This class implements rate independent perfect plasticity described by:

   The elastic trial state:

   .. math::

      \bm{\varepsilon}^{p}_{tr} = \bm{\varepsilon}^{p}_n

      \bm{\sigma}_{tr} = \mathbf{\mathfrak{C}}_{n+1} : 
         \left( \bm{\varepsilon}_{n+1} - \bm{\varepsilon}_{tr}^p  \right)

   The plastic correction:

   .. math::
      \bm{\sigma}_{n+1} = \mathbf{\mathfrak{C}}_{n+1} : 
         \left( \bm{\varepsilon}_{n+1} - \bm{\varepsilon}_{n+1}^p \right)

      \bm{\varepsilon}_{n+1}^p = 
         \begin{cases}
            \bm{\varepsilon}^{p}_{tr} & f\left(\bm{\sigma}_{tr}\right)\le0\\
            \bm{\varepsilon}^{p}_{tr}+\frac{\partial f_{n+1}}{\partial\bm{\sigma}_{n+1}}\Delta\gamma_{n+1} & f\left(\bm{\sigma}_{tr}\right)>0
         \end{cases}

   Solving for :math:`\Delta \gamma_{n+1}` such that

   .. math::
      f\left(\bm{\sigma}_{n+1} \right) = 0

In these equations :math:`f` is a yield function, parameterized by the yield
stress :math:`\sigma_0`.

If the step is plastic the stress update is solved through fully-implicit 
backward Euler integration.
The algorithmic tangent is then computed using an implicit function scheme.
The work and energy are integrated with a trapezoid rule from the final values
of stress and plastic strain.

This model does not maintain any history variables.

Parameters
----------

.. csv-table::
   :header: "Parameter", "Object type", "Description", "Default"
   :widths: 12, 30, 50, 8

   ``elastic``   , :cpp:class:`neml::LinearElasticModel`   , Temperature dependent elastic constants, No
   ``surface``   , :cpp:class:`neml::YieldSurface`         , The yield surface                      , No
   ``ys``        , :cpp:class:`neml::Interpolate`          , The yield stress as a function of T    , No
   ``alpha``     , :cpp:class:`neml::Interpolate`          , Temperature dependent instantaneous CTE, ``0.0``
   ``tol``       , :code:`double`               , Integration tolerance                  , ``1.0e-8``
   ``miter``     , :code:`int`                  , Maximum number of integration iters    , ``50``
   ``verbose``   , :code:`bool`                 , Print lots of convergence info         , ``false``
   ``max_divide``, :code:`int`                  , Maximum number of adaptive subdivisions, ``8``

Class description
-----------------

.. doxygenclass:: neml::SmallStrainPerfectPlasticity
   :members:
   :undoc-members: