Reference problem
=====================

The problem is detailed in reference [:ref:`1 <1>`], in particular obtaining the analytical solution. Only a brief description of its main characteristics is presented here. Precision now that only the viscoelastic branch of behavior model KICHENIN_NL is mobilized; the elastoplastic branch is limited to elasticity by the introduction of a sufficiently high plasticity threshold. The validation of this second branch is carried out in [:external:ref:`V6.04.268 <V6.04.268>`]. Since the two branches are parallel, there is no coupling other than the sum of the constraints and the tangent operators, which justifies this validation choice.


Geometry
---------

This is a "hardware point" test. It relies on the SIMU_POINT_MAT command.


Material properties
----------------------

The material obeys the law of viscoelastic behavior KICHENIN_NLavec as characteristics:


.. csv-table::

    ":math:`{E}^{p}=400\text{MPa}` ", "Young's modulus of the plastic branch"
    ":math:`{\mathrm{\nu }}^{p}=0.15` ", "Poisson's ratio in the plastic industry"
    ":math:`{E}^{v}=800\text{MPa}` ", "Young's modulus of the viscoelastic branch"
    ":math:`{\mathrm{\nu }}^{v}=0.3` ", "Poisson's ratio of the viscoelastic branch"
    ":math:`{\mathrm{\nu }}^{d}=0.25` ", "Poisson's ratio of the nonlinear shock absorber"
    ":math:`n=0.5` ", "Exponent of the nonlinear damper (:math:`\gamma =2`)"
    ":math:`\eta =200{\text{MPa.s}}^{1/2}` ", "Viscosity coefficient of the nonlinear shock absorber"



Boundary conditions and loads
-------------------------------------

This is a loading of imposed constraints, the expression of which will be seen in §2.1. It is based on two parameters: a characteristic time :math:`{t}_{c}` which determines the loading speed and a stress load on the :math:`{\sigma }^{0}` shock absorber.


.. csv-table::

    ":math:`{t}_{c}=2.5\text{s}` ", "Typical load time"
    ":math:`{\sigma }^{0}=\frac{10}{\sqrt{1.87}}\left[\begin{array}{ccc}1.0& 0.1& 0.3\\ 0.1& 0.2& 0.5\\ 0.3& 0.5& 0.4\end{array}\right]\text{MPa}` ", "Shock absorber stress (in MPa)"



Initial conditions
--------------------

The system is considered in its natural state at :math:`t=0`. In particular, the viscoelastic deformation is zero.