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


Plate geometry
----------------------


.. image:: images/Object_1.svg
    :width: 539
    :height: 389


.. _RefImage_Object_1.svg:


.. image:: images/Object_2.svg
    :width: 539
    :height: 389


.. _RefImage_Object_2.svg:


Characteristics of the models
----------------------------------

This test case is composed of 8 models. The table below summarizes their characteristics:


1. Orientation of the frames: longitudinal direction :math:`(L)`: :math:`\mathrm{OX}`; transverse direction :math:`(T)`: :math:`\mathrm{OY}`


2. Eccentricity = :math:`0m`


3. Section per linear meter = :math:`\mathrm{0,01}{m}^{2}/\mathrm{ml}` (same thickness in the transverse and longitudinal directions in case of presence of transverse reinforcements)


.. csv-table::

    "Modeling", "Law of behavior", "Presence of frame
    transversal", "Temperature application mode"
    "A", "linear isotropic", "Yes", "at the nodes"
    "B", "linear kinematics", "Yes", "at the nodes"
    "C", "Pinto Menegotto", "Yes", "At the knots"
    "D", "Pinto Menegotto", "Yes", "to the elements"
    "E", "Pinto Menegotto", "No", "to the elements"
    "F", "linear isotropic", "No", "at the nodes"
    "G", "linear kinematics", "No", "at the nodes"
    "H", "Pinto Menegotto", "No", "***"


*** For test case H, temperature loading is replaced by mechanical loading (displacement imposed on the nodes).


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


Properties common to all models
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "Young's module:", ":math:`E={2.10}^{11}\mathrm{MPa}`"
    "Poisson's ratio:", ":math:`\nu =0`"
    "Elastic limit:", ":math:`{\sigma }_{y}={2.10}^{8}\mathrm{MPa}`"


Coefficient of thermal expansion: :math:`\alpha ={10}^{-5}(°{C}^{-1})`


Isotropic and kinematic plastic behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For isotropic (GRILLE_ISOT_LINE) and kinematic (GRILLE_CINE_LINE) behaviors

Work hardening slope: :math:`{E}_{T}={2.10}^{10}\mathrm{MPa}`


.. _Ref473689253:


Pinto Menegotto's behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the PINTO MENEGOTTO (GRILLE_PINTO_MEN) behavior


.. csv-table::

    "EPSI_ULTM ", ":", "3.0. 10—2"
    "SIGM_ULTM ", ":", "2.58. 108"
    "EPSP_HARD ", ":", "0.0023"
    "R_PM", ":", "20,0"
    "EP_SUR_E ", ":", "0.01"
    "A1_PM", ":", "18.5"
    "A2_PM", ":", "0.15"
    "ELAN ", ":", "4.9"
    "A6_PM", ":", "620.0"
    "C_PM", ":", "0.5"
    "A_PM", ":", "0.008"



.. _Ref473028113:


Boundary conditions and loading
------------------------------------

The plate is fully embedded for models A to G (thermo-mechanical). For H modeling (mechanical), we block all the movements and all the rotations at the nodes except :math:`\mathrm{UX}` for the nodes :math:`\mathrm{NO2}` and :math:`\mathrm{NO3}`.

The loading is of thermal origin for models A to G. The evolution of the temperature as a function of time is given for each model in the following table. The temperature is applied to the nodes or to the elements, depending on the modeling.


.. csv-table::

    "Instant", "Evolution :math:`A`
    :math:`T°` ", "Evolution :math:`B`
    :math:`T°` ", "Evolution :math:`C`
    :math:`T°`"
    "0", "50", "50", "50"
    "1", "—50", "—50", "—300"
    "2", "—250", "—250", "—100"
    "3", "—150", "—150", "50"
    "4", "—250", "—250", "—150"
    "5", "—50", "—50", "—350"
    "6", "350", "350", "350", "—200"
    "7", "50", "150", ""
    "8", "—450", "—450", ""
    "9", "—110", "—250", ""
    "10", "550", "650", ""
    "11", "50", "450", ""



.. image:: images/1000297C000069D5000041D6CB32AC64A91ED196.svg
    :width: 539
    :height: 389


.. _RefImage_1000297C000069D5000041D6CB32AC64A91ED196.svg:

For all tests, a reference temperature of :math:`50°` was taken.

For the H modeling, a nodal force :math:`\mathrm{FX}` is applied to the nodes :math:`\mathrm{NO2}` and :math:`\mathrm{NO3}` (directed according to the vector :math:`\mathrm{UX}`) by driving the calculation by the displacement :math:`\mathrm{UX}` of :math:`\mathrm{NO3}` so that it follows the following evolution:


.. csv-table::

    "Instant", "0.001", "0.0023", "0.0023", "0.03", "0.2", "0.4", "6.4", "7.92"
    ":math:`\mathit{Ux}` (:math:`m`)", "0.001", "0.0023", "0.0023", "0.03", "0.0296667", "0.02", "-0.04", "-0.033"
    "Instant", "17", "19", "19", "20", "20", "21", "22", "25", "50"
    ":math:`\mathit{Ux}` (:math:`m`)", "0.01", "0.03", "0.03", "0.0298", "0.026", "0.022", "0.025", "0.05"