F modeling
==============


Characteristics of modeling
-----------------------------------


Simulation 1
~~~~~~~~~~~~

This is a thermomechanical test with zero imposed deformation along axis :math:`x`. The test is performed on an element of BARRE with the STAT_NON_LINE command. The temperature varies from :math:`{T}_{0}=20°C` to :math:`{T}_{\mathrm{max}}=100°C`. The material is able to be plasticized. The transition consists of NCAL steps. The reference temperature is :math:`{T}_{\mathrm{ref}}=20°C`.


Simulation 2
~~~~~~~~~~~~

It's about looping over NCAL mechanical calculations. For each calculation :math:`i`, the imposed load consists of the thermal deformation :math:`{\varepsilon }_{\mathrm{xx}}=-{\varepsilon }_{\mathrm{th}}=-\alpha (T)({T}_{i}-{T}_{\mathrm{Ref}})`. The initial load consists of the deformations, stresses and internal variables from the previous mechanical calculation.


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

The law of behavior tested is' PINTO_MENEGOTTO 'documented in doc [:ref:`R5.03.09 <R5.03.09>`]. This law is an elasto-plastic uniaxial isothermal law modeling the response of steel reinforcements in reinforced concrete under cyclic loading.

The elastic parameters are as follows:

:math:`E(T)`, :math:`\nu (T)`, and :math:`\alpha (T)`

The elastoplastic parameters are as follows:

:math:`{\sigma }_{y}^{0}`, :math:`{\varepsilon }_{u}`, :math:`{\sigma }_{u}`, :math:`{\varepsilon }_{h}`, :math:`b`,, :math:`\mathrm{R0}`, :math:`\mathrm{a1}`, :math:`\mathrm{a2}`,, :math:`L/D`, :math:`\mathrm{a6}`,,,,,,,,,,,, :math:`c` :math:`a`

Values of the parameters used:


.. csv-table::

    "Settings", ":math:`T=20°C` "," :math:`T=500°C`"
    ":math:`E(T)` "," :math:`2.1\mathrm{E11}` Pa", ":math:`1.E11` Pa"
    ":math:`\nu (T)` "," :math:`0.` "," :math:`0.`"
    ":math:`\alpha (T)` "," :math:`1.E-5` :math:`{K}^{-1}` "," :math:`2.E-5` :math:`{K}^{-1}`"



+--------------------------+-----------------++-------------------+-------------+
|Parameters                                  ||Parameters                       |
+--------------------------+-----------------++-------------------+-------------+
|:math:`{\sigma }_{y}^{0}` |:math:`2.E8` Pa  ||:math:`\mathrm{a1}`|:math:`18.5` |
+--------------------------+-----------------++-------------------+-------------+
|:math:`{\varepsilon }_{u}`|:math:`3.E-2`    ||:math:`\mathrm{a2}`|:math:`0.15` |
+--------------------------+-----------------++-------------------+-------------+
|:math:`{\sigma }_{u}`     |:math:`2.58E8` Pa||:math:`L/D`        |:math:`4.9`  |
+--------------------------+-----------------++-------------------+-------------+
|:math:`{\varepsilon }_{h}`|:math:`0.0023`   ||:math:`\mathrm{a6}`|:math:`620.` |
+--------------------------+-----------------++-------------------+-------------+
|:math:`b`                 |:math:`0.01`     ||:math:`c`          |:math:`0.5`  |
+--------------------------+-----------------++-------------------+-------------+
|:math:`\mathrm{R0}`       |:math:`20.`      ||:math:`a`          |:math:`0.008`|
+--------------------------+-----------------++-------------------+-------------+


Tested sizes and results
------------------------------

Validation is carried out by comparing the fields calculated at each step of the transient on the one hand and the result of a mechanical calculation on the other hand.

The command used is TEST_TABLE which tests the reference value against the calculated value.

The reference value being the field component extracted at a given time :math:`i` from the first thermo-mechanical simulation performed over NCAL times. The calculated value is the one obtained at the end of the mechanical calculation :math:`i+1` of the loop on the NCAL.


.. csv-table::

    "Result at order number :math:`i` ", "Name of the tested parameter", "Name of the tested parameter", "Reference type", "Tolerance"
    "RESU_i "," NOM_PARA ", "", "VALE_REF "," TOLE"
    "RESU_19 ", "N", "NON_REGRESSION ", "-2.58E8"," 0.1%"
    "RESU_19 ", "V4"," NON_REGRESSION ", "-9.6E-2", ""
    "RESU_19 ", "V5"," NON_REGRESSION ", "-7.08E-3", ""