C modeling
==============




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

3D modeling is used.

Two calculations are performed that differ only in the integration algorithm: 'NEWTON' and 'RUNGE_KUTTA'.

In this case, the coefficient :math:`{K}_{D}` is chosen only depending on the temperature.


.. csv-table::

    "T (°C)", "900", "1000", "1025"
    ":math:`{K}_{D}` ", "15", "15", "15"


**Table** 5.1-1 **: K_D of C** modeling

The discretization in time is quite fine:


.. code-block:: text

    (JUSQU_A = 2, NOMBRE = 10),

    (JUSQU_A = 2., NOMBRE = 10),

    (JUSQU_A = 20., NOMBRE = 10),

    (JUSQU_A = 200., NOMBRE = 10),

    (JUSQU_A = 2000., NOMBRE = 10),

    (JUSQU_A = 20000., NOMBRE = 10),

    (JUSQU_A = 200,000., NOMBRE = 10),

    (JUSQU_A = 1000000., NOMBRE = 30),

    (JUSQU_A = 1600000., NOMBRE = 30),

    (JUSQU_A = 1700000., NOMBRE = 40),

    (JUSQU_A = 1800000., NOMBRE = 40),

    (JUSQU_A = 1900000., NOMBRE = 40),

    (JUSQU_A = 2000000., NOMBRE = 40),

    (JUSQU_A = 2100000., NOMBRE = 40),

    (JUSQU_A = 2200000., NOMBRE = 40),

    (JUSQU_A = 2300000., NOMBRE = 40),

    (JUSQU_A = 2400000., NOMBRE = 40),

    (JUSQU_A = 2500000., NOMBRE = 40),




Characteristics of the mesh
----------------------------

It is chosen to represent the cylindrical test piece by a block in order to be able to perform a calculation on a single element. The modeling has 3 planes of symmetry.

Number of knots: 8

Number of stitches: 1 (HEXA8)


.. image:: images/100002010000027E000002A9D2D338CDC42533CB.png
    :width: 4in
    :height: 3.028in


.. _RefImage_100002010000027E000002A9D2D338CDC42533CB.png:

Figure 5.2-1: C modeling mesh




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


Case RUNGE_KUTTA + Matrix ELASTIQUE
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Evolution of the constraint, :math:`{\sigma }_{0}`, as a function of time. This value is tested at various times:


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "20", "'ANALYTIQUE'", "253.02000000000001", "0.5"
    "2000", "'ANALYTIQUE'", "164.36000000000001", "0.5"
    "200000", "'ANALYTIQUE'", "102.16", "0.5"
    "1000000", "'ANALYTIQUE'", "79.920000000000002", "0.5"
    "1600000", "'ANALYTIQUE'", "70.900000000000006", "1.0"


**Table** 5.3.1-1 **: C modeling results**

Evolution of the damage variable, :math:`D` as a function of time. This value is tested at various times depending on the modeling:


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "20", "'ANALYTIQUE'", "2.32E-4", "0.5"
    "2000", "'ANALYTIQUE'", "2.7399E-3", "0.5"
    "200000", "'ANALYTIQUE'", "0.027560999999999999", "0.5"
    "1000000", "'ANALYTIQUE'", "0.066266500000000006", "1.0"
    "1600000", "'ANALYTIQUE'", "0.090278800000000006", "1.0"


**Table** 5.3.1-2 **: C modeling results**

Evolution of the viscoplastic isotropic work hardening variable, :math:`r`, as a function of time. This value is tested at various times:


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "20", "'ANALYTIQUE'", "1.645999999999999999E-3", "0.5"
    "2000", "'ANALYTIQUE'", "2.2339E-3", "0.5"
    "200000", "'ANALYTIQUE'", "2.6282800000000002E-3", "0.5"
    "1000000", "'ANALYTIQUE'", "2.7522900000000001E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.799299999999999998E-3", "0.5"


**Table** 5.3.1-3 **: C modeling results**

Evolution of the isotropic visco-plastic work hardening variable, :math:`p`, as a function of time. This value is tested at various times:


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "20", "'ANALYTIQUE'", "1.6461E-3", "0.5"
    "2000", "'ANALYTIQUE'", "2.234499999999999999E-3", "0.5"
    "200000", "'ANALYTIQUE'", "2.6329000000000001E-3", "0.5"
    "1000000", "'ANALYTIQUE'", "2.762699E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.8137000000000001E-3", "0.5"


**Table** 5.3.1-4 **: C modeling results**


Case NEWTON + Matrix TANGENTE
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The quantities tested are the same as in the previous case. On the other hand, the tolerance is 4% (for all values tested).