D modeling
==============


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

We use a AXIS model.

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** 6.1-1 **:** **K_D from the D** 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 quadrangle with 8 knots in axisymmetry.

Number of knots: 8

Number of stitches: 1 (QUAD8)


.. image:: images/10000201000001BD0000022CAEC440B46B4E3C28.png
    :width: 2.6827in
    :height: 2.128in


.. _RefImage_10000201000001BD0000022CAEC440B46B4E3C28.png:

Image 6.2-1: D 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 (%)**"
    "1000000", "'ANALYTIQUE'", "79.92289999999999998", "0.5"
    "1600000", "'ANALYTIQUE'", "70.90219999999999993", "1.0"


**Table** 6.3.1-1 **: D 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 (%)**"
    "1000000", "'ANALYTIQUE'", "0.066266580000000005", "1.0"
    "1600000", "'ANALYTIQUE'", "0.090278800000000006", "1.0"


**Table** 6.3.1-2 **: D 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 (%)**"
    "1000000", "'ANALYTIQUE'", "2.747799999999999999E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.7993599999999999E-3", "0.5"


**Table** 6.3.1-3 **: D modeling results**

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


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "1000000", "'ANALYTIQUE'", "2.7522900000000001E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.8137437999999999E-3", "0.5"


**Table** 6.3.1-4 **: D 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).