Modeling A
==============




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

3D modeling is used.

The coefficient :math:`{K}_{D}` is chosen depending on the temperature and on :math:`{\sigma }_{0}`.


.. csv-table::

    "T (°C)", "0 MPa ", "100 MPa ", "200 ", "200 MPa"
    "900", "14,355", "14,855", "15,355"
    "1000", "14.5", "15", "15.5"
    "1025", "14.5363", "15.0363", "15.5363", "15.5363"
    "1050", "14.5725", "15.0725", "15.5725"


**Table** 3.1-1 **: K_D of modeling A**

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 3.2-1: Modeling A mesh



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

Two calculations are performed, the first with an explicit integration algorithm (ALGO_INTE =' RUNGE_KUTTA '), the second with an implicit integration algorithm (ALGO_INTE =' NEWTON').

**Calculation in explicit resolution:**

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


.. csv-table::

    "**Instant**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    "20", "'ANALYTIQUE'", "252.76091", "0.5"
    "2000", "'ANALYTIQUE'", "164.261", "0.5"
    "200000", "'ANALYTIQUE'", "101.596", "0.5"
    "1000000", "'ANALYTIQUE'", "75.97849999999999997", "0.5"
    "1600000", "'ANALYTIQUE'", "55.54209999999999998", "10.0"


**Table** 3.3-1 **: Results of modeling A**

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'", "2.3168400000000001E-4", "0.5"
    "2000", "'ANALYTIQUE'", "2.77144E-3", "0.5"
    "200000", "'ANALYTIQUE'", "0.032255100000000002", "0.5"
    "1000000", "'ANALYTIQUE'", "0.110134", "1.0"
    "1600000", "'ANALYTIQUE'", "0.28131600000000001", "10.0"


**Table** 3.3-2 **: Results of modeling A**

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'", "1.6445100000000001E-3", "0.5"
    "2000", "'ANALYTIQUE'", "2.2312600000000001E-3", "0.5"
    "200000", "'ANALYTIQUE'", "2.6251500000000001E-3", "0.5"
    "1000000", "'ANALYTIQUE'", "2.747799999999999999E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.79276E-3", "0.5"


**Table** 3.3-3 **: Results of modeling A**

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 (%)**"
    "20", "'ANALYTIQUE'", "1.644569999999999999E-3", "0.5"
    "2000", "'ANALYTIQUE'", "2.231879999999999999E-3", "0.5"
    "200000", "'ANALYTIQUE'", "2.6301200000000001E-3", "0.5"
    "1000000", "'ANALYTIQUE'", "2.7607899999999999E-3", "0.5"
    "1600000", "'ANALYTIQUE'", "2.814779999999999998E-3", "0.5"


**Table** 3.3-4 **: Results of modeling A**

**Calculation in resolution** implicit **:**

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'", "252.76091", "2.5"
    "2000", "'ANALYTIQUE'", "164.261", "2.5"
    "200000", "'ANALYTIQUE'", "101.596", "2.5"


Table 3.3-5: Results of modeling A

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'", "2.3168400000000001E-4", "3.5"
    "2000", "'ANALYTIQUE'", "2.77144E-3", "4.0"
    "200000", "'ANALYTIQUE'", "0.032255100000000002", "7.0"


Table 3.3-6: Results of modeling A

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'", "1.6445100000000001E-3", "2.5"
    "2000", "'ANALYTIQUE'", "2.2312600000000001E-3", "1.0"
    "200000", "'ANALYTIQUE'", "2.6251500000000001E-3", "1.0"


Table 3.3-7: Results of modeling A

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 (%)**"
    "20", "'ANALYTIQUE'", "1.644569999999999999E-3", "2.5"
    "2000", "'ANALYTIQUE'", "2.231879999999999999E-3", "1.0"
    "200000", "'ANALYTIQUE'", "2.6301200000000001E-3", "0.5"


Table 3.3-8: Results of modeling A

Note: This calculation does not converge beyond the instant 200000.