B modeling
==============


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

Volumic 3D mesh. Large displacements and deformations (but small rotations)

The applied surface force is equal here to :math:`–20\mathrm{MPa}` to :math:`t=\mathrm{1s}`, in order to pass through the critical point during the evolution of the load.

This load is applied in 10 equally divided time steps.

Two complete calculations are carried out: one with purely elastic behavior, in order to be able to compare the result with the reference elastic solution, and the other with elastoplastic behavior.


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

Number of knots: 600

Number of meshes and types: 90 HEXA20


Tested values
---------------

In elastic behavior

We test the final value of the critical coefficient: (non-regression test)


.. csv-table::

    "**Instant**", "**Reference**"
    "1", "19.0657"


In the case of large displacements or large deformations, the value of the critical coefficient must be interpreted differently from the case of small displacements: the structure becomes unstable when the "critical load" is cancelled out.

The evolution of this coefficient over time is as follows:


.. csv-table::

    "**No time**", "**Surface force**
    **(en** :math:`\mathrm{MPa}` **)**", "**Critical Coefficient** **Aster**", "**Critical Load** Euler"
    "1", "2", "-27.5797", "-27.5797", "12.9539"
    "2", "4", "-22.8250", "12.9539"
    "3", "6", "-17.9808", "12.9539"
    "4", "8", "-13.0407", "12.9539"
    "5", "10", "-7.9975", "-7.9975", "12.9539"
    "6", "12", "-2.8434", "-2.8434", "12.9539"
    "**7**", "14", "2.4301", "12.9539"
    "**8**", "16", "7.8324", "12.9539"
    "**9**", "18", "13.3738", "12.9539"
    "**10**", "20", "19.0657", "12.9539"


The critical coefficient therefore passes through 0 between the moments 6 and 7, and more precisely (cf. the following curve) around the instant 6.5, which corresponds well to the critical load in terms of elasticity.


.. image:: images/100002000000025D000001548CD4D3CEC54C9DF5.png
    :width: 6.0417in
    :height: 3.4008in


.. _RefImage_100002000000025D000001548CD4D3CEC54C9DF5.png:

In elastoplasticity, we test the moments when the critical coefficient changes sign. The tests are non-regression since there is no analytical solution in this case.


.. csv-table::

    "**Instant**", "**Reference**"
    "0.4", "-4.9917"
    "0.5", "1.3186"