E modeling
==============


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

It is a nonlinear elasticity calculation under the hypothesis of small displacements, in plane deformations.


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

The mesh is identical to that of modeling A.


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

The quantity tested is the reproduction rate :math:`G`, calculated for the first integration ring defined in modeling A and for ten values of the imposed horizontal displacement :math:`\delta` (:math:`\delta` varying from :math:`0` to :math:`\mathrm{0,2}\mathrm{mm}`).

Two distinct series of calculations are carried out:

first, a law of behavior with linear hardening is taken. Three calculations are carried out using the different methods available to define the same work-hardening curve (TRACTION, ECRO_LINE and ECRO_PUIS). The solutions obtained by these three calculations are compared with each other;

then we choose a law of behavior with hardening in power law and we compare the results obtained between ELAS_VMIS_TRAC and ELAS_VMIS_PUIS.

The test is of a non-regression type. It is verified that:

the three different calculations in linear work hardening lead to exactly the same result on :math:`G`;

the two power law hardening calculations lead to very similar results (difference less than 0.3%). The difference decreases if the calculation time step is reduced.


.. csv-table::

    "Imposed displacement :math:`\delta` (:math:`\mathrm{mm}`)", ":math:`G(N/\mathrm{mm})` - linear work hardening: ELAS_VMIS_TRAC "," :math:`G(N/\mathrm{mm})` - linear work hardening: ELAS_VMIS_LINE "," :math:`G(N/\mathrm{mm})` - linear work hardening: ELAS_VMIS_PUIS"
    "0.02", "3.59", "3.59", "3.59"
    "0.04", "14.47", "14.47", "14.47"
    "0.06", "32.88", "32.88", "32.88"
    "0.08", "59.42", "59.42", "59.42", "59.42"
    "0.10", "94.50", "94.50", "94.50"
    "0.12", "134.98", "134.98", "134.98", "134.98"
    "0.14", "178.95", "178.95", "178.95"
    "0.16", "226.24", "226.24", "226.24"
    "0.18", "276.47", "276.47", "276.47"
    "0.20", "329.61", "329.61", "329.61", "329.61"



.. csv-table::

    "Imposed displacement :math:`\delta` (:math:`\mathrm{mm}`)", ":math:`G(N/\mathrm{mm})` - work hardening power: ELAS_VMIS_TRAC "," :math:`G(N/\mathrm{mm})` - work hardening power: ELAS_VMIS_PUIS"
    "0.02", "3.59", "3.59"
    "0.04", "14.44", "14.44"
    "0.06", "32.76", "32.76"
    "0.08", "58.99", "58.98"
    "0.10", "93.76", "93.77"
    "0.12", "136.14", "136.14", "136.18"
    "0.14", "184.04", "184.04", "184.09"
    "0.16", "236.31", "236.31", "236.38"
    "0.18", "292.38", "292.46"
    "0.20", "351.82", "351.90"


This test validates the calculation of :math:`G` in linear elasticity for the various methods of defining the work hardening curve.