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


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

Behavior :math:`\mathrm{C4}`: non-linear kinematic work hardening (:math:`\mathrm{II}`) in :math:`\mathrm{3D}`. It is modelled in two ways:


* or using the VMIS_CIN1_CHAB behavior. All you have to do is then take:

:math:`{R}_{0}={R}_{I}=\mathrm{SIGY}`, :math:`b=0`, :math:`{C}_{I}=C(T)=(100+\mathrm{1.7.T})(50+\mathrm{2.T})`, :math:`{G}_{0}=D(T)`


* or using behavior VMIS_CIN2_CHAB, by choosing the parameters in such a way that the two kinematic variables are identical: All you have to do is then take:

:math:`{R}_{0}={R}_{I}=\mathrm{SIGY}`, :math:`b=0`, :math:`{\mathrm{C1}}_{I}={\mathrm{C2}}_{I}=C(T)/2`, :math:`{\mathrm{G1}}_{0}={\mathrm{G2}}_{0}=D(T)`

Temporal discretization: 40 time steps between :math:`t=\mathrm{0s}` and :math:`t=\mathrm{1s}` and 30 time steps between :math:`t=\mathrm{1s}` and :math:`t=\mathrm{2s}`.


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

The mesh includes a HEXA8 mesh.


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

Behavior VMIS_CIN1_CHAB:


.. csv-table::

    "**Instant (s)**", "**Plastic deformation by** :math:`Y` ", "**Reference**", "**Aster**", "**% difference**"
    "1"," :math:`\mathrm{EPYY}` ", "2.1072 10—03", "2.11 10—03", "2.10—03", "0.14"
    "1.26"," :math:`\mathrm{EPYY}` ", "4.18947 10—03", "4.231 10—03", "4.231 10—03", "0.99"
    "1.98"," :math:`\mathrm{EPYY}` ", "4.18947 10—03", "4.231 10—03", "4.231 10—03", "0.99"
    "2"," :math:`\mathrm{EPYY}` ", "4.12131 10—03", "4.163 10—03", "4.12131 10—03", "1.00"


Behavior VMIS_CIN2_CHAB:


.. csv-table::

    "**Instant (s)**", "**Plastic deformation by** :math:`Y` ", "**Reference**", "**Aster**", "**% difference**"
    "1"," :math:`\mathrm{EPYY}` ", "2.1072 10—03", "2.11 10—03", "2.10—03", "0.14"
    "1.26"," :math:`\mathrm{EPYY}` ", "4.18947 10—03", "4.231 10—03", "4.231 10—03", "0.99"
    "1.98"," :math:`\mathrm{EPYY}` ", "4.18947 10—03", "4.231 10—03", "4.231 10—03", "0.99"
    "2"," :math:`\mathrm{EPYY}` ", "4.12131 10—03", "4.163 10—03", "4.12131 10—03", "1.00"



note
--------

The difference with the reference solution (obtained for a very fine temporal discretization) comes from the temporal discretization. Here, a compromise was chosen between reasonable temporal discretization in terms of calculation time and nevertheless fairly accurate.