Reference problem
=====================

Geometry
---------

It is a square plate with side :math:`1\mathrm{mm}`


.. image:: images/10000000000001540000011994B642EC080F6174.png
    :width: 2.8228in
    :height: 2.3319in


.. _RefImage_10000000000001540000011994B642EC080F6174.png:

**Figure** 1.1-a **: Problem geometry.**


Material properties
--------------------

The various material properties are given below.

The material parameters used in this test case **should not be used for studies**. They do not correspond to real characteristics.

Young's module: :math:`E=210000.0–30.0T` in :math:`\mathrm{MPa}`

Poisson's ratio: :math:`\nu =0.30+5.0E-05T`

Coefficient of thermal expansion: :math:`\alpha =\left(15.0+0.002T\right)1.0E-06`

*Plastic part*

:math:`\kappa =0.8`

Elastic limit at :math:`\text{0.2\%}` in :math:`\mathrm{MPa}`: :math:`{R}_{02}={R}_{02}^{0}\mathrm{.}{C}_{w}\text{\_}{R}_{e}\mathrm{.}{I}_{r}\text{\_}{R}_{e}`

with

:math:`{R}_{02}^{0}=270.0-0.65T+0.001{T}^{2}`

:math:`{C}_{w}\text{\_}{R}_{e}=1.0`

:math:`{I}_{r}\text{\_}{R}_{e}=\left(4.0–3.0{e}^{\frac{-\mathit{IRRA}}{3}}\right)`

Ultimate compulsion in :math:`\mathit{MPa}`: :math:`{R}_{m}={R}_{02}(T,\mathrm{IRRA})+({R}_{m}^{0}–{R}_{02}^{0})\mathrm{.}{C}_{w}\text{\_}{R}_{m}\mathrm{.}{I}_{r}\text{\_}{R}_{m}`

with

:math:`{R}_{m}^{0}=600.0-1.5T+0.010{T}^{2}`

:math:`{C}_{w}\text{\_}{R}_{m}=0.50`

:math:`{I}_{r}\text{\_}{R}_{m}=0.005-0.002\left(1.0-{e}^{\frac{-\mathit{IRRA}}{4.0}}\right)+{e}^{\frac{-\mathit{IRRA}}{1.8}}`

Distributed elongation: :math:`{ϵ}_{u}=\mathrm{ln}(1.0+{ϵ}_{u}^{0}.{C}_{w}\text{\_}{ϵ}_{u}.{I}_{r}\text{\_}{ϵ}_{u}.1.0E-02)`

with

:math:`{\epsilon }_{u}^{0}\mathrm{=}5.0\mathrm{-}0.15T+0.0007{T}^{2}`

:math:`{C}_{w}\text{\_}{ϵ}_{u}=1.0`

:math:`{I}_{r}\text{\_}{\epsilon }_{u}\mathrm{=}{e}^{\mathrm{-}\mathit{IRRA}}`


.. csv-table::

    "*Irradiation part*", "*Swelling part*"
    ":math:`{A}_{\mathit{i0}}=3.0E-06{\mathit{MPa}}^{-1}.{\mathit{dpa}}^{-1}`
    :math:`{\eta }_{\mathit{is}}=1000\mathit{MPa.dpa}` "," :math:`R=0.0020{\mathit{dpa}}^{-1}`
    :math:`\alpha =1.0`
    :math:`{\phi }_{\mathrm{0 }}=40.0\mathrm{dpa}`"



Boundary conditions and loads
-------------------------------------

For edges :math:`\mathrm{AB}` and :math:`\mathrm{DC}`, :math:`\mathrm{DY}=0`

For edge :math:`\mathrm{AD}`, :math:`\mathrm{DX}=0`

In addition, a linear temperature ramp having a maximum :math:`400°C` as well as a linear irradiation ramp having a maximum of :math:`140\mathrm{dpa}` are applied.