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

Geometry
---------


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

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

It is a cylinder with radius :math:`1\mathrm{mm}` and height :math:`1\mathrm{mm}`.

The square in bold corresponds to axisymmetric modeling.


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

The material properties are dependent on the type of modeling and functions of the temperature in :math:`°C` and on the irradiation in :math:`\mathrm{dpa}` (displacement by atom).

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

*For all models*

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`

**For modeling a**

*Plastic part*

:math:`\kappa =1.0`

Elastic limit at 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}=3.0`

:math:`{I}_{r}\text{\_}{R}_{e}=\left(2.0–{e}^{\frac{-\mathit{IRRA}}{3}}\right)`

Ultimate compulsion in :math:`\mathrm{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}=0.25-0.10\left(1.0-{e}^{\frac{-\mathit{IRRA}}{10.0}}\right)+{e}^{\frac{-\mathit{IRRA}}{3.0}}`

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

with

:math:`{ϵ}_{u}^{0}=50.0-0.15T+0.0007{T}^{2}`

:math:`{C}_{w}\text{\_}{ϵ}_{u}=0.25`

:math:`{I}_{r}\text{\_}{ϵ}_{u}={e}^{\frac{-\mathit{IRRA}}{2}}`


.. csv-table::

    "*Irradiation part*", "*Swelling part*"
    ":math:`{A}_{\mathrm{i0}}=0.0{\mathrm{MPa}}^{-1}\mathrm{.}{\mathrm{dpa}}^{-1}`
    :math:`{\eta }_{\mathit{is}}=1.0E+50\mathit{MPa.dpa}` "," :math:`R=0.0{\mathrm{dpa}}^{-1}`
    :math:`\alpha =0.0`
    :math:`{\phi }_{\mathrm{0 }}=0.0\mathrm{dpa}`"


**For modeling b**

*Plastic part*

:math:`{R}_{02}=5.0E+09\mathit{Mpa}` :math:`{R}_{m}=5.0E+09\mathit{Mpa}` :math:`{\varepsilon }_{u}=0.0`


.. csv-table::

    "*Irradiation part*", "*Swelling part*"
    ":math:`{A}_{\mathit{i0}}=2.0E-06{\mathit{MPa}}^{-1}.{\mathit{dpa}}^{-1}`
    :math:`{\eta }_{\mathrm{is}}=1000.0\mathrm{MPa.dpa}` "," :math:`R=0.0{\mathrm{dpa}}^{-1}`
    :math:`\alpha =0.0`
    :math:`{\phi }_{0}=0.0\mathrm{dpa}`"


**For c modeling**

*Plastic part*

:math:`{R}_{02}=5.0E+09\mathit{Mpa}` :math:`{R}_{m}=5.0E+09\mathit{Mpa}` :math:`{\varepsilon }_{u}=0.0`


.. csv-table::

    "*Irradiation part*", "*Swelling part*"
    ":math:`{A}_{\mathrm{i0}}=0.0{\mathrm{MPa}}^{-1}\mathrm{.}{\mathrm{dpa}}^{-1}`
    :math:`{\eta }_{\mathit{is}}=1.0E+06\mathit{MPa.dpa}` "," :math:`R=0.0025{\mathit{dpa}}^{-1}`
    :math:`\alpha =1.0`
    :math:`{\phi }_{0}=1.0\mathrm{dpa}`"



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

**Modeling a**

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 with a maximum of :math:`400°C` is applied.

**Modeling b**

For edge :math:`\mathrm{AB}`, :math:`\mathrm{DY}=0`

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

For edge :math:`\mathrm{DC}`, apply a linear ramp of linear forces with a maximum value of :math:`\mathrm{FY}=\mathrm{200 }N/\mathrm{mm}`

In addition, a linear irradiation ramp with a maximum of :math:`10\mathrm{dpa}` and a temperature ramp with a maximum of :math:`400°C` are applied.

**C modeling**

For the :math:`\mathrm{AB}` :math:`\mathrm{DY}=0` edge

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