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


Geometry
---------


.. image:: images/Groupe 5.gif


.. _RefSchema_Groupe 5.gif:


.. image:: images/10000200000002FF0000041AB4B6F1BB61585083.png
    :width: 3.8161in
    :height: 5.2244in


.. _RefImage_10000200000002FF0000041AB4B6F1BB61585083.png:


.. csv-table::

    "Length", ":math:`L=400\mathit{mm}`"
    "Width", ":math:`W=100\mathit{mm}`"
    "Crack depth", ":math:`a=50\mathit{mm}`"



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

The material obeys a "non-linear" law of elastic behavior (Hencky's law) from von Mises with linear isotropic work hardening (EMAS_VMIS_LINE).

The properties of the material are as follows:


.. csv-table::

    "Young's module", ":math:`E=\mathrm{2,1}\times {10}^{5}\mathit{MPa}`"
    "Poisson's Ratio", ":math:`\mathrm{\nu }=0.3`"
    "Linear elastic limit", ":math:`{\mathrm{\sigma }}_{y}=150\mathit{MPa}`"
    "Traction curve slope", ":math:`\text{D\_SIGM\_EPSI}=\mathrm{5,25}\times {10}^{4}\mathit{MPa}`"
    "Expansion coefficient", ":math:`\mathrm{\alpha }={1.10}^{-5}`"
    "Reference temperature", ":math:`\text{VALE\_REF}=20°C` (AFFE_MATERIAU keyword)"



Boundary conditions and loading
------------------------------------

For models A and C, the model is limited to half of the structure, the horizontal plane of the crack being a plane of symmetry. For B and D models, the entire structure is represented.

**Boundary conditions**

A and C models:

Horizontal displacement :math:`\mathrm{UX}=0` at point :math:`F`

Vertical displacement :math:`\mathrm{UY}=0` in ligament :math:`\mathit{EF}` (symmetry condition)

B and D modeling:

Horizontal displacement :math:`\mathrm{UX}=0` on segment :math:`\mathit{AB}`

Vertical displacement :math:`\mathit{UY}=0` on segment :math:`\mathit{AB}`

**Loading** mechanical

A and C models:

Normal surface force applied to segment :math:`\mathit{CD}`

B and D modeling:

Horizontal displacement imposed on segment :math:`\mathit{CD}`: :math:`\mathit{UY}=\mathrm{\delta }`

Vertical displacement imposed on segment :math:`\mathrm{CD}`: :math:`\mathit{UX}=\mathrm{\delta }`

**Thermal load**

A to C modeling:

No thermal loading.

D modeling:

Stationary temperature dependent on the :math:`x` axis and increasing linearly between 0°C in :math:`x={x}_{A}=0` and 200°C in :math:`x={x}_{B}=w`. The reference temperature is fixed at 100° C.