Reference problem
====


Geometry
----

The crack is circular (penny shaped crack) with radius :math:`a=\mathrm{0,2}m` in the :math:`\mathit{OXY}` plane. The side of the cube is :math:`L=2m=10a` long. Thus, it is considered that the crack is in an infinite medium.


.. image:: images/100000000000034E000002C1E48DED210B8710B1.png
    :width: 4.3102in
    :height: 3.5909in


.. _RefImage_100000000000034E000002C1E48DED210B8710B1.png:

**Figure** 1.1-1 **: geometry** of the cracked cube
2222

Material properties
----

We consider a homogeneous isotropic elastic linear material whose characteristics are the following:


*   Young's module :math:`E=200000\mathit{MPa}`,


*   Poisson's ratio :math:`\nu \mathrm{=}\mathrm{0,3}`.


Boundary conditions and loads
----

Given the symmetries of the structure, the crack, and the loads, only half of the structure is modeled: half-space such as :math:`Y>0`. Symmetry conditions are therefore applied to the face in :math:`Y\mathrm{=}0`: on this face, the following movement :math:`Y` is blocked.

The structure is subject to a :math:`{\sigma }_{\mathit{xx}}\mathrm{=}1\mathit{MPa}` constraint. The local coordinate system :math:`(x,y,z)` is obtained by rotating an angle :math:`\alpha \mathrm{=}\pi \mathrm{/}4` around the axis :math:`\mathit{OY}`.


.. image:: images/10000000000002B20000022EE428EC6E1D6BBC34.png
    :width: 4.3311in
    :height: 3.389in


.. _RefImage_10000000000002B20000022EE428EC6E1D6BBC34.png:

**Figure** 1.3-1 **: Constraints in the global coordinate system**


.. image:: images/100000000000015B000000E8C71967D18D36A085.png
    :width: 2.952in
    :height: 1.9736in


.. _RefImage_100000000000015B000000E8C71967D18D36A085.png:

**Figure** 1.3-2 **: Local location**

So, we have:

on the upper side:


*   :math:`{\sigma }_{\mathit{ZZ}}=\sigma {\mathrm{sin}}^{2}(\alpha )`


*   :math:`{\sigma }_{\mathit{ZX}}=\sigma \mathrm{cos}(\alpha )\mathrm{sin}(\alpha )`

on the underside:


*   :math:`{\sigma }_{\mathit{ZZ}}\mathrm{=}\mathrm{-}\sigma {\mathrm{sin}}^{2}(\alpha )`


*   :math:`{\sigma }_{\mathit{ZX}}\mathrm{=}\mathrm{-}\sigma \mathrm{sin}(\alpha )\mathrm{cos}(\alpha )`

on the right side face:


*   :math:`{\sigma }_{\mathit{XX}}\mathrm{=}\sigma {\mathrm{cos}}^{2}(\alpha )`


*   :math:`{\sigma }_{\mathit{XZ}}\mathrm{=}\sigma \mathrm{cos}(\alpha )\mathrm{sin}(\alpha )`

on the left side face:


*   :math:`{\sigma }_{\mathit{XX}}\mathrm{=}\mathrm{-}\sigma {\mathrm{cos}}^{2}(\alpha )`


*   :math:`{\sigma }_{\mathit{XZ}}=-\sigma \mathrm{cos}(\alpha )\mathrm{sin}(\alpha )`

In order to block rigid body modes, the point :math:`\mathit{D1}` :math:`(L\mathrm{/}\mathrm{2,0}\mathrm{,0})` is blocked following :math:`X` and :math:`Z` and the point :math:`\mathit{D2}` :math:`(\mathrm{-}L\mathrm{/}\mathrm{2,0}\mathrm{,0})` is blocked following :math:`Z`.


.. image:: images/10000000000002C200000249CE148B0A9575D322.png
    :width: 3.6772in
    :height: 3.0472in


.. _RefImage_10000000000002C200000249CE148B0A9575D322.png:

**Figure** 1.3-3 **: Blocking rigid modes**