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


Geometry
---------


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Figure 1.1-1: geometry of the cracked cube

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


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

The material is isotropic elastic whose properties are:

:math:`E\mathrm{=}200000\mathrm{MPa}`

: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.


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Figure 1.3-1: Symmetry condition

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}`.


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Figure 1.3-2: Constraints in the global coordinate system


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Figure 1.3-3: Local coordinate system

So, we have:

on the upper side:


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


* :math:`{\sigma }_{\mathit{ZX}}\mathrm{=}\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}}\mathrm{=}\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`.


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Figure 1.3-4: Blocking rigid modes