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


Geometry
---------

Structure :math:`\mathrm{2D}` is a unitary square plate (:math:`\mathit{LX}\mathrm{=}\mathrm{0,1}`, :math:`\mathit{LY}\mathrm{=}\mathrm{0,1}`). A "straight" interface divides the domain into 2 sub-domains.

The interface branches off in the vicinity of the center of square :math:`(\mathrm{0,}0.5)` with an angle :math:`\theta \mathrm{=}\mathrm{67,5}°` from the horizontal.


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    :height: 3.3055in


.. _RefImage_10000201000000EE000000E060973FFD7F104D23.png:

**Figure** 1.1-1 **: Geometry of the cracked plate**

The interface is defined analytically by the union of 2 line segments, whose respective equations are:


* :math:`X=0` for the vertical segment.


* :math:`Y\mathrm{=}2x+0.55` for the oblique segment.

The intersection point of the 2 line segments is calculated to be located on the edge where the interface forks:


* :math:`(\mathrm{0,}0.55)` for :math:`A` and :math:`B` models, see


* :math:`(\mathrm{0,}0.6)` for :math:`C` and :math:`D` models, see


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    :height: 4.1535in


.. _RefImage_1000000000000360000002098B3E85B805699CF1.png:

Figure 1.1-2: "zoom" in the vicinity of the bifurcation point


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    :width: 6.889in
    :height: 4.1535in


.. _RefImage_10000201000003600000020958070A9972EF1609.png:

Figure 1.1-3: "zoom" in the vicinity of the bifurcation point


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

Young's module: :math:`E=210{10}^{9}\mathrm{Pa}`

Poisson's ratio: :math:`\nu =0`


.. _Ref193800026:


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

*Modeling* :math:`A`:

The interface divides the domain into 2 sub-domains. The solution is imposed on the move on each sub-domain, thanks to Dirichlet conditions at the edges:


* on edge :math:`\mathit{LIG2}`: :math:`\mathit{DX}=-1` and :math:`\mathit{DY}=0`


* on edge :math:`\mathit{LIG4}`: :math:`\mathit{DX}=\text{+}1` and :math:`\mathit{DY}=0`

There is no contact at the interface level. We are just interested in the movements of the 2 blocks, which can interpenetrate each other.

The imposed Dirichlet conditions also block rigid body movements in the 2 sub-domains.

*Models* :math:`B`, :math:`C` and :math:`D`:

Same load as in modeling :math:`A`.


Benchmark solution
---------------------

*Modeling* :math:`A`:

By construction, the field of movement is uniform on each sub-block.


* On the sub-domain on the right, we have: :math:`\mathit{DX}=-1` and :math:`\mathit{DY}=0`


* On the sub-domain on the left, we have: :math:`\mathit{DX}=\text{+}1` and :math:`\mathit{DY}=0`

*Modeling* :math:`B`:

Same analytical solution as modeling :math:`A`

*Modeling* :math:`C`:

Same analytical solution as modeling :math:`A`

*Modeling* :math:`D`:

Same analytical solution as modeling :math:`A`


Bibliographical references
---------------------------



.. _Ref112833260:


.. _Ref112833703:

1. GENIAUT S., MASSIN P.: eXtended Finite Element Method, *Code_Aster* Reference Manual, [:ref:`R7.02.12 <R7.02.12>`]