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


Geometry
---------

Structure :math:`\mathrm{2D}` is a rectangular plate (:math:`\mathrm{LX}=\mathrm{0,2}m`, :math:`\mathrm{LY}=\mathrm{0,5}m`), with a right central crack, inclined at a variable angle :math:`\theta` with respect to the horizontal axis [:ref:`Figure 1.1-1 <Ref94087906>`]. The crack length is constant (:math:`a=\mathrm{0,04}m`). In this test, the angle :math:`\theta` will successively take the values: :math:`0°`, :math:`15°`,, :math:`30°`,, :math:`45°`, :math:`60°` for modeling :math:`A` and :math:`0°`, :math:`45°` for modeling :math:`B`.

We call the "bottom line" the line in :math:`y=-\mathrm{LY}/2` and "top line" the line in :math:`y=\mathrm{LY}/2`.

The nodes marked :math:`A`, :math:`B`, :math:`C`, and :math:`D` on the :ref:`Figure 1.1-1 <Ref94087906>` serve to impose boundary conditions, which are explained in paragraph [§ :ref:`1.3 <Ref193800026>`].


.. image:: images/10000000000002B7000003C6BFA1BC58A85151FA.png
    :width: 4.1134in
    :height: 5.0752in


.. _RefImage_10000000000002B7000003C6BFA1BC58A85151FA.png:


.. _Ref94087906:

**Figure** 1.1-1 **: geometry of the cracked plate and loading of the modelling** :math:`A` **.**


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

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

Poisson's ratio: modeling A :math:`\nu =0.3`, modeling B :math:`\nu =0`

Density: B modeling: :math:`\rho =7800\mathrm{kg}/{m}^{3}`


.. _Ref193800026:


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

*Modeling* :math:`A` *:*

Loading consists of applying a force distributed over the lower and upper lines :math:`p={10}^{6}\mathrm{Pa}`.

In order to block rigid modes, we block the movements of the nodes :math:`A`, :math:`B`, :math:`C` and :math:`D` as follows:


* :math:`{\mathrm{DY}}^{A}={\mathrm{DY}}^{B}=0`;


* :math:`{\mathrm{DX}}^{C}={\mathrm{DX}}^{D}=0`.

*Modeling* :math:`B` *:*

Charging consists in applying to the plate an embedment on the upper line and a volume force of type FORCE_INTERNE or PESANTEUR. We make sure to apply the same load (following :math:`-Y`) for two successive calculations with these key words: for the first imposed force density of :math:`78000N/{m}^{3}` and for the second one chooses an acceleration due to gravity equal to :math:`10{\mathrm{m.s}}^{-2}` and therefore a force density of :math:`10.\rho =78000N/{m}^{3}`.

*Models* :math:`C` *,* :math:`D` *,* *,* :math:`E` *:*

Same assumptions as modeling :math:`A`

*Modeling* :math:`F` *:*

The tensile load on the upper and lower faces is replaced by pressure on the lips of the crack (the 2 loads are equivalent)


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

*Modeling* :math:`A` *:*

The analytic expressions for the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` are functions of the distributed force :math:`p`, the crack length :math:`a`, the plate width :math:`\mathrm{LX}`, and the angle :math:`\theta`:

:math:`\begin{array}{}{K}_{I}=p\sqrt{\pi \frac{a}{2}}F(\frac{a}{\mathrm{Lx}})\mathrm{cos²}\theta \\ {K}_{\mathrm{II}}=p\sqrt{\pi \frac{a}{2}}F(\frac{a}{\mathrm{Lx}})\mathrm{cos}\theta \mathrm{sin}\theta \end{array}`

where function :math:`F` can be determined in several different ways. We choose the one obtained by Brown in 1966 [:ref:`bib2, p41 <bib2, p41>`], whose precision is less than :math:`\text{0,5\%}` if the ratio between the length of the crack and the width of the plate is less than or equal to :math:`\mathrm{0,7}` (in our case, :math:`a/\mathrm{LX}=\mathrm{0,2}`):


.. image:: images/Object_3.svg
    :width: 246
    :height: 26


.. _RefImage_Object_3.svg:

With the numerical values of the test:


+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|Reference                                                                                                                                       |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|:math:`\alpha` (°)|:math:`{K}_{I}({\mathit{Pa.m}}^{\mathrm{-}1\mathrm{/}2})`|:math:`{K}_{\mathit{II}}({\mathit{Pa.m}}^{\mathrm{-}1\mathrm{/}2})`|
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|0                 |2.5725024656 105                                         |0                                                                  |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|15                |2,4001774761 105                                         |6,4312561642 104                                                   |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|30                |1.9293768492 105                                         |1,1139262432 105                                                   |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|45                |1,2862512328 105                                         |1,2862512328 105                                                   |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+
|60                |6,4312561641 104                                         |1,1139262432 105                                                   |
+------------------+---------------------------------------------------------+-------------------------------------------------------------------+

**Table** 1.4-1 **: reference values for** :math:`{K}_{I}` **and** :math:`{K}_{\mathrm{II}}`

Theoretically, these values are the same for both crack bottoms.

*Modeling* :math:`B` *:*

Non-regression tests are carried out.

*Models* :math:`C` *,* :math:`D` *,* *,* :math:`E` *,* :math:`F` *:*

We use the reference values from 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>`]


2. TADA H., PARIS P., IRWIN G.:The stress analysis of cracks handbook, 3rd ed., 2000