D modeling
==============


Characteristics of modeling
-----------------------------------

The load is of the distortion and pure shear type in the plane.


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+ .. image:: images/1000020000000171000001169669FCCAEC11AD67.png                                                              + .. image:: images/10000200000001540000011D5218302EC2AD09C1.png                                                             +
|    :width: 2.9236in                                                                                                         |    :width: 2.5874in                                                                                                        |
+    :height: 2.2075in                                                                                                        +    :height: 2.252in                                                                                                        +
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**Figure** 6.1-a **:** m**mesh and boundary conditions**

Modeling: DKT. :math:`L=1.0m`.

Boundary conditions (see figure above on the right) so that the plate is subject to pure distortion: :math:`{\varepsilon }_{\text{xy}}` must be constant or to pure shear so forces are applied. Therefore, the following displacement field is applied to the edges of the plate for distortion:

:math:`\mathrm{\{}\begin{array}{c}{u}_{x}\mathrm{=}{D}_{0}\mathrm{\cdot }y\\ {u}_{y}\mathrm{=}{D}_{0}\mathrm{\cdot }x\end{array}\mathrm{\Rightarrow }{\varepsilon }_{\mathit{xy}}\mathrm{=}\frac{1}{2}({u}_{x,y}+{u}_{y,x})\mathrm{=}{D}_{0}`

So:


* we impose an embedding in :math:`{A}_{1}`,


* :math:`{u}_{x}={D}_{0}\cdot y,{u}_{y}=0` on edge :math:`{A}_{1}-{A}_{3}`, :math:`{u}_{x}=0,{u}_{y}={D}_{0}\cdot x` on edge :math:`{A}_{1}-{A}_{2}`,


* :math:`{u}_{x}={D}_{0}\cdot y,{u}_{y}={D}_{0}\cdot L` on edge :math:`{A}_{2}-{A}_{4}`, :math:`{u}_{x}={D}_{0}\cdot L,{u}_{y}={D}_{0}\cdot x` on edge :math:`{A}_{3}-{A}_{4}`,

where :math:`{D}_{0}=3.3{10}^{-4}` and :math:`f(t)` represent the magnitude of cyclic loading as a function of the (pseudo-time) parameter :math:`t`, defined as:


.. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png
    :width: 3.7063in
    :height: 2.1181in


.. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png:

**Figure** 6.1-b **: loading function**

Integration increment: :math:`0.05s`.


Characteristics of the mesh
----------------------------

Knots: 121.

Stitches: 200 TRIA3; 40 SEG2.


Tested sizes and results
------------------------------

For the distortion, the shear force :math:`{N}_{\mathrm{xy}}` and :math:`B` obtained by the two models are compared; the tolerances are taken in absolute values based on these relative differences:


.. _DdeLink__15507_23373940:


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "DIST. **POS .** - PHASE CHAR. ELAS. :math:`t=\mathrm{0,25}` ", "", "", ""
    "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "2 10-1"
    "DIST. **POS. - PHASE** CHAR. **ENDO .** :math:`t=\mathrm{1,0}` ", "", "", ""
    "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "1 10-1"
    "DIST. **POS. - PHASE****** DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", ""
    "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "4 10-1"
    "DIST. **NEG. - PHASE** CHAR. **ELAS .** :math:`t=\mathrm{3,0}` ", "", "", ""
    "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "1 10-1"
    "DIST. **NEG. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{3,5}` ", "", "", ""
    "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "4 10-1"


Shearing force diagram :math:`{N}_{\mathrm{xy}}` **(in the plan)** as a function of time:


.. image:: images/10000201000002C800000235D0FB5D0038CDFC5B.png
    :width: 4.6457in
    :height: 3.4484in


.. _RefImage_10000201000002C800000235D0FB5D0038CDFC5B.png:

**shear force graph** :math:`{N}_{\mathit{xy}}` **** (in the plan) ****based on**:math:`{D}_{0}`**imposed: **


.. image:: images/10000201000002C80000021FC4C33D1BF8544398.png
    :width: 4.6457in
    :height: 3.3425in


.. _RefImage_10000201000002C80000021FC4C33D1BF8544398.png:


notes
---------

A similar shear behavior is observed for laws BETON_REGLE_PRet ENDO_ISOT_BETON under load: the shear stiffness is however higher for law BETON_REGLE_PR (~ 15%).

The landfill response is not taken into account by law BETON_REGLE_PR (elastic response).

The shear response for BETON_REGLE_PRest symmetric unlike the ENDO_BETON_ISOT law for which the memory of cracking in one direction is retained.


.. _RefNumPara__6725_1772789992:


.. _refheading__373859461: