E modeling
==============




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

Flexion-shear coupling in the plane.


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    :height: 2.2819in


.. _RefImage_1000020000000131000000F1568056C23582F307.png:

Figure 7.1-a: mesh

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

Boundary conditions (see figure below):


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    :width: 5.9043in
    :height: 2.4035in


.. _RefImage_10000200000002A9000001237C4BFB344F7F3898.png:

Figure 7.1-b: Boundary conditions


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


* :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` and :math:`\mathrm{DRY}=0.0` on edge :math:`{A}_{1}-{A}_{2}`


* :math:`{u}_{x}={D}_{0}\cdot y,{u}_{y}={D}_{0}\cdot L` and DRY = :math:`{R}_{0}` × :math:`f(t)` 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}=1.1{10}^{-4}\cdot f(t)`, :math:`{R}_{0}=6.0{10}^{-3}`, 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: 5.2508in
    :height: 3.0008in


.. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png:




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

Number of knots: 121.

Number of stitches: 200 TRIA3; 40 SEG2.




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

The results obtained by modeling GLRC_DM are evaluated by non-regression tests at various times:


.. csv-table::

    "**Identification**", "**Reference Type**", "**Reference Values**", "**Tolerance**"
    "At t=1,0", "", "", ""
    "*Movement* :math:`\mathrm{DX}` *in* :math:`\mathrm{A2}` "," NON_REGRESSION ", "0", "1 10-6"
    "*Move* :math:`\mathrm{DZ}` *in* :math:`\mathrm{A2}` "," NON_REGRESSION "," ", "-3.0 10-3", "1 10-4"
    "*Effort* :math:`{N}_{\mathrm{yy}}` *in* :math:`\mathrm{A2}` "," NON_REGRESSION "," ", "15058.8134864", "1 10-4"
    "Damage variable :math:`\mathrm{d1}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "1.178108", "1 10-4"
    "Damage variable :math:`\mathrm{d2}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "0.0", "1 10-4"
    "To :math:`t=\mathrm{2,8}` ", "", "", ""
    "*Move* :math:`\mathrm{DX}` *in* :math:`\mathrm{A4}` "," NON_REGRESSION "," ", "-8.8 10-5", "1 10-4"
    "*Move* :math:`\mathrm{DZ}` *in* :math:`\mathrm{A4}` "," NON_REGRESSION "," ", "1.9504378 10-3", "1 10-4"
    "*Effort* :math:`{N}_{\mathrm{yy}}` *in* :math:`\mathrm{A4}` "," NON_REGRESSION "," ", "-12047.0496585", "1 10-4"
    "Damage variable :math:`\mathrm{d1}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "1.178108", "1 10-4"
    "Damage variable :math:`\mathrm{d2}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "0.75196363", "1 10-4"
    "To :math:`t=\mathrm{3,0}` ", "", "", ""
    "Damage variable :math:`\mathrm{d1}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "1.178108", "1 10-4"
    "Damage variable :math:`\mathrm{d2}` in :math:`\mathrm{A1}` "," NON_REGRESSION "," ", "1.162571", "1 10-4"


Comparative diagrams multilayer model-model GLRC_DMmoment flexing :math:`{M}_{\mathit{yy}}` as a function of time:


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    :width: 5.3445in
    :height: 3.0008in


.. _RefImage_100000000000020100000120750D61E2D006F596.png:

Comparative diagrams multilayer model-model GLRC_DMeffort cutting edge :math:`{N}_{\mathit{xy}}` as a function of time:


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    :width: 5.2508in
    :height: 3.0008in


.. _RefImage_10000000000001F8000001202ECF3E416399CEA8.png:

Comparative diagrams multi-layer model GLRC_DMde the shear force bending moment :math:`{N}_{\mathit{xy}}` as a function of the distortion:


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    :width: 5.2508in
    :height: 3.0008in


.. _RefImage_10000000000001F80000012016266EE95B19F831.png:

Comparative diagrams multilayer model-model GLRC_DMdu bending moment :math:`{M}_{\mathit{yy}}` as a function of rotation:


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    :width: 5.3445in
    :height: 3.0008in


.. _RefImage_100000000000020100000120C9411B30A41011B8.png:



Diagram of the evolution of the damage of model GLRC_DM (:math:`{d}_{1},{d}_{2}`) over time:


.. image:: images/100000000000020100000120F47B63C6C1754EAF.png
    :width: 5.3445in
    :height: 3.0008in


.. _RefImage_100000000000020100000120F47B63C6C1754EAF.png:

We check, cf. [:external:ref:`R7.01.32 <R7.01.32>`], that with the data from the test case, we have: :math:`{k}_{0}=\mathrm{9,81138260345866}J/{m}^{2}`, hence the surface densities of energy dissipated:


.. csv-table::

    "Instant", ":math:`{d}_{1}` "," :math:`{d}_{2}` ", "energy dissipated :math:`J/{m}^{2}`"
    ":math:`t=\mathrm{2,0}s` ", "1.1781", "0.0", "11.5589"
    ":math:`t=\mathrm{4,0}s` ", "1.178", "1.161", "1.1626", "22.9653"