Modeling A
==============


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

Unsteady thermal calculation precedes mechanical calculation. Both calculations are done using the same mesh to avoid smoothing phenomena.


+---------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                                 |                                                                                                                                                         |
+ .. image:: images/1000131600000A06000028D38BD41485CA8956F6.svg                                                                  + .. image:: images/10000000000001040000009D241342BDB98FB4B3.png                                                                                          +
|    :width: 129                                                                                                                  |    :width: 3.5736in                                                                                                                                     |
+    :height: 526                                                                                                                 +    :height: 2.0618in                                                                                                                                    +
|                                                                                                                                 |                                                                                                                                                         |
+ **Full mesh**                                                                                                                   + **Zoom on the crack point**                                                                                                                             +
|                                                                                                                                 |                                                                                                                                                         |
+---------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------+



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

The mesh consists of 8651 nodes and 2772 elements, including 2732 QUA8 elements and 40 TRI6 elements.

The radial density of the mesh is determined by successive tests in order to reduce to :math:`\text{1\%}` the difference between the theoretical solution and the numerical solution, both from a thermal and thermomechanical point of view, in the case of the non-cracked bar.

The height of the half-model is set arbitrarily to 5 times radius :math:`R`. It is assumed a priory that the effect of the mesh size limitation in the :math:`Z` direction on the stress intensity factor is less than :math:`\text{1\%}`.

A non-deformable block, located under the lip, has been meshed in order to manage the frictionless contact induced by the closure of the lip.


Test values and modeling results A
-------------------------------------------------


.. csv-table::

    "**Identification**", "**Reference**", "**Aster**", ":math:`\text{\%}` **difference**"
    ":math:`G(\mathrm{Fo}=\mathrm{0,001})({\mathrm{J.m}}^{2})` ", "2,3E+2", "2,9449E+2", "30"
    ":math:`{K}_{I}(\mathrm{Fo}=\mathrm{0,001})({\mathrm{Pa.m}}^{\mathrm{0,5}})` ", "7,0E+6", "8,0438E6*", "14"
    ":math:`G(\mathrm{Fo}=\mathrm{0,04})({\mathrm{J.m}}^{2})` ", "1,0E+4", "1,25016E+4", "19"
    ":math:`{K}_{I}(\mathrm{Fo}=\mathrm{0,04})({\mathrm{Pa.m}}^{\mathrm{0,5}})` ", "4,8E+7", "5,24175E7*", "9"
    ":math:`G(\mathrm{Fo}=1)({\mathrm{J.m}}^{2})` ", "1,0", "1.2104864", "15"
    ":math:`{K}_{I}(\mathrm{Fo}=1)({\mathrm{Pa.m}}^{\mathrm{0,5}})` ", "4,8E+5", "5,1579E+5*", "7"



.. csv-table::

    "*", "Values obtained with the formula IRWIN in plane deformations, assuming that :math:`{K}_{\mathrm{II}}=0`, and taking the :math:`G` calculated by ASTER, which does not allow the automatic calculation of :math:`{K}_{I}` in axisymmetry."



notes
---------

To calculate :math:`{G}_{\mathrm{ref}}`, we use the formulas of IRWIN in plane deformations:

:math:`{G}_{\mathrm{ref}}=\frac{1-{\nu }^{2}}{E}({K}_{I}^{2}+{K}_{\mathrm{II}}^{2})`, :math:`{K}_{\mathrm{II}}=0`

The maximum difference recorded is :math:`\text{30\%}` out of :math:`G` (:math:`\mathrm{Fo}=1`), :math:`\text{14 \%}` out of :math:`{K}_{I}` (:math:`\mathrm{Fo}=1`).

The maximum relative difference in temperature at the bottom of the crack, compared to the analytical solution (summed over 900 terms), is less than :math:`\text{1\%}`.

The maximum relative deviation on :math:`{\sigma }_{\mathrm{zz}}` in the bar before cracking, compared to the analytical solution summed over 900 terms, at the location of the subsequent crack background, is less than :math:`\text{0,5 \%}`.

With ASTER, in axisymmetric mode, the stress field obtained is of the following form:


.. csv-table::

    "SIXX "," SIYY "," SIZZ "," "," SIXY ", "and the associated constraints are:"
    "", "", "", "", ""
    "SIRR "," SIZZ "," SITT "," SIRZ ", ""


To calculate the reference values, we use the curve in :math:`\mathrm{log}/\mathrm{log}` (page 5). Since the accuracy of reading the values is not very good, we can estimate that the results on the :math:`G` energy recovery rate are not too far from the reference.

It should be noted that the energy recovery rate :math:`G` is invariable on the computing cores.