Implemented in Code_Aster
=============================

The mechanical calculation is carried out under the hypothesis of thermo-elasto-plastic behavior associated with a Von Mises criterion with isotropic or linear kinematic hardening (VMIS_ISOT_TRAC, VMIS_ISOT_LINE, VMIS_CINE_LINE). We will specify what the calculation of :math:`{G}_{p}` consists of and how to identify the material parameters :math:`R=D/2` (radius of the cut) and :math:`{G}_{\mathit{pc}}` (break limit).

The methodology for calculating and identifying with Code_Aster is presented in [:external:ref:`U2.05.08 <U2.05.08>`]. The easy to use documentation is in [U4.82.31].




Gp calculation
------------

The calculation of :math:`{G}_{p}`, carried out using the CALC_GP macro command, is based on the use of POST_ELEM which allows the calculation of elastic energy on a group of elements. The models (finite elements, small deformations, etc.) and loads that can be used are those of the command POST_ELEM, keyword ENER_ELTR. More precisely, for each moment provided in the list of calculation moments, it is a question of carrying out the following two steps:

1/ First calculate quantity :math:`\stackrel{~}{{G}_{p}}(\mathrm{\Delta }a)` for increasing values of :math:`\mathrm{\Delta }a` by:

:math:`\stackrel{~}{{G}_{p}}(\mathrm{\Delta }a)=\frac{{\int }_{\mathrm{\Omega }}{\mathrm{\Phi }}_{t}^{\mathit{el}}d\mathrm{\Omega }}{\mathrm{\Delta }a}`

In 2D, it is therefore necessary to identify the elements of zone :math:`C\left(\mathrm{\Delta }a\right)` by a group of elements defined at the level of the mesh as presented in [:ref:`Figure 2-a <Figure 2-a>`], or by a geometric zone of Gauss points, then calculate the elastic energy on this zone and then divide it by :math:`\mathrm{\Delta }a`.

To identify the elements of zone :math:`C\left(\mathrm{\Delta }a\right)`, we will proceed as follows: the elements of the first chip will constitute a first group of elements, the elements of chips 1 and 2 will constitute a second group, the elements of chips :math:`\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...},i` will constitute a :math:`i` th group, etc. We must provide a number of chips large enough to be able to find the maximum of :math:`\stackrel{~}{{G}_{p}}(\mathrm{\Delta }a)`, which is most often located at a distance of approximately :math:`3R` from the bottom of the notch.


.. image:: images/10000201000002DA000001296C0DC404C366C84F.png
    :width: 4.0374in
    :height: 1.5543in


.. _RefImage_10000201000002DA000001296C0DC404C366C84F.png:

Figure 6.1-1- Definition of chips in the mesh

2/ Then it is a question of identifying the maximum of this function:

:math:`{G}_{p}=\underset{\mathrm{\Delta }a}{\mathit{max}}\stackrel{~}{{G}_{p}}(\mathrm{\Delta }a)`

Examples and usage tips can be found in document [:ref:` **U2.05.01**  < **U2.05.01** >`], in the ssnp131 tests (see [:external:ref:`V6.03.131 <V6.03.131>`] and ssnv218 (see [:external:ref:`V6.04.218 <V6.04.218>`]).


.. _RefHeading__2416_1933018034:


Identifying parameters
-----------------------------

The energy model is based on the pair of material parameters :math:`({G}_{\mathit{pc}},R)`, which must therefore be determined at each temperature. Note that :math:`{G}_{\mathit{pc}}` actually depends on the notch radius [:ref:`WAD 13 <WAD 13>`]. We will see that the prediction of the breakup does not depend on it.

On the one hand, the Young modulus :math:`E` and the critical stress :math:`{\sigma }_{c}` of a material are assumed to be known. On the other hand, it is hypothesized to know toughness :math:`{K}_{\mathit{Jc}}` evaluated experimentally from a tensile test on a CT specimen, for example. The application of the "minimum in relation to damage" at the scale of a material point in a state of simple tensile stress gives the relationship between the energy volume dissipated by a material point that is damaged :math:`{w}_{c}` and the critical stress [WAD 13] *(hypothesis H11)*:


.. math::
 :label: eq-24

    {w} _ {c} =\ frac {{\ mathrm {\ sigma}} _ {c} ^ {2}} {2E}

We then obtain the following equation which must be solved in order to identify the two parameters:


.. math::
: label: eq-25

    {G} _ {\ mathit {pc}} (R) =\ frac {{\ mathrm {\ sigma}} _ {c} ^ {2}} {2}} {E}} R

Left-hand member :math:`{G}_{\mathit{pc}}\left(R\right)` is a nonlinear function of :math:`R`. The right-hand side is a linear function of :math:`R`. Thus, to solve (25) it is necessary to calculate :math:`{G}_{\mathit{pc}}` for different values of :math:`R` as illustrated in figure [:ref:`Figure 6.2-1 <Figure 6.2-1>`].


.. image:: images/100002000000054C000003C4874D43DF0BCEBDFA.png
    :width: 3.9472in
    :height: 2.8063in


.. _RefImage_100002000000054C000003C4874D43DF0BCEBDFA.png:

Figure 6.2-1: identifying the :math:`({G}_{\mathit{pc}},R)` couple [:ref:`WAD 13 <WAD 13>`]

For each notch with a given radius :math:`R`, the parameter :math:`{G}_{\mathit{pc}}` is determined by simulating a test on a test specimen :math:`\mathit{CT}` in the following manner: for each value of the load, increasing from 0 to a critical value, the parameter :math:`{K}_{J}` is calculated on the one hand and the parameter :math:`{G}_{p}` on the other hand is calculated. For the critical load value corresponding to :math:`{K}_{J}={K}_{\mathit{Jc}}`, we get :math:`{G}_{p}={G}_{\mathit{pc}}` *(Hypothesis H12)*.

Solving equation (25) for various values of :math:`R` finally makes it possible to determine both the value of :math:`R` and the value of :math:`{G}_{\mathit{pc}}` at a given temperature. In the case of steel, values between 10 and 100 microns are found, which is representative of a real crack.