C modeling
==============


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


* Tested behavior: gursonTvergaardNeedlemanplasticflow_numericaljacobian.mfront

Elasto-plastic law with Gurson damage (model GTN in the literature).

Ref: "Analysis of the cup cone fracture in a round tensile bar".

V.Tvergaard, A.Needleman, Acat Metallurgica 32 (1984) 157-169

the criterion is of the shape (f is the porosity)

:math:`F(\sigma ,f)={(\frac{{\sigma }_{\mathit{eq}}}{{\sigma }_{0}})}^{2}+2{q}_{1}f\mathrm{cosh}(\frac{3}{2}{q}_{2}\frac{{\sigma }_{h}}{{\sigma }_{0}})-1-{({q}_{1}f)}^{2}\le 0`


*

Isotope work hardening is modified as follows (

:math:`R(p)=\mathit{R0}+{Q}_{1}(1–{e}^{-{b}_{1}p})+{Q}_{2}(1–{e}^{-{b}_{2}p})`


* Modeling: material point subject to imposed deformations:

:math:`{\epsilon }_{\mathit{xx}}=\mathrm{0,02}t`, :math:`{\epsilon }_{\mathit{yy}}=\mathrm{0,1}t` :math:`{\epsilon }_{\mathit{zz}}=0`


* Material properties:


.. csv-table::

    "Young", "200000"
    "Fish", "0.3"
    "R0", "500"
    "Q1", "1e9"
    "b1", "1-e-6"
    "Q2", "0"
    "b2", "0"
    "f0", "1-e-3"
    "fc", "1-e-2"
    "delta", "2"
    "beta", "0"
    "Cp", "1."
    "app", "1 e-5"
    "q1", "1.5"
    "q2", "1"



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

     Response curve :math:`{\sigma }_{\mathit{yy}}=f({\epsilon }_{\mathit{yy}})` is:


.. image:: images/10000000000007F100000823845EFB046544CF42.png
    :width: 4.2819in
    :height: 3.828in


.. _RefImage_10000000000007F100000823845EFB046544CF42.png:

     The tests are non-regression.


.. csv-table::

    "**Identification**", "**Instants**", "**Reference**", "**Tolerance**"
    ":math:`{\sigma }_{\mathit{xx}}` ", "0.3", "910.12"," 0.1%"
    ":math:`{\sigma }_{\mathit{yy}}` ", "0.3", "1069.37"," 0.1%"
    ":math:`{\sigma }_{\mathit{zz}}` ", "0.3", "870.308"," 0.1%"