Calculation of material parameters
==============================

The results of A.A. Almusallam's tensile tests [bib2] _ were used to identify the law of behavior of non-corroded steel and the dependence of plastic deformation at break on the corrosion rate.


Law of behavior of steel
------------------------------

Non-alloy steel is the main type of steel used in building structures. The one-dimensional behavior model of non-alloy steel under unidirectional monotonic loading should be determined from the results of a tensile test on a non-corroded bar or flat specimen. An example of numerical simulation required to determine this law of behavior is presented on [:ref:`Figure 3.2-a <Figure 3.2-a>`].


Taking corrosion into account
-------------------------------

The presence of corrosion has two effects on reinforcement in reinforced concrete structures:


* a reduction in the section;


* a reduction of :math:`{\epsilon }_{R}` based on :math:`{T}_{c}`:

The reduction in section results in a reduction in the diameter for the bars or in a decrease in thickness for the sheets:


.. _RefEquation 3.2- 1:

:math:`{T}_{c}\text{=}\text{100}\left(\frac{{d}_{\text{corrodé}}^{2}}{{d}_{\text{noncorrodé}}^{2}}\right)` or :math:`{T}_{c}\text{=}\text{100}\left(\frac{{e}_{\text{corrodé}}}{{e}_{\text{noncorrodé}}}\right)` eq 3.2- 1

Note: Section reduction is not treated at the behavior model level, it must be taken into account at the command file level in AFFE_CARA_ELEMpar example.

In the uniaxial case, the plastic deformation at break :math:`{\varepsilon }_{R}` depends on the corrosion rate. This evolution is presented on the [:ref:`Figure 3.2-b <Figure 3.2-b>`].

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+ .. image:: images/Forme2.gif                                                                                                                                         + .. image:: images/Object_46.svg                                                                          +
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| .. image:: images/10000000000001D70000015A6B29E8B483CE62B9.png                                                                                                       |                                                                                                          |
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|**Figure 3.2-a: Influence of corrosion on the behavior of steel as a function of the corrosion rate**                                                                 |**Figure 3.2-b: The evolution of plastic deformation at break as a function of the rate** **of corrosion**|
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From these experimental data (tensile tests), the variation in plastic deformation at break as a function of the corrosion rate is deduced:


.. _RefEquation 3.2- 2:

:math:`{T}_{c}<\text{15\%}\Rightarrow \text{}{\epsilon }_{R}\text{=-0}\text{.}\text{0111}\text{Tc+}0\text{.}\text{2345}` eq 3.2- 2


.. _RefEquation 3.2- 3:

:math:`{T}_{c}>\text{15\%}\Rightarrow {\epsilon }_{R}\text{=-0}\text{.}\text{0006}\text{Tc+}0\text{.}\text{051}` eq 3.2- 3

By analysis of various tensile tests, it is observed that the behavior of the corroded reinforcement is almost fragile and :math:`{\epsilon }_{D}\text{=}0\text{.}8\phantom{\rule{0.5em}{0ex}}{\epsilon }_{R}` (:math:`{\epsilon }_{D}` at peak).

In order to integrate the 3D model, the critical cumulative plastic deformation is calculated using :math:`{p}_{R}\text{=}{\left(\frac{2}{3}{\epsilon }_{R}\mathrm{:}{\epsilon }_{R}\right)}^{1/2}` taking into account that the uniaxial state is defined by a one-dimensional state under stress but three-dimensional in deformation [bib5] _:

:math:`{\epsilon }_{R}\text{=}\left[\begin{array}{ccc}{\epsilon }_{R}& 0& 0\\ 0& \text{-}{\nu }^{\text{*}}{\epsilon }_{R}& 0\\ 0& 0& \text{-}{\nu }^{\text{*}}{\epsilon }_{R}\end{array}\right]` eq 3.2-4

where :math:`{\nu }^{\text{*}}` is the contraction coefficient, equal to the Poisson's ratio :math:`\mathrm{\nu }` in elasticity:


.. _RefEquation 3.2-5:

:math:`{\nu }^{\text{*}}\text{=}\nu \frac{{\varepsilon }^{e}}{\varepsilon }\text{+}\frac{1}{2}\frac{{\varepsilon }^{p}}{\varepsilon }\text{=}\frac{1}{2}\text{-}\frac{{\varepsilon }^{e}}{\varepsilon }(\frac{1}{2}\text{-}\nu )` eq 3.2-5

here :math:`\epsilon \text{=}{\epsilon }_{R}` and we approximate :math:`{\mathrm{\varepsilon }}^{e}` by: :math:`{\varepsilon }_{y}\text{=}\frac{{\sigma }_{y}}{E}`


.. _RefEquation 3.2-6:

:math:`{\nu }^{\text{*}}\text{=}\frac{1}{2}\text{-}\frac{{\varepsilon }_{y}}{{\varepsilon }_{R}}(\frac{1}{2}\text{-}\nu )` eq 3.2-6

For the calculation of :math:`{p}_{D}`, it is considered that the triaxiality rate at the damage threshold is identical to that at breakage:


.. _RefEquation 3.2-7:

:math:`{p}_{D}\text{=}\text{0,8}{p}_{R}` eq 3.2-7