Benchmark solution
=====================


Tested sizes
-----------------

The quantities of interest are the cracking times obtained by the index method and by the BaBy (Barbier-Bystricky) model [:ref:`1 <1>`].

The crack time (in hours) of the index model is given by the following expression:

:math:`{t}_{f}=\frac{10000}{{i}_{T}\mathrm{.}{i}_{M}\mathrm{.}{i}_{\sigma }}`

where


* :math:`{i}_{T}={\mathrm{9,49.10}}^{15.}\mathrm{exp}(\frac{-22000}{T+\mathrm{273,15}})` with :math:`T` in °C. For :math:`T=350°C`, :math:`{i}_{T}=\mathrm{4,41}`


* :math:`{i}_{M}=1`


* :math:`{i}_{\sigma }={\mathrm{2,44.10}}^{-11}{\sigma }^{4}`, with :math:`\sigma` the biggest main constraint (expressed as :math:`\mathit{MPa}`)

The cracking time of the BaBy model is obtained by solving a coupled mechanical-corrosion problem. A simplified version of solving the problem consists in decoupling the equations and treating the corrosion equations only by post-processing a non-linear mechanical calculation. This is the strategy adopted in modeling A.

From the stress field :math:`\sigma`, the evolution of the cumulative plastic deformation :math:`p` and the concentration of corrosive agent :math:`{c}_{h}` is determined during post-treatment:

:math:`\begin{array}{c}\dot{p}={(\frac{{\Vert \sigma \Vert }_{\mathit{VM}}}{K(T)(1-{c}_{h})})}^{n}\mathrm{.}{p}^{-n/m}\\ \dot{{c}_{h}}=\frac{3}{2}\beta \dot{p}\frac{⟨1/3\mathit{tr}\sigma ⟩}{{\Vert \sigma \Vert }_{\mathit{VM}}}\end{array}`

then the time to crack is the time such as :math:`{c}_{h}({t}_{f})=1`.

Parameter :math:`\beta` is a material parameter identified from a creep experiment alone.

Here, :math:`\beta =\mathrm{889,5}`.


Bibliographical references
---------------------------


1. G. Barbier, P. Bystricky, Application of the BaBy model to the prediction of diffusion times by Stress Corrosion of Alloy 600 in the primary environment, Note EDF R&D HT-2C/97/048/A, March 1998