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


Calculation method
-----------------

Analytical solution for damage variable :math:`D`:

:math:`D(t)\mathrm{=}1\mathrm{-}{(1\mathrm{-}(1+k){(\frac{{\sigma }_{0}}{A})}^{R}t)}^{\frac{1}{1+k}}`


.. image:: images/Object_2.png
    :width: 4.3646in
    :height: 2.8752in


.. _RefSchema_Object_2.png:

Analytical solution for the viscoplastic isotropic work hardening variable, :math:`r`, in the case of a zero :math:`{\sigma }_{Y}` threshold:

:math:`r(t)\mathrm{=}{\left[\frac{(M+N)}{M(1+k\mathrm{-}N)}{(\frac{{\sigma }_{0}}{A})}^{\mathrm{-}R}{(\frac{{\sigma }_{0}}{K})}^{N}(1\mathrm{-}{(1\mathrm{-}(1+k){(\frac{{\sigma }_{0}}{A})}^{R}t)}^{\frac{1+k\mathrm{-}N}{1+k}})\right]}^{\frac{M}{M+N}}`


.. image:: images/Object_4.png
    :width: 4.3646in
    :height: 3.3335in


.. _RefSchema_Object_4.png:

In the previous expressions, :math:`D` is the damage variable corresponding to the internal variable :math:`\mathit{V9}` and :math:`r` is the multiplicative viscoplastic work hardening variable corresponding to the internal variable :math:`\mathit{V8}`.

We also have the following correspondence, in relation to the parameters of the VENDOCHAB keyword:

:math:`N\mathrm{=}{N}_{\mathit{VP}}`

:math:`M\mathrm{=}{M}_{\mathit{VP}}`

:math:`K\mathrm{=}{K}_{\mathit{VP}}`

:math:`A\mathrm{=}{A}_{D}`

:math:`R\mathrm{=}{R}_{D}`

:math:`k\mathrm{=}{K}_{D}`


Reference quantities and results
-----------------------------------

Evolution of the damage variable, :math:`D`, as a function of time. This value is tested at various times:


.. csv-table::

    "**Instant**", "**Reference**"
    "520000", "1.52596E-02"
    "1000000", "3.30676E-02"
    "2000000", "9.9465369E-02"
    "2250000", "1.37520763E-01"
    "2500000", "2.66018229E-01"


Evolution of the viscoplastic isotropic work hardening variable, :math:`r`, as a function of time. This value is tested at various times:


.. csv-table::

    "**Instant**", "**Reference**"
    "520000", "2.300147E-03"
    "1000000", "3.179469E-03"
    "2000000", "4.95103E-03"
    "2250000", "5.592847E-03"
    "2500000", "6.99749E-03"


The difference observed on :math:`D` for :math:`t=2.5{10}^{6}s` is due to the very high non-linearity of the evolution of the damage variable.


Uncertainties about the solution
----------------------------

Code accuracy