Reference problem
=====================

The damage :math:`D(t)` is calculated from the data of the stress tensor :math:`\sigma (t)` and the cumulative plastic deformation :math:`p(t)`.


.. csv-table::

    ":math:`\dot{D}=\frac{1}{{(1-D)}^{2}}\left[\frac{1}{\mathrm{3ES}}(1+\nu ){\sigma }_{\mathrm{eq}}^{2}+\frac{3}{\mathrm{2ES}}(1-2\nu ){\sigma }_{H}^{2}\right]\dot{p}` ", "if :math:`p>{p}_{d}`"
    ":math:`D=0` ", "otherwise"


:math:`{\sigma }_{\mathrm{eq}}` is the equivalent von Mises stress

:math:`{\sigma }_{H}` is the hydrostatic stress

:math:`{p}_{d}` represents the damage threshold

:math:`S` is a material characteristic (:math:`\mathrm{MPa}`)

The total damage :math:`D=\underset{i=1}{\overset{N}{\Sigma }}D({t}_{i})` is also calculated.


Material properties
-----------------------


.. csv-table::

    ":math:`\mathrm{Temp}(°C)` "," :math:`E(\mathrm{MPa})` "," :math:`\nu` "," :math:`S(\mathrm{MPa})`"
    "0. ", "2.E+5", "0. ", "12/07/09"
    "20. ", "2.E+5", "0. ", "7."
    "40. ", "2.E+5", "0. ", "7."


:math:`{p}_{d}=0.02`


Boundary conditions and loading
-----------------------------------

**Loading history:**


.. csv-table::

    ":math:`t` ", "43.11", "100. ", "100. ", "10000. ", "2000. ", "21000. ", "2000. ", "2200. ", "22400."
    ":math:`{\sigma }_{\mathrm{xx}}(t)` ", "300. ", "300. ", "300. ", "300. ", "300. ", "300. ", "300. ", "300. ", "300."
    ":math:`{\sigma }_{\mathrm{yy}}(t)` ", "0. ", "0. ", "0. ", "0. ", "0. ", "0. ", "0. ", "0. ", "0."
    ":math:`{\sigma }_{\mathrm{zz}}(t)` ", "", "", "", "", "", "", "", "", "", ""
    ":math:`{\sigma }_{\mathrm{xy}}(t)` ", "", "", "", "", "", "", "", "", "", ""
    ":math:`{\sigma }_{\mathrm{xz}}(t)` ", "", "", "", "", "", "", "", "", "", ""
    ":math:`{\sigma }_{\mathrm{yz}}(t)` ", "", "", "", "", "", "", "", "", "", ""
    ":math:`\mathrm{Temp}` ", "20. ", "20. ", "20. ", "20. ", "20. ", "20. ", "20. ", "20. ", "20."



.. csv-table::

    ":math:`t` "," :math:`p(t)` **(Cumulative plastic deformation)**"
    "43.11", "0.019996"
    "100. ", "0.046384"
    "1000. ", "0.46384"
    "10000. ", "4.6384"
    "20000. ", "9.2768"
    "21000. ", "9.74064"
    "2000. ", "10.20448"
    "22200. ", "10.297248"
    "22400. ", "10.390016"