Reference problem ===================== Geometry and boundary conditions ----------------------------------- .. image:: images/Object_1.svg :width: 288 :height: 192 .. _RefImage_Object_1.svg: **Figure** 1.1-a **: geometry of the studied structure** The structure studied is a :math:`1\mathit{mm}` long bar with a :math:`0.1\times 0.1\mathit{mm}` square cross section. One face is embedded, the opposite face is subjected to a :math:`\mathit{FZ}` force varying progressively from :math:`1` to :math:`5N`. Material properties ----------------------- +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |**Law of behavior** |**Elastic behaviour** |**Damaging behavior** **(healthy part)** |**Damaging behavior** | + + + + + | | | | **(weakened zone)** | + + + + + | | | | | +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |ENDO_ISOT_BETON |:math:`E\mathrm{=}20000\mathit{MPa}` |:math:`{\sigma }_{y}\mathrm{=}6\mathit{MPa}` |:math:`{\sigma }_{y}\mathrm{=}5\mathit{MPa}` | + + + + + | | :math:`\nu \mathrm{=}0` | :math:`{E}_{T}\mathrm{=}\mathrm{-}10000\mathit{MPa}` | :math:`{E}_{T}\mathrm{=}\mathrm{-}10000\mathit{MPa}` | + + + + + | | | | | +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |ROUSS_PR |:math:`E\mathrm{=}206400\mathit{MPa}` |:math:`D\mathrm{=}2` |:math:`D\mathrm{=}2` | + + + + + | and ROUSSELIER | :math:`\nu \mathrm{=}0.3` | :math:`{\sigma }_{1}\mathrm{=}490\mathit{MPa}` :math:`{f}_{0}\mathrm{=}5E-04` traction curve: | :math:`{\sigma }_{1}\mathrm{=}400\mathit{MPa}` :math:`{f}_{0}\mathrm{=}5E-04` traction curve: | + + + + + | | | .. image:: images/Object_2.svg | .. image:: images/Object_3.svg | + + + :width: 288 + :width: 150 + | | | :height: 192 | :height: 25 | + + + + + | | | | | + + + + + | | | :math:`p` *: cumulative plastic deformation* :math:`{r}_{i}\mathrm{=}1500\mathit{MPa}` :math:`{r}_{o}\mathrm{=}520\mathit{MPa}` :math:`b\mathrm{=}2.4` | :math:`p` *: cumulative plastic deformation* :math:`{r}_{i}\mathrm{=}1500\mathit{MPa}` :math:`{r}_{o}\mathrm{=}520\mathit{MPa}` :math:`b\mathrm{=}2.4` | + + + + + | | | | | +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |BETON_DOUBLE_DP |:math:`E\mathrm{=}31000\mathit{MPa}` |:math:`{f}_{c}\mathrm{=}38.3\mathit{MPa}` |:math:`{f}_{c}=38.3\mathit{MPa}` | + + + + + | | :math:`\nu \mathrm{=}0.22` | :math:`{f}_{t}\mathrm{=}4.0\mathit{MPa}` :math:`\beta \mathrm{=}1.16` :math:`{G}_{t}=2.83E-04\mathit{Nmm}/{\mathit{mm}}^{2}` :math:`{G}_{c}=2.83E-02\mathit{Nmm}/{\mathit{mm}}^{2}` | :math:`{f}_{t}=3.0\mathit{MPa}` :math:`\beta =1.16` :math:`{G}_{t}=2.83E-04\mathit{Nmm}/{\mathit{mm}}^{2}` :math:`{G}_{c}=2.83E-02\mathit{Nmm}/{\mathit{mm}}^{2}` | + + + + + | | | | | +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |ENDO_LOCA_EXP |:math:`E=34129\mathit{MPa}` |:math:`\kappa =5.842` |:math:`\kappa =5.842` | + + + + + | | :math:`\nu =0` | :math:`p=2` :math:`{\sigma }_{c}=3.034\ast 1.5\mathit{MPa}` :math:`{\sigma }_{0}=0.827\ast 1.5\mathit{MPa}` | :math:`p=2` :math:`{\sigma }_{c}=3.034\mathit{MPa}` :math:`{\sigma }_{0}=0.827\mathit{MPa}` | + + + + + | | | | | +--------------------------------------+------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+