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


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


Geometry
---------






* Cube size: :math:`\mathrm{1m}\mathrm{\times }\mathrm{1m}\mathrm{\times }\mathrm{1m}`


* Center of the cube: :math:`O:(0.\mathrm{,0}\mathrm{.}\mathrm{,0}\mathrm{.})`


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


* Elastic


    * :math:`E=5800.0\mathrm{E6}\mathrm{Pa}` Young's module


    * :math:`\nu =0.3` Poisson's Ratio


* DRUCK_PRAGER or DRUCK_PRAG_N_A with linear negative work hardening


    * :math:`\mathrm{\alpha }=0.33` Pressure Dependence Coefficient


    * :math:`{p}_{\mathrm{ultm}}=0.01` Ultimate cumulative plastic deformation


    * :math:`{\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}` Plastic constraint


    * :math:`h=-2.00\mathrm{E8}\mathrm{Pa}` Work hardening module


    * :math:`\mathrm{\beta }=0.33` Expansion coefficient (only for DRUCK_PRAG_N_A)




* DRUCK_PRAGER with parabolic negative work hardening


    * :math:`\mathrm{\alpha }=0.33` Pressure Dependence Coefficient


    * :math:`{p}_{\mathit{ultm}}=0.01` Ultimate cumulative plastic deformation


    * :math:`{\mathrm{\sigma }}^{Y}=2.57E6\mathit{Pa}` Plastic constraint


    * :math:`{\mathrm{\sigma }}_{\mathit{ultm}}^{Y}=0.57E6\mathit{Pa}` Ultimate constraint


    * :math:`\mathrm{\beta }=0.33` Expansion coefficient (only for DRUCK_PRAG_N_A)


* DRUCK_PRAG_N_Aavec exponential negative work hardening


    * :math:`\alpha =0.33` Pressure Dependence Coefficient


    * :math:`{p}_{c}=0.01` Ultimate cumulative plastic deformation


    * :math:`{\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}` Plastic constraint

:math:`{\sigma }_{\mathit{ultm}}^{Y}\mathrm{=}0.57\mathit{E6}\mathit{Pa}` Ultimate constraint


    * :math:`\mathrm{\beta }=0.33` Expansion coefficient


Boundary conditions and loads
-------------------------------------

The boundary conditions and loads applied are as follows:


* Step :math:`A`: :math:`t\in [\mathrm{0,1}\mathrm{.}]`

Compression :math:`p={2.10}^{6}\mathrm{Pa}` is progressively applied to 3 faces of the cube (top, front, right) according to the function shown in the figure below, and symmetry conditions are applied to the 3 other faces (bottom, back, left).


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:


* Step B: :math:`t\in \text{]}\mathrm{1,2}\mathrm{.}\text{]}`

Starting from the stress state at time :math:`t=1.s`, the following conditions are applied to the faces of the cube:

**Compulsory trips:**


*
    * the displacement varies progressively on the right face according to the function shown in the figure below:


.. image:: images/Shape3.gif


.. _RefSchema_Shape3.gif:


*
    * Symmetry conditions on all 3 sides (bottom, back, left).

**Imposed loads:**

A pressure of :math:`p={2.10}^{6}\mathrm{Pa}` is applied on the other 2 sides (front and top).