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

Geometry and boundary conditions
-----------------------------------

In these tests, a concrete plate (:math:`200\mathrm{\times }200\mathrm{\times }50\mathit{mm}`) is subjected to a loading where the main stress ratio :math:`{\sigma }_{2}\mathrm{/}{\sigma }_{1}` is fixed (:math:`{\sigma }_{3}\mathrm{=}0`). These tests are modelled in two dimensions under the condition of plane constraints (:math:`{\sigma }_{\mathit{zz}}\mathrm{=}0`) using a quadrangle element with 4 nodes (QUAD4).


.. image:: images/1000000000000357000001CA4236EF3D652EB857.jpg
    :width: 6.1516in
    :height: 3.1807in


.. _RefImage_1000000000000357000001CA4236EF3D652EB857.jpg:

Figure 1.1-a: 2D modeling and boundary conditions for biaxial tests

Only a quarter of the concrete plate is modelled. Thus, symmetry conditions are imposed:

• The following moves :math:`y` are stuck on the bottom edge.

• The following moves :math:`x` are stuck on the left edge.

Two pressures are imposed on the free edges: :math:`{P}_{1}` and :math:`{P}_{2}`. The relationship between the loads, and implicitly on the main constraints, is noted :math:`\omega`:

:math:`{\mathrm{\sigma }}_{2}=\mathrm{\omega }{\mathrm{\sigma }}_{1}` (Eq.1)

These pressures follow a linear law of evolution.

.. csv-table::

    "Instant", "0", "1"
    ":math:`{P}_{1}` (:math:`\mathit{MPa}`)", "0", "30"
    ":math:`{P}_{2}` (:math:`\mathit{MPa}`)", "0"," :math:`30\mathrm{\omega }`"


**Table** 1.1-1 **: Evolution of pressures**


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

For the MAZARS model, the following parameters were used:

Elastic behavior:

:math:`E\mathrm{=}\text{34}\text{000}\text{MPa},\nu \mathrm{=}0\text{.}19`

Damaging behavior:

:math:`{\varepsilon }_{\mathit{d0}}\mathrm{=}1\text{.}\text{1}\text{.}{\text{10}}^{\mathrm{-}3};\text{Ac}\mathrm{=}1\text{.}\text{25};\text{At}\mathrm{=}1\text{.}0;\text{Bc}\mathrm{=}\text{1965};\text{Bt}\mathrm{=}\text{9}\text{000};k\mathrm{=}0\text{.}\text{7}`

These material parameters induce a compression limit :math:`{f}_{c}` of the order of :math:`33\mathit{MPa}`.


Initial conditions
--------------------

Néant