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


Geometry
---------


.. image:: images/10000000000006F0000007C046D9570517C095E7.png
    :width: 5.9165in
    :height: 6.611in


.. _RefImage_10000000000006F0000007C046D9570517C095E7.png:

All ratings are in :math:`\mathrm{mm}`. The crack is located :math:`\mathrm{0,2}\mathrm{mm}` from the interface, in the upper part of the specimen.


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

**Material #1:** austenitic steel

Von Mises elastoplastic with isotropic work hardening

Young's modulus :math:`{E}_{1}\mathrm{=}{2.10}^{5}\mathit{MPa}`, Poisson's ratio :math:`{\nu }_{1}=\mathrm{0,3}`

Elastic limit :math:`{\sigma }_{\mathit{y1}}\mathrm{=}310\mathit{MPa}`

Uniaxial tensile curve:


.. csv-table::

    ":math:`\sigma (\mathrm{MPa})` ", "0", "310", "310", "600", "700"
    ":math:`\varepsilon` ", "0", "0.155", "0.155", "40", "100"


**Material #2:** ferritic steel

Von Mises elastoplastic with isotropic work hardening

Young's modulus :math:`{E}_{2}={2.10}^{5}\mathrm{MPa}`, Poisson's ratio :math:`{\nu }_{2}=\mathrm{0,3}`

Elastic limit :math:`{\sigma }_{\mathrm{y2}}=442\mathrm{MPa}`


.. csv-table::

    ":math:`\sigma (\mathrm{MPa})` ", "0", "442", "442", "600", "650"
    ":math:`\varepsilon` ", "0", "0.221", "40", "100"


**Material #3:** almost non-deformable pins

Isotropic linear elastic

Young's modulus :math:`{E}_{3}={6.10}^{10}\mathrm{MPa}`, Poisson's ratio :math:`{\nu }_{3}=\mathrm{0,3}`


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

Given the asymmetry of the materials, the entire specimen is modelled.

**Blocks:**


.. csv-table::

    ":math:`\mathrm{UX}=\mathrm{UY}=0` ", "at point :math:`B` (center of the lower pin)"
    ":math:`\mathrm{UX}=0` ", "at point :math:`A` (center of the top pin)"


**Loading per imposed displacement:**


.. math::
: label: EQ-None

    0\ le\ mathrm {UY}\ le 1\ mathrm {mm}

The load is therefore monotonous and increasing.