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


Geometry
---------

It is a material point, representative of a state of homogeneous stresses and deformations.

It can be simulated by a volume element represented by a single finite element.


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


Isotropic elasticity coefficients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`E=173000\mathrm{MPa}`, :math:`\nu =0.3` :math:`\mu =\frac{E}{2(1+\nu )}`


Crystal law coefficients MONO_DD_CFC (A, B, C, E models)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`A\mathrm{=}0.13`

:math:`B\mathrm{=}0.005`

:math:`Y\mathrm{=}2.5E\mathrm{-}7\mathit{mm}(2.5\mathit{Angström})`

:math:`{\tau }_{f}\mathrm{=}20.`

:math:`n\mathrm{=}50.`

:math:`\dot{{\gamma }_{0}}\mathrm{=}{10}^{\text{-3}}`

:math:`{\rho }_{\mathit{ref}}\mathrm{=}{10}^{6}{\mathit{mm}}^{\text{-2}}`

:math:`\alpha \mathrm{=}0.35`

:math:`\beta \mathrm{=}{2.5410}^{\text{-7}}(2.54\mathit{Angström})`

The interaction matrix is the one defined for MONO_DD_CFC [:ref:`R5.03.11 <R5.03.11>`].

The family of sliding systems is octahedral (:math:`\mathit{CFC}`)

The coefficients related to irradiation (modeling E) are:

:math:`\begin{array}{}{\alpha }^{\mathrm{loops}}=0{\phi }^{\mathrm{loops}}=0.001{\alpha }^{\mathrm{voids}}=0{\rho }^{\mathrm{voids}}=1.e3\\ {\rho }_{\mathrm{sat}}=4{\rho }_{0}{b}^{2}{\phi }_{\mathrm{sat}}=0.04{\xi }_{\mathrm{irra}}={10}^{7}{\zeta }_{\mathrm{irra}}={10}^{7}\end{array}` with :math:`{\rho }_{0}={10}^{\text{6}}{\mathrm{mm}}^{\text{-2}}`

The internal variables representing the dislocation density are initialized to :math:`{\rho }_{0}\ast {b}^{2}`

Those linked to irradiation have as initial values: :math:`{\rho }_{s}^{\mathrm{loops}}=2{\rho }_{0}{b}^{2}` :math:`{\phi }_{s}^{\mathrm{voids}}=0.001`


Crystal law coefficients MONO_DD_CC (D modeling)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`{D}_{\mathit{LAT}}\mathrm{=}1.0`, :math:`{K}_{\mathit{BOLTZ}}\mathrm{=}8.62E\mathrm{-}\mathrm{05,}` :math:`\mathit{GAMMA0}\mathrm{=}1.E\mathrm{-}\mathrm{3,}{\mathit{TAU}}_{F}\mathrm{=}2.E7,` :math:`{\mathit{TAU}}_{0}\mathrm{=}3.63,` :math:`{\mathit{RHO}}_{\mathit{MOB}}\mathrm{=}1.E11,` :math:`{K}_{F}\mathrm{=}30.0,{K}_{\mathit{SELF}}\mathrm{=}100.0,` :math:`B\mathrm{=}2.48E\mathrm{-}\mathrm{10,}\mathit{DELTAG0}\mathrm{=}0.84,D\mathrm{=}1.E\mathrm{-}\mathrm{08,}` :math:`N\mathrm{=}20.0,\mathit{BETA}\mathrm{=}0.2,` :math:`\mathit{GH}\mathrm{=}1.E11,{Y}_{\mathit{AT}}\mathrm{=}1.00E\mathrm{-}\mathrm{09,}`

The interaction matrix is based on the following values

:math:`\mathit{H1}\mathrm{=}0.1024,\mathit{H2}\mathrm{=}0.7,\mathit{H3}\mathrm{=}0.1,\mathit{H4}\mathrm{=}0.1,\mathit{H5}\mathrm{=}0.1\mathit{H6}\mathrm{=}0.1,`

The family of sliding systems is cubic1 (:math:`\text{CC}`).

The internal variables representing the dislocation density are initialized to :math:`{\rho }_{0}={10}^{\text{5}}{\mathrm{mm}}^{\text{-2}}`

The formulation used here is formulation 1 (chosen using the parameter DELTA1 =0) (cf. [:ref:`R5.03.11 <R5.03.11>`]


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

The cubic volume element on side :math:`\mathrm{1m}` is subjected to a simple homogeneous tensile test, using imposed deformations.1 HEXA8.


.. image:: images/100002010000010D0000012086D23B2BC458EA83.png
    :width: 1.2402in
    :height: 1.1929in


.. _RefImage_100002010000010D0000012086D23B2BC458EA83.png:

The imposed load is as follows:


* Face :math:`\mathrm{S1}` is blocked in the direction :math:`z`


* Face :math:`\mathrm{S3}` is moved from :math:`\mathrm{0,2}\mathrm{mm}` to :math:`0.2s` and in 100 increments.


* the movements following :math:`X` and :math:`Y` from the origin point are null


* a stiffness equal to :math:`{\mathrm{1O}}^{4}N/m` according to :math:`Y` is added to the origin point, via a discrete element, to allow an almost free rotation around :math:`Z`


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

Zero stresses and deformations. Initial dislocation density: :math:`{\rho }_{0}={10}^{6}{\mathrm{mm}}^{\text{-2}}`