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

Device description
-------------------------

The non-linear element is represented by the rheological model below.


.. image:: images/1000000000000200000000A8A4F1B7CFE1F12545.png
    :width: 2.7992in
    :height: 0.9189in


.. _RefImage_1000000000000200000000A8A4F1B7CFE1F12545.png:

**Figure** 1.1-a: Device model.

The equations governing behavior are in [:ref:`R5.03.17 <R5.03.17>`].


Modelizations
-------------

The models tested are on elements DIS_T then DIS_TR, cells SEG2 then cells POI1. The characteristics of the discrete elements are of the type: K_T_D_L, K_ TR_D_L, K_T_D_N, K_ TR_D_N.

*Note: The units of the parameters must agree with the unit of effort, the unit of lengths* [:ref:`R5.03.17 <R5.03.17>`]. For all models the units are homogeneous to [N], [mm].


Modeling A
~~~~~~~~~~~~~~~

This modeling makes it possible to test the non-linear static cyclic behavior of the law.


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

Modeling A
~~~~~~~~~~~~~~~

The only property is the non-linear function [:ref:`R5.03.17 <R5.03.17>`].

The figure shows the non-linear behavior used in the test case:

• Elastic behavior up to point :math:`(\mathrm{0.5mm},\mathrm{200.0N})`.

• Nonlinear behavior in the :math:`x` local direction of the discrete, governed by the following equation:


.. math::
: label: EQ-None

    R (p) =\ frac {{K} _ {\ mathit {elas}} .p} {\ left [1+ {\ left (\ frac {{k} _ {\ mathit {elas}} .p} .p} {{F} _ {u} _ {u}} — {F} _ {elas}}} .p} {{F} _ {elas}}} .p} {{F} _ {elas}} .p} {{F} _ {u}} — {F} _ {y}}\ right)} ^ {n}\ right]} ^ {(1/n)}} .p} {{F} _ {(1/n)}}


.. image:: images/100002010000041D00000355903461E673CA258E.png
    :width: 4.0193in
    :height: 2.8783in


.. _RefImage_100002010000041D00000355903461E673CA258E.png:

**Figure** 1.3.1-a **: Nonlinear behavior**


B modeling
~~~~~~~~~~~~~~~

The only property is the non-linear function [:ref:`R5.03.17 <R5.03.17>`].

The behavior is of the "isotropic work hardening" type in the tangential plane local to the element. It is defined by the following function:

fctsy2 **= DEFI_FONCTION (NOM_PARA = "DTAN"**,

**VALE =** **(** 0.0, 0.0, 0.0, 0.1, 0.1, 100.0, 0.2, 120.0, 20.2, 370.0),

**)**


C modeling
~~~~~~~~~~~~~~~

The only property is the non-linear function [:ref:`R5.03.17 <R5.03.17>`].

The behavior is of the "kinematic work hardening" type in the tangential plane local to the element. It is defined by the following function:

fctsy2 **= DEFI_FONCTION (NOM_PARA = "DTAN"**,

**VALE =** **(** 0.0, 0.0, 0.0, 0.1, 100.0, 20.1, 350.0),

**)**


.. _DdeLink__11456_1213890583:


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

Modeling A
~~~~~~~~~~~~~~~

When the discrete is a SEG2, one of the nodes is blocked, on the other a displacement condition is imposed. When the discrete is a POI1 the displacement condition is imposed on this node.

The condition while traveling is a function of time:

:math:`{U}_{0}.\mathrm{sin}(2\mathrm{\pi }.f.t)` with :math:`f=1\mathit{Hz}` and :math:`{U}_{0}=2.0\mathit{mm}`


B and C models
~~~~~~~~~~~~~~~~~~~~~~

When the discrete is a SEG2, one of the nodes is blocked, on the other a displacement condition is imposed. When the discrete is a POI1 the displacement condition is imposed on this node.

Travel conditions are functions of time:

:math:`\mathit{Depl}={U}_{1.}\mathrm{sin}(2\mathrm{\pi }.{f}_{1}.t)+{U}_{2.}\mathrm{sin}(2\mathrm{\pi }.{f}_{2}.t)+{U}_{3.}\mathrm{sin}(2\mathrm{\pi }.{f}_{3}.t)`

Following the direction :math:`Y`: :math:`(u,f)=[(\mathrm{0.20,0}.80),(\mathrm{0.15,1}.50),(\mathrm{0.10,3}.00)]`

Following the direction :math:`Z`: :math:`(u,f)=[(-\mathrm{0.20,0}.90),(\mathrm{0.15,2}.00),(-\mathrm{0.10,2}.80)]`