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


Geometry
---------



.. image:: images/Cadre1.gif


.. _RefSchema_Cadre1.gif:

**Figure** 1.1-a **: Diagram of the test case.**


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

The two masses are the same and equal :math:`{m}_{1}={m}_{2}=m=5\mathrm{kg}`. The stiffness placed between these masses is :math:`k={10}^{4}N/m`.


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

The masses :math:`{m}_{1}` and :math:`{m}_{2}` move along :math:`\overrightarrow{x}` only.

The modulus of normal force :math:`{F}_{N}` is constant equal to :math:`{F}_{N}={10}^{4}N`.

Mass :math:`{m}_{1}` is affected by a contact — friction condition. The coefficients of friction :ref:`5 <RefNumPara__37574621>` are:


* in the static case: :math:`{\mu }_{S}=\mathrm{0,3}`,


* in the dynamic case: :math:`{\mu }_{D}=\mathrm{0,2}`.

In this modeling, the system goes through three phases:


* An adhesion phase, during which the force exerted on mass :math:`{m}_{2}` is constant: :math:`F=3\cdot {10}^{3}N`.


* A sliding phase, starting at :math:`{t}_{1}`, when the tangential force can no longer compensate for the tensile force exerted by the spring on :math:`{m}_{1}`. After an arbitrary time (:math:`{t}_{2}=\mathrm{0,2}s`), the force imposed is set to 0 and the kinetics of the system then tends to 0.


* A second phase of grip, starting at :math:`{t}_{3}`, when the speed of the :math:`{m}_{1}` mass is cancelled out.


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

At time :math:`t=0`, both masses are at rest (zero displacement and zero speed).