Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

In 1881, Hertz established under certain hypotheses a solution to the problem that bears his name. Thus, assuming a contact without friction and for which the contact half width :math:`a` is very small compared to the radius of the spheres :math:`R` (:math:`a\ll R`), the contact pressure at points :math:`\mathrm{C1}` and :math:`\mathrm{C2}` is equal to:


+------------------------------------------------------------------+----------+

.. _RefEquation 2.1-1 |:

|:math: `{P} _ {0} =-\ frac {E} =-\ frac {E} {\ E} {\ pi (1- {\nu} ^ {2})}}\ sqrt {\ frac {2h} {2h} {R}} {R}}}` | eq 2.1-1 |
+------------------------------------------------------------------+----------+

where :math:`h` corresponds to the imposed overwrite, which is equal here to :math:`4\mathit{mm}`. Let's be :math:`{P}_{0}=-2798.3\mathit{Mpa}`.

The half-width of contact :math:`a` is expressed as a function of the imposed crushing and the radius of the spheres:


+----------------------------------------+----------+

.. _RefEquation 2.1-2 |:

|:math: `a=\ sqrt {\ frac {\ frac {\ mathrm {Rh}} {2}} `| eq 2.1-2 |
+----------------------------------------+----------+

In this test, for an overwrite of :math:`4\mathit{mm}`, :math:`a=10\mathit{mm}`.

The contact surface is a disk with radius :math:`a`, the pressure distribution in this zone is as follows:


+--------------------------------------------------------------------------+----------+

.. _RefEquation 2.1-3 |:

| if:math: `x\ le a` then:math: `P (x) = {P (x) = {P} _ {0}\ sqrt {1- {(\ frac {x} {a})}}}} ^ {2}}}} `| eq 2.1-3 |
+--------------------------------------------------------------------------+----------+


Reference quantities and results
-----------------------------------


.. image:: images/Object_5.svg
    :width: 26
    :height: 24


.. _RefImage_Object_5.svg:

at point :math:`G` (analytical solution).

Displacements at three points from the edge (non-regression except for analytical :math:`G`).


.. image:: images/Object_7.svg
    :width: 26
    :height: 24


.. _RefImage_Object_7.svg:

in meshes based on :math:`G` (non-regression).

Status and play at several points on the contact surface (analytical).


Uncertainties about the solution
----------------------------

The analytical calculation, although valid for :math:`a\ll R`, generally gives a very good approximation of the solution.


Bibliographical reference
-------------------------