Benchmark solution
====




Calculation method
----

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 half-width of contact :math:`a` is very small compared to the radius of the spheres :math:`R` (:math:`a\mathrm{\ll }R`), the contact pressure at points :math:`\mathit{C1}` and :math:`\mathit{C2}` is equal to:


.. math::
    :label: eq-None

    {P}_{0}\mathrm{=}\mathrm{-}\frac{E}{\pi (1\mathrm{-}{\nu }^{2})}\sqrt{\frac{2h}{R}}

where :math:`h` corresponds to the imposed overwrite, which is equal here to :math:`4\mathit{mm}`.

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


.. math::
    :label: eq-None

    a\mathrm{=}\sqrt{\frac{\mathit{Rh}}{2}}

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


.. math::
    :label: eq-None

    \textrm{Si}
    r\le a




Reference quantities and results
----

The pressure obtained at the center of the contact zone and the half-width of contact will be compared with the analytical solution above.




Uncertainties about the solution
----

None (under the assumptions given above).




References
----