Reference solution
=====================


.. _Ref518115239:


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

The reference results are from the book cited below [:ref:`bib1 <bib1>`].

:math:`{p}_{m}=3{\sigma }_{0}` with :math:`{p}_{m}` the contact pressure (page 171).

:math:`\mathrm{Rjohnson}={p}_{m}a=3{\sigma }_{0}a` if :math:`a` is the contact surface.

Gold in perfect plasticity :math:`\delta =\mathrm{0,368}{a}^{2}/R` according to Richmond analysis (page 200)

Finally, we get:

:math:`\mathrm{Rjohnson}=3\pi R{\sigma }_{0}\delta /\mathrm{0,368}`

:math:`\mathrm{Rjohnson}`: Normal contact reaction of the massif on the sphere

:math:`R`: Radius of the sphere

:math:`\delta`: Moving the summit of the massif

:math:`{\sigma }_{0}`: Elastic limit of the massif

This result is valid under the following hypotheses:


* axisymmetric problem,


* perfectly plastic material (the coefficient 0.368 is derived from this hypothesis)


* small deformations


* rigid sphere.


Benchmark results
----------------------

The reference results are obtained from the previous formula. It is valid for the complete 3D model.

**Note:**

*In our study, Rjohnson depends only on displacement, we can write the relationship in the following form using the data of the problem:* :math:`\mathrm{Rjohnson}=640270\delta` *with* :math:`\mathrm{Rjohnson}` *in newton and* :math:`\delta` *in millimeters.* :math:`\delta` *is directly linked to the moment of calculation.*

The value of the normal contact resultant from ASTER is given on a neighborhood of 1 radian of aperture in 2D axisymmetric and on a neighborhood of :math:`\pi /2` for the 3D model (by symmetry, it is enough to model a quarter of the problem).

Therefore, the reference values are:

.. csv-table::

    "in axisymmetric 2D", ":", ":math:`\mathrm{Rref}=\mathrm{Rjohnson}/2\pi =\mathrm{101902,1}\delta`"
    "in 3D", ":", ":math:`\mathrm{Rref}=\mathrm{Rjohnson}/4=\mathrm{160067,5}\delta`"



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

Analytical solution.


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


1. "Contact Mechanics" - K.L. JOHNSON - Cambridge University Press - chapter 6 p.153‑201