2. Reference solution#

2.1. Calculation method used for the reference solution#

The reference results are from the book cited below [bib1].

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

\(\mathrm{Rjohnson}={p}_{m}a=3{\sigma }_{0}a\) if \(a\) is the contact surface.

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

Finally, we get:

\(\mathrm{Rjohnson}=3\pi R{\sigma }_{0}\delta /\mathrm{0,368}\)

\(\mathrm{Rjohnson}\): Normal contact reaction of the massif on the sphere

\(R\): Radius of the sphere

\(\delta\): Moving the summit of the massif

\({\sigma }_{0}\): 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.

2.2. 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: \(\mathrm{Rjohnson}=640270\delta\) with \(\mathrm{Rjohnson}\) in newton and \(\delta\) in millimeters. \(\delta\) 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 \(\pi /2\) for the 3D model (by symmetry, it is enough to model a quarter of the problem).

Therefore, the reference values are:

in axisymmetric 2D

:

\(\mathrm{Rref}=\mathrm{Rjohnson}/2\pi =\mathrm{101902,1}\delta\)

in 3D

:

\(\mathrm{Rref}=\mathrm{Rjohnson}/4=\mathrm{160067,5}\delta\)

2.3. Uncertainties about the solution#

Analytical solution.

2.4. Bibliographical reference#

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