2. Benchmark solution#
2.1. 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 \(a\) is very small compared to the radius of the spheres \(R\) (\(a\ll R\)), the contact pressure at points \(\mathrm{C1}\) and \(\mathrm{C2}\) is equal to:
|:math: {P} _ {0} =-frac {E} =-frac {E} {E} {pi (1- {nu} ^ {2})}}sqrt {frac {2h} {2h} {R}} {R}}} | eq 2.1-1 | +——————————————————————+———-+
where \(h\) corresponds to the imposed overwrite, which is equal here to \(4\mathit{mm}\). Let’s be \({P}_{0}=-2798.3\mathit{Mpa}\).
The half-width of contact \(a\) is expressed as a function of the imposed crushing and the radius of the spheres:
|:math: `a=\ sqrt {\ frac {\ frac {\ mathrm {Rh}} {2}} `| eq 2.1-2 | +—————————————-+———-+
In this test, for an overwrite of \(4\mathit{mm}\), \(a=10\mathit{mm}\).
The contact surface is a disk with radius \(a\), the pressure distribution in this zone is as follows:
2.2. Reference quantities and results#
at point \(G\) (analytical solution).
Displacements at three points from the edge (non-regression except for analytical \(G\)).
in meshes based on \(G\) (non-regression).
Status and play at several points on the contact surface (analytical).
2.3. Uncertainties about the solution#
The analytical calculation, although valid for \(a\ll R\), generally gives a very good approximation of the solution.