2. Benchmark solution#

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

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

where \(h\) corresponds to the imposed overwrite, which is equal here to \(4\mathit{mm}\).

The half-width of contact \(a\) is expressed as a function of the imposed crushing and the radius of the spheres:

(2.3)#\[a\mathrm{=}\sqrt{\frac{\mathit{Rh}}{2}}\]

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

(2.3)#\[\textrm{Si} r\le a\]

2.2. 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.

2.3. Uncertainties about the solution#

None (under the assumptions given above).

2. Benchmark solution#

2.1. Calculation method#

2.2. Reference quantities and results#

2.3. Uncertainties about the solution#

2.4. References#