This section contains a few formulas, which made the listed assumptions, found in the Pre-Analysis & Start-Up page.
The analytical formula for computing the radius of contact zone (a) is given as follows:
The following command for the computation of the contact area can be downloaded here.
|Contact radius, a [mm]||1.00964||1.02517||1.538|
Using this value of contact radius, we can also compute the normal pressured induced at the contact zone. Theoretically, the maximum pressure (pmax) is induced along the y-axis, as expected, and is given by the following formula:
|Max Pressure, Pmax [MPa]||88.290||81.094||8.151|
Furthermore, we can derive the following formula for the normal stresses σz and σr = σθ along the z-axis.
Here we note that the principal normal stresses σ1 = σ2 = σr = σθ since the out-of-plane shear stresses, τrz = τθz = 0 and σ3 = σz. And we can deduce that τmax = |τ1|=|τ2|=|(σ1-σ2) / 2|. The effective stress (using the von-Mises criterion) along the y-axis can be computed as the following:
Lastly, we also confirm that the applied load at the top vertex of the sphere matches our numerical contact pressure, integrated along the interface.
|Mesh Size [m]||2.00E-04||1.00E-04||9.00E-05||Theoretical|
|Force Reaction [N]||187.95||188.32||188.52||188.50|
|Relative Error [%]||0.29||0.09||0.01||0.00|