Verification & Validation — Lesson 8

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.

• This command was generously provided by Mr. Sean Harvey. (Lead Technical Services Engineer at Ansys.)

Using this value of contact radius, we can also compute the normal pressured induced at the contact zone. Theoretically, the maximum pressure (p_max) is induced along the y-axis, as expected, and is given by the following formula:

Furthermore, we can derive the following formula for the normal stresses σ_z and σ_r = σ_θ along the z-axis.

Here we note that the principal normal stresses σ₁ = σ₂ = σ_r = σ_θ since the out-of-plane shear stresses, τ_rz = τ_θz = 0 and σ₃ = σ_z. And we can deduce that τ_max = |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.