Authors: Ju Hwan Shin and You Won Park

Problem Specification

1. Pre-Analysis & Start-Up

2. Geometry

3. Mesh

4. Physics Setup

5. Numerical Solution

6. Numerical Results

7. Verification & Validation

Exercises

Comments

# Verification and Validation

This section contains a few formulae, 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 Inc.)

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 σ_{1} = σ_{2} = σ_{r} = σ_{θ} since the *out-of-plane* shear stresses, τ_{rz} = τ_{θz} = 0 and σ_{3} = σ_{z}. And we can deduce that τ_{max} = |τ_{1}|=|τ_{2}|=|(σ_{1}-σ_{2}) / 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.