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:

$a = \left( \frac{3F \left[ \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2} \right]}{4 \left( \frac{1}{R_1} + \frac{1}{R_2} \right)} \right)^{1/3}%$

 

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.)

 

 

	Theoretical	Numerical	Relative Error(%)
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 (p_max) is induced along the y-axis, as expected, and is given by the following formula:

 

$p_{max}=\frac{3F}{2\pi a^2}$

 

 

	Theoretical	Numerical	Relative Error(%)
Max Pressure, Pmax [MPa]	88.290	81.094	8.151

 

 

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

$\sigma_z = -p_{max} \left( \frac{z^2}{a^2} + 1 \right)^{-1}$

$\sigma_r =\sigma_\theta = -p_{max} \left[ (1 + \nu_1) \left( 1 - \left| \frac{z}{a} \right| \tan^{-1} \left| \frac{a}{z} \right| \right) - \frac{1}{2 \left( \frac{z^2}{a^2} + 1 \right)} \right]$

 

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:

 

$\bar{\sigma}=\frac{1}{\sqrt{2}}\sqrt{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2}$

 

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